Distributed Control Methods and Cyber Security Issues in Microgrids 012816946X, 9780128169469

Table of contents :
Cover
Distributed Control Methods and Cyber Security Issues in Microgrids
Copyright
Contributors
Preface
1
Sliding mode control of grid-connected power converters for microgrid applications
Introduction of sliding mode control
First-order sliding mode control
Second-order sliding mode control
Mathematical models of power converters
Mathematical model of three-phase two-level AC/DC power converters
Mathematical model of DC/DC buck converters
SMC for power converters
SMC of three-phase AC/DC power converters
Mode uncertainties
Control objectives
Controller design
Extended state observer
Capacitor voltage regulation
Grid current tracking
Simulation results
SMC for DC/DC buck converters
Control objectives
Design of controller
Voltage regulation loop
Current tracking loop
Simulation results
Against load resistance variation
Against input voltage variation
Against reference voltage variation
Conclusion
References
2
Distributed voltage restoration and power allocation control in islanded DC microgrids
Introduction
Problem formulation and objectives
Droop control
Objectives
Distributed secondary control for voltage restoration
Controller design
Stability analysis
Distributed secondary control with pinning gain
Control parameters selection
Simulation results
Proposed distributed secondary control
Case A: Backup DG plug-and-play
Case B: Different pinning gains
Robustness test with respect to temporary fault
Comparison with existing methods
Experimental validation
Experimental results with resistant load
Experimental results with constant power load
Experimental results with communication time delay
Conclusion
References
3
Optimal distributed secondary control for a microgrid
Introduction
Preliminaries
Graph theory
Closed-loop optimal control algorithm
Proposed optimal distributed control strategy
Optimal distributed finite-time secondary frequency control and active power sharing
Optimal distributed secondary voltage control and reactive power sharing
Communication delay analysis
Algorithm implementation
Simulation analysis
Performance evaluation the proposed optimal distributed control strategy
Convergence analysis
Influence analysis of the communication delay
Robustness analysis against the uncertainties of parameters
Influence analysis of the bounded control input
Plug-and-play capability analysis
Scalability test
Conclusion
References
4
Distributed power control of flexible loads in microgrids
Coordinated active power dispatch control for a microgrid
Distributed pinning consensus on networks and corporation optimization
Distributed pinning consensus algorithm
Dispatch optimization of units in microgrid by λ-iteration algorithm
Corporation dispatch control of units in a microgrid
Distributed λ-iteration optimization of active power
Solution without power constraints
Solution with power constraints
Case studies
Case 1: The full participation of DGs and ESUs
Case 2: Exit of ESUs and time-varying demand
Case 3: The plugging-in DGs to share the active power
Conclusion
Demand response load following control of smart grids
Problem formulation and aggregate evaluation of TCLs
Basic model of a single TCL and an aggregator
Aggregate evaluation of TCL aggregator
Look-ahead economic dispatch to provide load following trajectories
Distributed pinning control of multiple aggregated TCLs
Simulation and results
Aggregate evaluation of the aggregator
Reference power trajectories solving
Demand response load following control of TCLs
Conclusion
References
5
False data injection attacks on inverter-based microgrid in autonomous mode
Introduction
Inverter-based microgrid structure
Physical layer
Cyber-communication layer
System dynamic model
Small signal model
Active power reference
System performance under FDI attacks
Distributed load sharing control under FDI attacks
Impacts of FDI attacks
Simulation examples
Stable region
System performance under attack strategy 1
System performance under attack strategy 2
Discussion
Conclusion
References
6
Distributed finite-time control of aggregated energy storage systems for frequency regulation in multiarea microg
Introduction
Background
Literature survey
Contributions
Proposed frequency control scheme
System overview
Multiarea microgrids
Proposed disturbance observer
System disturbance observer
Band-pass filter
Distributed finite-time control of ESA
Communication graph
Finite-time consensus control of ESA
Stability analysis
Numerical illustrations
Results and discussions
Case 1: System contingency
Case 2: Normal operation
Case 3: Multiarea microgrids
Case 4: Comparison with linear control algorithm
Conclusion
References
7
Distributed optimization algorithm for economic dispatch: A bisectional approach
Introduction
System modeling
Problem formulation
Centralized solution to the EDP
Introduction to consensus-like algorithm
Graph theory and nonnegative matrices
Consensus-like algorithm
Distributed bisection algorithm: Design and analysis
Distributed algorithm for aggregate demand
Distributed algorithm for feasibility test
Distributed bisection algorithm
Convergence analysis and stopping criteria
Numerical examples
Case 1: The EDP with quadratic cost functions only
Case 2: EDP with nonquadratic cost functions
Case 3: Convergence speed analysis
Case 4: The comparison with the algorithm in 6345156
Case 5: Implementation on IEEE 118-bus system
Conclusion and discussion
References
8
Scheduling of EV battery swapping in microgrids
Introduction
Background, motivation, and contributions
Literature
Problem formulation
Network model
DC power flow equations
Fix-point linearization of power flow equations
DistFlow equations and SOCP relaxation
Operational constraints
Battery swapping scheduling
Centralized solution
Distributed solutions
Relaxations
Distributed solution via ADMM
Distributed solution via dual decomposition
Numerical results
Setup
Centralized solution
Nearest-station policy
Optimal assignments
Optimality of generalized Benders decomposition
Exactness of SOCP relaxation
Computational effort
Benefit
Distributed solutions
Convergence
Suboptimality (comparison with centralized solution)
Exactness of SOCP relaxation
Scalability
Concluding remarks
Summary
Model limitations
Appendix: Proof of [TEO:NUMBER]Theorem 1
References
9
Dispatch strategy of energy bank system with hybrid energy storage
Introduction
EBS model structure
Definition in energy bank system
Energy bank system model and structure
Trading model
Call auction
Rule of maximum transaction volume
Rule of treaty violation
Listed-energy and listed-price (LELP) model
Selling listed-price
Buying listed-price
Assessment indices
Deposit system of energy
Figure of deposit energy currency
Operation model of DSE
Economic model of DSE
Case study
System description
Simulation
Conclusion
References
10
False data injection attacks and countermeasures in smart microgrid systems
Introduction
Preliminaries and problem formulation
Network model
Physical network
Communication network
Dynamic model
UIO-based detector
Problem formulation
Main results
Potential stealthy attacks
Attack impacts analysis
Countermeasures
Simulation
Conclusions
References
Index
Back Cover

Citation preview

Distributed Control Methods and Cyber Security Issues in Microgrids

Distributed Control Methods and Cyber Security Issues in Microgrids

Edited by

Wenchao Meng Xiaoyu Wang Shichao Liu

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-816946-9 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Joe Hayton Acquisitions Editor: Lisa Reading Editorial Project Manager: Joanna Collet Production Project Manager: Kamesh Ramajogi Cover Designer: Matthew Limbert Typeset by SPi Global, India

Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin. Jiming Chen (263), College of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China Peng Cheng (263), College of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China Ruilong Deng (263), College of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China; School of Computer Science and Engineering, Nanyang Technological University, Singapore Zhao Yang Dong (243), School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, Australia Minyue Fu (177), School of Electrical Engineering and Computing, University of Newcastle, Callaghan, NSW, Australia Yabin Gao (3), Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of China Fanghong Guo (29), Department of Automatic Control, Zhejiang University of Technology, Hangzhou, Zhejiang, People’s Republic of China Jianqiang Hu (83), Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Zhejiang, People’s Republic of China Zhiyun Lin (177), School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang, People’s Republic of China Jianxing Liu (3), Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of China Mengxiang Liu (263), College of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China Wensheng Luo (3), Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of China Wenchao Meng (125), Department of Control Science and Engineering, Zhejiang University, Hangzhou, China Junjian Qi (125), Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL, United States Jing Qiu (149, 243), School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW, Australia

xi

xii Contributors Guanghui Sun (3), Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of China Lingling Sun (243), School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, Australia Lei Wang (29), School of Electrical Engineering and Computing, University of Newcastle, Newcastle, NSW, Australia Wenhai Wang (263), College of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China Xiaoyu Wang (125), Department of Electronics, Carleton University, Ottawa, ON, Canada Yu Wang (149), Nanyang Technological University, Singapore Changyun Wen (29), School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore Hao Xing (177), School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang, People’s Republic of China Yan Xu (149), Nanyang Technological University, Singapore Yinliang Xu (59), Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School, Shenzhen, People’s Republic of China Zhongkai Yi (59), Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School, Shenzhen, People’s Republic of China Yunfei Yin (3), Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of China Pengcheng You (203), Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, United States Heng Zhang (125), School of Science, Jiangsu Ocean University, Lianyungang, Jiangsu, People’s Republic of China Chengcheng Zhao (263), College of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China Wei Xing Zheng (125), School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW, Australia

Preface Recent years have witnessed a growing interest in microgrids, since they can provide increased reliability and can facilitate the effective integration of distributed generators (DGs). However, the traditional centralized methods cannot guarantee the reliability and effective integration of DGs because they are subject to the well-known single point of failure and not flexible. As a result, distributed methods have been investigated in recent years. Meanwhile, microgrids integrate cyber components such as information and communication technologies and computer processing into physical components, which raises cyber security concerns such as Denial-of-Service, data integrity attacks, and replay attacks, and these cyber attacks can influence physical infrastructures because of their tight coupling. In order to improve the grid’s reliability and resilience, robust defense methods must be studied. In addition, the increasing sensing abilities of microgrids have also raised privacy concerns. Therefore, in this book, we investigate distributed control solutions and cyber security issues for microgrids. In Chapters 1–4, the distributed frequency control approaches and distributed voltage control approaches are studied. These methods are distributed in the sense that each participant only needs local information instead of the global information. In Chapter 5, the influence of false data injection on the distributed frequency control is investigated. In Chapter 6, the energy storage is used for frequency regulation in microgrids. In Chapters 7–9, distributed energy management including the economic dispatch, scheduling of EV battery swapping, and dispatch strategy for energy bank system is studied. In Chapter 10, the false data injection and its countermeasures in smart DC microgrid systems are investigated.

xiii

Chapter 1

Sliding mode control of grid-connected power converters for microgrid applications Jianxing Liu, Wensheng Luo, Yabin Gao, Yunfei Yin and Guanghui Sun Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang, People’s Republic of China

1 Introduction of sliding mode control Sliding mode control (SMC) was first proposed and elaborated by Emelyanov, Utkin, and other coresearchers in Soviet Union in the early 1950s [1]. The idea was inspired by the switching actions of electromagnetic relay. The main feature of SMC is that its control function does not remain the same; instead, it switches from one to the other. The system states are driven onto a particular surface (sliding surface) and maintained there. Once the sliding surface is reached, the system is invariant, meaning that the system is completely insensitive to parametric uncertainty and external disturbances. To overcome the undesirable chattering phenomenon of the traditional SMC (first-order sliding mode control [FOSMC]), a type of second-order sliding mode control (SOSMC) algorithm, which changes the discontinuous control function to a continuous one, was proposed. This suppresses the chattering while maintaining the robustness against the uncertainties [2–5]. The design of SMC consists of two steps. The first step is to design a sliding surface such that the sliding motion satisfies certain design specifications, that is, the dynamic behavior of the system can be tailored by the special choice of the sliding surface. The second step is to select a control law that will drive the system states to the sliding surface and maintain their subsequently, thus the required specifications are met, and the closed-loop system becomes totally insensitive to some particular uncertainties, including external disturbance, model parameter uncertainties, and nonlinearity that are bounded. Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00001-3 Copyright © 2020 Elsevier Inc. All rights reserved.

3

4 PART | I Frequency and voltage control

Consider a nonlinear system x˙ = a (x) + b (x, u), y = s (t, x),

(1) (2)

where x ∈ X ⊂ Rn is the state vector, u ∈ U ⊂ R is the input, s(t, x): Rn+1 → R is the sliding variable, and a(x) and b(x, u) are smooth uncertain functions. The control focus is to force the sliding variable s to zero. Geometrically, s = 0 is a hyperplane in error space, which is the so-called “sliding surface.” Hence, the system specifications can be taken into account when defining the sliding surface, consequently, the specifications are met as the system achieves the sliding surface. Now that the sliding surface has been defined, the next step is to design a control law steering the system trajectories to the sliding hyperplane in finite time. Here we present two algorithms: FOSMC and SOSMC.

1.1 First-order sliding mode control The FOSMC is the most conventional SMC, which takes the following form:  −K, s > 0 u = −Ksign(s) = (3) K, s < 0, where K is a positive constant sufficiently large. From Eq. (3), it can be seen that the control is switching between two constant values, crossing the hyperplane s = 0. The controlling mechanism is: control u affects the derivative of s, that is, when s > 0, u make the s˙ < 0, vice versa, when s < 0, u make the s˙ > 0, thus the case ss ˙ < 0 always holds, until the hyperplane s = 0 is reached. The control u will be switching at very high frequency, theoretically infinite in steady state. Since a real physical switch cannot take action that quickly, this would lead to system chattering. To deal with this problem, some other modified SMC algorithms have been proposed, among which the most attractive one is the SOSMC.

1.2 Second-order sliding mode control One classical form of SOSMC is the super twisting algorithm (STA). This section discusses the STA in a general case for system (1). The control objective is to force the sliding variable s and its time derivative s˙ to zero. By differentiating the sliding variable s(t, x) twice, the following relations are derived: ∂ ∂ (4) s˙ = s (t, x) + s (t, x) [a (x) + b (x, u)], ∂t ∂x ∂ ∂ s¨ = s˙ (t, x, u) + s˙ (t, x, u) [a (x) + b (x, u)] ∂t ∂x

Sliding mode control of grid-connected power converters Chapter | 1 5

∂ s˙ (t, x, u) u˙ ∂u = ϕ (t, x, u) + γ (t, x, u) u. ˙ =

(5)

∂ s˙ (t, x, u)  = 0, which means it has relative If sliding variable s satisfies ∂u degree one with respect to control input u, then there exist positive constant values Φ, Γm , and ΓM satisfying the following conditions:

0 < Γm < γ (t, x, u) < ΓM , −Φ ≤ ϕ (t, x, u) ≤ Φ.

(6) (7)

Applying Eqs. (6), (7) to Eq. (5), the following condition is satisfied: ˙ s¨ ∈ [−Φ, +Φ] + [Γm , ΓM ] u.

(8)

Then an STA controller is designed as follows: u = u1 + u2 , u˙1 = −αsign(s),

(9) (10)

u2 = −λ |s|1/2 sign(s),

(11)

where α and λ are design parameters which are decided based on the boundary conditions (6), (7). It can be seen that the STA consists of two terms: one is the integral of its discontinuous sign function while the other is a continuous function of sliding variable s. To ensure the finite time convergence to the sliding manifold s = s˙ = 0, the following conditions should be satisfied [2, 6]: α>

Φ , Γm

λ2 ≥

4Φ ΓM α + Φ . Γm2 Γm α − Φ

(12)

Remark 1. From Eqs. (3), (9) we can see that, rather than simply using the sign of the sliding surface function, STA adopts the sum of its integral and another continuous function. Thus STA is a continuous function, and the chattering phenomenon appearing in the FOSMC is greatly reduced. In this case, though the total invariance of sliding motion is routinely lost [7], it still presents very good robustness properties.

2 Mathematical models of power converters 2.1 Mathematical model of three-phase two-level AC/DC power converters Fig. 1 shows the topology of a three-phase two-level power converter under investigation. It operates in rectifier mode and is connected to a grid. van , vbn , and vcn are the balanced three-phase grid voltage, that is, van + vbn + vcn = 0.

6 PART | I Frequency and voltage control

Fig. 1

Topology of three-phase two-level power converter.

ia , ib , and ic are the three-phase grid current. L is the filtering inductor with internal resistance of r. RL is the assumed equivalent resistive load connected to the DC-link capacitor C. RL is not known a priori and is considered as an external disturbance. iload is the load current through RL . u = [ua ub uc ]T are the controlled switching signals for each phase, which are equal to +1 and −1, representing “ON” and “OFF” status of upper switches. The lower switches just take the opposite values as the upper ones. VDC is the DC-link voltage to be regulated. Through circuit analysis, the mathematical model of the converter under the natural abc coordinate frame is obtained as follows [8, 9]:   c 1 dia un + van , = −ria − VDC ua − L dt 3 n=a   c dib 1 L un + vbn , = −rib − VDC ub − dt 3 n=a (13)   c dic 1 L un + vcn , = −ric − VDC uc − dt 3 n=a C

dVDC = ia ua + ib ub + ic uc − iload . dt

To facilitate the controller design, the model (13) is transformed into a synchronous (d, q) reference frame through Park’s transformation with the matrix [10, 11]:



⎤  ⎡ 2 2 sin(ωt) sin ωt − π sin ωt + π 2⎣ 3 3 dq



⎦. (14) Tabc = 2 3 cos(ωt) cos ωt − 2 π cos ωt + π 3 3

Sliding mode control of grid-connected power converters Chapter | 1 7

Applying Eq. (14) to Eq. (13) yields the following system dynamics in the (d, q) frame: did = −rid + ωLiq + vd − ud VDC , dt diq = −riq − ωLid + vq − uq VDC , L dt 

dVDC C = ud id + uq iq − iload , dt L

(15)

where v = [vd , vq ]T , i = [id , iq ]T , and u = [ud , uq ]T are, respectively, grid voltage, grid current, and switching functions in (d, q) frame. ω is the angular speed of grid voltage. In (d, q) frame, the three-phase sinusoidal variables become constant variables, thus they are easier to control [11, 12].

2.2 Mathematical model of DC/DC buck converters The topology of the buck converter is presented in Fig. 2, which comprises an input DC voltage source vin , a switch device VT, a diode VD, an output capacitor C, a filter inductor L, and the equivalent load RL considered as the unknown load in this work. vo and iL represent the output voltage and inductor current, respectively. Here it should be pointed that we only study the converter operating in continuous conduction mode in this chapter. Denoting the output voltage vo and inductor current iL as the state variables, when the switch is ON, the model of the buck converter can be represented as follows [13, 14]: diL = −riL − vout + uvin , dt dvout vout C , = iL − dt RL L

(16) (17)

where iL is the inductor current, vo is the output voltage, and u is the control input.

Fig. 2

Topology of DC-DC buck converter.

8 PART | I Frequency and voltage control

3 SMC for power converters 3.1 SMC of three-phase AC/DC power converters 3.1.1 Mode uncertainties The mathematical models (13), (15) are ideal models, which are obtained under the assumption that the electrical and electronic devices and grid voltage are all ideal. However, in real applications, the grid angular speed ω, the smoothing inductor L, and its parasitic resistance r can vary in a small range, which form the model parametric uncertainties. These uncertainties would cause unexpected system behavior under some operating points. Therefore, in order to design a controller which is robust against these uncertainties, the corresponding parameters are defined as follows: ω = ω0 + Δω,

L = L0 + ΔL,

r = r0 + Δr,

(18)

where ω0 , L0 , and r0 are nominal values, and Δω, ΔL, and Δr are the parametric uncertainties which are slowly varying and unknown.

3.1.2 Control objectives As mentioned in Section 2, the three-phase two-level converter operates in rectifier mode, therefore the control objectives are as follows: ●

* : The DC-link voltage should be regulated to a desired reference value VDC * . VDC → VDC

●

(19)

The grid currents id and iq should be controlled to track their references i*d and i*q , respectively; where i*d is calculated based on DC-link voltage regulation, i*q is manually set to provide a desired power factor. id → i*d ,

iq → i*q .

(20)

3.1.3 Controller design A grid-connected three-phase two-level power converter system can be disturbed by parameter uncertainties and load variations. Therefore, the controller should be designed to have enough capability to suppress the disturbances. To achieve this, a cascaded control scheme is adopted for the system (15), which consists of a disturbance-observer-based voltage regulation loop and a current tracking loop. For the voltage loop, an extended state observer (ESO) is employed to estimate the external disturbance, which is practically the load power abruptly connected to the DC link. The estimated disturbance is then used to compensate the STA controller in the forward channel, thus the disturbance is actively rejected. Unlike conventional observers, such as high-gain observer [15], unknown input observer [16], and Luenberger observer [17], ESO regards the disturbances as new system states, therefore it is able to estimate both the

Sliding mode control of grid-connected power converters Chapter | 1 9

external disturbances and internal plant states [18–21]. For the current loop, the STA controller is adopted to quickly drive the currents id and iq to their references i*d and i*q . In the following section, the controller design of both loops is presented. Extended state observer First, the dynamics of the voltage loop need to be modified. Given that the current loop is much faster than the voltage loop [22], the voltage dynamic can be rewritten as

1 * dVDC = p − pload , (21) C dt VDC where p* = vd i*d + vq i*q and pload = VDC iload . 2 /2 is introduced in Eq. (21), To design the ESO, the new variable z = VDC yielding Cz˙ = p* − d(t),

(22)

with d(t) = pload , which is considered as an external disturbance. Thus the ESO is designed as ˆ + β1 (z − z), Cz˙ˆ = p* − d(t) ˆ ˙ˆ d(t) = −β2 (z − z), ˆ

(23) (24)

where β1 and β2 are positive gains and should satisfy that β1 β2 λ+ (25) C C is Hurwitz stable. Then, the natural frequency ωn and damping ratio ξ of the ESO are calculated as  β2 , (26) ωn = C  1 β1 . (27) ξ= 2 β2 C λ2 +

Thus the bandwidth of the ESO is determined by β1 and β2 ,   ωb = ωn 4ξ 4 − 4ξ 2 + 2 .

(28)

It can be deduced that the higher bandwidth of the ESO ensures higher observing performance, in terms of faster observing transient process and more accurate steady-state estimation. However, high bandwidth would increase the ESO’s sensitivity to noise and large oscillation during the transient state. Thus, there is a trade-off between the bandwidth and noise sensitivity when deciding the values of β1 and β2 .

10 PART | I Frequency and voltage control

ˆ as observation errors, then their dynamics Define z = z − z, ˆ d = d(t) − d(t) are as follows: C˙z = −β1 z − d , ˙d = β2 z + h (t) ,

(29) (30)

˙ is the changing rate of load disturbance. Rewrite Eqs. (29), where h(t) = d(t) (30) in the following compact form: ˙ = A + ψ, (31)   T − βC1 − C1 where  = [z , d ]T , A = , and ψ = 0 h(t) . β2 0 To demonstrate the finite-time convergence of ESO, the following lemma is given. 

Lemma 1. If h (t) is bounded, then there exists a finite time T > 0, such that when t ≥ T, the trajectory of the system (31) converges to a bounded area δ(T). Proof. The solution of Eq. (31) along time is  t (t−t0 )A  (t0 ) + e(t−τ )A ψ (τ ) dτ ,  (t) = e

(32)

t0

  where t0 is the initial time. Given that e(t−t0 )A  ≤ ke−β(t−t0 ) , where both k and β are positive, the following inequality can be obtained:  t  (t) ≤ ke−β(t−t0 )  (t0 ) + ke−β(t−τ ) ψ (τ ) dτ , t0

k ≤ ke−β(t−t0 )  (t0 ) + sup ψ (τ ) , β t0 ≤τ ≤t

(33)

= δ(t).

(34)

It can be seen that the system trajectory can enter a bounded area δ(T) when t ≥ T, and δ(T) is a positive constant which depends on k, β, and the upper bound of ψ (τ ). Capacitor voltage regulation As shown in Fig. 3, the controller for the voltage regulation loop consists of the STA and ESO. * )2 /2, then the regulation error is Define z* = (VDC z˜ = z* − z,

(35)

Cz˙˜ = −p* + d(t).

(36)

and it follows that

Sliding mode control of grid-connected power converters Chapter | 1 11

Fig. 3

Control structure of the voltage regulation loop.

Then the ESO-based STA controller for voltage regulation is proposed as ˆ p* = μDC (˜z) + d(t),

(37)

where dˆ is the estimated disturbance generated by the ESO and μDC (˜z) is the STA which takes the following form:  t μDC (˜z) = λDC |˜z|1/2 sign(˜z) + αDC sign(˜z)dτ , (38) 0

with positive gains λDC and αDC . Substituting Eq. (37) into Eq. (36), it follows that ˜ + d . Cz˙˜ = −μDC (z)

(39)

From Lemma 1, it can be obtained that ˙d  ≤ A δ + sup ψ (τ ) = Fd , t0 ≤τ ≤t

(40)

 where A = λmax (AT A) and Fd > 0. Then the sliding manifold z˜ = z˙˜ = 0 is reached if following conditions are satisfied [2, 6]: αDC > CFd ,

λ2DC ≥ 4C2 Fd

αDC + Fd . αDC − Fd

(41)

Grid current tracking Fig. 4 shows the control structure of the current tracking loop. As mentioned previously, the task of current control is to drive id , iq to i*d , i*q . i*d is generated by the voltage loop to realize the voltage regulation; i*q is manually set to provide a desired instantaneous reactive power q* : p* , vd q* i*q = . vd

i*d =

(42) (43)

12 PART | I Frequency and voltage control

Fig. 4

Control structure of the current tracking loop.

Define sliding mode variables for both currents as sd = i*d − id ,

(44)

sq =

(45)

i*q

− iq .

The first time derivative of sdq = [sd sq ]T is     ˙* r   id + L id − vLd − ωiq VDC ud s˙d = * r . + v s˙q L uq i˙q + L iq − Lq + ωid

(46)

The controllers ud and uq are designed as follows: ud = u¯ (md ), uq = u¯ (mq ), where

L0  −μd (sd ) + VDC L0  −μq (sq ) + mq = VDC

md =

(47) (48)

 vd r0 − id − i˙d* + ω0 iq , L0 L0  vq r0 − iq − i˙q* − ω0 id , L0 L0

μd (sd ) and μq (sq ) are STAs taking the following form:  t sign(sd )dτ , μd (sd ) = λd |sd |1/2 sign(sd ) + αd 0  t sign(sq )dτ , μq (sq ) = λq |sq |1/2 sign(sq ) + αq

(49) (50)

(51) (52)

0

where λi , αi , and i ∈ {d, q} are positive gains, and u¯ (x) is a saturation function limiting the signal to [−1, 1].

Sliding mode control of grid-connected power converters Chapter | 1 13

3.1.4 Simulation results In this section, the simulations are carried out to verify the effectiveness of the ESO-based STA control strategy. Table 1 shows the plant parameters for simulation. The filtering inductor is set as L¯ = L + 0.3 · L to simulate the parametric uncertainty. The load resistance is abruptly plugged to the DC link at t = 0.4 s. Furthermore, to demonstrate the advantage of the proposed control strategy, a well-tuned linear proportional-integral (PI) controller is used for comparing the baseline. The parameters of the ESO-based STA and PI controller are given in Table 2, which ensures that the current loop acts much faster than the voltage loop. Fig. 5 shows the transient performance of the voltage loop after an abrupt load (5.625 kW) connection. The controller starts operation at 0.15 s just after the precharge stage. It can be seen that the proposed strategy reaches the voltage reference with less overshoot and shorter transient process compared with PI. TABLE 1 Plant parameters for simulation. Parameter

Value

Description

f1

106

Simulation rate (Hz)

f2

104

Controller evaluation rate (Hz)

f3

104

Pulse width modulator rate (Hz)

RL

100

Load resistance ()

C

3300

DC-link capacitor (μF)

L

2

Phase inductor (mH)

w

50

Grid frequency (Hz)

vabc

400

Grid line voltage (V)

* VDC

750

Desired output voltage (V)

TABLE 2 Controller parameters. Gains

Value

Current tracking loop

(λd , αd ), (λq , αq )

(20, 150), (20, 150)

Voltage regulation loop

(λ, α), (β1 , β2 )

(15, 100), (6.6, 3300)

Current tracking loop

(kpd , kid ), (kpq , kiq )

(5, 200), (5, 200)

Voltage regulation loop

(kp , ki )

(0.2, 5)

STA-ESO

PI

14 PART | I Frequency and voltage control 800

DC−link voltage performance (V)

700 750

600

745 500

740 735

400 730 725 0.35

300

0.4

0.45

0.5

0.55

0.6

200

Reference ESO−based SOSM PI

100 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s) Fig. 5

DC-link voltage regulation performance.

When the load is connected at t = 0.4 s, the proposed strategy results in 3 V voltage drop and recovers to the reference value within 0.025 s, while the corresponding results obtained by PI are 25 V and 0.2 s. Fig. 6 shows one phase of grid current (ia ) and corresponding grid voltage (van ). It can be seen that both the controllers achieve unity power factor. However, from Fig. 7, the total harmonic distortion value obtained by the proposed controller 0.83% is much less than that obtained by PI 2.28%. Figs. 8 and 9 show the performance of the current tracking loop. It can be seen that the proposed strategy results in a shorter settling time (0.02 s) than PI (0.2 s). Fig. 10

(A)

(B)

Fig. 6 Grid current (ia ) and corresponding voltage (0.04 × van ). (A) ESO-based STA and (B) PI control.

Sliding mode control of grid-connected power converters Chapter | 1 15

1.5

1.5

(%)

2

a,k

1

0.5

0

(A)

1

i

ia,k (%)

2

0.5

0

2000

4000

6000

8000

Frequency (Hz)

10,000 12000

0

0

(B)

2000

4000

6000

8000

10,000 12,000

Frequency (Hz)

Fig. 7

Harmonic spectrum of grid current. (A) ESO-based STA and (B) PI control.

Fig. 8

id tracking performance. (A) ESO-based STA and (B) PI control.

16 PART | I Frequency and voltage control

Fig. 9

iq tracking performance. (A) ESO-based STA and (B) PI control.

shows that the ESO estimates the external disturbance accurately. To this end, the simulations show that the proposed strategy is advantageous over PI.

3.2 SMC for DC/DC buck converters 3.2.1 Control objectives The aims of DC/DC buck converters control are as follows: ●

●

The inductor current iL should track the reference signal i*L calculated in such a way that output DC voltage is driven to a desired value (i.e., iL → i*L ). The output DC voltage vout should be regulated to a constant reference vref (i.e., vout → vref ).

Sliding mode control of grid-connected power converters Chapter | 1 17

Fig. 10

Observed disturbance from the ESO.

3.2.2 Design of controller In this section, a control scheme will be designed for the DC-DC buck converter to regulate output voltage and drive inductor current tracking its reference by using the SMC method. As shown in Fig. 11, this control strategy is the cascade control structure including two control loops: a voltage regulation loop and a current tracking loop. An ESO-based second-order sliding mode controller is applied in the voltage regulation loop to regulate the output voltage and a second-order sliding mode controller is utilized in the internal loop to force the inductor current tracking its reference. Next, we will give the detailed design procedures.

vref

iL*

u

ESO-STA

vout

STA

-Voltage regulation i L

loop

Fig. 11

S PWM

Proposed cascade control structure.

Current tracking loop

DC-DC Buck Converter

18 PART | I Frequency and voltage control

Voltage regulation loop The goals of the voltage regulation loop (i.e., outer loop) are to regulate the output voltage to a certain desired value and provide the current command for the current tracking loop. Assume that the dynamic of current is faster than the dynamic of voltage, then one can get the dynamic of output voltage as dvout (53) = i*L − iload , dt in which load current iload = vRoutL is considered as an unknown perturbation. Next, we will design the linear ESO to estimate the unknown perturbation. An ESO is designed as follows:

 (54) Cv˙ˆout = i*L − iˆload + η1 vout − vˆout , ˙iˆ = −η v − vˆ , (55) C

load

2

out

out

in which vˆout is the estimation of vout , iˆload is the estimation of iload , and η1 and η2 are positive constants which can guarantee iˆload estimating iload . Define the estimate error v˜ out = vout − vˆout and ˜iload = iload − iˆload , and their dynamics can be expressed as Cv˙˜out = −η1 v˜ out − ˜iload , i˙˜load = −η2 v˜ out + i˙load .

(56) (57)

Moreover, one can use the matrix to represent the earlier system: ε(t) ˙ = Aε(t) + ζ (t), (58)    η1  v˜ 0 − − C1 , and φ(t) = ˙ . in which ε(t) = ˜ out , A = η2C i iload 0 load C It can be observed that A is Hurwitz matrix if η1 and η2 are proper parameters. Assuming that the i˙load is bounded, one can obtain the solution of Eq. (58):  t A(t−t0 ) ε(t0 ) + eA(t−t0 ) ζ (τ )dτ . (59) ε(t) = e 



t0

  λmax (A) Based on [23], one can obtain that eA(t−t0 )  ≤ ce 2 (t−t0 ) , in which λ is the max eigenvalue of A and c is a positive constant. Then Eq. (59) becomes  t λmax (A) λmax (A) ε(t) = ce 2 (t−t0 ) ε(t0 ) + e 2 (t−t0 ) t0

× ζ (τ ) dτ , ≤ ce

λmax (A) (t−t0 ) 2

ε(t0 ) −

2c sup ζ (τ ) . λmax (A) t0 ≤τ ≤t

(60)

Sliding mode control of grid-connected power converters Chapter | 1 19

Hence, one can conclude that estimate errors are bound, that is, ε(t) ≤ , where  > 0. Next, we will design a composite controller combined with the ESO and SOSMC. Define the voltage regulation error ev = vref − vout , and one can get its dynamic as Ce˙v (t) = −i*L + iload .

(61)

Design the following controller: i*L = uv (ev ) + iˆload , (62)  t in which uv (ev ) = μv |σ (ev )|1/2 sign(ev ) + αv t0 sign(ev )ds is the STA which belongs to one type of SOSMC, where μv and αv are the positive constants to be designed. Substituting Eq. (62) into Eq. (61) yields (63) Ce˙v (t) = −uv (ev ) + i˜load . Based on the result of [2], it can be concluded that the composite controller can regulate the output voltage to its reference if the μv and αv are selected as αv + Hv , (64) αv > CHv , μ2v ≥ 4C2 Hv αv − Hv     where Hv > 0 meeting i˙˜load  ≤ Hv . Fig. 12 shows the structure of the voltage regulation loop. Current tracking loop The STA controller is applied to the current tracking loop which ensures convergence of inductor current iL to its reference i*L calculated from the external loop to regulate output voltage. Define the current tracking error ei = i*L − iL , and from Eq. (16) can be obtained r vout vin − u. (65) e˙i = i˙*L + iL + L L L An STA controller is designed to guarantee that current tracking error converges to zero: vout r

L + iL , ui (ei ) + (66) u= vin L L vref

uv( ev

iL*

)

STA

–

+

iˆload

Voltage regulation loop

vout Fig. 12

Voltage regulation loop.

ESO

20 PART | I Frequency and voltage control

vout i

1

L

* L

ui ( ei ) + STA

–

iL Fig. 13

L

u vin

+ Current tracking loop r

L

Current tracking loop.

t where ui (ei ) = μi |σ (ei )|1/2 sign(ei ) + αi t0 sign(ei )ds. Substituting Eq. (66) in Eq. (65), the following equation is deduced: e˙i = −ui (ei ) + i˙L* .

(67)

From Eqs. (12), (67), one can obtain that regulation error ei converges to 0 if the parameters μi and αi are chosen such that αi + Hi , (68) αi − Hi   in which Hi is positive constant satisfying i¨L*  ≤ Hi . Fig. 13 shows the structure of the current tracking loop. αi > Hv ,

μ2i ≥ 4Hi

3.2.3 Simulation results Two simulation results (PI and the proposed SOSM approach), made by MATLAB/Simulink, are shown in this section to validate the effectiveness of the proposed strategy for DC-DC buck converters. The parameters and variables of the simulated system are presented in Table 3.

TABLE 3 Proposed system parameters. Parameters

Value

Description

f

2 × 103

Switching rate (Hz)

RL

40 → 20

Load resistance ()

C

4700

Output capacitor (μF)

L

2

Filter inductor (mH)

r

0.01

Parasitic resistance ()

vin

300 → 250

Input voltage (V)

vref

200 → 150

Output voltage reference (V)

Sliding mode control of grid-connected power converters Chapter | 1 21

In order to compare the robustness between the PI control and the proposed SOSM approach, three simulations have been carried out: controller against load resistance variation, input voltage variation, and reference voltage variation. Against load resistance variation In this simulation, a load resistance steps at the output link (from 40 to 20 ). The parameters of the PI and the proposed SOSM controllers are shown in Tables 4 and 5, respectively. Fig. 14 shows the dynamic of output voltage of

TABLE 4 SOSM controller design parameters. Controller

Variable

Value

Internal loop

(μi , αi )

External loop

(μv , αv )

(1 × 103 , 2 × 102 )

(η1 , η2 )

(39.6, 1.65 × 103 )

1.5, 10



TABLE 5 PI controller design parameters.

Fig. 14

Controller

Variable

Value

Internal loop

(Kip , Kii )



2.0 × 105 , 1.0 × 106

External loop

(Kvp , Kvi )

(0.8, 7)

Output voltage. (A) SOSM control and (B) PI control.

22 PART | I Frequency and voltage control

the buck converter when the load varies. It can be seen that both strategies can achieve the output voltage regulation. The dynamical characters of the buck converter are excellent, and dynamic excess, adjusting time, and the ripple of output voltage satisfy all the requirements. However, the dynamic performance of the proposed scheme is better than traditional PI control, especially the transient voltage drop and recovery time. Specifically, for our proposed control scheme, the voltage drop is only 0.05 V, and recovery time needs 80 ms after load steps at t = 0.5 s, while for PI control, the voltage drop is 5 V, and recover time needs 500 ms after the load is stepped at t = 0.5 s. This means that the proposed SOSM controller has a better robustness against load resistance variation. Fig. 15 presents the trajectories of inductor current along with its reference in the presence of load variation. It is easily remarked that both control laws can achieve the inductor current to track its reference signal coming from the external loop. Yet the faster dynamics are achieved with the proposed controller than the PI controller. The dynamic responses of load current and its corresponding estimation are pictured in Fig. 16, where the ESO can estimate the load current efficiently. Against input voltage variation In this simulation, the input voltage varies from 300 to 250 V at 0.5 s. The parameters of the PI and the proposed SOSM controllers are the same as the first simulation. Figs. 17 and 18 show the transient responses of the output voltage and inductor current of the buck converter when input voltage varies, respectively. One can see that both strategies can accomplish the voltage regulation and current tracking, and other performance indicators such as

(A)

(B) Fig. 15

Curves of iL and i*L . (A) SOSM control and (B) PI control.

Sliding mode control of grid-connected power converters Chapter | 1 23

Fig. 16

The dynamic of ESO.

Fig. 17

Output voltage. (A) SOSM control and (B) PI control.

dynamic excess, adjusting time, and the ripple of output voltage are all in an acceptable range. However, note that the performance of the proposed SOSM controller is obviously better than the PI control, and the input voltage variation hardly causes damage to the proposed SOSM method. Against reference voltage variation In this simulation, the reference voltage changes at 0.5 s from 200 to 150 V. The parameters of the PI and the proposed SOSM controllers are the same as the

24 PART | I Frequency and voltage control

Fig. 18

Curves of iL and i*L . (A) SOSM control and (B) PI control.

Fig. 19

Output voltage. (A) SOSM control and (B) PI control.

first simulation. Fig. 19 shows the output voltage of the buck converter when reference voltage changes. It can be seen that both strategies can achieve the output voltage regulation, while the proposed scheme has less dynamics excess and a quicker dynamic response. Specifically, for our proposed control scheme, the voltage drop is only 1 V, and recovery time needs 10 ms after the reference voltage steps at t = 0.5 s, while for PI control, the voltage drop is 3.5 V, and recovery time needs 300 ms. This means that the proposed SOSM controller can quickly regulate the voltage with low overshot. From Fig. 20, it can be seen

Sliding mode control of grid-connected power converters Chapter | 1 25

Fig. 20

Curves of iL and i*L . (A) SOSM control and (B) PI control.

that both control laws can fulfill current tracking, but the PI control has a quicker dynamic response.

4 Conclusion We proposed an observer-based sliding mode control for both three-phase two-level grid-connected power converters and DC/DC buck converters in this chapter. For three-phase AC/DC power converters, the proposed control strategy consists of the ESO and STA. An ESO is designed to estimate the load disturbance abruptly connected to the DC link, and the estimated value is used to compensate the STA in the voltage loop. Two STAs work in parallel in the current tracking loop to control the currents of the two axes. Stability analysis is provided to prove the convergence of the ESO and STA. Simulation results are carried out to verify the effectiveness of the ESO-based STA in the presence of load variation and parameter uncertainty. Simulation results show that the proposed ESO-based STA outperforms conventional PI control in terms of less overshoot, shorter settling time, and less sensitivity to external disturbances and parametric uncertainties. The proposed ESO-based STA shows obvious advantages over conventional PI control, in terms of both disturbance rejection and reactive power injection ability. For DC/DC buck converters, the problem of output voltage regulation has been investigated in this chapter. A sliding mode controller is proposed, which forces the input currents to track the desired values. A cascade control strategy is proposed, which consists of two control loops: a voltage regulation loop and a current tracking loop. An ESO-based STA is employed in the voltage regulation loop to regulate the output voltage, while an STA is adopted in the internal loop

26 PART | I Frequency and voltage control

to force the inductor current tracking its reference. Three simulations are carried out to compare the robustness between the PI control and the proposed ESObased STA approach, including controller against load resistance variation, input voltage variation, and reference voltage variation. Based on these simulation results, the proposed control strategy exhibits more excellent performance in comparison with the classical PI control under two conditions: reference voltage variation and unknown equivalent load variation.

References [1] V. Utkin, Variable structure systems with sliding modes, IEEE Trans. Autom. Control 22 (2) (1977) 212–222. [2] A. Levant, Robust exact differentiation via sliding mode technique, Automatica 34 (3) (1998) 379–384. [3] A. Levant, Principles of 2-sliding mode design, Automatica 43 (4) (2007) 576–586. [4] J. Davila, L. Fridman, A. Levant, Second-order sliding-mode observer for mechanical systems, IEEE Trans. Autom. Control 50 (11) (2005) 1785–1789. [5] I. Boiko, L. Fridman, A. Pisano, E. Usai, Analysis of chattering in systems with second-order sliding modes, IEEE Trans. Autom. Control 52 (11) (2007) 2085–2102. [6] A. Levant, Sliding order and sliding accuracy in sliding mode control, Int. J. Control 58 (6) (1993) 1247–1263. [7] S.K. Spurgeon, Sliding mode observers: a survey, Int. J. Syst. Sci. 39 (8) (2008) 751–764. [8] C.T. Pan, T.C. Chen, Modelling and analysis of a three phase PWM AC-DC convertor without current sensor, IEE Proc. B Electr. Power Appl. 140 (3) (1993) 201–208. [9] G. Escobar, R. Ortega, A.J. Van Der Schaft, A saturated output feedback controller for the three phase voltage sourced reversible boost type rectifier, in: Proceedings of the 24th Annual Conference of the IEEE Industrial Electronics Society, IECON ’98, vol. 2, 1998, pp. 685–690. [10] B.K. Bose, Modern Power Electronics and AC Drives, Prentice Hall, Upper Saddle River, New Jersey, U.S.A. 2002. [11] T.-S. Lee, Input-output linearization and zero-dynamics control of three-phase AC/DC voltage-source converters, IEEE Trans. Power Electron. 18 (1) (2003) 11–22. [12] Y. Shtessel, S. Baev, H. Biglari, Unity power factor control in three-phase AC/DC boost converter using sliding modes, IEEE Trans. Ind. Electron. 55 (11) (2008) 3874–3882. [13] V. Utkin, Sliding mode control of DC/DC converters, J. Franklin Inst. 350 (8) (2013) 2146–2165. [14] H. Sira-Ramirez, G. Escobar, R. Ortega, On passivity-based sliding mode control of switched DC-to-DC power converters, in: Proceedings of 35th IEEE Conference on Decision and Control, vol. 3, (1996) pp. 2525–2526. [15] A.N. Atassi, H.K. Khalil, A separation principle for the stabilization of a class of nonlinear systems, IEEE Trans. Autom. Control 44 (9) (1999) 1672–1687, https://doi.org/10.1109/9. 788534. [16] M.-S. Chen, C.-C. Chen, Unknown input observer for linear non-minimum phase systems, J. Franklin Inst. 347 (2) (2010) 577–588. [17] D.G. Luenberger, Observing the state of a linear system, IEEE Trans. Military Electron. 8 (2) (1964) 74–80, https://doi.org/10.1109/TME.1964.4323124. [18] J. Han, From PID to active disturbance rejection control, IEEE Trans. Ind. Electron. 56 (3) (2009) 900–906, https://doi.org/10.1109/TIE.2008.2011621.

Sliding mode control of grid-connected power converters Chapter | 1 27

[19] W.-H. Chen, D.J. Ballance, P.J. Gawthrop, J. O’Reilly, A nonlinear disturbance observer for robotic manipulators, IEEE Trans. Ind. Electron. 47 (4) (2000) 932–938. [20] L. Sun, D. Li, Q.C. Zhong, K. Lee, Control of a class of industrial processes with time delay based on a modified uncertainty and disturbance estimator, IEEE Trans. Ind. Electron. 63 (11) (2016) 7018–7028, https://doi.org/10.1109/TIE.2016.2584005. [21] X. Chen, S. Komada, T. Fukuda, Design of a nonlinear disturbance observer, IEEE Trans. Ind. Electron. 47 (2) (2000) 429–437. [22] F. Umbria, J. Aracil, F. Gordillo, F. Salas, J.A. Sanchez, Three-time-scale singular perturbation stability analysis of three-phase power converters, Asian J. Control 16 (5) (2014) 1361–1372. [23] W. Zhang, M.S. Branicky, S.M. Phillips, Stability of networked control systems, IEEE Control Syst. 21 (1) (2001) 84–99.

Chapter 2

Distributed voltage restoration and power allocation control in islanded DC microgrids Fanghong Guoa , Lei Wangb and Changyun Wenc a Department

of Automatic Control, Zhejiang University of Technology, Hangzhou, Zhejiang, People’s Republic of China, b School of Electrical Engineering and Computing, University of Newcastle, Newcastle, NSW, Australia, c School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore

1 Introduction In recent years, more renewable energy sources have penetrated into existing power systems to meet the rapidly increasing electricity demand [1]. One popular solution to integrate different types of renewable energy sources is the concept of the microgrid (MG). An MG is a small-scale power system that integrates a number of local distributed generator (DG) units, energy storage systems, and local loads together [2]. There are different types of MGs reported in the literature. According to the coupling bus type, the MG can be classified into three types: alternating current (AC) MG, direct current (DC) MG, and hybrid AC/DC MG. DC MG is gaining increasing attention due to its high efficiency, reliability, and scalability. In this chapter, we focus mainly on the DC MG, which usually operates in an islanded mode [3, 4]. In islanded DC MG, droop control is usually implemented in the local DG controller to realize the power sharing [5, 6]. It can force the current output of each DG to be proportional to the designed ratio, if the droop gains are chosen sufficiently larger than the line resistance. One of the advantages brought by the droop control method is that it is implemented in a fully decentralized way and no communication between each DG is required. However, as pointed out in [7, 8], when applying the droop control function, there is a trade-off between power sharing and voltage regulation. High droop gain leads to a high power sharing accuracy but a poor voltage regulation, that is, large DC bus voltage deviation. The DC bus voltage may vary under different load conditions, and it is hard to maintain at a desired reference value. In order to solve the voltage deviation problem, a secondary control is proposed in [9], where a centralized Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00002-5 Copyright © 2020 Elsevier Inc. All rights reserved.

29

30 PART | I Frequency and voltage control

secondary voltage compensation controller is designed. The voltage deviation error is fed to a standard proportional and integral (PI) controller, whose control signal is then sent to all the DGs. It is noted that such centralized control is usually implemented on an extra single controller, and this controller has some limitations, such as suffering easily from single-point failures and also being unable to meet the plug-and-play requirement of recent MG system [10–12]. In order to overcome the earlier limitations, distributed secondary voltage restoration control has been proposed in [13–17]. No extra centralized controller is needed and each secondary controller is located at each local DG and they can communicate with neighboring controllers, respectively. In [15], the average value of the current supplied by all the DGs is required to calculate the compensation signal, which is then added to the conventional droop control function. However, how to obtain the average value of the current is not discussed. Similarly, an average voltage sharing scheme is proposed in [16] to compensate for the voltage deviation caused by droop control. In [17], a distributed compensation method is proposed, which is based on the overall DGs’ droop gains and total load current. To some degree, this overall system information cannot be easily obtained in a distributed manner. It is also worth pointing out that all these methods are to compensate for the voltage deviation in a feed-forward fashion. In this chapter, a new distributed secondary control scheme is proposed, which is not only able to restore the DC bus voltage to the reference value, but also can maintain the power sharing accuracy brought by the droop control. Different from the above-mentioned methods, the proposed method is designed based on the feedback mechanism and no extra information such as the average current, average voltage, all the DGs’ droop gains are needed. It only requires measuring and feeding back the DC bus voltage to certain secondary controllers. Each secondary controller is located at a corresponding local DG, and its designed control output is fed to the droop control function in the primary layer, respectively. By allowing the secondary controller to communicate with its neighboring controllers, the proposed distributed secondary control method can achieve the same goal as the centralized one. In order not to destroy the power sharing accuracy brought by droop control, the designed control inputs from all the local secondary controllers should be equal at steady state. This imposes a constraint to the control inputs, which is a new challenge in design and analysis. Hence, to overcome the challenges, a secondary control error is defined by combining the voltage restoration error and control input consensus error, which are then fed to a PI controller to produce the secondary control input. The stability of the proposed distributed secondary control is analyzed by establishing a sufficient condition for the controller parameters. In addition, the proposed method can be further simplified by employing the idea of pinning control, that is, only the DGs with nonzero pinning gain get the DC bus voltage feedback. The main contributions of the proposed method can be summarized as follows: (1) The proposed secondary controllers are fully distributed, which

Distributed voltage restoration and power allocation Chapter | 2 31

can meet the plug-and-play requirement. (2) No overall system information is needed, which makes the designed controllers easy to implement and reliable in the sense that they do not suffer single-point failures. (3) Only DC bus voltage is needed to measure and feedback, which reduces the system cost. (4) A new control strategy, including employing the idea of pinning control, is proposed to overcome challenges caused by the constraints in ensuring the simplicity and flexibility of the controllers. (5) The effectiveness of the designed controllers is analyzed theoretically and verified by experimental results.

2 Problem formulation and objectives In an islanded DC MG, droop control is usually employed to realize the power sharing among all the DGs in a totally decentralized way. In this section, we first briefly review and analyze this decentralized power sharing method, and then point out some existing problems such as DC bus voltage deviation caused by this method.

2.1 Droop control The block diagram of decentralized power sharing method for each local DG is shown in Fig. 1, which is indeed the primary control layer of the islanded DC MG. Note that if the voltage and current control loops are well-designed, the converter output DC voltage Vi is fast enough to track the reference voltage ref Vi , that is, ref

Vi = Vi ref

The voltage reference Vi follows [18]:

is generated by the droop control function as ref

Vi

DC/DC converter

+

(1)

= V * − ki Ii

(2)

DC bus Ri

Vi

Li

Ii

Vb

Ci PWM Current Control loop

Fig. 1

Il

I

Vi

Voltage ref control loop

V ref

Decentralized primary control for power sharing.

Droop control

Primary control

32 PART | I Frequency and voltage control

where V * is nominal DC voltage, ki is the droop gain, and Ii is the output current of the ith DG. For an islanded DC MG with several DGs connected in parallel, the DC bus voltage Vb is Vb = Vi − Ri Ii

(3)

Combining Eqs. (1)–(3), we have Vb = V * − (Ri + ki )Ii

(4)

which implies that (Ri + ki )Ii = (Rj + kj )Ij ,

∀i, j

(5)

From Eq. (5), we can see that the power sharing ratio is inversely proportional to the sum of the line resistance Ri and designed droop gain ki , that is, Rj + kj Ii = , ∀i, j (6) Ij Ri + ki If the droop gain ki is set much larger than the line resistance Ri , that is, ki  Ri

(7)

then we have Rj + kj kj Ii = ≈ , Ij Ri + ki ki

∀i, j

(8)

which implies under condition (7), power sharing ratio (8) is dominated by the designed droop gain ki , which acts like a “virtual impedance,” as shown in Fig. 2. In other words, if the droop gain ki is properly chosen according the power rating ratio of each DG, then the proportional power sharing can be realized among all the DGs.

2.2 Objectives By further looking into Eq. (4), the DC bus voltage Vb will deviate from the nominal voltage V * as long as Ii  = 0. Furthermore, the larger the droop gain ki is, the more the voltage deviation V * − Vb is. Hence the objective of this chapter V ref

Droop control

V ref = V * − kI

V* V2

k2

V1

k1

Virtual I impedance

I1

Fig. 2

The diagram of proposed distributed secondary control.

Distributed voltage restoration and power allocation Chapter | 2 33

is to restore the DC bus voltage Vb to the nominal value V * while maintaining the power sharing accuracy (Eq. 8). In the next section, a fully distributed secondary control method is proposed to solve the voltage restoration problem.

3 Distributed secondary control for voltage restoration In this section, a new distributed secondary voltage restoration method is proposed for an islanded DC MG system, where several DC DGs are connected in parallel. Each DG is directly controlled by the droop function-based primary controller, where its detailed diagram is shown in Fig. 1. A secondary control signal ui is to be designed to add into the droop function (2) in the primary control, that is, ref

Vi

= V * − ki Ii + ui

(9)

which results in revising Eq. (4) as Vb = V * − (Ri + ki )Ii + ui

(10)

From Eq. (10), it can be easily concluded that if we need to maintain the power sharing accuracy brought by the droop control, that is, to make Eq. (8) still hold, there exists a constraint for the designed secondary control input ui . That is, in the steady state, all the secondary control input should be equal, that is, (ui )s = (uj )s ,

∀i, j

(11)

where (ui )s denotes the value of ui in the steady state.

3.1 Controller design From the previous analysis, we conclude that the objective of the proposed secondary controller is to restore the DC bus voltage Vb to the reference value V * , that is, lim eV (t) = 0, while meeting the control input constraint, that is, t→∞

lim eui (t) = 0, ∀i, where eV and eui are named as the voltage restoration error t→∞ and control input consensus error, which are defined as follows, respectively: eV = V * − Vb  eui = (uj − ui )

(12) (13)

j∈Ni

where Ni denotes the neighborhood set of the ith secondary controller. Then a simple distributed secondary control is designed as  ui = KPi ei + KIi ei dt

(14)

34 PART | I Frequency and voltage control

+

DC/DC converter Vi

DC bus Ri

Li

Ii

Ci

Primary control

V*

PWM

I ref

Current control

Secondary V* + control -

gi

+

ui

PI

Pinning gain

αi

+

Ii +

V ref

Voltage control

+ β i

Vb

-

Droop gain

ki

ui +

-

Vb

uj From neighboring converter

Fig. 3

The diagram of proposed distributed secondary control.

where KPi and KIi are the coefficients of the proportion and integral (PI) controller for the ith DG, ei is the combination error, which is defined as ei = αi eV + βi eui

(15)

where αi , βi ∈ R+ are the combination gains. The detailed diagram of proposed secondary control is shown in Fig. 3. Note that different from existing feed-forward methods [13–17], the proposed secondary voltage restoration is realized on a feedback mechanism. In addition, no overall system information is required. In addition, it is worth pointing out that the proposed distributed control methodology can be also applied to AC and AC/DC MGs with some modifications on the proposed scheme according to particular control tasks. The main common ideas behind these distributed control methods include distributed PI control and pinning control. In AC MG, the frequency and voltage restoration problem can be solved by applying this distributed control approach [19].

3.2 Stability analysis We now analyze the proposed distributed secondary controller and establish a sufficient condition for system stability. The closed-loop system consisting of Eqs. (10), (12)–(15) can be obtained as follows: eV 1n×1 = (R + k)I − u

(16)

Distributed voltage restoration and power allocation Chapter | 2 35

u˙ = KP e˙ + KI e

(17)

e = αeV 1n×1 − βLu

(18) T where R = diag(R1 , R2 , . . . , Rn ), k = diag(k1 , k2 , . . . , kn ), I = I1 I2 . . . In , T  u = u1 u2 . . . un , KP = diag(KP1 , KP2 , . . . , KPn ), KI = diag(KI1 , KI2 , . . . , T  KIn ), e = e1 e2 . . . en , α = diag(α1 , α2 , . . . , αn ), β = diag(β1 , β2 , . . . , βn ), L is the Laplacian matrix of the communication graph, 1n×1 denotes the ndimension vector with all the elements equal to 1. Assume that the resistance load is connected to the DC MG, then we have 

1Tn×1 I =

Vb RL

(19)

where RL is the load resistance. Before giving the main results, two important lemmas are presented as follows. Lemma 1. A = D + c1Tn×1 1n×1 ∈ Rn×n is invertible, and its inverse matrix A−1 is a strictly diagonally dominant matrix with the diagonal elements being positive, where D = diag(d1 , d2 , . . . , dn ) ∈ R+ n×n is a diagonal matrix, and c > 0 is some positive constant. Proof. Suppose ⎡ ⎢ ⎢ A=⎢ ⎣

c d1 + c c d2 + c .. .. . . c c

Doing elementary row operation of subtract row 1 (r1 ), then we have ⎡ d1 + c c · · · ⎢ −d1 d2 · · · ⎢ A = ⎢ .. .. .. ⎣ . . . −d1

0

···

··· ··· .. .

c c .. .

···

dn + c

⎤ ⎥ ⎥ ⎥ ⎦

(20)

A by letting each row (r2 , . . . , rn ) c 0 .. .

⎤ ⎥ A ⎥ 1 ⎥= ⎦ A 3

A 2 A 4

(21)

dn

T    where A 1 = d1 + c, A 2 = c · · · c ∈ R1×n−1 , A 3 = −d1 · · · −d1 ∈ Rn−1×1 , and A 4 = diag(d2 , d3 , . . . , dn ) ∈ Rn−1×n−1 . As det(A) = det(A ) = det(A 1 ) det(A1 )

(22)

where A1 = A 4 − A 3 A 1

−1 A2

= diag(d2 , d3 , . . . , dn ) +

d1 c T 1n−1×1 1 d1 + c n−1×1

(23)

36 PART | I Frequency and voltage control

Repeating the previous procedure, we finally have det(A) = (d1 + c)(d2 + c1 ) . . . (dn + cn−1 )

(24)

i ci−1 where ci = ddi +c , i = 1, . . . , n − 1 with c0 = c. i−1 Rewriting Eq. (24), we obtain

det(A) = d1 d2 . . . dn + cd2 d3 . . . dn + cd1 d3 . . . dn + · · · + cd1 d2 . . . dn−1 (25) It is obvious that det(A)  = 0, hence A is invertible, and its invert matrix is denoted as A−1 . Now we are going to prove A−1 is a strictly diagonally dominant matrix. Let adj(A) be the adjoint matrix of A, which is denoted as ⎤ ⎡ A11 A12 · · · A1n ⎢ A21 A22 · · · A2n ⎥ ⎥ ⎢ (26) adj(A) = ⎢ . .. .. .. ⎥ ⎣ .. . . . ⎦ An1 An2 · · · Ann Note that    d2 + c  c ··· c     + c · · · c c d 3   A11 =  (27)  .. .. .. ..   . . . .    c c · · · dn + c     c  c ··· c    c d3 + c · · ·  c   A12 =  . (28)  . . . .. .. ..  ..     c c · · · dn + c       c d2 + c · · ·    c c c ··· c      c    c d2 + c · · · c c · · · c     A13 =  . = − .   (29) . . . . . . .. .. .. .. .. ..  ..    ..      c   c c · · · dn + c  c · · · dn + c      c d2 + c · · ·  c  c c ··· c      ..    .. .. .. c c d2 + c · · · .    n−1 . . . A1n =    ..  = (−1) .. .. .. c    . . . c · · · dn−1 + c    . c   c ··· c c c · · · dn−1 + c  (30) Then it is easy to obtain that |A11 | = d2d3 .. . dn + cd3 d4 . . . dn + · · · + cd2 d3 . . . dn−1 , |A12 | = cd3 d4 . . . dn , . . ., A1j  = cd2 . . . dj−1 dj+1 . . . dn , j = 3, . . . , n − 1, and |A1n | = cd2 d3 . . . dn−1 . Hence we have |A11 | −  n  A  = d2 d3 . . . dn > 0. In a similar way, we also obtain |Aii | −  1j j=2 n   j=2 Aij > 0, ∀i = 1, . . . , n, which means the adjoint matrix adj(A) is a

Distributed voltage restoration and power allocation Chapter | 2 37 1 strictly diagonally dominant matrix. As the inverse matrix A−1 = det(A) adj(A), −1 then we conclude that A is also a strictly diagonally dominant matrix with the diagonal elements being positive (as det(A) > 0). This completes the proof of Lemma 1.

Lemma 2. The matrix Z = −αRL 1n×1 1Tn×1 A−1 − βL is Hurwitz, if the control parameters αi and βi are chosen that αβii = si > 0, ∀i = 1, . . . , n, where si is the sum of row i in matrix A−1 . Proof. Lemma 1 implies that the sum of each row or column (as A−1 is also a symmetric matrix) in A−1 is positive. Then we can rewrite the matrix Z as Z = −RL α1n×1 ST − βL 

(31)

T

where S = s1 s2 . . . sn with si > 0. In the following, we will show that Z is Hurwitz if the control parameters, αi and βi , are properly chosen. Consider a system with the following dynamics: x˙ = Zx where x ∈ Rn is the system state. A Lyapunov candidate is chosen as 1 V = xT β −1 x 2 where β is chosen as β = diag(β1 , . . . , βn ) > 0. Then we have V˙ = xT β −1 Zx = −xT β −1 (RL α1n×1 ST + βL)x

(32)

(33)

(34)

= −xT (RL β −1 α1n×1 ST + L)x If the designed control parameters αi and βi are chosen such that β −1 α1n×1 = S, that is, αi = si , ∀i = 1, . . . , n (35) βi then we have

 2 V˙ = −xT (RL SST + L)x = −RL ST x − xT Lx ≤ 0

Note that Eq. (36) becomes equality only when  T S x=0 xT Lx = 0

(36)

(37)

The second equation in Eq. (37) implies that x = m1n×1 , where m is an arbitrary value. Substituting it into the first equation in Eq. (37), it yields  mST 1n×1 = 0, that is, m ni=1 si = 0, which gives m = 0. This means that V˙ < 0, for x  = 0n×1 .

38 PART | I Frequency and voltage control

The above derivations conclude that the system (32) is global exponential stable (G.E.S.), which further implies that Z = −RL α1n×1 ST − βL is Hurwitz if the condition (35) is satisfied. Theorem 1. Consider the primary control system in Eq. (10). The proposed secondary controller (Eqs. 12–15) ensures the following results: (i) the DC bus voltage restores to its nominal value V * , that is, lim eV (t) = 0; and (ii) the t→∞ power sharing accuracy in Eq. (8) is guaranteed if the controller parameters αi and βi are chosen such that the matrix Z = −αRL 1n×1 1Tn×1 A−1 −βL is Hurwitz, and the PI parameters satisfy KPi ≥ 0, KIi > 0, where A = R+k +RL 1n×1 1Tn×1 . Proof. Substituting Eq. (19) into Eq. (12), we have eV = V * − Vb = V * − RL 1Tn×1 I

(38)

Substituting Eq. (38) into Eq. (16), we have eV 1n×1 = (V * − RL 1Tn×1 I)1n×1 = (R + k)I − u

(39)

which implies that (R + k + RL 1n×1 1Tn×1 )I = V * 1n×1 + u

(40)

According to Lemma 1, it is easy to conclude that A = R + k + RL 1n×1 1Tn×1 is invertible. Then I = A−1 V * 1n×1 + A−1 u

(41)

Substituting Eq. (41) into Eq. (16), we obtain eV 1n×1 = (R + k)I − u

(42)

−1 *

= (R + k)A

V 1n×1 + [(R + k)A

−1 *

= (R + k)A

−1

V

− E]u

1n×1 − RL 1n×1 1Tn×1 A−1 u

Then substituting Eq. (42) into Eq. (18), we have e = αeV 1n×1 − βLu

(43)

= α(R + k)A−1 V * 1n×1 − (αRL 1n×1 1Tn×1 A−1 + βL)u Differentiating both sides of Eq. (43) with respect to time t, e˙ = −(αRL 1n×1 1Tn×1 A−1 + βL)u˙

(44)

Substituting Eq. (17) into Eq. (44), we have e˙ = Z(KP e˙ + KI e)

(45)

−(αRL 1n×1 1Tn×1 A−1

+ βL). where Z = According to Lemma 2, if the control parameters αi and βi are properly chosen, then the matrix Z is Hurwitz. We rewrite Eq. (45) as (E − ZKP )e˙ = ZKI e where E is an identity matrix.

(46)

Distributed voltage restoration and power allocation Chapter | 2 39

From Eq. (46), it is easy to conclude that the following objective will be achieved if the PI parameters are chosen as KPi > 0, KIi > 0. lim e(t) = 0n×1

t→∞

(47)

In addition, it can be also obtained that if KPi is set as KPi = 0, then Eq. (45) becomes e˙ = ZKI e

(48)

And if KIi > 0, Eq. (47) is also achieved. Combining Eqs. (18), (47) implies that lim αeV (t)1n×1 = lim βLu(t)

t→∞

t→∞

(49)

Multiplying 1Tn×1 on both sides of Eq. (49), we get lim eV (t)

t→∞

n 

αi = lim 1Tn×1 βLu(t)

i=1

t→∞

(50)

If βi is chosen  such that βi = βj , ∀i, j, then the right-hand side of Eq. (50) is equal to 0. As ni=1 αi  = 0, thus lim eV (t) = 0

(51)

lim Lu(t) = 0

(52)

t→∞ t→∞

Eq. (51) implies that our proposed secondary controller can restore the DC bus voltage Vb to the reference value V * . In addition, Eq. (52) indicates that lim (ui (t) − uj (t)) = 0, ∀i, j, which satisfies the control input constraint and t→∞ further guarantee the power sharing accuracy in Eq. (8). This completes the proof of Theorem 1.

3.3 Distributed secondary control with pinning gain In the last section, the voltage restoration error (Eq. 12) is included in the controller (14) for all the DGs. In fact, as one advantage of distributed control, the DC bus voltage Vb could not be known by all the DGs [19]. Then motivated by the idea of pinning control, we modify Eq. (15) by multiplying the pinning gain gi in each controller, that is, ei = αi gi eV + βi eui

(53)

where gi is nonzero for the DG that has access to the DC bus voltage Vb . Theorem 2. Consider the primary control system in Eq. (10). The proposed secondary controller given in Eqs. (12)–(14), and Eq. (53) ensures the following results: (i) the DC bus voltage restores to its reference value V * , that is, lim eV (t) = 0; and (ii) the power sharing accuracy in Eq. (8) is guaranteed t→∞ if the controller parameters gi , αi , and βi , are chosen such that the matrix

40 PART | I Frequency and voltage control

Z = −αgRL 1n×1 1Tn×1 A−1 − βL is Hurwitz, and the PI parameters satisfy KPi ≥ 0, KIi > 0, where A = R + k + RL 1n×1 1Tn×1 . Proof. The stability analysis with pinning gain follows similarly to that of Theorem 1 by replacing α with αg, where g = diag(g1 , g2 , . . . , gn ). As long as the matrix Z = −(αgRL 1n×1 1Tn×1 A−1 + βL) is Hurwitz and the PI parameters satisfy KPi ≥ 0, KIi > 0, then the conclusions (51, 52) still hold. Remark 1. The proposed distributed secondary control method is further simplified by applying the idea of pinning control, which only needs to feedback the DC bus voltage Vb to certain DGs (in extreme case, only one DG is needed). This simplification can greatly reduce the number of communication links between the DC bus and local DGs. However, more DGs accessing the DC bus voltage will lead to a faster voltage restoration speed, which will be illustrated in the next section. Remark 2. In this chapter, similar to the work in [14–17], a DC MG with all the DGs connected in parallel is considered, where the main objective is to restore the common bus voltage Vb to the reference value. However, if the DGs are connected to different buses, the control objective is to maintain the average of all the bus voltages to certain reference value; see [10]. Due to different objectives, the latter case is out of the scope of this chapter and needs to be considered separately.

3.4 Control parameters selection According to the results in Theorems 1 and 2, one can follow the following general guidelines to design the control parameters. Step 1. Choose the control parameters gi , αi , and βi to make sure that the matrix Z or Z is Hurwitz. Step 2. Select proper PI parameters satisfying that KPi ≥ 0 and KIi > 0 to ensure system stability. For the system transient performances with respect to these two parameters, from our investigation based on simulation studies and experimental tests, it is found that larger KP and KI lead to shorter settling time but larger overshoot. Remark 3. Note that the condition (35) in Lemma 2 is just one sufficient condition to make Z Hurwitz. In practice, there are many other choices to select the control parameters gi , αi , and βi , as long as that the matrix Z and Z are Hurwitz. For example, the parameters we selected in our case studies section also meet the requirement of Z and Z being Hurwitz.

4 Simulation results In order to validate the proposed distributed control scheme, a simulation test model is built in a MATLAB/Simulink environment. The proposed secondary control with and without pinning gains are firstly verified, respectively. The plug-and-play property of the proposed method is also tested.

Distributed voltage restoration and power allocation Chapter | 2 41

The islanded DC MG system consists of three regular DGs (DG1, DG2, and DG3) and one backup DG (DG4). For simulation studies, different power ratings are chosen according to certain given ratios, for example, S1 :S2 :S3 :S4 = 1:2:3:3 in our simulation example, which is shown in Fig. 4. The DC/DC buck converter is employed in each DG and its detailed primary control loop is shown in Fig. 1, where two standard PI controllers are used in the voltage control loop and current control loop, respectively. The parameters of the primary control layer as well as the MG system are summarized in Table 1. The parameters of the proposed secondary controller are listed in Table 2. The DC bus voltage Vb is measured and then sent to certain DGs according to the definition of pinning gains. In this case study, we initially set the pinning gains as g1 = 1, g2 = g3 = g4 = 0, which means that we only need to send the DC bus voltage Vb to DG1. Suppose the droop gains of four DGs are chosen as the inverse proportion of the power rating of four DGs, that is, k1 :k2 :k3 :k4 = 6:3:2:2.

4.1 Proposed distributed secondary control The whole simulation can be divided into six stages: Stage 1 (0–2 s): Only the primary control is activated at t = 0 s. Stage 2 (2 s): The secondary control is in operation from t = 2 s. Stage 3 (5–12 s): Load 2 is connected to the MG system. Stage 4 (8–13 s): DG4 is plugged in and connected to the MG system. Stage 5 (12 s): Load 2 is disconnected from the DC bus. Stage 6 (13 s): DG4 is removed from the MG system.

Pinning gain

0/1 Pinning gain

0/1 Pinning gain

DC bus

DG1

0/1 Secondary controller

u1

Primary controller

R1

L1

R2

L2

I2

R3

L3

I3

I1

C1

DG2 u2

V2

Primary PWM controller

Load 1

DG3 u3

Primary PWM controller

V3 C3

DG4

0/1 Pinning gain

Secondary controller

u4

Primary PWM controller

R4

V4 C4

Vb

Fig. 4

Case study setup.

Vb

C2

u3

Secondary controller

V1

PWM

u2

Secondary controller

u2

u1

Voltage measurement

Vb

L4

I4

Load 2

42 PART | I Frequency and voltage control

TABLE 1 Parameters of the MG system and the primary controller. DG1/DG2 DG

DG3/DG4

VDC

100 V

VDC

100 V

fs

1.25 kHz

fs

1.25 kHz

LC filter

Lf

1e−2H

Lf

1e−2H

Cf

2200 μF

Cf

2200 μF

Line resistance

R1 /R2

0.01 

R3 /R4

0.01 

Voltage loop

KVP

4

KVP

4

KVI

800

KVI

800

Current loop

KIP

5

KIP

5

KII

110

KII

110

Droop gain

k1 /k2

6/3

k3 /k4

2/2

TABLE 2 Parameters of the secondary controller. DG1 Secondary controller

Reference Load

DG2

DG3

DG4

α1

1

α2

1

α3

1

α4

1

β1

1

β2

1

β3

1

β4

1

KP1

1

KP2

1

KP3

1

KP4

1

KI1

40

KI2

40

KI3

40

KI4

40

g1

1

g2

0

g3

0

g4

0

V*

= 48 V

RL1 = 5 ,

RL2 = 5 

The communication graph is shown in Fig. 4. The simulation results are presented in Figs. 4–6. As seen from Fig. 4, the bus voltage Vb drops to 39.94 V due to the effect of droop control during Stage 1, when only primary control is activated. However, when our proposed secondary control is activated at t = 2 s, the bus voltage Vb quickly restores to the reference value V ref = 48 V. The steady-state DC bus voltage remains at 48 V no matter Load 2 is connected to or disconnected from the bus, even though there are transient deviations. These results show that the proposed method is able to eliminate the bus voltage deviation caused by droop control. In addition, by further looking into the current output shown in Fig. 5, our proposed method is also able to maintain the power sharing ratio brought by the primary controller, that is, I1 :I2 :I3 = 1:2:3,

Distributed voltage restoration and power allocation Chapter | 2 43

regardless of the load increasing at Stage 3 or decreasing at Stage 5. The secondary control inputs are shown in Fig. 6. Clearly, these simulation results validate that the secondary control inputs are equal to each other in the steady state, as stated in Eq. (11). In the following, two different case studies will be discussed.

4.1.1 Case A: Backup DG plug-and-play In this case study, the plug-and-play property of the proposed method will be tested. The backup DG (DG4) is assumed to be connected to the DC bus from t = 8 s to t = 13 s, with the same power sharing ratio as DG3, and disconnected from t = 13 s. The simulation results are shown in Figs. 5–7. Before connecting to the bus, DG4 is running in a stand-by mode with the voltage output V4 = 47.9 V, and its secondary control input is 0. When DG4 is connected at t = 8 s,

48.8 47 48.4

45 48 43

47.6 4.6

4.8

5

5.2

7.8

5.4

8

8.2

8.4

8.6

55

Voltage (V)

50

45

DG1 DG2 DG3 DG4 Vb

52 50

40

48 46

35

0

12

12.5

5

13

13.5

10

15

Time (s) Fig. 5

Voltage output of test islanded DC MG with pinning gains g1 = 1, g2 = g3 = g4 = 0.

44 PART | I Frequency and voltage control 10

DG1 DG2 DG3 DG4

9 8

Current (A)

7 6 5 4 3 2 1 0

2

4

6

8

10

12

14

Time (s) Fig. 6

Current output of test islanded DC MG with pinning gains g1 = 1, g2 = g3 = g4 = 0.

the DC bus Vb increases to 48.6 V but then it quickly restores to 48 V. It is clearly seen that the secondary control input of DG4 reaches a consensus value along with those of all the other three regular DGs. When DG4 is disconnected at t = 13 s, both the voltage and current output are the same as those at Stage 2. These results validate the plug-and-play property of our proposed method. 20

DG1 DG2 DG3 DG4

18

Secondary control input

16 14 12 10 8

10

6

9

4

8

2

7 0.9

0

0

1.2

1.5

1.8

5

10

15

Time (s) Fig. 7 Secondary control input of test islanded DC MG with pinning gains g1 = 1, g2 = g3 = g4 = 0.

Distributed voltage restoration and power allocation Chapter | 2 45

4.1.2 Case B: Different pinning gains In this case study, we compare the performance of our proposed method under the condition of different pinning gains. Figs. 5–7 show the result with the pinning gains setting as g1 = 1, g2 = g3 = g4 = 0. In this case study, we set the pinning gains as g1 = g2 = g3 = g4 = 1, that is, all the DGs have access to the DC bus voltage Vb . In addition, compared to the previous case, in order to mitigate the current overshoot, smaller KPi = 0.1 is chosen. The simulation results are shown in Figs. 8–10. Comparing these to the results in Figs. 5–7, the transient voltage deviations in the case study are much smaller and the settling time is also shorter. However, these advantages are achieved at the expense of more cost in signal transmission, that is, the bus voltage needs to be transmitted to all the DGs. Hence there is a trade-off between the control performance and system cost.

4.2 Robustness test with respect to temporary fault In this section, in order to show the fault tolerance ability of our proposed method, a case study with a temporary phase to ground fault on the common bus during t = 5–6 s is conducted. The simulation results are shown in Figs. 11–13. During the fault period [5, 6 s], all the bus voltages drop down to almost zero and the common bus current output goes as high as 300 A. However, once the fault is cleared, after a transient process of 1.5 s, the whole system becomes stable and restored to the same prefault condition. These results demonstrate that our proposed method has the temporary fault tolerance ability.

4.3 Comparison with existing methods The comparison between the proposed method and existing distributed control methods is summarized in Table 3. It is noted that compared to the existing methods, our proposed method can achieve more precise voltage regulation with less communication burden and without requiring overall system information.

5 Experimental validation In this section, a scaled-down DC MG with two DGs is built in the laboratory to verify the proposed strategy, which is shown in Fig. 14. Each DG is represented by an ideal voltage source with a DC/DC boost converter. The control algorithms are executed on a dSPACE1006 control platform to generate PWM signals for two converters, with the sampling frequency synchronized to PWM frequency at 20 kHz. Two resistors are connected between the output of each converter and the common bus to emulate the line impedance. The resistive loads with two switches are used to generate different loading profiles. The detailed configuration parameters of the experiment setup are listed in Table 4.

46 PART | I Frequency and voltage control

48 48.5

47 48.3

48.1

45

47.9

43 47.7 4.7

4.9

5.1

5.3

8

5.5

8.1

8.2

8.3

8.4

55

Voltage (V)

50

45

DG1 DG2 DG3 DG4 Vb

52

50

40

48 46 11.6

35

0

12.2

12.8

5

13.4

10

15

Time (s) Fig. 8

Voltage output of test islanded DC MG with pinning gains g1 = g2 = g3 = g4 = 1.

5.1 Experimental results with resistant load First, we set the pinning gains as g1 = g2 = 1, which means that these two DGs both access the bus voltage Vb . At first Load 1 is connected to the bus. The experimental results are shown in Fig. 15. Initially the secondary control is not activated. It can be observed that the terminal voltages V1 and V2 drop to 90.38 and 90.36 V, respectively, and bus voltage Vb is 89.59 V, which is largely deviated from the reference voltage V ref = 100 V. When the secondary control is activated, it is obvious that the bus voltage Vb is restored to 100.74 V. In addition, the power sharing ratio is kept unchanged as I1 :I2 ≈ k2 :k1 = 1:1. In order to test the proposed method under different load conditions, an extra load,

Distributed voltage restoration and power allocation Chapter | 2 47 10

DG1 DG2 DG3 DG4

9 8

Current (A)

7 6 5 4 3 2 1 0

2

4

6

8

10

12

14

Time (s) Fig. 9

Current output of test islanded DC MG with pinning gains g1 = g2 = g3 = g4 = 1. 20

DG1 DG2 DG3 DG4

18

Secondary control input

16 14 12 10 8 10

6

9

4

8 7

2 1

0

0

1.2

1.4

1.6

5

10

15

Time (s) Fig. 10 g4 = 1.

Secondary control input of test islanded DC MG with pinning gains g1 = g2 = g3 =

that is, Load 2, is connected and disconnected to the bus. The experimental results are shown in Fig. 16. The bus voltage Vb remains at 100.74 V regardless of whether Load 2 is connected or disconnected. In addition, it is observed that the current outputs of DG1 and DG2 are increased almost by 50% when Load 2 is connected, which is the same proportion as the loads. When Load 2 is

48 PART | I Frequency and voltage control 500

DG1 DG2 DG3 Vb

450 400 100

350

Voltage (V)

80

300 60

250

40

200

20

150

0 5

7

6

8

9

100 50 0

0

1

2

3

4

5

6

7

8

9

10

Time (s) Fig. 11 Voltage output of test islanded DC MG with phase-to-ground fault occurred during t = 5–6 s.

350

DG1 DG2 DG3

300 250

Current (A)

DG1 DG2 DG3

200 20 150 10

100 0

50

2

4

6

8

1

2

3

4

10

0 −50 0

5

6

7

8

9

10

Time (s) Fig. 12 Current output of test islanded DC MG with phase-to-ground fault occurred during t = 5–6 s.

Distributed voltage restoration and power allocation Chapter | 2 49 700

DG1 DG2 DG3

600

Secondary control input

500

100 50

400 0

300

-50

200 -100 4.5

5.5

1

2

6.5

7.5

8.5

100 0 −100 −200 0

3

4

5

6

7

8

9

10

Time (s) Fig. 13 Secondary control input of test islanded DC MG with phase-to-ground fault occurred during t = 5–6 s.

TABLE 3 Comparison of different voltage restoration methods. Overall system information

Technique used

Voltage regulation

Thomas et al. [14]

Average current compensation

Good

Average current

All-to-all

Anand et al. [15]

Average voltage and current compensation

Good

Average voltage and current

All-to-all

Anand et al. [16]

Feed-forward

Good

Overall system droop gain

All-to-all

Proposed method

Feedback and pinning control

Precise

None

Neighborhood communication

Control method

Communication

50 PART | I Frequency and voltage control

Fig. 14

Experiment setup.

TABLE 4 Parameters of MG system and the controller in experiment. DG1 DG

DG2

VDC

50 V

VDC

50 V

fs

20 kHz

fs

20 kHz

LC filter

Lf

1.5e−3H

Lf

1.5e−3H

Cf

470 μF

Cf

470 μF

Line resistance

R1

1

R2

1

Voltage loop

KVP

0.1

KVP

0.1

KVI

1

KVI

1

Current loop

KIP

0.01

KIP

0.01

KII

1

KII

1

Droop gain

k1

10

k2

10

Secondary controller

α1

1

α2

1

β1

1

β2

1

KP1

0

KP2

0

KI1

10

KI2

10

Reference Load

V ref = 100 V RL1 = 50 ,

RL2 = 100 

Distributed voltage restoration and power allocation Chapter | 2 51

Fig. 15 Voltage and current output of each DG and DC bus before and after secondary control input is activated.

disconnected, both the voltage and current output of all DGs are the same as the previous one. Next we consider that only one DG (DG1) can access the bus voltage Vb , that is, setting the pinning gains as g1 = 1, g2 = 0. The experimental results are shown in Figs. 17 and 18. It is observed that the steady state of all the signals are the same as those in Figs. 15 and 16 except for a much slower transient response, which is the same as what is observed in the simulation section.

Fig. 16

Voltage and current output of each DG and DC bus under different load conditions.

52 PART | I Frequency and voltage control

Fig. 17 Voltage and current output of each DG and DC Bus before and after secondary control input is activated with pinning gains g1 = 1, g2 = 0.

Fig. 18 Voltage and current output of each DG and PCC under different load conditions with pinning gains g1 = 1, g2 = 0.

5.2 Experimental results with constant power load In this section, in order to verify the proposed scheme with nonlinear load, Load 2 is changed to a constant power load (CPL) PL = 100 W. The experimental results are shown in Fig. 19. It is observed that no matter whether this CPL is connected to or disconnected from the PCC, the bus voltage Vb

Distributed voltage restoration and power allocation Chapter | 2 53 T1 = 0.7 s

20 V/div

T2 = 0.7 s

100.74 V

V1

100.59 V

V2

99.86 V

Vb

V1 base line V2 base line

Vb base line CPL is disconnected

CPL is connected

(A)

1 s/div

1.58 A

0.5 A/div

1.13 A

1.13 A

I2

1.57 A 0.98 A

0.98 A

I2 base line

I1

I1 base line

1 s/div

(B)

CPL is connected

CPL is disconnected

Fig. 19 Voltage and current output of each DG and PCC with constant power load. (A) Voltage output and (B) current output.

remains unchanged as Vb = 100.59 V. In addition, the two DGs share the current output almost equally according to their designed droop gains (k1 :k2 = 1:1). These results validate the effectiveness of the proposed method under the CPL condition.

5.3 Experimental results with communication time delay In this section, the performance of the overall system is investigated by considering different communication time delay t among each secondary

54 PART | I Frequency and voltage control

V1 V2

20 V/div

Vb

V1 base line V2 base line Vb base line

(A)

CPL is connected

CPL is disconnected

1 s/div

I2

0.5 A/div

I1

I2 base line

I1 base line

(B)

CPL is connected

CPL is disconnected

1 s/div

Fig. 20 Voltage and current output of each DG and PCC with time delay t = 0.001 s. (A) Voltage output and (B) current output.

controller, namely 0.001, 0.01, and 0.1 s. A time delay block in dSPACE1006 is used to implement the delay of ui . Initially, Load 1 is connected to the bus and the secondary control is activated. A 100-W CPL is connected to and disconnected from the bus, respectively. The experimental results are shown in Figs. 20–22. From these results, it is observed that the performance of the proposed method deteriorates when the delay time becomes larger. By comparing the cases that t = 0.001 s and t = 0 s in Fig. 16, little difference is observed in their performances. Thus we can conclude that acceptable voltage restoration performances can be achieved with time delay t less than 0.001 s for this test example.

Distributed voltage restoration and power allocation Chapter | 2 55 T1 = 1.0 s

T2 = 0.9 s

V1 V2 20 V/div

Vb

V1 base line V2 base line Vb base line

(A)

CPL is connected

CPL is disconnected

1 s/div

I2

0.5 A/div

I1

I2 base line

I1 base line

(B)

CPL is connected

CPL is disconnected

1 s/div

Fig. 21 Voltage and current output of each DG and PCC with time delay t = 0.01 s. (A) Voltage output and (B) current output.

In practice, the distributed communication can be applied to wireless networks, such as ZigBee, Wi-Fi, and cellular communication networks [20]. For long-range low-delay networks such as a cellular communication network, the communication time delay is usually negligible, as pointed out in [20]. In addition, the proposed scheme is implemented in the secondary control layer, whose dynamics are much slower than those of the primary control layer. Furthermore, the communication information involved is only the control input ui and the bus voltage Vb . Hence networks with normal wireless communication speed and bandwidth are far more sufficient for implementing the proposed distributed scheme.

56 PART | I Frequency and voltage control T2 = 3.6 s

T1 = 4.0 s

V1 V2 20 V/div

Vb

V1 base line V2 base line

Vb base line

(A)

CPL is connected

CPL is disconnected

1 s/div

I2

0.5 A/div

I2 base line

I1

I1 base line

(B)

CPL is connected

CPL is disconnected

1 s/div

Fig. 22 Voltage and current output of each DG and PCC with time delay t = 0.1 s. (A) Voltage output and (B) current output.

6 Conclusion In this chapter, a distributed secondary control scheme is designed for power allocation and voltage restoration in islanded DC MGs. Compared to existing secondary control methods, which are usually designed in a feed-forward fashion, our proposed method is based on the feedback idea. By only measuring bus voltage and exchanging limited neighboring information, the distributed secondary controller can be designed for each local DG to realize the goal of bus voltage restoration as well as accurate power sharing. In addition, by applying the idea of pinning control, our proposed method can be further

Distributed voltage restoration and power allocation Chapter | 2 57

simplified by only sending the bus voltage to one DG. The effectiveness of the proposed controller is analyzed theoretically and verified by both simulation and experimental results.

References [1] F. Guo, C. Wen, Y.-D. Song, Distributed Control and Optimization Technologies in Smart Grid Systems, CRC Press, Boca Raton, FL, 2017. [2] F. Nejabatkhah, Y.W. Li, Overview of power management strategies of hybrid AC/DC microgrid, IEEE Trans. Power Electron. 30 (12) (2015) 7072–7089. [3] Y. Gu, W. Li, X. He, Frequency-coordinating virtual impedance for autonomous power management of DC microgrid, IEEE Trans. Power Electron. 30 (4) (2015) 2328–2337. [4] W. Yu, G. Wen, X. Yu, Z. Wu, J. Lu, Bridging the gap between complex networks and smart grids, J. Control Decis. 1 (1) (2014) 102–114. [5] X. Lu, K. Sun, J.M. Guerrero, J.C. Vasquez, L. Huang, State-of-charge balance using adaptive droop control for distributed energy storage systems in DC microgrid applications, IEEE Trans. Ind. Electron. 61 (6) (2014) 2804–2815. [6] A.A. Hamad, M.A. Azzouz, E.F. El-Saadany, Multiagent supervisory control for power management in DC microgrids, IEEE Trans. Smart Grid 7 (2) (2016) 1057–1068. [7] S. Augustine, N. Lakshminarasamma, M.K. Mishra, Control of photovoltaic-based low-voltage DC microgrid system for power sharing with modified droop algorithm, IET Power Electron. 9 (6) (2016) 1132–1143. [8] S. Augustine, M.K. Mishra, N. Lakshminarasamma, Adaptive droop control strategy for load sharing and circulating current minimization in low-voltage standalone DC microgrid, IEEE Trans. Sustain. Energy 6 (1) (2015) 132–141. [9] J.M. Guerrero, J.C. Vasquez, J. Matas, L.G. de Vicuna, M. Castilla, Hierarchical control of droop-controlled AC and DC microgrids: a general approach toward standardization, IEEE Trans. Ind. Electron. 58 (1) (2011) 158–172. [10] V. Nasirian, S. Moayedi, A. Davoudi, F.L. Lewis, Distributed cooperative control of DC microgrids, IEEE Trans. Power Electron. 30 (4) (2015) 2288–2303. [11] X.-K. Liu, H. He, Y.-W. Wang, Q. Xu, F. Guo, Distributed hybrid secondary control for a DC microgrid via discrete-time interaction, IEEE Trans. Energy Convers. 33 (4) (2018) 1865–1875. [12] J. Zhao, F. Dorfler, Distributed control and optimization in DC microgrids, Automatica 61 (1) (2015) 18–26. [13] T. Morstyn, B. Hredzak, G.D. Demetriades, V.G. Agelidis, Unified distributed control for DC microgrid operating modes, IEEE Trans. Power Syst. 31 (1) (2016) 802–812. [14] S. Thomas, S. Islam, S.R. Sahoo, S. Anand, Distributed secondary control with reduced communication in low-voltage DC microgrid, in: Proceedings of the 10th International Conference on Compatibility, Power Electronics and Power Engineering, 2016, pp. 126–131. [15] S. Anand, B.G. Fernandes, J.M. Guerrero, Distributed control to ensure proportional load sharing and improve voltage regulation in low-voltage DC microgrids, IEEE Trans. Power Electron. 28 (4) (2013) 1900–1913. [16] S. Anand, B.G. Fernandes, J.M. Guerrero, An improved droop control method for DC microgrids based on low bandwidth communication with DC bus voltage restoration and enhanced current sharing accuracy, IEEE Trans. Power Electron. 29 (4) (2014) 1800–1812.

58 PART | I Frequency and voltage control [17] F. Gao, S. Bozhko, G. Asher, P. Wheeler, C. Patel, An improved voltage compensation approach in a droop-controlled DC power system for the more electric aircraft, IEEE Trans. Power Electron. 31 (10) (2016) 7369–7383. [18] P.-H. Huang, P.-C. Liu, W. Xiao, M.S.E. Moursi, A novel droop-based average voltage sharing control strategy for DC microgrids, IEEE Trans. Smart Grid 6 (2) (2015) 1096–1106. [19] F. Guo, C. Wen, J. Mao, Y.-D. Song, Distributed secondary voltage and frequency restoration control of droop-controlled inverter-based microgrids, IEEE Trans. Ind. Electron. 62 (7) (2015) 4355–4364. [20] H. Liang, B.J. Choi, W. Zhuang, X. Shen, Stability enhancement of decentralized inverter control through wireless communications in microgrids, IEEE Trans. Smart Grid 4 (1) (2013) 321–331.

Chapter 3

Optimal distributed secondary control for an islanded microgrid Yinliang Xu and Zhongkai Yi Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School, Shenzhen, People’s Republic of China

1 Introduction With the increasing capacity of distributed generation, especially for renewable energy resources (RERs), the microgrid emerges as a popular approach to promote the RERs’ accommodation. However, the growing capacity and high penetration of RERs bring many challenges to an islanded microgrid [1, 2], such as the operation uncertainties, high order harmonics injection, frequency and voltage stability problems, etc. For the frequency control of an islanded microgrid, generally it can be divided into three levels: primary control, secondary control, and tertiary control [3, 4]. For the three-level frequency control of the microgrid, the droop control strategy can emulate the behavior of a synchronous generator without requiring any communication channels [5, 6]. The secondary control strategy is used to eliminate the frequency and voltage deviations that cannot be offset by primary control [7]. For the tertiary control, it focuses mainly on the economic dispatch and power flow optimization based on RERs and load forecasting [8]. In this chapter, we focus on the distributed secondary control strategy in an islanded microgrid. There are two objectives of the secondary control. The first object is to restore the frequency and voltage to the desired value after load or distributed generators (DGs) variation. The second object is to maintain accurate active and reactive power sharing of all the DGs. In the existing literature, the secondary control strategies can be divided into three categories: centralized control [9, 10], decentralized control [11, 12], and distributed control [2, 7, 13, 14]. For the traditional centralized secondary control strategy, it requires a central communication and computation unit to collect a large amount of DG information. This may suffer from a single point of failure problem and the computation burden is rather heavy for the centralized system operator. Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00003-7 Copyright © 2020 Elsevier Inc. All rights reserved.

59

60 PART | I Frequency and voltage control

For the decentralized secondary control strategy, it may not be effective to coordinate all the available RERs in a microgrid optimally because of lacking broader available information [15]. The distributed control strategy has been a hotspot in recent years, which is derived from the idea of multi agent coordination using the consensus algorithm, as in [16, 17]. The distributed control is to realize a common goal according to a peer-to-peer communication protocol, which has been used in the frequency restoration, voltage control, and active and reactive power sharing of the power system [18–20]. Nevertheless, the existing literatures on the distributed secondary control merely offer an asymptotical convergence speed for the frequency/voltage restoration and active/reactive power sharing [13, 14, 18–20]. Since the islanded microgrid often suffer from some disturbances because of the intermittency of loads and RERs. Therefore, the distributed control algorithm with finite-time convergence, fast response, robust performance, and high scalability is preferred. To realize the objective of the finite-time control, a finite-time approximate consensus approach is proposed in [21] to expedite the convergence of the distributed algorithm, but the voltage restoration and reactive power sharing are not investigated. The voltage and frequency control are considered simultaneously in [22], but some oscillations still exist in the active power response after the load variations. The voltage controller which is independent from the frequency controller is designed in [23, 24], but the accurate reactive power sharing of the proposed strategy is not studied. To facilitate the decoupled design of secondary frequency and voltage controllers in an islanded microgrid, two control strategies with different convergence times are proposed in this chapter. First, a finite-time secondary frequency control strategy to eliminate the frequency deviation and maintain the accurate active power sharing is proposed. Second, a secondary control strategy is proposed to adjust the average voltage magnitude of all the DG units to a desired value and realized the accurate reactive power sharing. Under the proposed optimal distributed control strategy, only the information from neighboring controllers is required for every local controller. The simulation studies conducted in MATLAB/Simulink validate that the proposed distributed control strategy is adaptive, scalable, strongly robust, and offers fast decisionmaking and plug-and-play functionality. In summary, the main points of this chapter are listed as follows: (1) It is demonstrated that the traditional asymptotically convergent distributed control method is just a special case of the proposed strategy under specific parameters of the controller. In addition, using the proposed distributed control strategy, the frequency restoration and active power sharing is converged in a finite-time manner. (2) To control the active power/frequency and reactive power/voltage effectively, a distributed secondary control strategy is proposed, only requiring the information of neighboring DGs in the sparse communication network, which can achieve a satisfactory effect.

Optimal distributed secondary control for a microgrid Chapter | 3 61

(3) The proposed distributed control strategy is implemented in an islanded 5-bus microgrid and a 42-bus distribution network, respectively, which validates the scalability of the proposed strategy.

2 Preliminaries To facilitate the understanding of the optimal distributed control strategy proposed in this chapter, some preliminaries are introduced in this section.

2.1 Graph theory Suppose that the DG i and DG j are connected by the transmission line ij. DG j is considered to be in the neighborhood set of DG i, which is denoted by j ∈ Ni . The DGs can only receive the information from the neighborhood that is directly connected to them. If DG i and DG j are directly connected, the communication coefficient dij > 0, otherwise dij = 0. If all the DG units are considered as the nodes and the transmission lines are considered as the edges, the whole topology of the microgrid can be considered as a graph. If dij = dji (∀i, j), the graph is called an undirected graph. If there exists a path that connects any two distinct DG units, the graph is called connected. dii represents the diagonal term of the graph, which is defined as follows:  dij , j = i (1) dii = − j∈Ni

  Here we define the D = dij as the Laplacian matrix. Suppose that λi is the eigenvalue of the D matrix and increased progressively, the following relationship can be obtained [25]. |λn | ≥ · · · ≥ |λ2 | > |λ1 | = 0,

λi ≤ 0,

∀i = 1, . . . , n

(2)

The following properties are valid for the undirected graph, which has been proved in [26]. n  i,j=1

n k 1+k  1   dij xi sign(xj − xi ) xj − xi  = − dij xj − xi  2

(3)

i,j=1

xT Dx =

n 1 dij (xj − xi )2 , 2

xT D2 x =

i,j=1

n 1 2 dij (xj − xi )2 2

xT Dx ≥ λ2 (D)xT x xT (D + C)x =

1 2

n 

(4)

i,j=1

dij (xj − xi )2 + ci

i,j=1

xT (D + C)x ≥ λ1 (D + C)xT x

(5) n 

xi2

(6)

i=1

(7)

62 PART | I Frequency and voltage control

where x = (x1 , . . . , xn ) is the state variable of the system, which is required to be exchanged with the neighboring DGs. C is a diagonal matrix, which is defined as C = diag {c1 , . . . , cn }. ci is the pinning gain. λ1 is the minimum eigenvalue of (D + C). λ2 is the second smallest eigenvalue of D, which is also the algebraic connectivity of D. sign(·) is the sign function.

2.2 Closed-loop optimal control algorithm Considering a dynamic non linear system represented by the following equation: x(t) ˙ = f (x) + G(x)u(t) = F(x),

x(0) = x0

(8)

The performance index of the dynamic system can be defined as follows:  ∞  ∞ J = min L(x, u)dt = min L1 (x) + uT (x)Ru(x)dt (9) x,u

x,u

0

0

where L1 is the performance criterion of the state variables, R is a diagonal matrix, which is defined as R = diag {r1 , . . . , rn }. ri is the weight coefficient of control input. The Hamiltonian function is defined as follows [27]: H(x, u) = L(x, u) + S(x)F(x)

(10)

where H(x, u) is the Hamiltonian function and S(x) is the set of costate variables. To minimize the Hamiltonian function, the optimal control can be designed as follows: u = arg min {L(x, u) + S(x)F(x)} x,u

(11)

The necessary condition for Eq. (11) to be valid is the first-order derivative of Eq. (10) with respect to u being equal to 0:  ∂  ∂H (12) = L1 (x) + uT Ru + S(x)( f (x) + G(x)u) = 0 ∂u ∂u Then, the optimal control law of u can be derived as follows: 1 (13) u = − R−1 [S(x)G(x)]T 2 For the controlled dynamic non linear system of Eq. (8), the HamiltonJacobi-Bellman equation in steady state can be presented as follows [27]: L(x, u) + S(x)F(x) = L1 (x) + S(x)f (x) −

1 [S(x)G(x)] · R−1 [S(x)G(x)] = 0 4 (14)

Thus, the performance criterion can be written as follows: L1 (x) =

1 [S(x)G(x)] · R−1 [S(x)G(x)] − S(x)f (x) 4

(15)

Optimal distributed secondary control for a microgrid Chapter | 3 63

Suppose that V(x) is a Lyapunov function candidate which is continuously differentiable and satisfies the following conditions: V(x) ≥ 0 dV(x) dx dV(x) = = S(x)F(x) dt dx dt

(16) (17)

dV(x) (18) ≤ −ρ (V(x))α , ρ > 0, 0 < α < 1 dt Then, the dynamic system of Eq. (8) is globally stable in a finite-time manner under the control law of Eq. (13), which satisfies the condition presented in Eq. (19) [27]:  limt→T V(x(t)) = 0 (19) V(x(t)) = 0, ∀t ≥ T And the finite settling time T is limited by T≤

1 (V(x0 ))1−α ρ(1 − α)

(20)

3 Proposed optimal distributed control strategy Consider an islanded AC microgrid that consists of multiple DGs connected by inverter. All of the DG units are composed of a DC voltage source, a voltage source DC-AC inverter (VSI), and an inductance-capacitance-inductance (LCL) filter. Fig. 1 provides more details. The LCL filter outputs impedance inductive dominant for each DG unit [28]. The active and reactive power sharing laws are based on the rated capacities of the droop settings, which can be written by the following equation [13]: ωni = ωi + mPi Pi Vni = Voi + mQi Qi

PWM

VDC

Current controller

i*li

Inverter voi jXfi

Fig. 1

voi

Voltage controller

ili

(21) (22)

Cfi

wni

Vni P i

Power controller

ioi

Qi

Power calculation

vbi MG

Rci + jXci Load i

Block diagram of the primary control system of the DG units [28].

64 PART | I Frequency and voltage control

where ωni is the nominal set points for the system frequency, Vni is the nominal set points for local voltage, ωi and Vi are the frequency and voltage of DG unit i, respectively, mPi and mQi are the frequency and voltage droop coefficients of DG unit i, respectively, and Pi and Qi are the active and reactive power injected to bus i by DG i, respectively. The droop coefficients mPi and mQi are set according to the acceptable deviations of frequency ω and voltage V as follows: mPi = ω/Pi max mQi = V/Qi max

(23) (24)

where Pi max and Qi max are the maximum active and reactive power, respectively. Essentially, frequency and voltage deviations are unable to be eliminated by primary control, so the secondary control is required. The secondary control is to change the ωni and Vni so that the ωi and Vi can be controlled to the desired value. According to the proposed distributed control strategy, for the primary control law presented in Eqs. (23), (24), only local information and the neighbors’ information of DG unit i is needed. According to the derivation of Eqs. (23), (24), the changing rate of frequency and active power output are chosen as the control variables. The following equations are obtained as follows: ω˙ni = ω˙i + mPi P˙i = uωi + uPi V˙ni = V˙oi + mQi Q˙ i = uVi + uQi

(25) (26)

Based on Eqs. (25), (26), the nominal value of the frequency and voltage magnitude can be derived as follows:  t uωi (τ ) + uPi (τ )dτ (27) ωni =  Vni =

t0 t t0

Q

uVi (τ ) + ui (τ )dτ

(28)

The objectives of the proposed distributed secondary control strategy are to restore the DGs’ frequency/voltage magnitude to the desired values and realize accurate active/reactive power sharing, which can be summarized as the following three points. (1) Restoring DGs’ frequency to the desired value in a finite-time manner:     ωi (t) = ωref , ∀t ≥ Tω (29) lim ωi (t) − ωref  = 0, t→Tω

(2) Achieving an accurate active power sharing in a finite-time manner:    P (t) Pj (t)  Pj (t) Pi (t)  i = max , ∀t ≥ TP (30) lim  max − max  = 0, max t→TP  Pi (t) Pj (t)  Pi (t) Pj (t)

Optimal distributed secondary control for a microgrid Chapter | 3 65

(3) A trade-off between the voltage magnitude restoring in Eq. (31) and accurate reactive power sharing in Eq. (32):     (31) lim Voi (t) − V ref  = 0 t→∞    Q (t) Qj (t)   i lim  max − max  = 0 (32) t→∞  Q Qj (t)  i (t) i (t) i (t) , Qmax are the active/reactive power utilization ratios of DG i where PPmax i (t) Qi (t) at time t, respectively. ωi (t) is the system frequency at time t. Voi (t) is the voltage output of DG i at time t. ωref is the nominal frequency value. V ref is the nominal voltage magnitude value.

3.1 Optimal distributed finite-time secondary frequency control and active power sharing Suppose that in an islanded microgrid, the number of DG units is equal to n. The state-space equations of the system in Eq. (25) can be written as follows:  x˙ω = uω (33) x˙P = uP where xω = (ω1 , . . . , ωn )T , xP = (mP1 P1 , . . . , mPn Pn )T , uω = (uω1 , . . . , uωn ), and uP = (uP1 , . . . , uPn ). The following two Lyapunov function candidates are defined: k/2 4  T 2 k/2 2   (34) xω Dω xω  + (xω − xωref )T Cω2 (xω − xωref ) k k   4 k/2 (35) VP = xPT D2P xP  k     where Dω and DP are defined as Dω = dωij and DP = dPij , respectively. Cω is a diagonal matrix defined as Cω = diag cω,i . k is the fractional power constant that satisfies 1 < k < 2. The partial derivatives of Vω and VP with regard to xω and xP can be expressed as follows: ⎡  k−1  ∂Vω = −2 ⎣ (dωij )k sign(ωj − ωi ) ωj − ωi  Sωi = ∂xωi j∈Ni ⎤ k−1    (36) + (cωi )k sign(ωref − ωi ) ωref − ωi  ⎦ Vω =

⎡ SPi =

∂VP = −2 ⎣ ∂xPi



⎤   k−1 (dPij )k mPj Pj − mPi Pi  sign(mPj Pj − mPi Pi )⎦

j∈Ni

(37)

66 PART | I Frequency and voltage control

According to Eqs. (36), (37), the optimal distributed control law in Eq. (13) can be derived as follows [29]: ⎡  k−1  −1 ⎣ (dωij )k sign(ωj − ωi ) ωj − ωi  uωi = rωi j∈Ni

+(cωi )k sign(ωref

⎤ k−1    ref ⎦ − ωi ) ω − ωi 

⎡ uPi =



−1 ⎣ rPi

(38)

⎤  k−1 (dPij ) mPj Pj − mPi Pi  sign(mPj Pj − mPi Pi )⎦ k

(39)

j∈Ni

The proof of the finite-time convergence of the proposed distributed algorithm is presented as the following steps, which were inspired by Ref. [23]. First, the errors of system global frequency and active power can be defined as follows:   (40) eω = (eω1 , . . . eωn ) = ω1 − ωref , . . . , ωn − ωref   n n 1 1 mPi Pi , . . . , mPn Pn − mPi Pi eP = (eP1 , . . . ePn ) = mP1 P1 − n n i=1

i=1

(41) The Lyapunov function candidate is chosen as follows: 1 T  1 T  V2 = Veω + VeP = e eω + e eP ≥ 0 2 ω 2 P The first-order derivative of Eq. (42) can be derived as follows: V˙2 =

n  i=1

=

n 

n 

eωi e˙ωi +

(42)

ePi e˙Pi

⎡

i=1



−1 ⎣ eωi rωi

k−1    (dωij )k sign(eωj − eωi ) eωj − eωi 

j∈Ni

i=1

⎤   k−1 ⎦ +(cωi )k sign(eωi ) eωi  + ≤− −

n 

k−1    −1 ePi rPi (dPij )k sign(ePj − ePi ) ePj − ePi 

i=1 n 

1 2

1 2

i=1 n  i=1

n  k   k   −1 −1 rωi (dωij )k eωj − eωi  − rωi (cωi )k eωi 

 k   −1 rPi (dPij )k ePj − ePi 

i=1

(43)

Optimal distributed secondary control for a microgrid Chapter | 3 67

The following relationship of V˙2 can be obtained based on the Cauchy inequality.  n k/2 n  1  − 2k − 2k 2 2 2 2 rωi (dωij ) (eωj − eωi ) + 2 rωi (cωi ) (eωi ) V˙2 ≤ − 2 i=1 i=1    −

 n 1  2



σI

(44)

k/2

− 2k

rPi (dPij )2 (ePj − ePi )2

i=1





σII

The two terms on the right-hand side of V˙2 in Eq. (44) can be rewritten as follows according to Eqs. (5), (7): −2

−2

σI = 2eTω Rω k (D2ω + Cω2 )eω ≥ 2λ1 (D2ω + Cω2 )Rω k eTω eω −2

(45)

= 4λ1 (D2ω + Cω2 )Rω k Veω −2

−2

−2

σII = 2eTP D2P RP k eP ≥ 2λ2 (D2P )RP k eTP eP = 4λ2 (D2P )RP k VeP Thus,

k/2 1  k/2 1 − 4λ1 (D2ω + Cω2 )Veω 4λ2 (D2P )VeP 2 2 k/2  k−1 −1 2 2 = −2 Rω λ1 (Dω + Cω ) (Veω )k/2 k/2  − 2k−1 R−1 (VeP )k/2 λ2 (D2P ) P

(46)

V˙2 ≤ −

(47)

According to Eq. (20), the finite time stability can be achieved for the frequency and active power control problem presented in Eqs. (29), (30). The settling time T = max {Tω , TP } can obtained as follows: ⎧ 1− k 1− k R Veω (eω (0)) 2 22−k Rω Veω (eω (0)) 2 ⎪ ⎪ = ⎨ Tω = k−1 ω 2 2 k k/2 (2−k)(λ1 (D2ω +Cω2 )) 2 (λ1 (Dω +Cω )) 2 (1− 2k ) (48) 1− k 2−k ⎪ ⎪ ⎩ T = 2 RP VeP (eP (0)) 2 P

 k/2 (2−k) λ2 (D2P )

3.2 Optimal distributed secondary voltage control and reactive power sharing If the power line impedance effect is considered, the control object of the accurate reactive power and the voltage restoration cannot be achieved simultaneously [28], and there is a trade-off between the accurate reactive power

68 PART | I Frequency and voltage control

sharing and the voltage regulation [20]. Therefore, in this section, an optimal distributed control strategy is proposed to ensure the voltage magnitude to the desired value and realize accurate reactive power sharing of all the DG units simultaneously. According to the state-space equations in Eq. (26), after the control strategy is implemented, the dynamics of the estimated average voltage (EAV) and reactive power can be described as follows:  x¯˙V = u¯ V (49) x˙Q = uQ T T   where x¯ V = V¯ o1 , . . . , V¯ on and xQ = mQ1 Q1 , . . . , mQn Qn . The following two Lyapunov function candidates for the EAV and reactive power sharing are designed as follows:  T     T ref ref ¯ (50) VV = 2 x¯ V DV¯ x¯ V + xV − x¯ V CV¯ xV − x¯ V   T DQ xQ (51) VQ = 2 xQ The partial derivatives of Eqs. (50), (51) with regard to xˆV and xˆQ can be calculated as follows:    ∂ V¯ V ref ¯ ¯ ¯ = = −2 d ( V − V ) + c − V V (52) SVi ¯ oj oi oi ¯ ¯ Vi oi Vij ∂ V¯ oi j∈Ni

SPi =

 ∂VQ = −2 dQij (mQj Qj − mQi Qi ) ∂ xˆQi

(53)

j∈Ni

According to Eqs. (13), (52), (53), the optimal distributed control law for EAV and reactive power sharing can be derived as follows: ⎤ ⎡    ref −1 ⎣ ¯ ¯ ¯ oi ⎦ dVij uVi = V˙¯ oi = rVi (54) ¯ Voi − V ¯ (Voj − Voi )+cVi ¯ j∈Ni

⎤

⎡



−1 ⎣ uQi = mQi Q˙ i = rQi

dQij (mQj Qj − mQi Qi )⎦

(55)

j∈Ni

The error terms can be defined as follows:   n n 1 1 eQ = (eQ1 , . . . , eQn ) = mQ1 Q1 − mQi Qi , . . . , mQn Qn − mQi Qi n n i=1



eV¯ = V¯ o1 − V ref , . . . , V¯ on − V ref

i=1



(56) (57)

Optimal distributed secondary control for a microgrid Chapter | 3 69

Based on Eqs. (49), (54), (55), the error terms of EAV and reactive power sharing can be derived, as shown in Eqs. (58), (59). e˙Q = RQ DQ eQ

(58)

e˙V¯ = RV¯ (DV¯ − CV¯ )eV¯

(59)

Since all of the eigenvalues of DO are always less than zero, the following analytical solution can be derived: T lim eQ (t) = lim exp(DQ t)RQ eQ (0) = In×1 In×1 RQ eQ (0) = 0

t→∞

t→∞

(60)

where In×1 = (1, . . . , 1)T . Therefore, the accurate reactive power sharing can be achieved as follows: 1 mQi Qi (∞) n n

mQ1 Q1 (∞) = · · · = mQi Qi (∞) = · · · = mQn Qn (∞) =

i=1

(61) Yield Eq. (60) into the frequency domain: sE¯ V¯ (s) − eV¯ (0) = RV¯ (DV¯ − CV¯ )E¯ V¯ (s)

(62)

According to the previous equation, the dynamic characteristics of the system in the frequency domain can be obtained as follows: E¯ V¯ (s) = [sI − RV¯ (DV¯ − CV¯ )]−1 eV¯ (0)

(63)

where I is the identity matrix. Since (DV¯ − CV¯ ) = 0, according to the final value theorem, Eq. (63) can be derived as follows:  −1 lim eV¯ (t) = lim sE¯ V¯ (s) = lim s sI − RV¯ (DV¯ − CV¯ ) eV¯ (0) = 0 (64) t→∞

s→0

s→0

Thus, the following condition can be obtained: V¯ o1 (∞) = · · · = V¯ oi (∞) = · · · = V¯ on (∞) = V ref

(65)

In addition, the voltage magnitude can be updated using the following law: uV = x˙V = x¯˙V − DV¯ x¯ V

(66)

where xV = (Vo1 , . . . , Von )T and x¯ V (0) = xV (0). sXV (s) − xV (0) = sX¯ V (s) − x¯ V (0) − DV¯ X¯ V (s)

(67)

Eq. (67) can be deformed as follows: X¯ V (s) = s(sI − DV¯ )−1 XV (s)

(68)

70 PART | I Frequency and voltage control

According to [25], the following condition can be obtained: lim s(sI − DV )−1 =

s→0

1 T In×1 In×1 n

(69)

Therefore, lim x¯ (t) t→∞ V

= lim sX¯ V (s) = lim s2 (sI − DV )−1 XV (s) s→0

s→0

= lim s(sI − DV¯ )−1 lim sXV (s) s→0

(70)

s→0

1 T xV (∞) In×1 In×1 n The following equation can then be easily derived: =

V¯ o1 (∞) = · · · = V¯ oi (∞) = · · · = V¯ on (∞) =

1 Voi (∞) n n

(71)

i=1

Combining Eqs. (65), (71) together, the following equation can be obtained: 1 Voi (∞) = V ref n n

(72)

i=1

Thus, from Eq. (72), it can be concluded that the average voltage magnitude values of all the DG units converge to the desired voltage value.

3.3 Communication delay analysis To receive and transmit the neighboring DGs’ information, a communication network should be employed for the proposed optimal distributed control strategy. Thus, a communication delay is unavoidable. In this section, the influence of the communication delay is analyzed in the following statement. According to Eq. (67), the communication delay for the microgrid under the proposed distributed voltage control strategy can be expressed as follows: V¯˙oi (t) = V˙oi (t) +



¯ oj (t − τj (t)) − V¯ oi (t − τi (t))] dVij ¯ [V

(73)

where τi (t) is the communication delay of time-varying which satisfies 0 ≤ τi (t) ≤ τ max . τ max is the maximum tolerable communication delay. Yield Eq. (73) into a matrix form: V¯˙oi (t) = (I ⊗ DV¯ )V¯ o (t) + (DV¯ ⊗ CV¯ RV¯ )V¯ o (t − τ (t))

(74)

It is worth pointing out that the system expressed by Eq. (74) is stable under a bounded communication delay, and the maximum tolerable communication delay can be calculated using the Ricatti equation. For detailed analysis, please refer to [30].

Optimal distributed secondary control for a microgrid Chapter | 3 71

3.4 Algorithm implementation The detailed implementation diagram of the proposed optimal distributed control strategy for multiple DGs is presented in Fig. 2. Using the proposed control strategy, the local secondary controller only requires the desired reference values and the information from the neighboring controllers with a peer-to-peer communication protocol. In addition, during the primary control, the nominal values of ωni and Vni of each DG are updated by the input signals of uωi + uPi and uVi + uQi , respectively. Under the proposed control strategy, the voltage and frequency regulations can be conducted in different timescales so that the active and reactive power controllers are decoupled.

4 Simulation analysis An islanded five-bus microgrid is presented in Fig. 3, composing of five DGs, several loads and transmission lines. The frequency is 50 Hz and the line-to-line AC voltage is 380 V. The parameters of the DG units, loads, and transmission lines are given in Table 1. The following simulation studies are conducted to demonstrate the effectiveness of the proposed optimal distributed secondary control strategy.

Fig. 2

Detailed block diagram of the proposed strategy.

72 PART | I Frequency and voltage control

DG5

L5

5 Z15

Z25

DG1

4

3

L2 Z23

Z14 DG4

Z34

DG3 Communication network

L4 Fig. 3

2

DG2

Z12

L1

1

L3 Diagram of the five-bus microgrid.

TABLE 1 Parameters for the five-bus islanded microgrid. Head DGs

Lines

Loads

DG1 and DG2

3.0 kW + 1.5 kVAr

DG3 to DG5

2.0 kW + 1.0 kVAr

mP

5 × 10−5

mP

6.5 × 10−5

mQ

6 × 10−4

mQ

8 × 10−4

ZC

0.04 + j0.25

ZC

0.03 + j0.15

Lf

1.5 mH

Lf

1.5 mH

Cf

47 μF

Cf

47 μF

Z14 and Z23

0.05 + j0.15

Z15 and Z25

0.04 + j0.1

Z12

0.08 + j0.2

Z34

0.1 + j0.25

L1

1.5 kW + 1.5 kVAr

L3 and L4

2.0 kW + 0.5 kVAr

L2

2.5 kW + 0.8 kVAr

L5

1.0 kW + 1.2 kVAr

4.1 Performance evaluation the proposed optimal distributed control strategy Initially, the conventional droop control strategy is implemented to the five-bus microgrid at t = 1 s. The proposed control strategy is activated at t = 2 s. An increment load of 1.5 kW + 0.9 kVAr is connected to the microgrid at t = 2 s and removed at t = 3 s. The simulation results are presented in Fig. 4. From Fig. 4, the frequency and voltage magnitude cannot reach the desired value under the primary droop control when the proposed distributed secondary

Optimal distributed secondary control for a microgrid Chapter | 3 73

Fig. 4 Performance analysis of the proposed distributed control strategy under load variation. (A) Frequency dynamics; (B) voltage magnitude dynamics; (C) active power sharing dynamics; and (D) reactive power sharing dynamics.

control strategy is not implemented at t < 1 s. After the proposed distributed secondary control is implemented, the frequency is converge to the desired value within 0.4 s and the voltage magnitude of all the DGs are regulated to 380 ± 1.5 V within 0.6 s, as shown in Fig. 4A and B, respectively. Also, after the proposed secondary control strategy is implemented, the active/reactive power sharing is close to the desired ration in spite of the load variations, as shown in Fig. 4C and D, respectively.

4.2 Convergence analysis In this simulation study, the proposed optimal distributed frequency controller designed in Eqs. (38), (39) is compared with the controller presented in [28] for the convergence analysis. For the convenience of observation, only the frequency response mean square error (MES) of DG 1 is shown in Fig. 5, which validates that the proposed strategy can achieve a faster convergence speed. ki can affect the convergence speed of the proposed control strategy. The MES of the frequency derivations under different ki is presented in Fig. 6. It can be indicated that the convergence speed is faster when ki is closer to 1 and the convergence speed is slower when ki is closer to 2. It is worth pointing out that when ki = 2, the proposed strategy is actually the same as the traditional asymptotical convergence control algorithm.

74 PART | I Frequency and voltage control

Fig. 5

Comparison of the convergence speed between the two strategies.

Fig. 6 Influences of the different control parameters on the convergence speed of the proposed strategy.

4.3 Influence analysis of the communication delay There always exists the unavoidable communication delay among the neighboring DGs, which can affect the dynamic performance of the system considerably. The influence of the communication delay on the proposed optimal distributed strategy is compared with that of the traditional centralized strategy. The simulation results are presented in Figs. 7 and 8, with communication latency bounds of 200 and 500 ms, respectively. In addition, during the simulation, the time delay is applied in the system by using a pure delay block e−τ (t)s . For simplicity, only the frequency dynamics are presented in Figs. 7 and 8. The centralized strategy is the PI-based method proposed in [31]. Compared with the centralized strategy, the proposed distributed control strategy can

Optimal distributed secondary control for a microgrid Chapter | 3 75

Fig. 7 Frequency dynamics using different control strategy with a communication delay bound of 200 ms. (A) The centralized strategy and (B) the proposed distributed strategy.

Fig. 8 Frequency dynamics using different control strategy with a communication delay bound of 500 ms. (A) The centralized strategy and (B) the proposed distributed strategy.

exhibit a more satisfactory performance when the communication time delay is 200 ms. Moreover, when the communication time delay is 500 ms, the centralized strategy fails to restore the frequency to the desired value within 1 s before the next event in the microgrid occurs, as shown in Fig. 8A. However, under the same condition, the proposed distributed control strategy still offers an acceptable performance, as shown in Fig. 8B. Therefore, compared with the traditional centralized strategy, the proposed distributed control strategy achieves a better control result under the same communication delay.

4.4 Robustness analysis against the uncertainties of parameters To analyze the robustness against the parameters’ uncertainties of the proposed distributed control strategy, the transmission line impedances are perturbed with a random error range of 10%–30% on the basis of the data presented in Table 1. Other parameters and conditions are unchanged. The simulation results are given in Fig. 9.

76 PART | I Frequency and voltage control

Fig. 9 Active/reactive sharing dynamic performance under the uncertainties of the line parameters. (A) Active power sharing dynamics and (B) reactive power sharing dynamics.

Fig. 9A and B indicates that the active and reactive power sharing are still acceptable only with very small fluctuations. These fluctuations are as a result of the model’s uncertainties, which can cause the inaccurate estimated states.

4.5 Influence analysis of the bounded control input Considering the engineering practical application and physical limitations, the bounded control input of the distributed controllers should be considered. The dynamic performance of the proposed strategy can be directly influenced by the parameter ri of the controllers. In this case study, the impact of ri on the dynamic performance of the proposed distributed control strategy is investigated. From Fig. 10A and B, larger ri can lead to the convergence speed being slower and the control effort being smaller. Thus, there is a trade-off for ri between the required control effort and the convergence speed. In addition, when ri is equal to 1.5, the frequency restoration time is approximately 0.55 s; however, the control inputs exceed the boundary during the time period from t = 1 s to t = 1.1 s. From Fig. 11A and B, when ri is equal to 3, the frequency is restored to the desired value with a longer time interval about 0.8 s, but

Fig. 10 Dynamic performance of the proposed distributed control strategy when ri = 1.5. (A) Frequency dynamics and (B) control inputs.

Optimal distributed secondary control for a microgrid Chapter | 3 77

Fig. 11 Dynamic performance of the proposed distributed control strategy when ri = 3. (A) Frequency dynamics and (B) control inputs.

at this time, the ramping rate is smaller than the case when ri is equal to 1.5 and the control inputs are limited within the desired boundary. It is worth pointing out that the settling time presented in Eq. (48) is the upper bound of the total convergence time, which is rather conservative. Therefore, the actual convergence time is not linearly related to ri . According to the analysis given here, it can be summarized that adjusting ri can be used to deal with the situation of bounded control inputs with certain ramping rate limitations.

4.6 Plug-and-play capability analysis Plug-and-play capability can ensure that when a new DG is added or the existing DG is removed, the microgrid control system does not need to be redesigned and centralized supervision is not required. Therefore, for a microgrid, it is desirable for it to support the plug-and-play functionality because of the availability of renewable generators. In this case study, the microgrid with five DGs and five loads is operated under the primary control at the initial state (t = 0 s). The proposed optimal distributed control strategy is implemented at t = 1 s. An additional DG 6 with the rated power of 2.0 kW + 1.0 kVAr and the line impedance Z16 of 0.08 = j0.18 is suddenly connected to the microgrid at t = 2 s. Then, DG3 is disconnected at t = 3 s. From Fig. 12A and B, after the plug-and-play operations of DG units, only small transient variations exist for DGs’ frequency and voltage magnitude, so the frequency and voltage magnitude remain stable. As shown in Fig. 12C and D, after DG 6 is plugged in at t = 2 s, all of the DGs are managed to share the active and reactive power to the desired ratios. After DG 3 is disconnected at t = 3 s, the active and reactive power outputs of DG 3 are decreased to 0 due to the RL filter. In summary, the proposed distributed control strategy can offer a satisfactory plug-and-play capability.

78 PART | I Frequency and voltage control

Fig. 12 System dynamic performance after the plug-and-play operation. (A) Frequency dynamics; (B) voltage magnitude dynamics; (C) active power sharing dynamics; and (D) reactive power sharing dynamics.

4.7 Scalability test The proposed optimal distributed control strategy is applied in a 42-bus distribution system to investigate scalability. The system diagram is presented in Fig. 13 [32]. Suppose that the communication network is the same with the transmission line connections. The algebraic connectivity of the communication graph (λ2 ) is equal to 0.066. In this case study, before t = 4 s, the 42-bus distribution system is operating in the islanded mode with all of the DGs are operated by the conventional primary droop control and the overall loads is 180 MW + 60 MVAr. The proposed optimal distributed control strategy is implemented at t = 4 s. The additional loads of 8.5 MW + 4.5 MVAr are then connected to the distribution system at t = 12 s. The utilization ratio of the active and reactive power sharing of the 42-bus test system are presented in Fig. 14A and B, respectively. Simulation results indicate that the proposed distributed strategy offers a possible application value in large systems since the active and reactive power utilization ratio can reach a consensus within 5 s. Notice that the maximum tolerable communication delay is essentially determined by the eigenvalues of the communication graph, which are jointly influenced by the communication channels’ number, the devices’ number,

Optimal distributed secondary control for a microgrid Chapter | 3 79

38

36

34

33 32

37 29 35

31 30

28

10

40 39

27 26

9

25

5

11 8

41

7

4

6

3

2

1

13 12

42

15

22 24

23 21

20

19

16

18 17

14 Fig. 13

Diagram of the 42-bus test system.

Reactive power utilization ratio

Active power utilization ratio

0.69 0.68 0.67 0.66 0.65 0.64 0.63 0

4

(A)

8

12 Time (s)

16

20

0.6

0.55

0.5 0

(B)

4

8

12

16

20

Time (s)

Fig. 14 Utilization ratio of active/reactive power sharing of the 42-bus distribution system. (A) Utilization ratio of active power and (B) utilization ratio of reactive power.

and the connection method of the devices [30]. For large-scale systems, the maximum tolerable communication delay is small so that the communication delay can limit the scalability of the distributed control strategy to some extent.

5 Conclusion In this chapter, an optimal distributed control strategy utilized in secondary frequency and voltage regulation for the islanded microgrid is proposed, which is based on a neighbor-to-neighbor communication protocol. The proposed distributed control strategy can ensure the distributed data processing to be more efficient, support the plug-and-play capability of the microgrid, and facilitate the scalable expansion of the system. Using the proposed finite-time controller,

80 PART | I Frequency and voltage control

the frequency regulation and active power sharing can be achieved in a finitetime manner. In addition, the voltage regulation and reactive power sharing at a different timescale can be designed separately. Simulation studies illustrate the effectiveness of the proposed optimal distributed strategy in spite of load variations and plug-and-play operations.

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[16] L. Lu, C. Chu, Consensus-based secondary frequency and voltage droop control of virtual synchronous generators for isolated AC micro-grids, IEEE J. Emerg. Sel. Top. Circuits Syst. 5 (3) (2015) 443–455. [17] H. Zhang, S. Kim, Q. Sun, J. Zhou, Distributed adaptive virtual impedance control for accurate reactive power sharing based on consensus control in microgrids, IEEE Trans. Smart Grid 8 (4) (2017) 1749–1761. [18] A. Bidram, A. Davoudi, F.L. Lewis, J.M. Guerrero, Distributed cooperative secondary control of microgrids using feedback linearization, IEEE Trans. Power Syst. 28 (3) (2013) 3462–3470. [19] J.W. Simpson-Porco, F. Dörfler, F. Bullo, Synchronization and power sharing for droop-controlled inverters in islanded microgrids, Automatica 49 (9) (2013) 2603–2611. [20] J.W. Simpson-Porco, Q. Shafiee, F. Dörfler, J.C. Vasquez, J.M. Guerrero, F. Bullo, Secondary frequency and voltage control of islanded microgrids via distributed averaging, IEEE Trans. Ind. Electron. 62 (11) (2015) 7025–7038. [21] S.T. Cady, A.D. Domínguez-García, C.N. Hadjicostis, Finite-time approximate consensus and its application to distributed frequency regulation in islanded AC microgrids, in: 2015 48th Hawaii International Conference on System Sciences, 2015, pp. 2664–2670. [22] F. Guo, C. Wen, J. Mao, Y. Song, Distributed secondary voltage and frequency restoration control of droop-controlled inverter-based microgrids, IEEE Trans. Ind. Electron. 62 (7) (2015) 4355–4364. [23] S. Zuo, A. Davoudi, Y. Song, F.L. Lewis, Distributed finite-time voltage and frequency restoration in islanded AC microgrids, IEEE Trans. Ind. Electron. 63 (10) (2016) 5988–5997. [24] N.M. Dehkordi, N. Sadati, M. Hamzeh, Distributed robust finite-time secondary voltage and frequency control of islanded microgrids, IEEE Trans. Power Syst. 32 (5) (2017) 3648–3659. [25] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520–1533. [26] L. Wang, F. Xiao, Finite-time consensus problems for networks of dynamic agents, IEEE Trans. Autom. Control 55 (4) (2010) 950–955. [27] W.M. Haddad, A. L’Afflitto, Finite-time stabilization and optimal feedback control, IEEE Trans. Autom. Control 61 (4) (2016) 1069–1074. [28] X. Lu, X. Yu, J. Lai, Y. Wang, J.M. Guerrero, A novel distributed secondary coordination control approach for islanded microgrids, IEEE Trans. Smart Grid 9 (4) (2018) 2726–2740. [29] Y. Xu, H. Sun, W. Gu, Y. Xu, Z. Li, Optimal distributed control for secondary frequency and voltage regulation in an islanded microgrid, IEEE Trans. Ind. Inf. 15 (1) (2019) 225–235. [30] S. Sahoo, S. Mishra, A distributed finite-time secondary average voltage regulation and current sharing controller for DC microgrids, IEEE Trans. Smart Grid 10 (1) (2019) 282–292. [31] J.A.P. Lopes, C.L. Moreira, A.G. Madureira, Defining control strategies for microgrids islanded operation, IEEE Trans. Power Syst. 21 (2) (2006) 916–924. [32] P.M.D.O.-D. Jesus, M.T.P. de Leao, J.M. Yusta, H.M. Khodr, A.J. Urdaneta, Uniform marginal pricing for the remuneration of distribution networks, IEEE Trans. Power Syst. 20 (3) (2005) 1302–1310.

Chapter 4

Distributed power control of flexible loads in microgrids Jianqiang Hu Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Zhejiang, People’s Republic of China

1 Coordinated active power dispatch control for a microgrid A microgrid is a controllable microsystem formed by distributed generators (DGs), energy storage units (ESUs), and local loads with the ability to operate connected to the main grid or as an island mode. When connecting to the main grid, a microgrid serves as a prosumer and can sell/buy electric power to/from the main grid by participating in the electricity market. In practice, a microgrid can be an alternating current (AC) microgrid, a direct current (DC) microgrid, or a hybrid AC/DC microgrid according to different transmission modes [1]. As can be seen from Fig. 1 (a typical structure for a microgrid), there is a microgrid central controller (MGCC) in the system, which is responsible for the maximization of the profit by optimizing power outputs of local DGs/ESUs and the power exchange with the main distribution grid on different timescales. The optimized operating scenario is achieved by sending control signals to the micro source controllers and load controllers in a centralized manner [2, 3]. One significant feature of microgrids is that the high penetration of flexible distributed micro units to the microsystem, and the traditional centralized dispatch control system becomes too complicated and experiences low efficiency when processing a diversity of optimization and control problems. Meanwhile, a bidirectional communication structure is needed between the MGCC and the terminal units in order to connect sampled data from micro units, and the MGCC is responsible for calculating and issuing control signals to all terminal units. The distributed technique has emerged with many merits, such as robustness, reliability, and lower cost to implement, which has good scalability and can survive single-point failures [4, 5]. Recently, distributed strategies have been utilized to solve different problems in microgrids. Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00004-9 Copyright © 2020 Elsevier Inc. All rights reserved.

83

84 PART | I Frequency and voltage control d

oa

l AC

Main grid

DC load

Fig. 1

A typical hybrid AC/DC microgrid structure.

Complex operational tasks managed at a centralized level can be decomposed into multiple undemanding operations implemented at a component level by distributed strategies. In this part, the operation optimization problem of a microgrid is considered by maximizing its profit via optimizing the outputs of DGs and ESUs by proposing a distributed λ-iteration algorithm. The basic idea of λ-iteration is that there is an independent system operator (MGCC) who is responsible for estimating the optimal incremental cost and broadcasting the estimated value to all units, and then collects the power outputs of all units to calculate the next estimation value, which has been utilized to solve economic dispatch problems (EDPs) in power systems. If the total power output is too low (high), λ value will be increased (decreased) until the desired operating point is found [6]. In the microgrid, MGCC is the central decision-maker and performs all calculations in a central level by the λ-iteration algorithm. Here, we introduce a distributed λ-iteration algorithm to reduce the communication and computation burden of MGCC, and mathematically prove the stability of the iteration algorithm.

1.1 Distributed pinning consensus on networks and corporation optimization Corporation optimization is intended to solve the optimization problem composed by multiple interactive units in a distributed way, which is based on distributed communication and computation. Suppose the communication network among multiple interactive units can be modeled by a digraph G = {V, E, A}, where the nodes’ set V = {1, 2, . . . , N} in the network denotes the set of individual units and the links’ set E denotes the set of communication lines. The edge eij = (i, j) ∈ E indicates that the jth unit can receive the information from ith unit. A graph is said to be undirected if eij ∈ E implies eji ∈ E. A directed

Distributed power control of flexible loads in microgrids Chapter | 4 85

graph is said to be strongly connected if there exists a path between any pair of two nodes with respect to the orientation of edges. A directed tree is a digraph, where every node, except the root, has exactly one parent node. A directed spanning tree of G is a directed tree whose node set is V and whose edge set is a subset of E. For a digraph G, the adjacency matrix A ∈ RN×N is defined as aij ≥ 0, in which aij = 1 ⇔ eji (j → i) ∈ E while aij = 0 if eji  ∈ E, and it is further required that self-links are not allowed (i.e., aii = 0). The Laplacian matrix L is defined as L = D − A, where D is a diagonal  a matrix with dii = j=i ij , thus L has nonnegative diagonal entries and zero row sums. Let dmax = max{dii } denote the maximal node in-degree of digraph G. Then, the matrix P = I − L is a nonnegative and row stochastic matrix for all  ∈ (0, 1/dmax ), in which P is called as the Perron matrix induced by digraph G.

1.1.1 Distributed pinning consensus algorithm For a large scale multiunit interactive system, there are always three kinds of optimization control strategies: centralized control, decentralized control, and distributed control [7, 8]. The following concerns the distributed control strategy of the interactive coupled system by spare communication links. The coupled system for a one-dimensional continuous-time integrator multiunit system is provided as x˙i (t) =

N 

  aij xj (t) − xi (t)

(1)

j=1

and its discrete-time counterpart is xi (k + 1) = xi (k) + 

N 

  aij xj (k) − xi (k) ,

(2)

j=1

for i = 1, 2, . . . , N, where  is the discrete-time step satisfying  ∈ (0, 1/dmax ). In the vector notation, the discrete-time multiunit system (2) takes the form x(k + 1) = P x(k), where P is the Perron matrix of the  communication topology G. Is has been shown that [9] xi (k+1) converges to N i=1 ωi xi (0) under the assumption that the communication topology is strongly connected, where ωT = [ω1 , . . . , ωN ] is the left eigenvalue vector of matrix P with the eigenvalue 1 (i.e., ωT P = ωT and 1TN ω = 1, here 1N = [1, . . . , 1]N ). The converged consensus value depends on the communication topology and the initial state values of each unit, that is, the well-known weighted average consensus for the first-order discrete-time system. However, the average consensus value in the previous formula is not always the desired final state in

86 PART | I Frequency and voltage control

practice. In order to drive the multiunit system to converge to a given objective value (Leader), the distributed pinning consensus protocol is introduced. The so-called “distributed pinning control” means that only a small fraction of the nodes in the network are pinned by the control center to the objective trajectory, and the rest of the nodes communicate with each other to reach the expected networked tracking. The following distributed pinning protocol is a special case for the continuous-time distributed system in [10]: x˙i (t) =

N 

    aij xj (t) − xi (t) − di xi (t) − θ (t) ,

(3)

j=1

where i = 1, 2, . . . , N; the pinning control gain di ≥ 0, in which di = 0 indicates that the ith unit is free of control; and θ is an expected consensus state which can be a static or dynamic trajectory. If a node is pinned, that is, di > 0, then it can access the global objective θ (t). That is, an additional communication link is built between the pinner and the pinned nodes. By denoting the objective trajectory θ (t) as the dynamics of an isolated node 0 and we use the union of the digraph G and the node {0} (G˜  G ∪ {0}) to denote the pinning joint communication topology. The Laplacian matrix of G˜ is   0 01×N , L˜ = −d˜ L + D ˜ is the pinning matrix. Before in which d˜ = [d1 , d2 , . . . , dN ]T and D = diag{d} proposing the main results, we need the following lemma. Lemma 1 (Song et al. [11]). For the multiunit system (3), if the pinning joint communication topology has a directed spanning tree, then L + D is a nonsingular M-matrix and the group value will be synchronized to the leader’s equilibrium θ0 (limt→∞ θ (t) = θ0 ), that is, limt→∞ (xi (t) − θ0 ) = 0.

1.1.2 Dispatch optimization of units in microgrid by λ-iteration algorithm The conventional EDP is introduced, which aims at minimizing the total generation cost of generating units and determining the power output levels of online generators. It can be formulated by the following optimization problem [12]: min Fcost (P) = ⎧ ⎨ s.t.

⎩

N 

Ci (Pi ),

(4)

i=1 N 

Pi = PD ,

i=1 Pmin ≤ i

Pi ≤

Pmax i ,

(5)

Distributed power control of flexible loads in microgrids Chapter | 4 87

for i  = 1, . . . , N, where Ci (Pi ) is the generation cost for the ith generating unit, N i=1 Pi is the total generated power which is consumed by the active load and Pmax are the lower power bound and upper power demand PD , and Pmin i i bound for each unit i, respectively. In each step of the λ iteration algorithm, the dispatch center estimates an incremental cost rate λ and sends it to all units, and then collects all the power outputs of units based on the issued λ. If the sum of all outputs of units is very low, then the dispatch center will increase the λ value and try another solution until it finds the desired operating point λ* , that is, |ε| ≤ TOLERANCE. Furthermore, the optimal output of each unit can be calculated. It has been shown that the λ-iteration procedure converges very rapidly for this particular type of cooperative optimization EDP. The actual computational procedure is slightly more complex than the steps in this algorithm, since it is necessary to observe the operating limits and the prohibited operating zones of each unit during the process of the computation [6].

1.1.3 Corporation dispatch control of units in a microgrid A microgrid is a mini autonomous source-grid-load system, which can operate in either a grid-connected mode or an islanded mode through a static transfer switch, such as the structure in Fig. 1. In the microgrid, MGCC serves as an independent system operator which is responsible for the maximization of total profit during interconnected operation by optimizing the active power outputs of local DGs/ESUs and the power exchange with the main grid. The fluctuation of the tie line power between the main grid and the microgrid reflects the dynamic power injection from the distribution system to the microgrid or the injection from the microgrid to the main grid. Suppose the positive direction of the exchanged power of the tie line is the power injection from the main grid to the microgrid. Therefore, each grid-connected microgrid needs to purchase (or sell) electric power from (or to) the distributed system and sell the electric power to customers in the microgrid. The tertiary frequency control of the microgrid will maximize its profit by determining the power outputs of all distributed energy resources and ESUs such that the supply and demand balance is maintained. Suppose there are N1 DGs and N2 ESUs in the microgrid and the microgrid is operated in a grid-connected mode, then the optimization model of the profit maximization can be expressed as max F1 (Pgi , Pbj ) = −ρe PE + ρd PD −

N1 

Cgi (Pgi ) −

i=1

N2 

Cbj (Pbj ),

(6)

j=1

subject to the active power balance constraint N1  i=1

Pgi +

N2  j=1

Pbj + PE = PD ,

(7)

88 PART | I Frequency and voltage control

where Cgi (·) and Cbj (·) are the cost functions of the ith DG and the jth ESU, which are always approximated by quadratic functions, provided as Cgi (Pgi ) =

(Pgi − α1,i )2 + γ1,i , 2β1,i

Cbj (Pbj ) =

(|Pbj | − α2,j )2 + γ2,j , 2β2,j

∀1 ≤ i ≤ N1 , ∀1 ≤ j ≤ N2 ,

where α1,i , β1,i , and γ1,i are the cost coefficients of the ith DG and α2,j , β2,j , and γ2,j are the cost coefficients of the jth ESU. The capacity constraint of the ith DG is given as max Pmin gi ≤ Pgi ≤ Pgi ,

(8)

with Pmin and Pmax (kW) being the minimum and maximum regulation gi gi capacities. Furthermore, each ESU has two operational states, that is, the discharging state as a generating unit and the charging state as a controllable load. Thus, the constraints are divided into two categories with the discharging power constraint being , 0 ≤ Pbj ≤ Pdch,max bj

(9)

and the charging power constraint being ≤ Pbj ≤ 0. − Pch,max bj

(10)

Based on the established profit maximization optimization problem, MGCC in microgrids is responsible for optimizing the active power output of all micro units so as to acquire the maximal profit via distributing the power demand among micro units or tie line between the microgrid and main grid.

1.2 Distributed λ-iteration optimization of active power 1.2.1 Solution without power constraints If there are no capacity constraints for the DGs and ESUs, then the optimization model reduces to be the objective function (6) with the equality constraint (7). By eliminating the variable PE in the objective function through the equality  1 N2 PE = PD − N i=1 Pgi − j=1 Pbj , one has the following minimum optimization function: N1 N2   [Cgi (Pgi ) − ρe Pgi ] + [Cbj (Pbj ) − ρe Pbj ], min F2 (Pgi , Pbj ) = i=1

(11)

j=1

subject to constraint (7). We furthermore simplify the optimization model by augmenting the optimization variable P = [Pg1 , . . . , PgN1 , Pb1 , . . . , PbN2 ] and denote N = N1 +N2 , then such an optimization problem (11), (7) is equivalent to

Distributed power control of flexible loads in microgrids Chapter | 4 89

⎧ N N   ⎪ ⎪ ˜ i (Pi ) = ⎪ (P) = [Ci (Pi ) − ρe Pi ], C min F 3 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩s.t.

N 

i=1

i=1

(12)

Pi + PE = PD .

i=1

The well-known solution to such an optimization problem (12) is the equal ˜

∂ C˜ (P )

j j i (Pi ) incremental cost criterion, that is, ∂ C∂P = ∂P = λ* , ∀ 1 ≤ i, j ≤ N and i j N * i=1 Pi + PE = PD , where λ is called the optimal incremental cost, which can be calculated by  PD − PE − N i=1 αi − ρe . λ* = N i=1 βi

If the optimal value λ* is shared with each unit, then the optimal power output for the ith unit can be calculated as P*i = βi (λ* + ρe ) + αi , ∀1 ≤ i ≤ N, where α = [α1 , . . . , αN ] = [α1,1 , . . . , α1,N1 , α2,1 , . . . , α2,N2 ], similarly for parameters βi , γi . The maximal profit F1max can be calculated by F1max = (ρd − ρe )PD − F3min through the optimal solution of the optimization problem (12). We introduce a distributed λ-iteration algorithm to solve the centralized optimization problem. Each unit cannot acquire the optimal incremental cost in the distributed scenario. In order to update its power output, units try to estimate the optimal value of λ* . Suppose the real-time estimation of the ith unit is λˆ i (t), which is characterized by the following distributed differential equation: λ˙ˆ i (t) = μ

N 

    ˆ i (t) − di μ λ ˆ i (t) − λ0 (t) , aij λˆ j (t) − λ

(13)

j=1

where μ is the coupling strength of the distributed protocol, aij is the element of the adjacent matrix A of the communication topology among the participating units, and di = 1 if the ith unit is pinned by the MGCC, otherwise di = 0. i − ρe , and Pi (0) is the initial The initial estimation value λˆ i (0) = Pi (0)−α βi state of the ith unit. λ0 (t) is the pinning signal generated from MGCC, which is updated by

⎧ N ⎪ ⎨λ˙ (t) = κ P − P −  P (t) , 0 D E i (14) i=1 ⎪ ⎩ ˆ Pi (t) = βi ( λi (t) + ρe ) + αi , and the initial value λ0 (0) is set to be the average value of all the pinned units,  pin  that is, λ0 (0) = ave λˆ i (0) . Remark 1. If all units are pinned by the MGCC, then di = 1, ∀1 ≤ i ≤ N, which reduces to be the centralized optimization algorithm (traditional λ-iteration algorithm). On the other hand, the optimal incremental cost λ*

90 PART | I Frequency and voltage control

can be calculated by MGCC under a cooperative scenario where the private cost information of all units is reported to MGCC, that is, λ0 (t) ≡ λ* . Then, the estimation value of each unit will converge to the optimal value even with the distributed algorithm (13) under the assumption that the joint communication topology has a directed spanning tree. In the implementation of the distributed iteration algorithm, continuoustime differential equations (13), (14) need to be transformed to discrete-time difference equations. Here, we utilize the Euler method to derive the discretetime system: ⎧ N ⎪      ⎪ ⎪ ⎪ aij λˆ j (k) − λˆ i (k) − di μh λˆ i (k) − λ0 (k) , λˆ i (k + 1) = λˆ i (k) + μh ⎪ ⎨ j=1

N ⎪  ⎪ ⎪ ⎪λ0 (k + 1) = λ0 (k) + κh PD − PE − Pi (k) , ⎪ ⎩ i=1

(15) and the output power of the ith unit is calculated by Pi (k) = βi ( λˆ i (k) + ρe ) + αi ,

(16)

where h is the discretization step, that is, the sampling period for the practical operation system. Algorithm 1 Distributed λ-iteration optimization

Theorem 1. Suppose the pinning joint communication topology between MGCC and participating units is connected for an undirected topology or has a directed spanning tree for a directed topology, the proposed distributed λ-iteration Algorithm 1 solves the optimization problem (12), that is, the

Distributed power control of flexible loads in microgrids Chapter | 4 91

estimated incremental cost λˆ i and output power Pi asymptotically converge to the optimal values λ* and P*i globally, respectively. Proof. To begin with, we define two error variables, the estimation error variable ei (t) = λˆ i (t) − λ0 (t) and the translation variable r(t) = λ0 (t) − λ* . According to the equilibrium equation of Eq. (14), one has N  [βi (λ* + ρe ) + αi ] = 0. PD − PE −

(17)

i=1

Furthermore, it is easy to derive the error system ⎧ N ⎪    ⎪ ⎪ ⎪ aij ej (t) − ei (t) − di μei (t) − r(t), ˙ e˙i (t) = μ ⎪ ⎨ j=1

N ⎪    ⎪ ⎪ ⎪ r(t) ˙ = −κ βi ei (t) + r(t) . ⎪ ⎩

(18)

i=1

By denoting e(t) = [e1 (t), . . . , eN (t)]T , the error system (18) can be transformed into the following augmented one:    e(t) ˙ = −μ(L + D)e(t) + κ1  N Be(t) + B1N r(t) , (19) r(t) ˙ = −κB 1N r(t) + e(t) . Then, consider the following Lyapunov candidate: 1 1 V(t) = r2 (t) + eT (t)Θe(t), (20) 2 2 where Θ is a positive definite matrix given in Algorithm 1. Then the time derivative of V(t) along the solution of error system (19) is ˙ = − κr(t)(B1N )r(t) − κr(t)Be(t) V(t) + κeT (t)(Θ1N B1N )r(t) + κeT (t)(Θ1N B)e(t)   μ − eT (t) Θ(L + D) + (L + D)T Θ e(t) 2 zT (t)Ωz(t), where z(t) = [rT (t), eT (t)]T and Ω is given as follows:   κ T −κB1N 2 [(Θ1N B1N ) − B] Ω= . * κΘ1N B − μ2 [Θ(L + D) + (L + D)T Θ]

(21)

By the Schur complement lemma, one knows that Ω < 0 is equivalent to κ > 0 and μ κM < 0, κΘ1N B − [Θ(L + D) + (L + D)T Θ] + 2 4B1N from which one can derive the scope of the parameter μ. This completes the proof, that is, the estimated incremental cost λˆ i asymptotically converge to the optimal values λ* globally.

92 PART | I Frequency and voltage control

1.2.2 Solution with power constraints When considering the regulation capacities of DGs and ESUs, expressed as ≤ Pi ≤ Pmax Pmin i i ,

(22)

the optimal incremental cost λ* satisfied the following optimum condition: ⎧ Pi − αi ⎪ ⎪ = λ* , for Pmin ≤ Pi ≤ Pmax ⎪ i i , ⎪ β ⎪ i ⎪ ⎨P − α i i < λ* , for Pi = Pmax (23) i , ⎪ β i ⎪ ⎪ ⎪ ⎪ ⎪ Pi − αi > λ* , for Pi = Pmin . ⎩ i βi By denoting Gp as the set of units whose optimal outputs are their maximal or minimal capacities, the optimal incremental cost λ* can then be calculated by   PD − PE − i∈GP Pm − i∈G / p αi i  − ρe . λ* = i∈G / p βi The distributed algorithm (15) considered the output constraint of Pi (k) by the following updating equation: ⎧ max if λˆ i (k) > λi ,   ⎨Pi , ˆ Pi (k) = gi λi (k) = βi ( λˆ i (k) + ρe ) + αi , if λi ≤ λˆ i (k) ≤ λi , (24) ⎩ min if λˆ i (k) < λi , Pi , − αi )/βi and λi = (Pmax − αi )/βi . where λi = (Pmin i i Algorithm 2 Distributed λ-iteration constrained optimization

Proposition 1. Suppose the joint communication topology between MGCC and participate units is connected for an undirected topology or has a directed spanning tree for a directed topology, the proposed distributed λ-iteration algorithm (1) based on Eq. (24) solves the optimization problem (12) with the

Distributed power control of flexible loads in microgrids Chapter | 4 93

additional constraint (22), that is, the output power Pi converge to the optimal value P*i with finite steps. This proposition is a generalization of Theorem 1, where the capacity constraints have been taken into account in the optimization iteration. When the microgrid is operated in a grid-connected mode, the coordinated active power dispatch problem of the microgrid is always solvable due to the existence of the exchanged power with the main grid. When the microgrid is operated in an isolated mode, the coordinated active power dispatch problem of the microgrid is solvable under the assumption that all the power supply from the sources is able to satisfy the demand of the load. Therefore, the optimization problem is always solvable. Proposition 1 under these two operating scenarios can be classified as the following two cases: (1) if the intersection of the incremental cost intervals  * ¯ 1≤i≤N [λi , λi ]  = ∅, then there exists a common optimal incremental cost λ , and all the incremental costs λi (1 ≤ i ≤ N) will converge to the optimal value within finite steps; and (2) if the intersection of the incremental cost intervals  ¯ 1≤i≤N [λi , λi ] = ∅, then partial incremental costs λi , i ∈ [1, N] converge to the boundary of the incremental cost interval, that is, some units with lower operating costs operate in their maximum output powers’ states. Thus, according to Theorem 1, the iteration updating in Proposition 1 is valid in the practical operation. Remark 2. The distributed algorithm (13) can be replaced with a fixed-time or a finite-time one, which can speed up the convergence speed of the algorithm, given as N    ˙ˆ =μ  a sigp  λ ˆj − λ ˆ i − λ0 ˆ i − di μ1 sigp λ λ i 1 ij j=1

+ μ2

N 

    ˆj − λ ˆ i − λ0 , ˆ i − di μ2 sigq λ aij sigq λ

j=1

where μ1 , μ2 are the coupling strengths of the distributed protocol, aij is the element of the adjacent matrix A of the communication topology among the participating units, and di = 1 if the ith unit is pinned by the MGCC, otherwise di = 0. The function sigp (·) = sign(·)| · |p . Especially, when p = 1, q = 0, and sig(x) = x, the previous algorithm reduces to be the previous one (Eq. 13).

1.3 Case studies In this section, the proposed distributed λ-iteration algorithms are tested on a seven-bus microgrid system and the physical topology structure is given in Fig. 2 (The dashed lines denote communication links and the full lines are physical links), where there are seven DGs, three ESUs, and seven-bus loads. The red dashed lines represent the communication links among MGCC and units. The cost coefficients and capacity constraints for these units are provided in Table 1.

94 PART | I Frequency and voltage control

DG2

ESU1 DG1

L1

L2

2

1

5 DG4

L3

ESU3

7

Fig. 2

L5

4

3 DG3

DG5

ESU2

DG6

L4

DG7

6

L6

Seven-bus microgrid with seven DGs and three ESUs.

TABLE 1 DG and ESU private parameters. DGs’ parameters (α, β, γ (×103 )) (kW) DG

α1,i

β1,i

γ1,i

min Pg,i

max Pg,i

DG1

−0.0593

0.0069

−0.2032

10

60

DG2

−0.0313

0.0050

−0.0502

10

60

DG3

−0.0219

0.0064

−0.0063

8

60

DG4

−0.0120

0.0048

0.0628

3.8

40

DG5

−0.0245

0.0061

−0.0073

5.4

45

DG6

−0.0345

0.0048

−0.0616

4.2

18

DG7

−0.0065

0.0053

0.0470

7.8

45

ESUs’ parameters

(α, β, γ (×103 ))

(kW)

ESU

α2,i

β2,i

γ2,i

dch,max Pb,i

ch,max

ESU1

−0.3170

0.0333

−1.4153

25

20

ESU2

−0.1331

0.0156

−0.4891

30

25

ESU3

−0.1677

0.0192

−0.6931

45

40

Pb,i

1.3.1 Case 1: The full participation of DGs and ESUs We consider the discharging mode of ESUs, that is, the storage units serve as generating units to provide the active power sharing and the communication topology (red dashed lines) is provided in Fig. 2, where only the third DG and the third ESU are pinned by MGCC. Suppose the power demand in this MG is PD = 370 kW and the injection power from the main grid is PE = 120 kW; the

Distributed power control of flexible loads in microgrids Chapter | 4 95

wholesale market price ρe = 1.2 $/kWh, and the retail price ρd = 1.8 $/kWh. The initial states of DGs are given by Pg (0) = [30, 20, 40, 15, 18, 6, 20] kW and the initial states of ESUs are given by Pb (0) = [10, 12, 21] kW. By setting κ = 0.01, one can derive that μ > 1.7773 and we set μ = 3.82 in the simulation. The sampling period is set to be h = 0.02 and the stop rule TOLERANCE = 0.4; one can solve the profit maximal problem by the proposed distributed λiteration algorithms. Then, it is easy to derive the optimal incremental cost λ* = 8.6431 (see Fig. 3) by utilizing the distributed Algorithm 1 without considering the capacities of each units and the corresponding active power outputs for DGs are P*g1 = 9.0418 kW, P*g2 = 17.3866 kW, P*g3 = 41.2305 kW, P*g4 = 34.8194 kW, P*g5 = 35.5010 kW, P*g6 = 12.8470 kW, P*g7 = 45.8619 kW, and the discharging power of ESUs are P*b1 = 11.0689 kW, P*b2 = 20.6586 kW, P*b3 = 21.5775 kW, which are shown in Fig. 4. Here, Figs. 3 and 4 illustrate the dynamic response curves for the distributed system (15), (16), but the distributed λ-iteration algorithm breaks out the circulation by the stop rule TOLERANCE = 0.4 (see Fig. 5). Furthermore, we utilize the distributed iterations (15), (24) by considering the capacity constraints of the units. It can be derived that the corresponding

l (t) and l (t)(1≤ i ≤ N)

12 10

i

8

X : 16 Y : 8.643

0

6 4

Fig. 3

0

2

4

6

8

10 Time (s)

12

14

16

18

20

14

16

18

20

Case 1: The estimated incremental costs by Algorithm 1.

P (t) and P (t )

60

B

40

G

20 0 −20

Fig. 4

0

2

4

6

8

10 Time (s)

12

Case 1: The active power outputs of DGs/ESUs by Algorithm 1.

96 PART | I Frequency and voltage control

E

250

P sum(t) and P −P

275

D

300

225

G

200

Stop here for TOLERANCE = 0.4

175 150

Fig. 5

0

2

4

6

8

10 Time (s)

12

14

16

18

20

16

18

20

Case 1: The supply and demand balance by Algorithm 1.

l 0(t) and li(t)(1≤ i ≤ N )

12 10 8 6 4

Fig. 6

0

2

4

6

8

10 Time (s)

12

14

Case 1: The estimated incremental costs by Algorithm 2.

active power outputs for DGs are P*g1 = 9.3565 kW, P*g2 = 17.4053 kW, P*g3 = 41.6542 kW, P*g4 = 34.7387 kW, P*g5 = 35.5236 kW, P*g6 = 12.8647 kW, P*g7 = 45 kW, and the discharging powers of ESUs are P*b1 = 11.8866 kW, P*b2 = 20.7166 kW, P*b3 = 20.8538 kW, which are shown in Fig. 7, and the incremental cost for each unit is given in Fig. 6. They illustrate the dynamic

PG (t) and PB (t )

50 40 30 20 10 0

Fig. 7

0

2

4

6

8

10 Time (s)

12

14

Case 1: The active power outputs of DGs/ESUs by Algorithm 2.

16

18

20

Distributed power control of flexible loads in microgrids Chapter | 4 97

E

250

G

P sum(t) and P −P

275

D

300

225 200

150

Fig. 8

Stop here for TOLERANCE = 0.4

175 0

2

4

6

8

10 Time (s)

12

14

16

18

20

Case 1: The supply and demand balance by Algorithm 2.

response curves for the distributed system (15), (24), but the distributed λ-iteration algorithm breaks out the circulation by the stop rule TOLERANCE = 0.4 (see Fig. 8). Employing the proposed algorithms, all units can coordinate with each other to minimize the total cost while converging to the optimal operation point subject to the power balance constraint. On the other hand, as can be seen from the simulation results, there is a common consensus state λ* for all participating units in the scenario without capacity constraints and all participating units do not have a common consensus state λ* due to the fact that part of the units operate at their boundaries.

1.3.2 Case 2: Exit of ESUs and time-varying demand In this simulation scenario, the cost and capacity parameters of DGs and ESUs are the same as in Case 1. The main difference is that we consider the case that the load demand is a time-varying one in different dispatch periods and the participating unit may be plugged-in or plugged-out. Here, we only consider two dispatch periods in the simulation. Suppose the load demand is PD = 370 kW and the injection power from the main grid is PE = 120 kW; the wholesale market price ρe = 1.2 $/kWh and the retail price ρd = 1.8 $/kWh in the first dispatch period, and in the second dispatch period (such as 15-min dispatch period), the load demand turns out to be PD = 400 kW and the injection power from the main grid is PE = 170 kW; the wholesale market price ρe = 1.4 $/kWh and the retail price ρd = 2.0 $/kWh. Meanwhile, suppose all DGs and ESUs participate the active power allocation in the first dispatch period and all ESUs exit in the second dispatch period and the load demand will be shared by all DGs. The communication topology of all DGs in this scenario is given in Fig. 9 and only the third DG is pinned by the MGCC. Next, the initial states of all DGs and ESUs in the first dispatch period are the same as in Case 1 and the initial states of all DGs in the second dispatch period are the steady state of the result of first dispatch period. By setting κ1 = 0.01 and μ1 = 3.82 in the first dispatch period, and κ2 = 0.01 and μ2 = 6.82

98 PART | I Frequency and voltage control

Fig. 9

The communication topology among seven DGs.

in the next dispatch period, and the sampling period h = 0.02, the stop rule TOLERANCE1 = 0.4 and TOLERANCE2 = 0.2, one can solve the profit maximal problem by the proposed distributed λ-iteration algorithms. The simulation results of all DGs without and with capacity constraints via the distributed λ-iteration Algorithms 3 and 4 are provided in Figs. 10–15.

X : 50 Y : 9.206

10 8

X : 15 Y : 8.643

No longer changes for ESUs

6

0

i

l (t ) and l (t )(1≤ i ≤ N)

12

4

Fig. 10

0

10

20

900 Time (s)

910

920

930

Case 2: The estimated incremental costs by Algorithm 1.

P (t ) and P (t)

60

B

40

G

20 0 −20 10

20

900 Time (s)

Fig. 11

Case 2: The active power outputs of DGs/ESUs by Algorithm 1.

910

Distributed power control of flexible loads in microgrids Chapter | 4 99

E

310

D

P sum(t) and P −P

280 250 Stop here for TOLERANCE = 0.2 220

G

Stop here for TOLERANCE = 0.4 190 160

Fig. 12

0

10

20

900 Time (s)

910

920

930

Case 2: The supply and demand balance by Algorithm 1.

l 0(t) and li (t)(1≤ i ≤ N )

12 10 8 6 4

0

10

20

900

910

920

930

920

930

Time (s)

Fig. 13

Case 2: The estimated incremental costs by Algorithm 2.

PG(t ) and PB(t )

60

40

20

0

Fig. 14

0

10

20

900 Time (s)

910

Case 2: The active power outputs of DGs/ESUs by Algorithm 2.

280

G

D

P sum(t ) and P −P

E

310

250 Stop here for TOLERANCE = 0.2 220 Stop here for TOLERANCE = 0.4 190 160

Fig. 15

0

10

20

900 Time (s)

910

Case 2: The supply and demand balance by Algorithm 2.

920

930

100 PART | I Frequency and voltage control

In the second dispatch period, all ESUs exit the dispatch system and therefore their output powers stay at the steady states of the previous dispatch period. The DGs will accomplish the active power demand by updating their output according to the distributed λ-iteration algorithms. As can be seen from the simulation results, all DGs operate to the new equilibrium under the timevarying load demand.

1.3.3 Case 3: The plugging-in DGs to share the active power In this mode, we consider the scenario that two additional DGs and bus loads are connected to the microgrid, labeled by DG8, DG9 and L8, L9. The new communication topology after plugging in the additional DGs is shown in Fig. 16, and the private coefficients for DGs are provided in Table 2. Suppose the load demand is PD = 410 kW and the injection power from the main grid is PD = 120 kW; the whole market price ρe = 1.2 $/kWh and the retail price ρd = 1.8 $/kWh are the same before and after the plugging-in DG2

ESU1 DG1

L1

ESU2 L2

2

1

5 DG4

L3

L5

4

3 Main grid

ESU3

DG3

L7

7

DG6

L4

DG7

L6

6

8

9 L9

L8 DG9

DG8

Fig. 16

DG5

The communication topology among nine DGs and three ESUs.

TABLE 2 The private parameters for DG8 and DG9. DGs’ parameters (α, β, γ , ×103 ) (kW) DG

α1,i

β1,i

γ1,i

min Pg,i

max Pg,i

DG8

−0.0550

0.0596

−0.1861

8

42

DG9

−0.0281

0.0379

−0.0456

8

40

Distributed power control of flexible loads in microgrids Chapter | 4 101

of the additional DGs and loads. After the connecting of DGs, the new load demand increases to PD = 445 kW and the injection power from the main grid reduces to PD = 115 kW. The initial states of the DGs and ESUs are the same as values in Case 1, and the initial states of DG8 and DG9 are Pg8 (0) = 10 kW, Pg9 (0) = 12 kW. Here, the dispatch period before the connecting of additional DGs and loads is labeled as the first dispatch period and the latter dispatch period is labeled as the second dispatch period. Along with the active power changes of load and units, the power output of each unit needs to adjust its outputs so as to achieve the new equilibrium. By setting κ1 = 0.01, one can derive that μ1 > 1.7773; by setting κ2 = 0.01, one can derive that μ1 > 1.4732 and we set μ1 = 3.82 in the first dispatch period and μ2 = 6.82 in the second dispatch period. By setting and the sampling period h = 0.02, the stop rule TOLERANCE1 = 0.4 and TOLERANCE2 = 0.2, one can solve the profit maximal problem by the proposed distributed λ-iteration algorithms. The optimal output power can be derived by the proposed distributed algorithm through the optimal incremental cost. The simulation results of all DGs without and with capacity constraints via the distributed λ-iteration Algorithms 3 and 4 are provided in Figs. 17–22.

l 0(t) and li (t)(1≤ i ≤ N)

12 10 8

X: 50 Y: 9.472

X: 20 Y: 9.2

6 4 2 0 0

Fig. 17

10

20

900 Time (s)

910

920

930

Case 3: The estimated incremental costs by Algorithm 1.

40 20

G

B

P (t ) and P (t)

60

0 0

Fig. 18

10

20

900 Time (s)

910

Case 3: The active power outputs of DGs/ESUs by Algorithm 1.

920

930

102 PART | I Frequency and voltage control

E

350 300 Stop here for TOLERANCE = 0.2

250

G

P sum (t) and P −P

400

D

450

200

Stop here for TOLERANCE = 0.4 0

l 0(t ) and li (t)(1≤ i ≤ N )

Fig. 19

10

20

900 Time (s)

910

920

930

920

930

920

930

Case 3: The supply and demand balance by Algorithm 1.

20 15 10 5 0 0

Fig. 20

10

20

900 Time (s)

910

Case 3: The estimated incremental costs by Algorithm 2.

PG (t) and PB (t)

60

40

20

0

Fig. 21

0

10

20

900 Time (s)

910

Case 3: The active power outputs of DGs/ESUs by Algorithm 2.

G

P sum (t ) and PD −PE

400 350 300 Stop here for TOLERANCE = 0.2 250 Stop here for TOLERANCE = 0.4 200 0

Fig. 22

10

20

900 Time (s)

910

Case 3: The supply and demand balance by Algorithm 2.

920

930

Distributed power control of flexible loads in microgrids Chapter | 4 103

In the second dispatch period, additional loads and DGs are connected to the microgrid suddenly, which has caused a shock to the microgrid, and the steady operation state of the microgrid is broken at this time. As can be seen from the simulation results, the proposed distributed λ-iteration algorithm has shown good scalability and robustness in the whole dispatch process. The proposed distributed algorithm can restore the steady operation of the microgrid and the new equilibrium is achieved after the disturbance.

1.4 Conclusion This section proposed a distributed λ-iteration algorithm to solve economic operation problem of microgrids with/without unit constraints, which could be used to deal with EDPs in power systems as well. Compared with the existing distributed optimization methods, the proposed algorithm considers the global active power constraint by adding a virtual pinner and it can deal with optimization problem with any initial states. The optimization algorithm is partially distributed such that each unit in the microgrid only communicates with its neighbors and part of the units have direct communications with MGCC as long as the joint communication topology between MGCC and participating units is connected for an undirected topology or has a directed spanning tree for a directed topology. The distributed algorithm can also enable the plug-andplay of some extra units in microgrids. Lastly, simulation results validated the effectiveness of the proposed algorithm.

2 Demand response load following control of smart grids Demand response (DR) in smart grids has been proposed and investigated for several decades, which can serve as a utility manager to redistribute the electricity demand for a certain period of time, for example, time-of-day, dayof-week. Several control techniques have been proposed for DR management, such as direct load control [13], dynamic demand control [14], and real-time pricing control [15, 16]. However, the basic problem for DR is how to make efficient energy consumption schedules for massive amounts of smart loads. Traditionally, DR is focused on the curtailment of some electrical loads during peak periods so as to alleviate the demand for peaking generation sources. On the other hand, consumers are more willing to participate in DR programs and try to shift some of their high-load household appliances to off-peak hours to reduce their energy expenses. Nowadays, with the development of fast communication and advanced control technologies, more and more electrical loads on the demand side can be equipped with communication and control modules, which are more active and can be deployed all the time instead of just during the peak periods. Therefore, demand-side resources can provide various ancillary services, such as regulation, load following, frequency responsive spinning reserve, and supplemental

104 PART | I Frequency and voltage control

reserve. Good candidates for these ancillary services include washers, dryers, water heaters, heating, ventilating, and air-conditioning (HVAC) systems with thermal storage, refrigerator, plug-in hybrid electric vehicles (PHEV) etc., among which aggregate populations of thermostatically controlled loads (TCLs), such as air conditioners, refrigerators, and electric water heaters can respond to power demand or be dispatched when subject to some control signals, such as pricing or thermostat set-point signals. In fact, electrical loads on the demand side can be grouped into two categories: rigid loads and flexible loads [4]. Flexible loads are composed of controllable loads and responsive loads, where controllable loads mean that the consumption behavior of the loads can be controlled to the expected objective designedly and responsive loads denote that the electrical loads can respond to the electricity price or system frequency by their own decisions, which may be uncertain and not controllable. In this chapter, we only consider the DR case of controllable loads and mainly focus on the cooling TCLs on the demand side, such as air-conditioning in the summer. Specifically, we utilize the traditional generating units and controllable load agents to follow the mismatched power caused by the load activities and the renewable power injection in real time. The controllable loads are mainly TCLs, and the bilinear aggregate model [17] for DR load follow multiple heterogeneous TCL aggregators is utilized. The homogeneous TCLs in an intelligent residential district are modeled as a TCL aggregator. Thus, there may be several heterogeneous aggregators in each district, and numerous districts are managed by a bus load agent. Generally, each bus load agent needs to collect all information of districts globally and process the huge amount of data, and then issue the control signal to each aggregator simultaneously. These centralized solutions [18] are costly to implement, especially when the number of the participants is large, and are often susceptible to single-point failures. We propose a distributed pinning control strategy to coordinate the operation of multiple generating units and multiple TCL aggregators such that these two kinds of controllable resources can provide the load following service together. Distributed pinning control is flexible, reliable, and less expensive to implement; it has a good scalability and can survive single-point failures. Pinning control means the global information is only shared with a small fraction of participants, and other ones communicate each other through a spare communication network, which can handle the optimization and control problems with global constraints. Here, the load agent just needs to send the global objective information to the adjacent districts and other districts communicate with their neighbors. This is a combined centralized-distributed control strategy, which is also a partially distributed one. On the other hand, the active power trajectory is analyzed and optimized by the dispatch center. The load agent must evaluate its regulation capacities and ramping rates exactly so as to obtain a feasible reference power trajectory. The dispatch center needs to collect all the private information of each aggregator to

Distributed power control of flexible loads in microgrids Chapter | 4 105

perform the optimal power calculation. Based on the bilinear aggregate mode, we first derive the regulation capacities and ramping rates of the TCL aggregator analytically, which is one of the main contributions in this chapter. On the other hand, we introduce a novel distributed pinning control solution for the coordination of multiple heterogeneous TCL aggregators in a load agent, which can provide a load following ancillary service with traditional generating units. Such a control strategy can guarantee the convergence of aggregate power of the load agents with any size of aggregators; meanwhile the distributed control can be implemented using a simple communication network, such as Wi-Fi connections. To realize the reference load following, the generating units can respond to the generating commands by regulating the speed of the units exactly by feedback control. As for TCL load aggregators, the temperature setpoint is regulated in a distributed way so as to change the aggregate power of all TCLs in the DR program.

2.1 Problem formulation and aggregate evaluation of TCLs In smart grids, renewable energy power generations (such as wind and photovoltaic) are developing rapidly. Connecting renewable energy sources to the power grid may result in some uncertainties or instability problems for the whole system. Therefore, more operation reserves are needed to provide ancillary services so as to balance the supply and demand [19]. Ancillary services are traditionally provided by regulating generators; that is, generating units follow the load activities. Some alternative techniques, for example, flywheels, distributed generations, electric vehicles, distributed energy storage, and DR resources, can also provide the same ancillary services, among which ancillary services contributed from the demand side are a promising technique since flexible loads can provide faster regulation or load following services without environmental contaminants [20]. Therefore, how to make efficient load management strategies is critical in DR control. We are concerned with how to control numerous TCL load aggregators to provide the load following service. The load following problem provided by both generating units and TCL aggregators is considered in this part. First, the dispatch center optimizes the optimal generating schedules for regulating units and planned utilizations for TCL aggregators. Furthermore, the generating units can be controlled in a centralized way to follow the generating command and TCL aggregators can be controlled in a distributed way to follow the reference power trajectory. The load following structure of interconnected source and load systems is given in Fig. 23. That is, we utilize the traditional units and DR TCL aggregators to follow the load activities caused by rigid and responsive loads and the injection of renewable energy power. In the proposed DR load following framework, the direction of the power flow is still from the generating units to loads. The difference lies in the fact that the actual load curve is reduced by incorporating the dynamic response of

Real-time dispatch

106 PART | I Frequency and voltage control

Dispatch center

Load and renewable energy power forecasting curves

G

G

G

Load following

Fig. 23

L

Bus load agents

Distributed pinning control

2

1

Power generation tracking

L

L

Power plants

Centralized control

Generators

Communication links

N-1

TCL aggregators Aggregate power tracking

Demand response load following structure of source and load systems.

TCL aggregators and the power flow is reduced slightly as well. The current research is focused on the active power distribution among generating units and flexible load agents and the load following service of multiple TCL aggregators in each load agent. Therefore, the active/reactive power flow constraints are not considered in the optimization dispatch layer.

2.1.1 Basic model of a single TCL and an aggregator A TCL aggregator is composed of a group of individual terminal TCLs and the aggregate power of all these devices is critical for analysis and synthesis. It has been shown in [21] that this kind of TCL can be modeled as a thermal capacitance C (kWh/◦ C) in series with a thermal resistance R (◦ C/kW), the dynamic behavior of which can be modeled by two state variables: the internal temperature of the conditioned mass θ (t) and the discrete operation state s(t) [0/1]. Suppose there are NL TCLs in the load aggregator and the basic model of the ith TCL is characterized by the following hybrid first-order ordinary differential equation [21]:  1  dθi (t) θa,i − θi (t) − si (t)Ri Pr,i , (25) = dt Ci Ri for i = 1, . . . , NL . Pr,i (kW) is the rate of energy transfer to or from the thermal mass, which is positive for cooling TCLs and negative for heating TCLs. The operation state si (t) is a discrete switching sequence governed by a thermostatic switching law with predetermined temperature deadband. For the cooling TCLs, one has the following switching law of the operation state: ⎧ 0 if si (t − ) = 1 and θi (t) < θi− , ⎨ (26) si (t) = 1 if si (t − ) = 0 and θi (t) > θi+ , ⎩ otherwise, si (t − )

Distributed power control of flexible loads in microgrids Chapter | 4 107

where  is the sampling period for the switching sequence, and the boundaries of the temperature deadband are given as δdb,i δdb,i ; θi+ = θset,i + . (27) 2 2 The on/off states of the TCL in the temperature deadband are not changed until the internal temperature hits the boundaries of the deadband, thus the operation period of the TCL can be divided into a cooling period and a heating time. The steady-state cooling time Tc,i and the heating time Th,i for the ith thermostatic load can be calculated according to the solution of Eqs. (26), (27) [22]:

Pr,i Ri + θi+ − θa,i θa,i − θi− , Th,i = Ci Ri ln . Tc,i = Ci Ri ln Pr,i Ri + θi− − θa,i θa,i − θi+ θi− = θset,i −

To illustrate the dynamic evolutions of a TCL, we simulate the hybrid dynamic equation with the parameters C = 1.8, R = 1/0.3, θa = 38, Pr = 16, θ (0) = 23, θset = 24, and δdb = 1. Therefore, it is easy to calculate the cooling period Tc = 9.153 min and the heating period Th = 25.725 min. The internal temperature and the operation state of a TCL are given in Fig. 24. Furthermore, the aggregate power output of the aggregator with NL TCLs can be calculated by PT (t) =

NL  1 si (t)Pr,i , ηi

(28)

i=1

where ηi is the power energy’s transmission efficiency.

25 q(t)

24.5 24 23.5 23

Tc

Th

0.5 0

0

0. 5

1

1.5

2

Time (h)

Fig. 24

Dynamic evolution for the temperature and on/off state of a thermostatic load.

2.5

s(t)

1

108 PART | I Frequency and voltage control

According to the hybrid model, one can conclude that the temperature setpoint θset,i is in fact the control signal for each TCL. Meanwhile, different customers have different comfort levels under control. That is, the control input must satisfy the customer’s constraint condition, − + ≤ θset,i ≤ θset,i , θset,i − θset,i

(29)

+ θset,i

where and are the lower and upper limits of the customers’ acceptable regulation thresholds. For a TCL aggregator, Eqs. (25)–(29) denote a hybrid physical aggregate model with θset,i and PT being the input and output variables, which is a multiple-input single-output system, which is difficult for application in practice. Bashash and Fathy [17] provide a single-input single-output bilinear system to approximate the aggregate response of Eqs. (25)–(29) for a group of homogeneous TCLs,  x(t) ˙ = Ax(t) + Bx(t)u(t), (30) ˜ y(t) = Cx(t), where x(t) = (x1 (t), . . . , xQ (t))T ∈ RQ is the state bin representing the average number of off state (x1 , . . . , xN ) or on state (xN+1 , . . . , xQ ) TCLs in each temperature interval. The control input u(t) is set to be u(t) = θ˙set ∈ [α¯ on , α¯ off ] and y(t) is the approximate aggregate power of the population of TCLs. Furthermore, the temperature setpoint θset,i of each terminal TCL is generated by des θset,i = θset,i + θset (t) and

max |θset (t)| ≤ θset ,

max = where the maximal bound of the comfort interval θset

+ max {|θset,i −

1≤i≤NL

des |, |θ − − θ des |} and θ (t) is the universal broadcast control signal for θset,i set set,i set,i the load aggregator which makes the temperature setpoints of all the TCLs in this load aggregator moved by the same amplitude. The coefficient matrices A ∈ RQ×Q , B ∈ RQ×Q , and C˜ ∈ RQ are given as follows:     A B A12 B12 A(α¯ on , α¯ off ) = 11 , B = 11 A21 A22 B21 B22 ⎤ ⎡ −α¯ off ⎤ ⎡

A11

⎢ ⎢ ⎢ =⎢ ⎢ ⎣

α¯ off −α¯ off

..

.

..

.

α¯ off −α¯ off

..

.

(12) RN×N , aM+1,N+1

..

.

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

α¯ off −α¯ off (21)

α¯ on −α¯ on

A22

⎢ ⎢ =⎢ ⎢ ⎣

..

.

..

.

α¯ on −α¯ on

..

.

..

.

α¯ on −α¯ on α¯ on

⎥ ⎥ ⎥, ⎥ ⎦

and A11 ∈ = α¯ on , aP,N = α¯ off , and other elements of matrices A12 , A21 are zeros. Then matrix B has the same structure as A, which can be

Distributed power control of flexible loads in microgrids Chapter | 4 109

obtained by setting α¯ on = α¯ off = −1, that is, B = A(−1, −1). C˜ = [C˜ 1 , C˜ 2 ]  [0, . . . , 0, P/η, . . . , P/η]. The parameters α¯ on and α¯ off are the average local load       N

Q−N

transport rates, which can be calculated by the following approximate values: 1 1 des des − RPr ), α¯ off = ), (θa − θset (θa − θset α¯ on = CRT CRT where T is the discretization step of the temperature deadband [θ − , θ + ], and N, Q are given in the variables xN and xQ of finite-difference discretization of the temperature interval.

2.1.2 Aggregate evaluation of TCL aggregator The dispatch center needs to analyze the dynamical response of TCLs under different ambient temperatures and evaluate the available maximal and minimal regulation capacities and the increasing/decreasing rates (ramping rates) for load agents so as to execute the optimal economic dispatch. According to the aggregate power equation (28), it can be derived that the aggregate power of a homogeneous TCL aggregator is bounded by 0 ≤ PT (t) ≤ Pη NL . However, the regulation objective of the load agent must ensure users’ comfort levels, that is, the temperature setpoint θset must lie in the − + , θset ]. For a homogeneous group of TCLs, the steady-state power interval [θset consumption PTs without control can be calculated as Pr Pr θa − θset NL Tc · NL ≈ · · NL ≈ (θa − θset ). PTs = · η Tc + Th η Pr R ηR Therefore, the available stable lower and upper bounds of the TCL aggregator can be obtained approximately based on the steady-state aggregate power as follows: NL NL + − ), Pmax ), (31) (θa − θset (θa − θset Pmin Ts = Ts = ηR ηR and we set the minimal and maximal capacities of the TCL aggregator to be max max min Pmin Agg = 0.5PTs and PAgg = 1.5PTs on the basis of simulation experiences. On the other hand, the ramping rate y(t) ˙ can be calculated as ˜ + B ⊗ u(t))x(t) y(t) ˙ = C(A    A11 + B11 ⊗ u(t) A12 + B12 ⊗ u(t)  ˜ x(t) = 0 C2 A21 + B21 ⊗ u(t) A22 + B22 ⊗ u(t)   = C˜ 2 (A21 + B21 ⊗ u(t)) C˜ 2 (A22 + B22 ⊗ u(t)) x(t) Pr Pr = [α¯ off − u(t)]xN + [α¯ on − u(t)]xN+1 . η η By the constraint α¯ on ≤ u(t) ≤ α¯ off , one can further derive that Pr Pr ˙ ≤ (α¯ off − α¯ on )xN . (α¯ on − α¯ off )xN+1 ≤ y(t) η η

110 PART | I Frequency and voltage control

Suppose the distribution of off-state and on-state TCLs in all temperature subintervals follows a uniform distribution, that is, xN + xN+1 =

NL T , δ

and the fact that α¯ off − α¯ on =

RPr CRT

and

xN+1

δ Pr = y, T η

where T is the length of the temperature subinterval. Furthermore, one has −

Pr (NL Pr − ηy) Pr y ≤ y(t) ˙ ≤ , Cδ ηCδ

which concludes that the increasing and decreasing rates of the TCL aggregator are, respectively, Rdn Agg = −

Pr y Cδ

and

up

RAgg =

Pr (NL Pr − ηy) . ηCδ

(32)

Since there are multiple TCL aggregators under a bus load agent, each TCL aggregator has to report its evaluation information to the load agent. Based on the information received, the load agent can calculate the total capacity and ramping rate values, which are summarized in Algorithm 3. Algorithm 3 Aggregate evaluation of a load agent

2.2 Look-ahead economic dispatch to provide load following trajectories Optimal EDP is one of the key problems in power system operation, which aims at minimizing the total economic cost of the stable operation for power systems subject to the supply-demand balance [23]. Traditionally, solving EDP can provide a solution for unit commitment and power output distributions for all generators. However, with the extensive application of renewable energy generation and the development of microgrid techniques, power resources and

Distributed power control of flexible loads in microgrids Chapter | 4 111

loads are becoming diversified. TCLs, as one kind of flexible adjustable load, account for a large proportion of prosumers which can provide services of regulation or load following for active power allocation. These aggregate TCLs, which are managed by the load agents, can participate in the load following service together with traditional generating units. Load agents and generating units can obtain their regulation capacities from bidding in a regulated market or from system operator in a deregulated system. Here, we consider the joint EDP for both flexible generating units and flexible load agents. Based on the load and renewable energy power forecasting curves, the look-ahead EDP can be solved for both participant units, that is, the reference power trajectories for flexible generating units and flexible load agents can be derived, respectively. Since generating units can carry out the generating schedules exactly based on the traditional centralized generating control of units, we mainly focus on the DR load following control and implementation problem of TCL aggregators in each load agent. The objective of the energy schedule is to minimize the total regulation cost for all the generating units and load agents: min F =

T   t=1 i∈G

     Cg,i PG,i (t) + Cl,j PL,j (t) ,

(33)

j∈L

(0) where PG,i (t) = PG,i (t) − P(0) G,i (t) and PL,j (t) = PL,j (t) − PL,j (t), and Cg,i (·) and Cl,j (·) are often described by quadratic functions, such as Cg,i (Pi ) = ai P2i + bi Pi + ci and Cl,j (Pj ) = a˜ j P2j + b˜ j Pj + c˜ j , in which ai (˜ai ), bi (˜ci ), and ci (˜ci ) are predetermined constants reported by the generating units and the load agents, and T is the number of the look-ahead periods. The energy mismatch can be derived by Ptotal (t) = PD (t) − Ppv (t) −  (0) Pwi (t) − i∈G PG,i (t) based on the prediction of the load and renewable energy power, and such a mismatched power will be counterbalanced by both  P generating units and load agents together in real time, that is, G,i (t) + i∈G  j∈L PL,j (t) = Ptotal (t). By denoting t as the optimization period (such as t = 5 min), then the power balance constraint and the inequality constraints for such an optimization problem can be summarized as follows:  ⎧ PG,i (t) + Pin (t) = PD (t) − PL,j (t), ⎪ ⎪ ⎪ ⎪ j∈L ⎪ i∈G ⎪ ⎪ ⎪ max ⎪ Pmin ⎪ ⎪ G,i ≤ PG,i (t) ≤ PG,i , ⎪ ⎨  1  up (34) −Rdn G,i ≤ t PG,i (t + 1) − PG,i (t) ≤ RG,i , ⎪ ⎪ ⎪ ⎪ ⎪ max ⎪ Pmin ⎪ L,j ≤ PL,j (t) ≤ PL,j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩−Rdn ≤ 1 P (t + 1) − P (t) ≤ Rup , L,j L,j L,j L,j t

112 PART | I Frequency and voltage control

for ∀i ∈ G, j ∈ L, and t = 1, . . . , T, where Pin (t) = Ppv (t) + Pwi (t). In this optimization problem, the dispatch cost F is the total regulation cost with respect to the active power regulation variable P(t) = [P1 (t), P2 (t), . . . , P|G|+|L| (t)]T (the vector of generation scheduling for generating units and consumption scheduling for load agents), which is a quadratic function. The regulation capacities and ramping rates for generating units are reported to the dispatch center; the regulation capacities and ramping rates for load agents are estimated by Algorithm 3 and then reported to the dispatch center as well. Such a look-ahead optimization problem (33) with the constraints (34) is a convex optimization and can be solved by the interior point method easily [24]. Therefore, one can obtain the energy scheduling P(t) = P (0) (t) + P(t) (0) for generating units and load agents, where P (0) (t) = [P(0) 1 (t), P2 (t), . . . , (0) P|G|+|L| (t)]T . In the following, we are concerned with the DR load following control and implementation problem of multiple TCL aggregators in each load agent. Based on the temperature setpoint control signal, multiple TCL aggregators can regulate their aggregate power consumption behavior to follow the reference power trajectory from the dispatch center in a distributed way.

2.3 Distributed pinning control of multiple aggregated TCLs TCLs have already accounted for a significant proportion of the residential load, and the aggregate power of a large population of TCLs can vary drastically under small changes of the temperature setpoint, which instead do not influence the comfort levels of customers. It is more suitable for them to provide minuteminute ancillary services due to the temperature field characteristics of this kind of load. In this section, we consider the DR load following the control problem of multiple heterogeneous TCL aggregators managed by a flexible load agent, where each aggregator is composed of numerous homogeneous TCLs. Here, we just need to control all TCL aggregators in each load agent to perform the active power tracking. The EDP has already provided a reference power trajectory for each load agent, and we will design a distributed pinning control algorithm to coordinate the operation of all aggregators inside the load agent such that the aggregate power response could track the reference power trajectory. Suppose there are Nj TCL aggregators in the jth load agent. For symbolic simplification, we consider the scenario that the reference power value Pref (t) is shared by N TCL aggregators with the ith aggregator modeled by the following bilinear model:  x˙i (t) = Ai xi (t) + Bi xi (t)ui (t), (35) yi (t) = C˜ i xi (t), i = 1, 2, . . . , N,

Distributed power control of flexible loads in microgrids Chapter | 4 113

where ui (t) = θ˙set,i (t) ∈ [α¯ on,i , α¯ off ,i ] is the control input for the ith aggregator and yi (t) is the approximated aggregate power of the aggregator. Since the reference power values are provided every optimization moment, which is a discrete power sequence. It is transformed to be a real time trajectory for control by the spline interpolation method. NThus, the total amount of the aggregate power under the load agent is i=1 yi (t), which can be measured by the transformer terminal unit installed at the terminal of the load agent. For a given reference power trajectory Pref (t), we need to design an efficient control strategy such that the aggregate response of all TCL aggregators under this load agent can track the reference power curve, that is, N i=1 yi (t) − Pref (t) → 0. To achieve a fair participation of all aggregators, we choose the relative amplitude of the temperature setpoint as a consensus variable. By utilizing the distributed pinning control, all the control inputs of the aggregators will be synchronized to common reference temperature setpoint dynamics, which can drive the aggregate power output to the reference power trajectory. Therefore, the main objective is to develop a reference temperature setpoint updating equation and a distributed control algorithm such that the aggregate output tracking can be achieved just by communications with the neighbors. First, we consider how to generate a reference temperature setpoint signal. For the load agent, the aggregate response deviation can be measured, based on which we can design a centralized pinner, which is utilized as a virtual leader for distributed pinning control, and the group of aggregators are called followers. Pinning means that the centralized pinner is only connected to a small fraction of the TCL aggregators and the remaining aggregators communicate with their neighbors based on the spare communication network. For the TCL aggregators in the load agent, we define a reference ref temperature setpoint variable θset , which is regulated by the response variation between the total aggregate power and the reference power N i=1 yi (t) − Pref (t). The changing rate of the reference temperature setpoint is closely related to the response variation. The following first-order regulation differential equation is used to model the reference temperature setpoint’s dynamics:

N N t   ˙ref yi (t) − Pref (t) + μ2 yi (t) − Pref (t) dt, (36) θset (t) =μ1 t0

i=1 ref θset (0)

i=1

ref |θset (t)|

max where the initial value = 0 and is bounded by the θset for the optimal comfort interval of the costumers and the rate is bounded ˙ref by max α¯ on,i ≤ θset (t) ≤ min α¯ off ,i . The coefficients μ1 and μ2 are 1≤i≤N

1≤i≤N

approximate positive regulation gains.

114 PART | I Frequency and voltage control

So far, the reference temperature setpoint variable is derived, which can guide the group of the aggregators to move toward the direction of the reference power trajectory and track the reference power. This reference signal can be issued out to all load aggregators in a centralized way. However, such a control strategy becomes both time-consuming and inefficient, especially when the number of the aggregators is very large and the distribution of these load aggregators is decentralized in different control areas. We consider the distributed pinning control for all TCL aggregators in the load agent. Distributed control has many advantages, such as easy implementation, low complexity, high robustness, and good scalability, which enable the participants to be plugged-in or plugged-out flexibly. The temperature variation θset,i (t) of the ith TCL aggregator is regulated by the following equation: θ˙set,i (t) =β

N 

    ref aij θset,j (t) − θset,i (t) − βdi θset,i (t) − θset (t) ,

j=1

(37) des , a is the element of the where the initial value θset,i (0) = θset,i (0) − θset,i ij adjacency matrix A of the communication topology among the TCL aggregators, and di = 1 if the ith TCL aggregator is pinned by the load agent (centralized pinner), otherwise di = 0. Generally, the communication topology needs to have a directed spanning tree for a directed communication structure or to be connected for an undirected communication structure. The reason why we utilize a distributed pinning strategy is at least threefold: (1) Compared with the limited number of generating units, the number of TCL aggregators is large and the distribution of these aggregators is decentralized in a wide range of areas, so the traditional centralized control strategy is timeconsuming and inefficient. (2) TCL aggregators have more flexibility; as they can choose to participate the DR or not by their own enabling condition, a distributed strategy can provide a more robust solution to handle this scenario and realize the plug-in and plug-out of TCL aggregators. (3) Different from the classical distributed average consensus strategy, there is a global coordination objective to fulfill the reference active power tracking in this problem. Therefore, the distributed pinning strategy is an efficient allocation strategy for coordinating multiple TCL aggregators. According to the distributed pinning communication protocol (37) and the bilinear model (35), it is easy to derive the control signal ui (t) for the ith TCL aggregator, that is,

ui (t) =β

N  j=1

    aij θset,j (t) − θset,i (t) − βdi θset,i (t) − θset (t) .

(38)

Distributed power control of flexible loads in microgrids Chapter | 4 115

In the implementation of the distributed protocol, the interaction among TCL aggregators and the centralized pinner occurs at discrete time steps. Therefore, continuous-time differential equations (36), (37) need to be transformed to discrete-time difference equations. Here, we utilize the Euler method to derive the discrete-time closed-loop system:

N ⎧  ⎪ (k+1) (k) ⎪ ⎪θset = θset + hμ1 yi (k) − Pref (k) ⎪ ⎪ ⎪ ⎪ i=1 ⎪

N ⎪ ⎪ k ⎪   ⎪ ⎪ ⎪ +hμ2 yi (l) − Pref (l) , ⎪ ⎪ ⎨ l=0 i=1 (39) N  ⎪  (k) ⎪ (k+1) (k) (k)  ⎪ θset,i = θset,i + hβ aij θset,j − θset,i ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪  (k)  ⎪ ref ⎪ ⎪ − θset (k) , −hβdi θset,i ⎪ ⎪ ⎪  (k+1) ⎩ (k)  ui (k) = θset,i /h, − θset,i where the output power yi (k) is updated from the discrete-time aggregate model:    xi (k + 1) = xi (k) + h Ai xi (k) + Bi xi (k)ui (k) , (40) yi (k) = C˜ i xi (k), i = 1, 2, . . . , N, where h is the discretization step, that is, the sampling period for the practical operation system. Remark 3. The TCL aggregator has the flexibility to switch on/off the DR program by setting/removing the communication links with its neighbors. New participants can be involved in the DR program by just setting up a communication link with the existing participants. The important and complex computational task of the load agent is greatly reduced, and now it just needs to receive the reference signal from the dispatch center and send out the control commands to the pinned TCL aggregators. Remark 4. The plugging-in and the plugging-out of the TCL aggregator will change the structure of the communication network. In fact, the consensus convergence speed is determined by the real part of the minimal nonzero eigenvalue of Laplacian matrix of the communication network. The larger the real part of the minimal nonzero eigenvalue, the larger the convergence rate of the convergence speed. On the other hand, the required condition for the communication network is that the communication network has a directed spanning tree for a directed communication network and connected for an undirected communication network. We summarize the steps for the load following of multiple TCL aggregators as an algorithm presented as follows.

116 PART | I Frequency and voltage control

Algorithm 4 Distributed load following of TCL aggregators

2.4 Simulation and results In this section, we validate the performance of the proposed load following strategy in a regional power system with three generating units and two DR load agents. The units and load agents are placed on a modified IEEE-9 bus system with one wind power plant and one photovoltaic power plant. The physical connections for the units and load agents and communication links for the TCL aggregators are given in Fig. 25. The DR load agents are utilized to balance the power generation and power utilization, especially the large power mismatch resulted from the intermittent injection of the renewable energy power. Suppose there are 26 TCL aggregators

Fig. 25

IEEE-9 bus system with DR load agents and renewable energy power injection.

Distributed power control of flexible loads in microgrids Chapter | 4 117

with 34,664 cooling air conditioners, which are distributed under 2 DR load agents and will participate in a real-time load following service with generating units. The reference active power trajectories for the load agents can be received from the dispatch center every 5 min. In the following, we illustrate the load following service in detail. We consider a short-term load following service, such as a peak load period in the afternoon. Taking a summer day in Nanjing, China as an example, suppose the average ambient temperature is 38◦ C from 13:00 until 15:00 and all the DR air-conditioners will be controlled in real time.

2.4.1 Aggregate evaluation of the aggregator Suppose all the cooling air-conditioners in one district are homogeneous TCLs and they are aggregated as an aggregator. The aggregator can be an energy management system [25] installed in this district. As for the DR load agent LA1, there are 10 TCL aggregators with a total number of 14,667 cooling air-conditioners. The width of all temperature deadbands is δdb = 1◦ C; the thermal resistance R of each load aggregator is chosen from [4.5, 5.5] uniformly; the thermal capacitance C of each load aggregator is chosen from [8, 12] uniformly; the output cooling energy Pra of each load aggregator is chosen from [16, 20] uniformly; and the coefficient of performance η of each load aggregator is chosen from [2.6, 3] uniformly. The preferred des = [23.15, . . . , 24.45]. The initial setpoint for each aggregator is given with θset,i proportion of off TCLs in each load aggregator is prooff = [0.472, . . . , 0.516], and the initial temperature for each air-conditioning in each load aggregator is chosen uniformly in the first temperature deadband. By Algorithm 3, one max can derive the private parameters for LA1: Pmin L,1 = 14 MW, PL,1 = 67 MW, up dn RL,1 = −76 MW/min, and RL,1 = 125 MW/min. On the other hand, for the DR load agent LA2, there are 16 TCL aggregators with a total number of 19,997 cooling air-conditioners. Suppose the width of the temperature deadband δdb = 1◦ C; and R ∈ [1.5, 2.5], C ∈ [8, 12], Pra ∈ [12, 16], η ∈ [2.6, 3] uniformly. The preferred setpoint for each aggregator is des = [23.15, . . . , 24.45]. The initial proportion of off-state TCLs in given with θset,i each load aggregator is prooff = [0.172, . . . , 0.116], and the initial temperature for each air-conditioning in each load aggregator is chosen uniformly in the first temperature deadband. By Algorithm 3, one can derive the private parameters up max dn for LA2: Pmin L,2 = 19 MW, PL,2 = 92 MW, RL,2 = −82 MW/min, and RL,2 = 95 MW/min. 2.4.2 Reference power trajectories solving Based on the reported aggregate evaluation results of the load agents and the dispatch cost functions of units and load agents, the dispatch center can perform the look-ahead optimization easily. The detailed private coefficients for units and load agents are given in Table 3. Meanwhile, the day-ahead generating plans

118 PART | I Frequency and voltage control

TABLE 3 Generation and demand units’ private parameters. Generation unit parameters (MW) up

G

ai

bi

Rdn G,i

RG,i

min PG,i

max PG,i

G1

0.1151

5.2034

20

25

10

250

G2

0.0856

1.2104

25

30

10

300

G3

0.1225

1.1518

22

27

10

270

Demand unit parameters (MW) L

a˜ j

b˜ j

Rdn L,j

RL,j

min PL,j

max PL,j

L1

0.1021

6.2015

76

125

14

67

L2

0.1265

6.3472

82

95

19

92

up

(estimated flexible utilization) for generating units (load agents), and the real time new energy power injection Pwi , Ppv and the real-time load forecasting PD for a 2h+ period are given in Table 4. Based on the centralized look-ahead optimization (look-ahead period T = 4) carried out by the dispatch center, the real-time reference power trajectories are issued to all participants every 5 min. Based on the earlier parameters’ setting, the reference power trajectories for the generating units and load agents can be calculated by the optimization problem (33), (34), which are shown in Figs. 26 and 27 versus the day-ahead scheduled/estimated power trajectories. Previously, we have discussed the real-time look-ahead optimal dispatch for generating units and load agents. Since the generating schedules are optimized by considering the physical constraints, the units can fulfill the generating commands by the existing centralized feedback control strategy. Next, we will illustrate the detailed load following problems for load agents based on distributed pinning control of multiple aggregate TCLs.

2.4.3 Demand response load following control of TCLs Suppose the comfortable levels of customers are ±2◦ C around the temperature setpoint and all the approximate aggregate models have the same dimensions (Q = 100 for models in LA1 and LA2) in the simulation. By calculations, one can derive the maximal and minimal bounds of the control inputs for models in ≤ 14.2868 and −9.0951 ≤ LA1 and LA2, which are given as −22.0334 ≤ uLA1 i ≤ 14.8180. uLA2 i In the following, the distributed pinning control strategy (39) is utilized to coordinate the operation of all TCL aggregators in the load agents LA1 and LA2.

Distributed power control of flexible loads in microgrids Chapter | 4 119

TABLE 4 The day-ahead scheduled (estimated) values for generating units (load agents) and the predicted values for Pwi , Ppv , and PD . Time (K)

(0) PG,1

(0) PG,2

(0) PG,3

(0) PL,1

(0) PL,2

Ppv

Pwi

PD

13:00

200

260

219

37

52

30.2

45.3

763

13:05

208

262

223

35

51

30.1

69.8

779

13:10

214

263

227

34

49

29.8

52.4

784

13:15

212

266

225

32

47

29.2

48.2

781

13:20

210

262

236

35

51

28.8

60.8

787

13:25

216

258

230

32

47

28.3

65.2

788

13:30

210

263

231

36

52

28.6

58.4

794

13:35

214

258

225

35

51

28.1

75.5

791

13:40

210

266

232

38

49

27.9

72.3

793

13:45

216

258

228

32

53

27.6

68.4

802

13:50

217

263

229

38

55

27.5

72.8

808

13:55

214

258

231

35

49

27.7

57.3

801

14:00

210

266

223

32

49

27.4

52.1

801

14:05

212

263

221

36

53

27.2

62.4

809

14:10

214

261

227

34

49

27.3

58.2

804

14:15

210

260

225

32

50

27.1

51.9

811

14:20

206

259

231

34

51

26.9

58.2

817

14:25

216

266

223

36

55

26.8

72.2

809

14:30

214

263

231

33

53

26.3

53.5

824

14:35

212

258

225

38

49

26.7

52.1

821

14:40

214

266

223

36

51

26.6

48.9

813

14:45

208

263

222

32

53

26.2

55.3

822

14:50

216

264

229

36

55

26.5

65.2

828

14:55

210

266

231

35

49

26.4

45.8

821

15:00

214

260

222

38

53

26.3

52.9

813

15:05

216

265

224

34

49

25.9

67.7

820

15:10

212

262

230

36

51

25.7

50.5

825

15:15

217

267

227

32

49

25.5

60.3

828

15:20

213

263

222

38

53

25.6

60.4

824

120 PART | I Frequency and voltage control 320 (0)

P G1

300

P

G1

(0)

PG2

P

G2

(0)

P G3

P

G3

PG (MW)

280 260 240 220 200 0

20

40

60 Time (min)

80

100

120

Fig. 26 The optimized power trajectories PGi and the day-ahead scheduled power trajectories P(0) Gi for generating units.

P (0)

60

L1

PL1

P (0) L2

PL2

PL (MW)

50

40

30

20 0

20

40

60 Time (min)

80

100

120

Fig. 27 The optimized power trajectories PLj and the day-ahead estimated power trajectories P(0) L for load agents.

The communication matrix (aij ) and the pinning links (di ) can be easily derived from Fig. 25 and the reference power values are provided by the dispatch center every 5 min, and we utilize the method of cubic spline interpolation to derive a more detailed reference power trajectory during each optimization period. For the real-time control of the TCL aggregators, the sampling period and the control commands are 10 s for a cycle. By setting the control gains μ1 = μ2 = 1.2 and the coupling strength β = 8 and running the simulation, one can obtain the relative incremental temperature θset,i and control input ui for each TCL aggregator in the load agent LA1,

5

1

4

Δq

3

i

0.5

Control input u (LA1)

1.5

2 0

set,i

(LA1)

Distributed power control of flexible loads in microgrids Chapter | 4 121

1 −0.5

0

−1 −1.5

Fig. 28

−1

0

20

40

60 Time (min)

80

−2 120

100

The relative incremental temperature θset,i and the control input ui in LA1.

40

Power tracking (MW)

35

30

25

Aggregate power Monte−Carlo power Reference power

20

15

Fig. 29

0

20

40

60 Time (min)

80

100

120

The power tracking curves for LA1.

which are given in Fig. 28. Furthermore, the power tracking curves followed by the approximate aggregate model and the Monte-Carlo method are given in Fig. 29 as well. For the 16 TCL aggregators in the load agent LA2, by setting the control gains μ1 = μ2 = 1.8 and the coupling strength β = 6 and performing the simulation, one can obtain the relative incremental temperature θset,i and control input ui for each TCL aggregator in load agent LA2, which are given in Fig. 30. Furthermore, the power tracking curves followed by the aggregate model and the Monte-Carlo method and the reference power trajectory are given in Fig. 31.

122 PART | I Frequency and voltage control 2

4

1.5

3 Control input ui (LA2)

Δq

2

0.5 0

set,i

(LA2)

1

1

−0.5

0

−1 −1

−1.5 −2

Fig. 30

0

20

40

60 Time (min)

80

−2 120

100

The relative incremental temperature θset,i and the control input ui in LA2.

55

Power tracking (MW)

50

45

40

Aggregate power Monte−Carlo power Reference power

35

30

Fig. 31

0

20

40

60 Time (min)

80

100

120

The power tracking curves for LA2.

Based on the load forecasting and the cost quotation of generating units and load agents, EDP aims at providing the reference power trajectories for generating units and load agents in real time (e.g., every 5 min in the simulation). Furthermore, the generating units can execute the generating schedules via the existing centralized control strategy, not covered in this chapter; the load aggregators respond to the load following control in a distributed way. From the simulation results of the Monte-Carlo method, the load following strategy proposed in this chapter can guide the aggregate power of the load agent in an orderly manner.

Distributed power control of flexible loads in microgrids Chapter | 4 123

2.5 Conclusion This section presented a DR load following algorithm to coordinate the operation of generating units and TCL aggregators in smart grids. Compared with the existing load following strategies, the proposed algorithm considers the aggregate evaluation and distributed pinning operation of load aggregators. Specifically, flexible load agents gain their regulation capacities from the dispatch center through bidding with generating units, and then the regulation capacity of load agent is shared by multiple TCL aggregators in a distributed way. The regulation capacity of the generating units is accomplished by its local controller. We also provide an implementation of aggregate control for the TCL aggregator by tuning the temperature setpoint of the terminal loads. As one demonstration, we apply the proposed strategy to a modified IEEE-9 bus system with DR load agents. The simulation results demonstrate that the proposed load following strategy is effective in driving the population of TCLs to the reference power trajectory without considerably changing the comfort level of customers.

References [1] L.-Y. Lu, C.-C. Chu, Consensus-based secondary frequency and voltage droop control of virtual synchronous generators for isolated AC micro-grids, IEEE J. Emerg. Sel. Top. Circuits Syst. 5 (3) (2015) 443–455. [2] A.G. Tsikalakis, N.D. Hatziargyriou, Centralized control for optimizing microgrids operation, in: IEEE Power and Energy Society General Meeting, San Diego, CA, 2011, pp. 1–8. [3] J. Hu, J. Cao, M.Z.Q. Chen, J. Yu, J. Yao, S. Yang, T. Yong, Load following of multiple heterogeneous TCL aggregators by centralized control, IEEE Trans. Power Syst. 32 (4) (2017) 3157–3167. [4] J. Hu, J. Cao, T. Yong, Multi-level dispatch control architecture for power systems with demand-side resources, IET Gener. Transm. Distrib. 9 (16) (2015) 2799–2810. [5] J. Hu, J. Cao, J.M. Guerrero, T. Yong, J. Yu, Improving frequency stability based on distributed control of multiple load aggregators, IEEE Trans. Smart Grid 8 (4) (2017) 1553–1567. [6] A.J. Wood, B.F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, New York, NY, 2012. [7] H. Su, Z. Rong, M.Z.Q. Chen, X. Wang, G. Chen, H. Wang, Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks, IEEE Trans. Cybernet. 43 (1) (2013) 394–399. [8] J. Hu, Y. Li, T. Yong, J. Cao, J. Yu, W. Mao, Distributed cooperative regulation for multiagent systems and its applications to power systems: a survey, Sci. World J. 2014 (2014) 139028. [9] W. Ren, Y. Cao, Distributed Coordination of Multi-Agent Networks: Emergent Problems, Models, and Issues, Springer Science & Business Media, Nairobi KE, Kenya, Nairobi, 2010. [10] F. Chen, Z. Chen, L. Xiang, Z. Liu, Z. Yuan, Reaching a consensus via pinning control, Automatica 45 (5) (2009) 1215–1220. [11] Q. Song, F. Liu, J. Cao, W. Yu, Pinning-controllability analysis of complex networks: an M-matrix approach, IEEE Trans. Circuits Syst. Regul. Pap. 59 (11) (2012) 2692–2701. [12] Z.-L. Gaing, Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Trans. Power Syst. 18 (3) (2003) 1187–1195.

124 PART | I Frequency and voltage control [13] K.-H. Ng, G.B. Sheble, Direct load control—a profit-based load management using linear programming, IEEE Trans. Power Syst. 13 (2) (1998) 688–694. [14] J. Short, D. Infield, L.L. Freris, Stabilization of grid frequency through dynamic demand control, IEEE Trans. Power Syst. 22 (3) (2007) 1284–1293. [15] P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V.W.S. Wong, J. Jatskevich, Optimal real-time pricing algorithm based on utility maximization for smart grid, in: Proceedings of 1st IEEE International Conference on Smart Grid Communications, Gaithersburg, MD, October, 2010, pp. 415–420. [16] J. Zhu, M.Z.Q. Chen, B. Du, A new pricing scheme for controlling energy storage devices in future smart grid, J. Appl. Math. 2014 (2014) 340842. [17] S. Bashash, H.K. Fathy, Modeling and control of aggregate air conditioning loads for robust renewable power management, IEEE Trans. Cont. Syst. Tech. 21 (4) (2013) 1318–1327. [18] K.T. Tan, X.Y. Peng, P.L. So, Y.C. Chu, M.Z.Q. Chen, Centralized control for parallel operation of distributed generation inverters in microgrids, IEEE Trans. Smart Grid 3 (4) (2012) 1977–1987. [19] Y.V. Makarov, C. Loutan, J. Ma, P. De Mello, Operational impacts of wind generation on California power systems, IEEE Trans. Power Syst. 24 (2) (2009) 1039–1050. [20] H. Hao, B.M. Sanandaji, K. Poolla, T.L. Vincent, Aggregate flexibility of thermostatically controlled loads, IEEE Trans. Power Syst. 30 (1) (2015) 189–198. [21] R.E. Mortensen, K.P. Haggerty, A stochastic computer model for heating and cooling loads, IEEE Trans. Power Syst. 3 (3) (1988) 1213–1219. [22] S. Kundu, N. Sinitsyn, S. Backhaus, I. Hiskens, Modeling and control of thermostatically controlled loads, in: Proceedings of 17th Power Systems Computation Conference, Stockholm, Sweden, August, 2011, pp. 1–7. [23] C. Li, X. Yu, W. Yu, T. Huang, Z.-W. Liu, Distributed event-triggered scheme for economic dispatch in smart grids, IEEE Trans. Ind. Inf. 12 (5) (2016) 1775–1785. [24] N. Duvvuru, K.S. Swarup, A hybrid interior point assisted differential evolution algorithm for economic dispatch, IEEE Trans. Power Syst. 26 (2) (2011) 541–549. [25] K.T. Tan, P.L. So, Y.C. Chu, M.Z.Q. Chen, Coordinated control and energy management of distributed generation inverters in a microgrid, IEEE Trans. Power Del. 28 (2) (2013) 704–713.

Chapter 5

False data injection attacks on inverter-based microgrid in autonomous mode Heng Zhanga , Wenchao Mengb , Junjian Qic , Xiaoyu Wangd and Wei Xing Zhenge a School

of Science, Jiangsu Ocean University, Lianyungang, Jiangsu, People’s Republic of China, of Control Science and Engineering, Zhejiang University, Hangzhou, China, c Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL, United States, d Department of Electronics, Carleton University, Ottawa, ON, Canada, e School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW, Australia b Department

1 Introduction Small distributed power systems, namely microgrids, integrate distributed generators (DGs), energy storing devices, energy converters, load monitors, etc. [1, 2]. They can either be affiliated to the main power grid, or work independently. When connected with the main grid, they cannot only consume power from the main grid, but also feed their redundant energy to the main grid. In the autonomous mode, they have to balance their own supply and demand by load sharing control. The proposed structure of microgrids is beneficial to take full advantage of distributed energy, and maintain the synchronization of various forms of distributed energy resources [3–6]. Recently, secure issues are becoming more critical from both technological and economic perspectives, especially for system operators due to the recent introduction of performance-based rules [7, 8]. Because of an increasing number of cyberattacks, the power systems, specifically microgrids, are becoming more and more vulnerable. Various efforts have been devoted to investigating secure power systems [7–13]. The performances of microgrids may be seriously deteriorated in the presence of attacks. Typical cyberattacks in microgrids include false data injection (FDI) attacks and Denial-of-Service (DoS) attacks [11–14]. FDI attacks can maliciously destroy the system performance by injecting false information into the original data, while DoS attacks may damage the system operations by breaking communications between the agents. Chlela et al. [12] demonstrated the effect of these attacks on the distributed energy resources’ Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00005-0 Copyright © 2020 Elsevier Inc. All rights reserved.

125

126 PART | I Frequency and voltage control

active power, network frequency, and load active power. An implementation example of FDI attacks in smart grid can be found in [11]. FDI attacks in microgrids have attracted considerable attention in recent years [15–22]. The existing literature has mainly focused on the evaluation of FDI attack effect [15, 16], intrusion detection technologies [17–19], and defense strategies [20–22]. Zhang et al. [15] studied the effect of FDI attacks on the dynamic microgrid partitioning process. Chlela et al. [16] developed a hardware platform to examine the impact of an FDI attack on the microgrid performance indices, including the total load lost, the frequency nadir, and latency time to achieve frequency stability. Li et al. [17] provided a conjunctive policy-based majority voting approach to detect the smart FDI attack actions in microgrids. Yang et al. [18] proposed a Gaussian-mixture model-based detection method to discover FDI attacks. A major advantage of this method is that there is no need to predefine a detection threshold. Based on recognizing the varying of inferred candidate invariants, Beg et al. [19] designed an intrusion detection method to judge the presence of FDI attacks. Hao et al. [20] considered the scenario that an FDI attacker injects false data into the intelligent voltage controller in a substation, which can negatively influence the performance of the microgrid. They provided an adaptive Markov strategy to defend against FDI attacks with unpredictable and dynamic behaviors. In order to eliminate an FDI attack that injects false data into the measurements of a microgrid, Rana et al. [21] presented a recursive systematic convolutional code to append redundancy in the states of microgrid, and a semidefinite programming-based optimal control policy to defend against FDI attacks. Wang et al. [22] designed a topology switch scheme to reduce the effect of FDI attacks on the measurements of microgrids. Bhattarai [A3] demonstrated a cyber attack that exploits the design flow of power system stabilizer (PSS) and manipulates the power outputs of distributed energy resources, and developed an attack source detection approach based on dissipative energy flow. However, the intrusion detection of cyberattacks and the implementation of defense strategies may result in increased operation cost and degraded system performances. Therefore, it would be important to be able to evaluate the effect of FDI attacks and decide whether it is necessary to implement defense measures. An important problem that has not been carefully studied is the theoretical analysis of FDI attacks on the stability of microgrids. Motivated by this, we investigate the system stability of inverter-based microgrid under FDI attacks. Specifically, in this chapter, we introduce the structure of inverter-based microgrid, and then present the model of FDI attacks which have access to inject false data into the bus agents (BAs). We then adopt a utilization level to define a stable region, and theoretically investigate the stability of microgrids with respect to the utilization level. In brief, the main contributions of this chapter can be summarized as follows: 1. We construct an FDI attack model in which the attacker is able to inject false data into the BAs.

False data injection attacks on inverter-based microgrid Chapter | 5 127

2. We define a utilization level of microgrid and evaluate the variation of the utilization level in the presence of FDI attacks with given injection strategy. 3. We define a stable region for the microgrid under FDI attack, and provide sufficient conditions for the system stability. The notations of the variables used in this chapter are listed in Table 1.

TABLE 1 Notations. Symbol

Explanation

m

DG number

ref iq,m

Set point of the q-axis (quadrature) component of the current for the mth DG

pp

λm

Proportional power control gain of the mth DG

ip λm

Integral power control gain of the mth DG

pi λm

Proportional current control gain of the mth DG

λiim pω λm

Integral current control gain of the mth DG

λiω m

Integral control gain of the mth DG for the secondary frequency control

λωm

Droop control gain of the mth DG

λω,m

Frequency droop control gain of the mth DG

ref PSF,m

mth DG supplementary real-power set point assigned by the secondary frequency controller of MGCC

ref PDC,m

Corrective real-power set point generated by the power control of the mth DG

Pm

Instantaneous real power of the mth DG

Pm

Instantaneous real power of the mth DG

Qm

Instantaneous reactive power of the mth DG

Qref m

Reactive power set point of the mth DG

ω0

Nominal frequency reference

ωm

Instantaneous frequency obtained from a phase-locked loop (PLL)

vd,m

Components of the voltage set points on d-axis of the mth DG

vq,m

Components of the voltage set points on q-axis of the mth DG

id,m

Instantaneous currents on d-axis of the mth DG

iq,m

Instantaneous currents on q-axis of the mth DG

Proportional control gain of the mth DG for the secondary frequency control

128 PART | I Frequency and voltage control

2 Inverter-based microgrid structure As shown in Fig. 1, the proposed microgrid structure is composed of two layers: a physical layer and a cyber-communication layer. The physical layer is an interconnected power grid which delivers power from the grid to consumers. The cyber-communication layer consists of a sparse communication network, which is the medium for exchanging data between physical elements in the physical layer.

2.1 Physical layer There are a number of DGs and local loads in the microgrid. In this study, we consider inverter-based DGs because their operation and control are more flexible than the conventional rotational machine-based generators. In fact, the inverter is an interface between the system and the DG which can be photovoltaic panels, fuel cells, or microturbines [23]. As shown in Fig. 2, a circuit breaker is usually utilized to connect the main grid and the microgrid. Hence, the microgrid can operate in either a grid-connected mode or an autonomous/islanded mode. In the autonomous mode, the microgrid has to maintain the power balance for a safe operation.

Fig. 1

The framework of a microgrid.

False data injection attacks on inverter-based microgrid Chapter | 5 129

Fig. 2

Canadian urban benchmark distribution system [26].

Assume the microgrid has n buses. If a bus is not equipped with DG, it can be viewed as one having a DG with zero available power generation. Similarly, if a bus is not equipped with load, it can be viewed as one having load with zero demand. Therefore, the microgrid has n DGs and n loads.

2.2 Cyber-communication layer The BA, which is installed at each bus, exchanges local information with its neighboring agents, and runs a distributed load sharing algorithm to collect the global microgrid information. The computing process of the distributed algorithm only requires a sparse communication network and very limited data. An apparent advantage of the proposed solution is the flexibility and adaptability to different operating conditions [24]. The communication network of the agents can be formulated as an undirected graph G = (V, E), where V = {v1 , v2 , . . . , vn } is the set of nodes which are the agents in the microgrid, and E ⊆ V × V is the set of edges [25]. The edge eij = (i, j) ∈ E means that there is a communication link between nodes i and j. For an undirected graph G, the statement that eij ∈ E ⇐⇒ eji ∈ E is true. The nodes i and j are called adjacent if eij ∈ E. Let A = (aij )n×n be the adjacency matrix, in which aij = 1 if eij ∈ E and aij = 0 otherwise. Define Ni = {j ∈ V | eij ∈ E} as the neighbor set of node i. The Laplacian operation of a graph G is defined as the positive  semidefinite matrix L = D − A, where D = diag{d1 , d2 , . . . , dn } with di = nj=1 aij . It can be easily seen that L1n = 0, where 1n = [1, 1, . . . , 1]T . A representative inverter-based microgrid is the Canadian urban benchmark distribution system (see Fig. 2) [26], which will be used as an example in this study.

130 PART | I Frequency and voltage control

3 System dynamic model 3.1 Small signal model For a microgrid in an autonomous mode, the small-signal model consists of three parts: the DG block, the network block, and the interface block [26]. The inverter-based DG block includes a local primary control loop and a secondary frequency control loop [26]. The local primary control loop works with a power controller and an inner current loop (see Fig. 3). It can manage the output power in terms of the preset power points. The controllers in the local primary loop follow the proportional-integral control law. The DG controller is given by   ip λm ref pp (1) (Qref iq,m = λm + m − Qm ), s   ip λ m pp ref iref (2) (Pref d,m = λm + SF,m + PDC,m − Pm ), s   λiim pi vd,m = λm + (3) (iref d,m − id,m ), s   λiim + (4) (iref vq,m = λpi m q,m − iq,m ), s where Pref DC,m refers to the ω−P characteristic of the frequency droop control and ref can be calculated by Pref DC,m = λω,m (ω0 − ωm ), and PSF,m is the supplementary

Fig. 3

Block diagram of local primary DG control loops.

False data injection attacks on inverter-based microgrid Chapter | 5 131

power set point of the mth DG assigned by the secondary frequency controller  pω

λiω

m and can be obtained by Pref SF,m = λm + s (ω0 − ωm ). The network block, the second part of the microgrid model, can be presented in a common reference frame x−y as follows:



Δix G −B ΔVx = , Δiy ΔVy B G

where ix = [ix1 , ix2 , . . . , ixn ]T , iy = [iy1 , iy2 , . . . , iyn ]T with ixm , iym , m = 1, · · · , n being the terminal current of the mth DG in the common x-axis and y-axis, respectively, Vx = [Vx1 , Vx2 , . . . , Vxn ]T , Vy = [Vy1 , Vy2 , . . . , Vyn ]T with Vxm , Vym , m = 1, . . . , n being the terminal voltage of the mth DG in the common x-axis and y-axis, respectively, and the matrices G and B are obtained from the network admittance matrix [26]. The third part, the interface block, can be modeled as follows: ΔVd = C0 ΔVx − Vx0 S0 Δδ + S0 ΔVy + Vy0 C0 Δδ, ΔVq = S0 ΔVx − Vx0 C0 Δδ + C0 ΔVy + Vy0 S0 Δδ, Δix = C0 Δid − id0 S0 Δδ − S0 Δiq − iq0 C0 Δδ, Δiy = S0 Δid + id0 C0 Δδ + C0 Δiq − iq0 S0 Δδ, where δi is the individual inverter terminal voltage phase angle in the x−y reference frame, and the matrices C0 = diag{cos(δi0 )} and S0 = diag{sin(δi0 )}. Then, according to Liu et al. [26], the whole system model can be expressed as follows: EΔx˙ = AΔx + Fr0 ,

(5)

where x = [δ, ω, id , iq , idref , iqref , ud , uq , P, Q, Pref , Vd , Vq , ix , iy , Vx , Vy ]T , r0 = [ω0 ]T , and the system matrix E is singular. Due to the limitations of space, the expressions of matrices E and A are omitted here, but can be found in Appendix A of [26].

3.2 Active power reference The active power setting depends on the total power demand  and power generation. The total active power demand is representable as Pd = nm=1 PmL +PLoss , where PmL is the active power demand of load at bus m, and PLoss is the total active power loss, which is only a small proportion of the total active power DG. Then demand. Denote by Pmax mG the maximum power generation of the mth n max we can present the total available power generation as Pmax m=1 PmG . Let G =  Pd ,1 (6) U = min Pmax G be a common utilization level for all DGs [24, 27, 28]. We assume that the load is less than the maximum available power generation. It can be seen that the

132 PART | I Frequency and voltage control

supply and demand are balanced if the active power generation reference of the max ref mth DG, that is, Pref mG , satisfies PmG = UPmG . In fact, the balance of supply and demand can be achieved when the load demand Pd is less than the maximum available power generation Pmax G . We have ≤ 1. Furthermore, we can see that U = Pd /Pmax G n

Pref mG =

m=1

n

UPmax mG =

m=1

n Pd max PmG = Pd . Pmax G m=1

However, if the load demand is more than the maximum available power generation (i.e., Pd > Pmax G ), then DGs should operate in the maximum power tracking mode, and the power storage needs to compensate for the power shortage.

4 System performance under FDI attacks 4.1 Distributed load sharing control under FDI attacks The global microgrid information includes the average power demands and available power generations of all BAs. It is the basis of designing the active power references of DGs. However, each BA only has its local information and cannot directly obtain the global information. Each agent can only exchange information with its neighbors. Hence, a distributed information processing law must be properly designed for the agents in order to obtain the global information. In the cyber layer, the computing process of information discovery at agent m can be represented by a linear time-invariant model





PmL (k) umL (k) PmL (k + 1) = + , PmG (k + 1) PmG (k) umG (k) where umL (k) and umG (k) are the control inputs of the load and the generation, respectively. The objective of information discovery is to find a distributed control law such that all the states converge to the average value of initial states, that is, lim PmL (k) = P¯ L ,

k→∞

lim PmG (k) = P¯ G ,

k→∞

m = 1, 2, . . . , n,

(7)

1 n 1 n ¯ PmG (0). where P¯ L = m=1 PmL (0), PG = n n m=1 The smart grid often suffers from FDI attacks [29, 30]. To analyze the impact of FDI attacks, we assume that the attacker has full knowledge of the power system [11]. The FDI attack on the information discovery processes can be modeled as follows:





a PmL (k) amL (k) PmL (k) = + , (8) PamG (k) PmG (k) amG (k)

False data injection attacks on inverter-based microgrid Chapter | 5 133

where amL (k) and amG (k) are the FDI data that are injected into the state of agent m at time k, and PamL (k) and PamG (k) denote the state of agent m at time k when the FDI attack is present. We focus on the discrete average consensus algorithm [31] under FDI attack. It can be seen that

wmjL [PajL (k) − PamL (k)] PamL (k + 1) = PamL (k) + j∈Nm (k)

= PmL (k) + amL (k) +

wmjL [PjL (k)

j∈Nm (k)

+ ajL (k) − PmL (k) − amL (k)] =

n

wmjL PjL (k) +

j=1

n

wmjL ajL (k),

j=1

where Nm (k) is the set of agent m’s neighbors at time k, wmjL is a positive weight with respect to load for j ∈ Nm (k) which represents importance degree of agent j’s information from the viewpoint of agent m, and wmmL (k) = 1 −  j∈Nm (k) wmjL . Its equivalent matrix form is given by PaL (k + 1) = WL (k)[PL (k) + AL (k)].

(9)

This means that the information discovery for the load under an FDI attack at time k + 1 is the linear combination of the information discovery without attack and the attack vector at time k. Similarly, we have PamG (k + 1) =

n

wmjG PjG (k) +

j=1

n

wmjG ajG (k),

j=1

and the equivalent matrix form PaG (k + 1) = WG (k)[PG (k) + AG (k)]

(10)

for the power generation. Notice that WL (k) and WG (k) are predefined Perron matrices which depend on the structure of graph G, that is, WL (k) = E − L (k)L and

WG (k) = E − G (k)L,

where L (k), G (k) are given  parameters that satisfy L (k) ∈ (0, 1/ρ), G (k) ∈ (0, 1/ρ) with ρ = max{ j =i aij } (more details can be found in [32, Section II.C]). Before investigating the impact of an FDI attack on the microgrid, two basic assumptions are presented. Assumption 1 (Nondegeneracy [31]). There exists w > 0 such that wmm (k) ≥ w for all m and w ≤ wmj (k) ≤ 1, or w = 0, for all m = j at any time k.

134 PART | I Frequency and voltage control

Assumption 2 (Balanced communication [33]). For any time k, 1T W(k) = 1T , and W(k)1 = 1. Notice that Assumption 1 can guarantee that each agent updates the states with its neighbors’ information, and Assumption 2 makes sure that all agents converge to the average initial states [34]. ∞ Lemma ∞1. If Assumptions 1 and 2 hold for WL and WG , k=1 |amL (k)| ≤ B¯ L , and k=1 |amG (k)| ≤ B¯ G , where B¯ L and B¯ G are constant FDI bounds on the load and generation of arbitrary agent m, respectively, then  n  1    a PmL (k) − P¯ L  ≤ nB¯ L , (11) lim   k→∞  n m=1  n  1    a lim  PmG (k) − P¯ G  ≤ nB¯ G . (12)  k→∞  n m=1

Proof. It is the direct result from Theorems 2 and 3 in [35]. Lemma 1 shows the property of difference between the convergence value of load under FDI attack and that without attack. The difference bound for load (generation) depends on the agents number, and the bound of the false data injected into the BA. When an FDI attack is present, the utilization level is  limk→∞ n1 nm=1 PamL (k) P¯ aL a . (13) U = a =  P¯ G limk→∞ n1 nm=1 PamG (k) ∞ Theorem ∞ 1. If Assumptions 1 and 2 hold for WL and WG , k=1 |amL (k)| ≤ B¯ L , and k=1 |amG (k)| ≤ B¯ G , then P¯ L − nB¯ L P¯ L + nB¯ L ≤ Ua ≤ . P¯ G + nB¯ G P¯ G − nB¯ G

(14)

Proof. According to Lemma 1, we have |P¯ L − P¯ aL | ≤ nB¯ L ,

|P¯ G − P¯ aG | ≤ nB¯ G ,

which is equivalent to P¯ L − nB¯ L ≤ P¯ aL ≤ P¯ L + nB¯ L , P¯ G − nB¯ G ≤ P¯ aG ≤ P¯ G + nB¯ G .

(15) (16)

Then Eq. (14) can be obtained from Eqs. (15) and (16).

4.2 Impacts of FDI attacks The characteristic equation of system (5) is det(λE − A) = 0, where λ is the eigenvalue to indicate the stability of system (5), that is, the system is stable if the real part of λ is less than 0, and it is unstable otherwise.

False data injection attacks on inverter-based microgrid Chapter | 5 135

The characteristic equation is often used to investigate the performance of microgrid systems [26]. In this chapter, we study the impact of FDI attacks on the microgrid performance with respect to the utilization level. Definition 1. A critical utilization interval denoted by U = (U, U) is called a stable region if system (5) is stable for U ∈ U , and it is unstable for U ∈ / U. Now it is ready to show the stability of a microgrid under FDI attack. ∞ Theorem ∞ 2. If Assumptions 1 and 2 hold for WL and WG , k=1 |amL (k)| ≤ B¯ L , and k=1 |amG (k)| ≤ B¯ G , then 1. system (5) is stable, if



P¯ L − nB¯ L P¯ L + nB¯ L , P¯ G + nB¯ G P¯ G − nB¯ G

 ⊂ U;

(17)

2. system (5) is unstable, if P¯ L − nB¯ L ≥ U, P¯ G + nB¯ G

or

P¯ L + nB¯ L ≤ U. P¯ G − nB¯ G

(18)

Proof. It can be directly obtained from Theorem 1 and Definition 1. Theorem 2 provides an important theoretical result for both the attacker and the defender. From the viewpoint of an FDI attacker, if they know the initial load information and generation information, they can design proper injection data to achieve their objective. If they want to make the system unstable, they can adopt the second statement of Theorem 2 to design an attack strategy. If they only aim at changing the average consensus values of load and power generation, the FDI attack strategy should satisfy the first statement of Theorem 2. From the viewpoint of a defender, if they have learned the attack quantitative characteristics satisfying boundedness assumption in this theorem, then they can design a new control policy to relieve the impact of the FDI attack.

5 Simulation examples In order to show the performance of distributed load sharing under FDI attacks, we provide an illustrative example based on the Canadian urban distribution system (Fig. 2) and implement it in MATLAB/SimPowerSystems. The main parameters of this microgrid are given in Table 2. In our simulation, the microgrid is disconnected with the main grid from time t = 0.2 s.

5.1 Stable region An illustrative example of a stable region is shown in Fig. 4. In this example, the utilization levels of all agents in Fig. 2 are varying in the interval [0.1, 0.7]. It can be observed that the stable region is U = (0, 0.38). In other words, when U ∈ U , the maximal real value of eigenvalues is less than 0, so the system is

136 PART | I Frequency and voltage control

TABLE 2 System parameters. Parameters

Value

Parameters

Value

Sbase

10 (MVA) √ √ 120 2/ 3 (kV) √ √ 12.5 2/ 3 (kV) √ √ 208 2/ 3 (kV)

Rs

1.73 × 10−6 (p.u.)

Xs

3.47×10−5 (p.u.)

Rf

0.0029 (p.u.)

Xf

0.0041 (p.u.)

Vbase,1 Vbase,2 Vbase,3

Largest real part of the eigenvalues

10 7.5 5 2.5 0

U=0.38

–2.5 –5 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

Utilization level Fig. 4

The variation of maximal real value of eigenvalues with respect to utilization level.

stable. When U ∈ / U, the system will become unstable. In general, the stable region is determined by the system parameters and it is still challenging to find the analytical expression of the stable region. At the moment, we can only derive the stable region for the given system by numerical computation.

5.2 System performance under attack strategy 1 In this section, we study the system performances under an FDI attack with Strategy 1: amL (k) = 0.05e−k−1 | sin[2π ξm (k)]|, amG (k) = 0.1e−k−1 | cos[2π ηm (k)]|, where ξm (k), ηm (k) are with independent identical uniform distribution U (0, 1).

False data injection attacks on inverter-based microgrid Chapter | 5 137

It is clear that |amL (k)| ≤ 0.05e−k−1 ,

|amG (k)| ≤ 0.1e−k−1 ,

k = 1, 2, . . . .

Thus, we can verify that ∞

|amL (k)| ≤ B¯ L =

k=1 ∞

|amG (k)| ≤ B¯ G =

k=1

∞

0.05e−k−1 =

k=1 ∞

0.1e−k−1 =

k=1

0.05e−2 , 1 − e−1 0.1e−2 . 1 − e−1

It means that the conditions in Theorems 1 and 2 hold for this strategy. We now investigate the microgrid system performances under FDI attack with strategy 1. According to the simulation results, we have P¯ L = 0.1418, P¯ G = 0.9137, and then it can be easily verified that   P¯ L − nB¯ L P¯ L + nB¯ L , = (0.0991, 0.2230) ⊂ U. P¯ G + nB¯ G P¯ G − nB¯ G Thus, the condition (17) holds when attack strategy 1 is implemented. Figs. 5–8 present the variations of voltage magnitudes, frequencies, loads, and generations under attack strategy 1, respectively. It can be seen that the indices can still reach steady states in a short time even when the agents are under FDI attacks. Our simulation results in Figs. 5–8 confirm the first statement of Theorem 2. In contrast to the microgrid performances in the absence of attack (see Figs. 9–12), although the performance indices, that is, voltage magnitudes, frequencies, loads, and powers, are still convergent under attack strategy 1, the convergence values are different from those in the absence of attack.

1.4 V1

1

V (p.u.)

V (p.u.)

1.5

0.5

V2

1.2 1 0.8

5

10

15

20

5

Time (s) V3

1.2

V (p.u.)

V (p.u.)

15

20

1.5

1.4

1 0.8

V4

1

0.5 5

10

Time (s) Fig. 5

10

Time (s)

15

20

5

10

Time (s)

Variations of voltage magnitudes under attack strategy 1.

15

20

138 PART | I Frequency and voltage control 400

(rad/s)

(rad/s)

400 1

390

0

380 370

2

390

0

380 370

5

10

15

20

5

Time (s) 3

390

0

380 370

4

390

0

380

10

15

20

5

Time (s)

15

20

Variations of frequency under attack strategy 1. 0.21 Load 1

0.2

0.15

Load (p.u.)

Load (p.u.)

10

Time (s)

0.25

Load 2

0.2 0.19 0.18

20

40

60

80

100

20

Iterations

40

60

80

100

Iterations 0.3 Load 3

0.2

0.15

Load (p.u.)

0.25

Load (p.u.)

20

370 5

Load 4

0.2 0.1 0

20

40

60

80

100

20

Iterations Fig. 7

15

400

(rad/s)

(rad/s)

400

Fig. 6

10

Time (s)

40

60

80

100

Iterations

Average load information discovery under attack strategy 1.

5.3 System performance under attack strategy 2 In this section, we study the system performances under an FDI attack with Strategy 2: amL (k) = 4.5e−k−1 | sin[2π ξm (k)]|, amG (k) = 0.1e−k−1 | cos[2π ηm (k)]|.

False data injection attacks on inverter-based microgrid Chapter | 5 139 0.95 Power 1

Power (p.u.)

Power (p.u.)

0.915 0.91 0.905

Power 2

0.9 0.85

0.9

0.8 0

50

100

0

50

Iterations 0.96 Power 3

Power (p.u.)

Power (p.u.)

0.93 0.92 0.91

Power 4

0.94 0.92

0.9

0.9 0

50

100

0

50

Iterations Fig. 8

100

Iterations

Average power generation information discovery under attack strategy 1.

1.2 V1

V2

V (p.u.)

V (p.u.)

1.2

1

0.8

1

0.8 5

10

15

20

5

Time (s)

15

20

1.2 V3

V4

V (p.u.)

V (p.u.)

10

Time (s)

1.2

1

0.8

1

0.8 5

10

15

20

5

Time (s) Fig. 9

100

Iterations

10

15

Time (s)

Variations of voltage magnitudes in the absence of attack.

Similar to attack strategy 1, we have ∞

|amL (k)| ≤ B¯ L =

k=1 ∞

k=1

|amG (k)| ≤ B¯ G =

∞

k=1 ∞

k=1

4.5e−k−1 =

4.5e−2 , 1 − e−1

0.1e−k−1 =

0.1e−2 . 1 − e−1

20

140 PART | I Frequency and voltage control 390

(rad/s)

(rad/s)

390 1 0

380

370

2 0

380

370 5

10

15

20

5

Time (s) 3 0

380

370

4

10

15

20

5

10

15

20

Time (s)

Variations of frequency in the absence of attack.

0.16 Load 1

0.14

Load (p.u.)

0.16

Load (p.u.)

0

380

Time (s)

0.12

Load 2

0.14

0.12 20

40

60

80

100

20

Iterations

40

60

80

100

Iterations

0.15

0.15 Load 3

0.14 0.13

Load (p.u.)

Load (p.u.)

20

370 5

0.12

Load 4

0.1

0.05 20

40

60

80

100

20

Iterations Fig. 11

15

390

(rad/s)

(rad/s)

390

Fig. 10

10

Time (s)

40

60

80

100

Iterations

Average load information discovery in the absence of attack.

P¯ L − nB¯ L = 1 > U = 0.38. Thus P¯ G + nB¯ G the condition (18) holds. Figs. 13–16 show the variations of voltage magnitudes, frequencies, loads, and generations under attack strategy 2, respectively. According to Figs. 15 and 16, it can be observed that the loads and generations can still reach steady states when all the agents are under FDI attacks with strategy 2. However, the performances of voltage magnitudes and frequencies For this attack strategy, we can derive

False data injection attacks on inverter-based microgrid Chapter | 5 141

Power 1

Power (p.u.)

Power (p.u.)

0.92

0.9

0.88

Power 2

0.9

0.85

0.8 0

50

100

0

50

Iterations 1

0.92 Power 3

Power (p.u.)

Power (p.u.)

100

Iterations

0.9

Power 4

0.95

0.9

0.88 0

50

100

0

50

Iterations Fig. 12

Average power generation information discovery in the absence of attack.

1.5

V (p.u.)

V (p.u.)

1.5

1

1 0.5

V2

V1

0.5

0 5

10

15

20

5

Time (s)

15

20

1.5

V (p.u.)

V (p.u.)

10

Time (s)

1.5

1

1 V4

V3

0.5

0.5 5

10

Time (s) Fig. 13

100

Iterations

15

20

5

10

15

20

Time (s)

Variations of voltage magnitudes under attack strategy 2.

are prominently influenced. Figs. 13 and 14 indicate that the voltage magnitudes and frequencies still drastically run up and down. Thus, the microgrid cannot achieve the steady states when attack strategy 2 is launched. In practice, p.u. voltage is around one, and will usually not get to a state with a p.u. voltage greater than 1.2. Since the frequency is usually maintained within a tight bound, the DG may have already been tripped after getting out of bound, either instantaneously or after a certain time delay depending on the actual

142 PART | I Frequency and voltage control 385 1

0

380 375

(rad/s)

(rad/s)

385

370

2

375 370

5

10

15

20

5

Time (s)

375

(rad/s)

(rad/s)

0

380

4

0

380 375

10

15

20

5

Time (s)

10

15

20

Time (s)

Variations of frequency under attack strategy 2.

1 Load 1

0.5

Load (p.u.)

1

0

Load 2

0.5

0 20

40

60

80

100

20

Iterations

40

60

80

100

Iterations 1 Load 3

0.5

0

Load (p.u.)

1

Load (p.u.)

20

370 5

Load 4

0.5

0 20

40

60

80

100

20

Iterations Fig. 15

15

385 3

370

Load (p.u.)

10

Time (s)

385

Fig. 14

0

380

40

60

80

100

Iterations

Average load information discovery under attack strategy 2.

frequency [36]. Thus, we can set a limit for voltage and frequency to stop the simulation when hitting that limit.

5.4 Discussion An FDI attacker may inject arbitrary false data to any node. However, the sufficiently large injected data may result in the false information seriously

False data injection attacks on inverter-based microgrid Chapter | 5 143 0.95 Power 1

0.91 0.905

Power (p.u.)

Power (p.u.)

0.915

0.9

Power 2

0.9 0.85 0.8

0

50

100

0

Iterations

100

0.96 Power 3

0.92 0.91 0.9

Power (p.u.)

Power (p.u.)

0.93

Power 4

0.94 0.92 0.9

0

50

Iterations Fig. 16

50

Iterations

100

0

50

100

Iterations

Average power generation information discovery under attack strategy 2.

deviating from the real value, and the attacked nodes would be suspected by their neighbors [37, 38]. Thus, in the case study, we assume that the injected data is decayed exponentially. Although the attack strength is very weak, the voltage magnitudes and frequency under attack strategy 2 are still becoming unstable. This simulation demonstrates that the distributed microgrid is very vulnerable to FDI attacks. Intuitively, a sustained attack may destroy the microgrid system even more significantly. Our simulations have evaluated the effect of FDI attacks on distributed load sharing. It is still challenging to design a proper active defense strategy to defeat FDI attacks. As is well known, a centralized operation system can be equipped with an intrusion detector at the fusion center side to detect FDI attacks. Unfortunately, every node in our considered distributed microgrids only knows its neighboring information and does not understand the global information. Therefore, the traditional intrusion detection methods cannot be applied to distributed microgrids. Inspired by the trust-aware defending method for distributed operation systems [39], we will design a trust-based distributed load sharing protocol against FDI attacks in the future.

6 Conclusion In this chapter, we have investigated the effect of FDI attacks on distributed load sharing of microgrids operating in autonomous mode. Each bus is supposed to be equipped with an agent and the power balance is achieved by a welldeveloped consensus protocol of multiagent systems. Under FDI attacks, the information among agents can be corrupted by attackers. Meanwhile, the impact

144 PART | I Frequency and voltage control

of FDI attacks on the utilization level has been investigated under different injection strategies. We have also defined the stable region and sufficient conditions for microgrids operating in stable regions. The theoretical results have been validated in MATLAB/SimPowerSystems on the Canadian urban distribution system. Future works include investigation of the impact of FDI attacks in more general form on microgrid performance, and designing proper active defense strategies against FDI attacks in microgrids.

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Chapter 6

Distributed finite-time control of aggregated energy storage systems for frequency regulation in multiarea microgrids Yu Wanga , Yan Xua and Jing Qiub a Nanyang

Technological University, Singapore, b School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW, Australia

1 Introduction 1.1 Background Nowadays, the dependence on traditional fossil fuels such as coal and oil continues to be a major concern in terms of climate change and air pollution all around the world. These environmental problems and the diminishing supply of fossil fuels accelerate the penetration of renewable energy sources (RESs) such as photovoltaic units (PVs) and wind turbines (WTs) [1, 2]. For example, the PV penetration level in Singapore is expected to increase to about 10% in less than a decade from now [3], and around 30 nations all over the world already have renewable energy contributing more than 20% of the total energy supply [4]. The Singapore power system is connected with the Malaysia power system by only two 200 MW power transmission lines. The power system with weak connection to another area can actually be viewed as an isolated microgrid. Although RESs are environmentally friendly, their high variability and uncertainty will bring new challenges for the control and operation of weak microgrid systems. In the meantime, the replacement of synchronous generators will also significantly reduce the system inertia as well as the spinning reserves [5, 6]. When a disturbance occurs, a system with less inertia and frequency regulation reserves tend to experience large frequency deviations, which will damage stable and secure operation. In this circumstance, it is important to find emerging, timely, and reliable frequency regulation resources in high RES-penetrated microgrid systems. Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00006-2 Copyright © 2020 Elsevier Inc. All rights reserved.

149

150 PART | II Energy management

Energy storage system (ESS) refers to the device of converting electrical energy from power systems into a form that can be stored for converting back to electrical energy when needed [7, 8]. As the development of renewable energy technologies, ESSs have also gained much attention and have been widely applied to utility grid and transportation applications [9]. For utility grid applications, energy storage plays a major role in the integration of renewable energy into power systems and provides ancillary services such as peak load shaving, system reserves, and frequency and voltage regulation [10–12]. According to the report of market research company HIS, grid connected energy storage will increase rapidly and reach 6 GW in 2017, and will be more than 40 GW in 2020 [13]. The Energy Market Authority of Singapore has raised the target to install 4.4 MWh island-wide ESSs by the year 2020 [14]. On the other hand, hybrid electric vehicles (EVs) and battery electric vehicles have been developed for decades and are already in service in order to reduce CO2 emissions and fossil fuel consumption. V2G capacities of plug-in EVs can offer a possible supplement for RESs including solar and wind energy, contributing integration of intermittent renewable energy generation in power systems. V2G technologies can also provide additional opportunities for grid operators, such as reactive power compensation, active power control, load leveling by valley filling/peak load shaving, and harmonic current filtering [15–17]. From the perspective of the size and location of ESSs, the large-scale centralized ESSs (CESSs) which are built by grid operators can have the capacity up to tens of megawatt hour, while the small-scale distributed ESSs (DESSs) owned by customers usually have a capacity below hundreds of kilowatt hour. As the number of DESSs such as community ESSs, plugin electric vehicles (PEV), and home ESSs keep growing, they will have large portions in the future frequency ancillary services market. However, the DESSs located at the demand side are usually larger in number but smaller in capacity. Therefore, their contributions to the power system are hard to account for individually. The concept of an aggregator offers the feasibility for a large number of DESSs to participate in ancillary services (such as frequency regulation and restoration) to gain additional income [18, 19]. In an aggregator, the geographically dispersed units can cooperate to achieve a common objective through communication networks. In particular, the energy storage aggregator (ESA) will have comparable power and capacity rating to a large CESS and it can be viewed as one entity for the system operator. In this chapter, the distributed control algorithms to enable the aggregation of ESSs will be studied, and the dynamic response including the external and internal behavior of the formed ESAs will be evaluated.

1.2 Literature survey It is well known that power system frequency is an indicator of instantaneous power matching between generation and load demand, which is maintained

Energy storage systems for frequency regulation Chapter | 6 151

hierarchically by synchronous generators in power plants [20]. The primary frequency control refers to the local control actions of turbine-generator systems, which aims to mitigate the system frequency deviations [21, 22]. In the secondary frequency control or automatic generation control (AGC), the power generation from plants is adjusted by the dispatching center to restore the system frequency back to nominal value (50 or 60 Hz) [23, 24]. These two control hierarchies will cooperate together in a real-time frequency regulation market. In addition, the power dispatching methods such as unit commitment and economic dispatch will function to adjust the amount of base generation to match the load demand in longer timescales [25, 26]. As for ESSs, how to incorporate them into existing frequency control architecture and effectively utilize the timely but limited capacity are important problems to investigate. In the context of primary frequency control, researchers have investigated the utilization of both CESSs and DESSs for local/decentralized frequency support [27, 28]. From the perspective of secondary frequency control by small-scale DESSs, the control architecture can be adopted in either a centralized or a distributed way. The centralized secondary control structures of DESSs have been reported by several existing works. In [29], a fuzzy-logicbased control for wind farms and ESSs within the load frequency control (LFC) scheme has been proposed. In [30], a master-slave-based centralized control of distributed ESS has been proposed for mitigating voltage and frequency deviations. Similarly, a centralized coordinated control for EVs and ESSs participating in LFC has been presented in [31]. The major concern of the centralized control structure is the requirement of a central communication infrastructure, and it is inherently vulnerable to any communication failure. The bottleneck limits the support for plug-and-play (PnP) functions, which is highly desired for widespread distributed devices. As an alternative approach, distributed control has raised considerable attention in power and energy research areas. Unlike centralized control, this control scheme does not need full communication access as each controllable device only communicates with its neighbors, which is more suitable for widespread distributed devices in power systems. The core idea of distributed control based on consensus algorithms is intended to achieve fair utilization of available devices by exchanging information through sparse communication networks. A major application of distributed control approaches is to coordinate PV units and ESSs for voltage regulation in power distribution networks [32–34]. In [32], based on leader-follower consensus algorithms, the authors propose a coordinated control scheme for DESSs to address voltage rise/drop issues in low-voltage networks. In [33], a two-stage distributed control architecture of distributed inverters is proposed for voltage/var control. In [34], a decentralizeddistributed hybrid control scheme for PV inverters is proposed for both network voltage fluctuation and violation issues. In another aspect, the distributed control methods for distributed devices such as responsible loads, ESSs, WTs to achieve aggregation are reported by

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recent research [35–38]. In [35], an average consensus algorithm is proposed for coordination of flywheel energy storage matrix systems. The leader-follower consensus algorithm is applied to distributed power dispatch for a group of WTs in [36]. This control algorithm is also applied to responsive load aggregators for LFC in [37]. The research work in [37] is further extended in [38], where a compound control strategy including model predictive control and consensusbased distributed control is designed to control ESSs in an LFC scheme. However, the state-of-charge (SoC), as the key performance indicator of ESSs is not considered in [38]. In addition, the convergence of the previous methods in finite time is not guaranteed [35–38]. As a result, the convergence time may be too long, and the frequency performance cannot be satisfactory. By contrast, the finite-time consensus can ensure the convergence of participants in finite time, which can effectively enhance the overall system dynamic performance [39]. The finite-time consensus algorithms are studied by plentiful research by multiagent system control theory [40, 41], and they are broadly applied for unmanned air vehicles [42], mobile robotics [43], as well as secondary control of microgrids [44, 45]. Yet so far, research has rarely studied the finitetime consensus approach for distributed aggregation control of ESSs while considering both the power outputs and SoCs.

1.3 Contributions Compared with previous research, the contribution of this chapter includes two parts: (1) A novel frequency control scheme is proposed to integrate the ESA into the multiarea microgrids. To exploit fully the capability of ESA for frequency regulation, a system disturbance observer is designed to supplement the original AGC signal. Then the high-frequency system disturbance in one control area can be extracted and compensated by the ESA. Compared with the conventional proportional-integral (PI)-based secondary frequency control methods, the proposed control scheme takes full advantage of the fast response speed from ESAs, thereby improving the system frequency recovery speed. (2) A leader-follower finite-time consensus algorithm is proposed to aggregate ESSs into the ESA. This algorithm ensures power tracking and SoC balancing of ESSs in finite time, allowing for desired dynamic performance to be obtained. Within the proposed ESA, first, a virtual leader will update the reference power and energy states based on the frequency control signal. Then the reference states are received by the pinning ESSs and transmitted to all ESSs through the predefined communication network. Both power and energy states of each ESS will reach consensus in steady state.

Energy storage systems for frequency regulation Chapter | 6 153

2 Proposed frequency control scheme 2.1 System overview An overview of the proposed frequency control scheme is shown in Fig. 1. As highlighted in this figure, there are two main contributions of this chapter: (i) a novel frequency control scheme that incorporates a system disturbance observer to supplement the secondary frequency control, and (ii) a leader-follower finitetime consensus algorithm to control ESSs within an ESA. With the proposed frequency control scheme, the frequency control signal of the ESA is comprised of two parts: (i) the AGC signal from the secondary frequency control with a predefined participation factor αESA . It should be noted that the AGC signal is shared by both generators and ESAs in each control area; and (ii) the system disturbance signal estimated by the system disturbance observer. The objective of this design is to accelerate the frequency recovery speed as compared to the conventional PI-based secondary frequency control methods. The frequency control signal of the ESA is further shared among each ESS through the proposed finite-time consensus algorithm. The proposed disturbance observer with the band-pass filter is introduced in Section 3. The detailed design of the ESA based on the finite-time consensus algorithm is presented in Section 4.

2.2 Multiarea microgrids Consider an interconnected microgrid with M control areas indexed by i = 1, 2, . . . , M. The LFC model, including RESs and ESAs, of the ith control area is presented in Fig. 1. The objective of primary control is to contain the

Fig. 1

An overview of the proposed frequently control scheme.

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frequency deviation caused by power imbalance in any control area, while the secondary control, that is, AGC, aims to recover the frequency back to the nominal value [20]. The system dynamics of the ith area can be represented by the following differential equations: Di 1  Pmi (t) − PL,i (t) fi (t) + f˙i (t) = − 2Hi 2Hi  + PRES,i (t) − Ptie,i (t) + PESA,i (t) (1) P˙mi (t) = −

1 1 Tai Pmi (t) + Pgi (t) + P˙gi (t) Tbi Tbi Tbi

1 1 1 Pgi (t) + Pci (t) − fi (t) Tgi Tgi Ri Tgi ⎡ ⎤ M  P˙tie,i (t) = 2π · ⎣ Tij (fi (t) − fj (t))⎦ P˙gi (t) = −

(2) (3)

(4)

j=1,j=i

where fi is the change in system frequency, Pci denotes control effort of secondary control, and Pmi and Pgi are the deviations of generator mechanical output and valve position, respectively. Hi , Di , Ri , Tgi Tai , and Tbi are the system inertia, load damping coefficient, speed droop, governor, and turbine time constants, respectively. PL,i , PRES,i , Ptie,i , and PESA,i are power variations of loads, RESs, tie-line, and ESAs, respectively, which can be viewed as external disturbances to the system. The control input for the secondary control is called area control error (ACE) and can be defined as ACEi (t) = Bi fi (t) + Ptie,i (t)

(5)

where Bi is the frequency bias factor. In the LFC model, PI control is commonly utilized for the secondary control, which is to eliminate the ACE for each control area. The AGC controller based on PI control is represented as (6) Pci (t) = −KP ACEi (t) − KI ACEi (t) where KP and KI are the PI gains, respectively. In practical condition, the frequency recovery time by secondary frequency control can be as long as 10 min. Remark 1. Traditionally, synchronous generators are responsible for frequency control caused by the load variation in the power system. As the penetration of RESs increases rapidly in power systems, additional disturbances have been added to the original LFC model. Given the high controllability, flexibility, and scalability of ESAs, additional frequency support can be provided by ESAs.

Energy storage systems for frequency regulation Chapter | 6 155

3 Proposed disturbance observer The proposed disturbance observer is designed to supplement the secondary frequency control for the ESA, therefore the system frequency response and recovery can be improved. The proposed disturbance observer is comprised of a system disturbance observer in series with a band-pass filter, as shown in Fig. 1. The following sections explain the detailed design of the proposed disturbance observer.

3.1 System disturbance observer The design objective of the disturbance observer is to estimate the real power variations of the control area within which it is deployed in real time. Based on system dynamical equations (1)–(6) in the ith control area, a state-space model ith control area can be obtained as follows: (7) x˙i (t) = Ai xi (t) + Bi μi (t) + Fi Pdi (t) T

where the state variable xi = fi Pmi Pgi , and the system input μi = Pci , the system disturbance Pdi = PRES,i − PL,i − Ptie,i + PESA,i . It is noted that the power output of ESA is also included in the system disturbance. Thus, with proper control of the ESA, the observed system disturbance can be reduced. ⎡ ⎡ ⎤ ⎤ −Di/2Hi 0 −1/2Hi 0 Ai = ⎣−Ta/Ri Tg Tb −1/Tbi (1/Tbi − Tai/Tgi Tbi )⎦ , Bi = ⎣ 0 ⎦, 1/T −1/Ri Tgi 0 −1/Tgi gi

T Fi = −1/2Hi 0 0 With the state-space model (7) of the ith area, the system disturbance observer can be designed as follows [46]: x˙ˆi (t) = Ai xˆi (t) + Bi μi (t) + Fi dˆi (t) + R(yi (t) − yˆi (t)) (8) yˆi (t) = Cxˆi (t) where C is an identity matrix, C = diag{1, 1, 1}, R is the constant gain matrix, and dˆi is the estimated system disturbance. With the estimated states xˆi , the system disturbance can be estimated as follows: dˆi (t) = ξi (t) + Mxˆi (t) (9) ˙ ξ˙i (t) = −M(Ai xˆi (t) + Bi μi (t) + Fi dˆi (t)) + dˆi (t) where ξi is an auxiliary variable and M is a constant gain matrix. Defining the state estimation error and disturbance estimation error as xi = xi − xˆi and di = di − dˆi , the output estimation error can be obtained with Cxi .

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With Eqs. (8), (9), the estimated state error dynamics become x˙i (t) = (Ai − RC)xi (t) + Fi di (t)

(10)

Combining the disturbance estimation error and state estimation error, we have (11) ω˙i (t) = Ai ωi (t) 



A − RC Fi . where ωi = xTi di and Ai = i −MRC 0 The eigenvalues of the matrix Ai can be placed arbitrarily by selecting the gain matrixes R and M as det[sI − Ai ] =

4 

(s + λi )

(12)

m=1

where λi are the desired eigenvalues of the system. Through the eigenvalue placement in Eq. (12), the desired dynamics of the system disturbance observer can be designed.

3.2 Band-pass filter A band-pass filter is designed to extract the magnitude of system disturbance to be compensated by the ESA within the control area. The function of the band-pass filter is illustrated in Fig. 2. The key idea is to decompose the estimated system disturbance, where the high-frequency components of system disturbance within the pass band [fL , fH ] will be sent to the ESA. Therefore, the ESAs can respond to sudden system disturbance and improve the system frequency response. The very high-frequency components of system disturbance [fH , ∞] can be viewed as system noise, while the low-frequency components within [0, fL ] can be compensated by conventional generators. Generally, fH is

Fig. 2

The function of the band-pass filter.

Energy storage systems for frequency regulation Chapter | 6 157

selected to be larger than the reciprocal of response time of the ESA and fL is selected to cover the most sensitive frequency band, as illustrated in [47].

4 Distributed finite-time control of ESA In a realistic power grid, the small-scale ESSs are usually owned by customers that lack direct communication with the AGC controller. In this condition, the ESSs are aggregated into the ESA with only peer-to-peer communications. The ESA is able to participate in system frequency services as an entity. In this section, detailed design of the ESA based on the finite-time consensus approach and its characteristics will be introduced.

4.1 Communication graph The communication network of an aggregated ESS (denoted by i = 0, 1, . . . , N) can be described by a graph, which is defined as G = (V , E ) with a set of nodes V = {v1 , v2 , . . . , vN } and a set of edges E = V × V . Each node is assigned to an ESS in the ESA, and edges represent communication links for data exchange. If communication links are bidirectional, (vi , vj ) ∈ E ⇒ (vj , vi ) ∈ E ∀i, j the graph is said to be undirected. Otherwise, it is directed. A directed graph is said to have a spanning tree, if there is a root node, such that there is a directed path from the root to any other node in the graph. In a matrix called an adjacency matrix, AG = [aij ] where aij can be defined as  1, if (vi , vj ) ∈ E aij = (13) 0, otherwise  The in-degree matrix of G is defined as  = diag{i} with i = N j=1 aij . The Laplacian matrix can be represented as  −aij , i = j (14) L =  − AG ⇒ lij = n a , i=j j∈Ni ij where the Laplacian matrix L has properties that all row sums are zero. For the system with a leader (labeled as node 0), the interaction topology can be expressed by directed graph G , which contains original graph G , node v0 , and edges (vi , v0 ) from node v0 to other nodes. The leader can send information to followers, but not vice versa. The matrix G = diag{g1 , g2 , . . . , gN } is used to describe whether each follower directly receives information from the leader, where  1, if ∃(vi , v0 ) (15) gi = 0, otherwise One important property to guarantee that the leader-follower consensus algorithm with graph G will converge is that the graph G contains a spanning tree rooted at node 0 and all the eigenvalues of matrix L + G have positive real parts [48].

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4.2 Finite-time consensus control of ESA In this section, a finite-time leader-follower consensus algorithm is proposed to control each ESS in the ESA. Compared to other consensus algorithms, the finite-time consensus approach guarantees the power tracking and SoC balancing of each ESS in finite time. The modeling and finite-time consensus control of ESA are illustrated as follows. The real power output of ith ESS is determined by PESS,i (t) = pi (t) · Pmax ESS,i

(16)

where Pmax ESS,i is the power rating (MW) of ith ESS. pi is the power state of the communication agent associated with ith ESS. pi ∈ [−1, 1] considering the power limits of ESSs. Based on the basic coulomb counting method [49], the SoC of the ith ESS can be estimated as follows: T ηi PESS,i (t) dt (17) SoCi (t) = SoCi (0) − t=0 3600 × CESS,i where CESS,i and SoCi (0) are the capacity (megawatt hour) and initial SoC of the ith ESS. ηi is the charging or discharging efficiency of the ith ESS, ηi = ηich if PESS,i (t) < 0, ηi = ηidis if PESS,i (t) > 0. It is considered that ηich = ηidis = 0.95 for all ESSs. CESS,i is multiplied by 3600 to convert the units from hours to seconds. Here we define an energy state ei which indicates the SoCi of the ith ESS. ei ∈ [0, 1] considering the capacity limits of ESSs. Combining Eq. (17) with Eq. (16) and differentiating it with respect to time, the relationship between pi and ei is expressed as e˙i (t) = −

ηi Pmax ESS,i pi (t) 3600 × CESS,i

= KESS,i pi (t)

(18)

where KESS,i is a coefficient between pi and ei . It is assumed that the power output of each ESS is rated at 1C, which means that Pmax ESS,i and CESS,i have the same value but different units. Considering the control input dynamics of the power state, the ESA can be modeled as a group of homogenous double integral systems as follows:  e˙i (t) = KESS pi (t) i = 1, 2, . . . , N (19) p˙i (t) = ui (t) where ei and pi are the energy and power states and ui is the control input of the ith ESS.

Energy storage systems for frequency regulation Chapter | 6 159

A leader is defined as a controller which can receive the upper level control signal (frequency control signal). The leader will update the reference power state and energy state of the ESA as follows: e˙0 (t) = KESS p0 (t) (20) P∗ (t) p0 (t) = PESA max ESA

where e0 and p0 are reference energy and power states of the leader. P∗ESA is the max system frequency  maxcontrol signal and PESA is the power rating of the ESA, which is equal to PESS,i . Therefore, the research problem can be formulated as designing a leaderfollower finite-time consensus control for a group of ESSs with model (19), (20) in an ESA. For an ESA formulated by Eqs. (19), (20), the leader-follower finite-time consensus is achieved if for any initial states, there exists a T0 ∈ [0, ∞], such that lim ei (t) − e0 (t) = 0,

t→T0

ei (t) = e0 (t),

lim pi (t) − p0 (t) = 0

t→T0

pi (t) = p0 (t),

∀t ≥ T0 , i = 1, 2, . . . , N

(21)

In order to achieve Eq. (21), the leader-follower finite-time consensus control protocol is designed as follows: ui (t) =

N 

aij (sig(ei (t) − ej (t))α ) − gi (sig(ei (t) − e0 (t))α )

j=1

−γ

N 

(22) aij (sig(pi (t) − pj (t)) ) − gi (sig(pi (t) − p0 (t)) ) β

β

j=1

where the function sig(x)α = |x|α sign(x), |x| denotes the absolute value of variable x, and sign(.) denotes the sign function. α, β, and γ are control gains to be selected, α, β, γ ∈ (0, 1), and β = 2 · α/(1 + α). The settling time T0 satisfies the inequality [50] T≤

V 1−α (0) c(1 − α)

(23)

where V(x) is a positive definite function defined in a neighborhood of the origin, and real number c > 0. The stability of systems (19), (20) with the protocol (22) can be proved using Lyapunov’s second method by properly selecting the Lyapunov function. Remark 2. The proposed ESA with finite-time power tracking and SoC balancing has the following advantages: (i) it avoids unintentional switching off of certain ESSs when they reach their SoC limits; (ii) the ESA can maintain a larger power rating for a longer time; (iii) it is convenient for the upper level controller to manage the SoCs of the entire aggregator; and (iv) it offers PnP

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capability, that is, when a new ESS joins the aggregator for frequency control, its SoC can autonomously be balanced with other ESSs.

4.3 Stability analysis In this section, the stability of the ESS represented by Eqs. (19), (20) with the finite control protocol in Eq. (22) is proved. First, the globally asymptotically stability is validated as follows. Let pˆi (t) = pi (t) − p0 (t), eˆi (t) = ei (t) − e0 (t), the error system of ESS depicted by Eqs. (19), (20) becomes  e˙ˆi (t) = KESS pˆi (t) i = 1, 2, . . . , N (24) pˆ˙i (t) = uˆi (t) The control protocol (22) becomes uˆi (t) =

N 

 aij (sig(eˆi (t) − eˆj (t))α ) − γ (sig(pˆi (t) − pˆj (t))β )

j=1

 − gi (sig(eˆi (t))α − γ (sig(pˆi (t))β ))

(25)

Select the Lyapunov function as follows: n n n n eˆi  1  2   eˆi −eˆj α pˆi + aij (sig(s) )ds + pˆi (sig(s)α )ds V= 2 0 0 i=1

i=1 j=1

(26)

i=1

Then take the derivative of V: ⎡ N N   ⎣ ˙ V= pˆi aij (sig(eˆi − eˆj )α ) − γ (sig(pˆi − pˆj )β ) i=1

j=1



− gi (sig(eˆi ) − γ sig(pˆi ) ) α

β

N N N  1  aij (pˆi − pˆj )sig(eˆi − eˆj )α + gi pˆi sig(eˆi )α 2 i=1 j=1 i=1 ⎡ ⎤ N N   = pˆi γ ⎣ aij sig(pˆi − pˆj )β − gi sig(pˆi )β ⎦

+

i=1

= ≤

1 2

N N   i=1 j=1

N N 1 

2

i=1 j=1

j=1

aij (pˆi − pˆj )sig(pˆi − pˆj )β −

n  i=1

aij (pˆi − pˆj )sig(pˆi − pˆj )β ≤ 0

gi pˆi sig(pˆi )β

(27)

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According to Lyapunov’s second method for stability, V ≥ 0, V˙ ≤ 0. Therefore, the system (24) with the control protocol (25) is stable. Furthermore, note that V˙ = 0, if and only if pˆi = pˆj = 0, which implies ˙ pˆi = 0, ∀i, j = 1, 2 . . . , N. p˙ˆi (t) =

N 

aij (sig(eˆi (t) − eˆj (t))α − γ sig(pˆi (t) − pˆj (t))β )

j=1

−gi (sig(eˆi (t))α − γ sig(pˆi (t))β ) =

N 

(28)

aij sig(eˆi (t) − eˆj (t))α − gi sig(eˆi (t))α = 0

j=1

It follows that ⎡ ⎤ N N   pˆi ⎣ aij sig(eˆi (t) − eˆj (t))α − gi sig(eˆi (t))α ⎦ i=1

j=1

 1  aij (eˆi − eˆj )sig(eˆi − eˆj )α − gi eˆi sig(eˆi )α = 0 2 N

=−

N

N

i=1 j=1

(29)

i=1

Equality above indicates that eˆi = eˆj = 0, which means the system will converge to ei → e0 , pi → p0 in steady state. Thus the system (24) with control protocol (25) is globally asymptotically stable. Next, continuing with the previous proof of globally asymptotic stability, the locally finite-time convergence of the system (24) with control protocol (25) is proved as follows. The following lemmas and definition from [51] are used in the proof. Lemma 1 (Lasalle’s invariance principle). Let x(t) be a solution of x˙ = f (x), x(0) = x0 ∈ Rk , where f : U → Rk is continuous with U an open subset of Rk , and let V: U → Rk be a locally Lipschitz function such that D+ V (x(t)) ≤ 0, where D+ denotes the upper Dini derivative. Then, denoting the positive limit set as Λ+ (x0 ), Λ+ (x0 ) ∩ U is contained in the union of all solutions that remain in S = {x ∈ U: D+ V (x) = 0}. Definition 1 (Homogeneity with dilation). A function V(x) of x ∈ Rk is homogeneous of degree σ ≥ 0 with dilation coefficients (r1 , . . . , rk ), if V(εr1 x1 , . . . , ε rk xk ) = εσ V(x),

ε>0

(30)

and r1 = · · · = rk = 1, then the dilation is called trivial. Consider the k-dimensional system x˙ = f (x),

x = (x1 , . . . , xk )T ∈ Rk

(31)

A continuous vector field f (x) = (f1 (x), . . . , fk (x))T is homogeneous of degree σ ∈ R with dilation (r1 , . . . , rk ), if fi (εr1 x1 , . . . , ε rk xk ) = εσ fi (x), ε > 0.

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System (31) is called homogeneous if its vector field is homogeneous. Moreover, x˙ = f (x) + fˆ(x),

fˆ(0) = 0,

x ∈ Rk

(32)

is called locally homogeneous if f is homogeneous of degree σ ∈ R with dilation (r1 , . . . , rk ) and fˆ is a continuous vector field satisfying fˆi (εr1 x1 , . . . , ε rk xk ) = 0, ∀x = 0 (33) ε→0 εσ +ri Lemma 2. Suppose system (31) is homogeneous of degree σ with dilation (r1 , . . . , rk ), f is continuous, and x = 0 is its asymptotically stable equilibrium. If homogeneity degree σ < 0, the equilibrium of system (31) is finite-time stable. Moreover, if Eq. (33) holds, then the equilibrium of system (31) is locally finitetime stable. lim

Then we can get a system with variables (eˆ1 , eˆ2 , . . . , eˆn , pˆ1 , pˆ2 , . . . , pˆn ) that is homogeneous of degree κ = α − 1 < 0 with dilation (2, 2, . . . , 2, 1 + α, 1 + α, . . . , 1 + α). Therefore, according to Lemma 2, the system with control law is locally finite-time stable. If the equilibrium of a control is globally asymptotically stable and locally finite-time convergent, then the control is globally finite-time stable. This follows the principle that globally asymptotical stability implies finite-time convergence to any given bounded neighborhood of the equilibrium. Therefore, system (24) with control protocol (25) is globally finite-time stable. In other words, we have eˆi − eˆj → 0, pˆi − pˆj → 0, ∀i, j = 1, . . . , N in finite time. This completes the proof.

4.4 Numerical illustrations This section aims to show the dynamics of power and energy states and validate control effectiveness under different communication graphs. The control algorithm is tested under two directed communication graphs (G 1 and G 2 ), as shown in Fig. 3. In G 1 , Node 1 (ESS 1) is considered as a pinning node which can

Fig. 3

Communication network represented by two directed graphs. (A) G 1 and (B) G 2 .

Energy storage systems for frequency regulation Chapter | 6 163

Fig. 4

Convergence speed of ESA with communication graph-a.

Fig. 5

Converge speed of ESA with communication graph-b.

receive the information from the leader labeled as 0. Other ESSs are connected in a directed line as shown in Fig. 3A. In G 2 , nodes 1 and 8 are both pining nodes while other ESSs are connected in two directed lines as shown in Fig. 3B. The convergence of each ESS with communication graphs G 1 and G 2 is shown in Figs. 4 and 5, respectively. The figures are drawn with Tca = 0.01 s, e0 = 0.5, α = 0.2, β = 0.33, and γ = 0.6. There is a step change in p0 from 0 to 0.5 at 1000 s. The initial energy states of each ESS are [0.46, 0.47, 0.48, 0.49, 0.51, 0.52, 0.53, 0.54] and the initial power state for all is 0.

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The results in Figs. 4 and 5 are further illustrated in time sequence: (1) 0 < t < 1000 s. At the beginning, the energy state of each ESS is not equal. Although the reference power state is 0 during this period, each ESS still needs to charge or discharge in order to balance the energy state. As shown in Figs. 4 and 5, both G 1 and G 2 will converge to consensus and satisfy Eq. (21) in finite time. In Fig. 4, the convergence of ESSs is not symmetrical as graph G 1 is unbalanced, while in Fig. 5, the convergence of ESSs 1–4 and ESSs 5–8 become symmetrical as these two subgroups have same topologies in graph G 2 . The sum of power state for all eight ESSs in graph G 2 remains 0, which is preferred as it is not desired that the SoC balancing influences the external characteristics of the ESA. (2) 1000 s < t < 2000 s. As the consensus is reached in the first time period, the power and energy states of each ESS are equalized. When there is a step change of p0 from 0 to 0.5 at 1000 s, all ESSs start to track the new reference power state. Because the energy states are balanced during this period, the power states of ESSs will converge to p0 much faster. The overshoot of G 2 is smaller than G 1 , as the connection of last node in G 2 is actually shorter than G 1 . Remark 3. From the numerical illustration, it is validated that both G 1 and G 2 can converge in the finite time. However, G 2 is preferred as it can reach consensus with smaller overshoot and does not influence the external power response of the ESA. G 2 will be adopted in the simulation tests in Section 5. Generally, the dynamics of the ESA are influenced by factors such as communication rate, graph, and control protocol.

5 Results and discussions In this section, a variety of scenarios are tested to validate the proposed frequency control scheme. The LFC model and the proposed ESA are implemented in MATLAB/Simulink. To evaluate the performance of the proposed method, both system contingency and normal operation scenarios have been considered. The simulation has been first conducted in the single LFC area, and then extended to multi-LFC areas. The LFC system under consideration is a perunit system with Sbase = 80 MVA. The parameters of the proposed method are shown in Table 1. The parameters of the three-area LFC are shown in Table 2. It is assumed that in one control area, all the ESSs are aggregated into one ESA. The capacity of the ESA (CESA ) in Area 1 is 3.2 MWh, with eight ESSs of capacity [0.68, 0.6, 0.52, 0.44, 0.36, 0.28, 0.2, 0.12] MWh, while the capacity of the ESA in Area 3 is scaled up to 4.8 MWh.

5.1 Case 1: System contingency In case 1, the dynamic performance of the proposed method is investigated under a system contingency condition in a single area microgrid. Area 1 with

Energy storage systems for frequency regulation Chapter | 6 165

TABLE 1 Parameters of proposed control method. Parameters

Value

Band-pass filter (fL , fH )

0.05 Hz, 10 Hz

Finite-time consensus control (α, β, γ )

0.2, 0.33, 0.6

Communication rate (Tca )

0.01 s

TABLE 2 Parameters of three-area microgrids. Area no.

1

2

3

2H (p.u./Hz)

0.1667

0.2

0.15

D (p.u./Hz)

0.0015

0.002

0.001

Ta (s)

5

4

4.5

Tb (s)

40

30

40

Tg (s)

0.4

0.3

0.35

R (Hz/p.u.)

3

3.6

2.5

B (p.u./Hz)

0.8675

0.795

0.87

Tij (p.u./Hz)

0.25

0.25

0.25

CESA (MWh)

3.2

0

4.8

3.2 MWh ESA, as in Table 2, is considered with initial SoCs of ESSs as 50%. A load disturbance of 6.4 MW at 5 s emulates the contingency. The system frequency responses under a conventional LFC scheme, LFC with ESA but without observer, and the proposed control scheme are compared in Fig. 6. Upon occurrence of the disturbance at 5, the frequency of the system begins to drop. The dynamics of the ESA are mainly governed by the proposed

Fig. 6

The frequency response under system contingency.

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Fig. 7

Performance of the proposed disturbance observer.

disturbance observer other than AGC signal in this condition. As a result, it can be observed that both the frequency nadir and the frequency deviation recovery within the allowable range (±0.2 Hz) are evidently improved with the proposed control scheme. The performance of the system disturbance observer is shown in Fig. 7. The parameters of matrixes M and R are chosen as M = [−1, −1, −1] and R = diag{10, 10, 10} according to Eqs. (11), (12). In this condition, the eigenvalues λ1 = −22.4, λ2 = −10, λ3,4 = −6.4 ± 4.3j are assigned to the disturbance observer and the system (11) is stable. It can be noticed that the estimated system disturbance can track the actual system disturbance after a short time. The participation of the ESA in conjunction with the proposed disturbance observer actually reduces the total system disturbance as compared to the load step of 6.4 MW, thereby improving the frequency response. The power output of each ESS in the ESA during this process is shown in Fig. 8. As shown in Fig. 8, each ESS can respond quickly to the load change and reach its maximum power output when there is a large system disturbance. The response speed of each ESS is determined by control protocol (22) as well as communication graphs. The maximum power rating of each ESS is also reflected in Fig. 8, which is the same as the capacity of each ESS mentioned earlier. As the ESA is designed to compensate the power variation rather than the steady-state disturbances, the band-pass filter will reduce the power output

Fig. 8

The power output of each ESS in the ESA.

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of each ESS from 14 s. Correspondingly, the disturbance in Fig. 7 will gradually increase after 14 s due to the reduction of power output from the ESA.

5.2 Case 2: Normal operation In Case 2, the normal operation study is conducted to further investigate the dynamic performance of the proposed method. The same single area system is considered as in Case 1. The 1-h PV and load variation profiles used in this case are shown in Fig. 9. The PV data with 1-s resolution was measured on June 2012 by EPRI [52]. The PV profile has very high fluctuations while the load profile is much smoother. The initial SoC of each ESS in the ESA is [46%, 47%, 48%, 49%, 51%, 52%, 53%, 54%]. In this case, the AGC participation factors of ESAs and generators are αESA = 0.1 and αG = 0.9, respectively. Fig. 10A–C shows system frequency deviations for a conventional LFC scheme, LFC with ESA but without observer, and with the proposed control scheme, respectively. Comparing Fig. 10A–C, it can be observed that frequency deviations are mitigated much more effectively with the proposed method. The frequency deviations can be regulated in an allowable range (±0.2 Hz) with the 3.2 MWh ESA using the proposed method. Furthermore, the total power output of the ESA with and without the proposed disturbance observer is compared in Fig. 11. The results correspond to (B) and (C) in Fig. 10. The ESA will react faster with the proposed disturbance observer; thus the system frequency deviation becomes smaller. The power output and SoC profiles of each ESS in the ESA in this case are shown in Figs. 12 and 13, respectively. Initially, the SoC differences between each ESS are large. Therefore, the SoC balancing effect dominates the ESS dynamics. The power sharing among each ESS is much influenced in order to balance the SoCs, while the total power output of the ESA still follows the frequency control signal. As the SoC of each ESS gradually become equalized during the operation, the power sharing among each ESS become proportional to its power rating, the same as Fig. 8 in Case 1. It can be also observed from Fig. 11 that each ESS in the ESA responds to sudden PV fluctuation to mitigate the frequency deviations.

Fig. 9

One-hour PV and load variation profiles.

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Fig. 10 Comparison of system frequency deviations under (A) conventional LFC scheme without ESA, (B) LFC with ESA without observer, and (C) proposed control scheme.

Fig. 11

Total power output of the ESA with and without the proposed disturbance observer.

Fig. 12

Power output of each ESS in case 2.

Energy storage systems for frequency regulation Chapter | 6 169

Fig. 13

SoC profile of each ESS in case 2.

5.3 Case 3: Multiarea microgrids In Case 3, the system contingency study is extended to multi-LFC areas to investigate further the performance of the proposed method. The structure of the studied three-area interconnected microgrid is depicted in Fig. 14. In Case 3, it is assumed that there is 3.2 MWh ESA in Area 1 and 4.8 MWh ESA in Area 3. There are step load disturbances of 6.4 MW at 5 s in Area 1 and 9.6 MW at 100 s in Area 3. The proposed disturbance observer will estimate the system distance in each control area for the corresponding ESA. Other simulation conditions are the same as in Case 1. Fig. 15 shows the total power output of ESAs under step load changes in Areas 1 and 3, respectively. The ESAs in Areas 1 and 3 will respond to the system disturbance in the respective control area, as shown in Fig. 15. Fig. 16 shows the frequency response in each control area of Case 3. As shown in

Fig. 14

The structure of the studied three-area power system.

Fig. 15

Total power output of ESAs in Area 1 and Area 3.

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Fig. 16

Frequency response of each control area. (A) Area 1, (B) area 2, and (C) area 3.

Fig. 16, the sudden load changes will influence the frequency not only in the same area, but also in the interconnected microgrids. The frequency response of each control area can be improved with the proposed observer and finite-time consensus-based ESAs.

5.4 Case 4: Comparison with linear control algorithm In this section, the proposed finite-time consensus control protocol is compared to a widely utilized linear consensus control protocol for the second-order multiagent system reported in [53]. Compared to the proposed finite-time approach, the linear control protocol in [53] cannot ensure consensus in finite time. For systems (19), (20), the linear control protocol in [53] can be expressed as follows: ui (t) = k1

N 

aij (ei (t) − ej (t)) − gi (ei (t) − e0 (t))

j=1

− k2

N 

(34) aij (pi (t) − pj (t)) − gi (pi (t) − p0 (t))

j=1

where k1 and k2 are control gains to be selected. First, the performance of the linear protocol (34) and the proposed protocol (22) are compared under the system contingency condition in Case 1. The comparison results of frequency response and ESA power output are shown in Fig. 17. The control gains of protocol (34) are selected as k1 = 35, k2 = 1.

Energy storage systems for frequency regulation Chapter | 6 171

Fig. 17

Comparison with the linear method. (A) Frequency response and (B) ESA power output.

As shown in Fig. 17A, it can be found that the proposed method improves the frequency nadir as compared to the linear method. Correspondingly, the power output of the ESA under the proposed method also reacts faster than the linear method, as shown in Fig. 17B. Second, the linear protocol (34) is compared against the proposed control protocol under the normal operation scenario in Case 2. The results of Section 5.2 serve as a base reference, that is, the convergence time for protocol (22) is 600 s (as seen in Fig. 13) and the maximum power overshoot observed is 0.33 MW (for ESS 1 as seen in Fig. 12). The study is undertaken in two steps and the performance in terms of convergence time and system overshoot are compared. In order to ensure the same convergence time (600 s) as in Case 2, the control gains of protocol (34) are tuned as k1 = 35, k2 = 1. As shown in Fig. 19, the SoC of each ESS can converge at 600 s, the same as in Fig. 13. However, the

Fig. 18

Power output of each ESS with linear protocol (k1 = 35, k2 = 1).

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Fig. 19

SoC profile of each ESS with linear protocol (k1 = 35, k2 = 1).

power overshoots in Fig. 18 during the convergence are much higher than in Fig. 12, which even reach the power limits of the ESSs. It can therefore be said that the proposed method presents a smaller system overshoot under the same convergence time. In order to ensure the same maximum power overshoot (0.33 MW of ESS-1) as in Case 2, the control gains of protocol (34) are tuned as k1 = 12, k2 = 1. As shown in Fig. 20, the maximum power overshoot is 0.33 MW, the same as in Fig. 12. However, the SoC (or system) in Fig. 21 converges much slower as compared to Fig. 13. That is, the proposed method can provide a faster convergence speed under the same system overshoot.

Fig. 20

Power output of each ESS with linear protocol (k1 = 12, k2 = 1).

Fig. 21

SoC profile of each ESS with linear protocol (k1 = 12, k2 = 1).

Energy storage systems for frequency regulation Chapter | 6 173

The advantages of the finite-time approach compared with the linear method in [53] are summarized as follows. In the case of a system contingency, the ESA with the proposed finite-time approach has a faster response of power output, thus it significantly improves the system frequency response. Under the normal operation, the ESS with the proposed finite-time approach presents a smaller power overshoot and faster convergence. Smaller power overshoot and faster convergence are desired for operation of ESA as: (i) ESSs within the ESA have a lower chance to reach the power limits during the operation, and (ii) any newly joined ESS can attain faster SoC balancing, ensuring PnP capability.

6 Conclusion In this chapter, a new frequency control scheme with finite-time consensus controlled ESA was proposed to improve the microgrid frequency response in the presence of stochastic renewable power generation. The total frequency control signal for ESA is determined by both AGC commands and the proposed disturbance observer. The control protocol using a leader-follower finite-time consensus algorithm was developed to aggregate ESSs in one control area into an ESA. The stability proof of the proposed method for ESSs was given. The internal and external characteristics of the ESA can be adjusted by its communication graph and control protocol settings. The simulation results have demonstrated that the system frequency response can be improved by using the proposed disturbance observer and finite-time consensus-based ESA. The proposed method is effective for both system contingency and normal operation conditions. The ESA can track the frequency control signal and the SoC of each ESS can be equalized in finite-time during the operation. The proposed method was further validated in a three-area microgrid. The comparative studies validate the advantages and necessity of the proposed method. In summary, the proposed method provides an effective way to aggregate widespread DESSs for system frequency regulation.

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Chapter 7

Distributed optimization algorithm for economic dispatch: A bisectional approach Hao Xinga , Minyue Fub and Zhiyun Lina a School

of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang, People’s Republic of China, b School of Electrical Engineering and Computing, University of Newcastle, Callaghan, NSW, Australia

1 Introduction The economic dispatch problem (EDP) has been actively studied in the electric power industry for optimal operation and planning of energy resources. This problem is usually formulated as an optimization problem [1]. The classic EDP is mainly concerned with the economic dispatch of fossil-fired power generation systems to achieve minimum operational costs within capacity limits. In this scenario, the operation and planning for power generation systems can be done by one or several central decision makers. Many types of cost functions are available. A convex and piecewise linear cost function is used in [2], but a quadratic cost function is usually preferred [1]. Many centralized solutions have been proposed to solve the EDP. In [1], the conventional Lagrangian relaxation approach and first-order gradient method are given. In [3], a strategy based on a direct search method with multilevel convergence is proposed to solve the EDP with transmission capacity constraints. In [4], an algorithm based on evolutionary programming, tabu search, and quadratic programming methods are proposed to solve the nonconvex EDP. A parallel microgenetic algorithm is employed in [5] to solve the ramp-rate constrained EDP with nonmonotonically and monotonically increasing incremental cost functions. Distributed algorithms for control, estimation, and optimization have been intensively investigated for large-scale systems [6]. Spatially distributed largescale systems interconnected by a communication network are ubiquitous in the real world, where the traditional centralized control algorithms are inefficient. A smart grid with distributed renewable power generation is a typical such large-scale system. A lot of work has been done about distributed optimization [7–9] using the distributed gradient method, distributed subgradient method, Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00007-4 Copyright © 2020 Elsevier Inc. All rights reserved.

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alternating direction method of multipliers (ADMM), and so on. In general, compared with centralized algorithms, distributed algorithms have many advantages, including enhanced robustness, reduction in communication between agents, and uniform power consumption for each agent. To meet environmental targets, to accommodate a greater emphasis on demand response [10], and to support plug-in hybrid electric vehicles [11, 12], distributed generation, and storage capabilities, traditional power grids need to become “smart grids.” This is an area that has been heavily studied in recent years [13]. In a smart grid integrating distributed generation, renewable power sources, and a communication network, it is desirable to solve the EDP in a distributed fashion. In fact, a lot of such work has been done so far. In [14, 15], the authors propose a consensus-based algorithm to realize decentralized economic dispatch, where a master node aware of the total power demand is required to ensure the equality between the total power supply and demand. In [16], the authors present a ratio consensus-based decentralized algorithm to find the optimal incremental cost, under the assumption that each node (i.e., generator) knows the parameters of all the nodes. In [17], an algorithm based on a consensus and innovation framework is proposed, where the consensus term makes all the nodes agree with each other to realize their common goal of estimating the global price index, while the innovation term makes all the nodes estimate the index according to the local knowledge of loads. In [18], the authors propose an algorithm for the EDP with a quadratic cost function, which can be treated as a distributed implementation of the standard Lambda-iteration method, without requiring other nodes’ parameters. In this chapter, a distributed bisection algorithm (DBA) based on a consensus-like iterative method is presented to solve the EDP. Compared with other algorithms, DBA has the following features. 1. DBA requires no global information of the system. In [18], a distributed algorithm is proposed, but global parameters including network topology and generators’ parameters are needed to design an appropriate learning gain to guarantee convergence. In [16], each node needs to know some parameters of all other nodes, which implies that the computation and communication package size grow at least linearly with the network size, while in DBA each node only needs to know its local parameters. 2. No master or leader node aware of the total power demand is needed in DBA, whereas such a node is required in [14–16, 19]. In DBA, none of the nodes knows the total demand, yet the demand and supply balance is guaranteed by the algorithm. For that purpose, every bus in the power grid, with pure load, pure generation, or both, is modeled as a node, which merely knows its local power demand (the demand from loads attached to it). However, in [18], the nodes only represent buses with a generator aware of its associated power demand (including the power demand of the pure generation buses in its neighborhood).

Distributed optimization algorithm for economic dispatch Chapter | 7 179

3. DBA only assumes that the communication network is a strongly connected directed graph, whereas in [14, 15, 17], an undirected graph is assumed. Communications may be subject to packet loss, device failure, or asymmetric bandwidth allocations, which makes the directed graph model more reasonable and general. However, it is well known in the field of distributed control and computation that convergence analysis is much more challenging in the directed graph setting. 4. The algorithm developed in this chapter can handle the EDP with general convex cost functions, whereas the algorithm proposed in [18] is hard to apply to the EDP with general convex functions, due to the difficulty in guaranteeing its convergence in such a situation. The spirit of this chapter is mostly technical. We start in Section 2 by presenting a coherent description of the problem formulation of the EDP and one of its existing centralized solutions. No prior knowledge is required except convex optimization theory and standard linear algebra. Great attention has been placed on the design of DBA, which, in the authors’ opinion, is of most relevance for electrical engineers. In Section 3, the consensus-like algorithm along with graph theory basics and nonnegative matrices is introduced, as the preliminaries for the design and analysis of DBA. We then elaborate the design of DBA as well as its stopping criteria and analyze the convergence in Section 4. Exhaustive numerical examples are presented in Section 5 to show the performance of DBA. We conclude this chapter in Section 6 by discussing possible extension of DBA in other scenarios.

2 System modeling 2.1 Problem formulation The EDP we study in this chapter is to minimize the aggregate cost of all the generators in the power grid on the premise that all the n generators cooperatively provide a required amount of power P within their individual generation capacities. We only consider active power in this chapter and we ignore power transmission loss and transmission capacity constraints, which are valid for many power networks. Each generator is associated with a local variable xi  0, that is, the (active) power generated by generator i, and a cost function Ci (xi ). In this chapter, we deal with the EDP with general cost functions satisfying the following assumption: Assumption 1. For every 1  i  n, Ci (xi ): R+ → R+ is strictly convex and twice continuously differentiable with d2 Ci (xi )  0, dxi2

∀x ∈ R+ ,

where R+ denotes the set of nonnegative real numbers, and the equality holds at isolated points only.

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One can easily verify that the commonly used quadratic cost functions, given as follows, are special cases satisfying Assumption 1, Ci (xi ) = ai xi2 + bi xi + ci ,

(1)

where ai > 0, bi , and ci are cost parameters. For simplicity of expression in the following sections, we use an equivalent function by changing a constant term: Ci (xi ) =

(xi − αi )2 2βi

(2)

with constants αi and βi > 0. The following is an example of a nonquadratic cost function with a natural exponential term [1]:   xi − ei Ci (xi ) = ai xi2 + bi xi + ci + di exp oi with ai > 0, di > 0, and oi > 0. We remark that many generators, especially distributed energy resources feeding on renewable energy, are often uncontrollable. For such a generator j, we may assume that a fixed amount of power xjfix [17]. These generators can be viewed as negative loads and added to the positive loads, that is, the load Pj for the associated bus j will be replaced with Pj − xjfix . With the previous convention, we can assume, without loss of generality, that every generator has variable generation capacity. Denote the total number of buses in the grid by m and the number of buses with power generators by n. In general, each bus can be with a generator only, loads only, or both. Since not all the buses are attached to power generators, we have m > n. Denoting by Pj the power demand (load) of bus j, the aggregate power demand P is given by m 

Pj = P ,

j=1

where Pj = 0, if bus j is a pure generation bus. For generators with variable generation capacities, denoting by xi and x¯i the lower and upper bounds of xi , we have 0  xi  xi  x¯i . In the framework assumed earlier, the EDP can be formulated as follows: min

n 

Ci (xi ),

(3)

i=1

s.t. xi  xi  x¯i , n  i=1

xi = P .

∀i = 1, 2, . . . , n,

(4) (5)

Distributed optimization algorithm for economic dispatch Chapter | 7 181

It is obvious that the EDP is feasible if and only if n 

xi  P 

i=1

n 

x¯i .

(6)

i=1

Note that the optimization problem (3)–(5) also finds application in other problems, including optimal resource allocation problem for parallel computing [7], and demand side management for power systems, especially for direct load control of smarter control systems [20]. Throughout the chapter, we assume that communication networks are imposed on the power grid so that each bus corresponds to a node in the communication network. Here we set up two communication networks, denoted by Gm = (Vm , Em ) and Gn = (Vn , En ), respectively. The node set Vm consists of all the m buses in the grid, while Vn consists of all the n generation buses, that is, Vm = {1, 2, . . . , n, n + 1, . . . , m} and Vn = {1, 2, . . . , n}. Define Em and En as the sets of directional communication paths between nodes in Gm and Gn , respectively. It is assumed that both Gm and Gn are strongly connected digraphs with self-loops. To make the distributed solution meaningful, we also assume that Gm and Gn are sparse graphs in the sense that + max dm,i  m,

1im

−  m, max dm,i

1im

+ max dn,i  n,

1in

− max dn,i  n.

1in

Except for the ability of exchanging information, the ability of local computation is also required for each node. In addition, every node i knows its local power demand Pi , cost function Ci (xi ), and capacity constraints xi and x¯i , but they do not need to know other nodes’ parameters.

2.2 Centralized solution to the EDP It is clear that the EDP is a convex optimization problem, and Assumption 1 guarantees a unique optimal solution for the problem (3)–(5). Many centralized algorithms have been developed for the convex optimization problem [21]. Furthermore, the cost function is strictly convex and the Slater condition holds due to affine constraint (5). Thus, strong duality is guaranteed, which allows us to solve the primal problem by solving its Lagrange dual problem [21]. The centralized solution is as follows. Denote the incremental cost of generator i by ui (xi ) =

dCi (xi ) , dxi

∀i ∈ Vn ,

which is continuous and strictly increasing with respect to xi according to Assumption 1. Thus the inverse function of ui (xi ), denoted by u−1 i , exists and −1 is also continuous and strictly increasing. Note that ui may not have a closed-

182 PART | II Energy management

form expression, but its numerical solution can be obtained using a bisection method due to its continuity and strict monotonicity. The Lagrange dual problem is given by max

n 

Ci (λ) + λP ,

(7)

i=1

where

⎧ ⎪ ⎨ Ci (xi ) − λxi  −1 Ci (λ) = Ci (u−1 i (λ)) − λui (λ) ⎪ ⎩ Ci (x¯i ) − λx¯i

and λ ∈ R is the Lagrange multiplier. From the above, we have ⎧ ⎪ − xi  dCi (λ) ⎨ gi (λ) = = − u−1 i (λ) ⎪ dλ ⎩ − x¯i

λ < ui (xi ), ui (xi )  λ < ui (x¯i ), ui (x¯i )  λ,

(8)

λ < ui (xi ), ui (xi )  λ < ui (x¯i ), ui (x¯i )  λ.

(9)

If the primal solution is feasible, the Lagrange dual problem (7) has a unique optimal solution λ* , satisfying P +

n 

gi (λ* ) = 0.

i=1

Accordingly, the primal EDP has a −gi (λ* ), i = 1, 2, . . . , n, that is, ⎧ ⎪ x ⎪ ⎨ i * * xi = u−1 i (λ ) ⎪ ⎪ ⎩ x¯ i

unique optimal solution given by xi* = λ* < ui (xi ), ui (xi )  λ* < ui (x¯i ),

(10)

ui (x¯i )  λ . *

3 Introduction to consensus-like algorithm 3.1 Graph theory and nonnegative matrices A directed graph (or just digraph) G = (V, E) consists of a nonempty finite set of nodes V = {1, 2, . . . , n} and a finite set of ordered edges E ⊆ V × V. For node i ∈ V, its in-neighbor set and out-neighbor set are denoted by Ni− = {j ∈ V − {i} : (j, i) ∈ E} and Ni+ = {j ∈ V − {i} : (i, j) ∈ E}, that is, node i receives information from its in-neighbors and sends out information to its outneighbors. The in-degree and out-degree of node i are the cardinalities of Ni− and Ni+ , denoted by di− = |Ni− | and di+ = |Ni+ |, respectively. A path in graph G is a finite sequence of edges in E connecting a sequence of distinct nodes in V, and the length of a path is the number of its edges.

Distributed optimization algorithm for economic dispatch Chapter | 7 183

The diameter of a connected directed graph G, denoted by D, is defined as the length of the longest among the shortest paths connecting any two nodes. The period d of G is defined as the greatest common divisor of all the lengths of cycles in G. We call the graph is d-periodic if d > 1 and aperiodic if d = 1. We assume that each node can communicate with itself, that is, ∀i ∈ V, (i, i) ∈ E, thus d = 1 and the graph is aperiodic. A graph is strongly connected if there is a path from any node to any other node in the graph, which is assumed throughout the chapter. We also say that a nonnegative matrix A ∈ Rn×n is associated with graph G, where [A]ij > 0 if and only if (j, i) ∈ E. For nonnegative matrices, we have the following lemma [22]. Lemma 1. A nonnegative matrix A ∈ Rn×n is primitive, if and only if its associated graph G is strongly connected and aperiodic.

3.2 Consensus-like algorithm For a strongly connected digraph G = (V, E), define a normalized adjacency matrix Q ∈ Rn×n as ⎧ 1 ⎪ ⎨ if (j, i) ∈ E, + (11) [Q]ij = qij = dj + 1 ⎪ ⎩ 0 otherwise. One can easily verify that Q is associated with G and is column stochastic, that is, QT is (row) stochastic. From the properties of the stochastic matrix, we have ρ(Q) = ρ(QT ) = 1 and QT 1 = 11, where 1 = [1, 1, . . . , 1]T . Since G is strongly connected with self-loops, from Lemma 1, Q is primitive, and thus QT is also primitive. From the Perron-Frobenius theorem [23], we have lim Qt = lim ((QT )t )T = (1ηT )T = η1T ,

t→∞

t→∞

(12)

where η = [η1 , η2 , . . . , ηn ]T is the right eigenvector associated to the eigenvalue 1 of Q, with the properties ηi > 0 for all i and 1T η = 1. Endow each node i in the graph G = (V, E) with a local variable πi ∈ R, and denote the global variable by the column vector π = [π1 , π2 , . . . , πn ]T . Let us consider the following consensus-like iterative algorithm with the iteration index denoted by t and initial value π(0): π(t + 1) = Qπ(t). This can be implemented in a distributed form, that is,  qij πj (t). πi (t + 1) = qii πi (t) + j∈Ni−

(13)

(14)

184 PART | II Energy management

From Eqs. (12), (13), the iterative algorithm has the following property: n

 * T πi (0) η, π = lim π(t) = η1 π(0) = t→∞

(15)

i=1

where π * denotes an equilibrium point of system (13). We call this algorithm “consensus-like” because, if the matrix Q is such that ηi = 1/n (which happens when Q is doubly stochastic, that is, both row and column stochastic), all the πi (t)s reach the average consensus asymptotically, that is, they all converge to the same average of the initial π(0) [24]. Remark 1. To guarantee the convergence of the “consensus-like” iteration, a strongly connected digraph with self-loops is required. In addition, each node i needs to know its out-degree di+ for the sake of the distributed implementation. As for communication, each node i sends out πi (t)/(di+ + 1), instead of πi (t), which is slightly different from the consensus algorithm. This consensus-like iterative algorithm is also used for the study of ratio consensus in [16, 25, 26].

4 Distributed bisection algorithm: Design and analysis In this section, we present a distributed bisection method to obtain the optimal Lagrange multiplier λ* for the problem (3)–(5). This is done based on the iterative algorithm (13), with no need for a central decision maker or a leader node. We first propose a distributed algorithm for gathering the aggregate power demand, then show distributed feasibility test of the EDP, and finally give a DBA for the EDP. Intuitively speaking,  to solve the EDP in a fully distributed fashion, the total power demand P = m j=1 Pj shall be obtained by all the generators using some distributed algorithm. We term such a problem as the sum consensus problem, that is, every node gets a common value equal to the sum of all the nodes’ initial values, using a distributed method. A special instance of the sum consensus problem is the so-called network size problem, where the goal is to use a fully distributed algorithm to find out the number of nodes in a connected network [27, 28]. This is a special instance of the sum consensus problem by setting all the initial values to 1. It is known that if each node has bounded memory, communication, and computation, and the network is anonymous (i.e., each node does not have a unique global identifier), then a sufficiently large network size cannot be computed using a fully distributed algorithm [28]. A challenge for us is to get over this technical difficulty.

4.1 Distributed algorithm for aggregate demand  The m first step of solving the EDP is to collect the aggregate power demand P = j=1 Pj . From our discussion earlier with regard to the sum consensus problem, we understand that it is a difficult task to compute P directly. Instead, DBA is to

Distributed optimization algorithm for economic dispatch Chapter | 7 185

 make every node i (generation bus) in Vn get a value yi such that ni=1 yi = P . As we will show in the next section, it turns out that such values of yi will be sufficient to solve the EDP. DBA is developed based on the aforementioned consensus-like algorithm, with novelty in how to transfer the aggregate power Gmto Gn using a fully distributed algorithm, that is, demand P from  the graph n   to  P = P we go from m j=1 j i=1 yi = P . For Gm = (Vm , Em ), define an associated normalized adjacency matrix Q ∈ Rm×m as ⎧ 1 ⎪ ⎨ if (j, i) ∈ Em , + d [Q]ij = qij = (16) m,j + 1 ⎪ ⎩ 0 otherwise, + is the out-degree of node j ∈ Gm . Similarly, for Gn = (Vn , En ), define where dm,j an associated normalized adjacency matrix R ∈ Rn×n as ⎧ 1 ⎪ ⎨ if (j, i) ∈ En , + (17) [R]ij = rij = dn,j + 1 ⎪ ⎩ 0 otherwise, + is the out-degree of node j ∈ Gn . where dn,j For every node i ∈ Vm , we first establish an auxiliary variable pi (t) with initial value pi (0) = Pi , and then run the following iterations until convergence:  qij pj (t), (18) pi (t + 1) = qii pi (t) + − j∈Nm,i

− denotes the in-neighbor set of node i in Gm . Denoting p = where Nm,i limt→∞ p(t), from Eq. (15), we have

pi = P ηi ,

∀i ∈ Vm ,

(19)

where η = [η1 , η2 , . . . , ηm ]T is the right eigenvector for the eigenvalue 1 of Q, with the properties ηi > 0 for all i and 1T η = 1. In words, pi s are the scaled local power demands. Using the auxiliary variables pi s, the demand information is gathered in Gm . Once Eq. (18) converges, for any node i ∈ Vm , we establish an auxiliary variable si (t) initialized with pi i = 1, 2, . . . , n, si (0) = 0 i = n + 1, n + 2, . . . , m, and then run the following iterations until convergence:  si (t + 1) = qii si (t) + qij sj (t). − j∈Nm,i

(20)

186 PART | II Energy management

Denoting s = limt→∞ s(t), from Eq. (15), we have ⎛ ⎞ n  ηj ⎠ P ηi , ∀i ∈ Vm . si = ⎝

(21)

j=1

Variables si s are the scaling ratios between graph Gm and Gn . We will use them to transfer the demand information from Gm to Gn . Next, for every node i ∈ Vn , we establish an auxiliary variable yi (t) with initial value yi (0) =

(pi )2 P ηi = n ,  si j=1 ηj

and then run the following iterations until convergence:  rij yj (t), yi (t + 1) = rii yi (t) +

(22)

(23)

− j∈Nn,i

− where Nn,i denotes the in-neighbor set of node i in Gn . Denoting y = limt→∞ y(t), from Eqs. (15), (22), we have ⎛ ⎞ n  yj (0)⎠ γi = P γi , ∀i ∈ Vn , (24) yi = ⎝ j=1

where γ = [γ1 , γ2 , . . . , γn ]T is the right eigenvector for the eigenvalue 1 of R, with the properties γi > 0 for all i and 1T γ = 1. Variables yi are the scaled power demand held by the generator buses only. Using the earlier procedures, we can get the needed demand information yi s in a distributed fashion. We summarize below the distributed algorithm. Algorithm 1 Distributed algorithm for P

Remark 2. It is clear that Algorithm 1 is fully distributed because each node only uses local information and information from its neighbors without any central processing or leader node. We will thereinafter show that based on Algorithm 1, DBA does not need a central information collector to compute P , and the total power demand P is not needed explicitly for solving the EDP.

Distributed optimization algorithm for economic dispatch Chapter | 7 187

4.2 Distributed algorithm for feasibility test Before proceeding to the distributed solution to the EDP, we propose a distributed iterative algorithm based on Eq. (14) for feasibility test. Recall the EDP is feasible if and only if Eq. (6) holds. For any node i ∈ Vn , consider two variables yi (t), y¯ i (t), with their initial values given by yi (0) = xi ,

y¯ i (0) = x¯i .

Run the following iterations simultaneously: yi (t + 1) = rii yi (t) + y¯ i (t + 1) = rii y¯ i (t) +



− j∈Nn,i



rij yj (t),

(25)

rij y¯ j (t).

(26)

− j∈Nn,i

It is clear that the previous converge. Denote their asymptotic values by yi and y¯ i , respectively. From Eq. (15), we have ⎛ ⎞ n  yi = ⎝ xj ⎠ γi , (27) j=1

⎛ ⎞ n  y¯ i = ⎝ x¯j ⎠ γi ,

(28)

j=1

for all i ∈ Vn . Since every γi > 0, the feasibility condition (6) holds if and only if yi  yi  y¯ i ,

(29)

for any i ∈ Vn , where is obtained from Eq. (24). Moreover, if the previous holds for one node, then it holds for all other nodes. For clarity, the result above is summarized in the following algorithm. yi

Algorithm 2 Distributed algorithm for feasibility test

188 PART | II Energy management

4.3 Distributed bisection algorithm We now present the DBA for the EDP, which is operated in graph Gn . Following  from Algorithm 1, we assume here that each node i contains yi with ni=1 yi = P . Let k  0 denote the iteration index for the bisection method. We let each node establish two variables λ− (k) and λ+ (k), representing the lower and upper bounds of the Lagrange multiplier. Their initial values are given such that λ− (0) is sufficiently small and λ+ (0) is sufficiently large, or alternatively given by λ− (0) = min ui (xi ), i∈Vn

+

λ (0) = max ui (x¯i ). i∈Vn

We will explain how to compute these initial values using a distributed algorithm later. Define a variable λ(k), which acts as an approximation of the Lagrange multiplier, as λ(k) = (λ+ (k) + λ− (k))/2.

(30)

xi (k) = −gi (λ(k)),

(31)

Each node i ∈ Vn takes establishes a local variable zi (t) initialized by zi (0) = xi (k), and runs the following iterations:  rij zj (t). (32) zi (t + 1) = rii zi (t) + − j∈Nn,i

Denoting z = limt→∞ z(t) and using Eq. (15), we have ⎛ ⎞ n  xj (k)⎠ γi , ∀i ∈ Vn , zi = ⎝

(33)

j=1

where zi s are the scaled generator outputs associated with λ(k). Every node updates λ+ (k + 1) and λ− (k + 1) by comparing yi and zi as follows: + λ (k + 1) = λ(k), λ− (k + 1) = λ− (k) for zi > yi , (34) λ+ (k + 1) = λ+ (k), λ− (k + 1) = λ(k) for zi  yi . Although the update of λ+ (k + 1) and λ− (k + 1) is done locally, every node computes the same λ+ (k + 1) and λ− (k + 1) because Eqs. (24), (33), and γi > 0 imply that sgn(zi − yi ) = sgn(zj − yj ), where sgn(·) is the sign function.

∀i, j ∈ Vn ,

Distributed optimization algorithm for economic dispatch Chapter | 7 189

It is clear from Eqs. (30), (34) that λ = limk→∞ λ(k) exists and that each node obtains locally its optimal solution from Eq. (31), that is, xi* = −gi (λ* ),

∀i ∈ Vn .

Algorithm 3 Distributed bisection method for the EDP

For clarity, we summarize the distributed bisection method for this scenario in Algorithm 3. The convergence property of Algorithm 3 is formally stated here. Theorem 1. Under the assumption that the EDP (3)–(5) is feasible, Algorithm 3 converges to the unique optimal solution as k → ∞. Proof. For all i ∈ Vn , the function gi (λ) is monotonically decreasing with respect to λ, therefore −gi (λ) is monotonically increasing. In particular, −gi (λ) is strictly increasing with respect to λ for ui (xi )  λ  ui (x¯i ). Define λ¯ = max ui (x¯i ), i∈Vn

λ = min ui (xi ). i∈Vn

Since the problem is feasible, the optimal Lagrange multiplier must satisfy Therefore, −

n

¯ λ  λ  λ. ¯

gi (λ) is strictly increasing with respect to λ ∈ [λ, λ]. Thus, i=1   n i, j=1 xj (k) γi is strictly increasing with respect to λ(k) ∈

for every node ¯ Therefore, Algorithm 3 converges. Moreover, since the optimal solution [λ, λ]. is unique, Algorithm 3 converges to the unique one.

190 PART | II Energy management

Remark 3. Algorithm 3 fully distributed due to the following properties. Information exchange between nodes occurs only when running the consensuslike iteration (32). All the computations are performed locally. In addition, each node only requires knowledge of local parameters αi and βi , without need for knowledge of other nodes’ parameters. Remark 4. In addition to getting around the difficulty of directly obtaining P in a distributed fashion, another benefit of Algorithm 1 is that it reduces the communication and computation burden of the nodes representing buses with pure loads. The utilization of the two networks Gm and Gn is motivated by the fact that in reality generator buses only account for a relatively small percentage in power systems. For instance, there are only 5 and 14 generator buses in the IEEE 14-bus and 118-bus test systems, respectively [29]. Intuitively, it seems unnecessary for the nongenerator buses to be involved in the overall process. But on the other hand, the power demand is spatially distributed at almost all the buses. To deal with this situation, two communication networks Gm and Gn are constructed and Algorithm 1 is developed. After yi s are computed using Algorithm 1, the remaining part of Algorithm 3 only involves generation buses, that is, the bisection steps are performed in Gn only. As we will show in the simulations, the utilization of the two networks greatly reduces the aggregate communication volume. We now address the issue of how to choose the initial values λ− (0) and Intuitively, the closer λ+ (0) and λ− (0) are to the optimal multiplier λ* , the fewer steps of bisection are needed. For Algorithm 3, if the EDP is feasible, we can initialize λ− (0) and λ+ (0) using a minimum/maximum consensus algorithm [30]. Note that for a strongly connected network, minimum/maximum consensus algorithm is a fully distributed iterative algorithm. λ+ (0).

Algorithm 4 Initialization of λ+ (0) and λ− (0)

4.4 Convergence analysis and stopping criteria Now we give analysis on convergence of Algorithm 3 and offer stopping criteria for a practical implementation of the algorithms. Two stopping criteria are needed, one for the consensus-like iterations at each bisection step, and the other for the bisection iterations.

Distributed optimization algorithm for economic dispatch Chapter | 7 191

First, we give the stopping criterion for the consensus-like iterations. The consensus-like iteration (13) converges asymptotically, but it is impossible to run it for infinite time. In most scenarios, when implementing iteration (13), a stopping criterion might be setting a fixed number of iterations t* such that π(t) − π * 2 < , π(0) − π * 2

∀t  t* ,

(35)

that is, for some preset > 0, which is small enough to assume the iteration converges to π * . Therefore, there accordingly exists t* , such that when t = t* , the iterations are assumed to have converged. The stopping criterion in a distributed manner to judge whether consensus is reached is also studied in [31]. However, such a stopping criterion is not suitable for the consensus-like iterations when used on Eq. (32) in Algorithm 3. According to Eq. (35), when convergence is assumed to be reached for t > t* , it is not necessarily that the nodes make the same decision on how to bisect their incremental cost intervals, which may happen when the iterations stop at some time t* , with zi > yi for some nodes, while zi  yi for the other nodes. Consequently, different decisions on bisection are made. In fact, at each bisection step, we only need to run iteration (32) till every node reaches agreement on the direction in which they shall bisect their incremental cost intervals. For clarity, we modify DBA 4 as follows. Reinitializing zi (0) = xi (k) − y , and then running iterations (32), it is easy to verify that Eq. (34) is equivalent to + λ (k + 1) = λ(k), λ− (k + 1) = λ− (k) for z > 0, (36) λ+ (k + 1) = λ+ (k), λ− (k + 1) = λ(k) for z  0. Define σi (t) = sgn(zi (t)), where sgn(·) is the sign function. We say that iteration (32) reaches sign consensus, if σi (t) = σj (t), ∀i, j. It is easily verified that for iterations (32), sign consensus can always be reached in finite time, that is, there exists t†  0, such that for all t > t† , σi (t) = σj (t), ∀i, j, unless z = 0. Now we give a distributed method to judge whether sign consensus is reached or not, which is based on the maximum/minimum consensus algorithm [30]. We assume each node has an estimate of (or knows) the diameter of Gn , denoted by Dn . At bisection, each node i establishes three auxiliary variables, † † σ i , σ¯ i , and σi . For σi , we define 1 if zi ( t/Dn Dn ) > 0, † σi (t) = (37) 0 if zi ( t/Dn Dn )  0, †

where · is the floor function. At t = 0, initialize σ i (0) = σi (0), σ¯ i (0) = † σi (0). Then for t  1, we update σ i (t) and σ¯ i (t) by ⎧ † ⎨ σi (t) if t/Dn Dn = t, σ i (t + 1) = (38) ⎩ min σ i (t) otherwise, − j∈{Nn,i ,i}

192 PART | II Energy management

σ¯ i (t + 1) =

⎧ † ⎨ σi (t)

if t/Dn Dn = t,

σ¯ i (t) ⎩ max −

otherwise.

(39)

j∈{Nn,i ,i}

The stopping criterion for iteration (32) is that for every node i, if there exists some positive integer τ * such that σ i (τ * Dn ) = σ¯ i (τ * Dn ),

(40)

then sign consensus is already reached. Then every node can stop the consensuslike iteration (32) and make the same decision at t = τ * Dn . In addition, σ i (τ * Dn ) = σ¯ i (τ * Dn ) = 1 implies that λ+ (k + 1) = λ(k) and λ− (k + 1) = λ− (k), while σ i (τ * Dn ) = σ¯ i (τ * Dn ) = 0 implies that λ+ (k + 1) = λ+ (k) and λ− (k + 1) = λ(k). The theoretical basis of the procedure (37)–(40) is that maximum/minimum consensus is bound to be reached within Dn steps, as Dn is the diameter of the strongly connected graph Gn (see [31] for more details). From Eq. (37), with † τ = 1, 2, . . . , we can see that the nodes in fact only update σi s at t = τ Dn , and † otherwise the σi s remain unchanged. Moreover, from Eqs. (38), (39), the nodes update σ i and σ¯ i at time t = (τ − 1)Dn + 1, (τ − 1)Dn + 2, . . . , τ Dn , using only a maximum/minimum consensus algorithm. Therefore, at t = τ Dn , the nodes are able to know whether they have reached sign consensus at t = (τ − 1)Dn . Remark 5. Another benefit of using the stopping criterion (40) is the probable reduction of iteration steps needed for convergence. A special situation is that the signs are the same initially, for example, for all i, zi (0) > 0. But due to the absence of a leader or master node, the nodes still have to run Dn steps of iteration (32). In comparison, even if the nodes can make the same decision when the consensus-like iteration (32) stops according to Eq. (35), it takes t* steps of iteration. Note that t* will be a large number when is sufficiently small, so it is often the case that t*  Dn . Next, we present the stopping criterion for the bisection steps. It is clear from Eq. (30) that each iteration of k halves the interval [λ− (k), λ+ (k)], thus the convergence of λ(k) is very rapid. Since solving the EDP is actually finding the optimal incremental cost λ* , a stopping criterion can be established by either setting a fixed number of iterations K, or using |λ(k) − λ* |  /2,

(41)

for some sufficiently small > 0. Since λ* is not available, an alternative can be |λ+ (k) − λ− (k)|  ,

(42)

which can be easily achieved in a distributed fashion, provided sign consensus is always reached in each bisection step.

Distributed optimization algorithm for economic dispatch Chapter | 7 193

5 Numerical examples In this section, we show the performance of the DBA using several numerical experiments based on the IEEE 14-bus and 118-bus systems [29]. Five cases are simulated. Case 1 demonstrates DBA using a quadratic cost function whereas Case 2 does so using a nonquadratic cost function. In Case 3, the convergence speed of DBA is analyzed. We compare DBA with the algorithm proposed in [17] in Case 4 with regard to the convergence speed, the total computation load, and the aggregate communication volume. The previous four cases use the IEEE 14-bus system. Finally, in Case 5, we apply DBA to the IEEE 118-bus system. For the numerical simulations on the IEEE 14-bus system, generator buses are {1, 2, 3, 6, 8} and load buses are {2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14}. Note that the power transmission grid is not necessarily the same with the information communication network, so we do not assign a node to bus 7. The two strongly connected directed graphs with self-loops Gm and Gn are established, as shown in Figs. 1 and 2. One can easily verify that Gm and Gn are sparse graphs.

5.1 Case 1: The EDP with quadratic cost functions only In this case, we solve the EDP with quadratic cost functions only, as it is most commonly assumed for the EDP. The generator parameters are adopted from Kar and Hug [17]. We set xi = 10 MW for all i, so they are not shown in Table 1. We take = 0.005 for the stopping criterion.

Fig. 1

An illustration of graph Gm , where self-loops are not shown.

Fig. 2

An illustration of graph Gn , where self-loops are not shown.

TABLE 1 Generator parameters (MU = monetary unit). Generator

Bus

αi (MW)

βi (MW2 /MU)

x¯i (MW)

1

1

−25

12.5

80

2

2

−50

16.67

90

3

3

−57.14

14.29

70

4

6

−66.67

16.67

70

5

8

−31.25

12.5

80

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The local power demands are: P1 = 0 MW, P2 = 9 MW, P3 = 56 MW, P4 = 55 MW, P5 = 27 MW, P6 = 46 MW, P8 = 0 MW, P9 = 8 MW, P10 = 24 MW, P11 = 53 MW,  P12 = 46 MW, P13 = 16 MW, and P14 = 40 MW. The total demand P = i∈Vm Pi = 380 MW, which is not known to the individual nodes. We set λ+ (0) = 20 MU/MW and λ− (0) = 0 MU/MW, which is sufficient to guarantee λ* ∈ [λ− (0), λ+ (0)]. The result is shown in Fig. 3. The upper subplot of Fig. 3 shows the evolution of λ(k), the middle subplotshows the evolution of each xi (k), and the lower subplot shows the evolution of i∈Vn xi . We artificially set the iteration step to be 20, while the stopping criterion is already satisfied at k = 12. Taking the results at k = 12 to be the optimal solution, we have x1* = 80.00 MW, x2* = 90.00 MW, x3* = 64.48 MW, x6* = 70.00 MW,  x8* = 75.35 MW, and i∈Vn xi* = 380.03 MW. The optimal incremental cost λ* = 8.5278 MU/MW, and the optimal solution x* stays within the capacity

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Results for the EDP with quadratic cost functions only.

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Distributed optimization algorithm for economic dispatch Chapter | 7 195

constraints, where xi* of generator 1, 2, and 4 take their upper bounds of capacity constraints, respectively. Note that DBA is based on the Lagrange dual method, so the equality constraint will not be truly satisfied until infinite bisections. Therefore,  using DBA, there is a tolerable gap between demand and supply, that is, i∈Vn xi* − P = 380.03 − 380 = 0.03 MW.

5.2 Case 2: EDP with nonquadratic cost functions We now demonstrate a case where some of the generators have nonquadratic cost functions and some generators have a fixed amount of power. In particular, fix we replace generator 4 (at bus 6) with a fixed generation of x4 = 100 MW and take   (x1 + 25)2 x1 + 40 + 50 exp , C1 (x1 ) = 25 100 C3 (x3 ) =

(x3 + 57.14)2 + 7 × 10−6 x34 . 28.58

Using the same λ+ (0), λ− (0), and total power demand P as those in Case 1, the result is shown in Fig. 4. After 13 steps of bisection, the algorithm converges to x1* = 64.86 MW, x2* = 90.00 MW, x3* = 48.62 MW, x6* = 100.00 MW, x8* =  76.46 MW, and i∈Vn xi* = 379.92 MW. The optimal incremental cost λ* = 8.6157 MU/MW.

5.3 Case 3: Convergence speed analysis Now we study how the starting values λ+ (0) and λ− (0) would affect the convergence speed. For simplicity, we still set λ− (0) = 0 MU/MW and only vary λ+ (0), that is, we run Case 1 again with different values of λ+ (0) from 10 to 150 MU/MW. The result is shown in Fig. 5. Apparently, a larger λ+ (0) leads to more bisection steps. To make convergence fast, Algorithm 4 can be adapted to choose good λ+ (0) and λ− (0). Furthermore, since λ is monotonically increasing with respect to P , historical data can be useful provided the current EDP and the previous EDP share the same configuration of generators and parameters, that is, the only difference must be the total power demand. If so, take the historical λ as λ− (0) of the current problem if the current total demand is larger than previous demand, and vice versa.

5.4 Case 4: The comparison with the algorithm in [17] In this case, we compare DBA with previous work, and show the benefits of using two communication networks and the stopping criterion based on the sign consensus. Although other references listed in the introduction are also highly related to this chapter, we mainly make comparisons with the algorithm

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proposed in [17] because Kar and Hug [17] has the most similarities with ours with regard to the problem setup. Specifically, both the algorithms in this chapter and in [17] are fully distributed without relying on a leader node, and all the buses in the grid are involved in both algorithms’ implementation. The comparison is primarily about the convergence speed, the computation load, and the communication volume. We first review the results in Case 1 using DBA. With = 0.005, the iteration steps needed for iterations (18), (20), and (23) are 29, 50, and 24, respectively. During the bisections, the consensus-like iteration steps needed for each bisection step are shown in Fig. 6. With the stopping criterion based on the sign consensus in Section 4.4, not many consensus-like iterations are needed for the bisections. Particularly, for the second and third bisection steps, only four consensus-like iterations are needed. Note that the diameter of graph Gn in Case 1 is four. According to Remark 5, it follows that in fact the sign consensus is

Distributed optimization algorithm for economic dispatch Chapter | 7 197 18 17 16

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already reached without the need of running the consensus-like iteration. The computation time is counted in terms of iteration steps rather than seconds, as the time which an iteration takes to compute depends on the CPU speed of the nodes. For DBA, the computation time of this simulation example is 351 steps.

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We then resolve the problem in Case 1 using the algorithm proposed in [17]. Learning gains of the algorithm are adopted from the numerical experiments in [17]. Since undirected graphs are assumed in [17], we use the undirected graph by replacing the unidirectional arrows in Fig. 1 with bidirectional ones to implement the algorithm in [17]. The results are shown in Fig. 7. One can see that the convergence of the algorithm in [17] is very slow. Though the λs held and updated by the buses reach consensus quickly, at that time the λs do not reach λ* . It still takes almost 1 × 105 steps to converge to the optimal Lagrange multiplier. Therefore, the algorithm in [17] converges much slower than DBA. In addition, in the algorithm in [17], there are in total 13 buses assigned with agents in this case. All those buses are involved in the algorithm’s implementation till the convergence. Since the iterations practically account for most of the computational load, we directly use the iterations needed in total to describe the computational load. Similarly, the total number of numerical values exchanged between nodes is regarded as the aggregate communication volume. Thus the total computation load using the algorithm in [17] is 1.3×106 . Each node communicates bidirectionally with other four nodes, so the total communication volume is 4 × 13 × 105 = 5.2 × 106 , which is huge. As for DBA, although there are also 13 buses assigned nodes, we only involve them all when running iterations (18), (20), and the remaining steps are implemented in the smaller graph Gn . Therefore, the total computation load using DBA is 13 × (29 + 50) + 5 × (24 + 248) = 2487. In DBA, each node in Gm and Gn communicates unidirectionally with other four and two nodes, respectively.

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Results for the EDP with quadratic cost functions only (Case 1) using the algorithm in [17].

Distributed optimization algorithm for economic dispatch Chapter | 7 199

TABLE 2 A summary of the comparisons between DBA and the algorithm proposed in [17]. Time

Computation load

Communication volume

DBA

3.5 × 102

2.5 × 103

2.3 × 103

The algorithm in [17]

1 × 105

1.3 × 106

5.2 × 106

Thus the total communication volume of DBA is 2 × 13 × (29 + 50) + 1 × (24 + 248) = 2326. The comparisons are summarized in Table 2. From the data above, one can see that DBA converges much faster than the algorithm in [17]. In addition, the computation load and communication volume using DBA are far less. The fast convergence and low operational cost of DBA are a cooperative feature of bisection, the utilization of the two communication networks, and the stopping criterion based on the sign consensus. We also remark that to guarantee the convergence of the algorithm in [17], one needs to design the gains properly, referred to as αt and βt in [17]. Those gains also affect the speed of convergence. However, the determination of αt and βt depends on global information and cannot be implemented in a distributed manner. As mentioned in Remark 3, no such gains in DBA need to be predetermined using global information.

5.5 Case 5: Implementation on IEEE 118-bus system In this case, we apply DBA to the IEEE 118-bus test system [29] to investigate DBA’s performance further. The generator parameters are adopted from Wu et al. [32]. The total power demand is 950 MW. We set λ+ (0) = 50 MU/MW and λ− (0) = 0 MU/MW, which is sufficient to guarantee λ* ∈ [λ− (0), λ+ (0)]. The result is shown in Fig. 8. We artificially set the iteration number to 20, but in the simulation the stopping criterion is already satisfied at k = 13. The optimal Lagrange multiplier λ* = 29.8332 MU/MW. From this case, we can find that DBA is also applicable to large power grids such as the IEEE 118-bus test system. Moreover, the convergence is still fast due to the nature of bisection, and is insensitive to the network size.

6 Conclusion and discussion In this chapter, we have proposed the DBA based on a consensus-like iteration to solve the EDP (3)–(5), where the cost functions can be general convex functions (i.e., they are not restricted to quadratic functions). DBA is fully distributed, with no need for a master node or leader, in which a strongly connected digraph with self-loops is sufficient for communication. In addition, each node only requires its local parameters, without knowledge of the global information. The convergence of DBA is proved, and by simulations we illustrate the performance of DBA.

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We would like to remind the reader that the design of fully distributed algorithms is usually of great difficulty, even if the target problem seems simple and has been well-studied in the traditional paradigm of centralized algorithms. The difficulties mainly stem from the absence of control center (leader node), lack of global information, limited communication capabilities, and the requirement of plug-and-play feature, etc. We hope the reader will appreciate the reasonability of our elaborating DBA using this entire chapter. The EDP (3)–(5) studied in this chapter is basic, in the sense that it only considers the generation capacity constraints and the demand-supply balance in a lossless power grid. However, when solving a complex EDP with other

Distributed optimization algorithm for economic dispatch Chapter | 7 201

factors, we can use decomposition techniques, for example, Lagrange dual decomposition, to decompose the problem into subproblems in the form of a basic EDP (3)–(5). Therefore, DBA can be used as a subalgorithm for a complex EDP or even optimal power flow/unit commitment problems. For instance, DBA can be embedded in the ADMM framework to solve a dynamic EDP with energy storage and generation ramp constraints [33]. In addition, DBA can be adopted for solving a lossy EDP and dealing with prohibited operating zones in the λ-iteration framework [34].

References [1] A.J. Wood, B.F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, New York, 1996. [2] R.A. Jabr, A.H. Coonick, B.J. Cory, A homogeneous linear programming algorithm for the security constrained economic dispatch problem, IEEE Trans. Power Syst. 15 (3) (2000) 930–936. [3] C.-L. Chen, N. Chen, Direct search method for solving economic dispatch problem considering transmission capacity constraints, IEEE Trans. Power Syst. 16 (4) (2001) 764–769. [4] W.-M. Lin, F.-S. Cheng, M.-T. Tsay, Nonconvex economic dispatch by integrated artificial intelligence, IEEE Trans. Power Syst. 16 (2) (2001) 307–311. [5] J. Tippayachai, W. Ongsakul, I. Ngamroo, Parallel micro genetic algorithm for constrained economic dispatch, IEEE Trans. Power Syst. 17 (3) (2002) 790–797. [6] L. Bakule, Decentralized control: an overview, Annu. Rev. Control 32 (1) (2008) 87–98. [7] L. Xiao, S. Boyd, Optimal scaling of a gradient method for distributed resource allocation, J. Optim. Theory Appl. 129 (3) (2006) 469–488. [8] A. Nedic, A. Ozdaglar, Distributed subgradient methods for multi-agent optimization, IEEE Trans. Autom. Control 54 (1) (2009) 48–61. [9] S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. 3 (1) (2011) 1–122. [10] H. Xing, Y. Mou, Z. Lin, M. Fu, Fast distributed power regulation method via networked thermostatically controlled loads, in: Proceedings of the 19th World Congress of the International Federation of Automatic Control, Cape Town, South Africa, 2014, pp. 5439–5444. [11] Y. Mou, H. Xing, Z. Lin, M. Fu, A new approach to distributed charging control for plug-in hybrid electric vehicles, in: Proceedings of the 33rd Chinese Control Conference (CCC), Nanjing, China, 2014, pp. 8118–8123. [12] Y. Mou, H. Xing, Z. Lin, M. Fu, Decentralized optimal demand-side management for PHEV charging in a smart grid, IEEE Trans. Smart Grid (2014), https://doi.org/10.1109/TSG.2014. 2363096. [13] A. Ipakchi, F. Albuyeh, Grid of the future, IEEE Power Energy Mag. 7 (2) (2009) 52–62. [14] Z. Zhang, M.-Y. Chow, Incremental cost consensus algorithm in a smart grid environment, in: Proceedings of the IEEE Power and Energy Society General Meeting, San Diego, CA, 2011, pp. 1–6. [15] Z. Zhang, X. Ying, M.-Y. Chow, Decentralizing the economic dispatch problem using a two-level incremental cost consensus algorithm in a smart grid environment, in: Proceedings of the IEEE North American Power Symposium (NAPS), Boston, MA, 2011, pp. 1–7. [16] A.D. Dominguez-Garcia, S.T. Cady, C.N. Hadjicostis, Decentralized optimal dispatch of distributed energy resources, in: Proceedings of the IEEE Conference on Decision and Control, Maui, Hawaii, 2012, pp. 3688–3693.

202 PART | II Energy management [17] S. Kar, G. Hug, Distributed robust economic dispatch in power systems: a consensus + innovations approach, in: Proceedings of the IEEE Power and Energy Society General Meeting, San Diego, CA, 2012, pp. 1–8. [18] S. Yang, S. Tan, J.X. Xu, Consensus based approach for economic dispatch problem in a smart grid, IEEE Trans. Power Syst. 28 (4) (2013) 4416–4426. [19] H. Xing, Y. Mou, M. Fu, Z. Lin, Consensus based bisection approach for economic power dispatch, in: Proceedings of the 53rd IEEE Conference on Decision and Control (CDC), Los Angeles, CA, 2014. [20] G. Strbac, Demand side management: benefits and challenges, Energy Policy 36 (12) (2008) 4419–4426. [21] S.P. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004. [22] Z. Lin, Distributed Control and Analysis of Coupled Cell Systems, VDM Publishing, Saarbrücken, Germany, 2008. [23] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. [24] A. Bemporad, M. Heemels, M. Johansson, Networked Control Systems, vol. 406, Springer, Heidelberg, 2010. [25] A.D. Dominguez-Garcia, C.N. Hadjicostis, Coordination and control of distributed energy resources for provision of ancillary services, in: Proceedings of the First IEEE International Conference on Smart Grid Communications (SmartGridComm), Gaithersburg, MD, 2010, pp. 537–542. [26] A.D. Dominguez-Garcia, C.N. Hadjicostis, Distributed algorithms for control of demand response and distributed energy resources, in: Proceedings of the IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, 2011, pp. 27–32. [27] I. Shames, T. Charalambous, C.N. Hadjicostis, M. Johansson, Distributed network size estimation and average degree estimation and control in networks isomorphic to directed graphs, in: Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing, IEEE, 2012, pp. 1885–1892. [28] J.M. Hendrickx, A. Olshevsky, J.N. Tsitsiklis, Distributed anonymous discrete function computation, IEEE Trans. Autom. Control 56 (10) (2011) 2276–2289. [29] R. Christie, Power Systems Test Case Archive, University of Washington, Electrical Engineering, 2000. https://www2.ee.washington.edu/research/pstca. [30] J. Cortés, Distributed algorithms for reaching consensus on general functions, Automatica 44 (3) (2008) 726–737. [31] V. Yadav, M.V. Salapaka, Distributed protocol for determining when averaging consensus is reached, in: Proceedings of 45th Annual Allerton Conference, 2007, pp. 715–720. [32] L.H. Wu, Y.N. Wang, X.F. Yuan, S.W. Zhou, Environmental/economic power dispatch problem using multi-objective differential evolution algorithm, Electr. Power Syst. Res. 80 (9) (2010) 1171–1181. [33] H. Xing, Z. Lin, M. Fu, B.F. Hobbs, Distributed algorithm for dynamic economic power dispatch with energy storage in smart grids, IET Control Theory Appl. 11 (11) (2017) 1813–1821. [34] H. Xing, P. Zeng, Y. Mou, Q. Wu, Consensus-based distributed approach to lossy economic power dispatch of distributed energy resources, Int Trans. Electr. Energy Syst. 29 (2019) e12041, https://doi.org/10.1002/2050-7038.12041.

Chapter 8

Scheduling of EV battery swapping in microgrids Pengcheng You Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, United States

1 Introduction 1.1 Background, motivation, and contributions We are at the cusp of a historic transformation of our energy system into a more sustainable form in the coming decades. Electrification of our transportation system will be an important component because vehicles today consume more than a quarter of energy in the United States and emit more than a quarter of energy-related carbon dioxide [3]. Electrification will not only greatly reduce greenhouse gas emission, but also have a large impact on the future grid because electric vehicles (EVs) are large but flexible loads [4]. The impact of EVs is especially significant for microgrids that group locally interconnected loads and distributed energy resources and act as an individual controllable entity either to interact with the main grid in the grid-connected mode or stand alone in the island mode. A microgrid is therefore an effective paradigm to integrate various sources of renewables by fully utilizing its dispatchable elements, among which EVs could play a momentous role. EVs, as moving storage in essence, readily realize energy/power shift in both temporal and spatial domains by their routine running and charging. It is foreseeable that a huge amount of flexibility can be exploited from EV operation to complement microgrid operation. As we will see in this chapter, there is a large literature on various aspects of EV charging. It is widely believed that uncontrolled EV charging may stress or even disrupt power grids, but well-controlled charging can help stabilize grids and integrate renewables. However, we look at EV operation from a different angle here (i.e., battery swapping). Instead of charging its battery when an EV is running out of energy, it could have the depleted battery swapped at a service station by a fully charged battery so as to avoid suffering the



This chapter basically summarizes and extends the work in [1, 2] to the context of a microgrid.

Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00008-6 Copyright © 2020 Elsevier Inc. All rights reserved.

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problem of long waits. All unloaded batteries are charged centrally at service stations to prepare for future battery swapping demand. Battery swapping as a refueling model for EVs, dating back to 2007 when it was first commercialized by a start-up Better Place in Israel, has been reemerging in recent years with major advances in battery charging and energy storage. Other than technological innovation, operational breakthrough is another critical factor that contributes to the bloom of battery swapping. Technically, this mechanism is implementable with pilot programs already established in Israel and China. Its advantages are fourfold. First, it takes only minutes to swap a battery but often hours to recharge it. Second, the aggregation of charging loads reduces demand uncertainty compared with individual EV charging, simplifying power system operation. Third, the aggregation of charging loads endows service stations with greater flexibility in scheduling battery charging and providing ancillary services. Fourth, batteries, as the most costly core of an EV, can be leased rather than purchased, tremendously lowering the expenditure for EV owners. In the meantime, battery swapping is also faced with unique challenges in popularization. First, it requires standardization of vehicles, batteries, and swapping infrastructure, which has proven to be difficult. Second, a business model is needed to address ownership, maintenance, and payment issues regarding shared batteries. However, the problem we will study in this chapter circumvents these obstacles and mainly focuses on an EV-station battery swapping system supplied by a microgrid and its interactions, motivated by a novel battery swapping model currently being pursued in China, especially for electric buses and taxis [5]. The State Grid (one of the two national utility companies) of China is experimenting with this model where it operates not only the power grid, but also service stations and a taxi service around a city, which constitute a vertically integrated system. When the state of charge of a State Grid taxi is low, it goes to one of the State Grid operated service stations to exchange its depleted battery for a fully charged one. While battery swapping takes only a few minutes, it is not uncommon for taxis to arrive at a service station, only to find that it runs out of fully charged batteries and there is a queue of taxis waiting to swap their batteries. The occasional multihour waits are a serious impediment that degrades the efficiency of battery swapping, which is predicated on having sufficient fully charged batteries at service stations. In fact, it is often the case that some service stations which EVs gather around run short of fully charged batteries quickly while others accrue more and more. Obviously it is neither economical nor practical to stock enough batteries at every service station to serve the worst-case EV arrival patterns. This indicates that EVs have the incentive to choose service stations so as to avoid long waits. On the other hand, microgrid operation is also tremendously influenced by EVs’ battery swapping decisions since battery charging loads are redistributed spatially in the network, which is rich in load flexibility and bound to improve the system operating efficiency if well managed.

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To this end, in this chapter we propose to coordinate battery swapping in a microgrid such that EVs can make the most efficient use of currently available batteries in the system and meanwhile the microgrid operation is jointly optimized. Specifically, we formulate in Section 2 an optimal scheduling problem for battery swapping in a microgrid that assigns to each EV a best station to swap its depleted battery based on its current location and state of charge. The station assignments not only determine EVs’ travel distance, but also impact significantly the power flows on the microgrid because batteries are large loads. The schedule aims to minimize a weighted sum of EVs’ travel distance and electricity generation cost over both station assignments and power flow variables, subject to EV range constraints, grid operational constraints, and AC power flow equations. This joint battery swapping and optimal power flow (OPF) problem is nonconvex and computationally difficult for two reasons. First, AC power flow equations are nonlinear. Second, the station assignment variables are binary. We address the first difficulty in Section 2.1 by summarizing several representative linearization/convexification methods to approximate/relax the nonlinear power flow equations that prove to be accurate and effective given different network topologies or parameterization. Fixing any station assignments, the remaining OPF problem is then convex. The second difficulty can be properly addressed in two fashions. The centralized solution in Section 3 applies generalized Benders decomposition to the current mixed-integer convex program, and is suitable for cases where the microgrid, service stations, and EVs are managed centrally by the same operator (e.g., the State Grid model). Supposing an exact underlying linearization/relaxation of the power flow equations is given, the generalized Benders decomposition computes a global optimum in reasonable time. In this centralized solution, the operator needs global information such as the grid topology, impedances, operational constraints, background loads, availability of fully charged batteries at each station, locations and states of charge of EVs, etc. It is implementable only in a vertically integrated system like the State Grid operated electric taxi program. As EVs proliferate and battery swapping matures, an equally (if not more) likely model will emerge where the microgrid is managed by a utility company, service stations are managed by a station operator (or multiple station operators), and EVs may be managed by individual drivers (or multiple EV groups, e.g., taxi companies in the electric taxi case). In particular, the set of EVs to be scheduled may include a large number of private cars in addition to commercial fleet vehicles. They may not be willing to share their private information. The centralized solution will fail to apply for these future scenarios. Moreover, generalized Benders decomposition solves a mixed-integer convex program centrally and is still computationally expensive. It is hard to scale to compute in real-time optimal station assignments and an (linearized/relaxed) OPF solution when the numbers of EVs and stations are large.

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To this end, we develop scalable distributed solutions that preserve private information and are more suitable for general scenarios. Instead of generalized Benders decomposition, we relax the binary station assignment variables to real variables in [0, 1]. With both the linearization/relaxation of power flow equations and the relaxation of binary variables, the resulting approximate problem of joint battery swapping and OPF is a convex program. This allows us to develop two distributed solutions where separate entities make their individual decisions but coordinate through information exchanges that do not involve their private information in order to solve jointly the global problem. The first solution, based on the alternating direction method of multipliers (ADMM), is for systems where the microgrid is managed by a utility company and all service stations and EVs are managed by a station operator. Here the utility company maintains a local estimate of some aggregate assignment information that is computed by the station operator, and they exchange the estimate and the aggregate information to attain consensus. The second solution, based on dual decomposition, is for systems where the microgrid is managed by a utility company, all service stations are managed by a station operator, and all EVs are individually operated. The utility company still sends its local estimate to the station operator while the station operator does not need to send the utility company the aggregate assignment information, but only some Lagrange multipliers. The station operator also broadcasts Lagrange multipliers to all EVs and individual EVs respond by sending the station operator their choices of stations for battery swapping based on the received Lagrange multipliers and their current locations and driving ranges. In both approaches, given the aggregate assignment information and Lagrange multipliers that are exchanged, different entities only need their own local states (e.g., power flow variables) and local data (e.g., impedance values, battery states, EV locations, and driving ranges) to compute iteratively their own decisions. See Fig. 1 for the distributed framework. Informaon exchanges

Utility company

Network

Staon operator

Locat ions,

Fig. 1

Distributed framework.

Scheduling of EV battery swapping in microgrids Chapter | 8 207

Suppose an exact underlying linearization/relaxation of the power flow equations is given; the proposed distributed algorithms, however, may return station assignments that are not binary due to the relaxation of binary variables, which suggest a probabilistic station assignment for an EV. We prove an upper bound on the number of such EVs with nonbinary station assignments. The bound guarantees that the discretization can be readily implemented and also justifies the final solution is close to optimum. In Section 5, we illustrate the performance of our centralized and distributed solutions through simulations on a real 56-bus test system from Southern California Edison (SCE). The simulation results suggest that the centralized solution is effective and computationally tractable for practical application, and the distributed solutions are scalable and usually achieve satisfactory station assignments under real conditions.

1.2 Literature There is a large literature on EV charging, for example, optimizing charging schedule for various purposes such as demand response, load profile flattening, or frequency regulation [6–9]; architecture for mass charging [10–13]; locational marginal pricing for EV charging [14, 15]; and the interaction between EV penetration and the optimal deployment of charging stations [16]. Sojoudi et al. [17] seem to be the first to optimize jointly EV charging and AC power flow spatially and temporally through semidefinite relaxation. Zhang et al. [18] extend the joint OPF-charging problem to multiphase distribution networks and propose a distributed charging algorithm based on ADMM. Chen et al. [19] decompose the joint OPF-charging problem into an OPF subproblem that is solved centrally by a utility company and a charging subproblem that is solved in a distributed manner by individual EVs through a coordinative valleyfilling signal from the utility company. De Hoog et al. [20] use a linear model and formulate EV charging on a three-phase unbalanced grid as a receding horizon optimization problem. It shows that optimizing the charging schedule can increase the EV penetration that is sustainable by the grid from 10%–15% to 80%. Linearization is also used in [21] to model EV charging on a threephase unbalanced grid as a mixed-integer linear program. The binarity arises from the fact that an EV is either being charged at its peak rate or off. These papers focus on jointly optimizing power flows and charging for EVs connected to given locations on the grid. A key feature of battery swapping scheduling is, however, the use of EV mobility to optimize explicitly the spatial redistribution of charging loads. The literature on battery swapping is much smaller. Tan et al. [22] propose a mixed queuing network that consists of a closed queue of batteries and an open queue of EVs to model the battery swapping processes, and analyze its steady-state distribution. Yang et al. [23] design a dynamic operation model of a battery swapping station and put forth a bidding strategy in power markets.

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You et al. [24] study the optimal charging schedule of a battery swapping station serving electric buses and propose an efficient distributed solution that scales with the number of charging boxes in the station. Sarker et al. [25] propose a day-ahead model for the operation of battery swapping stations and use robust optimization to deal with future uncertainty of battery demand and electricity prices. Zheng et al. in [26] study the optimal design and planning of a battery swapping station in a distribution system to maximize its net present value, taking into account life cycle cost of batteries, grid upgrades, reliability, operational cost, and investment cost. Zhang et al. [27] discuss several business models of battery swapping and leasing service in China. You et al. [28–30] present a series of work on station assignment for EV battery swapping to make better use of batteries in practical application. However, the impact of the assignments on power systems is not taken into account. To the best of our knowledge, joint optimization of battery swapping and power flows on microgrids has not been investigated, which is becoming an emerging practical issue. The distributed solutions are motivated by the need to preserve private information of different entities operating microgrids, stations, and EVs. Privacy in future grids is a key challenge facing both utilities and end users [31], for example, see [32–35] for privacy concerns on smart meters and [36–38] for privacy concerns on EVs. Distributed algorithms preserve privacy as global information is not needed for local computations. Liu et al. [34] schedule thermostatically controlled loads and batteries in a household to hide its actual load profiles such that no sensitive information can be inferred from electricity usage. Yang et al. [35] design an online control algorithm of batteries that only uses the current load requirement and electricity price to optimize the tradeoff between smart meter data privacy and users’ electricity cost. Liu et al. [39] propose a consensus-based distributed speed advisory system that optimally determines a common vehicle speed for a given area in a privacy-aware manner to minimize the total emission of fuel vehicles or the total energy consumption of EVs. Other applications can be found in data mining [40], cloud computing [41], etc. To the best of our knowledge, this work is the first to discuss the distributed scheduling of EV battery swapping in light of binary station assignments and microgrid operation.

2 Problem formulation We focus on the scenario where a fleet of EVs and a set of service stations operate in a region that is supplied by a microgrid. We assume the microgrid, service stations, and EVs are managed centrally by the same operator, for example, the State Grid in China. Periodically, say, every 15 min, the system determines a set of EVs that should be scheduled for battery swapping, for example, based on their current states of charge or their requests for battery swapping. At the beginning of the current control interval, the system assigns to each EV in the set a service station for battery swapping. It is reasonable

Scheduling of EV battery swapping in microgrids Chapter | 8 209

to assume that the EVs travel to their assigned service stations and finish swapping their batteries before the end of the current interval, since typically the geographic area served by a microgrid is limited. Under this assumption, station assignments are decoupled across control intervals and our work focuses on one such interval. Batteries returned by the EVs start to be charged at the service stations immediately (typically at service stations each battery is placed in a charging box before being swapped, thus a returned battery can immediately find its place in a charging box). Since we focus on the scheduling of battery swapping, we assume for simplicity that these batteries are charged at the constant rated power for the control interval under study, which contributes to better serving future battery swapping demand as well. Optimizing charging rates over multiple intervals can be integrated with battery swapping if more future information is available, but that is beyond the scope of the current work. Our goal is to design an assignment algorithm that minimizes a weighted sum of the distance traveled by the EVs for battery swapping and electricity generation cost, while respecting the EVs’ range constraints, the operational constraints of the microgrid, and AC power flow equations. In the following we formulate our optimal scheduling problem. For a finite set K that consists of some natural numbers, its cardinality is denoted as |K|. For a set of scalar variables yj , j ∈ K, its column vector is denoted as yK. yTK and yH K denote its transpose and Hermitian transpose, respectively. Sometimes the subscript K is dropped if the set is clear from the context. For a matrix Y, Y T and Y H denote its transpose and Hermitian transpose, respectively. Let Yi,j be the (i, j)th element of Y and YK1 K2 be a submatrix of Y composed of all the element Yij , i ∈ K1 and j ∈ K2 .

2.1 Network model Consider a single-phase microgrid network with a connected directed graph G = (N, E), where N := {0, 1, 2, . . . , N} and E ⊆ N × N. Each node in N represents a bus and each edge in E represents a power line. Let N+ := {1, 2, . . . , N}. Bus 0 is a slack bus if G is a mesh network, or a root bus if G is a radial (tree) network. We orient the graph, without loss of generality, by denoting a line in E by (j, k) or j → k if it points from bus j to bus k. Let zjk be the complex impedance of line (j, k) ∈ E, and yjk = z1jk be the corresponding complex admittance. Let Sjk := Pjk + iQjk denote the sending-end complex power from bus j to bus k where Pjk and Qjk denote the real and reactive power flows, respectively. Define Ijk as the complex current from bus j to bus k and Vj as the complex voltage phasor of bus j with its angle denoted by θj . Assume the voltage V0 of bus 0 is fixed and given. Each bus j has a base load sbj := pbj + iqbj (excluding the charging loads from stations), where pbj and qbj denote the real and reactive power, respectively.

210 PART | II Energy management g

g

g

Each bus j may also have distributed generation sj := pj + iqj . Let sj := pj + iqj denote the net complex power injection given by g s − sbj − sej if bus j supplies a station sj := jg sj − sbj otherwise where sej denotes the total charging load at bus j. We assume the base loads sbj g are given and the generations sj and charging loads sej are variables. We then summarize three representative linearization/convexification methods to model the power flows on the microgrid that are useful depending on network topologies and parameterization.

2.1.1 DC power flow equations Assumptions: 1. 2. 3. 4.

The bus voltage magnitude is constant as 1. Each line is lossless. Reactive power injections and flow are ignored. The bus phase angle difference across each line is small.

Given these standard assumptions for DC approximation, the generic AC power flow equations reduce to   Pjk = Pij + pj , j ∈ N (1a) k:(j,k)∈E

k:(i,j)∈E

Pjk = Bjk (θj − θk ),

j→k∈E

(1b)

where Bjk is the negative susceptance of line (j, k). Eq. (1a) enforces nodal power balance while Eq. (1b) defines line flows. This is the simplest linear model of power flows and it applies to systems where (a) the line resistance is negligible, and for normal operating points, (b) the bus phase angle difference across each line is small, and (c) the bus voltage magnitude is very close to 1 in the per-unit system. The complex notation of Eq. (1) is only a shorthand for a set of real equations in the real vector variables (p, P, θ ) := (pj , Pjk , θj , j, k ∈ N, (j, k) ∈ E).

2.1.2 Fix-point linearization of power flow equations Let IˆN+ := (Iˆj , j ∈ N+ ) be the vector of net current injections at all buses j ∈ N+ . The bus injection model of power flows can be written as IˆN+ = YN+ 0 V0 + YN+ N+ VN+ H N+ = diag(VN+ )IˆN+

(2a) (2b)

where Y isthe admittance matrix of the microgrid with Yij = −yij if i = j and Yii = j=i yij if i = j. Eq. (2a) imposes nodal balance of current (power)

Scheduling of EV battery swapping in microgrids Chapter | 8 211

while Eq. (2b) defines bus power injections. Substituting Eq. (2b) into Eq. (2a) to eliminate IˆN+ , we can attain the following fixed-point equation: −1 −1 −1 H (VN+ )sH VN+ = −YN + N+ YN+ 0 V0 + YN+ N+ diag N+

(3)

where the first term is a constant vector meaning zero-load voltage. Given a nominal operation point (Vˆ N+ , sˆN+ ) that is a solution to Eq. (3), we are able to linearize Eq. (2) in the following linear form based on one single iteration of Eq. (3): VN+ = J[pTN+ , qTN+ ]T + a

(4a)

−1 −1 −1 ˆ H −1 ˆ H where J := [YN (V N+ ), −iYN (V N+ )] and a := + N+ diag + N+ diag −1 − YN+ N+ YN+ 0 V0 . Let I := (Ijk , (j, k) ∈ E). It follows from [42] that |VN+ | and I can also be linearly approximated in terms of [pTN+ , qTN+ ]T :

|VN+ | = K[pTN+ , qTN+ ]T + b

(4b)

I = L[pN+ , qN+ ] + c

(4c)

s0 = D[pN+ , qN+ ] + d

(4d)

T

T

T

T

T

T

where K, L, D are also constant matrices dependent on Vˆ N+ and b, c, d are constant vectors. We refer readers to [42] for the detailed derivation and expressions of these parameters. The fixed-point linearization of power flow Eq. (4) interpolates between two power flow solutions (Vˆ N+ , sˆN+ ) and (a, 0), and is more computationally affordable than a lot of classical methods, for example, the first-order Taylor method. In most cases it also provides a better global approximation. The complex notation of Eq. (4) is only a shorthand for a set of real equations in the real vector variables (s, V, |V|, I) := (p, q, V, |V|, I) := (pj , qj , Vj , |Vj |, Ijk , j, k ∈ N, (j, k) ∈ E).

2.1.3 DistFlow equations and SOCP relaxation Suppose the microgrid is a radial network; the unique parent bus of each bus j (except bus 0) is indexed by i := ij . Define ljk := |Ijk |2 as the squared magnitude of the complex current from bus j to bus k and vj := |Vj |2 as the squared magnitude of the complex voltage phasor of bus j. We use the DistFlow equations proposed by Baran and Wu in [43] to model power flows on the network:  Sjk = Sij − zij lij + sj , j ∈ N (5a) k:(j,k)∈E 2 vj − vk = 2Re(zH jk Sjk ) − |zjk | ljk ,

vj ljk = |Sjk | , 2

j→k∈E

j→k∈E

(5b) (5c)

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The equations impose power balance at each bus in Eq. (5a), model the Ohm’s law in Eq. (5b), and define branch power flows in Eq. (5c). Note that Si0 := 0 and li0 := 0 when bus j = 0 is the root bus, and when bus j is a leaf node of G, all Sjk = 0 in Eq. (5a). The quantity zij lij is the loss on line (i, j), and hence Sij − zij lij is the receiving-end complex power at bus j from bus i. With the definition of v and l, Eqs. (5a), (5b) are both linear in variables. However, the DistFlow equations are still difficult to solve due to the nonconvex quadratic equality (5c). To deal with this nonconvexity, we adopt the recently developed second-order cone programming (SOCP) relaxation. Note that by relaxing the quadratic equality into inequality, that is,    2Pjk     vj ljk ≥ |Sjk |2 ←→   2Qjk  ≤ vj + ljk , j → k ∈ E vj − ljk  2 the nonconvex constraint (5c) is relaxed into a second-order cone. Specifically, we have the relaxed convex DistFlow equations:  Sjk = Sij − zij lij + sj , j ∈ N (6a) k:(j,k)∈E 2 vj − vk = 2Re(zH jk Sjk ) − |zjk | ljk ,

vj ljk ≥ |Sjk | , 2

j→k∈E

j→k∈E

(6b) (6c)

When we solve an OPF problem that satisfies Eq. (5), an alternative is to look at its relaxation with Eq. (6) replacing Eq. (5). If an optimal solution to the relaxation attains equality in Eq. (6c), then the solution is also feasible, and therefore optimal, for the original OPF problem. In this case, we say the SOCP relaxation is exact. Sufficient conditions are known that guarantee the exactness of the SOCP relaxation; see Refs. [44, 45] for a comprehensive tutorial and references therein. Even when these conditions are not satisfied, the SOCP relaxation for practical radial networks is still often exact, as confirmed also by our simulations in Section 5. Therefore, Eq. (6) is a computationally tractable power flow model for radial networks by assuming the underlying SOCP relaxation is exact. The complex notation of Eq. (6) is only a shorthand for a set of real equations in the real vector variables (s, v, l, S) := (p, q, v, l, P, Q) := (pj , qj , vj , ljk , Pjk , Qjk , j, k ∈ N, (j, k) ∈ E).

2.1.4 Operational constraints The operation of the microgrid must meet certain specifications. The voltage magnitudes must be maintained within stable regions: V j ≤ |Vj | ≤ V j

or

vj ≤ vj ≤ vj ,

j∈N

(7a)

Scheduling of EV battery swapping in microgrids Chapter | 8 213

where V j , vj and V j , vj are given lower and upper bounds on the (squared) voltage magnitude at bus j, respectively. The distributed real and reactive generations must satisfy pgj ≤ pj ≤ pj ,

g

g

j∈N

(7b)

g qj

g qj ,

j∈N

(7c)

qgj ≤ g

≤

g

where pgj , pj and qgj , qj are given lower and upper bounds on the real and reactive power generations at bus j, respectively. The thermal limit of line (j, k) must be satisfied: Pjk ≤ Pjk , or |Ijk | ≤ I jk , or |Sjk | ≤ Sjk ,

j→k∈E

(7d)

where Pjk , I jk , and Sjk denote different representations of the thermal limit of line (j, k). The model is quite general. If a quantity is known and fixed, then we set both its upper and lower bounds to the given quantity, for example, the voltage of the g substation bus. If there is no distributed generation at bus j, then pj = pgj = g

qj = qgj = 0.

2.2 Battery swapping scheduling Let Nw := {1, 2, . . . , Nw } ⊆ N denote the set of buses that supply electricity to stations, whose locations are fixed and known. For simplicity, assume there is only one station (or an ensemble of multiple stations) connected to each bus j ∈ Nw and we use j to index both the bus and the station. The batteries at each station are either charging at the constant rated power r or already fully charged and ready for swapping. Denote the total numbers of batteries and fully charged batteries at station j at the beginning of the current control interval by Mj and mj , respectively. Note that Mj is always fixed while mj is observed in each interval. Let A := {1, 2, . . . , A} denote the set of EVs in the service area that require battery swapping in the current interval. Denote their states of charge as (ca , a ∈ A). Let uaj represent the assignment:  1, if station j is assigned to EV a uaj = 0, otherwise and let u := (uaj , a ∈ A, j ∈ Nw ) denote the vector of assignments. The assignments u satisfy the following conditions:  uaj = 1, a ∈ A

(8a)

j∈Nw

 a∈A

uaj ≤ mj ,

j ∈ Nw

(8b)

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that is, exactly one station is assigned to every EV and every assigned station has enough fully charged batteries. The system knows the current location of every EV a and therefore can calculate the distance daj from its current location to the assigned station j, for example, by resorting to a routing application (like Google Maps). In the electric taxi case, if EV a is not currently carrying passengers and can go to swap its battery immediately, then daj is the travel distance from its current location to station j. If EV a must first complete its current passenger run before going to station j, then daj is the travel distance from its current location to the destination of its passengers and then to station j. The assigned station j must be within each EV a’s driving range, that is, uaj daj ≤ γa ca ,

j ∈ Nw , a ∈ A

(8c)

where ca is EV a’s current state of charge and γa is its driving range per unit state of charge. Denote the constraint set for u by U := {u ∈ {0, 1}ANw : u satisfies Eq. (8)} Assumption 1. U is nonempty. Under Assumption 1, there are enough fully charged batteries in the system for all EVs in A in the current interval. This can be enforced when choosing the candidate set A of EVs for battery swapping, for example, for EVs that can reach the same subset of stations, ranking them according to their states of charge, scheduling as many EVs as possible in an increasing order, upper limited by the number of fully charged batteries at those stations, and postponing remaining EVs to the next interval. Since every EV produces a depleted battery that needs to be charged at the rated power r, we can express the net power injection sj = pj + iqj at bus j in terms of the assignments u as  g    pj − pbj − r Mj − mj + a∈A uaj , j ∈ Nw (9a) pj = g pj − pbj , j ∈ N \ Nw g

qj = qj − qbj ,

j∈N

(9b)

Let fj : R → R model the generation cost at bus j, for example, for a distributed gas generator. We assume all fj s are increasing and convex functions, for example, quadratic functions [17–19]. Denote the power flow variables by φ. Given the previous three network models, φ ∈ {(p, P, θ ), (s, V, |V|, I), (s, v, l, S)}, and corresponds to the model used. We are interested in the following optimization problem:   g f (p ) + α daj uaj (10) min j j g u,s ,φ

j∈N

a∈A j∈Nw

s.t. Eq. (1) or (4) or Eqs. (6)–(9),

u ∈ {0, 1}ANw

Scheduling of EV battery swapping in microgrids Chapter | 8 215

  where a∈A j∈Nw uaj daj is the total travel distance of EVs and α > 0 is a weight that makes electricity generation cost and travel distance comparable, for example, the travel cost per unit of distance. The network model can be properly chosen based on the topologies and parameterization of the microgrid. Fixing any assignments u ∈ {0, 1}ANw , the problem (10) is a convex problem.

3 Centralized solution The joint battery swapping and OPF problem (10) is generally difficult to solve because the assignments u are discrete. Our centralized solution applies generalized Benders decomposition to deal with the discrete variables in Eq. (10). Benders decomposition was first proposed in [46] for problems where, when a subset of the variables are fixed, the remaining subproblem is a linear program. It is extended in [47] to problems where the remaining subproblem is a convex program. We now apply it to solving Eq. (10). Denote the continuous variables by x := (sg , φ) while the discrete variables are u. Denote the objective function by F(x, u) :=



g

fj (pj ) + α

j∈N



daj uaj

a∈A j∈Nw

Given any u, F(x, u) is convex in x since fj s are assumed to be strictly convex. Denote the constraint set for x by X := {x ∈ R(|N|+|φ|) : x satisfies Eq. (7) and one of Eq. (1), (4) or (6)} and the constraints (9) on (x, u) by G(x, u) = 0 while u ∈ U. Then the relaxation (10) takes the standard form for generalized Benders decomposition: min F(x, u)

(11)

x,u

s.t. G(x, u) = 0,

x ∈ X, u ∈ U

where F: R(|N|+|φ|) × {0, 1}ANw → R is a scalar-valued function, and G: R(|N|+|φ|) × {0, 1}ANw → R2|N| is a vector-valued constraint function. Fixing any u ∈ U, Eq. (11) is a convex subproblem in x. We now apply generalized Benders decomposition of [47] to Eq. (11). Write Eq. (11) in the following equivalent form: min W(u) s.t. u ∈ U ∩ W u

(12a)

where, for a fixed value of u, W(u) := min x∈X

s.t.

F(x, u) G(x, u) = 0

(12b)

216 PART | II Energy management

and W := {u : G(x, u) = 0

for some x ∈ X}

(12c)

The problem (12b), called the slave problem, is convex and much easier to solve than Eq. (11). The set W consists of all us for which Eq. (12b) is feasible and hence U ∩ W is the projection of the feasible region of Eq. (11) onto the u-space. The central idea of generalized Benders decomposition is to invoke the dual representations of W(u) and W to derive the following equivalent problem to Eq. (12) (see [47, Theorems 2.2 and 2.3]):  

T min sup min F(x, u) + μ G(x, u) u∈U μ∈R2|N|

x∈X



s.t. min λT G(x, u) = 0, x∈X

∀λ ∈ R2|N|

Note that we assume that Slater’s condition is always satisfied. Here λ and μ are Lagrangian multiplier vectors for W and W(u), respectively. This problem is equivalent to min

u∈U,u0 ∈R

u0



s.t. u0 ≥ min F(x, u) + μT G(x, u) , x∈X

min λT G(x, u) = 0, ∀λ ∈ R2|N|

(13) ∀μ ∈ R2|N|

x∈X

In summary, the series of manipulations have transformed the relaxation (10) into the master problem (13). Since Eq. (13) has uncountably many constraints with all possible λs and μs, it is neither practical nor necessary to enumerate all constraints in solving Eq. (13). Generalized Benders decomposition starts by solving a relaxed version of Eq. (13) that ignores all but a few constraints. If a solution to the relaxed version of Eq. (13) satisfies all the ignored constraints, then it is an optimal solution to Eq. (13) and the algorithm terminates. Otherwise, the solution process of the relaxed version of Eq. (13) will identify one μ or λ for which the corresponding constraint is violated. The violated constraint is then added to the relaxed version of Eq. (13), and the cycle repeats. Specifically, the generalized Benders decomposition algorithm for Eq. (10) (or equivalently, Eq. 11) is as follows. ●

Step 1. Pick any u¯ ∈ U ∩ W. Solve the dual problem of Eq. (12b) with u = u¯ to obtain an optimal Lagrangian multiplier vector μ. ¯ Let nμ = 1, nλ = 0, ¯ and UBD = W(¯u), where nμ , nλ are counters for the two types μ1 = μ, of constraints in Eq. (13), and UBD denotes an upper bound on the optimal value of Eq. (11).

Scheduling of EV battery swapping in microgrids Chapter | 8 217

●

Step 2. Solve the current relaxed master problem: min

u∈U,u0 ∈R

u0

(14)



 T s.t. u0 ≥ min F(x, u) + μi G(x, u) , x∈X

 T i min λ G(x, u) = 0, i = 1, . . . , nλ

i = 1, . . . , nμ

x∈X

●

Let (u, ˆ uˆ0 ) be the optimal solution to Eq. (14). Clearly uˆ0 is a lower bound on the optimal value of Eq. (11) since the constraints in Eq. (13) are relaxed to a smaller set of constraints in Eq. (14). Terminate the algorithm if UBD − uˆ0 ≤ , where  > 0 is a sufficiently small threshold. Step 3. Solve the dual problem of Eq. (12b) with u = u. ˆ The solution falls into the following two cases. 1. Step 3a. The dual problem of Eq. (12b) has a bounded solution μ, ˆ that is, W(u) ˆ is feasible and finite. Let UBD = min{UBD, W(u)}. ˆ Terminate the algorithm if UBD − uˆ0 ≤ . Otherwise, increase nμ by 1 and let ˆ Return to Step 2. μnμ = μ. 2. Step 3b. The dual problem of Eq. (12b) has an unbounded solution, that is, W(u) ˆ is infeasible. Determine λˆ through a feasibility check problem ˆ Return to Step 2. and its dual [48]. Increase nλ by 1 and let λnλ = λ.

We make three remarks. First, the slave problem (12b) is convex and hence can generally be solved efficiently. The relaxed master problem (14) involves discrete variables and is generally nonconvex, but it is much simpler than the original problem (11). Second, for our problem, Eq. (14) turns out to be a mixedinteger linear program in essence because both F and G are separable functions in (x, u) of the form F(x, u) =: F1 (x) + F2 (u) G(x, u) =: G1 (x) + G2 (u) where F2 and G2 are both linear in u. Indeed the constraints in Eq. (14) are

 T  T u0 − F2 (u) − μi G2 (u) ≥ min F1 (x) + μi G1 (x) , i = 1, . . . , nμ x∈X

 i T  T λ G2 (u) = − min λi G1 (x), x∈X

i = 1, . . . , nλ

where the left-hand side is linear in u and the right-hand side is independent of u. Hence, in each iteration, the algorithm solves Eq. (14), which is a simplified mixed-integer linear program (always with only one continuous auxiliary variable), and Eq. (12b), which is a convex program. Third, every time Step 2 is entered, one additional constraint is added to Eq. (14). This generally makes Eq. (14) harder to compute, but also a better approximation of Eq. (13). It is proved in [47, Theorem 2.4] that the algorithm will terminate in finite steps since U is discrete and finite.

218 PART | II Energy management

4 Distributed solutions 4.1 Relaxations The joint battery swapping and OPF problem (10) is computationally difficult since the assignment variables u are binary. To deal with this difficulty, we use generalized Benders decomposition in Section 3. This approach could compute an optimal solution in reasonable time but the computation is centralized and is suitable only when a single organization, for example, the State Grid in China, operates all of the microgrid, stations, and EVs. This section develops distributed solutions that are suitable for systems where these three are operated by separate entities that do not share their private information. To this end, we relax the binary assignment variables u to real variables u ∈ [0, 1]ANw . The constraints (8) are then replaced by uaj = 0 if daj > γa ca , j ∈ Nw , a ∈ A  uaj = 1, a ∈ A

(15a) (15b)

j∈Nw



uaj ≤ mj ,

j ∈ Nw

(15c)

a∈A

and we change to solve the following relaxation of Eq. (10):   g f (p ) + α daj uaj min j j g u,s ,φ

j∈N

(16)

a∈A j∈Nw

s.t. Eq. (1) or (4) or (6), (7), (9), (15),

u ∈ {0, 1}ANw

This problem has a convex objective and convex quadratic constraints. After an optimal solution (x* , u* ) to Eq. (16) is obtained, we discretize u*aj into {0, 1}, for example, by setting for each EV a a single large u*aj to 1 and the rest to 0 heuristically. An alternative is to randomize the station assignments using u* as a probability distribution. Whichever method is employed, it should guarantee the discretized station assignments are feasible. As we will show later, the discretization is readily implementable and achieves binary station assignments close to optimum.

4.2 Distributed solution via ADMM The relaxation (16) decomposes naturally into two subproblems, one on station assignments over u and the other on OPF over (sg , φ). The station assignment subproblem will be solved by a station operator that operates the network of stations. The OPF subproblem will be solved by a utility company. Our goal is to design a distributed algorithm for them to solve jointly Eq. (16) without sharing their private information.

Scheduling of EV battery swapping in microgrids Chapter | 8 219

These two subproblems are coupled only (9a) where  the utility  in Eq.  company needs the charging load sej = r Mj − mj + a∈A uaj of station j in order to compute the net real power injection pj . This quantity depends on the total number of EVs that each station j is assigned to and is computed by the station operator. Their computation can be decoupled by introducing an the utility company’s auxiliary variable wj at each bus (station)  j that represents estimate of the quantity r Mj − mj + a∈A uaj , and requiring that they be equal at optimality. Specifically, recall the station assignmentvariables u, and  denote the power flow variables by x := (w, sg , φ) where w := r Mj − mj + a∈A uaj , j ∈ Nw . Separate the objective function by defining  g fj (pj ) f (x) := j∈N

g(u) := α



daj uaj

a∈A j∈Nw

Replace the coupling constraints (9) by constraints local to bus j:  g pj − pbj − wj , j ∈ Nw pj = g pj − pbj , j ∈ N/Nw g

qj = qj − qbj ,

j∈N

(17a) (17b)

Denote the local constraint set for x by X := x ∈ R(|Nw |+|N|+|φ|) : x satisfies Eqs. (7), (17) and one of Eq. (1), (4)

or (6) Denote the local constraint set for u by

U := u ∈ RANw : u satisfies Eq. (15)  To simplify the notation, define uj := a∈A uaj for j ∈ Nw . Then the relaxation (16) is equivalent to min f (x) + g(u)

(18a)

x,u

s.t. x ∈ X, u ∈ U   wj = r Mj − mj + uj ,

(18b) j ∈ Nw

(18c)

We now apply ADMM to Eq. (18). Let λ := (λj , j ∈ Nw ) be the Lagrange multiplier vector corresponding to the current coupling constraint (18c), and define the augmented Lagrangian: Lρ (x, u, λ) := f (x) + g(u) + hρ (w, u, λ)

(19a)

220 PART | II Energy management

where hρ depends on (x, u) only through (wj , uj , j ∈ Nw ):      ρ  hρ (w, u, λ) := λj [wj − r Mj − mj + uj ] + [wj − r Mj − mj + uj ]2 2 j∈Nw

j∈Nw

(19b) and ρ is the step size for dual variable λ updates. The standard ADMM procedure is iteratively and sequentially to update (x, u, λ): for n = 0, 1, . . ., x(n + 1) := arg min f (x) + hρ (w, u(n), λ(n))

(20a)

u(n + 1) := arg min g(u) + hρ (w(n + 1), u, λ(n))

(20b)

x∈X

u∈U

λj (n + 1) := λj (n) + ρ[wj (n + 1) − r(Mj − mj + uj (n + 1))],

j ∈ Nw (20c)

Remark 1. 1. The x-update (20a) is carried out by the utility company and involves minimizing a convex objective with convex quadratic constraints. The (u, λ)updates (20b), (20c) are carried out by the station operator and the u-update minimizes a convex quadratic objective with linear constraints. Both can be efficiently solved. 2. The x-update by the utility company in iteration n + 1 needs (u(n), λ(n)) from the station operator. From Eq. (19b), the station operator does not need to communicate the detailed assignments u(n) = (uaj (n), a ∈ A, j ∈ Nw ) to the utility company, but only the charging load sej = r(Mj − mj + uj (n)) of each station j. 3. The (u, λ)-updates by the station operator in iteration n + 1 need the utility company’s estimate w(n + 1) of (r(Mj − mj + uj (n + 1)), j ∈ Nw ). 4. The reason why the x-update by the utility company needs (uj (n), j ∈ Nw ) and the u-update by the station operator needs w(n + 1) lies in the (quadratic) regularization term in hρ . This becomes unnecessary for the dual decomposition approach in Section 4.3 without the regularization term. The communication structure is illustrated in Fig. 2. In particular, private information of the utility company, such as network parameters (zjk , (j, k) ∈ E), network states (sg (n), φ(n)), cost functions f , and operational constraints, as well as private information of the station operator, such as the total numbers of batteries (Mj , j ∈ Nw ), the numbers of available fully charged batteries (mj , j ∈ Nw ), how many EVs or where they are or their states of charge, and the detailed assignments u(n), does not need to be communicated. When the cost functions fj are closed, proper, and convex, and Lρ (x, u, λ) has a saddle point, the ADMM iteration (20) converges in that, for any j ∈ Nw , the mismatch |wj (n) − r(Mj − mj + uj (n))| → 0 and the objective function f (x(n)) + g(u(n)) converges to its minimum value [49]. This does not automatically guarantee that (x(n), u(n)) converges to an optimal solution

Scheduling of EV battery swapping in microgrids Chapter | 8 221

Utility company:

updates

Station operator: updates Fig. 2

Communication between utility company and station operator.

to Eq. (16). In theory, ADMM may converge and circulate around the set of optimal solutions, but never reach one. In practice, a solution within a given error tolerance is acceptable. If (x(n), u(n)) indeed converges to a primal optimal solution (x* , u* ), u* may generally not be binary. We can use a heuristic to derive binary station assignments from u* , as mentioned earlier. Fortunately, the following result shows that the number of EVs with nonbinary assignments is bounded and small in u* . See Appendix for its proof. Theorem 1. It is always possible to find an optimal solution (x* , u* ) to the relaxation (16) in which the number of EVs a with u*aj < 1 for any j ∈ Nw is at most Nw (Nw − 1)/2. In practice, the number Nw of stations is much smaller than the number A of EVs that request battery swapping, and hence the number of nonbinary assignments that need to be discretized will be small. Simulations in Section 5 further suggest that the discretized assignments are close to optimum.

4.3 Distributed solution via dual decomposition The ADMM-based solution assumes the station operator directly controls the station assignments to all EVs. This requires that the station operator know the locations (daj ), states of charge (ca ), and performance (γa ) of EVs. Moreover, the charging load sej = r(Mj − mj + uj (n)) of each station j needs to be provided to the utility company. We now present another solution based on dual decomposition that is more suitable in situations where it is undesirable or inconvenient to share private information between the utility company, the station operator, and EVs. In the original relaxation (16), the update of the net power injections pj in Eq. (9) by the utility company involves uj which is updated by the station operator. These two computations are decoupled in the ADMM-based solution by introducing an auxiliary variable wj for each j ∈ Nw at the utility company and relaxing the constraint wj = r(Mj − mj + uj ). In addition, the station assignments u must satisfy uj ≤ mj in Eq. (15c). This is enforced in the ADMM-based solution by the station operator that computes u for all EVs. To distribute fully the computation to individual EVs, we dualize uj ≤ mj as well.

222 PART | II Energy management

Let λ := (λj , j ∈ Nw ) and μ := (μj ≥ 0, j ∈ Nw ) be the Lagrange multiplier vectors for the constraints wj = r(Mj −mj +uj ) and uj ≤ mj , j ∈ Nw , respectively. Intuitively, w and λ decouple the computation of the utility company and that of individual EVs through coordination with the station operator. Additionally, μ decouples and coordinates all EVs’ decisions so that EVs do not need direct communication among themselves to ensure that their decisions uaj collectively satisfy uj ≤ mj . Consider the Lagrangian of Eq. (18) with these two sets of constraints relaxed:   λj (wj −r(Mj −mj +uj ))+ μj (uj −mj ) (21) L(x, u, λ, μ) := f (x)+g(u)+ j∈Nw

j∈Nw

and the dual problem of Eq. (18): max D(λ, μ) :=

λ,μ≥0

min

ˆ x∈X,u∈U

L(x, u, λ, μ)

where the constraint set Uˆ on u is

Uˆ := u ∈ RANw : u satisfies Eqs. (15a), (15b) Let ua := (uaj , j ∈ Nw ) denote the vector of EV as decision on which station to swap its battery. Then the dual problem is separable in power flow variables x as well as individual EVs’ decisions ua :  Ua (λ, μ) (22a) D(λ, μ) = V(λ) + a∈A

where the problem V(λ) solved by the utility company is ⎛ ⎞  λj wj ⎠ V(λ) := min ⎝f (x) + x∈X

(22b)

j∈Nw

and the problem Ua (λ) solved by each individual EV a is   αdaj − rλj + μj uaj Ua (λ, μ) := min

(22c)

ˆa ua ∈U j∈N

w

where the constraint set Uˆ a on ua is ⎧ uaj ∈ [0, 1] , j ∈ Nw ⎪ ⎪ ⎪ ⎨ u aj = 0 if daj > γa ca , j ∈ Nw Uˆ a := ua ∈ RNw :  ⎪ ⎪ uaj = 1 ⎪ ⎩ j∈Nw

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

Note that Eq. (22c) has closed-form solutions. For instance, if there exists a unique optimal solution to Ua (λ, μ), that is, for any EV a there is a unique j*a (λ, μ) defined as j*a (λ, μ) := arg

min {αdaj − rλj + μj }

j: daj ≤γa ca

Scheduling of EV battery swapping in microgrids Chapter | 8 223

then the optimal solution can be uniquely determined as  1, if j = j*a (λ, μ) u*aj (λ, μ) := 0, if j = j*a (λ, μ) that is, it simply chooses the unique station j*a within EV as driving range that has the minimum cost αdaj − rλj + μj . From Eq. (21), the standard dual algorithm for solving Eq. (18) is, for j ∈ Nw , λj (n + 1) := λj (n) + ρ1 (n)[wj (n) − r(Mj − mj + uj (n))] μj (n + 1) := max{ μj (n) + ρ2 (n)(uj (n) − mj ), 0 }

(23a) (23b)

where ρ1 (n), ρ2 (n) > 0 are diminishing step sizes and, from Eq. (22), we have ⎛ ⎞  λj (n)wj ⎠ x(n) := arg min ⎝f (x) + (23c) x∈X

and, for a ∈ A, ua (n) := arg min

j∈Nw



ˆa ua ∈U j∈N

 αdaj − rλj (n) + μj (n) uaj

(23d)

w

Remark 2. 1. The x-update (23c) is carried out by the utility company and involves minimizing a convex objective with convex quadratic constraints. The only information that is nonlocal to the utility company for its x-update is one of the dual variables λ(n) computed by the station operator. 2. The ua -update (23d) is carried out by each individual EV. Each EV requires both the dual variables (λ(n), μ(n)) from the station operator for its update. 3. The dual updates (23a), (23b) are carried out by the station operator which uses a (sub)gradient ascent algorithm to solve the dual problem maxλ,μ≥0 D(λ, μ). It requires w(n) from the utility company and individual decisions ua (n) from EVs a. The communication structure is illustrated in Fig. 3. In particular, EVs are completely decoupled from the utility company and among themselves. Unlike the ADMM-based solution, the station operator knows only the battery swapping decisions of EVs, but not their private information such as locations (daj ), states of charge (ca ), or performance (γa ). Since the relaxation (16) is convex, strong duality holds if Slater’s condition is satisfied. Then, when the above (sub)gradient algorithm converges to a dual optimal solution (λ* , μ* ), any primal optimal point is also a solution to the corresponding x-update (23c) and ua -update (23d) [50, 51]. Suppose (x(n), ua (n), a ∈ A) indeed converges to a primal optimal solution (x* , u*a , a ∈ A), then typically (u*a , a ∈ A) is not binary. However, the bound in Theorem 1 still holds that guarantees easy discretization and suggests that the final discretized stations assignments are close to optimum.

224 PART | II Energy management

Utility company:

updates

Station operator: updates Broadcasts to EVs

EV 1: updates Fig. 3

• • • •

EV A: updates

Communication between utility company, station operator, and EVs.

Remark 3. The two solutions have their own advantages and can be adapted to different application scenarios. The ADMM-based solution requires a station operator that is trustworthy and can access EVs’ private information. Since the station operator optimizes station assignments on behalf of all EVs, no computation is required on each EV, and meanwhile communication is only required between the station operator and the utility company. In contrast, the solution based on dual decomposition does not require sharing EVs’ private information with the station operator. It does, however, necessitate computation capabilities on all EVs. In addition, communication is needed both between the station operator and the utility company and between the station operator and each EV.

5 Numerical results 5.1 Setup We now evaluate the proposed algorithms through simulations using a 56-bus test system from SCE with a radial structure. Therefore, we will adopt the DistFlow equations to model power flows on the network. Similar performance can be anticipated on the other two models and is therefore skipped here. A maximum voltage deviation of 0.05 p.u. is allowed and all line capacities are set to infinity. More details about the test system can be found in [52, Fig. 2 and Table I]. We add four distributed generators and four stations at different buses, with parameters given in Table 1A. Note that the units of the real power, reactive power, cost, distance, and weight are MW, Mvar, $, km, and $/km, respectively. The four stations are assumed to be uniformly located in a 4 km × 4 km square area supplied by the test system, as shown in Table 1B. Two cases with different

Scheduling of EV battery swapping in microgrids Chapter | 8 225

TABLE 1 Setup. (A) Distributed generator Bus

g pj

pj

g

qj

g

qj

g

Cost function

1

4

0

2

−2

0.3pg 2 + 30pg

4

2.5

0

1.5

−1.5

0.1pg 2 + 20pg

26

2.5

0

1.5

−1.5

0.1pg 2 + 20pg

34

2.5

0

1.5

−1.5

0.1pg 2 + 20pg

(B) Station Bus

Location

Mj

mj

5

(1, 1)

mj

(i) A; (ii) A/2

16

(3, 1)

mj

(i) A; (ii) A/10

31

(1, 3)

mj

(i) A; (ii) A/4

43

(3, 3)

mj

(i) A; (ii) A/4

mj s will be tested for illustration purposes. Suppose in a certain control interval, there are A EVs that request battery swapping (A will vary in our case studies). Their current locations are generated in a uniformly random manner within the square area while their destinations are ignored. We use the Euclidean distance for daj . We assume all EVs have sufficient battery energy to reach any of the four stations, which means that Eq. (8c) is readily satisfied. The extension to the general case where each EV has a limited driving range and can only reach some of the stations is straightforward. The constant charging rate is r = 0.01 MW [53] at all stations. We set the weight α to be 0.02 $/km first [54]. For each case, we conduct 10 simulation runs with random EV locations. All numerical tests are run on a laptop with Intel Core i7-3632QM [email protected] GHz, 8 GB RAM, and 64-bit Windows 10 OS.

5.2 Centralized solution We first fix Mj = mj = A, j ∈ Nw , which means that in each station, batteries are all fully charged and sufficient to serve all EVs. The centralized solution is applied to two test cases with different numbers of EVs, and scalability analysis follows.

5.2.1 Nearest-station policy Without optimization, the default policy is that all EVs head for their nearest stations to swap batteries. This is shown in Figs. 4A and 5A for two specific

226 PART | II Energy management Station 1

Station 2

Station 3

Station 4

EV

4

3

3 y (km)

y (km)

EV

4

2

1

0

Station 2

Station 3

Station 4

2

1

0

1

(A)

2

3

0

4

1

2

3

4

x (km)

#EVs = 100. (A) Nearest-station policy and (B) optimal assignments.

EV

Station 1

Station 2

Station 3

EV

Station 4

4

3

3

y (km)

4

2

Station 1

Station 2

Station 3

Station 4

2

1

1

0

0

(B)

x (km)

Fig. 4

y (km)

Station 1

0

(A) Fig. 5

1

2 x (km)

3

4

0

(B)

0

1

2 x (km)

3

4

#EVs = 300. (A) Nearest-station policy and (B) optimal assignments.

cases with 100 and 300 EVs, respectively. In practice this myopic policy can lead to a shortage in fully charged batteries at a station if many EVs cluster around that station due to correlations in traffic patterns. Moreover, it can cause voltage instability: the voltage magnitudes of some buses drop below the threshold 0.95 p.u. in the 300-EV case, as shown in Table 2, where the last column exhibits the resulting charging load at each bus.

5.2.2 Optimal assignments Figs. 4B and 5B show the optimal assignments computed using the centralized solution for the previous two cases, respectively. The nearest stations are not assigned to some of the EVs (highlighted with thicker circles in the figures) when grid operational constraints such as voltage stability are taken into account. The number of such EVs is larger in the 300-EV case than that in the 100-EV case. The tradeoff between the EVs’ travel distance and electricity generation cost is optimized. The OPF results of the 300-EV case are listed in Table 3 (compare

Scheduling of EV battery swapping in microgrids Chapter | 8 227

TABLE 2 Partial bus data under nearest-station policy (300 EVs). g

g



Bus

|Vj | (p.u.)

pj

qj

r

1

1.050

0.571

0.000

/

4

1.047

2.500

0.663

/

5

1.031

/

/

0.660

a∈A uaj

16

0.941

/

/

0.700

18

0.948

/

/

/

19

0.944

/

/

/

26

1.050

2.500

0.410

/

31

1.020

/

/

0.830

34

1.044

2.500

1.500

/

43

1.015

/

/

0.810

TABLE 3 Partial bus data under optimal assignments (300 EVs). g

g



Bus

|Vj | (p.u.)

pj

qj

r

1

1.050

0.520

0.000

/

4

1.048

2.500

0.590

/

5

1.025

/

/

0.990

15

0.981

/

/

/

16

0.974

/

/

0.300

17

0.980

/

/

/

18

0.973

/

/

/

19

0.969

/

/

/

26

1.050

2.500

0.439

/

31

1.019

/

/

0.840

34

1.044

2.500

1.500

/

43

1.013

/

/

0.870

a∈A uaj

with Table 2). As we can see from Table 3, the outputs (2.500 MW) of the distributed generators at buses 4, 26, and 34 have reached their full capacity (2.5 MW) while the injection (0.520 MW) at bus 1 (the substation bus) is far from its capacity (4 MW). This is consistent with our intuition that distributed

228 PART | II Energy management

generations that are closer to users and potentially cheaper than power from the transmission grid are favored in OPF. Under the optimal assignments, the deviations of voltages from their nominal value are all less than 5%.

5.2.3 Optimality of generalized Benders decomposition The upper and lower bounds on the optimal objective values for the previous two cases are plotted in Fig. 6 as the algorithm iterates between the master and slave problems. More iterations are required for larger-scale cases where the algorithm usually struggles longer to obtain an initial feasible solution. Once a feasible solution is found, the gap between the upper and lower bounds starts to shrink rapidly and the convergence to optimality is achieved within a few iterations. 5.2.4 Exactness of SOCP relaxation We check whether the solution computed by generalized Benders decomposition attains equality in Eq. (6c), that is, whether the solution satisfies power flow equations and is implementable. Our result confirms the exactness of the SOCP relaxation for most cases we have tested on, including the previous two. Due to space limit, only partial data of the 300-EV case are shown in Table 4. In summary, SOCP relaxation and generalized Benders decomposition seem to be effective in solving exactly our joint battery swapping and OPF problem (10). 5.2.5 Computational effort To demonstrate the potential of the centralized solution for practical application, we check its required computational effort by counting its computation time for different numbers of EVs and stations, since the number of discrete variables in the optimization problem is the computational bottleneck. We use Gurobi to

300

180

120

60

0

(A) Fig. 6

Infinity Upper bound Lower bound

240 Objective value ($)

240 Objective value ($)

300

Infinity Upper bound Lower bound

180

120

60

0

1

2 Iteration number

3

0

4

(B)

0

1

2

3

4

5

6

7

8

9

Iteration number

Convergence of generalized Benders decomposition. (A) #EVs = 100. (B) #EVs = 300.

10

Scheduling of EV battery swapping in microgrids Chapter | 8 229

TABLE 4 Exactness of SOCP relaxation (partial results for 300 EVs). Bus From

To

vj ljk

|Sjk |2

Residual

1

2

0.271

0.271

0.000

2

3

0.006

0.006

0.000

2

4

0.202

0.202

0.000

4

5

1.369

1.369

0.000

4

6

0.005

0.005

0.000

4

7

1.952

1.952

0.000

7

8

1.691

1.691

0.000

8

9

0.009

0.009

0.000

8

10

1.269

1.269

0.000

10

11

1.092

1.092

0.000

Average computation time (s)

200

160

120

80

40

0

Fig. 7

0

100 200 300 400 500 600 700 800 900 1000 Number of EVs

Average computation time as a function of #EVs.

solve the master problem (integer programming) and SDPT3 to solve the slave problem (convex programming) on the MATLAB R2012b platform. On the one hand, Fig. 7 shows the average computation time required by the centralized solution to find a global optimum for different numbers of EVs,

230 PART | II Energy management

given the four fixed stations (note that each data point in Figs. 7–11 is an average over 10 simulation runs with random EV locations). On the other hand, we fix the number of EVs at 100 and scale up stations that are located at different randomly picked buses. Fig. 8 shows the average computation time required grows accordingly, but its sensitivity to the number of stations is moderate as the iterations that struggle for an initial feasible solution (recall Fig. 6) do not increase significantly when the number of EVs is fixed. Therefore, overall the required computational effort is desirable.

5.2.6 Benefit Fig. 9 displays the average relative reduction in the objective value with different αs using optimal assignments, compared with the nearest-station policy. Scheduling flexibility is enhanced with more EVs, thus improving the savings. In addition, α expresses the system’s relative emphasis on the two objective components. Clearly the smaller the weight α on EVs’ travel distance is, the more benefit optimal assignments provide over the nearest-station policy. However, Fig. 9 also suggests that the improvement is small, that is, the neareststation policy is good enough if it is implementable. The nearest-station policy is sometimes infeasible either when there are more EVs nearest to a station than fully charged batteries at that station or when some operational constraints of the microgrid are violated. In our case studies, infeasibility is mainly due to some voltages dropping below the allowable√lower limit. Define a metric voltage drop violation as VDV := j∈N max{ vj − √ vj , 0} to quantify the degree of voltage violation. Fig. 10 shows the average

Average computation time (s)

100

80

60

40

20

0

Fig. 8

0

4

8

12

16 20 24 28 Number of stations

Average computation time as a function of #stations.

32

36

40

Average relative reduction in obj. value (%)

Scheduling of EV battery swapping in microgrids Chapter | 8 231 0.35

alpha = 0.01 alpha = 0.02 alpha = 0.05 alpha = 0.1

0.3 0.25 0.2 0.15 0.1 0.05 0 40

80

120

160

200

Number of EVs Fig. 9

Average relative reduction in objective value.

Average VDV (p.u.)

0.2

0.15

0.1

0.05

0 240

Fig. 10

280

320 Number of EVs

360

400

Average VDV under nearest-station policy.

VDV for the number of EVs ranging from 240 to 400 under the nearest-station policy. The voltage violation becomes more severe when the number of EVs increases. It is also interesting to look at cases where there are more EVs nearest to a station than fully charged batteries that station can provide, which, as far as we know, are common in practice. We reset M1 = m1 = M2 = m2 = 12 A and M3 = m3 = M4 = m4 = 18 A to simulate these situations. Hence the total number of fully charged batteries in the system is 54 A. Fig. 11A shows, for each station, the

232 PART | II Energy management 50 Nearest−station policy Optimal assignments

Average number of unserved EVs

Average ratio of #EVs / #batteries (%)

250

200

150

100

50

0

(A)

1

2

3 Station

4

40

30

20

10

(B)

0 40

80

120 Number of EVs

160

200

Fig. 11 (A) Average ratio of the number of forthcoming EVs to that of fully charged batteries. (B) Average number of unserved EVs under nearest-station policy.

average ratio of the number of EVs which go to the station for battery swapping to that of fully charged batteries at the station, under both the nearest-station policy and optimal assignments. In total, 99.40% of station 1’s batteries, 50.60% of station 2’s batteries, and all the batteries at stations 3 and 4 are used under the optimal assignments, thus they have collectively served all A EVs. Under the nearest-station policy, however, only 51.55% and 48.89% of stations 1 and 2’s batteries, respectively (i.e., a total of around 12 A batteries) are used for swapping. At either of stations 3 and 4, the number of EVs is approximately double that of available fully charged batteries (192.61% and 205.62%, respectively). Fig. 11B shows the average number of unserved EVs under the nearest-station policy as a function of the total number of EVs. On average, approximately one in four EVs cannot be served at their nearest stations, mainly due to congestion at stations 3 and 4, while available fully charged batteries at stations 1 and 2 are not fully utilized.

5.3 Distributed solutions We first fix the number of EVs that request battery swapping as A = 400, and test the two cases with different mj s, listed in Table 1B, using our distributed solutions, followed by scalability analysis.

5.3.1 Convergence The convergence of ADMM in case (i) is demonstrated in Fig. 12. Fig. 12A and B shows, respectively, that the Lagrange multiplier vector λ and the residual of the relaxed equality constraint (18c) converge rapidly. Case (ii) behaves similarly. Each iteration that computes the three steps of Eq. (20) takes on average 0.477 s by Gurobi. For the dual decomposition algorithm, Fig. 13A and B shows the convergence of its two Lagrange multiplier vectors λ and μ, respectively, in case (ii). λ maintains the consensus between the utility

Scheduling of EV battery swapping in microgrids Chapter | 8 233 0.1

0 Station 1 Station 2 Station 3 Station 4

−5

−15

0 −0.1 Residual

Lambda

−10

−20

−0.3

−25

−0.4

−30

−0.5

−35

−0.6

−40

(A) Fig. 12

−0.7 10

20 30 Iteration number

40

50

Station 1 Station 2 Station 3 Station 4 10

(B)

−10

Mu

30

−20

0.01

20

−25

0.005

−30 10

0 209

−35

Fig. 13

50

Station 1 Station 2 Station 3 Station 4

40

−15

(A)

40

50

Station 1 Station 2 Station 3 Station 4

−5

−40

20 30 Iteration number

Convergence of ADMM. (A) λ. (B) Residual of relaxed (18c).

0

Lambda

−0.2

50

100 150 Iteration number

200

0

(B)

50

100 150 Iteration number

210

211

200

Convergence of dual decomposition. (A) λ. (B) μ.

company and EVs at convergence, and μ guarantees (15c) is satisfied when it converges. Dual decomposition usually takes more iterations to converge due to the additionally required coordination among all EVs. For case (i), results are similar except that μ remains 0 during computation as Eq. (15c) is always satisfied. Each iteration of the dual decomposition algorithm involves the centralized update of Eqs. (23a), (23b) and the parallelized computation of Eqs. (23c), (23d). Each iteration takes on average 0.212 s by Gurobi.

5.3.2 Suboptimality (comparison with centralized solution) In case (i), both algorithms obtain a solution in which the station assignments to two EVs, highlighted with thicker circles in Fig. 14A, are nonbinary: u242 = [0.707 0.293 0.000 0.000] and u367 = [0.230 0.000 0.770 0.000]. This is consistent with Theorem 1. If we simply round u243 and u367 to binary values, the resulting solution turns out to coincide with a globally optimal solution computed using the centralized solution.

234 PART | II Energy management Station 1

Station 2

Station 3

EV

Station 4

4

3

3

y (km)

y (km)

EV

4

2

Station 3

Station 4

2

0

0 0

1

(A) Fig. 14

2 x (km)

3

0

4

1

2 x (km)

(B)

3

4

Suboptimality in different cases. (A) Case (i). (B) Case (ii).

140

Average computation time (s)

100

Average computation time (s)

Station 2

1

1

80

60

40

20

120 100 80 60 40 20 0

0 0 100 200 300 400 500 600 700 800 900 1000

(A)

Station 1

Number of EVs

(B)

0

4

8

12

16

20

24

28

32

36

40

Number of stations

Fig. 15 Average computation time of ADMM. (A) As a function of #EVs. (B) As a function of #stations.

In case (ii), we reduce available fully charged batteries at each station to activate Eq. (15c). Fig. 14B shows the solution achieved by both algorithms. The solution turns out to be globally optimal for the original problem (10); in particular, all station assignments are binary. EVs, to which the station assignments are altered due to the bound imposed on battery availability of each station, are highlighted with thicker circles in Fig. 14B. The intuition is that an active (Eq. 15c) can sometimes help eliminate nonbinary assignments to EVs. This is often the case in practice where battery availability is uneven across stations.

5.3.3 Exactness of SOCP relaxation In most cases that we have simulated, including cases reported here, the SOCP relaxation is exact, that is, the solutions computed by the two distributed algorithms attain equality in Eq. (6c) and therefore satisfy power flow equations. Partial data for case (ii) are listed in Table 5.

Scheduling of EV battery swapping in microgrids Chapter | 8 235

TABLE 5 Exactness of SOCP relaxation (partial results for case (ii)). Bus From

To

vj ljk

|Sjk |2

Residual

1

2

2.582

2.582

0.000

2

3

0.006

0.006

0.000

2

4

2.336

2.336

0.000

4

5

3.413

3.413

0.000

4

6

0.005

0.005

0.000

4

7

2.276

2.276

0.000

7

8

1.984

1.984

0.000

8

9

0.009

0.009

0.000

8

10

1.518

1.518

0.000

10

11

1.318

1.318

0.000

5.3.4 Scalability We follow the same principle earlier for the centralized solution to demonstrate the scalability of the two distributed solutions, that is, we first augment the number of EVs while the number of stations is fixed and then turn it the other way round. The computation time that is shown in Figs. 15 and 16 is averaged over 10 simulation runs with randomly generated cases. Approximately, the computational effort of both solutions increases linearly as EVs (or stations) scale up. Compared with the centralized solution, the required computation time

160

140

Average computation time (s)

Average computation time (s)

160

120 100 80 60 40 20

140 120 100

0

Number of EVs

60 40 20 0

0 100 200 300 400 500 600 700 800 900 1000

(A)

80

0

(B)

4

8

12

16

20

24

28

32

36

40

Number of stations

Fig. 16 Average computation time of dual decomposition. (A) As a function of #EVs. (B) As a function of #stations.

236 PART | II Energy management

of the distributed solutions is less sensitive to the EV scale, which is intuitive, but turns out to be more sensitive to the station scale. This results from the fact that the consensus that the distributed solutions strive toward has to be achieved at each station. Generally, more iterations are needed as more stations are involved.

6 Concluding remarks 6.1 Summary We formulate an optimal scheduling problem for battery swapping in a microgrid that assigns to each EV a best station to swap its depleted battery based on its current location and state of charge. The schedule aims to minimize a weighted sum of EVs’ travel distance and electricity generation cost over both station assignments and power flow variables, subject to EV range constraints, grid operational constraints, and AC power flow equations. Three representative linearization or convex relaxation methods are discussed for modeling the power flows on the microgrid, based on which we then propose both centralized and distributed solutions to handle the binary nature of station assignments. The centralized solution is applicable to vertically integrated systems where global information and controllability are available. The distributed solutions are more suitable for systems where the distribution grid, stations, and EVs are operated by separate entities that do not share their private information. They allow these entities to make individual decisions but coordinate through privacy-preserving information exchanges. Numerical case studies on the SCE 56-bus test system validate our analysis and reveal some interesting results in potential practical application.

6.2 Model limitations First, Assumption 1 is imposed by choosing a proper candidate set A of EVs when there is overwhelming demand of battery swapping, which significantly eases the model complexity at the sacrifice of a little performance. It will be interesting to model further the waiting cost of EVs when they cannot be immediately served at stations. Second, optimizing charging rates across intervals can be integrated to form a multiinterval scheduling problem if a good estimate of future information is available. Then it is worth evaluating the value of future information in improving the overall performance.

Appendix: Proof of Theorem 1 We refer to EV a as a critical EV if its station assignment satisfies uaj < 1 for all j ∈ Nw . We first show the following lemma and then prove Theorem 1. Let (u, y) := (u, sg , φ).

Scheduling of EV battery swapping in microgrids Chapter | 8 237

Lemma 1. It is always possible to find an optimal solution (u* , y* ) to the relaxation (16) where no critical EVs share two stations, that is, there do not exist a, b ∈ A and j, k ∈ Nw such that u*aj , u*ak , u*bj , u*bk > 0. Proof. Fix any (u, y) that is feasible for Eq. (16). If uaj , uak , ubj , ubk > 0 for some a, b ∈ A and j, k ∈ Nw , we will construct station assignments u that satisfy the lemma such that (u , y) is also feasible for Eq. (16) but has a lower or equal objective value. This proves the lemma. Let Ba := uaj +uak , Bb := ubj +ubk , Bj := uaj +ubj , and Bk := uak +ubk . The interpretation of these quantities is that rBa and rBb are the charging loads of EVs a and b, respectively, and rBj and rBk are their load distributions at stations j and k, respectively. Clearly, Ba + Bb = Bj + Bk . Without loss of generality, we can assume either case 1: Ba ≥ Bj ≥ Bk ≥ Bb or case 2: Bj ≥ Ba ≥ Bb ≥ Bk holds. We now construct u assuming case 1 holds. The construction is similar if case 2 holds instead. We consider four disjoint subcases and construct u for each subcase: 1.1 EV a is closer to station j but farther away from station k than b (daj ≤ dbj , dbk ≤ dak ): Let u aj = Bj , u ak = Bk −Bb , u bj = 0, u bk = Bb and the other variables remain the same as in (u, y). This means that the assignments u send EV b to station k but not station j, and also increase the likelihood of EV a going to station j while decreasing that to station k. Since u aj + u ak = Bj + Bk − Bb = uaj + uak u bj + u bk = Bb = ubj + ubk u aj + u bj = Bj = uaj + ubj u ak + u bk = Bk − Bb + Bb = uak + ubk (u , y) is feasible Eq. (16). Moreover,   dci u ci = daj Bj + dak (Bk − Bb ) + dbk Bb c=a,b i=j,k

= daj (uaj + ubj ) + dak (uak − ubj ) + dbk (ubj + ubk )   ≤ dci uci − ubj (dak − dbk ) c=a,b i=j,k

≤

 

dci uci

c=a,b i=j,k

where the first inequality uses daj ≤ dbj and the second inequality uses dbk ≤ dak . Therefore, (u , y) has a lower or equal objective value than (u, y). 1.2 EV b is closer to station j but farther away from station k than a (dbj ≤ daj , dak ≤ dbk ): This case is symmetric to subcase 1.1. 1.3 EV a is closer than b to both stations (daj ≤ dbj , dak ≤ dbk ): We have either dbj − dbk ≤ daj − dak or dbj − dbk > daj − dak . In the former case, let

238 PART | II Energy management

u aj = Bj − Bb , u ak = Bk , u bj = Bb , u bk = 0. Then     dci u ci = dci uci + (dak − dbk + dbj − daj )ubk c=a,b i=j,k

c=a,b i=j,k

≤

 

dci uci

c=a,b i=j,k

Similar to subcase 1.1, (u , y) is feasible and has a lower or equal objective value. In the latter case, let u aj = Bj , u ak = Bk − Bb , u bj = 0, u bk = Bb . Then (u , y) is feasible and has a lower objective value. 1.4 EV b is closer than a to both stations (dbj ≤ daj , dbk ≤ dak ): This case is symmetric to subcase 1.3. This completes the proof of the lemma. Proof. Fix an optimal solution (u* , y* ) to the relaxation (16) that satisfies Lemma 1. By definition, a critical EV splits its charging load between at least two different stations. An upper bound on the number of critical EVs is therefore the maximum number of critical EVs that we can assign the Nw stations to without violating Lemma 1. Consider the set C1 of critical EVs under the assignments u* that split their charging loads between station i = 1 and (at least) another station j = 2, . . . , Nw . Lemma 1 implies that there are at most Nw − 1 critical EVs in C1 since the assignments u* are optimal. Consider next the set C2 of critical EVs not in C1 that split their charging loads between station i = 2 and (at least) another station j = 3, . . . , Nw . There are at most Nw − 2 critical EVs in C2 . Similarly there are at most Nw − i critical EVs in the set Ci that are not in ∪i−1 k=1 Ck that split their charging loads between station i and (at least) another station j > i. Hence the maximum number of such critical EVs is (Nw − 1) + (Nw − 2) + · · · + 1 = 1 2 Nw (Nw − 1). This completes the proof of Theorem 1.

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Chapter 9

Dispatch strategy of energy bank system with hybrid energy storage for multiple microgrids Lingling Suna , Jing Qiu b and Zhao Yang Donga a School

of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, Australia, b School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW, Australia

1 Introduction Renewable energy is motivated by policy priorities because of environmental and energy security concerns [1]. However, renewable development cannot always rely on policies. In some cases, the favorable transmission access of renewable energy may not be allowed, or its charging price is high [2]. With the increasing of renewable distributed generation (RDG), more distribution generators of residential homes or business buildings are connected with the utility grid, but interconnection terms or standards of the grid are highly inconsistent or uncertain [1, 3]. As the technology of energy storage devices develops, storage devices play a more prominent role in the energy trading, and these devices can also help to smooth out the outputs of renewable generation system [4]. The storage system plays a major role in the energy trading system. Renewable energy is stored in the storage system during off-peak demand, and then it is used or exchanged in peak time [5]. For RDG, the storage system has been universal [6]. The smallsize storage devices are deployed for the consumption of the peak demand. For example, a hybrid energy storage system is proposed to mitigate the imbalance risk of electricity power [7]. However, the cost of installing new energy storage devices is expensive, and unaffordable for most common electricity users. This cost problem is one primary challenge of RDG development. The applications of energy trading have been raising interest in the past years [8]. Both electricity users and the generators are important participators in trading markets [9]. Studies of using RDG are growing in the field of energy trading [5, 10–15], and there are some energy trading applications. For instance, Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00009-8 Copyright © 2020 Elsevier Inc. All rights reserved.

243

244 PART | II Energy management

the authors propose an agent-based negotiation platform of energy trading, where the contract negotiation and the strategy optimization are combined by the negotiation platform [15]; the authors also set out a real-time trading system with an agent (rational aggregator) for distribution networks [5]. In most existing studies, the electrical trading mechanism is agent-based. In this agent-based trading mechanism, all players’ data/information is necessary for the central controller of the trading agent. However, in reality it is generally impossible to obtain clear data regarding the electrical market situation and opponents’ behaviors. The two-way communication technology determines trading success. First, the performance uncertainty of the communication system creates another problem for the energy trading system. Most current research assumes that the communication between all players is perfect; however, it is almost impossible. The effects of communication system reliability have been presented for energy demand users [16]. Second, for the central controller of the trading agent, the calculation amount is large, the mathematical algorithm is demanding, and the optimal period is long. Third, an appropriate optimization objective is used to all agent users for only one central controller. In most cases, there are usually multiple interactive users of one agent. The users cannot decide the optimization objective by themselves. For these reasons, there will be a cost due, charged by trading agents. To address this situation, a novel energy bank system (EBS) is presented in this chapter. First, as its name suggests, EBS is similar to an online bank of electrical energy. Power customers can trade energy with other users and receive gains. RDG can be more efficiently used. Second, there are significant advantages of EBS. It is nonprofit pursued. The trading costs for EBS users is lower. EBS can provide efficiently for the liquidity of energy similar to the financial liquidity in the bank system. Third, the capacity of storage devices can be traded or rented from other EBS users. There are no requirements to join in EBS, even EBS customers may not have storage devices. The agentbased trading mechanism follows the traditional trading mechanism (TTM). TTM is designed to suit traditional resources like thermal power generators. EBS is designed almost for all kind of energy resources, and can work well with traditional energy and renewable energy. Other salient features of the proposed EBS model are as follows: ●

●

For the trading model with the agent, the agent profit is listed in customers’ cost. EBS is the nonprofit trading platform without the agent, and the trading cost of users is lower. All information of all players is required for the agent-based trading model. In EBS, the previous calculation is operated on the customer side. The listedenergy and listed-price (LELP) of trading players are the only data to the central calculator. Thus, the duty of the communication system is lighter, and the calculation efficiency of the EBS central side is higher. Also, EBS customers can set the parameters like income and risk, according to their preference.

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 245

●

There is no required payment to start to use EBS, like costs of new DRG or storage equipment. However, if the customer wants to invest more later, EBS can recover the cost and gain money.

2 EBS model structure This section describes the nonprofit trading platform EBS with the operating requirements. It is inspired by how the current area of the energy market works with intelligent agents and multiple agents [5, 10–15].

2.1 Definition in energy bank system Some definitions in this EBS model are shown in Table 1. EBS is an energy trading platform. Energy users and generators are enabled to conduct several energy transactions with the EBS energy net as customers. This is a new kind of sharing energy, similar to the sharing economy. Typically, EBS will connect to or be a part of the core energy system. EBS contrasts with the traditional energy companies (TECs). EBS is a “virtual energy bank,” an internet-only institution. TECs are “brick-and-mortar,” there is cost for staff and branches. EBS has lower overhead costs than TECs. This is one important reason why EBS can be nonprofit. In EBS, electricity is seen as a digital/virtual currency; it is called “energy currency (EC)” to be stored or traded. EC functions as the operated “money.” A definition of virtual currency was provided by the European Central Bank in 2012: “a type of unregulated, digital money, which is issued and usually controlled by its developers, and used and accepted among the members of a specific virtual community” [17]. In EBS, EC is mostly generated by RDG. The utility grid and other energy companies are considered in the EBS model. All these kinds of connecting energy organizations are collectively called traditional energy modes (TEMs). As the online bank is the analogy of EBS,

TABLE 1 Definition comparison table between EBS, traditional trading system, and traditional bank system. Traditional trading system

Energy bank system

Traditional bank system

Electrical energy

Energy currency

Money

Energy storage system

Deposit system of energy

General deposit system

Central controller of a trading agent

Energy bank system central processing unit

Internet data center

246 PART | II Energy management

TEMs are comparable with the central bank or commercial banks. TEMs are responsible for the supervision of energy market system; they also receive energy from industries and individuals and sell energy to them. One EBS function is similar to the function of financial intermediaries. Financial intermediaries offer financial contracts to transfer risks in the form of derivatives and other securities. EBS offers EC contracts to transfer or reduce the RDG utilization risk for individual customers. However, EBS is a free trading platform; it has a not-for-profit purpose. This is the significant difference between financial intermediaries and EBS. Un-profit mode is a significant advantage of EBS. In EBS, EC is storage in the “deposit system of energy (DSE).” DSE includes most kinds of energy storage devices. EBS customers can be both depositor and trader. Every EBS customer has a “dedicated account,” including the deposit account (for EC saving) and trading account (TA, for EC trading). EBS customers can manage their EC by DSE. For example, EC deposited in DSE can supply power for loads to save money or be traded to gain profit. The profit can be saved in TA or transferred to be dollars in their personal bank account.

2.2 Energy bank system model and structure Fig. 1 shows the environment of EBS. TEMs include mostly kinds of TTMs and agents. DSE is the energy storage system. The charging energy is the extra EC generated by RDG; the discharging energy is used to trade or supply demand. The demand can be supplied by RDG, DSE, and TEMs. RDG can be generated by all kinds of generators; generated EC can supply demand, charge DSE, or trade with other EBS users.

Fig. 1

Complex environment of EBS.

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 247

The major abilities of EBS are specialized in EC trading, its operation safety, and DSE function. EC is transferred through the utility grid. Furthermore, EBS shows that a nonprofit trading platform can serve as one of the expanding conduits of energy trading. EBS takes full advantage of RDG energy to supply the load demands by the DSE. The utility grid can also supply energy to EBS customers. Along with users, EBS provides energy contracts that shift risks of RDG which is already installed and planned. The EBS participants can be generators or users. Customers role the generators and users in EBS. Different to other dispatch models with energy storage devices, there is no more required cost at the preliminary stage. Based on the current RDGs, transmission network, and storage devices, EBS can work for different users without any extra cost. Furthermore, further incomes correspond with the investments in the future. This is the most significant advantage of EBS. EBS can be used for most operating conditions of electricity markets; the advantages of EBS are reinspected in renewable energy using and energy trading. It can work for different kinds of demand. EBS is suitable for different kinds of power generators. It can run with different kinds of storage units, like energy batteries, supercapacitors, electric vehicles, etc. EBS can operate with different kinds of microgrids and electricity nets. As mentioned earlier, almost all kinds of electric consumers can be supplied with EBS. The pricing and trading models for each trading player in EBS are shown in Fig. 2. The overcapacity energy is traded or deposited. EBS cannot attack the utility grid. Before trading, customers can determine their LELP, and their profits can be maximized. Moreover, EBS can avoid vicious competition and monopoly of electricity prices. This is because in EBS the call-action model is the key trading method. The rule of maximum transaction volume is the basic trading principle. Energy bank system trading model Call-auction method

Trade energy & trade price

Maximumn transaction volume rule

Listed energy & listed price Listed-energy & listed-price (LELP) model Listed energy

Figure of deposit energy currency (FDEC)

Listed price

SOC

Cost

Treaty violation rule

Trade energy & trade price

Listed energy & listed price

Trade energy & trade price

Listed energy & listed price

Deposit system of energy (DSE)

Energy bank system customer-side (EBSCS) 1 Fig. 2

Introducing diagram of EBS model.

EBSCS 2

EBSCS n

248 PART | II Energy management

Fig. 3

Flow steps based on the rule of maximum transaction volume.

Fig. 3 shows the structure for EBS, which corresponds to multimicrogrid systems with four sections. EBS can work with most kinds of energy generators and demands. For predictive data, the forecast technique is not the main point in this chapter, and thus will not be further studied and discussed. All predictive data is real-world data from the AEMO website [18]. The goals of DSE are the same as those of energy storage systems, which are: (1) to improve utility grid optimization for bulk energy generation; (2) to balance power grid operation with RDG; (3) to defer capital-intensive upgrading in transmission and distribution grids; and (4) to offer ancillary serving for energy system operation [8]. DSE model bases on the previous methods [19–22]. DSE can operate with almost every kind of energy storage device. Figure of deposit EC (FDEC), SOC, and the DSE cost are expressed. FDEC is a setting for different needs of customers. It is useful to achieve the optimal power dispatch. The LELP model is one primary focus of EBS. The pricing model is based on the previous methods in Refs. [23, 24]. The pricing model is constrained by the physical models of microgrids with RDG. The customer can run on both islandoperation mode (no communication data) and grid-connected mode (transfer communication data). Only the information of trading players is transferred to the EBS central processing unit (EBSCPU). On the basic prediction data (energy

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 249

generation and demand), producers determine the forecast price. The differences are considered between the generation capacity, DSE cost, and risk return rate. The forecast price is set independent of the main grid. The LELP model does not discuss the energy price of the utility grid. The forecast data are submitted to EBSCPU for trading. One significant point is to determine the expected data to sell/buy by taking into account the relationship between risks and economies. However, the LELP model deals with only customer-self. The other main focus in EBS is the trading model. The trading model is operated in EBSCPU. It is based on the call-auction mode. The auction mechanism has become the basic method of the electric market [8]. Strong support has been provided by the literature for call-auction utilization. Through providing the same accessed price of all traders, the problems of the information asymmetry can be decreased by the call auction [25, 26]. For each auction, buyers and sellers list their expected transaction price and energy. The optimal allocation of distributed energy is obtained by the listed pricing. By this trading model, vicious competition and price monopoly can be avoided.

3 Trading model The call auction is the basic model of EBS in this project. The core function of the call auction is to calculate the transaction volume and trading price.

3.1 Call auction It is assumed that at the time interval of time t, all the data is given by the buy LELP model: buying and selling listed-price An , Asell n , buying and selling buy sell listed-energy En , En ; they are collected for n microgrid systems (there are buy no listed-price and listed-energy of island-operation microgrids). Based on An , sell An , for the maximum and minimum constraints of trading price, setting that      buy sell Atrade (1) n,max (t) = max max An (t) , max An (t)      buy sell (2) Atrade n,min (t) = min min An (t) , min An (t) trade Atrade (t) ≤ Atrade n,min ≤ An n,max

(3)

Irrespective of the principle of time preference currently, based on the rule of maximum transaction volume, the trading price is represented as   MG−1  MG−1   trade buy buy sell sell En (t)γn (t), En (t)γn (t) An (t) ∈ argmax min n

n

(4) where MG is the number of microgrid systems, the sum number of others buy is equal to (MG-1); γnsell and γn are the coefficients of selling and buying,

250 PART | II Energy management buy

γn (t) = 1, γnsell (t) = 1 denote the listed-energy (buying or selling) that is traded, which are given by buy 1, An (t) ≥ Atrade (t) n γnbuy (t) = (5) 0, others trade (t) 1, Asell n (t) ≤ An (6) γnsell (t) = 0, others

3.2 Rule of maximum transaction volume In this chapter, the trading principle is based on the rule of maximum transaction volume (Fig. 3). According to the previous calculated data, the trading price Atrade and transaction volume M trade can be provided by the following steps. First, sort out buying and selling listed-prices based on the numerical size   buy (t) (7) At (j) = descend Abuy n   sell (8) Asell t (j) = ascend An (t) where, according to the order of trading, the buying sequence is from highest to lowest, and the selling sequence starts from lowest. Second, as the same sequences of listed-prices, the corresponding buying and selling listed-energy sequences are  buy buy Et (j) (9) Mt (j) =  Mtsell (j) = Etsell (j) (10) buy

buy

where the buying and selling energy summations Mt (j) and Mt (j) correbuy spond the variates with the new sequence of At (j) and Asell t (j). At last, based on (j), the sum trading volume Mttrade (j) and the corresponding trading price Atrade t the maximum of Mttrade (j) is the final transaction volume in this time interval (11). The final trading price is the corresponding price of the final transaction volume (12).   (11) M trade (t) = max Mttrade (j) Atrade (t) = Atrade (t) (m) = Atrade t n

(12)

3.3 Rule of treaty violation The rule of treaty violation is for situations when selling or buying is unavailable. The violated treaty one needs to pay the liquidated damage Atrade ld (t), the liquidated damages can be used to trade further. If the seller violates the treaty, the liquidated damages will be set as to equal to the energy price of the utility grid, and the loss of the buyer can be minimized; and if the buyer violates

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 251

the treaty, the buying money is still required. The liquidated damages can be presented as grid Atrade n,ld (t) = An (t)

(13)

4 Listed-energy and listed-price (LELP) model Based on the deposit EC and FDEC in the DSE model, the LELP model is employed to predict the listed-energy and listed-price. The outputs of the LELP buy sell buy model are listed-prices and listed-energy (Asell n , En , An , En ), which are used DSE in the trading model. The inputs are the DSE cost (Cn ), deposit EC (EnDSE ), DSE ), which are calculated in Section 3. and FDEC (En,f According to FDEC, the selling/buying listed-energy can be predicted for different situations, as shown in Fig. 4. When there is remaining energy Enren (t)− Endem (t) ≥ 0: DSE (t) If EnDSE (t) ≥ En,f ⎧ DSE (t) Ensell (t) = Enren (t) − Endem (t) + En,dch ⎪ ⎪ ⎪ ⎨ buy En (t) = 0 (14) DSE ⎪ En,ch (t) = 0 ⎪ ⎪ ⎩ DSE (t) = E DSE (t) − E DSE (t) En,dch n n,f DSE (t) If EnDSE (t) < En,f ⎧ DSE (t) Ensell (t) = Enren (t) − Endem (t) − En,ch ⎪ ⎪ ⎪ ⎨ buy En (t) = 0 DSE (t) = E DSE (t) − E DSE (t) ⎪ E n ⎪ n,ch n,f ⎪ ⎩ DSE (t) = 0 En,dch

When RDG cannot supply demand Enren (t) − Endem (t) < 0:

Fig. 4

Situations of charging and discharging (ch&dch) energy and buying/selling energy.

(15)

252 PART | II Energy management

If EnDSE (t) ≥ Endem (t) − Enren (t) ⎧ ⎪ Ensell (t) = 0 ⎪ ⎪ ⎨ buy En (t) = 0 DSE (t) = 0 En,ch ⎪ ⎪ ⎪ ⎩EDSE (t) = Edem (t) − Eren (t) n n n,dch

(16)

If EnDSE (t) < Endem (t) − Enren (t) ⎧ ⎪ Ensell (t) = 0 ⎪ ⎪ ⎨ buy dem En (t) = En (t) − Enren (t) − EnDSE (t) DSE (t) = 0 En,ch ⎪ ⎪ ⎪ ⎩ DSE En,dch (t) = EnDSE (t)

(17)

The constraints of remaining energy to sell and buy are as follows: ⎧ sell DSE (t) En (t)≤ Enren (t) − Endem (t) + En,dch ⎪ ⎪ ⎪ ⎪ DSE (t) ≥ 0 ⎨ EnDSE (t) − En,f DSE (t) Ensell (t) ≤ Enren (t) − Endem (t) − En,ch ⎪ ⎪ ⎪ ⎪ ⎩ EDSE (t) − EDSE (t) < 0

⎧ ⎪ ⎪ ⎨

n

(18)

n,f

buy En (t) ≤ Enren (t) − Endem (t)   DSE En (t) − Endem (t) − Enren (t) > 0 buy ⎪ (t) ≤ Enren − Endem (t) − E nDSE (t) ⎪  (t)  ⎩En DSE dem En (t) − En (t) − Enren (t) ≤ 0

(19)

4.1 Selling listed-price Selling listed-price is the forecast selling price. EBS customers can decide different FDEC values. The different values of FDEC can impact the forecast price. Similarly, the selling listed-price coefficient (LPC) βn is one important term, which represents the relationship between the listed-price and the customerchosen FDEC. The selling listed-price is given by cost sell Asell n (t) = An (t) + βn (t)An,max (t)

(20)

is the sum of cost; the risked cost is related to the selling LPC where Acost n βn and the maximum selling price Asell n,max . βn is a percentage; βn ∈ [0, 1], βn = 0 represents the lowest risked ratio and the lowest payback. Here, the maximum cost is equal to the price difference between the total DSE cost and grid cost grid electricity bill, Asell n,max (t) = An (t) − An (t), in a time interval; this term is constant for a customer. The sum of cost is bank (t) + DCnDSE (t) = Abank (t) + Acost n (t) = An n

T  t=1

Cib

I  i=1

Pbn,i (t)

(21)

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 253

where Abank is the charging of the bank and DCnb is the DSE depreciation cost n (Eq. 38). In the nonprofit EBS, the charging of the bank is equal to the charging of the grid: bank (t) = Agrid Abank n n (t) = φn

T  



sell Pbuy n (t) + Pn (t)

(22)

t=1

where the charging of the bank relates to the coefficient of bank charging φnbank and energy throughput. Here, the coefficient of grid charging is constant for a grid customer, φnbank = φn .

4.2 Buying listed-price Buying listed-price is the forecast buying price, similar to Asell n ; the buying LPC δn is also a percentage, δn ∈ [0, 1], where δn = 1 represents the lowest risked buy ratio and the lowest gain. Here, the maximum buying price (An,max ) is simply set buy grid as equal to the price of the grid electricity bill, An,max (t) = An (t). The buying listed-price is buy grid Abuy n (t) = δn (t) ∗ An,max (t) = δn (t) ∗ An (t)

(23)

4.3 Assessment indices The selling and buying LPC (βn , δn ) are estimated with the existing real-data. In EBS, the coefficients βn , δn can be calculated by the forecast trading price based on the given data. Based on the empirical mean predictor method, the sophisticated optimization method is investigated as the guide of the current work, which is the minimization optimization of the empirical mean absolute percentage error (MAPE) [27]. Setting that, in the empirical range, the arbitrary forecast value is expressed by p, the probable discretization value of nth is expressed by pn . The prediction value is represented as the MAPE expectation of the chosen p,  fn,t (pn ) |p − pn/pn | (24) εn,t (p) = n

With the minimization optimization of empirical MAPE, the forecast value of nth at time t can be indicated as   (25) pt = min εn,t (p) From the call-auction method, the selling and buying listed-prices are the predictive trading prices. Based on the given data, the forecast trading price can be indicated as    trade A − A (t) = min fn,t (A) | n/An | (26) A n

254 PART | II Energy management

Thus, δn (t) = Atrade (t)/Agrid n (t) βn (t) =

(27) (28)

/

grid cost Atrade (t) − Acost n (t) An (t) − An (t)

5 Deposit system of energy In the DSE model, the storage devices can be any kind of energy storage units, and they can be arranged in different percentages. In EBS, the EM in DSE can be charged from RDG, and also can be discharged to self-demand or sell to others. Different deposit EC units belong to different customers, but they are managed uniformly to optimize them. The DSE capacity can be changed by customers at any time. DSE’s SOC and cost will also follow the capacity change.

5.1 Figure of deposit energy currency The energy model should be maintained to suit any trading condition. Customers can settle the risk-return ratio and different storage devices with any capacity size; EBS calculates FDEC based on these settings. FDEC is set as the minimum power of risked trade. For raising the economy efficiency, RDG can still be traded while the storage device has not been fully charged. The storage power must be higher than FDEC. DSE ) is one important coefficient; it is settled as a basic In EBS, FDEC (En,f line for DSE (Fig. 4). FDEC is decided by the risk-return ratio of EBS trading in the next time interval. T    DSE dem (t) = θn (t) (t + 1) − P (t + 1) + EnDSE (t) (29) Pren En,f n n t=1

where θn represents the relationship between FDEC and risk-return ratio. It is a percentage; θn ∈ [0, 1]; θn = 0 represents the highest risk-return ratio and payback.

5.2 Operation model of DSE The power statement of DSE in microgrid n is described by PDSE n (t) =

I  i=1

PDSE n,i (t) =

I  i=1

DSE PDSE n,ch,i (t)ηch,i −

PDSE n,dch DSE ηdch

(30)

where the charging power is positive (PDSE n,ch (t) > 0) and the discharging is DSE (t) denotes the negative (Pn,dch (t) < 0). There are I kinds of batteries; PbDSE n power statement of storage devices for microgrid n, which is the sum value of DSE in all different kinds of the energy storage devices; and the deposit EC

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 255

EnDSE (t) is the integration of PDSE n (t). The charging statement (SOC) of storage devices of microgrid n is given by SOCnDSE (t + 1) = SOCnDSE (t) + tPDSE n (t)

(31)

The energy system implements the power balance at any time. grid dem loss DSE Pren n (t) + Pn (t) = Pn (t) + Pn (t) + Pn (t)

(32)

Ploss n (t)

where the system loss power is assumed to be ignored. The grid grid supplying power Pn (t) is positive if EC is bought from the grid. EC can be charging and discharging (ch&dch), which is related to the produced and supplied power in the microgrid. The necessary conditions of ch&dch capacities with contribution to up and down reserves are represented by DSE DSE DSE RDSE n,up (t) ≤ Pn,max − Pn,dch (t) + Pn,ch (t)

(33)

DSE DSE DSE RDSE n,dn (t) ≤ Pn,max − Pn,ch (t) + Pn,dch (t)

(34)

DSE where the up and down reserves RDSE n,up (t) and Rn,dn (t) are related to the maximum DSE DSE power Pn,max and ch&dch power Pdch,n (t) and PDSE ch,n (t). The constraints consist of the limits of power capacity, DSE DSE PDSE n,min ≤ Pn (t) ≤ Pn,max

(35)

To avoid the overcharging and over-discharging of storage devices, complementarity constraint with contribution to up and down reserves is correspondingly represented by DSE + SOCn,min

tRDSE n,up (t) DSE ηch,i

DSE DSE DSE ≤ SOCnDSE (t) ≤ SOCn,max − tηdch,i Rn,dn (t)

(36)

where the SOC (SOCnDSE ) is characterized by its min and max figure, DSE ≤ SOCDSE (t) ≤ SOCDSE . SOCn,min n n,max

5.3 Economic model of DSE The cost of DSE includes the daily electricity cost (DEC) of the utility grid and the total daily operation cost (TDOC). DEC can be presented as grid DECnDSE (t) = Agrid n (t)Pn (t) t

(37)

Irrespective of the operation and maintenance cost (O and MC), TDOC is assumed to be the depreciation cost (DC). DCnDSE (t) = CiDSE

I  i=1

PDSE n,i (t) t

(38)

256 PART | II Energy management

where CiDSE denotes the ch&dch depreciation cost of the ith device, and DCnDSE (t) is related with the ch&dch DC and the corresponding ch&dch energy. The ch&dch DC is denoted by CiDSE =

DSE Ci,ref

QDSE lt,i

(39)

6 Case study In this chapter, we assume there are two kinds of renewable generators in microgrids; PV panels and wind turbines. The time interval of AEMO data is 15 min; to avoid error, the simulation time interval of trading is 30 min [18]. EBS has considered two kinds of battery: Li-ion and lead-acid. The full charging capacities are identical in three microgrids, 8 kWh. The ch&dch efficiency is assumed to be 95%. Three kinds of microgrid are load demands models: a business shopping center, an office building, and a residential area in Sydney. Their peak demands are 26, 59, and 17 kWh, respectively. RDGs are PV panels and wind turbines. RDG is the same for three microgrids. Two boundary situations are considered, which are the lowest-risked ratio with the lowest payback and the highest-risk situation. Monday, Thursday, and Sunday are simulated in this section based on the characteristic differences of three demands.

6.1 System description The real-world data of demand consumption and distribution generation come from the AEMO website [18]. For these three kinds of demand models, the peak demand of office building is from 8.30 a.m. to 5.30 p.m. on workdays, and it is at a low level on weekends. Correspondingly, the peak demand of the residential area is in the morning and evening on workdays, and is usually different on weekends. The peak demand of the business shopping center is from 9.30 a.m. to 6.30 p.m., but on Thursday and Sunday, there are some difference because the opening time are, respectively, from 9.30 a.m. to 9 p.m. and from 10 a.m. to 5 p.m. Therefore, from 10 a.m. to 16 p.m. on Monday and Thursday, the remaining energy of the residential area can be sold to the business center or office building or charge storage devices; and for the whole day on Sunday, the remaining energy of the office building can be sold or charging; the residential area and business center need to buy energy. Two kinds of storage batteries are discussed here, they are Li-ion battery and lead-acid battery. Li-ion battery having a high energy-to-weight ratio, a higher energy-to-volume ratio, the lower power-to-weight ratio, the tiny memory effect and low self-discharge; and higher cost. The lead-acid battery has a lower energy density, but also lower cost, higher storage time, and less maintenance requirements.

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 257

6.2 Simulation First of all, the EBS performance is evaluated by using multiple microgrids with hydride storage devices. The microgrids and batteries consider different demand properties and charging/discharging performances. The DSE will be charged to full between 0.00 and 4.00 by wind energy, it can be used to supply the morning peak demand; at day time, demand can be supplied by RDGs, and the remaining energy is listed to sell; the insufficient energy can buy from others EBS participants or buy from the grid. For Thursday night, because it is on off-peak-time of office building, the RDG remains and can be sold; and the remaining energy of the residential area is also listed to sell; but it is business hours of the shopping center, and the insufficient energy can be bought from the office building and residential area. The EBS platform allows the energy shifting across different microgrid systems, to trade energy with other EBS users, and based on their characteristics flexibly to use their DSE, greatly improving the utilization of RDG and the economic benefits. The demand profiles of three typical microgrids are considered at three typical days with a 30-min time interval. To highlight the EBS property of different demand situations, and decrease the difference of variables, RDG is similar in these 3 days. Fig. 5 illustrates the operation results of ch&dch energy of DSE for these three typical microgrids. These profiles indicate that, according to the significant difference of daily demand and peak-load time, the deposit EC varies to a large degree. Fig. 6 illustrates the operation results of selling and buying listed-energy and transaction energy and price. Diagrams (1–3) show that the RDG can be sold in 24 h (selling listed-energy) and the deficit of energy needs to be bought from other EBS users or the utility grid (buying listedenergy). Diagrams (4–6) illustrate the trading operation results of the final trade energy and trade price in EBS. The remaining RDG of the residential area can be sold to the office building or shopping center at Monday and Thursday daytime; the residential and shopping center can buy the remaining energy of the office building at Sunday. Table 2 shows the operation profits of these three EBS users on Monday, Thursday, and Sunday. EBS customers profit by trading their RDG. The profit includes two sides: one side is the selling proceeds, the other side is the saving money of buying energy from EBS. The operation results prove the profitability of using EBS. The “Renewable energy usage increase (%)” in Table 2 denotes the increase proportion of the renewable energy used by EBS, energy (selling and buying) by EBS × Renewable energy usage increase = The trading The total renewable energy usage 100%. In addition, EBS also allows customers to buy energy from the utility grid; in this way, the risks of the imperfect information can be offset. Namely, EBS can tolerate larger imperfect information and default risk. Simulation results are represented to verify the positives of EBS. First, the utilization ratio of renewable energy: remaining energy generated by RDGs supplies other microgrids; the higher utilization of renewable energy can be achieved. Second, the economic benefits: no extra cost of EBS, and the trading

–5 6

0

12

18

24

Time (h) Charging & discharging energy on Thursday (2) 5

5 0

0

123

–5

–5 6

0

12

18

24

Time (h) Charging & discharging energy on Sunday (3) 5

5 0

0

123

–5 0

–5 6

12

18

24

Discharging energy (–kWh)

0

–5

Discharging energy (–kWh)

5 123

Discharging energy (–kWh)

Charging energy (+kWh) Charging energy (+kWh) Charging energy (+kWh)

0

Time (h)

Fig. 5 Simulation results on Monday, Thursday, and Sunday. Left: Charging energy (positive); Right: discharging energy (positive). No. 1: shopping center; No. 2: office building; No. 3: residential area.

258 PART | II Energy management

Charging & discharging energy on Monday (1) 5

20

20 10

10

0

123

–10

–10

–20

–20 12

18

0.25 6

0.2

4

0.15 0.1

2 0

24

0.05 0

6

Time (h)

20

10

10 0

123

–10

–10 –20

–20

–30 0

6

12

18

–30 24

Trading energy (kWh)

30

20

Buying listed-energy (–kWh)

Selling listed-energy (+kWh)

40

30

0

0.3

8

0.25 6

0.2

4

0.15 0.1

2 0

0.05 0

6

20

10

10 0

123

6

12

Time (h)

18

–10 24

Trading energy (kWh)

20

Buying listed-energy (–kWh)

Selling listed-energy (+kWh)

30

–10 0

12

18

0 24

Time (h) 40

30

0

0 24

Trading energy & price Thursday (5)

10

Time (h)

Selling & buying listed energy on Sunday (3)

40

18

Time (h)

Selling & buying listed energy on Thursday (2)

40

12

Tranding price ($)

6

0

0.3

8

Trading energy & price Sunday (6)

5

0.3

4

0.25

3

0.2

2

0.15 0.1

1

0.05

0 0

6

12

18

Tranding price ($)

0

Trading energy & price Monday (4) 10

Tranding price ($)

30

Buying listed-energy (–kWh) Trading energy (kWh)

Selling listed-energy (+kWh)

40

30

0 24

Time (h)

Fig. 6 Simulation results. On Monday, Thursday, and Sunday, (1–3) Left: selling listed-energy (positive); Right: buying listed-energy (negative); No. 1: shopping center; No. 2: office building; No. 3: residential area. (4–6) Left: transaction energy; Right: transaction price.

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 259

Selling & buying listed energy on Monday (1)

40

260 PART | II Energy management

TABLE 2 Characteristic differences of demand and generation. Residential area

Shopping center

Office building

Buying energy (kWh)

N/A

1.523219

44.53091

Saved cost than utility grid ($)

N/A

0.13709

4.007782

Selling energy (kWh)

1.522176

47.38564

0.166612

Selling profit ($)

0.479485

14.92648

0.052483

Trading energy (kWh)

1.522176

48.90886

44.69752

Renewable energy use (%)

0.773361

0.705587

0.566484

Buying energy (kWh)

N/A

8.937452

47.21858

Saved cost than utility grid ($)

N/A

0.804371

4.249672

Selling energy (kWh)

12.79565

43.19377

0.166612

Selling profit ($)

4.043426

13.64923

0.056249

Trading energy (kWh)

12.79565

52.13122

47.38519

Renewable energy use (%)

0.81204

0.710061

0.574169

Buying energy (kWh)

25.815

N/A

N/A

Saved cost than utility grid ($)

2.32335

N/A

N/A

Selling energy (kWh)

N/A

25.815

N/A

Selling profit ($)

N/A

8.12012

N/A

Trading energy (kWh)

25.815

25.815

N/A

Renewable energy use (%)

0.852671

0.529682

0.47403

Time Monday

Thursday

Sunday

profit is almost the net profit. As a full configuration of multiple microgrids with RDG and a higher flexibility of energy scheduling, EBS makes a higher economic benefit and a higher social welfare. Third, the wide practicality: as the setting of FDEC, based on customers’ requirements, EBS can work for almost all kinds of microgrid systems.

7 Conclusion EBS is a platform for energy trading in multiple microgrid systems where customers trade energy with each other. FDEC is an energy index of energy management and energy trading in the DSE model. According to the results, customers show the listed-energy and listed-price, EC is dealt based on the

Dispatch strategy of energy bank system with hybrid energy storage Chapter | 9 261

rule of maximum transaction volume, and gives out the transaction volume and trading price. Simulation results prove that EBS can help to achieve high utilization efficiency of RDG. These characteristics encourage electricity users to become a member of EBS, and ensure the optimal allocation of multiple energy. In future work, the trading model can be optimized, the algorithm of the LELP model can be further improved, and the method of energy management can also be used in EBS.

References [1] F. Beck, E. Martinot, Renewable energy policies and barriers, Encycl. Energy 5 (7) (2004) 365–383. [2] C. Harabut, Legal and policy instruments to facilitate development for renewable energy, in: 2015 5th International Youth Conference on Energy (IYCE), Pisa, 2015, pp. 1–5. [3] M. Mendonça, Feed-In Tariffs: Accelerating the Deployment of Renewable Energy, Routledge, 2009. [4] J. Castaneda, J. Enslin, D. Elizondo, N. Abed, S. Teleke, Application of statcom with energy storage for wind farm integration, in: Proceedings of the IEEE PES Transmission and Distribution Conference and Exposition, 2010, pp. 1–6. [5] C. Zhang, Q. Wang, J. Wang, P. Pinson, J.M. Morales, J. Østergaard, Real-time procurement strategies of a proactive distribution company with aggregator-based demand response, IEEE Trans. Smart Grid 9 (2) (2018) 766–776. [6] B. Roberts, C. Sandberg, The role of energy storage in development of smart grids, Proc. IEEE 99 (6) (2011) 1139–1144. [7] H. Yang, J. Qiu, K. Meng, J. Zhao, Z. Dong, M. Lai, Insurance strategy for mitigating power system operational risk introduced by wind forecast uncertainty, Renew. Energy 89 (2016) 606–615. [8] I. Bayram, M. Shakir, M. Abdallah, K. Qaraqe, A survey on energy trading in smart grid, in: 2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP), Atlanta, GA, 2014, pp. 258–262. [9] Y. Okawa, T. Namerikawa, Distributed optimal power management via negawatt trading in real-time electricity market, IEEE Trans. Smart Grid 8 (6) (2017) 3009–3019. [10] Z. Wang, L. Wang, Adaptive negotiation agent for facilitating bidirectional energy trading between smart building and utility grid, IEEE Trans. Smart Grid 4 (2) (2013) 702–710. [11] W. Tushar, W. Saad, H.V. Poor, D.B. Smith, Economics of electric vehicle charging: a game theoretic approach, IEEE Trans. Smart Grid, 3 (4) (2012) 1767–1778. [12] C. Wu, H. Mohsenian-Rad, J. Huang, Vehicle-to-aggregator interaction game, IEEE Trans. Smart Grid 3 (1) (2012) 434–442. [13] Y. Wang, W. Saad, Z. Han, H. Poor, T. Basar, A game-theoretic approach to energy trading in the smart grid, IEEE Trans. Smart Grid 5 (3) (2014) 1439–1450. [14] W. Tushar, J.A. Zhang, D.B. Smith, H.V. Poor, S. Thiébaux, Prioritizing consumers in smart grid: a game theoretic approach, IEEE Trans. Smart Grid 5 (3) (2014) 1429–1438. [15] Y. Jia-Hai, Y. Shun-Kun, H. Zhao-Guang, A multi-agent trading platform for electricity contract market, in: International Power Engineering Conference, Singapore, vol. 2, 2005, pp. 1024–1029. [16] D. Niyato, P. Wang, E. Hossain, Reliability analysis and redundancy design of smart grid wireless communications system for demand side management, IEEE Wireless Commun. 19 (3) (2012) 38–46.

262 PART | II Energy management [17] Europa Central Bank, Available from: https://www.ecb.europa.eu/stats/ecb_statistics/sdw/ html/index.en.html. [18] Australian Energy Market Operator (AEMO), Available from: https://www.aemo.com.au/ Electricity/National-Electricity-Market-NEM/Data-dashboard. [19] M. Farrokhabadi, S. König, C.A. Cañizares, K. Bhattacharya, T. Leibfried, Battery energy storage system models for microgrid stability analysis and dynamic simulation, IEEE Trans. Power Syst. 33 (2) (2018) 2301–2312. [20] K. Meng, Z.Y. Dong, Z. Xu, S.R. Weller, Cooperation-driven distributed model predictive control for energy storage systems, IEEE Trans. Smart Grid 6 (6) (2015) 2583–2585. [21] F. Luo, K. Meng, Z.Y. Dong, Y. Zheng, Y. Chen, K.P. Wong, Coordinated operational planning for wind farm with battery energy storage system, IEEE Trans. Sustain. Energy 6 (1) (2015) 253–262. [22] C. Zhang, Y. Xu, Z.Y. Dong, J. Ma, Robust operation of microgrids via two-stage coordinated energy storage and direct load control, IEEE Trans. Power Syst. 32 (4) (2017) 2858–2868. [23] M. Roozbehani, M.A. Dahleh, S.K. Mitter, Volatility of power grids under real-time pricing, IEEE Trans. Power Syst. 27 (4) (2012) 1926–1940. [24] A.B. Kiani, A. Annaswamy, A dynamic mechanism for wholesale energy market: stability and robustness, IEEE Trans. Smart Grid 5 (6) (2014) 2877–2888. [25] C. Comerton-Forde, S.T. Lau, T. McInish, Opening and closing behavior following the introduction of call auctions in Singapore, Pacific-Basin Finance J. 15 (1) (2007) 18–35. [26] A. Madhavan, Trading mechanisms in securities markets, J. Finance 47 (2) (1992) 607–641. [27] W. Kong, Z.Y. Dong, D.J. Hill, F. Luo, Y. Xu, Short-term residential load forecasting based on resident behaviour learning, IEEE Trans. Power Syst. 33 (1) (2018) 1087–1088.

Chapter 10

False data injection attacks and countermeasures in smart microgrid systems Mengxiang Liua , Chengcheng Zhaoa , Ruilong Denga,b , Peng Chenga , Wenhai Wanga and Jiming Chena a College

of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China, b School of Computer Science and Engineering, Nanyang Technological University, Singapore

1 Introduction Microgrids are capable of connecting to the main grid (grid-connected mode) under normal operations, and proactively disconnecting from the main grid (islanded mode) when emergency situations occur. Due to the increasing penetration of distributed generation units (DGUs) such as solar photovoltaics (PVs), fuel cells, and microturbines into the power network, microgrids have become the most promising solution to make the power network stable, safe, resilient, and efficient [1]. Generally, microgrids can be divided into AC microgrids and DC microgrids (DCmGs). DCmGs have several advantages compared with AC ones, for example, loss reduction for DC loads, easier integration of DC DGUs, and cost reduction for synchronizing generators [2]. Therefore, significant attention has been paid to DCmGs. The basic objectives of DCmGs are voltage balancing and current sharing [3], and normally a hierarchical control strategy is deployed to achieve these objectives [4]. Specifically, the primary controller aims to track reference voltages and the secondary controller is usually designed with consensus algorithms to provide robust and scalable current regulation. However, the information flows in the secondary control are prone to cyber-attacks, like false data injection (FDI) attacks, which can cause economic losses or even destabilize the system voltages/currents. In the main grid, numerous works [5–7] on cyber security have been done in recent years since Liu et al. [8] proposed the FDI attack scheme against the state estimation of power networks. Meanwhile, due to the combination of characteristics like islanded operation mode [1], hierarchical Distributed Control Methods and Cyber Security Issues in Microgrids https://doi.org/10.1016/B978-0-12-816946-9.00010-4 Copyright © 2020 Elsevier Inc. All rights reserved.

263

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control scheme [4], and low inertial PV generators [9], small disturbances can cause devastating damage in DCmGs and it is hard to analyze theoretically the attack impacts considering the multilayer dynamics of the hierarchical control. Thus, considerable attention has also been paid to the cyber security of smart DCmG systems. Considering the injected malicious signals as constant values, Liu et al. [10] designed a distributed detection strategies for microgrids, which checks the dual-ascent update iterations. Then, Lu et al. [11] proposed a resilient decentralized secondary frequency control method for microgrids, which could mitigate the cyber-attacks injecting invalid commands and erroneous measurements into communication links. Moreover, Beg et al. [12] proposed a framework to detect cyber-attacks in DCmGs, which inject false data into the global variables randomly, by using candidate invariants. As for model-based detection methods, the unknown input observer (UIO)-based techniques, which allow the design of observers with the existence of unknown inputs (e.g., injected malicious signals), are widely adopted to detect and identify malicious nodes in unreliable networks [13]. Recently, a novel UIO-based detector has been proposed to detect and identify specific cyber-attacks in DCmGs [14], where each DGU estimates neighbors’ states and requires only neighbors’ knowledge. However, most existing works (like the aforementioned ones [10–14]) assume that attackers have little knowledge of the system model. Actually, intelligent attackers can learn critical system parameters by hiding themselves in the system, and launching the attack at an appropriate time, which can cause devastating damage like the Stuxnet [15] and even threaten human life. Therefore, it is necessary to investigate potential stealthy attacks, which can provide novel design criteria to enhance the security of DCmGs. In this chapter, we mainly investigate the zero-dynamics stealthy (ZDS) attack and the nonzero-dynamics stealthy (NDS) attack in DCmG. The ZDS attack, which injects the zero output signals into systems, has been deeply investigated in cyber-physical systems [16, 17]. Since the ZDS attack is invisible at the output, it can bypass the detector and cause disturbances to system states. Moreover, by appropriately designing the controller and detector, it is possible to eliminate the existence of the ZDS attack. However, it may not be enough to protect systems from the damage of stealthy attacks. By masking the malicious signals as the state estimation error and measurement noise, we are the first to explore the NDS attack against the UIO-based detector in DCmG, which can affect the detection residuals but remain stealthy. Thus, NDS attacks are one of the most threatening FDI attacks in DCmGs. By following the novel detector proposed in [14], we continue to investigate the potential stealthy attacks in DCmG. The main contributions of this chapter are as follows: 1. We prove that there exists no ZDS attack in current DCmG, and investigate the NDS attack by exploiting the state estimation error and measurement noise.

False data injection attacks and countermeasures Chapter | 10 265

2. By approximating the primary control loops as static unit gains [18], we theoretically analyze the dynamics of DCmG under the NDS attack, and obtain analytical expressions of point of common coupling (PCC) voltages. 3. The steady-state PCC voltages and steady-state output currents are analyzed to verify the voltage balancing and current sharing, respectively. 4. We prove that the NDS attack can guarantee exponential convergence rate in DCmG, and some countermeasures are mentioned. 5. Simulations are conducted in Simulink/PLECS to validate our theoretical analysis. The rest of this chapter is organized as follows. Section 2 presents the system model, UIO-based detector, and problem formulation. Then, we investigate the NDS attack and analyze its effects on the voltage balancing, current sharing, and convergence rates of voltages/currents, with corresponding countermeasures in Section 3. Extensive simulations are presented in Section 4. Section 5 concludes this chapter.

2 Preliminaries and problem formulation Notation. In this chapter, |·| denotes the cardinality of a finite set and the component-by-component absolute value of a matrix/vector,  ·  is the norm of a matrix/vector, inequalities of matrices/vectors are compared component-bycomponent, and y(∞) = limt→∞ y(t), where y(t) denotes a scalar or a vector. Moreover, 1n×n /1n and 0n×n /0n are matrices/vectors with all 1 and 0 entries, H1 denotes respectively, and In denotes a unit matrix with n × n dimension. 1 n n 1 a subspace of R and ∀v ∈ H , v = 0, where v = n i=1 vi returns the average of vector v and dim{H1 } = n − 1. H1⊥ is the orthogonal vector space of H1 , where v = α1n , α ∈ R, ∀v ∈ H1⊥ and dim{H1⊥ } = 1. Therefore, the   decomposition can be obtained as Rn = H1 H1⊥ , where denotes the direct sum of vector spaces.

2.1 Network model 2.1.1 Physical network DCmG consists of N DGUs interconnected through power lines, and a digraph Gel = {ν, εel , W} is used to characterize the electrical model of DCmG. ν = {1, 2, . . . , N} is the set of nodes (DGUs), and ε el ⊂ ν ×ν denotes the set of edges (power lines), whose orientations are defined arbitrarily for reference directions of positive currents. Moreover, W = diag{ R1ij } ∈ R|εel |×|εel | , where Rij denotes the resistance of power line eij ∈ ε el . The neighbor set of DGU i is denoted by Niel = {j|(i, j) ∈ ε el or (j, i) ∈ ε el , ∀j ∈ ν and j = i}. B ∈ R|ν|×|εel | denotes the incidence matrix, with which the Laplacian matrix can be calculated independently of edges’ orientations, that is, M = BWBT .

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2.1.2 Communication network As for the communication network in DCmGs, an undirected digraph Gc = {ν, ε c , W c } is used to model the bidirectional communication among DGUs. Here, ν is the set of DGU nodes, and ε c denotes the set of edges (communication links). Moreover, W c = diag{aij } ∈ R|εc |×|εc | , where aij is the weight of communication link eij ∈ εc . Besides, Nic = {j|(i, j) ∈ ε c , ∀j ∈ ν and j = i} denotes the neighbor set of DGU i. The Laplacian matrix of Gc is given by L. Furthermore, Gel and Gc should satisfy Assumption 1 to guarantee stable control of voltages/currents in DCmG [18]. Assumption 1. The digraph Gel is weakly connected and the graph Gc is undirected and connected. Moreover, Gel and Gc have the same topology, and L = γ M, γ > 0. 2.2 Dynamic model Each DGU is modeled as a DC voltage source and a buck converter to supply a local current load connected to the PCC through an RLC filter [19]. The dynamic of DGU i is described as ⎧ dV  1 1 1 i ⎪ I + (Vj − Vi ) − ILi , = ⎪ ti ⎪ ⎨ dt Cti C R C ti ij ti j∈Niel (1) ⎪ ⎪ R 1 1 dI ti ti ⎪ ⎩ Iti + Vti , = − Vi − dt Lti Lti Lti where Vti and ILi are the voltage input and current input to DGU i, and Vi and Iti are the states (i.e., the measured voltage at PCC i and output current of DGU i, respectively, see Fig. 1), Rti , Lti , Cti are the parameters of the RLC filter, and Vj is the PCC voltage of the neighbor node j ∈ Niel . A hierarchical control structure is adopted to achieve the regulation of voltages and currents [18], where the primary proportional integral (PI) controller tracks the reference voltage and the secondary controller achieves the global proportional load current sharing by the consensus algorithm (as Fig. 1 shows). Considering the state disturbances and noise, we can get the state-space model of DGU i, x˙i (t) = Aii xi (t) + bi ui (t) + gi ψi (t) + Mi di (t) + ξ i (t) + ωi (t), (2) yi (t) = Ci xi (t) + ρ i (t), where xi = [Vi , Iti , vi ]T is the local state vector and vi is the voltage error integral item. The dynamic of vi is v˙i = Vref ,i − Vi , where Vref ,i is the reference voltage. In addition, di = [ILi , Vref ,i ]T is the exogenous input vector, where ILi is the local load current unknown to DGU i, and yi ∈ R3 denotes the measurement output vector. The physical coupling with neighbor DGUs is modeled as ξ i =  3 j∈N el Aij xj ∈ R . Moreover, the process noise and measurement noise are i

modeled as |ωi (t)| ≤ ω¯ i ∈ R3 , |ρ i (t)| ≤ ρ¯ i ∈ R3 , ∀t ≥ 0, respectively, which

False data injection attacks and countermeasures Chapter | 10 267

Communication network

Line Electrical network

Line

Fig. 1 A hierarchical control structure of DGU i, where the communication network models its interactions with DGU neighbors Nic and the electrical network models the physical couplings with DGU neighbors Niel .

are bounded by some certain bounds. Interested readers can refer to [19] for detailed information of Aii , Aij , Ci , bi , gi , ξ i . The primary control input Vti = ui (t) = kTi yi (t), where the control gain vector ki ∈ R3 is designed by using local information to achieve the plugand-play operation [19]. The secondary control input ψi (t) is computed by the following consensus scheme to adjust the reference voltage, ψ˙i (t) = −[0, kI , 0]

 j∈Nic

aij

 c yi (t) yi,j (t) − s , Itis Itj

(3)

where yci,j (t) ∈ R3 is the output vector of DGU j transmitted to DGU i through link (i, j), Itis > 0 is the rated current of DGU i, kI > 0 is the consensus value common to all communication links, and aij = aji > 0 is the link weight. Under Assumption 2, the definitions of voltage balancing and current sharing are given, which are the control objectives of DCmG. Assumption 2. The reference voltages are equal among all DGUs, that is, Vref ,i = Vref , ∀i ∈ ν. Definition 1 (Voltage balancing). Voltage balancing is achieved if v(∞) = Vref , where v(∞) is the steady-state average voltage of all PCCs. Definition 2 (Current sharing). Current sharing is achieved if

Iti∞ Itis

=

Itj∞ Itjs ,

∀i, j ∈ ν, that is, the load currents are shared proportionally to DGUs’ rated currents.

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2.3 UIO-based detector We consider the cyber-attacks that inject attack vectors into the communication links. Since DGU j shares the full output measurement vector to its neighbors, we can model the attack compromising communication link from DGU j to i as yci,j (t) = yj (t) + τ (t − Ta )φ i,j (t),

(4)

where φ i,j (t) is the injected attack vector, yci,j (t) is the data DGU i receives from DGU j, and τ (t − Ta ) is a step function with Ta time delay, that is, the attack is started at t = Ta . The attacker only compromises communication data between DGUs, which means that the measurements inside DGUs are secure, and the primary control is not affected. The UIO techniques are adopted to detect and identify the cyber-attacks in DCmG, where each DGU estimates neighbor DGUs’ states without knowing their inputs [14]. First, the state-space model of DGU j, j ∈ Nic is rewritten as a typical UIO structure: ¯ j d¯ j (t) + ωj (t) + bj kj ρ j (t), x˙j (t) = Akj xj (t) + E (5) yj (t) = Cj xj (t) + ρ j (t), where Akj = Ajj + bj kTj ∈ R3×3 , d¯ j (t) is the input vector of DGU j unknown ¯ j is the corresponding parameter matrix. Then, a full to neighbor DGU i, and E order UIO is given as follows: z˙i,j (t) = Fj zi,j (t) + T j bj u¯ j (t) + Kˆ j yci,j (t), (6) xˆi,j (t) = zi,j (t) + Hj yci,j (t), where zi,j (t) denotes the state vector of the observer, xˆi,j (t) denotes the estimated state vector of DGU j by DGU i, u¯ j = 0, and Fj , T j , Kˆ j , Hj ∈ R3×3 are parameters of the observer. Lemma 1. If and only if the following conditions are satisfied, ¯ j ) = rank(E¯ j ) 1. rank(Cj E 2. the pair (Cj , T j Akj ) is detectable, it is guaranteed that the estimations of UIO (6) can converge to the actual system states. According to the characteristics of the system matrix Cj = I [19], the conditions in Lemma 1 are satisfied obviously. Therefore, matrices Fj , T j , Kˆ j , Hj are designed as that in [14], from which a stable matrix Fj is obtained. In the absence of cyber-attacks, the state estimation error vector i,j (t) = xj (t) − xˆi,j (t) can be obtained as t Fj t 1 eFj (t−τ ) σ 2i,j (τ )dτ , (7) i,j (t) = e σ i,j (0) − Hj ρ j (t) + 0

False data injection attacks and countermeasures Chapter | 10 269

where σ 1i,j (0) = i,j (0) + Hj ρ j (0) and σ 2i,j (t) = T j ωj (t) + (T j bj kj − Kˆ j )ρ j (t). Since Fj is stable, there always exists positive κ, μ such that ||eFj t || ≤ κe−μt . The estimation residual vector is ri,j (t) = yci,j (t)−Cj xˆi,j (t) = i,j (t)+ρ j (t), from which the upper bound of |ri,j (t)| can be obtained as t −μt 1 κe−μ(t−τ ) σ¯ 2i,j (τ )dτ , (8) r¯i,j (t) = κe σ¯ i,j (0) + |T j |ρ¯ j + 0

where = ¯ i,j (0) + = |T j |ω¯ j + |T j bj kj − Kˆ j |ρ¯ j , and ¯ i,j (0) is the bound of the initial state estimation error such that ¯ i,j (0) ≥ |i,j (0)| always holds. Then, we can obtain the new residual vector under attacks as r˜i,j (t) = ri,j (t) + rai,j (t), t ≥ Ta , where rai,j (t) is the attack impact on detection residuals as t a Fj (t−Ta ) Hj φ i,j (Ta ) + T j φ i,j (t) − eFj (t−τ ) Kˆ j φ i,j (τ )dτ . (9) ri,j (t) = e σ¯ 1i,j (0)

|Hj |ρ¯ j , σ¯ 2i,j (t)

Ta

Lemma 2. If there holds |rai,j (t)| > 2¯ri,j (t),

t > Td ,

(10)

the UIO-based detector (6) can detect the attack in link (i, j). Remark 1. The developed residual threshold r¯i,j (t) can ensure the absence of false alarm [14]. However, due to the restrictive trade-off between false-alarm and missed-alarm in model-based detection methods [20], it is obvious that the residual threshold r¯i,j (t) cannot simultaneously guarantee zero missed-alarm, which means that the UIO-based detector is vulnerable to stealthy attacks.

2.4 Problem formulation Since the stealthy attacks can cause devastating damage to the system without being perceived, it is very important to investigate the potential stealthy attacks in DCmG. If the attacker knows some static parameters of the DGU system model and UIO-based detector, namely Akj , Cj , E¯ j , T j , Hj , Fj , Kˆ j , j ∈ ν, it is possible for the attacker to design the ZDS attack, which can cause disturbances to system states without affecting the detection residuals. Moreover, if the attacker can infer the bound of initial estimation error ¯ i,j (0), i, j ∈ ν, i = j by eavesdropping the system’s output measurements, and obtain the bound of measurement noise ρ¯ j , j ∈ ν, the attacker is likely to design the NDS attack by exploiting the inherent error/noise (i.e., the state estimation error and system noise), which will affect the detection residuals while remaining stealthy. Thus, the first problem is to investigate the existence of ZDS and NDS attacks in DCmG. If there exist potential stealthy attacks, it is urgent to analyze how these stealthy attacks will affect the performances of DCmG, such as the voltage balancing, the convergence rates of currents/voltages, etc. We are mainly concerned about the performances of two aspects: the steady-state performances

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(i.e., the voltage balancing and current sharing) and dynamic performances (i.e., the convergence rate of currents/voltages and instantaneous impacts on currents/voltages). Therefore, the second problem is to investigate fully the impacts of the potential stealthy attacks on DCmG.

3 Main results In this section, we present the main results of this chapter, including the potential stealthy attacks, their impacts analysis, and corresponding countermeasures. We mainly consider stealthy attacks that can bypass the UIO-based detector and still cause disturbances to DCmG. The definitions of ZDS and NDS attacks are given as follows. Definition 3 (ZDS attack). The attack is ZDS if the estimation residual vector is unaffected and the injected attack vector is not always zero, that is, a |ri,j (t)| = 03 , ∀t ≥ Ta , (11) φ i,j (t) = 03 , ∃t ≥ Ta . Definition 4 (NDS attack). The attack is NDS if the estimation residual vector is affected but still within the detection threshold, and the attack vector is not always zero, that is, a |ri,j (t)| = 03 and |˜ri,j (t)| ≤ r¯i,j (t), ∃t ≥ Ta , (12) φ i,j (t) = 03 , ∃t ≥ Ta .

3.1 Potential stealthy attacks First, we prove that there exists no ZDS attack in DCmG, and then obtain the NDS attack by exploiting the state estimation error and the measurement noise. Theorem 1. Under the constraints of the UIO-based detector, there exists no ZDS attack in current DCmG. Proof. According to Definition 3, the constraints can be obtained as T j φ˙i,j (t) = (Fj T j + Kˆ j )φ i,j (t), (Hj + T j )φ i,j (Ta ) = 0.

(13)

Moreover, from the design rules of the UIO-based detector [14], we have Fj T j + Kˆ j = T j Akj , Hj + T j = I3 . Combining with Eq. (13), we can get φ i,j (t) = 03 ,

t ≥ Ta ,

(14)

which means that the ZDS attack does not exist in DCmG. Actually, a similar definition of the ZDS attack has been proposed in the existing work [14], but the existence of the ZDS attack was not provided. In this chapter, we find that there exists no ZDS attack in DCmG equipped with the UIO-based detector. The NDS attack is obtained in Theorem 2.

False data injection attacks and countermeasures Chapter | 10 271

Theorem 2. By exploiting the state estimation error and measurement noise, the NDS attack is given as φ i,j (t) = eAkj (t−Ta ) φ i,j (Ta ), t ≥ Ta (15) φ i,j (Ta ) = eFj Ta I˜ σ˜ 1i,j (0), where the nonzero vector |I˜ σ˜ 1i,j (0)| ≤ |σ 1i,j (0)| + σ¯ 1i,j (0), and I˜ ∈ R3 is derived from a unit matrix I3 with its (2, 2)th diagonal entry replaced by 0. Proof. Since T j φ i,j (Ta ) = 0 and T j = I − Hj , we can get Hj φ i,j (Ta ) = ˜ Fj Ta is satisfied,1 then we can obtain the following φ i,j (Ta ). If eFj Ta I˜ = Ie equation by substituting Eq. (15) into Eq. (9): rai,j (t) = eFj t I˜ σ˜ 1i,j (0),

t ≥ Ta ,

and the residual vector under the attack can be rewritten as t 

Fj t ˜ 1 1 eFj (t−τ ) σ 2i,j (τ )dτ , Iσ˜ i,j (0) + σ i,j (0) + T j ρ j (t) + r˜i,j (t) = e

(16)

t ≥ Ta .

0

(17) With |σ 1i,j (0)| ≤ σ¯ 1i,j (0), there always exists a nonzero vector |I˜ σ˜ 1i,j (0)| ≤ |σ 1i,j (0)| + σ¯ 1i,j (0) such that |I˜ σ˜ 1i,j (0) + σ 1i,j (0)| ≤ σ¯ 1i,j (0), and |rai,j (t)| = 03 , ∃t ≥ Ta also holds. Therefore, |˜ri,j (t)| ≤ r¯˜i,j (t) ≤ r¯i,j (t),

t ≥ Ta , (18)  t where r¯˜i,j (t) = κe−μt |I˜ σ˜ 1i,j (0) + σ 1i,j (0)| + |T j |ρ¯ j + 0 κe−μ(t−τ ) σ¯ 2i,j (τ )dτ , t ≥ Ta denotes the tighter residual threshold considering the attack impacts on detection residuals. Thus, the residual vector r˜i,j (t) under the attack is still within the detection threshold, and it is proved that the attack is NDS. Remark 2. Since Akj is stable, the magnitude of φ i,j (t), which determines the attack impacts on DCmG, can be approximated as the initial value φ i,j (Ta ). Under the constraints of Eq. (15), the bound of φ i,j (Ta ) is positive proportional to the bounds of the initial state estimation error and the measurement noise, negative proportional to the attack start time Ta . The bound of the initial state estimation error could be very large if the system cannot infer the initial states accurately, especially with the disturbances of unknown inputs in DCmG equipped with the UIO-based detector. Moreover, the detection redundancy2 r¯i,j (t)−|ri,j (t)|, which is related to the state estimation error, measurement noise,

1. Fj is designed to be stable simultaneously. 2. The detection redundancy denotes the gap between the detection residual and its threshold. On the one hand, an appropriate detection redundancy can decrease the false-alarm when some unexpected but normal operations happen, that is, the uncertainty of the initial estimation error. On the other hand, a high detection redundancy will increase the missed-alarm, which allows the existence of NDS attacks.

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and the parameters in residual thresholds (e.g., κ, μ, T j , Hj , etc.), may allow the existence of the NDS attack with |I˜ σ˜ 1i,j (0)| ≥ |σ 1i,j (0)| + σ¯ 1i,j (0).

3.2 Attack impacts analysis Under the NDS attack in Theorem 2, we theoretically analyze the attack impacts on the voltage balancing, current sharing and the convergence rates of voltages/currents in DCmG. The secondary inputs of DCmG can be obtained from Eq. (3) as ˙ = −LDi ˜ t (t), ψ(t)

(19)   where ψ(t) = [ψ1 (t), . . . , ψN (t)]T , L˜ = kI L, D = diag I1s , . . . , I1s , and t1

tN

it (t) = [It1 , . . . , ItN ]T . The primary control loops can be approximated as unit gains [18] v(t) = vref + ψ(t),

(20)

where v = [V1 , . . . , VN ]T , and vref = [Vref ,1 , . . . , Vref ,N ]T . By utilizing Kirchhoff’s laws, the dynamic of secondary control inputs can be obtained as ˙ = −Qψ(t) − LDi ˜ l − Qvref , ψ(t)

(21)

˜ integrate Laplacian matrices of Gc where il = [IL1 , . . . , ILN ]T and Q = LDM and Gel . Under Assumption 1, Q = kI γ MDM, aij = Rγij , and moreover, voltage balancing and current sharing are exponentially achieved [18]. Some properties of matrix Q are given in Lemma 3. Lemma 3. Under Assumption 1, some properties of Q are derived: 1. ker(Q)=H1⊥ , range(Q)= H1 ; 2. Q is diagonalizable and has nonnegative eigenvalues, and its algebraic multiplicity of zero eigenvalue is one. Therefore, pi = (λi , qi ), i ∈ ν is used to denote the eigenvalue eigenvector pairs of Q, where λ1 = 0, 0 < λ2 ≤ · · · ≤ λN , q1 ∈ H1⊥ and q2 , . . . , qN is a basis of H1 . Thus, ψ(t) = e−Qt ψ(0) +

N  αi i=2

λi

(1 − e−λi t )qi ,

(22)

 1 1 ˜ where N i=2 αi qi = −LDil − Qvref ∈ H . If ψ(0) ∈ H , then ψ(∞) = 0 and voltage balancing is achieved at an exponential rate λ2 [18]. Since the attacker only compromises communication data between DGUs, the measurements inside DGUs are not affected. Therefore, the primary control ˜ where v˜ (t) and loops can be also approximated as unit gains v˜ (t) = vref + ψ(t), ˜ ψ(t) are the PCC voltage vector and the secondary input vector, respectively,

False data injection attacks and countermeasures Chapter | 10 273

under the attack. Given the additivity of attack impacts when compromising multilinks (multiply communication links), we only analyze the scenario that only one communication link (i, j) is compromised, and specifically φ i,j (t) is injected into the data transmitted from DGU j to DGU i. Under the NDS attack on link (i, j), the dynamics of secondary inputs are modeled as follows: ˙˜ ˜ − LDi ˜ l − Qvref + Ca φ2 (t)l, ψ(t) = −Qψ(t) where Ca =

kI aij Itjs ,

(23)

l = 0˜ ni ∈ Rn , 0˜ ni is derived from a zero vector with its

ith entry replaced by 1 to denote the destination of the compromised data (i.e., DGU i), and φ2 (t) is the second entry of φ i,j (t). The secondary input vector is ˜ decomposed as ψ(t) = ψ(t) + ψ a (t), where vector ψ a (t) denotes the attack impacts on secondary inputs modeled by the last element Ca φ2 (t)l in Eq. (23). Lemma 4. If Akj is diagonalizable3 and λi + βj = 0, ∀i ∈ ν, j ∈ {1, 2, 3},4 then the analytical expression of ψ a (t) can be obtained by solving the differential equation (23), ψ a (t) =

N 3   Ca ηj aj2 δi j=1 i=1

βj + λi

 eβj (t−Ta ) − e−λi (t−Ta ) qi ,

(24)

where Re(β1 ) ≤ Re(β2 ) ≤ Re(β3 ) < 0 are eigenvalues of Akj , and ηj , aj2 , δi are parameters defined to describe the decomposition of the attack vector. Since Akj  is diagonalizable, then φ i,j (t) = 3j=1 ηj eβj (t−Ta ) aj , t ≥ Ta , where aj denotes  the corresponding eigenvector of eigenvalue βj and 3j=1 ηj aj = φ i,j (Ta ). Next,   φ2 (t) = k 3j=1 ηj eβj (t−Ta ) aj = 3j=1 ηj eβj (t−Ta ) aj2 , t ≥ Ta , where k = [0, 1, 0]  and aj2 = kaj . Furthermore, l can be decomposed as l = N i=1 δi qi . By utilizing the result in Lemma 4, we can theoretically analyze the steadystate PCC voltages and output currents, which are related to the voltage balancing and current sharing in DCmG. Moreover, convergence rates of the voltages/currents in DCmG are investigated. The results are given in Theorem 3 as follows. Theorem 3. When there exists an NDS attack in link (i, j), current sharing can still be achieved while voltage balancing is violated, that is, the steady-state average voltage of all PCCs ˜v(∞) is increased by ψ a (∞). Moreover, the currents/voltages in DCmG can still converge exponentially at rate min{|(β3 )|, λ2 }.

3. When Akj is not diagonalizable, the stability and convergence properties can be analyzed in the same way. The system states can still converge to a stable value, but not at an exponential rate. 4. When βj = −λi , the final integral result (24) will be different, and the system can still achieve convergence but not at an exponential rate.

274 PART | II Energy management

ψ a (∞) = −

kI aij −1 kAkj φ i,j (Ta ). NItjs

(25)

Remark 3. Once any link in DCmG is compromised by the NDS attack, the steady-state voltages at all PCCs will be increased by a same magnitude ψ a (∞), which is related to the initial value of attack vector φ i,j (Ta ), system matrix Akj , link weight aij , etc. Whereas current sharing can still be achieved and the convergence rates of currents/voltages in DCmG are still exponentially fast. Moreover, the steady-state voltage deviation at each PCC caused by the attack can change the operating point set by the tertiary control [21], which is related to economically optimal operations. Thus, the attack may cause economic losses like increasing the generation costs [22]. Practically, the controllable output voltage of DC-DC buck converter with a certain input voltage is bounded by an interval [Vmin , Vmax ] [23], which is limited by the internal reference voltage and real-world circuits losses. Thus, the attacker should also consider the limitations when designing attack vectors to guarantee that the output voltages Vti , ∀i ∈ ν of converters are always bounded by the physical feasible interval, so as to be stealthy. Corollary 1. When multilinks5 ε˜ a are compromised by NDS attacks, the steady-state voltage deviation at each PCC is obtained as  kI aij − s kA−1 φ (Ta ). (26) ψ a (∞) = NItj kj i,j (i,j)∈˜ε a

Corollary 1 implies that the attacker can compromise multilinks to manipulate the steady-state PCC voltages in DCmG. Moreover, the attacker should choose ε˜ a appropriately to maximize the steady-state voltage deviation at each PCC and simultaneously minimize the instantaneous impacts on each DGU. Since high instantaneous current/voltage may trigger the over-current/voltage protection mechanisms, this may destroy the stealthiness.

3.3 Countermeasures Intuitively, the most effective countermeasure is to adopt encryption methods in communication links [24], which can protect the communication data from being compromised. However, the encryption process usually cause large communication delay, which can degrade the control performance, for example, the convergence rate, and even crash the system. To balance the trade-off between the security level and control performance, it is strongly advised to encrypt less critical communication data in the network, which may be dependent on the network structure and the link weight, to increase the security level as much

5. We think that link (i, j) and link (j, i) represent the bidirectional communication links, and can be compromised independently.

False data injection attacks and countermeasures Chapter | 10 275

as possible. Meanwhile, the resilient dynamic consensus algorithm can also be adopted to effectively eliminate the attack impacts on voltage balancing caused by the NDS attacks [21]. Specifically, the dynamic consensus algorithm can estimate the average voltage of DCmG and thus the PI control algorithm is adopted to regulate the average voltage to the reference point. However, the dynamic consensus algorithm requires extra communication data to be transmitted, and these data may be compromised by the attackers. Therefore, by combining the methods above, a novel system-level defense countermeasure is proposed to protect DCmG from the NDS attacks. First, the dynamic consensus algorithm is exploited to detect the NDS attack by comparing the estimated average voltage with the reference voltage, and the extra communication data is transmitted through encryption communication links, which will not affect the control performance. Then, the compensation method is activated once any NDS attacks are detected, which can eliminate the voltage unbalancing caused by the NDS attacks. The detail of the system-level countermeasure will be further investigated in future work.

4 Simulation In this section, we demonstrate impacts of the NDS attack on detection residuals, PCC voltages, and output currents through extensive simulations. The simulation of DCmG is conducted in Simulink/PLECS [25], composed of four DGUs, and the topology is showed in Fig. 2. Specifically, electrical parameters and primary controllers of DGUs are designed following [19]. Process and measurement noise bounds are set as ρ¯ j = [0.001, 0.01, 0] and ω¯ j = [0.001, 0.01, 0], j ∈ ν = {1, 2, 3, 4}, respectively. According to [14], the second columns of Hj are chosen as [−0.02, 0.98, −0.02], and Fj = diag{−1, −1, −1}, j ∈ ν. Then, T j , Kˆ j can be calculated. We construct an NDS attack with the maximum attack vector magnitude, and the attack impacts on detection residuals are shown in Fig. 3. The isolated

DGU1 R

2

R1

DGU2

DGU4

R23

4

R3

13

DGU3

Fig. 2 Electrical coupling and communication link model of DCmG, where solid lines denote power lines with resistances and dotted lines denote bidirectional communication links. The attacker injects attack vectors into communication links between DGU 3 and its neighbor DGUs j, j ∈ N3c = {1, 2, 4}.

276 PART | II Energy management

0.3 0.2 0.1 0 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0.3 0.2 0.1 0 0.3 0.2 0.1 0

Time (s)

Fig. 3 Element-by-element comparison of estimation residuals |r3,j (t)|, j ∈ N3c with residual bounds r¯3,j (t), where |r3,j (t)|i denotes the ith element of |r3,j (t)|. Solid lines denote the estimation residuals and dotted lines denote the residual bounds; moreover, the colors of lines represent different DGUs as the legend shows. The secondary control is activated at Ts = 3 s, before which the residuals are all zero, and the attack is started at Ta = 6 s.

DGUs are interconnected by power lines at t = 0, the secondary control is activated at Ts = 3 s, and attack is started at Ta = 6 s. The attacker compromises communication data transmitted between DGU 3 and its neighbor DGUs j, j ∈ N3c = {1, 2, 4}, and the bound of initial state estimation error6 is ¯ 3,j (Ts ) = 0.3 × [1, 1, 1]T , ∀j ∈ N3c . Specifically, we investigate the maximum σ˜ 13,j (Ts ), ∀j ∈ N3c under the constraints of UIO detectors, which depend not only on the state estimation error and measurement noise, but also on the parameters of the detection thresholds (e.g., κ, μ, T j , Hj , j ∈ N3c , etc.). As shown in Fig. 3, when σ˜ 13,j (Ts ) = αmax σ¯ 13,j (Ts ) = 5.5σ¯ 13,j (Ts ), the estimation residuals are still within the bounds, from which the attack vector is obtained φ 3,j (t) = ˜ Fj (Ta −Ts ) σ˜ 13,j (Ts ), t ≥ Ta , j ∈ N c . eAkj (t−Ta ) Ie 3

6. The UIO is enabled at Ts .

A

False data injection attacks and countermeasures Chapter | 10 277

DGU 1 DGU 2 DGU 3 DGU 4

20 0

1

2

3

4

5

6

7

8

9

10

11

12

p.u.

1

0

(V)

DGU1 DGU2 DGU3 DGU4

0.5

1

2

3

4

5

6

7

8

9

10

11

12

41 DGU1 DGU2 DGU3 DGU4

40.5 40 39.5 1

2

3

4

5

6

7

8

9

10

11

12

(V)

40.5 Average voltage

40 1

2

3

4

5

6

7

8

9

10

11

12

Time (s)

Fig. 4 After DGUs are interconnected by power lines, evolutions of output currents in A (ampere), output currents in per-unit (p.u., i.e., Iti /Itis ), output voltages at PCCs and average output voltage in V (volt) are showed in sequential. The colors of lines represent different DGUs as the legend shows.

Under the NDS attack, the PCC voltages and output currents are depicted in Fig. 4, from which we analyze the voltage balancing, current sharing, and instantaneous attack impacts on currents/voltages. For the secondary control, we set kI = 1, aij = R1ij , ∀i, j ∈ ν, i = j according to Assumption 1. Moreover, the reference voltage is Vref = 40 V, the load currents are IL1 = 10 A, IL2 = 8 A, IL3 = 12 A, IL4 = 14 A, and the rated output currents are Itis = 20 A, s = 35 A. According to Fig. 4, the current sharing and voltage i ∈ {1, 2, 4}, It3 balancing are both achieved after activating the secondary control at Ts second. When the attack is launched at Ta second, the dynamic of currents/voltages are affected and eventually the steady-state average PCC voltage is increased  ∞ by |ψ˜ | = | j∈{1,2,4} C4a kA−1 kj φ 3,j (Ta )| = 0.4107 V, which increases the supply costs, whereas the current sharing is still achieved. The instantaneous attack impacts on currents/voltages can be huge if the DGU receives many compromised data, that is, DGU 3, which may trigger the over-current/voltage protection mechanisms. Therefore, the attacker should compromise multilinks connecting different nodes, which can decrease the instantaneous attack impacts on each DGU. Concrete analysis of the attack impacts on economic losses will be left as future work.

278 PART | II Energy management

5 Conclusions In this chapter, we mainly investigated the NDS attack, one of the most threatening FDI attacks against the UIO-based detector in DCmGs, and theoretically analyzed the attack impacts on current sharing, voltage balancing, and convergence rates of currents/voltages. Interestingly, we found that the attack impacts are closely related to the bounds of initial state estimation error and measurement noise. By appropriately choosing attack links ε˜ a , the steadystate average voltage of all PCCs could be deviated from the reference point, which may increase the output power of the DC-DC buck converter and thus cause more generation costs, whereas the current sharing is still achieved and currents/voltages in DCmG will still converge at an exponential rate. Moreover, the novel system-level countermeasure is briefly demonstrated against the NDS attack, which will be further investigated in future work. Compared with ZDS attacks, the state disturbances caused by the NDS attack are limited by the bounds of initial state estimation error and measurement noise. But it provides a novel perspective to investigate stealthy attacks in DCmGs, which exploits the common state estimation error and measurement noise in physical plants, and can cause economic losses. Moreover, new design criteria can be provided to enhance the security of DCmGs, such as appropriately designing the parameter matrices of UIOs to decrease the detection redundancy r¯i,j (t) − |ri,j (t)|, that is, the missed-alarm. Considering the uncertainty of real physical plants, it is difficult for attackers to acquire an exact system model. Therefore, investigating whether it is possible for attackers to design the NDS attack without an accurate system model will be another direction of future work.

References [1] N. Hatziargyriou, Microgrids: Architectures and Control, John Wiley & Sons, New York, NY, USA, 2014. [2] H. Lotfi, A. Khodaei, AC versus DC microgrid planning, IEEE Trans. Smart Grid 8 (1) (2017) 296–304. [3] T. Dragiˇcevi´c, X. Lu, J.C. Vasquez, J.M. Guerrero, DC microgrids—part I: a review of control strategies and stabilization techniques, IEEE Trans. Power Electron. 31 (7) (2016) 4876–4891. [4] A. Bidram, A. Davoudi, Hierarchical structure of microgrids control system, IEEE Trans. Smart Grid 3 (4) (2012) 1963–1976. [5] R. Deng, P. Zhuang, H. Liang, CCPA: coordinated cyber-physical attacks and countermeasures in smart grid, IEEE Trans. Smart Grid 8 (5) (2017) 2420–2430. [6] R. Deng, H. Liang, False data injection attacks with limited susceptance information and new countermeasures in smart grid, IEEE Trans. Ind. Inf. 15 (2018) 1619–1628. [7] R. Deng, P. Zhuang, H. Liang, False data injection attacks against state estimation in power distribution systems, IEEE Trans. Smart Grid 10 (2018) 2871–2881. [8] Y. Liu, P. Ning, M.K. Reiter, False data injection attacks against state estimation in electric power grids, ACM Trans. Inf. Syst. Secur. 14 (1) (2011) 13. [9] M.J. Hossain, H.R. Pota, M.A. Mahmud, M. Aldeen, Robust control for power sharing in microgrids with low-inertia wind and PV generators, IEEE Trans. Sustain. Energy 6 (3) (2015) 1067–1077.

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[10] H.J. Liu, M. Backes, R. Macwan, A. Valdes, Coordination of DERs in microgrids with cybersecure resilient decentralized secondary frequency control, in: Proceedings of the 51st Hawaii International Conference on System Sciences (HICSS), 2018. [11] L.-Y. Lu, H.J. Liu, H. Zhu, Distributed secondary control for isolated microgrids under malicious attacks, in: North American Power Symposium (NAPS), IEEE, 2016. [12] O.A. Beg, T.T. Johnson, A. Davoudi, Detection of false-data injection attacks in cyber-physical DC microgrids, IEEE Trans. Ind. Inf. 13 (5) (2017) 2693–2703. [13] A. Teixeira, H. Sandberg, K.H. Johansson, Networked control systems under cyber attacks with applications to power networks, in: American Control Conference (ACC), IEEE, 2010. [14] A.J. Gallo, M.S. Turan, P. Nahata, F. Boem, T. Parisini, G. Ferrari Trecate, Distributed cyber-attack detection in the secondary control of DC microgrids, in: European Control Conference (ECC), 2018. [15] T.M. Chen, Stuxnet, the real start of cyber warfare? [Editor’s Note], IEEE Netw. 24 (6) (2010) 2–3. [16] F. Pasqualetti, A. Bicchi, F. Bullo, Consensus computation in unreliable networks: a system theoretic approach, IEEE Trans. Autom. Control 57 (1) (2012) 90–104. [17] A. Teixeira, I. Shames, H. Sandberg, K.H. Johansson, A secure control framework for resource-limited adversaries, Automatica 51 (2015) 135–148. [18] M. Tucci, L. Meng, J.M. Guerrero, G. Ferrari-Trecate, Stable current sharing and voltage balancing in DC microgrids: a consensus-based secondary control layer, Automatica 95 (2018) 1–13. [19] M. Tucci, S. Riverso, J.C. Vasquez, J.M. Guerrero, G. Ferrari-Trecate, A decentralized scalable approach to voltage control of DC islanded microgrids, IEEE Trans. Control Syst. Technol. 24 (6) (2016) 1965–1979. [20] S. Attuati, M. Farina, F. Boem, T. Parisini, Reducing false alarm rates in observer-based distributed fault detection schemes by analyzing moving averages, IFAC-PapersOnLine 51 (24) (2018) 473–479. [21] V. Nasirian, S. Moayedi, A. Davoudi, F.L. Lewis, Distributed cooperative control of DC microgrids, IEEE Trans. Power Electron. 30 (4) (2015) 2288–2303. [22] C. Zhao, J. He, P. Cheng, J. Chen, Analysis of consensus-based distributed economic dispatch under stealthy attacks, IEEE Trans. Ind. Electron. 64 (6) (2017) 5107–5117. [23] J. Tucker, Understanding output voltage limitations of DC/DC buck converters, Analog Applications Journal Texas Instruments, 2008. [24] X. Li, M. Liu, R. Zhang, P. Cheng, J. Chen, An industrial control system testbed for the encrypted controller: demo abstract, in: Proceedings of the 9th ACM/IEEE International Conference on Cyber-Physical Systems, IEEE Press, 2018, pp. 343–344. [25] J. Allmeling, W. Hammer, PLECS-User Manual, 2013.

Index

Note: Page numbers followed by f indicate figures and t indicate tables.

A Active power reference, 131–132 Active/reactive power sharing, utilization ratio of, 78, 79f Adjacency matrix, 157 Alternating current (AC) microgrids, 29, 63, 83, 263 Alternating direction method of multipliers (ADMM), 206, 218–221, 236–238 Area control error (ACE), 154

B Backup DG plug-and-play, 43–44, 43–44f Band-pass filter, 156–157, 156f Battery swapping, electric vehicles (EVs) advantages, 203–204 aggregate assignment information, 206 binary variables, relaxation of, 206 centralized solution, 215–217 average ratio of the number of electric vehicles, 231–232, 232f computational effort, 228–230, 229–230f generalized Benders decomposition, optimality of, 228, 228f nearest-station policy, 225–226, 230–231, 231f optimal assignments, 226–228 relative reduction in objective value, 230, 231f second-order cone programming relaxation, exactness of, 228, 229t voltage drop violation (VDV), 230–231, 231f challenges, 204 convex relaxation method, 236 distributed framework, 206f distributed solutions, 208 convergence, 232–233, 233f relaxations, 218

scalability, 235–236 second-order cone programming relaxation, exactness of, 234, 235t suboptimality, 233–234 via ADMM, 218–221, 236–238 via dual decomposition, 221–224 generalized Benders decomposition, 205 Lagrange multipliers, 206 literature on, 207–208 model limitations, 236 network model DC power flow equations, 210 DistFlow equations and second-order cone programming relaxation, 211–212 fix-point linearization of power flow equations, 210–211 linearization/convexification methods, 210 operational constraints, 212–213 single-phase microgrid network, 209 numerical simulation setup, 224–225, 225t and optimal power flow (OPF) problem, 205 optimal scheduling problem, 209, 236 scheduling, 213–215 service stations, 208–209 State Grid, 204, 208–209 transpose and Hermitian transpose, 209 Bus agents (BAs), 126, 132 42-Bus test system diagram, 78, 79f

C Call-auction method, 249, 253 Canadian urban benchmark distribution system, 129, 129f Centralized-distributed control strategy, 104 Centralized energy storage systems (CESSs), 150–151

281

282 Index Closed-loop optimal control algorithm, 62–63 Communication delay analysis, 70, 74–75, 75f Communication network, 266, 267f aggregated energy storage system (ESSs), 157 Communication technology, 244 Consensus-based algorithm, 178 Convergence speed analysis, 195, 197f Convex cost function, 177 Convexification, 205, 210 Coordinated active power dispatch control, 83–103 Corporation dispatch control of units, 87–88 Corporation optimization, 84–85 Cyber-attacks, 263–264, 268–269 in DC microgrids (DCmGs), 264 intrusion detection of, 126 in microgrids, 125–126 Cyber-communication layer, 129, 129f

D DC. See Depreciation cost (DC) DC/DC buck converters controller design cascade control structure, 17, 17f current tracking loop, 19–20 voltage regulation loop, 18–19 control objectives, 16–17 mathematical model of, 7, 7f simulation results against input voltage variation, 22–23, 23–24f against load resistance variation, 21–23f, 21t, 22–23 against reference voltage variation, 23–25, 24–25f system parameters, 20, 20t DC microgrids (DCmGs), 263–264 cyber-attacks in, 264 electrical coupling and communication link model of, 275f false data injection (FDI) attacks in, 264–265 nonzero-dynamics stealthy (NDS) attack in, 264–265 objectives of, 263–264 Decentralized economic dispatch, 178 Decentralized power sharing method, 31, 31f

Demand response (DR) load, smart grids controllable loads, 104 (see also Thermostatically controlled loads (TCLs)) control techniques, 103 demand-side resources, 103–104 flexible loads, 104 look-ahead economic dispatch (see Economic dispatch problems (EDPs)) renewable energy power injection, 116, 116f rigid loads, 104 Denial-of-Service (DoS) attacks, 125–126 Deposit system of energy (DSE), 246, 254–256 economic model of, 255–256 figure of deposit energy currency (FDEC), 254 goals of, 248 in microgrid, 254–255 model, 248, 254 operation model of, 254–255 power statement of, 254–255 Depreciation cost (DC), 255–256 DESSs. See Distributed energy storage systems (DESSs) DGUs. See Distributed generation units (DGUs) Digraph, 84–85 Direct current (DC) microgrid, 29, 83 islanded DC MG (see Islanded DC MG) Directed graph, 84–85 Discrete-time closed-loop system, 115 Discrete-time difference equations, 115 Dispatch optimization of units, 86–87 DistFlow equations, 211–212 Distributed bisection algorithm (DBA), 188–190 for aggregate demand, 184–186 algorithm comparison, 195–199 consensus-like iterations, 196–197, 197f convergence analysis and stopping criteria, 190–192 for feasibility test, 187 features of, 178–179 IEEE 118-bus system implementation, 199, 200f Lagrange multiplier, 184, 188 Distributed energy storage systems (DESSs), 150 centralized secondary control structures, 151

Index 283

coordinated control scheme, 151 for local/decentralized frequency support, 151 Distributed finite-time control energy storage aggregator (ESA) communication graph, 157 finite-time consensus control, 158–160 numerical illustrations, 162–164 stability analysis, 160–162 Distributed generation units (DGUs), 263 hierarchical control structure, 267f Distributed generators (DGs), 59–60 active power outputs, 95–97, 95–96f, 98f capacity constraint, 88 communication topology, 97, 98f, 100, 100f cost and capacity parameters of, 97 full participation of, 94–97 graph theory, 61–62 plugging-in, 100–103 power output optimization, 83–84 private parameters, 93, 94t, 100, 100t profit maximization, 87–88 regulation capacities, 88 Distributed λ-iteration algorithm, 84 of active power solution without power constraints, 88–91 solution with power constraints, 92–93 cost coefficients and capacity constraints, 93, 94t dispatch optimization of units, 86–87 distributed generators and energy storage units full participation of, 94–97 private parameters, 93, 94t economic operation problem, 103 energy storage units and time-varying demand, exit of, 97–100 plugging-in DGs, 100–103 seven-bus microgrid system and physical topology structure, 93, 94f Distributed load sharing control, FDI attacks, 132–134 stable region, 135–136, 136f system performances under attack strategy 1 average load information discovery, 137, 138f, 140f average power generation information discovery, 137, 139f, 141f frequency, 137, 138f, 140f

voltage magnitudes, 137, 137f, 139f system performances under attack strategy 2 average load information discovery, 140–142, 142f average power generation information discovery, 140–142, 143f frequency, 140–142, 142f voltage magnitudes, 140–142, 141f Distributed optimization, economic dispatch problem (EDP), 177–178 Distributed pinning consensus algorithm, 85–86 Distributed pinning control, 85–86, 104 communication matrix, 118–120 of multiple aggregated TCLs, 112–116, 116b pinning links, 118–120 Distributed secondary control with pinning gain, 39–40 voltage restoration control, 30, 33 controller design, 33–34 stability analysis, 34–39 Distributed secondary control strategy closed-loop optimal control algorithm, 62–63 finite-time approximate consensus approach, 60 graph theory, 61–62 objectives, 59–60 peer-to-peer communication protocol, 60 secondary frequency control and active power sharing, 65–67 and reactive power sharing, 67–70 simulation analysis convergence analysis, 73, 74f five-bus microgrid diagram, 71, 72f five-bus microgrid parameters, 71, 72t influence analysis of bounded control input, 76–77, 76–77f influence analysis of communication delay, 74–75, 75f performance evaluation, 72–73, 73f plug-and-play capability analysis, 77, 78f robustness analysis against uncertainties, 75–76, 76f scalability test, 78–79, 79f Droop coefficients, 64 Droop control, 29–32, 59 DSE. See Deposit system of energy (DSE)

284 Index

E EBS. See Energy bank system (EBS) EC. See Energy currency (EC) Economic dispatch problem (EDP) centralized solution, 181–182 consensus-like algorithm, 183–184 cost functions, 177 distributed bisection algorithm (see Distributed bisection algorithm (DBA)) distributed optimization, 177–178 graph theory, 182–183 nonnegative matrices, 182–183 numerical examples, 193, 193f convergence speed analysis, 195, 197f with nonquadratic cost functions, 195, 196f with quadratic cost functions only, 193–195, 193t, 194f, 198f parallel micro genetic algorithm, 177 problem formulation assumption, 179 communication networks, 181 nonquadratic cost function, 180 quadratic cost functions, 180 Economic dispatch problems (EDPs), 84 convex optimization, 112 energy mismatch, 111–112 energy scheduling, 112 load agents and generating units, 111, 122 for power output distributions, 110–111 reference power trajectory, 112, 117–118, 119t, 120f total economic cost, 110–111 total regulation cost, 111 for unit commitment, 110–111 Economic model, of deposit system of energy (DSE), 255–256 Electrical trading mechanism, 244 Electric vehicles (EVs) battery, 150 battery swapping (see Battery swapping, electric vehicles (EVs)) charging, 203–204, 207 electrification, 203 hybrid, 150 impact of, 203 V2G capacities, 150 Electrification, 203 Energy bank system (EBS), 244–245, 247, 254, 256, 260–261 case study, 256–260 simulation, 257–260, 258–259f, 260t

system description, 256 complex environment of, 246f customers, 244, 246, 257 definition, 245–246 function, 246 model, 244–245, 247f structure, 246–249 performance, 257 structure for, 248 users, 244–245, 257 Energy bank system central processing unit (EBSCPU), 248–249 Energy currency (EC), 245, 247, 260–261 Energy Market Authority of Singapore, 150 Energy storage aggregator (ESA) distributed finite-time control communication graph, 157 finite-time consensus control, 158–160 numerical illustrations, 162–164 stability analysis, 160–162 external and internal behavior, 150 finite-time vs. linear consensus control protocol, 170–173 frequency control architecture multiarea microgrids, 153–154, 169–170 system overview, 153, 153f normal operation, 167, 167–169f proposed control method, parameters of, 164, 165t proposed disturbance observer band-pass filter, 156–157, 156f system disturbance observer, 155–156 system contingency, 164–167 three-area microgrids, parameters of, 164, 165t Energy storage devices, 243 Energy storage system (ESS) contributions, 152 definition, 150 energy storage aggregator (see Energy storage aggregator (ESA)) frequency control architecture, 151 large-scale centralized ESSs (CESSs), 150 literature survey, 150–152 small-scale distributed ESSs (DESSs), 150 transportation application, 150 utility grid, 150 Energy storage units (ESUs), 83 active power outputs, 95–97, 95–96f, 98f communication topology, 100, 100f full participation of, 94–97

Index 285

private parameters, 93, 94t regulation capacities, 88 and time-varying demand, exit of, 97–100 Energy trading, 243–245 applications of, 243–244 conduits of, 247 in multiple microgrid systems, 260–261 ESS. See Energy storage system (ESS) Extended state observer (ESO), 8–10

F False data injection (FDI) attacks, 125–126, 263–264 adaptive Markov strategy, 126 arbitrary false data, 142–143 attack impacts analysis, 272–274 bus agents (BAs), 126, 132 centralized operation system, 143 communication network, 266 countermeasures, 274–275 in DC microgrids (DCmGs), 264–265 distributed load sharing control, 132–134 stable region, 135–136, 136f system performances under attack strategy 1, 136–137, 137–141f system performances under attack strategy 2, 138–142, 141–143f on dynamic microgrid partitioning process, 126 dynamic model, 266–267 Gaussian-mixture model-based detection method, 126 impacts of, 134–135 intrusion detection method, 126 notations, 127t physical network, 265 potential stealthy attacks, 270–272 preliminaries and problem formulation, 265–270 problem formulation, 269–270 semidefinite programming-based optimal control policy, 126 simulation, 275–277 trust-aware defending method, 143 UIO-based detector, 268–269 FDEC. See Figure of deposit energy currency (FDEC) FDI attacks. See False data injection (FDI) attacks

Figure of deposit energy currency (FDEC), 248, 251, 260–261 Finite-time consensus algorithms, 151–152 Finite-time consensus control, 158–160 vs. linear consensus control, 170–173 First-order discrete-time system, 85–86 First-order sliding mode control (FOSMC), 4 Five-bus microgrid diagram, 71, 72f parameters, 71, 72t Fossil fuels, 149

G Gaussian-mixture model-based detection method, 126 Generalized Benders decomposition, optimality of, 228, 228f Global price index, 178 Graph theory distributed secondary control strategy, 61–62 economic dispatch problem (EDP), 182–183

H Hamiltonian function, 62 Hamilton-Jacobi-Bellman equation, 62 Hybrid AC/DC microgrid, 29, 83 structure, 83, 84f Hybrid energy storage system, 243

I IEEE-9 bus system, 116, 116f Inverter-based microgrid structure cyber-communication layer, 129, 129f framework of, 128, 128f physical layer, 128–129 Islanded DC MG control parameters selection, 40 DC bus voltage, 29–30 decentralized power sharing method, 31, 31f distributed secondary control with pinning gain, 39–40 distributed secondary voltage restoration control, 30, 33 controller design, 33–34 stability analysis, 34–39 droop control, 29–32

286 Index Islanded DC MG (Continued) experimental validation with communication time delay, 53–55, 54–56f with constant power load, 52–53, 53f experiment setup, 45, 50f MG system and the controller parameters, 45, 50t with resistant load, 46–51, 51–52f objectives, 32–33 robustness test with temporary fault, 45–49, 48–49f simulation results backup DG plug-and-play, 43–44, 43–44f communication graph, 41f, 42–43 current output of test islanded DC MG with pinning gains, 42–43, 44f different pinning gains, 43–44f, 45, 46–47f distributed secondary control stages, 41–45 primary controller parameters, 41, 42t secondary controller parameters, 41, 42t voltage output of test islanded DC MG with pinning gains, 42–43, 43f voltage restoration methods, comparison of, 45, 49t Islanded microgrid distributed secondary control strategy (see Distributed secondary control strategy) droop control, 59 frequency control of, 59 primary control, 59 secondary control, 59 tertiary control, 59

L Lambda-iteration method, 178 Laplacian matrix, 85, 157 Lead-acid battery, 256 Leader-follower consensus algorithm, 151–152 Leader-follower finite-time consensus control protocol, 159 LELP. See Listed energy and listed-price (LELP) Li-ion battery, 256 Linearization, 207, 210 fix-point linearization, 210–211

Listed energy and listed-price (LELP), 244 model, 248–249, 251–254 assessment indices, 253–254 buying listed-price, 253 selling/buying listed-energy, 251, 251f selling listed-price, 252–253 Listed-price coefficient (LPC), 252–253 selling and buying, 253 Load frequency control (LFC) model, 153–154, 164 LPC. See Listed-price coefficient (LPC) Lyapunov function, 63, 65–66, 68, 159–160

M Malaysia power system, 149 MAPE. See Mean absolute percentage error (MAPE) Mean absolute percentage error (MAPE), 253 MG. See Microgrid (MG) Microgrid (MG), 247, 249, 256–260, 263 alternating current (AC), 29, 63, 83, 263 in autonomous mode, 125, 130–131 small-signal model, 130–131 and batteries, 257 corporation dispatch control of units, 87–88 cyberattacks in, 125–126 definition, 29 deposit system of energy (DSE) in, 254–255 dispatch optimization of units in, 86–87 feature of, 83 isolated, 149 multiarea, 153–154, 169–170 multiple, 248, 257–261 operational constraints, 212–213 operation optimization problem, 84 physical models of, 248–249 power statement of deposit system of energy in, 254–255 storage devices of, 254–255 system parameters, 135, 136t three-area microgrids, parameters of, 164, 165t types, 29, 83 Microgrid central controller (MGCC), 83 bidirectional communication structure, 83 central decision-maker, 84 direct communications with, 103 Multiple microgrids, 248, 257–261

Index 287

N Nearest-station policy, 225–226, 230–231, 231f Network size problem, 184 Nonquadratic cost function, 180, 195, 196f Nonzero-dynamics stealthy (NDS) attack, 270, 272, 274–275, 277–278 in DC microgrids (DCmGs), 264–265 Normalized adjacency matrix, 183, 185

O One-dimensional continuous-time integrator multiunit system, 85 Optimal distributed control strategy algorithm implementation, 71, 71f communication delay analysis, 70 DGs’ frequency/voltage magnitude, 64–65 distributed secondary frequency control and active power sharing, 65–67 and reactive power sharing, 67–70 droop coefficients, 64 primary control system, block diagram of, 63, 63f simulation analysis convergence analysis, 73, 74f five-bus microgrid diagram, 71, 72f five-bus microgrid parameters, 71, 72t influence analysis of bounded control input, 76–77, 76–77f influence analysis of communication delay, 74–75, 75f performance evaluation, 72–73, 73f plug-and-play capability analysis, 77, 78f robustness analysis against uncertainties, 75–76, 76f scalability test, 78–79, 79f Optimal power flow (OPF) problem, 205 Optimal scheduling problem, 209, 236

P Parallel micro genetic algorithm, 177 economic dispatch problem (EDP), 177 Performance index, of dynamic system, 62 Perron matrix, 85 Physical network, false data injection (FDI) attacks, 265 Piecewise linear cost function, 177 Plug-and-play capability analysis, 77, 78f

Point of common coupling (PCC), 274 voltages, 265, 277 Power converters, mathematical model of DC/DC buck converters, 7, 7f sliding mode control for (see Sliding mode control (SMC)) three-phase two-level AC/DC power converters, 5–7, 6f Power customers, 244–245 Power flow equations DC, 210 fix-point linearization of, 210–211 Pricing model, 247–249 Proportional integral (PI) controller tracks, 266–267

Q Quadratic cost functions, 180, 193–195, 193t, 194f, 198f

R RDG. See Renewable distributed generation (RDG) Reference temperature setpoint’s dynamics, 113 Renewable distributed generation (RDG), 243–249, 254, 256–257 Renewable energy, 243 Renewable energy resources (RERs) growing capacity, 59 islanded microgrid (see Islanded microgrid) penetration of, 59 Renewable energy sources (RESs), 149

S Scalability test, 78–79, 79f Second-order cone programming (SOCP) relaxation, 211–212 exactness of centralized solution, 228, 229t distributed solutions, 234, 235t Second-order sliding mode control (SOSMC), 4–5 Singapore power system, 149 Sliding mode control (SMC) for DC/DC buck converters, 16–25 design steps, 3 feature of, 3 first-order sliding mode control (FOSMC), 4

288 Index Sliding mode control (SMC) (Continued) second-order sliding mode control (SOSMC), 4–5 sliding surface, 4 for three-phase AC/DC power converters, 8–16 Sliding surface, 4 Small-signal model interface block, 131 inverter-based DG block, 130–131 local primary DG control loops, 130–131, 130f network block, 131 whole system model, 131 Smart grids, 178 demand response (DR) load controllable loads, 104 (see also Thermostatically controlled loads (TCLs)) control techniques, 103 demand-side resources, 103–104 flexible loads, 104 look-ahead economic dispatch (see Economic dispatch problems (EDPs)) rigid loads, 104 State Grid, 204 State-of-charge (SoC), 151–152 Stealthy attacks, 269 damage of, 264–265 in DC microgrids (DCmGs), 278 potential, 264–265, 269–272 Storage devices, 243 capacity of, 244–245 energy, 243 Storage system, hybrid energy, 243 Sum consensus problem, 184–185 Super twisting algorithm (STA), 4 controller, 5 System contingency, 164–167 energy storage aggregator (ESA), 164–167 System disturbance observer, 155–156 System dynamic model active power reference, 131–132 small-signal model, 130–131

T TDOC. See Total daily operation cost (TDOC)

TECs. See Traditional energy companies (TECs) TEMs. See Traditional energy modes (TEMs) Thermostatically controlled loads (TCLs), 103–104 active power distribution, 105–106 aggregate evaluation, 109–110, 117 ancillary services, 105 basic model, 106–109 bilinear aggregate model, 104–105 control of, 118–122, 121–122f distributed pinning control, 112–116 heterogeneous aggregators, 104–105 power tracking curves, 120–121, 121–122f relative incremental temperature and control input, 120–121, 121–122f temperature setpoint, 105 Three-phase AC/DC power converters controller design capacitor voltage regulation, 10–11 cascaded control scheme, 8–9 extended state observer (ESO), 8–10 grid current tracking, 11–12 mathematical model of, 5–7, 6f mode uncertainties, 8 simulation results controller parameters, 13, 13t DC-link voltage regulation performance, 13–14, 14f grid current and corresponding voltage, 14–16, 14f harmonic spectrum of grid current, 14–16, 15f id tracking performance, 14–16, 15f iq tracking performance, 14–16, 16f observed disturbance from ESO, 14–16, 17f plant parameters, 13, 13t Total daily operation cost (TDOC), 255 Trading mechanism, electrical, 244 Trading model, 247, 249–251 call auction, 249–250 rule of maximum transaction volume, 250 rule of treaty violation, 250–251 Traditional bank system, 245t Traditional energy companies (TECs), 245 Traditional energy modes (TEMs), 245–246 Traditional trading mechanism (TTM), 244–246

Index 289

Traditional trading system, 245t TTM. See Traditional trading mechanism (TTM)

U Unknown input observer (UIO) detector, 268–269 techniques, 264

V Virtual energy bank, 245 Voltage drop violation (VDV), 230–231, 231f

Z Zero-dynamics stealthy (ZDS) attack, 264–265, 270, 278