Advanced Design and Control of Active Power Filters 2013009428

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Advanced Design and Control of Active Power Filters
 2013009428

Table of contents :
Cover
ADVANCED DESIGN AND CONTROL OF ACTIVE POWER FILTERS
ADVANCED DESIGN AND CONTROL OF ACTIVE POWER FILTERS
Library of Congress Cataloging-in-Publication Data
CONTENTS
ACKNOWLEDGMENTS
Chapter 1: INTRODUCTION
Chapter 2: NOVEL SLIDING MODE CONTROL OF ACTIVE POWER FILTER
2.1. INTRODUCTION
2.2. INDIRECT CURRENT CONTROL
2.3. SLIDING MODE CONTROL
2.4. SIMULATION ANALYSIS
CONCLUSION
Chapter 3: FEEDBACK LINEARIZATION BASED SLIDING MODE CONTROL OF ACTIVE POWER FILTER
3.1. INTRODUCTION
3.2. MATHEMATICAL MODEL OF ACTIVE POWER FILTER
3.3. FEEDBACK LINEARIZATION SLIDING MODE CONTROL
3.4. SIMULATION ANALYSIS
CONCLUSION
Chapter 4: ADAPTIVE CONTROL OF ACTIVE POWER FILTER USING PI-FUZZY COMPENSATOR
4.1. INTRODUCTION
4.2. DYNAMIC MODEL OF ACTIVE POWER FILTER
4.3. ADAPTIVE CURRENT CONTROL
4.4. PI-FUZZY COMPOUND VOLTAGE CONTROL
4.4. SIMULATION STUDY
CONCLUSION
Chapter 5: ADAPTIVE SLIDING MODE CONTROL OF ACTIVE POWER FILTER
5.1. INTRODUCTION
5.2. DESIGN OF ADAPTIVE SLIDING MODE CONTROLLER
5.3. SIMULATION ANALYSIS
CONCLUSION
Chapter 6: ADAPTIVE FUZZY CONTROL WITH SUPERVISORY COMPENSATOR OF ACTIVE POWER FILTER
6.1. INTRODUCTION
6.2. PRINCIPLE OF ACTIVE POWER FILTER
6.3. ADAPTIVE FUZZY CONTROL WITH SUPERVISORY COMPENSATOR
6.4. SIMULATION STUDY
CONCLUSION
Chapter 7: ADAPTIVE CONTROL WITH FUZZY SLIDING COMPENSATOR OF ACTIVE POWER FILTER
7.1. INTRODUCTION
7.2. PRINCIPLE OF ACTIVE POWER FILTER
7.3. DESIGN OF ADAPTIVE FUZZY SLIDING MODE CONTROL
7.4. SIMULATION STUDY
CONCLUSION
Chapter 8: ADAPTIVE NEURAL NETWORK CONTROL OF ACTIVE POWER FILTER
8.1. INTRODUCTION
8.2. BASIC PRINCIPLES OF ACTIVE POWER FILTER
8.3. DESIGN OF ADAPTIVE RBF NEURAL NETWORK CONTROLLER
8.4. SIMULATION STUDY
CONCLUSION
Chapter 9: CONCLUSION
REFERENCES
INDEX

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ENGINEERING TOOLS, TECHNIQUES AND TABLES

ADVANCED DESIGN AND CONTROL OF ACTIVE POWER FILTERS

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ENGINEERING TOOLS, TECHNIQUES AND TABLES

ADVANCED DESIGN AND CONTROL OF ACTIVE POWER FILTERS

JUNTAO FEI

New York

Copyright © 2013 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‟ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Advanced design and control of active power filters / editor: Juntao Fei (College of Computer and Information, Hohai University, Changzhou, P.R. China). pages cm Includes bibliographical references and index.

ISBN:  (eBook)

1. Electric power systems--Control. 2. Electric filters, Active--Design and construction. I. Fei, Juntao. TK1007.A38 2013 621.31'7--dc23 2013009428

Published by Nova Science Publishers, Inc. † New York

CONTENTS Acknowledgments

vii

Chapter 1

Introduction

1

Chapter 2

Novel Sliding Mode Control of Active Power Filter

5

Chapter 3

Feedback Linearization Based Sliding Mode Control of Active Power Filter

21

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator

35

Adaptive Sliding Mode Control of Active Power Filter

57

Adaptive Fuzzy Control with Supervisory Compensator of Active Power Filter

79

Adaptive Control with Fuzzy Sliding Compensator of Active Power Filter

95

Adaptive Neural Network Control of Active Power Filter

113

Conclusion

127

Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 References

131

Index

135

ACKNOWLEDGMENTS We wish to gratefully acknowledge the valuable help rendered by institutions and individuals in our conducting the research presented in this book. This research was partially supported by the National Science Foundation of China (2011-2017), Natural Science Foundation of Jiangsu Province (20112016), The Scientific Research Foundation of High-Level Innovation and Entrepreneurship Plan of Jiangsu Province, The Fundamental Research Funds for the Central Universities. We would like to thank their financial support that made this research possible. I am also thankful to Hohai University, University of Akron, and the University of Louisiana at Lafayette for a pleasant and supportive environment to do my research. I would like to express my thanks to Mr. Tianhua Li for his contribution to Chapter 2 and Chapter 3, Mr. Kaiqi Ma for his contribution to Chapter 4, Mr. Shenglei Zhang for his contribution to Chapter 4 and Chapter 5, Mr. Shixi Hou for his contribution to Chapter 6 and Chapter 7, and Mr. Zhe Wang for his contribution to Chapter 8. Finally, I am especially grateful to my family for the support to research my work, which made this project possible.

Chapter 1

INTRODUCTION In recent years there has been a substantial increase in the demand for controllable reactive power sources which can compensate for nonlinear loads. These requirements involve precise and continuous reactive power control with fast response time and avoidance of harmonic line current generation. Active power filter (APF) can be used for harmonic elimination, reactive current compensation and clean delivery of power. The basic principle of APF is to produce compensation current which is of the same amplitude and opposite phase with the harmonic currents to eliminate the unexpected harmonic currents. The shunt APF could compensate the harmonics generated by the load current through injecting compensation current to the grid. With the advantages of high controllability and fast response, it not only can compensate harmonics, but also can inhibit the flicker and compensate reactive power. Therefore, it is an effective approach to suppress the harmonic pollution. The dynamic models of APF can be established using various methods, and the behavior of reference current tracking can be improved using advanced control approaches. This book systematically studies the sliding mode control, feedback linearization, adaptive control, and intelligent control with application to active power filters, thereby significantly reducing APF‟s sensitivity to the nonlinear load and disturbance and improving the robust performance. The APF control systems are designed based on sliding mode control, adaptive control, and intelligent control. The Lyapunov stability theory makes the compensation current track the command current signal in real-time, thereby eliminating the harmonics, improving power quality, and enhancing the security of power transmission and distribution.

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Juntao Fei

The development of new semiconductor power components and advances in modern control theory have made this book possible for modern design engineers. This book begins with an overview of concepts from advanced control of active power filters. It then presents a comprehensive treatment of this analysis, and advanced control design of active power filters for the problem of harmonic elimination. The dynamical modeling procedure is introduced first, and then advanced control system design moves from sliding mode control of APF to robust adaptive sliding mode control of APF; a progression culminating in robust adaptive fuzzy control and adaptive neural control hierarchy. Main advanced control systems are presented: novel sliding mode control of indirect current control, feedback linearization based sliding mode control, adaptive control, adaptive sliding mode control, robust adaptive fuzzy control, adaptive fuzzy control with fuzzy sliding compensator, and adaptive neural network control. These methods generate solutions of guaranteed performance for problems of robustness and Lyapunov stability and parameter convergence. Case studies address application issues in the implementation of advanced control in active power filters. Examples, simulations, and comparative studies to the fundamental issues, illustrate the technical approaches and verify the performance of the various advanced control designs. With the fast development of the power electronics technology, more and more nonlinear and time-varying devices such as inverters, rectifiers and switching power supplies are used in grid casing power quality problems. The power quality problems contain low power factor, wave distortion, surge, phase distortion and so on. APF could compensate the harmonics generated by the load current through injecting compensation current to the grid, having the advantages of high controllability and fast response. It not only can compensate harmonics, but also can inhibit the flicker and compensate reactive power. Therefore, it is an effective approach to suppress the harmonic pollution. The basic principle of APF is to produce compensation current which is of the same amplitude and opposite phase with the harmonic currents to eliminate the unexpected harmonic currents. APFs are widely used in many applications to compensate the harmful harmonic currents produced by nonlinear loads on industrial, commercial and residential equipment such as diode rectifiers, thyristor converters, and some electronic circuits. In an APF connection, it is roughly classified as in series and in parallel. The shunt APF is the most widely used active filter because of its excellent performance characteristics and simplicity in implementation, as compared with the series

Introduction

3

APF requiring more high current and voltage. Shunt APF is an effective device to compensate the harmonic currents in power system. There are many current tracking control methods for APF, such as single cycle control, hysteresis current control, space vector control, sliding mode control [1-2], deadbeat control, repetitive control, predictive control, fuzzy control, adaptive control, iterative learning control, and artificial neural network control. At present, two traditional methods are mostly used in the current control for APF, hysteresis current control and triangle wave current control. However, the fluctuating switching frequency, low control accuracy, and slow response are the disadvantages for these two traditional methods respectively. Feedback linearization control is an effective method to establish the model of active power filter [3-4]. The models of active power filters have been established using various methods, and the behavior of reference signal tracking has been improved using advanced control approaches. Several control methods and harmonic suppression approaches for APF have been investigated [5-16]. In actual power system, the supply voltage may contain some harmonics and the parameters of the system model may be different from the actual value. In order to meet the requirements of the complex grid, the intelligent adaptive dynamic compensation and control of active power filter become important research project. Adaptive control approaches have been successfully applied in the active power filter [17-27]. In the last few years, fuzzy control has been extensively applied in a wide variety of industrial systems and consumer products because of its model free approach. Wang [28] proposed the universal approximation theorem and demonstrated that an arbitrary function of a certain set of functions can be approximated with arbitrary accuracy using fuzzy system on a compact domain. In this book, intelligent control approaches like fuzzy logic controller [29-34] and neural network controller [39-46] will be investigated to approximate nonlinear dynamic systems such as APF, since it is very hard to establish accurate mathematical models of APF, and classical linear control methods cannot achieve the ideal current tracking performance. Intelligent controllers are proposed to improve the current tracking performance and guarantee the Lyapunov stability of the closed-loop system. This book is organized into nine chapters: Chapter 1- Introduction Chapter 2- Novel Sliding Mode Control of APF Chapter 3- Feedback Linearization Based Sliding Control of APF

4

Juntao Fei Chapter 4Chapter 5Chapter 6Chapter 7-

Adaptive Control of APF using PI-Fuzzy Compensator Adaptive Sliding Mode Control of APF Adaptive Fuzzy Control with Supervisory Control of APF Adaptive Fuzzy Control with Fuzzy Sliding Compensator of APF Chapter 8- Adaptive Neural Network Control of APF Chapter 9- Conclusion

Chapter 2

NOVEL SLIDING MODE CONTROL OF ACTIVE POWER FILTER A novel sliding mode control (SMC) method for indirect current controlled three-phase parallel active power filter is presented in this chapter. There are two designed closed-loops in the system, one is the DC voltage controlling loop and the other is the reference current tracking loop. The first loop with a PI regulator is used to control the DC voltage approximating to the given voltage of capacitor. The output of the PI regulator through a low-pass filter is applied as the input of the power supply reference currents. The second loop implements the tracking of the reference currents using an integral sliding mode controller, which can improve the harmonic treating performance. Compared with the direct current control technique, it is convenient to be implemented with digital signal processing systems because of simpler system structures and better harmonic treating properties. Simulation results verify the generated reference currents have the same amplitude with the load currents, demonstrating the superior harmonic compensating effects with the proposed shunt active power filter compared with the hysteresis method.

2.1. INTRODUCTION Sliding mode control technique, which combines design and analysis closely, has robustness for model uncertainty as well as external disturbance. The sliding mode controller is composed of an equivalent control part that describes the behavior of the system when the trajectories stay over the sliding manifold, and a variable structure control part that enforces the trajectories to

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Juntao Fei

reach the sliding manifold and prevent them leaving the sliding manifold. Therefore, it is a suitable control method for reference current tracking of active power filter. In this chapter, a novel sliding mode control method is designed to implement the reference current tracking in the indirect current control. Two closed-loops are designed, one is the DC voltage controlling loop and the other is the reference current tracking loop. The outer voltage controlling loop with a PI regulator is used to control the DC voltage close to the given voltage of capacitor, in the current tracking loop. In the reference current tracking loop, a novel integral sliding mode controller is implemented to tracking the reference currents, thus improving harmonic treating performance. The contribution of this chapter can be summarized as: 1. The advantage of using novel sliding mode controllers for the shunt active power filter with indirect current control technique is that it does not need harmonic detection components, but supply grid voltage sensors, filter capacitor voltage sensors, and supply line current sensors. Compared with direct current control techniques, it has a simpler system structure and better harmonic treating performance. Therefore, it is easy to implement with digital signal processing systems. 2. A novel integral sliding mode control is proposed in reference with current tracking to reduce the tracking error. A PI regulator combined with a low-pass filter is used to generate the amplitude of the reference currents. The designed active power filter has superior harmonic treating performance and minimizes the harmonics for wide range of variation of load current under difference nonlinear load. Therefore, the proposed control scheme yields an improved THD performance compared with the hysteresis method.

2.2. INDIRECT CURRENT CONTROL The schematic block diagram of shunt active power filters is showed as Figure 1. It is supposed that IGBT is an ideal switch equipment of inverter parts in the shunt active power filter for the convenience of analysis, in other words, we ignore the death time and the equivalent resistance, invert rise time in „on,‟ and fall delay time in „off‟.

7

Novel Sliding Mode Control of Active Power Filter The system consists of the following modules:

1. Compensating currents generator A compensating currents generator can generate currents which have the same amplitude and opposite phase with the harmonic currents to offset the harmonic components in the power supply current.

2. Reference currents generator The DC voltage is controlled by a PI regulator whose output is applied as *

the input of power supply reference current I Sp . The power supply reference currents should have the same phase with the power supply voltage, so the unit sinusoidal signals (sin( wt ),sin( wt  120 ),sin( wt  120 )) can formed *

*

*

with Phase Locked Loop (PLL). Thus, supply reference currents (iSa , iSb , iSc ) *

can be obtained by multiplying I Sp and the unit sinusoidal signals.

3. Sliding mode controller In this module, a novel position tracking sliding mode control technique is applied. The output of the sliding mode controller is used to generate PWM signals ( g1 , g2 , g3 , g4 , g5 , g6 ) which control the status of IGBT „on‟ or „off‟.

4. Measuring devices This module contains devices for measuring supply voltages, supply currents, and DC voltage.

8

Juntao Fei ea

iSa

iLa

eb

iSb

iLb

ec

iSc

Vsa

Vsb

Vsc

Voltage Measurement

Nonlinear Load

iLc icc

icb

L

L

ica

isa isb isc

Current Measurement

L

1

2

3

Cdc Vsc Vsa Vsb Vdc Reference

Current Generater

isa isa* isb*

isb

isc

Sliding Mode isc* Controller

g1

PWM

g2 g3

g4

4

5

6

g5 g6

Voltage Measurement

Figure 2.1. Schematic block diagram of shunt active power filter with indirect current control technique.

A. Model of the Three-Phase Three-Wire Active Power Filter Define (iSa , iSb , iSc ) as the supply currents, (iLa , iLb , iLc ) as the load currents, (ica , icb , icc ) as the compensating currents. According to the Kirchhoff‟s current law, we can deduce the relation as follows:

iSa  iLa  ica  iSb  iLb  icb i  i  i  Sc Lc cc

(2.1)

The state equations of three-phase three-wire APF:  dica  L dt  uSa  ua   dicb  uSb  ub L  dt  dicc  L dt  uSc  uc 

(2.2)

9

Novel Sliding Mode Control of Active Power Filter Where

ua  udc (2 ja  jb  jc ) / 3  ub  udc ( ja  2 jb  jc ) / 3 u  u ( j  j  2 j ) / 3 dc a b c  c

(2.3)

The switching function ji  1 when the IGBT of i-phase‟s upper bridge is on and the below bridge‟s is off. Oppositely, ji  0 when the IGBT of iphase‟s upper bridge is off and the below bridge‟s is on.

B. Reference Currents *

*

*

Reference currents (iSa , iSb , iSc ) are produced from the reference currents generator. The schematic block diagram of the reference currents generator is shown as Figure 2. The three-phase unit sinusoidal signals should have the same phase with the supply voltage, so three PLL are used to get unit sin (sin wt ,sin(wt 120),sin(wt  120)) . The parameters of PI regulator determine the static and dynamic performance of the DC voltage. By proper parameters setting, the DC voltage can be well stabilized around the given value but not strictly equal to that value. Accordingly, the output of PI regulator is also stable. There is a fixed ratio between the PI regulator‟s output *

and load current amplitude, so I Sp can be formed by multiplying output of PI regulator and the ratio. Otherwise, a low-pass filter is added after the PI regulator, and its output, which is much more stable than PI regulator‟s, is *

*

applied as I Sp . Multiplying the unit sinusoidal signals and I Sp , the supply *

*

*

reference currents (iSa , iSb , iSc ) can be formed.

10

Juntao Fei

Vref Vdc

Lowpass Filter PI Controller

k Isp*

Vsa Vsb Vsc

PLL

sinwt

isa*

sin(wt-120)

isb*

sin(wt+120)

isc*

Figure 2.2. Schematic of the reference currents generator.

2.3. SLIDING MODE CONTROL The sliding mode control theory has been widely applied in the active power filter, due to its satisfactory operation characteristics such as fastness, robustness, and stability for large load variations. According to the position tracking sliding mode control theory, we define the reference current tracking error as e and the velocity error as e .

e  iS * iS , e  iS * i S .

(2.4)

Defining the tracking error e and velocity error e as state variable, we can obtain the sliding mode surface.

S  CE

(2.5)

11

Novel Sliding Mode Control of Active Power Filter

e 

Where C  [ke , kde ] , ke  0, kde  0 , and E     .

e 

When tracking random trajectory, traditional SMC method cannot avoid steady-state error in the presence of external disturbances. In this section, an improved position tracking sliding mode control technique with an integral portion is proposed to solve this problem. The switch function of the novel SMC can be described as: 

S  kee  kd e ki  e

(2.6)

The block diagram of the proposed sliding mode control is showed as Figure 2-3. The input is tracking error e and the output of sliding mode controller is applied to generate PWM pulse signals which control IGBT „on‟ or „off‟. ke

e

de / dt

kd

 edt

ki



Figure 2.3 Schematic of the Sliding Mode Control.

The voltage across the capacitor is stabilized by a PI regulator. The relationship between the switching function and control signal is shown as equation (2.7). So we can define the constant switching control as

ui  Ki udc sgn( Si ) Where K i =

1 2 i  a、b、c or 3 3

The control regulation is described in Table 2-1.

(2.7)

12

Juntao Fei Table 2-1. IGBT switching regulation Situation

Sa

Sb

Sc

ji mode

   Ⅳ Ⅴ Ⅵ

+ + +

+ + + -

+ + + -

(1,1,0) (0,1,0) (0,1,1) (0,0,1) (1,0,1) (1,0,0)

Remark 1. The PI controller is designed to control DC voltage stable, but not strictly equal to the given value. As the stable error exists, the output of PI regulator is also stable. There is a fixed ratio between the output of PI regulator and amplitude of load current, even if the load currents change. The reference currents amplitude can be formed by multiplying the output of PI regulator and the ratio. It is difficult to calculate the parameters of PI regulators in a mathematical way, so we merely adjust the parameters to make the DC voltage stable.

2.4. SIMULATION ANALYSIS In this section, the active power filter system control strategies are implemented on realistic models by using the SimPower Systems blockset of MATLAB. The non-linear load consists of three-phase Universal Bridge and Series RL Branch. The components and parameters are listed in Table 2-2. The APF begins to work at the time t  0.04s when the break is closed. In a practical power system, supply voltage may contain frequently varying harmonic and non-linear loads which deteriorate the effect of harmonic compensation. In order to analyze the performance of the active power filter system, we design the voltage of supply to contain 10th and 20th harmonic, and add another non-linear load separately at the time t  0.1s and t  0.2s .

13

Novel Sliding Mode Control of Active Power Filter Table 2-2. The components and parameters Supply voltage Harmonic of supply voltage

Vs Oder Amplitude

110 2 V 50 HZ 10th and 20th 20 5

Line‟s inductance & resistance

Ls, Rs

Inductance in compensator

L

3e-3 H

DC side capacitor Given voltage of Capacitance

C Vdef

320e-3 F 350V

Resistance in non-linear load

RL

0

Inductance in non-linear load

LL

2e-3H

Parameters of PI regulator

K p , Ki , K

100 , 1, 1.05

Parameters of sliding mode controller k e , k d , ki

5e-6H

1 , 0.9e-5 , 0.9

As shown in Fig.2.4 and Fig.2.5, the waves of supply voltage distort because of the 10th and 20th harmonic, and the total harmonic distortion (THD) is 7.07% according to the harmonic analysis. Unit sinusoidal signals should have the same phase with the power supply voltage. Fig.2.6 plots the waves of sinusoidal signals. Fig.2.7 shows the wave of APF connection point. The switching of IGBT makes the voltage of the connection vibrate frequently. With the switching of IGBT, the wave of connection voltages keeps buffeting with high frequency. The simulated result reveals that THD of APF connection point is also reduced. Fig.2.8 draws the waves of A-phase load current, I Sp

*

and A-phase reference current. Simulation results also reveal that the average *

value of I Sp is equal to that of the load current in each stage. The compensating currents are raised as the energy flows into the LC branch while decreased as the energy flows out, so the reference currents have the same amplitude with the load currents that can minimize the flowing of energy and improve the stability of system, as well as harmonic compensating effect. We set the initial value of the DC voltage at 350V to simplify the analysis. The DC voltage is stabilized by a PI regulator, but the steady error between the DC voltage and given value will increase as the load currents rise. Fig.2.9 is the waveform of the DC voltage. There is stable error between the DC voltage and the given value, which can be used to evaluate the amplitude of load currents.

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Juntao Fei

The waveforms of supply current, compensating current and tracking error in A-phase are shown in the Fig. 2.10. The formed supply currents have the same amplitude with the load currents.

Selected signal: 15 cycles. FFT window (in red): 1 cycles 100 0 -100 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 127 , THD= 7.07%

Mag (% of Fundamental)

5 4 3 2 1 0

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 2.4. The THD of supply voltage.

Selected signal: 15 cycles. FFT window (in red): 1 cycles 1

0

-1

0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 1.01 , THD= 3.41%

Mag (% of Fundamental)

3 2.5 2 1.5 1 0.5 0

0

2000

4000 6000 Frequency (Hz)

8000

Figure.2.5. The THD of unit sinusoidal signals.

10000

Novel Sliding Mode Control of Active Power Filter

15

sinwt

sin(wt-120)

sin(wt+120)

Figure 2.6. Waves of unit sinusoidal signals. Vcona

Vconb

Vconc

Figure 2.7. Waves of voltage on APF connect point.

The frequency range for THD evaluation (Fig.2.11 to Fig.2.16) is increased to 10000 in order to include HF components. Harmonic analysis results are shown from Fig. 2.11 to Fig.2.16. Comparison of harmonic compensating effects between hysteresis comparison method and the designed integral sliding mode control method is also implemented. Before active power filter compensating, the THD of supply current is 24.72%. The THD of supply current is reduced to 3.35% using indirect current control with hysteresis comparison method. Keeping the simulating parameters unchanged, the THD of supply current is reduced to 2.51% by applying the designed

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Juntao Fei

sliding mode controller. The THD of APF connection point is also reduced from 18.73% to 13.03%. According to the IEC-61000 issued by IEC in the year of 2000, harmonic content of the low voltage (  1kV ) should be under 5%, so indirect current controlled active power filter with sliding mode controller has better harmonic compensating effect. Wave of Load Current

Wave of Isp*

Wave of Reference Current

Figure 2.8. Waves of load current,

I Sp* and A-phase reference current in A-phase.

Vdc

Vdc

Figure 2.9. Waveform of the DC voltage.

Novel Sliding Mode Control of Active Power Filter

17

Wave of isa

Wave of ica

Wave of ea

Figure 2.10. Waveform of the supply current, compensating current and tracking error in A-phase.

Selected signal: 15 cycles. FFT window (in red): 1 cycles 20

0

-20 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Mag (% of Fundamental)

Fundamental (50Hz) = 12.12 , THD= 24.54% 20

15

10

5

0

0

2000

4000 6000 Frequency (Hz)

Figure 2.11. THD of load current.

8000

10000

18

Juntao Fei

Selected signal: 15 cycles. FFT window (in red): 1 cycles 50 0 -50 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 68.09 , THD= 18.73%

Mag (% of Fundamental)

14 12 10 8 6 4 2 0

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 2.12. THD of connection point before compensating.

Selected signal: 15 cycles. FFT window (in red): 1 cycles 50 0 -50 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 67.59 , THD= 13.32%

Mag (% of Fundamental)

10 8 6 4 2 0

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 2.13. THD of connection point with comparison controller.

Novel Sliding Mode Control of Active Power Filter

Selected signal: 15 cycles. FFT window (in red): 1 cycles

50 0 -50 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 67.81 , THD= 13.03%

Mag (% of Fundamental)

10 8 6 4 2 0

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 2.14. THD of connection point with designed sliding mode controller.

Selected signal: 15 cycles. FFT window (in red): 1 cycles 10 0 -10 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 11.91 , THD= 3.35%

Mag (% of Fundamental)

2.5 2 1.5 1 0.5 0

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 2.15. THD of supply current with hysteresis comparison controller.

19

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Juntao Fei

Selected signal: 15 cycles. FFT window (in red): 1 cycles 10 0 -10 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

Fundamental (50Hz) = 11.86 , THD= 2.51%

Mag (% of Fundamental)

1.5

1

0.5

0

0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 2.16. THD of supply current with designed sliding mode controller.

CONCLUSION In this chapter, a sliding mode control technique with integral portion is designed for indirect current control active power filter. This sliding mode control is applied in reference current tracking to reduce the tracking error. A PI regulator combined with a low-pass filter is used to generate the amplitude *

*

of the reference currents I Sp . Multiplying the I Sp with unit sinusoidal signals, reference currents having the same phase with power supply voltage can be formed. Simulations under the variable system load are carried out to show the robustness of the active power filter system. The result reveals that the reference currents have the same amplitude with the load currents and the designed active power filter has superior harmonic compensation effect. However, tracking error will increase when the load currents have high variance ratio. The comparative simulation shows a better reference current tracking performance of designed sliding mode control method, than the hysteresis comparison method.

Chapter 3

FEEDBACK LINEARIZATION BASED SLIDING MODE CONTROL OF ACTIVE POWER FILTER In this chapter, we present an approach using a novel feedback linearization sliding mode controlled parallel active power filter, with indirect current control. A feedback linearization technique and integral sliding mode control method are developed to implement harmonic compensation in a threephase three-wire grid. Since traditional unit synchronous sinusoidal signal calculating methods are not applicable when supply voltage contains harmonic, a novel unit synchronous sinusoidal signal computing method using synchronous frame transforming theory is presented to overcome this disadvantage. The feedback linearization indirect current controlled method is used to implement the DC side voltage regulation while a novel integral sliding mode controller is applied to reduce the total harmonic distortion of supply current. Simulation results verify that the DC side voltage is well stabilized at the given value and responds fast to the external disturbance. Comparisons are also made to show the advantages of the novel unit sinusoidal signal calculating method and super harmonic treating property of the designed active power filter. The simulation results verify the designed shunt active power filter dramatically reduces the THD of supply current, and has good dynamic and steady performance.

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Juntao Fei

3.1. INTRODUCTION In this chapter, the system consists of two control loops, the outer DC side voltage control and the inner reference current tracking control. The inputoutput feedback linearization is applied to implement the DC side voltage regulation in indirect current controlled active power filters. A novel sliding mode controller is incorporated into the reference current tracking control. The contribution of this chapter can be summarized as follows: 1. A novel adaptive sliding mode control is proposed in reference current tracking to reduce the tracking error. The designed APF has superior harmonic eliminating performance and minimizes the harmonics for wide range of variation of load current; therefore an improved THD performance can be achieved. 2. This chapter presents a nonlinear feedback linearization control method to implement the indirect current control of three-phase three-wire active power filters. A novel integral position tracking sliding mode controller is applied to implement reference current tracking control.

3.2. MATHEMATICAL MODEL OF ACTIVE POWER FILTER Suppose the supply voltage is (Vsp cos(wt ),Vsp cos(wt  2 / 3),Vsp cos(wt  4 / 3)) , and define ( Rca , Rcb , Rcc ) and ( Lca , Lcb , Lcc ) as resistances and inductances in compensation circuits. The DC side capacitor is named Cdc and its voltage is

Vdc . The supply currents, load currents and compensating currents are define as (iSa , iSb , iSc ) , (iLa , iLb , iLc ) and (ica , icb , icc ) separately. So the schematic block diagram of the designed shunt active power filter is shown as Figure 3.1.

A. Synchronous frame transforming theory The transformation formula from abc frame to dq frame is given as (3.2) . For the three-phase sinusoidal signals, the component in phase is a constant while the quadrature component is zero. So in order to compensate the

Feedback Linearization Based Sliding Mode Control …

23

harmonic in the load currents and make sinusoidal supply currents, compensating currents should eliminate the quadrature component of load currents, that is

jI cq   jI Lq

(3.1)

vd  cos(   / 6) sin   va  3 v p   v   2      2 0    sin(   / 6) cos   vb   q

(3.2)

B. Mathematical model based on average power balance principle The principle of average power balance is used to determine the approximate model of the compensator. The mathematical model is derived based on the following assumptions. 1. The supply voltages are balanced and contain little harmonics, so the quadrature component of the supply voltage can be ignored. 2. Only the fundamental components of the currents are considered, as the harmonic components do not affect the average power balance expressions. 3. The supply impedance has been ignored, and all losses of the system are lumped and represented by an equivalent resistance Rc . 4. IGBT is an ideal switch equipment of inverter. Load current in dq frame can be described as I L  I Ld  jI Lq , while the compensating current in dq frame is I c  I cd  jI cq . According to (3.1), the supply current

I S  I Ld  jI Lq  I cd  jI cq  I Ld  I cd

(3.3)

Vsa

iLa

a

isa

AC Vsb

L1 iLb

b

isb

AC Vsc

isc

R1

iLc

c

AC Nonlinear Load

I La , I Lb , I Lc

Rcc

Rcb

icc

icb

Rca ica

I ca , I cb , I cc

Vcona ,Vconb ,Vconc

Lca a`

abc / dq

abc / dq

abc / dq

Vdc b`

Lcc

VSa VSb VSc

I L d I Lq I cq

x1 Vdcref

Vdc

Feedback Linearization Control

u



Estimate Unit Sine Singnals

Isp*

Figure 3.1. Overall diagram of the proposed SAPF system.

Estimate Reference Current

cos( wt ) cos(wt  2 /3) iSa cos(wt  4 / 3)

iSareef iSbreef

iScreef

+ Cdc _

Lcb

c`

iSb iSc

Integral Sliding Mode Controller

Gating Signals Sa , Sb , Sc PWM Controller

Feedback Linearizatin Based Sliding Mode Control …

25

As shown in Figure 3. 1, the voltage at SAPF connection points is equal to the supply voltage, that is

va  VSp cos( wt )  vb  VSp cos( wt  2 / 3)  vc  VSp cos( wt  4 / 3)

(3.4)

By synchronous frame transforming, (3.5) can be derived.

 3 vcomd  VSp 2  v  comq  0

(3.5)

The compensator power Pcom  vcomd I cd  vcomq I cq  vcomd I cd (3.6) For a particular operating point I cq is constant, then the inductance power is:

PL  n

d 1  Lc I cd2   dt  2 

n3

(3.7)

Power loss in the resistor is given by (3.8).

PR  nRc ( I cd2  I cq2 )

n3

(3.8)

The capacitor average power:

PC  CdcVdc

dVdc dt

(3.9)

Where Vdc is the instantaneous DC side voltage. According to the power balance principle of compensator, the following equation can be established.

26

Juntao Fei

vcomd I cd  nRc ( I cd2  I cq2 )  n

dVdc d 1 2  L I  C V c cd dc dc  dt  2 dt

(3.10)

Then, the mathematical model of the active power filter can be derived. 2 vcond ( I S  I Ld )  nRc  I S2  2 I S I Ld  I Ld  I cq2   nLc ( I S  I Ld )

 nLc ( I S  I Ld )

dI Ld dV  CdcVdc dc dt dt

dI S dt

(3.11)

3.3. FEEDBACK LINEARIZATION SLIDING MODE CONTROL  x1  I S dx , and u  1 as input, the dt  x2  Vdc

Define two state variables 

mathematical model according to (3.3) can be written as:

 0   x   v ( x  I )  nR (2 x I  x 2  I 2  I 2 )  nL ( x  I ) dI Ld  cond 1 Ld c 1 Ld 1 Ld cq c 1 Ld dt  Cdc x2 

     

1    nL ( I  x )  u  c Ld 1   Cdc x2  y  x2

(3.12)

The former system is a single input single output (SISO) system. It can be described as:

Feedback Linearizatin Based Sliding Mode Control …

  x  f ( x)  g ( x)u   y  h( x) Where

the

state

27

(3.13)

variables

x  R2 ,

f ( x), g ( x) : R 2  R 2 ,

h( x) : R 2  R 2 and f (0)  0 , h(0)  0 . Then, 

y

h  h h x f ( x)  g ( x)u  f1 ( x)  g1 ( x)u x x x

(3.14)

The feedback linearization control law is designed as

u

R  f1 ( x) g1 ( x)

(3.15)

The (3.14) can be linearized as 

yR

(3.16)

Suppose position reference signal as yd (t ) , and define 

R  yd  k ( y  yd )

(3.17)

Where k  0 . Then, (3.17) can be transformed into 

e ke  0

(3.18)

28

Juntao Fei Where e  y  yd The (3.18) is the error dynamic equation and e(t ) will exponential 

converges to zero. If e(0)  e(0)  0 , e(t ) will be always zero when t  0 . For shunt active power filter, the reference value of the DC side voltage is constant, so 

y d  0 and R  k (Vdc  Vdcref )

(3.19)

We can deduce the feedback linearization control law as (3.9). dI vcond ( x1  I Ld )  nRc (2 x1I Ld  x12  I Ld2  I cq2 )  nLc ( x1  I Ld ) Ld  Cdc x2 (k (Vdcref  Vdc )) dt u nLc ( x1  I Ld ) (3.20) By integrating the control law u 

dx1 , the amplitude of supply reference dt

*

currents I sp can be derived and applied to calculate the reference supply currents of the indirect current control method. In this chapter, the integral sliding mode controller is also applied to implement the inner reference current tracking control to reduce the THD of supply currents and steady-state errors. The novel sliding mode control designs the switch function as 

S  kee  kd e ki  e

(3.21)

The voltage across the capacity is stabilized at 400V by feedback linearization control. The relationship between the switching function and input voltage is shown as (3.21), so the voltages of a ', b ', c ' are switching by constant.

ui '  Ki udc sgn(Si ) K i is

1 2 or , i '  a '、b '、c ' , i  a、b、c (3.22) 3 3

Feedback Linearizatin Based Sliding Mode Control …

29

The reference supply currents are produced by multiplying the amplitude *

signal I sp and the unit synchronized sinusoidal signals. However, in the actual grid, the unit sinusoidal signals produced by PLL may appear as wave distortion when the voltage contains harmonics. In this chapter, a novel unit sinusoidal signal calculating method is presented. The schematic of the novel unit synchronized sinusoidal signals calculating method is shown as Figure 3. 2. The amplitude of supply voltage is evaluated by synchronous frame transforming theory and the three-phase supply voltages are divided by the amplitude to calculate the unit supply voltage signals. Then the PLL is applied to produce the unit synchronized sinusoidal signals (cos(wt ),cos(wt  2 / 3),cos( wt  4 / 3) . The reference supply currents applied in the sliding mode controller is produced by multiplying the amplitude and unit sinusoidal signals. VSa

* I Sp

I Saref

PLL cos(wt ) ~

VSa V Sb

V Sd

abc/dq

Low Pass Filter

~

V Sp 2/3

V Sb

* I Sp

1/ V Sp

I Sbref

PLL

f 1/ u

cos(wt  2 / 3)

V Sc

* I Sp

PLL

I Scref

cos(wt  4 / 3)

Figure 3.2. Schematic of the designed reference currents calculating method.

3.4. SIMULATION ANALYSIS In this section, the active power filter system is implemented on a realistic model using the SimPowerSystem blockset of MATLAB. The nonlinear load consists of a three-phase Universal Bridge and Series RL Branch. The component and parameters are listed in Table 3.1. In actual power system, the supply voltage may contain some harmonics. According to the Electro Magnetic Compatibility (EMC) standards IEC-61000, the THD of supply voltage should be under 5% in the low-voltage grid (≤1kV). Simulation study

30

Juntao Fei

is implemented and comparisons are made between unit sinusoidal signals calculation by PLL and the designed method. Supposing the supply voltage contains 20th and 50th harmonics, THD of supply voltage is 8.91% by harmonic analysis. As unit sinusoidal signals derived by traditional sinusoidal signals calculation method appear distorted and cannot be used to calculate the supply reference currents, the novel method based on synchronous frame transforming theory is applied. The wave of unit sinusoidal signals calculated by the designed method is shown in Figure 3.3. In the actual power system, the nonlinear loads change obviously at the increase and decrease of the users. So the simulation is carried out with mutative loads. An extra nonlinear load is connected into the system when simulation time t=0.2s and cut off when t=0.4s to analyze the robustness to load disturbance. Table 3.1. The components and parameters Supply Voltage(Amplitude & Frequency) Harmonic of Supply Voltage(Orders & Amplitude) Inductance & Resistance in Nonlinear Load1 Inductance & Resistance in Nonlinear Load2 Inductance & Resistance in Compensator DC side Capacitor Reference value of DC side Voltage Parameter of Feedback Linearization Control Parameters of Sliding Mode Controller

220V 50HZ 20th 8 50th 4 2e-3H 5Ω 2e-3H 20Ω 3.3e-4H 0.05Ω 5e-3F 400V 1 1, 9.1e-7, 6e-2

Simulation comparison between the hysteresis comparison controller and designed sliding mode controller is also presented in Table 3.2. The THD of load currents when t=0.1s, 0.3s and 0.5s are separately 21.87%, 20.51% and 21.87%. Feedback linearization indirect current controlled APF with hysteresis comparison controller and designed sliding mode controller are both formed with the same components and parameters. Simulation results reveal that the THD of supply currents is reduced to 3.43%, 4.05%, and 3.67% by feedback linearization indirect current controlled APF with hysteresis comparison controller. It‟s decreased to 3.21%, 3.61%, and 3.24% after the designed sliding mode controller is applied in the system.

Feedback Linearizatin Based Sliding Mode Control …

31

Table 3.2. Comparison of compensation effect Time

0.1s 0.3s

THD of supply current with hysteresis comparison controller 3.43% 4.05%

THD of supply current with the designed sliding mode controller 3.21% 3.61%

0.5s

3.67%

3.24%

Static and dynamic performances of the designed feedback linearization sliding mode controlled APF are described in Figure 3.4-3.7. Figure 3.4 shows *

the evaluated value of the amplitude signal I Sp used to calculate the reference supply currents. The amplitude signal is equal with the amplitude of load currents by a short-time adjustment. The waves of load currents and supply currents are shown in Figure 3.5. Reference current tracking error, which is shown in Figure 3.6, will become a little bigger when the load current increases. DC side voltage, shown in Figure 3.7, is regulated by the feedback linearization control theory, which presents good dynamic and static characteristics. When load change happens, the voltage converges to the given value by a short-time adjustment. 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 0

0.1

0.2

0.3

0.4

0.5

Figure 3.3. Wave of the unit sinusoidal signals calculated by the designed method.

0.6

32

Juntao Fei

250

200

150

100

50

0

-50

-100

-150 0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 3.4. Amplitude of the reference supply current.

50

0

-50

50

0

-50

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 3.5. Enlarged view of the load current and supply current in A-phase.

0.5

Feedback Linearizatin Based Sliding Mode Control …

33

20

15

10

5

0

-5

-10

-15

-20 0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 3.6. Reference current tacking error in A-phase. 600

500

400

300

200

100

0 0

0.1

0.2

0.3

Figure 3.7. Voltage of the DC side capacity.

0.4

0.5

0.6

34

Juntao Fei

CONCLUSION In this chapter, the input-output feedback linearization has been applied to implement the DC side voltage regulation in the indirect current controlled active power filter. A novel sliding mode controller has also been designed and applied in the reference current tracking control. In the actual power system, supply voltage may contain harmonics, which limits the application of the APF. This chapter designs a novel unit synchronous sinusoidal signals calculating method based on synchronous frame transforming theory. Simulation results verify that the sinusoidal signals calculating method can produce ideal synchronous sinusoidal signals even when THD of the supply voltage is high. The designed active power filter shows good performance both in dynamic and static characteristics and the sliding mode controller reduces the THD of supply currents dramatically.

Chapter 4

ADAPTIVE CONTROL OF ACTIVE POWER FILTER USING PI-FUZZY COMPENSATOR In this chapter, an adaptive control technology and PI-Fuzzy compound control technology are proposed to control an active power filter (APF). AC side current compensation and DC capacitor voltage tracking control strategies are discussed and analyzed. Model reference adaptive controller for the AC side current compensation is derived and established based on the Lyapunov stability theory; the PI-Fuzzy compound controller is designed for the DC side capacitor voltage control. The adaptive current controller based on the PIFuzzy compound system is compared with the conventional PI controller for active power filter. Simulation results demonstrate the feasibility and satisfactory performance of the proposed control strategies. It is shown that the proposed control method has an excellent dynamic performance such as small current tracking error, reduced total harmonic distortion (THD), strong robustness in the presence of parameters variation, and nonlinear load.

4.1. INTRODUCTION In recent years, shunt active power filter (SAPF) is an effective device to implement the harmonic current in the grid and attracts more and more attention in the modern society, research studies on the APF including harmonic detection, topology studies, system modeling and control methods become promising topics, the new type of intelligent control and adaptive control methods get a lot of development. There are many current tracking

36

Juntao Fei

control methods, such as single cycle control, hysteresis current control, space vector control, sliding mode control, deadbeat control, repetitive control, predictive control, fuzzy control, adaptive control, iterative learning control, and artificial neural network control. However, most of the tracking issues for active power filter‟s DC voltage or AC current compensation are a unilaterally controlled study which cannot achieve accurate, rapid, and highly adaptable global control objectives. In this chapter, adaptive current tracking control methods for AC side current is developed for the current tracking. The proposed adaptive current control method based on PI-Fuzzy compound controllers for shunt power active filter can take advantage of the fuzzy control, which does not depend on the system dynamics. It also has good transient and steady-state behavior, great robust performance, and the adaptive control which has precise tracking performance, online real-time compensation of model uncertainties and external disturbances. Therefore, the proposed adaptive current control method based on PI-Fuzzy compound controller can greatly improve the current tracking and voltage control performance of the active filter compensation. The proposed control strategy has the following advantages. 1. The contribution of this chapter is the integration of the adaptive control and fuzzy control. A PI-Fuzzy controller is proposed to improve the voltage tracking performance for the DC side capacitor voltage control. A model reference adaptive controller for the AC side current compensation is derived based on Lyapunov analysis. 2. The proposed adaptive current control method based on PI-Fuzzy compound controller can deal with system nonlinearities and nonlinear load better, and improve the current tracking and robustness of the control system compared with the conventional control method. Fuzzy control has great ability to compensate for the nonlinear load and improve current tracking and total harmonic distortion (THD) performance.

4.2. DYNAMIC MODEL OF ACTIVE POWER FILTER This chapter studies parallel single-phase voltage active power filters. The dynamic model of APF is described referring to the linearization model [20]. The circuit diagram is shown in Figure 4-1.

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 37

is

iL Nonlinear Load

Us

ic L

Q1

Q3

C

R

Q4

Q2

Figure 4.1. Basic circuit structure of shunt APF.

The APF shown in Figure 4-1 can be decomposed into two work modes, shown in Figure 4-2. Assuming the switching converter frequency is f S , conversion cycle is TS  1 / f S , duty cycle is D  TON / TS . In mode (1), when 0  t  DTS , Q2 and Q3 are turned on, Q1 and Q4 are turned off; In mode(2), when DTs  t  TS , conversion process is as opposite mode 1,

Q2 and Q3 are turned off, Q1 and Q4 are turned on. L

L

+ Us -

iL

i1 Uc

(1)

+

C

i2 R

+ Us -

iL

+ Uc

i1

i2

C

R

-

(2)

Figure 4.2. Equivalent circuit diagram of the two switching state of the APF model.

According to Figure 4-2, we can establish the dynamic model of the single-phase shunt active filter:

38

Juntao Fei i L  i2   C   U  U ' c  i L (t )  s  L 

U c (t )   '

i L  i2   C   U  U ' c  i L (t )  s  L 

U c (t ) 

When 0  t  DTS

(4.1)

'

When DTS  t  TS

(4.2)

From (4.1), (4.2), average state equations of the inverter within one cycle can be obtained as:

i ' L (t ) 

U D 1 D 2D  1 (U s  U c )  (U s  U c )  Uc  s L L L L

U ' c (t )  

(4.3)

U D 1 D 1  2D (i L  i2 )  (i L  i2 )  i L  s (4.4) C L C RC

Where, D  [0,1] . Rewriting (4.3) and (4.4) yields the following form:

x '  Fx  GxD  EU s

where, x  [i L

T

Uc ]

 0 , F 1  C

(4.5)

 1  0 L , G   1   2   C RC 

2 1 L, E  L   0 

T

 0 . 

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 39 In order to simplify the controller design, an approximate linear model of the nonlinear model is derived around the equilibrium point. If x  x0 and

D  D0 satisfying:

f ( x0 , D0 )  Gx0  Gx0 D0  EU s  0 Then ( x 0 , D0 ) can be called the equilibrium point of the nonlinear model. Therefore, the right side of (4.5) is expanded into a Taylor series about ( x 0 , D0 ) and the high-order terms are neglected.

x '  f ( x, D ) 

f x

x  x0 D  D0

( x  x0 ) 

f D

x  x0 D  D0

( D  D0 )

(4.6)

Assuming that x  x  x0 , D  D  D0 , then the following linear APF model can be obtained:

x  ( F  GD0 ) x  (Gx0 ) D  Ap x  B p D '

(4.7)

Where, A p  F  GD0 , B p  Gx0 . The capacitor voltage and inductance current of the nonlinear APF mode at the point ( x 0 , D0 ) can be described as: x02 

Us 1  2 D0

(4.8)

x02 x01  R(1  2 D0 )

Where, x02 and x01 are the equivalent values of U c and iL respectively. Duty cycle can be expressed as:

D0 

U 1 (1  s ) 2 x 02

(4.9)

40

Juntao Fei

4.3. ADAPTIVE CURRENT CONTROL In this section, an adaptive current control for AC side current compensation is derived. A schematic diagram of the model reference adaptive control system is shown in Figure 3-1. We can obtain the state equation of the controlled models from (4.7)

x'

Ap x

Bp D

Where Ap  R

22

(4.10)

, Bp  R

21

, x  R 21 , D  R11 .

The reference model can be obtained as

xm'

Am xm Bm rp

(4.11) 11

Where Am  R 22 , Bm  R 21 , xm  R 21 , rp  R .

Reference Model xm'  Am xm  Bm rp

xm 

Adaptive System

rp

K



x'  Ap x  B p D



e

xp

F

 MB

 NB

T p

PexT dt

T p

PerpT dt

Adaptive Control Law Figure 4.3. MRAC system architecture based on the state variables.

Define tracking error as

e  x m  x

(4.12)

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 41 The adaptive controller is proposed as

D  Fx  Kr p

(4.13)

Where F and K are feedback and feed-forward gain of the closed loop system respectively. Substituting (4.13) into (4.10) yields

x  ( Ap  B p F ) x  B p Krp '

(4.14)

There exist optimal parameters F * , K matching condition (4.15) can be satisfied.

*

such that the following

Ap  B p F *  Am    * B p K  Bm  

(4.15)

Substituting (4.14), (4.15) into the derivative of tracking error e

xm x

yields

e  Ame  B p K (t )T x  B p F (t )T rp

(4.16)

Where K (t )  K (t )  K  , F (t )  F (t )  F  . Define a Lyapunov function







1 1 1 V  eT Pe  tr KM 1K T  tr FN 1F T 2 2 2



(4.17)

Where M 、 N are positive definite matrix, tr denoting the trace of a square matrix. Since Am is a Hurwitz stable matrix, there exists a unique positivedefinite symmetric matrix P satisfies the following:

AmT P  PAm  Q

(4.18)

42

Juntao Fei Where Q  QT  R 22 is positive definite. Substituting (4.16) into the derivative of V (t ) generates

1 V   eT Qe  eT PBp K T x  eT PB p F T r  trKM 1K T  trFM 1F T 2 (4.19) Making use of the properties of matrix trace xT Ax  tr ( xxT A) ,

tr ( A)  tr ( AT ) , (3.10) can be rewritten as 1 V   eT Qe  tr ( KBpT PT ex T )  tr ( FBpT PT er T )  trKM 1K T  trFM 1F T 2 (4.20) To make V  0 , we choose the adaptive laws as

K (t )T   MB pT Pex T   F (t )T   NBpT er T 

(4.21)

1 T e Qe  0 , according to Barbalat 2 lemma, e(t ) will asymptotically converge to zero, lim e(t )  0 . This adaptive laws yield V  

t 

4.4. PI-FUZZY COMPOUND VOLTAGE CONTROL In this section, PI-Fuzzy compound controller is designed for the DC side capacitor voltage control. The PI-Fuzzy compound controller structure is shown in Figure 4-1. When the system enters the transient state, the fuzzy controller can improve the system dynamic performance. On the contrary, if the system enters the steady state, PI controller can eliminate steady-state error of the system and improve steady-state performance of the system, where the switching of the controller is defined by the absolute value of the voltage error.

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 43

U ref

+



PI

Steady state

I p

-

U dc

Fuzzy

Transient state

Figure 4.4. PI-Fuzzy compound control structure.

In Figure 4-1, the parameters of PI controller are set by general tuning method. Using the voltage deviation of each sampling time and the fuzzy rule, fuzzy controller can judge quickly and effectively. The advantage of the PIfuzzy compound controller is that it can automatically switch between PI control and fuzzy control under different operating conditions, so it can take the advantages of both approaches, thereby improving system speed and enhancing the robustness under the premise of guaranteeing control precision. The switch between the two controllers depends on the indicators of the actual run-time system. The control program runs continuously and monitors the input and output characteristics of the control system, and coordinates between the two control law automatically. Since the two input fuzzy control is similar to proportional and derivative ( PD ) control, as an affine nonlinear kinematic system, small range of fluctuation of DC voltage in the regulation is inevitable. In order to reduce the interference brought by the derivative action of the controller, we use onedimensional fuzzy controller, and select the deviation eu (t ) between the actual voltage, and the reference voltage of the DC side as a fuzzy input variables, choose I p as the fuzzy output variables u , where I p is the control amount of the active current that grid injected into the APF main circuit. Fuzzy input eu (k ) is defined as eu (k )  U ref  U dc (k ) ,where U dc (k ) is the real value of the DC capacitor voltage for the first time k, U ref is the reference voltage. After scale changes, let the universe of input variables e(t ) and output variables u of the fuzzy controller are:

44

Juntao Fei

X  6 5 4 3 2 1 0 1 2 3 4 5 6 Select seven of linguistic variables in the universe: NB, NM, NS, ZO, PS, PM, PB. The input variables eu (t ) and output variables u are selected overlapping symmetrical triangle membership function, as shown in Figure 4.5.

NB 1.0

NM

NS

ZO

PS

PM

PB

-4

-2

0

2

4

6

0.5

0 -6

Figure 4.5. Triangular membership function.

The fuzzy control rules are the core of the fuzzy control, therefore, how to set up the fuzzy control rules become a crucial issue. The fuzzy control rule is the most natural way to describe the process of human behavior and decision analysis. It establishes the link between the fuzzy input variables and fuzzy output variables, multi-form of IF-THEN fuzzy conditional sentences.

U dc ( t )

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 45

U ref

0

t

Figure 4.6. The change curve of APF DC capacitor voltage.

Figure 4.6 is the APF DC voltage reference curve of the change process; we can develop fuzzy control rules based on this curve. Fuzzy control rules can be obtained according to the change process of the curve shown in Figure 4.6 and the existing experience of DC side capacitor voltage control is shown in Table 4.1. Table 4.1. DC side capacitor voltage fuzzy control rules

e(t ) u

NB

NM

NS

ZO

PS

PM

PB

NB

NM

NS

ZO

PS

PM

PB

The Mamdani type fuzzy inference system containing the fuzzy relationship such as “If e is A then u is C,” is adopted. For defuzzification method, the area of the center of gravity (centroid) is selected, then fuzzy controller output value can be obtained, that is grid controlled amount of active current I p to be injected into the APF main circuit.

4.4. SIMULATION STUDY The fuzzy controller is completed by using Matlab fuzzy control editor (FTS), fuzzy control rule table and fuzzy controller output curve are shown in Figure 4.7, Figure 4.8 respectively.

46

Juntao Fei

Figure 4.7. Fuzzy control rule table.

Figure 4.8. Fuzzy controller output curve.

According to the control schematic block diagrams shown in Figure 4.1 and Figure 4.3, a comprehensive model of active power filter based on reference adaptive control and PI-fuzzy compound control strategy can be established, including nonlinear load module (Nonlinear Load), harmonic current detection module (Harmonic Creator), the main circuit of filtering (APF Main Circuit), the PWM conversion module (PWM Creator), and adaptive current tracking compensation module (MARC Controller) and DC

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 47 side capacitor voltage fuzzy-PI the compound control module (Fuzzy-PI controller) and so on. In the simulations of adaptive current control based on PI-fuzzy compound control for the active power filter and hysteresis current control, the DC capacitor voltage PI-fuzzy compound control parameters are Kp = 0.2, Ki = 0.01. For comparison purpose, DC capacitor voltage conventional PI control parameters are the same as Kp = 0.2, Ki = 0.01. Table 4.2. Simulation Parameters parameters

value

power

Us=220Vrms/50HZ

DC capacitor voltage

Uc=600V

PWM switching frequency

fs=20KHZ

input inductor

L=6mH

Input capacitor

C=1000µF

output resistance

R=10000Ω

According to the parameters of Table 4.2, from (4.9-4.7), D0  0.24 T  86.4 and B  200000  231.5T x0  0.11573 600 , A p   0 p  518.5  0.1  can be obtained respectively. The positive definite matrix

2.4533 0.0833 1 0 . The adaptive gains are P M  0.000125 , Q  0 . 0833 2 . 2274   0 1  0 0.000183  ,N  . 0 0.000262   Regarding the reference model, it is designed as an over damped system, 2 damping ratio   1.4 , rise time t r  0.169s  1  1.5   , then the natural wn frequency

wn  30rad / s

can

be

obtained,

adjustment

time

t s  3.15 * T1  0.25s , the two poles of the reference model can be obtained

48

Juntao Fei

T  84  900 as -12.6061 and -71.3939. Then, Am   and Bm  5000 40  0   1 can be determined. During the simulation, the nonlinear load changes twice, first at the time of 0.4s, parallel nonlinear load is added to the APF system; second at the time of 0.72s, nonlinear load incorporated to the APF system at 0.4s is removed.





250 200 150 100

IL / A

50 0 -50 -100 -150 -200 -250 0

0.1

0.2

0.3

0.4

0.5 time / s

0.6

0.7 time / s

0.72

0.7

0.8

0.9

1

(a)

250 200 150 100

IL / A

50 0

-50 -100 -150 -200 -250 0.6

0.62

0.64

0.66

0.68

0.74

0.76

0.78

0.8

(b)

Figure 4.9. Grid current waveform without APF.

Figure 4.9 shows the grid current waveform without APF. It can be seen that due to the effects of nonlinear load, the grid current waveform has severe distortion. Figure 4.10 is the grid current waveform with adaptive current control based on PI-fuzzy compound control. As can be seen from Figure 4.10,

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 49 grid current distortion has been significantly improved with the incorporated APF. Figure 4.11 is the tracking waveform of DC capacitor voltage with PIfuzzy compound control. Figure 4.12 amplifies the waveforms which compare the DC capacitor voltage tracking between conventional PI control and fuzzyPI compound control with nonlinear load switching. It can be observed from Figure 4.12 that DC capacitor voltage with PI-fuzzy compound controller has smaller overshoot, better steady accuracy, robustness, and voltage setting than that with conventional PI control. 250 200 150 100

Is / A

50 0 -50 -100 -150 -200 -250 0

0.1

0.2

0.3

0.4

0.5 time / s

0.6

0.7 time / s

0.72

0.7

0.8

0.9

1

(a)

250 200 150 100

Is / A

50 0 -50 -100 -150 -200 -250 0.6

0.62

0.64

0.66

0.68

0.74

0.76

0.78

0.8

(b)

Figure 4.10. Grid current waveform with adaptive current control based on PI-fuzzy compound control.

50

Juntao Fei

1000

600 500

0

-500 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.11. Tracking waveform of DC capacitor voltage with PI-fuzzy compound control. 630 PI control

620

Udc / V

610

600

PI-Fuzzy control

590

580

570

560 1.5

2

2.5

3

3.5

4

4.5

5 (a)

5

x 10

Figure 4.12 (Continued)

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 51 680

660 PI control

Udc / V

640

620

600 PI-Fuzzy control

580

560 2.5

3

3.5

4

4.5 (b) 5

x 10

Figure 4.12. Amplification graph of DC capacitor voltage tracking waveform between conventional PI control and fuzzy-PI compound control with nonlinear load switching.

It can be seen from Figure 4.13 and Figure 4.14, the feed forward gain K and feedback gain F converge to the stable values after short time. These two adaptive parameters are updated online to make the current track reference model as close as possible. Figure 4.15 shows that nonlinear load leads grid current containing a large number of harmonics, where THD = 45.82%. Figure 4.16 plots the harmonic content with hysteresis control based on PI-fuzzy control, where THD = 4.26%. Figure 4.17 draws the total harmonic content with adaptive current control based on PI-fuzzy control, where THD = 3.84%. It is shown that adaptive current control with PI-Fuzzy compound control is effective in harmonic suppression of APF. It can be concluded that the current tracking and THD performance can be improved by using the proposed adaptive current control with PI-fuzzy control. Thus, the control performance and robustness to nonlinear load can be improved.

52

Juntao Fei

15

K(t)

10

5

0 0

0.1

0.2

0.3

0.4

0.5 time / s

0.6

0.7

0.8

0.9

Figure 4.13. Adaptation of MRAC controller‟s feed forward gains K. 0.02 0 -0.02

F1(t)

-0.04 -0.06 -0.08 -0.1 -0.12 -0.14 0

0.1

0.2

Figure 4.14 (Continued)

0.3

0.4

0.5 time / s

0.6

0.7

0.8

0.9

1

(a)

1

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 53 0.05 0 -0.05

F2(t)

-0.1 -0.15 -0.2 -0.25 -0.3 -0.35 0

0.1

0.2

0.3

0.4

0.5 time / s

0.6

0.7

0.8

0.9

1

(b)

Figure 4.14. Adaptation of MRAC controller „s feedback gain F. Fundamental (50Hz) = 11.29 , THD= 45.82% 100 90

Mag (% of Fundamental)

80 70 60 50 40 30 20 10 0 0

2

4

6

8

10 Harmonic order

12

14

Figure 4.15. Grid current spectrogram without active power filter.

16

18

20

54

Juntao Fei Fundamental (60Hz) = 8.373 , THD= 4.26% 100 90

Mag (% of Fundamental)

80 70 60 50 40 30 20 10 0 0

100

200

300

400

500 Frequency (Hz)

600

700

800

900

Figure 4.16. Grid current spectrogram with hysteresis control of active power filter based on PI-fuzzy compound control.

Fundamental (50Hz) = 43.16 , THD= 3.84% 100 90

Mag (% of Fundamental)

80 70 60 50 40 30 20 10 0 0

100

200

300

400

500 Frequency (Hz)

600

700

800

900

Figure 4.17. Grid current spectrogram with adaptive current control based on PI-fuzzy compound control.

1000

Adaptive Control of Active Power Filter Using PI-Fuzzy Compensator 55

CONCLUSION In this chapter, the approximate mathematical model of the active power filter is established, model reference adaptive current tracking control method for AC side current is developed, and PI-Fuzzy compound control is designed for the DC capacitor voltage regulation. Simulation studies prove that adaptive current control method based on PI-fuzzy compound control can not only regulate the DC capacitor voltage, but also track the AC current command signals, eliminate the power harmonics, and improve the power quality and system robustness.

Chapter 5

ADAPTIVE SLIDING MODE CONTROL OF ACTIVE POWER FILTER This chapter presents a thorough study of the adaptive sliding mode technique with application to single-phase shunt active power filter (APF). Based on the basic principle of single-phase shunt APF, the approximate dynamic model is derived. A model reference adaptive sliding mode control algorithm is proposed to implement the harmonic compensation for the singlephase shunt APF. This method will use the tracking error of harmonic and APF current as the control input, and adopt the tracking error of reference model and APF output as the control objects of adaptive sliding mode. In the reference current tracking loop, a novel adaptive sliding mode controller is implemented to tracking the reference currents, thus improving harmonic treating performance. Simulation results demonstrate the satisfactory control performance and rapid compensation ability of the proposed control approach under different conditions of the nonlinear load current distortion and the mutation load respectively.

5.1. INTRODUCTION In the presence of model uncertainties and external disturbance, sliding mode control is necessary to incorporate into the adaptive control system, since sliding mode control is a robust control technique which has many attractive features such as robustness to parameter variations and insensitivity to disturbance. Adaptive sliding mode control has the advantages of

58

Juntao Fei

combining the robustness of variable structure methods with the tracking capability of adaptive control. It is necessary to adopt adaptive sliding mode control that can on-line adjust the control parameter vector combined with the great robustness of sliding mode control for the harmonic suppression of single-phase shunt APF. This chapter will expand the model reference adaptive control (MRAC) and incorporate the sliding mode control into the adaptive system to design the adaptive sliding mode control algorithm and apply to the single-phase shunt APF. The contribution of this chapter can be summarized as: 1. A novel adaptive sliding mode control is proposed in reference current tracking to reduce the tracking error. The designed APF has superior harmonic treating performance and minimizes the harmonics for wide range of variation of load current under different nonlinear load. Therefore an improved THD performance can be achieved with the proposed control scheme. 2. It is the first time that adaptive sliding mode control is applied to the APF. The advantage of using adaptive controller for the shunt APF with sliding mode technique is that it has better harmonic treating performance and will improve the robustness of the APF under the nonlinear loads. 3. This chapter systematically and deeply studies the adaptive control and sliding mode technique with application to APF, comprehensively uses the adaptive control, sliding mode control with the APF, thereby significantly reducing the APF‟s sensitivity to the nonlinear load and disturbance, and improving the robust performance. The APF control system is designed to make the compensation current track the command signal in real-time, thereby eliminating the harmonics, improving the electric energy quality, and enhancing the security of the power transmission and distribution and power grid. Therefore, this research has great theoretical value and application potentials.

5.2. DESIGN OF ADAPTIVE SLIDING MODE CONTROLLER In this section, a detailed study of the shunt APF with parameter uncertainties is proposed. A new adaptive sliding mode control strategy for shunt APF using a proportional sliding surface is proposed, an adaptive sliding

59

Adaptive Sliding Mode Control of Active Power Filter

law to overcome the parameter uncertainties is also derived. The block diagram of the designed adaptive sliding mode controller for APF is shown in Figure 5.1. The goal of APF control is to design an adaptive sliding mode controller so that the APF output trajectory can track the reference model. The linear state equation of APF can be expressed as: 

X   AP X   BPu

(5.1)

Consider the system (5.1) with parametric uncertainties, i. e. the linearization errors such as high-order terms in Taylor series. 

X   ( AP  AP ) X  (t )  ( BP  BP )u  f d

(5.2)

Where AP the unknown parameter uncertainties of the matrix is AP ,

BP is the unknown parameter uncertainties of the matrix BP , f d is external disturbances such as nonlinear loads in the APF. We make the following assumptions. Assumption 1. There exist unknown matrices of appropriate dimension D, ~

~

G such that AP (t )  BP D(t )   AP (t ) , BP (t )  BPG(t )   BP (t ) ~

~

where BP D(t ) and BPG(t ) is matched uncertainty,  AP (t ) and  BP (t ) is unmatched uncertainty. From this assumptions, (5.2) can be rewritten as 

X   AP X  (t )  BPu  BP f m  fu

(5.3)

Where f m (t , X  , u ) represents the matched, lumped uncertainty and

fu (t , X  , u ) represents the unmatched, lumped uncertainty respectively which is given by

f m (t , X  , u )  D(t ) X  (t )  G(t )u ~

(5.4)

~

fu (t , X  , u )   AP (t ) X  (t )   BP (t )u

(5.5)

60

Juntao Fei Assumption 2. The matched and unmatched lumped uncertainty f m and f u

are bounded such as

f m (t , X  , u )   m and fu (t , X  , u )  u , where

 m 、  u are known positive constants. Assumption 3. There exist constant matrix K following matching condition AP  BP K

*T

*

and  * such that the

 Am and BP *T  Bm can

always be satisfied.

xr



x m  Am xm  Bm r



Reference Model



 

r

e

∑ 

u

u



 u0





x  Ap x  B p u

x

APF Model

x K T (t ) X  (t )

 ( BP ) 1

s s

( BP ) 1  Am e(t )

e e

e e

 T (t )r (t ) Adaptive Sliding Mode Controller

r

Figure 5.1. Block Diagram of Adaptive Sliding Mode Control Approach.

Then the adaptive sliding mode controller will be designed. The reference model is defined as: 

X m  Am X m  Bm r The tracking error and its derivative are:

(5.6)



 x0

x

61

Adaptive Sliding Mode Control of Active Power Filter

e  X  X m

(5.7)



e  Ame   AP  Am  X   BPu  Bm r  BP f m  f u

(5.8)

The proportional sliding surface is defined as:

s(t )  e Where

(5.9)

 is a constant matrix satisfying that  BP is a nonsingular

diagonal matrix. The derivative of the sliding surface is: 



s   e   Ame    AP  Am  X    BPu   Bm r   BP f m   f u (5.10) 

Setting s  0 to solve equivalent control ueq gives

ueq  ( BP )1  Ame  ( BP )1   AP  Am  X   ( BP )1  Bmr  f m  ( BP )1  f (5.11) From assumption 3, (5.11) can be rewritten as

ueq  ( BP )1  Ame  K *T X    *T r  f m  ( BP )1  f

(5.12)

From (5.12), the control signal u can be proposed as

u  ( BP )1  Ame  K *T X    *T r   ( BP )1

Where

 is constant,

s s

(5.13)

s is the sliding mode unit control signal. s

The adaptive sliding mode version of control input is

62

Juntao Fei

u (t )  ( BP )1  Ame(t )  K T (t ) X  (t )   T (t )r (t )   ( BP ) 1

s s

(5.14) Where K (t ) is the estimate of K * ,  (t ) is the estimate of  * . Define the estimation error as ~

K (t )  K (t )  K *

(5.15)

~

 (t )   (t )   *

(5.16)

Substituting (5.16) and (5.17) into (5.14) yields ~T  ~T  s u (t )  ( BP ) 1  Ame(t )   K (t )  K *T  X  (t )   (t )   *T  r (t )   ( BP ) 1 s     (5.17) From (5.17) and assumption 3, rewrite (5.3) as follows ~ T



X  (t )   BP ( BP ) 1  Am e(t )  BP K (t ) X  (t )  Am X  (t ) ~T

 BP  (t )r (t )  Bm r (t )  BP  ( BP ) 1

s  BP f m  f u s

(5.18)

Then, we have the derivative of the tracking error equation: 

~ T

~T

e(t )   BP ( BP ) 1  Am e(t )  BP K (t ) X  (t )  BP  (t )r (t )  Ame(t )  BP  ( BP ) 1 ~ T

~T

  I  BP ( BP ) 1   Am e(t )  BP K (t ) X  (t )  BP  (t )r (t )  BP  ( BP ) 1

s  BP f m  f u s

s  BP f m  f u s

(5.19) And the derivative of s (t ) is

63

Adaptive Sliding Mode Control of Active Power Filter ~ T



~T

s(t )  BP K (t ) X  (t )   BP  (t )r (t )  

s   BP f m   f u s (5.20)

Define a Lyapunov function

1 T 1  ~ 1 ~ T  1  ~ 1 ~ T  V  s s  tr  K M K   tr  N   2 2   2  

(5.21)

Where M , N are positive definite matrix, tr denoting the trace of a square matrix. Differentiating V with respect to time yields  T T  ~ ~ ~ ~ 1 1    V  s s  tr K M K  tr  N           

T



 T T  ~ ~ ~ ~T ~ 1 T 1       s  s  BP f m  s  f u  s  BP K (t ) X  (t )  tr K M K  s  BP  (t )r (t )  tr  N           T

T

T

~ T

(5.22)

For s

T

~ T

~T

 BP K (t ) X  (t ) and sT  BP  (t )r (t ) are scalar, from the

properties of matrix trace xT Ax  tr ( xxT A) 、 tr ( A)  tr ( AT ) there have ~ T ~ T   ~  sT  BP K (t ) X  (t )  tr  X  (t ) sT  BP K (t )   tr  K (t ) BPT  T sX  T (t )     

(5.23) ~T   ~  T s  BP  (t )r (t )  tr r (t )s  BP  (t )   tr  (t ) BPT  T sr T (t )      T

~T

(5.24) 

To make V  0 , we choose the adaptive laws as

64

Juntao Fei  T ~

 T

K (t )  K (t )   MBPT  T sX  T (t ) T ~

(5.25)

T

 (t )   (t )   NBPT  T sr T (t )

(5.26)

This adaptive sliding law yields:

V    s  sT  BP f m  sT  f u    s  s  BP

f m  s  fu

   s  s  BP  m  s   u   s     BP  m    u   0 (5.27) With

   BP  m   u  , where  is a positive constant, V

becomes negative semi-definite, i.e. V   s .

According to Barbalat

lemma, it can be proved that s (t ) will asymptotically converge to zero,

lim s(t )  0 . t 

5.3. SIMULATION ANALYSIS In this section, the single-phase shunt APF using adaptive sliding mode control is implemented with MATLAB/SIMPOWER Toolbox. The goal of adaptive sliding mode control is to make the APF output current track the detected harmonic current. Simulations on the APF system will verify the control effects of the proposed adaptive sliding approach. First of all, the nonlinear load of simulation model is described. Nonlinear load: A load branch is rectifier bridge connecting parallel RC load, R=15Ω, C=5e-3F. when 0-0.4s, a load works, the total harmonic distortion(THD) of nonlinear load current is 45.82%; when 0.4-0.8s, the breaker is switched on and another load branch which is the same as first one is injected, accompanied by a disturbance load (interference frequency f=1000Hz, square wave T=0.001s, pulse width 50%, parallel disturbance load L=0.2H and R=20Ω), the total harmonic

65

Adaptive Sliding Mode Control of Active Power Filter

distortion(THD) of nonlinear load current is 40.12%. The current mainly contains 3rd and 7th times, odd harmonic. The parameters of the proposed adaptive sliding mode controller are chosen as follows: AM =[-49.6 -351.8;519 0.21], BM =[7400;-8.6], CM =[1 0;0 1], DM = [0;0], M=5e-7, N=5e-5, ρ=200, λ=[0.04 0.05]. DC voltage VC =600V, it can be calculated that x0 = [0.11573; 600], u0 =0.24077. According to the APF circuit model it is calculated that

Ap = [0 -86.4; 518.5 -0.1], B p = [200000;-231.5]. Because the rectifier bridge and nonlinear load result in harmonic and reactive currents, serious distortions of the circuit current show up. Figure 5.2 draws the current waveform before and after the nonlinear load change. Mutations distortions of currents have serious effects on the power system and other electrical equipment that should be compensated for and eliminated. From the adaptive harmonic detection output ih , and the APF current iL obtained from measuring module, we can get the input signal of adaptive sliding mode controller (which is also the reference model input signal) r  ih  iL  iPI , where iPI is the compensation current of voltage PI control for APF DC side. Single phase harmonic detection method is used, where the input has only sine wave signals, implying the detection result ih contains the harmonic and reactive current. 100

80

60

Nonlinear Load Current(A)

40

20

0

-20

-40

-60

-80

-100 0.3

0.32

0.34

0.36

0.38

0.4 Time(Second)

0.42

0.44

Figure 5.2. Nonlinear Load Current (before and after load change).

0.46

0.48

0.5

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Juntao Fei

120

100

Control Input

80

60

40

20

0

-20

0

0.1

0.2

0.3

0.4 time(second)

0.5

0.6

0.7

0.8

Figure 5.3. Input of Adaptive Sliding Mode Controller.

According to the approximation process of APF dynamic model, near the equilibrium point ( x0 , u0 ) , the simulation variables x  x  x0 , u  u  u0 , so we can get x 、 u from x  x  x0 , u  u  u0 respectively. The input waveform of adaptive sliding mode controller (i.e. input waveform of the reference model) is drawn in Figure 5.3. The controller not only compensates and eliminates the harmonic, but it also can compensate the DC voltage and make it stay stable in near a predetermined value to improve the compensation effect. 400

350

300

Control Output

250

200

150

100

50

0

-50

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

Figure 5.4. Output of Adaptive Sliding Mode Controller.

0.6

0.7

0.8

Adaptive Sliding Mode Control of Active Power Filter

67

The output of the adaptive sliding mode controller is shown in Figure 5.4. The output u of the controller compares with triangular waves which have similar amplitude with u , then PWM generator output is the standard of fourphase control pulse, and can fully reflect the control function of controller output to the APF. PWM signal is generated using triangular wave comparison method, where the triangular wave amplitude is ±1, frequency is 1000Hz. By controlling on or off of the four thyristors in the full controlled bridge, the four phase PWM pulse control signal will control the charging and discharging process of the DC side capacitor C , thereby producing the inductor current iL , i.e. APF compensating current. The parameters of APF main circuit: L =0.006H inductor, capacitor C = 0.001F, R = 10KΩ, thyristors using the IGBT / Diode module, a default parameter. Figure 5.5 compares the harmonic and APF current, where the red curve represents APF output current (inductor current), and blue curve represents detected harmonic current. Compared with the detected harmonic current, the APF output current has equal amplitude but opposite direction. The APF output current is injected into the nonlinear load current, then the harmonics and reactive current caused by nonlinear loads can be eliminated, thus the current harmonic compensation can be achieved. After 0.4s the load and interference are increased, the APF current still can quickly track the harmonic current although there may have some errors between harmonic current and the APF output current even if the APF works stable. The errors mainly contain compensation current from PI control circuit of the DC voltage for the DC voltage compensation of APF. Simulation of the PI control signal is shown in Figure 5.6. The initial moment due to the DC voltage is zero; the output of PI controller is a big compensation signal, so that the DC voltage can quickly arrive at the setting value. After 0.4s, the load and interference are increased, the harmonic is increased, and the DC voltage is reduced. The PI control output could adjust in time, and then quickly recover a normal working state. Figure 5.7 shows, after the circuit starts to work, the APF capacitor voltage quickly rises to more than 500V, and basically stays at a setting value near 600V, rapidly adjusted by the PI control of DC voltage. The load changed at 0.4s, the DC voltage could be quickly stabilized to the set value also.

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Juntao Fei 80

60

Harmonic and APF Output Current(A)

40

20

0

-20

-40

-60

-80

-100

-120

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.6

0.7

0.8

0.4 Time(Second)

0.5

0.6

0.7

0.8

Figure 5.5. Harmonic and APF Current. 40

20

PI Control Output

0

-20

-40

-60

-80

-100

0

0.1

0.2

0.3

Figure 5.6. Output of DC Voltage PI Control.

69

Adaptive Sliding Mode Control of Active Power Filter

700

600

APF DC Voltage(V)

500

400

300

200

100

0

-100

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.6

0.7

0.8

0.4 Time(Second)

0.5

0.6

0.7

0.8

Figure 5.7. APF DC Voltage.

200

150

Power Current(A)

100

50

0

-50

-100

0

0.1

0.2

0.3

Figure 5.8.A. Power Current(0-0.8s)

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Juntao Fei

80

60

40

Power Current(A)

20

0

-20

-40

-60

-80 0.35

0.4

0.45 Time(Second)

0.5

0.55

Figure 5.8.B. Power Current(0.35-0.55s) (Nonlinear Load Changes in 0.4 seconds).

Figure 5.8 shows the power current after APF has been compensated, where the Figure 5.8B zoom in on 0.4s when the loads change. It can be seen from the Figure 5.9 that the current waveform has been improved and adjusted rapidly to approximate standard sine wave within 0.08s after the circuit started to work and load changed. Compared with the nonlinear load current waveform, it can be seen that before compensation the load current has serious distortion, and after compensation, the current quality has been obviously improved. Thus, the proposed purposes of nonlinear load current compensation through APF can be successfully achieved. Through the FFT Analysis tools of SIMLINK powergui module, the load current and the power current is acquired. Figure 5.10 is the nonlinear load current waveform and its harmonic analysis. The serious waveform distortions can be observed. FFT analysis show that, nonlinear current harmonic is mainly 3rd and 7th harmonic interference, and with a certain degree of 5th and higher harmonic interference. Power current waveform after APF is shown in Figure 5.11. After compensation, the 3rd harmonic is greatly reduced, the 7th harmonic is almost eliminated, all other higher harmonics are reduced, and the power current is approximate to sine wave. The compensated current THD is 3.5%, more than the national standard level of 5%. After load changed, nonlinear load current THD changed from 45.81% to 40.12%. This is due to the percentage of nonlinear part load (bridge, capacitance, inductance) relative

Adaptive Sliding Mode Control of Active Power Filter

71

to the linear portion (resistance) decreased. But we can still see the obvious distortion of nonlinear load current. Figure 5.12 shows the current after load changed, the same bridge and nonlinear load (with certain high frequency interference) paralleled in the circuit. It can be seen that although the nonlinear load and the load current increase, the current waveform distortion is relatively minor as of 40.12% THD. After compensation, the quality of power current is greatly improved with THD reaching at 3.28%. It demonstrates good compensation capability of APF using adaptive sliding mode controller. Because of the increase of load disturbance and the changes of currents, the compensated current has some minor glitches, but the power current quality remains within the national standard. The control parameters K, e in the adaptive sliding mode controller are analyzed in the following steps. From Figs. 5.13-5.14, it can be seen that the parameters can be adjusted to a stable value soon and the stability of the control system can be maintained, the parameters K and Ө can converge to constant and stay in stable state within 0.1s, i.e. within 4-5 circuit cycles.

FFT window: 3 of 40 cycles of selected signal 20 0 -20 0.2

0.21

0.22

0.23 Time (s)

0.24

0.25

Mag (% of Fundamental)

Fundamental (50Hz) = 22.59 , THD= 45.81%

40 30 20 10 0

0

200

400 600 Frequency (Hz)

800

1000

Figure 5.9. Nonlinear Load Current Analysis (before load change).

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Juntao Fei

FFT window: 5 of 40 cycles of selected signal 20

0

-20 0.2

0.21

0.22

0.23

0.24 0.25 0.26 Time (s)

0.27

0.28

0.29

Fundamental (50Hz) = 20.58 , THD= 3.49%

Mag (% of Fundamental)

3 2.5 2 1.5 1 0.5 0

0

200

400 600 Frequency (Hz)

800

1000

Figure 5.10. Power Current after APF Compensation (before load changes).

FFT window: 3 of 40 cycles of selected signal 50 0 -50 0.6

0.61

0.62

0.63 Time (s)

0.64

0.65

Fundamental (50Hz) = 48.92 , THD= 40.12%

Mag (% of Fundamental)

40

30

20

10

0

0

100

200

300

400 500 600 Frequency (Hz)

700

800

900

1000

Figure 5.11. Nonlinear Load Current (after load change).

Adaptive Sliding Mode Control of Active Power Filter

73

FFT window: 5 of 40 cycles of selected signal 40 20 0 -20 -40 0.6

0.61

0.62

0.63

0.64 0.65 0.66 Time (s)

0.67

0.68

0.69

Fundamental (50Hz) = 46.71 , THD= 3.28%

Mag (% of Fundamental)

2.5 2 1.5 1 0.5 0

0

200

400 600 Frequency (Hz)

800

1000

Figure 5.12. Power Current after APF Compensation (after load changes). 0.1

K1

0.05

0

-0.05

-0.1

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.6

0.7

0.8

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.6

0.7

0.8

0.6

K2

0.4

0.2

0

-0.2

Figure 5.13. Controller Parameters K.

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Juntao Fei 25

20

theta

15

10

5

0

-5

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.6

0.7

0.6

0.7

0.8

Figure 5.14. Controller Parameter. 3000

2000

Current Error

1000

0

-1000

-2000

-3000

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.8

Figure 5.15. Current Error of Reference Current item and APF Inductance Current.

75

Adaptive Sliding Mode Control of Active Power Filter 3000

2000

Voltage Error

1000

0

-1000

-2000

-3000

-4000

0

0.1

0.2

0.3

0.4 Time(Second)

0.5

0.6

0.7

0.8

Figure 5.16. Voltage Error of Model Reference Voltage item and APF DC Voltage.

Figure 5.17. APF DC Voltage Using Adaptive Control.

The tracking error e   ei

eu  as parameters of adjustment process T

exists within the adaptive sliding mode process shown in Figs. 5.15-5.16. It will decrease in a certain way, and maintain in a certain range finally. The simulation is based on the actual APF model with nonlinear load and model uncertainties. Because of the complexity of the circuits and the approximation

76

Juntao Fei

of the model, the parameters of APF model have some uncertainties. The tracking error can be quickly adjusted to dozens magnitude of dimensionless parameters, and to be maintained in a certain range.

Figure 5.18. APF DC Voltage Using Adaptive Sliding Mode Control.

Table 5.1. Comparison of THD after Compensation Time Period/ THD of Current Controller Type

Current Type

Without controller

Nonlinear Load Current

Adaptive Controller Adaptive sliding mode Controller

Power Current after Compensation

0-0.4s

0.4-0.8s

45.81%

40.12%

4.16%

3.54%

3.49%

3.29%

It can be observed from Table 5.1 that adaptive sliding mode controls have better harmonic compensation performance than adaptive control; the index of THD has been reduced with the adaptive sliding control. If you compare the waveforms before and after APF works, it can be seen that the adaptive sliding mode control not only has good compensation effect, but also has quick compensation ability. At the same time, the controller could optimize the adaptive parameters to make it more reasonable, and make it

Adaptive Sliding Mode Control of Active Power Filter

77

possible for the investigation of the intelligent control. Simulation and analysis verified the feasibility of adaptive sliding mode control and showed good control effect in single-phase SAPF.

CONCLUSION This chapter studied the principle and dynamic model of single-phase shunt APF, and proposed a new adaptive sliding mode control algorithm. The simulation results proved that for nonlinear load current the adaptive sliding mode controller has successful compensation effect, i.e. it can compensate the most current harmonic and eliminate certain reactive current, which can recover sine wave from severely detuned current waveform and improve the power factor. The reference currents tracking behavior has been improved and the power supply current harmonic has been reduced with novel adaptive sliding mode control. The proposed control system has the satisfactory adaptive and robust ability in the presence of the changing disturbances and nonlinear loads.

Chapter 6

ADAPTIVE FUZZY CONTROL WITH SUPERVISORY COMPENSATOR OF ACTIVE POWER FILTER In this chapter, an adaptive fuzzy control system with supervisory controller is proposed to improve dynamic performance of the three-phase active power filter (APF). The proposed adaptive fuzzy controller for APF does not need to establish accurate mathematical model and has the ability to approximate the nonlinear characteristics of APF. The adaptive law based on the Lyapunov analysis can adaptively adjust the fuzzy rules, therefore the asymptotical stability of the adaptive fuzzy control system can be guaranteed. Simulation results show that this control method has an excellent dynamic performance such as small current tracking error, reduced total harmonic distortion (THD), strong robustness in the presence of parameters variation and nonlinear load.

6.1. INTRODUCTION In the last few years, fuzzy control has been extensively applied in a wide variety of industrial systems and consumer products because of its model free approach. However, systematic stability analysis and controller design of the adaptive fuzzy controller with application to APF have not been investigated in the literature. Therefore, it is necessary to utilize the adaptive fuzzy control scheme to improve the current tracking and filtering performance. In this chapter, a novel adaptive fuzzy control with supervisory controller is

80

Juntao Fei

developed to improve the control performance and guarantee the Lyapunov stability of the close-loop system. The contribution of this chapter is the integration of the adaptive control, the nonlinear approximation of fuzzy control and supervisory compensator. The control strategy proposed here has the following advantages: 1. The proposed control strategy does not depend on an accurate mathematical model of APF which is difficult to obtain, and may not give satisfactory performance under parameter variations. Adaptive fuzzy control with a supervisory controller has great ability to compensate for the system nonlinearities and improve the power dynamic performance such as current tracking and total harmonic distortion (THD) index. 2. In order to eliminate the nonzero problem of the fuzzy approximation errors, a supervisory compensator is incorporated into the adaptive fuzzy control scheme in the Lyapunov framework. A novel adaptive fuzzy control with supervisory compensator is developed to improve the control performance and guarantee that the closed loop system is globally stable and current tracking error is as small as possible. 3. An adaptive fuzzy control with supervisory compensator is proposed to deal with system nonlinearities and nonlinear load in order to improve the current tracking and robustness of the control system compared with conventional control method. The robust adaptive fuzzy control method has been extended to the control of APF in this chapter. This is the successfully application example using adaptive control, fuzzy control and robust compensator with the APF. Both of these features are the innovative development of intelligent robust adaptive control methods incorporated with conventional control for the APF.

6.2. PRINCIPLE OF ACTIVE POWER FILTER The shunt APF can be considered to be the most basic structure of APF. This chapter mainly studies the most widely used parallel voltage type of APF. In the practical application, the three-phase is the most widely used shunt APF because of its excellent performance characteristics and simplicity in implementation, therefore the three-phase three wire system will be

Adaptive Fuzzy Control with Supervisory Compensator …

81

investigated in this section and the dynamics of three-phase APF will be described. In practical operation, APF is equivalent to a flow control current source. The whole APF system consists of three sections, harmonic current detection module, and a current tracking control module and compensation current generating circuit. A harmonic current detection module usually uses instantaneous reactive power theory based on rapid detection of harmonic current. Three-phase three-wire APF produces compensation currents with three bridge-arm circuit. In order to eliminate the harmonic components in the currents from power supply, a compensation circuit produces compensation currents which have the same amplitude and opposite phase with the harmonic currents. The block diagram of the three-phase three wire active power system is given in Figure 1. The principle of APF is to detect the voltage and current of *

the compensation object, get command signal i c of the compensation current by using operation circuit for instructions current, and then obtain compensation current

ic

by PWM generator in order to offset harmonic

current, and achieve ideal source current. The mathematical model of APF can be described in the following paragraph. According to circuit theory and Kirchhoff's theorem, we can get the following state equations:

ica  

rica  vsa vdc  s L L

(6.1)

icb  

ricb  vsb vdc  s L L

(6.2)

icc  

ricc  vsc vdc  s L L

(6.3)

Where s is the switching function, denoting the ON/OFF status of the devices in the two legs of the IGBT Bridge. We can define s as

1 QN  1 s . The main voltage of vsa 、 vsb 、 vsc supplies the power for 0 QN  0

82

Juntao Fei

the APF and the nonlinear loads. The parameter of L and r are the inductance and resistance of the APF respectively. Vsa Nonlinear loads

Vsb Vsc

L

iL

Vsa

is

Operation circuit for instructions current

r ica-icc

Vdc

ic i*c

Tracking PWM and control signal circuit

Driving circuit

Vdc Figure 6.1. Block diagram for main circuit of APF.

6.3. ADAPTIVE FUZZY CONTROL WITH SUPERVISORY COMPENSATOR In this section, an adaptive fuzzy control is derived and Lyapunov analysis is implemented to guarantee the stability of the closed-loop system. The adaptive control will be approximated by adjusting the parameters of an adequate fuzzy logic system. A fuzzy controller is composed of the following four elements: fuzzier, some fuzzy IF-THEN rules, a fuzzy inference engine, and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping

Adaptive Fuzzy Control with Supervisory Compensator … from an input linguistic vector

x   x1 , x2 ,

83

, xn  U  R n to an T

output variable y V  R . The fuzzy rule base consists of a collection of fuzzy if-then rules which can be expressed as

Rl : If x1 is A1l and … x n is Anl ,then y is B l l Where Ai and B are fuzzy sets, and l  1, l

,M , M denotes the number

of fuzzy if-then rules. The fuzzy inference engine performs a mapping from fuzzy sets in U to a crisp point in V . In this chapter, the singleton fuzzifier mapping is adopted, xi and y have the same kind of member functions that are all Gaussian membership functions defined as

  xi  ci 2   Al  xi   exp    2 i   2  i   Where ci and  i are the centre and width of the i

(6.4)

th

l

fuzzy set Ai ,

respectively. From the knowledge of the fuzzy systems, the output of the fuzzy system can be expressed using center-average defuzzifier, product inference and singleton fuzzifier.

 n  h  l    Ail  xi      T ( x ) y( x )  l 1M  ni 1       Ail  xi   l 1  i 1  M

(6.5)

Where  Al  xi  is the membership function value of the fuzzy variable i

xi ,    h1 , h2 , T

, hM  is adaptive parameter vector,

T  x  1 ( x ),  2 ( x ),  ,  M ( x ) is the vector of the fuzzy basis

functions.

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Juntao Fei

xm u  uD ( x  )  us ( x)

+

Active Power Filter 被控对象 x  f ( x)  bu

-

+

Fuzzy controller 模糊控制器 uD ( x  )   T  ( x)



 (0)

Adaptive law    ep ( x)

Supervisory controller us ( x)  ks sgn(eT Pb) Figure 6.2. Adaptive fuzzy control block for APF.

We will show how to construct adaptive fuzzy-sliding control in the next steps. The block diagram of the adaptive fuzzy control system with supervisory controller for APF is shown in Figure 2. Systematic stability analysis is performed in the design of proposed adaptive fuzzy control with supervisory controller. We can transform the dynamic model of (6.1-6.3) into the following form:

x  f ( x)  bu



Where x  ica

(6.6)

icb icc  , f ( x)  

rick  vsk , k =1,2,3 , L

b

vdc , L

control target is to make currents x tracking the given reference current signal

xm .The tracking error is defined as e  xm  x . We choose the control law as

u  uD ( x  )  us ( x)   T  ( x)  ks sgn(eT Pb)

(6.7)

Adaptive Fuzzy Control with Supervisory Compensator … Where fuzzy controller uD ( x  ) 

n

  ( x)   i i

T

85

 ( x) as in (6.7),

i

 ( x)

is

fuzzy

basis

function,

the

supervisory

controller

us ( x)  ks sgn(e Pb) . T

Substituting (6.7) into (6.6) yields:

e  ke  b[u  uD ( x  )  us ( x)]

(6.8)

Define optimal parameter vector:

   arg min[sup u  uD ( x  ) ]  Rm

(6.9)

xR

Define fuzzy approximation error:

  uD ( x   )  u 

(6.10)

Then (6.8) becomes:

e  ke  b[uD ( x   )  uD ( x  )]  bks sgn(eT Pb)  b  ke  b(    )T  ( x)  bks sgn(eT Pb)  b



Theorem: A feedback control u  uD x 



(6.11)

and adaptive law for

adjusting parameters vector  ( t ) are designed to satisfy that the closed loop system must be globally stable in the sense that all variable, x( t ), ( t ) and

uD  x   must be uniformly stable and the current tracking error e( t ) should be as small as possible. Proof: Define Lyapunov function candidate

V

1 2 b T pe    2 2

(6.12)

86

Juntao Fei Where

 is a positive constant,    *   , p is the positive constant to

satisfy the following condition (k T ) p  p(k )  q , that is 2 pk  q . Differentiating V with respect to time yields:

1 b V   qe2   T [ ep ( x)   ]  epbks sgn(eT Pb)  epb 2  (6.13) Therefore the adaptive law is chosen as

   ep ( x) Where

(6.14)

 is the adaptive gain.

Substituting (6.14) into (6.13) yields

1 1 V   qe2  epbks sgn(eT Pb)  epb   qe 2  epb (supt 0   ks ) 2 2 (6.15) Choosing ks  supt 0  , (6.15) becomes

1 V   qe2  0 2

(6.16)

Then we can obtain V  0 , V is negative definite implies that V , s,  converge to zero. The fact that V is negative semi-definite ensures that

V , e, are all bounded. e is also bounded. The inequality (6.16) implies that t 1 e is integrable as  e2 dt  [V (0)  V (t )] . Since V (0) is bounded and 0 q t

V (t ) is non-increasing and bounded, it can be concluded that lim  e2 dt is t  0

Adaptive Fuzzy Control with Supervisory Compensator … bounded. Since lim

t

87

 e dt is bounded and e is also bounded, according to 2

t  0

Barbalat lemma, e( t ) will asymptotically converge to zero, lim e(t )  0 . t 

Remark 1: From the universal approximation theorem,  can be made to be arbitrary small using fuzzy system on a compact domain. Because of fuzzy approximation error,  cannot always be equal to zero. The fact that  is equal to zero can only be realized in the ideal situation. This will result that the stability of the control system cannot be guaranteed. In order to solve such problem, a supervisory controller u s is added with uC to eliminate the negative influence of the fuzzy approximation errors and guarantee the stability condition.

6.4. SIMULATION STUDY In this section, the performance of the proposed adaptive fuzzy control will be testified using Matlab/Simulink package with SimPower Toolbox. Simulation results are presented to verify the effectiveness of the proposed adaptive fuzzy control. We choose six membership functions as

exp [ x 4 2(i 1)]2 , i 1,

,6.

We define membership function of sliding function s as

NM

( s)

1 , 1 exp(5( s 3))

ZO

(s) exp( s 2 ) ,

PM

( s)

1 . 1 exp(5( s 3))

The parameters in the simulation of adaptive fuzzy control of APF are chosen as:

k  2 , q  50 ,adaptive gain   500 in (6.14), ks  2.5 in (6.15), PI control is adopted for DC Voltage in active power filter and the parameters of PI controller are chosen as k p  0.05 and ki  0.01 to achieve satisfactory performance, the inductance in the circuit of APF is 10mh and the capacitance

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Juntao Fei

is 100 F . The nonlinear load branch is the rectifier bridge connecting parallel RC load, where R=15Ω, C=5e-3F. A phase source current before and after APF works is shown in Figure 6.4. Current harmonic analysis for the first two circles and last two cycles are depicted in Figure 6.5-6.6. When t=0.4s, the switch of compensation circuit is closed and APF begins to work. It can be seen that there are a lot of harmonics before 0.04s. The source currents recover steady state after half cycle about 0.01 s, the THD is 24.71% before harmonic compensation and 1.72% after harmonic compensation. That is within the limit of the harmonic standard of IEEE of 5%. It can be observed that the supply current is close to sinusoidal wave and it remains in phase with the supply voltage, demonstrating that APF performs well in the steady state operation. Instruction current and compensation current are drawn in Figure 6.7, and compensation current tracking error is drawn in Figure 6.8. They show that compensation current can track the instructions current very well. This means that the proposed adaptive fuzzy control with supervisory controller can guarantee asymptotic output tracking. Therefore, the harmonic current can be effectively compensated and harmonic distortion of source current can be reduced. The adaptation values of θ1、θ2 and θ3 of adaptive fuzzy controller in (6.14) are depicted in Figure 6.9. It is demonstrated that the parameters of the proposed adaptive fuzzy controller converge to stable constant values. As can be seen from Figure 6.10, DC capacitor voltage is stable by using PI controller. 1

0.9

0.8

Membership function degree

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -15

-10

-5

0 x

Figure 6.3. Membership function degree of x.

5

10

15

Adaptive Fuzzy Control with Supervisory Compensator … 40 30

source current(A)

20 10 0 -10 -20 -30 -40

0

0.02

0.04

0.06 time(s)

0.08

Figure 6.4. A phase source current.

Selected signal: 5 cycles. FFT window (in red): 2 cycles 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Mag (% of Fundamental)

Fundamental (50Hz) = 31.98 , THD= 24.71%

20 15 10 5 0

0

500

1000 Frequency (Hz)

1500

2000

Figure 6.5. Current harmonic analysis for the first two circles.

89

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Juntao Fei

Selected signal: 5 cycles. FFT window (in red): 2 cycles 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Fundamental (50Hz) = 32.5 , THD= 1.72%

Mag (% of Fundamental)

2

1.5

1

0.5

0

0

500

1000 Frequency (Hz)

1500

2000

Figure 6.6. Current harmonic analysis for the last two circles. 60 reference current compensation current 40

current(A)

20

0

-20

-40

-60

0

0.01

0.02

0.03

0.04

0.05 time(s)

0.06

Figure 6.7. Instructions current and compensation current.

0.07

0.08

0.09

0.1

Adaptive Fuzzy Control with Supervisory Compensator …

91

Figure 6.8. Compensation current tracking error.

value of θ1

2000 1000 0 -1000

0

0.01

0.02

0.03

0.04

0.05 time(s)

0.06

0.07

0.08

0.09

0.1

0

0.01

0.02

0.03

0.04

0.05 tims(s)

0.06

0.07

0.08

0.09

0.1

0

0.01

0.02

0.03

0.04

0.05 tims(s)

0.06

0.07

0.08

0.09

0.1

value of θ2

1000 0 -1000

value of θ3

1000 0 -1000

Figure 6.9. Adaptive values θ1、θ2 and θ3 of adaptive fuzzy controller.

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Juntao Fei

900 800

reference voltage DC capacitor voltage

700

voltage(v)

600 500 400 300 200 100 0 -100

0

0.01

0.02

0.03

0.04

0.05 time(s)

0.06

0.07

0.08

0.09

0.1

Figure 6.10. DC capacitor voltage.

In order to demonstrate that the adaptive fuzzy control system has strong robustness in the presence of parameters variation, we will testify the APF with the parameters variation. The performance is listed in Table 6.1. We can see that THD is still in the normal range with the parameters variation. Therefore, the adaptive fuzzy control system has good robustness to the parameter uncertainties. Table 6.1. Performance for variation in filter inductance and DC capacitor L(mH)

C(uF)

THD(%)

10

100

1.72

10

200

1.51

10

1000

2.58

8

100

1.65

5

100

1.68

Adaptive Fuzzy Control with Supervisory Compensator …

93

It can be concluded that the current tracking and THD performance can be improved by using the proposed adaptive fuzzy control with supervisor compensator. Thus, the control performance and robustness to external disturbance can be improved, and the stability of the closed loop system can be guaranteed.

CONCLUSION An improved direct adaptive fuzzy control system with supervisory controller is applied to the three-phase APF. The proposed adaptive fuzzy controller can effectively eliminate the reactive and harmonic component of the load current. The designed controller can guarantee the asymptotic output tracking of the closed-loop system, and the compensation current can follow the tracks of instruction current. The designed APF has superior harmonic treating performance and minimizes the harmonics for wide range of variation of load current under difference nonlinear load. Therefore, the proposed control scheme yields an improved THD performance.

Chapter 7

ADAPTIVE CONTROL WITH FUZZY SLIDING COMPENSATOR OF ACTIVE POWER FILTER In this chapter, an adaptive fuzzy-sliding control system is proposed to improve dynamic performance of the three-phase active power filter (APF). The adaptive fuzzy systems are employed to approximate both the equivalent control term and the switching control term in the sliding mode controller. An on-line adaptive tuning algorithm for the consequent parameters in the fuzzy rules is also designed. The switching control becomes continuous and the chattering phenomena can be attenuated. Simulation demonstrated that the proposed control method has an excellent dynamic performance such as small current tracking error, reduced total harmonic distortion (THD), strong robustness in the presence of parameters variation, and nonlinear load.

7.1. INTRODUCTION In this chapter, a novel adaptive fuzzy control with fuzzy sliding term is developed to improve the current tracking performance and guarantee the Lyapunov stability of the close-loop system. The control strategy proposed here has the following advantages: 1. It integrates adaptive control, sliding mode control and the nonlinear approximation of fuzzy control. Fuzzy controllers are proposed to approximate the equivalent control term and the switching control term in the sliding mode controller. A fuzzy

96

Juntao Fei switching part is employed to approximate the sliding mode controller. The sliding controller that is approximated by the fuzzy system is designed to compensate the approximation error between the fuzzy controller and optimal fuzzy control law. 2. The proposed adaptive fuzzy-sliding controller for APF does not need to establish an accurate mathematical model and has the ability to approximate the nonlinear characteristics of APF. Adaptive fuzzy control has great ability to compensate for the system nonlinearities and improve the power dynamic performance, such as current tracking and improved total harmonic distortion (THD) performance. 3. An adaptive fuzzy control with fuzzy sliding term is proposed to deal with system nonlinearities and nonlinear load in order to improve the current tracking and robustness of the control system compared with conventional control method. The proposed adaptive fuzzy sliding mode controller can guarantee the stability of the closed loop system and improve the robustness for external disturbances and model uncertainties.

7.2. PRINCIPLE OF ACTIVE POWER FILTER The block diagram of the three-phase three wire active power system is given in Figure 7.1. The principle of APF is to detect voltage and current of compensation object, get command signal i

* c

of compensation current by

using operation circuit for instructions current, and then obtain compensation current

ic by PWM generator in order to offset harmonic current, and achieve

ideal source current. The mathematical model of APF can be described in the following paragraph. According to circuit theory and Kirchhoff's theorem, we can get following state equations:

Adaptive Fuzzy Control with Fuzzy Sliding Compensator …

di1   v1  Lc dt  Rci1  v1M  vMN  di2   Rci2  v2 M  vMN v2  Lc dt  di3   v3  Lc dt  Rc i3  v3 M  vMN 

97

(7.1)

The parameter of Lc and Rc are the inductance and resistance of the APF respectively. By summing the three equations in (7.1), and taking into account the absence of the zero-sequence in the three wire system currents, and assuming that the AC supply voltages are balanced, one obtains:

vMN  

1 3  vmM 3 m1

(7.2)

The switching function ck denotes the ON/OFF status of the devices in the two legs of the IGBT Bridge. We can define ck as

1, if Sk isOn andSk+3 isOff ck   0, if Sk isOff andSk+3 isOn

(7.3)

Where k =1,2,3 . Hence, by writing vkM =ck vdc , then (7.1) becomes

 di1 Rc v1 vdc 1 3   i   ( c  cm )   1 1 dt L L L 3 m  1 c c c   di2 Rc v2 vdc 1 3   i   ( c  cm )   2 2 dt L L L 3 m  1 c c c   di R v v 1 3  3   c i3  3  dc (c3   cm ) Lc Lc Lc 3 m 1  dt

(7.4)

Vs1

N

Ls

is1

Vs2

is2

Vs3

is3

iL1

v1

iL2

v2

iL3

v3 Lc

idc

Rc S1

S2

S3

+

Vdc

i3 i2

C

i1

S4

S5

M

Figure 7.1. Block diagram for main circuit of APF.

S6

v3M

v2M

v1M

Nonlinear loads

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

99

7.3. DESIGN OF ADAPTIVE FUZZY SLIDING MODE CONTROL In this section, an adaptive fuzzy-sliding control is derived and Lyapunov analysis is implemented to guarantee the stability of the closed-loop system. An adaptive fuzzy control method based on the sliding-mode control is proposed to approximate the unknown equivalent control and sliding term. First, a fuzzy logic system is introduced. We will show how to construct adaptive fuzzy-sliding control in the next steps. The block diagram of the adaptive fuzzy-sliding control system for APF is shown in Figure 7.2. Systematic stability analysis is performed in the design of proposed adaptive fuzzy-sliding control. The detailed design procedure of the adaptive fuzzy-sliding control system can be described in the following steps. 监督 d 控制 dt 器

1 u  [fˆ(x)  xm hˆ(s)] b

Active Power Filter

xm

-

x  f (x) bu d +

+

fˆ (x  f )   f T (x)

-

 f (0)

 h (0)

-

监督控制器 f  r1 s ( x )

监督控制器 h  r2  s ( s )

hˆ(s监督控制器 h ) hT(s)

Figure 7.2. Adaptive fuzzy-sliding control block for APF.

监督控制 s 器 e

+

100

Juntao Fei We can transform the dynamic model of (7.4) into the following form:

x=f ( x) bu d



Where x  i1

(7.5)

i2 i3  , f ( x)  -

v Rc v ik + k , k =1,2,3 , b  - dc , Lc Lc Lc

because vdc is not constant, so b is unknown. d is an unknown disturbance, and the model of (7.4) is proposed in an ideal situation, so in order to demonstrate that the control system has strong robustness, we add the disturbance d to the model, d (t )  D , D  0 . The control target is to make currents x to track the given reference current signal xm . The tracking error is defined as e  x  xm and the sliding function is designed as

s(t )

e

Where

(7.6) is positive constant.

We choose the control law as

u

1 b

f ( x) xm usw

(7.7)

Where 0 Substituting (7.7) into (7.5) yields:

s(t )=

sgn(s) d

(7.8)

Then

s(t ) s(t )=

s sd

s

s d

s

sD

s

D

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

If we choose



D



, then s(t ) s(t )

101

0 shows that the asymptotical

stability can be guaranteed. If f ( x) and b are unknown, the control law (3.5) cannot be directly implemented. However, fuzzy system fˆ and hˆ can be utilized to approximate

f and sgn( s) . The fuzzy controller can be expressed as

1 u = [ fˆ ( x) xm hˆ( s)] b Where fˆ ( x  f )   f T  ( x) , hˆ( s fuzzy basis function,

(7.9)

h

)

T h

( s),

( x) and

( s) are

 f T and  hT can be updated by the following adaptive

laws

 f  r1 s ( x)

(7.10)

h  r2 s (s)

(7.11)

Where r1 and r2 are positive constants. Then we will give the proof for the adaptive laws in (7.10) and (7.11).

1 Theorem: A feedback controller u = [ fˆ ( x) xm b

laws (7.10) and (7.11) for adjusting parameters vector

hˆ( s)] and adaptive

 f ( t ) and  h ( t ) are

designed to satisfy that the closed loop system must be globally stable in the sense that all variables, x(t ), f (t ),h (t ) and u must be uniformly stable and the current tracking error e( t ) should be as small as possible. Proof: Define optimal parameter vector:

 *f  arg min[sup fˆ ( x  f )  f ( x) ]  f  f

xR n

(7.12)

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Juntao Fei

 h*  arg min[sup hˆ( s  h )  usw ] h h

(7.13)

xR n

Where  f and  h are assemble for

 f and  h .

Define fuzzy approximation error  :

  f ( x)  fˆ ( x  *f )

(7.14)

Where  is bounded by positive constant max that is

  max

(7.15)

Then the derivative of sliding surface can be derived as

s   ( x  xm )  [ f ( x)  bu  d  xm ]  [ f ( x)  fˆ ( x)  hˆ( s  h )  d ]  [ fˆ ( x  *f )  fˆ ( x)  hˆ( s  h )    hˆ( s  h* )  hˆ( s  h* )  d ]  [ Tf  ( x)  hT  ( s)    hˆ( s  h* )  d ] Where

(7.16)

 f   f   *f , h  h  h* .

Define Lyapunov function candidate:

1 1 1 V  ( s 2   Tf  f  hT h ) 2 r1 r2 Then the derivative of V becomes

(7.17)

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

103

1 1 V  ss   Tf  f  hT  h r1 r2

1 1 (7.18)  s[ Tf  ( x)  hT  ( s)    hˆ( s  h* )  d ]   Tf  f   hT  h r1 r2 1 T 1 T T T  s f  ( x)   f  f  sh  ( s)  h  h   s   sd   shˆ( s  h* ) r1 r2 * Because hˆ( s | h )   sgn( s) ,   D   , d (t )  D , D  0 ,

Thus: 1 T 1  f [ f  r1 s ( x)]  hT [h  r2 s ( s)]  [ s  sd  ( D   ) s ] (7.19) r1 r2 1 T 1 T   f [ f  r1 s ( x)]  h [h  r2 s ( s)]  [ s   s ] r1 r2

V

Substituting (7.10), (7.11) into (7.19) yields

V    s  s     s   s    s     Assume the approximation error  is bounded as

(7.20)

max

, if we

choose   max , then V  0 . This implies that V is negative semi-definite,

s( t )  f and  h are bounded. Since V (0) is bounded and V (t ) is nonincreasing

and

bounded

function,

then

we

have

 s   w d  V( 0 )  V( t )   , that is s  L . From (3.14), it can be t

1

0

known that s( t ) is bounded. If s( t ) is bounded, then s( t ) is uniformly continuous. According to the Corollary of Barbalat‟s Lemma, we have lim s( t )  0 . Then lim e( t )  0 . t 

t 

7.4. SIMULATION STUDY The performance of the proposed adaptive fuzzy-sliding control will be tested using Matlab/Simulink package with SimPower Toolbox. Simulation

104

Juntao Fei

results are presented to verify the effectiveness of the proposed adaptive fuzzy-sliding control. We choose six membership functions as

exp [ x 4 2(i 1)]2 , i 1,

,6.

Where i is number of fuzzy rules, i  6 which means that there are 36 fuzzy rules to approximate the unknown system. The selected 36 fuzzy rules can cover the entire space and approximate any system states and nonlinear functions. The membership function is shown in Figure 3. The three membership functions over the interval [-3, 3] for sliding function s are selected as

NM

( s)

1 , 1 exp(5( s 3))

ZO

(s) exp( s 2 ) ,

PM

( s)

1 . 1 exp(5( s 3))

The initial values of fuzzy parameters are chosen randomly in the interval, the vector of fuzzy basis functions were constructed by (3.2). e , where 100 , adaptive gain Sliding function is chosen as s

r1 10000 , r2 1000

 fa   fa1  fa 2  fa 3  fa 4  fa5  fa 6       f   fb    fb1  fb 2  fb3  fb 4  fb5  fb6   fc   fc1  fc 2  fc3  fc 4  fc5  fc 6     

 ha   ha1  ha 2  ha 3   h   hb    hb 2  hb 2  hb 3   hc   hc1  hc 2  hc 3  Vs1 =Vs 2 Vs 3 110V ,f=50Hz .The inductance in the circuit of APF is

10mh and the capacitance is 100 F . Nonlinear load branch is rectifier bridge connecting parallel RC load, where R=15Ω, C=5e-3F. PI control is adopted for DC Voltage in active power filter and the parameters of PI controller are chosen as k p  0.005 and ki  0.02 to achieve satisfactory performance.

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

105

The difference between DC capacitor voltage and reference voltage is the control input for PI control, then the control input is added to the DC component of instantaneous active current in order to get the reference current, and the compensation current includes the active component of fundamental current which leads to energy exchange between AC side and DC side. In this way, DC capacitor voltage can follow the reference voltage. A phase source current before and after APF works is shown in Figure 7.3. Current harmonic analysis for the first two circles and last two cycles are depicted in Figures 7.4-7.5. When t =0.04s , the switch of compensation circuit is closed and APF begins to work. It can be seen that there are a lot of harmonics before 0.04s. The source currents recover steady state after half cycle about 0.01 s, THD is 24.71% before harmonic compensation, and it decreases to 1.59% after harmonic compensation that is within the limit of the harmonic standard of IEEE of 5%. It can be observed that the supply current is close to sinusoidal wave and it remains in phase with the supply voltage, demonstrating that APF performs well in the steady state operation. 80

source current(A)

60

40

20

0

-20

-40

0

0.02

0.04

0.06

0.08

0.1

time(s)

Figure 7.3. A phase source current.

Instruction current and compensation current are drawn in Figure 7.6, and compensation current tracking error is depicted in Figure 7.7. It can be observed that compensation current can track the instruction current well which demonstrates that the proposed adaptive fuzzy-sliding control can guarantee asymptotic state tracking. Thus, the harmonic current can be

106

Juntao Fei

effectively compensated and harmonic distortion of source current can be reduced. Adaptive parameters of

 f and  h are depicted in Figure 7.8 and

Figure 7.9. It is demonstrated that the parameters of the proposed adaptive fuzzy-sliding controller converge to stabilize constant values. As can be seen from Figure 7.10, DC capacitor voltage is stable by using PI controller. It is noted that DC voltage is zero before 0.04s, because the initial capacitor voltage is not set, and there is no energy exchange between AC side and DC side before APF works, then DC capacitor voltage cannot change from 0 to 1000.

Selected signal: 5 cycles. FFT window (in red): 2 cycles 60 40 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Mag (% of Fundamental)

Fundamental (50Hz) = 31.98 , THD= 24.71%

20 15 10 5 0

0

500

1000 Frequency (Hz)

1500

2000

Figure 7.4. Current harmonic analysis for the first two circles. Selected signal: 5 cycles. FFT window (in red): 2 cycles 60 40 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Mag (% of Fundamental)

Fundamental (50Hz) = 32.68 , THD= 1.59%

1.5

1

0.5

0

0

500

1000 Frequency (Hz)

1500

2000

Figure 7.5. Current harmonic analysis for the last two circles.

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

107

30

reference current compensation current

20 10

current(A)

0 -10 -20 -30 -40 -50

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time(s)

30

reference current compensation current

20 10

current(A)

0 -10 -20 -30 -40 -50

0

0.01

0.02

0.03

0.04

0.05

0.06

time(s)

Figure 7.6. Instructions current and compensation current.

0.07

0.08

0.09

0.1

108

Juntao Fei

Figure 7.7. Compensation current tracking error. 6

value of θfa

10

θfa2 θfa3 θfa4

0

θfa5

4 value of θfb

θfa1

5

-5

0 6 x 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

θfb2 θfb3

0

θfb4

-2

5

θfa6 θfb1

2

-4

value of θfc

x 10

θfb5 0 6 x 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

θfb6 θfc1 θfc2 θfc3

0

θfc4 θfc5

-5

0

0.01

0.02

0.03

0.04

0.05

time(s)

Figure 7.8. Adaptive law

f

.

0.06

0.07

0.08

0.09

0.1

θfc6

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

109

5

x 10

value of θha

2 0

θha1

-2

θha2 θha3

-4 -6

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

6

x 10

value of θhb

5 0

θhb1

-5

θhb2 θhb3

-10 -15

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

6

x 10

value of θhc

5 0

θhc1

-5

θhc2 θhc3

-10 -15

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time(s)

Figure 7.9. Adaptive law

h .

1200

1000

reference voltage DC capacitor voltage

voltage(V)

800

600

400

200

0

-200

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time(s)

Figure 7.10. DC capacitor voltage.

In order to demonstrate that the adaptive fuzzy-sliding control system has strong robustness in the presence of parameters variation, APF with the parameters variation is tested. As shown in Table 7.1, THD is still in the normal range with the parameters variation. It can be concluded that the

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adaptive fuzzy-sliding control system has good robustness to the parameter uncertainties. In order to demonstrate that the adaptive fuzzy-sliding control system can achieve better performance than the conventional method, APF using the hysteresis control is also tested. It can be seen from Table 7.2 that the adaptive fuzzy-sliding control system has better robustness compared with conventional method. Table 7.1. Performance for variation in filter inductance and DC capacitor using adaptive fuzzy sliding control

L(mH)

C(uF)

THD(%)

10

100

1.59

10

200

1.58

10

1000

2.32

8

100

1.53

5

100

1.66

Table 7.2. Performance for variation in filter inductance and DC capacitor using hysteresis controller

L(mH)

C(uF)

THD(%)

10

100

1.72

10

200

1.78

10

1000

4.15

8

100

1.78

5

100

2.05

In summary, the current tracking, THD performance, and the control performance and robustness to external disturbance, can be improved with the proposed adaptive fuzzy-sliding controller.

Adaptive Fuzzy Contol with Fuzzy Sliding Compensator …

111

CONCLUSION An improved adaptive fuzzy-sliding control system has been applied to the three-phase APF in this paper. Universal approximation property of fuzzy system is employed to approximate the unknown equivalent control and sliding mode control. The parameters of fuzzy system in both the fuzzy control part of unknown equivalent control and fuzzy sliding mode part can be adaptively updated based on the Lyapunov analysis. The asymptotic stability of the closed-loop system can be guaranteed with the proposed adaptive fuzzy control strategy with fuzzy switch term. The proposed adaptive fuzzy-sliding controller can make the compensation current follow the instruction current, and effectively eliminate the reactive and harmonic component of the load current. The designed APF control system has superior harmonic suppression performance and yields an improved THD performance. Simulation results demonstrated the excellent dynamic performance, asymptotic stability, and strong robustness with the proposed APF control system.

Chapter 8

ADAPTIVE NEURAL NETWORK CONTROL OF ACTIVE POWER FILTER This chapter presents an adaptive radial basis function (RBF) neural network control system for the three-phase active power filter (APF). It is designed to generate compensation current to track command current so as to eliminate the harmonic current of non-linear load and improve the quality of the power system. The stability of the control system can be guaranteed with the proposed adaptive neural network algorithm, which has a fast learning speed and can improve the accuracy, robustness, and adaptivity of the system. The simulation results demonstrate the good performance such as small current tracking error, reduced total harmonic distortion (THD) and strong robustness in the presence of parameters variation and nonlinear load. It is shown that the adaptive RBF neural network control system for three-phase active power filter (APF) has better control effect than hysteresis control.

8.1. INTRODUCTION The neural network has the capability to approximate any nonlinear function over the compact input space. Therefore, neural network‟s learning ability to approximate arbitrary nonlinear functions makes it a useful tool for adaptive application. The neural network control for the nonlinear dynamic system has become a promising research topic. By properly choosing neural network structures and training the weights, researchers may use neural networks for special tasks. The conventional trainings of neural networks are

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mainly based on optimization theory. In order to search the optimal weights for neural networks, a number of algorithms have been developed. The gradient-based training algorithms are probably the most popular ones. Therefore, it is necessary to utilize the adaptive neural control to improve the current tracking and filtering performance. In this chapter, a Lyapunov adaptive control method based on RBF neural network is proposed to overcome the shortcomings of traditional methods and improve the current tracking performance and guarantee the Lyapunov stability of the closed-loop system. This method is featured by high control accuracy, real-time, and wide range of application, and it can reduce current total harmonic distortion effectively. The control strategy proposed here has the following advantages: 1. The contribution of this chapter is the integration of the adaptive control, sliding mode control, and the nonlinear approximation of RBF neural control. An adaptive neural network controller is used to compensate the system nonlinearities and improve the tracking performance. An RBF neural network controller which can be trained on line is incorporated into the adaptive control scheme in the Lyapunov framework to guarantee the stability of the closed loop system. 2. It is unnecessary to establish an accurate mathematical model with the proposed adaptive neural controller since it has the ability to approximate the nonlinear characteristics of APF. Adaptive neural control has great ability to compensate for the system nonlinearities and improve the power dynamic performance such as current tracking and THD performance. 3. Adaptive neural control is proposed to deal with system nonlinearities and nonlinear load in order to improve the current tracking and the robustness for parameters variations and nonlinear loads compared with conventional control method.

8.2. BASIC PRINCIPLES OF ACTIVE POWER FILTER The basic structure of the three-phase three wire shunt active power filter is shown in Figure 8.1, where vsa vsb vsc are voltages of three-phase power system, r is the resistance from power source to inductance on the AC side

115

Adaptive Neural Network Control of Active Power Filter of APF, L is the inductance on the AC side of APF,

vdc

is the capacitors

voltage on the DC side, is is line current, iL is non-linear load current, ic is *

compensate current, ic is command current as the basis of compensate current. The system of APF can be divided into two parts, the first part is the command current operating circuit used to detect the harmonic and reactive *

components of load current, which is defined as command current i c . This part is usually realized with the harmonic current detecting method based on instantaneous reactive theory. The second part is compensation current generating circuit made up of three parts: following control circuit, driving circuit and main circuit. This part is used to generate compensate current ic with the basis of command current i

* c

from the first part, the process is that

the compensate current generating circuit enlarges the command current and gets the compensate current. Then the compensate current is injected into the line to compensate the harmonic and reactive components of the load current, and the line current is forced to become sine wave. Based on the circuit theory and Kirchhoff‟s current law, we can get the expressions as follows:

ica  

rica  vsa vdc  s L L

(8.1)

icb  

ricb  vsb vdc  s L L

(8.2)

icc  

ricc  vsc vdc  s L L

(8.3)

Where S is the switching function to indicate the working state of IGBT.

1 QN  1 ,it is equal to 1 when the switch is turn on, and it 0 QN  0

We define s  

is equal to 0 when the switch is turn off. This is the mathematical model of shunt APF.

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Juntao Fei Vsa iL

is

Vsb

non-linear load

Vsc ic r

L

Vdc

ic iL Vsa

is Command current operation circuit

i*c

Tracking control circuit

PWM Signal

drive circuit

V dc Figure 8.1. The basic structure of shunt APF.

8.3. DESIGN OF ADAPTIVE RBF NEURAL NETWORK CONTROLLER In this section, adaptive RBF neural network controller will be designed and the weights of the RBF neural controller will be updated according to the error between command current and compensate current based on Lyapunov analysis. RBF neural networks simulate the structure of human beings‟ neural network to adjust locally. The basic architecture of a RBF is a 3-layer network, including input layer, hidden layer, and output layer. The hidden layer performs a non-linear mapping from the input space into a higher dimensional space in which the patterns become linearly separable. The output layer performs a simple weighted sum with a linear output. RBF neural network is a kind of local approximate neural networks which can approximate any continuous function if there are enough neurons. Therefore, the RBF neural network can accelerate learning speed and avoid the local

Adaptive Neural Network Control of Active Power Filter

117

minimum problem, and it‟s suitable for the real-time control requirement of APF. The output of RBF controller can be expressed as follows:

u(k )  h11  ...  h j j  ...  hmm Where m the number of hidden layer is neural,  j is the weight between hidden layer and output layer, and h j is the output of hidden layer. In RBF neural network, Z  [ z1 ,...zn ] is the input vector, and we take T

the tracking error e as z . The radial basis vector of RBF is H  [h1 ,...hm ]T ,

h j is Gaussian function as follows: h j  exp(

 Z  cj

2

2b j 2

), j  1, 2,..., m

Where b j is the base width of the node j , B  [b1 ,...bm ] ; C j is the T

centric vector of the node j , C j  [c j1 ,...c ji ,...c jn ] .The weight vector of T

the RBF neural network is

  [1 ,...m ]T

u

x

Plant x  f ( x )  bu

Integral

y =x -

y * =x * +

z=e

Neural network controllr u ( z  )   T h( z )

Gaussian function h(z)



Adaptive law    eT ph(z)

Figure 8.2. Block diagram of adaptive RBF neural network control system.

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The block diagram of adaptive RBF neural network is shown in Fig. 8.2. We use the tracking error between compensate current ic and command *

current i c as the input of controller, and PWM signal is generated by the controller to control the switches of main circuit. Then compensate current is controlled to eliminate harmonic current. The detailed design process can be described as follows: Functions of (8.1-8.3) can be written as:

x  f ( x)  bu

(8.4)



icb icc  , f ( x)  

Where x  ica

rick  vsk , k =a,b,c , L

b

vdc , L

* * * , icb and icc . x* is ica

If y  x , y  =x* , the control objective is to make y track y  , and the tracking error is e  y   y , u represents the switching function which is the output of RBF neural network, u  u (z /  ) . The derivative of tracking error can be expressed as:

e  y *  y  x*  x  ( f ( x)*  bu* )  ( f ( x)  bu )  

r * (ic  ic )  b(u *  u )  ke  b(u *  u (z /  )) L (8.5)

With the optimal weight  * , the approximate error of real output to expected error is minimum, and the tracking effect of compensate current y to command current y  is the best. The optimal weight can be defined as follows:

   arg min[sup u  u(z /  ) ] Rm

xR

Where sup * represents the upper bound of the error.

(8.6)

119

Adaptive Neural Network Control of Active Power Filter The least tracking error of RBF neural network is defined

m  u( z   )  u

(8.7)

Substituting (8.7) into (8.5) yields

e  ke  b[u ( z /  * )  u ( z /  )  m]  ke  b( *   )T h( z )  bm

(8.8)

Define the Lyapunov function candidate

1 b V  eT pe  (    )T (    ) 2 2 Where,

(8.9)

 is a positive constant, p is the positive constant to satisfy the

following equation

(  k T ) p  p ( k )   q

(8.10)

Where q is a positive constant. Differentiating V with respect to time yields:

1 b V   eT qe  (    )T [ eT ph(z)   ]  eT pbm 2 

(8.11)

The adaptive law is chosen as

   eT ph(z) Where  is adaptive gain. Substituting (8.12) into (8.11) yields

(8.12)

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Juntao Fei

1 V   eT qe  eT pbm 2

(8.13)

Because q, p, b>0 ,from the universal approximation theorem, m can be made to be arbitrary small using fuzzy system on a compact domain, that is m can be equal to zero. m is the least tracking error and it can be designed to

1 T e qe  0. . This implies that V is 2 * negative semi-definite, e( t ) ,      is bounded. According to the Corollary of Barbalat‟s Lemma, we have lim e( t )  0 . It can be concluded be sufficiently small, so we have V  

t 

that the system is asymptotically stable.

8.4. SIMULATION STUDY The performance of the proposed adaptive RBF neural network control will be tested using Matlab/Simulink package with SimPower Toolbox. Simulation results are presented to verify the effectiveness of the proposed adaptive neural control. The parameters of RBF neural network are chosen as follows: Input node n  3 , hidden layer node m=27 , output node is 3, centric vector

c  13:1:13 , base width b  2 , learning rate   80 . The parameters of APF are selected as follows: We use PI controller to control the DC side voltage, k p  0.08 , ki  0 , inductance on the AC side L  5mH , capacitors voltage on the DC side

vdc =100uF . Nonlinear load branch is rectifier bridge connecting parallel RC load, where R=15Ω, C=5e-3F. Figure 8.3 describes the wave graph of A phase current before and after controlling the APF. It can be seen from Figures 8.4-8.5, before 0.04 second

Adaptive Neural Network Control of Active Power Filter

121

the A phase current contains numerous harmonic, the total harmonic distortion (THD) is 24.71%. After 0.04 second, APF begins to take effect, the current wave is close to sine wave in no more than 0.01 second and it tends to be stable. Therefore, it can be concluded that APF can compensate the harmonic current effectively with the proposed adaptive neural network controller. 40

source current(A)

30

20

10

0

-10

-20

-30

-40

0

0.025

0.05

0.075

0.10

time(s)

Figure 8.3. A phase current.

Selected signal: 5 cycles. FFT window (in red): 2 cycles 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Mag (% of Fundamental)

Fundamental (50Hz) = 31.98 , THD= 24.71%

20 15 10 5 0

0

500

1000 Frequency (Hz)

1500

2000

Figure 8.4. Current harmonic analysis for the first two circles.

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Juntao Fei

Selected signal: 5 cycles. FFT window (in red): 2 cycles 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Fundamental (50Hz) = 32.43 , THD= 1.81%

Mag (% of Fundamental)

2

1.5

1

0.5

0

0

500

1000 Frequency (Hz)

1500

2000

Figure 8.5. Current harmonic analysis for the last two circles.

80

reference current compensation current

60

current(A)

40

20

0

-20

-40

-60

-80

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time(s) Figure 8.6. Command current and compensation current.

Figure 8.6 draws the graph of compensate current tracking command current. As we can see, compensate current can track the command current very well before 0.045 s. So harmonic current is eliminated and the line

123

Adaptive Neural Network Control of Active Power Filter

current is forced to be approximate to sine wave, and THD is largely reduced. Figure 8.7 is the dynamic curve of voltage on the DC side, the PI controller can make it track the reference voltage quickly, and maintain in a relative stable state. 900

800

reference voltage DC capacitor voltage

voltage(V)

700

600

500

400

300

200

100

0

-100

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time(s) Figure 8.7. Voltage wave of the DC side.

Table 8.1. Performance for variation in filter inductance and DC capacitor L(mH) 10 2 5 5 5 10

C(uF) 100 100 50 200 1000 1000

THD(%) 2.01 2.38 2.2 1.51 1.84 2.15

In real practical application, some physical parameters of APF may be changed. In order to demonstrate that, the adaptive fuzzy-sliding control system has strong robustness in the presence of parameters variation, we test the APF with the parameters variation as shown from Table 8.1. We can see

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Juntao Fei

that THD is still in the acceptable range with the parameters variation. It is proved that the adaptive neural control system has good robustness to the parameter variations.

Selected signal: 5 cycles. FFT window (in red): 2 cycles 60 40 20 0 -20 0

0.02

0.04 0.06 Time (s)

0.08

0.1

Mag (% of Fundamental)

Fundamental (50Hz) = 32.31 , THD= 2.10% 2

1.5

1

0.5

0

0

500

1000 Frequency (Hz)

1500

2000

Figure 8.8. Current harmonic analysis for the last two circles using hysteresis control.

Moreover, in order to demonstrate that the adaptive neural control system has achieved better than the conventional method, we also tested the APF with the hysteresis control. It can be observed from Figure 8.8 that the THD is 2.1% using hysteresis control, which is higher than the 1.81% using adaptive neural control and the big distortion occurs from 0.04 to 0.05 second, which is worse than the case using adaptive neural control.

CONCLUSION This chapter presents an adaptive neural network control method towards three-phase active power filter to make compensate current compensate harmonic current, and improve the quality of power system. The parameters of neural control system can be adaptively updated based on the Lyapunov analysis. The stability of the closed-loop system can be guaranteed with the

Adaptive Neural Network Control of Active Power Filter

125

proposed adaptive neural control strategy. Simulation results prove that this control method can reduces total harmonic distortion effectively. Comparing with traditional method, the adaptive neural network control method has better harmonic elimination performance.

Chapter 9

CONCLUSION This book develops adaptive sliding mode control strategies for a MEMS z-axis gyroscope. A direct adaptive sliding mode controller with proportional and integral sliding surface for a general linear system with single and multiple inputs are also developed. The proposed adaptive sliding mode controllers for MEMS z-axis gyroscope make real-time estimates of the angular velocity as well as all gyroscope parameters including coupling stiffness and damping parameters. Therefore, fabrication imperfection and time varying noise and disturbance can be compensated. The reference model trajectory is designed to satisfy the persistent excitation to enable that the estimations of parameters to converge to their true values. Chapter 2 investigates a sliding mode control technique with integral portion design for indirect current control APF. This sliding mode control is applied in reference current tracking to reduce the tracking error. A PI regulator combined with a low-pass filter is used to generate the amplitude of the reference currents. Simulations under the variable system load are carried out to show the robustness of the APF system. The result reveals that the reference currents have the same amplitude with the load currents and the designed APF has superior harmonic compensation effect. However, tracking error will increase when the load currents have high variance ratio. The comparative simulation shows the better reference current tracking performance of designed sliding mode control method than hysteresis comparison method. Future research direction includes the advanced control methodologies such as adaptive control and intelligent control with application to the direct current control APF.

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Juntao Fei

Chapter 3 develops an input-output feedback linearization to implement the DC side voltage regulation in indirect current controlled APF. A novel sliding mode controller has also been designed and applied in the reference current tracking control. In the actual power system, supply voltage may contain harmonics, which limits the application of the APF. This chapter designs a novel unit synchronous sinusoidal signals calculating method based on synchronous frame transforming theory. Simulation results verify that the sinusoidal signals calculating method can produce ideal synchronous sinusoidal signals even when THD of the supply voltage is high. The designed APF shows good performance both in dynamic and static characteristics, and the sliding mode controller reduces the THD of supply currents dramatically. Modeling approach of APF based on feedback linearization theory can be applied to develop an adaptive sliding mode control or intelligent control of APF system in the future research. Chapter 4 proposes adaptive current tracking control method with PIFuzzy compound control for the current and voltage tracking of APF, approximate mathematical model of the APF is established. Model reference adaptive current tracking control method for AC side current is developed and PI-Fuzzy compound control is designed for the DC capacitor voltage regulation. Simulation studies prove that adaptive current control method based on PI-fuzzy compound control not only can regulate the DC capacitor voltage but also track the AC current command signals, eliminate the power harmonics and improve the power quality and system robustness. Chapter 5 discusses the principle and dynamic model of single-phase shunt APF, and proposed a new adaptive sliding mode control algorithm. The simulation results proved that for nonlinear load current the adaptive sliding mode controller has successful compensation effect, i.e. it can compensate the most current harmonic and eliminate certain reactive current, which can recover sine wave from severely detuned current waveform and improving the power factor. From Figure 9 we can see that the APF current can quickly track the harmonic current, thus to achieve the harmonic compensation. The reference currents tracking behavior has been improved and the power supply current harmonic has been reduced with novel adaptive sliding mode control. The proposed control system has the satisfactory adaptive and robust ability in the presence of the changing disturbances and nonlinear loads. Chapter 6 analyzes an improved direct adaptive fuzzy control system with supervisory controller for the three-phase APF. The proposed adaptive fuzzy controller can effectively eliminate the reactive and harmonic component of the load current. The designed controller can guarantee the asymptotic output

Conclusion

129

tracking of the closed-loop system, and the compensation current can follow the tracks of instruction current. The designed APF has superior harmonic treating performance and minimizes the harmonics for wide range of variation of load current under difference nonlinear load. Therefore, the proposed control scheme yields an improved THD performance. Simulation results demonstrated the excellent dynamic performance, stability and strong robustness with the proposed controller. Chapter 7 analyzes an improved adaptive fuzzy-sliding control system with application to the three-phase APF. Universal approximation property of fuzzy system is employed to approximate the unknown equivalent control and sliding mode control. The parameters of fuzzy system both in the fuzzy control part of unknown equivalent control, and fuzzy sliding mode part, can be adaptively updated based on the Lyapunov analysis. The stability of the closed-loop system can be guaranteed with the proposed adaptive fuzzy control strategy with fuzzy switch term. The proposed adaptive fuzzy-sliding controller can make the compensation current follow the instruction current, and effectively eliminate the reactive and harmonic component of the load current. The designed control system of APF has superior harmonic treating performance and yield an improved THD performance. Simulation results demonstrated the excellent dynamic performance, stability, and strong robustness with the proposed controller. Chapter 8 presents an adaptive neural network control method towards three-phase APF to make compensate current compensate harmonic current, and improve the quality of power system. The parameters of neural control system can be adaptively updated based on the Lyapunov analysis. The stability of the closed-loop system can be guaranteed with the proposed adaptive neural control strategy. Simulation results prove that this control method can reduces total harmonic distortion effectively. Comparing with traditional method, the adaptive neural network control method has better harmonic elimination performance.

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INDEX A Active power filter (APF), 1 adaptation, 88 adaptive control, 35, 36, 40, 46, 57, 58, 80, 82, 114, 127, 132 adaptive fuzzy control, 2, 79, 80, 82, 84, 87, 88, 91, 92, 93, 95, 96, 99, 111, 128, 129, 133 adaptive neural network control, 2, 114, 121, 124, 129 adjustment, 31, 47, 75 algorithm, 57, 58, 77, 95, 113, 128, 134 amplitude, 1, 2, 5, 6, 7, 9, 12, 13, 20, 28, 29, 31, 67, 81, 127

complexity, 75 computing, 21, 133 convergence, 2 cycles, 71, 88, 105

D damping, 47, 127 DC side voltage, 21, 22, 25, 28, 31, 34, 120, 128 detection, 6, 35, 46, 65, 81 deviation, 43 distortions, 65, 70 distribution, 1, 58 disturbance, 1, 64, 71, 93, 100, 127

B base, 83, 117, 120

C commercial, 2 compensation, 1, 2, 3, 12, 20, 21, 22, 31, 35, 36, 40, 46, 57, 58, 65, 66, 67, 70, 71, 76, 77, 81, 88, 90, 93, 96, 105, 107, 111, 113, 115, 122, 127, 128, 129, 131, 132, 133, 134

E electronic circuits, 2 energy, 13, 58, 105 equilibrium, 39, 66 equipment, 2, 6, 23 equivalent control, 5, 61, 95, 99, 111, 129 excitation, 127

F fabrication, 127

136

Index

feedback linearization, 1, 2, 21, 22, 27, 28, 30, 31, 34, 128 FFT, 70 filters, 1, 2, 3, 6, 22, 36, 132 formula, 22 fuzzy controller, 43, 44, 45, 79, 82, 85, 88, 93, 96, 101, 128 fuzzy sets, 83

G Gaussian membership, 83 graph, 51, 120, 122 gravity, 45

H

L laws, 42, 63, 101 learning, 3, 36, 113, 116, 120 legs, 81, 97 linear model, 39 Luo, 132, 133 Lyapunov function, 41, 63, 85, 102, 119, 132 Lyapunov function candidate, 102 Lyapunov stability, 1, 2, 3, 35, 80, 95, 114, 134

M

harmonic elimination, 1, 2, 125, 129 harmonics, 1, 2, 3, 6, 22, 23, 29, 30, 34, 51, 55, 58, 67, 70, 88, 93, 105, 128, 129, 131, 133 human, 44, 116 human behavior, 44 hybrid, 132, 133 hysteresis, 3, 5, 6, 15, 19, 20, 30, 31, 36, 47, 51, 54, 109, 110, 113, 124, 127 hysteresis current control, 3, 36, 47

magnitude, 76 mapping, 82, 83, 116 mathematical model, 23, 26, 55, 79, 80, 81, 96, 114, 115, 128 matrix, 41, 42, 47, 59, 60, 61, 63 membership, 44, 83, 87, 104 MEMS, 127 mobile robots, 134 models, 1, 3, 12, 40 modern society, 35 modules, 7 mutation, 57

I

N

indirect current control, 2, 5, 6, 8, 15, 20, 21, 22, 28, 30, 34, 127, 128, 134 inductor, 47, 67 inequality, 86 input signal, 65 integration, 36, 80, 114 intelligent control, 1, 3, 35, 77, 127, 128 interference, 43, 64, 67, 70, 71

J Japan, 133

neural network, 2, 3, 36, 113, 114, 116, 117, 118, 119, 120, 121, 125, 129, 134 neurons, 116 nonlinear dynamic systems, 3

P parallel, 2, 5, 21, 36, 48, 64, 80, 88, 105, 120 pollution, 1, 2 power quality, 1, 2, 55, 128, 133 power transmission and distribution, 1, 58 project, vii, 3

137

Index propagation, 134

R radial basis function, 113 reactive current compensation, 1 reactive power, 1, 2, 81, 131, 133 reference current, 1, 5, 6, 7, 9, 10, 12, 13, 16, 20, 22, 28, 29, 30, 34, 57, 58, 77, 84, 100, 105, 127, 128 reference model, 40, 47, 51, 57, 59, 60, 65, 66, 127 requirements, 1, 3 researchers, 113 resistance, 6, 13, 23, 47, 71, 82, 97, 114 response, 1, 2, 3 response time, 1 rules, 44, 45, 79, 82, 83, 95, 104

S SAPF, 24, 25, 35, 77 security, 1, 58 semiconductor, 2 sensitivity, 1, 58 sensors, 6 signals, 7, 9, 11, 13, 14, 15, 20, 22, 29, 30, 31, 34, 55, 65, 128 simulation, 2, 20, 21, 30, 47, 48, 64, 66, 75, 77, 87, 113, 127, 128, 132 sine wave, 65, 70, 77, 115, 121, 123, 128 single cycle control, 3, 36 sliding mode control, 1, 2, 3, 5, 6, 7, 10, 11, 13, 15, 19, 20, 21, 22, 28, 29, 30, 31, 34, 36, 57, 58, 59, 60, 64, 65, 66, 67, 71, 76, 77, 95, 96, 111, 114, 127, 128, 129, 131, 133, 134 sliding surface, 58, 61, 102, 127 space vector control, 3, 36

stability, 1, 2, 3, 10, 13, 35, 71, 79, 82, 84, 87, 93, 95, 96, 99, 101, 111, 113, 114, 125, 129, 134 state, 8, 10, 11, 26, 27, 28, 36, 37, 38, 40, 43, 59, 67, 71, 81, 88, 96, 104, 105, 115, 123 structure, 5, 6, 37, 42, 43, 58, 80, 114, 116 supply voltage, 3, 7, 9, 12, 13, 14, 20, 21, 22, 23, 25, 29, 30, 34, 88, 97, 105, 128 suppression, 3, 51, 58, 111, 132

T target, 84, 100 techniques, 6, 131, 133 technology, 2, 35 topology, 35 total harmonic distortion, 13, 21, 35, 36, 64, 79, 80, 95, 96, 113, 114, 121, 125, 129 tracking error, 10, 11, 14, 17, 20, 41, 57, 60, 62, 75, 80, 84, 85, 88, 100, 101, 105, 117, 118, 119, 120, 127 tracks, 93, 129 training, 113 trajectory, 11, 59, 127 transformation, 22 transmission, 1, 58 variations, 10, 57, 80, 114, 124 vector, 3, 36, 58, 83, 85, 101, 104, 117, 120 velocity, 10, 127

W weights, 114, 116

Y yield, 42, 129