Peak Load Pricing and Reliability - Contributions to Theory and Method 9783659134760

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CONTENTS

Preface

Chapter 1: Introduction 1.1 Electric utility

1

1.2 Some Techno-Economic Characteristics of Electric Utility

2

1.2.1 Load duration Curve

2

1.2.2 Reserve Margin

3

1.2.3 Cost Structure

4

1.2.4 Load Factor

5

1.2.5 Optimum Plant Mix

5

1.2.6 Tariffs

6

1.3 Seasonal Time of Day Pricing

8

1.4 Objective and Structure of the Study

12

Chapter 2: Peak Load Pricing: A Review 2.1 Introduction

17

2.2 Public Utilities

18

2.3 The Welfare Foundations: A Review

21

2.3.1 Marginal Cost Pricing

21

2.3.2 Second-Best Dilemma

23

2.3.3 Monopoly Pricing

25

2.3.4 Regulated Monopoly Pricing

26

2.3.5 Ramsey Pricing

27

2.4 The Peak-Load Pricing Theory: A Review

29

2.4.1 A Basic Peak-Load Model

30

2.4.2 Peak-Load Pricing Under Uncertainty

33

2.4.3 Empirical Studies on Peak-Load Pricing

34

2.5 Conclusion

36

i

Chapter 3: Revisiting the Peak Load Model 3.1 Introduction

37

3.2 The Modified Peak Load Pricing Model

38

3.2.1 The Welfare Function and the Constraints

38

3.2.2 The Optimum Prices: Homogeneous technology

40

3.2.3 The Optimum Prices: Heterogeneous technology

42

3.2.4 The Optimum Prices: Generalisation

45

3.3 Conclusion

45

Chapter 4: An Illustrative Application of the Modified Model 4.1 Introduction

47

4.2 Kerala Power system

48

4.3 Methodology

51

4.4 Increment Quotient Method and Results

54

4.5 Conclusion

57

Chapter 5: Reliability and Peak Load Pricing in a Power System 5.1 Introduction

59

5.2 Reliability Indices

62

5.3 Normal (Gaussian) Distribution

66

5.4 Application to Power System Reliability

67

5.5 Reliability Rationing Cost and Peak Load Pricing under Uncertainty

70

5.6 Numerical Applications

75

5.7 Conclusion

78

Appendices

80

Chapter 6: Steady State Probability and Loss of Load Probability of a Power System 6.1 Introduction

91

6.2 Availability and Outage Measures

93

6.3 Markov Chain

96

6.4 A New Estimator for Steady State Probability

100

6.5 Forced Outage Rate and Availability Rate

110

6.6 Maximum Likelihood Estimators

112 ii

6.7 Propensity to Down (Mobility)

113

6.8 Capacity Outage Distribution

114

6.9 Loss of Load Probability (LOLP)

117

6.10 Conclusion

119

Appendix: The Distributional Properties of Reliability Rate

121

Chapter 7: Estimating Loss of Load Probability: A Case Study of the Kerala HydroPower System 7.1 Introduction

125

7.2 Availability and Forced Outage Rates

126

7.3 Capacity-Outage Probability and LOLP

130

7.4 Loss of Load Probability

136

7.5 Conclusion

137

Chapter 8: Conclusion 8.1 Summarising the Study

139

8.2 Practical Problems in Implementing Peak Load Pricing

142

8.3 Empirical Evidence

144

References

149

iii

iv

List of Figures

Figure 1.1: Chronological Load Curve and the Derived Load Duration Curve

3

Figure 2.1: Traditional Peak Load Pricing

31

Figure 4.1: The relevant techno-economic variables of the Kerala power system over the last 4 decades

50

Figure 6.1: Frequency Distribution of Transitions from the Previous State i to the Present State j

99

Figure 6.2: Capacity Outage: Case of a 2-unit Plant

115

Figure 6.3: Capacity Outage: Case of a 3-unit Plant

116

Figure 6.4: Load Duration Curve

117

Figure 6.5: Estimation of LOLP

118

v

vi

List of Tables

Table 4.1: Some Techno-Economic Characteristics of the Kerala Power System

52

Table 4.2: Marginal Costs and Peak Load Prices for the Kerala Power System

56

Table 5.1: Distribution of the daily maximum demand for electricity on the Kerala Power System during 2010-11

68

Table 6.1: Monthly FOR of the First Unit of the Pallivasal Power Plant and the Corresponding States (for the 1980s)

103

Table 6.2(a, b and c): Frequency Distribution of FOR States (based on Table 6.1)

107

Table 6.3: Frequency Distribution of FOR States (based on Table 6.2b)

108

Table 7.1: Characteristics of the 10 Hydropower Stations

126

Table 7.2: Long run Availability and Forced Outage Rates

129

Table 7.3: Transition Probabilities and Propensity to Down

131

Table 7.4: Estimation of LOLP by Capacity States

134

Table 7.5: Loss of Load Probability and Expected Available Capacity

137

vii

viii

Preface

The present work, based on the revised version of some of my earlier papers at Centre for Development Studies, Kerala, India, seeks to contribute modestly to peak load pricing and reliability analysis, in both theory and application. The peak load pricing theory was once a fertile field for analytical exercises with The Bell Journal of Economics being its major breeding centre. The general result from the traditional theory of peak load pricing charged the off peak consumers marginal operating costs only and the on-peak users marginal operating plus marginal capacity costs, since it is the on-peakers who were assumed to press against capacity. The theoretical refinements have since somehow ceased to attract attention, possibly because the classical framework and the inevitable result have been taken for granted, and the research interest has shifted from theory to empirics. However, the distinction has haunted me and I have sought to take issue with the classical conclusion with the result that I have revisited the model and shown that if the off-peak period output is explicitly expressed in terms of capacity utilization of that period, the result will be an offpeak price including a fraction of the capacity cost in proportion to its significance relative to total utilization. An important property of our pricing model is that it easily lends itself to generalization in practical application.

Analyzing the implications of the relationship between reliability and rationing cost involved in a power supply system in the framework of the standard inventory analysis, instead of the conventional marginalist approach of welfare economics, I have also formulated indirectly a peak period price in terms of rationing cost. I have fioud a relationship among loss of load probability (LOLP), loss of energy probability (LOEP) and percent reserve margin in the context of normally distributed power demand, and derived an expression for the rationing cost (applicable when demand presses upon capacity) in terms of LOLP. The relationship thus estimated has also been analysed for its implications for the peak load pricing rule under uncertainty. Also proposed has been a very simple and easy method in the context of Markov chain for computing long run probabilities from micro data arranged in a contingency table of current and previous states. I have also sought to model the maximum likelihood estimation of reliability rate and the related statistical properties.

ix

The present work is based on these refinements.

It goes without saying that any work, howsoever privately conceived it is, is a concrete outcome of the collective consciousness. This also has thus a social foundation. I am grateful that the academic atmosphere at the Centre for Development Studies has significantly contributed to that social basis of this study.

A good deal of my grateful appreciation and love is due to my near and dear ones; and this I dedicate with an unsatiated sense of unfulfilled obligations to the burning memory of those who departed and to those who still survive to suffer.

And finally, as Rju as usual smiles away my excuses for my absences from her little kingdom, as Kala as usual resigns herself to my non-corrigible only-the-weekend return to family care, I just seek to balance all, bring all to mind … In balance with this life, this academic pursuit!

x

Chapter 1 Introduction

“Bread has been made (indifferent) from potatoes; And galvanism has set some corpses grinning, But has not answer’d like the apparatus Of the Humane Society's beginning, By which men are unsuffocated gratis: What wondrous new machines have late been spinning.”

— Lord George Gordon Byron (Don Juan 1819, Canto I, CXXX, 35)

1.1 Electric utility

Three distinct functions are involved in supplying electricity in its usable form to the customers: generation, transmission and distribution, corresponding to production, transportation to market and retail distribution of many other products, the chief differences being that i) electricity moves from the generator to the end-use equipment in a continuous flow at a speed approaching that of light, and ii) it cannot be stored in its original form.

Generation, the production of electrical energy from mechanical energy, takes place at central stations normally far away from consumers necessitating the other two processes of transmission and distribution. Transmission is the moving of this electrical energy from generating plants through wire at high voltage to bulk delivery points called substations where it is transformed down to low voltage ready for distribution through low voltage lines to individual meters.

1

There are only two basic sources for driving electric generators: hydro and thermal. The energy source in hydro-plant is water-driven turbines and that in the thermal plant, steam-driven turbines, the steam being produced either by burning fossil fuel (coal, oil or natural gas) or by a nuclear reactor.

The electric utility is unique in that its product is one that must be generated at the instant it is to be used. If the utility has excess generating capacity, it can usually meet any anticipated demand; but an over-abundance of excess capacity entails increasing cost for idle hours. At the same time few products have a greater need for quality and reliability — cases of brown-outs and black-outs. As a matter of practical economics, electric power systems are so designed as to keep both the black-outs and brown-outs within tolerable limits by means of reserves.

1.2 Some Techno-Economic Characteristics of Electric Utility 1.2.1 Load duration Curve

As already noted, electric utility is characterized by an economically non-storable product and a periodically fluctuating load. The load on a utility is the varying sum of all the residential, commercial, and industrial loads, each varying by time of day in its own way. A typical (smoothed) system load curve (as a plot of load, in kilowatt (kW), against the time at which it occurs) is given in the first part of Fig. 1.1. There is a pronounced valley in the curve during early morning hours and a peak in the evening. The area under a (daily) chronological load curve measures the total energy consumption during the day, evaluated by kW

,

expressed in kilowatt-hour (kWh) terms. From the load curve is derived the load duration curve (LDC), by rearranging all the loads of the chronological curve in the order of

2

Figure 1.1 Chronological Load Curve and the Derived Load Duration Curve Load (kW)

Load (kW)

P Peak load M kW

kW

B

Intermediate load

Base load 12

6 AM

12

6 PM

12

0

6

12

18

24

Hours of duration

Time of day

descending magnitude; thus it plots the load against the number of hours (or duration, θ ) during the day for which it occurs. A typical LDC also is shown in Fig. 1.1. Note that the areas under the chronological curve and the corresponding LDC are equal. Annual LDC is generated from the aggregation of all the daily load curves, and is used for planning purposes, which we will consider later on.

1.2.2 Reserve Margin

The excess demand situation in the case of any good or service is a major problem. If demand were uniform, it would be just enough to gear supply or stock to the expected demand. However, random fluctuations in demand (L) about its mean value [E(L) ≡ µ, where E is the expectation operator] involve some chances of shortage and necessitate a further safety margin or buffer or reserve (R). Thus the standard inventory analysis posits

3

the supply or stock (S) at

S = µ + R = µ (1 + PRM),

…..(1)

where PRM is the percent reserve margin. Thus with a usual 10 percent reserve margin, installed capacity equals 1.1 times the expected maximum demand. The value of the reserves determines the reliability of service. If R is too small, excess shortages result; if R is too large, excessive holding costs have to be borne. Hence the significance of an optimum level of reserves.

1.2.3 Cost Structure

The cost of supplying electricity to consumers may be divided into demand and energy costs, comparable to the common industrial classification of fixed and variable costs. Demand (or load or capacity) costs are the capacity related costs for generation, transmission and distribution, and vary with the quantity of plant and equipment and the associated investment. Energy (or unit or output) costs are those which vary directly with the quantity of units (kilowatt-hours) generated. They are largely made up of the costs of fuel, fuel handling and labor. Determination of energy costs is straightforward, once number of units of energy generated is known. On the other hand, demand costs are determined on the basis of pro-rating of the annuitized cost of installing and maintaining the plant over its useful life. Thus, if the basic cycle is one year and the life of a plant of 100 kW capacity is 25 years, then demand costs will be equal to 1/100th of the annuity sufficient to maintain and replace the plant after 25 years. See Turvey (1969) for more details.

One of the technical characteristics of electric utilities is that they operate under common cost conditions; electricity supply involves joint utilization of all or most of the facilities. Under such conditions, only a part of the total supply costs if possible can be identified as applicable entirely to a certain customer class. The substantial portion of the costs thus

4

left unaccountable for must be allocated to different consumer classes in proportion of their contribution to the relevant cost causation factors, such as peak load.

1.2.4 Load Factor

An important concept that has an overwhelming bearing on common cost allocation is load factor (LF), defined as the ratio of the average load (in kilowatts) to the peak or maximum load during a given period (say, a year). If we disregard reserve margin, assuming capacity as equal to peak load, then the ratio (LF) yields capacity factor (CF), a measure of capacity utilization, rather than demand variability as implied in the former. Plant load factor (PLF) defined in the same vein measures capacity utilization of a given plant. It goes without saying that cost per unit (kilowatt-hour) generated is inversely related to capacity utilization and thus to LF. That is, at cent per cent LF, installed capacity is put to the best possible use, and the maximum possible amount of energy is produced during a given period; capacity cost distributed over this maximum amount of energy would be a minimum in this respect. On the other hand, at a low LF, the same capacity cost is spread over a less number of units generated, yielding higher unit cost. Thus a poor LF implies cost inefficiency also. It is this techno-economic characteristic that we make use of in our model; that is, the capacity cost is distributed on an average basis according to the PLF.

1.2.5 Optimum Plant Mix

Given this background, let us now turn to the optimum planning of plant mix. With reference to Fig. 1.1, suppose that the peak demand on the system is P kW. If there is only one generating plant in the station, with a capacity equal to the peak load, then the prime mover and generator will be running under-loaded most of the time, thus rendering the operation uneconomical. A better method is to divide the load into three parts, referred to as base load (B), intermediate load (M) and peak load (P), as shown in Fig. 1.1, each being supplied from separate plants. Thus the base load plant, with a capacity of B kW, is run continuously for all the time (i.e., on full load), and the peak load plant, with 5

a capacity of (P – M) kW, only for a short time. Between these two is the duration of operation of the intermediate load plant, with a capacity of (M – B) kW.

The most economical operation of an electric utility requires that the plant having the minimum operating cost be used to meet the base load, e.g., run-of-river-flow-type (or reservoir-type) hydroelectric plant or nuclear power plant, and that the plant with the highest operating cost, to supply the peak load, e.g., gas turbine plant or pumped-hydro plant. The logic is simple – the total running cost will be a minimum, if the plants are operated inversely to their running costs; remember the base load plants are run for the longest time (with full load, i.e., at cent per cent PLF) and the peak load plants for the shortest time (at lower PLF). Evidently, the total operating cost will be a minimum, when a low-running-cost plant (rather than a high-cost one) is used as the base load plant. At the same time, optimum planning also requires that the capacity cost of the base load plant be the highest and that of the peak load plant, the lowest, as it is so in practice: nuclear or hydropower plants are much costlier to install than the gas turbines. The cost minimization in this respect evidently follows from the inverse cost-PLF relationship, explained above. It should also be pointed out here that in actual practice, hydro- or diesel-power plant is used as peak load plant, since these sets are quick to respond to load variations, as the control required is only for the prime mover, whereas in steam-turbine plant, control is needed for the turbines as well as for the boilers.

1.2.6 Tariffs

Tariff is the rate of payment or schedule of rates on which charges to be recovered from the consumer of electrical energy are computed. A number of tariff structures has been designed and put in use with various types of consumers.1 Usually cost differences have been the primary justification for rate structure differences. The traditional approach involves division of costs into three categories: 1

For the different tariffs in use, see Edison Electric Institute (1976, Ch. 9), Partab (1970, Ch.

1) and Gupta (1983, Ch. 4); and for criteria for well-designed rates, see Bonbright (1961).

6

i)

capacity, demand or load costs,

ii)

energy (unit), output or volumetric costs, and

iii)

consumer costs.2

The first of these (‘kilo watt (Kw) costs’), related to investments in generation, transmission and distribution, vary with the speed and time with which customers use electricity. The second (‘kilo watt hour (Kwh) costs’) vary directly with the number of units generated; they are mainly fuel costs and operating and maintenance (O & M) costs. And the last are those costs varying directly with the number of customers served rather than units consumed. They include expenses on connection, meter reading, billing, collection and consumer services. Then prices are set so as to recover historical (accounting) costs over these three categories with the fair contribution from the several customer classes usually grouped in terms of diversity and load factor.

This backward-looking embedded (accounting) costs approach, concerned mainly with recovering sunk costs, ignores some very vital issues especially from the angle of efficient resource allocation.3 The prices should be related to the true value of additional resources required for an extra unit of supply and this necessitates a forward-looking estimate, i.e., pricing according to marginal costs (MC), which are calculated on the basis of expansion plans4 and operating schedules of the power system in line with demand variation. 2

See Edison Electric Institute (1976, Ch.9), Turvey and Anderson (1977, Ch. 2) and Crew and

Kleindorfer (1986, Ch. 8). 3

For a detailed discussion see Turvey and Anderson (1977, Ch.2) and Munasinghe and Warford

(1982, Ch.1). 4

There has been a controversy about the dichotomy of choice between short-run marginal cost

(SRMC) and long-run marginal cost (LRMC) pricing [see Foster (1963, Appendix 1), Walters (1969, pp. 16-22) and Munasinghe and Warlord (1982, pp. 23-24)]. However, a simple reflection upon the temporal nature of these costs can clarify the issue. SRMC relates to the costs of an 7

1.3 Seasonal Time of Day Pricing

Demand for the product of most public utilities, such as electric utility industry and telecommunications, varies by time of day and season of year. Since the product is economically non-storable, and the demand is time varying, sufficient capacity must be on hand to meet the demand at all times. These characteristics in turn tend to result in non-uniform utilisation of capacity – demand might press hard against or even cross the capacity at times of peak, while a good part of the capacity might remain unutilised at other times. The literature on time of day (or peak load) pricing essentially emerged in response to such peak load problems.5 Here time of day pricing offers an indirect load management mechanism that meets the dual objectives of i) reducing growth in peak load (‘peak clipping’), by charging a higher peak price, and ii) shifting a portion of the load from the peak to the base load plants (‘valley filling’), by charging a lower off peak price. The first objective serves to cut the need for capacity expansion, while the second secures some savings in peaking fuels. This thus ensures an improvement in capacity utilisation as well as a cut in operating and capacity costs. The context of public utilities in such peak load problem led the economists (Boiteux 1949; Steiner 1957; Hirshleifer 1958; Williamson 1966; to name a few) to model pricing rules based on maximisation of social welfare rather than profits.

Thus peak load pricing of electricity is an indirect form of load management that prices electricity according to differences in the cost of supply by time of day and season of year. It reflects the costs in a more accurate manner than do the traditional extra unit of consumption in the short-run with fixed capacity, while LRMC is the cost of meeting an incremental increase in consumption in the long-run with required capacity adjustments. To be more specific, SRMC is nothing but operating costs and LRMC, capacity costs. 5

It is in fact the peak load problem in electricity supply that motivated much of the early work on

the peak load pricing theory (Crew and Kleindorfer 1976).

8

block rate structures, as it logically stems from the marginal cost pricing theory, yet is compatible with the historical accounting costs. Again, compared to the block rate structures, the seasonal time-of-day (STD) pricing offers more potential for improving system load factors; its cost-based price signal motivates customers to modify their usage patterns, which in turn will move the system toward attaining the above twin goals.

It has long been argued and advocated that the sale of electricity and other services, in which periodic variations in demand are jointly met by a common plant of fixed capacity, should be at time-differential tariffs. Time of day (TOD) or peak load electricity rates have thus been widely used in Europe for several decades to reflect time of use cost variations. In particular, Electricité de France (EDF) has been responsible for innovations in implementing TOD pricing and load management techniques. French tariffs departed from the traditional pattern of Hopkinson rates common in the USA and the UK some years after the nationalisation of the electricity industry in 1946, with the introduction (in 1958) of le tarif vert (the green tariff), that applied to high voltage (HV) customers. At present in most of the European countries, where thermal generation is dominant, the HV tariff structure reflects the significance of seasonal and TOD variations in demand and costs of the marginal plants operated on specific load scheduling. In the Scandinavian countries with substantial hydel generation, seasonal variations in the availability of water also tend to influence the tariff level. The low voltage (LV), domestic, tariffs in general have been less complicated, using less complex metering. By contrast, the US regulatory bodies and utilities began to consider TOD rates only following the 1973-74 Arab oil embargo. It was in 1978 only that a provision was included in the Public Utilities Regulatory Policies Act (PURPA) of 1978, to charge TOD rates to each class of customers “unless such rates are not cost-effective with respect to each class” (PURPA, Section III d). Developing countries have started to introduce TOD tariffs at the HV and MV level only recently and that too in simpler terms primarily due to metering problems. In India some crude attempts at TOD tariffs have been made since the early eighties, but with only marginal effects of peak reduction and in some cases with revenue losses also (Parikh, et al., 1994: 222). Non-availability of suitable meters has been the main problem in India. 9

Implementation of peak-load pricing involves substantial capital expenditure in changing meters and increasing customer service as well as transition costs of moving from one rate schedule to another. As already noted, though the STD electricity rates have widely been in use in some of the advanced countries for several decades, it was applied initially only to large industrial customers where metering costs constituted a trivial fraction of the total electric bills. The reforms in the electricity sector have given a fillip to this initiative as spot markets for electricity have come up, rendering the price of electricity on the wholesale market to vary each hour and thus opening up opportunities for electricity distribution companies to apply a real-time pricing scheme to the customer. The progress in solid state technology has now introduced smart meters with many advantages over simple automatic meter reading, such as real-time or near real-time readings, power outage notification, and power quality monitoring. The smart meters have now helped these countries to extend STD pricing to almost all consumers.

The general result from the traditional theory charges the off peak consumers marginal operating costs only and the peak users marginal operating plus marginal capacity costs, since it is the on-peakers who press against capacity. Following Turvey (1968), Crew and Kleindorfer (1971) relax the assumption of homogeneous production capacity, and considers diverse technology, as efficient provision for a periodic demand generally implies an optimal plant mix of different types of capacity with different relative energy and capacity costs. They show that the traditional conclusion holds only for homogeneous plant capacity (e.g., in one plant case), and in economic loading of two or more plants, the off peak price also includes a part of capacity costs. Wenders (1976) also shows that with heterogeneous technology, off peak marginal cost prices almost always should include some marginal capacity costs a la marginal capacity cost savings under certain circumstances. But Joskow (1976), in his comment on the paper, clarifies that these off peak prices can also be rewritten in terms of marginal energy costs only, in a way to validate the traditional result. Panzar (1976), on the other hand, proposes that the usual peak load pricing result is due to the fixed proportions technological assumption employed in the traditional theory and is not a consequence of the fundamental nature of 10

the peak load problem. He shows in particular that in a framework of neoclassical technology of short run decreasing returns to scale, consumers in all periods make a positive contribution toward the cost of capital inputs.

It goes without saying that the equity norms are violated in the traditional peak load pricing, whereby off-peak users pay no capacity charges, but are supplied output out of the capacity, ‘bought/hired’ by the on-peakers. Weintraub (1970) sees a ‘free ride’ problem in the peak load pricing, and argues that “The P-H [peak hour] buyers have every reason to claim that the ‘property’ – the capital facility – is theirs, that they pay for it and that others can use it only at a price in order to reduce the net price to them – the PH users. An outcome which allocates common costs to only the peak-users thus has some disquieting equity features which go to the roots of private property, income distribution, and the diffusion of consumer well-being.” (Weintraub 1970: 512). He therefore suggests “an alternative solution”, (“output maximization”) that is, setting prices such that peak plus off-peak output are maximised, subject to the constraint that costs are covered. For him it is possible that peak price is greater than or less than or equal to off-peak price (p. 513). But this would detract from the time of day pricing as a load management strategy: the peak price must always be greater than the off-peak one in order to improve capacity utilisation at a desirable uniform level through ‘peak clipping’ and ‘valley filling’; at the same time it should be so structured as to ensure equity concerns by apportioning capacity costs, (which are common to all periods), to both the peak and off-peak users by their importance relative to total use. Our task in this work is to seek such a solution, especially in the context of electricity supply.

There is yet another, technical, reason why off-peakers also should bear capacity charges. Power consumption rises over time, with increasing number of consumers and of electrical gadgets in use, as well as increasing intensity of their use. Additional plants are required to meet not only the rising peak load but also the expanding base load. Thus the additional capacity costs involved in installing new base load plants must be borne by all the consumers, irrespective of the period of use, as the base load plants are continuously used in both the periods. This is why in the diverse technology framework, implied in 11

economical load scheduling, off-peakers are also charged a part of capacity costs. As already stated, in Crew and Kleindorfer (1971) and Wenders (1976) this appears in terms of an expression for capacity and running cost savings in line with the logic of optimum plant mix, without yielding a practical rate structure in a format like that of peak price. Our methodology does yield such a one.

1.4 Objective and Structure of the Study

The present study is a modest attempt at a contribution to peak load pricing and reliability analysis, in both theory and application. The general result from the traditional theory of peak load pricing charges the off peak consumers marginal operating costs only and the on-peak users marginal operating plus marginal capacity costs, since it is the on-peakers who press against capacity. Our contribution consists in showing that if the off-peak period output is explicitly expressed in terms of capacity utilization of that period, the result will be an off-peak price including a fraction of the capacity cost in proportion to its significance relative to total utilization. An important property of our pricing model is that it easily lends itself to generalization in practical application. We also give an illustration by estimating marginal costs and peak load prices using time series data on the Kerala power system, further contributing to practical methods of estimation.

Our next contribution is to the power system reliability theory; here we employ a novel approach by utilising the results on reliability in the standard inventory analysis, making use of particular (normal) demand distribution for the daily internal maximum (peak) demand of the Kerala power system during 2010-11. We find a relationship among loss of load probability (LOLP), loss of energy probability (LOEP) and percent reserve margin in the context of normally distributed power demand, and derive an expression for the rationing cost (applicable when demand presses upon capacity) in terms of LOLP. The relationship thus estimated is also analysed for its implications for the peak load pricing rule under uncertainty.

12

Our contribution to the reliability theory continues further in terms of a method to evaluate the LOLP of a power system. We also propose a very simple and easy method in the context of Markov chain for computing long run probabilities from micro data arranged in a contingency table of current and previous states. We also seek to model the maximum likelihood estimation of reliability rate and the related statistical properties.

What follows is divided into seven chapters.

The next chapter provides a detailed review of the peak load pricing theory. We start with the economic characteristics of the public utility, and taking up its traditional natural monopoly status, we review the welfare foundations of pricing: the two extreme cases of marginal cost and monopoly pricing and their regulated cases. A review of the peak load pricing theory under certainty followed by an illustration of the classical model brings out the conventional conclusion that the on-peakers are to bear the entire capacity costs, and the off-peakers are favored by charging them only operating costs. The theory of peak load pricing under uncertainty is also reviewed. It is further noted that the theoretical interest on peak load pricing has waned over time and given way to empirical analysis of residential electricity demand by time of use and a critique of these studies is also given.

In chapter three, we revisit the peak load pricing model, contributing to the literature a significant challenge to the conventional conclusion that charges the off peak consumers marginal operating costs only and the on-peak users marginal operating plus marginal capacity costs, since it is the on-peakers who press against capacity. This has already been called into question. It has also been shown that the equity norms are violated in the traditional peak load pricing, whereby off-peak users pay no capacity charges, but are supplied output out of the capacity, ‘bought/hired’ by the on-peakers. It has been traditionally assumed that whenever a unit of capacity is installed at a cost, it becomes available for demand in all periods; off-peak demand also is met from this capacity; yet this relationship has not been explicitly incorporated into the cost equation. And thus the off-peak price has come out without the capacity cost component! Our study, however, seeks to show that if the off-peak period output is explicitly expressed in terms of 13

capacity utilization of that period, the result will be an off-peak price including a fraction of the capacity cost in proportion to its significance relative to total utilization. This would appear as a general case, irrespective of the nature of generation technology, that is, even when there is only one plant.

An important property of our pricing model is that it easily lends itself to generalisation in practical application and thus in chapter four we give an illustration by estimating marginal costs and peak load prices using time series data on the Kerala power system. The pricing structure we are interested in essentially consists in the long run marginal cost, which includes short run marginal energy cost and marginal capital cost, where the former is obtained as ‘variable cost’ from marginal generating and purchasing costs, adjusted for loss of energy in the transmission and distribution processes, and the latter is the cost of additional capacity to meet one kW increase in demand. We find that the marginal-plant costing method, especially in the forward-looking context, is ineffectual. Hence an alternative method is followed in practice, viz., taking a weighted average of all the ex post costs associated with different sources of marginal output to the system. While the statistical estimator of marginal cost is given by the differential quotient, the estimation of which by means of regression is fraught with non-stationarity problems, we use an analogous discrete difference quotient, simple but involving no statistical problems. Using these estimates of the marginal operating and energy import costs, we obtain an estimate of marginal energy (or running) cost, adjusted with the marginal loss factor. An alternative estimate also is obtained by ignoring the marginal loss factor and dividing the total costs by total sales; and we find the two estimates hardly differ. Next we apportion the estimated marginal capital cost between peak and off-peak periods and estimate all the prices using different estimates of marginal costs, based on different base periods. We find that the recent base period marginal cost estimates lead to higher prices. The illustration clearly brings home the advantage of our method, its simplicity, manageability and at the same time effectiveness.

Our next contribution is to the theory of reliability. In chapter five we seek to establish a relationship between loss of load probability and loss of energy probability on the one 14

hand, and based on this relationship, we derive an estimator of rationing cost and thence a peak load pricing rule in general and also in terms of the actual power demand distribution, found to be normal. We deviate from the conventional comparative static analysis of welfare economics, involving long and tedious derivations, and, instead, make use of the simple results on reliability from the standard inventory analysis, making use of particular (normal) demand distribution for the daily internal maximum (peak) demand of the Kerala power system during 2010-11. Thus the concepts of buffer stock, shortage probability and unit loss function are extended to power system reliability in terms of percentage reserve margin, LOLP and LOEP respectively. This novel approach surprisingly and most assuredly yields the same results on rationing cost as the former method. The relationship thus estimated is also analysed for its implications for the peak load pricing rule under uncertainty.

The next chapter continues with the reliability theory, now enriched with a contribution in terms of a method to evaluate the reliability of a power system. A detailed illustration of a simple method to estimate the loss of load probability is provided. We also propose a very simple and easy method in the context of Markov chain for computing long run probabilities from micro data arranged in a contingency table of current and previous states. Our new long run probability estimator has very significant advantages in consideration of the increasing computer costs involved in the solution of the simultaneous equations when the number of states becomes larger. An Appendix to this chapter purports to model the maximum likelihood estimation of reliability rate and the related statistical properties. The major conclusion here is that the expression we have derived for analyzing the distributional properties of the reliability rate is less amenable to empirical estimation and thus remains only as an intellectual exercise. We feel that simulation might be a better alternative to direct analytic methods to study the distributional characteristics of the reliability rate.

An empirical substantiation of the reliability analysis thus developed is illustrated in chapter seven for the ten major hydro-power stations of the Kerala power system. We calculate the maximum likelihood estimates of availability and forced outage rates as 15

well as loss of load probability (LOLP) measures for these 10 hydropower plants. It is significant to note that a study of this dimension (theory-informed methodology) is the first of its kind in the case of most of the State Electricity Boards in India, especially the Kerala State Electricity Board; the technical appraisal of these power systems in general is confined to examining the plant load factor (PLF) as a measure of capacity utilization only. It goes without saying that PLF is by no means directly comparable with LOLP, as the two methodologies totally differ from each other; this precludes us from attempting at any comparison with the official measure.

Finally the concluding chapter summarises the study and discusses the practical problems in implementing peak load pricing with empirical evidences.

16

Chapter 2 Peak Load Pricing: A Review “We call that fire of the black thunder-cloud ‘electricity’, and lecture learnedly about it, and grind the like of it out of glass and silk: but what is it? What made it? Whence comes it? Whither goes it?”

— Thomas Carlyle (The Carlyle Anthology 1876, 230)

2.1 Introduction

The traditional examples of public utilities include industries such as electricity, telecommunications, oil, natural gas, airlines, trucking, cable television and railroads, that share a common ‘network’ structure, that is an extensive distribution system of lines, pipes, or routes requiring the use of public rights of way, usually with vertically integrated components. They are typically characterised by substantial sunk costs in terms of extensive infrastructure.

In this chapter we start with an attempt at defining public utility in terms of its natural monopoly position and proceed to analyze its welfare implications under the two extreme assumptions of welfare and profit maximization in both unconstrained and constrained domains. Then follow a review of the peak load pricing theory, under both certainty and uncertainty, and a review of the empirical attempts.

17

2.2 Public Utilities

The technical features defining a public utility are those giving rise to economies of scale in public utilities and their consequent ‘natural monopoly’ position. Traditional definition of monopoly viewed as a single-product industry runs in terms of everywhere decreasing average cost curve, i.e., C( λ x) < λC(x), 1 Q), is known as loss of load probability (LOLP). The expected quantum of energy not served, i.e., energy shortage, as a ratio to mean demand is referred to as loss of energy probability (LOEP). Of these two indices, LOLP is the most commonly used one in planning exercises; see for detailed theory and application Pillai (2010). 59

Note that LOLP implies the expected accumulated amount of time in any given period during which load exceeds available capacity. Thus during a one year period, LOLP expressed in terms of days per year is LOLP = 365 × P(D > Q). When expressed as a fraction of time, LOLP gives the probability that there will be a shortage of power (or loss of load) of any magnitude in a given period. Hence the name. It has been a common practice in some of the advanced countries to adopt in electricity supply an arbitrary reliability target, such as a one-day-in-ten-years LOLP. This does not mean a full day of shortages once every 10 years, rather, it refers to the total accumulated time of shortages that should not exceed one day in 10 years, or, equivalently, 0.03 per cent of a day. Telson (1975) has criticized this reliability criterion as too high from an economic standpoint, and suggested a five-day-in-ten-years LOLP (about 0.14 per cent of a day) as more appropriate. In some of the Asian and African countries (e.g., Korea, Hong Kong, Thailand, and Zimbabwe), LOLP criteria are found to vary from 12 to 24 hours per year, equivalent to five to ten days in ten years (Government of India 1997:2).

In India, a reliability target planning criterion of 5 per cent (i.e., 18.25 days a year) was adopted in the First National Power Plan (NPP, 1983) and in the Second NPP (1987). Such a very high level of LOLP target was justified in view of the inability, in terms of funds, to bring in a very large quantum of new capacity additions required for a desirably low target. Later on the general improvement in the technical performance of the generating plants, as also the introduction of new vintage larger size plants with higher efficiency lent sufficient force to decide, during the exercises for evolving the Third NPP (1991), on an LOLP target of 2 per cent (i.e., 7.3 days a year). The same level is now proposed to be retained during the 9th Five Year Plan (FYP) period in the Fourth NPP (1997) also, in view of the unexpected wide gap between the target and the achievement in capacity addition. At the same time, substantial capacity addition in the near future through independent power producers (IPP) has also been expected. It is hence proposed to improve the reliability target planning criterion further to 1 per cent (i.e., 3.65 days a year) LOLP by the end of the 10th FYP. The LOEP, on the other hand, is accepted to be 60

targeted for less than 0.15 per cent.

Random fluctuations in demand imply that a situation may occur where quantity demanded exceeds available capacity. The capacity shortage or excess demand in turn implies that the unserved electricity be rationed among the consumers. The rationing schemes, ranging from simple rotating blackouts to sophisticated load shedding, affect consumers’ surplus differently. In addition to surplus loss, rationing involves some administrative costs also. The sum of these two gives the short run social cost of rationing (or outage), the magnitude of which depends on the particular rationing scheme.

The perfect load shedding scheme, proposed originally by Brown and Johnson (1969) assumes costless rationing according to willingness to pay. Taking this as a base case, Crew and Kleindorfer (1976; 1986) defined (incremental) rationing cost as any surplus losses and administrative costs, that are incurred over and above those under the Brown and Johnson scheme, and that depend only on the level of excess demand. Random rationing and rationing in order of lowest willingness to pay are other forms of rationing examined by Visscher (1973) and Carlton (1977). With the former, consumers are served in random order until capacity is exhausted, without incurring any additional penalty costs. Our experiences of load shedding in India corresponds with this scheme. The latter assumes that when consumers have to queue up for service, those with the lowest willingness to pay may be willing to stand in line the longest, leading to rationing in order of lowest willingness to pay until capacity is exhausted, without any social costs. The simplest rationing scheme (with the most convenient, linear, functional form) assumes that each unit demanded but not supplied involves a constant marginal outage cost (Anderson and Turvey 1977: Chapter 14). This can be viewed as a special case of random rationing (Chao 1983).

Though rationing cost is crucial for determining both price and capacity at optimal level, actual numerical estimates of it are very rare. One way of estimating it is from LOLP

61

itself, as a certain level of rationing cost is implied in a given LOLP target (Chao 1983; Pillai 1991). Derived from a maximised expected net social welfare (gross welfare less capital and operating costs less rationing cost), the estimate of the rationing cost is found to vary inversely with the LOLP target chosen, and to depend on the capital and operating costs of a given technology.

In this chapter we seek to establish a relationship between LOLP and LOEP on the one hand, and based on this relationship, we derive an estimator of rationing cost and thence a peak load pricing rule in general and also in terms of the actual power demand distribution, found to be normal. We deviate from the conventional comparative static analysis of welfare economics, involving long and tedious derivations, and, instead, make use of the simple results on reliability from the standard inventory analysis – a novel approach, that surprisingly and most assuredly yields the same results on rationing cost as the former method. The relationship thus estimated is also analysed for its implications for the peak load pricing rule under uncertainty.

In the next section, we discuss the most common reliability indices of LOLP and LOEP in the framework of the standard results of inventory analysis, and bring out the relationship between them; and in section 3, we apply these rules/results to power system reliability. Section 4 seeks to establish a relationship among the two reliability criteria and outage cost. Some numerical examples are given to illustrate the implications of the relationship for different technologies based on the techno-economic parameters of some representative power plants. The final section is a brief summary.

5.2 Reliability Indices

The excess demand situation in the case of any good or service is a major problem. If demand were uniform, it would be just enough to gear supply or stock to the expected demand. However, random fluctuations in demand (D) about its mean value [E(D) ≡ µ, 62

where E is the expectation operator] involve some chances of shortage and necessitate a further safety margin or buffer or reserve (R). Thus the standard inventory analysis, as mentioned earlier, posits the supply or stock (Q) at

Q=µ+R

…..(5.1)

The value of the reserves determines the reliability of service. If R is too small, excess shortages result; if R is too large, excessive holding costs have to be borne. Hence the significance of an optimum level of reserves.

The simplest and the most frequently used approximation of reserve levels is based only on the mean (µ) and the standard deviation (σ) of the demand distribution. Here R is equated to the demand deviation from its mean, (D − µ), which in turn is set equal to some value ZQ times the standard deviation, i.e., R = D − µ = ZQσ; the value of ZQ is chosen such as to fix at some predetermined level, the probability that demand exceeds supply. Thus for the normal distribution, one σ demand deviation, (ZQ = 1) results in the expected demand exceeding supply for 15.9 per cent of the time; and 2σ deviation (ZQ = 2), for 2.3 per cent of the time only. In other words, if demand deviates from its mean by one σ point, then for 84.1 per cent of the time, we are confident, supply is sure to meet demand. The risks, however, are different, sometimes substantially, for other distributions. For instance, when ZQ = 1, the risk of the expected demand exceeding supply is 13.5 per cent for exponential distribution, and 21.1 per cent for uniform distribution; when ZQ = 2, the risks are 5 per cent and 6.7 per cent respectively. Thus the reserve margin required is defined to be equal to ZQσ, and (5.1) becomes

Q

µ.R

µ . ZQ σ.

….. (5.2)

The probability of shortage (or LOLP) and hence the reliability of service varies with demand distribution. Let the demand for a good follow a probability function f(D), the

63

probability that the demand lies between D and (D + dD). Then the probability of shortage, i.e., the probability that demand (D) will exceed available supply (Q), is given by

LOLP

I T‚U

„ ƒ o

T

T,

…..(5.3)

and the expected quantum of shortage (S ) is D

E T

U

„ o

T

U ƒ T

T.

….. (5.4)

The reliability of service, (ρ), can be defined as the ratio of mean value of supply to mean value of demand, and is given by

µ−S /µ

ρ

1– S/µ

1 – LOEP,

….. (5.5)

where LOEP is the loss of energy probability (also called shortage factor)26 in power system reliability analysis, defined as LOEP = E(D – Q)/E(D) ≡ S/µ. From (5.4) and (5.5), we have an inverse relationship at the margin between shortage/LOEP and available supply expressed through LOLP: "‡ "o

26

=ˆ

" "o

(LOEP) =

„ ƒ(T) o

T=

LOLP.

…..(6.5)

This roughly corresponds to unit loss function in inventory analysis. Usually it is given as S/σ,

rather than as S/µ, that refers to our LOEP. Unit loss function, S/σ, appears more as a convenient formulation, than as a definitional one, as, for example, the unit loss function under the normal distribution can thus be made equal to the term within the parentheses of equation (5.14), that is, independent of ν, the demand variability, as σ is cancelled out in the formulation.

64

Noting that "‡

‰"o‰

"‡

"o

is negative, "

‰ˆ "o LOEP ‰

LOLP.

….. (5.7)

Using (5.2), we can also express (5.7) as Š

"

"‹Œ

LOEP Š = LOLP σ/µ = LOLP ν,

…..(5.8)

where ν = σ/µ is the coefficient of variation (CV) of demand. Thus, LOLP may be defined as a marginal rise in shortage for a one unit fall in supply. This leads to two (equivalent) implications. i) It implies from (5.7) that LOLP may also be expressed as a fraction of the expected demand, the fraction being determined by the marginal change in LOEP for a unit change in the available supply; it also shows, for example, for a 10 per cent LOLP, that the marginal rise in LOEP associated with a one unit fall in supply is equivalent to 10 per cent of the inverse of the expected demand. ii) LOLP may also be interpreted [from (5.8)] as a fraction of demand variability, the fraction being determined by the marginal change in LOEP with respect to the standardised supply; it also follows that for a 10 per cent LOLP, the marginal rise in LOEP for a unit fall in standardized supply corresponds to 10 per cent of the coefficient of variation of demand. The second implication is significant, as it establishes a direct relationship between LOLP and marginal change in shortage factor (LOEP) in terms of demand variability; and we will make use of it in the estimation of rationing cost in Section 5.4.

To evaluate the reliability criteria, we need to consider the actual demand distribution, since the risks are different for different distributions, as we have already seen. For illustration, we take up the distribution of the daily maximum (peak) demand for electricity on the Kerala power system during 1995-96, presented in Table 5.1. When the 65

demand series is distributed in suitable class intervals, the tail (lower and higher) values are found to be fewer in frequency, and hence we have tried to explore whether the data fit a normal distribution well. We have found that this closely approximates a normal distribution with mean = 1425 MW and standard deviation = 97.7 MW, giving a coefficient of variation (CV) of 6.86 per cent.27 Hence below we consider the relevant properties of the reliability criteria in the context of the normal distribution.

5.3 Normal (Gaussian) Distribution

We have the normal distribution ƒ T Defining Z

T

•√2•

?

T − ˆ !.

exp −1/2•

….. (5.9)

D − µ)/σ ,

….. (5.10)

we find that the probability of shortage (LOLP) is

LOLP

P(D > Q)

√2•

?

„ exp ‹“

−‘ /2 ‘

’ ‘o ,

….. (5.11)

which is a well-tabulated function.

The expected shortage is given by D 27

• √2•

?

exp −‘o /2 − •‘o √2•

?

„ exp ‹Œ

−‘o /2

‘

For a long time, Kerala has been reeling under severe power shortage, and power cut/load

shedding has become the rule of the day. The very low variability in the maximum demand distribution obtained here might be a reflection of the ironed-out pattern of the supply-constrained demand. Thus the very reliability of the data is in question, but we use the same just for illustration.

66

= σ[Ψ( ZQ)− ZQ (LOLP)], where Ψ( ZQ) = √2•

?

….. (5.12)

exp −‘o /2 ,

….. (5.13)

is the standard normal distribution N(0, 1), again a well-tabulated function; and LOLP = φ (ZQ). The shortage factor (LOEP), expressed in terms of LOLP, is LOEP where ν

ν [Ψ( ZQ) − ZQ (LOLP)],

….. (5.14)

σ/µ is the CV of demand distribution.28 Note that LOEP < LOLP, and falls

with demand variability. Thus, given the normally distributed maximum demand, we can estimate LOLP and LOEP for the corresponding standard variate, ZQ.29 But how shall we determine ZQ in the context of a power system? The next section discusses our method of estimation of ZQ.

5.4 Application to Power System Reliability

Now we turn to applying the above results from the standard inventory analysis to the reliability analysis of a power system. Sufficient redundancy and excess capacity throughout the system designed to meet contingent exigencies are the major safeguards against power supply shortages. At the generation level, excess capacity is expressed in the form of buffer or reserve margins. In determining the capacity to be installed to meet an expected maximum demand (peak load), due account is taken of the possible 28

Also see (5.8).

29

See the Appendix 1 for the results on other two distributions, viz., exponential and uniform,

presented only for comparative illustration of the relationship between LOLP and LOEP.

67

fluctuations in load from its expectation (µ). If demand were uniform, installed capacity could correspond just to the expected maximum demand. However, the random deviations of demand as well as the day-to-day variations in the available capacity necessitate some reserve margin to account for them. Thus installed capacity required in a power supply system is determined in relation to the expected maximum demand with due considerations for a certain reserve margin to ensure reliability in meeting the contingent demand deviations. Thus with 10 per cent reserve margin (PRM), installed

Table 5.1: Distribution of the daily maximum demand for electricity on the Kerala power system during 2010-11 Maximum Demand (MW)

Actual Frequency

Theoretical Frequency

2800

2850

6

6

2850

2900

18

17

2900

2950

42

43

2950

3000

70

75

3000

3050

91

89

3050

3100

79

72

3100

3150

38

42

3150

3200

16

16

3200

3250

5

5

Mean = 3023.77 MW Standard Deviation = 80.698 MW Chi Square value = 1.522 Chi Square Critical value for 8 degrees of freedom at 5 per cent level = 15.51 Hence we cannot reject the null hypothesis that the data fit the theoretical distribution well. ------------------------------------------------------------------------------------------

68

capacity (Q ) equals 1.1 times the expected maximum demand (µ), as already seen. That is,

Q

µ.R

µ 1 . PRM .

….. (5.15)

Now comparing (5.15) with (5.2), we find that

µ PRM ZQ

or where ν

ZQ σ,

PRM µ/σ

PRM/ν,

…… (5.16)

σ/µ is the CV of maximum demand.

Thus the standardized variate, ZQ, that determines LOLP and reliability, as shown above, is in turn determined, in the reliability analysis of a power system, by both demand and supply factors, i.e., demand variability and PRM. Here we consider three possible cases:30 i) If PRM is just enough to contain demand variability, i.e., PRM = ν, then ZQ = 1, and for the normal distribution, installed capacity falls short of the expected maximum demand only for 15.9 per cent of the time; i.e., LOLP = 0.159 [from (5.11)], LOEP = 0.083 times CV [from (5.14)], and the reliability of service is ρ = 1 – 0.083ν. For example, for the Kerala power system with ν = 2.67 per cent as in 2010-11, assuming PRM = ν, we have LOEP = 0.0022 and ρ = 0.9978, so that the expected peak power shortage (with µ = 3023.77 MW) is 6.7 MW only [from (5.12)]. It can also be seen that for every one unit drop in available supply, LOEP increases by 0.0022 per cent [from (5.7)]. 30

See Appendix 2 for simulation results on LOLP and LOEP (as also the multiplier for the

capacity charge component of the rationing cost) for different values of PRM and ν, under the three distributions of normal, exponential and uniform.

69

ii) With PRM > ν, we have ZQ > 1, that reduces the chances of shortage and raises the service reliability. Thus, considering the overall reserve margin of the Kerala power system in 2010-11 (with a maximum available capacity of 3267 MW, including the Central share of capacity, provided the Kerala system has undisturbed access to its share of Central allocation) of about 7.5 per cent of the mean maximum demand (3023.77 MW), we get ZQ = 2.81, and hence LOLP= 0.0025 and LOEP = 0.0264 – evidently, the largesse of a very low demand variability.

iii) On the other hand, if the reserves are inadequate in relation to demand variability, i.e., PRM < ν, then ZQ < 1, and the probability of demand exceeding available supply increases and reliability decreases. In this case, disregarding the Central share (of 1150 MW), Kerala’s installed capacity in 2010-11 (2117 MW) is incapable of meeting even the mean maximum demand, leaving very high LOLP and LOEP. The significance of our formulation of the relationship (5.16) is very much evident from this discussion.

5.5 Reliability Rationing Cost and Peak Load Pricing under Uncertainty Now that we have determined LOLP and LOEP, as also ZQ, in terms of PRM and ν, required in their estimation in the context of a normally distributed maximum demand in a power supply system, we now derive a relationship among rationing cost, and reliability (LOLP, LOEP) in electricity supply in the event of excess (maximum) demand. Instead of the conventional marginalist approach of analysing the expected net social welfare, we seek to minimise the total cost made up of costs of capacity, output (generation) and shortage that yields exactly the same result as does the former. We assume the simplest rationing scheme, with a constant marginal penalty cost of excess demand, that is a special case of random rationing, that we usually experience in our country. Thus the rationing costs we estimate later at the end of this section approximate the reality. For sake of simplicity, again, we dispense with subscripts and symbols for diverse technology and periods.

70

From the preceding sections, we have the fraction of expected energy shortage, LOEP, as LOEP = Q/µ, giving the availability factor as (1 – LOEP). During the length (θ) of a certain period (i.e., peak period),31

the energy shortage is θS

θ µ LOEP units,

involving a penalty price of ‘r’ per unit short. The energy available for supply, then, is

θ µ (1 – LOEP) units with an operating cost of ‘b’ per unit. The capacity cost is ‘β’ per kW of capacity, Q. Thus the total cost is: TC = βQ + b θ µ (1 – LOEP) + r θ µ LOEP.

….. (5.17)

Minimizing the total cost, "–5 "o

0

N . — − O Sˆ

"

"o

LOEP.

….. (5.18)

Now, using (5.10), Z = (D – µ)/σ → (Q – µ)/σ = ZQ,

….. (5.19)

we can rewrite (5.18) as N. —−O

t "

˜ "‹Œ

LOEP

0.

….. (5.20)

where ν = σ/µ.

31

The analysis can be extended to multiple period case also; for example, shortage and non-

shortage periods as well as peak and off-peak periods. Only during the shortage duration (in peak period) is the third term in (5.17), r θ µ LOEP, active; and this is the only case we consider here.

71

"

Note that

"‹Œ

LOEP, the rate of change in LOEP with respect to standardized capacity,

depends on the particular demand distribution. Using (8) and the definition of LOLP in (5.11), we obtain for normally distributed demand: "

"‹Œ

LOEP = – νLOLP = – νφ (ZQ),

….. (5.21)

Now, from (20) and (21), we get LOLP as LOLP = φ(ZQ) = β/θ(r – b).

….. (5.22)

Equation (5.22) presents a number of significant implications:

The denominator in (5.22) represents the net expected social cost of shortage. Each unit of power cut imposes a penalty of ‘r’, but saves a marginal operating cost of ‘b’. As the net social cost of rationing increases, LOLP falls. The rationale for this is clear. As (nonprice) rationing (in the event of excess demand) becomes more inefficient, rationing cost increases; this necessitates to place more reliance on price rationing (price rise) to reduce excess demand, which in turn leads to increased profits and capacity expansion. Thus reliability also increases. Equation (5.22) shows that for any given LOLP target criterion, a certain level of rationing cost is implied in it. This then provides an estimate of rationing cost:

in general, —

O.

i

t ™š™›

,

….. (5.23)

or, in particular,32 for normally distributed demand:

32

See Appendix 1 for the results on other two distributions, viz., exponential and uniform.

72

—

i

O . t œ

‹Œ

r = b + β/θ φ(ZQ),

….. (5.24)

where ZQ = PRM/ ν.

The general expression, (5.23), for ‘r’ that explicitly states the relationship between the rationing cost and LOLP, is a significant result in that it brings out an economic justification in setting an outage cost vis-à-vis an optimal reliability target planning criterion (LOLP); it also marks the effects on this reliability criterion of the assumptions of capital costs, generation costs and rationing costs (see Chao 1983; Pillai 2010).

Equation (5.22) is significant again in that it sets the optimal marginal capacity cost as a fraction (equal to LOLP) of the net social cost of rationing implied in that level of LOLP. That is, for a 10 per cent LOLP, the marginal capacity cost corresponds to 10 per cent of the net rationing cost. Thus we obtain an optimal investment rule in shortage period. The denominator in (5.22) may also be interpreted as the net expected benefits from a unit increase in capacity. Each unit increase in capacity implies that with each unit of power cut avoided, a rationing cost of ‘r’ is escaped, but a marginal operating cost of ‘b’ is incurred. Thus the investment rule in (5.22) states that the marginal capacity cost (β /θ) equals the net expected benefits from a unit increase in capacity in shortage period times the probability of that period, i.e., (r – b)LOLP (Turvey and Anderson 1977: Chapter 14).

Another significant implication of (5.22) is its potential in yielding the stochastic equivalent of the deterministic peak load pricing rule, whereby peak period price equals b + β /θ. Now substituting here for β /θ from (5.22), we have the stochastic pricing rule:

P b (1 – LOLP) + r LOLP,

….. (5.25)

73

a probability-weighted average of the marginal operating cost and the marginal rationing cost (Turvey and Anderson 1977: Chapter 14). Since (1 – LOLP) is the probability of meeting demand and LOLP, that of power cut, the average price is applicable to both the situations. This is in contrast to the deterministic pricing rule that charges the off-peak customers only the marginal operating cost (b).

Compared with the peak-period price, rationing cost is much higher through the effect of LOLP on the unit capacity cost. If, for example, PRM is just sufficient to meet demand variability (ν) such that ZQ = 1, then as we have already seen, LOLP for normal distribution is 0.159; for exponential distribution, it is 0.135 and for uniform distribution, 0.211. Then from (23), we find that the capacity charge component of the rationing cost is about (1/0.159 =) 6.3 times higher than that of peak-load tariff rate if demand follows normal distribution; it is about 7.4 times higher, if demand is exponentially distributed, and about 4.7 times higher for uniformly distributed demand. If PRM > ν, then LOLP falls, with a much higher implied rationing cost; and if PRM < ν, reliability falls, with a lower implied rationing cost.33

Some numerical examples will illustrate the implications of the relationship between LOLP and rationing cost. Let us consider the following parameters of different power plants, and estimate the outage costs for different LOLP targets, as well as peak- and offpeak-period prices (representing generation cost only, and assuming a peak period of 3.5 hours a day, i.e., from 6 to 9.30 in the evening). The basic data (as in 1996) are taken from International Energy Initiative (1998).

33

See foot note 30. 74

5.6 Numerical Applications

I A Coal-Based Thermal Power Plant of 350 MW Capacity:

(a)

Capital cost

: Rs. 1310 crores

Annuitised capital cost (at 12 per cent discount rate and for 25 years of plant life) : Rs. 4772.14/kW/year Marginal capital cost at peak demand (with 10 per cent transmission and distribution loss and 20 per cent reserve margin): Rs. 6299.23/kW/year.

(b)

(c)

Fuel costs Coal consumption norm

: 3192 kg./kW

Price of coal

: Rs. 1000/tonne

Oil consumption norm

: 72000 ml./kW

Price of oil

: Rs. 6396/kl.

Total fuel cost

: Rs. 3652.5/kW

Operation and maintenance cost (2.5 per cent of capital cost)

: Rs. 935.7/kW

(d)

Total operating cost (off-peak price)

: Rs. 0.524/kWh

(e)

Peak period price

: Rs. 5.45/kWh

(f)

Average (accounting) price

: Rs. 1.24/kWh.

(g)

Rationing cost (Rs./kWh) with 10 per cent LOLP

: 49.83

5 per cent LOLP

: 99.14

2 per cent LOLP

: 247.07 75

Five-day-in-ten-years LOLP

: 3600.08

One-day-in-ten-years LOLP

: 17998.32

II A Diesel-Based Thermal Power Plant of 5 MW Capacity

(a)

Capital cost

: Rs. 7.5 crores

Annuitised capital cost (at 12 per cent discount rate and for 25 years of plant life) : Rs. 1912.5/kW/year Marginal capital cost at peak demand (with 10 per cent transmission and distribution loss and 20 per cent reserve margin): Rs. 2524.5/kW/year.

(b)

(c)

Fuel costs Oil consumption norm

: 438 litres/kW

Price of oil

: Rs. 7/litre

Total fuel cost

: Rs. 3066/kW

Operation and maintenance cost (4.5 per cent of capital cost)

: Rs. 675/kW

(d)

Total operating cost (off-peak price)

: Rs. 0.427/kWh

(e)

Peak period price

:Rs. 2.40/kWh

(f)

Average (accounting) price

: Rs. 0.72/kWh

(g)

Rationing cost (Rs./kWh) with 10 per cent LOLP

: 20.19

5 per cent LOLP

: 39.95

2 per cent LOLP

: 99.23

Five-day-in-ten-years LOLP

: 1443.0

One-day-in-ten-years LOLP

: 7213.28 76

III A Gas-Based Thermal Power Plant of 300 MW Capacity

(a)

Capital cost

: Rs. 900 crores

Annuitised capital cost (at 12 per cent discount rate and for 25 years of plant life) : Rs. 3825/kW/year Marginal capital cost at peak demand (with 10 per cent transmission and distribution loss and 20 per cent reserve margin): Rs. 5049/kW/year.

(b)

(c)

Fuel costs Gas consumption norm

: 1740.3 gms./kW

Price of gas

: Rs. 3.2/gm.

Total fuel cost

: Rs. 5569/kW

Operation and maintenance cost (4.5 per cent of capital cost)

: Rs. 1350/kW

(d)

Total operating cost (off-peak price)

: Rs. 0.7898/kWh

(e)

Peak period price

: Rs. 4.74/kWh

(f)

Average (accounting) price

: Rs. 1.37/kWh

(g)

Rationing cost (Rs./kWh) with 10 per cent LOLP

: 40.31

5 per cent LOLP

: 79.83

2 per cent LOLP

: 198.40

Five-day-in-ten-years LOLP

: 2885.93

One-day-in-ten-years LOLP

: 14426.50

77

IV. A Hydro-Electric Power Plant of 120 MW Capacity

(a)

Capital cost

: Rs. 180 crores

Annuitised capital cost (at 12 per cent discount rate and for 25 years of plant life) : Rs. 1912.5/kW/year Marginal capital cost at peak demand (with 10 per cent transmission and distribution loss and 20 per cent reserve margin): Rs. 2524.5/kW/year.

(b)

Operation and maintenance cost (10 per cent of capital cost)

: Rs. 1500/kW

(c)

Total operating cost (off-peak price)

: Rs. 0.171/kWh

(d)

Peak period price

: Rs. 2.15/kWh

(e)

Average (accounting) price

: Rs. 0.46/kWh

(f)

Rationing cost (Rs./kWh) with 10 per cent LOLP

: 19.93

5 per cent LOLP

: 39.69

2 per cent LOLP

: 98.98

Five-day-in-ten-years LOLP

: 1442.74

One-day-in-ten-years LOLP

: 7213.03

5.7 Conclusion

The present chapter employs a novel approach to power system reliability study by utilising the results on reliability in the standard inventory analysis, making use of particular (normal) demand distribution for the daily internal maximum (peak) demand of the Kerala power system during 2010-11. Thus the concepts of buffer stock, shortage probability and unit loss function are extended to power system reliability in terms of 78

percentage reserve margin, LOLP and LOEP respectively. We find that the inverse relationship at the margin between LOEP and available supply corresponds to LOLP weighted by the inverse of the expected demand. It is found that in the case of normally distributed demand, LOEP < LOLP, and falls with demand variability.

Rationing cost involved in power shortage includes loss of consumers’ surplus and cost of administering a certain rationing scheme. Minimising the total cost incurred in power supply in a shortage period yields a significant inverse relationship between LOLP and rationing cost along with other (capacity and operating) cost components. Various implications of this relationship are examined, for optimal investment rule, stochastic version of peak load pricing, etc. Rationing cost implied in different LOLP target criteria are also estimated, based on the techno-economic parameters of different types of power plants. The assumption in our model of a random rationing scheme brings these estimates up as representing the actual penalty costs of the excess demand for power that we exert on the system in India. It is found that the rationing cost, as also the peak-period price (representing generation cost only), associated with a hydro-electric power plant is lower than that with thermal plants.

79

APPENDIX 1 In this appendix we present, for comparative illustration, the results on the relationship between LOLP and LOEP as also that between the reliability indices and rationing cost in the case of exponential and uniform distributions.

Exponential distribution

The exponential distribution is

f D dD

λ exp −λD dD,

….. (A1)

1/µ, and standard deviation, σ

where λ

µ

1/λ .

…..(A2)

Here the shortage probability (LOLP) is LOLP

P D>Q

exp −λQ .

….. (A3)

Now from (2) and (A2), we get LOLP

P D>Q

exp −λQ

exp[− 1 . ZQ ],

….. (A4)

and the expected shortage,

S

σ exp[− 1 . ZQ ].

….. (A5)

Thence the LOEP is LOEP

ν exp[− 1 . ZQ ]

ν LOLP

LOLP,

….. (A6)

since ν =1; hence the reliability of service is not influenced by CV and LOEP = LOLP, unlike in the case of normal distribution. 80

For exponential distribution, we have from (8) and (A4): "

"‹Œ

LOEP

−LOLP

hence the rationing cost: r

−expŸ− 1 . ‘o

;

b . β /θ exp[−(1 . ZQ )].

Uniform (Rectangular) distribution

The uniform distribution is

f(D)dD = 1/(D1 – D0), D0 < D < D1 = 0 otherwise.

…..(A7)

We have

µ = (D1 + D0)/2, and •=

(Vj ?V¡ ) √¢

and the limits on the distribution are

b = ˆ

•√3

and b = ˆ + •√3

81

Considering the definition of ZQ implied in (2), we get LOLP

√¢?‹Œ

P T‚U

√¢

and directly from (2)

LOLP

Vj ?o

I T‚U

¤√¢

.

…..(A8)

Note that LOLP is not defined for values of ZQ greater than the square root of 3. The expected quantum of shortage is D

Vj ?o w ¤√¢

¤

√¢

√3 − ‘o ,

so that LOEP is ˜

LOEP

√¢

√3 − ‘o

˜

√3 − ‘o LOLP.

Like the normal distribution, LOEP < LOLP, and falls with demand variability.

Using the definition of LOLP in (A8), we have the rationing cost as —

i

O.t

√¢ √¢?‹Œ

.

Again, r also is not defined for values of ZQ greater than the square root of 3.

82

APPENDIX 2

In this Appendix we present the simulation results for LOLP, LOEP and multiplier for the capacity charge component of the rationing cost for different values of percentage reserve margin (PRM) and coefficient of variation (ν), when demand is assumed to follow different distributions, viz., normal, exponential, and uniform When Demand is Normally Distributed 1. Loss of Load Probability (LOLP) Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

0.15866

0.30854

0.3707

0.40129

0.42074

0.43251

0.44433

0.44828

0.4562

0.46017

20

0.02275

0.15866

0.25143

0.30854

0.34458

0.3707

0.38591

0.40129

0.41294

0.42074

30

0.0013499

0.066807

0.15866

0.22663

0.27425

0.30854

0.3336

0.35197

0.3707

0.38209

Percent

40

0.000031671

0.02275

0.091759

0.15866

0.21186

0.25143

0.28434

0.30854

0.32997

0.34458

Reserve

50

0.0000003

0.0062097

0.04746

0.10565

0.15866

0.20327

0.23885

0.26435

0.28774

0.30854

Margin

60

0

0.0013499

0.02275

0.066807

0.11507

0.15866

0.19489

0.22663

0.25143

0.27425

70

0

0.00023263

0.0099031

0.040059

0.080757

0.121

0.15866

0.18943

0.2177

0.24196

80

0

0.000031671

0.0037926

0.02275

0.054799

0.091759

0.12714

0.15866

0.18673

0.21186

90

0

3.3977E-06

0.0013499

0.012224

0.03593

0.066807

0.098525

0.12924

0.15866

0.18406

100

0

0.00000025

0.00043423

0.0062097

0.02275

0.04746

0.076359

0.10565

0.1335

0.15866

83

2. Loss of Energy Probability (LOEP) Coefficient of Variation

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.008326204

0.039544899

0.076122191

0.11450713

0.153408

0.1927653

0.2319359

0.271778

0.311147

0.350856

20

0.00084801

0.016652409

0.045529121

0.0790898

0.115182

0.1522444

0.1908548

0.229014

0.267633

0.306816

30

3.81257E-05

0.005856208

0.024978613

0.05244174

0.0843038

0.1186347

0.1546245

0.191833

0.228367

0.266684

Percent

40

7.1193E-07

0.001696021

0.012489523

0.03330482

0.0600726

0.0910582

0.1234098

0.15818

0.193228

0.230364

Reserve

50

2.3642E-08

0.000400105

0.006107138

0.02021994

0.041631

0.0674781

0.0969123

0.130301

0.163771

0.197724

Margin

60

6.07466E-10

7.62514E-05

0.002544031

0.01171242

0.0280315

0.0499572

0.0764337

0.104883

0.136587

0.168608

70

9.13288E-13

1.16604E-05

0.000932814

0.00646268

0.0183188

0.0364748

0.0582834

0.084999

0.11289

0.142819

80

5.05125E-16

1.42386E-06

0.000384028

0.00339204

0.0116101

0.024979

0.0435994

0.06661

0.092436

0.120145

90

1.02777E-19

1.38175E-07

0.000114377

0.00169171

0.0071301

0.0175686

0.0334971

0.053151

0.074936

0.100378

100

7.69305E-24

0.00000005

2.83606E-05

0.00080021

0.0042401

0.0122143

0.0242793

0.04044

0.060134

0.083262

84

3. Multiplier for the Capacity Charge Component of the Rationing Cost Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

6.303

3.241

2.698

2.492

2.377

2.312

2.251

2.231

2.192

2.173

20

43.956

6.303

3.977

3.241

2.902

2.698

2.591

2.492

2.422

2.377

30

740.796

14.968

6.303

4.412

3.646

3.241

2.998

2.841

2.698

2.617

Percent

40

31574.627

43.956

10.898

6.303

4.720

3.977

3.517

3.241

3.031

2.902

Reserve

50

4000000.000

161.038

21.070

9.465

6.303

4.920

4.187

3.783

3.475

3.241

Margin

60

Not Defined

740.796

43.956

14.968

8.690

6.303

5.131

4.412

3.977

3.646

70

Not Defined

4298.672

100.978

24.963

12.383

8.264

6.303

5.279

4.593

4.133

80

Not Defined

31574.627

263.671

43.956

18.249

10.898

7.865

6.303

5.355

4.720

90

Not Defined

294316.744

740.796

81.806

27.832

14.968

10.150

7.738

6.303

5.433

100

Not Defined

4000000.000

2302.927

161.038

43.956

21.070

13.096

9.465

7.491

6.303

85

When Demand is Exponentially Distributed 1. Loss of Load Probability (LOLP) = Loss of Energy Probability (LOEP) Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

0.13534

0.22313

0.26360

0.28650

0.30119

0.31140

0.31891

0.32465

0.32919

0.33287

20

0.04979

0.13534

0.18888

0.22313

0.24660

0.26360

0.27645

0.28650

0.29457

0.30119

30

0.01832

0.08208

0.13534

0.17377

0.20190

0.22313

0.23965

0.25284

0.26360

0.27253

Percent

40

0.00674

0.04979

0.09697

0.13534

0.16530

0.18888

0.20775

0.22313

0.23588

0.24660

Reserve

50

0.00248

0.03020

0.06948

0.10540

0.13534

0.15988

0.18009

0.19691

0.21107

0.22313

Margin

60

0.000912

0.01832

0.04979

0.08208

0.11080

0.13534

0.15612

0.17377

0.18888

0.20190

70

0.000335

0.01111

0.03567

0.06393

0.09072

0.11456

0.13534

0.15335

0.16901

0.18268

80

0.000123

0.00674

0.02556

0.04979

0.07427

0.09697

0.11732

0.13534

0.15124

0.16530

90

4.5400E-05

0.00409

0.01832

0.03877

0.06081

0.08208

0.10170

0.11943

0.13534

0.14957

100

1.6702E-05

0.00248

0.01312

0.03020

0.04979

0.06948

0.08816

0.10540

0.12110

0.13534

86

2. Multiplier for the Capacity Charge Component of the Rationing Cost Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

7.389

4.482

3.794

3.490

3.320

3.211

3.136

3.080

3.038

3.004

20

20.086

7.389

5.294

4.482

4.055

3.794

3.617

3.490

3.395

3.320

30

54.598

12.182

7.389

5.755

4.953

4.482

4.173

3.955

3.794

3.669

Percent

40

148.413

20.086

10.312

7.389

6.050

5.294

4.814

4.482

4.239

4.055

Reserve

50

403.429

33.115

14.392

9.488

7.389

6.255

5.553

5.078

4.738

4.482

Margin

60

1096.633

54.598

20.086

12.182

9.025

7.389

6.405

5.755

5.294

4.953

70

2980.958

90.017

28.032

15.643

11.023

8.729

7.389

6.521

5.917

5.474

80

8103.084

148.413

39.121

20.086

13.464

10.312

8.524

7.389

6.612

6.050

90

22026.466

244.692

54.598

25.790

16.445

12.182

9.833

8.373

7.389

6.686

100

59874.142

403.429

76.198

33.115

20.086

14.392

11.343

9.488

8.257

7.389

87

When Demand is Uniformly Distributed 1. Loss of Load Probability (LOLP) Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

0.21132

0.3557

0.4038

0.4278

0.4423

0.4519

0.4588

0.4639

0.4679

0.4711

20

Not Defined

0.2113

0.3075

0.3557

0.3845

0.4038

0.4175

0.4278

0.4358

0.4423

30

Not Defined

0.0670

0.2113

0.2835

0.3268

0.3557

0.3763

0.3917

0.4038

0.4134

Percent

40

Not Defined

Not Defined

0.1151

0.2113

0.2691

0.3075

0.3350

0.3557

0.3717

0.3845

Reserve

50

Not Defined

Not Defined

0.0189

0.1392

0.2113

0.2594

0.2938

0.3196

0.3396

0.3557

Margin

60

Not Defined

Not Defined

Not Defined

0.0670

0.1536

0.2113

0.2526

0.2835

0.3075

0.3268

70

Not Defined

Not Defined

Not Defined

Not Defined

0.0959

0.1632

0.2113

0.2474

0.2755

0.2979

80

Not Defined

Not Defined

Not Defined

Not Defined

0.0381

0.1151

0.1701

0.2113

0.2434

0.2691

90

Not Defined

Not Defined

Not Defined

Not Defined

Not Defined

0.0670

0.1288

0.1752

0.2113

0.2402

100

Not Defined

Not Defined

Not Defined

Not Defined

Not Defined

0.0189

0.0876

0.1392

0.1792

0.2113

88

2. Loss of Energy Probability (LOEP) Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

0.00774

0.04382

0.08472

0.12681

0.16939

0.2122

0.2552

0.2982

0.3413

0.3845

20

0.00104

0.01547

0.04915

0.08764

0.12805

0.1694

0.2114

0.2536

0.2961

0.3388

30

0.02321

0.00155

0.02321

0.05568

0.09249

0.1315

0.1717

0.2126

0.2541

0.296

Percent

40

0.07424

0.00207

0.00688

0.03094

0.06269

0.0983

0.1361

0.1753

0.2154

0.2561

Reserve

50

0.15415

0.01702

0.00019

0.01342

0.03868

0.0699

0.1047

0.1415

0.1798

0.2191

Margin

60

0.26292

0.04641

0.00311

0.00311

0.02043

0.0464

0.0773

0.1114

0.1474

0.185

70

0.40056

0.09023

0.01566

0.00002

0.00796

0.0277

0.0541

0.0848

0.1183

0.1537

80

0.56706

0.14848

0.03782

0.00415

0.00126

0.0138

0.0351

0.0619

0.0924

0.1254

90

0.76244

0.22117

0.06962

0.01549

0.00033

0.0047

0.0201

0.0426

0.0696

0.0999

100

0.98668

0.30829

0.11103

0.034049

0.00518

0.0004

0.0093

0.0268

0.0501

0.0774

89

3. Multiplier for the Capacity Charge Component of the Rationing Cost Coefficient of Variation 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

4.732

2.812

2.477

2.337

2.261

2.213

2.180

2.156

2.137

2.123

20

Not Defined

4.732

3.252

2.812

2.601

2.477

2.395

2.337

2.294

2.261

30

Not Defined

14.928

4.732

3.527

3.060

2.812

2.658

2.553

2.477

2.419

Percent

40

Not Defined

Not Defined

8.688

4.732

3.717

3.252

2.985

2.812

2.690

2.601

Reserve

50

Not Defined

Not Defined

52.981

7.186

4.732

3.854

3.404

3.129

2.944

2.812

Margin

60

Not Defined

Not Defined

Not Defined

14.928

6.511

4.732

3.959

3.527

3.252

3.060

70

Not Defined

Not Defined

Not Defined

Not Defined

10.432

6.127

4.732

4.042

3.630

3.357

80

Not Defined

Not Defined

Not Defined

Not Defined

26.233

8.688

5.879

4.732

4.108

3.717

90

Not Defined

Not Defined

Not Defined

Not Defined

Not Defined

14.928

7.761

5.706

4.732

4.163

100

Not Defined

Not Defined

Not Defined

Not Defined

Not Defined

52.981

11.415

7.186

5.579

4.732

90

Chapter 6 Steady State Probability and Loss of Load Probability of a Power System “And God said, ‘Let there be light’ and there was light, but the Electricity Board said ‘He would have to wait until Thursday to be connected’.” — Spike Milligan

6.1 Introduction

Electric power is vital to both our economic and personal well-being and hence a power system is expected to supply electrical energy as economically as possible, and with a high degree of quality and reliability. In fact, few products have a greater need for quality and reliability. Reliability in its broad sense refers to the probability that a component or system comprising components is able to perform its intended function satisfactorily during a specified period of time under normal operating conditions. Thus the reliability assessment of a power system is mainly concerned with its capability, which is related to the existence and availability of sufficient facilities to satisfy customer load. The basic facilities of a system are in the three sectors of its function, viz., generation, transmission and distribution, which are usually vertically integrated. Electric power produced at the generation end is carried to the consumers via transmission and distribution facilities. In this paper our focus is only on the generation sector.

A modern power system is very large and complex, composed of n power generating stations, where power is generated from fuels (fossil or nuclear) or by hydroelectric stations. Each generating station or plant consists of M plant units or generators, each with a rated capacity. Each of the N stations has an installed capacity Ki megawatts (mw), which is the sum of the rated capacities of its M units, and the system installed capacity to supply power is the sum of the

91

installed capacities of all the stations. In the case of a hydropower system, each power station has usually associated with it a big reservoir behind a dam that supplies hydraulic power to drive each of the M generators.

As already explained power system is unique in that its product is one that must be generated the instant its service is demanded and that the demand for electricity varies greatly at random according to the time of the day and the season of the year. Therefore a power system is designed to supply instantaneously the power demanded by consumers. However, failures in the system do occur when demand exceeds supply as in the case of any other goods and services.

Demand can exceed supply for two main reasons. One is the random deviations of the demand from its expected level such that a very high peak demand exceeds the installed capacity of the system. Capacity of a power system is in general determined after taking due considerations of such unforeseen fluctuations in demand. This is effected by means of reserve or standby capacity over and above the expected peak period demand that is to be met.

Shortage may still occur, even if the load is not far from its expectation; a high demand that does not exceed the installed capacity of the system can exceed the available capacity at that moment. This is due to generator deratings, scheduled preventive maintenance and forced outages of generators. Generator deratings result from equipment problems and changes in operating conditions, and are a function of the age of the equipment. Outage refers to a certain state of a unit when it becomes unavailable to perform its intended function due to some event directly associated with it. An outage may be either a scheduled one or a forced one. Scheduled outage (or maintenance outage) is a planned event, whereby a component/unit is deliberately taken out of service at a chosen time for preventive maintenance or overhaul or repair; this is to keep the generating units in proper running condition. Forced outage, on the other hand, results when a unit falls out of service due solely to random events such as breakdown, malfunction of equipment, etc.

92

In the case of a hydropower system, besides these two scenarios, shortage can still occur if the hydraulic power in any storage is not sufficient to turn the concerned generator. The plant unit is then shut down, and the system capacity falls accordingly.

A modest attempt is made here to evaluate the reliability of the Kerala power system. Following a detailed discussion in this chapter of the methodology used in this study, the maximum likelihood estimates of availability and forced outage rates as well as loss of load probability measures are calculated for the 10 hydropower plants of Kerala in the next chapter.

6.2 Availability and Outage Measures

In a Markov process, the life history of a repairable electric power system component during its useful life period is represented by a two-state model, the two possible states being labeled ‘up’ or ‘functioning’ and ‘down’ or ‘unavailable’, denoted by 1 and 0 respectively. Thus when the component fails, it is said to undergo a transition from the up to the down state, and conversely, when repairs are over, it is said to return from the down to the up state. This idea then facilitates to interpret the concept of reliability in terms of the fraction of total time the component remains in the up state. The length of functioning period is also referred to as the time-to-failure, and that of the period under repair as the downtime.

The probabilistic approach to power system reliability analysis views the system as a stochastic process evolving over time. At any moment the system may change from one state to another because of events such as component outages or planned maintenance. Corresponding to a pair of states, say i, j), there is a conditional probability of transition from the state i to the state j. Suppose the performance of a power plant is continuously monitored to record the sequence of failures and repairs during sustained operation in order to assess its performance. During each failure-repair cycle, the time to failure (when the plant is in ‘up’ state) and the time to repair (when the plant is in ‘down’ state) are recorded. The number of failures per unit of time is known as the failure (or hazard) rate, and the number of repairs per unit of time, the repair rate. The reliability of a power plant is often measured in terms of two availability indices, viz., 93

instantaneous availability, A(t), and steady-state (long-run) availability, A(∞). The former refers to the probability that the power plant is available for operation at any time t and the latter to its avaiability for large values of t, that is, in long run. Thus,

A t = Probability(available at time t), and A(∞) = lim A(t ) . t →∞

….(6.1)

The first step in an availability study is to specify certain probability models for the two variables, time-to-failure, denoted by X and time-to-repair, denoted by Y. The second step is the derivation of the availability indices, which in general are the functions of the parameters of the statistical models specified for X and Y. Usually the failure and repair rates are assumed to be constant; this leads to the assumption that the time-to-failure and the time-to-repair variables follow exponential distribution. The exponential distribution is one of the two (the other being the geometric distribution) unique distributions with the memoryless or no-ageing property. That is, future lifetime of a component remains the same irrespective of its previous use, if its lifetime distribution is exponential. Thus we assume that the time-to-failure, X, is an exponential variable with parameter λ, so that its density function, viz., failure (hazard) density function, f x , is given by

f(x) =

− x exp  , for x > 0. λ  λ  1

….(6.2)

The parameter 1/λ is the constant failure (hazard) rate. For an exponential distribution of the above form, the mean is given by λ. Hence the mean-time-to-failure (MTTF) of the power plant is equal to λ; this is also known as the expected survival time. The probability of a plant surviving at time t in a constant failure rate environment, i.e., its survival function, denoted by

R t), is then obtained by integrating the failure density function, f x , and is given by R t

94

exp −x/λ . The complement of this survival probability is the probability of failure in time t,

given by 1−exp −x/λ .

Similarly we assume an exponential model with parameter µ for the time-to-repair variable Y, so that the density function of Y, viz., the repair density function, g y , is

g(y) =

− y  , for y > 0. exp µ  µ  1

….(6.3)

In this model, 1/µ is the constant repair rate and its reciprocal, µ, is the mean down (repair) time (MDT) or the expected outage time. The sum of MTTF and MDT is termed the mean-timebetween-failures (MTBF) or cycle time.

Shooman (1968, Chapter 6) and Gnedenko, Belyayev and Solovyev (1969, Chapter 2) have shown that for the above exponential models, the instantaneous availability of a power plant is A(t) =

λ

µ

 1 1  exp − ( + )t  . µ  (λ + µ ) (λ + µ )  λ +

….(6.4)

The steady-state availability is obtained by taking the limit of A t as t approaches infinity. This gives

A(∞) =

λ (λ + µ )

=

MTTF . MTBF

….(6.5)

Corresponding to these availability measures, we can also define two down-state probabilities, instantaneous forced outage, denoted by R t and steady-state forced outage, denoted by R ∞ (Pillai,1992, Chapter 4). Thus the instantaneous forced outage rate of a plant is

95

R(t) =

µ (λ + µ )

+

λ

 1 1  exp− ( + )t  , (λ + µ )  λ µ 

….(6.6)

and the long-run (steady-state) forced outage is

R(∞) =

µ λ+µ

=

MDT . MTBF

….(6.7)

Now in the context of a Markov chain, we can find that the instantaneous availability and instantaneous forced outage rate, as obtained above, are nothing but the same state transition probabilities. So we turn to a discussion of Markov chain.

6.3 Markov Chain

Since the turn of the last century, probability or non-deterministic models have been increasingly recognised as more realistic than the deterministic ones in many contexts. The probabilistic approach to time series analysis has thus led to ‘dynamic indeterminism’ (Neyman, 1960). Physicists have played a leading role in its development; and the innumerable variety of its applications in the realms of physical, biological, economic, social and behavioural sciences has made the approach all the more significant. Markov chain models are useful in analysing situations where the fluctuations of a process are either up or down or constant at a time compared with the preceding period. The scope of its applications has been on the increase in almost every context: among others, in the stellar dynamics and solid state physics (Chandrasekhar, 1943); in hydrologic time series simulation models (Moran, 1959; Klemes, 1970); in meteorological predictions (Gabriel and Neumann 1957, 1962; Medhi, 1976); in changes in the distribution of farm sizes (Krenz 1964); the movement of rental housing areas (Clark 1965); the land-use zoning (McMillen and McDonald 1991); market research in consumer brand-switching behaviour (Whitaker, 1978); occupational mobility (Blumen et al. 1955; Sampson 1990); quality test processes at different stages of manufacturing (Clark and Disney 1970); income and wage distributions (Solow 1951; Champernowne 1953); size distribution of 96

firms and business concentration (Hart and Prais 1956; Adelman 1958; Steindl 1965); in the dynamics of inter-regional savings and capital growth (Richardson 1973); convergence tendency in labour productivity (Temel et al. 1999; Webber 2001) and in income (Kawagoe 1999; Cheshire and Magrini 2000); income distribution dynamics (Dardanoni 1995); the forecasting of the extent of the HIV/AIDS epidemic (Harris et al. 1992; Sendi et al. 1999) and of tuberculosis (Debanne et al. 2000); the random walk modelling of stock prices (McQueen and Thorley 1991); the exchange rate regime transitions (Masson 2001); the diffusion of reserves and the money supply multiplier (Tsiang 1978; Brody 2000); the monopoly game (Ash and Bishop 1972; Stewart 1996); in the faculty replacement strategies (Hackett and Magg 1999); in the options pricing (Duan and Simonato 2001).

Pillai (2002 c) applies Markov chain for the first time to monthly price movements in India. He modified the method and results (Pillai 2006) by proposing a new methodology, both conceptually simple and computationally easy, for estimating the steady state probabilities of a Markov chain from sample micro-unit data when the number of states considered is very large. His study also includes a novel interpretation of the dynamics of state transition and steady state probabilities in the framework of an error correction process.

The vast scope of applications of Markov chains in diverse fields has initiated many studies into the inference problems, such as estimation and hypothesis testing, about Markov chains (see Anderson and Goodman, 1957; Billingsley, 1961; Lee, Judge, and Zellner, 1970; Collin, 1974). Different methods have been suggested for estimation of transition probabilities under different situations – for instance, one based on linear and quadratic programming procedures to obtain least squares estimates (Lee, et al., 1970) and another on maximum likelihood (ML) method to estimate transition probabilities from individual or micro-unit data (Anderson and Goodman, 1957; Collin, 1974). We make use of the definition of the ML estimator in this study. Consider a time-homogeneous Markov chain with a finite number, m, of states (1, 2, ……, m) and having transition probability matrix P

Pij), i, j = 1, 2, ..…, m. Let the number of observed

direct transitions from the state i to the state j is nij and the total number of observations be N ( see Fig. 1). The transition probability, Pij, the probability of transition from the preceding state i 97

(that is, St – 1

i , to the current state j, (St

j), is the conditional probability of the current state

j, given the information on the previous state i, that is P j | i = P{St = j | St – 1 = i},

….(6.8)

which is given by

P( j | i ) = P( j , i ) / P(i ) .

….(6.9)

Now the ML estimate of the joint probability

P( j, i) = nij / N ,

….(6.10)

and the mean probability of a certain previous period state i (or the marginal probability)

P ( i ) = ni • / N ,

….(6.11)

m

where ni• = ∑ nij , the number of times a particular state i is occupied irrespective of the current j

state, that is, the corresponding row sum (see Figure 1). This gives the ML estimate of transition probability Pij, denoted by Pˆij

Pˆij = nij / ni• ,

……(6.12)

98

Figure 6.1: Frequency Distribution of Transitions from the Previous State i to the Present State j

1

1

n11

Previous

2

n21

state

:

:

i

i

:

m Total

where ni.

ni1 :

nm1 n.1

2

n12

Current state j …. ….

j

….

n1j

….

m

Total

n2.

n1m

n1.

n22

….

n2j

….

n2m

ni2

….

nij

….

nim

ni.

nm2

….

nmj

….

nmm

nm.

n.2

n.j

n.m

N

m

∑ nij . This ML estimator (6.12) is consistent, but not generally unbiased (Kendall j =1

and Stuart 1961: 39-40, 42); however, as the sample size increases, the bias tends to zero (Kendall and Stuart 1961: 42). It is also shown that the estimates are asymptotically normally distributed (Kendall and Stuart 1961: 43-44; Anderson and Goodman 1957: 95). Given the ML estimates, Anderson and Goodman (1957: 96-103) also provide likelihood ratio tests and χ2 tests for testing various hypotheses.

Given the sample estimates of the state transition probabilities, we derive the steady state (long run) probabilities as the limiting values reached after a large number of transitions. These state probabilities are independent of the initial state j. This means that regardless of the initial state of price changes or its probability vector, we could predict, subject to the underlying assumptions, the probabilities which the states will eventually take for the system to settle down and become stable. A direct solution for finding the limiting state probabilities follows from the following fact: if the state probability vector has attained its limiting value, say π, it must then satisfy the equation

99

π

Pπ Pπ

…….(6.13)

where P is the (one-step) state transition probability matrix. This enables us to solve the steady

state vector π. In order to estimate the π i), i = 1, 2, …, m, a set of m simultaneous equations

is obtained from equation (6.13), but this dependent set of m equations has an infinite number of solutions. We can have m independent equations if one of these is replaced by

m

∑ π (i) = 1 .

…….(6.14)

i =1

6.4 A New Estimator for Steady State Probability When the number of states, m, becomes larger (more than 2 or 3), solution of the simultaneous equations involves computer costs that increase with m. Though a simple graph theoretic formula for finding the steady state probabilities is available (Solberg 1975), this too requires computer cost for larger m. Remember these methods make use of one-step transition probabilities in the computation of steady state probabilities. However, here we propose a very simple and easy method of computing long run probabilities from sample micro data arranged as in a contingency table framework. Our new estimator of the steady state probability essentially follows from the simple fact that the corresponding marginal probabilities are equal. The long run implies identical probability for the current as well as for the previous state; i.e., Pj

Pi, which in turn

suggests that the corresponding marginal totals are equal. For example, in Figure 6.1, nij gives the number of observed direct transitions from the previous state i to the current state j as culled

out from a micro data set, and the marginal total, n..j, is the number of times a particular state j is occupied irrespective of the previous state. What distinguishes such a table from the usual contingency table is that in general

100

nij

nji ∀ i, j = 1, 2, …, m,

…. (6.15)

such that34

n..j = ni..

…. (6.16)

Tables 6.1 and 6.2 given below illustrate this property.

Now in analogy with a contingency table, assuming the marginal totals to be fixed, the probability of any state j is given by

Pj

n..j/N

ni./N

Pi.

…. (6.17)

We can easily prove that this is the expected (and hence steady state) probability. We know that the theoretical frequency corresponding to nij is obtained as

nij*

n..j × ni./N,

…. (6.18)

and the associated probability estimator, from (6.12), is

Pij*

nij*/n..j

Pj

…. (6.19)

such that the given state probability also is the expected probability. The vector [Pij*] gives the steady state probability vector [Pj]. Thus

Pj

n..j/N

…. (6.20)

gives the long run probability for any state j = 1, 2, …, m. Below we prove it for two particular cases of j = 2 and 3.

34

In some cases, it might be that is insignificant for large N.

nij = nji – 1 or nij – 1 = nji and n.j = ni. − 1, or ni. = n.j − 1.The difference 101

Now with reference to Fig. 6.1, the unconditional or marginal probability of current state j,

denoted by P j is the probability Pr{St estimate is given by P j

j}irrespective of the preceding state i, and its ML m

n..j/N, where n• j ≡ ∑ nij , the number of times a particular state j is i =1

occupied irrespective of the previous state. Note that this is also the expected probability: we know that in a contingency table as Fig. 6.1, the theoretical frequency corresponding to nij is

obtained as nij*

Pij*

nij*/ni.

n..j × ni./N, and the associated expected probability estimator, from (6.12), is

P j . And this must also be equal to the steady state probability of Markov chain.

Indeed it is so in a special case of Markov chain defined particularly for most of the economic time series data arranged in a contingency table framework of current versus previous states. Here we find, for example, with reference to Fig.1, that P j and ni • ≡

P i , for i

j, where P i

ni./N

m

∑ nij .

This is easy to explain in terms of the general result in economics that the

j =1

long-run equilibrium is independent of time such that in our case Pr{St equilibrium, and thus P j

P i , for i

above, in turn, implies that n..j

ni., for i

j}

Pr{St – 1

i}in

j. Now this, together with the definition of P j given j, and hence nij

nji ∀ i, j = 1, 2, …, m.

This result might appear strange, but is as true as a mathematical regularity in a special case of a contingency table of time series data on current versus previous states. For an illustration,35 a part of our data set is reproduced in Table 6.1, which shows the monthly FOR of the first unit of the Pallivasal power plant for the decade of the 1980s. We consider two states of nature of positive FOR (St = 1) and zero FOR(St = 0).

35

For similar empirical results in meteorology, see Gabriel and Neumann (1962) and Medhi (1976).

102

Table 6.1: Monthly FOR of the First Unit of the Pallivasal Power Plant and the Corresponding States (for the 1980s) Monthly FOR % Current 1981-82

April

0.28

June

0.15

August

1.44

May July

September

1982-83

0.34

0.55 0

0.55

0

February

1.47

1.47

0

December January

February March

0

0

June

November

0

5.18

0.41

October

0.18

3.74

April

September

0

0.05

5.18

August

0.13

0.18

December

July

0

0

September

May

0

0.54

0.05

March

0.5

0.13

July

January

1983-84

0.16

0.34

November

0.15

0

May

October

0.12

29.61

0.54

August

0

0.5

March

June

1

1.44

0

April

0

0.15

February

3.74 0

0

0.41

0

0

0 0 0 0 0 0 0 0

St – 1 0

0.15

0.16

St

Previous

1

0.12

December

Current

0

0.28

29.61

January

Previous

0

October

November

State of the Nature

0 0 0 0 0 0 0 0 0

0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0

103

Monthly FOR % Current 1984-85

April

0.88

June

0.79

August

0.14

May July

September October

November December

1985-86

0

0.88

0.51

0.79

0.33

0.14

0.19 0

0.35

11.88

March

0.3

0.2

0.51 0.33 0.19 0

0.35

0

11.88

April

2.24

0.3

June

0.3

May July

August

September October

November December

2.24

0.28

0.3

0.28

0

0

0 0 0

March

100

May

June July

August

0 0

100

100

100 100 100 100

November

0

100 100 100 100 100 100

0

5.68

0

0

January

2.96

March

0

February

0

72.28

5.68

December

0.37

100

September October

1.1

0.37

72.28

April

0

1.1

January

February 1986-87

Previous

0.2

January

February

State of the Nature

0.92

0 0

2.96 0.92

Current

St 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0

Previous

St – 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1

104

Monthly FOR %

1987-88

Previous

Previous

April

0.28

0

1

0

June

0

0

0

0

May

August

0

0.28

0

0

0

September

1.41

November

0

October

December January

February March April

0

0

0 0 0

0.97

July

0.59

September

April

November December January

February March

5.11 0.1

0.02

0.03 0 0

1.56

October

0.01

0.61

July

September

0.97

13.71

0.18

August

0

0.28

May

June

0

0.59

0.1

0.02

March

0

5.11

December February

0

94.49

13.71

January

0

0.01

October

November

0

1.41

94.49

August

0

0

May

June

1989-90

Current

Current

July

1988-89

State of the Nature

0.28 0.61 0.03 0 0

0.12

0.18

0.16

1.56

1.51 0.96 0.02 0.02 0 0

5.12

0.12 0.16 1.51 0.96 0.02 0.02 0 0

St 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1

St – 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0

105

Monthly FOR %

1990-91

State of the Nature

Current

Previous

April

0.08

5.12

June

36.92

6.6

August

0.98

May July

September October

November

6.6

36.92

0.14

0.98 0.14

0

February

0

March

8.99

0.59 3.31

January

0.08

8.99

December

Current

0.59 0

0.58

3.31

0

0

0.58

Previous

St

St – 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

0

1

1

0

1

0

0

Note: St = 0 for nil FOR, and = 1 for otherwise.

A cross tabulation of the current (St) versus previous (St–1) price inflation states yields a 2 x 2

contingency table (Table 6.2) that confirms the above result: n..j where n10 is the total counts of the event {St

0 | St – 1

ni., for i = j, and n10

1} and n01 that of {St

n01,

1 | St – 1

0}. It is not difficult to see that this result is due to the close association between n10 and n01.

When St

St

1 in the case of n01, then St – 1

0 in the case of n10, then St – 1

1 for n10 becomes predetermined. Similarly, when

0 gets determined for n01. This regularity almost results in

n10 = n01, as in Table 6.2. In some cases, it might be that nij

nji ± 1, ∀ i, j = 1, 2, …, m, and

n..j = ni. ± 1, for i = j. The difference is insignificant for large N. We get invariably similar results for any number of states of price change.

For example, Table 6.3 reports the short-run state transition probabilities and the long-run steady state probabilities, estimated using our method from Table 6.2(b).

106

Table 6.2 6.2 a : Frequency Distribution of FOR States Based on Table 6.1 6.1

Previous Month's FOR % Total

R

R

0 0

Current Month's FOR 0 0

28

0 < R ≤ 100 1

Total

0 < R ≤ 100 1

17

17

45

75

120

58

45

75

Table 6.2 6.2 b : Frequency Distribution of FOR States Based on Table Table 6.1 6.1

R

Previous Month's FOR % Total

0

Current Month's FOR %

R

28

0 < R ≤ 10

0

0 < R ≤ 10

10 < R ≤100

41

5

16

16

10 < R ≤100

1

5

45

62

Total

1

45

7

13

62

13

120

Table 6.2 6.2 c : Frequency Distribution of FOR States Based on Table Table 6.1 6.1

R

0

Previous

0 < R ≤ 0.5

FOR %

1 0,     2  2 n / 2 Γ( n / 2 )  1

we find the joint density of these two independent chi-square variables as

 1 U n −1V n −1 exp  − (U + V  2 2 2 n ( Γ n )2 1

 )  , U, V >0. 

……(A3)

Now using the transformation technique and (16), which gives ∂U 1 ω = V, ∂R (1 − R)2 ρ

(V is taken here as a constant; in (A3) V or U does not affect each other thru R.) we get the joint density of R and V:

n

 1 ω    R n −1 (1 − R ) − n −1V 2 n −1 exp −  2 2 2 n ( Γn )2  ρ  1

 R ω  1 + V  , 1 − R ρ   

0