Reliability pricing of electric power service: A probabilistic production cost modeling approach

E,trrgv Vol. 21, No. 2, pp. X7-97, 1996 Copyright 0 1996 Elsevw Science Ltd PrInted in Great Britain. All rights reserved (X60-5442/96 $15.00 + 0.00

Pergamon

RELIABILITY PRICING OF ELECTRIC POWER SERVICE: A PROBABILISTIC PRODUCTION COST MODELING APPROACH YOUSSEF tEnergy Parkway,

Management NW. Atlanta,

HEGAZY?

and JEAN-MICHEL

GULDMANNfg

Associates, Utilities Division of EDS, 100 Northcreek, Suite 800, 3715 GA 30080 and $City and Regional Planning, The Ohio State University, 17th Avenue, Columbus, OH 43210, U.S.A. (Received

30 June

Northside 190 West

1995)

Abstract-We develop a pricing model for electric power service that accounts for customers’ reliability preferences as well as the randomness of outages in the power-supply system. The model combines features of both reliability pricing and real-time pricing. A production cost simulation submodel is used to estimate expected total and marginal production costs on an hourly basis, as well as system reliability measured by the loss-of-load probability. These estimates are inputs to a welfare-maximization model with revenue constraints differentiated by customer class. The methodology is illustrated by a numerical application with typical power system and market data. The results show that the proposed pricing approach is significantly superior to spot and Ramsey pricing in terms of economic efficiency, energy conservation, and generation-capacity requirements.

I.

INTRODUCTION

electricity pricing models have been developed recently that allow customers to choose among different levels of service reliability, with the authors arguing that such a choice results in improved efficiency and benefits to both customers and utilities. I4 Reviews of the models, including other related issues (e.g. real-time pricing, peak load pricing, interruptible/curtailable dispatching), may be found in Refs. l-8. Our model differs from this research stream in our attempt to overcome shortcomings of these earlier models pertaining to the nature of the supply cost functions and complexity of pricing structures. The first issue is that supply systems have been customarily represented by predetermined cost functions independent of any reliability parameter, thus failing to represent the random nature of powersystem operations and interruptions. In reality, an electric power network is a highly complex and operationally uncertain system. At any moment, there is a possibility of total or partial outage leading to immediate changes in system topology and, hence, in the form of the cost function. For example, to represent a supply system with N generating units accurately, one needs 2N different cost functions to represent the actual supply system, each function representing a different possible system configuration because each unit is either available or out of service. To account for such factors, we introduce a production-cost simulation model that estimates the expected system marginal costs and the associated reliability levels, while accounting for all possible random changes in supply and demand. The second issue is that the proposed pricing structures are cumbersome to implement. The commonly used paradigm to characterize consumers’ choices is that willingness to pay reflects differences in tastes across the population of consumers, leading to optimal schedules that are implicitly defined, and to a continuum of options’ which represent an extremely difficult scheme to implement from a dispatching and operational point of view. In contrast, we introduce here a customer’s choice model based on the idea that the reliability of service represents the risk involved in the customers’ benefit from it, and we propose a pricing mechanism that combines priority (reliability differentiation) pricing with real-time pricing of electricity. Under this scheme, customers with different reliability preferences are charged different prices. The utility is assumed to be a regulated, welfare-maximizing firm, that has direct control over customer loads and applies a rationing method to customers willing to accept power interruptions. Customers can choose either to be served with a high-reliability firm service or to be subject to interruption. The first choice means that customers are served at all times with loo’% continuous service. The Several

$To whom all correspondence should be addressed. 87

88

Youssef Hegazyand Jean-MichelGuldmann

second choice means that they are allowed to select their preferred interruption level (probability and magnitude of interruption) in advance from a priority-price menu provided by the utility. The model is applied to the case of four customer classes, with actual and synthetic data pertaining to price-demand relationships, outage costs, consumption patterns, and reliability preferences. These applications enable us to examine the welfare gain and energy and reserve savings resulting from alternative pricing schemes such as spot and Ramsey pricing, in addition to reliability-based pricing, and thus to carry out a comprehensive assessment of the proposed approach. The remainder of the paper is organized as follows. The modeling methodology, including the customer choice, supply cost, and welfare maximization/pricing models, is presented in Section 2. The data used to apply the methodology are discussed in Section 3, and the results of its application are analysed in Section 4. Section 5 contains conclusions and outlines for further research. 2. MODELINGMETHODOLOGY 2.1.

Customer choice model

We first consider residential electricity customers and represent the impact of service reliability on customer satisfaction by assuming that unreliable service directly reduces utility. If R is the probability that the amount X of electricity is served, the utility is U = U ( X , R , . . ) with OU/OR > 0. The rationale that reliability is a desirable attribute of electricity is just the same as the rationale that efficient fuel burning is a desirable attribute of a car. Consequently, greater reliability increases the utility obtained from any given electricity consumption. Electricity is an experience good with regard to reliability of service. However, it is not always the case that reliability is learned after electricity is bought. Learning in this case is a stochastic process because service may or may not be interrupted. For such an experience good, the main issue is information. The probability of interruption is the most important piece of information that a consumer needs in advance, and it is the most developed index a power company can provide. The consumer's utility function, therefore, is expressed as U(X,R) = R . U,(X, . .) + (1 - R ) .

U2(X, . .) .

(l)

The first term of the rhs U, represents the satisfaction in consuming the amount X of electricity with reliability R. The second part is the loss or dissatisfaction caused by an interruption of X with a probability 1 - R. We assume a quadratic, additive, quasi-linear with respect to the composite good, utility function with no income effect, i.e.

U, = R, (a, X, - ~ - - )

- ( 1 - R,) . X, OCt + m

(t = 1, ..., 24),

(2)

where U, = utility function of the residential class at period t, X, = residential class demand for electricity at period t, R, = reliability level or probability that the load will be served at period t, OC, = customer class outage cost at period t, a,, b, = parameters of the utility function, m = composite (numeraire) good. Equation (2) leads to the following inverse demand function for the residential class: P,(X,, R,) = R , . (a, - b, X,) - ( 1 - R , ) . OC,

(t = 1, ..., 24).

(3)

We next consider the second type of electricity customer: the firm. We assume a short-term-rationale competitive firm whose benefit from electricity usage at any time depends on consumption at that time only, implying that the profits are maximized at each time t independently. The firm produces an output q, sold at price r,, and it has a set of non-electricity inputs denoted by the vector L, with prices w,; and an electricity input X, with price P,. The firm has a production function q, = h,(X,,L,), and maximizes its profits, given factor and output prices, and its production technology. The profit maximization problem is analytically separable into two problems. First, find the cost minimizing combination of nonelectricity inputs L, for producing any given output using a fixed amount of electricity. This gives rise to the demands L,(q,,w,,X,) for the non-electricity inputs, and hence to the cost function C*,(q,,w,,X,),

Reliability pricing of electric power service

89

representing the cost of purchasing those non-electricity inputs. Second, produce the output level q, and select the amount of electricity X, that maximizes profits: II, = r,h,(X,,L*,) - C , ( q , , w , , X , ) - P,X,. L e t V, be the value-added or benefit function for the firm's use of electricity at time t. It is equal to the firm's revenue minus the cost of all non-electricity variable inputs, with: V , ( X , , w , , r , ) = C , ( q , , w , , X , ) . Assuming that the firm is facing a competitive market for its output and all non-electricity inputs, V, can be expressed as a function of the electricity input only. Then maximizing profits is equivalent to (4)

Maximize II(XI) = V,(X,) - P,X, ,

which yields the identity OV,/OX, = P,, where P, is the price-demand function of electricity at time t. In order to estimate the consumer surplus of a firm consuming electricity, the demand function for the good produced by the firm must be estimated. As a practical matter, it may be difficult to obtain the necessary data for estimating the demand for the output of industrial or commercial firms. However, Schmalensee 9 has demonstrated that valuations of consumer surplus are identical, using either input or output markets, provided that the output market is competitive. Under the assumption that the framework of reliability impact on consumers holds for firms and that V,(X,) has the same quadratic form as is assumed for the residential class utility function, we use the following demand function form for any non-residential class i: Pi,( X~,,R~,) = Ri, • ( a~, - b~,Xi,) - ( 1 - R~,) • OC/t 2.2.

Supply-cost

(t = 1, ..., 24).

(5)

model

The proposed stochastic production cost simulation extends the method developed by Rau et al n°'~j to include estimation of the expected marginal cost (the probability distribution of system incremental cost) at each level of demand. The concept is illustrated in Fig. 1, where the probability density function (PDF) of loadflLo) is shown in the top right-hand side. The probability of outage of a generation unit f ( C ) is represented in a binary fashion on the left of the vertical line. The forced outage rate (FOR) of the ith unit is qi, and pi is its availability. A capacity of [-C] is shown to be available with probability p to reduce the demand. If no unit is available, the PDF of the load is the unsatisfied demand (residual demand RDo), and the zeroth moment of the PDF is the probability that the demand is unserved ( 1.00 in this case). If only unit #1 were to be considered, the convolution of its binary representation f l C ~ ) with load would result in the density shown in Fig. l(b). That density, compared to the PDF of load, is shifted to the left by Cn, due to the negative sign of capacity in its binary representation, meaning that a maximum of C~ can be reduced from the residual demand RDo by the operation of unit #1. In Fig. 1(b) the density to the right of the vertical line illustrates the PDF of residual or unmet demand (RDj). Similarly, the density shown in Fig. l(c) is obtained by considering the second generation unit in the merit order of loading. Rau and Necsulescu n° have shown that the difference between the means of any two residual demands is the mean (or expected) value of the energy production from the relevant unit. For example, the difference between the means of the densities to the right of the vertical line in Fig. l(b) and l(a) is the mean energy generated by unit #1, and the corresponding difference in Fig. l(c) and l(b) represents the mean energy generated by unit #2. A repetition of this procedure gives the expected production from each unit. The expected production cost can be obtained by multiplying the expected production from each unit by its generation cost. We extend this analysis to obtain the expected marginal cost of the system by estimating the probability distribution of system lambda (incremental cost). The zeroth moment of the density to the right of the vertical line in Part (a) of Fig. 1 is the probability that the demand is unserved before committing any unit, and the zeroth moment of the density to the right of the vertical line in Part (b) is the probability that the demand is unserved after committing unit #1; the higher the value of this probability, the higher the probability that the system needs more units to be committed in order to serve the residual demand. Therefore, the difference between the zeroth moment of the densities to the right of the vertical line in Parts (b) and (a) is the probability that unit #1 was needed to cover the total residual demand RDo, or the probability that unit # ! is the marginal unit. If the residual demand is larger than the available capacity of the upcoming unit, then the unit is not the marginal one (and the probability is zero). On the other hand, if the available unit capacity is greater than or equal to the residual demand, this probability is larger than

90

Youssef Hegazy and Jean-Michel Guldmann

I

I

f(G) PDF of unit 1

P2

! I

Ii

Probability

qa

f(Lo)

PDF of Load

O

a

•

I

Load

1

q:

I,

t

"C2

f(L) = F(LO) x f(C,) o

b

ECI, Excess

Capacity

i

'1

! I f(L) = f(Lo) x f(C,) x f(C~ I I I

C ~., ~.,.'f~.." ~.i,,~:~'.1.~'i,.;,

}~,'~):,,:j~,

:i i

Fig. 1. Convolution of residual demand and unit availability.

zero and represents the probability that the incremental cost of this particular unit is the system incremental cost. The list of these probabilities represents the probability distribution of the system marginal cost. The summation of the products of each unit incremental cost by the probability that the unit is the marginal unit gives an accurate estimation of the expected value of the system marginal cost. The mathematical description of these processes is presented in detail in Hegazy.s In summary, the supply cost model provides estimates of the expected total production cost, the expected marginal cost, and the system reliability level at each level of consumption and time period. These parameters are defined as follows: RDi = residual demand after committing generating unit i, LOLPi = loss-of-load probability after committing unit i [= the zeroth moment of (RDi_~ - RD;), i being the last unit committed], LOLP(X) = system loss-of-load probability at load level X (= the zeroth moment of RDn, n being the last unit available in the system), IQ = fuel cost of unit i, TC(X) = expected total production cost at load level X {= X~ IC~ [the first moment of (RD~_~ -Rd~)] }, MC(X) = expected marginal cost at load level X [= X~. IC~. (LOLP~_t - LOLPg)]. The major advantage of this algorithm is that it uses the probability density functions (PDF) of demand and unit availabilities. Since the PDFs can be obtained for any random distribution, it is possible to simulate demand and supply systems with any contingency form.

Reliability pricing of electric power service

91

2.3. The welfare maximization model We consider a regulated electric utility assumed to be welfare maximizing, with, as criterion, the sum of consumers' and producer's surpluses. This utility is allowed to consider an additional dimension of service, namely its reliability, and to serve its various customer classes with different reliability levels, in line with each customer class's own choice. No regulatory oversight is assumed to take place over reliability, although there is such oversight with regard to its cost allocation implications. The utility is in charge of designing the menu of choices which fits its system-specific limitations. To allow for service reliability differentiation among customers, we rank customer classes into two groups: a high-reliability customer group and a low-reliability one. High-reliability customer classes are assumed to opt for firm power service, that is, a 100% continuous service. Customer classes in the low-reliability group are ranked in the order of the interruption level they would opt for. In any outage situation, the utility interrupts low-reliability customers in accordance with their chosen rank. The mechanism to encourage customers to make rational choices is such that, while high-reliability customers pay for the high quality firm services they opt for, their incentive to make this choice is that their payment is less than the expected outage costs they may suffer if they are not protected. In turn, lowreliability customers are compensated by receiving payments higher than their expected outage costs. The payment/compensation mechanism is designed to ensure that the electric utility maximizes social welfare within the regulated revenue constraint, and that customers are not cross-subsidizing each other. The value of the payment/compensation for any kWh of outage that fits the specified criteria is then the average value of all customers' outage costs, defined as the social outage cost SOC, and set equal to the sum of all customers' outage costs divided by the total number of customer classes. Because the high-reliability customers are protected from outages, the total expected shortages must be allocated to the low-reliability customers. The total expected shortage, at any hour, is estimated by using the production cost simulation model, and is taken as the total expected demand at that hour multiplied by the loss-of-load probability. Assuming that shortages happen mainly as a result of excessive demand (excessive demand leads to excessive use of generating units, which leads to higher probabilities of unit failures), we divide the total expected shortages into: (a) the expected shortages due to the high-reliability demands and (b) the expected shortages due to the low-reliability demands. Then, at any hour, each customer class in the low-reliability group is expected to suffer an "obligatory curtailment" equal to its expected demand multiplied by the loss of load probability, plus a "voluntary curtailment" equal to a portion of the expected shortages caused by the high-reliability demands. The voluntary portion is, of course, the customer's choice. The payment made by the high-reliability customers is then the total expected shortages multiplied by the social outage cost. Each low-reliability customer, on the other hand, is compensated by a payment equal to its total expected curtailments multiplied by the social outage cost. This allocation mechanism minimizes the total expected outage damages for society. The regulated utility is entrusted to select the optimal set of prices (quantities) which maximize economic welfare subject to the revenue constraint, and to allow for reliability differentiation within the above incentive mechanism. We consider the general case where there are N customer classes, partitioned into two groups: classes 1, 2 . . . . . S, which opt for low-reliability service, and classes S + 1, S + 2 . . . . . N, which opt for high-reliability service. The problem is then to maximize the sum of the aggregate consumers' and producer's surplus, under the constraint that the total revenue must not exceed total costs. The total revenue constraint is then divided into two sets of constraints. The first set has S constraints, each allocating part of the required revenue to a low-reliability customer class, whose payment covers that class's share of the total variable and fixed costs of service (TC + FC) less a compensation payment. The compensation payment includes two parts: (a) one related to the expected average cost of interruptions caused (directly or indirectly) by the customer's own consumption, and set equal to [SOC x LOLP x customer class consumption], and (b) one related to the expected average cost of interruptions caused by the total consumption of the high-reliability group and set equal to [SOC x LOLP x total consumption of ( S + 1, S + 2 . . . . . N) classes] multiplied by or, which is defined as the share of "voluntary" interruptions that the customer class is willing to shoulder. It is a customer choice, and the sum of all c~ must equal to 1.0. The second set includes only one constraint, that allocates the required revenue to the high-reliability classes so as to cover their share of the total cost of service, plus a penalty payment equal to the total expected average cost of interruptions

92

Youssef Hegazy and Jean-Michel Guidmann

[SOC × LOLP × total demand]. Note that this compensation/payment mechanism is purely redistribu= tive (the sum of the compensatory terms over all customers classes is zero) and ensures that the utility covers exactly its fixed and variable costs. The mathematical formulation of the model for period t is then

Max W ( X t , , ..., Xi,, ..., XN,) = "~u=l

P~, (yi,,Ri,) dy

- TC, (Xr,) - FC ,

(6)

subject to the following (S + 1 ) constraints: Pk, (Xk,,R,) X~, = Xk, [TC,(Xr,) + F C ] / X r , - LOLP, SOC, [Xk, + Ctk," (Xk+,., + - - . + Xu,)]

(7)

for low-reliability customers' revenue constraints (k = 1 ---* S), and

ENi=s+1 Pi,

(Xi,,1) Xi, = ~Uk=s+l Xk, [TC,(Xrt) + F C ] / X . + LOLP, • SOCt • Xr,

(8)

for high-reliability customers' revenue constraints, where Xi, = demand of class i at time t (i = 1. . . . . N, t = 1. . . . 24), Xr, = total demand at time t, Pa,(X~,,RI,) = inverse-demand function for customer class i [= R i , . (ai, - b~,X~,) - ( 1 - R~,) • OCi,], LOLP, = loss-of-load probability (probability that the generation system is unable to serve the load at time t), R~, = level of reliability chosen by/made available to customer class i.t (For the high-reliability classes, R;,= 1. For the low-reliability ones, overall R, = 1 - L O L P , . However, specific low-reliability classes will enjoy higher or lower reliability level than 1 - L O L P , , depending on their choices for the parameter a). OCi, = interruption losses for class i, SOC, = average value of social outage costs (= 2~OC~,/N), FC = total fixed costs of the electric power system, TC, = total variable costs estimated at each load level by the production cost simulation model, a~, = an allocation parameter related to the level of interruptions selected in advance by customer class i; it represents the share of the voluntary interruptions that customer i class is willing to accept

(]~~=l c~= 1). The model defined by Eqs. (6)-(8) is a classical equality-constrained optimization problem, which is solved using the method of Lagrange multipliers with the variables X;, as basic unknowns. The resulting set of equations is solved iteratively using the Newton-Raphson algorithm. However, this procedure is further complicated by the need to use, at each iteration, the production cost simulation model to estimate the expected total production cost and marginal cost, and the loss-of-load probability (LOLP), which all vary with the total load. When the algorithm converges, a partial market equilibrium is reached, with prices reflecting customers' valuations of the service. Further details on this solution procedure are available in Hegazy. 8 3. DATA The model is applied to a power system identical to the IEEE reliability test system (Ref. 12), which' is well known to accurately represent a typical electric utility generation system with power plants of different sizes, types, costs, and availability rates, as presented in Table 1. The total installed capacity of this test system is 3405 MW. Four customer classes are considered: large industrial, small industrial, commercial, and residential users. Their 24-hour load patterns are obtained from a typical utility (Ohio Edison) and are scaled to match the IEEE generation system, with a total system peak load of 2850 MW, as presented in Table 2. These data are first used to calibrate the proposed demand functions, by relating the assumed loads to the resulting average costs of providing these loads. To do so, we first compute the variable cost of production using the production cost simulation model. The expected production cost of supplying the load is estimated at $12.63/MWh. We assume that fixed costs represent one-third of the variable pro-

tR = 1 - LOLP is a function of both total demand and system configuration. However, there is no closed functional form that can express that relation, thus its estimation through simulations. While R is not considered as a decision variable in our model, it varies with variations of other decision variables X.

93

Reliability pricing of electric power service Table I. Description of the generation system.

Unit size (MW) 400 350 197 155 100 76 50 20 12

Number of units Fuel type

Fuel cost ($/MWh)

Forced outage rate

Mean time failure (h)

Mean repair (h)

2 I 3 4 3 4 6 4 5

5.59 11.40 19.87 11.16 22.08 14.88 16.30 37.50 28.56

0.12 0.08 0.05 0.04 0.04 0.02 0.01 0.10 0.02

1100 1150 950 960 1200 1960 1980 450 2940

150 100 50 40 50 40 20 50 60

Nuclear Coal # 6 Oil Coal # 6 Oil Coal Hydro # 2 Oil # 6 Oil

Table 2. Initial hourly demand (MW). Class 1 = large industrial customers, Class 2 = small industrial customers, Class 3 = residential customers. Class 4 = commercial customers. Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Class 1 91.5 90.4 89.2 88.9 89.6 92.2 99.9 110.9 115.9 I 18.1 120.2 120.2 120.2 122.5 120.5 115.1 109.2 103.6 I 01.4 100.1 99.1 99.6 97.2 94.8

Class 2 734.8 726.1 729.2 716.8 711.2 733.6 787.6 851.6 873.4 880.8 888.3 878.3 887.1 890.7 870.3 831.7 808.8 787.6 785.8 774.6 769.6 784.5 778.3 754.1

Class 3 426.9 363.1 314.7 291.1 285.9 305.5 388.3 475.7 558.1 628.1 703.1 750.9 722.1 743.2 760.7 862.5 1058.5 1089.9 1032.8 874.3 858.9 708.2 634.7 511.7

Class 4 411.4 387.9 371.6 360.7 359.1 367.4 406.4 529.9 685.4 806.5 867.2 915.1 901.6 943.5 940.6 905.4 873.5 734.1 680.8 645.2 634.3 564.7 502.3 445.3

Total 1664.6 1567.6 1504.9 1457.6 1445.8 1498.8 1682.3 1968.3 2232.8 2433.4 2578.9 2664.5 2630.9 2700.0 2692.1 2714.8 2850.0 2715. I 2600.8 2394.3 2361.9 2157.2 2(112.6 1806.1

duction costs ($12.63/3), and we allocate them among customer classes according to their shares of the total system peak demand. The resulting average cost/price in $/MWh for each class is then: P~= 12.79 (large industrial customers), P c = 13.83 (small industrial), P3 = 14.49 (residential), P4 = 13.92 (commercial). These prices are assumed to be constant over the 24-hour period and are used with hourly demand elasticity data for each customer class to estimate the required demand function parameters a and b (under assumptions of perfect reliability, i.e., R -- 1 ). The hourly elasticities reported in Table 3 are based, with some adjustments, on elasticities reported by Pacific Gas & Electricity Company in its annual real time pricing report. Outages are assumed to occur without prior notification, with short and equal durations. Outage cost data for the four classes are drawn from studies and reports prepared by the Electric Power Research Institute (EPRI) over the last 10 years. Residential outage costs are drawn from Ref. 13, conveniently provided for each hour of the day. Cost information for the three other classes is taken from Ref. 14. Since these data were not provided on an hourly basis, we used the hourly distribution (percentage) curves, as provided in EPRI, ~4 to estimate the hourly outage costs for each class. The final outage cost data are presented in Table 4. EGY 21-2-C

94

Youssef Hegazy and Jean-Michel Guldmann

Table 3. Demand elasticities. Class I = large industrial customers, Class 2 = small industrial customers, Class 3 = residential customers, Class 4 = commercial customers. Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Class 1

Class 2 -0.70 -0.70 --0.70 -0.70 -1.00 -I.25 -1.25 -I.25 -I.50 -I.50 -I.45 -I.40 "1.40 -1.00 -1.00 -1.00 -1.00 -1.20 - 1.40 -1.20 -1.00 -0.70 -0.70 -0.70

-0.050 -0.081 -0.090 -0.101 -0.100 -0.087 -0.084 -0.069 -0.047 -0.054 -0.085 -0.085 -0.095 -0.064 -0.050 -0.036 -0.036 --0.036 -0.040 -0.040 -0.036 -0.042 -0.050 -0.050

Class 3 --0.81 -0.81 --0.81 --0.80 -0.81 -0.80 -0.80 -0.18 -0.15 -0.15 -0.15 --0.18 -0.20 -0.20 -0.19 -0.20 -0.11 -0.11 -0.08 -0.13 -0.14 -0.14 -0.14 -0.14

Class 4 -0.200 -0.200 -0.200 -0.010 -0.050 -0.011 -0.011 -0.020 -0.011 -0.011 -0.020 -0.020 -0.015 -0.010 -0.011 -0.010 -0.015 -0.015 -0.100 -0.100 -0.190 -0.170 -0.150 -0.190

Table 4. Hourly outage costs (t~/kWh). Class 1 = large industrial customers, Class 2 = small industrial customers, Class 3 = residential customers, Class 4 = commercial customers. Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Class 1

Class 2

14.91 14.91 14.76 14.49 14.76 14.76 14.76 14.49 14.49 14.49 14.76 14.76 14.76 14.49 14.49 14.49 14.49 14.49 14.49 14.49 14.49 14.49 14.49 14.49

14.44 13.98 14.14 14.16 14.16 14.16 14.68 14.38 14.06 14.38 14.74 14.74 14.84 14.68 14.54 14.52 14.42 13.50 13.50 13.50 13.50 13.50 13.50 13.50

4.

Class 3 9.51 9.58 9.56 9.92 9.54 9.81 9.90 10.03 19.26 10.43 10.61 10.70 10.97 10.41 11.59 10.70 10.08 9.36 8.47 7.85 7,58 7.49 7.40 7,31

Class 4 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 21.6 21.4 21.2 20.2 20.2 20.2 20.2 20.2 28.6 28.4 38.8 23.6 22.4 23.0 22.0 22.6

APPLICATIONS

We simulate the effects of three pricing schemes on the four customer classes: (i) spot pricing (SPOT), which reflects the optimal "first best" pricing, and corresponds to the maximization of social welfare [Eq. (6)] without any constraint; (ii) Ramsey pricing (RAMSEY), which extends SPOT by

Reliability pricing of electric power service

95

adding a unique revenue requirement constraint; and (iii) reliability pricing (RELIAB), which is a special case of the general model proposed in Section 2, with only two reliability options. In SPOT and RAMSEY, we assume that all customers are subject to the same level of reliability, that is, the overall system reliability level. In RELIAB, we assume that Classes 1 and 4 are opting for high (firm) reliability and Classes 2 and 3 for low reliability levels. Classes 1 and 4 are assumed to have more inelastic demands and higher outage costs than the other two classes. The two-options reliability model (RELIAB) for each period t (= 1 . . . 24) is formulated as follows: Max W(X1, ..., X4,) = Ei4,

Pi, (Yir,R,) dy

]

(9)

- TC, (Xr,) - FC ,

subject to P2, (X2,,R,) X2, + P3t (X~,,R,) X3, = [(X2, + X3,)/Xr,] [TC, (Xr,) + FC] - LOLP/• SOC, • X r , ,

(10)

P,, (X~,,1) X,, + P4, (X4,,I) X4, = [(XI, + Xa,)/Xr,] [TC, (Xr,) + FC] + LOLP, • SOC,. Xr, • ( 1 1 ) The criteria used to compare reliability pricing with spot and Ramsey pricing include (a) the impact on consumer surplus and total economic welfare, and (b) the impact on power demand patterns, whereby reductions in peak demand and total energy requirements imply lesser operating costs, capacity expansion, environmental pollution, and financial burden on the utility. The results are presented in Tables 5-7. The values of the evaluation criteria are presented in Table 5, including percentage deviations from the benchmark SPOT pricing case. RELIAB yields the highest economic welfare, total consumer's surplus, and energy savings, with SPOT ranked second and RAMSEY third. RELIAB peak demand is slightly lower than RAMSEY, though higher than SPOT. Reliability-based pricing, in the proposed form, appears uniformly superior to Ramsey pricing. Table 6 presents system hourly reliability levels, as measured by loss-of-load probabilities (LOLP), and confirms that the perfect reliability world does not exist. Reliability under Ramsey pricing is higher (smaller LOLP) than under reliability pricing, because prices under the Ramsey framework are generally higher (see Table 7), and, consequently, consumption is lower, and the production cost simulation model yields higher reliability for lower demand.t The differences are further accentuated by the fact Table 5. Comparison of SPOT, RAMSEY, and RELIAB pricing schemes.

Total economic welfare ($) Deviation from SPOT (%)

SPOT

RAMSEY

RELIAB

7,028,047 0

7,027,265 -0.01

7,061,132 0.48

280,855 82,449 629,968 5,714,371 6,707,645 0

280,126 80.718 638,466 5,739,956 6,739,268 0.47

282,956 69.893 658.371 5,762,500 6,773,722 0.98

2440.1 0

2458.8 0.77

2457.2 0.70

2464 14.344 14,753 15,066 46,627 0

2450 14,109 14,795 15,046 46,400 --0.52

2464 13.283 14.708 15,066 45.521 -2.40

Consumer surplus ($)

Large industries Small industries Residential Commercial Total

Deviation from SPOT (%) Peak demand (MW)

Deviation from SPOT (%) Energy consumed (MWh)

Small Industrial Large Industrial Residential Commercial Total

Deviation from SPOT (%)

tThis result follows because the forced outage rates (FOR) of generation units are assumed fixed, are not functions of the unit operating time, and all units are assumed to be available at the initial stage (t = 0). Using Markovian analysis to account for the status of each unit in relation to its operation history could improve the accuracy of the relations between reliability and demand levels.

Youssef Hegazy and Jean-Michel Guldmann

96

Table 6. Reliability under different pricing schemes. (Unit = LOLP x 103). Hour

SPOT

I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 0 0 0 0 0 0 0.15 0.78 2.50 4.90 6.70 2.80 11.10 6.95 8.06 22.60 11.30 6.39 2.75 1.45 1.00 0.15 0

RAMSEY

RELIAB

0 0 0 0 0 0 0 0.07 0.56 1.68 3.49 5.20 4.38 9.48 9.59 10.8 21.00 10.00 5.25 2.48 2.54 0.96 0.26 0

0 0 0 0 0 0 0 0.07 0.58 2.01 4.07 6.27 5.48 13.00 13.10 15.00 25.00 11.30 5.23 2.49 2.65 0.98 0.26 0

Table 7. Price menus of different pricing schemes ($/MWh). Class 1 = large industrial customers, Class 2 = small industrial customers, Class 3 = residential customers, Class 4 = commercial customers. RELIAB

RAMSEY

Hour

SPOT

Class I

Class 2

Class 3

Class 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

14.78 14.43 14.18 13.99 13.93 14.07 14.63 15.96 16.82 17.64 18.24 18.54 17.73 19.02 18.57 18.71 19.65 19.05 18.51 17.72 17.22 16.99 16.07 15.41

21.01 20.67 21.12 12.21 13.81 9.66 13.42 15.91 16.51 17.12 17.61 17.94 17.93 18.73 18.61 18.59 18.84 18.15 14.82 14.96 13.39 14.91 16.11 17.59

16.89 17.28 17.59 17.71 17.81 17.38 16.49 15.88 16.54 17.22 17.83 18.21 18.04 18.61 18.63 18.76 19.45 18.89 18.24 17.55 17.38 16.74 16.19 15.69

16.64 16.94 17.19 17.34 19.11 19.84 18.08 16.73 15.67 15.12 14.48 14.72 14.73 13.99 14.05 14.29 14.21 14.94 15.21 15.27 15.29 15.54 16.11 17.17

15.86 16.36 16.61 18.93 19.28 20.29 17.91 16.29 16.12 16.12 16.21 16.31 16.26 16.62 16.65 16.77 17.48 16.74 16.82 16.52 16.77 16.43 16.18 16.04

Class 1 26.1 25.5 26.4 14.9 18.6 15.3 15.6 16.1 16.3 16.8 17.1 17.3 17.4 18.1 17.7 17.4 16.8 16.4 13.5 13.1 I I.I 13.8 15.9 18.6

Class 2

Class 3

15.4 15.3 15.2 14.1 14.3 14.2 14.9 15.9 16.4 17.1 17.4 17.8 17.8 18.6 18.5 18.7 18.6 18.1 15.7 16.1 15.8 15.9 16.1 16.8

15.4 15.3 15.2 14.1 14.3 14.2 14.9 15.9 16.4 17.1 17.4 17.8 17.8 18.6 18.5 18.7 18.6 18.1 15.7 16.1 15.8 15.9 16.1 16.8

Class 4 17.7 19.1 19.9 25.1 24.4 24.5 21.1 17.0 15.5 14.5 14.1 13.8 13.8 12.9 13.0 12.6 12.3 12.4 16.3 15.7 16.3 16.1 16.1 16.3

t h a t , u n d e r R a m s e y , c u r t a i l m e n t s a r e a p p l i e d to all c u s t o m e r c l a s s e s w i t h o u t d i s c r i m i n a t i o n , w h e r e a s , u n d e r r e l i a b i l i t y p r i c i n g , t h e y a r e a p p l i e d o n l y to t h o s e w h o o p t f o r t h e l o w e r r e l i a b i l i t y l e v e l , w i t h t h e other customer classes served with firm power. Table 7 presents the price menus. Several points are worthy of note. First, RELIAB

yields higher

Reliability pricing of electric power service

97

prices for the high-reliability customers during peak and shoulder hours than RAMSEY. This result is consistent with the basic economic premises of this research, namely, that higher prices should be paid for higher-quality commodities or services. Second, RELIAB yields lower prices for the high-reliability customers during off-peak hours than RAMSEY. As shown in Table 5, the LOLP values during offpeak hours are zeros, implying perfect reliability. Therefore, reliability plays no role in the determination of prices during these hours. When reliability is 100%, the role of cost allocation becomes more important in price design. Since high-reliability classes have lower demands, their prices during offpeak hours become lower than they are under RAMSEY, where only the inverse elasticity rule affects prices. A similar analysis can be applied to explain price differences for low-reliability customers. 5. CONCLUSIONS We have presented a reliability pricing model for electric power service, that overcomes some problems associated with earlier models. First, the problem of accurately representing the electric power system contingencies and disturbances, has been solved by using a probabilistic production cost simulation technique that can account for any form of generation contingency. The technique is extended to estimate the expected value of system marginal cost associated with the expected total demand and reliability level. Second, the problem associated with the difficulties of observing the relationship between customer satisfaction and reliability level, is accounted for by assuming that customers' preferences are contingent upon the level of service reliability. The results of numerical applications of the methodology suggest that there is considerable advantage to unbundle power services in terms of reliability, and to allocate costs accordingly. Reliability pricing is uniformly superior to Ramsey pricing in terms of economic efficiency, energy conservation, and capacity requirements. We should also mention that the implementation of our pricing approach is technically very much similar to the implementation of real-time pricing, which recent advances in electronic metering and control devices have made feasible. This research could be extended in several directions. First, one could consider and incorporate transmission outages, in addition to generation outages, into reliability-based pricing. Outages due to transmission failures are very significant. Measuring transmission reliability is usually conducted using stochastic load flow and transient stability models. The integration of such models with a reliabilitypricing model would allow unbundling of service on a much larger scale. Second, one could use the history of a generating unit operations and failures when estimating its forced outage rate. The assumption that the FOR is fixed at all time is inaccurate over the long term, and the use of a Markovian process to represent changes in unit performance could enhance the model's applicability. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

H.-P. Chao, S. S. Oren, S. A. Smith, and R. B. Wilson, Energy Econ. 8, 199 (1986). H.-P. Chao, S. S. Oren, S. A. Smith, and R. B. Wilson, The EnergyJ. 9, 77 (1989). H.-P. Chao and R. B. Wilson, Am. Econ. Rev. 77, 899 (1987). S. A. Smith, J. Regul. Econ. 1, 203 (1989). "Priority Service: Unbundling the Quality Attributes of Electric Power," EPRI Report EA 485 I, Electric Power Research Institute, Palo Alto, CA (1986). M. A. Crew and P. R. Kleindorfer, The Economics of Public Utility Regulation, MIT Press, Cambridge, MA (1986). N. Viswanathan and E. Tse, lnternat. Econ. Rev. 1, 153 (1989). Y. Hegazy, "Reliability-Based Pricing of Electricity Service," Ph.D. Dissertation, Ohio State University, Columbus, OH (1993). R. Schmalensee, Am. Econ. Rev. 66, 239 (1976). N. S. Rau and C. Necsulescu, 1EEE Trans. PowerApp. Syst. 102, 2543 (1983). N. S. Rau and Y. Hegazy, IEEE Trans. Power App. Syst. 5, 367 (1989). IEEE, IEEE Trans. Power App. Syst. 98, 2047 (1979). "Environmental and Socioeconomic Consequences of a Shortage in Installed Generation Capacity," EPRI Report EA-2642, Electric Power Research Institute, Palo Alto, CA (1982). "Customer Demand for Service Reliability: A Synthesis of Outage Cost Literature," EPRI Report P-6510, Electric Power Research Institute, Palo Alto, CA (1989).