Estimating operational costs in an electric utility

O~.IEG4 The Int JI of Mgmt S,:~ Vol I0. No 4 ~ . 3"3 to 38-.2. 19~2 PrTntett an Great gru,t~n All r~ght, reserved

0305-O,~83 g2 0,.tO373-1050300 0 CopYright "~ 1932 Pergamon Press Lid

Estimating Operational Costs in an Electric Utility KW CAMPBELL Virginia Polytechnic Institute and State University, USA

FH MURPHY Department of Energy, Washington DC, USA

AL SOYSTER Pennsylvania State University, USA

(Receired July 1981: in reri.sed form November 1981) Generation expansion planning in the electric utility industry requires consideration of uncertainties in both the demand and supply of electric power. The expected demand is usually expressed via a load-duration curve, while, on the supply side, each generating unit has a given nameplate capacity and a predicted reliability. This paper focuses on coruideratioos of the supply.side uncertainties and their effects on estimating operating costs in electric utility planning. However, the methods and analysis developed in this paper may be applicable to a wider class of production planning problems which deal with any nonstorahle product with time varying demand. Two methods for estimating the energy generation from each generating unit are compared. The first is the method of probabilistic simulation, while the second involves a heuristic technique usually denoted the derating method. A bias inherent in the derating method is examined by comparing it with a prohabilistic simulation method. The bias is examined for various load curve shapes, in certain cases, a closed form expression for the bias is obtained. However, a closed form expression of the bias for an arbitrary load curve is difficult to achieve. In these situations some examples are studied in which the trend of the relative bias among plants in the loading order is examined. Finally, the bias is examined using actual 1977 load and supply data for some New England utilities.

INTRODUCTION

represents the cumulative number of units produced in [tt, t2]. For the example in Fig. 1, a production capacity with a m a x i m u m output rate of at least P is required to ensure that all demand is satisfied. However, with such a production capacity, note that much of it will be idle for substantial periods of time. An important example of a product that embodies the foregoing characteristics is electricity. The production and consumption of electricity occurs instantaneously; it cannot be (economically) stored. D e m a n d varies by time of day as well as the season of the year. Enough generating capacity is always maintained to meet the peak load. Due to the everincreasing air conditioning load, most US elec-

I N T H I S P A P E R, we compare two alternative methods for estimating the operational cost of satisfying the demand for some product that exhibits a time varying pattern of sales and one that cannot be inventoried. This latter characteristic means that the product must be produced at the precise instant of demand. Furthermore, since the demand is time varying, if one is to satisfy demand at all times, then the production capacity must be large enough to meet the peak demand. Figure 1 illustrates a typical case. Observe that since demand is measured in terms of a units per time, the area under the rate function in the interval [t t, t2] 373

Campbell, Murphy, Soyster--Estimatin9 Operational Costs in an Electric Utility

374

t~

t,,,

Time

FIG. 1. Time raryiny demand.

tric utilities incur the peak load demand during some midsummer afternoons, i.e. in Fig. 1, tv is most likely at or around 3 pm during some August afternoons. Another context in which this general framework is appropriate deals with the transport of information, people or things. Figure 1 could just as well describe the demand for long distance telephone service between two cities or the demand for buses in a transit district. Clearly, these demands for transport are a function of time and the product, in this case, is a service in which the production and consumption occur simultaneously, i.e. the service cannot be inventoried. The discussion thus far deals with the demand-side characteristics of the problem. On the supply-side, we assume that the product under consideration can be produced from N alternative sources and the unit cost of source n, n = 1,2 . . . . . N is a,. Assume that the sources are numbered so that at < a2 < . . . < au. Further, let C, be the capacity of source n (units/time) and denote by f ( t ) the demand at time t. At each instant of time t enough capacity must be allocated to satisfy f(t). To minimize the operational cost at time t one first allocates source i, then source 2, and so on until the demand is satisfied. In the case of electricity, this straightforward procedure is termed 'merit order dispatching'. At time t + At the allocation of capacity is appropriately adjusted. Note that for each t one first dispatches all capacity type n before dispatching any capacity type n + I. The same situation occurs with trucks and buses. Here one uses the newer, more efficient and reliable buses first to minimize transit delays. Less re-

liable and/or efficient vehicles are used during peak demands or when the better vehicles are inoperative. Since the context in which we have studied this problem is electric utilities, the discussion in the remainder of the paper is in terms of electric utility planning. The issue we address in this paper is well known in the electric power industry. Given a certain mix of facilities and a demand function, how does one estimate the operational cost of satisfying demand over some interval of time [0, T] when the N alternative sources are subject to failure. In particular, suppose the reliability of each source n is characterized by a single parameter q, which represents the probability that source n is down and unavailable for production (termed an outage in the electric utility industry). We asssume q,, n = 1,2 . . . . . N is applicable for all time t; furthermore, we assume that a failure of source n does not affect the status of sources m 4: n, i.e. the stochastic processes describing the state of each source (either up or down) are independent of one another. Given the stochastic nature of the supplyside facilities, the operational cost over a given time period is itself a random variable. At any instant t, the available sources may be a set [nl, n2 . . . . . nM] with a,, < a,: < ... < a,,,. In this case all capacity type nl is dispatched before any type n2 and so on. The deterministic question of estimating the operational cost over an interval of time [0, T] becomes the task of estimating the expected value of the operational cost in [0, T]. In the electric utility industry the problem of estimating operational costs has traditionally made use of the so-called load-duration curve. The load-duration curve is what results if the chronological demand (usually one entire year) of Fig. 1 is reordered to obtain a nonincreasing function. A typical load-duration curve is illustrated in Fig. 2. The rescaling of the time axis to unit length simplifies the subsequent discussion. To understand the straightforward use of the load-duration curve, consider an electric utility with N generating plants where, as before, q, represents the probability that plant n, n = 1,2. . . . , N is unavailable for generation, i.e. q, is the forced outage rate. (Other important considerations in the electric utility industry such as planned maintenance, spin-

Omega, Vol. 10, No. 4

375

L.

Peok,

MW Mm

MW

Time Time

FIG. 3. Merit order loading by nameplate capacities. FIG. 2. Load-duration curve.

ning reserve, unit commitments and other complicating factors are not considered in this paper.) The energy generation of a particular plant, say plant n, at any given time is dependent on three factors: (1) whether or not plant n is available, i.e. not on forced outage, (2) plant n's position in the merit order, and (3) the operating status of the ( n - 1) plants before plant n in the merit order. Each possible combination of operating plants defines a generating configuration. With N plants, there are 2 N possible generating configurations, each with a certain probability of occurrence. This probability of a given configuration is obtained in a straightforward manner as the product of the reliabilities of the operating plants and the forced outage rates for the nonoperating plants. The energy generation for each plant in a given configuration is obtained by stacking the plants in merit order along the vertical axis of the load duration curve. Each operating plant is assigned a horizontal slice of this curve. The width of the slice is equal to the plant's full nameplate capacity. A typical configuration is shown in Fig. 3. The area of the slice associated with each plant represents the plant's energy generation for that particular configuration. If 7~ is the probability of configuration j and A] is the area (energy) associated with plant n in configuration j, then the expected energy generation of plant n is

reliability of 0.90. The third plant in this system operates in one of the four generating configurations as shown in Fig. 4. The first configuration occurs when the first three plants are operating. Thus, :q = PlP2P3 = (0-90)3 = 0.729 and A 3 = 7 5 . The other (vi , ~ ' and f~A3', are computed in a similar fashion so that E3 = '~. -,jA~ =

(0.729)(75) + (0.081)(100) +

(0.081)(100)

(0.009)(100)

The method just described is conceptually straightforward, although in practice, the strategy of enumerating all possible operating configurations is not practical. However, this brute force enumeration strategy obviously results in

400 300

~23=1~ ~ "

200

I00 a t • 0.72 9

a2-0.081

,q.loo ].

2~

E. = y. =sAT. j=l

As an example, consider a system with five plants, each with capacity of 100MW and a

+

= 71.775 M W h r .

a3-0.081

':'4

• 0,009

FIG. 4. Possible configurations.for third plant in mertt order.

376

Campbell..~lurphy. So)'ster--Estimatiny Operational Costs in

unbiased estimates of the expected energy generation of each plant. Hence. we denote this procedure as the "expected value method'. The expected value method is more widely known as probabilistic simulation, a name which stems from the method used to compute the expected energy associated with each plant [2,3,4]. This method is a highly efficient, recursive routine that has been successfully used by utility engineers and is often incorporated directly into capacity planning models such as WASP [9] and O G P [10]. However, for the purposes of this paper, the computational advantages of probabilistic simulation over the brute force expected value method are not relevant. The derating method is a deterministic means of estimating the expected energy generation from each plant. In this method, the capacity of each plant is decreased, or derated, by a factor equal to its reliability. Thus, a 100 MW plant with a reliability factor of 0.90 has a derated capacity of 90 MW. As before, the plants are dispatched in merit order, but now the derated capacity is used to define the width of the horizontal slice associated with each plant. The resultant area associated with each slice is the generation estimate for that plant. Clearly, however, the plants which lie above the peak load will have zero area associated with them, and hence, for these plants the derating method yields an estimate of zero MWhr. Figure 5 illustrates the results of the derating method for estimating the energy generation associated with each plant of the five plant example used earlier. Obviously, the derating method is much simpler and easier to use than determining the actual expected values. Because of its simplicity, the derating method is used to approximate the operational costs in many capacity expansion models such as in M E M M [1]. However, the derating method has been shown in practice to provide a biased estimate of the expected energy generation [12]. For example, in Fig. 5, the derating method yields an energy estimate of 77.75 MWhr for plant n = 3. In contrast, the expected energy for plant n = 3, as discussed previously, is 71.775. As will be shown in this paper, the extent of this bias varies according to the plant's position in the merit order. In fact, the motivation, and to some extent the usefulness, of this paper

an

Electric Utility

lies in the establishment of what sort of systematic bias results from using the derating method in capacity expansion models. After an understanding of what bias occurs, one can develop a scheme to attempt to ameliorate any systematic error associated with derating. These subsequent design modifications are considered in [5]. A comparison of the derating method with the expected value method will be used to measure the bias associated with the derating method. For plant n, let D, be the energy estimate of the derating method. Similarly, as before, let E, be the energy associated with the expected value method from the nth plant. As a measure of the bias, consider the ratio, R,, defined by R. =

o. --.

E,

(l)

If R, > 1, the derating method overestimates the expected energy generation of plant n. Conversely, if R, < i, the derating method underestimates the expected energy generation of plant n. Hence, the value R, indicates how well the derating method approximates the true expected energy generation, with R, = 1 implying no bias exists. The main emphasis in this paper is the study of R, for various assumptions about load curves and supply parameters. The remainder of this paper is divided into three sections. The next two sections build progressively more sophisticated models of the load-duration curve and the bias ratio is then examined with each model. In the fourth

Plont 5 400 plant 4

58.25

Plant 3

77.75

Hont 2

90

Plent I

90

FIG. 5. Deratin(I method.

Omega. Vol. 10. No. 4

section, bias ratios are computed for a US utility using 1977 load-duration curves and supply data. The load curve approximation in the next section is linear, i.e. the load curve is simply a triangle. Clearly, with such a representation, the minimum (or base) load is zero. For this idealized situation, a closed form expression for the bias ratio is derived. This expression is one in which R, is written explicitly in terms of the peak load, the nth plant's capacity and reliability, and the sum of the previous (n - 1) plants' derated capacities. The load curve in the following section includes a minimum base load. Graphically, this is characterized by a triangle resting on a rectangle, i.e. a trapezoid. This general shape appears to be a reasonable approximation to actual load curves. The last section presents the results of measuring the bias for a US electric utility using 1977 load data. The load data is from New England (US Power Supply Area 2) and the supply data is from Connecticut Light and Power Company. All the supply data has been obtained from public sources such as [6], [7], [8] & [ l l ] while the demand data has been obtained from the Department of Energy.

Let D, represent the energy estimate for the. nth plant using the derating method. For example, consider the nth plant in the triangular load curve shown in Fig. 6, where p, is the

MW B

d,, + p,,C,,

d.

t2

MW 8

nC {n-I)C t I I I II I I

4C 3C !

c

1 Time

FIG. 6. Dc'ratiml method u'idl linear load curce.

I

i i 1

t,. &_,

Time

FIG. 7. Operating positions for nth plant,

reliability of plant n, C, is the capacity of plant n, d, is the sum of the derated capacities of the previous (n - 1) plants and B is the peak load. As evident in Fig. 6, D, is equal to the sum of the areas of the rectangle and triangle associated with the nth plant. Thus, D, = (d. + p,C. - d , ) t t + ½ ( d . + p , C . - d~ltt,. - it).

Substituting for tL and t2 yields, after some simplification, D. = (p.C.)

LINEAR L O A D CURVES

tf

377

1

2B

"

(2)

However, as emphasized in the previous section, the derating method produces a value of zero for those plants whose derated capacity lies above the peak load, B. To determine an expression for E,. first consider the special case in which all the N plants have the same reliability, p, and the same capacity, C. These assumptions, in effect, reduce the number of different operating positions for the nth plant (1 _< n _< N) from 2" to ;7. Thus, plant n can operate in exactly n different positions as illustrated by the n horizontal slices in Fig. 7. Plant n operates in position n if and only if all n plants are operating. This occurs with probability p". Plant n operates in position (n - I ) i f and only if exactly one of the (n - 1) plants dispatched prior to plant n is down and all other plants (1 to n) are operating. This occurs with probability p(~"_-~)p"--'(1 -- p ) .

In general, plant n operates in the k + 1st

Campbell, Murphy, Soysrer--Estimatinq Operational Costs in an Electric Utility

378

position, k = 0, 1. . . . . n - 1, with probability p(.-ljpk(l _ p).-l-~

(3)

Let A~ be the area associated with position k. A straightforward algebraic analysis yields

A~=c I - - g + ~ .

p(.- z)p~(t p).-l-~

xcl ,k+l,CT+ 2B

B ~=o

The sum appearing in the last term in this equation is simply the expected value of a binomial random variable. Hence,

E" = PC[ 1 c 2~

(a) decreasing linear function of p.,

(c) greater than 1,2. . . . . ~V.

or equal

to

1 for

n =

Furthermore, R. is invariant to scale multiples of plant capacities and load, i.e. if plant capacities and the load-duration curve are uniformly scaled by a common factor, then R., n = 1 , . . . , N, is not affected.

k=O

-- pC 1

Assume the load-duration curve is triangular. Then R. is a

(b) non-decreasing function of C.,

The expected energy generation for plant n is E.

Theorem 1

c ]. -~(n-l)p

Since ( n - 1) pC is the sum of the derated capacities for the first (n - 1) plants, which is denoted d., one obtains C

In the general case, where the reliability, p., and the capacity, C., are plant specific, the above equation can be easily generalized to

Proof Straightforward analysis of equation (5). Theorem 1 indicates that low reliability plants have a larger bias than high reliability plants, while it indicates that large capacity plants have a larger bias than small capacity plants. From these two results, it follows that large capacity, small reliability plants are more prone to a larger bias than small capacity, good reliability plants. Define L. as the difference between plant n's full nameplate capacity and its derated capacity, i.e. L, = ( 1 - p,)C,. This quantity is useful in specifying a sufficient condition for the bias ratio to increase between successive plants in the merit order.

Theorem 2 E. = p.C.

1

2B

'

(4)

From equations (2) and (4), the bias ratio for the triangular load curve is 1

R. = 1

p.O.

d.

2B

B

c.

d.

2B

B

(5)

This expression applies to those plants which, when dispatched according to their derated capacities, fall below the peak load. For those plants lying above the peak (B), the bias ratio is zero, since D. = 0. Some important properties characterizing R. are summarized in Theorem 1. For this theorem, let A7 be the highest numbered plant for which equation (5) is applicable, i.e. those plants in which the derated capacity lies below the peak load.

If L.÷ t _> L., then R.+ t _> R..

Proof Given in [5]. Note that an important special case of Theorem 2 occurs when p. = p and C . = C for all plants n. In this case L.+I = L. which then means R.+I _> R.. Hence, if all plants are identical, then R. will be a non-decreasing function of the merit order dispatching policy, i.e. the plant dispatched first will have the least bias.

BASELOAD C O N S I D E R A T I O N S In this section, the triangular load curve of the previous section is generalized to provide a positive minimum base load, as shown in

Omeqa, Vol. 10, No. 4

379

Pe~k

MW

3

Min

,{i

Z I

Oen~le(lcapacity

Nornep~m capoc~ Time

FtG. $. Trapezoidal

FIG. 9. Illustration of five class types.

approximation to load curve.

Fig. 8. For this class of load curves, certain characteristics of the bias ratio are examined. However, no closed form expression for R, has been determined. For the trapezoidal load curve of Fig. 8, each plant in the merit order is assigned to one of five classes. These classes are defined as follows.

MW I012 II

MW

800

800

ClassPlonls

12 II

1,2,3,4 5 6,7, 8 9,10 11,12

I0

",,

6

400

400

I 2 5 4 .5

Class 1 : Includes plants whose derated and full capacities both lie within the baseload part of the load curve.

4

I

I

Class 2: Includes plants whose derated capacity lies within the baseload part and full capacity lies within the triangular part of the load curve.

Nan'~-p~ate cap(}oh/

gerate¢l cop(x:,ty

Class 3' Includes plants whose derated and full capacities both lie within the triangular part of the load curve. Class 4: Includes plants whose derated capacity lies within the triangular part and whose full capacity is above the peak load.

14,,, '3 .2

3

FIG. 10. Twelve plant example.

to Class 1, plant 5 is the single Class 2 plant, plants 6-8 fall into Class 3, plants 9 & 10 are Class 4 plants, and finally, plants 11 & 12 are Class 5 plants. As suggested in Fig. 11, the bias ratio displays a definite trend as one proceeds from Class 1 to Class 5 plants. The only exception is

Class 5: Includes plants whose derated and full capacities both lie above the peak load. (For the sake of simplicity, plants whose capacities straddle two different parts of the load curve are neglected.) A graphical illustration of these five load regions determined by these five classes is shown in Fig. 9. An example consisting of a 12 plant utility is presented in Fig. 10. All plants in this utility have a nameplate capacity of 100 MW and a reliability of 0.80. The peak load is 800 MW and the minimum load is 400 MW. In this example, plants I-4 belong

1.0 .9 0,5 I

I

I

I

I

i. M W n-

Cioss

I

FIG.

2

3

4

$

l 1, Ratio trend orer five classes.

380

Campbell. Murphy, So vster--Estimating Operational Costs in an Electric Utility

Class 3. The ratio is uniformly one for Class 1 plants, monotone increasing for Class 2 plants, monotone decreasing for Class 4 plants and zero for Class 5 plants. The following theorem describes the trend of the bias ratio for plants in these five classes under the assumption that all plants have equal reliability and capacity.

8,

a 2

Theorem 3

Assuming a trapezoidal load curve, the bias ratio

71

FIG. 12. Piecewise linear load curve.

(i) is 1.0 for every plant in Class 1, (ii) is greater than one for all plants in Class 2 and increases between successive Class 2 plants, (iii) decreases plants,

between

successive

a~J Trne

Class

4

(iv) is zero for every plant in Class 5. Proof

Given in [5]. The ratio in Class 3 is not easily characterized. Case studies have shown that in various instances the trend between successive plants can be increasing, decreasing or both, i.e. U-shaped. The behavior seems to depend upon various supply and demand factors such as peak and minimum loads as well as plant specific information. However, as suggested by Fig. 11, the bias ratio appears to reach its maximum somewhere in the general region of Classes 2 and 3. This characteristic is consistent with the observation in [12] that the derating method overestimates the generation from plants at the so-called 'knee' of the load-duration curve. To further analyse this observation, we consider a more general class of load-duration curves, a class which encompasses the triangular and trapezoidal load curves as special cases. Figure 12 illustrates a load curve which is piecewise (two pieces) linear and concave. To simplify matters, assume that the first two plants in the merit order have unit nameplate capacities and that the breakpoint i is such that f(i) = 1. We intend to determine the bias value of the second plant as a function of the shape of the curve. In particular, note that if

a./a2 = 1, then the curve is linear, and if al/a2 = ~ , then a trapezoid results. The fundamental result is that as al/az proceeds from 1 toward vo (maintaining f ( i ) = 1), then Rz becomes ever larger, i.e. the bias increases. As shown in Fig. 12, a, and a,_ are the slopes of the first and second line segments, respectively, Since the peak load is assumed greater than 2 (and only concave curves are considered), the slope a~ must be algebraically less than - 2 . With these assumptions, the bias value for the second unit is a function of p (the c o m m o n reliability), at (slope of first line segment) and a2 (slope of second line segment). Under these conditions the bias value, R2, can be shown to be

p<½

(6)

R 2 =

a2 a t - ~ + 2 -

+at 2 p - 2 + p>"

a2(a, + -~

(7) In both (6) and (7), for fixed at, the bias is the ratio of two linear functions of a 2. Furthermore, as required, both (6) and (7) reduce to (5) in the triangular case, i.e. a I = a2 = - B . Next, observe that when la2l < Jail the load curve (see Fig. 12) is concave. The question we address is as follows: for some fixed value of al, what happens to 82 as la2J becomes smaller? Observe that as la2J becomes smaller,

381

Omequ. Vol. I0, No. 4

i.e. a,_ increases, the load curve becomes rectangular in shape. We now show that as the load curve in Fig. 12 becomes "more' concave. i.e. rectangular, the bias ratio R2 increases. For fixed a l, it is straightforward to show that the sign of ?R2/'ga, for p > ½ is 5,-" 4

1 1 p-~+~+a~

(3p ]-

2+~ )

(

')

c

t81

')

2 + _Tp > - 2 3 p / 2 -

2 +~pp .

Hence, the value of (8) is at least as large as I

1

5/4p 2 - p + ~ + ~ , p - 3p + 4 -

f(t) = 1.0130 - 2.351r + 11.23t" - 26.659/"~ + 27.713 't - 10.437r 5

.

To obtain a concave curve and maintain a peak load greater than or equal to 2, one must have a t < - 2. Since (3p)/2 - 2 + 1/(2p) < 0 for p > ½, it follows that for al < - 2 a I 3p,2-

Energy. is used to represent an "actual" load curve. PSA 2 represents the six New England States. A fifth degree (scaled by 1/2515) polynomial representation of this load curve (obtained through ordinary curve fitting) is

l/p.

(9)

Multiplying (9) by p yields the cubic 5p 3 - 16p 2 + 14p - 3. Since this cubic is nonnegative for ½ _< p < i, it follows that ORz/~a2 > 0 for p in this region. An analogous argument shows that gR2/ga2 > 0 for 0_ 1 when al = a2. Hence, as a2 increases ([a21 decreases) it follows that the ratio becomes more biased, i.e. R, exceeds 1 by an ever-increasing margin. CASE STUDY In this section a comparison between the derating and expected value methods is made in a more realistic context. Actual 1977 load data for the United States Power Supply Area (PSA) 2, obtained from the Department of

for 0 < t _< 1. Observe that the actual load curve is f(t) = Pf(t), where P is the peak load, i.e. f ( 0 ) = Pf(O)= P. One electric utility operating in PSA 2 is the Connecticut Light and Power Company, which is estimated to have 3105 MW of nameplate capacity in 1977. The mix of plants which constitute the generating capability of this utility is shown in Table 1. A comparison between the two methods of estimating energy generation utilized the 1977 PSA 2 load curve. In particular, the load curve f(t) was apportioned under the assumption that the load served by Connecticut Light and Power Company was proportional to its relative share of New England generating capacity. Supply-side parameters were estimated (from [6, 7, 8, 11]). Hence, exact nameplate capacities for each of the utility's plants were known. The merit order dispatch policy, in all cases, is assumed to be simply in decreasing order of plant size. The reliability (availability) of the individual plant types are assumed to be Hydro Turbines Coal/oil steam Nuclear

98% 90% 80% 70%.

TABLE I. UTILITY: CONNECTICUT LIGHT AND POWER COMPANY (Peak load: 2515 MW, Min. load: 840 MW)

Plant Millstone no. 2 Millstone no. 1 Montville Devon Norwalk Harbor Cos Cob Shepang Stevenson Bran ford Enfield Tunnel Devon Norwalk Harbor Tracy

Type Nuclear Nuclear Oil Oil Oil Oil Water Water Oil Oil Oil Oil Oil Oil

Capacity (MW)

Availability

Class

Derated

Expected

Ratio

900 660 550 430 325 65 40 30 20 20 20 15 15 15

70 70 80 80 80 90 98 98 90 90 90 90 90 90

2 3 3 4 4 4 4 4 4 4 4 4 4 4

630.00 446.28 307.58 124.36 32.88 2.55 1.23 0.70 0.34 0.28 0.21 0.12 0,09 0.05

629.57 378.62 256.67 126.21 62.78 11.14 6.89 4.87 2.85 2.76 2.67 1.94 1.89 1.84

1.0007 1.1787 1.1983 0.9853 0.5238 0.2293 0, 1791 0.1440 0.1193 0.10013 0.0799 0.0619 0.0460 0.0297

382

Campbell, .Vlurphy, Soyster--Estimating Operational Costs in an Electric Utility

Ratio

1.5

Plant I

Mm" 840MW P~:~k = 2 5 ~ 5 MW

io 0.5-

t,

iooo Min

,

2(300

I

3000

Peak

FIG. 13. Connecticut Light and Power Company.

With this load and supply data, the bias ratios were determined for each plant and are presented in the last column of Table 1. In addition, the ratios are presented graphically in Fig. 13. Observe that these bias ratios follow the general trend illustrated in Fig. 11 of the previous section. REFERENCES 1. Annual Administrators' Report for 1977 US Department of Energy, Washington. 2. BALERIAUX H, JAMOULLE E •

DE GUERTECHIN. L Fr

(1967) Simulation de I'exploitation d'un parc de machines thermiques de production d'electricite coupl6 :~ des stations de pompage Rev. E (Ed SRBE) V (7). 3-24.

3. BOOTH RR (1972) Power system simulation model based on probability analysis IEEE Trans. Power Appar. Syst. PAS-91 (1). 4. BOOTH RR (1972) Optimal generation planning considering uncertainty IEEE Trans. Power Appar. Syst. PAS.-91 ( 1). 5. CAMPBELLKW (1981) A comparison between probabilistic simulation and the derating method of operational costing in electric utility modeling. MS Thesis. Virginia Polytechnic Institute and State University. 6. Gas Turbine Electric Plant Construction Cost and Annual Production Expenses (1975) Third Annual Supplement. US Department of Energy, Washington. 7. Hydroelectric Plant Construction Cost and Annual Production Expenses (1978) 22nd Annual Supplement. US Department of Energy, Washington. 8. Inventory; of Power Plants in the United States (1977) Federal Energy Administration, Washington. 9. JENKINSRT & Joy DS (1974) WIEN Automatic System Planning Package (WASP)--An Electric Utility Optimal Generation Expansion Planning Computer Code. Oak Ridge National Laboratory, ORNL--4945. 10. Optimized Generation Planning (OGP) User Manual (1975) General Electric, Schenectady, New York. 11. Steam-Electric Plant Construction Cost and Annual Production Expenses (1977) 30th Annual Supplement. US Department of Energy, Washington. 12. STREMELJ (1980) New concerns and new methods for generation expansion planning in the electric utility industry Energy Modelling 111: Dealing with Energy Uncertainty. Institute of Gas Technology, Chicago, Illinois. ADDRESS FOR CORRESPONDENCE: Professor

AL Soyster, Industrial and Management Systems Engineering Department, The Pennsylvania State University, 207 Hammond Building, University Park, PA 16802, USA.