A New Approach to Electric Power System Planning — Explicit Consideration of System Uncertainty

© I F.-\C Large Scale S\stellls. Berlin. (;OR. 19H9

Cop\Tj~ h t

A NEW APPROACH TO ELECTRIC POWER SYSTEM PLANNING - EXPLICIT CONSIDERATION OF SYSTEM UNCERTAINTY Y. Kaya and L. Xinnyi DrpartlllFllt of ElfCtrical Enginenillg. Unil'enity uf Tukyo. HUllgu i-J-J. Bllnk)·u-ku. Tokyu . japan J J J

Abstrac~. A new method for electric power system planning is proposed which deals w1th uncertainties of various system parameters such as fuel and f~cility costs as a part of the criterion function. Namely the criterion function is chosen a~ the s~m of.the average system cost squared and its weighted variance. T~e key 1dea 11es 1n the way of determining variances of cost parameters of d1fferent power plants . The authors propose first to evaluate these variances from the ~ata of existing plans and then to modify them in an orderly way for the plann1ng of future power systems . The first process is no other than I]. socal~ed . 1nverse optimization process whereas the second one is the or dinary ophm1~ahon . The results of applying this method to the planning of Japanese electr1c power system demonstrate usefulness of the method .

Powe r system plonnin~i inverse optimization; optimization ; Keywords . uncertainty ; variance

includes variances of various cost parameters , which are not known explicitly . Also the weighting parameter w should be given in the modeli n g process . The authors employ the following method for the determination of these paramete r s. First , choose a typical system plan for reference. Then apply the inverse optimization procedure to find the values of the above paramete r s in the plan. In other words these pa r ameters are determined so that the distance between the plan and the output of the model in a multi - dimens i onal space of the system variables is minimized . By doing this cost variances and the weighting parameter having been used implicitly in the reference plan are identified. It is to be noted that the cost variances thus derived are not necessarily r eal cost variances ; rather they are parameters integrating various types of uncertainties and constraints subjectively introduced int o the plan as described before . In the further planning process the de r ived parameters are so modified as to reflect planner ' s future image . This process is still subjective but transparent as others can easily see the difference between parameters in the reference plan and those modified by the planner . The model optimization is then executed and resul ts can be analyzed together with the values of cos~ variances . In this paper the above procedure is formulated and then applied to the electric powe r system plan of Japan const r ucted by Central Institute of Electric Power Industry (Japan) .

INTRODUCTION One of the significant characteristics of the "lectric power system planning is that the time cange is mo r e than 20 years and so various uncertainties in the system environment and system paramete r s shou l d be taken into account . Future rise in oil prices , change in public acceptance of nuclear powe r plants, emergence of the greenhouse effect and its impact on use of coal are typical uncertainties which may influence the power system planning significantly. There I however I have been few methods to deal with these uncertainties in an orderly way in the planning process ( 1 )(2) . A rather common process is just to modify the outputs of planning models so that these may r eflect the intuitive image of the planners which is formed with the above uncertainties implicitly taken into ac count, and/or to introduce inequality constraints of which limit values are given subjectively . If we agree with the opinion that credibility of a plan heavily depends upon the degree of its transparency, the above process clearly reduces the value of t he plan significantly. The authors thereby p r opose to build a planni ng model with uncertainties explicitly taken into account in a part of the criterion function :>f the model. Namely the criterion function has the following form : J

=

E**2 + w*VAR

(1)

where the first term is the average system cost squared and the second is the variance of the system cost with a weighting paramete r w. The above criterion function can be derived from the assumption that the utility for a power company is the quadratic function of the system cost and the behavior of the power company is r isk- averse( 3) . The seco nd term of equation (1)

MODEL FORMULATION Criterion Function The planning model proposed here is an optimiza tion model with the quadratic criterion function of i quation(2) .

425

426

y, Kava and L. Xinmi

{ E +E}

+ W' VAR -

2

min (2)

lk;:number of oprating plants in k-th demand/j-th time zone

o i(k):capacity expectation of the total system cost C E expectation of the power shortage cost VAR variance of C In the model the following assumptions are

utilization ratio of i-th kind plants in k-th demand

where E(C>

made. 1) Total demand patterns ( hourly, daily and monthly) are given. 2) Costs other than power plant construction and fuel costs are negligible. In other words) power transmission cost and labor cost are not included in the optimization model. 3) Power plants are categorized in several types due to differences in fuels.i.e.petroleum, natural gas(in Japan liquefied one, Le.LNG), coal, nuclear and hydro. Since the contribution of hydro-power plants are comparatively small and only to base load they are neglected in the model except pumping-up stations which are used only in peak load period. The cost of plants of each kind consists of fixed cost and operation cost. The former is mainly the plant construction cost counted on yearly basis (¥/kw/year) and the latter is the fuel cost(¥/kwh). 4) All uncertainties involved in power system planning are replacable in terms of variances of the above cost parameters. Pattern of use of power plants for given demand The first problem prior to the model formulation is how to determine the usage pattern of power plants for the demand in a given time zone ( say , from 10 to 11 am of a certain day). The ordinary procedure being used in power system modeling is here employed. That is, 1) For plants other than pumping-up stations, the cheaper the operation cost the higher the priority. In other words the plants with the lowest operation cost are used for the base load, and those plants with the second cheapest operation cost are for the next load portion. This is illustrated in Figure 1.in which the load duration curve is divided into several zones. 2) Pumping-up stations are for simplicity assumed to pump-up water at night and generate power in the daytime of the same day. The total available power is the power spent for pumping-up times total efficiency (normally 70 %). These stations are put into use only when these use is economically profitable or when the total demand exceeds the total capacity of power plants other than pumping-up stations. Cost eguation , The system cost C is then formulated ln equation (3), with use of the above assumptions. (Derivation of the equation is omitted because it is a lengthy procedure.)

C

I k ·-I

=IP tig(77)Ki+L{LPViO k. i

J

I

i(k)K i

where

n:number of kinds of plants k:index for kinds of demand j:index for time zone of a day ( I hour ki:total capacity of i-th kind plants g(77):interest rate P, i :uni t construction cost of i-th kind plant (V/kw) Pvi:unit operation cost of i-th kind plant (V/kwh)

Dk;: total demand in k-th demand/j-time zone Tk:number of days of k-th demand qk; = [1Ir :when pumping-up stations pump up water -I :.hen pumping-up stations generate power Power plants of different kinds are numbered as: 1 :nuclear 2:coal-fired 3:natural gas-fired 4:petroleum-fired(already installed) 5:petroleum-fired(new) 6:pumping-up stations Supply shortage cost E in Equation (2) is the expected supply shortage cost. There have been a few ideas to estimate this cost. One is to consider this cost as the blackout damage cost and evaluate it from real data obtained from experiences of blackout. However only few data are available on blackout damages and the size of damage is very different from case to case. The other idea is to utilize the concept of the loss in consumer surplus when power shortage. When power supply is e~pected ~o be lower than demand, consumers will adjust thelr power demands so that they do not exceed power supply. It means that consumers loose ~ome pa~t of utility of the electricity by reduClng then demands. This loss is illustrated in Fig.2. The consumer surplus is defined as the shaded area in Fig.2(5), when supply is equal to demand and the price of electricity is Po($/kwh). Suppose that the supply is expected to be So which is lower than demand D. Then consumers are requested to reduce their demands to So, given the price Po. After reduction of demand the consumer surplus is equal to cs

=J ~OP(X)dx - PoSo

So the loss of consumer surplus by demand adjustment is the double hatched area in Fig.2. This concept seems much more reasonable than the concept of blackout damage cost, as in most cases of power shortage utility companies ask customers to reduce their demands so as not to lnduce blackout. The authors utilize this concept and utilized the real data of demand adjustment contracts made between Japanese power companies and industrial customers. Although details of the cost evaluation are not described due to space limitation, the results are plausible and the evaluated power shortage cost is of the order of several hundred yen/kwh. Inverse optimization problem In the preceding sections all terms of iquation (2) have been identified as functions of the plant capacities, . Since t~e criterion function J is almost a quadratlc function of , the optimum of J can be derived from the following equation.

B { E + E<6CS> } 2

BJ

BK

i

BK

i

BVAR +W

=0(4)

BK

i

The first step of the proposed method is to assume that the existent plan(or reference plan) is the result of the above optimization and then to evaluate the values of cost variances from the existent planning data. The above equations are simply first order simultaneous equations of cost variances so that cost variances are easily calculated by solving the equations.

A \iew Approach

to

Electric Power System Planning

Planning model: Ordinary optimization model Once the cost variances are derived from the existent plan, we can modify these to reflect our understanding about the future.(see the example in the later section) Then the final plan will be derived by optimizing the criterion function of equation (2). The optimization process is more complex than the process of inverse optimization, as the equations are non-linear. By proper selection of the initial values of variables, however, use of Newton-Raphson method for iterative optimization is very successful and we can finally reach the optimal point of J and the corresponding values of . APPLICATION TO THE PLANNING OF "APANESE ELECTRIC POWER SYSTEM The method thus formulated has been applied to the planning ~f Japanese electric power system. A few plans are already known, among which we chose the plan by Central Research Institute of Electric Power Indus try( CRIEPI) as the basis, as it is most widely known and also its detail data are easily available. Table 1 is the outline of the CRIEPI plan in the year 2000. According to the government data the total number of the customers which have the special load adjustment contract with power companies in Japan is 1292 and the total adjustable capacity is 8.4 GW ( measured in July 1,1986). These data are used to evaluate parameters of the supply shortage cost. Inverse optimization The result of inverse optimi za tion, i. e. of evaluating cost variances is shown in Table 2. In this table only 4 equations are shown and those equations corresponding to optimization with regard to existent oil-fired plants and to pumping-up stations are omitted. The reason of this omission is that capacities of these two types of plants are determined not on the basis of ordinary economic optimization. Here, we use normalized variances as shown below.

427

future. 2) Coal and nuclear plants are cheap in terms of estimated costs/but planners have to worry about pub 1 i c a c c e pta n c e 0 f the s e p 1 ant s t,..om GIrl environmental point of view. It means that planners are quite uncertain about costs to improve public acceptance of these plants. Ordinary optimization: Planning process On the basis of the estimated cost variances we can now proceed to the new process of the power system planning. Table 1 is the plan made by CRIEPI,and we can modify the plan by changing the cost variances. For example we can assume that the public acceptance of nuclear plants will be worsened in future.(The present trend of antinuclear movement is serious not only in Europe but also in Japan.) To reflect this possibility we double the cost variance of nuclear power pl&nt (in this case operation cost) and see the result of an ordinary optimization with the same model. Table 3 shows the results when the cost variance of plants of a type is doubled. In each case the total size of the plants concerned is reduced, as is expected. The most significant in the results is the behavior of oil-fired plants:the total size of these plants is very sensitive to any of cost variances, and in case the variance of oil-fired plant operation cost is doubled the oil-fired plants vanish from the entire plan. CONCLUSION A novel approach to electric power system planning has been proposed. It consists of two processes, i. e. inverse optimization to evaluate cost variances from the existent plans and ordinary optimization to obtain future plans based on planner's own view point on future uncertainty, which is expressed explicitly in the mo?el as values of cost variances. This methodology will help the plan more transparent and then users rely more on the plan. REFERENCES

a

where

v

if j.1.

11

vi

vi ;

-+

a

v i

and a

E

and 11,

f i

if /.l ;

a

f i

E
f i

i>.

tne

Table 2 that 1number of cost variances is 8 whereas the number of equations is 4. Therefore some of cost variances should be determined from the other data or consideration. We assumed here that the variances of fixed plant costs are negligibly small compared to those of operation costs. Then/from Table 2,we obtain It is seen from

av,' Ot1

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a

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a

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The resultant variances seem compara ti vely large but these are actually the products of cost variances and the weighting parameter of the criterion function w. In other words/the above result indicates that w is large and only values of parameters relative to others are significant. We see from the above result that the cost variance of LNG is the lowest while those of coal and nuclear are high. This is interpreted as follows: 1) Construction of LNG plants has been planned under the present take-or-pay contract. This contract means the supply of LNG is fully assured for the contract period so that the planners do not have to worry about availability of LNG in

1. W.D.Dapkus: Planning for New Electric Generation Technologies - A Stochastic Dynamic on PAS-103,6, Programming Approach, IEEE Trans. pp.1447-1453,1985 2. Y.M.Park et al: New Analytical Approach for Long-term Generation Expansion Planning Based on Maximum Principle and Gaussian Distribution Function, IEEE Trans. on PAS 104,2, pp.390-397, 1986 3. J.Tobin: Liquidity Preference as Behavior Toward Risk, Review of Economic Studies, 27, pp.152-166,1960 4. R.Anderson and L.Taylor: The Social Cost of Unsupplied Electricity A Critical Review, Energy Economics,8,3 pp.139-146,1986 5. M.A.Crew and P.R.Kleindorfer Public Utility Economics,1979, Macmillan Press

428

Y. Kava and L. Xinnyi

demand

price

I ~- -I. __ _____n~ I

p

.'

\

I

\

;----~ ---">.,

Po

2

~---------'---

~---~---~--.deman(

hour s

So

8760 Fig . I

D

Fig. 2 Loss of Consumer Surplus

load duration curve and plant allocation

Table I. A view of power supply in the year 2000 by CR IEPI plan t cost(nominal pr i c es) cosntruct ion co s t operation cos t (10 ' ¥/ kw)

nuclear coal oil (0 I d) oil (new) pumping-up

(GW)

( ¥/ kwh )

386 286 243 88 186 143

LNG

system s turucture

5. 13 11. 85 21. 52 24.59 24 . 59 0

65 14 36 26 5 19

Tab le 2. Es timation resu I t of cos t variances O f 12

1 2 3 5

a

f 22

a

f 3

(J f 4

2

o

f 5

2

Of.

2

(J .1

O. 76

1. 00 I. 00

I. 00 I. 00

2

0 ., 2

0 .,

-0. 04 -3. 9. 48 -31. 10. -37.

2

22 88 34 10

0 ••

-0. 98 -9. 73 -4. 76 9. 47

Table 3. Change in total capicity of power generation plants of each kind when the variances double "a

changes in capac i ty

nuclear

i nuclear .... ! coa I

I. 00

: LNG : 0

iI

1 I

I

I

v, 2

-f

20 •. '

coal

0. 716

I

1.068

I. 319

I

0.541

I. 190

2. 034

; :

I. 042 I. 310

1 1

I I

I 1

!

LNG I. 118

I. 155 O. 648 2.012

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. I

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oil 1. 031 I. 044

I. 132 0. 0

2

-10.46

+ -63.98 5. 03 96. 38

=0