Compel. & F./ect. ,~ / ,g Vol. 6, pp. 27-34 0045...7906/79/0301-0027/$02.00/0 © Pcrlpunon Press Ltd., 1979. Printed in Great Britain

L O N G R A N G E S T O C H A S T I C P L A N N I N G O F G E N E R A T I O N C O N S I D E R I N G O V E R D E S I G N

K. RAMACHANDRAN and JAYDEV SHARMA

Department of Electrical Engineering, University of Roorkee, Roorkee, India

(Received 19 July 1978; received [or publication 12 October 1978)

Abstract--This paper presents a new model for the optimal expansion planning of generation facilities in Power Systems under uncertainties of demand forecasts and generator outages. The objective is to find an expansion strategy requiring minimum present worth of revenue requirements so that electric power can be supplied to consumers at a reasonably low price with high degree of reliability. System utility is evaluated in terms of its capacity to meet demand requirements under uncertain conditions. The design also aims at reducing the over design of the system which requires a large capital outlay. A method based on the preference order dynamic programming is presented to determine the long-range optimal expansion strategy. This avoids the drawbacks of the conventional dynamic programming methods and requires lesser number of alternatives to be considered for selection of the optimal expansion policy.

NOTATION n number of planning stages

NG number of plant sites Sm state variable set at stage m

D, G expected values of demand and generation ~rooc standard deviations of demand and generation

PF probability of failure AI,A2 penalty costs

r, i interest and inflation rates j jth plant or unit type

NT number of types of units

I. I NTRODUCTION

Power system generation planning determines the expansion of generation facilities in an optimal way so as to provide a reliable supply to consumers at the lowest possible cost. When planning for future, the effect of uncertainty, is an important factor. The planner, while designing the system, is confronted with the uncertainties associated with future demand forecasts, generator unit availabilities, fuel costs, money values, construction costs and availability of capital[l]. Earlier models proposed in this field are based on either heuistic approaches [2] or conventional dynamic programming methods [3].

A model, considering the uncertainties in future demands and generator outages, is presen- ted. This model is based on the concept of engineering design under risk[4]. The risks involved in a power system may be defined as its capacity to meet the system demands at all times without any failure or curtailment of load. The utility of a power supply undertaking may therefore be evaluated in terms of its capacity to meet demand on all conditions. The system utility is interpreted in equivalent money values by assigning proper costs for any failure to meet design requirements.

Usual practice in power system design, is to provide adequate reserve capacity to meet design requirements under exigent conditions. Calculation of reserve capacity based on consideration of uncertainty may result into underdesign or overdesign of the system, Over design of the systemrequires large capital investments--a good portion of which remains idle for long periods, whereas a small reserve capacity results in an unreliable power supply to the consumers. Hence a trade-off between the utility and over design of the system may be sought, while planning the systems.

Objective of the model is to determine the sequence of decisions pertaining to when, where and how much generation capacity is to be added to meet the future demands. Using dynamic

27

28 K. RAMACHANDRAN and J. SHARMA

programming technique, the system at each stage of the planning period, is represented by a set of possible values for generation levels. These generation levels form the state variable set in the dynamic programming formulation. The decision and hence an optimal partial policy associated with element of the state variable set, is then determined with the help of preference order dynamic programming method [5].

The cost function associated with each element of the state variable set is non-linear, its expected value is not suitable for the evaluation of the system in the presence of uncertainties since adverse deviation from the average value might mean an unacceptable or hazardous performance [4]. An objective function, which contains in addition to the fixed, operating and penalty costs, the deviation of the cost, is therefore defined. Optimization of this function yields the decision associated with the elements of the state variable set.

Size of the problem can be considerably reduced by the choice of the elements of the state variable set at each sub-period. This results in the reduction of computer memory and time required for the solution of the problems.

2. MATHEMATICAL FORMULATIONS

(i) Uncertainties Uncertainties in the future demand are due to factors such as the nature of loads, seasonal

changes, socio-economic growth rate, historical data, model chosen to describe the growth rate etc. The future demand may therefore be described by an appropriate probability function f(D), which defines the probability distribution of the demands over the entire range. If the system consists of a number of independent demands, each one being governed by appropriate distribution function, the total demand probability distribution f(D) is obtained by the con- volution of the individual density functions [6]. Expected value of the demand and its variations are then given by,

if) = f ~ Dr(D) dD (la)

crff = f~= (D - / ) ) 2 f(D) dD. (lb)

Many different types of generators are in service in a power system and they are randomly forced off-line because of technical reasons. Random outages or availability of a generator is described by a discrete probability distribution function. Since a power plant consists of a number of same or different type of units, the availability of the plant is determined by the convolution of the distribution function of its individual units. Finally, the total installed generation capacity can be described by a distribution function I(G) which is obtained by convolving the distribution functions describing the outputs of the power plants in the system. Let t~ and tro be the expected value and standard deviation respectively, of the total installed capacity.

(ii) Dynamic programming formulation Total planning period is subdivided into n stages and the decisions are taken during these

subperiods. New demands are assumed to materialise at the end of each stage. At each stage, planner has to decide the amount of expansion to be made at each power plant site to meet design requirements.

At any stage m, the system may be described by a set-S m of state variables. Each element of the set S ~' represents a feasible generation level of the system. Each X ~ $" is a vector whose cardinality is equal to the number of power plants in the system and the components of X represent the individual plant capacities. X can therefore be defined to belong to the Cartersian product space Sz of the plant sizes available at each generation site. That is,

where X e S~ (2a)

$~ = $) x S~ 2 . . . . . x S~ Na. (2b)

Long range stochastic planning of generation considering overdesign 29

(iii) Probability of failure Reliability of a power system can be defined as its capacity to meet consumer demands.

Accordingly, the failure probability gives the total probability of demand to exceed generation. Let,

Y = DIG. (3)

The probability distribution of Y is determined from the distributions of D and G. In the present formulation, Y is expressed by an equivalent normal distribution. The parameters of the normal curve namely the mean and variance are obtained from the Taylor's Series evaluation of f(Y) about the mean values/5 and ¢~0) and are given by,

Y = D/(~ -[- (,J~/(~3)O'G2 (4a)

oe2 = (G)2O'D2 + (D2/G2)2oeG 2. (4b)

The probability distribution of Y is expressed as

[ i t r - f(Y) = try'X/(2cr) exp [ -~ \----~y ] j . (5)

Probability of failure (PF) is defined as,

PF= f~ f(Y)dy= l - f~=f(Y)dy. (6)

(iv) Objective function The present model uses preference order dynamic programming technique for the formula-

tion and solution of the multi-stage expansion planning problem. Total planning period consists of a number of stages (n), at the end of each one, new demands are assumed to occur. As in the case of conventional dynamic programming technique, at each stage, a set of state variables are defined. In the present model, these state variables represent the possible generation levels of the system. Decision associated with each element then corresponds to determining optimally its predecessor state in the previous stage, from which an optimal partial policy which defines the optimal path from the initial stage to the present stage generation level, is constructed.

The model, at each stage, defines an objective (cost) function associated with the elements of the state variable set. Minimization of this function yields the decision regarding its predecessor state. The objective function defined, reflects not only the various costs but also the types of generators available for the expansion. In general it has the following components:

(a) Capital cost. For every point x belonging to the space Sz defined by (2b), preference ordering determines the state variable X E S m by combining x with the elements of S "-t of stage m - l. The capital cost therefore gives the present worth of the capital required for the installation of plant capacities represented by vector x. It is given by

{l + i~t ~ (l + r)TJ+'- l, C~ = \ l- '~r] i=l r(l + r)T/+l "~j (7)

where t = year at which investment is made; Tj = life expectancy (years) of jth plant: and lcj = annual cost of capital investment due to jth type of unit in the alternative represented by x. The plant expansions are selected from discrete sizes, but the operating cost is determined from the operating policy of the system. For a given installed capacity, the operating policy may be the one which corresponds to minimum operating cost subject to constraints of plant capacities and demand requirements. In other works, the operating costs is determined from [8], CAEE Vog 6. No. I--C

30 K. RAMACHANDRAN and J. SHARMA

"1 + i ~t u~ minimize Co =/1---~r) ~ "fJ(P~)

N G

subject to ~ / ~ j _>/5. i=l

(8)

where f(Pgj) = operating costs of the jth plant for a mean output of fi~; (7~ = capacity of the ]th unit or plant associated with the State variable x; and/5,, = mean demand at stage m.

(b) Cost of overdesign. Effect of overdesign in the system design is an increase in the capital cost of the system, In order to curb the tendency for large overdesign, a penalty is imposed on the installed capacity in excess of the demand requirements. Accordingly, cost of overdesign is defined as,

Ca={Al((~x-/Sm)2 if (3x->Dm] Otherwise/' (9)

(c) Penalty for [aUure. Utility of the system is obtained by assigning proper penalty for failure to meet design requirements. Hence it is defined as,

where Cs = A2(PF). (10)

(d) Deviation o1: cost. Since the utility of the system is defined as its failure to meet to design requirements, it is non-linear with cost. Therefore the planner's view of risk or failure is considered through the utilities assigned to costs which deviate from mean. If there is appreciable risk involved, a design which minimizes the expected net cost may result in unacceptable failure levels. Under such conditions Wiezman et al. [4] concludes that the utility is maximized by minimizing the expected value of the cost and its variance.

Deviation of the total cost due to (a)-(d) is obtained by expanding it by Taylor's series about the expected values of the random variables, viz. generation and demand. Let

Then C(X) = Cc + Co+ Ca + Q.

(ac(x) (ac(x) °'c'x'=i~--I k-~'~'~] o'~,,+ "= \ aD, ]

(11)

(12)

Preference ordering determines for any unit combination, the state variable X E S m by determining a suitable predecessor state y E S m-~. For every x E Sz, the objective function is therefore defined as:

Z(x) = E[C(x)] + ao,~q~ + jym-I (13)

where E[C(x)] = expected value of C(x); a = scaling factor; jym-1 = cost of the predecessor state of y E Sin-I; and 2 - variance of the cost. O" C ( x ) - -

3. SOLUTION TECHNIQUE The model determines for every x E $,, the state variable X at stage m. This obtained by

minimizing the objective function (13) with respect to y E S m-I. This optimal solution gives the decision 8(x) associated with x. Let

where

J/"= min Z(x) (14) y~Ss m-I

Sx"-I -- predecessor states of x.

Let Yo = Y gives the optimal solution of (15). Then X E S m is given by,

X = xUyo. (15)

Long range stochastic planning of generation considering overdesign

The partial policy associated with x is defined as

31

D"(X) = 8(x) XD"-'(y). (16)

Cost of the partial policy is given by Jm(X). Repeating the procedure for other elements of the space S.,, and the complete state variable

set at stage m of the planning period S" and the partial policies are determined. The expansion strategy for the system is then selected from the completed partial policies.

The following general steps are involved in the solution procedure. Step 1. The size of the problem depends upon the size of the state variable set at each stage.

Any x ~ Sz is admissible if and only if there exists an element y E S ~-m (S O being the initial state of the system) so that the resulting state x plus y satisfies the following requirements

Gxi <~ Gi m (17)

where ¢~ = expected value of the installed capacity due to state x plus y and Gi m = maximum available capacity at ith site.

The probability distribution of generation associated with x is obtained by convolution of the distribution of the components. The resultant distribution is expressed by an equivalent gaussian distribution. This simplifies the determination of the distribution of the total generation due to x and the predecessor states without materially sacrificing the accuracy.

Step 2. For any feasible point x, the capital cost Cc is calculated from (7). Form the set of predecessor states S~ =-~ from the elements o f S "-~ using conditions

described by (17). Step 3. For every y E S, ~-~, the operating costs, cost of overdesign, failure cost and cost of

deviation are calculated and the objective function Z(x) is determined. Step 4. Minimize Z(x) with respect to y. The solution determines the state variable X and

the partial policy associated with X at stage m. This is repeated for other every x E S.,. Step 5. Repeat step 1 to 4 for m = 1,2 . . . . . n. The optimal expansion strategy is then

determined from

min J"(X), X E S". (18)

Let X0 = X solves (18). Then D"(Xo) determines optimal expansion strategy and J"(Xo) the minimal cost.

4. EXAMPLE

Two problems (A) and (B) are chosen for the application of the above method. Problem A. The sample power system shown in Fig. I is considered for expansion. The

details of plant sizes, costs, outage rates, etc. are given in Table I(a). The operating costs of

G ° ® G

®

Fig. I.

32 K. RAMACHANDRAN and J. SHARMA

Table I(a).

Operating cost ' parameters Percentage

Sizes available Cost a b Max. outage Site 1 (MW)t MU* (MU/MW) (MU/MW 2) capacity rate

40 50 I 80 80 1.2178 0.0055 120 6

120 100 2 25 60

50 I10 0.21 100 3 75 150

100 190

? M W = mesawatt. *MU = money units.

Table 2(a).

Demand

Table 3(a). Penalty costs At = 0.01, A2 = 10.000: Interest rate r = 0; Inflation rate i = 0: Opti-

mal expansion policy

Mean Percentage Capacity Time period value deviation

Stage Site I Site 2

1 70 4 2 IO0 4 I 0 IO0 3 150 4 2 40 I00

3 80 100

Optimal cost: 472.088 MU.

plants I and 2 are assumed to be of the form (9).

f l ( P 1 ) = a t p l + b l p t 2 (19a)

f 2 ( P 2 ) - a2P2 (19b)

where ata2 are expressed in money unitslMW output and bt in money units[(MW) 2. Number of stages in the planning period is taken as 3 and Table 2(a) gives the demand

forecasts for the stages. The penalty factors, interest and inflation rates, and the optimal expansion strategy are given

in Table 3(a). Problem B. Table l(b) gives the system composition at the initial stage and Table 2(b) gives

the demand requirements for a ten stage expansion problem. There are 4 types of units available for expansion. The unit sizes and other relevant data are

given in Table 3(b). The interest and inflation rates, penalty factors are given in Table 4. The operating cost of a unit of type j is calculated from:

lot = 8760*/3~CFfjn i

where nl = number of units of type j ; / / i = cost per M W H for unit of type j; CF~ = capacity factor for a unit of type j; Pj = rating capacity in M W for a unit of type j; and

C (1 + i) t ~-, (n + r)T ~+t - 1 o = ~ ~ _ I r ( l+r )T i+ t loj

where NT = number of types of units. Optimal expansion policy which satisfies the demand and requiring minimum cost is given in

Table 5.

Long range stochastic planning of generation considering overdesign

Table I(b). Initial system composition

Unit capacity Number of Forced outage (MW) units rate Load factor

2 3 3 8

I0 17 17 27 38 38 44 44 48 48 48 69 69

109 109 109 255 440 455

0.1 0.03 0.022 0.03 0.I 0.03 0.022 0.03 0.009 0.03 0.076 0.03 0.077 0.03 0.1 0.03 0.056 0.03 0.01 0.03 0.032 0.03 0,005 0.03 0.022 0.03 0.004 0.03 0.01 O.25 0.041 0.25 0.015 0.25 0.029 0.25 0.007 0.25 0.008 0.25 0.019 0.8 0.050 0.8 0.038 0.8

Table 2(b). Energy and demand forecast

Energy Mean demand Standard deviation Stage GWH (MW) (MW)

I 4275 1522 46 2 4639 1633 49 3 4994 1752 53 4 5373 1880 56 5 5778 2017 61 6 6218 2164 65 7 6692 2322 70 8 7203 2492 75 9 7751 2673 8O

10 8341 3036 90

Table 3(b).

33

Type of unil

Capacity (MW)

Cost Life expectancy Capacity Carrying (mills/ $/kW (years) factor charge kWh) FOR

I 100 150 20 0.05 0.2 35 0.1 2 200 200 30 0.2 0.175 20 0. I 3 400 350 30 0.70 0.175 10 0.05 4 600 300 30 0.70 0.175 10 0.07

Table 4.

AI=10 -6 A2=I0 r=lO% i=5%

Table 5. Optimal cost $1160.42 x i0 s

Type Type Type Type Stage 1 2 3 4

1 0 0 0 0 2 0 0 0 0 3 0 0 I 0 4 0 0 0 0 5 0 0 1 0 6 0 0 0 0 7 0 0 0 0 8 0 0 I 0 9 I 0 0 0

10 3 0 0 0

34 K. RAMACHANDRAN and J. SHARMA

5. CONCLUSIONS

Model presented here for the long-term planning of power systems gives an expansion strategy which provides a reliable power supply under uncertainties of demand forecasts and unit availabilities. It also considers the effect of interest and inflation rates in the determination of long-term policies. Assigning proper penalty costs, the design aims at a trade-off between system unreliability and reserve requirements. The formulation is capable of determining the planning of the system based on maximum power demand or maximum power and energy requirements and with different type of cost models.

The present model is better than the models based on conventional dynamic programming method, as it does not have the drawbacks such as limited scope, computational burden imposed by the multiple uncertainties and reliance upon experienced planning engineer to produce particular configurations of the latter.

Constraining the choice of the state variables at each stage, the size of the problem can be considerably reduced. Preference order dynamic programming requires the evaluation of lesser number of alternative policies in comparison with the dynamic programming method. These results in the reduction of computer time and memory requirements for the solution of the

problem.

REFERENCES I. R. L. Sullivan, Power System Planning, pp. 61. McGraw-Hill, New York 0977). 2. J. Peschon and P. H. Henault, Long-term power system expansion planning by dynamic programming and production

cost simulation. IEEE Syrup. Adaptive Process Design and Control, Austin. Texas (Dec. 1970). 3. R. R. Booth. Optimal generation planning considering uncertainity. IEEE Trans. on Power Apparatus and Systems PAS

91, 70-71 (1971). 4. J. Weinsner and A. G. Holzman, Engineering design under risk. Management Sci. 19(2), 235-249 (1972). 5. L. G. Mitten, Preference order in dynamic programming. Management Sci. 21(I), 34-46 (1974). 6. R. N. Adams. B. Borkowska and C. H. Grigg, Probabilistic analysis of power flows. Proc. of lEE 121(2), 1551 0974). 7. A. Papoulis. Probability, Random Variables and Stochastic Process, pp. 165. McGraw-Hill, New York (1965). 8. L. K. Kirchmayer. Economic' Operation of Power Systems. Wiley, New York (1958). 9. R. Billinton and S. S, Sachdeva, Optimal real and reactive power operation in hydro-thermal system. IEEE Trans. on

Power Apparatus and Systems PAS 91, 1405-1411 (1972).