Multistage stochastic programming model for electric power capacity expansion problem - Shiina and...

Japan J. Indust. App!. Math., 20 (2003), 379-397 Area (3)

Multistage Stochastic Programming Model for

Electric Power Capacity Expansion Problem

Takayuki SHIINAt and John R. BIRGE1

t Communication and Information Research Laboratory,Central Research Institute of Electric Power Industry,2-11-1, Iwado -Kita, Komae-Shi, Tokyo 201-8511, JapanE-mail: [email protected]

t Department of Industrial Engineering and Management Sciences,McCormick School of Engineering and Applied Science, Northwestern University,21.¢5 Sheridan Road, Evanston, Illinois 60208, U.S.A.E-mail: [email protected]

Received April 26, 2002

Revised June 9, 2003

This paper is concerned with power system expansion planning under uncertainty. In ourapproach, integer programming and stochastic programming provide a basic framework.We develop a multistage stochastic programming model in which some of the variables arerestricted to integer values. By utilizing the special property of the problem, called blockseparable recourse, the problem is transformed into a two-stage stochastic program withrecourse. The electric power capacity expansion problem is reformulated as the problemwith first stage integer variables and continuous second stage variables. We propose anL-shaped algorithm to solve the problem.

Key words: stochastic programming, optimization under uncertainty, electric powercapacity expansion problem, L-shaped method, block separable recourse

1. Introduction

This paper is concerned with the capacity expansion planning of power systemsunder uncertainty. The basic objective of the capacity expansion planning is todetermine an investment schedule for the installation of new generation plants andeconomic operations which ensure a reliable supply to the electricity demand.

Since the long-term planning problems involve uncertain data, methods whichcan deal with stochastic factors have been developed. In Murphy et al. [14], theuncertainty in demand is incorporated into stochastic programming with recourse.Dapkus and Bowe [6] developed a stochastic dynamic programming method forexpansion planning. Gorenstin et al. [10] examined an approach to expansion plan-ning based on stochastic programming and decision analysis. Shiina [15] developeda chance-constrained programming model for capacity expansion, where electricitydemand is defined by a random variable. Electric power industry is undergoingrestructuring and deregulation. It is necessary for electric power utilities or powergenerators to incorporate uncertainty into power generation planning. Hence, thedevelopment of an effective algorithm to solve the capacity expansion problem un-der uncertainty is required.

380 T. SHIINA and J.R. BIRGE

Stochastic programming (Birge and Louveaux [4]) deals with optimization un-der uncertainty. Birge [3] is a state-of-the-art survey in this field. A stochastic pro-gramming problem with recourse is referred to as a two-stage stochastic problem.In the first stage, a decision has to be made without complete information on ran-dom factors. After the value of random variables are known, recourse action can betaken in the second stage. For the continuous stochastic programming problem withrecourse, an L-shaped method (Van Slyke and Wets [18]) is well-known. It camefrom the shape of the non-zeros in the constraint matrix. The L-shaped method wasused to solve the stochastic concentrator location problems (Shiina [16, 17]). For amultistage stochastic programming with recourse, nested decomposition methodshave been studied by Birge [2] in the linear case and Louveaux [12] in the quadraticcase. Louveaux [13] introduced the concept of block-separable recourse. This prop-erty is essential for capacity expansion as described later. Birge et al. [5] exploreda parallel implementation of the nested decomposition algorithm.

An important problem has been left unsolved in modeling the capacity expan-sion problem. The capacity expansion models we mentioned above are based on astochastic linear programming problem with continuous decision variables. How-ever, decision variables are restricted to integer values in some real problems. Forexample, the decision to build new plants or not is represented by a binary variable.To evaluate the exact investment and operation cost of power generation, we de-velop a multistage stochastic programming problem in which some of the decisionsare binary variables.

A stochastic integer programming problem is a difficult problem to solve. Ifinteger variables are involved in a recourse problem, optimality cuts do not pro-vide facets of the epigraph of recourse function. It is difficult to approximate therecourse function of a multistage problem, since it is necessary to consider the nest-ing of integer programming problems. In the capacity expansion problem, integervariables are involved only in the decisions about the investment of new technology.By utilizing the special property called block separable recourse (Louveaux [13]),the decision variables are classified into two categories, the aggregate level decisionand the detailed level decision. Binary decision variables are involved only in ag-gregate level decision. The algorithm we develop exploits this property and solvesthe problem efficiently.

In Section 2, we review the definition of multistage stochastic programmingproblems and block separable recourse. In Section 3, we show that the electric powercapacity expansion problem is reformulated as the problem with first stage integervariables and continuous second stage variables. In Section 4, we present numericalresults obtained from applying our solution approach to a power generating system.

Multistage Stochastic Programming for Electric Power 381

2. Multistage Stochastic Programming Problem

We consider a multistage stochastic linear programming problem with stagest=0,1,...,Has

(Multistage Stochastic Linear Programming Problem)

min c°x° + E£ i. [min cl x' + • • • + EeuIe1...^H_1 [min cHxH] .. .

subject to W° x° = h°T° x° + W lx l = h 1 , a.s.

TH-1xH-1 +WHxH = hH , a.s.

x° > 0, xt > 0, t = 1, ... , H, a.s.,

where c° is a known vector in R'" 0 , h° is a known vector in mO, and each W t isa known mt x nt matrix. Bold face vectors and matrices are possibly stochastic,where ct , ht , Tt are in nt, amt, J2mt x Fnt, respectively. Let xt denote decision

vector in frt for stage t, t = 1, ... , H. They are chosen so that the constraints

hold almost surely (denoted a.s.).We assume the stochastic elements are defined over a finite discrete probability

space (E, a(S ), P), where S = 1 x ... x 5H is the support of the random data

in each stage with S t = {6s = (Ts, hs, cs), s = 1, ..., kt} and (Ti, hs, cs) is a

realization of (Tt, ht , ct). The possible sequences of the realization of random

variables (6 1 , ... , 6H ) are called scenario. The scenarios are often described usinga scenario tree as shown in Fig. 1. In stages t < H, we have limited number ofpossible realizations which we call the stage t scenarios.

Scenario

S

0 1 2 Stage

Fig. 1. Scenario tree.

In a scenario tree, the stage t scenario connected to the stage t — 1 scenario

s is referred to as a successor of stage t — 1 scenario s. The set of all successors

of stage t — 1 scenario s is denoted by D t (s). Similarly, the predecessor of stage t


scenario s is denoted by a(s, t). These relationships are illustrated in Fig. 2.

s ^ 0 De(s)

^is'tl

Stage t — 1 ^` OStage t — 1

Stage t

Stage t

Fig. 2. Successor and predecessor.

We can formulate a deterministic equivalent problem for the multistage stochas-tic linear programming problem by replicating the constraints for each possibleevent in E t , t = 1, ... , H since the probability space is finite and discrete. To solvethe deterministic equivalent problem, the nested decomposition method (Birge [2],Birge et al. [5]) can be used. But this problem becomes large as the number ofstages or the number of realizations increase. However in some problems, we canavoid difficulty in solving the problem if the problem has a special structure. Theconcept of block separable recourse was introduced by Louveaux [13].

DEFINITION 2.1. A multistage stochastic linear program has block separablerecourse if for all stages t = 1, ... , H the decision vectors x t can be written as x t =(w t , yt ) where wt represents aggregate level decisions and y t represents detailedlevel decisions. The constraints also follow these partitions:

1. The stage t objective contribution is ctxt = rt wt + gtyt

2. The constraint matrix W t is block diagonal: W t = (At 0 I , where A t is

associated to the vector w t and Bt to the vector y t . JJJ

3. The other components of the constraints are random but we assume that Ttt t

and h t can be writte: Tt = I / St 0) and ht =

( ,) to conform with the

(w t , yt ) separation.

If the problem has block separable recourse, the problem can be rewritten asfollows.

(Multistage Stochastic Linear Programming Problemwith Block Separable Recourse)

min rowo + go yo

+ E£1 [min(rlwl + ql y')

+ E^H1^1...^H-1 [min(r Hw H + gHyh')] ... ]


subject to A°w° = b°Bo y° = do

R°w° + A'w l = b', a.s.

S°w° + B l y 1 = d', a.s.

RH-1 wH-1 + AHwH = bH , a.s.

SH-1 wH-1 + BHyH = dH , a.s.

w0 > 0, wt > 0, t = 1, ... , H, a.s.y° > 0, y t > 0, t = 1, ..., H, a.s.

The equivalence of multistage programs with block-separable recourse and two-stage programs was shown in Louveaux [13].

PROPOSITION 2.1 (Louveaux [13]). A multistage stochastic program withblock separable recourse is equivalent to a two stage stochastic program, where thefirst-stage is the extensive form of the aggregate level problems, and the value func-tion of the second stage is the sum (weighted by the appropriate probabilities) of thedetailed level recourse functions for all stage t scenarios, t = 1, ... , H.

From the proposition, the multistage stochastic program with block separablerecourse can be transformed into the two stage stochastic program with recourse.The electric power capacity expansion problem turns out to be the problem thathas first stage integer variables and continuous second stage variables.

The deterministic equivalent problem for the multistage stochastic program-ming problem with block-separable recourse can be written as

(Deterministic Equivalent for Multistage StochasticLinear Programming Problem with Block Separable Recourse)

min rl'0w1,0 + g l,oy l,o

Kl KH+ 1r51ws1 + , , . + HrsHwsH

ps pss=1 s=1

KI KH+ E psQ9(w

1,0) + ... + Ep

H(̂ H (wa(S H) H-1 )s=1 s=1

subject to Ao w 1,o = bo

Boyl,o = do

Ra(s,l),°ws0 + A 1 ws 1 = bs l s = 1 K

Ra(s,H),H-1 ws,H-1 + AH wsH = bsH S = 1 KHr..., Hw°>0, w5t>0, s=1,...,Kt , t=1,...,HQs(wa(s,t),t-1) = min{gstyst I sa(s,t),t—l wa(s,t),t-1 + Bt yst = dst

yst > 0}, s = 1, ... , Kt, t = 1, ... , H,


where:

Kt = Cumulative number of realizations through stage t,Kt=koxk,x...xkt,t=1,...,H,ko=1,

(rst, qst) = Realization of random objective coefficents (r t , qt ) for stage t scenarios, s = 1, ... , Kt , t = 0,1, .. . , H, (r1,o , q"°) _ (r°, qo) for the sake ofsimplicity,

(Rst , Sst) = Realization of random technology matrix (Rt , St ) for stage t scenario

ws t = Aggregate decision vector to take in stage t for stage t scenario s,

yst = Detailed level decision vector to take in stage t for stage t scenario s,

ps = Probability that stage t scenario s occurs, s = 1, ... , Kt , t = 1, . .. , H,

a(s, t) = The predecessor or parent of stage t scenario s, s = 1, ... , Kt , t =1,...H.

For the scenario tree of Fig. 1, the matrix structure of block separable recourseand the transformed two-stage stochastic programming problem are illustrated inFig. 3 and Fig. 4, respectively.

Stage 0 Stage 1 Stage 2w '0 y'° w 11 y" w21 y21 w 12 y 12 w22 y22 w32 y32 w42 y42

A° b°

B° d°

R io Al b"

s io B' d"Rio Al bei

Sio B' d2I

R" A2 b12S '1 B2 d'2

R'1 A2 b22

S11 B2 d22

Rei A2 b32

Sei B2 d32

R21 A2 b42

Sei B2 d42

Fig. 3. Matrix structure of block separable recourse.

In Fig. 4, it is shown that the aggregate level decision w and the detailed leveldecision y are divided into stage 0 and stage 1, respectively.


Stage 0 Stage 1w 10 w" w21 w 12 w22 w 32 w42 y lo y ll y2, y 12 y22 y32 y42

A°Rlo Al b"

Rlo Al b21

R'' A2 b12

R"l A2 b22

R21 A2 b32

R21 A2 b42

B° d°Slo B' l

d"S10 B'

d21

S11 B2 d12

5.11 B2 d22

S21 B2 d32

S21 B2 d42

Fig. 4. Matrix structure of transformed two-stage problem.

3. Electric Power Capacity Expansion

We consider the application of the multistage stochastic programming prob-

lem to the electric power capacity expansion problem. The basic objective of the

problem is to determine an investment schedule of new technology and to operate

power plants to ensure an economic and reliable supply to electricity demand. As

an aid in long range system planning, production cost models are widely used to

calculate a generation system's costs, requirements for new generation plants and

fuel consumption. In expansion planning, load models which cover weeks, months

or years are required. Wood and Wollenberg [19] classified the energy cost programs

and load models as shown in Table 1.

Table 1. Energy production cost programs.

Load Model Total Energyor Load Duration

Load Durationor Load Cycles

Load Durationor Load Cycles

Load Cycle

Long-Range Seasons Months Months, Weeks WeeksPlanning or Years or weeks or Days or Days

Operation Months Months, Weeks WeeksPlanning or weeks or Days or Days

The load patterns are modeled by load cycles or load duration curves. For

long-range planning, block approximations of load duration curves are used. A

load duration curve represents the number of hours in which the load equals or


exceeds the given load value. A typical load duration curve is illustrated in Fig. 5.

Load

f̂í1

0 8760 hour

Fig. 5. Yearly load duration curve.

This types of problems have been considered many times over the past years.Anderson [1] is a review on the models used in electric power industry for deter-mining investments. Delson and Shahidehpour [7] reviews how linear programmingand mathematical programming methods are applied to the planning of capitalinvestments in electric power generation. Kleinpeter [11] is a survey of the differ-ent energy models and energy planning methods. Frauendorfer [9] covers modelingsubjects and mathematical programming methodologies in electric power systems.

In this problem, the investment cost, the operation cost or the electricity de-mand are regarded as random. The problem is formulated as a multistage stochas-tic programming problem in which the decision to install new technology or not isrepresented by a binary variable. Let

• t = 0, 1, . .. , H index the period of stages;

• i = 1, ... , n index the available types of plants;

• j = 1, ... , m index the operating modes in the load duration curve.

We also define the following:

• a ti = availability factor of plant i;

• gi = existing capacity of plant i at stage t, decided before t = 0;• C^ = maximum capacity of plant i that can be installed at stage t;• ri = unit investment cost of plant i at stage t;• f i = fixed investment cost of plant i at stage t;• qi = unit production cost of plant i at stage t;• d^ = maximal power demanded in mode j at stage t;

• T^ = duration of mode j at stage t.


We consider the set of decisions as:

• xi = new capacity made available for plant i at stage t;• wi = total capacity of plant i available at stage t;

1, if new capacity for technology i is installed at stage t;• v.= <

0, otherwise;

• yij = generation level of plant i at stage t in mode j.

The multistage stochastic electric power capacity expansion problem is formulatedas follows.

(Multistage Stochastic Electric Power Capacity Expansion Problem)n

min ^(r°w° + f°v°)i=1

n m

+ [min z (riwi + 4Z E Iy j + f à v% Iz-1 j-1

n m

+ EEHI^1...CH - 1 [min> (r'w2 + qH ^ THyH + f Hv 1) ] ...l

i=1 j-1 J

subject to w° = x°, i = 1, ... , nw? = w° + xi, i = 1, ... , n, a.s.

wi = wi -1 + x, i = 1, ... ,n, t = 2, ... ,H, a.s.x° < C°v°, i = 1, ... , nxi < Civi, i = 1, ... , n, t = 1, ... , H, a.s.

n

=d, j = 1, ... , m, t= 1, ... , H, a.s.z-1

yíj < aj(gí + w á -1 ) , i = 1, ..., n, t = 1, ..., H, a.s.j=1

v° E {0, 1}, vi E {0,1}, i = 1, ... , n, t= 1, ... , H, a.s.w° > 0, wi > 0, i = 1, ... , n, t = 1, ... , H, a.s.x°>0, xi>0, i= 1,...,n, t=1,...,H, a.s.

yzj >0, i= 1,...,n, j=1,...,m, t=1,...,H, a.s.

Since the problem has the property of block separable recourse from Proposition 1,

the decision variables v°, v2, w°, wi, x°, xi, i = 1, ... , n, t = 1, ... , H and y2 j > 0,i = 1, ... , n, j = 1, ... , m, t = 1, .. . , H correspond to the aggregate level decisionsand the detailed level decisions, respectively. The structure of constraint matrix at

stage t is shown in Fig. 6, where I denotes an identity matrix and e T = (1, ..., 1).From Fig. 6, it is evident that the problem has block separable recourse.

—I I 0 0 ••• 0

Ci 0

0-I 0 ... 0d Cl.

00 0 1 •.. i

00 0 —e T T

zut 0Xt > 0V t = dt

Y1 —a1g

Yn, —angn


1 0 0 0 ... 0

0 0 0 0 ... 00 0 0 0 ... 0

al 0

0 an

0 0 0 ... 0

wt-1

xt-1aggregate level decisions

Vt-1

t1

detailed level decisions

ynl

Fig. 6. Structure of constraint matrix at stage t.

We assume the stochastic elements in bold face are defined over a finite discreteprobability space (S, a(S ), P), where ° = - 1 x • • • x 5H is the support of therandom data in each period with 5t = { fis = (rit, f, qit, d^, Tj ), s = 1, ... , k} and(ri , f, qi, dj, Tj) is a realization of (r, f t, qi, dd, r ) .

The deterministic equivalent problem for the multistage stochastic electricpower capacity expansion problem can be written as

(Deterministic Equivalent forMultistage Stochastic Electric Power Capacity Expansion Problem)

nmin E( 1,0 1,0 10 10ri wi +f vi )

i=1K I n KH n

+ E 1 s s s s s s sH sHps (ri

l wi

' + f

1i vi

1 ) ... + pHs ^(ri

H wi

H +

s=1 i=1 s=1 i=1Kl KH

+ psQs(w l ,0 ) + ... + EpHQH(wa(s,H),H-1 )

s=1 s=1

subject to wi ' 0 = xi' 0 , i = 1, ... , nst a(5,t),t-1 st =wi =wi +xi , i =1,...,n, s=1 ...,Kt , t=1,...,H

mit < CZvit, i = 1,...,n, s = 1,..., Kt , t = 0,1,..., H

vitE{0,1}, i =1,...,n, s=1,...,Kt , t=0,1,...,H


wi t > 0, i = 1,...,n, s = 1,..., Kt, t = 0,1,..., HxZt > 0, i = 1,...,n, s= 1,..., Kt, t= 0,1,..., H

n m

Qs (wa ( s' t) ' t-1) = min { E qi tT̂ tyi=1 j=1n

i=1rn

E st t 8 < 2i (9i + w (8t)t_1 ),

i i = 1, ... , nj=1

y^ > 0, i= 1,...,n, j = 1 ,...,m}

s = 1,...,Kt , t= 1,...,H,

where:

Kt = Cumulative number of realizations through stage t,Kt=koxkl x...xkt,t=1,...,H,ko=1,

(rst , q8t , fst) = Realization of random objective coefficent vectors (r t , qt , f t )

for stage t scenario s, s = 1, ... , Kt , t = 0, 1, . .. , H,(r"°, f1,o) _ (ro fo) for the sake of simplicity,

(Tst , dst) = Realization of random duration and demand vectors (rrst, dt)

for stage t scenario s, s = 1, ... , Kt , t = 1, ... , H,wst = Total capacity decision vector to take in stage t for stage t

scenario s, s = 1, ... , Kt , t = 0,1, . .. , H,

xs t = New capacity decision vector to take in stage t for stage t

scenario s, s = 1, ... , Kt , t = 0, 1, . .. , H,

vst = Binary decision vector to take in stage t for stage t scenario s,

ySt = Generation level decision vector to take in stage t for stage tscenario s, s = 1, ... , Kt , t = 1, ... , H,

ps = Probability that stage t scenario s occurs,s = 1,...,Kt , t = 1,...,H,

a(s, t) = The predecessor or parent of stage t scenario s,

The electric power capacity expansion problem can be transformed to the prob-lem that has first stage integer variables and continuous second stage variables. Tosolve the problem, the following master problem is formulated.


(Master Problem forMultistage Stochastic Electric Power Capacity Expansion Problem)

n

min ( l,o 1,ori wi + fZ v^ )i=1

Kl n KH n1 s s1 sl s s sH+ ps (r2

l wi + f, v ' )) ... + psH E(rtH w^ + f

sHz vz

sH )

s=1 i=1 s=1 z-1

Kl KH

+ pres + ... p8 8

s=1 s=1

subject to wi'o = x2'0 , i = 1, ..., nSt -(s,t),t-1 S twt =w2 +x , i =1,...,n, s=1,....Kt, t=1,...,H

xit < Civz t , i = 1,...,n, s= 1,..., Kt , t = 0,1,..., Hv2 , E{0,1}, i= 1,...,n, s= 1,...,Kt , t = 0,1,...,H

we t > 0, i = 1,...,n, s= 1,..., Kt , t = 0,1,..., Hx2t > 0, i = 1,...,n, s= 1,..., Kt , t = 0,1,..., H

BS > Qs(wa(S ' t) ' t-1), s = 1, ... , Kt, t = 1, ... , H

In this formulation, the recourse functions Qs(wTh(8, t) , t -1) s = 1, ..., Kt , t =1, ... , H are not known explicitly in advance. Therefore, the optimality cuts areadded to approximate Bs >_ Qs(wa(s't) ,t -1 ) The optimal solution to the mas-ter problem is obtained by solving the mixed integer programming problem. Letv*St w *st x*st i= 1 n s= 0 1 K t= 1 H 0* t s= 1 KZ , 2 , ,^ , ,...> > > >...> t> >...> > s > >..., t,

t = 1, ... , H, be the optimal solution to the master problem. Then the recourseproblem for stage t scenario s is solved at the optimal solution of the master prob-

*a(s,t),t-11em, wi

(Recourse Problem for Stage t Scenario s)

Qt

n m ns s st st=min{gi t

Tt

J yij ^yij = dj t ,j=l,...,mi =1 j=1 i=1

yzt < az(g +wt),t-1) i = 1,...,nj

j=1

y^ > 0, i= 1,...,n, j= 1 ,...,m},

s = 1,..., Kt , t = 1,..., H

If the recourse problem is infeasible, the feasibility cut is added to the formulationof the master problem. If the solution of the master problem and the recourseproblem do not satisfy the inequality Ost > Qs(w*a(s,t),t-1) the optimality cutwhich approximates Qt(w«(8,t),t-1) is added to the master problem. The dual


problem to the recourse problem for stage t scenario s is described as follows.

(Dual Problem to Recourse Problem for Stage t Scenario s)m n

st st — [^ t a(s,t),t-1 stmax dj Aj L ai(gi + wi )µij=1 i=1

subject to pt < qz t'ij , j = 1, ... , m, i = 1, ... , n

µ i > 0, i = 1,...,n

If QS(w*«(s , t) , t -1 ) = +00 , then w*a(8,t),t-1 is not feasible with respect to all con-straints of the multistage stochastic electric power capacity expansion problem.By the duality theory, we have , j = 1, ... , m, µ? t (> 0), i = 1, ... , n so that

- Li-i ai gi + wi «(s ' t) 't-1 st > 0 and ást - " 0. For any feasible^. ^. ( )µi ^ Nst <i - Y fe

(s,t),t-1 there must exist a ys t > 0 such that >n= st = dt ,

j = 1 m— i 1 yij ^ Ej 1 y j < ai(git + w«(s't)'t-1), i = 1, ... , n. Scalar multiplication of these con-

straints by a j t , j = 1, ... , m, Ai t , i= 1, ... ,n yields

m n m n n mE dstÄst - E a. (

g t + w«(s,t),t-1

) f st < E Äst st - E i t st

^ zli i bi — ^ M.7 µi 13ijj=1 i=1 j=1 i=1 i=1 j=1

m n

j=1i=1

which cuts off w*« (s,t),t-1 since m-1 dstêt _

> i lai(gz + w «(s

> t),t-1)ît > 0.j e x=

m n

Feasibility Cut: d^t5 t - ai(gi + w«(s ' t) ' t-1 )µi t < 0 (1)

j=1 i=1

If Qs(w'«(s , t) , t -1) is finite, we have the optimal primal solution y t , i =

1, ... , n, j = 1, ... , m and the optimal dual solution )^ st j = 1 , m, µ*st

i = 1, ... ‚ii. The following inequality holds.

Bs > Qs(w*a(s,t),t-1)

m fl st st st st

= max {^d^tAlt- ai(gz +wi «(s't)'t-1)îti) <q Tj ,j 1,... ,

j= 1x=1 1

m nst *st t *a(s,t),t-1 *st_ dj %^j —> ai(gi +wi )lzi

j=1 i=1

To approximate the recourse function, the optimality cut which cuts off(w*«(8,t),t-1 0*') so that Bst < Qt (w*«(s,t),t-1) is added to the formulation of the


master problem.

m nt st *st t a(s,t),t-1 *stOptimality Cut: Bs > dj Aj - E ai(g i + wi )µi (2)

j=1 i=1

The algorithm of the L-shaped method for multistage stochastic electric power ca-pacity expansion problem is shown in Fig. 7. Since all of the integer variables areaggregate level decision variables, the master problem becomes a mixed integer pro-gramming problem. The upper bound for the optimal objective value is calculatedas follows.

Upper Boundn

_ ( 1,0 *1,0 1,0 '01,0— (r wi +

i=1

Kl n ICH n

+ E 1 sl *sl {s1 H sH *sHps ^(ri wi + Ji v

*s1 )... + ps (ri w

*sH + J

{sHi vi )

s=1 i=1 s=1 i=1Kl KH

+ p1 Q1(w*1,o) + ... + pHQH(w*a(s,H),H-1) (3)s s

s=1 s=1

The value of upper bound for the optimal objective value can be adopted as theapproximate optimal objective value.

• Step 1. Solve Master ProblemSolve the mixed integer programming master problem by branch and boundmethod. Let wt, i = 1 n s = 0 , 1,.. K t 1 H 0* t s =1, ... , Kt , t = 1, ... , H be the optimal solution to the master problem.

• Step 2. Solve Recourse ProblemSolve the recourse problem for stage t scenario s, s = 1, ... , Kt , t = 1, ... , H.

• Step 3. Add Feasibility CutsIf the recourse problem for stage t scenario s is infeasible, the feasibility cut(1) is added to the formulation of the master problem. Go to Step 1.

• Step 4. Add Optimality CutsCalculate Q8, (w*s,t-1), Vs ' E Dt(s), s = 1,.. .,K, t = 0, ... , H — 1. If6s, < (1 - e)Q t , (w* 8, t -1) s' E Dt (s), the optimality cut (2) is added to theformulation of the master problem (E > 0: tolerance). Go to Step 1.

• Step 5. Convergence CheckIf no optimality cuts are added, then stop.

Fig. 7. Algorithm of the L-shaped method for multistage stochastic electric

power capacity expansion problem.


4. Numerical Experiments

The L-shaped method for the multistage stochastic electric power capacity ex-pansion problem was implemented using C language on SUN Enterprise 420R. Thewhole framework of the algorithm was coded in C. The mathematical programmingproblems were described using AMPL [8] and solved by linear programming/branch-and-bound solver CPLEX 7.0. We consider the following two problems.• Problem 1. n = 20, m = 4, H = 4, K4 = 8.• Problem 2. n = 60, m = 4, H = 6, K6 = 16.

In both problems, the load patterns are modeled by yearly load duration curves.The scenarios are generated from the power demand of stage 1 scenario 1 by addingan increase in demand as shown in Table 2.

Table 2. Demand Increases for different scenarios.

Problem 1 stageScenario Probability 1 2 3 4

1 0.125 0 0 0 +10%2 0.125 0 0 +10% +20%3 0.125 0 +10% 0 +10%4 0.125 0 +10% +10% +20%5 0.125 +10% 0 0 +10%6 0.125 +10% 0 +10% +20%7 0.125 +10% +10% 0 +10%8 0.125 +10% +10% +10% +20%

Problem 2Scenario Probability 1

stage2 3 4 5 6

1 0.0625 0 0 0 +10% +10% +20%2 0.0625 0 0 0 +10% +30% +30%3 0.0625 0 0 +10% +20% +10% +20%4 0.0625 0 0 +10% +20% +30% +30%5 0.0625 0 +10% 0 +10% +10% +20%6 0.0625 0 +10% 0 +10% +30% +30%7 0.0625 0 +10% +10% +20% +10% +20%8 0.0625 0 +10% +10% +20% +30% +30%9 0.0625 +10% 0 0 +10% +10% +20%10 0.0625 +10% 0 0 +10% +30% +30%11 0.0625 +10% 0 +10% +20% +10% +20%12 0.0625 +10% 0 +10% +20% +30% +30%

13 0.0625 +10% +10% 0 +10% +10% +20%14 0.0625 +10% +10% 0 +10% +30% +30%15 0.0625 +10% +10% +10% +20% +10% +20%16 0.0625 +10% +10% +10% +20% +30% +30%


The structure of the scenario trees of Problem 1 and 2 are shown in Fig. 8.

Scenario Scenario

0 1 2 3 4 Stage 0 1 2 3 4 5 6 Stage

Problem 1 Problem 2

Fig. 8. Scenario tree of Problem 1 and 2.

We compare the L-shaped method with the branch-and-bound method for thedeterministic equivalent problem. The L-shaped method is applied to the masterproblem. We set E = 0.01(%) in step 4 of the L-shaped method. In solving themaster problem or the deterministic equivalent problem using branch-and-bound,the search is terminated when the best value of lower bound times (1 + 10 -4 ) isgreater than or equal to the best integer objective value. From the optimal solutionof the master problem, the upper bound for the optimal objective value can becalculated. The results of the numerical experiments are shown in Table 3 and 4.From the numerical experiments, it appears that sufficient results were obtained. Itis noticed that the L-shaped method requires far less solution time than the exactbranch-and-bound method, and the same objective value is obtained. The resultsshow that the integer L-shaped method performs reasonably well on relatively largeproblems.


Table 3. Results for Problem 1.

Master Problem

Optimal Objective Value

66375.0

69610

98043

98099

Number of

Added Cuts

4 feasibility cuts

22 optimality cuts

11 optimality cuts

0 cuts

Iteration

Number

1

2

3

4

Approximate Cost by L-shaped 98103 CPU time 152 (sec)

Optimal Cost by branch-and-bound 98103 CPU time 355 (sec)

Table 4. Results for Problem 2.

Iteration

Number

1

2

3

4

Master Problem

Optimal Objective Value

118799

142859

170384

170391

Number of

Added Cuts

28 feasibility cuts

54 optimality cuts

2 optimality cuts

0 cuts

Approximate Cost by L-shaped 170393 CPU time 322 (sec)

Optimal Cost by branch-and-bound 170393 CPU time 1898 (sec)

As for the total capacity, the same solution is obtained by branch-and-boundand L-shaped method as shown in Table 5.


Table 5. Total capacity.

Stage Scenario

OptimalCapacityby B & B

Problem 1Approximate

Capacityby L-shaped Demand

OptimalCapacityby B & B

Problem 2Approximate

Capacityby L-shaped Demand

1 1 255 255 230 315 315 2801 2 255 255 253 315 315 3082 1 255 255 230 315 315 2802 2 255 255 253 315 315 3082 3 255 255 230 315 315 2802 4 255 255 253 315 315 3083 1 255 255 230 315 315 2803 2 255 255 253 315 315 3083 3 255 255 230 315 315 2803 4 255 255 253 315 315 3083 5 255 255 230 315 315 2803 6 255 255 253 315 315 3083 7 255 255 230 315 315 2803 8 255 255 253 315 315 3084 1 255 255 253 335 335 3084 2 276 276 276 339 339 3364 3 255 255 253 335 335 3084 4 276 276 276 339 339 3364 5 255 255 253 335 335 3084 6 276 276 276 339 339 3364 7 255 255 253 335 335 3084 8 276 276 276 339 339 3365 1 364 364 3085 2 364 364 3645 3 364 364 3085 4 364 364 3645 5 364 364 3085 6 364 364 3645 7 364 364 3085 8 364 364 3645 9 364 364 3085 10 364 364 3645 11 364 364 3085 12 364 364 3645 13 364 364 3085 14 364 364 3645 15 364 364 3085 16 364 364 3646 1 364 364 3366 2 364 364 3646 3 364 364 3366 4 364 364 3646 5 364 364 3366 6 364 364 3646 7 364 364 3366 8 364 364 3646 9 364 364 3366 10 364 364 3646 11 364 364 3366 12 364 364 3646 13 364 364 3366 14 364 364 3646 15 364 364 3366 16 364 364 364


5. Concluding Remarks

We developed a multistage stochastic programming model for the electricpower capacity expansion problem. By utilizing the property of block separablerecourse, the L-shaped method solves the capacity expansion problem effectively.The following point is left as a future problem. In the short term planning of powergeneration, it is necessary to incorporate a setup cost for the operation of a powergenerating unit. To model such a system, we have to consider the stochastic pro-gramming problem in which binary decision variables are involved in the detailedlevel decisions. It is a worthwhile problem to investigate.

References

[ 1 ] D. Anderson, Models for determining least-cost investments in electricity supply. Bell J.Econom. Management Sci., 3 (1972), 267-301.

[ 2 ] J.R. Birge, Decomposition and partitioning methods for multistage stochastic linear pro-grams. Oper. Res., 33 (1985), 989-1007.

[ 3 ] J.R. Birge, Stochastic programming computation and applications. INFORMS J. Comput.,9 (1997), 111-133.

[4] J.R. Birge and F. Louveaux, Introduction to Stochastic Programming. Springer- Verlag,1997.

[ 5 ] J.R. Birge, C.J. Donohue, D.F. Holmes and O.G. Svintsitski, A parallel implementationof the nested decomposition algorithm for multistage stochastic linear programs. Math.Programming, 75 (1996), 327-352.

[ 6 ] W.D. Dapkus and T.R. Bowe, Planning for new electric generation technologies— a stochas-tic dynamic programming approach —. IEEE Trans. Power Appar. Syst., PAS-103 (1984),1447-1453.

[ 7] J.K. Delson and S.M. Shahidehpour, Linear programming applications to power systemeconomics, planning and operations. IEEE Trans. Power Syst., 7 (1992), 1155-1163.

[ 8] R. Fourer, D.M. Gay and B.W. Kernighan, AMPL: A Modeling Langage for MathematicalProgramming. Scientific Press, 1993.

[ 9 ] K. Frauendorfer, H. Glavitsch and R. Bacher, Optimization in Planning and Operation ofElectric Power Systems. Springer-Verlag, 1993.

[10] B.G. Gorenstin, N.M. Campodonico, J.P. Costa and M.V.F. Pereira, Power system expan-sion planning under uncertainty. IEEE Trans. Power Syst., 8 (1993), 129-136.

[11] M. Kleinpeter, Energy Planning and Policy. John Wiley & Sons, 1995.[12] F.V. Louveaux, A solution method for multistage stochastic programs with recourse, with

application to an energy investment problem. Oper. Res., 28 (1980), 889-902.[13] F.V. Louveaux, Multistage stochastic programs with block-separable recourse. Math. Pro-

gramming Stud., 28 (1986), 48-62.[14] F.H. Murphy, S. Sen and A.L. Soyster, Electric utility capacity expansion planning with

uncertain load forecasts. IIE Trans., 14 (1982), 52-59.[15] T. Shiina, Numerical solution technique for joint chance constrained programming problem

— an application to electric power capacity expansion—. J. Oper. Res. Soc. Japan, 42 (1999),128-140.

[16] T. Shiina, L-shaped decomposition method for multi-stage stochastic concentrator locationproblem. J. Oper. Res. Soc. Japan, 43 (2000), 317-332.

[17] T. Shiina, Stochastic programming model for the design of computer network (in Japanese).Trans. Japan Soc. Indust. Appl. Math., 10 (2000), 37-50.

[18] R. Van Slyke and R.J.-B. Wets, L-shaped linear programs with applications to optimalcontrol and stochastic linear programs. SIAM J. Appl. Math., 17 (1969), 638-663.

[19] A.J. Wood and B.J. Wollenberg, Power Generation, Operation and Control. John Wiley &Sons, 1996.

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