An Analytic Approach to Unit Commitment

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AN ANALYTIC APPROACH TO UNIT COMMITMENT F. D. Galiana, F. Zhuang, A. Meriched and R. Calderon D l'/JI/I '!III/' II! of ElI'/'Irim/ ElIgill/' nillg, .\/ r (;i// L'lIi"I'ni!y, ,\lOIl!rNl/, (21/(·hl'(, (;//1/(1//0

ABSTRACT: Unit commitment in electric po~er systems is a mixed integer programming problem ~hich because of its high complexity has been approached primarily through numerical methods. Fe~ general analytic results exist ~hich shed light on the manner in ~hich units are committed. In this paper the unit commitment problem is studied from an analytic point of vie~ , A set of analytic conditions governing the s~itching mechanism of the system units is derived for the static case in terms of the load, reserve margins, load incremental cost, and reserve incremental cost. This analytic approach, termed the s~itching curve concept, provides ne~ and unique insight into the unit s~itching mechanism not available from purely numerical approaches, as ~ell as laying the ground~ork for further related ~ork on more general unit commitment formulations. Ke~ords:

Po~er

generation; unit commitment; analytic solution. (1) The concept of s~itching curves is developed as a set of analytic conditions for the optimum static unit commitment.

INTRODUCTION The thermal unit commitment (UC) problem is one of various daily tasks faced by utility operators in addition to load forecasting, inter-utility economic transa ctions , and system security analysis. The UC should minimize the total cost over the period of study usually ranging bet~een 24 and 168 hours, ~here the cost includes fixed and variable operational costs as ~ell as start-up and banking costs. The optimum schedule should, in general, also satisfy mInImum up and do~n times for each unit, output ramp limits, area reserve limits and security constraints (Yood & Yollenberg, 1984).

(2) This concept systematically reduces the high dimensional combinatorial complexity of the UC problem by explicitly characterizing a relatively small number of regions of constant unit commi tmen t. (3) A natural physical interpretation of the conditions for optimum unit selection is demonstrated in terms of the average unit cost and the system incremental cost . (4) The nature of the optimum schedules in terms of the problem parameters can be analytically studied. Thus, the effect of system load, and reserve margins on the optimum solutio n can be studied for specific or for general cases.

Mathematically, UC can be formulated as a mixed-integer programming problem. The various approaches available for its solution fall roughly into the categories of heuristic methods (Le et al,1983, Pang & Chen,1976), dynamic programming (van den Bosch & Honderd,1985, Pang et al,1981), integer programming (Cohen & Yoshimura,1983, Dillon et aI, 1978), and Lagrangian relaxation techniques (Merlin & Sandrin, 1983, Lauer et aI, 1982, Muckstadt & Koenig, 1977). Many more references are available in the literature, the above being just a representative sample.

(5) Some very fast solution strategies are being developed for the static UC problem. These can be practically useful as lo~er bound solutions or as the basic solution engine of the more general dynamic UC problem. (6) This paper also has tutorial value since it clearly illustrates the properties and limitations of some integer optimization techniques as applied to unit commitment.

A common thread among the above mentioned approaches to UC is that their output consists of a single numerical solution to the given UC problem. Although such numeri cal methods are necessary to solve specific cases, they do not easily provide insight of an analytic nature about the mechanism governing the s~itching of units. The purpose of this paper is therefore to present an analytic approach for the study and solution of the a class of unit commitment problems, namely the static case, ~hich complements the purely numerical approaches mention"d above. Al though the s ta tic UC problem treated here does not consider various practical dynamic constraints ~e feel that the analytical insight gained is unique and is of value for both the static and dynamic case. Some of the results of this ~ork are:

LIST OF SYMBOLS output of unit i

P

Po~er

u,

Unit s~itching function and = 0 if unit is off.

1

~hen

=

unit is on.

1 if unit is on

Vector of P 'so i

Vector of u 's o i

Number of available units. C/P)

~hen

unit is on.

C

System cost per hour.

P

Specified system load.

R ,

Specified minimum reserve margin.

R

Actual reserve margin.

d

mln

285

Cost per hour of unit i

2S6

F. D. Galiana

~I

al.

the well known classical equal incremental cost criterion solution, the solution to step A then takes the form given below:

Minimum generation output. Maximum generation output. System load incremental cost.

pmin

Reserve incremental cost.

.

P

p . (A)

min

Minimize ~

C

i=1

, P

..

C (P)

(1)

subject to the power balance equation, n

E u. P i=1 •

'

..

i

The basic static UC problem can be formulated as follows: For each time interval of the unit commitment period, given the load, Pd , and the minimum reserve margin, R , solve problem 51:

(2)

P

d

and to the reserve margin requirement

.

(A-a )/b . pmax

PROBLEM FORHULATION

n E u

i

(9)

.

'

It should be noted that the economic dispatch solution in terms of A has the very simple analytic (piece-wise linear) form given by the function Pi(A) in equation (9). Although step (B) of the solution calls for A to be eliminated and replaced by a function of u (which can be done from equations (2) and (9», it was observed that a much more elegant and better behaved problem results in step (B) if the optimization is carried over u and A with the power balance equation (2) as an explicit constraint. A major simplification has ne vertheless still been achieved over the original problem, 51, since the n unknown variables P are now all known but for the scalar A. Optimization over u and A

P

d

.

wi th

u.

+

R > Pd

+

Rmin

(3)

(4)

0 or 1

Step B of the general solution methodology consists now of minimizing the cost over the 0-1 variables u and the continuous variable A. This step is denoted problem 52:

(5)

Minimize ~,

SOLUTION OF THE STATIC UC PROBLEM

A

n E u

i::l

subject to the power balance equation

General solution methodology

n

Pi (A)

E u

i=l

The solution of problem 51 can be approached by the following standard decomposition method: A) Minimize the cost with respect to P holding ~ constant. Find the corresponding optimum P as a function of ~. Denote it as ~(~). B) Replace P in the cost function by ~(~). Minimize the cost over u. Step (A) is the well known economic dispatch problem where the unit commitment, u, is given, and the generations, P, are to be found. This step can be solved analytically for an explicit function, P(u). Step (B) is a more difficult nonlinear minimization over the vector of integer valued variables, u. The contributions of this paper are centered prImarily around step B.

The solution of step A, the economic dispatch step, assumes that, within the permissible generation range, the unit operating costs are convex and quadratic, of the form, + 112 b

p2 i

(6)

This assumption could be easily generalized to a convex and piece-wise quadratic function. Outside of the permissible interval of Pi' the cost function is assumed to be infinitely large. The mlnlmum and maximum unit incremental costs are defined by,

max

X.

i

a

+ b

ai + b

pmin

(7)

pmax

(8)

i

•

i

P

(11)

d

and to the reserve margin constraint (3). Note that the limits on Pi will be automatically satisfied if

P

=

.

P.(A) and A satisfies, max

max

Xi

(12)

Solution Methods In this paper we are primarily interested in obtaining results of an analytic nature about the solution of problem 52. These results can be obtained in a number of ways. One way is to assume that the variables u . are continuous over [0,1) followed by an appiication of the Kuhn-Tucker optimality conditions. This approach will however yield some optimum values of u i which are not or

°

Analytic Solution of the Economic Dispatch Step

A"'10

(10)

C (P (A» i l l

If the symbol A is used to denote the system incremental cost or the so-called system Lambda in

1 and are therefore infeasible, nevertheless the Kuhn-Tucker approach is still useful as it will yield a lower bound solution to problem 52. Such a lower bound plays an important role in branch and bound solution methods. The Lagrangian relaxation approach (Fischer, 1981) is another powerful methodology which can be applied to mixed integer problems. Some authors (Merlin & Sandrin, 1983, Lauer et al, 1982, Muckstadt & Koenig, 1977) have applied it with success to numerically solve the UC problem. The authors are presently investigating it to obtain analytical results to the more complex dynamic UC problem. The approach used in this paper to derive analytic properties of the solution of problem 52 is the generalized Lagrange multiplier method (Everett, 1963), which can also be thought of as the minimization step in the min-max procedure of Lagrangian relaxation methods.

Analytic Approach

to

Unit Commitment

287

SVITCHING CURVES

The switching curve concept is an analytic result which characterizes the optimum unit commitment combination,u, for a specified pair of Lagrange multipliers (~,A). The first one (~) can be physically interpreted as the incremental cost of the minimum reserve margin, while the second (A) is the system load incremental cost. The switching curve concept is derived from the generalized Lagrange multiplier theorem (Everett,1963) which is a sufficient condition guaranteeing that the UC schedule found for a given ~ and A is the optimum solution of problem S2 for the system load and mInImum reserve margin corresponding to the given pair of Lagrange multipliers (~,A). In the following sections we show how to find the optimum value of u given (~,A), as well as how to select the correct values of ~ and A in order to satisfy a given load

R .

P

'd'

o

if \ (~, A)

>0

(19)

(~, A)

<0

(20)

if 5

k

If ~ and A are such that they lie right on the switching curve, Sk = 0, then the optimum value of u

will be either 0 or 1. Ve call equations (19) (20) the switching curve law. A typical ~witching curve is shown in Figure 1, where u = 1 In the space above the curve, while u =0 belo~ the k curve.

a~d

a

and a given minimum reserve margin,

mln

Derivation of the Switching Curve Law In optimum

this section we show how to obtain the unit commitment u, for a given (~,A). Referring to problem -S2 we first form the corresponding Lagrangian, L, for given positive values of the Lagrange multipliers, ~ and y, n

Analytic Properties of the Switching Curves

n

E u . C. (P (A» -y ( E u . P (A) i=l l. 1 .1 i=l 1 1. _~ {

n E u . P~ax

i",l

1.

}

Vhen the unit costs are quadratic, the switching curve takes on the following simple analytic form, (13)

1.

o:Pma x k

L is then minimized with respect to the continuous variable A and with respect to the 0-1 variables, U . ' The values of u and A thus obtained will define c6rresponding values of reserve margin and load through equations (3) and (11). Everett/s theorem then guarantees that u and A are the optimum solution to problem 52 -with the load and reserve margins found above. The minimum with respect to A can be found by solving dL

FIGURE 1. Switching curve example.

min A > Ak

C (pmin)_APmin k k k a' -2Aa +A' C

k

kO

Ck(p:

2b ax

)

_

(21)

k

A P:'

x

(22)

From this result we can deduce properties of the switching curves: (i) They are composed of linear and one quadratic in A.

o

(14)

dA

(ii) The ~-axis non-negative value,

the following

three segments, two

intersect

occurs

which can easily be shown to be true if and only if

A=

(15)

y

To minimize L over the quantities ~, we must then solve min

integer

value

subject to u = 0 or 1. i

Sk(~,A) = ~ P:'

x

+ A Pk(A) - Ck(Pk(A»

(16)

and the switching curve of unit k as (17)

so that

L(~,A,~,A)

can be written as n

L(~,A,~,A)

from which the optimum

PSPPC- J ·

Sk(~,A)

-E u

k=l

(iii) The A-axis intersect non-negative value given by, C (pmin)/pmin k k k a k + (2 c Ok b )1 / ' k Ck{p:ax)/p:.

This minimization over u follows simply if we first define the following -switching function for unit k, Sk(~,A), as

k

~

is given by

(18)

at

the (23)

A k

L(~,A,~,A)

(20)

x

occurs at another min

k -< Ak

if A

min

if A

k

< Ak -< k max

k -> k

if A

(24) ).,max

A

(25) (26)

(iv) If C (pmin) = 0 , both intersects ~ and A will be zer8, ~o that the switching curve \ill b~ below the first quadrant. The switching curve law then states that such a unit would always be on. This may correspond to a gas fired unit with zero fixed running costs, which would be on, but producing zero power until the system A would be high enough to be economically advantageous to draw power from such a unit.For large units with non-zero fixed operating costs, the switching curve will have the characteristic shown in Figure 1, that is, dividing the first quadrant of the ~ plane into two parts, one where the unit is on and the other where the unit is off. v) Right on the switching curve, the ~ versus A behaviour is such that ~ decreases mono tonically with increasing A.

unit,

vi) Increasing the fixed operat i ng costs of a COk ' shifts the entire switching curve

PHYSICAL INTERPRETATION OF THE SVITCHING MECHANISM

upward . Increased fixed operating costs therefore increase the size of the operating region where the unit s hould be turned off. A unit with a large C

A physical interpretation can be given to the switching mechanism as follow s : The switching curve characteristics can be written as,

would therefore be on only if A is large (large load), or if a is large (large reserve margin).

If

Ok

C

k

pmax

(P (A» k

- - ---------

<

A + a

Pk (A)

REGIONS OF CONSTANT UNIT COMMITHENT Uhen the switchi ng cur ves of all available units are combined on the same a vs A plane, a finite number of regions are defined inside which the optimum unit selection, u, is constant and integer valued. See Figure 2 f ~ r an exam ple. Thes e curves repre s ent the conditions which make it economically preferable to turn a unit on or off rather than continue with the present UC. For instance, in the exa mple of Figure 2, by co ntinuous ly increa s ing A from ze ro with a=O, the units turn on in the order 1,3,5,2,4. If, on the other hand, a i s in cr eased from a t A . 0.5, the units turn on in the sequen ce 1 ,3,2, 4,5 .

°

-- - -- - - -- - P (A) k

then

U

(28)

k

pm!. >:.

Ck(Pk(A))

If

k

Pk (A)

)

A + a

k

Pk ( A)

then U k

(29)

°

°

Consider first the case vhere a (this corresponds to having no active reserve limits), then inequalities (28) and (2 9) s tate that if the av erage cost of operating unit k, Ck/P k , is less (greater) than the system incremental cost, A, then unit k should be on (off) . Thi s conc lu s ion is reasonable fro m an economics point of vi ev. Thus , a unit with a la r ge fi xed cos t (co,) will have a r e latively l arge average cost for l ov load s , and v iII not be turned on unl ess the system A i s suffi ciently large. The introduction of minimum reserve margin generally require s that a> O, so that from (28) and (29), it can be seen that a unit will be on at a higher average cost than A. This is so since,in spite of it s high average cost, the unit is required on to s ati s fy the higher reserve requirement s .

.75 .5

U

4

ANALYTIC PROPERTIES OF THE OPTIMUM UNIT COMMITMENT IN TERMS OF LOAD AND RESERVE MARGIN

.25 U1

Uo

0

U.

0.5

1.0

U3

A

1.5

FIGURE 2: Regions of constant unit commitment in the aA plane. See Appendix A for example data. The number of regions of co nstant unit commitment varies de pending on the operating cost data. If the data are such that the var ious unit curves do not intersect, then there will be at mos t n r eg ions . At th e other extreme, if ea ch cur ve intersects every other curve once then the re will be n' such regions. In the example of Figure 2, there exist 8 such regions denoted by the unit commitment ve c tors, u , as sociated with each region uC k ' k.O , 1, ~~ . ,7, given by: uC o . >

~o · (O,O,O,O,O)

uc uc uc

3 4

5

UC 6

-,

=>

~ 3= (1

.>

~4 •

,0, 1 ,0, 1)

°,

°

(27)

(1 , 1 , 1 , 1 ) u . (1,1,1,0,0) => -5 .> ~6=(1,1,1,1,0)

uc 7 .)

Behaviour of (Pd,R) vs A at constant a Consider first the case where a • 0 , that is, where th ere i s no active reserve limit. In moving to A. Amox along the axis a • 0, it can from A •

uC 1 . > ~1·(1,0,0,0,0) uc .> u =(1,0,1,0,0)

,

The sw itching curve conce pt presented above helps explain th e swit ching mechanism in terms of the quantities A and a. These Lagrange multipliers can be phys i ca ll y inte r preted respectively as the increased cost in S/hour needed to supply the next MU of load, and as a measure of the cost needed to supply the minimum reserve margin, both quantities of signifi can ce to the system operator. Normally, however, it is more common to spec ify t~e s y;!~m load, P, and the minimum reserve margln, R , ins tead dof A and a. As will be see n in this sec tion, there exists an equivalence between ( A,a) and (P ,R) which permit s the analysis to be carried out i~ eit he r plane (Note that R st ands for the actual res er ve margin which should exceed, but viII not necessarily be equal to the specified minimum reserve).

~7·(1,1,1,1,1)

In general, the enumeration and characterization of all the region s of constant unit commitment (satisfying the sufficient conditions required by the switching curves) is not practical or desirable, since these will number between nand n'. A more useful result is the characterization of the vec t ors ~k' from among all possible integer so luti ons, which satisfy a given system load, and whic h meet a minimum reserve requirement. This approach will be discussed in a subsequent section in more detail.

be seen from Figure 2 that the units turn on in the sequence 1,3,5,2,4 . The behaviour of the system load, P , and the reserve, R, as given by equations (11), ~nd (3) respectively, therefore undergo dis continuities whenever the system A reaches a new switching curve in the aA plane requiring a unit to swi tch on . This behaviour i s depicted in Figure 3 . Thus the load is a non-de creasing mono tonic function of A. The reserve, R, on the other hand jumps to a higher le vel whenever a new unit gets turned on, and decreases monotonically until the next switc hing takes place. Note that in general, v ith a-O ,the re serve margin can be zero or ve ry small,an unacceptable operating condition (refer to the dotted line in Figure 3). In Figures 4a and 4b, tbe effect of increasing the fixed value of a on the behaviour of P and R versus A is illustrated. The most apparent e1fect is that the entire reserve margin curve is shifted upwards indicating a higher

.-\nah·lic Approach reserve margin for all operating points . This higher reserve is achieved by utilizing a different s~itching sequence ~hich from Figure 2 can be seen to be 1,3,2,5,4 for ~0.1.

10

L·nit Commitlllent and

if A=l, at ~O only units increases units 5,2 and order.

3

are

on. As u

4 get turned on in this

Optimum Unit Commitment for a Specified Load

MW

A typical behaviour of u versus A for a constant specified load, Pd , is shoyn in Figure 5,

500

indicating that the given load can be satisfied anyyhere on the solid line dra~n. Hoyever, only those portions of the constant P trajectory

400

d

300

entirely inside a region of constant UC produce an integer solution. Those portions of the trajectory on a syitching curve correspond to a non-integer value of the corresponding u i ' and are therefore

fd

200 R

100

, . ... .!.I

~'.

....

0

0

A

lO

0.5

1.5

FIGURE 3: Behaviour of P and R versus A for d

~O.

infeasible. In the example of Figure 5, there are four sections corresponding to integer solutions (the vertical ones). It should be observed that the constant Pd trajectory is not a straight line . Instead, as the trajectory reaches a switching curve and the corresponding unit begins to s~itch on, the system A decreases. This is necessary to maintain the constant P requirement . d

500

.75 /

400

/

[a=.1)

,//

300

( r~.J

200

[a ~ o)

/

100

I

A OL-____-L~~L-~~----_7~ r---...I

o

FIGURE

0.5

4a: Behaviour

Pd

of

~0.1.

R

1.0

1.5

versus A for

~O

and

100 Y-_-LL--4~L_~~+_---A

o

to

0.5

FIGURE 4b: Behaviour of

1.5

R versus

A

for

~O

and

~0.1.

Behaviour of At

P

d

and R vs

constant

A,

FIGURE 5: Constant load trajectory on the UA plane

u

at constant A

the

quantities

d

of increasing u (or R) through the integer solutions ~2' ~s' ~" and ~7 given by equation (27). From Figure 6, one can observe that the first integer solution ~2 yields a value of reserve margin, R, ~hich for a selected minimum reserve margin requirement of 50 M~ ~ould be infeasible. The remaining three integer solutions are hoyever feasible, yith ~s giving the lo~est

P (A) 1

are

feasible reserve margin found by this method .

constant, thus, from equations (11) and ( 3), it is apparent that Pd and R ~ill vary only if the s~itching

variables, u;' change. Along the constant A trajectory, it is clear from Figure 2, that this ~ill happen only ~henever a s~itching curve is reached in ~hich case the corresponding u; goes from 0 to 1, and

both

amount corresponding respectively.

P

d

and to

1.5

to its maximum value. It is observed that, in general, R ~ill increase ~ith increasing a and that. R ~ill pass through at most n values corresponding to at most n possible integer solutions. By comparing Figures 5 and 2, one can see that ,in the example, the constant P trajectory passes in order

200

OL-_ __

A

to

0.5

Figure 6 sho~s, for the same example, the behaviour of the reserve margin, R, as one proceeds along the constant Pd trajectory starting ~ith ~O

- ---a:.1 - - a :O

300

0 0

... ...

... ...

~

R

increase

p . (A) 1

by an

and

R

300 200

[P :290) d

100

U7 ·", U .. - 4

\

Us" "

R · : 50

- ----mm A 0L-______- 4________ -+~·'·~·U~2____~~--

o

0.5

to

1.5

It is important to observe here (constant A) that both P and R remain constant for a finite d

range of u as long as u remains inside one of the constant UC regions. In addition, as in the case of constant u, there can be at most n s~itchings over the entire trajectory . For the example of Figure 2,

FIGURE 6: Behaviour of R vs . A for constant Pd

290

F. D. Galiana et al.

The main conclusion summarized as follows:

of

this

section

is

The trajectory in the ~A-plane corresponding to a specified load Pd , generates at most n integer unit commitment solutions satisfying the load constraint. These solutions can be listed in order of increasing ~, or equivalently in order of increasing reserve margin, R. Since the switching curve law is based on a sufficient condition of optimality, we can only guarantee optimality if the minimum reserve margin is equal to one of the reserve margins found as above. If, as in the e~ample of Figure 6, the minimum reserve margin lIes between two of the reserve margins found, then a better solution can be searched for using branch and bound techniques. CONCLUSIONS The unit commitment problem has been examined from an analytic view point . This approach provides some new and unique insight into the mechanism governing the switching on and off of generating units, not possible through purely numerical approaches, by analytically characterizing certain sufficient optimality conditions derived from the method of generalized Lagrange multipliers. The following are the principal results presented in this paper: concept of switching curves is (1) The developed set of analytic sufficient as a conditions for the optimum static unit commitment. (2) This concept systematically reduces the high dimensional combinatorial complexity of the UC problem by explicitly characterizing a relatively small number of regions of constant unit commi tmen t. (3) A natural physical interpretation of the conditions for optimum unit selection is demonstrated in terms of the average unit cost and the system incremental cost. (4) The nature of the optimum schedules in terms of the problem parameters can be readily analyzed. Thus, the effect of system load, and reserve margins on the optimum solution can be studied for specific or for general cases. (5) Fast numerical implementations of the switching curve law are possible for the static UC problem. These can be practically useful as lower bound solutions or as the basic solution engine of the more general dynamic UC problem.

REFERENCES Cohen, A.I., Yoshimllra, M. , "A Branch-and-Rollnd Algorithm for Unit Commitment", IEEE Transactions on Power Apparatus and Systems, Vol . PAS-102, No.2, February 1983, pp.444-451. Dillon, T.S. et aI, Integer Programming Approach to the Problem of Optimal Unit Commitment with Probabilistic Reserve Determination", IEEE Transactions on Power Apparatus and Syst~ Vol. PAS 97, No . 6, November/December 1978, Opp.2154-2166. Everett, H., "Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources", Operations Research, Vol.11, 1963, pp . 399-417. Fisher,M.L., "The Lagrangian Relaxation Method for Solving Integer Programming Problems", Management Science, Vol.27, No.1, January 1981, pp.1-17. Lauer, G.S. et aI, "Solution of Large Scale Unit Commitment Problems", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No.1, January 1982, pp. 79-86. Le, K.D. et aI, "A Global Optimization Method for Scheduling Thermal Generation, Hydro Generation and Economic Purchases", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No.7, July 1983, pp. 1986-1993. Merlin,A. , Sandrin,P. , "A New Method for Unit Commitment Electrici te France" , IEEE at de Transactions on Power and Sl::stems, Apparatus Vol. PAS-102, No.5, May 1983, 1218-1225. Muckstadt, J., Koenig,S.A., "An Application of Lagrangian Relaxation to Scheduling in Power Generation Systems", Operations Research, Vol.25, 1977, pp . 387-403. Pang, C.K., Chen, H.C., "Optimal Short-Term Uni t Commitment", IEEE Transactions on Power Apparatus and Sl::stems, Vol. PAS-95, July/August 1976, pp . 1336-1346 . Pang, C.K. et aI, "Evaluation of Dynamic Multiple Area Programming Based Methods and Representation for Thermal Unit Commitment", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-lOO, No . 3, March 1981, pp. 1212-1218. van den Bosch, P.P.J., Houderd, G., "A Solution of the Unit Commitment Problem via Decomposition and Dynamic Programming", IEEE Transactions on Power Apparatus and Sl::stems, Vol. PAS-104, No.7, July 1985, pp . 1684-1690. lIood, A. J. , lIollenberg, B. F. , Power Generation Operation & Control, John lIiley & Sons, 1984.

ADKNOVLEDGEHENT : This research was funded by the Natural Sciences and Engineering Research Council, Canada, and by Fonds pour la Formation de Chercheurs et l'Aide a la Recherche, Quebec.

APPENDIX A GENERATOR COST DATA Unit Number 1 2 3 4 5

pmin

prnax

(MII)

(MII)

50 40 40 20 40

175 125 125 75 70

a ($/h) ($/Mllh) 0.63 7 0.90 11 0.65 13 1. 20 12 0.84 4

C

0

b ($/MII 2 h) 0.001 0 . 003 0.004 0.002 0.009