Solving Unit Commitment Problems in Power Production ... - CiteSeerX

Solving Unit Commitment Problems in Power Production Planning Stefan Feltenmark1, Krzysztof C. Kiwiel2 and Per-Olov Lindberg3 1 2 3

Royal Institute of Technology, Stockholm, Sweden Systems Research Institute, Warsaw, Poland Linkoping Institute of Technology, Linkoping, Sweden

Abstract

Unit commitment in production planning of power systems involves dynamic, mixed-integer programming problems. We show that specialized bundle methods applied to Lagrangian relaxations provide not only lower bounds on the optimal value (as do subgradient methods), but also certain relaxed primal solutions. These solutions can be used for constructing good primal feasible solutions. We present computational experience for large-scale examples.

1 Introduction The unit commitment (UC) problem in the daily operation of power systems determines on/o schedules and power outputs of the generators, so as to minimize the system operating cost over a planning horizon of 24 to 168 hours. It is a large-scale mixed integer programming problem. Many solution methodologies [ShF94] have been proposed for the UC problem. Among the most promising is Lagrangian relaxation [MuK77, MoR94, LPRS96], in which a dual function is formed by minimizing the cost augmented with coupling constraints weighted by Lagrange multipliers. The purpose of this paper is to describe an optimization methodology based on recent advances in bundle methods [Kiw90, HUL93] and the fact [FeK96] that these methods not only solve the dual problem, but also nd certain multipliers that solve the convexi ed primal problem [Sha79, x5.3]. These multipliers may be used to generate good primal feasible solutions [BLSP83]. Such multipliers could also be found by the smoothing methods of [Ber82, x5.6] and [BLSP83], but these methods are restricted to problems with piecewise linear costs. In x2 we formulate the UC problem and its dual problem. In x3 we review convergence properties of our bundle method. In x4 we show how to nd a good primal feasible solution by making use of the solution of the relaxed problem. In x5 we report on the performance of the bundle method on fairly large examples.

2 Unit commitment model Our mathematical model of the UC problem is given by min u;p s.t.

T I (X X

)

uitCi(pit) + Si (ui) ; i=1 t=1 I I X X uitri(pit )  Rt; t = 1: T; uitpit  Dt; i=1 i=1 uitpi  pit  uitpi; t = 1: T; i = 1: I; ui 2 Ui;

F (u; p) :=

(2.1a) (2.1b)

i = 1: I;

(2.1c)

Research supported by the Swedish Research Council for Engineering Sciences, the Goran Gustafsson Foundation and the Polish Academy of Sciences. 

where I is the number of units, T is the number of time periods, Dt and Rt are the demand and reserve in period t, and for each unit i, Ci is a convex cost-power generation function, Si is the startup/shutdown cost, uit = 1 (0) if unit i is operating (shut down, resp.) at time t, pit is the output power in period t, pi and pi are the minimum and maximum output powers, ri(pit) = minfpi ? pit; p
i g is the reserve function, where p
i is the maximumincrease in power, ui = (ui1; : : :; uiT ) is the schedule, and Ui represents minimum up/down times and must on/o constraints. It is convenient to treat the UC problem as an instance of the optimization problem s.t. j (z)  0; j = 1: N; z 2 Z; (2.2) with Z  IRm compact and j : Z ! IR upper semicontinuous, j = 0: N . Suppose the dual function 0(z )

max 0 = max

f (x) = maxf 0(z) + hx; (z)i : z 2 Z g (2.3) can be evaluated at each x 2 S := IRN+ = fx 2 IRN : x  0g by nding z(x) 2 Z (x) := Arg maxf 0(z) + hx; (z)i : z 2 Z g; (2.4) where = ( 1; : : :; N ) and h
;
i is the usual inner product, and that the dual problem minS f is solvable, i.e., X := Arg minS f 6= ;. Since problem (2.2) may be nonconcave (e.g., for (2.1)), consider its relaxed convexi ed version M rel = max X  (z j ) j 0 0 j ;zj M j=1 j =1 f

g

M X

s.t.

j =1

j

(zj )  0;

M X j =1

j = 1; zj 2 Z; j  0;

(2.5)

where M = N + 1. In other words, we choose decisions zj and their probabilities j that solve the problem maxfE 0(z) : E (z)  0g, where E denotes expected value [BLSP83]; see [Sha79, x5.3] for alternative interpretations. Both (2.2) and (2.5) have the same dual function (2.3). If, as is the case for (2.1), (2.2) has the separable form n X

n X

ji (zi )  0;

j = 1: N; (2.6a) z := (z1; : : : ; zn) 2 Z := Z1  : : :  Zn ; (2.6b) with each Zi  IRmi compact and ji : Zi ! IR upper semicontinuous, letting fi(x) = max f 0i(zi) + hx; i(zi)i : zi 2 Zi g ; (2.7) zi(x) 2 Zi (x) := Arg max f 0i(zi) + hx; i(zi)i : zi 2 Zig ; (2.8) Pn where i = ( 1i; : : :; Ni), we have z(
) = (z1(
); : : :; zn(
)) in (2.4) and f (
) = i=1 fi(
). The relaxed problem (2.5) becomes max

0 (z ) :=

i=1

0i(zi )

j (z ) :=

s.t.

i=1










max

M n X X i=1 j =1

ij 0i(zij ) s.t.

M n X X i=1 j =1

ij i(zij )  0;


M X j =1

ij = 1; zij 2 Zi ; ij  0:

Our UC problem is an instance of (2.6) with N = 2T , n = I + 1, zi = (ui; pi ),

(2.9) 0i(zi )

=

? PTt=1 uitCi(pit) ? Si(ui), ti(zi) = uitpit, T +t;i(zi) = uitri(pit), t = 1: T , Zi = fzi : uitpi  pit  uitpi; t = 1: T; ui 2 Ui g, i = 1: I , 0n(zn) = 0, tn(zn) = ?Dt zn, T +t;n(zn) = ?Rtzn, t = 1: T , Zn = f1g. Then (cf. (2.7){(2.8)) fi(x) = ? umin i Ui 2

(X T t=1

[C (p ) ? xtpit ? xT +tri(pit)] + Si(ui) uit pi min pit pi i it 



)

(2.10)

may be evaluated by nding a minimizer zi(x) = (ui(x); pi(x)) of (2.10) via dynamic programming, for i = 1: I , whereas fn(x) = ? hx; (D; R)i.

3 Bundle relaxation The proximal bundle method of [Kiw90, Kiw95] for the dual problem minX f generates a sequence fxk gk=1  S converging to some x 2 X , and trial points yk 2 S for evaluating linearizations D E f k (
) = 0(z(yk )) +
; (z(yk)) of f s.t. f k (xk ) = f (xk ) and (zk) 2 @f (xk ) (cf. (2.3){(2.4)). Iteration k uses the cutting plane model fk = maxj J k f j of f with J k  f1; : : :; kg, jJ k j  N + 2, for nding yk+1 = arg minffk (x) + uk jx ? xk j2=2 : x 2 S g; (3.1) where uk > 0 is a proximity weight. A descent step to xk+1 = yk+1 occurs if f (yk+1 )  f (xk )+ mL vk, where mL 2 (0; 1) is xed and vk = fk (yk+1) ? f (xk )  0 predicts descent (if vk = 0 then xk 2 X and the method stops). Otherwise, a null step xk+1 = xk improves the next model fk+1 . The QP method of [Kiw94] solves (3.1) by nding yk+1 and  k = fk (yk+1) that solve D E min uk jx ? xk j2=2 +  s.t. 0(z(yj )) + x; (z(yj ))  ; 8j 2 J k : (3.2) It also nds Lagrange multipliers jk  0 of (3.2) s.t. Pj J k jk = 1 and J^k = fj : jk 6= 0g satis es jJ^k j  N + 1. To save storage without impairing convergence, one may let J k+1 = J k [ fk + 1g and then, if necessary, drop from J k+1 an index j 2 J k n J^k with the largest linearization error f (xk+1) ? f j (xk+1). We assume for simplicity that the uk -updating technique of [Kiw90, Kiw95] generates fuk g  [umin; umax] for some 0 < umin  umax < 1. Then [Kiw95, Thm 3.1] yields Theorem 3.1. We have xk ! x 2 X , f (xk ) # minS f and vk ! 0. The following two results of [FeK96] show that fjk ; z(yj )gj J^k solve (2.5) asymptotically. Lemma 3.2. Let pk = ?uk dk , ~kp = ?vk ? uk jdk j2  0, 1 = (1; : : : ; 1) 2 IRN . Then E D X k j )) = f (xk ) ?  k ? pk ; xk  f (xk ) + v k ? juk v k j1=2jxk j; (3.3a)  ( z ( y ~ p j j 0 X k k k 1=2 j k k (3.3b) j j (z (y ))  p  ?jp j1  ?ju v j 1; where ~ kp ! 0, pk ! 0, xk ! x, f (xk ) # f (x) = 0rel and v k ! 0. Theorem 3.3. Each cluster point of the bounded sequence ffjk ; z(yj )gj J^k g solves (2.5). P k j ))  rel ? k and P  k (z (y j ))  ?jpk j1 with k =  ~kp ? Remark 3.4. Since  ( z ( y 0 j j j f j 0 f D k kE p ; x ! 0 and pk ! 0 (cf. Lem. 3.2), fjk ; z(yj )gj J^k is a generalized solution of (2.5), and one may stop if kf and jpk j are small enough. The preceding results extend [FeK96] to the separable form of (2.6), i.e., for our problem (2.1). To exploit the additive structure of f = Pni=1 fi, we may use the models fk = Pni=1 fik and fik = maxj Jik fij constructed from the linearizations fij (x) = 0i(zi(yj ))+ hx; i(zi(yj ))i of fi. The sets Jik are selected [Kiw90] by nding Lagrange multipliers ijk  0, j 2 Jik , i = 1: n, of the corresponding extension of (3.2) 1

2

2

2

2

2




2

min uk jx ? xk j2=2 + such that

n X i=1

i s.t.

n X jJ^ikj  N + n; i=1

D

E

j j k 0i (zi(y )) + x; i (z (y ))  i ; 8j 2 Ji ; i = 1: n;


X j J^ik

ijk = 1; i = 1: n;

(3.4) (3.5)

2

and Pi Pj ijk i(zi(yj )) replace the left sides of (3.3), and we have the following extension of Theorem 3.3 [FeK96]. Theorem 3.5. Each cluster point of ffijk ; zi(yj )gj J^ik ;i=1:ng solves (2.9). Remarks 3.6. (a) (3.4)khas at most N + 2n kconstraints if Jik+1 = J^ik [ fk + 1g 8i, 8k. (b) The fact that jfi : jJ^i j > 1gj  N , jfi : jJ^i j = 1gj  n ? N will be exploited in the next section.

where J^ik = fj 2 Jik : ijk = 6 0g. Then Pi Pj ijk

0i(zi (y j )) 2




Case I T best dual Bard 10 24 5.409528e+05 Paci c 19 24 1.889766e+06 EPRI 48 48 2.843720e+06 Emod 48 168 9.909559e+06 Bard168 100 168 3.761877e+07

best primal gap (%) 5.451702e+05 0.77 1.893419e+06 0.19 2.849151e+06 0.19 9.939505e+06 0.30 3.754104e+07 0.21

Origin [Bar88] [LJS96] [ZWC+77] scaled EPRI scaled Bard

Table 5.1: Test problems and their best known dual and primal values

4 Obtaining a feasible primal solution

First, we note that if the schedules ui 2 Ui, i = 1: I , satisfy I X i=1

I X

uitpi  Dt + Rt;

i=1

uitp
i  Rt

(4.1)

for t = 1: T , then we may solve T continuous optimization problems in p to obtain a feasible solution to (2.1). Conversely, any feasible solution to (2.1) must satisfy (4.1). In view of Thm 3.5, we may suppose that, for k large enough, ij = ijk and zij = (uji ; pji ) = zi(yj ), j 2 J^i = J^ik , i = 1: I , form a relaxed solution to (2.1) treated as an instance of (2.9). Then we have uji 2 Ui, j 2 J^i, i = 1 : I , and I X X i=1 j J^i

ij ujitpi  Dt + Rt;

2

I X X i=1 j J^i

ij ujitp
i  Rt t = 1: T:

(4.2)

2

Thus we may use the interpretation of u~it = Pj J^i ij ujit 2 [0; 1] as the probability of unit i to be on-line at time t in various ways to generate feasible solutions to the original problem (2.1). An important observation is that, in view of Rem. 3.6(b), for problems with many more units than time periods, only relatively few u~i can be fractional. A simple way to utilize the relaxed schedules fu~igIi=1 is to put uit = 1 in order of decreasing u~it until (4.1) is satis ed for all periods t, while respecting the requirement ui 2 Ui. This heuristic will be referred to as PSF0 in the computational results presented in the next section. Another approach is to try to introduce some variation into the generated solutions by sampling schedules according to the probability distribution fu~igIi=1, combined with some heuristic that, if necessary, ensures that (4.1) is satis ed. We implemented two such heuristics. The rst simply samples all schedules, treating fu~igIi=1 as probabilities. The second heuristic only allows change of state if some uji changes state at a particular period. The two heuristics will be called PFS1 and PFS2, respectively. 2

5 Computational results In this section we report on our preliminary numerical experience. Table 5.1 gives some details on our testing problems. The nal two problems are fairly large. Table 5.2 compares the quality of primal feasible solutions generated via the various heuristics. Both PFS1 and PFS2 seem to perform quite well, and they tend to improve on PFS0 on larger problems. Table 5.3 gives the running times of our bundle code, obtained on a SPARC-20 machine. The optimality tolerance opttol is used in a stopping criterion of the code. Usually, when opttol = 1e-m is used, upon termination the dual objective value has m correct digits. It can be seen that the running time may grow quite rapidly when higher solution accuracy is required. However, it seems that in practice the accuracy of about four digits (corresponding to opttol = 1e-4) will suce. Further, we expect improvements in the running times once we gather enough experience to incorporate suitable modi cations into our code.

Case IT Bard 1024 Paci c 1924 EPRI 4848 Emod 48168 Bard168 100168

dual value 5.409528e+05 1.889766e+06 2.843720e+06 9.909559e+06 3.761877e+07

PSF0 5.604361e+05 1.893419e+06 2.951717e+06 1.051116e+07 3.848490e+07

PFS1 5.451702e+05 1.894647e+06 2.855289e+06 1.002365e+07 3.754104e+07

PFS2 5.454657e+05 1.896404e+06 2.849151e+06 9.939505e+06 3.761883e+07

Table 5.2: Dual and primal objectives from dierent randomized PFS heuristics

Case Bard

IT 1024

opttol iter time (min:sec) 1e-3 10 0.48 1e-4 24 2.72 1e-5 31 4.18 1e-6 132 20.45 Paci c 1924 1e-3 14 1.52 1e-4 23 4.82 1e-5 38 10.70 1e-6 57 19.02 EPRI 4848 1e-3 16 10.77 1e-4 25 28.50 1e-5 42 57.20 1e-6 60 96.13 Emod 48168 1e-3 16 28.58 1e-4 27 44.88 1e-5 43 1:17 1e-6 96 2:35 Bard168 100168 1e-3 7 57.31 1e-4 44 8:37 1e-5 96 25:51 1e-6 336 58:37

Table 5.3: Performance of the disaggregate bundle method

References [Bar88] [Ber82] [BLSP83] [FeK96] [HUL93] [Kiw90] [Kiw94] [Kiw95] [LJS96] [LPRS96] [MoR94] [MuK77] [Sha79] [ShF94] [ZWC+ 77]

J. F. Bard, Short-term scheduling of thermal-electric generators using Lagrangian relaxation, Oper. Res. 36 (1988) 756{766. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982. D. Bertsekas, G. S. Lauer, N. R. Sandell and T. A. Posberg, Optimal short-term scheduling of large-scale power systems, IEEE Trans. Automat. Control AC-28 (1983) 1{11. S. Feltenmark and K. C. Kiwiel, Lagrangian relaxation of nonconvex problems via bundle methods, Report TRITA/MAT-96-OS3, Department of Mathematics, Royal Institute of Technology, Stockholm, April 1996. J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, 1993. K. C. Kiwiel, Proximity control in bundle methods for convex nondierentiable minimization, Math. Programming 46 (1990) 105{122. , A Cholesky dual method for proximal piecewise linear programming, Numer. Math. 68 (1994) 325{340. , Approximations in proximal bundle methods and decomposition of convex programs, J. Optim. Theory Appl. 84 (1995) 529{548. C. Li, R. B. Johnson and A. J. Svoboda, A new unit commitment method, Tech. report, IEEE Winter Meeting, 1996. C. Lemarechal, F. Pellegrino, A. Renaud and C. Sagastizabal, Bundle methods applied to the unitcommitment problem, in System Modelling and Optimization, Proceedings of the Seventeenth IFIP TC7 Conference on System Modelling and Optimization, 1995, J. Dolezal and J. Fidler, eds., Chapman & Hall, London, 1996, pp. 395{402. A. Moller and W. Romisch, A dual method for the unit commitment problem, Tech. report, Institut fur Mathematik, Humboldt-Univ. Berlin, Berlin, Germany, 1994. J. A. Muckstadt and S. A. Koenig, An application of Lagrangian relaxation to scheduling in thermal power-generation systems, Oper. Res. 25 (1977) 387{403. J. F. Shapiro, Mathematical Programming: Structures and Algorithms, Wiley, New York, 1979. G. B. Sheble and G. N. Fahd, Unit commitment literature synopsis, IEEE Trans. Power Systems 9 (1994) 128{135. H. W. Zeminger, A. J. Wood, H. K. Clark, T. F. Laskowski and J. D. Burns, Synthetic electric utility systems for evaluating advanced technologies, Final Rep. EM-285, Electrical Power Research Institute (EPRI), 1977.