Algorithms to solve Unit Commitment Problem

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1 Algorhms to solve Un Commment Problem Takayuki SHIINA The electric power industry is undergoing restructuring and deregulation. This paper reviews mathematical programming models for the un commment. The un commment problem consists of determining the schedules for power generating uns and the generating level of each un. The decisions concern which uns to comm during each time period and at what level to generate power to meet the electricy demand. The problem is a typical scheduling problem in an electric power system. For the stochastic un commment problem, is assumed that demand and price uncertainty can be represented by a scenario tree. Various types of optimization methods can be applied to solve the problems. : 1,.,. (Shahidehpour [10]), (power system) ( [9], Delson Shahidehpour [3], Sheble Fahd [11], Wood Wollenberg[3], Momoh[6]),,,.,, ( []).,,,, ( [16])., (un) (commment) (un commment problem).,, (Sheble Fahd [11]). (Bard [1], Muckstadt Koenig [7]), Takri Birge Long [0], Takri Birge [18] [15],,., (Lagrangean relaxation method) (Bertsekas []),,, 1

2 3 t = 1,..., T t d t, d t. d t T d = (d 1,..., d T ) (scenario)., S, s d s = (d s 1,..., d s T ) ( S s=1 = 1). 1 (directed graph) (scenario tree),. 1 B(1, 1) = B(, 1) 3 B(3, 1) = B(4, 1) 3 4 d 1 d 0 1 1: d s 1, d s (s 1 s ) t, (d s 1 1,..., ds 1 t ) = (d s 1,..., ds t ), t d s1, d s, t, t d s 1 d s t t + 1, t d t (nonanticipativy). {1,..., S},. t s B(s, t), (scenario bundle). 1, B(1, 1) = B(, 1) = {1, }, B(3, 1) = B(4, 1) = {3, 4}. B(s, t) = B(s, t) B(s, t + 1) B(s, t + 1), s < s,, s s t + 1 s s < s T (s), s (spl point). 1,T (1) = 1. 1, T () = T (4) =, T (3) = 1. d s t, t = 1,..., T, s = 1,..., S, (list) (Takri Krasenbrink Wu [1]). s, d s T (s),..., ds T d s T (s) B(s), d s T (s),..., ds T B(s),..., B(s + 1) 1 1 s s T... B(s).. Ḅ(s + 1) 1 B(s + 1)... B(s + )... d 1 1 d 1... d 1... d s T T (s)... d s... 1 T ds+1 T (s+1) d s+1... T : [15],, x s, t = 1,..., T, s = 1,..., S, xs, t = T (s),..., T, s = 1,..., S, t s. 4 (SUC). (Takri Birge Long [0], Takri Birge [18]), 0-1, I, (Yamayee [4], [17]), u i t 0-1 i s t x s 0-1 u, x s f i (x s ) i x s g i(u i,t 1, u i,t ) i, (u i,t 1, u i,t ) = (0, 1), 0

3 (SUC): S min s=1 t=1 T f i (x s )u + T g i (u i,t 1, u i,t ) t=1 x s d s t, t = 1,..., T, s = 1,..., S u u i,t 1 u iτ, τ = t + 1,..., min{t + L i 1, T }, i = 1,..., I, t =,..., T u i,t 1 u 1 u iτ, τ = t + 1,..., min{t + l i 1, T }, i = 1,..., I, t =,..., T q i u x s Q iu, i = 1,..., I, t = 1,..., T, s = 1,..., S x s 1 = xs, i = 1,..., I, t = 1,..., T, s 1, s {1,..., S}, s 1 s, B(s 1, t) = B(s, t) u {0, 1}, i = 1,..., I, t = 1,..., T,,. 1,, i L i 3, i l i 4 Q i, q i i, 5 5 Takri Birge Long [0], Takri Birge [18],, (Rockafellar Wets [8]) [15] u,., (SUC) I xs ds t (Lagrangean multiplier) λ s t ( 0) (Lagrangean relaxation problem) (LSUC). (LSUC): L(λ) = min s=1 t=1 S s=1 t=1 S T λ s t ( x s d s t ) T f i (x s )u + T g i (u i,t 1, u i,t ) t=1 u u i,t 1 u iτ, τ = t + 1,..., min{t + L i 1, T }, i = 1,..., I, t =,..., T u i,t 1 u 1 u iτ, τ = t + 1,..., min{t + l i 1, T }, i = 1,..., I, t =,..., T q i u x s Q iu, i = 1,..., I, t = 1,..., T, s = 1,..., S x s 1 = xs, i = 1,..., I, t = 1,..., T, s 1, s {1,..., S}, s 1 s, B(s 1, t) = B(s, t) u {0, 1}, i = 1,..., I, t = 1,..., T (LSUC) i = 1,..., I, I (LSUC(i)), i = 1,..., I. (LSUC(i)) : S T min f i (x s )u + s=1 t=1 T g i (u i,t 1, u i,t ) t=1 S T λ s t x s s=1 t=1 u u i,t 1 u iτ, τ = t + 1,..., min{t + L i 1, T }, t =,..., T u i,t 1 u 1 u iτ, τ = t + 1,..., min{t + l i 1, T }, t =,..., T q i u x s Q iu, t = 1,..., T, s = 1,..., S x s 1 = xs, i = 1,..., I, t = 1,..., T, s 1, s {1,..., S}, s 1 s, B(s 1, t) = B(s, t) u {0, 1}, t = 1,..., T (LSUC(i)) x s, (GO(i, s, t): generation optimization for un i under scenario s at time t) s = 1,..., S, t = T (s),..., T (LSUC(i)) I xs ds t (4), x s, u (GO(i, s, t)), x s. x s s t T (s), t s = (GO(i, s, t)):min f i (x s ) λs t x s q i x s Q i, u x s (dynamic programming) (1) 3

4 C i (t, k) = C i (t + 1, k + 1) { ( + s {s T (s ) t} [ min C i (t + 1, k) { ( + s {s T (s ) t} C i (t + 1, k + 1) { ( + s {s T (s ) t} ) } fi (x s ) λ s t x s if 1 k < L i ) } fi (x s ) λ s t x s, ) }] fi (x s ) λ s t x s if k = L i C i (t + 1, k + 1) if L i < k < L i + l i min{c i (t + 1, k), C i (t + 1, 1) + g i (0, 1)} if k = L i + l i (1) i k L i + l i,k = 1,..., L i, k = L i + 1,..., L i + l i C i (t, k), i t k, t T., t s {s T (s ) t}, =, t = T t = 1, (LSUC(i)) 3,. t t + 1,,, t, t + 1, C i (t, L i + l i ) C i (t, L i + 1) C i (t, L i ) C i (t, 1) 0 C i (t + 1, L i + l i ) 0 C i (t + 1, L i + 1) C i (t + 1, L i ) 7 g i (0, 1) C i (t + 1, 1) ) λs t x s } s {s T (s ) t} {( )fi(x s 3: (L i = l i = ) (LSUC) (LSUC(i)), (ELD: economic load dispatching problem) s = 1,..., S, t = T (s),..., T (ELD): min f i (x s )u x s = d s t q i u xs Q iu, i = 1,..., I q i u xs Q iu, i = 1,..., I, (ELD) (Lagrange s method of undetermined multipliers). λ ( λ ), L = I f i(x s )u λ( I xs ds t ), f i (x s ), () (3). L x s L λ = df i(x s ) u λ = 0, i = 1,..., I () = dx s x s d s t = 0 (3) (), u = 1 (incremental fuel cost) dfi(xs ) dx s λ (binary search) (lambda eration method) I 0. q iû d s t < 0 I Q iû d s t > 0 λ = f i (q i)u, λ = f i (Q i)u. 1. ˆx s f i (xs )u λ+λ = 0. ˆx s < q i, ˆx s = q i, ˆx s > Q i, ˆx s = Q i.. I ˆxs ds t < 0 λ = λ + λ λ, I ˆxs ds t > 0 λ = λ λ+λ. 1. 4: (LSUC) (SUC), (LSUC) l l + 1, λ l+1 = λ l + α l ξ l (4) α l ξ l L(λ) λ = λ l, L(λ) L(λ l ) + (λ λ l )ξ l (5) 4

5 I xs u ds t λ 4 λ λ 3 1 5: 0. l = 0, λ sl t, s = 1,..., S, t = T (s),..., T. 1. (LSUC(i)), i = 1,..., I. (GO(i, s, t)), i = 1,..., I, s = 1,..., S, t = T (s),..., T x s, i = 1,..., I, s = 1,..., S, t = T (s),..., T, (1) u, i = 1,..., I, t = 1,..., T.. x s, t = T (s),..., T, s = 1,..., S 3. (4), 4. l = l L(λ) (LSUC), ξt l = ( x s d s t ) (6)., α l ξ l 0 l α l ξ l 0 (7). α l = L L(λ l ) ξ l (8), L (LSUC),, α l = θl (UB L(λ l )) ξ l (9) UB (SUC), θ l 0 < θ l < (SUC) 6. 6 I = 10, T = 168 (7 ) 1 1, 1, S = 1, ū, (ASCOP: average supply cost optimization problem) 6: [15], (ASCOP) AMPL [4], ILOG CPLEX10.0 (ASCOP): S T min f i (x s )ū + s=1 t=1 t=1 x s d s t, t = 1,..., T, s = 1,..., S T g i (ū i,t 1, ū i,t ) q i ū x s Q iū, i = 1,..., I, t = 1,..., T, s = 1,..., S x s1 = xs, i = 1,..., I, t = 1,..., T, s 1, s {1,..., S}, s 1 s, B(s 1, t) = B(s, t) S = 16, :, (5%, 10%, 10%, 5%) ( 1 16 )

6 1: % % % +10% % % 0 +10% % +0% % +0% +10% % % % % 0 +0% % 0 +0% +10% % +0% % +0% 0 +10% % +0% +0% % +0% +0% +10% (10%, 0%, 0%, 10%) 1 16 ) (1%, %, %, 1%) 6. (SUC), 0-1, (ASCOP) = 100 (10), (5%, 10%, 10%, 5%) (8%, 16%, 16%, 8%) 4 (ASCOP), 16 (9%, 18%, 18%, 9%), (10%, 0%, 0%, 10%), (ASCOP) (ASCOP), 4.89% = 100 ( /385835) : , ,,, (Takri Birge [19]). Shiina Birge [13], (column generation), Shiina [1],,, ( [5]).,,,,,,,, Hobbs Rothkopf O Neill Chao [5], 4 1. Shiina Watanabe [14],,, (mathematical programming), 6

7 : (ASCOP) (%) (5%,10%,10%,5%) (6%,1%,1%,6%) (7%,14%,14%,7%) (8%,16%,16%,8%) (9%,18%,18%,9%) (10%,0%,0%,10%) [1] J. F. Bard. Short-term scheduling of thermal-electric generators using Lagrangian relaxations. Operations Research, Vol. 36, pp , [] D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 198. [3] J. K. Delson and S. M. Shahidehpour. Linear programming applications to power system economics, planning and operations. IEEE Transactions on Power Systems, Vol. 7, pp , 199. [4] R. Fourer, D. M. Gay, and B. W. Kernighan. AMPL: A Modeling Langage for Mathematical Programming. Scientific Press, [5] B. F. Hobbs, M. H. Rothkopf, R. P. O Neill, and H.- P. Chao, edors. The Next Generation of Electric Power Un Commment Models. Kluwer Academic, 001. [6] J. A. Momoh. Electric Power Applications of Optimization. Marcel Dekker, New York, NY, 001. [7] J. A. Muckstadt and S. A. Koenig. An application of lagrangian relaxation to scheduling in powergeneration systems. Operations Research, Vol. 5, pp , [8] R. T. Rockafellar and R.J.-B. Wets. Scenarios and polycy aggregation in optimization under uncertainty. Mathematics of Operations Research, Vol. 16, pp , [9],,,,.,, [10] M. Shahidehpour, H. Yamin, and Z. Li. Market Operations in Electric Power Systems -forecasting, scheduling and risk management-. John Wiley & Sons, 00. [11] G. B. Sheble and G. N. Fahd. Un commment lerature synopsis. IEEE Transactions on Power Systems, Vol. 11, pp , [1] T. Shiina. A Lagrangian relaxation and column generation algorhm for stochastic un commment problem. Journal of Statistics & Management Systems, Vol. 7, pp , 004. [13] T. Shiina and J. R. Birge. Stochastic un commment problem. International Transactions in Operational Research, Vol. 11, pp. 19 3, 004. [14] T. Shiina and I. Watanabe. Lagrangian relaxation method for price-based un commment problem. Engineering Optimization, Vol. 36, pp , 004. [15]., Vol. 13, pp , 003. [16]. ( ), Vol. 16, pp , 004. [17],., Vol. 8, pp , [18] S. Takri and J. R. Birge. Lagrangian solution techniques and bounds for loosely coupled mixed-integer stochastic programs. Operations Research, Vol. 48, pp , 000. [19] S. Takri and J. R. Birge. Using integer programming to refine lagrangian-based un commment solutions. IEEE Transactions on Power Systems, Vol. 15, pp , 000. [0] S. Takri, J. R. Birge, and E. Long. A stochastic model for the un commment problem. IEEE Transactions on Power Systems, Vol. 11, pp , [1] S. Takri, B. Krasenbrink, and L. S. Y. Wu. Incorporating fuel constraints and electricy spoy prices into the stochastic un commment problem. Operations Research, Vol. 48, pp , 000. [], [3] A. J. Wood and B. J. Wollenberg. Power Generation, Operation and Control. John Wiley & Sons, New York, NY, second edion, [4] Z. A. Yamayee. Maintenance scheduling: Description, lerature survey and interface wh overall operations scheduling. IEEE Transactions on Power Apparatus and Systems, Vol. PAS101, pp , 198. [5],

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