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
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 + 1 1... 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
(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
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). 4 5. 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
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 + 1. 1.. 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, 16 1 1 0-1 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, 7 15 16 0 5 49 73 97 168 7:, (5%, 10%, 10%, 5%) ( 1 16 ) 1 3 4 5 6 7 8 9 10 11 1 13 14 5
1: 5-48 49-7 73-96 97-10 1 0.065 0 0 0 0 0.065 0 0 0 +10% 3 0.065 0 0 +0% 0 4 0.065 0 0 +0% +10% 5 0.065 0 +0% 0 0 6 0.065 0 +0% 0 +10% 7 0.065 0 +0% +0% 0 8 0.065 0 +0% +0% +10% 9 0.065 +10% 0 0 0 10 0.065 +10% 0 0 +10% 11 0.065 +10% 0 +0% 0 1 0.065 +10% 0 +0% +10% 13 0.065 +10% +0% 0 0 14 0.065 +10% +0% 0 +10% 15 0.065 +10% +0% +0% 0 16 0.065 +10% +0% +0% +10% (10%, 0%, 0%, 10%) 1 16 ) (1%, %, %, 1%) 6. (SUC), 0-1, (ASCOP) 10000 = 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 (1 3669641/385835) 8. 0 4 48 7 96 10 144 168 8: 7 1 3 4 5 6 7 -, 8-9 10 -,,, (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
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