Siripatanakulkhajorn Sakchai Study on Stochastic Optimal Electric Power Procurement Strategies with Uncertain Market Prices Sakchai Siripatanakulkhajorn,StudentMember,YuichiSaisho, Student Member, Yasumasa Fujii,Member,KenjiYamaji,Member The player in deregulated electricity markets can be categorized into three groups of GENCO (Generator Companies), TRNASCO (Transmission Companies), DISCO (Distribution Companies). This research focuses on the role of Distribution Companies, which purchase electricity from market at randomly fluctuating prices, and provide it to their customers at given fixed prices. Therefore Distribution companies have to take the risk stemming from price fluctuation of electricity instead of the customers. This entails the necessity to develop a certain method to make an optimal strategy for electricity procurement. In such a circumstance, this research has the purpose for proposing the mathematical method based on stochastic dynamic programming to evaluate the value of a long-term bilateral contract of electricity trade, and also a project of combination of the bilateral contract and power generation with their own generators for procuring electric power in deregulated market. Keywords: electricity market, uncertainty of electric power price, bilateral contract, own generator, procurement cost 1. 3 113-8656 7-3-1 Graduate School of Engineering, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656 113-8656 7-3-1 Graduate School of Frontier Science, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656 2. (1) (3) d ln p(t) =a t (ln b t ln p(t))dt + σ t dz + ξ m dq m (1) B 124 3 24 413
t p(t) a t t b t t σ t t dz ξ m m dq m m M dz dt dz = ε dt (2) ε dq m ξ m µ m δ m N(µ m,δ m ) dt { 1 λ m dt dq m = (3) (1 λ m )dt λ m (1) 3. 3 1 t p [ /kwh] dt π(t, p)dt = {p(t)d t F (d t )}dt (4) π(t, p) t p [ /h] d t t [kwh/h] F (d t ) d t [ /h] π(t, p)dt + e rdt E t,p [V (t + dt, p + dp)] < Pen t (5) r E t,p [V ] t p V V (t, p) t p t [ ] Pen t t [ ] t V (t, p) V (t, p) =max{ Pen t,π(t, p)dt + e rdt E t,p [V (t + dt, p + dp)]} (6) 3 2 (1) x =lnp p π(t, p) =π x (t, ln p) =π x (t, x) V (t, p) =V x (t, ln p) = V x (t, x) x (3) (4) dv x dv x = V x (t + dt, x + dx) V x (t, x) [ Vx (t, x) = + a t (ln b t x) V x(t, x) + 1 2 ] V x (t, x) V x (t, x) 2 σ2 t 2 dt + σ t dz + {V x (t, x + ξ m ) V x (t, x)}dq m (6) (7) V x (t, x) =max{ Pen t,π x (t, x)dt + e rdt E t,x [V x (t + dt, x + dx)]} (8) E t,x [V x (t + dt, x + dx)] = V x (t, x)+e t,x [dv x ] V x (t, x) > Pen t (7) (8) e rdt (1 rdt) dt 2 = V x (t, x) =π x (t, x)dt +(1 rdt)v x (t, x) [ Vx (t, x) + + a t (ln b t x) V x(t, x) + 1 2 ] V x (t, x) 2 σ2 t 2 dt [ ] V x (t, x) + E t,x σ t dz + E t,x [{V x (t, x+ξ m ) V x (t, x)}dq m ] (9) E t,x [dq m ]=λ m dt ξ m 414 IEEJ Trans. PE, Vol.124, No.3, 24
p m (ξ) N(µ m,δ m ) E t,x [V x (t, x + ξ m )dq m ] ( = V x (t, x + ξ)p m (ξ)dξ ) λ m dt (1) E t,x [dz]= (9) (1) ( ) r + λ m V x (t, x) = π x (t, x)+ V x(t, x) +a t (ln b t x) V x(t, x) + 1 2 V x (t, x) 2 σ2 t 2 + λ m V x (t, x + ξ)p m (ξ)dξ (11) t = T V x (T,x)= ( <x< ) (12) Pen t (8) V x (t, x) Pen t ( <t<t)( <x< ) (13) 4. 3. (11) (12) (13) (5) (1) p(t) 4 1 1 (1) b t 5.5 [ /kwh].72 [ /kwh] (1) a t b t σ t µ t δ t λ t t (11) b t r 1% 2 1 Table 1. Table 2. 2 Assumed parameters. Settings of calculation cases. 1 µ.5 δ.5 (1) 211 45 4 2 3 1 t = V (,p) p 2 1 V (t, p) p V (t, p) 1 V (t, p) = Pen t 2 2 1 3 2 4 3 σ µ δ λ V (,p) 3 4 4 V (,p) B 124 3 24 415
(a) Case 1 (a) Sensitivity to variation in σ (b) Case 2 (b) Sensitivity to variation in µ (c) Case 3 1 Fig. 1. Value of bilateral contract by case. (c) Sensitivity to variation in δ 2 Fig. 2. Value of bilateral contract at initial time point. (d) Sensitivity to variation in λ 3 V (,p) Fig. 3. Sensitivity analysis of V (,p). (4) F (d t ) Pen t p V (,p) F (d t ) Pen t p =1 1 /kwh V (,p) > V (,p) V (,p) < V (,p) 416 IEEJ Trans. PE, Vol.124, No.3, 24
5. 5 1 1 2 kw kwh 3 1 c a E [c a ] c f c v E [c v ] E [c a ]=c f + E [c v ] (14) c f T c f = (f b k p + f g k g )e rt (15) f b [ /kw/day] k b f g [ /kw/day] k g c v T c v = {p m (t, h)d m (t, h) + p b d b (t, h)+p g (t)d g (t, h)}dhe rt (16) Tp m (t, h) t h [ /kwh] d m (t, h) t h [kw] p b [ /kwh] d b (t, h) t h [kw] p g (t) t [ /kwh] d m (t, h) t h [kw] d m (t, h) d m (t, h)+d b (t, h)+d g (t, h) load t,h (17) d b (t, h) k b (18) d g (t, h) k g (19) load t,h t h p m (t, h) p g (t) (i) (ii) (iii) (iv) p m (t, h) <p b p m (t, h) <p g (t) d m (t, h) =load t,h d b (t, h) = d g (t, h) = p m (t, h) >p b p m (t, h) <p g (t) d m (t, h) =load t,h k b d b (t, h) =k b d g (t, h) = p m (t, h) <p b p m (t, h) >p g (t) d m (t, h) =load t,h k g d b (t, h) = d g (t, h) = k g p m (t, h) >p b p m (t, h) >p g (t) d m (t, h) = load t,h k b k g d b (t, h) = k b d g (t, h) =k g t c v,t c v,t = [load t,h p m (t, h)+k b min(,p b p m (t, h)) + k g min(,p g (t) p m (t, h))]dh (2) (2) 2 3 t p m (t, h) p d (t, h) p b = p d (t, h b ) p g (t) =p d (t, h g ) (2) c v,t = load t,h p m (t, h)dh hb k b (p d (t, h) p b )dh hg k g (p d (t, h) p g (t))dh (21) (21) 1 2 3 k b k g load t,h k b k g load t,h B 124 3 24 417
c v,t = load t,h p m (t, h)dh π(t, p; k) (23) p =(p max (t),p g (t)) k =(k b,k g ) (22) π(t, p; k) 4 S b S g 4 Fig. 4. Assumed price duration curve model. π(t, p; k) =k b S b + k g S g (24) S b S g S b = f(p max (t),p b ) (25) S g = f(p max (t),p g (t)) (26) f Fig. 5. 5 Approximation result of duration curve. 5 2 t p m (t, h) p d (t, h) (21) k b k g p m (t, h) (1) p d (t, h) p d (t, h) t p max (t) h p d (t, h) =p max (t)(1 h/h ) (22) H H 4 5 Nord Pool 2 1 3 6 (22) 5 Nord Pool (22) H 16 (22) 5 3 π(t, p; k) (21) f(x, y) {( 24 1 12 ) } x y H = H {x y} 2 2x x y H H 24 y<x y<x< H H 24 y (14) E[c a ] E [c a ]=c f + E [c v ]= + E [ T T (k f)e rt load t,h p m (t, h)dhe rt ] [ T ] E π(t, p; k)e rt (27) f =(f b,f g ) (27) 2 E[c a ] k =(k b,k g ) load t,h 5 4 (27) 3 t p t π(t, p; k) V (t, p; k) [ ] T V (t, p; k) =E t π(t, p; k)e rτ dτ (28) t t = (28) [ } T V (, p; k) =E π(t, p; k)e rτ dτ (29) 418 IEEJ Trans. PE, Vol.124, No.3, 24
(27) 3 t dt V (t, p; k) V (t, p; k)=π(t, p; k)dt+e rdt E t [V (t+dt, p+dp; k)] (3) p = (p max (t),p g (t)) p max (t) p g (t) 4. d ln p max (t) =a 1 (ln b 1 ln p max (t))dt + σ 1 dz 1 (31) d ln p g (t) =a 2 (ln b 2 ln p g (t))dt + σ 2 dz 2 (32) a 1 a 2 b 1 b 2 σ 1 σ 2 dz 1 dz 2 x 1 =lnp max x 2 =lnp g p π(t, p; k) =π x (t, x; k) V (t, p; k) = V x (t, x; k) x = (x 1,x 2 ) ρ ij dt = dz i dz j V x (t, x; k) dv x [ V x (t, x; k) dv x = + a i (ln b i x i ) V x(t, x; k) i=1 i + 1 2 V x (t, x; k) ρ ij σ i σ j dt 2 i j + i=1 j=1 i=1 σ i V x (t, x; k) i dz i (33) (33) (3) e rdt (1 rdt) E[dZ i ]= dt 2 = V x (t, x; k) rv x (t, x; k) = π(t, x; k)+ V x(t, x; k) + a i (ln b i x i ) V x(t, x; k) i + 1 2 i=1 i=1 j=1 ρ ij σ i σ j 2 V x (t, x; k) i j (34) t = T V x (t, x; k) = ( <x 1 < )( <x 2 < ) (35) (34) V x (t, x; k) V (, p; k) (27) 3 6. (27) load t,h 3 MW 18 MW load t,h 3 4 15 MW p max (t) p g (t) 6 p g () 15 [ /kwh] a 2 a 1 7 σ 1 3 Table 3. Parameter of electric power price on peak and generation cost. 4 Table 4. Others parameters. B 124 3 24 419
7. 6 Fig. 6. Procurement cost in consideration period. 7 Fig. 7. Sensitivity of procurement cost..5 σ 1 σ 1 (26) S g S g p max (t) σ 1 S g (31) σ 1 dz 1 σ 1 S g DSM Demand Side Management 15 3 26 15 8 19 1 S. Yamada: Financial Engineering in New Electricity Markets, Toyo Keizai Inc (21) (in Japanese), (21) 2 S. Deng: Stochastic Models of Energy Commodity Prices and Their Applications: Mean-reversion with Jumps and Spikes, University of California Energy Institute (2-2) 3, (1998) 4 A.K. Dixit and R.S. Pindyck: INVESTMENT UNDER UN- CERTAINTY, Princeton University Press (1994) 5, (1999) Siripatanakulkhajorn Sakchai 1977 11 4 21 3 23 3 42 IEEJ Trans. PE, Vol.124, No.3, 24
1979 11 27 22 3 4 195 2 25 1972 3 1977 EPRI 1965 4 12 1988 3 1993 3 4 B 124 3 24 421