ECE 421/599 Electric Energy Systems 7 Optimal Dispatch of Generation. Instructor: Kai Sun Fall 2014

Transcription

1 ECE 4/599 Electrc Eergy Systems 7 Optmal Dspatch of Geerato Istructor: Ka Su Fall 04

2 Backgroud I a practcal power system, the costs of geeratg ad delverg electrcty from power plats are dfferet (due to fuel costs ad dstaces to load ceters) Uder ormal codtos, the system geerato capacty s more tha the total load demad ad losses. Thus, there s room to schedule geerato wth capacty lmts Mmzg a cost fucto that represets, e.g. Operatg costs Trasmsso losses System relablty mpacts Ths s called Optmal ower Flow (OF) problem A typcal problem s the Ecoomc Dspatch (ED) of real power geerato

3 Itroducto of Nolear Fucto Optmzato Ucostraed parameter optmzato Costraed parameter optmzato Equalty costrats Iequalty costrats

4 Ucostraed parameter optmzato Mmze cost fucto f x x x (,,, ). Solve all local mma satsfyg two codtos (ecessary & suffcet) Codto-: Gradet vector f f f f (,,, ) 0 x x x Codto-: Hessa matrx H s postve defte Statoary pot (where f s flat all drectos) Local mmum (a pure source f vector feld). Fd the global mmum from all local mma 4

5 f(x,y) (cos x + cos y) 5

6 Mmze f(x, y)x +y y f f f (, ) ( x, y) 0 x y f x 0, y 0 H f f x xy 0 f 0 yx y 0 x 6

7 arameter Optmzato wth Equalty Costrats Mmze Subject to f x x x (,,, ) g ( x, x,, x ) 0 k k,,, K Itroduce Lagrage Multplers λ ~ λ K K L f λ g + Necessary codtos for the local mma of L (also ecessary for the orgal problem) k k L f g + x x x K λk k 0 L λ k g k 0 7

8 Mmze f(x, y)x +y Subject to (x-8) +(y-6) 5 g(x, y)(x-8) +(y-6) -50 y L x + y + λ[( x 8) + ( y 6) 5] f L x + λ(x 6) 0 x L y + λ(y ) 0 y L + λ ( x 8) ( y 6) x Solutos (from the N-R method): λ, x4 ad y (f5) λ, x ad y9 (f5) 8

9 arameter Optmzato wth Iequalty Costrats Mmze Subject to : f x x x (,,, ) g ( x, x,, x ) 0 k k,,, K Itroduce Lagrage Multplers λ ~ λ K ad µ ~ µ m u ( x, x,, x ) 0 j,,, m j K L f + λ g + µ u k k j j k j m Necessary codtos for the local mma of L L f g u + + x x x x K m k j λ k µ j j 0,, Kuh-Tucker (KKT) ecessary codto L λ k g k 0 L u j 0 µ j µ u 0 µ 0 j j j k,,, K j,, m 9

10 Mmze f(x, y)x +y Subject to (x-8) +(y-6) 5 g(x, y)(x-8) +(y-6) -50 x+ y uxy (, ) x y 0 L x y x y x y + + λ [( 8) + ( 6) 5] + µ ( ) L x + λ (x 6) µ 0 x L y + λ ( y ) µ 0 y L ( x 8) + ( y 6) 5 0 λ L L x y < 0 µ 0 0, µ µ L µ ju j 0, µ j 0 or x y 0 µ > 0 µ y f x Solutos: µ0, λ, x ad y9 (f5) µ5.6, λ-0., x5 ad y (f9) µ, λ-.8, x ad y6 (f) 0

11 Operatg Cost of a Thermal lat Fuel-cost curve of a geerator (represeted by a quadratc fucto of real power) C α + β + γ Icremetal fuel-cost curve: dc λ γ + β d

12 A real case FUELCO MAX MIN HEMIN X Y X Y X Y Ge ID RIOR ($/MBtu) (MW) (WM) (MBtu/hr) (MW) (Btu/kWh) (MW) (Btu/kWh) (Btu/kWh) (Btu/kWh) A B Btu/h X Y 000 $/h Btu/h $/MBtu / 000,000 $/MWh Y /000 $/MBtu Cost ($/h) (MW) Lambda ($/MWh) (MW)

13 ED Neglectg Losses ad No Geerator Lmts If trasmsso le losses are eglected, mmze the total producto cost: C t g C α + β + γ subject to g Apply the Lagrage multpler method ( g + ukows to solve): g L Ct + λ( D ) L 0 L 0 λ Ct g D λ 0 D g Ct All plats must operate at equal cremetal cost dc β + γ λ d λ β γ D λ D + g β g γ γ λ β γ,, g Solve

14 Example 7.4 The fuel-cost fuctos for three thermal plats are C ~C $/h., ad are MW. D 800MW. Neglectg le losses ad geerator lmts, fd the optmal dspatch ad the total cost $/h λ D + g λ β γ β g γ γ $ / MWh dc λ d C C C dc λ d (0.004) (0.006) (0.009) dc λ d Equal cremetal cost λ C t $ / h 4

15 Solvg λ by the N-R Method g λ β γ g D λ g ( ) For a geeral case: g f( λ) ( k) df ( λ) ( k) ( k) f( λ) + ( ) λ dλ D D λ λ + λ ( k+ ) ( k) ( k) λ df ( λ) df ( λ) d ( ) ( ) ( ) dλ dλ dλ D D f( λ) ( k) ( k) g utl ( k+ ) ( k) λ λ ε 5

16 Apply the R-N Method Example 7.4 λ () λ 6.0 () () () γ β (0.004) (0.006) (0.009) 800 ( ) () () λ + + (0.004) (0.006) (0.009) ( k) ( k) λ d ( ) dλ γ $/MWh (MW) () λ + () () () (0.004) (0.006) (0.009) () + + C t ( ) $ / h 6

17 ED Neglectg Losses but Icludg Geerator Lmts Cosderg the maxmum (by ratg) ad mmum (for stablty) geerato lmts, Mmze Subject to C t g C g α + β + γ D (m) (max),,,, g g g L C + λ ( ) + [ µ ( ) + γ ( )] t D (max) (m) L 0 L 0 λ L 0 µ L γ 0 C 0 λ + µ γ g µ ( ) 0, µ 0 D (max) γ ( ) 0 γ 0 (m) (m) (max),,,, g (m) (max) ( µ γ 0) ( 0) (max) γ ( 0) (m) µ 4 dc d dc d dc d λ λ µ λ λ+ γ λ Excludg the plats that reach ther lmts, the other plats stll operate at equal cremetal cost 7

18 Example 7.6 Cosder geerator lmts ( MW) for Example 7.4, let D 975MW dc d λ D 975MW dc d λ D 906MW D 800MW dc d λ Soluto: 450MW 5MW 00MW λ9.4$/mwh C t 86.5 $/h 8

19 λ γ β ( k) ( k) λ d ( ) dλ γ () λ 6.0 () () () (0.004) (0.006) (0.009) () ( ) 84.7 () λ (0.004) (0.006) (0.009) () λ + () () () (0.004) (0.006) (0.009) () + + >, so let 450 (max) 975 ( ) () λ (0.006) (0.009) () λ + () () () (0.006) (0.009) () ( )

20 Solve the optmal dspatch for D 550MW C C C dc d λ or 0 or 0 or 0 Ut Commtmet roblem (mxed-teger optmzato) 0 dc d λ Soluto : 80MW Soluto : 40MW 0 dc d λ 70MW 00MW λ7.54$/mwh C t 4676 $/h 0MW 0MW λ8.0$/mwh C t 4584 $/h 0 Soluto : Soluto 4: 0MW 400MW 40MW 0MW 0MW 50MW λ9.58$/mwh λ8.5$/mwh C t $/h C t 45 $/h 0

21 Trasmsso Loss Whe trasmsso dstaces are very small ad load desty the system s very hgh Trasmsso losses may be eglected All plats operate at equal cremetal producto cost to acheve optmal dspatch of geerato However, a large tercoected etwork ower s trasmtted over log dstaces wth low load desty areas Trasmsso losses are a major factor affectg the optmal dspatch.

22 Calculato of Trasmsso Losses I V cosφ L I R R V cosφ R V cos φ V R+jX I D L B V V I I I R + I R + I + I R L R + V cos cos φ V φ cosα cos β + + V cos φ V cosφ R R R +jx R +jx I R +jx D I β L I + + α B B B I

23 More geeral loss formulas: Quadratc form Kro s loss formula L L g j g g g g B j j B B B j j 0 00 j B j, B 0 ad B 00 are Loss coeffcets or B-coeffcets. See Chapter 7.7 for detals of calculatg B-coeffcets Chages wth power flows but usually assumed costat ad estmated for a power-flow base case

24 Ecoomc Dspatch Icludg Losses Cost fucto g g t C C dc d β + γ Icremetal fuel cost Subject to g L D L g g g B B B j j 0 00 j g L B B j j j 0 Icremetal trasmsso loss,, (m) (max) g 4

25 Usg Lagrage Multplers g g g g L C ( ) ( ) ( ) where D L (max) (max) (m) (m) g g g B B B L j j 0 00 j g g t C C L 0 λ L 0 dc d g g g g D L D j j 0 j L dc + λ(0 + ) 0 d B B B L + λ λ 00 L dc d ealty factor: L L,, g dc d β + γ g L B B j j j 0 g BB,, j j j 0 g 5

26 BB j j 0 j g g ( B) B ( B ) j j 0 j j B B B B g 0 B B B B0 g g g B g Bg B B g g 0g Solve λ ad (~ g ) for the optmal dspatch: g g g g B B B D L D j j 0 00 j Check equalty costrats: If < (m), let (m) If > (max), let (max) Remove that from the equatos Use the N-R Method: f( ) g D L ( B ) ( B ) 0 j j j B df ( ) f( ) ( ) d ( k) ( k) ( k) ( k) D L f( ) ( k) ( k) ( k) D L df ( ) ( ) d g ( k) ( k) df ( ) g ( ) d 6

27 Ital ad (0) (0) ( B ) ( B ) 0 j j j B ( B ) ( k) ( k) ( k) 0 j j j ( k) ( B ) B f( ) ( k) ( k) ( k) D L g g g g ( k) ( k) ( k) ( k) ( k) D Bjj B0 B00 j g ( B ) B B 0 j j ( k) ( B ) j ( k) ( k) df ( ) g ( ) d g ( k) ( k) Utl (k) <ε 7

28 Example 7.7 Solve the optmal dspatch of three thermal plats a power system. The total system load s 50MW. The base for per ut values s 00MVA. C $ / h C $ / h C $ / h 0 MW 85 MW 0 MW 80 MW 0 MW 70 MW L(pu) (pu) (pu) (pu) (oly B 0) L MW 0.08( ) 0.08( ) 0.079( ) 00 MW 8

29 ( k) ( k) ( B ) (0) λ 8.0 g g B ( k) ( B ) g 0 MW 85 MW 0 MW 80 MW 0 MW 70 MW () () () MW ( ) MW ( ) MW ( ) (5.6) (78.59) () L (7.575).886 () ( ) 48.9 () (4) (4) (4) MW ( ) MW ( ) MW ( ) (5.0907) (64.7) (4) L (4) (5.4767) () ( ) () Ct $/h 9