Counterexamples to commonly held Assumptions on Unit Commitment and - PowerPoint PPT Presentation

Counterexamples to commonly held Assumptions on Unit Commitment and Market Power Assessment NAPS Conference Tempe, 14. October 2002 Wolfgang Gatterbauer, Marija Ilic Presented by Audun Botterud Massachusetts Institute of Technology

2 Topics • Comparison of Theoretical Efficiency of Centralized and Decentralized Unit Commitment (PoolCo vs. PX) • Determination of Market Power revisiting the fundamental Economic Assumption of Marginal Costs being the baseline of competitive prices 2

Agenda 1: PoolCo vs PX • Background Information • The commonly used Argument • Counterexample • Conclusions 3

Background Information 1: PoolCo vs PX • Unit Commitment: Technological constraints (minimum up-time, starting costs) • ISO: Independent System Operator • PoolCo vs. PX (Power Exchange) 4

Agenda 1: PoolCo vs PX • Background Information • The commonly used Argument • Counterexample • Conclusions 5

Conventional Centralized Unit Commitment • Minimize the total generation cost n ∑ min u C Q ( ) i i i u Q i , = i i 1 • So that total generation equals total load n ∑ = Q Q i D = i 1 • Lagrangian relaxation method n ( ) ∑ λ = − λ + λ L u Q ( , , ) u C Q ( ) Q Q i i i i D = i 1 6

Conventional Centralized Unit Commitment • Minimized over Q δ δ C C = = = λ 1 n ... δ δ Q Q 1 n • Plug back n ( ) ( ) ∑ λ = λ − λ λ + λ L u ( , ) u C Q ( ) Q ( ) Q i i i i D = i 1 • Minimized with respect to u i -> Switching Law − λ >  0 if C Q 0 C i i =  < λ i u i − λ <  1 if C Q 0 Q i i i 7

Decentralized Unit Commitment • Maximize the individual profit π π = − max ( Q ) PQ C Q ( ) i i i i i i Q i • Decide in advance whether to turn on the unit � � u Q p k k k • Expected Profit � � � � π = ⋅ − p Q C Q ( ) on i i i 8

Decentralized Unit Commitment • Decision � π > 0 on • Switching Law � C Q ( ) � < i i p � Q i • Conclusion: a centralized system operator would schedule the same units as the individual power producers would in a decentralized way 9

Agenda 1: PoolCo vs PX • Background Information • The commonly used Argument • Counterexample • Conclusions 10

Counterexample: 2 Generators G 1 , G 2 = 2 + + • Quadratic Cost Function: C Q ( ) a Q b Q c i i i i i i = + • Linear increasing MC: MC Q ( ) 2 a Q b i i i i i P, MC • Supply Functions: G 2 G 1 2a 2 2a 1 P 1 a 1+2 P 1+2 Q • = f( ) a b c Q , , , i b 1 b 2 Q Demand Q min Q 11

Counterexample: Conditions We search Parameters so as to: a b c Q , , , • Generator 1 makes profits: 2 + + < a Q b Q c P Q + 1 1 1 1 1 1 2 1 • Generator 2 loses money if switched on: + + > 2 a Q b Q c P Q + 2 2 2 2 2 1 2 2 • Total costs are lower with both generators on: + + > + + + + + 2 2 2 a Q b Q c a Q b Q c a Q b Q c 1 1 1 1 1 1 1 1 2 2 2 2 2 12

Counterexample: Numerical Values • Typical Parameters: G 1 G 2 a 1 2 b 1 1.6 c 1.1 0.7 • Differences: Q 2 G 1 G 1 and G 2 P 5 3.87 G 1 G 1 + G 2 G 1 G 2 % Q 100% 100% 72% 28% C 7.1 6.84 4.59 2.25 Rev 10 7.73 5.54 2.19 � 2.9 0.9 0.95 -0.06 13

Agenda 1: PoolCo vs PX • Background Information • The commonly used Argument • Counterexample • Conclusions 14

Conclusion 1 (PoolCo vs PX) • A centralized Unit Commitment can lead to higher efficiency • Explanation: It is possible that several generators can supply the demand with lower costs than the sub- group of generators that would obtain a profit in a free competitive market – assuming bidding marginal costs (!) 15

Agenda 2: Market Power • Background Information • Illustrative Example • Numerical Values • Conclusions 16

Background Information 2 (Market Power) • „Offering power at a price significantly above marginal production (or opportunity) cost, or failing to generate power that has a production cost below the market price, is an indication of the exercise of market power…“ [Borenstein00] • “Market power exists when a supplier or consumer influences prices ... If suppliers exercise market power, prices could be higher than marginal costs.” [DOE97] 17

Background Information 2 (Market Power) • „Economic withholding occurs when a supplier offers output to the market at a price that is above both its full incremental costs and the market price (and thus, the output is not sold)” [FERC01] [Borenstein00]: Borenstein S., Bushnell J., Wolak F.; Diagnosing Market Power in California’s Restructured Wholesale Electricity Market; NBER Working Paper 7868 [DOE97]: Department of Energy; Electricity Prices in a Competitive Environment: Marginal Cost Pricing of Generation Services and Financial Status of Electric Utilities. [FERC01]: Federal Energy Regulatory Commission; Investigation of Terms and Conditions of Public Utility Market-Based Rate Authorizations; Order E-47, 18

Agenda 3: Market power • Background Information • Illustrative Example • Numerical Values • Conclusions 19

Illustrative Example • MC=const, t up,min =2, SU+SD=FOC • Price taker P(k) P k+1 P k P k+1 f(P k+1 ) MC k k+1 t (hours) 20

Illustrative Example { } ∈ • Discrete Prices: P P ,..., P ,..., P 1 11 1 i 15 { } ∈ P P ,..., P ,..., P 2 21 2 j 25 P(k) P 1 P 2 P 2 p(P 2 ) MC 1 2 t (hours) 21

Illustrative Example • Correlation between Hours possible P 2 |P 1 = a = = = | = p( P P ) p( P P P P ) 2 2 j 2 2 j 1 1 i p(P 2 ) P 2 |P 1 = b P(k) a p(P 2 ) b P 1 P 2 P 2 |P 1 = c P 2 p(P 2 ) p(P 2 ) MC P 2 |P 1 = d p(P 2 ) P 2 |P 1 = e 1 2 t (hours) p(P 2 ) 22

Agenda 3: Market Power • Background Information • Illustrative Example • Numerical Values • Conclusions 23

Numerical Values – Example • MC=50, Q=1, FOC=10 Bid Sequence Exp.Profit Prices (58,52), (60,54) 1.1720 independent (58,54) 1.1538 (50,50) 1.0798 (56,50),(56,52), (60,56),(62,56) (60,52) 0.9266 Prices correlated (60,52) 1.7650 (58,52) 1.6838 (60,54) 1.6834 (50,50) 1.0798 24

Agenda 3: Market Power • Background Information • Illustrative Example • Numerical Values • Conclusions 25

Conclusion 2 (Market Power) • Market Prices above MC of the last unit do not prove the exercise of Market Power (!) • In order to determine the optimal bidding sequence, the price correlations between hours have to be included in the algorithms 26

Summary • A decentralized Unit Commitment is not always as efficient as the centralized one – even in the theoretical case. • Marginal Costs cannot be used as the baseline from which Market Power is measured. 27

Contact the authors • Wolfgang: flow@alum.mit.edu • Marija: ilic@mit.edu 28

Backup – 3 – Formula • Expected Profit of Bidding (b 1 ,b 2 ): ( )   + − P P 2 MC Q ∑ ∑ i j   = = ⋅ = ⋅ J b b ( , ) p( P P ) p( P P ) 1 2 1 i 2 j   − FOC | ≥ | ≥ P P b P P b i i 1 j j 2 ( ( ) ) ∑ ∑ + = ⋅ = ⋅ − − p( P P ) p( P P ) P MC Q FOC 1 i 2 j i | ≥ | < P P b P P b i i 1 j j 2 ( ( ) ) ∑ ∑ + = ⋅ = ⋅ − − p( P P ) p( P P ) P MC Q FOC 1 i 2 j j | < | ≥ P P b P P b i i 1 j j 2 29

Backup – 3 – Numerical Values P 1 ∈ {56,58,60,62,64,66} MC =50; P 2 ∈ {46,48,50,52,54,56} Q =1; FC =10; With p i =p( P 1 =P 1 i ) = p( P 2 =P 2 i ) and p i  j =p( P 2 =P 2 j  P 1 =P 1 i ): p 1 =0.1888 p 1  1 =0.45 p 1  2 =0.20 p j  3 = p j p 2 =0.1624 p 2  1 =0.20 p 2  2 =0.32 p 3 =0.2978 p 3  1 =0.27 p 3  2 =0.33 p j  4 = p 5-j  2 p 4 =0.1624 p 4  1 =0.06 p 4  2 =0.08 p 5 =0.1888 p 5  1 =0.02 p 5  2 =0.08 p j  3 = p 5-j  1 30