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

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

3

Agenda 1: PoolCo vs PX

• Background Information
• The commonly used Argument
• Counterexample
• Conclusions

4

Background Information 1: PoolCo vs PX

• Unit Commitment: Technological constraints

(minimum up-time, starting costs)

• ISO: Independent System Operator
• PoolCo vs. PX (Power Exchange)

5

Agenda 1: PoolCo vs PX

• Background Information
• The commonly used Argument
• Counterexample
• Conclusions

6

Conventional Centralized Unit Commitment

• Minimize the total generation cost
• So that total generation equals total load
• Lagrangian relaxation method

, 1

min ( )

=

∑

i i

n i i i u Q i

u C Q

1 =

=

∑

n i D i

Q Q

( )

1

( , , ) ( ) λ λ λ

=

= − +

∑

n i i i i D i

L u Q u C Q Q Q

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Conventional Centralized Unit Commitment

• Minimized over Q
• Plug back
• Minimized with respect to ui -> Switching Law

1 1

... δ δ λ δ δ = = =

n n

C C Q Q

( )

( )

1

( , ) ( ) ( ) λ λ λ λ λ

=

= − +

∑

n i i i i D i

L u u C Q Q Q 0 if 1 if λ λ − >  =  − < 

i i i i i

C Q u C Q

i i

C Q λ <

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Decentralized Unit Commitment

• Maximize the individual profit
• Decide in advance whether to turn on the unit
• Expected Profit

max ( ) π

i

i i Q

Q ( ) π = −

i i i i

PQ C Q

• k

Q

k

u

• k

p

• (

) π = ⋅ −

• n

i i i

p Q C Q

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• Conclusion: a centralized system operator would

schedule the same units as the individual power producers would in a decentralized way

Decentralized Unit Commitment

• Decision
• Switching Law
• π

>

• n
• (

)

i i i

C Q p Q <

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Agenda 1: PoolCo vs PX

• Background Information
• The commonly used Argument
• Counterexample
• Conclusions

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Counterexample: 2 Generators G1, G2

• Quadratic Cost Function:
• Linear increasing MC:
• Supply Functions:
• = f( )

2

( )

i i i i i i

C Q a Q b Q c = + +

Q P, MC P1+2 P1 G2 b1 a1+2 G1 b2 2a2 2a1 Qmin QDemand

( ) 2

i i i i i

MC Q a Q b = + , , , a b c Q

i

Q

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Counterexample: Conditions

We search Parameters so as to:

• Generator 1 makes profits:
• Generator 2 loses money if switched on:
• Total costs are lower with both generators on:

2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2

a Q b Q c a Q b Q c a Q b Q c + + > + + + + +

2 2 2 2 2 2 1 2 2

a Q b Q c P Q

+

+ + >

2 1 1 1 1 1 1 2 1

a Q b Q c P Q

+

+ + < , , , a b c Q

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Counterexample: Numerical Values

• Typical Parameters:
• Differences:

0.7 1.6 2 G2 2 1.1 1 1 G1 c Q a b

• 0.06

2.19 2.25 28% G2 0.95 5.54 4.59 72% G1 0.9 7.73 6.84 100% G1+ G2 3.87 G1 and G2 2.9 10 7.1 100% G1 5 G1 P

• Rev

% Q C

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Agenda 1: PoolCo vs PX

• Background Information
• The commonly used Argument
• Counterexample
• Conclusions

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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 (!)

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Agenda 2: Market Power

• Background Information
• Illustrative Example
• Numerical Values
• Conclusions

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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]

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Background Information 2 (Market Power)

• „Economic withholding occurs when a supplier offers
• utput 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,

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Agenda 3: Market power

• Background Information
• Illustrative Example
• Numerical Values
• Conclusions

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Illustrative Example

MC P(k) t (hours) k k+1 f(Pk+1) Pk+1 Pk+1 Pk

• MC=const, tup,min=2, SU+SD=FOC
• Price taker

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Illustrative Example

MC P(k) t (hours) 1 p(P2) P2 P1 2 P2

{ }

1 11 1 15

,..., ,...,

i

P P P P ∈

• Discrete Prices:

{ }

2 21 2 25

,..., ,...,

j

P P P P ∈

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Illustrative Example

• Correlation between Hours possible

2 2 2 2 1 1

p( ) p( )

j j i

P P P P P P = = = | =

MC P(k) t (hours) 1 p(P2) P2 P1 2 P2 b p(P2) P2 |P1= b P2 |P1= a a p(P2) P2 |P1= d P2 |P1= c p(P2) p(P2) P2 |P1= e p(P2)

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Agenda 3: Market Power

• Background Information
• Illustrative Example
• Numerical Values
• Conclusions

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Numerical Values – Example

Prices correlated 1.0798 (50,50) (60,52) 1.6834 (60,54) 1.6838 (58,52) 0.9266 1.0798 (50,50) (56,50),(56,52), (60,56),(62,56) 1.1538 (58,54) 1.7650 (60,52) 1.1720 (58,52), (60,54) Prices independent Exp.Profit Bid Sequence

• MC=50, Q=1, FOC=10

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Agenda 3: Market Power

• Background Information
• Illustrative Example
• Numerical Values
• Conclusions

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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

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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.

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Contact the authors

• Wolfgang: flow@alum.mit.edu
• Marija: ilic@mit.edu

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Backup – 3 – Formula

• Expected Profit of Bidding (b1,b2):

( )

( ) ( )

( ) ( )

1 2 1 2 1 2

1 2 1 2 1 2 1 2

2 ( , ) p( ) p( ) p( ) p( ) p( ) p( )

| ≥ | ≥ | ≥ | < | < | ≥

+ − = = ⋅ = ⋅ − + = ⋅ = ⋅ − − + = ⋅ = ⋅ − −

     

∑ ∑ ∑ ∑ ∑ ∑

i i j j i i j j i i j j

i j i j P P b P P b i j i P P b P P b i j j P P b P P b

P P MC Q J b b P P P P FOC P P P P P MC Q FOC P P P P P MC Q FOC

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Backup – 3 – Numerical Values

MC=50; P1∈{56,58,60,62,64,66} Q=1; P2∈{46,48,50,52,54,56} FC=10; With pi=p(P1=P1i) = p(P2=P2i) and pij=p(P2=P2jP1=P1i): p1=0.1888 p11=0.45 p12=0.20 pj3= pj p2=0.1624 p21=0.20 p22=0.32 p3=0.2978 p31=0.27 p32=0.33 pj4= p5-j2 p4=0.1624 p41=0.06 p42=0.08 p5=0.1888 p51=0.02 p52=0.08 pj3= p5-j1