Strategic Equilibria in Competitive Electricity Markets

1. Strategic Equilibria in Competitive Electricity Markets Pedro Correia Power Affiliates Program Meeting University of Illinois at Urbana-Champaign May 11th, 2001

2. Outline • Introduction • Objectives • Model assumptions and solutions • Individual Welfare Maximization • Multiple equilibria • Example • Future work

3. Introduction • Competitive electricity markets in which the participants submit their preferred schedules and the IGO centrally sets the trading solution based on them have been put in practice • This centralized model is usually known as PoolCo, of which PJM Interconnection is a specific implementation example

4. Introduction (contd) • The trading solution set by the IGO aims usually at short-term efficiency (max. social welfare) and incorporates rules to enforce feasibility • Participants are rational, therefore they seek individual profit maximization; they may do that by, if allowed, untruthfully reporting their cost/benefits in their offer/bid curves

5. Objectives • Given specific market rules, it is of great interest to be able to model the strategic behavior (gaming) of the market participants and to identify solutions (equilibria) for those games • Identify opportunities for market power abuse based on gaming by the participants • Improve market rules either by preventing gaming, or by controlling its consequences

6. Model Assumptions • The game is modeled as a static non-cooperative continuous-kernel game under complete information • (static) simultaneous for all players • (non-cooperative) each individual pursues his or her own interests • (continuous) each player has at his or her disposal a continuum of choices and the utility functions are also continuous

7. Model Assumptions (contd) • (incomplete information) players’ true costs/benefits constitute common knowledge among players • players’ rationality also constitutes common knowledge • The game is run for each time period of the market (one snapshot); typically one hour • The players game by changing one or more parameters of their reported schedules

8. Model: Solutions • The solutions prescribed by the proposed game are the so-called Nash equilibria, either in pure or mixed strategies • They constitute the strategically stable or self-enforcing points in the bidding space from which the players have no incentive to deviate • stability is defined with respect to the readjustment scheme employed by the players

9. Individual Welfare Maximization* (IWM) in a PoolCo market • The IGO runs a centralized economic dispatch subject to the system constraints (OPF), based on bids and offers freely submitted by participants • The OPF sets the nodal prices (Lagrangian multipliers) that are used to charge/pay consumption/generation on every node of the grid * by James Weber

10. IWM • It is a nested optimization problem; each player maximizes his or her profits given the fact that the IGO runs an OPF • Each player also mimics the other players’ behavior; the solutions are the same for all players • The readjustment is done using Newton’s method and if a stationary point is found then it constitutes, by definition, a Nash equilibrium

11. Multiple equilibria: example by James Weber

12. Utility function for player controlling 7&8

13. Searching for multiple equilibria • If the cost functions are quadratic and the constraints are linear the problem is well behaved on each region of the bidding space defined by the constraints • Implies the use of D.C. load flow equations • If we initialize the IWM algorithm inside each region then we may infer on the existence of a Nash equilibrium in that particular region

14. Enforcing the regions • However, the problem of initializing the algorithm inside a particular region is not easy to solve • Instead, we run the IWM algorithm considering the linear constraints defining each region as equality constraints • Since we’re modifying the problem, we have to test the solutions

15. Utility function: P78max 200MW

16. Utility function: P78max= 200MW

17. Utility function: P78max 240MW

18. Utility function: P78max= 240MW

19. Problem Size Reduction • The algorithm runs in O(3m) time • (non-polynomial): may be huge • - The problem may be reduced • Historical record (known congested lines) • Unfeasible regions (some cases are not supported by any load flow solution) • Previously visited regions

20. Example – 4 bus system; 2 players aP=0.05 $/MW2h bP=15.0 $/MWh aP=0.05 $/MW2h bP=6.0 $/MWh aD=-0.01 $/MW2h bD=80.0 $/MWh ~ ~ 1 2 aD=-0.10 $/MW2h bD=82.0 $/MWh x=0.790 pu Pmax = 1.87 Line state  {0,1,-1} 34 = 81 cases 27 feasible x=0.116 pu Pmax = 1.81 x=0.122 pu Pmax = 2.11 x=1.030 pu Pmax = 1.93 3 4 aD=-0.50 $/MW2h bD=440.0 $/MWh ~ ~ aD=-0.50 $/MW2h bD=440.0 $/MWh aP=0.10 $/MW2h bP=1.0 $/MWh aP=0.10 $/MW2h bP=1.0 $/MWh

21. Example: solutions • 4 confirmed pure strategy equilibria • For the remaining cases the algorithm oscillates between ; this may indicate the existence of mixed strategy equilibria • For this example, 3 of the equilibria are strictly dominated; this is not necessarily always the case

22. Future work • What is the solution when players face multiple equilibria that cannot be eliminated by domination? • The answer may be in super-games of dynamic nature (where we assume that the market conditions repeat themselves after some time)