1. Nash equilibria in Electricity Markets: A comparison of different approaches Seminar in Electric Power Networks, 12/5/12 Magdalena Klemun Master Student, Earth Resources Engineering Department of Earth and Environmental Engineering

2. Content 1)Liberalization and new structure of electricity market 2)The Nash equilibrium 3)Application in electricity market 4)Problem: Finding all equilibria in multiplayer games 5)Solution strategies: Polynomial equations Conversion to maximization problem

3. A new market structure Vertical integration  horizontal integration ONE UTILITY = MONOPOLY GENERATION TRANSMISSION DISTRIBUTION Independent Power Producers (IPP) Independent Systems Operator (ISO)/Con Edison Distribution

4. Idea: More competition – more efficiency + More efficient utilities + Prices showing economic production cost of different generation technologies -More power producers  more difficult power allocation

5. Pricing mechanism: Bidding and clearing Wholesale market for power >1MW: IPPs submit bid How much (MW)? What price? Source: NYISO Graphics: New York ISO, 2012

6. The Nash equilibrium = solution concept in non-competitive strategic games Defines choice of action that leaves no incentive to “move” unless all other players move  Nash equilibrium = stable state Players take single actions, no coalitions formed

7. Mathematical formulation A strategic game is comprised of… …a finite set of players …the non-empty set of actions available for each player …a preference relation for each player, defined over, the set of possible outcomes.  Definition of a PAYOFF FUNCTION: such that for given actions a,b if

8. Nash equilibrium A Nash equilibrium is a profile of actions in a strategic game of players such that for each player i  for the payoff function Competitor’s choice in Nash equ. Player i’s choice in Nash equilibrium Player i choice outside Nash equ.

9. Application in Electricity Markets IPP = player Power output = action Payoff function = profit GOAL: Find Nash equilibria to… …monitor the market …optimise bidding strategies

10. Major challenges: More Independent Power Producers  multiplayer game Transmission constraints complicate payoff functions Mixed-strategy Nash-equilibria (probabilistic), not only pure-strategy (deterministic) QUESTIONS: - How many equilibria? Multiple equilibria create strategic uncertainty! - Which conditions for Nash equilibrium? -Which equilibria are best?

11. Strategy: Polynomial equations as conditions for Nash Equilibrium, Yang et al. -Reformulate Nash equilibrium condition as set of polynomial equations, based on 2 ideas: 1)Construct set of pure strategies that are undominated and therefore “stable”  Same payoff functions  1 st condition: where profit 2) The probabilities of all the strategies in the subset must add up to 1:

12.  Solve polynomial equations numerically  Case study for three bus system & three generators, one transmission line has constraint: Number of NE sensitive to transmission constraint! Remaining problem: Which NE to choose?

13. Strategy: Convert to maximization problem (Contreras J. et al.) How to find Nash equilibrium? Maximise Nikaido-Isoda function: Sum of gains if player changes strategy from xi to yi while other players do not Nash Equilibrium at,  Relaxation algorithm using N. Isoda function used to compute NE

14. Understand different approaches for calculating Nash equilibria Identifying problems Look at including cap and trade system for CO2 emissions for power producers GOALS