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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

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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

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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

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

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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

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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

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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

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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.

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

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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?

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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:

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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?

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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

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

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