Computing Equilibria in Electricity Markets Tony Downward Andy Philpott Golbon Zakeri University of Auckland

Overview (1/2) Electricity Electricity Network Electricity Market Game Theory Concepts of Game Theory Cournot Games Issues No Equilibrium Multiple Equilibria

Overview (2/2) Computing Equilibria Sequential Best Response EPEC Formulation Example Simplified Version of NZ Grid Equilibrium over NZ Grid

Electricity

Electricity Network Nodes At each node, there can be injection and/or withdrawal of electricity. Lines The nodes in the network are linked together by lines. The lines have the following properties: Capacity – Maximum allowable flow Loss Coefficient – Affects the electricity lost Reactance – Affects the flow around loops Electricity

Electricity Market (1/3) Generators The electricity market in New Zealand is made up of a number of generators located at different nodes on the electricity grid. We will assume there exist two types of generator: Strategic Generators – Submit quantities at price 0 Tactical Generators – Submit linear supply curve Demand Initially we will assume that demand, at all nodes, is fixed and known. Electricity

Electricity Market (2/3) Electricity

Electricity Market (3/3) Dispatch Model Electricity

Game Theory

Concepts of Game Theory Players Each player in a game has a decision which affects the outcome of the game. Payoffs Each player in a game has a payoff; this is a function of the decisions of all players. Each player seeks to maximise their own payoffs. Nash Equilibria A Nash Equilibrium is a point in the game’s decision space at which no individual player can increase their payoff by unilaterally changing their decision. Game Theory

Cournot Game (1/4) Situation Let there be n strategic players and one tactical generator, all situated at one node where there is a given demand d. The tactical generator’s offer curve slope is b. The price seen by all players is the same. This effectively reduces the game to a Cournot model. Residual Demand Curve From the point of view of the competing strategic generators, the above situation leads to a demand response curve with intercept db and slope –b. Therefore the nodal price is given by b(d – Q). Where Q is the sum of the strategic generators’ injections q i. Game Theory

Best Response Correspondences In an n player Cournot game it can be shown that: For a two player game this reduces to: Cournot Game (2/4) Game Theory

Cournot Game (3/4) Best Response Correspondences These previous functions are known as best response correspondences; they are the optimal quantity a player should offer in response to given quantities for the other players. Nash Equilibrium Game Theory

Cournot Game (4/4) Game Theory Nash Equilibrium

Issues

No Equilibrium (1/2) | f | ≤ K q1q1 q2q2 dd Q1Q1 Q2Q2 Issues Borenstein, Bushnell and Stoft Competitive Effects of Transmission Capacity.

No Equilibrium (2/2) Issues

Multiple Discrete Equilibria Issues Two Equilibria

Continuum of Equilibria | f | ≤ K q1q1 d Q q2q2 Continuum of Equilibria Issues

Computing Equilibria

Sequential Best Response (1/2) Best Response We need to be able to calculate the global optimal injection quantity. To do this we can perform a bisection search. Computing Equilibria Cournot, A Recherchés sur les principes mathematiques de la theorie des richesses.

Sequential Best Response (2/2) SBR Algorithm 1.Set starting quantities for all players. 2.For each player, choose optimal quantity assuming all other players are fixed. 3.If not converged go to step 2. Computing Equilibria

EPEC Formulation (1/2) Formulate KKT System Dispatch Problem Player’s Revenue Maximisation Formulate KKT System Solve all players’ revenue maximisation KKTs simultaneously; a Nash equilibrium will be a feasible solution to these equations. Computing Equilibria

EPEC Formulation (2/2) Non-Concave The issue with the EPEC formulation is that the revenue maximisation problems are not concave. This means that there will exist solutions to the EPEC system which are only local, not global equilibria. Candidate Equilibria The non-concavity stems from capacity constraints, which give rise to orthogonality constraints in the KKT. Solving this problem for a specific regime yields a candidate equilibrium. Checking Equilibria Once a candidate equilibrium is found, it still needs to be verified. Computing Equilibria

Example

PERM Grid This is a cut-down version of the New Zealand electricity network. It has 18 nodes and 25 lines. The actual New Zealand network has 244 nodes and over 400 lines. Example

Equilibria over NZ Grid (1/2) Example Q2Q2 Q1Q1 d d q1q1 q2q2

Equilibria over NZ Grid (2/2) Example

Thank You Any Questions?

Electricity Market Dispatch Example 1 node with demand equal to 1 1 tactical generator with offer curve, p = q t 2 strategic generators, which offer q 1 and q 2 If q 1 + q 2 ≤ 1, then the tactical generator is dispatched for, q t = 1 – q 1 – q 2 The tactical generator sets the price, p = q t Electricity

Sequential Best Response Bi-Section Search It can be shown that price at node i is non-increasing with injection at node i. This allows bounds to placed upon revenue to speed up search process. Computing Equilibria

Multi-nodal Best Response So far we have considered a player to be a generator situated at a single node. Some New Zealand generators have plants situated at multiple nodes around the grid; these plants may receive different prices. The challenge is therefore to maximise their combined profit, when changing the offer at one node impacts other nodes’ prices. An extension of the bi-section method can be used. Future Work

Supply Function Equilibria Until now we have assumed demand to be fixed, however a more realistic situation is demand being a random variable. This means an offer at price 0 is no longer the best response in expectation. As there now exist multiple residual demand curves, which each have an associated probability. If we confine our decision space to piecewise linear offer curves, we can parameterise the curve by the end of each piece (p,q). It is then possible to perform a multi-dimensional bisection method to find a best response. Future Work

Supply Function Best Response Future Work