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Competitive Capacity Sets Existence of Equilibria in Electricity Markets A. Downward G. Zakeri A. Philpott Engineering Science, University of Auckland 7 September 2007
Motivation EPOC Winter Workshop 2007 It has been shown that transmission grids can affect the competitiveness of electricity markets. It is important for grid investment planners to understand how expanding lines in a transmission grid can facilitate competition. Borenstein et al. (2000) showed that pure-strategy Cournot equilibria do not always exist in electricity markets with transmission constraints. We wanted to derive a set of conditions on the transmission capacities which guarantee the existence of an equilibrium.
Outline • Assumptions / Simplifications • Competitive Play • Competitive Capacity Set • Impact of Losses • Loop Effects EPOC Winter Workshop 2007
Assumptions / Simplifications Generation & Demand / Transmission Grid Strategic Generator Strategic Generator Tactical Generator Tactical Generator Demand Demand EPOC Winter Workshop 2007 Generators The electricity markets consist of a number of generators located at different locations. We will assume that there exist two types of generator: Strategic Generators:Submit quantities at price $0. Tactical Generators:Submit linear offer curves. Demand At each node demand is assumed to be fixed and known. We approximate the grid using a DC power flow model, consisting of nodes and lines. Nodes Each generator is located at a GIPand each source of demand is located at a GXP; these are combined into nodes. Lines The lines connect the nodes and have the following properties: Capacity: Maximum allowable flow. Loss Coefficient: Affects the electricity lost. Reactance:Affects the flow around loops.
Pricing & Dispatch – Single Node Assumptions / Simplifications π Price q1 q2’ q2 d Quantity EPOC Winter Workshop 2007 Aggregating Offers Suppose that there are two strategic and one tactical generator at a node, • the tactical generator submits an offer with slope 1, • the strategic generators offer a quantities, q1 and q2, • the demand as the node is d. We get the following combined offer stack,
Pricing & Dispatch – Radial Network Assumptions / Simplifications EPOC Winter Workshop 2007 Simplified Dispatch Model xiis the MW of electricity injected by the tactical generator i. fij is the MW sent directly from node i to node j. qi is the MW of electricity injected by the strategic generator i. diis the demand at node i. 1/ai is the tactical offer slope of tactical generator i. Kij is the capacity of line ij.
Competitive Play Definitions Cournot Game A Cournot game is played by generators selecting quantity of electricity to sell and being paid a price /MW for that electricity based on the total amount offered into the market. Players The players in the game are the strategic generators. Each player has a decision which affects the payoffs of the game. Decision The players’ decision is the amount of electricity they offer. Payoffs Each player in a game has a payoff, in this case, revenue; this is a function of the decisions of all players. Each player seeks to maximize its own payoff. Nash Equilibrium A Nash Equilibrium is a point in the game’s decision space at which no individual player can increase its payoff by unilaterally changing its decision. EPOC Winter Workshop 2007
Formulation as an EPEC Competitive Play KKT System for Dispatch Problem This is embedded as the constraint system in each player’s optimization problem Simultaneously satisfying the above problem for all players will be a Nash-equilibrium, however each problem is non-convex, so using first order conditions will not necessarily find an equilibrium, as only local maxima are being found. EPOC Winter Workshop 2007
Unconstrained Equilibrium Competitive Play Unlimited Capacity In a network with unlimited capacity on all lines, the Nash equilibrium is identical to that of a single node Cournot game. This is because the network cannot have any impact on the game. Hence, if the capacities of the lines are ignored, it is possible to calculate the Nash-Cournot equilibrium. This is the most competitive equilibrium in a Cournot context. Single-Node Nash-Cournot Equilibrium Candidate Equilibrium However, the capacities of the lines can potentially create incentive to deviate. This means that the equilibrium is not necessarily valid and it is only a candidate equilibrium, which needs to be verified. EPOC Winter Workshop 2007
Competitive Play q1 q2 | f |≤ K x1 x2 1 1 Line Capacity’s Effect on Equilibrium Two Node Example Borenstein, Bushnell and Stoft considered a symmetric two node network, with a strategic generator and a tactical generator at each node. The revenue attained by the strategic generator at node 1 (g1), when the injection of g2 is set to the unconstrained equilibrium quantity, qU = 2/3 , is shown below. Cournot Quantity EPOC Winter Workshop 2007 10/22
Competitive Capacity Set Properties of Residual Demand Curve d1 d2 d3 EPOC Winter Workshop 2007 11/22
Conditions for Existence of Equilibrium Competitive Capacity Set Definitions Dn is the set of decompositions containing node n. δis a decomposition, which divides the network into two sections. Nδ is the set of nodes in the super-node associated with decomposition δ. Lδ is the set of lines connecting decomposition δ, to other parts of the network. Nδ Lδ Generalized Formulation EPOC Winter Workshop 2007
Competitive Capacity Set q1 q2 q3 |f12| ≤ K12 |f23| ≤ K23 d1=100MW d2=320MW d3=180MW Example Three Node Linear Network Competitive Capacity Set K23 1 2 3 K12 EPOC Winter Workshop 2007 13/22
Impact of Losses Effect on Existence of Equilibria arrive sent f = 1/2r Losses are a feature of all electricity networks and need to be considered. The inclusion of losses raises two main questions: • Does the unconstrained equilibrium still exist? • How is the Competitive Capacity Set affected? We treat the loss as being proportional to what it sent from a node, i.e. if f MW is sent from node 1, the amount arriving at node 2 is f – r f 2. The presence of these losses creates an effective constraint on the flow: EPOC Winter Workshop 2007 14//22
Impact of Losses Effect on Existence of Equilibria Loss = r f 2 1 2 d1 = 1 d2 = 1 In the economics literature it has been stated that for large values of the loss coefficient, r, that no pure strategy equilibrium exists. The reasoning was that as the loss coefficient becomes large the effective capacity on the line tends to zero. Consider a two node example, with a demand of 1 at each node, We have shown, for this example, that there exists a pure strategy equilibrium for any value of the loss coefficient. EPOC Winter Workshop 2007
Loop Effects Impact of Kirchhoff's Laws on Competition |f13| ≤ K13 Three Node Loop q1 1 3 q3 f12 f23 d1=100MW d3=180MW q2 2 d2=320MW Capacity of Added Line If a new line is added connecting nodes 1 and 3 directly, we may not longer be able to achieve a pure strategy equilibrium. As Kirchhoff’s Law governs the flow around a loop, the new line must have a capacity of at least 26 2/3 MW to support the flows on the lines at equilibrium. EPOC Winter Workshop 2007
Loop Effects Convexity of Competitive Capacity Set Now considering a loop consisting of three nodes and three lines of equal reactance. Lines 12 and 23 each have a capacity, line 13 does not. Residual Demand Curve 1 2 3 With the loop, we are no longer guaranteed that the competitive capacity set will be convex. EPOC Winter Workshop 2007 17/22
Non-Convexity of Player’s Non-deviation Set Loop Effects K23 K12 Non-Deviation Set of Player 1 For a three node loop with capacities on the lines as shown, there are a number of congestion regimes player 1 can deviate to attempt to increase revenue. 1 2 3 EPOC Winter Workshop 2007
Non-Convexity of Player’s Non-deviation Set Loop Effects EPOC Winter Workshop 2007 19/22
Conclusions • The electricity grid can affect the competitiveness of electricity markets. • For radial networks, we have derived a convex set of necessary and sufficient conditions for the existence of the unconstrained equilibrium – the Competitive Capacity Set. • The capacity imposed by the loss on a line does not impact the existence of the unconstrained equilibrium. • When there is a loop, the set of conditions ensuring the existence of the unconstrained equilibrium is not necessarily convex. EPOC Winter Workshop 2007 20/22
Thank You Any Questions? EPOC Winter Workshop 2007