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Competitive Capacity SetsExistence of Equilibria in Electricity

Markets

A. Downward

G. Zakeri A. Philpott

Engineering Science, University of Auckland

7 September 2007

Motivation

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.

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• Assumptions / Simplifications

• Competitive Play

• Competitive Capacity Set

• Impact of Losses

• Loop Effects

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Outline

Generators The electricity markets consist of a number of generators located atdifferent 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 GIP and 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.

Assumptions / SimplificationsGeneration & Demand / Transmission Grid

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StrategicGenerator

TacticalGenerator

Demand

StrategicGenerator

TacticalGenerator

Demand

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,

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q1 q2Quantity

Price

d

π

q2’

Pricing & Dispatch – Single Node

Assumptions / Simplifications

Simplified Dispatch Model

xi is 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.

di is the demand at node i.

1/ai is the tactical offer slope of tactical generator i.

Kij is the capacity of line ij.

Pricing & Dispatch – Radial Network

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212

, ,

min

s.t. Node Balance

Line Capacities

i iai N

i ij ji i ij ij L j ji L

ij ij

x

x f f d q i N

f K ij L

Assumptions / Simplifications

Competitive PlayDefinitions

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Cournot GameA 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.

PlayersThe players in the game are the strategic generators. Each player has a

decision which affects the payoffs of the game.

DecisionThe players’ decision is the amount of electricity they offer.

PayoffsEach 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 EquilibriumA 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.

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.

Formulation as an EPEC

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, ,

010

0 0

0 0

i ij ji i ij ij L j ji L

i i i

i j ij ij

ij ij ij

ij ij ij

x f f d q i N

x a i N

ij L

K f ij L

K f ij L

Competitive Play

max

s.t. 1n nq

Unconstrained Equilibrium

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1 1

1 1

iU U i N

ii N i

i N

dq d

N N a

Unlimited CapacityIn 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 EquilibriumHowever, 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.

Competitive Play

| f | ≤ K

q1 q2

1 1

x1 x2

Two Node ExampleBorenstein, Bushnell and Stoft considered a symmetric two node network, with a strategic generator and a tactical generator at each node.

Line Capacity’s Effect on Equilibrium

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Competitive Play

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

Properties of Residual Demand Curve

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Competitive Capacity Set

d1 d2 d3

Conditions for Existence of Equilibrium

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Generalized Formulation

Competitive Capacity Set

DefinitionsDn is the set of decompositions containing node n.δ is 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.

Lδ

Nδ

1 2 3

Three Node Linear Network

Example

K12

K23

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Competitive Capacity Set

d1=100MW d2=320MW d3=180MW

q1 q2 q3

|f12| ≤ K12 |f23| ≤ K23

Competitive Capacity Set

Impact of LossesEffect on Existence of Equilibria

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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:1

2f

r

sent

arrive

f = 1/2r

Impact of LossesEffect on Existence of Equilibria

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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,

1 2Loss = r f 2

d1 = 1 d2 = 1Equilibrium Price vs. Loss Coefficient

0.3

0.35

0.4

0.45

0.5

0.55

0 1 2 3 4 5

Loss Coefficient

Pri

ce

Equilibrium Price Duopoly Price Monopoly Price

Competitive Capacity vs. Loss

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 1 2 3 4 5

Loss Coefficient

Min

imu

m L

ine

Cap

acity

We have shown, for this example, that there exists a pure strategy equilibrium for any value of the loss coefficient.

|f13| ≤ K13

Loop EffectsImpact of Kirchhoff's Laws on Competition

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1

2

3

d1=100MW

d2=320MW

d3=180MW

q1

q2

q3

f12 f23

Three Node Loop

Capacity of Added LineIf 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.

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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.

With the loop, we are no longer guaranteed that the competitive capacity set will be convex.

n

nqUnq

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Convexity of Competitive Capacity Set

Loop Effects

1212 Kf

1

2 3

Residual Demand Curve

Non-Convexity of Player’s Non-deviation Set

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Loop Effects

1212 Kf

Non-Deviation Set of Player 1For 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

K12

K23

Non-Convexity of Player’s Non-deviation Set

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Loop Effects

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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.

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THANK YOU

Any Questions?

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