Competitive Capacity Sets
Existence of Equilibria in Electricity Markets
A. Downward
G. Zakeri
A. Philpott
Engineering Science, University of Auckland
7 September 2007
1/22
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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Outline
• Assumptions / Simplifications
• Competitive Play
• Competitive Capacity Set
• Impact of Losses
• Loop Effects
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Assumptions / Simplifications
Generation & Demand / Transmission Grid
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.
Strategic
Tactical Submit linear
Strategic
Tactical
Tactical Generators:
offer curves.
Demand
Generator
Generator
Generator
Generator
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.
Demand
Demand
Lines
The lines connect the nodes and have the following properties:
Capacity:
Loss Coefficient:
Reactance:
Maximum allowable flow.
Affects the electricity lost.
Affects the flow around loops.
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Assumptions / Simplifications
Pricing & Dispatch – Single Node
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,
Price
π
q1
q2’
q2
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d
Quantity
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Assumptions / Simplifications
Pricing & Dispatch – Radial Network
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.
min

iN
s.t.
1
2 ai
xi 
xi 2

j ,ijL
fij  Kij
fij 

j , jiL
f ji  di  qi
i  N
 Node Balance 
ij  L
 Line Capacities 
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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.
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Competitive Play
Formulation as an EPEC
KKT System for Dispatch Problem
xi 

j ,ijL
fij 

j , jiL
f ji  di  qi
i  N
xi  ai  i  0
i  N
 i   j  ij  ij  0
0  ij  Kij  fij  0
0  ij  Kij  fij  0
ij  L
1
ij  L
ij  L
This is embedded as the constraint system in each player’s optimization problem
max qn   n
s.t.
1
Simultaneously satisfying the above problem for all players will be a Nashequilibrium, however each problem is non-convex, so using first order conditions
will not necessarily find an equilibrium, as only local maxima are being found.
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Competitive Play
Unconstrained Equilibrium
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 NashCournot equilibrium. This is the most competitive equilibrium in a Cournot context.
Single-Node Nash-Cournot Equilibrium
1
qU 
di

N  1 iN
U 
d

1
N 1  a
iN
iN
i
i
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.
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Competitive Play
Line Capacity’s Effect on Equilibrium
Two Node Example
Borenstein, Bushnell and Stoft considered a
symmetric two node network, with a strategic
x1
generator and a tactical generator at each
node.
q1
q2
|f|≤K
x2
1
1
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.
Revenue Functions
0.25
revenue
0.2
0.15
K = 0.04
K = 0.057
K = 0.08
0.1
0.05
Cournot Quantity
0
0
0.2
0.4
0.6
0.8
1
1.2
q1
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Competitive Capacity Set
Properties of Residual Demand Curve
d1
d2
d3
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Competitive Capacity Set
Conditions for Existence of Equilibrium
Definitions
Dn 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.
Nδ
Lδ
Generalized Formulation
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Competitive Capacity Set
Example
Three Node Linear Network
q1
1
q2
|f12| ≤ K12
d1=100MW
2
q3
|f23| ≤ K23
d2=320MW
3
Competitive Capacity Set
K23
d3=180MW
K12
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Impact of Losses
Effect on Existence of Equilibria
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: f 
1
2r
arrive
f = 1/2r
sent
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Impact of Losses
Effect on Existence of Equilibria
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,
Loss = r f 2
1
2
d1 = 1
d2 = 1
Equilibrium Price vs. Loss Coefficient
Competitive Capacity vs. Loss
0.55
0.07
Minimum Line Capacity
0.5
Price
0.45
0.4
0.35
0.06
0.05
0.04
0.03
0.02
0.01
0
0.3
0
1
2
3
4
5
0
1
Loss Coefficient
Equilibrium Price
Duopoly Price
2
3
4
5
Loss Coefficient
Monopoly Price
We have shown, for this example, that there exists a pure strategy equilibrium for any
value of the loss coefficient.
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Loop Effects
Impact of Kirchhoff's Laws on Competition
Three Node Loop
|f13| ≤ K13
q1
1
d1=100MW
3
f12
f23
q3
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.
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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
n
1
f12  K12
2
3
qnU
qn
With the loop, we are no longer guaranteed that the competitive capacity set will be
convex.
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Loop Effects
Non-Convexity of Player’s Non-deviation Set
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.
K23
1
f12  K12
2
3
K12
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Loop Effects
Non-Convexity of Player’s Non-deviation Set
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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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