Competition Benefits of Line Upgrades in Electricity Markets

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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
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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.
• 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.

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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
q1
q2
Quantity
d
q2
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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.

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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 games 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 This is
embedded as the constraint system in each
players 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.
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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 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.
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Competitive Play
Line Capacitys 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.
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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. d is a decomposition, which
divides the network into two sections. Nd is the
set of nodes in the super-node associated with
decomposition d. Ld is the set of lines
connecting decomposition d, to other parts of the
network.
Nd
Ld
Generalized Formulation
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Competitive Capacity Set
Example
Three Node Linear Network
Competitive Capacity Set
K23
1
2
3
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

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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,
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
1
3
q1
q3
f12
f23
d1100MW
d3180MW
q2
2
d2320MW
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 Kirchhoffs 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
1
2
3
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 Players 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.
1
2
3
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Loop Effects

• Non-Convexity of Players 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?