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Computing Equilibria in Electricity Markets
Tony DownwardAndy PhilpottGolbon 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
NodesAt each node, there can be injection and/or withdrawal of electricity.
LinesThe 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)
GeneratorsThe 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
DemandInitially we will assume that demand, at all nodes, is fixed and known.
Electricity
Electricity Market (2/3)
Strategic Generator
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
quantity
pri
ce
Tactical Generator
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
quantity
pri
ce
Electricity
Electricity Market (3/3)
Dispatch Model
Electricity
212
2
1 2
1 2
min
/
0
0 ,
,
Tb x
s t Mx Af Bf d
Lf
x q
K f K
Amount of electricity dispatched
Flows along lines
Demand at nodes
Slope of offer curve
Matrix mapping generation to nodes
Node-Arc incidence matrix
Loss Coefficients
Impedance Values
Quantities offe
x
f
d
b
M
A
B
L
q red by generators
Capacities on the linesK
Game Theory
Concepts of Game Theory
PlayersEach player in a game has a decision which affects the outcome of the game.
PayoffsEach 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 EquilibriaA 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)
SituationLet 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 CurveFrom 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 qi.
Game Theory
Best Response CorrespondencesIn an n player Cournot game it can be shown that:
For a two player game this reduces to:
Cournot Game (2/4)
Game Theory
2
jj i
i
d q
q
2 11 22 2
d q d qq q
arg maxi
i i jq j
q q b d q
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
1 2 3
dq q
Game Theory
Cournot Game (4/4)
Game Theory
Best Response Correspondences
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
q1
q2 Player 1
Player 2
Nash Equilibrium
Issues
No Equilibrium (1/2)
| f | ≤ K
q1 q2
d d
Q1 Q2
Profit, q2 = 0.75
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1 1.2
q1
Pro
fit
Profit, q2 = 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1 1.2
q1
Pro
fit
Issues
Borenstein, Bushnell and Stoft. 2000. Competitive Effects of Transmission Capacity.
No Equilibrium (2/2)
Issues
No Intersection of Best Response Curves
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
q1
q2 Player 1
Player 2
Multiple Discrete Equilibria
Issues
Two Nash Equilibria
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
q1
q2 Player 1
Player 2
Two Equilibria
Continuum of Equilibria
| f | ≤ K
q1
d
Q
q2
Continuum of Equilibria
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
q1
q2 Player 1
Player 2
Continuum of Equilibria
Issues
Computing Equilibria
Sequential Best Response (1/2)
Best ResponseWe need to be able to calculate the global optimal injection quantity. To do this we can perform a bisection search.
Computing Equilibria
Residual Demand Curve
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Offer
Pri
ce
Revenue as Function of Offer
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Offer
Rev
enu
e
Cournot, A. 1838. 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
Sequential Best Response
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
q1
q2
Player 1
Player 2
SBR
EPEC Formulation (1/2)
212
2
1 2
1 2
min
/
0
0 ,
,
Tb x
s t Mx Af Bf d
Lf
x q
K f K
Formulate KKT System
Dispatch Problem
Player’s Revenue Maximisation
max
max
/ Optimal Dispatch
0
i i
i i
x
s t
q q
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-ConcaveThe 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 EquilibriaThe 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 EquilibriaOnce 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
Q2
Q1
d
d
q1
q2
Price at Benmore
0
10
20
30
40
50
60
70
0 200 400 600 800
Benmore's Offer /MW
Pri
ce /
$
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 = qt
• 2 strategic generators, which offer q1 and q2
If q1 + q2 ≤ 1, then the tactical generator is dispatched for,
qt = 1 – q1 – q2
The tactical generator sets the price, p = qt
Combined Offers
0
0.25
0.5
0.75
1
0 0.25 0.5 0.75 1 1.25 1.5
QuantityP
ric
e
Electricity
Iteration 1
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
q
p
Sequential Best Response
Bi-Section SearchIt 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.
Iteration 2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
q
p
Computing Equilibria
Iteration 3
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
q
p
Iteration 4
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
q
p
Iteration 7
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
q
p
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
Supply Function Best Response
0
50
100
150
200
250
0 20 40 60 80 100 120 140 160
q
p
1 Piece
2 Pieces
3 Pieces
Future Work
