A numerical method to calculate Nash-Cournot equilibria in
electricity markets
Javier Contreras Jacek B. KrawczykUCLM Victoria University of WellingtonCiudad Real WellingtonSpain New Zealand
2
Summaryn Introductionn Definitions and conceptsn The relaxation algorithmn Duopoly examplen Case study 1n Case study 2n Conclusions
3
Introductionn Power system restructuring fosters
competitionn Cost minimization schemes replaced by
bidding algorithmsn Nash Equilibria in network-constrained
electricity markets
4
Introductionn Nikaido-Isoda Relaxation Algorithm
(NIRA)n Convergence to a Nash-Cournot
equilibriumn Unique solution under certain conditionsn NIRA solves coupled constraint gamesn Applicable to electricity markets
5
Definitions and conceptsn A game represents:
– a set of individuals interacting – strategic interdependence
n To describe a game you need:– players– rules of the game– outcomes– payoffs
6
Definitions and conceptsn Information set: information about his
own and other players’ actions. It constains information on what is observable
n Strategy: A rule that tells the player wich action he should take, according to his information set
n Action: a choice made by a player as a result of his strategy
7
Definitions and conceptsn i = 1, ..., n players in a gamen Individual actions given by vector xi
n Collective action given by vector x = (x1, ..., xn)
n Xi is the action set of player in Øi : X is the payoff functionn X is the collective action set
→ ℜ
8
Definitions and conceptsn x = (x1, ..., xn) and y = (y1, ..., yn) are
elements of the collective action set Xn An element
of X is a set of actions where the ith player plays yi while the remaining agents play xj
n A point x* = (x1*, ..., xn
*) is called the Nash equilibrium point if, for each i,
( ) ),,,,,,(| 111 niiii xxyxxy LL +−≡x
)(max)()(
xx*x
*ii
Xxi x
i
φφ∈
=|
9
Definitions and conceptsn Nikaido-Isoda function (NI):
n Each summand of the NI function represents the improvement in payoff that a player will receive if he changes his action from xi to yi
n Each summand can be at most zero at Nash equilibrium
])()([),(1
∑=
−=Ψn
iiii y xxyx φφ 0),( ≡Ψ xx
10
Definitions and conceptsn An element is referred to as
Nash normalized equilibrium point if
n Given the concavity conditions, a Nash normalized equilibrium is also a Nash equilibrium point
n An iterative algorithm using the NI function can converge to a Nash equilibrium
X∈*x
0),(max * =Ψ∈
yxy X
11
Definitions and conceptsn The optimum response function at
the point x is
n All players try to unilaterally maximize their payoffs
n By playing actions Z(x), rather than x, the players approach the equilibrium
XZZX
∈Ψ=∈
)( , ),,(max arg)( xxyxxy
12
The relaxation algorithmn It is applied to the optimum response
functionn Given concavity conditions, single-
valuedness of Z(x), and an initial estimate x0 then
K0,1,2,s ),()1(1 =+−=+ ss
ss
s Z xxx αα
10 ≤< sα
13
The relaxation algorithmn An iterative step s+1 is constructed as a
weighted average of the improvement point Z(xs) and the current point xs
n The optimum response function Z(xs) is a result of maximizing the NI function
n The averaging ensures convergence of the algorithm under certain conditions
14
The relaxation algorithmn Taking a sufficient number of iterations
the NIRA converges to the Nash equilibrium x*
n The problem calculates the successive actions taken by the players at each stage when optimizing their response unilaterally
15
The relaxation algorithmn To ensure convergence of NIRA, the most
important condition in coupled constraint games is Diagonal Strict Concavity (DSC)
n DSC means that each player has more control over his payoff than the other players have over it
n Coupled constraint games can be due, for example, to Kirchhoff’s laws
16
Duopoly examplen Two identical firms selling a productnMarket price:n Profit made by firm i:
n The Nikaido-Isoda function :
)()( 21 xxp +−= ραx
iiii xxxxxp ])([)()( 21 +−−=−= ρλαλφ xx
221221
121121
])([])([ ])([])([xxxyyxxxxyxy
+−−−+−−++−−−+−−
ρλαρλαρλαρλα
),( yxΨ
17
Duopoly examplen The optimum response function Z(x):
),(21)1 ,1(
2),(max arg 12 xx
X−
−=Ψ
∈ ρλαyx
y
22 ;
22
e wher),,(),(
12
21
2121
xyxy
yyxxZ
−−
=−−
=
=
ρλα
ρλα
18
Duopoly examplen Convergence to Nash equilibrium if:
is positive definite
xyyyxyxxQ == Ψ−Ψ= ),(),(),( yxyxxx
19
Duopoly example
2112
2ρ =
xy
yy
yy
xy
xx
xx
xyyyxyxx
yyxxy
yyxxy
yyxxy
yyxxy
yyxxx
yyxxx
yyxxx
yyxxx
Q
=
=
==
Ψ∂∂
Ψ∂∂
Ψ∂∂
Ψ∂∂
−
Ψ∂∂
Ψ∂∂
Ψ∂∂
Ψ∂∂
=Ψ−Ψ=
),,,(),,,(
),,,(),,,(
),,,(),,,(
),,,(),,,(
),(),(),(
212122
212121
212112
212111
212122
212121
212112
212111
yxyxxx
20
Case study 1n Three-bus systemn Unlimited capacity generators at buses
1 and 2n Demand at buses 1, 2 and 3n Each pair of buses connected by a
transmission linen No transmission line flow limits
21
Case study 1n Demand functions:
pi(qi) = 40 – 0.08*qi for buses i = 1, 2p3(q3) = 32 – 0.0516*q3 for bus 3
n Generator constant marginal costs are $15/MWh and $20/MWh, respectively
n Bilateral market: sij means that generator (company) i sells s MW to consumers at node j
n Pfgi is the production of generator g of company f placed at node i
22
Case study 1n Each generator (company) maximizes profit:
sales revenue – generation cost
subject to
fgigi
fgifjfk
kjfjjojoj
jo PPCsssQPP ∑∑∑ −+−≠ ,
)(]))(/([max
0
generators , nodes ,
,
,
max
≥∀
=
=
∀≤
∑
∑∑
fgifj
jf
fj
gifgi
jfj
fgifgi
Ps
qs
Ps
giPP
23
Case study 1Firm 1
Firm 2
subject to:
}15])(0516.032[
])(08.040[])(08.040[ { max
1,1,1132313
122212112111
Psss
ssssss
−+−+
+−++−
}20])(0516.032[
])(08.040[])(08.040[ { max
2,2,2232313
222212212111
Psss
ssssss
−+−+
+−++−
24
Case study 1
0,,,,,
,)()(
,)()(
,)()(
,,,
,,
232221131211
323
32
13
31
223
23
12
212,2,2
113
13
12
121,1,1
23133
22122
21111
2322212,2,2
1312111,1,1
≥
=−
+−
=−
+−
+
=−
+−
+
+=+=+=
++=
++=
ssssss
qx
Sx
S
qx
Sx
SP
qx
Sx
SP
ssqssqssq
sssPsssP
basebase
basebase
basebase
θθθθ
θθθθ
θθθθ
26
Results case study 1
n q1 = 187.5 MW, price1 = $25/MWh n q2 = 187.5 MW, price2 = $25/MWh n q3 = 187.3 MW, price3 = $22.3/MWhn Flow 1-2: 73.95 MWn Flow 1-3: 130.65 MWn Flow 2-3: 56.7 MWn Profit1 = $/h 3542.1n Profit2 = $/h 730.6
27
Case study 2n Transmission line flow limit of 25 MW on
line 1-2 (binding constraint)n New constraints added to the ones in
case study 1:
baseSx25
12
21 ≤−θθ baseSx
25
12
21 ≤−θθ
baseSx25
12
12 ≤−θθ
30
Results case study 2
n q1 = 199.1 MW, price1 = $24.1/MWh n q2 = 175.9 MW, price2 = $25.9/MWh n q3 = 187.3 MW, price3 = $22.3/MWhn Flow 1-2: 25 MWn Flow 1-3: 106.15 MWn Flow 2-3: 81.15 MWn Profit1 = $/h 2985n Profit2 = $/h 956.9
31
Case study 2: constraint enforcementn Assume an authority is empowered to
charge agents from deviations of the line flow limit
n The Lagrange multipliers of the activeconstraints (Kirchhoff’s law and line flow limits) calculated by NIRA can be used to enforce constraints
32
Case study 2: constraint enforcementn The Lagrange multipliers decouple a
game where the players’ actions are coupled through constraints
n The steps of the procedure are as follows:
33
Case study 2: constraint enforcement1. The regulator solves a coupled constraint
game, i.e. computes production levels (which do not violate constraints) and the Lagrange multipliers
2. The regulator modifies the players’ payoffs by telling them the value of the Lagrange multipliers
3. The players solve the new game with new payoffs and “miraculously” obtain the same production levels as the regulator
34
Case study 2: constraint enforcementn The authority announces that for a unit
of constraint violation each player will be charged:
)25,0max(
))()(,0max(
))()(,0max(
12
214
323
32
13
313
113
13
12
121,1,11
−
−+
−−
+−
+
−−
+−
+
xS
qx
Sx
S
qx
Sx
SP
base
basebase
basebase
θθλ
θθθθλ
θθθθλ
35
Case study 2: constraint enforcementn The previous extra term is added to the
original individual payoff functions of firms 1 and 2 (slide 23)
n The same result as in case 2 is obtained, without coupling constraints (decoupled Nash equilibrium)
36
Conclusions
n A Nikaido-Isoda Relaxation Algorithm to find unique Nash-Cournot equilibria with coupled constraints
n Either a centralized or a distributed optimization viewpoint
n Applications to electricity tradingn Enforcement of constraints decouples
the problem and finds the same solution