Supply function equilibrium models
Examples
Application
Supply function equilibrium models for electricity
markets1
Andy Philpott
Electric Power Optimization Centre
University of Auckland
www.epoc.org.nz
(joint work with Par Holmberg and Tony Downward)
1 ICSP,
Bergamo, July 11, 2013
Conclusion
Supply function equilibrium models
Examples
Application
This talk
Some recent results in supply function models.
Basic concepts
New results on SFE in transmission networks
Examples
Application to a problem in transmission charging
Conclusion
Supply function equilibrium models
Examples
Application
Supply function auction (single location)
Supply-function auction with two suppliers and inelastic demand
Suppliers o¤er supply functions qi (p ) indicating what they will
supply at price p. Denote the inverse supply curves by pi (q ).
Demand is D (p ) + h, where h is a random shock. When h is
realized the markets clear at a price p de…ned by
∑ qi (p ) = D (p ) + h.
i
Conclusion
Supply function equilibrium models
Examples
Application
Conclusion
Market clearing conditions for radial (tree) networks
The topology of the network is described by a node-arc incidence
matrix A.
KKT: 0
0
ρ?K
t
0
σ ? K+t
0
A> p = ρ
σ
At + s(p) = ε.
ρ, σ are non-negative and only positive
when their constraints bind
shadow prices ) price di¤erence
net-supply=net-exports in every node
Nodes connected by uncongested lines have the same price. We
say that they are completely integrated. Each node is completely
integrated with itself.
Supply function equilibrium models
Examples
Application
Best response (single node)
If other suppliers o¤er supply functions qj (p ) for j 6= i, then in
demand shock h, supplier i faces a residual demand curve
D ( p ) ∑ j 6 = i qj ( p ) + h
Conclusion
Supply function equilibrium models
Examples
Application
Conclusion
Supply function equilibrium (SFE)
Wilson, 1979 (auctions of shares)
Klemperer and Meyer, 1989 (uncertain demand)
Green and Newbery, 1992 (UK market)
Rudkevich, Duckworth, and Rosen, 1998 (inelastic demand )
Anderson and Philpott, 2002 (inelastic demand)
Holmberg, 2008 (bounds and price caps)
Anderson and Hu, 2008 (asymmetry)
Wilson, 2008 (networks)
Holmberg and Newbery, 2010 (survey)
Anderson, 2013 (existence)
SFE becoming increasingly relevant in electricity auctions because
of demand uncertainty.
Supply function equilibrium models
Examples
Application
Conclusion
Ex-post optimality
Klemperer and Meyer, 1989
In a single node with n suppliers and one demand shock h, a
symmetric SFE q (p ) satis…es the following di¤erential equation
j residual demand slope j
q (p ) = p
z
C 0 (q ) ((n
}|
1)q 0 (p )
{
D 0 (p ))
The solution does not depend on the probability distribution of h.
For each realization of h the supply function speci…es an optimal
quantity and price. This is a wait-and-see solution or ex-post
optimal supply function.
Supply function equilibrium models
Examples
Application
Conclusion
Ex-ante optimality
In transmission networks with many nodes, each with a demand
shock, the solution to an equilibrium cannot be ex-post optimal. It
must be a here-and-now supply curve that depends on the
probability distribution of the demand shock, i.e. ex-ante optimal.
Supply function equilibrium models
Examples
Application
Conclusion
Market distribution function
Anderson and Philpott (2002), Wilson (1979)
The market distribution function ψ(q, p ) de…nes the probability
that a supplier is not fully dispatched if they o¤er the quantity q at
price p. It can be interpreted as the measure of residual demand
curves that pass below and to the left of the point (q, p ).
Random residual demand curves faced by a supplier. Here ψ(q, p )
is the probability of a curve being red.
Supply function equilibrium models
Examples
Application
Conclusion
Pro…ts at uniform prices
Anderson and Philpott (2002)
The optimal o¤er curve p (q ) for a supplier with pro…t R (q, p )
facing a market distribution function ψ(q, p ) maximizes
E[R ] =
Z qM
0
R (q, p (q ))d ψ(q, p (q )) +
+R (qM , P )(1
ψ(qM , P ))
Z P
p (q M )
R ( qM , p ) d ψ ( qM , p )
Supply function equilibrium models
Examples
Application
Conclusion
Optimality conditions for p(q) under uniform pricing
p
q
Example showing contours of ψ(q, p ) (black). Isopro…t lines for R (q, p )
are red. The optimal curve q = S (p ) (blue) passes through the points
where these curves have the same slope.
Blue curve is Z (q, p ) = 0 where
Z (q, p ) =
∂R (q, p ) ∂ψ(q, p )
∂q
∂p
∂R (q, p ) ∂ψ(q, p )
∂p
∂q
Supply function equilibrium models
Examples
Application
Conclusion
Example (ex-post optimal duopoly)
Suppose other player o¤ers supply curve S (p ) and h has
distribution function F .
R (q, p ) = pq
C (q )
ψ(q, p ) = Pr(h < q + S (p )
D (p )) = F (q + S (p )
D (p ))
then
0 =
=
∂R (q, p ) ∂ψ(q, p ) ∂R (q, p ) ∂ψ(q, p )
∂q
∂p
∂p
∂q
0
0
0
p C (q ) (S (p ) D (p ))f (q + S(p)
qf (q + S(p)
D(p))
gives (setting S (p ) = q (p ) from symmetry)
q= p
C 0 (q ) (q 0 (p )
D 0 (p ))
D(p))
Supply function equilibrium models
Examples
Application
Conclusion
Two node networks
Holmberg and Philpott, 2013
Symmetric two node network. For symmetric duopoly with small K ,
Cournot equilibria don’t exist (Borenstein et al, 1998, Downward et al
2010). But SFE exist for some demand shock densities (e.g. uniform).
Supply function equilibrium models
Examples
Application
Conclusion
Market distribution function at node 1
Suppliers have capacity q, a price cap of P, and inelastic demand
with shocks uniform with density V1 on
f(ε1 , ε2 ) 2 [ K , 2q + K ] [ K , 2q + K ] : 0 ε1 + ε2 4q
ψ (q, p ) = ψ (0, 0) + q
2K + 2qM
2K
2q
+ 3q (p )
+ q (p ) M
V
V
V
Supply function equilibrium models
Examples
Application
Supply-function equilibrium (inelastic demand)
q= p
C 0 (q )
q 0 (p )
q
K
+ 3q 0 (p )
K +q
K +q
Conclusion
Supply function equilibrium models
Examples
Application
Conclusion
Extension to radial (tree) networks
For given demand realization, only net-supply in nodes that are
completely integrated contribute to slope of residual demand.
Optimal output q of …rm g in node i at price p:
!
q= p
Cg0 (q )
Si0,
|
g
(p ) +
∑ Sk0 (p ) Pik (p, q )
k 6 =i
{z
Expected residual demand slope
Pik (p, q ) = conditional probability node i and k
completely integrated
0
Sk (p ) = total net-supply slope in node k
(including D (p ))
Si0, g (p ) = total net-supply slope node i,
(including
D (p ) not supplier g )
.
}
Supply function equilibrium models
Examples
Application
Conclusion
Symmetric …rms in symmetric radial network
Optimal output setting
q = (p
0
Expected residual demand slope
z
C (q )) (µ (p, q ) n
}|
1) q 0 (p )
µ (p, q ) D 0
µ (p, q ) = 1 + ∑k 6=i Pik (p, q )
= Expected number of completely integrated nodes
n = number of …rms per node
q (p ) = optimal supply of each …rm
D 0 (p ) = non-strategic demand slope per node
{
The market integration function µ (p, q ) is independent of q (p ) for
inelastic demand or uniformly distributed shocks. For M nodes
1
µ
M.
Supply function equilibrium models
Examples
Application
Conclusion
Optimal production in radial (tree) networks
Ex-post optimal SFE in single-node network (one shock):
Residual demand slope
q = (p
z
C 0 (q )) ((n
}|
1)S 0 (p )
{
D 0 (p ))
Ex-ante optimal SFE in radial network (multiple demand shocks):
q = (p
z
C 0 (q )) Si0,
Expected residual demand slope
g
(p ) +
}|
∑ Sk0 (p ) Pik (p, q )
k 6 =i
!{
Supply function equilibrium models
Examples
Application
Conclusion
SFE in 2-node network
Assume n suppliers per node each with capacity q and shocks are
multi-dimensional uniform distribution. Market integration factor is
µ=
q = (p
q= p
nq + 4K
q + 2K
=
nq + 2K
q+K
C 0 (q ))
C 0 (q )
2
q 0 (p )
q + 2K
q+K
1 q 0 (p )
q
K
+ 3q 0 (p )
K +q
K +q
Supply function equilibrium models
Examples
Application
Conclusion
SFE in star network
Assume n suppliers per leaf node each with capacity q and shocks
are multi-dimensional uniform distribution. Market integration
factor is
3 (nq )2 + 12K (nq ) + 12K 2
µ=
3 (nq )2 + 8K (nq ) + 4K 2
Supply function equilibrium models
Examples
Application: transmission charges
Application
Conclusion
Supply function equilibrium models
Examples
Application
Conclusion
Bene…ciary-pays pricing
1
2
Run dispatch software with transmission asset and record
dispatch and price.
Compute bene…ts Π0 (i ) of agent i, for each i.
3
Re-run software with transmission asset derated to represent
previous state.
4
Record new dispatch and price.
5
Compute counterfactual bene…ts Πc (i ) of agent i, for each i.
6
Charge a proportion of (Π0 (i )
Πc (i ))+ to agent i.
Supply function equilibrium models
Examples
Application
Conclusion
Bene…ciary-pays pricing
The supplier bene…t (Π0 (i ) Πc (i )) in a demand outcome q̂ shown
shaded for two candidate o¤er curves. The counterfactual (constrained
line) reduces dispatch to q̂c .
Supply function equilibrium models
Examples
Application
Example
Model line expansion as a change in market distribution function
from ψc to ψ.
E[B (q, p )] = Eψ [qp ]
Z qM
0
Eψc [qp ]
p (q )(1
Z qM
ψ(q, p (q )))dq
p (q ) (1
0
Π (p (q )) = E[R (q, p (q ))]
ψc (q, p (q ))) dq
αE[B (q, p (q ))]
Conclusion
Supply function equilibrium models
Examples
Application
Conclusion
Example
Z (q, p ) =
p
∂ψ
∂ψ
q
∂p
∂q
∂ψ ∂ψc
+ ψ (q, p )
∂q
∂q
C 0 (q )
+α q
ψc (q, p )
Supply function equilibrium models
Examples
Application
Recall market distribution function
ψc (q, p ) = ψ (0, 0) + q
2K + 2qM
2K
2q
+ 3q (p )
+ q (p ) M
V
V
V
Conclusion
Supply function equilibrium models
Examples
Application
Conclusion
Equilibrium example
Suppose C (q ) = 0, D (p ) = 0, two suppliers at each node, each
agent has capacity qM , hi uniform on [0, 4qM ]
p
q ( p ) = qM ( ) t ( α ) ,
P
p
t (α) =
α (10K + 6qM
3
5) + 1
4
2
0
0.0
0.1
Supply function equilibria when K =
25% bene…ts tax (blue) and
0.2
3
8,
q
qM = 14 , with no bene…ts tax,
100% bene…ts tax (red).
Supply function equilibrium models
Examples
THE END
Application
Conclusion