Risk averse equilibrium
in electricity market
Henri Gerard
Andy Philpott, Vincent Leclère
CERMICS - EPOC
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
1 / 25
Why do we look for equilibria in electricity market ?
In electricity market, agents are
more and more subject to
I
I
uncertainty (e.g. weather forecast)
risks (e.g. black out)
Taking risk into account, their optimal
decison influence the market equilibrium
We want to
I
I
I
understand the impact of risk on equilibrium
manage large scale problem
study the distribution of welfare
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Outline
1
Numerical results on a toy problem
2
Abstract framework and results on risk averse equilibrium
3
Ongoing work and open questions
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
3 / 25
Numerical results on a toy problem
Outline
1
Numerical results on a toy problem
2
Abstract framework and results on risk averse equilibrium
3
Ongoing work and open questions
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
4 / 25
Numerical results on a toy problem
Statement of the toy problem
This problem is largely inspired from Philpott, Ferris, and Wets (2013)
The problem has the following
features
two stage problem
uncertainty = inflows
scenario tree structure
Thermal
production
Hydro
production
D
thermal producer
hydro producer
an exogenous demand D
Price
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Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Expectation
puts weight on all scenari
EP [X ]
Henri Gérard (CERMICS - EPOC)
= 0.25X (ω1 ) + 0.5X (ω2 ) + 0.25X (ω3 )
=2
Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Worst Case risk measure
puts all the weight on the worst cost
WC (X ) = X (ω3 )
=3
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Risk averse equilibrium in electricity market
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Numerical results on a toy problem
CVARβ
puts the weight on a fraction β of the worst scenari
1 0.5X (ω2 ) + 0.25X (ω3 )
1−β
= 2.3333 with β = 0.75
CV@Rβ [X ] =
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Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Statement of the toy problem Producer/Producer
min
x1a ,u2a ,x2a ,u2a
La1 (x1a , u1a ) +
(1 − λ)
| {z }
Proportion of risk
|




EP La2 (x2a , u2a ) + λ CV@R1−α La2 (x2a , u2a )
| {z }
risk measure
{z
Objective cost
z}|{
Production
cost:
Lai
=
subject to
a
a
 Dynamics: x2 (ω) = x1 − u1a + ω


Bounds: xia ∈ Xai , uia ∈ Uai
0≤
}
numerical evalutaion of second stage
instantaneous cost
z}|{
Cia
price production
z}|{
gia
z}|{
− πi
X
Di −
gia (xia , uia ) ⊥ πi ≥ 0
|{z}
a∈A
demand
{z
}
|
supply
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Computing an equilibrium
We have computed equilibria using two tools
GAMS and EMP: generation of a system of KKT conditions
Ferris et al. (2009)
Julia and JuMP: implementation of an iterative algorithm (Uzawa
algorithm, Walras tâtonnement, ...)
Cohen (2004)
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Finding equilibrium by Walras tatonnement Uzawa (1960)
(Trial and error)
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Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Numerical results on a toy problem
Mean error
Max error
Standard deviation error
0.01%
0.14%
0.01%
GAMS vs Julia
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Numerical results on a toy problem
How price evolves with risk aversion ?
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
13 / 25
Numerical results on a toy problem
Conclusion on the toy problem
So far we have
defined a risk averse toy problem with two producers
presented an iterative algorithm to compute equilibrium prices
shown that risk aversion can have an impact on equilibrium
We will now
state the problem in a more general framework
give a theorem of existence
discuss on efficiency of the solution
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
14 / 25
Abstract framework and results on risk averse equilibrium
Outline
1
Numerical results on a toy problem
2
Abstract framework and results on risk averse equilibrium
3
Ongoing work and open questions
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
15 / 25
Abstract framework and results on risk averse equilibrium
Statement of the problem
Cost function
We look for a competitive equilibrium in electricity market
Agent act as if their actions have no influences on the price π
They are said to be price takers
We endowed each agent by
I
I
I
a decision space Xa
a cost function Ca (xa )
a production function ga (xa )
and she tries to solve the problem
min Ca (xa ) − πga (xa )
xa ∈Xa
| {z }
Costs
Henri Gérard (CERMICS - EPOC)
| {z }
Incomes
Risk averse equilibrium in electricity market
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Abstract framework and results on risk averse equilibrium
Statement of the problem
Demand
There is an exogenous demand D
We want the equality supply = demand so we had the
complementarity constraint (coupling constraint)
0≤D−
X
ga (xa ) ⊥ π ≥ 0
a
Price is bounded by πmax and we use the rule
D(πmax ) = min D,
X
ga (xa] (πmax ))
a∈A
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
17 / 25
Abstract framework and results on risk averse equilibrium
Existence of an equilibrium
Proposition (Rosen (1965))
If
the decision spaces Xa are closed, convex and bounded
the cost functions Ca are continuous and convex in xa
the production functions ga are continuous and concave in xa
then there exist a competitive equilibrium in the electricity market
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Abstract framework and results on risk averse equilibrium
Pareto efficiency of an equilibrium
Definition
A feasible production is said to be Pareto efficient if we cannot improve
the welfare of some agent without deteriorating the welfare of an other
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Risk averse equilibrium in electricity market
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Abstract framework and results on risk averse equilibrium
Pareto efficiency of an equilibrium
Definition
A feasible production is said to be Pareto efficient if we cannot improve
the welfare of some agent without deteriorating the welfare of an other
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
19 / 25
Abstract framework and results on risk averse equilibrium
Pareto efficiency of an equilibrium
Definition
A feasible production is said to be Pareto efficient if we cannot improve
the welfare of some agent without deteriorating the welfare of an other
Proposition (Levin (2006))
Assume that agent are price takers
A competitve equilibrium is Pareto efficient
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
19 / 25
Abstract framework and results on risk averse equilibrium
Recall on the theory
In this section we have
given a theorem of existence
shown that equilibria are Pareto efficient
We present now some on going work
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
20 / 25
Ongoing work and open questions
Outline
1
Numerical results on a toy problem
2
Abstract framework and results on risk averse equilibrium
3
Ongoing work and open questions
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
21 / 25
Ongoing work and open questions
On large scale problems,
exact implementation is slow
EMP is unefficient for multistage problem
On a big scenario tree, calculating the optimal price
at each node is inefficient
We want to approximate the price using iterative algorithm
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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Ongoing work and open questions
What is the impact of risk aversion on the distribution of
Welfare
If we had contracts to trade risk, we know there exists a social
planning problem which gives the same solution than the market
clearing problem
We want to study how evolves the distribution of welfare
with risk aversion
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
23 / 25
Ongoing work and open questions
Conclusion
In this talk, we have
studied impact on risk on equilibirum with a toy problem
given some abstract results
given clues to scale the problem
Henri Gérard (CERMICS - EPOC)
Risk averse equilibrium in electricity market
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References I
G. Cohen. Optimisation des grands systemes. 2004.
M. C. Ferris, S. P. Dirkse, J.-H. Jagla, and A. Meeraus. An extended
mathematical programming framework. Computers & Chemical
Engineering, 33(12):1973–1982, 2009.
J. Levin. General equilibrium. San Francisco, 2006.
A. Philpott, M. Ferris, and R. Wets. Equilibrium, uncertainty and risk in
hydro-thermal electricity systems. Mathematical Programming, pages
1–31, 2013.
J. B. Rosen. Existence and uniqueness of equilibrium points for concave
n-person games. Econometrica: Journal of the Econometric Society,
pages 520–534, 1965.
H. Uzawa. Walras’ tatonnement in the theory of exchange. The Review of
Economic Studies, 27(3):182–194, 1960.
Bibliography
Walras’s tâtonemment
Initialize π
For i from 1 to maximum_iteration do
1
√
I
update the step size: τ =
I
compute optimal decision: xan= arg minxa ∈Xa Ca (xa ) − πgao
(xa )
P
]
update multiplier : π = max 0, π + τ D − a∈A ga (xa )
i
]
I
Lemma
The tâtonnement process leaves the price invariant if and only if it is an
equilibrium price
Proposition
If there exist an equilibrium, then the tâtonnement process converges to an
equilibrium price
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Risk averse equilibrium in electricity market
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