Energy-mix equilibrium in a two stage electricity market
with risk averse producers
Risk Management in Electricity Generation
Paolo Falbo1
1 Department
Carlos Ruiz Mora2
of Economics and Management, University of Brescia
of Statistics, Universidad Carlos III de Madrid
2 Department
May 30 - June 1, 2017
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Motivation
Controlling the volatility of pro…ts for an electricity producer is a big
challenge.
key variables of daily pro…t:
1
Electricity price
2
Fuel costs (gas, coal)
3
Generation from renewable sources
4
Demand level, ...
However
These variables do not move independently
Dependencies change from market to market, and from time to time
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Correlations among key factors
Economic and structural facts originate links between factors:
high concentration,
the uniform auction mechanism on the spot market and, last but not
least,
technical impossibility to store large amount of energy,
the high inelasticity of the demand.
positive link direct generation costs ) electricity prices
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Some empirical evidence - Spain
Spline interpolation of Spot electricity price against Demand and
Gas price (Spanish market, daily data, period 2015-2016)
Gas price clearly leads Electricity price much more directly than
Demand
SPAIN
ElecPri
62.32
45.93
29.54
13.15
850000
800000
Falbo and Mora (Institute)
750000
700000
650000
Demand
600000
550000
500000
Risk Management in Electricity Generation
1.4
1.6
1.8
2.0
2.2
3.2
3.0
2.8
2.6
2.4
GasCont
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Some empirical evidence - Spain
Spline interpolation of Demand against Renewable source
generation (Spanish market, daily data, period 2015-2016)
Very weak (if any) negative dependency
Spain
Demand
850000
800000
750000
700000
650000
600000
550000
500000
30000
90000
150000
210000
270000
330000
Solar+Wind
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Some empirical evidence - Germany
Spline interpolation of Spot electricity price against Demand and
Gas price (German market, daily data, period 2015-2016)
Here even observe a negative impact of Demand on Spot electricity
price. Gas price still shows a postive in‡uence overall.
GERMANY
ElecPriDE
43.67
30.63
17.58
2.4
2.6
2.8
3.0
2.2
4.54
2.0 GasCont
1000000 950000
1.8
900000 850000
800000 750000
1.6
700000 650000
Demand
600000 550000 1.4
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Some empirical evidence - Germany
Spline interpolation of Demand against Renewable source
generation (German market, daily data, period 2015-2016)
Completely di¤erent type of dependency (w.r.t. Spain), waving from
negative to positive.
Germany
Demand
1000000
950000
900000
850000
800000
750000
700000
650000
600000
550000
500000
0
50000
100000
150000
200000
250000
300000
Solar+Wind
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Risk management in the electricity market
Two major channels to sell the generation In the electricity market:
spot market and
bilateral contracts
Bilateral contracts:
large family with many examples
common feature: …xed price to supply energy over a period of time.
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Risk management problem
Basic idea
Set up an optimization problem for a risk averse producer (decision
maker) where
the dependency between random Demand and Renewable
generation is introduced and
next to the (classical) decision on quantities and price, he can decide
the optimal sales channel-mix between Spot market and Bilateral
contracts
interacting with competitor producers
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Model
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Shaping the demand function
A price cap (p s ,max ) is placed to bound the inelastic component of the
demand
Figure: Inverse demand curve for the Spanish day-ahead electricity market (12:00,
07/03/2016) and its piecewise linear approximation
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Essential notation
Indices
i Index for competitor producers running, 1 to I .
j Index for energy blocks (e.g. plants) owned by the
decision maker, 1 to J. In particular j = 1 is referred to
solar+wind (RES) plant
p Index for competitor producer energy blocks, 1 to P.
ω Index for scenarios running from 1 to Ω. Random variables
are identi…ed with a tilde ˜.
Decision variables
qjb Bilateral (futures) quantity signed by decision maker for
energy block j
s
qj Spot quantity for decision maker for energy block j.
qisr,j Spot quantity assigned by competitor producer i for block j
Output variables
λ Spot price of electricity
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Essential notation
Random variables
q
ecd max demand level.
e
cj
max
q
ej
direct unit cost.for energy block (plant) j
max generation from renewable energy block (plant) j = 1
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Bilevel formulation
1
First level: risk averse producer
2
Second level: cost minimizer competitors
max
q jb ,u j
s.t.
0
F p b ∑j qjb + λ ∑j qjs
qjb
(1
uj 2 f0, 1g
fqjs , qisr,p , qcd , qed g 2 arg min
s.t.
qed
+ qdd
=∑
0
qjs
(continues...)
Falbo and Mora (Institute)
qs
j j
8j
ui ,j )q
ejmax
8j
e
cj qjs + ∑ e
cir,p qisr,p
i ,p
∑j ecj (qjb + qjs )
β 2
(qcd + qed )p s max + qed
2
∑
+ i ,p qisr,p (λ)
max
uj q
ejmax
(µmin
)
j , µj
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8j
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Bilevel formulation
(... continued)
0
qisr,p
0
qcd
0
qed
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r
q
eimax
,p
q
ecd max
p s max
β
r
max r
(µmin
i ,p , µi ,p )
∑j qjb
8ip
max
(γmin
c , γc )
max
(γmin
e , γe )
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Single level MPEC
Four …rst-order KKT conditions
F p b ∑j qjb + λ ∑j qjs
max
q jb ,u j
s.t.
qjb
0
(1
uj 2 f0, 1g
q1d
e
cj
+ q2d
uj )q
ejmax
8j
+λ+
8j
r
µmin
=0
i ,p
p s max + λ + γmax
1
p
8j
µmin
=0
j
r
λ + µmax
i ,p
s max
(1)
= ∑j qjs + ∑i ,p qisr,p
λ + µmax
j
e
cir,p
∑j ecj (qjb + qjs )
βq2d
(2)
8i, 8p
(3)
γmin
=0
1
+ γmax
2
γmin
2
(4)
=0
(5)
(continues...)
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Single level MPEC
(... continued)
eight complementarity constraints
0
0
0
0
0
0
0
0
Falbo and Mora (Institute)
uj q
ejmax
qjs ? µmax
j
qjs ? µmin
j
r
q
eimax
,p
qisr,p ?
q
ecd max
qisr,p
r
µmin
i ,p
8j
0
?
r
µmax
i ,p
0
∑j qjb
0
8i, p
qcd ? γmax
c
qcd ? γmin
0
c
s
max
p
qed ? γmax
e
β
qed ? γmin
e
8j
0
8i, p
0
0
0
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Treatment of non linear terms
The complementarity constraints can be linearized by introducing
r
min r
d max ,
auxiliary binary variables (ujmax , ujmin , uimax
,p , ui ,p , uc
ucd min ,ued max and ued min ).
As for the bilinear products λ ∑j qis,j in the objective function, it is
possible to use the …rst KKT condition:
λ=e
cj + µmax
j
µmin
j
8j
multiplying all terms by qjs :
λqjs = e
cj qjs + µmax
qjs
j
s
µmin
j qj
8j
and adding 8j, turns out:
λ ∑j qjs =
Falbo and Mora (Institute)
uj q
ejmax
∑j ecj qjs + ∑j µmax
j
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MILP formulation
F p b ∑j qjb + ∑j vjmax qejmax
max
s.t.
qjb
0
(1
0 vjmax
0 µmax
j
d
q1 + q2d =
e
cj
uj )q
ejmax
uj M
vjmax
qs
j j
∑
λ + µmax
j
e
cir,p
8j
(1
+∑
p
+λ+
uj )M
8j
βqed
(7)
8i, p
(8)
γmin
=0
c
+ γmax
e
(6)
8j
q sr
i ,p i ,p
r
µmin
=0
i ,p
p s max + λ + γmax
c
s max
8j
µmin
=0
j
r
λ + µmax
i ,p
∑j ecj qjb
γmin
e
(9)
=0
(10)
(continues...)
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MILP formulation
(... continued)
0
0
0
0
0
0
0
0
uj q
ejmax
µmax
j
qjs
(1
ujmax M
8j
ujmax )M
qjs
ujmin M
8j
µmin
(1 ujmin )M
j
r
r
q
eimax
qisr,p uimax
,p
,p M
r
r
µmax
(1 uimax
,p )M
i ,p
r
qisr,p uimin
8i, j
,p M
min r
min r
µi ,p
(1 ui ,p )M
8j
8j
8i, p
8i, p
8i, p
(continues...)
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MILP formulation
(... continued)
0
0
0
0
0
q
ecd max
γmax
c
qcd
∑j qjb
(1
qcd
ucd max M
ucd max )M
ucd min M
γmin
(1 ucd min )M
c
p s max
qed
ued max M
β
0
γmax
e
0
qed
0
γmin
e
(1
ue )d max M
ued min M
(1
ued min )M
r
max d max d min d max d min
ujmin , ujmax , uimin
, uc
, ue
, ue
2 f0, 1g
,p , ui ,p , uc
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8j, i, p
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A risk averse formulation
A speci…cation of the functional F in the previous MILP formulation, that
is in line with a risk averse decision maker, is the linear combination of the
expected pro…t and CVAR (see [1], [2], and [3] for MILP application).
(1
max
q jb ,ξ,η ω
ϕ) ∑ω σω Πω + ϕCVaR
s.t.
CVaR =
1
ξ
Ω
σω η ω
α ∑ ω =1
1
Πω + ξ, η ω
ηω
0,
max
Πω = p b ∑j qjb + ∑j vjmax
,ω qj ,ω
0
qjb
0
vjmax
,ω
(1
uj )qjmax
,ω
uj M
8j, ω
8ω
∑j cj ,ω qjb
8ω
8j, ω
(continues...)
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A risk averse formulation
(... continued)
µmax
j ,ω
0
vjmax
,ω
d
qcd,ω + qe,ω
=
λω + µmax
j ,ω
cir,p,ω
r
λω + µmax
i ,p,ω
s max
0
0
0
uj )M
∑ qjs,ω + ∑ qisr,p,ω
j ,ω
cj ,ω
(1
i ,p
µmin
j ,ω = 0
8j, ω
8ω
8j, ω
r
µmin
i ,p,ω = 0
8i, p, ω
p
+ λω + γmax
γmin
c ,ω
c ,ω = 0 8 ω
d
p s max + λω + βqe,ω
+ γmax
γmin
e,ω
e,ω =
max
s
max
uj qj ,ω
qj ,ω uj ,ω M
8j, ω
max
max
µj ,ω
(1 uj ,ω )M
8j, ω
s
min
qj ,ω uj ,ω M
8j, ω
0
8ω
(continues...)
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A risk averse formulation
(... continued)
0
µmin
j ,ω
0
r
qimax
,p,ω
0
r
µmax
i ,p,ω
qisr,p,ω
r
µmin
i ,p,ω
max
qcd,ω
0
0
0
ujmin
,ω )M
(1
qisr,p,ω
(1
r
uimax
,p,ω M
r
uimax
,p,ω )M
r
uimin
,p,ω M
(1
∑
8j, ω
8i, p, ω
8i, p, ω
8i, p, ω
r
uimin
,p,ω )M
qjb qcd,ω
8i, p, ω
max
ucd,ω
M
j
0
γmax
c ,ω
0
qcd,ω
0
γmin
c ,ω
(1
max
ucd,ω
)M
min
ucd,ω
M
(1
8ω
min
ucd,ω
)M
8ω
8ω
8ω
(continues...)
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A risk averse formulation
(... continued)
0
p s max
β
0
γmax
e,ω
d
qe,ω
(1
d max
ue,ω
M
d max
ue,ω
)M
8ω
8ω
d
d min
0 qe,ω
ue,ω
M
8ω
d min
0 γmin
(1 ue,ω
)M
8ω
e,ω
min max min r
max
d max d min d max d min
uj ,ω , uj ,ω , ui ,p,ω , ui ,p,ω , uc ,ω , uc ,ω , ue,ω , ue,ω
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2 f0, 1g
8j, i, p, ω
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Application
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Generation of scenarios and Dependencies among random
variables
To keep the analytical tractability, no variable appears to model the
dependencies among the random variables.
However, a rank correlation matrix M can be manipulated and placed
at the heart of a copula method for the generation of scenarios.
In this application a scenario (ω) is a realization of 10 random
variables: demand level (e
qcd max ), renewable generation (solar+wind,
max
q
ej , j = 1), unit cost of gas (e
cj , j = 2, .., 5), unit cost of coal
(e
cj , j = 6, .., 9)
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Generation of scenarios and Dependencies among random
variables
The rank correlation matrix M has been …rst estimated from
historical data (Spanish market)
Focus on the rank correlation index between Demand of electricity
and Renewable generation (ρ12 and ρ21 ) .
For this parameter a grid of values has been
considered:ρ12 = 0.9, 0.6, ..., 0.9.
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Generation of scenarios and Dependencies among random
variables
Scenario plan: for every distinct value of ρ12 )100 instances of
the problem, each based on 50 scenarios.
In the end, the procedure generates a distribution of 100 optimal
solutions for every value of ρ12
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Results
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Basics for interpretation
1
Observe that increasing renewable generation has three (strong)
competing e¤ects: lower spot price, lower probability to hit max
spot price, sustain pro…tability (RES have the highest pro…t margin
per MWh)
2
Question: how high and low levels of Demand and RES generation
interact on sport price, sales and risk of pro…t,...
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Basics for interpretation
Negative correlation
It favors scenarios where high Demand meets low Renewable generation
and viceversa
1
1
2
3
) spot price can be very high or very low
) sales can be large in volume and price level, but with low
pro…tability (because of low contribution form RES), or low volume and
low price (because of high renewable gen.)
) impact on pro…t unclear, it depends on which e¤ect dominates
between price and RES sales can switch from average to very low ()
High risk)
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Basics for interpretation
Positive correlation
It favors scenarios where high Demand meets high Renewable generation
or viceversa
1
1
2
3
) spot price could remain average in both cases because of reciprocal
compensating e¤ects
) sales can be large in volume and good in pro…tability (because large
sales of RES) or low in volume and average pro…tability (high RES
generation lowers spot price but increase pro…tability)
) pro…t, like prices, should remain around average level because of
compensating e¤ects of RES on pro…tablity and level of spot price. ()
Low risk)
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Basics for interpretation
The following …gure synthesizes the various impacts.
No obvious solution can be advanced even at a qualitative level.
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Risk averse solution (risk weight 50%)
Futures trading
Max futures trading when risk increases (with negative ρ12 ).
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Risk averse solution (risk weight 50%)
Spot market sales
With negative covariance, slight increase of spot trading
Figure: Left panel distrib. of optimal solutions; right panel distrib. of singleton
solutions
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Risk averse solution (risk weight 50%)
Spot market sales for competitors
At negative correlation, competitors reduce their trading on the spot
market. Futures contracts (of the decision maker) shrink the demand
left to the spot market (i.e. demand is …rst met by futures contracts).
Figure: Left panel distrib. of optimal solutions; right panel distrib. of singleton
solutions
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Risk averse solution (risk weight 50%)
Spot price
Spot prices increase and reduce in volatility as correlation parameter
ρ12 increases. Volatility of spot price tends to grow as ρ12 becomes
negative, however it …nally reduces again at ρ12 = 0.6 and 0.9
because of the increase in futures trading.
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(Institute)
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Management
in Electricity Generation
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2017
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Figure:
Left
distrib. of
solutions;
right panel distrib.
of 1,singleton
Risk averse solution (risk weight 50%)
Expected pro…ts
Negative values of ρ12 reduce expected pro…ts. Volatility of pro…ts is
not so clear.
Figure: Left panel distrib. of optimal solutions; right panel distrib. of singleton
solutions
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Risk averse solution (risk weight 50%)
CVar
CVar worsen as correlation increases (let’s recall that CVar here
measures the expectation of the worst α-percent cases, so, the higher
the CVar, the better for the decision maker).
In the end, higher values of ρ12 improve expected pro…t, but increase
risk, and viceversa. So the optimal solutions drive the electricity
producer to a standard …nancial framework
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Risk neutral solution (risk weight 0%)
Similar solutions obtain for a risk-neutral producer.
The noticeable di¤erences consist of:
the quantities traded in the futures are smaller than in the risk
averse case.
more energy is traded in the spot ) higher average spot prices.
CVaR lowers (lower pro…ts, so higher risk)
So overall the risk neutral solutions re‡ects theoric expections.
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Conclusions and directions for future research
Including correlation of Demand-RES generation has an strong
impact on the optimal solution.
In particular, a risk averse producer should prefer the spot market
channel for his sales when ρ12 is positive and increase bilateral
contracts in situations of negative correlation.
Such results sustain the opportunity for the development of e¢ cient
…xed price markets
It is necessary to extend the analysis to include the correlations
between other key variables.
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Essential bibliography
Rockafellar, R.T., and S. Uryasev, 2000. Optimization of conditional
value-at-risk, J. Risk 2:21–41.
Rockafellar, R. T., and S. Uryasev, 2002. Conditional value-at-risk for
general loss distributions. J. Banking Finance 26(7):1443–1471.
Schultz, R., and S. Tiedemann, 2006. Conditional value-at-risk in
stochastic programs with mixed-integer recourse, Mathematical
Programming 105(2-3):365–386.
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Thank you...
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