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Short-Term Trading for a Wind
Power Producer
Antonio J. Conejo
Juan M. Morales
Juan Pérez
Univ. Castilla – La Mancha
Spain
2010
1/29/2010
1/29/2010 2
3
What
1. Aim
2. Motivation
3. Problem description
4. Mathematical formulation
5. Stochastic programming approach
6. Numerical simulations
7. Conclusions
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4
Aim
Designing a
stochastic programming model
to build offers for wind producers
to trade in different short-term markets
(pool) within a fully-fledged electricity
market.
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5
Wind producer problem
• Wind power technology maturing and
reaching break-even costs
• Subsides decreasing or even being
suspended
• Interest in participating in electricity
markets to maximize profit
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6
Wind producer problem
• Competitiveness damaged by uncertain
wind availability
• Imbalances covered by expensive energy
sources through a balancing mechanism
Imbalance cost!1/29/2010
7
Wind producer problem
The wind producer offer strategy should:
1. Be profit effective
2. Limit profit variability
3. Minimize need for balancing energy
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8
Market framework
• Three independent and successive markets:
1. Day-ahead market
2. Adjustment market
3. Balancing market
• No market power capability by the wind producer in any of the markets.
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9
Market framework
Day d-1 Day d
Price uncertainty
...
10 am
Day-ahead Market
Day d, hours: 1-24
11 pm
Adjustment Market
Day d, hours: 1-24
Balancing Market
before each hour
Wind uncertainty
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10
Market framework
Day d-1 Day d
Price uncertainty
...
10 am
Day-ahead Market
Day d, hours: 1-24
11 pm
Adjustment Market
Day d, hours: 1-24
Balancing Market
before each hour
Wind uncertainty
Knowledge gained on wind
power stochastic behavior1/29/2010
11
Balancing mechanism
• A balancing mechanism is required to
cope with unexpected
production/consumption deviations
• Positive (negative) deviation: higher
(lower) production or lower (higher)
consumption than scheduled
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12
Imbalance prices
• A price for positive deviation:
• A price for negative deviation:
• Settled in a balancing market organized for each time period (hour)
• Cost of the energy required to counteract the system imbalance
tλ
tλ
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13
Mechanism for imbalance prices
If the system imbalance > 0 (excess of generation):
D
tt
DN
t
D
tt
λλ)λ,min(λλ
Dt
DNt
λ
λ
: Day-ahead market price
: Price for the downward energy required to restore
system balance
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14
Mechanism for imbalance prices
If the system imbalance < 0 (deficit of generation):
)λ,max(λλ
λλUPt
Dtt
Dtt
balance system
restore to requiredenergy upward the for price :λUP
t
It holds:Dtt
Dtt λλ and λλ
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15
Mechanism for imbalance prices
Dt-Dt
+
Price
Energy
ltD
l tUP
ltDN
Aggregated power
supply curve
Actual demand
Demand considered
by producers
Excess of
Generation
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16
Mechanism for imbalance prices
Dt-Dt
+
Price
Energy
ltD
l tUP
ltDN
Aggregated power
supply curve
Actual demand
Demand considered
by producers
Deficit of
Generation
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17
Imbalance cost
An opportunity cost
Loss of profit resulting from not having traded the energy deviation in the day-ahead market
If we define (no zero price):
t
t tDt
t
t tDt
λr , r 1
λ
λr , r 1
λ
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18
Imbalance cost
Then, the imbalance cost is given by:
deviationenergy Total :tD
0Δ,1)Δ(rλ
0Δ,)Δr(1λI
ttt
D
t
ttt
D
t
t
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19
Uncertainty sources
• Price uncertainty in (i) day-ahead, (ii) adjustment, and (iii) balancing markets
• Wind generation availability
makes the wind producer problem unique and
is responsible for its profitability loss
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20
Uncertainty characterization
Seasonal ARIMA models to characterize prices
Variable Time series model
ARIMA(2,0,1)(1,1,1)24
ARIMA(1,0,11)(1,0,1)24
ARIMA(2,0,1)(1,0,1)24
)log(λD
t
D
t
A
t λλ -
1rrtt
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0 5 10 15 20 250.4
0.6
0.8
1
1.2
1.4
Period (h)
rt
++r
t
--1
0 5 10 15 20 250.4
0.6
0.8
1
1.2
1.4
Period (h)
rt
-
0 5 10 15 20 250.4
0.6
0.8
1
1.2
1.4
Period (h)
rt
+
Easier to model!
superimposing
1/29/2010
22
Uncertainty characterization
0 5 10 15 20 2520
40
60
80
100
120
Period (h)
€/M
Wh
ND
scenarios for lt
D
0 5 10 15 20 25-20
-10
0
10
20
30
Period (h)
€/M
Wh
NA scenarios for l
t
A-l
t
D
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
Period (h)
€/M
Wh
NI scenarios for r
t
++r
t
--1
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23
Uncertainty characterization
ARMA model (filtered with the marginaldistribution) to characterize wind speeduncertainty
Wind speed
scenarios
Hypothesis: The same wind
characteristics all over the plant at
each instant
Wind turbine
power scenarios
Wind plant power
scenarios
Aggregating Power curve
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24
Wind data
1/29/2010
Wind data in Spain…
25
Wind data
1/29/2010
in Kansas:http://eece.ksu.edu/~gjohnson/
in Massachusetts:http://www.ceere.org/rerl/publications/resource_data/index.html
in Denmark:http://www.energinet.dk/en/menu/Market/Download+of+Market+
Data/Download+of+Market+Data.htm
in New Zealand:http://www.electricitycommission.govt.nz/opdev/modelling/central
iseddata/
Excellent wind data available
26
Uncertainty characterization
• The generation of wind power scenarios should consider the certainty gain phenomenon
• Nw1 scenarios between day-ahead and adjustment market
• Nw2 scenarios after the adjustment market
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0 10 20 30 400
20
40
60
80
100
Period (h)
MW
Nw
1
x Nw
2
= 2 x 50
27
Uncertainty characterization
Average wind power values
observed from the day-ahead
market (and from the adjustment
market if the certainty gain is not
considered)
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0 10 20 30 400
20
40
60
80
100
Period (h)
MW
Nw
1
x Nw
2
= 2 x 50
28
Uncertainty characterization
Average wind power values
observed from the adjustment
market if the certainty gain
effect is modeled
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29
Stochastic programming approach
Three-stage stochastic programming model:
– First-stage variables: Energy sold in the day-
ahead market
– Second-stage variables: Energy traded in the
adjustment market
– Third-stage variables: Deviations and their
associated cost
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30
Stochastic
programming
approach
)(market ahead-Day DtP
trrP ttttt ,,,,,
:yUncertaint
AD ll
TTt
ttt
NNtP
trr
,,1,
,,,
:yUncertaint
1
A
l trr tt ,,
:yUncertaint
)(market Adjustment AtP )(market Balancing tI
1,,1,;,D
Ttt NtPt l TTtt NNtPt ,,1,;,1
A l
Three stages, each one
representing a market
1/29/2010
31
Risk management
1. Tradeoff between expected profit and risk
due to profit variability
2. Risk measure: CVaR (average profit in
scenarios with lowest profit)
3. CVaR advantage: linear formulation
1/29/2010
32
Risk management
Probability
1 - α
CVaR VaR1/29/2010
33
Problem formulation
• Maximize Expected profit + × CVaR
• Subject to constraints associated with:
– Linearization of imbalance income
– Non-anticipativity of information
– CVaR
The tradeoff between expected profit and risk is
enforced through the weighting factor ϵ [0,∞)
1/29/2010
34
Expected profit
E{profit} = E{
Revenue from trading in the day-ahead market
+ Revenue from trading in the adjustment market
+ Imbalance income } =
I
tωt
A
tω
A
tωt
D
tω
D
tω
N
1ω
N
1t
ω IdPλdPλπT
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35
Expected profit
E{profit} = E{
Revenue from trading in the day-ahead market
+ Revenue from trading in the adjustment market
+ Imbalance income } =
= I
tωt
A
tω
A
tωt
D
tω
D
tω
N
1ω
N
1t
ω IdPλdPλπT
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36
Expected profit
E{profit} = E{
Revenue from trading in the day-ahead market
+ Revenue from trading in the adjustment market
+ Imbalance income } =
= I
tωt
A
tω
A
tωt
D
tω
D
tω
N
1ω
N
1t
ω IdPλdPλπT
1/29/2010
37
Expected profit
E{profit} = E{
Revenue from trading in the day-ahead market
+ Revenue from trading in the adjustment market
+ Imbalance income } =
= I
tωt
A
tω
A
tωt
D
tω
D
tω
N
1ω
N
1t
ω IdPλdPλπT
1/29/2010
38
Linearization of imbalance income
0Δ,Δrλ
0Δ,ΔrλI
tωtωtω
D
tω
tωtωtω
D
tωI
tω
Linearization
t
max
tω
ttωtω
tωtωtω
A
tω
D
tωtωttω
tωtωtωtω
D
tω
I
tω
dPΔ0
dPΔ0
ΔΔΔ
)PP(PdΔ
)ΔrΔ(rλI
Total deviation decomposed into positive
and negative deviations
1/29/2010
39
Non-anticipativity constraints
ω'ω,t,,PPD
ωt
D
tω
1/29/2010
40
Non-anticipativity constraints
ω'ω,t,,PPD
ωt
D
tω
Relaxed to obtain offer curves!
Power traded in day-ahead market can be
different for different price realizations!
curve offer an up making Pairs )λ,(PD
tω
D
tω
D
ωt
D
tω λλ:
1/29/2010
41
Non-anticipativity constraints
)N ,2, 1,t,P(P and
t)λ(λ:ω'ω,t,,PP
1Tωttω
D
ωt
D
tω
A
ωt
A
tω
Relaxed to model the certainty gain effect!
Power traded in the adjustment market can be different
depending on the wind realization between the closure
of the day-ahead and adjustment markets!
1/29/2010
42
Risk measure (CVaR)
ω0η
ω0ηξ
)]ΔrΔ(rλdPλdP[λ
ηπα1
1ξCVaR
ω
ω
tωtωtωtω
D
tωt
A
tω
A
tωt
D
tω
N
1t
D
tω
Ωω
ωω
T
1/29/2010
43
Risk hedging
• No risk hedging instruments!
• Risk hedging strategies are possible
1/29/2010
44
Other constraints
• Non-decreasing condition
)λ(1)(λ:ω'ω,t,0,PPD
ωt
D
tω
D
ωt
D
tω OO
• Bounds
ωt ,,PPP0
ωt ,,PP0
maxA
tω
D
tω
maxD
tω
1/29/2010
45
Numerical simulations
• Price data from the electricity market of the
Iberian Peninsula
• Wind speed data from a location in Kansas
(U.S.)
• 1 day
1/29/2010
Some results
46
0 2 4 6 8 10600
800
1000
1200
1400
1600
1800
2000
Risk aversion P
ow
er
tra
de
d in
th
e d
ay-a
he
ad
ma
rke
t (M
W)
0 1 2 3 4 50
500
1000
MW
h
0 1 2 3 4 50
500
1000
MW
h
0 1 2 3 4 50
500
1000
Risk aversion ()
MW
h
Expected total deviation
Expected positive deviation
Expected negative deviation
No adjustment market
Reducing the
energy traded in the
day-ahead market
in the hope of
selling in the
balancing market at
a competitive price
1/29/2010
}E{Δ
}E{Δ}Ε{Pβ
D
47
Some results
Single stacks of energy for = 0 …
Period t T8 T12 T13 T14 T15 T23
Bid (MW) 61.86 98 92.14 49.58 94.88 98
VSS = 2.3 %
1/29/2010
48
Some results
…turning into curves for 0
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
MWh
€/M
Wh
= 0.3
T8
T12
T13
T14
T15
T23
To reduce the
energy traded
in the day-
ahead market
in the hope of
selling in the
balancing
market at a
competitive
price
1/29/2010
0 5 10 15 20 250
5
10
15
20
25
Hour
MW
Reduction in expected total deviation per hour
= 0
Analysis of certainty gain effect
49
Adjustment market included !
0 1 2 3 4 5700
750
800
850
900
950
Risk aversion ()
MW
h
Expected total deviation
Without certainty gain
With certainty gain
A significant reduction for the first few hours1/29/2010
0 0.5 1 1.5 2 2.5 3200
400
600
800
Risk aversion ()
MW
h
Energy traded in the day-ahead market
0 0.5 1 1.5 2 2.5 3
500
1000
1500
2000
Energy traded in the adjustment market
MW
h
0 0.5 1 1.5 2 2.5 3
1000
1500
2000
MW
h
Resulting energy schedule
Without certainty gain
With certainty gain
Analysis of certainty gain effect
50
Transfer of trading
from the day-ahead
market to the
adjustment market
(better wind-behavior
knowledge)
1/29/2010
Analysis of certainty gain effect
51
The key figure: EFFICIENT FRONTIER
2 2.2 2.4 2.6 2.8
x 104
7.2
7.25
7.3
7.35
7.4x 10
4
= 0
= 0.1
= 0.2
= 0.3
= 0.4
= 0.5
CVaR (€)
Exp
ecte
d p
rofit (€
) = 0
= 0.1
= 0.2
= 0.3
= 0.4
= 0.5
Considering certainty gain
Not considering certainty gain
1/29/2010
Analysis of certainty gain effect
52
The key figure: EFFICIENT FRONTIER
1.8 2 2.2 2.4 2.6 2.8 3
x 104
6.7
6.8
6.9
7
7.1
7.2
7.3
7.4x 10
4
= 0 = 0.1
= 0.2 = 0.3
= 0.5
= 1
= 2
= 5
= 50
CVaR (€)
Exp
ecte
d P
rofit (€
)
= 0 = 0.1
= 0.2
= 0.3
= 0.5
= 1
= 2
= 5
= 10 = 50
Reflecting certainty gain
Without reflecting certainty gain
1/29/2010
2 2.2 2.4 2.6 2.8
x 104
7.2
7.25
7.3
7.35
7.4x 10
4
= 0
= 0.1
= 0.2
= 0.3
= 0.4
= 0.5
CVaR (€)
Exp
ecte
d p
rofit (€
)
Analysis of certainty gain effect
53
The key figure
DCVaR
DE
{pro
fit}
% 31.36100CVaR
ΔCVaR
% 1.66100profit}{E
profit}{ΔE
23.5ΔE{profit}
ΔCVaR
0)(β
0.5)0(β
0)(β
0.5)0(β
0.5)0(β
0.5)0(β
1/29/2010
Conclusions
54
Risk management is possible A high decrease in the risk of profit variability for a
comparatively low reduction in expected profit
Adjustment markets are important These markets allow designing offering strategies with
a reduced uncertainty level, which results in a higher
profit and a smaller risk. Policy implication!
1/29/2010
Future work
55
• Wind production (not speed) scenarios?
• Risk hedging instruments?
1/29/2010
References
56 29/01/2010
• G. N. Bathurst, J. Weatherill, and G. Strbac, “Trading Wind Generation inShort Term Energy Markets,” IEEE Trans. Power Syst., vol. 17, no. 3, pp. 782–789, August 2002.
• J. Matevosyan and L. Söder, “Minimization of Imbalance Cost Trading WindPower on the Short-Term Power Market,” IEEE Trans. Power Syst., vol. 21,no. 3, pp. 1396–1404, August 2006.
• J. M. Morales, A. J. Conejo, J. Pérez Ruiz “Short-Term Trading for a WindPower Producer”. IEEE Trans. Power Syst. Vol. 25, No. 1, pp. 554-564,February 2010.
• J. M. Morales, R. Mínguez, A. J. Conejo, “A Methodology to GenerateStatistically Dependent Wind Speed Scenarios”. Applied Energy. Vol. 87, No.3, pp. 843-855, March 2010.
57
Thanks for your attention!
GSEE: http://www.uclm.es/area/gsee/
1/29/2010
