Competition and Cooperation Among
Wind Farms
Baosen Zhang
Stanford
CMU Electricity Conference
Ramesh Johari
Ram Rajagopal
Wind Power
•
•
•
•
Significant increase in wind power
30% renewable by 2030 in CA, 50% by 2050
Taken as negative load
Must participate in the market if subsides end
Uncertainty limits wind participation
1/8
Overcoming Randomness
• Spatial diversity: coalitions
100
550
90
500
80
450
70
400
60
350
50
300
40
250
30
200
20
150
10
100
0
0
10
20
30
40
50
60
70
80
50
0
10
20
30
40
50
60
70
80
5 Wind Farm
1 Wind Farm
4
1100
4.5
x 10
4
1000
3.5
900
3
800
2.5
700
2
1.5
600
1
500
400
0
0.5
10
20
30
40
50
60
10 Wind Farm
70
80
0
0
10
20
30
40
50
60
70
80
50 Wind Farm
2/8
Overcoming Randomness
• Spatial diversity: coalitions
• What about market power?
CA Energy Crisis
2/8
Overcoming Randomness
• Spatial diversity: coalitions
• What about market power?
• Gil & Lin,PJM, Trans. Power Systems, 2013
– 10% of wind can reduce the day-ahead price by half
2/8
Overcoming Randomness
• Spatial diversity: coalitions
• What about market power?
Contribution:
• Tradeoff between uncertainty and market power
• Optimal size of coalitions
2/8
Market Setup
• Two-stage market
Day-Ahead
Real-Time
• N wind farms with output
𝑊1 , … , 𝑊𝑁
• Day-ahead profit for farm i
Revenue
Penalty
𝜋𝑖 𝑊𝑖 = 𝑝 𝑊1 , … , 𝑊𝑁 𝑊𝑖 − 𝑞𝐸[ 𝑊𝑖 − 𝑊𝑖
Bid
Day-ahead Price
+
]
Real-time Shortfall
3/8
Market Setup
• Two-stage market
Day-Ahead
Real-Time
• N wind farms with output
𝑊1 , … , 𝑊𝑁
• Day-ahead profit for farm i
𝜋𝑖 𝑊𝑖 = 𝑝 𝑊1 , … , 𝑊𝑁 𝑊𝑖 − 𝑞𝐸[ 𝑊𝑖 − 𝑊𝑖
Bid
Day-ahead Price
+
]
Real-time Shortfall
3/8
Aggregate Price Curve
• Two kinds of generators: Traditional and Wind
• Wind farms bid quantity
• Aggregate price curve for traditional generators
Fraction of Total Demand
20% Wind
PJM 2008 Data
Price ($/MW)
4/8
Linear Approximation
• Price as function of wind
𝑁
𝑝 𝑊1 , … , 𝑊𝑁 = 1 − 𝛼
𝑊𝑖
𝑖=1
Fraction of Total Demand
𝛼 = 3.2
0
0.17
0.34
0.51
0.68
0.83
1
Normalized Price ($/MW)
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Independent Wind Farms
• Wind farms are i.i.d.
• Large N regime, 𝛼 = 3.2
• Grand coalition
max 1 − 𝛼 𝑊 𝑊 − 𝑞𝐸[(∑ 𝑊𝑖 − 𝑊)+ ]
• Optimal solution:
𝑊∗
≤
1
2𝛼
= 15%
• Withholding because of market power
6/8
Independent Wind Farms
• Wind farms are i.i.d.
• Large N regime, 𝛼 = 3.2
• Grand coalition≤
1
2𝛼
= 15%
• Individual competition
max 1 − 𝛼∑𝑊𝑖 𝑊𝑖 − 𝑞𝐸[( 𝑊𝑖 − 𝑊𝑖 )+ ]
• Optimal solution: total bid ≤ 15%
• Withholding because of uncertainty
6/8
Independent Wind Farms
• Wind farms are i.i.d.
• Large N regime, 𝛼 = 3.2
• Grand coalition ≤
1
2𝛼
= 15%
• Individual competition ≤
1
2𝛼
= 15%
• Intermediate size
Group size of 𝑁 achieves a penetration level of
1
𝛼
= 30%
• Socially optimal
6/8
Correlated Wind Farms
• Identical but not independent
• Can still achieve
1
𝛼
• Solve an optimization to find the optimal group size
NREL Data
Optimal Size of 30
7/8
Conclusion
• Tradeoff between market power and uncertainty
• Optimal size of coalitions
• Future directions: more detailed models
8/8