Risk-Limiting Dispatch
for Power Networks
David Tse, Berkeley
Ram Rajagopal
(Stanford)
Baosen Zhang
(Berkeley)
Motivation
• Traditional power generators slow to ramp up and down.
• Have to be dispatched in advance based on predicted
demand.
• Increased penetration of renewables comes increased
uncertainty.
Questions:
• How to do dispatch in face of uncertainty?
• How to quantify the impact of uncertainty?
1/15
Motivation
• Currently: 3𝜎 rule
• Error~2%
$1 Billion
$300 Million
𝜎
• Add 25% wind, 20% error
• Total Error~2+5=7%
Reserve
Reserve
Error
Error
Forecasted
load
Forecasted
net demand
1% is about $50 Million/yr (for CAISO)
2/15
Notation
• Three types of devices in the power system:
Renewables:
Random,
High Uncertainty
Generators:
Controllable
Loads:
Random,
Low Uncertainty
𝑑=net demand=Load-Renewable
Prediction
Error Gaussian in this talk
3/15
Two-Stage Formulation
• Two-stage problem
Actual net-demand: 𝑑 = 𝑑 + 𝑒
Predicted net-demand: 𝑑
Stage 2 (real-time)
Stage 1 (day ahead)
Set slow generators: 𝑔
Price 𝛼 ($/MW)
Set fast generators
<
Price 𝛽 ($/MW)
• Dynamic programming problem: numerical solution
possible but offers little qualitative insight.
• Make small ¾ assumption.
4/15
Nominal Problem
𝑑
𝑑
𝛼
Stage 1
𝑔
𝛽
𝛼
𝑑
Nominal Problem
𝑔: nominal generation
nominal congestion
Stage 1
Stage 2
optimal under
small ¾ assumption
Stage 2
5/15
Impact of uncertainty
• We want to find (as a function of 𝑑)
– Optimal cost
– Optimal control
• Also want
Cost of uncertainty= Optimal Cost − Clairvoyant Cost
• Intrinsic impact of uncertainty
– Depend on 𝛼, 𝛽
6/15
Nominally Uncongested Network
• Networks are lightly congested
Nominally
Uncongested
New England ISO
Single Bus Network
Result:
Price of uncertainty
7/15
Single-bus network
• No congestion => single bus network
• Easy to get the optimal control
Reserve/σ
3.5
3σ
3
Q-1 (/)
2.5
~$100 Million/yr
2
1.5
optimal
1
0.5
0
-0.5
0
5
10
15
/
20
25
30
8/15
Price of Uncertainty
• Price of uncertainty is a function of 𝑑
• Small Error
renewable>load
renewable<load
0
9/15
Nominally Congested Network
• One nominally congested line
?
Midwest ISO
10/15
Dimensionality Reduction
• One congested line
• Single bus?
KVL
x
Result:
Reduction to an equivalent twobus network always possible.
x
IEEE 13 Bus Network
11/15
Two-bus network: Further reduction?
• Nominally congested line from 1 to 2
2
x
?
1
2
Two isolated
buses?
1
• Congestion is nominal
2
Nominal
Supply > expected
Real-time
x
1
x
Back-flow
Supply < expected
• Errors still average
12/15
Nominal solution regions
𝑐
x
𝐶
𝐶
−𝐶
−𝐶
13/15
Prices of uncertainty
x
𝐶
𝑐
𝐶
−𝐶
−𝐶
14/15
Conclusion
• Management of risk in the presence of renewables
• Price of uncertainty
– Intrinsic impact of uncertainties
• Dimension reduction for congested networks
15/15