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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?
• How to hedge against risks from randomness?
2/15
• Add 25% wind, 20% error• Total Error~2+5=7%
Motivation
• Currently: 3 rule• Error~2%
3/15
Forecasted load
ErrorReserve
𝜎
Forecasted net demand
Error
Reserve
1% is about $50 Million/yr (for CAISO)
$1 Billion$300 Million
Notation
• Three types of devices in the power system:
4/15
=net demand=Load-Renewable
Generators:
Controllable
Renewables:
Random,
High Uncertainty
Loads:
Random,
Low Uncertainty
Prediction Error Gaussian in this talk
Two-Stage Formulation
• Two-stage problem
• Dynamic programming problem: numerical solution possible but offers little qualitative insight.
• Make small ¾ assumption.
5/15
Stage 1 (day ahead)
Predicted net-demand:
Set slow generators:
Stage 2 (real-time)
Actual net-demand:
Set fast generators
Price ($/MW) Price ($/MW)¿
Nominal Problem
6/15
Stage 1 Stage 2
�̂� 𝑑𝑔𝛼 𝛽
Nominal Problem
�̂�𝛼
Stage 1 Stage 2
optimal under
small ¾ assumption
Impact of uncertainty
• We want to find (as a function of )– Optimal cost– Optimal control
• Also want
• Intrinsic impact of uncertainty– Depend on
7/15
Cost of uncertainty= Optimal Cost Clairvoyant Cost
Nominally Uncongested Network
• Networks are lightly congested
Result:
8/15
New England ISO
Nominally
Uncongested
Single Bus Network
Price of uncertainty
Single-bus network
• No congestion => single bus network• Easy to get the optimal control
9/15
0 5 10 15 20 25 30-0.5
0
0.5
1
1.5
2
2.5
3
3.5
/
Q-1
(
/) ~$100 Million/yr
3
optimal
Res
erve
/
Price of Uncertainty
• Price of uncertainty is a function of • Small Error
10/15
0
renewable>load renewable<load
Dimensionality Reduction
• One congested line• Single bus?
Result:
Reduction to an equivalent two-bus network always possible.
12/15IEEE 13 Bus Network
KVL
x
x
Two-bus network: Further reduction?
• Nominally congested line from 1 to 2
• Congestion is nominal
• Errors still average
13/15
?
1
2
x1
2Two isolated
buses?
x1
2 Supply > expected
Supply < expected
Real-timeNominal x Back-flow