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

Nominally Congested Network

• One nominally congested line

11/15

Midwest ISO

?

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

Nominal solution regions

14/15

𝑐

𝐶

−𝐶

−𝐶

𝐶

x

Prices of uncertainty

15/15

𝑐

𝐶

−𝐶

−𝐶

𝐶

x

Conclusion

• Management of risk in the presence of renewables• Price of uncertainty

– Intrinsic impact of uncertainties

• Dimension reduction for congested networks

16/15