Two-stage stochastic programming and SAA

Importance Sampling with Analog Scenarios for Stochastic Economic 

Dispatch

Ignas Satkauskas; Matthew Reynolds; Devon Sigler; Wes Jones

National Renewable Energy Laboratory

Increasing penetrations of renewable energy sources such as wind 

into power grids motivates the investigation of new approaches to 

computing 5-minute economic dispatch.

We investigate extending previous work on two-stage stochastic 

economic dispatch with importance sampling [1] to use historical 

data as a source of uncertainty scenarios. 

Introduction

Objective

Methods

In our experiments, using importance sampling to select scenarios 

for the SAA leads to significant second-stage savings, 

corresponding to less loss-of-load.

First stage costs from the standard Monte Carlo sampling and 

importance sampling experiments are not significantly different.

In cases where there is a significant amount of wind-power 

decrease (see Fig. 4), as occurs at timestamp (b), importance 

sampling will not prevent loss-of-load.

However, the loss-of-load will be less severe than two-stage 

stochastic programming using standard MC sampling.

Discussion

Importance sampling for stochastic economic dispatch using 

analog scenarios shows promise; our experiments demonstrate 

improved performance for 5-minute dispatch decisions in terms of 

cost over standard SAA approaches. 

Future work will investigate higher-fidelity approaches to 

approximating average costs of bins.

Future work will also investigate different metrics for data binning. 

For example, binning by costs output from deterministic economic 

dispatch experiments would be ideal, but costly.

Conclusions & Future Work

To investigate the benefits of using importance sampling with 

analog scenarios in two-stage stochastic economic dispatch. 

References

1. King, R., Reynolds, M., Sigler, D., Jones, W. “Advanced Scenario Creation Strategies for Stochastic 

Economic Dispatch with Renewables”, https://arxiv.org/abs/1806.10530 (submitted for publication).

2. C. Draxl, A. Clifton, B.-M. Hodge, and J. McCaa, “The wind integration national dataset (WIND) toolkit,” 

Applied Energy, vol. 151, 2015. 

3. Grid Modernization Lab Consortium, “Reliability test system – Grid Modernization Lab Consortium,” 

2017, [Online; accessed 11-December-2017]. [Online]. Available: https://github.com/GridMod/RTSGMLC

Experiment 2: repeated stochastic economic dispatch computations at timestamps (a) and (b)

Sampling with multiple wind sites:

1. Bin wind scenario deviations from persistence using percentiles defined by ∑ii ξξii.

2. Compute average deterministic costs jj̃cc (sample with replacement) for each bin.

3. Set probability of selecting bin jj by jj̃cc / ∑jj′ ̃cc

jj′ .

4. Select bin according to probability, draw scenario uniformly from bin, compute weight. 

Table 1: Comparing sums of first and second stage costs from stochastic 

economic dispatch experiments in Fig. 2. These experiments used standard and 

importance sampling for different sampling rates.

This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and 

the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

Linear two-stage stochastic program:

where,

x – first stage variables (generator setpoints)

y – second stage variables (e.g. wind dispatched, slack)

ξξ – uncertain variable (wind deviation from persistence)

Second stage constraints include power flow constraints, power 

balance at nodes, constraints on 2nd stage variables.

Sample average approximation

Importance sampling

Methods cont’d

Experiments and Results

Experiment 1:two-stage stochastic economic dispatch over varying dates and times

Figure 2: Importance sampling (orange) and 

standard Monte Carlo (blue) 2nd stage costs 

for datetimes in our test suite. (Top) 10 

scenario, (bottom) 20 scenario experiments.

(a) (b)

Figure 2: Comparing importance sampling (orange) and standard Monte Carlo 

(blue) 2nd stage costs. 

10 scenario experiments 20 scenario experiments

(b)

(a) (a)

Sampling method # of scenarios First stage costs ($) Second stage costs ($)

MC 10 620418 58556

MC 20 620937 34512

IS 10 621374 31700

IS 20 622311 18317

Figure 1: RTS-GMLC system with 120 lines, 73 buses, and 118 

generators, 22 of which are wind plants. 

Figure 3: 2nd stage costs for std. MC sampling (blue) and IS (orange) on timestamps (a) and (b) defined in Figure 2.

Sampling 

method

Mean Std.

10 MC 4639 2275

20 MC 3289 2072

10 IS 2746 1851

20 IS 1648 1519

Sampling 

method

Mean Std.

10 MC 1033 1278

20 MC 512 813

10 IS 168 452

20 IS 17 97

(b)

Table 2: Statistics on 200 

economic dispatch 

experiments, 2nd stage costs 

for (b).

Table 3: Statistics on 200 

experiments, economic 

dispatch 2nd stage costs for 

(a).

Figure 4: Deviation from persistence forecasts of wind power (in MW) for 

timestamps (a) red and (b) blue at 22 wind plants on RTS. Total 

deviations from persistence: (b) -41.5 MW (a) -13.9 MW.

Six years of WIND Toolkit [2] data are used as synthetic historical 

wind data, and 1 year of WIND Toolkit data is used to simulate 

actuals on the RTS-GMLC network [3].

Network constraints on the RTS-GMLC are approximated by a 

transport network.

The investigation is conducted by comparing the standard Monte 

Carlo approach to the sample average approximation (SAA) of 

two-stage stochastic programs with our importance sampling 

approach.

Continuous importance sampling

Since we use a dataset to build our scenarios, we employ a 

discrete version of importance sampling over bins of analog 

scenarios.

Acknowledgement:

Funding for this project provided by the Exascale Computing Project , 

ExaSGD project.

Energy Systems and Optimization Workshop 2018

Georgia Tech, Atlanta, GA

November 15-16, 2018

NREL/PO-2C00-72784