(715c) Comparison of Risk-Averse Stochastic Programming and Adaptive Robust Optimization: A Virtual Power Plant Scheduling Application

In this presentation, we compare two-stage risk-averse stochastic programming (SP) and two-stage adaptive robust optimization (RO) to address the self-scheduling and electricity market involvement of a virtual power plant (VPP). We investigate both approaches concerning formulations, uncertainty and risk, and decomposition algorithms and their computational performance. To quantify the risk in SP, we use the Conditional Value at Risk (CVaR) because it can resemble a worst-case measure, which naturally links to RO.

The VPP comprises a thermal plant, a pumped-storage hydro unit, and a wind farm. The objective of the VPP is to maximize the operating profit for a time horizon of one week. The self-scheduling of the VPP involves the commitment and dispatch of the thermal unit and the hydro plant. The determination of the commitment and dispatch of the thermal unit is subjected to minimum up and down times, limits to ramp up and down rates, startup costs depending on the previous state of the unit, and upper and lower limits on the output generation. Regarding the hydro plant, the commitment and dispatch include the determination of generation and consumption with its region of operation constrained by the power generation function, limits on the output generation, and limits on the volumes of water in the reservoir; see Lima et al. (2013) and Lima and Novais (2016) for further details on these constraints.

The electricity market involvement includes the selection of forward contracts to sell or buy electricity and the option to sell or buy electricity in an hourly pool.

The deterministic VPP problem is represented by a Mixed-Integer Linear Programming model. However, the decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast, leading to an MILP stochastic problem with mixed-integer first-stage variables.

The electricity prices forecasts are sampled from an Auto-Regressive Integrated Moving Average model, and the wind speed forecasts are based on a 51-member wind ensemble obtained from weather forecast models from the European Centre for Medium-Range Weather Forecasts. Using the ensemble, we construct a Karhunen-Loève expansion for sampling the wind speed.

We use two efficient implementations of the decomposition algorithms for SP and RO (Lima et al. 2015, Lima et al. 2017) and we assess 1) their performance taking into consideration different sample sizes and risk management parameters; and 2) the operational results regarding first-stage decision variables. Furthermore, we use a Monte Carlo Sample Average Approximation methodology (Lima et al. 2018) to evaluate inference statistics on the expected profit and CVaR of the profit to compare both approaches.

The results show that the SP approach is computationally competitive with the RO approach in specific circumstances, and that there is a set of instances that poses convergence difficulties to both approaches. In these problematic instances, the RO decomposition algorithm cannot meet the stop criteria, whereas the SP decomposition method meets the stop criteria for a large percentage of the samples tested but at a high computational cost. A relevant insight is also discussed regarding the risk management: similar first-stage solutions are obtained with SP and RO depending on the risk parameterizations used in each formulation.

References

Lima, R. M., Marcovecchio, M. G., Novais, A. Q., Grossmann, I. E., 2013. On the Computational Studies of Deterministic Global Optimization of Head Dependent Short-Term Hydro Scheduling. IEEE Transactions on Power Systems 28 (4), 4336–4347

Lima, R. M., Novais, A. Q., Conejo, A. J., 2015. Weekly self-scheduling, forward contracting, and pool involvement for an electricity producer. An adaptive robust optimization approach. European Journal of Operational Research 240 (2), 457–475.

Lima, R. M., Novais, A. Q., 2016. Symmetry breaking in MILP formulations for Unit Commitment problems. Computers and Chemical Engineering 85, 162–176.

Lima, R. M., Conejo, A., Langodan, S., Hoteit, I., Knio, O., 2017. Risk-averse formulations and methods for a virtual power plant. Computers & Operations Research (Accepted).

Lima, R. M., Conejo, A. J., Giraldi, L., Le Maître, O., Hoteit, I., Knio, O., 2018. Sample average approximation for risk-averse problems: A virtual power plant scheduling application. (To be submitted).