(544c) On the Optimal Sizing and Dispatch Problem for Microgrids with Stochastic Generators – When Is the Expected-Cost Differentiable?

This presentation concerns the problem of jointly optimizing the component sizes and dispatching rules for microgrids incorporating intermittent renewable resources such and wind and solar. The stochasticity introduced by these renewable resources, and to a lesser extent by the loads, plays a central role in this problem.  First, it is well known that sizing components (e.g., generators and storage units) based on average resource availabilities can lead to designs that are highly unreliable or suboptimal in reality. Moreover, dispatching decisions must be made under large uncertainties about the future demand and resource availabilities, despite the fact that the relative costs and benefits of the competing options may change radically across alternative futures.

Thus, the optimization problem at hand involves complex trade-offs between cost and reliability arising from the interplay of both design and operational decisions over time.  To capture these trade-offs, a number of detailed simulation models have recently been developed.  A given design is typically evaluated through an hourly time-series simulation over the course of a year. Hourly load, irradiance, and wind speed data are supplied by the user or stochastically generated from monthly averages. In each hour, these data are used to compute the net available power and dispatch decisions are made, typically based on a fixed set of logical rules parameterized by user-defined thresholds. Finally, model equations are solved for each component to update the system state and evaluate performance over the hour. The high temporal resolution of these simulations makes it possible to incorporate accurate performance and operational cost models for the system components. Moreover, these simulations can be driven with randomized load and generation profiles to accurately evaluate the expected cost of a system, including both stochastic operational costs and reliability metrics.

In light of the preceding discussion, the optimal microgrid sizing and dispatch problem can be formulated as a nonlinear program in which the objective is defined as the expected-value of randomized time-series simulations. In fact, this simulation-optimization formulation has been used extensively in conjunction with heuristic optimization algorithms that view the stochastic simulation as a black-box. However, in the interest of enabling more efficient optimization approaches, we consider here the important question of whether or not the expected-cost is a continuously differentiable function of the decision variables (i.e., the component sizes and parameters influencing the dispatching rules). We begin by formulating time-series microgrid simulations generically as discrete-time stochastic hybrid systems (DTSHS) with associated stage and terminal costs. Alarmingly, for any fixed realization of uncertainty, we find that discrete dispatching decisions can introduce discontinuities in the system cost at every time-step along the simulation horizon. This observation carries over to finite sample averages of the cost, so that formally optimizing such an average would require the introduction of very many binary variables (there is a relation to the conventional multistage stochastic programming formulation of the problem here). Remarkably, our findings show that the true expected-value of the cost function is nonetheless continuously differentiable under very general conditions, and hence is amenable to gradient-based optimization strategies. We provide sufficient conditions for continuous differentiability and demonstrate their verification for some representative microgrid models. We also highlight particular model features and dispatching strategies that may lead to violations. Finally, we present improved optimization results using a stochastic gradient-descent algorithm.