(637f) Effective Variance Reduction and Gradient Estimation Techniques for Stochastic Simulation and Optimization of Microgrids

It is well known that renewable energy resources such as wind and solar have many economic and environmental benefits over conventional energy sources. However, efficiently and reliably generating power from renewable resources is extremely challenging due to their distributed and intermittent availability. Microgrids are small, autonomous power systems that address the geographical distribution of renewable power sources by intelligently pairing local loads with locally available resources. However, managing intermittency remains a key challenge, often necessitating complex systems that must coordinate multiple generation and storage technologies in real-time. In turn, this leads to challenging design and control problems that require efficient methods for both simulating and optimizing detailed microgrid models subject to the highly stochastic behavior of the renewable generators.

This presentation concerns the problem of jointly optimizing the generator/storage capacities and dispatching rules for microgrids under uncertainties in both renewable resources and loads. In existing literature, this problem has been addressed using: (1) mixed-integer programming formulations that must assume linear component models and deterministic load/resource profiles; (2) two-stage stochastic programming formulations that do not enforce non-anticipativity; (3) multistage stochastic programming formulations that are severely limited in the number of scenarios that can be addressed, even for linear models; and (4) trial-and-error or heuristic optimization approaches applied to black-box microgrid simulation codes. The later category is very attractive because it enables the use of much higher-fidelity models than the other approaches, particularly with respect to nonlinear component models and the essential discrete behavior introduced by dispatching rules. However, the heuristic methods currently used to optimize such models suffer from slow convergence and a lack of theoretical performance guarantees.

Our recent work has been addressing these shortcomings by developing gradient-based approaches for optimizing detailed microgrid simulation models that provide theoretical convergence guarantees and exhibit much faster convergence than black-box approaches in practice. As a first step, we have recently proven conditions under which a cost function computed through a stochastic microgrid simulation is differentiable, and hence amenable to efficient gradient-based optimization. This is nontrivial because of the discrete behavior introduced by component dispatching decisions in microgrid simulations, and interestingly our results show that only the true expected value of the cost is differentiable, whereas a finite sample-average is likely to be highly discontinuous. This result lays the theoretical groundwork for a gradient-based approach, but leaves a number of significant implementation challenges to be addressed.

This presentation will address the outstanding challenge of efficiently and accurately estimating the gradient of the expected cost from a finite number of stochastic simulations. As a starting point, we consider a straightforward finite difference (FD) approach based on direct Monte Carlo sampling, and demonstrate that this leads to very high variance estimates due to artificial â??rare eventsâ?? created by the differencing scheme. Compensating for this high variance requires very large numbers of samples, which significantly deteriorates the performance of our gradient-based optimization algorithm. To address this, we next present three novel sample path estimators with substantially reduced variance compared to FD. Respectively, these estimators are based on (1) a custom conditional expectation expansion for the FD estimator; (2) an automated change-of-variables technique combined with automatic differentiation (AD); and (3) a method for evaluating a derivative formula in terms of surface integrals using so-called `phantom simulationsâ??. We discuss conditions under which each estimator is preferable to the others, and present performances comparisons for some representative microgrid examples. Our results show much lower variance estimates relative to naïve FD, leading to very substantial speed-ups in the overall optimization scheme.