An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

Abstract

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.

Cost‐efficient monitoring of continuous‐time stochastic processes based on discrete observations

“…A finite difference scheme for numerical discretization of the problem (30)‐(34) is presented. Our scheme is based on the previously established monotone and stable numerical schemes, relying on exact solutions to local boundary value problems 7,42,51 . Similar numerical schemes can also be found in and Mickens et al 59 and Gyulov et al 60 Our numerical contribution is to demonstrate that the scheme is stable, monotone, and can potentially approximate the viscosity solution to the problem (30)‐(34) if it exists.…”

“…This kind of estimates follows under the standard assumptions 49 . In applications, there exist many almost surely bounded processes 35,50,51 . For such cases, we can assume a sharper bound under an appropriate normalization of the variables: …”

“…Appropriate boundary conditions have to be specified when necessary in a standard setting, but we assume that the PDE (30) is satisfied over Ω for each t > 0 as in some degenerate parabolic PDEs 55 . This assumption is satisfied in our applications where the target SDEs have degenerate diffusion and/or drift coefficients 7,42,51 such that the PDE (30) is naturally satisfied over Ω for each t > 0. This is also true in some other problems in biology and ecology 50,56 .…”

“…The final model application is monitoring cloud cover dynamics in the sky as an index for evaluating power generation by solar panels 51 . The cloud cover X t at each time t is a nondimensional quantity valued in the unit interval [0, 1], where X t = 0 corresponds to the fully clear sky and X t = 1 to the fully clouded sky.…”

“…Yoshioka et al 51 identified the cloud cover dynamics in a model site in Matsue City, Japan as with the reverting rate A = 0.59 (1/day), the mean value M = 0.75, and the volatility σ = 0.53 (1/day 1/2 ). The stationary probability density function with the identified model parameters is weakly bimodal having a larger and smaller maximum values around X t = 0.10 and X t = 0.95, respectively.…”