Optimal consumption analysis for a stochastic growth model with technological shocks

Appl Stoch Models Bus & Ind

Yano

2019

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.

Section: Mathematical Modelmentioning

confidence: 99%

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

Appl Stoch Models Bus & Ind

Yano

2019

“…Proposition 2.3 is a consequence of Propositions 2.1 and 2.2. Uniqueness of viscosity solutions to HJB equations is not a trivial issue even for seemingly simple problems [6,67,69].…”

Section: Proposition 23: the Value Functionmentioning

confidence: 99%

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Journal of Biological Dynamics

2019

A simplified stochastic control model for optimization of logistic dynamics with the control-dependent carrying capacity, which is motivated by a recent algae population management problem in the river environment, is presented. Solving the optimization problem reduces to finding a solution to a non-local first-order differential equation called tt-Jacobi-Bellman (HJB) equation. It is shown that the HJB equation has a unique viscosity solution and that the solution can be approximated with a finite difference scheme. Asymptotic estimates of the solution and the optimal control are derived and compared with numerical solutions. Finally, parameter dependence of the optimal control is examined numerically with implications to river environmental management.

“…In general, solutions to HJBI and related equations are not sufficiently smooth and should be dealt with in the framework of weak solutions called viscosity solutions . This concept has successfully been applied to analyzing existence, uniqueness, regularity, parameter dependence, and approximation of solutions to HJB and HJBI equations . Application of the stochastic control, especially HJBI equations and viscosity solutions, is a new challenge for management of river environment.…”

Section: Introductionmentioning

confidence: 99%

Modeling stochastic operation of reservoir under ambiguity with an emphasis on river management

Optim Control Appl Methods

2019

An optimization problem of controlling a dam installed in a river is analyzed based on a stochastic control formalism of a diffusion process under model ambiguity: a new mathematical approach to this issue. The diffusion process is a pathwise unique solution to a water balance equation considering the inflow, outflow, water loss in the reservoir, and direct rainfall. Finding the optimal reservoir operation policy reduces to solving a degenerate parabolic partial differential equation: a Hamilton-Jacobi-Bellman-Isaacs equation. A monotone finite difference scheme is constructed for discretization of the equation, successfully generating nonoscillatory and reasonably accurate numerical solutions. Stability analysis of the resulting water balance dynamics is finally carried out for both environmentally friendly and not friendly reservoir operations. KEYWORDS finite difference scheme, Hamilton-Jacobi-Bellman-Isaacs equation, reservoir operation, stochastic control, viscosity solution INTRODUCTIONThis paper contributes to providing a mathematical framework for modeling and control of reservoirs created in rivers.Since the problem has an engineering background, it is first presented, and then, the issues to be addressed and our contributions are explained.Many rivers have dams serving as central infrastructures to supply essential water resources for human lives. 1-3 The stored water in the reservoir created by a dam is utilized for multiple purposes, such as drinking water, irrigation, and hydropower generation. 4 Several dams are used for flood mitigation as well. 5,6 On the other hand, controlling a dam critically affects its downstream river environment because the original flow regime is altered. 7-9 Sediment transport and attached algae population dynamics in dam downstream experience significant changes because dams trap sediment and the algae growth is highly flow-dependent. 10 Dam operation is thus desired to be environmentally friendly such that its impacts on the downstream are minimized, while it should be fit-for-purpose at the same time.Management of environment and ecology in rivers has been a long-standing engineering problem, 11-13 and mathematical tools and concepts for their modeling, analysis, and control are still developing. Several researchers considered river management problems based on stochastic process models such as stochastic differential equations (SDEs) 14 that can naturally consider stochasticity involved in environmental and ecological processes. 15 Interactions between aquatic species, such as riparian vegetation and fishes, and river discharges have been described with SDEs. 16 Miyamoto and Kimura 17 analyzed seedling, growth, and mortality dynamics of riparian trees subject to stochastic flood disturbances. Vesipa et al 18 764