Singular stochastic control model for algae growth management in dam downstream

Yaegashi

Appl Stoch Models Bus & Ind

et al. 2019

Self Cite

A new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.

Section: Mathematical Modelmentioning

confidence: 99%

“…The management cost would stem from the extermination of predators such as the waterfowls 42,44,45,51 and cleaning up of the river environment. 52,53 A recent model by Pinheiro 54 focuses on a similar harvesting problem in a finite horizon, but subject to a different SDE and a performance index.…”

Section: Performance Index and Value Functionmentioning

confidence: 99%

Optimal harvesting policy of an inland fishery resource under incomplete information

Yaegashi

Appl Stoch Models Bus & Ind

et al. 2019

Self Cite

“…The target discharge

{\overset{true‾}{q}}_{s}

is prescribed considering the operational purpose of the dam. For example, some dams conventionally employ the rule

{\overset{true‾}{q}}_{s} = Q_{s}

. The parameter l measures the sensitivity of the performance index on the deviation

(||, q_{s} - {\overset{true‾}{q}}_{s})

.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The parameter l measures the sensitivity of the performance index on the deviation

(||, q_{s} - {\overset{true‾}{q}}_{s})

. For operating the dam considering its downstream environment and ecosystems, the outflow discharge is often required to be larger than some threshold below which the growth rate of the harmful benthic algae becomes positive . An example of the term for the environmentally friendly reservoir operation can be constructed using the quantity

\max ({}, 0, \underline{q} - q_{s})

with

\underline{q} > 0

the threshold value.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…In the research areas of environment and ecology, HJB equations are modern mathematical tools for effective system management . On river management, very recently, some progresses based on stochastic control have been made for cost‐effective management of benthic algae population in dam downstream with mathematical and numerical approaches . Their approaches can be a foundation of modeling, analysis, and control of river environment and ecology.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Optim Control Appl Methods

2019

Self Cite

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

Stochastic Control of Dam Discharges

Wiley StatsRef: Statistics Reference Online

2021

Dam is a dynamical system receiving stochastic inflows while releasing controlled outflows. Its optimal control requires considering a state constraint on the stored water volume and should use an objective function considering both human activities and downstream environmental management. In addition, the system dynamics are not always known exactly. We briefly explain a robust modeling framework of dam discharges based on a dynamic programming approach and constrained viscosity solutions with a demonstrative computational example. We focus on a single problem rather than reviewing diverse examples because it has key features that stochastic control in real applications encounters.