Robust stochastic control modeling of dam discharge to suppress overgrowth of downstream harmful algae

Abstract

The mathematical concept of multiplier robust control is applied to a dam operation problem, which is an urgent issue on river water environment, as a new industrial application of stochastic optimal control. The goal of the problem is to find a fit-for-purpose and environmentally sound operation policy of the flow discharge from a dam so that overgrowth of the harmful algae Cladophora glomerata Kützing in its downstream river is effectively suppressed. A minimal stochastic differential equation for the algae growth dynamics with uncertain growth rate is first presented. The performance index to be maximized by the operator of the dam while minimized by nature is formulated within the framework of differential games. The dynamic programming principle leads to a Hamilton-Jacobi-Bellman-Isaacs equation whose solution determines the worst-case optimal operation policy of the dam, ie, the policy that the operator wants to find. Application of the model to overgrowth suppression of Cladophora glomerata Kützing just downstream of a dam in a Japanese river is then carried out. Values of the model parameters are identified with which the model successfully reproduces the observed population dynamics. A series of numerical experiments are performed to find the most effective operation policy of the dam based on a relaxation of the current policy.

Optimal harvesting policy of an inland fishery resource under incomplete information

“…The model parameters for the performance index J are not identified here, but their impact on the resulting optimal harvesting policy is investigated numerically. Our field surveys and numerical computation of the algae dynamics for C glomerata imply that the harmful algae are almost detached from the riverbed when the river discharge from the Obara Dam exceeds 15 m 3 /s . The statistical analysis revealed that such events reasonably follow the Poisson process with the intensity of 1/10.4 (1/day) .…”

“…In addition, our survey results and observations by union members of the HRFC suggest that the harmful algae dominate rock‐attached diatoms, which are the main food of P altivelis . The survey results at just downstream of the Obara Dam indicated that C glomerata is removed from the riverbed when the discharge exceeds 15‐20 m 3 /s in the Hii River . As indicated in the next subsection, such events can be modeled within the context of the regime‐switching mechanism.…”

“…Harvesting cost is not considered here assuming that the harvesting is carried out with relatively cheap fishing gears, which is satisfied at least in the model application demonstrated later. The management cost would stem from the extermination of predators such as the waterfowls and cleaning up of the river environment . A recent model by Pinheiro focuses on a similar harvesting problem in a finite horizon, but subject to a different SDE and a performance index.…”

“…The dam regulates the discharge of the mainstream of the river, and available public data from 2016 suggest that the base discharge from the dam is about 1 (m 3 /s) . This value is too low to suppress the overgrowth of the harmful algae Cladophora glomerata Kützing . The experimental study by Uchisda et al showed that the algae cannot be digested by P altivelis .…”

“…The statistical analysis revealed that such events reasonably follow the Poisson process with the intensity of 1/10.4 (1/day) . In addition, the algae grow sufficiently within a few days since their intrinsic growth rate has been identified as O (10 −1 ) (1/day) . The diatoms grow when the harmful algae are sparse or absent.…”