Optimal control of hazardous waste disposals in rivers and estuaries using the adjoint equation solution.

Abstract

Along with industrialization and technological development, public concern for environmental protection has risen and the necessity for guidelines, laws and control mechanisms has become apparent. We study two tasks that arise from the obligation to protect the environment. First, it is necessary to determine the optimum location of a discharge point in a river, so the allowable maximum concentration of some toxic solute is not exceeded. Second, we want to compute an optimum strategy in real time, when a deliberate release seems unavoidable. Finally, neither the design nor the control problem can be accurately resolved without an efficient method for field parameter estimation. In particular, the questions of optimal dimension of the parameter space and uncertainty of parameter values are of great importance in the development of a robust control method. The present control procedure consists of a finite element model for the hydrodynamic processes encountered in vertically mixed rivers and estuaries. On the same computational grid, the adjoint equation of the mass transport problem is solved in order to provide all necessary gradient information needed by the optimization algorithm. The latter is comprised by a conjugate gradient and a variable metric procedure used alternately to find the minimum of a specific performance measure. This is based on a least-squares approach incorporating both actual and desired state variable values corresponding to the concentration of an arbitrary conservative solute. The results obtained show that the adjoint solution of the transport equation can provide very efficiently and accurately the gradient information needed for optimal control. In the case of the optimum location of a discharge node, great savings of computer execution time can be achieved, since a single solution of the adjoint problem produces the same result with as many repetitions of the direct problem as the number of computational nodes in the system. The results are confirmed for steady and unsteady state solutions, in both regular and arbitrary flow domains. In the case of the optimal release history needed to control the concentration at some point in the flow domain, the increase in efficiency is equal to the number of time steps in the control mechanism. The algorithm produces excellent results for steady and unsteady flow regimes, including flood and tidal waves. Finally, the adjoint equation theory is used to compute sensitivities for parameter uncertainty and estimation. It is shown that the alternative formulation of the objective function, i.e., incorporating the adjoint variable, leads to satisfactory results in parameter estimation, provided that the physics of the problem is carefully accounted for, especially in the case of spatially distributed and nonisotropic field properties.