This work presents an approach to solve stochastic optimal control problems in the application of flow quality management in reservoir systems. These applications are challenging because they require real-time decision-making in the presence of uncertainties such as wind velocity. These uncertainties must be accounted for as stochastic variables in the mathematical model. In addition, computational costs and storage requirements increase rapidly due to the stochastic nature of the simulations and optimisation formulations. To overcome these challenges, an approach is developed that uses the combination of a reduced-order model and an adjoint-based method to compute the optimal solution rapidly. The system is modelled by a system of stochastic partial differential equations. The finite element method together with collocation in the stochastic space provide an approximate numerical solution—the “full model”, which cannot be solved in real-time. The proper orthogonal decomposition and Galerkin projection technique are applied to obtain a reduced-order model that approximates the full model. The conjugate-gradient method with Armijo line-search is then employed to find the solution of the optimal control problem under the uncertainty of input parameters. Numerical results show that the stochastic control yields solutions that are above the bound of the set solutions of the deterministic control. Applying the reduced model to the stochastic optimal control problem yields a speed-up in computational time by a factor of about 80 with acceptable accuracy in comparison with the full model. Application of the optimal control strategy shows the potential effectiveness of this computational modeling approach for managing flow quality.
A Reduced-order Model for Real-time NMPC of Ethanol Steam Reformers
