A simultaneous dynamic optimization approach for addressing the control problem of algae growth in water reservoirs through biogeochemical models

V. ESTRADA; E. PARODI; M.S. DIAZ

Massachusetts, USA

Conferencia; Foundations of Computer-Aided Process Operations 2008; 2008

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Biogeochemical processes that take place in water bodies can be represented through a set of complex nonlinear partial differential algebraic equations resulting from dynamic mass balances for phytoplankton (algae) and nutrients. Once calibrated, the model can be integrated into a dynamic optimization framework for the determination of optimal control policies to prevent eutrophication of the water body. The process of eutrophication is accelerated by increasing concentrations of nutrients, with a strong development of the phytoplankton community and a decrease in the water depth caused by sediment accumulation. In this work, we develop a dynamic optimization model for a water body to determine optimal limiting nutrient inflow profiles to the lake. This is required because the lake tributaries bring nutrients into the water body and part of their flows has to be derived to a nearby wetland for nutrients elimination. The model is based on horizontally averaged concentrations and its main parameters have been previously estimated (Estrada et al., 2007) based on collected data sets corresponding to an entire year (Parodi et al., 2004). Dynamic mass balances in phytoplankton (in the form of diatoms, green algae and cyanobacteria), dissolved oxygen and nutrients, which include nitrate, ammonium, organic nitrogen, phosphate and organic phosphorus, have been formulated. Biogeochemical processes include growth, respiration and death processes for phytoplankton, as well as mineralization, nitrification and uptake and release of nutrients. Algebraic equations represent profiles for temperature, solar radiation and river inflows, in addition to the calculation of most factors that affect rate equations. The resulting partial differential algebraic equations (PDE) model is transformed into an ordinary differential equations system by spatially discretizing the PDE into sets of ordinary differential-algebraic equations (DAE). The objective function is the minimization of water flowrate to treatment in the wetland along the time horizon, subject to the previously described DAE system and upper bounds on phytoplankton and nutrients concentrations, to remain below eutrophication levels. The control variable is inflow profile of limiting nutrients. The dynamic optimization problem is solved through a simultaneous approach by transforming it into a large-scale nonlinear programming (NLP) problem discretizing state and control variables applying collocation over finite elements (Raghunathan et al., 2004). The discretized NLP model has more than 17,000 variables and it has been solved with a reduced successive quadratic programming algorithm (Kameswaran and Biegler, 2006). The study has been performed on Paso de las Piedras Lake, which supplies drinking water for more than 400,000 inhabitants in Bahia Blanca (Argentina). References Estrada V., E. Parodi, S. Diaz, Ecological Studies and Dynamic Parameter Estimation for Eutrophication Models, ECCE-6, Denmark, 16-21 September 2007. Kameswaran, S., L.T. Biegler (2006), Simultaneous dynamic optimization strategies: Recent advances and challenges, Comp. & Chem.Eng., 30, 1560–1575 Parodi, E.., Estrada, V., Trobbiani, N., Argañaraz Bonini, G., 2004, Análisis del estado trófico del Embalse Paso de las Piedras. Ecología en tiempos de Cambio. 178. Raghunathan A., S. Diaz, L.T. Biegler, 2004, An MPEC Formulation for Dynamic Optimization of Distillation Operations, Comp. & Chem.Eng., 28, 2037.