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Dynamic Optimization Approaches for the Determination of Bottom-up and Top-Down Control in Lakes and Reservoirs

Vanina Estrada, Chemical Engineering, Planta Piloto de Ingenieria Quimica - Universidad Nacional del Sur, Camino La Camino La Carrindanga Km 7, Bahia Blanca, 8000, Argentina, Elisa Parodi, Biology, Biochemistry and Pharmacy, Instituto Argentino de Oceanografía (IADO) – Universidad Nacional del Sur, CONICET, Camino La Carrindanga Km 7, Bahia Blanca, 8000, Argentina, and Maria Soledad Diaz, Chemical Engineering, Planta Piloto de Ingenieria Quimica (PLAPIQUI), Universidad Nacional del Sur - CONICET, Camino La Carrindanga Km 7, Bahia Blanca, 8000, Argentina.

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 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.

In this work, we develop a dynamic optimization model for a water body to determine optimal concentrations of zooplankton required to decrease phytoplankton concentration in the lake, associated to fish removal, as well as optimal limiting nutrient inflow profiles to the lake, with the associated flowrate of the inflows that has to be derived to a nearby wetland for remediation.

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 the offset between phytoplankton concentration and the desired concentration in the water body along the first year, subject to the previously described DAE system and upper bounds on phytoplankton and nutrients concentrations, to remain below eutrophication levels.

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.