Ecological water quality models are being increasingly required to predict water quality evolution as a result of progressive eutrophication of lakes throughout the world. Eutrophication models provide a representation of major physical, chemical and biological processes that affect the biomass of phytoplankton and nutrients. They represent ecological processes through a set of complex nonlinear partial differential algebraic equations, with rate coefficients that require calibration to suit site-specific conditions. Consequently, the first step in the development of an eutrophication model is the formulation of a parameter estimation problem. The parameter estimation problem in eutrophication models has been addressed through different approaches. Zhang et al. (2004) have proposed a sequential procedure to determine phytoplankton and zooplankton parameters using exergy as the objective function and calibrating both physical and chemical parameters by trial and error. Shen and Kuo (1998) used the variational method for estimating unknown kinetic parameters. More recently, Shen (2006) proposed a least-squares objective function and the resolution of the dynamic parameter estimation problem through the application of a modified Gauss-Newton method capable of handling upper and lower bounds on parameters and the Hessian being approximated with information from the sensitivity matrix calculated by finite differences.

In this work, we formulate a parameter estimation problem with a least-squares objective function subject to a partial differential algebraic equations (PDE) model resulting from temporal and spatial (one dimensional, along water column) dynamic mass balances in phytoplankton (in the form of diatoms, green algae and cyanobacteria), dissolved oxygen and nutrients, such as nitrate, ammonium, organic nitrogen, silica, phosphate and organic phosphorus. Algebraic equations represent profiles for temperature, solar radiation and river inflows, in addition to the calculation of most factors that affect rate equations, such as effect of solar radiation, nutrients, etc. The PDE is transformed into an ordinary differential equation system by spatially discretizing the PDE into sets of ordinary differential-algebraic equations (DAE) (Rodriguez and Diaz, 2006). The DAE optimization problem is solved through a simultaneous approach by transforming it into a large-scale nonlinear programming (NLP) problem by representing state and control variables profiles by polynomial functions over finite elements in time. The discretized NLP model has more than 10,000 variables. Data sets from an entire year, with a frequency of twice a week have been included. The present eutrophication model has been formulated for Lake Paso de las Piedras, an artificial lake which supplies drinking water for more than 400,000 inhabitants in Bahia Blanca (Argentina) and for industrial purposes at a petrochemical complex nearby. The high content of phosphorus and nitrogen in Paso de las Piedras Lake is consequence of agricultural activities. The discretized NLP problem has been solved with a reduced successive quadratic programming algorithm (Biegler et al., 2002). Numerical results show good agreement with values from the literature. The model is currently being validated with recently obtained additional data from the lake. References Biegler L.T., Cervantes A, Waechter A., 2002; Advances in Simultaneous Strategies for Dynamic Process Optimization. Chem. Eng. Sci. 57: 575-593. Parodi, E. R., Estrada, V., Trobbiani, N., Argañaraz Bonini, G., 2004, Análisis del estado trófico del Embalse Paso de las Piedras (Buenos Aires, Argentina). Ecología en tiempos de Cambio. p. 178. Rodríguez, M., M. S. Diaz, 2007, Dynamic modelling and optimisation of cryogenic systems, Applied Thermal Engineering, 27, 1182-1190. Shen, J., 2006, Optimal estimation of parameters for aestuarine eutrophication model, Ecological Modelling 191, 521–537. Shen, J., Kuo, A.Y., 1998, Application of inverse model to calibrate estuarine eutrophication model. J. Environ. Eng. 124 (5), 409–418. Zhang, J.J., Jorgensen, S.E., Mahler, H., 2004, Examination of structurally dynamic eutrophication model. Ecol. Model. 173, 313–333.