Dynamic Parameter Estimation Problem For A Water Quality Model

VANINA G. ESTRADA; ELISA PARODI; M. SOLEDAD DIAZ

Chemical Engineering Transactions Vol 11

AIDIC

Lugar: Milan; Año: 2007; p. 247 - 252

The need of modelling for predictive ecological water quality assessments has arisen as a result of the increasing eutrophication of lakes throughout the world.  Eutrophication models provide a representation of the major physical, chemical and biological processes that affect the biomass of phytoplankton and nutrients. They represent ecological processes through a set of complex nonlinear differential algebraic equations, with rate coefficients that require calibration to suit site-specific conditions. Consequently, the first step in an eutrophication model development is the resolution 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 large-scale partial differential algebraic equations (PDE) model resulting from temporal and spatial 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 applying the Method of Lines to spatially discretize the PDE into sets of ordinary differential-algebraic equations (DAE) (Rodriguez and Diaz, 2006). The DAE optimization problem is then transformed into a large nonlinear programming (NLP) problem by representing state and control variables profiles by polynomial functions over finite elements in time. The present study has been performed on Paso de las Piedras Lake (38° 22´ S and 61° 12´ W), which supplies drinking water to the cities of Bahía Blanca and Punta Alta (Argentina), with a population of more than 400,000 inhabitants. At present, the trophic level of this lake corresponds to eutrophic category and it undergoes repeated blooms of algae (Parodi et al., 2004). The high content of phosphorus and nitrogen in Paso de las Piedras Lake is consequence of agricultural activities. Field data were obtained throughout a year with high frequency, twice a week (Trobbiani et al., 2005). The model has been formulated within the GAMS modelling environment and the NLP problem has been solved with a successive quadratic programming algorithm. Numerical results show good agreement with values from the literature. The model is currently being validated with recently obtained additional data from the lake.