Model re-parameterization and output prediction for a bioreactor system Kartik Surisetty, Hector De la Hoz Siegler, William C. McCaffrey, Amos Ben-Zvi n Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2V4 article info Article history: Received 17 July 2009 Received in revised form 17 March 2010 Accepted 16 April 2010 Available online 7 May 2010 Keywords: Parameter identification Model reduction Mathematical modeling Optimal experimental design Control Bioreactors abstract Microalgal bioprocesses are of increasing interest due to the possibility of producing fine chemicals, pharmaceuticals, and biofuels. In this work, the parameter estimability of a first principles ODE model of a microalgal bioreactor, containing 6 states and 12 unknown parameters, is investigated. For this purpose, the system input trajectories are computed using the D-optimality criterion. Even by using a D-optimal input, not all parameters were found to have a significant effect on model predictions. Linear and non-linear transformations are used to partition the parameter space into estimable and inestimable subspaces. For the linear re-parameterization, a set of four directions in the twelve dimensional parameter space, along which a significant change in the output occurs, is identified using singular value decomposition of the parameter covariance matrix. The non-linear re-parameterization utilizes the three system rate functions as pseudo-outputs in order to perform a non-linear transformation which reduces the dimension of the parameter space from twelve to three. Both the proposed re-parameterization methods achieve a good degree of output prediction at a greatly decreased computational cost. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Global warming and depletion of fossil fuels has increased the need for cleaner and sustainable energy production. Biodiesel, a proven alternative fuel, provides a 67% reduction of greenhouse gases emission when compared to fossil fuels (USEPA, 2002). Microalgae have the ability to produce large amounts of oil that can be directly used as high value single-cell oils (Chen and Chen, 2006), or be converted into biodiesel (Li et al., 2007). Good control of culture conditions is critical for the economic viability of large-scale production of microalgae. Model-based control strategies have been successfully applied to biochemical reaction systems to improve their economic performance (Bastin and Dochain, 1990). In order to implement a model-based control strategy, one must identify a model that properly captures the biochemical dynamics of microalgae, yet simple enough to allow its implementation for controller design. Biochemical systems are highly non-linear, and models of such systems typically contain parameters that are not directly accessible to measurement (Audoly et al., 2001). Therefore, the use of a model to develop a controller for microalgal processes requires the estimation of the model parameters using well designed experiments. A mathematical model is not estimable if the data collected for parameter identification and model validation is not sufficient for the accurate estimation of every parameter in the model. Inestimability implies that several parameter values will lead to statistically indistinguishable predictions (Ben-zvi, 2008). A large amount of literature has been devoted to identifying and dealing with inestimable parameters (Yao et al., 2003; Ben-Zvi et al., 2004; Sidoli et al., 2005). If model parameters cannot be estimated from available data, the experimenter may invest in obtaining additional data or, alternatively, the inestimable parameters can be removed from the model or be fixed at some nominal value. Even if one is able to obtain additional data, the additional expense may not be justified if the new information does not significantly alter the model predictions in the region of interest. As a result, one is often interested in estimating only a subset of the unknown model parameters. In this work, a first-principles based microalgal bioreactor model consisting of six ordinary differential equations and 12 unknown parameters is studied for estimability. It is shown that even under an optimal experimental design many of the process parameters do not have statistically significant effect on model predictions. Two model reparameterizing algorithms are pro- posed in order to reduce the number of parameters that must be estimated for accurate predictions. The first algorithm is based in a linear transformation of the parameter space while the second algorithm is based on a non-linear one. Both algorithms greatly decrease computational time while achieving a good degree of output prediction and significantly reducing computational complexity. It should be noted that, even though the re-parameterization is done off-line, a lower computational time is a desirable characteristic as it is preferable to have a simpler ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.04.024 n Corresponding author. Tel.: + 1 780 492 7651; fax: + 1 780 492 2881. E-mail address: abenzvi@ualberta.ca (A. Ben-Zvi). Chemical Engineering Science 65 (2010) 4535–4547