DYNAMIC OPTIMIZATION OF BIOPROCESSES:

Transcription

1 DYNAMIC OPTIMIZATION OF BIOPROCESSES: DETERMINISTIC AND STOCHASTIC STRATEGIES Eva Balsa-Canto 1, Antonio A. Alonso 2 and Julio R. Banga 1,* 1 Chem. Eng. Lab, IIM (CSIC). Eduardo Cabello 6, Vigo, SPAIN 2 Dept. of Chem. Eng., Universidad de Vigo. Apto 874, Vigo, SPAIN * julio@iim.csic.es, Fax: Abstract The general problem of dynamic optimization of bioprocesses with unspecified final time is considered. Several solution strategies, both deterministic and stochastic, are compared based on their results for three bioprocess case studies. A hybrid (stochastic-deterministic) method is also presented and evaluated, showing significant advantages in terms of robustness and computational effort. INTRODUCTION In recent years, many efforts have been devoted to the optimization and control of bioprocesses. An example of a problem that has received major attention is the dynamic optimization (open loop optimal control) of fed-batch bioreactors (e.g. van Impe, 1996; Roubos et al, 1997; Banga et al., 1997; Tholudur and Ramirez, 1997). Most bioprocesses have highly nonlinear dynamics, and constraints are also frequently present on both the state and the control variables. Thus, efficient and robust dynamic optimization methods are needed in order to successfully obtain their optimal operating policies. OPTIMAL CONTROL PROBLEM The general open loop optimal control problem (OCP) with free (unspecified) terminal time, considering a lumped parameter bioprocess, can be stated as: Find u ( t ) and t f over t [ to, t f ] to minimize (or maximize): subject to: t f J [ x, u] = θ ~ [ x{ t }] + ~ φ [ x{ t }, u{ t },t ]dt (1) dx dt f t0 = ψ~[ x{ t }, u{ t },t ], x ( t 0 ) = x0 (2) h [ x( t), u( t) ] = 0 (3) g [ x( t), u( t) ] 0 (4) x u L L U x( t) x (5) U u( t) u (6) ACoFoP IV (Automatic Control of Food & Biological Processes), Göteborg, Sweden, September Correspondence to: Dr. Julio R. Banga, IIM-CSIC. Eduardo Cabello 6, Vigo, SPAIN. julio@iim.csic.es

2 where J is the performance index, x is the vector of state variables, u the vector of control variables, eqns. (2) are the system of ordinary differential equality constraints with their initial conditions, eqns. (3) and (4) are the equality and inequality algebraic constraints and eqns. (5) and (6) are the upper and lower bounds on the state and control variables. Free final time problems are easily incorporated into the form of a fixed terminal time OCP. The idea is to specify a nominal time interval for the problem and to use a scale factor, adjustable by the optimization procedure, to scale the system dynamics and hence, in effect, scale the duration of the time interval. SOLUTION METHODS Basically, existing methods for the solution of optimal control problems can be classified in two classes: indirect and direct approaches. Indirect approaches use the necessary conditions of Pontryagin, while direct approaches transform the original problem into a non-linear programming (NLP) problem, either using control parameterization (Vassiliadis, 1993; Barton et al, 1998) or complete (control and state) parameterization (Cuthrell and Biegler, 1987). These NLPs are frequently multimodal, so deterministic (gradient based) optimization techniques may converge to local optima, especially if they are started far away from the global solution. Therefore, stochastic methods might be a good alternative to surmount these difficulties, as they are usually able to escape from local solutions, locating the vicinity of the global optimum in reasonable computation times, as shown by Banga and Seider (1996) using an adaptive stochastic algorithm. Genetic algorithms (GAs) are another class of stochastic methods which have become very popular in recent years, and several researches have used them for optimal control (Michalewicz et al., 1992; Seywald et al., 1995; Yamashita and Shima, 1997; Lee et al, 1997; Wang and Chiou, 1997). However, further refinement of the solution is usually made at a large computational cost using these methods. Although there is always a trade-off between convergence speed and robustness in both stochastic and deterministic methods, the latter usually have the opposite behavior, i.e. they converge very fast if they are started close to the global solution. Clearly, a convenient approach would be to combine both methodologies in order to compensate for their weaknesses while enhancing their strengths. STOCHASTIC, DETERMINISTIC AND HYBRID METHODS Several strategies for the dynamic optimization of bioprocesses with unspecified terminal time are considered and compared here, all based on the control parameterization approach. Regarding stochastic methods, four recently proposed methods were considered to solve the resulting NLP problem: one adaptive stochastic method, ICRS/DS (Banga and Seider, 1996), and three GA-based 2

3 methods, GENOCOP III (Michalewicz, 1996), DE (Storn and Price, 1997) and DHC (de la Maza and Yuret, 1994). Regarding deterministic methods, gopt (Vassiliadis et al, 1994a,b), which runs in connection with the gproms 1.4F process modelling package (Barton and Pantelides, 1994), was considered. In order to surmount the typical difficulties of each class of methods, in this study a hybrid (stochastic-deterministic) approach was also considered. This approach was developed by adequately combining the key elements of an stochastic (ICRS/DS) and a deterministic (gopt) method, taking advantage of their complementary features (Carrasco and Banga, 1998). The resulting Hybrid Method for Dynamic Optimization (HyMDO) operates in two sequential steps. In the first, ICRS/DS is used to locate the vicinity of the global solution. This information is then used to initialize gopt in the second step. CASE STUDIES A comparison of the above mentioned methods was made based on their results for three challenging case studies. Cases I and II consider the dynamic optimization of fed-batch bioreactors with unspecified final time, one for ethanol production (Bojkov and Luus, 1996) and the other for penicillin production (Banga et al, 1997). Case III is a time optimal drug displacement problem (Bojkov and Luus, 1994), where two drugs in a patient s bloodstream must reach a desired level in minimum time. The mathematical statements are not included here for the sake of brevity, but the interested reader will find them in the above references. All computations were done in double precision, using a PC Pentium/200MHz. Computation times reported in the literature were often obtained using different platforms. In order to make meaningful comparisons here, they were transformed to Pentium/200 CPU times using the Linpack benchmark tables. In each case study, the same initial control profiles and integration and optimization tolerances were used for all the methods. RESULTS AND DISCUSSION The results obtained for the three case studies using the different methods are summarized in Tables 1-3. The overall conclusion is that the hybrid method (HyMDO) presented here provided the best solutions in all cases, with very competitive CPU times. In fact, regarding case I, it is indeed interesting to see how both ICRS and gopt converged to local solutions, while their combination in HyMDO was able to arrive to the same global solution reported by Bojkov and Luus (1996) but 20 times faster. With respect to the other GA-based methods, only DE provided a good solution, although with a much larger computational effort than the hybrid. Regarding case II, both gopt and ICRS arrived to very good solutions (note that the one obtained by ICRS corresponds to a shorter batch 3

4 time, i.e. greater productivity), but the hybrid was able to slightly improve those results with a similar CPU time. Again, DE was the only GA-based method that arrived close to the best solution. Method J t f t CPU (s) ICRS/DS GENOCOP III DE DHC gopt HyMDO IDP (1) Table 1.- Results for case study I (1) Bojkov and Luus (1996) Method J t f t CPU (s) ICRS/DS GENOCOP III DE DHC gopt HyMDO a HyMDO b Table 2.- Results for case study II a,b runs started from different initial u profiles Considering case III, both the hybrid and gopt obtained the same minimum time, but the former was able to satisfy tighter tolerances for the state constraints. In fact, the hybrid method obtained better values than the IDP of Bojkov and Luus (1994) in only a small fraction of their CPU time, as it also happened with the first case study. Also in agreement with previous results, DE was able to arrive to a good solution. However its CPU time, although quite reasonable, was significantly larger than that of HyMDO. Method Minimum t f Final tolerances for t CPU (s) state constraints ICRS/DS , GENOCOP III , DE , DHC , gopt , HyMDO a , HyMDO b , IDP (1) , 10-4 N.A. IDP (2) , Table 3.- Results for case study III (1,2) Bojkov and Luus (1994) a,b different requested tolerances for state constraints In order to evaluate the robustness of the hybrid method, case I was solved considering three different initial constant control profiles, and the results are shown in Table 4, where the solutions obtained with gopt are also reported. It can be seen that HyMDO is not sensitive to initialization, arriving to essentially the same solutions in all the runs, while gopt, although a bit faster, converged to worse results in two runs. The best optimal control found by the hybrid method is shown in Figure 1. 4

5 gopt HyMDO u 0 J 0 J t f t CPU (s) J t f t CPU (s) Table 4.- Comparison of results considering three different initial control profiles (u 0 ) for case study I U(t) Time, hr Figure 1.- Optimal control for case study I obtained with HyMDO CONCLUSIONS Several different strategies were considered for the solution of the dynamic optimization of three bioprocesses, including a hybrid stochastic-deterministic method. This hybrid combines the ICRS/DS and the gopt methods sequentially, so their weaknesses are compensated and their strengths enhanced, thus showing significant advantages over other approaches. ACKNOWLEDGMENTS This work was supported in part by the EC (project FAIR CT ) and the Spanish Government (CICyT project ALI CE). The Centre for Process Systems Engineering (Imperial College, London) is gratefully acknowledged for an academic licence of gopt 1.5/gPROMS 1.4F. 5

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