Optimization of Fed-Batch Reactors by the Luus-Jaakola Optimization Procedure Rein Luus* and Denis Hennessy Department of Chemical Engineering, University of Toronto, Toronto, Ontario M5S 3E5, Canada The presence of numerous local optima makes optimization of fed-batch reactors a challenging problem. In addition, the low sensitivity of the performance index on the control policy makes it very difficult to determine the optimal control policy accurately. As an alternative to iterative dynamic programming, we consider the application of the direct search optimization procedure of Luus and Jaakola (AIChE J. 1973, 19, 760-766) with the recent refinements involving the use of a quadratic penalty function with a shifting term to handle equality constraints. In using the optimization procedure in multipass fashion, we use the extent of variation of the variables in a pass to provide the region size for the beginning of the subsequent pass. In establishing the optimal control policies for three fed-batch reactor models, we show that this approach provides a good way of solving such optimization problems. 1. Introduction For design and operation of chemical engineering systems, it is important to determine the maximum possible economic benefit that can be realized under physical constraints. In fed-batch reactors, we are concerned with determining the feed rate to the reactor that will give the maximum amount of the desired product without violating any constraints. Although having a single control variable in the form of the feed rate may appear to present a simple optimal control problem, considerable difficulties have been reported in the determination of the optimal feed-rate policy for fed- batch reactors. 2-6 Advantages of using iterative dynamic programming 7 for establishing the optimal control of fed-batch reactors were pointed out by Hartig et al. 8 and by Bojkov and Luus. 9 Because there is always a pos- sibility of obtaining a local optimum, it is useful to cross- check results with a different optimization procedure. One such means is to use direct search optimization, where the optimal control policy is obtained by optimiz- ing the policy over all of the time stages simultaneously, rather than using stage to stage optimization as in iterative dynamic programming (IDP). The availability of very fast computers at low cost makes such a procedure feasible for solving very difficult optimal control problems such as the cancer chemotherapy scheduling problem. 10 By using the Luus-Jaakola (LJ) direct search opti- mization procedure 1 in a multipass fashion with the search region determined according to the extent of the change in the variables during a pass, a nonseparable optimal control problem involving three state variables and three control variables over 100 stages was suc- cessfully solved by Luus 11 as a 300-variable nonlinear optimization problem. Using the same approach, very accurate parameter estimates for the Lotka-Volterra problem were obtained. 12 Because the LJ optimization procedure exhibits a high reliability of obtaining the global optimum 13 and it is easy to program, the goal of this paper is to examine the viability of this approach for establishing the optimal control of fed-batch reactors. 2. Models of Typical Fed-Batch Reactors In fed-batch reactors, the reactors are operated in fed- batch mode, where the feed rate into the reactor is used for control. Because there is no outflow, the feed rate must be chosen so that the batch volume does not exceed the physical volume of the reactor. There is also an upper limit for the feed rate. For the present work we consider models of three fed-batch reactors that have been used recently for optimal control studies. Model 1. For the first model we choose the fermenta- tion process involving ethanol fermentation as formu- lated and used for optimal control by Hong, 14 and used for optimal control studies by Chen and Hwang, 15 Luus, 5 Hartig et al., 8 and Bojkov and Luus. 9 The differential equations describing the ethanol fermentation are with where x 1 represents the cell mass concentration, x 2 is the substrate concentration, x 3 is the desired product * To whom correspondence should be addressed. E-mail: luus@ecf.utoronto.ca. Fax: 416-978-8605. Phone: 416-978- 5200. dx 1 dt ) Ax 1 - u x 1 x 4 (1) dx 2 dt )-10Ax 1 + u ( 150 - x 2 x 4 29 (2) dx 3 dt ) Bx 1 - u x 3 x 4 (3) dx 4 dt ) u (4) A ) ( 0.408 1 + x 3 /16 29( x 2 0.22 + x 2 29 (5) B ) ( 1 1 + x 3 /71.5 29( x 2 0.44 + x 2 29 (6) 1948 Ind. Eng. Chem. Res. 1999, 38, 1948-1955 10.1021/ie980731w CCC: $18.00 © 1999 American Chemical Society Published on Web 04/10/1999