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11 Batch Fermentor
Dynamic optimization of bioprocesses: efficient and robust numerical strategies 2003, Julio R. Banga, Eva Balsa-Cantro, Carmen G. Moles and Antonio A. Alonso
Case Study I: Optimal Control of a Fed-Batch Fermentor for Penicillin Production
11.1 Problem description
This problem considers a fed-batch reactor for the production of penicillin, as studied by Cuthrell and Biegler (1989). This problem has also been studied by many other authors (Dadebo & McAuley 1995, Banga & Seider 1996, Banga et al. 1997). We consider here the free terminal time version where the objective is to maximize the amount of penicillin using the feed rate as the control variable. It should be noted that the resulting NLP problem (after using CVP) does not seem to be multimodal, but it has been reported that local gradient methods do experience convergence problems if initialized with far-from-optimum profiles, or when a very refined solution is sought. Thus, this example will be excellent in order to illustrate the better robustness and efficiency of the alternative stochastic and hybrid approaches. The mathematical statement of the free terminal time problem is:
Find u(t) and t_f over t in [t0; t_f ] to maximize
subject to:
| h2 = 0.0055 * x3 * (0.0001 + x3*(1 + 10*x3)) |
where x1, x2, and x3 are the biomass, penicillin and substrate concentrations (g=L), and x4 is the volume (L). The initial conditions are:
There are several path constraints (upper and lower bounds) for state variables (case III of Cuthrell and Biegler, 1989):
The upper and lower bounds on the only control variable (feed rate of substrate) are:
Reference: [3]
11.2 Solving the problem on multiple grids.
The problem is solved in two stages. First, a solution is computed for a small number of collocation points, then the number of collocation points is increased, and the problem is resolved. This saves time, compared to using the fine grid immediately.
toms t
toms tfs
% Scale time by factor 100
t_f = 100*tfs;
for n=[10 35 70]
p = tomPhase('p', t, 0, t_f, n);
setPhase(p);
tomStates x1s x2s x3s x4s
tomControls us
% Create scaled states and control
x1 = 20*x1s;
x2 = 4*x2s;
x3 = 0.5*x3s;
x4 = 10*x4s;
u = 10*us;
% Initial guess
% Note: The guess for t_f must appear in the list before
% expression involving t.
if n==10
x0 = {t_f == 126
icollocate(x1 == 1)
icollocate(x2 == 1)
icollocate(x3 == 1)
icollocate(x4 == 1)
collocate(u==11.25)};
else
% Copy the solution into the starting guess
x0 = {t_f == tf_init
icollocate(x1 == x1_init)
icollocate(x2 == x2_init)
icollocate(x3 == x3_init)
icollocate(x4 == x4_init)
collocate(u == u_init)};
end
% Box constraints
% Setting the lower limit for x1 and x4 to slightly more than zero
% ensures that division by zero is avoided during the optimization
% process.
cbox = {t_f <= 256
1e-8 <= icollocate(x1) <= 40
0 <= icollocate(x2) <= 50
0 <= icollocate(x3) <= 25
1e-8 <= icollocate(x4) <= 10
0 <= collocate(u) <= 50};
% Boundary constraints
cbnd = initial({x1 == 1.5; x2 == 0
x3 == 0; x4 == 7});
% Various constants and expressions
h1 = 0.11*(x3./(0.006*x1+x3));
h2 = 0.0055*(x3./(0.0001+x3.*(1+10*x3)));
% ODEs and path constraints
ceq = collocate({
dot(x1) == h1.*x1-u.*(x1./500./x4)
dot(x2) == h2.*x1-0.01*x2-u.*(x2./500./x4)
dot(x3) == -h1.*x1/0.47-h2.*x1/1.2-x1.*...
(0.029*x3./(0.0001+x3))+u./x4.*(1-x3/500)
dot(x4) == u/500});
% Objective
objective = -final(x2)*final(x4);
% Solution for a small number of collocation points
options = struct;
options.name = 'Batch Fermentor';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05
=====================================================================================
Problem: --- 1: Batch Fermentor f_k -89.724083587647414000
sum(|constr|) 0.000000299524330643
f(x_k) + sum(|constr|) -89.724083288123083000
f(x_0) -0.000000000000007105
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 97 ConJacEv 97 Iter 64 MinorIter 1061
CPU time: 0.390625 sec. Elapsed time: 0.437000 sec.
Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05
=====================================================================================
Problem: --- 1: Batch Fermentor f_k -88.107470653415817000
sum(|constr|) 0.000000000336700538
f(x_k) + sum(|constr|) -88.107470653079119000
f(x_0) -89.100205205521547000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 125 ConJacEv 125 Iter 103 MinorIter 1311
CPU time: 1.687500 sec. Elapsed time: 1.922000 sec.
Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05
=====================================================================================
Problem: --- 1: Batch Fermentor f_k -88.031062605951576000
sum(|constr|) 0.000001353348234583
f(x_k) + sum(|constr|) -88.031061252603337000
f(x_0) -88.107470653415135000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 200 ConJacEv 200 Iter 143 MinorIter 3044
CPU time: 9.359375 sec. Elapsed time: 9.515000 sec.
11.3 Plot result
subplot(2,1,1);
ezplot([x1; x2; x3; x4]);
legend('x1','x2','x3','x4');
title('Batch Fermentor state variables');
subplot(2,1,2);
ezplot(u);
legend('u');
title('Batch Fermentor control');
drawnow
% Copy solution for initializing next round
x1_init = subs(x1,solution);
x2_init = subs(x2,solution);
x3_init = subs(x3,solution);
x4_init = subs(x4,solution);
u_init = subs(u,solution);
tf_init = subs(t_f,solution);
end
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