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82 Penicillin Plant
Fed-batch Fermentor Control: Dynamic Optimization of Batch Processes II. Role of Measurements in Handling Uncertainty 2001, B. Srinivasan, D. Bonvin, E. Visser, S. Palanki
Illustrative example: Nominal Optimization of a Fed-Batch Fermentor for Penicillin Production.
82.1 Problem description
This particular example was featured in the work of B. Srinivasan et al. 2001. The optimal trajectories for the problem was provided in the work.
In this problem, the objective is to maximize the concentration of penicillin, P, produced in a fed-batch bioreactor, given a finite amount of time.
Reactions: S -> X, S -> P
Conditions: Fed-batch, isothermal.
Objective: Maximize the concentration of product P at a given final time.
Manipulated variable: Feed rate of S.
Constraints: Input bounds; upper limit on the biomass concentration,
which is motivated by oxygen-transfer limitation typically occurring at
large biomass concentration.
subject to:
Programmer: Wee Kiat Lim (Nanyang Technological University)
82.2 Problem setup
Penalty for variations in u
penalty_constant = 0.001;
% Various constants
miu_m = 0.02; Km = 0.05; Ki = 5;
Yx = 0.5; Yp = 1.2; v = 0.004;
Sin = 200; umin = 0; umax = 1;
Xmin = 0; Xmax = 3.7; Smin = 0;
% no. of collocation points to use
narr = [30 100];
for n=narr
toms t1
toms tcut
p1 = tomPhase('p1', t1, 0, tcut, n);
setPhase(p1);
tomStates X1 S1 P1 V1 %Vs %Scaling is disabled here
tomControls u1
% Initial guess
if n == narr(1)
x01 = {tcut == 75
icollocate({X1 == 1+2.7*t1/tcut; S1 == 0.5;
P1 == 0.6*t1/tcut; V1 == 150})
collocate(u1 == 0.03+0.06*t1/tcut)};
else
x01 = {tcut == tcutg
icollocate({X1 == Xg1; S1 == Sg1; P1 == Pg1; V1 == Vg1})
collocate(u1 == ug1)};
end
% Box constraints
cbox1 = {75 <= tcut <= 85
0 <= icollocate(X1) <= Xmax
Smin <= icollocate(S1) <= 100
0 <= icollocate(P1) <= 5
1 <= icollocate(V1) <= 300
umin <= collocate(u1) <= umax};
% Boundary constraints
cbnd1 = initial({X1 == 1; S1 == 0.5;
P1 == 0; V1 == 150});
miu1 = (miu_m*S1)/(Km + S1 + S1^2/Ki);
% ODEs and path constraints
temp11 = miu1*X1;
temp21 = u1/V1;
temp31 = v*X1;
ceq1 = collocate ({
dot(X1) == temp11 - u1/V1*X1
dot(S1) == -temp11/Yx - temp31/Yp + temp21*(Sin - S1)
dot(P1) == temp31 - temp21*P1
dot(V1) == u1});
if n == narr(1)
% No objective in first phase
objective = 0;
else
% Variation penalty
objective = penalty_constant*integrate(dot(u1)^2);
end
toms t2
p2 = tomPhase('p2', t2, tcut, 150-tcut, n);
setPhase(p2);
tomStates X2 S2 P2 V2 %Vs %Scaling is disabled here
tomControls u2
% Initial guess
if n == narr(1)
x02 = {icollocate({X2 == Xmax; S2 == 0; P2 == 0.6+t2/150;
V2 == 150}); collocate(u2 == 0.01)};
else
x02 = {icollocate({X2 == Xg2; S2 == Sg2; P2 == Pg2; V2 == Vg2})
collocate(u2 == ug2)};
end
% Box constraints
umax2 = 0.03;
cbox2 = {0 <= icollocate(X2) <= Xmax
Smin <= icollocate(S2) <= 100
0 <= icollocate(P2) <= 5
1 <= icollocate(V2) <= 300
umin <= collocate(u2) <= umax2
initial(S2) <= 0.2};
miu2 = (miu_m*S2)/(Km + S2 + S2^2/Ki);
% ODEs and path constraints
temp12 = miu2*X2;
temp22 = u2/V2;
temp32 = v*X2;
ceq2 = collocate ({
dot(X2) == temp12 - u2/V2*X2
dot(S2) == -temp12/Yx - temp32/Yp + temp22*(Sin - S2)
dot(P2) == temp32 - temp22*P2
dot(V2) == u2});
% Phase links
links = {initial(X2) == final(p1,X1)
initial(S2) == final(p1,S1)
initial(P2) == final(p1,P1)
initial(V2) == final(p1,V1)};
if n == narr(1)
% Objective (Negative sign is added to 'maximize' P)
objective = -final(P2);
type = 'lpcon';
solver = 'snopt';
else
objective = objective-final(P2)+penalty_constant*integrate(dot(u2)^2);
type = 'con';
solver = 'snopt';
end
% Solve the problem
options = struct;
options.name = 'Penicillin Plant';
Prob = sym2prob(type, objective, {cbox1, cbnd1, ceq1, cbox2, ceq2, links}, {x01, x02}, options);
Result = tomRun(solver, Prob, 1);
solution = getSolution(Result);
t1 = subs(collocate(p1,t1),solution);
t2 = subs(collocate(p2,t2),solution);
ug1 = subs(collocate(p1,u1),solution);
Xg1 = subs(collocate(p1,X1),solution);
Sg1 = subs(collocate(p1,S1),solution);
Pg1 = subs(collocate(p1,P1),solution);
Vg1 = subs(collocate(p1,V1),solution);
ug2 = subs(collocate(p2,u2),solution);
Xg2 = subs(collocate(p2,X2),solution);
Sg2 = subs(collocate(p2,S2),solution);
Pg2 = subs(collocate(p2,P2),solution);
Vg2 = subs(collocate(p2,V2),solution);
tcutg = solution.tcut;
np=length(t1) + length(t2);
end
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05
=====================================================================================
Problem: --- 1: Penicillin Plant f_k -1.682687588966484700
sum(|constr|) 0.000001056696736974
f(x_k) + sum(|constr|) -1.682686532269747700
f(x_0) -1.600000000000001400
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 126 ConJacEv 126 Iter 68 MinorIter 3674
CPU time: 2.375000 sec. Elapsed time: 2.390000 sec.
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05
=====================================================================================
Problem: --- 1: Penicillin Plant f_k -1.677945102060630000
sum(|constr|) 2.202586683090413700
f(x_k) + sum(|constr|) 0.524641581029783750
f(x_0) -1.157798971399910400
Solver: snopt. EXIT=1. INFORM=31.
SNOPT 7.2-5 NLP code
Iteration limit reached
FuncEv 61 GradEv 59 ConstrEv 59 ConJacEv 59 Iter 23 MinorIter 10090
CPU time: 43.328125 sec. Elapsed time: 44.235000 sec.
82.3 Plot result
Optimal states and control trajectories
t = [t1;t2];
uopt = [ug1;ug2];
Xopt = [Xg1;Xg2];
Sopt = [Sg1;Sg2];
Popt = [Pg1;Pg2];
Vopt = [Vg1;Vg2];
Pfinal=subs(final(P2),solution);
% Plots of the trajectories
figure(1)
subplot(3,1,1);
plot(t,Popt,'*-');
title(['Final Penicillin concentration is ',num2str(Pfinal),' g/L.'])
ylabel('Penicillin Conc')
xlabel('Time (hrs)')
subplot(3,1,2);
plot(t,Xopt,'*-');
ylabel('Cell Mass Conc')
xlabel('Time (hrs)')
subplot(3,1,3);
plot(t,Sopt,'*-');
ylabel('Substrate Conc')
xlabel('Time (hrs)')
figure(2)
subplot(2,1,1);
plot(t,Vopt,'*-');
title(['Final Penicillin concentration is ',num2str(Pfinal),' g/L.'])
ylabel('Volume of medium')
xlabel('Time (hrs)')
subplot(2,1,2);
plot(t,uopt,'*-');
ylabel('Feed flowrate')
xlabel('Time (hrs)')
fprintf('\n')
fprintf('Optimization completed... \n')
fprintf('Final Penicillin concentration is %5.4f g/L.\n',Pfinal)
Optimization completed...
Final Penicillin concentration is 1.6788 g/L.
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