PROPT Oil Shale Pyrolysis: Difference between revisions
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for n=[4 10 20 30 35] | for n=[4 10 20 30 35] | ||
p = tomPhase('p', t, 0, t_f, n); | p = tomPhase('p', t, 0, t_f, n); | ||
setPhase(p); | setPhase(p); |
Latest revision as of 11:49, 21 January 2015
This page is part of the PROPT Manual. See PROPT Manual. |
Dynamic Optimization of Batch Reactors Using Adaptive Stochastic Algorithms 1997, Eugenio F. Carrasco, Julio R. Banga
Case Study II: Oil Shale Pyrolysis
Problem description
A very challenging optimal control problem is the oil shale pyrolysis case study, as considered by Luus (1994). The system consists of a series of five chemical reactions:
A1 -> A2
A2 -> A3
A1+A2 -> A2+A2
A1+A2 -> A3+A2
A1+A2 -> A4+A2
This system is described by the differential equations
where the state variables are the concentrations of species, Ai, i = 1, ..., 4. The initial condition is
The rate expressions are given by:
where the values of ki0 and Ei are given by Luus (1994). The optimal control problem is to find the residence time t_f and the temperature profile T(t) in the time interval 0 <= t <= t_f so that the production of pyrolytic bitumen, given by x2, is maximized. Therefore, the performance index is
The constraints on the control variable (temperature) are:
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
toms t
toms t_f
ai = [8.86; 24.25; 23.67; 18.75; 20.70];
bi = [20300; 37400; 33800; 28200; 31000]/1.9872;
for n=[4 10 20 30 35]
p = tomPhase('p', t, 0, t_f, n);
setPhase(p);
tomStates x1 x2 x3 x4
tomControls T
% Initial guess
if n == 4
x0 = {t_f == 9.3
collocate(T == 725)};
else
x0 = {t_f == tfopt
icollocate({
x1 == x1opt; x2 == x2opt
x3 == x3opt; x4 == x4opt
})
collocate(T == Topt)};
end
% Box constraints
cbox = {9.1 <= t_f <= 12
icollocate({0 <= x1 <= 1; 0 <= x2 <= 1
0 <= x3 <= 1; 0 <= x4 <= 1})
698.15 <= collocate(T) <= 748.15};
% Boundary constraints
cbnd = initial({x1 == 1; x2 == 0; x3 == 0; x4 == 0});
% ODEs and path constraints
ki1 = exp(ai(1)-bi(1)./T);
ki2 = exp(ai(2)-bi(2)./T);
ki3 = exp(ai(3)-bi(3)./T);
ki4 = exp(ai(4)-bi(4)./T);
ki5 = exp(ai(5)-bi(5)./T);
ceq = collocate({
dot(x1) == -ki1.*x1-(ki3+ki4+ki5).*x1.*x2
dot(x2) == ki1.*x1-ki2.*x2+ki3.*x1.*x2
dot(x3) == ki2.*x2+ki4.*x1.*x2
dot(x4) == ki5.*x1.*x2});
% Objective
objective = -final(x2);
Solve the problem
options = struct;
options.name = 'Oil Pyrolysis';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
x1opt = subs(x1, solution);
x2opt = subs(x2, solution);
x3opt = subs(x3, solution);
x4opt = subs(x4, solution);
Topt = subs(T, solution);
tfopt = subs(final(t), solution);
Problem type appears to be: lpcon Time for symbolic processing: 0.42383 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Oil Pyrolysis f_k -0.357327805323273240 sum(|constr|) 0.000000000957541036 f(x_k) + sum(|constr|) -0.357327804365732190 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 93 ConJacEv 93 Iter 50 MinorIter 197 CPU time: 0.062400 sec. Elapsed time: 0.064000 sec.
Problem type appears to be: lpcon Time for symbolic processing: 0.42239 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Oil Pyrolysis f_k -0.354368283541904860 sum(|constr|) 0.000000002719365042 f(x_k) + sum(|constr|) -0.354368280822539790 f(x_0) -0.357327805323273070 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 208 ConJacEv 208 Iter 112 MinorIter 305 CPU time: 0.156001 sec. Elapsed time: 0.153000 sec.
Problem type appears to be: lpcon Time for symbolic processing: 0.43305 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Oil Pyrolysis f_k -0.351747594492437030 sum(|constr|) 0.000000280199856233 f(x_k) + sum(|constr|) -0.351747314292580830 f(x_0) -0.354368283541905970 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 124 ConJacEv 124 Iter 71 MinorIter 281 CPU time: 0.140401 sec. Elapsed time: 0.142000 sec.
Problem type appears to be: lpcon Time for symbolic processing: 0.42125 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Oil Pyrolysis f_k -0.352833701465704920 sum(|constr|) 0.000000018165177960 f(x_k) + sum(|constr|) -0.352833683300526950 f(x_0) -0.351747594492436640 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 409 ConJacEv 409 Iter 194 MinorIter 697 CPU time: 0.639604 sec. Elapsed time: 0.645000 sec.
Problem type appears to be: lpcon Time for symbolic processing: 0.42103 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Oil Pyrolysis f_k -0.352618526247765740 sum(|constr|) 0.000016167955157926 f(x_k) + sum(|constr|) -0.352602358292607830 f(x_0) -0.352833701465704590 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 64 ConJacEv 64 Iter 46 MinorIter 364 CPU time: 0.187201 sec. Elapsed time: 0.180000 sec.
end
t = subs(collocate(t),solution);
x1 = subs(collocate(x1opt),solution);
x2 = subs(collocate(x2opt),solution);
x3 = subs(collocate(x3opt),solution);
x4 = subs(collocate(x4opt),solution);
T = subs(collocate(Topt),solution);
Plot result
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Oil Pyrolysis state variables');
subplot(2,1,2)
plot(t,T,'+-');
legend('T');
title('Oil Pyrolysis control');