PROPT Catalytic Cracking of Gas Oil

This page is part of the PROPT Manual. See PROPT Manual.

Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY

Problem Formulation

Find theta over t in [0; 0.95] to minimize

$J=\sum _{j=1}^{2}\sum _{i=1}^{21}(y_{j,i}-y_{j,i,meas})^{2}$

subject to:

${\frac {dy_{1}}{dt}}=-(theta_{1}+theta_{3})*y_{1}^{2}$ ${\frac {dy_{2}}{dt}}=theta_{1}*y_{1}^{2}-theta_{2}*y_{2}$ $theta>=0$

Where the data is given in the code.

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t theta1 theta2 theta3
p = tomPhase('p', t, 0, 0.95, 100);
setPhase(p);

tomStates y1 y2

% Initial guess
x0 = icollocate({
    y1 == 1-(1-0.069)*t/0.95
    y2 == 0.01*t/0.95});

% Box constraints
cbox = {0 <= theta1; 0 <= theta2; 0 <= theta3};

% Various constants and expressions
y1meas = [1.0;0.8105;0.6208;0.5258;0.4345;0.3903;...
    0.3342;0.3034;0.2735;0.2405;0.2283;0.2071;0.1669;...
    0.153;0.1339;0.1265;0.12;0.099;0.087;0.077;0.069];
y2meas = [0;0.2;0.2886;0.301;0.3215;0.3123;0.2716;...
    0.2551;0.2258;0.1959;0.1789;0.1457;0.1198;0.0909...
    ;0.0719;0.0561;0.046;0.028;0.019;0.014;0.010];
tmeas = [0;0.025;0.05;0.075;0.1;0.125;...
    0.15;0.175;0.2;0.225;0.25;0.3;0.35;0.4;...
    0.45;0.5;0.55;0.65;0.75;0.85;0.95];

y1err = atPoints(tmeas,y1) - y1meas;
y2err = atPoints(tmeas,y2) - y2meas;

% ODEs and path constraints
ceq = collocate({
    dot(y1) == -(theta1+theta3)*y1.^2
    dot(y2) == theta1*y1.^2-theta2*y2});

% Objective
objective = sum(y1err.^2)+sum(y2err.^2);

Solve the problem

options = struct;
options.name = 'Catalytic Cracking';
solution = ezsolve(objective, {cbox, ceq}, x0, options);
t  = subs(collocate(t),solution);
y1 = subs(collocate(y1),solution);
y2 = subs(collocate(y2),solution);
theta1 = subs(theta1,solution);
theta2 = subs(theta2,solution);
theta3 = subs(theta3,solution);

Problem type appears to be: qpcon
Time for symbolic processing: 0.13569 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Catalytic Cracking             f_k       0.004326020490938554
                                       sum(|constr|)      0.000000000508288956
                              f(x_k) + sum(|constr|)      0.004326020999227510
                                              f(x_0)      0.165642328947365690

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   38 ConJacEv   38 Iter   29 MinorIter  238
CPU time: 0.561604 sec. Elapsed time: 0.598000 sec. 

Plot result

figure(1);
tm  = tmeas;  y1m = y1meas; y2m = y2meas;
t1  = theta1; t2  = theta2; t3  = theta3;
plot(t,y1,'*-',t,y2,'*-',tm,y1m,'ro',tm,y2m,'ro');
legend('y1','y2','y1meas','y2meas');
title(sprintf('Catalytic Cracking state vbls, theta = [%g %g %g]',t1,t2,t3));