PROPT Methanol to Hydrocarbons

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; 1.122] to minimize

$J=\sum _{j=1}^{3}\sum _{i=1}^{21}(y_{j,i}-y_{j,i,meas})^{2}$

subject to:

${\frac {dy_{1}}{dt}}=-(2*theta_{2}-{\frac {theta_{1}*y_{2}}{(theta_{2}+theta_{5})*y_{1}+y_{2}}}+theta_{3}+theta_{4})*y_{1}$

${\frac {dy_{2}}{dt}}={\frac {theta_{1}*y_{1}*(theta_{2}*y_{1}-y_{2})}{(theta_{2}+theta_{5})*y_{1}+y_{2}}}+theta_{3}*y_{1}$

${\frac {dy_{3}}{dt}}={\frac {theta_{1}*y_{1}*(y_{2}+theta_{5}*y_{1})}{(theta_{2}+theta_{5})*y_{1}+y_{2}}}+theta_{4}*y_{1}$

$theta>=0$

Where the data is given in the code.

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

Problem setup

toms t theta1 theta2 theta3 theta4 theta5

% Various constants and expressions
y1meas = [0.7085;0.5971;0.5537;0.3684;0.1712;...
    0.1198;0.0747;0.0529;0.0415;0.0261;0.0208;...
    0.0085;0.0053;0.0019;0.0018];
y2meas = [0.1621;0.1855;0.1989;0.2845;0.3491;...
    0.3098;0.3576;0.3347;0.3388;0.3557;0.3483;...
    0.3836;0.3611;0.3609;0.3485];
y3meas = [0.0811;0.0965;0.1198;0.1535;0.2097;...
    0.2628;0.2467;0.2884;0.2757;0.3167;0.2954;...
    0.295;0.2937;0.2831;0.2846];
tmeas = [0.05;0.065;0.08;0.123;0.233;0.273;...
    0.354;0.397;0.418;0.502;0.553;...
    0.681;0.75;0.916;0.937];

Solve the problem, using a successively larger number collocation points

for n=[20 80]

    p = tomPhase('p', t, 0, 1.122, n);
    setPhase(p);

    tomStates y1 y2 y3

    % Initial guess
    if n == 20
        x0 = {theta1 == 1; theta2 == 1
            theta3 == 1; theta4 == 1
            theta5 == 1
            icollocate({
            y1 == 1-(1-0.0006)*t/1.122
            y2 == 0.3698*t/1.122
            y3 == 0.2899*t/1.122})};
    else
        x0 = {theta1 == theta1opt; theta2 == theta2opt
            theta3 == theta3opt; theta4 == theta4opt
            theta5 == theta5opt
            icollocate({
            y1 == y1opt
            y2 == y2opt
            y3 == y3opt})};
    end

    % Box constraints
    cbox = {sqrt(eps) <= theta1; sqrt(eps) <= theta2
        sqrt(eps) <= theta3; sqrt(eps) <= theta4
        sqrt(eps) <= theta5};

    y1err = sum((atPoints(tmeas,y1) - y1meas).^2);
    y2err = sum((atPoints(tmeas,y2) - y2meas).^2);
    y3err = sum((atPoints(tmeas,y3) - y3meas).^2);

    % Start and end points cannot be interpolated
    y1end = (1-initial(y1)).^2 + (0.0006-final(y1))^2;
    y2end = (0-initial(y2)).^2 + (0.3698-final(y2))^2;
    y3end = (0-initial(y3)).^2 + (0.2899-final(y3))^2;

    % ODEs and path constraints
    ceq = collocate({
        dot(y1) == -(2*theta2-(theta1*y2)./((theta2+theta5)*y1+y2)+theta3+theta4).*y1
        dot(y2) == (theta1*y1.*(theta2*y1-y2))./((theta2+theta5)*y1+y2)+theta3*y1
        dot(y3) == (theta1*y1.*(y2+theta5*y1))./((theta2+theta5)*y1+y2)+theta4*y1});

    % Objective
    objective = y1err+y2err+y3err+y1end+y2end+y3end;

Solve the problem

    options = struct;
    options.name = 'Methanol to Hydrocarbons';
    solution = ezsolve(objective, {cbox, ceq}, x0, options);

    % Optimal x, theta for starting point
    y1opt = subs(y1, solution);
    y2opt = subs(y2, solution);
    y3opt = subs(y3, solution);
    theta1opt = subs(theta1, solution);
    theta2opt = subs(theta2, solution);
    theta3opt = subs(theta3, solution);
    theta4opt = subs(theta4, solution);
    theta5opt = subs(theta5, solution);

Problem type appears to be: qpcon
Time for symbolic processing: 0.41616 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Methanol to Hydrocarbons       f_k       0.008301664004164877
                                       sum(|constr|)      0.000000001050824926
                              f(x_k) + sum(|constr|)      0.008301665054989802
                                              f(x_0)     -0.959232294294469990

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

FuncEv    1 ConstrEv   42 ConJacEv   42 Iter   41 MinorIter   88
CPU time: 0.046800 sec. Elapsed time: 0.052000 sec. 

Problem type appears to be: qpcon
Time for symbolic processing: 0.4199 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Methanol to Hydrocarbons       f_k       0.008301664004167542
                                       sum(|constr|)      0.000000988209788045
                              f(x_k) + sum(|constr|)      0.008302652213955587
                                              f(x_0)     -5.007954925995837100

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

FuncEv    1 ConstrEv    1 ConJacEv    1 MinorIter  159
CPU time: 0.046800 sec. Elapsed time: 0.042000 sec. 

end

t  = subs(collocate(t),solution);
y1 = collocate(y1opt);
y2 = collocate(y2opt);
y3 = collocate(y3opt);
t1 = subs(theta1,solution);
t2 = subs(theta2,solution);
t3 = subs(theta3,solution);
t4 = subs(theta4,solution);
t5 = subs(theta5,solution);

Plot result

figure(1);
tm  = [0;tmeas;1.122];
y1m = [1;y1meas;0.0006];
y2m = [0;y2meas;0.3698];
y3m = [0;y3meas;0.2899];
plot(t,y1,'*-',t,y2,'*-',t,y3,'*-',tm,y1m,'ko',tm,y2m,'ko',tm,y3m,'ko');
legend('y1','y2','y3','ymeas');
title(sprintf('Methanol to Hyd, theta = [%g %g %g %g %g]',t1,t2,t3,t4,t5));