next up previous contents

Subsections

13 Methanol to Hydrocarbons

Determine the reaction coefficients for the conversion of methanol into various hydrocarbons. The nonlinear model [12,17] that describes the process is

$\displaystyle y_1'$ $\textstyle =$ $\displaystyle -\left(2 \theta_2 - \frac{\theta_1 y_2}{(\theta_2+\theta_5) y_1 + y_2} + \theta_3 + \theta_4 \right) y_1$  
$\displaystyle y_2'$ $\textstyle =$ $\displaystyle \frac{\theta_1 y_1 (\theta_2 y_1 - y_2)}{(\theta_2 + \theta_5) y_1 + y_2} + \theta_3 y_1$ (10)
$\displaystyle y_3'$ $\textstyle =$ $\displaystyle \frac{\theta_1 y_1 (y_2 + \theta_5 y_1)}{(\theta_2 + \theta_5) y_1 + y_2} + \theta_4 y_1$  

with coefficients $\theta_i \ge 0 $ for $ i = 1, \ldots , 5 $. Initial conditions for (13.1) are known. The problem is to minimize

\begin{displaymath} \ \sum_{j=1}^{16} \Vert y(\tau_j;\theta)-z_j \Vert^2, \end{displaymath}

where $ z_j $ are concentration measurements for $y$ at time points $\tau_1,\ldots,\tau_{16}$.

Formulation

Our formulation of the methanol-to-hydrocarbons problem as an optimization problem follows [21,3]. We use a $k$-stage collocation method, a uniform partition of the interval $[0,\tau_{16}]$ with $ n_h $ intervals, and the standard [2, pages 247-249] basis representation,

\begin{displaymath} v_{i} + \sum_{j=1}^k \frac{(t-t_i)^j}{j! \, h^{j-1}} w_{ij}, \qquad t \in [t_i,t_{i+1}], \end{displaymath}

for the components of the solution $ ( y_1 , y_2, y_3 )$ of . The constraints in the optimization problem are the initial conditions in (13.1), the continuity conditions, and the collocation equations. The continuity equations are a set of $3(n_h - 1)$ linear equations. The collocation equations are a set of $3 k n_h$ nonlinear equations obtained by requiring that the collocation approximation satisfy (13.1) at the collocation points. Data for this problem appears in Table 13.1.



Table 13.1: Methanol-to-hydrocarbons data
Variables $3 (k+1) n_h + 5$
Constraints $ 3 (k+1) n_h$
Bounds 5
Linear equality constraints $3 n_h$
Linear inequality constraints 0
Nonlinear equality constraints $3 k n_h$
Nonlinear inequality constraints 0
Nonzeros in $ \nabla ^2 f(x) $ $48k^2$
Nonzeros in $ c'(x) $ $7k (k+1) n_h$

Performance

We provide results for the AMPL formulation with $k=3$ in Table 13.2. The initial values for the $\theta$ parameters are $ \theta_i = 1.0 $. The initial basis parameters are chosen so that the collocation approximation is piecewise constant and interpolates the data. Data is generated by solving numerically using $\theta= (2.69 , 0.5 , 3.02 , 0.5 , 0.5 )$ as given in Maria [17] and applying a relative random perturbation of size $10^{-1}$. Figure 13.1 shows the solution and the data.



Table 13.2: Performance on methanol-to-hydrocarbons problem
Solver $n_h=50$ $n_h=100$ $n_h=200$ $n_h=400$
LANCELOT 196.62 s 1792.75 s $\ddagger$ $\ddagger$
$f$ 9.02300e-03 9.00563e-03 $\ddagger$ $\ddagger$
$c$ violation 4.92130e-06 4.78630e-06 $\ddagger$ $\ddagger$
iterations 251 622 $\ddagger$ $\ddagger$
LOQO 2.13 s 5.45 s 18.78 s 45.2 s
$f$ 9.02229e-03 9.02229e-03 9.02229e-03 9.02229e-03
$c$ violation 3.5e-07 4.7e-08 1.7e-07 1.9e-08
iterations 19 21 30 26
MINOS 5.05 s 13.49 s 41.83 s 263.67 s
$f$ 9.02228e-03 9.02229e-03 9.02228e-03 9.02228e-03
$c$ violation 9.2e-13 9.8e-13 4.4e-12 3.5e-13
iterations 9 9 9 34
SNOPT 12.92 s 32.38 s 131.99 s 512.16 s
$f$ 9.02228e-03 9.02229e-03 9.02228e-03 9.02228e-03
$c$ violation 6.8e-09 9.8e-11 1.6e-09 1.3e-09
iterations 64 60 40 71
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

Figure 13.1: Solution and data for the methanol-to-hydrocarbons problem
\includegraphics[width=3.25in]{ps/methanol.eps}

next up previous contents
Liz Dolan
2001-01-02