Subsections

• Formulation
• Performance

12 Catalytic Cracking of Gas Oil

Determine the reaction coefficients for the catalytic cracking of gas oil into gas and other byproducts. The nonlinear model [21] that describes the process is

$$y_1' = -(\theta_1 + \theta_3) y_1^2$$ (9)

$$y_2' = \theta_1 y_1^2 - \theta_2 y_2$$

with coefficients $\theta_i \ge 0$ for $i = 1, \ldots , 3$. Initial conditions for (12.1) are known. The problem is to minimize

$$\ \sum_{j=1}^{21} \Vert y(\tau_j;\theta)-z_j \Vert^2,$$

where $z_j$ are concentration measurements for $y$ at time points $\tau_1,\ldots,\tau_{21}$.

Formulation

Our formulation of the catalytic cracking of gas oil problem as an optimization problem follows [21,3]. We use a $k$-stage collocation method, a uniform partition of the interval $[0,\tau_{20}]$ with $n_h$ intervals, and the standard [2, pages 247249] basis representation,

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

for the components of the solution $( y_1 , y_2 )$ of . The constraints in the optimization problem are the initial conditions in (12.1), the continuity conditions, and the collocation equations. The continuity equations are a set of $2(n_h - 1)$ linear equations. The collocation equations are a set of $2 k n_h$ nonlinear equations obtained by requiring that the collocation approximation satisfy (12.1) at the collocation points. Data for this problem appears in Table 12.1.

Table 12.1: Catalytic cracking of gas oil data

Variables	$2 (k+1) n_h + 3$
Constraints	$2 (k+1) n_h$
Bounds	3
Linear equality constraints	$2 n_h$
Linear inequality constraints	0
Nonlinear equality constraints	$2 k n_h$
Nonlinear inequality constraints	0
Nonzeros in $\nabla ^2 f(x)$	$40k^2$
Nonzeros in $c'(x)$	$3k (k+1) n_h$

Performance

We provide results for the AMPL formulation with $k=4$ in Table 12.2. The initial values for the $\theta$ parameters are $\theta_i = 0.0$. 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 the Tjoa and Biegler [21] values $\theta= ( 12 , 8 , 2 )$ and applying a relative random perturbation of size $10^{-1}$. Figure 12.1 shows the solution and the data.

Table 12.2: Performance on catalytic cracking of gas oil problem

Solver	$n_h=50$	$n_h=100$	$n_h=200$	$n_h=400$
LANCELOT	918.28 s	3502.88 s	$\ddagger$	$\ddagger$
$f$	5.23633e-03	5.23471e-03	$\ddagger$	$\ddagger$
$c$ violation	2.51920e-07	6.72780e-07	$\ddagger$	$\ddagger$
iterations	575	993	$\ddagger$	$\ddagger$
LOQO	1.37 s	3.36 s	11.6 s	49.62 s
$f$	5.23664e-03	5.23659e-03	5.23659e-03	5.23659e-03
$c$ violation	2.8e-09	1.8e-09	1.9e-09	1.1e-09
iterations	21	22	31	43
MINOS	5.04 s	14.66 s	49.56 s	161.99 s
$f$	5.23664e-03	5.23659e-03	5.23659e-03	5.23659e-03
$c$ violation	2.5e-10	9.3e-12	1.3e-08	5.9e-09
iterations	24	28	35	42
SNOPT	5.25 s	14.41 s	48.56 s	179.71 s
$f$	5.23664e-03	5.23659e-03	5.23659e-03	5.23659e-03
$c$ violation	2.0e-10	1.4e-08	2.2e-08	4.2e-07
iterations	35	28	24	18
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

Figure 12.1: Solution and data for the catalytic cracking of gas oil problem