# A Study of Explorative Moves during Modifier Adaptation with Quadratic Approximation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modifier Adaptation with Quadratic Approximation

- Step 1.
- Choose an initial set-point ${\mathbf{u}}^{\left(0\right)}$ and probe the plant at ${\mathbf{u}}^{\left(0\right)}$ and ${\mathbf{u}}^{\left(0\right)}+h{\mathbf{e}}_{i}$, where h is a suitable step size and ${\mathbf{e}}_{i}\in {\mathbb{R}}^{{n}_{u}}(i=1,\dots ,{n}_{u})$ are mutually orthogonal unit vectors. Use the finite difference approach to calculate the gradients at ${\mathbf{u}}^{\left(0\right)}$ and run the IGMO approach [3] until $k\ge {C}_{2}^{{n}_{u}+2}$ set-points have been generated. Run the screening algorithm to define the regression set ${\mathcal{U}}^{\left(k\right)}$. Initialize ${\rho}_{m}^{\left(k\right)}=0$ and ${\rho}_{\varphi}^{\left(k\right)}=0$.
- Step 2.
- Calculate the quadratic functions ${J}_{\varphi}^{\left(k\right)}$ and ${\mathbf{C}}_{\varphi}^{\left(k\right)}$ based on ${\mathcal{U}}^{\left(k\right)}$. Determine the search space ${\mathcal{B}}^{\left(k\right)}$ by (5).
- Step 3.
- Compute the gradients from the quadratic functions. Adapt the model-based optimization problem and determine the optimal set-point ${\widehat{\mathbf{u}}}^{\left(k\right)}$ as follows:
- Step 4.
- If $\parallel {\widehat{\mathbf{u}}}^{\left(k\right)}-{\mathbf{u}}^{\left(k\right)}\parallel <\Delta \mathbf{u}$ and there exists one point ${\mathbf{u}}^{\left(j\right)}\in {\mathcal{U}}^{\left(k\right)}$ such that $\parallel {\mathbf{u}}^{\left(j\right)}-{\mathbf{u}}^{\left(k\right)}\parallel >2\Delta \mathbf{u}$, set ${\widehat{\mathbf{u}}}^{\left(k\right)}=\left({\mathbf{u}}^{\left(j\right)}+{\mathbf{u}}^{\left(k\right)}\right)/2$.
- Step 5.
- Evaluate the plant at ${\widehat{\mathbf{u}}}^{\left(k\right)}$ to acquire ${J}_{p}\left({\widehat{\mathbf{u}}}^{\left(k\right)}\right)$ and ${\mathbf{C}}_{p}\left({\widehat{\mathbf{u}}}^{\left(k\right)}\right)$. Prepare the next step as follows
- (a)
- If ${\widehat{J}}_{p}^{\left(k\right)}<{J}_{p}^{\left(k\right)}$, where ${\widehat{J}}_{p}^{\left(k\right)}={J}_{p}\left({\widehat{\mathbf{u}}}^{\left(k\right)}\right)$, this is a performance-improvement move. Define ${\mathbf{u}}^{(k+1)}={\widehat{\mathbf{u}}}^{\left(k\right)}$ and run the screening algorithm to define the next regression set ${\mathcal{U}}^{(k+1)}$. Update the quality indices ${\rho}_{m}^{(k+1)}$ and ${\rho}_{\varphi}^{(k+1)}$. Increase k by one and go to Step 2.
- (b)
- If ${\widehat{J}}_{p}^{\left(k\right)}\ge {J}_{p}^{\left(k\right)}$, this is an explorative move. Run the screening algorithm to update the regression set for ${\mathbf{u}}^{\left(k\right)}$. Go to Step 2.

## 3. Analysis of the Explorative Moves

- Distribution of the regression set (quantified by Λ and the distance to the current point)
- Non-quadratic nature of the plant (quantified by G, the upper bound on the third derivative)
- Measurement noise (quantified by ${\delta}_{noise}$).

## 4. Simulation Studies

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

RTO | Real-time optimization |

MAWQA | Modifier adaptation with quadratic approximation |

TMS | Thermomorphic multicomponent solvent system |

DMF | Dimethylformamide |

## References

- Chen, C.Y.; Joseph, B. On-line optimization using a two-phase approach: An application study. Ind. Eng. Chem. Res.
**1987**, 26, 1924–1930. [Google Scholar] [CrossRef] - Roberts, P.D. An algorithm for steady-state system optimization and parameter estimation. Int. J. Syst. Sci.
**1979**, 10, 719–734. [Google Scholar] [CrossRef] - Gao, W.; Engell, S. Iterative set-point optimization of batch chromatography. Comput. Chem. Eng.
**2005**, 29, 1401–1409. [Google Scholar] [CrossRef] - Marchetti, A.; Chachuat, B.; Bonvin, D. Modifier-adaptation methodology for real-time optimization. Ind. Eng. Chem. Res.
**2009**, 48, 6022–6033. [Google Scholar] [CrossRef] - Bunin, G.A.; François, G.; Bonvin, D. From discrete measurements to bounded gradient estimates: A look at some regularising structures. Ind. Eng. Chem. Res.
**2013**, 52, 12500–12513. [Google Scholar] [CrossRef] - Gao, W.; Wenzel, S.; Engell, S. A reliable modifier-adaptation strategy for real-time optimization. Comput. Chem. Eng.
**2016**, 91, 318–328. [Google Scholar] [CrossRef] - Conn, A.R.; Scheinberg, K.; Vicente, L.N. Introduction to Derivative-Free Optimization; SIAM: Philadelphia, PA, USA, 2009. [Google Scholar]
- Ciarlet, P.G.; Raviart, P.A. General lagrange and hermite interpolation in R
^{n}with applications to finite element methods. Arch. Ration. Mech. Anal.**1972**, 46, 177–199. [Google Scholar] [CrossRef] - Zagajewski, M.; Behr, A.; Sasse, P.; Wittman, J. Continuously operated miniplant for the rhodium catalyzed hydroformylation of 1-dodecene in a thermomorphic multicomponent solvent system (TMS). Chem. Eng. Sci.
**2014**, 115, 88–94. [Google Scholar] [CrossRef] - Hernández, R.; Engell, S. Modelling and iterative real-time optimization of a homogeneously catalyzed hydroformylation process. In Proceedings of the 26th European Symposium on Computer Aided Process Engineering, Portorož, Slovenia, 12–15 June 2016.
- Hentschel, B.; Kiedorf, G.; Gerlach, M.; Hamel, C.; Seidel-Morgenstern, A.; Freund, H.; Sundmacher, K. Model-based identification and experimental validation of the optimal reaction route for the hydroformylation of 1-Dodecene. Ind. Eng. Chem. Res.
**2015**, 54, 1755–1765. [Google Scholar] [CrossRef] - Schäfer, E.; Brunsch, Y.; Sadowski, G.; Behr, A. Hydroformylation of 1-dodecene in the thermomorphic solvent system dimethylformamide/decane. Phase behavior-reaction performance-catalyst recycling. Ind. Eng. Chem. Res.
**2012**, 51, 10296–10306. [Google Scholar] [CrossRef]

**Figure 2.**Illustration of one MAWQA iteration with noisy data. Surface plot: real profit mapping, mesh plot: quadratic approximation, : regression set-point, : not chosen set-point, ${\u25cf}$: measured profit, : next set-point move, blue contours: model-predicted profit, magenta contours: modifier-adapted profit, dash-dot line: constrained search space.

**Figure 5.**Illustration of one MAWQA iteration with noise-free measurements. Surface plot: real profit mapping, mesh plot: quadratic approximation, : regression set-point, : not chosen set-point, ${\u25cf}$: measured profit, : next set-point move, blue contours: model-predicted profit, magenta contours: modifier-adapted profit, dash-dot line: constrained search space.

**Figure 6.**Illustration of the MAWQA iteration after two explorative moves. Surface plot: real profit mapping, mesh plot: quadratic approximation, : regression set-point, : not chosen set-point, ${\u25cf}$: measured profit, : next set-point move, blue contours: model-predicted profit, magenta contours: modifier-adapted profit, dash-dot line: constrained search space.

**Figure 7.**(

**a**) Thermomorphic multicomponent solvent (TMS) system; (

**b**) Reaction network of the hydroformylation of 1-dodecene. Adapted from Hernández and Engell [10].

**Figure 8.**Modifier-adaptation optimization of the thermomorphic solvent system using finite-difference approximation of the gradients, the left figures show the evolutions of the normalized optimization variables $\mathbf{u}$ (the additional set-point perturbations are represented by the small pulses which are superimposed on the set-point evolutions; the star symbols at the end mark the real optima), the right figures show the evolutions of the cost and the number of plant evaluations (the inset figure zooms in on the cost evolution, and the dashed line marks the real optimum).

**Figure 9.**MAWQA optimization of the thermomorphic solvent system, the left figures show the evolutions of the normalized optimization variables (the additional plant evaluations, i.e., initial probes and unsuccessful moves, are represented by the small pulses which are superimposed on the set-point evolutions; the star symbols at the end mark the real optima), the right figures show the evolutions of the cost and the number of plant evaluations (the inset figure zooms in on the cost evolution, and the dashed line marks the real optimum).

Symbol | Description |
---|---|

k | Index of iteration |

${\mathbf{u}}^{\left(k\right)}$ | Current set-point |

$\nabla {J}_{p}^{\left(k\right)}$ | Gradient vector of the plant objective function at ${\mathbf{u}}^{\left(k\right)}$ |

$\nabla {J}_{m}^{\left(k\right)}$ | Gradient vector of the model-predicted objective function at ${\mathbf{u}}^{\left(k\right)}$ |

${\mathbf{C}}_{p}^{\left(k\right)}$ | Vector of the plant constraint values at ${\mathbf{u}}^{\left(k\right)}$ |

${\mathbf{C}}_{m}^{\left(k\right)}$ | Vector of the model-predicted constraint values at ${\mathbf{u}}^{\left(k\right)}$ |

$\nabla {\mathbf{C}}_{p}^{\left(k\right)}$ | Gradient matrix of the plant constraint functions at ${\mathbf{u}}^{\left(k\right)}$ |

$\nabla {\mathbf{C}}_{m}^{\left(k\right)}$ | Gradient matrix of the model-predicted constraint functions at ${\mathbf{u}}^{\left(k\right)}$ |

Symbol | Description |
---|---|

${C}_{i}$ | Concentration of 1-dodecene, n-tridecanal, iso-dodecene, n-dodecane, iso-aldehyde, decane, and DMF |

${C}_{j}$ | Concentration of CO and ${\mathrm{H}}_{2}$ |

${C}_{j}^{eq}$ | Equilibrium concentration of CO and ${\mathrm{H}}_{2}$ at the G/L interface |

${V}_{R}$ | Reactor volume |

${\dot{V}}_{in}$ | Inflow rate |

${\dot{V}}_{out}$ | Outflow rate |

${C}_{cat}$ | Concentration of the active catalyst |

${M}_{cat}$ | Molar mass of the catalyst |

${\nu}_{i,l}$ | Coefficients of the stoichiometric matrix [11] |

${r}_{l}$ | Reaction rate of the ${l}^{\mathrm{th}}$ reaction |

${k}_{eff}$ | Mass transfer coefficient |

${P}_{j}$ | Partial pressure |

T | Reaction temperature |

${H}_{j,0}$ | Henry coefficient |

${C}_{Rh,precursor}$ | Concentration of the catalyst precursor |

${K}_{cat,1\backslash 2}$ | Equilibrium constants |

${T}_{decanter}$ | Decanter temperature |

${A}_{i,0\backslash 1\backslash 2}$ | Coefficients regressed from experimental data [12] |

${n}_{i,product}$ | Molar flows of the components in the product stream |

${n}_{i,catalyst}$ | Molar flow of the components in the recycled catalyst stream |

${n}_{i,decanter}$ | Molar flow of the components in the decanter inlet stream |

Operating variable | Operating Interval | Initial Set-Point | Real Optimum | Model Optimum |
---|---|---|---|---|

Reactor temperature (${}^{\circ}\mathrm{C}$) | 85∼105 | 95.0 | 88.64 | 85.10 |

Catalyst dosage (ppm) | 0.25∼2.0 | 1.1 | 0.51 | 0.49 |

Gas pressure (bar) | 1.0∼3.0 | 2.0 | 3.0 | 3.0 |

CO fraction | 0.0∼0.99 | 0.5 | 0.55 | 0.61 |

Cost (Euro/kmol) | 899.04 | 761.33 | 818.88 |

Description | Symbol | Value |
---|---|---|

Screening parameter | $\Delta \mathbf{u}$ | 0.1 |

Search space parameter | γ | 3 |

Perturbation step size | $\Delta h$ | 0.1 |

$\mathbf{\Delta}\mathbf{u}$ | Δh | γ | Cost after 30 Eval. | Deviation from the True Optimum (%) | |
---|---|---|---|---|---|

Initial | 0.1 | 0.1 | 3 | 761.5 | 0.02% |

$\uparrow \Delta \mathbf{u}$ | 0.15 | 0.1 | 3 | 762.5 | 0.15% |

$\uparrow \uparrow \Delta \mathbf{u}$ | 0.2 | 0.1 | 3 | 762.9 | 0.21% |

$\uparrow \Delta h$ | 0.1 | 0.15 | 3 | 761.4 | 0.01% |

$\uparrow \uparrow \Delta h$ | 0.1 | 0.2 | 3 | 763.3 | 0.26% |

$\downarrow \gamma $ | 0.1 | 0.1 | 2 | 763.2 | 0.24% |

$\downarrow \downarrow \gamma $ | 0.1 | 0.1 | 1 | 772.1 | 1.14% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gao, W.; Hernández, R.; Engell, S.
A Study of Explorative Moves during Modifier Adaptation with Quadratic Approximation. *Processes* **2016**, *4*, 45.
https://doi.org/10.3390/pr4040045

**AMA Style**

Gao W, Hernández R, Engell S.
A Study of Explorative Moves during Modifier Adaptation with Quadratic Approximation. *Processes*. 2016; 4(4):45.
https://doi.org/10.3390/pr4040045

**Chicago/Turabian Style**

Gao, Weihua, Reinaldo Hernández, and Sebastian Engell.
2016. "A Study of Explorative Moves during Modifier Adaptation with Quadratic Approximation" *Processes* 4, no. 4: 45.
https://doi.org/10.3390/pr4040045