Toward a Comprehensive and Efficient Robust Optimization Framework for (Bio)chemical Processes
Abstract
:1. Introduction
2. Background of ProbabilityBased Robust Optimization
3. Point Estimate Method
3.1. Basics of the Point Estimate Method
3.2. Sampling Strategy for Independent/Correlated Random Variables of Arbitrary Distributions
Algorithm 1 Sampling for correlated random variables 
Initialization: Random variables $\mathbf{\xi}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$, $I\in {\mathbb{R}}^{d\times d}$; $\mathbf{\theta}$ have marginal cumulative density functions $[{F}_{1}({\theta}_{1}),\dots ,{F}_{d}({\theta}_{d})]$ and correlation matrix $\Sigma \in {\mathbb{R}}^{d\times d}$;

4. Moment Method for Approximating Robust Inequality and Equality Constraints
4.1. Categorization of the Constraints
4.2. Robust Formulation of Soft Inequality Constraints
4.3. Robust Formulation of soft Equality Constraints
5. Robust Optimization with the PEM
6. Global Sensitivity Analysis
7. Case Studies
7.1. Case Study 1: A Jacket Tubular Reactor
7.1.1. Robust Design with Parameter Correlation
7.1.2. Performance of the Fourth Moment Method
7.1.3. Impact of Robust Equality Constraints
7.2. Case Study 2: FedBatch Bioreactor for Fermentation of Penicillin
7.2.1. Global Sensitivity Analysis
7.2.2. Robust Optimization
8. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Parameters  Unit  Nominal Value  Uncertainty 

${x}_{1}(0)$    0   
${x}_{2}(0)$    0   
${\alpha}_{kin}$  s^{−1}  0.058  $\mathcal{N}(0.058,{0.0058}^{2})$ 
$\beta $  s^{−1}  0.2  $\mathcal{N}(0.2,{0.02}^{2})$ 
v  ms^{−1}  0.1   
$\gamma $    16.66   
$\delta $    0.25   
Second Moment Method  Fourth Moment Method  

Number of  Independent  Correlated  Independent  Correlated  
violations  470  357  440  385  
Probability  0.047  0.036  0.044  0.039 
Parameters  Unit  Nominal Value  Parameters  Unit  Nominal Value 

${\mu}_{m}$  1/h  0.11  ${m}_{x}$  1/h  0.029 
${K}_{x}$    0.006  ${S}_{f}$  g/L  400 
${\theta}_{m}$  1/h  0.004  t  h  0–80 
${K}_{p}$  g/L  0.0001  $X(0)$  g/L  1 
${K}_{i}$  g/L  0.1  $S(0)$  g/L  0.5 
K  1/h  0.01  $P(0)$  g/L  0 
${Y}_{x}$    0.47  $V(0)$  L  250 
${Y}_{p}$    1.2 
Independent  Correlated  

${\epsilon}_{nq}=1\%$  X  146  35 
S  572  554  
performance  3.63  3.76  
${\epsilon}_{nq}=0.14\%$  X  19  2 
S  378  369  
performance  3.53  3.67 
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Xie, X.; Schenkendorf, R.; Krewer, U. Toward a Comprehensive and Efficient Robust Optimization Framework for (Bio)chemical Processes. Processes 2018, 6, 183. https://doi.org/10.3390/pr6100183
Xie X, Schenkendorf R, Krewer U. Toward a Comprehensive and Efficient Robust Optimization Framework for (Bio)chemical Processes. Processes. 2018; 6(10):183. https://doi.org/10.3390/pr6100183
Chicago/Turabian StyleXie, Xiangzhong, René Schenkendorf, and Ulrike Krewer. 2018. "Toward a Comprehensive and Efficient Robust Optimization Framework for (Bio)chemical Processes" Processes 6, no. 10: 183. https://doi.org/10.3390/pr6100183