Max Planck Institute for Dynamics of Complex Technical Systems
Model Order Reduction for Chromatographic Processes
Preparative liquid chromatography is an efficient separation technique, and has gained great popularity in many industrial fields such as petrochemical, fine chemical and pharmaceutical industries. In this project, we focus on two types of chromatography: batch chromatography and simulated moving bed (SMB) chromatography. They pose great challenges to (parametric) model order reduction ((P)MOR), such as switching behavior, parameter dependence, and nonlinearity. We have proposed three types of (P)MOR methods for chromatography processes, namely proper orthogonal decomposition (POD) methods, Krylov methods and reduced basis (RB) methods. These methods prove to be both accurate and efficient in accelerating analysis of preparative liquid chromatography.
For both cases, the process is generally governed by highly coupled and (non)linear partial differential equations (PDEs) posing a significant challenge to model-based research activities, such as simulation, parameter estimation, optimization as well as control problems. In cooperation with the group of PCF headed by Prof. Seidel-Morgenstern, this project focuses on systematic studies of model order reduction (MOR) schemes for both batch and SMB chromatographic processes.
1. POD approach
Proper orthogonal decomposition (POD) is a powerful MOR tool for linear and nonlinear systems and has been successfully employed in a wide variety of applications to accelerate model-based analysis. This motivates us to employ it to construct ROMs for the SMB process. We mainly concern the computation of the cyclic steady state (CSS), which is of central interest for chemical engineers since it is used for production purposes. We propose to first collect snapshots at the CSS of the original full-order model. Then based on these snapshots, we derive POD-based ROMs. Numerical results show that a ROM with a significantly lower order is capable of reproducing the original CSS dynamics [1,5].
2. Krylov subspace method
Currently, we restrict our efforts to SMBs with linear adsorption isotherms. We have proposed a straightforward method with two strategies for further speedup.
The Straightforward Method [2,3,4] builds a ROM for each period, whose initial condition was carefully considered to avoid projection error during switching.
The Partial-Update Strategy [3,4] precomputes most of the basis, and update only a few basis vectors for each period.
The Subspace-Exploiting Strategy [2,4] employs the Pseudo-CSS's to achieve fast convergence, thanks to the much fewer ROMs required to be built to compute the CSS.
3. Reduced basis method
Reduced basis method (RBM) is a robust parametric MOR technique widely used for linear or nonlinear problems in many applications. The main task in the development of RBM for specific applications is to propose an a posteriori error estimation, which is used to indicate the accuracy of the ROM to guide the parameter sampling during the basis extension process and to certify the ROM for any given parameter value.
An adaptive technique of snapshot selection is proposed to reduce the cost of the generation of the ROM [7,8].
A primal-only output error bound is derived in the vector space for nonlinear evolution equations .
A primal-dual output error estimation is proposed for nonlinear evolution equations, which is sharper than the primal-only one and is inexpensive .
4. ROM-based applications
We apply the proposed (P)MOR methods to three types of applications:
Simulation [1-9]. State-space simulation of the high-order state-space chromatography models is very expensive, especially for CSS computation of SMB processes. All proposed methods show high accuracy and significant speedup.
Operating parameter optimization [1-3,5-9]. The proposed methods accelerate locating the best operating parameters, which maximize the yield rate under the constraints of physics and purity. Operating parameter optimization requires simulations repeatedly and therefore, is computationally more expensive. We embed the proposed methods to both gradient-based approaches and gradient-free approaches.
Uncertainty Quantification [2,4]. Uncertainty quantification studies the influence of the uncertain parameters on the outputs. Like operating parameter optimization, non-intrusive uncertainty quantification is also multi-query analysis, which can benefit a lot from (P)MOR in reducing the analysis time.
S. Li, L. Feng, P. Benner, and A. Seidel-Morgenstern, Efficient optimization of simulated moving bed processes using reduced order models, Comput. Aided Chem. Eng., vol. 30, pp. 1232-1236, 2012.
S. Li, Y. Yue, L. Feng, P. Benner, and A. Seidel-Morgenstern, Model reduction for linear simulated moving bed chromatography systems using Krylov-subspace methods, AIChE J., vol. 60, no. 11, pp. 3773–3783, 2014.
Y. Yue, S. Li, L. Feng, A. Seidel-Morgenstern, and P. Benner, Accelerating cyclic steady state optimization of linear simulated moving bed chromatography models by Krylov-type model order reduction methods, in Engineering Optimization IV, Lisbon, Portugal, 2014, pp. 453–458.
Y. Yue, S. Li, L. Feng, A. Seidel-Morgenstern, and P. Benner, Efficient model reduction of SMB chromatography by Krylov-subspace method with application to uncertainty quantification, presented at the 24th European Symposium on Computer Aided Process Engineering, Budapest, 2014, pp. 925–930.
S. Li, L. Feng, P. Benner, and A. Seidel-Morgenstern, Using surrogate models for efficient optimization of simulated moving bed chromatography, Computers & Chemical Engineering, vol. 67, pp. 121–132, 2014.
P. Benner, L. Feng, S. Li, and Y. Zhang, Reduced-order modeling and ROM-based optimization of batch chromatography, In A. Abdulle, S. Deparis, D. Kressner, F. Nobile, and M. Picasso (Eds.), Numerical Mathematics and Advanced Applications - ENUMATH 2013, Lecture Notes in Computational Science and Engineering, Vol. 103, pp. 427-435, Springer International Publishing, 2015.
Y. Zhang, L. Feng, S. Li, and P. Benner, An efficient output error bound for model order reduction of parametrized evolution equations, FAC-PapersOnLine, 8th Vienna International Conference on Mathematical Modelling MATHMOD 2015, 18-20 February, 2015. Vienna, Austria, Vol. 48, pp. 9-10, 2015.
Y. Zhang, L. Feng, S. Li, and P. Benner, Accelerating PDE constrained optimization by the reduced basis method: application to batch chromatography, Int. J. Numer. Meth. Engng, 104 (2015), pp. 983-1007, DOI: 10.1002/nme.4950, available online 3 June 2015.
Y. Zhang, L. Feng, S. Li, and P. Benner, An efficient output error estimation for model order reduction of parametrized evolution equations, SIAM J. Sci. Comput., accepted on 22 September 2015.
Y. Zhang, L. Feng, S. Li, and P. Benner. A new error estimation for model order reduction of parametrized evolution equations; 8th Vienna International Conference on Mathematical Modelling (MATHMOD 2015) MATHMOD 2015; Vienna, Austria; February 18 - 20, 2015.
Y. Yue, S. Li, L. Feng, A. Seidel-Morgenstern, and P. Benner. (2014). Accelerating uncertainty quantification of linear simulated moving bed chromatography models by Krylov-type (parametric) model order reduction methods. The 18th European Conference on Mathematics for Industry. Taormina, Italy, 9-13 June 2014.
Y. Yue, S. Li, L. Feng, A. Seidel-Morgenstern, and P. Benner. (2014). Efficient model reduction of SMB chromatography by Krylov-subspace method with application to uncertainty quantification. The 24th European Symposium on Computer Aided Process Engineering (ESCAPE), Budapest, Hungary, 15-18 June 2014.
Y. Yue, S. Li, L. Feng, A. Seidel-Morgenstern, P. Benner and Karl Meerbergen. (2014). Accelerating optimization of dynamical systems by Krylov-type model order reduction methods. 4-th International Conference on Engineering Optimization (EngOpt), Lisbon, Portugal, 8-11 September, 2014.
Y. Zhang, L. Feng, S. Li, and P. Benner. Reduced-order modeling and ROM-based optimization of batch chromatography; European Numerical Mathematics and Advanced Applications (ENUMATH), Lausanne, Switzerland; August 26 - 30, 2013
Y. Zhang, L. Feng, and P. Benner. An adaptive technique for snapshot selection during model reduction for parameterized evolution equations; Reduced Basis Summer School 2012; Freudenstadt; August 28 - 31, 2012.
Suzhou Li, Lihong Feng, Peter Benner and Andreas Seidel-Morgenstern, Using surrogate models for efficient optimization of simulated moving bed chromatography, 14th International Symposium on Preparative and Industrial Chromatography and Allied Techniques, Brussels, Belgium, 2012.