Optimization-based predictive control of a simulated moving bed process using an identified model
Introduction
The simulated moving bed (SMB) process was patented in 1961 by Universal Oil Products. It was designed to simulate the solid phase movement of the corresponding true moving bed (TMB) process, in which the fluid and solid phases flow counter-currently to each other. The fundamental features of SMB process have been described in our previous work (Song et al., 2006a), which also included a list of related references.
Several methods have been developed for the determination of a suitable operating condition of the SMB process (Hidajat et al., 1986, Mazzotti et al., 1997, Cauley et al., 2004). Among them, the “local equilibrium theory of chromatography” (Rhee et al., 1971, Rhee et al., 1989, Mazzotti et al., 1997) is now commonly used for the design of SMB processes. The key design parameters are the flow rate ratios , defined as the ratio between net fluid and solid flow rates in each section j of the unit. These four dimensionless parameters arise naturally when the equilibrium theory is applied to the SMB process and are defined asin which denotes the internal flow rate in section , and V, and represent the volume and total porosity of each column, and the switching time, respectively.
In recent years, a number of studies have been reported on the dynamic control strategies and optimization techniques implemented on SMB processes (Marteau et al., 1994, Kloppenburg and Gilles, 1999, Schramm et al., 2001, Klatt et al., 2000, Wang et al., 2003). Off-line optimization cannot be a solution to the safe operation of an SMB process in the presence of unmeasured disturbances, and dynamic control alone cannot optimize the process under changing conditions even though it may guarantee the product purity specifications. These off-line optimization and dynamic control were combined together but no synergistic effect was observed, i.e., if a disturbance is introduced, the operating condition given by off-line optimization is not meaningful any more.
On-line optimization-based automatic control has been proposed (Natarajan and Lee, 2000, Abel et al., 2004; Erdem et al., 2004a, Erdem et al., 2004b) and implemented in an SMB process experimentally (Abel et al., 2004, Erdem et al., 2005). In this work, the production cost of an SMB unit was optimized with the fulfilment of product purity specifications by manipulating the flow rates of solvent, feed, raffinate and extract, respectively. This may be recognized as a novel strategy and exhibits a remarkable performance. Afterwards, another optimization-based control technique has also been applied to the control of an SMB reactor based on the Varicol process (Toumi and Engell, 2004). The first principles models, however, are to be solved numerically with much computational effort although such computational burden can be alleviated by virtue of the model reduction technique (Natarajan and Lee, 2000).
Following a rather different approach, the SMB process was identified by the method of subspace identification (Verhaegen and Dewilde, 1992, Van Overschee and De Moor, 1993, Favoreel et al., 1999) and a predictive controller was designed on the basis of the identified model (Song et al., 2006a). In this last work, input and output variables were chosen with the on-line measurement in mind; i.e., average purities over one switching period were considered as output variables whereas the flow rate ratios in Sections 2 and 3 were selected as input variables. In simulation studies the controller performed quite satisfactorily for the purpose of purity control but its application would certainly be limited in practice, being unable to optimize the separation by accounting for economic considerations.
The present work, therefore, aims to expand the scope of the above approach by including the reciprocal productivity and solvent consumption averaged over one switching period in the output variables whereas our previous work simply aimed at purity control (Song et al., 2006a). In practice, one can maximize the productivity and minimize the solvent consumption in addition to the purity control by choosing the four flow rate ratios as input variables and thus exploit the full economic potential of the SMB unit. This leads to a 4 by 4 multiple-input-multiple-output (MIMO) control problem and requires some additional effort for the design of the predictive controller. A case study dealing with separation of the enantiomers of 1--bi-2-naphthol is considered with the first principles model as the virtual process to generate input/output data to be used in process identification and to carry out simulation studies for the evaluation of the performance of the designed controller.
Section snippets
Mathematical model of SMB process as a virtual plant
The lumped solid diffusion model is used in this work for the description of a standard SMB unit with four sections. A detailed description of the mathematical model has been reported elsewhere (Abel et al., 2004). As a model system, we shall consider the separation of enantiomers of 1--bi-2-naphthol on a 3,5-dinitrobenzoyl phenylglycine bonded to a silica gel stationary phase, using a mixture of heptane–hexane (78:22) as the mobile phase (Migliorini et al., 1999). For the parameters
Brief description of the identified model
In this work an extension of the “model-free approach” suggested by Kadali et al. (2003) is employed because among various subspace identification techniques this approach provides the desired integral action. These authors presented how to get incremental variables in the prediction model by using an integrated noise model. The formulation of a linear input/output data-based prediction model has been well discussed together with its unique features in their article.
The identified prediction
Design of the predictive controller
It is desired to operate the SMB process at an optimum point maintaining constant product purities and/or concentrations in the extract and raffinate, regardless of the occurrence of disturbances, and to adapt the process in case of changes of product specifications. For such control purposes, the property control, a quadratic objective function is usually adopted for model predictive control. The conventional quadratic programming (QP) method is then exploited to calculate the control input
Control performances
Our aim here is to exploit the full potential of the SMB process, which can be fulfilled by maximizing the productivity and minimizing the solvent consumption, without sacrificing the product specification even in the presence of disturbances. The performance of the controller will be evaluated in two typical control problems of practical interest for the SMB process; namely, rejection of disturbances and tracking of set-point changes.
For the control purpose, the prediction and control horizons
Conclusions
A novel scheme for on-line optimization and predictive control is developed and applied to an SMB process operated for the chiral separation of enantiomers of 1--bi-2-naphthol, whose adsorption equilibrium is described by nonlinear isotherms. First, a prediction model is obtained by employing the subspace identification technique with input/output data generated by the first principle model, which is considered as the actual SMB plant. Second, a predictive controller is designed with a
Notation
concentration of component i in mobile phase averaged concentration of species A over one switching period at extract port averaged concentration of species B over one switching period at raffinate port k kth sampling time gain matrix of output prediction model for past data gain matrix of output prediction model for input flow rate ratio in section j averaged purity of component A over one switching period at extract port Pr productivity of SMB unit averaged purity of
Acknowledgement
One of the authors (In-Hyoup Song) gratefully acknowledges the partial support provided by the Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Research Promotion Fund) (KRF-2004-214-D00243).
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