(256h) Nonlinear Model Predictive Control Of Cryogenic Air Separation Columns

Cryogenic distillation is used to produce large quantities of purified nitrogen, oxygen, argon and rare gases for the steel, chemical, food processing, semiconductor and health care industries. Current state-of-the-art control technology in the air separation industry is based on linear dynamic models and linear model predictive control [1]. Despite the very high product purities required, linear control technology has proven to be successful because cryogenic distillation columns traditionally operate over a small range of production rates. Deregulation of the electric utility industry is expected to lead to frequent and unpredictable changes in the cost of electricity, which will mandate fundamental changes in the operating philosophy of air separation plants. Large changes in production rate and more frequent startups/shutdowns will be required to take full advantage of time-varying utility rates. Column nonlinearities will become much more pronounced under these operating conditions, and some type of nonlinear control will be necessary to achieve satisfactory performance.

Nonlinear model predictive control (NMPC) is a well known extension of linear model predictive control where a nonlinear model is used to describe the process dynamics. A variety of numerical algorithms are available to solve the NMPC optimization problem. The simultaneous solution approach involves temporal discretization of the dynamic model equations to produce a set of nonlinear algebraic equations (AEs) that are posed as equality constraints in the NMPC optimization problem [2]. The decision variables are current and future values of the manipulated inputs and state variables. Because control moves are generated by real-time solution of the resulting nonlinear program at each sampling period, computational effort is inextricably linked to the complexity of the controller design model.

Fundamental models of distillation columns are comprised of stage-by-stage mass and energy balances combined with vapor-liquid equilibrium relations expressed for each stage. A distillation column with N equilibrium separation stages and n chemical species is commonly modeled with N(n+2) nonlinear ordinary differential equations (ODEs) describing the species compositions, liquid and vapor holdups, and enthalpy on each stage. For example, the 59-stage cyrogenic distillation column consider in this contribution has been modeled with 180 ODEs and 137 AEs. Such nonlinear dynamic models are generally viewed as being too complex to be effectively utilized for real-time control due to their high dimensionality. While multiple-shooting solution techniques allow the application of NMPC to distillation column models of moderate complexity [3], there remains considerable motivation to develop reduced-order dynamic models that provide a more favorable tradeoff between prediction accuracy and computational effort.

In this contribution, a nonlinear model predictive control strategy based on a reduced-order model of a cryogenic distillation column is developed and evaluated. The column considered has multiple feed and product withdrawal streams, further complicating the design of an effective control system. A detailed stage-by-stage balance model is derived to serve as a surrogate plant in our simulation studies and to provide a rigorous basis for reduced-order model development. A nonlinear compartmental model [4] is derived by exploiting time-scale separations in the detailed model. The temporally discretized compartmental model has increased sparsity as compared to the detailed model, which facilitates efficient solution of the NMPC problem. A number of real-time implementation strategies are employed to achieve converged NMPC solutions within the controller sampling interval. A nonlinear receding horizon estimator is used to generate predictions of unmeasured stage compositions and temperatures such that the NMPC controller is implementable with industrially available measurements. State-feedback and output-feedback simulations are performed to compare the performance of the NMPC controller to a conventional linear model predictive controller and to assess the potential utility of NMPC technology for cryogenic and other high-purity distillation columns.

Acknowledgements

Financial support from the Praxair and the National Science Foundation (CTS-0241211) is gratefully acknowledged.

References

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2. E. S. Meadows and J. B. Rawlings. Model predictive control. In M. A. Henson and D ~E. Seborg, editors, Nonlinear Process Control, chapter 5, pages 233-310. Prentice-Hall, Englewood Cliffs, NJ, 1997.

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4. S. Bian, S. Khowinij, M. A. Henson, P. Belanger, and L. Megan. Compartmental modeling of high purity air separation columns. Computers and Chemical Engineering, 25:2096-2109, 2005.