(16a) Dynamic Discrepancy Reduced-Order Modeling and Advanced Control of a Fischer-Tropsch Slurry Bubble Column Reactor

High-fidelity first-principles models have been widely used in industrial process applications due to their ability to forecast dynamic behavior based on the underlying process physics. However, solving these models is computationally expensive, and thus, they may not be suitable for online model predictive control (MPC) applications. To reduce the MPC computational burden, the dynamic system model developed can be represented using a reduced-order model with a discrepancy function, and the model mismatch errors minimized by formulating and solving a parameter estimation problem. However, under stochastic disturbances, the output probability distribution of a reduced-order model can be different from the distribution of the respective high-fidelity model, which can lead to inaccurate predictions and suboptimal controller performance.

To address the challenge of reduced-order models in advanced control under uncertainty, a novel grey-box model identification algorithm for process control is developed by integrating dynamic operability mapping [1] and Bayesian calibration. In the developed framework, the plant is represented by a high-fidelity model, and the reduced-order model dynamic discrepancy function is added to reduce the output dimensions of the dynamic calibration problem [2]. The dynamic discrepancy terms are in the form of Gaussian processes with Bayesian smoothing spline, so the uncertainty propagation can be decomposed into independent discrepancy terms [3]. The reduced-order model is calibrated using a Markov Chain Monte Carlo algorithm, and a Bayesian model selection criterion is implemented to avoid model underfit/overfit.

For the application in focus, a setpoint tracking and disturbance rejection MPC is implemented for a Fischer-Tropsch synthesis process that takes place in a slurry bubble column reactor. The Fischer-Tropsch process is a collection of reactions that process syngas to produce higher-valued hydrocarbons in the liquid products. Because the products of this process include a wide range of hydrocarbons from C1 to C50+ and the product distribution is influenced by the reactor pressure and temperature, controlling such process conditions simultaneously and accurately is crucial to yield the product with desired specifications. The high-fidelity dynamic model is developed based on a hydrodynamic model from the literature [4] and extended to paraffin and olefins products in the slurry phase. The proposed reduced-order model predictive controller is compared to traditional and nonlinear MPC formulations to evaluate its performance, considering different scenarios of setpoint tracking and disturbance rejection.

References:

[1] V. Gazzaneo, J. C. Carrasco, D. R. Vinson, and F. V. Lima, “Process Operability Algorithms: Past, Present, and Future Developments,” Ind. Eng. Chem. Res., vol. 59, no. 6, pp. 2457–2470, Feb. 2020, doi: 10.1021/acs.iecr.9b05181.

[2] K. S. Bhat, D. S. Mebane, P. Mahapatra, and C. B. Storlie, “Upscaling Uncertainty with Dynamic Discrepancy for a Multi-Scale Carbon Capture System,” Journal of the American Statistical Association, vol. 112, no. 520, pp. 1453–1467, Oct. 2017, doi: 10.1080/01621459.2017.1295863.

[3] B. J. Reich, C. B. Storlie, and H. D. Bondell, “Variable Selection in Bayesian Smoothing Spline ANOVA Models: Application to Deterministic Computer Codes,” Technometrics, vol. 51, no. 2, pp. 110–120, May 2009, doi: 10.1198/TECH.2009.0013.

[4] C. Maretto and R. Krishna, “Modelling of a Bubble Column Slurry Reactor for Fischer–Tropsch Synthesis,” Catalysis Today, vol. 52, no. 2–3, pp. 279–289, Sep. 1999, doi: 10.1016/S0920-5861(99)00082-6.