(433g) A Multi-Scenario Stochastic Framework for Dynamic Real Time Optimization Under Uncertainty with Embedded Closed-Loop MPC

Modern chemical plants operate in a paradigm where adaptability is paramount to ensure smooth, safe, and effective operation. This is often the result of uncertainty in the plant behavior, which can arise from different sources depending on the specific application, including model uncertainty, parametric uncertainty, upstream disturbances, or uncertainty in economic trends such as product demand. A common technique to attempt to maintain effective control and operation of a chemical plant is the use of a control hierarchy. This allows tasks of different fidelity and frequencies to be performed separately at different levels of the hierarchy, with more frequent and immediate problems handled at the lower levels and longer time scale problems at the upper levels. These control layers can include, among others, (Dynamic) Real-Time Optimization (D-RTO), Model Predictive Control (MPC), and Proportional-Integral-Derivative (PID) Control. This paper is primarily concerned with the RTO and MPC layers.

The use of multiple control layers and its effects on the overall performance of a plant has been studied extensively in recent years. Tosukhowong et. al. (2004) took advantage of time scale differences between the DRTO and MPC layers by performing a reduced order model DRTO less frequently than an underlying linear MPC and found that this two-layer system outperformed an MPC in standalone operation. Kadam et. al. (2002) employed a trigger analysis strategy where the RTO problem was only solved under certain plant conditions when a new steady state needed to be calculated and it was appropriate to do so. Ellis and Christofides (2014) included a dynamic model and control criteria in the DRTO to improve performance in unstable systems. Engell (2007) provides an excellent review on feedback control methods, with particular emphasis on multi-level control schemes. Swartz and Kawajiri (2019) review more recent advancements in dynamic plant operation and optimization.

A common strategy for handling uncertainty in process operation is to directly model that uncertainty in one of the control layers. Robust MPC is the most common version of this strategy. Bemporad and Morari (1999) provide an excellent review of many of the relatively early strategies in this field. Of particular interest to this work is the more recent advancement in multi-scenario robust MPC, specifically the paradigm developed by Dadhe and Engell (2008) and Engell (2009). In these works, the uncertain plant behavior is modelled as discrete scenarios. Lucia et. al. (2012) employ this approach with a nonlinear MPC to improve the fidelity and performance of the controller.

With multiple levels of control being active in a given chemical plant, the true plant behavior will depend on both the plant dynamics and the specifics of the control strategies being used. Jamaludin and Swartz (2015) attempted to account for this by modelling the plant behavior and the behavior of an underlying MPC directly in a single DRTO problem. The resulting closed-loop (CL) DRTO was solved by converting the MPC into its first order Karush-Kuhn-Tucker optimality conditions and including these conditions as constraints into the DRTO problem. It was then shown that this CL DRTO was able to outperform a DRTO which did not account for the underlying MPC behavior. Li and Swartz (2019) extended this approach to distributed MPC systems.

In this work, the robust multi-scenario approach used previously at the MPC level is instead employed at the DRTO level in conjunction with CL prediction of an underlying MPC. This transfers the uncertainty analysis to the DRTO level, allowing for economic optimization under uncertainty. It also predicts the MPC behavior under different plant scenarios, allowing the DRTO to determine optimal set-points for the MPC-plant system for a range of possible plant behaviors. The CL DRTO uses a nonlinear plant model for improved plant prediction while it is assumed the MPC uses a linear plant model for faster solution times. This allows the DRTO to exactly predict the MPC behavior with its first order KKT conditions as the linear MPC is a convex quadratic problem. The inclusion of the KKT conditions as constraints in the DRTO results in a single-level Mathematical Program with Complementarity Constraints (MPCC). In this work, the complementarity constraints are solved using an exact penalty approach (Ralph and Wright, 2004).

The robust multi-scenario CL DRTO is then tested in two case studies against a single-scenario CL DRTO in the presence of parametric uncertainty. The first case study is single-input-single-output (SISO) with one irreversible reaction occurring in a jacketed CSTR. The robust CL DRTO showed small but significant economic improvements in this case study, with more substantial improvements in reduction of constraint violation. Additionally, this case study tested both a rigorous MPC prediction at the DRTO level and an input clipping approximation of the MPC (Jamaludin and Swartz, 2017; Li and Swartz, 2018). It is shown that the input clipping approximation method resulted in identical behavior to the rigorous method when the inputs do not saturate, which is expected given the nature of the approximation, and in very similar behavior to the rigorous method when the inputs do saturate. Therefore, this approximation is determined to be a viable method for improved computation time.

The second case study is multi-input-multi-output (MIMO) with a polymerization reaction occurring in a jacketed CSTR. The input clipping approximation introduced in the SISO case study is used for this entire case study. The robust CL DRTO shows substantially improved economic performance over the single-scenario CL DRTO as well as significantly reduced constraint violation when the robust CL DRTO considers one uncertain parameter with three possible plant scenarios. This is considered a standard methodology as three scenarios with one source of uncertainty has been commonly used in robust MPC algorithms, such as in Lucia et. al. (2012). Moving beyond this standard, the robust CL DRTO is also tested with more scenarios with the total range of possible parameter values remaining constant. In these cases, the robust CL DRTO economic performance continues to improve, though with diminishing returns, at greatly increased computational cost. Finally, the effect of multiple sources of uncertainty is tested, specifically with two uncertain parameters considered by the robust CL DRTO. For this version, nine scenarios are used so that every possible pairing of three scenarios per uncertain parameter is represented. The robust CL DRTO is once again shown to outperform the single-scenario CL DRTO in terms of economics and constraint violation. Furthermore, it does so by a much larger margin than in the single uncertain parameter version, suggesting that the performance of the robust CL DRTO relative to the single-scenario improves as the degree of uncertainty in the plant behavior increases.

The results of these case studies indicate that the robust multi-scenario approach for a CL DRTO algorithm is an effective method for mitigating plant uncertainty at the DRTO level. This furthers the field of robust real-time optimization and increases the available methods for effective control and optimization of chemical plants with uncertain behavior.

References

Bemporad, A. and Morari, M., 1999. Robust model predictive control: A survey. In Robustness in Identification and Control (pp. 207-226). Springer, London.

Dadhe, K. and Engell, S., 2008. Robust nonlinear model predictive control: A multi-model nonconservative approach. In Book of Abstracts, Int. Workshop on NMPC, Pavia (Vol. 24).

Ellis, M. and Christofides, P.D., 2014. Optimal time-varying operation of nonlinear process systems with economic model predictive control. Industrial & Engineering Chemistry Research, 53(13), pp.4991-5001.

Engell, S., 2007. Feedback control for optimal process operation. Journal of Process Control, 17(3), pp.203-219.

Engell, S., 2009. Online optimizing control: The link between plant economics and process control. In 10th International Symposium on Process Systems Engineering (Vol. 27, pp. 79-86). Elsevier.

Jamaludin, M.Z. and Swartz, C.L.E., 2015. A bilevel programming formulation for dynamic real-time optimization. IFAC-PapersOnLine, 48(8), pp.906-911.

Jamaludin, M.Z. and Swartz, C.L.E., 2017. Approximation of closed-loop prediction for dynamic real-time optimization calculations. Computers & Chemical Engineering, 103, pp.23-38.

Kadam, J.V., Schlegel, M., Marquardt, W., Tousain, R.L., Van Hessem, D.H., van den Berg, J. and Bosgra, O.H., 2002. A two-level strategy of integrated dynamic optimization and control of industrial processes—a case study. In Computer Aided Chemical Engineering (Vol. 10, pp. 511-516). Elsevier.

Li, H. and Swartz, C.L.E., 2018. Approximation techniques for dynamic real-time optimization (DRTO) of distributed MPC systems. Computers & Chemical Engineering, 118, pp.195-209.

Li, H. and Swartz, C.L.E., 2019. Dynamic real-time optimization of distributed MPC systems using rigorous closed-loop prediction. Computers & Chemical Engineering, 122, pp.356-371.

Lucia, S., Finkler, T., Basak, D. and Engell, S., 2012. A new robust NMPC scheme and its application to a semi-batch reactor example. IFAC Proceedings Volumes, 45(15), pp.69-74.

Ralph, D. and Wright, S.J., 2004. Some properties of regularization and penalization schemes for MPECs. Optimization Methods and Software, 19(5), pp.527-556.

Swartz, C.L.E. and Kawajiri, Y., 2019. Design for dynamic operation-A review and new perspectives for an increasingly dynamic plant operating environment. Computers & Chemical Engineering, 128, pp.329-339.

Tosukhowong, T., Lee, J.M., Lee, J.H. and Lu, J., 2004. An introduction to a dynamic plant-wide optimization strategy for an integrated plant. Computers & chemical engineering, 29(1), pp.199-208.