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261483 Online Integration of Scheduling and Control for Cyclic Production in CSTR

Integration of scheduling and control achieves a better overall performance than the conventional sequential approaches because a good tradeoff can be made between the two problems [1-3]. The integrated problem is a mixed integer dynamic optimization (MIDO) problem [4, 5] where the differential equations describing the control problem are incorporated into the scheduling problem that includes discrete variables. The resulting MIDO problem might be quite computationally expensive, and as a consequence, most integration methods can only be applied offline. However, online implementation of the integration method is critical to deal with various uncertainties and disturbances in the production system. Yet the online solution is rather more challenging because there are stringent requirements on the computational efficiency. It is the goal of the paper to propose a novel framework for integrating scheduling and control and a fast computational strategy capable for online applications.

Different from most existing integration methods where the trajectory of the control variable (the process input) is determined directly in the integrated problem, a parameterized controller is considered in the proposed method. The controller designs, rather than the values of the control variables, are determined simultaneously with the scheduling decisions. The direct calculation of the control variable will incur the risk of open loop if the control variable cannot be updated immediately by solving the integrated problem in the sampling period. In contrast, the proposed integration structure is more robust since there is always a closed-loop controller working in real time even if the controller parameters are not updated by the integrated problem. Due to the existence of the real-time controller, the integrated problem does not need to be solved in real time, but can be solved in a relatively large time scale, e.g. once per production period. The most popular parameterized controller in the process industry is the PI controller [6, 7] and it is adopted in this work.

Efficient solution to the formulated MIDO problem is the key to the online implementation of the integrated method. Therefore, we propose an efficient computational strategy, which significantly reduces the model complexity by decomposing the dynamic optimization for the controller design from the scheduling model. For each potential transition, the range of the transition time is discretized by a number of sampling points and a controller is designed to minimize the transition cost at each transition time point. A set of controller candidates is then calculated and stored. Based on the controller candidates generated offline, the integrated MIDO problem is approximated by a simultaneous scheduling and controller selection problem, which is a mixed-integer nonlinear fractional programming problem. The objective function of this problem is the ratio of a convex quadratic function to a linear function and all constraints are linear, because the equations related to the dynamic system are all removed after the decomposition. The derived problem has a much smaller size than the original integrated problem and it can be globally optimized using a fast computational method based on the Dinkelbach’s algorithm [8, 9].

The novelties in the proposed integration framework are summarized as

- A novel MIDO framework for scheduling and control which calculates the controller parameters in the integrated problem rather than the direct value of the control variable.
- A fast computational strategy based on the decomposition of control problems from the integrated problem where all dynamic optimization problems are solved offline.
- An efficient global optimization algorithm based on the model properties and Dinkelbach’s algorithm.

The advantage of the proposed method is demonstrated through four case studies [10]. The proposed method based on the Dinkelbach’s algorithm is compared with the solvers of BARON, SBB, and DICOPT. The results show that the proposed method globally optimizes the simultaneous scheduling and controller selection problem with the shortest computational time.

Comparisons with the traditional sequential method show that the proposed integration method can reduce the overall production cost rate because it provides a better balance between the inventory cost rate and the transition cost rate. Comparing with another integration method which solves the MIDO using the simultaneous approach, the proposed method is computationally much more efficient, especially for cases with a large number of products. The proposed method addresses the challenges in the online implementation of the integrated scheduling and control problems. The results also show that the online solution allows the production system to quickly respond to various uncertainties and disturbances.

**Reference**

[1] I. Grossmann, "Enterprise-wide optimization: A new frontier in process systems engineering," *AIChE Journal, *vol. 51, pp. 1846-1857, 2005.

[2] I. Harjunkoski, R. Nystrom, and A. Horch, "Integration of scheduling and control-Theory or practice?," *Computers & Chemical Engineering, *vol. 33, pp. 1909-1918, 2009.

[3] E. Munoz, E. Capon-Garcia, M. Moreno-Benito, A. Espuna, and L. Puigjaner, "Scheduling and control decision-making under an integrated information environment," *Computers & Chemical Engineering, *vol. 35, pp. 774-786, 2011.

[4] R. J. Allgor and P. I. Barton, "Mixed-integer dynamic optimization I: problem formulation," *Computers & Chemical Engineering, *vol. 23, pp. 567-584, 1999.

[5] V. Bansal, V. Sakizlis, R. Ross, J. D. Perkins, and E. N. Pistikopoulos, "New algorithms for mixed-integer dynamic optimization," *Computers & Chemical Engineering, *vol. 27, pp. 647-668, 2003.

[6] K. J. Astrom, H. Panagopoulos, and T. Hagglund, "Design of PI controllers based on non-convex optimization," *Automatica, *vol. 34, pp. 585-601, 1998.

[7] S. Skogestad, "Simple analytic rules for model reduction and PID controller tuning," *Journal of Process Control, *vol. 13, pp. 291-309, 2003.

[8] W. Dinkelbach, "On nonlinear fractional programming," *Management Science, *vol. 13, pp. 492-498, 1967.

[9] F. Q. You, P. M. Castro, and I. E. Grossmann, "Dinkelbach's algorithm as an efficient method to solve a class of MINLP models for large-scale cyclic scheduling problems," *Computers & Chemical Engineering, *vol. 33, pp. 1879-1889, 2009.

[10] Y. Chu and F. You, "Integration of scheduling and control with online closed-loop implementation: Fast computational strategy and large-scale global optimization algorithm," submited to* Computers & Chemical Engineering, *2012.

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