(745e) A Hybrid Dynamic Optimization Strategy for Systems with Steep Fronts

Dynamic optimization problems are often solved using either the sequential or simultaneous approach. In the sequential method only the control variables are discretized and the states are calculated using a DAE solver [1] while with the simultaneous strategy both states and controls are fully discretized [2]. The discretization of the state trajectories is often performed using an implicit Runge-Kutta method, of which the simplest one is the Backward Euler method; however more complex schemes with variable step size can also be implemented [2]. The controls can be discretized at the same level as the state variables or using a coarser grid, and within each interval the control values can be approximated using a low order polynomial [1]. A third strategy used for solving dynamic optimization methods is multiple shooting, which sits between the two aforementioned methods. In this approach the time horizon is again divided into intervals and within each interval the states trajectories are obtained using a DAE solver, while continuity of the states at the interval boundaries is enforced via equality constraints [2], [3]. Multiple shooting is normally classified as a simultaneous strategy [3].

Sequential methods are easier to implement and yield smaller, denser NLPs. Because the DAE system and sensitivity equations have to be integrated at every NLP iteration, the solution time can increase prohibitively for large problems [3]. Another pitfall of the sequential method is that constraints can not be imposed directly on the states, since the they are not available to the NLP [4]. Due to the full discretization, simultaneous methods yield large and sparse NLPs that can be efficiently solved using state-of-the-art NLP solvers [3]. Furthermore, bounds can be directly imposed on the discretized states [5] and the DAE system is only solved once at the optimal point decreasing the time expended evaluating non-optimal solutions compared with sequential methods [5]. Despite several advantages, the simultaneous strategy requires initialization of all the states and not only the controls as with the single shooting method, and for systems with steep profiles a finer discretization may be necessary leading to very large NLPs that may become prohibitively time consuming to solve.

Here, we propose a hybrid optimization approach for systems with steep fronts or fast changing dynamics. The interval with rapid change of the states is integrated using a DAE solver as with the single shooting method, and the remaining intervals, where the states trajectories are smoother, are fully discretized as with the simultaneous strategy. A motivating application for this approach is the Basic Oxygen Furnace (BOF), a semi-batch process that is widely used in steel processing, where fast-changing dynamics occur in the first few seconds of operation. The steep front is due to the temperature gradient at the interface between hot metal at 1600 K and solid scrap metal which is initially at ambient temperature (300 K), and also due to the small mass of slag at the beginning of the process relative to the mass of flux and iron ore additions. More details about the mathematical model can be found in [6]. Dynamic optimization with integration over an initial time interval performed using a variable step size DAE solver [7], and the remainder of the time horizon fully discretized using the Backward Euler method was found to outperform both the sequential and simultaneous optimization approaches. The hybrid method was implemented using the Python front-end of CasADi [8] and the resulting optimization problem solved using IPOPT [9]. In this paper, we present a general formulation for this hybrid sequential-simultaneous strategy, and demonstrate its implementation on a simple batch reactor problem as well as the more complex BOF system. We describe its implementation using readily available software tools such as CasADi, and investigate it performance relative to the single-shooting and fully discretized simultaneous approaches under various conditios.

References

[1] V. S. Vassiliadis, R. W. H. Sargent, and C. C. Pantelides, “Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints,” 1994.

[2] S. Kameswaran and L. T. Biegler, “Simultaneous dynamic optimization strategies: Recent advances and challenges,” Comput. Chem. Eng., vol. 30, no. 10–12, pp. 1560–1575, Sep. 2006.

[3] L. T. Biegler, “An overview of simultaneous strategies for dynamic optimization,” Chem. Eng. Process. Process Intensif., vol. 46, no. 11, pp. 1043–1053, Nov. 2007.

[4] V. S. Vassiliadis, R. W. H. Sargent, and C. C. Pantelides, “Solution of a Class of Multistage Dynamic Optimization Problems. 2. Problems with Path Constraints,” 1994.

[5] L. T. Biegler and I. E. Grossmann, “Retrospective on optimization,” Comput. Chem. Eng., vol. 28, no. 8, pp. 1169–1192, Jul. 2004.

[6] D. Dering, C. Swartz, and N. Dogan, “Dynamic Modeling and Simulation of Basic Oxygen Furnace (BOF) Operation,” Processes, vol. 8, no. 4, p. 483, Apr. 2020.

[7] R. Serban, C. Petra, and A. C. Hindmarsh, “User Documentation for IDAS v1.2.2 (Sundials v2.6.2),” vol. 1, 2012.

[8] J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, “CasADi: a software framework for nonlinear optimization and optimal control,” Math. Program. Comput., 2018.

[9] A. Wächter and L. T. Biegler, “On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,” Math. Program., vol. 106, no. 1, pp. 25–57, 2006.