(364d) Comparing Formulations for Global Flowsheet Optimization with Simultaneous Heat Integration

Conventional design methodologies typically first conduct flowsheet optimization and subsequently heat integration. However, this results in suboptimal solutions to the overall problem of designing an optimal, heat-integrated process. The seminal work of Duran and Grossman [1] provides a way to integrate pinch analysis [2] in the flowsheet optimization problem: it includes constraints that describe the grand composite curve of the heat-integrated flowsheet, and thereby allows to consider the minimum hot and cold utility demand in the objective function (e.g., regarding their costs) for a given minimum temperature approach. This method has the potential to provide processes with better performance than when conducting flowsheet optimization and pinch analysis separately.

However, considering heat integration during flowsheet optimization comes with computational challenges: Even flowsheet optimization itself is almost inevitably a nonconvex optimization problem [3]. As such, it requires deterministic global optimization to ensure globally optimal solution, which is, however, computationally expensive. Including heat integration via pinch analysis in flowsheet optimization problems makes the problem even more challenging in two respects: First, when formulated in an equation-based way as originally proposed [1], it results in additional variables and constraints. This is a drawback regarding the poor scaling of deterministic global solvers with problem size. Second, the constraints describing the grand composite curve introduce a potentially large number of nonsmooth terms, namely the maximum of two functions. This prohibits or at least complicates the use of gradient-based local solvers. In the context of global optimization, these are often used for providing upper bounds on the optimal objective. Additionally, the nonsmooth terms are also additional nonconvexities and thus may cause weak relaxations in global solvers.

In this contribution, we compare several existing and new formulations for simultaneous heat integration and flowsheet optimization regarding their computational performance in deterministic global optimization. This includes the original formulation [1] as well as equivalent nonsmooth reformulations, smoothing approaches that approximate the max function with differentiable functions [1,4], and mixed-integer formulations [5,6]. Additionally, we investigate the effect of reformulating the problem from an equation-oriented, full-space formulation to reduced-space formulations, which we have shown to be beneficial for flowsheet optimization (without heat integration) [7]. In these reduced-space formulations, optimization variables are eliminated from the problem using equality constraints. This way, heat integration can be considered in flowsheet optimization with few, one, or even no additional variables beyond the pure flowsheeting problem.

We compare the formulations on a number of case studies using different global solvers. Depending on the solver and case study, either nonsmooth or mixed-integer formulations were the fastest to solve. This suggests that for global optimization, where the performance of the local, gradient-based solvers may not be as critical, smoothing approaches do not necessarily facilitate solution. For isothermal streams (e.g., condensing or evaporating single-component streams), mixed-integer formulations tended to perform best. For the considered case studies, the reduced-space formulations significantly reduced computational time, especially for our open-source solver MAiNGO [8] that can exploit them efficiently via the use of McCormick relaxations [9,10].

References

[1] M.A. Duran & I.E. Grossmann, AIChE J. 32 (1986), 123.

[2] B. Linnhoff, J.R. Flower, AIChE J. 24 (1978), 633.

[3] L.T. Biegler, I.E. Grossmann, A.W. Westerberg, Systematic Methods of Chemical Process Design (1997), Pearson Education.

[4] A.W. Dowling, L.T. Biegler, Comput. Chem. Eng. 72 (2015), 3.

[5] I.E. Grossmann, H. Yeomans, Z. Kravanja, Comput. Chem. Eng. 22 (1998), 157.

[6] A. Wechsung, A. Aspelund, T. Gundersen, P.I. Barton, AIChE J. 57 (2011), 2090.

[7] D. Bongartz, A. Mitsos, J. Global Optim. 69 (2017), 761.

[8] D. Bongartz, J. Najman, S. Sass, A. Mitsos, MAiNGO – McCormick-based Algorithm for mixed-integer Nonlinear Global Optimization. Technical Report, RWTH Aachen University (2018). http://permalink.avt.rwth-aachen.de/?id=729717. The open-source version is available at https://git.rwth-aachen.de/avt-svt/public/maingo.

[9] G.P. McCormick, Math. Prog. 10 (1976), 147.

[10] A. Mitsos, B. Chachuat, P.I. Barton, SIAM J. Optim. 20 (2009), 573.