# Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations

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## Abstract

Deterministic global methods for flowsheet optimization have almost exclusively relied on an equation-oriented formulation where all model variables are controlled by the optimizer and all model equations are considered as equality constraints, which results in very large optimization problems. A possible alternative is a reduced-space formulation similar to the sequential modular infeasible path method employed in local flowsheet optimization. This approach exploits the structure of the model equations to achieve a reduction in problem size. The optimizer only operates on a small subset of the model variables and handles only few equality constraints, while the majority is hidden in externally defined functions from which function values and relaxations for the objective function and constraints can be queried. Tight relaxations and their subgradients for these external functions can be provided through the automatic propagation of McCormick relaxations. Three steam power cycles of increasing complexity are used as case studies to evaluate the different formulations. Unlike in local optimization or in previous sequential approaches relying on interval methods, the solution of the reduced-space formulation using McCormick relaxations enables dramatic reductions in computational time compared to the conventional equation-oriented formulation. Despite the simplicity of the implemented branch-and-bound solver that does not fully exploit the tight relaxations returned by the external functions but relies on further affine relaxation at a single point using the subgradients, in some cases it can solve the reduced-space formulation significantly faster without any range reduction than the state-of-the-art solver BARON can solve the equation-oriented formulation.

## Keywords

Global optimization Process design Sequential modular Branch-and-bound Relaxation of algorithms## Notes

### Acknowledgements

This work received funding through the “Competence Center Power to Fuel” of RWTH Aachen University and project “Power to Fuel” of JARA Energy, both of which are funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities, as well as from the German Federal Ministry of Education and Research (BMBF) under grant number 03SFK2A. The responsibility for the content lies with the authors. The authors would also like to thank Jaromił Najman, Hatim Djelassi, and Wolfgang Huster for helpful discussions.

## Supplementary material

## References

- 1.Adjiman, C.S., Androulakis, I.P., Maranas, C.D., Floudas, C.A.: A global optimization method, \(\alpha \)BB, for process design. Comput. Chem. Eng.
**20**, S419–S424 (1996)CrossRefGoogle Scholar - 2.Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs-I. Theor. Adv. Comput. Chem. Eng.
**22**(9), 1137–1158 (1998)CrossRefGoogle Scholar - 3.Ahadi-Oskui, T., Vigerske, S., Nowak, I., Tsatsaronis, G.: Optimizing the design of complex energy conversion systems by branch and cut. Comput. Chem. Eng.
**34**(8), 1226–1236 (2010)CrossRefGoogle Scholar - 4.Ahmetović, E., Grossmann, I.E.: Global superstructure optimization for the design of integrated process water networks. AIChE J.
**57**(2), 434–457 (2011)CrossRefGoogle Scholar - 5.Androulakis, I.P., Maranas, C.D., Floudas, C.A.: A global optimization method for general constrained nonconvex problems. J. Glob. Optim.
**7**(4), 337–363 (1995)CrossRefzbMATHMathSciNetGoogle Scholar - 6.Balendra, S., Bogle, I.D.L.: A comparison of flowsheet solving strategies using interval global optimisation methods. In: Kraslawski, A., Turunen, I. (eds.) European symposium on computer aided process engineering, vol. 13, pp. 23–28. Elsevier Science B.V., Amsterdam (2003)Google Scholar
- 7.Balendra, S., Bogle, I.D.L.: Modular global optimisation in chemical engineering. J. Glob. Optim.
**45**(1), 169–185 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - 8.Baliban, R.C., Elia, J.A., Misener, R., Floudas, C.A.: Global optimization of a MINLP process synthesis model for thermochemical based conversion of hybrid coal, biomass, and natural gas to liquid fuels. Comput. Chem. Eng.
**42**, 64–86 (2012)CrossRefGoogle Scholar - 9.Bendtsen, C., Stauning, O.: FADBAD++, a flexible C++ package for automatic differentiation. Version 2.1. (2012). http://www.fadbad.com. Accessed 18 October 2016
- 10.Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. MOS-SIAM, Philadelphia (2010)CrossRefzbMATHGoogle Scholar
- 11.Biegler, L.T., Grossmann, I.E., Westerberg, A.W.: Systematic Methods of Chemical Process Design. Prentice Hall PTR, Upper Saddle River (1997)Google Scholar
- 12.Biegler, L.T., Hughes, R.R.: Infeasible path optimization with sequential modular simulators. AIChE J.
**28**(6), 994–1002 (1982)CrossRefGoogle Scholar - 13.Bogle, I.D.L., Byrne, R.P.: Global optimisation of chemical process flowsheets. In: Dzemyda, G., Saltenis, V., Zilinskas, A. (eds.) Stochastic and Global Optimization, pp. 33–48. Springer, Dordrecht (2002)CrossRefGoogle Scholar
- 14.Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Glob. Optim.
**52**(1), 1–28 (2012)CrossRefzbMATHMathSciNetGoogle Scholar - 15.Bongartz, D., Mitsos, A.: Infeasible path global flowsheet optimization using McCormick relaxations. In: Espuña, A., Graells, M., Puigjaner, L. (eds.) Proceedings of the 27th European Symposium on Computer Aided Process Engineering - ESCAPE 27, in press (2017)Google Scholar
- 16.Bracco, S., Siri, S.: Exergetic optimization of single level combined gas-steam power plants considering different objective functions. Energy
**35**(12), 5365–5373 (2010)CrossRefGoogle Scholar - 17.Byrd, R.H., Nocedal, J., Waltz, R.A.: KNITRO: an integrated package for nonlinear optimization. In: Di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization, pp. 35–59. Springer, Berlin (2006)Google Scholar
- 18.Byrne, R.P., Bogle, I.D.L.: Global optimisation of constrained non-convex programs using reformulation and interval analysis. Comput. Chem. Eng.
**23**(9), 1341–1350 (1999)CrossRefGoogle Scholar - 19.Byrne, R.P., Bogle, I.D.L.: Global optimization of modular process flowsheets. Ind. Eng. Chem. Res.
**39**(11), 4296–4301 (2000)CrossRefGoogle Scholar - 20.Chachuat, B.: MC++ (version 2.0): Toolkit for Construction, Manipulation and Bounding of Factorable Functions. (2014). https://omega-icl.bitbucket.io/mcpp/?. Accessed 18 October 2016
- 21.Chen, J.J.J.: Comments on improvements on a replacement for the logarithmic mean. Chem. Eng. Sci.
**42**(10), 2488–2489 (1987)CrossRefGoogle Scholar - 22.Diwekar, U.M., Grossmann, I.E., Rubin, E.S.: An MINLP process synthesizer for a sequential modular simulator. Ind. Eng. Chem. Res.
**31**(1), 313–322 (1992)CrossRefGoogle Scholar - 23.Drud, A.S.: CONOPT–a large-scale GRG code. ORSA J. Comput.
**6**(2), 207–216 (1994)CrossRefzbMATHGoogle Scholar - 24.Edgar, T.F., Himmelblau, D.M., Lasdon, L.: Optimization of Chemical Processes. McGraw-Hill, New York (2001)Google Scholar
- 25.Epperly, T.G.W., Pistikopoulos, E.N.: A reduced space branch and bound algorithm for global optimization. J. Glob. Optim.
**11**(3), 287–311 (1997)CrossRefzbMATHMathSciNetGoogle Scholar - 26.Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci.
**15**(9), 550–569 (1969)CrossRefzbMATHMathSciNetGoogle Scholar - 27.GAMS Development Corporation: General Algebraic Modeling System (GAMS) Release 24.8.4. Washington, DC (2016)Google Scholar
- 28.Gunasekaran, S., Mancini, N.D., Mitsos, A.: Optimal design and operation of membrane-based oxy-combustion power plants. Energy
**70**, 338–354 (2014)CrossRefGoogle Scholar - 29.Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
- 30.International Business Machines Corporation: IBM ILOG CPLEX v12.5. Armonk, NY (2009)Google Scholar
- 31.Johnson, S.G.: The NLopt nonlinear-optimization package. http://ab-initio.mit.edu/nlopt. Accessed 18 October 2016
- 32.Jüdes, M., Tsatsaronis, G.: Design optimization of power plants by considering multiple partial load operation points. In: Proceedings of IMECE2007. ASME International Mechanical Engineering Congress and Exposition. November 11–15, 2007, Seattle, WA, pp. 217–225 (2007)Google Scholar
- 33.Kehlhofer, R., Hannemann, F., Stirnimann, F., Rukes, B.: Combined-Cycle Gas & Steam Turbine Power Plants, 3rd edn. PennWell Corporation, Tulsa (2009)Google Scholar
- 34.Khan, K.A., Watson, H.A., Barton, P.I.: Differentiable McCormick relaxations. J. Glob. Optim.
**67**(4), 687–729 (2017)CrossRefzbMATHMathSciNetGoogle Scholar - 35.Kocis, G.R., Grossmann, I.E.: Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problems in process synthesis. Ind. Eng. Chem. Res.
**27**(8), 1407–1421 (1988)CrossRefGoogle Scholar - 36.Kraft, D.: A software package for sequential quadratic programming. Tech. Rep. DFVLR-FB 88-28, Institut für Dynamik der Flugsysteme, Oberpfaffenhofen (1988)Google Scholar
- 37.Kraft, D.: Algorithm 733: TOMP-Fortran modules for optimal control calculations. ACM T. Math. Softw.
**20**(3), 262–281 (1994)CrossRefzbMATHGoogle Scholar - 38.Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications, vol. 15. MOS-SIAM, Philadelphia (2013)CrossRefzbMATHGoogle Scholar
- 39.Manassaldi, J.I., Arias, A.M., Scenna, N.J., Mussati, M.C., Mussati, S.F.: A discrete and continuous mathematical model for the optimal synthesis and design of dual pressure heat recovery steam generators coupled to two steam turbines. Energy
**103**, 807–823 (2016)CrossRefGoogle Scholar - 40.McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I–convex underestimating problems. Math. Program.
**10**, 147–175 (1976)CrossRefzbMATHGoogle Scholar - 41.Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim.
**59**, 503–526 (2014)CrossRefzbMATHMathSciNetGoogle Scholar - 42.Mistry, M., Misener, R.: Optimising heat exchanger network synthesis using convexity properties of the logarithmic mean temperature difference. Comput. Chem. Eng.
**94**, 1–17 (2016)CrossRefGoogle Scholar - 43.Mitsos, A., Chachuat, B., Barton, P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim.
**20**(2), 573–601 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - 44.Najman, J., Mitsos, A.: Convergence analysis of multivariate McCormick relaxations. J. Glob. Optim.
**66**, 597–628 (2016)CrossRefzbMATHMathSciNetGoogle Scholar - 45.Najman, J., Mitsos, A.: Convergence order of McCormick relaxations of LMTD function in heat exchanger networks. In: Kravanja, Z. (ed.) Proceedings of the 26th European Symposium on Computer Aided Process Engineering, pp. 1605–1610 (2016)Google Scholar
- 46.Quesada, I., Grossmann, I.E.: Global optimization algorithm for heat exchanger networks. Ind. Eng. Chem. Res.
**32**(3), 487–499 (1993)CrossRefGoogle Scholar - 47.Reneaume, J.M.F., Koehret, B.M., Joulia, X.L.: Optimal process synthesis in a modular simulator environment: new formulation of the mixed-integer nonlinear programming problem. Ind. Eng. Chem. Res.
**34**(12), 4378–4394 (1995)CrossRefGoogle Scholar - 48.Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng.
**19**(5), 551–566 (1995)CrossRefGoogle Scholar - 49.Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim.
**8**(2), 107–138 (1996)CrossRefzbMATHMathSciNetGoogle Scholar - 50.Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim.
**51**, 569–606 (2011)CrossRefzbMATHMathSciNetGoogle Scholar - 51.Silveira, J.L., Tuna, C.E.: Thermoeconomic analysis method for optimization of combined heat and power systems. Part I. Prog. Energ. Combust.
**29**(6), 479–485 (2003)CrossRefGoogle Scholar - 52.Smith, E.M.B., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng.
**21**, S791–S796 (1997)CrossRefGoogle Scholar - 53.Smith, E.M.B., Pantelides, C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng.
**23**(4), 457–478 (1999)CrossRefGoogle Scholar - 54.Stuber, M.D., Scott, J.K., Barton, P.I.: Convex and concave relaxations of implicit functions. Optim. Method. Softw.
**30**, 424–460 (2015)CrossRefzbMATHMathSciNetGoogle Scholar - 55.Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Progam.
**99**(3), 563–591 (2004)CrossRefzbMATHMathSciNetGoogle Scholar - 56.Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program.
**103**(2), 225–249 (2005)CrossRefzbMATHMathSciNetGoogle Scholar - 57.Tsoukalas, A., Mitsos, A.: Multivariate McCormick relaxations. J. Glob. Optim.
**59**, 633–662 (2014)CrossRefzbMATHMathSciNetGoogle Scholar - 58.Turton, R., Bailie, R.C., Whiting, W.B.: Analysis, Synthesis and Design of Chemical Processes, 4th edn. Prentice Hall PTR, Upper Saddle River (2012)Google Scholar
- 59.U.S. Energy Information Administration: United States Natural Gas Industrial Price. https://www.eia.gov/dnav/ng/hist/n3035us3m.htm. Accessed 6 September 2016
- 60.Valdés, M., Duran, M.D., Rovira, A.: Thermoeconomic optimization of combined cycle gas turbine power plants using genetic algorithms. Appl. Therm. Eng.
**23**(17), 2169–2182 (2003)CrossRefGoogle Scholar - 61.Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program.
**106**(1), 25–57 (2006)CrossRefzbMATHMathSciNetGoogle Scholar - 62.Wechsung, A., Barton, P.I.: Global optimization of bounded factorable functions with discontinuities. J. Glob. Optim.
**58**(1), 1–30 (2014)CrossRefzbMATHMathSciNetGoogle Scholar - 63.Wechsung, A., Scott, J.K., Watson, H.A., Barton, P.I.: Reverse propagation of McCormick relaxations. J. Glob. Optim.
**63**(1), 1–36 (2015)CrossRefzbMATHMathSciNetGoogle Scholar - 64.Zamora, J.M., Grossmann, I.E.: Continuous global optimization of structured process systems models. Comput. Chem. Eng.
**22**(12), 1749–1770 (1998)CrossRefGoogle Scholar - 65.Zebian, H., Mitsos, A.: A double-pinch criterion for regenerative Rankine cycles. Energy
**40**(1), 258–270 (2012)CrossRefGoogle Scholar