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


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Note that the recent extension of this approach to multivariate outer functions [57] also enables the use of functions that are not typically considered part of this class, such as, e.g., \(\min (x,x^2)\).

  2. 2.

    Note that it might be more reasonable not to restrict the area itself, but rather set the heat exchanger cost to a constant below a certain threshold. However, this would lead to a nonsmooth problem requiring special care for upper bounding. Also, BARON does not currently support the max function.

  3. 3.

    Note also that while Epperly and Pistikopoulos [25] give a convergence proof for a selective branching strategy, Stuber et al. [54] argue that this is limited to linear equations for equality constrained problems.

  4. 4.

    While the gas enthalpy can in principle still be solved for temperature, e.g., using Cardano’s method, this is challenging without the use of conditional statements and is thus not attempted here.

  5. 5.

    We decided to use Aspen rather than a basic thermodynamic model to avoid mistakes in setting up the equations and to utilize the temperature dependent property data (in particular heat capacities) that get more important at the high temperatures involved.


  1. 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)

    Article  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. 4.

    Ahmetović, E., Grossmann, I.E.: Global superstructure optimization for the design of integrated process water networks. AIChE J. 57(2), 434–457 (2011)

    Article  Google Scholar 

  5. 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)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

  7. 7.

    Balendra, S., Bogle, I.D.L.: Modular global optimisation in chemical engineering. J. Glob. Optim. 45(1), 169–185 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. 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)

    Article  Google Scholar 

  9. 9.

    Bendtsen, C., Stauning, O.: FADBAD++, a flexible C++ package for automatic differentiation. Version 2.1. (2012). Accessed 18 October 2016

  10. 10.

    Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. MOS-SIAM, Philadelphia (2010)

    Google Scholar 

  11. 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. 12.

    Biegler, L.T., Hughes, R.R.: Infeasible path optimization with sequential modular simulators. AIChE J. 28(6), 994–1002 (1982)

    Article  Google Scholar 

  13. 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)

    Google Scholar 

  14. 14.

    Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Glob. Optim. 52(1), 1–28 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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)

  16. 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)

    Article  Google Scholar 

  17. 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)

  18. 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)

    Article  Google Scholar 

  19. 19.

    Byrne, R.P., Bogle, I.D.L.: Global optimization of modular process flowsheets. Ind. Eng. Chem. Res. 39(11), 4296–4301 (2000)

    Article  Google Scholar 

  20. 20.

    Chachuat, B.: MC++ (version 2.0): Toolkit for Construction, Manipulation and Bounding of Factorable Functions. (2014). Accessed 18 October 2016

  21. 21.

    Chen, J.J.J.: Comments on improvements on a replacement for the logarithmic mean. Chem. Eng. Sci. 42(10), 2488–2489 (1987)

    Article  Google Scholar 

  22. 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)

    Article  Google Scholar 

  23. 23.

    Drud, A.S.: CONOPT–a large-scale GRG code. ORSA J. Comput. 6(2), 207–216 (1994)

    Article  MATH  Google Scholar 

  24. 24.

    Edgar, T.F., Himmelblau, D.M., Lasdon, L.: Optimization of Chemical Processes. McGraw-Hill, New York (2001)

    Google Scholar 

  25. 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)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci. 15(9), 550–569 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    GAMS Development Corporation: General Algebraic Modeling System (GAMS) Release 24.8.4. Washington, DC (2016)

  28. 28.

    Gunasekaran, S., Mancini, N.D., Mitsos, A.: Optimal design and operation of membrane-based oxy-combustion power plants. Energy 70, 338–354 (2014)

    Article  Google Scholar 

  29. 29.

    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)

    Google Scholar 

  30. 30.

    International Business Machines Corporation: IBM ILOG CPLEX v12.5. Armonk, NY (2009)

  31. 31.

    Johnson, S.G.: The NLopt nonlinear-optimization package. Accessed 18 October 2016

  32. 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)

  33. 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. 34.

    Khan, K.A., Watson, H.A., Barton, P.I.: Differentiable McCormick relaxations. J. Glob. Optim. 67(4), 687–729 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  35. 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)

    Article  Google Scholar 

  36. 36.

    Kraft, D.: A software package for sequential quadratic programming. Tech. Rep. DFVLR-FB 88-28, Institut für Dynamik der Flugsysteme, Oberpfaffenhofen (1988)

  37. 37.

    Kraft, D.: Algorithm 733: TOMP-Fortran modules for optimal control calculations. ACM T. Math. Softw. 20(3), 262–281 (1994)

    Article  MATH  Google Scholar 

  38. 38.

    Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications, vol. 15. MOS-SIAM, Philadelphia (2013)

    Google Scholar 

  39. 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)

    Article  Google Scholar 

  40. 40.

    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I–convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MATH  Google Scholar 

  41. 41.

    Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59, 503–526 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  42. 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)

    Article  Google Scholar 

  43. 43.

    Mitsos, A., Chachuat, B., Barton, P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim. 20(2), 573–601 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  44. 44.

    Najman, J., Mitsos, A.: Convergence analysis of multivariate McCormick relaxations. J. Glob. Optim. 66, 597–628 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  45. 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)

  46. 46.

    Quesada, I., Grossmann, I.E.: Global optimization algorithm for heat exchanger networks. Ind. Eng. Chem. Res. 32(3), 487–499 (1993)

    Article  Google Scholar 

  47. 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)

    Article  Google Scholar 

  48. 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)

    Article  Google Scholar 

  49. 49.

    Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  50. 50.

    Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51, 569–606 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  51. 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)

    Article  Google Scholar 

  52. 52.

    Smith, E.M.B., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, S791–S796 (1997)

    Article  Google Scholar 

  53. 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)

    Article  Google Scholar 

  54. 54.

    Stuber, M.D., Scott, J.K., Barton, P.I.: Convex and concave relaxations of implicit functions. Optim. Method. Softw. 30, 424–460 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  55. 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)

    Article  MATH  MathSciNet  Google Scholar 

  56. 56.

    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  57. 57.

    Tsoukalas, A., Mitsos, A.: Multivariate McCormick relaxations. J. Glob. Optim. 59, 633–662 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  58. 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. 59.

    U.S. Energy Information Administration: United States Natural Gas Industrial Price. Accessed 6 September 2016

  60. 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)

    Article  Google Scholar 

  61. 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)

    Article  MATH  MathSciNet  Google Scholar 

  62. 62.

    Wechsung, A., Barton, P.I.: Global optimization of bounded factorable functions with discontinuities. J. Glob. Optim. 58(1), 1–30 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  63. 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)

    Article  MATH  MathSciNet  Google Scholar 

  64. 64.

    Zamora, J.M., Grossmann, I.E.: Continuous global optimization of structured process systems models. Comput. Chem. Eng. 22(12), 1749–1770 (1998)

    Article  Google Scholar 

  65. 65.

    Zebian, H., Mitsos, A.: A double-pinch criterion for regenerative Rankine cycles. Energy 40(1), 258–270 (2012)

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Alexander Mitsos.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (txt 3 KB)

Supplementary material 2 (txt 12 KB)

Supplementary material 3 (txt 7 KB)

Supplementary material 4 (txt 8 KB)

Supplementary material 5 (txt 5 KB)

Supplementary material 6 (txt 13 KB)

Supplementary material 7 (txt 7 KB)

Supplementary material 8 (txt 16 KB)

Supplementary material 9 (txt 7 KB)

Supplementary material 10 (txt 4 KB)

Supplementary material 11 (txt 4 KB)

Supplementary material 12 (txt 9 KB)

Supplementary material 13 (txt 6 KB)

Supplementary material 14 (txt 6 KB)

Supplementary material 15 (txt 2 KB)

Supplementary material 16 (txt 2 KB)

Supplementary material 17 (txt 5 KB)

Supplementary material 18 (txt 4 KB)

Supplementary material 19 (txt 5 KB)

Supplementary material 20 (txt 5 KB)

Supplementary material 21 (txt 29 KB)

Supplementary material 22 (txt 19 KB)

Supplementary material 23 (txt 22 KB)

Supplementary material 24 (txt 13 KB)

Supplementary material 25 (txt 17 KB)

Supplementary material 26 (txt 19 KB)

Supplementary material 27 (txt 12 KB)

Supplementary material 28 (txt 10 KB)


Formulation (RS*) does not introduce multimodality

In the following, we show that (RS*) does not introduce additional local minima compared to (FS). For ease of analysis, we recast the formulations in slightly different (and more general) form.

Let \(\mathbf {x} \in X \subset {\mathbb {R}}^{n_x}\), \(\mathbf {y} \in Y \subset {\mathbb {R}}^{n_y}\), with XY nonempty compact convex sets, and the functions \(f: X \times Y \rightarrow {\mathbb {R}}\), \(\mathbf {h}_\text {exp}: X \rightarrow Y\), \(\mathbf {h}_\text {imp}: X \times Y \rightarrow {\mathbb {R}}^{n_{h,\text {imp}}}\), \(\mathbf {g}: X \times Y \rightarrow {\mathbb {R}}^{n_g}\) continuous. Consider the NLP


Let \(\mathbf {y}\) and \(\mathbf {h}_\text {exp}\) be selected such that the equality constraints \(\mathbf {y}=\mathbf {h}_\text {exp}(\mathbf {x})\) are of the form

$$\begin{aligned} y_1= & {} {\hat{h}}_{\text {exp},1}(\mathbf {x}),\\ y_i= & {} {\hat{h}}_{\text {exp},i}(\mathbf {x},\hat{h}_{\text {exp},1}(\mathbf {x}),...,\hat{h}_{\text {exp},i-1}(\mathbf {x})),~~~~~~~~~~i=2,...,n_y, \end{aligned}$$

where \({\hat{h}}_\text {exp,i}, i=1,...,n_y\) consist only of compositions of binary sums, binary products, or univariate or multivariate functions from a given library as discussed in Sect. 2. In total, the constraints \(\mathbf {y}=\mathbf {h}_\text {exp}(\mathbf {x})\) can thus be evaluated sequentially to compute a unique vector \(\mathbf {y} \in Y\) for a given \(\mathbf {x} \in X\). Note, however, that this mapping need not be injective nor surjective.

The full-space formulation (A-FS) can be converted to the following reduced-space formulation, which is a generalization of (RS*) :


Note that in (A-RS*) the substitution of \(\mathbf {y}\) by \(\mathbf {h}_\text {exp}(\mathbf {x})\) need not be done symbolically, but can rather be deferred to the time of function evaluation.

Proposition 1

A point \(\mathbf {x}^* \in X\) is a local solution of (A-RS*) if and only if there is a \(\mathbf {y}^* \in Y\) such that \((\mathbf {x}^*,\mathbf {y}^*)\) is a local solution of (A-FS).


The feasible regions of (A-FS) and (A-RS*) are

$$\begin{aligned} {\mathscr {F}}_\text {FS}:= & {} \{(\mathbf {x},\mathbf {y})\in X \times Y :~ \mathbf {y}=\mathbf {h}_\text {exp}(\mathbf {x}),~\mathbf {h}_\text {imp}(\mathbf {x},\mathbf {y})=\mathbf {0}, ~\mathbf {g}(\mathbf {x},\mathbf {y})\le \mathbf {0}\},\\ {\mathscr {F}}_\text {RS}:= & {} \{\mathbf {x}\in X :~ \mathbf {h}_\text {imp}(\mathbf {x},\mathbf {h}_\text {exp}(\mathbf {x}))=\mathbf {0}, ~\mathbf {g}(\mathbf {x},\mathbf {h}_\text {exp}(\mathbf {x}))\le \mathbf {0}\},\\ \end{aligned}$$

respectively, and it holds that

$$\begin{aligned} \mathbf {x} \in {\mathscr {F}}_\text {RS} \wedge \mathbf {y}=\mathbf {h}_\text {exp}(\mathbf {x}) \iff (\mathbf {x},\mathbf {y}) \in {\mathscr {F}}_\text {FS}. \end{aligned}$$

Assume a point \((\mathbf {x}^*,\mathbf {y}^*)\) is a local solution of (A-FS). Then by definition (cf., e.g., [38]) there exists \(\epsilon >0\) such that

$$\begin{aligned} \begin{aligned}&(\mathbf {x}^*,\mathbf {y}^*) \in {\mathscr {F}}_\text {FS},\\&f(\mathbf {x}^*,\mathbf {y}^*)\le f(\mathbf {x},\mathbf {y})~~~~\forall ~~(\mathbf {x},\mathbf {y}) \in {\mathscr {F}}_\text {FS} \cap {\mathscr {N}}_{\text {FS},\epsilon ,(\mathbf {x}^*,\mathbf {y}^*)},\\&{\mathscr {N}}_{\text {FS},\epsilon ,(\mathbf {x}^*,\mathbf {y}^*)} := \{(\mathbf {x},\mathbf {y})\in X \times Y :~ ||(\mathbf {x},\mathbf {y}) - (\mathbf {x}^*,\mathbf {y}^*) ||\le \epsilon \}. \end{aligned} \end{aligned}$$

From (1) it follows that \(\mathbf {x}^* \in {\mathscr {F}}_\text {RS}\) and \(\mathbf {y}^*=\mathbf {h}_\text {exp}(\mathbf {x}^*)\). Thus, from (2) it follows

$$\begin{aligned} f(\mathbf {x}^*,\mathbf {h}_\text {exp}(\mathbf {x}^*))\le f(\mathbf {x},\mathbf {h}_\text {exp}(\mathbf {x}))~~~~\forall ~~(\mathbf {x},\mathbf {y}) \in {\mathscr {F}}_\text {FS} \cap {\mathscr {N}}_{\text {FS},\epsilon ,(\mathbf {x}^*,\mathbf {y}^*)}. \end{aligned}$$

Take \({\mathscr {N}}_{\text {RS},\hat{\epsilon },\mathbf {x}^*} := \{\mathbf {x} \in X :~ ||\mathbf {x}-\mathbf {x}^* ||\le \hat{\epsilon }\}\) with \(\hat{\epsilon } := \max _{(\mathbf {x},\mathbf {y}) \in {\mathscr {F}}_\text {FS} \cap {\mathscr {N}}_{\text {FS},\epsilon ,(\mathbf {x}^*,\mathbf {y}^*)}} ||\mathbf {x}-\mathbf {x}^*||\). Note that \(\hat{\epsilon }\) exists by continuity of \(\mathbf {h}_\text {exp}\), \(\mathbf {h}_\text {imp}\), and \(\mathbf {g}\) and compactness of X and Y. It follows

$$\begin{aligned} f(\mathbf {x}^*,\mathbf {h}_\text {exp}(\mathbf {x}^*))\le f(\mathbf {x},\mathbf {h}_\text {exp}(\mathbf {x}))~~~~\forall ~~\mathbf {x} \in {\mathscr {F}}_\text {RS} \cap {\mathscr {N}}_{\text {RS},\hat{\epsilon },\mathbf {x}^*}, \end{aligned}$$

which together with \(\mathbf {x}^* \in {\mathscr {F}}_\text {RS}\) shows that \(\mathbf {x}^*\) is a local solution of (A-RS*).

Assume, on the other hand, that \(\mathbf {x}^*\) is a local solution of (A-RS*), thus satisfying \(\mathbf {x}^* \in \mathscr {F}_\text {RS}\) and (3) for some \(\hat{\epsilon }>0\). We define the vector \(\mathbf {y}^* \in Y\) as \(\mathbf {y}^*=\mathbf {h}_\text {exp}(\mathbf {x}^*)\) and by (1) we obtain that \((\mathbf {x}^*,\mathbf {y}^*) \in {\mathscr {F}}_\text {FS}\) and furthermore

$$\begin{aligned} f(\mathbf {x}^*,\mathbf {y}^*)\le f(\mathbf {x},\mathbf {h}_\text {exp}(\mathbf {x}))~~~~\forall ~~\mathbf {x} \in {\mathscr {F}}_\text {RS} \cap {\mathscr {N}}_{\text {RS},\hat{\epsilon },\mathbf {x}^*}. \end{aligned}$$

Since for any \((\mathbf {x},\mathbf {y}) \in X \times Y\) we have \(||(\mathbf {x},\mathbf {y}) - (\mathbf {x}^*,\mathbf {y}^*) ||\ge ||\mathbf {x}-\mathbf {x}^*||\), using (1) it follows for \({\mathscr {N}}_{\text {FS},\hat{\epsilon },(\mathbf {x}^*,\mathbf {y}^*)} := \{(\mathbf {x},\mathbf {y})\in X \times Y :~ ||(\mathbf {x},\mathbf {y}) - (\mathbf {x}^*,\mathbf {y}^*) ||\le \hat{\epsilon }\}\) that

$$\begin{aligned} f(\mathbf {x}^*,\mathbf {y}^*)\le f(\mathbf {x},\mathbf {y})~~~~\forall ~~(\mathbf {x},\mathbf {y}) \in {\mathscr {F}}_\text {FS} \cap {\mathscr {N}}_{\text {FS},\hat{\epsilon },(\mathbf {x}^*,\mathbf {y}^*)}, \end{aligned}$$

which together with \((\mathbf {x}^*,\mathbf {y}^*) \in {\mathscr {F}}_\text {FS}\) shows that \((\mathbf {x}^*,\mathbf {y}^*)\) is a local solution of (A-FS). \(\square \)

Process models

The following sections provide details on the model equations and the calculation sequences for simulating the cycles by sequential evaluation of these equations to obtain the desired thermodynamic quantities, in particular the power output \(\dot{W}_\text {net}\). Enthalpies and entropies in the process models are computed using the ideal gas and ideal liquid equations of state with constant heat capacities, and saturation temperatures are computed using the Antoine equation (cf., e.g., [11]). Pressure losses in components other than the pumps and turbines are neglected. Tables 13 and 14 summarize the fixed model parameter values for the thermodynamic calculations used in the case studies. A list of all symbols and subscripts used is given in Table 15.

Table 13 Fixed model parameters for the cycles in the case studies. The gas outlet temperature is only fixed for Case Study I. The condenser pressure is 0.2 bar for Case Study I, and 0.05 bar for Case Studies II and III
Table 14 Fixed model parameters for the thermodynamic properties of water
Table 15 List of symbols and subscripts used in the process models of the case studies

Case Study I: basic Rankine cycle

For every stream i, the saturation temperature is computed from the pressure \(p_i\) of the stream via the Antoine equation:

$$\begin{aligned} T_\text {sat,pi} = \frac{B}{{A-\log _{10}}(\frac{p_{i}}{bar})}-C. \end{aligned}$$

Where required, the saturated vapor and liquid enthalpy and entropy can be computed via:

$$\begin{aligned} h_\text {sat,vap,pi}&= \Delta h_\text {evap,p0} + c_\text {p,ig} \cdot (T_\text {sat,pi} - T_0)\\ h_\text {sat,liq,pi}&= c_\text {if} \cdot (T_\text {sat,p1} - T_0) + v_\text {if} \cdot (p_1 - p_0)\\ s_\text {sat,vap,pi}&= \frac{\Delta h_\text {evap,p0}}{T_0} + c_\text {p,ig} \cdot \ln \left( \frac{T_\text {sat,pi}}{T_0}\right) - R \cdot \ln \left( \frac{p_i}{p_0}\right) \\ s_\text {sat,liq,pi}&= c_\text {if} \cdot \ln \left( \frac{T_\text {sat,pi}}{T_0}\right) , \end{aligned}$$

where the reference temperature \(T_0\) is also obtained from the Antoine equation.

To simulate the cycle, we start at the condenser outlet. Since the condenser pressure \(p_1\) is treated as a fixed parameter and by assumption the fluid leaves the condenser in the saturated liquid state, we have

$$\begin{aligned} h_1 = h_\text {sat,liq,p1}. \end{aligned}$$

For the pump, we can compute the specific pump work and power consumption using the isentropic efficiency \(\eta _\text {P}\):

$$\begin{aligned} w_\text {P}&= \frac{v_\text {if}\cdot (p_2 - p_1)}{\eta _\text {P}}\\ \dot{W}_\text {P}&= \dot{m} \cdot w_\text {P}.\\ \end{aligned}$$

The outlet state then follows from the energy balance:

$$\begin{aligned} h_2 = h_1 + w_\text {P}. \end{aligned}$$

Since the gas outlet temperature is fixed, we first calculate the overall heat transfer rate in the boiler, which is assumed to be isobaric (\(p_5=p_4=p_3=p_2\)):

$$\begin{aligned} \dot{Q}_\text {B} = (\dot{m}c_\text {p})_\text {G} \cdot (T_\text {G,in}-T_\text {G,out}). \end{aligned}$$

In the superheater, the live steam enthalpy can be computed from the energy balance since the gas inlet temperature is known:

$$\begin{aligned} h_5 = h_2 + \frac{\dot{Q}_{B}}{\dot{m}}. \end{aligned}$$

In the economizer, the outlet enthalpy of the water can be computed using the known outlet temperature \(T_3 = T_\text {sat,p3} - \Delta T_\text {ap}\):

$$\begin{aligned} h_3 = c_\text {if} \cdot (T_3-T_0) + v_\text {if} \cdot (p_3-p_0). \end{aligned}$$

Using an energy balance around the economizer, the gas temperature between economizer and evaporator can be calculated as

$$\begin{aligned} T_\text {G3} = T_\text {G,out} + \frac{\dot{m}\cdot (h_3-h_2)}{(\dot{m}c_\text {p})_\text {G}}. \end{aligned}$$

In the evaporator, the water leaving the steam drum towards the superheater is in the saturated vapor state, so that we obtain similarly:

$$\begin{aligned} T_4&= T_\text {sat,p4}\\ h_4&= h_\text {sat,vap,p4}\\ T_\text {G2}&= T_\text {G,in} - \frac{\dot{m}\cdot (h_5-h_4)}{(\dot{m}c_\text {p})_\text {G}}. \end{aligned}$$

In the turbine, the inlet temperature and entropy can be computed as

$$\begin{aligned} T_5&= T_0 + \frac{h_5 - \Delta h_\text {evap,p0}}{c_\text {p,ig}}\\ s_5&= \frac{\Delta h_\text {evap,p0}}{T_0} + c_\text {p,ig} \cdot \ln \left( \frac{T_5}{T_0}\right) - R \cdot \ln \left( \frac{p_5}{p_0}\right) . \end{aligned}$$

From this, the specific turbine work and power output are computed using the isentropic efficiency \(\eta _\text {T}\), similar to the pump, with the hypothetical isentropic turbine outlet state 6s being in the two-phase region.

$$\begin{aligned} p_6&= p_1\\ p_{6s}&= p_1\\ x_\text {6s}&= \frac{s_5 - s_\text {sat,liq,p6}}{s_\text {sat,vap,p6} - s_\text {sat,liq,p6}}\\ h_\text {6s}&= h_\text {sat,liq,p6} + x_\text {6s} \cdot (h_\text {sat,vap,p6}-h_\text {sat,liq,p6})\\ w_\text {T}&= \eta _\text {T} \cdot (h_5 - h_\text {6s})\\ \dot{W}_\text {T}&= \dot{m} \cdot w_\text {T}.\\ \end{aligned}$$

An energy balance then yields the true outlet state 6:

$$\begin{aligned} h_6&= h_5 - w_\text {T}\\ x_6&= \frac{h_6-h_\text {sat,liq,p6}}{h_\text {sat,vap,p6}-h_\text {sat,liq,p6}}. \end{aligned}$$

Note that when computed this way, the vapor quality \(x_{6}\) is greater than unity if the enthalpy \(h_6\) is greater than the saturated vapor enthalpy at \(p_6\). Therefore, the condition \(x_{6}\le 1\) can be used to ensure the validity of the assumption of 6 (and hence also 6s) being in the two-phase region.

Finally, the net power output of the cycle is

$$\begin{aligned} \dot{W}_\text {net} = \dot{W}_\text {T} - \dot{W}_\text {P}. \end{aligned}$$

Case Study II: regenerative Rankine cycle

In this case, the simulation of the cycle starts at the turbine inlet since the pressure (\(p_7=p_4\)) and enthalpy (\(h_7\) is an optimization variable) are known. The turbine with bleed extraction can be modeled as two separate turbines in parallel, each of which is treated as described in Sect. B.1. The one associated with the bleed stream expands to the bleed pressure that is equal to the deaerator pressure \(p_2\), while the other one expands to the condenser pressure \(p_1\). The power output of the turbine can be obtained as the sum of these two parts (\(\dot{W}_\text {T} = \dot{W}_\text {T,Bl} + \dot{W}_\text {T,Cond}\)). The condenser outlet state and the condensate (CD) pump can be modeled as above, with the exception that only the mass flow that is not extracted as a bleed contributes to the pump power consumption:

$$\begin{aligned} \dot{W}_\text {P,CD} = \dot{m} \cdot (1-k_\text {Bl}) \cdot w_\text {P}.\\ \end{aligned}$$

The deaerator is assumed to be isobaric and its outlet enthalpy follows from the energy balance:

$$\begin{aligned} p_3&= p_2\\ h_3&= k_\text {Bl} \cdot h_8 + (1-k_\text {Bl}) \cdot h_2. \end{aligned}$$

The calculation of the feedwater (FW) pump is analogous to the condensate pump, but with the entire cycle mass flow. Since in this case study the gas outlet temperature is not fixed any more, the overall heat transfer rate is determined using the known inlet and outlet enthalpies of the water:

$$\begin{aligned} \dot{Q}_\text {B}&= \dot{m} \cdot (h_7 - h_4)\\ T_\text {G4}&= T_\text {G,in} - \frac{\dot{Q}_\text {B}}{(\dot{m} c_\text {p})_\text {G}}. \end{aligned}$$

From this, the missing quantities for the economizer and evaporator can be calculated as described above.

Finally, the net power output of the cycle is

$$\begin{aligned} \dot{W}_\text {net} = \dot{W}_\text {T} - \dot{W}_\text {P,CD} - \dot{W}_\text {P,FW}. \end{aligned}$$

Case Study III: two-pressure cycle

For convenience, variables for the mass flow rates through the LP and HP parts of the HRSG and the ones of the turbine bleed and that being expanded to the condenser are defined as

$$\begin{aligned} \dot{m}_\text {LP}&= \dot{m} \cdot k_\text {LP}\\ \dot{m}_\text {HP}&= \dot{m} \cdot (1-k_\text {LP})\\ \dot{m}_\text {Bl}&= \dot{m} \cdot k_\text {Bl}\\ \dot{m}_\text {Cond}&= \dot{m} \cdot (1-k_\text {Bl}). \end{aligned}$$

We start at the HP turbine inlet, since its state is known (cf. above). For the HP turbine, the outlet state is assumed to be in the vapor region (which is ensured by the constraint \(h_\text {12s} \ge h_\text {sat,vap,p12}\)). Therefore, the temperature and enthalpy of the isentropic outlet state is calculated based on the corresponding ideal gas equations, while the rest of the equations remains the same as above:

$$\begin{aligned} p_{12}&= p_4\\ p_\text {12s}&= p_4\\ s_\text {12s}&= s_{11}\\ T_\text {12s}&= T_0 \cdot \exp \left( \frac{s_\text {12s} + R \cdot \ln (p_{12s}/p_0) - \Delta h_\text {evap,p0}/T_0}{c_\text {p,ig}}\right) \\ h_\text {12s}&= \Delta h_\text {evap,p0} + c_\text {p,ig} \cdot (T_\text {12s}-T_0)\\ w_\text {T,HP}&= \eta _\text {T} \cdot (h_{11} - h_\text {12s})\\ h_{12}&= h_{11} - w_\text {T,HP}\\ \dot{W}_\text {T,HP}&= \dot{m}_\text {HP} \cdot w_\text {T,HP}.\\ \end{aligned}$$

The mixing of the HP turbine outlet stream 12 with the outlet stream 7 of the LP superheater is analogous to the model of the deaerator described above. The models for the LP turbine, condenser, condensate pump, deaerator, and LP pump (corresponding to the feedwater pump) are the same as described above and can be evaluated in this order. The HP pump is analogous to the LP pump but uses only the mass flow of the HP part of the cycle. The HRSG can then be evaluated in a similar manner as described above, starting from the HP superheater and working back to the LP economizer. The net power output finally follows as

$$\begin{aligned} \dot{W}_\text {net} = \dot{W}_\text {T,HP} + \dot{W}_\text {T,LP} - \dot{W}_\text {P,CD} - \dot{W}_\text {P,LP} - \dot{W}_\text {P,HP}. \end{aligned}$$

Case Study I with temperature-dependent sub-models

Case Study I was also repeated using temperature-dependent sub-models for the ideal gas heat capacity and enthalpy of vaporization:

$$\begin{aligned} c_\text {p,ig}(T)&= c_1 + c_2 \cdot T + c_3 \cdot T^2, \\ \Delta h_\text {evap} (T)&= \Delta h_\text {evap,Tref} \cdot \left( \frac{1-T/T_\text {crit}}{1-T_\text {ref}/T_\text {crit}} \right) ^a. \end{aligned}$$

The corresponding parameters are given in Table 16. Note that since we decided to use the ideal gas heat capacity (rather than liquid), for convenience the reference state is shifted to the dew curve at \(p_0\). Using these sub-models, the enthalpy and entropy for streams in the gas phase can be computed for given \(p_i\) and \(T_i\) as

$$\begin{aligned} h_\text {i}&= \int _{T_0}^{T_i} c_\text {p,ig}(T) dT,\\ s_\text {i}&= \int _{T_0}^{T_i} \frac{c_\text {p,ig}(T)}{T} dT - R \cdot \ln \left( \frac{p_i}{p_0}\right) ,\\ \end{aligned}$$

while for liquid streams they are given by

$$\begin{aligned} h_\text {i}&= \int _{T_0}^{T_i} c_\text {p,ig}(T) dT - \Delta h_\text {evap} (T_i) + v_\text {if} \cdot (p_i - p_\text {sat,Ti}),\\ s_\text {i}&= \int _{T_0}^{T_i} \frac{c_\text {p,ig}(T)}{T} dT - R \cdot \ln \left( \frac{p_\text {sat,Ti}}{p_0}\right) - \frac{\Delta h_\text {evap} (T_i)}{T_i}.\\ \end{aligned}$$

The vapor pressure at the stream temperature is again obtained from the Antoine equation:

$$\begin{aligned} p_\text {sat,Ti} = 10^{A-\frac{B}{C+T_i}}\mathrm{{bar}}. \end{aligned}$$
Table 16 Fixed model parameters for the thermodynamic properties of water using temperature dependent sub-models

Unlike for the simple thermodynamic model used in the other case studies, these expressions for enthalpy and entropy cannot be solved for the temperature analyticallyFootnote 4. Therefore, some additional variables and equality constraints have to be introduced when optimizing the cycle using formulation (RS*) . This is the case for Streams 2 and 5, the state of which is determined from energy balances (cf. Sect. B.1). Thus, their temperatures \(T_2\) and \(T_5\) are handed to the optimizer as additional module variables (\(\mathbf {x}_m\)), and Eqs. (4) and (5) are added as additional module equations (\(\tilde{\mathbf {h}}_m=\mathbf {0}\)). Note that for the two-phase streams 6 and 6s, the vapor fraction can still be computed from the given enthalpies or entropies as described in Sect. B.1 so that no additional variables are needed.

Economic analysis

The LCOE of the CCPP is calculated according to the equation [33, 51]

$$\begin{aligned} \text {LCOE} = \frac{TCI \cdot \varPsi \cdot \varphi }{\dot{W}_\text {CCPP} \cdot T_\text {eq}} + \frac{C_\text {Fuel}}{\eta _\text {CCPP}} + u_\text {var}, \end{aligned}$$

where \({ TCI}\), \(\dot{W}_\text {CCPP}\) and \({\eta _\text {CCPP}}\) denote the total capital investment, the net power output, and the efficiency of the CCPP, respectively. The remaining quantities are constant parameters that can be found in Table 17. The annuity factor was determined as described in Ref. [51] assuming a depreciation period of 20 years and a construction time of 2 years, as well as their values for interest and inflation rates.

While the gas turbine is not considered in the optimization itself since its design is assumed to be fixed, some data is required for evaluating the aforementioned quantities. To this end, it was simulated in AspenPlus\(^{\circledR }\) assuming a pressure ratio of 20 with a turbine inlet temperature of 1620K and isentropic compressor and turbine efficiencies of 0.8 and 0.9, respectivelyFootnote 5. These conditions and the mass flow rate through the gas turbine were selected to be in a typical range while matching the assumptions on the exhaust gas flow rate and heat capacity made for the simulation of the bottoming cycle. The resulting net power output of the gas turbine is \(\dot{W}_\text {GT} =69.7 \mathrm{{MW}}\) while consuming \(\dot{Q}_\text {Fuel} = 182 \mathrm{{MW}}\) (based on lower heating value) of natural gas. From this, the power output of the CCPP follows as the sum of net power output of the gas turbine and that of the steam cycle obtained as described in “Appendix B”, and the CCPP efficiency can be calculated as \(\eta _\text {CCPP} = \dot{W}_{ CCPP} / \dot{Q}_\text {Fuel}\).

The capital investment for the gas turbine as well as the steam cycle is calculated using the cost correlations given in Ref. [51] for pumps, steam turbines, and generators, as well as the gas turbine including compressor and combustor which are based on their power and mass flow rates as well as pressures and temperatures. For heat exchangers and deaerators, the more detailed correlations from Ref. [58] are used. To this end, the heat transfer areas \(A_j\) of the heat exchangers involved (i.e., condenser, economizer, evaporator, and superheater) are computed based on their heat flows \(\dot{Q}_j\), inlet and outlet temperature differences \(\Delta T_\text {in,j}\) and \(\Delta T_\text {out,j}\), and heat transfer coefficients \(k_j\) using Chen’s approximation of the logarithmic mean temperature difference (LMTD) [21]:

$$\begin{aligned} \text {LMTD}_{\text {Chen},j}&= \left( \Delta T_{\text {in},j} \cdot \Delta T_{\text {out},j} \cdot \frac{\Delta T_{\text {in},j}+\Delta T_{\text {out},j}}{2}\right) ^{1/3} \\ A_j&= \frac{\dot{Q}_j}{k_j \cdot \text {LMTD}_{\text {Chen},j}} \end{aligned}$$

For the heat transfer coefficients, average values are used that depend on the state of the fluids involved (cf. Table 17). For the condenser, water cooling is assumed with specified cooling water inlet and outlet temperatures. The investment cost \(\text {Inv}_j\) of the heat exchangers is calculated via the base purchase cost \(C_{p,j} \) and the pressure factor \(F_{p,j}\) [58]:

$$\begin{aligned} C_{p,j}&= 10^{K_1 + K_2 \cdot \log _{10}(A_{j}/\mathrm{{m}}^{2}) + K_3 \cdot \log _{10}(A_j/\mathrm{{m}}^{2})^2}\\ F_{p,j}&= 10^{C_1 + C_2 \cdot \log _{10}(p_j/\mathrm{{bar}}) + C_3 \cdot \log _{10}(p_{j}\mathrm{{bar}})^2}\\ \text {Inv}_{j}&= 1.18 \cdot (1.63+1.66 \cdot 2.75\cdot F_{p,j} ) \cdot C_{p,j} \end{aligned}$$

The deaerator is treated as a process vessel, the volume of which is determined for a 10min liquid holdup with another 50% added for vapor in the vessel [33]:

$$\begin{aligned} V_\text {Dae}&= 1.5 \cdot \dot{m}_\text {Dae,out} \cdot v_\text {if} \cdot 600s\\ C_{p,\text {Dae}}&= 10^{K_4 + K_5 \cdot \log _{10}(V_\text {Dae}/m^3) + K_6 \cdot \log _{10}(V_\text {Dae}/m^3)^2}\\ F_{p,\text {Dae}}&= 1.25\\ \text {Inv}_\mathrm{Dae}&= 1.18 \cdot (1.49+1.52 \cdot F_{p,\text {Dae}}) \cdot C_{p,\text {Dae}} \end{aligned}$$
Table 17 Parameters for the economic evaluation taken from Refs. [33, 51, 58, 59]

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bongartz, D., Mitsos, A. Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations . J Glob Optim 69, 761–796 (2017).

Download citation


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