Deterministic global optimization of steam cycles using the IAPWS-IF97 model

Established methods for deterministic global optimization are based on spatial branch-and-bound (B&B) (Falk and Soland 1969) and rely on the availability of valid convex and concave relaxations (Locatelli and Schoen 2013).

In Sects. 2.3 and 3.4, we make use of non-standard under- and overestimators to derive tighter relaxations.

The original McCormick technique (McCormick 1976) provides rules for computing relaxations of so-called factorable functions consisting of finite compositions of binary sums, binary products, and a library of univariate functions. The latter are called intrinsic functions, and for these functions both convex and concave relaxations and range bounds need to be known.

Tsoukalas and Mitsos (2014) extended the McCormick technique to allow for multivariate intrinsic functions. Note that such intrinsic functions are also used in the context of the AVM, such that the relaxations and range bounds developed herein could also be applied in that context. Unlike the AVM (Smith and Pantelides 1997; Tawarmalani and Sahinidis 2002), the McCormick technique provides convex and concave relaxations in the original variable space. A challenge in the application of the multivariate McCormick method is that the evaluation of the relaxation requires the solution of a convex but possibly nonlinear and nonsmooth problem (Theorem 2, Tsoukalas and Mitsos (2014)). To this end, it is highly desirable for the developed relaxations to have specific monotonicity properties that allow us to either determine closed-form solutions of the problem described by Theorem 2 of Tsoukalas and Mitsos (2014) analytically, or compute it numerically with relatively little effort, e.g., by solving a one-dimensional nonlinear equation, which is solved via Newton’s method.

Componentwise convexity of multivariate functions enables the construction of tight relaxations.

For twice continuously differentiable functions, a convenient way of confirming componentwise convexity or concavity with respect to a variable consists in examining the second derivative of the univariate function \({\hat{f}}\) in (1) (and hence the second partial derivative with respect to the variable of interest) and proving that it is non-negative or non-positive over the considered set (cf. Rockafellar 1970), either analytically or through global maximization or minimization. However, since some of the functions considered herein are only piecewise continuously differentiable, we use a specific procedure that enables an analysis for functions where a partial derivative need not exist at all points. The functions we consider herein are of the formwhere \({\mathcal {X}}_1,{\mathcal {X}}_2 \in {\mathbb {I}}{\mathbb {R}}\), \(f_1,f_2\) are both smooth on \({\mathcal {X}}_1\times {\mathcal {X}}_2\), f is continuous on \({\mathcal {X}}_1\times {\mathcal {X}}_2\), and \({\tilde{x}}_2(x_1)\) is a strictly monotonic function. To determine whether functions of form (2) are componentwise convex, we proceed as described in the following. First, we solve the optimization problemsto global optimality. Note that these problems are much smaller and less complex than the design problems within which the functions are intended to be used and can be solved with general purpose methods. Furthermore, since we only consider certain known functions, the problems need to be solved only once in order to construct tighter relaxations for later use in design problems. If the solution values of both problems (3) and (4) are non-negative, we know that \(f_1\) is componentwise convex with respect to \(x_2\) on \({\mathcal {S}}_1:=\{(x_1,x_2)\in {\mathcal {X}}_1\times {\mathcal {X}}_2 | x_2 \ge {\tilde{x}}_2(x_1)\}\) and \(f_2\) is componentwise convex with respect to \(x_2\) on \({\mathcal {S}}_2:=\{(x_1,x_2)\in {\mathcal {X}}_1\times {\mathcal {X}}_2 | x_2 \le {\tilde{x}}_2(x_1)\}\). To show that f is componentwise convex with respect to \(x_2\) on \({\mathcal {X}}_1\times {\mathcal {X}}_2\), it remains to show that the difference between the right and left derivatives of f with respect to \(x_2\) at the boundary between the subdomains \({\mathcal {S}}_1\) and \({\mathcal {S}}_2\) is non-negative. To this end, we solve a third optimization problem given asand examine the sign of the optimal objective value. When applying the above procedure for testing componentwise convexity with respect to \(x_1\), the order of the derivatives of \(f_1\) and \(f_2\) in (5) needs to be exchanged in case \({\tilde{x}}_2(x_1)\) is increasing. An analogous procedure can be used for testing for componentwise concavity.

Once it has been established that a function is componentwise convex (concave), its concave (convex) envelope is known to be vertex polyhedral (Tardella 2004). Since the multivariate functions considered herein are only bivariate, this envelope can be computed efficiently using the method of Meyer and Floudas (2005). A convex (concave) relaxation, on the other hand, can be computed using the method of Najman et al. (2019a) if the derivatives of the function fulfill certain additional requirements. However, since some of the functions herein are nonsmooth, we need to adapt the latter method. First, we define partial subderivatives in analogy to the definition of subgradients for convex functions (cf. Rockafellar 1970).

Using this definition of the partial subdifferential, we introduce the following adapted version of Theorem 1 from Najman et al. (2019a):

In particular, Theorem 1 can be applied to componentwise convex functions of form (2). Similar to Najman et al. (2019a), up to 2 valid relaxations can be obtained from Theorem 1 by reordering the variables and changing the sign of the coordinates. Furthermore, an analogous way for constructing concave relaxations of componentwise concave functions is given by considering \(-f\) instead.

In Corollary 1 of Najman et al. (2019a), a convenient procedure based on mixed second-order partial derivatives is provided for confirming that the conditions of Theorem 1 therein are satisfied. This procedure is not directly applicable for confirming the validity of (7) because functions of form (2) are possibly not twice differentiable, despite the fact that \(f_1\) and \(f_2\) are both twice differentiable. Therefore, we describe an alternative way of confirming the validity of (7) for functions considered of form (2), which is analogous to the test for componentwise convexity described above: First, we solve the two optimization problemsto global optimality. If the solution values of both problems (8) and (9) are non-negative, Corollary 1 of Najman et al. (2019a) suggests that the corner \({\mathbf {x}}^{c } =(x_1^{L },x_2^{L })\) can be used in Theorem 1 in order to construct the convex relaxation. Assume w.l.o.g. that f equals \(f_2\) at \({\mathbf {x}}^{c }\). In order to guarantee that the relaxation resulting at \({\mathbf {x}}^c\) is also valid for f, a sufficient condition is thatis non-negative for all \(x_1\in {\mathcal {X}}_1\) if \({\tilde{x}}_2(x_1)\) is decreasing and non-positive if \({\tilde{x}}_2(x_1)\) is increasing. Since this condition is equivalent to the one for confirming componentwise convexity via problem (5) or concavity via the equivalent maximization problem, respectively, the condition is automatically satisfied for componentwise concave functions in case \({\tilde{x}}_2(x_1)\) is increasing and componentwise convex functions in case \({\tilde{x}}_2(x_1)\) is decreasing. An analogous procedure can be used for testing whether the corner point \({\mathbf {x}}^{c } =(x_1^{L },x_2^{U })\) can be used in Theorem 1.

The \(\alpha\)BB method was introduced as a general-purpose method for constructing relaxations of twice continuously differentiable functions (Androulakis et al. 1995). The basic idea is to add a quadratic function with parameters \(\pmb {\alpha }\) to construct convex underestimators, where the \(\pmb {\alpha }\) parameters are chosen such that the quadratic function offsets the smallest eigenvalue of the Hessian of the function over the considered domain. These \(\pmb {\alpha }\) parameters can either be precomputed and thus be independent of the interval, or they can be computed using interval methods when constructing the relaxation for a specific interval (Adjiman et al. 1998).

Hasan (2018) recently introduced a variant of \(\alpha\)BB that does not use the \(\pmb {\alpha }\) parameters to construct convex underestimators directly, but rather to construct a componentwise concave underestimator, and then uses the vertex polyhedral convex envelope of the latter (cf. Sect. 2.2) as convex underestimator. Herein, we use a similar approach in the sense that we only aim at achieving componentwise convexity or concavity and not convexity or concavity of the function directly.

Since both the original \(\alpha\)BB method and the variant of Hasan (2018) require the function to be twice continuously differentiable, we consider the following procedure for two dimensional functions of form (2): If the solution of problem (5) is non-negative, we use the minimum of the solution values of problems (3) and (4) to construct under- and overestimators as in Lemma 1 that are componentwise convex with respect to \(x_1\). Instead, if the solution value of the maximization problem analogous to (5) is non-positive, we solve the maximization problems analogous to problems (3) and (4) and use the maximum of their solution values to construct under- and overestimators as in Lemma 2 that are componentwise concave with respect to \(x_1\). The same procedure is used to construct under- and overestimators that are componentwise convex or concave with respect to \(x_2\).

In the present work, we only use \(\alpha\)BB to compute relaxations of intrinsic functions within larger factorable functions, which are then relaxed using the multivariate McCormick theorem (cf. Sect. 2.1). Therefore, we need to be able to compute minima of convex relaxations and maxima of concave relaxations, respectively, over boxes, either analytically or with little effort. However, the \(\alpha\)BB-type relaxations (10)–(13) may not be monotonic even when the original functions are, thus complicating the application of the multivariate McCormick theorem. We therefore add a linear term to functions described in (10)–(13) to make the final \(\alpha\)BB relaxation monotonic. While this makes the final relaxations slightly weaker, it greatly simplifies the application of the multivariate McCormick theorem.

In order to apply Theorem 1, it may additionally be necessary to alter the mixed second-order partial derivatives of a function f, which we achieve with mixed\(\alpha\)BB terms. For example, to obtain an underestimator that has a non-negative mixed second-order partial derivative with respect to \(x_i\) and \(x_j\), where \(i,j\in \{1,\dots ,n\}\), throughout its domain, we usewith \(\alpha ^{\mathrm{pos}}_{i,j}\ge \max \left\{ 0,-\frac{1}{2}\min _{{\mathbf {x}}\in {\mathcal {X}}} \frac{\partial ^2 f}{\partial x_i \partial x_j}\right\}\) in analogy to Lemma 1. This procedure can be used analogously to achieve a non-positive mixed second-order partial derivative and for overestimators.

For the bivariate functions, we first determine the physical domains and box domains according to Definitions 7 and 8. For the functions \(h_1(p,T)\), \(s_1(p,T)\), \(h_2(p,T)\), and \(s_2(p,T)\), which are obtained as derivatives of the basic equations of the IAPWS-IF97, the domains can be taken directly from the model definition (IAPWS 2007a). For most other functions, the physical domains are not explicitly given in the IAPWS-IF97. In this case, the physical domains need to be determined through an analysis of the monotonicity properties of related functions from the IAPWS-IF97 and the box domains need to be determined via globally maximizing and minimizing each variable over the physical domain. Examples for this procedure both for a straightforward case and a more involved case can be found in “Appendix 1” along with a summary of all box and physical domains in Tables 5 and 6.

We analyze the model functions and their derivatives to determine whether the functions exhibit excessive peaks or other undesired behavior in those regions of their box domains that are outside the physical domain. In case of undesired behavior, we replace the functions with a suitable extrapolation outside (a relaxation of) the physical domain. The resulting piecewise defined functions are of form (2) and will be called intermediate modified functions in the following and denoted with the superscript \(int\). The extrapolations are chosen such thatBeyond these intermediate modifications, we restrict the range of each function to that achieved over its physical domain. This helps to avoid domain violations when considering compositions of functions from the IAPWS-IF97. The resulting functions will be called final modified functions in the following and denoted with the superscript \(mod\).

The final modified functions are the ones that will be used in the optimization problems. The relaxations, however, will be constructed based on the intermediate modified functions that have more useful convexity properties (cf. Sect. 3.4). Based on these relaxations of the intermediate modified functions, relaxations for the final modified functions are obtained using the rules for composition with the \(\max\) and \(\min\) functions (Tsoukalas and Mitsos 2014).

Figure 2 shows the application of the above procedure to the function \(h_2(p,T)\). A more detailed description for this function can be found in “Appendix 2”. Similar modifications are conducted for the functions \(h_1(p,T)\), \(s_1(p,T)\), \(T_1(p,h)\), and \(s_2(p,T)\). The intermediate modifications are summarized in Table 7 in “Appendix 2”. The remaining bivariate functions are only cut at the minimum and maximum values occurring on their physical domains but not extended otherwise since their properties outside their physical domains are already satisfactory.

Note that the modifications of the model functions themselves are only conducted in parts of the box domain where the model has no physical meaning and that are excluded via suitable constraints when using the functions in an optimization problem. They thus do not alter the solutions of meaningful process models using these functions. Note also that the modifications do make the functions nonsmooth outside of the physical domains (both the replacement of the function with an extrapolation for some of the intermediate modified function and the restriction of the range of the functions to that over their physical domains when constructing the final modified functions). However, we did not find this to be an issue in practice so far (cf. discussion in Sect. 4).

To derive range bounds, we analyze the intermediate modified functions for monotonicity properties by globally maximizing and minimizing their first partial derivatives over the corresponding box domains using our open-source global solver MAiNGO (Bongartz et al. 2018). For functions that are replaced by an extrapolation outside their physical domains and that are hence of form (2), the maximization and minimization is done separately within the relaxed physical domain and the domain of the extrapolation. In both subdomains, the functions are differentiable by construction. Even though the functions may be nonsmooth at the boundary between the subdomains, monotonicity results within the subdomains can still be translated into global monotonicity properties since the functions are continuous. These properties also hold for the final modified functions since taking the maximum or minimum with a constant does not change the monotonicity properties.

For functions with monotonicity properties over the entire (box) domain, we immediately obtain exact range bounds. For other functions, exact range bounds can only be obtained in certain cases. In case a function is only monotonic with respect to one variable, an exact upper or lower bound can sometimes still be obtained in case of componentwise convexity or concavity with respect to the other variable (cf. Sect. 3.4). As a last resort, we obtain natural interval extensions from the FILIB++ library (Lerch et al. 2011) by evaluating the factorable representation of the function using the interval datatypes from FILIB++ either with respect to one or both variables over a suitable part of its domain. Examples for both a straightforward case and a more involved case can be found in “Appendix 3”, and all exploited monotonicity properties are summarized in Table 8.

Similar to the monotonicity analysis described above, we analyze the intermediate modified functions for (componentwise) convexity by globally maximizing and minimizing their second partial derivatives over their box domains using MAiNGO. For bivariate functions with piecewise definition of form (2), we proceed as described in Sect. 2.2. The identified properties are summarized in Table 9 in “Appendix 4”. These properties are used to derive relaxations as described in the following.

Compared to the original reduced-space formulation (Bongartz and Mitsos 2017), we need to make some slight changes to the model. These changes are required because the functions \(T_2(p,h)\) and \(T_2(p,s)\) for computing temperatures for given pressure and enthalpy or entropy in the gas phase are not available (cf. discussion in Sect. 3). Specifically, we need to use steam temperatures at the superheater outlets as optimization variables, either instead of the corresponding enthalpies as degrees of freedom (for Case Studies 2 and 3) or as an additional variable with a corresponding equality constraint (for Case Study 1). For Case Study 3, we also need to add the steam temperature at the inlet of the LP turbine, i.e., after mixing the HP turbine outlet with the LP superheater outlet, as well as the temperature of the hypothetical isentropic state at the outlet of the HP turbine. Additionally, we use mass flow rates of all branches of the flowsheet as design variables instead of the overall mass flow rate and split fractions, which we found to improve the relaxations.

We implement the models in C++ using two different ways of handling the IAPWS-IF97 functions: using their factorable representation, and considering them as intrinsic functions. In the former case, the factorable representation of each IAPWS-IF97 function is incorporated into the directed acyclic graph (DAG) built in MAiNGO by the MC++ library (Chachuat et al. 2015). Range bounds are then obtained via natural interval extensions by FILIB++ (Lerch et al. 2011), relaxations and their subgradients via the multivariate McCormick technique by MC++ , and gradients via automatic differentiation by FADBAD++ (Bendtsen and Stauning 2012). In the latter case, each IAPWS-IF97 function occurs as a single node in the DAG. In this case, the range bounds and relaxations derived in the previous sections are used instead. For gradients, we use automatic differentiation via FADBAD++ for each of the piecewise defined regions of the functions and arbitrarily use one of the one-sided limits at the nonsmooth kinks (the same is done for the \(\max\) and \(\min\) functions). While we are aware that this could potentially cause difficulties for the local subsolvers that rely on gradients, we did not experience such difficulties in the case studies (cf. Sect. 4.2). This is likely due to the fact that the performance of the local subsolvers for upper bounding is often not as crucial for the performance of global solvers. Nevertheless, in the future, it would be desirable to use recently published methods for generalized derivatives (Khan and Barton 2015) instead. Finally, we use custom relaxations for the equipment cost correlations and pressure factors for heat exchangers and deaerator vessels (Najman et al. 2019b), and for the logarithmic mean temperature difference in heat exchangers (Mistry and Misener 2016; Najman and Mitsos 2016). The process models are available via our website,² while the functions from the IAPWS-IF97 and the proposed relaxations are available in the MC++ library used by MAiNGO.

All problems are solved with MAiNGO v0.2.0 (Bongartz et al. 2018) using CLP 1.17.0 (Forrest et al. 2019) for the linear lower bounding problems constructed on the basis of the subgradients of the relaxations, Ipopt 3.12.12 (Wächter and Biegler 2006) as local NLP solver during pre-processing, and no local solver (pure function evaluation of the lower-bounding solution point) during the B&B. For range reduction, we use optimization-based bound tightening with trivial filtering (Gleixner et al. 2017), duality-based bound-tightening (Ryoo and Sahinidis 1995), and basic constraint propagation. The feasibility-tolerances are set to \(10^{-6}\) and the relative optimality tolerance to \(10^{-2}\). All calculations are conducted on an Intel^® Core^™ i5-3570 CPU with 3.4 GHz running Fedora Linux 30.

The optimal solution points and objective values of the three case studies are summarized in Tables 1, 2 and 3. To avoid confusion with the symbols introduced in the previous sections, the stream indices are prefixed by S in the tables. For comparison, the results with the very simple ideal thermodynamic models of Bongartz and Mitsos (2017) are also shown. Note that these deviate slightly from the original values (Bongartz and Mitsos 2017) when minimizing LCOE because the cooling water temperature in the condenser needed to be modified for the chosen condenser pressure to remain feasible when using the IAPWS-IF97. While the optimal objective values differ by less than 6% between the results with the IAPWS-IF97 model and the results with the ideal model, the optimal solution points and hence the optimal cycle designs differ significantly. For example, for Case Study 1, at the solution for maximum power output the cycle pressure \(p_{S2 }\) is at its upper bound (note that the bound \(p_{S2 }\le {10}\,{\hbox {MPa}}\) was chosen for consistency with the original problem of Bongartz and Mitsos (2017); therein, it was chosen because of the simplistic model that was expected to perform best at moderate pressures) when using the IAPWS-IF97 model, whereas with the simple model, it is approximately 50% lower (cf. Table 1). This highlights the importance of using detailed thermodynamic models for the design of steam power cycles.

When using McCormick relaxations for the IAPWS-IF97 functions, only Case Study 1 can be solved (rather quickly) within the given time limit of 24 h (see Table 4). For Case Studies 2 and 3, the global solution and thus the correct final upper bound (UBD; all problems are cast as minimization problems when implementing them for MAiNGO) is found during pre-processing as well, but the lower bound (LBD) barely improves from the values for the root node, resulting in large or even undefined (for \({\dot{W}}_{\mathrm{net}}\) in Case Study 3; the lower bound remains at minus infinity, i.e., no valid overall lower bound was identified) relative optimality gaps, defined as \(({\text {UBD}}-{\text {LBD}})/{\text {UBD}}\). Additionally, numerous numerical difficulties are encountered by CLP when solving the linear lower bounding problems (e.g., false infeasibility claims or solution points that are not actually feasible). These are likely due to the extremely large function values encountered outside the physical domains as well as the very weak relaxations (cf. Sect. 3).

When using the proposed relaxations, the solution of Case Study 1 takes almost 97% less B&B iterations and 86% (for \({\dot{W}}_{\mathrm{net}}\)) to 95% (for LCOE) less CPU time than the solution with McCormick relaxations. Unlike with McCormick relaxations, Case Study 2 can be solved quickly as well with either objective. For Case Study 3, only the problem of maximizing \({\dot{W}}_{\mathrm{net}}\) can be solved to the desired relative optimality tolerance of 1% within a few minutes, while the minimization of LCOE terminates at the CPU time limit of 24 h with a remaining relative optimality gap of 3.7%. Nevertheless, the optimality gap is closed much faster with the proposed relaxations than with McCormick relaxations (cf. Fig. 6). Furthermore, this problem can not be solved to the desired accuracy with the current version of MAiNGO when using the very simple ideal model of Bongartz and Mitsos (2017) either (cf. Table 4 and Fig. 6). This indicates that the difficulty with this problem is not purely due to the complexity or relaxations of the IAPWS-IF97.