## 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.

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## Notes

- 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.
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.
- 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.
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.

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## 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.

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## Appendices

### 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 *X*, *Y* 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

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).

### Proof

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

respectively, and it holds that

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

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

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

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

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

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.

### 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:

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

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

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

The outlet state then follows from the energy balance:

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

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

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

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

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

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

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 6*s* being in the two-phase region.

An energy balance then yields the true outlet state 6:

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 6*s*) being in the two-phase region.

Finally, the net power output of the cycle is

### 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:

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

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:

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

### 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

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:

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

### 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:

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

while for liquid streams they are given by

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

Unlike for the simple thermodynamic model used in the other case studies, these expressions for enthalpy and entropy cannot be solved for the temperature analytically^{Footnote 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 6*s*, 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]

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, respectively^{Footnote 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]:

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]:

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]:

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Bongartz, D., Mitsos, A. Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations .
*J Glob Optim* **69, **761–796 (2017). https://doi.org/10.1007/s10898-017-0547-4

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### Keywords

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