Unlocking the Potential of Synthetic Fuel Production: Coupled Optimization of Heat Exchanger Network and Operating Parameters of a 1 MW Power-to-Liquid Plant

The use of synthetic fuels is a promising way to reduce emissions significantly. To accelerate cost-effective large-scale synthetic fuel deployment, we optimize a novel 1 MW PtL-plant in terms of PtL-efficiency and fuel production costs. For numerous plants, the available waste heat and temperature level depend on the operating point. Thus, to optimize efficiency and costs, the choice of the operating point is included in the heat exchanger network synthesis. All nonlinearities are approximated using piecewise linear models and transferred to MILP. Adapting the epsilon constraint method allows us to solve the multi-criteria problem with uniformly distributed solutions on the Pareto front. The results show that compared to the conventional design process, the production cost can be reduced to 1.83 EUR/kg and the PtL-efficiency can be increased to 61.30 %. By applying the presented method, climate-neutral synthetic fuels can be promoted and emissions can be reduced in the long term.


Introduction
The transport sector contributed 7.98 Gt of CO 2 , equivalent to about 23 % of global CO 2 emissions, in 2022 [1].Reducing these emissions is crucial for achieving climate goals.Electrification of the transport sector will enable a significant reduction in climate-damaging emissions.Projections by the IEA indicate that the share of electric vehicles will increase from 1.5 % in 2021 to about 30 % by 2030 and more than 60 % by 2050 [2].Electrification is not possible for the entire transport sector.There is still no viable alternative to liquid fuels for maritime and aviation.Heavy machines such as snow groomers or agricultural machinery are similarly affected by a lack of climate-neutral powertrain systems.Synthetic fuels, or e-fuels, can be used directly in existing combustion engines as a drop-in solution to achieve climate neutrality.Compared to other climate-neutral fuels like hydrogen, using e-fuels offers the advantages of cheap transport and storage costs and existing infrastructure can be continued to be used.

Synthetic Fuel Production
The production of synthetic fuels involves multiple chemical conversion steps of mainly CO 2 , H 2 O and renewable electricity sources like wind, solar or photovoltaic.The CO 2 can be provided through different routes like capturing CO 2 from the atmosphere with direct air capture (DAC), water, biomass, or flue gas [3].Simplified, electrolysis reduces H 2 O to a hydrogen-rich synthesis gas.In a subsequent step, the gas is synthesized with purified CO 2 in a Fischer-Tropsch (FT) reactor.The hydrocarbons from the FT-reactor are then separated into different fractions such as naphtha, diesel and waxes.
Even though the key components such as electrolysis and FT-reactor are already commercially available, only a few plants are still in operation [4,5].The first commercial PtL-plant build was the Haru Oni in Punta Arenas, Chile [6].Methanol is produced using DAC and a polymer electrolyte membrane (PEM) electrolysis system powered by renewable electricity from a 3.4 MW wind turbine.During the pilot phase in December 2022, the first barrel of synthetic fuel could already be filled.The production is to be increased to 130 000 L in March 2023.The Norsk e-fuel plant in Mosjøen Norway is expected to produce up to 50 million liters of kerosene for aviation with renewable electricity from wind and hydropower as early in 2026 [7].The CO 2 required for the solid oxide electrolysis cell (SOEC) is supplied by DAC.According to Norsk e-fuel, three production facilities are expected to produce more than 250 million liters annually by the end of 2030.The George Olah plant in Svartsengi, Iceland, can produce up to 4000 tons (about 5 million liters) of methanol annually [8].A nearby geothermal power plant's off-gas provides the CO 2 feedstock.The alkaline water electrolysis is fed entirely renewably from the Icelandic power grid.INERATEC's PtL-plant in Karlsruhe, Germany, is expected to produce about 3500 tons (about 4.6 million liters) of synthetic kerosene and diesel per year, starting in 2023 [9].Feedstocks are up to 10 000 tons of biogenic CO 2 per year and renewable electricity.
The production of e-fuels is still in its earliest stage and significant challenges still need to be overcome [10].The central problem is that production costs are currently too high, which means that the fuel cannot be used in an economically viable way for end users [11].Ueckerdt et al. [12] have estimated production costs of 3.2 e/L for gasoline from 2020 to 2050.Ram et al. [13] have estimated production costs of 1.14 e/L for the year 2050.This forecast refers to all FTproducts.However, depending on the FT-reactor, the share of gasoline is only about a third, which again leads to an estimate similar to that of Ueckerdt et al.Based on the investment costs of the demonstration plant in Haru Oni, Chile, a current production price of about 50 e/L results.In contrast, the average wholesale price of fossil gasoline in 2021 was about 0.5 e/L [14].A CO 2 price of about 1000 e/tCO 2 would have to be introduced to create cost equality [15].According to Ueckerdt et al. [15], production costs of about 1 e/L will occur in the long term, which will make the fuel interesting for end users.Large and, above all, plants with low production costs and high efficiency must be built to achieve this price target.

Plant Design & Operation
Complex plants like a PtL-plant are composed of several units.At the design stage of a plant, the size of the units is primarily determined by the intended capacity of the plant.Standard manufacturers' sizes are used to reduce costs by buying off-the-shelf units.To guarantee the proper operation of the plant and the units, process engineers use commercial software such as Aspen HYSYS or IPSEpro.Due to the large number of units interacting with each other, many independent operating parameters result, which makes it challenging to find an optimal operating point empirically.Therefore, the plant's performance is highly dependent on the knowledge and experience of the process engineers.From an engineering point of view, it is necessary to use optimization methods to utilize the system's full potential.In contrast to commercial software, operational optimization enables the minimization of certain criteria like costs, efficiency, or emissions to find an optimal operation point.Al-Rashed & Afrand [16], for example, used a genetic algorithm (GA) to optimize the exergetic and economic efficiency of a combined gas turbine and supercritical CO 2 cycle for power production.They were able to find an optimal operating point with 25.3 % higher exergetic efficiency and achieved 24.6 % cost savings resulting from optimized inlet cooling of the compressor.Cao et al. [17] compared biomass gasification and digestion for a combined biomass to power and hydrogen plant.The units' characteristics were modeled to represent emissions, efficiency and levelized cost of product (LCOP) criteria and optimized with a GA.The results do not indicate a preferred system design.However, modeling the components depending on the operating point could create a holistic basis for decision-making.Hai et al. [18] optimized the operation of a solar-geothermal energy system providing electricity and hydrogen using GA.Optimizing the operating point of the components increased the power output by 500 kW to 4.099 MW.At the same time, the production of H 2 was increased from 8 g/s to 29 g/s.Wang et al. [19] optimized the operation strategy of a solar tower power plant using particle swarm optimization.They increased the daily power generation by 13.4 %.
Recent literature shows the great potential of optimizing the operating point.However, these methods are limited because the heat exchanger network (HEN) design is neglected.The HEN design is a crucial factor for cost and energy savings, especially in plants where much energy is necessary for heating and cooling process streams.Commercial software can be used to simulate an existing HEN.However, the optimal interconnection must be specified by the process engineers.The empirical rules of pinch analysis are often used [20].A promising approach for finding an optimal HEN is the heat exchanger network synthesis (HENS).With HENS, an optimal heat exchanger configuration can be found using mathematical programming subjected to given stream parameters such as temperatures and heat capacity flows.Some representative use cases and the achievable potential for cost and energy savings can be found in the publications of, for example, Yee & Grossmann [21], Escobar et a. [22] and Liu et al. [23].
A significant limitation of HENS is that a defined operating point and, as a result, constant stream parameters are assumed.In cases where the operating point affects the temperatures and flow capacities of the streams, the operating behavior cannot be covered by classical HENS.In contrast, when the operating point is optimized, a constant configuration of the HEN is necessary.Since the operating point influences the HEN and vice versa, optimizing HEN and the operating point simultaneously will lead to better results.

Novelty & Contribution
In this work, we close a research gap by coupling the optimization of the HEN and operating parameters.Our method allows us to overcome design and operational optimization weaknesses and enforce both approaches' strengths.To exploit the potential of modern MILP solvers, we use piecewise linear approximations and efficient logarithmic coding.Applying an adapted epsilon constraint method enables the consideration of the trade-off between high process efficiency and low production costs.Therefore, decision-makers and process engineers can be provided with a holistic foundation for the process design.Our method is demonstrated on a novel 1 MW PtL-plant.We further present how to formulate the efficiency and production cost objective functions.To evaluate our method's potential, the results from coupled optimization are compared to results from the traditional plant design approach, where only the HEN is optimized.The results show that synthetic fuels can be produced with minimal production costs of 1.83 e/kg or at the highest efficiency of 61.84 %.The decarbonization of the transport sector can thus be accelerated and emissions can be avoided.

Paper Organization
The novel 1 MW PtL-plant studied is described in Section 2.1.In Section 2.2, we show the basic concept of the adapted HENS superstructure, the piecewise linear approximation, the logarithmic coding approach for an efficient transfer to MILP, and our adaptions to the epsilon constraint method.Since implementing the operating point dependent stream parameters is specifically tailored to the use case, the modeling is discussed in detail in Section 3. In Section 4, we show the results of our method in comparison to an optimization of the HEN only.

System Description
As part of the IFE (de.: Innovation Flüssige Energie, eng.: Innovation Liquid Energy) research project, a novel PtL-plant is designed and basically engineered.
The PtL-plant will be designed to a maximum electrolysis capacity of approximately 1 MW and will produce climate-neutral fuels for the transport sector from the feedstocks water, renewable electricity, exhaust gas from a cement plant and air.In Figure 1, the PtL-plant is schematically shown with its five main components: steam generation, CO 2 conditioning, high-temperature solid oxide co-electrolysis (co-SOEC), Fischer-Tropsch (FT) reactor with upgrading and combustion system.For simplicity, no valves, pumps, or compressors are shown.Heat exchangers indicate the heat transfer points of the HEN.The cold process streams, which must be heated, are shown in blue.Analogous, the hot process streams that must be cooled are shown in red.In the following sections, the central components are briefly discussed.

Steam Generation
The steam generator is fed with pure water at 20 • C. The conditioned water is preheated, evaporated and superheated by the heat exchangers in streams C7, C4 and C5.

CO 2 Conditioning
Due to the high energy consumption for the supply of CO 2 in sufficient purity, the quality of the CO 2 source is an essential parameter for the design.In this case, exhaust gas from a cement plant is used due to the high CO 2 concentration.The exhaust gas enters the conditioning with 15.17 wt% CO 2 , 81.11 wt% N 2 and 3.72 wt% H 2 O and a temperature of 40 • C. To control the temperature for adsorption and desorption, the hot streams H7 and H11 must be cooled and the cold stream C2 heated.At the end of the conditioning process, the CO 2 has a purity of 98.73 wt% and a low residual content of water.

Co-SOEC
The central element of the PtL-process is the co-SOEC.The conditioned CO 2 is superheated within stream C6 and mixed with the superheated steam.Within stream C1, the mixture is further superheated to the operating temperature of the co-SOEC.Both are reformed with preheated air at temperatures between 800 and 900 • C to an H 2 -rich gas and CO.Before entering the FT reactor, the synthesis gas leaving the co-SOEC is cooled in four stages and condensed constituents are separated.A compressor-based cooling system supports the second cooling stage.
With a maximum power of approx. 1 MW, the co-SOEC is the largest electricity consumer in the system.The cell voltage has a strong influence on the overall power consumption and subsequently on the efficiency and production costs.

FT-Reactor & Upgrading
In the FT-reactor, the synthetic gas from the co-SOEC is passed over a catalyst at high temperature and pressure.In the subsequent upgrading process, the FT-syncrude is separated into the fractions FT-wax, diesel and naphtha and prepared for final use.The unreacted synthesis gas is partially recirculated and fed to the combustion system.The product properties downstream of the upgrading are given in Table 1.The combustion system serves as an internal hot utility and provides energy to heat the cold process streams.The CS consists of three serially connected combustion chambers.The offgas from the separation, which can no longer be recirculated, is used as fuel.Combustion takes place at high air surplus (λ > 150 for 1 st CS; λ > 30 for 3 rd CS).The first combustion chamber is supplied with the entire exhaust air from the co-SOEC.The other two combustion chambers are each fed with the exhaust gas from the combustion chamber in advance.
Since the CS is used as an internal hot utility, the optimal design is a decisive factor for the efficiency of the HEN and the overall process.The selection of the combustion parameters inlet and outlet temperature into the combustion chamber and fuel mass flow are, in contrast to other processes, not closely linked to technical limitations.Only the material-specific temperature limit of 900 • C must not be exceeded.The inlet temperature of the air mainly influences the outlet temperature from the combustion chamber and can be adjusted by the fuel quantity.The inlet temperature results from the streams connected within the HEN.Due to the many freely selectable parameters and the strong influence on the efficiency of the overall process, the design of the CS is a significant challenge for process engineers.

Methods
In this section, we present the underlying idea behind the coupled optimization of the HEN and operating point.The concept is shown schematically in Figure 2. The upper third represents the physical system composed of two units with two hot and two cold streams.For simplicity, the heat exchanger network is not illustrated.The operating characteristics of each unit can be reduced to multiple independent operational parameters.It is assumed that a change of the operating parameters leads only to a change of the stream parameters T in , T out and F .Depending on the process, all, none, or only some of the three stream parameters can be affected.In the middle of Figure 2, the HEN with N st stages and all possible interconnections are shown for the same system.This graph-theoretic representation of the HEN is based on the superstructure formulation of Yee & Grossman [21].The heat exchange between the hot and cold streams can occur in N st stages with stream splits.The hot and cold utilities are located at the stream ends.
The coupling between the optimization of the HEN design and the operating point is achieved by implementing models that describe the operating characteristics of the units.In Figure 2, at the bottom, the models of the two units are shown schematically.Each unit has an independent operating parameter; in this case, only inlet or outlet temperatures change.The model of unit 1 represents the relationship between the outlet temperature of stream H1 and the inlet temperature of stream C1.The model of unit 2 illustrates the correlation between the outlet temperatures of stream C1 and C2 along with the inlet temperature of stream H2.The objective links the operating parameters.The coupled HEN design problem with variable stream parameters is solved using the adapted superstructure formulation of Huber et al. [24].

Linearization
All nonlinearities of the HENS are piecewise linear approximated with superpositioned planes in the two-dimensional.Plane Simplices are used for threedimensional correlations.Detailed information regarding the methodology can be found in the paper by Huber et al. [24].
Analogously, the units' operating and stream parameters correlations are linearly approximated.The advantage of this method is that the characteristics of the units can be represented with the help of black-, grey-or white-box models.The resulting flexibility in choosing the modeling approach reduces limitations regarding data availability and increases the proposed method's applicability.

Transfer to MILP
All piecewise linear functions are transferred to MILP with the least possible number of binary variables to reduce computation time.One-dimensional, mainly convex curved functions are transferred to MILP without binary variables.All other functions require the use of binary variables.Applying a logarithmic coding approach, according to Vielma and Nemhauser [25], can reduce the number of binary variables to a minimum.Further information about the transfer to MILP can be taken from [24].

Multi-Objective Optimization
The two-objective optimization problem is solved with an adapted epsilon constraint method.As shown in Equation (1), one objective function is minimized and the second is constrained with an upper and lower bound.The epsilon parameters are chosen to be in the range of f min 2 In contrast to the conventional epsilon constraint method, overhanging regions of the Pareto front can be covered.Equidistantly distributed points on the Pareto front can be calculated with minimal computational effort.

Modeling
The models are created based on data from steady-state process simulations by project partners using Aspen HYSIS.Modeling is based on the following assumptions: • The system was simulated with seven different cell voltages between lower (U min cell = 1.275V) and the upper technical limit (U max cell = 1.305V). • The stream parameters T in , T out and F are only dependent on the cell voltage U cell .
• The stream parameters of the CS are independent of the cell voltage.
Temperatures and heat capacity flows are limited by the amount of offgas available.
• The sizes and parameterization of the units are independent of the cell voltage.Identical system costs are assumed.

Feedstock
In addition to climate-neutral electricity, the primary feedstocks of the PtLprocess are H 2 O, CO 2 and air.All three mass flows depend only on the cell voltage.Figure 3 shows the piecewise linear models of the feedstock flows.

System Power & FT-products
The parameters which significantly influence the overall process's performance are the system power P sys and the massflow of FT-products v ṁprod,v .Figure 4 shows the simulated data points and the model with piecewise linear approximated lines as a function of cell voltage U cell .Both P sys and product output increase with higher cell voltage.The model on the left side in Figure 4 describes the system's power consumption.P sys represents the required power to run the co-SOEC, circulation pumps, valves, and control equipment, and to cover losses.The power consumption of utilities is considered in the modeling of the objectives in Section 3.5.Figure 4 on the right shows the product flow downstream of the FT-reactor as the sum of the fractions FT-wax, diesel and naphtha.The composition of the fractions changes depending on the cell voltage.Figure 5 shows the absolute and relative share of the product flows.The percentage of FT-wax increases with increasing cell voltage.With an almost unchanged share of diesel, the share of naphtha is reduced simultaneously.Since we subsequently relate the production costs to all FT-products, the relative share of the fractions is not considered further.

Streams
The stream parameters T in , T out and F are depending on the cell voltage.For each parameter, a piecewise linear approximation is generated.Figure 6 shows the results from process simulation and the piecewise linear approximation for stream H9.In this case, all parameters increase with increasing cell voltage.All other stream models can be found in the supplementary material.Table 2 summarizes the stream parameters' bounds for all streams.Values without brackets are independent of cell voltage and constant.Especially for processes that require defined inlet and outlet conditions, the temperatures remain constant and only the flow capacity changes.This can be seen in the stream data of streams H12 and H4-H6 of the FT-reactor and upgrading.Stream parameters in square brackets are variable and depend on the cell voltage.The minimum and maximum values from the process simulation are given.The overall heat transfer coefficients were assumed to be U = 0.5 kW/(m 2 K) for all streams.

Combustion System
The stream parameters of the CS are independent of the cell voltage; thus, no piecewise linear approximation is needed.Table 3 summarizes all stream parameters and heat transfer coefficients.The exhaust stream's inlet temperature corresponds to the combustion chamber's outlet temperature.An upper technical limit of 900 • C was chosen for all three streams.The outlet temperature can vary between 100 • C and 890 • C. The boundaries of the heat capacity flows were selected to guarantee that always enough offgas is available for combustion.All overall heat transfer coefficients were assumed to be U = 0.5 kW/(m 2 K).

Objectives
Considering the objective functions efficiency and cost simultaneously makes sense when optimizing energy conversion systems [16].These objective functions are commonly antagonistic, allowing multiple optimal solutions to be represented as a Pareto front.The fuel production costs are minimized and the PtL-efficiency is maximized.The two objective functions reflect the trade-off between efficiency and productivity.

PtL-Efficiency
The objective to maximize the PtL-efficiency η PtL is defined by Equation (2) as the ratio of chemically bounded energy to electrical energy input.
The numerator describes the chemically bounded energy of the FT-products downstream of the separation.The denominator represents the total electrical energy input P el .Both the hot and cold utilities are electrified.The coefficient of performance ε describes the electrical-to-thermal energy input ratio.
The Ptl-efficiency from Equation ( 2) forms a non-linear correlation of the optimization variables for the chemically bounded energy in the product Ḣprod and the electrical energy required P el .The points in Figure 7 on the right show the non-linear function of the PtL-efficiency.Using MILP requires a piecewiselinear approximation.In this case, we used simplices on a regular grid with 4 x 4 points, see Figure 7 on the left.With 18 simplices, the objective can be approximated with sufficient accuracy at an RMSE of 0.61 %.

Production Costs
The objective to minimize the specific production costs c prod is defined according to Equation ( 3) and describes the ratio of total annual costs to product output.
The T AC are composed of annual capital expenses CAPEX and operational expenditures OPEX .According to Equation ( 4), the CAPEX comprises the investment costs for the system and the heat exchanger network.The investment costs C sys include all relevant investments related to the PtL-plant, excluding the costs for the heat exchanger network.The other terms in Equation ( 4) represent the investment costs for the heat exchanger network, according to Yee & Grossman [21].Fixed costs for all heat exchangers and variable costs proportional to the heat exchanger area are considered.The investment annualization factor can be expressed as AF inv = 1 /a.Where a is the depreciation period in years.In contrast to linear depreciation, other options are also possible [26].
q hu,j U hu,j LMTD hu,j β variable HEX hot utility costs According to Equation ( 5), the operational expenses are composed of feedstock and electricity costs.The OPEX depend on the annual full load hours t and are usually depreciated within one year, which results in an operational annualization factor of AF op = 1.
The nonlinear objective of the minimum fuel production costs is piecewiselinear approximated and transferred to MILP. Figure 8 shows the approximation in its valid domain on a 3 x 3 grid.With only eight simplices, an RMSE of 0.37 % can be achieved.The three nonlinear terms of the variable heat exchanger costs in Equation ( 5) are piecewise-linear approximated analogously to the procedure of Huber et al. [24].Accordingly, the stream and utility heat exchanger area correlations are approximated by superpositioned planes and transferred to MILP without additional binary variables.

Costs
As the number of full load hours of the plant increases, the net production costs decrease [27].Like the authors in [27,28,29], we also assume t = 8000 h/y full load hours per year.All components are depreciated linearly within 20 years resulting in an annualization factor of AF inv = 1 /20.
The fixed investment costs have been estimated by the project partners responsible for the economic viability to be C sys = 10 000 000 e.This estimate assumes that the cost of the central components will decrease significantly due to technological advances [12].In the papers of G. Herz et al. [28] and D.H. König et al. [30], costs in the range of 9 600 000 e and 22 000 000 e have been predicted, which legitimizes the intra-project estimation.
The feedstock for the production of synthetic fuels can be reduced to H 2 O, CO 2 rich exhaust gas and air in a simplified form.Table 4 lists the corresponding cost factors c f .The goal of producing CO 2 -neutral fuels can only be achieved by using electricity produced in a CO 2 -neutral way.The electricity required for this will be generated from a mix of wind and solar PV.At a projected plant location in Europe, an average electricity price of c el = 20 e/(MW h) is assumed, according to J.L.L.C.C. Janssen et al. [33].

Heat Exchanger Network
The heat exchanger network was defined with N st = 3 stages for heat exchange and a minimum temperature difference of ∆T min = 1 K.
Air coolers are used as cold utilities.Since the environment is used as a heat sink, a coefficient of performance of ε uc = 0.05 is assumed to cover evaporation losses and mechanical work.Electrical heating elements with a coefficient of performance of ε uh = 1.05 are used as hot utilities.
The cost parameters for the heat exchangers in the heat exchanger network have been determined based on the DACE Price Booklet [34].We have selected stainless plate heat exchangers made of AISI 316.The cost parameters are c f,hex = 1013.6e/y and c v,hex = 61.8e/(m 2 β y) with a degressive cost exponent of β = 0.8.

Piecwise-linear Approximation
Superpositioned lines, planes or simplices were added to all piecewise-linear models until an RMSE of less than 1 % was achieved.When approximating with simplices, care was taken to fully utilize the logarithmic encoding with respect to the selected grid points.

Implementation
All optimization problems were coded in MATLAB 2020b [35].YALMIP R20210331 has been used as an interface between MATLAB and the MILP solver [36].Gurobi 10.0.0 is used as the MILP Solver [37].A MIP of less than 1 % was defined as a termination criterion for the solver.The calculations were performed on a 64-core server (AMD EPYC 7702P) with 265 GB of RAM.

Results
The coupled optimization results are compared with two conventionally designed plants without optimization to quantify the presented method's potential.The following assumptions are made for the conventional plant design: • The two operating points are set at the extreme cell voltage values.
• The HEN is designed empirically following the PINCH rules.
• The stream parameters of the CS are chosen empirically.
Figure 9 shows the two solutions of the conventional design without optimization and the Pareto front of the coupled optimization as a function of the cell voltage.The different cell voltages are color coded.Figure 9 shows the two solutions of the conventional design without optimization and the Pareto front of the coupled optimization as a function of the cell voltage.The Pareto front is constructed with 42 points.The gaps in areas of high production costs result from the solver timeout.Within 12 hours, no solutions with a MIP gap of less than 1 % could be found.A high cell voltage generally leads to low efficiency and production costs.A low cell voltage, on the other hand, leads to increased efficiency and, at the same time, high production costs.The two objective functions are intensely antagonistic, at least near the optimum.Table 5 shows the cell voltages, efficiencies, and production costs of the extreme points of the Pareto front and the conventional approach.The results of the coupled optimization dominate the results of the conventional approach.At a cell voltage of 1.305 V, the production cost can be reduced by 5.22 ct (2.77 %) while increasing the efficiency by 0.504 percentage points (0.88 %).Production costs at a cell voltage of 1.275 V are almost identical at 2.355 e/kg compared to 2.358 e/kg.In contrast, the efficiency can be increased by 0.514 percentage points (0.84 %).The coupled optimization yields better results regarding both efficiency and production costs.The shape of the Pareto front, whether convex or non-convex, provides information about the nature of the underlying problem and the trade-offs between the different objectives.A convex Pareto front indicates clear dominant solutions where an improvement in one objective necessarily leads to a degradation of the other.A distinct optimal solution usually combines the best values for all objectives in such cases.In contrast, in this case, a non-convex Pareto front indicates no unambiguously dominant solution and different combinations of objective functions represent equivalent alternatives.Looking at the Pareto front from Figure 9, we find that the efficiency remains nearly constant in the low-production cost region at about 58 %.For a cell voltage of U min cell = 1.305V, the production costs vary between 1.834 e/kg and 2.057 e/kg, while the efficiency varies between 57.803 % and 58.371 %.In our case, an identical cell voltage means that the solutions differ only in the HEN design and the parameterization of the CS.It can be concluded that, especially in this range, the influence of the design variables has significant effects on the production costs with negligible effects on efficiency.Similarly, in the high production cost range, a significant increase in efficiency can seen with a minor increase in production cost.From 2.068 e/kg to 2.297 e/kg, the PtL efficiency increases by 3.963 percentage points to 61.785 %.However, solutions show different cell voltages in this interval.It can be concluded that the choice of cell voltage has a more significant impact on the objective functions than the choice of HEN and the parameterization of the CS.In the range of highest production costs between 2.297 e/kg and 2.355 e/kg, four solutions with a cell voltage of U min cell = 1.275V are found.The efficiency varies only by about 0.096 % in the nearly horizontal zone increasing from 61.785 % .The production costs, on the other hand, differ by up to 5.863 ct/kg.
The results show that in our use case, the cell voltages of the Pareto optimal solutions do not change in the horizontal section at the beginning and end of the Pareto front.In this region, the HEN design and the parameterization of the CS are of crucial importance.However, in the near-vertical part of the Pareto front, the choice of cell voltage significantly affects the objective functions.The HEN design and the parameterization of the CS have only a minor influence.Therefore, the coupled optimization of the operating parameters and the HEN is essential if the impact of different design parameters cannot be estimated a priori.

Conclusion
In this paper, we present a method that enables coupled optimization of design and operating parameters in the heat exchanger network (HEN) problem.The coupling is realized by an operating point dependent behavior of stream parameters inlet, outlet temperature and flow capacity.All nonlinearities are approximated with piecewise linear approximations to keep the problem traceable and to leverage the potential of fast MILP solvers.The transfer to MILP is done highly efficiently using logarithmic coding.The method is applied to a novel 1 MW PtL-process.Selecting the cell voltage of the co-SOEC and the combustion system's (CS) parameterization is crucial for the process design.These complex processes cannot be evaluated purposefully based on only one objective.Therefore, multi-criteria optimization of PtL-efficiency and production costs provides a robust decision basis for comprehensive process assessment.
Our results show that coupled optimization leads to better results than the conventional design approach.Higher efficiencies can be achieved with lower production costs at the same time.By modeling the systems' operational and design parameters, it is possible to optimize the overall process comprehensively and to calculate a Pareto front that reflects the trade-off of the objective functions.The shape of the Pareto front provides valuable insight into the nature of the problem.In this paper, a non-convex Pareto front was observed, suggesting that no uniquely dominant solution and different combinations of objectives represent equivalent alternatives.Efficiency remained approximately constant in the low-production cost region.This indicates that the influence of HEN and CS parameters significantly affects production costs, while the effects on efficiency are negligible.On the other hand, in the high production cost area, a slight increase in production cost resulted in a significant improvement in efficiency.The choice of cell voltage was found to have a more substantial impact on the objective functions than the choice of HEN and the parameterization of the CS.The results underline the relevance of coupled optimization, especially when the effects of the different operational and design parameters cannot be estimated a priori.
With our method, we close a research gap by coupled optimization of operating and design parameters.Simultaneously considering several objective functions enables a comprehensive analysis of the solution space based on Pareto fronts.This allows synergy effects to be exploited and optimal solutions to be identified highly efficiently.Further research and analysis could focus on implementing additional operational or design variables to improve the overall performance and sustainability of the PtL-plant.With this work, we can contribute to the highly efficient and cost-effective production of synthetic fuels, which will promote timely large-scale industrial production and reduce emissions in the long run.

Figure 1 :
Figure 1: Schematic representation of the 1 MW PtL-plant with the five main components: steam generation, CO 2 conditioning, co-SOEC, FT-reactor with upgrading and combustion system.

Figure 2 :
Figure 2: Schematic representation of the method for two units with two hot and two cold streams.

Figure 5 :
Figure 5: Absolute and relative product composition as a function of cell voltage U cell .

Figure 7 :
Figure 7: Piecewise-linear approximation with 18 simplices of the PtL-efficiency η PtL as a function of chemically bounded energy in the produced fuels and electricity demand P el .RM SE = 0.61 %.

Figure 8 :
Figure 8: Piecewise-linear approximation with eight simplices of production costs c prod as a function of the total annual costs T AC and that total product flow v ṁprod,v .RM SE = 0.37 %.

Figure 9 :
Figure 9: Non-dominated solutions and Pareto front of coupled optimization (circles).Dominated solutions of the conventional approach (diamonds).

Table 1 :
Chemical and physical product properties at 40 • C and 101 324.97 Pa downstream the upgrading.

Table 2 :
Stream data with limits for inlet, outlet temperature and flow capacity.

Table 3 :
Stream data with limits for inlet, outlet temperature and flow capacity.Stream T in / • C T out / • C

Table 4 :
Feedstock cost factors.CO 2 treatment is considered in Csys and Psys.2Air is available free of charge.The process was designed to use air with ambient conditions.

Table 5 :
Comparison of extreme values from coupled optimization and conventional approach without optimization.