Flexible operation of modular electrochemical CO 2 reduction processes

: Electrochemical CO 2 reduction (eCO 2 R) is an emerging technology that is capable of producing various organic chemicals from CO 2 , but its high electricity cost is a big economic obstacle. One solution to reduce the cumulative electricity cost is demand side management, i.e., to adjust the power load based on time-variant electricity prices. However, varying the power load of CO 2 -electrolyzers often leads to changes in Faraday eﬃciency towards target components and thereby inﬂuences the product composition. Such deviations from the target product composition may be undesired for downstream processes. We tackle this challenge by proposing a ﬂexible operating scheme for a modular eCO 2 R process. We formulate the economically optimal operation of an eCO 2 R process with multiple electrolyzer stacks as a parallel-machine scheduling problem. Adjusting the power load of each sub-process properly, we can save electricity costs while the desired product composition is met at any time. We apply an algorithm based on wavelet transform to solve the resulting large-scale nonlinear scheduling problem in tractable time. We solve each optimization problem with a deterministic global optimization software MAiNGO. We examine ﬂexible operation of a modular eCO 2 R process for syngas production. The case studies show that the modular structure enables savings in the cumulative electricity cost of the eCO 2 R process via ﬂexible operation while deviations in the syngas composition could be reduced. Also, the maximum ramping speed of the entire process is found to be a key parameter that strongly inﬂuences the cost saving.


INTRODUCTION
Electrochemical carbon dioxide (CO 2 ) reduction (eCO 2 R) is an emerging technology that is capable of producing various organic chemical compounds out of CO 2 . Being integrated with renewable energy systems, eCO 2 R has been known as a promising alternative to conventional fossilbased chemical production. Recent research on eCO 2 R has focused on synthesis of novel catalysts for the production of, e.g., carbon monoxide, formic acid, ethylene, and ethanol (De Luna et al., 2019). In addition, electrode and reactor design (Vennekoetter et al., 2019), conceptual process design (Chae et al., 2020), and techno-economic and sustainability analysis (Roh et al., 2020) are of interest as well.
The electricity cost is one of the major cost drivers of eCO 2 R technologies (Jouny et al., 2018). A straightforward solution to the electricity cost reduction is to lower the overpotential of, particularly, the anode oxygen evolution reaction (Sun et al., 2017). On the other hand, if an eCO 2 R process is powered by grid electricity, one can adjust the plant load to fluctuations in electricity (spot) prices. This flexible operation, namely demand side management (DSM) or demand response (DR), leads to savings in overall electricity costs (Daryanian et al., 1989). In academia, DSM has been applied to various industrial plants, such as air separation units, chlor-alkali plants, and aluminium plants .
A major challenge of flexible operation of eCO 2 R processes is that the composition of the effluent stream depends on the current density (or applied voltage). For example, increasing the current density of CO 2 -electrolyzers dedicated to syngas (a mixture of hydrogen and carbon monoxide) production leads to a higher H 2 /CO ratio due to the changes in the Faraday efficiencies toward hydrogen and carbon monoxide (Vennekötter et al., 2019). The deviation of the product composition from the nominal level is typically undesired for downstream processes. This feature is distinct from other conventional electrolysis, such as water electrolysis and chlor-alkali electrolysis, in which the product concentrations do not significantly change with respect to the current density. To meet the product specification, a part of the components should be separated, which would complicate the process configuration and worsen the economic viability as well.
To tackle this challenge, we propose a flexible operation of modular eCO 2 R processes. Motivated from the parallel machine scheduling (Jans, 2009), an eCO 2 R process consists of N identical sub-processes that can be operated independently from one another. Adjusting the power load of each sub-process properly according to the instantaneous electricity prices rather than keeping the power load fixed, we can save the overall electricity costs while the desired product concentration can be met.
We formulate scheduling problems to determine the optimal load profile of each sub-process. In this study, syngas is chosen as a target product of an eCO 2 R process. Since the syngas production rate and power requirement are represented as nonlinear functions of the current density, we solve nonlinear scheduling problems. In order to reduce the computational cost, we apply our recently proposed algorithm based on wavelet transform (Schäfer et al., 2020). We conduct a sensitivity analysis that perturbs the maximum ramping speed of each sub-process to see how much it influences the electricity cost savings. Figure 1 depicts the block flow diagram of an eCO 2 R process, which is modularized and thus capable of running flexibly. The entire system consists of N identical subprocesses that are independently operable. Herein, we consider a moderate size of N (2 " 8). Carbon dioxide and water are fed into electrolyzer stacks in each sub-process and then target chemicals are synthesized at the cathode chambers via electrochemical CO 2 reduction reactions. Meanwhile, oxygen, a byproduct, is generated in the anode chambers.

SYSTEM DESCRIPTION
Recent studies on electrochemical CO 2 reduction have reported per-pass conversion of CO 2 below 35% (Jouny et al., 2018). Thus, the significant amount of unreacted CO 2 remains in the cathode effluent streams and has to be separated by a CO 2 separation unit(s). In this study, we introduce N identical CO 2 separation units connected to N CO 2 -electrolyzer stacks to treat the respective cathode effluent. The separated CO 2 is then preferably recycled for reducing the CO 2 feed cost.
To enable DSM, the process requires overcapacity, meaning that we install more than the minimum number of electrolyzer cells needed to achieve the target production rate. In addition, the products from all the stacks are mixed and stored in a storage tank. This intermediate storage is mandatory for the constant supply of the product to the downstream process.

MATHEMATICAL FORMULATION
The scheduling model presented below is developed to determine the optimal operation of each sub-process of an eCO 2 R process. The following assumptions are made: ‚ quasi-steady-state model with discrete-time variables; ‚ perfect forecast of electricity prices in the day-ahead market; ‚ no side-product generated; ‚ unreacted CO 2 in the cathode effluent is completely removed; ‚ considering power demand for electrolysis only.

Mass balances
The molar amount of a component c produced by stack s at time step t (n s,c,t ) is calculated by @s P S, c P C, t P T, (1) where FE s,c,t denotes the Faraday efficiency toward component c of stack s, j s,t the current density of stack s, A Cell the active electrode area, N Cell the number of cells per stack, ∆t the length of time step, F the Faraday constant, and z c the number of electrons transferred. Note that FE s,c,t of CO 2 -electrolyzers often sharply varies with respect to j s,t . For example, when a silver catalyst is applied to the cathode, the CO 2 reduction reaction that generates CO is superior at low current density due to its low onset potential while the hydrogen evolution reaction becomes dominant at high current density (Vennekoetter et al., 2019). Therefore, this dependency should be properly represented by exploiting either experiment or simulation data.
The total amount of each component and the target product is calculated by To maintain the product quality, an additional constraint is imposed: Qpn c,t q " q Prod , @t P T.
(4) where Qpn c,t q calculates the product quality as a function of n c,t , e.g., the H 2/ CO molar ratio of syngas, and q Prod is the target value.
We consider the amount of the stored product limited to the maximum storage capacity C Prod : where n N Prod is the nominal production level. Initially, half of the storage tank is assumed to be filled by the product.

Power consumption
The power consumption at time step t can be calculated by where U s,t denotes the cell potential of stack s at time step t. As the current density j s,t increases, the cell potential U s,t rises due to, for example, the activation and ohmic overpotentials.

Ramping constraints
Suitable ramping constraints should be imposed in order to make the quasi-steady state assumptions adequate for optimizing the operation of the dynamic systems, such aś ∆j ď j s,t´js,t´1 ď ∆j, @s P S, t P T, where ∆j denotes the maximal ramping speed. ∆j is calculated by where j max and j min denote the maximal and minimal allowable current density, respectively. T Ramp denotes the minimal ramping duration between the two extreme operating points.

Symmetry-breaking constraints
We impose the lexicographic ordering constraints (Sherali and Smith, 2001) that exclude alternative solutions to break the symmetry and shorten the computation time accordingly: j s,t ě j s`1,t , @s " t1...S T´1 u, t P T, (9) where S T is the total number of stacks. This constraint forces, for example, the current density of Stack 1 to be higher than those of other stacks over the simulation horizon.

Objective function
We minimize the sum of the electricity costs over the time horizon: min where p t denotes an electricity spot price.

CASE STUDY
We demonstrate the proposed operation concept on a syngas production system based on co-electrolysis of CO 2 and H 2 O. Four identical and independently operable subprocesses produce syngas via the following electrochemical reactions over Ag catalysts: 2 H``2 e´ÝÝÑ H 2 CO 2`2 H``2 e´ÝÝÑ CO`H 2 O.
We choose a desired H 2/ CO molar ratio of 1, which is suitable for liquid fuel synthesis via the Fisher-Tropsch process with iron-based catalysts (de Smit and Weckhuysen, 2008) and oxo-synthesis for isomeric aldehydes production (Billig and Bryant, 2005). Therefore, where q Syn " 1 and n H 2 ,t and n CO,t are the total molar amount of H 2 and CO produced at time step t. Each sub-process comprises one CO 2 -electrolyzer stack and one CO 2 separation unit. Each stack consists of 30 electrolyzer cells, so the total number of electrolyzer cells is 120. The total syngas production rate is 5.87 kmol{h, which can be manufactured by 100 electrolyzer cells operated at the nominal current density (117 mA/cm 2 ) determined by Brée et al. (2020). The effective area of one electrolyzer cell is assumed to be 2.7 m 2 , which is the size of the typical chlor-alkali electrolyzer cell (O'Brien et al., 2005). The syngas storage is capable of supplying syngas to a downstream for maximum of six hours (C Syn ). The time step size is an hour. The original dynamic model for calculation of Faraday efficiency and cell potential are highly nonlinear, so they are not suitable to the scheduling model. We derive the cubic polynomial functions f FE s,c,t and f U-j s,t as the surrogate functions of FE s,c,t j s,t (effective current density) in (1) and U s,t j s,t (power density) in (6) as follows: f FE s,c,t " FE s,c,t j s,t " α FE,c j 3 s,t`βFE,c j 2 s,t`γFE,c j s,t δ FE,c , @s P S, c P tH 2 , COu, t P T, (12) f U-j s,t " U s,t j s,t " α P j 2 s,t`βP j s,t`γP , @s P S, t P T,  2019), the current density is measured up to 100 mA/cm 2 . In this study, we extrapolate the Faraday efficiency and cell potential at the current density of 100 to 150 mA/cm 2 with the rigours dynamic model of Brée et al. (2020).
Regarding the removal of bulk CO 2 from syngas, several technical candidates are available, such as chemical absorption, adsorption and membrane gas separation. These options differ in energy demand, operating and capital expenses, and dynamic responses. In this study, we assume that the unreacted CO 2 is separated via an arbitrary separation technology. As mentioned in Section 3, we exclude the energy demand for CO 2 separation because it is much smaller than the energy demand for electrolysis. For example, CO 2 is assumed to be separated by pressure swing adsorption (PSA) that demands approximately 1.75 GJ elec {t CO 2 (Bui et al., 2018). This amounts to only 11.5% of the power demand of co 2 -electrolysis at the nominal current density (assuming 30 % of CO 2 per-pass conversion). Note that the energy demand of PSA given in Bui et al. (2018) is required for capturing CO 2 from flue gases, of which the CO 2 concentration is below 20 mol.%. As the CO 2 concentration of the cathode effluent of the CO 2 -electrolyzer stack would be much higher (up to 50 mol.%), the specific energy demand for the CO 2 separation would be even lower (see Hasan et al. 2014).
Dynamic responses of electrochemical reactions are generally fast, e.g., water electrolysis (Buttler and Spliethoff, 2018), so the CO 2 separation unit is likely to limit the ramping speed of the sub-processes (T Ramp ). Possible technologies for CO 2 separation would differ in the ramping speed. For instance, membrane gas separation and PSA allow fast ramping (DiMartino et al., 1988;Sinha and Padhiyar, 2019) while chemical absorption (Jung et al., 2020) takes longer time to change the operation level. In order to investigate how much the maximal ramping speed affects the optimal operation as well as the cost savings, we assume different values of T Ramp (one to three hours).
An hourly electricity spot price profile (Figure 3) for three days is taken from the German EPEX SPOT market, recorded end of May in 2018 (Agora Energiewende, 2019).
Our optimization problem is a nonlinear program due to the nonlinear surrogate functions f FE s,c,t and f U-j s,t . Nonlinear scheduling problems are rarely solved because they are computationally challenging to obtain global solutions. To tackle this challenge, linear approximation is usually applied to reformulate the problem into mixed-integer linear programs, e.g., Zhang et al. (2016), however, such an approximation could generate inaccurate results. Instead, we apply the wavelet-based grid adaptation algorithm (Schäfer et al., 2020). We can find near-optimal solutions of (nonlinear) scheduling problems in a tractable time by using a few optimization variables only. Moreover, this algorithm always creates feasible schedule as the correct nonlinear models can be used. Herein, the current density j s,t are the optimization variables. The entire horizon (144 h) comprises two time intervals (128 and 16 steps) concatenated. As a result, the number of degrees of freedom is greatly reduced compared to 574 (1444 ) in the original problem. The mathematical models are implemented in our in-house open-source software for deterministic global optimization MAiNGO (Bongartz et al., 2018) based on McCormik relaxation (McCormick, 1976;Mitsos et al., 2009).
The simulation results are illustrated in Figure 3. The results show that the optimal current density of each stack is adjusted to the time-variable electricity prices while at anytime meeting the H 2 /CO ratio of the mixed syngas stream entering the syngas storage. Because of the symmetry-breaking constraints (9) imposed, Stack 1 and Stack 4 always operate at the highest and lowest current densities, respectively. Interestingly, more than two stacks occasionally follow the same trajectory, e.g., Stack 2 and 3 in the case of 2 hr of the maximal ramping duration. This kind of optimal trajectories would facilitate the process control in practice. During the period of low electricity prices, some stacks are operated at relatively high current density, which results in not only the higher syngas production rate but also the higher H 2 /CO ratio than the nominal levels (see Figure 2-(a)). Other stacks, however, run at below the nominal current density due to the syngas ratio constraint (11).
As shown in Figure 3, the shorter the maximal ramping duration, the more dynamical the operation of the subprocesses. As a result, more dynamic operation ends up with higher savings in the cumulative electricity cost compared to the steady operation at the nominal current density. It indicates that employing a CO 2 separation unit that allows fast changes in the operation level will considerably improve the overall economics of the modular eCO 2 R system by DSM.

CONCLUSION
We proposed the modularization of an electrochemical CO 2 reduction process for the purpose of reducing the electricity costs by demand side management. The modularization is an effective operation strategy to meet a certain product quality while properly shifting the power demand of individual sub-processes. We formulated a nonlinear scheduling problem to minimize the cumulative electricity costs of an electrochemical system while optimizing the operation of each sub-process. The case study demonstrated our operation idea on co-electrolysis of CO 2 -H 2 O for the production of syngas. The entire system is made of four sub-processes, each of which produces syngas and separates unreacted CO 2 independently. We used the wavelet-based grid adaptation algorithm to obtain nearoptimal solutions in a reasonable computation time. The simulation results suggest that the flexible operation of the modular process can reduce the electricity costs while maintaining the syngas ratio. Moreover, the maximum ramping speed of the sub-process, which is likely to be determined by the CO 2 separation unit, is found to be a key factor that has a considerable impact on cost savings.
For future research, we should investigate the influences of other parameters, e.g., the degree of modularization (i.e., the number of sub-processes) and the strength of electricity price fluctuation, on cost savings. Also, additional capital costs required for employing flexible operation and modular configuration should be analyzed. Particularly, installing a fewer number of large CO 2 separation units is worth investigating because of benefits from economies-ofscale.