Method for optimizing thickness of electrolyte and cathode and improving energy efficiency of SOECs

Abstract

The invention discloses a method for optimizing the thickness of an electrolyte and a cathode and improving the energy efficiency of SOECs. The method specifically comprises the steps of electrochemical modeling of SOEC considering current leakage, prediction of working potential needed by the SOEC during operation, building of an electrolytic cell energy efficiency model, building of a residual stress model based on the SOEC and building of a thermal-electric-chemical-mechanical coupling stress model. Two optimization problems are formulated by considering the volatility of renewable energy sources, the optimization problems are combined with a nonlinear quadratic programming SQP algorithm and a particle swarm optimization PSO algorithm to be converted into a wanted standard optimization problem, the optimal electrolyte thickness and cathode thickness are obtained, the optimal electrolytic cell energy efficiency is obtained, and the optimal energy efficiency of the SOEC is achieved. According to the method, hydrogen production and energy efficiency improvement can be reasonably balanced, stress safety is guaranteed, and valuable guidance is provided for design and development of SOEC.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of new energy system optimization, and particularly relates to a method for optimizing electrolyte and cathode thickness and improving SOECs energy efficiency. BACKGROUND

[0002] In view of the increasingly serious environmental pollution problem, promoting energy saving and emission reduction is an important content to realize green development and sustainable development. Hydrogen, as a promising green energy, provides a feasible way to maximize carbon emission reduction and maximize the use of renewable energy, which is in line with the long-term global energy strategy. Among various hydrogen production technologies, water electrolysis is the most promising technology to use renewable energy to produce hydrogen on a large scale. However, the existing methods are low in efficiency, which limits their competitiveness in market application. When comparing the performance characteristics of proton exchange membrane (PEM) electrolyzer and alkaline electrolyzer, solid oxide electrolyzer (SOECs) has significant advantages, including lower energy consumption and higher hydrogen production rate. In addition, SOEC overcomes the limitations of proton exchange membrane (PEM) and alkaline electrolyzer caused by the increase of operating temperature. Therefore, the environmental protection and high efficiency of steam electrolysis are increasingly concerned by people, especially high temperature water electrolysis. However, their high working temperature brings challenges to durability. Mechanical stress caused by structural size is a key factor of various failures, and the comprehensive effects of these factors must be considered.

[0003] During the high-temperature operation of solid oxide electrolysis cell (SOEC), the structural parameters and operating conditions interact with each other, which directly determines the electrolysis efficiency and structural stability. The thickness of electrolyte and cathode is the most critical design variable, which not only affects the electrochemical behavior such as ion transport and concentration polarization, but also is closely related to stress distribution and interface stability.

[0004] Firstly, the change of electrolyte thickness has two effects. On the one hand, thinner electrolyte can reduce ohmic resistance, enhance ion conduction, and thus significantly improve hydrogen production rate and overall energy efficiency. On the other hand, too thin electrolyte will lead to enhanced current leakage and increased risk of gas permeation. At the same time, thicker electrolyte, although it can reduce leakage current, will cause residual stress and thermoelectrochemical mechanical coupling stress to rise sharply, exceeding the safety threshold, which will cause interface delamination and crack propagation, leading to failure. This shows that the electrolyte thickness must be balanced between high energy efficiency and residual stress and thermoelectrochemical mechanical coupling stress safety. Secondly, the design of cathode thickness also has inherent contradictions. Thinner cathode can effectively reduce concentration polarization and improve energy efficiency; however, too thin cathode is more susceptible to tensile stress and damage under high temperature operating conditions, especially under the combined action of residual stress and thermoelectrochemical mechanical (TECM) coupling stress, the failure probability increases significantly. Therefore, the optimization of cathode thickness must strike a balance between reducing polarization loss and ensuring structural integrity. Further, the residual stress in the manufacturing process and the TECM coupling stress caused by the mismatch between thermal expansion and chemical expansion during operation will form a complex stress distribution in the electrolyte and electrode layers. These stresses are difficult to accurately evaluate by experimental means, but have a decisive influence on the service life and safe operation. Therefore, optimization from the perspective of electrochemical performance alone is far from enough, and stress constraint models must be introduced to avoid structural failure while achieving high-efficiency hydrogen production.

[0005] Based on the above mechanism logic, the electrolyte thickness and cathode thickness are selected as optimization variables, the hydrogen production and energy efficiency maximization are selected as optimization objectives, and the current leakage model, oxygen partial pressure safety constraint model, and residual and coupling stress model are jointly introduced to fully characterize the contradictory relationship between performance and safety. This optimization framework not only reveals the coupling mechanism of structural parameters on electrolysis efficiency and failure risk, but also finds a balance point between efficiency improvement and structural reliability, thus providing a scientific basis for the efficient and long-term stable operation of SOEC. Therefore, the necessity of optimizing the thickness of electrolyte and cathode lies in that they are the core parameters that determine the electrochemical performance and mechanical safety of SOEC. Only by considering current leakage and residual / coupling stress can hydrogen production rate and energy efficiency be maximized and structural reliability be ensured.

[0006] In the existing research on the performance optimization of SOEC electrolyzers, although a large number of studies on the parameter characterization and optimization of SOEC have been carried out in recent years, most of the studies mainly focus on the optimization of efficiency, and the consideration of safety is limited. More importantly, there is a significant lack of research on the safety of SOEC, especially the residual stress failure in the manufacturing process and the thermal-electric-chemical-mechanical coupling induced stress failure in the operation process. Effective management of these stresses is needed to prevent failures and ensure the long-term reliability of the electrolyzer. And the existing research is not fully clear about the influence of electrochemically induced stress on the electrode failure in SOFCs. A comprehensive mechanical model is developed, which considers the interaction between the PEN components and incorporates mechanical effects at different stages of SOEC manufacturing and operation. In addition, preventing tensile stress that can cause structural damage in the PEN structure of SOECs is still a key challenge in optimizing the PEN structure size to improve the performance of SOECs.

[0007] Therefore, by improving energy efficiency to improve overall performance, while addressing leakage current and stress safety issues. By optimizing the electrolyte and cathode thickness, the contradiction between energy efficiency, leakage current and stress-induced delamination is systematically addressed. In the current optimization method, it is difficult to consider thinner electrolyte layers to reduce ohmic resistance and enhance ionic conductivity, but they also increase the risk of gas leakage and leakage current. This creates a design conflict because the optimal electrolyte thickness must strike a balance between maximizing energy efficiency, minimizing leakage current, and ensuring cathode stress safety. Similarly, thinner cathodes help reduce concentration polarization, but they are more prone to tensile stress and mechanical failure, especially under high-temperature operating conditions. This brings another conflict, achieving high energy efficiency by minimizing concentration polarization, while maintaining the structural integrity of the cathode under stress constraints. Balancing these conflicting requirements - reducing electrolyte and cathode thickness to improve energy efficiency, while reducing leakage current and ensuring structural integrity. Therefore, existing research lacks an optimization method that can provide an efficient solution for SOEC electrolyzer electrolyte and cathode thickness optimization. SUMMARY

[0008] To solve the above problems, the present application provides a method for optimizing electrolyte and cathode thickness and improving the energy efficiency of SOECs.

[0009] The method for optimizing electrolyte and cathode thickness and improving the energy efficiency of SOECs of the present application comprises the following steps:

[0010] Step 1: Electrochemical modeling of SOEC considering current leakage to predict the required operating potential of SOEC when in operation.

[0011] The SOEC solid oxide electrolysis cell as a whole contains anode and cathode side electrode plates, anode flow channel, cathode flow channel, and dense electrolyte layer, porous anode and cathode, etc. structural components. In this invention, the impact of the SOEC structure along the thickness direction is mainly considered.

[0012] First, the working principle of SOEC and its planar structure are shown as follows: Figure 1 When an input power is applied to the SOEC through an external circuit, a high-temperature steam / hydrogen mixture (600-1000 ◦ C) enters the SOEC through the gas channel and diffuses through the porous cathode electrode layer. Inside the cathode, water vapor is electrolytically decomposed into hydrogen gas ( ) and oxygen ions ( ). Oxygen ions migrate through the dense electrolyte layer to the anode, where they lose electrons ( ) and are oxidized to form oxygen gas. Oxygen ions are transported through the dense electrolyte layer to the anode, where they lose electrons and are oxidized to form oxygen gas. The generated oxygen gas diffuses through the porous anode electrode to the gas channel, where it is carried away by the sweep gas to prevent accumulation, ensuring the efficient operation of the SOEC. At the same time, the electrons produced by the anode are transmitted back to the cathode through the external circuit, completing the circuit closure required for the electrolysis process, thereby achieving continuous production of hydrogen gas on the cathode side. In addition, the redox reaction between oxygen vacancies ( ) and stable oxygen lattice sites ( ) at the interface promotes the migration of electron-hole pairs ( ), resulting in a hole leakage current ( ). The complete electrochemical redox reaction of water electrolysis can be described as follows:

[0013] Cathode reaction (reduction) at the cathode-electrolyte interface:

[0014]

[0015] Anode reaction (oxidation) occurs at the anode-electrolyte interface:

[0016]

[0017] Step 1.1: Calculate the operating potential of SOEC:

[0018] (1)

[0019] where , , and represent the equilibrium voltage, activation overpotential, concentration overpotential, and ohmic overpotential, respectively; and represent the thickness of the electrolyte and the cathode, respectively.

[0020] Step 1.2: Activation overpotential is expressed as:

[0021] (2)

[0022] where, and are the exchange current densities for the anode and cathode, respectively; is the gas constant, is the temperature, is the Faraday constant.

[0023] Step 1.3: Concentration overpotential is related to the cathode thickness and is expressed as:

[0024] (3)

[0025] where, and represent the gas partial pressure at the electrode surface and electrode-electrolyte interface, respectively.

[0026] Step 1.4: Ohmic overpotential is related to the electrolyte thickness and is defined as:

[0027] (4)

[0028] where, is the electrolyte area, is the oxygen ion conductivity.

[0029] Step 1.5: Leakage current model.

[0030] The leakage current is based on the transport mechanism of electrons and oxygen vacancies in the electrolyte, and a numerical model is established; the leakage current density is calculated and expressed as follows Expression:

[0031] (5)

[0032] where, is the pre-exponential factor, and represent the oxygen partial pressure at the cathode / electrolyte interface and the anode / electrolyte interface, respectively; in addition,

[0033] (6)

[0034] Step 2: Establishment of electrolytic tank energy efficiency model.

[0035] Step 2.1: Establishment of energy efficiency model:

[0036] (7)

[0037] where, is the lower heating value of hydrogen gas.

[0038] Step 2.2: Hydrogen production rate model setup:

[0039] (8)

[0040] Step 2.3: External current density corresponding to input power is expressed as:

[0041] (9)

[0042] Step 3: Residual stress model setup based on SOEC and thermo-electro-chemical-mechanical coupled stress model setup.

[0043] Based on the elastic multilayer system stress analytical model proposed by Hsueh, the residual stress caused by the mismatch of thermal expansion coefficients in the supported multilayer structure is considered as the linear superposition of the stress of each single layer. The model is adopted in the present invention because it can accurately solve the stress of each layer through the closed-form analytical equation. During the manufacturing stage, the residual stress caused by the difference in the thermal expansion coefficients between the layers will threaten the structural integrity. When the PEN structure cools down to room temperature, the residual stress reaches a peak, thereby increasing the risk of micro-crack formation in the PEN inner layer and the layer with weak mechanical properties. These defects will significantly increase the failure probability of the material, becoming a key challenge for the long-term reliability of the electrolytic cell.

[0044] Step 3.1: For the elastic multilayer system of the electrolytic cell, the residual stress is expressed as:

[0045] (9)

[0046] where D is the elastic matrix of the isotropic material, , is the strain at the neutral axis, is the bending strain.

[0047] It is assumed that the stress is in an equi-biaxial state , and the stress in the thickness direction can be ignored ; due to the Poisson effect, the stress in the direction will cause the strain in the direction; therefore, , , the strain in the three directions is expressed as:

[0048] (10)

[0049] (11)

[0050] (12)

[0051] where, E is the Young's modulus, ν is the Poisson's ratio.

[0052] The residual stress distribution in the i-th layer along the x-direction is given by:

[0053] (13)

[0054] where, and are the effective Young's modulus and the total strain, respectively, and .

[0055] The thermal strain in the i-th layer depends on the cooling temperature range and is given by:

[0056] (14) where,

[0057] αiis the thermal expansion coefficient of the i-th layer, T0is the stress-free reference temperature, Tis the room temperature; assuming the PEN thickness along the x-direction, then , hiis the i-th layer thickness; the total thermal strain is given by . .

[0058] According to the force and moment balance conditions, we have:

[0059] (15)

[0060] (16)

[0061] (17)

[0062] The thickness of the i-th layer is given by where the strain components are calculated by:

[0063] (18)

[0064] where,​​​​​ represents the layer number, and the thermal expansion coefficients of each layer are .

[0065] According to the basic bending theory of elastic multilayer plates, the bending strain of PEN is expressed as:

[0066] (19)

[0067] where is the curvature, is the distance from the bending axis to the free surface of the first layer, which is obtained by the following formula:

[0068] (20)

[0069] The curvature is expressed as:

[0070] (21)

[0071] Step 3.2: Build based on SOEC thermal-electric-chemical-mechanical coupling stress model.

[0072] During the electrolysis cell operation phase, in the mixed ionic electronic conductor (MIEC) electrolyte (such as gadolinium-doped ceria GDC), cerium ions can be reduced from to , thereby affecting the oxygen activity inside the electrolyte. This reduction process is realized by defect diffusion controlled by electrochemical driving force. Mechanical stress will affect this diffusion behavior, and the change in defect concentration deviating from the stoichiometric ratio during the diffusion process will cause volume chemical expansion, thereby generating mechanical stress in the solid. Therefore, the present application establishes a thermal-electric-chemical-mechanical (TECM) coupling stress model to describe the interaction between mechanical stress and charged defect diffusion.

[0073] According to Hooke's law, the constitutive relationship of the stress tensor is expressed as:

[0074] (22)

[0075] where, where represents the stress component, and represent the axes of the Cartesian coordinate system, respectively, is the strain tensor; is the chemical strain of the PEN structure, is the thermal strain of the PEN structure.

[0076] Assuming that the defect concentration only changes along the thickness direction of the electrolyte, , ; the strain of each layer is written as then the first layer stress is written as

[0077] (23)

[0078] During operation, the thermal stress-strain relation of the first layer depends on the operating temperature range and is expressed as

[0079] (24)

[0080] where is the thermal expansion coefficient of the arbitrary layer; is the chemical strain, which is expressed as

[0081] (25)

[0082] where is the chemical expansion coefficient, is the change in the oxygen vacancy concentration, which is a function of the electrolyte thickness ; and represents the chemical induced strain; it is assumed that the non-stoichiometry ratio is linearly related to the oxygen vacancy concentration, and is expressed as

[0083] (26)

[0084] where is a constant.

[0085] The non-stoichiometry difference in the electrolyte thickness direction is calculated as

[0086] (27)

[0087] where represents the non-stoichiometry ratio on the anode side, and ; is the Nernst voltage across the mixed conductor, which deviates from the equilibrium voltage , and is expressed as ; is the working potential drop across the mixed electrolyte conductor, and ; is the total electrode overpotential, which is calculated as

[0088] (28)

[0089] where and respectively represent the overpotential at the cathode and anode electrodes.

[0090] is modeled as a function of electrolyte thickness, denoted as:

[0091] (29)

[0092] where, is the penetration depth.

[0093] Since the electrode ohmic resistance is usually very small, the ohmic resistance of the anode and cathode can be ignored; therefore, is approximately equal to the applied working potential .

[0094] Similarly, based on the analytical model of stress of elastic multilayer system according to the theory of Timoshenko, according to the force balance and bending moment balance equation: , and

[0095] then the thermal strain and the chemical induced stress in the running stage are calculated as:

[0096] (30)

[0097] In the operating stage, the bending strain of the PEN is represented as:

[0098] (31)

[0099] where, is the curvature, is the distance from the bending axis to the free surface of the first layer, which is obtained in the operating stage and represented as:

[0100] (32)

[0101] and is represented as:

[0102] (33)

[0103] Step 3.3: Residual stress safety modeling in the SOEC manufacturing stage.

[0104] Since ceramic materials have brittle characteristics, the PEN structure of SOEC is more prone to failure under the action of residual stress; assuming that the failure of the PEN is only caused by the residual stress accumulated in the process of cooling from high temperature stress-free state to room temperature, the probability of stress-induced failure is usually represented by the Weibull statistical model as:

[0105] (34)

[0106] wherein, represents the principal tensile stress, represents the index of the respective principal tensile stress component; the parameter is the fracture strength of the material, corresponding to the stress level at which 63% of the samples fail; is the Weibull modulus; represents the volume of the PEN layer subjected to the respective stress level, is the reference volume.

[0107] To prevent residual stress-induced failure, the maximum allowed probability of stress-induced failure must satisfy:

[0108] (35)

[0109] wherein, represents the safety threshold set to avoid stress-induced failure during the manufacturing phase, and .

[0110] Step 3.4: Thermal-electro-chemical-mechanical coupled stress safety modeling for the SOEC operating phase.

[0111] The probability of stress-induced failure of the PEN material when subjected to thermal-electro-chemical-mechanical coupled stresses during the operating phase is represented as:

[0112] (36)

[0113] wherein, represents the principal tensile stress during the operating phase, is the fracture strength of the material during this phase.

[0114] To ensure the structural safety of the electrolyzer during the operating phase, the maximum probability of TECM stress-induced failure should be lower than a specified safety threshold, represented as:

[0115] (37)

[0116] wherein, represents the safety threshold during the operating phase, taken as .

[0117] Step 4: Comprehensive analysis of the combined effect of stress safety during the manufacturing phase and the operating phase on the optimization of the electrolyte thickness and the cathode thickness for the SOEC electrolyzer.

[0118] To reduce the computational complexity, only the parameter distribution in the thickness direction of electrolyte and cathode is concerned, and the inhomogeneity in the radial direction is ignored, and a one-dimensional model is adopted for simulation; for the electrolyte layer, thinner electrolyte can reduce ohmic polarization loss and improve energy efficiency, but will increase current leakage and cathode mechanical stress; similarly, thinning the cathode thickness can minimize the concentration polarization loss and improve energy efficiency, but will exacerbate the mechanical stress of the electrode; the increase of residual stress and coupled TECM stress may lead to electrode failure, which is manifested as crack and peeling phenomenon, and finally shortens the service life of PEN structure in SOEC; the strong coupling relationship between electrochemical reaction, current leakage and stress-induced damage further increases the complexity of the optimization of electrolyte layer and cathode thickness.

[0119] Based on the defined evaluation index, the maximum residual stress and TECM induced stress are quantified, and the failure probability of PEN in the manufacturing and running process is calculated. These evaluations can determine whether the stress exceeds the safety threshold, so as to ensure the applicability of the electrolytic cell. By quantitative analysis, the optimal electrolyte and cathode thickness are determined, aiming at maximizing the energy efficiency while maintaining the mechanical integrity of the PEN.

[0120] Step 5: Mathematical modeling of SOEC electrolytic cell energy efficiency optimization problem.

[0121] Step 5.1: Problem analysis.

[0122] For cathode-supported solid oxide electrolysis cell (SOEC), a thicker cathode will increase the gas diffusion resistance of the fuel, thereby reducing the mass transfer efficiency. On the contrary, a too thin cathode layer may not provide sufficient mechanical support for the electrolysis cell, thereby affecting its stress characteristics.

[0123] It should be noted that the higher residual stress on the cathode side and the thermal-electric-chemical-mechanical (TECM) coupled stress often lead to structural failure problems such as cracking and delamination of the oxygen electrode. In addition, to improve the energy efficiency of SOEC , the electrolyte thickness and the cathode thickness should be reduced to minimize ohmic loss and concentration polarization loss. However, thinning the electrolyte layer will significantly increase the leakage current density , which in turn will lead to a decrease in SOEC energy efficiency. Therefore, there are two main conflicts in the system optimization process:

[0124] (1) Determine the optimal electrolyte thickness to maximize energy efficiency and minimize leakage current under the premise of ensuring the safety of cathode stress.

[0125] (2) Balance between high energy efficiency and low concentration polarization loss under the constraint of cathode stress.

[0126] To ensure system safety and improve hydrogen production rate Energy efficiency The output power of the electrolysis cell must be balanced by a proper choice of electrolyte and cathode thickness, while minimizing the risk of stress-induced failure; denoted as and .

[0127] In addition, the total current density is inversely proportional to and , indicating that reducing the thickness of the electrolyte layer helps to improve the hydrogen production rate; however, when the electrolyte layer is too thin, the leakage current will increase sharply, so it is not feasible to improve the hydrogen production rate by simply thinning the electrolyte layer; it is worth noting that and the thickness are not a monotonic relationship; therefore, a reasonable trade-off must be made between and to simultaneously consider stress safety, leakage current control, and energy efficiency optimization.

[0128] To solve the above two types of conflicting problems, the present application combines the electrochemical model, the leakage current model, and the stress constraint model to establish a unified optimization framework. In addition, the water electrolysis hydrogen production process can also be used as an effective means of "peak shaving and valley filling" for the power grid. Although the power grid can usually provide stable power input, the output of renewable energy has intermittency and volatility, thereby bringing additional challenges to system optimization.

[0129] Step 5.2: Formulation of the optimization problem.

[0130] Considering the fluctuation characteristics of renewable energy, two types of hydrogen production optimization problems are proposed: one based on constant power input from the power grid, and the other based on variable power input from photovoltaic renewable energy; the specific optimization problems are described as follows:

[0131] Define the decision variable vector as:

[0132] (38)

[0133] The objective is to maximize the energy efficiency ; obviously, the optimization objective is a function of all decision variables.

[0134] Step 5.3: Mathematical model of optimization problem (1).

[0135] Constant power input (power grid power supply)

[0136] When the input power is provided by the power grid and remains constant, the optimization objective is to maximize the energy efficiency of the electrolysis cell, and its mathematical expression is:

[0137] (39)

[0138] where, and are the safety thresholds for residual stress induced failure and TECM coupled stress induced failure, respectively; hydrogen production rate is obtained from equation (8).

[0139] Step 5.4: Mathematical model for optimization problem (2).

[0140] Renewable energy variable power input (photovoltaic power supply)

[0141] When the input power is from photovoltaic renewable energy and varies with time, let be the discrete time step, be the total number of sampling points; at this time, the maximum energy efficiency is

[0142] (40)

[0143] where, , and represent the hydrogen production rate, residual stress failure probability and TECM coupled stress failure probability at the th sampling time, which are all dependent on the input power at this time.

[0144] The annual cumulative hydrogen production is determined by the following formula:

[0145] (41)

[0146] where, represents the sampling time interval, satisfying ; if or , it indicates that the current operating condition is not safe, and the electrolysis cell will stop running.

[0147] Step 6: Based on the SOEC electrochemical model, stress model and its safety modeling, the optimization problem is converted into the standard optimization problem desired by combining the nonlinear quadratic programming SQP and particle swarm optimization PSO algorithm, to obtain the optimal electrolyte thickness and cathode thickness, to obtain the optimal electrolysis cell energy efficiency, and to calculate the efficiency of the electrolysis cell under multiple working conditions.

[0148] Step 6.1: For optimization problem (1): constant power input (grid power supply).

[0149] The nonlinear programming problem with inequality constraints is expressed as:

[0150] (42)

[0151] Subsequently, the nonlinear programming problem with inequality constraints is rewritten as a standard inequality-constrained quadratic programming form:

[0152] (43)

[0153] If satisfies the preset convergence condition , it is considered as the optimal solution, and the corresponding is the optimization result; otherwise, let , and re-solve the sub-problem to further update the electrolyte thickness and the cathode thickness .

[0154] Based on this, the MATLAB built-in function fmincon is used to obtain the solution form of the corresponding energy efficiency optimization problem.

[0155] Step 6.2: For the optimization problem (2): renewable energy variable power input (photovoltaic power supply).

[0156] The PSO algorithm is used to solve the optimization problem defined by equation (40).

[0157] Let:

[0158] (44)

[0159] Equation (40) is converted to the expression:

[0160] (45)

[0161] Then the optimization problem (40) is represented as a standard optimization problem expression:

[0162] (46)

[0163] Further, based on the PSO method, it is ensured that all inequality constraints in equation (44) are satisfied. By repeatedly running the global search capability combined with PSO, it is ensured that the obtained solution has stability and practical reliability, thereby reducing the risk of missing the global optimal solution. The optimal electrolyte thickness and cathode thickness in SOEC applications are determined.

[0164] Step 7: the optimal variable of the full power interval searched based on the SQP and PSO optimization algorithm, for the stable power grid long power input, the selection from low to power point is verified, and the optimal electrolyte and cathode thickness combination from low to high power input is given; at the same time, for the variable power of the renewable energy photovoltaic power input, the optimal energy efficiency of the whole year is obtained by the whole year photovoltaic input of six typical regions in China, the optimal electrolyte and cathode thickness combination searched by the PSO is verified, and whether the result realizes the optimal energy efficiency of the SOEC under the premise of meeting the stress safety constraint is guaranteed.

[0165] The beneficial technical effects of the present application are:

[0166] 1. The present application proposes a method for electrolyte and cathode thickness optimization and improvement of SOEC water electrolysis cell energy efficiency considering residual stress and thermal-electric-chemical-mechanical coupling stress, residual stress and thermal-electric-chemical-mechanical coupling stress of SOEC are taken as constraint conditions, SOEC energy efficiency is taken as objective function, SQP and PSO algorithm are used to determine the optimal combination of electrolyte and cathode thickness under different application scenarios and different input power. And through the verification of the seen model and the optimization result, it is proved that the method can reasonably balance the hydrogen production and the improvement of energy efficiency, and guarantee the stress safety, which shows that the optimization result of the optimization framework has good engineering application prospect.

[0167] 2. The present application includes manufacturing stage and operation stage, wherein the working environment of the equipment is also dynamically changed, which will affect the risk of the equipment and change the risk of the equipment, the method can realize the optimization technology closest to the actual application through different models according to the dynamic change of the equipment. BRIEF DESCRIPTION OF DRAWINGS

[0168] Figure 1 It is a principle diagram of SOEC solid oxide electrolysis cell.

[0169] Figure 2 It is a verification diagram of SOEC solid oxide electrolysis cell electrochemical model.

[0170] Figure 3 It is a verification diagram of SOEC residual stress model.

[0171] Figure 4 It is a verification diagram of SOEC thermal-electric-chemical-mechanical coupling stress model accuracy.

[0172] Figures 5-8 It is a diagram of the influence of SOEC electrolyte / cathode thickness on stress failure probability.

[0173] Figures 9-12 It is a diagram of the influence of SOEC electrolyte / cathode thickness on hydrogen production rate and energy efficiency.

[0174] Figures 13-16 Optimized results of electrolyte and cathode thickness at different constant power densities and the best energy efficiency and hydrogen production rate map.

[0175] Figure 17 Annual photovoltaic power input power change data map.

[0176] Figure 18 、 Figure 19 Annual energy efficiency and hydrogen production map of YSZ electrolyte / cathode thickness optimal value in different regions.

[0177] Figure 20 、 Figure 21 Annual energy efficiency and hydrogen production map of GDC electrolyte / cathode thickness optimal value in different regions.

[0178] Figure 22 、 Figure 23 Annual energy efficiency and hydrogen production comparison map of optimal YZS and GDC cells in different regions. DETAILED DESCRIPTION

[0179] The application will be further described in detail below in combination with the drawings and specific implementation methods.

[0180] The environmental friendliness and high efficiency of steam-based electrolyzers are challenged by high operating temperatures. Mechanical stresses induced by electrolyte and cathode thickness dimensions are identified as key factors for various failures, and the combined effects of these factors must be considered. In particular, when combined with renewable energy sources such as solar energy. Balancing energy efficiency and structural stress safety remains a key challenge. Achieving high electrolysis efficiency and strong stability in constructed SOECs requires maintaining appropriate electrolyte and cathode thickness dimensions and operating parameters. These factors are mainly influenced by electrolyte and cathode parameter characteristics and pressure conditions, and are crucial for improving SOEC performance. In particular, the structural dimensions of the electrolyte and cathode have a crucial impact on the energy efficiency of the electrolyzer. The focus of the present invention is to improve the overall performance by improving energy efficiency while addressing leakage current and stress safety issues. By optimizing the electrolyte and cathode thickness, the conflict between energy efficiency, leakage current, and stress-induced delamination is systematically addressed. Based on the established integrated residual stress and electrochemically induced stress model, the influence of electrode and electrolyte microstructure evolution on cell performance is investigated. The failure probability of PEN material in electrolysis mode is analyzed, and targeted optimization issues are proposed. According to the designed optimization framework, a more comprehensive perspective is provided by balancing efficiency and structural safety to determine the key PEN electrolyte and cathode thickness parameters to achieve high energy efficiency while incorporating residual and coupled stress-related safety constraints. Further resolving two major conflicts, the optimal electrolyte and cathode thickness are determined to maximize energy efficiency while minimizing leakage current and ensuring cathode stress safety. Under cathode stress constraints, high energy efficiency is achieved by reducing concentration polarization losses.

[0181] Embodiments:

[0182] According to the electrochemical theory of SOEC. The calculation steps are as follows:

[0183] Step 1: Electrochemical modeling of SOEC considering current leakage: predicting the operating potential required for SOEC during operation, Step 2: building an electrolyzer energy efficiency model, and Step 3: building a SOEC residual stress model and a thermo-electro-chemical-mechanical coupled stress model. The operating potential and leakage current are calculated.

[0184] The polarization curves obtained based on the established SOEC model are compared with experimental data at temperatures of 900℃, 950℃, and 1000℃. As Figure 2 shown, the model results agree well with the experimental data. The high agreement between model predictions and experimental results confirms the high accuracy of the theoretical model established in this study, effectively validating its ability to characterize SOEC behavior.

[0185] During the manufacturing process of SOEC electrolysis cell, the cathode / electrolyte composite material is subjected to plastic deformation at 1360 °C to eliminate warping. Therefore, it is assumed that the cathode and electrolyte have the same stress relief temperature of 1360 °C, while the stress relief temperature of the anode is 900 °C. The residual stress is originated from the difference in the thermal expansion coefficient between the materials during the manufacturing process. Figure 3 The residual stress distribution along the PEN thickness direction of the SOEC is shown. To verify the accuracy of the electrolysis cell stress calculation, Figure 2 The residual thermal stress distribution curve obtained from the simulation model is compared with the experimental data at 25 °C and 800 °C, which is in good agreement with the simulation results.

[0186] To verify the calculated chemical-mechanical coupling stress in the GDC cell, Figure 4 The comparison results of the simulation stress distribution and the experimental data at 25 °C and 800 °C are shown. The simulation results are in good agreement with the experimental measurement data, which confirms the accuracy of the model.

[0187] Based on the stress theory, the residual stress and thermal-electric-chemical-mechanical coupling stress model of the SOEC is built. The probability of residual stress and thermal-electric-chemical-mechanical coupling stress-induced failure is usually adopted by Weibull distribution. The failure probabilities of the electrolyte layer and the cathode layer are shown in Figures 5-8 Based on the theoretical model of the electrolysis cell, the research on the residual stress and thermal-electric-chemical-mechanical coupling stress-induced stress is carried out to intuitively explain the changes of the decision variables, i.e. the electrolyte thickness and the cathode thickness, on the electrolysis cell cathode. It is a small probability.

[0188] Based on the completed solid oxide electrolysis cell electrochemical model and the residual stress and thermal-electric-chemical-mechanical coupling stress model, the hydrogen production of the electrolysis cell and the corresponding energy efficiency of the electrolysis cell are further given as shown in Figures 9-12 Among them, the changes at the input power of 200 W, 400 W and 600 W can be seen. The electrolyte and cathode play a key role in determining the hydrogen production rate and energy efficiency of SOECs.

[0189] Mathematical modeling of SOEC electrolysis cell energy efficiency optimization problem.

[0190] Step 1: Based on the previous analysis of the influence of electrolyte and cathode thickness on leakage current, total external input current, and residual stress and thermal-electric-chemical-mechanical (TECM) coupling stress-induced failure probability, the influence of electrolyte and cathode thickness on the hydrogen production of the electrolysis cell and the corresponding energy efficiency of the electrolysis cell is analyzed.

[0191] Step 2: Considering that the energy efficiency and hydrogen production rate of the electrolyzer are significantly affected by the electrolyte and cathode thickness, and improving the energy efficiency and hydrogen production rate of the electrolyzer faces the following contradictory challenges: (a) determining the optimal electrolyte thickness to maximize energy efficiency and minimize leakage current while ensuring cathode stress safety; (b) achieving a balance between high energy efficiency and low concentration polarization loss under cathode stress constraints. Based on this, a mathematical modeling of the SOEC electrolyzer energy efficiency optimization problem is proposed.

[0192] Step 3: Based on two common application scenarios, two electrolytic hydrogen optimization problems are proposed: one based on constant power input from the power grid, and the other based on variable power input from photovoltaic renewable energy. The variable power input is shown in Figure 17

[0193] Based on the established SOEC electrochemical model, stress model and safety modeling, and after verifying the accuracy of the electrochemical model and stress model, the optimization problem is combined with the nonlinear quadratic programming (SQP) and particle swarm optimization (PSO) algorithm to convert it into the corresponding standard optimization problem, obtaining the optimal electrolyte thickness and cathode thickness, and obtaining the optimal electrolyzer energy efficiency. The energy efficiency of the electrolyzer under multiple working conditions based on constant power input from the power grid and variable power input from photovoltaic renewable energy is calculated, as shown in Figures 13-16 and 18- Figure 23 It can be seen from Figures 13-16 that when the electrolyte material is YSZ and GDC material, under the condition of constant input power provided by the power grid, the input power range is 0 to 600 W. Using the SQP optimization algorithm, combined with the constructed model and safety constraints, the optimal electrolyte thickness and cathode thickness that can maximize the energy efficiency of the YSZ electrolyzer and the GDC electrolyzer are determined. Higher hydrogen production rate and energy efficiency are achieved. In addition, from Figures 18-23 ​It can be seen that for variable power input scenarios based on photovoltaic renewable energy, where the power input is fluctuating input power from photovoltaic sources, it is common to deviate from the optimal operating conditions when directly using renewable energy, which can cause significant stress on the cathode and can lead to serious cracking, thereby reducing energy efficiency. The particle swarm optimization (PSO) based algorithm is used to solve the energy efficiency optimization problem of the electrolytic cell under six photovoltaic hydrogen production scenarios. The six scenarios correspond to different regions in China, namely southwest, central China, northeast, northwest, north China and southeast. The optimal electrolyte and cathode thickness corresponding to the overall energy efficiency are calculated. The results show that for YSZ electrolyte material, the energy efficiency of the electrolytic cell is improved by 2.38%, 2.64%, 4.27%, 1.18%, 2.72% and 9.54%, respectively, and the annual hydrogen production is increased by 1.06%, 0.79%, 0.88%, 1.10%, 1.08% and 3.94%, respectively. For GDC electrolyte material, the relative improvement of the energy efficiency of the electrolytic cell is 17.95%, 18.27%, 18.91%, 17.15%, 18.06% and 19.20%, respectively, and the relative improvement of the annual hydrogen production is 18.75%, 18.28%, 18.91%, 17.15%, 18.06% and 20.31%, respectively. Compared with GDC-Cell, the optimized YSZ-Cell electrolyte and cathode thickness increases the annual hydrogen production by 11.84%, 11.08%, 9.22%, 13.71%, 11.52% and 2.96%, respectively; in terms of energy efficiency, the optimized YSZ-Cell is 14.78%, 13.43%, 13.09%, 14.73%, 13.93% and 13.43% better than GDC-Cell, respectively. This study provides valuable insights for the optimal structural design of solid oxide electrolytic cells (SOEC) and proposes an effective strategy to balance energy efficiency, current leakage and stress-induced failure risk.

[0194] The present application aims to maximize the energy efficiency of SOEC electrolytic cells, ensure safe operation and prevent electrolyte cracking. The optimization results based on the electrochemical model of SOEC and the residual stress and thermal-electric-chemical-mechanical coupling stress model have been verified to meet the expected requirements. This method provides valuable insights for the optimal structural design of SOEC and proposes an effective strategy to balance energy efficiency, current leakage and stress-induced failure risk. This research not only lays a theoretical foundation for promoting the commercialization of SOEC, but also provides practical guidelines.

[0195] Although the present work only obtains the optimized thickness value and is only applicable to the structural optimization configuration, the modeling framework and optimization procedure can be generalized to other SOEC designs by replacing the corresponding material properties and geometric parameters. Future work will focus on establishing more accurate three-dimensional models, optimizing material selection, and fine-tuning SOEC structural parameters (including anode thickness, electrolyte, and cathode thickness) to further improve energy efficiency. In addition, precise control of electrolyte thickness and improved electrode structure can significantly reduce stress-induced failure, thereby improving device durability. The development of composite multi-layer electrolyte and electrode structures also opens up promising avenues for gradually improving SOEC performance and reliability, enhancing its market competitiveness. These additional contents clearly highlight the high engineering application value of this research and demonstrate that the proposed methodology can effectively guide the design of electrolyte and cathode thickness in actual SOEC systems.

Claims

1. A method for optimizing electrolyte and cathode thickness and improving SOECs energy efficiency, characterized in that, Includes the following steps: Step 1: Consider the electrochemical modeling of SOEC with current leakage and predict the operating potential required for SOEC during operation; Step 1.1: Calculate the SOEC operating potential: （1） in, , , and These represent the equilibrium voltage, activation overpotential, concentration overpotential, and ohmic overpotential, respectively. and These represent the thicknesses of the electrolyte and the cathode, respectively. Step 1.2: Activation overpotential Represented as: （2） in, and These are the exchange current densities of the anode and cathode, respectively. The gas constant is For temperature, Where is Faraday's constant; J is the electrolytic current density; Step 1.3: Concentration overpotential With cathode thickness The relevant expression is: （3） in, and These represent the partial pressures of the gas at the electrode surface and the electrode-electrolyte interface, respectively. Step 1.4: Ohmic overpotential With electrolyte thickness Related, defined as: （4） in, For the electrolyte surface area, The conductivity of oxygen ions; Step 1.5: Leakage current model; The leakage current is modeled based on the transport mechanism of electrons and oxygen vacancies in the electrolyte; the leakage current density is calculated as follows: expression: （5） in, For the former exponential factor, and These represent the oxygen partial pressures at the cathode / electrolyte interface and the anode / electrolyte interface, respectively; furthermore, （6） Step 2: Building an energy efficiency model for the electrolyzer; Step 2.1: Building the energy efficiency model: （7） in, It has the lowest calorific value of hydrogen. Step 2.2: Building the hydrogen production rate model: （8） Step 2.3: Corresponding to input power external current density Represented as: （9） Step 3: Building the SOEC residual stress model and the thermo-electric-chemical-mechanical coupled stress model; Step 3.1: For the elastic multilayer system of the electrolytic cell, the residual stress is expressed as: （9） Where D is the elastic matrix of the isotropic material. , Strain at the neutral axis For bending strain; Assuming the stress is in an isobiaxial state Furthermore, the stress along the thickness direction is negligible. Due to the Poisson effect, Stress in the direction will cause Strain in the direction; therefore, , , The strain in the three directions is expressed as follows: （10） （11） （12） in, For Young's modulus, Poisson's ratio; along The direction of the first The residual stress distribution of the layer is expressed as: （13） in, and These are the effective Young's modulus and the total strain, respectively. ; No. The thermal strain of the layer depends on the cooling temperature range , is represented as: （14） in, The coefficient of thermal expansion of each layer, The stress-free reference temperature At room temperature; assuming the PEN thickness is along... Direction, then , For the first Layer thickness; total thermal strain is expressed as ; According to the conditions for force equilibrium and bending moment equilibrium, we have: （15） （16） （17） No. The thickness of the layer is Among them, strain components Calculate using the following formula: （18） in, Indicates the layer number, and the coefficients of thermal expansion for each layer are as follows: ; According to the basic bending theory of elastic multilayer plates, the bending strain of PEN is expressed as: （19） in For curvature, The distance from the bending axis to the first free surface is obtained by the following formula: （20） curvature Represented as: （21） Step 3.2: Building a coupled stress model based on SOEC thermo-electric-chemical-mechanical stress; According to Hooke's law, the stress tensor The constitutive relation is expressed as: （22） Among them, Represents stress components, and These represent the axes of the Cartesian coordinate system. It is the strain tensor; For PEN structure chemical strain, Thermal strain of the PEN structure; Assuming the defect concentration varies only along the electrolyte thickness direction, then , The strain of each layer is written as Then the first Layer stress is written as: （23） During operation, the first The thermal stress-strain relationship of the layer depends on the operating temperature range. , is represented as: （24） in, It is the coefficient of thermal expansion of any layer; It is chemical strain, and its expression is: （25） in, The coefficient of chemical expansion, This is the change in oxygen vacancy concentration, which is the amount of electrolyte thickness. The function; Represents chemically induced strain; hypothesis Non-stoichiometry They are linearly correlated, as shown below: （26） in, It is a constant; Nonstoichiometric difference in the electrolyte thickness direction The calculation is as follows: （27） in, This indicates the non-stoichiometric ratio on the anode side, and ; It is the Nernst voltage across the two ends of the mixed conductor, and the equilibrium voltage. There is a deviation, and its expression is: ; It is the working potential drop across the mixed electrolyte conductor, and ; This is the total electrode overpotential, and its calculation formula is: （28） in, and These represent the overpotentials at the cathode and anode electrodes, respectively. The function modeled as electrolyte thickness is expressed as: （29） in, The depth of penetration; Since the electrode ohmic resistance is typically extremely small, the ohmic resistance of the anode and cathode can be ignored; therefore... Approximately equal to the applied working potential ; Similarly, based on the analytical model of stress in an elastic multilayer system using Scheres' theory, and according to the force equilibrium and moment equilibrium equations: , as well as The thermal strain during the operation phase Calculation of chemically induced stress for: （30） During the operation phase, the bending strain of PEN is expressed as: （31） in, For curvature, The distance from the bending axis to the first free surface is obtained and expressed during the operation phase as follows: （32） and Represented as: （33） Step 3.3: Residual stress safety modeling during SOEC manufacturing; Due to the brittle nature of ceramic materials, SOEC PEN structures are more prone to failure under residual stress. Assuming that PEN failure is caused solely by residual stress accumulated during cooling from a high-temperature stress-free state to room temperature, the probability of stress-induced failure is typically expressed using the Weibull statistical model as follows: （34） in, Indicates the principal tensile stress. Indicates the index of each principal tensile stress component; parameters The fracture strength of the material corresponds to the stress level at which 63% of the specimens fail. It is the Weibull modulus; This represents the volume within the PEN layer that bears the corresponding stress level. For reference volume; To prevent residual stress-induced failure, the maximum permissible probability of stress-induced failure must satisfy the following: （35） in, This represents a safety threshold set to prevent stress-induced failure during the manufacturing process, and ; Step 3.4: Safety modeling of coupled thermal-electrical-chemical-mechanical stress during SOEC operation; When PEN material is subjected to coupled thermo-electro-chemical-mechanical stress during operation, its stress-induced failure probability is expressed as: （36） in, Indicates the principal tensile stress during the operation phase. This represents the fracture strength of the material at this stage. To ensure the structural safety of the electrolytic cell during operation, the maximum probability of TECM stress-induced failure should be lower than a specified safety threshold, expressed as: （37） in, This represents the safety threshold during the operational phase, and is taken as... ; Step 4: Comprehensive analysis of the combined impact of stress safety on electrolyte thickness and cathode thickness optimization during the manufacturing and operation phases of the SOEC electrolyzer; To reduce computational complexity, we focus only on the parameter distribution along the thickness of the electrolyte and cathode, ignoring radial inhomogeneities, and employ a one-dimensional model for simulation. For the electrolyte layer, a thinner electrolyte can reduce ohmic polarization losses and improve energy efficiency, but it increases current leakage and cathode mechanical stress. Similarly, while reducing the cathode thickness can minimize concentration polarization losses and thus improve energy efficiency, it exacerbates electrode mechanical stress. Increased residual stress and coupled TECM stress may lead to electrode failure, manifesting as cracks and peeling, ultimately shortening the lifetime of the PEN structure in SOEC. The strong coupling between electrochemical reactions, current leakage, and stress-induced damage further increases the complexity of optimizing the electrolyte layer and cathode thickness. Step 5: Mathematical modeling of the SOEC electrolyzer energy efficiency optimization problem; Step 5.1: Problem Analysis; There are two main types of conflicts in the system optimization process: (1) Under the premise of ensuring cathode stress safety, determine the optimal electrolyte thickness to maximize energy efficiency and minimize leakage current; (2) Achieving a balance between high energy efficiency and low concentration polarization loss under cathode stress constraint; To ensure system safety while simultaneously increasing hydrogen production With energy efficiency The output power of the electrolytic cell must be balanced by rationally selecting the electrolyte and cathode thicknesses, while minimizing the risk of stress-induced failure; these are denoted as follows: and ; In addition, total current density and and The two conditions show an inverse relationship, indicating that reducing the electrolyte thickness helps improve hydrogen production; however, when the electrolyte thickness is too thin, leakage current increases. The yield will increase dramatically, therefore increasing hydrogen production simply by thinning the electrolyte layer is not feasible; it is worth noting that... The relationship between thickness and other parameters is not monotonic; therefore, it is necessary to... and A reasonable balance needs to be struck between these factors to simultaneously ensure stress safety, leakage current control, and energy efficiency optimization. Step 5.2: Optimization problem statement; Considering the fluctuating characteristics of renewable energy, two optimization problems for hydrogen production via electrolysis are proposed: one based on constant power input from the power grid, and the other based on variable power input from photovoltaic renewable energy. The specific optimization problems are as follows: Define the decision variable vector as follows: （38） The goal is to maximize energy efficiency. Clearly, the optimization objective It is a function of all decision variables; Step 5.3: Optimize the mathematical model of problem (1); Constant power input, i.e., grid power supply When input power When the power supply is provided by the grid and remains constant, the optimization objective is to maximize the energy efficiency of the electrolyzer, and its mathematical expression is: （39） in, and Safety thresholds for residual stress-induced failure and TECM-coupled stress-induced failure, respectively; hydrogen yield. It is obtained from formula (8); Step 5.4: Optimize the mathematical model of problem (2); Renewable energy variable power input, i.e., photovoltaic power supply When input power When the energy source is photovoltaic renewable energy and varies over time, let For discrete time steps, This represents the total number of sampling points; at this point, the maximum energy efficiency is... The expression is: （40） in, , and They represent the first time. The hydrogen yield, residual stress failure probability, and TECM coupled stress failure probability at each sampling time point all depend on the input power at that time point. ; Annual cumulative hydrogen production Determined by the following formula: （41） in, Indicates the sampling time interval, satisfying ;like or If this is the case, it indicates that the current operating conditions are unsafe, and the electrolytic cell will stop operating. Step 6: Based on the SOEC electrochemical model, stress model and safety modeling, the optimization problem is transformed into a standard optimization problem by combining nonlinear quadratic programming (SQP) and particle swarm optimization (PSO) algorithms to obtain the optimal electrolyte thickness and cathode thickness, obtain the optimal electrolyzer energy efficiency, and calculate the efficiency of the electrolyzer under multiple operating conditions. Step 6.1: For optimization problem (1): constant power input, i.e., grid power supply; The nonlinear programming problem with inequality constraints can be expressed as: （42） Subsequently, the nonlinear programming problem with inequality constraints is rewritten into the standard quadratic programming form with inequality constraints: （43） like Meets the preset convergence conditions If it is the optimal solution, then it is considered the corresponding solution. That is the optimized result; otherwise, let And resolve the subproblem to further update the electrolyte thickness. With cathode thickness ; Based on this, the built-in MATLAB function fmincon is used to obtain the solution form of the corresponding energy efficiency optimization problem; Step 6.2: For optimization problem (2): renewable energy variable power input, i.e. photovoltaic power supply; The PSO algorithm is used to solve the optimization problem defined in equation (40); make: （44） Transform equation (40) into an expression: （45） The optimization problem equation (40) is then expressed as a standard optimization problem expression: （46） Furthermore, based on the PSO method, it is ensured that all inequality constraints in equation (44) are satisfied. By repeatedly running the solution and combining it with the global search capability of PSO, the stability and practical reliability of the solution are ensured, thereby reducing the risk of missing the global optimal solution; and the optimal electrolyte thickness and cathode thickness are determined in SOEC applications. Step 7: Based on the optimal variables obtained from the SQP and PSO optimization algorithms across the entire power range, for stable long-term grid power input, the optimal combination of electrolyte and cathode thickness is selected from low to high power input for verification. Simultaneously, for variable-power renewable energy photovoltaic power input, the optimal energy efficiency for the entire year is obtained using the annual photovoltaic input in six typical regions of China. The optimal combination of electrolyte and cathode thickness obtained from the PSO search is verified to ensure that the results achieve the optimal energy efficiency of SOEC under the premise of satisfying stress safety constraints.

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