Surrogate model-based multiobjective design optimization for air-cooled battery thermal management systems

ABSTRACT A well-designed battery thermal management system (BTMS) can achieve optimal cooling performance with less power consumption than a poorly-designed system. However, it is difficult to use the computational fluid dynamics (CFD) method to perform an effective and optimal design of BTMSs when there are several structural design parameters and multiple evaluation criteria. In this paper, instead of CFD, a compound surrogate model based on the mixture of experts (MoE) method is developed to accurately approximate the BTMS performance of different structural configurations. Then, the multiple criteria evaluation of the structural design is transformed into a multiobjective optimization (MOO) problem, which is solved by the nondominated sorting genetic algorithm II (NSGA-II). To address the nonuniqueness of the optimal solutions and the contradiction between evaluation criteria, the entropy weight method (EWM) and criteria importance through the intercriteria correlation (CRITIC) method are applied to analyze the weight of each evaluation criterion. Finally, the optimal structural parameters are obtained for the corresponding weights. The results show that the surrogate-based MOO can find a structural design that meets expectations, and this approach can provide guidelines for the design of BTMSs.


Introduction
Lithium-ion batteries (LIBs) have superior features of high efficiency, high specific energy, long cycle life, and no memory effects Xia et al., 2020). Currently, LIBs are widely employed as the primary power source of electric vehicles (EVs) and hybrid electric vehicles (HEVs) to replace traditional internal combustion engines, thus solving increasingly severe environmental pollution and energy shortage problems . However, LIBs are extremely sensitive to the operating temperature, which will significantly affect the cycle life, degradation rate, safety, and reliability of the battery. In general, the permissible temperature range for battery discharge is from −20 to 60°C and from 0 to 45°C for charging; the optimal operating range is from 20 to 40°C (Qian et al., 2019). In battery packs, the temperature differences between batteries should be limited to 5°C to prevent rapid degradation of individual batteries. Therefore, a well-designed battery thermal management system (BTMS) is essential for maintaining the temperature CONTACT Xiaojun Tan tanxj@mail.sysu.edu.cn and temperature difference of the batteries in a battery pack within the desired range (Shi et al., 2021).
In the past decade, numerous studies have proposed various cooling strategies for BTMSs, and they can be mainly divided into air cooling , liquid cooling Yang et al., 2020), phase change material (PCM) cooling (Safdari et al., 2020;Zandie et al., 2022), heat pipe cooling (Liang et al., 2018) and a combination of these methods (Kong et al., 2020). Every BTMS has merits and shortcomings and is suitable for specific application scenarios. For example, liquid cooling can ensure the temperature uniformity of battery packs, while having the problem of high weight and risk of leakage. In PCM cooling, the maximum temperature of the battery can be maintained below a safe value, and the temperature uniformity can be guaranteed . However, PCM is essentially an energy storage medium. If the heat absorbed by the PCM does not diffuse rapidly into the environment, then the PCM-based thermal management systems fail, especially during long periods of continuous charging and discharging . Therefore, PCM cooling is often combined with other cooling strategies, such as air cooling (He et al., 2019) and liquid cooling (Ling et al., 2018), which increase the complexity and weight of the combined system. Heat pipe cooling transfers the heat from the evaporator to the condenser without an external energy supply. However, similar to PCM cooling, heat pipe cooling must be combined with other cooling systems to dissipate the heat from the condenser to the environment and ensure its continuous operation . Air cooling has the characteristics of light weight, low cost, simple structure, and no risk of liquid leakage, and it has been widely employed in BTMSs for EVs and energy storage systems (ESSs) . However, there are some inadequacies for air-cooled BTMSs, such as poor temperature uniformity and limited heat dissipation capability, that are caused by the low thermal conductivity of air . Therefore, an enormous amount of research focusing on structure design optimization and flow configuration improvement has been conducted to improve the cooling performance of air-cooled BTMSs. Na et al. (2018) proposed an air-cooled BTMS structure with internal partitions and compared the effect of different flow configurations for cylindrical battery packs, i.e. unidirectional airflow and reversed layered or stratified air flow. They found that reversed layered or stratified air flow significantly improved temperature uniformity. The results also indicated that the distance between adjacent batteries had a significant impact on the maximum temperature and temperature uniformity. A similar conclusion was obtained for laminated batteries (Wang et al., 2019). Chen et al. (2018) improved the cooling efficiency of parallel air-cooled BTMSs with Utype flow by optimizing the widths of the plenums and the inlet and outlet. Liu and Zhang (2019) proposed a new J-type airflow structure for prismatic battery packs and conducted structure parameter optimization based on Jtype, U-type and Z-type structures, which significantly reduced the maximum temperature rise of the battery packs.
However, as Akinlabi and Solyali (2020) indicated, most current BTMS optimization problems are carried out with numerous computational fluid dynamics (CFD) calculations, which require extensive computing resources, especially for structure optimization. To reduce the computational burden, Zhang, Wu, et al. (2021) developed a flow resistance network model for I-type BTMSs as the basis for performing optimization. However, this method involves an increased amount of calculation when the model scale increases, which increases the difficulty of solving optimization problems. Therefore, a more mainstream method is to use surrogate models to approximate CFD simulation data and carry out optimization based on surrogate models (Sun et al., 2021). According to Zhang, Tezdogan, et al. (2021), a well-tuned surrogate model can use a small amount of original data to provide good predictions.  adopted a Kriging surrogate model for battery modules with herringbone fins and long sleeves, which was well adapted to the studied problem, and optimized it for maximum temperature rise, maximum temperature difference and power consumption. Cheng et al. (2020) also applied a Kriging surrogate model for a new type of finned forced air-cooled BTMS and reduced the average temperature and pressure drop of the battery pack through multiobjective optimization. Liu and Zhang (2019) utilized a larger library of models to search for the best surrogate model and, on this basis, applied a two-stage genetic algorithm to optimize the structure of a battery pack to regulate the maximum temperature rise.
The BTMS structure optimization problem is a multiobjective optimization problem that should include design goals from multiple disciplines. However, most of the previous research has comprehensively taken thermal characteristics into consideration while neglecting power consumption. When the flow rate is fixed, the most statistically significant influencing factor for power consumption is not a change in the cell spacing but the height of the flow channel. Multiobjective optimization is essential to design an efficient BTMS with a partial-height channel that balances cooling performance and power consumption.
In this paper, the influence of the critical design parameters (cell space combination and channel height) on cylindrical battery air-cooled systems is numerically investigated to address the cooling performance and power consumption problems for air-cooled BTMSs. A surrogate model based on the mixture of experts (MoE) method is proposed to separate the optimization effort from the complicated CFD simulation and avoid dependense on a large amount of simulation data. On this basis, multiobjective optimization based on the entropy weight method (EWM) and criteria importance through the intercriteria correlation (CRITIC) method is performed to determine the optimal design parameters for air-cooled BTMSs that balance cooling performance and power consumption.

Air-cooled battery pack structural design
An energy storage battery pack (ESBP) with air cooling is designed for energy transfer in a fast-charging pile with a positive-negative pulse strategy. The key characteristics of the ESBP are listed in Table 1, and a structural diagram is shown in Figure 1(a). An air-cooled ESBP comprised of eight battery blocks, each of which consists of 4 × 16 cylindrical batteries in parallel and series. Battery blocks are separated by a ceramic-fiber aerogel pad for electrical and heat insulation. Therefore, the thermal and flow fields between blocks are independent and do not influence each other. Coupled with reducing the computational burden, a block is chosen as the object of numerical simulation, as illustrated in Figure 1(b). Twelve cells in the battery block are chosen for typical temperature characteristic analysis, and the temperature monitoring points are located on the back surface of the cells along the direction of the fluid and near the cathodes of the cells, as displayed in Figure 1 In this study, the space between cells along the flow direction and the height of a channel are the critical points for promoting the maximum temperature, the maximum temperature difference, and power consumption. Therefore, battery blocks with various cell spaces and channel heights are designed for computational fluid dynamics (CFD) studies. Side views of battery blocks with different channel heights are provided in Figure 1(c), and the design parameters are displayed in Table 2. Notably, in this paper, to reduce the computational complexity, 15 cell spaces along the flow direction of one   block are divided into three groups represented by X 1 , X 2 and X 3 . As the channel height is one of the critical parameters, both the radial conduction and axial conduction of the batteries are considered, and the thermal conductivities are provided in Table 3.

Battery heat generation
The heat generationQ gen of LIBs during discharging or charging is approximately the sum of the Joule (irreversible) and entropic (reversible) heats, which are given by:Q whereQ irr andQ rev are the irreversible and reversible heats, I is the discharge/charge current, and E eq and T b are the electromotive force (EMF) and temperature of the battery, respectively. R e (SoC, T) is the equivalent internal resistance (EIR) of the battery, which is a function of the state of charge (SoC) and temperature.
dE eq dT b is the entropy coefficient. In this study, the critical parameters ofQ gen , such as the EIR and entropy coefficient, are obtained through experiments.

Energy conservation equation and control equations
The energy conservation equation of the battery can be expressed as follows: where m is the mass of the battery and C p is the battery specific heat capacity. h denotes the convective heat transfer coefficient, and A represents the effective surface area. T a is the free-stream temperature of the air. τ is the time.
In this paper, all blocks have the same airflow velocity of 5 m/s, which means that their Reynolds numbers are all greater than 4000 (varying from 6100 to 2.373 × 10 10 ), so a viscous model should be set to realizable k-epsilon for all blocks. The governing equations of air are as follows: Continuity equation: Momentum equation: Turbulence kinetic energy equation: Dissipation rate equation: Turbulent viscosity equation: Energy equation: where T, ρ, C a , λ, and μ are the temperature, density, specific heat capacity, thermal conductivity, and dynamic viscosity of air, respectively. k is the turbulent kinetic energy, and is the turbulent kinetic energy dissipation rate. D k and D are the effective diffusivities for k and , respectively. G is the turbulent kinetic energy production rate. V is the velocity vector of the air. P is the pressure. S is the momentum source term, S k is the internal source term for k and S is the internal source term for . τ is the time. S is the symmetric tensor, which is formulated as:

Development and verification of CFD models
To reduce the cost, CFD simulations are chosen instead of experiments. Three-dimensional (3D) CFD simulation models of air-cooled battery packs with modified structures are built in ANSYS Fluent 17.0. According to our previous work, the number of mesh elements in the models is set to approximately 1,200,000 to simultaneously ensure calculation accuracy and efficiency. Furthermore, polyhedral meshes are adopted in the models. The major benefit of polyhedral mesh is that each individual cell has many neighbors, which ensures that the gradients can be well approximated, and polyhedral mesh is less sensitive when stretching than tetrahedron, which results in better mesh quality with a faster mesh generation speed than hexahedral mesh (Sosnowski et al., 2018). Thus, polyhedral meshes can provide more efficient and accurate results for model simulation. The grids near the boundary layer are thickened to improve the calculation accuracy. The time step is set to 1 s, and the number of iterations per time step is 20. The total calculation time is 840 s (SoC 10% to 80%). The initial and environmental temperature is 294 K. The heat source of the battery cell in this model is defined by a user-defined function according to Eq. (1), in which the EIR and entropy coefficient are obtained by experiments. A sketch of the experimental system is shown in Figure 2(a). The details of the experimental process and results can be found in Supplementary Materials S.1.
Then, the verification of the heat transfer model is conducted by establishing an experimental platform, as shown in Figure 2(b). The experimental model of the air-cooled BTMS is established according to the original value, same as the CFD model. Three thermocouples are attached to the battery surface to measure the temperature, as shown in Figure 2(c). The maximum relative error between the experimental and simulation data is 6.9%, and the average relative error is 2.8%. Overall, the heat generation and heat transfer models used in this study are reasonable and have sufficient calculation accuracy.

Surrogate model and optimization
The BTMS structural optimization problem is a multiobjective optimization problem based on scientific evaluation criteria. It is difficult to obtain the optimal structural parameters by CFD simulation, which involves extensive computation and data processing. To address this problem, a multiobjective optimization problem based on a surrogate model is proposed, as illustrated in Figure 3.

Evaluation indicators
A well-designed BTMS can limit the battery temperature within a desired range and maintain temperature uniformity with minimum energy consumption during charging/discharging. Accordingly, in this paper, the maximum temperature rise (T r,max ), maximum temperature difference (T d ), and power consumption (P) are utilized to comprehensively evaluate the BTMS performance. The temperature characteristics are defined as: where T cell,i is the temperature of each monitored cell; T cell, max and T cell, min are the maximum temperature and minimum temperature of the cells, respectively; and i is the index of the monitored cells.
For power consumption characteristics, only power consumption by fans is taken into consideration. In the numerical simulation, the power consumption of fans for supplying air to the cells can be calculated based on the volumetric flow rate and pressure drop between the inlet and outlet of the air-cooled battery block and is formulated as: where P is the fan power consumption,V is the volumetric flow rate of the air, and p is the pressure drop between the inlet and outlet.  (S Bester et al., 2004). The design parameters of the CFD simulation include three groups of cell spaces (X 1 , X 2 , and X 3 ) and the channel height (X h ), ranging from 2 mm to 6 mm for cell spacing and from 15 mm to 60 mm for channel height. A total of 80 simulations are carried out based on the DoE for the air-cooled battery pack, 75% of which are treated as the training data, and the rest are taken as the test data. To improve the prediction accuracy, a compound surrogate model based on the MoE method is proposed. The main working principle of the MoE method is to divide the global data space by applying clustering algorithms and utilize multiple surrogate models, i.e. experts, instead of one single model, to perform local approximation in the divided subspaces and finally combine them to obtain the overall model (Trevor et al., 2001). In this paper, MoE is implemented by the expectation−maximization (EM) algorithm and Gaussian mixture model (GMM), which can improve the accuracy of regression (Bettebghor et al., 2011). Multiple surrogate models involve Kriging, least squares (LS), quadratic polynomials (QP), radial basis function (RBF), Kriging combined with partial least squares (KPLS) (Bouhlel et al., 2016a), improved KPLS (KPLS + K) (Bouhlel et al., 2016b), and inversedistance weighting (IDW).

Development of a compound surrogate model
The criteria for evaluating the effect of regression include the root mean square error (RMSE) and mean relative error (MRE), which are determined by: where y and r are the values that are estimated based on the surrogate model and the actual value from CFD simulations, respectively. N is the data scale of the test set.
For learning data mapping X → Y and (x, y) ∈ (X, Y), the Gaussian mixture model can be expressed as: where π k , μ k and k are the weights, means and covariances of each mixture component, respectively, and K is the number of components. Phase 1: At the beginning of the MoE method, initializes the Gaussian parameters π k , μ k and k . For a given (x, y) ∈ (X, Y), the probability that it belongs to the k-th component can be expressed as follows based on Bayes' formula: Phase 2: By applying the EM algorithm, π k , μ k and k are corrected iteratively until convergence: At the end of the iteration, the data space is divided into several clusters. For each cluster, each surrogate model in the pool is used to fit to capture the RMSE and MRE of different surrogate models on that cluster. Then, the surrogate model with the minimized RMSE and MRE for each cluster is selected as the best expert for the local model. Now, the best experts f best,k are found over each subset k.
Phase 3: The local model is merged. However, it is inappropriate to directly linearly combine the local models, while the weight of the experts should be adjusted with the distance between x and the center of cluster k; in other words, the smaller the Mahalanobis distance between them is, the greater weight the expert should have. Then, the global model and derivation of the local weights can be expressed as: where w k indicates the local weight of the expert, which depends on x. Law x is the projection of the law (x, y) on the input space; in other words, it can be derived for specified x and unknown y.
Thus, the global model obtained by combining the local models in the abovementioned method is completely smooth. Moreover, for the function Y = f ( X) and all the components of vector X, i.e. the structural parameter, there is a continuous partial derivative of Y = f ( X) since the evaluation indicators are continuously differentiable for any single structural parameter. Thus, Y = f ( X) is continuous, and the smooth combination methods can minimize the error.

Multiobjective optimization based on surrogate model
The nondominated sorting genetic algorithm II (NSGA-II) is adopted to solve this multiobjective optimization problem. NSGA-II determines the Pareto optimal front, i.e. the optimal solution set in a multiobjective sense, by an elite strategy with consideration of congestion (Park et al., 2020). For this problem, only the result of the objective value is considered, so the decision to terminate the algorithm is made by observing the change in the target space. In the objective space, changes in the nadir point z nadir , ideal point z * , and Pareto front z pf need to be considered, which are defined as follows: For feasible set Y = f ( X),z pf = {y ∈ Y : {y * ∈ Y : y * y, y = y * } = ∅} For the n dimension function f pf ∈ z pf , x i extreme = argmax where subscript lg indicates the solution of a certain generation in the past. ED is the Euclidean distance between two point sets. The schematic of the objective space, Pareto front, ideal point and nadir point can be seen in Supplementary Materials S.2. The optimization goal is to minimize the maximum temperature rise T r, max , maximum temperature difference T d and power consumption P within the required range and meet the structural parameter constraints, i.e. cell spacing X 1 , X 2 and X 3 and channel height X h . The description of the multiobjective optimization problem is as follows: Moreover, to improve the robustness of the iteration in NSGA-II, the sliding window method is introduced. That is, the last 30 generations are taken into consideration, and the maximum value of the distance between each generation of the last 30 generations and the current generation is taken as the characteristic value.

Multiobjective decision making
To find a representative Pareto optimal solution, it is necessary to introduce weights on the basis of the existing Pareto optimal solutions. Unlike the method of directly using nonnegative weights to transform a multiobjective problem into a single-objective problem, solving the multiobjective problem first and then determining the weights can avoid the risk of not being able to search for a nonconvex solution set after transformation into a single-objective problem (Cho et al., 2017).
In this section, two approaches, the EWM and CRITIC methods, are used to determine the weight of the evaluation indicators using the Pareto optimal solutions obtained from the MOO as the inputs.

Entropy weight method
The EWM is based on Shannon entropy (C., 1948) and is always adopted to measure the uncertainty in information. Entropy can measure the relative intensities of contrast criteria well, which can represent the information transmitted for decision-making (Huang et al., 2021). Then, information entropy can characterize the uncertainty of data. Generally, if the information entropy of a certain criterion is lower, it indicates that the discreteness of that criterion value is greater; in other words, it can provide more information and can play a greater role in the comprehensive evaluation and should have a greater weight. The detailed stepwise procedure of the EVM can be seen in Supplementary Material S.3.

Criteria importance through intercriteria correlation method
The CRITIC method is an objective weight determination method that is commonly used on datasets when it is necessary to consider both the standard deviation and correlation. By applying the CRITIC method, it is possible to assess the discreteness and conflicts between criteria (Panah et al., 2021). For the CRITIC method, when the standard deviation is constant, the smaller the conflict between indicators is, the smaller the weight; and the greater the conflict is, the greater the weight. In addition, when the degree of positive correlation between two indicators is greater, in other words, the closer the correlation coefficient is to 1, the smaller the conflict, which indicates that the information reflected by the two indicators in the evaluation has greater similarity. The detailed stepwise procedure of the CRITIC method can be seen in Supplementary Material S.4.

Effects of the structural parameters
As mentioned above, four structural parameters, three cell spacings (X 1 , X 2 and X 3 ) and the channel height (X h ), are selected to modify the geometric size of the flow field structure. The influence of these parameters is evaluated by the maximum temperature rise, maximum temperature difference, and power consumption, as follows. Figure 4(a) shows the battery temperature characteristics and system power consumption of the air-cooled battery module with various cell spacing configurations at the end of charging with a 35 mm channel height and 5 m/s airflow velocity. The data come from the face-weighted average temperature of the monitoring points. As shown in Figure 4(a), the various cell spacing configurations have different performances of the evaluation indicators and cannot simultaneously achieve the best performance.

Effect of the cell spacing configuration
For instance, the 2-2-2 mm configuration performs best for the maximum temperature control, but its power consumption is the highest. The 5-4-3 mm configuration has the best performance for the maximum temperature difference, but it is not optimal for the other two aspects.

Effect of the channel height
As depicted in Figure 4(b), the flow channel height has a significant influence on the battery temperature and power consumption characteristics. Figure 4(b) shows that the maximum temperature decreases exponentially with increasing channel height, but the benefit weakens gradually. Meanwhile, the power consumption increases linearly. When the channel height increases from 15 to 60 mm, the maximum temperature rise and the maximum temperature difference are reduced by 71.4% and 67%, respectively, at the expense of a 210.6% increase in power consumption. When analyzing the design goal of a maximum temperature of 318 K, it is found that the cooling performance can be satisfied at a channel height of 30 mm or more. Thus, the traditional full-size channel height design has great redundancy in cooling performance and power consumption.

Performance and error analysis of the compound surrogate model with the MoE method
To further explore the relationship between the parameters (cell spacing configurations and channel height) and find a performance-balanced structure, a compound surrogate model based on the MoE method is utilized. Notably, applying multiple experts can perform better than applying only one expert for this problem. However, too many experts will split the data space into clusters, which causes there to be too few examples to train reliable surrogate models. In this paper, the number of experts for maximum temperature rise, maximum temperature difference and power consumption regressions are set to 3, 3, and 2, respectively, to obtain the best fitting accuracy. The performance of each expert on local models with three outputs is listed in Tables 4-6, in which the best experts are shown in boldface.
With the utilization of the MoE method, the surrogate model can achieve the optimal fitting performance for the maximum temperature rise, maximum temperature difference, and power consumption, as shown in Figure 5(a). Compared with the traditional fitting method of a single surrogate model without the MoE method (as shown in Table 7), the compound surrogate model with the MoE method can more accurately fit the relationship between the inputs and outputs. As shown in Figure 5(b), the compound surrogate model with the MoE method reduces the global RMSE and MRE by up to 76.1% and 82.1%, respectively, for the maximum temperature aspect.

Solution of the multiobjective optimization problem and performance discussion
Since the change trends of the maximum temperature rise and the maximum temperature difference are not the same, it is necessary to consider the maximum temperature rise and the maximum temperature difference at the same time when selecting the optimization objectives. As shown in Figure 6(a), the distance between generations is below the termination tolerance threshold of 0.0025 after the 31st generation, and the iteration terminates at the 61st generation.
At the end of the iteration, a total of 1000 Pareto optimal solutions were obtained, as illustrated in Figure 6(b). However, these solutions are considered equally good; in other words, on the premise that the weights of the three evaluation indicators are the same, these solutions are all optimal design values for the system.
For the set of solutions mentioned above, the weights calculated by the EWM and CRITIC method for each criterion are shown in Table 8.
The optimal solutions obtained by ranking the scores using the weights obtained from the EWM and CRITIC methods are shown in Table 9, which also gives the baseline used for comparison.
The performance of these three structures is shown in Figure 7. Compared to the baseline, the optimal solution sorted by the EWM decreases the maximum temperature rise by 7.8% and the maximum temperature difference by 17.6% at the cost of increasing the power consumption by 4.2%. With regard to the optimal solution sorted by the CRITIC method, the maximum temperature rise     increases by 4.8%, the maximum temperature difference decreases by 9.0% and the power consumption decreases by 8.9% compared to the baseline. In general, the performance of the optimal solutions screened by the different weighting methods are consistent with the expectations.

Flow inlet velocity independence verification of the optimal design
Baseline independence verification is carried out to ensure the generalization ability of the optimized design structure. As the flow inlet velocity increases, the absolute improvement obtained by the optimal structures also increases in terms of power consumption while decreasing in terms of the maximum temperature rise and maximum temperature difference, as shown in Figure 8(a). However, the relative increase does not change much with the flow rate, and the weight characteristics of the EWM and CRITIC methods remain unchanged. It can be concluded that the optimization effect of the optimal structures selected by the EWM and CRITIC methods are independent of the flow inlet velocity.

Performance verification under discharge conditions
To ensure the generalizability of the optimized structure, it is essential to verify its performance under discharge conditions. Therefore, in this section, the performance of the optimized structure is verified using the US06 driving schedule condition (i.e. two US06 driving schedules connected in series) with a flow inlet velocity of 1 m/s. Under discharge conditions, the average C-rate of the battery is much smaller than that under fast charging conditions, which results in a smaller temperature rise and slower  flow inlet velocity required for cooling than under fast charging conditions. Using the US06 driving schedule as an example, a flow inlet velocity of 1 m/s can regulate the cell pack temperature within the ideal range. As shown in Figure 8(b), although the overall heat generation and power consumption of the battery pack under the above conditions are not statistically significant, the optimized structure can still obtain the expected optimization effect.

Conclusions
In this paper, a combination of a structural parameter optimization design approach, including the design of experiments (CFD numerical simulation), multiobjective optimization based on a compound surrogate model and a criterion weight-evaluation method, is proposed to perform an effective and optimal design of BTMSs when there are several structural design parameters and multiple evaluation criteria.
Compared with the traditional surrogate model, the compound surrogate model based on the mixture of experts with less training data has higher approximation accuracy for battery temperature distribution and power consumption of the BTMS; the global RMSE and MRE are reduced by up to 76.1 and 82.1%, respectively.
Based on the compound surrogate model, multiobjective optimization is conducted. Two weight-evaluation methods, the entropy weight method (EWM) and criteria importance through the intercriteria correlation (CRITIC) method, are introduced to determine the optimal structure. Compared to the baseline, the optimal solution provided by the EWM decreases the maximum temperature increase by 7.8% and the maximum temperature difference by 17.6%. The one provided by the CRITIC method, decreases the maximum temperature difference by 9.0% and the power consumption by 8.9%. Overall, the performance of the optimal solutions screened by the two weighting methods are consistent with the expectations.
In conclusion, the MoE-based surrogate model shows promise for use in applications for structural optimization problems. However, when the number of design parameters is further increased, the data obtained by the DoE will increase significantly, and more computational time will be needed.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that may appear to influence the work reported in this paper.

Disclosure statement
No potential conflict of interest was reported by the author(s).