1. Introduction
Due to their enhanced capacity to withstand impurities [
1], simplified water management [
2], and more convenient heat rejection [
3] relative to low-temperature proton exchange membrane fuel cells (LT-PEMFCs), HT-PEMFCs present substantial advantages for future commercialization. LT-PEMFCs typically operate from 60 to 80 °C [
4], and HT-PEMFCs generally operate within 120 to 200 °C [
5]. Numerous studies have been undertaken to enhance their performance through various properties, including durability [
6,
7,
8], corrosion [
9,
10], degradation [
11,
12,
13], and lifetime [
14]. Furthermore, in conjunction with the factors above, a crucial area of research is the optimization of design and operational parameters for HT-PEMFC performance, especially considering the heightened global attention directed towards HT-PEMFC applications in stationary micro combined heat and power systems [
15], the automotive industry [
16], backup power applications [
17], and auxiliary power units [
18]. Therefore, to realize the commercialization of HT-PEMFC technology, it is imperative to ascertain and optimize relevant parameters.
To date, many studies have been carried out to examine the environmental, economic, and thermodynamic evaluation of HT-PEMFCs. Unlike single-objective optimization, the multi-objective optimization methodology has garnered increasing interest in recent years due to its enhanced practicality and relevance [
19,
20,
21]. However, there is currently only a limited amount of research being conducted on improving the performance of a single HT-PEMFC by using the multi-objective optimization methodology. Li et al. [
19] introduced a novel power system in their study, which utilizes an HT-PEMFC system that combines methanol steam reforming and the organic Rankine cycle. The NSGA-II method was employed to optimize the system, and the findings demonstrate that the optimized system attains a net output power of 36.98 kW and a levelized energy cost of 0.2138
$/kWh. In their research, Mamaghani et al. [
21] undertook a multi-objective optimization investigation on a micro combined heat and power system that relies on a steady-state HT-PEMFC system. The study examined two distinct sets of objective functions, including thermal efficiency, net electrical output, thermal power generation, and net electrical efficiency. Consequently, by employing the primary energy-saving index, the researchers could identify the optimal operating conditions with electrical and thermal efficiency. The combined system concept described by Guo et al. [
20] comprises an HT-PEMFC system, a regenerator, and a thermoelectric generator. The integrated system exhibits a significant improvement in maximum power density of 19.1% compared to a standalone HT-PEMFC. This improvement is accompanied by similar energy and exergy efficiency advances, which increase by 12.4% and 12.6%, respectively. It is worth mentioning that the observed rise in exergy destruction rate density amounts to a mere 8.6%. In their study, Sarabchi et al. [
22] introduced a novel cogeneration system that combines an HT-PEMFC system with a Kalina cycle and a solar methanol steam reformer. This integrated system aims to generate both electricity and heat. The findings of the optimization study revealed that the average daily exergy efficiency has the potential to improve by a maximum of 29.3%. Nevertheless, it is essential to note that the performance of the system can be greatly influenced by the attributes of the individual HT-PEMFC [
23,
24,
25].
The predominant optimization approach for the single HT-PEMFC involves analyzing the design and operating parameters through single-objective analysis. Factors like material selection, operating parameters, and component geometry [
26] play a significant role in determining the performance of the HT-PEMFC. Consequently, the ultimate determination of diverse parameters is inherently complex. The main research factors currently include operating temperature [
27], operating pressure [
28], relative humidity [
29], doping level [
30], thickness of membrane [
31], and Pt loading [
32]. Thus, a significant obstacle arises in the optimization process of the HT-PEMFC: conducting a comprehensive assessment encompassing various factor groups is unrealistic. Most related studies within our review have primarily focused on independently clarifying the impact of select factors. For example, Bayat et al. [
29] obtained valuable results on the impact of membrane thickness, operating temperature, and relative humidity on exergy and energy performance. In their study, Guo et al. [
30] employed a single-factor analysis to demonstrate the significance of elevated operating temperatures and increased doping levels in enhancing the performance of HT-PEMFC. Haghighi et al. [
33] analyzed the exergy of a HT-PEMFC. They employed a genetic algorithm to compute and optimize various performance parameters. However, it is worth noting that the study did not specifically address the individual impact of a single factor on a single HT-PEMFC. Furthermore, the genetic algorithm employed in the analysis only considered three factors. Although numerous investigations have examined the impacts of various parameters on the energy and exergy performance of the HT-PEMFC, there is a lack of multi-objective analysis integrating evaluations, particularly those that focus on the trade-off of the parameters on a single HT-PEMFC energy efficiency, power, and exergy efficiency.
Hence, to address the research gap, an effective method was proposed to investigate the parameter–performance relationship and optimize the parameters utilizing a multi-objective optimization for a single HT-PEMFC. Firstly, a steady-state model developed in MATLAB R2021b is demonstrated as the base model. Secondly, mathematical statistics methods are used to compare and verify the model with experimental data in the literature. The initial results were compared with two experimental investigations at different operating temperatures and validated with two statistical techniques. Thirdly, assessments are carried out to determine how parameters affect performance. Lastly, optimizing three objective functions for the HT-PEMFC was conducted utilizing the NSGA-II [
34] to obtain improved performance compared with the base case.
3. Generic Performance Characteristics
To evaluate the validity of the proposed model, a comparison was made between the predicted and experimental results [
53,
54,
55] at various current densities and operating temperatures. This comparison is depicted in
Figure 3. The net output voltage was calculated and compared independently at three different operating temperatures (423 K, 438 K and 453 K).
The parameters used for validating the HT-PEMFC model can be found in
Table 4. During the process of model selection, validation, and comparison with experimental data, variations in the consistency of parameters provided in different literature sources were observed. Ensuring the model’s accuracy and its universality across three temperatures necessitated alignment with the existing model framework. This approach involved aligning key parameters, such as operating temperature and membrane thickness. In cases where complete numerical details were not available, representative values were chosen for input to enhance the reliability and applicability of the model.
The discrepancy between the experimental values and the modeling predictions in
Figure 3 can be primarily attributed to the following factors. First, the model validated the polarization curves at 423 K, 438 K, and 453 K. Extensive literature research has shown that experiments validating these three temperatures under identical conditions are scarce. In the papers that were accessed for these temperatures, the provided parameters were not entirely consistent. Therefore, representative values were assigned to parameters without specific values, while ensuring that key parameters remained aligned with experimental data. This approach not only maintains the correlation between the polarization curves and experimental values but also explains the characteristics of the three temperatures from a universality perspective. Second, the deviation in the high current density region is primarily due to the concentration overpotential, which is predominantly influenced by the limiting current density. Factors affecting the limiting current density, including pressure, operating temperature, fuel flow rate, and reactant concentration [
56], were not considered in this study.
The outcomes of the proposed model were assessed utilizing two mathematical statistical methods to quantify the disparity between the experimental and predicted data. The values of the root-mean-square error (
RMSE) for operating temperatures at 423 K, 438 K, and 453 K were 0.043, 0.032, and 0.040, respectively. Moreover, the R-squared (
R2) was calculated to be 0.991, 0.996, and 0.989 for operating temperatures at 423 K, 438 K, and 453 K, respectively. Additionally,
Figure 4 illustrates
R2 correlation coefficient distribution between the predicted and experimental data. Consequently, there is a remarkable correlation between the two data sets, confirming the high validation of the proposed model across nearly the entire current density range. The values of the
RMSE and
R2 can be obtained by the following formulas [
56,
57]:
where
n stand for the number of data points, and
,
, and
represent the experimental, predicted net output voltage values, and the mean of experimental net output voltage values, respectively.
Variations in the internal resistance,
, net output voltage,
U, reversible potential,
and three overpotentials, including concentration overpotential,
, activation overpotential,
, ohmic overpotential
, are illustrated based on Equations (4)–(21), as shown in
Figure 5.
remains constant regardless of
i, whereas the three overpotentials increase with
i.
and
increase hyperbolically and linearly with
i, respectively, whereas
increases logarithmically with
i. At a high current density,
U falls sharply due to a rapid increase in
, whereas at a low current density,
U falls suddenly due to a rapid increase in
.
RI* decreases monotonically as
i increases and approaches zero owing to the three overpotentials cumulative influence, where
RI*=
RI ·
A. Subsequent appearances of parameters with an asterisk have similar meanings, that is, they involve multiplying the corresponding parameter by the activated area (
A).
The performance parameters concerning energy and exergy can be derived from Equations (22)–(33) and their respective relationships with current density are illustrated in
Figure 6. It is essential to highlight that parameters marked with an asterisk represent equivalents based on the effective polarization area of the fuel cell. According to Refs. [
58,
59,
60], the regular current density is selected at
i =
ib = 0.60 A/cm
2 at base case 1 (BC-1), and the net output voltage (
Ub), equivalent power density (
P*
b), exergy efficiency (
φb), exergy destruction rate (
ExD*
b), energy efficiency (
ηb), and entropy production rate (
*
b) are 0.489 V, 2.932 kW/m
2, 33.96%, 3.623 kW/m
2, 37.50%, and 0.011 kW/m
2K, respectively. It should be highlighted that the ultimate optimization in this study is grounded on the performance metrics of the BC-1. Another base case 2 (BC-2) is when
P* reaches its maximum
P*
p of 3.942 kW/m
2, at
i =
ip = 1.23 A/cm
2, and
Up,
φp,
ExD*
p,
ηp, and
*
p are 0.321 V, 22.22%, 9.447 kW/m
2, 24.54%, and 0.028 kW/m
2K, respectively.
5. Optimization Analysis
In this section, the HT-PEMFC is optimized using the NSGA-II method for multi-objective purposes. The Pareto solution reveals an equilibrium among power density, energy efficiency, and exergy efficiency. Notably, all optimal solutions are represented on the Pareto surface, derived by weighting these three objectives.
Figure 12 demonstrates the Pareto surface used to identify the optimal solution. At Point A, while the equivalent power density peaks at 5.50 kW/m
2, energy and exergy efficiency are minimized at 25.03% and 26.88%. Conversely, Point B prioritizes energy and exergy efficiency at 50.15% and 49.55% but sees a marked dip in power density to 1.38 kW/m
2. Point C, being nearest to the ideal point, emerges as the optimal choice, registering values of 3.42 kW/m
2, 41.13%, and 41.30% for power density, energy, and exergy efficiencies, respectively. Fluctuations around Point C have minimal impact on the evaluation indicators.
Figure 13 depicts the optimization of HT-PEMFC parameters, highlighting the distribution of different parameters. As presented in
Figure 13a, the optimal range for the operating current density lies between 0.2–1.4 A/cm
2. This result could be attributed to the optimization process involving uniform varying of the weights for the three objectives. Therefore, the derived solutions represent the optimal trade-off among the objectives. As depicted in
Figure 13b, the preponderance of optimal operating temperature values is approximately 447.91K. The optimal operating pressure, as shown in
Figure 13c, is uniformly distributed between 2.3 and 3 atm. In addition, as illustrated in
Figure 13d, the optimal membrane thickness is predominantly around 0.0070 cm, whereas the optimal
DL is mostly around 7.95, as depicted in
Figure 13e.
Table 5 compares optimization outcomes, encompassing design and operating parameters, along with performance indicators of the HT-PEMFC at points A, B, C, BC-1, and BC-2. By employing parameters associated with Point A, the equivalent power density rose by 87.71% in comparison to BC-1. Meanwhile, Point B stands out as the best solution for energy and exergy efficiency. Point C is identified as the ultimate optimal point, exhibiting commendable performance across all assessment criteria. By adopting the operating parameters of Point C, there was a 17.72% rise in equivalent power density, a 21.11% enhancement in energy efficiency, and a 10.37% boost in exergy efficiency relative to BC-1. These findings indicate that employing the NSGA-II algorithm for optimization leads to diverse enhancements in the efficiency of the HT-PEMFC and equivalent power density at the ultimate optimal point, balancing between power density and efficiency.
6. Conclusions
In this research, a zero-dimensional, isothermal steady-state model was employed to probe the thermodynamic and electrochemical attributes of the HT-PEMFC. The multi-objective optimization strategy was proven effective in the trade-off of the optimizing process between power and efficiency. This methodology can be helpful during cell and system designing to optimize cell dimensions and operation parameters. Initially, the accuracy of the established model was validated using two mathematical statistical techniques. Subsequently, two primary cases were given based on the model results, namely BC-1 and BC-2. After evaluating their performances, BC-1 was ultimately chosen as the baseline case. After conducting parameter studies, it was established that the performance of HT-PEMFC is primarily affected by factors such as operating temperature, membrane thickness, and doping level. In contrast, variations in operating pressure were observed to have minimal influence on improving the performance of HT-PEMFC. Lastly, the NSGA-II approach was employed to optimize power and efficiency. The optimization results show that the optimal point significantly increased the power density by 17.72%, the energy efficiency by 21.11%, and the exergy efficiency by 10.37% compared with BC-1.