2. Model of PEMFC
PEMFCs are devices that facilitate converting chemical energy into electricity, as represented by Equation (1):
Figure 1 illustrates the polarization curve of a PEMFC, which consists of three distinct regions [
27]. The first region is the activation loss region, where the chemical response delay and electron excitation take place. The second region corresponds to concentration loss, which is attributed to losses caused by mass transfer. The third region represents the ohmic loss resulting from the material characteristics of PEMFC. The electrical characteristics of PEMFC can be modeled using Equation (2):
The activation polarization is denoted as
, the ohmic polarization as
, and the concentration loss as
. The thermodynamic potential is represented as
[
40]. This can be deduced from Equation (3):
The Faraday constant is denoted as
F, and the Gibbs free energy is represented by Δ
G. The entropy is denoted as Δ
S, and the gas constant is
R. The temperature of the FC is symbolized by
T, while the reference temperature is denoted as
. The partial pressure of hydrogen and oxygen are represented as
and
, respectively. These quantities can be described by Equations (4) and (5):
The anode and cathode humidities are captured as
and
, respectively. The pressures at the inlet for the anode and cathode are represented as
and
, respectively [
27]. The electrode area is denoted as A, and the current from the cell is denoted as
i. The water vapor saturation pressure is represented as
, and
is calculated as
. The activation polarization (
can be determined using Equation (7).
The semi-empirical constants are denoted as
. The oxygen concentration at the cathodic electrode is represented as
and can be calculated using Equation (8).
The mass concentration loss is represented by Equation (9).
The diffusion parameter is represented as
, and the limiting current for the fuel cell is presented as
The ohmic loss (
can be presented by Equation (10)
Resistance within the connectors is denoted as
, and ohmic membrane resistance is represented as
and is determined using Equation (11).
The membrane thickness is denoted as
, and the surface area is represented as
. The membrane resistivity, denoted as
, is determined using Equation (12). The membrane material water content is represented as
.
3. Parameter Identification Process
The power conversion system of PEMFCs based on a boost topology DC/DC power converter is depicted in
Figure 2. The boost topology allows for stepping up the output terminal voltage of the PEMFC by utilizing a single inductor, capacitor, semiconductor switch, and diode. The FLC MPPT utilizes the measured output terminal voltage and current of the PEMFC to track the MPP under various operating conditions. This is achieved by adjusting the duty cycle input to the power semiconductor switch, thereby controlling the system. Therefore, by employing this topology with PEMFC, it can simultaneously track MPPT and step up the low terminal voltage of the PEMFC. The relationship between the input and output voltages in the boost topology is dependent on the operating duty cycle and may be mathematically formulated using the following:
where
Vo represents the converter output voltage and
Vin represents the input voltage coming from the PEMFC side. The duty cycle of the converter, as denoted by
D, is adjusted and controlled using the FLC MPPT controller. The value of
D is typically varied based on the defined MPP of the PEMFC, and for each operating condition, an optimum
D needs to be searched for to ensure MPPT. Therefore, different operating conditions, such as variations in membrane water content and temperature, require different values of
D, which are dynamically adjusted through FLC MPPT control.
In MPPT control schemes, the terminal voltage and current of PEMFCs are measured to calculate the slope of the output power with respect to voltage (ΔPFC/ΔVFC), which serves as an effective criterion for determining the location of the MPP. The slope is continuously evaluated to track changes in the operational circumstances of the PEMFC. The controller regulates the duty cycle to track the MPP, aiming for a zero slope at the MPP.
The proposed controller in this study utilizes only the measured PEMFC output voltage (VFC) and current (IFC). The FLC is designed with two inputs based on the estimated error (E), derived from the calculations of Δ
PFC/Δ
VFC for MPPT control, and the change in error (Δ
E) in each sampling period. These two inputs are mathematically modeled for any sampling period (
k) as follows [
38]:
where
E(
k) represents the inputted error at instant
k, Δ
E(
k) represents the change in error at instant
k,
PFC(
k) and
PFC (
k − 1) represent the measured output power at sampling instants
k and (
k − 1), respectively, and
VFC(
k) and
VFC(
k − 1) represent the measured terminal voltage at sampling instants
k and (
k − 1), respectively. The FLC MPPT scheme solely relies on the measured electrical quantities of current and voltage from the PEMFCs. This simplifies the implementation and eliminates the need for expensive sensors, distinguishing it from other existing MPPT controllers. The principal stages of the FLC MPPT scheme are illustrated in
Figure 3 and can be defined as follows:
Fuzzification stage:
In this stage, the MFs are calculated for the inputs of the FLC. The calculation of these MFs depends on the selected shape and type of MFs for each input. In the proposed method, triangular MFs are chosen and designed for the inputs. The purpose of this stage is to convert the physically measured electrical signals into a suitable form for the FLC method.
Fuzzy-rule based evaluation stage:
In this step, a set of fuzzy rules is applied to evaluate the calculations obtained from the previous fuzzification stage.
Defuzzification stage:
In this stage, the output of the fuzzy rule-based evaluation is inputted to the defuzzification step. Afterward, it outputs the MF for the converter duty cycle by utilizing the calculated values and increments for the duty cycle.
Figure 3.
Main stages of FLC in MPPT applications.
Figure 3.
Main stages of FLC in MPPT applications.
The suggested FLC utilizes MFs that are divided into seven levels for both the input and output sides. These levels consist of three positive levels (PLS3, PLS2, and PLS1), three negative levels (NLS3, NLS2, and NLS1), and a single zero level (ZLS0). The shape of the employed MFs for the input and output sides is depicted in
Figure 4a.
Table 1 presents the fuzzy rules, which comprise a total of 49 rules based on the designed MFs. The output MF consists of seven different levels, allowing for more adaptive control of the step size. This adaptive control enables fast tracking and reduces fluctuations in steady-state.
The classical FLC design, as depicted in
Figure 4a, lacks the ability to adjust and optimize the MFs as fixed divisions are used in its design. However, in the proposed FLC design, the shape of the MFs is adjusted, allowing for better flexibility and optimization. This enhanced flexibility enables the optimization of the FLC design, leading to improved performance using the proposed method.
Figure 4b illustrates the flexibility in designing and adjusting the various parts of the MFs in both directions. In the proposed method, the optimization process is carried out simultaneously for the input and output MFs, allowing for a better and more optimized design compared to conventional approaches. As a result of the optimization, the distances between the points of the MFs will vary to achieve their optimized values.
Figure 4c displays the resulting shape of the MFs in the proposed method. During the optimization process, each MF is designed with seven points, resulting in a total of 21 points that are simultaneously optimized using the proposed gradient-based optimizer (GBO) method. In the proposed design, a symmetrical design is maintained for the MFs around the zero value, ensuring a balanced representation of the input and output variables. Specifically, each MF has 3 adjustable points, resulting in a total of 9 adjustable points in the proposed process.
Figure 5 depicts the different stages involved in the proposed FLC optimization process. The authors have implemented the FLC in Matlab, which allows for greater flexibility in optimizing the FLC. In this scheme, the objective function is established using the integral-absolute error (IAE), which corresponds to the MPPT criterion based on Δ
PFC/Δ
VFC and presented by the following relation.
where
Tsim stands for total simulation time during optimization run for evaluating desired objective function. The targeted output of the proposed optimization scheme is to obtain optimized points for the MFs. The implementation of the suggested FLC is shown in
Figure 6.
5. Results
In this case study, the use of a GBO-FLC optimization method is proposed to improve the performance of a PEMFC. The specifications of the PEMFC, as outlined in
Table 2, serve as the basis for this optimization process. The PEMFC is connected to a boost converter with specific components, including an inductor of 0.5 mH, a capacitor of 500 μF, and a switching frequency of 20 kHz. The effectiveness of the proposed GBO-FLC is evaluated under two different operating conditions: Case 1, representing normal operation with a constant membrane water content and temperature, and Case 2, involving variable water content and temperature.
The GBO controller parameters selected for the study are as follows: a population size of 10, a maximum iteration limit of 30, and 30 independent runs. The performance of the proposed GBO-FLC is compared against other algorithm approaches for both Case 1 and Case 2. In the following subsections, a more detailed explanation of each case is provided, and the comparative analysis of the different algorithms is presented.
During the normal operation of the PEMFC, a constant membrane water content of 12 and a steady temperature of 343 K were maintained. To ensure a fair comparison, the population size and iterations are 10 and 20, respectively. The optimization process focused on determining the values of the nine unknown parameters of the FLC, with the objective of minimizing the integral absolute error.
Table 3 provides limits of these unknown parameters, along with their optimal values obtained after 30 runs. Statistical evaluation of the results is presented in
Table 4.
Based on the results shown in the table, the mean cost function values range from 1.86315 to 1.93269. The GBO algorithm achieves the minimum mean cost function value of 1.86315, followed by SCA with a value of 1.88947, while SSA obtains the highest mean cost function value of 1.93269. The standard deviation values vary between 0.01152 and 0.13314. The GBO algorithm achieves the minimum standard deviation value of 0.01152, followed by SCA with a value of 0.01831, while WOA obtains the largest standard deviation value of 0.13314. Therefore, for the investigated cases, the GBO algorithm outperforms the OOA, SCA, SSA, and WOA algorithms.
Table 5 provides detailed data over 30 runs.
ANOVA and Tukey tests were accomplished to verify the superiority of GBO in comparison with other methods. ANOVA measures mean differences between groups and compares the variability between groups and within groups. This test helps clarify whether the differences between group means are statistically significant.
Table 6 presents the ANOVA test results, while
Figure 7 illustrates the corresponding ranking. The results confirm that GBO consistently produces the best results, as evidenced by its lowest mean fitness and fluctuations compared to the other algorithms. The statistical analysis further supports the conclusion that GBO outperforms the competing algorithms in terms of optimization performance.
The results of ANOVA are further validated via the Tukey test, as depicted in
Figure 8. The mean of the GBO group is observed to be lower compared to the OOA and SCA groups, and significantly lower compared to the SSA and WOA groups. This provides strong evidence that the GBO algorithm exhibits superior performance compared to the other algorithms under consideration.
Figure 9 illustrates the variation of the objective function. As depicted in
Figure 9a and
Table 5, the IAE values converge to 1.86015, 1.88457, 1.89635, 1.9615, and 1.93018, respectively, for GBO, OOA, SCA, SSA, and WOA. GBO achieves rapid convergence to the optimal solution, while OOA requires more time to reach its best solution. In run number 15, as shown in
Figure 9b and
Table 5, the IAE values converge to 1.8575, 1.97045, 1.86067, 1.92974, and 1.8985, respectively, for GBO, OOA, SCA, SSA, and WOA. Once again, GBO demonstrates fast convergence to the optimal solution, while OOA requires more time to reach its best solution.
Figure 10 displays the time responses of power, current, voltage, and duty cycle for a normal case study using both classical FLC and FLC-GBO. The dynamic performance of FLC-GBO is noticeably superior to classical FLC.
The PEMFC’s temperature and membrane water content are altered in the second scenario. At first, the water content is 10, and the temperature is set to 40 °C. The water content rises to 14 at 0.1 s while the temperature stays at 40 °C. The temperature then rises to 70 °C while the water content stays at 14 at 0.25 s. In this case, the PEMFC’s power climbs from 5000 W to 6880 W in 0.1 s and then from 6880 W to 9180 W in 0.25 s.
Figure 11 displays the PEMFC’s power, voltage, current, and duty cycle responses in detail. In comparison to the standard FLC, the optimized FLC delivers a maximum power of 6880 W with a short tracking time.
6. Conclusions
An improved design methodology and optimization approach have been proposed in this study for FLC-based MPPT in PEMFC applications. The flexible and accurate tracking ability of fuzzy FLC has been effectively optimized using the advanced GBO method. The utilization of the GBO optimization method has enabled the simultaneous optimization of FLC membership functions (MFs) for both the input and output sides. This approach offers enhanced design flexibility, allowing for precise control over the positioning of points within the triangular MFs in the proposed FLC design. The proposed FLC design has been implemented and tested under various operating conditions, including normal operation and scenarios with variable temperature and water content. Extensive comparisons were conducted using statistical analyses to assess the performance of the GBO optimizer in comparison to other recent optimizers, namely OOA, SCA, SSA, and WOA. The results demonstrated that the FLC system utilizing the GBO optimizer exhibited a superior response compared to classical MPPT designs. Moreover, the statistical analysis, including mean cost function values and standard deviation, revealed the superior performance of the GBO optimizer in comparison to the studied optimizers. The obtained objective functions over 30 runs showed that GBO has a minimum mean cost function value of 1.86315, followed by the SCA with a value of 1.88947, while the WOA has a mean cost function of 1.92757 and the SSA has the highest mean cost function value of 1.93269. Meanwhile, the standard deviation value of the proposed GBO is 0.01152, which is only 8.65% of the standard deviation of the WOA method. ANOVA and Tukey tests were conducted to validate the superiority of the GBO algorithm over other algorithms. The ANOVA test results confirmed that GBO consistently produced the best results with the lowest mean fitness and fluctuations. The Tukey test further confirmed the significant difference in performance, showing that the GBO group achieved lower means compared to other groups. These findings validate the effectiveness and competitiveness of the proposed FLC design with GBO optimization in maximizing the power point tracking (MPPT) capability of PEMFCs. The proposed method can also help extend the applications of PEMFCs with better efficiency and faster response. Moreover, it can contribute to the extensive use of green hydrogen, power to hydrogen technology, and, in general, the energy transition process. Future research will include comparisons with more recent optimization techniques, the extension to type-2 FLC systems, and the application of the proposed method for PV systems MPPT control.