Accurate Key Parameters Estimation of PEMFCs’ Models Based on Dandelion Optimization Algorithm
<p>Flowchart illustrating the Research methodology.</p> "> Figure 2
<p>Electrical model of a PEM fuel cell.</p> "> Figure 3
<p>Convergence curve of fitness function: (<b>a</b>) 250 W fuel cell; (<b>b</b>) NedStack PS6 fuel cel.</p> "> Figure 4
<p>The statistical study of the 250 W after 30 repetitions: (<b>a</b>) best values of the objective function; (<b>b</b>) boxplot of SSE.</p> "> Figure 5
<p>The statistical study of the NedStack PS6: (<b>a</b>) best values of the objective function; (<b>b</b>) boxplot of SSE.</p> "> Figure 6
<p>The current–voltage curve: (<b>a</b>) 250 W fuel cel; (<b>b</b>) NedStack PS6 fuel cell.</p> "> Figure 7
<p>The current–power curve: (<b>a</b>) 250 W fuel cell; (<b>b</b>) NedStack PS6 fuel cell.</p> "> Figure 8
<p>PEMFC simplified dynamic model.</p> "> Figure 9
<p>Characteristic and dynamic simulations of the PS6 stack: (<b>a</b>) I–V polarisation; (<b>b</b>) stack load current changes; (<b>c</b>) stack voltage responses; (<b>d</b>) voltage error.</p> ">
Abstract
:1. Introduction
1.1. Motivation
1.2. Literature Review
1.3. Contribution
- Low computational complexity: The evolution strategy used by the DO requires fewer computations than most other metaheuristic algorithms;
- Property of self-adaptation: The DO can self-adapt its parameters, allowing it to converge quickly and accurately to local minima and maximize the solution’s performance;
- Minimization of stagnation of convergence: The DO utilizes a memory-based mechanism to identify stagnation of convergence and then takes steps to move away from a local minimum and towards a better solution;
- Ability to refer to an entire population: Unlike many other metaheuristic algorithms, the DO is able to refer to the entire population of solutions to best determine the optimal solution;
- Decentralized nature: The DO is a decentralized algorithm, meaning its components are distributed among multiple nodes or machines, making it less prone to failure than centralized algorithms.
- To adapt the DO algorithm, and benefit from its high exploitation and exploration abilities to address, for the first time, the issue of the PEMFC unidentified parameter estimation;
- Conducting a deep comparative study between DO and three recently published optimizers (Grey wolf optimizer (GWO), Gradient-based optimizer (GBO), and Harris hawks optimizer (HHO)) simulated under the same hypothesis;
- Comparing the DO performance with three competitive metaheuristic optimizers from the literature such as (Improved artificial ecosystem optimizer (IAEO), Vortex search-differential evolution (VSDE), and Artificial bee colony differential evolution shuffled complex (ABCDESC));
- The experimentation of two commercialized PEMFCs (EMFC 250 W, and NedStack PS6) has confirmed the superiority of the proposed DO in terms of stability, convergence speed, and absolute accuracy.
2. General Model of PEM Fuel Cells
3. Problem Formulation
4. Proposed DO-Based Approach
4.1. Initialization
4.2. Rising Stage
- Condition 1:On clear days without weather fluctuations, the wind speed is characterized by what can be modeled by a logarithmic distribution according to Equation (19):Under these conditions, the transmission by seeds is remote randomly because the distribution is mainly along the y-axis, which triggers the process of DO exploration. In the search area, the dispersion of dandelion seeds is closely matched to the wind speed, which influences their height and dispersion. Under this impact, the vortices above the seeds are continuously adjusted to force them to spiral upward, according to the following equation:and denote the location of the dandelion seed and that of the search space for the iteration number t, respectively. As such, the location obtained by random is expressed by:It is important to mention that is a lognormal distribution obeying the conditions = 1 and = 0, and so mathematically translated as:is an adjustment parameter to adapt the search step length, and y is defined as the normal standard distribution . Accordingly, is given by:It was shown in [35] that the parameter behaves as a function of the number of iterations as a random fluctuation in the range [0, 1] that decreases nonlinearly toward 0. When defining and as the dandelion lift parameter coefficients under the effect of the whirlwind action, the force calculation on the variable dimension obeys:
- Condition 2:During rainy days, the air resistance is increased due to the high humidity, and therefore the buoyancy of dandelion seeds and their height in space are restricted, which involves the need to process it in their local proximities, according to the following equation Equation (25):At this stage, the seeds that are undergoing the rising phase are approximated by Equation (27):
4.3. Descending Stage
4.4. Landing Stage
4.5. Pseudo Code of the Proposed DO Algorithm
Algorithm 1 Pseudo-code of the proposed DO optimizer. |
|
5. Validation of Simulation Results
5.1. Identification of PEMFCs’ Stack Parameters, Performance Measures and Comparisons
5.2. Transient Response to Current Changes
- Regarding the parameter settings, the population size, the mutation rate and the convergence criteria have been set appropriately for the present problem without any difficulties to be mentioned;
- For the implementation complexity, the complex operations involved in the whole process, i.e., the rising, descending, and landing stages, and the steps of generating the seeds and updating them, etc., the simplicity of the DO has allowed its implementation to be successful;
- Concerning memory usage, the dimensionality of the solved optimization problem did not cause any problems in this respect;
- The convergence speed of the DO has been satisfactory since it runs for a short period of time and achieves better results than the other tested algorithms.
6. Conclusions and Future Works
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PEMFC | Proton exchange membrane fuel cell |
DO | Dandelion optimizer |
GWO | Grey wolf optimizer |
GBO | Gradient-based optimizer |
HHO | Harris Hawks optimizer |
IAEO | Improved artificial ecosystem optimizer |
VSDE | Vortex search-differential evolution |
ABCDESC | Artificial bee colony differential evolution shuffled complex |
HR | Hail region |
CO2 | Carbon dioxide |
CO | Carbon monoxide |
H2O | Water molecule |
H2 | Hydrogen gas |
O2 | Dioxygen gas |
RE | Renewable energy |
KSA | Kingdom of Saudi Arabia |
UNWTO | World Tourism Organization |
COVID-19 | Coronavirus disease |
EV | Electric vehicle |
ZEVs | Zero-emission vehicles |
AC | Alternative current |
GTO | Gorilla troops optimizer |
MGTO | Modified GTO |
HBO | Honey badger optimizer |
SSE | Sum of squared errors |
BO | Bonobo optimizer |
QOBO | Quasi oppositional bonobo optimizer |
EBES | Enhanced bald eagle search |
DA | Dandelion algorithm |
ELM | Extreme learning machine |
NNA | Neural network algorithm |
ELMD | Dandelion algorithm with ELM |
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Parameter | b | ||||||
---|---|---|---|---|---|---|---|
Min | −1.19969 | 10 | 0.0136 | ||||
max | −0.8532 | 24 | 0.5 |
Parameters | A(cm) | (m) | (A cm) | (bar) | (bar) | (atm) | (atm) | T(K) | P(kW) | |
---|---|---|---|---|---|---|---|---|---|---|
250 W | 24 | 27 | 127 | 0.86 | 3 | 5 | - | - | 353.15 | 0.25 |
NedSstack | 65 | 240 | 178 | 1.2 | - | - | 1 | 1 | 343 | 6 |
Parameters | DO | GWO | GBO | HHO |
---|---|---|---|---|
−0.961592 | −0.871596 | −0.974457 | −0.895253 | |
0.00253443 | 0.00255737 | 0.00334035 | 0.00232372 | |
5.90919 | ||||
−0.000138245 | −0.000141236 | −0.000136735 | −0.000100082 | |
13.3372 | 13.3735 | 13.9459 | 10.0273 | |
0.000423201 | 0.000136504 | 0.0007994 | 0.000100273 | |
b | 0.0149649 | 0.0157339 | 0.0148931 | 0.0136372 |
Minimum | 0.158400329 | 0.160086164 | 0.158527784 | 0.276840933 |
Maximum | 0.166811864 | 0.353987785 | 0.272967021 | 2.136931329 |
Average | 0.160609673 | 0.194413051 | 0.18248663 | 0.983809404 |
Std | 0.002660149 | 0.046728642 | 0.033055337 | 0.488586136 |
Parameters | DO | GWO | GBO | HHO |
---|---|---|---|---|
−1.10823 | −0.981556 | −1.19841 | −0.8532 | |
0.00348488 | 0.0029638 | 0.00435526 | 0.00274411 | |
23.0714 | 14.9218 | 23.95 | 14.3893 | |
0.000127527 | 0.000198012 | 0.000327086 | 0.000186993 | |
b | 0.0835662 | 0.0236393 | 0.0539602 | 0.0195792 |
Minimum | 2.077565321 | 2.288386949 | 2.278642183 | 2.286126251 |
Maximum | 2.627548699 | 4.705647651 | 3.756412117 | 9.447348766 |
Average | 2.501037792 | 3.038447633 | 2.873919639 | 4.517318823 |
Std | 0.076297062 | 0.618991833 | 0.248397126 | 1.865763408 |
Parameters | DO | IAEO [18] | VSDE [39] | ABCDESC [40] |
---|---|---|---|---|
−0.961592 | −0.9991 | −1.1921 | −1.12806 | |
0.00253443 | 0.002825 | 0.0031990 | 0.00394667 | |
−0.000138245 | −0.00017 | −0.000187 | -0.000174889 | |
13.3372 | 19.99358 | 22.817 | 19.9358326 | |
b | 0.0149649 | 0.0145 | 0.0290 | 0.014526 |
Minimum | 0.158400329 | 0.3360 | 1.0526 | 0.33597 |
Parameters | DO | IAEO [18] | VSDE [39] | ABCDESC [40] |
---|---|---|---|---|
−1.10823 | −0.9822 | −1.1212 | −0.9350526 | |
0.00348488 | 0.0035957 | 0.00033487 | 0.0035035 | |
23.0714 | 13.4650 | 13.0000 | 13.094707 | |
0.0001 | ||||
b | 0.0835662 | 0.0136 | 0.0494 | 0.0136 |
Minimum | 2.077565321 | 2.1459 | 2.088 | 2.079204604 |
N | |||||
---|---|---|---|---|---|
1 | 0.2729 | 23.5410 | 23.436884 | 0.1041 | 0.0108 |
2 | 1.2790 | 21.4756 | 21.552742 | −0.0771 | 0.0060 |
3 | 2.6603 | 20.3484 | 20.542713 | −0.1943 | 0.0378 |
4 | 3.9734 | 19.8969 | 19.913082 | −0.0162 | |
5 | 5.3547 | 19.4642 | 19.385395 | 0.0788 | 0.0062 |
6 | 6.7190 | 19.0127 | 18.934608 | 0.0781 | 0.0061 |
7 | 8.0321 | 18.5049 | 18.470406 | 0.0345 | 0.0012 |
8 | 10.7265 | 17.8835 | 17.792614 | 0.0909 | 0.0083 |
9 | 13.4720 | 17.2808 | 17.135401 | 0.1457 | 0.0449 |
10 | 16.1664 | 16.2089 | 15.996914 | 0.2120 | 0.0212 |
11 | 17.4966 | 15.8701 | 15.988641 | −0.1185 | 0.0141 |
12 | 18.8608 | 15.5312 | 15.594024 | −0.0628 | 0.0039 |
13 | 20.1910 | 15.1923 | 15.175456 | 0.01686 | |
14 | 21.5553 | 14.6282 | 14.586637 | 0.0416 | 0.0017 |
15 | 22.9195 | 13.7450 | 13.693124 | 0.0519 | 0.0027 |
SSE | 0.158400329 |
N | |||||
---|---|---|---|---|---|
1 | 2.25 | 61.64 | 62.452443 | −0.8124 | 0.6600 |
2 | 6.75 | 59.57 | 59.873841 | −0.3038 | 0.0923 |
3 | 9.00 | 58.94 | 59.140399 | −0.2003 | 0.4877 |
4 | 15.75 | 57.54 | 57.582298 | −0.04229 | 0.0017 |
5 | 20.25 | 56.80 | 56.799747 | 0.0002 | |
6 | 24.75 | 56.13 | 56.122583 | 0.0074 | |
7 | 31.50 | 55.23 | 55.729928 | 0.0003 | |
8 | 36.00 | 54.66 | 54,589203 | 0.0707 | 0.0050 |
9 | 45.00 | 53.61 | 53.604175 | 0.0058 | |
10 | 51.75 | 52.86 | 52,799762 | 0.0602 | 0.0036 |
11 | 67.50 | 51.91 | 51.48410 | 0.4258 | 0.1813 |
12 | 72.00 | 51.22 | 51.068878 | 0.1511 | 0.0228 |
13 | 90.00 | 49.66 | 49,452415 | 0.2075 | 0.04309 |
14 | 99.00 | 49.00 | 48.659847 | 0.3401 | 0.1157 |
15 | 105.80 | 48.15 | 48.064054 | 0.0859 | 0.0073 |
16 | 110.30 | 47.52 | 47.670360 | −0.1503 | 0.0022 |
17 | 117.00 | 47.10 | 47.084098 | 0.0159 | 0.0002 |
18 | 126.00 | 46.48 | 46.294582 | 0.1854 | 0.0343 |
19 | 135.00 | 45.66 | 45.500518 | 0.1594 | 0.0254 |
20 | 141.80 | 44.85 | 44.799173 | 0.0508 | 0.0025 |
21 | 150.80 | 44.24 | 44,088817 | 0.1511 | 0.0228 |
22 | 162.00 | 42.45 | 43.069394 | −0.6193 | 0.3836 |
23 | 171.00 | 41.66 | 42,235973 | −0.5759 | 0.3317 |
24 | 182.30 | 40.68 | 40.708183 | −0.0281 | 0.0007 |
25 | 189.00 | 40.09 | 40.102165 | −0.0121 | 0.0001 |
26 | 195.80 | 39.51 | 39.455396 | 0.0546 | 0.0.0030 |
27 | 204.80 | 38.73 | 38.553613 | 0.1763 | 0.0311 |
28 | 211.50 | 38.15 | 38.696963 | 0.1837 | 0.0337 |
29 | 220.50 | 37.38 | 37.18800 | 0.1919 | 0.0368 |
SSE | 2.257565321 |
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Abbassi, R.; Saidi, S.; Abbassi, A.; Jerbi, H.; Kchaou, M.; Alhasnawi, B.N. Accurate Key Parameters Estimation of PEMFCs’ Models Based on Dandelion Optimization Algorithm. Mathematics 2023, 11, 1298. https://doi.org/10.3390/math11061298
Abbassi R, Saidi S, Abbassi A, Jerbi H, Kchaou M, Alhasnawi BN. Accurate Key Parameters Estimation of PEMFCs’ Models Based on Dandelion Optimization Algorithm. Mathematics. 2023; 11(6):1298. https://doi.org/10.3390/math11061298
Chicago/Turabian StyleAbbassi, Rabeh, Salem Saidi, Abdelkader Abbassi, Houssem Jerbi, Mourad Kchaou, and Bilal Naji Alhasnawi. 2023. "Accurate Key Parameters Estimation of PEMFCs’ Models Based on Dandelion Optimization Algorithm" Mathematics 11, no. 6: 1298. https://doi.org/10.3390/math11061298