Proton Exchange Membrane Fuel Cell Parameter Extraction Using a Supply–Demand-Based Optimization Algorithm
Abstract
:1. Introduction
2. PEMFC Modeling
2.1. Basic Operation of PEMFC
2.2. Theoretical Modeling
2.3. Objective Function and Constraints
3. Optimization Method and Implementation
3.1. Preliminary Concepts
3.2. Supply–Demand-Based Optimization (SDO) Algorithm
4. Experimental Results and Discussion
4.1. Case Study 1 (Different Operational Conditions)
4.2. Case Study 2 (Different Types of PEMFC Stacks)
4.2.1. BCS-500W
4.2.2. Horizon-500W
4.3. Average Convergence Rate
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Abbreviations | SBO | satin bowerbird optimizer | |
ABS | artificial bee swarm algorithm | TGA | tree growth algorithm |
ABC | artificial bee colony algorithm | TS | tabu search |
ALO | ant lion optimizer | TLBO | teaching learning-based optimizer |
ASO | atom search optimizer | VSA | vortex search algorithm |
AIS | artificial immune system | WOA | whale optimization algorithm |
BBBC | big bang–big crunch algorithm | ||
BSA | backtracking search algorithm | Variables | |
BBO | biogeography-based optimization | A | active area of membrane |
BMO | bird mating optimizer | concentration of dissolved oxygen | |
BA | bat algorithm | nernst theoretical voltage | |
CSA | crow search algorithm | PEMFC current | |
CS | cuckoo search algorithm | , | current density and maximum current density |
DEA | differential evolution algorithm | membrane thickness | |
DA | dragonfly algorithm | number of fuel cells | |
FC | fuel cell | partial gas pressures of hydrogen and oxygen | |
FFA | fruit fly algorithm | pressure at anode side and cathode side | |
FPA | flower pollination algorithm | saturation pressure of water | |
FFO | firefly optimization | lower and higher cell connections resistance | |
GWO | grey wolf optimizer | relative humidity at cathode node and anode node | |
GOA | grasshopper optimization algorithm | membrane resistance against transfer of protons | |
GA | genetic algorithm | equivalent resistance of membrane | |
HHO | harris hawks optimization | temperature of cell | |
HS | harmony search | output voltage of PEMFC stack | |
ICA | imperialist competitive algorithm | experimental and simulation output voltage of PEMFC | |
MVO | multi-verse optimizer | activation voltage losses | |
OSA | owl search algorithm | ohmic voltage losses | |
PSO | particle swarm optimization | concentration voltage losses | |
PEMFC | proton exchange membrane fuel cell | output voltage of one fc | |
SDO | supply–demand-based optimization | semi-empirical coefficients | |
SA | simulated annealing | specific resistivity of membrane | |
SSA | salp swarm algorithm | empirical parameter of membrane preparation | |
SSE | sum of the squared error | parametric coefficient | |
SSO | shark smell optimizer | , | lowest and highest values of parametric coefficient |
SFLA | shuffled frog-leaping algorithm | lowest and highest values of empirical coefficients | |
SOA | seagull optimization algorithm | lowest and highest values of preparation parameter |
Appendix A
No. | 3/5 bar 353.15 K | 1/1 bar 343.15 K | 2.5/3 bar 343.15 K | 1.5/1.5 bar 343.15 K | ||||
---|---|---|---|---|---|---|---|---|
Current (A) | Voltage (V) | Current (A) | Voltage (V) | Current (A) | Voltage (V) | Current (A) | Voltage (V) | |
1 | 0.2729 | 23.5410 | 0.2046 | 21.5139 | 0.2582 | 23.2710 | 0.2417 | 22.6916 |
2 | 1.2790 | 21.4756 | 1.2619 | 19.6737 | 1.3340 | 21.0280 | 1.3177 | 20.1869 |
3 | 2.6603 | 20.3484 | 2.6433 | 18.7154 | 2.6471 | 20.0748 | 2.6819 | 19.2897 |
4 | 3.9734 | 19.8969 | 3.9734 | 17.9449 | 4.0281 | 19.4019 | 4.0118 | 18.5607 |
5 | 5.3547 | 19.4642 | 5.3206 | 17.5497 | 5.3919 | 18.8972 | 5.3755 | 18.1682 |
6 | 6.7190 | 19.0127 | 6.7019 | 17.1545 | 6.7726 | 18.5047 | 6.7563 | 17.7196 |
7 | 8.0321 | 18.5049 | 8.0491 | 16.6843 | 8.0852 | 18.0561 | 8.0689 | 17.2710 |
8 | 10.7265 | 17.8835 | 10.7265 | 15.8752 | 10.8297 | 17.2897 | 10.8134 | 16.4299 |
9 | 13.4720 | 17.2808 | 13.4720 | 15.1411 | 13.5230 | 16.5047 | 13.4556 | 15.7009 |
10 | 16.1664 | 16.2089 | 16.1494 | 14.4634 | 16.1652 | 15.7196 | 16.1488 | 14.9907 |
11 | 17.4966 | 15.8701 | 17.4795 | 14.0870 | 17.5459 | 15.3271 | 17.5295 | 14.6542 |
12 | 18.8608 | 15.5312 | 18.8438 | 13.5792 | 18.8584 | 14.9907 | 18.8423 | 14.0374 |
13 | 20.1910 | 15.1923 | 20.1739 | 12.6772 | 20.2733 | 14.5421 | 20.2234 | 13.1963 |
14 | 21.5553 | 14.6282 | 21.5382 | 10.8743 | 21.5523 | 13.5888 | 21.6049 | 12.0187 |
15 | 22.9195 | 13.7450 | 22.9025 | 8.92130 | 22.9337 | 12.5234 | 22.9189 | 10.1308 |
No. | Current (A) | Voltage (V) | No. | Current (A) | Voltage(V) |
---|---|---|---|---|---|
1 | 0.60 | 29 | 10 | 15.73 | 21.09 |
2 | 2.10 | 26.31 | 11 | 17.02 | 20.68 |
3 | 3.58 | 25.09 | 12 | 19.11 | 20.22 |
4 | 5.08 | 24.25 | 13 | 21.20 | 19.76 |
5 | 7.17 | 23.37 | 14 | 23.00 | 19.36 |
6 | 9.55 | 22.57 | 15 | 25.08 | 18.86 |
7 | 11.35 | 22.06 | 16 | 27.17 | 18.27 |
8 | 12.54 | 21.75 | 17 | 28.06 | 17.95 |
9 | 13.73 | 21.45 | 18 | 29.26 | 17.30 |
No. | Current (A) | Voltage (V) |
---|---|---|
1 | 0.6 | 29.370000 |
2 | 2.5 | 26.777390 |
3 | 5 | 25.290250 |
4 | 7.5 | 24.281859 |
5 | 10 | 23.418000 |
6 | 12 | 22.739103 |
7 | 14 | 22.058523 |
8 | 16 | 21.386148 |
9 | 18 | 20.721728 |
10 | 20 | 20.026000 |
11 | 21 | 19.636350 |
12 | 22 | 19.191807 |
13 | 23 | 18.663630 |
14 | 24 | 18.015227 |
15 | 25 | 17.201250 |
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Parameter | Constraints | |
---|---|---|
Upper | Lower | |
−0.8532 | −1.19969 | |
5.00 × 10−3 | 1.00 × 10−3 | |
9.8 × 10−5 | 3.6 × 10−5 | |
−9.54 × 10−5 | −260 × 10−4 | |
(Ω) | 8.00 × 10−4 | 1.00 × 10−4 |
(V) | 0.5 | 0.0136 |
24 | 10 | |
178 | 51 | |
1500 | 850 |
Parameter | Value | Parameter | Value |
---|---|---|---|
n | 24 | Pa (bar) | 1.0–3.0 |
A (cm2) | 27 | Pb (bar) | 1.0–5.0 |
Power (w) | 250 | RHa | 1.0 |
T(K) | 343.15–353.15 | RHb | 1.0 |
SSE | Optimization Algorithms | |||||
---|---|---|---|---|---|---|
WOA | GWO | SSA | HHO | GA | SDO | |
Minimum | 0.154639186 | 0.146884822 | 0.152883434 | 0.167631296 | 0.350360131 | 0.145167469 |
Mean | 0.419153062 | 0.170258026 | 0.178072981 | 0.491306301 | 1.561131011 | 0.148407841 |
Median | 0.397898527 | 0.157920235 | 0.155409751 | 0.424590144 | 1.360612054 | 0.147780145 |
Maximum | 1.073495308 | 0.346056858 | 0.419199433 | 1.288663305 | 7.448856801 | 0.154441127 |
STD | 0.196771275 | 0.033980005 | 0.062370696 | 0.262778766 | 1.266038132 | 0.002718387 |
Variable | Algorithm | |||||
---|---|---|---|---|---|---|
WOA | GWO | SSA | HHO | GA | SDO | |
−0.884007645 | −0.944234024 | −1.07199885 | −1.156237962 | −0.90822 | −0.964599939 | |
0.002484898 | 0.002893255 | 0.002880452 | 0.002958249 | 0.00295 | 0.002440458 | |
6.06253 × 10−5 | 7.86346 × 10−5 | 4.99592 × 10−5 | 3.73411 × 10−5 | 0.00009 | 3.96338 × 10−5 | |
−0.000139704 | −0.000137972 | −0.000137754 | −0.000144177 | −0.00013 | −0.000138061 | |
0.000178974 | 0.000172751 | 0.000158811 | 0.000559279 | 0.00045 | 0.000100003 | |
0.051607571 | 0.019026091 | 0.026518418 | 0.016646806 | 0.089330 | 0.015471 | |
16.02236023 | 10.46643873 | 15.5096387 | 10.00002765 | 12.14717 | 10.00039 | |
0.008914346 | 0.010066463 | 0.014396789 | 0.008537823 | 0.00815 | 0.010209 | |
0.939531061 | 0.873824589 | 0.886530415 | 0.864155406 | 1.27485 | 0.86617 | |
0.154639186 | 0.146884822 | 0.152883434 | 0.167631296 | 0.350360131 | 0.145167 |
SSE | Algorithms | |||||
---|---|---|---|---|---|---|
WOA | GWO | SSA | HHO | GA | SDO | |
Minimum | 0.300165193 | 0.295994285 | 0.309587261 | 0.29652213 | 0.699327274 | 0.287824529 |
Mean | 0.577755836 | 0.377546219 | 0.399386254 | 0.936986404 | 2.553703279 | 0.291280092 |
Median | 0.463634598 | 0.371127461 | 0.40787451 | 0.544159818 | 2.515251534 | 0.290517966 |
Maximum | 2.037256664 | 0.458303674 | 0.456663355 | 2.774235022 | 6.340709893 | 0.300122801 |
STD | 0.305965318 | 0.04832738 | 0.040168552 | 0.736897544 | 1.213128222 | 0.003306558 |
Variable | Algorithms | |||||
---|---|---|---|---|---|---|
WOA | GWO | SSA | HHO | GA | SDO | |
−0.9715 | −0.890402371 | −0.935709319 | −1.188920972 | −0.971500 | −1.108875289 | |
0.00251 | 0.002491237 | 0.003094492 | 0.003832179 | 0.002510 | 0.00344834 | |
0.000036 | 4.91 × 10−5 | 7.92 × 10−5 | 7.78 × 10−5 | 0.000036 | 6.85 × 10−5 | |
−0.00015 | −0.000178916 | −0.000178501 | −0.000177945 | −0.000150 | −0.00018002 | |
0.00047 | 0.000185305 | 0.000125487 | 0.000100056 | 0.000470 | 0.000100079 | |
0.24611 | 0.128317025 | 0.116173722 | 0.132384143 | 0.246110 | 0.133014223 | |
15.32527 | 21.56036185 | 19.20514021 | 19.76775546 | 15.32527 | 23.99819531 | |
0.00807 | 0.005306549 | 0.008197421 | 0.005272008 | 0.008070 | 0.005101562 | |
1.40447 | 0.85 | 0.85 | 0.850476147 | 1.404470 | 0.850000467 | |
0.699327274 | 0.295994285 | 0.309587261 | 0.29652213 | 0.699327274 | 0.287824529 |
SSE | Algorithm | |||||
---|---|---|---|---|---|---|
WOA | GWO | SSA | HHO | GA | SDO | |
Minimum | 0.625844193 | 0.573184966 | 0.57403609 | 0.615881844 | 0.819730928 | 0.56426671 |
Mean | 1.061565396 | 0.636443826 | 0.736015304 | 1.896482889 | 1.628325244 | 0.567576781 |
Median | 1.011063049 | 0.619793268 | 0.697756101 | 0.788884976 | 1.395475534 | 0.565250928 |
Maximum | 4.841737291 | 0.887128302 | 0.952794404 | 10.23959957 | 4.949883945 | 0.58254602 |
STD | 0.559655929 | 0.065968875 | 0.108664932 | 2.287173428 | 0.810383099 | 0.004466494 |
Variable | Algorithm | |||||
---|---|---|---|---|---|---|
WOA | GWO | SSA | HHO | GA | SDO | |
−1.199166641 | −0.887481442 | −1.155052815 | −0.854760973 | −0.975720 | −0.902401475 | |
0.003060175 | 0.002252875 | 0.003202954 | 0.001875009 | 0.003110 | 0.002101847 | |
4.64 × 10−5 | 5.56 × 10−5 | 6.62 × 10−5 | 3.60 × 10−5 | 0.000098 | 4.16 × 10−5 | |
−0.000109916 | −0.000112573 | −0.00011186 | −0.000106249 | −0.000100 | −0.000112065 | |
0.000100072 | 0.000119116 | 0.000101536 | 0.000100031 | 0.000550 | 0.000100019 | |
0.160910761 | 0.195372815 | 0.194323477 | 0.177260104 | 0.220680 | 0.199832662 | |
10.01525068 | 21.55450995 | 18.31630737 | 10.003083 | 18.90174 | 23.995893 | |
0.006712446 | 0.00554807 | 0.0051 | 0.005101572 | 0.011480 | 0.005100055 | |
0.85106307 | 0.85 | 0.85 | 0.850262055 | 1.017500 | 0.850000594 | |
0.625844193 | 0.573184966 | 0.57403609 | 0.615881844 | 0.819730928 | 0.56426671 |
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Al-Shamma’a, A.A.; Ali, F.A.A.; Alhoshan, M.S.; Alturki, F.A.; Farh, H.M.H.; Alam, J.; AlSharabi, K. Proton Exchange Membrane Fuel Cell Parameter Extraction Using a Supply–Demand-Based Optimization Algorithm. Processes 2021, 9, 1416. https://doi.org/10.3390/pr9081416
Al-Shamma’a AA, Ali FAA, Alhoshan MS, Alturki FA, Farh HMH, Alam J, AlSharabi K. Proton Exchange Membrane Fuel Cell Parameter Extraction Using a Supply–Demand-Based Optimization Algorithm. Processes. 2021; 9(8):1416. https://doi.org/10.3390/pr9081416
Chicago/Turabian StyleAl-Shamma’a, Abdullrahman A., Fekri Abdulraqeb Ahmed Ali, Mansour S. Alhoshan, Fahd A. Alturki, Hassan M. H. Farh, Javed Alam, and Khalil AlSharabi. 2021. "Proton Exchange Membrane Fuel Cell Parameter Extraction Using a Supply–Demand-Based Optimization Algorithm" Processes 9, no. 8: 1416. https://doi.org/10.3390/pr9081416