Application of Multiple Linear Regression and Artificial Neural Networks in Analyses and Predictions of the Thermoelectric Performance of Solid Oxide Fuel Cell Systems
Abstract
:1. Introduction
2. Materials and Methods
2.1. The SOFC System
2.2. Dataset
2.3. Data Processing
- (1)
- General
- (2)
- Preliminary data processing
- (3)
- PCA method
2.4. Calculation of the System’s Performance
2.5. Predictive Models of Performance
- (1)
- MLR
- (2)
- ANN
2.6. Evaluation of Predictive Accuracy
3. Results
3.1. Statistical Results of Thermoelectric Performance Variables
3.2. Analysis of Related Variables Affecting the System’s Performance
- (1)
- According to the operating data of the SOFC system for five experiments, shown in Figure 4, we extracted the parameters of the fuel/air flow rate and fuel/air pressure of multiple components during each operation, as well as the load power, the stack’s temperature, the combustion chamber’s temperature, and the change in the cooling water’s temperature.
- (2)
- We drew a chart of the fluctuation trend of each parameter according to the time series (the trend chart has been omitted in this article) and preprocessed the data. The standard deviation method was used to eliminate outliers in during preprocessing, and the nearest neighbor interpolation method was used for missing values.
- (3)
- We carried out a correlation analysis of the data, sorted out the variables with large correlations and the variables with small correlations, and prepared for reducing the data’s dimensionality.
- (4)
- Data normalization was realized by minimum–maximum normalization to obtain data that could be used for reducing the data’s dimensionality.
- (5)
- Principal component analysis (PCA) was used to reduce the dimensions; the specific method is shown in Equations (5)–(7).
3.3. Quantitative Analysis of the Variables of Interest
3.4. Results of the Predictive Model of the SOFC System’s Performance
3.4.1. MLR
3.4.2. ANN
3.4.3. Comparison of the Predictive Accuracy
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | Value or Type |
---|---|
Number of cells | 27 |
Coverage (m3) | 20 |
CH4 flow rate (NL/min) | 0–10.0 |
water-carbon ratio | 1–2 |
Operating temperature of the stack (°C) | 650–850 |
Air flow (NL/min) | 0–400 |
Stack’s nominal power (kW) | 1 |
Catalyst | Pt |
Catalytic processor | Allothermic |
Electrolyte | Precious metals |
Type of cell | Flat-paneled |
Fuel | CH4 |
variable voltage range (V) | 0–27 |
Name | Interpretation | Name | Interpretation |
---|---|---|---|
I | The stack’s current | Premix air pressure | Value of air pressure in the straight-through exhaust gas combustion chamber |
Taferburner | The temperature of the exhaust gas combustion chamber | TC17 | Temperature of the exhaust gas combustion chamber’s front end |
Burner CH4 PV | The methane flow rate of the reforming combustion chamber | TC19 | Middle temperature of the exhaust gas combustion chamber |
Stack air PV | Air velocity of the reactor cathode | AnIn Pressure | Reactor cathode’s inlet pressure |
Feed water pump SV | Set flow rate of the water | TC9 | Exhaust gas temperature at the inlet of the exhaust gas heat exchanger |
Premix air PV | Air flow rate directly through the exhaust gas combustion chamber | TC4 | Temperature of the reformer outlet’s fuel |
Burner air PV | Air flow rate of the reforming combustion chamber | Heater PV | The electric heater’s pressure value and feedback value |
Feed CH4 PV | Feedback value for the flow rate of methane supply | TC1 | Temperature of the reformer’s inlet |
TC22 | Temperature of the reformer’s exhaust gas | TC24 | Temperature of the stack’s anode inlet |
TC3 | Temperature of the reformer’s combustion chamber | TC14 | Temperature of the cold gas premixing chamber of the exhaust gas combustion chamber |
TC2 | Temperature of the reformer’s reforming chamber | TC6 | Temperature of the exhaust gas outlet of the tail gas heat exchanger |
TC10 | Temperature of the combustion heat exchanger’s inlet | TC0 | Reformer’s ignition temperature |
Stack air pressure | The stack’s air pressure | TC5 | Temperature of the exhaust gas combustion chamber’s outlet |
TC21 | Temperature of the exhaust gas combustion chamber’s end | TC23 | Final temperature of exhaust gases |
Forming gas pressure | Pressure of hydrogen and nitrogen (95% nitrogen, 5% hydrogen) | TC13 | Temperature of the combustion heat exchanger’s air outlet |
CH4 pressure | Pressure of methane | TC11 | Temperature of the fuel outlet of the combustion heat exchanger |
TC15 | Exit temperature of the cold gas premixing chamber of the exhaust gas combustion chamber | Tstack | The stack’s temperature |
TC20 | Temperature in the middle and back of the exhaust gas combustion chamber | U | The stack’s voltage |
CaOut pressure | Pressure of the stack’s cathode outlet | P | Power |
TC12 | Temperature of the combustion heat exchanger’s air inlet |
Name | Units and Measuring Instruments | Typology |
---|---|---|
I | A, electronic load | Input variable |
TAFERBURNER | °C, PT 100 thermometer | Input variable |
BURNER CH4 PV | NL/min, flow meter | Input variable |
STACK AIR PV | NL/min, flow meter | Input variable |
FEED WATER PUMP SV | NL/min, flow meter | Input variable |
PREMIX AIR PV | NL/min, flow meter | Input variable |
BURNER AIR PV | NL/min, flow meter | Input variable |
FEED CH4 PV | NL/min, flow meter | Input variable |
TC22 | °C, PT 100 thermometer | Input variable |
TC3 | °C, PT 100 thermometer | Input variable |
TC2 | °C, PT 100 thermometer | Input variable |
TC10 | °C, PT 100 thermometer | Input variable |
TSTACK | °C, PT 100 thermometer | Output variable |
U | V, electronic load | Output variable |
P | J, P = UI | Output variable |
Time | Median Power Generation (W) | Quantity of Heating Water (L) | Peak Water Temperature (°C) | Running Time (h) |
---|---|---|---|---|
1 | 720 | 3000 | 65 | 298 |
2 | 755 | 2900 | 70 | 382 |
3 | 910 | 3800 | 68 | 503 |
4 | 810 | 3100 | 65 | 438 |
5 | 730 | 2900 | 65 | 357 |
PCC | I | Tafterburner | Burner CH4 PV | Stack Air PV | Feed Water Pump SV | Premix Air PV | Burner Air PV | Feed CH4 PV | TC22 | TC3 | TC2 | TC10 | TC25 | TC18 | Premix Air Pressure | TC17 | TC19 | AnIn Pressure | TC9 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Tstack | 0.974 | 0.969 | 0.901 | 0.883 | 0.842 | 0.81 | 0.808 | 0.806 | 0.78 | 0.768 | 0.765 | 0.749 | 0.708 | 0.651 | 0.636 | 0.631 | 0.588 | 0.537 | 0.536 |
U | 0.919 | 0.841 | 0.87 | 0.843 | 0.892 | 0.805 | 0.845 | 0.909 | 0.587 | 0.598 | 0.581 | 0.519 | 0.491 | 0.5 | 0.635 | 0.691 | 0.579 | 0.427 | 0.564 |
P | 0.999 | 0.972 | 0.903 | 0.921 | 0.908 | 0.824 | 0.825 | 0.875 | 0.701 | 0.683 | 0.677 | 0.658 | 0.703 | 0.614 | 0.58 | 0.696 | 0.585 | 0.653 | 0.571 |
PCC | TC4 | Heater PV | TC1 | TC24 | TC14 | TC6 | TC0 | Stack Air Pressure | TC21 | Forming Gas Pressure | CH4 Pressure | TC15 | TC20 | CaOut Pressure | TC12 | TC5 | TC23 | TC13 | TC11 |
Tstack | 0.533 | 0.528 | 0.514 | 0.496 | 0.495 | 0.414 | 0.389 | 0.377 | 0.364 | 0.301 | 0.225 | 0.224 | 0.221 | 0.182 | 0.169 | 0.147 | 0.141 | 0.118 | 0.116 |
U | 0.303 | 0.555 | 0.292 | 0.531 | 0.555 | 0.268 | 0.334 | 0.314 | 0.461 | 0.156 | 0.349 | 0.388 | 0.279 | 0.272 | 0.342 | 0.296 | 0.091 | 0.188 | 0.03 |
P | 0.431 | 0.562 | 0.401 | 0.367 | 0.578 | 0.419 | 0.377 | 0.37 | 0.416 | 0.223 | 0.243 | 0.193 | 0.237 | 0.295 | 0.264 | 0.261 | 0.062 | 0.072 | 0.147 |
Times | Value | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
---|---|---|---|---|---|---|---|---|---|---|
Time 1 | Stack temperature | 259.3013 | −25.2129 | −9.7954 | 0.0393 | −1.5448 | 2.0382 | −3.6983 | 2.5273 | 1.0347 |
Voltage | 17.5669 | 0.1855 | 1.9084 | −0.0103 | −0.1083 | 0.0656 | −0.1474 | −0.2017 | 0.0116 | |
Power | 136.0201 | −25.7975 | −105.1033 | 1.2978 | −3.3728 | 1.4678 | −1.3815 | 17.0507 | 0.5091 | |
Time 2 | Stack temperature | 525.5366 | −29.2521 | 14.3873 | 0.2598 | −0.1497 | 0.7433 | 0.9772 | 1.7931 | 0.0621 |
Voltage | 22.6174 | −1.596 | 1.3378 | 0.0164 | −0.0582 | −0.0146 | 0.2514 | −0.0276 | −0.0008 | |
Power | 197.8473 | −3.0413 | 30.2764 | 0.03 | −3.1234 | −0.3156 | 1.0768 | 13.6352 | −0.057 | |
Time 3 | Stack temperature | 1798.0234 | 13.1742 | −78.1279 | −1.3623 | 0.0566 | −1.1655 | −18.1553 | 0.9617 | −0.2705 |
Voltage | 33.7386 | −0.0398 | −0.2589 | −0.0128 | 0.001 | −0.0306 | −0.511 | −0.0357 | −0.0021 | |
Power | 978.5543 | 1.5459 | −15.9767 | −0.8382 | 0.1815 | −1.0462 | −33.1387 | 12.5398 | −0.0507 | |
Time 4 | Stack temperature | 277.1368 | −5.7635 | 14.5393 | −1.3032 | 0.0498 | 3.3466 | −1.205 | 1.473 | 0.624 |
Voltage | 32.7131 | 1.2716 | −2.0299 | −0.0676 | −0.0384 | 0.0829 | 0.4116 | 0.095 | −0.022 | |
Power | 596.7318 | 47.0304 | −66.1091 | −2.7356 | −1.5946 | 3.7635 | 14.7878 | 19.7956 | −0.8187 | |
Time 5 | Stack temperature | 269.2489 | −21.5247 | 20.8656 | −0.0221 | 0.3768 | 1.0195 | 5.187 | 0.7194 | 0.4849 |
Voltage | 2.5948 | −0.3633 | −0.3121 | −0.0158 | −0.044 | 0.009 | 0.0676 | −0.07 | 0.0407 | |
Power | −411.928 | 10.4022 | −12.8524 | −0.1902 | −1.153 | 0.4861 | −10.7301 | 13.44 | 1.1031 |
Model: MLR | Statistics | Stack Temperature | Voltage | Power |
---|---|---|---|---|
1 | MAE | 2.28 | 0.19 | 2.93 |
NME | 0.33 | 0.95 | 0.63 | |
NRMSE | 0.44 | 1.45 | 0.86 | |
2 | MAE | 2.03 | 0.18 | 5.64 |
NME | 0.33 | 1.22 | 1.18 | |
NRMSE | 0.42 | 1.59 | 1.49 | |
3 | MAE | 4.3 | 0.13 | 6.46 |
NME | 0.65 | 0.91 | 0.98 | |
NRMSE | 0.98 | 1.28 | 1.44 | |
4 | MAE | 2.04 | 0.56 | 20.99 |
NME | 0.31 | 3.51 | 3.64 | |
NRMSE | 0.52 | 5.20 | 5.46 | |
5 | MAE | 2.62 | 0.37 | 11.21 |
NME | 0.41 | 2.45 | 2.61 | |
NRMSE | 0.60 | 3.23 | 3.38 |
Model: ANN | Statistics | Stack Temperature | Voltage | Power |
---|---|---|---|---|
1 | MAE | 0.70 | 0.07 | 1.45 |
NME | 0.10 | 0.34 | 0.31 | |
NRMSE | 0.14 | 0.48 | 0.44 | |
2 | MAE | 0.98 | 0.07 | 2.82 |
NME | 0.16 | 0.51 | 0.59 | |
NRMSE | 0.22 | 0.73 | 0.88 | |
3 | MAE | 0.91 | 0.03 | 2.10 |
NME | 0.14 | 0.22 | 0.32 | |
NRMSE | 0.24 | 0.34 | 0.52 | |
4 | MAE | 0.94 | 0.16 | 5.98 |
NME | 0.14 | 0.97 | 1.04 | |
NRMSE | 0.22 | 1.50 | 1.67 | |
5 | MAE | 1.14 | 0.10 | 3.06 |
NME | 0.18 | 0.69 | 0.71 | |
NRMSE | 0.27 | 1.14 | 1.18 |
CH4 Flow Rates (NL/min) | Combustion Chamber’s Temperature (°C) | Stack’s Temperature (°C) | P (J) |
---|---|---|---|
6.1 | 688 | 550 | 18.974 |
6.3 | 683 | 549 | 68.284 |
6.6 | 691 | 555 | 100.643 |
6.9 | 685 | 567 | 134.159 |
7 | 684 | 564 | 133.735 |
7.3 | 765 | 679 | 806.654 |
7.4 | 690 | 564 | 159.621 |
7.7 | 751 | 702 | 788.277 |
7.8 | 767 | 639 | 765.666 |
8 | 756 | 580 | 265.082 |
8.2 | 801 | 640 | 654.654 |
8.9 | 800 | 604 | 478.241 |
9 | 798 | 607 | 493.068 |
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Lai, M.; Zhang, D.; Li, Y.; Wu, X.; Li, X. Application of Multiple Linear Regression and Artificial Neural Networks in Analyses and Predictions of the Thermoelectric Performance of Solid Oxide Fuel Cell Systems. Energies 2024, 17, 4084. https://doi.org/10.3390/en17164084
Lai M, Zhang D, Li Y, Wu X, Li X. Application of Multiple Linear Regression and Artificial Neural Networks in Analyses and Predictions of the Thermoelectric Performance of Solid Oxide Fuel Cell Systems. Energies. 2024; 17(16):4084. https://doi.org/10.3390/en17164084
Chicago/Turabian StyleLai, Meilin, Daihui Zhang, Yu Li, Xiaolong Wu, and Xi Li. 2024. "Application of Multiple Linear Regression and Artificial Neural Networks in Analyses and Predictions of the Thermoelectric Performance of Solid Oxide Fuel Cell Systems" Energies 17, no. 16: 4084. https://doi.org/10.3390/en17164084