Electro-Hydraulic Transient Regimes in Isolated Pumps Working as Turbines with Self-Excited Induction Generators
Abstract
:1. Introduction
2. Self-Excited Induction Generator Model
2.1. Magnetizing Inductance
2.2. Capacitor and Electrical Load Models
3. Pump as a Turbine (PAT) Model
4. PAT-SEIG Model Validation
- I1. Nash–Sutcliffe index (NSI). This index is a fit indicator, which is used in temporal series. NSI value oscillates between − and 1. When values are below 0, the fit is considered poor. When the values are above 0, the model is considered good. Table 1 shows the used ranges to define the fit according to NSI values. NSI is defined by (25) where is the experimental value in each interval, is the average of the observed values and is the simulated value in each interval.
- I2. Root Relative squared error (RRSE). It measures the error of the model by normalizing the variable. Perfect fits are defined when the RRSE value is zero. The efficiency of the simulation is better when the RRSE value is low. This index is defined by (26).
- I3. Mean relative deviation (MRD). The index defines the significance of the error concerning variable value (27). The fit is good when MRD has values close to 0.
- I4. Bias (BIAS). This index compares the tendency of the simulated values, determining if the simulated values are lower or higher than experimental data (28). The model overestimates if BIAS is negative. When BIAS is positive, the variable is underestimated by the model. The optimal value is zero when BIAS is analyzed.
4.1. Self-Excited Induction Generator: d–q Model Validation
4.2. PAT Model Validation
5. Impact of Electric and Hydraulic Perturbations in the PAT-SEIG Stability
5.1. Variation of Excitation Capacitance
5.2. Variation of Resistive Load
5.3. PAT Head Variation
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Goodness Fit | NSI | RRSE | BIAS |
---|---|---|---|
Very Good | NSI > 0.6 | 0.00 ≤ RRSE ≤ 0.50 | |
Good | 0.40 < NSI ≤ 0.60 | 0.50 < RRSE ≤ 0.60 | |
Satisfactory | 0.20 < NSI ≤ 0.40 | 0.60 < RRSE ≤ 0.70 | |
Unsatisfactory | NSI < 0.20 | RRSE > 0.70 |
Frequency | 50 Hz |
---|---|
Voltage | 400 V |
Current | 1.6 A |
Output Power | 0.55 kW |
Power factor | 0.73 |
Speed | 910 rpm |
Experimental | Model | MRD | ||
---|---|---|---|---|
C = 50 μF | N (rpm) | 750 | 758 | +0.010 (1.0%) |
(Hz) | 35.2 | 35.0 | −0.006 (0.6%) | |
(Vrms) | 144 | 145 | +0.007 (0.7%) | |
C = 80 μF | N (rpm) | 597 | 603 | +0.010 (1.0%) |
(Hz) | 27.6 | 27.2 | −0.015 (1.5%) | |
(Vrms) | 113 | 108 | −0.044 (4.4%) |
Experimental | Model | MRD | |||||
---|---|---|---|---|---|---|---|
Initial | Final | Initial | Final | Initial | Final | ||
= 600 Ω | N (rpm) | 839 | 834 | 842 | 835 | +0.004 (+0.4%) | +0.001 (+0.1%) |
(Hz) | 41.0 | 40.0 | 40.9 | 39.3 | −0.000 (−0.0%) | −0.018 (−1.8%) | |
(Vrms) | 183 | 141 | 191 | 141.9 | −0.044 (−4.4%) | −0.006 (−0.6%) | |
(Arms) | 1.6 | 1.05 | 1.53 | 1.12 | −0.023 (−2.3%) | +0.023 (+2.3%) | |
(V) | 4.46 | 3.53 | 4.67 | 3.61 | −0.005 (−0.5%) | +0.003 (+0.3%) | |
= 300 Ω | N (rpm) | 848 | 843 | 849 | 851 | +0.001 (0.1%) | +0.010 (+1.0%) |
(Hz) | 41.2 | 40.3 | 41.5 | 40.0 | +0.007 (+0.7%) | −0.007 (−0.7%) | |
(Vrms) | 181 | 90 | 184.6 | 84.8 | +0.002 (+0.2%) | +0.058 (+5.8%) | |
(Arms) | 1.6 | 0.8 | 1.62 | 0.87 | +0.013 (+1.3%) | +0.088 (+8.8%) | |
(V) | 4.4 | 2.2 | 4.49 | 2.22 | +0.020 (+2.0%) | +0.009 (+0.9%) |
Indexes | Stator Voltage (Figure 15a) | Frequency (Figure 15b) | Stator Voltage (Figure 15c) | Frequency (Figure 15d) |
---|---|---|---|---|
NSI | 0.960 (VG) | 0.797 (VG) | 0.802 (VG) | 0.841 (VG) |
RRSE | 0.200 (VG) | 0.451 (VG) | 0.441 (VG) | 0.399 (VG) |
BIAS | 0.039 (VG) | −0.071 (VG) | 0.095 (VG) | −0.075 (VG) |
MRD | −0.0244 (−2.44%) | 0.062 (6.2%) | 0.010 (1.0%) | 0.0664 (6.64%) |
Indexes | Stator Voltage (Figure 16a) | Frequency (Figure 16b) | Stator Voltage (Figure 16c) | Frequency (Figure 16d) |
---|---|---|---|---|
NSI | 0.569 (G) | 0.787 (VG) | 0.761 (VG) | 0.793 (VG) |
RRSE | 0.648 (S) | 0.462 (VG) | 0.429 (VG) | 0.455 (VG) |
BIAS | 0.063 (VG) | −0.041 (VG) | −0.034 (VG) | −0.037 (VG) |
MRD | 0.0360 | 0.0269 | −0.054 | 0.281 |
C. | N (rpm) | ||||||
---|---|---|---|---|---|---|---|
−50% | 1005 | 1365 | 139.7 | 1.16 | 331 | −392 | 33.0% |
−40% | 1223 | 1252 | 152.6 | 1.34 | 392 | −515 | 32.0% |
−30% | 1333 | 1173 | 157.4 | 1.47 | 416 | −592 | 31.1% |
−20% | 1401 | 1114 | 158.8 | 1.58 | 427 | −658 | 30.5% |
−10% | 1460 | 1057 | 155.5 | 1.60 | 403 | −661 | 27.6% |
0% | 1501 | 1010 | 150.2 | 1.65 | 384 | −664 | 25.6% |
+10% | 1512 | 997 | 157.0 | 1.83 | 417 | −783 | 27.6% |
+20% | 1535 | 969 | 155.2 | 1.90 | 412 | −810 | 26.9% |
+30% | 1553 | 944 | 154.4 | 1.95 | 408 | −833 | 26.3% |
+40% | 1568 | 922 | 152.1 | 2.01 | 403 | −851 | 25.7% |
+50% | 1581 | 903 | 149.4 | 2.07 | 387 | −866 | 24.4% |
N (rpm) | |||||||
---|---|---|---|---|---|---|---|
−30% | 1311 | 1190 | 119.1 | 1.54 | 342 | −466 | 26.1% |
−20% | 1412 | 1105 | 135.6 | 1.61 | 387 | −565 | 27.5% |
−15% | 1439 | 1078 | 139.6 | 1.60 | 375 | −588 | 26.0% |
−10% | 1460 | 1056 | 141.5 | 1.60 | 366 | −601 | 25.0% |
0% | 1501 | 1010 | 150.2 | 1.65 | 384 | −664 | 25.6% |
+10% | 1499 | 1013 | 153.5 | 1.66 | 365 | −698 | 24.3% |
+15% | 1503 | 1007 | 157.9 | 1.66 | 360 | −722 | 24.0% |
+20% | 1507 | 1003 | 159.2 | 1.69 | 353 | −749 | 23.4% |
+30% | 1507 | 1003 | 168.1 | 1.75 | 359 | −827 | 23.9% |
N (rpm) | |||||||
---|---|---|---|---|---|---|---|
−50% | 457 | 887 | 75.0 | 0.74 | 96 | −145 | 21.1% |
−40% | 642 | 912 | 100.2 | 1.01 | 163 | −267 | 25.3% |
−30% | 842 | 942 | 118.8 | 1.23 | 221 | −386 | 26.2% |
−20% | 1057 | 970 | 132.9 | 1.41 | 277 | −499 | 26.2% |
−10% | 1297 | 995 | 144.8 | 1.55 | 334 | −601 | 25.3% |
0% | 1501 | 1010 | 150.2 | 1.65 | 384 | −664 | 25.6% |
+10% | 1782 | 1059 | 170.2 | 1.92 | 453 | −885 | 25.5% |
+20% | 2047 | 1086 | 180.6 | 2.08 | 521 | −1017 | 25.5% |
+30% | 2320 | 1091 | 181.3 | 2.11 | 569 | −1040 | 25.3% |
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Madeira, F.C.; Fernandes, J.F.P.; Pérez-Sánchez, M.; López-Jiménez, P.A.; Ramos, H.M.; Costa Branco, P.J. Electro-Hydraulic Transient Regimes in Isolated Pumps Working as Turbines with Self-Excited Induction Generators. Energies 2020, 13, 4521. https://doi.org/10.3390/en13174521
Madeira FC, Fernandes JFP, Pérez-Sánchez M, López-Jiménez PA, Ramos HM, Costa Branco PJ. Electro-Hydraulic Transient Regimes in Isolated Pumps Working as Turbines with Self-Excited Induction Generators. Energies. 2020; 13(17):4521. https://doi.org/10.3390/en13174521
Chicago/Turabian StyleMadeira, Filipe C., João F. P. Fernandes, Modesto Pérez-Sánchez, P. Amparo López-Jiménez, Helena M. Ramos, and P. J. Costa Branco. 2020. "Electro-Hydraulic Transient Regimes in Isolated Pumps Working as Turbines with Self-Excited Induction Generators" Energies 13, no. 17: 4521. https://doi.org/10.3390/en13174521