# Electro-Hydraulic Transient Regimes in Isolated Pumps Working as Turbines with Self-Excited Induction Generators

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## Abstract

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## 1. Introduction

## 2. Self-Excited Induction Generator Model

#### 2.1. Magnetizing Inductance

_{L}= 120, 200 and 600 Ω), the deviation between experimental and numerical induction generator model decreased by about 37%, to an average deviation of 10.7%. The evolution of ${L}_{M}$ with the magnetization flux was obtained experimentally in [5] and was approximated by the polynomial Equation in (10). This equation shows the experimentally calculated magnetizing inductance as a function of ${\varphi}_{m}$ for the range of frequencies 20–60 Hz.

#### 2.2. Capacitor and Electrical Load Models

## 3. Pump as a Turbine (PAT) Model

_{rh}) imposed by the differential pressure tanks to the PAT terminals (PS1-PS2 in Figure 1), the PAT head can be computed using (20), where $\rho $ is the water density (1000 kg/m

^{3}) and $g$ is the standard acceleration due to gravity (9.81 m/s).

## 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 −$\infty $ 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 ${E}_{i}$ is the experimental value in each interval, ${\overline{E}}_{i}$ is the average of the observed values and ${S}_{i}$ is the simulated value in each interval.$$NSI=1-\frac{{\sum}_{i=1}^{N}{\left[{E}_{i}-{S}_{i}\right]}^{2}}{{\sum}_{i=1}^{N}{\left[{E}_{i}-{\overline{E}}_{i}\right]}^{2}}$$
- 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).$$RRSE=\sqrt{\frac{{\sum}_{i=1}^{N}{\left[{E}_{i}-{S}_{i}\right]}^{2}}{{\sum}_{i=1}^{N}{\left[{E}_{i}-{\overline{E}}_{i}\right]}^{2}}}$$
- 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.$$MRD={\displaystyle \sum}_{1}^{x}\frac{\left|{O}_{i}-{P}_{i}\right|/{P}_{i}}{x}$$
- 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.$$BIAS=\frac{{\sum}_{i=1}^{N}{\left[{O}_{i}-{P}_{i}\right]}^{}}{{\sum}_{i=1}^{N}{\left[{O}_{i}\right]}^{}}$$

#### 4.1. Self-Excited Induction Generator: d–q Model Validation

_{L}of 600 Ω and 300 Ω, respectively. These resistive loads mimic an electric demand to the SEIG from a consumer. Since in the experimental setup, a DC motor provides the SEIG mechanical torque, its speed decreases with the increase of electric load. Notice that this is similar behavior to the PAT. When decreasing the resistive load, the electric current increases and, if the stator voltage remains (or has a very small change), the requested active power ${P}_{a}=3\frac{{U}_{s}{}^{2}}{{R}_{L}}$ would increase. However, Figure 11 shows that this is not true. When the requested active power increases, the speed of the DC motor decreases and thus also the SEIG electrical frequency and stator voltage. Decreasing the stator voltage will also decrease the stator current, Figure 11a,b, decreasing the active power, Figure 11c, and increasing the machine’s speed. Also, a reduction of electrical frequency will influence the reactive power, as shown in Figure 11d, provided by the capacitor to the SEIG, thus changing the SEIG operation point, per Figure 11e. In conclusion, this process will be iterative until a new steady state is obtained. The SEIG starts without electric load, with a capacitor bank of C = 35 μF, and the current at its rated value, I = 1.6 Arms. After the inclusion of the resistive load, ${R}_{L}$ = 600 Ω, the rms stator voltage and current drop, from 183 Vrms to 141 Vrms and from 1.6 Arms to 1.05 Arms, Figure 11a,b, respectively. This is due to the reduction of speed, N, and electrical frequency, ${f}_{s}$, resulting into new active and reactive powers, shown in Figure 11c,d, and a new magnetizing flux inside the machine, $E/f$, shown in Figure 11e. As shown in Table 4, MRDs between the simulation and experimental results are less than 0.088 (8.8%). The highest deviation of 0.088 occurs for low magnetization points of the SEIG, $E/f$ = 2.2. This is related to the magnetizing inductance, ${L}_{M}$, obtained experimentally. For lower values of magnetic flux, the stator leakage cannot be neglected and, thus, the associated error to the determination of ${L}_{M}$ increases.

#### 4.2. PAT Model Validation

## 5. Impact of Electric and Hydraulic Perturbations in the PAT-SEIG Stability

_{h}vs. Q) for the original PAT (PAT1 in blue) and the new one (PAT2 in red). PAT2 is now able to produce enough hydraulic power to set the SEIG rated conditions (rated power of 550 W). The specific speed of PAT2 was 20 rpm (m,kW), and the best efficiency point (BEP) was defined in the operation point 3.48 L/s and 12.38 m w.c.

#### 5.1. Variation of Excitation Capacitance

_{i}, of 35 μF and an electric load of ${R}_{L}$ = 200 Ω. Next, the capacitors’ value was changed, and the new steady-state electromechanical and hydraulic conditions were recorded. Table 7 shows the final steady-state hydraulic and electric parameters after a sudden change of the capacitor’s value.

#### 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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**Figure 9.**Experimental setup to test the SEIG, isolated from the PAT: (

**a**) DC motor coupled mechanically to the SEIG, (

**b**) variable capacitor and resistor banks both connected in parallel to the stator coils of the induction generator, (

**c**) scheme of two experimental setup components.

**Figure 10.**SEIG stator voltage during the excitation process with: (

**a**) C = 50 μF and (

**b**) C = 80 μF. In blue, the experimental results. In red, the simulation ones.

**Figure 11.**SEIG stator voltage when connecting a resistive load of ${R}_{L}$ = 600 Ω with the SEIG excited with a capacitor bank having C = 35 μF: (

**a**) stator voltage, (

**b**) stator current, (

**c**) active power, (

**d**) reactive power and (

**e**) magnetizing flux.

**Figure 12.**SEIG stator voltage when connecting a resistive load of ${R}_{L}$ = 300 Ω with the SEIG excited with a capacitor bank having C = 35 μF.

**Figure 13.**Experimental hydraulic circuit: (

**a**) circuit schematic and (

**b**) PAT used (Etanorm 32-125 KSB 4.8).

**Figure 14.**Experimental hydraulic circuit developed for validation of the PAT model: in (

**a**) the PAT-SEIG, control flow tanks and recirculating pump and, in (

**b**) the pressure tank and the resistive load and capacitors.

**Figure 15.**PAT-SEIG stator voltage and frequency during excitation with: C = 17.4 μF, ${R}_{L}$ = $\infty $ Ω and $H$ = 5.82 m w.c: (

**a**) stator voltage and (

**b**) electric frequency; and with C = 34.7 μF, ${R}_{L}$ = 300 Ω and $H$ = 7.5 m w.c.: (

**c**) stator voltage and (

**d**) electric frequency.

**Figure 16.**PAT-SEIG stator voltage and electric frequency after a sudden change: (

**a**,

**b**) change of electric load from ${R}_{L}$ = 600 Ω to ${R}_{L}$ = 300 Ω, with C = 17.4 μF and $H$ = 7.5 m w.c. and (

**c**,

**d**) change of capacitor from C = 17.4 μF to C = 34.7 μF with ${R}_{L}$ = 300 Ω and $H$ = 7.5 m.w.c.

**Figure 17.**Hydraulic characteristic curves of the original PAT (PAT1, blue color) and the new one (PAT2, red color), for N = N

_{ref}: (

**a**) head curve and (

**b**) hydraulic power curve.

**Figure 18.**Variation of hydraulic and electromechanical quantities after a change in the capacitor, C: (

**a**) variation of hydraulic (black), active (red) and reactive (blue) powers, and (

**b**) variation of speed (black) and system (red) efficiency.

**Figure 19.**Variation of hydraulic and electromechanical parameters after a change in the resistive load, ${R}_{L}$: (

**a**) variation of hydraulic (black), active (red) and reactive (blue) powers, and (

**b**) variation of speed (black) and system (red) efficiency.

**Figure 20.**Variation of hydraulic and electromechanical parameters after a change in the PAT head, $\u2206H$: (

**a**) variation of hydraulic (black), active (red) and reactive (blue) powers, and (

**b**) variation of speed (black) and system (red) efficiency.

Goodness Fit | NSI | RRSE | BIAS |
---|---|---|---|

Very Good | NSI > 0.6 | 0.00 ≤ RRSE ≤ 0.50 | $\mathrm{BIAS}<\pm 0.10$ |

Good | 0.40 < NSI ≤ 0.60 | 0.50 < RRSE ≤ 0.60 | $\pm 0.10\le \mathrm{BIAS}<\pm 0.15$ |

Satisfactory | 0.20 < NSI ≤ 0.40 | 0.60 < RRSE ≤ 0.70 | $\pm 0.15\le \mathrm{BIAS}<\pm 0.25$ |

Unsatisfactory | NSI < 0.20 | RRSE > 0.70 | $\mathrm{BIAS}>\pm 0.25$ |

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%) |

${f}_{s}$ (Hz) | 35.2 | 35.0 | −0.006 (0.6%) | |

${U}_{s}$ (Vrms) | 144 | 145 | +0.007 (0.7%) | |

C = 80 μF | N (rpm) | 597 | 603 | +0.010 (1.0%) |

${f}_{s}$ (Hz) | 27.6 | 27.2 | −0.015 (1.5%) | |

${U}_{s}$ (Vrms) | 113 | 108 | −0.044 (4.4%) |

**Table 4.**Comparison between steady-state experimental and model’s results after the application of resistive loads.

Experimental | Model | MRD | |||||
---|---|---|---|---|---|---|---|

Initial | Final | Initial | Final | Initial | Final | ||

${R}_{L}$ = 600 Ω | N (rpm) | 839 | 834 | 842 | 835 | +0.004 (+0.4%) | +0.001 (+0.1%) |

${f}_{s}$ (Hz) | 41.0 | 40.0 | 40.9 | 39.3 | −0.000 (−0.0%) | −0.018 (−1.8%) | |

${U}_{s}$ (Vrms) | 183 | 141 | 191 | 141.9 | −0.044 (−4.4%) | −0.006 (−0.6%) | |

${I}_{s}$ (Arms) | 1.6 | 1.05 | 1.53 | 1.12 | −0.023 (−2.3%) | +0.023 (+2.3%) | |

$E/f$ (V${\mathrm{Hz}}^{-1}$) | 4.46 | 3.53 | 4.67 | 3.61 | −0.005 (−0.5%) | +0.003 (+0.3%) | |

${R}_{L}$ = 300 Ω | N (rpm) | 848 | 843 | 849 | 851 | +0.001 (0.1%) | +0.010 (+1.0%) |

${f}_{s}$ (Hz) | 41.2 | 40.3 | 41.5 | 40.0 | +0.007 (+0.7%) | −0.007 (−0.7%) | |

${U}_{s}$ (Vrms) | 181 | 90 | 184.6 | 84.8 | +0.002 (+0.2%) | +0.058 (+5.8%) | |

${I}_{s}$ (Arms) | 1.6 | 0.8 | 1.62 | 0.87 | +0.013 (+1.3%) | +0.088 (+8.8%) | |

$E/f$ (V${\mathrm{Hz}}^{-1}$) | 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 |

**Table 7.**Steady-state hydraulic and electromechanical quantities after a sudden change in the capacitors’ value.

$\mathsf{\u2206}$C. | ${\mathit{P}}_{\mathit{h}}\left(\mathbf{W}\right)$ | N (rpm) | ${\mathit{U}}_{\mathit{s}}{}_{\mathit{r}\mathit{m}\mathit{s}}\left(\mathbf{V}\right)$ | ${\mathit{I}}_{\mathit{s}}{}_{\mathit{r}\mathit{m}\mathit{s}}\left(\mathbf{A}\right)$ | ${\mathit{P}}_{\mathit{a}}\left(\mathbf{W}\right)$ | ${\mathit{Q}}_{\mathit{r}}\left(\mathbf{Var}\right)$ | ${\mathit{\eta}}_{\mathit{s}\mathit{y}\mathit{s}}$ |
---|---|---|---|---|---|---|---|

−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% |

**Table 8.**Steady-state hydraulic and electromechanical parameters after a sudden change in the resistance value.

$\mathsf{\u2206}{\mathit{R}}_{\mathit{L}}$ | ${\mathit{P}}_{\mathit{h}}\left(\mathbf{W}\right)$ | N (rpm) | ${\mathit{U}}_{\mathit{s}}{}_{\mathit{r}\mathit{m}\mathit{s}}\left(\mathbf{V}\right)$ | ${\mathit{I}}_{\mathit{s}}{}_{\mathit{r}\mathit{m}\mathit{s}}\left(\mathbf{A}\right)$ | ${\mathit{P}}_{\mathit{a}}\left(\mathbf{W}\right)$ | ${\mathit{Q}}_{\mathit{r}}\left(\mathbf{Var}\right)$ | ${\mathit{\eta}}_{\mathit{s}\mathit{y}\mathit{s}}$ |
---|---|---|---|---|---|---|---|

−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% |

**Table 9.**Steady-state hydraulic and electromechanical quantities after a sudden change in the PAT head

$\mathsf{\u2206}\mathit{H}$ | ${\mathit{P}}_{\mathit{h}}\left(\mathbf{W}\right)$ | N (rpm) | ${\mathit{U}}_{\mathit{s}}{}_{\mathit{r}\mathit{m}\mathit{s}}\left(\mathbf{V}\right)$ | ${\mathit{I}}_{\mathit{s}}{}_{\mathit{r}\mathit{m}\mathit{s}}\left(\mathbf{A}\right)$ | ${\mathit{P}}_{\mathit{a}}\left(\mathbf{W}\right)$ | ${\mathit{Q}}_{\mathit{r}}\left(\mathbf{Var}\right)$ | ${\mathit{\eta}}_{\mathit{s}\mathit{y}\mathit{s}}$ |
---|---|---|---|---|---|---|---|

−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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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Madeira, 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