# A Rotor Winding Internal Short-Circuit Fault Protection Method for Variable-Speed Pumped Storage Units

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## Abstract

**:**

## 1. Introduction

## 2. The Harmonic Circulating Current between Stator Branches-Based Protection Method

_{s}is the fundamental wave frequency of the stator side, equal to 50 Hz.

_{k}of the circulating current of phase A, phase B, and phase C transformed by FFT respectively.

_{k}. Therefore, the larger amplitude values on both sides of f

_{k}in FFT transformation results should be selected as ${I}_{\mathrm{A}.{f}_{k}}$, ${I}_{\mathrm{B}.{f}_{k}}$ and ${I}_{\mathrm{C}.{f}_{k}}$ for calculation. This processing method can also overcome the adverse effect of slight rotational speed measurement error.

## 3. Model of the Variable-Speed Pumped Storage Unit

**U**and

_{s}**U**are the branch voltage matrices of stator and rotor, respectively,

_{r}**I**and

_{s}**I**are the branch current matrices of stator and rotor, respectively,

_{r}**R**and

_{s}**R**are the branch resistance matrices of stator and rotor, respectively, and ${\psi}_{s}$ and ${\psi}_{r}$ are the branch flux linkage matrices of stator and rotor, respectively, which can be expressed as:

_{r}**L**is 3 m × 3 m stator inductance matrix,

_{ss}**L**is 3 n × 3 n rotor inductance matrix, and

_{rr}**L**and

_{sr}**L**are mutual inductance matrices of stator and rotor. Since the relative motion is between the coils of stator and rotor, the mutual inductance matrices

_{rs}**L**and

_{sr}**L**are time-varying function matrices. Combining Equation (11) and Equation (9), Equation (9) is transformed into a time-varying ordinary differential equation system:

_{sr}**L**,

_{ss}**L**,

_{sr}**L**, and

_{sr}**L**are as follows.

_{rr}_{s}is the number of turns of a single stator coil, $\tau $ is the pole pitch, l is the core length, ${\mu}_{0}$ is the permeability of vacuum, P is the number of pole-pairs, k is the harmonic order, $k=1/P,2/P,3/P,\dots $, $\delta $ is the equivalent air gap length, ${\alpha}_{s}$ is the stator electric angle, and ${k}_{sk}$ is the stator side short distance coefficient of kth harmonic.

_{r}is the number of turns of a single rotor coil, ${\alpha}_{r}$ is the rotor electric angle, and ${k}_{rk}$ is the rotor side short distance coefficient of kth harmonic. Similarly, the mutual inductance coefficient ${M}_{HJ}^{r}$ between the rotor loop H and loop J is expressed as:

**L**,

_{ss}**L**,

_{sr}**L**, and

_{sr}**L**.

_{rr}## 4. Batch Simulation and Verification

#### 4.1. Effectiveness Analysis and Verification of the Proposed Protection Method

_{1}. The upper connecting line of the upper bar conductor in slot n is overlapped with the upper connecting line of the lower bar conductor in slot (n + 1) − (n + y

_{1}− 1), which may lead to a total of (y

_{1}− 1) short-circuit faults. The lower connecting line of the upper bar conductor in slot n is overlapped with the lower connecting line of the lower bar conductor in the slot (n − 1) − (n − y

_{2}+ 1), which may lead to a total of (y

_{2}− 1) short-circuit faults. There are (y

_{1}+ y

_{2}− 2) faults in total. Therefore, the number of possible short-circuit faults at the end of the whole rotor winding is (y

_{1}+ y

_{2}− 2) Z.

#### 4.2. Analysis and Verification of Protection Method against Maloperation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Equivalent circuit diagram of stator and rotor windings; (

**a**) stator windings; (

**b**) rotor windings.

**Figure 3.**The transient current of waveform stator branches (the inter-turn short circuit fault occurs at t = 35.0 and in rotor branch A2).

**Figure 4.**The transient circulating current of each stator phase (the inter-turn short circuit fault occurs at t = 35.0 and in rotor branch A2).

**Figure 6.**The finite element analysis model; (

**a**) the distribution of magnetic density and magnetic force line during normal operation; (

**b**) the distribution of magnetic density and magnetic force line after the rotor winding has a short-circuit fault.

**Figure 9.**When s = 0.1, the simulation results of all possible internal short-circuit faults in rotor windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

**Figure 10.**When s = −0.1, the simulation results of all possible internal short-circuit faults in rotor windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

**Figure 11.**When s = 0.05, the simulation results of all possible internal short-circuit faults in rotor windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

**Figure 12.**When s = −0.05, the simulation results of all possible internal short-circuit faults in rotor windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

**Figure 13.**When s = 0.01, the simulation results of all possible internal short-circuit faults in rotor windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

**Figure 14.**When s = −0.01, the simulation results of all possible internal short-circuit faults in rotor windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

**Figure 16.**When s = 0.1, the simulation results of all possible internal short-circuit faults in stator windings: (

**a**) in-slot faults; (

**b**) end faults in the same branch with different short turns; (

**c**) end faults between two branches in the same phase; (

**d**) end faults between phases.

Parameter | Stator | Rotor |
---|---|---|

Slot number (Z) | 252 | 294 |

Winding form | Double-layer lap winding | Double-layer wave winding |

Number of parallel branches | 4 | 2 |

Number of coils per branch | 21 | 49 |

First pitch (y_{1}) | 15 | 21 |

Second pitch (y_{2}) | 14 | 21 |

Rated current/A | 12317 | 6400 |

**Table 2.**The theoretical characteristic frequency of the circulating current between stator branches.

The Harmonic Magnetic Field with Order k | Frequency of the Circulating Current/Hz $\mathbf{50}\left[\mathit{k}(\mathbf{1}-\mathit{s}\mathbf{)}\mathbf{+}\mathit{s}\mathbf{\right]}$ | Frequency of the Circulating Current/Hz $\mathbf{50}\mathbf{\left[}\mathit{k}(\mathbf{1}\mathbf{-}\mathit{s})\mathbf{-}\mathit{s}\mathbf{\right]}$ |
---|---|---|

1/7 | 11.43 | 1.43 |

2/7 | 17.86 | 7.86 |

3/7 | 24.28 | 14.28 |

4/7 | 30.71 | 20.71 |

5/7 | 37.14 | 27.14 |

6/7 | 43.57 | 33.57 |

8/7 | 56.43 | 46.43 |

9/7 | 62.86 | 52.86 |

10/7 | 69.29 | 59.29 |

11/7 | 75.71 | 65.71 |

12/7 | 82.14 | 72.14 |

13/7 | 88.57 | 78.57 |

**Table 3.**Characteristic frequency amplitude obtained by multi-loop simulation method and finite element simulation method.

The Characteristic Frequency (Hz) | Amplitude Obtained by Multi-Loop Method (p.u) | Amplitude Obtained by Finite Element Method (p.u) | Relative Error (%) |
---|---|---|---|

1.43 | 0.0729 | 0.0738 | −1.2195 |

7.86 | 0.0839 | 0.0841 | −0.2378 |

11.43 | 0.0726 | 0.0735 | −1.2245 |

14.28 | 0.0195 | 0.0209 | −6.6986 |

17.86 | 0.0807 | 0.0818 | −1.3447 |

20.71 | 0.0172 | 0.0185 | −7.0270 |

24.28 | 0.0163 | 0.0172 | −5.2326 |

27.14 | 0.0244 | 0.0264 | −7.5758 |

30.71 | 0.0183 | 0.0194 | −5.6701 |

33.57 | 0.0388 | 0.0443 | −12.4153 |

37.14 | 0.0235 | 0.0271 | −13.2841 |

43.57 | 0.0373 | 0.0434 | −14.0553 |

46.43 | 0.0217 | 0.0248 | −12.5000 |

52.86 | 0.0073 | 0.0078 | −6.4103 |

56.43 | 0.0254 | 0.0251 | 1.1952 |

59.29 | 0.0078 | 0.0089 | −12.3596 |

62.86 | 0.0064 | 0.0071 | −9.8592 |

65.71 | 0.0070 | 0.0079 | −11.3924 |

69.29 | 0.0067 | 0.0075 | −10.6667 |

72.14 | 0.0121 | 0.0123 | −1.6260 |

75.71 | 0.0075 | 0.0086 | −12.7907 |

78.57 | 0.0179 | 0.0196 | −8.6735 |

82.14 | 0.0121 | 0.0126 | −3.9683 |

88.57 | 0.0184 | 0.0199 | −7.5377 |

**Table 4.**Protection action values obtained by the multi-loop method and finite element method under various faults.

Fault Condition | Protection Action Values Obtained by Multi-Loop Method (p.u) | Protection Action Values Obtained by Finite Element Method (p.u) | Relative Error (%) |
---|---|---|---|

A1_13 and A1_30 | 0.355 | 0.368 | −3.46% |

A1_15 and A2_4 | 0.258 | 0.266 | −2.93% |

A1_6 and B1_21 | 0.751 | 0.779 | −3.71% |

B2_11 and C2_17 | 0.824 | 0.859 | −4.14% |

Faults in the Same Branch with Different Short Turns | Faults between Two Branches in the Same Phase | Faults between Phases | Total | |
---|---|---|---|---|

In-slot faults | 252 | 42 | 0 | 294 |

End faults | 1584 | 1944 | 8232 | 11,760 |

Total | 1836 | 1986 | 8232 | 12,054 |

Slip Ratio | Number | Proportion |
---|---|---|

0.1 | 11,875 | 98.52% |

−0.1 | 11,858 | 98.37% |

0.05 | 11,767 | 97.62% |

−0.05 | 11,780 | 97.73% |

0.01 | 11,577 | 96.04% |

−0.01 | 11,546 | 95.79% |

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## Share and Cite

**MDPI and ACS Style**

He, R.; Qiao, J.; Peng, Y.; Yin, X.; Wang, Y.; Zhang, H.; Wang, W.
A Rotor Winding Internal Short-Circuit Fault Protection Method for Variable-Speed Pumped Storage Units. *Appl. Sci.* **2022**, *12*, 7783.
https://doi.org/10.3390/app12157783

**AMA Style**

He R, Qiao J, Peng Y, Yin X, Wang Y, Zhang H, Wang W.
A Rotor Winding Internal Short-Circuit Fault Protection Method for Variable-Speed Pumped Storage Units. *Applied Sciences*. 2022; 12(15):7783.
https://doi.org/10.3390/app12157783

**Chicago/Turabian Style**

He, Rufei, Jian Qiao, Yumin Peng, Xianggen Yin, Yikai Wang, Hao Zhang, and Wenhui Wang.
2022. "A Rotor Winding Internal Short-Circuit Fault Protection Method for Variable-Speed Pumped Storage Units" *Applied Sciences* 12, no. 15: 7783.
https://doi.org/10.3390/app12157783