# Symmetrical Components and Sequence Networks Connections for Short-Circuit Faults in Five-Phase Electrical Systems

## Abstract

**:**

## 1. Introduction

## 2. Classification of Short-Circuit Faults

## 3. Investigation Methodology

_{q}and I

_{q}are the symmetrical components of voltage and current, and Z

_{q}is their equivalent sequence impedance.

- −
- step 1: write the boundary conditions for the selected type of short-circuit fault assuming no load conditions;
- −
- step 2: derive the symmetrical components of voltage and/or current at fault location, as well as the relevant relationships between them;
- −
- step 3: determine the mathematical constraints imposed on the sequence networks connections by the selected fault;
- −
- step 4: draw the equivalent sequence networks connections diagram based on the mathematical constraints previously determined;
- −
- step 5: check the obtained results using computer simulations.

## 4. Short-Circuit Analysis in Five-Phase AC Systems

#### 4.1. Single-Phase-to-Ground Fault

_{F}is the fault resistance, i.e., the resistance between phase A and the ground.

_{A}is in-phase with I

_{1}, this is also the case with the other symmetrical components in which I

_{A}decomposes.

_{0}, Z

_{1}, Z

_{2}, Z

_{3}, Z

_{4}are its equivalent sequence impedances. Figure 1a shows the five-phase electrical network with an Ag short-circuit.

_{A}is expressed in Equation (10) on the basis of the sequence networks diagram shown in Figure 1b.

#### 4.2. Two-Phase Faults

_{F}is the fault resistance, i.e., the resistance between phase B and phase E.

_{B}can be expressed as given in Equation (14).

_{B}is then replaced in Equation (13) with its expression from Equation (14), thus obtaining, after some rearrangements, the relation (15).

_{0}, Z

_{1}, Z

_{2}, Z

_{3}, Z

_{4}are its sequence impedances.

_{1}and I

_{4}, and I

_{2}and I

_{3}, are the primary and the secondary winding currents of an ideal transformer, respectively, with a turns-ratio of m:1 (m ≅ 0.618), with I

_{1}in opposition to I

_{4}, and I

_{2}in opposition to I

_{3}. The polarity of this ideal transformer, indicated in Figure 2b using a dot convention, ensures that all the constraints given in (16) are met by the depicted sequence networks diagram for a five-phase system under a BE fault.

_{1}can be written as given in Equation (18).

_{2}with its expression from Equation (16), the fault current I

_{B}can be rewritten as a function of I

_{1}only, with the final result given in Equation (19).

_{1}and that it is sufficient to calculate I

_{1}to determine I

_{B}. Of course, I

_{B}can also be expressed as a function of I

_{2}, I

_{3}, or I

_{4}, if desired, and according to Equation (16), I

_{B}is in quadrature with these currents as well.

_{B}.

_{F}is the fault resistance, i.e., the resistance between phase C and phase D.

_{C}can be expressed as given in Equation (24).

_{C}is replaced in Equation (23) with its expression from Equation (24) and, after some rearrangements, the relation Equation (25) is obtained.

_{1}can be written as given in Equation (28).

_{2}in Equation (24) with its expression from Equation (26), the fault current I

_{C}can be calculated as a function of I

_{1}only, with the final result given in (29).

_{1}, as well as with I

_{2}, I

_{3}, and I

_{4}. Moreover, it is sufficient to calculate I

_{1}to determine I

_{C}.

_{C}.

## 5. Simulation Results

_{1}= Z

_{2}= Z

_{3}= Z

_{4}= Z

_{0}= Z.

_{f}. For each fault, the symmetrical components of the current given by the described MATLAB Simulink model were compared with the symmetrical components given by the equivalent sequence networks diagram determined previously in this paper. Figure 5 and Figure 6 present the symmetrical components of the current for Ag, BE, and CD faults, for R

_{f}= 0 Ω and for R

_{f}= 2 Ω, respectively. As can be seen in both figures, there is virtually no difference between the phasors obtained using the MATLAB Simulink model and those obtained analytically using the short-circuit analysis and the equivalent sequence networks connections of the five-phase system.

_{1}= I

_{2}= I

_{3}= I

_{4}= I

_{0}. In the case of BE and CD faults, regardless of the fault resistance, I

_{1}= I

_{4}, I

_{3}= I

_{4}, I

_{0}= 0 and I

_{2}= m·I

_{1}, with m ≅ 0.618 for BE faults and m ≅ 1.618 for CD faults. Again, these results were observed both for the phasors obtained by simulations and for those obtained analytically. Because the relationships between the symmetrical components of the fault current are independent of the fault resistance but depend on the fault type, they can be used in practice for the classification of short-circuit faults in a five-phase electrical system. Accordingly, a protection system could first detect a short-circuit fault by detecting that a symmetrical component, such as I

_{2}, has exceeded a predefined threshold value, and then it could identify the type of fault by comparing the numerical values of I

_{1}, I

_{2}, I

_{3}, I

_{4}and I

_{0}, and by establishing the relationships between these quantities.

_{1}= Z

_{2}= Z

_{3}= Z

_{4}= Z

_{0}, which are in fact the impedance parameters of the five-phase network that has been simulated.

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Symmetrical components of current, denoted I

_{1}, I

_{2}, I

_{3}, I

_{4}, I

_{0}, during Ag, BE, and CD faults for R

_{f}= 0 Ω.

**Figure 6.**Symmetrical components of current, denoted I

_{1}, I

_{2}, I

_{3}, I

_{4}, I

_{0}, during Ag, BE, and CD faults for R

_{f}= 2 Ω.

Fault Type | Short-Circuit Faults in Three-Phase Systems | Short-Circuit Faults in Five-Phase Systems |
---|---|---|

Single-phase-to-ground | Ag, Bg, Cg | Ag, Bg, Cg, Dg, Eg |

Two-phase-to-ground | ABg, ACg, BCg | ABg, ACg, ADg, AEg, BCg, BDg, BEg, CDg, CEg, DEg |

Two-phase | AB, AC, BC | AB, AC, AD, AE, BC, BD, BE, CD, CE, DE |

Three-phase-to-ground | ABCg | ABCg, ABDg, ABEg, ACDg, ACEg, ADEg, BCDg, BCEg, BDEg, CDEg |

Three-phase | ABC | ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE |

Four-phase-to-ground | - | ABCDg, ABCEg, ABDEg, ACDEg, BCDEg |

Four-phase | - | ABCD, ABCE, ABDE, ACDE,BCDE |

Five-phase-to-ground | - | ABCDEg |

Five-phase | - | ABCDE |

Total/significant faults | 11/5 | 57/13 |

**Note**: The significant short-circuit faults are written in bold.

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**MDPI and ACS Style**

Ciontea, C.I.
Symmetrical Components and Sequence Networks Connections for Short-Circuit Faults in Five-Phase Electrical Systems. *Electricity* **2022**, *3*, 251-263.
https://doi.org/10.3390/electricity3030015

**AMA Style**

Ciontea CI.
Symmetrical Components and Sequence Networks Connections for Short-Circuit Faults in Five-Phase Electrical Systems. *Electricity*. 2022; 3(3):251-263.
https://doi.org/10.3390/electricity3030015

**Chicago/Turabian Style**

Ciontea, Catalin Iosif.
2022. "Symmetrical Components and Sequence Networks Connections for Short-Circuit Faults in Five-Phase Electrical Systems" *Electricity* 3, no. 3: 251-263.
https://doi.org/10.3390/electricity3030015