It is starting from Fig. 1 when forming the criteria on which the operation of the algorithm is based. Figure 1 shows a three phase line where two fault locations are considered. The impedance of the three phase line is Z. F1 fault corresponds to a location inside the length of the line (Z*F1**<*Z), while F2 fault corresponds to a location outside the length of the line (Z *F2**>*Z).

If we assume that there was the phase to phase fault in the line, based on Fig. 1, equations for auxiliary voltages can be written at the beginning of the line for faults *a-b, b-c*, and *c-a*, respectively:

$${\underset{\_}{U}}_{ab}^{\text{'}}={\underset{\_}{U}}_{a}-{\underset{\_}{U}}_{b}-\underset{\_}{Z}\left({\underset{\_}{I}}_{a}-{\underset{\_}{I}}_{b}\right)$$

1

$${\underset{\_}{U}}_{bc}^{\text{'}}={\underset{\_}{U}}_{b}-{\underset{\_}{U}}_{c}-\underset{\_}{Z}\left({\underset{\_}{I}}_{b}-{\underset{\_}{I}}_{c}\right)$$

2

$${\underset{\_}{U}}_{ca}^{\text{'}}={\underset{\_}{U}}_{c}-{\underset{\_}{U}}_{a}-\underset{\_}{Z}\left({\underset{\_}{I}}_{c}-{\underset{\_}{I}}_{a}\right)$$

3

The previous equations can be written in reduced form:

$${\underset{\_}{U}}_{ab}^{\text{'}}={\underset{\_}{U}}_{ab}-\varDelta {\underset{\_}{U}}_{ab}$$

4

$${\underset{\_}{U}}_{bc}^{\text{'}}={\underset{\_}{U}}_{bc}-\varDelta {\underset{\_}{U}}_{bc}$$

5

$${\underset{\_}{U}}_{ca}^{\text{'}}={\underset{\_}{U}}_{ca}-\varDelta {\underset{\_}{U}}_{ca}$$

6

At the occurrence of phase to phase fault *a-b*, the following equations apply for currents:

$${\underset{\_}{I}}_{c}=0$$

7

$${\underset{\_}{I}}_{a}+{\underset{\_}{I}}_{b}=0$$

8

$${\underset{\_}{I}}_{ab}={\underset{\_}{I}}_{a}-{\underset{\_}{I}}_{b}={2\underset{\_}{I}}_{a}$$

9

thus reaching the equations:

$${\underset{\_}{I}}_{bc}=-\frac{{\underset{\_}{I}}_{ab}}{2}$$

10

$${\underset{\_}{I}}_{ca}=-\frac{{\underset{\_}{I}}_{ab}}{2}$$

11

Figure 2 shows the vector diagram of the voltages in case of phase to phase fault *a-b*. From the figure, it is observed that for faults outside the line (Z*F2**>*Z), the direction of voltage \({\underset{\_}{U}}_{ab}^{\text{'}}\) remains unchanged relative to \({\underset{\_}{U}}_{ab}\). When there is the fault on the line (Z*F1**<*Z), there is a change in the direction of the voltage \({\underset{\_}{U}}_{ab}^{\text{'}}\). Phase to phase voltages \({\underset{\_}{U}}_{bc}^{\text{'}}\) and \({\underset{\_}{U}}_{ca}^{\text{'}}\) do not change direction, regardless of the fault position.

Similarly, equations for auxiliary voltages for the single phase to ground fault of phases *a, b and c*, respectively, can be written:

$${\underset{\_}{U}}_{a}^{\text{'}}={\underset{\_}{U}}_{a}-\underset{\_}{Z}\left({\underset{\_}{I}}_{a}+{k}_{0}{\underset{\_}{I}}_{0}\right)$$

12

$${\underset{\_}{U}}_{b}^{\text{'}}={\underset{\_}{U}}_{b}-\underset{\_}{Z}\left({\underset{\_}{I}}_{b}+{k}_{0}{\underset{\_}{I}}_{0}\right)$$

13

$${\underset{\_}{U}}_{c}^{\text{'}}={\underset{\_}{U}}_{c}-\underset{\_}{Z}\left({\underset{\_}{I}}_{c}+{k}_{0}{\underset{\_}{I}}_{0}\right)$$

14

The previous equations can be written in abbreviated form:

$${\underset{\_}{U}}_{a}^{\text{'}}={\underset{\_}{U}}_{a}-\varDelta {\underset{\_}{U}}_{a}$$

15

$${\underset{\_}{U}}_{b}^{\text{'}}={\underset{\_}{U}}_{b}-\varDelta {\underset{\_}{U}}_{b}$$

16

$${\underset{\_}{U}}_{c}^{\text{'}}={\underset{\_}{U}}_{c}-\varDelta {\underset{\_}{U}}_{c}$$

17

At the occurrence of the single phase to ground fault in phase *a*, the following terms apply for currents:

$${\underset{\_}{I}}_{a}=3{\underset{\_}{I}}_{0}$$

18

$${\underset{\_}{I}}_{b}={\underset{\_}{I}}_{c}=0$$

19

In Fig. 3, the vector diagram of the voltages in the case of the fault in phase *a* is shown. It can be seen from the figure that in the case of an internal fault (Z*F1**<*Z), the voltage \({\underset{\_}{U}}_{a}^{\text{'}}\) changes direction concerning Ua, while in the case of an external fault, the direction remains unchanged. Voltages in healthy phases \({\underset{\_}{U}}_{b}^{\text{'}}\) and \({\underset{\_}{U}}_{c}^{\text{'}}\) do not change direction, regardless of the fault position.

Based on the conducted analysis, it can be concluded that the angle *δ* in phases with fault will have the value of approximately 180°, while the angle *δ* in healthy phases ranges from 0° to approximately 30°. To increase the sensitivity and speed of fault detection, a value of 120° can be adopted instead of the threshold angle of 180°, without compromising the safety of the protection operation.

Table 1 summarizes the criteria for the operation of the proposed algorithm.

Table 1

Threshold angles for detecting different faults types

Phase | *Threshold angles* |

δa | δb | δc | δab | δbc | δca |

**a-G** | 120o | / | / | / | / | / |

**b-G** | / | 120o | / | / | / | / |

**c-G** | / | / | 120o | / | / | / |

**a-b** | / | / | / | 120o | / | / |

**a-b-G** | 120o | 120o | / | 120o | / | / |

**b-c** | / | / | / | / | 120o | / |

**b-c-G** | / | 120o | 120o | / | 120o | / |

**c-a** | / | / | / | / | / | 120o |

**c-a-G** | 120o | / | 120o | / | / | 120o |

**a-b-c** | 120o | 120o | 120o | 120o | 120o | 120o |

**a-b-c-G** |