k-factor (power engineering)

In power engineering , especially in electrical energy networks , the k-factor or earth factor is the relationship between line and earth impedance . When protecting overhead lines or underground cables , the k-factor is of particular importance as a decisive factor in the setting of distance protection relays . Figure 1 schematically shows the impedance of a conductor-conductor loop. Figure 2 shows the impedance loop of a conductor-earth fault.

Figure 1: Schematic representation of a ladder-ladder loop

K-factor in protection technology

Figure 2: Schematic representation of a conductor-earth loop

Figure 3: Schematic representation of an energy network with optimally set distance protection relays

Figure 4: Schematic representation of an energy network with not optimally set distance protection relays

The task of the protective relay is to disconnect the relevant network section in the event of a phase-to-earth fault (Figure 2). If the relay settings are correct, a consumer that is usually supplied from two sides (Figure 3) continues to be supplied with energy via one line if the other line has to be switched off.

However, if the k-factors or zones of a relay are not set correctly, the zones will overreach or underreach (Figure 4). In this case the fault is seen by three relays in zone 1 instead of just 2 relays (DIS), i.e. H. all three relays trip. The result is that a second supply line is switched off. The consumer is now cut off from the supply.

Different formats

Figure 5: Equivalent circuit diagram of conductor impedances

There are several formats for representing the k-factor, which are shown in equations 1-5. Here, make , and the ground impedance, , and the circuit impedance is, and the k-factor. For all of them, however, it is true that they are constants of the line, regardless of the line length. ${\displaystyle {\underline {Z}}_{0}}$ ${\displaystyle {\underline {R}}_{E}}$ ${\displaystyle {\underline {X}}_{E}}$ ${\displaystyle {\underline {Z}}_{1}}$ ${\displaystyle {\underline {R}}_{L}}$ ${\displaystyle {\underline {X}}_{L}}$ ${\displaystyle {\underline {k}}_{0}}$ ${\displaystyle {\underline {k}}_{L}}$

${\displaystyle {\underline {k}}_{0}={\frac {{\underline {Z}}_{0}}{{\underline {Z}}_{1}}}\,}$ Equation 1

(Comment on equation 1: This equation 1 is certainly misleading, since the indices 0 and 1 often stand for positive and negative system. The error often occurs that zero impedance and earth impedance in general are equated. If one expresses earth impedance and conductor impedance correctly in relation to zero impedance and positive sequence impedance, one finds equation 5)

${\displaystyle {\underline {k}}_{0}={\frac {{\underline {R}}_{E}}{{\underline {R}}_{L}}}\,}$ Equation 2

${\displaystyle {\underline {k}}_{0}={\frac {{\underline {X}}_{E}}{{\underline {X}}_{L}}}\,}$ Equation 3

${\displaystyle {\underline {k}}_{L}={\frac {{\underline {Z}}_{E}}{{\underline {Z}}_{L}}}\,}$ Equation 4

${\displaystyle {\underline {k}}_{0}={\frac {1}{3}}\cdot ({\frac {{\underline {Z}}_{0}}{{\underline {Z}}_{1}}}-1)}$ Equation 5 with index 1 = positive sequence system and index 0 = zero system (see also the comment on equation 1)

The k-factors describe the relationship between the impedance of a conductor-conductor loop and a 3-conductor-earth loop. Half the impedance of a balanced ladder-ladder loop, i.e. H. the impedance of a conductor is referred to as positive sequence impedance , and three times the impedance of a 3-conductor-earth loop is referred to as zero-sequence impedance (Figure 5). ${\displaystyle {\underline {Z}}_{1}}$ ${\displaystyle {\underline {Z}}_{0}}$

calculation

Figure 6: Schematic representation of the line geometry of a high-voltage line

The parameters required to calculate the conductor impedance are varied. On the one hand, information about the geometric configuration is required (Figure 6). These are:

• Height above the ground and horizontal distance for each conductor and ground wire,
• average slack of conductor and grounding wires in the middle between two masts,

on the other hand, various electrical parameters must be known:

• specific earth resistance , ${\displaystyle \rho }$
• DC resistance of all lead wires,
• Twist build-up of the line wires,
• mean radius of the lead wires, and
• Total diameter of the lead wires.

A general problem is that a large number of parameters play a role in calculating line impedances. A single incorrect parameter can lead to a significant error.

Measurement

Compared to the calculation, a measurement of the line parameters including the k-factors is relatively easy.

For a long time, the problem with determining the line parameters was that very high performance was required. Either currents around or above the rated current had to be used in order to get well above the interference level caused by coupling in from other systems or to measure with a large diesel generator just above the interference level using the beat method . This is equipment that typically weighs several tons.

In the age of electronic signal sources, there is a third way of measuring k-factors. A current with a frequency well below the mains frequency is fed into an overhead line or a cable, and by means of a frequency-selective measurement, exactly the portion with the generated frequency can be filtered out of the measurement signal. Interference with a frequency other than the one generated - including interference with mains frequency - is masked out. If the line impedance is measured and averaged once above and once below the line frequency, the impedance at line frequency can be determined very precisely. With this method, it is possible to work with currents well below the interference levels coupled in from other systems.

Surge arresters reliably protect such a measurement against voltages above the expected levels.

credentials

1. ↑ Klapper, Ulrich: Transmission security through measurement of line impedance and k-factor , EW Magazin für Energiewirtschaft special print no. 6170, vol. 1006 (2007), issue 4, pp. 36–41