# 1. Introduction

The basic parameter of the grid performance at ground fault is the zero-sequence voltage component *U*_{0}, which, with the line’s longitudinal impedances neglected, represents the voltage between the grid’s neutral point and the ground. However, the level of this voltage is determined not only by the conditions at the ground fault location, but also by the unbalances of to-ground capacitances and conductances, as well as to-ground voltages, in various points in the grid. Phenomena induced by such unbalances are particularly evident in compensated grids and must be considered in determining the operating conditions of earth fault protections and follow-up ground fault compensation controllers. Therefore, in the theoretical analysis of disturbances due to the to-ground unbalance the grid’s equivalent circuit shown in Fig. 1 is typically used.

Fig. 1. Equivalent circuit of compensated MV grid for ground fault calculations, where: *C _{0i}*

_{ }– to-ground capacitance of line i,

*G*– to-ground conductance of line

_{0i}*i, I*– to-ground current in line

_{0i}*i*(resulting from

*C*and

_{0i}*G*),

_{0i}*L*– compensation coil reactance in grid’s neutral point,

_{N}*G*– conductance of grid’s neutral point grounding circuit

_{N}and:

(1)

(2)

where: * U_{asci} *– voltage due to to-ground unbalance of phase capacities of line

*i*,

*voltage due to to-ground unbalance of phase conductances of line*

__U___{asgi}–*i*,

__E___{l}_{1}

*,*source voltages in phases L1, L2 and L3 of the grid,

__E___{L2},__E___{L3}–*C*

_{0L1i}

*, C*

_{0l2i}

*, C*

_{0L3i}

*–*to-ground capacities of phases L1, L2 and L3 of line

*i*,

*G0*

_{L}_{1i}

*, G*

_{0L2i}

*, G*

_{0L3i}– to-ground conductances of phases L1, L2 and L3 of line

*i*.

The next chapter presents a relationship that allows one to assess the impact on *U*_{0} if effects of to-ground unbalance of the grid’s individual lines, depending on the ground fault compensation conditions of (Petersen coil tuning).

# 2. *U*_{0 }voltage components

According to the diagram in Fig. 1, *U*_{0 }voltage is expressed by the following relationship:

(3)

where: *ω *– operating grid pulsation.

Assuming that:

and that:

the following is obtained:

(4)

where: *d*_{0} – grid attenuation factor, *s *– ground fault compensation detuning factor

After entering the equations:

and

(5)

where:

the following *U*_{0} formula is obtained:

(6)

The above dependence shows that the grids voltage’s zero-sequence component is determined by:

- resultant unbalance of to-ground capacities
- resultant unbalance of the grid’s to-ground phase conductances
- compensation detuning factor s
- attenuation level in to-ground circuit (coefficient
*d*_{0}).

It’s easy to notice that voltage *U*_{0} has the following two components:

- resulting from unbalance of the grid’s to-ground phase capacitances

(7)

- resulting from unbalance of the grid’s to-ground phase conductances

(8)

The grid voltage’s zero-sequence component can therefore be expressed by the equation:

(9)

Upon introduction of the concepts of relative capacitance unbalance and conductance unbalance voltages in the form of:

(10)

and upon appropriate transformations, the formulas for relative values of U_{0} voltage components take the form:

(11)

A graphic picture of the above dependences is presented in Fig. 2 and 3.

Fig. 2. *U _{0asc}* voltage component’s dependency on compensation detuning for grid’s attenuation factor d0 = 0.03

Fig. 3. *U _{0asg}* zero-sequence voltage component’s dependency on compensation detuning for grid’s attenuation factor d0 = 0.03

Since *U*_{0asc} is usually much greater than *U*_{0asg}*, *only the capacitance unbalance component will be considered in the further analysis of the ground fault compensation tuning conditions.

# 3. Follow-up compensation in grids with significant to-ground unbalance

In grids with a relatively high to-ground unbalance, the problem appears of *U*_{0} voltage reduction while the Petersen coil is tuned to the required level. In practice, the accurate compensation is often abandoned, and detuning is set at the level reducing *U*_{0} voltage to a pre-set value, e.g. *U*_{0}* < *0.05 of the grid phase voltage. To fulfil this requirement for *U*_{0} voltage an algorithm is needed, whereby the minimum detuning factor *s _{x}* is determined according to the following relationship:

** **(12)

where: *U _{d}*

_{ (%)}– allowable percentage share of the zero-sequence voltage component due to resonance,

*U*– grid’s phase voltage (rated) (e.g.

_{f}*U*8660

_{f}=*V), X*– actual measured reactances,

_{d}, X_{cs}*Id*– actual coil current due to grid’s to-ground unbalance.

Under these criteria, the Petersen’s coil changes the reactance only in the areas of the grid’s obvious under-compensation or over-compensation. This is shown in Fig. 4, with the grid attenuation factor assumed at *d*_{0}* =* 0.025 and the natural unbalance at X_{c} = 0.5%.

Fig. 4. *U _{0}* zero-sequence voltage component’s dependency on ground-fault compensation detuning;

*S*– compensation detuning at ng adjustment at

_{x}*U*voltage reduction to 5% of the grid phase voltage

_{0}This type of criterion can be used by dispatchers in field grids with a large share of overhead lines and high to-ground unbalance. However, the effects of such detuning reduce the coil’s arc-suppression capability. It is easy to find examples of grids, in which this criterion detunes the compensation even to *K *= 1.3 or further.

In addition, such a grid is still exposed to an increase in *U*_{0} voltage during regulation and seeking the required detuning level. This will be particularly noticeable when an additional recommendation is given, that requires adjustment always to over-compensation. An extreme case may be assumed, whereby the tuning process starts from the factor’s negative values and ends on positive values. The grid will then be exposed to an increase in voltage *U*_{0}, which at a point in the adjustment can reach the maximum (resonant point voltage). It has been demonstrated in laboratory tests that an effective solution to reduce such effects is to increase attenuation of the to-ground circuit for the duration of the adjustment process. The effect of this measure is shown in Fig. 5, for over-compensation pre-set at 15%.

Fig. 5. Petersen coil adjustment in the area of large resonant voltages with transient grid attenuation increase

It can be shown by analysing the results of MV grid models’ testing that a better solution in this respect is to permanently increase the to-ground circuit attenuation. Compensation can then be very accurate, reducing the fault current’s reactive component to the minimum. Degrading the reactor (Petersen’s coil) quality factor can increase coefficient *d*_{0} up to three or four times its natural value and thus continuously reduce the resonant voltages. Fig. 6 and 7 show the effects of such a solution. From the comparison of the *U*_{0} voltage courses in both compensation states, it is clear that Petersen coil’s detuning in the grid more adversely effects the recovery of the ground-faulted phase (L1) voltage than the state of the grid’s increased to-ground conductance. Moreover, a grid with a detuned coil generates a larger fault current with the predominant reactive component and in this way additionally worsens the conditions for the fault’s spontaneous suppression. In a grid with increased conductivity, the smaller fault current with the strongly reduced reactive component facilitates fast recovery of the fault space’s isolation.

Fig. 6. *U _{0}* zero-sequence voltage component’s dependency on compensation detuning and attenuation factor

*d*

_{0}Therefore, the probability of re-ignitions at the fault location is many times greater in a grid with detuned compensation (example of such a grid in Fig. 7a) than for the grid condition described in Fig. 7b.

Fig. 7. Waveforms of phase voltages in relation to the grid and its zero-sequence voltage component *U _{0}* during L1 phase’s ground fault: a) in a grid with detuned compensation, b) in a compensated grid

# 4. Methods of measuring to-ground capacitances

In follow-up reactors the role of measuring systems that evaluate grid’s to-ground parameters and control tuning is important. Generally, such systems use natural to-ground unbalances or operate by introducing additional sources to the grid. In a grid with a relatively large unbalance, to evaluate its to-ground parameters usually an algorithm is used, whereby *U*_{0} voltage and the coil current are measured. Fig. 8 presents a simplified grid to-ground circuit’s diagram, with *U _{as}* unbalance voltage impact location marked.

Fig. 8. Equivalent circuit of MV grid’s to-ground circuit L, C and G denote the grid’s to-ground circuit parameters: coil inductance, capacitance and conductance. Uas represents the natural unbalance voltage

If *U _{as}* voltage is sufficient (e.g.

*U*

_{0}> 0,2%

*U*

_{f}) to determine the grid to-ground capacitance, it is enough to measure

*U*

_{0}voltage and

*I*current before and after a small change in the coil reactance. Then capacitance

_{s }*C*is calculated from the formula:

(13)

where: *U*_{01} and I_{s}_{1} – measurements before coil reactance change, *U*_{02} and *I*_{s2} – measurements after coil reactance change

Based on these measurements, the actual *U _{as}* voltage may be determined by simple transformation to:

(14)

where:

(15)

With the grid to-ground capacitance known, it is enough to adjust the Petersen coil to the desired compensation level, while controlling the inductive reactance in accordance with the formula:

(16)

# 5. Summary

The ground fault compensation principle has been known for 100 years. Capacitive ground fault currents’ compensation devices were first used in power grids in 1917 [^{4}]. They were developed by Waldemar Petersen. Despite design alterations and use of automatic control devices, the name of Petersen coil is commonly used (Photo 1). Grounding the grid neutral point through such a coil is troublesome at a large unbalance of individual lines’ to-ground capacitances. They relate to a significant increase in the grid’s zero-sequence voltage when tuning the coil to the accurate compensation level. The study has shown that a good solution in such a situation is to artificially increase the grid to-ground circuit’s attenuation by inserting properly selected resistors. They may be permanently connected in parallel to the coil (in a grid with a relatively large unbalance) or switched on only during the coil adjustment in the area of the resonant point (in a grid with smaller to-ground unbalance).

Photo 1. The original Waldemar Petersen’s coil