Fault theory

In spite of the careful building, faults take place in the distribution network from time to time. These are usually caused by weather conditions or faults in the network components. [2] The fault situations are seldom symmetrical which is why their handling and analysis require a specific theory. The network behaves differently during each fault, which is why also some mathematical methods are needed in order to receive accurate results. In the following sections the pi-section and the symmetrical components are introduced.

2.1.1. Pi-section

Normally, the loads per phase are assumed to be equal. Therefore transmission lines are analyzed on a per phase basis. A short transmission line can be represented with its series impedance alone. The shunt admittance is negligible, which means that the equivalent circuit is according to the one in Figure 2.1. This model is however accurate only for short transmission lines, which usually are defined as lines less than 100 km. In Figure 2.1 US and UR are voltages in the sending (S) and in the receiving (R) end, IS and IR are currents in the sending and in the receiving end. Z is the line impedance, which consists of resistance R and reactance jX. [3]

Figure 2.1 The single-phase equivalent of a short transmission line [3]

When the transmission line lengths are more than 100 km the model introduced above is not adequate. More accurate results are gained with the usage of pi-section.

This model gives also more accurate calculation results for shorter transmission lines and cables, which is why it is used throughout this work. It must, however, be noticed that most of the calculation methods developed to analyze transmission lines are simplifications. Pi-section is illustrated in Figure 2.2. [3] In the pi-section the Y is the shunt admittance, which practically means the line capacitance to earth. [4]

Figure 2.2 The transmission line represented with a pi-section [3]

When modeling long overhead lines (OHL) with pi-sections the correction factors must be used in order to model the line correctly. The behavior of the shunt capacitance and the series impedances are non-linear and the usage of correction factors compensates this non-linearity. The correction factors are frequency dependent and easy to calculate for a line examined on a fundamental frequency. If one does not wish to calculate and use correction factors, the model can as well be completed with several pi-connections representing the long line. The correction factors can be calculated according to equations (1) and (2). [4]

G Z G

Z sinh

π = (1)

( )

2 tanh 2

2* G

Y G

Y_π = (2)

Where

G is the line conductance Y is the line admittance

Y_π is the corrected shunt admittance Z is the line impedance

Zπ is the corrected series impedance 2.1.2. The symmetrical components

In normal operating conditions, the electricity network is almost symmetrical. This means that the load impedances and the transmission line impedances are the same in every phase and the phase voltages are equal with 120° phase shift to each other.

Because of the symmetry, the network can be described with a single-phase equivalent, which simplifies the network analysis and calculation. If, for example, the current in one phase is calculated, it can be concluded that in normal operating conditions the currents in the two other phases are in the same magnitude with 120° phase-shift to each other.

The symmetrical voltage phasors are illustrated in Figure 2.3. In the figure the termsU_A, U_Band U_C represent the phase voltage phasors in phases A, B and C. [5]

Figure 2.3 Voltage phasors in normal operating conditions. [5]

Some of the network faults, however, are not symmetrical and these kinds of faults cannot be described with single-phase equivalents. Asymmetric situations can be described with symmetrical components and sequence networks. Representing the network with symmetrical components is a mathematical method for network calculation where the phasor coordinates are transformed into sequence coordinates.

This is shown in Figure 2.4. [4; 5]

Figure 2.4 Symmetrical components. Positive sequence network a₁, negative sequence network a2 and the zero sequence network a0. [4]

The idea is that by connecting these sequence phasors, the phasor diagram of the fault can be represented. The asymmetrical phase voltages are thereby formed as a combination of three symmetrical networks. [3] Figure 2.5 shows how the sequence networks are connected when representing an asymmetric fault. In Figure 2.5 U'a1, U’b1

and U’c1 represent the positive sequence network, U’a2, U’b2 and U’c2 represent the negative sequence network and U’a0, U’b0 and U’c0 represent the zero sequence network.

The sequence network phasors are drawn in different colors for clarification. Terms U’a, U’_b and U’_c represent the real phase voltage phasors during the fault. [3]

Figure 2.5 The positive sequence network components (red), negative sequence network components (blue), zero sequence network components (green) and total phase voltage phasors (black) during an asymmetric fault. [3]

So the three-phased network will be transformed into sequence networks. The sequence networks, on the other hand, can be represented with two-terminal equivalents. All the voltages, represented as U1eq, are generated in the positive sequence network, and the two other networks contain only the equivalent impedances Z_2eq and Z_0eq. This is shown in Figure 2.6. In the figure U₁ is the phase-to-earth voltage in positive sequence network, U₂ is the phase-to-earth voltage in negative sequence network and U0 is the phase-to-earth voltage in zero sequence network. Z1eq is the equivalent impedance in positive sequence network. [4]

Figure 2.6 Sequence networks represented as two-terminal equivalents. [4]

By connecting these sequence network equivalents, the asymmetric fault can be represented as an equivalent coupling of the three sequence networks. This helps the analysis and calculation of the network’s asymmetric situations and enables more accurate calculation results. This is illustrated in Figure 2.7 representing a single-phase earth-fault. [4]

Figure 2.7 Equivalent coupling of the sequence network equivalents in an earth-fault. [4]

For three-phase short-circuit fault the single-phase equivalent mentioned earlier, can be used to simplify the calculation but for example with a single-phase earth-faults and two-phase short-circuit faults need to be analyzed with symmetrical components for those are asymmetric faults. Other asymmetrical network situations are cross-country faults, line breaks and asymmetrical loadings. [3; 4]