Fault studies

Conductors in distribution networks are separated from earth and from each other by a variety of insulating materials: air, paper or polymers. Occasionally, an unpredictable event ruptures this insulation, creating a short circuit between conductors or between conductors and

Figure 3.13 Power flow for minimum load and embedded generation at 0.95 power factor lagging on a voltage controlled bus

64 Embedded generation

earth. This abnormal conducting path is called a fault. Being able to predict the value of the current in faults is very important for two main reasons. First, this current may be so large that it could damage the distribution plant or exceed the rating of the breakers that are supposed to interrupt it. Paradoxically, the second reason for calculating fault cur-rents is to check that they are not too small for the fault to be detected.

Devising a protection system capable of discriminating between a large (but normal) load current and a small fault current is difficult. Since failing to detect a fault is an unacceptable safety risk, the distribution system must be designed in such a way that fault currents are large enough to be detected under all operating conditions.

A distinction must be made between balanced and unbalanced faults.

Balanced faults affect all three phases of the network in a similar manner and the symmetry between the voltages and currents in the three phases is not altered. A single-phase representation of the network can therefore be used when studying such faults. On the other hand, unbalanced faults create an asymmetry in the network and require a more complex analysis based on symmetrical components.

This section begins with an explanation of balanced fault calculations using a simple two-bus example. These calculations are then generalised to networks of arbitrary size and complexity and then to unbalanced faults. Finally, these concepts are illustrated using examples from an embedded generation scheme.

3.4.1 Balanced fault calculations

Let us consider again a simple two-bus power system such as the one shown in Figure 3.14. We will assume that no load is connected to this system. For balanced fault calculations, this system must be modelled as shown in Figure 3.15. The main difference between this model and the model used for power flow studies is that the generator is now repre-sented as an ideal voltage source behind a source impedance ZS. The total impedance of the line is represented by ZL.

If a balanced fault occurs at the far end of the line, Figure 3.15 shows that the fault current is given by

I_f= E Z_S+ ZL

(3.25) On the other hand, if the fault occurs at a distance d down the line, the fault current is given by

I_f= E Z_S+ l

d Z_L

(3.26) System studies 65

From these equations, we conclude that the amount of impedance between the fault and the source determines the fault current. A short circuit at the end of a long distribution feeder will therefore result in a much smaller fault current than a fault near a primary substation.

In this very simple example, the expression for the fault current could be written by inspection. This is not the case in an actual system where the fault current might be supplied from multiple generators through a meshed network. In addition, it may be necessary to consider the effect of the prefault load current. As with the power flow, a solution to this problem arises from a more advanced technique of circuit analysis, namely Thevenin’s theorem. This theorem states that an electrical circuit can always be modelled as an ideal voltage source in series with an impedance. For fault analysis, we need to model the network as seen from the location of a potential fault. The source voltage, which is called the Thevenin voltage and is denoted by Vth, is equal to the voltage at that point before the occurrence of the fault. The series impedance, which is Figure 3.14 Two-bus power system for fault calculations

Figure 3.15 Model of the two-bus power system for fault calculations 66 Embedded generation

called the Thevenin impedance, is the impedance seen from the location of the fault looking back into the network. When we are interested only in what happens at a given location, a network of arbitrary complexity can therefore always be represented as shown in Figure 3.16. The fault cur-rent is then given by

I_f=V_th

Z_th (3.27)

If the network under consideration is not too large or complex, the Thevenin impedance can be calculated by hand through network reduc-tion. Furthermore, if a high degree of accuracy is not required, the Thevenin voltage can be assumed equal to the nominal voltage. On the other hand, for large networks where the prefault load current is not negligible, determining this voltage requires a power flow calculation. A systematic determination of the Thevenin impedance is best achieved through a partial inversion of the admittance matrix of the network.

Specialised software packages have been developed to carry out such computations accurately and efficiently. Using such a package, determin-ing the fault current for any number of potential fault locations, for several configurations of the network, and for various combinations of generating units is relatively easy.

3.4.2 Concept of fault level

Describing the effect of faults on a system in terms of the current that would flow in a fault at various locations could be somewhat confusing.

This fault current must indeed be compared to the normal load current, and this load current is inversely proportional to the nominal voltage. To compensate for the effect of the voltage level, the magnitude of potential faults in the system is given in terms of the fault level. This quantity is usually expressed in MVA and is defined as

FL= √3VnominalI_f (MVA) (3.28)

Figure 3.16 Thevenin equivalent of a network of arbitrary complexity

System studies 67

The base quantities for MVA, voltages and currents are usually chosen in such a way that they obey the following equation:

MVA_B= √3 VBI_B (3.29)

If we divide eqn. (3.28) by eqn. (3.29) and assume that the base voltage is equal to the nominal voltage, we find that the per unit value of the fault level is equal to the per unit value of the fault current:

FL^pu= I^pu (3.30)

Finally, if we assume that the voltage was equal to its nominal value prior to the fault, combining eqns. (3.27) and (3.30) gives

FL^pu= I^pu= 1

|Zthpu| (3.31)

The fault level therefore gives an indication of ‘how close’ a particular point is from the sources of a system. For example, fault levels in an EHV transmission system can be three orders of magnitude larger than in an LV distribution system. The configuration of the system can have a significant effect on the fault level. In particular, synchronising additional generators or connecting lines in parallel reduces the equivalent Thevenin impedance of the network and hence increases its fault level. Rather than providing a detailed model of their network to the designers of an embedded gen-eration scheme, distribution utilities will usually give them the fault level at the connection point and the ratio X/R of the source impedance. Eqn.

(3.31) shows that this information is sufficient to create a Thevenin equivalent of their network and hence to carry out simple fault studies.

Example 3.2: Figure 3.17 shows an embedded synchronous generator connected to a large power system through a distribution network. The fault level at the near end of this network is 1000 MVA at zero power factor (i.e. X/R= ∞). The impedance of the feeder is j0.1 p.u. while the source impedance of the synchronous generator is j0.2 p.u. All these impedances have been calculated using a 100MVA basis. If the network is operating at nominal voltage and the load current is negligible, what is

Figure 3.17 Power system of Example 3.2 68 Embedded generation

the magnitude of the current that would flow if faults were to occur at the ends of the feeder?

Figure 3.18 shows the equivalent circuit that will be used for this calcu-lation. The large power system has been replaced by its Thevenin equiva-lent with the impedance obtained from eqn. (3.31). Since the network operates at nominal voltage and the load current is assumed negligible, both voltage sources have been set at 1.0 p.u. The magnitude of the fault current at bus A is given by

I_f,A=1.0 X_A= 1.0

0.075= 13.3 p.u.

where XA is the parallel combination of 0.1 p.u. and 0.3 p.u.

The magnitude of the fault current at B is I_f,B=1.0

X_B=1.0

0.1= 10.0 p.u.

where X_B is the parallel combination of 0.2 p.u. and 0.2 p.u.

In addition to its role in fault current calculations, the concept of fault level is also useful when analysing a network operating under normal conditions. As eqn. (3.31) indicates, larger fault levels correspond to smaller Thevenin impedances. The voltage drop caused by a given load current will therefore be smaller at buses where the fault level is high.

Fault levels are thus sometimes used to quantify the ‘strength’ or ‘stiff-ness’ of a network at a given location.

3.4.3 Application to an embedded generation scheme

We will now use the small system introduced in Section 3.3.5 to illustrate the concepts developed in the previous sections. Let us first consider the case of Figure 3.19 where the embedded generator is disconnected from the system. The fault levels have been calculated (using a commercial-grade program) for faults at various buses in the system and are shown

Figure 3.18 Model of the power system of Example 3.2

System studies 69

next to the bus names. It is clear that the fault level decreases as the distance between the fault and the source increases. This figure also shows the flows that would result from a fault at busbar C. Note that the sum of the branch flows at bus C is only roughly equal to the fault level at this bus because the flows are expressed in MVA and the corresponding currents have slightly different phases. Figure 3.20 shows that the pres-ence of an embedded generation significantly increases the fault levels in the system. In particular, 58 MVA would be drawn from this generator by a fault at bus C.

3.4.4 Unbalanced faults

Since balanced faults affect all three phases in an identical manner, the symmetry of the network is preserved and the fault current is balanced, albeit larger than a normal load current. The network impedances that must be considered in the calculation of balanced fault currents are therefore the ‘normal’ impedances of the network. A majority of the faults that occur in a network, however, do not affect all three phases in the same manner. Such faults are called unbalanced because they destroy the three-phase symmetry of the network. Unbalanced faults usually involve a short circuit between one line and the earth, between two lines or between two lines and the earth. Since the currents that result from such faults are not symmetrical, carrying out the analysis in terms of the Figure 3.19 MVA fault levels for balanced three-phase faults at the various buses in the system. The arrows show the MVA flows for a fault at bus C.

The system is assumed to be unloaded prior to the fault 70 Embedded generation

actual phase quantities is extremely difficult. Transforming the actual phase quantities into a set of abstract variables called symmetrical com-ponents considerably simplifies the calculation of unbalanced fault cur-rents. It can be shown [5] that any set of unbalanced phase quantities (voltages or currents) can be decomposed into three components:

1 a positive sequence component consisting of three balanced voltages or currents in a normal (positive) phase sequence

2 a negative sequence component consisting of three balanced voltages or currents in a reverse (negative) phase sequence

3 a zero sequence component consisting of three voltages or currents of equal magnitude and phase.

The symmetrical component currents are related to the phase currents by the following matrix transformation:

where a is the complex number,

Figure 3.20 MVA fault levels for balanced three-phase faults at the various buses in the system, including the contribution of the embedded generator at bus E. The arrows show the MVA flows for a fault at bus C. The system is assumed to be unloaded prior to the fault

⎧

^I0

⎪

^I1

⎩

^I2

⎫ ⎧ ⎪

^{= –¹}_³

⎪

⎭ ⎩

1 1 1 1 a a² 1 a² a

⎫ ⎧ ⎪

^·

⎪

⎭ ⎩

I_a

⎫

I_b

⎪

^(3.32)

I_c

⎭

System studies 71

a= 1∠120° = − –¹² +j√3 2

(3.33) The inverse transformation gives the phase currents in terms of the symmetrical component currents:

An identical transformation relates phase voltages and symmetrical component voltages.

The impedance that lines, cables, generators and transformers present to the flow of currents in a balanced system is the positive sequence impedance.

The impedances that these devices offer to negative and zero sequence cur-rents are called, respectively, the negative and zero sequence impedances.

In our discussion of balanced faults, we argued that, when observed from a given location, a network of arbitrary complexity could be replaced by its Thevenin equivalent. In that case, the Thevenin voltage source and impedance can be calculated from the positive sequence volt-ages produced by the generators and the positive sequence impedances of the components of the network. When unbalanced conditions are considered, we also need to take into account the equivalent circuits for the negative and zero sequence components. Since generators produce only a positive sequence voltage, there are no sources in the negative and zero sequence equivalent circuits. Figure 3.21 illustrates these concepts.

A network of arbitrary complexity is examined from a given location (e.g. a potential fault location). Instead of considering the A, B and C phases of this network, we replace it by its positive, negative and zero sequence equivalent networks as seen from this location. Under normal conditions, no currents flow in these circuits and a voltage is present only in the positive sequence circuit. As discussed in Section 3.4.1, a balanced fault to the system at that location can be modelled by applying a short circuit on the equivalent circuit of the positive sequence. Each type of unbalanced fault can be modelled by a particular connection of the three equivalent circuits. For example, suppose that we would like to calculate the fault current that would result from a fault between phase A and earth. Since phases B and C are unaffected by this fault, we have:

I_b= 0 (3.35)

I_c= 0

Inserting these values in eqn. (3.32), we obtain:

⎧

^Ia

from which we deduce that

I₀= I1= I2= –¹³I_a (3.37) Figure 3.22a shows how eqn. (3.37) is translated into a connection between the symmetrical component circuits. From this figure it is then easy to see that the fault current is given by

I_a= 3I0= 3I1= 3I2= 3Vth

Z₁+ Z2+ Z0

(3.38) It can be shown [5] that line-to-line and line-to-line-to-earth faults can be modelled using the connections shown in Figures 3.22b and 3.22c. Using these diagrams and eqn. (3.34), we get the following relation for a fault between phases B and C:

I_a= 0

(3.39) I_b= −Ic=−j√3 Vth

Z₁+ Z2

Similarly, for a fault involving phases B, C and earth:

Figure 3.21 Equivalent circuits for unbalanced faults

System studies 73

I_a= 0

I_b= V_th

Z₁Z₂+ Z1Z₀+ Z0Z₂[(1− a)Z2+ (1 − a²)Z0] (3.40) I_c= V_th

Z₁Z₂+ Z1Z₀+ Z0Z₂[(a− a²)Z₂+ (a − a²)Z₀]

Effect of neutral earthing

The connection of generator and transformer windings and the earthing of neutral points play a particularly important role for line-to-earth faults. Figure 3.23 shows the three possible combinations of winding configurations and neutral earthing. For the delta and ungrounded Y connections, applying Kirchhoff’s current law around the windings gives

I_a+ Ib+ Ic= 0 (3.41)

while for the grounded Y connection we get

Figure 3.22 Connection of the symmetrical component equivalent circuits for line-to-earth, line-to-line and line-to-line-to-earth faults

74 Embedded generation

I_a+ Ib+ Ic= In (3.42) The first row of the matrix eqn. (3.32) gives

I₀= –¹³(I_a+ Ib+ Ic) (3.43) Comparing eqns. (3.41) and (3.43) we conclude that a zero sequence current cannot flow in windings that are in a delta or ungrounded Y connection. The zero sequence equivalent Thevenin impedance of gener-ators and transformers whose windings are delta- or Y-connected is therefore infinite. On the other hand, eqn. (3.42) shows that a zero sequence current can flow in a grounded Y connection. Therefore, only the parts of the network that allow the circulation of a current through the earth should be included in the zero sequence equivalent circuit. If a fault occurs in a part of the network where there is no neutral-to-earth connection, the zero sequence equivalent impedance is infinite. Eqn.

(3.38) and Figure 3.22a show that in such a case, a line-to-earth fault would not result in any fault current. Since such a fault would be difficult to detect, such a situation is extremely hazardous. In most countries, networks must therefore be designed in such a way that a neutral point is connected to earth under all conceivable configurations.

Unbalanced faults are more common but usually less severe than three-phase faults. However, when the neutral point of a generator is connected to earth through a zero or low-impedance connection, the current in a single line-to-earth fault can exceed the current in a balanced three-phase fault. This can also occur on the Y-grounded side of a Δ–Y grounded transformer.

3.4.5 Application to an embedded generation scheme

To compute the fault levels for unbalanced faults, we must know the configuration of the transformer windings. Figure 3.24 gives that infor-mation for the small system that we have already used in previous examples. This combination of winding configurations is typical of what Figure 3.23 Types of transformer and generator winding connections

System studies 75

we might find in a distribution system with embedded generation. The low-voltage side of the 132/33 kV transformers connecting buses A and B is star-connected and the neutral point is connected to earth through a resistance R₀. The value of R₀ is selected so that a line-to-ground fault on the low-voltage side would give rise to a 1000 A fault current. In add-ition, we must know, for each system component, not only its positive sequence impedance but also its negative and zero sequence impedance.

While the negative sequence impedance of lines, cables and transformers is often assumed equal to the positive sequence impedance, the zero sequence impedance is usually significantly different. Table 3.3 gives these values for the relevant components of the small system of our example. Figure 3.24 gives two values of the fault level at each bus. The upper one corresponds to the case where the embedded generator at bus E is not connected to the system and the lower one to the case where it is connected. We observe that, contrary to the situation for balanced faults, the embedded generator does not significantly increase the fault level, except in the close vicinity of bus E. This is due to the fact that the fault current at buses B, C and D is limited mostly by the equivalent zero sequence impedance, which is dominated by the resistance R₀. A single line-to-ground fault at bus A would not produce any fault current as there is no path to earth represented in that part of the network. Finally, Figure 3.24 MVA fault levels for single line-to-ground faults at various buses in

the system. The upper number corresponds to the case where the embedded generator is not connected to the system and the lower number to the case where it is. The system is assumed to be unloaded prior to the fault

76 Embedded generation

it is interesting to note that the fault level at bus E is higher for a single line-to-ground fault than for a balanced three-phase fault.

3.4.6 Standards for fault calculations

The purpose of the previous sections was to introduce the basic concepts

The purpose of the previous sections was to introduce the basic concepts

In document Embedded Generation (Page 83-96)