4

Fault Analysis in Power Systems

Charles A. Gross

Auburn University

4.1    Simplifications in the System Model

4.2    The Four Basic Fault Types

The Balanced Three-Phase Fault • The Single Phase-to-Ground Fault • The Phase-to-Phase Fault • The Double Phase-to-Ground Fault

4.3    An Example Fault Study

Balanced Three-Phase Fault • Single Phase-to-Ground Fault

4.4    Further Considerations

4.5    Summary

4.6    Defining Terms

References

Further Information

A fault in an electrical power system is the unintentional and undesirable creation of a conducting path (a short circuit) or a blockage of current (an open circuit). The short-circuit fault is typically the most common and is usually implied when most people use the term fault. We restrict our comments to the short-circuit fault.

The causes of faults include lightning, wind damage, trees falling across lines, vehicles colliding with towers or poles, birds shorting out lines, aircraft colliding with lines, vandalism, small animals entering switchgear, and line breaks due to excessive ice loading. Power system faults may be categorized as one of four types: single line-to-ground, line-to-line, double line-to-ground, and balanced three-phase. The first three types constitute severe unbalanced operating conditions.

It is important to determine the values of system voltages and currents during faulted conditions so that protective devices may be set to detect and minimize their harmful effects. The time constants of the associated transients are such that sinusoidal steady-state methods may still be used. The method of symmetrical components is particularly suited to fault analysis.

Our objective is to understand how symmetrical components may be applied specifically to the four general fault types mentioned and how the method can be extended to any unbalanced three-phase system problem.

Note that phase values are indicated by subscripts, a, b, c; sequence (symmetrical component) values are indicated by subscripts 0, 1, 2. The transformation is defined by

$\left[\begin{array}{c}{\overline{V}}_a\\ {\overline{V}}_b\\ \overline{V}\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{c}{\overline{V}}_0\\ {\overline{V}}_1\\ {\overline{V}}_2\end{array}\right]=\left[T\right]\left[\begin{array}{c}{\overline{V}}_0\\ {\overline{V}}_1\\ {\overline{V}}_2\end{array}\right]$

4.1    Simplifications in the System Model

Certain simplifications are possible and usually employed in fault analysis.

•  Transformer magnetizing current and core loss will be neglected.

•  Line shunt capacitance is neglected.

•  Sinusoidal steady-state circuit analysis techniques are used. The so-called DC offset is accounted for by using correction factors.

•  Prefault voltage is assumed to be 1∠0° per-unit. One per-unit voltage is at its nominal value prior to the application of a fault, which is reasonable. The selection of zero phase is arbitrary and convenient. Prefault load current is neglected.

For hand calculations, neglect series resistance is usually neglected (this approximation will not be necessary for a computer solution). Also, the only difference in the positive and negative sequence networks is introduced by the machine impedances. If we select the subtransient reactance ${X^{″}}_d$ for the positive sequence reactance, the difference is slight (in fact, the two are identical for nonsalient machines). The simplification is important, since it reduces computer storage requirements by roughly one-third. Circuit models for generators, lines, and transformers are shown in Figures 4.1 through 4.3, respectively.

Our basic approach to the problem is to consider the general situation suggested in Figure 4.4a. The general terminals brought out are for purposes of external connections that will simulate faults. Note carefully the positive assignments of phase quantities. Particularly note that the currents flow out of the system. We can construct general sequence equivalent circuits for the system, and such circuits are indicated in Figure 4.4b. The ports indicated correspond to the general three-phase entry port of Figure 4.4a. The positive sense of sequence values is compatible with that used for phase values.

FIGURE 4.1  Generator sequence circuit models.

FIGURE 4.2  Line sequence circuit models.

FIGURE 4.3  Transformer sequence circuit models.

FIGURE 4.4  General fault port in an electric power system. (a) General fault port in phase (abc) coordinates; (b) corresponding fault ports in sequence (012) coordinates.

4.2    The Four Basic Fault Types

4.2.1 The Balanced Three-Phase Fault

Imagine the general three-phase access port terminated in a fault impedance $\left({\overline{Z}}_f\right)$ as shown in Figure 4.5a. The terminal conditions are

$\left[\begin{array}{c}{\overline{V}}_a\\ {\overline{V}}_b\\ {\overline{V}}_c\end{array}\right]=\left[\begin{array}{ccc}{\overline{Z}}_f& 0& 0\\ 0& {\overline{Z}}_f& 0\\ 0& 0& {\overline{Z}}_f\end{array}\right]\left[\begin{array}{c}{\overline{I}}_a\\ {\overline{I}}_b\\ {\overline{I}}_c\end{array}\right]$

Transforming to [Z₀₁₂],

$\left[Z_{012}\right]={\left[T\right]}^{-1}\left[\begin{array}{ccc}{\overline{Z}}_f& 0& 0\\ 0& {\overline{Z}}_f& 0\\ 0& 0& {\overline{Z}}_f\end{array}\right]\left[T\right]=\left[\begin{array}{ccc}{\overline{Z}}_f& 0& 0\\ 0& {\overline{Z}}_f& 0\\ 0& 0& {\overline{Z}}_f\end{array}\right]$

The corresponding network connections are given in Figure 4.6a. Since the zero and negative sequence networks are passive, only the positive sequence network is nontrivial.

${\overline{V}}_0={\overline{V}}_2=0$

(4.1)

${\overline{I}}_0={\overline{I}}_2=0$

(4.2)

${\overline{V}}_1={\overline{Z}}_f{\overline{I}}_1$

(4.3)

FIGURE 4.5  Fault types. (a) Three-phase fault; (b) single phase-to-ground fault; (c) phase-to-phase fault; (d) double phase-to-ground fault.

4.2.2    The Single Phase-to-Ground Fault

Imagine the general three-phase access port terminated as shown in Figure 4.5b. The terminal conditions are

$\begin{array}{ccc}{\overline{I}}_b=0& {\overline{I}}_c=0& {\overline{V}}_a={\overline{I}}_a{\overline{Z}}_f\end{array}$

Therefore,

${\overline{I}}_0+a^2{\overline{I}}_1+a{\overline{I}}_2={\overline{I}}_0+a{\overline{I}}_1+a^2{\overline{I}}_2=0$

or

${\overline{I}}_1={\overline{I}}_2$

Also,

${\overline{I}}_b={\overline{I}}_0+a^2{\overline{I}}_1+a{\overline{I}}_2={\overline{I}}_0+\left(a^2+a\right){\overline{I}}_1=0$

or

${\overline{I}}_0={\overline{I}}_1={\overline{I}}_2$

(4.4)

Furthermore, it is required that

$\begin{array}{l}{\overline{V}}_a={\overline{Z}}_f{\overline{I}}_a\hfill \\ {\overline{V}}_0+{\overline{V}}_1+{\overline{V}}_2=3{\overline{Z}}_f{\overline{I}}_1\hfill \end{array}$

(4.5)

FIGURE 4.6  Sequence network terminations for fault types. (a) Balanced three-phase fault; (b) single phase-toground fault; (c) phase-to-phase fault; (d) double phase-to-ground fault.

In general then, Equations 4.4 and 4.5 must be simultaneously satisfied. These conditions can be met by interconnecting the sequence networks as shown in Figure 4.6b.

4.2.3    The Phase-to-Phase Fault

Imagine the general three-phase access port terminated as shown in Figure 4.5c. The terminal conditions are such that we may write

$\begin{array}{ccc}{\overline{I}}_0=0& {\overline{I}}_b=-{\overline{I}}_c& {\overline{V}}_b={\overline{Z}}_f{\overline{I}}_b+{\overline{V}}_c\end{array}$

It follows that

${\overline{I}}_0+{\overline{I}}_1+{\overline{I}}_2=\overline{0}$

(4.6)

${\overline{I}}_0=0$

(4.7)

${\overline{I}}_1=-{\overline{I}}_2$

(4.8)

In general then, Equations 4.6 through 4.8 must be simultaneously satisfied. The proper interconnection between sequence networks appears in Figure 4.6c.

4.2.4    The Double Phase-to-Ground Fault

Consider the general three-phase access port terminated as shown in Figure 4.5d. The terminal conditions indicate

$\begin{array}{ccc}{\overline{I}}_a=0& {\overline{V}}_b={\overline{V}}_c& {\overline{V}}_b=\left({\overline{I}}_b+{\overline{I}}_c\right){\overline{Z}}_f\end{array}$

It follows that

${\overline{I}}_0+{\overline{I}}_1+{\overline{I}}_2=\overline{0}$

(4.9)

${\overline{V}}_1={\overline{V}}_2$

(4.10)

and

${\overline{V}}_0-{\overline{V}}_1=3{\overline{Z}}_f{\overline{I}}_0$

(4.11)

For the general double phase-to-ground fault, Equations 4.9 through 4.11 must be simultaneously satisfied. The sequence network interconnections appear in Figure 4.6d.

4.3    An Example Fault Study

Case: EXAMPLE SYSTEM

Run:

System has data for two Line(s); two Transformer(s);

four Bus(es); and two Generator(s)

Transmission line data

Transformer data

Generator data

Zero sequence {Z} matrix

0.0 + j(0.1144)	0.0 + j(0.0981)	0.0 + j(0.0163)	0.0 + j(0.0000)
0.0 + j(0.0981)	0.0 + j(0.1269)	0.0 + j(0.0212)	0.0 + j(0.0000)
0.0 + j(0.0163)	0.0 + j(0.0212)	0.0 + j(0.0452)	0.0 + j(0.0000)
0.0 + j(0.0000)	0.0 + j(0.0000)	0.0 + j(0.0000)	0.0 + j(0.1700)

Positive sequence [Z] matrix

0.0 + j(0.1310)	0.0 + j(0.1138)	0.0 + j(0.0862)	0.0 + j(0.0690)
0.0 + j(0.1138)	0.0 + j(0.1422)	0.0 + j(0.1078)	0.0 + j(0.0862)
0.0 + j(0.0862)	0.0 + j(0.1078)	0.0 + j(0.1422)	0.0 + j(0.1138)
0.0 + j(0.0690)	0.0 + j(0.0862)	0.0 + j(0.1138)	0.0 + j(0.1310)

The single-line diagram and sequence networks are presented in Figure 4.7.

Suppose bus 3 in the example system represents the fault location and ${\overline{Z}}_f=0$. The positive sequence circuit can be reduced to its Thévenin equivalent at bus 3:

$\begin{array}{cc}{E}_{T1}=1.0\angle 0°& {\overline{Z}}_{T1}=j0.1422\end{array}$

FIGURE 4.7  Example system. (a) Single-line diagram; (b) zero sequence network; (c) positive sequence network; (d) negative sequence network.

FIGURE 4.8  Example system faults at bus 3. (a) Balanced three-phase; (b) single phase-to-ground; (c) phase-to-phase; (d) double phase-to-ground.

Similarly, the negative and zero sequence Thévenin elements are

$\begin{array}{cc}{\overline{E}}_{T2}=0& {\overline{Z}}_{T2}=j0.1422\\ {\overline{E}}_{T0}=0& Z_{T0}=j0.0452\end{array}$

The network interconnections for the four fault types are shown in Figure 4.8. For each of the fault types, compute the currents and voltages at the faulted bus.

4.3.1    Balanced Three-Phase Fault

The sequence networks are shown in Figure 4.8a. Obviously,

$\begin{array}{l}{\overline{V}}_0={\overline{I}}_0={\overline{V}}_2={\overline{I}}_2=0\hfill \\ {\overline{I}}_1=\frac{1\angle 0°}{j0.1422}=-j7.032;\text{also}{\overline{V}}_1=0\hfill \end{array}$

To compute the phase values,

$\begin{array}{l}\left[\begin{array}{c}{\overline{I}}_a\\ {\overline{I}}_b\\ {\overline{I}}_c\end{array}\right]=\left[T\right]\left[\begin{array}{c}{\overline{I}}_0\\ {\overline{I}}_1\\ {\overline{I}}_2\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{c}0\\ -j7.032\\ 0\end{array}\right]=\left[\begin{array}{c}7.032\angle -90°\\ 7.032\angle 150°\\ 7.032\angle 30°\end{array}\right]\hfill \\ \left[\begin{array}{c}{\overline{V}}_a\\ {\overline{V}}_b\\ {\overline{V}}_c\end{array}\right]=\left[T\right]\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\hfill \end{array}$

4.3.2    Single Phase-to-Ground Fault

The sequence networks are interconnected as shown in Figure 4.8b.

$\begin{array}{l}{\overline{I}}_0={\overline{I}}_1={\overline{I}}_2=\frac{1\angle 0°}{j0.0452+j0.1422+j0.1422}=-j3.034\hfill \\ \left[\begin{array}{c}{\overline{I}}_a\\ {\overline{I}}_b\\ {\overline{I}}_c\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{c}-j3.034\\ -j3.034\\ -j3.034\end{array}\right]=\left[\begin{array}{c}-j9.102\\ 0\\ 0\end{array}\right]\hfill \end{array}$

The sequence voltages are

$\begin{array}{l}{\overline{V}}_0=-j0.0452\left(-j3.034\right)=-1371\hfill \\ {\overline{V}}_1=1.0-j0.1422\left(-j3.034\right)=0.5685\hfill \\ {\overline{V}}_2=-j0.1422\left(-j3.034\right)=-0.4314\hfill \end{array}$

The phase voltages are

$\left[\begin{array}{c}{\overline{V}}_a\\ {\overline{V}}_b\\ {\overline{V}}_c\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{c}-0.1371\\ 0.5685\\ -0.4314\end{array}\right]=\left[\begin{array}{c}0\\ 0.8901\angle -103.4°\\ 0.8901\angle -103.4°\end{array}\right]$

Phase-to-phase and double phase-to-ground fault values are calculated from the appropriate networks (Figure 4.8c and d). Complete results are provided.

 

Faulted Bus	Phase a	Phase b	Phase c
3	G	G	G

Sequence voltages

Phase voltages

Sequence currents

 

Faulted Bus	Phase a	Phase b	Phase c
3	G	G	G

Phase currents

Faulted Bus	Phase a	Phase b	Phase c
3	G	0	0

Sequence voltages

Phase voltages

Sequence currents

Faulted Bus	Phase a	Phase b	Phase c
3	G	0	0

Phase currents

Faulted Bus	Phase a	Phase b	Phase c
3	0	C	B

Sequence voltages

Phase voltages

Sequence currents

Faulted Bus	Phase a	Phase b	Phase c
3	0	C	B

Phase currents

Faulted Bus	Phase a	Phase b	Phase c
3	0	G	G

Sequence voltages

Phase voltages

Sequence currents

Faulted Bus	Phase a	Phase b	Phase c
3	0	G	G

Phase currents

4.4    Further Considerations

Generators are not the only sources in the system. All rotating machines are capable of contributing to fault current, at least momentarily. Synchronous and induction motors will continue to rotate due to inertia and function as sources of fault current. The impedance used for such machines is usually the transient reactance $X_d^{\prime }$ or the subtransient $X_d^{″}$, depending on protective equipment and speed of response. Frequently, motors smaller than 50 hp are neglected. Connecting systems are modeled with their Thévenin equivalents.

Although we have used AC circuit techniques to calculate faults, the problem is fundamentally transient since it involves sudden switching actions. Consider the so-called DC offset current. We model the system by determining its positive sequence Thévenin equivalent circuit, looking back into the positive sequence network at the fault, as shown in Figure 4.9. The transient fault current is

$i\left(t\right)=I_{\text{AC}}\sqrt{2}\cos \left(\text{ω}t-\text{β}\right)+I_{\text{DC}}{e}^{-t/\text{τ}}$

This is a first-order approximation and strictly applies only to the three-phase or phase-to-phase fault. Ground faults would involve the zero sequence network also.

$\begin{array}{l}{I}_{\text{AC}}=\frac{E}{\sqrt{R^2+X^2}}=\text{rms AC current}\hfill \\ I_{\text{DC}}\left(t\right)=I_{\text{DC}}{e}^{-t/\text{τ}}=\text{DC offset current}\hfill \end{array}$

The maximum initial DC offset possible would be

$\text{Max}{I}_{\text{DC}}=I_{\max }=\sqrt{2}{I}_{\text{AC}}$

The DC offset will exponentially decay with time constant τ, where

$\text{τ}=\frac{L}{R}=\frac{X}{\text{ω}R}$

The maximum DC offset current would be I_DC(t)

$I_{\text{DC}}\left(t\right)=I_{\text{DC}}{e}^{-t/\text{τ}}=\sqrt{2}{I}_{\text{AC}}{e}^{-t/\text{τ}}$

The transient rms current I(t), accounting for both the AC and DC terms, would be

$I\left(t\right)=\sqrt{I_{\text{AC}}^2+I_{\text{DC}}^2\left(t\right)}=I_{\text{AC}}\sqrt{1+2e^{-2t/\text{τ}}}$

FIGURE 4.9  Positive sequence circuit looking back into faulted bus.

Define a multiplying factor k_i such that I_AC is to be multiplied by k_i to estimate the interrupting capacity of a breaker which operates in time T_op. Therefore,

$k_i=\frac{I\left(T_{\text{op}}\right)}{I_{\text{AC}}}=\sqrt{1+2e^{-2T_{\text{op}}/\text{τ}}}$

Observe that the maximum possible value for k_i is √3.

The Thévenin equivalent at the fault point is determined by normal sinusoidal steady-state methods, resulting in a first-order circuit as shown in Figure 4.9. While this provides satisfactory results for the steady-state component I_AC, the X/R value so obtained can be in serious error when compared with the rate of decay of I(t) as measured by oscillographs on an actual faulted system. The major reasons for the discrepancy are, first of all, that the system, for transient analysis purposes, is actually high-order, and second, the generators do not hold constant impedance as the transient decays.

4.5    Summary

Computation of fault currents in power systems is best done by computer. The major steps are summarized below:

•  Collect, read in, and store machine, transformer, and line data in per-unit on common bases.

•  Formulate the sequence impedance matrices.

•  Define the faulted bus and Z_f. Specify type of fault to be analyzed.

•  Compute the sequence voltages.

•  Compute the sequence currents.

•  Correct for wye-delta connections.

•  Transform to phase currents and voltages.

For large systems, computer formulation of the sequence impedance matrices is required. Refer to Further Information for more detail. Zero sequence networks for lines in close proximity to each other (on a common right-of-way) will be mutually coupled. If we are willing to use the same values for positive and negative sequence machine impedances,

$\left[Z_1\right]=\left[Z_2\right]$

Therefore, it is unnecessary to store these values in separate arrays, simplifying the program and reducing the computer storage requirements significantly. The error introduced by this approximation is usually not important. The methods previously discussed neglect the prefault, or load, component of current; that is, the usual assumption is that currents throughout the system were zero prior to the fault. This is almost never strictly true; however, the error produced is small since the fault currents are generally much larger than the load currents. Also, the load currents and fault currents are out of phase with each other, making their sum more nearly equal to the larger components than would have been the case if the currents were in phase. In addition, selection of precise values for prefault currents is somewhat speculative, since there is no way of predicting what the loaded state of the system is when a fault occurs. When it is important to consider load currents, a power flow study is made to calculate currents throughout the system, and these values are superimposed on (added to) results from the fault study.

A term which has wide industrial use and acceptance is the fault level or fault MVA at a bus. It relates to the amount of current that can be expected to flow out of a bus into a three-phase fault. As such, it is an alternate way of providing positive sequence impedance information. Define

$\begin{array}{ll}\text{Fault level in MVA at bus }i\hfill & =V_{i_{{\text{pu}}_{\text{nominal}}}}{I}_{i_{{\text{pu}}_{\text{fault}}}}{S}_{3{\varphi }_{\text{base}}}\hfill \\ \hfill & =\left(1\right)\frac{1}{Z_{ii}^1}{S}_{3{\varphi }_{\text{base}}}=\frac{S_{3{\varphi }_{\text{base}}}}{Z_{ii}^1}\hfill \end{array}$

Fault study results may be further refined by approximating the effect of DC offset.

The basic reason for making fault studies is to provide data that can be used to size and set protective devices. The role of such protective devices is to detect and remove faults to prevent or minimize damage to the power system.

4.6    Defining Terms

DC offset—The natural response component of the transient fault current, usually approximated with a first-order exponential expression.

Fault—An unintentional and undesirable conducting path in an electrical power system.

Fault MVA—At a specific location in a system, the initial symmetrical fault current multiplied by the prefault nominal line-to-neutral voltage (×3 for a three-phase system).

Sequence (012) quantities—Symmetrical components computed from phase (abc) quantities. Can be voltages, currents, and/or impedances.

References

Anderson P.M., Analysis of Faulted Power Systems, Ames, IA: Iowa State Press, 1973.

Elgerd O.I., Electric Energy Systems Theory: An Introduction, 2nd edn., New York: McGraw-Hill, 1982.

El-Hawary M.E., Electric Power Systems: Design and Analysis, Reston, VA: Reston Publishing, 1983.

El-Hawary M.E., Electric Power Systems, New York: IEEE Press, 1995.

General Electric, Short-circuit current calculations for industrial and commercial power systems, Publication GET-3550.

Gross C.A., Power System Analysis, 2nd edn., New York: Wiley, 1986.

Horowitz S.H., Power System Relaying, 2nd edn., New York: Wiley, 1995.

Lazar I., Electrical Systems Analysis and Design for Industrial Plants, New York: McGraw-Hill, 1980.

Mason C.R., The Art and Science of Protective Relaying, New York: Wiley, 1956.

Neuenswander J.R., Modern Power Systems, Scranton, PA: International Textbook, 1971.

Stagg G. and El-Abiad A.H., Computer Methods in Power System Analysis, New York: McGraw-Hill, 1968.

Westinghouse Electric Corporation, Applied Protective Relaying, Newark, NJ: Relay-Instrument Division, 1976.

Wood A.J., Power Generation, Operation, and Control, New York: Wiley, 1996.

Further Information

For a comprehensive coverage of general fault analysis, see Paul M. Anderson, Analysis of Faulted Power Systems, New York: IEEE Press, 1995. Also see Chapters 9 and 10 of Power System Analysis by C.A. Gross, New York: Wiley, 1986.