Short Circuit Current Calculation Theory & Techniques (3).pdf

CHAPTER 2. SHORT CIRCUIT CURRENT CALCULATION THEORY AND TECHNIQUES

2.1 Introduction Power system short circuit levels must be determined in order to specify the required interrupting capacity at various points on the system and also to evaluate the earthing provisions at substations. Specifically fault current calculations are required: „

To establish short circuit ratings of interrupting devices, such as circuit breakers and fuses.

„

To determine the short-time or withstand thermal rating of system components, such as cables, transformers, reactors, etc.

„

To calculate settings of short circuit protective devices, such as direct-acting trips, fuses and relays.

„

To evaluate voltage levels in the system under the fault condition.

„

To determine the amount of current flow through the earthing structure and to calculate the ground potential rise (GPR).

Generally, short-circuit practices are established to solve the first four problems, particularly circuit-breaker sizing. A well-known IEEE/ANSI standard [1] specially caters for this. This standard enables suitable circuit-breakers and interrupting devices to be selected on the basis of the magnitude of symmetrical short circuit current only. The asymmetric nature of the short circuit current has been analysed in various IEEE papers [2, 3, 4]. The IEEE/ANSI standards [1, 5, 6] set out a methodology for considering the total asymmetrical capability of the switchgear. In addition to that, one of the IEEE/ANSI standards [7] is specially designed to deal with total (asymmetrical) current based circuit-breakers.

Short Circuit Current Calculation Theory and Techniques

On the other hand, IEC 909 [8] is a standard devoted to calculating the short circuit current itself and pays particular attention to establishing the dynamic wave shape of the current. It also takes into account asymmetry as a function of the system R/X ratio. Both the IEC standard and IEEE/ANSI approach are based on simple quasi steadystate calculation techniques and employ various empirical factors. Recent studies [9, 10, 11] have shown that both the IEC standard and the IEEE/ANSI approach provide highly conservative results when compared to detailed dynamic calculation procedures. The need for more accurate fault calculation algorithms has been the subject of some recent publications [12, 13]. In general, the three-phase fault is accepted to be the most severe one. The severity of fault depends on the fault location. However it is possible for the magnitude of the single line-to-earth fault current to be greater than the three-phase fault. Generally, line-to-line fault currents on the system are about 87% of the three-phase value, while line-to-earth fault currents can range from about 25% to 125% of the three-phase value [14]. For that reason, three-phase and single line-to-earth fault analyses are the most important for the purpose of the interrupting duty calculation. The faults which occur most frequently in power systems are unbalanced and predominantly the single line-to-earth fault. The typical occurrence frequency for threephase, double line-to-earth, line-to-line and single line-to-earth faults is 5%, 10%, 15% and 70%, respectively [15]. In the literature [9, 10, 11], the three-phase fault is by far the most analysed and the accuracy of the standards in terms of predicting three-phase fault currents is well tested. Additionally, single line-to-earth faults that are ‘far from the generator’ (i.e. without decrement of the symmetrical component of the short circuit current) have been analysed and various comparisons made [16]. Additionally, both IEC and IEEE/ANSI standards have been compared in various papers [17, 18, 19]. ψψ Recent developments in computing technology have facilitated the implementation of more detailed fault calculation methods, and consequently a series of new methods which are aimed to represent the network and its components in more detail have been proposed. Some only determine the d.c. component of fault current for single or three phase faults [20]. On the other hand, other methods are limited to three-phase balanced faults [21]. An approach which is advocated by Halpin et al [22, 23] is claimed to be valid for all type of faults. Unfortunately, no results are given in the papers for single line-to-earth types of fault.

Power System Transients and Earthing Systems

2-2

Short Circuit Current Calculation Theory and Techniques

Accordingly, this chapter will examine the short circuit calculation in terms of „

basic calculation methodology,

„

the various factors which give rise to and affect the transient nature of a.c. and d.c. components of short circuit current,

„

the IEC and IEEE/ANSI standard approaches.

2.2 Steady-State Short Circuit Current Calculation Methodology In the reviewed literature [24, 25, 26], two main methods are available to calculate the rms value of the symmetrical short circuit current. These are called method one and two. Both methods are applications of the classical network theorem of superposition. Generally, method one is used to evaluate the generator current of the faulted system and its distribution over the network’s branches while method two is used to calculate the fault current at the faulted point directly. An overview of the methods is given here considering a simple network which consists of a generator, a delta/star connected step-up transformer and a passive load fed by an overhead line as shown in Figure 1. one Source

two

three Line

D/Y

Load

V Fault

Figure 1

Example network.

Both methods establish positive, negative and zero sequence networks according to Fortescue [27] transformation due to the unbalanced nature of the single line-to-earth type of fault. The sequence networks for Figure 1 are shown in Figure 2.

Power System Transients and Earthing Systems

2-3

Short Circuit Current Calculation Theory and Techniques

Zg1

Positive sequence network

Zg2

one

Zt1

two

V1.

one

Zl1

three

V2

Zt2

two

V3

Zl2

three

Negative sequence network

Zt0

two

Zl0

Zload

three

Zero sequence network

Figure 2

Zload

Zload

Sequence networks relating to network in Figure 1.

2.2.1 Method One The impedances of the negative and zero sequence networks are lumped as Z2 and Z0, respectively as shown in the Figure 3.

Power System Transients and Earthing Systems

2-4

Short Circuit Current Calculation Theory and Techniques

Zg

Zt

E

one

Zl three

two

Zload

Ig

If

Z2

Z0

Figure 3

The network according to method one.

Internal voltages of all generators and motors are determined from the pre-fault network motor/generator currents, which may be calculated by running a load flow calculation, and then employing

VN = E + I1ZG -1 where VN is the pre-fault voltage at the generator terminal, I1 is the generator or motor current prior to the fault. ZG is the impedance of the generator and E is the internal voltage of the generator. Ig and If refer to the total current of the generator and the fault current of the particular nodes, respectively. The generator current, fault current and load current can be calculated by employing suitable network solution methods.

2.2.2 Method Two In this method, it is necessary to determine the pre-fault voltage at the faulted node in order to calculate the Thevenin equivalent voltage source of the faulted node. The prefault voltage of the node can be determined by running a load flow study before the short circuit current calculation. According to this method, the pre-fault voltage acts as the only voltage source of the system. Therefore, internal voltages of generators and motors are ignored. However, impedances of the system sources should be considered and motors whose contributions to the short circuit current exist up to the end of the transient period, can be taken into account for the initial short circuit calculation. As a result, the network is greatly simplified as shown in Figure 4.

Power System Transients and Earthing Systems

2-5

Short Circuit Current Calculation Theory and Techniques

Zt

Zg one

Zl two

three

Zload If

Z2 V2 Z0

Figure 4

The network for method two.

The single line-to-earth fault current may be calculated using

If = 3 (V2/Zeq) -2 where V2 is the phase to neutral voltage of the faulted node prior to the fault, Zeq is the equivalent impedance of the system as seen by the faulted point, and If is the symmetrical fault current. Since the calculation of internal voltages of generators and motors is not necessary, this method is preferable to method one.

2.2.3 The Simplified Method Although method two is much simpler than method one, it is still necessary to run a load flow program before the short circuit calculation in order to determine pre-fault network conditions. Obviously, this requires extra calculation effort and knowledge of pre-fault network load conditions. It has been shown [28] that if pre-fault network loads are considered, a pessimistic load flow calculation should be introduced in order to calculate the maximum possible short circuit current. If it is wished to avoid the need for running a pre-fault load flow calculation, a simplified approach is established which builds in conservative assumptions. According to this approach, all non-rotating system loads and shunts, such as line capacitors and transformer magnetising impedances are ignored. Also, pre-fault loading of motors is neglected. Accordingly, there is zero pre-fault current and therefore Power System Transients and Earthing Systems

2-6

Short Circuit Current Calculation Theory and Techniques

equal voltages at all system nodes. Thus, internal voltages of generators and motors are chosen to be equal to one per unit with zero phase angle for the initial value of the short circuit current. Depending upon the time constants of the particular devices and the fault duration, the contribution of motors may be excluded from the steady-state value of the short circuit current. If more conservative results are required, viz. the calculation of maximum short circuit current values, a voltage factor is introduced by taking into account the voltage tolerance of the system and the network structure such as tap settings. Table 1 presents the voltage factor, c, [29] from IEC 909. On the other hand, a nominal one per unit voltage is assumed to be the short circuit voltage for most ANSI/IEEE [1] calculations. However, it is also stated in the IEEE Buff book [5] that a correction can be made by considering the operating voltage of the faulted bus bar. As a result of these assumptions the calculation procedure is greatly simplified and a load flow study is not essential. Hence, in IEC 909 the short circuit current is given by the following equation.

I "k1 =

3cU n Z1 + Z 2 + Z 0 -3

Where I”k1 is the initial single line-to-earth symmetrical short circuit current, Z1, Z2 and

Z0 are the total sub-transient short circuit impedances of the sequence networks as seen by the faulted point, c is the voltage factor and Un is the rated voltage of the system. Table 1

Voltage factor, c, according to IEC 909. Voltage factor c for the calculation of Nominal voltage (Un)

Maximum short circuit current (cmax)

Minimum short circuit current (cmin)

Low voltage 100 V to 1000 V 230/400 V

1.00

0.95

Other voltages

1.05

1.00

Medium voltage > 1 kV to 35 kV

1.10

1.00

High voltage > 35 kV to 230 kV

1.10

1.00

Power System Transients and Earthing Systems

2-7

Short Circuit Current Calculation Theory and Techniques

2.3 The Transient Variations of Short Circuit Current Short-circuits in electrical power systems cause electromechanical and electromagnetic transient disturbances, which depend on the magnitude and time variation of the current. Generally, the transient nature of the short circuit current depends on: „

Available short circuit current sources, which are principally the synchronous machines. Asynchronous generators and motors also have a significant effect on the initial short circuit currents.

„

The instant at which the short circuit occurs, relative to the voltage wave form. This has a considerable effect upon the peak value of the short circuit current.

„

The location of the short circuit, which has influence mainly on the severity of the short circuit current, and also determines whether the synchronous machines have a greater or lesser effect on the short circuit current time variations.

„

The duration of the short circuit, which is significant with regard to determining the effects of additional control systems upon the short circuit current.

„

The structure of the system, whether meshed or radial, which defines the path of the short circuit current and the effectiveness of the parallel impedances.

„

The system impedance, which not only determines the magnitude of the short circuit current but also has a significant effect on the transient nature of the current.

„

The initial loading condition of the system, which determines the effective voltage of the system and also establishes possible short circuit current sources.

The extent of the transient nature of the short circuit current is mainly determined according to the location of short circuit with respect to the synchronous machines of the system. Therefore in IEC 909, the evaluation of short circuit current is classified in two groups related to fault location. These are the ‘near to generator/motor’ and ‘far from generator/motor’ position. Also, the IEEE/ANSI approach defines the fault regarding its location as local or remote. Through this classification, the IEC standard

Power System Transients and Earthing Systems

2-8

Short Circuit Current Calculation Theory and Techniques

and IEEE/ANSI approach simplify and consequently introduce a further approximation to the calculation of short circuit current. On the other hand, ER G74 [30] proposes a computer-based method and treats the short circuit current without such positional discrimination when a greater precision is needed.

2.3.1 Short Circuit Far from the Generator The IEC standard defines the ‘far from the generator’ short circuit in clause 3.18 as; “A short circuit during which the magnitude of the symmetrical a.c. component of prospective (available) short circuit current remains constant.” In this case, the short circuit voltage source is assumed to be constant. Therefore, the rms and peak to peak values of symmetrical short circuit current are constant during the short circuit. An illustrative waveform of the short circuit current is shown in Figure

Current in A.

5.

Time in s.

Symmetrical Figure 5

DC

Total

Short circuit current as a function of time for far from the generator fault.

The rms value of the symmetrical component of the short circuit current may be determined by either method one or two, which have been presented in section 2.2. Since the short circuit impedance proportionally has a large amount of inductive reactance, the X/R ratio of the system is high and there will be a transient unidirectional current. Additionally, the initial value of this d.c. component may be of the same order of magnitude as the initial peak current of the short circuit. The basic mathematical analysis of a circuit containing linear resistance and reactance gives the d.c. or asymmetrical component of the current by the following equation [24, 31]

Power System Transients and Earthing Systems

2-9

Short Circuit Current Calculation Theory and Techniques

idc = 2 I " k [ e ( − t / Ta ) sin(ψ − ϕ k )] -4 where, I”k is the rms value of the initial symmetrical short circuit current. ψ is the initiation angle related to the initiation time of the short circuit and measured with respect to voltage zero crossing. φk is the phase angle, t is time, and lastly, Ta is the d.c. time constant which is related to the system impedance and expressed by the following equation [14]

Ta = L2/R1 ≈ L1/R1 -5 where, L2 is the negative sequence inductance of the system. R1 is the effective positive sequence resistance of the system, and L1 is the positive sequence system inductance. A typical value of d.c. time constant for a synchronous machine is 0.015s [14, 25, and 34]. It is assumed in one reference that the value of this constant remains unchanged for all types of fault [25]. However, a different reference calculates this constant for each type of fault [32]. The total instantaneous value of the short circuit current may be written by the following equation

i k = i ac + i dc =

2 I " k [ sin(ω t + ψ − ϕ k ) − e ( − t / Ta ) sin(ψ − ϕ k )] -6

Referring to the assumption that the short circuit voltage is constant, it can be stated that all the parameters of equation - 6, except initiation angle ψ, angular velocity ω and time t, are a function of the short circuit impedance of the system. However the initiation angle ψ depends upon the instant of occurrence of the fault. The maximum d.c. offset occurs for an initiation angle given by

ψDCmax = φk – π/2 -7 Generally, short circuits occur when a system is loaded. Therefore, for accurate calculation, pre-fault load current and the effect of passive load impedance as part of the short circuit impedance should be considered. Therefore, equation - 6 should be generalised to consider pre-fault load conditions. Hence

Power System Transients and Earthing Systems

2-10

Short Circuit Current Calculation Theory and Techniques

i k = iac + idc + ib

= 2 [ I " k sin(ωt + ψ + ϕ k ) + I "b sin(ψ + ϕ )e ( − t / Ta ) − I " k sin(ψ +ϕ k )e ( − t / Ta ) ] -8

where, Ib and φ are the rms value of the load current and the pre-fault phase angle, respectively. The first term of equation - 8 is related to the instantaneous value of the symmetrical a.c. short circuit current, the second term corresponds to the instantaneous value of the load current and the third term represents the d.c. component. The last two terms decay with the related d.c. time constant.

2.3.1.1

Effects of the Initiation Angle

Although the effect of the initiation angle is well known [25, 31], there does not appear to be in the published literature details of the proportion of the d.c. component to the total asymmetrical current as a function of initiation angle ψ. Accordingly, a simple parametric study has been carried out, considering a fault on the network shown in Figure 1. A single line-to-earth fault is assumed to occur at bus-bar two. The d.c. component of fault current is presented in Figure 6 for different initiation angles.

6.0E+04

Current [A]

3.0E+04

0.0E+00 0

0.05

0.1

0.15

0.2

0.25

0.3

-3.0E+04

-6.0E+04 Time [s]

Initiation Angle 0 Initiation Angle 90

Figure 6

Initiation Angle 30 Initiation Angle 120

Initiation Angle 60 Initiation Angle 180

D.C. component of single line-to-earth fault current for different initiation

angles. Since the short circuit angle φk, generally is close to 90 degrees, the maximum asymmetry occurs around zero degrees of initiation angle. In order to consider more precisely the effect of initiation angle, a passive load with a constant 0.8 power factor has been modelled and short circuit current assumed to be

Power System Transients and Earthing Systems

2-11

Short Circuit Current Calculation Theory and Techniques

five times higher than nominal load current. The short circuit current angle is kept constant at 83 degrees. The initiation angle is varied from zero to 180 degrees. The dependence of the initial value of d.c. component of the current on the initiation angle ψ of the fault is presented in Figure 7.

Initial d.c. component of short circuit current [pu]

1

0.5

0 0

1.5

3

-0.5

-1 Initiation angle [rad]

Figure 7

Effects of the initiation angle on initial value of fault current.

The initiation angle at which the highest d.c. offset occurs is related to the load condition and may be determined by the following equation [25].

ψ DC max = tan −1

sin(ϕ k − ϕ ) I " k / I b − cos(ϕ k − ϕ ) -9

In the case shown in Figure 7, ψDCmax is equal to 9.5 degrees. However, it should be noted that the maximum d.c. transient does not necessarily cause a maximum peak in the total asymmetrical current. It has been shown [33] that the maximum peak in total asymmetrical current always occurs for fault initiation at zero crossing voltage. On the other hand, maximum d.c. transient current occurs for the fault initiation time at which the symmetrical component of the short circuit current is at its peak. This angle ψDCmax is always different from zero and is a function of the short circuit angle as given in equation - 7 or - 9.

Power System Transients and Earthing Systems

2-12

Short Circuit Current Calculation Theory and Techniques

To illustrate this difference, Figure 8 shows the total asymmetrical current and d.c. component of current for the two particular fault initiation angles viz. ψ = zero and ψ =

ψDCmax. Although the d.c. component is higher for the later condition this does not result in a higher peak in total current.

Current [A]

5.0E+03

3.0E+03

1.0E+03

0

0.005

0.01

0.015

0.02

-1.0E+03 Time [s] Total Current with maximum peak Total Current with maximum transient

Figure 8

DC component with maximum peak DC component with maximum transient

Comparison of peak currents.

It is evident that the evaluation of fault current magnitude at ψDCmax is not of concern for fault interruption assessment. However when the maximum rms current is required for earthing assessments it would be necessary, if the worst case is to be considered, to calculate the current at ψDCmax.

2.3.2 Short Circuit Near to Generator In the case of ‘near to generator’ short circuits, the alternating component of the short circuit current also decays with time since it is under the influence of near-by generators or motors. This effect is recognised in the fault current calculation standards and hence a broad treatment of this phenomenon based on semi-empirical tables and graphs is provided for the three-phase balanced fault. The ‘near to generator’ type of fault is defined as: “A short circuit to which at least one synchronous machine contributes a prospective initial symmetrical short circuit current which is more than twice the generator’s rated current, or a short circuit to which synchronous or asynchronous motors contribute more than 5% of the initial

Power System Transients and Earthing Systems

2-13

Short Circuit Current Calculation Theory and Techniques

symmetrical short circuit current without motors.” However in IEC 909 this effect is completely neglected for the single line-to-earth type of fault. Illustrative waveforms of current for this condition can be seen in Figure 9 where the a.c. component of the current is under the influence of the near-by generator and it has

Current in A.

a decaying component in relation to the generator time constants.

Time in s.

Symmetrical Figure 9

DC

Total

Time dependent short circuit current with a.c. and d.c. components.

The d.c. component of the current may be analysed as in the previous section of this chapter. Additionally, equation - 5 which is given in section 2.3.1 remains valid. The symmetrical decrement of the short circuit current, which is mainly due to the synchronous machines of the system [14, 25] will be analysed in the following section.

2.3.2.1

Short Circuit on Generator Terminal Without Load

The transient behaviour of the short circuit current of a synchronous machine which is subject to a sudden short circuit is a well known phenomenon and is dealt with in detail by Kimbark [34] and Concordia [35]. In order to characterise the transient behaviour of single line-to-earth fault current, the three-phase balanced fault will be first investigated. The short circuit currents of a synchronous machine which is subject to a sudden three-phase short circuit are presented in Figure 10.

Power System Transients and Earthing Systems

2-14

Short Circuit Current Calculation Theory and Techniques

Figure 10 Short circuit current of synchronous machine (reproduced from 34). Each phase current in Figure 10 has a symmetrical and asymmetrical or d.c. component. The envelope of the symmetrical component is large at the first instant and decays eventually to a sustained (steady state) value. If the sustained component of the symmetrical current is subtracted, the remaining current can be found to have two exponential components namely; sub-transient and transient components. The rms symmetrical short circuit current can be formulated with a close approximation to

I ac = ( I "k − I ' k ) e − t / T "d + ( I ' k − I k ) e − t / T 'd + I k - 10 where Iac is the symmetrical rms fault current, I’k is the symmetrical transient current, I”k is the symmetrical sub-transient current, T’d and T”d are the transient and subtransient d axis short circuit time constants of the machine, respectively. The time constants of the machine will be discussed later in section 2.3.2.2. Since there is no load on the machine terminal, the effective voltage of each stage, namely the transient, sub-transient and steady state voltage, is equal to the phase to neutral voltage of the machine. Additionally, since the resistance of the machine is assumed to be constant throughout each stage, the decaying characteristics of short

Power System Transients and Earthing Systems

2-15

Short Circuit Current Calculation Theory and Techniques

circuit current may be explained by an increase of machine reactance. The associated reactance in each stage is known as: „

Synchronous reactance

„

Transient reactance

„

Sub-transient reactance.

Thus, the rms value of currents may be expressed in terms of machine reactances

Ik = Un/Xd

I’k = Un/X’d

I”k = Un/X”d - 11

where Un is the terminal voltage and equal to the internal voltage of the machine. Xd,

X’d and X”d are the machine reactances of corresponding stages. Thus, the total instantaneous short circuit current with symmetrical and asymmetrical components, can be expressed by t   − Ta i k = 2  I ac sin( ωt + ψ − ϕ k ) − I " k e sin( ψ − ϕ k )   

- 12 Substituting equation - 11 into equation - 12 yields t t t  1  1  − T "d  1 1  − T 'd 1  1 − Ta i k = 2U n  − + − + e sin ϕ k  e e  sin( ωt − ϕ k ) + Xd  X "d  X 'd X d   X " d X ' d  

- 13 The first term of the equation is the sub-transient component of the short circuit current, the second term is the transient component, the third is the sustained component and the last one is the d.c. component. Therefore, the waveform of the short circuit current can be expressed completely only using the machine reactances. In equation - 13, the initiation angle, ψ is assumed to be zero.

2.3.2.2

Time Constants

It has been shown that the time constants of the machine are dependent on machine impedances [36] and may be determined by plotting the short circuit current on logarithmic graph [34].

Power System Transients and Earthing Systems

2-16

Short Circuit Current Calculation Theory and Techniques

The typical value of the sub-transient short circuit time constant is approximately 0.04s. On the other hand, the transient time constant is much longer than the sub-transient one and its value is about 1.5s [14, 25]. Each constant can be calculated by

T”d = T”d0 (X”d/X’d) T’d = T’d0 (X’d/Xd) - 14 where T”d0 and T’d0 are sub-transient and transient open circuit time constants of the synchronous machine. However, all the time constants are highly dependent upon external impedances if the fault occurs remote from the machine’s terminal. The given time constants are valid for the direct axis representation of the machine, while a similar treatment may be used for the quadrature axis of the machine. Concardia [35] calculates these time constants for the single line-to-earth type fault and provides the following equations.

T”d = T”d0 [(X”d+X2+X0)/(X’d+X2+X0)] T’d = T’d0 [(X’d+X2+X0)/(Xd+X2+X0)] - 15 where X2 and X0 refer to negative and zero sequence reactances of the machine, respectively.

2.3.2.3

Short Circuit with Load

While the generator is feeding a load at a given power factor with a constant terminal voltage, the effective (internal) voltage of the generator differs from the voltage available on the machine terminal due to the voltage drop across the machine impedances. Therefore, the effective voltage of the generator needs to be related to sub-transient, transient and steady state stage values of the machine reactances. Internal voltages of the machine may be calculated for each stage by employing related reactances behind the terminal voltage. The machine internal voltages corresponding to each stage are called the sub-transient, transient and steady state voltages, and are symbolised as E”, E’ and E. In Figure 11, the general vector diagram of the synchronous machine for the transient stage is presented.

Power System Transients and Earthing Systems

2-17

Short Circuit Current Calculation Theory and Techniques

E Ed

Eq

E’q

E’d

Quatrature axis

E’ (X’q-X’d)Iq Ei

IqXq

IbX’d

Un Direct axis

IqX’q IdXd

IqX’d

IbRa

Figure 11 The general vector diagram of the synchronous machine for the transient state. Fortunately, since direct and quadradure axis reactances are equal to each other for turbo (cylindrical rotor) generators, internal voltages can be derived easily by equating

E to Ei, E’ to E’i and lastly E” to E”i. Additionally, ignoring the machine resistance the following equations can be formulated

E” = Un + jIbX”d E’ = Un + jIbX’d E = Un + jIbXd - 16 where Ib is the load current and the related phasor diagram is given in Figure 12.

Direct axis

Quadrature axis

E E’ E” Ib

Un

IbXd IbX’d IbX”d

Figure 12 Approximate phasor diagram

Power System Transients and Earthing Systems

2-18

Short Circuit Current Calculation Theory and Techniques

Nevertheless, it should be appreciated in the case of salient pole machines, much more calculation effort is required for accurate calculation of the internal voltages, due to the differing d and q axis reactances. However, it is claimed in some published literature that the same procedure may be applied with sufficient accuracy for short circuit calculation, especially in the initial period of fault [14, 25]. It is possible to determine the actual voltages for each stage by knowing the direct and quadrature axis currents of the machine for the initial loading condition. The d and q axis currents can be determined by using Park’s [37] transformations. The same phasor diagram can be used to represent the sub-transient state of the machine by substituting transient reactance values by sub-transient reactances. Thus the internal voltages of the machine can be determined by

E” = Un + IbRa + IbX”d + jIq(X”q – X”d) E’ = Un + IbRa + IbX’d + jIq(X’q – X’d) - 17 where the last term of each equation becomes negligible for round rotor machines, since the d and q axis reactances are equal each other in this case. Therefore, the equivalent circuit of the machine can be arranged by considering equation - 17 as shown in Figure 13; X’d

Ra

+ Un

E’ -

-

j(X’q-X’d)Iq

+

Figure 13 Equivalent circuit of the transient state The phasor diagram of the machine is also given in Figure 14 for the steady-state condition.

Power System Transients and Earthing Systems

2-19

Short Circuit Current Calculation Theory and Techniques (Xd-Xq)Id IbXq Direct axis Ei

E

IqXq

Un

Ib

Quadrature axis

IbRa

IdXd

Figure 14 Steady-state phasor diagram of the machine The internal voltage of the machine can be determined by;

E =Eq = Un + IbRa + jIbXq + jId(X’d – X’q) - 18 Substituting the appropriate voltages into equation - 11 yields more accurate results [21].

2.3.3 The Effect of the External Network Impedance In general, there is always an impedance between the short circuit point and the machine terminal. Since this impedance can be proportionally larger than the machine impedances, it can not only have considerable influence upon the rms value of the symmetrical short circuit current but also has a large effect on the time constants of the short circuit. Therefore, equation - 11 may be modified as

E" E" = Z " k Ra + Rs + j ( X " d + X s ) E' E' I 'k = = Z ' k Ra + Rs + j ( X ' d + X s ) E E Ik = = Z k Ra + Rs + j ( X d + X s )

I "k =

- 19 where subscript s refers to system values, and Ra represents the machine armature resistance. The time constants of the short circuit current which are given in equations - 5 and - 14 may be rewritten as

Power System Transients and Earthing Systems

2-20

Short Circuit Current Calculation Theory and Techniques

T"d =

X "d + X s T" X 'd + X s d 0

T 'd =

X 'd + X s T' Xd + X s d0

Ta =

L2 + Ls RG + Rs - 20

If the system impedance has a relatively high resistance, the time constants need to be extended to consider the system’s resistance. This effect is particularly important for the transient time constant since it is significantly larger. Hence

T 'd =

Rs2 + ( X ' d + X s )( X d + X s ) T 'd 0 Rs2 + ( X d + X s ) 2 - 21

The external resistance is the main element that determines whether there is significant decrement or not.

2.4 Overview of Standards The short circuit current calculation standards are intended to give simple but conservative estimates of short circuit current magnitude. The standards employ a quasi steady state calculation technique to enable particular short circuit values to be determined related to short circuit duties. In general, the machine model adopted has been kept very simple and referred to as the ‘ constant impedance behind the constant voltage source’. IEC 909 uses the sub-transient reactance of the machine in this respect. IEC 909 and IEEE/ANSI approaches calculate similar values, however each uses a particular terminology for each duty. The duties are presented in Table 2 Table 2

Comparison of short circuit current quantities and terminologies. Initial Short circuit currents

IEC 909

Symmetrical

Peak

I”K

ip

First cycle duty IEEE/ANSI Symmetrical

Isc

Peak

Breaking currents Asymmetrical Symmetrical

Ibasym

Ibsym

Steady-state short circuit current

Ik

Short circuit current for time delaying relays Asymmetrical Symmetrical Contact parting duty

ip

Iasym

Isym

Ik

The above duties coincide with the specific values of the short circuit current and can be visualised on the wave shape of the short circuit current as presented in Figure 15. Power System Transients and Earthing Systems

2-21

Short Circuit Current Calculation Theory and Techniques

Figure 15 Short circuit current reproduced from [8]. The relevant details of each standard are described in the following sections.

2.4.1 IEC 909 Standard IEC 909 classifies the short circuit with respect to its location as ‘near-to-generator’ and ‘far from the generator’ or motor. The main objective of the standard is ‘to establish a general, practicable and concise procedure leading to conservative results with sufficient accuracy’. The short circuit current is treated as the sum of a decaying unidirectional (d.c.) and a symmetrical component (a.c.). For near-to-generator types of short circuit, the symmetrical component of the current is treated as a decaying current while for far-from-generator faults it is considered constant. It should be noted that the standard treats the single line-to-earth fault as a far-from-generator type without paying any attention to the location of the fault. Therefore, no symmetrical decrement is considered for this type of fault. IEC 909 in particular is interested in the following current duties and describes calculation techniques for each of them.

Power System Transients and Earthing Systems

2-22

Short Circuit Current Calculation Theory and Techniques

2.4.1.1

Initial Symmetrical Short Circuit Current

The standard employs an approximate superposition method to calculate this current as explained in section 2.2.3. A series of correction factors are employed to improve the accuracy of the results while erring on the safe side. These factors are employed to calculate the effective impedance of the power system components. The magnitude of the factors depend upon the network configuration and fault location and include the „

voltage factor which is given in Table 1,

„

generator correction factor, (KG),

„

transformer correction factors (KT),

„

power station units correction factors, (KPS).

Once the short circuit network is established by ignoring all non-rotating loads and system shunts, the initial symmetrical short circuit current can be calculated using equation - 3. The short circuit impedance in equation - 3 is calculated by considering the above given correction factors for the specific elements. For the other elements of the network such as; overhead lines, cables, external network infeed, the standard also provides appropriate formulae and techniques. If the fault is considered as far-fromgenerator, rotating machine impedances are ignored. For unbalanced faults, the standard provides an appropriate equation to calculate the short circuit current by considering the connection of the sequence networks. The single line-to-earth type of fault is calculated using equation - 3.

2.4.1.2

Peak Short Circuit Current

The peak short circuit current is shown in Figure 15 and defined by the standard as “The maximum possible instantaneous value of the prospective (available) short circuit current.” The standard also notes that “The magnitude of the peak short circuit current varies in accordance with the moment at which the short circuit occurs.” Consequently the standard calculates the “greatest possible short circuit current” by considering the worst case scenario. In the case of calculation of the peak short circuit current, it is necessary to distinguish between the cases of a radial or meshed network to calculate this current. In the case of a radial network, the total peak current is the sum of the contributions from the converging path to the faulted bus. The summation of each converging branch is advised to be done by algebraic sum. To employ an algebraic sum instead of a vector

Power System Transients and Earthing Systems

2-23

Short Circuit Current Calculation Theory and Techniques

one will result in a conservative result. However this greatly simplifies the calculation process. Each branch’s peak current is calculated using the equation.

ipi = Ki √2 I”ki -

22

where coefficient Ki depends on the X/R ratio of the related branch. It is also possible to evaluate this value by empirical formulae. In the case of a meshed network the standard describes three different methods (viz. Method A, B and C) to compute the equivalent X/R ratio for the system. The same methods can be used for near-to generator or far-from-generator types of faults. For unbalanced faults, the peak short circuit current is calculated using the same techniques.

2.4.1.3

Symmetrical and Asymmetrical Breaking Current

The standard adopts different approaches for the treatment of near-to-generator and far-from-generator short circuits to compute this current. In the case of the far-fromgenerator fault, Ib is assumed to be the initial symmetrical current without any decrement. For the near-to-generator case, the standard distinguishes between the types of network as meshed or radial in a similar way to the treatment of peak short circuit current. In a radial network, the total symmetrical breaking current is expressed as the sum of the contributing branches’ currents. Each branch current is calculated by multiplying the initial symmetrical current of the branch by a multiplication factor, namely µ. The factor µ, is given by a series of empirical formulae which depend on a number of factors including „

time delay,

„

the partial symmetrical rms short circuit current at the machine terminal,

„

the highest symmetrical rms current of the machine with locked rotor fed with rated voltage at rated frequency.

In the case of a meshed network, with a conservative approximation, the factor µ is assumed equal to unity. Asymmetrical breaking current is not separately considered in the main body of the standard. However, in the appendix of the standard, this current is calculated with the given formulae, which enables the asymmetrical current to be determined for a given time interval.

Power System Transients and Earthing Systems

2-24

Short Circuit Current Calculation Theory and Techniques

I basym = I b2 + idc2 idc = 2 I k" e ( −2πftR / X ) - 23 In the case of a radial network idc is computed by the summation of branch currents. The X/R ratio of related branches is calculated by the methods used for the determination of peak short circuit current. In a meshed network, the total X/R ratio is computed by either method B or C.

2.4.1.4

Steady-State Short Circuit Current

For the computation of the steady-state short circuit current in the case of the near-togenerator fault, the standard supplies several graphs for each type of machine. These graphs provide a factor λ to calculate the current injected from generators. The contribution from motors is accepted to fade away rapidly and hence neglected. The factor λ depends on; the generator type (viz. turbo or salient pole machine), steady state reactance of the generator and the ratio between the partial short circuit current on the generator terminal and the highest symmetrical rms current of the generator with locked rotor fed with rated voltage at rated frequency. In the case of a meshed network, the standard re-structures the network diagram and re-calculates the symmetrical short circuit current without considering motors. This new calculated value is assumed equal to the steady-state current. For far-from-generator type of faults, no decrement is considered and therefore this current is assumed equal to the initial symmetrical one. Since no decrement is foreseen for the single line-toearth types of faults in the standard, no calculation technique is provided.

2.4.2 IEEE/ANSI Approach The main scope of this standard is to allow the installation of suitable sizes of medium and high voltage circuit breakers for systems at 1000kV and above [1]. The standard specifies a simplified calculation of short circuit current based on the method described in section 2.2.3. According to this method the standard reduces the network to an equivalent voltage source and reactance. The voltage source is the typical operating voltage of the system. The nominal voltage of the system could also be employed for this purpose. In general, only a reactance network is considered to determine the first cycle current. However, it is also recommended that two different reactance and

Power System Transients and Earthing Systems

2-25

Short Circuit Current Calculation Theory and Techniques

resistance networks should be used to evaluate the total short circuit impedance. The networks are built by considering all rotating loads without taking into account whether they are remote from or near to the fault location. However, depending upon duty types a series of impedance multiplication factors are introduced. Particular guidelines are provided for the determination of single line-to-earth fault current levels.

2.4.2.1

First Cycle Duty

To compute this duty, two separate first cycle networks, namely resistance and reactance networks, are built. These networks consist of branch impedances and corrected rotating machine sub-transient impedances. All passive loads are neglected. Correction factors for rotating machines are provided in the standard. These factors depend upon rated power, number of poles and type of the machine. All machines are modelled in the first cycle network whether the fault is remote or not. The first cycle duty can be then evaluated as E/X. This is the symmetrical rms value of the current in the first cycle. The asymmetrical rms current can be calculated by multiplying this current by the factor 1.6. In addition, peak current is given as 2.7 times the first cycle symmetrical rms current. The factors 1.6 and 2.7 are not empirically determined values as they may appear. The calculation procedure for arriving at these values has been described in references [33, 38] in detail. The multiplication factors for the asymmetrical first cycle current Srms and for the peak current Speak can be derived by using the given formulae below

S rms = 1 + 2e

−4 πft

R X

R  −2 πft  X S peak = 2 1 + e   

- 24 Where, f is the frequency of the system and t is the half cycle time. The above factors of 1.6 and 2.7 correspond to a ratio of X/R ≈ 25. However, the standard pays particular attention to calculating the contact parting currents by considering different X/R ratios. These factors are also called ‘capability factors’ and it has been recognised that they predict a ‘pessimistic’ total fault current value [38]. For the single line-to-earth fault, the standard provides multiplication factors to calculate the asymmetrical and peak values from the symmetrical value.

Power System Transients and Earthing Systems

2-26

Short Circuit Current Calculation Theory and Techniques

2.4.2.2

Contact Parting Duty

To calculate this current value, the standard rebuilds the resistance and reactance network by considering different machine reactances and related multiplication factors. A table which lists machine reactances and multiplication factors according to IEEE/ANSI C37 [1] is provided below. Table 3

IEEE/ANSI machine reactance and multipliers First cycle network

Contact parting

Factor

Reactance

Factor

Reactance

Turbo alternators

1.00

X”d

1.00

X”d

Hydro with dampers

1.00

X”d

1.00

X”d

Hydro no dampers

0.75

X’d

0.75

X’d

Condenser

1.00

X”d

1.00

X”d

Synchronous Motor

1.00

X”d

1.50

X”d

Larger than 1000 HP 1800 rpm or less

1.00

X”d

1.50

X”d

Above 250 HP at 3600 rpm

1.00

X”d

1.50

X”d

All others 50 HP and above

1.20

X”d

3.00

X”d

Smaller than 50 HP

1.67

X”d

Neglect

-

Induction motors

Motors

Generators

Machine Type

The standard also provides several curves to determine the multiplication factor for each time interval, (viz. Two, three, five and eight cycles.) Using these factors the asymmetrical breaking current can be calculated. For remote faults this current could also be calculated using equation- 24. In the case of a local fault empirical ‘look up tables’ are provided in the appendix of the standard. However it is claimed that a curve fit equation is also available for this purpose [39, 40]. Similar curves are also provided for the single line-to-earth fault.

2.4.2.3

Short Circuit Current for Time Delaying Relays

In order to calculate the sustained value of short circuit current a different network should be built by considering only generators and passive elements of the system. All loads including rotating ones are neglected. The generators are modelled by their transient reactance or by a larger reactance that takes into account a.c. decrement.

Power System Transients and Earthing Systems

2-27

Short Circuit Current Calculation Theory and Techniques

The d.c. component of the current is supposed to be zero. Symmetrical decrement on the single line-to-earth fault is also treated in the standard.

2.4.3 Comparison of the Standards It has been shown in many publications [11, 16, 17, 18, 19, 41] that IEC 909 provides more conservative results than IEEE/ANSI. This is mainly due to the higher pre-fault voltages recommended by IEC 909. In addition, smaller machine reactances (mainly sub-transient) are employed in IEC 909 for all types of duties. IEC 909 is probably more complicated to use than the IEEE/ANSI, particularly, in the treatment of meshed networks. On the other hand the IEC standard treats the effects of machine excitation systems for the sustained value of short circuit current while IEEE/ANSI does not. However, in terms of data requirements, both standards are similar. For comparison, both standards have been applied to an example network which is detailed in the IEC 909 appendices. It should be noted that only balanced three phase faults are considered. The circuit is a high voltage system fed by an infinite bus which is modelled by a constant impedance behind a voltage source and hence no decrement of a.c. component of the short circuit current is simulated. The network also includes two induction machines. Resulting current values from each standard are reproduced in Table 4. Table 4

Comparison of standards for three-phase fault current magnitudes [8, 19]. IEC 909 current in kA

IEEE/ANSI current in kA

Percentile difference in %

Initial symmetrical

18.82

16.74

11.10

Peak

47.18

42.50

9.92

Breaking symmetrical

16.33

15.60

4.47

Breaking asymmetrical

17.51

15.69

10.39

Steady-state

14.32

13.32

6.98

It is clear from the above figures, the IEC 909 calculation over-estimates all current values. Differences in percentile rank suggest that differences between the two standards reduce in the later period of the fault.

Power System Transients and Earthing Systems

2-28

Short Circuit Current Calculation Theory and Techniques

2.4.4 Engineering Recommendation G.74 The realisation of conservative modelling in IEC 909 has prompted the employment of more precise calculation practices. For example, ER G.74 [30, 42] ‘Procedure to Meet the Requirements of IEC 909 for the Calculation of Short Circuit Currents in Three Phase ac Systems’ sets out a ‘Good Industry Practice’ for a computer-based method of calculating short-circuit currents, which can be used as an alternative to the methods presented in IEC 909. This engineering recommendation does not provide or recommend any particular software for short circuit calculation but describes a computer-based technique and details the models of the power system components. The main principle of the recommendation is the consideration of pre-fault network conditions and inclusion of all loads including passive ones and rotating plants. ER G.74 pays particular attention to induction machine contributions in the case of three phase faults. However, the scope of the document is limited to circuit breaker sizing in three phase ac power systems having a nominal voltage range 380V to 400kV.

2.5 Conclusions The factors which give rise to and affect the transient nature of fault current have been identified in terms of a.c. and d.c. components. A simple parametric study has shown how a particular fault initiation angle results in the maximum d.c. component of fault current but this does not coincide with the maximum peak total asymmetrical current. It has been shown that the IEC and IEEE/ANSI standards characterise a.c. and d.c. transients in an approximate way. Both standards and particularly IEC 909 appear to over-estimate short circuit current magnitudes. In view of this apparent overestimation through the use of simple models and simplifying assumptions, and the fact that IEC 909 does not correctly deal with single line-to-earth faults, a more detailed treatment of the fault condition is required which is in line with the consensus of the reviewed literature. Accordingly, chapter 3 investigates different a.c. machine models in an attempt to identify the appropriate level of complexity required for short circuit current evaluation for earthing design applications.

2.6 References

Power System Transients and Earthing Systems

2-29

Short Circuit Current Calculation Theory and Techniques

1 IEEE/ANSI C37: ‘IEEE standard Rating Structure for AC High-Voltage Circuit Breakers Rated on a Symmetrical Current Basis’, (The IEEE Inc. November, 1985) 1986 edn. 2 AIEE COMMITTEE REPORT: ‘Calculated symmetrical and asymmetrical short-circuit current decrement rates on typical power systems’, AIEE Trans. on PAS, June 1956, part 3, vol. 75, pp. 274-285 3 AIEE COMMITTEE REPORT: ‘Calculation of electric power system short-circuits during the first few cycles’, AIEE Trans. on PAS, April 1956, part 3, vol. 75, pp. 120127 4 LANTZ, M.: ‘Power System Fault Current Analysis Including Study of Transient Offset’, AIEE Trans. on PAS, October 1954, vol. 73, part 3, pp. 1073-1078 5 IEEE Std. 242-1975: ‘IEEE Recommended Practice for Protection and Co-ordination of Industrial and Commercial Power Systems, Buff Book’, (IEEE Press, New York, 1980) 6 IEEE Std. 399-1980: ‘IEEE Recommended Practise for Power System Analysis, Brown Book’, (IEEE Press, New York 1980) 7 IEEE/ANSI, C37.12-1981: ‘Guide to Specifications for AC High-Voltage Circuit Breakers Rated on a Symmetrical Current Basis and a Total Current Basis’, Approved on August 1981, (The IEEE Inc. November, 1985) 8 IEC 909: ‘Short Circuit Calculation in Three Phase ac Systems’, IEC, International Electrotechnical Commission publication, 1988, First edn. 9 DUNKI-JACOBS, J.R., LAM, B.P., STRATFORD, R.P.: ‘A comparison of ANSI-based and dynamically rigorous short-circuit current calculation procedures’, IEEE Trans. on Industry Applications, November/December 1988, Vol. 24, No. 6, pp. 1180-1194 10 ROENNSPIESS, O.E., EFTHYMIADIS, A.E.: ‘A comparison of static and dynamic short-circuit analysis procedures’, IEEE Trans. on Industry Applications, May/June 1990, Vol. 26, No. 3, pp. 463-475 11 BERIZZI, A., MASSUCCO, S., SILVESTRI, A., ZANINELLI, D.: ‘Short-circuit current calculation: A comparison between methods of IEC and ANSI standards using dynamic simulation as reference’, IEEE Trans. on Industry applications, July/August 1994, Vol. 30, No. 4, pp. 1099-1106 12 PROFESSIONAL GROUP P9: ‘IEE Colloquium on Fault level assessment-guessing with greater precision?’, The IEE, London, 30 January 1996, Digest No: 1996/016 13 ARRIILAGA, J., ARNOLD, C.P.: ‘Computer Analysis of Power Systems’ (J. Willey and Sons, London, 1990) 14 ANDERSON, P.M.: ‘Analysis of Faulted Power Systems’ (Iowa State University Press Ames, Iowa, 1973) 15 WESTINGHOUSE ELECTRIC CORPORATION: ‘Electrical Transmission and distribution Reference Book’, (East Pittsburgh, PA., 1950) 4th edn. 16 CASTELLI-DEZZA, F., SILVESTRI, A., ZANINELLI, D.: ‘The IEC 909 standard and dynamic simulation of short-circuit currents’, ETEP (European Transactions on Electrical Power), May/June 1994, Vol. 4, No. 3, pp. 213-221

Power System Transients and Earthing Systems

2-30

Short Circuit Current Calculation Theory and Techniques

17 BRIDGER, B.: ‘All amperes are not created equal: A comparison of current of high voltage circuit breakers rated according to ANSI and IEC standards’, IEEE Trans. on Industry Applications, January/February 1993,Vol. 29, No. 1, pp. 195-201 18 KNIGHT, G., SIELING, H.: ‘Comparison of ANSI and IEC 909 short-circuit current calculation procedures’, IEEE Trans. on Industry Applications, May/June 1993, Vol. 29, No. 3, pp. 625-630 19 BERIZZI, A., MASSUCCO, S., SILVESTRI, A., ZANINELLI, D.: ‘ANSI/IEEE and IEC standards for short-circuit current evaluation: methodologies, computed values and results’, 6th International symposium on short-circuit currents in power systems, September 1994, Liege Belgium, pp. 1.1.1-1.1.8 20 MORCHED, A.S., TENCH, G.A., KUNDUR, P.: ‘Accurate calculation of asymmetrical fault currents in complex power systems’, IEEE Trans. on PAS, August 1981, Vol. 100, No. 8, pp. 3875-3790 21 GIUSEPPE, P.: ‘A new approach to calculate the decaying AC contributions to short-circuit: The “characteristics” current method’, IEEE Trans. on Industry Applications, January/February 1995, Vol. 31, No. 1, pp. 214-221 22 HALPIN, S.M., GROSS, C.A., GRIGSBY, L.L.: ‘An improved method of including detailed synchronous machine representations in large power system models for fault analysis’, IEEE Trans. on Energy conversion, December 1993, Vol. 8, No. 4, pp. 719-725 23 HALPIN, S.M., GRIGSBY, L.L., GROSS, C.A., NELMS, R. M.: ‘An improved fault analysis algorithm for unbalanced multi-phase power distribution systems’, IEEE Trans. on PWD., July 1994, Vol. 9, No. 3, pp. 1332-1338 24 CLARKE, E.: ‘Circuit Analysis of A-C Power Systems’, Volume I and II, (J. Willey and Sons, London, 1950) 25 ROEPER, R.: ‘Short-circuit Currents in Three-phase Systems’, (J. Willey and Sons, the Bath press. Avon, 1985) 2nd edn. 26 STAGG, W.G., EL-ABIAD, A.: ‘Computer Methods in Power System Analysis’, (McGraw-Hill Book Company, 1968) 27 FORTESCUE, C. L.: ‘Method of symmetrical co-ordinates applied to the solution of polyphase networks’, Trans. AIEE ,1918, Vol. 37, pp. 1027-1140 28 OEDING, D., SCHEIFELE, J.: ‘Maximum short-circuit current at pessimal load flow’, 18th Universities Power Engineering Conference, April 1983, University of Surrey Guildford UK, pp. 545-550 29 IEC 909-1: ‘Short Circuit Calculation in Three Phase ac Systems Part 1. Factors for the calculation of short-circuit currents in three-phase a.c. systems according to IEC 909’, International Electrotechnical Commission publication, 1991 First edn. 30 EA ER-G74, (Electricity Association, Engineering Recommendation G74) 1992: ‘Procedure to meet the requirements of IEC 909 for the calculation of short circuit currents in three phase ac systems’, Electricity Association Services Limited, London 1992 31 ROBERTSON, D. ed.: ‘Power System Protection Manual’, (Oriel Press Ltd., Northumberland UK, 1982) pp. 47-65

Power System Transients and Earthing Systems

2-31

Short Circuit Current Calculation Theory and Techniques

32 KAI, T, TAKEUCHI, N., FUNABASHI, T., SASAKI, H.: ‘A simplified Fault Current Analysis Method Considering Transient of Synchronous Machine’, IEEE Trans. on Energy Conversion, Sep 1997, Vol. 12, No. 3, pp.225-231 33 HERMANN, W.R., JUAN, C.G.: ‘Relationship of X/R, Ip, and Irms’ to asymmetry in resistance/reactance circuits’, IEEE Trans. on Industry Applications, March/April 1985, Vol. IA-21, No.2, pp. 481-492 34 KIMBARK, W.E.: ‘Power System Stability, Volume III Synchronous machines’, (John Willey & Sons, INC. New York, 1956) 35 CONCORDIA, C.: ‘Synchronous Machines, Theory and Performance’, (J. Willey and Sons Inc. New York, Chapman & Hall Ltd. London, 1951) 36 ADKINS, B., HARLEY, R.G.: ‘The General Theory of Alternating Current Machines: Application to Practical Problems’ (Chapman and Hall, London, 1975) 37 PRENTICE, B.R.: ‘Fundamental concepts of synchronous machine reactances’, AIEE. Trans. 1937, vol. 56 supplement, pp. 1-21 38 CRAIG, N.H.: ‘Understanding Asymmetry’, IEEE Trans. on Industry Applications, July/August 1985, Vol. IA-21, No. 4, pp. 842-848 39 CONRAD, R. ST. PIERRE: ‘Sample System for Three-Phase Short Circuit Calculations’, IEEE Trans. on Industry Applications, March/April 1990, Vol. 26, No. 2, pp. 204-211 40 SIMPSON, R.H.: ‘Multivoltage Short-Circuit Duty Calculation for Industrial Power Systems’, IEEE Trans. on Industry Applications, March/April 1986, Vol. IA-22, No. 2, pp. 365-381 41 RODALAKIS, A.: ‘A Comparison of North American (ANSI) and European (IEC) Fault Calculation Guidelines’, IEEE Trans. on Industry Applications, May/June 1993, Vol. 29, No. 3, pp. 515-521 42 EA ETR-120, (Electricity Association, Engineering Technical Report No. 120) 1995: ‘Application Guide To Engineering Recommendation G74 for calculation of fault currents in three phase ac power systems’, Electricity Association Services Limited, London 1995

Power System Transients and Earthing Systems

2-32