Network transformers (factor K_T)

Impedance correction factors are applied to the nominal rated impedance values of two-winding and three-winding network transformers. The correction factor covers transformers with and without on-load tap-changers if the transformer’s voltage ratio is different from the ratio of the base voltages of the network. The corrected PPS impedance is given by

     (7.1a)

and the correction factor K_T is given by

     (7.1b)

where Z_T = R_T + jX_T is the uncorrected rated transformer PPS impedance in Ω and x_T is the rated transformer PPS reactance in pu. This correction factor is also applied to the transformer’s NPS and ZPS impedances. For three-winding transformers, the correction factor is applied to the three reactances between windings. For example, if the three windings are denoted A, B and C and the pu reactance between windings A and B with C open is considered, the correction factor is given by

     (7.2)

and similarly for the two other windings.

No correction is applied to the star winding neutral earthing impedance, where present. Similarly, this correction factor is not applied to generator step-up transformers.

Directly connected synchronous generators and compensators (factor K_G)

Where the short-circuit location is on a low or medium voltage network directly fed from generators without step-up transformers, such as in industrial power systems, a correction factor is applied to the PPS subtransient impedance of the generators. This correction is introduced to compensate for the use of the equivalent voltage source instead of the subtransient voltage E″ behind subtransient reactance calculated at rated operation. The corrected PPS subtransient impedance is given by

     (7.3a)

where the correction factor K_G is given by

     (7.3b)

where is the uncorrected generator subtransient impedance, U_rG is the rated phase-phase voltage of the generator, ϕ_rG is the phase angle between and the rated current I_rG and is the generator subtransient reactance in pu. To calculate the peak make current using the factor κ (this is covered in Section 7.2.5), a fictitious generator resistance R_Gf is used such that:

The correction factor K_G is also applied to the NPS and ZPS generator impedances but not to the generator neutral earthing impedance, where present. If the terminal voltage of the generator is different from U_rG by a factor p_G, then U_rG in Equation (7.3b) is replaced by U_rG (+ p_G). Synchronous motors equipped with voltage regulation equipment are treated as synchronous generators.

Power station units with and without on-load tap-changers (factors K_S; K_G,S; K_T,S; and K_SO; K_G,SO; K_T,SO)

A power station unit is defined as the combined generator and its step-up transformer is shown in Figure 7.1. Two cases are considered: the first where the step-up transformer is equipped with an on-load tap-changer and the second where it is not.

Figure 7.1 IEC 60909 power station unit for calculation of impedance correction factors for short-circuit faults at F1 and F2

Step-up transformer with an on-load tap-changer (factors K_S; K_G,S and K_T,S)

For a short-circuit fault on the high voltage side of the step-up transformer, i.e. at F₁, the corrected impedance of the entire power station unit referred to the high voltage side of the step-up transformer is given by

     (7.4a)

and the correction factor K_S is given by

     (7.4b)

where Z_THV is the uncorrected transformer’s impedance referred to the high voltage side, U_NQ is the nominal system voltage at the power station unit connection point, x_T is the transformer reactance in pu at nominal tap position and t_r = U_rTHV/U_rTLV is the rated voltage ratio of the transformer. may be replaced with U_nQU_{Q min} if experience shows that the minimum operating voltage is such that U_{Q min} > U_nQ and the worst-case current is not required. U_rG may be replaced with U_rG(1 + p_G) if the actual generator voltage is always higher than the rated value. In Equation (7.4b), the absolute value of () is taken.

For unbalanced short circuits, K_S is also applied to the NPS and ZPS impedances of the power station unit provided overexcited generator operation is assumed. The correction is not applied to the transformer’s neutral impedance, if present. However, for underexcited generator operation, the application of K_S may give non-conservative results in which case the superposition method may be used. In calculating the current contribution of the entire power station unit on the transformer’s high voltage side, it is not necessary to take into account the contribution of the motors within the auxiliary system.

When calculating the partial short-circuit currents from the generator and through the transformer for a fault between the generator and the transformer, i.e. at F₂, the generator and transformer impedance correction factors are, respectively, given by

     (7.5a)

and

     (7.5b)

Step-up transformer without an on-load tap-changer (factors K_SO; K_G,SO and K_T,SO)

For a short-circuit fault on the high voltage side of the step-up transformer, i.e. at F₁, the corrected impedance of the entire power station unit referred to the high voltage side of the step-up transformer is given by

     (7.6a)

and the correction factor K_SO is given by

     (7.6b)

1 ± p_T is used if the transformer has off-load taps and one tap position is permanently used otherwise p_T = 0. If the highest current is required, then 1 – p_T is used.

For unbalanced short-circuit faults, K_SO is also applied to the NPS and ZPS impedances of the power station unit but not to the transformer’s neutral earthing impedance, if present. In calculating the current contribution of the entire power station unit on the transformer’s high voltage side, it is not necessary to take into account the contribution of the motors within the auxiliary system.

When calculating the partial short-circuit currents from the generator and through the transformer for a fault between the generator and the transformer, i.e. at F₂, the generator and transformer impedance correction factors are, respectively, given by

     (7.7a)

and

     (7.7b)

Calculation of peak make current

From the initial rms symmetrical current , the peak make current for a single radial circuit is calculated using a peak factor κ and is given by

     (7.10a)

where

     (7.10b)

and (X/R) is the system X/R ratio at the fault point calculated using generator fictitious resistance R_Gf as described in Section 7.2.3. The variation of κ with the X/R ratio is shown in Figure 7.2.

Figure 7.2 IEC 60909 factor K for the calculation of initial peak short-circuit current

For parallel independent radial circuits, the total peak make current is the sum of the individual make currents of each radial circuit. For meshed systems, Equation (7.10a) is also used but Equation (7.10b) changes to

     (7.11)

where (X/R)_i is calculated in accordance with one of the following three methods:

Calculation of dc current component

The maximum dc component of the short-circuit current is calculated using

     (7.14)

where f is power frequency and (X/R) is the break X/R ratio calculated using Method C equivalent frequency depending on the minimum time f_min from the instant of fault. In calculating (X/R)_b, the actual resistance of generators, R_G, is used. The equivalent frequency f_c to be used in calculating Z_C is determined as shown in Table 7.2.

Table 7.2 Equivalent frequency f_c for the calculation of the break (X/R)_b ratio used to estimate the magnitude of the break dc current component at t_min

Calculation of rms symmetrical breaking current

For a near to generator three-phase short-circuit fault, consider first either a singly fed short circuit or a non-meshed network consisting of parallel independent short-circuit sources. In the case of two sources namely a synchronous generator and an asynchronous motor, the symmetrical short-circuit breaking current is given by

     (7.15a)

where

     (7.15b)

and are the synchronous generator and asynchronous motor initial symmetrical short-circuit current contributions measured at their terminals. Denoting I_rG and I_rM as the generator and motor rated currents, the multiplying factors μ and q are given by

     (7.16)

For asynchronous motors, in Equation (7.16) is replaced with . and

     (7.17)

where P_rM and p are the motor rated power in MW and number of pole pairs. The maximum value for q is unity. Equations (7.16) and (7.17) are plotted in Figures 7.3 and 7.4, respectively. Equation (7.16) for μ applies for synchronous generators employing rotating or static exciters provided, for the latter, that t_min < 250 ms and the maximum excitation voltage is < 1.6 pu of the rated value. μ is set equal to unity if the excitation system is unknown.

Figure 7.3 IEC 60909 factor μ used for calculation of short-circuit breaking current from synchronous generators

Figure 7.4 IEC 60909 factor q used with factor μ for calculation of short-circuit breaking current from asynchronous motors

For a far from generator three-phase short circuit, i.e. ≤ 2, or for unbalanced short circuits, μ = 1 and . Similarly, for far from motor three-phase short circuit or for unbalanced short circuits μq = 1 and .

For meshed networks, the approximation can be used. Alternatively, for improved accuracy, the decay in short-circuit currents from synchronous machines and asynchronous motors can be estimated from the partial step changes in voltage at the connection locations of the machines due to the fault and the following equation can be used:

     (7.18a)

where

     (7.18b)

are the voltage drops at the terminals of the synchronous machine (i) and the asynchronous motor (j). is the corrected subtransient reactance of synchronous machine (i), i.e. with K_v = K_G, K_S or K_SO as appropriate and is asynchronous motor j subtransient or transient reactance. and are the initial symmetrical current contributions from synchronous machine (i) and asynchronous motor (j) as measured at their terminals. For a far from motor short circuit, μ_j = 1 and 1 – μ_jq_j = 0 irrespective of the value of q_j.

Calculation of rms symmetrical steady state current

IEC 60909–0 recognises that the calculation of the steady state current I_k is less accurate than the calculation of because it depends on factors such as the generator initial operating point, machine saturation, excitation system characteristics and automatic voltage regulator action, etc. Worst-case assumptions are made to calculate maximum and minimum steady state short-circuit currents using factors expressed in terms of the generator rated current. For a single-fed near to generator three-phase short circuit, the maximum and minimum steady state short-circuit current is given by

     (7.19a)

     (7.19b)

λ_max is a factor that is dependent on the saturated direct axis synchronous reactance and for turbo-generators and salient-pole generators is obtained from IEC 60909–0 Figures 18(a and b) and 19(a and b), respectively. For turbo-generators, these figures apply for maximum excitation voltages of 1.3 and 1.6 pu whereas for salient-pole generators, they apply for maximum excitation voltages of 1.6 and 2 pu. The voltage factor c = c_max is used. λ_m;_n is calculated under constant no-load excitation, i.e. without voltage regulator action and the c factor used is c = c_min.

For meshed networks, and and similarly for unbalanced short-circuit faults.

Synchronous machines

Synchronous machines are represented by a voltage source behind an impedance model where the PPS reactance is time dependent and is given by

     (7.20)

The time constants are affected by the external network between the machine and the point of fault, as discussed in Chapter 6. Where the machines are connected to the faulted network by step-up transformers having comparable reactances to the machine subtransient reactances, unsaturated subtransient and transient reactances are used. However, saturated reactances are used where the machines are directly connected to the faulted system. For extended fault clearance times, the effect of excitation control systems, and in particular those that have high speed response, may need to be considered.

Asynchronous machines

Asynchronous machines or groups of such machines are represented by a voltage source behind an impedance model where, for a single-cage or single-winding rotor, the time-dependent PPS reactance is given by

     (7.21)

where X′ and T′ are calculated from the machine parameters or tests as discussed in Chapter 5. Like synchronous machines, the time constant is modified by the external network impedance. Where neither the internal parameters of the machines nor its starting characteristics are known, the IEC 60909–0 approach of using the ratio of locked rotor current to rated current shown in Equation (7.9a) can be used to calculate the initial PPS impedance. If ac current decrement cannot be directly represented, the asynchronous machine’s current adjustment factor (μq) of IEC 60909 of Equation (7.15b) is used to calculate the PPS reactance corresponding to any elapsed time t after the instant of fault as follows:

     (7.22)

X/R ratios for induction motors and static converter drives are considered as per IEC 60909–0 and is given in Section 7.2.4. The NPS impedance is assumed equal to the initial motor PPS impedance.

Small induction motors forming part of the general power system load

Numerous small induction motors that form part of the general load in the power system and hence are not individually identifiable are represented as an equivalent motor. The initial PPS impedance and time constant of the equivalent motor should be obtained by measurement. In the absence of measured data, indicative empirical estimates are suggested. For calculating the initial three-phase rms short-circuit current contribution at a 33 kV bulk supply substation busbar from induction motors in the general load supplied from that busbar the induction motor contribution is represented as follows:

These short-circuit infeeds relate to a complete loss of supply voltage to the motors.

The time variation of the PPS reactance of the equivalent motor is given by Equation (7.21) and the ac time constant is taken as T′ = 40 ms. It is therefore assumed that the contribution to a three-phase short circuit from motors in the general load decays to a negligible value at around 120 ms. The effective X/R ratio of the equivalent motor at the 33 kV busbar is taken equal to 2.76.

Passive loads

Both the real and reactive power components of passive loads are represented in the analysis. In particular, ER G7/4 recognises that the reactive load may have an appreciable effect on short-circuit currents because of its effect on transformer tap positions and rotating plant internal voltages.

Short-circuit making current

This is the peak asymmetric current or the maximum instantaneous value of the prospective short-circuit current. If the analysis method used does not allow the initial peak current to be directly calculated, IEC 60909 Equation (7.10) is used.

Short-circuit breaking current

The short-circuit breaking current allows for both ac and dc current decay. Both the ac and dc currents are calculated at a specified break time t_b. The magnitude of the dc current component at t = t_b, is given by

     (7.23)

To determine the ability of a circuit-breaker to interrupt these component currents or a combination of them, reference should be made to the test data of the circuit-breaker and the standards on which the test data is based, e.g. BS 116:1952, ESI 41–10:1987 or IEC 62271–100 Standard.

X/R ratio for estimation of the magnitude of the dc current component

The system X/R ratio at the fault point is calculated and used to estimate the magnitude of the required dc current component where explicit calculation of the asymmetric initial peak and break currents cannot be made. If some network elements have X/R ratios below 3.3, as may be the case in low voltage networks, then a 1.15 safety factor is used in calculating the peak asymmetric currents unless the equivalent frequency Method C of IEC 60909 is used to calculate the X/R ratio. Since the time constants of the dc current decrements for three-phase and earth faults, e.g. single-phase to earth fault, are different, different X/R ratios will need to be evaluated. Where a conservative estimate is adequate, the initial peak current can be assumed to be equal to for low voltage networks (i.e. 1 kV or below) and for networks operating above 1 kV thereby obviating the need to calculate the X/R ratio.

General

IEEE C37.010 is a fixed impedance short-circuit calculation method. The initial generator and motor prefault loading conditions are not modelled nor are the explicit decay rates of individual generators and motors. The Standard recommends the use of a voltage source at the short-circuit location whose magnitude is equal to the highest typical prefault operating voltage E at the fault location.

The system X/R ratio at the fault point is calculated from separate X and R networks. The standard assumes that the reduction of separate X and R networks tends to correct for the multiple time constants and associated decay rates due to short-circuit currents passing through multiple paths to the fault location. In practical cases, the resultant error is on the conservative side.

Both a simplified E/X method and a more accurate, semi-rigorous, E/X method that accounts for ac and dc current decrements are described. Where the use of impedances is preferred to the use of reactances in the calculation of the magnitude of short-circuit current, the E/X calculation may be replaced with E/Z calculation.

E/X simplified method

The E/X simplified method calculates the symmetrical short-circuit current and is very simple to use. The ‘E’ represents the prefault voltage at the short-circuit location. The ‘X’ is the equivalent system reactance at the fault point calculated with the resistances of network elements set to zero and machine reactances set to their subtransient values. The equivalent reactance at the fault point is equal to X^p for three-phase short circuits and 2X^P + X^Z for single-phase to earth short circuits. The calculated short-circuit currents are thus E/X^P for three-phase short circuits and 3E/(2X^P + X^Z) for single-phase short circuits. The Standard considers this simplified procedure to be sufficient provided that:

Otherwise, the more accurate E/X procedure, described below, should be used.

E/X method with correction for ac and dc decrements

This more accurate procedure involves applying multiplying factors to the E/X calculated short-circuit results to account for ac and dc current decrements. The factors to be used depend on the calculated system X/R ratio seen at the fault point and on whether the fault is ‘local’ to or ‘remote’ from generation. The applicable factors are taken from three figures in the IEEE Standard namely ‘Figures 8, 9 and 10’. The curves of ‘Figures 8 and 9’ give the K_acdc factor that includes the effects of both ac and dc decrements and applies for ‘local’ three-phase and single-phase to earth short circuits. ‘Figure 8’ gives a maximum value of K_acdc of 1.25 which supports the 80% criterion used in the application of the E/X simplified method for three-phase faults. ‘Figure 9’ gives a maximum multiplying factor of K_acdc of 1.41 which supports the 70% criterion used in the application of the E/X simplified method for one-phase faults. ‘Figure 10’ gives the K_dc factor that includes the effect of dc decrement only and applies for ‘remote’ three-phase and single-phase short-circuit faults. These three figures also include multipliers for longer circuit-breaker contact parting times than the minimum. The corrected E/X short-circuit current result should not exceed the symmetrical interrupting capability of the circuit-breaker. A more accurate multiplying factor can be determined by using the ratio of remote generation contributions to total fault current known as the no ac decrement or NACD ratio. The new multiplying factor K_Interpolated may be obtained by interpolation between K_acdc and K_dc using the NACD ratio as follows:

     (7.29)

First cycle symmetrical current

This is the half cycle rms current and is calculated using the closing and latching impedance network and the appropriate machine reactances shown in Table 7.4.

Closing and latching current

The closing and latching current is equivalent to the half cycle peak current. If the first cycle symmetrical current including motor infeed is below the symmetrical current rating of the circuit-breaker and the system X/R ratio is less than 17 at 60 Hz, then no further calculations are required. If the system X/R ratio is greater than 17, the closing and latching current, expressed as a peak current, may be calculated using a peak factor similar to the IEC 60909–0 peak factor κ of Equation (7.10) as follows:

     (7.30)

where the X/R ratio is calculated from the closing and latching impedance network.

If the value of the closing and latching current duty is calculated as a total rms value of the asymmetrical peak current, a simplified rule using a factor equal to 1.6 times first cycle symmetrical current, which assumes an X/R ratio of 25 at 1/2 cycle, may be used. Alternatively, the total rms value may be calculated using a more accurate multiplying factor of the rms symmetrical current based on the calculated X/R at the fault location as follows:

     (7.31)

Symmetrical interrupting current

This current is calculated using the interrupting impedance network and the appropriate machine reactances shown in Table 7.4.

Asymmetrical interrupting current

The asymmetrical interrupting current is calculated using the symmetrical interrupting current and an applicable multiplying factor which is a function of the circuit-breaker contact parting time and the system X/R ratio at the fault location. The factor is obtained from C37.010 ‘Figures 8, 9 and 10’ depending on whether the fault is fed by local or remote sources and the short-circuit type. The asymmetrical interrupting current duty is calculated as an rms value of an asymmetrical current.

Time-delayed 30 cycle (steady state) current

The time-delayed 30 cycle (steady state) short-circuit current applies when transient effects have completely decayed. In calculating this current, C37.010 recommends the use of a network that comprises only generators represented by either their transient reactance or a higher reactance that takes into account the ac component decay. The dc component is assumed to have decayed to zero.

Breaking or interrupting rating

IEC 62271–100 defines a rated short-circuit breaking current in kA and this is characterised by two values; the rms value of its ac component and the percentage dc component. Standard values of the ac component are selected from the R1O series 6.3, 8, 10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80 and 100 kA.

The IEEE Standard defines a symmetrical interrupting capability equal to rated short-circuit current × K where K = 1 for most modern circuit-breakers. For some older circuit-breakers, K = rated maximum voltage/operating voltage except that for single-phase to earth faults, the symmetrical interrupting capability is 15% higher. The IEEE rated short-circuit current in amperes is equivalent to the IEC rms ac component of the rated short-circuit breaking current.

dc current rating

In both IEC and IEEE Standards, the value of the percentage dc component can be determined from an envelope that consists of a single decaying exponential having a time constant T_dc. The percentage value of the dc component (%dc) in per cent of the peak ac current component or peak symmetrical interrupting capability is given by

     (7.32a)

where

     (7.32b)

and (T_r + T_cb) is the circuit-breaker contact parting time measured from the fault instant. T_cb, is the minimum opening time of the circuit-breaker and T_r is the protection relay time and is equal to half cycle of power frequency (10 ms at 50 Hz and 8.33 ms at 60 Hz). T_r is equal to zero for self-tripping circuit-breakers.

If the rms component of the short-circuit breaking current or the symmetrical current component is denoted I_Sym in kA, the dc current component in kA is given by

     (7.33)

Both IEC and IEEE specify a standard dc time constant T_dc = 45 ms. In addition, IEC provides special case time constants related to the rated voltage of the circuit-breaker. These recognise the higher time constants that may be encountered at locations close to generating stations, transformer or series reactor fed faults and extra high voltage transmission lines, etc. The special case time constants are:

The dc current envelope rating is shown in Figure 7.8.

Figure 7.8 Percentage dc current component for standard IEC and IEEE dc time constants, and special IEC cases

From Equation (7.32b), a dc time constant of 45 ms is equivalent to a system X/R ratio of 14.14 at 50 Hz and 17 at 60 Hz. For IEEE Standard, the dc current value is taken at the time of primary arcing contact parting.

Asymmetrical ratings

IEC specifies a rated short-circuit making current (peak current at half cycle) of 2.5 times the rms value of the ac component of the rated short-circuit breaking current for T_dc = 45 ms and a nominal frequency of 50 Hz. The factor is 2.6 for a nominal frequency of 60 Hz. For all special case time constants, the factor is 2.7 irrespective of the nominal frequency.

IEEE specifies a peak current capability at half cycle of 2.6 times the rated rms symmetrical short-circuit current. In addition, IEEE defines a total rms (asymmetric) fault current as follows:

     (7.34a)

and using Equations (7.31) and (7.32), we obtain

     (7.34b)

where c is the circuit-breaker contact parting time in cycles.