# According to the IEC 61660

### Introduction

Figure 1. Equivalent circuit diagram for calculating the partial short-circuit currents

The scope of IEC 61660 is to describe a method for calculating short-circuit currents in DC auxiliary systems in power plants and substations. Such systems can be equipped with the following equipment, acting as short-circuit current sources:

• rectifiers in three-phase AC bridge connection for 50 Hz;
• smoothing capacitors;
• DC motors with independent excitation.

NOTE – Rectifiers in three-phase AC bridge connection for 60 Hz are under consideration. The data of other equipment may be given by the manufacturer.

This standard is only concerned with rectifiers in three-phase AC bridge connection. It is not concerned with other types of rectifiers.

The purpose of the standard is to provide a generally applicable method of calculation which produces results of sufficient accuracy on the conservative side. Special methods, adjusted to particular circumstances, may be used if they give at least the same precision. Short-circuit currents, resistances and inductances may also be ascertained from system tests or measurements on model systems. In existing DC systems the necessary values can be ascertained from measurements taken at the assumed short-circuit location. The load current is not taken into consideration when calculating the short-circuit current. It is necessary to distinguish between two different values of short-circuit current:

• the maximum short-circuit current which determines the rating of the electrical equipment;
• the minimum short-circuit current which can be taken as the basis for fuse and protection ratings and settings.

### Calculating the Total Short-Circuit Current

Each DC source during the fault shall contribute to the total short-circuit current. The superposition principle is being applied. When one source is observed then the other ones are being disconnected and ignored. The potential DC sources are battery, rectifier, capacitor and machine.

The partial short-circuit currents are calculated for each of those sources as follows:

• for 0$\le$t$\le$tp:

${{i}_{1}}\left( t \right)={{i}_{p}}\frac{1-{{e}^{-\frac{t}{{{\tau }_{1}}}}}}{1-{{e}^{-\frac{{{t}_{p}}}{{{\tau }_{1}}}}}}$

Where tp is the time to peak of the partial current and τ1 is the rise time constant for the partial current source.

• for tp$\le$t$\le$Tk:

${{i}_{1}}\left( t \right)={{i}_{p}}\frac{1-{{e}^{-\frac{t}{{{\tau }_{1}}}}}}{1-{{e}^{-\frac{{{t}_{p}}}{{{\tau }_{1}}}}}}$

Where Tk is the fault duration time and τ2 the decay time constant for the partial current source.

And the total short-circuit current is the sum as follows:

$i_{shc}(t) = \sum_{j=1}^{n_{DC}} i_{1j}(t) + i_{2j}(t)$

for 0$\le$t$\le$Tk. And nDC is the number of the DC sources contributing the fault current, j is the observed DC source.

### Partial Fault Currents

#### Fault Current from Batteries

Figure 2. Time to peak and rise time constant (Figure 10. IEC 61660:1997)

The peak short-circuit current is calculated as:

${{i}_{pB}}=\frac{{{E}_{B}}}{{{R}_{BBr}}}$

The quasi steady-state short-circuit current is calculated as follows:

${{I}_{kB}}=\frac{0.95{{E}_{B}}}{{{R}_{BBr}}+0.1{{R}_{B}}}$

The decay component is calculated as:

$\frac{1}{\delta }=\frac{2}{\frac{{{R}_{BBr}}}{{{L}_{BBr}}}+\frac{1}{{{T}_{B}}}}$

The rise-time constant (τ1B) and time-to-peak of short-circuit currents of batteries is taken from the diagram (Figure 10. in IEC 61660:1997). The time constant of the battery TB is assumed to be 30 ms. The decay-time constant (τ2B) is assumed to 100 ms. RBBr is the sum of the battery internal resistance and the line (path) resistance up to the fault location (RBBr=0,9RB+RBr). LBBr is the sum of the battery internal inductance and the line (path) inductance up to the fault location.

• Rise-time current , for 0 ≤ t ≤ tpB:

${{i}_{1B}}\left( t \right)={{i}_{pB}}\frac{1-{{e}^{-\frac{t}{{{\tau }_{1B}}}}}}{1-{{e}^{-\frac{{{t}_{pB}}}{{{\tau }_{1B}}}}}}$

• Decay-time current, for tpB ≤ t ≤ Tk:

${{i}_{2B}}\left( t \right)={{i}_{pB}}\left[ \left( 1-\frac{{{I}_{kB}}}{{{i}_{pB}}} \right){{e}^{-\frac{t-{{t}_{pB}}}{{{\tau }_{2B}}}}}+\frac{{{I}_{kB}}}{{{i}_{pB}}} \right]$

And the total current from the battery is:

${{i}_{B}}\left( t \right)={{i}_{1B}}\left( t \right)+{{i}_{2B}}\left( t \right)$

#### Fault Current from Capacitors

Figure 3. Factor k1C to determine rise-time constant (Figure 14. IEC 61660:1997)
Figure 4. Factor k2C to determine decay-time constant (Figure 15. IEC 61660:1997)

The peak short-circuit current is calculated using:

${{i}_{pC}}={{\kappa }_{C}}\frac{{{E}_{C}}}{{{R}_{CBr}}}$

Where EC is the voltage of the capacitor terminal before the fault, and RCBr is the sum of capacitor and branch resistance, up to the fault location. The factor κC depends on the eigen-frequency ω0 and the decay coefficient δ, as follows:

${{\omega }_{0}}=\frac{1}{\sqrt{{{L}_{CBr}}C}}$

$\frac{1}{\delta }=\frac{2{{L}_{CBr}}}{{{R}_{CBr}}}$

LCBr is the inductance of the capacitor and common branch up to the fault location.

• a) If δ > ω0:

${{\kappa }_{C}}=\frac{2\delta }{{{\omega }_{d}}}{{e}^{-\delta {{t}_{pC}}}}\sinh ({{\omega }_{d}}{{t}_{pC}})$

${{t}_{pC}}=\frac{1}{2{{\omega }_{d}}}\ln \frac{\delta +{{\omega }_{d}}}{\delta -{{\omega }_{d}}}$

${{\omega }_{d}}=\sqrt{{{\delta }^{2}}-\omega _{0}^{2}}$

• b) If δ < ω0:

${{\kappa }_{C}}=\frac{2\delta }{{{\omega }_{d}}}{{e}^{-\delta {{t}_{pC}}}}\sin ({{\omega }_{d}}{{t}_{pC}})$

${{t}_{pC}}=\frac{1}{{{\omega }_{d}}}{{\tan }^{-1}}\frac{{{\omega }_{d}}}{\delta }$

${{\omega }_{d}}=\sqrt{\omega _{0}^{2}-{{\delta }^{2}}}$

• c) If δ = ω0:

${{i}_{pC}}=0.736\cdot {{i}_{pmax}}$

${{t}_{pC}}=\frac{1}{\delta }$

Where the time-to-peak is tpC. And the rise-time constant is:

${{\tau }_{1C}}={{k}_{1C}}\cdot{{t}_{pC}}$

And the decay-time constant is:

${{\tau }_{2C}}={{k}_{2C}}\cdot{{R}_{CBr}}C$

And coefficients k1C and k2C are taken from the diagrams/tables (defined in Figure 14. IEC 61660). The quasi steady-state current of the capacitor is considered to be 0.

• Rise-time current , for 0 ≤ t ≤ tpC:

${{i}_{1C}}\left( t \right)={{i}_{pC}}\frac{1-{{e}^{-\frac{t}{{{\tau }_{1C}}}}}}{1-{{e}^{-\frac{{{t}_{pC}}}{{{\tau }_{1C}}}}}}$

• Decay-time current, for tpC ≤ t ≤ Tk:

${{i}_{2C}}\left( t \right)={{i}_{pC}}{{e}^{-\frac{t-{{t}_{pc}}}{{{\tau }_{2C}}}}}$

And the total current from the battery is:

${{i}_{C}}\left( t \right)={{i}_{1C}}\left( t \right)+{{i}_{2C}}\left( t \right)$

#### Fault Current from Rectifiers

The quasi steady-state short-circuit current IkD of a rectifier in three-phase AC bridge connection is:

${{I}_{kD}}={{\lambda }_{D}}\frac{3\sqrt{2}}{\pi }\frac{c{{U}_{n}}}{\sqrt{3}{{Z}_{N}}}\frac{{{U}_{nTLV}}}{{{U}_{nTHV}}}$

Where Un is the nominal system voltage on AC side of rectifier, ZN is the network impedance AC side, UnTLV and UnTHV are transformer rated voltages of low and high voltage side, respectively. The factor λD is calculated using:

${{\lambda }_{D}}=\sqrt{\frac{1+{{\left( \frac{{{R}_{N}}}{{{X}_{N}}} \right)}^{2}}}{1+{{\left( \frac{{{R}_{N}}}{{{X}_{N}}} \right)}^{2}}{{\left( 1+\frac{2}{3}\frac{{{R}_{DBr}}}{{{R}_{N}}} \right)}^{2}}}}$

The peak short-circuit current is calculated using:

${{i}_{pD}}={{\kappa }_{D}}\cdot{{I}_{kD}}$

And the factor κD and ${{\varphi }_{D}}$ is calculated using:

${{\kappa }_{D}}=1+\frac{2}{\pi }{{e}^{-\left( \frac{\pi }{3}+{{\varphi }_{D}} \right)\cot {{\varphi }_{D}}}}\cdot \sin {{\varphi }_{D}}\left( \frac{\pi }{2}-{{\tan }^{-1}}\frac{{{L}_{DBr}}}{{{L}_{N}}} \right)$

${{\varphi }_{D}}={{\tan }^{-1}}\frac{1}{\frac{{{R}_{N}}}{{{X}_{N}}}\left( 1+\frac{2}{3}\frac{{{R}_{DBr}}}{{{R}_{N}}} \right)}$

The time-to-peak is calculated for all values κD ≥1,05 as follows:

• for $\frac{{{L}_{DBr}}}{{{L}_{N}}}\le 1$ it is ${{t}_{pD}}=\left( 3{{\kappa }_{D}}+6 \right)$ (ms)
• for $\frac{{{L}_{DBr}}}{{{L}_{N}}}\gt1$ it is ${{t}_{pD}}=\left[ \left( 3{{\kappa }_{D}}+6 \right)+4\cdot \left( \frac{{{L}_{DBr}}}{{{L}_{N}}}-1 \right) \right]$ (ms)

The rise-time constant for rectifiers is:

• For κD >= 1.05 :

${{\tau }_{1D}}=\left[ 2+\left( {{\kappa }_{D}}-0.9 \right)\left( 2.5+9\frac{{{L}_{DBr}}}{{{L}_{N}}} \right) \right]$

• For κD < 1.05 :

The suitable approximation is given as:

${{\tau }_{1D}}=\frac{{{t}_{pD}}}{3}$

The decay-time constant is calculated using:

${{\tau }_{2D}}=\frac{2}{\frac{{{R}_{N}}}{{{X}_{N}}}\left( 0.6+0.9\frac{{{R}_{DBr}}}{{{R}_{N}}} \right)}$

#### Fault Current from DC Machines

The quasi steady-state short-circuit current is calculated using:

${{I}_{kM}}=\frac{{{L}_{F}}}{{{L}_{OF}}}\left( \frac{{{U}_{M}}-{{I}_{M}}{{R}_{M}}}{{{R}_{MBr}}} \right)$

Where LF is the field inductance and LOF is the unsaturated field inductance at no-load. This equation is valid only if the motor speed remains constant during the duration of the short-circuit fault. Otherwise IkM = 0.

 Figure 5. Factor κM for determining the peak short-circuit current ipM (Figure 17. IEC 61660:1997) Figure 6. Factors for determining tpM, τ1M for nominal and decreasing speed (Figure 18. IEC 61660:1997) Figure 7. tpM for decresing speed (Figure 19. IEC 61660:1997)

The armature time constant is calculated as:

${{\tau }_{M}}=\frac{{{L}_{MBr}}}{{{R}_{MBr}}}$

The time constant of the field circuit is calculated as:

${{\tau }_{F}}=\frac{{{L}_{F}}}{{{R}_{F}}}$

And the mechanical time constant is calculated as:

${{\tau }_{Mec}}=\frac{2\pi J{{n}_{0}}{{R}_{MBr}}{{I}_{M}}}{{{M}_{M}}{{U}_{M}}}$

The eigen frequency is calculated as:

${{\omega }_{0}}=\sqrt{\frac{1}{{{\tau }_{Mec}}{{\tau }_{M}}}\left( 1-\frac{{{R}_{M}}{{I}_{M}}}{{{U}_{M}}} \right)}$

The decay coefficient is calculated from:

$\frac{1}{\delta }=2{{\tau }_{M}}$

The peak short-circuit current:

${{i}_{pM}}={{\kappa }_{M}}\cdot \left( \frac{{{U}_{M}}-{{I}_{M}}{{R}_{M}}}{{{R}_{MBr}}} \right)$

The factors k1M, k2M, k3M and k4M are taken from the diagrams (Figure 18, 20, 21 in IEC 61660). The factor κM is taken from the diagram (Figure 17 in IEC 61660).

 Figure 8. Factor k3M for determining the rise-time constant t1M for decreasing speed (Figure 20. IEC 61660:1997) Figure 9. Factor k4M for determining the decay-time constant t2M for decreasing speed (Figure 21. IEC 61660:1997)

The time-to-peak in case when τMec≥10τF:

${{t}_{pM}}={{k}_{1M}}\cdot {{\tau }_{M}}$

• And the rise-time constant:

${{\tau }_{1M}}={{k}_{2M}}\cdot {{\tau }_{M}}$

• The decay-time constant:

τ2M = τF when n=nn=const.

τ2M = (k4M)(τMec)(LOF/LF) when n→0

In case when τMec<10τF then the time-to-peak is taken from the diagram/table (Figure 19. IEC 61660).

The rise-time constant and the decay-time constant τ1M and τ2M are calculated using:

${{\tau }_{1M}}={{k}_{3M}}\cdot {{\tau }_{M}}$

${{\tau }_{2M}}={{k}_{4M}}\cdot {{\tau }_{Mec}}$

• Rise-time current , for 0 ≤ t ≤ tpM:

${{i}_{1M}}\left( t \right)={{i}_{pM}}\cdot \frac{1-{{e}^{-\frac{t}{{{\tau }_{1M}}}}}}{1-{{e}^{-\frac{{{t}_{pM}}}{{{\tau }_{1M}}}}}}$

Where tp is the time to peak of the partial current and τ1 is the rise time constant for the observed voltage source.

• Decay-time current, for tpM ≤ t ≤ Tk:

${{i}_{2M}}\left( t \right)={{i}_{pM}}\cdot \left[ \left( 1-\frac{{{I}_{kM}}}{{{i}_{pM}}} \right){{e}^{-\frac{t-{{t}_{pM}}}{{{\tau }_{2M}}}}}+\frac{{{I}_{kM}}}{{{i}_{pM}}} \right]$

• And the total current from the DC machine is:

${{i}_{M}}\left( t \right)={{i}_{1M}}\left( t \right)+{{i}_{2M}}\left( t \right)$

#### Correction Factors

Corrected resistance for the each source

Due to the fact that all non-observed sources at the time are neglected along with their branches it is suggested to use correction factors, which are supposed to improve total results. Each calculated correction factor is multiplied with the partial fault current of the each source, as follows:

${{I}_{jcor}}={{\sigma }_{j}}\cdot {{I}_{j}}$

${{i}_{pjcor}}={{\sigma }_{j}}\cdot {{i}_{pj}}$

Where Ij is the initial partial fault current and ${{\sigma }_{j}}$ is the correction factor, both for the source "j".

${{\sigma }_{j}}=\frac{{{R}_{resj}}\cdot \left( {{R}_{ij}}+{{R}_{Br}} \right)}{{{R}_{ij}}\cdot {{R}_{Br}}+{{R}_{resj}}\cdot \left( {{R}_{ij}}+{{R}_{Br}} \right)}$

Y refers to the branch (Br).

### References

IEC 61660: Short-circuit currents in d.c. auxiliary installations in power plants and substations - Part 1: Calculation of short-circuit currents.