Introduction to Short Circuit Current Calculations Introduction to Short Circuit
Current Calculations
Course No: E08-005
Credit: 8 PDH
Velimir Lackovic, Char. Eng.
Continuing Education and Development, Inc.
9 Greyridge Farm Court
Stony Point, NY 10980
P: (877) 322-5800
F: (877) 322-4774
info@cedengineering.com
Introduction to Short Circuit Current Calculations
Introduction and Scope
Short circuits cannot always be prevented so system designers can only try to
mitigate their potentially damaging effects. An electrical system should be designed
so that the occurrence of the short circuit becomes minimal. In the case short circuit
occurs, mitigating its effects consists of:
-
Isolating the smallest possible portion of the system around the faulted area in
order to retain service to the rest of the system, and
-
Managing the magnitude of the undesirable fault currents.
One of the major parts of system protection is orientated towards short-circuit
detection. Interrupting equipment at all voltage levels that is capable of withstanding
the fault currents and isolating the faulted area requires considerable investments.
Therefore, the main reasons for performing short-circuit studies are as follows:
-
Defining system protective device settings and that is done by quantities that
describe the system under fault conditions.
-
Verification of the adequacy of existing interrupting equipment.
-
Assessment of the effect that different kinds of short circuits of varying
severity may have on the overall system voltage profile. These calculations
identify areas in the system for which faults can result in unacceptable voltage
depressions.
-
Defining effects of the fault currents on various system components such as
cables, overhead lines, buses, transformers, capacitor banks and reactors
during the time the fault persists. Mechanical stresses from the resulting fault
currents are compared with the corresponding short-term withstand
capabilities of the system equipment.
-
Compliance with codes and regulations governing system design and
operation.
-
Design and sizing of system layout, neutral grounding, and substation
grounding.
Electrical power systems are systems composed of a wide range of power
equipment used for generating, transmitting, and distributing electrical power to
consumers. Complexity of these systems indicates that breakdowns and faults are
unavoidable, no matter how carefully these systems have been designed. An
electrical system can be designed with zero failure rate, however that is
economically unjustifiable. From the perspective of short-circuit analysis, system
faults manifest themselves as insulation breakdowns. These breakdowns lead to one
or more phenomena:
-
Currents of excessive magnitudes that usually cause equipment damage
-
Undesirable power flow
-
Voltage depressions
-
Excessive over-voltages
-
Cause conditions that could harm personnel
Extent and requirements of short-circuit studies
Short circuit studies are as necessary for any power system as other fundamental
system studies such as power flow studies, transient stability studies, harmonic
analysis studies, etc. Short-circuit studies can be performed at the planning stage in
order to help finalize the single line diagrams, determine and set voltage levels, and
network equipment such as cables, transformers, and conductors. For existing
systems, fault studies are necessary in the cases of added generation, installation of
extra rotating loads, network topology modifications, rearrangement of protection
equipment, verification of the adequacy of existing breakers, relocation of already
acquired switchgear, etc. Short-circuit studies can also be performed in order to
duplicate the reasons and system conditions that led to the system’s failure.
The requirements of a short-circuit study will depend on the set objectives. These
objectives will dictate what type of short-circuit analysis is required. The amount of
data required will also depend on the extent and the nature of the study. The majority
of short-circuit studies in industrial and commercial power systems address one or
more of the following four kinds of short circuits:
-
Line-to-line fault. Any two phases shorted together.
-
Double line-to-ground fault. Any two phases connected together and then to
ground.
-
Single line-to-ground fault. Any one, but only one, phase shorted to ground.
-
Three-phase fault. May or may not involve ground. All three phases shorted
together.
Fault types are graphically presented in the figures below:
L3
L3
L2
L2
L1
L1
Three Phase Fault
Line to Line Fault
L3
L3
L2
L2
L1
L1
Double Line to
Ground Fault
Single Line-to-Ground Fault
These types of short circuits are also referred to as “shunt faults,” since all four are
associated with fault currents and MVA flows diverted to paths different from the prefault “series” ones. Three-phase short circuits often turn out to be the most severe of
all. It is thus customary to perform only three phase-fault simulations when searching
for the maximum possible magnitudes of fault currents. However, exceptions exist.
For instance, single line-to-ground short-circuit currents can exceed three-phase
short-circuit current levels when they occur in the electrical vicinity of:
-
The solidly grounded wye side of a delta-wye transformer of the three-phase
core (three-leg) design
-
The grounded wye side of a delta-wye autotransformer
-
A solidly grounded synchronous machine
-
The grounded wye, grounded wye, delta-tertiary, three-winding transformer
For electrical systems where any or more of the above conditions exist, it is
advisable to perform a single line-to-ground fault simulation. Line-to-line or double
line-to-ground fault studies may also be required for protective device coordination
requirements. Also, since only one phase of the line-to-ground fault can experience
higher interrupting requirements, the three-phase fault will still contain more energy
because all three phases will need the same interrupting requirements. Other types
of fault conditions that may be of interest include the “series faults” and they refer to
one of the following types of system unbalances:
-
Two lines open. Any two of the three phases may be open.
-
One line open. Any one of the three phases may be open.
-
Unequal impedances. Unbalanced line impedance discontinuity.
Series fault types are graphically presented in the figures below:
L3
L3
L2
L2
L1
L1
One line open
Two lines open
The term “series faults” is used because these faults are associated with a
redistribution of the pre-fault load current. Series faults are of interest when
assessing the effects of snapped overhead phase wires, failures of cable joints,
blown fuses, failure of breakers to open all poles, inadvertent breaker energization
across one or two poles and other situations that result in the flow of unbalanced
currents.
System modelling and computational techniques
AC and DC decrement
Physical phenomena that determine the magnitude and duration of the short-circuit
currents are:
-
The operation of the rotating machinery in the electrical system
-
The electrical proximity of the rotating machinery to the short-circuit location
-
The fact that the pre-fault system currents cannot change instantaneously,
due to the signiﬁcant system inductance
The ﬁrst two can be conceptually linked to the AC decrement, while the third, to the
DC decrement.
AC decrement and rotating machinery
For modelling purposes, these impedances increase in magnitude from the minimum
post fault subtransient value 𝑋𝑑" , to the relatively higher transient value 𝑋𝑑′ , and
ﬁnally reach the even higher steady-state value 𝑋𝑑 , assuming that the fault is not
cleared before. The rate of increase of machine reactance is different for
synchronous generators/motors and induction motors. Rate of increase for induction
motors is higher than for synchronous generators. This modelling approach is
fundamental in properly determining the symmetrical RMS values of the short-circuit
currents furnished by the rotating equipment for a short circuit anywhere in the
system.
AC decrement is determined by the fact that the magnetic ﬂux inside the windings of
the rotating machinery cannot change momentarily. For that reason, synchronous
machines, under fault conditions, show different ﬂux variation patterns as compared
to induction machines. The ﬂux dynamics dictate that a short-circuit current decays
with time until a steady-state value is reached. Machine software models present
rotating machines as constant voltages behind time-varying impedances.
Fault current DC decrement and system impedances
Short circuit currents cleared by circuit breakers must consider this unidirectional
component, especially for shorter interrupting periods. Same DC component is
important when verifying the capability of a circuit breaker to reclose against or
withstand fault currents. Fault currents containing high current DC offsets, usually
present no zero crossings in the ﬁrst several cycles right after fault introduction and
are especially burdensome to the circuit breakers of large generators.
Fault current DC decrement is also impacted by the fact that because the current
existing in the system before the fault, cannot change instantaneously, a
considerable unidirectional component may exist in the fault current which actually
depends on the exact occurrence of the short circuit. This unidirectional component
of the fault current is often referred to as DC current offset as it reduces with time
exponentially. The rate of decay is related to the system total reactance and
resistance. Although this decay is quick, the DC current component could last
enough time to be detected by the protective relay equipment, particularly when fast
fault clearing is very needed to maintain system stability or prevent the damaging
effects of the fault currents.
Modelling requirements of the power system
Fault currents have dynamic aspect that is necessary to associate calculated short
circuit currents to a specific moment in time from the onset of the short circuit. AC
current decrement assessment is used to properly determine the symmetrical RMS
values of the short circuit currents, while DC decrement calculations provide the
necessary DC current component of the fault current, hence affording a correct
approximation of the total short circuit current. The total fault current, must be used
for breaker and switchgear sizing and in some specific scenarios for protective relay
device coordination. Electrical system topology conditions are evenly significant
because the system arrangement and electrical closeness of the rotating machinery
to the fault location will influence the total order of magnitude of the fault current. It is
therefore essential to come up with an electrical system model as a whole and
examine it in an accurate and computationally convenient manner.
Modern power systems are usually compromised of multiple generators and motors.
They are interlinked using other equipment like transformers, overhead lines and
cables. Also there is usually one or more locations at which a local, smaller power
system is connected to a larger electrical grid. These locations are referred to as
“point of common coupling”. The main goal of the short-circuit study is to calculate
the short-circuit currents and voltages at various locations throughout the system.
Representation of the three-phase vs. symmetrical components
It is a customary practice for conventional three-phase electrical systems to be
interpreted on a single-phase basis. Mentioned simpliﬁcation, successfully applied
for power ﬂow and transient stability studies, leans on the assumption that the
electrical system is equally balanced or can be accepted to be so for practical
purposes. However, electrical system modelling, on a single-phase basis is
insufficient for examining processes that take into account serious system
imbalances. From the short-circuit analysis point of view, three-phase fault lends
itself to single-phase analysis, because the fault is balanced and asks for all three
phases, presuming a balanced three-phase electrical system. Other short circuit
current conditions will bring in imbalances that need the analysis of the remaining
unaffected two phases. There are two options to address this problem:
-
Representation using symmetrical components. Analysis using symmetrical
components is a method that, instead of asking for assessment of the
imbalanced electrical system, provides provision for the creation of three
electrical subsystems: the positive, the negative, and the zero-sequence systems that are correctly connected at the short circuit point which depends on
the type of the electrical system imbalance. Once fault currents and voltages
are modelled anywhere in the network, they can be obtained by properly
aggregating findings of the analysis of the three-sequence networks.
-
Representation of the system using all three-phases. If the system is
represented on a three-phase basis, the identity of all three phases is
retained. The advantage of this approach is that any kind of short circuit
current imbalance can be promptly assessed, including coincidental faults.
Moreover, the short circuit current condition is defined with bigger ﬂexibility,
especially for arcing faults. The main disadvantages of the technique are:
o If the computer program is used, it can be data-intensive.
o It is not convenient for manual calculations, even for small electrical
systems.
The distinguishable advantage of the approach that uses symmetrical
components is that it gives provision for representing imbalanced short circuit
conditions, while it still holds the conceptual simplicity of the single-phase
assessment. Additional significant advantage of the symmetrical components
technique is that impedances of the system equipment can be measured in
the symmetrical components reference frame.
This reduction is true only if the system is balanced in all three phases
(excluding fault location which becomes the connection point of the sequence
networks), the premise that can be entertained without bringing in
considerable modelling errors for most electrical systems.
The main weakness of the method is that for complex short circuit current
conditions, it may bring in more problems than it resolves. The method of
symmetrical components continues the favoured analytical tool for short
circuit current analysis for hand and computer-based assessments.
Impedances of the electrical system and analysis of symmetrical components
Theory of the symmetrical components prescribes that for a three-phase electrical
system, it needs to be established for the assessment of imbalanced short circuit
current conditions. The ﬁrst part is the positive sequence system, that is determined
by a balanced set of voltages and currents of equal magnitude, following the phase
sequence of a, b, and c.
The second part is the negative sequence system, that is similar to the positive
sequence system, but is determined by a balanced set of voltages and currents with
a reverse phase sequence of a, c, and b.
Lastly, the zero sequence system is deﬁned by a group of voltages and currents that
are in phase with each other and not displaced by 120 degrees, as it is the case with
the other two systems. Electrical connectivity of the zero sequence system can be
different from the positive and negative sequence systems. This is due to the fact
that it is influenced by the power transformer winding connections and system
neutral grounding; components that are not important when ascertaining the
topology of the positive and the negative sequence networks.
Three-phase fault analysis for the balanced systems requires only the positive
sequence system components impedances Z 1 = (R 1 + jX 1 ). For calculation of the
line-to-line faults, negative sequence impedances Z 2 = (R 2 + jX 2 ) are required. For
all faults involving connection to the ground, such as line-to-ground and double lineto-ground faults, the zero sequence system impedances Z 0 = (R 0 + jX 0 ) are required
in addition to the positive and negative systems. System neutral grounding
equipment components such as grounding resistors or reactors and grounding
transformers constitute an inherent part of the impedance data for the zero sequence
system.
Fault current AC decrement conditions prescribe that rotating electrical equipment
impedances differ from the onset of the short circuit. This is applicable only to
positive sequence impedances that range from sub-transient through transient to
steady-state values. The negative and zero sequence impedances for the rotating
electrical equipment are considered unaltered. The same is valid for the electrical
impedances of the static system components.
Electrical system equipment components such as transformers, overhead lines,
cables, bus bars, and static loads, under balanced system conditions can be
considered as static and have the same impedances that are used for calculating
positive and negative sequence currents. In principle, same components present
different electrical impedances for determining the ﬂow of zero sequence currents.
Rotating electrical equipment like electrical synchronous generators and motors
have different electrical impedances for all three phase sequence networks. The
positive sequence electrical impedances are usually used for balanced power ﬂow
calculations. Sequence impedances must be calculated, measured, specified by the
manufacturers of the equipment, or estimated based on the standard engineering
practice. The zero sequence electrical impedance may not exist for particular
rotating equipment which depends on the machine grounding system.
Quasi-steady-state short circuit current assessment
Quasi-steady-state short circuit current assessment relates to methods that interpret
the system at steady state. Phasor vectors are used to present voltages across the
system, currents, and electrical impedances at basic, fundamental frequency.
Electrical system modelling and the resulting calculation methods are based on the
premise that the electrical system and its associated electrical components can be
comprised of linear models. Keeping electrical system linearity greatly simpliﬁes the
calculations. Moreover, linear algebra and the numerical advancements in matrix
calculations make it possible to enforce practical computer solutions for large
electrical systems. These methods have been preferred by the many industry
standards.
Time domain short circuit current analysis – Calculation methods
Time-domain short circuit current assessment refers to methods that give provision
for the computation of the fault currents as a function of time from the instant of the
fault origin. For large electric power systems, which consist of numerous electrical
machines and generators that jointly contribute to the total fault current, the
contributions of many electrical machines will have to be considered concurrently.
Electrical machine models were formulated that let predictions of significant accuracy
be made with respect to behaviour of any electrical machine for a short circuit
current occurrence either at or beyond its terminals. These models are complex
because they represent in detail not only the electrical machine itself but also
nonlinear controllers including excitation systems and their related stabilization
electronic equipment with associated nonlinearities. It can therefore be noted that the
computational necessities could be colossal, because the task is cut down to
simultaneously solving a huge number of differential equations. Despite its
underlying power, the usage of time-domain short circuit current analysis is not
widespread and is only utilized for special calculations because it is data and time
intensive (required data can be at least as requiring as transient stability studies) and
it asks for a special software.
Industry standards for short circuit current calculations
Certain analytical techniques are defined by industry standards that adhere to
speciﬁc guidelines and are specifically accommodated to address the problems of
AC and DC current decay in practical multi-machine systems that are in conformity
with established, practices accepted by the power industry. They are also associated
to and accord with adopted, existing switchgear rating structures. Typical industry
standards are:
-
International standard, IEC 60909
-
North American ANSI
The analytical and calculation framework in the analytical processes prescribed by
the aforementioned standards stays algebraic and linear, and the computations are
kept easily managed by hand for small systems. The extent of the information base
necessities for computerized solutions is kept to a necessary maximum for the
solutions to be acceptably precise. This type of analyses presents the best
compromise between solution accuracy and simplicity of the simulation. The vast
majority of commercial fault analysis programs fall under this category.
IEC 60909 - International standard
Standard IEC 60909 (published in 1988) distinguishes four duty types resulting in
four different calculated short circuit currents:
-
The initial short-circuit current I" k
-
The peak short-circuit current I p
-
The breaking short-circuit current I b
-
The steady-state fault current I k
Although, the breaking and steady-state short circuit currents are in principal similar
to the interrupting and time-delayed short circuit currents, respectively, the peak
short circuit currents are the maximum fault currents reached during the ﬁrst cycle
from a beginning of a fault’s and are importantly different from the ﬁrst-cycle fault
currents described in IEEE standards, which are total asymmetrical RMS short circuit
currents. The initial short-circuit current is determined as the symmetrical RMS short
circuit current would inflow to the point of the fault if there are no changes in network
impedances.
AC current decrement is addressed by considering contribution from every
generation source, which depends on the voltage at generator terminals during the
short circuit. AC decrement of induction motor is represented in a different way from
synchronous machines decrement, because an extra decrement factor that
represents the more rapid ﬂux decay is included in induction motors. AC decrement
is considered and modelled only when breaking currents are calculated.
DC current decrement is addressed in IEC 60909, by using the principle of
superposition for the contributing electrical sources in conjunction with topology of
the network and the locations of the contributing sources with respect to the location
of the fault. Standard IEC 60909 prescribes that different calculation steps need to
be used when the contribution converges to a fault location via a meshed or radial
path. These conditions are applicable to the calculation of peak and asymmetrical
breaking short circuit currents.
Standard IEC 60909 gives calculation methodology of the maximum and minimum
short circuit currents. Maximum short circuit currents are used for sizing circuit
breakers while minimum short circuit currents are used for setting protective relays.
The main factor for the calculation of the short circuit currents is pre-fault voltage at
the point of the fault and the number of generators in service.
Short circuit currents calculated for the steady state take into account the fact that
the short circuit currents do not contain DC component and that all short circuit
current contributions from induction motors have decayed to zero. Synchronous
motors also have to be taken into account. Provisions are taken for salient and round
rotor synchronous machines and for different excitation system settings.
Loading conditions before the fault are considered with due attention in IEC 60909.
In order to account for system loads leading to higher voltages before the fault, the
standard advocates that voltages before the fault at the fault location point can be
different from 1.00 per unit. This means that a load flow solution is not required in
order to calculate short circuit currents. IEC 60909 suggests impedance correction
factors for the generators. These correction factors can also be applicable to their
step up transformers.
ANSI standards – North American standard
IEEE standards covering short circuit current calculations for low voltage electrical
systems (below 1000 V), are:
-
IEEE Standard 242-1986
-
IEEE Standard 241-1990
-
IEEE Standard C37.13-1990
-
IEEE Standard 141-1993
IEEE standards dealing with short circuit current calculations for medium and high
voltage electrical networks are:
-
IEEE Standard 141-1993
-
IEEE Standard C37.5-1979
-
IEEE Standard 241-1990
-
IEEE Standard 242-1986.
-
IEEE Standard C37.010-1979
Three types of fault currents are determined, depending on the time frame of interest
considered from the origin of the fault, as first-cycle fault currents, also called
momentary fault currents, are the currents at 1/2 cycle after fault initiation. These
currents pertain to the duty circuit breakers face when “closing against” or
withstanding fault currents. These currents usually contain DC offset and are
computed on the assumption of no AC decay in the contributing sources. Bearing in
mind that low voltage circuit breakers operate in the ﬁrst cycle, their breaking ratings
are compared to these currents.
Differences between the IEEE C37 and IEC 60909
There are numerous and signiﬁcant differences between IEEE C37 and IEC 60909
short circuit calculation standards. System modelling and computational techniques
are different in the two standards. Because of this results obtained by IEEE and IEC
standards can be different, with IEC 60909 generally providing higher short circuit
current values. The essential differences between IEEE and IEC short circuit
calculation standards can be summarized as follows:
-
Short circuit DC current decrement described in IEC 60909 does not always
rely on a single X/R ratio. Generally, more than one X/R ratio has to be taken
into account. In addition, the notion of separate X and R networks for
obtaining the X/R ratio at the location of the fault is not applicable to IEC
60909.
-
Short circuit AC current decrement considered by IEC 60909 depends on the
fault location and the standard quantifies rotating machinery’s proximity to the
fault. IEEE standard recommends system-wide modelling of the AC
decrement.
-
Steady-state short circuit current calculation in IEC 60909 considers excitation
settings of the synchronous machines.
Considering these important differences, numerical simulations performed using
IEEE C37 standards cannot be used to account for the computational requirements
of IEC 60909 and vice versa.
Calculated short-circuit currents and interrupting equipment
Previously discussed calculation procedures are used to perform fault calculations
on industrial and commercial power systems that are comprised of several voltage
levels including low, medium and high voltage systems. Fault currents that appear in
the first cycles of the faults are usually used to determine interrupting requirements
of low voltage fuses and breakers. These fault currents are used for the calculation
of the:
-
First cycle currents
-
Time delayed currents
-
Interrupting currents
Currents that are the result of short circuit current calculations are used for medium
and high voltage systems since they operate with a time delay that is introduced by
protective relaying and operating requirements. Since IEEE Standard C37.13-1990
has adopted the symmetrical rating structure and calculates symmetrical RMS, fault
currents and X/R ratio can be considered as sufficient in the case calculated X/R
ratio is less than the X/R ratio of the test circuit of the circuit breaker. This procedure
is suitable for calculations in systems with a low voltage fuse and circuit breakers as
defined by IEEE Standard C37.13-1990.
IEEE Standard C37.010-1979 and IEEE Standard C37.5-1979 contain coefficients
that can be applied to symmetrical RMS short circuit currents in order to get
asymmetrical RMS currents. IEEE Standard C37.5-1979 defines them as total
asymmetrical short circuit currents while IEEE Standard C37.010-1979 describes
fault currents that are compared against circuit breaker interrupting capabilities. The
above mentioned coefficients are obtained from curves normalized against the circuit
breaker contact opening time. In order to be in line with IEC standards, ANSI
C37.06-1987 introduced peak fault current to the preferred ratings as an alternative
to total asymmetrical currents.
It is important to mention that distinction needs to be made between ratings of
medium and high voltage circuit breakers. Circuits breakers that are described in
IEEE Standard C37.5-1979 and that are based on the older rating structure, are
assessed on the total asymmetrical short circuit current, or total fault MVA, and short
circuit current calculations are bounded by minimum parting time. The newer circuit
breaker rating structure that was introduced by IEEE Standard C37.0101979, defines
breakers on their symmetrical basis. The symmetrical short circuit currents
calculated using this method can be sufficient since certain degree of asymmetry is
included in the rating structure of the breaker depending on the actual operational
conditions and overall system X/R ratio.
Remote contribution is defined in IEEE Standard C37.010-1979, IEEE Standard
C37.5-1979, IEEE Standard 141-1993 and IEEE Standard 242-1986 as the current
is produced by a generator that:
-
Has a per unit X"d that is 1.5 times less than the per unit external reactance
on a common MVA base, and
-
Is located two or more transformations away from the point of the fault.
Generally, it needs to be pointed out that the most important step in the calculations
of the total fault currents for the medium and high voltage circuit breakers is deciding
which part of the total short circuit current comes from “local” and “remote” sources
in order to obtain a meaningful estimate of the circuit breaker interrupting
requirements. This distinction is reasoned by the fact that currents from remote
sources introduce slower AC current decay or do not introduce it comparing to short
circuit currents from the local sources.
Factors that affect short-circuit studies results
The accuracy of the calculated short circuit currents depends on the modelling
accuracy, system configuration and equipment impedances. Other factors include
modelling of the electrical machines, generators, grounding point of the system,
other system components and different operating conditions.
Electrical system configuration
Configuration of the electrical system is comprised of the following:
-
Network arrangement that defines how fault current sources are
interconnected
through
transmission
lines,
underground
cables,
power
transformers and bus bars.
-
Location of the potential sources of the fault currents including
synchronous generators, synchronous and connection points to utility network
Good practice indicates that more than one single line diagram should be consulted
for the studied system which depends on the actual operating conditions and the
final study objective. If the short circuit current study is done to determine the rating
of the switchgear, maximum short circuit currents in the system are calculated. This
implies that short circuit currents need to be calculated with all available generators
in service with all bus couplers and bus ties closed, while utility interconnectors
should be attained to their highest values. If the system study is performed to
determine protection relay requirements, some of these conditions do not need to be
considered. Various system conditions may require the study of one or more
alternative network arrangements especially for the purpose of determining
protective relay settings.
Grounding of the system neutral points
Faults that include zero sequence data of the electrical equipment such as line to
ground faults, double line-to-ground faults, and series faults, the ﬂow of short circuit
currents is affected by the grounding conditions of the electrical system. Presence of
multiple grounding points is of critical importance as well as the values of electrical
system grounding impedances. These grounding impedances can be used to limit
the magnitudes of the line-to-ground faults to minimum values, to inhibit over
voltages caused by line-to-ground faults, and to provide a reference point for line-toground fault protection. Grounding of the electrical system is also very important for
properly simulating zero sequence response of the electrical system. More
importantly, for solidly, or low-impedance grounded electrical systems, it is enough to
account for only the occasional current limiting transformer and/or electrical
generator grounding impedances and at the same time ignoring zero sequence
impedances of the transmission lines and underground cables. However, this will
have to be taken into account for high-impedance grounded, ﬂoating, and/or
resonant-grounded electrical systems (per IEC 60909).
Electrical Impedances of the System
Electrical impedances that are assumed should not by any means give lower short
circuit current results than those actually experienced in an electrical system. If this
is done and the system fault levels are underestimated, this can lead to under sizing
of the circuit breakers that may not be able to interrupt short circuit currents.
Contrary to that, over estimating anticipated short circuit levels can lead to over
sizing the required electrical equipment which makes system design impractical,
uneconomical and with less sensitive protection relay settings. Network utility
interconnection must be represented with impedance of the adequate value in order
to reflect expected fault level rating. All doubts regarding system impedances should
be solved in a way that higher fault levels are obtained for the sake of safety in the
design of the overall system. Electrical impedances of bus ducts or busways must be
also be considered for low voltage systems because they may limit short circuit
currents. Also, it is a common practice to use saturated impedance values of the
synchronous generators.
Consideration for the AC and DC current decrement modelling are important factors
when it comes to selection of the impedances of the rotating equipment for fault
calculations. It is crucial to refer to manufacturer’s catalogues and datasheets and, if
required, to conduct calculations to determine and check reliable impedance values.
Standard impedance values can be used in the case no other information is
available but these assumptions should be made on the safe side.
Finally, resistive components of the system electrical impedances should be
modelled considering working temperature, especially if considerable lengths of
underground cables exist in the system. Although resistances of the system
equipment may be neglected for calculation of the magnitude of the fault currents,
they are very critical for calculation of the system X/R ratio at the point of the fault. In
general, the total system impedance is a complex number comprised of its active
and reactive parts, namely resistance and impedance. It has to be calculated at the
point of the fault to afford a correct estimate of the short circuit current. This is
especially important for low voltage electrical systems, in which system resistance
can be compared in magnitude to the system reactance and it can help limit the
short circuit current.
Zero sequence of mutual coupling
This phenomenon is very important when parallel circuits share the same corridor
and their geometrical arrangement is such that current ﬂow in one circuit causes a
voltage drop in the other circuit. A typical example is overhead lines that share the
same support structure. It should be pointed out that mutual coupling exists between
phases in the positive sequence.
This phenomenon of mutual coupling, known as “interphase coupling,” is not
explicitly modelled in a positive sequence because it is limited within the same circuit
of which only one phase is modelled. Coupling of the zero sequence is extended
between two or more electrical circuits and needs to be explicitly modelled in zero
sequence. Neglecting or incorrectly modelling this phenomenon leads to incorrect
ground short circuit current results and erroneous performance assessment of distance protection relays. Although this phenomenon is infrequent for industrial power
system assessments, it should be properly considered and treated accordingly.
Electrical system loads and shunts before the fault
It is usual practice to presume that the electrical system is at steady state condition
before a short circuit occurs. Introduced simplification that neglects the load before
the fault is based on the fact that the magnitude of the system load current before
the fault is, usually, much smaller than the short circuit current. The importance of
the load current before the fault in the system increases along with rated voltage of
the system and particular loading patterns of the system. For this reason it is
justiﬁable to assume a 1.00 per-unit voltage before the fault for every bus.
This is particularly applicable for typical industrial power system short circuit current
calculations. For electrical systems in which loading before the fault is a concern, a
load ﬂow analysis should be conducted before short circuit current calculations in
order to assure that the system voltage profile will be coherent with the existing
system loads, shunts, and transformer tap settings. If the actual system condition
before the fault is modelled, it is crucial to keep all the system static loads and
capacitive line/cable shunts for the short circuit current calculation.
Standard IEC 60909 addresses this issue by using elevated voltages before the fault
and impedance correction factors for the synchronous generators. The ANSI and
IEEE C37 practice is focused around considering voltage before the fault as being
the nominal system voltage with the except for the assessment of the interrupting
requirements of circuit breakers.
Delta-wye transformer phase shifts
It is usually assumed that the phase of the short circuit current from the primary to
the secondary winding remains the same going through the transformer when
calculating the distribution of the three phase fault in the electrical system. However,
this is only true when transformers are Wye-wye and Delta-delta connected. If a
transformer is Delta-wye connected, phase shift is introduced between the primary
and secondary winding voltages and currents. Phase shift exists only in positive and
negative sequence values. Zero sequence values are not affected by the phase shift.
Existing practice in North America indicates that the positive sequence line-to-ground
voltage on the high voltage side of the transformers has to lead the positive
sequence line-to-ground voltage on the transformer low voltage side by 30 degrees.
This is also applicable for transformers following IEC standards. If this phase shift is
not taken into account for the unbalanced faults, different short circuit current
magnitudes are obtained when going through a delta-wye transformer since
sequence currents are treated as vectors in order to obtain phase short circuit
currents. Consequently, this can lead to misleading relay protection settings which
finally can compromise the selectivity and sensitivity of an overcurrent protection
scheme.
Computer solutions
General
Fault calculations are not that computationally demanding as other power system
studies such as load flow or harmonic analyses. Since short circuit current
calculations are linear, results for a small to medium sized system can be obtained
manually, particularly if the system electrical resistances are neglected, which
simplifies the overall complex calculation procedure.
Short circuit current
calculations are further simplified for radial systems. Practical industrial systems can
contain several hundred to thousand buses especially if low voltage bus bars are
considered and modelled. In those circumstances, numerical solutions can be
obtained only using computer solutions. It should be pointed out that the speed and
reliability of computer based calculations are significantly better even for the small
electrical systems.
Numerical network solutions
Hand short circuit current calculations are based on a series of combinations and
transformations of the impedances of the system branches until the electrical system
can be represented by an equivalent Thevenin impedance. This process is repeated
for every fault location. These hand performed calculations rely on the intuition of the
system analyst. Computer systems do not have this intuition, however they use
various techniques. These techniques neither rely on analyst abilities nor they
consider electrical system topology. This is why these techniques are applicable for
radial and meshed systems and can accommodate systems of almost any size.
Notions of admittance and impedance matrices are central in realizing all numerical
computer schemes.
The bus impedance matrix
Bus impedance matrix which is usually referred to as a Z-matrix is deﬁned as the
inverse of the admittance matrix. This matrix is also complex, square and symmetric;
in other words the entries Z ij and Z ji are equal for passive electrical networks. Since
the Z-matrix is inverse of the sparse Y-matrix it is full and does not have zero entries.
It can be shown that diagonal elements Z jj for bus j are the equivalent Thevenin
impedances used for short circuit current calculations. On the other side, entry Z ij ,
does not necessarily present the value of the physical impedance between buses i
and j. Actually there will always be an impedance Z ij although there may not be
branches between buses i and j. Entries on the diagonal of the Z-matrix are used in
calculations of the short circuit currents, whereas entries of the diagonal are useful
for the calculation of the branch contributions and voltage profiles across the system
under fault conditions.
The bus admittance matric, which is usually referred to as the Y-matrix, is a square
complex matrix (a matrix that is populated with complex numbers) where the number
of rows and columns are equivalent to the number of system bus bars. Elements of
this matrix are component admittances or sums of component admittances. The
component admittance represents inverse component complex impedance where
the component can be a branch, generator, motor, etc. When system buses are
identified, the admittance matrix can be formed as follows:
-
Allot a non-diagonal element to all the matrix elements that represent an
electrical
system
branch.
For
example,
if
an
electrical
branch
is
interconnected between buses A and B, the matrix entry YAB will be different
from zero and equal to the negative sum of the admittances of all electrical
components directly interconnected between buses A and B.
-
Allot a diagonal element of a matrix to every bus bar in the system. The value
of the diagonal elements in the matrix is the sum of the admittances of all
electrical system equipment connected to that bus bar.
Electric systems are passive and they have few branches compared to all of the
possible bus connections. As a result, typical electrical power system bus admittance
matrices are symmetric. (Keep in mind the assumption that power transformers are
not modelled in off-nominal TAP position). That practically means that Y AB = Y BA ,
and they are sparse in the case they feature a lot of zero entries.
Short circuit current solution algorithms and system topology
Sparsity of the admittance matrix calls for special computational techniques that are
used for storing the data about the system since simple data storage techniques can
be insufficient. Storing and keeping the entire impedance matrix is not only
impractical but also unneeded because only a few of its elements may be necessary.
Therefore solution algorithms have focused on effective recovery of the needed
impedance matrix entries with the least possible requirements for storage and
calculations. Modern computer software uses calculation and system data storage
techniques that focus around the so-called “sparse vector” and/or “sparse matrix”
calculation techniques which deliver very fast and precise solutions.
Computer software
Commercial software tools are widely used these days although advanced software
existed for more powerful hardware platforms and mainframes even in the 1960s.
Power system related calculations can now be performed using personal computers
and they are recognized as a credible analytical tool due to significant advances in
their overall performance, speed, stability and memory as well as user friendly
operating system environment. Analytical computer programs were one of the first
that were developed for all computer platforms. These numerical programs rely on
matrix techniques and ask from system analyst to provide accurate and exact
system data so that the computer can facilitate analysis and produce the required
results.
Software selection
There are numerous commercially available computer programs used for short
circuit current calculations. They can be differentiated based on a wide variety of the
analytical and calculation tasks they perform, sophistication degree of the userinterface and user-friendliness, and the computer platform for which they are made.
Different computer software tools and capabilities introduce a wide variety of prices;
therefore, it is very important to choose and acquire the software that is suitable for
most of the engineering tasks for which it was procured. From the investment point
of view, it is doubtful to acquire sophisticated and powerful software tool whose
capabilities may be rarely or never used. However, it can be problematic if the user
requirements quickly outgrow software capabilities which can compromise the
accuracy of the study, or end in a consistent waste of time and resources to cover
some of the inherited insufficiencies. It is also important to determine to what extent
the software is user friendly as well as the extent of the computer-literacy of the user
who will work with the software. Usually, it is worth working with software that
introduces easy data entry, meaningful and helpful error and warning notices, and
comprehensive readable reports. Finally, it is important to purchase a software tool
that is properly documented, well supported, and regularly updated and upgraded by
its manufacturers.
Short-circuit current calculation software features
Previously mentioned important guidelines for the selection of the adequate software
tool can be extended by a number of additional features that are particularly
applicable and suitable for fault level analyses. One of the most important steps in
the short circuit current studies is preparation of the input data. This stage can also
be numerically demanding especially if the software tool accepts inputs only on per
unit basis. It can be considered very important for a computer program to help the
analyst to prepare data for the study and allow provision for identifying and
correcting obvious errors and typos. Whenever international standards are used, it is
important that the software presents the calculation method in a transparent way and
to provide sufficient information and results that can be easily interpreted.
The table below summarizes some of the very important features of the computer
programs that are used for short circuit current studies. These features are, or may
not be, supported by all computer programs. These features are labelled as
“Important” and “Optional”. “Very important” mark means that a particular feature is
frequently encountered in a daily operation and can be considered as indispensable.
Category “Important” marks features that will show as very useful for demanding
studies while the category “Optional” marks computer software features that may
bring additional value for special studies.
Analytical feature
Line monitors
Currents in all three phases
Summary reports
Series faults
Arcing faults
Ground faults (LG and LLG)
Systems with more than one voltage level
Looped and radial system topology
Simultaneous faults
Complex arithmetic
Explicit negative sequence
Very
important
Important Optional
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Analytical feature
Interface with power ﬂow
Per-unitization of equipment data
Currents in all three sequences
One-bus-away fault contributions
Voltages in nonfaulted buses
Input data reports
Separate X and R reduction for X/R ratios
(IEEE Std C37.010-1979)
Currents in all branches
Interrupting fault currents (IEEE Std
C37.010-1979)
Protection coordination interface
Total current multiplying factors (IEEE Std
C37.5-1979 factors)
X/R—dependent peak multipliers (IEEE Std
C37.010-1979)
Mutual coupling in zero sequence
Multiplying factors (IEEE Std C37.13-1990)
Transformer phase shifts
Methodology in accordance with IEC 60909
(1988)
Symmetrical current multiplying factors
(IEEE Std C37.010-1979)
Rotating equipment impedance adjustment
(IEEE Std C37.010-1979)
X/R—dependent 1/2 cycle multipliers (IEEE
Std C37.010)
Remote and local fault contributions (IEEE
Std C37.010)
First cycle fault currents (IEEE Std C37.0101979)
Time-delay fault currents (IEEE Std
C37.010-1979)
Very
important
Important Optional
Yes
Yes
Yes
Yes
Yes
Yes
Mandatory
Yes
Mandatory
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Mandatory
Mandatory
Yes
Practical example
In the following sections short circuit current calculations are carried on a typical
industrial system. The objective of this example is to demonstrate typical steps,
calculation requirements and the way results are obtained. A studied electrical
system is composed of circuits operating on different voltage levels, local generation,
utility interconnection and different rotating loads.
Determination of the scope and extent of the study
Short circuit current calculation results may be used for recommending changes to
the existing system or proposing completely new plant design for a system which is
in its planning or expansion stage. There are a number of important questions whose
answers may be helpful for conducting fault level studies:
1. Is non-interrupting equipment, such as reactors, underground cables,
power transformers, and bus ducts, properly rated to withstand short-circuit
currents until they are cleared by the interrupting equipment?
2. Is this switchgear sufficient for line-to-ground faults? If not, should a new
switchgear be obtained or can some system modifications be effected to
avoid the extra capital expenditure?
3. What will be the effect on the calculated short-circuit currents in the plant
electrical system if there is an increase in the utility’s fault level?
4. Do load circuit breakers or disconnecting switches have adequate momentary
bracing and/or close-and-latch capacities?
5. Do the voltages during the fault, on unaffected buses drop to levels that can
cause motor-starter contactors to drop out or under voltage relays to operate?
6. Is particular protective relay equipment necessary to allow protective device
selectivity and sensitivity for both maximum and minimum value of fault
currents?
7. Is electrical circuit breaking equipment sufficient for the system interrupting
requirements at all voltage levels?
8. Can the medium and high voltage switchgears withstand the momentary
and interrupting duties enforced by the system?
9. Is there any provision in the interrupting capacity of the circuit breakers and
switchgears for accommodating planned system expansion and upgrades? If
not, is it essential to have a safety margin for future developments? If so, how
can the electrical system be modified to allow these concerns?
Every short circuit current study has to be analysed on different merits and output
results should be considered only for the purpose of the conducted study. It is not
unusual for these types of short circuit current calculations to consider only three
phase short circuit currents since they give more severe fault breaking requirements
when compared to other shunt fault types especially when it is known that many of
the electrical systems are impedance grounded. Single line-to-ground faults are
necessary to determine the adequacy of the switchgears if the electrical system is
such that line-to-ground faults may exceed three phase short circuit currents.
It is critical to determine the scope and extent of the short circuit calculations as well
as the desired accuracy since these factors will influence what types of faults need to
be simulated and the level of details that need to be considered during system
modelling. The number and type of short circuit current studies for a given system
are decided based on engineering judgment and common engineering practice. This
implies that various network topologies need to be assessed depending on the
specific purpose of the study.
Step by step procedures for short circuit current calculation
The following steps identify the basic considerations in making short circuit current
calculations. In the simpler systems, several steps may be combined; for example,
use of a combined one-line and impedance diagram.
1. Prepare the system one-line diagram. Include all significant system
components.
2. Decide on the short circuit current calculations required based on the type of
equipment being applied. Consider the variation of system operating
conditions required to display the most severe duties. Assign bus numbers or
suitable identification to the short-circuit locations.
3. Prepare an impedance diagram. For systems above 600 volts, two diagrams
are usually required to calculate interrupting and momentary duty for high
voltage circuit breakers. Select suitable kVA and voltage buses for the study
when the per-unit system is being used.
4. For the designated short circuit locations and system conditions, resolve the
impedance network and calculate the required symmetrical currents. When
calculations are being made on a computer, submit impedance data in proper
form as required by the specific program. For the high voltage equipment,
apply appropriate multipliers to the calculated symmetrical values so that the
short circuit currents will be in terms of equipment rating.
Type and location of faults required
All buses should be numbered or otherwise identified. The location where short
circuits are required should be selected. In many studies, all buses are faulted. The
type of short-circuit currents required is based on the short circuit rating of the
equipment located at the faulted bus.
A system one-line diagram
The system one line diagram is fundamental to short circuit analysis. It should
include all significant equipment and components and show their interconnections.
System conditions for most severe duty
It is sometimes difficult to predict which of the intended or possible system conditions
should be investigated to reveal the most severe duties for various components.
Severe duties are those that are most likely to tax the capabilities of components.
Future growth and change in the system can modify short circuit currents. For
example, the initial utility available short circuit duty for a system may be 150 mVA
but further growth plans may call for an increase in available duty to 750 mVA
several years hence. This increase could substantially raise the short circuit duties
on the installed equipment. Therefore, the increase must be factored in the present
calculations so that adequate equipment can be selected. In a similar manner, future
expansions very often will raise short circuit current duties in various parts of the
power system so that future expansions must also be considered initially. The most
severe duty usually occurs when the maximum concentration of machinery is in
operation and all interconnections are closed. The conditions most likely to influence
the critical duty include:
1. Which machines and circuits are to be considered in actual operation?
2. Which switching units are to be open or closed?
3. What future expansions or system changes will affect in plant short circuit
currents?
Preparing impedance diagrams
The impedance diagram displays the interconnected circuit impedances that control
the magnitude of short circuit currents. The diagram is derived from the system oneline diagram, showing impedance for every system component that exerts a
significant effect of short circuit current magnitude. Not only must the impedances be
interconnected to reproduce actual circuit conditions, but it will be helpful to preserve
the same arrangement pattern used in the one-line diagram.
Component impedance values
Component impedance values are expressed in terms of any of the following units:
1. Ohms per phase
2. Percent on rated kVA or a reference kVA base
3. Per unit on a reference kVA
In formulating the impedance diagram, all impedance values must be expressed in
the same units, either in Ohms per phase or per unit on a reference kVA base
(percent is a form of per unit).
Use of per unit of ohms
Short circuit calculations can be made with impedances represented in per unit or
ohms. Both representations will yield identical results. Which should be used?
In general, if the system being studied has several different voltages levels or is a
high voltage system, per unit impedance representation will provide the easier, more
straightforward calculation. A per unit system is ideal for studying multi voltage
systems. Also, most of the components included in high voltage networks (machines,
transformers and utility systems) are given in per unit or percent values and further
conversion is not required. On the other hand, where few or no voltage
transformations are involved and for low-voltage systems where many conductors
are included in the impedance network, representation of system elements in ohms
may provide the easier, more straightforward calculation.
Per unit representations
In the per unit system there are four base quantities: base kVA, base volts, base
ohms and base amperes. When any two of the four are assigned values, the other
two values can be derived. It is a common practice to assign study base values to
kVA and voltage. Base amperes and base ohms are then derived for each of the
voltage levels in the system. The kVA base assigned may be the kVA rating of one
of the predominant pieces of system equipment such as a generator or transformer
but more conveniently a number such as 10,000 is selected as base kVA. The latter
selection has some advantage of commonality when many studies are made while
the former choice means that the impedance or reactance of at least one significant
component will not have to be converted to a new base.
The nominal line-to-line system voltages are normally used as the base voltages.
Conversion of impedances to per unit on an assigned study kVA base will be
illustrated for various equipment components. A summary of frequently used per unit
relationships follows:
𝑃𝑒𝑟 − 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑡𝑠 =
𝑃𝑒𝑟 − 𝑢𝑛𝑖𝑡 𝑎𝑚𝑝𝑒𝑟𝑒𝑠 =
𝑃𝑒𝑟 − 𝑢𝑛𝑖𝑡 𝑜ℎ𝑚𝑠 =
Assigned values for three phase systems:
𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑜𝑙𝑡𝑠
𝐵𝑎𝑠𝑒 𝑣𝑜𝑙𝑡𝑠
𝐴𝑐𝑡𝑢𝑎𝑙 𝑎𝑚𝑝𝑒𝑟𝑒𝑠
𝐵𝑎𝑠𝑒 𝑎𝑚𝑝𝑒𝑟𝑒𝑠
𝐴𝑐𝑡𝑢𝑎𝑙 𝑜ℎ𝑚𝑠
𝐵𝑎𝑠𝑒 𝑜ℎ𝑚𝑠
Base volt = line-to-line volts
Base kVA = three phase kVA
Derived values:
𝐵𝑎𝑠𝑒 𝑎𝑚𝑝𝑒𝑟𝑒𝑠 =
𝐵𝑎𝑠𝑒 𝑘𝑉𝐴 (1000)
√3(𝐵𝑎𝑠𝑒 𝑣𝑜𝑙𝑡𝑠)
𝐵𝑎𝑠𝑒 𝑜ℎ𝑚𝑠 =
𝐵𝑎𝑠𝑒 𝑜ℎ𝑚𝑠 =
=
𝐵𝑎𝑠𝑒 𝑣𝑜𝑙𝑡𝑠
𝐵𝑎𝑠𝑒 𝑘𝑉𝐴
√3 𝐵𝑎𝑠𝑒 𝑘𝑉
√3 𝑏𝑎𝑠𝑒 𝑎𝑚𝑝𝑒𝑟𝑒𝑠
𝐵𝑎𝑠𝑒 𝑘𝑉 2 (1000)
𝐵𝑎𝑠𝑒 𝑘𝑉𝐴
Changing from per unit on an old base to per unit on a new base
𝑛𝑒𝑤 𝑏𝑎𝑠𝑒 𝑘𝑉𝐴
New 𝑋𝑝𝑢 = 𝑜𝑙𝑑 𝑋𝑝𝑢 � 𝑜𝑙𝑑 𝑏𝑎𝑠𝑒 𝑘𝑉𝐴 �
Neglecting resistance
All system components have impedance (Z) consisting of resistance (R) and
inductive reactance (X) where:
𝑍 = �𝑅 2 + 𝑋 2
Many system components such as rotating machines, transformers and reactors
have high values of reactance compared to resistance. When the system impedance
consists mainly of such components, the magnitude of a short circuit current as
derived by the basic equation I=E/Z is primarily determined by the reactance such
that the resistance can practically be neglected in the calculation. This allows a much
simpler calculation because then I=E/X.
Conductors (cables, buses and open wire lines) however, have a significant
resistance compared to their reactance so that when the system impedance contains
considerable conductor impedance, the resistance may have an effect on the
magnitude of the short circuit current and should be included in the calculation.
The result is the appearance of using Z or X interchangeably. The proper concept is
that whenever the resistance does not significantly affect the calculated short circuit
current, a network of reactance alone can be used to represent the system
impedance. When the ratio of the reactance to the resistance (X/R ratio) of the
system impedance is greater than 4, negligible errors (less than 3%) will result from
neglecting resistance. Neglecting R introduces some error but always increases the
calculated current. On systems above 600 volts, circuit X /R ratios are usually
greater than 4 and resistance can generally be neglected in short circuit current
calculations. However, on systems below 600 volts, the circuit X/R ratio at locations
remote from the supply transformer can be low and the resistance of circuit
conductors should be included in the short circuit current calculations. Because of
their high X/R ratio, rotating machines, transformers and reactors are generally
represented by reactance only, regardless of the system voltage, except
transformers with impedances less than 4%.
Transformers
Transformer reactance (impedance) will almost commonly be expressed as a
percent value (%XT or %ZT) on the transformer rated kVA.
Example: A 500 kVA transformer with an impedance of 5% on its kVA rating
(assume impedance is all reactance). Conversion to per unit on a 10,000 kVA base
(kVAb ):
𝑋𝑝𝑢 =
%𝑋𝑇
𝑘𝑉𝐴𝑏
5
10,000
�
�=
∙�
�=1
100 𝑇𝑟𝑎𝑛𝑠𝑓 𝑘𝑉𝐴
100
500
Cables and conductors
The resistance and reactance of a cable and a conductor will most frequently be
available in terms of ohms per phase per unit length.
Example: 250 ft. of a three conductor copper 500 mcm cable (600 volt) is installed in
steel conduit on a 480 volt system. Conversion to per unit on a 10,000 kVA base
(kVAb ):
R=0.0287 ohms/1000 ft.
R=0.00718 ohms/ft.
X=0.0301 ohms/ft.
X=0.00753 ohms/ft.
𝑘𝑉𝐴𝑏
10,000
𝑅𝑝𝑢 = 𝑅 �
� = 0.00718 ∙ �
� = 0.312
2
1000 𝑘𝑉
1000 ∙ 0.482
𝑋𝑝𝑢 = 𝑋 �
𝑘𝑉𝐴𝑏
10,000
� = 0.00753 ∙ �
� = 0.327
2
1000 𝑘𝑉
1000 ∙ 0.482
For high voltage cables (above 600 volts) the resistance of cables can generally be
omitted, in fact for short high voltage cable runs (less than 1000 feet) the entire
impedance of the cable can be omitted with negligible error.
The electric utility system
The electric utility system is usually represented by a single equivalent reactance
referred to the user’s point of connection which is equivalent to the available short
circuit current from the utility. This value is obtained from the utility and may be
expressed in several ways:
1. Three phase short circuit amperes available at a given voltage.
2. Three phase short circuit kVA available.
3. Reactance in ohms-per phase (sometimes R+jX) at a given voltage.
4. Percent or per unit reactance on a specified kVA base.
The X/R ratio of a utility source varies greatly. Sources near generating plants have
higher X/R ratio (15-30) while short circuit current levels of long open wire lines have
lower X/R ratios (2-15). Typically, the X/R value of a utility source is from 5 to 12. As
explained previously, R may be neglected with a small error (less than 3%) when the
X/R ratio is greater than 4. However, it is always more accurate to include R. If the
X/R ratio is known or estimated, then R may be determined by dividing X by the
value of the X/R ratio.
X
If X/R=10 and X=0.0025 ohms per phase then R = 10 =
phase.
0.0025
10
= 0.00025 ohms per
Example: Conversion to per unit on a 10,000 kVA base (𝑘𝑉𝐴𝑏 )
1. Available three phase short circuit kVA 500,000 kVA (500 MVA)
𝑋𝑝𝑢 =
𝑘𝑉𝐴𝑏
10,000
=
= 0.02
𝑘𝑉𝐴𝑠𝑐 500,000
2. Available three phase short circuit amperes = 20,94 at 13.8 kV
𝑋𝑝𝑢 =
𝑘𝑉𝐴𝑏
√3𝐼𝑠𝑐 𝑘𝑉
=
10,000
√3 ∙ 20,94 ∙ 13.8
= 0.02
3. Equivalent utility reactance = 0.2 per unit on a 100,000 kVA base
𝑋𝑝𝑢 = 𝑋𝑝𝑢 𝑜𝑙𝑑 �
𝑘𝑉𝐴𝑏
10,000
� = 0.2 ∙ �
� = 0.02
𝑘𝑉𝐴𝑜𝑙𝑑
100,000
4. Equivalent utility reactance = 0.38 ohms per phase at 13.8 kV
𝑘𝑉𝐴𝑏
10,000
𝑋𝑝𝑢 = 𝑋 �
� = 0.38 ∙ �
� = 0.02
2
1000 𝑘𝑉
1000 ∙ 13.82
Rotating machines
Machine reactances are usually expressed in term of per cent reactance (%Xm) or
per unit reactance Xpu on the normal rated kVA of the machine. Either the subtransient reactance (X”) or the transient reactance (X’) should be selected,
depending on the type of short circuit calculation required.
Motors rated 600 volts or less
In systems of 600 volts or less, the large motors (of several hundred horsepower)
are usually few in number and represent only a small portion of the total connected
horsepower. These large motors can be represented individually, or they can be
lumped in with the smaller motors, representing the complete group as one
equivalent motor in the impedance diagram. Small motors are turned off and on
frequently, so it is practically impossible to predict which ones will be on the line
when a short circuit occurs. Therefore, small motors are generally lumped together
and assumed to be running. Where more accurate data are not available, the
following procedures may be used in representing the combined reactance of a
group of miscellaneous motors:
1. In all 240 volt systems, a substantial portion of the load consists of lighting, so
assume that the running motors are grouped at the transformer secondary
bus and have a reactance of 25% on a kVA rating equal to 50% on the
transformer rating.
2. In systems rated 600 or 480 volts, assume that the running motors are
grouped at the transformer secondary bus and have a reactance of 25% on a
kVA rating equal to 100% of the transformer rating.
3. Groups of small induction motors as served by a motor control center can be
represented by considering the group to have a reactance of 25% on a kVA
rating equal to the connected motor horsepower.
Example: A 500 HP, 0.8 PF, synchronous motor has a sub-transient reactance (𝑋𝑑" )
of 15%. Conversion to per unit on a 10,000 kVA base (kVAb ):
𝑋𝑑" =
%𝑋𝑑"
𝑘𝑉𝐴𝑏
15 10,000
�
�=
∙�
�=3
100 𝑀𝑜𝑡𝑜𝑟 𝑘𝑉𝐴
100
500
Motors rated above 600 volts
Motors rated above 600 volts are generally high in horsepower rating and will have a
significant bearing on short circuit current magnitudes. Very large motors of several
thousand horsepower should be considered individually and their reactances should
be accurately determined before starting the short circuit study. However, in large
plants where there are numerous motors of several hundred horsepower, each
located at one bus, it is often desirable to group such motors and represent them as
a single equivalent motor with one reactance in the impedance diagram.
Other circuit impedances
There are other circuit impedances such as those associated with circuit breakers,
current transformers, bus structures and connections, which for ease of calculation
are usually neglected in short circuit current calculations. Accuracy of the calculation
is not generally affected because the effects of the impedances are small, and
omitting them provides conservative short circuit currents. However, on low voltage
systems, there are cases where their inclusion in the calculation can result in a lower
short circuit current and allow the use of lower rated circuit components. The system
designer may want to include these impedances in such cases.
System driving voltage (E)
The system driving voltage (E) in the basic equation can be represented by the use
of a single over-all driving voltage rather than the array of individual, unequal
generated voltages acting within individual rotating machines. This single driving
voltage is equal to the prefault voltage at the point of fault connection. The equivalent
circuit is a valid transformation accomplished by Thevenin’s theorem and permits an
accurate determination of a short circuit current for the assigned values of system
impedance. The prefault voltage referred to is ordinarily taken as the system nominal
voltage at the point of fault as this calculation leads to the full value of short circuit
current that may be produced by the probable maximum operating voltage.
In making a short circuit calculation on three phase balanced systems, a single
phase representation of a three phase system is utilized so that all impedances are
expressed in ohms per phase, and the system driving voltage (E) is expressed in
line-to-neutral volts. Line-to-neutral voltage is equal to line-to line-voltage divided by
the √3.
When using the per unit system, if the system per unit impedances are established
on voltage bases equal to system nominal voltages, the per-unit driving voltage is
equal to 1. In the per unit system, both line-to-line voltage and line-to-neutral voltage
have equal values. That is, both would have a value of 1. When system impedance
values are expressed in ohms per phase rather than per unit, the system driving
voltage would be equal to the system line-to-neutral voltage.
Shunt connected impedances
In addition to the components already mentioned, every system includes other
components or loads that would be represented in a diagram as shunt connected
impedances. A technically accurate solution requires that these impedances be
included in the equivalent circuit used in calculating a short circuit current, but
practical considerations allow the general practice of omitting them. Such
impedances are relatively high values and their omission will not significantly affect
the calculated results.
Short circuit current calculations
After the impedance diagram is prepared, the short circuit currents can be
determined. This can be accomplished by longhand calculation, network analyser or
digital computer. In general, the presence of closed loops in the impedance network,
such as those found in large industrial plant high voltage systems, and the need for
short circuit duties at many system locations will favor using a digital computer.
Simple radial systems, such as those used in most low-voltage systems, can be
easily resolved by longhand calculations though digital computers can yield
significant time savings particularly when short circuit duties at many locations are
required and when resistance is being included in the calculation.
A longhand solution requires the combining of impedances in series and parallel
from the source driving voltage and Z (or X) is the single equivalent network
impedance. The calculation to derive the symmetrical short circuit current is I = E/Z
where E is the system driving voltage and (or X) is the single equivalent network
impedance. When calculations are made in per unit, the following formulas apply:
Epu
Symmetrical three phase short circuit current in per unit Ipu = Z
Symmetrical three phase short circuit current in kVA kVA =
pu
kVAb
Zpu
I
Symmetrical three phase short circuit current in amperes I = Z b
pu
where Ipu - per unit amperes, Zpu - equivalent network per unit impedance, Epu − per
unit volts, Ib - base amperes, kVAb - base kVAb .
A new combination of impedances to determine the single equivalent network
impedance is required for each fault location. For a radial system, the longhand
solution is very simple. For systems containing loops, simultaneous equations may
be necessary, though delta-wye network transformations can usually be used to
combine impedances.
Use of estimating tables and curves
There are many times when a short circuit current duty is required at the secondary
of a transformer or at the end of a low voltage conductor. Curves and tables, which
give the estimated short circuit current duty, are available for commonly used
transformers and for various conductor configurations. Use of these tables may
eliminate the need for a formal short circuit current study and can be used where
appropriate.
Means for reducing short-circuit currents
There is a natural reduction of a short circuit current duty due to the impedance of
the conductors from the power source to the loads. For example, the short circuit
duty at the terminals of a 1500 kVA, 480 volt transformer may be 37,000 amperes,
while at the end of a 600 amp cable run, the duty may be 13,000 amperes. But
beyond this natural reduction in short circuit duty, it is sometimes desired or
necessary to insert additional impedance in the form of reactance to achieve a lower
required duty for application of some specific equipment. This can be done with
current limiting reactors (all voltages) or current limiting busways (600 volts and
below).
For instance, the available short circuit duty from a utility service supplying a plant or
building may be 850 mVA at 13.8 kV. This would require 1000 mVA circuit breakers
for the in-plant equipment. A more economical approach might be to apply current
limiting reactors on the incoming line to reduce the available duty to less than 500
mVA so that lower cost 500 mVA circuit breakers can be applied.
The general procedure is to determine the additional reactance required to reduce
the short circuit duty to the desired level as follows:
𝑋=
𝐸
𝐼𝑑𝑒𝑠𝑖𝑟𝑒𝑑
−
𝐸
𝐼𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒
Example of AC short circuit current calculation
The provided example illustrates how short circuit components are calculated using
previously described procedures.
It is clear that the selection of the calculation
method must be coordinated with specific component requirements.
Step A – The system one-line diagram
The figure below contains basic details of system components and the way they are
connected. Also the basic system parameters are shown. The schematic diagram
presents the necessary data as follows:
-
The voltage, short circuit and X/R ratio from the utility system
-
The kVA, voltage, connection, impedance and X/R ratio for transformers T1,
T2 and T3.
-
The type, HP, RPM, reactance and X/R ratios for motors M1, M2, M3 and
motor summation designated as M4.
-
The length and impedance of the underground cables
Utility 1500
MVA, X/R=15
115 kV Bus 1
10 Ω
T1/15 MVA
Z=7%
X/R=20
13.8 kV Bus 2
100 ft Cu cable
Z=0.0977+j0.0385
Ω/1000 ft
Bus 3
300 ft Cu cable
Z=0.0614+j0.0359
Ω/1000 ft
Bus 4
M1
1.0 P.F Syn
4000 HP
1800 rpm
Xd”=15%
X/R=28.9
6Ω
Bus 6
T2/3750 kVA
Z=5.5%
X/R=11
4.16 Y / 2.4 kV
200 ft Cu cable
Z=0.3114+j0.0472
Ω/1000 ft
T3/1500 kVA
Z=5.75%
X/R=6.5
Bus 5
480 Y/277 V
M2
M3
Ind
500 HP
1800 rpm
Xd”=16.7%
X/R=19.3
Ind
2000 HP
3600 rpm
Xd”=16.7%
X/R=30
Bus 7
250 ft Cu cable
Z=0.0534+j0.0428
Ω/1000 ft
F
M4
Bus 8
Σ 1500 HP
Xd”=16.7%
4-100 HP
8-50 HP
28-25 HP
Step B – Type and location of short circuits
Protective devices are located at buses 2, 5, 7 and 8 and at these locations short
circuit currents need to be calculated. High voltage power circuit breakers and
associated equipment are located at buses 2 and 5 while low voltage circuit breakers
are located at buses 7 and 8. Three phase bolted short circuits are needed for
device selection and they will be calculated since the line-to-ground short circuit
currents are limited by the grounding resistors. In addition, the most severe duty
occurs when utility is connected, motors are operating, and breakers are closed.
Step C – System impedance diagrams
System impedance diagrams should be presented according to the one line diagram.
The arrangement of network elements should allow easy identification of given
components in the two types of diagrams (one line vs. impedance) although
identification of system components and significant points in the network circuit may
become difficult or even impossible as the network is resolved and converted into a
single value impedance. The per unit system lends itself to an analysis of this system
because of the several voltage levels. In this particular example, a base power of
15,000 kVA is selected. The assigned base voltages will be the nominal system
voltages of 13,800, 4,160 and 480 volts. Base amperes and base ohms for each of
the voltage levels can be derived as shown below:
Assigned Values Calculated Values
kVA B
kV B
IB
ZB
15,000
15,000
15,000
13.8
4.16
0.48
628
2,084
18,064
12.7
1.1539
0.0153
The figure below is an impedance diagram for the electrical network shown above.
The assigned impedance values are based on the ANSI and IEEE recommended
machine modified sub-transient Xd” values for-multi voltage systems. The figure
shows first-cycle impedance presentation.
E
Ut
0.0007+j0.01
Bus 1
T1
0.0035+j0.0699
M1
0.0251+j0.
703
N
System
Driving
Voltage
M2
0.3331+j6.3284
Bus 3
C1
0.0008+j0.0003
Bus 2 13.8 kV
M3
0.0449+j1.3917
Bus 5
T2
0.0199+j0.2191
Bus 4
C2
0.0014+j0.0008
4-M4 (100)
0.9054+j7.515
8-M4 (50)
1.3664+j7.515
Bus 7
T3
0.0874+j0.5683
28-M4 (25)
1.5727+j5.9763
C4
0.8691+j0.6966
Bus 6
C3
0.0049+j0.0007
Bus 8
The per unit values for all component impedances for the above network impedance
diagrams are calculated as shown below:
First cycle Z=R+jX
Utility Z=15,000/1,500,000=0.01 pu
X/R=15, tan-115=86.19°
R=(cos 86.19)(0.01), X=(sin 86.19)(0.01)
Transformer T1
T1=(7x15,000)/(100x15,000)=0.07 pu
X/R=20, , tan-120=87.14°
R=(cos 87.14)(0.07), X=(sin 87.14)(0.07)
Motor M1 cable
Z=0.0977+j0.0386 Ω/1000 ft
Z=(100/1000)x(0.0977+j0.0385)x15/(13.8)^2
Motor M1
X=(15x15,000)/(100x4,000x0.8)=0.703
X/R=28, R=X/28=0.0251
1x(R+jX)
1.5(R+jX)
Transformer T2 cable
Z=0.0614+j0.0359 Ω/1000 ft
Z=300/1,000x(0.0614+j0.0359)x15/(13.8)^2
Transformer T2
T2=(5.5x15,000)/(100x3750)=0.22 pu
X/R=11, , tan-111=84.80°
R=(cos 84.8)(0.22), X=(sin 84.8)(0.22)
Motor M2
X=(16.7x15,000)/(100x500x0.95)=5.2737
X/R=19, R=X/19=0.2776
1.2x(R+jX)
3.0(R+jX)
Motor M3
X=(16.7x15,000)/(100x2000x0.9)=1.3917
X/R=31, R=X/31=0.0449
1.0x(R+jX)
1.5(R+jX)
Transformer T3 cable
Z=0.3114+j0.0472 Ω/1000 ft
Z=200/1,000x(0.3114+j0.0472)x15/(13.8)^2
Transformer T3
T2=(5.75x15,000)/(100x1500)=0.575 pu
X/R=6.5, , tan-16.5=81.25°
R=(cos 81.25)(0.575), X=(sin 81.25)(0.575)
Motor M4 Σ1500 HP
Assumed 25% 100 HP = 4-100 HP
35% 50 HP = 8-50 HP
Remainder 25 HP = 28 – 25 HP
100 HP
X = (16.7x15,000)/(100x4x100)=6.2625
X/R=8.3, R=X/8.3=0.75457
0.0007+j0.01
0.0035+j0.0699
0.0008+j0.0003
0.0251+j0.703
0.0014+j0.0008
0.0199+j0.2191
0.3331+j6.3284
0.0449+j1.3917
0.0049+j0.0007
0.0874+j0.5683
1.2 (R+jX)
3.0 (R+jX)
First cycle Z=R+jX
0.9054+j7.515
50 HP
X = (16.7x15,000)/(100x8x50)=6.2625
X/R=5.5, R=X/5.5=1.1386
1.2 (R+jX)
3.0 (R+jX)
25 HP
X = (16.7x15,000)/(100x28x25)=3.5786
X/R=3.8, R=X/3.8=0.9417
1.67 (R+jX)
Cable (Bus 7 to Bus 8)
Z=0.0534+j0.0428 Ω/1000 ft
Z=250/1,000x(0.0534+j0.0428)x15/(0.48)^2
1.3664+j7.515
1.5727+j5.9763
0.8691+j0.6966
Step D – Calculation of short circuit current
The presented results depend on the method used to resolve the impedance
network. Accurate results will be obtained if network resolution is treated as a
complex quantity. If the network is treated as separate R and X networks, the result
will provide slightly higher short circuit currents. In the case the system impedance
has a large resistance compared to the reactive component, then the resultant
current calculation will increase. A completely separate R and X calculation is
performed to calculate the short circuit X/R ratios in line with the ANSI standard. The
base voltages were assigned values and they are equal to the nominal system
voltages. These voltages are the same as the pre-short circuit or operating voltages.
Basically, the system per-unit driving voltage (E) equals 1.
Two cases are presented by:
-
Pointing out an applicable network
-
Pointing out the network resolution
-
Solving the network to a single value impedance
-
Calculating the symmetrical fault current
The First Cycle Short Circuit Current Calculation at location F
The impedances shown in the system impedance diagrams can be resolved into a
single impedance value that limits the value of the three phase short circuit current at
fault location F. The resolution methods that are applied are:
Method A: Neglecting resistive components of the impedances except for low
voltage cables and neglecting resistive and reactive components for high voltage
cables. This can be justified since neglected components have smaller values than
considered components.
Method B: Consider both resistive and reactive components, however resolve each
independently. This can be done to consider all resistances and reactances but to
simplify the calculation relative to the calculation in Method C.
Method C: Consider all components resistive and reactive and solve the network as
a complex quantity.
Method A – Network resolution
Branch
X
1/X
Ut+T1
0.01+0.07
12.5
M1
0.703
1.4225
Ut+M1+T1
0.0718
13.9225
M2
6.3284
M3
1.3917
0.7185
M2+M3
1.1409
0.8765
M2+M3+T2
1.1409+0.22 0.7348
(Ut+T1+M1)+(M2+M3+T2)
0.0682
14.6573
(Ut+T1+M1+M2+M3+T2)+T3 0.0682+0.575 1.5547
ΣM4
2.3068
0.4335
Net X
0.503
1.9882
Equivalent Z=0+j0.503 per unit
Method B – Network resolution
Branch
Ut+T1
M1+C1
Ut+M1+T1+C1
M2
M3
M2+M3
M2+M3+T2+C2
(Ut+T1+M1+C1)+(M2+M3+T2+C2)
R
1/R
0.0007+0.0035
238.0952
0.0251+0.0008
38.61
0.0036
276.7052
0.3331
3.0021
0.0449
22.2717
0.0396
25.2738
0.0396+0.0199+0.0014 16.4294
0.0034
293.1346
Branch
R
(Ut+T1+M1+C1+M2+M3+T2+C2)+C3+T3 0.0034+0.0049+0.0874
ΣM4
0.4045
Net R
0.0774
1/R
10.4493
2.4722
12.9215
Branch
X
1/X
Ut+T1
0.01+0.0699
12.5156
M1+C1
0.703+0.0003
1.4219
Ut+M1+T1+C1
0.0717
13.9375
M2
6.3284
0.158
M3
1.3917
0.7185
M2+M3
1.1409
0.8765
M2+M3+T2+C2
1.1409+0.2191+0.0008 0.7349
(Ut+T1+M1+C1)+(M2+M3+T2+C2)
0.0682
14.6724
(Ut+T1+M1+C1+M2+M3+T2+C2)+C3+T3 0.0682+0.0007+0.5683 1.5694
ΣM4
2.3068
0.4335
Net X
0.4993
2.0029
Equivalent Z=Net R + Net X=0.0774+j0.4993 per unit
X/R ratio = 0.4993/0.0774=6.45
Method C – Network Resolution
Ut+T1+M1+C1+M2+M3+T2+C2=0.0034+j0.0682
ΣM4=0.4469+j2.3149
(Ut+T1+M1+C1+M2+M3+T2+C2)+(C3+T3)=(0.0034+).0049+0.0874)+j(0.0682+0.00
07+0.5683)= 0.0957+j0.6372
Net Z =
(Ut+T1+M1+C1+M2+M3+T2+C2+C3+T3)+ΣM4=(0.0957+j0.6372)x(0.4469+j2.3149)/
((0.0957+j0.6372)+(0.4469+j2.3149))=0.0796+j0.5 per unit
The symmetrical short circuit current at location F is:
𝐼 = 𝐼𝑏 (𝐼 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡) = 𝐼𝑏 ∙
𝐸
1
18064
= 18064 ∙
=
𝑍𝑛𝑒𝑡
𝑍𝑛𝑒𝑡
𝑍𝑛𝑒𝑡
Method Z net (per unit) Symmetrical RMS (A)
A
0+j0.503
35912
B
0.0774+j0.4993
35751
C
0.0796+j0.5
35679
It can be concluded that calculated currents differ only by 0.65% depending on the
calculation method which justifies neglecting high voltage resistance as well as cable
reactance, thus simplifying calculations.
Summary
This course presented fundamental facts about calculating short circuit currents in
the electrical power systems. Fault types are described and how these fault types
are treated by various international standards is illustrated. Important considerations
about network equipment that need to be taken into account when calculating fault
levels were mentioned. Basic conversion to per units was described for various
electrical network elements. Two different calculation methods were demonstrated
on the practical electrical system along with a step-by-step calculation procedure.