Calculation of short circuit current

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 Calculation of short circuit current

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مُساهمةموضوع: Calculation of short circuit current   2011-11-15, 17:10

. Calculation of short circuit current

Calculation of short circuit curren

[/size].The current that flows through an element of a power system is a parameter which
can be used to detect faults, given the large increase in current flow
when a short circuit occurs.

For this reason a review of the concepts and
procedures for calculating fault currents will be made in this chapter,
together with some calculations illustrating the methods used.

Although the use of these short-circuit
calculations in relation to protection settings will be-considered in
detail, it is important to bear in mind that these calculations are also
required for other applications, for example calculating the substation
Earthing grid, the selection of conductor sizes and for the
specifications of equipment such as power-circuit breakers.
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. 1 Mathematical derivation of fault currents
The treatment of electrical faults should be carried out as a function of time,
from the start of the event at time
until stable conditions are reached, and therefore it is necessary to
use differential equations when calculating these currents. In order to
illustrate the transient nature of the current,




consider an RL circuit as a simplified equivalent of the circuits in
electricity-distribution networks. This simplification is important
because all the system equipment must be modeled in some way in order to
quantify the transient values which can occur during the fault
condition.
For the circuit shown in Figure 1, the mathematical expression which defines the behaviour of the current is:
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.

e(t) = L di + Ri(t) 2.1


Figure 1 RL, circuit for transient analysis studyThis is a differential equation with constant coefficients, of which the solution is in two parts:



Where:
ih(t) Is the solution of the homogeneous equation correspond ing to the transient period and ip(t) is the solution to the particular equation corresponding to the steady-state period.
By the use of differential equation theory, which will not be discussed
in detail here, the complete solution can be determined and expressed
iii the following form:



Where:


α = the closing angle which defines the point on the source sinusoidal voltage when the fault occurs and
It
can be seen that, in eqn. 2.2, the first term varies sinusoidally,
while the second term decreases exponentially with a time constant of
L/R. The latter term can be recognised as the DC component of the
current, and has an initial maximum
value when, and zero value when Φ=α, see Figure 2.
It is impossible to predict at what point the fault will be applied on
the sinusoidal cycle and therefore what magnitude the DC component will
reach. If the tripping of the circuit, owing to a fault, takes place
when the sinusoidal component is at its negative peak, the DC component
reaches its theoretical maximum value half a cycle later.





Figure 2 Variation of fault current with time

a (α–Φ) =0
b (αΦ)=π/2
An approximate formula for calculating the effective value of the total asymmetric current,
including the AC and DC components, with acceptable accuracy can be obtained from the following expression:


The fault current which results when an alternator is short circuited
can easily be analysed since this is similar to the case which has
alreadybeen analysed, i.e. when voltage is, applied to an RL circuit.
The reduction in current from its value at the onset, owing to the
gradual decrease in the magnetic flux caused by the reduction of the
e.m.f. of the induction current, can be seen in Figure 3. This effect is
known as armature reaction.
The physical situation that is presented to a generator, and which makes
the calculations quite difficult, can be interpreted as a
reactancewhich varies with time. Notwithstanding this, in the majority
of practical applications it is possible to take account of the
variation of reactance in only three stages without producing
significant errors. In Figure 4 it will be noted that the variation of
current with time, 1(t), comes close to the three discrete levels of
current, I", 1' and I, the subtransient, transient and steady-state
currents, respectively. The corresponding values of direct axis
reactance are denoted by and Xd,

Figure 3 Transient short-circuit currents in a synchronous generator


Figure 4 Variation of current with time during a fault

Figure 5 Variation of generator reactance with time during a fault
And the typical variation with, time for each of these is illustrated in
Figure 5.
To sum up, when calculating short-circuit currents it is necessary to
take into account two factors which could result in the currents varying
with time:
· the presence of the DC component;
· the behaviour of the generator under short circuit conditions.

In studies of electrical protection some adjustment
has to be made to the values of instantaneous short circuit current
calculated using subtransient reactance's which result in higher values
of current.
Time delay units can be set using the same values but, in some cases,
short-circuit values based on the transient reactance are used,
depending on the operating speed of the protection relays. Transient
reactance values are generally used in stability studies.
Of necessity, switchgear specifications require reliable calculations of
the short-circuit levels which can be present on the electrical
network. Taking into account the rapid drop of the short-circuit current
due to the armature reaction of the synchronous machines, and the fact
that extinction of an electrical arc is never achieved instantaneously,
ANSI Standards C37.010 and C37.5 recommend using different values of
subtransient reactance when calculating the so-called momentary and
interrupting duties of switchgear.
Asymmetrical or symmetrical r.m.s. values can be defined depending on
whether or not the DC component is included. The peak values are
obtained by multiplying the R.M.S. values by.

The asymmetrical values are calculated as the square
root of the sum of the squares of the DC component and the r.m.s. value
of the AC current, i.e.:

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Calculation of short circuit current
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