Calculation of short circuit current |
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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.
1 Mathematical derivation of fault currents
The treatment of electrical faults
should be carried out as a function of time, For the circuit shown in Figure 1, the mathematical expression which defines the behaviour of the current is: e(t) = L di + Ri(t) 2.1
This is a differential equation with constant
coefficients, of which the solution is in two parts:
Where: i_{h}(t) Is the solution of the homogeneous equation corresponding to the transient period and i_{p}(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:
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
b (α–Φ)=π/2
An approximate
formula for calculating the effective value of the
total asymmetric current,
The fault current which results when an alternator is short circuited can easily be analysed since this is similar to the case which has already been 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 reactance which 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 X_{d,}
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.
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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