Calculation of short circuit current

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,
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:

e(t) = L di + Ri(t)        2.1

Figure 1 RL, circuit for transient analysis study

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:
 

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 (α–Φ)=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 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,

Figure 3 Transient short-circuit currents in a synchronous generator

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.: