 # Problems

• 7.1 Four 11 kV generators designated A, B, C, and D each have a subtransient reactance of 0.1 p.u. and a rating of 50 MVA. They are connected in parallel by means of three 100 MVA reactors which join A to B, B to C, and C to D; these reactors have per unit reactances of 0.2,0.4, and 0.2, respectively. Calculate the volt-amperes and the current flowing into a three-phase symmetrical fault on the terminals of machine B. Use a 50 MVA base.
• (Answer: 937.5 MVA; 49 206 A)
• 7.2 Two 100 MVA, 20 kV turbogenerators (each of transient reactance 0.2 p.u.) are connected, each through its own 100 MVA, 0.1 p.u. reactance transformer, to a common 132 kV busbar. From this busbar, a 132 kV feeder, 40 km in length, supplies an 11 kV load through a 132/11 kV transformer of 200 MVA rating and reactance 0.1 p.u. If a balanced three-phase short circuit occurs on the low-voltage terminals of the load transformer, determine, using a 100 MVA base, the fault current in the feeder and the rating of a suitable circuit breaker at the load end of the feeder. The feeder impedance per phase is (0.035 + j0.14) V/km.
• 7.3 Two 60 MVA generators of transient reactance 0.15 p.u. are connected to a busbar designated A. Two identical machines are connected to another busbar B. Busbar A is connected to busbar B through a reactor, X. A feeder is supplied from A through a step-up transformer rated at 30 MVA with 10% reactance.

Calculate the reactance X, if the fault level due to a three-phase fault on the feeder side of the 30 MVA transformer is to be limited to 240 MVA. Calculate also the voltage on A under this condition if the generator voltage is 13 kV (line).

• (Answer: X = 0.075 p.u.; VA = 10.4 kV)
• 7.4 A single line-to-earth fault occurs on the red phase at the load end of a 66 kV transmission line. The line is fed via a transformer by 11 kV generators connected to a common busbar. The line side of the transformer is connected in star with the star point earthed and the generator side is in delta. The positive-sequence reactances of the transformer and line are /10.9 V and /44 V, respectively, and the equivalent positive and negative-sequence reactances of the generators, referred to the line voltage, are /18 V and /14.5 V, respectively. Measured up to the fault the total effective zero sequence reactance is /150 V. Calculate the fault current in the lines if resistance may be neglected. If a two- line-to-earth fault occurs between the blue and yellow lines, calculate the current in the yellow phase.
• (Answer: 391 A; 1486 A)
• 7.5 A single-line-to-earth fault occurs in a radial transmission system. The following sequence impedances exist between the source of supply (an infinite busbar) of voltage 1 p.u. to the point of the fault: Z: = 0.3 + /0.6 p.u., Z2 = 0.3 + /0.55 p.u., Z0 = 1 + /0.78 p.u. The fault path to earth has a resistance of 0.66 p.u. Determine the fault current and the voltage at the point of the fault.
• (Answer: If = 0.74 p.u.; Vf = (0.43 — /0.23) p.u.)
• 7.6 Develop an expression, in terms of the generated e.m.f. and the sequence impedances, for the fault current when an earth fault occurs on phase (A) of a three-phase generator, with an earthed star point. Show also that the voltage to earth of the sound phase (B) at the point of fault is given by Two 30 MVA, 6.6 kV synchronous generators are connected in parallel and supply a 6.6 kV feeder. One generator has its star point earthed through a resistor of 0.4 V and the other has its star point isolated. Determine: (a) the fault current and the power dissipated in the earthing resistor when an earth fault occurs at the far end of the feeder on phase (A); and (b) the voltage to earth of phase (B). The generator phase sequence is ABC and the impedances are as follows:

 Generator p.u. Feeder V/p.h. To positive-sequence currents j0.2 j0.6 Figure 7.30 System for Problem 7.7

 To negative-sequence currents j0.16 J0.6 To zero-sequence currents j0.06 j'0.4

Use a base of 30 MVA.

• (Answer: (a) 5000ff - 58.45° A; 10 MW; (b) -2627 - /1351 V
• 7.7 An industrial distribution system is shown schematically in Figure 7.30. Each line has a reactance of /0.4 p.u. calculated on a 100 MVA base; other system parameters are given in the diagram. Choose suitable short-circuit ratings for the oil circuit breakers, situated at substation A, from those commercially available, which are given in the table below.
 Short circuit (MVA) 75 150 250 350 Rated current (A) 500 800 1500 2000

The industrial load consists of a static component of 5 MVA and four large induction motors each rated at 6 MVA. Show that only three motors can be Figure 7.31 Circuit for Problem 7.9

started simultaneously given that, at starting, each motor takes five times fullload current at rated voltage, but at 0.3 p.f.

7.8 Explain how the Method of Symmetrical Components may be used to represent any 3 p.h. current phasors by an equivalent set of balanced phasors.

A chemical plant is fed from a 132 kV system which has a 3 p.h. symmetrical fault level of 4000 MVA. Three 15MVA transformers, connected in parallel, are used to step down to an 11 kV busbar from which six 5 MVA, 11 kV motors are supplied. The transformers are delta-star connected with the star point of each 11 kV winding, solidly earthed. The transformers each have a reactance of 10% on rating and it may be assumed that X = X2 = Xq. The initial fault contribution of the motors is equal to five times rated current with 1.0 p.u. terminal voltage.

Using a base of 100 MVA,

a. calculate the fault current (in A) for a line-to-earth short circuit on the 11 kV busbar with no motors connected;

b. calculate the 3 p.h. symmetrical fault level (in MVA) at the 11 kV busbar if all the motors are operating and the 11 kV busbar voltage is 1.0 p.u.

• (Answer: (a) 22 kA; (b) 555 MVA)
• (From Engineering Council Examination, 1996)
• 7.9 Describe the effect on the output current of a synchronous generator following a solid three-phase fault on its terminals.

For the system shown in Figure 7.31 calculate (using symmetrical

components):

a. the current flowing in the fault for a three-phase fault at busbar A;

b. the current flowing in the fault for a one-phase-to-earth fault at busbar B;

c. the current flowing in the faulted phase of the overhead line for a one-phase- to earth fault at busbar B.

Generators G1 and G2: Xf = X2 = /0.1 p.u.; 11 kV

Transformers T1 and T2: X = X2 = Xq = /0.1 p.u.; 11/275 kV

(Earthed star-delta)

Line: Z = Z2 = j0.05 p.u., Zo = j0.1 p.u.; 275 kV (All p.u. values are quoted on a base of 100 MVA)

Assume the pre-fault voltage of each generator is 1 p.u. and calculate the symmetrical fault currents (in amps) immediately after each fault occurs. (Answer: (a) 1.89 kA; (b) 2.18 kA; (c) 0.89 kA)

• (From Engineering Council Examination, 1997)
• 7.10 Why is it necessary to calculate short-circuit currents in large electrical systems?

A generator rated at 400 MW, 0.8 power factor, 20 kV has a star-connected stator winding which is earthed at its star point through a resistor of 1V. The generator reactances, in per unit on rating, are: The generator feeds a delta-star-connected generator transformer rated at 550 MVA which steps the voltage up to a 275 kV busbar. The transformer star- point is solidly earthed and the transformer reactance is 0.15 p.u. on its rating. The 275 kV busbar is connected only to the transformer. Assume that for the transformer X1 = X2 = X0.

Using a base of 500 MVA calculate the base current and impedance of each voltage level.

Calculate the fault current in amperes for:

a. a 275 kV busbar three-line fault;

b. a 275 kV single-line-to earth fault on the busbar;

c. a 20 kV three-line fault on the generator terminals;

d. a 20 kV single-line-to-earth fault on the generator terminals.

• (Answer: (a) 3.125kA; (b) 4.1 kA; (c) 72.15kA; (d) 11.44kA)
• (From Engineering Council Examination, 1995) 