# Problems

- 8.1 A round-rotor generator of synchronous reactance 1 p.u. is connected to a transformer of 0.1 p.u. reactance. The transformer feeds a line of reactance
- 0.2 p.u. which terminates in a transformer (0.1 p.u. reactance) to the LV side of which a synchronous motor is connected. The motor is of the round-rotor type and of 1 p.u. reactance. On the line side of the generator transformer a three-phase static reactor of 1 p.u. reactance per phase is connected via a switch. Calculate the steady-state power limit with and without the reactor connected. All per unit reactances are expressed on a 10 MVA base and resistance may be neglected. The internal voltage of the generator is 1.2 p.u. and of the motor 1 p.u.
- (Answer: 5 MW and 3.13 MW for shunt reactor)
- 8.2 In the system shown in Figure 8.20, investigate the steady state stability. All per unit values are expressed on the same base and the resistance of the system (apart from the load) may be neglected. Assume that the infinite busbar voltage is 1 p.u.
- 8.3 A generator, which is connected to an infinite busbar through two 132 kV lines in parallel, each having a reactance of 70 V/phase, is delivering 1 p.u. to the infinite busbar. Determine the parameters of an equivalent circuit, consisting of a single machine connected to an infinite busbar through a reactance, which represents the above system

a. Pre-fault.

b. When a three-phase symmetrical fault occurs halfway along one line.

c. After the fault is cleared and one line isolated.

**Figure 8.20 **Line diagram of system in Problem 8.2

If the generator internal voltage is 1.05 p.u. and the infinite busbar voltage is 1.0 p.u, what is the maximum power transfer pre-fault, during the fault and post-fault?

Determine the swing curve for a fault clearance time of 125 ms.

The generator data are as follows:

Rating 60 MW at power factor 0.9 lagging.

Transient reactance 0.3 p.u.

Inertia constant 3 kWs/kVA.

- (Answer: 1.62 p.u., 0.6 p.u., 1.24 p.u.)
- 8.4 An induction motor and a generator are connected to an infinite busbar. What is the equivalent inertia constant of the machines on 100 MVA base? Also calculate the equivalent angular momentum.

Data for the machines are:

Induction motor |
Rating 40MVA; Inertia constant 1 kWs/kVA. |

Generator: |
Rating 30 MVA; Inertia constant l0 kWs/kVA. |

- (Answer 3.4Ws/VA, 0.00038 p.u.)
- 8.5 The P-V, Q-V characteristics of a substation load are as follows:

V |
1.05 |
1.025 |
1 |
0.95 |
0.9 |
0.85 |
0.8 |
0.75 |

P |
1.03 |
1.017 |
1 |
0.97 |
0.94 |
0.92 |
0.98 |
0.87 |

Q |
1.09 |
1.045 |
1 |
0.93 |
0.885 |
0.86 |
0.84 |
0.85 |

The substation is supplied through a link of total reactance 0.8 p.u. and negligible resistance. With nominal load voltage, * P =* 1 and

*1 p.u. By determining the supply voltage-received voltage characteristic, examine the stability of the system by the use of*

**Q =***All quantities are per unit.*

**dE/dV.**- 8.6 A large synchronous generator, of synchronous reactance 1.2 p.u., supplies a load through a link comprising a transformer of 0.1 p.u. reactance and an overhead line of initially 0.5 p.u. reactance; resistance is negligible. Initially, the voltage at the load busbar is 1 p.u. and the load P +
is (0.8 + j0.6) p.u. regardless of the voltage. Assuming the internal voltage of the generators is to remain unchanged, determine the value of line reactance at which voltage instability occurs.**jQ** - (Answer: unstable when
2.15 p.u.)**X =** - 8.7 A load is supplied from an infinite busbar of voltage 1 p.u. through a link of series reactance 1 p.u. and of negligible resistance and shunt admittance. The load consists of a constant power component of 1 p.u. at 1 p.u. voltage and a per unit reactive power component (Q) which varies with the received voltage (V) according to

All per unit values are to common voltage and MVA bases.

Determine the value of *X* at which the received voltage has a unique value and the corresponding magnitude of the received voltage.

Explain the significance of this result in the system described. Use approximate voltage-drop equations.

- (Answer: X = 0.25 p.u.;
*V**=*0.67 p.u.) - 8.8 Explain the criterion of stability based on the equal-area diagram.

A synchronous generator is connected to an infinite busbar via a generator transformer and a double-circuit overhead line. The transformer has a reactance of 0.15 p.u. and the line an impedance of 0 + j0.4 p.u. per circuit. The generator is supplying 0.8 p.u. power at a terminal voltage of 1 p.u. The generator has a transient reactance of 0.2 p.u. All impedance values are based on the generator rating and the voltage of the infinite busbar is 1 p.u.

a. Calculate the internal transient voltage of the generator.

b. Determine the critical clearing angle if a three-phase solid fault occurs on the sending (generator) end of one of the transmission line circuits and is cleared by disconnecting the faulted line.

- (Answer: (a) 1.035 p.u. (b) 64°
- (From
*Engineering Council Examination, 1995)* - 8.9 A 500 MVA generator with 0.2 p.u. reactance is connected to a large power system via a transformer and overhead line which have a combined reactance of
- 0.3 p.u. All p.u. values are on a base of 500 MVA. The amplitude of the voltage at both the generator terminals and at the large power system is 1.0 p.u. The generator delivers 450 MW to the power system.

Calculate

a. the reactive power in MVAr supplied by the generator at the transformer input terminals;

b. the generator internal voltage;

c. the critical clearing angle for a 3 p.h. short circuit at the generator terminals. (Answer: (a) 62MVAr; (b) 1.04 p.u.; (c) 84°)

*(From Engineering Council Examination, 1996)*