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PROBLEMS / Mörsky & Harrison
1.
Show that the ratio of phase-to-phase-voltage and phase-to-earth-voltage is
3.
2.
Derive the equation for delta-star transformation.
3.
a)
Derive the following equation (Harrison eq. 5.1)
U h  I R  cos   I X  sin 
b) Can this equation be applied in all situations?
c)
Draw the phasor diagrams related to the equation in cases where the reactive power is
inductive and capacitive.
4.
Calculate voltage drop for the case in H5.3 presuming that the given voltage 132 kV is at the end of the
line.
H2.4
A star-connected three-phase load absorbs 40 kW at a power factor of 0.875 lagging when supplied
from a 6.6 kV line. Calculate:
a) the phase voltage
b) the line current.
H2.5
A delta-connected three-phase load of (80+j60)  per phase is connected to a 440 V three-phase
supply. Calculate:
a) the phase current
b) the line current
c) the total real power consumed.
H4.9
A 50 Hz, 500 MVA transformer with a 400 kV primary and a 275 kV secondary has a total leakage
reactance per star phase referred to the primary of 38.4 . Calculate:
a) the leakage reactance referred to the secondary
b) the per-unit reactance referred to the primary
c) the per-unit reactance referred to the secondary.
H5.3
A 50 Hz, 132 kV line is 5 km long. At the receiving end there is a load of 100 MVA. The line can be
represented by a resistance of 0.16  per phase per km and an inductance of 1.3 mH per phase per km.
Calculate the voltage drop along the line if the load power factor is:
a) 0.8 lagging
b) 0.95 lagging
c) 0.93 leading.
M10
A synchronous generator is synchronized through a transformer to a bus. At the bus the short circuit
power is 1000 MVA and the voltage is 15 kV. After synchronizing the generator’s power is increased
to 100 MW without changing the excitation. Calculate the generator’s terminal voltage U g and reactive
power Q.
M118
Draw the sequence networks for transient state.
M122
In the middle of the line there is a zero resistance three-phase short circuit. What is the voltage at the
fault location prior to the fault? Calculate
a) sub-transient
b) transient
c) steady state
short-circuit current.
M134
Line
Properties reduced to Line voltage:
G:
M:
Line:
X’’d = X2 = 15 , X0 = 10 
Zk = 10 
X1 = 5 , X0 = 16 
A two phase zero impedance fault occurs at point A. The distance between phases in the bus bar system
is 2.5 m. Calculate the maximum peak force affecting each phase (per length) in area 1.
M181
Calculate the maximum active power Pmax that can be transferred from busbar 1 to busbar 2 using the
voltages shown in the picture.
M184
The power that can be transferred by a line is, usually limited by the reactive power resources. A line
has a series reactance X=100  and the voltage at the beginning of the line is U1=115 kV. The reactive
power at the end of the line is Q2=0. Derive the expressions for power transferred and reactive power
Q1 as the function of voltage angle . How much smaller is the maximum transferred power compared
to a case where the necessary reactive power could be fed to the end of the line?
M186
In the picture, a transmission line is modeled as an impedance Z = (10 + j67) .
a) Define the maximum transferred active power when U1 = U2 = 230 kV.
b) How many percent can the transferred power be increased by compensating the line with serial
capacitors?
M2
A transmission system shown in the picture. Calculate generator’s load current and terminal voltage
a) by reducing the network to generator’s voltage level
b) by using per-unit values
M21
A generator is feeding a synchronous motor through a transmission line. Nominal values for both
electrical machines are: SN=30 MVA, UN=13.2 kV, xd’’=0.2. The transmission line has a reactance of
0.1 pu (machines’ base power). Motor load power is 20 MW, cos=0.8cap. A zero resistance threephase fault occurs near the motor when the motor terminal voltage is 12.8 kV. Calculate the short
circuit currents for the generator and motor.
M213
A generator is feeding an infinite bus through two parallel transfer lines. In one of the lines occurs a
three-phase short-circuit which decreases the transfer capacity of the lines by 75 %. At which value of
the power-angle should the faulty line’s breakers be opened to maintain stability?
The maximum loading capacity of one line is 40 MW and 60 MW for both lines. Generator’s active
power prior to the fault is 30 MW. There is no re-closing action.
M216
A generator is feeding an infinite bus through transfer line J. The generator’s power is p=0.8 and
cos=0.85ind. In point A there is a zero resistant fault that causes the breakers K1 and K2 to open 0.1 s
after the fault. We shall presume that the fault is cleared while the breakers are open. Calculate:
a) the power-angle at which the breakers must be re-closed at the latest to maintain stability
b) the maximum time from breaker opening to re-closing
M24
The 400 kV bus A has short circuit power of 4600 MVA (at 415 kV). What is the short circuit power if
one of the lines connected to the infinite bus is disconnected?
M248
Al
Fe
4,25 mm
A three-phase overhead line’s conductors are in the same plane and the distance between phases is 1.3
m. Phase conductor is of type Al/Fe and it consists of a steel sub-conductor (r=50) and six aluminium
sub-conductors wound around the steel conductor. The sub-conductor diameter is 4.25 mm. Calculate
the line’s capacitance per length.
M25
Power plant’s 420 kV bus’ sub-transient short circuit power is Sk’’=3800 MVA when the plant consists
of one unit (generator and transformer). Calculate the increment of the short circuit power if there are
three similar units connected to the 420 kV bus.
M268
Transform load S=(200 + j150) kVA into equivalent resistance and reactance which are connected
a) in series
b) in parallel
Voltage is 20 kV.
M270
A certain load curve can be presented with the following equation:

t


Pt   1  0.5 sin 
  MW
 12months  

Calculate:
a) Annual energy
b) mean power for one year Pk
c) load factor
d) load duration time tk
M272
10 MW of active power is transferred using a three-phase line shown in the picture. Line-to-line
voltages in the both ends of the line are 110 kV. Calculate the power factor (cos) of the load.
M285
Calculate the loss load factor f for the case in M270 when cos = 1.
M294
At the end of a three-phase line the voltage is 20 kV. The line has following properties: resistance 0.8
 and reactance 0.6  per phase. The load at the end of the line has a load duration curve as shown in
the picture above. The power factor of the load is constant cos=0.8ind. Calculate:
a) active power losses Phmax and Phmin
b) reactive power losses Qhmax and Qhmin
c) energy losses Wh
M320
a=0.3 m
A short circuit occurs in a 24 kV bus bar system. The phase current instantaneous values are i R= 30 kA,
iS=15 kA and iT=15 kA.
Calculate the forces (per length) that affect each bus bar for the
a) upper system
b) lower system.
M324
In an isolated neutral network there is a high resistance earth fault. The network’s earth capacitance is
C0, voltage U and the fault resistance is Rf. Derive expressions for
a)
b)
c)
d)
earth fault current
neutral point voltage
faulty phase’s voltage
healthy phases’ voltage
M326
Show that in an isolated neutral network, that has an earth capacitance c 0=6.13 nF/km, the zero
resistance earth fault current can be approximately expressed as:
Ie=U*l/300 where [U]=1 kV, [Ie]=1 A, [l]=1 km
M37
The resistance of an aluminium conductor is known at 15C temperature. Calculate the resistance at
40C.
The parameters for the conductor:
20=0.00403/C
R=R20(1+20(T-20C))
M4
A no loaded generator is connected to a network. Its terminal voltage is 20 kV and synchronous
reactance 2.5 . Calculate the terminal voltage when a load (P=20 MW and Q=0 MVAr) is connected
and the excitation is not changed.
M400
A transformer station is connected to a 20 kV overhead line. The total length of the medium voltage
network is 500 km. At the station there occurs an earth fault with fault resistance 500  between one
phase and protective grounding (see figure).
a)
Calculate voltage strain the earth fault causes between transformer’s casing and low voltage
winding.
b) The maximum allowed voltage in a grounded part of a network during a fault can be for example
defined as
U e max 
750
t
V . For how long is the earth fault allowed to last in this case? The earth
capacitance of the 20 kV overhead network is 6.13 nF/km (per phase)
M437
Choose the smallest possible fuse to the transformer’s low-voltage side so that the protection is
selective considering the IN=35 A fuse. The two cases are:
a) l=150 m, xj=0.075 /km, rj=0.103 /km
b) l=600 m, xj=0.104 /km, rj=0.868 /km
Selectivity can be considered sufficient when the major fuse’s melting time is at least ten times the
melting time of the minor fuse plus the maximum arcing time (10 ms). When we have short melting
times (t<1 ms) fusing is selective enough when the major fuse’s melting energy is at least three times
the minor fuse’s operation energy. The melting times and energies are presented in the following
pictures.
Nominal current
Melting
time
current
Melting times of fuses with different nominal current as a function of current.
I2tS melting energy, I2ta operation energy, melting time is less than 1 ms.
M438
Show that a 230 V protection system is selective. The system consists of two circuit breakers I N=35 A
(PhN=2.45 W) and IN=1A (PhN=1.62 W) connected in series. PhN is the breakers’ resistive loss at
nominal loading. Both breakers work rapidly when the current is 5*I N < I < 10*IN.
M47
A transposed 400 kV line’s conductors are in the same plane and the distance between phases is 11.0
m. The bundled conductors consist of two sub-conductors which are 45.0 cm apart. The sub-conductors
are of type Al/Fe 565/72 mm2 and their geometric mean diameter is 32.9 mm. Calculate the mean
phase inductance. The sub-conductor’s internal inductance can be taken into consideration by
multiplying the conductor’s geometric mean diameter by 0.987.
M48
Calculate the phase conductor mean capacitance (per length) for the case presented in M47.
M5
The generator is feeding the network. P=250 MW and Q=0 MVAr. Calculate the generator terminal
voltage when it is disconnected from the network.
M59
A 220 kV overhead line has the following properties:
l=200 km
r=0.07 /km
x=0.32 /km
b=3.6 S/km
Define the parameters for short line
a) -model
b) T-model
and calculate the open circuit voltage at the end of the line if the voltage at the beginning is 220 kV.
M75
Calculate per-unit values for all the parameters in the picture above. Use base values:
Sb=50 MVA
Ub=110 kV
M77
Based on the results obtained from M75, calculate the voltage in busbar A (per-unit value and in volts).
M79
Generator
Transformer M1
Transformer M2
Calculate the load voltage UK in the network shown in the picture above. Generator’s terminal voltage
is Ug = 9.8 kV. Use generator’s voltage level.
Derive the equations for active and reactive power as the function of power angle  (that is, the powerangle equations) at the
a) beginning of the line
b) end of the line.
M91
Derive the exact power-angle equations for the
a) beginning
b) end of the line.
M93
Generator
Transformer
Starting from the power-angle equations, define the maximum line reactance X so that the power plant
can feed its full active power PN without exceeding the maximum apparent power SN. Voltage of the
transmission network is constant U2 = 410 kV and it doesn’t consume any reactive power, that is Q 2 =
0.