1. Exercise Session 11
Power systems

2. Question 1
In an isolated neutral network there is a high-resistance 1-phase earth fault. The network’s earth capacitance is C0, voltage U and the fault resistance is Rf. Derive expressions for
earth fault current
neutral point voltage
faulty phase’s voltage
healthy phases’ voltage

3. Question 1
Positive sequence network (positive sequence impedances small compared to the zero sequence network impedances):
Negative sequence network (negative sequence impedances small compared to the zero sequence network impedances):
Zero sequence network: U1 U2 U0
As we have learned previously (see E5Q3), in a single phase-to-earth fault,
the sequence networks are in series and the circuit is
closed by an impedance 3 times the fault impedance 

4. Question 1 Single phase-to-earth fault: earth fault current
b) neutral point voltage (zero sequence voltage is the voltage between neutral and ground)

5. Question 1 Single phase-to-earth fault: с) faulty phase’s voltage
d) healthy phases’ voltage
Correspondingly:

6. Question 2
Show that in an isolated neutral network, that has an earth capacitance c0 = 6.13 nF/km, the zero resistance earth fault current can be approximately expressed as:
𝐼 e = 𝑈×𝑙 , where [U] = 1 kV, [Ie] = 1 A, [l] = 1 km

7. Question 2 If , then Single phase-to-earth fault:
If [U] = 1 kV and [l] = 1 km, then

8. major
Question 3
minor
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:
1) l = 150 m, xj = /km, rj = /km
2) l = 600 m, xj = /km, rj = /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 I2tS is at least three times the minor fuse’s operation energy I2ta. The melting times and energies are presented in the following pictures.

9. Question 3 major minor If t>10ms: If t<1 ms:
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:
1) l = 150 m, xj = /km, rj = /km
2) l = 600 m, xj = /km, rj = /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 I2tS is at least three times the minor fuse’s operation energy I2ta. The melting times and energies are presented in the following pictures.
If t>10ms:
𝑡 major ≥10 𝑡 minor + 𝑡 minor arcing
If t<1 ms:
𝐼 2 𝑡 s,major ≥3 𝐼 2 𝑡 a,minor

10. Question 3
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.

11. Question 3
The worst case from the selectivity point of view is when a short circuit happens at the 35-A fuse terminals. Let us compute the 3-phase short circuit current in this point:
Assuming that the voltage is at its rated value.
In 400-V voltage level, transformer parameters:

12. Question 3: Part 1 Line characteristics: Short circuit current:
From figures shown on the previous slide: the operation time of a 35-A fuse with the short circuit current 𝐼 k =7.05 kA, fuse melting time corresponds to a value below 1 ms, therefore, we use the melting/operating energy diagram. From the second diagram the 35-A fuse corresponds to a operating energy value of 𝐼 2 𝑡 𝑎 =5∗ A 2 s at 220 V. According to our initial selectivity criteria (if t<1ms, then 𝐼 2 𝑡 𝑠,𝑚𝑎𝑗𝑜𝑟 ≥3 𝐼 2 𝑡 𝑎,𝑚𝑖𝑛𝑜𝑟 ),
the 𝐼 2 𝑡 𝑠,𝑚𝑎𝑗𝑜𝑟 ≥3∗5∗ 10 3 =15 ∗ 10 3 A 2 s .
On the diagram the next fuse that satisfies this condition is a 100 A fuse.
The major fuse size

13. Question 3
Melting times of fuses with different nominal current as a function of current.
𝐼 k =7.05 kA, 𝐼 N =35 A
I2tS melting energy, I2ta operation energy, melting time is less than 1 ms.

14. Question 3: Part 2 Line characteristics: Short circuit current:
From figures shown on the previous slide: the operation time of a 35-A fuse with the short circuit current 𝐼 𝑘 =436 A, fuse melting time corresponds to a value 20 – 30 ms. Therefore, we use the same diagram. According to our initial selectivity criteria (if t >10 ms, then 𝑡 major ≥ 10𝑡 minor +10 ms),
the 𝑡 major ≥10∗20 ms+10 ms=210 ms. On the diagram, the next fuse that satisfies this condition is a 63-A fuse.
The major fuse size

15. Question 3
Melting times of fuses with different nominal current as a function of current.
𝐼 k =436 A, 𝐼 N =35 A

16. Exercises based on Lecture 12
Power systems

17. Question 4
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).
Calculate voltage strain the earth fault causes between transformer’s casing and low voltage winding.
The maximum allowed voltage in a grounded part of a network during a fault can be, for example, defined as
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)

18. Question 4
The voltage strain between the winding and the casing includes the earth resistance of the grounding of the casing:
a) Total fault resistance is

19. Question 4
The maximum allowed voltage in a grounded part of a network during a fault can be defined as: b)
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)
Therefore:

20. Question 5
A 33-kV, 60-Hz overhead line has a capacitance to ground per phase of approximately 0.7 F. Calculate the reactance of a suitable arc-suppression coil.

21. Question 5
Serious overvoltages may occur with arcing faults because of the inductance and shunt capacitance of the system.
Electric Power Systems, 5th ed., Weedy et al.
Petersen coil can be used to overcome this condition, for compensation of the arc current.
The voltage at the faulty phase at the fault location can be considered to match the ground potential. The line-to-line voltage between the healthy phases and the faulty phase is
𝑈 ph−ph = 3 𝑈 ph
Figure 7.29 (a) System with arc suppression coil, (b) Phasor diagram of voltages and
currents in part (a)
Electric Power Systems, 5th ed., Weedy et al.

22. Question 5 𝑈 ph−ph = 3 𝑈 ph 𝐼 𝑎 = 𝐼 𝑏 = 3 𝑈 ph 𝜔𝐶
Thus, there are incoming phase-to-phase currents equivalent to
As before, there exists a 120-degree phase shift between the currents 𝐼 𝑎 and 𝐼 𝑏 , and, thus, the sum vector is
𝐼 𝑎 = 𝐼 𝑏 = 3 𝑈 ph 𝜔𝐶
𝐼 𝑎 + 𝐼 𝑏 = 𝑈 ph 𝜔𝐶=3 𝑈 ph 𝜔𝐶,
The voltage of the common neutral is a phase voltage higher than at the faulted point. This induces a current through the Petersen coil equivalent to
𝐼 𝐿 = 𝑈 𝑝ℎ 𝜔𝐿
Figure 7.29 (a) System with arc suppression coil, (b) Phasor diagram of voltages and
currents in part (a)
Electric Power Systems, 5th ed., Weedy et al.

23. Question 5
Therefore, to compensate the sum of the currents 𝐼 𝑎 and 𝐼 𝑏 currents,
the Peterson coil reactance should be