High Phase Order Transmission Line Fault Types

Figures & data

TABLE 1 The number of fault types, significant fault types, and symmetrically significant fault types for specified number of phases.

Number of phases	Fault types	Significant fault types	Symmetrically significant fault types
1	1	1	1
2	4	3	2
3	11	5	4
4	26	9	7
5	57	13	12
6	120	23	19
7	247	33	32
8	502	57	52
9	1013	89	86
10	2036	153	145
11	4083	249	248
12	8178	445	428
15	65,519	2445	2436

TABLE 2 The Phi function for d = 1 to d = 24.

d	Numbers relatively prime to d	phi(d)
1	1	1
2	1	1
3	1, 2	2
4	1, 3	2
5	1, 2, 3, 4	4
6	1, 5	2
7	1, 2, 3, 4, 5, 6	6
8	1, 3, 5, 7	4
9	1, 2, 4, 5, 7, 8	6
10	1, 3, 7, 9	4
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	10
12	1, 5, 7, 11	4
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	12
14	1, 3, 5, 9, 11, 13	6
15	1, 2, 4, 7, 8, 11, 13, 14	8
16	1, 3, 5, 7, 9, 11, 13, 15	8
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	16
18	1, 5, 7, 11, 13, 17	6
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	18
20	1, 3, 7, 9, 11, 13, 17, 19	8
21	1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20	12
22	1, 3, 5, 7, 9, 13, 15, 17, 19, 21	10
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	22
24	1, 5, 7, 11, 13, 17, 19, 23	8

FIGURE 1. Symmetric significant fault type example. The four-phase fault and the four-phase fault to ground have the same fault analysis in balanced systems.

TABLE 3 The summation over d|n examples.

n	Positive integers that divide into n evenly
1	1
2	1, 2
3	1, 3
4	1, 2, 4
5	1, 5
6	1, 2, 3, 6
7	1, 7
8	1, 2, 4, 8
9	1, 3, 9
10	1, 2, 5, 10
11	1, 11
12	1, 2, 3, 4, 6, 12
13	1, 13
14	1, 2, 7, 14
15	1, 3, 5, 15
16	1, 2, 4, 8, 16
17	1, 17
18	1, 3, 6, 9, 18
19	1, 19
20	1, 2, 5, 10, 20
21	1, 3, 7
22	1, 2, 11, 22
23	1, 23
24	1, 2, 3, 4, 6, 8, 12, 24

FIGURE 2. The SFTs for three-phase transmission lines. The fault with a “star” next to it is symmetric, i.e., the fault is the same as the fault to ground.

FIGURE 3. The four-phase SFTs. Five SFTs are not shown. Four not shown are identical to these four but also to ground. The last not shown is one phase to ground. The faults with a “star” next to it are symmetric, i.e., the fault is the same as the fault to ground.

FIGURE 4. The five-phase SFTs. Seven SFTs are not shown. Six not shown are identical to these six but also to ground. The last not shown is one phase to ground. The fault with a “star” next to it is symmetric, i.e., the fault is the same as the fault to ground.

FIGURE 5. The six-phase SFTs. 12 SFTs are not shown. 11 not shown are identical to these 11 but also to ground. The last not shown is one phase to ground. The faults with a “star” next to it are symmetric, i.e., the fault is the same as the fault to ground.

FIGURE 6. The nine-phase significant fault types. 45 significant fault types are not shown. 44 not shown are identical to these 44 but also to ground. The last not shown is one phase to ground. The faults with a “star” next to it are symmetric, i.e., the fault is the same as the fault to ground.

FIGURE 7. The twelve-phase significant fault types. 223 significant fault types are not shown. 222 not shown are identical to these 222 but also to ground. The last not shown is one phase to ground. The faults with a “star” next to it are symmetric, i.e., the fault is the same as the fault to ground.

FIGURE 8. Fault analysis phasor diagram of prefault voltages and postfault voltages after a phase to neutral fault on phase F for a simple six-phase fault analysis example. Phase variables are depicted.