ELEC-E8406 Electricity Distribution and Markets
Research Assignment: Voltage Drop
Instructor: John Millar
Group #2
Mathias Westerholm 294515
Markus Ukkonen 226415
Verner Püvi 466505
Kennet Bexar 296801
Camila Barragán 545604
The task was to determine the percentage error in the voltage drop calculation between using the
approximate formula and the more accurate formula, as defined below. This was to be done for a bare
overhead line section of varying lengths, with a single load at the end, and a power factor of 0.9.
Approximate formula
Lagging power factor
𝑉𝑑 ≈ √3 ∙ 𝐼(𝑅 cos 𝜑 + 𝑋𝑠𝑖𝑛𝜑)
Leading power factor
𝑉𝑑 ≈ √3 ∙ 𝐼(𝑅 cos 𝜑 − 𝑋𝑠𝑖𝑛𝜑)
More accurate formula
Lagging power factor
(𝑋𝐼 cos 𝜑 − 𝑅𝐼 sin 𝜑)2
𝑉𝑑 = √3 ∙ (𝑅𝐼 cos 𝜑 + 𝑋𝐼𝑠𝑖𝑛𝜑 +
)
2𝑉𝐿𝑁,𝑠
Leading power factor
𝑉𝑑 = √3 ∙ (𝑅𝐼 cos 𝜑 − 𝑋𝐼𝑠𝑖𝑛𝜑 +
(𝑋𝐼 cos 𝜑 + 𝑅𝐼 sin 𝜑)2
)
2𝑉𝐿𝑁,𝑠
To perform the calculations, a Raven conductor with the following characteristics was selected:
DC @20°C
(ohm/km)
Ampacity
240 (A)
Resistance
AC @25°C
AC @50°C
(ohm/km)
(ohm/km)
AC @75°C
(ohm/km)
0.521679
0.534803
0.646357
0.708696
Reactance
Inductive @25°C
Inductive @50°C
Inductive @75°C
(ohm/km)
(ohm/km)
(ohm/km)
0.341224
0.374034
0.380596
The resistance and reactance values used were those at 75°C, since the cable will be running close to
its thermal limits, and thus temperatures are expected to be on the high end of the operating range.
The load at the end of the line was determined under the assumptions that it put the cable close to its
thermal limit (I = 240 A), that the supply (bus-bar) end voltage was 20.5 kV, and that the line length
gave a voltage drop of 7% (Vr = 20.5 kV – 7%∙20.5 kV). This means that voltage drops above 7%
(namely 10%, 15% and 20%) will cause overloading of the cable. The formulas used can be seen below:
𝑆 = √3𝑉𝐿𝐿 𝐼𝑉
𝑃 = √3𝑉𝐿𝐿 𝐼𝑉 cos 𝜑
𝑃 = √3 ∙ 19.065 ∗ 103 ∙ 240 ∙ 0.9 = 7.13 𝑀𝑊
A load of 7.13 MW would put the conductor at its thermal limit at 7% voltage drop.
Voltage drop calculations
Once the load was determined, the next step was to determine the line lengths that gave voltage drops
of 1%, 3%, 5%, 7%, 10%, 15% and 20%. This was done for a lagging power factor of 0.9, and using
the approximate formula. The results can be seen in the following table.
Line lengths (km)
Voltage drop
0.65
1%
1.92
3%
3.135
5%
4.295
7%
5.94
10%
8.41
15%
10.555
20%
Using these particular line lengths, voltage drop was calculated using approximate and more accurate
formula, for a lagging and a leading power factor of 0.9
The final results (not including the iterations) are shown in the following table. Calculations can be
found in Appendix 1 (Excel file).
Lagging power factor / simple formula
Vr (V)
I (amps)
Vd (V)
Vd(%)
Lagging power factor / accurate formula
Vr
I (amps)
Vd (V)
Vd(%)
Error
Vd (V)
20295.9955
225.4544
204.0045
1.00%
20295.9937
225.4544
204.0063
1.00%
0.0009%
19884.9460
230.1149
615.0546
3.00%
19884.9293
230.1151
615.0712
3.00%
0.0027%
19474.5762
234.9639
1025.4308
5.00%
19474.5289
234.9644
1025.4782
5.00%
0.0046%
19065.0022
240.0116
1435.0372
7.00%
19064.9072
240.0128
1435.1322
7.00%
0.0066%
18449.3592
248.0206
2050.8883
10.00%
18449.1574
248.0233
2051.0902
10.01%
0.0098%
17426.1640
262.5834
3074.1924
15.00%
17425.6749
262.5908
3074.6821
15.00%
0.0159%
16400.4575
279.0057
4099.5776
20.00%
16400.4555
279.0058
4099.5796
20.00%
0.0000%
Leading power factor / simple formula
Vr (V)
I (amps)
Vd (V)
Vd(%)
Leading power factor / accurate formula
Vr
I (amps)
20380.7111
224.5173
119.2889
0.58%
20380.0459
224.5246
20143.4895
227.1613
356.5106
1.74%
20137.4724
227.2292
19911.0912
229.8127
588.9092
2.87%
19894.4534
230.0049
19683.8739
232.4655
816.1284
3.98%
19651.4710
19351.9443
236.4528
1148.0683
5.60%
18829.4383
243.0142
1670.5693
8.15%
18348.3755
249.3857
2151.6246
10.50%
Vd (V)
Vd(%)
119.9541
Error
Vd (V)
0.59%
0.5546%
362.5277
1.77%
1.6598%
605.5471
2.95%
2.7476%
232.8488
848.5321
4.14%
3.8188%
19286.3737
237.2567
1213.6465
5.92%
5.4034%
18684.7961
244.8955
1815.2218
8.85%
7.9689%
18097.3641
252.8446
2402.6362
11.72%
10.4473%
The errors between the approximate formula and the accurate formula can be observed in the following
figure.
Percentage error
Percentage error in voltage drop calculation between
approximate formula and more accurate formula.
15.0%
10.0%
5.0%
Lagging power factor
0.0%
0.65
1.92
3.135
Leading power factor
4.295
5.94
8.41
10.555
Length of line (km)
Corresponding to Vd of 1%, 3%, 5%, 7%, 10%, 15%, and 20%
Observations and conclusions
Various things can be pointed out from the results. The more obvious one is that, contrary to what was
expected, the difference between the approximate and the accurate formula is practically negligible for
a lagging power factor, even at voltage drops higher than 10%. The error does slightly increase as the
voltage drop increases, except for the voltage drop of 20%, which is an anomaly in the pattern.
For a leading power factor, on the other hand, the difference between the results of both formulas is
bigger, reaching even an error of 10% for a leading power factor voltage drop of around 11% (length
line corresponding to 20% voltage drop in a line with lagging power factor). Thus the approximate
equation should be used with a lot of caution, if ever be used at all, for a leading power factor.
It must also be reminded that as voltage drops, the voltage at the receiving end is lower, and can easily
reach limits where the current goes higher than the rated ampacity of the cable (see figure below). This
fact should cautiously be considered during the design phase of the network.
Voltage at receiving end (V)
Drop in Vr as a function of line length: lagging PF
21000.0000
20000.0000
19000.0000
18000.0000
17000.0000
Ampacity limit crossed
16000.0000
15000.0000
0
2
4
6
Line length
8
10
12