Analysis of Electric Power Transmission Line Wave Voltage Components in Polar Form

Analysis of Electric Power Transmission Line Wave Voltage Components in Polar Form

Georgios Leonidopoulos 

Electrical Engineering Dept., University of West Attica, Thevon 250, Aegaleo 12244, Greece

Corresponding Author Email: 
gleon@uniwa.gr
Page: 
1-5
|
DOI: 
https://doi.org/10.18280/mmc_a.922-401
Received: 
15 March 2019
|
Accepted: 
10 July 2019
|
Published: 
30 December 2019
| Citation

OPEN ACCESS

Abstract: 

In this paper, a well-known mathematical model of electric power transmission line under steady state conditions is considered. From this model, the mathematical expressions that describe the two components of the resultant voltage i.e. voltage travelling and refracted waves along a power transmission line have been developed taking as starting point the end of the line.

We use the fore-mentioned mathematical expressions and the data of a typical electric transmission line to calculate how the voltage travelling and refracted waves vary. The results are also graphed in order to have an optical view of how the voltage travelling and refracted waves behave. Finally, the results are analysed and the relative conclusions are drawn.

Keywords: 

voltage, wave, electric power transmission line, travelling voltage wave, refracted voltage wave, voltage refraction co-efficient

1. Introduction

Most people think of the voltage as an element that when it is put on, it is applied immediately. They cannot imagine that the voltage is a wave (an electromagnetic wave) that travels and refracts with almost the speed of light. This understanding is due to the length of line and the inability that people have to perceive the very small time intervals (psecs, μsecs, msecs depending on the line length) that the wave needs to cover these distances.

In this paper, the length under consideration is that of a power transmission line of an electric power system [1-9], a length of some hundred kilometers. The equivalent electric circuit under steady state conditions is drawn and the respective differential equations are extracted from it using as independent variable the distance x from either the rears of the line. The above mathematical model already exists in the literature and can easily be found [1-5].

Solving the differential equations, the mathematical expressions describing the voltage travelling and refracted waves are obtained (section 2). The proof that the above voltages are the travelling and refracted wave respectively is the mathematical expressions themselves. They are the mathematical expressions of a travelling and refracted wave respectively.

As far as I know and search in the literature, I could not find calculation and graphical representation of the voltage travelling and refracted waves along an electric power transmission line. Thus, in this paper, the above mathematical expressions are tested on a typical electric power transmission line and the results are presented in section 3. Furthermore, in section 3, the above results are graphed in order to have an optical image of how the voltage travelling and refracted waves along the line behave. Finally, in section 4, a discussion is developed, the results are studied, analysed and in section 5, the relative conclusions are drawn.

2. Development and Analysis of the Mathematical Expressions of Voltage Travelling and Refracted Waves

In Figure 1, the electric equivalent representation of power transmission line under steady state conditions and using divided elements has been drawn.

Where z dx=the infinitesimal long-wise complex impedance of dx

y dx=the infinitesimal transversal complex conductance of dx

From the infinitesimal element dx, the following equations are drawn:

1st law of Kirchhoff: [I(x)+dI(x)]=I(x) + dI(x)

2nd law of Kirchhoff: [V(x)+dV(x)]=V(x) + dV(x)

Voltage drop on element zdx:

dV(x)=[I(x)+dI(x)]zdx$\cong$I(x)zdx→$\frac{d V(x)}{d x}$ =I(x) z   (1)

Voltage drop on element ydx:

dI(x)=V(x) ydx  →$\frac{d I(x)}{d x}$ =V(x) y  (2)

Figure 1. Electric equivalent representation of electric power transmission line

Differentiating Eq. (1) and replacing it into Eq. (2), we get:

$\frac{\mathrm{d}^{2} \mathrm{V}(\mathrm{x})}{\mathrm{d} \mathrm{x}^{2}}$​ =yz V(x)  (3)

Differentiating Eq. (2) and replacing it into Eq. (1), we also get:

$\frac{\mathrm{d}^{2} \mathrm{I}(\mathrm{x})}{\mathrm{dx}^{2}}$=yz I(x)  (4)

From Eqns. (3) and (4), V(x) and I(x) are described by the same differential equations. The above implies that V(x) and I(x) are described by similar mathematical functions.

We take as initial conditions:

V(x=0)=V(5)

and

I(x=0)=IR   (6)

i.e. we take as x=0 the end of electric power transmission line

Then, from Eqns. (3), (4), (5) and (6), we extract the following mathematical expressions of voltage travelling and refracted wave respectively:

Vtrav(x)= $\frac{V_{R}+I_{R} z_{C}}{2}$  eγx    (7)

Vrefr(x)= $\frac{V_{R}-I_{R} z_{C}}{2}$  e-γx    (8)

The above Eqns. (7) and (8) are the mathematical expressions of a wave.

Then, the voltage refraction co-efficient ρV(x) can be defined as a function of distance x. The voltage refraction co-efficient is set as :

$\rho_{\mathrm{V}(\mathrm{x})}=\frac{\mathrm{V}_{\mathrm{refr}}(\mathrm{x})}{\mathrm{V}_{\mathrm{trav}}(\mathrm{x})}$  (9)

3. Calculation and Graphical Presentation of Voltage Travelling and Refracted Waves

We consider a typical electric power transmission line with the following parameters:

R=0.107 Ω/km                 L=1.362 mH/km

G=0  S/km                       C=0.0085 μF/km

f=50 Hz                            l=360 km

VR=115470 $<0^{\circ}$  V          IR=360.844 $<0^{\circ}$  A

Then using the list of symbols and the analysis of section 2, we can calculate the other complex parameters of the above line in polar and/or cartesian form:

γ=1.085x10-3<82.98° km-1=(0.1326x10-3 + j 1.07687x10-3) km-1

α=0.1326x10-3 neper/km              β=1.07687x10-3 rad/km

zC=406.41<-7.02° Ω

$\frac{\mathrm{V}_{\mathrm{R}}+\mathrm{I}_{\mathrm{R}} \mathrm{z}_{\mathrm{C}}}{2}$ =130817.935 <-3.93 °  V

$\frac{\mathrm{V}_{\mathrm{R}}-\mathrm{I}_{\mathrm{R}} \mathrm{z}_{\mathrm{C}}}{2}$ =17507.97 < 149.213 °  V

υ=291733.696 km/sec τ=1.234 msecs

λ=5834.674 km

Δ=22.212°  Δ/l=0.0617° /km

Then, Eqns. (7), (8) and (9) using the above parameters become:

Vtrav(x)=130817.935<-3.93° e(0.1326x10-3 + j 1.07687x10-3)x     V   (10)

Vrefr(x)=17507.97<149.213° e-(0.1326x10-3 + j 1.07687x10-3)x     V   (11)

$\rho_{\mathrm{V}}(\mathrm{x})=\frac{17507.97<149.213^{\circ} \mathrm{e}-(0.1326 \times 10-3+\mathrm{j} 1.07687 \times 10-3) \mathrm{x}}{130817.935<-3.93^{\circ} \mathrm{e}(0.1326 \times 10-3+\mathrm{j} 1.07687 \times 10-3) \mathrm{x}}$  (12)

Using Eqns. (10), (11) and (12) and taking step Δx=10km, we calculate the values of voltage travelling and refracted wave as well as voltage refraction co-efficient and the results are presented in Table 1. Since the voltages are vectors, the results are complex numbers and are given in polar form i.e. in voltage magnitude (Volts) and voltage phase (°) representation. The voltage refraction co-efficient ρV(x) is a pure complex number since is derived from the division of the voltage waves and is also given in table 1 in polar form ie. in magnitude(pure real number) and phase (°) form.

The graphical presentations of results obtained in table 1 are given in Figures 2 to 4.

Figure 2. Absolute value (intensity) and phase (angle) of voltage travelling wave from the beginning towards the end of line i.e. along the direction the travelling wave moves (direction right to left of electric power transmission line of figure 1)

Figure 3. Absolute value (intensity) and phase (angle) of voltage refracted wave from the end towards the beginning of line i.e. along the direction the refracted wave moves (direction opposite to that of graph 1, i.e. left to right of electric power transmission line of figure 1)

Figure 4. Absolute value and phase (angle) of voltage refraction co-efficient from the end where the refraction occurs towards the beginning of line (direction opposite to that of graph 1, i.e. left to right of electric power transmission line of figure 1)

Table 1. Calculation results of voltage travelling and refracted wave

α/α

x (km)

Vtrav(x) (Volts)

φVtrav(x) (°)

Vrefr(x) (Volts)

φVrefr(x) (°)

ρV(x)

φρV(x) (°)

1

0

130818.0

-3.928010

17507.96

149.2129

0.1338345

153.1409

2

10

130991.6

-3.310870

17484.75

148.5958

0.1334799

151.9066

3

20

131165.5

-2.693720

17461.58

147.9786

0.1331263

150.6723

4

30

131339.6

-2.076570

17438.43

147.3615

0.1327736

149.4381

5

40

131513.9

-1.459430

17415.32

146.7443

0.1324219

148.2038

6

50

131688.4

-0.842280

17392.23

146.1272

0.1320711

146.9695

7

60

131863.2

-0.225130

17369.18

145.5100

0.1317212

145.7352

8

70

132038.2

0.392014

17346.16

144.8929

0.1313722

144.5009

9

80

132213.5

1.009160

17323.17

144.2757

0.1310242

143.2666

10

90

132389.0

1.626307

17300.21

143.6586

0.1306771

142.0323

11

100

132564.7

2.243454

17277.28

143.0415

0.1303309

140.7980

12

110

132740.6

2.860600

17254.37

142.4243

0.1299856

139.5637

13

120

132916.8

3.477747

17231.50

141.8072

0.1296413

138.3294

14

130

133093.2

4.094893

17208.67

141.1900

0.1292979

137.0951

15

140

133269.8

4.712040

17185.86

140.5729

0.1289553

135.8608

16

150

133446.7

5.329186

17163.08

139.9557

0.1286137

134.6265

17

160

133623.8

5.946333

17140.33

139.3386

0.1282730

133.3922

18

170

133801.2

6.563480

17117.61

138.7214

0.1279332

132.1579

19

180

133978.8

7.180626

17094.92

138.1043

0.1275942

130.9237

20

190

134156.6

7.797773

17072.26

137.4871

0.1272562

129.6894

21

200

134334.6

8.414919

17049.63

136.8700

0.1269191

128.4551

22

210

134512.9

9.032066

17027.03

136.2528

0.1265829

127.2208

23

220

134691.5

9.649212

17004.46

135.6357

0.1262475

125.9865

24

230

134870.2

10.266360

16981.93

135.0185

0.1259131

124.7522

25

240

135049.2

10.883510

16959.42

134.4014

0.1255795

123.5179

26

250

135228.5

11.500650

16936.94

133.7843

0.1252468

122.2836

27

260

135408.0

12.117800

16914.49

133.1671

0.1249150

121.0493

28

270

135587.7

12.734950

16892.07

132.5500

0.1245841

119.8150

29

280

135767.6

13.352090

16869.68

131.9328

0.1242541

118.5807

30

290

135947.8

13.969240

16847.32

131.3157

0.1239249

117.3464

31

300

136128.3

14.586390

16824.99

130.6985

0.1235966

116.1121

32

310

136308.9

15.203530

16802.69

130.0814

0.1232692

114.8778

33

320

136489.8

15.820680

16780.42

129.4642

0.1229426

113.6436

34

330

136671.0

16.437820

16758.17

128.8471

0.1226169

112.4093

35

340

136852.4

17.054970

16735.96

128.2299

0.1222921

111.1750

36

350

137034.0

17.672120

16713.78

127.6128

0.1219681

109.9407

37

360

137215.9

18.289260

16691.62

126.9956

0.1216450

108.7064

4. Discussion

The curves of graphs 1, 2 and 3 may appear common but they are not. Some of them may look straight lines or almost straight lines but they are not. The above quantities have an exponential behavior as someone can verify from the respective equations in section 2. Their graphical representations depend on the values of their exponential constant factors (α and β). If their values are small and as variable x increases, the values αx and βx do not change enough in order their exponential behavior to appear on the graphs. This is the reason they seem to be straight or almost straight lines.

The above explanation is given regarding their form. Regarding now their variation, the following reasoning is developed.

On one hand, the terms (VR+IRzC) and (VR-IRzC) of Eqns. (7) and (8) in section 2 are constant complex numbers since VR, IR and zC are constant complex numbers. That implies that they have a constant absolute value and a constant phase as shown in section 3.

On the other hand, the terms eγx and e-γx vary with distance x from the end of power transmission line.

The term eγx can be written as e(α+jβ)x=eαx ejβx=eαx[cos(βx) + j sin(βx)]

The values of α and β are real positive numbers for a typical real power transmission line. This will be understood from the following analysis.

The term eαx is the absolute value of the above term while the ejβx is the phase (angle) of the above term.

The term eαx increases as x increases i.e. the absolute value of voltage travelling wave increases as we approach the beginning of line. In other words, the absolute value (intensity) of voltage travelling wave (Eq. (7)) diminishes as the wave travels from the beginning of line (where the voltage is applied and the voltage travelling wave starts) to the end of line as one expects in real world (the intensity of signal diminishes as it moves away from source).

The term βx similarly increases as x increases. With similar as above reasoning, the term βx i.e. the phase of voltage travelling wave (Eq. (7)) diminishes as the wave travels from the beginning of line and moves to the end of line.

Similarly, the term e-γx can be written as e-(α+jβ)x=e-αx e-jβx=e-αx[cos(-βx) + j sin(-βx)]

At the end of electric power transmission line a part of voltage travelling wave is refracted and moves in the opposite direction of that of the voltage travelling wave ie. from the end towards the beginning of the line. This is implied by the negative value of -γx. With similar as above reasoning, the term e-αx decreases as x increases. In other words, the absolute value (intensity) of voltage refracted wave (Eq. (8)) decreases as the wave moves from the end towards the beginning of line as one expects. It is really the part of voltage travelling wave that arrives at the end of line and refracts travelling in the opposite direction of line.

Additionally, the term -βx decreases as x increases i.e. the phase (angle) of voltage refracted wave (Eq. (8)) decreases as the wave moves from the end towards the beginning of line.

Using similar thinking, the term e-2γx of Eq. (9) regarding the voltage refraction co-efficient can be written as follows:

e-2(α+jβ)x=e-2αx e-j2βx=e-2αx[cos(-2βx) + j sin(-2βx)]

Thus, using similar as above reasoning, both the magnitude and the phase angle of the voltage refraction co-efficient decrease as we move from the end (where the refraction occurs) towards the beginning of line as one expects.

5. Conclusions

Studying the results presented in Table 1 and their graphs 1 to 3 of section 3, we can observe and conclude the following:

(1) the intensity (absolute value) of voltage travelling wave decreases as the wave travels from the beginning towards the end of line i.e. along the direction the voltage travelling wave moves

(2) the phase (angle) of voltage travelling wave decreases as the wave travels from the beginning towards the end of line i.e. along the direction the voltage travelling wave moves

(3) the intensity (absolute value) of voltage refracted wave decreases as the wave moves from the end towards the beginning of line i.e. along the direction the current refracted wave moves

(4) the phase (angle) of voltage refracted wave decreases as the wave moves from the end towards the beginning of line i.e. along the direction the current refracted wave moves

(5) the percentage (absolute value) of voltage refraction co-efficient decreases from the end (where the refraction occurs) towards the beginning of line

(6) the phase (angle) of voltage refraction co-efficient also decreases from the end towards the beginning of line

Regarding now the information that is drawn from the graphs 1 to 2 is discussed in the following paragraphs.

Looking at graphs 1 and 2, the magnitude of voltage travelling and refracted wave decreases as one moves from the left rear of the line where the power source is towards the right rear of the line where the load is and then back to the beginning. This observation implies that both line and load present an ohmic-inductive behaviour. In other words, we have a reactive power flow from the source to line and load. Regarding the load is pure ohmic as one can see in section 3 from the data of the typical power line given. Thus, the above statement is right.

The line from the data given in section 3 has an ohmic (R) as well as an inductive (L) long-wise elements plus a capacitive (C) transversal element. The above statement that the line presents an ohmic-inductive behaviour means that the capacitive element of the line does not produce enough reactive power to cover the needs of the inductive long-wise element of the line and thus the source comes to cover the rest reactive power needed.

Looking again at graphs 1 and 2, we can see that the phase of voltage travelling and refracted wave also decrease as one moves from the left rear of the line where the power source is towards the right rear of the line and then back to the beginning. The above observation implies and cannot be otherwise that we have an active power flow from the left rear of the line where the power source is towards the right rear of the line where the load is and back to the beginning in order to cover the needs in active power of both the ohmic element of the line and load.

Then, we can conclude that the above observations verify the analysis and discussion developed in section 4 of the paper. For better understanding of electric transmission line voltage as a wave, we propose to study it using cartesian co-ordinates. This will be the subject of a future paper.

List of Symbols

R=long-wise omhic resistance of power transmission line (under sinusoidal voltage) per unit length of line (Ω/km)

L=long-wise inductance of power transmission line (under sinusoidal voltage) per unit length of line (H/km)

C=transversal capacitance of power transmission line (under sinusoidal voltage) per unit length of line (F/km)

G=transversal conductance of power transmission line (under sinusoidal voltage) per unit length of line (S/km)

l=length of power transmission line (km)

z=R+jωL=long-wise complex impedance of power transmission line per unit length of line (Ω/km)

y=G+jωC=transversal complex conductance of power transmission line per unit length of line (S/km)

Z=z.l=total long-wise complex impedance of power transmission line (Ω)

Y=y.l=total transversal complex conductance of power transmission line (S)

VS=complex line to earth voltage at the beginning of power transmission line, Sending voltage (V)

VR=complex line to earth voltage at the end of power transmission line, Receiving voltage (V)

IS=complex phase current at the beginning of power transmission line, Sending current (A)

IR=complex phase current at the end of power transmission line, Receiving current (A)

γ=$\sqrt{\mathrm{zy}}$=α+jβ=transmission co-efficient of power transmission line (km-1)

α=reduction co-efficient of power transmission line (neper/km)

β=phase co-efficient of power transmission line (rad/km)

zC=$\sqrt{\frac{z}{y}}$=characteristic impedance of power transmission line (Ω)

e=cosφ +jsinφ=Euler’s equation

λ=$\frac{2 \pi}{\beta}$=wave length of power transmission line (km)

υ=wave transmission velocity of power transmission line (km/sec)

τ=wave travelling time in order to cover the length of power transmission line (sec)

Δ=electric phase (angle) of power transmission line (rad)

$\frac{\Delta}{I}$=electric phase (angle) of power transmission line per unit length of line (rad/km)

Vtrav(x)=voltage travelling wave as a function of distance x (V)

Vrefr(x)=voltage refracted wave as a function of distance x (V)

ρV(x)=$\frac{V_{\text {refr }}(x)}{V_{\operatorname{trav}}(x)}$=voltage refraction co-efficient as a function of distance x

φ(x)=electric phase(angle) of respective complex quantity as function of distance x (°)

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