Analysis of Electric Power Transmission Line Presenting Only Long-Wise Inductance

Analysis of Electric Power Transmission Line Presenting Only Long-Wise Inductance

Georgios Leonidopoulos

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

Corresponding Author Email: 
gleon@uniwa.gr
Page: 
94-97
|
DOI: 
https://doi.org/10.18280/mmc_a.922-409
Received: 
5 March 2019
|
Accepted: 
8 June 2019
|
Published: 
30 December 2019
| Citation

OPEN ACCESS

Abstract: 

In this paper, a simple electric power transmission line presenting only long-wise inductance is studied electrically and is analysed mathematically in order to find the mathematical equations, regarding mainly voltage, current and power flowing in the line in steady state condition, that describe it. The information that one and particularily an electrical engineer can dig out regarding its electric behaviour are obtained and analysed in detail.

Keywords: 

electric power transmission line, modelling, power, electrical analysis, mathematical analysis

1. Introduction

The simplest representation of an electric power transmission line [1-8] is the electric equivalent that presents only long-wise inductance.

In next section 2, the above power transmission line is examined carefully and thoughtfully from the electrical point of view that means what an electrical engineer expects from this type of line having the above mentioned characteristics and to write them down.

Then, in section 3 that follows, the above line is analyzed mathematically and the equations drawn are coming to verify or not the expectations of the previous section 2.

In section 4, the experimental results obtained from a low voltage laboratory model of an electric power transmission line presenting only long-wise inductance are presented. The results verify the theoretical results obtained in section 3.

Finally, in section 5, the relative discussion is developed and the respective conclusions are drawn.

2. Electric Analysis of the Electric Power Transmission Line

In Figure 1, the electric equivalent of the above mentioned electric power transmission line is presented. The data given are the voltages at the beginning and at the end of line as well as the inductance of line.

Figure 1. Electric equivalent representation of electric power transmission line

If the above power transmission line (Figure 1) is examined carefully and thoughtfully from the electrical point of view, an electrical engineer expects the following:

(1) the active power P1 at the beginning of the line and the active power P2 at the end of the line must be both positive since the active power is actually Joule losses i.e. ohmic losses or heat losses that are consumed.

(2) the active power P1 at the beginning of the line must be equal to the active power P2 at the end of the line. That is because the line has no ohmic resistance and therefore no active power is consumed in the line. In other words, the active power of the line Pline is zero.

(3) the reactive power of the line Qline must be always positive. The above statement is based on the fact that the line presents only inductance and the inductance absorbs reactive power.

(4) the reactive power Q1 at the beginning of the line must be equal to the addition of the reactive power Q2 at the end of the line plus the reactive power of the line Qline. In other words, the reactive power Q1 minus the reactive power Q2 must be equal to Qline.

(5) the apparent resistance in complex form Z1 at the beginning of the line must be equal to the apparent resistance in complex form Z2 at the end of the line plus the resistance in complex form of the line Zline. This is due to the connection in series of all the above complex resistances. In other words, the complex resistance Zline is equal to Z1 minus Z2.

(6) either from the mathematical expression of the apparent power S2 at the end of the line or the apparent complex resistance Z2 at the end of the line a criterion must and can be developed i.e. a mathematical relationship among the quantities V1, θ1, V2, θ2, ω, Lline to indicate when the load is ohmic-inductive, ohmic-capacitive or pure ohmic. Logically, the criterion drawn by either of the above S2 and Z2 must be the same. We expect to draw the above criterion from either expression S2 or Z2 because they refer not only to the end of the line but also to the load which is connected to the end of the line.

In the following section, the mathematical expressions of all the above quantities are going to be developed in order to verify or not all the electrical conclusions drawn logically in this section. The relative mathematical expressions will be based on the electric equivalent of the electric power transmission line (Figure 1).

3. Mathematical Analysis of the Electric Power Transmission Line

From Figure 1, using Kirchoff’s 2nd law and Ohm’s law, we have:

$I=\frac{V_{1}<\theta_{1}-V_{2}<\theta_{2}}{Z_{\text {line}}}=\frac{\left(V_{1} \sin \theta_{1}-V_{2} \sin \theta_{2}\right)-j\left(V_{1} \cos \theta_{1}-V_{2} \cos \theta_{2}\right)}{\omega L}$      (1)

$S_{1}=V_{1}<\theta_{1} I^{*}=\frac{V_{1} V_{2} \sin \left(\theta_{1}-\theta_{2}\right)+j\left[V_{1}^{2}-V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)\right]}{\omega L}$      (2)

$P_{1}=\frac{V_{1} V_{2} \sin \left(\theta_{1}-\theta_{2}\right)}{\omega L}$      (3)

It must be:

$P_{1}>0 \rightarrow \sin \left(\theta_{1}-\theta_{2}\right)>0 \rightarrow \theta_{1}>\theta_{2}$      (4)

$Q_{1}=\frac{V_{1}^{2}-V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)}{\omega L}$      (5)

$S_{2}=V_{2}<\theta_{2} I^{*}=\frac{V_{1} V_{2} \sin \left(\theta_{1}-\theta_{2}\right)+j\left[V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)-V_{2}^{2}\right]}{\omega L}$      (6)

$P_{2}=\frac{V_{1} V_{2} \sin \left(\theta_{1}-\theta_{2}\right)}{\omega L}$      (7)

It must be:

$P_{2}>0 \rightarrow \sin \left(\theta_{1}-\theta_{2}\right)>0 \rightarrow \theta_{1}>\theta_{2}$      (8)

$Q_{2}=\frac{V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)-V_{2}^{2}}{\omega L}$      (9)

Vline=V1<θ1 –V2<θ2=(V1cosθ1-V2cosθ2)+j(V1sinθ1-V2sinθ2)      (10)

V2line,magnitude=V12+V22-2V1V2cos(θ1-θ2)      (11)

$S_{l i n e}=V_{l i n e} I^{*}=\frac{V^{2} \text {line,magnitude}}{Z^{*} _\text {line}}=j \frac{V_{1}^{2}+V_{2}^{2}-2 V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)}{\omega L}$      (12) 

Pline=0      (13)

$Q_{\text {line}}=\frac{V_{1}^{2}+V_{2}^{2}-2 V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)}{\omega L}$      (14)

It must be:

Qline>0→V12+V22-2V1V2cos(θ12)>0→V12+V22>? 2V1V2cos(θ12)

This is true due to Eq. (15) below:

(V1-V2)2≥ 0 → V12+V22-2V1V2 ≥ 0→ V12+V22 ≥ 2V1V2 ≥ 2V1V2cos(θ1-θ2)→ V12+V22 ≥ 2V1V2cos(θ1-θ2)      (15)

$S_{l i n e}=S_{1}-S_{2}=j \frac{V_{1}^{2}+V_{2}^{2}-2 V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)}{\omega L}$      (16)

$Z_{1}=\frac{V_{1}<\theta_{1}}{I}=\omega L \frac{V_{1} V_{2} \sin \left(\theta_{1}-\theta_{2}\right)+j\left[V_{1}^{2}-V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)\right]}{V_{1}^{2}+V_{2}^{2}-2 V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)}$      (17)

$Z_{2}=\frac{V_{2}<\theta_{2}}{I}=\omega L \frac{V_{1} V_{2} \sin \left(\theta_{1}-\theta_{2}\right)+j\left[V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)-V_{2}^{2}\right]}{V_{1}^{2}+V_{2}^{2}-2 V_{1} V_{2} \cos \left(\theta_{1}-\theta_{2}\right)}$      (18)

Zline=Z1–Z2=jωL      (19)

(1) If the load is ohmic-inductive, it must be: Q2 > 0 (Eq. (9)) or the imaginary part of Z2 must be > 0 (Eq. (18))

then V1V2cos(θ1-θ2)-V22 > 0 →$\frac{V_{2}}{V_{1}}$ <cos(θ1-θ2)      (20)

since 0 ≤ cos(θ12) ≤1, then  $\frac{V_{2}}{V_{1}}$ < 1 → V2<V1      (21)

(2) If the load is ohmic-capacitive, it must be: Q2 < 0 (Eq. (9)) or the imaginary part of Z2 must be < 0 (Eq. (18))

then V1V2cos(θ1-θ2)-V22 < 0 → cos(θ1-θ2) < $\frac{V_{2}}{V_{1}}$      (22)

(3) If the load is omhic, it must be: Q2=0 (Eq. (9)) or the imaginary part of Z2 must be=0 (Eq. (18))

then V1V2cos(θ1-θ2)-V22=0 → cos(θ1-θ2)= $\frac{V_{2}}{V_{1}}$      (23)

4. Experimental Examination of Theoretical Analysis

A low voltage laboratory model of an electric power transmission line presenting only long-wise inductance is utilized in order to obtain experimental results. These results will then be compared to the theoretical ones in order to verify or not them.

The electric power transmission line model in steady state condition gave the following experimental measurements:

Zline=400< 90° Ω

V1=142.4V

V2=132V

I=0.12A

θ1=22°

θ2=0°

P1=18W

Q1=6Var

P2=18W

Q2=0Var

Then, calculating the following equations, we find:

Eq. (1): I=0.13<-0.03°A

Eq. (3): P1=17.6W

Eq. (4): θ1 > θ2  → 22° > 0°

Eq. (5): Q1=7.12Var

Eq. (7): P2=17.6W

Eq. (8): θ1 > θ2  → 22° > 0°

Eq. (9): Q2=0.01Var

Eq. (13): Pline=P1 – P2=0

Eq. (14): Qline=7.11Var

Eq. (15): 37701.76 > 34856.18

Eq. (23): 0.927=0.927

The values of Eqns. (3) and (7) are the same and very close to that of experimental result. The positive value of equations (3) and (7) are verified by the inequalities (4) and (8). The values of Eq. (5) and (14) and their positive value verify inequality (15) and with the value of Eq. (13) also indicate the inductive character of the line. Any small differences are due to the rounding of numbers and the precision of the instruments. The equality of Eq. (23) implies the ohmic character of the load.

5. Discussion and Conclusions

Comparing the electric analysis developed logically in section 2 to the mathematical analysis of the electric equivalent circuit of the electric power transmission line under discussion in section 3, we can draw the following:

(1) the active power P1 at the beginning of the line is positive as we expect (section 2, case 1) only if θ12 (Eqns. (3), (4)). Otherwise, there is no flow of active power from the source to the line. The same applies to the active power P2 at the end of the line (Eq. (7)).

(2) the active power P1 at the beginning of the line is equal to the active power P2 at the end of the line (Eqns. (3), (7)) as stated in section 2, case 2. That means Pline=0 (Eq. (13)) as expected.

(3) the reactive power Qline of the line is positive as expected (section 2, case 3) and shown in Eq. (14) and (15). In the case that V1=V2 and θ12, there is no flow of reactive power in the line and no flow of active power from the source to the line (case 1 above) since complex voltage V11 at the beginning of the line is equal in magnitude and phase (angle) to the complex voltage V22 at the end of the line and the current flow in the line is zero (Eq. (1)).

(4) the difference between the reactive power Q1 and Q2 (Eqns. (5), (9)) at the beginning and the end of the line respectively gives as result the reactive power of the line Qline (Eq. (14)) as expected in section 2, case 4.

(5) similarly, the difference between the apparent complex resistance Z1 and Z2 (Eqns. (17), (18)) at the beginning and the end of the line respectively gives as result the complex resistance of the line Zline (Eq. (19)) as expected in section 2, case 5.

(6) looking at the expressions of the apparent power of the load S2 (Eq. (6)) and the load Z2 in complex form (Eq. (18)) since ωL>0 and the denominator in Eq. (18) is always positive (Eq. (15)), one can see that the same complex numerator in both Eq. (6) and (18) define the criterion regarding the type of load as expected in section 2, case 6. The real part of the numerator is always positive (Eq. (4)) showing the existence of the ohmic part of the load. If the imaginary part of the numerator is positive implying the presence of load with inductive character, both inequalities of Eq. (20) and (21) must be valid. If the above imaginary part is negative implying the presence of load with capacitive character only the inequality in Eq. (22) must be valid. If the above imaginary part is zero implying the presence of no imaginary load or the presence of both inductive and capacitive loads that nullify each other the equality of Eq. (23) is valid.

Finally, the experimental results of section 4 as analyzed and discussed in detail in that section come to verify the theoretical results and the relative analysis.

List of Symbols

V1<θ1=voltage in complex form (polar) at the beginning of electric power transmission line

V1=voltage magnitude (V)

θ1=voltage phase(angle) (°)

<θ=ejθ=cosθ +jsinθ=Euler’s equation

S1=apparent power (VA) in complex form at the beginning of electric power transmission line

S1=P1+jQ1

P1=active power (W) at the beginning of electric power transmission line

Q1=reactive power (VAr) at the beginning of electric power transmission line

Z1=apparent resistance in complex form at the beginning of electric power transmission line

V2<θ2=voltage in complex form (polar) at the end of electric power transmission line

V2=voltage magnitude (V)

θ2=voltage phase(angle) (°)

S2=apparent power (VA) in complex form at the end of electric power transmission line

S2=P2+jQ2

P2=active power (W) at the end of electric power transmission line

Q2=reactive power (VAr) at the end of electric power transmission line

Z2=apparent resistance in complex form at the end of electric power transmission line

Vline=voltage drop in complex form along the electric power transmission line

Sline=apparent power (VA) in complex form of electric power transmission line

Sline=Pline+jQline

Pline=active power (W) of electric power transmission line

Qline=reactive power (VAr) of electric power transmission line

Zline=jωLline=resistance in complex form of electric power transmission line

Lline=inductance of electric power transmission line

Z*line=complex conjugate of Zline

I=current in complex form of electric power transmission line

  References

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