Georgios Leonidopoulos*
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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In this paper, a medium length electric line that presents only longitudinal ohmic resistance and longitudinal inductance is modelled, is studied mathematically and is tested experimentally in order to verify the findings.
medium length electric line, model, theoretical analysis, experimental testing
An electric power transmission line is one of the basic parts of an electric power system [1-8]. In a previous paper [1], a short length electric power transmission line was studied in depth. The characteristic element of a short line is the long-wise inductance. The other electric elements of the line such as the long-wise ohmic resistance and the transversal ohmic conductance and capacitance due the shortness of the line have very small value and are not considered in the electric analysis. In this paper, the study is extended to a medium length transmission line where the value of the long-wise ohmic resistance is increased and therefore is taken into account. The equations that are developed from the electric analysis of the line are therefore longer and more complicated than those of the short line that developed in [1].
In section 2, the above line is examined electrically while in section 3 the above line is modelled and the findings are compared with those of section 2.
In section 4, what is expected and stated in section 2 and the equations drawn in section 3 are tested experimentally.
At the end, in section 5, discussion and conclusions follow.
In Figure 1, the electric circuit of the medium length line is presented [1].
Examining electrically the mentioned line, the following are expected [1]:
1) P1, P2 and Pline must be positive;
2) P1=Pline+P2;
3) Qline must be positive;
4) Q1=Qline+Q2;
5) Z1=Zline+Z2;
6) A criterion drawn by either S2 or Z2 can be developed to indicate the kind of the load. The criterion logically must be the same whatever element (S2 or Z2) is used.
Figure 1. Electric circuit of medium length line
From Figure 1, using the basic electric laws and carrying out the relative mathematical operations, the following are derived:
$\begin{gathered}\mathrm{I}=\frac{\mathrm{V}_1<\theta_1-\mathrm{V}_2<\theta_2}{\mathrm{Z}_{\text {line }}} \\ =\frac{\left[\mathrm{R}\left(\mathrm{V}_1 \cos \theta_1-\mathrm{V}_2 \cos \theta_2\right)+\omega \mathrm{L}\left(\mathrm{V}_1 \sin \theta_1-\mathrm{V}_2 \sin \theta_2\right)\right]}{\mathrm{Z}^2} \\ +\frac{\mathrm{j}\left[\mathrm{R}\left(\mathrm{V}_1 \sin \theta_1-\mathrm{V}_2 \sin \theta_2\right)-\omega \mathrm{L}\left(\mathrm{V}_1 \cos \theta_1-\mathrm{V}_2 \cos \theta_2\right)\right]}{\mathrm{Z}^2}\end{gathered}$ (1)
$\begin{gathered}\mathrm{S}_1=\mathrm{V}_1<\theta_1 \mathrm{I}^* \\ =\frac{\left[\mathrm{R}\left[\mathrm{V}_1{ }^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]+\omega\mathrm{L}\mathrm{V}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2} \\ +\frac{\mathrm{j}\left[\omega \mathrm{L}\left[\mathrm{V}_1{ }^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]-\mathrm{RV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}\end{gathered}$ (2)
$\mathrm{P}_1=\frac{\left[\mathrm{R}\left[\mathrm{V}_1{ }^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]+\omega\mathrm{L} \mathrm{V}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}$ (3)
$\mathrm{Q}_1=\frac{\left[\omega \mathrm{L}\left[\mathrm{V}_1{ }^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]-\mathrm{RV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}$ (4)
$\begin{gathered}\mathrm{S}_2=\mathrm{V}_2<\theta_2 \mathrm{I}^*\\= \frac{\left[\mathrm{R}\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2^2\right]+\omega \mathrm{LV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}\\+ \frac{\mathrm{j}\left[\omega \mathrm{L}\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2^2\right]-\mathrm{RV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}\end{gathered}$ (5)
$\mathrm{P}_2=\frac{\left[\mathrm{R}\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2^2\right]+\omega\mathrm{L} \mathrm{V}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}$ (6)
$\mathrm{Q}_2=\frac{\left[\omega \mathrm{L}\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2^2\right]-\mathrm{RV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]}{\mathrm{Z}^2}$ (7)
$\begin{gathered}\mathrm{V}_{\text {line }}=\mathrm{V}_1<\theta_1-\mathrm{V}_2<\theta_2=\left(\mathrm{V}_1 \cos \theta_{1^{-}}\right. \\ \left.\mathrm{V}_2 \cos \theta_2\right)+\mathrm{j}\left(\mathrm{V}_1 \sin \theta_1-\mathrm{V}_2 \sin \theta_2\right)\end{gathered}$ (8)
$\mathrm{V}^2{ }_{\text {line,magn. }}=\mathrm{V}_1^2+\mathrm{V}_2^2-2 \mathrm{~V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)$ (9)
$\begin{aligned} & \mathrm{S}_{\text {line }}=\mathrm{V}_{\text {line }} \mathrm{I}^*=\frac{\mathrm{V}^2 \text { line,magn. }}{\mathrm{Z}^* \text { line }}=\mathrm{S}_1-\mathrm{S}_2 \\ & =\frac{\mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{~V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)}{\mathrm{Z}^2}(\mathrm{R}+\mathrm{j} \omega \mathrm{L})\end{aligned}$ (10)
$\mathrm{P}_{\text {line }}=\frac{\mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{~V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)}{\mathrm{Z}^2} \mathrm{R}$ (11)
$Q_{\text {line }}=\frac{\mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)}{\mathrm{Z}^2} \omega \mathrm{L}$ (12)
$\begin{gathered}\mathrm{Z}_1=\frac{\mathrm{V}_1<\theta_1}{\mathrm{I}} \\ =\mathrm{Z}_{\text {line }} \frac{\left[\mathrm{V}_1{ }^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]-\mathrm{jV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)}{\mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)}\end{gathered}$ (13)
$\begin{gathered}\mathrm{Z}_2=\frac{\mathrm{V}_2<\theta_2}{\mathrm{I}} \\ =\mathrm{Z} \text { line } \frac{\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2{ }^2\right]-\mathrm{jV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)}{\mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{V}_1 \mathrm{V}_7 \cos \left(\theta_1-\theta_2\right)}\end{gathered}$ (14)
$Z_{l i n e}=Z_1-Z_2=Z_{\text {line }}$ (15)
Furthermore, the following inequalities must always be valid:
a) $\begin{aligned} \mathrm{P}_1>0 & \rightarrow\left[\mathrm{R}\left[\mathrm{V}_1^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]\right. \\ & \left.+\omega\mathrm{L}\mathrm{V}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]>0 \\ & \rightarrow \frac{\cos \left(\theta_1-\theta_2+\varphi_{\mathrm{L}}\right)}{\cos \varphi_{\mathrm{L}}}<\frac{\mathrm{V}_1}{\mathrm{~V}_2}\end{aligned}$ (16)
b) $\begin{aligned} \mathrm{P}_2>0 \rightarrow & {\left[\mathrm{R}\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2^2\right]\right.} \\ & + \left.\omega \mathrm{LV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]>0 \\ & \rightarrow\frac{\cos \left(\theta_1-\theta_2-\varphi_{\mathrm{L}}\right)}{\cos \varphi_{\mathrm{L}}}>\frac{\mathrm{V}_2}{\mathrm{~V}_1}\end{aligned}$ (17)
c) $\begin{gathered}\mathrm{P}_{\text {line }}>0 \rightarrow \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{~V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)>0 \rightarrow \\ \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2>2 \mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right) ?\end{gathered}$ (18)
d) $\begin{gathered}\mathrm{Q}_{\text {line }}>0 \rightarrow \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{~V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)>0 \rightarrow \\ \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2>2 \mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right) ?\end{gathered}$ (19)
The above (c) and (d) are always true due to Eq. (20).
$\begin{array}{r}\left(\mathrm{V}_1-\mathrm{V}_2\right)^2 \geq 0 \rightarrow \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2-2 \mathrm{V}_1 \mathrm{V}_2 \geq 0 \rightarrow \\ \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2 \geq 2 \mathrm{V}_1 \mathrm{~V}_2 \geq \\ 2 \mathrm{~V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right) \rightarrow \\ \mathrm{V}_1{ }^2+\mathrm{V}_2{ }^2 \geq 2 \mathrm{V}_1 \mathrm{~V}_2 \cos \left(\theta_1-\theta_2\right)\end{array}$ (20)
In addition, the following inequalities are always valid when the character of the respective load is ohmic-inductive:
a) $\begin{gathered}\mathrm{Q}_1>0 \rightarrow\left[\omega \mathrm{L}\left[\mathrm{V}_1^2-\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)\right]-\right. \\ \left.\mathrm{RV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]>0 \rightarrow \\ \frac{\sin \left(\theta_1-\theta_2+\varphi_{\mathrm{L}}\right)}{\cos \varphi_{\mathrm{L}}}<\frac{\mathrm{V}_1}{\mathrm{V}_2} \frac{\omega \mathrm{L}}{\mathrm{R}}\end{gathered}$ (21)
b) $\begin{aligned} \mathrm{Q}_2>0 \rightarrow & {\left[\omega \mathrm{L}\left[\mathrm{V}_1 \mathrm{V}_2 \cos \left(\theta_1-\theta_2\right)-\mathrm{V}_2^2\right]-\right.} \\ & \left.\mathrm{RV}_1 \mathrm{V}_2 \sin \left(\theta_1-\theta_2\right)\right]>0 \rightarrow \\ & \frac{\sin \left(\theta_1-\theta_2-\varphi_{\mathrm{L}}\right)}{\cos \varphi_{\mathrm{L}}}<\frac{\mathrm{V}_2}{\mathrm{V}_1}\left(-\frac{\omega \mathrm{L}}{\mathrm{R}}\right)\end{aligned}$ (22)
If the opposite inequalities are true, the character of the respective load is ohmic-capacitive. If the equality is true, the character of the respective load is pure ohmic.
Using a laboratory model of the electric line, the following measurements were taken:
$Z_{\text {line }}=400<90^{\circ} \Omega$
$\mathrm{V}_1=142.4<22^{\circ} \mathrm{V}$
$\mathrm{V}_2=132<0^{\circ} \mathrm{V}$
$\mathrm{I}=0.133<0^{\circ} \mathrm{A}$
P1=18W
Q1=6.5Var
P2=18W
Q2=0Var
Then, calculating the following equations, we find:
Eq. (1): $\mathrm{I}=0.133<-0.033^{\circ} \mathrm{A}$
Eq. (3): P1=17.6W
Eq. (4): Q1=7.12Var
Eq. (6): P2=17.6W
Eq. (7): Q2=0.01Var
Eq. (11): Pline=P1 – P2 = 0
Eq. (12): Qline=7.11Var
Eq. (16): $-\infty<1.0788$
Eq. (17): $+\infty>0.927$
Eq. (18): 37701.76>34856.18
Eq. (19): 37701.76>34856.18
Eq. (21): 0.927=0.927
Eq. (22): 0.927=0.927
The value of Eq. (1) is very close to that of the experimental result. The values of Eqs. (3) and (6) are the same and very close to that of the experimental result implying that the line has no resistance. Their positive value is verified by inequalities (16) and (17). The values of Eqs. (4) and (7) are very to those of experimental results and indicate the inductive character of the line. Their positive value is verified by Eqs. (21) and (22) that also prove the ohmic character of the load. The value of Eq. (11) is the same to that of the experimental result indicating a line without resistance and is verified by inequality (18). The value of Eq. (12) is almost the same to that of the experimental result indicating a line with inductance and is verified by inequality (19). Any small differences are due to the rounding of numbers, the precision of the instruments and the fact that the meters measuring power and phase are not digital.
If you compare the analysis and findings in sections 2, 3 and 4, the following can be drawn:
1) the P1 and P2 are positive as expected,
2) the Pline was proved to be always positive and equal to P1 minus P2 as stated,
3) the Qline proved to be positive and equal to Q1 minus Q2 as suggested,
4) it was proved that the Z2 plus the Zline is equal to the Z1,
5) a criterion, inequality (22), was developed and tested successfully to indicate the kind of the load,
6) the inequalities (16) and (17) regarding that active power P1 and P2 are always positive were developed and tested successfully,
7) the inequalities (21) and (22) regarding reactive power Q1 and Q2 were developed and tested successfully.
At the end, the experimental values come to verify the theoretical findings.
$\mathbf{Z}_{\text {line }}=R_{\text {line }}+j \omega L_{\text {line }}=Z<\varphi_L$
For all other symbols see [1].
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