Pi-equivalent Line Model

Authors: Carlos Alegre (eRoots)

Reviewers: Eduardo Prieto Araujo (UPC), Josep Fanals Batllori (eRoots)

Context

Transmission lines carry the electric power from one substation of the grid to another. There are overhead lines or underground cables, and they can be classified as low-range, middle-range or long-range depending on the distance between the two ends. Modelling precisely their behavior is essential to calculate the voltage drops, the phase shifts due to inductive behavior and the losses that occur when transporting energy from one point to another.

Model use, assumptions and validity

The model described is intended to be used in transmission lines for any range, making appropriate assumptions depending on their length. For lower distances, more simplifications can be made without loosing precision. For longer distances, the equivalent circuit proposed can be used, but its parameters will need to be obtained considering the distribution of impedance across the whole line, instead of the grouped admittance proposed for middle range lines.

The derivation of the equations has been done starting from the distributed parameter model as in references [1], [2], [3] and [4]. It then is transformed into a $\pi$-equivalent circuit for the general case of transmission lines.

The assumptions made are the following:

The line is represented phase by phase with a series impedance and two parallel capacitors, distributing the total capacitance of the line in two equal blocks situated at each end of the line. The resulting circuit is named after the $\pi$-shape it has.
In the case of three-phase transmission lines, the currents are assumed to be balanced (no neutral cable), and there exists transposition of phases in order to mitigate the inductive unbalanced due to unsymmetrical disposition. This allows the description of the line as a set of three single-phase lines.
The conductance effects are neglected in front of the inductive and capacitive effects of the line.
Skin effect and corona effect are not considered in the model.
The model does not consider the dependency on the temperature for its impedance values.
The frequency of the system is considered constant.

To determine the equivalent parameters of the line, the user may provide their values calculated according to the characteristics of the line. Depending on the distance of the line, some simplifications can be made. There is no consensus on the distance ranges in which each model should be used, the values provided are the ones stated in reference [1].

Depending on the number of phases, conductors per phase and geometry of the lines, the values for capacitance and inductance can be calculated differently. The three-phase calculations for asymmetrical disposition are developed in the model description as it is a more general case.
For long range transmission lines (over 200 kilometers of length), the equivalent parameters calculated from the exact solutions should be used.
For medium range transmission lines (between 80 and 200 kilometers of length), the direct calculation of the $\pi$-equivalent values using the total series and shunt impedance and admittance can be used.
For short range transmission lines (below 80 kilometers of length), the shunt admittance can be neglected, and the model assumes that all the current travels from the source to the receiver, only having heat losses.

Its validity is for steady-state analysis of transmission lines in any configuration, given the appropriate values for the line parameters. It does not consider the transient behavior of the lines. For such analysis, the Constant Parameters Line Model and the Frequency Dependent Line Model should be used.

Model description

Line parameters

Parameters	details	Unit
$l$	Length of the line	$m$
$r$	Resistance of the line per meter	$\Omega m^{-1}$
$x_L$	Reactance of the line per meter	$\Omega m^{-1}$
$z$	Impedance of the line per meter $z = r + jx_L$	$\Omega m^{-1}$
$Z$	Total impedance of the line $Z = zl$	$\Omega$
$Z'$	Equivalent impedance of the line	$\Omega$
$g$	Conductance of the line per meter	$\Omega^{-1} m^{-1}$
$b$	Susceptance of the line per meter	$\Omega^{-1} m^{-1}$
$y$	Equivalent admittance of the line per meter $y = g + jb$	$\Omega^{-1} m^{-1}$
$Y$	Total admittance of the line $Y = yl$	$\Omega^{-1}$
$Y'$	Equivalent admittance of the line	$\Omega^{-1}$
$Z_C$	Characteristic impedance of the line $Z_C = \sqrt(\frac{z}{y})$	$\Omega$
$\gamma$	Propagation constant of the line $Z_C = \sqrt(zy)$	$m^{-1}$

Line variables

Variable	details	Unit
$V_{S}$	Complex phase-to-ground voltage of the source terminal	$V$
$I_{S}$	Complex current injected by the source terminal into the line	$A$
$V_{R}$	Complex phase-to-ground voltage of the receiver terminal	$V$
$I_{R}$	Complex current injected by the receiver terminal into the line	$A$

Equations

$$ V_S = (1 + \frac{1}{2}Y'Z')V_R - Z' I_R $$ $$ I_S = Y'(1 + \frac{1}{4}Y'Z')V_R - (1 + \frac{1}{2}Y'Z') I_R $$

where $Z'= Z$ and $Y' = 0$ for short-range lines, $Z' = Z$ and $Y' = Y$ for medium-range lines, and $Z'= Z(\frac{\sinh(\gamma l)}{\gamma l})$, $Y' = Y \frac{\tanh(\frac{\gamma l}{2})}{\frac{\gamma l}{2}}$ for long-range lines.

Operational principles

Line parameters

The line parameters are obtained from the per-unit length values of resistance, inductance, capacitance and conductance, as described in references [2] and [4]. The total series impedance and shunt admittance of the line are calculated as:

$$ Z = zl = (r + jx) l = (r + j\omega l_{ind})l $$ $$ Y = yl = (g + jb)l = (g + j\omega c)l $$

where $r$ is the resistance in $\Omega m^{-1}$, $x$ is the reactance in $\Omega m^{-1}$, $z$ is the impedance in $\Omega m^{-1}$, $g$ is the conductance in $\Omega^{-1} m^{-1}$, $b$ is the susceptance in $\Omega^{-1} m^{-1}$, $l_{ind}$ is the inductance in $H m^{-1}$, $c$ is the capacitance in $F m^{-1}$, $\omega$ is the angular frequency of the system in $rad/s$ and $l$ is the length of the line in $m$.

Resistance

Resistance is the opposition to the flow of current in the line. For transmission lines, its effects can be neglected in front of the inductive and capacitive behavior of the line, although it represents the main source of losses of the power lines, hence why it can be useful to include it in the models. Its value is calculated as the product of the resistance per unit length and the length of the line. The resistance per unit length is obtained from the resistivity of the material and the cross-sectional area of the conductor. The expression is given by:

$$ r = \frac{\rho}{A} $$

where $\rho$ is the resistivity of the material in $\Omega m$ and $A$ is the cross-sectional area of the conductor in $m^2$. The total resistance of the line can be obtained multiplicating by the total length of the line.

Although the model developed does not take into account thermal changes, it has to be noted that the resistance of the line is temperature-dependent, and it can be calculated as:

$$ R = R_0 (1 + \alpha (T - T_0)) $$

where $R_0$ is the resistance at a reference temperature $T_0$, $\alpha$ is the temperature coefficient of the material in $K^{-1}$ and $T$ is the temperature in $K$.

Some other possible sources of losses are due to the skin effect, which is the tendency of the current to flow on the surface of the conductor, and the corona effect, which is the ionization of the air surrounding the conductor due to the high voltage gradients.

Inductance

Inductive effects dominate the behavior of the transmission lines, and arise from the interaction between the parallel conductors (considered to be infinite for the purpose of this subsection) due to the magnetic flux linkages. The inductance of the line can be calculated using different methods depending on the geometry and configuration of the transmission lines.

For a single-phase line, the inductance of one of its phases can be calculated using the expression:

$$ L_p = 2 \cdot 10^{-7} \ln \left( \frac{D_{eq}}{r_p} \right) $$

where $\mu$ is the permeability of the medium in $H m^{-1}$, $D$ is the distance between the conductors in $m$, and $r$ is the radius of the conductor in $m$. The inductance of the line is calculated as the product of the inductance per unit length and the length of the line. The total inductance can be done by adding the inductances of the phases as $L_1 + L_2 = L$.

For three-phase lines, the inductance of a phase can be calculated using the expression:

$$ L_p = 2 \cdot 10^{-7} \ln \left( \frac{D}{r_p} \right) $$

where $D_{eq}$ is the equivalent distance between the conductors in $m$, calculated as $D_{eq} = (D_{12}D_{23}D_{31})^{\frac{1}{3}}$ using the distances between conductors. The total inductance can be found again by adding all the inductances.

Conductance

Conductance is present in the transmission lines in the form of a shunt admittance, being result of the leakage of current over the surface of the insulators. It is negligible in comparison to the conductive effects, which are also a form of shunt admittance of the line.

Capacitance

Capacitive effects are the most relevant form of shunt admittance of transmission lines. They are a result of the electric fields between different conductors and between conductors and ground, creating a charging current that is present even when there are no loads in the line. It is specially relevant for longer lines (it starts to be important around 100 km).

For three-phase unsymmetrical spaced lines, which is the more general case, and assuming uniform charge distribution in the surface of the conductors, the line to neutral capacitance can be calculated as:

$$ C = \frac{0.0242}{\log \left( \frac{D_{eq}}{r} \right)} $$

in $\mu F/km$, where $D_{eq}$ is the equivalent distance between the conductors in $m$ calculated as $D_{eq} = (D_{12}D_{23}D_{31})^{\frac{1}{2}}$ using the distances between conductors, and $r$ is the radius of the conductor in $m$. The capacitance of the line is calculated as the product of the capacitance per unit length and the length of the line.

Considering the conductor to earth capacitance, the expression can be expanded to:

$$ C = \frac{0.0242}{\log \left( \frac{D_{eq}}{r} \right) - \log \left( \frac{(h_{12}h_{23}h_{31})^{\frac{1}{3}}}{(h_1h_2h_3)^{\frac{1}{3}}}\right)} $$

with $h_{ij}$ being the distance between the conductor $i$ to the image reflection with respect to the ground of conductor $j$ in $m$, and $h_i$ being the distance of the conductor $i$ to its ground reflection.

Distributed parameters line model

The distributed parameters line model considers the distribution of the impedance along the line. The model is based on the transmission line equations, which are a set of partial differential equations that describe the behavior of the line. It is used to obtain the accurate values of the impedance, admittance, and other parameters of the line at any point of the line. Equivalent circuits such as the $\pi$-equivalent are used to describe the performance as seen from the terminals.

Suppose a line with its circuit parameters $z = r + jx$ and $y = g + jb$ calculated as the shown in the previous section. An infinitesimal section of the line, of length $dx$, can be represented by the following schematic:

Figure 1. Schematic of the distributed parameter transmission line.

Using the Kirchhoff laws, the following equations can be obtained:

$$ V(x+dx) = V(x) + zdxI(x) $$ $$ I(x+dx) = I(x) + ydxV(x)$$

Which can be expressed in their differential form for $dx \rightarrow 0$ as:

$$ \frac{dV}{dx} = zI(x) $$ $$ \frac{dI}{dx} = yV(x) $$

Performing the second derivative of the voltage expression, the expression obtained is:

$$ \frac{d^2V}{dx^2} = z\frac{dI}{dx} = zyV(x) $$

This is a linear differential equations the solution of which can be written as:

$$ V(x) = C_1e^{\gamma x} + C_2e^{-\gamma x} $$

with $\gamma = \sqrt{yz}$ being the propagation constant, with $z$ and $y$ being the per-length impedance and shunt admittance respectively. Deriving this general expression, the current can be obtained as:

$$ I(x) = \frac{1}{z} \frac{dV}{dx} = \frac{C_1}{Z_c} e^{\gamma x} - \frac{C_2}{Z_c}e^{-\gamma x} $$

with $Z_c$ being the characteristic impedance of the line, defined as $Z_c = \sqrt{\frac{z}{y}}$. To determine the values of the constants $C_1$ and $C_2$, the values of the voltages have to be evaluated at the ends of the line, i.e., $V(0) = V_R$ and $I(0) = - I_R$, changing the sign of the current to use the positive sign for current injected to the line from the bus. The resulting expressions are:

$$ V_R = C_1 + C_2 $$ $$ -I_R = \frac{C_1}{Z_c} - \frac{C_2}{Z_c} $$ $$ C_1 = \frac{V_R - Z_cI_R}{2} $$ $$ C_2 = \frac{V_R + Z_cI_R}{2} $$

Substituting this in the general expression for the voltage and current and rearranging, the resulting expressions along the line are:

$$ V(x) = \frac{V_R}{2} \left( e^{\gamma x} + e^{-\gamma x} \right) - \frac{Z_cI_R}{2} \left( e^{\gamma x} - e^{-\gamma x} \right) $$ $$ I(x) = \frac{V_R}{2Z_c} \left( e^{\gamma x} - e^{-\gamma x} \right) - \frac{I_R}{2} \left( e^{\gamma x} + e^{-\gamma x} \right) $$

Which can be expressed in terms of hyperbolic functions as:

$$ V(x) = V_R \cosh(\gamma x) - Z_cI_R \sinh(\gamma x) $$ $$ I(x) = \frac{V_R}{Z_c} \sinh(\gamma x) - I_R \cosh(\gamma x) $$

For $x=l$, the matrix relationship between the voltage and currents of the terminals are:

$$ \begin{bmatrix} V_S \\ I_S \end{bmatrix} = \begin{bmatrix} \cosh(\gamma l) & -Z_c \sinh(\gamma l) \\ \frac{1}{Z_c} \sinh(\gamma l) & -\cosh(\gamma l) \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix} $$

Pi-equivalent circuit

The pi-equivalent circuit corresponds to a representation of the line with the line admittance equally lumped into the ends of the line. The equivalent circuit proposed is the following:

Figure 2. Pi-equivalent circuit of the lumped admittance transmission line.

In this case, applying the Kirchhoff laws yields the following equations:

$$ I_S = -I_R + \frac{1}{2} V_R Y' + \frac{1}{2} V_S Y' $$ $$ V_S = V_R + (-I_R + \frac{1}{2} V_R Y')Z' = V_R (1 + \frac{1}{2}Y'Z') -I_R Z' $$

which rearranged in matrix form are:

$$ \begin{bmatrix} V_S \\ I_S \end{bmatrix} = \begin{bmatrix} 1 + \frac{1}{2} Y'Z' & -Z' \\ Y'(1 + \frac{1}{4}Y'Z') & -(1 + \frac{1}{2} Y'Z') \end{bmatrix} \begin{bmatrix} V_R \\ I_R \end{bmatrix} $$

Long-range transmission lines

For long-range transmission lines, the exact solution has already been derived, and it can be represented as a $\pi$-equivalent by finding the equivalent values of its impedance and lumped admittance. To obtain the equivalence, the following relationship have to hold:

$$ Z' = Z_C \sinh(\gamma l) $$ $$ 1 + \frac{1}{2}Y'Z' = \cosh(\gamma l) $$

Developing the first expression:

$$ Z' = Z_C \sinh(\gamma l) = \sqrt{\frac{z}{y}} \sinh(\gamma l) = zl \frac{\sinh(\gamma l)}{\gamma l} = Z(\frac{\sinh(\gamma l)}{\gamma l})$$

where $Z$ is the series impedance of the line, and $\frac{sinh(\gamma l)}{\gamma l}$ is the factor to obtain the $Z'$ for the $\pi$-equivalent circuit. The second expression can be developed as:

$$ 1 + \frac{1}{2}Y'Z_C \sinh(\gamma l) = \cosh(\gamma l) $$ $$ \frac{1}{2}Y' = \frac{1}{Z_C} \frac{\cosh(\gamma l) - 1}{\sinh(\gamma l)} = \sqrt{\frac{y}{z}} \tanh(\frac{\gamma l}{2}) $$ $$ \frac{1}{2}Y' = \frac{Y}{2} \frac{\tanh(\frac{\gamma l}{2})}{\frac{\gamma l}{2}}$$

where $Y$ is the shunt admittance of the line and $\frac{\tanh(\frac{\gamma l}{2})}{\frac{\gamma l}{2}}$ is the factor to obtain the $Y'$ for the $\pi$-equivalent circuit.

The expression $Y'(1 + \frac{1}{4}Y'Z') = \frac{1}{Z_C}\sinh(\gamma l)$ is consistent with the values for $Z'$ and $Y'$ obtained.

Medium and short range transmission lines

For lower values of $l$, $\frac{\sinh(\gamma l)}{\gamma l} \approx 1$ and $\frac{\tanh(\frac{\gamma l}{2})}{\frac{\gamma l}{2}} \approx 1$. In this case, the direct expression of the $\pi$-equivalent circuit can be used substituting by the series and shunt values for impedance and capacitance respectively, with $Z' = Z$ and $Y' = Y$.

Open-source implementations

This model has been successfully implemented in :

Software	URL	Language	Open-Source License	Last consulted date	Comments
dynawo	Dynawo public library	modelica	MPL v2.0	24/05/2024	-
OpenIPSL	Link	modelica	BSD-3-Clause	24/05/2024	-

References

[1] Kundur, Prabha. “Power System Stability and Control” New York, USA, 1994, McGraw-Hill.

[2] Kothari, D. P.; Nagrath, I. J. “Modern Power System Analysis”, 4th ed., New Delhi, India, 2011, Tata McGraw-Hill.

[3] Salam, A. “Fundamentals of Electrical Power Systems Analysis”, Singapore, 2020, Springer.

[4] Duncan Glover, J.; Overbye, T. J.; Sarma, M. “Power System Analysis and Design”, 6th ed., 2015, USA, Cengage Learning.

Evaluate

Parameters	details	Unit
\(l\)	Length of the line	\(m\)
\(r\)	Resistance of the line per meter	\(\Omega m^{-1}\)
\(x_L\)	Reactance of the line per meter	\(\Omega m^{-1}\)
\(z\)	Impedance of the line per meter \(z = r + jx_L\)	\(\Omega m^{-1}\)
\(Z\)	Total impedance of the line \(Z = zl\)	\(\Omega\)
\(Z'\)	Equivalent impedance of the line	\(\Omega\)
\(g\)	Conductance of the line per meter	\(\Omega^{-1} m^{-1}\)
\(b\)	Susceptance of the line per meter	\(\Omega^{-1} m^{-1}\)
\(y\)	Equivalent admittance of the line per meter \(y = g + jb\)	\(\Omega^{-1} m^{-1}\)
\(Y\)	Total admittance of the line \(Y = yl\)	\(\Omega^{-1}\)
\(Y'\)	Equivalent admittance of the line	\(\Omega^{-1}\)
\(Z_C\)	Characteristic impedance of the line \(Z_C = \sqrt(\frac{z}{y})\)	\(\Omega\)
\(\gamma\)	Propagation constant of the line \(Z_C = \sqrt(zy)\)	\(m^{-1}\)

Variable	details	Unit
\(V_{S}\)	Complex phase-to-ground voltage of the source terminal	\(V\)
\(I_{S}\)	Complex current injected by the source terminal into the line	\(A\)
\(V_{R}\)	Complex phase-to-ground voltage of the receiver terminal	\(V\)
\(I_{R}\)	Complex current injected by the receiver terminal into the line	\(A\)