Editing Electric power transmission (section)

== Modeling ==
{{Main|Performance and modelling of AC transmission}}{{Unreferenced section|date=November 2022}}[[File:Transmission Line Black Box.JPG|thumb|upright=1.6|"Black box" model for transmission line]]The terminal characteristics of the transmission line are the voltage and current at the sending (S) and receiving (R) ends. The transmission line can be modeled as a [[black box]] and a 2&nbsp;by&nbsp;2 transmission matrix is used to model its behavior, as follows:

:<math>
\begin{bmatrix}
	V_\mathrm{S}\\
	I_\mathrm{S}\\
\end{bmatrix}
=
\begin{bmatrix}
	A & B\\
	C & D\\
\end{bmatrix}
\begin{bmatrix}
	V_\mathrm{R}\\
	I_\mathrm{R}\\
\end{bmatrix}
</math>

The line is assumed to be a reciprocal, symmetrical network, meaning that the receiving and sending labels can be switched with no consequence. The transmission matrix '''T''' has the properties:
* <math>\det(T) = AD - BC = 1</math>
* <math>A = D</math>

The parameters ''A'', ''B'', ''C'', and ''D'' differ depending on how the desired model handles the line's [[Electrical resistance and conductance|resistance]] (''R''), [[inductance]] (''L''), [[capacitance]] (''C''), and shunt (parallel, leak) [[Electrical conductance|conductance]] ''G''.

The four main models are the short line approximation, the medium line approximation, the long line approximation (with distributed parameters), and the lossless line. In such models, a capital letter such as ''R'' refers to the total quantity summed over the line and a lowercase letter such as ''c'' refers to the per-unit-length quantity.

===Lossless line===
The lossless line approximation is the least accurate; it is typically used on short lines where the inductance is much greater than the resistance. For this approximation, the voltage and current are identical at the sending and receiving ends.
[[File:Losslessline.jpg|thumb|Voltage on sending and receiving ends for lossless line]]
The characteristic impedance is pure real, which means resistive for that impedance, and it is often called surge impedance. When a lossless line is terminated by surge impedance, the voltage does not drop. Though the phase angles of voltage and current are rotated, the magnitudes of voltage and current remain constant along the line. For load&nbsp;>&nbsp;SIL, the voltage drops from sending end and the line ''consumes'' VARs. For load&nbsp;<&nbsp;SIL, the voltage increases from the sending end, and the line ''generates'' VARs.

===Short line===
The short line approximation is normally used for lines shorter than {{cvt|80|km}}. There, only a series impedance ''Z'' is considered, while ''C'' and ''G'' are ignored. The final result is that A&nbsp;=&nbsp;D&nbsp;=&nbsp;1 per unit, B&nbsp;=&nbsp;Z&nbsp;Ohms, and C&nbsp;=&nbsp;0. The associated transition matrix for this approximation is therefore:
:<math>
\begin{bmatrix}
	V_\mathrm{S}\\
	I_\mathrm{S}\\
\end{bmatrix}
=
\begin{bmatrix}
	1 & Z\\
	0 & 1\\
\end{bmatrix}
\begin{bmatrix}
	V_\mathrm{R}\\
	I_\mathrm{R}\\
\end{bmatrix}
</math>

===Medium line===
The medium line approximation is used for lines running between {{cvt|80 and 250|km}}. The series impedance and the shunt (current leak) conductance are considered, placing half of the shunt conductance at each end of the line. This circuit is often referred to as a ''nominal [[Pi (letter)|''π'' (pi)]]'' circuit because of the shape (''π'') that is taken on when leak conductance is placed on both sides of the circuit diagram. The analysis of the medium line produces:

:<math>
\begin{align}
A &= D = 1 + \frac{G Z}{2} \text{ per unit}\\
B &= Z\Omega\\
C &= G \Big( 1 + \frac{G Z}{4}\Big)S
\end{align}
</math>

Counterintuitive behaviors of medium-length transmission lines:

* voltage rise at no load or small current ([[Ferranti effect]])
* receiving-end current can exceed sending-end current

===Long line===
The long line model is used when a higher degree of accuracy is needed or when the line under consideration is more than {{cvt|250|km}} long. Series resistance and shunt conductance are considered to be distributed parameters, such that each differential length of the line has a corresponding differential series impedance and shunt admittance. The following result can be applied at any point along the transmission line, where <math>\gamma</math> is the [[propagation constant]].
:<math>
\begin{align}
A &= D = \cosh(\gamma x) \text{ per unit}\\[3mm]
B &= Z_c \sinh(\gamma x) \Omega\\[2mm]
C &= \frac{1}{Z_c} \sinh(\gamma x) S
\end{align}
</math>

To find the voltage and current at the end of the long line, <math>x</math> should be replaced with <math>l</math> (the line length) in all parameters of the transmission matrix. This model applies the [[Telegrapher's equations]].