Transmission-Line Modeling
UW - Electrical Engineering
APTTransmission lines are non-ideal. When a power system is modeled it is necessary to consider the non-ideal nature of transmission lines. It cannot be assumed that the wires that transmit electricity are loss less. In fact there are both resistive and reactive components to transmission line models. When modeling an electric power transmission line it is useful to use per phase analysis rather than trying to model all three phases at once. The information that follows introduces the transmission line model without mathematical rigor. For more mathematical rigor take EE361 and EE454.
Real wires have resistance - they are not perfect conductors. To say that a wire has no resistance is the same as saying that it has infinite conductance. Of course this is not true. No wire can carry an infinite amount of current. As a result, a wire is in fact a resistive element. Thus, resistance is a component of the transmission line model. The figure below shows a model of a transmission line as a resistive element.
If an AC signal is injected at one end of a transmission line, and the signal is then observed at a distance away from the injection point, it will be apparent that as the distance from the injection point increases, the voltage waveform will begin to lead the current. As the distance increases, the the amount of voltage phase lead will slowly increase. This implies that the transmission line model requires an inductive element. Another way to think of a wire is to think of it as an inductor with only one coil. The inductive "quality" of a power line is referred to as inductive reactance - XL.
Where:
If we do the same experiment as above - injecting an AC signal onto a transmission line - and measure the magnitude of the current flow along the transmission line at some distance from the source, we will find that the magnitude of the current and thus the amount of power flowing along the line decreases as distance increases. How is this possible? This seems to violate Kirchhoff's law. It does not appear that the sum of the current entering the line is the same as the current leaving the line yet there is no other explicit current path. What happened?
Recall that an AC current flows across a capacitor. The current phase angle shifts across a capacitor. Current will lead voltage by 90 degrees. If the frequency is high enough, the capacitor acts like a short circuit. Now notice that in there there are two parallel components in per-phase transmission line model. The return path has a very low potential compared to the "outgoing" line. Thus a distributed capacitance is set up. Think of the two wires as two conductive plates placed close together (but not touching). Opposing charges tend to be attracted toward one another near the opposing surfaces of the "plates"(or wires). If voltage applied to the wires were DC, then current would only flow across the gap until the plates charged up to the potential of the source. Since the applied voltage is AC some current constantly flows across the distributed capacitance.
The result is that the model must include a shunt capacitance and shunt conductance.
The capacitive reactance will add leading power factor to the circuit model while the inductive reactance will add lagging power factor to the circuit. The two reactances tend to cancel each other to some degree. Typically the inductive reactance dominates and the net result will be voltage phase lag induced by the transmission line.
Both the series impedance and the shunt admittance are distributed along the entire length of the transmission line; they are not lumped or discrete elements. The use of discrete components is purely for the sake of modeling the system.
For short lines, the effect of the shunt admittance is usually too small to be of much consequence, so it is often ignored. Often for high voltage transmission lines, the series resistance is much smaller than the reactance and it is ignored. The result is that transmission line models can sometimes be modeled as a simple series reactance. For longer lines this is not true at all. The entire system must be modeled.
The model shown above leads to another model if some principles from electromagnetic are applied. The following is offered without proof.
For the full proof see Power System Analysis, by Aurthur R. Bergen and Vijay Vittal, 1986, Prentice-Hall, Inc.
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