Transmission Line Models Part 1

Unlike the electric machines studied so far, transmission lines are characterized by their distributed parameters: distributed resistance, inductance, and capacitance. The distributed series and shunt elements of the transmission line make it harder to model. Such parameters may be approximated by many small discrete resistors, capacitors, and inductors. However, this approach is not very practical, since it would require to solve for voltages and currents at all nodes along the line. We could also solve the exact differential equations for a line but this is also not very practical for large power systems with many lines.

Fortunately, certain simplifications can be used Overhead transmission lines shorter than 80 km (50 miles) can be modeled as a series resistance and inductance, since the shunt capacitance can be neglected over short distances. The inductive reactance at 60 Hz for overhead lines is typically much larger than the resistance of the line. For medium-length lines (80-240 km), shunt capacitance should be taken into account. However, it can be modeled by two capacitors of a half of the line capacitance each. Lines longer than 240 km (150 miles) are long transmission lines and are to be discussed later.

The total series resistance, series reactance, and shunt admittance of a transmission line can be calculated as R X Y rd xd yd where r, x, and y are resistance, reactance, and shunt admittance per unit length and d is the length of the transmission line. The values of r, x, and y can be computed from the line geometry or found in the reference tables for the specific transmission line.

The per-phase equivalent circuit of a short line V S and V R are the sending and receiving end voltages; I S and I R are the sending and receiving end currents. Assumption of no line admittance leads to We can relate voltages through the Kirchhoff s voltage law I S VS VR ZI VR RI jx LI VR VS RI jx LI which is very similar to the equation derived for a synchronous generator. I R

A transmission line can be represented by a 2- port network a network that can be isolated from the outside world by two connections (ports) as shown. If the network is linear, an elementary circuits theorem (analogous to Thevenin s theorem) establishes the relationship between the sending and receiving end voltages and currents as V AV BI S R R I CV DI S R R Here constants A and D are dimensionless, a constant B has units of, and a constant C is measured in siemens. These constants are sometimes referred to as generalized circuit constants, or ABCD constants.

The ABCD constants can be physically interpreted. Constant A represents the effect of a change in the receiving end voltage on the sending end voltage; and constant D models the effect of a change in the receiving end current on the sending end current. Naturally, both constants A and D are dimensionless. The constant B represents the effect of a change in the receiving end current on the sending end voltage. The constant C denotes the effect of a change in the receiving end voltage on the sending end current. Transmission lines are 2-port linear networks, and they are often represented by ABCD models. For the short transmission line model, I S = I R = I, and the ABCD constants are A 1 B C Z 0 D 1

AC voltages are usually expressed as phasors. Load with lagging power factor. Load with unity power factor. Load with leading power factor. For a given source voltage V S and magnitude of the line current, the received voltage is lower for lagging loads and higher for leading loads.

In real overhead transmission lines, the line reactance X L is normally much larger than the line resistance R; therefore, the line resistance is often neglected. We consider next some important transmission line characteristics 1. The effect of load changes Assuming that a single generator supplies a single load through a transmission line, we consider consequences of increasing load. Assuming that the generator is ideal, an increase of load will increase a real and (or) reactive power drawn from the generator and, therefore, the line current, while the voltage and the current will be unchanged.

1) If more load is added with the same lagging power factor, the magnitude of the line current increases but the current remains at the same angle with respect to V R as before. The voltage drop across the reactance increases but stays at the same angle. Assuming zero line resistance and remembering that the source voltage has a constant magnitude: V V jx I S R L voltage drop across reactance jx L I will stretch between V R and V S. Therefore, when a lagging load increases, the received voltage decreases sharply.

2) An increase in a unity PF load, on the other hand, will slightly decrease the received voltage at the end of the transmission line.

3) Finally, an increase in a load with leading PF increases the received (terminal) voltage of the transmission line. In a summary: 1. If lagging (inductive) loads are added at the end of a line, the voltage at the end of the transmission line decreases significantly large positive VR. 2. If unity-pf (resistive) loads are added at the end of a line, the voltage at the end of the transmission line decreases slightly small positive VR. 3. If leading (capacitive) loads are added at the end of a line, the voltage at the end of the transmission line increases negative VR. The voltage regulation of a transmission line is VR V nl V V fl fl 100% where V nl and V fl are the no-load and full-load voltages at the line output.

2. Power flow in a transmission line The real power input to a 3-phase transmission line can be computed as P 3V I cos 3V I cos in S S S LL, S S S where V S is the magnitude of the source (input) line-to-neutral voltage and V LL,S is the magnitude of the source (input) line-to-line voltage. Note that Y-connection is assumed! Similarly, the real output power from the transmission line is P 3V I cos 3V I cos out R R R LL, R R R The reactive power input to a 3-phase transmission line can be computed as Q 3V I sin 3V I sin in S S S LL, S S S

And the reactive output power is Q 3V I sin 3V I sin out R R R LL, R R R The apparent power input to a 3-phase transmission line can be computed as And the apparent output power is S 3V I 3V I in S S LL, S S S 3V I 3V I out R R LL, R R

If the resistance R is ignored, the output power of the transmission line can be simplified A simplified phasor diagram of a transmission line indicating that I S = I R = I. We further observe that the vertical segment bc can be expressed as either V S sin or X L Icos. Therefore: V I cos S sin X L Then the output power of the transmission line equals to its input power: P 3VV sin S Therefore, the power supplied by a transmission line depends on the angle between the phasors representing the input and output voltages. R X L

The maximum power supplied by the transmission line occurs when = 90 0 : P max 3VV S R This maximum power is called the steady-state stability limit of the transmission line. The real transmission lines have non-zero resistance and, therefore, overheat long before this point. Full-load angles of 25 0 are more typical for real transmission lines. X L

Few interesting observations can be made from the power expressions: 1. The maximum power handling capability of a transmission line is a function of the square of its voltage. For instance, if all other parameters are equal, a 220 kv line will have 4 times the power handling capability of a 110 kv transmission line. Therefore, it is beneficial to increase the voltage However, very high voltages produce very strong EM fields (interferences) and may produce a corona glowing of ionized air that substantially increases losses.

2. The maximum power handling capability of a transmission line is inversely proportional to its series reactance, which may be a serious problem for long transmission lines. Some very long lines include series capacitors to reduce the total series reactance and thus increase the total power handling capability of the line. 3. In a normal operation of a power system, the magnitudes of voltages V S and V R do not change much, therefore, the angle basically controls the power flowing through the line. It is possible to control power flow by placing a phase-shifting transformer at one end of the line and varying voltage phase.

3. Transmission line efficiency The efficiency of the transmission line is P out P in 100%

4. Transmission line ratings One of the main limiting factors in transmission line operation is its resistive heating. Since this heating is a function of the square of the current flowing through the line and does not depend on its phase angle, transmission lines are typically rated at a nominal voltage and apparent power. 5. Transmission line limits Several practical constrains limit the maximum real and reactive power that a transmission line can supply. The most important constrains are: 1. The maximum steady-state current must be limited to prevent the overheating in the transmission line. The power lost in a line is approximated as P loss 3I R The greater the current flow, the greater the resistive heating losses. 2 L

2. The voltage drop in a practical line should be limited to approximately 5%. In other words, the ratio of the magnitude of the receiving end voltage to the magnitude of the sending end voltage should be V V R S 0.95 This limit prevents excessive voltage variations in a power system. 3. The angle in a transmission line should typically be 30 0 ensuring that the power flow in the transmission line is well below the static stability limit and, therefore, the power system can handle transients. Any of these limits can be more or less important in different circumstances. In short lines, where series reactance X is relatively small, the resistive heating usually limits the power that the line can supply. In longer lines operating at lagging power factors, the voltage drop across the line is usually the limiting factor. In longer lines operating at leading power factors, the maximum angle can be the limiting f actor.

Considering medium-length lines (50 to 150 mile-long), the shunt admittance must be included in calculations. However, the total admittance is usually modeled ( model) as two capacitors of equal values (each corresponding to a half of total admittance) placed at the sending and receiving ends. The current through the receiving end capacitor can be found as I C2 V And the current through the series impedance elements is R Y 2 Y I V I 2 ser R R

From the Kirchhoff s voltage law, the sending end voltage is YZ VS ZIser VR Z I I V V ZI 2 The source current will be C2 R R 1 R R Y Y ZY ZY I IC1 Iser IC1 IC2 IR VS VR IR Y 1 V 1 I 2 2 4 2 S R R Therefore, the ABCD constants of a medium-length transmission line are If the shunt capacitance of the line is ignored, the ABCD constants are the constants for a short transmission line. ZY A 2 B Z 1 ZY C Y 1 4 ZY D 1 2