AC TRANSMISSION Copyright © P. Kundur This material should not be used without the author's consent

Performance Equations and Parameters of Transmission Lines A transmission line is characterized by four parameters: series resistance (R) due to conductor resistivity shunt conductance (G) due to currents along insulator strings and corona; effect is small and usually neglected series inductance (L) due to magnetic field surrounding the conductor shunt capacitance (C) due to the electric field between the conductors These are distributed parameters. The parameters and hence the characteristics of cables differ significantly from those of overhead lines because the conductors in a cable are much closer to each other surrounded by metallic bodies such as shields, lead or aluminum sheets, and steel pipes separated by insulating material such as impregnated paper, oil, or inert gas

For balanced steady-state operation, the performance of transmission lines may be analyzed in terms of single-phase equivalents. The general solution for voltage and current at a distance x from the receiving end (see book: page 202) is: Fig. 6.1 Voltage and current relationship of a distributed parameter line (6.8) (6.9) where

The constant ZC is called the characteristic impedance and K is called the propagation constant. The constants K and ZC are complex quantities. The real part of the propagation constant K is called the attenuation constant α, and the imaginary part the phase constant β. If losses are completely neglected,

At SIL, Equations 6.17 and 6.18 further simplify to For a lossless line, Equations 6.8 and 6.9 simplify to When dealing with lightening/switching surges, HV lines are assumed to be lossless. Hence, ZC with losses neglected is commonly referred to as the surge impedance. The power delivered by a line when terminated by its surge impedance is known as the natural load or surge impedance load. where V0 is the rated voltage At SIL, Equations 6.17 and 6.18 further simplify to (6.17) (6.18) (6.20) (6.21)

Hence, for a lossless line at SIL, V and I have constant amplitude along the line V and I are in phase throughout the length of the line The line neither generates nor absorbs VARS As we will see later, the SIL serves as a convenient reference quantity for evaluating and expressing line performance Typical values of SIL for overhead lines: nominal (kV): 230 345 500 765 SIL (MW): 140 420 1000 2300 Underground cables have higher shunt capacitance; hence, ZC is much smaller and SIL is much higher than those for overhead lines. for example, the SIL of a 230 kV cable is about 1400 MW generate VARs at all loads

Typical Parameters Table 6.1 Typical overhead transmission line parameters Note: 1. Rated frequency is assumed to be 60 Hz 2. Bundled conductors used for all lines listed, except for the 230 kV line. 3. R, xL, and bC are per-phase values. 4. SIL and charging MVA are three-phase values. Table 6.2 Typical cable parameters * direct buried paper insulated lead covered (PILC) and high pressure pipe type (PIPE)

Voltage Profile of a Radial Line at No-Load With receiving end open, IR = 0. Assuming a lossless line from Equations 6.17 and 6.18, we have At the sending end (x = l), where θ = βl. The angle θ is referred to as the electrical length or the line angle, and is expressed in radians. From Equations 6.31, 6.32, and 6.33 (6.31) (6.32) (6.33) (6.35) (6.36)

As an example, consider a 300 km, 500 kV line with β = 0 As an example, consider a 300 km, 500 kV line with β = 0.0013 rads/km, ZC = 250 ohms, and ES = 1.0 pu: Base current is equal to that corresponding to SIL. Voltage and current profiles are shown in Figure 6.5. The only line parameter, other than line length, that affects the results of Figure 6.5 is β. Since β is practically the same for overhead lines of all voltage levels (see Table 6.1), the results are universally applicable, not just for a 500 kV line. The receiving end voltage for different line lengths: - for l = 300 km, VR = 1.081 pu - for l = 600 km, VR = 1.407 pu - for l = 1200 km, VR = infinity Rise in voltage at the receiving end is because of capacitive charging current flowing through line inductance. known as the "Ferranti effect".

(a) Schematic Diagram (b) Voltage Profile (c) Current Profile Figure 6.5 Voltage and current profiles for a 300 km lossless line with receiving end open-circuited

Voltage - Power Characteristics of a Radial Line Corresponding to a load of PR+jQR at the receiving end, we have Assuming the line to be lossless, from Equation 6.17 with x = l Fig. 6.7 shows the relationship between VR and PR for a 300 km line with different loads and power factors. The load is normalized by dividing PR by P0, the natural load (SIL), so that the results are applicable to overhead lines of all voltage ratings. From Figure 6.7 the following fundamental properties of ac transmission are evident: There is an inherent maximum limit of power that can be transmitted at any load power factor. Obviously, there has to be such a limit, since, with ES constant, the only way to increase power is by lowering the load impedance. This will result in increased current, but decreased VR and large line losses. Up to a certain point the increase of current dominates the decrease of VR, thereby resulting in an increased PR. Finally, the decrease in VR is such that the trend reverses.

Figure 6.7 Voltage-power characteristics of a 300 km lossless radial line

Voltage - Power Characteristics of a Radial Line (cont'd) Any value of power below the maximum can be transmitted at two different values of VR. The normal operation is at the upper value, within narrow limits around 1.0 pu. At the lower voltage, the current is higher and may exceed thermal limits. The feasibility of operation at the lower voltage also depends on load characteristics, and may lead to voltage instability. The load power factor has a significant influence on VR and the maximum power that can be transmitted. This means that the receiving end voltage can be regulated by the addition of shunt capacitive compensation. Fig. 6.8 depicts the effect of line length: For longer lines, VR is very sensitive to variations in PR. For lines longer than 600 km (θ > 45°), VR at natural load is the lower of the two values which satisfies Equation 6.46. Such operation is likely to be unstable.

Figure 6.8 Relationship between receiving end voltage, line length, and load of a lossless radial line

Voltage-Power Characteristic of a Line Connected to Sources at Both Ends With ES and ER assumed to be equal, the following conditions exist: the midpoint voltage is midway in phase between ES and ER the power factor at midpoint is unity with PR>P0, both ends supply reactive power to the line; with PR<P0, both ends absorb reactive power from the line. Fig. 6.8 (developed for a radial line) may be used to analyze how Vm varies with PR. with the length equal to half that of the actual line, plots of VR shown in Figure 6.8 give Vm. Fig. 6.9 Voltage and current phase relationships with ES equal to ER, and PR less than Po

Power Transfer and Stability Considerations Assuming a lossless line, from Equation 6.17 with x = l, we can show that where θ = βl is the electrical length of line and is the angle by which ES leads ER, i.e. the load angle. If ES = ER = rated voltage, then the natural load is and Equation 6.51 becomes The above is valid for synchronous as well as asynchronous load at the receiving end. Fig. 6.10(a) shows the δ - PR relationship for a 400 km line. For comparison, the Vm - PR characteristic of the line is shown in Fig. 6.10(b). (6.51)

Figure 6.10 PR-δ and Vm-PR characteristics of 400 km lossless line transmitting power between two large systems

Reactive Power Requirements From Equation 6.17, with x = l and ES = ER = 1.0, we can show that Fig. 6.11 shows the terminal reactive power requirements of lines of different lengths as a function of PR. Adequate VAR sources must be available at the two ends to operate with varying load and nearly constant voltage. General Comments Analysis of transmission line performance characteristics presented above represents a highly idealized situation useful in developing a conceptual understanding of the phenomenon dynamics of the sending-end and receiving-end systems need to be considered for accurate analysis.

Figure 6.11 Terminal reactive power as a function of power transmitted for different line lengths

Loadability Characteristics The concept of "line loadability" was introduced by H.P. St. Clair in 1953 Fig. 6.13 shows the universal loadability curve for overhead uncompensated lines applicable to all voltage ratings Three factors influence power transfer limits: thermal limit (annealing and increased sag) voltage drop limit (maximum 5% drop) steady-state stability limit (steady-state stability margin of 30% as shown in Fig. 6.14) The "St. Clair Curve" provides a simple means of visualizing power transfer capabilities of transmission lines. useful for developing conceptual guides to preliminary planning of transmission systems must be used with some caution Large complex systems require detailed assessment of their performance and consideration of additional factors

Figure 6.13 Transmission line loadability curve "St. Clair Curve" Figure 6.13 Transmission line loadability curve

Stability Limit Calculation for Line Loadability Figure 6.14 Steady state stability margin calculation

Factors Influencing Transfer of Active and Reactive Power Consider two sources connected by an inductive reactance as shown in Figure 6.21. representation of two sections of a power system interconnected by a transmission system a purely inductive reactance is considered because impedances of transmission elements are predominately inductive effects of shunt capacitances do not appear explicitly (a) Equivalent system diagram δ = load angle Φ = power factor angle (b) Phasor diagram Figure 6.21 Power transfer between two sources

The complex power at the receiving end is Hence, Similarly, Equations 6.79 to 6.82 describe the way in which active and reactive power are transferred Let us examine the dependence of P and Q transfer on the source voltages, by considering separately the effects of differences in voltage magnitudes and angles (6.79) (6.80) (6.81) (6.82)

Figure 6.22 Phasor diagrams with δ = 0 (a) Condition with δ = 0: From Equations 6.79 to 6.82, we have With ES > ER, QS and QR are positive With ES < ER, QS and QR are negative As shown in Fig. 6.22, transmission of lagging current through an inductive reactance causes a drop in receiving end voltage transmission of leading current through an inductive reactance causes a rise in receiving end voltage Reactive power "consumed" in each case is (a) ES>ER (b) ER>ES Figure 6.22 Phasor diagrams with δ = 0

(b) Condition with ES = ER and δ  0 From Equations 6.79 to 6.82, we now have With δ positive, PS and PR are positive, i.e., active power flows from sending to receiving end In each case, there is no reactive power transferred from one end to the other; instead, each end supplies half of Q consumed by X. (a) δ > 0 (b) δ < 0 Figure 6.23 Phasor diagram with ES = ER

(c) General case applicable to any condition: We now have If, in addition to X, we consider series resistance R of the network, then The reactive power "absorbed" by X for all conditions is X I 2. This leads to the concept of "reactive power loss", a companion term to active power loss. An increase in reactive power transmitted increases active as well as reactive power losses. This has an impact on efficiency and voltage regulation. (6.83) (6.84) (6.85) (6.86)

Conclusions Regarding Transfer of Active and Reactive Power The active power transferred (PR) is a function of voltage magnitudes and δ. However, for satisfactory operation of the power system, the voltage magnitude at any bus cannot deviate significantly from the nominal value. Therefore, control of active power transfer is achieved primarily through variations in angle δ. Reactive power transfer depends mainly on voltage magnitudes. It is transmitted from the side with higher voltage magnitude to the side with lower voltage magnitude. Reactive power cannot be transmitted over long distances, since it would require a large voltage gradient to do so. An increase in reactive power transfer causes an increase in active as well as reactive power losses. Although we have considered a simple system, the general conclusions are applicable to any practical system, In fact, the basic characteristics of ac transmission reflected in these conclusions have a dominant effect on the way in which we operate and control the power system.

Appendix to Section on AC Transmission Copy of Section 6.4 from the book “Power System Stability and Control” provides background information related to power flow analysis techniques