Transmission-Line Parameters and Performance

Introduction

Transmission lines connect generating stations, substations, distribution systems, and large consumers. Their electrical behavior is governed by four distributed parameters: series resistance, series inductance, shunt capacitance, and shunt conductance. These parameters determine voltage drop, losses, reactive-power exchange, transmission efficiency, stability, and equipment ratings.

A line cannot always be represented as a simple lumped impedance. For longer lines, voltage and current vary continuously with position, requiring distributed-parameter equations. Engineers therefore select short-, medium-, or long-line models according to line length, frequency, voltage, and required accuracy.

Learning Objectives

After studying this chapter, the reader should be able to:

• calculate line resistance, inductance, capacitance, and conductance;
• account for conductor temperature, transposition, spacing, and bundling;
• derive and apply transmission-line ABCD parameters;
• analyze short, nominal-π, nominal-T, and long-line models;
• calculate voltage regulation, losses, efficiency, and power transfer;
• evaluate surge-impedance loading, corona, insulation, stability, and thermal limits; and
• interpret line-rating results in planning and operational contexts.

1. Distributed Transmission-Line Parameters

For a balanced three-phase line, analysis is normally performed per phase. Over an incremental length dx, the line has:

At angular frequency ω = 2πf, the distributed series impedance and shunt admittance are

Here, j = √{-1} and f is frequency in hertz.

Figure: Incremental series impedance and shunt admittance of a transmission line.

2. Series Resistance

2.1 Direct-current resistance

For a uniform conductor,

where ρ is resistivity in Ω·m, l is conductor length in metres, and A is conducting area in m². The result is in ohms.

Resistance varies approximately linearly with temperature over normal operating ranges:

Here, R_T0 is resistance at reference temperature T₀, and α_T0 is the temperature coefficient in °C⁻¹. This relation is an engineering approximation; tabulated conductor data should be used for high-accuracy studies.

2.2 Alternating-current effects

AC resistance exceeds DC resistance because of:

• skin effect: current concentrates near the conductor surface;
• proximity effect: nearby conductors distort current density;
• stranding and spiraling: actual current path exceeds axial length;
• magnetic-core losses: relevant to steel-reinforced conductors.

Manufacturers normally provide AC resistance at specified frequencies and temperatures. A simplified correction may be written as R_ac=k_acR_dc, where k_ac>1 must be obtained from calculation or test data rather than assumed universally.

Worked Example 1: Conductor resistance

An aluminium conductor has area 300 mm², resistivity ρ=2.826×10^-8 Ω·m at 20°C, and temperature coefficient α₂₀=0.00403 °C⁻¹. Estimate its DC resistance at 75°C and its AC resistance if the verified AC correction factor is 1.03.

For 1 km,

The temperature increase is more influential than the assumed 3% AC correction.

3. Line Inductance

Current produces magnetic flux both inside and outside a conductor. Flux linkages per ampere define inductance. For a solid round nonmagnetic conductor with uniform low-frequency current distribution, internal flux can be represented by the geometric mean radius (GMR):

where r is physical conductor radius. For stranded conductors, the manufacturer's GMR should be used.

3.1 Three-phase transposed line

In an unsymmetrically spaced line, each phase would otherwise have a different inductance. Transposition makes every phase occupy each physical position for one-third of the line, producing equal average parameters.

The geometric mean distance is

For a fully transposed, balanced, single-conductor-per-phase overhead line,

This expression assumes conductor spacing is large compared with conductor radius and earth effects are neglected.

3.2 Bundled conductors

Bundling reduces inductive reactance and surface electric stress. For a two-subconductor bundle with subconductor GMR r' and spacing d, the bundle GMR is

Bundle expressions differ with bundle geometry and number of subconductors; the appropriate geometric mean must therefore be used.

Worked Example 2: Inductance and capacitance

A transposed three-phase line has D_ab=4 m, D_bc=4 m, D_ca=8 m, and conductor radius 1.5 cm. Assume a solid nonmagnetic conductor.

Thus,

At 50 Hz,

4. Capacitance and Shunt Conductance

4.1 Capacitance

The voltage between conductors produces an electric field and stored electric energy. For a transposed three-phase overhead line with one conductor per phase, neglecting earth effects,

where ε=ε₀ε_r, ε₀=8.854×10^-12 F/m, and ε_r≈1 for air. Unlike inductance, capacitance uses physical radius r, not GMR.

For Worked Example 2,

At 50 Hz, the charging susceptance is

Earth, shield wires, multiple circuits, and conductor sag alter the electric field. Potential-coefficient or electromagnetic-field methods are used when these effects cannot be neglected.

4.2 Shunt conductance

Shunt conductance represents real-power loss through insulation and dielectric media. It is usually very small for clean overhead lines but can become significant with contamination, wet insulators, cables, or deteriorated insulation.

If the measured insulation resistance across a line section of length l is R_ins,l in ohms, and leakage is uniformly distributed, then

This is dimensionally valid because the total conductance of the section is Gl=1/R_ins,l. Thus, R_ins,l must be explicitly associated with the measured section; writing merely G=1/R_ins would give total conductance, not conductance per unit length.

For a dielectric represented by a loss tangent \tan δ_d,

provided G and C refer to the same unit length and the dielectric admits this sinusoidal equivalent model. Corona loss is nonlinear and should not generally be absorbed into a constant G.

5. Transmission-Line Equations and ABCD Parameters

Applying Kirchhoff's laws to the incremental model gives the telegrapher equations:

Differentiation and substitution produce

Define propagation constant γ and characteristic impedance Z_c as

Here, α is attenuation in nepers per unit length and β is phase constant in radians per unit length.

The terminal relation is

or

For a uniform long line of length l,

A and D are dimensionless, B is in ohms, and C is in siemens. A reciprocal passive line satisfies AD-BC=1. A geometrically and electrically symmetrical line also has A=D. Cascaded networks are analyzed by multiplying their ABCD matrices in physical order.

6. Short-, Medium-, and Long-Line Models

Length ranges are engineering conventions rather than physical discontinuities. At 50 or 60 Hz, the following classification is common:

Higher voltage, cables, or stringent accuracy may require a more detailed model at shorter lengths.

6.1 Short line

With total series impedance Z,

Thus V_s=V_r+ZI_r and I_s=I_r.

6.2 Nominal-π model

The total admittance Y is divided equally between the two line ends, while total impedance Z is placed in series:

6.3 Nominal-T model

Half the impedance is placed on each side of the total shunt admittance:

Both approximations retain first-order charging behavior, but their B and C parameters differ.

7. Worked Nominal-π Performance Analysis

A balanced 132 kV, three-phase line supplies 100 MW at 0.90 power factor lagging. Its total per-phase parameters are

Take receiving phase voltage as reference:

The load current is

The negative angle is essential because the current lags the receiving voltage.

For the nominal-π model,

and

Using I_r=437.38-j211.83 A,

Therefore,

The sending current is

so |I_s|≈474.75 A. Sending real power must be calculated from the complex phasors:

Hence,

With sending voltage held constant, the no-load receiving voltage is V_r0=V_s/A. Therefore,

and the full-load voltage regulation is

This definition holds sending-end voltage constant and compares no-load with specified full-load receiving voltage. Alternative regulatory definitions must be identified explicitly.

8. Long-Line Behavior

For a low-loss line, R≈0 and G≈0, giving

Consequently,

At light or zero load, line capacitance supplies charging current. The receiving voltage may then exceed the sending voltage—the Ferranti effect. Shunt reactors are commonly installed to absorb excess capacitive reactive power.

For a lossless line, three-phase receiving-end real power is

where voltages are per-phase RMS magnitudes and δ is the angle by which V_s leads V_r. The theoretical steady-state maximum occurs at δ=90°, but practical stability margins require operation well below this point.

9. Voltage Regulation, Losses, and Efficiency

Receiving power is

and sending power is

Transmission efficiency is

The difference P_s-P_r includes conductor, dielectric, leakage, and modeled corona losses. For a short line with negligible leakage, copper loss is approximately 3|I|²R. This special-case expression must not replace terminal complex-power calculations when charging current is appreciable.

Voltage regulation is influenced by load power factor. Lagging current normally increases voltage drop, whereas sufficiently leading current can produce negative regulation. Compensation options include shunt capacitors, shunt reactors, synchronous condensers, series capacitors, and power-electronic FACTS devices.

10. Line Loading and Operating Limits

A transmission line's permissible loading is the lowest limit imposed by thermal capability, voltage performance, stability, insulation, protection, and equipment ratings.

10.1 Surge-impedance loading

The surge-impedance loading (SIL), or natural loading, of an approximately lossless three-phase line is

where V_LL is line-to-line RMS voltage in volts and Z_c is positive-sequence surge impedance in ohms. The result is watts.

Below SIL, an overhead line is typically net capacitive and supplies reactive power; above SIL, it is net inductive and absorbs reactive power. SIL is a reactive-balance benchmark, not a universal thermal or stability rating.

10.2 Thermal limit

Conductor temperature is determined by a heat balance:

where Q_s is solar heat gain, Q_c convective cooling, and Q_r radiative cooling, all expressed consistently per unit conductor length. Maximum temperature is constrained by annealing, sag, ground clearance, connector capability, and safety standards.

Static line ratings use conservative weather assumptions. Dynamic line rating uses measured wind, ambient temperature, solar radiation, conductor temperature, or sag. Machine-learning forecasts may assist rating prediction, but operators must retain validated physical models, uncertainty margins, cybersecurity controls, and fail-safe fallback ratings.

10.3 Stability and voltage limits

Long lines may reach angular- or voltage-stability limits before their conductors reach thermal limits. Series compensation lowers effective series reactance and increases transfer capability, but it can create subsynchronous resonance and protection challenges. Shunt compensation controls voltage and reactive power.

11. Corona and Insulation Performance

11.1 Corona

Corona begins when conductor-surface electric stress ionizes surrounding air. Effects include power loss, audible noise, radio interference, ozone formation, and conductor deterioration.

A commonly used empirical estimate of disruptive critical voltage to neutral is Peek's expression:

In this form, r and D are in centimetres, g₀≈21.1 kV/cm RMS is the disruptive gradient under reference conditions, m₀ is the surface-irregularity factor, and δ is relative air-density factor. The formula is empirical and applies approximately to overhead conductors with appropriate spacing interpretation; bundled conductors require bundle-specific surface-gradient treatment.

Example. Let m₀=0.85, δ=1.0, r=1.5 cm, and D/r=200. Then

The corresponding line-to-line value is approximately

Actual corona inception varies with rain, altitude, conductor condition, geometry, and measurement criterion. Larger conductors, smooth surfaces, and conductor bundles reduce surface stress.

11.2 Insulation coordination

Insulation must withstand continuous power-frequency voltage, temporary overvoltage, switching surges, and lightning impulses. Coordination involves:

• selecting equipment withstand levels, including basic lightning impulse insulation level;
• controlling surges with shielding, grounding, and surge arresters;
• providing adequate air clearance and creepage distance;
• accounting for altitude, pollution, humidity, and contamination;
• coordinating protective levels with statistical overvoltage distributions.

Excess insulation raises cost and structure size, while inadequate insulation threatens life, reliability, and public safety. Applicable grid codes and standards must govern final design.

12. Illustrative Planning Case: Reinforcing a 220 kV Corridor

Background

A fictional regional utility operates a 220 kV overhead corridor whose load is increasing because of new renewable generation. Operators report high peak-load voltage drop and light-load overvoltage.

Problem

A proposal based only on conductor ampacity concludes that the line can carry substantially more power. The conclusion ignores reactive power, stability, insulation, and weather-dependent sag.

Analysis

The engineering team develops positive-sequence ABCD models using verified conductor geometry, temperature-corrected resistance, and transposition. Long-line calculations reveal that charging capacitance causes light-load receiving voltage to rise. Peak transfer requires substantial reactive support, while contingency studies show that voltage stability—not normal ampacity—is the binding limit. Corona and insulation checks also show reduced margins under wet, high-altitude conditions.

Approach and solution

The utility adopts a coordinated package:

1. switched shunt reactors for light-load voltage control;
2. dynamic reactive support near the receiving bus;
3. weather stations and validated dynamic line rating;
4. replacement of thermally weak connectors;
5. revised protection settings and contingency limits; and
6. condition monitoring of insulators and surge arresters.

Outcome

The corridor accommodates additional renewable transfer without presenting the dynamic thermal rating as guaranteed capacity. The operational limit is continuously selected as the minimum of thermal, voltage, stability, and equipment constraints.

Learning points

A single line rating is an oversimplification. Reliable expansion requires electrical modeling, environmental data, equipment condition, contingency analysis, and transparent operating margins.

13. Engineering Practice, Limitations, and Governance

Parameter formulas rely on assumptions such as balanced operation, transposition, sinusoidal steady state, uniform geometry, and known conductor temperature. Real networks contain unbalance, harmonics, tower variation, sag, earth-return effects, frequency-dependent parameters, and uncertain weather. Electromagnetic-transients software is needed for lightning, switching, traveling waves, and wideband cable behavior.

Professional best practice includes:

• maintaining consistent per-phase and three-phase quantities;
• distinguishing line-to-neutral from line-to-line voltage;
• using the negative current angle for lagging loads;
• retaining complex quantities until final magnitudes are required;
• validating ABCD parameters through AD-BC≈1 for reciprocal models;
• documenting parameter sources, temperature, frequency, and units;
• checking results against power balance and equipment ratings; and
• applying statutory safety clearances, grid codes, environmental rules, and reliability criteria.

Utilities must also address land rights, electromagnetic exposure requirements, visual and ecological impacts, wildfire risk, equitable service reliability, and cybersecurity of sensor-based rating systems.

Chapter Summary

Transmission-line performance originates in distributed resistance, inductance, capacitance, and conductance. Resistance depends on material, dimensions, temperature, and AC effects. Inductance and capacitance depend strongly on conductor geometry, transposition, and bundling. Shunt conductance must be defined per unit length and should not be confused with total insulation conductance.

Short lines neglect shunt admittance, medium lines use nominal-π or nominal-T approximations, and long lines use hyperbolic distributed-parameter equations. ABCD parameters provide a unified method for calculating sending voltage, current, power, regulation, and efficiency. Practical loading is governed by the most restrictive thermal, voltage, stability, insulation, corona, protection, or equipment limit. Sound engineering therefore combines accurate phasor analysis with environmental, operational, regulatory, and safety considerations.

Key Terms

ABCD parameters: Two-port constants relating sending- and receiving-end voltage and current.

Characteristic impedance: Ratio Z_c=√{z/y} governing traveling-wave voltage and current.

Corona: Partial ionization of air around a high-voltage conductor.

Dynamic line rating: Time-varying thermal rating based on measured or forecast conditions.

Ferranti effect: Rise of receiving-end voltage above sending-end voltage under light load.

Geometric mean distance: Equivalent phase spacing used in transposed-line calculations.

Geometric mean radius: Effective conductor radius used in inductance calculations.

Insulation coordination: Selection of withstand levels and protective devices for expected overvoltages.

Nominal-π model: Medium-line model with half the shunt admittance at each end.

Nominal-T model: Medium-line model with total shunt admittance between two half impedances.

Propagation constant: Complex quantity γ=α+jβ describing attenuation and phase shift.

Shunt conductance: Distributed leakage or dielectric-loss conductance per unit length.

Skin effect: Concentration of AC current near a conductor's surface.

Surge-impedance loading: Loading at which an approximately lossless line's reactive generation and absorption balance.

Thermal limit: Maximum loading consistent with permissible conductor and equipment temperatures.

Transposition: Periodic exchange of phase positions to equalize average line parameters.

Voltage regulation: Change between defined no-load and full-load receiving voltage at fixed sending conditions.

Review Questions

Short Answer Questions

1. Why is physical conductor radius used for capacitance while GMR is used for inductance?
2. What is the physical meaning of shunt conductance per unit length?
3. State the conditions under which a nominal-π model is appropriate.
4. Explain the Ferranti effect.
5. Why is surge-impedance loading not necessarily the operating rating of a line?

Long Answer Questions

1. Derive the inductance of a transposed three-phase overhead line and discuss the effects of bundling.
2. Derive the long-line ABCD parameters from the telegrapher equations.
3. Compare short-line, nominal-π, nominal-T, and exact long-line models, including their assumptions.
4. Explain how thermal, voltage, stability, corona, and insulation constraints jointly determine line loading.
5. Discuss transmission-line voltage-control methods and their technical limitations.

Analytical / Application Questions

1. A 150 km line has specified per-kilometre R, L, C, and G. Form its nominal-π ABCD parameters and calculate sending-end voltage, current, regulation, and efficiency for a lagging load.
2. Compare the SIL, thermal rating, and steady-state stability limit of a proposed line, and identify the binding constraint under two operating conditions.
3. Evaluate whether dynamic line rating is suitable for a renewable-energy corridor, considering measurement uncertainty, communications failure, cybersecurity, and statutory clearances.

Multiple Choice Questions with Answers

Question 1

Which distributed parameter primarily represents real-power loss in the phase conductor?

A. Series resistance

B. Series inductance

C. Shunt capacitance

D. Phase constant

Correct Answer: A. Series resistance

Question 2

A lagging load current referenced to a voltage at 0° must have which angle?

A. A positive angle

B. A negative angle

C. Exactly 90°

D. Exactly 0°

Correct Answer: B. A negative angle

Question 3

For a reciprocal transmission-line model, which relationship should its ABCD parameters satisfy?

A. A+B=C+D

B. AB-CD=1

C. AD-BC=1

D. A=D=1

Correct Answer: C. AD-BC=1

Question 4

For a nominal-π line having total impedance Z and admittance Y, what is parameter C?

A. Y

B. Y(1+ZY/4)

C. Z(1+ZY/4)

D. 1+ZY/2

Correct Answer: B. Y(1+ZY/4)

Question 5

Using Z=8+j24 Ω and Y=j3.00×10^-4 S, the nominal-π parameter C is approximately:

A. -1.80×10^-7+j2.9946×10^-4 S

B. 8+j24 S

C. 0.9964+j0.0012 S

D. j3.00×10^-4 Ω

Correct Answer: A. -1.80×10^-7+j2.9946×10^-4 S

Question 6

A 132 kV line supplies 100 MW at 0.90 lagging power factor. What is the receiving-current magnitude?

A. 243 A

B. 437 A

C. 486 A

D. 875 A

Correct Answer: C. 486 A

Question 7

What phenomenon can cause the receiving voltage of a long, lightly loaded line to exceed its sending voltage?

A. Proximity effect

B. Ferranti effect

C. Skin effect

D. Subsynchronous resonance

Correct Answer: B. Ferranti effect

Question 8

An approximately lossless three-phase line has V_LL=220 kV and Z_c=400 Ω. What is its SIL?

A. 55 MW

B. 110 MW

C. 121 MW

D. 220 MW

Correct Answer: C. 121 MW

Question 9

Using Peek's expression with m₀=0.85, δ=1, g₀=21.1 kV/cm, r=1.5 cm, and D/r=200, what is the disruptive critical phase voltage?

A. 82.7 kV

B. 142.5 kV

C. 200.0 kV

D. 246.8 kV

Correct Answer: B. 142.5 kV

Question 10

Which statement about dynamic line rating is correct?

A. It eliminates voltage- and stability-limit studies.

B. It guarantees the forecast ampacity under all weather conditions.

C. It adjusts thermal capability using environmental or conductor measurements.

D. It makes statutory ground-clearance requirements unnecessary.

Correct Answer: C. It adjusts thermal capability using environmental or conductor measurements.