Transmission lines and cables

In short

A transmission line carries bulk power from generating stations to load centres using four distributed parameters — series resistance and inductance, shunt capacitance and conductance. Because these are spread along the length, line behaviour is modelled by short, medium or long-line equivalents and by the ABCD two-port constants, which decide voltage regulation, efficiency and power transfer limits. Indian transmission runs at 66, 132, 220, 400, 765 kV AC (and ±800 kV HVDC), planned and operated by POWERGRID, state transcos and CTUIL under CEA and CERC rules. The topic is heavily examined in GATE EE (parameter calculation, ABCD, SIL, Ferranti effect) and in interviews at POWERGRID, NTPC, state boards, Sterlite, KEC, L&T, Siemens, Hitachi Energy and Tata Power.

Explained

Line constants: R, L, C and G

• Resistance R (Ω/km): R = ρl/A at DC; AC resistance is higher because of skin effect (current crowds to the conductor surface as frequency rises) and proximity effect (field of a neighbouring conductor redistributes current). Hollow/stranded ACSR conductors exploit this — the steel core carries little current.
• Resistance rises with temperature: R₂ = R₁[1 + α(T₂ − T₁)], α ≈ 0.00403/°C for aluminium at 20 °C.
• Inductance L (H/km): arises from flux linking the conductor — internal flux (giving the 1/2 term, hence GMR = 0.7788r for a solid round conductor) plus external flux up to the return conductor.
• Capacitance C (F/km): charge on conductors with respect to each other and to earth; raises charging current and causes voltage rise on lightly loaded lines.
• Conductance G (S/km): leakage over insulator surfaces and corona loss; so small and erratic that it is normally neglected.
• GMR (self GMD) replaces the physical radius to account for stranding and bundling; GMD (mutual GMD) replaces the simple spacing for unsymmetrical or transposed lines.
• Transposition: rotating conductor positions over equal lengths of an unsymmetrically spaced line equalises the three phase inductances and capacitances — the line is then treated as equivalent-equilateral with Deq = (D₁₂D₂₃D₃₁)^1/3.
• Bundled conductors (twin, triple, quad — twin or quad "Moose" on Indian 400 kV, quad "Bersimis" on 765 kV, octuple at the 1,200 kV test station) raise effective GMR: lower inductance, higher capacitance, higher SIL, much lower corona and radio interference.
• Effect of earth on capacitance: modelled by image conductors below the earth plane; it slightly increases capacitance. Inductance is practically unaffected.

Line models: short, medium, long

• Short line (up to about 80 km, below 66 kV): shunt capacitance neglected; series Z = R + jX only. VS = VR + I Z, I is the same at both ends.
• Medium line (80–250 km, 66–230 kV): capacitance lumped — nominal π (half of Y at each end) or nominal T (whole Y at the middle). π is preferred in practice.
• Long line (above 250 km, EHV): parameters must be treated as distributed; solution of the differential equations gives hyperbolic (rigorous) ABCD constants.
• Long-line solution: VS = VR cosh γl + IR ZC sinh γl; IS = (VR/ZC) sinh γl + IR cosh γl.
• Propagation constant γ = √(zy) = α + jβ: α is attenuation (neper/km), β is phase shift (rad/km); wavelength λ = 2π/β ≈ 6,000 km at 50 Hz, so a 50 Hz line is "electrically short" compared with a wavelength.
• Characteristic (surge) impedance ZC = √(z/y); for a lossless line ZC = √(L/C), about 400 Ω for a single-circuit overhead line, 250–300 Ω for bundled EHV lines, and only 30–60 Ω for cables (because C is large).
• Equivalent π (exact, for long lines): series arm ZC sinh γl, each shunt arm (1/ZC) tanh(γl/2) — unlike nominal π, it gives no error.

ABCD constants and power transfer

• Any passive, linear, bilateral two-port: VS = A·VR + B·IR and IS = C·VR + D·IR.
• Units: A and D dimensionless, B in ohms, C in siemens. For a symmetrical line A = D; for any passive reciprocal network AD − BC = 1 — the standard check in exams.
• Open-circuit (IR = 0): A = VS/VR, C = IS/VR. Short-circuit (VR = 0): B = VS/IR, D = IS/IR.
• Receiving-end power: PR = (VS VR /B) cos(β − δ) − (A VR²/B) cos(β − α), where α, β are the angles of A and B and δ the load angle; sending-end PS = (A VS²/B) cos(β − α) − (VS VR/B) cos(β + δ), using A = D for a symmetrical line.
• For a lossless line, the simple and most-quoted form is P = VS VR sin δ / X, maximum at δ = 90°.
• Practical limits: short lines are limited by thermal rating (conductor temperature, sag), medium lines by voltage regulation, long lines by steady-state stability (δ kept near 30–40°).
• Surge Impedance Loading (SIL): the load, at ZC and unity power factor, at which the line's own reactive generation equals its reactive absorption — flat voltage profile, no net reactive flow. Roughly 500–600 MW for a 400 kV line and 2,000–2,400 MW for 765 kV (values depend on bundle and spacing).
• Below SIL a line is capacitive (exports VARs, voltage rises); above SIL it is inductive (absorbs VARs, voltage sags). St Clair curve gives loadability versus length as a multiple of SIL.

Voltage regulation, Ferranti effect and compensation

• Voltage regulation = (|VR,no-load| − |VR,full-load|)/|VR,full-load| × 100. It can be negative at leading power factor — a standard trick question.
• Ferranti effect: on a long line open-circuited or lightly loaded, the charging current through the series inductance produces a voltage rise so the receiving-end voltage exceeds the sending-end voltage; rise ≈ (1/2)ω²LCl² per unit, so it grows with the square of length and with frequency. Countered by shunt reactors (standard at Indian 400 and 765 kV substations), switched capacitor removal, and by not energising a long line open-ended.
• Shunt compensation: shunt capacitors/STATCOM at heavy load raise voltage and improve power factor; shunt reactors at light load absorb the excess charging VARs.
• Series compensation: series capacitors cancel part of X, raising power transfer and improving stability; degree of compensation 30–70%; risks sub-synchronous resonance (SSR) with turbo-generators and complicates protection (distance relay under/over-reach). Fixed series capacitors are used on Indian 400 kV corridors; TCSC at Raipur and Kanpur.
• Interconnected / parallel lines share load in inverse proportion to impedance; a line's loadability is often set by the N−1 contingency, not by its own rating.

Corona, travelling waves and insulation coordination

• Corona: partial breakdown of air at the conductor surface when the surface gradient exceeds the breakdown strength of air (about 30 kV/cm peak, 21.1 kV/cm rms at NTP). Visible as a violet glow, audible hiss, smell of ozone.
• Disruptive critical voltage (corona starts) Vd = 21.1 mo δ r ln(d/r) kV/phase rms; visual critical voltage Vv = 21.1 mv δ r (1 + 0.3/√(δr)) ln(d/r). δ = air density factor = 3.92b/(273 + t) with b in cm of Hg.
• Corona advantages: reduces steepness of surge fronts, provides a virtual larger conductor diameter. Disadvantages: power loss, radio and TV interference, audible noise, ozone and nitric acid attack on hardware.
• Reduction: larger diameter or bundled conductors, corona rings/grading rings on hardware, smooth stranding, wider spacing.
• Travelling waves: a switching or lightning surge travels with velocity v = 1/√(LC) (≈ 3 × 10⁵ km/s overhead, roughly half that in cables) and sees ZC. At a junction, reflection coefficient for voltage = (Z₂ − Z₁)/(Z₂ + Z₁), refraction (transmission) coefficient = 2Z₂/(Z₂ + Z₁).
• Open end (Z₂ = ∞): voltage doubles, current becomes zero. Short-circuited end (Z₂ = 0): voltage becomes zero, current doubles. Line into cable: voltage falls because the cable's ZC is much smaller — the junction is protected but the cable sees a current surge.
• Protection against surges: shield/earth wires at the tower top, good tower-footing resistance, surge arresters (metal-oxide ZnO gapless type) at terminals and transformers, insulation coordination per IS/IEC 60071.

Underground cables for transmission and their comparison

• Construction: conductor (stranded Cu or Al) → conductor screen → insulation → insulation screen → metallic sheath/earth screen → armour → outer sheath. HV/EHV cables are XLPE insulated today; older designs were paper-insulated lead-covered (PILC), oil-filled and gas-pressure types.
• Insulation resistance of a single-core cable R = (ρ/2πl) ln(R₂/R₁); capacitance C = 2πε₀εr l / ln(R₂/R₁).
• Stress grading: electric stress is maximum at the conductor surface and minimum at the sheath; stress is minimum overall when R₂/R₁ = e (so the most economical conductor radius is R₂/e). Intersheath grading and capacitance (dielectric) grading make the stress distribution more uniform.
• Charging current and dielectric loss limit AC cable length; a long AC cable can be fully loaded by its own charging current, which is why long submarine and city links (for example the ±320 kV Pugalur–Thrissur HVDC link and the Mumbai 220/400 kV supply cables) are HVDC or need reactive compensation.
• Dielectric loss = V²ωC tan δ, where tan δ is the loss tangent (dissipation factor) of the insulation.
• Testing: very low frequency / damped AC withstand, partial discharge measurement, tan δ, DC sheath test; type and routine tests per IS 7098 (Part 2 for 3.3–33 kV, Part 3 for above 33 kV) and IEC 60840 / 62067.
• Faults and location: Murray loop and Varley loop bridges for earth and short-circuit faults; TDR (time-domain reflectometry) and thumper in modern practice.

Indian standards, practice and ratings

• Standard transmission voltages (IS 12360 / CEA practice): 11, 33, 66, 110, 132, 220, 400, 765 kV AC; 1,200 kV AC test station at Bina; ±320, ±500 and ±800 kV HVDC.
• Conductors: ACSR (Zebra, Moose, Panther, Bersimis) per IS 398; AAAC and increasingly HTLS conductors (ACCC, ACSS, INVAR core) for re-conductoring existing corridors.
• Statutes and codes: Electricity Act 2003; CEA (Measures relating to Safety and Electric Supply) Regulations, 2023 (which replaced the 2010 regulations and carry the clearance, earthing and safety requirements formerly in the IE Rules 1956); CEA (Technical Standards for Construction of Electrical Plants and Electric Lines) Regulations, 2022; Indian Electricity Grid Code (IEGC) 2023 by CERC; transmission pricing and ISTS charges under CERC Sharing Regulations.
• Institutions: CTUIL (central transmission utility), POWERGRID, NLDC/RLDC under Grid-India with SLDCs under the states, state transcos; TBCB (tariff-based competitive bidding) is now the normal route for new ISTS lines.
• Typical line parameters at 50 Hz, per km per phase: R ≈ 0.03–0.16 Ω, X ≈ 0.3–0.4 Ω, B ≈ 2.7–5.2 µS. Exact figures vary with conductor and geometry — use the data given in the question.

Key points

• Four line parameters - R and L in series (distributed along the line), C and G in shunt; G is normally neglected.
• Skin effect - AC current crowds to the surface, so R(AC) > R(DC); worse at higher frequency and larger diameter.
• GMR = 0.7788r - self geometric mean radius of a solid round conductor; used in place of r in inductance formulae.
• Transposition - equalises phase inductance and capacitance on unsymmetrically spaced lines; Deq = cube root of the product of the three spacings.
• Bundling - raises GMR, lowers X, raises C and SIL, and is the main defence against corona at 400 kV and above.
• Short/medium/long - below ~80 km (no shunt), 80–250 km (nominal π or T), above 250 km (hyperbolic/distributed).
• ABCD constants - VS = AVR + BIR, IS = CVR + DIR; A = D for symmetrical lines and AD − BC = 1 always.
• Surge impedance ZC = √(L/C) - about 400 Ω overhead, 30–60 Ω for cables; independent of line length.
• SIL = VL²/ZC - the loading at which the line neither absorbs nor generates reactive power; the natural yardstick of line capacity.
• Ferranti effect - receiving-end voltage exceeds sending-end voltage on a lightly loaded long line; rise ∝ length².
• Voltage regulation can be negative at leading power factor, because the load's capacitive current reduces voltage drop.
• Series capacitors raise transfer capability (cancel X) but can cause sub-synchronous resonance; shunt reactors control light-load voltage rise.
• Loading limits - thermal for short lines, voltage drop for medium, stability (δ ≈ 30–40°) for long lines.
• Corona - starts above about 21.1 kV/cm rms surface gradient; causes loss, radio noise and audible hum.
• Travelling waves - reflection coefficient (Z₂ − Z₁)/(Z₂ + Z₁); voltage doubles at an open end, vanishes at a short circuit.
• Cable stress - maximum at the conductor surface; most economical when the ratio of sheath to conductor radius is e (≈ 2.718).
• Cables versus overhead - cables cost several times more, have much higher capacitance and are used for cities, river/sea crossings and airport approaches.
• Indian framework - Electricity Act 2003, CEA Safety Regulations 2023, IEGC 2023, CTUIL/POWERGRID, IS 398 conductors, IS 7098 cables.

Formulas and examples

Example 1 — Inductance and capacitance of a transposed 220 kV line. Conductors of radius 1.0 cm are spaced 4 m, 4 m and 8 m. Find L and C per phase per km.

• Deq = (4 × 4 × 8)^1/3 = (128)^1/3 = 5.04 m.
• GMR = 0.7788 × 0.01 = 0.007788 m. Deq/GMR = 5.04/0.007788 = 647.2; ln = 6.473.
• L = 2 × 10⁻⁷ × 6.473 = 1.295 × 10⁻⁶ H/m = 1.295 mH/km; XL = 2π × 50 × 1.295 × 10⁻³ = 0.407 Ω/km.
• For C use the actual radius: Deq/r = 5.04/0.01 = 504; log₁₀ 504 = 2.702.
• C = 0.0242/2.702 = 0.00896 µF/km; charging susceptance B = 2π × 50 × 0.00896 × 10⁻⁶ = 2.81 µS/km.

Example 2 — Short line: regulation and efficiency. A three-phase 33 kV, 20 km line has R = 0.2 Ω/km, X = 0.4 Ω/km and delivers 6 MW at 0.8 pf lagging at 33 kV.

• Z = 4 + j8 Ω per phase. Line current I = 6 × 10⁶/(√3 × 33,000 × 0.8) = 131.2 A.
• Approximate drop per phase = I(R cos φ + X sin φ) = 131.2 (4 × 0.8 + 8 × 0.6) = 131.2 × 8.0 = 1,049.6 V.
• VR per phase = 19,053 V → VS ≈ 20,103 V per phase (34.82 kV line). Regulation ≈ 5.5%.
• Loss = 3 I²R = 3 × 131.2² × 4 = 206.6 kW. η = 6,000/(6,000 + 206.6) = 96.7%.

Example 3 — Nominal π constants and the AD − BC check. A 150 km medium line: Z = 40 + j125 Ω, total Y = j1.0 × 10⁻³ S.

• ZY = (40 + j125)(j1 × 10⁻³) = j0.040 − 0.125 = −0.125 + j0.040.
• A = D = 1 + ZY/2 = 1 + (−0.0625 + j0.020) = 0.9375 + j0.020 (|A| = 0.9377).
• B = Z = 40 + j125 Ω (|B| = 131.2 Ω, β = 72.3°).
• C = Y(1 + ZY/4) = j1 × 10⁻³ (0.96875 + j0.010) = −1.0 × 10⁻⁵ + j9.6875 × 10⁻⁴ S.
• Check: AD − BC = A² − BC. A² = 0.8785 + j0.0375 (since 0.9375² − 0.020² = 0.8785, 2 × 0.9375 × 0.020 = 0.0375). BC = (40 + j125)(−1.0 × 10⁻⁵ + j9.6875 × 10⁻⁴) = (−4 × 10⁻⁴ − 0.12109) + j(0.03875 − 1.25 × 10⁻³) = −0.1215 + j0.03750. A² − BC = 1.000 + j0.000 = 1 — constants are consistent.

Example 4 — Surge impedance, SIL and loadability. A 400 kV line has L = 1.0 mH/km and C = 0.0112 µF/km.

• ZC = √(1.0 × 10⁻³ / 0.0112 × 10⁻⁶) = √(89,286) = 298.8 Ω.
• SIL = 400²/298.8 = 160,000/298.8 = 535 MW.
• Velocity v = 1/√(LC) = 1/√(1.0 × 10⁻³ × 0.0112 × 10⁻⁶ per km²) = 1/(3.347 × 10⁻⁶) = 2.99 × 10⁵ km/s — close to the speed of light, as expected.
• If this line is 300 km long and X ≈ 0.314 × 300 = 94.2 Ω, then with VS = VR = 400 kV and δ limited to 30°, P = 400 × 400 × 0.5/94.2 = 849 MW, i.e. about 1.6 × SIL — consistent with the St Clair curve for this length.

Example 5 — Travelling wave at a line–cable junction. A 500 kV surge travels on an overhead line of ZC = 400 Ω into a cable of ZC = 50 Ω.

• Γ = (50 − 400)/(50 + 400) = −350/450 = −0.778 → reflected voltage = −389 kV back along the line.
• Transmitted voltage = 2 × 50 × 500/450 = 111.1 kV into the cable.
• Transmitted current = 111.1/50 = 2.22 kA, against an incident current of 500/400 = 1.25 kA — the cable sees a far lower voltage but a much larger current surge.

Where it is used

• Transmission planning and design: POWERGRID, CTUIL, state transcos and EPC firms (KEC International, Kalpataru, Sterlite Power, L&T) size conductors, compute SIL, plan compensation and run PSS/E, PSCAD and ETAP studies.
• Grid operation: NLDC/RLDC/SLDC engineers manage voltage profiles with shunt reactors and capacitors, watch line loading against thermal and stability limits, and handle light-load Ferranti conditions at night.
• Cable engineering: Polycab, KEI, Apar, Finolex and Universal Cables design and test 33–400 kV XLPE cable systems; Mumbai, Delhi, Bengaluru and Hyderabad city networks depend on them.
• Protection and insulation coordination: distance relay reach settings depend on line impedance per km; surge arrester and BIL selection depends on travelling-wave behaviour.
• PSU recruitment and exams: GATE EE, ESE (Electrical), NTPC/POWERGRID/BHEL/NHPC executive trainee tests and state board AE exams all test line parameters, ABCD constants and SIL.
• Industry and consultancy roles: HTLS re-conductoring studies, right-of-way-constrained city links, renewable evacuation corridors from Rajasthan and Gujarat solar/wind parks.

Interview questions

• Q: Why are shunt conductance and leakage usually ignored in line models? A: Leakage over insulators and corona conductance are very small and highly variable with weather, so including them adds complexity without improving accuracy.
• Q: What is the Ferranti effect and when does it appear? A: On a long, lightly loaded or open-ended line the charging current flowing through the series reactance raises the receiving-end voltage above the sending-end voltage; the rise grows roughly as the square of line length.
• Q: Why is GMR used instead of the actual radius for inductance but the actual radius for capacitance? A: GMR accounts for internal flux linkage inside the conductor, which affects inductance; charge resides on the conductor surface, so capacitance depends on the physical radius.
• Q: What does surge impedance loading tell a planner? A: It is the loading at which the line's reactive generation and absorption balance, giving a flat voltage profile; it is used as the natural measure of a line's capacity and of how much compensation will be needed.
• Q: Why are bundled conductors used at 400 kV and above? A: They increase the effective conductor radius, reducing the surface voltage gradient (so less corona and radio interference) and reducing reactance, which raises SIL and power transfer.
• Q: Can voltage regulation be negative? A: Yes — with a leading power factor load the capacitive current partly cancels the inductive drop, so the receiving-end voltage can exceed the no-load value.
• Q: What limits the power a line can carry? A: Thermal rating (and sag) for short lines, voltage drop for medium lines, and steady-state/transient stability for long lines; δ is normally held around 30–40°.
• Q: What happens to a surge at an open-circuited line end? A: The voltage reflects with coefficient +1 and doubles, while the current reflects with −1 and becomes zero at that point.
• Q: Why is a long AC cable impractical for bulk transmission? A: Its capacitance is 20–30 times that of an overhead line, so charging current can consume the whole rating; HVDC or heavy reactive compensation is required.
• Q: What is the risk of series capacitor compensation? A: Sub-synchronous resonance, where the electrical resonance interacts with turbine-generator torsional modes, plus distance-relay under- or over-reach and ferroresonance concerns.
• Q: Why is the most economical cable conductor radius given by R₂/e? A: Maximum stress Emax = V/(R₁ ln(R₂/R₁)) is minimised when ln(R₂/R₁) = 1, i.e. R₂/R₁ = e, so for a given sheath radius the insulation is used most efficiently.
• Q: Does transposition change the total line impedance? A: No — it redistributes the phases so each sees the same average inductance and capacitance, removing unbalance; the average value is what the equivalent-spacing formula gives.

Common mistakes

• Using GMR in the capacitance formula or the actual radius in the inductance formula.
• Mixing per-phase and line quantities — putting line-to-line voltage into a per-phase equation, or forgetting the factor 3 in 3I²R losses.
• Treating nominal π and the exact equivalent π as the same thing on lines longer than about 250 km, where the error becomes significant.
• Forgetting that B has units of ohms and C of siemens, and so failing the AD − BC = 1 check.
• Assuming voltage regulation is always positive, and overlooking the Ferranti rise on open-ended lines.
• Writing SIL with phase voltage instead of line-to-line voltage, giving a figure three times too small.
• Using 30 kV/cm (peak) where 21.1 kV/cm (rms) is required in corona formulae, or omitting the air-density factor δ.
• Quoting one "standard" value of ZC or line reactance per km instead of computing from the data given in the question.

Related topics

Overhead line design: sag, tension and insulators · Underground cables · HVDC transmission · FACTS devices · Load flow and fault analysis · High-voltage engineering and insulation