Q4Power System - I

(a) Three-Phase Line Calculation

Given: P = 3600 kW, cos φ = 0.8 lagging (so sin φ = 0.6), Vs (line) = 33 kV, R = 5.31 Ω/conductor, X = 5.54 Ω/conductor.

Since the receiving-end voltage Vr is unknown but the line current I also depends on Vr (through P = √3·Vr·I·cosφ), the problem must be solved iteratively: assume a trial Vr, compute I, compute the voltage drop, find a new Vr, and repeat until convergence.

Starting with an initial estimate Vr ≈ Vs = 33 kV: I = 3,600,000/(√3×33,000×0.8) = 78.74 A. Approximate drop = √3×78.74×(5.31×0.8+5.54×0.6) = √3×78.74×(4.248+3.324) = √3×78.74×7.572 ≈ 1031 V. New Vr = 33,000-1031 = 31,969 V. Repeating with Vr=31,969 V: I = 3,600,000/(√3×31,969×0.8) = 81.24 A; drop = √3×81.24×7.572 ≈ 1064 V; new Vr = 33,000-1064 = 31,936 V. Iterating once more converges to:

(i) Receiving end voltage: Vr ≈ 31.93 kV (line-to-line).

(ii) Line current: I ≈ 81.36 A.

(iii) Transmission efficiency: the total line loss (in all three conductors) is:

Thus the transmission efficiency of the line is approximately 97.15%, meaning about 2.85% of the transmitted power (roughly 105.4 kW) is dissipated as I²R loss in the line's resistance.

Voltage regulation (supplementary check): the percentage voltage regulation of the line, defined as the rise in receiving-end voltage when full load is thrown off (with sending-end voltage held constant), can be estimated using the same current and impedance values: %Regulation ≈ (Vs-Vr)/Vr × 100 = (33,000-31,930)/31,930 × 100 ≈ 3.35%. This relatively low regulation value is consistent with the fairly short/medium transmission distance implied by the given resistance and reactance values, and confirms that the line is operating well within acceptable voltage-drop limits for the assumed load and power factor.

Sanity check via power balance: as a verification, the sending-end apparent power can also be estimated as Ss = √3×Vs×I = √3×33,000×81.36 ≈ 4,650 kVA, and the sending-end real power Ps = Pr + Ploss = 3600+105.4 = 3705.4 kW, giving an implied sending-end power factor of cosφs = Ps/Ss = 3705.4/4650 ≈ 0.797, which is reasonably close to the receiving-end power factor of 0.8 — the small difference arises because the line's series reactance draws additional reactive power that slightly alters the power factor angle between the sending and receiving ends, confirming the internal consistency of the iterative solution obtained above.

(b) ACSR Conductors

ACSR (Aluminium Conductor Steel Reinforced) is a composite stranded conductor consisting of a central core of galvanized steel strands (providing high tensile strength) surrounded by one or more outer layers of aluminium strands (providing high electrical conductivity), widely used for overhead transmission and distribution line conductors.

Advantages of ACSR for overhead lines: (1) High tensile strength-to-weight ratio — the steel core bears most of the mechanical tension, allowing longer spans between towers with acceptable sag, reducing the number of supporting towers required over a given route length; (2) Good conductivity at lower cost and weight — aluminium is a much lighter and cheaper conductor material than an equivalent all-copper conductor of the same current-carrying capacity, and although aluminium alone has lower tensile strength than copper, the steel core compensates for this mechanical weakness; (3) Corrosion resistance — the steel core, being fully enclosed within the outer aluminium strands, is well protected from atmospheric corrosion, while the aluminium surface itself readily forms a thin, stable, protective oxide layer; (4) Economical for long-distance, high-voltage transmission — the overall reduction in conductor weight (compared to all-copper) directly reduces tower/support structure costs, foundation costs and transportation costs, which is significant for the very long conductor lengths used in transmission systems; (5) Flexibility in design — different ACSR conductor codes (with varying steel-to-aluminium strand ratios, e.g., standard 6/1, high-strength 30/7, or extra-high-strength constructions) allow the conductor to be optimized for the specific mechanical (span, ice/wind loading) and electrical (ampacity) requirements of a particular line design.

Construction detail: an ACSR conductor is built up in concentric stranded layers, with the central steel strand(s) forming the core and successive layers of aluminium strands wound around it in alternating (opposite) lay directions to maintain overall roundness, flexibility and mechanical stability of the finished conductor. The exact ratio of aluminium cross-sectional area to steel cross-sectional area is chosen to balance conductivity requirements against strength requirements — a higher steel content increases strength (suited to long spans, river crossings, or high ice/wind-loading regions) at some cost in overall conductivity per unit weight, while a lower steel content maximizes conductivity for a given weight but sacrifices some tensile strength and sag performance.

Why ACSR is preferred over plain aluminium or all-copper conductors for transmission lines: all-aluminium conductors (AAC), while cheaper and lighter than copper, have insufficient tensile strength for long transmission spans without excessive sag or an impractical number of intermediate towers; all-copper conductors have excellent conductivity and reasonable strength but are considerably heavier and far more expensive than aluminium for the same current rating, substantially increasing both material cost and the structural cost of supporting towers designed to carry that extra weight and the associated wind/ice loading. ACSR conductors combine the light weight and low cost of aluminium (for conductivity) with the mechanical robustness of steel (for strength), achieving a more optimal overall economic and mechanical design for the very long spans and exposure conditions typical of overhead transmission and distribution networks, which is why ACSR remains the standard, most widely used conductor type for overhead lines across most of the world's power transmission and distribution systems, including in India.

Skin effect consideration: because the steel core is ferromagnetic and carries essentially negligible current compared to the surrounding aluminium layers (steel's resistivity is far higher than aluminium's, so current strongly prefers the aluminium path), ACSR conductors also benefit indirectly from a favorable current distribution at power frequency — nearly all the AC current flows in the outer aluminium strands, which is actually advantageous since it keeps the current away from the higher-resistance steel core, minimizing additional core losses that would otherwise arise from hysteresis and eddy currents induced in the ferromagnetic core by the surrounding AC magnetic field.