Q1Power System - I

(a) Inductance of a Three-Phase Transmission Line

Consider a 3-phase line with conductors a, b, c carrying currents Ia, Ib, Ic (with Ia+Ib+Ic=0 for a balanced system), separated by distances Dab, Dbc, Dca, each conductor having radius r (self-GMR r' = 0.7788r). The total flux linkage of conductor a is the sum of its own self-flux linkage and the mutual flux linkage contributions from conductors b and c:

For a symmetrically spaced line (equilateral triangular spacing, Dab=Dbc=Dca=D), substituting Ib+Ic=-Ia (balanced system) gives, after simplification:

For an unsymmetrically spaced but transposed line, the same result holds with the geometric mean distance GMD=(Dab·Dbc·Dca)^(1/3) substituted for D, and more generally with the conductor's GMR (accounting for bundled/stranded construction) substituted for r':

This is the standard working formula for computing the series inductance per unit length of a 3-phase transmission line, applicable to both symmetrically spaced and transposed unsymmetrical configurations, and to single or bundled conductor arrangements by using the appropriate GMR value.

(b) Sending-End Quantities, Voltage Regulation and Efficiency

Given: A = D = 0.98182+j0.0012447, B = 4.035+j58.947 Ω, C = j0.00061137 S. Line delivers S = 400 MVA at cosφ=0.8 lagging, at Vr(line) = 345 kV.

Taking Vr(phase) = 345,000/√3 = 199,186 V as reference (0°), the receiving-end current magnitude is Ir = S/(√3·Vr,line) = 400×10⁶/(√3×345,000) = 669.39 A, at angle -36.87° (since cosφ=0.8 lagging, φ=36.87°): Ir = 669.39∠-36.87° = (535.51 - j401.63) A.

Substituting the given complex values and carrying out the complex arithmetic (multiplying A×Vr, B×Ir, then summing for Vs; similarly for Is):

Voltage regulation: at no load (Ir=0), Vr,NL = Vs/A, so |Vr,NL|(phase) = |Vs|/|A| = 223,450/0.98182 ≈ 227,587 V, giving:

Transmission efficiency: received real power Pr = S×cosφ = 400×0.8 = 320 MW. Sending-end real power Ps = 3×Re(Vs·Is*) ≈ 324.93 MW (computed from the sending-end phase voltage and current phasors found above).

Summary of results: sending-end line voltage ≈ 387.0 kV, sending-end current ≈ 592.4 A, voltage regulation ≈ 14.26%, and transmission efficiency ≈ 98.48% — showing that although this 345 kV line has fairly low transmission losses (as expected for EHV transmission), it exhibits a fairly substantial voltage regulation, characteristic of a long, series-reactance-dominated EHV line delivering a large block of power at a lagging power factor, which in practice would typically be improved through shunt/series reactive compensation to keep the actual operating voltage swing within tighter, more desirable limits.

Extended Detail: Internal (Self) Flux Linkage Term and the Origin of 0.7788

The self-GMR factor r'=0.7788r used in the inductance formula above accounts specifically for the internal flux linkage within the conductor's own cross-section, which conventional external-flux-only reasoning would otherwise omit. For a solid round conductor carrying uniformly distributed current, integrating the internal magnetic flux linkage (weighted by the fraction of current enclosed at each internal radius, since flux inside the conductor links only a fraction of the total current) contributes an internal inductance of exactly μ0/(8π) H/m, independent of the conductor radius; when this constant internal-inductance contribution is combined with the standard external-flux logarithmic term, the algebra shows it is mathematically equivalent to simply replacing the physical radius r with r'=r·e^(-1/4)=0.7788r inside the same external-flux-only logarithmic formula, which is precisely why GMR (0.7788r for a solid conductor, or the appropriately computed geometric mean for a stranded/bundled conductor) can be substituted directly for r in the single unified inductance formula L=2×10⁻⁷ln(GMD/GMR) without needing to separately add an internal-inductance term.

Sanity check on the sending-end power balance: the difference between sending-end real power (≈324.93 MW) and receiving-end real power (320 MW) is the total line loss of about 4.93 MW; dividing this by the receiving-end power gives a loss fraction of 4.93/320 ≈ 1.54%, consistent with the very high transmission efficiency (98.48%) computed above, and reflecting the fact that at 345 kV EHV transmission for a 400 MVA block, the line current (under 700 A) is modest relative to typical conductor thermal ratings, keeping I²R losses low even over a long transmission distance — this is a direct illustration of the same inverse-square relationship between transmission voltage and I²R loss discussed in Part A of this paper (Q.1), now confirmed quantitatively for a specific, realistic EHV line case using full ABCD-constant-based sending/receiving-end power computation rather than the simplified proportionality argument alone.

Comparison with the medium/short-line approximations: for a 345 kV line of this length (implied to be in the long-line category given the non-negligible imaginary parts of A and D, i.e., A,D ≠ purely real ≈1), a nominal-π or short-line approximation would introduce a small but non-negligible error in the computed regulation and efficiency values compared to the given, presumably long-line-derived, exact ABCD constants — reinforcing why real-world EHV transmission studies always use the given (or rigorously computed, hyperbolic-function-based) ABCD constants for such lines rather than a short/medium-line approximation, since even a few percent error in B directly translates into a comparable error in the computed voltage regulation figure.