Q2Power System - I

(a) Per Unit System

In the per-unit (p.u.) system, every quantity (voltage, current, power, impedance) is expressed as a fraction (or ratio) of a chosen base quantity of the same dimension: quantity in p.u. = actual value / base value. Typically, a base MVA (Sbase, common to the whole system) and a base voltage (Vbase, chosen separately for each voltage level/zone) are selected, from which base current and base impedance are derived: Ibase = Sbase/(√3·Vbase) for three-phase systems, and Zbase = Vbase²/Sbase.

Advantages over the absolute (actual-unit) method: (1) it eliminates the need to refer impedances across transformers using the turns-ratio squared, since if bases are chosen consistently on each side of a transformer (related by the transformer's nominal turns ratio), the transformer's per-unit impedance is automatically the same when viewed from either side — this greatly simplifies multi-voltage-level network analysis; (2) manufacturers typically specify equipment impedance directly in per-unit (or percentage) on the equipment's own rating, making data directly usable without unit conversion; (3) per-unit impedances of similar apparatus (e.g., transformers of different sizes) fall within a narrow, characteristic numerical range (e.g., 0.05-0.10 p.u. for transformer leakage reactance), making data-checking and estimation easier; (4) three-phase power/voltage relations simplify since the factor of √3 (present in actual line quantities) disappears in balanced per-unit equations.

Proof that per-unit transformer impedance is the same referred to either side: consider a two-winding transformer with primary (1) and secondary (2) turns N1, N2, and let the actual leakage impedance referred to the primary side be Z1 (ohms). The impedance referred to the secondary side is, by the standard transformer referring relation, Z2 = Z1·(N2/N1)².

Now choose base impedances consistently: on the primary side, Zbase1 = Vbase1²/Sbase; on the secondary side, Zbase2 = Vbase2²/Sbase (same Sbase throughout, and Vbase2/Vbase1 = N2/N1, the transformer's nominal turns ratio — this is the crucial consistency condition). Then the per-unit impedance referred to the primary side is:

and the per-unit impedance referred to the secondary side is:

since (N2/N1)² cancels out exactly (because Vbase2 = Vbase1·(N2/N1) was chosen consistent with the turns ratio). This proves Z1,pu = Z2,pu — the per-unit impedance of a two-winding transformer is identical whether computed referred to its high-voltage side or its low-voltage side, provided the base voltages on the two sides are chosen in the same ratio as the transformer's own turns ratio. This is precisely why, in per-unit network analysis, a transformer can simply be represented as a single series impedance connecting the two per-unit voltage zones, without any separate ideal transformer or turns-ratio device required in the equivalent circuit.

(b) Surge Impedance

Surge (or characteristic/natural) impedance, Zc, of a transmission line is defined as the ratio of voltage to current in a traveling wave propagating along a lossless (or low-loss) line, given by Zc = √(L/C), where L and C are the series inductance and shunt capacitance per unit length of the line. It represents the impedance that the line itself presents to a traveling voltage/current surge (such as a lightning or switching surge), independent of the line's length or termination, and is a purely resistive quantity (has no reactive/imaginary part) for a lossless line.

Evaluation for an overhead line: overhead lines have relatively high series inductance per unit length (due to wide conductor spacing, giving high self/mutual inductance) and relatively low shunt capacitance per unit length (due to the wide spacing and the low permittivity of air as the dielectric medium between conductors), so the ratio L/C is large, and typical overhead line surge impedance values are around 300-400 Ω (single circuit) — for example, a typical EHV overhead line might have L ≈ 1 mH/km and C ≈ 0.01 μF/km, giving Zc = √(1×10⁻³/0.01×10⁻⁶) ≈ 316 Ω.

Evaluation for an underground cable: underground cables have closely spaced conductors separated by a high-permittivity solid dielectric (XLPE, paper-oil), giving much lower series inductance per unit length (closer conductor spacing reduces flux linkage) but much higher shunt capacitance per unit length (closer spacing and higher dielectric permittivity both increase capacitance substantially). This makes the ratio L/C for a cable much smaller than for an overhead line, so cables have a much lower surge impedance, typically only 30-60 Ω — for example, with L ≈ 0.3 mH/km and C ≈ 0.2 μF/km, Zc = √(0.3×10⁻³/0.2×10⁻⁶) ≈ 39 Ω. This much lower surge impedance of cables means that, for the same amount of energy in a traveling surge, cables experience higher surge current but lower surge voltage than overhead lines, and it also means that the natural (surge impedance) loading power of a cable, being inversely proportional to Zc, is much higher than that of an equivalent-voltage overhead line.

Practical measurement/evaluation methods: in practice, Zc is not always computed purely from nominal L and C formulas; it can also be evaluated experimentally by an open-circuit and short-circuit test on the line/cable at power frequency, analogous to transformer OC/SC tests — Zc = √(Zoc·Zsc), where Zoc is the input impedance measured with the far end open-circuited and Zsc is the input impedance measured with the far end short-circuited. This experimental approach automatically captures the true, as-installed L and C (including any stray effects) without needing to separately calculate L and C from geometric first principles, and is especially useful for cables, where the exact geometric and dielectric parameters (multiple layers of semi-conducting screen, insulation and sheath) make an analytical L, C calculation more involved than the well-known GMD/GMR overhead-line formulas.

Significance in surge/lightning studies: knowledge of Zc is essential in insulation coordination and lightning/switching-surge studies because it determines both the initial voltage and current of a traveling wave launched onto the line (V=Zc·I at the point of wave initiation) and the reflection/refraction coefficients at any junction between line sections of different characteristic impedance (such as the overhead-line-to-cable transition common at substations, where the sharp drop in Zc from ~400 Ω to ~40 Ω causes a significant negative voltage reflection back into the overhead line and a much-reduced transmitted voltage surge into the cable, which is a physical phenomenon partly exploited to protect substation equipment from incoming lightning surges).