Power Systems

Syllabus

Power generation concepts; AC and DC transmission; models and performance of transmission lines and cables; economic load dispatch (with and without losses); series and shunt compensation; electric field distribution and insulators; distribution systems; per-unit quantities; bus admittance matrix; Gauss-Seidel and Newton-Raphson load flow; voltage and frequency control; power factor correction; symmetrical components; symmetrical and unsymmetrical faults; over-current, differential, directional and distance protection; circuit breakers; stability and the equal area criterion.

Key concepts and formulas

Per-unit system. Z_base = V_base²/S_base, and Z_pu = Z/Z_base. Changing base: Z_pu,new = Z_pu,old (S_new/S_old)(V_old/V_new)². A transformer has the same per-unit impedance on both sides.

Transmission lines.

• Short line (< 80 km): series impedance only; A = D = 1, B = Z, C = 0.
• Medium line: nominal π or T models. Long line: distributed parameters, characteristic impedance Z_c = √(z/y) ≈ √(L/C).
• Surge impedance loading SIL = V_L²/Z_c. Above SIL a line absorbs reactive power; below it, a lightly loaded long line shows a voltage rise at the receiving end (Ferranti effect).
• Voltage regulation = (V_{R,no load} − V_{R,full load})/V_{R,full load}.

Power transfer. Between two buses connected by a reactance X: P = (V₁V₂/X) sin δ, maximum V₁V₂/X at δ = 90°. Reactive power flows from the higher voltage to the lower voltage bus.

Economic dispatch. Without losses, generators run at equal incremental cost: dC₁/dP₁ = dC₂/dP₂ = λ, subject to ΣP = load.

Load flow. Bus types: slack (V, δ given), PV or generator (P, |V| given), PQ or load (P, Q given). Y_bus: diagonal Y_ii = sum of admittances connected to bus i; off-diagonal Y_ij = −(admittance between i and j). Newton-Raphson converges quadratically and needs few iterations; Gauss-Seidel is simpler but slower.

Faults. Symmetrical components: a = 1∠120°. For a fault at a point with Thevenin pre-fault voltage E = 1 pu:

Three-phase fault MVA = base MVA/Z_pu.

Stability. Swing equation (2H/ω_s) d²δ/dt² = P_m − P_e. Equal area criterion: the system stays stable if the accelerating area can be matched by an equal decelerating area. Inertia constant H = stored kinetic energy/machine rating (MJ/MVA).

Protection. Differential relays protect transformers and generators; distance (impedance) relays protect lines; over-current relays with time grading protect feeders. Plug setting multiplier = fault current/(relay setting × CT ratio).

Worked example

A generator rated 50 MVA, 11 kV has X = 0.2 pu. Express it on a 100 MVA, 11 kV base, and find the three-phase fault MVA and current at its terminals. Then find the SIL of a 400 kV line with Z_c = 400 Ω, and dispatch 300 MW between two units with incremental costs 20 + 0.1P₁ and 16 + 0.08P₂ (₹/MWh, losses ignored).

import math

Xpu = 0.2 * 100 / 50
print(f"X on 100 MVA base = {Xpu:.2f} pu, fault MVA = {100 / Xpu:.0f} MVA")
Ibase = 100e6 / (math.sqrt(3) * 11e3)
print(f"fault current = {Ibase / Xpu / 1000:.3f} kA")
print(f"SIL = {400e3 ** 2 / 400 / 1e6:.0f} MW")
# 20 + 0.1 P1 = 16 + 0.08 P2 = lambda, P1 + P2 = 300
P1 = (16 - 20 + 0.08 * 300) / (0.1 + 0.08)
P2 = 300 - P1
print(f"P1 = {P1:.2f} MW, P2 = {P2:.2f} MW, lambda = {20 + 0.1 * P1:.3f} per MWh")

X on 100 MVA base = 0.40 pu, fault MVA = 250 MVA
fault current = 13.122 kA
SIL = 400 MW
P1 = 111.11 MW, P2 = 188.89 MW, lambda = 31.111 per MWh

Practice questions