Per-Unit System and Load Flow

Large networks are analysed in per unit on common bases, and their steady-state operation is found by load flow (power flow) studies.

Per-unit system

• Choose S_base (three-phase MVA) and V_base (line kV) in one zone; V_base in other zones follows the transformer ratios.
• Z_base = V_base²/S_base; I_base = S_base/(√3 V_base).
• Change of base: Z_new = Z_old (S_new/S_old)(V_old/V_new)².
• Advantages: transformer impedances are the same on both sides, values fall in a narrow range, and √3 factors disappear.

Bus admittance matrix (Y_bus)

For a network with line admittances y_ij:

• Diagonal Y_ii = sum of all admittances connected to bus i (including shunt charging).
• Off-diagonal Y_ij = −y_ij (zero if not connected).
• Y_bus is sparse and symmetric (without phase shifters). Z_bus = Y_bus⁻¹ is full.

Bus types

Power flow equations

P_i − jQ_i = V_i* Σ Y_ij V_j

These are non-linear, so they are solved iteratively:

• Gauss–Seidel: simple, little memory per iteration, but many iterations (convergence slows with system size); acceleration factor about 1.6.
• Newton–Raphson: uses the Jacobian; quadratic convergence, few iterations (3–5) independent of size; standard for large systems.
• Fast decoupled: uses P–δ and Q–V decoupling with constant matrices; very fast.
• DC load flow: lossless, flat voltages, P = B′δ; linear and approximate.

Voltage and frequency control

• Real power ↔ frequency (governors, load frequency control).
• Reactive power ↔ voltage (generator excitation, tap-changing transformers, shunt capacitors and reactors, SVCs).

Worked example

A 50 MVA, 11 kV generator has X = 0.2 pu. On a 100 MVA, 11 kV base:

• X_new = 0.2 × (100/50) × (11/11)² = 0.4 pu.

Practice