Per unit & load flow formulas

Per-unit system, bus admittance matrix and load flow. 11 formulas for GATE EE Power Systems, each with when to use it. Each is explained in the topic notes.

All 11 Per unit & load flow formulas

BASE CURRENT AND BASE IMPEDANCE

I_{ba se} = \frac{S _{ba se} \times 1 0 ^{3}}{3 V _{ba se}} A, Z_{ba se} = \frac{V _{ba se}^{2}}{S _{ba se}} Ω

Three-phase MVA base $S_{ba se}$ in MVA and line-to-line kV base $V_{ba se}$ in kV; the impedance is per phase of the equivalent star.

PER-UNIT IMPEDANCE

Z_{p u} = \frac{Z _{Ω}}{Z _{ba se}} = Z_{Ω} \frac{S _{ba se}}{V _{ba se}^{2}}

Per-unit voltage, current and power follow in the same way: $V_{p u} = V / V_{ba se}$ , $I_{p u} = I / I_{ba se}$ , $S_{p u} = S / S_{ba se}$ .

CHANGE OF BASE

Z_{p u, n e w} = Z_{p u, o l d} \times \frac{S _{ba se, n e w}}{S _{ba se, o l d}} \times (\frac{V _{ba se, o l d}}{V _{ba se, n e w}})^{2}

Used when an equipment rating (given on its own MVA and kV) must be moved to the system base; the rated MVA and kV are the old base.

BASE VOLTAGE ACROSS A TRANSFORMER

\frac{V _{ba se, 1}}{V _{ba se, 2}} = \frac{V _{r a t e d, 1}}{V _{r a t e d, 2}}

Line-to-line rated voltages of the transformer; $S_{ba se}$ is unchanged across it.

COMPLEX POWER INJECTED AT BUS $I$

S_{i} = P_{i} + j Q_{i} = V_{i} I_{i}^{*}, I_{i} = k = 1 \sum n Y_{ik} V_{k}

Phasors in per unit; $Y_{ik}$ are elements of the bus admittance matrix.

REAL AND REACTIVE POWER EQUATIONS

P_{i} Q_{i} = ∣ V_{i} ∣ k = 1 \sum n ∣ V_{k} ∣ (G_{ik} cos δ_{ik} + B_{ik} sin δ_{ik}) = ∣ V_{i} ∣ k = 1 \sum n ∣ V_{k} ∣ (G_{ik} sin δ_{ik} - B_{ik} cos δ_{ik})

Here $Y_{ik} = G_{ik} + j B_{ik}$ and $δ_{ik} = θ_{i} - θ_{k}$ . The equations are non-linear, so they are solved by iteration.

ELEMENTS OF THE BUS ADMITTANCE MATRIX

Y_{ii} = y_{i 0} + k \neq = i \sum y_{ik}, Y_{ik} = Y_{k i} = - y_{ik}

$y_{i 0}$ is the total shunt admittance from bus $i$ to ground; $y_{ik}$ is the series admittance between $i$ and $k$ . Valid with no mutual coupling and nominal-ratio transformers.

GAUSS-SEIDEL UPDATE (PQ BUS)

V_{i}^{(r + 1)} = \frac{1}{Y _{ii}} \frac{P _{i} - j Q _{i}}{( V _{i}^{(r)} ) ^{*}} - k = 1 \sum i - 1 Y_{ik} V_{k}^{(r + 1)} - k = i + 1 \sum n Y_{ik} V_{k}^{(r)}

$P_{i}$ and $Q_{i}$ are net injections in per unit. Buses already updated in this iteration use their new values.

NEWTON-RAPHSON CORRECTION EQUATION

[Δ P Δ Q] = [J_{1} J_{3} J_{2} J_{4}] [Δ θ Δ∣ V ∣], J_{1} = \frac{\partial P}{\partial θ}, J_{2} = \frac{\partial P}{\partial ∣ V ∣}, J_{3} = \frac{\partial Q}{\partial θ}, J_{4} = \frac{\partial Q}{\partial ∣ V ∣}

Mismatch $Δ P = P_{s p ec} - P_{c a l c}$ , $Δ Q = Q_{s p ec} - Q_{c a l c}$ ; update $θ \leftarrow θ + Δ θ$ , $∣ V ∣ \leftarrow ∣ V ∣ + Δ∣ V ∣$ and repeat.

SIZE OF THE JACOBIAN

order = (n - 1) + (n - 1 - m) = 2 n - m - 2

$n$ buses, one slack, $m$ PV buses (polar form). $J_{1}$ is $(n - 1) \times (n - 1)$ , $J_{4}$ is $(n - 1 - m) \times (n - 1 - m)$ , $J_{2}$ is $(n - 1) \times (n - 1 - m)$ and $J_{3}$ is $(n - 1 - m) \times (n - 1)$ .

DC LOAD FLOW

P_{ik} = \frac{θ _{i} - θ _{k}}{x _{ik}}, P = B^{'} θ

Assumes $∣ V ∣ = 1$ pu at all buses, $r ≪ x$ , small angle differences, so $sin δ \approx δ$ . Angles in radians. $B^{'}$ is the same matrix as in the fast decoupled method (from $1/ x$ , slack row and column removed), and $P$ holds the net real-power injections of the non-slack buses. It gives real power flows only, and no losses.

Where this was asked: 2018 (2), 2020 (2), 2023 (2), 2024 (1), 2025 (1), 2026 (2). See the 10 previous-year questions on Per unit & load flow. This shows what was asked, not what will be asked.

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Per unit & load flow: the notes and 10 solved previous-year questions