Protection formulas

Protection and circuit breakers. 11 formulas for GATE EE Power Systems, each with when to use it. Each is explained in the topic notes.

All 11 Protection formulas

RELAY (SECONDARY) QUANTITIES

I_{sec} = \frac{I _{pri}}{N _{C T}}, V_{sec} = \frac{V _{pri}}{N _{V T}}, Z_{sec} = Z_{pri} \frac{N _{C T}}{N _{V T}}

N_CT = CT ratio and N_VT = VT ratio, each as primary over secondary. Z_pri is the impedance in primary ohms, from the primary voltage and current.

PLUG SETTING MULTIPLIER

PSM = \frac{relay current}{pickup current} = \frac{I _{fault} / N _{C T}}{PS \times I _{CT,rated sec}}

Pickup current = plug setting (as a fraction) times the CT rated secondary current. The relay operates only if PSM > 1.

IDMT OPERATING TIME (IEC STANDARD CURVES)

t = TMS \cdot \frac{k}{( PSM ) ^{α} - 1}

k and alpha depend on the characteristic (table below). Time is in seconds, and the TMS scales the TMS = 1 curve linearly. If a problem gives its own curve, or a table of operating times against PSM for TMS = 1, read the time from it and multiply by the TMS.

PERCENTAGE DIFFERENTIAL RELAY

I_{o p} = ∣ I_{1} - I_{2} ∣, I_{r} = \frac{∣ I _{1} + I _{2} ∣}{2}, trip if I_{o p} > K I_{r}

I_1 and I_2 are the CT secondary current phasors at the two ends, both taken in the same sense through the zone. K is the slope (bias) setting, for example 0.2 for 20 %. A small minimum pickup is usually added.

CT RATIO BALANCE, DELTA-STAR TRANSFORMER

\frac{I _{L, Δ}}{N _{C T, Δ side}} = 3 \frac{I _{L, Y}}{N _{C T, Y side}}

I_L,Delta = line current on the power transformer's delta-winding side, whose CTs are in star. I_L,Y = line current on the star-winding side, whose CTs are in delta. N = ratio of each CT set, primary over secondary. Both sides refer to the same through load.

MHO CHARACTERISTIC

∣ Z - \frac{1}{2} Z_{r} e^{j τ} ∣ < \frac{1}{2} Z_{r}, Z_{reach} (θ) = Z_{r} cos (θ - τ)

Z_r = relay reach (the diameter) along the maximum-torque angle tau. theta = angle of the measured impedance. The reach shrinks as the impedance angle moves away from tau. j is the imaginary unit.

UNIVERSAL RELAY TORQUE EQUATION

T Impedance: Reactance (τ = 9 0^{\circ}) : Mho: = K_{1} I^{2} + K_{2} V^{2} + K_{3} V I cos (θ - τ) + K_{4} K_{1} I^{2} - K_{2} V^{2} > 0 \Rightarrow ∣ Z ∣ < K_{1} / K_{2} K_{1} I^{2} - K_{3} V I sin θ > 0 \Rightarrow X < K_{1} / K_{3} K_{3} V I cos (θ - τ) - K_{2} V^{2} > 0 \Rightarrow ∣ Z ∣ < \frac{K _{3}}{K _{2}} cos (θ - τ)

theta = angle by which I lags V, so Z = V/I at angle theta. K_4 (spring restraint) is neglected in the three special cases. Here a negative constant restrains and a positive one operates.

IMPEDANCE SEEN FOR PHASE AND EARTH FAULTS

\frac{V _{b} - V _{c}}{I _{b} - I _{c}} = x Z_{1}, \frac{V _{a}}{I _{a} + k _{0} I _{res}} = x Z_{1}, k_{0} = \frac{Z _{0} - Z _{1}}{3 Z _{1}}, I_{res} = I_{a} + I_{b} + I_{c} = 3 I_{0}

x = distance to a bolted fault, Z_1 and Z_0 = positive- and zero-sequence line impedance per unit length, phasor quantities measured at the relay. The phase pair (b, c) works for any fault involving those two phases. The earth-fault element on phase a needs the residual compensation k_0, or it measures the wrong distance.

RESTRIKING VOLTAGE (UNDAMPED LC)

v (t) = V_{m} (1 - cos ω_{n} t), ω_{n} = \frac{1}{L C}, peak = 2 V_{m}, RRRV_{m a x} = V_{m} ω_{n}

Assumes resistance is neglected and the system voltage is constant during the transient (f_n is much higher than the supply frequency). V_m = system voltage across the pole at current zero; for a purely inductive fault circuit this is the peak phase voltage. L and C = inductance and capacitance of the circuit seen from the breaker. The peak occurs at t = pi/omega_n and the maximum RRRV at t = pi/(2 omega_n). The average RRRV up to the peak is 2V_m/(pi/omega_n) = 4 V_m f_n.

CRITICAL DAMPING RESISTANCE

R_{crit} = \frac{1}{2} \frac{L}{C}

For the source inductance L in series, with a resistor R across the contacts in parallel with the capacitance C. If R <= R_crit, the restriking voltage is non-oscillatory.

BREAKING AND MAKING CAPACITY (THREE-PHASE)

Breaking capacity (MVA) = 3 V_{L} I_{b}, Making current (peak) \approx 2.55 I_{b}

V_L = rated line voltage in kV. I_b = rated symmetrical (rms) breaking current in kA. Breaking capacity is in MVA. The making current is the peak of the first current loop, about 1.8 x sqrt(2) = 2.55 times the symmetrical rms breaking current. If the current at contact separation has a dc component I_dc, the asymmetrical rms breaking current is sqrt(I_b^2 + I_dc^2).

Where this was asked: 2020 (1), 2022 (1), 2023 (1), 2026 (2). See the 5 previous-year questions on Protection. This shows what was asked, not what will be asked.

KEEP GOING

Protection: the notes and 5 solved previous-year questions