Relay protection relies on formulas for current, time, and impedance to ensure accurate fault detection and system coordination.

Overcurrent Relay Formulas

Overcurrent relays operate when the current exceeds a preset value. Key formulas include:

• Relay Pickup Current (Ip): $I_p=K\times I_n$ Where $I_n$ is the rated current of the protected equipment and $K$ is a multiplier based on relay type and coordination requirements .
• Time-Current Characteristic (Inverse Time Relays): $T=\frac{K·T_d}{(I/I_p{)}^n-1}$ Where $T$ is the operating time, $I$ is the fault current, $I_p$ is the pickup current, $T_d$ is the time dial setting, $K$ and $n$ are constants depending on the relay type (e.g., standard inverse, very inverse), .

Differential Relay Formulas

Differential relays protect transformers, generators, and buses by comparing currents at two or more points:

• Operating Current: $I_d=|I_1-I_2|$ Where $I_1$ and $I_2$ are currents entering and leaving the protected zone. The relay operates if $I_d>I_{pickup}$, .
• Percentage Restraint Differential Relay: $I_d>I_{pickup}+k·(I_1+I_2)/2$ Where $k$ is the restraint factor to prevent false tripping during inrush or CT saturation .

Distance Relay Formulas

Distance relays operate based on line impedance:

• Impedance Seen by Relay: $Z_{relay}=\frac{V_{relay}}{I_{relay}}$ Where $V_{relay}$ and $I_{relay}$ are the voltage and current measured by the relay.
• MHO Relay Operating Condition: $|Z_{line}|\le Z_{set}$ The relay trips if the measured impedance falls within the characteristic circle defined by $Z_{set}$, .
• Quadrilateral Relay Characteristic: Combines resistive and reactive components: $R+jX\le Z_{set}$ Where $R$ and $X$ are the line resistance and reactance, and $Z_{set}$ defines the relay reach .

Directional Relay Formulas

Directional relays determine fault direction using voltage and current phase angles:

• Phase Angle Calculation: $\theta =\angle V-\angle I$ The relay operates if $\theta$ falls within the forward or reverse zone .

Coordination and Backup Protection

• Time Coordination: $T_{backup}\ge T_{primary}+T_{margin}$ Ensures the primary relay operates first, and backup relays operate only if the primary fails .
• Current Transformer Ratio: $I_{relay}=\frac{I_{line}}{CTratio}$ Ensures the relay receives a scaled current suitable for its pickup setting .

Summary

Relay protection formulas are essential for:

• Determining pickup currents and operating times for overcurrent relays.
• Calculating differential currents and restraint factors for differential relays.
• Measuring line impedance and setting reach for distance relays.
• Evaluating phase angles for directional relays.
• Coordinating primary and backup protection to maintain system reliability . These formulas, combined with CT/PT ratios and relay characteristics, form the foundation for designing and setting protective relays in power systems.