Understanding the Per-Unit System in Power Systems

The per-unit (pu) system is fundamental to power systems analysis. While it may seem like unnecessary complexity at first, it dramatically simplifies calculations involving transformers, generators, and interconnected systems at different voltage levels.

Why Per-Unit?

Consider a simple example: a 13.8 kV generator connected through a transformer to a 138 kV transmission line. Without per-unit:

• You must track voltage levels across every component
• Transformer turns ratios appear in every calculation
• Comparing equipment ratings becomes cumbersome

With per-unit, all quantities are normalized to base values. A 1.0 pu voltage means "rated voltage"—whether that's 13.8 kV or 138 kV.

Defining Base Values

You choose two base quantities; the rest are derived:

1. Base Power ($S_{base}$): Typically system MVA (e.g., 100 MVA)
2. Base Voltage ($V_{base}$): Line-to-line voltage at each level

Derived values:

$$I_{base} = \frac{S_{base}}{\sqrt{3} \times V_{base}}$$ $$Z_{base} = \frac{V_{base}^2}{S_{base}}$$

Converting to Per-Unit

For any quantity $X$ with base $X_{base}$:

$$X_{pu} = \frac{X_{actual}}{X_{base}}$$

Example: Transformer Impedance

A 50 MVA, 138/13.8 kV transformer has $Z = 8\%$ on its nameplate.

This means $Z = 0.08$ pu on a 50 MVA, 138 kV base.

To convert to a 100 MVA system base:

$$Z_{new} = Z_{old} \times \frac{S_{base,new}}{S_{base,old}} \times \left(\frac{V_{base,old}}{V_{base,new}}\right)^2$$ $$Z_{new} = 0.08 \times \frac{100}{50} \times \left(\frac{138}{138}\right)^2 = 0.16 \text{ pu}$$

Common Pitfalls

1. Forgetting to convert bases: Equipment ratings use nameplate bases; system studies use a common base.

2. Voltage base mismatch: Each voltage level needs its own $V_{base}$, related by transformer ratios.

3. Single-phase vs. three-phase: Be consistent. Most systems use three-phase bases.

Practical Application

In a fault study:

1. Convert all impedances to common base
2. Build the impedance network
3. Calculate fault current in per-unit
4. Convert back to amps using $I_{base}$ at the fault location

The per-unit system eliminates transformer ratios from the network—they're absorbed into the base conversions.

Summary

The per-unit system is a normalization technique that simplifies power system calculations. Master the base conversions, and complex multi-voltage networks become straightforward.