Understanding the Per Unit System

The Per Unit (PU) system is a normalized system of quantities used in power system analysis. It simplifies calculations by expressing electrical quantities (like voltage, current, power, impedance) as fractions of a chosen base value. This normalization is particularly useful for comparing equipment of different ratings and for analyzing large, interconnected power systems.

Why Use the Per Unit System?

The Per Unit system offers several advantages for power system engineers, especially in competitive exams like GATE:

Simplification of Calculations: It reduces the need for frequent conversions between different voltage levels and ratings, making complex network analysis more manageable.
Standardization: Equipment manufacturers often provide ratings in per unit values, making it easier to compare components.
Consistency: Impedances of transformers and rotating machines remain relatively constant regardless of the base chosen, simplifying fault calculations and system studies.

The Per Unit value of a quantity is its actual value divided by its base value.

The fundamental formula for converting any electrical quantity into its per unit equivalent is: Per Unit Value = Actual Value / Base Value. This applies to voltage, current, power, and impedance.

The per unit value of a quantity is defined as the ratio of the actual value of the quantity to its base value. Mathematically, this is expressed as:

$PU = \frac{\text{Actual Value}}{\text{Base Value}}$

For example, if a voltage is 132 kV and the base voltage is 132 kV, its per unit value is 1. If the base voltage were 220 kV, the per unit value would be $132/220 = 0.6$ PU.

Choosing Base Quantities

In a power system, we typically choose two base quantities, and the rest are derived from them. The most common choices are:

Base Apparent Power (Sbase): This is usually chosen as a common value for the entire system or a significant portion of it. It's often the kVA or MVA rating of the largest transformer or generator.
Base Voltage (Vbase): This is usually chosen as the nominal voltage of a particular section of the power system. For example, the line-to-line voltage for a 220 kV bus.

Derived Base Quantities

Once Sbase and Vbase are chosen, other base quantities can be derived using fundamental power system relationships:

Base Current (Ibase): $I_{base} = \frac{S_{base}}{\sqrt{3} V_{base}}$ (for three-phase systems)
Base Impedance (Zbase): $Z_{base} = \frac{V_{base}^2}{S_{base}}$ (for three-phase systems)

What are the two most common base quantities chosen in a power system for per unit calculations?

Base Apparent Power (S_base) and Base Voltage (V_base).

Per Unit Impedance of Transformers

A key advantage of the per unit system is that the impedance of a transformer, when expressed in per unit on its own rating, remains the same regardless of whether it's referred to the primary or secondary side. This significantly simplifies the analysis of systems with multiple transformers.

Quantity	Formula (3-Phase)	Per Unit Formula
Apparent Power	$S = \sqrt{3} V_L I_L$	$PU_S = \frac{S}{S_{base}}$
Voltage	$V$	$PU_V = \frac{V}{V_{base}}$
Current	$I$	$PU_I = \frac{I}{I_{base}}$
Impedance	$Z = \frac{V}{I}$	$PU_Z = \frac{Z}{Z_{base}}$

The per unit impedance of a transformer can be calculated using its actual impedance value and the chosen base values. For a transformer with primary voltage $V_{p}$ , secondary voltage $V_{s}$ , and impedance $Z_{actual}$ , the per unit impedance referred to the primary side is $Z_{pu,p} = \frac{Z_{actual}}{V_{p}^2 / S_{base}}$ , and referred to the secondary side is $Z_{pu,s} = \frac{Z_{actual}}{V_{s}^2 / S_{base}}$ . Crucially, $Z_{pu,p} = Z_{pu,s}$ if the base values are chosen appropriately for each side ( $S_{base,p} = S_{base,s}$ and $V_{base,p}/V_{base,s} = V_p/V_s$ ). This consistency is a major benefit.

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Changing Base Values

Often, different parts of a power system may have different base values (e.g., different voltage levels or different system MVA ratings). To analyze the system as a whole, it's necessary to convert per unit impedances from one base to another. The formula for changing the base of an impedance is:

$Z_{new\_base} = Z_{old\_base} \times \left(\frac{V_{old\_base}}{V_{new\_base}}\right)^2 \times \left(\frac{S_{new\_base}}{S_{old\_base}}\right)$

If a transformer has a per unit impedance of 0.1 on a 100 MVA, 220 kV base, what is its per unit impedance on a 50 MVA, 110 kV base?

0.1 * (220/110)^2 * (50/100) = 0.1 * (2)^2 * 0.5 = 0.1 * 4 * 0.5 = 0.2 PU

Remember that for DC machines, the per unit system is applied similarly, but the focus is on quantities like voltage, current, torque, and power. The base values are chosen based on the machine's rating.

Learning Resources

Per Unit System in Power Systems - Electrical Engineering(blog)

Provides a clear explanation of the per unit system, its advantages, and how to calculate base values and per unit impedances.

Per Unit System - GATE Electrical Engineering(blog)

A GATE-focused resource that breaks down the per unit system with examples relevant to competitive exams.

Per Unit System - Electrical Engineering(blog)