Per Unit (PU) System in Power System

Imagine you are comparing two students. One scored 45 out of 50. The other scored 90 out of 100. Who did better?

It is hard to compare marks directly. But if you convert both into percentage, comparison becomes easy. Both become 90%.

The Per Unit System in electrical engineering works on the same idea. It converts big and different values of voltage, current, and power into one simple, common scale. This makes calculations in power systems much easier.

If you are a electrical student studying power systems, transformers, or transmission lines, this topic is very important. It appears almost every semester in exams. In this article, we will explain the Per Unit System, its formulas, and calculations in very simple English, step by step.

What is Per Unit (PU) System?

Definition:

The Per Unit System is a method of expressing electrical quantities like voltage, current, power, and impedance as a fraction (or ratio) of a chosen base value.

In simple words, instead of writing the actual value, we write:

$\textbf{Per unit value}=\frac{\textbf{Actual value}}{\textbf{Base Value}}$

For example, if the base voltage is 100 V and the actual voltage is 90 V, then:

Per Unit Voltage = 90 / 100 = 0.9 PU

So instead of saying “90 volts,” we say “0.9 per unit.” This small number is easier to use in calculations, especially when working with big power systems that have many different voltage levels.

Real-Life Analogy: Think about currency exchange. If you travel to different countries, each country has its own currency. To compare prices easily, you convert everything into one common currency, like the US Dollar. The Per Unit System does the same thing for electrical quantities. It converts different voltage and power levels into one common “currency” called the per unit value.

Remember:

• Per unit value has no unit. It is just a number (or ratio).
• Per unit value can also be written in percentage by multiplying by 100.
  • Example: 0.9 PU = 90%

Why do we use the Per Unit System?

Power systems have many parts: generators, transformers, and transmission lines. Each part may work at a different voltage level. For example:

• Generator: 11 kV
• Transformer output: 132 kV
• Transmission line: 132 kV
• Another transformer: 33 kV

If we do calculations using actual values, we must keep converting units again and again. This becomes confusing and increases chances of mistakes.

The Per Unit System solves this problem. Once we convert all values to a common base, the numbers become simple, small, and easy to handle.

Main reasons to use Per Unit System:

• It simplifies calculations across transformers with different voltage ratios.
• It removes the need for constant unit conversion.
• It gives a quick idea of how a machine is performing compared to its rated value.
• It helps in comparing equipment of different sizes and ratings.
• It reduces calculation errors in large power system networks.

Base Quantity

Before calculating any per unit value, we must first choose some base values. Base values are reference values that we select ourselves. They act like a “measuring scale.”

Definition:

A base value is a fixed reference quantity, chosen by the engineer, which is used to convert actual values into per unit values.

There are four main base quantities in the Per Unit System:

1. Base Voltage
2. Base Current
3. Base Power
4. Base Impedance

Let us understand each one clearly.

1. Base Voltage (V_base)

Base voltage is the voltage we choose as our reference. Usually, the rated voltage of the equipment (like a transformer or generator) is selected as the base voltage.

• Symbol: V_base
• Unit: Volts (V) or kilovolts (kV)

Example: If a transformer is rated at 11 kV, we can choose 11 kV as the base voltage.

2. Base Power (S_base)

Base power (also called base MVA) is the reference power value. It is usually chosen based on the rating of the main equipment in the system, like a generator or transformer.

• Symbol: S_base
• Unit: VA, kVA, or MVA

Example: If the system has a 100 MVA transformer, we can choose 100 MVA as the base power.

Remember:

Base power is normally kept the same for the entire power system, even if voltage levels change at different points (like before and after a transformer). Only base voltage changes according to the transformer ratio.

3. Base Current (I_base)

Base current is calculated using base power and base voltage. It is not chosen directly; we calculate it using a formula.

For single-phase systems:

$I_{\textbf{base}}=\frac{S_{\textbf{base}}}{V_{\textbf{base}}}= \frac{\text{Base KVA}}{\text{Base KV}} = \frac{\mathrm{KVA}_{B}}{\mathrm{KV}_{B}}$

For three-phase systems:

$\mathbf{I}_{\mathbf{base}}=\frac{\mathbf{S}_{\mathbf{base}}}{\boldsymbol{\sqrt{3}}\,\mathbf{V}_{\mathbf{base}}}$

Here, V_base is the line-to-line voltage, and S_base is the total three-phase power.

4. Base Impedance (Z_base)

Base impedance is also calculated, not chosen directly. It tells us the reference resistance/reactance value for the system.

For single-phase systems:

$\mathbf{Z}_{\mathbf{base}} = \frac{\mathbf{V}{\mathbf{base}}}{\mathbf{I}_{\mathbf{base}}} = \frac{\mathbf{V}_{\mathbf{base}}^{2}}{\mathbf{S}_{\mathbf{base}}}= \frac{\textbf{Base KV}\times 1000} {\textbf{Base current}}= \frac{\mathrm{KV}_{B}^2}{\mathrm{KVA}_{B}}$

For three-phase systems:

$\mathbf{Z}_{\mathbf{base}} = \frac{\mathbf{V}_{\mathbf{base}}^{2}\,(\text{line})} {\mathbf{S}_{\mathbf{base}}\,(\text{three-phase})}$

Here, both V_base and S_base must be in matching units (like kV and MVA) for the formula to give correct results.

Exam Tip:

Students often forget that only Base Voltage and Base Power are chosen freely. Base Current and Base Impedance are always calculated from these two. Never choose current or impedance base directly.

Per Unit Formula

Now that we understand base values, let us look at the general Per Unit formula.

General Formula (Important):

Per Unit Value = Actual Value /Base Value (of the same quantity)

This same formula applies to voltage, current, power, and impedance. Only the base value changes depending on which quantity we are converting.

Quantity	Formula
Per Unit Voltage	$V_{\text{pu}}=\frac{V_{\text{actual}}}{V_{\text{base}}}$
Per Unit Current	$I_{\text{pu}}=\frac{I_{\text{actual}}}{I_{\text{base}}}= \frac{\text{Actual value of current}} {\mathrm{KVA}_{B}/\mathrm{KV}_{B}}= \frac{\text{Actual value of current}\times \mathrm{KV}_{B}} {\mathrm{KVA}_{B}}$
Per Unit Power	$S_{\text{pu}}=\frac{S_{\text{actual}}}{S_{\text{base}}}$
Per Unit Impedance	$Z_{\text{pu}}=\frac{Z_{\text{actual}}}{Z_{\text{base}}} = \frac{(\mathrm{KV}_{B})^{2}\times1000} {\mathrm{KVA}_{B}}$

Remember:

The actual value and the base value must always be in the same unit. For example, if base voltage is in kV, actual voltage must also be converted to kV before dividing.

How to Calculate Per Unit Value (Step-by-Step)

Here is a simple step-by-step method that works for almost every per unit calculation problem.

Step 1: Choose the base power (S_base). This is usually given in the problem, or you can choose the rating of the main equipment.

Step 2: Choose the base voltage (V_base). Usually, the rated voltage of the transformer or generator zone is used.

Step 3: Calculate base current using the formula: I_base = S_base / (√3 × V_base) for 3-phase systems.

Step 4: Calculate base impedance using the formula: Z_base = (V_base)² / S_base

Step 5: Convert the actual value (voltage, current, or impedance) into the per unit value by dividing it by its matching base value.

Step 6: If needed, convert the final per unit answer back to percentage by multiplying by 100.

Remember:

Always keep units consistent (kV with kV, MVA with MVA) throughout all steps. Mixing units is the most common mistake in per unit calculations.

Easy Numerical Example

Let us solve a simple problem so the method becomes crystal clear.

Question: A single-phase transformer is rated at 200 kVA, 400 V. Its actual impedance is 0.5 ohm. Find the per unit impedance of the transformer, taking its own rating as the base.

Given Data:

• S_base = 200 kVA = 200 × 10³ VA
• V_base = 400 V
• Z_actual = 0.5 ohm

Step 1: Find Base Current

I_base = S_base / V_base

I_base = 200 × 10³ / 400

I_base = 500 A

Step 2: Find Base Impedance

Z_base = V_base / I_base

Z_base = 400 / 500

Z_base = 0.8 ohm

(You can also verify using Z_base = (V_base)² / S_base = (400)² / (200×10³) = 160000 / 200000 = 0.8 ohm. Same answer.)

Step 3: Find Per Unit Impedance

Z_(pu) = Z_(actual) / Z_(base)

Z_(pu) = 0.5 / 0.8

Z_(pu) = 0.625 PU

The per unit impedance of the transformer is 0.625 PU, or 62.5%.

Exam Tip:

In numerical questions, always write all three steps clearly: finding base current, finding base impedance, and then finding the per unit value.

Advantages of Per Unit System

• It makes calculations simple, especially in systems with many transformers.
• Transformer connections (star or delta) do not affect the per unit impedance value, which reduces confusion.
• Per unit values usually fall in a small, predictable range (like 0.8 to 1.2 PU), so mistakes are easy to spot.
• Manufacturers often give equipment ratings directly in per unit or percentage, so no extra conversion is needed.
• It is very helpful for computer-based power system analysis, like fault calculations and load flow studies.

Remember:

A transformer’s per unit impedance stays almost the same on both the primary and secondary side, even though actual ohmic values are very different. This is one of the biggest benefits of the PU system.

Disadvantages of Per Unit System

• Beginners often find it confusing at first because it looks abstract.
• Choosing wrong base values can lead to wrong results.
• It requires extra initial steps (finding base current and base impedance) before actual calculation.
• If different parts of a large system use different base values without proper conversion, errors can occur.

Remember:

The per unit system is not difficult once you practice a few numerical problems. The confusion usually comes only from silly mistakes in choosing or converting base values.

Applications of Per Unit System

The Per Unit System is widely used in real power system work, such as:

• Transformer analysis: Comparing impedance of transformers of different ratings.
• Short circuit / fault analysis: Calculating fault current in large networks.
• Load flow studies: Analyzing how power flows in a network with many buses.
• Power system stability studies: Understanding how a system behaves during disturbances.
• Protection system design: Setting relay values based on per unit fault levels.
• Equipment rating comparison: Comparing machines of very different sizes on the same scale.

Common Mistakes Students Make

Mistake	Correction
Mixing kV and V while calculating	Always convert to the same unit before applying formulas
Forgetting the √3 factor in 3-phase base current formula	Use √3 only for 3-phase systems, not single-phase
Choosing base current or base impedance directly	These must always be calculated, never chosen directly
Forgetting to change base values when moving across a transformer	Base voltage changes according to transformer ratio; base MVA usually stays the same
Not converting the final answer to percentage when asked	Multiply per unit value by 100 to get percentage

Exam Tip:

If the question mentions percentage impedance (like “5% impedance”), simply convert it to per unit by dividing by 100. Example: 5% = 0.05 PU.

Quick Revision Points

• Per Unit Value = Actual Value ÷ Base Value
• Base Voltage and Base Power are chosen; Base Current and Base Impedance are calculated.
• I_base = S_base / (√3 × V_base) for three-phase systems.
• Z_base = (V_base)² / S_base
• Per unit values have no unit; they can be converted to percentage by multiplying by 100.
• The per unit system makes transformer and network calculations much easier.
• Always keep base units consistent throughout the calculation.

Frequently Asked Questions (FAQ)

Q1. What is the Per Unit System in simple words?

The Per Unit System is a method where we express voltage, current, power, or impedance as a ratio of a chosen base value, instead of using their actual values. It makes power system calculations simpler.

Q2. Why is base impedance important in the Per Unit method?

Base impedance acts as the reference value for converting actual impedance into per unit impedance. Without it, we cannot calculate the per unit impedance of any equipment.

Q3. Does changing base values change the actual per unit impedance of a transformer?

Yes. If you change the base values, the per unit value will also change. That is why it is important to always mention which base values were used for a calculation.

Q4. What is the difference between per unit value and percentage value?

They represent the same information in different scales. Per unit value is on a scale of 0 to 1 (approximately), while percentage value is on a scale of 0 to 100. You can convert per unit to percentage by multiplying by 100.

Q5. Why do we not choose base current and base impedance directly?

Because base current and base impedance are dependent quantities. They are mathematically related to base voltage and base power through fixed formulas, so choosing them independently would create inconsistency in the system.