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Introduction to Per-Unit Systems Part 1b. Per unit conversion formulas and base values.
In the previous video, Part 1a, we saw what per unit systems are and where they are used?
In this video, Part 1b, we will be discussing the formulas for converting values
into their per unit equivalents and we will also have a look at the term ‘Base values’.
The general formula that is used to calculate per unit values is that
The per unit value is equal to the Actual value in any unit divided by the base value in any unit.
For example, consider taking an actual value of 15 and a base value of 100,
then the per unit value becomes 0.15.
This is just a simple example and 15 and 100 are just numbers.
In power engineering, these numbers can take the form of
voltages, currents, impedances and power.
Which means that the per unit value of voltage is equal to the
Actual value of voltage divided by the base value of voltage.
Similarly, The per unit value of current is equal to the actual value of current divided by the
base value of current. And the same goes on for impedances and power.
Let’s dig deeper into these parameters. Actual values and Base values,
both are in the same units which means that per unit
values have no units because the original units were cancelled out with each other because of division,
leaving the answer, essentially dimensionless,
and so we call them per unit values p.u depicting the per unit.
Now, taking one of the formulas above, for instance, voltage.
Both the numerators and denominators have units as ‘volts’
which get cancelled out, leaving us with a dimensionless per unit value.
So, we can write whatever the value is like 0.1, 0.2, 0.6 etc. in p.u or per units.
Coming to the base values.
Now, base values are rated values of the equipment, like
ratings of a generator or a transformer.
Normally, we select base values of voltage and the apparent power to carry out per unit calculations.
In power systems that consists of many equipment like
generators, transformers, transmission lines, capacitor banks and so forth,
we take the largest equipment rating as the base value.
Now, to convert a power system into a per unit equivalent,
we require four values that are the
voltage base, the current base, the impedance base and the power base.
However, at many instances, we may come across the ratings of
generators or any other equipments as 100 V, 330 VA or 200 V or 100 A,
meaning only two base values are given.
But we don’t have to worry. By using these given base values we can actually calculate all four base values!
Now, let’s understand this by way of an example from the book called Electric Machinery Fundamentals by
Stephen Chapman (Edition no. 4).
Now, in this example we are given a generator in Region 1 which is at
480 volts at 0 degrees and there is a step-up transformer in
Region 2 that has a transmission line with a 20 ohms resistance and 60 ohms reactance.
And then there is step-down transformer which leads us to Region 3
which has a load of 10 ohms at 30 degrees.
Now, in this example, we are told that the base values of the system are
480 volts and 10 KVA at the generator
and in Part (a), we are asked to find the base voltage, current, impedance and the apparent power at every point in the power systems
and Part (b) is to convert the system into per unit equivalents.
Part (c) find the power supplied to a load in this system and
Part (d) find the power lost in the transmission line.
Now, since this video is served as an introduction, we will solve only Part (a) and Part (b) for Region 1 only.
The remaining parts will be solved in the next video.
Part (a) asks us to calculate the base values for current and impedances for Region 1.
As given in the question, we know that Vbase is equal to 480 volts and Sbase is equal to 10 KVA.
We also know that to calculate Ibase and Zbase, we just have to use the following formulas:
Base current is equal to Base power divided by the Base voltage and
Base impedance is equal to the Base voltage squared divided by Base power.
By putting the values of voltage base and power base in these equations,
we can easily calculate current base and impedance base.
So the base current is equal to the power base divided by the base voltage
which equals Sbase divided by Vbase,
which we know that to be 10 KVA divided by 480 volts and that gives us 20.83 Amps.
Now that’s the base current.
The base impedance is equal to the Base voltage squared divided by the power base.
Vbase squared divided by Sbase is equal to 480 volts squared divided by 10 KVA and that gives us 23.04 ohms.
So, our answers are that the current base is equal to 20.83 Amps and the impedance base is equal to 23.04 ohms.
By solving these above parts, we conclude that a minimum of two base quantities
can help us find all of the four base values
and then we can convert the system into per unit equivalent system.
Now, moving on to Part (b).
In Part (b), we are asked to convert Region 1 into the per unit equivalent.
Since Region 1 consists of generation station, we would have voltage in per units
and we will use the per unit formula to find out Vpu.
And Vp.u or V per unit is equal to the actual voltage of voltage source in Region 1 divided by the base voltage in region 1.
Since Region 1 is the generator side and the base voltage is equal to 480V,
Vp.u simply equals 480 volts divided by 480 volts that gives us 1 per unit
and that is for Region 1.
Finally, we have our answers to the easy part of the question.
Now, in this video we saw how to calculate base values and per unit values.
And we solved an easy example to clear our concepts.
In Part 1c, we will complete this example.
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