Per-Unit System: What It Is and Why It Is Used

Short answer

The per-unit system expresses voltages, currents, powers and impedances as a fraction of reference values called bases. In a network with transformers at several voltage levels, pu values do not change when crossing the transformer, which simplifies short-circuit and load flow calculations and allows equipment of very different sizes to be compared.

What the per-unit system is

The per-unit (pu) system is a way of writing the electrical quantities of a network as a fraction of a reference value, called the base. A voltage of 0.98 pu is 98 % of the base voltage; a current of 2.5 pu is two and a half times the base current.

It is used because a real network mixes voltage levels and equipment of very different sizes. A 20 MVA transformer and a 200 MVA transformer have very different impedances in ohms, but in pu they tend toward similar values. A pu value tells you at once whether a number is reasonable. A value in ohms requires knowing which side of the transformer it is on. To convert values and change base you can use the short-circuit tool.

The bases and the four quantities

Electrically, voltage, current, power and impedance are linked. For that reason two bases are chosen freely and the other two are calculated. The usual practice is to fix the base power and the base voltage. In a three-phase system, with three-phase power and line-to-line voltage:

Base current: the base power divided by √3 and by the base voltage.
Base impedance: the square of the base voltage divided by the base power.

Each quantity in pu is the real value divided by its base. The pu impedance of an element with impedance Z in ohms is Z divided by the base impedance. These relationships are those of the classic power system analysis textbooks, such as those by Grainger and Stevenson, or by Glover, Sarma and Overbye.

One base per voltage zone

The base power is single for the whole network. The base voltage changes in each voltage zone and is obtained from the rated ratio of the transformers that separate the zones. That is the central trick of the system: referred to their base, a voltage of 1 pu on one side of the transformer corresponds to 1 pu on the other, and the transformer impedance in pu is the same seen from either of its windings.

The nameplate impedance of a transformer

A transformer nameplate shows the short-circuit impedance as a percentage. IEC 60076-1 defines that impedance as the equivalent series impedance measured at one winding with the other short-circuited, and expresses it in relative form, as the ratio between it and a reference impedance. With the rated power and the winding voltage as the reference, that ratio is the pu impedance of the transformer on its own base.

The same standard warns that percentage values can lead to confusion because of different reference practices. It therefore advises always stating the reference power and voltage that accompany the percentage (clause 5.5). An impedance of «8 %» with no base is incomplete.

From the nameplate to the study

Nameplate impedance in %. Referred to the rated power and voltage of the transformer
Change to the study base. Adjusted by the power ratio and the square of the voltage ratio
Study impedance in pu. Can now be added to lines, networks and other equipment

Changing base

Equipment data arrives on different bases: each transformer on its own, the network as a short-circuit power, cables in ohms. Before combining them, they are brought to the common base of the study. The rule is the one in the reference textbooks: the new pu impedance is the old one multiplied by the square of the ratio of the old to the new base voltage, and by the ratio of the new to the old base power.

For an impedance given in ohms, divide by the base impedance of the zone where it is located. That base is derived from the base voltage of the zone, not from the rated voltage of the equipment, and that is why it is legitimate to express ohms from different zones on one common base.

An example with numbers

A 20 MVA transformer with 8 % impedance feeds a 34.5 kV bus from a network considered infinite. With a 100 MVA base:

Its impedance on the study base is 8 % times 100/20, that is 40 %, or 0.40 pu.
The base current at 34.5 kV is about 1.67 kA.
The three-phase fault at the bus is 1/0.40 = 2.5 pu, that is, about 4.2 kA.

The same result is obtained without pu: the rated current of the transformer, about 0.33 kA, divided by 0.08. The per-unit system arrives at the same figure with a sum of impedances, which is what matters when there are many elements.

Where it is used and what to watch for

The per-unit system is the basis of the short-circuit study, the load flow study and the network models used by analysis software. It also appears in equipment data: the impedance of a transformer, the reactance of a generator, the voltage of a regulator.

Three mistakes recur:

Mixing bases. Adding a transformer impedance on its own base to a line impedance on the study base gives a meaningless result, even though both look reasonable.
Forgetting the zone. An impedance in ohms is converted only with the base voltage of the zone where it is located.
Leaving the base out. A pu number without its base cannot be reproduced or audited. Every report must state it.

The per-unit system is a calculation language. Final results, for specifying a circuit breaker or setting a protection, are converted to amperes. If you want to see how the X/R ratio and the first peak influence the result, see the short-circuit peak factor and the transformer short-circuit impedance test.

Two frequent applications are the nameplate impedance of a transformer, which is specified according to Power transformer selection and specification, and the voltage drop in a cable, which is calculated in AC cable voltage drop.

Calculate it here

To change the base of an impedance or convert it from ohms to pu, use our free short-circuit and base change tool.

We can help with your studies

At Tiza Group we carry out short-circuit, load flow and protection coordination studies, with a documented model and explicit bases. See our protection coordination service and our primary engineering service.

References

IEC 60076-1, power transformers, general: short-circuit impedance of a pair of windings (clause 3.7.1) and impedance specification (clause 5.5).
J. Grainger and W. Stevenson, Power System Analysis, chapter 2.
J. Glover, M. Sarma and T. Overbye, Power System Analysis and Design, chapter 3.

Frequently asked questions

What is the difference between a percentage and a per-unit value?

It is the same quantity written two ways: the percentage is the per-unit value multiplied by 100. An impedance of 8 % is 0.08 pu. Both refer to a base, which must be known to convert them to ohms.

Can I add pu impedances of different equipment?

Only if they are on the same base. The nameplate impedance of a transformer is referred to its own rated power and voltage; before adding it to that of a line or of the network it has to be brought to the common base of the study.

What is the correct base power?

Any value works: the final result in amperes or ohms does not depend on it. A practical value is chosen, for example one close to the rating of the main equipment, and it is kept throughout the study. What you cannot do is mix bases.