9.8: Per-Unit Systems

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James Kirtley
Massachusetts Institute of Technology via MIT OpenCourseWare

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Before going on, we should take a short detour to look into per-unit systems, a notational device that, in addition to being convenient, will sometimes be conceptually helpful. The basic notion is quite simple: for most variables we will note a base quantity and then, by dividing the variable by the base we have a per-unit version of that variable. Generally we will want to tie the base quantity to some aspect of normal operation. So, for example, we might make the base voltage and current correspond with machine rating. If that is the case, then power base becomes:

\(\ P_{B}=3 V_{B} I_{B}\)

and we can define, in similar fashion, an impedance base:

\(\ Z_{B}=\frac{V_{B}}{I_{B}}\)

Now, a little caution is required here. We have defined voltage base as line-neutral and current base as line current (both RMS). That is not necessary. In a three phase system we could very well have defined base voltage to have been line-line and base current to be current in a delta connected element:

\(\ V_{B \Delta}=\sqrt{3} V_{B} \quad I_{B \Delta}=\frac{I_{B}}{\sqrt{3}}\)

In that case the base power would be unchanged but base impedance would differ by a factor of three:

\(\ P_{B}=V_{B \Delta} I_{B \Delta} \quad Z_{B \Delta}=3 Z_{B}\)

However, if we were consistent with actual impedances (note that a delta connection of elements of impedance \(\ 3 Z\) is equivalent to a wye connection of \(\ Z\)), the per-unit impedances of a given system are not dependent on the particular connection. In fact one of the major advantages of using a per-unit system is that per-unit values are uniquely determined, while ordinary variables can be line-line, line-neutral, RMS, peak, etc., for a large number of variations.

Perhaps unfortunate is the fact that base quantities are usually given as line-line voltage and base power. So that:

\(\ I_{B}=\frac{P_{B}}{\sqrt{3} V_{B \Delta}} \quad Z_{B}=\frac{V_{B}}{I_{B}}=\frac{1}{3} \frac{V_{B \Delta}}{I_{B \Delta}}=\frac{V_{B \Delta}^{2}}{P_{B}}\)

Now, we will usually write per-unit variables as lower-case versions of the ordinary variables:

\(\ v=\frac{V}{V_{B}} \quad p=\frac{P}{P_{B}} \quad \text { etc }\)

Thus, written in per-unit notation, real and reactive power for a synchronous machine operating in steady state are:

\(\ p=-\frac{v e_{a f}}{x_{d}} \sin \delta \quad q=\frac{v^{2}}{x_{d}}-\frac{v e_{a f}}{x_{d}} \sin \delta\)

These are, of course, in motor reference coordinates, and represent real and reactive power into the terminals of the machine.