Per-unit system
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In power transmission and distribution, a per-unit system is a system-specific set of units of measurement chosen for convenient analysis of complex power systems. Note that there is no universal set of units for per-unit analysis, and that the units are meaningless outside the context of the intended power system. In particular, converting from per-unit to SI requires additional information about the system.
A per-unit system provides units for power, voltage, current, impedance, and admittance. Only two of these are independent, usually power and voltage. All quantities are specified as multiples of selected base values. For example, the base power might be the rated power of a transformer, or perhaps an arbitrarily selected power which makes power quantities in the system more convenient. The base voltage might be the nominal voltage of a bus. Different types of quanities are labeled with the same symbol (pu); it should be clear from context whether the quantity is a voltage, current, etc.
Per-unit is used primarily in power flow studies; however, because parameters of transformers and machines (motors and generators) are often specified in terms of per-unit, it is important for all power engineers to be familiar with the concept.
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Purpose
There are three main reasons for using a per-unit system:
- Quantities in power systems are often so different in order of magnitude from those seen in other electrical systems that SI units are often inconvenient. For example, units such as MW, kV, kA, and mΩ are commonplace.
- Between different levels of a power system, quantities vary widely. For example, 10 kV is ridiculously low for transmission, but ridiculously high for supply to a residence.
- Per-unit analysis allows different parts of a power system to be analyzed in terms of their respective nominal voltages. In terms of per-unit quantities, a transformer can be analyzed the same way as a transmission line.
The per unit system was developed to make manual analysis of power systems easier. Since power system analysis is now done by computer the use of this system is rare except for analysis of a single machine or manual analysis of small (3-4 node) power systems.
Relationship between units
The relationship between units in a per-unit system depends on whether the system is single phase or three phase.
Single phase
Assuming that the independent base values are power and voltage, we have:
- <math>P_{base} = 1 pu<math>
- <math>V_{base} = 1 pu<math>
Alternatively, the base value for power may be given in terms of reactive or apparent power, in which case we have, respectively,
- <math>Q_{base} = 1 pu<math>
or
- <math>S_{base} = 1 pu<math>
The rest of the units can be derived from power and voltage using the equations <math>P = IV<math> and <math>V = IZ<math> (Ohm's law). We have:
- <math>I_{base} = \frac{P_{base}}{V_{base}} = 1 pu<math>
- <math>Z_{base} = \frac{V_{base}}{I_{base}} = 1 pu<math>
- <math>Y_{base} = \frac{1}{Z_{base}} = 1 pu<math>
Three phase
Power and voltage are specified in the same way as single phase systems. However, due to differences in what these terms usually represent in three phase systems, the relationships for the derived units are different. Specifically, power is given as total (not per-phase) power, and voltage is line to line voltage.
- <math>I_{base} = \frac{P_{base}}{V_{base} \times \sqrt{3}} = 1 pu<math>
- <math>Z_{base} = \frac{V_{base}}{I_{base} \times \sqrt{3}} = 1 pu<math>
- <math>Y_{base} = \frac{1}{Z_{base}} = 1 pu<math>
Example of per-unit
As an example of how per-unit is used, consider a three phase power transmission system that deals with powers on the order of 500 MW and uses a nominal voltage of 138 kV for transmission. We arbitrarily select <math>P_{base} = 500 MW<math>, and use the nominal voltage 138 kV as the base volage <math>V_{base}<math>. We then have:
- <math>I_{base} = \frac{P_{base}}{V_{base} \times \sqrt{3}} = 2.09 kA<math>
- <math>Z_{base} = \frac{V_{base}}{I_{base} \times \sqrt{3}} = 50 \Omega<math>
- <math>Y_{base} = \frac{1}{Z_{base}} = 26.3 mS<math>
If, for example, the actual voltage at one of the buses is measured to be 136 kV, we have:
- <math>V_{pu} = \frac{V}{V_{base}} = \frac{136 kV}{138 kV} = 0.9855 pu<math>
An advantage of this system can be seen immediately; the per-unit voltage e nominal voltage.