Per Unit Values
Instead of physical values of ohm, ampere, kV, MVA, the per-unit values can be used for computer-based calculation in particular.
The impedance value (Z) is important for the power system studies such as short circuit calculation, transient analysis. To obtain the pu quantity of impedance, base values of electrical quantity are of importance.
Line - neutral (LN) and line-line (LL) voltages are involved in the calculation with the R,S,T currents.
Base and pu values for single phase system
Base apparent power for single phase system,
\[{S_{base(1\phi )}} = {V_{base(LN)}}.{I_{base}}\]
Impedance can be obtained by using voltage, current and apparent power bases.
\[{Z_{base}} = \frac{{{V_{base(LN)}}}}{{{I_{base}}}} = \frac{{{V_{base(LN)}}}}{{\frac{{{S_{base(1\phi )}}}}{{{V_{base(LN)}}}}}} = \frac{{{V_{base(LN)}}^2}}{{{S_{base(1\phi )}}}}\]
\[{Z_{base}} = {X_{base}} = {R_{base}}\]
Unit can be determined in a simple level:
\[\frac{{{V_{base(LN)}}^2}}{{{S_{base(1\phi )}}}} = \frac{{Vol{t^2}}}{{VoltAmper}} = \frac{{k{V^2}.1000.1000}}{{MVA.1000.1000}} = \frac{{k{V^2}}}{{MVA}}\]
Per-unit value of Z is based on rated value of power system equipment.
\[{Z_{pu}} = \frac{{{Z_{rated}}}}{{{Z_{base}}}}\]
There is no change for pu of reactance.
\[{X_{pu}} = \frac{{{X_{rated}}}}{{{Z_{base}}}}\]
For instance, a single phase, 30 kVA, 10 kV transformer with 110 ohm random values.
\[{Z_{base}} = \frac{{{{10}^2}(k{V^2})}}{{{{30.10}^{ - 3}}(MVA)}} = 3333\Omega\]
\[{X_{pu}} = \frac{{{X_{rated}}}}{{{Z_{base}}}} = \frac{{110}}{{3333}} = 0.033pu\]
Each equipment have their own ratings and own impedance base.
\[{S_{base(3\phi )}} = \sqrt 3 {V_{base(LL)}}.{I_{base}}\]
\[{Z_{base}} = \frac{{{V_{base(LL)}}}}{{\sqrt 3 {I_{base}}}} = \frac{{{V_{base(LL)}}}}{{\frac{{{S_{base(3\phi )}}}}{{{V_{base(LL)}}}}}} = \frac{{{V_{base(LL)}}^2}}{{{S_{base(3\phi )}}}}\]
In other way,
\[{S_{base(1\phi )}} = \frac{1}{3}{S_{base(3\phi )}}\]
\[{V_{base(LN)}} = \frac{{{V_{base(LL)}}}}{{\sqrt 3 }}\]
we know,
\[{Z_{base}} = \frac{{{V_{base(LN)}}^2}}{{{S_{base(1\phi )}}}}\]
And for the three phase,
\[{Z_{base}} = \frac{{{{(\frac{{{V_{base(LL)}}}}{{\sqrt 3 }})}^2}}}{{\frac{1}{3}{S_{base(3\phi )}}}} = \frac{{\frac{{{V_{base(LL)}}^2}}{3}}}{{\frac{{{S_{base(3\phi )}}}}{3}}}\]
\[{Z_{base}} = \frac{{{V_{base(LL)}}^2}}{{{S_{base(3\phi )}}}}\]
\[\sqrt 3\]
Each equipment have their own ratings and own impedance base.
Base and pu values for three phase system
\[{S_{base(3\phi )}} = \sqrt 3 {V_{base(LL)}}.{I_{base}}\]
\[{Z_{base}} = \frac{{{V_{base(LL)}}}}{{\sqrt 3 {I_{base}}}} = \frac{{{V_{base(LL)}}}}{{\frac{{{S_{base(3\phi )}}}}{{{V_{base(LL)}}}}}} = \frac{{{V_{base(LL)}}^2}}{{{S_{base(3\phi )}}}}\]
In other way,
\[{S_{base(1\phi )}} = \frac{1}{3}{S_{base(3\phi )}}\]
\[{V_{base(LN)}} = \frac{{{V_{base(LL)}}}}{{\sqrt 3 }}\]
we know,
\[{Z_{base}} = \frac{{{V_{base(LN)}}^2}}{{{S_{base(1\phi )}}}}\]
And for the three phase,
\[{Z_{base}} = \frac{{{{(\frac{{{V_{base(LL)}}}}{{\sqrt 3 }})}^2}}}{{\frac{1}{3}{S_{base(3\phi )}}}} = \frac{{\frac{{{V_{base(LL)}}^2}}{3}}}{{\frac{{{S_{base(3\phi )}}}}{3}}}\]
\[{Z_{base}} = \frac{{{V_{base(LL)}}^2}}{{{S_{base(3\phi )}}}}\]
\[\sqrt 3\]
is eliminated to facilitate the analysis.
\[{Z_{pu}} = \frac{{{Z_{rated}}}}{{{Z_{base}}}}\]
Impedance base will written as below.
\[{Z_{pu}} = \frac{{{Z_{rated}}}}{{\frac{{{V_{base(LL)}}^2}}{{{S_{base(3\phi )}}}}}} = {Z_{rated}}.\frac{{{S_{base(3\phi )}}}}{{{V_{base(LL)}}^2}}\]
It can be translated into by adding "rated".
\[{Z_{pu(rated)}} = {Z_{rated}}.\frac{{{S_{base(3\phi )(rated)}}}}{{{V_{base(LL)}}{{_{(rated)}}^2}}}\]
\[{Z_{rated}} = {Z_{pu(rated)}}.\frac{{{V_{base(LL)}}{{_{(rated)}}^2}}}{{{S_{base(3\phi )(rated)}}}}\]
New value of Z impedance can be obtained if we connect equipment each other.
\[{Z_{pu(new)}} = {Z_{rated}}.\frac{{{S_{base(3\phi )(new)}}}}{{{V_{base(LL)}}{{_{(new)}}^2}}}\]
\[{Z_{rated}}\] and
New base values in power system with different voltage and apparent power bases
Synchronous generators, transformers, transmission lines, distribution lines etc. can have different ratings. After connection of each other, base values are defined. Base value can be the voltage of transmission line or any generator, or arbitrarily chosen value 100 MVA, 1000 MVA based on network size.
We already know,\[{Z_{pu}} = \frac{{{Z_{rated}}}}{{{Z_{base}}}}\]
Impedance base will written as below.
\[{Z_{pu}} = \frac{{{Z_{rated}}}}{{\frac{{{V_{base(LL)}}^2}}{{{S_{base(3\phi )}}}}}} = {Z_{rated}}.\frac{{{S_{base(3\phi )}}}}{{{V_{base(LL)}}^2}}\]
It can be translated into by adding "rated".
\[{Z_{pu(rated)}} = {Z_{rated}}.\frac{{{S_{base(3\phi )(rated)}}}}{{{V_{base(LL)}}{{_{(rated)}}^2}}}\]
\[{Z_{rated}} = {Z_{pu(rated)}}.\frac{{{V_{base(LL)}}{{_{(rated)}}^2}}}{{{S_{base(3\phi )(rated)}}}}\]
New value of Z impedance can be obtained if we connect equipment each other.
\[{Z_{pu(new)}} = {Z_{rated}}.\frac{{{S_{base(3\phi )(new)}}}}{{{V_{base(LL)}}{{_{(new)}}^2}}}\]
Note that there will be no change in rated Z value, such as physical properties of ohm of transformers, or generators.
\[{Z_{rated}}\] and
\[{Z_{pu(new)}}\]
can be merged.
\[{Z_{pu(new)}} = {Z_{pu(rated)}}.\frac{{{V_{base(LL)}}{{_{(rated)}}^2}}}{{{S_{base(3\phi )(rated)}}}}.\frac{{{S_{base(3\phi )(new)}}}}{{{V_{base(LL)}}{{_{(new)}}^2}}}\]
\[{Z_{pu(new)}} = {Z_{pu(rated)}}{\left[ {\frac{{{V_{base(LL)}}{{_{(rated)}}^{}}}}{{{V_{base(LL)}}{{_{(new)}}^{}}}}} \right]^2}.\frac{{{S_{base(3\phi )(new)}}}}{{{S_{base(3\phi )(rated)}}}}\]
Generator itself has 0.19 pu synchronous reactance.
In case of 100 MVA network connection, new per-unit value will be as below.
\[{X_{pu(rated)}} = {Z_{pu(rated)}}\]
\[{Z_{pu(new)}} = {Z_{pu(rated)}}{\left[ {\frac{{{V_{base(LL)}}{{_{(rated)}}^{}}}}{{{V_{base(LL)}}{{_{(new)}}^{}}}}} \right]^2}.\frac{{{S_{base(3\phi )(new)}}}}{{{S_{base(3\phi )(rated)}}}}\]
\[{X_{pu(new)}} = 0.19.{\left[ {\frac{{13.8}}{{15}}} \right]^2}.\frac{{100}}{{50}} = 0.32pu\]
The new value can be used for equivalent circuit to analyse power system.
\[{Z_{pu(new)}} = {Z_{pu(rated)}}.\frac{{{V_{base(LL)}}{{_{(rated)}}^2}}}{{{S_{base(3\phi )(rated)}}}}.\frac{{{S_{base(3\phi )(new)}}}}{{{V_{base(LL)}}{{_{(new)}}^2}}}\]
\[{Z_{pu(new)}} = {Z_{pu(rated)}}{\left[ {\frac{{{V_{base(LL)}}{{_{(rated)}}^{}}}}{{{V_{base(LL)}}{{_{(new)}}^{}}}}} \right]^2}.\frac{{{S_{base(3\phi )(new)}}}}{{{S_{base(3\phi )(rated)}}}}\]
Now consider 50 MVA, 13.8 kV generator which will be connected to 100 MVA, 15 kV network.
New impedance values are calculated.
In case of 100 MVA network connection, new per-unit value will be as below.
\[{X_{pu(rated)}} = {Z_{pu(rated)}}\]
\[{Z_{pu(new)}} = {Z_{pu(rated)}}{\left[ {\frac{{{V_{base(LL)}}{{_{(rated)}}^{}}}}{{{V_{base(LL)}}{{_{(new)}}^{}}}}} \right]^2}.\frac{{{S_{base(3\phi )(new)}}}}{{{S_{base(3\phi )(rated)}}}}\]
\[{X_{pu(new)}} = 0.19.{\left[ {\frac{{13.8}}{{15}}} \right]^2}.\frac{{100}}{{50}} = 0.32pu\]
The new value can be used for equivalent circuit to analyse power system.
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