Transformer per unit transformations

The per-unit (p.u.) system is a fundamental tool in power system analysis, especially when dealing with transformers. It simplifies calculations by normalizing all quantities (voltage, current, power, impedance) to a common base. This effectively "retains" the ideal transformer from the circuit diagram because the per-unit impedance of a transformer remains the same when referred from one side to the other. This page is not a comprehensive guide on transformer per-unit calculations, a more in depth explanation can be found in this link or basic power system literature.

Establishing Base Values

For a multi-voltage system with transformers, you need to establish consistent base values across different voltage zones.

• Transformer Voltage Base ($V_{base, LL}$):

  • The voltage base is determined by the transformer's nominal line-to-line voltage ratio: $V_{base, \text{secondary}} = V_{base, \text{primary}} \times \frac{V_{\text{rated, secondary}}}{V_{\text{rated, primary}}}$ Where $V_{\text{rated, secondary}}$ and $V_{\text{rated, primary}}$ are the transformer's nominal (rated) line-to-line voltages on its secondary and primary sides, respectively.
  • This value can be slightly different that the attached bus voltage value. In certain low voltage systems, transformers with a higher base voltage can be connected to buses with lower voltage set points. As of PowerSystems v5 transformers now have field for the base voltage.

• How is the data stored?: Transformer impedance (usually reactive impedance, $X_{pu}$) is typically given on its own nameplate ratings (rated MVA and rated voltages). The data in PowerSystems.jl is stored in the device base and transformer to the system base when using the correct getter functions.

• Derived Base Impedance ($Z_{base}$):

  • Once $S_{base, 3\phi}$ and $V_{base, LL}$ are established for each voltage zone, the base impedance can be calculated as follows:

    \[Z_{base} = \frac{(V_{base, LL})^2}{S_{base, 3\phi}}\]

    Where $V_{base, LL}$ is in kV and $S_{base, 3\phi}$ is in MVA, resulting in $Z_{base}$ in Ohms ($\Omega$).

Transformer Impedance Transformations

The most significant advantage of the per-unit system for transformers is that the per-unit impedance of a transformer is the same on both sides, provided the base voltages are chosen according to the transformer's turns ratio and the base power is consistent.

• Changing Base for Transformer Impedance: If the system-wide base $S_{base, 3\phi}$ and the zone-specific voltage bases ($V_{base, \text{primary zone}}$ and $V_{base, \text{secondary zone}}$) differ from the transformer's ratings, you need to convert the transformer's per-unit impedance to the new system base.The formula for changing base of an impedance is:

  \[Z_{pu, \text{new}} = Z_{pu, \text{old}} \times \left(\frac{S_{base, \text{new}}}{S_{rated, \text{old}}}\right) \times \left(\frac{V_{rated, \text{old}}}{V_{base, \text{new}}}\right)^2\]

  • Here, $S_{base, \text{new}}$ is your chosen system-wide base MVA.

  • \[S_{rated, \text{old}}\]

    is the transformer's rated MVA (from nameplate).

  • \[V_{rated, \text{old}}\]

    is the transformer's rated voltage on the side you are considering (e.g., if you're transforming the impedance to the primary side's base, use the primary rated voltage).

  • \[V_{base, \text{new}}\]

    is the new system base voltage for that side of the transformer.
  When calculating the transformer's impedance on the system base, you only need to perform this calculation once. Since the per-unit impedance of a transformer is the same when referred from one side to the other (given correct base voltage selection), the $Z_{pu, \text{new}}$ calculated for the transformer will be used regardless of which side you are viewing it from in the per-unit circuit diagram.