Convert Transformer Impedances Between Per-Unit Bases

Transformer impedance data is typically given on the transformer's own nameplate ratings (its rated MVA and rated voltages). When building a power system model you usually need that impedance expressed on the system-wide base instead. This guide walks through the manual calculation and explains how PowerSystems.jl handles it automatically through getter functions.

For the underlying theory, see Transformer per unit transformations in the Per-unit Conventions explanation.

Step 1: Establish base values for each voltage zone

Every voltage zone in the network needs a consistent pair of base values: a base power $S_{base, 3\phi}$ (system-wide, in MVA) and a base voltage $V_{base, LL}$ (zone-specific, in kV).

The base voltage on the secondary side of a transformer follows from its turns ratio:

\[V_{base,\,\text{secondary}} = V_{base,\,\text{primary}} \times \frac{V_{\text{rated,\,secondary}}}{V_{\text{rated,\,primary}}}\]

where $V_{\text{rated,\,primary}}$ and $V_{\text{rated,\,secondary}}$ are the transformer's nameplate line-to-line voltages.

Once both base quantities are known for a zone, the base impedance is:

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

with $V_{base,\,LL}$ in kV and $S_{base,\,3\phi}$ in MVA, giving $Z_{base}$ in Ohms.

Step 2: Convert the transformer impedance to the new base

If the transformer's nameplate ratings differ from the system base values, convert the per-unit impedance using:

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

where:

• $S_{base,\,\text{new}}$ — your chosen system-wide base MVA.
• $S_{rated,\,\text{old}}$ — the transformer's nameplate rated MVA.
• $V_{rated,\,\text{old}}$ — the transformer's nameplate rated voltage on the side being considered.
• $V_{base,\,\text{new}}$ — the system base voltage for that same side.

This calculation only needs to be done once per transformer. Because the per-unit impedance of a transformer is identical when referred from either side (provided base voltages are chosen consistently with the turns ratio), the result applies to the transformer as a whole.

Example: a transformer rated 50 MVA, 115/13.8 kV with $X_{pu} = 0.10$ on its own base, on a system with $S_{base} = 100\,\text{MVA}$ and $V_{base} = 115\,\text{kV}$ on the primary side:

\[X_{pu,\,\text{new}} = 0.10 \times \frac{100}{50} \times \left(\frac{115}{115}\right)^2 = 0.20\,\text{p.u.}\]

Step 3: Read impedance values in PowerSystems.jl

PowerSystems.jl stores transformer impedance on the device base internally. Getter functions such as get_x automatically apply the MVA-ratio part of the base conversion ($S_{base,\,\text{new}} / S_{rated,\,\text{old}}$) and return the value on whichever base the System is currently set to, following the conventions described in Per-unit Conventions.

The example below builds the same transformer used in Step 2 (50 MVA, 115/13.8 kV, $X_{pu} = 0.10$ on its own nameplate base):

using PowerSystems

# Two buses — primary at 115 kV, secondary at 13.8 kV.
bus_primary = ACBus(;
    number = 1,
    name = "primary",
    available = true,
    bustype = ACBusTypes.REF,
    angle = 0.0,
    magnitude = 1.0,
    voltage_limits = (min = 0.9, max = 1.05),
    base_voltage = 115.0,   # kV
)

bus_secondary = ACBus(;
    number = 2,
    name = "secondary",
    available = true,
    bustype = ACBusTypes.PQ,
    angle = 0.0,
    magnitude = 1.0,
    voltage_limits = (min = 0.9, max = 1.05),
    base_voltage = 13.8,    # kV
)

arc = Arc(; from = bus_primary, to = bus_secondary)

xfmr = Transformer2W(;
    name = "xfmr_50MVA",
    available = true,
    active_power_flow = 0.0,
    reactive_power_flow = 0.0,
    arc = arc,
    r = 0.0,
    x = 0.10,               # p.u. on device base (nameplate: 50 MVA, 115/13.8 kV)
    primary_shunt = 0.0 + 0.0im,
    rating = 1.0,           # p.u. on the device base: 50 MVA / 50 MVA nameplate rating
    base_power = 50.0,      # MVA — transformer nameplate rating
)

# System base power = 100 MVA (matches the Step 2 example).
sys = System(100.0)
add_component!(sys, bus_primary)
add_component!(sys, bus_secondary)
add_component!(sys, arc)
add_component!(sys, xfmr)

# get_x returns the reactance on the current unit base (SYSTEM_BASE by default).
# Expected: 0.10 × (100/50) × (115/115)² = 0.20 p.u.
x_system_base = get_x(xfmr)

0.2

To inspect the raw device-base value, switch the unit system first:

set_units_base_system!(sys, "DEVICE_BASE")
x_device_base = get_x(xfmr)   # returns 0.10 — the original nameplate value

set_units_base_system!(sys, "SYSTEM_BASE")  # restore

[ Info: Unit System changed to UnitSystem.DEVICE_BASE = 1
[ Info: Unit System changed to UnitSystem.SYSTEM_BASE = 0

See Read Component Values in Different Unit Systems for a full description of the available unit system settings.