Units & Conventions

Power systems use specialized unit systems and conventions that can confuse newcomers. This reference explains the per-unit system, sign conventions, and standards used in GAT.

The Per-Unit System

Why Per-Unit?

Power systems span enormous ranges:

• Voltages from 120 V to 765,000 V
• Powers from kW to GW
• Impedances from milliohms to thousands of ohms

The per-unit (p.u.) system normalizes all quantities to dimensionless ratios, providing:

1. Simplified calculations: Transformer ratios disappear; values are comparable across voltage levels
2. Numerical stability: All quantities are O(1), avoiding floating-point issues
3. Quick sanity checks: Normal voltages are ~1.0 p.u., impedances are ~0.01-0.1 p.u.
4. Standard data formats: MATPOWER, PSS/E, and GAT all use per-unit

Base Quantities

The per-unit system requires choosing base values. Once you pick two, the rest are determined:

Derived bases:

I_base = S_base / (√3 · V_base)     [for three-phase]
Z_base = V_base² / S_base
Y_base = S_base / V_base² = 1/Z_base

Converting to Per-Unit

X_pu = X_actual / X_base

Example: A 345 kV line with X = 50 Ω on a 100 MVA base:

Z_base = (345 kV)² / 100 MVA = 1190.25 Ω
X_pu = 50 / 1190.25 = 0.042 p.u.

Converting from Per-Unit

X_actual = X_pu · X_base

Example: A generator producing 0.8 p.u. real power on a 100 MVA base:

P_actual = 0.8 × 100 MVA = 80 MW

Multi-Voltage Networks

Each voltage level has its own V_base (the nominal voltage), but S_base is the same everywhere. This makes transformer modeling elegant:

• Impedances referred to either side use that side's Z_base
• Ideal transformers have 1:1 ratio in per-unit (tap ratio handles off-nominal)

GAT Base Conventions

System Base

GAT reads base MVA from the network's system.arrow table:

All per-unit quantities in GAT use this base.

Voltage Bases

Each bus has a nominal voltage in buses.arrow:

Impedance Convention

Branch impedances in branches.arrow are in per-unit on the system base:

For transformers, impedances are typically given on the transformer's own MVA rating. GAT expects them converted to the system base:

Z_pu_system = Z_pu_nameplate × (S_base / S_nameplate)

Sign Conventions

Generator Convention (Source)

Generators use generator convention: positive P and Q mean power flowing out of the device into the network.

P_gen > 0 → producing real power (normal operation)
Q_gen > 0 → producing reactive power (overexcited, exporting VARs)
Q_gen < 0 → absorbing reactive power (underexcited)

Load Convention (Sink)

Loads use load convention: positive P and Q mean power flowing into the device from the network.

P_load > 0 → consuming real power (normal operation)
Q_load > 0 → consuming reactive power (inductive load)
Q_load < 0 → producing reactive power (capacitive load, rare)

Net Injection

Power flow equations use net injection at each bus:

P_inj = P_gen - P_load
Q_inj = Q_gen - Q_load

Positive injection = net generation at the bus.

Branch Flow Direction

Branch flows use the from-to convention:

P_ij > 0 → power flowing from bus i to bus j
P_ij < 0 → power flowing from bus j to bus i

Due to losses, P_ij + P_ji ≠ 0 (the difference is I²R loss).

Angle Conventions

Reference Angle

The slack bus (reference bus) has angle θ = 0. All other angles are relative to this reference.

• Positive angles: bus leads the reference
• Negative angles: bus lags the reference

Units

GAT stores angles in radians internally (angle_rad column). Output may be converted to degrees for display.

θ_deg = θ_rad × (180/π)

Typical transmission angles: ±30° (±0.52 rad) under normal operation.

Angle Differences

Power flow depends on angle differences, not absolute angles:

P_ij ∝ sin(θ_i - θ_j)

The choice of reference only affects absolute values, not flows.

Power Conventions

Three-Phase vs. Single-Phase

Power system quantities are typically three-phase totals unless noted:

S_3φ = √3 · V_LL · I_L

GAT uses three-phase quantities throughout. Single-phase equivalents (common in textbooks) differ by factors of 3 or √3.

Complex Power

Complex power S combines real (P) and reactive (Q):

S = P + jQ

Power Factor

pf = P / |S| = cos(φ)

where φ is the angle between voltage and current.

• Lagging pf: Current lags voltage (inductive load, Q > 0)
• Leading pf: Current leads voltage (capacitive load, Q < 0)
• Unity pf: P = |S|, Q = 0

Impedance Conventions

Series vs. Shunt

Series elements (lines, transformers) have:

• R: resistance (causes real power loss)
• X: reactance (limits power transfer)
• Combined: Z = R + jX

Shunt elements (capacitors, reactors, line charging) have:

• G: conductance (rare, represents corona/leakage)
• B: susceptance (main component)
• Combined: Y = G + jB

Inductive vs. Capacitive

Lines and transformers are inductive (X > 0). Line charging is capacitive (B > 0).

The π-Model

Transmission lines use the π-equivalent circuit:

    Bus i                    Bus j
      o──────┬──[R+jX]──┬──────o
             │          │
            jB/2       jB/2
             │          │
            ═╧═        ═╧═

• Series impedance: Z = R + jX
• Shunt admittance: B/2 at each end (line charging)

B is the total line charging; each end gets half.

Transformer Conventions

Tap Ratio

The tap ratio a relates primary and secondary voltages:

V_primary = a · V_secondary (ideal transformer)

• a = 1.0: Nominal tap position
• a > 1.0: Step-up (or boost on regulated side)
• a < 1.0: Step-down (or buck on regulated side)

GAT's tap_ratio in branches.arrow uses this convention.

Off-Nominal Taps in Y-bus

For a transformer from bus i to bus j with tap ratio a and impedance Z:

Y_ii += y/a²
Y_jj += y
Y_ij = Y_ji = -y/a

where y = 1/Z.

Note: Off-nominal taps make Y-bus asymmetric if the tap is not at 1.0.

Phase Shifters

Phase-shifting transformers add an angle shift:

V_i = a·e^(jφ) · V_j

The phase shift φ (in radians) controls real power flow direction. GAT stores this in phase_shift_rad.

Common Pitfalls

Mixing Bases

Problem: Combining data from different sources with different S_base.

Solution: Always convert to a common base:

Z_new_base = Z_old_base × (S_new / S_old)

Forgetting √3

Problem: Using single-phase formulas for three-phase systems.

Solution: Remember:

• Line-to-line voltage = √3 × line-to-neutral voltage
• Three-phase power = 3 × single-phase power

Sign Errors

Problem: Confusing generator and load convention.

Solution:

• Generators: positive = producing
• Loads: positive = consuming
• Injections = generation - load

Angle Units

Problem: Mixing radians and degrees.

Solution: GAT uses radians internally. Convert explicitly:

radians = degrees × π/180
degrees = radians × 180/π

Quick Reference Tables

Unit Prefixes

Common Units

Typical Per-Unit Values

See Also

• Glossary — Term definitions
• Power Flow Theory — Equations using these conventions
• Arrow Schema — How GAT stores these values