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Per-unit system

The per-unit system is a normalization technique widely used in electrical power engineering to express system quantities such as voltage, current, power, and impedance as dimensionless ratios relative to selected base values, typically derived from the nominal ratings of equipment like apparent power and line-to-line voltage.^[1] This method simplifies the analysis of interconnected power networks by eliminating the need for explicit conversions between different voltage levels and equipment scales.^[2] Base values form the foundation of the per-unit system, with apparent power (often in MVA) chosen as a common system-wide reference and voltage bases set to the nominal line-to-line voltages at each bus or winding.^[1] Derived bases include current, calculated as the base apparent power divided by the base voltage, and impedance, obtained as the square of the base voltage divided by the base apparent power.^[2] For example, per-unit impedance is the actual impedance divided by the base impedance, resulting in values that typically cluster around unity for healthy equipment, aiding in fault detection and performance evaluation.^[1] A primary advantage of the per-unit system lies in its handling of transformers, where a uniform base power is applied across all windings while base voltages adjust according to turns ratios, allowing impedances to be directly added without ratio corrections.^[2] This normalization keeps per-unit parameters within a narrow range (e.g., transformer leakage reactance between 0.01 and 0.12 pu), independent of equipment size, which streamlines short-circuit calculations, load flow studies, and stability assessments in large-scale power systems.^[1] Additionally, it reduces computational complexity by treating ideal transformers as unity ratios and ensures voltage magnitudes remain near 1 pu under normal conditions, facilitating quick validation of results.^[2]

Fundamentals

Purpose

The per-unit system is a normalization technique in electrical engineering that expresses quantities such as voltage, current, power, impedance, and admittance as dimensionless ratios relative to selected base values.^[3]^[4] This approach yields per-unit values typically ranging from 0 to 1 or slightly above, concentrating parameters within a narrow numerical range regardless of the absolute scales involved.^[3] The primary purpose of the per-unit system is to facilitate comparisons across diverse power systems with varying nominal ratings, thereby reducing scaling errors and simplifying complex analyses such as fault studies and load flow calculations.^[4]^[5] By normalizing to base quantities like power and voltage, it enables engineers to model interconnected networks without adjusting for disparate equipment sizes or voltage levels, enhancing accuracy in system behavior predictions.^[3] Originating in early 20th-century power engineering, the per-unit system addressed the challenges of handling diverse voltage levels in expanding electrical grids without introducing dimensional inconsistencies during manual computations. It eliminates the need for explicit unit conversions in interconnected networks by inherently absorbing transformer ratios and phase factors into the base framework, streamlining overall system evaluation.^[4]^[5]

Base Quantities

In the per-unit system, the base power S_{\text{base}}, typically expressed in megavolt-amperes (MVA), serves as the fundamental reference quantity for normalizing all power-related values across the electrical system, ensuring a unified scale for analysis.^[2]^[6] This choice of S_{\text{base}} is common because it aligns with apparent power ratings in power engineering, often selected as round numbers such as 100 MVA to simplify computations.^[7] The base voltage V_{\text{base}} is chosen based on the nominal voltages at various system levels, such as line-to-line voltages for transmission (e.g., 138 kV or 345 kV) or generation and distribution (e.g., 13.8 kV).^[2]^[6] Selection prioritizes the rated voltages of key equipment like generators or transformers to reflect actual operating conditions without introducing scaling errors.^[7] From these primary bases, other quantities are derived to maintain dimensional consistency. For three-phase systems, the base current is calculated as

I_{\text{base}} = \frac{S_{\text{base}}}{\sqrt{3} V_{\text{base}}},

where V_{\text{base}} is the line-to-line voltage.^[2]^[6] Similarly, the base impedance is given by

Z_{\text{base}} = \frac{V_{\text{base}}^2}{S_{\text{base}}},

which facilitates the normalization of circuit parameters like reactance and resistance.^[7]^[2] Criteria for selecting base quantities emphasize system-wide consistency, particularly by using the same S_{\text{base}} throughout to avoid mismatches in power flow calculations.^[6] Bases are often rounded to convenient values, such as 100 MVA for S_{\text{base}}, and aligned closely with the rated capacities of major components like transformers to minimize computational complexity while preserving accuracy.^[2]^[7] In zoned or multi-voltage-level systems, multiple base sets may be required, with V_{\text{base}} changing at transformer interfaces according to the turns ratio to account for voltage transformations, while S_{\text{base}} remains constant to ensure continuity in per-unit power representations.^[6] This approach prevents discontinuities in analysis across subsystems, such as from high-voltage transmission to lower-voltage distribution.^[2]

Per-Unit Conversions

Single-Phase Systems

In the per-unit system for single-phase circuits, electrical quantities are normalized by dividing their actual values by corresponding base values to obtain dimensionless per-unit values, facilitating analysis across different voltage levels and system sizes. For instance, the per-unit voltage is defined as V_{pu} = \frac{V_{actual}}{V_{base}}, where V_{base} is the selected base voltage, typically the nominal or rated voltage of the system.^[1] This normalization applies similarly to other quantities such as current and power, ensuring consistency in calculations for AC single-phase systems.^[8] The base current for single-phase systems is derived from the base apparent power and base voltage as I_{base} = \frac{S_{base}}{V_{base}}, where S_{base} represents the chosen base power, often the rated capacity of the circuit or equipment.^[9] The base impedance then follows as Z_{base} = \frac{V_{base}}{I_{base}} = \frac{V_{base}^2}{S_{base}}, providing a reference for normalizing impedances in the circuit.^[10] These base quantities, selected based on system ratings, allow direct conversion without dimensional inconsistencies.^[11] Conversions for resistance and reactance in per-unit terms are performed by dividing the actual values by the base impedance: X_{pu} = \frac{X_{actual}}{Z_{base}} and similarly for resistance R_{pu} = \frac{R_{actual}}{Z_{base}}.^[10] A key advantage is that, when S_{base} is held constant across voltage levels, the per-unit values of percentage impedances remain unchanged; for example, a reactance representing 10% of the base impedance in actual ohms equates to 0.1 per unit, regardless of whether the system operates at 120 V or 480 V.^[8] This scaling property simplifies comparative analysis and fault studies in single-phase networks.^[1]

Three-Phase Systems

In three-phase power systems, the per-unit system extends the single-phase foundations by accounting for the relationships between line and phase quantities in balanced configurations. The total three-phase power base S_{\text{base}} is defined as the product of the line-to-line voltage base V_{\text{line base}} and the line current base I_{\text{line base}} multiplied by \sqrt{3}, yielding S_{\text{base}} = \sqrt{3} \, V_{\text{line base}} \, I_{\text{line base}}.^[1] Equivalently, it can be expressed per phase as S_{\text{base}} = 3 \times (V_{\text{phase base}} \times I_{\text{phase base}}), where the phase quantities align with the balanced system's symmetry.^[12] This choice of base ensures that per-unit values remain consistent across phases for balanced operation, facilitating analysis of interconnected equipment like transformers and lines. The line voltage base V_{\text{line base}} is selected as the nominal line-to-line voltage of the system, while the phase voltage base is derived as V_{\text{phase base}} = V_{\text{line base}} / \sqrt{3}.^[1] For currents, the line current base is calculated as I_{\text{base}} = S_{\text{base}} / (\sqrt{3} \, V_{\text{line base}}), which equals the phase current base in a balanced wye-connected system.^[13] These definitions normalize voltages and currents to unity under nominal conditions, simplifying scaling across different voltage levels in multi-machine systems. The impedance base in three-phase per-unit systems is given by Z_{\text{base}} = V_{\text{phase base}}^2 / (S_{\text{base}} / 3), which simplifies to Z_{\text{base}} = V_{\text{line base}}^2 / S_{\text{base}}.^[14] This formulation ensures that per-unit impedances match those from single-phase equivalents when applied to balanced three-phase circuits, as the \sqrt{3} factors in voltage and current cancel out in the impedance ratios.^[1] A key advantage of the per-unit system in three-phase applications arises in symmetrical component analysis for fault studies, where it preserves the per-unit impedances across positive, negative, and zero sequence networks without additional scaling.^[15] This normalization simplifies the connection of sequence networks for unbalanced faults, such as line-to-ground or line-to-line types, enabling straightforward computation of fault currents in per-unit terms that directly translate to actual values using the chosen bases.^[16]

Key Formulas and Relations

Voltage, Current, and Impedance

In the per-unit system, voltage is normalized by dividing the actual voltage by a chosen base voltage, yielding the per-unit voltage V_{pu} = \frac{V_{actual}}{V_{base}}.^[8]^[2] Similarly, the per-unit current is obtained as I_{pu} = \frac{I_{actual}}{I_{base}}, where the base current is typically derived from the base power and base voltage as I_{base} = \frac{S_{base}}{V_{base}} for single-phase systems or adjusted for three-phase equivalents.^[8]^[17] These normalizations express voltages and currents as dimensionless ratios relative to system bases, facilitating comparisons across different voltage levels without dimensional inconsistencies.^[2] Impedance in per-unit form is defined as Z_{pu} = \frac{Z_{actual}}{Z_{base}}, with the base impedance given by Z_{base} = \frac{V_{base}^2}{S_{base}}.^[8]^[17] This relation ensures that Ohm's law preserves its form in per-unit terms: V_{pu} = I_{pu} \cdot Z_{pu}, as the scaling factors cancel out due to the consistent bases.^[8]^[2] For admittance, the per-unit value is Y_{pu} = \frac{Y_{actual}}{Y_{base}}, where Y_{base} = \frac{S_{base}}{V_{base}^2} = \frac{1}{Z_{base}}, thus equivalently Y_{pu} = Y_{actual} \cdot Z_{base}.^[17] This inverse scaling maintains the reciprocity between impedance and admittance in normalized circuits. The per-unit framework ensures that fundamental circuit equations, such as Kirchhoff's voltage and current laws, hold identically without additional scaling factors.^[8]^[17] For instance, the sum of per-unit voltages around a loop equals zero, and the sum of per-unit currents at a node is zero, mirroring the actual system's behavior but in normalized values.^[8] This property simplifies analysis in complex networks by eliminating the need for base conversions during equation setup.^[17] A key advantage of the per-unit system is its consistency across interconnected subsystems, such as those separated by transformers.^[8]^[2] When a uniform base power S_{base} is selected throughout the system, the per-unit impedance remains unchanged even if the base voltage varies, as in voltage step-up or step-down scenarios; the actual impedance scales with the square of the voltage ratio, matching the base impedance scaling.^[17] This invariance allows seamless integration of components operating at different voltage levels into a single per-unit model.^[8]

Power and Admittance

In the per-unit system, apparent power is normalized by dividing the actual apparent power by the chosen base apparent power, yielding S_{\pu} = \frac{S_{\actual}}{S_{\base}}. This normalization preserves the complex relationship between per-unit voltage and current, such that S_{\pu} = V_{\pu} I_{\pu}^* for single-phase systems, where the asterisk denotes the complex conjugate.^[2] For three-phase systems, the expression is analogous but accounts for the three-phase base power, typically defined as the total three-phase VA rating, ensuring consistency in balanced per-phase equivalent circuits.^[18] Active and reactive powers are similarly normalized using the apparent power base, with P_{\pu} = \frac{P_{\actual}}{S_{\base}} and Q_{\pu} = \frac{Q_{\actual}}{S_{\base}}. The magnitude of the per-unit apparent power then follows from the Pythagorean relation |S_{\pu}| = \sqrt{P_{\pu}^2 + Q_{\pu}^2}, which maintains the fundamental power triangle geometry in normalized form.^[2] The power factor, defined as \cos \phi = \frac{P_{\pu}}{|S_{\pu}|}, remains unchanged by the per-unit transformation because it is a dimensionless ratio unaffected by the common scaling of power quantities.^[19] Admittance in per-unit is obtained by Y_{\pu} = G_{\pu} + j B_{\pu} = \frac{Y_{\actual}}{Y_{\base}}, where the admittance base is Y_{\base} = \frac{S_{\base}}{V_{\base}^2}, derived from the reciprocal of the impedance base. This formulation is particularly valuable in load flow analysis, as it allows the construction of the per-unit Y-bus matrix for efficient computation of bus voltages and power injections under steady-state conditions.^[20] Conductance and susceptance components convert accordingly, with G_{\pu} = G_{\actual} \cdot \frac{V_{\base}^2}{S_{\base}} and B_{\pu} = B_{\actual} \cdot \frac{V_{\base}^2}{S_{\base}}; the susceptance normalization is especially useful for modeling shunt elements like capacitor banks, where per-unit reactive power compensation can be directly assessed relative to system bases without dimensional inconsistencies.^[18]

Applications

Transformers

In the per-unit system applied to transformers, a common base apparent power S_{\text{base}} is selected for both the primary and secondary windings, typically based on the transformer's rated capacity. The base voltage for the primary winding V_{\text{base, primary}} is chosen as the nominal primary line-to-line voltage, while the base voltage for the secondary winding V_{\text{base, secondary}} is calculated as V_{\text{base, secondary}} = \frac{V_{\text{base, primary}}}{a}, where a is the nominal turns ratio a = \frac{N_{\text{primary}}}{N_{\text{secondary}}}. This approach ensures that the per-unit voltages align with the transformer's voltage transformation, simplifying analysis across multi-voltage networks without explicit conversion factors.^[21]^[22] Impedance referral in the per-unit system for transformers leverages the scaling properties of base values. The base impedance for each winding is Z_{\text{base}} = \frac{V_{\text{base}}^2}{S_{\text{base}}}, which varies between primary and secondary due to the differing V_{\text{base}} but constant S_{\text{base}}. Consequently, the per-unit leakage reactance X_{\text{pu}} (and impedance Z_{\text{pu}}) remains identical when referred from one side to the other, as the referral factor a^2 for actual impedances matches the ratio of base impedances \frac{Z_{\text{base, primary}}}{Z_{\text{base, secondary}}} = a^2. This invariance facilitates direct combination of impedances in network models.^[21]^[22] The per-unit equivalent circuit of a transformer simplifies the ideal transformer component to a 1:1 ratio, eliminating the need for voltage or current scaling in simulations. Series impedances from the primary and secondary—such as leakage reactance and resistance—are expressed in per-unit on the common base and added directly in the equivalent circuit, while shunt magnetizing and core loss branches are similarly normalized. This representation streamlines power flow and fault studies by treating the transformer as a unified impedance element. For tap changers, which adjust the effective turns ratio for voltage regulation, the model incorporates an off-nominal ratio \alpha = \frac{a_{\text{nominal}}}{a_{\text{actual}}}, altering per-unit voltages on the adjusted side (e.g., V_{\text{pu, secondary}} = V_{\text{pu, primary}} / \alpha) without changing the per-unit impedances.^[21]^[23] The short-circuit impedance of a transformer, expressed in per-unit as \%Z_{\text{pu}}, quantifies its ability to limit fault currents and is defined on the transformer's own base values. It is determined experimentally by short-circuiting the secondary and applying voltage to the primary until rated current flows, yielding \%Z_{\text{pu}} = \left( \frac{V_{\text{applied}}}{V_{\text{rated, primary}}} \right) \times 100\%, which equates to \%Z_{\text{pu}} = \left( \frac{I_{\text{rated}}}{I_{\text{short-circuit}}} \right) \times 100\% since I_{\text{short-circuit, pu}} = 1 / Z_{\text{pu}} at rated voltage. This per-unit value is independent of system-wide base choices when converted appropriately, providing a standardized metric for fault level assessment across different transformer sizes.^[24]

Rotating Machines

In rotating machines, the per-unit system normalizes electrical and mechanical quantities relative to the machine's rated values, facilitating analysis of dynamic behavior in power systems. For synchronous machines, the base apparent power S_{\text{base}} is selected as the machine's rated MVA rating, typically the total three-phase value, while the base voltage V_{\text{base}} is the rated terminal line-to-line voltage. The base angular speed \omega_{\text{base}} is defined as the synchronous speed in electrical radians per second, given by \omega_{\text{base}} = 2\pi f / (p/2), where f is the rated frequency and p is the number of poles. These bases ensure that per-unit quantities remain consistent across different machine sizes and configurations, as outlined in standard normalization practices for AC rotating machinery.^[25]^[26] Reactances in synchronous machines are expressed in per-unit form to capture steady-state, transient, and subtransient behaviors, particularly in stability studies. The synchronous reactance X_{s,\text{pu}} (along the direct axis X_{d,\text{pu}}), transient reactance X'_{d,\text{pu}}, and subtransient reactance X''_{d,\text{pu}} are all referred to a common base, computed as the actual reactance divided by the base impedance Z_{\text{base}} = V_{\text{base}}^2 / S_{\text{base}}. This referral allows these parameters to be directly incorporated into multimachine models without scaling, where typical per-unit values range from 0.1 to 0.3 for subtransient and transient reactances in large generators, highlighting their relative magnitudes to rated conditions. Similarly, quadrature-axis reactances follow the same normalization.^[25]^[26] Mechanical quantities like torque and inertia are also normalized in per-unit terms for dynamic simulations. The per-unit torque is defined as T_{\text{pu}} = T_{\text{actual}} / (S_{\text{base}} / \omega_{\text{base}}), where the base torque represents the torque at rated power and speed, ensuring torque expressions align with electrical power relations. The inertia constant H_{\text{pu}}, measured in seconds, is given by H_{\text{pu}} = \frac{1/2 J \omega_{\text{base}}^2}{S_{\text{base}}}, quantifying the stored kinetic energy relative to the base power; this value remains invariant for similar machine designs regardless of rating, typically ranging from 2 to 10 seconds for synchronous generators.^[25] For induction motors, the per-unit system employs the same stator bases S_{\text{base}} and V_{\text{base}} as for synchronous machines, with \omega_{\text{base}} at synchronous speed. The slip s_{\text{pu}} is directly the actual slip s_{\text{actual}} = ( \omega_{\text{base}} - \omega_r ) / \omega_{\text{base}}, a dimensionless quantity that simplifies torque-speed characteristics. Rotor quantities, such as resistance and reactance, are referred to the stator base using the effective turns ratio a = V_{\text{stator}} / V_{\text{rotor}} (or approximately the square root of the impedance ratio), yielding an equivalent circuit where rotor parameters appear as R_{2,\text{pu}} = R_{2,\text{actual}} / Z_{\text{base}} and similarly for reactance, enabling unified analysis of stator-referred performance.^[27] The primary advantage of per-unit parameters in multimachine systems lies in their invariance to machine ratings, allowing direct comparison and interconnection of synchronous and induction machines in stability and fault studies without conversion factors. This uniformity ensures that reactances, torques, and slips from diverse units yield consistent numerical values—typically clustering around 0.1 to 2.0 for key impedances—reducing computational complexity and error risks in large-scale simulations.^[25]

Numerical Examples

Simple Circuit

Consider a simple single-phase R-L circuit connected to a 10 kV source, consisting of a 5 Ω resistor in series with an inductor having a reactance of 10 Ω. To demonstrate per-unit conversions, select base values of 100 kVA for apparent power and 11 kV for voltage, consistent with single-phase system conventions where the voltage base is the line-to-line nominal value.^[2] The base current is calculated as I_{\text{base}} = \frac{S_{\text{base}}}{V_{\text{base}}} = \frac{100{,}000}{11{,}000} \approx 9.09 A, and the base impedance as Z_{\text{base}} = \frac{V_{\text{base}}^2}{S_{\text{base}}} = \frac{(11{,}000)^2}{100{,}000} = 1{,}210 Ω.^[2] These bases normalize the actual quantities for analysis. The per-unit voltage is V_{\text{pu}} = \frac{10{,}000}{11{,}000} \approx 0.909 pu. The per-unit resistance and reactance are R_{\text{pu}} = \frac{5}{1{,}210} \approx 0.004 pu and X_{\text{pu}} = \frac{10}{1{,}210} \approx 0.008 pu, yielding the per-unit impedance Z_{\text{pu}} = 0.004 + j0.008 pu. The per-unit current is then I_{\text{pu}} = \frac{V_{\text{pu}}}{Z_{\text{pu}}}, with magnitude |I_{\text{pu}}| \approx 98.4 pu.^[2] This conversion simplifies arithmetic, as the per-unit values are dimensionless decimals that avoid handling large actual magnitudes; for instance, the power factor is \cos(\theta) = \frac{R_{\text{pu}}}{|Z_{\text{pu}}|} \approx 0.45 lagging, and real power loss in the resistor is approximately |I_{\text{pu}}|^2 R_{\text{pu}} \approx 40 pu, directly interpretable relative to the base power without unit conversions.^[2]

Transformer Network

In a typical application of the per-unit system to transformer networks, consider a three-phase step-down transformer rated at 100 MVA with a primary voltage of 220 kV and a secondary voltage of 11 kV, exhibiting an impedance of 0.1 pu (10%) on its rated base.^[28] The transformer supplies a load of 50 MW at a power factor of 0.8 lagging on the secondary side. This scenario illustrates how the per-unit method handles multi-voltage levels by maintaining consistent base values across the system, facilitating analysis of voltage drops and power flows without repeated conversions.^[29] To apply per-unit values, the system base power is selected as S_{\text{base}} = 100 MVA, matching the transformer rating for direct compatibility. The base voltages are set to the nominal ratings: V_{\text{base, primary}} = 220 kV and V_{\text{base, secondary}} = 11 kV. The corresponding base impedances are calculated as Z_{\text{base, primary}} = \frac{V_{\text{base, primary}}^2}{S_{\text{base}}} = \frac{(220)^2}{100} = 484 \, \Omega on the primary side and Z_{\text{base, secondary}} = \frac{(11)^2}{100} = 1.21 \, \Omega on the secondary side. These bases ensure that the transformer's turns ratio aligns with the voltage base ratio, allowing impedances and other quantities to be referred seamlessly between sides without altering their per-unit magnitudes in an ideal transformer.^[28]^[30] The load apparent power is S_{\text{load}} = \frac{50}{0.8} = 62.5 MVA, yielding S_{\text{load, pu}} = \frac{62.5}{100} = 0.625 pu on the secondary side, assuming nominal voltage. The per-unit load current is thus I_{\text{pu}} = 0.625 pu (since S_{\text{pu}} = V_{\text{pu}} \cdot I_{\text{pu}} and V_{\text{pu}} \approx 1 at nominal conditions). To analyze the primary side, the load is referred through the transformer using the turns ratio a = \frac{220}{11} = 20; however, in per-unit terms, the referred load power remains 0.625 pu, and the per-unit current on the primary is also 0.625 pu due to the scaling of current bases by the same ratio. The transformer impedance of 0.1 pu is directly applicable in this normalized framework.^[29] Voltage regulation across the transformer is approximated as \text{VR}_{\text{pu}} = I_{\text{pu}} \cdot Z_{\text{pu}} = 0.625 \cdot 0.1 = 0.0625 pu, representing the per-unit voltage drop from no-load to full-load conditions. This results in a secondary terminal voltage of approximately V_{\text{sec, pu}} = 1 - 0.0625 = 0.9375 pu under load, highlighting the voltage sag due to the impedance. For efficiency, the input power is the output power plus losses; assuming negligible resistance and core losses proportional to rated conditions, the efficiency is roughly \eta \approx \frac{50}{50 + (0.625)^2 \cdot P_{\text{cu, rated}}}, where copper losses at rated load are typically 1-2% of rated power, yielding \eta \approx 98\% here—demonstrating how per-unit scaling reveals the benefits of base consistency in quantifying performance without side-specific conversions.^[30]^[29]

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### Summary of Per-Unit System for Transformers from Lesson 10_et332b.pdf