Power-System Modeling and Per-Unit Analysis

Introduction

Electrical power systems are too large and interconnected to analyze conveniently as physical drawings containing every conductor, winding, switch, and protection device. Laboratory and engineering studies therefore replace the physical system with mathematical models. A useful model preserves the electrical behavior relevant to the study while omitting unnecessary detail.

This chapter develops the main modeling tools used in laboratory power-system analysis: single-line diagrams, per-unit representation, transformer and line models, sequence networks, and simulation validation. The central engineering practice is to compare three forms of evidence:

1. calculated results from equations,
2. simulated results from a software model, and
3. measured results from laboratory equipment.

Agreement among these results increases confidence in the model. Differences must be interpreted rather than simply discarded, because they may reveal parameter errors, instrument limitations, incorrect connections, or neglected physical effects.

Learning Objectives

After studying this chapter, the learner should be able to:

• construct and interpret a single-line diagram;
• select appropriate equivalent models for generators, transformers, transmission lines, loads, and faults;
• convert electrical quantities to a consistent per-unit base;
• calculate fault currents and voltage changes using per-unit quantities;
• explain positive-, negative-, and zero-sequence networks;
• implement a basic power-system model in a simulation environment;
• compare analytical, simulated, and measured results;
• identify modeling assumptions, limitations, and sources of error.

1. Representing a Power System

A single-line diagram represents a balanced three-phase system with one line for the three phases. It shows the topology and principal equipment, but normally does not show every phase conductor. Busbars are identified by numbers or names, and generators, transformers, lines, loads, breakers, and measuring points are connected between buses.

A single-line diagram should communicate:

• electrical connectivity;
• nominal voltage levels;
• equipment ratings;
• transformer ratios and connections;
• locations of circuit breakers and measurement points;
• the operating or fault condition being studied.

For a balanced steady-state study, the three-phase system can be represented by a single-phase equivalent. This is valid when the phase system is symmetrical and the loads and equipment are adequately balanced. Fault studies involving unsymmetrical faults require sequence components or an explicit three-phase model.

Figure: A single-line representation of a generator, transformer, transmission line, and load.

Common diagram conventions

A generator is commonly shown as a circle marked G, a transformer as coupled windings, a line as an impedance or a transmission-line section, and a load as an impedance or a power-consumption symbol. The diagram should state whether impedances are in ohms, per unit, or percentage form. A diagram without voltage bases and equipment ratings is incomplete for quantitative analysis.

2. Equivalent Circuit Models

2.1 Generator model

For a preliminary balanced short-circuit study, a synchronous generator may be represented by an internal voltage behind a subtransient reactance:

Here E'' is the internal subtransient voltage, V is the terminal voltage, I is current, and X''_d is the direct-axis subtransient reactance. Resistance is often neglected in short-circuit calculations because reactance is much larger, but resistance must be retained when losses, damping, or accurate voltage drops are important.

The appropriate reactance depends on the time scale. Subtransient reactance is used immediately after a fault, transient reactance during the following interval, and synchronous reactance for longer-term steady-state analysis. Using the wrong reactance can produce a large error in calculated fault current.

2.2 Transformer model

A practical transformer model includes series leakage impedance and a shunt magnetizing branch. For many short-circuit studies, the magnetizing branch is neglected and the transformer is represented by its leakage impedance. The series impedance is

where R_T represents copper-loss resistance and X_T represents leakage reactance. The ideal transformer ratio transfers voltage, current, and impedance between voltage levels. The impedance-reflection rule is

where a = V₁/V₂ is the turns or voltage ratio, assuming an ideal transformer and consistent phase quantities.

2.3 Transmission-line models

For a short line, shunt capacitance may be neglected and the series impedance is sufficient. A medium line is commonly represented by the nominal-π model, containing one series impedance and two shunt admittances, each equal to half the total line charging admittance. Long lines require distributed-parameter equations or a suitable equivalent model.

The series impedance of a line is

where resistance is measured in ohms and reactance is inductive reactance in ohms. The corresponding shunt admittance is

where G_L is conductance and B_L is susceptance in siemens. In an overhead line, capacitance can become important at higher voltage and longer distance.

2.4 Load models

A load can be represented as constant impedance, constant current, constant power, or a composite model. These models are not interchangeable. For a constant impedance load,

where V is RMS voltage and S^* is the complex conjugate of apparent power. For a balanced three-phase system, V is the line-to-line RMS voltage when the corresponding three-phase formula is used:

The units are ohms when voltage is in volts and apparent power is in volt-amperes. A constant-power load changes its current as voltage changes and is therefore more realistic for some electronic and controlled loads, but it introduces nonlinear behavior.

3. Per-Unit Analysis

Per-unit analysis expresses a quantity as the ratio of its actual value to a selected base value:

The quantity is dimensionless. Per-unit notation is particularly useful because transformer impedance conversion becomes simpler and quantities at different voltage levels can be compared consistently.

For a three-phase system, choose a common apparent-power base S_base and a voltage base V_LL,base. The associated current and impedance bases are

The voltage in the first equation is line-to-line RMS voltage, power is three-phase apparent power in volt-amperes, current is line current in amperes, and impedance is ohms. If phase voltage and per-phase apparent power are used instead, the corresponding equations must be applied consistently.

A voltage base at one bus determines voltage bases throughout a transformer-connected system. For an ideal transformer,

provided the selected base follows the transformer rated voltage ratio. The same power base is normally used on all buses.

Changing a per-unit impedance base

If an impedance is given as Z_pu,old on base S_old, V_old, and a new base is selected, then

This formula assumes that the voltage quantities refer to the same side of the transformer and use the same line-to-line or phase convention. If voltage bases are selected in the transformer ratio, the voltage factor is often one.

Worked Example 1: Base quantities and impedance conversion

A 50 MVA, 132 kV three-phase system uses a common base. Calculate the current and impedance bases. Then convert a line impedance of 20 Ω to per unit.

Using the three-phase equations,

I_base = 50×10⁶/(√3×132×10³) = 218.7 A

and

Z_base = (132×10³)²/(50×10⁶) = 348.48 Ω.

Therefore,

Z_pu = 20/348.48 = 0.0574 pu.

The calculation is dimensionally consistent: amperes result from VA divided by volts, and ohms result from volts squared divided by VA.

Worked Example 2: Converting transformer impedance

A transformer is rated 25 MVA, 11/132 kV and has a leakage reactance of 8% on its own rating. Find its reactance on a 100 MVA system base, using voltage bases of 11 kV and 132 kV.

Because the voltage bases follow the transformer ratio, the voltage conversion factor is one. Thus,

X_pu,new = 0.08(100/25) = 0.32 pu.

The result is 0.32 pu on the 100 MVA base. This illustrates why a common MVA base is essential when combining equipment data.

4. Balanced Fault Analysis

For a three-phase fault through impedance Z_f, the fault current in per unit is

where E_th is the Thevenin voltage at the fault location and Z_th is the Thevenin impedance seen from that location. For a bolted fault, Z_f = 0. If the prefault voltage is approximately 1.0 pu,

The actual current is obtained by multiplying the per-unit current by the current base at the fault bus. This approximation is suitable for preliminary studies, but generator operating state, prefault voltage, fault resistance, motor contribution, and network configuration may need to be included in a detailed study.

Worked Example 3: Three-phase fault

Suppose a network has a Thevenin reactance of j0.25 pu at a 13.8 kV, 50 MVA bus. For a bolted fault and prefault voltage of 1.0 pu,

I_f = 1/(j0.25) = -j4.0 pu.

The magnitude is 4.0 pu. The base current is

I_base = 50×10⁶/(√3×13.8×10³) = 2091.8 A.

Therefore the fault-current magnitude is

[[MATH{|I_f| = 4×2091.8 = 8367 A ≈ 8.37 kA]].

The negative-j sign indicates the phase angle relative to the selected voltage reference; the magnitude is used for interrupting-current estimation.

5. Symmetrical Components and Sequence Networks

An unsymmetrical three-phase set can be resolved into three balanced sets called positive-, negative-, and zero-sequence components. Positive-sequence currents have equal magnitudes, are separated by 120°, and have the normal phase order. Negative-sequence currents also have 120° separation but the reverse phase order. Zero-sequence currents are equal in magnitude and phase.

If a = 1∠120°, the phase currents are related to sequence currents by

where I₀, I₁, and I₂ are zero-, positive-, and negative-sequence currents. The inverse transformation is

The matrix notation is a compact representation; each element is a phasor and all quantities must use the same RMS convention.

Sequence networks are formed by replacing each component with its positive-, negative-, and zero-sequence impedance. Generator positive-sequence impedance is normally the relevant internal reactance for a balanced fault. Negative-sequence impedance is usually different because reverse-rotating fields induce additional rotor currents. Zero-sequence current depends strongly on transformer winding connections, grounding impedance, and generator neutral grounding.

Fault-network connections

• A three-phase fault uses only the positive-sequence network.
• A single-line-to-ground fault connects positive-, negative-, and zero-sequence networks in series.
• A line-to-line fault uses positive- and negative-sequence networks, with no zero-sequence current when the ground is not involved.
• A double-line-to-ground fault uses all three sequence networks in a configuration determined by the fault conditions.

The exact interconnection must be derived from the boundary conditions at the fault. Memorizing a connection without checking those conditions is unsafe, particularly when fault impedance is present.

6. Simulation and Laboratory Implementation

A robust simulation workflow is as important as the equations. The following sequence is recommended:

1. Define the study: load flow, voltage drop, fault current, transient response, or another objective.
2. Draw the single-line diagram and label buses, ratings, phase connections, and measurement points.
3. Select a common MVA base and bus voltage bases.
4. Enter component data and convert all parameters to a consistent system.
5. Reduce or assemble the network, checking transformer orientation and grounding.
6. Run the simulation using realistic initial conditions and solver settings.
7. Independently calculate at least one expected result.
8. Measure the corresponding laboratory quantity using calibrated instruments.
9. Compare values, phasor angles, waveforms, and percentage differences.
10. Investigate discrepancies before changing parameters.

For a measured quantity x_m and calculated quantity x_c, a simple percentage difference relative to the calculated value is

This is not an uncertainty estimate. It only quantifies the difference between two values. If a reference or accepted value is available, percentage error should be defined relative to that reference.

Measurement practice

Voltage transformers and current transformers introduce ratio and phase errors. Digital meters may report RMS values over a bandwidth that differs from the simulation. Oscilloscopes require correct probes, grounding, sampling rate, and scaling. A current probe must not be placed around multiple conductors whose currents cancel unless that arrangement is intentional. Safety procedures require isolation, correct fusing, discharge of stored energy, and verification of absence of voltage before modifying a circuit.

7. Illustrative Case: Laboratory Fault-Level Validation

Background

Consider an illustrative 50 MVA system containing a 13.8 kV generator, a 13.8/132 kV transformer, a short transmission line, and a 132 kV receiving bus. The purpose is to estimate and validate the three-phase fault current at the receiving bus.

Problem

The analytical network reduction gives a Thevenin reactance of 0.25 pu on a 50 MVA base. The prefault voltage is assumed to be 1.0 pu. A simulation is constructed using the same topology and component parameters. A laboratory scaled model is then operated at a safe low voltage, and its current is measured through an appropriate current transformer.

Analysis and approach

The analytical result is 4.0 pu fault current. At 132 kV on the 50 MVA base, the base current is

I_base = 50×10⁶/(√3×132×10³) = 218.7 A.

The corresponding full-scale current is approximately 4×218.7 = 874.8 A. The laboratory current is not expected to equal this value directly if the apparatus is a scaled model; comparison must be made in per unit or by using the laboratory base quantities.

Suppose the simulation reports 3.92 pu and the measured result is 3.80 pu. Relative to the analytical value, the differences are 2.0% and 5.0%, respectively. These values might be acceptable for an instructional model, but acceptance criteria must be stated before the experiment.

Interpretation

The simulation and measurement are lower than the ideal analytical result. Possible causes include generator resistance, transformer losses, nonzero fault impedance, instrument burden, conductor resistance, voltage sag in the source, and uncertainty in the actual reactance. If the measured result were substantially higher, incorrect voltage-base conversion, a shorted transformer section, or an incorrectly scaled current transformer would be investigated.

Learning points

The case demonstrates that per-unit comparison allows systems of different physical ratings to be compared. It also shows that agreement is meaningful only when topology, bases, parameter definitions, and measurement locations are identical. A simulation is not automatically a truth source; it is an implementation of assumptions.

8. Limitations, Professional Practice, and Contemporary Tools

No model is universally correct. A constant-impedance load is convenient for a voltage-drop calculation but may not represent induction motors or power-electronic converters. Neglecting resistance simplifies fault calculations but removes real power losses and changes phase angles. A positive-sequence model is efficient for balanced studies but cannot represent ground-current paths in an earth fault.

Professional studies should document data sources, assumptions, base values, transformer winding connections, neutral grounding, frequency, temperature, and software version. Results should be traceable and reproducible. Protection and equipment ratings must follow applicable electrical safety standards and utility or regulatory requirements; laboratory models must not be used to justify field equipment ratings without an appropriate validated study.

Modern software can automate network assembly, power-flow solution, short-circuit analysis, and waveform visualization. Artificial-intelligence tools may assist with parameter checking, anomaly detection, or report generation, but they do not replace engineering judgment. Any automatically generated model or conclusion must be checked against conservation laws, dimensional consistency, manufacturer data, and independent calculations. Sensitive grid data should also be handled according to cybersecurity and organizational governance requirements.

Best-Practice Checklist

• Draw the single-line diagram before entering software data.
• State whether voltages are line-to-line or phase-to-neutral.
• Use one clearly identified MVA base and propagate voltage bases correctly.
• Convert every impedance to the same base before combining it.
• Check transformer winding connections and grounding paths.
• Use the reactance appropriate to the time scale of the event.
• Compare analytical, simulated, and measured quantities at the same location.
• Report magnitude and angle when phasors are relevant.
• Distinguish model error from measurement uncertainty.
• Preserve safety, data integrity, and result traceability.

Chapter Summary

Power-system modeling converts a physical network into an analyzable representation. Single-line diagrams expose topology, while equivalent circuits represent generators, transformers, lines, and loads at the level required by the study. Per-unit analysis normalizes quantities, simplifies transformer calculations, and permits meaningful comparison across voltage levels and equipment ratings. Three-phase fault current can be found from a Thevenin equivalent, but the selected generator reactance, prefault voltage, and fault impedance must match the study objective.

Symmetrical components separate unbalanced conditions into positive-, negative-, and zero-sequence systems. Their networks provide a systematic method for analyzing unsymmetrical faults and reveal the importance of grounding and transformer connections. Simulation should be supported by independent calculations and laboratory measurements. Differences among results are valuable evidence about assumptions, parameter quality, instrumentation, and implementation. Sound engineering practice therefore combines mathematical correctness, model awareness, safe experimentation, and transparent validation.

Key Terms

Per-unit quantity: A normalized quantity equal to actual value divided by its selected base value.

Base apparent power: The common MVA or VA reference selected for a per-unit system.

Voltage base: The reference voltage used to calculate base current and impedance.

Single-line diagram: A symbolic representation of a three-phase power system using one line.

Bus: An electrical node at which equipment and circuits are interconnected.

Thevenin equivalent: An equivalent voltage source and impedance representing a network at two terminals.

Subtransient reactance: Generator reactance governing the earliest interval after a disturbance.

Transient reactance: Generator reactance used during the intermediate period after a disturbance.

Synchronous reactance: Generator reactance used for longer-term steady-state analysis.

Leakage reactance: Transformer reactance representing magnetic flux that does not link both windings.

Nominal-π model: A medium-line model with a series impedance and two shunt admittances.

Constant-power load: A load model maintaining specified active and reactive power as voltage changes.

Symmetrical components: Positive-, negative-, and zero-sequence balanced sets used to represent unbalanced phasors.

Positive sequence: Components with the normal phase rotation.

Negative sequence: Components with reverse phase rotation.

Zero sequence: Three equal phasors having the same phase angle.

Sequence network: A network model containing component impedances for one sequence system.

Fault impedance: Impedance between a fault point and the conductors or ground involved.

Percentage difference: A numerical comparison between two values relative to a selected reference.

Model validation: Comparison of model predictions with independent calculations or measurements.

Review Questions

Short Answer Questions

1. What information should be included on a power-system single-line diagram?
2. Why is a common MVA base used in per-unit analysis?
3. Distinguish subtransient, transient, and synchronous reactance.
4. What physical conditions determine the zero-sequence network?
5. Why must measured and simulated quantities be compared at the same electrical location?

Long Answer Questions

1. Explain the construction and use of a per-unit system for a transformer-connected network, including base current and impedance equations.
2. Describe suitable equivalent models for a generator, transformer, transmission line, and load, and state when each model may be inadequate.
3. Derive the procedure for calculating a three-phase fault current using a Thevenin equivalent.
4. Explain symmetrical components and the sequence-network arrangements for common fault types.
5. Develop a complete laboratory workflow for validating a simulated power-system model against analytical and measured results.

Analytical / Application Questions

1. A 100 MVA, 230 kV bus has a Thevenin reactance of 0.20 pu and a prefault voltage of 1.02 pu. Calculate the bolted three-phase fault current in per unit and amperes.
2. A transformer has 10% reactance on a 20 MVA base. Determine its per-unit reactance on a 50 MVA base when voltage bases follow the transformer ratio. Explain why the result changes.
3. A measured fault current is 3.76 pu while the analytical result is 4.00 pu. Calculate the percentage difference and identify at least three technically plausible causes of the discrepancy.

Multiple Choice Questions with Answers

Question 1

Which quantity is dimensionless in a per-unit system?

A. Base current

B. Per-unit impedance

C. Physical resistance

D. Line-to-line voltage

Correct Answer: B. Per-unit impedance

Question 2

For a three-phase system, the current base is calculated from which expression?

A. I_base = S_base/V_base

B. I_base = V_base/S_base

C. I_base = S_base/(√3 V_LL,base)

D. I_base = √3 S_base/V_LL,base²

Correct Answer: C. I_base = S_base/(√3 V_LL,base)

Question 3

A transformer has 8% reactance on a 25 MVA base. What is its reactance on a 100 MVA base when voltage bases follow the transformer ratio?

A. 0.02 pu

B. 0.08 pu

C. 0.16 pu

D. 0.32 pu

Correct Answer: D. 0.32 pu

Question 4

Which sequence network is used alone for an ideal balanced three-phase fault?

A. Positive-sequence network

B. Negative-sequence network

C. Zero-sequence network

D. Positive- and zero-sequence networks in parallel

Correct Answer: A. Positive-sequence network

Question 5

What is the principal role of a single-line diagram?

A. To display every physical conductor and insulation layer

B. To represent three-phase system topology with one line

C. To replace all laboratory measurements

D. To show only mechanical equipment

Correct Answer: B. To represent three-phase system topology with one line

Question 6

If the Thevenin reactance at a bus is 0.25 pu and prefault voltage is 1.0 pu, what is the magnitude of bolted three-phase fault current?

A. 0.25 pu

B. 1.25 pu

C. 2.5 pu

D. 4.0 pu

Correct Answer: D. 4.0 pu

Question 7

Which condition most strongly affects zero-sequence current?

A. Only the positive-sequence generator voltage

B. Transformer connections and grounding

C. The color of the busbar insulation

D. The software screen resolution

Correct Answer: B. Transformer connections and grounding

Question 8

Which model maintains specified active and reactive power as voltage changes?

A. Constant-impedance model

B. Constant-current model

C. Constant-power model

D. Open-circuit model

Correct Answer: C. Constant-power model

Question 9

A 20 Ω line impedance is connected to a 50 MVA, 132 kV base. If Z_base = 348.48 Ω, what is the per-unit impedance?

A. 0.0057 pu

B. 0.0574 pu

C. 5.74 pu

D. 17.42 pu

Correct Answer: B. 0.0574 pu

Question 10

Which practice best supports model validation?

A. Changing parameters until the result looks reasonable

B. Comparing results at different buses without conversion

C. Comparing independent calculations, simulation, and measurements using consistent bases

D. Ignoring instrument uncertainty

Correct Answer: C. Comparing independent calculations, simulation, and measurements using consistent bases