## Introduction

This article describes the most common overhead line (OHL) models used in power systems analysis. By line models, we mean equivalent circuits composed of elementary series and shunt impedances (i.e. resistances, inductances, capacitances and conductances). The calculation of these impedances (i.e. based on geometrical and material considerations) is covered elsewhere in the section on Overhead Line Constants.

So why even bother with having a range line models - isn't one enough? The problem is that there are several characteristics of transmission lines that complicates the modelling of lines, for instance:

• The line impedances are not lumped, but distributed continuously over the length of the line. The interaction of inductances and capacitances is therefore different when considering a line as having lumped parameters versus distributed parameters.
• The geometry of overhead line conductors / sub-bundles (phase and earth) are not always symmetrical and therefore, a single-phase, single conductor equivalent circuit is not always appropriate. In such cases, multi-conductor models may be required.
• For transient studies (especially those covering higher frequencies), the line parameters can have significant frequency dependencies (in particular the series resistance and inductance). For example, the skin effect leads to increased resistances at higher AC frequencies. This necessitates the use of frequency-dependent line models.

The factors listed above imply that the most accurate line model would have distributed parameters, be fully multi-conductor to allow for asymmetrical geometries and imperfect line transpositions, and capture the frequency dependencies of the line. Such line models exist, but they are relatively complicated and often, much simpler alternatives can be used depending on the application.

## Taxonomy of Overhead Line Models

Line Model Applications Properties Remarks
Steady-State (Frequency Domain) Models Single-Phase Models Lossless (L) Line
• Hand calculations
• Teaching purposes
• Very short lines
• Lumped parameters
• Lossless
Lossless (LC) Line
• Rarely used
• Lumped parameters
• Lossless
RL Line
• Low voltage networks
• Short lines (<10km)
• Lumped parameters
• Includes losses
• Shunt capacitance not included
Nominal $\pi$ Line
• Balanced systems
• Medium length transposed lines (<250km)
• Preliminary EMT studies on short lines
• Lumped parameters
• Includes losses
• A "T" circuit is also sometimes used instead of a "$\pi$ " circuit
Distributed Parameter Line
• Balanced systems
• Long transposed lines
• Distributed parameters
• Includes losses
Multi-Conductor Models Nominal $\pi$ Line
• Unbalanced systems
• Medium length untransposed lines (<250km)
• EMT studies on short lines
• Lumped parameters
• Includes losses
• Kron reduced models are typically used to reduce impedances and admittances to 3 x 3 matrices
Cascaded Nominal $\pi$ Line
• Unbalanced systems
• Longer untransposed lines
• EMT studies on medium length lines
• Lumped parameters
• Includes losses
• For EMT studies, cascaded $\pi$ lines are computationally more expensive than travelling wave models
Distributed Parameter Line
• Unbalanced systems
• Long untransposed lines
• Distributed parameters
• Includes losses
• Modal transformation
• Cannot be used directly in EMT studies since propagation constant and characteristic impedance are complex and frequency dependent
Travelling Wave (Time Domain) Models Single-Phase Models Bergeron Line Model
• Preliminary EMT studies
• Not frequency dependent
• Losses are captured as lumped resistances at the middle and ends of the line
• Single-phase model does not capture full three-phase nature of the line, e.g. inter-phase coupling and unbalances
Frequency Dependent Line Model
• Rarely used
• Frequency dependent
• Single-phase model does not capture full three-phase nature of the line, e.g. inter-phase coupling and unbalances
Multi-Conductor Models Bergeron Line Model
• EMT studies with slow-medium transient frequencies
• Not frequency dependent
• Losses are captured as lumped resistances at the middle and ends of the line
• Model is initialised with a representative frequency / travel time of the transient under investigation
Frequency Dependent Line Model
• EMT studies where frequency dependence is significant (e.g. very fast transients)
• Frequency dependent
• Can be a modal or phase-domain model
• Modal formulations can have frequency-dependent transformation matrices (e.g. L. Marti model)

## Selecting an Appropriate Line Model

Key considerations for selecting a suitable line model:

• Application: what kind of analysis is being done? e.g. power flow, stability, EMT, etc. For static studies such as power flows, simple fault studies, etc, then steady-state (frequency domain) models are appropriate. Steady-state models are also suitable for dynamic simulations that are only concerned with electro-mechanical transients (e.g. transient stability and motor starting) rather than electromagnetic transients (e.g. switching and lightning studies).
• Line Length: simpler models can be used for shorter lines
• Load unbalance and Transpositions: multi-conductor line models should be used for longer untransposed lines and systems with significant imbalances
• Level of detail: is it a preliminary or a detailed study?
• Accuracy of input data: what kind of input data is available?