Harmonic Studies & Modelling

Cable Modelling for Harmonic Studies

Cables carry much more shunt capacitance than overhead lines, so they tend to pull network resonance down into the low-order harmonics that matter most. This page covers the choice between nominal-π, cascaded and distributed-parameter cable models; the influence of cable length, design and laying formation; sheath bonding and cross-bonding; skin and proximity effects; losses and damping; zero-sequence behaviour; and how to handle the uncertainty that comes with planning-stage cable data.

Reading time ≈ 23 min · Part Three of the series

Cables are critical components in harmonic studies because their electrical characteristics can strongly influence the harmonic impedance of the network. Compared with overhead lines, cables normally have lower series impedance and higher shunt capacitance. This means cables are more likely to create resonance at lower harmonic frequencies — often the most important frequencies, because damping is usually lower at the low end of the spectrum and many harmonic sources inject significant current at low-order harmonics such as \(h=5,\ 7,\ 11,\ 13\).

For this reason, cable modelling must be carried out carefully when assessing harmonic distortion, resonance, filter performance or harmonic compliance. The key practical point is simple: cable capacitance can shift network resonance into important harmonic orders.

Key idea

Cables add a lot of shunt capacitance, pulling resonance toward low-order harmonics where damping is low.
A single nominal-π section captures roughly one resonance; long cables need cascaded or distributed models.
Length, layout, sheath bonding and loss modelling usually dominate the harmonic result.
Planning-stage cable data is uncertain — use sensitivity studies rather than a single case.

Key terms used on this page

01Series impedance, \(Z\)

The impedance along the cable conductor and return path.

02Shunt admittance, \(Y\)

The admittance between conductor and screen/sheath/earth, mainly due to cable capacitance.

03Cable capacitance

The capacitance formed by the conductor, insulation, screen/sheath and surrounding return path.

04Sheath or screen

The metallic layer around the insulation, used for fault-current return, electric-field control and bonding.

05Sheath bonding

How the cable sheath is connected to earth — single-point, both-end or cross-bonding.

06Cross-bonding

Sectional bonding of cable sheaths to reduce sheath circulating current.

07Skin effect

The tendency of high-frequency current to flow near the surface of a conductor.

08Proximity effect

The change in current distribution caused by nearby conductors carrying current.

09Sequence impedance

Positive-, negative- and zero-sequence impedance used to represent balanced and unbalanced behaviour.

10Distributed-parameter model

A cable model where impedance and admittance are represented along the cable length rather than as one lumped element.

Section 1

Why cables matter for harmonics

A cable behaves differently from an overhead line. It has a conductor, insulation layers, a screen or sheath, possible armour, a bonding arrangement, and a surrounding medium such as soil, duct, concrete or seabed. These physical features affect the series impedance, shunt admittance, losses, damping, resonance frequency, zero-sequence behaviour and phase coupling.

Cables are often more important than overhead lines in harmonic resonance studies because their capacitance is much higher. That capacitance interacts with transformer and system inductance to create resonance at relatively low harmonic orders. So even if a cable is short from a load-flow point of view, it may still have a strong influence on the harmonic impedance of the network.

The harmonic voltage at a bus is still governed by the same relationship, and because cable systems can significantly change \(Z_h\) they can have a major effect on the calculated distortion:

\[ V_h=Z_h\,I_h \]

\(V_h\): harmonic voltage at harmonic order \(h\)
\(Z_h\): harmonic impedance seen at harmonic order \(h\)
\(I_h\): harmonic current injected at harmonic order \(h\)

This equation shows why cable modelling matters. A cable can significantly change \(Z_h\), especially near resonance — so if \(Z_h\) is calculated incorrectly, the harmonic voltage distortion will be incorrect too.

Table 1 — Network conditions where cable modelling is especially important.

Network Condition	Harmonic Relevance
Long underground cables	High capacitance and low damping
Submarine export cables	Strong influence on low-frequency resonance
Offshore wind connections	Long cable networks and converter harmonics
Cable-rich urban networks	Higher shunt capacitance
Cross-bonded cable systems	Phase and sequence coupling
Closely spaced single-core cables	Proximity effects
Multi-core cables	Core-core and core-armour interaction
Strong grid connection	Resonance peaks can be sharp
Passive filters or capacitor banks	Additional tuned paths

A poor cable model can therefore lead to an incorrect prediction of harmonic amplification or damping.

Section 2

Cable models used in harmonic studies

Cable systems can be represented using the same general modelling families as overhead lines — a nominal-π model, cascaded nominal-π sections, or an equivalent-π distributed-parameter model. The main difference between overhead lines and cables is not the final circuit form, but how the series impedance and shunt admittance are calculated.

For cables, the calculation must consider the conductor geometry, insulation thickness, screen or sheath, bonding arrangement, armour or pipe, laying formation, soil or seabed return path, and skin and proximity effects. The cable model contributes to the harmonic admittance matrix:

\[ [Y_h][V_h]=[I_h] \]

\([Y_h]\): network admittance matrix at harmonic order \(h\)
\([V_h]\): vector of bus harmonic voltages at harmonic order \(h\)
\([I_h]\): vector of harmonic current injections at harmonic order \(h\)
\(h\): harmonic order

The cable model contributes to \([Y_h]\) through its frequency-dependent series impedance and shunt admittance.

If the cable model does not represent the frequency-dependent impedance and admittance correctly, the calculated harmonic voltages and currents may be inaccurate.

The three model families at a glance

Nominal-π model — represents the whole cable as one series impedance with half of the shunt admittance placed at each end.
Cascaded nominal-π model — divides the cable into several shorter π sections to better represent the voltage and current variation along the cable.
Distributed-parameter model — represents the cable parameters continuously along the length, and is more suitable for long cables and higher harmonic frequencies.

Section 3

Nominal-π cable model

The nominal-π model represents the cable using lumped series impedance and shunt admittance. It is simple and computationally efficient, but for harmonic studies its use is limited: a single nominal-π section can normally represent only one resonance frequency, and cannot accurately represent the distributed nature of long cables or multiple resonance points.

A nominal-π model may be acceptable for short cables, preliminary screening and low-frequency approximate studies, but it should be used with caution for long cables, strong grid connections, filter design, compliance studies and resonance-sensitive networks.

When is a single nominal-π model acceptable?

A single nominal-π cable model may be acceptable for short cable connections, low harmonic orders and early screening. It should not normally be relied on for long cables, offshore export cables, resonance studies, filter design or compliance studies where the exact resonance frequency and peak impedance matter. It may show the approximate resonance region, but it may not give reliable harmonic levels.

Section 4

Cascaded nominal-π cable sections

A better approximation is obtained by splitting the cable into multiple nominal-π sections — one long cable represented as several shorter sections. For example, a 20 km cable may be modelled as 10 × 2 km sections. As the number of cascaded sections increases, the model better represents the distributed nature of the cable: the advantage is improved resonance representation, the disadvantage a larger model with more intermediate nodes.

The number of sections required increases with frequency. A model that is adequate up to the 5th harmonic may not be adequate up to the 50th. The section length should therefore be selected based on the highest harmonic order of interest and the required accuracy.

Section 5

Distributed-parameter cable model

The distributed-parameter model is normally the preferred option for harmonic studies, especially for cables longer than a few kilometres. It represents the cable parameters as continuously distributed along the length, and can capture long-cable effects, multiple resonance points, wave propagation, frequency-dependent impedance and admittance, and phase and sheath interactions.

Practical recommendation

Use distributed-parameter (equivalent-π) cable models for detailed harmonic studies — especially when the cable length is greater than about 2–5 km, or when resonance, filter performance or compliance margins are important.

Section 6

Cable length

Cable length is usually the most important single parameter affecting harmonic resonance. Increasing length increases the total shunt capacitance and changes the electrical length of the circuit, which shifts resonances to lower frequencies — a longer cable gives a lower resonance frequency, and often more resonance points within the range of interest. Length affects both the positive-sequence and the zero-sequence harmonic impedance.

This matters during planning, when the final route may not be fixed. A route-length change of even a few percent can move a resonance closer to or further from a characteristic harmonic order. If the resonance shifts close to the 5th harmonic, a converter or nonlinear load injecting 5th-harmonic current may produce much higher voltage distortion.

Practical recommendation

Perform sensitivity studies for cable length, especially during planning stages where the final route is uncertain.

Section 7

Cable design and layout

Cable design characteristics

Cable design affects harmonic impedance because it changes the capacitance, resistance, inductance and losses of the cable.

Table 2 — Cable design parameters and their harmonic relevance.

Cable Design Parameter	Harmonic Relevance
Conductor radius	Affects resistance, inductance and capacitance
Conductor material	Copper and aluminium have different resistance
Insulation thickness	Strong effect on capacitance
Screen or sheath dimensions	Affects return current and losses
Armour construction	Affects impedance and proximity effects
Layer thicknesses	Affects capacitance and losses
Material properties	Affects dielectric and conductor losses

The conductor radius can influence resonance frequency, particularly in the positive sequence. Insulation thickness affects capacitance, and because capacitance is central to resonance behaviour it influences both resonance frequency and magnitude. For a cable capacitance \(Y_C=j\omega C\), the shunt admittance increases with harmonic order:

\[ Y_C(h)=j\,h\,\omega_1 C \;=\; h\,Y_C(1) \]

\(Y_C(h)\): cable capacitive admittance at harmonic order \(h\)
\(j\): imaginary operator
\(h\): harmonic order
\(\omega_1\): fundamental angular frequency, equal to \(2\pi f_1\)
\(f_1\): fundamental frequency, normally 50 Hz in the UK and Ireland
\(C\): cable capacitance

As the harmonic order increases, the capacitive admittance increases. This is one reason why long cable circuits can strongly affect harmonic resonance.

This is why accurate cable capacitance is important in harmonic studies.

Cable layout

Cable layout has a significant effect on harmonic impedance. Single-core cables may be installed in flat formation, trefoil (touching or spaced), duct banks, trenches or submarine arrangements. The physical spacing between conductors affects mutual impedance and capacitance, and also proximity effects and phase coupling — in short, layout → mutual coupling → harmonic impedance. Flat and trefoil formations can produce different resonance frequencies and magnitudes.

For underground and submarine cables, the layout may be constrained by civil design, installation method, duct arrangement, trench spacing or seabed conditions. Model the actual layout wherever possible; if the final layout is not known, study credible options as sensitivity cases.

Section 8

Sheath bonding

Sheath bonding is one of the most important modelling aspects for cable harmonic studies. Common arrangements are solid bonding, single-point bonding and cross-bonding. The configuration affects sheath currents, losses, impedance and damping, and introduces non-continuous impedance along the route. Cross-bonding divides the cable into major and minor sections, and the number of joints and the section lengths can change the harmonic impedance.

It is worth explaining how bonding changes the model. The bonding arrangement sets the return-current path, and the return path sets the cable impedance. With both-end bonding, sheath circulating currents can flow, which increases losses and damping. With single-point bonding, circulating current is reduced, but the induced sheath voltage may become important. With cross-bonding, the sheath path is divided into sections, so the impedance becomes phase- and section-dependent. For harmonic studies, the bonding arrangement should therefore match the actual cable installation wherever possible.

Table 3 — Effect of sheath bonding arrangement on harmonic cable modelling.

Sheath Bonding Arrangement	Harmonic Modelling Relevance
Single-point bonded	Reduced sheath circulating current; induced sheath voltage may be important
Both-end bonded	Sheath circulating current and additional losses can affect damping
Cross-bonded	Phase and sequence coupling may matter; section-by-section modelling may be required
Solidly bonded with armour / pipe	Return path and loss distribution may affect zero sequence and damping

The practical effects include a change in resonance frequency, a change in resonance magnitude, a change in damping, and phase and sequence coupling. Sheath bonding has a significant effect on positive-sequence harmonic impedance; its effect on zero-sequence impedance may be smaller in some cases, but this depends on the cable arrangement and return path.

Practical recommendation

Include the actual sheath bonding and cross-bonding details in the model. If these details are unknown, treat them as a modelling uncertainty.

Section 9

Skin and proximity effects, losses and damping

At harmonic frequencies, current does not distribute uniformly across the conductor. Skin effect pushes current towards the conductor surface; proximity effect further distorts the current distribution because of the magnetic fields from nearby phase conductors, screens, armour or adjacent cables. Both effects increase the AC resistance and therefore affect damping — and because damping controls the height of resonance peaks, the selected loss model can significantly change the harmonic study result.

Skin effect in cables

Skin effect is the tendency of AC current to concentrate near the conductor surface as frequency increases, so the effective AC resistance rises with frequency, \(R(h)>R_1\). This matters because cable losses provide damping: if skin effect is not modelled correctly, harmonic damping may be wrong — incorrect resistance → incorrect damping → incorrect resonance peak. Analytical methods using Bessel functions give high accuracy for the internal impedance; approximate correction factors may be acceptable at low frequencies but can become inaccurate at harmonic frequencies. Skin effect should be included in cable harmonic impedance calculations for both single-core and multi-core cables.

Proximity effect

Proximity effect occurs when the current distribution in a conductor is influenced by the magnetic fields of nearby conductors. It is important for closely spaced single-core cables, multi-core cables, armoured cables, pipe-type cables and touching-trefoil arrangements. The effect increases AC resistance and changes losses, and can affect damping even at lower harmonic frequencies: proximity effect → higher AC resistance → more damping. If it is ignored, cable losses may be underestimated, leading to overestimated resonance peaks or incorrect stability margins — but inaccurate correction factors can also mislead, so where proximity effects are important a more accurate method should be used.

Modelling skin and proximity effects

Table 4 — Methods for modelling skin and proximity effects.

Method	Accuracy	Comment
Simple correction factors	Low to medium	May be inaccurate at harmonic frequencies
IEC-based proximity factors	Useful for losses	May not suit harmonic-emission studies
Analytical formulae	Medium to high	Common for standard cable arrangements
Finite element method	High	Accurate but computationally intensive
MoM-SO method	High	Efficient advanced method where available
Sub-conductor representation	Medium to high	Approximates conductor current distribution

For detailed harmonic studies, the most accurate available method should be used where resonance damping or stability margins are critical. In power-park module, HVDC, offshore-wind or long-cable studies, inaccurate cable damping can lead to either under-designed mitigation or unnecessary over-design — both undesirable.

Cable losses and damping

Cable losses are important because they provide damping, and damping controls the amplitude of resonance peaks. A resonance peak can be represented conceptually as inversely proportional to the damping resistance:

\[ Z_{peak}\propto\frac{1}{R_{\text{damping}}} \]

\(Z_{peak}\): approximate impedance magnitude at resonance
\(R_{\text{damping}}\): effective damping resistance in the network
\(\propto\): “is proportional to”

This is a conceptual relationship, not a detailed design equation. It is used to show that lower damping can produce higher resonance peaks.

If damping is underestimated, the calculated peak may be too high; if overestimated, too low. Both errors can change the engineering decision, so accurate harmonic damping requires accurate cable loss modelling. This is especially important for submarine cables, where modelling losses can be difficult and manufacturer-specific design data may not be available during early project stages.

Section 10

Sequence impedance and planning uncertainty

Cable modelling should consider the sequence components relevant to the study. For balanced harmonic studies, positive-sequence impedance may be sufficient. For unbalanced studies, zero-sequence and negative-sequence behaviour may be important. The zero-sequence impedance depends on return paths through the sheath, screen, armour, earth, parallel metallic paths and the bonding system. If the study includes triplen harmonics, unbalanced sources, phase-wise limits or cable sheath effects, zero-sequence modelling becomes important — use a phase-domain or multi-conductor model where sequence coupling matters.

Sequence quantities defined

Impedances. \(Z_1\) (positive sequence) — the impedance seen by a balanced set of phase quantities with normal phase rotation. \(Z_2\) (negative sequence) — the same for reverse phase rotation. \(Z_0\) (zero sequence) — the impedance seen when the three phase quantities are in phase and return through sheath, screen, neutral, earth or other return paths.
Currents. \(I_0,\ I_1,\ I_2\) — the zero-, positive- and negative-sequence current components. Voltages. \(V_0,\ V_1,\ V_2\) — the zero-, positive- and negative-sequence voltage components.

Notation: \(h\) is harmonic order, not a sequence index

In sequence notation, the subscripts 0, 1 and 2 mean the zero-, positive- and negative-sequence components — they do not mean harmonic order. On this page \(h\) is always used for harmonic order, and the two are combined where needed: for example \(Z_{0,h}\) means the zero-sequence impedance at harmonic order \(h\), and \(I_{1,h}\) means the positive-sequence current at harmonic order \(h\). (By contrast, a bare \(Z_5\) would normally mean impedance at the 5th harmonic.)

Planning-stage uncertainty

Cable harmonic studies are often carried out during planning, before final manufacturer data is available, which introduces uncertainty.

Table 5 — Sources of planning-stage uncertainty and their effect.

Uncertain Item	Effect
Final route length	Changes resonance frequency
Cable design	Changes capacitance and losses
Conductor material	Changes resistance and radius
Insulation thickness	Changes capacitance
Laying formation	Changes coupling and resonance
Bonding design	Changes losses and impedance
Soil or seabed properties	Affects return path and losses
Manufacturer tolerances	Data-sheet values may differ from physical values

Because of these uncertainties, a single harmonic study case may not be sufficient. Use sensitivity studies during planning — varying length, capacitance, losses, bonding arrangement, layout, source impedance and operating condition.

Planning-stage uncertainty — what to vary

At feasibility or grid-connection stage, the final cable manufacturer, sheath-bonding design, cable formation, installation depth, soil thermal/resistivity data and exact route length may not yet be fixed. The harmonic study should test reasonable sensitivities, especially for:

cable length;
capacitance;
sheath bonding arrangement;
laying formation;
soil or seabed return path;
frequency-dependent resistance;
number of parallel circuits;
filter or capacitor-bank status.

Section 11

Model selection and comparison

Table 6 — Selecting a cable model for harmonic studies.

Study Condition	Recommended Model
Very short cable, low harmonic order	Nominal-π may be acceptable
Cable longer than a few kilometres	Distributed equivalent-π
Resonance assessment	Distributed model
Filter design	Distributed model with accurate losses
Offshore or submarine export cable	Distributed model with detailed cable data
Cable-rich network	Frequency-dependent cable model
Cross-bonded cable system	Sectional model with bonding details
Multi-core cable	Model skin and proximity effects
Closely spaced single-core cables	Include proximity effects
Unbalanced harmonic study	Phase-domain or multi-conductor model
Early planning study	Sensitivity cases around uncertain parameters

Practical rule

The more important the resonance result, the more detailed the cable model must be.

Comparison with overhead-line modelling

Cables and overhead lines are both represented using series impedance and shunt admittance, but their harmonic behaviour differs.

Table 7 — Overhead line versus cable for harmonic behaviour.

Aspect	Overhead Line	Cable
Series impedance	Higher	Lower
Shunt capacitance	Lower	Higher
Resonance frequency	Usually higher	Often lower
Earth-return importance	Important for zero sequence	Important through sheath/earth paths
Phase geometry	Tower-dependent	Laying-formation-dependent
Loss modelling	Skin effect important	Skin and proximity effects important
Long-line effects	Important for long OHLs	Important even for shorter lengths
Bonding	Usually not applicable	Very important
Armour / sheath	Usually not applicable	Important

The key difference is that cables introduce much more capacitance into the network, which is why cable systems can cause resonance at lower frequencies than overhead lines.

Section 12

Workflow and sensitivity

A practical cable modelling workflow for harmonic studies can be structured as follows:

Define the study objective — resonance screening, compliance, filter design, harmonic stability or active filtering.
Identify the harmonic frequency range — e.g. \(h=2\) to \(50\), or the project-specific range.
Collect the cable data (see Table 8).
Choose the model type — nominal-π, cascaded nominal-π, or distributed equivalent-π.
Include frequency-dependent losses.
Include sheath bonding and cross-bonding.
Run frequency scans to identify resonance.
Run harmonic penetration studies to calculate distortion.
Perform sensitivity studies where cable data is uncertain.

Table 8 — Cable data required and its purpose.

Data	Purpose
Cable length	Resonance frequency and number of resonances
Conductor material and radius	Resistance and inductance
Insulation thickness	Capacitance
Screen or sheath details	Return path and losses
Armour or pipe details	Proximity and losses
Laying formation	Mutual coupling
Spacing and depth	Impedance and admittance
Bonding arrangement	Sheath current and damping
Cross-bonding section lengths	Resonance and phase coupling
Soil or seabed properties	Return path and loss modelling
Manufacturer frequency-dependent data	Improves accuracy

Sensitivity priorities

Table 9 — Relative sensitivity of harmonic results to cable modelling choices.

Parameter	Typical Impact	Comment
Cable length	Very high	Strongly affects resonance frequency and count
Model type	Very high	Nominal-π may miss resonance behaviour
Cable layout	High	Affects resonance frequency and magnitude
Sheath bonding	High	Affects impedance, damping and resonance
Skin effect	High	Affects losses and damping
Proximity effect	High for close / multi-core	Affects AC resistance and damping
Insulation thickness	Medium	Affects capacitance
Conductor radius	Low to medium	Affects frequency and impedance
Pipe thickness	Low	Usually minor for harmonic studies

The practical message: length, model type, layout, bonding and loss modelling are usually the most important.

Minimum data for a defensible cable model

Because cable model quality depends heavily on input data, a robust cable harmonic model should state:

cable type and voltage level;
conductor size and material;
cable length;
single-core or three-core construction;
insulation type and thickness;
screen/sheath material and cross-section;
armour or pipe details, if applicable;
laying formation: trefoil, flat, duct, tunnel, direct-buried or submarine;
phase spacing and circuit spacing;
sheath bonding arrangement;
cross-bonding section lengths, if used;
soil or seabed return assumptions;
cable capacitance and frequency-dependent impedance;
selected model type: nominal-π, cascaded π, or distributed;
whether the model is balanced, unbalanced or phase-domain;
frequency range and highest harmonic order assessed.

Section 13

Reporting and summary

A harmonic study report should clearly state how cables were modelled. A weak statement — “cables were included in the model” — tells the reader nothing about reliability.

Examples of a clear modelling statement

“Cables were represented using frequency-dependent distributed-parameter models, including cable capacitance, conductor losses and sheath-bonding arrangement.”

“Skin and proximity effects were included to improve the representation of harmonic damping; cross-bonding sections were represented explicitly where data was available.”

“Sensitivity cases were performed for cable length, capacitance, bonding arrangement and loss assumptions.”

Common modelling mistakes

Using a 50 Hz cable impedance without checking frequency dependency.
Ignoring cable capacitance in resonance studies.
Using a single nominal-π section for a long cable.
Ignoring sheath bonding or cross-bonding.
Ignoring skin and proximity effects when damping is important.
Treating a cross-bonded cable as a simple balanced positive-sequence element.
Not testing sensitivity where final cable data is uncertain.
Not stating whether cable losses are manufacturer-specific or assumed.

To summarise: cable modelling is one of the most important parts of harmonic analysis. Cables have lower series impedance and much higher shunt capacitance than overhead lines, which means they can shift network resonance to lower harmonic frequencies. The governing relationship is \(V_h=Z_h I_h\), and cables can significantly change \(Z_h\), especially near resonance. Cable models should represent frequency-dependent series impedance and shunt admittance, distributed-parameter behaviour, skin effect, proximity effect, sheath bonding, cable layout, and zero-sequence and phase coupling.

A nominal-π model may be acceptable only for short cables and approximate screening. For detailed harmonic studies, an equivalent-π distributed-parameter model should normally be used, especially for cables longer than a few kilometres. The most sensitive parameters are cable length, cable layout, sheath bonding, model type, and skin and proximity effects.

In short: cable modelling is critical because cable capacitance can shift network resonance to lower harmonic orders, and cable losses strongly affect resonance damping. For short cables and early screening a nominal-π model may be acceptable; for resonance studies, compliance studies, offshore connections, long underground cables or filter design, a distributed and frequency-dependent model should normally be used. A clear report should then state:

the cable length and layout;
the sheath-bonding arrangement;
the cable capacitance and frequency-dependent impedance;
the loss model used;
the sequence representation;
the selected model type;
the sensitivity cases assessed.

Without these details, the harmonic results may not be technically defensible.

Key message

Use distributed, frequency-dependent cable models for resonance and compliance studies, and perform sensitivity studies when the final cable data is uncertain. Only with an appropriate cable model can harmonic resonance, damping, voltage distortion and mitigation requirements be assessed reliably.

Part Three Reading now

Cable Modelling for Harmonic Studies

Nominal-π, cascaded and distributed cable models; cable capacitance and resonance; length, layout and sheath bonding; skin and proximity effects; losses, damping and zero-sequence behaviour.

Series progress 3 of 11

Part Four

Transformer Modelling for Harmonic Studies

Frequency-dependent impedance and damping; winding connections and zero-sequence paths; tap position; high-frequency capacitances; and harmonic heating factors.