Harmonic Studies & Modelling

Harmonic Study Domains & Modelling Approaches

A harmonic study can be carried out in several modelling domains, and the choice shapes the accuracy, the input data, the calculation time and the physical behaviour that can be represented. This page sets out the frequency, time, hybrid and harmonic domains; explains the frequency-domain family — frequency scan, harmonic penetration and harmonic load flow; weighs balanced against unbalanced modelling; and gives a practical guide to selecting the right method for the question being asked.

Reading time ≈ 22 min · Part One of the series

Harmonic studies can be carried out using different modelling domains. The selected domain affects the accuracy of the results, the required input data, the calculation time and the type of physical behaviour that can be represented. In practice, the engineer must decide whether the study should be performed in the frequency domain, the time domain, a hybrid domain, or the harmonic domain.

Each approach has a different purpose, and no single method is best for all harmonic studies. The correct method depends on the study objective, the required accuracy, the available data and the behaviour of the equipment being modelled. This guide builds the picture from the four domains, through the frequency-domain family of methods, to a practical rule for choosing between them.

Key idea

A harmonic study can run in the frequency, time, hybrid or harmonic domain — each a different trade-off of speed against physical detail.
The frequency domain is the usual starting point for planning and compliance: fast, robust and efficient for steady-state harmonics.
Its three methods — frequency scan, harmonic penetration and harmonic load flow — answer different questions about resonance and distortion.
Move to time, hybrid or harmonic domains only when nonlinear control, saturation or cross-frequency coupling must be represented.

Key terms used on this page

01Harmonic order, \(h\)

The integer multiple of the fundamental frequency. For a 50 Hz system the 5th harmonic is 250 Hz.

02Fundamental component, \(V_1\)

The voltage component at the power-system fundamental frequency, normally 50 Hz in the UK and Ireland.

03Harmonic component, \(V_h\)

The voltage component at harmonic order \(h\).

04Frequency scan

Calculation of network impedance versus frequency to identify resonances.

05Harmonic penetration

Calculation of how injected harmonic currents, or background harmonic voltages, propagate through the network.

06Harmonic load flow

A fuller harmonic calculation in which harmonic sources, network conditions and operating-point dependency are solved together.

07Balanced model

A simplified model that assumes the three phases behave symmetrically.

08Unbalanced model

A phase-by-phase model used when asymmetry, zero-sequence, negative-sequence or line/cable geometry matters.

Section 1

The four study domains

Before any modelling begins, the engineer chooses a domain in which to solve the problem. There are four practical choices, and they differ in what they can represent and what they cost to run.

Frequency domain — the network is solved separately at each harmonic frequency. Fast and robust for steady-state harmonics.
Time domain — the waveform is simulated as a function of time using differential equations. Captures nonlinear and switching behaviour directly.
Hybrid domain — a combination that solves the network in the frequency domain while representing nonlinear devices in the time domain.
Harmonic domain — an advanced steady-state framework that explicitly represents coupling between harmonics, phases and nonlinear devices.

The practical question

What do we need to know from the study? The answer decides the domain — not habit, and not whichever tool happens to be open.

If the objective is to identify resonance, calculate harmonic impedance, estimate harmonic voltage distortion or design passive filters, frequency-domain methods are usually the preferred starting point. If the objective is to study nonlinear control interaction, transformer saturation, converter switching behaviour or cross-frequency coupling, time-domain, hybrid-domain or harmonic-domain methods may be required.

Section 2

Why different domains exist

A distorted waveform can be viewed in two different ways. In the time domain, the waveform is represented as a function of time, \(v(t)\). This shows the actual waveform shape, including switching events, transient behaviour and nonlinear device operation.

In the frequency domain, the same waveform is represented by its frequency components \(V_1,\ V_2,\ V_3,\ \ldots,\ V_h\), where \(V_1\) is the fundamental component and \(V_h\) is the harmonic component at order \(h\). The relationship between the two views is provided by Fourier analysis: a distorted periodic waveform may be written as a sum of sinusoids.

\[ v(t)=V_1\sin(\omega t+\phi_1)+\sum_{h=2}^{H}V_h\sin(h\omega t+\phi_h) \]

\(v(t)\): instantaneous distorted voltage waveform as a function of time
\(V_1,\ \phi_1\): magnitude and phase angle of the fundamental component
\(V_h,\ \phi_h\): magnitude and phase angle of the harmonic of order \(h\)
\(\omega\): angular frequency of the fundamental component, \(\omega=2\pi f_1\)
\(f_1\): fundamental frequency, normally 50 Hz in the UK and Ireland
\(h\): harmonic order
\(H\): highest harmonic order considered in the study

Peak vs RMS: as written, \(V_1\) and \(V_h\) are sine-wave amplitudes (peak values). Harmonic studies usually report and limit RMS harmonic values, related by \(V_{\text{rms}}=V_{\text{peak}}/\sqrt{2}\). The two differ only by a constant scale factor, so the form of the equation is unchanged — but a result should always state whether the quoted harmonic magnitudes are peak or RMS.

This equation shows why both domains are useful. The time-domain waveform shows the shape of the distortion, while the frequency-domain representation shows which harmonic orders are present and how large they are. For harmonic planning and compliance studies, the frequency-domain representation is often more convenient because harmonic limits, network impedance and filter tuning are all naturally expressed as a function of frequency.

Section 3

Overview of frequency-domain methods

Frequency-domain harmonic methods can be grouped into two main study objectives. The first is network impedance calculation — principally the frequency scan, where the network impedance is calculated over a range of frequencies. The second is the calculation of harmonic voltages and currents — harmonic penetration studies and harmonic load-flow methods. Figure 1 sets out the full family.

Figure 1 — Overview of frequency-domain methods used in harmonic studies. The two objectives — network impedance calculation (green) and calculation of harmonic voltages and currents (blue) — each resolve down to balanced or unbalanced representations.

The simplified structure is a single root that branches into the two objectives: frequency-domain methods → network impedance calculation and frequency-domain methods → calculation of harmonic voltages and currents. Each branch answers a distinct engineering question.

Table 1 — The two frequency-domain objectives and the questions they answer.

Objective	Question It Answers
Network impedance calculation	Where can resonance occur?
Calculation of harmonic voltages and currents	How do harmonic emissions propagate through the system?

Both are normally needed in a complete harmonic study: one tells you where the network is sensitive, the other tells you what the sources actually do to it.

Which question does each method answer?

Frequency scan — “Where are the network resonances?”
Harmonic penetration — “If a harmonic source injects current, what voltage distortion appears at each bus?”
Harmonic load flow — “What is the harmonic distortion under a defined operating condition?”
Time-domain simulation — “How do switching, control action, saturation or nonlinear behaviour affect the waveform?”
Hybrid / harmonic-domain methods — “Do nonlinear devices or converters create coupling between different harmonic frequencies?”

Section 4

Frequency-domain analysis

Frequency-domain analysis solves the network separately at each harmonic frequency. At each harmonic order the network is represented using an admittance matrix, and the harmonic voltages follow from the harmonic current injections.

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

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

For each harmonic order the network is solved using its frequency-dependent admittance matrix. The result gives the harmonic voltage at each bus produced by the specified harmonic current injections.

This equation is the basis of both frequency scans and harmonic penetration studies. At each harmonic order, the network elements are represented by their impedance or admittance at that frequency. An inductive reactance increases with frequency, while a capacitive reactance decreases with frequency:

\[ X_L(h)=h\,X_{L1} \qquad\qquad X_C(h)=\frac{X_{C1}}{h} \]

\(X_L(h)\): inductive reactance at harmonic order \(h\)
\(X_C(h)\): capacitive reactance at harmonic order \(h\)
\(X_{L1},\ X_{C1}\): reactances at the fundamental frequency

This frequency dependence is essential, because resonance occurs when the inductive and capacitive parts of the network interact. The major advantage of the frequency-domain method is that it is fast, stable and easy to apply to large networks — many operating conditions, outage cases and harmonic source scenarios can be studied efficiently.

The limitation is that nonlinear device behaviour is normally simplified. Transformer saturation, converter control interaction, switching dead-time and harmonic cross-coupling are not fully represented unless special models are used.

Section 5

Frequency scan

A frequency scan is usually one of the first studies performed in a harmonic assessment. It calculates the harmonic impedance of the network as seen from a selected busbar, and its purpose is to identify resonance conditions. A current injection is applied at the selected busbar, usually \(I_h=1\ \text{A}\) or \(I_h=1\ \text{pu}\), and the resulting voltage is calculated at each frequency. The driving-point impedance is then:

\[ Z_h=\frac{V_h}{I_h} \]

\(Z_h\): network harmonic (driving-point) impedance at order \(h\)
\(V_h\): harmonic voltage calculated at the study bus at order \(h\)
\(I_h\): harmonic current injection at order \(h\)

This relationship is the basis of frequency-scan studies: a small test current is injected at each frequency, and the resulting voltage is used to calculate the impedance seen from the study bus.

If a 1 A current injection is used, the calculated voltage directly represents the impedance in ohms. A frequency scan therefore answers: what impedance does a harmonic source see at each frequency? The shape of that curve reveals the resonances:

A high impedance peak (\(Z_h\uparrow\)) indicates a possible parallel resonance — a risk to be avoided near harmonic orders.
A low impedance trough (\(Z_h\downarrow\)) indicates a possible series resonance or a filter-tuning point.

This matters because a small harmonic current injected at a frequency where the system impedance is high can produce a large harmonic voltage:

\[ V_h=Z_h\,I_h \]

\(V_h\): harmonic voltage produced at the busbar
\(Z_h\): network harmonic impedance at order \(h\)
\(I_h\): harmonic current injected by the source

Therefore harmonic voltage distortion is not determined only by the harmonic current source — it is also determined by the harmonic impedance of the network.

Practical use of frequency scan

Frequency scans are used across a range of applications, especially before adding capacitor banks or harmonic filters, because these elements can significantly shift the network resonance frequency.

Table 2 — Typical applications of a frequency scan.

Application	Purpose
Resonance screening	Identify parallel or series resonance frequencies
Filter design	Select tuning frequency and damping
Capacitor bank assessment	Check resonance caused by capacitor installation
Connection studies	Check how a new harmonic source interacts with the network
Network sensitivity	Compare operating cases and outage conditions
Harmonic impedance envelope	Establish the range of possible network impedances
Compliance studies	Support calculation of harmonic voltage distortion

For example, adding a capacitor bank changes the network capacitance and may move the parallel resonance close to the 5th, 7th or 11th harmonic. If a harmonic source injects current near that frequency, voltage distortion may be amplified.

Worked example — scan then penetration

Suppose a new capacitor bank is to be connected to an industrial network. The first step is normally a frequency scan to check whether the capacitor creates a resonance near a characteristic harmonic such as the 5th or 7th. If a resonance is identified, a harmonic penetration study is then used to calculate whether the existing harmonic sources will produce excessive voltage distortion at that frequency. The scan finds the danger; the penetration study quantifies it.

Practical rule

Always check the frequency response before trusting a harmonic distortion result.

Limitation of frequency scan

A frequency scan is a linear analysis. It assumes that the network admittance matrix does not depend on voltage or current magnitude. This is acceptable for most linear network components.

Table 3 — Linear components and their frequency-domain representation.

Component	Frequency-Domain Representation
Lines	Frequency-dependent impedance and admittance
Cables	Frequency-dependent impedance and capacitance
Transformers	Leakage impedance and winding connection
Reactors	Inductive impedance
Capacitors	Capacitive impedance
Passive filters	Tuned RLC branches

The method is limited, however, for nonlinear components whose behaviour depends on voltage, current or control state.

Table 4 — Nonlinear components and their behaviour.

Component	Nonlinear Behaviour
Saturated transformers	Magnetising current depends on voltage
Voltage transformers	Saturation and ferroresonance effects
Surge arresters	Highly nonlinear voltage–current characteristic
Power electronic converters	Harmonic emission depends on control and operating point
Active filters	Current injection depends on the measurement and control loop

For most planning-level harmonic studies these nonlinear effects may be neglected or approximated. For specialised studies, time-domain or hybrid-domain modelling may be required.

Section 6

Harmonic penetration

A harmonic penetration study calculates how harmonic current injections propagate through the network and what harmonic voltages appear at different buses. It uses the same basic equation, \([Y_h][V_h]=[I_h]\), but instead of injecting a test current at one bus over many frequencies, it applies actual or assumed harmonic current injections from harmonic sources. The results give:

harmonic voltage at network buses;
harmonic current in branches;
harmonic loading of transformers, cables and filters;
THD and individual harmonic distortion.

The question it answers

If these harmonic sources inject these currents, what distortion appears in the network? This makes harmonic penetration one of the most important methods for harmonic compliance studies.

The propagated distortion is most often summarised by the voltage total harmonic distortion, which collects every harmonic order into a single index relative to the fundamental:

\[ \text{THD}_V=\frac{\sqrt{\displaystyle\sum_{h=2}^{H}V_h^{2}}}{V_1}\times 100\% \]

\(\text{THD}_V\): voltage total harmonic distortion
\(V_h\): RMS voltage of harmonic order \(h\)
\(V_1\): RMS fundamental voltage
\(H\): highest harmonic order included

Individual harmonic limits and total harmonic distortion limits are normally assessed separately, depending on the applicable grid code, planning level or power-quality standard.

Direct harmonic penetration

In the direct method, the admittance matrix is built for each harmonic order, the harmonic current injections are applied, and the linear equations are solved directly — \([Y_h][V_h]=[I_h]\), solved once per harmonic order. It is called “direct” because it does not iterate the harmonic current injections based on the calculated harmonic voltages. The assumption is that \(I_h\) is known and fixed: the harmonic source is represented as a defined current injection (or voltage source) whose harmonic behaviour does not change with the calculated distortion.

The direct method is suitable for many practical studies because harmonic emissions are often provided as fixed spectra — \(I_5,\ I_7,\ I_{11},\ I_{13},\ \ldots\). The method is efficient and can be applied to large systems.

Iterative harmonic penetration

The iterative method extends the direct method by allowing the harmonic current injection to depend on the harmonic voltage, \(I_h=f(V_h)\). The solution is then repeated until the harmonic voltages and harmonic currents are consistent:

\[ I_h^{(1)}\ \rightarrow\ V_h^{(1)}\ \rightarrow\ I_h^{(2)}\ \rightarrow\ V_h^{(2)}\ \rightarrow\ \cdots\ \rightarrow\ \text{converged solution} \]

\(I_h^{(k)}\): harmonic current injection at iteration \(k\)
\(V_h^{(k)}\): harmonic voltage computed at iteration \(k\)

This is more realistic where the harmonic source is not independent of the network distortion.

Table 5 — Where the iterative method is useful.

Application	Reason
Converter studies	Harmonic emission may depend on terminal distortion
Active filter studies	Compensation current depends on measured current/voltage
Nonlinear load studies	Current waveform may depend on supply voltage
Background distortion	Source emission may change with pre-existing distortion

The disadvantage is that more detailed models are required, and convergence may become more difficult.

Section 7

Balanced vs unbalanced modelling

The choice between balanced and unbalanced modelling is one of the most important decisions in a harmonic study. It applies to penetration and load-flow studies alike.

Balanced harmonic penetration

In a balanced study, the network is assumed to be symmetrical. The three phases are represented by a single equivalent phase, or by positive-sequence quantities: the phase voltages \(V_a,\ V_b,\ V_c\) are equal in magnitude and separated by \(120^\circ\), and only the positive-sequence network is represented. This is simpler and computationally efficient.

Table 6 — When a balanced model may be adequate.

Condition	Reason
Network is symmetrical	Single-phase representation is sufficient
Loads and sources are balanced	No significant phase-wise distortion
Filters are symmetrical	Equal response in all phases
Study is for screening	An approximate result is acceptable
Converter station is symmetrical	Injection can be represented by sequence assumptions

A balanced method may be adequate, for example, for a single HVDC or industrial converter installation where the station arrangement and filters are symmetrical and the objective is to calculate approximate harmonic distortion and filter rating. The advantage is simplicity; the limitation is that it cannot fully represent phase asymmetry, unbalanced lines, untransposed circuits, cable sheath asymmetry or unequal mutual coupling.

Unbalanced harmonic penetration

In an unbalanced study, the network is represented phase-by-phase, which allows differences between phases and coupling between phases. The three-phase admittance matrix is used, and harmonic quantities can be calculated in each phase, \(V_{a,h},\ V_{b,h},\ V_{c,h}\), and transformed into sequence components \(V_{1,h},\ V_{2,h},\ V_{0,h}\).

Table 7 — Causes of asymmetry that call for an unbalanced model.

Cause	Effect
Untransposed overhead lines	Different phase impedances
Flat-formation cables	Unequal mutual coupling
Cross-bonded cable systems	Frequency-dependent phase asymmetry
Unequal sheath bonding	Asymmetric impedance
Single-phase or two-phase loads	Unbalanced harmonic injection
Transformer connections	Sequence-dependent harmonic propagation
Triplen harmonics	Zero-sequence behaviour may be important
Phase-wise converter emissions	Different harmonic injection in each phase

The practical reason for using an unbalanced model is that the balanced model may underestimate or misrepresent harmonic voltages in individual phases. At some frequencies, especially near resonance, small phase asymmetries can create significant inter-sequence coupling.

Practical rule

Near resonance, asymmetry matters more.

Balanced vs unbalanced — the comparison

A balanced model assumes \(Z_a=Z_b=Z_c\) with symmetrical phase coupling; an unbalanced model allows \(Z_a\neq Z_b\neq Z_c\) with phase-wise mutual coupling.

Table 8 — Balanced versus unbalanced modelling.

Aspect	Balanced Method	Unbalanced Method
Network representation	Single-phase or sequence-based	Full phase-wise representation
Calculation time	Faster	Slower
Data requirement	Lower	Higher
Phase asymmetry	Not represented accurately	Represented
Inter-sequence coupling	Usually ignored	Included
Suitable for	Screening and symmetrical systems	Detailed studies and asymmetric systems
Risk	May miss phase-specific resonance	More complex and data-intensive

A balanced model is not automatically wrong — it is often suitable for early-stage screening or symmetrical installations. However, if the study involves long cables, untransposed overhead lines, phase-wise harmonic limits, cable sheath effects or resonance near important harmonic orders, an unbalanced model may be necessary.

Section 8

Harmonic load flow

The term “harmonic load flow” is often used loosely in commercial software. In many cases, what is called harmonic load flow is actually a harmonic penetration study based on a fundamental-frequency load flow. A true harmonic load flow is more advanced: it solves the network at harmonic frequencies while considering the dependency of harmonic quantities on the operating state of network devices.

In a true harmonic load flow, harmonic currents and voltages may depend on the fundamental load flow, the harmonic voltages, the device operating point, the converter controls and other nonlinear device behaviour. This means the solution may require iteration, similar to a conventional load flow.

A harmonic load flow is therefore used when harmonic quantities depend on the operating condition of the system. It may bring together fundamental-frequency load-flow results, harmonic source models, network impedance, background harmonic voltages and load damping. Compared with a simple harmonic penetration study, it is more suitable when the harmonic sources or the network response depend on loading, voltage level, converter operating point or filter status.

The practical distinction

Harmonic penetration → a linear solution with specified harmonic injections. True harmonic load flow → an iterative solution with voltage/current dependency. Harmonic load flow can itself be balanced or unbalanced, depending on the modelling approach.

Frequency scan vs penetration vs load flow

The three methods should not be confused. A frequency scan is mainly a network impedance study; a harmonic penetration study is mainly a distortion propagation study; a harmonic load flow is a more detailed, operating-point-dependent harmonic solution.

Table 9 — The three frequency-domain methods compared.

Method	Main Purpose	Main Input	Main Output
Frequency scan	Identify resonance and impedance	Test current injection	Harmonic impedance vs frequency
Harmonic penetration	Calculate harmonic distortion	Harmonic current spectrum	Harmonic voltages and currents
Harmonic load flow	Solve harmonic interaction with operating point	Device models and operating state	Harmonic voltages/currents with dependency effects

The simplest interpretation is a chain of increasing depth: frequency scan → network response; harmonic penetration → distortion propagation; harmonic load flow → distortion interaction. A complete harmonic study often uses more than one method — the engineer may first run a frequency scan to identify resonance, then a harmonic penetration study to calculate harmonic voltages, and finally a time-domain or detailed harmonic load-flow study if nonlinear interaction is suspected.

Section 9

Time-domain methods

Time-domain methods represent the system using differential equations and simulate the waveform as a function of time. Instead of solving each harmonic frequency separately, the time-domain method simulates \(v(t)\) and \(i(t)\) directly; the harmonic spectrum is then obtained by applying Fourier analysis to the steady-state waveform. The main advantage is that nonlinear and time-varying behaviour can be represented more accurately.

Table 10 — Where time-domain analysis is useful.

Application	Reason
Power electronic converter controls	Control loops can be represented directly
Transformer saturation	Nonlinear magnetising behaviour included
Switching dead-time effects	Important for low-order harmonics in VSCs
Harmonic instability	Interaction can be observed in time
AC/DC interaction	Cross-modulation can be represented
Active filter control	Measurement and control loops can be modelled
Incident investigation	Actual waveform behaviour may be reproduced

The limitation is that time-domain studies can be slow, especially for large networks. The simulation must run long enough for transients to decay and steady-state harmonic behaviour to be reached — higher accuracy comes at the cost of higher modelling effort and computation time. Time-domain results are only reliable if the simulation has reached steady state before harmonic quantities are extracted.

Limitation of time-domain methods

Time-domain methods are powerful, but they are not automatically more accurate for every harmonic study. They require detailed input data and accurate models.

Table 11 — Data and modelling requirements for time-domain studies.

Requirement	Why It Matters
Converter control model	Harmonic behaviour depends on controller details
Switching dead-time	Affects low-order harmonics in VSCs
Transformer saturation curve	Needed for nonlinear magnetic behaviour
Frequency-dependent network model	Difficult to represent in the time domain
Small time step	Needed for switching accuracy
Long simulation time	Needed for steady-state harmonic extraction
Model validation	Required for OEM black-box models

If important details are missing, a time-domain model may give a misleading result. Time-domain analysis should therefore be used when the added modelling detail is necessary and supported by reliable data.

Section 10

Hybrid and harmonic-domain methods

Hybrid methods

Hybrid methods combine frequency-domain and time-domain approaches. The purpose is to benefit from the efficiency of the frequency-domain network solution while still representing nonlinear or time-varying components in the time domain — a frequency-domain network plus a time-domain nonlinear device model. The interface between domains may be achieved by converting frequency-domain quantities to time-domain waveforms, or by iterating between the two domains until a consistent solution is obtained.

Table 12 — Where hybrid methods are useful.

Application	Reason
Nonlinear converter interaction	Needs device detail and network frequency response
Large system with a nonlinear local device	Avoids a full time-domain model of the entire network
Harmonic instability studies	Combines network and controller effects
Research applications	Allows a flexible modelling framework

Hybrid methods can represent nonlinear components more accurately than pure frequency-domain methods while avoiding the full computational burden of pure time-domain simulation. The main disadvantage is practical implementation: they are complex, often require specialist knowledge and careful interfacing between software tools, and are not widely available in standard commercial tools.

Frequency domain vs harmonic domain

It is easy to confuse the two terms, but they are not the same. The frequency domain usually treats each harmonic frequency independently — for example, the 5th, 7th and 11th harmonics are solved separately using the network impedance at those frequencies. The harmonic domain is more advanced because it can represent coupling between harmonic orders: a disturbance or control action at one harmonic frequency may influence another. This matters for some converter-based systems, but it is not normally required for routine planning studies.

Harmonic domain

The harmonic domain is a more advanced steady-state framework that explicitly represents coupling between harmonics, phases and nonlinear devices. In simple terms, it allows one harmonic frequency to influence another. This matters because some devices do not behave independently at each harmonic order — power electronic converters, for example, can create coupling between AC-side and DC-side harmonics, or between different harmonic orders, due to modulation and control action. The harmonic domain can represent:

phase coupling;
harmonic coupling;
nonlinear components;
time-varying components;
converter control effects.

It is a powerful method for studying harmonic interaction between AC systems and large converters. However, harmonic-domain modelling is complex: it requires specialised models, high engineering skill and careful implementation, and is therefore less commonly used in routine industrial harmonic studies.

Section 11

Comparing the domains

The four study domains can be compared directly. Each has a clear strength, a clear limitation and a typical use.

Table 13 — Comparison of the four study domains.

Domain	Main Strength	Main Limitation	Typical Use
Frequency domain	Fast, robust, efficient for steady-state harmonics	Limited nonlinear and control modelling	Harmonic impedance, resonance, penetration, filters
Time domain	Detailed nonlinear and control modelling	Slow and data-intensive	Converter control, saturation, switching, incident analysis
Hybrid domain	Combines nonlinear modelling with frequency-domain efficiency	Complex and not widely available	Advanced studies and research
Harmonic domain	Represents cross-frequency and phase coupling	Complex and specialist	Converter interaction and advanced steady-state studies

The practical selection follows from that table: the frequency domain for most planning and compliance studies; the time domain for nonlinear and control-sensitive studies; the hybrid domain for advanced mixed-domain studies; and the harmonic domain for cross-frequency interaction studies.

Which method should be used?

The selected method should match the study objective. For early-stage screening, the best starting point is usually a frequency scan; for compliance assessment, harmonic penetration is usually required; for filter design, both are normally needed. For active filter control or converter interaction, time-domain or hybrid-domain analysis may be required; for large converter systems with harmonic cross-coupling, harmonic-domain or detailed time-domain modelling may be needed.

Table 14 — Practical method-selection guide.

Study Objective	Recommended Method
Identify resonance	Frequency scan
Check impact of capacitor banks	Frequency scan
Calculate harmonic voltage distortion	Harmonic penetration
Check harmonic compliance	Harmonic penetration
Estimate filter rating	Harmonic penetration and frequency scan
Study unbalanced cable propagation	Unbalanced harmonic penetration
Study converter control interaction	Time domain or hybrid domain
Study transformer saturation	Time domain
Study active filter controller behaviour	Time domain or hybrid domain
Study AC/DC harmonic coupling	Time domain or harmonic domain
Study a large number of operating cases	Frequency domain

Section 12

Practical workflow and network modelling

A practical harmonic study often follows a staged workflow. Each stage builds the data the next one needs:

Build the fundamental-frequency load-flow model — operating condition, loading, voltage levels and network configuration.
Perform a frequency scan at the relevant buses to identify resonances and sensitive harmonic orders.
Define the harmonic sources — converters, drives, rectifiers, furnaces or background harmonic voltage.
Perform harmonic penetration studies to calculate harmonic voltages and currents.
Assess compliance against the applicable objectives.
Design or verify mitigation — passive filters, active filters, detuned capacitor banks or network changes.

The workflow in one line

Load flow → frequency scan → harmonic source model → harmonic penetration → compliance check → mitigation design.

If the results indicate nonlinear interaction or controller sensitivity, the study may then be extended to time-domain, hybrid-domain or harmonic-domain analysis.

Importance of network modelling

The accuracy of any harmonic study depends strongly on the network model. A simple model may be adequate for screening; a detailed model is required when resonance, unbalance, cable systems or compliance margins are important.

Table 15 — Network modelling elements and their harmonic relevance.

Element	Harmonic Relevance
Lines and cables	Frequency-dependent impedance and capacitance
Transformers	Leakage impedance, connection group, saturation if relevant
Loads	Damping and frequency dependence
Capacitor banks	Resonance and reactive compensation
Filters	Tuned impedance paths
Converter stations	Harmonic sources and impedance
Generators	Source impedance and damping
Background distortion	Existing harmonic voltage
Earthing and zero sequence	Important for triplen and unbalanced harmonics
Cable sheaths and bonding	Can create asymmetry and coupling

Balanced and unbalanced modelling decision

The decision between balanced and unbalanced modelling should be based on the network and the study objective. A balanced model may be acceptable when the network is symmetrical, the sources are balanced, the filters are symmetrical, and only positive-sequence performance is needed.

An unbalanced model should be considered when line or cable geometry is asymmetrical, the network is untransposed, phase-wise harmonic limits are important, zero-sequence or negative-sequence harmonics may propagate, resonance occurs near a critical harmonic order, or the study involves long cable systems or cross-bonded cables.

Practical recommendation

As a practical rule, a balanced model is acceptable for early screening where the network is symmetrical and the study is focused on positive-sequence harmonic behaviour. An unbalanced model should be used where the result may be affected by phase asymmetry, untransposed overhead lines, long cable circuits, cross-bonding, single-phase loads, zero-sequence paths or phase-specific compliance limits.

Section 13

Reporting and summary

The choice of harmonic study method should be justified in the report. A statement such as “a harmonic study was performed” is not sufficient. A better statement names the domain, the representation and the source model — for example:

Examples of a clear method statement

“A balanced frequency-domain harmonic penetration study was performed using fixed harmonic current injections.”

“An unbalanced three-phase frequency scan was performed to capture cable-system asymmetry and inter-sequence coupling.”

In short, a robust harmonic study should clearly state:

the selected study domain;
whether the model is balanced or unbalanced;
the harmonic source representation;
the frequency range and harmonic orders assessed;
the network operating cases;
the equipment models used;
the assumptions and limitations.

This tells the reader what the study can and cannot represent. The report should clearly state the study domain; the balanced or unbalanced representation; the harmonic source model; the frequency range; the network operating cases; the equipment models; and the limitations. Without this information, harmonic study results can be misunderstood or over-interpreted.

To summarise: harmonic studies may be performed in the frequency, time, hybrid or harmonic domain. The frequency domain is the most widely used approach for practical planning studies because it is fast, robust and efficient for steady-state harmonic assessment. Its methods are the frequency scan (network impedance versus frequency, mainly to identify resonance, \(Z_h=V_h/I_h\)), harmonic penetration (how harmonic current injections propagate and create harmonic voltages, \([Y_h][V_h]=[I_h]\)), and harmonic load flow (the operating-point-dependent solution). Balanced studies are simpler and may be adequate for symmetrical systems; unbalanced studies are more detailed and should be used where phase asymmetry, cable geometry, untransposed lines, zero-sequence behaviour or resonance effects are important. Time-domain methods suit nonlinear devices, converter controls, transformer saturation and active-filter control; hybrid and harmonic-domain methods are powerful but more complex and less common in routine work.

Key message

Start with frequency-domain methods for planning and screening, and move to time-domain, hybrid or harmonic-domain methods only when the required physical behaviour cannot be represented accurately in the frequency domain. A robust harmonic study should specify the study domain, the method, the balanced or unbalanced representation, the source model, the network cases, the frequency range and the limitations — only then can the results be interpreted correctly and used confidently for compliance assessment, resonance screening, filter design and harmonic mitigation.

Part One Reading now

Harmonic Study Domains & Modelling Approaches

The four study domains, the frequency-domain family of methods, balanced versus unbalanced modelling, and how to choose a method for the question being asked.

Series progress 1 of 11

Part Two

Overhead Line Modelling for Harmonic Studies

Lumped, cascaded and distributed line models; skin effect and frequency-dependent parameters; zero-sequence impedance; shunt capacitance and compensation.