IEC TS 63222-4:2026

IEC TS 63222-4 ®
Edition 1.0 2026-04
TECHNICAL
SPECIFICATION
Power quality management -
Part 4: Harmonic analysis on public electric power network
ICS 29.240.01; 91.140.50 ISBN 978-2-8327-1185-9

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CONTENTS
FOREWORD . 4
1 Scope . 6
2 Normative references . 6
3 Terms and definitions . 6
4 Harmonic models of public electric power network . 8
4.1 General . 8
4.2 Typical electric equipment models . 8
4.2.1 Overhead lines . 8
4.2.2 Underground cables . 9
4.2.3 Power transformers . 9
4.2.4 Shunt capacitors/reactors . 9
4.2.5 Synchronous generators . 10
4.2.6 Loads . 10
4.3 System equivalence requirements . 11
5 Methods for analysis of harmonic transfer characteristics . 12
5.1 General . 12
5.2 Methods of harmonic resonance analysis . 12
5.2.1 General . 12
5.2.2 Frequency scanning . 12
5.2.3 Modal analysis method [23], [24] . 14
5.3 Harmonic flow calculation . 17
5.3.1 General . 17
5.3.2 Alternating iteration method . 17
5.3.3 Decoupling method [26] . 19
5.3.4 Direct method [27] . 19
5.3.5 Indicators based on harmonic flow calculation . 19
6 Harmonic analysis on public electric power network . 20
6.1 General . 20
6.2 Background data collection and constraints . 20
6.2.1 Background data of system topology . 20
6.3 Harmonics analysis objectives . 20
6.4 Procedures of harmonics analysis . 21
6.4.1 Assessment of harmonic levels and power quality . 21
6.4.2 Planning connection of significant harmonic sources and/or resonant
plant . 21
6.4.3 Design of reconstruction schemes and determination of new operation
modes of public electric power network. 22
6.4.4 Analysis of post-accident issues related to harmonics . 22
Annex A (informative) Harmonic models of public electric power network . 23
A.1 Overhead lines. 23
A.2 Underground cables . 26
A.3 Power transformer . 27
A.4 Shunt capacitors/reactors . 32
A.5 Synchronous generators . 33
A.6 Loads . 35
Annex B (informative) Cases of harmonic analysis . 41
B.1 Resonance analysis by frequency scanning method . 41
B.2 Modal analysis method for resonance analysis . 43
B.3 Harmonic flow calculation . 46
Bibliography . 48

Figure A.1 – Lumped π model of an overhead line . 23
Figure A.2 – Distributed π model of an overhead line . 24
Figure A.3 – Comparison of equivalent parameters of overhead lines under different
lengths . 26
Figure A.4 – Comparison of equivalent parameters of underground cables under
different lengths . 27
Figure A.5 – Power transformer model 1: Electra 167 [7] . 27
Figure A.6 – Power transformer model 2: IEEE Std. 399 [8] . 28
Figure A.7 – Power transformer model 3: Electra-164 [9] . 29

Figure A.8 – Power transformer model 4: Arrillaga [10] . 30

Figure A.9 – Power transformer model 5: Funk [10] . 30
Figure A.10 – Power transformer model 6: IEEE [12] . 31
Figure A.11 – Calculated transformer resistance and reactance . 32
Figure A.12 – The equivalent circuit of a capacitor . 32
Figure A.13 – The equivalent circuit of a reactor . 32
Figure A.14 – Synchronous generator harmonic impedance . 33
Figure A.15 – Synchronous generator model 1: IEEE . 34
Figure A.16 – Synchronous generator model 2: Electra 167 . 34
Figure A.17 – Calculated generator resistance and reactance . 35
Figure A.18 – Load model 1: (IEEE; "RL Series"). Series . 36
Figure A.19 – Load model 2: (IEEE; "RL//"). Parallel . 36
Figure A.20 – Load model 3: (IEEE). Skin effect . 37
Figure A.21 – Load model 4: (IEEE). Induction motors . 37
Figure A.22 – Load model 5: (IEEE). CIGRE/EDF . 38
Figure A.23 – Load model 6: (IEEE; "2RL//") . 39
Figure A.24 – A typical 11 kV distribution system . 40
Figure A.25 – System harmonic impedance seen at the load bus . 40
Figure B.1 – 3-Bus network . 41
Figure B.2 – Magnitude-frequency curves of harmonic self-impedance . 41
Figure B.3 – AHU and AHC when j = 1. 42
Figure B.4 – AHU and AHC when j = 2. 42
Figure B.5 – AHU and AHC when j = 3. 43
Figure B.6 – Curves of modal impedance with frequency . 44
Figure B.7 – Comparison of component sensitivity under mode 3 . 45
Figure B.8 – Comparison of component normalized sensitivity under mode 3 . 45
Figure B.9 – Comparison of voltage waves . 46
Figure B.10 – 5-bus network . 46

Table A.1 – Values for Coefficients a , a , a and b (requirement: a + a + a = 1) . 30
0 1 2 0 1 2
Table A.2 – Coefficients for Transformer Model 5 [11] . 31
Table B.1 – Resonance frequencies of the 3-bus network . 43
Table B.2 – Participation factors of the three-node network . 44
Table B.3 – Modal sensitivity and normalized sensitivity under mode 3 . 44
Table B.4 – Voltage indicators . 46
Table B.5 – Current and power loss indicators . 47

INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
Power quality management -
Part 4: Harmonic analysis on public electric power network

FOREWORD
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IEC TS 63222-4 has been prepared by IEC technical committee 8: System aspects of electrical
energy supply. It is a Technical Specification.
The text of this Technical Specification is based on the following documents:
Draft Report on voting
8/1747/DTS 8/1791/RVDTS
Full information on the voting for its approval can be found in the report on voting indicated in
the above table.
The language used for the development of this Technical Specification is English.
This document was drafted in accordance with ISO/IEC Directives, Part 2, and developed in
accordance with ISO/IEC Directives, Part 1 and ISO/IEC Directives, IEC Supplement, available
at www.iec.ch/members_experts/refdocs. The main document types developed by IEC are
described in greater detail at www.iec.ch/publications.
A list of all parts in the IEC 63222 series, published under the general title Power quality
management, can be found on the IEC website.
The committee has decided that the contents of this document will remain unchanged until the
stability date indicated on the IEC website under webstore.iec.ch in the data related to the
specific document. At this date, the document will be
– reconfirmed,
– withdrawn, or
– revised.
1 Scope
This part of IEC 63222 is a Technical Specification. IEC TS 63222-4 specifies the requirements
of the models, methods and procedures for harmonic analysis on the public electric power
th
network. This document is applicable to harmonic analysis up to 40 harmonic at high, medium
and low voltage of the public electric power network with nominal frequency of 50 Hz or 60 Hz.
NOTE 1 The boundaries between the various voltage levels can be different in different countries/regions. In this
document, the following terms for system nominal voltage U are used:

N
– Low voltage (LV) refers to U ≤ 1 kV;
N
– Medium voltage (MV) refers to 1 kV < U ≤ 35 kV;
N
– High voltage (HV) refers to 35 kV < U ≤ 230 kV.
N
NOTE 2 Because of existing network structures, the boundary between medium and high voltage can be different
in some countries/regions.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies.
For undated references, the latest edition of the referenced document (including any
amendments) applies.
IEC TR 61000-3-6:2008, Electromagnetic compatibility (EMC) - Part 3-6: Limits - Assessment
of emission limits for the connection of distorting installations to MV, HV and EHV power
systems
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology databases for use in standardization at the following
addresses:
– IEC Electropedia: available at https://www.electropedia.org/
– ISO Online browsing platform: available at https://www.iso.org/obp
3.1
harmonic order
h
integral number given by the ratio of the frequency of a harmonic to the fundamental frequency
[SOURCE: IEC 60050-161:1990, 161-02-19]
3.2
harmonic frequency
f
frequency which is an integer multiple greater than one of the fundamental frequency or of the
reference fundamental frequency
[SOURCE: IEC 60050-551-20:2001, 551-20-05]
3.3
total harmonic distortion
THD
THD
Y
ratio of the RMS value of the sum of all the harmonic components ( Y ) up to a specified order
Hh,
( h ) to the RMS value of the fundamental component ( Y ):
max H,1
h
max
Y
Hh,
THD = ()
Y ∑
Y
H,1
h=2
Note 1 to entry: The symbol Y is replaced, as required, by the symbol I for currents or by the symbol U for
voltage.
Note 2 to entry: The value of h is 2 and that of h is 40 for the purpose of this document.
min max
[SOURCE: IEC 61000-4-7:2009, 3.2.1]
3.4
fundamental component
component whose frequency is the fundamental frequency
[SOURCE: IEC 61000-4-30:2021, 3.7]
3.5
fundamental frequency
frequency in the spectrum obtained from a Fourier transform of a time function, to which all the
frequencies of the spectrum are referred
Note 1 to entry: In case of any remaining risk of ambiguity, the fundamental frequency may be derived from the
number of poles and speed of rotation of the synchronous generator (s) feeding the system.
[SOURCE: IEC 61000-4-30:2021, 3.8]
3.6
harmonic component
any of the components having a harmonic frequency
Note 1 to entry: Its value is normally expressed as an RMS value. For brevity, such component may be referred to
simply as a harmonic.
[SOURCE: IEC 61000-2-2:2002, 3.2.4]
3.7
amplification index of harmonic voltage
AHU
AHU
fij
 
ratio of the harmonic voltage U of node i to the harmonic voltage U of node j when the
fi fj
harmonic current in per unit or the harmonic voltage in per unit is injected into node j at
frequency f:

U
fi
AHU =
fij

U
fj
3.8
amplification index of harmonic current
AHC
AHC
fab
ratio of the harmonic current on the branch between two nodes (node a, b) to the injected
harmonic current of node j when the harmonic voltage or the harmonic current in per unit at
frequency f is applied to node j:

I
fab
AHC =
fab

I
fj
3.9
participation factors
PF
value calculated by modal analysis method to find the resonance centre
Note 1 to entry: The node with the largest value is the resonance centre, which means the node is with the largest
harmonic voltage at the resonance frequency and the highest degree of participating the resonance.
4 Harmonic models of public electric power network
4.1 General
The harmonic models of classical network elements, such as overhead lines, cables and power
transformers, have been studied for years and a good summary can be found in CIGRE
Technical Brochure C4/B4 [1] . This clause provides a recommendation on the model selection
for different applications of harmonic studies.
4.2 Typical electric equipment models
4.2.1 Overhead lines
An overhead line comprises series parameters (resistance and reactance) and shunt
parameters (conductance and susceptance), which are distributed over the entire length of the
line and are affected by frequency. The model of the overhead line for harmonic studies
considering long-line effect and skin effect are given in Clause A.1.
A simple representation of an overhead line is a lumped π model shown in Figure A.1. When
the length of the line increases, the representation of long-line effects (i.e. distributed
parameters) need to be improved by using hyperbolic functions, and the distributed π model is
shown in Figure A.2. The following should be followed to determine whether the long-line effect
needs to be considered.
• For the harmonic flow analysis: The selection of the lumped π model and the distributed π
model depends on the length of the line. If the length of the overhead line is less than
12 000/f km, the lumped π model can be used. Otherwise, the distributed π model should
be adopted.
• For the harmonic resonance analysis: The lumped π model is only capable of representing
the first resonant point, and the frequency of this resonance is always slightly below the first
resonant frequency obtained with the accurate distributed π model. Therefore, the use of a
distributed π model (including long line effects) is recommended as the default option in the
resonance analysis.
___________
Numbers in square brackets refer to the Bibliography.
The model of an overhead line considering skin effect can be calculated by equations (A.7) and
(A.8). Recommendations for whether to consider skin effect in the overhead line modeling are
as follows:
• For the harmonic flow analysis: The conductor resistance is much smaller than the
reactance, so the skin effect does not have an obvious impact on the flow of harmonics.
This means that the skin effect may be ignored if a precise harmonic flow analysis is not
required.
• For the harmonic resonance analysis: At (or near) resonance conditions, where the line
impedance is dominated by the resistive component, changes in the resistive component of
the line have a significant impact on the calculated resonance peak. Neglecting skin effect
leads to underestimation of circuit damping at resonant frequencies, and the error
introduced will increase with the frequency. In addition, skin effect would result in a slight
upward shift of resonant frequencies caused by an effective reduction in the internal
inductance of the phase conductors, and this effect is getting noticeable at high frequencies.
4.2.2 Underground cables
The underground cable model is essentially similar to the overhead line model shown in
Clause A.1. The consideration of skin effect is consistent. In terms of long-line effect, the
lumped π model shown in Figure A.1 can be used if the length of the underground cable is less
than 3 000/ f km. Otherwise, the distributed π parameter model shown in Figure A.2 needs to
be used [2]. Compared with overhead lines, cables are more likely to cause resonance at low
frequencies.
4.2.3 Power transformers
Accurate representation of power transformer is paramount for performing meaningful harmonic
studies in power systems. A transformer can be represented by a series combination of
resistance and inductive reactance, and both components are frequency dependent. Clause A.3
gives six models to capture this feature. For Models 2 to 6, the demagnetization effect is
considered to be weak and the reactance approximately increases linearly with frequency,
which means that the selection of the transformer models does not affect the location of the
resonant frequencies, but only changes the amplitude of the resonance peak. For Model 1, the
reactance does not increase linearly with frequency. Model 1 takes into account the effect of
demagnetization of induced eddy current in the wire on the decrease of leakage inductance
with the increase of frequency. The reactance can be calculated by equation (A.14).
All models provide increasing damping with frequency. Larger differences in performance are
observed when the main resonances appear at the higher frequency range. Each model has
different accuracy in different working conditions. If high accuracy is needed in the harmonic
studies, it is recommended to obtain frequency dependent characteristics for R and X from the
transformer manufacturer. In the absence of measurements to validate models, the analysis in
[1] suggests the use of Model 5 with default parameters or Model 1 with 50/60 Hz datasheet
X/R value as options with reasonable results.
Note the influence of the transformer models on the harmonic studies is related to the distance
between the node and the transformer. Therefore, the choice of transformer model is of key
importance for nodes where the harmonic impedance is dominated or influenced by power
transformers. For nodes that are electrically distant from power transformers, the selection of
a transformer model does not have material impact on the calculation results.
4.2.4 Shunt capacitors/reactors
The parameters of shunt capacitors and reactors in power systems are independent of
harmonics. Their harmonic models are specified in Clause A.4.
4.2.5 Synchronous generators
Two harmonic impedance models are commonly used to represent synchronous generators, as
shown in Clause A.5.
The reactance of both models is assumed to increase linearly with frequency and can be
calculated based on the negative sequence reactance X at the fundamental frequency. For
neg
α
Model 1, the change of resistance with frequency is represented by h ; thus, the value of α
should be appropriately selected (α is in the range of 0,5 to 1,5. α = 0,5 is theoretically correct,
and α = 1,0 was verified through limited tests). For Model 2, the change of resistance with
frequency is represented by √h. Therefore, the model selection does not affect the frequency
of the resonance, but there is a difference in the damping provided at the resonant point. In
general, Model 1 with α = 1,5 provides the highest level of damping, while Model 2 provides the
lowest damping.
Note the influence of synchronous generator model on the harmonic studies is related to the
distance between the node and the generator. For a transmission node near large synchronous
generators, the selected model affects the amplitude of the harmonic resonance. Therefore, the
use of frequency-dependent data from manufacturers is recommended. For a transmission node
where the synchronous generator is far away, the selection of a synchronous generator model
has no material effect on the harmonic impedance at that point.
4.2.6 Loads
Loads have a considerable influence on the harmonic characteristics of the network in terms of
location of resonances and level of damping, and their appropriate modelling is a very important
consideration.
The load models for harmonic analysis of power systems are shown in Clause A.6 [3], where
the fraction (or participation) K of the induction motors into the total load demand P is defined
as the ratio P /P, and P is the induction motor demand. For the application of the load models,
M M
this standard recommends:
• K < 0,1: If K is relatively small, the aggregate linear load behaves as constant impedance
load. In this case, Models 1, 2, and 3 provide a good approximation of the load harmonic
impedance, with Models 2 and 3 resulting in higher damping. Model 3, in addition, provides
frequency dependent parameters (i.e. skin effect).
• 0,1 < K < 0,3: Model 4 and 5 can be adopted. The harmonic impedance of the motor load in
Models 4 and 5 is jhX . Note that this approximation is not accurate for harmonic orders
IM1
below 5. These models neglect the harmonic attenuation and therefore are to be used if the
participation of the induction motors is moderate.
• K > 0,3: If the load consists predominately of motors, a more accurate representation results
by including a resistor in series with X . The harmonic impedance of the motor load
IM1
becomes R + jhX . The value of R is found directly from the resistance of the rotor
1 IM1 IM1
circuit. Thus, if the locked rotor circuit has a quality factor Q = X / R = K , then
IM1 IM1 o
R = X / K . K assumes values of approximately 8.
IM1 IM1 o o
Depending on where the models are connected and which models are used, different load
models will have a significant impact on the harmonic characteristics. For a highly accurate
assessment of the damping effect of loads, detailed time-domain modelling might occasionally
be preferable. Neglecting loads may result in a pessimistic evaluation of harmonic distortion.
4.3 System equivalence requirements
In order to analyse the harmonic transfer characteristic through the network accurately, it is
important to have a good representation of all network elements. For many studies, however,
the extent of the network model may be limited for a number of practical reasons. In such cases,
the rest of the network for which the system elements (lines, cables, transformers, loads, etc.)
are not represented in detail, can be represented by an equivalent network. The equivalents
are generally provided as either Thévenin or Norton circuit, which are effectively identical in
their performance and can be converted from one to another.
To get the equivalent impedance of the Thévenin circuit, a rough estimation can be achieved
by using power frequency network equivalents (short circuit impedance or other variations).
However, such an equivalent is only adequate when the external network does not exhibit any
resonances within the frequency range of interest or when the external network is electrically
so far from the area of study that does not have significant impact on the harmonic impedance
at the buses of interest. Therefore, it is recommended to obtain the equivalent impedance of
the Thévenin circuit using the frequency dependent network equivalent from a detailed network
model. However, in most cases, the detailed network model is not available in a simulation tool.
Then the network can be modelled in detail up to a certain distance from the point of interest
and with power frequency network equivalent, like Thévenin voltage source, beyond this
distance. In terms of minimum number of nodes or distance to be accurately represented, a
sensitivity study progressively extending the network model until the results from two
consecutive iterations converge is suggested [4].
The extent of the network represented in detail must be sufficient to incorporate all
contingencies and operation conditions that need to be studied. The following should be
considered in the study:
• In the case of a planning study, it is recommended to consider at least the year of connection
as well as that corresponding to some years later or the last year of the planning horizon;
In the case of an operational study, it is important to replicate as much as possible the
conditions that can be the cause of trouble or investigation.
• To reduce the number of cases, the contingencies up to the third node away from the point
of interest are recommended, unless there is evidence that other contingencies beyond that
boundary could result in more onerous conditions within the range of harmonic frequencies.
• At least two or three system demand levels: minimum, intermediate and maximum should
be considered. The objective is to capture the changes in system resonances and damping
associated with the various levels and composition of the load as well as the status of
reactive compensation equipment.
• Only realistic operating conditions should be considered in order to limit the computational
burden and, more importantly, to avoid overly-pessimistic results.
The calculated harmonic impedances under different contingencies, demand levels and
operating conditions can be presented as envelopes in an R-X plane encompassing all possible
values. The harmonic impedance data can then be displayed graphically in a single envelope
including all frequencies of interest or as a family of envelopes each comprising one or more
harmonic frequencies. To derive the harmonic impedance envelopes, the following should be
considered.
• A suitable frequency step resolution needs to be selected in order to capture all resonances
in the frequency scan calculations. In general, a frequency step no larger than 5 Hz should
be adopted for a 50 Hz system.
• Since low harmonic orders do not tend to exhibit large resonances, the impedance can be
adequately defined with individual harmonic order envelopes or with narrow frequency
bands. At higher frequencies, it is necessary to widen the size of the frequency bands to
confidently capture possible resonance shifts and/or modelling uncertainties.
• The introduction of some overlapping between subsequent envelopes is recommended as
a safety margin to account for uncertainties.
• The shape of the envelopes, e.g., circles, sectors and polygons, should be selected to
minimize "empty" areas without realistic impedance points. Polygons are recommended as
they can provide a very close fit to a set of calculated impedance points.
The background harmonic voltages in the Thévenin circuit should be measured for a
"representative measuring period" to derive a representative signature of the background
distortion at that location. The minimum duration requirement is system dependent, but in
general it can be stated that measuring should be conducted for as long as possible, ideally for
not less than three months, including measurements of all three phases. As it is often the case,
some nodes cannot be monitored (or have not been built yet), therefore background harmonic
estimation is needed, which can be done by harmonic state estimation approach or "Constant

[5].
Voltage behind Thévenin Equivalent" approach
5 Methods for analysis of harmonic transfer characteristics
5.1 General
The harmonic currents caused by non-linear equipment will spread in the power system.
Harmonic currents lead to increased losses and heating in numerous electrical
devices/installations. High levels of harmonic voltage and current distortion may occur in the
resonant condition, which will endanger the safety of the system and equipment.
This document aims for harmonic analysis oriented to engineering situations of public electric
power network balanced, where frequency-domain methods based on the
impedance/admittance matrixes are employed, including frequency scanning, modal analysis
and harmonic flow calculation, to analyse harmonic transfer and resonance characteristics,
node voltage distortion and branch harmonic current flows, etc.
5.2 Methods of harmonic resonance analysis
5.2.1 General
The method of harmonic resonance analysis is not only to obtain the resonant frequencies,
quantify the severity of harmonic resonance amplification but also to investigate the
participation degrees of nodes for the resonances. Two kinds of methods are often used for the
harmonic resonance analysis.
• Frequency scanning method, is to obtain all the resonant frequencies and calculate the
resonant amplification indices, which can represent the resonant influence degree.
• Modal analysis method is to obtain all the parallel resonant frequencies and participation
factors of resonance buses and sensitivity of influencing components on the resonances.
5.2.2 Frequency scanning
5.2.2.1 Magnitude-frequency curves of harmonic self-impedance
The resonant frequency can be obtained from magnitude-frequency curves of harmonic self-
impedances. The steps of frequency scanning method are as follows:
a) Based on harmonic models in chapter 4, the model in a single-line diagram of public electric
Y
power network with n nodes is established and the node admittance matrix is
fn×n
Z
obtained in per unit. Obtain the nodal impedance matrix at frequency f calculated by
fn×n
Y ;
the inverse of
fn×n
b) Take the appropriate frequency resolution as the change step in the frequency band to scan
Z
the self-impedance at node j for frequency f in ;
fn×n
NOTE The change step can be selected based on actual engineering requirements. The range from 1 Hz to
10 Hz is recommended;
Z
c) Calculate (j=1,2,…,n) and plot the magnitude-frequency curve of self-impedance of
fjj
each node in the frequency band scanned.
The frequencies corresponding to the peak values of the magnitude-frequency curves of self-
impedance are the parallel resonance frequencies and the frequencies corresponding to the
valley values of the magnitude-frequency curves of self-impedance are the series resonance
frequencies in the analyzed public electric power network.
5.2.2.2 Calculation of harmonic amplification index
Z
AHU can be calculated based on , as shown in equation (1).
fn×n

UZ
fi fij
AHU
(1)
fij

UZ
fj fjj
where
Z
is the transfer impedance between nodes i (i=1,2,…n, and i≠j ) and j for frequency f,
fij
p.u., which shows the mutual impedance between two different nodes;
Z
fjj is the self-impedance at node j for frequency f, p.u.
The harmonic voltage at node i is greater than that at node j where the harmonic current injected
AHU > 1
and a non-local resonance has occurred if . The severity of harmonic resonance and
fij
influence area of resonance can be confirmed according to the harmonic voltage amplification
indices.
Z
AHC can be calculated based on , as shown in equation (2).
fn×n

I ZZ−
an∈ {1,., }
fab
faj fbj
AHC
(2)
fab

b ∈≠{0,1,.,n} and b j
Z
I
fab
fj
where

I is the harmonic current phasor at node j for frequency f, p.u.;
fj
Z is the impedance of the branch between node a ( ) and node b
an∈ {1,., }
fab
(b ∈≠{0,1,.,n} and b j where 0 means reference node), p.u.;

I
is the harmonic current phasor on the branch between node a and node b at
fab
frequency f.
The harmonic current on the branch between node a and node b is greater than that at the node
AHC > 1
j injected by the harmonic voltage, which the harmonic current is amplified if . The
fab
influence area and the corresponding severity of the harmonic resonance can be confirmed
according to the amplification index of harmonic current.
==
==
5.2.3 Modal analysis method [23], [24]
5.2.3.1 Resonance model analysis and participation factors
The parallel resonance analysis based on modal analysis is as follows:
a) Obtain the network structure and the impedance or admittance of each branch under the
frequency band which needs to be paid attention;
b) Establish the node admittance matrix Y (n is the node number of the analysed public
nn×
electric power network) at the lowest frequency under the analysed frequency band;
c) Diagonalize the admittance matrix Y , as equation (3);
Y = LTΛ (3)
where
L is the transformation matrix for the matrix diagonalization, each column of which the
eigenvector is corresponding to an eigenvalue of
Y
−1
T is the inverse matrix of transformation matrix, TL=
Λ is a diagonal matrix, diagonal elements λ (i=1,2,…,n) of which are the eigenvalues
i
of , as equation (4);
Y
λ 0 . 0
 
 
0 λ . 0
 
Λ TYL
(4)
 
  
 
0 0 . λ
 n 
−1
Z
d) Define Λ as modal impedance matrix denoted by shown as equation (5);
λf,
−1

λ 00
Z 00 
λ1

 
−1
00Z  
00λ 
λ2 −1
 
Z Λ
λf,  (5)
 
   
  

 

00  Z −1
 λn 

f 00 λ
n
f
e) Set frequency resolution Δf as the step size, repeat steps b)~e) and calculate the modal
impedance matrix Z of the system under the observed frequency band. The magnitude-
λ
frequency curves of modal impedance are obtained. The frequencies corresponding to the
peak values of the magnitude-frequency curves are parallel resonance frequencies f . The
r
Z
peak Z of diagonal value in the m-th row and the m-th column of is
λf,
λm,f
r r
corresponding to the critical mode m;
== =
==
f) Calculate the sensitivity matrix P
...