Chapter 1: An Overview of
Power System Harmonic Analysis
Tutorial on Harmonics Modeling and Simulation
Contributors: W. Xu and S. Ranade
1
IEEE PES General Meeting, Tampa FL
June 24-28, 2007
Conferência Brasileira de Qualidade de Energia
Santos, São Paulo, Agosto 5-8, 2007
Chapter 1: An Overview of Power System Harmonic Analysis
Outline
•
Status and methods of harmonic analysis
• Modeling of power system components
• Algorithms for harmonic analysis
•
New challenges of harmonic analysis
• Analysis of systems with distributed harmonic sources
• Modes of harmonic resonance
• Analysis of interharmonics
•
Summary
2
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
Santos, São Paulo, Agosto 5-8, 2007
Chapter 1: An Overview of Power System Harmonic Analysis
Status and methods of harmonic analysis
Methods:
1) Frequency scan
2) Harmonic power flow
Models:
1) Harmonic source: current source model
2) Non H-source: linear impedance model
Variations:
1) Single-phase versus multiphase
2) Iterative versus non-iterative H power flow
Applications: Systems with limited number of H-sources and
the sources are typically large in size
3
IEEE PES General Meeting, Tampa FL
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Chapter 1: An Overview of Power System Harmonic Analysis
Modeling of harmonic loads as current sources
h=2,H
I
V1 I1
+
VFD
=
-
P+jQ
V
+
I h  I1
+
-
-
VFD load
Vh Ih
+
VFD model at
60Hz
I h _ spc
VFD model at
harmonic freq.
h  h _ spc  h1_ spc
I1_ spc
spc = given spectrum data
4
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
Example of source modeling
SVFD  P  jQ  100  j 20(kVA) V1  25/ 30  14.430(kV )
Harmonic
Order
1
5
7
11
13
17
19
23
25
Typical Spectrum
I1 
SVFD
3V
I h _ spc (%)
 h _ spc ()
Harmonic Current
Source
I h ( A)
 h ()
100
18.24
11.9
5.73
4.01
1.93
1.39
0.94
0.86
0
-55.68
-84.11
-143.56
-175.58
111.39
68.30
-24.61
-67.64
2.355
0.4296
0.2802
0.1349
0.0944
0.0455
0.0327
0.0221
0.0203
I5  I1  I5_ spc %  2.355*18.24%  0.4296 A
 2.355  11.31 ( A)
-11.31
-112.23
-163.28
-267.97
-322.61
-80.88
-146.59
-284.74
-350.39
5  5_ spc  51_ spc  55.68  5  (11.31)  112.23
5
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
Harmonic analysis methods
Objectives
• Check if resonance exists
• Check harmonic distortion levels (safe equipment operation)
• Filter design
• Compliance with standards
Two types of assessments:
 Frequency response check
resonance
filter design
(Frequency scan)
 Distortion level calculation
(harmonic power flow)
compliance check
equipment operating conditions
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Chapter 1: An Overview of Power System Harmonic Analysis
Frequency scan analysis
1
Frequency Scan:
Determine the frequency response of a
network at a given bus
 V1   Y11 . . . Y1N 
V   Y . . . Y 
2N 
 2   21
 ..    .. .. .. .. .. 
  

..
..
..
....
....
......
  

VN  YN 1 .. .. .. YNN 
1
Network
Z
1
0
 
.. 
 
.. 
0
f
7
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
Santos, São Paulo, Agosto 5-8, 2007
Chapter 1: An Overview of Power System Harmonic Analysis
Harmonic power flow analysis
Objective: compute harmonic distortion levels for a given operating
condition
There are many harmonic power flow algorithms proposed. Here we discuss
the most useful algorithm.
• Current source model for harmonic sources
• Frequency domain
• Non-iterative
What is known for solving the problem
• Fundamental frequency power flow results (I1 and q1).
• Typical spectrum of harmonic sources (Ih-spc, qh-spc)
• System Y(h) matrix, h=harmonic number
• Current source model described earlier
8
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
Harmonic power flow analysis
Solution steps
1) Compute 60Hz power flow
2) Determine drive current (I1 and q1)
3) Determine drive harmonic current I(h) using the formula and
typical drive spectrum
4) With known Y(h) matrix and drive current I(h), compute
nodal voltage V(h) and branch current IB(h)
5) Compute harmonic indices (THD, IHD) using the V(h), IB(h)
results.
9
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Chapter 1: An Overview of Power System Harmonic Analysis
Harmonic power flow analysis - other algorithms
• Time domain algorithm (e.g. EMTP simulation) or hybrid algorithm
• Iterative algorithms (frequency domain)
F( [V1], [V2],...,[Vn], [I1], [I2], ..., [In],C) =0
1) Newton method
2) Harmonic iteration method (see the diagram below)
Linear network
(including power
flow constraints)
Bus voltages
Harmonic Source
(non-linear)
Current source
10
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges
Distributed harmonic sources
Fluctuation of harmonic distortions with time
Concerns on interharmonics
Need to identify system deficiency more efficiently
Need to revisit some of the modeling assumptions
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IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 1 - distributed harmonic sources
The harmonic-production characteristics of the sources will affect each other.
(attenuation and diversity effects)
50
45
Traditional method
40
Iterative method
Voltage THD (%)
35
30
25
20
15
10
5
Actual results
0
0
20
40
60
80
100
120
140
Bus number
The harmonic sources may also vary randomly.
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 1 - distributed harmonic sources
Harmonic attenuation effect
100
Ih/I1 (%)
80
60
40
20
0
3
5
7
9
Harmonic Order
11
13 0
5
10
15
20
25
30
VTHD (%)
New harmonic analysis methods need to take into account the characteristics
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 2 - analysis of harmonic resonance
Impedance (pu)
4
Bus 9
3
Bus 10
2
Bus 7
1
Bus 5
0
0
5
10
15
20
25
30
Frequency (pu)
· Which bus can excite a particular resonance more easily?
· Where the resonance can be observed more easily?
· What are the components involved in the resonance?
· How far the resonance can propagate in a system?
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
Santos, São Paulo, Agosto 5-8, 2007
Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 2 - analysis of harmonic resonance
XL
V  Y 1I  (
XC
V
I
1
1 1

) I
jX L jX C
If this term = 0 => Resonance
[V ]  [Y ]1[ I ]
• Some elements of [Y]-1 are large (the extreme case is [Y]-1= )
• Implies that [Y] approaches singularity (something like [Y]=0)
• The singularity of [Y] can only be caused by one or more
eigenvalues of the [Y] matrix = 0.
IEEE PES General Meeting, Tampa FL
June 24-28, 2007
Conferência Brasileira de Qualidade de Energia
Santos, São Paulo, Agosto 5-8, 2007
Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 2 - analysis of harmonic resonance
Eigen-decomposition
of the Y matrix:
1
[Y ]  [T ] [][T ]
Left
eigenvector
matrix
1
[V ]  [Y ] [ I ]

Eigenvalue
matrix
Right
eigenvector
matrix
1
[U ]  [] [ J ]
[U]=[T][V] -- called modal voltage
[J] =[T][I] -- called modal current
[L] -- can be called modal Y matrix
IEEE PES General Meeting, Tampa FL
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 2 - analysis of harmonic resonance
1
U1  1
0
0
0   J1 
 
U  
1
0 2
0
0  J2 
2
1


[U ]  [ ] [ J ] 

 ...   0
0 ... 0   ... 
  

1  
U
J
0
0 n   n 
 n   0
• Assume l1 is the eigenvalue approaching zero
• modal current J1 will lead to a large modal voltage U1
• Other modal voltages are not affected (since they are
decoupled from l1)
IEEE PES General Meeting, Tampa FL
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 2 - analysis of harmonic resonance
20
Bus 9
3
Modal impedance (pu)
Impedance (pu)
4
Bus 10
2
Bus 7
1
Bus 5
0
Resonance mode
15
10
5
0
0
5
10
15
20
Frequency (pu)
Physical domain
25
30
0
5
10
15
20
25
30
Frequency (pu)
Modal domain
Summary: In the modal domain, it is much easier to find the ‘locations’ or ‘buses’
(i.e. the modes) that are related to a resonance
Once we know the resonance mode, we can find the buses most affected by the
reassurance - based on the eigenvector information
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 2 - analysis of harmonic resonance
Harmonic=5.9
13
Participation of components
in a resonance
14
12
11
C
G
1
6
10
9
8
SVC
5
4
2
G
7
3
Converter
Participation of buses
in a resonance
IEEE PES General Meeting, Tampa FL
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Conferência Brasileira de Qualidade de Energia
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 3 - analysis of interharmonics
DC link reactor
Source
Inverter
Converter
60Hz ripple
60Hz
Motor
50Hz ripple
50Hz
• Interharmonics produce flicker
• Frequency of interharmonic varies with the drive operating condition
IEEE PES General Meeting, Tampa FL
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 3 - analysis of interharmonics
An interharmonic-producing drive cannot be modeled as an interharmonic
current source
IDC2
Source
Converter
IAC2
60Hz
VDC2
Inverter
Motor
50Hz
•VDC2 has ripples associated with the motor frequency
•VDC2 produces IDC2 through some impedances (including supply system Z)
•IDC2 is rectified (or penetrate) into the AC side to produce IAC2
•Therefore, interharmonic current of IAC2 is affected by some impedances
IEEE PES General Meeting, Tampa FL
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Chapter 1: An Overview of Power System Harmonic Analysis
New challenges 3 - analysis of interharmonics
Sequence characteristics of interharmonics
800
Positive sequence
Negative sequence
Interharmonic Frequency (Hz)
700
Practical operating range
600
500
B
400
X
300
Y
A
200
100
0
0
10
20
30
40
50
60
70
Drive Output Frequency (Hz)
IEEE PES General Meeting, Tampa FL
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Chapter 1: An Overview of Power System Harmonic Analysis
Summary
• Harmonic analysis has become a relatively mature
area. This tutorial will focus on the well-established
methods
• It is important to note that there are still many subjects
remaining to be explored. Three examples have been
used to demonstrate the possible developments in the
area
IEEE PES General Meeting, Tampa FL
June 24-28, 2007
Conferência Brasileira de Qualidade de Energia
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