Chapter 8: Procedure of Time-Domain
Harmonics Modeling and Simulation
Tutorial on Harmonics Modeling and Simulation
Presenter: Paulo F Ribeiro
Contributors: C. J. Hatziadoniu, W. Xu, and G. W. Chang
1
IEEE PES General Meeting, Tampa FL
June 24-28, 2007
Conferência Brasileira de Qualidade de Energia
Santos, São Paulo, Agosto 5-8, 2007
OUTLINE
1.
Introduction: Relevance of the Time Solution Procedures
2.
The Modeling Approach
•
•
•
•
Harmonic Sources in the Time Domain
Apparatus Modeling
Formulation of the Network State Equation
Harmonic Solution Procedure
3.
Software Demonstration of Harmonic Simulation
4.
Summary and Conclusion
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IEEE PES General Meeting, Tampa FL
June 24-28, 2007
Conferência Brasileira de Qualidade de Energia
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INTRODUCTION
•
Why Time Domain Solution?
•
When is Time Domain Solution Appropriate?
•
How Accurate is Time Domain Solution
Compared to Direct Methods?
•
What are the General Characteristics of a Time
Domain Solution Procedure?
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Conferência Brasileira de Qualidade de Energia
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Why Time Domain Solution?
•
“Time Domain Simulation is preferable to direct
methods in certain line varying conditions
involving power converters and non-linear
devices.”
– It allows detail modeling, especially of non-linear
network elements;
– It allows the assessment of non-linear feedback loops
onto the harmonic output (e.g. study of harmonic
instability in line commutated converters).
•
Example of Direct Methods
– PCFLOH;
– SuperHarm.
4
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When is Time Domain Solution
Appropriate?
•
Calculations of non-characteristic harmonics from power
converters.
•
Calculation of harmonic instability and harmonic interactions
between power converters and the converter control.
•
Harmonic filter design and harmonic mitigation studies.
•
The effect of harmonics on equipment and protection
devices.
•
Real time digital simulations-RTDS of harmonics such as
hardware-in-loop simulations.
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Accuracy of Time Domain Simulation v.
Direct Methods
•
The time response of the system must arrive at a periodic
steady state.
– Quasi periodic or aperiodic response possible under
non-linear feedback control.
•
Sampling and integration errors. The sampling step is
dictated by the highest harmonic order of interest.
•
Modeling errors approximating the non-linear characteristic
of certain apparatuses (e.g. transformer magnetization and
arrester v-i characteristics)
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What are the General Characteristics of a
Time Domain Solution Procedure?
•
Slow Transient Modeling. May use programs
such as EMTP, PSCAD, and SIMULINK. May
incorporate local controls of power converters.
•
Describe a limited part of the system around the
harmonic source.
•
Run simulation until steady state  Use FFT
within the last simulation cycle to compute
harmonics.
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Modeling Approach
•
Harmonic Sources
– Power Converters

Detail representation including grid control and, possibly,
higher level control loops.

Equivalency: Represent as rigid source.
– Non-Linear Devices

Transformer magnetizing and inrush current.

Arrester current in over-voltage operation.
– Background harmonics: Rigid source representation.
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Power Converters: Detail Representation
•
Detail Valve model
•
Surge arrester
representation in studies
of harmonic overvoltages
Ls
Surge
Arrester
•
Snubber
Representation of the grid
control
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Power Converters: Switching Function
id
va
vb
vc
ia
ib
+
vd
ic
-
•
Voltage-Sourced inverters are more suitable for this
representation.
•
Switching function approach:
– Voltage: v a  s a (t )  v d , vb  sb (t )  v d , v c  s c (t )  v d
– Current:
i d  s a (t )  i a  sb (t )  ib  s c (t )  ic
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Non-Linear Devices: Transformer
i
L2
lA
L1
Saturation Characteristic
v
Lo
SA
SB
LA
LB
Lo
iA
•
•
Aircore Inductance
lB
Unsaturated Segment
Saturated
Segment
l(i)
iB
i
Piece-wise Linear representation of the core inductance.
Switching inductance model (flux controlled switches).
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Formulation of the Network Equations
•
•
Pre-integrated Components: Algebraic Equations
State Equations: Numerical Integration
– Piece-wise Linear Equations
– Time Varying Equations
 x L   ALL
 x    A
NL
 N 
 xC   ACL ( x, t )
ALN
ANN
ACN ( x, t )
ALC ( x, t )   x L   0   BL 
ANC ( x, t )   x N    f ( x)   BN   u (t )
ACC ( x, t )   xC   0   BC ( x, t )
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Summary of The Time Domain Procedure
Sart
Network
and
converter
data
Slow
Transient
Modeling
Run Slow
Transient
Program
Run FFT
Fine-Tune
Model
No
Yes
Steady
State?
Met
Criteria?
No
Yes
End
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SIMULINK Demonstrations
•
Converter Simulation Using the Switching
Function
•
Non-Linear Resistor
•
Rigid Harmonic Source
•
Impedance Measurement
•
Network Equivalency
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Converter Simulation Through the
Switching Function
(a)
A
A
B
B
•
Linear Network.
•
Insert the converter as:
C
C
– Voltage source on ac
side.
– Current source on dc
side.
(b)
Inverter DC Side
P1
powergui
P2
Discrete 3-phase
PWM Generator
[Sa]
[Sb]
+
+
Vd
-
Cd
V
Id
s
Continuous
-
Switching Function Generation
•
[Sc]
[Ia]
[Sa]
[Sb]
[Sc]
[Ib]
[Ic]
Inverter AC Side
Incorporate high level
converter controls.
Va
[Sa]
s
[Vd]
+
-
Vb
[Ia]
[Sb]
s
+
4
Multimeter
-
A
A
B
C
B
C
A
B
C
[Ib]
Vc
[Sc]
s
+
[Ic]
Measurements from network
-
[Vd]
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Example of Non-Linear Resistor Using
User-Defined Functions
Continuous
Network Thévenin Equivalent
pow ergui
node 1
From
RT
[i]
i(v)
Series RLC Branch
-
s
C'
v
+
i(v)
+
+
v
-
Voltage Controlled Resistor
+
v
-
v
Non-linear voltage
controlled resistor
i(v )
[i]
v   RT  i(v)  V ( x)
Goto
node 0
•
•
+
-
-
Scope
node 1
V(x)
Lookup Table
Describing i(v)
Voltage Controlled Element: Parasitic capacitance C’
User-defined function describing the i(v) function
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Rigid Harmonic Source Using the sFunction
A
s
+
1
-
Phase A
sfHarm_3ph
s
Rigid Source
-
B
+
2
Phase B
C
s
+
3
-
Phase C
•
S-Function: Calculation of the harmonic current:
i a (t )  I1 cos(t  1 ) 
•
 I n cos(nt   n  n1 )
n 3,5,..,N
Simulation time slows down with increasing order N
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Impedance Scans Using Rigid Harmonic
Sources
•
Basic assumptions:
Procedure:
– Linear Network Model.
– Single driving point (e.g.
location of harmonic
source).
– The harmonic source is
represented by a rigid
current source at predefined harmonic orders.
•
•
1. Inject positive, negative, or
zero sequence current
separately at unit amplitude;
2. Arrive at steady state
3. Obtain bus voltage
4. Apply FFT
1.
Driving point impedance
Driving point impedance
Zkk  jn1  
Transfer impedance
2.
Transfer Impedance
Z mk ( jn1 ) 
18
Vk ( jn1 )
 Vk ( jn1 )
I k ( jn1 )
Vm ( jn1 )
 Vm ( jn1 )
I k ( jn1 )
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Impedance Scan: Transfer Function
Method
•
•
Basic Assumptions
– The impedance is
defined as a current-tovoltage network (transfer)
function:
Procedure
1. Define network as a
subsystem;
2. Define the controlling
signals of the current
sources as the inputs;
Vm ( s)
Z mk ( s) 
I k ( s)
3. Define the voltages at
the buses of interest as
the outputs;
– Network is driven by a
signal-controlled current
source. More than one
inputs can be used.
4. Use the LTI tool box to
obtain the driving and
transfer impedances.
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Impedance Scan: Transfer Function
Method—Example
V_bus1
V_bus2
V_bus3
Injection Source
1
Bus 1
+
Vb1
-
Input
1
s
+
I1
Vb2
aA
I1a
bB
-
I1b
cC
aA
A
B
A
B
bB
C
C
Feeder 1-2: 2 mi
A
B
C
bB
A
B
C
cC
700kVAR
750kVAR
A
B
C
A
B
C
A
B
C
A
B
C
Vb3
Line Impedance data
r'=0.278 Ohm/mi
x'=0.733 Ohm/mi
+
Load 1
2MW
0.7MVAR
A
B
C
Feeder 2-3: 2mi
cC
-
Continuous
Bus 3
aA
A
B
C
s
I1c
3
2
Bus 2
A
B
C
s
Load 2
2MW
0.7MVAR
Load 3
4MW
1.4MVAR
•
Inputs: Signal node 1 (array input: number of input signals is three).
•
Outputs: Voltage at network nodes 1, 2, and 3 (each is an array of three).
Voltage is measured by the voltmeter or the multimeter block
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Network Equivalency
•
It is often desirable to represent a part of the
network (referred to as the external network) by a
reduced bus/element equivalent preserving the
impedance characteristic at one or more buses
(interface or interconnection buses).
•
The part of the network that is of interest can be
represented in detail.
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Network Equivalency Using SIMULINK
•
The procedure replaces the external network by
a TF block representing the driving point
impedance at the interface bus.
•
The TF block is embedded into the network of
interest:
1. Drive the block input by the interface bus voltage;
2. Connect the block output to the input of a signal driven
current source;
3. Connect the current source to the interface bus;
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Network Equivalency: Example
Admittance Phase a
Injection Source
Bus 1
(s+0.1)(s+100)
(s+100)(s+40-800i)(s+40+800i)
s
+
aA
Admittance Phase b
-
I1a
Ih
(s+0.1)(s+100)
+
bB
V_bus1
(s+100)(s+40-800i)(s+40+800i)
s
A
B
C
Admittance Phase c
-
I1b
(s+0.1)(s+100)
(s+100)(s+40-800i)(s+40+800i)
s
cC
+
-
Load 1
2MW
0.7MVAR
A
B
C
A
B
C
I1c
Continuous
700kVAR
•
Method becomes cumbersome for multiple interface buses.
•
Mutual phase impedances are omitted.
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Summary
1.
Time domain harmonic computation is useful in
cases where detail modeling of the harmonic
source is required;
2.
The modeling approach is the same as the slow
transient modeling approach;
3.
The size of the network simulated is limited to a
few buses around the harmonic source;
4.
Software like SIMULINK combine several useful
features that can provide insight into a problem,
especially for educational purposes.
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IEEE PES General Meeting, Tampa FL
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