1 Chapter 8: Procedure of Time-Domain Harmonics Modeling and Simulation Contributors: C. J. Hatziadoniu, W. Xu, and G. W. Chang Organized by Task Force.

1

Chapter 8: Procedure of Time-Domain Harmonics Modeling and Simulation

Chapter 8: Procedure of Time-Domain Harmonics Modeling and Simulation

Contributors: C. J. Hatziadoniu, W. Xu, and G. W. Chang

Organized by

Task Force on Harmonics Modeling & Simulation

Adapted and Presented by Paulo F Ribeiro

AMSC

May 28-29, 2008

2

OUTLINEOUTLINE

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

3

INTRODUCTIONINTRODUCTION

• 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?

4

Why Time Domain Solution?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.

5

When is Time Domain Solution Appropriate?

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.

6

Accuracy of Time Domain Simulation v. Direct Methods

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)

7

What are the General Characteristics of a Time Domain Solution Procedure?

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.

8

Modeling ApproachModeling 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.

9

Power Converters: Detail RepresentationPower Converters: Detail Representation

• Detail Valve model

• Surge arrester representation in studies of harmonic overvoltages

• Representation of the grid control

SurgeArrester

Ls

Snubber

10

Power Converters: Switching FunctionPower Converters: Switching Function

• Voltage-Sourced inverters are more suitable for this representation.

• Switching function approach:

– Voltage:

– Current:

va

vc

vb

ia

ic

ib

id

+vd

-

dccdbbdaa vtsvvtsvvtsv )(,)(,)(

ccbbaad itsitsitsi )()()(

11

Non-Linear Devices: TransformerNon-Linear Devices: Transformer

• Piece-wise Linear representation of the core inductance.

• Switching inductance model (flux controlled switches).

l(i)

i

Lo

L2

L1

iA iB

lA

lB

Aircore Inductance

Saturation Characteristic

Uns

atur

ated

Seg

men

t

Sat

urat

ed

Seg

men

t

i

v Lo LA LB

SA SB

12

Formulation of the Network EquationsFormulation of the Network Equations

• Pre-integrated Components: Algebraic Equations

• State Equations: Numerical Integration– Piece-wise Linear Equations

– Time Varying Equations

)(

),(0

)(

0

),(),(),(

),(

),(

tu

txB

B

B

xf

x

x

x

txAtxAtxA

txAAA

txAAA

x

x

x

C

N

L

C

N

L

CCCNCL

NCNNNL

LCLNLL

C

N

L

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Summary of The Time Domain ProcedureSummary of The Time Domain Procedure

Run Slow Transient Program

Network and

converter data

Steady State?

Sart

Slow Transient Modeling

No

Run FFT

Met Criteria?

Fine-Tune Model

End

Yes

No

Yes

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SIMULINK DemonstrationsSIMULINK Demonstrations

• Converter Simulation Using the Switching Function

• Non-Linear Resistor

• Rigid Harmonic Source

• Impedance Measurement

• Network Equivalency

15

Converter Simulation Through the Switching Function

Converter Simulation Through the Switching Function

Switching Function Generation Inverter DC Side

Inverter AC Side

Measurements from network

Continuous

powerguiV

s

-+

Vc

s

-+

Vb

s

-+

Va

ABC

A

B

C

A

B

C4

Multimeter

s -+

Id

[Ic]

[Ib]

[Ia]

[Vd]

[Sc][Sb][Sa]

[Ic][Ib][Ia] [Sc][Sb][Sa]

[Sc]

[Sb]

[Sa]

[Vd]

P1

P2

Discrete 3-phasePWM Generator

Cd

A

B

C

A

B

C

(a)

(b)

+Vd

-

• Linear Network.

• Insert the converter as:– Voltage source on ac

side.

– Current source on dc side.

• Incorporate high level converter controls.

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Example of Non-Linear Resistor Using User-Defined Functions

Example of Non-Linear Resistor Using User-Defined Functions

+v-

Voltage Controlled Resistor

Continuous

pow ergui

s -+

i(v)

v+-

Series RLC Branch

Scope

node 0

node 1

node 1

Lookup TableDescribing i(v)

[i]

Goto

[i]From

C'

v i(v )

RT

+-

+

v

-

i(v)

V(x)

Network Thévenin Equivalent

Non-linear voltage controlled resistor

• Voltage Controlled Element: Parasitic capacitance C’

• User-defined function describing the i(v) function

)()( xVviRv T

17

Rigid Harmonic Source Using the s-Function

Rigid Harmonic Source Using the s-Function

• S-Function: Calculation of the harmonic current:

• Simulation time slows down with increasing order N

Nn

nna ntnItIti,..,5,3

111 )cos()cos()(

3

C

2

B

1

A

sfHarm_3ph

Rigid Source

s

-+

Phase C

s

-+

Phase B

s

-+

Phase A

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Impedance Scans Using Rigid Harmonic Sources

Impedance Scans Using Rigid Harmonic Sources

• Basic assumptions:– Linear Network Model.

– Single driving point (e.g. location of harmonic source).

– The harmonic source is represented by a rigid current source at pre-defined harmonic orders.

• Driving point impedance

• Transfer impedance

Procedure:1. Inject positive, negative, or

zero sequence current separately at unit amplitude;

2. Arrive at steady state

3. Obtain bus voltage

4. Apply FFT1. Driving point impedance

2. Transfer Impedance

)()(

)(1

1

11

jnV

jnI

jnVjnZ k

k

kkk

)()(

)()( 1

1

11

jnV

jnI

jnVjnZ m

k

mmk

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Impedance Scan: Transfer Function Method

Impedance Scan: Transfer Function Method

• Basic Assumptions– The impedance is

defined as a current-to-voltage network (transfer) function:

– Network is driven by a signal-controlled current source. More than one inputs can be used.

• Procedure

1. Define network as a subsystem;

2. Define the controlling signals of the current sources as the inputs;

3. Define the voltages at the buses of interest as the outputs;

4. Use the LTI tool box to obtain the driving and transfer impedances.

)(

)()(

sI

sVsZ

k

mmk

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Impedance Scan: Transfer Function Method—Example

Impedance Scan: Transfer Function Method—Example

• 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

Line Impedance datar'=0.278 Ohm/mix'=0.733 Ohm/mi

Injection Source

Input

3

Vb32

Vb2

1

Vb1

Continuous

ABC

ABC

A B C

Load 34MW1.4MVAR

A B C

Load 22MW0.7MVAR

A B C

Load 12MW

0.7MVAR

s

-+

I1c

s

-+

I1b

s

-+

I1a

V_bus1 V_bus3V_bus2

ABC

ABC

Feeder 2-3: 2mi

A

B

C

A

B

C

Feeder 1-2: 2 mi

A

B

C

a

b

c

Bus 3

A

B

C

a

b

c

Bus 2

A

B

C

a

b

c

Bus 1

A B C

750kVAR

A B C

700kVAR

1

I1

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Network EquivalencyNetwork 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 SIMULINKNetwork 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: ExampleNetwork Equivalency: Example

• Method becomes cumbersome for multiple interface buses.

• Mutual phase impedances are omitted.

Injection Source

ContinuousA B C

Load 12MW

0.7MVAR

s

-+

I1c

s

-+

I1b

s

-+

I1aV_bus1

A

B

C

a

b

c

Bus 1

(s+0.1)(s+100)

(s+100)(s+40-800i)(s+40+800i)

Admittance Phase c

(s+0.1)(s+100)

(s+100)(s+40-800i)(s+40+800i)

Admittance Phase b

(s+0.1)(s+100)

(s+100)(s+40-800i)(s+40+800i)

Admittance Phase a

A B C

700kVAR

AB

C

Ih

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SummarySummary

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.