Paper ID SGD5451 EMD-Prony for phasor estimation in harmonic and - - PowerPoint PPT Presentation

Jalal Khodaparast Ghadikolaei

Postdoctoral researcher

Jalal.khodaparast@ntnu.no Olav.fosso@ntnu.no Marta.molinas@ntnu.no

Professor Professor

Olav Bjarte Fosso Marta Molinas

Paper ID SGD5451 EMD-Prony for phasor estimation in harmonic and noisy conditions

Signal Analysis Research Lab

2

Phasor Estimation Prony Algorithm

Empirical Mode Decomposition (EMD)

EMD-Prony

Denoising signal Order of Prony

3

First-Order Prony algorithm

 

t f j t f j

e p e p t s

1 1

2 * 2

2 1 ) (

  

 

 j

ae p 

Main signal phasor First step of Prony Second step of Prony Third step of Prony

                                                     

    * ) 1 ( 1 1 1 1 1

. . . . . . . . . . . 1 1 ] 1 [ . ] [ . . . . ] [ p p Z Z Z Z N s n s s

N N n n

 

2 1 2 1 1

) ).( ( a z a z Z z Z z z F      



                                                 

2 1

. ] 3 [ ] 2 [ . . . . . . ] 3 [ ] 2 [ ] 2 [ ] 1 [ ] 1 [ ] [ ] 1 [ . . . ] 4 [ ] 3 [ ] 2 [ a a N s N s s y s s s s N s s s s

4

Frequency analysis of Prony







10 1

) (

k t jk

e t s



Empirical Mode Decomposition (EMD)

A time domain algorithm for separating a non-linear and non-stationary signal into its individual components. Sifting process Intrinsic Mode Function (IMF) stopping index





 

L k k

IMF R t s

1

) ( Determination of Prony’s order Hurst Index

5

Mode mixing in EMD

Decomposition procedure is done with different noisy data and finally mean (ensemble) of the corresponding IMFs provides the final result.

Detection of Mode mixing Ensemble EMD (EEMD)

) ( ) ( ) (

2 1

t s t s t s   ) 2 cos( ) (

1

t t s   ) 2 cos( ) (

2

    t a t s 15 . ) 300 2 cos( 2 . ) 50 2 cos( ) (    t t t t s   15 . ) 90 2 cos( 2 . ) 50 2 cos( ) (    t t t t s  

6

Simulation Results

Hurst Index

Denoising using EMD

HI_1=0.3576 HI_2=0.2743 HI_3=0.2022 HI_4=0.3010 HI_5=0.7347 Method Amplitude Error Phase Error Prony EMD-Prony 0.2851 0.2601 0.0171 0.0157

3 2

10 50 ) ( ) 5 . 2 cos( ) (



       f t w t f t s

7

Simulation Results

Prony's model order

) ) 8 ( 2 cos( 15 . ) ) 5 ( 2 cos( 3 . ) 5 . 2 cos( ) ( t f t f t f t s        

Method Amplitude Error Phase Error Prony EMD-Prony 0.4636 1.4553 5ˣ10-8 4.8ˣ10-7 EMD performance

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Simulation Results

Prony's model order

) ) 6 . 5 ( 2 cos( 15 . ) ) 5 ( 2 cos( 3 . ) 5 . 2 cos( ) ( t f t f t f t s        

Method Amplitude Error Phase Error Prony EMD-Prony 0.5845 1.2230 3.4ˣ10-7 3.02ˣ10-6 EEMD performance Hurst Index IMF1=0.1739 IMF2=0.2993 IMF3=0.2022 IMF4=0.5721 IMF5=0.6468 IMF6=0.7927

9