A New Approach for Prediction of Harmonic Currents Generated by a Cluster of AC/DC Converters

Gary W. Chang Paulo F. Ribeiro Department of Electrical Engineering National Chung Cheng University TAIWAN Engineering Department Calvin College USA

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Outline • • • • • Introduction Proposed Approach Probabilistic Model of Multiple Converters Case Study Conclusions

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Introduction

• • • In the past, most probabilistic or statistical approaches to estimate harmonic currents generated by a group of converters are based on deriving an analytical probabilistic harmonic current model for a single converter.

To obtain an equivalent probabilistic model of a cluster of converters with more accurate operations becomes tedious and timing.

Most literature show that ripple currents at the converter dc side and its effects on the ac side are ignored when predicting harmonic currents generated by multiple converters.

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Introduction

(Continued)

• This paper presents a new approach to predict harmonic currents generated by multiple converters while taking the effects of ripple currents at the converter dc side into account.

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

 Switching function-based approach to obtain the AC side equivalent circuit for harmonic analysis of a converter [4]

i V dc

  

G

 ,

dc E

  

G

 ,

ac i dc

 

a

,

b

,

c

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

(Continued)

 Voltage and current transfer functions with firing angle variation and commutation overlap  The ideal switching function,

G

, includes both fundamental and harmonic components NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

(Continued)

 The

ac

input current at phase

a

of the converter which includes effects of ripple components at its dc side becomes

i a

   

I

[cos 1  

n h

   1  1 6

n

1 (cos(

I wt

6  6

h h

 

n

   1 6 6

n n

1 

{[cos(

1

6

1 [(cos( 6

h

 1 cos( )

h

 6

wt

6 

n n

1

 

)

  1 )

wt

1 ) 6

n wt



wt

] )]}   6

h

 6

n

]

) where

I 1

and

I 6h

are calculated based on the equivalent circuits representing effects of dc side current components on the input current at the ac side that proposed in [4].

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

(Continued)

 Equivalent circuit of a single converter at fundamental frequency [4]

Z

1   18 2

R d

cos 

E

1  3 

E d

3 cos  NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

(Continued)

 Equivalent circuit of a single converter at each dc ripple frequency [4] NCCU

Z

6

h

 1 2 

R d



j

6

hwL d

 E 6

h

 9 2  2

e

 ( 6

h j

  1 

e

6

h j

  1 ) E Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

(Continued)

•

Equivalent circuit of a cluster of converters at fundamental frequency

I

1 

i M

  1

I si I si

 1 (

E s

 

E i j M

  1 (

Z s

) / /

Z ci Z cj

) NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Proposed Solution Method

(Continued)

•

Equivalent circuit of a cluster of converters at each dc ripple frequency

I

6

h



i M

  1

I h si I h si

 1 

E h ci

/

Z j M

  1 (

Z h s h ci

/

Z h cj

) NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Probabilistic Model of Multiple Converters •

The bivariate normal distribution model of a converter harmonic current can be obtained by calculating mean value and variance of the pdf for real part and imaginary part of the harmonic current.

•

To avoid the complicated derivation of theoretical probabilistic model of harmonic currents generated by a cluster of converters, a statistical approach is proposed.

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Probabilistic Model of Multiple Converters

(Continued)

•

Bivariate normal distribution model of a cluster of converter harmonic currents

Pdf

(

X r

,

X i

)  2 

r



i

1 1   2 exp(

Q

)

Q

  2 ( 1  1  2 )       

X r

 

r



r

   2  2  (

X r

  

r r

 )(

i X i

 

i

)    

X i

 

i



i

   2      

E

[

X r X



r i

]  

i



r



i

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Probabilistic Model of Multiple Converters

(Continued)

•

Statistical approach to find the mean value and variance of harmonic currents generated by a cluster of converters

X i

 1

N k N

  1

X k S i

2 

N

1  1

k N

  1 (

X k



X

) 2

X r S

2

r

  1

N k N

  1

X k N

1  1

k N

  1 (

X k



X

) 2 as long as the number of samples,

N

, is sufficiently large, 

r



X r



i



X i



r

2 

S r

2 

i

2 

S i

2 NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Case Study

• Test system (10 converters) • Simplified circuit of a converter NCCU • Basic converter data

-------------------------------

R d =50

Ω

L d =1 mH L c =10 mH C d =10 uF

-------------------------------

Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Case Study

(Continued)

•

All converters of the test system are assumed identical.

•

The range of each firing angle is assumed uniformly distributed from 10 º to 40º.

•

Each converter firing angle is randomly sampled 500 times.

•

The proposed approach is validated by using Monte Carlo simulation.

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Case Study

(Continued)

•

Comparison of harmonic current pdf -- proposed approach (-) and Monte Carlo simulation (*) NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Case Study

(Continued)

•

Comparison of the harmonic current cdf with ( o ) and without ( + ) considering dc ripple using the proposed approach .

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

Conclusions

 Solutions obtained by the proposed approach and by Monte Carlo simulation have a good agreement.

 The effect of dc ripples on harmonic currents is significant.

 Ignoring dc ripples may result in underestimation of

5-th

harmonic current and overestimation of both

7-th

and

13-th

harmonic currents.

 The proposed statistical approach is simple and computationally efficient.

NCCU Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro

NCCU

Thank You!

Department of Electrical Engineering National Chung Cheng University, Taiwan IEEE ICHQP 10, 2002, Rio de Janeiro