2004-pedase-herold-1

Diagnostics of asymmetrical nonlinear loads using Symmetrical Space Phasor Components

Gerhard Herold, University of Erlangen-Nürnberg, Germany

1. Zero component and space phasor. If there is given any 3-phase ac system, we can transform it into its zero component and the space phasor.

(013

)R S Tv v v v= + + (1.1)

( ) j2 32 with e3 R S Tv v a v a v a= + + =

2π

(1.2)

Using these two components instead oft the origin system we have many advantages in describing and computing of 3-phase networks and our results are of a higher clarity.

The instantaneous power of a 3-phase ac system with-out zero components is given by

( ) { } { }3 3Re Re2 2

p t u i pω ∗= = (1.3)

The mean value of (1.3) is the active Power P. The r.m.s. value of a space phasor is given by

22

0

1 d2

V v v tω∗= ∫π

π (1.4)

With Equation (1.4) we get the apparent power.

32

S U= I (1.5)

At last we get the power factor and the wattless power. 2 2andP Q S P

Sλ = = − (1.6)

They include the wattless power caused by phase shift, asymmetry and distortion.

2. Space phasors of symmetrical nonlinear loads At first we have a look to the magnetizing currents of a 3-phase transformer. The magnetization should be free, that means without any constraint. In this case we get three non-sinusoidal symmetrical currents like those in figure 2.1.

0i

i

tω

1

Ri Si Ti

Figure 2.1: Magnetizing currents of a transformer

The space phasor of these magnetizing currents is given in figure 2.2. It is called a 6-pulse space phasor, because it owns the following qualities:

( ) ( ) ( )0

266 6

61 with

3

h hh

v t h a v tω γ ωγ

∈⎧⎪+ = − ⎨

=⎪⎩π (2.1)

( ) ( )( )( ) ( )( )

66 6

6 66 6arg arg

v t h v t

v t h v t h

ω γ ω

ω γ ω

+ =

+ = + γ (2.2)

All the information about the space phasor is included in an intervall of 6 6tω γ= . Therefore we can calculate its harmonics by the followoing equation.

( )2

j

0

ˆ2 e n tnV v t ω d tω ω−= ∫

π

π (2.3)

( ) j6

0

ˆ2 e n tnV c v t t

γω dω ω−= ∫

6

π (2.4)

66

6 for 6 10,1, 2,3,

0 otherwisen k

c k= ± +⎧

= ⎨⎩

= (2.5)

Equation (2.5) specifies the characteristic harmonics of a nonlinear load of a 3-phase ac network.

{ }Re i

{ }Im i

6i

j1̂ e tI ω

Figure 2.2: 6-pulse space phasor of the magnetizing currents

Sometimes it could be useful to have a look at the space phasors in another coordinate system which is turned by any angle x in relation to the first one. The transformation into it is very simple by using figure 2.3 for orientation:

j j2 1 e ex t

rv v x t v v ωω− −= ⇒ = ⇒ = (2.6)

If x is a linear time function depending on the angular operating frequency of the 3-phase network, we get the space phasor in a synchronous rotating coordinate

G. Herold: Diagnostics of asymmetrical nonlinear loads using Symmetrical Space Phasor Components 2

system (SRC) by equation (2.6). Then our point of observation is the rotor of a synchronous machine.

v

x Re v1l q

j Im v1l q

R ev 2lqj Im

v 2lq

γ

Figure 2.3: Transformation of a space phasor into another coordinate system

From the equationes (2.5) and (2.6) we get the harmonics of a 6-pulse space phasor in a SRC:

6 6 1 6 0,1, 2,3,rn n k k= − = ± = (2.7)

Figure 2.4 shows the space phasor of the magnetizing currents in a SRC. It pulsates six times during an operating period in the complex number area. Its mean value (marked by a small circle in figure 2.4) is the first harmonic, the positive-sequence component.

{ }Re ri

{ }Im ri

0

0 Figure 2.4: Space phasor of the magnetizing currents in SRC

All space phasors of symmetrical 3-phase ac systems with converter loads (symmetrical converter systems, sCS) are of the 6-pulse type, too. We consider such a system with the single-line diagramm in figure 2.5.

Ni SRi

SKi

Lu

Cu

SRu

SK L Cu u u= +

powersystem

Figure 2.5: Single-line diagramm of a simple sCS

tωpu

SKi

SRiNi

u,i

Figure 2.6: Currents in the phase R of the sCS

SRi

SKi

Ni{ }Im v

{ }Re v

Figure 2.7: 6-pulse space phasors of the currents

The currents and the no-load voltage of phase R are shown in figure 2.6. In Figure 2.7 the space phasors of the currents in the sCS are shown. At last in figure 2.8 are shown the 6-pulse space phasors of the no-load voltage and the filter voltages. The no-load voltage consists of one harmonic only, the positive-sequence system. Therefore its space phasor is circle. The power phasor p from Equ. (1.3) is also of the 6-pulse type.

{ }Im v

pu

SKu

Cu

Lu

{ }Re v

Figure 2.8: 6-pulse space phasers of the voltages

3. Space phasors of asymmetrical nonlinear loads If the 3-phase ac system is asymmetrical, the two half-bridges of the converter are different or the firing control of the converter is imprecise, then the space phasors become asymmetrically and we have to detect the causes. For this purpose we are able to split up any space phasor into some sets of symmetrical compo-nents. In imitation of the Symmetrical Components for sinusoidal 3-phase ac systems I call them Symmetrical


Space Phasor Components (SSPC´s). At first we have to choose a base for splitting up. The simplest one is the base 2. For

with 0, 1hx x h t h hπ ω π= − = − = (3.1)

we can formulate

( )( )

( )( )

0220

1221

1 111 12

v xv xv xv x

⎛ ⎞ ⎛⎛ ⎞=⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜−⎝ ⎠⎝ ⎠ ⎝

⎞⎟⎟⎠

(3.2)

The SSPC 20v is a zero component of the space phasor to the base 2. It includes all the harmonics with even ordinal numbers.

20 2n = ± k (3.3)

The SSPC 21v is a 2-pulse space phasor. It includes all The harmonics of odd ordinal numbers.

21 2n k= ± +1 (3.4)

This SSPC owns the qualities

( ) ( ) ( )21 211 hv t h v tω ω+ = −π (3.5)

( ) ( )( )( ) ( )( )

21 21

21 21arg arg

v t h v t

v t h v t h

ω ω

ω ω

+ =

+ = +

π

π π (3.6)

In figure 3.1 an asymmetrical and distorted 3-phase ac system is shown. The asymmetrical space phasor and its SSPC are represented in figure 3.2.

0

1

2v

tω

1

2

RvSv Tv

Figure 3.1:Asymmetrical and distorted 3-phase system

0 1 2

0

1

{ }Re v

{ }Im vv

21v

20v

( ) ( )1 1v v+ −+1

2 1 Figure 3.2: Asymmetrical space phasor and its SSPC to the base 2

We could also choose the base 3 for SSPC. For

2 2 with 0, 1, 23 3hx x h t h hπ π

ω= − = − = (3.7)

we can write down

( )( )( )

( )( )( )

030 32

131 32 232 3

1 1 11 13

1

v x v xv x a a v xv x v xa a

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜

= ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎝ ⎠ ⎝⎝ ⎠

⎟⎟⎟⎠

(3.8)

The SSPC 30v is a zero component of the space phasor to the base 3. It includes all the harmonics with ordinal numbers which are divisble by 3.

30 3n k= ± (3.9)

The SSPC 31v is a 3-pulse space phasor. It includes all the harmonics of the following ordinal numbers.

31 3n k 1= ± + (3.10)

The SSPC 32v is also a 3-pulse space phasor. It includes all the harmonics of the ordinal numbers:

32 3n k 2= ± + (3.11)

31v is a positive-sequence and 32v a negative-sequence space phasor. They own the qualities

( )31,2 31,2hv t h a v tω ω⎛ ⎞+ =⎜ ⎟

⎝ ⎠

2π

3 (3.12)

( )

( )( )

31,2 31,2

31,2 31,2

3

arg arg3 3

v t h v t

v t h v t h

ω ω

ω ω

⎛ ⎞+ =⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞+ = +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

2π

2π 2π

(3.13)

The SSPC to the base 3 of the 3-phase system in figure 3.1 are represented in figure 3.3.

-1 0 1 22

1

0

1

2

{ }Re v

{ }Im v

v

31v

32v

2

0 0.1

0

0.1

{ }Re v

{ }Im v

0.1

30v

Figure 3.3: Asymmetrical space phasor and its SSPC to the base 3


It is also possible to split any space phasor into SSPC to the base 6. For

6 with 0 53hx x h t h hπ

ω γ= − = − = … (3.14)

the SSPC are given by

( )( )( )( )( )( )

( )( )( )( )( )( )

060 2 2161

2 2262363 2 2464 2 2565

1 1 1 1 1 11 1

1 1 11 1 1 1 1 161 11 1

v x v xv x a a a a v xv x v xa a a a

v xv xa a a a v xv x

v xv x a a a a

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟− − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜⎜ ⎟= − − −⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜⎜ ⎜ ⎟⎜ ⎟ − − −⎝ ⎠⎝ ⎠⎝ ⎠

⎟⎟

⎟⎟

5

(3.15)

The ordinal numbers of the SSPC to the base 6 are

6 3 with 0hn k h h= ± + = … (3.16)

The SSPC to the base 6 are

60

61

62

63

64

65

zero-sequence6-pulse positive-sequence2 3-pulse positive-sequence3 2-pulse 2 3-pulse negative-sequence6-pulse positive-sequence

vvvvvv

×××

(3.17)

61v and 65v are of the 6-pulse type. Their qualities are given by the equations (2.1) and (2.2). We obtain the qualities of 62v and 64v from the equations (3.12) and (3.13)

( ) ( )662,4 62,4hv t h a v tω γ+ = ω (3.18)

We get the qualities of 63v from the equations (3.5) and (3.6)

( ) ( ) ( )63 631 hv t h v tω γ+ = −6 ω

2

(3.19)

0 1

0

1

2{ }Im v

{ }Re v

v

61v

65v

122

1

Figure 3.4: Asymmetrical space phasor and its SSPC to the base 6 61v and 65v

The SSPC´s of the asymmetrical space phasor are represented in the figures 3.4 to 3.6. We have obtained from the space phasor 6 symmetrical components.

0.05 0 0.05 0.1

0

0.05

60v63v{ }Im v

{ }Re v0.05

Figure 3.5: SSPC of the assymetrical space phasor to the base 6 60v and 63v

0.2 0.1 0 0.1 0.20.2

0.1

0

0.1

0.2

{ }Re v

{ }Im v

64v

62v

Figure 3.6: SSPC of the assymetrical space phasor to the base 6 62v and 64v

It is possible to create SSPC´s for higher-pulse conver-ter systems too, for instance for 12- or even 24-pulse systems. 4. Relations between the SSPC of different bases Between the SSPC of the bases 2 and 6 we find the following relations:

20 60 62 64v v v v= + + (4.1)

21 61 63 65v v v v= + + (4.2)

The realtions between the SSPC of the bases 3 and 6 are given by:

30 60 63v v v= + (4.3)

31 61 64v v v= + (4.4)

32 62 65v v v= + (4.5)

In our example all components in the equations (4.1) to (4.5) are different from zero. That means the 3-phase system in figure 3.1 is totally asymmetrical but it does not own any zero component (equ. (1.1)). Therefore the space phasor in figure 3.2 and 3.3 is also totally asymmetricall. But normally some of the components are zero or can be ignored. Then we are able to diagnose the kind of asymmetry.


5. Asymmetrical space phasor in a converter system The 3-phase system in figure 5.1 measured in a converter system is given as an example. Obviously it is distorted, but to see anything more is not easy.

0

2

2

tω

uRv Sv Tv

Figure 5.1: 3-phase ac system in a converter system

The space phasor of the 6-pulse type belonging to it is shown in figure 5.2. But it is not symmetrically, because its positive-sequence space phasor component is different from it. We also see the first harmonic, the positive- sequence component, in figure 5.2.

0 1 2 3

0

1

2

3

{ }Re v

{ }Im v

v

61v

( )1v +1

2

33 2 1

Figure 5.2: Space phasor to figure 5.1

Analysing the space phasor we find two other SSPC´s of the 6-pulse type represented in figure 5.3. There is also shown the (–1)st harmonic, the negative-sequence component.

0.1

0

0.1{ }Im v

65v

( )1v −

62v

0.1 0.1 { }Re v0

Figure 5.3: 6-pulse SSPC´s to figures 5.1 and 5.2

The other 6-pulse SSPC can be neglected.

60 63 64 0v v v= = ≈ (5.1)

Now the problem is indicated much more clearly. We get

20 62v v= (5.2)

21 61 65v v v= + (5.3)

from the equations (4.1) to (4.2) and

30 0v = (5.4)

31 61v v= (5.5)

32 62 65v v v= + (5.6)

from the equations (4.3) to (4.5). This result represents a 3-pulse asymmetry in the system and that means one of the two halfe-bridges of the converter is asymmetri-cal. This asymmetry is indicated by the SSPC of the 3-pulse type 32v shown in figure 5.4.

0 0.1 0.2

0

0.1

0.2

0.10.20.2

0.1

{ }Im v

{ }Re v

32v

( )1v −

Figure 5.4: 3-pulse SSPC to figure 5.1, 5.2 and 5.3

6. Dc-magnetizing of a converter transformer If the firing control of a converter in a dc-link between to 3-phase ac networks, especially a PWM-converter in a back-to-back dc-link between two medium volta-ge power systems, is not precisely enough there is the possibility of dc-premagnetization of the converter transformers. There are two extreme situations we can find:

( )dc dc dc1 02R S Ti i i+ + = (6.1)

dc dc 0S Ti i+ = (6.2)

The first one (6.1) is a premagnetization in the α- and the second (6.2) in the β-axis. We can find all kinds of dc-premagnetization between these two situations. The magnetizing currents for premagnetization in the α-axis are shown in figure 6.1. They are related to the magnetizing currents of the transformer in normal operation. The space phasor belonging to them is given in figure 6.2. There the 6-pulse space phasor in normal operation is also shown. In addition to the space phasor a zero current flowing in the ∆-winding of the transformer also exists.


0

2

4

6

8

10

2

4

6

hRi

hTihSi tω

i

Figure 6.1: Magnetizing currents belonging to premagnetization in the α-axis

0 2 4 6 8 10

0

2

4

6

2

4

62

hi αhni

{ }Re i

{ }Im i

Figure 6.2: Space phasor of the currents belonging to premagnetization in the α-axis

The space phasor is completely asymmetrical. Its 6-pulse SSPC are given in the figures 6.3 to 6.5.

0 1 2

{ }Re i

63hi 60hi

Figure 6.3: SSPC´s 60 63andh hi i

0 2

0

2

2 2

62hi

0 0

0

0.5

.50.5

0.5

65hi

Figure 6.4: SSPC´s 62 65andh hi i The SSPC´s 60 63andh h pulsate in the α-axis only. The small imaginary part of

i i63hi in figure 6.3 is set for

better demonstration. The SSPC 65hi does not include a (-1)st harmonic, a negative-sequence component. That means the fundamental mode of the space phasor is symmetrical. It is shown in figure 6.4.

0 1 2 3

0

1

2

1

2

2 13

{ }Re i

{ }Im i61v

( )1v +

64v

Figure 6.5: SSPC´s 61 64andh hi i The positive-sequence SSPC belonging to premagne-tization in the α-axis is of the 3-pulse type, because

64hi is not zero.

The space phasor of the currents belonging to pre-magnetization in the β-axis is represented in figure 6.6. The most important differences to the magneti-zation in the α-axis is shown in figure 6.7. The SSPC

60hi pulsates in the β- and 63hi pulsates in the α-axis only. In addition to that we detect

64 31 610h hi iβ β hi β≈ ⇒ ≈ (6.3)

The positive-sequence SSPC belonging to premagne-tization in the β-axis is of the 6-pulse type, because

64hi is almost zero.

0 2 4

0

2

4

6

8

62

4 2

hi β

hni

{ }Im i

{ }Re i

Figure 6.6: Space phasor of the currents belonging to premagnetization in the β-axis

Premagnetization is not a problem for the transformer itself. But it could be a problem for the power qualitiy, because there is the posibility of excitation the filters of the CS by the total asymmetrical space phasor.


0 1

0

1

2

1

{ }Re i

{ }Im i 60hi

63hi

30hi 30hi

Figure 6.7: SSPC´s 60 63 30, andh h hi i i belonging to premagnetization in the β-axis 7. Summary To evaluate the power quality of 3-phase ac power systems we have to create an overall view of the system currents, voltages and powers. Looking only at one phase or at the three phases separately will not lead to sufficient results. Coordinate transformations are very helpful for this purpose.

From the theory of Symmetrical Components we know, that any sinusoidal 3-phase ac system can split-ted off into three Symmetrical Components, the zero-, positive- and negative-sequence component. If the 3-phase systems are not sinusoidally, we can transform them into a zero component and the space phasor of its instantaneous values.

The space phasors of nonlinear, but symmetrical loads are symmetrical and of the 6-pulse type. They own well defined quantities, which can be used effectively for analysing of 3-phase networks with nonlinear loads. The space phasors of symmetrical converter systems also can be of a higher pulse type.

The space phasors of nonlinear, but asymmetrical loads are also asymmetrical. At least they are of a 1-pulse type. It is possible to split off asymmetrical space phasors into Symmetrical Space Phasor Components (SSPC´s) of different bases. The most important bases for 3-phase systems are 2, 3 and 6. But for converter systems we also can choose other bases, for instance 12 or 24.

In special cases the SSPC allows us to diagnose the kind of assymmetry of a 3-phase system. If digital measurement equipment is available, using SSPC is only a matter of software.

References 1. Strobl, Bernhard:

Symmetrische Raumzeiger-Komponenten zur Beschreibung und Analyse unsymmetrischer Stromrichterschaltungen Dissertation, Universität Erlangen-Nürnberg, Technische Fakultät 2001

2. Weindl, Christian: Beschreibung und Berechnung von Drehstromsy-stemen mit leistungselektronischen Anlagen im Zustandsraum Dissertation, Universität Erlangen-Nürnberg, Technische Fakultät 1999

3. Strobl, B.; Heinrich, W.; Herold, G.: Diagnosis of Asymmetries in Bridge Converters Using a Novel Transform to Symmetrical Space-Phasor-Components 8th International Power Electronics & Motion Control Conference, Prague, Czech Republic; 8.-10. September 1998, Proceedings Vol 2 pp 18-23

4. Strobl, B.; Heinrich, W.; Herold, G.: Analysis and Identification of Faults in Bridge Converters Using the Symmetrical Space Phasor Components. EPE '99, Lausanne, Switzerland, 1999, Proceedings S.1-9, Paper 140

5. Strobl, B.: A Diagnostic System for Asymmetries in Bridge Converters 9th Internat. Conf. on Power Electronics and Motion Control - EPE-PEMC 2000, Kosice (Slowakia), 5.-7. September 2000, Proceedings Vol. 2 pp. 160-165

6. Strobl, B.: Harmonic Transfer through Converters 9th European Conference on Power Electronics and Applications, Graz (Österreich), 27.-29. August 2001, Proceedings Paper No 210

7. Strobl, B.: A Diagnostic System for Power Electronic Devices based on the Symmetrical Space Phasor Components 3rd IEEE SDEMPED 2001: International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives, Grado (Italien), 1.-3. September 2001, Proceedings pp 211-216

8. Herold, G.: Elektrische Energieversorgung I: Drehstrom-systeme – Leistungen – Wirtschaftlichkeit J. Schlembach Fachverlag 2002

Author: Prof. Dr.-Ing. Gerhard Herold Chair of the Institute of Electrical Power Systems University Erlangen-Nürnberg, Germany

2004-pedase-herold-1

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