Electric Machines and Drives Part 14 doc

Figures 13 and 14 are presenting two general cases, where: • Figures noted as ‘a’ present the reference three phase voltage system; • Figures noted as ‘b’ present the space vector trajec

6.2 Determination of the tetrahedrons

In each TP there are six vectors, these vectors define four tetrahedrons Each tetrahedron

contains three active vectors from the six vectors found in the TP The way of selecting the

tetrahedron depends on the polarity changing of each switching components included in

one vector The following formula permits the determination of the tetrahedron in which the

voltage space vector is located

1

Where:

, ,

i a b c

=

To clarify the process of determination of the TP and T h for different three phase reference

system voltages cases which may occurred Figures 13 and 14 are presenting two general

cases, where:

• Figures noted as ‘a’ present the reference three phase voltage system;

• Figures noted as ‘b’ present the space vector trajectory of the reference three phase

voltage system ;

• Figures noted as ‘c’ present the concerned TP each sampling time, where the reference

space vector is located;

• Figures noted as ‘d’ present the concerned T h in which the reference space vector is

located

Case I: unbalanced reference system voltages

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-300

-200

-100

0

100

200

300

Time (s)

Vc

-400 -200 0 200 400 -400

-200 0 200

400-30 -20 -10 0 10 20 30

Valp ha ( V)

Vbeta (V)

(a) (b)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time (s)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0

2 4 6 8 10 12 14 16 18 22

Time (s)

(c) (d)

Fig 13 Presentation of instantaneous three phase reference voltages, reference space vector,

TP and T h

Case II Unbalanced reference system voltages with the presence of unbalanced harmonics

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

-500

-400

-300

-200

-100

0

100

200

300

400

500

Time (s)

-400 -200

0 200 400

-400 -200 0 200 400 -150 -100 -50 0 50 100 150

Valph a (v)

Vbeta (V)

(a) (b)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time (s)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0

5 10 15 20 25

Time (s)

(c) (d) Fig 14 Presentation of instantaneous three phase reference voltages, reference space vector,

TP and T h

6.3 Calculation of duty times

To fulfill the principle of the SVPWM as it is mentioned in (9) which can be rewritten as

follows:

3

0

i

=

Where:

3

0

i

=

In this equation the a b c− − frame components can be used, either than the use of the

α β γ− − frame components of the voltage vectors for the calculation of the duty times, of

course the same results can be deduced from the use of the two frames The vectors V1,V2

and V3 present the edges of the tetrahedron in which the reference vector is lying So each

vector can take the sixteen possibilities available by the different switching possibilities On

the other hand these vectors have their components in the α β γ− − frame as follows:

ai fi i

V

α β γ

=⎢ ⎥= ⋅⎢ − ⎥⋅

From (30), (31) and (32) the following expression is deduced:

3

1 1

1

g

V

−

In the general case the following equation can be used to calculate the duty time for the

three components used in the same tetrahedron:

t

refa

refc

V

σ

Where:

3

1

1

σ=

Variable j and k are supposed to simplify the calculation where:

j i= + − ⋅INT i ; k i= + − ⋅2 3 INT i(( +1 / 3)) i =1,2,3

A question has to be asked From one tetrahedron, how the corresponding edges of the

existing switching vectors can be chosen for the three vectors used in the proposed SVPWM

Indeed the choice of the sequence of the vectors used for V1, V2 and V3 in one tetrahedron

depends on the SVPWM sequencing schematic used [108],[115], in one sampling time it is

recommended to use four vectors, the fourth one is corresponding to zero vector, as it was

shown only two switching combination can serve for this situation that is V16(0000) and V1

(1111) On the other hand only one changing state of switches can be accepted when passing

from the use of one vector to the following vector For example in tetrahedron 1 the active

vectors are: V11(1000), V3(1001) and V4(1101), it is clear that if the symmetric sequence

schematic is used and starts with vector V1 then the sequence of the use of the other active

vectors can be realized as follow:

1

V ,V11,V3,V4,V10,V4,V3,V11,V1

Active

vector V 1 V 11 V 3 V 4 V 10 V 4 V 3 V 11 V 1

a

b

c

f

S

S

S

S

1

1

1

1

1

1

0

1

1

0

0

1

1

0

0

0

0

0

0

0

1

0

0

0

1

0

0

1

1

1

0

1

1

1

1

1 1

0

a

T

1

0

b

T

1

0

c

T

1

0

f

T

0 4

2

2

2

2

2

2

2

4

t

Otherwise, if it starts with vector V16 then the sequence of the active vectors will be presented as follow Tab.9:

16

V ,V4,V3,V11,V1,V11,V3,V4,V16 Active

a

b

c

f

S

S

S

S

0

0

0

0

1

0

0

0

1

0

0

1

1

1

0

1

1

1

1

1

1

1

0

1

1

0

0

1

1

0

0

0

0

0

0

0 1

0

a

T

1

0

b

T

1

0

c

T

1

0

f

T

0 4

2

2

2

2

2

2

2

4

t

6.4 Applications

To finalize this chapter two applications are presented here to show the effectiveness of the four-leg inverter The first application is the use of the four-leg inverter to feed a balanced resistive linear load under unbalanced voltages The second application is the use of the four-leg inverter as an active power filter, where the main aim is to ensure a sinusoidal balanced current circulation in the source side In the two cases an output filter is needed between the point of connection and the inverter, in the first case an “L” filter is used, while for the second case an “LCL” filter is used as it is shown in Fig 15 and Fig 20

6.4.1 Applications1

g

V

Ff R

1

F

Ff L

3D-SVM

axis

β −

axis

γ −

8

V

7

V

4

V V3 V6 5

V

2

V

1

V V

15

V

12

V

14

V

13

V

9

V

10

V

11

V

V g V

g V

+

g V

+

g V

g V

−

g V

−

g V

−1

axis

f c b

S

a

a

V

b V

c V

f

bf V cf V

a

T

a

f T c T b T a T

3-phase balnced linear load

La I Lb I Lc I Ln I

Fig 15 Four-leg inverter is used as a Voltage Source Inverter ‘VSI’ for feeding balanced linear load under unbalanced voltages

In this application, the reference unbalanced voltage and the output voltage produced by the four leg inverter in the three phases a, b and c are presented in Fig 16 The currents in the four legs are presented in Fig 17, it is clear that because of the voltage unbalance the fourth leg is handling a neutral current To clarify the flexibility of the four leg inverter and the control algorithm used, Fig 18 shows the truncated prisms and the tetrahedron in which the reference voltage space vector is located

-500

0

500

-1000

0

1000

-1000

0

1000

-1000

0

1000

Time (s)

Fig 16 Presentation of three phase reference voltages and the output voltage of the three legs

0 0.01 0.02 0.03 0.04 0.05 0.06

-5

0

5

-5

0

5

-5

0

5

-5

0

5

Time (s)

Fig 17 Presentation of instantaneous load currents generated by the four legs

0 0.01 0.02 0.03 0.04 0.05 0.06

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time (s)

0 0.01 0.02 0.03 0.04 0.05 0.06 0

2 4 6 8 10 12

Time (s)

Fig 18 Determination of the Truncated Prism TP and the tetrahedron T h in which the reference voltage space vector is located

The presentation of the reference voltage space vector and the load current space vector are presented in the both frames α β γ− − and a b c− − ,where the current is scaled to compare the form of the current and the voltage, just it is important to keep in mind that the load is purely resistive

-40 -20

40

-40 -20 0 20

40

-200

-100

0

100

200

Valpha

Vbeta

-40 -20

40

-40 -20 0 20 40 -200 -100 0 100 200

A axis

B axis

Vo ltag e

C u rre n t

Cu rre n t

Vo l ta g e

Fig 19 Presentation of the instantaneous space vectors of the three phase reference system voltages and load current in α β γ− − and a b c− − frames ( the current is multiplied by 10,

to have the same scale with the voltage)

6.4.2 Applications2

The application of the fourth leg inverter in the parallel active power filtering has used in the last years, the main is to ensure a good compensation in networks with four wires, where the three phases currents absorbed from the network have to be balanced, sinusoidal

and with a zero shift phase, on the other side the neutral wire has to have a nil current circulating toward the neutral of power system source Figures 21, 22, 23 and 24 show the behavior of the four leg inverter to compensation the harmonics in the current The neutral current of the source in nil as it is shown in Fig 24 Finally the current space vectors of the load, the active filter and the source in the both frames α β γ− − and a b c− − are presented

a

S

g

V

b

a

V

b V c V f

V V af bf V cf V

a

T

a

f T c T b T a

T

3-phase Non-linear load

1-phase Non-linear load

3-phase unbalnced linear load

3-phase unbalanced non-linear load

sa e sb e sc e

s

Power Supply

b

c

N

a

Ff R

1

F R

2

F R

1

F L

2

F L

Ff L

F C FC R

sa I

sb I

sc I

sN I

Fa I Fb I

Fc I

FN I

La I

Lb I

Lc I

LN I

3D-SVM

axis

β −

axis

γ −

V

V

V V3 V6

V

V

1 16

V V

15 12

13

V

V

+3

+3

⋅0

−3

−3

−1

axis

α −

f c b

a S S S S

Fig 20 Four-leg inverter is used as a Parallel Active Power Filter ‘APF’ for ensuring a sinusoidal source current

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-50

0

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -30

-20 -10 0 10 20

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-30

-20

-10

0

10

20

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -30

-20 -10 0 10 20

Time (s)

Fig 21 Presentation of the instantaneous currents of Load, reference, active power filter and source of phase ‘a’

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-40

-20

0

10

30

ILb

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -30

-20 -10 0 10 20

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-30

-20

-10

0

10

20

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -30

-20 -10 0 10 20

Time (s)

Fig 22 Presentation of the instantaneous currents of Load, reference, active power filter and source of phase ‘b’

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-30

-20

-10

0

10

20

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -20

-10 -5 0 10

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-30

-20

-10

0

10

20

30

-30 -20 -10 0 10 20

Time (s)

Fig 23 Presentation of the instantaneous currents of Load, reference, active power filter and source of phase ‘c’

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-15

-10

-5

0

5

10

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -15

-10 -5 0 5 10

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

Time (s)

1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 -15

-10 -5 0 5 10

Time (s)

Fig 24 Presentation of the instantaneous currents of Load, reference, active power filter and source of the fourth neutral leg ‘f’

-50 0 50

-40 -20 0 20 40

-30

-20

-10

0

10

20

30

40

A axis

B axis

-50 0 50

-50 0 50 -8 -6 -4 -2 0 2 4 6 8

Alpha ax is

Beta axis

Source Source

Fig 25 Presentation of the instantaneous currents space vectors of the load, active power filter and the source in α β γ− − and a b c− − frames

7 Conclusion

This chapter deals with the presentation of different control algorithm families of four leg inverter Indeed four families were presented with short theoretical mathematical explanation, where the first one is based on α β γ− − frame presentation of the reference

space vector, the second one is based on a b c− − frame where there is no need for matrix transformation The third one which was presented recently where the determination of the space vector is avoided and there is no need to know which tetrahedron is containing the space vector, it is based on the direct values of the three components following the three phases, the duty time can be evaluated without the passage through the special location of the space vector The fourth method in benefiting from the first and second method, where the matrix used for the calculation of the duty time containing simple operation and the elements are just 0,1 and -1 As a result the four methods can lead to the same results; the challenge now is how the method used can be implemented to ensure low cost time calculation, firstly on two level inverters and later for multilevel inverters But it is important to mention that the SVMPWM gave a great flexibility and helps in improving the technical and economical aspect using the four leg inverter in several applications

8 References

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[10] A Julian, R Cuzner, G Oriti, and T Lipo, “Active filtering for common mode

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[11] Z Lin, L.Mei, Z Luowei, Z Xiaojun, and Y Yilin, “Application of a fourleg ASVG based

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[15] P Rodriguez, R Pindado, J Bergas, “Alternative Topology For Three-Phase Four-Wire

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[16] N Mendalek, “Modeling and Control of Three-Phase Four-Leg Split-Capacitor Shunt

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