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0.03

0.04

0.05

0.06

ILb

-50

0.01

0.02

0.03

0.04

0.05

0.06

c 0

-50

0.01

0.02

0.03

0.04

0.05

0.06

n 0

-50

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

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

5.5

4.5

rism TP

n Th

3.5

hedro

uncated P

2.5

Tetra

1.5

0.01

0.02

0.03

0.04

0.05

0.06

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

Fig. 18. Determination of the Truncated Prism TP and the tetrahedron h

T 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.

200

100

C u rre n t

xic

Cu rre n t

Vga

C a

-100

Vo l ta g e

Vo ltag e

-200

b 0

-20

Valpha

-20

-40

A axis

-40

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

The Space Vector Modulation PWM Control Methods Applied on Four Leg Inverters

255

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.

3-phase unbalanced non-linear load

Power Supply

3-phase

Non-linear

load

1-phase

esb

Non-linear

I Fb

load

3-phase

unbalnced

linear load

I FN

LF 2

RF 2

F 1

Vcf

3D-SVM

γ − axis

Vγ

β − axis

1⋅ Vg

V 8

V 7

+ ⋅ Vg

V 4 V 3

V 6

V 5

+ ⋅ Vg

S S S S

V 2

α −

axis

0⋅ Vg

V V

V 15

V 12

− 1 ⋅ Vg

V 11

V 13

− ⋅ Vg

V 10

V 9

−1⋅ Vg

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

sinusoidal source current.

ILa

Iaref

-10

-20

-50

-30

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Ias

Iaf

-10

-20

-30

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Fig. 21. Presentation of the instantaneous currents of Load, reference, active power filter and

source of phase ‘a’

256

Electric Machines and Drives

ILb

Ibref

-10

-20

-30

-40

-30

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Ibs

Ibf

-10

-20

-30

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Fig. 22. Presentation of the instantaneous currents of Load, reference, active power filter and

source of phase ‘b’

ILc

Icref

-5

-10

-20

-15

-30

-20

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Ics

Icf

-10

-20

-30

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Fig. 23. Presentation of the instantaneous currents of Load, reference, active power filter and

source of phase ‘c’

ILn

Ifref

-5

-10

-15

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

-14

Isn

Iff

x 10

1.5

0.5

-0.5

-5

-1

-10

-1.5

-15

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Time (s)

Fig. 24. Presentation of the instantaneous currents of Load, reference, active power filter and

source of the fourth neutral leg ‘f’

The Space Vector Modulation PWM Control Methods Applied on Four Leg Inverters

257

Load

xis

a a

C a

-2

-10

Gam -4

-20

-6

Source

-30

Filter

-8

Filter

Source

-20

A axis

Alpha axis

-40

-50

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

[1] Ionel Vechiu, Octavian Curea, Haritza Camblong, “Transient Operation of a Four-Leg

Inverter for Autonomous Applications With Unbalanced Load,” IEEE

TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 2, FEBRUARY 2010

[2] L. Yunwei, D. M. Vilathgamuwa, and L. P. Chiang, “Microgrid power quality

enhancement using a three-phase four-wire grid-interfacing compensator,” IEEE

Trans. Power Electron., vol. 19, no. 1, pp. 1707–1719, Nov./Dec. 2005.

[3] T. Senjyu, T. Nakaji, K. Uezato, and T. Funabashi, “A hybrid power system using

alternative energy facilities in isolated island,” IEEE Trans. Energy Convers, vol. 20,

no. 2, pp. 406–414, Jun. 2005.

[4] M. N. Marwali, D. Min, and A. Keyhani, “Robust stability analysis of voltage and current

control for distributed generation systems,” IEEE Trans. Energy Convers., vol. 21,

no. 2, pp. 516–526, Jun. 2006.

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Electric Machines and Drives

[5] C. A. Quinn and N. Mohan, “Active filtering of harmonic currents in three-phase, four-

wire systems with three-phase and single-phase nonlinear loads,” in Proc. IEEE-

APEC’93 Conf., 1993, pp. 841–846.

[6] A. Campos, G.. Joos, P. D. Ziogas, and J. F. Lindsay, “Analysis and design of a series

voltage unbalance compensator based on a three-phase VSI operating with

unbalanced switching functions,” IEEE Trans. Power Electron., vol. 10, pp. 269–274,

May 1994.

[7] S.-J. Lee and S.-K. Sul, “A new series voltage compensator scheme for the unbalanced

utility conditions,” in Proc. EPE’01, 2001.

[8] D. Shen and P. W. Lehn, “Fixed-frequency space-vector-modulation control for three-

phase four-leg active power filters,” in Proc. Inst.Elect. Eng., vol. 149, July 2002, pp.

268–274.

[9] Zhihong Ye; Boroyevich, D.; Kun Xing; Lee, F.C.; Changrong Liu “Active common-mode

filter for inverter power supplies with unbalanced and nonlinear load” Thirty-

Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE, Vol., pp. 1858-

1863, 3-7 Oct. 1999.

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

conducted EMI reduction in voltage source inverters ” Applied Power Electronics

Conference APEC 98 , Anaheim, CA, 1998, pp. 934–939.

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

on 3D SVPWM in compensating the harmful currents of unbalanced system,” in

Proc. IEEE Power Syst. Technol., 2002, vol. 2, pp. 1045–1050.

[12] P. Lohia, M. K. Mishra, K. Karthikeyan, and K. Vasudevan, “A minimally switched

control algorithm for three-phase four-leg VSI topology to compensate unbalanced

and nonlinear load,” Trans. Power Electron., vol. 23, no. 4, pp. 1935–1944, Jul. 2008.

[13] C. A. Quinn, N. Mo