Space Vector PWM for a Three-phase to Sixphase Direct AC/AC Converter Atif Iqbal, Senior Member, IEEE, Khaliqur Rahman, Rashid Alammari, Member IEEE
[email protected],
[email protected],
[email protected] Department of Electrical Engineering, Qatar University, Doha, Qatar
Haitham Abu-Rub, Senior Member, IEEE, Department of Electrical & Computer Engineering, Texas A&M University at Qatar, Doha, Qatar
[email protected], Abstract—In this paper a generalized multi-phase space vector theory is considered for developing the space vector modulation of a three-phase to six-phase AC to AC converter. The modulation is based on the control of the voltage vectors in the first d-q plane, while imposing the remaining voltage vectors in the second and the third planes (x-y, 0+-0-), being zero. The duty cycles of the bidirectional switches are obtained using space vector modulation theory. The proposed converter system offers full control of the input power factor, no limitation on the output frequency range and nearly sinusoidal output voltage. The proposed space vector algorithm can be fully implemented on a digital platform. The theoretical analysis is confirmed by digital simulations which is further verified using real time implementation. Index Terms—AC to AC converter, six-phase, space vector PWM, duty cycles. Multi-phase
I.
INTRODUCTION
Bi-directional power flow using direct ac-ac conversion with power semiconductor switches arranged in the form of matrix array, called a matrix converter, has been investigated extensively as a possible alternative to back-to-back converter. Matrix converter (MC) is known to offer some attractive features such as operation at unity power factor for any load, controlled bi-directional power flow, sinusoidal input and output currents and elimination of bulky dc link capacitors etc. A comprehensive overview of the development in the field of matrix converter research is presented in [1]. It is to be noted here that the most common configuration of the matrix converter discussed in the literature is three-phase to threephase [2]. Recently, topologies of matrix converter with threephase input and multi-phase outputs have been reported [416]. This paper is motivated from the recent development in three-phase input and multi-phase output matrix converter. The conventional structure for multi-phase variable-speed drives consists of a multi-phase motor supplied by a threephase power electronic converter. However, when the machine is connected to a modular power electronic converter, such as a voltage source inverter, ac-dc-ac converter with active front end or a matrix converter then the need for a specific number of phases, such as three, no longer exists since simply adding one leg of converter increases the number of output phases. Nowadays, the development of modern power electronic converters makes it possible to consider the number of phases 978-1-4799-7800-7/15/$31.00 ©2015 IEEE
a degree of freedom, i.e., an additional design variable. This has led to the tremendous interests in multi-phase drives. Multi-phase motor drives offers some advantages over the traditional three-phase motor drives such as reducing the amplitude and increasing the frequency of torque pulsations, reducing the rotor harmonic currents losses and lowering the dc link current ripples. In addition, owing to their redundant structure, multi-phase motor drives improve the system reliability. Detail reviews on the development in the area of multi-phase drives are presented in [7]. Since multi-phase drive systems have gained popularity, a need is felt to develop power electronic converters to supply such multi-phase systems. Modulation methods of the converters are complex and the complexity highly increases with the higher number of phases. Scalar control method was proposed at the initial stage of development of matrix converters [8,17] limiting the output to half the input voltage. This limit was subsequently raised to 0.866 i.e. output magnitude is 86.6% of input magnitude by taking advantage of third harmonic injection [9],[18] and it was realized that this is the maximum output that can be obtained from a three-to-three phase matrix converter in the linear modulation region. Indirect control method assumes a matrix converter as a cascaded virtual three-phase rectifier and a virtual voltage source inverter with imaginary dc link. With this representation, space vector PWM method of voltage source inverter was extended to a three-to-three phase matrix converter [2],[8]. Although the space vector PWM method is suited to three-phase system but the complexity of implementation increases with the increase in the number of switches/phases. Motivated from the simple implementation, carrier-based PWM scheme was introduced recently for threeto-three phase matrix converter [2],[8] and three-to-five phase matrix converter [3],[4]. Carrier-based PWM scheme was also investigated for three-to-seven phase [11]. Direct duty ratio based PWM scheme is presented for a generalized topology of three-to-k phase matrix converter [13] which is a modular approach. The output voltage magnitude is limited to nearly 79% in three-to-five phase and 77% in three-to-six phase while in three-to-nine phase it is limited to 76.2% in the linear mode of operation [3]. This limit can be raised by overmodulation, however the complexity will be too high. In this paper the space vector modulation scheme of threeto-five phase has been extended to a three-to-six phase AC to AC converter, considering reference space vectors in the d-q
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plane [11]. The aim of the modulation is to obtain sinusoidal output voltage. In particular, the proposed SVM strategy selects the switches configurations among the 36 = 729 possibilities by favoring the space vector on the first d-q, plane and eliminating vectors on x-y and 0+-0- plane. Simulation results are supported by preliminary experimental verification. II.
MODELING OF A THREE-PHASE TO SIX-PHASE AC TO AC CONVERTER
It is a direct power converter, generating variable amplitude voltage and frequency from a rigid entry. An intermediate DC-link circuit is not necessary as in the classical inverter case. The principle of such a converter is based on a matrix topology connecting each input phase with each output phase by a two-way power switch, allowing the flow of power in both directions and therefore, operating in all four quadrants.
Clamp circuit
Computer
Filter
MC
1: {6,0,0}:- in this group all the output phases are connected to the same input phase. Following it there are three possible combinations that are either all are connected to input phase a or b or c. {6,0,0}, {0,6,0},{0,0,6} there are 3 possible swithings. 2. {5,0,1}:- in this group 5 output phases are connected to the same input phase and remaining phase is connected to any of the remaining input phases. Thus there are 6C5 (5 out of 6 output phases are connected to same input) and 1C1 (remaining one output from any one input) (6!/5!*1!)=6 combinations. In this way {5,0,1},{5, 1,0},{0,5,1},{1,5, 0},{0,1,5},{1,0,5} 6!/(5!*1!)=6 combinations for every state combining all total 6*6=36 switching combinations. 3. {4,0,2},{4, 2,0},{0,4,2},{2,4, 0},{0,2,4},{2,0,4} 6 C4 *2C2 =(6!/(4!*2!)) * (2!/(2!*0!)) =15*6=90 4. {3,3,0},{3,0,3},{0,3,3} 6 C3 *3C3 =(6!/(3!*3!)) * (3!/(1!*3!))=20*3=60 5. {4,1,1},{1,4,1},{1,1,4} 6 C4 *2C1*1C1 =(6!/(4!*2!)) * (2!/(1!*1!))=15*2*3=90 6. {3,1,2},{3,2,1},{1,3,2},{1,3,2},{2,1,3},{1,2,3} 6 C3 *3C2 * 1C1 =(6!/(3!*3!)) * (3!/(2!*1!)) =30*3*6 =360 7.{2,2,2},6C2 *4C2 * 2C2 =(6!/(2!*4!)) * (4!/(2!*2!)) =15*6 =90 Thus total switching vectors are given as: Group 1 :{ 6, 0, 0} consists of 3 vectors, Group 2 :{ 5, 1, 0} consists of 36 vectors. Group 3 :{ 4, 2, 0} consists of 90 vectors. Group 4 :{ 3, 3, 0} consists of 60 vectors. Group 5 :{ 4, 1, 1} consists of 90 vectors, Group 6 :{ 3, 2, 1} consists of 360 vectors. Group 7 :{ 2, 2, 2} consists of 90 vectors.
Fig.1. three phase to six phase matrix converter
Source
and z are no of output phases connected to input phase a, b and c.
Table.1: Output voltage space vectors in d-q plane. Vectors Magnitude number of vectors
Six phase load
27
1 to 27
108
28 to 135
dSPACE
0 V ab,bc,ca
0.333 V ab,bc,ca
Fig.2. Six-phase Drive system topology
36
There are 218 switching combinations. Considering the constraints that “any two input phases cannot be shorted due to consideration of source as current source and output phase cannot be opened due to inductive load” [1] switching combination reduces to 36=729 only. These 729 switching combinations are analyzed in six groups that shows how many output phases are connected from input phase a, b, c. it is represented as {x,y,z} where x, y
136-171
1 3
18
*e
V ab,bc,ca * e
3
j ( 2 k 1)
172-189 0.666 V ab,bc,ca
j ( k 1)
*e
j ( k 1)
6
3
k is sector no that varies from 1st to 12th sector. In this way there are total= 3+36+90+60+90+90+360=729 switching combinations in which vectors mentioned in 1, 2, 3 and 4th group are having varying amplitude and fixed frequency. Such
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switching combinations are 3+36+90+60=189 which are used as active switching. Remaining are having varying frequency so they are not helpful to find out the output voltage. Thus for our conversion operation we will use group 1 to 4. For our operation we will use line to line voltage having minimum phase difference. Model for matrix converter Vo / p M T V I / p
(1)
I i MI o
(2)
S Aa M S Ab S Ac
S Ba
S Ca
S Da
S Ea
S Bb
S Cb
S Db
S Eb
S Bc
S Cc
S Dc
S Ec
S Aa S Ba M T S Ca S Da S Ea S Fa
S Fa S Fb S Fc
S Ac S Bc S Cc S Dc S Ec S Fc
S Ab S Bb S Cb S Db S Eb S Fb
Vx y
V0 0
SAb SBb SAc SBc SBb SCb SBc SCc SCb SDb SCc SDc SDb SEb SDc SEc SEb SFb SEc SFc SFb SAb SFc SAc
Vo / p V AB
VCD
VDE
V EF
V FA T
V I / P Va Vb Vc T I o / p I A
I I / P Ia
IB
IC
Ib
IE
IF
T
T
(3)
120
S Bb SCb SCb S Db S Db S Eb S Eb S Fb S Fb S Ab
S Ac S Bc S Bc SCc V a SCc S Dc V b S Dc S Ec V c S Ec S Fc S Fc S Ac
S Ba
SCa
S Da
S Ea
S Bb S Bc
SCb SCc
S Db S Dc
S Eb S Ec
IA S Fa I B S Fb I C S Fc I D IE I F
240
(5)
60
0 d
-q
300
(a) (6)
120
y 1 0.5
(10)
60
0 x
180 (11)
The obtained output line voltage from equation no can be used to find out voltages and currents in d-q and x-y plane after changing it in polar form as Vd-q, and Vx-y are the output voltages in different orthogonal plain for adjacent phases. 2 Vdq (VAB aVBC a2VCD a3VDE a4VEF a5VFA) 6
0.8 0.6 0.4 0.2
180
(9)
S Ab S Bb
(14)
j 2 pi / 6
(7) (8)
Ic
V AB S Aa S Ba V S S Ca BC Ba VCD SCa S Da VDE S Da S Ea VEF S Ea S Fa VFA S Fa S Aa
I a S Aa I S b Ab I c S Ac
ID
2 (V AB a 3V BC a 6VCD a 9V DE a12V EF a15V FA ) 6
(4)
SAa SBa S S Ba Ca T SCa SDa M ' SDa SEa SEa SFa SFa SAa
(13)
Where, a e , The space vectors obtained are illustrated in Fig. 3. The zero sequence vectors are not considered as the load assumed is with isolated neutral point. The space vectors are generated in pairs such that they have equal magnitude but opposite polarity (180 ̊ phase shift). Further they are represented in terms of Vab, Vbc and Vca (input line-line voltages).
SAa….. shows that the switch connect the out phase A with input phase a and so on. T shows the transpose of the transformation matrix. Moreover “ ’ ” shows the shifts of rows for output line voltage calculation.
VBC
2 (V AB a 2V BC a 4VCD a 6V DE a 8V EF a10V FA ) 6
(12) 1181
240
300 (b)
0-
2 60 1.5 1 0.5
120
180
240
vm vm jvm and vl vl jvl
and t s tl t m t sh to III.
0 0+
(c)
ts
*
t sh 2(1 k ) tm
vl v* vl v* ts vl vm vl vm
source
[Volts], I
source
V
-3 0.04
0.05
0.06
0.07
0.08
0.09
0.08
0.09
0.1
[Volts] An
V
Spectrum of V
An
100 0 -100 0.04
0.05
0.06
0.07 Time [s]
40
Fundamental = 34.9312
20 0 0
0.1
100
200
300
400 500 600 Frequency [Hz]
700
800
900
1000
[Volts]
200 0
AB
V
(15)
-200 0.04
0.05
0.06
0.07 Time [s]
0.08
0.09
0.1
40 Fundamental = 34.9542
20 0 0
100
200
300
400 500 600 Frequency [Hz]
700
800
900
1000
(c) 1
I
Load
[Amp]
0.5 0 -0.5 -1 0.04
*
vl vm vl vm
-2
(b)
(16)
vm v vm v
0 -1
(a)
Where tl, tm and tsh are the time for large, medium and short length vectors. Expressions for these time for corresponding space vectors are obtained in (sir chapter). The proportion of the time is adjusted for modification of the PWM pattern. vl vm vl vm
1
AB
and the medium space vectors are of length 1 / 3 of VLL and the short space vectors are of length 1/3 of VLL. The two orthogonal planes are composed of three different hexagons. There are 12 sectors in the d-q plane, each sector have spanning of 30 deg. In the proposed scheme of this paper we are using five active and two zero vectors in which two vectors of short, two vectors of medium and one vector of large length are taken for modulation. The performance of the scheme has been investigated in terms of THD in output voltage. There are different methods to form the reference voltage vectors but our aim is to have only d-q component and eliminate x-y component to reduce the THD. For this the voltseconds principle is applied that is,
2
Spectrum of V
It is clear from the space vector diagram that all the 162 active and 27 zero vectors are placed at 19 different positions. In which 6 positions are for 108 small, 6 positions are for 36 medium and 6 positions are for 18 large vectors, which is shown in the Fig.3. It is to be noted that the vectors lying on the same position have different switching combinations. Line voltage space vectors in sector 1st with x-y and 0+-0- plane is shown in Fig.3. The lengths of the outer large space vectors are 2/3 of VLL
tl k
3
[Amp]
Fig. 3. Three to six phase space vectors in (a) d-q, (b) x-y and (c) 0+-0- plane
vm v* vm v*
SIMULATION RESULTS
To validate the modulation scheme, Matlab/Simulink model is developed with the following parameters: Three-phase input voltage magnitude = 100V, Load element: R =75 Ω; L = 193 mH, Switching frequency = 6 kHz, Input supply frequency = 50 Hz. Output frequency is taken as 50 Hz, simulation and experiment is performed at 0.45 percent of maximum transfer ratio. The resulting waveforms from the simulation model are presented in Fig. 4. The output currents are sinusoidal. The source side current is having harmonics within acceptable limit.
300
v*s t s vl tl vmt m vsht sh vo to
(19)
here k is constant that varies from 0 to 1. Results are presented at 2/3.
0.05
0.06
0.07 Time [sec]
0.08
0.09
(d)
ts
(17)
Fig. 4. (a) Source voltage and current, (b) Output phase voltage with spectrum(c) Output line to line voltage with spectrum (d) Load currents
(18)
Where 1182
0.1
Table.2 Output voltage with their THD at full range of operation. S.N % of maximum Percentage THD Fundamental transfer 1 0.1 16.06 8.1764 2 0.2 14.40 17.2358 3 0.3 14.10 26.1508 4 0.4 13.95 34.94 5 0.5 13.80 43.5241 6 0.6 13.49 51.76 7 0.7 13.71 58.6956 8 0.8 14.67 64.2707 9 0.9 15.34 69.2479 10 1 16.16 74.4143
From the table it is clear that at maximum output is around the same as calculated in [3] i.e 77% at the maximum transfer ratio. Moreover THD is almost same for full range of operation. IV.
(b)
EXPERIMENTAL RESULTS
A laboratory prototype of a matrix converter is developed for testing the three-to-six phase converter. The developed structure of matrix converter is modular in nature, since one module is one leg. Each bidirectional switch is having two IGBTs connected in antiparallel. Clamp circuit is connected at both input and output sides. The control code developed in Matlab/Simulink was uploaded to the dSpace 1006 working in conjunction with FPGA board DS 5203 in order to produce PWM signals. Experimental parameters are as follows: switching frequency is kept at 6 kHz, sampling time of the code is 30 µs, the ac input voltage is 100 V (rms), 50 Hz and the output is commanded at 50 Hz. A dead band of 2 µs is kept between the turning on and off of the two adjacent switches of a leg. The dead band is created using an external Xilinx FPGA board which is programmed in VHDL. The resulting waveforms are presented in Fig.5. The output voltage is seen as expected from the simulation result, the current is almost sinusoidal. The THD in voltage at 50 Hz is 13.8%. The reason of not getting perfect sine wave is due to the presence of the dead band and some non-linarites, and more over the limitation of the sampling time that is limited to 10 micro second.
(c)
(d) Fig. 5. (a) Source voltage and current, (b) Output phase voltage (c) Output line to line voltage (d) Load currents
V.
(a)
CONCLUSION
A Space Vector Modulation control strategy for a three to six-phase AC to AC converter has been proposed in this paper. The modulation is based on the extension of the space vector approach to three and five phase circuits, leading to triple d-q planes representation. The duty cycles of both active and null vector configurations are calculated on the basis of a detailed space vector approach, leading to the analytical determination of the modulation limits. The numerical simulations carried out with reference to a six-phase converter supplying a sixphase balanced load confirm the effectiveness of the proposed SVM strategy.
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Acknowledgment: This publication was made possible by NPRP grant # [NPRP 4-152-2-053], from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors.
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[14]
[15]
[16]
[17] [18]
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