energies Article
A Switching Frequency Optimized Space Vector Pulse Width Modulation (SVPWM) Scheme for Cascaded Multilevel Inverters Xiongmin Tang 1, *, Junhui Zhang 1 , Zheng Liu 2 and Miao Zhang 1 1 2
*
School of Automation, Guangdong University of Technology, Guangzhou 510006, China;
[email protected] (J.Z.);
[email protected] (M.Z.) School of Electrical and Information Engineering, Changsha University of Science and Technology, Changsha 410004, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-020-3932-2553
Academic Editor: Gabriele Grandi Received: 21 March 2017; Accepted: 12 May 2017; Published: 21 May 2017
Abstract: This paper presents a novel switching frequency optimized space vector pulse width modulation (SVPWM) scheme for cascaded multilevel inverters. The proposed SVPWM is developed in a α0 β0 coordinate system, in which the voltage vectors have only integer entries and the absolute increment of coordinate values between adjacent vectors is equal to dc-bus voltage of power cells (1 pu). The new SVPWM scheme is built with three categories of switching paths. During each switching path, the change of one phase voltage is limited in 1 pu. This contributes to decrease the number of commutations of switches. The proposed SVPWM scheme is validated on a 7-level cascaded inverter and the results show that it significantly outperforms traditional SVPWM schemes in terms of decreasing the number of switch commutations. Keywords: space vector pulse width modulation (SVPWM); switching frequency optimized; cascaded multilevel inverter; switching path planning
1. Introduction Multilevel inverters have been selected as a preferred power converter topology for high voltage and high power applications [1,2]. They are widely used in medium voltage motor driver [3], high-voltage direct current (HVDC) transmission systems [4], static var generator (SVG) [5], active power filter (APF) [6] and other applications [7,8]. Several topologies for multilevel inverters have been proposed in the past decades, including diode-clamped [9], flying capacitor [1], cascaded H-bridge [10] and modular multilevel [11]. For a given multilevel converter topology, modulation scheme is a key factor [12–17]. Among various modulation schemes, space vector pulse width modulation (SVPWM) scheme is an attractive candidate due to its flexibility to optimize switching waveforms and its convenience to be implemented in digital signal processors [1,12–14]. Generally, most SVPWM schemes can be implemented with the following three steps: (1) (2) (3)
Locate the reference voltage vector and select three nearest space vectors. Calculate the duty cycles for the three space vectors. Generate the switching sequences with specific constraint.
Because the space vector diagram of multilevel inverters is comprised of equilateral triangles in traditional αβ coordinate system, the location of reference voltage vector and the duty cycles of the three nearest vectors are obtained with extensive computation. Consequently, the implementation of
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SVPWM schemes in αβ coordinate system usually becomes more complex as the number of levels increases. To speed up Step 1 and Step 2, various coordinate systems have been developed [18–21]. The determination of redundancies with specific constraint in Step 3 remains a big challenge for SVPWM schemes. In order to decrease the number of commutations of switches, some switching Energies 2017, 10, 725 2 of 18 techniques have been developed in [22,23]. By carefully selecting redundant switching positions in The determination of redundancies withthe specific constraint in Step 3 remains a loss big challenge forscheme upper type triangle, lower type triangle and section number, a minimum SVPWM SVPWMin schemes. In order to as decrease the number of commutations of switches, some switching of the was proposed [22]. However, the number of level increases, the selecting complexity techniques have been in [22,23]. By carefully selecting redundant in redundancies increases and developed no generalized technique for redundancies and switching switchingpositions state transitions upper type triangle, lower type triangle and the section number, a minimum loss SVPWM scheme is developed. Based on two mappings, a SVPWM scheme is proposed in [23]. Because the different was proposed in [22]. However, as the number of level increases, the selecting complexity of the “modes”redundancies need to be determined andnothegeneralized region number of theformodulation triangle should bestate calculated increases and technique redundancies and switching in the SVPWM scheme, the rule of determining theaswitching sequences is still so far. transitions is developed. Based on two mappings, SVPWM scheme is proposed in challenging [23]. Because the different “modes” needswitching to be determined and theoptimized region number of the modulation shouldand the In this paper, a novel frequency SVPWM scheme istriangle proposed be calculated are in the SVPWM scheme, determining the switching is still main contributions summarized below:the (1)rule the of proposed SVPWM schemesequences is developed in a α0 β0 challenging so far. coordinate system; (2) The change of one phase voltage is limited in 1 pu during each switching path; In this paper, a novel switching frequency optimized SVPWM scheme is proposed and the main (3) The experimental results show that the proposed method significantly outperforms the traditional contributions are summarized below: (1) the proposed SVPWM scheme is developed in a α′β′ SVPWMcoordinate schemes system; in terms decreasing of commutations (2)ofThe change of the one number phase voltage is limited in 1 of puswitches. during each switching Thepath; paper organized as follows: Section describes concepts of basicoutperforms switching mode and (3) isThe experimental results show that2the proposedthe method significantly the traditional SVPWM 3schemes in terms of decreasing of commutations of experimental switches. switching path. Section presents the principle of the thenumber SVPWM scheme. The results The paper is organized as follows:to Section 2 describes the concepts of basic switching mode and and comparative analysis are presented validate the theoretical analysis in Section 4. Finally, the switching path. Section 3 presents the principle of the SVPWM scheme. The experimental results and conclusions are presented in Section 5. comparative analysis are presented to validate the theoretical analysis in Section 4. Finally, the conclusions are presented in Section 5.
2. Basic Switching Modes and Switching Paths
2. Basic Switching Modes and Switching Paths
2.1. α0 β0 Coordinate System
2.1. α′β′ Coordinate System
A α0 β0 coordinate system is our developed coordinate system in [24]. In the α0 β0 coordinate A α′β′ coordinate is entries our developed in [24]. In the α′β′between coordinate system, voltage vectors havesystem integer and thecoordinate absolutesystem coordinate increment adjacent system, voltage vectors have integer entries and the absolute coordinate increment between adjacent 0 0 vectors is equal to 1. The transformation matrix among the α β coordinate system, the traditional αβ vectors is equal to 1. The transformation matrix among the α′β′ coordinate system, the traditional αβ coordinate systemsystem [1] and system inEquation Equation coordinate [1]abc andcoordinate abc coordinate systemare arewritten written in (1):(1):
√2 3#" √2 2 2 2 3 2
2#
# a " # a α 1 0 −1 a−c c 1 0 1 b a = , (1) = 2 = (1) β0 β − 1 1 0 b ,− a + b c 1 1 0 a b 2 c 2 where, a, b and c are the normalized three phase voltages of multilevel inverters. [α β]T and [α0 β0 ]T where, a, b and c are the normalized three phase voltages of multilevel inverters. [α β]T and0 [α′ β′]T are the coordinate values of [a b c]T inT the traditional αβ coordinate system and the α β0 coordinate are the coordinate values of [a b c] in the traditional αβ coordinate system and the α′β′ coordinate system, system, respectively. respectively. 0 coordinate system are shown in The projections of vectors for for 3-level inverter αβand and α0 βcoordinate The projections of vectors 3-level inverterin in the the αβ α′β′ system are shown in Figure 1,Figure where, each vector is represented bybyone 1, where, each vector is represented onedot. dot. "
α0
#
"
√
3 2√ − 23
"
2 3 3
3 3
3 3
2 3 3
(a)
(b)
Figure 1. Space vector diagram for 3-level inverter in different coordinate systems, (a) in the traditional
Figure 1. Space vector diagram for 3-level inverter in different coordinate systems, (a) in the traditional αβ coordinate system (b) in the α′β′ coordinate system. αβ coordinate system (b) in the α0 β0 coordinate system.
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2.2. Three Basic Switching Modes 2.2. Three Basic Switching Modes The basic switching mode and switching asfollows: follows: The basic switching mode and switchingpath pathare are defined defined as (1) Basic switching mode: the is changed changedwith with1 1pupu between adjacent (1) Basic switching mode: thevalue valueof of a,a, bb or c is between twotwo adjacent switching states. There three kindsofofbasic basic switching switching modes vector diagram, and and the the switching states. There areare three kinds modesininspace space vector diagram, of basic switching modes is named as diagonal switching mode, vertical switching three three kindskinds of basic switching modes is named as diagonal switching mode, vertical switching mode, mode,and horizontal switching mode in the paper, respectively. The three kinds of basic switching and horizontal switching mode in the paper, respectively. The three kinds of basic switching modes modes arein illustrated are illustrated Figure 2.in Figure 2.
0 1
0
V2
V0
0
V3
V1
0 1
Figure Locusofofthree three basic basic switching Figure 2. 2. Locus switchingmodes. modes.
(2) Switching path: a trajectory is formed by a finite number of basic switching modes in turn.
(2) Switching path: a trajectory is formed by a finite number of basic switching modes in turn. The increment expression of Equation (1) is described in Equation (2).
The increment expression of Equation (1) is described in Equation (2). a c ( , 0 ∆α −a∆c b ∆a = , ∆β0 = −∆a + ∆b
(2)
(2)
where, △a, △b and △c are the increment of a, b and c in the abc coordinate system. △α′ and △β′ are the increment of α′ and β′. where, ∆a, ∆b and ∆c are the increment of a, b and c in the abc coordinate system. ∆α0 and ∆β0 are the If phase A voltage is changed with 1 pu in one basic switching mode, subject to:
increment of α0 and β0 . ain one 1; basic b cswitching 0, If phase A voltage is changed with 1 pu mode, subject to:
(3)
Substituting Equation (3) in Equation (2), the expression of the diagonal switching mode is ∆a = ±1; ∆b = ∆c = 0, (3) written in Equation (4):
a 1 of the diagonal switching mode is written expression Substituting Equation (3) in Equation (2), the , (4) a 1 in Equation (4): ( 0 = ∆a = ±1 If phase B voltage is changed with 1∆α pu in one basic switching mode, subject to: , (4) ∆β0 = −∆a = ∓1
b 1; a c 0 ,
(5)
If phase B voltage is changed with 1 pu in one basic switching mode, subject to:
Substituting Equation (5) in Equation (2), the expression of the vertical switching mode is written in Equation (6):
∆b = ±1; ∆a = ∆c = 0, (5) 0 (6) Substituting Equation (5) in Equation (2),the expression, of the vertical switching mode is written b 1 in Equation (6): ( If phase C voltage is changed with 1 pu ∆α0in=one 0 basic switching mode, subject to: , (6) 0 =1;∆b ∆β c a = ±b1 0 , (7) Substituting Equation (7) inwith Equation (2),one the basic expression of themode, horizontal switching mode is If phase C voltage is changed 1 pu in switching subject to: written in Equation (8):
∆c = ±1; ∆a = ∆b = 0,
(7)
Substituting Equation (7) in Equation (2), the expression of the horizontal switching mode is written in Equation (8):
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c 1 , ( 0 0 ∆α = −c = ∓1
(8)
, (8) 0 =0 As shown in Figure 2, the locus of∆β diagonal switching mode is from V2 to V1, the locus of vertical switching mode is from V2 to V0 or V1 to V3, and the locus of vertical switching mode is from As shown in Figure 2, the locus of diagonal switching mode is from V 2 to V 1 , the locus of vertical V3 to V2 or V0 to V1. The increment relationships between a, b, c and α′, β′ are listed in Table 1. switching mode is from V 2 to V 0 or V 1 to V 3 , and the locus of vertical switching mode is from V 3 to V 2 or V 0 to V 1 . The increment relationships between a, b, c and α0 , β0 are listed in Table 1. Table 1. Three basic switching modes.
Three basic ∆b switching modes. Switching ModeTable 1. ∆a ∆c diagonal ±1 0 0 Switching Mode ∆b ∆c 0 ∆α0 vertical 0 ∆a ±1 diagonal 00 0 ±1 ±1 horizontal 0 ±1 vertical horizontal
2.3. Characteristics of Switching Paths
0 0
±1 0
0 ±1
0 ∓1
∆α′ ±1 0 0∆β ∓1∓1
∆β′ ∓1
1 0
±1 0
It is shown of in Switching Figures 1bPaths and 2 that any triangle is formed by the above-mentioned three basic 2.3. Characteristics switching modes. If a switching path is composed of three sides of a triangle by end to end, the It is shown in Figuresof1bthe and 2 that anyoftriangle is formed bybefore the above-mentioned three basic constraint relationships realization the starting vector and after the switching path switching modes. If a switching path is composed of three sides of a triangle by end to end, the can be written-in Equation (9): constraint relationships of the realization of the starting vector before and after the switching path can ( aˆ 0 , bˆ0 , cˆ0 ) ( a 0 , b0 , c0 ) (1,1,1) , (9) be written-in Equation (9): ˆ ( aˆ 0 , b0 , cˆ0 ) = ( a0 , b0 , c0 ) ± (1, 1, 1), (9) where, ( a 0 , b0 , c0 ) and ( aˆ 0 , bˆ0 , cˆ 0 ) are the realization of the starting vector before and after where, ( a0 , b0 , c0 ) and ( aˆ 0 , bˆ 0 , cˆ0 ) are the realization of the starting vector before and after switching switching path, respectively. The sign of (1, 1, 1) is determined by the direction of switching path. path, respectively. The sign of (1, 1, 1) is determined by the direction of switching path. If the direction If the direction of switching path is clockwise, the sign is positive, otherwise the sign is negative. of switching path is clockwise, the sign is positive, otherwise the sign is negative. However, if the switching path is composed of four sides of a quadrangle by end to end, the However, if the switching path is composed of four sides of a quadrangle by end to end, the constraint relationships of the realization of the starting vectors before and after switching path can constraint relationships of the realization of the starting vectors before and after switching path can be be written as: written as: ˆ0 )( ( aˆ 0 ,(bˆaˆ00,,cˆbˆ00), c= a0(, ab0 ,, bc0 ), ,c0 ) , (10) (10) There areare three kinds of quadrangles formed by right triangle 4A △A andand 4B,△B, andand the three kinds of There three kinds of quadrangles formed by right triangle the three kinds quadrangles are shown in Figure 3. of quadrangles are shown in Figure 3.
(a)
(b)
(c)
Figure Threekinds kindsofofquadrangles, quadrangles, the first kind; the second kind; the third kind. Figure 3.3.Three (a)(a) the first kind; (b)(b) the second kind; (c)(c) the third kind.
As described above, if switching paths consist of the four sides of the quadrangles, the proposed As described above, if switching paths consist of the four sides of the quadrangles, the proposed SVPWM scheme will be achieved. SVPWM scheme will be achieved.
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Energies 10, 725 SVPWM Scheme 3. The 2017, Proposed 3. The Proposed SVPWM Scheme
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3.1. Trajectory and Traversing Graphs of Reference Voltage Vector 3.1. Trajectory and Traversing Graphs of Reference Voltage Vector 3. The Proposed SVPWM Scheme Generally, the three normalized phase voltages ar, br and cr for reference voltage vector Vr can be Generally, theTraversing three voltages ar, bVector r and cr for reference voltage vector Vr can be 3.1. Trajectory and Graphs ofphase Reference Voltage defined in Equation (11):normalized defined in Equation (11): Generally, the three normalized phase art,)brEand cr for reference voltage vector Vr can be V cos( ar voltages ar Vm mcos(t ) E defined in Equation (11): t 2 3) E , (11) br Vm cos( (ωt arbr=VVmm cos t)/E2 3) E , cos( (11) co(s( t 2π/3 4 3) (11) ωt− )/EE , br c=r VmVmcos cos(t 4 3) E c cr=VVm cos ( ωt − 4π/3 ) /E r m where, Vm and ω are the peak voltage and angular frequency. E is dc-bus voltage of power cells. m and ω are the peak voltage and angular frequency. E is dc-bus voltage of power cells. where, V where, Vm and ω are the peak and angular frequency. E is dc-bus voltage of power cells. Substituting Equation (11)voltage in Equation (1): Substituting Equation (11) in Equation (1): Substituting Equation (11) in Equation (1): sin(t ) # sin(ωt √ 3Vr sin( r t + π 3) ) 3V αr0 r V 3 3 , 3 r r = π , , β0r r E E sin (ωt t − )) E sin( r si n(t 33)
"
T
(12) (12) (12)
3
0 0 T the where, coordinate values of of VrV . r. the coordinate values where, [α[ r , rβ, r] rT] is is is can the coordinate values of V r. where,Equation [ r , r ] (12) Equation (12) can be be rewritten rewritten as: as: Equation (12) can be rewritten ( as: √ αr0 +r β0r =r 3V (ωt 3V sin ()t/E ) E r sin r , α0 − 3Vr sin ( t ) r β 0 r= 3Vr cos (ωt)/E E , , r r
(13) (13) (13)
r r 3Vr cos(t ) E r r 3Vr cos(t ) E
The is shown shown in in Figure Figure 4. It can can be be seen seen that that the the trajectory trajectory of is an an ellipse. ellipse. The trajectory trajectory of of V Vrr is 4. It of V Vrr is The trajectory of Vr is shown in Figure 4. It can be seen that the trajectory of Vr is an ellipse.
Figure 4. 4. Trajectory Trajectoryof ofVVr.. Figure Figure 4. Trajectory of Vr.r
Figure 5 demonstrates the different traversing graphs of reference voltage vector for 7-level Figure55 demonstrates demonstratesthe thedifferent differenttraversing traversinggraphs graphsof ofreference referencevoltage voltagevector vectorfor for7-level 7-level Figure inverter with different modulation indexes. inverterwith withdifferent differentmodulation modulationindexes. indexes. inverter
(a) (a)
(b) (b)
(c) (c)
Figure 5. Traversing graphs of reference voltage vector with different modulation indexes for 7-level Figure 5.5.Traversing voltage vector with modulation Figure Traversing graphs ofreference reference voltage vector withdifferent different modulationindexes indexesfor for7-level 7-level inverter, (a) 0.9632
