Research on Matrix Converter Based on Space Vector Modulation

Transcription

1 Research Journal of Applie Sciences, Engineering an Technology 6(16): , 2013 ISSN: ; e-issn: Maxwell Scientific Organization, 2013 Submitte: September 15, 2012 Accepte: October 31, 2012 Publishe: September 10, 2013 Research on Matrix Converter Base on Space Vector Moulation 1 Zhanjun Qiao an 2 Wei Xu 1 Department of Mechanical an Electronic Engineering, North China Institute of Science an Technology, Beijing , China 2 University of Macau, Macao SAR , China Abstract: In this stuy, we stuy the control strategy for matrix converter. Through absorbing ieas about virtual rectification put forwar by P.D.Ziogas, the common control iea of AC-DC-AC convertor is introuce into control of matrix converter; SPWM moulation strategy an space vector moulation strategy for matrix converter are stuie respectively. In terms of SPWM moulation strategy, it has some avantages: for example, main circuit structure is simple; control program is simple; control switch function oes not nee complex mathematical erivation an calculation. In terms of space vector moulation strategy, it has avantages as follows: the physical conception is clear; input power factor can be ajuste; output voltage an current sine egree are high. In this chapter, an in-epth analysis will be conucte for this control metho. Keywors: Matrix convertor, SPWM moulation, simulation, space vector moulation INTRODUCTION The transformation process of matrix converter can be consiere to be compose of rectification an inversion (Brenna et al., 2009). Sinusoial input current an ajustable input power factor are obtaine by using space vector moulation for input phase current in the rectification part an the amplitue value an sinusoial output voltage with an ajustable frequency are obtaine by using space vector moulation for output line voltage in the inversion part (Brooks, 1986); then the mile DC link is eliminate through the principle of equal mile DC voltage an DC current; thus, the metho of space vector moulation of matrix converter is obtaine (Brooks an Kaupp, 2007). The classical SPWM moulation mainly focuses on making the output voltage of inverter approaching sinusoial wave as much as possible, or expecting the funamental wave component of output PWM voltage to be as large as possible an the harmonic component to be as small as possible (Cao an Dai, 2008). If PWM voltage is controlle by tracking the circular magnetic fiel an a constant electromagnetic torque is prouce, it will have a better effect. As the trajectory of flux linkage is obtaine by voltage space vectors versus time integration, it is also calle voltage space vector moulation (Carmine et al., 2006). It is prove that the maximum amplitue value of output voltage of inverter is DC sie voltage, which is 15% higher than general SPWM inverter output voltage an the utilization rate of DC voltage is relatively high while using voltage space vector moulation (Chella et al., 2010). Voltage space vectors can be obtaine by Park transformation of three-phase output voltage in complex plane: 2 ( 2 AB BC CA ) U = U + U + U (1) 3 where, = ej 120. When the power of a motor with symmetric threephase wining is supplie by three-phase equilibrium sinusoial voltage, the voltage equation for using blene space vectors to represent stator is as follows: Ψ U = RI + (2) t where, U is a blene space vector of three-phase voltage of stator, I is a blene space vector of threephase current of stator an Ψ is a blene space vector of three-phase flux linkage. When the revolving spee is not very low, the pressure rop of stator resistance occupies a small proportion which can be ignore. So the approximate relationship between stator voltage an flux linkage is: Ψ U, i.e., Ψ= Ut (3) t Corresponing Author: Wei Xu, University of Macau, Macao SAR , China 2899

2 The equation above shows that voltage space vector U equals to the change rate of Ψ as a function of time, while its irection is consistent with the irection of motion of Ψ. Therefore, the motion trail of flux linkage can be controlle by voltage space vectors. It can be known from the above that a matrix converter can equal to a virtual AC-DC-AC converter. Therefore, a relatively mature space vector moulation moe with superior performance can be aopte. N P A B C (a) Virtual inversion structure SPACE VECTOR MODULATION OF OUTPUT LINE VOLTAGE OF MATRIX CONVERTER (VSI SVM) U BC Im ( p, pn, ) U 2 Now space vector moulation is conucte for the inversion part on the right half of the Fig. 1. The power of this part is supplie by a DC voltage. Enable U pn = U c. There are 8 permissible switch combination moes in VSI, among which 6 will prouce non-zero output voltage an 2 will prouct zero output voltage. Voltage space vector of output line is efine as: ( npn,, ) ( npp,, ) U 3 U 4 Ⅳ Ⅲ Ⅴ Ⅱ U0( U7) Ⅰ Ⅵ U 1 U AB U 6 ( pnn,, Re ( pnp,, 2 ( j120 j120 ol = AB + BC + CA ) u u u e u e (4) 3 Assume that U 0 ~ U 6 are seven voltage switch state vectors as shown in Fig. 1b. U 0 ~ U 6 six vectors ivie a cycle into six sectors, i.e., I, II, III, IV, V an VI. Each sector accounts for 60. U 0 an U 7 represent zero voltage vectors. Table 1 shows the relationship between U 0 ~ U 7 space vectors an the switch state. Any voltage space vector of output line UU RoL can be expresse by two ajacent switch state vectors UU R an UU Rβ an zero voltage vector UU R0, as shown in Fig. 3. Assume that the inclue angle between UU RoL an the subsequent component is θ SV, therefore: U ol = U + U β + U ov (5) β ov where,, ββ an ov are respectively the uty cycle ofuu R, UU Rβ an zero vector UU Rov. It can be obtaine through sine theorem within the switching perio T s that: o = T / TS = mv sin(60 θsv ) β = Tβ / Ts = mv sin( θsv ) = T / T = 1 0v ov s β (6) where: T s refers to a PWM switching perio; m v is voltage moulation factor. Moulation factor 0 m v = 3UU oooo /UU 1; θ sv is the inclue angle between 2900 U CA U 5 ( nn,, p) (b) Diagram of sectors of voltage space vectors of output line o θ SV U β β U U U OL U β (c) Vector synthesis schematic iagram Fig. 1: Diagram of voltage space vectors of output line Table 1: Relationship between VSI space vectors an switch state Switch state Space vectors S AP S BP S CP S AN S BN S CN (1: represents switch on, 0: represents switch off)

3 output voltage vector an the initial position of sectors an U PN is the voltage between buses. When the voltage space vector of output line is locate in sector I, the local average of output line voltage is: P N a b c u AB Uc Uc + β u U 0 = + β = U u ca 0 U c β cos( θsv 30 ) = mv sin(60 θsv ) Uc sin( θsv ) Bc c c (7) -30 ω oo tt-φ o in the first sector an θ sv = (ω oo tt-φ o +30 ) +30. Therefore, Eq. (3-7) changes into: u AB cos( ωot ϕo + 30 ) u = m cos( ω t ϕ ) U TVSI + = U u ca cos( ωot ϕo ) Bc v o o c c (8) (a)virtual rectification structure (, ba) (, ca) I 3 I 4 I b I c Ⅳ Ⅲ Ⅴ Im (,) bc I 2 Ⅱ I0( I7) Ⅰ I 5 Ⅵ I 1 I 6 ( ac, ) (, cb) (b) Diagram of sectors of input phase current space vectors o ν I ν I ν I ip I a Re ( ab, ) where, TTVVVVVV represents the transfer matrix of voltage source inverter at a low frequency. The local average of input current of VSI is: (9) ii RP = TT TT VVVVVV i ol = 3 I om m 2 v cos(φ L ) = CONST SPACE VECTOR MODULATION OF INPUT PHASE CURRENT OF MATRIX CONVERTER (VSR SVM) θ SC µ I µ I µ (c) Vector synthesis schematic iagram Fig. 2: Diagram of input phase current space vectors o = T / T = m sin(60 θ ) µ µ s c i = T / T = m sin( θ ) v ν s c i = T / T = 1 0c oc s µ v (11) The left half of Fig. 2 to 4 is a virtual rectifier (VSR). Enable it to prouce irect current i P = I c, similar to VSR input current space vector an VSI output voltage space vector as shown in Fig. 3. If II μμ an II vv are use to represent two ajacent phase current space vectors an II 0 is use to represent zero current space vectors, in each VSR switching perio: IiP = µ I µ + νiν + 0c I0c (10) where, μμ, vv an 0cc are respectively the uty cycle of II μμ, II vv an II 0cc an: 2901 where, m c is current moulation factor an 0 mm cc = II iiii /II 1. θ SC is the inclue angle between output voltage vector an the initial position of sectors (Table 2). For example, switch state vector is locate in the first sector of VSR hexagon. The local average of input phase current is: ia Ic Ic µ + ν i I 0 = µ + ν = µ I i c 0 I c ν cos( θsc 30 ) = mc sin(60 θsc ) Ic sin( θsc ) b c c (12)

4 Substitute Eq. (3-12) with -30 ω ii tt-φi +30 an θ SC = (ω ii tt-φi)+30. It can be obtaine that: ia cos( ωit ϕ) i m cos( ωt ϕ 120 ) = I TVSR = I i c cos( ωit ϕ+ 120 ) b c i c c a b c A B C A B C A B C Fig. 4: Three switch states of zero vectors where, T VSR represents the transfer matrix of voltage source rectifier at a low frequency. While the local average output voltage of VSR is: U 3 U 2 U 1 I 3 I 2 I 1 uu Rpn (13) = TT TT VVVVVV.uu iiiih = 3 2. mm cc.uu iiii. cos (φφ ii ) = CONST U 4 U 6 I 4 I 6 EQUIVALENT DIRECT SPACE VECTOR MODULATION U 5 I 5 It is inferre from the above that VSR space vector moulation an VSI space vector moulation can be irectly linke from the perspective of local average since the local average output voltage in VSR space vector moulation an the local average input current of VSI space vector moulation are constants. An equivalent (i.e., actual) irect space vector moulation metho can be obtaine by combining VSR an VSI space vector moulation an eliminating the mile DC link. The synthesis metho of switch control law is as shown in Fig. 3 to 5. Since input current an output voltage respectively have six (a) Sectors where voltage vectors an current vectors locate (b) General (stanar) vector transformation orer a b c P S cp N S bn S NB S NC a b c S ca S bb S bc Fig. 3: Synthesis of switch control law S PA A B C A B C 2902 (c) Optimize vector transformation orer Fig. 5: Space vector moulation orer (1) effective static vectors, 36 possible switch combination states can be obtaine accoring to the law shown in Fig. 3 to 5 an Table 3 each state correspons to a combination moe of 9 switches, as shown in Table 3. In the table, acc represents that input phase a an output phase A, I nput phase c an output phase B, input phase c an output phase C respectively connect through break-over switch. Three zero vectors not inclue in the table are aaa, bbb an ccc. It is require to first fin the switch combination of matrix converter corresponing to the equivalent AC- DC-AC structure in orer to realize space vector moulation in real matrix converter an accomplish

5 Table 2: Relationship between VSR space vectors an the switch state Switch state Space vectors S Pa S Pb S Pa S Na S Nb S Nc (1 represents switch on, 0 represents switch off, an vectors 7, 8 an 9 are zero vectors) Table 3: 36 kins of switch combination I1 I2 I3 I4 I5 I6 V1 acc bcc baa caa cbb abb V2 aac bbc bba cca ccb aab V3 caa cbc aba aca bcb bab V4 caa cbb abb acc bcc baa V5 cca ccb aab aac bcc bba V6 aca bcb bab cac cbc aba part are both ivie into 6 sectors. Therefore, there are 36 possible kins of combination. The virtual VSI space vector moulation an virtual VSR space vector moulation in matrix converter are inepenently conucte in the virtual AC-DC-AC equivalent structure. However, in the real matrix converter, the same switch conucts not only output line voltage space vector moulation but also input phase current space vector moulation an one PWM perio correspons to 5 uty cycles of voltage an current an each comprehensive uty cycle correspons to one switch combination, respectively: T1 = Tµ = µ Ts T2 = Tβµ = β µ Ts T3 = Tv = v Ts T4 = Tβµ = β v Ts T = T T T T T 0 s (14) irect transformation of AC-AC power. There are 27 kins of switch combination of matrix converter through which input voltage is not subject to short circuit an output current is not subject to suen open circuit, as shown in Table 3. Six of them cannot fin a corresponence in the equivalent AC-DC-AC transformation, which are switch combinations where three output phases respectively connect to three input phases. Therefore, 21 kins of switch combination are available, which are ivie into two groups. In group I, three output phases respectively connect to two input phases; in group II, three output phases only connect to one in out phase, resulting in short circuit. Matrix converter has an only switch state corresponing to each legal switch state of the equivalent AC-DC-AC transformation. After obtaining the switch combination of matrix converter corresponing to the equivalent AC-DC-AC structure, it is also necessary to obtain the action time (uty cycle) of each switch combination so as to realize AC-AC irect transformation control of matrix converter. AC-AC irect transformation control moe of matrix converter (i.e., ouble space vector PWM moulation) can be obtaine accoring to the corresponing relation of switch functions after aopting output line voltage space vector moulation for the inversion part of the equivalent AC-DC-AC transformation an aopting input phase current space vector moulation for the rectification part. In the implementation of ouble space vector PWM moulation, the ieal output line voltage reference vector circle of the inversion part an the ieal input phase current reference vector circle of the rectification 2903 Take the following example to explain the highfrequency synthesis steps of ouble space vector moulation of matrix converter, in which the output line voltage space vector is locate in the first sector an the input phase current space vector is locate in the first sector. It can be obtaine through Eq. (3-8) an Eq. (3-13) that: cos( θsv 30 ) cos( θsc 30 ) T mm v c sin(60 θsv ) sin(60 θsc ) = TVSIT (15) VSR sin( θsv ) sin( θsc ) Enable m= mm v c an T PhL = T VSI TT VVVVVV, thus T cos( θsv 30 ) cos( θsc 30 ) = msin(60 θ ) sin(60 θ ) sin( θsv ) sin( θsc ) PhL SV SC TT T T (16) To simplify work, enable m c = 1 an m = mv an substitute the equation above with Eq. (3-5 an 3-10). Thus, the local average of output line voltage is: u AB + β µ + ν uao ubc µ u = bo u CA β ν u co (17) Because: uab = uao ubo an uac = uao uco, therefore,

6 Table 4: Switches of matrix converter corresponing to equivalent AC-DC-AC transformation G S Aa S Ab S Ac S Ba S Bb S Bc S Ca S Cb S Cc A B C U AB U BC U CA i a i b i c I a b c U ab U bc U ca i A i B i C a c b -U ca - U bc - U ab i A i C i B b a c -U ab - U ca - U bc i B i A i C b c a U bc U ca U ab i B i C i A c a b U ca U ab U bc i C i A i B c b a -U bc - U ab - U ca i C i B i A II a c c -U ca 0 U ca i A 0 -i A b c c U bc 0 -U bc 0 i A - i A b a a -U ab 0 U ab i A i A c a a U ca 0 -U ca i A 0 i A c b b -U bc 0 U bc 0 - i A i A a b b U ab 0 -U ab i A - i A c a c U ca - U ca 0 i B 0 -i B c b c -U bc U bc 0 0 i B -i B a b a U ab -U ab 0 -i B i B a c a -U ca U ca 0 -i B 0 i B b c b U bc -U bc 0 0 -i B i B b a b -U ab U ab 0 -i B i B c c a 0 U ca - U ca i C 0 -i C c c b 0 -U bc U bc 0 i C - i C a a b 0 U ab - U ab -i C i C a a c 0 -U ca U ca -i C 0 i C b b c 0 U bc -U b 0 - i C i C b b a 0 -U ab U ab i C - i C 0 III a a a b b b c c c u AB µ + βµ ν + βν u = u + u u CA βµ βν BC µ ab ν ac where, = = msin(60 θ )sin(60 θ ) = T / T µ µ SV SC µ S βµ = β µ = msin( θsv )sin(60 θsc ) = Tβµ / TS = = sin(60 θ )sin( θ ) = T / T (18) ν ν SV SC ν S = = msin( θ )sin( θ ) = T / T βν β ν SV SC βν S = 1 = T / T 0 µ βµ ν βν 0 S etermination of 5 uty cycles an their corresponing 5 kins of switch combination state. Mutually neste ouble space vector PWM moulation strategies can not only ensure goo sinusoial property of output line voltage but also ensure that of input phase current, which realize the purpose of applying space vector moulation in control strategies of matrix converter an provie matrix converter with the effect of ouble PWM converter. The analysis above actually reflects the process of AC-AC irect transformation control of matrix converter (Table 4). OPTIMIZED SPACE VECTOR MODULATION OF MATRIX CONVERTER It can be seen from the analysis above that each output line voltage can be mae up of input line voltage combination within each switching perio. It can be obtaine from equation 3-18 that: u AB µ βµ ν βν u µ u 0 u = + + ν u + 0 u u CA 0 βµ 0 βν BC ab ab ac ac (19) There are five ifferent kins of combination corresponing to five uty cycles in each PWM perio. The control of matrix converter can be conclue as the 2904 The section above introuces space vector synthesis methos of matrix converter. Assume that the ajacent basic vectors in the section where output voltage vectors of matrix converter locates are V M (vector M for short) an V N (vector N for short) an the ajacent basic vectors in the section where input current vectors of matrix converter locates are I (vector for short) an I β (vector β for short). So the comprehensive moulation of both space vectors is realize by using mutual nesting metho. The overall vector synthesis process of input phase current an output line voltage has 5 kins of combination in total, i.e., -M, -N, β-n, β-m an zero vector I 0 - U 0.

7 U 3 U 4 U 2 U 5 U 1 U 6 (a) Sectors where voltage vectors an current vectors locate (b) General (stanar) vector transformation orer Fig. 6: Space vector moulation orer (2) I 3 I 4 The action time of each vector combination is expresse by uty cycle. Since matrix converter is working uner highfrequency PWM moulation moe an the switching loss in the switching process cannot be neglecte, how to reuce the switching loss of matrix converter has become a significant problem. We aopt a new ninesection optimize moulation strategy on the basis of common space vector moulation methos introuce above. Using the common space vector moulation metho, the vector transformation orer in one switching perio is: M- N - βn - βm- 0-βM-βN -NM, while there are three switch states of zero vector: aaa, bbb an ccc, as shown in Fig. 3 to 6. In the process of realizing the transformation of a vector sequence, the selection of zero vectors is relate to the require switching times. Up to now, the selection principle of zero vectors is still focusing on reucing the switching times of MC. As shown in Fig. 3 to 5a, assuming that voltage vectors an current vectors are both in their respective first section, first consier output voltage vector moulation an the orer of voltage vectors is M -N - N - M 0 in the first half of switching perio base on the general (stanar) vector transformation orer; an then consier input current vector moulation an its orer of vectors (in the first half of switching perio) is - β- 0. As the last effective vector (in the first half of switching perio) βm is aca, zero vector aaa is often selecte because only one switching is require from vector βm to zero vector aaa. It can be known from 2905 I 2 I 5 I 1 I 6 Fig. 3 to 5b that the switching times necessary for realizing the orer of vectors is 10. Assuming that the orer of voltage vectors in the first half of switching perio is change into N-M-M-N-0, the orer of input current vectors remains unchange. As the last effective vector βn is acc, select ccc as zero vector similarly, as shown in Fig. 3 to 5c. The switching times necessary for realizing the orer of vectors is 8. For a vector sequence with nine-section an eight-step transformation, this is the least times that can be achieve. However, it can be known from the analysis above that such times cannot always be ensure in any situation. As shown in Fig. 3 to 6a, the current vector enters into the secon sector an the voltage vector is still in the first sector. Only 8 times of switching is require base on the general (stanar) vector transformation orer, that is, the minimum switching times. The reason that the switching times are not the minimum one is that the transformation from the secon vector to the thir vector requires two times of switching instea of one. Similar situation occurs in the secon half of switching perio, making the total switching times twice more than the minimum times. It can be known from the analysis above that it is only necessary to change the orer of voltage vectors, change the general (stanar) transformation orer M - N -N - M 0 into N - M - M N-0 an meanwhile change the selection of zero vector switch corresponingly when the switching times is not the minimum one. Thus, the switching times necessary for realizing vector transformation can be ecrease to the minimum value-8 times. Figure 3 to 6b shows the transformation sequence iagram after the orer of voltage vectors is change. In short, as long as an appropriate vector sequence an zero vector switches is selecte, the switching times can be the minimum one. Through careful observation an analysis of all switching combinations using the metho above, it is known that the minimum switching times can be ensure in any situation if the vector sequence an zero vector switches are selecte accoring to the following rules. The orer of input current vectors is maintaine as -β- 0 If the sum of input sector number an output sector number is o, the sequence of output vectors is M - N - N - M 0 If the sum of input sector number an output sector number is even, the sequence of output vectors is N - M - M - N 0 When the current sector number is 4 an 1, the output zero vector is aaa; When the current sector is 5 an 2, the output zero vector is bbb; otherwise, it is ccc

8 It is thus clear that the moulation of voltage vectors an current vectors are no longer mutually inepenent an that the selection of zero vectors is etermine by current vectors. If the traitional (general) orer of vectors is aopte for moulation, the switching perio in which 10 transformation times occur will account for 50% among all switching perios, making the average switching times in a switching perio as 9. If the optimize orer of vectors is aopte, the average switching times is 8, with exceptions. When the sector where a reference vector locates in the switching perio changes, the switching state of the first vector no longer equals to that of the last vector; thus, the switching times increases. The switching times corresponing to both moulation strategies above can be expresse as: General metho : N = t[9ƒ SS +6(ƒ OO + ƒ ii )] Optimize metho : NN oooooo = t[8ƒ SS +6(ƒ OO + ƒ ii )] where, ƒ i is input frequency, ƒ o is output frequency. As ƒ i an ƒ i are much less than ƒ s (moulation frequency), the switching times increase ue to sector change can be neglecte. Then, the switching times corresponing to both moulation strategies above can be expresse as: General metho : N = t 9 ƒ SS Optimize metho : NN oooooo = t 8 ƒ SS It is thus clear that the total switching times ecreases by 11% (1/9) compare to the traitional strategy after the optimize moulation strategy is aopte an that it s switching loss will also ecrease by the same proportion corresponingly. CONCLUSION DC-AC converters into the control of matrix converter through absorbing the virtual rectification iea put forwar by P.D.Ziogas et al. an respectively stuies SPWM moulation strategy an space vector moulation strategy of matrix converter. It also proposes the optimize metho of space vector moulation strategy, that is, selecting an applicable vector sequence an zero vector base on a certain law can make the switching times reaching the minimum, thus achieving the purpose of reucing switching loss. The analysis in this stuy has significant theoretical meanings. REFERENCES Brenna, A., C. Sonia, V. Manuela an B. Browning, A survey of robot learning from emonstration. Robot. Autonom. Sys., 57: Brooks, R., A robust layere control system for a mobile robot. IEEE J. Robot. Auto., 2(1): Brooks, R. an M. Kaupp, ORCA: A Component Moel an Repository. Software Engineering for Experiment Robotics. Springer, Berlin, New York, pp: Cao, L. an R. Dai, Funamentals Concepts, Analysis, Design an Implementation. Post an Telecom Press, Beijing, China. Carmine, G., L. Lppolito an L. Vicenza, Agentbase architecture for esigning hybri control system. Info. Sci., 176: Chella, A., C. Massimo an G. Salvatore, Agentoriente software patterns for rapi an afforable robot programming. J. Sys. Software, 83(4): This stuy stuies the control strategies of matrix converter, introuces the control iea of general AC- 2906