Simple On-line Step Pulse PWM Method for

Unbalanced Voltage Sources in Cascade MultilevelInverters

Nguyen-Van Nho1, Hong-Hee Lee2, Nguyen-Dinh Tuyen1,Phan Quoc Dzung1'HochiminhCity University Of Technology-VNUHCM, Department of Electrical and Electronics Engineering

268 LyThuongKiet, District 10, HochiminhCity, Vietnam2Department of Electrical Engineering, University of Ulsan680 - 749 San 29, Mugeo 2-Dong, Nam-Ku, Ulsan Korea.

Abstract - Nowadays, an intensive development of multilevelinverters has been seen in hybrid and fuel cell electric vehicles.For this, a high voltage cascade inverter with step pulse PWMstrategy is an applicable topology as it can reduce electromagneticinterference as well as switching losses on power devices. Theconverter is usually controlled with a look-up table ofcommutation angles, which are computed off-line base on anassumption of equal dc voltage sources. However, in reality, theinequality of these dc voltage sources can occur due to differentcharging/discharging processes and this reduces the quality ofoutput voltages such as the existence of low order harmonics andan error in fundamental voltage.

To overcome this problem, PWM angles in the look-up tablecan be changed according to different cases of dc source variation.However, this solution may require a large memory for the look-up table. Thus, off-line PWM stratergy in case of dc voltagesource variation may not be a suitable approach to control theconverter.

In this paper, an on-line step pulse PWM method will beproposed to control fundamental voltage and reduce theamplitudes of low-order harmonics to possibly low values.Similar to the original on-line method, the control principlebetween two defined trajectories will be appropriately modifiedfor solving the problem. The principle is mathematicallyformulated, and demonstrated by simulation results.

I. INTRODUCTION

In recent years, it has been seen a very fast development ofmultilevel inverters in high power applications. One offavorable topologies is cascade multilevel inverters. The highvoltage converters have been considered for use in hybridelectric vehicle. For low number of switchings in fundamentalperiod, a step pulse PWM method becomes a possible solutionfor low switching losses. Normally, step pulse PWM isimplemented off-line with use of look-up table. For storingcomputated commutation angles as function of modulationindex and obtaining high precision, it is required a largememory. To avoid the problem, on-line step pulse methodintroduces a flexible solution [1]. In recent work, an on-linealgorithm to calculate reference commutation angles with theuse of two limit angle sets has presented several advantages:linear control in entire modulation range, low harmonic

distortion factor for large midddle modulation index range,possibility for applications respectiveless to number of levels[1].

Van

Ct 264; 5A

2

vl n (--a) (2-a5)

.2n

(yr+a4) (2~-eO

I 2I )

Or IT 2la2 2T*v5n

Figure 1: Cascade inverter. Diagram of a) one phase circuit and b) PWMscheme.

The previously discussion concerns mainly to PWM schemefor balance dc voltage sources. In practice, because of severalfactors as variable load power factor, different power sharingof H-bridge cells, the dc sources can not be supposed asconstant As a result, commutation angles will be varieddifferently for individual voltage source. To satisfy theprinciple as eliminate harmonics and control fundamentalvoltage, look-up table may require too big memory.This paper study a method to overcome the describedproblem. The solution can give some advantages as: 1) no largememory is needed, 2) linear control of fundamental voltage forentire range of modulation index, 3) flexible control for variousvoltage source conditions and 4) possible low harmonicscontent in the output.

II. PRINCIPLE OF PROPOSED PWM METHOD

Conventional step pulse PWM for cascade inverter withbalanced dc-voltages: commutation angles are calculated to

1-4244-0427-4/06/$20.00 ©2006 IEEE - 43 - Oct. 18 -Oct. 20, 2006 FOST2006

control fundamental voltage and eliminate several unrequired sources Vd . These commutation angles perform a standardharmonics. For modulation index of m, the mathematical data stored in a look-up table for proposed method.statement of these conditions is then expressed by equations asfollows:

cos(01) + cos(02)+ +cos(s) = 5mcos(kO1) + cos(kO2)+ .+cos(kOs) = 0k=5,7,11,13

(1 a)(lb)

Exact solutions for number of modulation indexes of0.4,0.5,..,0.8 will be used for implementing on-line step pulsePWM by applying principle control between two limittrajectories. To get data for lower modulation indexes,solutions can be obtained by reducing the number ofeliminated harmonics. In this case, some commutation angleswill be set equal to 2T/2 [1 ]. In overmodulation, simplesingle-mode overmodulation has upper limit modulation indexof m=1.1, corresponding six-step mode and all commutationangles are set equal to 0.

O0 IO O 4

Undermodulation

m Ua1:ia, U CUC

m U. Ul, a a a; Normal modulation

.... ......... .. .. .........

.... ...,C ,,,,,C ==t = w =. ........................ ...=|. v rmod lahom Overmodulafion

II

a tGf a -a 4 fa 0

Figure 2: Principle control between limit trajectories

The described approach enables to determine referencecommutation angles for entire modulation index range,including under- and over-modulation. Related diagrams aredrawn in Fig.3.

For proposed method, the commutation angles at selectedmodulation indexes deduced on condition of balanced dcvoltages will perform standard angle sets for generatingreference commutation angles on condition of unbalanced dcvoltage sources.

Principle of proposed PWM method for unbalanced dcvoltage sources can be described in three stages.Stage 1. Define number of selected limit modulation indexes{M} = {O, MIn,m2 I...Immax } * For each modulation index

mi e {M} , compute off-line commutation angles

{0, } = {0li , 02i ...0 Ni } on conditions of balance dc voltage

Figure 3: Diagram of commutation angles, Harmonic amplitudes, controlcharacteristic and THD factor for case of balanced dc voltage sources

Stage 2. For reference modulation index m 0 < m < MMax,select two limit modulation indexes ma mb closest to m, i.e.

{ma, mb } C {M}, ma <m <mb . Normalize related limitcommutation angle setsfor unbalanced dc voltage sources.

Define Vdl ,Vd2 ,..., Vdk unbalanced dc voltage sources on

N H-bridge cells and 0ik commutation angles normalized in

unbalanced dc voltage sources Vdk . The condition is that thegenerated step pulse waveform produces the same fundamentalvoltage of case ofbalance dc voltages, i.e.

Vd s 0,'1 + Vd2 cO,'2 +. + VdN COS IN = nkmm (2)

mi e{M}, i=a,b.One of possible solutions is deduced from conditions asVdkCOS0ik = Vd COSOk; i=a, b; k 1,2,. N (3)

Where Vd dc source voltage and 0ik commutation angle for k-

th H-bridge cell and modulation index of mi under balance

condition. For any limit modulation index of {M},

commutation angles for dc voltage Vdk can be computed as:

Oik =Cos (Vd cos Oik / Vdk),i =a,b; k =1,2,..,N (4)If (Vd cos 0ik / Vdk) > 1, then the angle Oikwill be set equaltoO' = 0 (5)In this case, an error of fundamental voltage will occur.Stage 3. Applying principle control between limit trajectories,from normalized limit commutation angle sets.

- 44 -

For reference modulation index m, ma < m < mb , the

reference angular set 0.k on condition of the voltagecan be derived from normalized angles as:

(1-)VA COS 0ak+i7VdkcoSbk = Vdk cosk (6)

or Ok = COS [(1-h) COSOak +h COSObk ]where 77 = (m-ma ) I(mb -ma )

VA

(7)(8)

III. CHARACTERISTICS OF PROPOSED METHODS

Maximum modulation index. Maximum modulation indexvaries depending on the total dc voltage source and can bedefined as:

7, ( 11T *(Vdl +Vd2 + Z *)=(Vdl +Vd2 +" VS*)%ax= ~- - .(9)(2/;1T).*Vd + Vd +. . .) NVd (n-l)Vd /2

Where Vd is a standard voltage on each dc source on

condition of balance voltages. In overmodulation, themaximum modulation index can get a higher value than 1.1 iftotal dc voltage Vs > 0.5(n-l)Vd.Linearity ofcontrol characteristic.

It can be proved that the linear control of fundamentalvoltage can be obtained for the entire range of modulationindex until mMax.

Solution existence. For existence of solution 0tkI it is requiredthat° < (Vd COSOik /VAk).1 (10)

In case of Vdl > Vd, the solution will be always found.

The problem may be unsolvable for small dc source Vdk and

commutation angle 0xk . To avoid this, H-bridge cell of lower

dc voltage source Vdk should be designed with larger

commutation angle Oxk .

Define a minimum value of the voltage drop, for instance500, then commutation angle determined from diagrams ofbalance case should meet the condition as: COS0xk < 0.95

and Oxk >18 . For example, if all 5-dc voltage sources

decrease to values of Vd = Vd2 Vd5 = 0.95Vd, to

meet the condition (10), it is required limit modulation indexmi <0.65. For m>0.65, single-mode overmodulation can be

established with ma = 0.65 and mb = mMaxI If only some

from dc-source decrease to the mentioned limit, thenovermodulation will occur with higher limit modulation index.

For extending linear modulation range beforeovermodulation, for example when some reduced voltagesource Vdk = 0.95Vd, to satisfy condition (10), it is required

that corresponding commutation angle must be higher than 180.The condition can be satisfied if the value of correspondingcommutation angle is selected higher from diagram ofcommutation angles. From the previous consideration, H-bridge cell of higher dc voltage source should have gotcommutation sooner than a cell of lower dc-voltage.From previous consideration, in the algorithm, the measured dcvoltage sources will be arranged in decreasing sequence as

VdlI Vd2 .. VdN (11)

Sequences of corresponding commutation angles for two limitmodulation indexes ma and mb will be:

(12)Oal < Oa2 < <OaN

Obl -.0b2 -....<ObNBalancing possibility ofdc voltage sources

Each k-th H-bridge cell gives at output the power as:

(14)Pkout V,,Im cOS COS17T

Where Im and are amplitude and phase displacement of

load current. For simplifying consideration, suppose thatswitching losses is negligible, the power output is suppliedfirom dc souce ofpkin =VdkIdk' where Idk is dc input currentdetermined as follow:

(15)idk = COSOkCOS9

From previous algorithm, a H-bridge cell of higher dc voltage

Vdj is proposed with smaller commutation angle O,k. From

(15), higher dc current Idk will be produced. It is expected a

decreasing of voltage on relevant capacitor and favorable dcsource balancing.As a result, commutation angles of limit modulation indexescan be properly arranged in a reverse sequence to that ofamplitudes of corresponding dc voltage sources.

In the braking mode of motor, the load power is regeneratedand returned to dc sources. Because of possible increasing ofvoltages on dc capacitors, the relationship between sequence ofcommutation angles and voltage amplitudes in the algorithmwill be expectively reverse for voltage balancing purpose.

IV. RESULTSTotal harmonic distortion (THD) factor is calculated as

0x 31

2 v1, 2

TH-ID k=5,7,11,... k=5,7,11,...VI VI

Diagrams of commutation angles, THD factor, linearcontrol characteristic and harmonic amplitudes for variousratios of dc voltage sources (Vdl:Vd2:Vd3:Vd4:Vd5) are shown inFig.5-10. In Fig.5, all dc voltages are equal to 0.95 Vd . For

- 45 -

(13)

detail, diagram of THD factor is redrawn in Fig.6 for0.36<m<0.75. In Fig.7-8, diagrams are drawn for the secondlimit case, whose all dc sources are increased by a factor of1.05.

number of inverter levels, better balance of harmonic contentwill be expected.

0.4 0.45 0.5 0.5 0.6 0.65 0.7 0.75 06Modulation Index rn

Figure 7: (1.05,1.05, 1.05, 1.05, 1.05)

Figure 4: Diagram of commutation angles, Harmonic amplitudes, controlcharacteristic and THD factor for dc voltage ratios as ( 0.95, 0.95, 0.95, 0.95,0.95)

04medwMon htdal.

- C ---

3m-------- i----------- --------------

im

------- ----------

- - - - Q 6 ti CIA (.3 a C aWum.m Mdw. ft WdU

----------- L ------

Figure 8: Diagram of commutation angles, Harmonic amplitudes, control0.4 0.5 0.6 0.7 0. a characteristic and THD factor for dc voltage ratios as (1.05,1.05,1,0.95,0.95)Modulation Index m

Figure 5 ( 0.95,0.95,0.95,0.95,0.95)

0.4 0.45 0.5 0.55 0.6 0.65Modulation Index mn

Figure 9: (1.05,1.05,1,0.95,0.95)

AcKNOWLEDGMENT

Figure 6: Diagram of commutation angles, Harmonic amplitudes, controlcharacteristic and THD factor for dc voltage ratios as (1.05,1.05, 1.05, 1.05,1.05)

V. CONCLUSIONS

A novel PWM method for cascade inverter for unbalanceddc voltage sources has been proposed. The simplicity of on-line algorithm and flexibility for unbalanced voltage conditionsare advantages of proposed method. Appropriate values ofTHD factor have been shown for different source unbalances.It has been proved that THD factor can be improved for lowmodulation index (m<0.2) by a similar scheme as for SHE two-level inverter. For lower unbalance of voltages and higher

0.7 0.75 06

We would like to thank University of Technology-VietnamNational University HCMCity for partial support and Ministryof Commerce, Industry and Energy and Ulsan MetropolitanCity, which partly supported the research through the Network-based Automation Research Center (NARC) at University ofUlsan, Korea.

REFERENCES

[1 ]N.V.Nho,P.Q.Dung,H.H,Lee," Novel On-Line Step Pulse PWMAlgorithm For Cascade Multilevel Inverters ",Proceeding of the 37tIEEE Power Electronics Specialists Conference PESO 18-22n June2006, Jeju, Korea

- 46 -

1 --I

B

r-li

1--