Development of an unbalanced switching scheme for a current source inverter

H.C.Tay and M.F.Conlon

Abstract: The development of an unbalanced switching scheme for the power conditioning system of a superconducting magnetic energy storage (SMES) device is described. The pulse-width modulation switching scheme for a current-source inverter, based on the unified switching algorithm, is presented. A device sequencing scheme and pulse merging arrangement is developed which minimises low-order harmonics. The balanced switchmg scheme is extended to an unbalanced scheme where the positive and negative sequence components of the output AC currents are controlled directly by their respective modulation indices and delay angles. The switching scheme is implemented on an experimental SMES device and its performance is assessed with respect to modulation linearity, harmonic generation, independent control of active and reactive power, and independent control of sequence currents.

List of symbols

iR(t), idt), iB(t) = AC output phase currents Id = DC source current A4 = modulation index w = network frequency in rads mRl(t), mR2(t) = fractional ON durations for switched 1

mwl(t), mm(t) = fractional ON durations for switched 3

m,,(t), mm(t) = fractional ON durations for switched 5

ia(t), ikt) = space-vector representation of phase cur-

ml = positive sequence modulation index m2 = negative sequence modulation index a1 = positive sequence delay angle a2 = negative sequence delay angle I1 = positive sequence current phasor I2 = negative sequence current phasor Prefi Qrej = reference active and reactive power flow Po = net active power for maintaining coil

Vnl, = line voltage of the AC side of the con-

Pd, Qd = active and reactive demanded power

and 4 respectively

and 6 respectively

and 2 respectively

rents

. current

verter

flows from SMES

0 IEE, 2000 IEE Proceedkgs online no. 20000026 DOL 10.1049iiggtd:20000026 Paper fmt received 29th January and in revised form 1 lth August 1999 The authon are with VENCorp, PO Box 1721, Collingwood, Victoria 3066, Australia The authorj were previously with the Centre for Electrical Power Engineering, Monash University, Clayton, Victoria, Australia

1 Introduction

With advances in semiconductor switching devices, con- verters operating with pulse-width modulation (PWM) switching schemes have become a relatively well established technology. Since the early 1980s, there have been many switching schemes developed for both voltage-source (VS) and current-source (CS) converters. However, the majority of the switchmg schemes are developed for balanced opera- tion [ 1 4 As described in an earlier paper [5], a supercon- ducting magnetic energy storage (SMES) system has been developed as a fluctuating load compensator. One of the objectives of the work is to operate the power conditioning system (PCS) of the SMES in an unbalanced mode to com- pensate for unbalanced loads. The development of an unbalanced switchmg scheme for the SMES is described in this paper.

If there is more than one converter in the PCS, it is rela- tively simple to achieve unbalanced operation. For exam- ple, with a PCS consisting of two converters in parallel, one converter can operate with positive sequence switchmg and the other with negative sequence switchmg using existing PWM schemes. The combined effect of the two converters is to produce the required unbalanced operation. However, t h s may result in one or other converter not being fully uti- lised, depending on the degree of unbalance required. Fur- thermore, the system cost increases with the number of converters. The aim here is to develop an unbalanced switching scheme for operation using a single converter, thereby making maximum use of the available capacity of the PCS.

There are three requirements imposed on the unbalanced switching scheme: 1 The modulation can be expressed in terms of positive and negative sequence components. 2 It must produce single sequence output under single sequence switching (e.g. positive sequence switching should produce positive sequence output only). 3 It must be implemented on-line for incorporation in a compensation device.

23 IEE Proc-Gener. Transm. Distrib.. Vol. 147, No. I , January 2000

There are two general categories of PWM techniques for CS converters: the carrier-modulated PWM and pre-calcu- lated programmed PWM. The third requirement above essentially eliminates the application of pre-calculated pro- grammed PWM for ths application because of the com- plexity in implementing it with three degrees of freedom [4, 61. The degrees of freedom are the positive sequence modulation, negative sequence modulation and the angle between the positive and negative sequence components. Therefore, the PWM techniques which are applicable to this study are confined to carrier-modulated PWM schemes. There are only a few techniques that could satisfy the three requirements above within the carrier-modulated PWM scheme. These include the sine-triangle PWM, the space-vector PWM and the unified switching algorithm PWM.

The sine-triangle PWM is usually a 2-level PWM. It requires translation to a 3-level PWM in order to be appli- cable for a 3-phase current-source converter. The transla- tion can be achieved by the methods described in Ledwich [A and Wang et al. [8]. The main disadvantage of this PWM scheme is that its transfer ratio is approximately 14% less than the transfer ratio of a space-vector PWM and a unified switching algorithm PWM [3, 9, lo].

Compared to the unified switching algorithm PWM, space-vector PWM produces lower low-order harmonics and has lower switching losses. The effective switching fre- quency of the devices for space-vector PWM is considera- bly lower than for the unified switching algorithm PWM and thus the space-vector based approach has reduced switching losses. Although space-vector PWM is theoreti- cally superior in performance when compared to the uni- fied switching algorithm in terms of harmonic generation and switching losses, implementation in an unbalanced mode for a current-source converter is considerably more complicated than in a balanced mode. Additionally, evalu- ation of the reference rotating vector for unbalanced cur- rents in the space-vector PWM approach requires various transcendental and square root mathematical operations. On the other hand, the unified switching algorithm can be easily implemented by a single stage look-up table and with integer arithmetic. The desired accuracy can also be achieved with relatively small look-up tables. This issue can be considered to be platform dependent.

The typical SMES system is usually synchronised to the power network. Pulse-dropping can occur at any level of modulation depth in space-vector PWM as the space vector rotates through the power frequency cycle. @he-dropping is when the width of an individual pulse of the pulse train derived from the PWM algorithm is less than the minimum achievable on-time of the devices. This is determined largely by the switching frequency and the characteristics of the devices.) It is difficult to avoid small pulse-widths in the space-vector PWM. This is particularly pronounced in the low modulation range where active pulse-widths are small. The unified switching algorithm is virtually free of this problem except for modulation depths close to 1.0, in which case the effect of pulse-dropping can be ignored since the pulses that would drop are very small compared to the other active pulses. For the SMES system, the operation of the converter may be required to extend down to zero modulation depth when the system is in stand-by mode. Therefore, to minimise the effect of pulse-dropping, and to reduce the complexity of this study, the unified switching algorithm was selected. This issue can be considered to be application dependent.

Therefore, the unified switching algorithm PWM rather than the space-vector PWM was selected.

24

2 Unified switching algorithm PWM

The unified switchng algorithm PWM is sirmlar to a regu- lar-sampled PWM, except that it has a modified moclula- tion wave. The algorithm origmates from the AC-AC matrix converter switching algorithm proposed by Alesina and Venturini [ll]. It was extended by Holmes [lo] to include the algorithm for AC-DC converters.

’R _I

- 1 I I I

i B _ J

5 4 ’6 5 2

Fig. 1 Current source converter for SMES

The unified switching algorithm essentially defines the turn-on duration of each switching device in each switching period for the current source converter shown in Fig. 1. The output AC current waveform is synthesised by sequen- tial piecewise sampling of the DC input current. The dura- tion of each sample is controlled in such a way that the average value of the output waveform with each sample period tracks the desired 50Hz waveform. The sampling/ switching frequency is much higher than the power fre- quency of 50Hz (typically between 500Hz and 10kHz) and the resultant synthesised waveform contains a fundamkntal component which is the desired 50Hz waveform [12]. Eqns. 1-7 describe the algorithm.

cos (wt + y )

(3)

mwl( t )=m -cos w t - - +-cos 2wt+-- {: ( 7) 376 ( 2:) -L cos ( 4 d - F)} + 1

36 (4)

IEE ProcGener . Transm. Distrib., Vol. 147, No. I . January 2000

(5)

(6)

(7)

mB1 ( t ) = 1 - mRl ( t ) - m W l ( t )

mL?z(t) = 1 - mI22(t) - mwz(t) where iR(t), iW(t) and idt) are the AC output currents; Id is the DC source current; m is the modulation index which ranges between 0.0 and 1.0; w is the network frequency in radian. The algorithm computes the fractional ON dura- tion M(t) in every switchmg period. mRl(t) and mR2(t) cor- respond to the fractional ON duration of devices 1 and 4, respectively (see Fig. 1 regarding the numbering of the devices); mm(t) and mm(t) relate to devices 3 and 6, respectively, etc.

The difference between the fractional ON-durations of the top and bottom devices for each phase, as shown in eqn. 1, gives the line currents which track the required sinusoids. The output fundamental component of the cur- rent (in rms) is mZJd2. However, this algorithm does not define the order in which the devices have to switch on. If a futed sequence, as initially used in [lo], is used for the whole cycle, (for example in a red-white-blue sequence as shown in Fig. 2) the resultant 3-phase line currents have the unde- sirable feature of being slightly unbalanced because the switching order is not equally distributed across the three phases (ths effect is insignlficant for high switching fre- quencies). Therefore to satisfy requirement 2, a more elabo- rate sequencing arrangement is needed.

R W B R W B R W B R W B R W B R W B

B0t:oOmp ;i 00 180° 360°

I

I

I

Fig. 2 Unijied switching ulgorithm m fmed sequence

3 Sequencing for unified switching algorithm PWM

Some methods of sequencing of devices have been pro- posed [13], but they either violate the above requirement 2, or produce low order even harmonics that the filter net- works are not normally designed to fiter. These low order even harmonics are caused by combining adjacent pulses, whch is a feature of the technique described in [13]. The following method for defining the switching sequence elimi- nates the above two problems and is based on the concept of a rotating vector. The required output currents are rep- resented as a single rotating vector on the a-p space by the transformation described in eqn. 8.

The space vector defined by iJt) and i&t) can lie in any of the six sectors from (I) to (VI) as shown in Fig. 3. Depend- ing on the sector in which the vector lies, the corresponding

IEE Proc-Cener. Trunsm. Distrib., Vol. 147, No. I , Junuury 2000

sequence is selected for the particular switching period. If the vector lies between the two sectors qs shown by I, to I,, then the sequence of the sector in the anticlockwise direc- tion is selected. For example, if the vector lies on I,, the sequence of sector (11) is used. For ease of analysis, a switchmg frequency of six times a multiple of the power frequency was selected. If the sequence is evenly distributed in the whole cycle as described above, the output currents produced would be balanced and contain no even harmon- ics. Fig. 4 shows the ideal output currents generated by a 1.5 kHz switching frequency with the sequencing suggested for a unit DC current and a modulation index of 0.8.

P

Sequence I B R W 1 a -

Fig. 3 Determination of switching sequence

L ' L , , , ~, , , , , 0 2 L 6 8 10 12 1L 16 18 20

W

-

b i 1 6 8 I b ;2 li 1'6 1'8 io W

L na 0 s! -1 -

0 2 L 6 8 10 12 1L 16 18 20 t ime ,ms

Fig. 4 3-phare current wuvefims Modulation index = 0.8

a

..... C .... ... .... ............. ..... i i i i ) .... ....... ...................... ..... $ ....... ...

(;vi. 1

L -

c 2 - U (ii)

L

6 0 1 a , 8 a 8 a a I

,\" 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 modulation index

Fig. 5 ages of 5th, 7th, l l th Lmrl13th h n i c s (i) 5th harmonic (ii) 7th harmonic (iii) llth harmonic (iv) 13th harmonic

Lowarder hurmonics of m f ~ d switching P WM showing the percent-

A closed form representation of the line currents in terms of its harmonic components using the technique described by [14] is extremely complex to derive for this PWM strat- egy. However, a hybrid method was used instead which evaluates the Fourier series of the PWM waveforms as out- lined in the Appendix (Section 10). Fig. 5 shows the signifi-

25

cant low-order harmonics of the PWM for modulations ranging from 0.1 to 1.0. It is obvious that the 5th and 7th harmonics are relatively large compared to a space-vector or optimised PWM [3, 61. This has led to some efforts in reducing the harmonics by modifying the sequence of firing the 3-phase devices, and altering the pulse production [13]. One of these methods which was used to reduce harmonic generation of the PWM is described below.

4 Pulse merging

In Fig. 6u, there are two pulses generated in the red phase within a switching period. The two pulses are separated by a null state of mR2 duration. If the pulses are merged by removing the null state and adding half of mR2 to each of the durations of the other phases, as shown in Fig. 6b, the PWM effectively becomes a space-vector PWM. Therefore it will have harmonic generation characteristics similar to space-vector PWM. The harmonic generation of the pulse- merging PWM is analysed and shown in Fig. 7. The har- monics are sigmfkantly reduced. As the modulation increases to 1 .O, the harmonic generation approaches that of the non-pulse merging scheme because the null state becomes very small at hgh modulation, and thus the pulses also appear to merge for non-pulse merging PWM. The effective switchmg frequency is also reduced, resulting in a lower switching loss. The following development of unbal- anced switchmg PWM is based on pulse merging.

I W I W

U *B

U U

'B

a ----U-- b

Fig. 6 a Non-pulse merging b Pulse merging mRl' = niRI - mR2 mW' = mW + 0.5" mBl' = mBI + 0.5" mW' = mW + 0.5" mm' = mm + 0.5"

Allocation of pukes within a switching period

,\" 0 0.1 0.2 0.3 0.L 0.5 0.6 0.7 0.8 0.9 1.0 m o d u l a t i o n index

Fig. 7 of 5th, 7th. 11th and 13th harmonics (i) 5th harmonic (i) 7th harmonic (iii) I 1 th harmonic (iv) 13th harmonic

26

Loworder harmonics of pulse merging P WM showing the percentages

5 Unbalanced switching algorithm

Sections 2 and 3 have described the method for calculating switching intervals using the unified switching algorithm for production of balanced converter output. Section 4 described how the 5th and 7th harmonics can be minimised by pulse merging. This Section deals with the combination of two such patterns to produce controllable unbalanced operation.

Using the Fortescue transformation, the unbalanced 3- phase currents in a 3-wire system can be decomposed into two sets of balanced 3-phase currents, namely positive sequence and negative sequence components. Since the uni- fied switching algorithm effectively produces a 50Hz posi- tive sequence waveform, an unbalanced switchmg P'WM can be similarly generated by adding a negative sequence component to the original algorithm, as shown in eqns. 9-12.

where ml is the positive sequence modulation index, in2 is the negative sequence modulation index, a, is the positive sequence delay angle and a2 the negative sequence delay angle. These are the fundamental control variables for defining the combined positive and negative sequence oper- ation. The modulation indices can range from 0.0 to 1.0. The resultant fundamental components of the line currents are:

i ~ ( t ) = mlId sin(wt - a1) + m2Id sin(wt - a:l) zw(t) = mlId sin(wt - 120" - a1)

z B ( t ) = mlId sin(wt + 120" - a l ) + m2Id sin(& + 120" - a2)

+ m2Id sin(& - 120" - a2)

1: 18) In phasor notation, the sequence components are:

3 9 )

IEE Proc.-Gener. Transm. Disrrih., Vol. 147, No. I , January 2000

where I , is the positive sequence current phasor and I2 the negative sequence current phasor. To generate positive sequence currents only, m2 is set to zero and vice versa The amounts of sequence component currents can be varied by their corresponding modulation index and delay angle. However, there is an upper limit for the combined sequence modulations. The sum of the positive and negative sequence modulation indices of each phase must not exceed 1 .o.

ml +m2 I 1 (21) This is a recognition of the fact that the maximum peak current for each phase is Id, the coil current.

To ensure that the generated sequence currents can be totally decoupled as shown in eqns. 19 and 20, the interac- tion between the positive and negative sequence modula- tions was analysed. Fig. 8 shows the effect on the negative sequence current at m2 = 0.1, resulted from the variation in ml. Three curves, which correspond to three different angles between the positive and negative sequence modula- tions, vary within a 2% error band (the error can be reduced by increasing the switching frequency). This result shows that the assumption that ml and m2 are decoupled is valid. The sharp changes in the curves were the result of changes in the selection of sequencing as the locus of the space vector changes with the varying amount of unbal- ance.

3 -

c 2 - I

- 3 -1

0 2 L 6 8 10 12 1L 16 18 20 ttme,ms

Unbalrmed P WM witching Fig .9 a, = az = 0" ml = mz = 0.5

-

Fig. 9 shows the idealised waveforms of the line currents generated by the unbalanced switching PWM with ml = m2 = 0.5 and aI = a2 = 0" for a unit DC current. The funda-

IEE Proc -Gener Trunsrn Dirtrrb, Vol 147, No l , Junuury 2000

mental components of the line currents indicated by the dotted curves are as expected; the red phase current is dou- ble in amplitude and opposite in phase to the currents of the other phases.

6 Experimental results

The performance of the unbalanced switching PWM was experimentally verified. Four important characteristics of the modulation were investigated, namely the modulation linearity, harmonic contents, independent control of modu- lation amplitude and phase, and validity of the decoupling relationship between the positive and negative sequence modulations.

The circuit was assembled as shown in Fig. 10. The SMES system comprises a 4.5H superconducting coil (SC), a cryogenic system to maintain the temperature of the coil at 4.2"K, a 15kVA IGBT-based AC-DC converter, associ- ated microprocessor controller and a transformer for matching the system and SMES voltages. The coil conduc- tor is multifilamental NbTi/Cu wire and the coil capacity is 16k.J when carrying a maximum current of 86A.

I I transformer ' varioc 31p.F converter SC coil

capoci tors Fig. 1 0 Experrinentd set-up for testing the Unbalanced switching scheme

The variac ratio was set to unity and was energised from a 415V source. The superconducting coil current was main- tained at about 50A in the experiment. The method used to maintain the coil current varies for each part of the experi- ment and is explained in the following Sections.

6. I Modulation linearity and harmonic contents The modulation hearity and harmonic contents of the PWM were investigated together. The negative sequence modulation was set to zero, and the coil current was main- tained by varying the delay angle al. The red phase current of the converter was analysed for a range of positive sequence modulation values, and the fundamental compo- nents were extracted and plotted in Fig. 11 against the desired linear relationship (as described in eqn. 19). The experimental results are close to the desired linear curve and thus the fundamental component of the modulation can

Q

c ? 3 v

W '0 3 .- .- - a

5 s W a

be approximated by the linear relationship.

50r

LO

30

20

10-

-

-

-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 modulation

2

1.0

Fig. 1 1 - desired values 0 measured values

Lineurity oj'the developed unbuhced P WM

The measured 5th and 7th harmonics of the PWM are plotted in Fig. 12 and the measured 1 lth and 13th harmon-

21

ics in Fig. 13. Both experimental points and theoretical curves agree well for a modulation index higher than 0.3. For low modulations, the harmonics were so small as to be dificult to measure. Comparing this harmonic generation method with that of an optimal PWM method [6], the low- order harmonics generated by this PWM method are rela- tively high.

:I 1

5~

2 + .: .I-.. . .c ' .. .+ . . ''.+.. "4- ... . _................ ..

modulat ion Fig. 12

~ 5th harmonic desired value 0 5th harmonic measured value

~~~~ 7th harmonic desired value + 7th harmonic measured value

5th and 7th harmonics ofthe developed unbalanced P W M

5

n 4 * .- & I

0 0.1 0.2 0.3 0.1 0.5 0.6 0.7 0.8 0.9 1.0 modulot ion

11th and 13th harmonics ofthe developed mbalunced P W M Fig. 13 ~ I Ith harmonic desired value 0 I 1 th harmonic measured value _ _ _ _ 13th harmonic desired value + 13th harmonic measured value

6.2 Independent control of active and reactive power If the amplitude of the output current is only controlled by the modulation index and its phase angle by the delay angle, then the active and reactive power flows in the con- verter can be simultaneously and independently controlled. This is an important feature for a SMES device as it allows both active and reactive power to flow. To demonstrate this, the active and reactive power flows of the converter were varied according to two independent reference signals. The negative sequence modulation was zero. (Since m2 is zero, the angle a2 need not be defined.) (a2 defines the phase relationshp of the negative sequence with respect to the positive sequence.) The coil current was maintained by a constant net active power flow. The modulation index m, and delay angle al were evaluated according to the follow- ing expressions such that the desired power flow was imple- mented:

P d = PTt?f f Po (22)

where Pref.is the reference of the active power flow; .Po is the net active power for maintaining coil current; ere,, is the reference of the reactive power flow; V,,, is the line voltage of the AC side of the converter and Id is the coil DC cur- rent. The response of the converter is plotted in Fig. 14. Pref was set to vary sinusoidally and ere, as a saw-tooth. The active and reactive power flows of the converter followed almost exactly the reference values, thus verifying the inde- pendent control of modulation amplitude and phase angle. The delay between the reference signals and the measured output power was due to the delay in the controller.

6r 5 - i

0 0.1 0.2 0.3 0.L 0.5 0.6 0.7 0.9 0.9 1.0 t ime,s I

z 4 Indeprmdrmt control of active und reactive power wmg the developed

(i) reactive (ii) reference (iii) reference (iv) active

ced P W M switchmg

2

0 0.1 0.2 0.3 0.L 0.5 0.6 0.7 0.8 0.9 ' 1 1.0 time.s

Fig. 15 the developed tolbalanced P W M switchmg

11) re erence (iii) Ipos , I

Independent control ofpositive und negative sequence currents wmg

IN?

6.3 Independent control of positive and negative sequence modulations The final part of the experiment is to c o n f i i the dt:cou- pling relationshp between the positive and negative sequence modulations. The positive sequence modulation was set to a constant level with ml = 0.23 and a1 =. 80" which maintained the coil current at approximately 50A. The negative sequence modulation was varied according to a reference signal with a2 kept constantly at zero. Both sequence currents generated by the converter were moni- tored. Th~s was carried out on the primary side of the SMES transformer because of measurement limitations. The components of the sequence currents due to the mag- netising branch of the transformer, the filter capacitors and maintenance of coil current were recorded when the SlMES was in a quiescent mode. These recorded values were then used to remove these components from the primary current and the resulting waveshapes are shown in Fig. 15. The plot shows only the variation in the sequence currents of the converter. The reference signal of the negative sequence current was a 2Hz sinusoid. The phase shift between the reference signal and the output current was mainly due to the slow response of the instrumentation used to measure the sequence components. The output negative sequence

IEE Proc -Gener Transm Distrih , Vol 147. No 1. Jannarv 2000 28

current tracked the reference signal very well while the pos- itive sequence current remained virtually constant. This ver- ifies the decoupling relationship between the positive and negative sequence modulations.

7 Conclusion

An unbalanced switchmg scheme has been developed and its performance has been experimentally verified. Using this switching scheme, a linear relationship can be assumed between the modulation variables, ml and m2, and the out- put currents for the SMES system. Also, the modulation of the positive and negative sequences are virtually decoupled. In other words, the converter operation with this unbal- anced switching scheme can be viewed as two independent sinks/sources of positive and negative sequence currents. Independent control of the active and reactive power flows, and independent control of the positive and negative sequence current flows for the SMES are achieved. This switching scheme is not restricted to a SMES system and is applicable to all current source converters with a Graetz bridge topology. Furthermore, this switching scheme allows output currents to change w i t h a switchmg period, thus allowing a rapid response to changes. However, the main drawback of ths switching scheme is the relatively high 5th and 7th harmonic generation at high modulation depth. With a 1.5kHz switching frequency, the maximum 5th har- monic was 5%. The harmonic generation may be reduced by rearrangmg the pulse generation. This would require separate harmonic reduction studies, similar to those reported by other researchers.

8 Acknowledgments

This work was supported by the Australian Electricity Sup- ply Industry Research Board (AESIRB) and the Energy Research and Development Corporation (ERDC). The authors would also like to thank the other members of the SMES Research Group at Monash University and the group leader, Prof. W.J. Bonwick.

9

1

2

3

4

5

6

7

8

9

References

OHNISHI, T., and OKITSU, H.: ‘A novel PWM technique for three- phase inverterlconverter’, Truns. Inst. Electr. Eng. Jpn. E, 1984, 104, (%IO), pp. 193-201 BOWES, S.R., and MIDOUN, A.: ‘Suboptimal switching strategies for microprocessor-controlled PWM inverter drives’, IEE Proc. B, 1985, 132, (3), pp. 133-148 VAN DER BROECK, H.W., SKUDELNY, H.-C., and STANKE, G.: ‘Analysis and realisation of a pulse width modulator based on voltage space vectors’. IEEE Industrial Application Society conference record, 1986, pp. 244251 ENJETI, P.N., ZIOGAS, P.D., and LINDSAY, J.F.: ‘Programmed PWM techniques to eliminate harmonics - A critical evaluation’. IEEE Industrial Application Society conference record, 1988, pp. 418- 430 TAY, H.C., and CONLON, M.F.: ‘Development of a SMES system as a fluctuating load compensator’, IEE Proc., Gener. Transm. Distrib., 1998, 145, (6), pp. 700-708 JIANG, Q., GIESNER, D.B., and HOLMES, D.G.: ‘A linearised optimal modulation strategy for a GTO converter to allow real-time control of active and reactive power flows into a SMES system’. 4th European conference on Power electronics and upplicutions, 1991, Vol. 1, pp. 504509 LEDWICH, G.: ‘Current source inverter modulation’, IEEE Trum. Power Electron., 1991, 6, (4), pp. 618423 WANG, X., and 001, B.T.: ‘Utility PF current-source rectifier based on dynamic trilogic PWM’, IEEE Truns. Power Electron., 1993,8, (3), pp. 288-294 FUKUDA, S., and IWAJI, Y.: ‘A singlechip microprocessor-based PWM technique for sinusoidal inverters’. IEEE Industry Applications Society conference record, 1988, pp. 921-926

11 ALESTNA, A., and VENTURN, M.: ‘Intrinsic amplitude limits and optimum design of 9-switches direct PWM AC-AC converter’. IEEE Power Electronics Specialists conference record, 1988, Vol. 2, B-6, pp. 1284129 1

12 ALESINA, A., and VENTURNI, M.: ’Solid-state power conversion: Fourier analysis approach to generalised transformer synthesis’, IEEE Trans. Circuits Syst., 1981, 28, (4), pp. 31S330

13 HOLMES, D.G.: ‘The general relationshp between regular-sampled pulse-width-modulation and space vector modulation for hard switched converter’. IEEE Industry Applications Society conference record, 1992, pp. 1002-1009

14 BOYS, J.T., and HANDLEY, P.G.: ‘Harmonic analysis of space vec- tor modulated PWM waveforms’, IEE Proc. B, 1990, 137, (4), pp. 197-204

10 Appendix

This Appendix presents the numerical method used for evaluation of harmonic components of ideal PWM wave- forms. The PWM waveform y(8), of period 275 is repre- sented by a series of N pulses as shown in Fig. 16. Each pulse is defined by three parameters: A , a, and A,. Ai repre- sents the height of pulse i, ai is the phase shlft from a nom- inal reference and Ai is the pulse width.

U

AN

I

Fig. 16 PWhf wavgom

The Fourier series of y(8) can be obtained by applying the Principle of Superposition to the Fourier series of the N pulses. The result is given below:

N 2 mr

N

i=l m=l

i + sin m(a; + A,/2) sin me]

(26) The first term in eqn. 26 is the average value of y(0). The second term relates to the fundamental and harmonic com- ponents of y(8). Therefore, to evaluate the harmonic com- ponents, the amplitudes of the harmonic components in the second term are extracted. Thus the harmonics can be expressed as:

x [cos m(ai + &/a) cos mf3

+ s inm(a , + &/a) sinme]

n z = - { S1 cos mf3 + Sz sin me} mr = P, cos(m0 + pn) (27)

IO HOLMES, D.G.: ‘A unified modulation algorithm for voltage and current source inverters based on ac-ac matrix converter theory’, ZEEE Trans. Ind Appl., 1992, 28, (I), pp. 31-40

where is the harmonic the mth harmonic and is given by:

‘rn is the amplitude Of

IEE Proc-Gener. Transm. Distrib., Vol. 147, No. I , January 2000 29

and N m& Fm = q= mr (28)

and 6, is the phase shift of the mth harmonic component s1 = ~i sin (?) [cos m(ai + ~ i / 2 ) 1 i=l and is given by: ._

Pm = - tar-1 (22) N m a ;

Sz = Ai sin (1) [sinm(ai + & / a ) ] (30) i= l

30 IEE Proc.-Gener. Transm. Distrib.. Vol. 147, No. 1, January 2000