Parallel Inverter

76 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 1, JANUARY 2014

Minimization of Grid Current Distortionin Parallel-Connected Converters

Through Carrier InterleavingJ. S. Siva Prasad and G. Narayanan

Abstract—Identical parallel-connected converters with unequalload sharing have unequal terminal voltages. The difference interminal voltages is more pronounced in case of back-to-backconnected converters, operated in power circulation mode forthe purpose of endurance tests. In this paper, a synchronousreference-frame-based analysis is presented to estimate the gridcurrent distortion in interleaved grid-connected converters withunequal terminal voltages. Influence of carrier interleaving angleon rms grid current ripple is studied theoretically as well asexperimentally. Optimum interleaving angle to minimize the rmsgrid current ripple is investigated for different applications ofparallel converters. The applications include unity power factorrectifiers, inverters for renewable energy sources, reactive powercompensators, and circulating-power test setup used for thermaltesting of high-power converters. Optimum interleaving angle isshown to be a strong function of the average of the modulationindices of the two converters, irrespective of the application. Thefindings are verified experimentally on two parallel-connectedconverters, with a circulating reactive power of up to 150 kVAbetween them.

Index Terms—Carrier interleaving, circulating-power setup,grid-connected converter, harmonic analysis, parallel converters,pulsewidth-modulated converter, space vector, synchronously re-volving reference frame, waveform quality.

I. INTRODUCTION

INPUT waveform quality is an important issue in grid-connected converters [1]–[3]. The quality of grid current

waveform can be improved by increasing the filter size orswitching frequency [2], [3]. In case of parallel-connectedconverters shown in Fig. 1 [4]–[10], carrier interleaving is aviable option for improving the waveform quality [9]–[14].

Carrier interleaving, i.e., use of phase-shifted carriers for thetwo converters as illustrated in Fig. 2, leads to phase shifting ofthe harmonics of the two converters and possible cancellation ofcertain harmonics [9]–[11]. An interleaving angle of 180◦ leadsto cancellation of the first-sideband harmonics [9], [10] whichare dominant at high modulation indices close to the maximummodulation index when space vector modulation (SVPWM)is used [15]. A phase shift of 90◦ between the two carriersresults in cancellation of the second-sideband harmonics [9],[10] which are dominant at low modulation indices [15].

Manuscript received August 27, 2012; revised November 8, 2012; acceptedDecember 18, 2012. Date of publication February 6, 2013; date of currentversion July 18, 2013.

The authors are with the Department of Electrical Engineering, Indian Insti-tute of Science, Bangalore 560012, India (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TIE.2013.2245620

Fig. 1. Parallel-connected converters with common dc bus. (a) Schematicdiagram. (b) Phasor diagram for rectification with unequal dc load sharing.(c) Phasor diagram for power circulation mode. (d) Phasor diagram for activepower circulation.

Grid-connected converters mostly operate in the interme-diate range of modulation indices [19], where both the firstand second sidebands are comparable. Hence, the problem of

0278-0046/$31.00 © 2013 IEEE

SIVA PRASAD AND NARAYANAN: GRID CURRENT DISTORTION IN PARALLEL-CONNECTED CONVERTERS 77

Fig. 2. Phase-shifted carrier waves of two parallel-connected converters.

improving the waveform quality does not readily boil downto elimination of one specific sideband; the first and secondsidebands both need to be reduced. In other words, the rmscurrent ripple should to be reduced.

The rms line current ripple in a voltage source converter(VSC) is usually evaluated by first obtaining the voltage har-monic spectrum using double Fourier series [10], [11], [17],[19]. Based on the voltage harmonics, the current harmonicsand the rms current ripple are then calculated [10]. A moredirect approach to calculate the rms current ripple would be tointegrate the error voltage vector in the space vector domain[20]–[22]. This approach has been adopted extensively to eval-uate line current ripple in motor drives [23]–[26]. A similarapproach has been used for evaluating the grid current ripplein identical parallel-connected converters with equal terminalvoltages [9]. In this paper, this method of analysis is extendedto parallel-connected converters with unequal terminal voltagessince such converters are found widely in different applicationsas discussed in the following.

Identical parallel-operated converters are commonly used inunity power factor (upf) rectification [4]–[10]. The focus in theliterature has largely been on such rectifiers with equal loadsharing, where the terminal voltages of the two converters areequal [4]–[10]. However, when the load sharing is unequal, theterminal voltages differ both in terms of amplitude as well asphase, as shown by the phasor diagram in Fig. 1(b). Further-more, the terminal voltages could differ more significantly inamplitude and/or phase when active and/or reactive power iscirculated between two high-power converters for the purposeof heat-run test [27]. In such case, the two converter currentsare out of phase and so are their inductor voltage drops, as seenfrom Fig. 1(c). A particular case of Fig. 1(c) is active powercirculation [27] [see Fig. 1(d)], where the terminal voltagesdiffer in phase, although of the same amplitude.

There are also applications of parallel-connected converterswith separate dc buses, as shown in Fig. 3. When two parallelconverters are injecting unequal amounts of reactive powerinto the grid [28], [29], the terminal voltages are of unequalamplitude, as shown in Fig. 3(b). When parallel converters,connected to different dc sources (e.g., renewable energy ap-plications), are feeding unequal amounts of active power intothe grid, the terminal voltages differ in terms of amplitude aswell as phase [29]–[31]. When reactive power is circulatedbetween the two converters [32], [33] to carry out thermaltests on converters [32], the filter inductor drops are out ofphase, resulting in substantial difference in the amplitudes ofthe converter terminal voltages [Fig. 3(c)].

Fig. 3. Parallel-connected converters with separate dc buses. (a) Schematicdiagram. (b) Phasor diagram for unequal reactive power sharing. (c) Phasordiagram for reactive power circulation mode [32], [33].

Thus, the terminal voltages differ whenever 1) load sharingbetween the two converters is unequal or 2) power is circulatedbetween the two converters. The difference in terminal voltagesis more pronounced in the power circulation mode than in caseof unequal sharing in the power-sharing mode.

When the terminal voltages of the converters are different,their harmonic spectra also differ. The order of the domi-nant harmonic in the two converters itself may differ. Hence,cancellation of dominant sideband harmonics through carrierinterleaving is not a clear option. The choice of interleavingangle could rather be based on minimization of rms grid currentripple of the parallel-connected converters.

This paper investigates the influence of carrier interleavingangle β on the rms grid current ripple and the optimal value ofβ to minimize the rms current ripple for parallel converters withunequal terminal voltages.

Section II analyzes and reviews the optimal interleav-ing angle for parallel-connected converters with equal ter-minal voltages from a space vector perspective. Section IIIpresents a method to evaluate the rms grid current ripple in


Fig. 4. Voltage vectors of a VSC. I to VI are sectors.

parallel-connected converters with unequal terminal voltages.Section IV investigates optimal β for different applicationsof parallel-connected converters. The findings are validatedthrough simulations and experiments, reported in Section V.The conclusion is presented in Section VI.

II. PARALLEL-CONNECTED CONVERTERS

WITH EQUAL TERMINAL VOLTAGES

The total rms harmonic distortion in the grid current, drawnby parallel-connected converters with equal terminal voltages,is evaluated in a synchronously revolving “dq” reference frame.The effect of interleaving on rms grid current ripple is studiedfor such converters. Optimum interleaving angle, which mini-mizes the rms grid current ripple, is obtained as a function ofthe modulation index of the two converters.

A. Analysis of Grid Current Ripple in SynchronouslyRevolving Reference Frame

The high-frequency ripple in the three-phase converter cur-rents can be represented as a current ripple vector in a syn-chronously revolving “dq” reference frame. The coordinateaxes are defined in Fig. 4. The q-axis is chosen to align withthe voltage vector VS, which represents the three-phase fun-damental components of terminal voltages of either converter.The modulation index of either converter can be defined as

Vref =|VS|Vdc

(1)

where Vdc is the dc bus voltage.The q-axis and d-axis components (iq1 and id1) of the current

ripple vector of converter-1 can be obtained by integratingthe corresponding error-voltage components (vq1 and vd1),as detailed in [24] and [25]. The error-voltage components(vq1, vd1) themselves can be obtained from the correspondingcomponents of the applied voltage vector, namely, (vq1, vd1)[24], [25], as shown in the following:

vq1 = vq1 − |VS|; vd1 = vd1. (2)

Fig. 5. Trajectory of the tip of the current ripple vector of converter-1 (solidlines) and of converter-2 when β = 180◦ (dashed lines) for Vref = 0.655 andα = 15◦ [33].

The q- and d-axes components of the current ripple vector ofconverter-2 (iq2 and id2) can be obtained in a similar fashion.

The current ripple of converter-1 gets added to that ofconverter-2 to yield the grid current ripple, as indicated by

iq = iq1 + iq2; id = id1 + id2. (3)

The rms q-axis grid current ripple (iq,rms), the rms d-axisgrid current ripple (id,rms), and the total rms grid current ripple(irms) are computed as follows:

iq,rms =

⎡⎢⎣ 6

T

T6∫

0

i2qdt

⎤⎥⎦1/2

id,rms =

⎡⎢⎣ 6

T

T6∫

0

i2ddt

⎤⎥⎦1/2

irms =

√(i2q,rms + i2d,rms

)(4)

where T is the fundamental period.The rms grid current ripple, thus computed in the dq refer-

ence frame, can be converted into rms per-phase grid currentripple (IHF,rms) as shown in the following:

IHF,rms =

(2

3

)(1√2

)irms. (5)

B. Influence of Interleaving Angle on RMS Current Ripple

The variation of the current ripple vector over a subcycle,pertaining to converter-1, is illustrated in solid lines in Fig. 5.The modulation index Vref is assumed to be 0.655. The terminalvoltage vector VS is considered to be at an angle α = 15◦,measured from the starting boundary of the sector (see Fig. 4).The current ripple vector of converter-2 is the same when thesame carrier is used for both converters, i.e., β = 0◦. However,when the carriers are phase shifted by 180◦, i.e., β = 180◦, thecurrent ripple vector of converter-2 is shown in dashed linesin Fig. 5. Thus, the grid current ripple vector, being the sumof the individual converter current ripple vectors, is stronglyinfluenced by β.

The influence of β on the rms values of the q-axis, d-axis, andtotal grid current ripples is brought out by Fig. 6, considering


Fig. 6. Influence of carrier interleaving angle β on iq,rms (dotted lines),

id,rms (dashed lines), and irms (solid lines) of identical parallel converters withequal modulation index (Vref) of 0.655.

Vref = 0.655. The figure presents the rms quantities, defined in(4), normalized with respect to the following base quantity:

Ibase =

(VdcTS

L

)(6)

where L is the effective series inductance per phase.As can be observed from Fig. 6, the rms value of the d-axis

current ripple reduces monotonically and is lowest at β = 180◦.The rms value of the q-axis ripple has a minimum at β = 90◦.The total rms current ripple has a minimum at β = 114◦. Thisrepresents the optimum interleaving angle (βopt) to minimizethe rms grid current ripple, when both the converters have amodulation index of 0.655.

Fig. 7(a)–(c) shows the variations of rms q-axis currentripple, rms d-axis current ripple, and total rms current ripple, re-spectively, against Vref for selected values of β. For β = 0◦, therms q-axis ripple dominates at low modulation indices, whilethe rms d-axis ripple is high at high modulation indices.

Fig. 7(a) shows that β = 90◦ results in substantial reductionin rms q-axis ripple, compared to both β = 0◦ and β = 180◦,over the entire range of Vref . Fig. 7(b) shows that β = 180◦

reduces the rms d-axis current ripple considerably over thewhole range of Vref , compared to β = 0◦ and β = 90◦. Thetotal rms current ripple is low with β = 90◦ in the lower rangeof Vref and with β = 180◦ at the higher range of Vref , as seenfrom Fig. 7(c).

C. Variation of βopt With Respect to Vref

The optimal value of β which minimizes the rms q-axiscurrent ripple (i.e., βq,opt) is investigated and is shown plottedagainst Vref in Fig. 8(a). At low modulation indices, βq,opt isclose to 90◦, while it increases at high modulation indices toreach 108◦ at Vref = 0.866. Optimal β to minimize the rmsd-axis current ripple (i.e., βd,opt) is found to be close to 180◦

over a wide range of Vref , as shown in Fig. 8(b).However, the rms q-axis current ripple with βq,opt is very

close to that with β = 90◦ in Fig. 7(a) and hence not shownplotted. Similarly, the rms d-axis ripple with βd,opt (not shownplotted) is almost indistinguishable from that with β = 180◦,shown in Fig. 7(b). Thus, β = 90◦ and β = 180◦ are goodapproximations for βq,opt and βd,opt, respectively, to minimizethe rms values of the q-axis and d-axis current ripples. While it

Fig. 7. Variation of rms values of (a) q-axis, (b) d-axis, and (c) total gridcurrent ripples against Vref for different values of β.

is obvious that any reduction in d-axis or q-axis current ripplewould reduce the grid current ripple, the decrease in q-axiscurrent ripple also results in reduced dc capacitor voltage ripplein PWM rectifiers [20].

The optimal interleaving angle βopt, which minimizes thetotal current ripple, is found to vary between 90◦ and 180◦

against Vref , as presented in Fig. 8(c). Understandably (sincethe q-axis ripple dominates at low modulation indices), βopt

is close to 90◦ at low values of Vref , while it is equal to 180◦

at high values of Vref , close to 0.866 (where the d-axis rippledominates). In the range of Vref from 0.5 to 0.82, βopt increasesfrom 90◦ to 180◦ (since the relative values of the d- and q-axisripples change with Vref ).

Fig. 7(a)–(c) also presents the rms values of the q-axis,d-axis, and total grid current ripples with β = βopt. In termsof the q-axis current ripple, βopt is better than other valuesof β except β = 90◦ in Fig. 7(a). Similarly, in terms of thed-axis current ripple, it is worse only compared to β = 180◦

in Fig. 7(b). However, it is the best in terms of total rms currentripple as seen from Fig. 7(c). Compared to β = 0◦, substantial


Fig. 8. Optimum interleaving angles for minimizing (a) iq,rms, (b) id,rms,

and (c) irms as functions of modulation index Vref of the two converters.

reduction in rms grid current ripple is achieved by βopt over theentire range of Vref . The percentage reduction observed in totalrms current ripple at Vref = 0.4, 0.7, and 0.866 are 65, 47, and70, respectively.

The results based on the analysis in the dq reference framepresented in Fig. 8(c) agree with the findings based on doubleFourier series in [10]. The analysis also gives an understandingof the rms values of the d-axis and q-axis current ripples, andtheir minimization using carrier interleaving. This is helpfulto correlate the frequency domain and synchronous reference-frame-based analyses as will be brought out in the next section.

III. PARALLEL-CONNECTED CONVERTERS

WITH UNEQUAL TERMINAL VOLTAGES

In this section, the rms grid current ripple, drawn by identicalparallel-connected converters with unequal terminal voltages, isevaluated. The influence of carrier interleaving angle β on therms value of the grid current ripple is studied.

Fig. 9. Synchronously revolving (d1, q1), (d2, q2), and (d, q) referenceframes, and the stationary (a, b) reference frame.

A. Estimation of Grid Current Ripple in SynchronouslyRevolving Reference Frame

The fundamental components of the terminal voltages of thetwo converters are represented by voltage vectors VS1 and VS2

in Fig. 9. These are separated by an angle δ. The magnitudesof these two vectors, normalized with respect to Vdc, give themodulation indices of the two converters (Vref1 and Vref2) asshown in the following:

Vref1 =|VS1|Vdc

=3

2

VR1N

Vdc

Vref2 =|VS2|Vdc

=3

2

VR2N

Vdc(7)

where VR1N and VR2N are the peak values of the R-phasefundamental voltages of the two converters.

The current ripple of converter-1 can be estimated in thesynchronously revolving (d1, q1) reference frame (see Fig. 9)using the procedure explained in the previous section, whereq1 is aligned with VS1. Similarly, the line current ripple drawnby converter-2 can be analyzed in the (d2, q2) reference frame,where q2 is aligned with VS2.

The components of the grid current ripple vector in the dqreference frame (which is chosen to be the same as d1, q1 framehere) are calculated using (iq1, id1) and (iq2, id2) as shown inthe following:

i′q2 = (iq2 cos δ) + (id2 sin δ)

i′d2 =(−iq2 sin δ) + (id2 cos δ)

iq = iq1 + i′q2; id = id1 + i′d2. (8)

The orthogonal components of the current ripple vectors ofthe two converters over a subcycle are illustrated in Fig. 10,assuming both vectors VS1 and VS2 to be at an angle α = 15◦

from the a-axis, i.e., in sector-I (see Fig. 4). For the purpose ofillustration, Vref1 = 0.58, Vref2 = 0.73, and δ = 0◦ are consid-ered. These correspond to reactive power circulation, illustratedin Fig. 3(c). The dc bus voltages and reactive drops of bothconverters are 700 V and 11%, respectively. The mains voltage(rms line to neutral) is assumed to be 216 V. Since δ = 0◦,the components of the grid current ripple vector are simplyalgebraic sums of the respective components of the individualconverter current ripple vectors, as in (3).

Carrier interleaving phase shifts the ripple current of oneconverter (dotted lines) with respect to the other (solid lines)as illustrated in Fig. 10. When the carriers are not phase shifted(i.e., β = 0◦), the q-axis and d-axis components of the ripplecurrent vectors of the two converters get added up to yield highvalues of the q-axis and d-axis ripples in the grid current asillustrated in Fig. 10(a).


Fig. 10. d- and q-axis components of the current ripple vectors over asubcycle. (a) β = 0◦, (b) β = 90◦, and (c) β = 180◦ for Vref1 = 0.58,Vref2 = 0.73, and α = 15◦.

However, with β=90◦, the q-axis current ripples of the twoconverters are of opposite signs for most part of the subcycleas illustrated in Fig. 10(b). Similarly, with β=180◦, the d-axiscurrent ripples of the converters have opposite signs, as seenfrom Fig. 10(c). Hence, the q-axis and d-axis grid currentripples are low with β=90◦ and β=180◦, respectively, in thegiven subcycle.

In fact, β = 90◦ reduces the q-axis current ripple over theentire sector, compared to β = 0◦, as brought out by Fig. 11(a)and (b). Similarly, β = 180◦ leads to substantial reduction in

Fig. 11. Components of the grid current ripple vector for Vref1 = 0.58 andVref2 = 0.73. (a) β = 0◦. (b) β = 90◦. (c) β = 180◦.

the d-axis current ripple over the entire sector, as demonstratedby Fig. 11(a) and (c).

Supporting evidence is also presented by the harmonic spec-tra of the q-axis and d-axis ripple currents in Fig. 12. Whileβ = 90◦ reduces the dominant second-sideband harmonics inthe q-axis current ripple, β = 180◦ reduces the dominant first-sideband harmonics in the d-axis current ripple as shown byFig. 12(a)–(c).

As seen from Fig. 10(a), the q-axis current ripple alternatesover a subcycle TS (i.e., half a switching cycle), while thed-axis current ripple alternates over two subcycles (i.e., aswitching cycle). This can also be seen from Fig. 11(a). Hence,the q-axis ripple has a dominant second sideband, while thed-axis ripple has a dominant first sideband as seen fromFig. 12(a). β = 90◦ reduces the q-axis current ripple andthe second sideband. Similarly, β = 180◦ reduces the d-axiscurrent ripple and the first sideband. Thus, the d-axis andq-axis current ripples in the synchronous reference-frame-based


Fig. 12. Harmonic spectra of grid current ripple vector for Vref1 = 0.58 andVref2 = 0.73. (a) β = 0◦. (b) β = 90◦. (c) β = 180◦.

Fig. 13. Variation of the rms values of the q-axis, d-axis, and total grid currentripples against β (Vref1 = 0.58 and Vref2 = 0.73).

Fig. 14. Harmonic spectra of the grid current ripple vector for Vref1 = 0.58and Vref2 = 0.73 with β = βopt = 114◦. (a) q-axis component. (b) d-axiscomponent.

analysis can be related to the first and second sidebands, respec-tively, in the frequency domain study.

B. Influence of β on Current Ripple

For Vref1 = 0.58, Vref2 = 0.73, and δ = 0◦, the rms values ofthe q-axis, d-axis, and total grid current ripples are calculatedfor all values of β. These are shown plotted in Fig. 13. Itis observed that the q-axis current ripple is minimum aroundβ = 90◦. The d-axis current ripple is lowest at β = 180◦. Thetotal rms current ripple is minimum at β = 114◦. This is theoptimum interleaving angle (βopt) for Vref1 = 0.58, Vref2 =0.73, and δ = 0◦.

Fig. 14 shows the harmonic spectrum of the grid currentripple for β = βopt. Figs. 14 and 12(a) show that βopt reducesthe first as well as second-sideband harmonics.

More analytical results are presented in Section V alongwith experimental results for quick comparison with the latter.


Fig. 15. Effect of difference in filter inductances of the two converters on rmsgrid current ripple (Vref1 = 0.575 and Vref2 = 0.72).

Fig. 16. Effect of difference in dc voltages of the two converters on rmscurrent ripple (Vref1 = 0.61 and Vref2 = 0.69).

The optimal values of β for different applications of parallel-connected converters with unequal terminal voltages are studiedin Section IV.

C. Variation of Filter Inductor and DC Bus Voltage

Fig. 13 assumes the per-phase inductances of the two con-verters to be equal. However, while modular converters areusually designed to have equal values of line inductance [9],tolerance in inductance values of up to ±10% is quite common.If the filter inductances of converter-1 and converter-2 are1.1L and 0.9L, respectively (where L represents 10% filterinductance), the corresponding rms current ripple drawn bythe two converters is slightly higher than that of the equalinductance case (Fig. 13), as shown by Fig. 15. However, theoptimum interleaving angle is not changed significantly.

Once again, Fig. 13 assumes the dc voltages of the twoconverters to be equal. However, in case of parallel converterswith separate dc buses, the dc bus voltages of the two couldbe different. If the dc bus voltages of the two converters are inthe ratio 0.95 : 1.05, the corresponding rms grid current ripple isquite close to that of the equal dc bus voltage case, as shown byFig. 16. Also, βopt is almost unchanged as seen from Fig. 16.Further study in this regard is presented in Section IV-G.

IV. OPTIMUM INTERLEAVING ANGLE FOR PARALLEL-CONNECTED CONVERTERS WITH UNEQUAL

TERMINAL VOLTAGES

In case of converters with unequal terminal voltages, theoptimum interleaving (βopt) depends on Vref1, Vref2, and δ,where δ is the phase angle between the converter terminal

voltages; Vref1 and Vref2 are the modulation indices of the twoconverters, which can be redefined as

Vref1 =3

2

V1,pu

Vdc,pu; Vref2 =

3

2

V2,pu

Vdc,pu(9)

where the normalized converter terminal voltages V1,pu andV2,pu and the normalized dc bus voltage Vdc,pu are defined asfollows:

V1,pu =VR1N

VRN; V2,pu =

VR2N

VRN; Vdc,pu =

Vdc

VRN. (10)

Rather than investigating the optimal value of β for eachset of (Vref1, Vref2, δ), βopt can be calculated only for sets ofvalues of (Vref1, Vref2, δ) that are of practical relevance. Forexample, low values of Vref1 and Vref2 are of no interest inline-side converters. Furthermore, the value of δ cannot be veryhigh. The regions of practical importance in the (Vref1, Vref2, δ)space can be understood from the various practical applicationsof parallel-connected converters. Furthermore, Vref1, Vref2, andδ are related in a specific manner in each application, as seenfrom the phasor diagrams in Figs. 1 and 3. Hence, the optimalvalue of β is studied here by considering each application.

A. Real Power Circulation

During real power circulation between the converters, thereactive power drops of both converters are proportional to P .Furthermore, considering 10% line reactance, the p.u. reactivedrop is given by 0.1P . The reactive drops are added in quadra-ture with the mains voltage [see Fig. 1(d)] to obtain V1,pu andV2,pu. The modulation indices Vref1 and Vref2 are calculatedusing (9) and (10). The angle δ is given by

δ = 2 tan−1(0.1P ). (11)

For a given (Vref1, Vref2, δ), the rms grid current ripple isevaluated for different values of β as explained in Section III;βopt, which minimizes the rms grid current ripple, is deter-mined at each operating point. Fig. 15(a) presents βopt for allvalues of P between 0 and 1 p.u. at Vdc,pu = 1.88, 2.14, and2.5, in solid lines.

As seen from Fig. 17(a), βopt is practically unchanged withP at any given dc bus voltage. Any change in P at a given dcbus voltage corresponds to only a change in δ from the forego-ing discussion. Thus, βopt is practically independent of δ withinthe range of practical relevance of the latter. Furthermore, asseen from the figure, βopt increases with the decrease in Vdc,pu.Also, Vref1 and Vref2 increase with the decrease in Vdc,pu, asseen from (9) and (10). Hence, one can see that βopt increaseswith the increase in Vref1 and Vref2.

B. Real Power Sharing

When the parallel-connected converters are operated as upfrectifiers, sharing a dc load unequally, their reactive drops areproportional to their individual share of load. Assuming 10%line reactance and unequal load sharing in the ratio 80 : 20, the


Fig. 17. Variation of βopt with respect to Vdc,pu. (a) Real power circulation.(b) Real power sharing. (c) Reactive power sharing. (d) Reactive power circulation.

reactive drops of the two converters are equal to 0.08 and 0.02,respectively, for P = 1 p.u. The angle δ is given by

δ = tan−1(0.08P )− tan−1(0.02P ). (12)

As P varies between 0 and 1 p.u., δ varies over a shorterrange (0◦–3.43◦) than in Section IV-A (where the correspondingrange is 0◦–11.42◦). However, V1,pu and V2,pu are unchangedas before. Also, Vref1 and Vref2 remain unchanged with P butvary with Vdc.

TABLE IRANGE OF Vdc,pu CONSIDERED FOR DIFFERENT APPLICATIONS

Fig. 17(b) shows βopt plotted for a range of P between 0and 1 p.u. and different dc bus voltages. As in previous case,βopt is found to be practically unchanged against P , while itincreases with the decrease in Vdc,pu. This confirms that βopt ispractically independent of δ and that βopt increases with Vref1

and Vref2.

C. Reactive Power Sharing

When two parallel converters share the reactive power sup-plied to the grid unequally, their reactive drops are proportionalto their share of reactive power. Assuming 10% inductance and80 : 20 sharing as before, the reactive drops are 0.08 and 0.02at Q = 1 p.u. As seen from the phasor diagram in Fig. 3(b), theangle δ is always zero; the modulation index increases with Q.

Fig. 17(c) shows βopt against the total reactive power (Q) forvarious values of Vdc,pu in solid lines. As seen, βopt increaseswith Q as well as Vdc,pu. Once again, this indicates that βopt

increases with Vref1 and Vref2.

D. Reactive Power Circulation

When reactive power is circulated between the converters,the reactive drops are proportional to the amount of reactivepower circulated. As Q increases from 0 to 1 p.u., the reactivedrops of the two converters vary from 0% to 10%. Whilethe reactive drop of one converter is in phase with the mainsvoltage, the reactive drop of the other is out of phase. Hence,the terminal voltages of the two converters differ in amplitudebut not in phase, as seen from Fig. 3(c).

The values of βopt, calculated at various operating con-ditions, are shown plotted in solid lines in Fig. 17(d). Asbefore, βopt is seen to increase with the decrease in Vdc,pu.This confirms that βopt increases with the increase in Vref1

and Vref2. Furthermore, βopt is reasonably unchanged with Q,which is equivalent to (Vref1 ∼ Vref2), except when Vref1 orVref2 is close to 0.866. Hence, in other words, βopt is reason-ably independent of the difference between Vref1 and Vref2,except at high modulation indices. This suggests that βopt isa strong function of the average of Vref1 and Vref2.

E. Optimum Interleaving Angle as a Function of theDC Voltage

In Fig. 17(a)–(d), βopt is presented for specific values of thedc bus voltage. These figures suggest that βopt is a decreasingfunction of the dc bus voltage. This is verified by investigatingβopt over a range of dc bus voltages, assuming P = 1 p.u.or Q = 1 p.u., as the case may be. The range of Vdc,pu foreach application is specified in Table I. The minimum Vdc,pu

considered is the minimum possible dc bus voltage for the givenapplication. In all cases, this is slightly higher than 1.732 p.u.,


Fig. 18. Variation of βopt with respect to Vdc,pu for different applications(P = 1 p.u. or Q = 1 p.u.).

Fig. 19. Variation of βopt with respect to 0.5 (Vref1 + Vref2) for differentapplications (P = 1 p.u. or Q = 1 p.u.).

which corresponds to the peak line–line voltage. The maximumVdc,pu is restricted to 3.75 p.u.

The optimal interleaving angle βopt is found to be adecreasing function of Vdc,pu for all four applications, asshown by Fig. 18. Furthermore, the βopt versus Vdc,pu curvesfor all of the applications are close to one another. This showsthat βopt is a strong function of the common dc bus voltage ofthe two converters.

F. Optimum Interleaving Angle as a Function of the AverageModulation Index

In Section IV-E, βopt is studied as a function of the dc busvoltage for various applications. In this section, effort is madeto relate βopt to ac side quantities of the parallel converters. Theresults in Section IV-D indicate that βopt is a strong functionof the average modulation index of the two converters. In thissection, this is verified over wide ranges of Vref1 and Vref2.

In Fig. 19, βopt is plotted against 0.5 (Vref1 + Vref2) forthe aforementioned applications, assuming P = 1 p.u. or Q =1 p.u. It can be observed that the curves corresponding tovarious applications are close to one another. These curvesare close to one another even for other values of P and Q,although not presented here. This confirms that βopt (for allof the applications) is a function of the average modulationindex except when one or both modulation indices are closeto 0.866. Hence, βopt can be approximated as a function of asingle variable, namely, 0.5 (Vref1 + Vref2), instead of beingconsidered as a function of three variables—Vref1, Vref2, and δ.

Fig. 20. Influence of difference in filter inductances of the two converters onβopt for reactive power circulation (Q = 1 p.u.).

In parallel converters with equal terminal voltages, the aver-age modulation index of the two converters is the same as themodulation index of the individual converters. Hence, the βopt

versus Vref curve in Fig. 8(c) is represented in thick solid linesin Fig. 19. As seen from the figure, this curve is quite close tothose of the four applications.

Hence, βopt for the aforementioned applications cannot onlybe approximated as a function of the average modulation index,but this function is the same as that between βopt and Vref inthe equal-terminal-voltage case.

In Fig. 17(a)–(d), βopt based on the aforementioned approxi-mation is plotted in dashed lines for different applications for arange of P or Q from 0 to 1 p.u. The dashed lines are very closeto the solid lines, indicating that the approximation is valid fordifferent amounts of P or Q circulated.

The aforementioned approximation simplifies the evaluationof βopt for parallel converters operating with unequal terminalvoltages.

G. Effect of Unequal Filter Inductances and Unequal DCVoltages on βopt

To study the effects of unequal line inductances and unequaldc bus voltages of the two converters on βopt, the two convert-ers are first assumed to operate in the reactive power circulationmode. It may be recalled that the difference between the mod-ulation indices of the two converters is most pronounced in thisapplication. Furthermore, the deviation of βopt from the equalterminal voltage case is also the highest in this case, as seenfrom Figs. 18 and 19.

When the two converters have per-phase inductances of 1.1Land 0.9L, respectively, as in Fig. 15, the optimum interleavingangle βopt varies with the average modulation index, as shownin dashed lines in Fig. 20. This is found to be close to the βopt

curve for converters with equal inductances, which is shown insolid lines in the same figure.

Similarly, when the dc bus voltages of the two converters arein the ratio 0.95 : 1.05, as in Fig. 16, βopt is again a function ofthe average modulation index of the two converters, as shown(in dashed lines) in Fig. 21. Once again, this βopt curve isreasonably close to that for the equal-dc-bus-voltage case.

Thus, the conclusion regarding βopt in the previous sectionsis reasonably valid even if there is considerable differencebetween 1) the filter inductances of the two converters and2) the dc voltages of the two converters.


Fig. 21. Influence of difference in dc bus voltages on βopt for reactive powercirculation (Q = 1 p.u.).

Fig. 22. Variation of βopt with respect to 0.5 (Vref1 + Vref2) for convertersrated for different values of kVA, sharing a dc load.

A more serious case of difference in filter inductances of thetwo converters arises when converters of significantly differentratings are operated in parallel for sharing power. Fig. 22 illus-trates the optimum interleaving angle for such a case, wherea common dc load is shared by two converters whose kVAratings are in the ratio 2 : 1. Furthermore, the filter inductancesare assumed to be 10% of their respective kVA base. Hence,their filter inductors are in the ratio 1 : 2. For modulation indicesup to 0.6, this βopt curve closely follows the βopt curve corre-sponding to identical converters with equal terminal voltages.The two curves deviate significantly at high modulation indices.This deviation becomes more pronounced as the differencebetween the filter inductances of the two converters increases.

V. SIMULATION AND EXPERIMENTAL RESULTS

In this section, simulation and experimental results, demon-strating the influence of carrier interleaving on grid currentdistortion, are presented. The reduction in grid current ripplewith optimal carrier interleaving is shown through simulationand actual measurements.

A. Simulation Results

Different configurations of parallel-connected converters inFigs. 1 and 3 are simulated using MATLAB/Simulink in thissection. The converter details are shown in Table II. The dc busvoltage is assumed to be 655 V. The semiconductor devices areassumed to be ideal for the purpose of simulation.

When 150 kVA of reactive power is circulated betweenthe two converters [see Fig. 1(d)], the simulated grid current

TABLE IIDETAILS OF THE EXPERIMENTAL CONVERTERS

TABLE IIIANALYTICAL AND SIMULATED VALUES OF RMS GRID CURRENT

RIPPLE IHF,rms (A), FOR CASES 1 TO 5 WITH Vdc = 655 V

waveforms for different values of β are shown in Fig. 23(a)–(d).Carrier interleaving reduces the peak grid current ripple as canbe seen from Fig. 23.

The rms grid current ripple is evaluated analytically fordifferent applications at various operating conditions. Fromthe simulated grid current waveform, its rms current ripple iscalculated. Analytical and simulation results for the followingcases are tabulated in Table III:

1) 150 kW of power delivered to the dc load at upf with thetwo converters sharing the power equally (50 : 50);

2) 150 kW of power delivered to the dc load at upf with thetwo converters sharing power in the ratio 80 : 20;

3) 150 kW of active power circulated between theconverters;

4) 150 kVA of reactive power supplied to the grid with theconverters sharing the same in the ratio 80 : 20;

5) 150 kVA of reactive power circulated between theconverters.

The analytical and simulation results match closely. Inter-leaving angles of 90◦ and 180◦ improve the grid current dis-tortion over β = 0◦ in all cases. Furthermore, β = βopt yieldsthe lowest rms grid current ripple in all of the cases, with areduction in the range of 44%–55%, as compared with β = 0◦.

Compared to β = 0◦, the amplitudes of the dominant har-monic current in the first sideband and that in the secondsideband are both reduced with βopt as brought out by Table IV.The amplitudes of the dominant components in the first andsecond sidebands, obtained through simulations, are presentedin ordered pairs in the table. As seen, the dominant harmoniccurrent is significantly reduced with βopt in all five cases.

Table V confirms the reduction in total harmonic distortionin the grid current (ITHD), which is the ratio of the rms current


Fig. 23. Simulated grid current waveform under 150-kVA reactive powercirculation at different values of β(Vdc = 655 V). (a) β = 0◦. (b) β = 90◦.(c) β = 180◦. (d) β = βopt = 114◦.

ripple to the fundamental current, with optimum interleavingangle. The simulated values of ITHD are obtained as follows:

ITHD =

√I2rms − I21

I1(13)

TABLE IVSIMULATED VALUES OF DOMINANT HARMONIC AMPLITUDES

IN THE FIRST AND SECOND SIDEBANDS OF GRID CURRENT

SPECTRA (A) FOR CASES 1 TO 5 WITH Vdc = 655 V

TABLE VSIMULATED VALUES OF % ITHD FOR CASES 1,

2, AND 4 WITH Vdc = 655 V

Fig. 24. Measured grid voltage under switching action.

where Irms is the rms grid current and I1 is the rms value offundamental component of the grid current. The fundamentalcurrents, corresponding to power circulation, i.e., cases 3 and 5,are quite low and are difficult to be obtained through simulation.Hence, ITHD values corresponding to cases 1, 2, and 4 onlyare presented in Table V. Measured ITHD values pertaining toreactive power circulation are presented in Section V.

Considering that the fundamental grid current is close to655 A (peak) for the cases of power sharing, one can clearly seefrom Table IV that the high-frequency grid current harmonicsare very low. Furthermore, ITHD is also low as seen fromTable V. Thus, in such case, the requirements of IEEE-519-1992 are adequately complied with [34].

B. Estimation of Grid Inductance

In case of line-side high-power converters, the value of gridinductance is comparable to that of filter inductance [31], [35].While the grid inductance is considerable, a switching wave-form can be seen superimposed over the sinusoidal grid voltagewaveform. A zoomed version of the measured grid voltage,when a single converter is operating, is shown in Fig. 24.

The step change (ΔvRN ) in this waveform is due to thecorresponding step change (ΔvRIN ) in the converter terminalvoltage at the switching instants of the converter. When theconverter switches from zero state 0 to active state 1 (see Fig. 4),ΔvRIN is equal to (2/3) Vdc i.e., 467 V. The correspondingΔvRN measured is 44 V, as seen from Fig. 24. The grid


Fig. 25. Single-phase equivalent circuit of a PWM converter at switchingfrequency.

Fig. 26. Supply voltage (vRN ) and line currents of the two converters at150-kVA reactive power circulation (Vdc = 700 V) (experimental results).Ch1-iR1 and Ch2-iR2, 250 A/div.; Ch3: vRN , 240 V/div [32].

inductance can be estimated from these two values, based onthe per-phase equivalent circuit shown in Fig. 25, as shown inthe following:

ΔvRN = ΔvR1NLgrid

Lf + Lgrid. (14)

With the filter inductance being 330 μH, as indicated inTable II, the estimated value of Lgrid is 34 μH.

C. Measured Grid Waveforms and Harmonic Spectra

The experimental setup consists of two parallel-connectedconverters with specifications as mentioned in Table II. Fig. 26shows the measured current and supply voltage waveformsunder 150-kVA reactive power circulation, with Vdc = 700 V.As can be observed, iR1 lags the supply voltage by 90◦, whileiR2 leads the voltage by 90◦. Fig. 27 shows the grid currentwaveforms under 150-kVA reactive power circulation for differ-ent values of β. The corresponding measured harmonic spectraare presented in Fig. 28. It can be observed from Fig. 28 thatthe fundamental grid current magnitude is approximately 13 A,which corresponds to the losses in the semiconductor devicesas well as other passive components.

The harmonic spectra confirm that β = 90◦ and β = 180◦,respectively, reduce the second- and first-sideband harmonicsconsiderably, compared to β = 0◦. Furthermore, βopt reducesboth the first and second sidebands such that the rms currentripple is minimized.

The experimental results in Figs. 27 and 28 show that low-order harmonics are present in the grid current for all valuesof β. These can be attributed to the harmonic distortion in thegrid voltage, which is brought out by the spectrum of the mainsvoltage in Fig. 29, which is captured when the converters arenonfunctional.

Fig. 27. Experimental grid current (iR) at 150-kVA reactive power circula-tion for different values of β (fsw = 5 kHz and Vdc = 700 V). (a) β = 0◦.(b) β = 90◦. (c) β = 180◦. (d) β = 114◦. Scale: 50 A/div.

The optimum interleaving angle brings down the amplitudeof the dominant harmonic in the grid current from 10 to 7 A asseen from Fig. 28. Furthermore, this also reduces the THD ingrid current (ITHD), as indicated in Fig. 28. Thus, the ability ofoptimal carrier interleaving to reduce the dominant harmonicas well as THD in grid current makes it a useful step towardcompliance with grid standards.

With the individual converters handling a power of 150 kVA,if the converters are operated in power-sharing mode, thepeak fundamental grid current would be 655 A as opposedto 13 A in power circulation mode. Hence, the grid stan-dards would be readily complied with, as already shown bythe simulation results in Section V-A. In case of the powercirculation mode, additional filtering may be required to meetthe standards. However, the current rating of such filter wouldbe very small, compared to the size and ratings of individualconverters.

D. Measured RMS Grid Current Ripple

The rms grid current ripple drawn by two parallel converters,operated in the reactive-power circulation mode, is estimatedanalytically using the procedure discussed in Section III. Thenormalized rms current ripple is converted into its naturalunits using (10), where L = (Lgrid + Lf ). The variation of theanalytically evaluated rms current ripple against β is shown inFig. 30 for different values of the dc bus voltage.


Fig. 28. Measured harmonic spectrum of grid current at 150-kVA reactivepower circulation with fsw = 5 kHz(Vdc = 700 V). (a) β = 0◦. (b) β = 90◦.(c) β = 180◦. (d) β = 114◦ (x-axis: 2.5 kHz/div.; y-axis: 5 A/div.).The measured %ITHD values are (a) 163, (b) 137, (c) 129, and (d) 125.

Fig. 29. Measured harmonic spectrum of grid voltage (vRN ) when theconverters are nonfunctional (x-axis: 200 Hz/div.; y-axis: 2.5 V/div.).

High-frequency rms current ripple (IHF,rms) is consideredto be the total rms harmonic content of the first four sidebands(ignoring higher sidebands). This is obtained by consideringthe harmonic components between 0.5fsw and 4.5fsw in themeasured current harmonic spectra (as in Fig. 28) as shown inthe following:

IHF,rms =

√√√√4.5fsw∑0.5fsw

I2h. (15)

These results are presented in Fig. 30 for various operatingconditions in the form of dots. As seen from Fig. 30(a)–(d),there is a reasonably good agreement between the analyticaland experimental results.

Fig. 30. Comparison between analytically evaluated and experimentally mea-sured distortion factors of grid current, mains voltage = 216 V phase-neutralrms. (a) Vdc = 700 V and fsw = 5 kHz. (b) Vdc = 650 V and fsw = 5 kHz.(c) Vdc = 625 V and fsw = 5 kHz. (d) Vdc = 700 V and fsw = 3.75 kHz.

Fig. 30 clearly shows the influence of β on the rms gridcurrent ripple. The theoretically estimated βopt is validatedexperimentally. In all of the presented cases, the rms value ofthe grid current ripple is reduced approximately by 44%–50%(analytically), with βopt, compared to β = 0◦. Experimentally,


although the reduction is slightly less (38%–47%), the improve-ment is still very much significant.

As can be observed, at a given dc bus voltage, the rms gridcurrent ripple, as well as βopt, is almost unchanged with kVAloading on the converters. This is in line with the analyticalresults in Fig. 17(d).

In Fig. 30(a)–(c), the rms grid current ripple is plotted fordifferent values of Vdc. As Vdc decreases, the average Vref andhence βopt increase.

In Fig. 30(d), the rms grid current ripple is presented forVdc = 700 V at a switching frequency of 3.75 kHz. Comparedto Fig. 30(a), the rms current ripple is increased as the switchingfrequency is reduced, as should be expected. However, βopt

remains the same, which shows that βopt is independent ofswitching frequency.

If the system is to be operated as in Fig. 1, with a common dcbus, the resulting circulating currents have to be mitigated eitherby using an isolation transformer [36] or by using common-mode chokes [27] or interphase chokes [10], [11]. This willsignificantly add to the cost of the system at higher kVA rating.Hence, such results are not presented in this paper. In case ofseparate dc buses, the circulating currents are absent. Further-more, as voltage inequalities are higher in power circulationmode than those in power-sharing mode, results pertaining toreactive power circulation alone are presented here.

Higher order LCL filters are increasingly employed in grid-connected converters [2], [3]. The procedure for determiningthe optimal interleaving angle, presented for converters with Lfilters in this paper, can certainly be extended to converters withLCL filters. Needless to say, the value of optimum interleavingangle for the LCL case should be expected to differ signifi-cantly from that for the L filter case.

VI. CONCLUSION

It is convenient to evaluate the rms grid current ripple ofparallel-connected converters with equal/unequal terminal volt-ages in a synchronously revolving dq reference frame.

This paper has established a correlation between the d-axisand q-axis ripple currents in the dq reference frame and thefirst- and second-sideband harmonics in the frequency domain.The effect of interleaving angle on the rms values of thed-axis, q-axis, and total current ripples is presented. Optimuminterleaving angles which minimize the q-axis, d-axis, andtotal current ripples in parallel converters with equal terminalvoltages are presented. These are expressed as functions ofconverter terminal voltage.

Optimum interleaving angle (βopt) is investigated for paral-lel converters with unequal terminal voltages. βopt is presentedfor parallel converters under upf rectification with unequal loadsharing, real power circulation, unequal reactive power sharing,and reactive power circulation. βopt is reasonably independentof the phase angle between the two converter terminal voltagesand the difference in their modulation depths. However, βopt

is shown to be a strong function of the average of the twomodulation depths.

It is further shown that βopt with unequal terminal voltages(e.g., Vref1 and Vref2) can be approximated by βopt corre-

sponding to the average of the converter voltage amplitudes[0.5 ∗ (Vref1 + Vref2)] in the equal converter terminal voltagecase.

It is shown theoretically as well as experimentally that βopt

results in substantial reduction in grid current ripple as com-pared to β = 0◦ (i.e., no interleaving).

Carrier interleaving is a cost-effective option for improvingthe input waveform quality of parallel-connected converters,particularly with separate dc buses.

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J. S. Siva Prasad received the B.Tech. degree fromthe S.V.H. College of Engineering, Nagarjuna Uni-versity, Guntur, India, in 2000 and the M.E. degreefrom the PSG College of Technology, Coimatore,India, in 2002. He is currently working toward thePh.D. degree at the Indian Institute of Science,Bangalore, India.

Since October 2012, he has been with Delta PowerSolutions, Bangalore. He was with the Departmentof Energy Systems, Indian Institute of Technology,Bombay, India, from 2002 to 2005 and was with

Vellore Institute of Technology, Vellore, India, from 2005 to 2006. His researchinterests are ac drives, pulsewidth modulation, and design of solar converters.

G. Narayanan received the B.E. degree from AnnaUniversity, Chennai, India, in 1992, the M.Tech.degree from the Indian Institute of Technology,Kharagpur, India, in 1994, and the Ph.D. degree fromthe Indian Institute of Science, Bangalore, India, in2000.

He is currently an Associate Professor with theDepartment of Electrical Engineering, Indian Insti-tute of Science, Bangalore. His research interestsinclude ac drives, pulsewidth modulation, multilevelinverters, and protection of power devices.

Dr. Narayanan received the Innovative Student Project Award for his Ph.D.work from the Indian National Academy of Engineering in 2000 and the YoungScientist Award from the Indian National Science Academy in 2003.

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