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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014 5229 Closed-Form Expressions for Minimizing Total Harmonic Distortion in Three-Phase Multilevel Converters David Hong, Sanzhong Bai, Student Member, IEEE, and Srdjan M. Lukic, Member, IEEE Abstract—The total harmonic distortion (THD) of a waveform is a standard way to quantify its deviation from a sinusoid. In three-phase systems, we are interested in minimizing 3THD: the component of THD produced by odd nontriplen harmonics. How- ever, its definition involves an infinite sum, making it difficult to evaluate and analyze. This paper solves the problem of finding equivalent closed-form expressions for the 3THD of a staircase waveform. In particular, two expressions are rigorously derived, which reveal 3THD to be a piecewise differentiable function. One expression is shorter but describes the pieces implicitly. The other is longer but describes the pieces explicitly. We minimize 3THD using the closed-form expression and provide a comparison with previous techniques. Finally, we provide experimental results that show close agreement with the theoretical results. Index Terms—Closed-form solution, infinite sum, staircase waveform, three-phase electric power, total harmonic distortion (THD). I. INTRODUCTION M ULTILEVEL converters are used in numerous high volt- age, high-power systems for three-phase applications such as static synchronous compensators (STATCOM) [1], [2], energy storage system integration [3], active filters [4], photo- voltaic systems [5], [6] and electric/hybrid electric vehicles [7]. Multilevel converters have the benefit of allowing direct inte- gration of low-voltage power sources into medium-voltage and high-voltage applications, while producing high-quality wave- forms with the state-of-the-art switching devices. The cascaded topology (see Fig. 1) is of particular interest due to its modular- ity and simplicity of control. An open question that has received considerable attention recently is determining the optimal mod- ulation technique. Though many modulation techniques are pos- sible, staircase modulation (see Fig. 2) is of particular interest Manuscript received June 3, 2013; revised September 6, 2013; accepted October 24, 2013. Date of current version May 30, 2014. Recommended for publication by Associate Editor R. Burgos. D. Hong was with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA. He is now with the Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). S. Bai was with the FREEDM Center, North Carolina State University, Raleigh, NC 27695 USA. He is now with Eaton Corporation, Raleigh, NC 27616 USA (e-mail: [email protected]). S. M. Lukic is with the FREEDM Center, North Carolina State University, Raleigh, NC 27695 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2013.2290377 Fig. 1. Three-phase cascaded topology (s =3 sources). Fig. 2. Staircase waveform. due to the minimum number of switching instances, resulting in reduced switching losses. One approach to determining the switching angles of the stair- case modulation is selective harmonic elimination (SHE). This requires solving a system of transcendental equations, which is done using various techniques: Newton–Raphson [8], symmet- ric polynomials and resultants [9]–[11], neural networks [12], genetic algorithms [13], and bee algorithms [14]. Typically, the first few odd nontriplen harmonics are eliminated since they have the highest influence on the overall harmonic distortion and are difficult to filter. In the case of three-phase systems, SHE im- plementations do not eliminate the triplen harmonics [9]–[13], choosing instead to eliminate the low-order nontriplen harmon- ics. Triplen harmonics are considered to be eliminated automat- ically in the line voltages. It should be noted here that in many applications triplen harmonics are a source of common mode currents. However, these issues are often dealt with separately from the modulation strategy [8]–[18], for example by using common-mode filters or other filter structures. Another approach minimizes the total harmonic distortion (THD) [15]–[19] of the staircase waveform. The goal is to reduce the low-order harmonics to a level below the pertinent IEEE standard, while producing a lower overall THD compared to the SHE approach. However, the definition of THD involves an infinite sum over all odd harmonics, making it challenging to 0885-8993 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

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Page 1: Closed-Form Expressions for Minimizing Total Harmonic Distortion in Three-Phase Multilevel Converters

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014 5229

Closed-Form Expressions for Minimizing TotalHarmonic Distortion in Three-Phase

Multilevel ConvertersDavid Hong, Sanzhong Bai, Student Member, IEEE, and Srdjan M. Lukic, Member, IEEE

Abstract—The total harmonic distortion (THD) of a waveformis a standard way to quantify its deviation from a sinusoid. Inthree-phase systems, we are interested in minimizing 3THD: thecomponent of THD produced by odd nontriplen harmonics. How-ever, its definition involves an infinite sum, making it difficult toevaluate and analyze. This paper solves the problem of findingequivalent closed-form expressions for the 3THD of a staircasewaveform. In particular, two expressions are rigorously derived,which reveal 3THD to be a piecewise differentiable function. Oneexpression is shorter but describes the pieces implicitly. The otheris longer but describes the pieces explicitly. We minimize 3THDusing the closed-form expression and provide a comparison withprevious techniques. Finally, we provide experimental results thatshow close agreement with the theoretical results.

Index Terms—Closed-form solution, infinite sum, staircasewaveform, three-phase electric power, total harmonic distortion(THD).

I. INTRODUCTION

MULTILEVEL converters are used in numerous high volt-age, high-power systems for three-phase applications

such as static synchronous compensators (STATCOM) [1], [2],energy storage system integration [3], active filters [4], photo-voltaic systems [5], [6] and electric/hybrid electric vehicles [7].Multilevel converters have the benefit of allowing direct inte-gration of low-voltage power sources into medium-voltage andhigh-voltage applications, while producing high-quality wave-forms with the state-of-the-art switching devices. The cascadedtopology (see Fig. 1) is of particular interest due to its modular-ity and simplicity of control. An open question that has receivedconsiderable attention recently is determining the optimal mod-ulation technique. Though many modulation techniques are pos-sible, staircase modulation (see Fig. 2) is of particular interest

Manuscript received June 3, 2013; revised September 6, 2013; acceptedOctober 24, 2013. Date of current version May 30, 2014. Recommended forpublication by Associate Editor R. Burgos.

D. Hong was with the Department of Electrical and Computer Engineering,Duke University, Durham, NC 27708 USA. He is now with the Department ofElectrical and Computer Engineering, University of Michigan, Ann Arbor, MI48109 USA (e-mail: [email protected]).

S. Bai was with the FREEDM Center, North Carolina State University,Raleigh, NC 27695 USA. He is now with Eaton Corporation, Raleigh, NC27616 USA (e-mail: [email protected]).

S. M. Lukic is with the FREEDM Center, North Carolina State University,Raleigh, NC 27695 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2013.2290377

Fig. 1. Three-phase cascaded topology (s = 3 sources).

Fig. 2. Staircase waveform.

due to the minimum number of switching instances, resulting inreduced switching losses.

One approach to determining the switching angles of the stair-case modulation is selective harmonic elimination (SHE). Thisrequires solving a system of transcendental equations, which isdone using various techniques: Newton–Raphson [8], symmet-ric polynomials and resultants [9]–[11], neural networks [12],genetic algorithms [13], and bee algorithms [14]. Typically, thefirst few odd nontriplen harmonics are eliminated since theyhave the highest influence on the overall harmonic distortion andare difficult to filter. In the case of three-phase systems, SHE im-plementations do not eliminate the triplen harmonics [9]–[13],choosing instead to eliminate the low-order nontriplen harmon-ics. Triplen harmonics are considered to be eliminated automat-ically in the line voltages. It should be noted here that in manyapplications triplen harmonics are a source of common modecurrents. However, these issues are often dealt with separatelyfrom the modulation strategy [8]–[18], for example by usingcommon-mode filters or other filter structures.

Another approach minimizes the total harmonic distortion(THD) [15]–[19] of the staircase waveform. The goal is to reducethe low-order harmonics to a level below the pertinent IEEEstandard, while producing a lower overall THD compared tothe SHE approach. However, the definition of THD involves aninfinite sum over all odd harmonics, making it challenging to

0885-8993 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

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5230 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014

Fig. 3. 3THD of staircase waveform with two switching angles (θ1 ,θ2 ).

find its global minimum. In [19], a closed form for the THDof staircase waveforms was discovered and used to developan algorithm that computes the optimal switching angles in realtime. This produces the minimum THD in single-phase systems,where triplen harmonics cannot be eliminated through topology.

In the case of three-phase systems, the minimization shouldinstead consider a form of THD we call three-phase THD(3THD) that ignores the triplen harmonics, as is done in SHE.As was the case in the definition of THD, the definition of 3THDinvolves an infinite sum. In [18], Huang formulated the problemfor the nontriplen harmonics, noting that the 3THD expressionis nonlinear and that there were no standard ways to evaluatethe expression. To solve the 3THD minimization problem, theauthors of [17] apply a genetic algorithm. Recently, the authorsof [15] presented a closed-form expression for the 3THD ofstaircase waveforms in a five-level cascaded topology, whichreveals the 3THD to be piecewise differentiable (as can be seenin Fig. 3). The expression is found by writing the 3THD forfive levels in terms of a well-known infinite sum, which can bereplaced by its closed-form expression. This approach can alsobe applied to the 3THD for any number of levels, but doing sorequires repeating the procedure of writing the 3THD in termsof the well-known infinite sum and replacing it with its closed-form expression. Furthermore, the expression initially obtainedhas implicitly defined pieces. Hence, further work must then bedone to explicitly identify the pieces. Without doing so, the ex-pression is better suited for 3THD evaluation than minimization.

In this paper, we present general closed-form expressions for3THD that can be used for an arbitrary number of levels. Twoclosed-form expressions are provided, which also reveal 3THDto be piecewise differentiable. One expression is shorter but de-scribes the pieces implicitly. This expression is well-suited forthe evaluation of 3THD. Note that evaluating 3THD without aclosed-form expression requires truncation of the infinite sum,compromising accuracy for efficiency. The resulting error isuncertain and thus choosing where to truncate is a tricky issue.The other expression is longer but describes the pieces explicitly.This facilitates the implementation of optimization algorithmsthat use off-the-shelf optimizers to minimize over each pieceand then take the minimum among the pieces. Thus, this ex-pression is well-suited for the minimization of 3THD. The two

expressions give the 3THD for equal sources. However, they canboth be trivially generalized to handle unequal source voltagesby incorporating the source voltages in the same way as wasdone in [15]. The extension amounts to incorporating the sourcevoltages into a single line of each closed-form expression.

The main contribution of this paper is the formulation ofclosed-form expressions for 3THD. This study adds to the bodyof knowledge on this topic by clearly identifying 3THD as apiecewise differentiable function and by explicitly identifyingthe boundaries of each piece. This in turn gives fundamental in-sight into the source of nonlinearities in the 3THD expression.In addition, the approach provides a computationally efficientformulation of 3THD that can be integrated into an optimizationroutine; the formulation can be handled by off-the-shelf opti-mizers in place of the complex tools proposed in [9]–[13], [18].Furthermore, optimization on each piece can be done in isola-tion, and thus in parallel. This makes consideration of 3THDminimization in systems with large numbers of levels computa-tionally feasible.

The remainder of the paper is organized as follows. Sec-tion II contains the statement of the problem and the definitionof 3THD, which involves an infinite sum. Section III providestwo closed-form expressions for 3THD: implicit and explicit,with the corresponding proofs presented in Appendices A and B,respectively. Section IV describes how to minimize 3THD usingthe closed-form expression, and Section V compares the resultsobtained using the closed-form expression with those obtainedusing previous methods. Section VI presents the experimentalverification showing a strong agreement between experimentsand theory.

II. PROBLEM

The staircase modulation, as defined in Fig. 2, results froma cascade topology with s sources, resulting in a 2s + 1 levelwaveform. The waveform is quarter wave symmetric and thusunambiguously determined by s switching angles θ1 , . . . , θs . Itis well known that all the even harmonics of these waveformsare 0 and the odd nth harmonic is given by

Vn =4

πn

s∑

k=1

cos(nθk ). (1)

In three-phase systems, triplen harmonics cancel and thus areignored. Hence, the three-phase THD is defined as

3THD =

√∑n=5,7,11,... V

2n

V 21

(2)

where the sum is over all odd n that are not multiples of 3. Notethat this expression involves an infinite sum. The problem is tofind an equivalent closed-form expression.

III. CLOSED-FORM EXPRESSIONS FOR 3THD

In this section, we present two closed-form expressions for3THD. The first expression allows the evaluation of 3THD for agiven set of switching angles. Due to the closed-form nature ofthe expressions, the 3THD evaluation considers all harmonics

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HONG et al.: CLOSED-FORM EXPRESSIONS FOR MINIMIZING TOTAL HARMONIC DISTORTION IN THREE-PHASE MULTILEVEL CONVERTERS 5231

and makes no truncation in the calculation of THD. Furthermore,the closed-form expression is actually easier to compute than atruncated sum. After computing the fundamental harmonic, it re-quires only arithmetic operations. On the other hand, calculatingthe truncated sum requires several trigonometric computationsper harmonic. As a result, the closed-form expression gives anexact result more efficiently than the truncated sum.

The expression also reveals that 3THD is piecewise differ-entiable. To explicitly evaluate the pieces of the function, wederive a second closed-form expression that explicitly identifiesthe boundaries of the piecewise differentiable function. Thisexpression allows for the minimization of 3THD subject to adesired modulation index.

A. Implicit Closed-Form Expression

It can be shown that 3THD can be expressed in a closed-formexpression as

3THD =

√E

V 21

− 1 (3)

where

E =s∑

i=1

s∑

j=1

Ri,j

Ri,j = S(θi + θj ) + S(θi − θj )

S(φ) = T (φ) − 132 T (3φ)

T (ψ) =

⎧⎪⎪⎨

⎪⎪⎩

1 − 2π

ψ, if 0 ≤ ψ ≤ π

−T (ψ + π), if ψ < 0

−T (ψ − π), if ψ > π

(4)

and V1 is the fundamental of the sinusoidal waveform. As pre-sented, (3) can be used to directly evaluate the component ofTHD produced by all of the nontriplen harmonics. Note that theexpression is for the equal source voltage case. However, it canbe trivially generalized to handle unequal source voltages bymultiplying the ith and jth source voltages to Ri,j in (4). Thederivation of the presented implicit closed-form expression forequal sources is provided in Appendix A.

Note also that T (ψ) is a piecewise linear function. HenceS(φ), Ri,j , and E are also piecewise linear functions. Finally3THD is a piecewise differentiable function. However it is notimmediately clear what pieces 3THD will have because thisclosed-form expression describes the pieces implicitly. Hence,we call this closed-form expression the implicit closed-form ex-pression, and also derive the explicit form expression that iden-tifies the boundaries of the piecewise differentiable function.

B. Explicit Closed-Form Expression

This section provides another closed-form expression thatdescribes the pieces explicitly. Specifically, the 3THD is the

Fig. 4. Pieces of the piecewise differentiable function Rb,pi,j .

piecewise function

3THD =

⎧⎪⎪⎨

⎪⎪⎩

......

...

Db,p if Cb,p

......

...

(5)

where1) b = (b1 , . . . , bs) with bi ∈ {0, 1, 2}

p = (p1 , . . . , ps) is a permutation of (1, . . . , s).2) Cb,p is the condition

0 ≤ γp1 ≤ · · · ≤ γps≤ π

6where

γi =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

θi if bi = 0π

3− θi if bi = 1

θi −π

3if bi = 2

.

3) Db,p is given by

Db,p =

√Eb,p

V 21

− 1

where

Eb,p =s∑

i=1

s∑

j=1

Rb,pi,j (6)

and Rb,pi,j is one of the following

r1 =169

− 83

θi

πr1 =

169

− 83

θj

π

r2 =209

− 123

θi

π− 4

3θj

πr2 =

209

− 123

θj

π− 4

3θi

π

r3 =249

− 163

θi

πr3 =

249

− 163

θj

π

r4 =129

− 83

θi

πr4 =

129

− 83

θj

π

chosen according to the diagram in Fig. 4.Note that the expression is for the equal source voltage case.

However, it can be trivially generalized to handle unequal sourcevoltages by multiplying the ith and jth source voltages to Rb,p

i,j

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5232 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014

Fig. 5. Path of optimal switching angles (s = 2 sources).

in (6). The derivation of the presented explicit closed-form ex-pression for equal sources is provided in Appendix B.

IV. MINIMIZATION OF 3THD

In this section, we illustrate the usefulness of the explicitclosed-form expression by describing how it may be used forglobal optimization. We note that the main contributions of thispaper are the formulations of 3THD given in Section III, andwe simply provide global optimization as a possible use of theexpressions. In particular, consider the problem of minimizing3THD as defined by (2) subject to a given modulation index ma

ma =m

s=

π

4sV1 (7)

where ma ≤ 1. We will use the more convenient form m asdefined in (7).

The explicit closed form is used to generate all the pieces of3THD. Then, an off-the-shelf global constrained optimizer isused to find the minimum on each piece. The overall minimumis then the minimum among the piece minimums. We imple-mented this optimization using the global constrained optimizerin Maple 16 to realize results for s = 2, 3, 5. In the case of s = 5,the computation of the optimal angles for 499 modulation in-dices ranging from 0.01 to 4.99 takes around 80 s to execute ona single core of a 3.4 GHz Intel Core i7-2600 CPU with 4GB ofRAM.

The implementation takes advantage of the structure of 3THDto speed up the computation. First, note that minimizing 3THDamounts to minimizing E, which is piecewise linear. Next, notethat 3THD in the case of equal sources is symmetric in the an-gles, and so we consider only angles that are in increasing order.This reduces the number of pieces to consider from 3ss! to 3s .Finally, we precompute the range of modulation indices achiev-able on each piece. This allows further pruning of the pieces tobe considered: only pieces containing angles that achieve thegiven modulation index can contain the global minimum.

To visualize the results of the minimization of 3THD, a plotfor the path of the optimal angles as m changes from 0.01 to1.99 for s = 2 is shown in Fig. 5. Note that the optimal angleschange discontinuously over m when the optimal angle jumpsfrom one piece to another.

Fig. 6. Comparison of 3THD evaluation for s = 5: numerical approximationup to 31 harmonics (red), and closed-form expression evaluation (blue).

V. COMPARISON WITH PREVIOUS METHODS

In this section, we compare the results obtained using theclosed-form expressions with those obtained using previousmethods. A natural previous approach is to minimize the trun-cated 3THD. However, this approach is not practical because theobjective function involves many trigonometric functions. As aresult, there are numerous local minima making the global min-imization problem prohibitively time-consuming. On the otherhand, minimization using the closed-form expressions amountsto minimizing piecewise linear objective functions, which ismuch easier.

Recently, there have been more sophisticated techniques thatcan be used to reduce the 3THD [10], [19]. The authors in [10]choose switching angles to eliminate several specific harmon-ics and calculate the 3THD approximately using the first 31harmonics. The authors in [19] choose switching angles to min-imize the THD (which includes the triplen harmonics) and donot calculate the 3THD. We now illustrate that using the closed-form expressions has the following advantages: 1) the closed-form expressions calculate 3THD exactly and 2) minimizationusing the explicit closed-form expression accounts for all non-triplen harmonics and hence achieves lower 3THD than previ-ously proposed approaches.

We first compare the exact calculation of 3THD obtainedusing the closed-form expressions with the approximate calcu-lation of 3THD obtained using the first 31 harmonics, as wasdone in [10]. Fig. 6 shows a comparison of the two calculations.This shows that truncating the infinite sum to the first 31 har-monics significantly underestimates the 3THD. Furthermore, itis unclear how to a priori determine the number of harmonicsthat should be considered to be accurate enough. Hence, we willuse the exact 3THD to carry out the comparisons that follow. Asa side note, it may be argued that high-order harmonics are moreeasily filtered and hence can be ignored. Still, the exact 3THDhas important applications including filter design, where it pro-vides a way to calculate the THD produced by all the high-ordernontriplen harmonics.

Now, we compare the switching angles and 3THD obtainedby the minimization described in Section IV with those ob-tained by the approaches described in [10] and [19]. In order tomake the comparison, we consider an 11-level converter (s = 5).Fig. 7 presents the switching angles obtained using the three

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HONG et al.: CLOSED-FORM EXPRESSIONS FOR MINIMIZING TOTAL HARMONIC DISTORTION IN THREE-PHASE MULTILEVEL CONVERTERS 5233

Fig. 7. Comparison of switching angles for s = 5: approach described in [10](blue), approach described in [19] (green), and approach described in Section IV(red).

Fig. 8. Comparison of 3THD for s = 5: approach described in [10] (blue),approach described in [19] (green), and approach described in Section IV (red).

approaches. Note that the approach described in [10] providesmultiple sets of switching angles for some modulation indices,and hence requires another step to choose which set of switch-ing angles to use. On the other hand, the approach describedin Section IV provides only one set of optimal switching an-gles. Next note that the switching angles obtained using theapproach described in [19] only exist over modulation indicesm > 3.397, where the optimization can be accomplished in realtime with the Lagrange multiplier method. These switching an-gles vary smoothly with the modulation index in this range. Onthe other hand, the switching angles obtained using the approachdescribed in Section IV exist over the full range of modulationindices, but are piecewise differentiable functions of the mod-ulation index. Fig. 8 compares the 3THD corresponding to theswitching angles obtained using all three approaches. We seethat the approach described in Section IV results in the mini-mal 3THD. This is expected because the approach described inSection IV aims to minimize the 3THD.

Further insight can be gained from the harmonic content ofthe phase voltages obtained in the three approaches. The phasevoltages contain the triplen harmonics, which cancel in the linevoltages. Hence, the phase voltage harmonics reveal how thethree approaches handle both nontriplen and triplen harmonics.Figs. 9–11 show the first 39 harmonics of the phase voltagesobtained from the three approaches for a modulation index ofm = 4.12. The harmonics are shown as fractions of the fun-damental harmonic. From the plots, we see that the approachdescribed in [10] is effective at completely eliminating the 5th,

Fig. 9. Phase voltage harmonics (normalized by the fundamental harmonic)for modulation index m = 4.12 with switching angles determined using theapproach described in [10]. Blue bars show triplen harmonics, and red barsshow nontriplen harmonics.

Fig. 10. Phase voltage harmonics (normalized by the fundamental harmonic)for modulation index m = 4.12 with switching angles determined using theapproach described in [19]. Blue bars show triplen harmonics, and red barsshow nontriplen harmonics.

Fig. 11. Phase voltage harmonics (normalized by the fundamental harmonic)for modulation index m = 4.12 with switching angles determined using theapproach described in Section IV. Blue bars show triplen harmonics, and redbars show nontriplen harmonics.

7th, 11th, and 13th harmonic. Due to the limited degrees offreedom when s = 5, there is no ability to eliminate the 17thharmonic, which is relatively high as a result. The approachdescribed in [19] is effective in reducing harmonics throughoutthe spectrum. This includes the triplen harmonics, which lim-its its ability to control the nontriplen harmonics and hence the3THD. Finally, the approach described in Section IV is effectiveat limiting the nontriplen harmonics throughout the spectrum,with none of the nontriplen harmonics up to the 39th harmonicexceeding 1.3% of the fundamental harmonic.

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5234 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014

Fig. 12. Phase voltage harmonics (normalized by the fundamental harmonic)for modulation index m = 2.25 with switching angles determined using theapproach described in [10]. Blue bars show triplen harmonics, and red barsshow nontriplen harmonics.

Fig. 13. Phase voltage harmonics (normalized by the fundamental harmonic)for modulation index m = 2.25 with switching angles determined using theapproach described in Section IV. Blue bars show triplen harmonics, and redbars show nontriplen harmonics.

Figs. 12 and 13 show the first 39 harmonics of the phase volt-ages obtained from two of the approaches for a lower modulationindex (m = 2.25). Note that the approach described in [19] isnot included in this comparison because it does not provideswitching angles for modulation indices this low. As before,we see from the plots that the approach described in [10] iseffective at completely eliminating the 5th, 7th, 11th, and 13thharmonic. Again, due to the limited degrees of freedom whens = 5, there is no ability to eliminate higher harmonics, someof which are relatively high as a result. The approach describedin Section IV again aims to minimize the 3THD by allowinglow-order harmonics to remain. At lower modulation indices, itis generally more difficult to reduce the 3THD (as can be seenin Fig. 8). As a result, this approach is less effective at limitingthe nontriplen harmonics, which are relatively high. However,the approach still achieves the minimum 3THD (as was seen inFig. 8). Furthermore, none of the nontriplen harmonics up to the39th harmonic exceed 3.7% of the fundamental harmonic.

Note that for the two modulation indices considered above,the approaches described in [10] and Section IV generallyachieve control of nontriplen harmonics by allowing triplen har-monics to grow large. As noted in the introduction, the triplenharmonics can cause common mode currents to exist, which aredetrimental to system operation in many applications. In thesecases, the minimization proposed in [19] or other optimization

Fig. 14. Physical system setup (s = 3 sources).

TABLE IPARAMETERS OF THE EXPERIMENTAL SETUP

criterion may be more appropriate than the approaches describedin [10] and Section IV.

VI. EXPERIMENTAL VALIDATION

We have constructed a three-phase seven-level cascaded H-Bridge topology (s = 3). We have tested the system for a numberof modulation indices and compared the calculated and mea-sured 3THD values, and found a close agreement between thecalculated and measured line voltage harmonics.

We first describe the physical system (shown in Fig. 14). Eachphase is made up of three H-bridges that are Y-connected. Weuse a resistive delta-connected load to eliminate the triplen har-monics, while preserving the nontriplen harmonics present inthe voltage waveform through the use of a resistive load (ratherthan an inductive load). The electric circuit diagram of the sys-tem is presented in Fig. 1 and the experimental parameters arelisted in Table I. The system is powered by nine 12−V batter-ies, one powering each H-Bridge. The delta-connected resistorshave a nominal resistance of 10 Ω, resulting in a nominal systemdelivering 100 W to the load.

We now describe how the experiment was carried out on thesystem.

1) First the optimal switching angles for sufficiently manymodulation indices were computed offline as described inSection IV (See Fig. 15).

2) Next, we fed the optimal switching angles for several mod-ulation indices into the system and measured the resultingwaveforms. Fig. 16 shows the measured waveforms form = 2.1: line voltage Vab (45V/div), phase current iab

(3.75A/div), and line phase voltage Van (45V/div). Notethat the line voltage Vab and the phase current iab are in

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HONG et al.: CLOSED-FORM EXPRESSIONS FOR MINIMIZING TOTAL HARMONIC DISTORTION IN THREE-PHASE MULTILEVEL CONVERTERS 5235

Fig. 15. Optimal switching angles (s = 3 sources) computed by using theclosed-form expressions.

Fig. 16. Experimental waveforms for modulation index m = 2.1. From topto bottom: line voltage Vab (45V/div), phase current ia b (3.75A/div), and linephase voltage Van (45V/div).

Fig. 17. Line voltage harmonics (normalized by the fundamental harmonic)for modulation index m = 2.1 with switching angles determined using the ap-proach described in Section IV. Blue bars (on left) show theoretically predictedharmonics, and red bars (on right) show experimentally measured harmonics.

phase. This is expected because we used a delta-connectedresistive load.

3) We also measured the first 39 line voltage harmonics form = 2.1, which are shown in Fig. 17 as red bars. Thetheoretically predicted harmonics are shown in blue bars.The harmonics are shown as fractions of the fundamentalharmonic. Note that the experimental measurements andtheoretical predictions closely agree. Note also that thetriplen harmonics are not present. This is expected becausethe triplen harmonics cancel in the line voltage.

4) Finally, we approximately computed the 3THD for sev-eral modulation indices using the measured harmonics,which are shown in Fig. 18 as red crosses. We also showthe theoretically predicted 3THD computed exactly byusing the closed-form expressions (blue) and computedapproximately by using the first 39 theoretically predictedharmonics (black). Note that the 3THD computed from the

Fig. 18. Optimal 3THD (s = 3 sources) computed exactly by using the closed-form expressions (blue) and computed approximately by using the first 39theoretically predicted harmonics (black). Overlaid is the 3THD calculated byusing the first 39 experimentally measured harmonics (red crosses).

experimentally measured harmonics (red crosses) stronglymatches the approximately computed 3THD from the the-oretically predicted harmonics (black). Furthermore, notethat the 3THD exactly computed by using the closed-formexpressions (blue) is slightly greater than both the exper-imental 3THD (red crosses) and the approximate theoret-ical 3THD (black). This is expected, because the exactcomputation includes the harmonics beyond the 39th.

In summary, the experiment and theory strongly agree witheach other.

VII. DISCUSSION

The main contribution of this paper is the explicit closed-formformulation. It gives insight into the piecewise differentiable na-ture of 3THD and explicitly identifies the pieces. It should benoted that the 3THD minimization trades off an increase in low-order nontriplen harmonics for an overall decrease in the 3THDcompared to the SHE approach. Since IEEE standards allowfor the presence of moderate low-order harmonics in the volt-age and current waveform, the resulting low-order harmonicsfrom the 3THD minimization need to be evaluated against thepertinent standard. As shown in this paper, for the five-source(i.e., nine level) H-Brige inverter, the low-order harmonics canbe kept very low for relatively high modulation indices. For amodulation index of m = 4.12, all nontriplen harmonics up tothe 39th harmonic are kept below 1.3% of the fundamental har-monic; for the same modulation index, SHE results in a 17thharmonic above 4% of the fundamental harmonic. As a result,the 3THD minimization approach may be best suited for grid-connected applications, where the operating modulation indexis relatively high and changes in a relatively small range.

To deal with abnormal conditions such as input voltage sagand unbalance, different control strategies have been proposed[20]–[22]. These control strategies would be implemented in thecontroller that provides a modulation index to the lookup tableobtained from the optimization described in Section IV. For thestaircase modulation approach chosen in this paper, the systemresponse to any change in the modulation index is relativelyslow, because the modulation index can only be updated everyhalf cycle. Nonetheless, cascaded converters with single-pulsemodulation have been implemented in photovoltaic [12] andSTATCOM [23] applications. We refer the reader to these worksfor an in-depth presentation of the converter dynamics.

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5236 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014

VIII. CONCLUSION AND FUTURE WORK

In this paper, we derive general closed-form expressions forthe 3THD of staircase waveforms for an arbitrary number ofsources. The closed-form expressions reveal that 3THD is piece-wise differentiable. Two closed-form expressions are presented:one describes the pieces implicitly and the other describes thepieces explicitly. The implicit form is ideally suited for effi-ciently evaluating the 3THD including all nontriplen harmonics,and we show that truncating the calculation of the 3THD sub-stantially underestimates the 3THD. The explicit closed formpresented in this paper is used to choose switching angles thatminimize the 3THD subject to a given modulation index. Wenote that this optimization criterion differs from previous ap-proaches, and we compare the results of this minimization withthose obtained using the approaches described in [10] and [19].Finally, we provide experimental verification that shows goodagreement between the theoretical predictions and experimentalmeasurements.

The derived expressions open other interesting research di-rections. The piecewise linear nature of E could lead to anefficient control law that can be implemented in real time inthe place of a lookup table, which is typically used. Recall thatminimizing 3THD subject to a given modulation index amountsto minimzing E subject to the given modulation index. Anotherinteresting research avenue would be to use the modified orweighted THD (WTHD) definition [24], which gives low-orderharmonics greater weight and is suitable for inductive loads. Per-haps a similar approach can be used to derive an explicit closedform for the WTHD, which could be used to find switchingangles that yield even smaller low-order harmonics.

APPENDIX APROOF OF IMPLICIT CLOSED-FORM EXPRESSION

In this section, we provide a proof of the implicit closed-formexpression (see Section III-A). From the definition of 3THD,we have

3THD =

√∑n=5,7,11,... V

2n

V 21

=

√∑n=1,5,7,... V

2n

V 21

− 1

Therefore

3THD =

√E

V 21

− 1 (8)

where

E =∑

n=1,5,7,...

V 2n .

By substitution, we have

E =∑

n=1,5,7,...

(4

πn

s∑

k=1

cos(nθk )

)2

(9)

=∑

n=1,5,7,...

(4

πn

)2 s∑

i=1

s∑

j=1

cos(nθi) cos(nθj ) (10)

=s∑

i=1

s∑

j=1

(4π

)2 ∑

n=1,5,7,...

1n2 cos(nθi) cos(nθj ) (11)

Therefore,

E =s∑

i=1

s∑

j=1

Ri,j (12)

where

Ri,j =(

)2 ∑

n=1,5,7,...

1n2 cos(nθi) cos(nθj ).

Recall that

cos(α) cos(β) =12(cos(α + β) + cos(α − β)).

Hence, we have

Ri,j =12

(4π

)2 ∑

n=1,5,7,...

1n2 cos(n(θi + θj ))

+12

(4π

)2 ∑

n=1,5,7,...

1n2 cos(n(θi − θj )).

Therefore

Ri,j = S(θi + θj ) + S(θi − θj ) (13)

where

S(φ) =12

(4π

)2

S(φ)

S(φ) =∑

n=1,5,7,...

1n2 cos(nφ).

Note that we can write S as

S(φ) =∑

n=1,3,5,...

1n2 cos(nφ) −

n=3,9,15,...

1n2 cos(nφ)

=∑

n=1,3,5,...

1n2 cos(nφ) −

n=1,3,5,...

1(3n)2 cos((3n)φ)

=∑

n=1,3,5,...

1n2 cos(nφ) − 1

32

n=1,3,5,...

1n2 cos(n(3φ)).

Therefore

S(φ) = T (φ) − 132 T (3φ) (14)

where

T (ψ) =12

(4π

)2 ∑

n=1,3,5,...

1n2 cos(nψ).

It is well known that

T (ψ) =∑

n=1,3,5,...

12

(4π

)2 1n2 cos(nψ)

is the fourier series of the triangle waveform

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HONG et al.: CLOSED-FORM EXPRESSIONS FOR MINIMIZING TOTAL HARMONIC DISTORTION IN THREE-PHASE MULTILEVEL CONVERTERS 5237

Written recursively, this is

T (ψ) =

⎧⎪⎨

⎪⎩

1 − 2π

ψ if 0 ≤ ψ ≤ π

−T (ψ + π) if ψ < 0−T (ψ − π) if ψ > π

. (15)

Finally, by collecting (8), (12), (13), (14), and (15), we have

3THD =

√E

V 21

− 1

E =s∑

i=1

s∑

j=1

Ri,j

Ri,j = S(θi + θj ) + S(θi − θj )

S(φ) = T (φ) − 132 T (3φ)

T (ψ) =

⎧⎪⎪⎨

⎪⎪⎩

1 − 2π

ψ if 0 ≤ ψ ≤ π

−T (ψ + π) if ψ < 0

−T (ψ − π) if ψ > π

.

We have completed the proof.This expression can be trivially generalized to handle unequal

source voltages by slightly modifying a few lines of the abovederivation. In particular, the kth source voltage is multiplied tocos(nθk ) in (1) and (9). Hence, the ith and jth source voltagesare multiplied to cos(nθi) cos(nθj ) in (10). Finally the ith andjth source voltages are multiplied to (4/π)2 in (11), and so theyare also multiplied to Ri,j in (12).

APPENDIX BPROOF OF EXPLICIT CLOSED-FORM EXPRESSION

In this section, we provide a proof of the explicit closed-formexpression (see Section III-B). The proof has three parts. First,we expand the expression for Ri,j . Next, we find a systematicway to identify all the pieces of 3THD. Finally, we reexpress3THD with the piece identifications.

A. Expanded Expression for Ri,j

Recall that Ri,j is given as

Ri,j = S(θi + θj ) + S(θi − θj )

S(φ) = T (φ) − 132 T (3φ)

T (ψ) =

⎧⎪⎪⎨

⎪⎪⎩

1 − 2π

ψ if 0 ≤ ψ ≤ π

−T (ψ + π) if ψ < 0

−T (ψ − π) if ψ > π

.

Note that 0 ≤ θi, θj ≤ π2 . Thus, it is sufficient to expand S(φ)

over −π ≤ φ ≤ π. Evaluating T (ψ) recursively gives

S(φ) = T (φ) − 132 T (3φ)

={−T (φ + π) −π ≤ φ ≤ 0

T (φ) 0 ≤ φ ≤ π

− 132

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−T (3φ + 3π) −33π ≤ φ ≤ −2

T (3φ + 2π) −23π ≤ φ ≤ −1

−T (3φ + π) −13π ≤ φ ≤ 0

T (3φ) 0 ≤ φ ≤ +13π

−T (3φ − π) +13π ≤ φ ≤ +

23π

T (3φ − 2π) +23π ≤ φ ≤ +

33π

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−T (φ + π) +132 T (3φ + 3π) −3

3π ≤ φ ≤ −2

−T (φ + π) − 132 T (3φ + 2π) −2

3π ≤ φ ≤ −1

−T (φ + π) +132 T (3φ + π) −1

3π ≤ φ ≤ 0

T (φ) − 132 T (3φ) 0 ≤ φ ≤ +

13π

T (φ) +132 T (3φ − π) +

13π ≤ φ ≤ +

23π

T (φ) − 132 T (3φ − 2π) +

23π ≤ φ ≤ +

33π

.

Substituting and simplifying, we obtain

S(φ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

49

+43

φ

π−3

3π ≤ φ ≤ −2

129

+83

φ

π−2

3π ≤ φ ≤ −1

89

+43

φ

π−1

3π ≤ φ ≤ 0

89− 4

π0 ≤ φ ≤ +

13π

129

− 83

φ

π+

13π ≤ φ ≤ +

23π

49− 4

π+

23π ≤ φ ≤ +

33π

.

Substituting this equation to get an expression for Ri,j we mayanticipate obtaining 6 × 6 = 36 pieces. However, most of thesepieces are inconsistent, resulting in only 8 feasible pieces shownin the following diagram.

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5238 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014

Expanding and simplifying the consistent pieces, we obtain

Ri,j =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

169

− 83

θi

π0 ≤ θi + θj ≤ +

26π

0 ≤ θi − θj ≤ +26π

169

− 83

θj

π0 ≤ θi + θj ≤ +

26π

−26π ≤ θi − θj ≤ 0

209

− 123

θi

π+

26π ≤ θi + θj ≤ +

46π

−43

θj

π0 ≤ θi − θj ≤ +

26π

209

− 123

θj

π+

26π ≤ θi + θj ≤ +

46π

−43

θi

π−2

6π ≤ θi − θj ≤ 0

249

− 163

θi

π+

26π ≤ θi + θj ≤ +

46π

+26π ≤ θi − θj ≤ +

36π

249

− 163

θj

π+

26π ≤ θi + θj ≤ +

46π

−36π ≤ θi − θj ≤ −2

129

− 83

θi

π+

46π ≤ θi + θj ≤ +

66π

0 ≤ θi − θj ≤ +26π

129

− 83

θj

π+

46π ≤ θi + θj ≤ +

66π

−26π ≤ θi − θj ≤ 0

.

B. Systematic Identification of Pieces

Recall that

E =s∑

i=1

s∑

j=1

Ri,j (16)

Thus, E is a sum involving terms of the form Rk,k and Ri,j ,where i �= j. The partitioning of E into pieces comes from thesummation of these terms.

First consider terms of the form Rk,k , which simplify to

Rk,k =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

169

− 83

θk

π0 ≤ θk ≤ 1

209

− 163

θk

π

16π ≤ θk ≤ 2

129

− 83

θk

π

26π ≤ θk ≤ 3

Note that there are s of these terms: R1,1 , . . . , Rs,s . These termspartition E into boxes given by

π

6b1 ≤ θ1 ≤ π

6(b1 + 1)

...π

6bs ≤ θs ≤ π

6(bs + 1)

where bk ∈ {0, 1, 2}. Note that b = (b1 , . . . , bs) uniquely iden-tifies a box.

Now we consider terms of the form Ri,j where i �= j. Over-laying the box identification on the feasible pieces, we have thefollowing diagram.

Thus, these terms further partition each box into pieces alongone of the diagonal boundaries

θi + θj =26π,

46π

θi − θj = −26π, 0,

26π

We need to determine which diagonals pass through a givenbox b. Observe that when bi + bj is even, the box is divided bythe diagonal

θi − θj =π

6(bi − bj )

When bi + bj is odd, the box is divided by the diagonal

θi + θj =π

6(bi + bj + 1)

This can be expanded into the table

bi bj

even even 0 =(θi −

π

6bi

)−

(θj −

π

6bj

)

even odd 0 =(θi −

π

6bi

)−

6(bj + 1) − θj

)

odd even 0 =(π

6(bi + 1) − θi

)−

(θj −

π

6bj

)

odd odd 0 =(π

6(bi + 1) − θi

)−

6(bj + 1) − θj

).

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HONG et al.: CLOSED-FORM EXPRESSIONS FOR MINIMIZING TOTAL HARMONIC DISTORTION IN THREE-PHASE MULTILEVEL CONVERTERS 5239

It can also be written compactly for bk ∈ {0, 1, 2} as

0 = γi − γj

where

γk =

⎧⎪⎪⎨

⎪⎪⎩

θk bk = 0π

3− θk bk = 1

θk − π

3bk = 2

.

Hence, each box is partitioned into pieces by the diagonal bound-aries

γi = γj .

Each piece of a box is then identified by the ordering ofγ1 , . . . , γs . Thus, each permutation p of (1, . . . , s) identifiesa piece in each box described by

γp1 ≤ · · · ≤ γps.

Therefore, each piece of E is uniquely identified by its box band permutation p. Note that 3THD is partitioned into the samepieces as E and thus, each piece of 3THD is also uniquelyidentified by its box b and permutation p.

C. Expression for 3THD in Terms of b and p

Note that the piece identification also enables us to enumeratethe pieces, writing 3THD as

3THD =

⎧⎪⎪⎨

⎪⎪⎩

......

...

Db,p if Cb,p

......

...

where the condition Cb,p describes the piece b, p and Db,p is theexpression for 3THD over the piece. Thus, we need to find Cb,p

and Db,p .Recall that the permutation identifies pieces described by

γp1 ≤ · · · ≤ γps.

Note that the box itself can be described by

0 ≤ γ1 , . . . , γs ≤ π

6.

Thus, the piece b, p is fully described by the condition

Cb,p : 0 ≤ γp1 ≤ · · · ≤ γps≤ π

6.

Now, we need to find Db,p . Note that

Db,p =

√Eb,p

V 21

− 1

Eb,p =s∑

i=1

s∑

j=1

Rb,pi,j . (17)

Hence, we need an expression for Rb,pi,j . Namely, we need an

expression for Ri,j over the piece b, p. Note that this piecewiseexpression can be represented graphically. Namely, Ri,j is one

of the following

r1 =169

− 83

θi

πr1 =

169

− 83

θj

π

r2 =209

− 123

θi

π− 4

3θj

πr2 =

209

− 123

θj

π− 4

3θi

π

r3 =249

− 163

θi

πr3 =

249

− 163

θj

π

r4 =129

− 83

θi

πr4 =

129

− 83

θj

π

chosen according to the following diagram.

Note further that the domain for θi, θj is partitioned as shownin the following diagram.

Note that γi < γj when i occurs before j in the list p. Hence,combining the two diagrams gives us

We have completed the proof.

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5240 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 10, OCTOBER 2014

This expression can be trivially generalized to handle unequalsource voltages by slightly modifying a few lines of the abovederivation. In particular, the ith and jth source voltages aremultiplied to Ri,j in (16). Hence, the ith and jth source voltagesare also multiplied to Rb,p

i,j in (17).

ACKNOWLEDGMENT

The authors would like to thank L. Wang for helpful discus-sions on the subject matter. In particular, he observed that theclosed-form expressions in a draft could be trivially generalizedto unequal sources by incorporating the source voltages in oneline of each closed-form expression. The authors would also liketo thank the anonymous reviewers for their very helpful sugges-tions to consider the growth of triplen harmonics, to considerthe spectrum under low modulation indices and to describe theexperimental setup in greater detail.

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David Hong received the B.S. degree in electricaland computer engineering and mathematics in 2013from Duke University, Durham, NC, USA. He is cur-rently working toward the Ph.D. degree in electricalengineering from the University of Michigan, AnnArbor, MI, USA.

His research interests include algorithms, statisti-cal signal processing, and big data.

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HONG et al.: CLOSED-FORM EXPRESSIONS FOR MINIMIZING TOTAL HARMONIC DISTORTION IN THREE-PHASE MULTILEVEL CONVERTERS 5241

Sanzhong Bai (S’09) received the B.S. and M.S. de-gree in electrical engineering from Sichuan Univer-sity, Chendu, China, in 2003 and 2006, respectively,and the Ph.D. degree in electrical engineering fromNorth Carolina State University, Raleigh, NC, USA,in 2013.

Between 2006 and 2008, he was a Research Assis-tant in the Electric and Technology Institute, SichuanUniversity. He is currently a Senior Electrical Engi-neer at Eaton Corporation, Raleigh, NC, USA. Hisresearch interests include power converters, energy

storage systems, and electric power qualities.

Srdjan M. Lukic (S’02–M’07) received the Ph.D.degree in electrical engineering from the Illinois In-stitute of Technology, Chicago, IL, USA, in 2008.

He is currently an Assistant Professor in the De-partment of Electrical and Computer Engineering,North Carolina State University, Raleigh, NC, USA.He serves as the Distributed Energy Storage DevicesSubthrust Leader in the National Science Founda-tion Future Renewable Electric Energy Delivery andManagement Systems Engineering Research Center.His research interests include design, and control of

power electronic converters and electromagnetic energy conversion with appli-cation to wireless power transfer, energy storage systems, and electric automo-tive systems.

Dr. Lukic serves as an Associate Editor of the IEEE TRANSACTIONS ON IN-DUSTRY APPLICATIONS; he has served as a Guest Editor for the Special Sectionof the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS on Energy StorageSystems—Interface, Power Electronics and Control, and is a Distinguished Lec-turer with the IEEE-Vehicular Technology Society during the 2011–2015 term.