A Method for the Calculation of the AC-Side …kth.diva-portal.org/smash/get/diva2:1262497/FULLTEXT01.pdfIndex Terms—Modular multilevel converters, admittance, lin-earization techniques,

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A Method for the Calculation of the AC-SideAdmittance of a Modular Multilevel ConverterLuca Bessegato, Student Member, IEEE, Lennart Harnefors, Fellow, IEEE, Kalle Ilves, Member, IEEE,

Staffan Norrga, Member, IEEE,

Abstract—Connecting a modular multilevel converter to anac grid may cause stability issues, which can be assessed byanalyzing the converter ac-side admittance in relation to the gridimpedance. This paper presents a method for calculating theac-side admittance of modular multilevel converters, analyzingthe main frequency components of the converter variables indi-vidually. Starting from a time-averaged model of the converter,the proposed method performs a linearization in the frequencydomain, which overcomes the inherent nonlinearities of theconverter internal dynamics and the phase-locked loop usedin the control. The ac-side admittance obtained analytically isfirstly validated by simulations against a nonlinear time-averagedmodel of the modular multilevel converter. The tradeoff posedby complexity of the method and the accuracy of the result isdiscussed and the magnitude of the individual frequency compo-nents is shown. Finally, experiments on a down-scaled prototypeare performed to validate this study and the simplification onwhich it is based.

Index Terms—Modular multilevel converters, admittance, lin-earization techniques, frequency-domain analysis, stability.

I. INTRODUCTION

MODULAR multilevel converters (MMCs) [1], [2] are aclass of topologies that is currently preferred in many

high-power applications, such as voltage source converter(VSC) based HVDC systems, static synchronous compensators(STATCOMs), medium-voltage motor drives, and static trac-tion converters [3]–[6]. The main advantages that make MMCssuitable for high-power applications are modularity, scalability,low power losses, and low harmonic distortion.

In recent years, MMCs have been intensively studied, withfocus mainly on dynamic performance [7], [8], modulationtechniques [9]–[11], modeling and control [4], [5], steady-stateanalysis [12]–[14], operation under unbalanced grid conditions[15], and submodule implementations [16]. However, a topicwhere research is still limited is system stability. Similarly totwo-level VSCs, stability problems may arise when connectingan MMC to an ac grid, leading to amplification or destabiliza-tion of resonances and other stability issues. The stability ofthe system formed by the ac grid and the power convertercan be studied using frequency-domain methods, such asthe impedance-based stability criterion [17], the passivity-based stability assessment [18], or the net-damping criterion[19]. For instance, when using the impedance-based stability

Manuscript received February 16, 2018; revised June 20, 2018; acceptedJuly 30, 2018.

L. Bessegato and S. Norrga are with the Royal Institute of Technology(KTH), Stockholm 100 44, Sweden (e-mail: [email protected]; [email protected]).

L. Harnefors and K. Ilves are with ABB Corporate Research, Vasteras 72178, Sweden (e-mail: [email protected]; [email protected]).

I

VgZg

IcZc

Zc

Zg

I

Zc Ic

ac grid converter Vg

Fig. 1. Small-signal representation of the system formed by the ac grid andthe power converter (left) and its expression as a negative feedback system(right).

criterion, the ac grid is modeled by its Thevenin equivalentcircuit, Vg and Zg , while the converter is modeled by itsNorton equivalent circuit, Ic and Zc, as shown in Fig. 1. Usingthis linear representation, valid only for small-signal analysis,the current I flowing from the converter to the grid can beexpressed as

I(s) =

[Ic(s)−

Vg(s)

Zc(s)

]1

1 + Zg(s)/Zc(s). (1)

Assuming that Ic, Vg , and 1/Zc are stable, the stability of theinterconnected system depends on the stability of the secondterm on the right-hand side of (1), which can be expressedas a negative feedback system, shown in Fig. 1. By linearcontrol theory, this feedback system is stable if and only ifZg(s)/Zc(s) satisfies the Nyquist stability criterion. Thus, theexpression of the converter a-side impedance, or admittance,can be used for stability analysis.

The converter ac-side admittance has been extensively doc-umented for the two-level VSC topology [20], [21]. Still,these results cannot be directly extended to MMCs, mainlydue to their internal-dynamic behavior being inherently morecomplex than the two-level VSCs. Reference [22] presentsa model of the MMC impedance; however, the submodulecapacitor-voltages are treated as constant, effectively modelingthe MMC as a two-level VSC. Later in [23], the same authorsimprove the study by including the capacitor-voltage dynamicsand propose a new impedance model based on harmonicstate-space modeling, a method that involves extensive ma-trix manipulation. Still, the impact of the phase-locked loop(PLL), needed for synchronization with the ac grid, is notinvestigated. Additionally, in [24], the same authors proposean MMC impedance model based on harmonic linearization,deriving the solution analytically by elimination of variables.The proposed derivation is mathematically intricate; therefore,it cannot be easily adjusted to include additional frequencycomponents, different control strategies, or the PLL dynamics.The authors of [25] and [26] derive an analytical model



of the MMC ac-side admittance by developing a small-signal model of the MMC, including its internal dynamicsand considering different ac-side control strategies. Comparedto small-signal modeling, methods that analyze the singlefrequency components of the MMC variables are an interestingalternative, as they not only allow to obtain the expressionof the ac-side admittance, but also describe how the MMCinternal variables are impacted by a perturbation in the ac-gridvoltage. Reference [27] develops a sequence impedance modelfor MMCs, using multiharmonic linearization. The derivationis accurate and thorough; yet, it involves extensive matrixmanipulation and it requires obtaining the steady-state solutionby solving nonlinear equation, or by simulation.

This paper aims to, at least partially, bridge the mentionedgaps in the state of the art. Specifically, a combination ofharmonic linearization and frequency-domain representation isused to overcome the system nonlinearities, building a linearsystem which describes the main frequency components of theMMC internal variables individually. This method providesinsight on how the spectra of the MMC internal variables areimpacted by a perturbation term in the ac-grid voltage andhow the converter parameters shape the ac-side admittance.This study is aimed at achieving good accuracy while avoidinglaborious mathematical manipulation, in order to keep thecomplexity to a minimum level. The MMC internal dynamicsand the PLL, which are sources of nonlinearities, are includedin the proposed admittance model. However, in order tocontain the volume of this study, the MMC is operated with afixed sinusoidal modulation pattern; further investigation, e.g.,evaluating the impact of different ac-side control strategies onthe admittance, will be covered in future studies. The proposedmethod is verified experimentally on a down-scaled MMCprototype, in order to assess the validity of this study, andthe simplifications on which it is based, in relation to a realMMC.

An early version of this study is presented in [28]. Com-pared with the previously published results, the main addi-tions and improvements presented in this paper are: 1) arevised analytical derivation of the ac-side admittance, basedon Fourier coefficients instead of phasors, which is moresuitable when using linearization in the frequency domain;2) a frequency-domain model of the PLL, which allows forquantifying the impact of the PLL dynamics on the ac-side admittance; 3) the MMC operated with fixed sinusoidalmodulation pattern, which exposes how the MMC internaldynamics shape the admittance in absence of feedback control,providing foundation for future work; 4) an analysis of themain perturbation frequency components resulting from thesystem nonlinearities and their impact on the accuracy ofthe results; 5) the experimental verification of the proposedmethod.

The paper is organized as follows. Section II describesthe models used in this study: the MMC model, based ontime averaging, and the PLL model. Section III presents theproposed method for ac-side admittance calculation, based onlinearization in the frequency domain. Section IV shows theresults used to validate the study, obtained from simulationsand experiments. Finally, Section V summarizes this work and

L

R

ila(t)

vla(t)

R

L

iua(t)

vua(t)

L

R

ilb(t)

vlb(t)

R

L

iub(t)

vub(t)

L

R

ilc(t)

vlc(t)

R

L

iuc(t)

vuc(t)

vdu(t)

vdl(t)

isa(t)

isb(t)

isc(t)

ea(t) eb(t) ec(t)

N s

ub

mo

du

les

E.g.:

Fig. 2. Modular multilevel converter topology; a half-bridge submodule isshown as example of submodule implementation.

its conclusions.

II. SYSTEM MODEL

The MMC topology, depicted in Fig. 2, consists of threeupper arms and three lower arms, each comprised of Nsubmodules, typically half-bridge or full-bridge submodules.Each submodule includes switching elements, which allow theinsertion or bypassing of an energy-storage element, typicallya capacitor. An arm inductor, with inductance L, is usedfor limiting the switching harmonics in the arm current. Inaddition, a parasitic resistance R is included to account for thelosses in the arm. Stray inductances in the converter arms canbe neglected, as they appear in series with the arm inductor,which is considerably larger.

In this paper, a time-averaged model of the MMC is used,i.e., the switching operations are neglected [29]; furthermore,it is assumed that the submodule capacitances are balancedwithin the arm, meaning that the individual submodule-capacitance dynamics are also neglected. The converter ismodeled on a per-phase basis, dropping the subscript denotingthe phase when not needed.

In this study, the converter is operated in rectifier mode, witha resistive load on the dc side. Although having a resistive loadis uncommon for MMCs, it allows for validating the study



using an experimental setup, which is configured likewise,as shown in Section IV-B. This choice of configuration doesnot jeopardize the validity of this study, which can easily bereverted to the common case of a stiff dc-bus voltage.

The point-of-common-coupling (PCC) voltage,

e(t) = e1 cos[ϑ(t)], with ϑ(t) = ω1t = 2πf1t, (2)

is a pure cosine wave with no phase shift, because ϑ(t) ischosen as reference angle.

A. Dynamics of the Converter

The arm-current dynamics are described using Kirchhoff’svoltage laws,

Ldiu(t)

dt+Riu(t) = vdu(t)− vu(t)− e(t) (3)

Ldil(t)

dt+Ril(t) = vdl(t)− vl(t) + e(t), (4)

where iu(t) is the upper-arm current, il(t) is the lower-armcurrent, vu(t) is the upper-arm voltage, vl(t) is the lower-armvoltage, vdu(t) is the upper-dc-side voltage, vdl(t) is the lower-dc-side voltage. As the converter is connected to a resistiveload on the dc side, the dc-side voltages are functions of thearm currents, i.e.,

vdu(t) = −Rd2[iua(t) + iub(t) + iuc(t)] (5)

vdl(t) = −Rd2[ila(t) + ilb(t) + ilc(t)], (6)

where Rd is the resistance of the load.Modeling the converter using its time-averaged model al-

lows to express the arm voltages as

vu,l(t) = nu,l(t)vΣCu,l(t), (7)

where nu,l(t) are the insertion indices and vΣCu,l(t) are the

sum-capacitor-voltages, defined as

vΣCu,l(t) =

1

C

∫nu,l(t)iu,l(t)dt+ vΣ

C0, (8)

where vΣC0 is the average sum-capacitor-voltage and C is

the arm capacitance, defined as the submodule capacitancedivided by N . The converter is operated with fixed sinusoidalmodulation pattern, i.e.,

nu(t) =1

2− m1

2cos(ω1t) (9)

nl(t) =1

2+m1

2cos(ω1t), (10)

where m1 is the modulation index and the estimated PCCvoltage angle ω1t is provided by the PLL, as shown in Fig. 3.

e(t)

PLL cos( )∙ϑ(t)

m1/2nu

nl

1/2

Fig. 3. Block diagram of the MMC control scheme.

e(t)1

e1

a(t) abc

Hf (s) αp

ω1

dq cos( )∙a(t)ϑ(t)

∫ ∙( ) dt

Fig. 4. PLL block diagram.

B. Phase-Locked Loop

The purpose of the PLL is to provide the MMC with theestimation ϑ(t) of the PCC voltage angle ϑ(t), which is usedin (9) and (10), i.e.,

ϑ(t) = ω1tPLL−−−→ ϑ(t) = ω1t. (11)

The PLL employs a abc/dq transformation, using the q-signalto drive a feedback loop built around the transformation itself,as shown in Fig. 4. The input of the transformation is, forphase a,

a(t) = e(t)/e1 = cos[ϑ(t)], (12)

resulting in the q-signal

cos[ϑ(t)]abc/dq−−−−→ sin[ϑ(t)− ϑ(t)]. (13)

As shown in Fig. 4, the feedback loop includes a low-passfilter

Hf (s) =ω2f

s2 +√2ωfs+ ω2

f

, (14)

a proportional gain αp, a feedforward of the nominal angularfrequency ω1, and an integrator.

The PLL is a nonlinear system due to the abc/dq transfor-mation. However, since ϑ(t) tracks ϑ(t), the q-axis signal canbe linearized as

sin[ϑ(t)− ϑ(t)] ' ϑ(t)− ϑ(t). (15)

This allows to model the PLL as a linear feedback system, asshown in Fig. 5.

A second source of nonlinearities is the cosine operator in(9) and (10). Expressing ϑ(t) as

ϑ(t) = ω1t+ ε(t), (16)

where ω1t is the integral of the feedforward term and ε(t) isthe integral of the output of the proportional gain, allows toexpress the estimated signal a(t) as

a(t) = cos [ω1t+ ε(t)]

= cos(ω1t) cos [ε(t)]− sin(ω1t) sin [ε(t)] . (17)



ω1

ϑ(t)ϑ(t)Hf (s) αp ∫ ∙( ) dt

Fig. 5. PLL model with linearized abc/dq transformation.

Since ϑ(t) tracks ϑ(t), the previous equation is linearized forε(t) ' 0, resulting in

a(t) ' cos(ω1t)− sin(ω1t)ε(t). (18)

The linearized equations (15) and (18) will prove useful inSection III-B, when deriving the frequency response of thePLL.

III. AC-SIDE ADMITTANCE CALCULATION

The ac-side admittance of the MMC is defined as the ratiobetween the frequency-domain representation of current andvoltage at the MMC ac terminals, with the current flowingtowards the MMC, i.e.,

Yac(f) = −Is(f)

E(f). (19)

In other words, Yac(f) measures the current response of theMMC when a voltage is applied to its ac terminals.

Because of the nonlinear behavior of the MMC, caused bythe multiplications in (7) and (8), Yac(f) cannot be calculatedusing linear methods. Classic linearization methods, either instationary coordinates or in synchronous coordinates, cannotbe applied either, because there are several frequency com-ponents that needs to be considered when calculating Yac(f).Instead, harmonic linearization [30] [31] is used for tacklingthe problem.

Harmonic linearization is a technique that develops a small-signal linear model for a nonlinear system along a periodicallytime-varying operation trajectory. By this method, a harmonicperturbation [i.e., E(fp)] is superimposed on the excitation ofthe system, the resulting response of the variable of interest isdetermined, and the corresponding component at the pertur-bation frequency is extracted [i.e., Is(fp)].

Because of the nonlinearities of the converter internaldynamics, applying the perturbation term E(fp) affects thefrequency spectra of the converter variables; this generatesfrequency components at fp, multiples of fp, and combinationsof these with the steady-state frequency components throughaddition and subtraction. Under the assumption that the appliedperturbation is small, i.e.,

E(fp) E(f1), (20)

harmonic linearization assures that the multiples of fp can beneglected without compromising the accuracy of the result.

Before proceeding with the calculation of the ac-side admit-tance, a brief review of time- and frequency-domain represen-tation, highlighting their usefulness in this study, is presented.

A. Time- and Frequency-Domain Representation

The signals analyzed in this study are real-valued periodicfunctions, represented in the time domain as, e.g.,

x(t) = x0 + x1 cos(2πf1t+ ϕ1)

= x0 +1

2x1

[ej(2πf1t+ϕ1) + e−j(2πf1t+ϕ1)

]. (21)

These signals can be compactly represented by using complexcoefficients, which encapsulate the amplitude and the phase atspecific frequencies, i.e.,

x(t) = X(0) +X(f1)ej2πf1t +X(−f1)e

−j2πf1t, (22)

where

X(f) =

12x1e

jϕ1 for f = f1

x0 for f = 012x1e

−jϕ1 for f = −f1.

(23)

These coefficients effectively describe the frequency compo-nents of x(t) and, in addition, they coincide with the Fouriercoefficients of x(t).

Since x(t) is a real-valued function, X(f) features Hermi-tian symmetry about f = 0, i.e.,

X(−f) = X(f)∗, (24)

where X(f)∗ is the complex conjugate of X(f). This propertyensures that the negative frequency components of x(t) canalways be obtained from their positive counterparts, meaningthat it is sufficient to analyze the positive spectra of thevariables of interest.

A recurring situation in this study is the multiplication oftwo real-valued signals, e.g.,

z(t) = x(t)y(t) = [x1 cos(2πf1t+ ϕ1)][y2 cos(2πf2t+ ϕ2)]

=1

2x1y2cos[2π(f1 + f2)t+ ϕ1 + ϕ2]

+ cos[2π(f1 − f2)t+ ϕ1 − ϕ2], (25)

which is expressed in the frequency domain as

Z(f) =

X(f1)Y (f2) for f = f1 + f2

X(f1)Y (f2)∗ for f = f1 − f2

Z(f1 + f2)∗ for f = −(f1 + f2)

Z(f1 − f2)∗ for f = −(f1 − f2),

(26)

where X(f1) = 12x1e

jϕ1 and Y (f2) = 12y2e

jϕ2 . Noticethat when calculating Z(f), if X(f1) is known, then Z(f)becomes a linear function of Y (f2).

Situations similar to (26) are often encountered when cal-culating the perturbation terms in Section III-B. Perturba-tion frequency components resulting from a multiplicationare always the product of a steady-state and a perturbationvalue. Therefore, knowing the steady-state solution allows theperturbation solution to be obtained using linear methods.



TABLE IANALYZED FREQUENCY COMPONENTS

Steady-State PerturbationFrequency Sequence Frequency Sequence

0 zero fp positivef1 positive f1 − fp zero2f1 negative f1 + fp negative

2f1 − fp positive2f1 + fp zero3f1 − fp negative3f1 + fp positive

B. Perturbation Solution

Adding to the PCC voltage (2) the perturbation term

e(t) = ep cos(2πfpt) =⇒ E(fp) =ep2

(27)

generates new frequency components in the converter vari-ables, referred to as perturbation frequency components, dueto the nonlinearities of the converter internal dynamics. Thechoice of frequency components to consider in the analysisimpacts the complexity of the study and the accuracy of the re-sult. An intuitive choice is the minimum amount of frequencycomponents necessary to achieve the desired accuracy in theresults. The perturbation frequency components considered inthis study are listed in Table I; notice that multiples of fp arenot considered when using the harmonic linearization method.

In the following, linear expressions describing the pertur-bation frequency components of the converter variables arederived. Due to the inherent symmetries of the MMC topology,the ac-side admittance can be obtained analyzing only a singlearm, e.g., the upper arm of phase a. Hence, the attribute“upper-arm” is omitted for simplicity.

For arm current, arm voltage, and sum-capacitor-voltage, thederivations of the different perturbation frequency componentsare similar. Therefore, only the expressions at fp are presentedin this section, while the expressions at the remaining frequen-cies are given in the Appendix.

Insertion Index: The perturbation frequency components ofthe insertion index are derived by analyzing the PLL stage-by-stage. Firstly, (13) shows that, in the frequency domain, theabc/dq transformation produces a frequency shift of −f1 anda phase shift of −π/2, i.e.,

A(f)abc/dq−−−−→ A(f + f1)

j, (28)

where A(f) is the frequency-domain representation of a(t) =e(t)/e1. Notice that the frequency shift by −f1 adds theterm +f1 in the argument of A because, e.g., the frequencycomponent at f = fp is shifted to f = fp − f1, thereforeA(f + f1) evaluated for f = fp − f1 coincides with A(fp).Then, using the linearized PLL model according to (15),shown in Fig. 5, the closed-loop frequency response of thePLL results

HPLL(s) =αpHf (s)

s+ αpHf (s). (29)

For the last stage, the linearized expression for the cosineoperator (18) is used. That is, the signal ε(t), which con-tains the perturbation frequency component, is multiplied by− sin(ω1t). In the frequency domain, this results in

E(f)cosine−−−−→ − E(f − f1)− E(f + f1)

2j, (30)

i.e., the frequency spectrum E(f) is scaled by −1/2j and splitinto two components, one shifted by f1 and a second shiftedby −f1. For clarity, the frequency shifts from (28) and (30)are summarized

fpabc/dq−−−−→ fp − f1

cosine−−−−→

fp

fp − 2f1.(31)

Equations (27)–(30) are combined together to obtain the firstperturbation frequency component, i.e.,

Nu(fp) = −m1ep8e1

HPLL[j(ωp − ω1)], (32)

where HPLL(s) is evaluated for s = j(ωp − ω1), due to thefrequency shift by −f1, and (9) adds the factor −m1/2. Thesecond perturbation frequency component of the insertion in-dex is obtained from (30), observing that the two componentsdiffer by a factor −1, i.e.,

Nu(fp − 2f1) = −Nu(fp). (33)

However, in order to align with the perturbation values selectedin Table I, it is chosen to use the negative counterpart of (33)instead, i.e.,

Nu(2f1 − fp) = [Nu(fp − 2f1)]∗ = [−Nu(fp)]∗. (34)

Arm Current: Kirchhoffs voltage law (3) expresses thearm current as a linear function of the arm voltage, meaningthat each frequency component of Iu(f) is only affected byVu(f) at the same frequency. Therefore, expressing (3) in thefrequency domain at fp leads to

Iu(fp) +Vu(fp)

jωpL+R= − E(fp)

jωpL+R. (35)

Arm Voltage: Equation (7) contains a nonlinearity con-sisting of a multiplication between nu(t) and vΣ

Cu(t), whichgenerates frequency components in the arm voltage spectrumaccording to (26). When calculating a specific perturbationfrequency component of Vu(f), one must consider all possiblecombinations of steady-state components and perturbationcomponents of Nu(f) and V Σ

Cu(f), through addition andsubtraction, that result in the desired frequency component(see Table II). Thus, expressing (7) in the frequency domainusing (26) leads to

Vu(fp)−Nu(0)V ΣCu(fp)−Nu(f1)V

ΣCu(f1 − fp)∗

−Nu(f1)∗V ΣCu(f1 + fp) = V Σ

Cu(0)Nu(fp)

+ V ΣCu(2f1)Nu(2f1 − fp)∗. (36)

Notice that the perturbation frequency component of interestalways result from the product of a steady-state and a perturba-tion value, meaning that (36) is linear, because the steady-statevalues are known (see Appendix).



TABLE IIFREQUENCY-COMPONENT COMBINATIONS; ANALYZED COMPONENTS

ARE HIGHLIGHTED

−2f1 −f1 0 f1 2f1

fp −2f1 + fp −f1 + fp fp f1 + fp 2f1 + fp

f1 − fp −f1 − fp −fp f1 − fp 2f1 − fp 3f1 − fp

f1 + fp −f1 + fp fp f1 + fp 2f1 + fp 3f1 + fp

2f1 − fp −fp f1 − fp 2f1 − fp 3f1 − fp 4f1 − fp

2f1 + fp fp f1 + fp 2f1 + fp 3f1 + fp 4f1 + fp

3f1 − fp f1 − fp 2f1 − fp 3f1 − fp 4f1 − fp 5f1 − fp

3f1 + fp f1 + fp 2f1 + fp 3f1 + fp 4f1 + fp 5f1 + fp

−fp −2f1 − fp −f1 − fp −fp f1 − fp 2f1 − fp

−f1 + fp −3f1 + fp −2f1 + fp −f1 + fp fp f1 + fp

−f1 − fp −3f1 − fp −2f1 − fp −f1 − fp −fp f1 − fp

−2f1 + fp −4f1 + fp −3f1 + fp −2f1 + fp −f1 + fp fp

Sum-Capacitor-Voltage: Similarly to the arm voltages, sum-capacitor-voltages also result from a nonlinearity consistingof a multiplication, as shown in (8). Therefore, the samereasoning can be applied when calculating the perturbationfrequency components of V Σ

Cu(f), leading to

V ΣCu(fp)−

1

jωpC[Nu(0)Iu(fp) +Nu(f1)Iu(f1 − fp)∗

+Nu(f1)∗Iu(f1 + fp)] =

1

jωpC[Iu(0)Nu(fp)

+ Iu(2f1)Nu(2f1 − fp)∗] (37)

AC-Side Admittance: Equations (35)–(37) and (43)–(60)consist of a linear system of the form

Apxp = Bp, (38)

with 21 equations and the 21 unknown variables

xp =[Iu(fp), Iu(f1 − fp), Iu(f1 + fp), Iu(2f1 − fp),Iu(2f1 + fp), Iu(3f1 − fp), Iu(3f1 + fp),

Vu(fp), Vu(f1 − fp), Vu(f1 + fp), Vu(2f1 − fp),Vu(2f1 + fp), Vu(3f1 − fp), Vu(3f1 + fp),

V ΣCu(fp), V

ΣCu(f1 − fp), V Σ

Cu(f1 + fp), VΣCu(2f1 − fp),

V ΣCu(2f1 + fp), V

ΣCu(3f1 − fp), V Σ

Cu(3f1 + fp)]T.(39)

The matrix Ap contains the coefficients of the linear system,while the vector Bp contains the constants terms, which arerelated to the applied perturbation E(fp). For example, (35)yields

A(1, 1) = 1 A(1, 8) =1

jωpL+R

B(1) = − E(fp)

jωpL+R. (40)

Notably, the steady-state values of the converter variables areknown (see Appendix); therefore, they are included into Apand Bp.

The linear system (38) can be solved to obtain the perturba-tion solution of arm current, arm voltage, and sum-capacitor-voltage. From this solution, Iu(fp) is used to obtain the ac-side

admittance, as the ac-side current can be expressed as

Is(fp) = Iu(fp)− Il(fp) = 2Iu(fp), (41)

due to the inherent symmetries of the MMC topology. There-fore, using (19), we obtain

Yac(fp) = −2Iu(fp)

E(fp). (42)

This expression can be extended for to an arbitrary frequency,i.e., fp → f , matching the definition (19).

Remark: The focus of this paper is on the method usedto derive the ac-side admittance of MMCs. When MMCs areoperated in a different way than described in this study, e.g.,with a different control strategy or with a different network atthe dc side, the analysis presented in this section needs to beadjusted accordingly. For example, operating the MMC withcurrent control makes the insertion index linearly dependenton the arm current through the current-controller transferfunction. Consequently, perturbation frequency componentsappear in the insertion index spectrum, which need to beincluded into xp. The effect of these additional frequencycomponents on arm voltages and sum-capacitor-voltages canbe added into the analysis similarly to how N(fp) is includedinto (36) and (37), i.e., as linear combination of steady-state and perturbation frequency components of the convertervariables. An exhaustive assessment of the impact of differentcontrol strategies on the ac-side admittance will be covered infuture studies.

IV. VERIFICATION

The validation of this study is made in two parts. Firstly,SIMULINK simulations are used to evaluate how the methodoutlined in Section III performs in relation to the time-averaged model described in Section II. Then, experiments ona down-scaled prototype are used to assess the validity of thisstudy, and the simplifications on which it is based, in relationto a real MMC.

A. Simulation Results

The MMC time-averaged model described in Section IIis implemented in SIMULINK, using the parameters of theexperimental setup, listed in Table III. In order to measurethe ac-side admittance, the system is simulated multiple times,sweeping the perturbation frequency through the desired range.At every iteration, the current Is(fp), produced by the voltageE(fp), is measured using MATLAB’s fast Fourier transformalgorithm and the admittance is calculated using (19).

The admittance plot resulting from simulations is comparedwith the analytical model outlined in Section III, which isevaluated for different degrees of accuracy. Namely, the modelis firstly evaluated excluding the impact of the PLL andcalculating only two perturbation frequency components (fpand f1 + fp), then calculating three perturbation frequencycomponents (fp, f1 − fp, and f1 + fp), and finally includingthe impact of the PLL and calculating all the seven pertur-bation frequency components listed in Table I. The result ofthis comparison is presented in Fig. 6, which shows that,



101

102

−20

−10

0

10

Frequency [Hz]

Magnitude[dB]

101

102

−100

−50

0

50

Frequency [Hz]

Phase

[deg]

Fig. 6. Bode diagram of the MMC ac-side admittance: SIMULINK simulations (blue dots); lowest-order analytical model, i.e., no PLL impact and only twoperturbation frequency components (green line); mid-order analytical model, i.e., no PLL impact and only three perturbation frequency components (yellowline); highest-order analytical model, i.e., including PLL impact and all seven perturbation frequency components (red line).

TABLE IIIMMC PARAMETERS - SIMULATION AND EXPERIMENT

PCC voltage amplitude e1 48 VFundamental frequency f1 50 HzArm inductance L 5.7 mHArm resistance R 0.55 Ω

Arm capacitance C 0.54 mFResistive load Rd 25 Ω

Modulation index m1 0.9PLL bandwidth αp 25 rad/sPLL low-pass filter ωf 250 rad/sPerturbation amplitude ep 0.8 VSubmodules per arm N 5 (experiment only)Carrier frequency fc 763 Hz (experiment only)

evidently, the accuracy of the analytical model improves asmore perturbation terms are added to it. The lowest-ordermodel can capture the admittance qualitative behavior; then,the mid-order model shows an improved fit, especially aroundthe fundamental frequency; finally, the highest-order modelshows an excellent fit at all frequencies. Judging which orderis the most appropriate ultimately depends on the applicationfor which the admittance model is required.

Observation: As seen throughout Section III-B, a singleperturbation term in the PCC voltage generates multiple fre-quency components in the converter variables, due to thenonlinearity of the MMC internal dynamics. Solving thelinear system (38) not only allows obtaining the expressionof the ac-side admittance, but also provides the values of allthe main perturbation frequency components in the convertervariables produced by E(fp). As an example, Fig. 7 shows

101

102

−60

−50

−40

−30

−20

−10

0

Frequency [Hz]

Magnitude[dB]

Fig. 7. Magnitude of the perturbation frequency components of the armcurrent generated by E(fp): Iu(fp) (blue line); Iu(f1 − fp) (purple line);Iu(f1 + fp) (red line); Iu(2f1 − fp) (yellow line); Iu(2f1 + fp) (blackline); Iu(3f1 − fp) (green line); Iu(3f1 + fp) (dashed black line).

the magnitude of the perturbation frequency components inthe arm current generated by E(fp), giving an insight on theimpact that each frequency component has.

B. Experimental Results

The down-scaled MMC prototype used for experimentalverification employs five full-bridge submodules per arm, i.e.,a total of 30 submodules, and has a rated power of 10 kW. Thecontrol system is based on Xilinx Zynq-7000 system on chip,which integrates a processing system with a programmablelogic. The programmable logic performs low-level control, i.e.,communication with the submodules and phase-shifted carriersmodulation, which ensures the balancing of the individualcapacitor voltages within the arm [32]. The processing systemperforms high-level control, such as insertion indices compu-tation, PLL, and a MATLAB-based graphic user interface.



Perturbation

inverter

LC filter MMC

prototype

PCC

ea(t) isa(t)

eb(t)

ec(t)

isb(t)

isc(t)

Yac( f )

Vdc

Fig. 8. Configuration of the experimental setup for MMC ac-side admittancemeasurement.

Fig. 9. Photograph of the experimental setup.

The experiment is designed as follows. The device undertest, i.e., the MMC prototype, is connected at the PCC to atwo-level inverter, which is designed to generate a perturbationterm e(t) superimposed onto the fundamental-frequency three-phase voltage e(t). In the experiment, the MMC is operatedwith fixed sinusoidal modulation pattern, as shown in Fig. 3.The system configuration of the experimental setup is shownin Fig. 8, with a photograph in Fig. 9, and Table IV lists thespecifications of the perturbation inverter. The PCC voltage

TABLE IVPERTURBATION INVERTER PARAMETERS

DC side voltage Vdc 105 VPCC voltage amplitude e1 48 VFundamental frequency f1 50 HzPerturbation amplitude ep 0.8 VPerturbation frequency fp 2 Hz – 2 kHzSwitching frequency fsw 24 kHzLC filter resonant frequency fLC 4.1 kHz

e(t) and the ac-side current is(t) are measured using the dataacquisition system of the prototype, exported in MATLAB, andanalyzed in the frequency domain (using the discrete Fouriertransform) to extract the ac-side admittance of the prototypeat fp using the definition (19). This operation is iterated,sweeping the perturbation frequency through the desired rangeto obtain Fig. 10, showing the frequency-domain plot of theadmittance. Fig. 11 and Fig. 12 show two examples of thewaveforms of the converter variables during the experiment,with fp = 406 Hz and fp = 4 Hz. In these figures, theamplitude of the perturbation is increased to ep = 3.5 V, inorder to make the perturbation visible in the time domain.However, ep should be small compared to e1 (see assumption20), because harmonic linearization is based on small-signalmodeling.

The admittance plot resulting from experiments is comparedwith the highest-order analytical model derived in SectionIII. The result of this comparison, presented in Fig. 10,shows remarkable agreement between the two curves in thewhole frequency range. This agreement validates the devel-oped analytical model and allows to conclude that using theMMC time-averaged model, which neglects the switchingoperations and the individual capacitor-voltage dynamics, doesnot compromise the validity of this study.

Observation: In order to produce accurate results, the MMCparameters L, R, and C need to be known with good precision.In typical MMC applications, R is the only parameter subjectto variations. Fortunately, as MMCs are designed for highefficiency, R is normally small. Hence, imprecise values of Rare also acceptable, as the predominant contributors in shapingthe MMC ac-side admittance are L and C.

V. CONCLUSION

In this paper, a method for calculating ac-side admittanceof MMCs is presented. The nonlinearities introduced by theMMC internal dynamics and the PLL are included in themodel and are handled using linearization in the frequencydomain. Through simulations and experiments, the proposedmethod is verified, confirming the validity of this study andthe simplifications on which it is based. The tradeoff posedby complexity of the method and accuracy of the resultsis discussed, concluding that the optimal choice ultimatelydepends on the application. Furthermore, it is shown that, inaddition to the ac-side admittance, the proposed method showsin detail how a perturbation frequency component in the PCCvoltage affects the spectra of the converter variables, whichprovides additional useful insight.



101

102

−20

−10

0

10

Frequency [Hz]

Magnitude[dB]

101

102

−100

−50

0

50

Frequency [Hz]

Phase

[deg]

Fig. 10. Bode diagram of the MMC ac-side admittance: measurements on a MMC down-scaled prototype, phase a (blue dots), phase b (yellow dots), andphase c (green dots); analytical model (red line).

The method proposed in this paper provides foundation forfuture work, which will focus on evaluating the impact ofdifferent control strategies on the ac-side admittance, offeringuseful design tools for analyzing grid-converter interaction anddesigning the system.

APPENDIX

A. Perturbation Solution

The linear expressions describing arm current, arm voltage,and sum-capacitor-voltage at f1 ± fp, 2f1 ± fp, and 3f1 ± fpare given in this subsection.

Arm Current: The following expressions are obtained sim-ilarly to (35).

Iu(f1 − fp) +Vu(f1 − fp)

j(ω1 − ωp)L+R+ 32Rd

= 0 (43)

Iu(f1 + fp) +Vu(f1 + fp)

j(ω1 + ωp)L+R= 0 (44)

Iu(2f1 − fp) +Vu(2f1 − fp)

j(2ω1 − ωp)L+R= 0 (45)

Iu(2f1 + fp) +Vu(2f1 + fp)

j(2ω1 + ωp)L+R+ 32Rd

= 0 (46)

Iu(3f1 − fp) +Vu(3f1 − fp)

j(3ω1 − ωp)L+R= 0 (47)

Iu(3f1 + fp) +Vu(3f1 + fp)

j(3ω1 + ωp)L+R= 0. (48)

Notably, (43) and (46) include the term 3/2Rd, as zero-sequence components of the arm current flow into the load.

Arm Voltage: The following expressions are obtained sim-ilarly to (36).

Vu(f1 − fp)−Nu(0)V ΣCu(f1 − fp)−Nu(f1)V

ΣCu(fp)

∗

−Nu(f1)∗V ΣCu(2f1 − fp) = V Σ

Cu(f1)Nu(fp)∗

+ V ΣCu(f1)

∗Nu(2f1 − fp) (49)

Vu(f1 + fp)−Nu(0)V ΣCu(f1 + fp)−Nu(f1)V

ΣCu(fp)

−Nu(f1)∗V ΣCu(2f1 + fp) = V Σ

Cu(f1)Nu(fp) (50)

Vu(2f1 − fp)−Nu(0)V ΣCu(2f1 − fp)

−Nu(f1)VΣCu(f1 − fp)−Nu(f1)

∗V ΣCu(3f1 − fp)

= V ΣCu(0)Nu(2f1 − fp) + V Σ

Cu(2f1)Nu(fp)∗ (51)

Vu(2f1 + fp)−Nu(0)V ΣCu(2f1 + fp)

−Nu(f1)VΣCu(f1 + fp)−Nu(f1)

∗V ΣCu(3f1 + fp)

= V ΣCu(2f1)Nu(fp) (52)

Vu(3f1 − fp)−Nu(0)V ΣCu(3f1 − fp)

−Nu(f1)VΣCu(2f1 − fp) = V Σ

Cu(f1)Nu(2f1 − fp) (53)

Vu(3f1 + fp)−Nu(0)V ΣCu(3f1 + fp)

−Nu(f1)VΣCu(2f1 + fp) = 0. (54)

Sum-Capacitor-Voltage: The following expressions are ob-tained similarly to (37).

V ΣCu(f1 − fp)−

1

j(ω1 − ωp)C[Nu(0)Iu(f1 − fp)

+Nu(f1)Iu(fp)∗ +Nu(f1)

∗Iu(2f1 − fp)]

=1

j(ω1 − ωp)C[Iu(f1)Nu(fp)

∗ + Iu(f1)∗Nu(2f1 − fp)]

(55)



−4

−2

0

2

Upper-A

rmCurrent[A

]

0 0.01 0.02 0.03 0.04

−4

−2

0

2

Time [s]

Low

er-A

rmCurrent[A

]

85

90

95

100

105

Upper-A

rmSum-Cap.Volt.[V]

0 0.01 0.02 0.03 0.04

85

90

95

100

105

Time [s]

Low

er-A

rmSum-Cap.Volt.[V]

−50

0

50

PCC

Voltage[V

]

0 0.01 0.02 0.03 0.04

−6

−4

−2

0

2

4

6

Time [s]

AC-SideCurrent[A

]

Fig. 11. Waveforms of the converter variables during the experiment at fp = 406 Hz: upper- and lower-arm currents (left); upper- and lower-arm sum-capacitor-voltages (center); PCC voltage (upper-right); ac-side current (lower-right). Three-phase waveforms are shown: phase a (blue), phase b (red), andphase c (yellow).

V ΣCu(f1 + fp)−

1

j(ω1 + ωp)C[Nu(0)Iu(f1 + fp)

+Nu(f1)Iu(fp) +Nu(f1)∗Iu(2f1 + fp)]

=Iu(f1)Nu(fp)

j(ω1 + ωp)C(56)

V ΣCu(2f1 − fp)−

1

j(2ω1 − ωp)C[Nu(0)Iu(2f1 − fp)

+Nu(f1)Iu(f1 − fp) +Nu(f1)∗Iu(3f1 − fp)]

=1

j(2ω1 − ωp)C[Iu(0)Nu(2f1 − fp) + Iu(2f1)Nu(fp)

∗]

(57)

V ΣCu(2f1 + fp)−

1

j(2ω1 + ωp)C[Nu(0)Iu(2f1 + fp)

+Nu(f1)Iu(f1 + fp) +Nu(f1)∗Iu(3f1 + fp)]

=Iu(2f1)Nu(fp)

j(2ω1 + ωp)C(58)

V ΣCu(3f1 − fp)−

1

j(3ω1 − ωp)C[Nu(0)Iu(3f1 − fp)

+Nu(f1)Iu(2f1 − fp)] =Iu(f1)Nu(2f1 − fp)j(3ω1 − ωp)C

(59)

V ΣCu(3f1 + fp)−

1

j(3ω1 + ωp)C[Nu(0)Iu(3f1 + fp)

+Nu(f1)Iu(2f1 + fp)] = 0. (60)

B. Steady-State Solution

Insertion Index: The MMC is operated with fixed sinusoidalmodulation pattern as (9), which can be expressed in thefrequency domain, using (23), as

Nu(0) =1

2(61)

Nu(f1) = −m1

4. (62)

Arm Current: Due to the MMC topology, only the zero-sequence component of the arm current flows into the dc-sideload, i.e., Vdu(f) = Vdu(0) = −3/2RdIu(0). The steady-statearm current is obtained from (3) as

Iu(0) +Vu(0)

32Rd +R

= 0 (63)

Iu(f1) +Vu(f1)

jω1L+R= − E(f1)

jω1L+R(64)

Iu(2f1) +Vu(2f1)

j2ω1L+R= 0, (65)

where E(f1) = e1/2 from (2) and (23).Arm Voltage: Expressing (7) in the frequency domain using

(26) leads to

Vu(0)−Nu(0)V ΣCu(0)− Re[Nu(f1)V

ΣCu(f1)

∗] = 0 (66)

Vu(f1)−Nu(0)V ΣCu(f1)−Nu(f1)V

ΣCu(0)

−Nu(f1)∗V ΣCu(2f1) = 0 (67)

Vu(2f1)−Nu(0)V ΣCu(2f1)−Nu(f1)V

ΣCu(f1) = 0. (68)



−6

−4

−2

0

2

4Upper-A

rmCurrent[A

]

0 0.1 0.2 0.3 0.4 0.5

−6

−4

−2

0

2

4

Time [s]

Low

er-A

rmCurrent[A

]

80

90

100

110

Upper-A

rmSum-Cap.Volt.[V]

0 0.1 0.2 0.3 0.4 0.5

80

90

100

110

Time [s]

Low

er-A

rmSum-Cap.Volt.[V]

−50

0

50

PCC

Voltage[V

]

0 0.1 0.2 0.3 0.4 0.5

−5

0

5

Time [s]

AC-SideCurrent[A

]

Fig. 12. Waveforms of the converter variables during the experiment at fp = 4 Hz: upper- and lower-arm currents (left); upper- and lower-arm sum-capacitor-voltages (center); PCC voltage (upper-right); ac-side current (lower-right). Three-phase waveforms are shown: phase a (blue), phase b (red), and phase c(yellow).

Sum-Capacitor-Voltage: Expressing (8) in the frequencydomain using (26) leads to

Nu(0)Iu(0) + Re[Nu(f1)Iu(f1)∗] = 0 (69)

V ΣCu(f1)−

1

jω1C[Nu(0)Iu(f1) +Nu(f1)Iu(0)

+Nu(f1)∗Iu(2f1)] = 0 (70)

V ΣCu(2f1)−

1

j2ω1C[Nu(0)Iu(2f1) +Nu(f1)Iu(f1)] = 0,

(71)

where (69) is written using (8) with the constraint that theproduct iu(t)nu(t) has zero dc component.

Solution: Equations (63)–(71) consist of a linear system ofthe form

Asxs = Bs, (72)

with 9 equations and the 9 unknown variables

xs =[Iu(0), Iu(f1), Iu(2f1), Vu(0), Vu(f1),

Vu(2f1), VΣCu(0), V

ΣCu(f1), V

ΣCu(2f1)

]T. (73)

The linear system (72) can be solved to obtain the steady-state solution of arm current, arm voltage, and sum-capacitor-voltage.

REFERENCES

[1] A. Lesnicar and R. Marquardt, “An innovative modular multilevelconverter topology suitable for a wide power range,” in Power TechConf. Proc., Bologna, Italy, 2003.

[2] S. Allebrod, R. Hamerski, and R. Marquardt, “New transformerless,scalable modular multilevel converters for HVDC-transmission,” in 2008IEEE Power Electron. Spec. Conf. IEEE, Jun. 2008, pp. 174–179.

[3] H. Akagi, “Classification, terminology, and application of the modularmultilevel cascade converter (MMCC),” IEEE Trans. Power Electron.,vol. 26, no. 11, pp. 3119–3130, Nov. 2011.

[4] S. Debnath, J. Qin, B. Bahrani, M. Saeedifard, and P. Barbosa, “Oper-ation, control, and applications of the modular multilevel converter: Areview,” IEEE Trans. Power Electron., vol. 30, no. 1, pp. 37–53, Jan.2015.

[5] M. A. Perez, S. Bernet, J. Rodriguez, S. Kouro, and R. Lizana, “Circuittopologies, modeling, control schemes, and applications of modularmultilevel converters,” IEEE Trans. Power Electron., vol. 30, no. 1, pp.4–17, Jan. 2015.

[6] M. Glinka and R. Marquardt, “A new ac/ac multilevel converter family,”IEEE Trans. Ind. Electron., vol. 52, no. 3, pp. 662–669, Jun. 2005.

[7] M. Saeedifard and R. Iravani, “Dynamic performance of a modularmultilevel back-to-back HVDC system,” IEEE Trans. Power Deliv.,vol. 25, no. 4, pp. 2903–2912, Oct. 2010.

[8] L. Harnefors, A. Antonopoulos, S. Norrga, L. Angquist, and H.-P. Nee,“Dynamic analysis of modular multilevel converters,” IEEE Trans. Ind.Electron., vol. 60, no. 7, pp. 2526–2537, Jul. 2013.

[9] M. Hagiwara and H. Akagi, “Control and experiment of pulsewidth-modulated modular multilevel converters,” IEEE Trans. Power Electron.,vol. 24, no. 7, pp. 1737–1746, Jul. 2009.

[10] M. Hagiwara, R. Maeda, and H. Akagi, “Control and analysis of themodular multilevel cascade converter based on double-star chopper-cells (MMCC-DSCC),” IEEE Trans. Power Electron., vol. 26, no. 6,pp. 1649–1658, Jun. 2011.

[11] S. Rohner, S. Bernet, M. Hiller, and R. Sommer, “Modulation, losses,and semiconductor requirements of modular multilevel converters,”IEEE Trans. Ind. Electron., vol. 57, no. 8, pp. 2633–2642, Aug. 2010.

[12] Q. Tu, Z. Xu, and L. Xu, “Reduced switching-frequency modulationand circulating current suppression for modular multilevel converters,”IEEE Trans. Power Deliv., vol. 26, no. 3, pp. 2009–2017, Jul. 2011.

[13] K. Ilves, A. Antonopoulos, S. Norrga, and H.-P. Nee, “Steady-stateanalysis of interaction between harmonic components of arm andline quantities of modular multilevel converters,” IEEE Trans. PowerElectron., vol. 27, no. 1, pp. 57–68, Jan. 2012.

[14] Q. Song, W. Liu, X. Li, H. Rao, S. Xu, and L. Li, “A steady-stateanalysis method for a modular multilevel converter,” IEEE Trans. PowerElectron., vol. 28, no. 8, pp. 3702–3713, Aug. 2013.

[15] M. Guan and Z. Xu, “Modeling and control of a modular multilevelconverter-based HVDC system under unbalanced grid conditions,” IEEETrans. Power Electron., vol. 27, no. 12, pp. 4858–4867, Dec. 2012.

[16] A. Nami, J. Liang, F. Dijkhuizen, and G. D. Demetriades, “Modularmultilevel converters for HVDC applications: Review on converter cellsand functionalities,” IEEE Trans. Power Electron., vol. 30, no. 1, pp.18–36, Jan. 2015.



[17] J. Sun, “Impedance-based stability criterion for grid-connected invert-ers,” IEEE Trans. Power Electron., vol. 26, no. 11, pp. 3075–3078, Nov.2011.

[18] L. Harnefors, X. Wang, A. G. Yepes, and F. Blaabjerg, “Passivity-basedstability assessment of grid-connected VSCs—An overview,” IEEE J.Emerg. Sel. Top. Power Electron., vol. 4, no. 1, pp. 116–125, Mar. 2016.

[19] G. Stamatiou and M. Bongiorno, “Stability analysis of two-terminalVSC-HVDC systems using the net-damping criterion,” IEEE Trans.Power Deliv., vol. 31, no. 4, pp. 1748–1756, Aug. 2016.

[20] L. Harnefors, M. Bongiorno, and S. Lundberg, “Input-admittance cal-culation and shaping for controlled voltage-source converters,” IEEETrans. Ind. Electron., vol. 54, no. 6, pp. 3323–3334, Dec. 2007.

[21] X. Wang, L. Harnefors, and F. Blaabjerg, “Unified impedance model ofgrid-connected voltage-source converters,” IEEE Trans. Power Electron.,vol. 33, no. 2, pp. 1775–1787, Feb. 2018.

[22] J. Lyu, X. Cai, and M. Molinas, “Frequency domain stability analysisof MMC-based HVdc for wind farm integration,” IEEE J. Emerg. Sel.Top. Power Electron., vol. 4, no. 1, pp. 141–151, Mar. 2016.

[23] ——, “Optimal design of controller parameters for improving thestability of MMC-HVDC for wind farm integration,” IEEE J. Emerg.Sel. Top. Power Electron., pp. 1–1, 2017.

[24] J. Lyu, Q. Chen, and X. Cai, “Impedance modeling of modular multilevelconverters by harmonic linearization,” in 2016 IEEE 17th Work. ControlModel. Power Electron. IEEE, Jun. 2016.

[25] M. Beza, M. Bongiorno, and G. Stamatiou, “Analytical derivation of theac-side input admittance of a modular multilevel converter with open-and closed-loop control strategies,” IEEE Trans. Power Deliv., vol. 33,no. 1, pp. 1–1, Feb. 2017.

[26] J. Khazaei, M. B. Beza, and M. Bongiorno, “Impedance analysis ofmodular multi-level converters connected to weak ac grids,” IEEE Trans.Power Syst., pp. 1–1, 2017.

[27] J. Sun and H. Liu, “Sequence impedance modeling of modular multilevelconverters,” IEEE J. Emerg. Sel. Top. Power Electron., vol. 5, no. 4, pp.1427–1443, Dec. 2017.

[28] L. Bessegato, S. Norrga, K. Ilves, and L. Harnefors, “Ac-side admittancecalculation for modular multilevel converters,” in 2017 IEEE 3rd Int.Futur. Energy Electron. Conf. ECCE Asia, IFEEC - ECCE Asia 2017.IEEE, Jun. 2017, pp. 308–312.

[29] K. Sharifabadi, L. Harnefors, H.-P. Nee, S. Norrga, and R. Teodorescu,Design, control and application of modular multilevel converters forHVDC transmission systems. Chichester, UK: John Wiley & Sons,Ltd, 2016.

[30] J. Sun, “Small-signal methods for ac distributed power systems—Areview,” IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2545–2554,Nov. 2009.

[31] J. Sun and K. J. Karimi, “Small-signal input impedance modeling ofline-frequency rectifiers,” IEEE Trans. Aerosp. Electron. Syst., vol. 44,no. 4, Nov. 2008.

[32] K. Ilves, L. Harnefors, S. Norrga, and H. P. Nee, “Analysis and operationof modular multilevel converters with phase-shifted carrier PWM,” IEEETrans. Power Electron., vol. 30, no. 1, pp. 268–283, Jan. 2015.

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A Method for the Calculation of the AC-Side …kth.diva-portal.org/smash/get/diva2:1262497/FULLTEXT01.pdfIndex Terms—Modular multilevel converters, admittance, lin-earization techniques,