TPEL2480845_AcceptedVersion.pdf - Sites at Penn State

transcript

Stability Analysis of Vector-Controlled ModularMultilevel Converters in Linear Time Periodic

FrameworkNilanjan Ray Chaudhuri,Member, IEEE, Rafael Oliveira, and Amirnaser Yazdani,Senior Member, IEEE

Abstract—Stability analysis of average value models (AVMs) ofvector-controlled Modular Multilevel Converters (MMCs) i s thesubject matter of this paper. Stability analysis of fundamentalfrequency phasor-based AVMs of MMCs can be conducted ina traditional linear time-invariant (LTI) framework throu gheigenvalue computation. This class of models do not considercirculating current control loop and hence fails to capturesysteminstability that occurs in a certain range of gains of the circulatingcurrent controller. We propose stability analysis in a linear time-periodic (LTP) framework to solve this issue. To that end, anonlinear AVM is presented that considers the submodule (SM)capacitor insertion dynamics and takes into account the outputand the circulating current control schemes in vector controlapproach. Upon linearization, an LTP model is derived fromthis averaged model. It is shown that the Poincare multipliersare indicative of system instability corresponding to a certainrange of gains of the circulating current controller.

Index Terms—Linear Time Periodic, Linear Time Varying,Modular Multilevel Converter, Stability, State Transitio n Matrix.

I. I NTRODUCTION

T HE Modular Multilevel Converter (MMC) has gainedimmense popularity since it was invented [1], [2]. The

focus of this paper is on the stability analysis of averagevalue models (AVMs) of MMCs. It presents a comprehensivestability analysis framework for MMCs under closed-loopcontrol that takes into account the circulating current controlloops and the output current control loops. As an example,vector control, which is popular in the industry, is consideredas the control methodology. The proposed analytical approachis critical in developing insight into the zones of stabilityof MMC controller gains, and therefore, can have significanttheoretical and practical importance.

Literature review shows that a lot of attention has beenfocused on the modeling and control of MMCs [3] - [30]. Onthe contrary, only a few papers [31] - [34] presented stabilityanalysis framework of the MMC. We divide the literature intotwo parts: (A) papers that presented only modeling and controlphilosophy, and (B) papers that presented stability analysis.

N. R. Chaudhuri is with Department of Electrical & Computer Engi-neering, North Dakota State University, Fargo, ND, USA (e-mail: nilanjan-ray.chaudhur@ndsu.edu).

R. Oliveira and A. Yazdani are with Department of Electrical&Computer Engineering, Ryerson University, Toronto, ON, Canada (e-mail:rafael.oliveira@ryerson.ca, yazdani@ryerson.ca).

Financial support from NSF ND EPSCoR New Faculty Startup Grant(Award # FAR0021960) is gratefully acknowledged.

In the following sections, we will conduct a comprehensivereview of this literature in order to distinguish our contribution.

A. Literature on MMC Modeling & Control

Xiaofeng et-al [3] proposed a model to represent the cir-culating current in the MMC. The paper did not include thecontrollers in their model and did not present any frameworkfor stability analysis. Kolb et-al [4] focused on a novel controlstrategy for MMCs, which allows feeding a three-phase ma-chine over its complete frequency range. Two operating modeswere proposed in this paper: a low frequency mode for startupand low speed operation, and a high frequency mode for higherspeeds. The same authors proposed a cascaded control systemfor MMCs for variable-speed drives [5]. The decoupled currentcontrol strategy proposed in this paper transforms theabc

frame quantities intoαβ0 quantities for the phase currents andthe circulating currents (referred as ‘e’ currents). The proposedcontrol system ensures a dynamic balancing of the energies inthe MMC cells at minimum internal currents over the completefrequency range. However, none of these papers presenteda comprehensive modeling framework that can be used forstability analysis of MMCs. Munch et-al [6] wrote a veryimportant paper that showed that MMC can be modeled asa periodic bilinear time-varying system capturing all currentsand energies. It assumed that the submodule (SM) capacitorvoltages are balanced and focused on horizontal and verticalenergy balancing among the group of phase modules (PMs).The state-space model, although insightful, leads to a controldesign that requires a p-periodic Linear Quadratic Regulator(PLQR), which possesses periodic time-varying gains. Noobvious advantage was established over the existing constantgain controllers that are more popular due to their simplicity.Moreover, the paper did not offer any insight into the stabilityanalysis using such models. Reference [7] proposed a Linear-Time-Invariant (LTI) state-apace model of the MMC, which isnot adequate to analyze the interaction between the circulatingcurrent control and output current control loops.

Siemaszko et-al [8] presented a comparison between fourmodulation strategies in MMCs. These strtegies are: directmodulation, closed-loop control, open-loop control and Phase-shifted Carrier-based Pulse-width modulation (PSCB-PWM).Similar analysis was also reported in [9]. A fundamental-frequency control of the circulating current of the MMCwas proposed in [10]. Although these three papers focus ondifferent control methodologies of MMC, they do not present

This is the author’s version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available athttp://dx.doi.org/10.1109/TPEL.2015.2480845

any modeling and stability analysis. A nonlinear switchingfunction-based model of MMC was presented in [11], whichwas used for time-domain simulation. Marcelo et-al presenteda model of the MMC based on switching function-drivencontrollable voltage sources in [12] and proposed a vectorcontrol strategy in [13] for output current control, circulatingcurrent control, and average dc voltage control. Unfortunately,due to the switched nature of the model, it is not straight-forward to do stability analysis. Yan et-al [14] presented anaveraged model in rotatingd− q reference frame. The modelis oversimplified and does not consider circulating current.Stefan et-al [15] proposed the reduction of cell capacitance byinjecting harmonic current in the circulating current. Steffenet-al [16], [17] presented a nonlinear time-varying state-spaceAVM of MMCs. The model described in [17] is of particularinterest, and has some similarities with the model developedin this paper. The key difference however, is that, in this paperthe closed loop control is also considered within the model.More importantly, an approach for stability analysis has alsobeen presented in our paper, which was not done in [17]. Manyother papers including [18] - [26] also focused on modelingand control of MMCs without considering any framework forstability analysis. Teeuwsen et-al [27] presented a positive-sequence fundamental frequency model of MMCs for phasorsimulation with large AC systems. This model cannot capturethe circulating current and corresponding control loops. Afundamental frequency AVM, called the Type6 model, wasreported in [28] and [29]. These models can also be usedfor phasor-based simulations. In this paper, we shall call suchmodels ‘phasor-based AVMs.’ Phasor-based AVMs neglect thedynamics of the submodule (SM) capacitors and, therefore,do not capture the circulating current. Upon linearization,this class of models can be treated as linear time-invariant(LTI) and traditional eigenvalue analysis can be performedtoascertain stability. We will demonstrate that there are differentregions of gains of the circulating current controller thatcandestabilize the MMC. Since the phasor-based AVMs do notconsider the circulating current control loop, it can not indicatesuch instability. The objective of this work is to present amodeling and stability analysis framework that solves thisproblem.

It is important to note that the operating principle of MMCsis fundamentally different from other VSC topologies. Unlikethe conventional VSC topologies (e.g.,2 or 3-level), the MMCoperates based onphysical modificationof its circuit, i.e.,insertion and bypassing of its SMs in a discrete manner. TheAVM presented in [30] approximates the SM insertion andbypassing as a continuous function, and, thus, captures thecirculating currents flowing through the arms.

B. Literature on Stability Analysis of MMC

Although a lot of work has been done on the modelingand control aspect of MMCs, very little has been reported onestablishing a comprehensive stability analysis framework ofMMCs that considers closed-loop control, e.g. vector controlapproach. The global asymptotic stability of MMCs was ana-lyzed in [31], which did not consider any closed-loop control

strategy for stability analysis. In [32] and [33], the authorsstudied the stability of the MMC as an open-loop system. Theymade multiple simplifying assumptions to convert the lineartime-varying (LTV) model into an LTI model. Hagiwara et-al [34] used Routh-Hurwitz stability criterion for conductingstability analysis of one arm of the MMC converter where onlythe circulating current was considered as the state variable.

C. Motivation, Contribution & Application of this paper

In view of the above literature review, it is clear that

• Papers that presented stability analysis of MMCs eithermodeled systems under open-loop conditions or madequite a few simplifying assumptions to avoid complexitiesof a rigorous stability analysis.

• A modeling framework of MMCs considering closed-loop control, which is suitable for stability analysis, hasnot been presented.

• An analytical method for rigorous stability study is neces-sary. It will be shown in this paper that traditional eigen-value analysis fails to detect the regions of instability ofMMCs under closed-loop control.

The AVM introduced in this paper, while complex, considersthe closed-loop control system in the modeling framework.Moreover, it presents a rigorous stability analysis approachwithout simplifying assumptions.

The key contributions of this paper are:

• It presents a comprehensive modeling framework thataugments the AVM that considers SM insertion dynamicswith widely-used vectorial control for the output currentcontrol loop and the circulating current control loop.

• It proposes a stability analysis methodology of MMCs ina linear time-periodic (LTP) framework.

• It demonstrates through case studies that the proposedapproach can indicate a range of compensator gains of thecirculating current control scheme which can destabilizethe MMC.

The proposed analytical approach is critical in developinginsight into the zones of stability of MMC controller gains,andtherefore, can have significant theoretical and practical impor-tance. It should be mentioned that the proposed technique isapplicable for any control philosophy (not limited to vectorcontrol), as long as the closed-loop system can be representedin the form of an LTP model.

II. OVERVIEW OF MMC CONTROL SYSTEM

Figure 1 shows a circuit diagram of thejth phase ofan MMC. The MMC is connected to the host ac systemthrough a transformer represented by its series resistanceandleakage inductance. The MMC control system has three keyfunctionalities:

• Balancing Control:The capacitor voltages across all SMsshown in Fig. 1 must be balanced and kept equal. Dif-ferent voltage balancing techniques have been proposedin literature, e.g. [35] - [42].

• Circulating Current Control:Second harmonic circulat-ing current originates from unbalance in the arm voltages.

1NSM +

2NSM +

diffji

( )1s Nv

( )2s Nv

PCC giv

Fig. 1. Schematic of thejth phase (j = a, b, c) of the MMC.

This distorts the arm current and increases the dc voltageripple in SMs. References [43] - [48] are a few papersfrom the vast literature in this area that have proposedcirculating current control strategies.

• Output Current Control:The output current or phasecurrents are not affected by the circulating current andcan be controlled by decoupled current control approach.

Throughout this paper, it is assumed that the voltages across allthe SMs are balanced and the dc-side voltagevdc is constant.

III. PHASOR-BASED AVM OF MMC

The phasor-based AVM [28] is derived with the followingassumptions:

• SM capacitor insertion dynamics is neglected.• Second harmonic circulating current is completely sup-

pressed.• The model is derived in a synchronously rotatingd − q

reference frame assuming no harmonic content in the armvoltagesv1j andv2j .

From Fig. 1, applying KCL in phasej, one obtains

i1j =ij

2+ idiffj

i2j = − ij

2+ idiffj (1)

Applying KVL in phasej one obtains

vdc − v1j − v2j = 2Ldidiffj

dt+ 2Ridiffj

2− v1j

2− vgj = L′

dt+R′i1j (2)

whereL′ = Lt +L2

, andR′ = Rt +R2

. As shown in Fig. 1,v1j and v2j are the voltages across the upper and the lowerarm SMs that are inon-state.

As mentioned before, the second harmonic component ofthe circulating current is assumed to be perfectly suppressed.

Expressing the second equation of (2) in a synchronouslyrotatingd− q reference frame, one can write:

L′did

dt= −R′id + L′ωiq + ed − vgd

L′diq

dt= −R′iq − L′ωid + eq − vgq (3)

where,ed = v2d−v1d2

, eq =v2q−v1q

2. We consider widely-used

vector control strategy for the output current control loop,which is described next.

A. Vector Control: Output Current Control Scheme

A VSC is commonly current-controlled through a vectorialcontrol strategy in a rotatingd − q reference frame [49].Figure 2 shows the current control scheme of the MMC inthe d − q-frame with the decoupling feed-forward signals. APhase-Lock-Loop (PLL) ensures that thed-axis of the rotatingd− q reference frame is aligned with the grid voltage vector~vg.

Fig. 2. The current control scheme of the MMC in a rotatingd − q frameof reference.

As Fig. 2 indicates, the reference voltage commands in thed− q frame are given by

e∗de∗q

i∗di∗q

+ωL′

−iqid

vgdvgq

wherexd1 andxq1 are the state variables of the Proportional-Integral (PI) compensators in thed−q frame. Thed andq-axiscomponents of the ac grid voltage vector~vg and those of thecurrent~i are denoted byvgd, vgq, id, andiq, respectively. Thestate-space equations of the PI compensators can be writtenas

i∗di∗q

It is assumed that the converter delay is negligible and thefollowing control law as described in [43] is considered:

ed = e∗d =v2d − v1d

2, eq = e∗q =

v2q − v1q

B. State-space Model of Phasor-based AVM

Combining equations (3), (4), (5), and (6) one can write thephasor-based AVM in the following state-space form:

idiqxd1

−Kp+R′

L′0 KI

0 −Kp+R′

L′0 KI

−1 0 0 00 −1 0 0

idiqxd1

i∗di∗q

(7)It can be seen that this model has4 state-variables and2control variables. Next, we will discuss the stability analysisof the same.

IV. STABILITY ANALYSIS: PHASOR-BASED AVM

Equation (7) shows that

• The model is linear time-invariant (LTI) in nature• The state matrix is independent of the circulating current

controller gains

Traditional eigenvalue computation can be performed to ana-lyze the stability of this system. Clearly, any instabilitycausedby the circulating current controller will not be captured.InSection VII, we will do eigenvalue analysis for a test systemto highlight this point.

This sets up the motivation for developing a comprehensivemodeling and stability analysis framework that will be abletosolve such issues.

V. PROPOSEDAVM OF MMC CONSIDERING SMINSERTIONDYNAMICS

Unlike the phasor-based AVM, the insertion of SM capac-itances was considered in this model, as proposed in [30].In practice, the change in the total capacitance of one armof the converter shown in Fig. 1 will happen in a discretemanner. As the number of SMs increase, this variation can beapproximated using a continuous function.

Let η∗Uj andη∗Lj denote the fractions of the total number ofSMs in the upper and the lower arm of phasej, which are inon-state. Variablesη∗Uj andη∗Lj are control commands, whichare obtained from the output current control scheme and thecirculating current control scheme. SinceC is the capacitanceof each SM andN is the total number of SMs in each arm,the equivalent capacitance of the modules inon-stateis givenby C

Nη∗

for the upper arm and CNη∗

for the lower arm. The

capacitor voltage dynamics of the upper and the lower armSMs of phasej can be written as

Nη∗Uj

dt= i1j

Nη∗Lj

dt= i2j (8)

where vUj and vLj are the voltages across all SMs in theupper arm and the lower arm of thejth phase, respectively.

In this work, the dc-side voltagevdc is assumed constant.From (1), (2), and (8) a nonlinear AVM can be formulated forthe MMC, as:

f1j =dvUj

2+ idiffj

η∗Uj (9)

f2j =dvLj

− ij

2+ idiffj

η∗Lj (10)

f3j =didiffj

vdc − 2Ridiffj − η∗UjvUj − η∗LjvLj

f4j =dij

−vgj −R′ij +η∗LjvLj

η∗UjvUj

We shall proceed from where we left in Section III-A anddevelop the framework for including the output current controlscheme into our model, which is described next.

A. Vector Control: Output Current Control Scheme

Continuing from where we were in Section III-A, let the acsystem phase voltages be

vga = Vgm cosωt

vgb = Vgm cos

ωt− 2π

vgc = Vgm cos

ωt− 4π

Assumingρ = ωt, the reference voltage corresponding tophasea can be derived as:

e∗a =[

cos ρ − sin ρ] [

e∗d e∗q]T

Pre-multiplying (4) by[

cos ρ − sin ρ]

, we get:

e∗a = Kp

i∗d cos ρ− i∗q sin ρ

−Kpia +KIxa1 + vga

− ωL′ id sin ρ+ iq cos ρ (15)

wherexa1 = [ cos ρ − sin ρ ][

xd1 xq1

]T. We as-

sume that no zero-sequence current can flow in the system,i.e.,

ia + ib + ic = 0 (16)

With this assumption, we can write:

id sin ρ+ iq cos ρ =1√3(ib − ic) =

1√3(ia + 2ib) (17)

Keeping the identities of the reference input quantitiesi∗dand i∗q , one can derive the equation fore∗a as

e∗a = Kp

i∗d cos ρ− i∗q sin ρ)

Kp +ωL′

ia +KIxa1

+ vga −2ωL′

ib (18)

It can be observed that the voltage referencee∗a is cross-coupled with phase-b current. Fundamentally, this couplingimplies that the reference voltage of one phase is not onlydetermined by the voltage, current, and PI-controller statevariable of that phase, but also by the current of the otherphase.

Following similar steps, the reference voltage for phase-b

can be derived as

e∗b = Kp

i∗d cos

ρ− 2π

− i∗q sin

ρ− 2π

Kp −ωL′

ib +KIxb1 + vgb +2ωL′

ia (19)

The voltage reference for phase-c can be expressed in termsof those of phasea and b. Therefore, from now on, only themodels for phasesa andb will be analyzed.

The state-space model of the PI compensators in a syn-chronously rotatingd − q reference frame was described inequation (5). Transforming thed-q frame quantities to phasequantities, one obtains

f5a =dxa1

dt= − ω√

3(xa1 + 2xb1)− ia +

i∗d cos ρ− i∗q sin ρ)

f5b =dxb1

ω√3(2xa1 + xb1)− ib + i∗d cos

ρ− 2π

− i∗q sin

ρ− 2π

Equations (20) and (21) will be augmented with equations(9) through (12) while formulating the nonlinear state-spacemodel described in Section V-D.

Next, we will include the circulating current controller inthe model.

B. Vector Control: Circulating Current Control Scheme

The circulating current [43],idiffj, is given by

idiffa =idc

3+ I2f cos (2ωt+ ϕ)

idiffb =idc

3+ I2f cos

2ωt+ ϕ− 4π

idiffc =idc

3+ I2f cos

2ωt+ ϕ− 2π

The circulating current is controlled in ad − q referenceframe that rotates with an angular speed of2ω [43], as Fig. 3illustrates.

Fig. 3. Scheme for regulation of circulating current.

As Fig. 3 indicates, the reference voltage commands in thed− q reference frame can be written as[

e∗2fde∗2fq

i∗2fdi∗2fq

−Kpf

i2fdi2fq

i2fq−i2fd

(23)Assumingξ = 2ωt, one can derive the corresponding voltagereferences for phasea andb as

e∗fa = Kpf

i∗2fd cos ξ − i∗2fq sin ξ)

Kpf +2ωL√

idiffa −idc

+KIfxa2

− 4ωL√3

idiffb −idc

e∗fb = Kpf

i∗2fd cos

ξ − 4π

− i∗2fq sin

ξ − 4π

Kpf − 2ωL√3

idiffb −idc

+KIfxb2 +4ωL√

idiffa −idc

The state variables of the PI compensators, Fig. 3, in thed−q

frame are related to the corresponding phase values as[

cos ξ − sin ξcos

ξ − 4π3

− sin(

ξ − 4π3

(26)The state-space equations of the PI compensators can be

written as[

i∗2fdi∗2fq

i2fdi2fq

Transforming thed-q frame quantities toa andb-phase quan-tities we get

f6a =dxa2

2ω√3(xa2 + 2xb2)−

idiffa −idc

i∗2fd cos ξ − i∗2fq sin ξ)

f6b =dxb2

dt= − 2ω√

3(2xa2 + xb2)−

idiffb −idc

i∗2fd cos

ξ − 4π

− i∗2fq sin

ξ − 4π

Equations (28) and (29) will be augmented with equations(9) through (12), (20), and (21) while formulating the nonlin-ear state-space model.

C. Control Law

The control commandsη∗Uj andη∗Lj are produced from thecontrol commandse∗j ande∗fj based on the relations

η∗Uj =1

e∗j + e∗fj

vdc(30)

η∗Lj =1

e∗j − e∗fj

vdc(31)

The following constraints need to be imposed on the controlcommands

0 ≤ η∗Uj ≤ 1, 0 ≤ η∗Lj ≤ 1 (32)

D. State-space Model of AVM with SM Insertion Dynamics

Equations (9) through (12), (20), (21), (28), and (29)constitute a nonlinear state-space model for the MMC of theform x = f (x, u, z). These equations were put into boxes inprevious sections for ease of identification. Variablesη∗Uj andη∗Lj in equations (9)-(12) are replaced by expressions ofe∗jande∗fj as in equations (18), (19), (24), (25), (30), and (31).Since there are6 equations for each phase (phasesa andb), itleads to12 state variables (x). In addition, there are4 controlvariables (u) and 4 algebraic variables (z). These equationsare expressed in a compact form as shown below:

x = f (x, u, z)

x = [ vUa vUb vLa vLb idiffa idiffb · · ·ia ib xa1 xb1 xa2 xb2]T

i∗d i∗q i∗2fd i∗2fq]T

vga vgb vdc idc]T

As mentioned before, equations for phasesa andb are suf-ficient for the dynamic model, in view of the assumption (16).

VI. PROPOSEDSTABILITY ANALYSIS: AVMCONSIDERING SM INSERTIONDYNAMICS

As described in Section V-D, the state-space model isnonlinear in nature. To do stability analysis, we will linearizethis model around an operating point, as described next.

A. Linearized State-Space Model

The nonlinear state-space model (33) can be linearizedaround a nominal operating point(x0, u0, z0), and expressedin the form

∆x(t) = A(t)∆x(t) +B(t)∆u(t) + Γ(t)∆z(t),

A(t) ∈ ℜn×n, B(t) ∈ ℜn×m,Γ(t) ∈ ℜn×p

A(t) =∂f

, B(t) =∂f

,Γ(t) =∂f

The subscript‘0′ is used to signify values at the currentoperating condition. The non-zero elements of the matrixA(t),B(t), andΓ(t) are given below.

Elements of A(t) matrix:

A(1, 5) =N

η∗Ua0 +1

2+ idiffa0

Kpf +2ωL√

A(1, 6) =N

4ωL√3vdc0

2+ idiffa0

A(1, 7) =N

η∗Ua0

2+ idiffa0

Kp +ωL′

A(1, 8) =N

2ωL′

√3vdc0

2+ idiffa0

A(1, 9) = −N

2+ idiffa0

A(1, 11) = −N

2+ idiffa0

A(2, 5) = −N

4ωL√3vdc0

2+ idiffb0

A(2, 6) =N

η∗Ub0 +1

2+ idiffb0

Kpf − 2ωL√3

A(2, 7) = −N

2ωL′

√3vdc0

2+ idiffb0

A(2, 8) =N

η∗Ub0

2+ idiffb0

Kp −ωL′

A(2, 10) = −N

2+ idiffb0

A(2, 12) = −N

2+ idiffb0

A(3, 5) =N

η∗La0 +1

− ia0

2+ idiffa0

Kpf +2ωL√

A(3, 6) = −N

4ωL√3vdc0

− ia0

2+ idiffa0

A(3, 7) =N

−η∗La0

2− 1

− ia0

2+ idiffa0

Kp +ωL′

A(3, 8) = −N

2ωL′

√3vdc0

− ia0

2+ idiffa0

A(3, 9) =N

− ia0

2+ idiffa0

A(3, 11) = −N

− ia0

2+ idiffa0

A(4, 5) = −N

4ωL√3vdc0

− ib0

2+ idiffb0

A(4, 6) =N

η∗Lb0 +1

− ib0

2+ idiffb0

Kpf − 2ωL√3

A(4, 7) =N

2ωL′

√3vdc0

− ib0

2+ idiffb0

A(4, 8) =N

−η∗Lb0

2− 1

− ib0

2+ idiffb0

Kp −ωL′

A(4, 10) =N

− ib0

2+ idiffb0

A(4, 12) = −N

− ib0

2+ idiffb0

A(5, 1) = −η∗Ua0

2L,A(5, 3) = −η∗La0

A(5, 5) =1

−R− 1

Kpf +2ωL√

(vUa0 + vLa0)

A(5, 6) = − 2ω√3vdc0

(vUa0 + vLa0)

A(5, 7) =1

Kp +ωL′

(−vUa0 + vLa0)

A(5, 8) =1

ωL′

√3vdc0

(−vUa0 + vLa0)

A(5, 9) =1

vdc0(vUa0 − vLa0)

A(5, 11) =1

vdc0(vUa0 + vLa0)

A(6, 2) = −η∗Ub0

2L,A(6, 4) = −η∗Lb0

A(6, 5) =2ω√3vdc0

(vUb0 + vLb0)

A(6, 6) =1

−R− 1

Kpf − 2ωL√3

(vUb0 + vLb0)

A(6, 7) =1

ωL′

√3vdc0

(vUb0 − vLb0)

A(6, 8) =1

Kp −ωL′

(−vUb0 + vLb0)

A(6, 10) =1

vdc0(vUb0 − vLb0)

A(6, 12) =1

vdc0(vUb0 + vLb0)

A(7, 1) = −η∗Ua0

2L′, A(7, 3) =

η∗La0

A(7, 5) =1

Kpf +2ωL√

(vLa0 − vUa0)

A(7, 6) =1

2ωL√3vdc0

(vLa0 − vUa0)

A(7, 7) =1

−R′ − 1

Kp +ωL′

(vLa0 + vUa0)

A(7, 8) = − ω√3vdc0

(vLa0 + vUa0)

A(7, 9) =1

vdc0(vLa0 + vUa0)

A(7, 11) =1

vdc0(−vLa0 + vUa0)

A(8, 2) = −η∗Ub0

2L′, A(8, 4) =

η∗Lb0

A(8, 5) = − 1

2ωL√3vdc0

(vLb0 − vUb0)

A(8, 6) =1

Kpf − 2ωL√3

(vLb0 − vUb0)

A(8, 7) =ω√3vdc0

(vLb0 + vUb0)

A(8, 8) =1

−R′ − 1

Kp −ωL′

(vLb0 + vUb0)

A(8, 10) =1

vdc0(vLb0 + vUb0)

A(8, 12) =1

vdc0(−vLb0 + vUb0)

A(9, 7) = −1, A(9, 9) = − ω√3, A(9, 10) = − 2ω√

A(10, 8) = −1, A(10, 9) =2ω√3, A(10, 10) =

ω√3

A(11, 5) = −1, A(11, 11) =2ω√3, A(11, 12) =

4ω√3

A(12, 6) = −1, A(12, 11) = − 4ω√3, A(12, 12) = − 2ω√

Elements of B(t) matrix:

B(1, 1) = −NKp cos ρ

2+ idiffa0

B(1, 2) =NKp sin ρ

2+ idiffa0

B(1, 3) = −NKpf cos ξ

2+ idiffa0

B(1, 4) =NKpf sin ξ

2+ idiffa0

B(2, 1) = − NKp

2+ idiffb0

ρ− 2π

B(2, 2) =NKp

2+ idiffb0

ρ− 2π

B(2, 3) = −NKpf

2+ idiffb0

ξ − 4π

B(2, 4) =NKpf

2+ idiffb0

ξ − 4π

B(3, 3) = −NKpf cos ξ

− ia0

2+ idiffa0

B(3, 4) =NKpf sin ξ

− ia0

2+ idiffa0

B(4, 1) =NKp

− ib0

2+ idiffb0

ρ− 2π

B(4, 2) = − NKp

− ib0

2+ idiffb0

ρ− 2π

B(4, 3) = −NKpf

− ib0

2+ idiffb0

ξ − 4π

B(4, 4) =NKpf

− ib0

2+ idiffb0

ξ − 4π

B(5, 1) =Kp cos ρ

2Lvdc0(vUa0 − vLa0)

B(5, 2) =Kp sin ρ

2Lvdc0(−vUa0 + vLa0)

B(5, 3) =Kpf cos ξ

2Lvdc0(vUa0 + vLa0)

B(5, 4) =Kpf sin ξ

2Lvdc0(−vUa0 − vLa0)

B(6, 1) =Kp

2Lvdc0(vUb0 − vLb0) cos

ρ− 2π

B(6, 2) =Kp

2Lvdc0(−vUb0 + vLb0) sin

ρ− 2π

B(6, 3) =Kpf

2Lvdc0(vUb0 + vLb0) cos

ξ − 4π

B(6, 4) =Kpf

2Lvdc0(−vUb0 − vLb0) sin

ξ − 4π

B(7, 1) =Kp cos ρ

2L′vdc0(vLa0 + vUa0)

B(7, 2) =Kp sin ρ

2L′vdc0(−vLa0 − vUa0)

B(7, 3) =Kpf cos ξ

2L′vdc0(−vLa0 + vUa0)

B(7, 4) =Kpf sin ξ

2L′vdc0(vLa0 − vUa0)

B(8, 1) =Kp

2L′vdc0(vLb0 + vUb0) cos

ρ− 2π

B(8, 2) =Kp

2L′vdc0(−vLb0 − vUb0) sin

ρ− 2π

B(8, 3) =Kpf

2L′vdc0(−vLb0 + vUb0) cos

ξ − 4π

B(8, 4) =Kpf

2L′vdc0(vLb0 − vUb0) sin

ξ − 4π

B(9, 1) = cos ρ,B(9, 2) = − sin ρ

B(10, 1) = cos

ρ− 2π

, B(10, 2) = − sin

ρ− 2π

B(11, 3) = cos ξ, B(11, 4) = − sin ξ

B(12, 3) = cos

ξ − 4π

, B(12, 4) = − sin

ξ − 4π

Elements of Γ(t) matrix:

Γ(1, 1) = − N

2+ idiffa0

Γ(1, 3) =Ne∗a0Cv2dc0

2+ idiffa0

Γ(1, 4) =Nη∗Ua0

Γ(2, 2) = − N

2+ idiffb0

Γ(2, 3) =Ne∗b0Cv2dc0

2+ idiffb0

Γ(2, 4) =Nη∗Ub0

Γ(3, 1) =N

− ia0

2+ idiffa0

Γ(3, 3) = −Ne∗a0Cv2dc0

− ia0

2+ idiffa0

Γ(3, 4) =Nη∗La0

Γ(4, 2) =N

− ib0

2+ idiffb0

Γ(4, 3) = − Ne∗b0Cv2dc0

− ib0

2+ idiffb0

Γ(4, 4) =Nη∗Lb0

Γ(5, 1) =1

2Lvdc0(vUa0 − vLa0)

Γ(5, 3) =1

1 +e∗a0v2dc0

(−vUa0 + vLa0)

Γ(5, 4) = − R

Γ(6, 2) =1

2Lvdc0(vUb0 − vLb0)

Γ(6, 3) =1

1 +e∗b0v2dc0

(−vUb0 + vLb0)

Γ(6, 4) = − R

Γ(7, 1) =1

−1 +1

2vdc0(vUa0 + vLa0)

Γ(7, 3) =e∗a0

2L′v2dc0(−vLa0 − vUa0)

Γ(8, 2) =1

−1 +1

2vdc0(vUb0 + vLb0)

Γ(8, 3) =e∗b0

2L′v2dc0(−vLb0 − vUb0)

Next, we will analyze the stability properties of this lin-earized model.

B. Linear Time Periodic Framework & Analysis

From the expressions ofA(t), B(t), andΓ(t), it is clearthat the elements of these matrices are functions of instan-taneous values of different variables (e.g. voltages, currents,and controller state variables). From the physical properties ofMMC, it is known that some of these have only fundamentalfrequency component while others have dc, fundamental,second harmonic components, or a combination thereof. Letthe nominal operating condition for phase-a variables be:

e∗a0 = Em0 cos(ωt+ θea0)e∗fa0 = Emf0 cos (2ωt+ θefa0)

ia0 = Im0 cos(ωt+ θia0)idiffa0 = idc0

3+ I2f0 cos(2ωt+ ϕ0)

vUa0 = VU00 + VU10 cos(ωt+ θU10) + VU20 cos(2ωt+ θU20)vLa0 = VL00 + VL10 cos(ωt+ θL10) + VL20 cos(2ωt+ θL20)

Similar expressions can be written for the phaseb variables.Substituting these variables in the expressions ofA(t), B(t),

and Γ(t), it is clear that the linear model is time-varying.Stability analysis of such a linear time-varying (LTV) system ischallenging since modal analysis methods, such as eigenvalueanalysis used for linear time-invariant (LTI) systems, cannotbe applied.

It can be seen that the state-space model (37) is primarilyTA-periodic with a time-period ofTA = ω

2π. Hence, we

can categorize this as a Linear Time-Periodic (LTP) system.Therefore, the stability properties of LTP systems [50], [51]can be applied in this case.

For the sake of completeness, one can recall the followingdefinitions and fundamental concepts relating the LTP sys-tems [50], [51]:

1) Fundamentals of Linear Time-Periodic (LTP) Systems:

• Periodic Function:A function is primarilyTA-periodicor periodic with primary periodTA if TA ∈ ℜ+∗ is thesmallest number such that

f(t) = f(t+ TA), ∀t (38)

whereℜ+∗ denotes the set of real strictly positive num-bers.

• Linear Time-Periodic (LTP) system:A Linear Time-Periodic (LTP) system is characterized by the followingrepresentation

∆x(t) = A(t)∆x(t) +B(t)∆u(t) + Γ(t)∆z(t),

∆x(t) ∈ ℜn,∆u(t) ∈ ℜm,∆z(t) ∈ ℜp

Here, the elements of matricesA(t), B(t), andΓ(t) areknown, real-valued, piecewise continuous, primarilyTA-periodic functions defined onℜ+.

• Fundamental Matrix:Any nonsingular solution of thehomogeneous differential system∆x (t) = A(t)∆x (t)is known as its Fundamental Matrix.

• State Transition Matrix (STM):There exists a uniquefundamental matrixΦ(t, t0) of the homogeneous systemmentioned above, such thatΦ(t0, t0) = I. This matrix iscalled the State Transition Matrix (STM) of the system.Without any loss of generality, we shall assumet0 = 0.

• Monodromy Matrix:The STM computed after one time-period TA, i.e. Φ(TA, 0) is known as the MonodromyMatrix.

• Poincare multipliers:The eigenvalues of the Monodromymatrix are called the Poincare multipliers.

For the stability analysis of the LTP systems, it is essentialto calculate the state-transition matrix (STM)Φ(·, 0). In mostcases, the STM can not be computed in closed form. For-tunately, the computation of a monodromy matrixΦ(TA, 0),which is essentially the STM after one time-periodTA, sufficesfor stability analysis.

The monodromy matrixΦ(TA, 0) can be calculated bynumerically solving the equation∆x(t) = A(t)∆x(t) with n

different initial conditionsxr(0) = ǫr, r = 1, 2, . . . n, whereǫr = [δri] is the rth column of the identity matrixI [50].Let xr(TA), r = 1, 2, ..., n, be then independent solutionsobtained for each initial condition. Then the monodromy

matrix is given by:

Φ (TA, 0) =[

x1(TA) x2(TA) . . . xn(TA)]

As mentioned above, the eigenvalues of the monodromy matrixare called the Poincare multipliers. For an asymptoticallystable system, the Poincare multipliers lie within the unitcircle.

Therefore, the steps for conducting stability analysis of theproposed MMC model are summarized as follows:

• Step I: Derive the nonlinear state-space AVM of MMCas described in equation (33).

• Step II: Develop the Linear Time-Periodic (LTP) modelof the MMC by linearizing the nonlinear AVM obtainedfrom Step I around an operating condition(x0, u0, z0) asshown in equaion (37).

• Step III: Calculate the monodromy matrixΦ(TA, 0) bynumerical integration of∆x(t) = A(t)∆x(t) with n

different initial conditionsxr(0) = ǫr, r = 1, 2, . . . n,whereǫr = [δri] is therth column of the identity matrixI [50]. This has been described before in more details.

• Step IV:Compute the Poincare multipliers, i.e. the eigen-values of the monodromy matrixΦ(TA, 0).

• Step V:If the largest magnitude of the Poincare multiplieris less than unity, then the MMC is considered asymp-totically stable.

VII. C ASE STUDY

A. Test System

The test system consists of a401-level, 1000-MW, ±320-kV MMC, Fig. 1, with the following parameters:

Rated MV A = 1059MVA,N = 400, C = 10 mF,

L = 50 mH,R = 0.5236 Ω,Lt = 60 mH,Rt = 0.5236 Ω.

B. Benchmarking the AVM with SM Insertion Dynamics

The proposed nonlinear AVM of the MMC was describedin Section V and the state-space model was presented ina compact form in subsection V-D. These differential equa-tions was implemented using basic math blocks from MAT-LAB/Simulink library and its response was benchmarkedagainst a detailed model built in PSCAD/EMTDC. The de-tailed model considers individual SMs and the voltage balanc-ing control for all 400 SMs per arm. The MMC simulationmodel was developed in PSCAD/EMTDC in the followingway:

• The converter leg was modeled by two dependent voltagesources, two resistances and two inductances. The termi-nal voltage is connected to the ac grid through a leakagereactance;

• Each dependent voltage source in the arm is controlled bya hosted code that reads the arm currents and computesthe voltage of each submodule, generating the controlsignal for the dependent voltage sources and formingthe final arm voltage. The hosted code has a500 kHzsample frequency, allowing a high accuracy in the digitalcomputation;

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1 3.12i diffa[kA]

2PSCADMATLAB

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1 3.12

i a[kA]

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1 3.12

v 1a[kV]

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1 3.12

v 2a[kV]

time [s]2.96 2.98 3 3.02 3.04 3.06 3.08 3.1 3.12

Fig. 4. Comparison of the response between the detailed model in PSCAD/EMTDC against a nonlinear averaged model in MATLAB/Simulink. The circulatingcurrent controller is enabled att = 3.0 s.

2.96 2.97 2.98 2.99 3 3.01 3.02 3.03 3.04

v 1a[kV]

2.994 2.995 2.996 2.997 2.998 2.999 3 3.001 3.002 3.003

v 1a[kV]

300 PSCADMATLAB

time [s]2.96 2.97 2.98 2.99 3 3.01 3.02 3.03 3.04

v 2a[kV]

time [s]2.994 2.995 2.996 2.997 2.998 2.999 3 3.001 3.002 3.003

v 2a[kV]

Fig. 5. Zoomed view of the response from the detailed model inPSCAD/EMTDC, compared against a nonlinear averaged model in MATLAB/Simulink.The circulating current controller is enabled att = 3.0 s.

• The simulation model results were compared to othersimulation models where each submodule was created byconventional electronic components from PSCAD library,for M = 4, 6, 8, 10, and20, showing negligible error;

• Then, the simulation model was augmented forM = 400,in order to simulate the system mentioned in the presentpaper.

Such a benchmarking analysis gives us the confidence inthe accuracy of the AVM and any stability analysis that isperformed by linearizing such models.

The converter was assumed to control the real power (P )and the reactive power (Q) at the PCC, Fig. 1. The PIcompensator parameters for the output current control schemeand the circulating current control scheme were calculated

using the following equations:

Kp =L′

τ, KI =

τ, Kpf =

τf, KIf =

For benchmarking studies,1τ= 500 s−1 and 1

τf= 2000 s−1

was chosen.

1) Enabling Circulating Current Control:The response ofdifferent variables obtained from the nonlinear averaged model(black trace) is compared against the detailed model (red trace)in Fig. 4. During t = 0 − 3.0 s, the circulating currentcontroller is not activated. It can be seen that the converteris controlling the real powerP at 1000 MW at unity powerfactor. The circulating currentidiffa has a dc component anda double frequency component. As expected, the phase current

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

time[s]0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

−1000

time[s]

Fig. 6. Time periodic nature of the elements of the A matrix inthe linear model. Trajectories of the elements of the first, third, fifth, and seventh rows areshown.

ia does not have any harmonics. The voltages across the upperand lower arm SMs that are turned ON,v1a and v2a, havesecond and third harmonics. It can be seen that, under steadystate, the dotted traces and the solid traces overlap almostindistinguishably.

At t = 3.0 s, the circulating current controller is enabled.The controller suppresses the second harmonic componentand only the dc component remains inidiffa. Following thetransients att = 3.0 s, harmonic content in voltagesv1aand v2a are significantly reduced. It can also be observedthat the circulating current control scheme is not completelydecoupled from the output current control scheme. Transientsin real powerP , reactive powerQ, and phase currentia canbe observed. It can be seen that the response of the averagedmodel very closely matches that of the detailed model.

The zoomed view ofv1a andv2a are shown in Fig. 5. Onecan appreciate the close match between these models from thisfigure. It can also be observed that the detailed model insertsthe SMs in a discrete manner as opposed to the averaged modelthat treats the insertion of SMs as a continuous function.

C. Stability Analysis

1) Eigen Analysis of Phasor-based AVM:As describedin sections III-B and IV, stability of phasor-based AVMcan be analyzed through eigenvalue analysis. Considering1

τ= 500 s−1, eigenvalues were obtained from the state

matrix shown in equation (7). The eigenvalues are:λ =[−500.0000− 36.9599 − 382.2588 − 154.7011]. Please notethat the eigenvalues are real, stable, and independent of thecirculating current controller gains.

2) Stability Analysis of AVM Considering SM InsertionDynamics: The nonlinear averaged model was linearized forstability analysis. As mentioned in Section VI, the first step inthis process is to obtain a steady-state operating condition. Inthis case, the steady-state condition(x0, u0, z0) was obtainedby numerical integration of (33) underP = 1000 MW andQ = 0 MVAr with the circulating current controller enabled.

The values of the variables needed for computing matrixA ofthe linearized model are:e∗a0 = 276.60cos(ωt+ 0.14)e∗fa0 = 19.35 cos (2ωt− 4.63)ia0 = 2.45cos(ωt)idiffa0 = 0.5250vUa0 = 634.37+50.01cos(ωt−1.70)+16.95cos(2ωt−4.52)vLa0 = 634.37+50.01cos(ωt+1.44)+16.95cos(2ωt−4.52)where the angles are expressed in radians, voltages are in kV,and currents are in kA. The values of the phase-b quantitiescan be determined considering appropriate phase difference.

Under this nominal condition, the periodic nature of theelements of the first, third, fifth, and seventh row of matrixA

are illustrated in Fig. 6. The monodromy matrix was computedthrough numerical integration as mentioned in Section VI-B.The corresponding Poincare multipliers are:

µ= [0.8717 + 0.0000j 0.8427 + 0.0000j . . .

0.8380± 0.0655j 0.1437 + 0.0000j . . .

0.0066± 0.1032j 0.0130 + 0.0000j . . .

−0.0007± 0.0006j − 0.0000± 0.0000j]T

Therefore, the system is stable as the multipliers lie withinthe unit circle.

Next, the value of the PI controller parameters were changedby varying the value of1

τf. The locus of the maximum value

of the magnitude of the Poincare multipliers with respect to1

τfis shown in Fig. 7. The operating condition corresponds to

P = 1000 MW, Q = 0 MVAr. It can be seen that an increasein the value from2000 s−1 to 4400 s−1 moves the maximumvalue of the magnitude of the Poincare multipliers towards theperimeter of the unit circle.

The value of 1τf

was set to5000 s−1 and the simulation wasrun for the conditionP = 1000 MW, Q = 0 MVAr. At t = 3.0s, the circulating current controller is enabled. Figure 8 showsthe instability under such a scenario. The averaged model and

1/τf500 1000 1500 2000 2500 3000 3500 4000

NominalValue

Fig. 7. Locus of the maximum value of the magnitude of the Poincare multipliers with respect to 1

τf. The operating condition corresponds toP = 1000

MW, Q = 0 MVAr.

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5

i diffa[kA]

4.3 4.32 4.34 4.36 4.38 4.4 4.42 4.44 4.46 4.48 4.5

i diffa[kA]

time [s]3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5

PSCADMATLAB

time [s]4.3 4.32 4.34 4.36 4.38 4.4 4.42 4.44 4.46 4.48 4.5

1000.5

1001.5ZoomedView

ZoomedView

Fig. 8. Comparison of response between the detailed model inPSCAD/EMTDC against a nonlinear averaged model in MATLAB/Simulink. Instability isobserved in the response when1

τfis set to5000 s−1. The operating condition corresponds toP = 1000 MW, Q = 0 MVAr. At t = 3.0 s, the circulating

current controller is enabled.

the detailed model both demonstrate this phenomena.

When the value of1τf

is reduced from2000 s−1, it can beseen from Fig. 7 that the maximum value of|µ| reduces andbecomes minimum just above1500 s−1. As 1

τfis reduced

further, the system approaches instability. Figure 9 showsthe response obtained from the averaged model. Instabilityis observed in the response when1

τfis set to150 s−1, with

the operating point corresponding toP = 1000 MW, Q = 0MVAr. At t = 3.0 s, the circulating current controller isenabled.

VIII. C ONCLUSION

A framework for stability analysis of the MMC is presentedbased on the Linear Time-Periodic nature of the proposedmodel. It has been shown that the proposed framework canindicate the zones of instability for certain gains of thecirculating current controller, which can not be captured bythe traditional eigenvalue analysis of phasor-based AVM.

IX. A CKNOWLEDGEMENT

The authors would like to thank the anonymous reviewersfor their constructive comments, which were very helpful inimproving the quality of the paper.

6.5 7 7.5 8 8.5 9 9.5 10

i diffa[kA]

8.5 8.6 8.7 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5

i diffa[kA]

time [s]6.5 7 7.5 8 8.5 9 9.5 10

v 1a[kV]

time [s]8.5 8.6 8.7 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5

v 1a[kV]

ZoomedView

Fig. 9. Response obtained from the averaged model. Instability is observed in the response when1τf

is set to150 s−1. The operating condition correspondsto P = 1000 MW, Q = 0 MVAr. At t = 3.0 s, the circulating current controller is enabled.

REFERENCES

[1] A. Lesnicar and R. Marquardt, “An Innovative Modular MultilevelConverter Topology Suitable for a Wide Power Range“IEEE PTCConference 2003, vol. 3, Jun. 2003.

[2] Perez, M.A.; Bernet, S.; Rodriguez, J.; Kouro, S.; “Editorial SpecialIssue on Modular Multilevel Converters,” IEEE Transactions on PowerElectronics, vol.30, no.1, pp. 1-3, Jan. 2015.

[3] Xiaofeng Yang; Jianghong Li; Xiaopeng Wang; Wenbao Fan;Zheng,T.Q., “Circulating Current Model of Modular Multilevel Con-verter,” 2011 Asia-Pacific Power and Energy Engineering Conference(APPEEC), vol., no., pp.1-6, 25-28 March 2011.

[4] Kolb, J.; Kammerer, F.; Braun, M., “Straight forward vector control ofthe Modular Multilevel Converter for feeding three-phase machines overtheir complete frequency range,” IECON 2011 - 37th Annual Conferenceon IEEE Industrial Electronics Society , vol., no., pp.1596-1601, 7-10Nov. 2011.

[5] Kolb, J.; Kammerer, F.; Gommeringer, M.; Braun, M., “Cascaded Con-trol System of the Modular Multilevel Converter for FeedingVariable-Speed Drives,” IEEE Transactions on Power Electronics, vol.30, no.1,pp.349-357, Jan. 2015.

[6] Munch, P.; Gorges, D.; Izak, M.; Liu, Steven, “Integrated current control,energy control and energy balancing of Modular Multilevel Converters,”IECON 2010 - 36th Annual Conference on IEEE Industrial ElectronicsSociety , vol., no., pp.150-155, 7-10 Nov. 2010.

[7] Munch, P.; Liu, Steven; Ebner, G., “Multivariable current control ofModular Multilevel Converters with disturbance rejectionand harmonicscompensation,” 2010 IEEE International Conference on Control Appli-cations (CCA), vol., no., pp.196-201, 8-10 Sept. 2010.

[8] Siemaszko, D.; Antonopoulos, A.; Ilves, K.; Vasiladiotis, M.; Angquist,Lennart; Nee, H.-P., “Evaluation of control and modulationmethods formodular multilevel converters,” 2010 International PowerElectronicsConference (IPEC), vol., no., pp.746-753, 21-24 June 2010.

[9] Lachichi, A.; Harnefors, L., “Comparative analysis of control strategiesfor modular multilevel converters,” 2011 IEEE Ninth InternationalConference on Power Electronics and Drive Systems (PEDS), vol., no.,pp.538-542, 5-8 Dec. 2011.

[10] Ilves, Kalle; Antonopoulos, A.; Harnefors, Lennart; Norrga, Staffan;Nee, H.-P., “Circulating current control in modular multilevel converterswith fundamental switching frequency,” 2012 7th International PowerElectronics and Motion Control Conference (IPEMC), vol.1,no., pp.249-256, 2-5 June 2012.

[11] Solas, E.; Abad, G.; Barrena, J.A.; Carcar, A.; Aurtenetxea, S., “Mod-elling, simulation and control of Modular Multilevel Converter,” 201014th International Power Electronics and Motion Control Conference(EPE/PEMC), vol., no., pp.T2 90-96, 6-8 Sept. 2010.

[12] Perez, M.A.; Rodriguez, J., “Generalized modeling andsimulation of amodular multilevel converter,” 2011 IEEE International Symposium onIndustrial Electronics (ISIE), vol., no., pp.1863-1868, 27-30 June 2011.

[13] Perez, M.A.; Lizana F, R.; Rodriguez, J., “Decoupled current controlof modular multilevel converter for HVDC applications,” 2012 IEEEInternational Symposium on Industrial Electronics (ISIE), vol., no.,pp.1979-1984, 28-31 May 2012.

[14] Zhao Yan; Hu Xue-hao; Tang Guang-fu; He Zhi-yuan, “A study onMMC model and its current control strategies,” 2010 2nd IEEEInter-national Symposium on Power Electronics for Distributed GenerationSystems (PEDG), vol., no., pp.259-264, 16-18 June 2010.

[15] Engel, S.P.; De Doncker, R.W., “Control of the Modular Multi-LevelConverter for minimized cell capacitance,” Proceedings ofthe 2011-14th European Conference on Power Electronics and Applications (EPE2011), vol., no., pp.1-10, Aug. 30 2011-Sept. 1 2011.

[16] Rohner, S.; Bernet, S.; Hiller, M.; Sommer, R., “Modelling, simulationand analysis of a Modular Multilevel Converter for medium voltageapplications,” 2010 IEEE International Conference on Industrial Tech-nology (ICIT), vol., no., pp.775-782, 14-17 March 2010.

[17] Rohner, S.; Weber, J.; Bernet, S., “Continuous model ofModularMultilevel Converter with experimental verification,” 2011 IEEE EnergyConversion Congress and Exposition (ECCE), vol., no., pp.4021-4028,17-22 Sept. 2011.

[18] U.N. Gnanarathna, A. M. Gole and R. P. Jayasinghe, “Efficient Modelingof Modular Multilevel HVDC Converters (MMC) on ElectromagneticTransient Simulation Programs,”IEEE Transactions on Power Delivery,vol. 26, no. 1, pp. 316-324, Jan. 2011.

[19] F. Ajaei and R. Iravani, “Enhanced Equivalent Model of the ModularMultilevel Converter,” IEEE Transactions on Power Delivery, 2014.

[20] O. Venjakob, S. Kubera, R. Hibberts-Caswell, P.A. Forsyth, T.L.Maguire, “Setup and Performance of the Real-Time Simulatorused forHardware-in-Loop-Tests of a VSC-Based HVDC scheme for OffshoreApplications”, Proceedings of the International Conference on PowerSystems Transients (IPST’13) in Vancouver, Canada July 18-20, 2013.

[21] H. Saad, S. Dennetiere, J. Mahseredjian, P. Delarue, X.Guillaud, J.Peralta, and S. Nguefeu, “Modular multilevel converter models forelectromagnetic transient,” IEEE Trans. on Power Delivery, vol. 29, no.3, pp. 1481 - 1489, July 2013.

[22] Ludois, D.C.; Venkataramanan, G.; “Simplified Terminal BehavioralModel for a Modular Multilevel Converter,” IEEE Transactions on PowerElectronics, vol.29, no.4, pp. 1622-1631, April 2014.

[23] Vasiladiotis, M.; Cherix, N.; Rufer, A.; “Accurate Capacitor VoltageRipple Estimation and Current Control Considerations for Grid- Con-nected Modular Multilevel Converters,” IEEE Transactionson PowerElectronics, vol.29, no.9, pp. 4568-4579, Sep. 2014.

[24] Riar, B.S.; Geyer, T.; Madawala, U.K.; “Model Predictive Direct CurrentControl of Modular Multilevel Converters: Modeling, Analysis, and

Experimental Evaluation,” IEEE Transactions on Power Electronics,vol.30, no.1, pp. 431-439, Jan. 2015.

[25] Riar, B.S.; Madawala, U.K.; “Decoupled Control of Modular MultilevelConverters Using Voltage Correcting Modules,” Power Electronics,IEEE Transactions on, vol.30, no.2, pp. 690-698, Feb. 2015.

[26] S. Norrga, L. Angquist, K. Ilves, L. Harnefors, and H.-P. Nee,“Frequency-domain modeling of modular multilevel converters,” in Proc.38th Annu. Conf. IEEE Ind. Electron. Soc., Oct. 2528, 2012, pp.49674972.

[27] Teeuwsen, S.P., “Simplified dynamic model of a voltage-sourced con-verter with modular multilevel converter design,” IEEE/PES PowerSystems Conference and Exposition, 2009. PSCE ’09. vol., no., pp.1-6,15-18 March 2009.

[28] Working Group on Modeling and Analysis of System Transients UsingDigital Programs, “Dynamic Averaged and Simplified Models for MMC-Based HVDC Transmission Systems,”IEEE Transactions on PowerDelivery, vol. 28, no. 3, pp. 1723-1730, July. 2013.

[29] Cigre Working group B4.57 guide for MMC modelling (document 604).[30] A. Antonopoulos, L. Angquist and H.P. Nee,“On Dynamicsand Voltage

Control of the Modular Multilevel Converter,”in Proc. Eur. Conf. PowerElectron. Appl., Barcelona, Spain, 2009.

[31] A. Antonopoulos, L. Angquist, L. Harnefors, K. Ilves and H.P.Nee,“Global Asymptotic Stability of Modular Multilevel Converter,”IEEE Transactions on Industrial Electronics, vol. 61, no. 2, pp. 603-612, Feb. 2014.

[32] Harnefors, L.; Antonopoulos, A.; Norrga, S.; Angquist, L.; Nee, H.-P.,“Dynamic Analysis of Modular Multilevel Converters,”IEEE Transac-tions on Industrial Electronics, vol.60, no.7, pp.2526-2537, July 2013.

[33] Harnefors, L.; Norrga, S.; Antonopoulos, A.; Nee, H.-P., “Dynamicmodeling of modular multilevel converters,” Proceedings of the 2011-14th European Conference on Power Electronics and Applications (EPE2011), vol., no., pp.1-10, Aug. 30 2011-Sept. 1 2011.

[34] Hagiwara, M.; Maeda, R.; Akagi, H., “Control and Analysis of the Mod-ular Multilevel Cascade Converter Based on Double-Star Chopper-Cells(MMCC-DSCC),” IEEE Transactions on Power Electronics, vol.26,no.6, pp.1649-1658, June 2011.

[35] M. Hagiwara and H. Akagi, “PWM control and experiment ofmodularmultilevel converter,” in Proc. IEEE Power Electron. Specialists Conf.,Tokyo, Japan, Jun. 2008, pp. 154161.

[36] M. Saeedifard and R. Iravani, Dynamic performance of a modularmultilevel back-to-back HVDC system, IEEE Trans. Power Del., vol.25, no. 4, pp. 29032912, Oct. 2010.

[37] E. Solas, G. Abad, J. A. Barrena, A. Carcar, and S. Aurtenetxea,“Modulation of modular multilevel converter for HVDC application,” inProc. 14th Int. Power Electron. Mot. Control Conf., Mondragon, Spain,Sep. 2010, pp. 8489.

[38] Meshram, P.M.; Borghate, V.B.; “A Simplified Nearest Level Control(NLC) Voltage Balancing Method for Modular Multilevel Converter(MMC),” IEEE Transactions on Power Electronics, vol.30, no.1, pp.450-462, Jan. 2015.

[39] Siemaszko, D.; “Fast Sorting Method for Balancing Capacitor Voltagesin Modular Multilevel Converters,” IEEE Transactions on Power Elec-tronics, vol.30, no.1, pp. 463-470, Jan. 2015.

[40] Fujin Deng; Zhe Chen; “A Control Method for Voltage Balancing inModular Multilevel Converters,” IEEE Transactions on Power Electron-ics, vol.29, no.1, pp. 66-76, Jan. 2014.

[41] Shengfang Fan; Kai Zhang; Jian Xiong; Yaosuo Xue; “An ImprovedControl System for Modular Multilevel Converters with New Modu-lation Strategy and Voltage Balancing Control,” IEEE Transactions onPower Electronics, vol.30, no.1, pp. 358-371, Jan. 2015.

[42] Fujin Deng; Zhe Chen; “Elimination of DC-Link Current Ripple forModular Multilevel Converters With Capacitor Voltage-Balancing Pulse-Shifted Carrier PWM,” IEEE Transactions on Power Electronics, vol.30,no.1, pp. 284-296, Jan. 2015.

[43] Q. Tu, Z. Xu, and L. Xu “Reduced Switching-Frequency Modulationand Circulating Current Suppression for Modular Multilevel Converters,”IEEE Transactions on Power Delivery, vol. 23, no. 3, pp. 2009-2017,July. 2011.

[44] Darus, R.; Pou, J.; Konstantinou, G.; Ceballos, S.; Agelidis, V.G.,“Circulating current control and evaluation of carrier dispositions inmodular multilevel converters,” 2013 IEEE ECCE Asia Downunder(ECCE Asia), vol., no., pp.332-338, 3-6 June 2013.

[45] Shaohua Li; Xiuli Wang; Zhiqing Yao; Tai Li; Zhong Peng;“CirculatingCurrent Suppressing Strategy for MMC-HVDC Based on NonidealProportional Resonant Controllers Under Unbalanced Grid Conditions,”IEEE Transactions on Power Electronics, vol.30, no.1, pp. 387-397, Jan.2015.

[46] Pou, J.; Ceballos, S.; Konstantinou, G.; Agelidis, V.G.; Picas, R.;Zaragoza, J., “Circulating Current Injection Methods Based on In-stantaneous Information for the Modular Multilevel Converter,” IEEETransactions on Industrial Electronics, vol.62, no.2, pp.777-788, Feb.2015.

[47] Liqun He; Kai Zhang; Jian Xiong; Shengfang Fan; “A Repetitive ControlScheme for Harmonic Suppression of Circulating Current in ModularMultilevel Converters,” IEEE Transactions on Power Electronics, vol.30,no.1, pp. 471-481, Jan. 2015.

[48] Ming Zhang; Long Huang; Wenxi Yao; Zhengyu Lu; “Circulating Har-monic Current Elimination of a CPS-PWM-Based Modular MultilevelConverter With a Plug-In Repetitive Controller,” IEEE Transactions onPower Electronics, vol.29, no.4, pp. 2083-2097, April 2014.

[49] A. Yazdani, and R. Iravani “Voltage-sourced converters in power systems: modeling, control, and applications,” IEEE press, John Wiley and Sons,1st edition, 2010.

[50] M. Montagnier, R.J. Spiteri, and J. Angeles,“The Control of LinearTime-Periodic Systems with Floquet-Lyapunov Theory,”InternationalJournal of Control, vol. 77, no. 5, pp. 472-490, Jan. 2004.

[51] Wilson J. Rugh “Linear System Theory,” Prentice-Hall information andsystems sciences series, Prentice Hall,1993.

Nilanjan Ray Chaudhuri (S’08-M’09) NilanjanRay Chaudhuri received his Ph.D. degree from Im-perial College London, London, UK in 2011 inPower Systems. From 2005-2007, he worked in Gen-eral Electric (GE) John F. Welch Technology Center.He came back to GE and worked in GE Global Re-search Center, NY, USA as a Lead Engineer during2011-2014. Presently, he is an Assistant Professorwith North Dakota State University (NDSU), Fargo,ND, USA. He is a member of theIEEE, IEEE PES,andSigma Xi. Dr. Ray Chaudhuri is the lead author

of the bookMulti-terminal Direct Current Grids: Modeling, Analysis,andControl (Wiley/IEEE Press, 2014), and an Associate Editor of the IEEETRANSACTIONS ON POWER DELIVERY.

Rafael Oliveira received his BSc in Control andAutomation Engineering and MSc in Electrical En-gineering in 2003 and 2005, respectively, from Pon-tifica Univeridade Catolica do Rio Grande do Sul,Brazil. He is currently a PhD candidate in theElectrical Engineering Program at Ryerson Univer-sity, Canada. His main area of research is powerconverters for power systems applications, includingsimulation, control and modelling.

Amirnaser Yazdani (SM’09) received the Ph.D.degree in electrical engineering from the Universityof Toronto, Toronto, ON, Canada, in 2005. Hewas an Assistant Professor with the University ofWestern Ontario (UWO), London, ON. He is cur-rently an Associate Professor at Ryerson University,Toronto. His research interests include modeling andcontrol of electronic power converters, renewableelectric power systems, distributed generation andstorage, and microgrids. He is a co-author of thebook Voltage-Sourced Converters in Power Systems

(IEEE-Wiley Press, 2010), and an Associate Editor of the IEEE TRANSAC-TIONS ON SUSTAINABLE ENERGY.

TPEL2480845_AcceptedVersion.pdf - Sites at Penn State

Documents