1、模块化多电平换流器中电压应用的建模、仿真与分析

Modelling,Simulation and Analysis of a Modular Multilevel Converter for Medium V oltageApplicationsSteffen Rohner and Steffen Bernet Dresden University of Technology D-01069Dresden,Germany Email:steffen.rohner@tu-dresden.de Email:steffen.bernet@tu-dresden.de Marc Hiller and Rainer Sommer Siemens AG,I DT LDD-90443Nuremberg,Germany Email:marc.hiller@ Email:rainer.sommer@Abstract—The Modular Multilevel Converter(M2C)is an emerging multilevel converter for HVDC and FACTS as well as for future AC-fed traction vehicles or medium voltage drives (MVD).A suitable converter model of the Modular Multilevel Converter(M2C)is presented.A7.2kV,7.5MV A converter is taken as an example to analyze characteristic waveforms of the converter.The cause of the circulating currents-a phenomenon unique to this topology-is investigated.Furthermore,particular attention is paid to the current distribution in the converter cells for characteristic load phase angles.I.I NTRODUCTIONMedium voltage(MV)power electronics is a technology of continuously growing importance for a wide range of applications.Cycloconverters,grid or load commutated con-verters applying conventional thyristors are used especially in applications with very high power demands,which can not be met with state of the art self commutated converters at comparable prices.Today the clear majority of medium voltage converter and drive manufacturers offer PWM voltage source converters in a power range of S C=300kVA–30MVA[1],[2].In special applications for energy systems converter ratings up to several 100MVA have been realized on the basis of PWM-VSCs.Re-garding the circuit configuration PWM-VSCs can be classified in Two-Level V oltage Source Converters(2L VSC)and Multi-Level-V oltage Source Converters(ML-VSC).The well-known Two-Level V oltage Source Converter(2L VSC)features a simple structure,a high reliability and power density.The main disadvantages are the difficult converter voltage increase (requiring power semiconductors with increased blocking volt-age or a series connection of devices)and the high switching losses.Thus,the topology is mainly used for low and medium switching frequency traction converters for which reliability and compactness are the most important criteria[3].On the other hand,there are the multilevel converter topolo-gies which generate multiple voltage levels in the quantized line-to-line voltages.The three most popular topologies are the Three-Level Neutral Point Clamped V oltage Source Converter (3L NPC VSC)[4],the Cascaded H-Bridge V oltage Source Converter(CHB VSC)[5]and the Flying Capacitor V oltage Source Converter(FC VSC)[6].The3L NPC VSC is the most widely used multilevel con-verter topology for MVDs.With the use of active switches such as IGCTs and IGBTs,a line-to-line voltage range of 2.3kV–4.16kV withfive voltage levels is achieved without series connection of power semiconductors.Typical applica-tions for this topology are industrial converters for oil and gas, metals,mining,water,and marine[7].The CHB VSC consists of a series connection of identical power cells.Each cell can generate three voltage levels.Thus, the number of voltage levels in the line-to-line voltage depends on the number of power cells used.Applications of this topology are retrofitting,power generation,water/wastewater, oil and gas,and pulp/paper[8].The FC VSC also consists of a series connection of identical commutation cells-the number of voltage levels in the line-to-line voltage depends on the number of commutation cells. One particular challenge is balancing the capacitor voltages, which is accomplished by the choice of the converter switching states.Applications for this topology are traction converters for electrical locomotives,drive test benches,and high speed compressor drives[9].This paper considers the emerging multilevel converter topology:the Modular Multilevel Converter(M2C)[10].The structure of this converter combines elements of both the CHB VSC and the FC VSC:the series connection of identical power cells resembles a CHB VSC and the fact that the capac-itor voltages are determined by the phase currents resembles a FC VSC.This topology is applicable in thefield of HVDC and FACTS[11],[12]as well as for future AC-fed traction vehicles or medium voltage drives(MVD)[13]–[15].Up to the present,there is one commercial application in industry:the new self-commutated voltage source converter with IGBTs HVDC PLUS[16].The modular cell design of this topology features several advantages,such as cost reduction resulting from mass production,distributed locations of energy storage elements,simple protection schemes,simple voltage scaling,and simple realization of redundancy.Furtheramenities are high front end flexibility (e.g.12p,18p,24p diode or active),grid connection via standard transformers (or transformerless),and the possibility of common d.c.bus configurations.Space-vector modulation [17],[18]and pulse width modulation [19]schemes for the M2C have been investigated and first experimental results for two different prototype converters show the practicability of this topology [20]–[22].In order to investigate the competitive ability of the M2C,semiconductor losses and efficiencies of the M2C and the 2L VSC have been calculated and compared [23].In this paper,an appropriate converter model is derived.Simulation results of a 7.2kV ,7.5MV A M2C using the PWM scheme from [19]are presented,showing typical voltage and current waveforms of this converter.A special phenomenon unique to this topology is the so-called circulating currents,which circulate within the converter circuit.Their mathemat-ical definition is given in [24].The cause of their appearance during the converters operation are highlighted for the first time.The converter analysis describes effects of varying phase currents and line-to-line voltages.Furthermore the current distribution of the semiconductors in the converter cells (also referred to as submodules)is shown for four characteristic load phase angles.II.M ODELLING AND P ARAMETER S ELECTIONA simulation model of an M2C circuit configuration con-taining a d.c.connection,the M2C converter,and a three-gwFig.1.Simulation model of the M2Cphase a.c.connection is depicted in Fig.1.The d.c.side (between points A and B)is modelled by two d.c.voltage sources V d /2,two inductors L d /2,and two resistors R d /2.The three-phase a.c.side (at terminals U,V ,W)is model by a series connection of resistor R g ,inductor L g ,and a.c.voltage source v gu (t )(phase U).This equivalent network can either represent a three-phase a.c.grid or an electrical machine.The M2C is marked with a red dashed rectangle.It consists of six arms,each of which contains a series connection of n identical submodules and an inductor L .Each submodule (SM)contains a half-bridge of two active switches and inverse diodes (in this case IGBT modules)and a capacitor C SM .In the simulation model,the submodule switches are considered ideal.The principle voltage generation with the submodules works as follows.Each submodule can be toggled between two different states:the on-state when the upper IGBT is switched on and the lower one is switched off,and the off-state when the lower IGBT is switched on and the upper one is switched off.With these states,the cell terminal voltage v SM (t )(from terminal x to y)can be toggled between v SM (t )=v C (t )(on-state)and v SM (t )=0(off-state).The number of voltage levels in the line-to-line voltages u uv (t ),u vw (t ),u wu (t )depends on the number of submodules per arm n and the modulation scheme.The resistor R represents ohmic losses which occur in real electrical connections and semiconductors.For preparing simulations,values for the electrical elements such as resistors,inductors,and capacitors as well as variable simulation settings are taken from Table I.The number of submodules per arm is n =12,thus,the converter contains 72submodules and 144IGBT modules including inverse diodes.The simulation parameter V LL,rms,1defines the maxi-mum rms-value of the fundamental line-to-line voltages u uv (t ),u vw (t ),u wu (t ),and the parameter I ph,rms,1the rms-value of the fundamental phase currents i u (t ),i v (t ),i w (t ).With param-eter ω,the angular frequency of both the fundamental line-to-line voltage and the fundamental phase current is defined.The load phase angle φis the phase displacement of the phase current fundamental and the phase voltage fundamental in the corresponding converter phase (e.g.phase current i u (t )and phase voltage v u (t )in phase U).It is defined such that phase currents and phase voltages are in-phase at φ=0,inductive at φ=π/2,inversely phased at φ=π,and capacitive at φ=–π/2.With modulation index m ,the amplitude of the line-to-line voltages is controlled.Due to a third harmonic injection (in order to increase the fundamental of the line-to-line voltages at the border of the linear modulation range),the maximum amplitude of the fundamental line-to-line volt-age √2V LL,rms,1is achieved with the maximum modulation in-dex of m =2/√3.The PWM period T PWM is a parameter for the modulation scheme.The smaller the PWM period T PWM ,the better the approximation of the line-to-line voltages and the higher the resulting switching frequency of the IGBTs.Furthermore,the voltages for the applied voltage sources are required for the simulation process.Due to the third harmonicinjection,the d.c.voltage source V d is set toV d =√2V LL,rms,1≈10182V .(1)In order to adjust the a.c.voltage sources at the three-phase a.c.side,the generated phase voltages must be taken into account.With the PWM scheme described in [19],the fundamental of the phase voltage v u (t )isv u (t )=mV LL,rms,1√2sin ωt −16π.(2)The phase voltages cause the phase currents.The fundamentals of the phase currents define the parameter I ph,rms,1and φwithi u (t )=√2I ph,rms,1sin ωt −16π−φ(3)i v (t )=√2I ph,rms,1sin ωt −56π−φ.(4)With these definitions,the a.c.voltage source v gu (t )can be calculated with (2)and (3)byv gu (t )=−R g i u (t )−L gd i u (t )d t+v u (t )(5)=abs (δ)·sin (ωt +arg (δ))(6)withδ=mV LL,rms,1√2exp−j 16π−√2I ph,rms,1(R g +jωL g )exp −j φ+16π(7)where abs (δ)and arg (δ)are the magnitude and the phase angle of the complex value δ,respectively.With these con-siderations,the other a.c.voltage sources are calculated with respect to the phase displacement of the phase currents withv gv (t )=abs (δ)·sin ωt −23π+arg (δ)(8)v gw (t )=abs (δ)·sin ωt +23π+arg (δ).(9)TABLE IS IMULATION S PECIFICATIONSDescriptionSymbol Value D.C.resistor R d 1mΩD.C.inductor L d 1mH Arm resistor R 60mΩArm inductorL 100µH Submodule capacitor C SM 6mF Machine resistor R g 300mΩMachine inductorL g 11.5mH Number of submodules per arm n 12Maximum line-to-line voltage V LL,rms,17.2kV Phase currentI ph,rms,1300A ,600A Angular frequency ω2π50s –1Load phase angle φπ6and 0,π2,π,–π2Modulation indexm 1/√3,2/√3Reciprocal PWM period1/T PWM5400s –1Finally,initial values (at t =0)must be assigned for the simulation.The approach for calculating the initial value for the d.c.inductor is to assume power balance of the d.c.side and the three-phase a.c.side.Thus,the initial value for the inductor L d is set toi d (0)=3mV LL,rms,1I ph,rms,1cos (φ)2V d.(10)The initial currents for the d.c.inductors L are calculated with (3),(4),(10),and at t =0byi z1(t )=13i d (t )+12i u (t )(11)i z2(t )=13i d (t )−12i u (t )(12)i z3(t )=1i d (t )+1i v (t )(13)i z4(t )=13i d (t )−12i v (t )(14)i z5(t )=13i d (t )−12(i u (t )+i v (t ))(15)i z6(t )=13i d (t )+12(i u (t )+i v (t )).(16)The initial currents for the inductors L g at the three-phase a.c.side are assigned with (3)and (4)as well.Due to the modulation scheme,where at every point of time n of the 2n submodules of one phase (an upper and a lower arm)are in the on-state,the initial voltages for the capacitors are set with (1)tov C (0)=V dn.(17)III.S IMULATION OF M ODULAR M ULTILEVEL C ONVERTERSimulations were done with MATLAB/Simulink and the toolbox Plecs with the PWM scheme introduced in [19],which will be described briefly here.The PWM scheme uses reference arm voltages v z k ,ref (t )for the six arm voltages v z k (t ).The switching states of the converter at every point in time are calculated such that the moving average of the corresponding arm voltages follows the reference arm voltages.Simulation results in steady-state are depicted in the fol-lowing figures:Fig.2shows simulated voltage and current waveforms for a phase current I LL,rms,1=600A and the maximum modulation index m =2/√3.A load phase angle of φ=π/6is typical for an electrical machine.In Fig.2(a),the three line-to-line voltages v uv (t ),v vw (t ),v wu (t )are depicted.They are congruent and symmetrically phase-shifted.With n =12submodules per arm,the number of voltage level is 2n +1=25for this PWM scheme.Due to the relatively high number of voltage levels in comparison e.g.to the 3L NPC VSC,the quality of the generated line-to-line voltages is very high.The line-to-line voltages result from the six arm voltages v z k (t )with k =1,2,...,6,which are the sum of all submodule voltages of one arm.Two of these,v z1(t )and v z2(t ),are depicted in Fig.2(b).They are congruent,but phase-shifted by a half fundamental period π/ω=10ms .With n =12,the number of voltage levels in the arm voltages is n +1=13.The other four arm voltages,which are not shown,V o l t a g e (V o l t a g e (C u r r e n t (k A )Phase CurrentsTime (ms)C u r r e n t (k A )Arm Currents Time (ms)C u r r e n t (k A ) D.C. Link CurrentTime (ms)Fig.2.Simulated voltage and current waveforms for V LL,rms,1=7.2kV ,I LL,rms,1=600A ,φ=π6,and m =2√3are congruent to the plotted ones.They are phase-shifted symmetrically to each other (e.g.v z3(t )and v z5(t )by one-third and two-thirds of a fundamental period to v z1(t ),respectively).The arm voltages show the influence of the the capacitor voltage ripples,which are depicted in Fig.2(c).This diagram shows all 72capacitor voltages of the converter,where the twelve capacitor voltages of one converter arm are labeled with v Cak (t ).Due to the implemented balancing algorithm (described in [11],[17],[18]),the capacitor voltages of one converter arm remain within a small voltage band.These waveforms reveal the symmetrical operation of the converter,because the voltage waveforms v Ca k (t )are congruent and symmetrically phase-shifted.In Fig.2(d),phase currents i u (t ),i v (t ),i w (t )are presented.They are symmetrical and fulfill the requirements regarding the parameters I ph,rms,1,ω,and φfrom Table I.The waveforms of the arm currents i z1(t )and i z2(t )in Fig.2(e)are particularly interesting.In this symmetrical operating point,the waveforms are all congruent and phase-shifted in the same way as the arm voltages v z k (t )or thecapacitor voltages v Ca k (t ).The expected waveforms of the arm currents should only include two components described by (11)-(16):one component from the current of the d.c.side (13i d (t ),depicted in Fig.2(f))and another component of the currents from the three-phase a.c.side (±12i u (t ),±12i v (t )or ±12i w (t ),depicted in Fig.2(d)).But Fig.2(e)shows that there is a third component:the so-called circulating current.This additional current component is referred to as circulating because it circulates within the converter circuit and has no effect on the d.c.side or on the three-phase a.c.side.In order to investigate the reasons for these circulating currents,the waveforms and spectra of the resistor and the inductor arm voltages of arm 1are depicted in Fig.3.In Fig.3(a),the sum voltage v R1(t )+v L1(t )of the arm resistor R and the arm inductor L of arm 1are shown.With a maximum value of the sum voltage ofmax (abs {v R1(t )+v L1(t )})<100V(18)the voltage drops across the arm resistor and the arm inductorTime (ms)Resistor and Inductor Arm VoltageV o l t a g e (V )Time (ms)Resistor Arm VoltageV o l t a g e (V )Time (ms)Inductor Arm VoltageV o l t a g e (V )Frequency (Hz)V o l t a g e (V )Spectrum of Resistor and Inductor Arm VoltageFrequency (Hz)V o l t a g e (V )Spectrum of Resistor Arm VoltageFrequency (Hz)V o l t a g e (V )Spectrum of Inductor Arm VoltageFig.3.Waveforms and spectra of resistor and inductor arm voltage of arm 1for V LL,rms,1=7.2kV ,I LL,rms,1=600A ,φ=π6,and m =2√3V o l t a g e (V o l t a g e (C u r r e n t (k A )Phase CurrentsTime (ms)C u r r e n t (k A )Arm Currents Time (ms)C u r r e n t (k A ) D.C. Link CurrentTime (ms)Fig.4.Simulated voltage and current waveforms for V LL,rms,1=7.2kV ,I LL,rms,1=300A ,φ=π6,and m =2√3are very low in comparison to the maximum arm voltage v z1(t )(voltage drops across the submodules)max (abs {v R1(t )+v L1(t )})max (abs {v z1(t )})<1%.(19)The waveform of Fig.3(a)has both low and high fre-quency components,which is confirmed by the spectrum in Fig.3(d).There are low-frequency components up to 500Hz and higher-frequency components at harmonics of the PWM frequency 1/T PWM =5400Hz .The distribution of the sum voltage v R1(t )+v L1(t )over arm resistor R and arm inductor L are shown in Figs.3(b)and (c).The arm resistor solely takes the low-frequency components (Fig.3(e))and the arm inductor takes both the low-frequency components and the higher-frequency sidebands (Fig.3(f)).The waveform of the arm resistor v R1(t )is inherently proportional to that of the arm current i z1(t )(Fig.2(e)).The voltage v L1(t )across the arm inductor L is finally the reason for the dynamic properties ofthe arm current i z1(t )d i z1(t )d t =v L1(t )L(20)and thus,the origin of the circulating currents.It is interesting to note,that circulating currents can even be generated with the use of constant d.c.voltages sources instead of the capacitors in the submodules.Thus,the circulating currents strongly de-pend on the converter modulation and control scheme as well as the operating point and not necessarily of the fluctuating capacitor voltages.Simulated voltage and current waveforms for a lower phase current of I ph,rms,1=300A are given in Fig.4,while other parameters remain the same.The line-to-line voltages v uv (t ),v vw (t ),v wu (t )in Fig.4(a)and the arm voltages v z1(t ),v z2(t )in Fig.4(b)do not show large differences to those of Figs.2(a)and (b).Phase currents i u (t ),i v (t ),i w (t )in Fig.4(d)fulfill the demand of the lower rms-value.Thus,arm currents i z1(t ),i z2(t )in Fig.4(e)are also lower in comparison to those of Fig.2(e).Resultingly,voltage ripples ∆v C of the capacitor voltages v Ca k (t ),depicted in Fig.4(c),decrease from aboutV o l t a g e (k V )Line-to-Line VoltagesArm VoltagesV o l t a g e (V )Capacitor VoltagesC u r r e n t (k A )Phase CurrentsTime (ms)C u r r e n t (k A )Arm CurrentsTime (ms)C u r r e n t (k A ) D.C. Link CurrentTime (ms)Fig.5.Simulated voltage and current waveforms for V LL,rms,1=7.2kV ,I LL,rms,1=300A ,φ=π6,and m =1√3Lower Diode Current (φ = 0)Lower IGBT Current (φ = 0)Upper Diode Current (φ = 0)Upper IGBT Current (φ = 0)C u r r e n t (A )C u r r e n t (A )Time (ms)Time (ms)Time (ms)Time (ms)Fig.6.IGBT and diode current waveforms for V LL,rms,1=7.2kV ,I LL,rms,1=600A ,m =2√3,φ=0(upper row)and φ=π(lower row)54V (Fig.2(c))to 27V .The average of the d.c.link cur-rent i d (t )is half that of Fig.2(f),because the converters real output power decreases in the same way.Hence,the stored energy of the converter in the main energy storage elements –the submodule capacitors C SM –remain balanced within a fundamental period 2π/ω.The effect of halving the modulation index to m =1/√3is shown in Fig.5.Due to the proportional relation of the modulation index m and the fundamental of the line-to-line voltages v uv (t ),v vw (t ),v wu (t )in Fig.5(a),the fundamental rms-value of the line-to-line voltages is about 3.6kV .With this reduced line-to-line voltage,the number of voltage levels is reduced from 25to 13as well.The line-to-line voltages result initially from arm voltages v z k (t ).Two of these,v z1(t )and v z2(t ),are depicted in Fig.5(b).The number of voltage levels of the arm voltages v z k (t )is reduced from 13to 7,but the average value V d /2≈5091V remains the same.The phase currents i u (t ),i v (t ),i w (t )fulfill the specification of I ph,rms,1=300A .Due to the reduced converters real outputpower,the average of d.c.link current i d (t )in Fig.5(f)is only half of that of Fig.4(f)or one-fourth of that of Fig.2(f).Arm currents i z1(t ),i z2(t )in Fig.5(e)are only little changed in the average value and the circulating currents to those of Fig.4(e).In Fig.5(c),voltage ripples ∆v C of the capacitor voltages v Ca k (t )increase from about 27V (Fig.4(c))to 109V ,due to the decrease of the modulation index m from 2/√3to 1/√3.A mathematical method for the calculation of the current distribution of the submodules semiconductors will now be derived.With respect to the current definitions in Fig.1,only two current waveforms are required for the calculation for one submodule:the submodule current i SM (t )and the capacitor current i C (t ).The four current components upper IGBT and diode currents i TU (t )and i DU (t )as well as the lower IGBT and diode currents i TL (t )and i DL (t )are calculated withi TU (t )=abs {i C (t )}−i C (t )2(21)Upper IGBT Current (φ = π/2)C u r r e n t (A )Upper Diode Current (φ = π/2)Lower IGBT Current (φ = π/2)Lower Diode Current (φ = π/2)C u r r e n t (A )Time (ms)Time (ms)Time (ms)Time (ms)Fig.7.IGBT and diode current waveforms for V LL,rms,1=7.2kV ,I LL,rms,1=600A ,m =2√3,φ=π2(upper row)and φ=–π2(lower row)i DU(t)=abs{i C(t)}+i C(t)2(22)i TL(t)=abs{i C(t)−i SM(t)}−{i C(t)−i SM(t)}2(23)i DL(t)=abs{i C(t)−i SM(t)}+{i C(t)−i SM(t)}2(24)where abs{x(t)}is the magnitude at each point in time of the function x(t).Submodule current i SM(t)equates to the corresponding arm current i z k(t)i SM(t)=i z k(t)(25) which must be measured for the modulation algorithm any-way.Capacitor current i C(t)can be calculated with capacitorvoltage v C(t)asi C(t)=C SM d v C(t)d t(26)which must be measured for the modulation algorithm as well.With the use of(25)and(26)with(21)–(24),current distributions in the semiconductors can be calculated without additional effort,in order to approximate e.g.semiconductor losses or junction temperatures online.The current distribution of the upper and lower semicon-ductors inside the submodules is depicted in Figs.6and7for four different load phase anglesφ.Due to the symmetrical modulation scheme and the symmetrical operation of the con-verter,the current distribution of all submodules is similar,but not exactly congruent and they are phase-shifted in the other arms.The current distribution of submodule3in arm1is taken as an example.In Fig.6,the current distribution forφ=0 (upper row)andφ=π(lower row)is given,where the three-phase a.c.side operates as a motor and a generator,mutations between the upper IGBT and the lower diode(i TU3(t)↔i DL3(t))as well as commutations between the upper diode and the lower IGBT(i DU3(t)↔i TL3(t))are clearly recognizable.Afirst interesting fact is that the current distribution of the four component parts is rather unequal.For a load phase angle ofφ=0,the upper IGBT current i TU3(t) (Fig.6(a)),the upper diode current i DU3(t)(Fig.6(b)),and the lower diode current i DL3(t)(Fig.6(d))have small average val-ues(I TU3,A V=15A,I DU3,A V=15A,I DL3,A V=5A)and small rms-values(I TU3,eff=34A,I DU3,eff=64A,I DL3,eff=22A). Compared to these values,the lower IGBT current i TL3(t) (Fig.6(c))has a higher average value(I TL3,A V=250A)and a higher rms-value(I TL3,eff=419A).Although the semi-conductor losses depend on the conduction and switching losses,which depend again on the corresponding capacitor voltage v C(t),the simulation results show,that the lower IGBT component is the most stressed component for this load phase angle.The second interesting point is that with the change of the load phase angleφfrom0toπ,the current wave-forms change within one IGBT module(iφ=0TU3(t)≈iφ=πDU3(t),iφ=0 DU3(t)≈iφ=πTU3(t),iφ=0TL3(t)≈iφ=πDL3(t),iφ=0DL3(t)≈iφ=πTL3(t)).Thus,the considerations for the semiconductor losses for a load phase angle ofφ=πcan be done in an analogous way. In this case,the most stressed semiconductor is the lower diode (Fig.6(h)).For load phase anglesφofπ/2and–π/2,the three-phase a.c.side operates inductively and capacitively,respectively. Simulated current distributations are depicted in Fig.7,where the upper row represents the results forφ=π/2and the lower rowφ=–π/2.The currents in the four semiconductor components are more homogenized,as for load phase angles ofφ=0andφ=π,but still are not uniformly distributed. For both load phase anglesφ,the lower semiconductor com-ponents(Figs.7(c),(d),(g),(h))are more stressed than the upper semiconductor components(Figs.7(a),(b),(e),(f)), because their average and rms-values are increased by a factor of two.The same symmetries of the current wave-forms arise from the change of the load phase angleφfromπ/2to–π/2,as decribed for the load phase an-gles0andπ:iφ=π/2TU3(t)≈iφ=–π/2DU3(t),iφ=π/2DU3(t)≈iφ=–π/2TU3(t),iφ=π/2TL3(t)≈iφ=–π/2DL3(t),iφ=π/2DL3(t)≈iφ=–π/2TL3(t).In summary, for all investigated load phase anglesφ,the current stress of the lower semiconductor is higher than that of the upper one.IV.C ONCLUSIONSThis paper derives a suitable converter model of the Modular Multilevel Converter(M2C).Utilizing a given PWM scheme and the derived converter model simulated voltage and current waveforms were generated and presented.These waveforms show the behaviour of the converter for varied line-to-line voltages and phase currents.A special characteristic of the M2C is the generation of so-called circulating currents,which circulate solely within the converter phases.These currents result from the generated voltages across the arm inductors. Furthermore,these voltages are influenced by the modulation scheme and by the switching states of the converter.The current distribution within a half bridge of a submodule was investigated for four characteristic load phase angles.For this,mathematical equations were derived which are based on quantities,which are otherwise already available,as they are required for the modulation.The simulated current distribution revealed that,for the four different load phase angles,the lower IGBT module of a submodule clearly carries the highest current and with it,experiences the largest conduction and switching losses.R EFERENCES[1]J.Rodr´ıguez,S.Bernet,B.Wu,J.Pontt,and S.Kouro,“MultilevelVoltage-Source-Converter Topologies for Industrial Medium-Voltage Drives,”IEEE Trans.Ind.Electron.,vol.54,no.6,pp.2930–2945,Dec.2007.[2] B.Wu,J.Pontt,J.Rodr´ıguez,S.Bernet,and S.Kouro,“Current-SourceConverter and Cycloconverter Topologies for Industrial Medium-Voltage Drives,”IEEE Trans.Ind.Electron.,vol.55,no.7,pp.2786–2797,July 2008.[3]H.Eckel,M.Bakran,E.Krafft,and A.Nagel,“A new Family ofModular IGBT Converters for Traction Applications,”in Proc.Power Electronics and Applications Conference(EPE),Dresden,Germany, Sept.2005,p.10.[4] A.Nabae,I.Takahashi,and H.Akagi,“A New Neutral-Point-ClampedPWM Inverter,”IEEE Trans.Ind.Applicat.,vol.IA-17,pp.518–523, Sept.1981.[5]J.Rodr´ıguez,J.Pontt,P.Correa,P.Cort´e s,and C.Silva,“A NewModulation Method to Reduce Common-Mode V oltages in Multilevel Inverters,”IEEE Trans.Ind.Electron.,vol.51,pp.834–839,Aug.2004.。