BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 62, No. 2, 2014 DOI: 10.2478/bpasts-2014-0038

Selected aspects of Modular Multilevel Converter operation

M. ZYGMANOWSKI1, B. GRZESIK1, M. FULCZYK2, and R. N ALEPA2*

1 Department of Power Electronics, Electrical Drives and Robotics, Silesian University of Technology, 2B Krzywoustego St., 44-100 Gliwice, Poland 2 ABB Sp. z o.o., Corporate Research Centre, 13A Starowislna St., 31-038 Krakow, Poland

Abstract. The operation of the Modular Multilevel Converter (MMC) is the main subject of this paper. Selected operation aspects are discussed on the basis of the averaged model, with a special focus on power section parameters and control. The direct modulation method has been chosen for the control of the MMC.

Key words: Modular Multilevel Converter MMC, power electronic converter modeling.

Nomenclature

In this paper the following symbols are adopted for description of the modular multilevel converter operation and its parameters:

Carm - arm capacitance,

CSM - submodule capacitance,

EV - amplitude of fundamental component of

output voltage eV, eV - output voltage related to converter medi-

um point of dc-link, EV,max - maximum value of the output voltage, fm - fundamental frequency,

iarmu, iarmL - upper (U) and lower (L) arm currents, icc - circulating current, being a part of the dif-

ference current idiff, iCU, iCL - submodule capacitor current, idc - Dc-link current,

Idc,ac - alternating component of the dc-link cur-

1dc,AV - average value of the dc-link current,

idiff - difference current, idiff = (iarmU + iarmL)/2,

idiff,ac - alternating component of the difference

current idiff (the rms value of which is

1diff,ac,RMs),

Idiff,dc - Dc component of the difference current

idiff,

1V - amplitude of the output current iV funda-

mental component, iV - output current,

Larm - arm inductance,

ma - modulation index,

n - number of submodules in converter arm,

nu, nL

rpU, rpL rr, rD

snu, snl TpwM

VdisU, VdisL Vpu, VpL

vto, vdo Vcufc, VcLk vcu, VCL

^armU varmL varm, max

Aidiff

modulating signals in upper and lower arms,

arm resistance,

equivalent series resistance of a single

submodule capacitor,

average on-state resistance of a single

switch,

non-ideal parameter arm resistances, on-state resistance of single transistor and diode,

carrier signals for upper and lower arm, switching period, TPWM = 1/fpwM, rated value of dc-link voltage, Dc-link voltage,

discrepancy arm voltage components, non-ideal parameter arm voltages, on-state threshold voltage of a single transistor and diode,

k submodule capacitor voltage in the upper (U) and lower arm (L), arm capacitor voltage - the sum of n capacitor voltages in the upper (U) and lower arm (L),

upper and lower arm voltages, maximum value of arm voltage, phase angle between carriers signals SNU and Snl,

peak-to-peak value of voltage ripple in a sum of voltages across capacitors in a single arm,

maximum difference current ripple, output phase-shift angle, phase angle of modulating signals in phase A,

angular frequency, w = 2nfm.

•"e-mail: marcin.zygmanowski@polsl.pl

1. Introduction

The modular multilevel converter (MMC) is one of the newest and most promising power electronic converters intended for very and ultra high voltage and power applications, particularly for HVDC systems [1,2]. The converter converts electrical energy from ac voltage to dc voltage or from dc voltage to ac voltage. Its specific topology, without the need for the dc-link capacitor bank, and full modularity, make it suitable also for medium voltage applications, e.g. medium voltage electric drives [3, 4]. Such a wide range of application fields for a single topology makes it challenging to understand all important circuit selection and operation aspects. The challenge does not disappear even if advanced numerical circuit simulations are deployed - simulation is not a substitute for sufficient understanding of this advanced circuit.

The topology relies on the well-known half-bridge (or full-bridge) switching cell with a capacitor inside - called in this paper "SubModule", SM. SMs can be connected in series to satisfy application voltage requirements. SMs connected in this way, under a certain control scheme, turn into a technology enabler for high voltage power electronic circuits.

One of the most popular topologies with SMs can be seen in Fig. 1. In this circuit the capacitors of SMs are subject to voltages (vCUXY, vCLXY). Here, X = A, B or C, Y = 1, 2, ...n, where n is the number of SMs in each converter arm. Two arms - an upper (U), and a lower arm (L) are connected in series and constitute one converter phase.

have led to a mathematical, linear but time-varying model of the MMC. Using this model, the MMC operation is explained on the basis of a system configuration example. Furthermore, the submodule capacitor voltage ripple is discussed together with currents circulating inside the MMC. These currents are named circulating currents, and are parts of a difference current which flows through both arms of each converter phase.

The second MMC model presented in the paper is a detailed simulation model, which allows verifying the correctness of results obtained for the averaged model.

The MMC control method has an impact on the converter operation and its components selection, not only the capacitor CSM [5], but also the arm resistance Rarm (as a resultant design value rather than a design parameter) and the inductance Larm [5, 6]. For the purpose of this article, the direct modulation method [7, 8] is assumed to be used in both MMC models. In this method transistors in submodules are switched according to sinusoidal modulating signals, without any feedback from capacitor voltages.

2. Averaged model of MMC

The averaged model of the three-phase MMC (Fig. 2) is based on the approach that output arm voltages varmUX and varmLX are continuous rather than switched in the PWM manner. An analysis of the converter operation can be performed for each phase separately, and phase denotations (X = A, B or C) for all values are omitted for the sake of simplicity. This is possible due to the assumption that the dc-link circuit is ideal and modelled as a dc voltage source.

Fig. 1. Three-phase modular multilevel converter, connected to the grid, with n submodules SM in each converter arm

In this paper the MMC operation is explained by means of state equations. The equations have been derived on the basis of an averaged model for each converter phase [2] - they

Fig. 2. Three-phase modular multilevel converter averaged model with arm voltage controlled sources as per Eqs. (1)-(3)

Bull. Pol. Ac.: Tech. 62(2) 2014

Converter arm voltages are

VarmU = nUVcU,

VarmL = nLVCL,

where nU and nL are the normalized modulating signals raging from 0 (corresponding to the state when all arm submodules are turned off) to 1 (when all submodules in the converter arm are turned on). Upper and lower arm voltages (vCU, vCL) are sums of adequate n-capacitors voltages, (2).

VCU = ^2 VCUk, VCL = ^2 VCLfc. (2)

k=1 k=1 In the converter averaged model all submodules in one arm can be replaced by controlled voltage sources according to (1).

This replacement alone is not sufficient because the arm voltages varmU and varmL are built up of the capacitor voltages vCUk and vCLk. Thus, it is necessary for each converter arm to include a modulated capacitance, which can operate according to (3), as presented in Fig. 2.

_ Carm dt'cu . _ Carm dt'ÇL

^armU — ^armL — lf ? (jJ

nU dt nL dt

where Cajm is the capacitance of the series connection of all n arm submodule capacitances, C^m = CSM/n; iarmU and iarmL are upper and lower arm currents respectively. Both Eqs. (3) are state equations, describing the dynamics of the converter. With the modulated capacitances the MMC can be simplified to the form presented in Fig. 2.

In accordance with the Kirchhoff voltage law, two equations can be written:

Kic _ „ . , T d

2 — -ikarm^armU

l^d (4)

— ^Aii'iii 'iu'iiil. I LarmTT i' 'iil'm[. ) I i'armL 1 V• 2 dt

Summing up both Eqs. (4) leads to a third state equation as

follows:

arm ^ ( ' ;il'm I ,) I i'armU I * V -

rMiff —

varmU + varmL

«v , .

«armU — — + idiff

1- armL =--^ + Miff, (6)

where ¿v is the converter output current as in Fig. 2.

Defining the ¿diff reduces the number of the state variables down to 3 per each converter phase. The Eqs. (3), (5) and (6) can be written as

_ ^'',1111 «U nL

d idiff Larm 2Larm 2Larm

dt Vcu = nu Carm 0 0

VCL "L Carm 0 0

idiff Vrfc 2Larm

vcu + nuiv 2Carm

VCL nLiy 2Carm

Equation (7) is linear and time-variant because the state matrix has four coefficients nu and nL, which both are time-varying. To find out the exact form of the modulation signals nu and nL, Eq. (4) are subtracted from each other to give the output voltage ev (8)

varmL varmU

Larm div

i . ^arm "»v

2 2 "iv 9 dt~' (8)

here according to (6) ¿armU - ¿armL = ¿v. Assuming that voltage drops across Rarm and L^m are smaller than VarmL and varmU, the output voltage can be simplified to

varmL varmU 2 '

Assuming that ev = EV sin(wt) and varmL + varmU = Vdc, the arm voltages, generated by submodules, have to fulfil the following equations:

î'armU = "Tp - Ey sin (ut - 1p) , î'armL = sin (ut - 1p) .

Due to the fact that the arm voltages are positive (varmL > 0, varmU > 0), the maximum value of the output voltage is Ev,max = Vdc/2. This assumption also allows determining the maximum arm voltage.

varmU,max — varmL,max

— + -iiV.max

Vdc. (11)

From (11) one can see that, due to the maximum value of the modulating signals equal to one, nUjmax = nL,max = 1, the sum of capacitor voltages in each arm should be set at

vcu,AV = VCL,AV = Vdc. (12)

Modulating signals nu and nL are determined from (10) according to (13)

VarmU 1 - ma sin (wt -

dt diff Larm 2Larm 2Larm

where ¿diff can be calculated as half of a sum of arm currents, ¿diff = (¿armU + ¿armL)/2. The ¿diff is the arm current component, which is the same in both converter arms and can be written as:

t'armL _ 1 + Wa sin (ujt - j))

where ma is the modulation index and its value is between 0 and 1. Equations (13) represent the direct modulation method used for controlling the MMC. In the converter analysis presented in this paper the modulating signals nu and nL are without the third harmonic injection.

Difference current. Assuming that the sums of capacitor voltages are constant at a given Vdc, vCU = vCL = Vdc, the modulating signals, nu and nL, are as per (13), and from Kirchhoff voltage law VarmU + VarmL = Vdc, the derivative of the difference current (5) is equal to (14).

idiff =

"Miff-

The solution to this equation is non-periodic - as is expected for the converter operating in the steady state. It means that the assumption VarmU + Vaj-mL = Vdc is no longer valid,

and when modulating signals nu and nL are given as (13), the sums of capacitor voltages deviate from their rated values. In other words - the capacitor voltage ripple always exists in the MMC, as per (3). The issue of the capacitor voltage ripple is investigated later in this paper.

Capacitor voltage ripple together with other non-ideal parameters of the MMC (represented by voltage sources vpU and vpL) causes difference current oscillations. In the best case the difference current is constant and equals to 1/3 of the dc current idc. Achieving this best case state is the aim of the MMC control method known as open-loop control [7, 9] and other control methods [10, 11]. Reducing the difference current to merely the dc component minimizes MMC power losses [12]. On the other hand, allowing the difference current oscillation simplifies the MMC control. A compromise based on the understanding of the MMC nature is highly recommended at this point. The difference current idiff, can be divided into two components. The first one equals 1/3idc and the second one is the circulating current as in (15):

idc , . '¿diff = — + ¿cc =

^dc,AV . ¿dc,ac . .

H-----h le

¿CU — nU ¿armü,

¿CL — nL ¿armL.

in Matlab/Simulink. Similar results have been obtained in a model developed in GeckoCIRCUITS.

3. Selection of the MMC parameters

In this section the operation of the converter is presented for selected parameters collated in Table 1. With such parameters and the assumption that output currents are sinusoidal as in (17), the selected waveforms for phase A are shown in Fig. 3.

2 ' 2 "CC = ^diff,dc+ ^diff,ac- (15)

The circulating current icc flows between MMC phases and is composed of negative and positive sequence components (excluding fundamental and zero-sequence components). Typically the second harmonic is a dominant part of the circulating current. The idc is composed of a dc component 1dc,AV, delivering the active power from/to the dc circuit, and of an alternating component idc,ac, which consists of zero-sequence components (harmonics multiples of the third harmonic e.g. sixth).

The sum of non-constant components of the difference current, idc,ac/3 + icc = idiff,ac is an ac component which generates extra power losses in the MMC - without transmission of the effective power between the ac side and dc circuit. The ac component of the difference current, idiff,ac, is minimized (ideally down to zero) by the open loop control method [3, 7] - as opposed to the MMC operating under the direct modulation method governed by (13), where the idiff,ac can be large.

Capacitor voltage ripple. When the direct modulation method is applied, the capacitor voltage ripple can be evaluated by using the averaged MMC model described by Eqs. (1)-(13). The peak-to-peak value of the capacitor voltage, AVC is treated as a design criterion in this paper.

Capacitor voltage ripple, AVC, depends on the capacitor currents iCU and iCL, which depend on the modulation signals nU and nL and the arm currents iarmU and iarmL according to:

iVA — /V sin (^t — 0 — , ¿VB — /y sin (^t — 0 — ^ — 2n/3).

Table 1

Parameters of the MMC simulation model

Name of the parameter

Symbol

DC circuit voltage

output current amplitude

angular frequency

314.15 rad/s

phase angle

modulating signal phase angle

modulation index

arm inductance

750 ^H

arm resistance

arm capacitance

Arm currents iarmU and iarmL in the MMC operating under the direct modulation control consist of circulating current icc [13] which, together with other components of the arm currents, influences capacitor voltage ripples. Moreover these voltage ripples directly influence circulating currents. The results presented in the next section are obtained by using simulations of the averaged model of single-phase converter

Fig. 3. Waveforms of the MMC averaged model operating with the direct modulation control method - parameters are given in Table 1

Figure 3 presents the waveforms of modulating signals nU and nL (13); arm currents, iarmU, iarmL, with their components as per (6) and (15); capacitor currents, iCU, iCL (as in (16)); and the peak-to-peak value, AVC, of the capacitor voltage sums, vCU, vCL. The rms value of the difference current ac

Bull. Pol. Ac.: Tech. 62(2) 2014

component, and the peak-to-peak value of the capacitor voltage are considered in this paper as a criterion for selecting the arm capacitances Carm - in the presence of changing arm resistance Rarm. The arm inductance for the purpose of this paper has been set at Larm = 0.75 mH.

The first analysis concerns the influence of the Rarm on the rms value of the ac component of the difference current, 1diff,ac,RMS, see Fig. 4. It can be seen in this figure that a larger Rarm gives a smaller /diff,ac,RMS. Exemplary characteristics of the current /diff,ac,RMS are presented for capacitance Carm raging from 0.05 mF to 1 mF - equivalent to CSM = nx0.05 mF and nx1 mF in n-submodule converter. Such capacitances in the three-phase converter correspond to the energy-to-power ratio raging from 25 J/kW to 500 J/kW, where the converter rated power is calculated from P = (3/2)maEvJv = 150 kW, which is obtained for parameters from Table 1 and the amplitude of the phase voltage Ev = Vdc/2. In practice a lower energy-to-power ratio decreases the cost of the converter. However, for the generality of the analysis presented in this paper higher energy-to-power ratios are considered.

Fig. 6 four distinguished points (A-D) are selected, for which harmonic spectra are given in Fig. 7.

Fig. 5. Harmonic spectra (rms values in amperes) of the difference current alternating component idiff,ac at points A-D (as per Fig. 4)

Fig. 6. The rms values of the difference current AC component, 1diff,ac,RMs as a function of Carm parameterized with Rarm

Fig. 4. The rms values of the difference current AC component, 1diff,ac,RMs as a function of Rarm parameterized with Carm

As can be seen in Fig. 4 the influence of the arm capacitances on the difference current ac component /diff,ac,RMS is nonmonotonic. It is discussed later in this paper as an effect of resonances between Larm and modulated Carm - mostly at second and fourth harmonics [5, 13]. During the selection of the MMC components these resonances should be taken into consideration.

Harmonic spectra of the idiff,ac for the MMC with different capacitances, which correspond to points A-D in Fig. 4, are presented in Fig. 5. In the case of point A, corresponding to parameters given in Table 1, the even harmonics exist - with a dominant second harmonic. At point B, both second and fourth harmonics are explicit.

Selection of arm capacitance Carm demands a similar analysis to the idiff,ac assessment in the function of arm resistance Rarm. In Fig. 6 the results of such an analysis are presented for four different values of Rarm.

Resonances in circulating current occur for the fixed value of Larm = 0.75 mH, with different values of Carm [5, 13]. This confirms that the mathematical description of the converter operating under direct modulation is quite complex. In

Fig. 7. Harmonic content (rms values in amperes) of the difference current AC component idiff,ac at points B and E-G (as in Fig. 6)

At point E resonance at the second harmonic exists, at point F resonance occurs at fourth harmonic. Point C lies between points E and F. At point G another resonance occurs having distributed spectrum among second and sixth harmonics, with smaller fourth and eighth. In a three-phase converter the sixth harmonic in the difference current is a zero-sequence component and flows through the dc circuit (observed in idc) while the other harmonics (positive and negative-sequence) close through the MMC phases (as circulating current icc).

Resonances occurring in the difference current are much more evident in Fig. 8, which presents the same characteristics as in Fig. 6 but for smaller arm capacitances Carm. Arm capacitances smaller than 0.1 mF (Carm < 0.1 mF) allow avoiding resonances at small order harmonics (2nd, 4th and 6th). It should be noted that such arm capacitance Carm results from the resonance with arm inductance and for other values of Larm the results can be different. In summary, it can be concluded that arm capacitance Carm should be determined together with other converter parameters. Possible resonances have to be taken into consideration.

In Fig. 10 arm capacitor voltage ripple AVC is shown as a function of arm capacitance Carm. From this figure it can be seen that resonances still influence voltage ripple AVC.

Fig. 10. Arm capacitor voltage ripple AVC as a function of Carm

The influence of arm resistance Rarm on arm capacitor voltage ripple AVC is presented in Fig. 11. This ripple is nearly constant for a wide range of Rarm.

Fig. 8. The rms values of the difference current AC component, 1diff,ac,RMs as a function of Carm parameterized with Rarm with indicated resonances of fourth, sixth and eighth harmonics

Another criterion for arm capacitance selection is arm capacitor voltage ripple, AVC. Here, in the converter averaged model the AVC is defined as a peak-to-peak value of ripples observed in sums of capacitor voltages vCU or vCL. For a single submodule capacitor the voltage ripple is n times smaller than AVC. This is because of the assumption that all capacitor voltages inside a particular converter arm are equal due to the balancing operation of a submodule selector [1]. Analyses of the capacitor voltage ripples have been performed for the same MMC parameters as in the previous case, except the phase angle which is changing or set to 80o or 0o. The first value (^ = 80o ) has been dictated by the largest voltage ripple AVC = 623 V (Fig. 9 - point H). Point A in Fig. 9 corresponds to the ripple AVC shown in Fig. 3, where AVC = 406 V. This means that the largest voltage ripples, for the converter operating with parameters given in Table 1, are below 13%Vdc = 650 V. In many publications, [8, 14], authors report even larger values of voltage ripples e.g. 20% Vdc.

Fig. 11. Arm capacitor voltage ripple AVC as a function of Rarm

Similarly to the analysis of the circulating current, the arm capacitor voltage ripple AVC analysis shows that the parameter selection for modular multilevel converter is a multidimensional optimization problem. The presented results show the general behaviour of the converter with partially selected parameters. During the converter design process some aspects presented in this section can be useful. It should be noted that for different converter parameters the achieved results can be slightly different.

4. Influence of non-ideal parameters on MMC operation

The transistor and diode on-state resistances, capacitor series equivalent resistances, threshold on-state voltages in switching devices and the dead time effect in switching signals influence the MMC operation. In Fig. 2 the non-ideal parameters are represented by equivalent resistances rpU, rpL and voltage sources vpU and vpL. All those parameters are not constant and can be presented as a function of the modulating signals nu and nL and arm current direction. The rpU can be written as (18).

Fig. 9. Arm capacitor voltage ripple AVC as a function of phase-shift

angle ^

n [nu (rD + Resr) + (1 - nu) rT] for i n [nu (rT + Resr) + (1 - nu) rD] for i

'arm > 0, 'arm < 0,

Bull. Pol. Ac.: Tech. 62(2) 2014

where n is the number of submodules in a single arm, rT and rD are on-state resistances of transistor and diode respectively, Resr is the equivalent series resistance of a single submodule capacitor CSM . The transistor and diode resistances can be comparable; they can therefore be equal to on-state resistance ron, rT « rD = ron, which in return simplifies Eqs. (18) to (19).

rpU = n (nuResr + ron) . (19)

A similar simplification can be made for rpL as in (20).

rpL = n (nLResr + ron) . (20)

Both non-ideal parameter resistances rpUrpL are time varying due to nu and nL, which are given in (13).

Voltage sources vpU and vpL depend on transistor and diode threshold on-state voltages VT0, VF0 according to (21) and (22).

n [nU Vf0 + (1 - nU ) Vt0 ] for

-n [nUVt0 + (1 - nU) Vf0] for

n [nL Vf0 + (1 - nL ) Vt0 ] for

-n [nLVt0 + (1 - nL) Vf0] for

¿armU > ° ¿armU < °

¿armL > 0, ¿armL < 0.

The dependence of voltage sources vpU and vpL on the direction of the arm current complicates the analysis of the influence of such voltage sources on converter operation. Here in this paper the analysis of non-ideal parameters is limited to presenting their major influence. This can be done by using one of the state equations (5), which is now rewritten to (23) -including the non-ideal parameters. The non-ideal parameters change both equations (3) too - this is due to the existence of series equivalent resistance in all capacitors. However this impact is not discussed in the paper.

TT^diff = dt

^arm - rpU - rpL . VarmU + VarmL

¿diff

Vdc VpU + VpL

TT^diff = dt

2Rarm + rpU + rpL .

VdisU + VdisL VpU + VpL

It can be seen from (25) that the effect of the non-ideal parameters on the difference current is similar to the effect of arm resistance and discrepancy arm voltages. This effect cannot be neglected if voltages vpU and vpL are comparable with voltages vdisU and vdisL or resistances rpU, rpL are comparable with Rarm.

5. Simulation results

In this section the operation of a detailed simulation model of 5-submodule MMC is presented. The results of MMC operation are shown for the converter with ideal parameters.

Among many PWM strategies, which can be used in the MMC, the triangle carrier based PWM technique is typically applicable. This strategy can differ in the phase shift between triangular carrier signals SN, which are compared with modulating signals nU, nL. In the modulation strategy of five-submodule MMC, five carrier signals are used in each converter arm. In the paper the two following PWM strategies are analyzed:

a) the method with zero degree upper and lower arm carrier phase shift, 7 = 0°, which is very similar to the phase disposition PWM method (PD PWM) - the method commonly used in other multilevel converters. Here, all 5 upper arm carriers, SNU and all 5 lower arm carriers, SNL, are in phase as per Fig. 12,

b) the method with zero degree upper and 180° lower arm carrier phase-shift, 7 = 180° which is similar to the method known as phase opposition disposition PWM (POD PWM). Here, all 5 upper arm carriers SNU are out of phase with those in the lower arm, SNL, as per Fig. 14.

where arm voltages can now be expressed by (24)

Vdc Vdc . / . I\ . ^arrnU = —--ma — Sin {vt ~ 1/J) + VdisU,

Vdc . Vdc . / . I \ rarmL = — + ma — Sin (CJi - -0) + Vdish

and voltages vdisu and vdisL are the discrepancy arm voltage components - resultant from the existence of ripples in capacitor voltages. The exact analytical form of discrepancy voltages is not the subject of this paper, but can be determined only after determining the analytical form of the difference current. After substituting (24) for (23) the derivative of the difference current can be expressed as

Fig. 12. Construction of the MMC arm voltages with the direct modulation method and the phase disposition PWM strategy with a carrier phase-shift angle 7 = 0o

Both modulation strategies are based on the direct modulation method with modulating signals nU, nL given as per (13).

In Fig. 12 arm voltages varmU and varmL during three switching periods Tpwm are presented. In this case the carrier phase angles are the same (y = 0o), therefore switchings occurs in both arm voltage waveforms in the same direction - the first one with a falling slope (when all carriers increase), and the second one with a rising slope (when all carriers decrease). In such a case, the sum of both arm voltages varmU + varmL is not equal to Vdc but is either Vdc + Vdc /n or Vdc — Vdc/n (assuming a constant sum of capacitor voltages vCU = vCL = Vdc). The difference between the arm voltages sum, varmU + varmL, and dc circuit voltage, Vdc, equals to +/ — Vdc/n and occurs across both arm inductors. Voltages across the arm inductances (vLarmU and vLarmL) are the same, thus VLarmU = VLarmL = +/ — Vdc/(2n) - Fig. 12.

Arm voltages lead to the occurrence of relatively high difference current ripples (Fig. 12). The highest ripples exist in periods when modulating signals are in the middle of each carrier signal (e.g. first switching period TPWM in Fig. 12). In such a case the maximum difference current ripple Aidiff max is

riers. Signals from the PWM modulator together with voltage and current waveforms are shown in Figs. 14 and 15.

Aidiff,max —

1 Vdc TPWM

The maximum difference current ripple Aidiff? max is relatively high, for instance when Vdc = 5000 V, n = 5 and Larm = 750 ^H, Tpwm = 200 ^s (1/Tpwm = 5 kHz), (26) returns Aidiffmax = 66.7 A. This particular current ripple compared to the output current amplitude 1V = 40 A is relatively high.

Figure 13 presents waveforms in the MMC operating with a zero degree phase-shift in all carriers SNU and SNL, Y = 0o. These results are obtained from the three-phase simulation model of the converter, which has been developed in Mat-lab/Simulink. Converter parameters are listed in Table 1 with the switching frequency equal to fPWM = 5 kHz. In Fig. 13 it can be seen that in the difference current (and in both arm currents) relatively high ripples occur, the maximum value of which is equal to the value from (26). The output voltage eV, (as in Fig. 2) consists of 11 levels maximally. Such a number comes from the achievable values of the voltage, varmL — varmU (9), where six levels are present (n+1 per single n-submodule MMC arm) in each arm voltage.

Maximum ripples in iarmU, iarmL are equal to Aidiffmax = 66.7 A. Depending on the phase-shift angle between carrier signals in different phases of the converter, the ripples in the dc-circuit current idc can be even higher than Aidiff max. The capacitor voltage ripple of a single capacitor (peak-to-peak value) is approximately equal to 90 V, thus the ripple in arm capacitor voltage is AVC = 450 V. This value is higher than the value obtained for the averaged model of the converter (where AVC = 406 V). Such a difference can be explained by a higher ripple in the difference current and the operation of the capacitor voltage balancing circuit.

The occurrence of such high ripples in the arm currents practically disqualifies this modulation method (Y = 0o ). Therefore, the second PWM strategy, where carrier signals are with opposite phase angles, is chosen. In this method the upper arm carriers are y = 180o away from the lower arm car-

Fig. 13. Current and voltage waveforms of the MMC operating with the direct modulation method and zero degree carrier phase shift,

Y = 0o

Fig. 14. Construction of the MMC arm voltages under the direct modulation method and the PWM strategy where the upper arm carriers are opposite to carriers from the lower arm, y = 180°

Bull. Pol. Ac.: Tech. 62(2) 2014

Fig. 15. Current and voltage waveforms of the MMC with the direct modulation method and the PWM strategy, where the upper arm carriers are opposite to carriers from the lower arm, 7 = 180°

From Fig. 14 one can see that due to the complementary shape of the varmU and varmL their sum is equal to Vdc (varmU + varmL = Vdc). This leads to reduction of the ripple in the difference current because both arm inductor voltages equal zero vLarmU = vLarmL = 0. In practice the capacitor voltage ripple and other non-ideal parameters of the converter can cause small ripples during the switching period TPWM. However, they are relatively small.

In the example from Fig. 15 voltage and current waveforms are obtained from the three-phase converter detailed model which has been developed in Matlab/Simulink. The parameters of such a converter are the same as in Table 1 and the phase-shift angle of carrier signal is 7 = 180o. The differences existing between waveforms in Figs. 13 and 15 are as follows:

• the output voltage eV has six voltage levels rather than 11 in the first case,

• no significantly high harmonic ripple exists in arm currents

^armU ^armLü

• the difference current consists of harmonic components which are predicted in the averaged model. The current waveform is similar to the one in Fig. 3. The harmonic spectrum of this current is depicted in Fig. 5 (point A),

• ripples in the arm capacitor voltages, when summed up, are equal to AVC = 5 x 80 V = 400 V - this is similar to the value presented in Figs. 9-11 at point A.

The simulation results presented in Fig. 15 prove the correctness of the analyses based on the averaged mathematical model of the MMC.

6. Simulation model performance

This paragraph is devoted to a comparison of six simulation models of the modular multilevel converter, the results of which have been presented in this paper. These models have been developed in Matlab/Simulink and GeckoCIRCUITS and they are as follows:

• averaged model of a single-phase MMC (1AM or 1AG),

• averaged model of a three-phase MMC (3AM or 3AG),

• detailed model of a three-phase MMC (3DM or 3DG),

where letters M and G correspond to Matlab/Simulink and GeckoCIRCUITS.

All the results obtained from the models implemented in GeckoCIRCUITS are similar to those obtained by using Matlab/Simulink models. However, the computational time of all models' simulations differ between these two simulators. To compare all the developed simulation models, the computational time of simulation is chosen as a criterion.

All the simulation models, the parameters of which are the same as in Table 1, are executed with the simulation step equal to 1 ^s (1e-6) for the simulation time ranging from 0 to 1.5 seconds. Models in Matlab/Simulink are executed using a discrete fixed-step solver, guaranteeing the smallest computational time among other available solvers. GeckoCIRCUITS models are executed with a Backward Euler solver, which guarantees accurate simulation results.

Each model has been simulated for 20 iterations to find a mean value of computational time. This method offers more reliable results, which can be next used to compare different models - Fig. 16. Computational times of all the models have been obtained by using personal computer with a quad-core microprocessor Intel® Core™ i7-3630QM @ 2.4 GHz, whose number of floating point computations per second is 1724 MFLOPS for a single core and 6650 MFLOPS for all four cores.

The simulation computational time strongly depends on the number of variables observed in the models. Therefore, in all the models only a limited number of variables are observed and they are the same as presented in Figs. 3, 13 and 15. For three-phase models only one converter phase is measured. For increasing the computational time, the simulation results are transferred to memory rather than the computer display unit.

It should be noted that the results presented in Fig. 16 cannot be directly transferred to other computers. However, there is a close relationship between computer performance, measured in MFLOPS, and simulation computational time.

In Fig. 16 it can be clearly seen that averaged models are computed much faster than detailed ones. For Matlab/Simulink three-phase models the averaged one is comput-

ed more than 14.6 times faster and for GeckoCIRCUIT this relationship is even higher and equals to 21.6. The simulation computational time can be further decreased by simulating merely a single-phase converter, which gives similar results in the case of supplying the converter from an ideal dc voltage source.

— Intel® Core™ Í7-3630QM @ 2.4 GHz 88.28

_ —3.600—--^¡30 — 1.573------3.023—■

1AM 3AM 3DM 1AG 3AG 3DG

Fig. 16. Simulation computational time for averaged (A) and detailed (D) models of a single-phase (1) or three-phase (3) MMC implemented in Matlab/Simulink (M) or GeckoCIRCUITS (G)

The analyzed models developed in Matlab/Simulink or GeckoCIRCUITS cannot be easily compared with each other because the computational time strongly relies on the number of variables being observed in the simulation. And this relationship is different in Matlab/Simulink, which allows observing more variables without increasing computational time as drastically as in the case of GeckoCIRCUITS. The authors are aware of the fact that all the presented models can be computed by using many different simulators or computational methods, which allow achieving even faster computational times. It is, however, true that averaged models are computed much faster than detailed ones.

A shorter computational time allows performing different analyses on converter operation. They include a power loss analysis (without switching losses due to a lack of switchings in the model) [12]; the second important analysis can be used for converter parameter selection [5].

On the other hand, some effects can be modelled only by means of a detailed model with switches. These include dc voltage balancing, dead time effects or the effect of nonideal parameters on converter control. It is up to the user which model is chosen to observe a required feature of the converter.

7. Conclusions

The influence of two main MMC parameters, arm resistance and arm capacitance, on converter operation has been investigated in the paper. This influence is strong, particularly on the circulating current and capacitor voltages ripple. The analysis was confined to 5 kV DC circuit voltage, 40 A of output current amplitude and 0.75 mH of arm inductance with a unity modulation index. The direct modulation strategy was assumed. In the analysis variable parameters are within the range usually quoted in the literature of the subject as representative.

The main influence of the presented parameters is observed as resonances in circulating currents and capacitor

voltages. The arm parameters, resistance, capacitance and also inductance should be carefully selected in order to avoid resonances. This needs further investigation, where circulating currents and capacitor voltage ripple are minimized within the range constrained by a given application.

The presented results have been obtained using two computer models, the averaged and the detailed one. The first one is relatively simple and demands less computational time. This model does not reflect all of the operational details, like switching. However, it allows one to carry out an analysis in a much broader range. The model can be applied for an introductory analysis, the results of which could be used by the detailed model. The second detailed model embraces transistor switching and PWM strategy, which significantly influences MMC behaviours. The price of a more precise picture of the converter operation is the much larger computational time.

Both models can be extended by non-ideal parameters like transistor or diode, threshold voltage and its dynamical resistance, to name just a few.

The models are not dependent on the selected modulation strategy (direct, open loop control).

In conclusion, the presented analysis is recommended for MMC designing.

Acknowledgements. This paper has been prepared under a project financed by ABB Corporate Research Center in Krakow, Poland.

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