Generalized Modeling and Simulation of a Modular Multilevel Converter Marcelo A. P´erez and Jos´e Rodr´ıguez Electronics Engineering Dept. Universidad T´ecnica Federico Santa Mar´ıa Valpara´ıso, Chile Email:
[email protected],
[email protected]
Abstract—This paper proposes a generalized dynamic model of the modular multilevel converter, which is valid for single-phase or three-phase loads and for DC or single-phase source. The proposed model is simple and easy to simulate, it gives a powerful tool to analyze the entire converter including the circulating currents and simplify the controller design. An analysis of the converter model applied to circulating currents and simulations results are presented.
I. I NTRODUCTION Multilevel converters can reach high power levels and medium voltage operation [1], [2]. Among multilevel converter topologies Neutral Point Clamped (NPC) converters is a mature technology widely used in industry applications [3]. The commonly used NPC configuration has only three levels because the complexity of power topology and its control is greatly increased when the number of levels is higher. For high power systems, the Cascaded H-Bridge (CHB) converter is the main alternative. This converter is composed by several identical cells connected in series to form an output phase voltage [4]. CHB converters exhibit a high degree of modularity and can reach higher power levels and voltages than NPC converters. However, they require isolated DC sources for each H-bridge, which are usually provided by a multipulse transformer, reducing the scalability of the output power because the transformer sets a fixed number of cells in the converter. Recently, a new multilevel converter topology called Modular Multilevel Converter (MMC) has been proposed. It is composed, similarly to the CHB, by identical cells connected in series, but in this case each cell does not deliver active power to the load. Therefore they do not require isolated DC sources being replaced by floating DC capacitors [5]. The transformer-less operation allows to easily scale the output voltage and, therefore, the power delivered to the load. The MMC has been extensively investigated during the last decade, particularly its control and modulation. In the MMC there are three main control objectives: input current, output current and voltage in the floating capacitors. Hence, the simplest approach to control the MMC is to compose the modulation signals by several terms, each one related to one control objective. The same approach has been already proposed in [6] but with a slightly modification: using two
978-1-4244-9312-8/11/$26.00 ©2011 IEEE
terms to control the DC voltage, one related to the average DC voltage and the second one related to the balance among the DC voltages. The control reference can be modulated using carrier based phase shifted PWM [7]. An alternative approach uses a modified PWM modulation to balance the DC voltage capacitors by itself [8]. A modified selective harmonic elimination (SHE) which minimizes the THD of the output voltages generated by the MMC has been proposed in [9]. The DC voltage balance can be done using a logic function based on a combination of voltages and currents which enables and disables the PWM[10]. The control of average DC voltage using a energy balance based approach has been also proposed [11]. Additionally, several studies on MCC have been carried out during the last years including: model and current control [12], control and modulation [13], modulation and associated losses [14] and operation at low frequency [15]. The high modularity that these converters exhibit make them well suited for HVDC applications [16], [17] where the DC voltage can be easily scaled changing the number of cells in series. It can also be used in wind turbines [18] as well as a motor drive [19]. Also, an interesting application in traction has been proposed in [20] where a single-phase input / threephase output MMC interfaces a high-voltage low-frequency line to the low-voltage adjustable-frequency motor drives. The main drawback of this converter is the circulating current which circulate across the converter phases but it is not viewed from the source nor the load. This current increases the losses and changes the relationship between input and output power, degrading the performance of DC-voltage control. Several approaches to model and control this current have been proposed [21], [22]. In this work a generalized dynamic model of the MMC is proposed. This model is valid for single-phase and threephase loads and also for DC or single-phase Ac sources. The proposed model are intended to simplify the analysis of the converter, particularly the circulating currents, how are there produce and evolve. The model can also be use to easily simulate the converter and for controller design purposes, particularly when complex control strategies are tested. The following sections contain the description of the MMC, the mathematical background required to obtain the converter
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Ls
is
is Cap1
Cbp1
CapN iap
vs +
CbpN
-
Can1
CanN
Fig. 1.
iao ibo ico
Lo
Cbn1
-
ian
CcnN
l Cdc
Rdc
s1
ix
l
s1
iao ibo ico
Lo
Ro
icn
L
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L
R
R
R
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van
Fig. 3.
+ vdcxi
s2
vbn
+ -
vcn
Circuital Model of the MMC
III. DYNAMICAL M ODELING
Cdc s2
-
Fig. 2.
ibn
+ -
Rdc s1
L
R R R vap + vbp + vcp ibp icp
+ -
Ccn1
CbnN
L
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+ vdcxi
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iap
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Three-phase Modular Multilevel Converter
s1 ix
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CcpN icp
ibn
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ian
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MMC Power Cells a) DC Source b) Single-phase AC Source
model and simulations results using the proposed dynamic model. II. M ODULAR M ULTILEVEL C ONVERTER The power topology of a three-phase MMC is shown in Fig. 1. Each phase of the MMC is composed by two arms, namely positive and negative. Each one of these arms is composed by N identical cells connected in series. The load is connected to the mid-point of each phase, between both arms, and the source is connected at the top and bottom of each phase. An inductive-resistive (Lo and Ro ) load is considered in the analysis. An optional inductance Ls can be used as a filter to improve the input current waveform. The MMC can be designed to work with DC or singlephase AC sources, the only change required is related to the cell topology. When a DC source is used a DC-DC boost cell is used as shown in Fig.2 a) because only positive voltages in each cell terminals are required. When an AC source is used an H-bridge must be used as shown in Fig. 2 b) to produce positive and negative voltages at each cell terminals. In both cases, the DC-side is connected to a floating capacitor Cdc and a DC resistance Rdc which models the losses in the cell. An inductor l is placed at the input terminals of each cell.
The dynamic model of the MMC is obtained simplifying each arm of the converter to a controlled voltage source, an inductance and a resistor as shown in Fig 3. The controlled voltage is given by the sum of all the DC voltages multiplied by the switching function, i.e. 0 and Vdc if the cells are DC or −Vdc , 0 and +Vdc if AC cells are used. The inductance L is the sum of the inductances l of all cells and the resistor R is used to model the losses in the arm. Both values are considered to be equal in all arms and very small in comparison with the input filter Ls , Rs and load Lo , Ro parameters. All the model equations given in the following sections are valid for AC and DC sources and single-phase and three-phase configurations. In the cases that single-phase and three-phases configurations have any difference, it will be remarked. A. Controlled Arm Voltages As mentioned earlier, each controlled arm voltage is given by the combination of DC-link voltage and switching function of each series connected cells vx =
N X
sxi vdcxi = sTx vdcx
(1)
i=1
where x = {an, bn, cn, ap, bp, cp} and i = {1..N }. The dynamic equation of the DC voltage is given by Cdc
d vdcxi vdcxi + = sxi ix dt Rdc
(2)
or using vectorial notation Cdc
d vdcx vdcx + = sx ix dt Rdc
(3)
where vdcx and sx are the array of DC voltages and switching functions of each arm. Hence, the DC model of each arm is composed by N dynamical equations.
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B. Arm Currents Model A vectorial notation which greatly simplifies the analysis is used along this section. This notation considers the arm currents and voltages are defined by iap ian vap van ip = ibp in = ibn vp = vbp vn = vbn icp icn vcp vcn where the currents and voltages of the positive arms are labelled with a p subscript and for the negative arms with a n subscript. The input voltages and currents, the output currents and the circulating currents are defined respectively by vs is iao iac vs = vs is = is io = ibo ic = ibc vs is ico icc If the converter is connected to a single-phase load only a and b variables are considered. According to Fig. 3 the output voltage equation considering io = ip − in gives d d in + (Ro + QR)in + Qvn = Lo ip + Rip (4) dt dt On the other hand, from the input voltage equation, considering is = nPip = nPin gives (Lo + QL)
vs
=
(nLs P + L)
d in + (nRs P + R)in + vn dt
and 1 1 1
1 1 1
2 1 −1 Q= 3 −1
−1 2 −1
−1 −1 2
for single-phase and three-phase respectively. It is worth to notice that matrices Q and P are related by P+Q = I and PQ = 0. Additionally, they have the following interesting properties that help to simplify the analysis: Qio = io ,
Qvs = 0,
Pio = 0,
Pvs = vs
d 1 in = (vs − vp − vn − Rip − Rin ) dt (nLs + 2L) 1 ((2Ro + R)(ip − in ) + Q(vp − vn )) + 2(2Lo + L) 1 nLs − (QR(ip + in ) + Q(vp + vn )) 2(nLs + 2L) L 1 PnRs in (8) − (nLs + 2L) These dynamic equations completely define the behavior of all the currents inside the MMC. C. Output Currents Model The output currents are defined by io = ip − in
(9)
replacing the derivatives from the arm currents (7) and (8), the model of the output current is given by d io + (2Ro + R)io = −Q(vp − vn ) (10) dt Therefore, the dynamic response of the load current is mainly defined by the load parameters and the difference of the positive and negative controlled voltages multiplied by Q. (2Lo + L)
D. Input Current Model
d +L ip + Rip + vp (5) dt The constant n is related to the number of phases, being n = 2 for single-phase and n = 3 for three-phase. The matrices P and Q are defined by 1 1 1 1 1 −1 P= Q= 2 1 1 2 −1 1 1 1 1 P= 3 1
and for the negative arm.
Multiplying the positive arm current model by P the following relation is obtained: 2L d 2R Ls + Pip + Rs + Pip = vs − P(vp + vn ) n dt 2 (11) This equation contains the same equation in each element vector, therefore it is possible to consider it only a scalar model. The dynamic of the input current is mainly defined by the input filter inductance Ls . Also, it depends on the difference between the input voltage vs and the sum of all the controlled voltages P(vp + vn ). E. Circulating Currents Model According to Fig. 4, the circulating currents do not appear in the input current, therefore
(6)
The dynamic model of the arm currents can be obtained combining equations (4) and (5) and solving for arm currents derivatives, resulting for the positive arm 1 d ip = (vs − vp − vn − Rip − Rin ) dt (nLs + 2L) 1 − ((2Ro + R)(ip − in ) + Q(vp − vn )) 2(2Lo + L) 1 nLs (QR(ip + in ) + Q(vp + vn )) − 2(nLs + 2L) L 1 − PnRs in (7) (nLs + 2L)
Picirc,p = Picirc,n = 0
(12)
and also they do not appear in the output current, hence icirc,p − icirc,n = 0
(13)
Therefore the circulating currents can be defined as ic = icirc,p = icirc,n . The positive and negative arm currents are composed by terms related to the input, output and circulating currents as
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ip
=
in
=
1 Pin + io + 2 1 Pin − io + 2
1 ic 2 1 ic 2
(14) (15)
TABLE I MMC SIMULATION PARAMETERS
is icirc,ap
icirc,bp
iac ibc icc icirc,an
is Fig. 4.
icirc,bn
icirc,cp iao ibo ico
Parameter Input filter inductance Input filter resistance Load inductance Load resistance Arm inductance Arm resistance Input voltage DC capacitance DC resistance DC voltage reference
icirc,cn
Circulating currents in the MMC
Adding up these two equations and solving for circulating currents ic = ip + (I − 2P)in (16) Replacing the derivatives of the arm currents in the previous equation, the model of the circulating currents is obtained d ic + Ric = −Q(vp + vn ) (17) dt In this case, the dynamic response of the circulating current is entirely defined by the arm inductance and resistance and the combination of controlled voltages. As the value of L is usually smaller than Ls and Lo , the dynamic response of the circulating currents is faster than input and output currents, which could lead to controllability problems. L
IV. C IRCULATING C URRENT A NALYSIS In this section the circulating current is analyzed using the proposed model. The MMC is simulated and controlled using linear controllers as proposed in [19]. The DC voltages are controlled using a PI controller which changes the input current reference amplitude, the output current is not controlled. The reference is modulated using a phase shifted carrier at 1kHz. The simulation parameters are given in Table I. Figures 5 and 6 show simulation results of a three-phase DC-AC MMC converter using the model proposed in this paper. From Fig. 5 the input current has a DC value of 14A, the output currents show a sinusoidal waveform with an amplitude of 45A and 50Hz and the circulating currents show high harmonic content. Fig. 6 shows the input filter voltage with a very low DC value but with high peak voltages. The output voltages show a 5-level waveform. The circulating voltages show a small DC value but high voltage peaks. The DC voltages show an average value of 1000V but a second order harmonic of 20V peak. Using equations (3), (10), (11) and (17) the complete dynamic behavior of the converter can be modelled. Each current model is fed by a different combination of arm voltages vp and vn , namely output voltage, input filter voltage and circulating voltage and defined respectively by vo vLs vc
Value Ls = 2mH Rs = 0.1Ω Lo = 20mH Ro = 10Ω L = 0.1mH R = 10mΩ vs = 1000V Cdc = 1000µF Rdc = 10kΩ Vdc = 1000V
= −Q(vp − vn )
(18)
= vs − P(vp + vn )
(19)
= −Q(vp + vn )
(20)
Considering symmetrical operation (i.e. the parameters L and R in all arms are equal), the circulating current is produced only when the circulating voltage is different from zero. Carrier based PWM can not, by itself, avoid the use of nonzero circulating voltages as shown in 6 c). Moreover, the switching commutations produce high circulating current as can be shown in Fig. 5 c). A first approach to minimize the circulating current could be to consider the DC voltages constants and simply avoid the switching states that produce circulating voltage. However, as the DC voltages show a small second harmonic, the circulating current will still remains. To visualize this effect, the MMC model developed in the previous sections is used. The model is fed by a continuous modulation index m instead of switching states sx . Figures 7 and 8 show simulation results of the MMC without modulation. From Fig. 7 the input current is almost flat with a DC value of 14A. The output current is sinusoidal and there is still a circulating current. This current shows a fundamental frequency of 100Hz and an amplitude of 10A. Fig. 8 shows the input filter voltage wit a DC value of 0.6V. The output voltage which is completely sinusoidal, the circulating voltages which have a fundamental frequency of 100Hz and an amplitude of 2V. The DC voltages have an average value of 1000V and also shows a second order harmonics. The relationship between the circulating voltage and current amplitudes is given by Ic = p
Vc R2
+ (2ωo L)2
(21)
It is important to note that, because the arm impedance is small, circulating current amplitude Ic has a considerable amplitude even if the circulating voltage amplitude Vc is small. V. C ONCLUSION A generalized dynamic model of the MMC has been proposed. The model can be used for single-phase and threephase loads and for DC or singe-phase AC sources. The model defines the behavior of the input, output and circulating currents as simply first order models. Using the proposed model it is possible to perform a straightforward analysis of the MMC, simplify the simulation of the converter and bring a powerful tool to design complex control schemes.
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Fig. 6. Simulation results with PWM. a) input filter voltage b) output voltage c) circulating voltage d) DC voltages
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Fig. 5. Simulation results with PWM. a) input current b) output current c) circulating current
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ACKNOWLEDGMENT The authors gratefully acknowledge the financial support provided by the Chilean Research Fund CONICYT FONDECYT (Grant 11090253), the Basal project FB0821 CCTVal and by Universidad T´ecnica Federico Santa Mar´ıa. R EFERENCES [1] J. Rodriguez, L. Franquelo, S. Kouro, J. Leon, R. Portillo, M. Prats, and M. Perez, “Multilevel converters: An enabling technology for highpower applications,” Proceedings of the IEEE, vol. 97, no. 11, pp. 1786 –1817, 2009. [2] S. Kouro, M. Malinowski, K. Gopakumar, J. Pou, L. G. Franquelo, B. Wu, J. Rodriguez, M. A. Perez, and J. I. Leon, “Recent advances and industrial applications of multilevel converters,” Industrial Electronics, IEEE Transactions on, vol. 57, no. 8, pp. 2553 –2580, 2010. [3] J. Rodriguez, S. Bernet, P. Steimer, and I. Lizama, “A survey on neutralpoint-clamped inverters,” Industrial Electronics, IEEE Transactions on, vol. 57, no. 7, pp. 2219 –2230, 2010. [4] M. Malinowski, K. Gopakumar, J. Rodriguez, and M. Perez, “A survey on cascaded multilevel inverters,” Industrial Electronics, IEEE Transactions on, vol. 57, no. 7, pp. 2197 –2206, 2010. [5] A. Lesnicar and R. Marquardt, “An innovative modular multilevel converter topology suitable for a wide power range,” in Power Tech Conference Proceedings, 2003 IEEE Bologna, vol. 3, 2003, p. 6 pp. Vol.3. [6] M. Hagiwara and H. Akagi, “Control and experiment of pulsewidthmodulated modular multilevel converters,” Power Electronics, IEEE Transactions on, vol. 24, no. 7, pp. 1737 –1746, 2009. [7] S. Rohner, S. Bernet, M. Hiller, and R. Sommer, “Pulse width modulation scheme for the modular multilevel converter,” in Power Electronics and Applications, 2009. EPE ’09. 13th European Conference on, 2009, pp. 1 –10. [8] G. Adam, O. Anaya-Lara, G. Burt, D. Telford, B. Williams, and J. McDonald, “Modular multilevel inverter: Pulse width modulation and capacitor balancing technique,” Power Electronics, IET, vol. 3, no. 5, pp. 702 –715, 2010. [9] L. Qiang, H. Zhiyuan, and T. Guangfu, “Investigation of the harmonic optimization approaches in the new modular multilevel converters,” in Power and Energy Engineering Conference (APPEEC), 2010 AsiaPacific, 2010, pp. 1 –6. [10] K. Wang, Y. Li, and Z. Zheng, “Voltage balancing control and experiments of a novel modular multilevel converter,” in Energy Conversion Congress and Exposition (ECCE), 2010 IEEE, 2010, pp. 3691 –3696. [11] A. Antonopoulos, L. Angquist, and H.-P. Nee, “On dynamics and voltage control of the modular multilevel converter,” in Power Electronics and Applications, 2009. EPE ’09. 13th European Conference on, 2009, pp. 1 –10. [12] Z. Yan, H. Xue-hao, T. Guang-fu, and H. Zhi-yuan, “A study on mmc model and its current control strategies,” in Power Electronics for Distributed Generation Systems (PEDG), 2010 2nd IEEE International Symposium on, 2010, pp. 259 –264. [13] D. Siemaszko, A. Antonopoulos, K. Ilves, M. Vasiladiotis, L. A andngquist, and H.-P. Nee, “Evaluation of control and modulation methods for modular multilevel converters,” in Power Electronics Conference (IPEC), 2010 International, 2010, pp. 746 –753. [14] S. Rohner, S. Bernet, M. Hiller, and R. Sommer, “Modulation, losses, and semiconductor requirements of modular multilevel converters,” Industrial Electronics, IEEE Transactions on, vol. 57, no. 8, pp. 2633 –2642, 2010. [15] A. Korn, M. Winkelnkemper, and P. Steimer, “Low output frequency operation of the modular multi-level converter,” in Energy Conversion Congress and Exposition (ECCE), 2010 IEEE, 2010, pp. 3993 –3997. [16] S. Allebrod, R. Hamerski, and R. Marquardt, “New transformerless, scalable modular multilevel converters for hvdc-transmission,” in Power Electronics Specialists Conference, 2008. PESC 2008. IEEE, 2008, pp. 174 –179. [17] R. Marquardt, “Modular multilevel converter: An universal concept for hvdc-networks and extended dc-bus-applications,” in Power Electronics Conference (IPEC), 2010 International, 2010, pp. 502 –507.
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