Scholarly article on topic 'Modelling and Control of the Modular Multilevel Converter (MMC)'

Modelling and Control of the Modular Multilevel Converter (MMC) Academic research paper on "Electrical engineering, electronic engineering, information engineering"

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{"Modular Multilevel Converter (MMC)" / "HVDC transmission" / "Voltage Source Converter (VSC)" / "Converter Control" / "Mathematical modelling"}

Abstract of research paper on Electrical engineering, electronic engineering, information engineering, author of scientific article — Elisabeth N. Abildgaard, Marta Molinas

Abstract The Modular Multilevel Converter (MMC) represents an emerging topology with a scalable technology making high voltage and power capability possible. The MMC is built up by identical, but individually controllable submodules. Therefore the converter can act as a controllable voltage source, with a large number of available discrete voltage steps. This characteristic complicates the modelling both mathematically and computational. A mathematical model of the MMC is presented with the aim to develop a converter control system and the model is converted into the dq reference frame. Block diagrams for control of active power and AC voltage magnitude are shown.

Academic research paper on topic "Modelling and Control of the Modular Multilevel Converter (MMC)"

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Energy Procedia 20 (2012) 227 - 236

Technoport RERC Research 2012

Modelling and Control of the Modular Multilevel Converter (MMC)

Elisabeth N. Abildgaarda, Marta Molinasa

aDepartment of Electrical Power Engineering, Norwegian University of Science and Technology, O.S. Bragstads plass 2E, 7491

Trondheim, Norway


The Modular Multilevel Converter (MMC) represents an emerging topology with a scalable technology making high voltage and power capability possible. The MMC is built up by identical, but individually controllable submodules. Therefore the converter can act as a controllable voltage source, with a large number of available discrete voltage steps. This characteristic complicates the modelling both mathematically and computational. A mathematical model of the MMC is presented with the aim to develop a converter control system and the model is converted into the dq reference frame. Block diagrams for control of active power and AC voltage magnitude are shown.

© 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre for Renewable Energy.

Keywords: Modular Multilevel Converter (MMC), HVDC transmission, Voltage Source Converter (VSC), Converter Control, Mathematical modelling

1. Introduction

With new renewable energy production, HVDC is more applicable than ever. More stochastic energy production calls for solutions that can transport power from areas with high generation to areas with lower generation. Offshore wind farms far from the coast require HVDC transmission to the shore and compact and reliable converter technology with large power capability. Connecting the converter to a DC grid should be feasible and the converter should be able to handle fault situations. To gain compactness, the need for filters should be minimized. The emerging topology, the Modular Multilevel Converter (MMC) might address these aims.

1.1. HVDC Converter Technologies

LCC. The thyristors based Load Commutated Converters (LCCs) were introduced during the 1970s. LCC is still the converter that can be built with highest power rating and hence is the best solution for bulk power transmission. Another advantage of LCC is the low losses, typically 0.7 % per converter [1]. The largest

Email address: (Elisabeth N. Abildgaard) URL: (Elisabeth N. Abildgaard)

1876-6102 © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre for Renewable Energy. doi:10.1016/j.egypro.2012.03.023

disadvantage is that both the inverter and the rectifier absorb a varying amount of reactive power from the grid, and accordingly adjustable reactive compensation is needed [2]. The LCC will also need an AC voltage source at each terminal to be able to succeed with commutation. In order to minimize the harmonic content, the standard LCC design is made with two 6-pulse bridges in parallel on the AC side and in series on the DC side. The two bridges are phase shifted 30 degrees on the AC side, using transformers [3].

VSC. Classical Voltage Source Converter (VSC) utilizing Insulated Gate Bipolar Transistors (IGBTs) for HVDC applications was introduced in 1997 by the ABB concept HVDC Light [4]. Classical VSC for HVDC applications is based on two-level or three-level converters [4]. With this concept it is not possible to adjust the voltage magnitude at AC terminals, but the voltage can be either ±V with two-level or ±V or zero voltage with three-level VSC [2]. Pulse Width Modulation (PWM) is used to approximate the desired voltage waveform and the difference between the desired and implemented waveform is an unwanted distortion which has to be filtered [2]. Because IGBTs have limited voltage blocking capability, they need to be connected in series in two-level and three-level VSCs [4]. In order to limit the voltage across each semiconductor, series connected IGBTs must be switched absolutely simultaneously. This requires sophisticated gate drive circuits to enforce voltage sharing under all conditions [5].

Comparison of LCC and VSC. With VSCs, both active power flow and reactive power flow can be controlled, independently [2], and accordingly no reactive compensation is needed. A VSC station is therefore more compact than a LCC station as the harmonic filters are smaller and no switch yards and capacitor banks are needed [4]. Other advantages with the VSC is that the converter can be connected to weak systems and even to networks lacking generation [4], and as no phase shift is needed, the VSC can use ordinary transformers. A disadvantage is that the VSC has larger losses than LCC, typically 1.7 % per converter [1]. Using LCC, the current direction is fixed and power reversal is done by changing the voltage polarity. With VSCs power reversal is done by changing of the current direction. This makes the VSC technology more suitable for a DC grid application [3]. Cross-linked polyethylene (XLPE) cables can be used with VSCs, but cannot handle the stress from a polarity change. XLPE cables are advantageous as they are less costly, lighter, and smaller in diameter than traditional mass impregnated cables [6]. The power reversal with VSCs can be done gradually because the full range of active power is available, even zero active power can be combined with a positive or negative reactive power. Because both active and reactive power can obtain positive and negative values, the converter is said operate in all four quadrants of the PQ plane [7]. LCCs normally have a minimum active power output 5% below rated power [8]. This makes VSC more favourable for power transmission with varying power e.g. power generated from a wind farm. But an advantage with LCC HVDC is that DC pole to pole short circuit faults can be cleared in the converter station. This is not the case with classical VSC HVDC where in most cases the fault currents must be suppressed by opening the AC breaker feeding the converter [5].

MMC. In 2010 the first Siemens HVDC PLUS system was commissioned, a multilevel VSC technology called MMC [2]. At the same time, ABB updated their HVDC Light product to make use of approximately the same technology [4]. MMCs are built up by a number of identical, but individually controllable submodules. The submodules in the MMC can either be two-level half-bridge converters, each capable of producing +V or zero voltage, or two-level full-bridge converters, producing ±V or zero voltage [5]. This means that the converter acts as a controllable voltage source with a high number of possible discrete voltage steps. The multilevel topology prevents generation of any major harmonic content [4].

The MMC is a scalable technology. The voltage level determines the number of submodules needed, and the technology can be used up to the highest transmission voltages [9]. The configuration is without series connection of semiconductor switches, and hence problems with simultaneous switching are irrelevant. Losses are lower than for two-level and three-level VSCs, about 1 % per converter [4]. The low losses are obtained by low switching frequency in each submodule and low voltage across each switch [9]. However, as the submodules are switched at different points in time, the effective switching frequency of the converter is high, giving a low harmonic distortion [4].

A MMC with two-level half-bridge submodules requires twice the number of IGBTs of to a two-level VSC of the same rating. For a MMC with two-level full-bridge submodules, the need for IGBTs is twice

as high as with half-bridge submodules [5]. The MMC has no DC link capacitance, but one capacitor in each submodule and these capacitors require both large voltage capacity and large capacitance. The result of many semiconductor switches and capacitors with high ratings is a heavy and bulky circuit, giving a converter that is less compact than the classical VSC, but still more compact than the LCC [5].

The MMC with two-level half-bridges cannot block fault currents during a DC pole to pole fault. With two-level full-bridge submodules the MMC is capable of suppressing the fault current and therefore no AC breaker opening is needed [5]. It can be discussed whether this advantage is large enough to defend the increased number of semiconductors. As both vendors delivering MMC solutions uses two-level half-bridges [2, 4], only this solution will be described in the following.

An advantage with MMCs compared to classical VSC is that the dj on the AC side is reduced as the voltage steps at the terminals are smaller. This enables the use of transformers with lower insulation requirement [10]. Compared to LCC the MMC uses ordinary transformers, no phase shift is needed.

Planned installations in 2011 shows that LCC HVDC can be built with 7200 MW and ±800 kV, while MMC projects are planned with 1000 MW and ±320 kV [11, 12].

Other Converter Technologies. A number of other possible converter topologies has been purposed, such as other multilevel converters and the hybrid converters. Among the most important multilevel topologies are the neutralpoint clamped converter [13], the diode-clamped multilevel converter [14], and flying capacitor multilevel converter [15], in addition to the MMC. Hybrid converters can be constructed by combining the advantages of classical VSC and MMC [5]. The aim is to achieve a better output signal than with classical VSC combined with using fewer semiconductor devices than with MMCs. Small MMCs can be used as active filters or wave shaping circuits. Connections can be done in different manners. The MMC is the only one of these topologies applied in commissioned HVDC projects.

1.2. Outline

This paper first describes the operation principle of the MMC. Further the mathematical modelling is shown. Subsequently an approach for simulation of MMCs in an electromagnetic transient (EMT) simulation program is presented. This approach is applicable even for MMCs with a large number of submodules. Finally a control system of the MMC is discussed. This section includes block diagrams for current controllers, active power, and AC voltage Submodule Multivalve Phase unit magnitude control.

Fig. 1. The MMC Structure

2. The Operation Principle of the MMC

In a three phase MMC, each of the phase units consists of two multivalves, and each multivalve consists of N submodules connected in series (Fig. 1) [9]. With a DC voltage of ±320 kV N=38 is typically required [4]. The half-bridge submodule consists of two valves (T1 and T2) and a capacitor (Fig. 2). The valves are made up of an IGBT and a freewheeling diode in antiparallel. In normal operation, only one of the valves is switched on at a given instant in time. Depending on the current direction the capacitor can charge or discharge [9].

When only one IBGT is switched on, either that IGBT or the freewheeling diode in the same valve will conduct, depending on the current direction, and for this reason it makes sense to define a valve as on, indicating that either the IGBT or the diode is conducting [9].

Three possible switching states can be defined [4]:

• In the ON or inserted state T1 is on, and T2 is off. The submodule output voltage, VS M, equals the capacitor voltage, VC, and the capacitor charges if the multivalve current is positive and discharges otherwise.

• In the OFF or bypassed state T2 is on, and T1 is off. The submodule output voltage, VSM, is zero and the capacitor voltage is constant, i.e. the capacitor will not charge nor discharge.

• In the blocked state, both valves are off, and the current can only conduct through the freewheeling diodes. The capacitor will charge if the current is positive, but ideally it cannot discharge.

The blocking voltage in each phase unit is twice the DC voltage. This can be explained from the situation when all the submodules in the upper multivalve are bypassed, giving a phase voltage equal to the DC voltage. The lower multivalve must be able to block the voltage across itself, i.e. the DC voltage. The result is that each switch must be able to block the DC voltage, UD, divided by the number of submodules in each multivalve, N, giving Vblock = N. The capacitors in the lower multivalve will also share the DC voltage and must be dimensioned in the same way as the IGBTs. Considering the same case and a negative ISM relative to Fig. 2, each IGBT in the upper valve must be able to block the voltage across the capacitor in the same submodule. This is one of the reasons why capacitor voltage balancing is important. Both the upper and the lower multivalves should always have half the DC link voltage as average value in order to get a phase output with zero DC offset. The multivalves may take any amplitude between zero and the DC voltage. The sum of inserted submodules in a phase is constant, so inserting a submodule on one multivalve is done simultaneously as bypassing one in the other multivalve of the same phase.

3. Mathematical Modelling of the MMC

Using thyristors, the only controllable parameter is the firing angle, and therefore modelling of the LCC is quite straight forward. For VSC schemes using series connected IGBTs, all the series connected switches are either conducting or blocking. This is utilized in the modelling by defining the share of time the switches are on, the duty ratio [16]. This method cannot be applied for MMCs as some submodules in the multivalve are inserted while others are bypassed. The selection of which submodule to insert or bypass is made on basis of measurements of the capacitor voltages [9]. The capacitor voltages must be kept in a narrow band and this is done through the submodule selection algorithm, using the knowledge of whether a capacitor will charge or discharge given the present current direction.

The following circuit model is developed assuming infinite switching frequency in the converter and infinitive number of submodules per multivalve. These assumptions are made in order to enable the development of a continuous model [17].

Using Kirchhoff's current law in Fig. 3:

iu + ih = iv (1)

iU = Is1 + idiff (2)

Fig. 2. The Submodule Circuit

iL = Is2 - idiff

Table 1. The Parameters

e _ ULUCL-UUUCU eV — -2-

iV — iU + iL

; _ iu-iL

läiff — —

DC pole to pole voltage Output AC voltage

Sum of capacitor voltages, upper multivalve Sum of capacitor voltages, lower multivalve

Inner alternating voltage Current in the upper multivalve Current in the lower multivalve Output AC current Circulating current

Insertion index, upper multivalve, on interval [0,1] Insertion index, lower multivalve, on interval [0,1]

Inserting equations (2) and (3) into equation (1) gives:

iv — Is1 + iäiff + Is2 - iäiff — Is1 + Is2

- CIV,V nu

The difference between the two multivalve currents is:

iU - iL = Isl + idiff - (Is2 - idiff ) = Isl - Is2 + 2idiff (5)

If the converter consists of N submodules per multivalve, and nm = 0 means that all the N submodules are bypassed, while nm = 1 means that all N submodules are inserted, then the available voltage in a multivalve m, i.e. sum of all the inserted capacitor voltages, is given as:

UCm — nmUCm

where u^m is the total capacitor voltage in the multivalve and m=U,L.

The sum of the two insertion indexes should be kept equal to 1, as an insertion in one multivalve corresponds to a bypassing in the other multivalve in the phase, expressed mathematically as:

Fig. 3. Continuous Equivalent of a Phase Leg Using Kirchhoff's voltage law in Fig. 3:

nu + nL — l

Assuming that:

UD y diäiff diV dIs1

-f - nuUCu - Uv - (Riäiff + L-^ ) - Lgriä — RI si + L-d-

UD y diäiff diV dIs2

-~f + nLUCL - UV + Riäiff + L-g- - Lgriä-V — RIs2 + L^f

Is1 — Is2

Combining this assumption with the fact that UV = UV in equations (8) and (9) gives:

Ud - nuUCu - nLuCL = 2(Ridiff + L^)

(8) (9)

In the perfectly balanced case U^U = U^L = UD. This shows that the circulating current is a result of not perfectly balanced multivalve voltages. If the deviation from UD is zero, the steady state value of idiff will also be zero.

Using the assumption in equation (10) on equation (4) gives:

Isl - Is2 - 2 (12)

And using the assumption on equation (5) gives:

iU - iL

idiff (13)

Using that U2L = U2L in equations (8) and (9) gives:

R(iu + Il) + Ld(iUd+ iL) + 2Lgrid^ + 2Uy - nLUCL - nuU^ (14)

L' can be defined as:

L - 2 + Lgrid (15)

Inserting from equation (1):

Uy - ^(UlUCl - nuUCu) - Riy - Lddy (16)

This shows that the output voltage, UV, is only dependent on the output current, iV, and the difference between the two multivalve voltages nUuCU and nLU^L [17]. The difference between nUuCU and nLU^L can be considered as an inner alternating voltage. This voltage will be denoted eV :

ey - 2 (nL UCL - nu UCu)

Uv = ev - 2 iv - L (18)

Equation (16) can be rewritten in the dq reference frame and the Laplace domain as:

Vd = evd - (2 + sL')id - ^L'iq (19)

Vq = evq - (2 + sL')iq + uL'iq (20)

4. Simulation Model of the MMC in an EMT Simulation Program

The challenge when developing a simulation model of the MMC is the large number of switches. In simulation models of LCCs and VSCs with series connected IBGTs, only two switches per phase and bridge are needed, leading to a model with few nodes. When modelling the switching operation properly, an admittance matrix with size equal to the number of nodes in the network must be inverted every time a switch operates [9]. This requires large computational efforts when every MMC submodule consists of three nodes. At the same time, if a model is to be valid during abnormal operation, every level down to each valve must be modelled independently. Gnanarathna et al. [9] describes a model where all the levels are included, and hence it is invariably valid, but by using a Thevenin equivalent, the sizes of the admittance matrixes that need to be inverted are drastically reduced. This is made possible by dividing the solution into two parts; the valve operation and capacitor balancing control is solved separately. Each multivalve is expressed as a

specially designed Thevenin equivalent. This implementation requires reduced computational effort, but is mathematically exactly equivalent to conducting a traditional simulation.

The Thevenin equivalent is deduced using the trapezoidal integration method. vC is the voltage across the capacitor and IC the current through it (Fig. 2).

Vc (t) =

CIcdt « Vc(t - AT) + C

Defining RC and V

Ic(t - AT) + Ic(t) 2

VcEQ(t - AT) = —Ic(t - AT) + Vc(t - AT)


Vc(t) = RcIc(t) + VcEQ(t - AT) (24)

The valves can be treated as two-state resistive devices with low resistance when switched on and high resistance in the off state. The values of resistors R1, for valve 1, and R2, for valve 2 in Fig. 4, depend on the switch state of the valves and are either RON or Roff .

The Thevenin equivalent is developed using Kirchhoff 's voltage law:

Vsm = Ic (R1 + Rc ) + Vceq

VSM = R2(ISM - IC)

Equation (26) gives:

r2ism - vs

Fig. 4. Submodule Thévenin Equivalent

Inserting equation (27) into equation (25) gives:

R2(RI + RC ) R2

VsM(t) = IsM(t)n 1 , „ + VcEQ(t - AT)

R2 + Ri + Rc

R2 + Ri + Rc

Defining RsMEQ(t) and VsMEQ(t - AT):


R2(Ri + RC ) R2 + Ri + RC

VsMEQ(t - AT) = VcEQ(t - AT)

R2 + R1 + R

Insertion into equation (28) gives:

VsM(t) = IsM(t)RsMEQ(t) + VsMEQ(t - AT)

This calculation only requires values from the last time step, the resistance values and the submodule current, which is the same current for all submodules in the multivalve. The voltage across the multivalve is given as:

Vmv (t) = 2 VsMt (t) = Imv(t) 2 RsMEQt (t) + Yj i=1 i=1 i=1


(t - AT) = ImvReq + Veq(î - AT)

where N is the number of submodules in the multivalve, IMV is the current through the multivalve, Req is the equivalent multivalve resistance, and VEQ is the equivalent voltage source.

The Thevenin equivalent is shown in Fig. 5 . Going into the equivalent is FP, the firing pulses, one for each valve. These are necessary for determining the value of R1 and R2. The capacitor voltage, VC, for each submodule goes out of the Thevenin equivalent and is made available for the capacitor voltage balancing controller. VC is found by combining equations (24) and (27). The firing pulses are determined based on the capacitor voltage values.

In contrast to averaged models, this model is capable of representing the exact behaviour of the converter during abnormal operation, e.g. control system failure and module failure.

Fig. 5. The Multivalve Thévenin Equivalent

5. Control of the MMC

The control of the LCC is done by controlling the firing angles. In a DC link, one converter controls the DC voltage while the other controls the DC current. Transformer tap changers can be used to obtain the desired combination of voltage and current [7]. With VSCs it's possible to control both the delay angle and the voltage magnitude, the first influencing the active power and the latter influencing the reactive power [7]. The voltage magnitude is manipulated with the modulation index. The control of the VSC is normally done in a dq reference frame with one active power control loop and one reactive power control loop. The active power control loop can control either active power or DC voltage, while the reactive power loop can control the reactive power or the AC voltage magnitude [7]. The possibilities of the MMC control system is generally equal to those of the two-level and three-level VSCs: Both can successfully be implemented in a dq reference frame controlling two out of the four parameters mentioned above. However, as the mathematical modelling is quite different, the blocks representing the converter system will differ. In addition, the MMC will need a capacitor voltage controller, keeping the capacitor voltages as equal and as close to the reference value as possible.

The dq reference frame controllers use a cascaded structure with a fast inner current loop and an outer loop controlling active power and reactive power or the AC voltage magnitude. Equations (19) and (20) will be used to develop the controllers.

The Current Control Loops. Fig. 6 shows the d axis current control loop. It consists of a PI controller, a time delay representing the converter and a block representing the electrical system given by equation (19). From the symmetry of equations (19) and (20) it can be seen that the q axis current control loop will have the same structure and parameters and this loop is therefore not shown here. The PI controller in the control loop can be tuned using modulus optimum [18]. Using modulus optimum, the PI controller's zero should cancel the largest time constant in the system transfer function. In this case that will be the time delay in the block representing the electrical system.

Fig. 6. The D Axis Current Control Loop The open current loop transfer function is found by multiplying all the block transfers functions:

1 + T„s 1

Tics 1 + TasR + L's

where kc is the gain in the PI controller, Tic is the integral time constant, Ta is the converter time delay, and R and L are the electrical system parameters.

Using modulus optimum [18] the parameters of the PI controller are determined as kc = —a and Tic =

The Active and Reactive Power Control Loops. The active power and reactive power controllers use the dq reference frame expressions that are obtained when the grid voltage vector is defined to be aligned with the d axis. With this alignment vq = 0 and active power and reactive power are given as [19]:

P = vdid

Q = Vdiq

From the similarity of these two equations, it can be seen that the active power controller and the reactive power controller will have the same structure and parameters. The reactive power control loop will contain the q axis current control loop. This loop has the same closed loop transfer function as the d axis current control loop. Due to these similarities only the active power control loop is shown here (Fig. 7). It consist of a PI controller, the d axis current control loop, and a gain given by equation (34). Tuning of the PI controller must be done to ensure a sufficiently large phase margin combined with a high crossover frequency. Plotting of the transfer function shows that the gain must be kept under a certain value and that the integral time constant, TiP, must be kept a number of times higher than the time delay in the converter Ta.

Fig. 7. The Active Power Control Loop

The AC Voltage Magnitude Control Loop. The AC voltage magnitude controller uses the relation between dq quantities and rms values given as:

The controller in Fig. 8 consist of a PI controller, the q axis current control loop, a block representing the electrical system given by equation (20), and a function representing the relationship between dq quantities and phase quantities given by equation (36). The control loop is stable with any parameters in the PI controller.

Fig. 8. The AC Voltage Magnitude Control Loop

6. Conclusion

The mathematical and computational modelling of a MMC has been presented. These enable respectively analytical evaluations and simulations, and are therefore important tools when the MMC is introduced in the power system. Due to the complexity of the MMC topology, simulation models turn out to be quite different from classical VSC models. The mathematical modelling also needs to be done differently to account for the fact that some submodules are inserted while others are bypassed. Assumptions were made to enable development of a continuous mathematical model. For the simulation model, a Thevenin equivalent was introduced to obtain a voltage value for each multivalve at every instant. This model must be combined with a capacitor voltage balancing algorithm. The Thevening equivalent is important as it reduces the computational efforts a lot, and hence makes realistic simulations possible. Regarding control, the MMC has the same advantages as two-level and three-level VSCs, d axis and q axis control can be done independently. This can be used to control either DC voltage or active power and either AC voltage magnitude or reactive power. The presented control loops use a cascaded structure with a fast inner current loop and an outer loop controlling active power and reactive power or the AC voltage magnitude. The equations resulted in similar id and iq control loops. The structure and parameters of the active power and reactive power control loops also became quite similar. Tuning of the PI controllers in the current loops can be done using modulus optimum. The PI controllers in the outer control loops must be tuned to achieve a reasonable crossover frequency combined with suitable phase and gain margins. In the future, simulations should be carried out to identify the appropriateness of the controllers.


[1] S. Norrga, VSC HVDC - Past , Present and Future, Presentation at the 14th European Conference on Power Electronics and Applications, EPE'11 (2011).

[2] K. Friedrich, Modern HVDC PLUS Application of VSC in Modular Multilevel Converter Topology, in: IEEE International Symposium on Industrial Electronics (ISIE), 2010.

[3] S. Cole, R. Belmans, Transmission of Bulk Power: The History and Applications of Voltage-Source Converter High-Voltage Direct Current Systems, IEEE Ind. Electron. Mag (2009) 6.

[4] B. Jacobson, P. Karlsson, G. Asplund, L. Harnefors, T. Jonsson, VSC-HVDC Transmission with Cascaded Two-Level Converters, in: CIGRE B4-110, 2010.

[5] C. Davidson, D. Trainer, Innovative Concepts for Hybrid Multi-Level Converters for HVDC Power Transmission, in: 9th IET International Conference on AC and DC Power Transmission (ACDC 2010), 2010.

[6] Alstom Grid, Multi-Terminal HVDC System for Large Offshore Wind Park Grid Integration, (2010).

[7] C. Bajracharya, Control of VSC-HVDC for Wind Power, Master's thesis, Norwegian University of Science and Technology Department of Electrical Power Engineering (2008).

[8] ABB, Differences Between HVDC Light and Classic HVDC, b8a81e7c435f37bac125755800373a31.aspx (November 2011).

[9] U. N. Gnanarathna, A. M. Gole, R. P. Jayasinghe, Efficient Modeling of Modular Multilevel HVDC Converters (MMC) on Electromagnetic Transient Simulation Programs, IEEE TRANSACTIONS ON POWER DELIVERY 26 (1) (2011) 316-324.

[10] A. Cross, D. Trainer, R. Crookes, Chain-Link Based HVDC Voltage Source Converter Using Current Injection, in: 9th IET International Conference on AC and DC Power Transmission (ACDC 2010), 2010, p. 5.

[11] ABB HVDC Reference Projects, Jinping - Sunan, 545527721af2bf14c12578690049fea4.aspx (November 2011).

[12] Siemens HVDC PLUS - References. (November 2011).

[13] A. Nabae, I. Takahashi, H. Akagi, A New Neutral-Point-Clamped PWM Inverter, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS lA-17 (5) (1981) 518-523.

[14] N. S. Choi, J. Cho, G. H. Cho, A General Circuit Topology of Multilevel Inverter, in: Proc. IEEE 22nd Power Electron. Sepc. Conf., PESC'91, 1991.

[15] A. Schiop, P. Scortaru, Simulink Model of Flying Capacitor Multilevel Inverter, in: 11th International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2008. 2008, pp. 203-208.

[16] V. Blasko, V. Kaura, A New Mathematical Model and Control of a Three-Phase AC-DC Voltage Source Converter, IEEE TRANSACTIONS ON POWER ELECTRONICS 12 (1) (1997) 116-123.

[17] A. Antonopoulos, L. Angquist, H.-P. Nee, On Dynamics and Voltage Control of the Modular Multilevel Converter, in: 13th European Conference on Power Electronics and Applications, 2009. EPE '09.

[18] J. W. Umland, M. Safiuddin, Magnitude and Symmetric Optimum Criterion for the Design of Linear Control Systems: What Is It and How Does It Compare with the Others?, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS (1990) 26 (3).

[19] J. Machowski, J. W. Bialek, J. R. Bumby, Power System Dynamics, Stability and Control, 2nd Edition, Wiley, 2008.