Optimal Control Design for Modular Multilevel
Converters Operating on Multi-Terminal DC Grid
Mohamed Moez Belhaouane∗ , Julian Freytes∗ , Mohamed Ayari† , Frderic Colas‡ ,
Franois Gruson‡ , Naceur Benhadj Braiek† , and Xavier Guillaud∗
∗
L2EP, Ecole Centrale de Lille, 59650, Villeneuve dAscq, France.
{mohamed-moez.belhaouane, xavier.guillaud}@ec-lille.fr
† LSA, Ecole Polytechnique de Tunisie, BP. 743, La Marsa, Tunisie.
{mohamed.ayari, naceur.benhadj}@ept.rnu.tn
‡ L2EP, Arts et Mtiers ParisTech, 8 Bv Louis XIV, 59046 Lille, France.
{frederic.colas}@lille.ensam.fr, {françois.gruson}@ensam.eu
Abstract—This paper proposes an advanced control strategy
for Modular Multilevel Converters (MMC) integrated in Multiterminal DC grid. In this present work, a three terminal
MMC-MTDC system connecting onshore AC systems with an
offshore wind farm is setup. Firstly, the voltage droop control
associated to the conventional cascaded controllers for MMC
stations is studied, the dynamic behavior of the DC voltage is
analyzed and some drawbacks are outlined. In order to improve
the dynamic behavior of the controlled DC bus voltage and
the stability of MTDC system, an optimal multivariable control
strategy of each MMC converter is proposed and integrated
in a voltage droop controller strategy. The designed advanced
controller allows to improve the overall DC grid stability and
to reach the droop values designed on static considerations with
acceptable dynamic behavior. By means of numerical simulations
in EMTP-RV software, it is shown that the proposed control
strategy performs well the stability of MTDC grid with 400level model for MMC compared with the classic existing control
methods.
Index Terms—Modular Multilevel Converters (MMCs), MultiTerminal DC grids (MTDC), Droop voltage controller, Multivariable state space controller, linear optimal quadratic controller.
I. I NTRODUCTION
Currently, the need to transmit high amounts of electrical
energy to different places, located at a great distances and
the ability to connect more than two HVDC station by a
DC grid, make the emergence of MTDC systems suitable
for the integration of large scale renewable energy sources.
Hence, the development of multiterminal DC grids (MTDC)
to exploit and connect remotely located offshore wind energy
resources to onshore AC systems is an important step towards
the integration of renewable energy in the grid [1] thanks to the
improvement of reliability, the decrease of conversion losses
and the minimization of environmental and energetic cost
[2]. Before, the converter substations of HVDC systems were
equipped with classical Voltage Source Converters (VSCs)
such as for Cross Sound Project [3]. However, according to a
typical three level converters, it is difficult to reach the desired
DC bus voltage levels, e.g. 640 KV, due to the large voltage
stress on each switch. For this reason, MMC converters have
gained popularity and are currently an appropriated solution
since a higher number of voltage levels help to decrease the
AC harmonic content, avoid the need of passive filters and
decrease switching losses [4], [5]. Then, MMC has prodigious
potential mainly in transmission applications, such as wind
farm connection, point to point HVDC links, multiterminal
operation and passive network power supply.
Furthermore, some research concerning the modeling and
control of multiterminal DC systems on VSCs converters have
been proposed in literature [6]. Many studies have been carried
out on the control design of MTDC systems using the Voltage
Droop Control strategy [7]. The main idea of the droop control
method, when more than one converter are controlling the
DC voltage at the same time, is inspired from the primary
frequency control [8].The implementation of power voltage
droop suspends the requirement to have a master converter in
the MTDC network and ensure the correct power sharing in
case of a power variation or, worse, a converter tripping. So,
different droop controller techniques have been investigated
[2], regarding the control of MTDC grids. In [9], an analytical
expression for estimating the distribution of balancing power
which accounts for dc line voltage drops is derived. In [10], a
scheme for adapting the droop coefficients to share the burden
according to the available headroom of each converter station
is proposed. However, in most of the suggested approaches,
the droop is set to a value close to 5%, i.e., typical value
for frequency droop controllers. Nevertheless the design of
droop parameters is based on steady states criterions [11].
The dynamic behaviors have not been accorded a considerable attention in these previous studies for the sizing of
the droop value. The use of a low droop constant value
may give high dynamics for the DC bus voltage inducing
important oscillations which can deteriorate the stability of the
MTDC system. It is important to remark that for the results
mentioned above, the control methodologies of MTDC-based
converters are subjected to conventional controllers such as
PI/PID controllers.
Thus, in order to overcome these problems, the proposed idea
in this paper consists to associate a classical droop control
with advanced controllers for each MMC converter which
constitutes the DC grid. This allows improving the overall
DC grid stability and reaching the droop values designed
on static considerations with acceptable dynamic behavior.
The proposed multivariable optimal control method for MMC
converters allows transferring the desired power, minimizing
the circulating currents and controlling the stored energy of
the MMC converter. The optimal control technique provides
a uniform control law for all states variables to overcome the
known drawbacks of conventional cascaded control structure
based on PI/PID controllers, like stability problem which can
be caused by the effects of interaction between the inner and
outer loops.
The paper is organized as follows: Section II presents the
studied three terminal MMC-HVDC systems. In section III,
the description and modeling of the used MMC converter in
MTDC system are presented. Based on this, Section IV mainly
focuses on the conventional controllers and the proposed
control strategy of MMC converter integrated in MTDC transmission system. In section V, performance of the proposed
control strategy based on optimal control method associated
to the droop technique, is evaluated for a multi-terminal DC
system Finally, Section VI concludes this paper.
II. T HE S TUDIED THREE TERMINAL MMC-HVDC
S YSTEM
The proposed MMC-MTDC test system used to assess
control performance of the DC grid is presented in Fig. 1.
This studied system is composed of three converter stations,
two stations are onshore and one is offshore. The two onshore
stations are directly linked by a 75 km long DC cable. An
offshore wind farm is connected to the two onshore stations
through DC cables with different lengths (see Fig. 1). In
order to deliver the wind power to the DC grid, only grid
side converters are equipped with voltage droop controllers,
offshore station is controlling its power flow to supply the wind
power into DC grid. Thus, the control system must perform the
transfer of the wind farm power to the onshore grid and also
a power transfer between the two onshore injection points. As
already said above, the voltage droop controller is inspired by
the primary frequency control [8]. The frequency counterpart
in the DC system is the DC voltage level. Hence, according
to the same principle, the power flow can be controlled by a
power-voltage droop. However, the voltage droop controller is
designed by a simple proportional controller depicted in Fig.
2. Furthermore, different droop techniques within MTDC grids
such as dead band droop method or piecewise droop method
are proposed in literature [7]. In this study, the classical droop
technique is adopted which has some advantages and gives
interesting results by increasing the reliability and reducing
the stress on the DC grid.
From the voltage droop controller presented by Fig. 2, it yields
1
∆vdc , where ∆pg (in W
the following relation ∆pg = −
Kdroop
or pu) is the deviation power injected into the AC grid, ∆p∗g0
is the power set point, ∆vdc (in V or pu) is the deviation of
DC voltage and Kdroop (in V /W or pu/pu) is the droop value.
Figure 1. MMC converter based 3-terminal MTDC system
Figure 2. DC voltage Droop Controller
III. MTDC BASED C ONVETER : MMC CONVERTER
A. Description and Topology of MMC Converter
Figure 3. MMC topology
The MMC topology is recalled in Fig. 3. Each phaseleg consists of two arms, i.e., an upper arm (represented by
subscript ”u”) and a lower arm (represented by subscript ”l”).
Each arm is comprised of N series- connected, nominally
identical, half-bridge SMs, one inductor, and one resistor.
Each SM can provide two voltage levels at its terminals,
i.e., zero or vcmi, j ,m=u,l; j=a,b,c. , depending on the state of its
complementary switches S1 and S2 [12]. The index ” j = a, b, c”
are designated, throughout the present paper, as the three-phase
variables and asterix symbol for the reference signal. Each
arm includes also an arm inductance Larm and arm resistance
Rarm . In the AC-side, the filter is modeled by the inductance
L f and the resistance L f . So, regarding the published works,
different models can be used according to the type of study
and required accuracy. Since the detailed IGBT-based model
is nonlinear and quite complex to be used in control task, a
simplified model is proposed in order to simplify the study. In
this model, each MMC arm is averaged using the switching
function concept of a half-bridge converter.
IV. C ONTROL T ECHNIQUES OF MMC C ONVERTER
A. Classical control strategy of MMC converter [4]
In this study, an energy based control is considered. This
supposes to control the AC and DC power independently. On
the AC side, the grid currents are regulated using a classical
(d − q) frame control to decouple active and reactive powers,
and ensure the stability of grid currents.
B. Modeling of MMC Converter
Based on the average model widely presented in [13] and
applying the Kirchhoff theorem, the mathematical equations
that govern the dynamic behavior of MMC converter in phase
j are given as follows:
diu j
dig j
vdc
− vmu j − Rarm iu j − Larm
= vg j + Rig j + L
2
dt
dt
(1)
dil j
dig j
vdc
− vml j − Rarm il j − Larm
= vg j + Rig j + L
2
dt
dt
(2)
By adding and subtracting the equations (1) and (2), the
dynamics of the main currents of MMC converter ig j and idi f f j
are obtained as
dig j
R0
1
1
= − 0 ig j + 0 vv j − 0 vg j
dt
L
L
L
didi f f j
Rarm
1
=−
idi f f j +
vdi f f j
dt
Larm
Larm
(3)
(4)
dc
where
i +i
uj l j
Larm
0
R0 = R + Rarm
2 ; L = L + 2 ; ig j = iu j − il j ; idi f f j =
2 ;
vml j +vmu j
vml j −vmu j
vdc
;
v
=
;
v
−
v
=
vv j =
di f f j
di f f j dc
di f f j
2
2
2
Applying the Park transformation on the differential equation
given by (3), we get:
digd
R0
1
= − 0 igd + 0 vd
dt
L
L
digq
R0
1
= − 0 igq + 0 vq
dt
L
L
(5)
with vd = L0 ωigq + vvd − vgd ; vq = −L0 ωigd + vvq − vgq .
As discussed previously in [13], the total energy per phase
Wtot j stored in an arm is depending on the instantaneous
AC power pac j and the power exchanged with the DC bus
pdc j which depends on the DC component of the differential
currents. Then the dynamics of total stored energy per phase
is expressed by
2
dvctot
dWtot j
j
= Ctot
= pdc j − pac j
dt
dt
(6)
where pac j = 31 vgd igd and pdc j = vdc idi f f j .
The total capacitor per arm is defined by Ctot = C N, with C
and N are respectively the capacitor and the number of submodules (SM) per arm.
Figure 4. AC power, Sum Energy and DC bus voltage control via conventional
controllers.
On the DC side, the differential currents controllers allow
regulating the DC components of idi f f j and can eliminate
the circulating current [20]. When the MMC converter is
integrated in MTDC system illustrated by Fig. 1, two control
modes are identified. In the power control mode concerning
the offshore station, the DC bus voltage is imposed (selector
in position 1, see Fig. 4). The regulation of the stored energy
inside MMC converters is settled via the differential currents
references given in (abc) frame. Three independent energy
control loops have to be implemented to define the three
independent references for each differential current loop. The
power flowing through the converter is controlled by the
reference on the direct component of grid current [4]. As
shown in Fig. 4, the structure of the control strategy is deduced
by the model inversion method. However, in the DC voltage
control mode concerning the onshore stations (selector in
position 2, see as well Fig. 4), the DC voltage is variable
and consequently a voltage droop control presented above
is required in order to regulate the DC bus voltage to its
operating point. Note that all the control loops of the global
control scheme given by Fig. 4 are based on PI- controllers
(Proportional-Integral) with poles placement control design
technique. It is important to underline that in this study, the
references of the stored energy are considered as constants. In
addition, the transfer functions Cv (s) in the Laplace domain,
of the designed ∗PI-controllers in ”abc” frame, are expressed
∆p
by Cv (s) = 2 ∗di f f2 = k pv (1 + k1I ).
(vctot −vctot )
v
B. The proposed advanced control strategy
This section, based on the developed state equations,
presents a state space controller design to control the MMC
dynamics.
1) Linear state space deviation model: According to equations (4), (5) and (6) a linear state space model of MMC
converter can be written according to the
T following state
vector x ∈ ℜ8 = igd igq idi f f j v2ctot j
and input vector
T
5
u ∈ ℜ = vd vq vdi f f jdc
:
ẋ = Ax + Bu
(7)
where the parameters matrix A and the input matrix B,
are of appropriate dimensions and depend on MMC system
parameters, illustrated in Table I. In the equilibrium point Xe ,
the system (7) can be expressed as follows:
Ẋe = AXe + BUe
i∗gd =
= R0 i
v∗q
where Xe =
v∗d
Ps
vgd
i∗gq
= R0 igq
(8)
v2∗
ctot j
T
= Rarm i∗di f f j
T
i∗di f f j =
v∗di f f jdc
Ps
3vdc
and
Ue =
.
gd
Around the equilibrium point Pe = Xe Ue , the relations
(7) and (8) lead to the following linear small signal model:
∆ẋ(8×1) = A(8×8) ∆x(8×1) + B(8×5) ∆u(5×1)
(9)
where ∆x = x − Xe and ∆u = u −Ue .
The output vector are defined as:
y ∈ ℜ5 =
igd
igq
v2ctot j
T
(10)
The small-signal of (10) can be written as:
∆y = C.∆x
(11)
Finally, from (13) and (15), the augmented state space model
is given as:
ż = Āz + B̄u + ξ¯
(16)
w = C̄z
T
T
where z = ∆x v
and w = y v
B
012×1
A 08×5
¯
; B̄ =
;ξ =
Ā =
;
−C 05×5
05×5
∆y∗


C
05×5 05×1
I5×5 05×1 
C̄ =  05×8
0
1 01×5
0
2) Optimal Control Design: This section, based on LQR
control method, provides the optimal control algorithm of the
MMC converter. The considered continuous-time quadratic
cost function is given as follows [15]:
Z +∞ J=
(w − w∗ )T Q̄ (w − w∗ ) + uT Ru u dt
(17)
0
where Q̄ = diag Qy Qv
with Qy ≥ 0; Ru > 0 are the
weights matrices which specifies the desired performances.
The minimization of the criterion (17) expresses adequately
the desired performance of the designed control law providing
the best compromise between the references tracking problem
with good performances and the minimization of the control
energy cost. The resolution of this multivariable optimal control problem leads to the following linear state-space feedback
control law:
∆u = −Kz + Ny∗ ;K = KP KI
(18)
where KP and KI are respectively the proportional and integral
gains matrices, and N is a compensator gain to be selected for
reference tracking.
Fig. 5, sum up the linear multivariable optimal controller
proposed for both control modes. In the position 1, the
”AC power control mode”, which the AC power reference
is constant. In position 2, the optimal control is combined
with ”power-droop voltage control”. For more details about the
development of augmented state space model and the linear
optimal control design, the readers are encouraged to consult
[26].
where
∆y ∈ ℜ5 = y −Ye =
∆igd
∆igq
∆v2ctot j
T
V. MMC INTEGRATION IN MTDC SYSTEMS
(12)
Then, taking the equations (9) and (11), yields to:
∆ẋ = A∆x + B∆u
∆y = C∆x
(13)
Considering now the intermediate variable denoted v which
designs the integrators outputs where the dynamic equation is
given by
v̇ = y∗ − y
(14)
which can be expressed as:
v̇ = ∆y∗ −C∆x
(15)
A. Description and simulation data
In this section, a simulation study is proposed for the MMCMTDC test system given by Fig. 1, where the all stations
are controlled with both classic and advanced MMC control
strategies.
Some basic specifications of the MMC converter and DC cable
are provided respectively in TABLES I and II. Furthermore,
all the MMC conversion stations are 1000 MVA with 400 submodules per arm and a DC-link voltage of 640 KV. The DC
capacitors are sized in order to have an electrostatic constant
1
2
CarmVdc
2
, where the equivalent
value, expressed by HC =
Pbase
Figure 5. Advanced control structure associated to the droop controller
value is indicated in TABLE I.
The MMC stations are connected by three ±320 KV pairs
of DC cables where the DC cable capacitor of each MMC
converter is equal to Cdc = 15µF. The DC cable is modeled
using a wideband line model (reference model in EMTPRV software) which the required data will be summarized in
TABLE II.
For simulation, all model developments and simulation studies
are performed using EMTP-RV [16]. For significance and
credibility of simulation results, it is important to note that the
control implementation and the simulation study are based on
Detailed Equivalent Circuit-based Model (DECM) of MMC
converter with 400 SM/arm [17].
TABLE I.
PARAMETERS THE STUDIED POWER SYSTEM .
Parameter
Value
Rated voltage, Vg
Rated power, Sg b
Rated angular frequency, ωb
Filter inductance, L f
Filter resistance, R f
Arm inductance, Larm
Arm resistance, Rarm
Arm capacitance, Carm
Arm capacitance, Cdc
SM electrostatic constant, Hc
DC electrostatic constant, HDC
Filter gain, T f il
325 kV
1 GVA
2.π.50 Hz
0.18 pu
0.005 pu
0.15 pu
0.01 pu
32.5 µF
15 µF
40 ms
3.1 ms
0.01 s
TABLE II.
DC CABLE SPECIFICATION [19].
Parameter
Value
Cable length
Outer radius of sheath
Outer radius of insulation
Resistivity of core
Resistivity of sheath
Insulator relative permittivity
Insulator loss factor
70 km
5.82 cm
6.39 cm
1.72e-08 Ωm
2.83e-08 Ωmm
2.5
0.0004
In this model the SM power switches are replaced by
ON/OFF resistors with a small value of RON (inmilliohms) and
a large value of ROFF (inmegaohms).
This approach allows performing an arm circuit reduction
for eliminating internal electrical nodes and allowing the
creation of a Norton equivalent for each MMC arm. Indeed,
the algorithm considers each SM separately and maintains a
record for individual voltages. This allows to implement the
lower level control composed of Nearest Level Control (NLC)
modulation and Capacitor Balancing Algorithm (CBA). For
more details, the reader can refer to [18].
The PI controllers gains of the classic control approach are
synthesized in order to ensure a settling time around 5 ms
for both MMC converter currents. For the outer control loops
concerning the active power and the total stored energy,
settling time is equal to 50 ms. The initial state of each MMC
stations are 0.5 pu (station #1), 0.5 pu (station #2) and -1 pu
(station #3). The simulated scenario used to assess the control
methods performances is a sudden 20% power decrease. This
event may occur if a feeder breaker trips in the wind farm.
B. Simulation results and discussion
For the simulation study and analysis, two different droop
values have been chosen, as Kdroop = 0.6pu/pu and 0.1pu/pu
to adjust the power deviation portion which is following
through a converter station. The same droop value is considered for the two MMC stations.
• Case 1: Kdroop = 0.6pu/pu
As depicted in Fig. 6, the loss of production is equally
shared between the two onshore stations since they have the
same droop value. From Fig. 7(a), the DC bus voltage starts
decrease until it reaches 601.6 KV at about 0.53s. Fig. 7(b)
shows the DC-side power of the MMC: the optimal controller
reduces clearly the transient ripple and cutback the overshoot
of the power and voltage on DC side. Using the optimal state
feedback control law which all state variables contribute to
reject the effect of the disturbance, the internal MMC variables
are controlled with a fast dynamics and good performances
which improve the MTDC system stability. In this way, it is
clear from Fig. 7(c) that through advanced control strategy,
the three-phase differential currents are regulated reducing
the transient oscillations Fig. 7(d) reveals a correct energy
dynamic behavior even in the transient.
However, the advanced controller is able to converge the
initial states of the MMC to their desired values in the steady
state operating point. In Fig. 9, compared to the conventional
PI controller, the multivariable optimal allows regulating the
differential currents of MMC with less overshoot and ripples.
From the comparative analysis, these results show that even
with a low droop value , the optimal control approach is
able to regulate the DC bus voltage with good performances
and to improve MTDC system stability. In addition, since the
obtained control gains are not high and the control structure
is simple, the optimization control design algorithm of linear
feedback controller is fast and its implementation is easy compared to other advanced control methods given in literature.
Figure 6. AC power responses for 0.6 pu step change on wind farm station.
•
Case 2: Kdroop = 0.1pu/pu
Figure 8. Dynamic response with 10% of droop: (a) dc bus voltage of station
n◦ 1 and (b) dc-side power of station n◦ 1.
Figure 7. Simulation results and comparative study with 60% of droop.
In order to show the influence of the droop sizing on the DC
voltage dynamics and illustrate the capability of the advanced
controller to improve dynamics and MTDC grid stability even
for low droop values, a second simulation test is proposed.
Figs. 8(a) and (b) show respectively the wind power change
in DC bus voltage and DC power of station n◦ 1 with a lower
droop value. It is clear from Fig. 8(a), that with 10% of
droop value, the conventional PI controllers based on cascaded
structure allow to regulate the DC bus voltage but with poor
dynamics given by a fast frequency oscillations through the
DC voltage which results a negative consequences on the DC
cable characteristics where resonance phenomena could be
generated. The result of Fig. 8(a) shows that compared to the
conventional method, the advanced control strategy associated
to the droop voltage controller, provides a more satisfactory
dynamic response in dc side which highly reduces oscillations.
Figure 9. Dynamic response of phase-a differential current of station n1 with
10% of droop
VI. C ONCLUSIONS
In this study, an advanced control strategy of MMC converters integrated in MTDC system is proposed. The novelty of
this approach consists in the combination between the optimal
state space control design of MMC stations and voltage droop
control technique. An optimization problem is formulated and
solved for the design of a state feedback control law to enlarge
the region of stability for the MTDC grid. The proposed control method allows to reduce the oscillations and improves the
DC bus voltage dynamics even for a lower droop parameters
designed on static considerations. All power and control model
developments have been performed using EMTP-RV software.
Simulation results are provided to confirm and validate the
effectiveness and robustness of the proposed strategy. From the
simulations results, the advanced controllers associated to the
classic droop control method improves MMC-MTDC system
stability and provides disturbance rejection.
R EFERENCES
[1] C. E. Spallarossa, T. C. Green, C. Lin, and X. Wu, ”A DC Voltage Control
Strategy for MMC MTDC Grids incorporating Multiple Master Stations”
IEEE PES T&D Conference and Exposition, Birmingham, April 14-17.
2014.
[2] B. Jacobson,”Developments in multiterminal HVDC”, IEEE EPEC, Winnipeg Manitoba, Canada, 2011.
[3] B. D. Railing, J. J. Miller, P. Steckley G. Moreau, P. Bard, L. Ronstrom
and J. Lindberg, ”Cross Sound Cable Project: Second generation VSC
Technology for HVDC,” CIGR 2004, Paper no. B4-102.
[4] S. Samimi, F. Gruson, P. Delarue, and X. Guillaud, ”Synthesis of different
types of energy based controller for a modular multilevel converter integrated in a HVDC link,” Conference on AC and DC power Transmission
(ACDC), Birmingham, UK, Feb 10-12. 2015.
[5] X. Yao, L. Herrera, and J. Wang,”Modulation and control of MMC based
multiterminal HVDC”, IEEE Energy Conversion Congress and Exposition
(ECCE), Pittsburgh, pp. 499-504, Sept. 14-18, 2014.
[6] P. Rault, ”Dynamic Modeling and Control of Multi-Terminal HVDC
Grids,” PhD in electrical engineering, Ecole central de lille, 20 March
2014.
[7] K. Rouzbehi, A. Miranian, A. Luna, and P. Rodriguez, ”A generalized
voltage droop strategy for control of multi-terminal DC grids,” IEEE
Energy Conversion Congress and Exposition (ECCE), Denver, CO, Sept.
15-19. 2013.
[8] C. D Barker, and R. Whitehouse, ”Autonomous converter control in
a multi-terminal HVDC system,” In Proc. ACDC AC and DC Power
Transmission 9th IET Int. Conf, pp. 1-5, 2010.
[9] T. M. Haileselassie and K. Uhlen,”Impact of DC line voltage drops on
power flow of MTDC using droop control”, IEEE Transactions on Power
Systems, vol. 27, no. 3, pp. 1441-1449, 2012.
[10] N. R. Chaudhuri and B. Chaudhuri,”Adaptive droop control for effective
power sharing in multi-terminal DC (MTDC) grids”, IEEE Transactions
on Power Systems, vol. 28, no. 1, pp. 21-29, 2013.
[11] O. Despouys, ”Offshore DC Grids. Impact of Topology on Power Flow
Control,” In proc. in the 10th International Conference on AC and DC
Power Transmission, Birmingham 2012.
[12] H. Saad, X. Guillaud, J. Mahseredjian, S. Dennetiere, and S. Nguefeu,
”MMC capacitor voltage decoupling and balancing controls,” IEEE Trans.
Power Delivery, vol. PP, no. 99, July. 2014.
[13] P. Delarue, F. Gruson, and X. Guillaud, ”Energetic macroscopic representation and inversion based control of a modular multilevel converter,”
15 th European Conference on Power Electronics Applications, Lille,
France, Sept. 2013.
[14] T. Qingrui, X. Zheng, and X. Lie, ”Reduced switching-frequency
modulation and circulating current suppression of modular multilevel
converter,” IEEE Trans. Power Delivery, vol. 26, no. 3, pp. 2009-2017,
July. 2011.
[15] W. Cebuhar, and V. Costanza, ”Approximation procedures for the
optimal control of bilinear and nonlinear systems,” J. of Optimization
Theory and Applications, vol. 43, no. 4, pp. 615-627, 1984.
[16] J. Mahseredjian, S. Dennetiere, L. Dube, B. Khodabakhchian and L.
Gerin-Lajoie, ”On a New Approach for the Simulation of Transients in
Power Systems,” Electric Power Systems Research, vol. 77, issue 11,
September 2007, pp. 1514-1520.
[17] H. Saad, S. Dennetiere, J. Mahseredjian, , Ph. Delarue, X. Guillaud,
J. Peralta and S. Nguefeu, ”Modular Multilevel Converter Models for
Electromagnetic Transients,” IEEE Trans. Power Delivery, vol. 29, n. 99,
June 2014.
[18] H. Saad, ”Modélisation et Simulation temps réel d’une liaison HVDC
de type VSC-MMC,” PhD in Electrical Engineering, Ecole Polytechnique
de Montréal, March 2015.
[19] O. Ramos-Leanos, J. Luis Naredo, J. Mahseredjian, C. Dufour, J. Alberto
Gutiérrez-Robles, and I. Kocar, ”A Wideband Line/Cable Model for RealTime Simulations of Power System Transients,” IEEE TRANSACTIONS
ON POWER DELIVERY, vol. 27, issue 4, October 2012, pp. 2211-2218.
[20] M.M. Belhaouane, H. Saad, and X. Guillaud, ”Control and performance
of modular multilevel converter using resonant controller,” 40th Annual
conference on IEEE industrial electronics society (IECON), USA, Oct.
29- Nov. 1. 2014.
[21] M.A. Perez, F. R. Lizana, and J. Rodriguez, ”Decoupled current control
of modular multilevel converter for HVDC applications” IEEE International Symposium on Industrial Electronics (ISIE), Hangzhou, May 28-31.
2012.
[22] M. Guan , Z. Xu, and H. Chen, ”Control and modulation strategies
for modular multilevel converter based HVDC system”, 37th Annual
Conference on IEEE Industrial Electronics Society (IECON), Melbourne,
pp. 849-854, Nov. 7-10 2011.
[23] A. Lesnicar and R. Marquardt,”An innovative modular multilevel converter topology suitable for a wide power range”, in Proc. IEEE Bologna
Power Tech, vol. 3, 2003.
[24] R. Lizana, and M.A. Perez, ”Control of MMC-HVDC transmission system operating with local variables,” IEEE 23rd International Symposium
on Industrial Electronics (ISIE), Istanbul, pp. 1490-1495, June 1-4 2014.
[25] D. Görges, M. Izák, and S. Liu, ”Optimal control of systems with
resource constraints,” in Proceedings of the 46th IEEE Conference on
Decision and Control, New Orleans, LA, USA, 2007, pp. 1070-1075.
[26] F.L. Lewis, D.Vrabie and V.L. Syrmos, ”Optimal Control,” John Wiley
& Sons, 3rd Edition, March 2012.