Grégoire Le Goff
email: legoff@laplace.univ-tlse.fr

Maurice Fadel
email: fadel@laplace.univ-tlse.fr

Marc Bodson
email: marc.bodson@utah.edu

Scalable Control-Oriented Model of the Modular Multilevel Converter for Polyphase Systems

Keywords: Current Model, MMC, Polyphase System, Scalable Model, State-Space Model

This paper presents a new state-space model (SSM) to represent the four types of currents flowing through a Modular Multilevel Converter (MMC). In line with the converter topology, the model itself is modular and scalable. In other words, its configuration adapts itself to the number of AC phases, to the active or passive nature of the AC-side grid, to the connection between the AC and DC neutral points, and to the number of submodules. The model describes the influence of the MMC arm voltages on the currents using an average behavior of the submodules. An analysis of the operating space for the currents flowing in the MMC is proposed. Simulations performed covering the region of operation and using a software dedicated to electrical engineering systems demonstrate the value and the accuracy of the proposed model with negligible differences observed for all test conditions. A computational time comparison between the proposed model and the commercial software shows that the model runs about 2 times faster. Hardware-In-the-Loop (HIL) testing demonstrates the capability of the scalable model to be used in order to derive a current-control law on a 5-phase AC-system.

I. INTRODUCTION

A. Modular multilevel converters

As for any type of static converter, properly managing the power flow to satisfy given performance criteria requires a model of the converter having a sufficient accuracy for control purposes [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF]. Multilevel converters have significant advantages over classical converters for high-power applications. They make it possible to lower the stress on the switching cells as well as the total harmonic distortion [START_REF] Arrillaga | Flexible Power Transmission: The HVDC Options[END_REF]. The original modular multilevel converter (MMC) has a topology that is made of identical series-connected submodules (SM) [START_REF] Lesnicar | An innovative modular multilevel converter topology suitable for a wide power range[END_REF]. The possibility of adding submodules gives scalability to the MMC for high-power and high-voltage applications. Therefore, the converter has enabled an acceleration of the deployment of HVDC transmission systems [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF]. The versatility benefit of this topology for rapid energy supply was reported in [START_REF] Elsanabary | Medium Voltage Large-Scale Grid-Connected Photovoltaic Systems Using Cascaded H-Bridge and Modular Multilevel Converters: A Review[END_REF]. The more SMs an MMC has, the more degrees of freedom are available to control the power flowing through the MMC.

B. Research objective

This paper contributes to an overall research objective aimed at developing control allocation methods for the MMC. The principle of this control method is to take advantage of the large number of redundant control variables to operate the system optimally [START_REF] Bodson | Evaluation of optimization methods for control allocation[END_REF], [START_REF] Johansen | Control allocation-A survey[END_REF]. In the case of the MMC, several SM connection configurations are possible to achieve the same voltage level in one arm. Control allocation thus seems particularly suitable if a state-space model (SSM) can be developed that makes it possible to fully exploit this redundancy.

Therefore, the first goal of this paper is to develop a statespace model of the currents flowing through the MMC for control purposes. Assumptions will be made since a detailed physical model of the converter would increase the computational burden of the control algorithm for the currents [START_REF] Bacha | Power Electronic Converters Modeling and Control: with Case Studies, ser. Advanced Textbooks in Control and Signal Processing[END_REF]. The second goal of this paper is to confirm that the simulations using the model guarantee a sufficient accuracy for the control purpose. For the generality of the method, the state-space model should also be scalable and modular, reflecting the nature of the converter that it represents. Accordingly, the model is developed to be scalable to AC networks having any number of phases.

The MMC is characterized by the quantity N of SMs present in each arm and by the number m of phases in the AC network to which it is connected, as shown in Fig. 1. The model is thus derived to be scalable and modular according to these two quantities in order to allow the modeling, simulation and control of any MMC.

C. Assumptions about the submodules

Each SM is itself a DC-DC converter. There is a diversity of submodule topologies [START_REF] Marquardt | Modular multilevel converters: state of the art and future progress[END_REF], but the principle remains the same: each SM contains a voltage source in the form of a capacitor (or capacitors) able to impose a voltage level within the arm to which the SM belongs. The imposed voltage level can be modulated by acting on the switches of the SM to connect it to the arm [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF]. The half-bridge (chopper cell) and the full-bridge cell [START_REF] Marquardt | Modular multilevel converters: state of the art and future progress[END_REF] are the main topologies used in the SMs composing most MMCs. In these two cases, the equilibrium voltage of a capacitor is V DC /N , with V DC = v p -v n the DC bus voltage, [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF]. Note that the subscript p refers to the positive pole of the DC bus and the subscript n to the negative one.

Since each SM contains a capacitor, the impedance encountered by the current flowing through an arm depends on the In the approach used here, the choice is made to work with an average model for the SMs. Thus, the SMs of an entire arm are modeled by the equivalent capacitance C eq = C/N resulting from the series connection of the N SMs, as is done in [START_REF] Freytes | Small-signal state-space modeling of an HVDC link with modular multilevel converters[END_REF], [START_REF] Bergna-Diaz | Generalized voltage-based state-space modeling of modular multilevel converters with constant equilibrium in steady state[END_REF] using the principle of average model detailed in [START_REF] Bacha | Power Electronic Converters Modeling and Control: with Case Studies, ser. Advanced Textbooks in Control and Signal Processing[END_REF]. In this approach, capacitor balancing is assumed to be ensured independently. As a consequence, the set of SMs of an arm is replaced by a variable voltage source, as shown in [START_REF] Lizana | Decoupled current model and control of modular multilevel converters[END_REF]- [START_REF] Zama | Modeling and control of modular multilevel converters (MMCs) for HVDC applications[END_REF]. The representation of the voltage sources is shown on Fig. 1 as v x,y (x ∈ {p, n} and y ∈ {y 1 , . . . , y m }). Note that y k refers to the k-th phase on the AC-side.

L s R s L o R o v y1 L o R o v ym L R v p,y1 L R v p,ym L R v n,y1 L s R s L R v n,ym v n v p n DC n AC i p i n i p,y1

D. Novelty of the proposed model

From a high-level point-of-view, there are at most four types of currents that can appear during the operation of the MMC: the DC source current i s , the AC output currents i oy 1 , . . . , i oy m , the circulating currents between the arms of the MMC i cy 1 , . . . , i cy m , and the common mode current i h when the AC grid and the DC bus neutral points are connected. The different types of currents are grouped as: I s = [i s ] the single component vector for the DC source current, I h = [i h ] the single component vector for the common mode current, I c = [i cy 1 , . . . , i cy m ] T the circulating current vector with m components and I o = [i oy 1 , . . . , i oy m ] T the output current vector with m components. The behavior of these different currents is determined by the voltage levels in each arm of the MMC (v x,y , x ∈ {p, n} and y ∈ {y 1 , . . . , y m }), the DC bus voltage (represented by v p and v n ), and the AC voltages (v y1 to v ym ).

Previous works have decoupled the dynamics of these different current types for three-phase electrical systems like in [START_REF] Lizana | Decoupled current model and control of modular multilevel converters[END_REF], [START_REF] Perez | Decoupled capacitor voltage control of modular multilevel converters[END_REF] where a matrix model is derived. Taking inspiration from these studies and [START_REF] Karwatzki | Generalized control approach for a class of modular multilevel converter topologies[END_REF], a first novelty of the present model is to derive a non-matrix decoupled model of the currents extended to the polyphase (m-phase) case. The approaches of [START_REF] Freytes | Small-signal state-space modeling of an HVDC link with modular multilevel converters[END_REF], [START_REF] Bouarfa | Méthodes de commande par allocation de convertisseurs statiques polyphasés, multi-niveaux : de la modélisation à la mise en oeuvre temps-réel[END_REF], [START_REF] Karwatzki | Generalized control approach for a class of modular multilevel converter topologies[END_REF] produce a state-space model representing the dynamics of only three out of the four possible current types. Compared to [START_REF] Karwatzki | Generalized control approach for a class of modular multilevel converter topologies[END_REF], the contribution of [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF] presents a more limited, but also more explicit derivation of the converter model, in addition to considering the presence of a common mode current. To put it in a nutshell, the interest of the statespace model introduced by [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF], and extended in the current paper, is to have a state-space model which readily adapts to m without loss of generality on the possibilities of the DC bus, the AC network and the connection between the two. Table I summarizes the comparison of characteristics from the proposed model with the literature.

Preliminary results were presented by the authors in [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF]. Additional features of the present paper include:

• A minimal state-space realization which is fully controllable and observable is introduced from the previous state-space model. • An analysis of the operating zone of the currents flowing within the MMC is performed. • New open-loop simulation results are presented using the simulation tool introduced in [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF], enabling the validation of the proposed model across the entire operating zone. • A comparison of computational times is provided between the simulation tool and PLECS®1 . • A closed-loop current control method is applied using the model and then tested with a HIL procedure.

E. Outline

Throughout the article, attention is focused on the high-level model (i.e. current-level model) of the MMC. After the model is derived and before simulating the model, the operating zone is analyzed. The proposed model is then tested across the operating zone in a parallel simulation using PLECS® and MATLAB®-Simulink®. Simulation computational time of the state-space model and PLECS® are compared. Finally a closed-loop control of the currents tuned with the proposed model is tested with a HIL procedure.

II. CURRENT STATE-SPACE MODEL A. Coupled current dynamics

In order to explain the polyphase model, Fig. 2 focuses on a given leg, and thus phase, of the MMC structure. To simplify the notation, any time variable x(t) will be denoted x and the derivative operator is replaced by the LAPLACE variable, s. Applying the Kirchhoff voltage law on the current path flowing through the positive pole of the DC bus gives the following equation:

∀y ∈ {y 1 , . . . , y m } : v p = v y + (R o + L o s)i y + (R + Ls)i p,y +v p,y + (R s + L s s)i p + v n AD (1)
L s R s L o R o v y k L R v p,y k L R v n,y k L s R s v n v p n DC n AC i p i n i p,y k i y k 2m i h 2m i h i n,y k
v n = v y + (R o + L o s)i y + (R + Ls)i n,y +v n,y + (R s + L s s)i n + v n AD (2)
Thus, (1) and ( 2) are summarized by the following equation to which the name (α x,y ) is given:

∀x ∈ {p, n}, ∀y ∈ {y 1 , . . . , y m } :

(α x,y ) : 3) brings to light the fact that the variable voltage sources, represented by v x,y , control different types of currents: i s , i y and i x,y . In other words, the behaviors of different current types are coupled. In the following, current decoupling will be addressed and the definition of the four types of currents, previously mentioned in paragraph I-D, will be stated.

v y + (R o + L o s) i y + (R + Ls)i x,y + v x,y + (R s + L s s) i x -v x = -v n AD (3) Equation (

B. Current decomposition

The currents flowing in the MMC can be split as shown in four different curent types [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF]. Indeed, a first contribution will come from the common mode current i h , a second from the DC bus i s , a third from the currents flowing between the arms i cy , and a fourth from the AC side output i oy [START_REF] Perez | Decoupled capacitor voltage control of modular multilevel converters[END_REF]. According to the orientation convention of each type of current in the MMC legs, using the superposition of the four current types in each arm gives ∀y ∈ {y 1 , . . . , y m }:

i p,y = i h + i s + i cy + i oy (4) i n,y = i h -i s -i cy + i oy (5)
The sum and difference of ( 4) and ( 5) yield:

∀y ∈ {y 1 , . . . , y m }, i p,y + i n,y = 2(i h + i oy ) i p,y -i n,y = 2(i s + i cy ) (6)
A decomposition of the currents is then achieved in order to explicitly obtain each of the four types of current as a function of the currents i p,y and i n,y whose dynamic behaviors are already described by [START_REF] Lesnicar | An innovative modular multilevel converter topology suitable for a wide power range[END_REF]. The decomposition process, derived in detail in [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF], produces four useful equations, one for each type of current, that will help deriving the decoupled dynamics of all four currents independently. These equations are ( 7)- [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF]. To facilitate the forthcoming derivations, these equations are copied below. Thus, ∀y ∈ {y 1 , . . . , y m }:

2m i h = i p + i n = ym y=y1 (i p,y + i n,y ) (7)
2m i s = i p -i n = ym y=y1 (i p,y -i n,y ) (8)
2m i oy = m(i p,y + i n,y ) -(i p + i n ) (9) 2m i cy = m(i p,y -i n,y ) -(i p -i n ) (10)
Note that, in (1) (or (α p,y )), the arm current i p,y appears similarly to i n,y in (2) (or (α n,y ))2 . Accordingly, [START_REF] Johansen | Control allocation-A survey[END_REF] suggests that in order to extract the dynamics of the common mode current i h , the formula ym y=y1 [(α p,y ) + (α n,y )] should be developed. Following the same logic, [START_REF] Bacha | Power Electronic Converters Modeling and Control: with Case Studies, ser. Advanced Textbooks in Control and Signal Processing[END_REF] suggests that the dynamics of the DC source current i s will be found from 1) and ( 2), gives the following expression for (α p,y ) + (α n,y ):

v n + v p = 2v y + [(R + 2R o ) + (L + 2L o )s](i p,y + i n,y ) +(R s + L s s)(i p + i n ) + 2v n AD + (v p,y + v n,y ) (11)
Summing [START_REF] Freytes | Small-signal state-space modeling of an HVDC link with modular multilevel converters[END_REF] on the entire set of phases gives ym y=y1 [(α p,y ) + (α n,y )]: [START_REF] Bergna-Diaz | Generalized voltage-based state-space modeling of modular multilevel converters with constant equilibrium in steady state[END_REF] and dividing by 2m gives:

m(v n + v p ) = 2 ym y=y1 v y +[(R + 2R o ) + (L + 2L o )s](i p + i n ) +m(R s + L s s)(i p + i n ) +2mv n AD + ym y=y1 (v p,y + v n,y ) (12) From (7), i p + i n = 2mi h . Substituting i p + i n in
(vn+vp) 2 = 1 m ym y=y1 v y +[(mR s + R + 2R o ) + (mL s + L + 2L o )s]i h +v n AD + 1 2m ym y=y1 (v p,y + v n,y ) (13
) As i h is the only current appearing in this equation, ( 13) is the differential equation governing the common mode current. A first decoupled differential equation was obtained. To simplify the equation, the following definitions are introduced:

                       R eq h = mR s + R + 2R o and L eq h = mL s + L + 2L o N Σ (m) = 1 m [1, . . . , 1] ∈ M 1,m (R) I h = [i h ] V x = [v p , v n ] T V y = [v y1 , . . . , v ym ] T V n AD = [v n AD ] = [v n AC -v n DC ] V p,y = [v p,y1 , . . . , v p,ym ] T V n,y = [v n,y1 , . . . , v n,ym ] T (14)
Using these definitions, (13) becomes:

(R eq h + L eq h s)I h = -1 2 N Σ (m) V p,y + V n,y + N Σ (2)V x -N Σ (m)V y -N Σ (1)V n AD (15)
In the case where the DC and AC neutrals are disconnected, the common mode current is structurally forced to zero, transforming the previous equation into:

V n AD = -1 2 N Σ (m) V p,y + V n,y + N Σ (2)V x -N Σ (m)V y (16)
2) Source current (DC-side): For the source current i s , the objective is to expand

ym y=y1 [(α p,y ) -(α n,y )].
First of all, using (3) or ( 1) and (2) gives the following equation for (α p,y ) -(α n,y ):

v p -v n = (R + Ls)(i p,y -i n,y ) +(R s + L s s)(i p -i n ) + (v p,y -v n,y ) (17)
Summing [START_REF] Karwatzki | Generalized control approach for a class of modular multilevel converter topologies[END_REF] on the entire set of phases yields

ym y=y1 [(α p,y ) -(α n,y )]: m(v p -v n ) = (R + Ls)(i p -i n ) +m(R s + L s s)(i p -i n ) + ym y=y1 (v p,y -v n,y ) (18)
From ( 8), i p -i n = 2mi s . Substituting i p -i n in [START_REF] Rosenbrock | State-space and Multivariable theory[END_REF] and dividing by 2m implies that:

(vp-vn) 2 = [(mR s + R) + (mL s + L)s]i s + 1 2m ym y=y1 (v p,y -v n,y ) (19)
A single type of current (i s ) appears in equation [START_REF] Siemaszko | Evaluation of control and modulation methods for modular multilevel converters[END_REF], so that the second decoupled differential equation has been obtained. As was done previously for the common mode current, specific notations are introduced in order to simplify the equation:

   R eq s = mR s + R and L eq s = mL s + L N ∆ (m) = 1 m [mI m -J m ] ∈ M m (R) I s = [i s ] (20)
where I m is the identity matrix of size m, J a,b is the (a × b) matrix filled with 1 ′ s, and J m = J m,m . Using the definitions, [START_REF] Siemaszko | Evaluation of control and modulation methods for modular multilevel converters[END_REF] becomes:

(R eq s + L eq s s)I s = -1 2 N Σ (m) V p,y -V n,y + [1 0]N ∆ (2)V x (21)
3) Circulating currents: For the circulating currents, the same reasoning applied for the common mode current and the source current will be repeated. However, the procedure will be slightly more complex. For a given circulating current i cy , the objective is to transform m [START_REF] Rosenbrock | State-space and Multivariable theory[END_REF], m [(α p,y ) -(α n,y )] is expressed using ( 1) and ( 2), which yields the following equation for m [(α p,y ) -(α n,y )]: 23) and dividing by 2m yields:

[(α p,y ) -(α n,y )] - ym y=y1 [(α p,y ) -(α n,y )]. Since ym y=y1 [(α p,y ) -(α n,y )] was already developed in
m(v p -v n ) = m(R + Ls)(i p,y -i n,y ) +m(R s + L s s)(i p -i n ) + m(v p,y -v n,y ) (22)
Subtracting ym y=y1 [(α p,y ) -(α n,y )] in (18) from m [(α p,y ) -(α n,y )] in (22) gives m [(α p,y ) -(α n,y )] - ym y=y1 [(α p,y ) -(α n,y )]: 0 = (R + Ls) (m(i p,y -i n,y ) -(i p -i n )) + m(v p,y -v n,y ) - ym y=y1 (v p,y -v n,y ) (23) From (10), m(i p,y -i n,y ) -(i p -i n ) = 2mi cy . Substituting m(i p,y -i n,y ) -(i p -i n ) in (
0 = (R + Ls)i cy + 1 2m m(v p,y -v n,y ) - ym y=y1 (v p,y -v n,y ) (24)
As was done previously, notations are introduced to simplify equation (24):

R eq c = R and L eq c = L I c = [i c1 , . . . , i cm ] T (25)
Since ( 24) is true for any leg (i.e., any y), the following equation encapsulates the dynamics of all circulating currents into a single vector differential equation with m elements:

(R eq c + L eq c s)I c = -1 2 N ∆ (m) V p,y -V n,y (26)
4) Output currents (AC-side): For the output currents, the same procedure used for the circulating currents is applied. For a given output current i oy , the objective is to develop m [(α p,y ) + (α n,y )] -ym y=y1 [(α p,y ) + (α n,y )]. Since ym y=y1 [(α p,y ) + (α n,y )] was already expanded in [START_REF] Bergna-Diaz | Generalized voltage-based state-space modeling of modular multilevel converters with constant equilibrium in steady state[END_REF], m [(α p,y ) + (α n,y )] is expressed using (1) and (2): 28) and dividing by 2m gives:

m(v p + v n ) = 2mv y +m[(R + 2R o ) + (L + 2L o )s](i p,y + i n,y ) +m(R s + L s s)(i p + i n ) +2mv n AD + m(v p,y + v n,y ) (27) Subtracting ym y=y1 [(α p,y ) + (α n,y )] in (12) from m [(α p,y ) + (α n,y )] in (27) gives m [(α p,y ) + (α n,y )] - ym y=y1 [(α p,y ) + (α n,y )]: 0 = 2 mv y - ym y=y1 v y +[(R + 2R o ) + (L + 2L o )s] (m(i p,y + i n,y ) -(i p + i n )) + m(v p,y + v n,y ) - ym y=y1 (v p,y + v n,y ) (28) From (9), (m(i p,y + i n,y ) -(i p + i n )) = 2mi oy . Substituting (m(i p,y + i n,y ) -(i p + i n )) in (
-1 m mv y - ym y=y1 v y = [(R + 2R o ) + (L + 2L o )s]i oy + 1 2m m(v p,y + v n,y ) - ym y=y1 (v p,y + v n,y ) (29)
The following notation is introduced to simplify equation (29):

R eq o = R + 2R o and L eq o = L + 2L o I o = [i o1 , . . . , i om ] T (30)
Since ( 29) is true for any leg (i.e., any y), the following equation encapsulates the dynamics of all output currents into a single vector differential equation with m elements:

(R eq o + L eq o s)I o = -1 2 N ∆ (m) V p,y + V n,y + -N ∆ (m)V y (31)
At this point, all four types of currents have been shown to satisfy independent differential equations driven by the MMC arm voltages. Thus, the currents dynamics are decoupled by current type and are governed by [START_REF] Zama | Modeling and control of modular multilevel converters (MMCs) for HVDC applications[END_REF] for the common mode current, by [START_REF] Hagiwara | Control and Experiment of Pulsewidth-Modulated Modular Multilevel Converters[END_REF] for the DC source current, by (26) for the circulating currents and by (31) for the AC output currents.

D. Full-order current state-space model

The state variables of the model are the components of the current vectors, so the state vector is:

X HL = [I h , I s , I c T , I o T ] T (32)
Equations ( 15) and ( 21) both represent the behavior of a single current while (26) and (31) describe the dynamics of m different current components, with a total of 2m + 2 currents. The state vector X HL contains all these currents, so the order of the state-space model is 2m + 2 and varies according to the number of phases of the AC network.

The state-space model is given by:

ẊHL = A HL X HL + B HL U HL + E HL Y HL = C HL X HL + D HL U HL (33)
where:

A HL =       - R eq h L eq h - R eq s L eq s - R eq c L eq c I m - R eq o L eq o I m       ∈ M 2m+2 (R) B HL =-1 2     N Σ (m)/L eq h N Σ (m)/L eq h N Σ (m)/L eq s -N Σ (m)/L eq s N ∆ (m)/L eq c -N ∆ (m)/L eq c N ∆ (m)/L eq o N ∆ (m)/L eq o     ∈ M 2m+2,2m (R) C HL =     1 1 I m-1 O m-1,1 I m-1 O m-1,1     ∈M 2m,2m+2 (R) D HL = O 2m,2m ∈ M 2m (R) U HL = [v p,y1 , . . . , v p,ym , v n,y1 , . . . , v n,ym ] T ∈ M 2m,1 (R) E HL =     N Σ (2)/L eq h [1 0] • N ∆ (2)/L eq s 0 0     V x +     -N Σ (m)/L eq h 0 0 -N ∆ (m)/L eq o     V y +     -N Σ (1)/L eq h 0 0 0     V n AD ∈ M 2m+2,1 (R) (34)
and O a,b is the (a × b) null matrix. As the sum of the circulating currents is equal to zero, one of the currents is redundant. The same is true for the output currents. The choice was made to drop the last circulating current i cm and the last output current i om out of the output vector Y HL . Hence the form taken by C HL .

The control variables are the arm voltages v x,y . It is readily possible to adapt equations (32)-(34) to the number m of phases and the equations are independent of N . All together, they embody the General Polyphased Full Order Current State-Space Model (GPFOCSSM).

E. Controllable and observable current state-space model

Considering the controllability property for multi-input multi-output systems [START_REF] Rosenbrock | State-space and Multivariable theory[END_REF] applied to {A HL , B HL }, it is possible to show that only 2m states among 2m+2 are controllable. Indeed, the sum of the circulating and output currents is zero, so that they cannot be independently controlled. However, a reduced form of GPFOCSSM can easily be obtained that is controllable. The new state-space vector XHL is the same as X HL , but with i oy m and i cy m removed, which is obtained by multiplying the state vector by C HL with: This reduced-order state-space realization of the GPFOCSSM with matrices { ÃHL , BHL , CHL , DHL } is controllable, observable, and therefore minimal. Thus, this generalized statespace model can direclty be used for control design of the currents for the MMC.

III. OPEN-LOOP MODEL SIMULATION AND TESTING

A. Test procedure

To ensure that the model developed in this paper properly represents the behavior of the MMC currents, a verification procedure is set-up using MATLAB®. The state-space model is coded in Simulink® using a Matlab-Function able to adapt itself depending on the number of MMC legs and neutral points connection, as shown in Fig. 3. In parallel, the electrical diagram of Fig. 1 is also simulated with the PLECS®-Blockset in Simulink®, as shown in Fig. 4.

The model as well as the electrical circuit are simulated jointly, both are fed with the same input voltage signals. At the end of the simulation, the difference between the currents computed by PLECS® and computed by the model is evaluated as shown on Fig. 5. In the accuracy tests to follow, the plots will show the four types of currents, including those computed by the state-space model as well as those computed by the circuit simulator. Finally, the plots will highlight the difference between the two sets of electrical signals.

To verify the model, the most general simulation option is considered. This option is characterized by an active AC Fig. 3: State-space model Matlab Function used in Simulink®. The question then arises as the choice of non-zero input voltage signals v x,y . In a first approach, the currents are not controlled and the direct modulation method is used, as in [START_REF] Siemaszko | Evaluation of control and modulation methods for modular multilevel converters[END_REF]. Accordingly:

∀y ∈ {y 1 , . . . , y m }, v p,y = V DC 2 [1 -M cos(ω o t -φ y )] v n,y = -V DC 2 [1 + M cos(ω o t -φ y )] (39)
The modulation index M is set to 1. The choice of control signals makes it possible to sweep the entire voltage range that can be reached by the MMC arms. However, using directly those control signals will not produce all four types of currents. For example, [START_REF] Lizana | Decoupled current model and control of modular multilevel converters[END_REF] implies that the common mode current is governed by

∀y ∈ {y 1 , . . . , y m }, v p,y = V DC 2 [m p -d p cos(ω o t -φ y )] v n,y = -V DC 2 [m n + d n cos(ω o t -φ y )]
(40) where the parameters m p , d p , m n and d n have to be carefully chosen in order for v p,y and v n,y to remain within their boundaries. In [START_REF] Goff | Modular Polyphased Full Order Current State-Space Model of the Modular Multilevel Converter[END_REF], those parameters took the values 1, M , 1/2 and M/2, respectively, enabling a preliminary accuracy assessment on particular operating points. To improve the accuracy assessment, an objective of this paper is to simulate the model on a set of points covering the entire operating space.

B. Analysis of the operating zone limits -from the control variables point of view

It is possible to identify two main kinds of operating limits for a given system. A system can be limited from a physical point of view (e.g. current and voltage limits from the semiconductors, temperature limit in a copper wire,..) but also from a control point of view (e.g. duty cycle limits,..). Thus, it is possible to define an operating zone that is reachable from a physical point of view and another operating zone that is reachable from a control point of view. In the analysis presented below, the decision was made to focus on the operating zone that can by reached from the control point of view, because the developed state-space model has for final objective to be used for control.

According to [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF], the nominal operating voltage of the SM capacitors is V DC /N . Having N SMs per arm, the maximum voltage across an arm is V DC , i.e.,

max (|v x,y |) = V DC (41)
Evaluating the minimum value of the arm voltages is less obvious. Indeed, this value depends on the chosen type of SM.

With SM-HB (half-bridge submodules), the lowest voltage that each SM can impose within the arm to which it belongs is 0 V. In contrast, with SM-FB (full-bridge submodules), the lowest voltage is -V DC /N . Therefore, the minimum voltage reached by a given arm containing N SMs is 0 V is the case of SM-HB and -V DC in the case of SM-FB. This fact can be summarized by:

min (|v x,y |) = (1 -q)V DC (42)
with q = 1 for the case of SM-HB and q = 2 for SM-FB. Then, according to the voltage orientation of v p,y and v n,y relatively to the SMs in Fig. 1, it can be concluded that: Evaluating the maximum and minimum values of v p,y and v n,y over time using (39), gives:

∀y ∈ {y 1 , . . . , y m }, (1 -q)V DC ≤ v p,y ≤ V DC -V DC ≤ v n,y ≤ (q -1)V DC (43) m p |d p | m n |d n | 0 1 2 -1 -2 2 0 1 2 -1 -2 2 SM-HB SM-FB
∀y ∈ {y 1 , . . . , y m },                min t (v p,y ) = V DC 2 (m p -|d p |) max t (v p,y ) = V DC 2 (m p + |d p |) min t (v n,y ) = V DC 2 (-m n -|d n |) max t (v n,y ) = V DC 2 (-m n + |d n |) (44)
Thus, boundaries on m p , d p , m n and d n can be obtained when combining (44) with (43):

         mp+|dp| 2 ≤ 1 mp-|dp| 2 ≥ 1 -q -mn+|dn| 2 ≤ q -1 -mn-|dn| 2 ≥ -1 ⇐⇒        m p + |d p | ≤ 2 m p -|d p | ≥ 2(1 -q) -m n + |d n | ≤ 2(q -1) -m n -|d n | ≥ -2 (45)
Finally, the operating zone of the MMC from the control variables point of view and as functions of the SM type can be represented graphically in Fig. 6, based on (45). Using a different model and different notations, similar figures were obtained to describe the operating limits of the MMC in [START_REF] Sharifabadi | Design, control, and application of modular multilevel converters for HVDC transmission systems[END_REF].

C. Accuracy assessment of the model -Set of simulations on several operating points

Simulations were carried out for several operating cases and, in particular, for several values of the number of m of phases. To highlight the interest of the model developed here, the curves obtained for the case where m = 7 are shown. The parameters of the MMC correspond to those of a converter available at the LAPLACE laboratory (see Fig. 7) [START_REF] Serbia | Modular multilevel converters for HVDC power stations[END_REF], and are listed in Table II.

The control signals defined in (40) were applied in simulation for seven phases with m p = 1, d p = M = 1, m n = 1/2 and d n = 0.97, resulting in the curves of Fig. 8. In order to trigger transient conditions, the v x,y signals take 50% of the value given by (40) over the first half of the simulation before a sudden change is applied in the middle of the simulation to make them reach 100% of the expected value. Note that important circulating currents are due to the unusual forms of v p,y and v n,y using different modulation indices. The reader is reminded that this behavior is intended in order to test the accuracy of the model with non-zero circulating currents.

For each type of current, the bottom plot shows the difference between the current from the state-space model The results shown in Fig. 8 are obtained using a particular set of operating points. In order to validate the proposed model in a more comprehensive manner, it is necessary to simulate the model for a set of operating points that covers the entire operating zone as shown in Fig. 6. This objective will be adressed in the next subsection.

D. Accuracy assessment of the model -Set of simulations covering the entire operating zone

A set of operating points is chosen to cover the operating zone from the control point of view, as shown on Fig. 9. Simulations are performed for 5 different number of phases: 3, 7, 10, 14 and 21 phases.

Having 9 operating points covering the entire zone chosen in the (m p , |d p |) space and 9 others chosen in the (m n , |d n |) space gives a total of 81 possible combinations. Thus, 81 simulations are done for each number of phases in the selected list of phase numbers, which makes a total of 405 simulations. Each simulation will output figures like those from Fig. 8 and will, for each type of current, compute the maximum difference between GPFOCSSM and PLECS® over the entire time. Once those 405 simulations are completed, the maximum difference is extracted for each type of current. Finally, it is possible to represent the maximum difference between GPFOCSSM and PLECS® over the entire operating zone on figures gathering the maximum differences for each type of current after the 405 simulations, see Fig. 10.

Each plot shows the relative difference between PLECS® and GPFOCSSM versus the number of phases. There are four different plots, each corresponding to one of the four types of currents. It can be concluded that, for any given type of current and any given number of phases in the selected set, the difference between PLECS® and GPFOCSSM is very small. Indeed, the maximum reaches 7.4 • 10 -10 % (see Fig. 10(c) for m = 10). Further, the evolution of the difference versus the number of phases shows that maximum difference stays very small even for higher number of phases.

IV. COMPUTATIONAL TIME ASSESSMENT

A comparison of the simulation time for both tools is an interesting indicator of the computational burden when used from a control perspective.

Fig. 11 shows that the dedicated simulation tool -in the form of a Matlab-Function -requires less time to simulate the system than PLECS®. Indeed, for 3 to 21 phases, GPFOCSSM is 1.8 to 2.2 times faster than PLECS® and, according to Fig. 11, this computational time difference increases with the number of phases.

V. CLOSED-LOOP ASSESSMENT OF THE MODEL

Testing the state-space model in open-loop showed that the proposed model is accurate. To demonstrate that this controloriented model has sufficient accuracy for control purposes, a current control law is developed and tuned using the model (32)-(38). The decoupled current control introduced by [START_REF] Lizana | Decoupled current model and control of modular multilevel converters[END_REF] is applied here and extended to the case of any number of phases in the AC-side network. This generalization of the current controllers to polyphase systems is readily done by merging the approach of [START_REF] Lizana | Decoupled current model and control of modular multilevel converters[END_REF] with the state-space model of this paper. Even though the capacitor voltage balancing is not the main topic of this paper, it is needed for Hardware-In-the-Loop (HIL) testing and voltage control within the capacitors is achieved using a method based on [START_REF] Lizana | Decoupled current model and control of modular multilevel converters[END_REF], [START_REF] Hagiwara | Control and Experiment of Pulsewidth-Modulated Modular Multilevel Converters[END_REF]. The current and voltage control scheme is shown in Fig. 12. Control assessment is done using an HIL testing procedure with the Opal-RT OP5600 HIL Box. The parameters of this control testing are gathered in Table III, while Fig. 13 displays the laboratory setup with the OP5600 HIL simulation hardware. During the test, currents are logged over time, Fig. 14 shows the current control performance in the case of a 5-phase AC-system.

m p |d p | m n |d n | 0 1 2 -1 -2 2 0 1 2 -1 -2 2
Selected Operating Points to Cover the Operating Zone Fig. 14 shows the current and capacitor voltage response to several steps in output current reference. In order to ease the reading of Fig. 14, references for phases #2 to #5 are the same than in phase #1 but shifted by 2π/5 rad between each phase.

In Fig. 14, one observe excellent current control reference tracking, since the maximum control error on the different types of currents is smaller than 1.0 • 10 -3 A in steady-state. Moreover, after a step in output current reference, the 5% settling time is about 1 ms, which corresponds to the expecta-

MMC Converter

VI. CONCLUSIONS

A. Presented work

A state-space model of the MMC was derived to describe the dynamics of the currents with the arm voltages as input variables. A first feature of the model was the decomposition into the four independent current components for the general case of electrical polyphase systems. A major contribution of the model was also its modularity both in the number m of phases and in the number N of submodules, and the model was developed to be adaptive to the active or passive nature of the AC-grid as well as to the connection between the AC and DC neutrals. The model was evaluated in openloop simulations and the results were found consistent with a circuit-based simulation software. Assessment was also done in closed-loop using a HIL procedure with a decoupled control of the currents and a balancing control of the capacitors, thus validating the state-space model's accuracy of the MMC for control purposes.

The model was derived using simplifying assumptions concerning the influence of the capacitors of each submodules. It could be interesting to remove these assumptions in order to obtain a more general model.

B. New capabilities made possible by the proposed model

The developed state-space model that is readily scalable to any number of phases paves the way for new possibilities: This last feature is currently being developed by the authors so that, to control a given MMC, it will be sufficient to specify to the algorithm the parameters of the MMC, of the AC network and of the DC bus without having to undergo major changes in the system model and control algorithm structure.

Fig. 2 :

 2 Fig. 2: Electrical diagram of the MMC's y k -th arm where v n AD = v n AC -v n DC is the voltage between the DCside and AC-side neutral points. Applying the same voltage law to the path linking to the negative pole of the DC bus gives: ∀y ∈ {y 1 , . . . , y m } : v n = v y + (R o + L o s)i y + (R + Ls)i n,y +v n,y + (R s + L s s)i n + v n AD

  ym y=y1 [(α p,y ) -(α n,y )]. For the two remaining current types, the reasoning will lead to more complex formulas: m [(α p,y ) + (α n,y )] -ym y=y1 [(α p,y ) + (α n,y )] for the output current i oy behavior according to (9) and m [(α p,y ) -(α n,y )] -ym y=y1 [(α p,y ) -(α n,y )] for the circulating current i cy according to (10). C. Decoupled current dynamics 1) Common mode current: For the common mode current i h , the objective is to expand ym y=y1 [(α p,y ) + (α n,y )]. In a first step, using (3) or (

  XHL = C HL X HL (35) Note that: A HL X HL = A HL C T HL XHL (36) Therefore: ẊHL = C HL ẊHL = C HL A HL C T HL XHL + (C HL B HL ) U HL + C HL E HL = ÃHL XHL + BHL U HL + ẼHL (37) Keeping the same output vector gives: Y HL = C HL X HL + D HL U HL = I 2m XHL + D HL U HL = CHL XHL + DHL U HL (38)

Fig. 4 : 3 StateFig. 5 :

 435 Fig. 4: Electrical circuit built with PLECS® Blockset -here for m = 3 State-Space Model M atlab F unction

  ym y=y1 (v p,y + v n,y ), and according to the choice made in (39), v p,y +v n,y = -V DC M cos(ω o t-(k -1)2π/m). Generally, ym y=y1 cos(ω o t -(k -1)2π/m) = 0, and thus ym y=y1 (v p,y +v n,y ) = 0. In other words, the control signals do not influence the common mode current. Other control signals waveforms are therefore necessary to fully test the model. A slight update in the definitions of (39) will enable this feature for all curents:

Fig. 6 :

 6 Fig. 6: Operating Zone from the Control Variables Point of View in the (m p , |d p |) and (m n , |d n |) spaces.

Fig. 7 :

 7 Fig. 7: Picture of the 18-submodule MMC at the LAPLACE laboratory. and the same current from PLECS®. Fig. 8 shows that the maximum difference between GPFOCSSM and PLECS® is on the circulating current I c . The deviation is of the order of 7.43 • 10 -12 A which is very low compared to the magnitude of the current signals. Note that transient conditions have very little effect on the model accuracy. This result validates the state-space model for the operating points reached during the simulation.The results shown in Fig.8are obtained using a particular set of operating points. In order to validate the proposed model in a more comprehensive manner, it is necessary to simulate the model for a set of operating points that covers the entire operating zone as shown in Fig.6. This objective will be adressed in the next subsection.

Fig. 8 :

 8 Fig. 8: Open-loop comparison of the proposed model with PLECS® for m = 7. (a) Common mode current, (b) Source current, (c) Circulating current #4 out of 7, (d) Output current #4 out of 7.Each simulation will output figures like those from Fig.8and will, for each type of current, compute the maximum difference between GPFOCSSM and PLECS® over the entire time. Once those 405 simulations are completed, the maximum difference is extracted for each type of current. Finally, it is possible to represent the maximum difference between GPFOCSSM and PLECS® over the entire operating zone on figures gathering the maximum differences for each type of current after the 405 simulations, see Fig.10.Each plot shows the relative difference between PLECS® and GPFOCSSM versus the number of phases. There are four different plots, each corresponding to one of the four types of currents. It can be concluded that, for any given type of

Fig. 9 :

 9 Fig. 9: Selected operating points in the in the (m p , |d p |) and (m n , |d n |) spaces for further simulations.

Fig. 10 :

 10 Fig. 10: Summary of the 405 open-loop simulations: current maximum relative difference with PLECS® for the (a) Common mode current, (b) Source current, (c) Circulating current, (d) Output current.

Fig. 11 :

 11 Fig. 11: Simulation average CPU computation time (per simulation step) comparing GPFOCSSM & PLECS®.

  Fig. 12: Control architecture developped with GPFOCSSM and taking advantage of [13].

Fig. 13 :

 13 Fig. 13: Picture of the experimental Hardware-In-the-Loop (HIL) setup. Monitoring computers above and on the left, Opal-RT OP5600 HIL box on the right. tions with controllers tuned for a 500 Hz bandwidth. Overall, the results prove that the use of the control-oriented scalable state-space model for the development of a current control law ensures a very good control performance in transient as well as in steady-state conditions.

•Fig. 14 :

 14 Fig. 14: Closed-loop response of the system to changes in the output current reference. (a) Common mode current, (b) Source current, (c) Circulating current, (d) Output current, (e) Capacitors voltages.types of currents (maximum reachable values of the currents) vs the number of phases.• Current-level simulations can be faster than PLECS®, a circuit-based simulation software dedicated to systems in electrical engineering. • The developed model enables the ability to design generic control algorithms for the currents within the MMC, i.e. algorithms that ensure current-control for any number of phases in the AC-side.

TABLE I :

 I Comparison of the characteristics of the proposed scalable state-space model with the literature

	Criterion	[11]	[12]	[13]	[14]	[16]	[17]	Proposed model (See sections II-D, II-E)
	State-space model	Yes	Yes	No	Yes	No	Yes	Yes
	Model derivations	Explicit Explicit	Explicit	Explicit	Explicit	Implicit	Explicit
	Scalability to higher number of phases	No	No	No	No	No	Yes	Yes
	Currents represented	No I h	No I h	All	No I h	All	No I h	All
	Neutral points connection	Fixed	Fixed	Versatile	Fixed	Versatile Versatile	Versatile
	Possible active nature of AC-side	Yes	Yes	Yes	Yes	No	Yes	Yes
	Readily adaptive to any MMC	No	No	No	No	No	No	Yes

TABLE II :

 II Open-Loop Simulation & MMC Parameters

	Meaning	Symbol	Value
	MMC Hardware Parameters
	Bus voltage	V DC	600 V
	Half-bus voltage	vp = -vn	V DC /2 = 300 V
	Bus resistance and inductance	Rs, Ls	50 mΩ, 2 mH
	Switching frequency and period	fs, Ts	4 kHz, 250 µs
	Arm resistance and inductance	R, L	10 mΩ, 5 mH
	AC active voltage	VAC	150 V
	AC grid frequency and period	fo, To	50 Hz, 20 ms
	AC grid pulsation	ωo	2πfo ≃ 314 rad/s
	AC load resistance and inductance	Ro, Lo	40 Ω, 5 mH
	Open-Loop Simulation Parameters
	CPU Type	-	Intel Xeon
	CPU Clock	-	3.00 GHz
	CPU Cache / RAM	-	11.6 Gb / 64 Gb
	Simulation time step	Tstep	Ts/25 = 10 µs
	Simulation end time	t end	7 • To = 140 ms

TABLE III :

 III Closed-Loop HIL & Opal-RT HIL Box Parameters

	Meaning	Symbol	Value
	Opal-RT 5600 HIL Box Parameters
	CPU Type	-	Intel Xeon X5690
	CPU Clock	-	3.46 GHz
	CPU Cache / RAM	-	12 Mb / 4 Gb
	Closed-Loop Parameters
	Current control poles, band-width pcc, f bw	-3142 rad/s, 500 Hz
	Control sampling time	Tc	250 µs
	HIL data logging duration	t log	7 • To = 140 ms

PLECS® (Piecewise Linear Electrical Circuit Simulation) is a circuitbased simulation software for systems in electrical engineering. As it embeds a detailed description of electrical components, its accuracy has already been proven. It will be considered as the reference allowing us to evaluate the accuracy of the proposed model.

Equation (1) is also named (αp,y) and (2) is also named (αn,y), see[START_REF] Lesnicar | An innovative modular multilevel converter topology suitable for a wide power range[END_REF].

Connecting neutrals implies that vn AC = vn DC , thus Vn AD = 0 for the model and the circuit simulator, so that ϵ Vn AD = 0. This zero error is not shown in the upcoming curves, so that

out of

error curves are presented.