Common-Mode-Voltage Regulation of Modular Multilevel Converters Through Model Predictive Control

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by IEEE Transactions on Power Electronics (TPEL) available at: 10.1109/TPEL.2024.3377203 “© 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” 1 Common-Mode-Voltage Regulation of Modular Multilevel Converters Through Model Predictive Control Jiapeng Yin, NingYi Dai, Senior Member, IEEE, Jose I. Leon, Fellow, IEEE, Marcelo A. Perez, Senior Member, IEEE, Sergio Vazquez, Fellow, IEEE, and Leopoldo G. Franquelo, Life Fellow, IEEE Abstract—Modular multilevel converters (MMCs) have emerged as an attractive converter topology for renewable energy integration systems. A smooth operation of an MMC depends on proper regulation of multiple control variables, e.g., phase current, circulating current, capacitor voltage, etc. The model predictive control (MPC) is an excellent candidate for such a scenario since different control objectives can be simultaneously handled in a unified cost function. Though extensive research has been conducted in the literature, a computationally efficient MPC implementation capable of regulating the common-mode voltage (CMV) of the AC-side neutral point of an MMC as a control target has not been properly investigated. This article proposes such an MPC strategy that can flexibly regulate the CMV by incorporating the CMV tracking error into the cost function. The biggest challenge is the dramatically increased number of available control actions and the consequent computational burden as the three phases are controlled as a whole, which are addressed employing a quadratic programming technique, leading to only 64 cost-function evaluations per control period regardless of the number of submodules of the MMC. Several experiments and simulations are carried out considering the regulation of the CMV in different scenarios, and the results validate the effectiveness of the proposed method. Index Terms—Common-mode voltage (CMV), modular multilevel converters (MMCs), model predictive control (MPC), quadratic programming. I. INTRODUCTION AS a relatively recent topology of multilevel voltage source converters (VSCs), the modular multilevel converter (MMC) [1] is gaining increasing applications in modern power grids with ever-growing integration of renewable energy sources [2]. Figure 1 shows the topology of a three-phase MMC for AC-to-DC or DC-to-AC power conversion, where each phase contains two arms of a set of (the number of which is given by N) submodules (SMs) connected in series, plus the arm inductors necessary for current buffering. This high modularity enables the input/output voltage and power of the MMC to be easily scalable to high ratings by simply adding more cascaded SMs, facilitating direct connection to a medium-voltage grid without the need of a step-up transformer. In addition, satisfying power quality can be guaranteed without high device-switching frequency or bulky filters. These features make the MMC a very cost-effective solution for modern power conversion systems, such as high-voltage direct current (HVDC) [3], [4], motor drives [5], [6], traction power supply systems [7], among others. DC AC DC AC DC AC C S ... ... S _ DC AC DC AC DC AC C S ... ... S _ DC AC DC AC DC AC S ...... S _ Fig. 1. Circuit configuration of the MMC. The unique structure of the MMC also brings challenges to its operation. Apart from the phase current or active/reactive power tracking as the primary control objective for most VSCs, voltages of floating SM capacitors need to be well regulated around the rated value and balanced among SMs in the same arm, with the unavoidable voltage ripples reduced as much as possible. The circulating current, induced by the discrepancy between the dc-link voltage and the sum of all the SM output voltages of each phase, also needs to be controlled to reduce losses or to aid in shaping the capacitor voltages [8]. Other issues include switching frequency regulation [3], semiconductor power routing [9], fault-tolerant operation [10], just to name a few. All these challenges need to be addressed by the control and modulation techniques of the MMC. Additionally, the common-mode voltage (CMV) of the AC terminals of an MMC has to be considered in the control stage. For conventional VSCs, the CMV has been suppressed as a detrimental effect in most cases to enhance the system reliability [11], for example, in motor drive applications [12] and PV systems [13]. When it comes to the MMC, the CMV reduction has also been investigated [6], [14], [15], which is normally achieved through the rearrangement of gate-driving This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 2 pulses [6] or the selection of space vectors [15] that are well suited for an MMC with a low number of SMs. In addition, the CMV has been widely utilized as a degree of freedom to meet certain control requirements, such as the reduction of SM capacitor voltage ripples especially in low-frequency operation [5], [16], countermeasures against grid voltage imbalance [17], arm clamping in discontinuous modulation [18], etc. Normally, in such cases, a CMV of a certain form needs to be injected, such as a third-order harmonic [16] or highfrequency sinusoidal/square-wave/trapezoidal CMV [5], [19] for reducing the capacitor voltage ripples. As for the control and modulation stages, most techniques for the conventional VSC topologies can also be extended to an MMC. Two of the most commonly used schemes are the pulsewidth modulation (PWM) [20] and nearest level control (NLC) [4], plus the conventional proportional integral (PI) controllers and/or resonant (R) controllers for phase current, circulating current, SM capacitor voltage, etc. [8]. In addition, the model predictive control (MPC) is gaining increasing attention for the control of an MMC, considering its ability to handle multiple control objectives, fast dynamic response, amongst other features [21]. It includes two categories, the finite control set (FCS)-MPC and the continuous control set (CCS)-MPC. The former evaluates the different available control actions through a cost function and selects the optimal one, while the latter generates the phase/arm voltage reference and requires an additional modulation stage [22]. The FCS-MPC has drawn more attention for its intuitive implementation that matches the discrete nature of semiconductors [21], [22]. For an MMC, the FCS-MPC can derive the gate signals directly [10], [17], [21], [23], or the arm voltage levels followed by capacitor voltage balancing to reduce computational burden [3], [24]– [29]. Applying the FCS-MPC, satisfying performance can be achieved with different control variables well regulated. However, in the literature, existing MPC methods for MMCs still present the following limitations regarding the CMV: 1) The majority of relevant research ignores the CMV and treats the three phases of the MMC independently [3], [24]–[28], causing an inaccuracy of prediction models and subsequently a degradation of control performance, especially in unbalanced grid conditions [17]. 2) Though the CMV is considered in the prediction models in [17], [21], [23], [29], [30], a large number of control actions may have to be evaluated, leading to a heavy computational cost which makes the MPC technique infeasible when the number of SMs is high. 3) Significant difficulties exist in CMV regulation due to the following reasons and few work has investigated this issue. a) In conventional control schemes of an MMC, the reference corresponding to the CMV generated by the outer-loop controller can be directly added to the arm voltage references [5], [16], [18], [19], which is incompatible with the existing MPC techniques. b) The CMV can be regulated by confining the control actions [23], which is only feasible for an MMC with very few SMs due to the high computational cost, and (b) (c) + _ + _ + _ +_ _+ (a) + _ _+ + _ Fig. 2. Per-phase models of a grid-connected MMC system: (a) circuit of phase-x. (b) equivalent phase-current model. (c) equivalent circulating-current model. becomes intractable when the SM capacitor voltages present non-ignorable differences between arms. To overcome the above limitations, a new MPC technique is developed in this work which considers the CMV in the prediction model and the squared CMV tracking error in the cost function. A significant challenge in this scenario is the extremely high number of possible control actions as three phases of the MMC are coupled through the CMV. This is addressed using quadratic programming and a subsequentlyderived FCS-MPC technique. Contributions of this work are summarized as follows: 1) The CMV is considered to construct a more accurate prediction model of the MMC system, and its tracking error is incorporated into the cost function as a penalty term. In this way, the CMV can be easily and effectively regulated considering the discrepancies of SM capacitor voltages between arms. A flexible trade-off can be achieved between the CMV and other control objectives. 2) The high number of control actions resulted from the three-phase MPC are addressed by properly transforming the cost function minimization into a positive definite quadratic programming problem extending the technique in [28] from per-phase MPC to three-phase MPC. The high-dimensional control inputs are derived through a subtle reformulation of the optimization problem, leading to an explicit solution serving as the arm voltage references of the MMC. 3) An FCS-MPC technique is subsequently derived employing a generalized version of the variable rounding level control (VRLC) method proposed in [25], leading to only 64 control actions per control period, which represents an affordable computational burden for existing microprocessors, regardless of the number of SMs of the MMC. II. CONVENTIONAL MPC TECHNIQUES A. MPC for phase current & circulating current tracking As the first step of an MPC technique, the system models have to be derived for all the control variables. In most previous MPC techniques for an MMC, each phase is controlled independently to simplify the implementation. Throughout this work, a grid-connection or load-connection system is studied, This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 3 and the equivalent model of each phase can be derived from Fig. 1 as shown in Fig. 2(a) (in the case of load connection the grid voltage is equivalently set as zero), where the identifiers of phase, arm and SM are respectively given as x=a, b, c, y=u, l and z= 1, ..., N. The dc-link voltage, grid phase voltage, phase current, arm voltage (modeled by a controllable voltage source) and arm current are denoted by Vdc,vSx,ix, vxy and ixy, respectively. Larm denotes the arm inductance and Rarm is defined as the arm resistance to quantify the conduction loss. Similarly, Land Rdenote the grid-side filter parameters (or load parameters in the case of load connection). The CMV denoted by vcm is defined as the voltage of the AC-side neutral point of the MMC relative to the DC-side neutral point. As mentioned in the introduction section, vcm is neglected in most previous MPC techniques to study each phase independently. Thus, from Fig. 2(a), the mathematical models of phase current and circulating current (as main control variables of an MMC) can be easily derived through KVL and KCL principles [25] as (Larm +2L)dix dt = (−Rarm −2R)ix−2vSx −vxu +vxl (1) 2Larm dicirc,x dt =−2Rarmicirc,x +Vdc −vxu −vxl (2) respectively, corresponding to the equivalent circuits illustrated in Fig. 2(b) and 2(c). In the case of load connection, the term related with vSx is omitted. For prediction purpose, the current models (1) and (2) need to be discretized through forward Euler approximation (from time step kto k+ 1 with sampling period of Ts), leading to the prediction models as ˆ ix(k+ 1) = ix(k) + Ts Larm + 2L[(−Rarm −2R)ix(k) −2vSx(k)−vxu(k) + vxl(k)] (3) ˆ icirc,x(k+ 1) = icirc,x(k) + Ts 2Larm [−2Rarmicirc,x(k) +Vdc −vxu(k)−vxl(k)] (4) where the hat operator is used to denote the predicted variables. In the above current models, the arm voltage represents the sum of the output voltages of all the SMs of arm xy, and can be approximated (provided the capacitor voltages are well balanced with negligible differences) as vxy(k) = N X z=1 vCxyz(k)SCxyz(k) ≈PN z=1 vCxyz(k) N | {z } vCxy (k) N X z=1 SCxyz(k) | {z } Nxy(k) (5) where vCxyz,Sxyz (= 0,1) and vCxy denote the SM capacitor voltage, SM gate signal and arm-average SM capacitor voltage, respectively. Nxy denotes the arm insertion index, which refers to the number of inserted SMs of the corresponding arm. In each control period, different possible combinations of Sxyz of the 2NSMs or those of Nxy of both arms are substituted into the arm voltage equation (5) and subsequently into the prediction models (3) and (4). Then, the predicted currents are evaluated by a cost function defined as Jx,i(k) = p1[i∗ x(k+ 1) −ˆ ix(k+ 1)]2 +p2[i∗ circ,x(k+ 1) −ˆ icirc,x(k+ 1)]2(6) where the symbols with "*" represent the corresponding references, and p1,2>0are the weighting factors that tune the importance of different control objectives. Once completing the above prediction-evaluation procedures for all the possible control actions, the MPC selects the candidate that minimizes the cost function to be implemented during the next control period. In this article, arm insertion indexes (Nxy) are studied as the control actions to reduce the involved calculation, as has been extensively adopted in the literature [21]. After determining Nxu(k)and Nxl(k), the gate signals are generated through the simple sorting-and-selecting method [1]. B. MPC for arm energy regulation Considering the regulation of arm energy, two control variables can be defined as wx∆=wxu −wxl (7) wxΣ=wxu +wxl (8) where wxy denotes the arm energy, i.e., the total energy of all SM capacitors of arm xy. Their mathematical model can be easily derived as dwx∆ dt =ixuvxu −ixlvxl (9) dwxΣ dt =ixuvxu +ixlvxl.(10) Similarly, through Euler approximation, the corresponding prediction models can be derived as ˆwx∆(k+ 1) = wx∆(k) + Ts[ixu(k)vxu(k)−ixl(k)vxl(k)] (11) ˆwxΣ(k+1) = wxΣ(k)+Ts[ixu(k)vxu(k)+ixl(k)vxl(k)] (12) where the arm energy can be calculated as wxy(k) = C 2 N X z=1 v2 Cxyz(k)≈NCv2 Cxy(k) 2(13) neglecting the imbalances of SM capacitor voltages within the same arm, similar to (5). Correspondingly, the cost function is defined as Jx,w(k) = p3[w∗ x∆(k+ 1) −ˆwx∆(k+ 1)]2 +p4[w∗ xΣ(k+ 1) −ˆwxΣ(k+ 1)]2(14) where p3,4are the weighting factors. It should be noted that Jx,w is normally applied as an optional and ancillary term as Jx,iw =Jx,i +Jx,w.(15) This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 4 If Jx,i is adopted alone, the arm-voltage normalization has to be achieved by the capacitor voltage reference for asymptotic stability [27], i.e., vCxy in (5) has to be substituted by VC=Vdc N.(16) III. PROPOSED MPC TECHNIQUE CONSIDERING CMV REGULATION As discussed in [21], [23], the consideration of the CMV (vcm) in the prediction models helps to regulate the SM capacitor voltage, circulating current and phase voltage/current. Moreover, in some cases vcm is expected to track a specific reference for certain purposes, as summarized below: 1) CMV reduction or suppression, i.e., v∗ cm = 0 [6], [14], [15]. 2) Min-max or third-harmonic injection, for higher utilization of the dc-link voltage [4]. 3) Third-harmonic injection, as countermeasures against grid voltage imbalance [17]. 4) Third-harmonic injection, along with second-harmonic injection of circulating current, for capacitor voltage ripple reduction [16]. 5) High-frequency injection, along with circulating current injection of the same frequency, for capacitor voltage ripple attenuation in low-frequency MMC operation, e.g., motor drive applications [5], [19]. 6) CMV injection for arm clamping in discontinuous modulation method [18]. In all these cases, the conventional MPC schemes are unable to fulfill the control requirements. A. MPC with CMV consideration To overcome the above limitations, the phase-current model (3) is modified as ˆ ix(k+ 1) = ix(k) + Ts Larm + 2L[(−Rarm −2R)ix(k) −2vSx(k)−2vcm(k)−vxu(k) + vxl(k)] (17) considering the existence of vcm. It is worth noting that the circulating current and arm energy models remain the same. Adding (17) for all the three phases (i.e., for x=a, b, c), vcm can be obtained as vcm(k) = 1 6X x=a,b,c [−vxu(k) + vxl(k)] | {z } vcmc(k) −1 3X x=a,b,c vSx(k) | {z } vcmg(k) (18) where vcmc and vcmg are components induced from the converter side and grid side, respectively. Correspondingly, three phases of the MMC have to be controlled together through a unified cost function as Jiw =Ja,iw +Jb,iw +Jc,iw.(19) With vcm taken into account, the modified model (17) possesses improved prediction accuracy of phase current compared with (3), and enables the MPC technique to function in unbalanced operations, as have been verified in [17]. !Vdc 20Vdc 2 vcmc 0 0.5 1 1.5 2 2.5 3 3.5 Number of candidates #104 N = 4 N = 5 N = 6 N = 7 N = 8 Fig. 3. Number of control action candidates corresponding to different CMV values. TABLE I NUMBER OF CONTROL ACTION CANDIDATES FOR SELECTED VALUES OF vcmc AND N. N4 5 6 7 8 Total 15625 46656 117649 262144 531441 vcmc = 0 1751 4332 9331 18152 32661 vcmc =VC/61686 4221 9156 17892 32292 vcmc =VC/31506 3906 8652 17136 31212 vcmc =VC/21246 3431 7872 15946 29492 However, to implement an MPC technique with consideration of the CMV, combinations of all control actions of the three phases have to be evaluated through the unified cost function (19), leading to a drastic increase in the computational cost of control implementation compared with the conventional per-phase MPC schemes. For an MMC with NSMs per arm, (N+1)6control action candidates (combinations of Nxy) need to be considered for predictions and cost-function evaluations, which means a large computational burden even for an MMC with low number of SMs, e.g., 15625 candidates when N= 4. This challenge hinders the widespread application of three-phase MPC methods for MMC. B. Challenges in CMV regulation using MPC The CMV regulation has not received much attention in the literature regarding the application of the MPC to an MMC. In conventional control schemes of MMCs [5], [16], [18], [19], the CMV is regulated by adding the corresponding reference directly to the arm voltage references, which, however, are generated by conventional phase/circulating current controllers that result in zero CMV. Thus, such a method is infeasible for the existing MPC methods. As an intuitive solution, confining the control actions is an effective scheme investigated in the literature [23]. However, this technique suffers from significantly increased complexity as Ngrows. To illustrate this point, the CMV is analyzed in details as follows. For simplicity, all the SM capacitor voltages are assumed to be VC, and a focus is given to vcmc neglecting vcmg generated in unbalanced grid conditions. From (16) and (18), it can be inferred that vcmc has (6N+1) discrete voltage levels ranging from −Vdc/2to Vdc/2, with a voltage step of This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 5 VC/6. Figure 3 shows the number of possible control action candidates for different CMV levels, with the detailed profile of several selected CMV values summarized in Table I. Taking CMV suppression as an example, thousands of candidates need to be evaluated even for an MMC with very few SMs, and the number of candidates increases drastically with N. In addition, the enumeration of such candidates is not a trivial task, and the total (N+ 1)6number of candidates may have to be searched exhaustively. Though in some previous works this number can be reduced based on certain assumptions and restrictions, the resulted number of candidates is still very high. For example, through forcing both arm insertion indexes in the same phase to be complementary without controlling the circulating currents, the number of candidates can be reduced to (N+ 1)3[23], [30]. Furthermore, if the discrepancies of SM capacitor voltages between arms need to be considered, the CMV regulation would be rather complex. C. Proposed MPC technique To address the above challenges and provide the MPC technique with CMV-regulating capability, a new cost function is proposed as Jiwv(k) = Jiw(k) + p5[v∗ cm(k)−vcm(k)]2 | {z } Jv(k) (20) with a newly added term that quantifies the CMV tracking error. As can be inferred from (18), the CMV is a linear function of the six arm voltages. Thus, the dynamics of the CMV and arm voltages are similar, and both will present a typical staircase waveform. Different from the other control objectives (phase/circulating currents and arm energies) which rely on the predictions considering the evolutions during each control period, the CMV is instantly determined once the control actions are applied (neglecting the small SM capacitor voltage fluctuations). Therefore, this added CMV error term can also be treated as a control input regulating term and the CMV regulation has no requirements for the control frequency. In the proposed technique, the control frequency is determined considering factors such as circulating current ripples, phase current distortion, implementation time of the control algorithm and SM switching frequency. This measure has previously been adopted for other converter topologies such as the two-level converter [31], neutralpoint-clamped converter [32], matrix converter [33], multilevel diode-clamped converter [34], etc., but not yet for an MMC. Using this strategy, the actual values of SM capacitor voltages can be taken into account, leading to a more accurate control of the CMV. In addition, in certain cases when the SM capacitor voltages present apparent differences between arms, e.g., under SM faulty condition when the DC capacitor voltage reference has to be adjusted accordingly, the proposed technique also works properly. Last but not least, the control of the CMV and other control objectives can be flexibly balanced through weighting factor tuning to achieve a trade-off. Further experimental and simulation results in section VI verify the effectiveness of the proposed method in CMV regulation while maintaining or improving the control performance in terms of phase current, circulating current and arm energy. IV. COST-FUNCTION OPTIMIZATION THROUGH QUADRATIC PROGRAMMING In order to drastically reduce the computational cost involved in the proposed method, a modification has to be performed. A solution on the basis of quadratic programming is proposed in this article to optimize the cost function (20) instead of enumerative evaluations of all control actions. A. Solution for conventional per-phase MPC techniques In [28], a quadratic-programming approach is developed to optimize the cost function directly for a per-phase MPC implementation neglecting the CMV of the MMC. Several key points of this method are briefly summarized in this subsection to make this work self-contained. For more details regarding this control method, please refer to [28]. Firstly, the cost function (15) can be reformulated as a quadratic function of the arm voltage vector vxy = vxu vxlTas (all variables without explicit indication denote those of time instant k): Jx,iw(vxy) = vT xyQx,iwvxy + 2rT x,iwvxy +dx,iw.(21) Relevant matrices are given as (here the original form in [28] is further decomposed according to the control objectives) Qx,iw =λ11−1 −1 1  |{z } Q1 +λ21 1 1 1 | {z } Q2 +λ3i2 xu −ixuixl −ixuixl i2 xl  | {z } Q3x +λ4i2 xu ixuixl ixuixl i2 xl  | {z } Q4x (22) rx,iw =λ1f1x1 −1 |{z} r1 +λ2f2x1 1 |{z} r2 +λ3f3xixu −ixl | {z } r3x +λ4f4xixu ixl  |{z} r4x (23) where f1x(k)=2vSx(k)+(Larm + 2L)i∗ x(k+ 1) −ix(k) Ts + (Rarm + 2R)ix(k)(24) f2x(k) = −Vdc + 2Larm i∗ circ,x(k+ 1) −icirc,x(k) Ts + 2Rarmicirc,x(k)(25) f3x(k) = wx∆(k)−w∗ x∆(k+ 1) Ts (26) f4x(k) = wxΣ(k)−w∗ xΣ(k+ 1) Ts (27) and the weighting factors are scaled by system parameters as λ1=T2 sp1 (Larm + 2L)2, λ2=T2 sp2 4L2 arm , λ3=T2 sp3, λ4=T2 sp4. (28) This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 6 dx,iw is a constant and can be neglected since it has no influence on the cost-function optimization. Further investigation reveals that Qiw is positive definite for arbitrary weighting factors that satisfy λ1,2>0and λ3,4≥0. Thus, the minimization of Jx,iw is a positive-definite quadratic programming problem, and an unconstrained global solution can be easily derived as vxy_0=vxu_0vxl_0T=−Q−1 iwriw (29) leading to the arm voltage references that can be combined with any modulation method to generate the SM gate signals for the MMC [35]. However, the above global solution cannot be directly applied since the arm voltage should be bounded as 0≤vxy ≤NvCxy.(30) This linear constraint can be considered to form a constrained optimization problem resulting in a rigorous solution of arm voltage references, or simply handled by a saturation block as v∗ xy =     0vxy_0<0 vxy_00≤vxy_0≤NvCxy NvCxy vxy_0> NvCxy (31) B. Solution for proposed three-phase MPC technique The strategy reviewed in the previous subsection can also be extended to the proposed MPC technique. In such a case where three phases are evaluated collectively, the cost function should be formulated as a quadratic function of a 6-dimensional optimization vector v=vT ay vT by vT cyTas Jiwv(v) = vTQiwvv+ 2rT iwvv+diwv.(32) Relevant matrices are given as Qiwv =  Qa,iw Qb,iw Qc,iw   | {z } Qiw −λ1 3  Q1Q1Q1 Q1Q1Q1 Q1Q1Q1   | {z } ∆Qiw +λ5  Q1Q1Q1 Q1Q1Q1 Q1Q1Q1   | {z } Qv (33) riwv =  ra,iw rb,iw rc,iw   | {z } riw −λ1(f1a+f1b+f1c) 3  r1 r1 r1   | {z } ∆riw +λ5f5  r1 r1 r1   | {z } rv (34) where f5(k)=6v∗ cm(k)−6vcmg(k), λ5=p5 36 (35) and related matrices are defined in (22) and (23). diwv is a constant and can be neglected similar to dx,iw in (21). Since the CMV is taken into account, f1xin (24) has to be revised as f1x(k)=2v¯ Sx(k)+(Larm + 2L)i∗ x(k+ 1) −ix(k) Ts + (Rarm + 2R)ix(k)(36) where v¯ Sx(k) = vSx(k) + vcmg(k).(37) Then, riwv can be simplified as riwv =riw +rv(38) as ∆riw equals zero. The system matrix Qiw is positive definite if λ1,2,5>0 and λ3,4≥0(the proof is given in the appendix). Thus, the global solution can be directly given, similar to (29), through ∇Jiwv(v0) = 2(Qiwvv0+riwv) = 06×1(39) where 0denotes a zero vector of the corresponding dimension, leading to the unconstrained solution as v0=vT ay_0vT by_0vT cy_0T=−Q−1 iwvriwv (40) However, it is not a straightforward task to derive the inverse matrix of Qiwv. To overcome this obstacle, (39) is divided (by phase) into three equations as (Qa,iw +Q∆)vay_0+Q∆vby_0+Q∆vcy_0 +ra,iw +r∆=02×1(41) Q∆vay_0+ (Qb,iw +Q∆)vby_0+Q∆vcy_0 +rb,iw +r∆=02×1(42) Q∆vay_0+Q∆vby_0+ (Qc,iw +Q∆)vcy_0 +rc,iw +r∆=02×1(43) where Q∆= (−λ1 3+λ5)Q1(44) r∆=λ5f5r1.(45) Subtracting (41) from (42) and (43) leads to Qa,iwvay_0+ra,iw =Qb,iwvby_0+rb,iw =Qc,iwvcy_0+rc,iw (46) respectively. Thus, vby_0and vcy_0can be obtained as vby_0=Q−1 b,iwQa,iw | {z } Qba vay_0+Q−1 b,iw(ra,iw −rb,iw) | {z } rba (47) vcy_0=Q−1 c,iwQa,iw | {z } Qca vay_0+Q−1 c,iw(ra,iw −rc,iw) | {z } rca (48) Substituting (47) and (48) into (41), the solution of vay_0can be finally derived as vay_0=−[Qa,iw +Q∆+Q∆(Qba +Qca)]−1[ra,iw +r∆+Q∆(rba +rca)].(49) Then, vby_0and vcy_0can also be derived from (47) and (48), respectively. Then, the constraint (30) has to be considered. A linearly constrained optimization problem can be formulated to derive This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 7 NLC Sxuz SM-CVB SM-CVB Phase-x Of MMC //.../ Nxu Nxl Optimizer of (20) *N/Vdc N/Vdc Round{ } . Round{ } .Sxlz //.../ + + + _ 0.5 vCxuzixu ixl vCxlz icir,x ix icir,x * ixvxl * vxu * Quadratic programming optimizer VRLC Sorting MMC Fig. 4. Overall control structure of the proposed technique. the rigorous solution as [28]. However, as the arm voltage vector of the three-phase MPC has six dimensions, the enumeration of cases in which the above-derived optimal solution violates the constraint is much more complex to implement than the two-dimensional per-phase MPC in [28]. For a low computational burden and a simple implementation, constraint (30) is dealt with in this article through the saturation block (31), though it unavoidably introduces a compromise in the control performance (i.e., increased reference tracking errors in the control objectives) when the system has to operate with control inputs (arm voltages) close to their physical limits. Considering that this situation seldom occurs, the negative influences can be deemed limited. The resulted arm voltage references v∗ xy will be subsequently modulated to generate the SM gate signals. The overall control structure of the proposed strategy is outlined in Fig. 4. Remark: It is worth noting that the proposed method can be deemed a CCS-MPC technique. The optimizationproblem formulation adopted in this article is tailored to an MPC with single step prediction horizon. For a generic H- horizon implementation of the proposed technique, as the first step, the bilinear arm energy models (11) and (12) have to be linearized [36]. Then, the unconstrained solution can be derived through the conventional multi-step formulation based on discrete-time state-space model [37] or through the generalized predictive control technique [22], leading to the references of six arm voltages which are modulated by the same modulation stage to get the final switching states of all the SMs. Correspondingly, the control inputs of the quadratic programming optimization problem will have a dimension of 6H(only the first six elements will be applied during the next control period according to the receding horizon principle [36]), which means that the computational burden still increases significantly with the prediction horizon H. For constraint handling, numerical methods such as the active-set technique would be required [38]. V. MODULATION STRATEGY FOR GATE SIGNAL GENERATION In terms of modulation, any modulation methods can be applied, such as PWM, NLC, among others. As mentioned in section II, this article focuses on the pseudomodulation techniques, meaning that a modified version of the NLC is implemented. For a proper determination of the insertion indexes, the VRLC method [25] is employed in this article. A. Insertion-index determination using VRLC The operating principle of VRLC is simple: instead of using the same rounding method as conventional NLC scheme, the ceiling( ) floor( ) Combination 1 ceiling( ) floor( ) Combination 4 . . . . Phase-a Combination 2 Combination 3 Phase-b Phase-c Cost function evaluation of 64 candidates Fig. 5. Schematic diagram of the generalized VRLC technique. TABLE II NUMBER OF CONTROL ACTION CANDIDATES OF DIFFERENT MPC SCHEMES. MPC scheme Per-phase Three-phase Exhaustive search [24] 3(N+ 1)2(N+ 1)6 Quadratic programming + VRLC 12 [28] 64 (Proposed) rounding method is independently selected from ceiling and floor functions for each arm as N∗ xy = ceiling/floor v∗ xy vCxy (50) leading to different combinations of rounding methods that are posteriorly evaluated by the cost function. Details of this pseudomodulation method can be found in [25]. In [25], each phase is treated independently as conventional MPC schemes, thus resulting in 4 control actions per phase. Since this article controls three phases of the MMC in a combined way, the implementation of the VRLC strategy leads to 43= 64 control action candidates, which are then evaluated by the cost function, i.e., (32) in this article, and the candidate corresponding to the minimum cost function value (among the 64 candidates) is selected for implementation. Figure 5 illustrates the schematic diagram of the generalized threephase VRLC scheme. It is worth noting that constraints (on such as the capacitor voltages, arm currents, etc.) can also be taken into account in this control stage which is by nature an FCS-MPC technique. The number of control action candidates of different MPC schemes are summarized in Table II. It should be noted that the number of control actions of the proposed method is always fixed as 64 for an MMC with any number of SMs, even for an HVDC application where hundreds/thousands of SMs are involved. This represents a low computational burden for existing microprocessors, making the proposed MPC technique feasible in not only low/medium-voltage but also high-voltage scenarios. It is worth noting that for a potential multi-horizon MPC implementation, as the prediction horizon Hgrows, the increased calculation is only related to the quadratic programming optimization stage, while the number of control action candidates to be evaluated by the FCS-MPC stage is always 64, regardless of H. This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. 8 B. SM Gate signal generation The SM gate signals are generated considering the capacitor voltage balance among SMs in each arm. In this article, the conventional sorting scheme proposed in [1] is adopted for its simplicity. Specifically, in each control period, the SMs in the same arm have to be sorted by their capacitor voltages, and the SMs to be inserted are selected according to the corresponding arm current direction: when the arm current charges the capacitors of the inserted SMs, the SMs with lower capacitor voltages will be selected; when the arm current discharges the inserted capacitors, the SMs with higher capacitor voltages will be inserted. Previous stages of the proposed method have derived the number of SMs that need to be inserted (insertion index) in each arm. The adoption of this sorting method will ensure that the SMs are switched properly, maintaining well balanced SM capacitor voltages in each arm. It is worth noting that many other capacitor voltage balancing schemes [2] can also be applied in combination with the proposed MPC strategy, leading to a reduced switching frequency and thus lower losses. In addition, in high-power applications where the MMC contains a large number of SMs, hardware complexity and overhead also need to be considered. For capacitor voltage balancing, actually fewer voltage sensors in combination with properly-designed algorithms [39], [40] (instead of the default one-sensor-per-SM configuration) can be employed to simplify the hardware of the system, without any compromise on the control performance. This possibility can also be well incorporated into the proposed control structure if necessary. C. Implementation procedures with delay compensation As a merit of MPC techniques, the delay of one sampling/control period (Ts) can be (and should be) compensated through an extra step of prediction [25]. Through both steps of predictions, the arm-average capacitor voltages vCxy are treated as invariant considering their relatively low dynamic. Detailed implementation procedures during control period [k, k + 1] with delay compensation are described as follows: 1) Apply the SM switching signals S∗ xyz(k)obtained in the previous control period [k−1, k]at sampling instant k. 2) Obtain the measurements of arm currents ixy(k)and SM capacitor voltages vCxyz(k)at sampling instant k. 3) Calculate the phase currents ix(k), circulating currents ix,circ(k), arm-average SM capacitor voltages vCxy(k), difference/sum of both arm energies (in the same phase) wx∆(k)/wxΣ(k), and arm insertion indexes Nxy(k), of sampling instant k, according to their definitions. 4) Get the predicted values of phase currents ˆ ix(k+ 1), circulating currents ˆ ix,circ(k+ 1) and arm energy terms ˆwx∆(k+1)/ˆwxΣ(k+1), of sampling instant k+1, using their prediction models (17), (4), (11) and (12). 5) Calculate the relevant matrices Qx,iw,Q∆,rx,iw and r∆, of sampling instant k+ 1, according to (22), (23), (25)-(28), (35)-(37), (44) and (45). Fig. 6. The experimental MMC setup with an RTU-BOX204 controller (the top enclosure). TABLE III PARAMETERS OF THE MMC-BASED SYSTEMS UNDER STUDY. Parameter Value in Experiment Simulation Number of SMs per arm, N4 20 SM capacitance, C0.41 mF 1.1 mF Arm inductance, Larm 2.5 mH 5 mH Arm resistance, Rarm 0.5 Ω0.5 Ω DC-link voltage, Vdc 200 V 2000 V Load inductance, L10 mH 18 mH Load resistance, R30.5 Ω50 Ω AC frequency, f50 Hz 50 Hz Sampling/control time, Ts125 µs 200 µs 6) Obtain the six arm voltage references v∗ xy(k+ 1), to be implemented during control period [k+1, k+2], applying (47)-(49), considering the saturation block (31). 7) Get the six arm insertion indexes N∗ xy(k+1), to be implemented during control period [k+1, k+2], employing the generalized VRLC technique which selects the optimum among the 64 candidates. 8) Obtain the SM switching signals S∗ xyz(k+ 1), to be implemented during control period [k+ 1, k + 2], using the sorting scheme. VI. EXPERIMENTAL AND SIMULATION VALIDATIONS A. Experimental results considering an MMC with N= 4 To experimentally verify the effectiveness of the proposed technique, a three-phase laboratory MMC setup is employed as shown in Fig. 6, which contains 4 SMs per arm. The AC side is connected to a resistive-inductive load. Detailed system parameters are summarized in Table III. The proposed control strategy is implemented on an RTU-BOX204 platform as shown in Fig. 6. This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2024.3377203 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.