Model Predictive Control of Modular Multilevel Converters Using Quadratic Programming

1 Model Predictive Control of Modular Multilevel Converters using Quadratic Programming Jiapeng Yin, Student Member, IEEE, Jose I. Leon, Fellow, IEEE, Marcelo A. Perez, Senior Member, IEEE, Leopoldo G. Franquelo, Life Fellow, IEEE, Abraham Marquez, Member, IEEE, and Sergio Vazquez, Senior Member, IEEE Abstract—The finite control set-model predictive control (FCSMPC) has been adopted as an excellent choice for the applications of multilevel converters during the last two decades for its salient performance. However, in the case of modular multilevel converters (MMCs), a high amount of calculation is always involved in the implementation, making the FCS-MPC less suitable especially for an MMC with high number of submodules (SMs). To cope with the issue, this paper proposes an MPC technique for the MMC with very low calculation cost. In each sampling period, the arm voltage references of each phase are determined analytically by solving a constrained quadratic programming problem formulated from the cost function. Both a rigorous and a simplified procedure are provided to solve the optimization problem. Then, the four nearest candidates around the arm voltage references are evaluated, leading to a proper selection of arm voltage levels. Several experimental tests on an MMC prototype are carried out to validate the effectiveness of the proposed method. Results show that compared with the conventional FCS-MPC method which evaluates all voltage-level combinations, the proposed scheme presents apparent advantage in terms of calculation cost while achieving similar performance. Index Terms—Model predictive control (MPC), modular multilevel converters, quadratic programming. NOMENCLATURE NNumber of SMs per arm of the MMC. x∈ {a, b, c}. Phase identifier. y∈ {u, l}(upper, lower). Arm identifier. z∈ {1, ..., N}. SM identifier. Larm Inductance of arm inductor. Manuscript received June 8th, 2020; revised August 31st, 2020; accepted October 22nd, 2020. This work was supported in part by the Spanish Science and Innovation Ministry under project TEC2016-78430-R and Consejeria de Economia, Conocimiento, Empresas y Universidad, in part by the Secretaria General de Universidades, Investigacion y Tecnologia under the project PY18- 1340, in part by the 2020 horizon SPARTAN project (ID 821381), in part by the Fondecyt Project 1181839, in part by the Advanced Center for Electrical and Electronics Engineering AC3E (ANID/FB0008), and in part by the Chilean Solar Energy Research Center (ANID/FONDAP/15110019). (Corresponding author: Jiapeng Yin.) J. Yin, A. Marquez and S. Vazquez are with the Department of Electronic Engineering, Universidad de Sevilla, 41092 Seville, Spain (e-mail: [email protected]; [email protected]g; [email protected]). J. I. Leon and L. G. Franquelo are with the Laboratory of Engineering for Energy and Environmental Sustainability, Department of Electronic Engineering, Universidad de Sevilla, 41092 Seville, Spain, and also with the Department of Control Science and Engineering, School of Astronautics, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]). M. A. Perez is with the Department of Electronics, Universidad Técnica Federico Santa María, Valparaiso 2390123, Chile (e-mail: marce- [email protected]). Rarm Parasitic resistance of arm. CCapacitance of SM floating capacitor. Vdc DC-link voltage. idc DC-link current. vSx Grid voltage of phase-x. LGrid-side inductance (or load inductance in case of load connection). RGrid-side resistance (or load resistance in case of load connection). vCxyz Capacitor voltage of SM-xyz. vxy Arm voltage of arm-xy. vdcxy Sum of SM capacitor voltages of arm-xy. ixy Arm current of arm-xy. ixPhase current of phase-x. icirc,x Circulating current of phase-x. Wxy Arm energy of all SMs of arm-xy. Wx∆Difference of both arm energies of phase-x. WxΣSum of both arm energies of phase-x. Nxy Number of switched on SMs (insertion index) of arm-xy. Sxyz Switching state of SM-xyz. fAC side frequency. TsSampling/control period. kInteger representing a certain time step in the discrete time domain. JiCost function considering current tracking. JwCost function considering arm-energy regulating. Jiw Cost function considering both current tracking and arm-energy regulating. p1,2,3,4Weighting factors of the original cost functions. λ1,2,3,4Weighting factors of the reformed cost functions. f1,2,3,4, g1,2Intermediate variables for cost-function reformulation. vxy Arm-voltage vector of phase-x. vxy_0Arm-voltage vector of phase-xcorrespond- ing to the minimum value of the unconstrained cost function. vxy_0Element of vxy_0. Qi,ciReal-valued coefficient matrices of quadratic function derived from Ji. Qw,cwReal-valued coefficient matrices of quadratic function derived from Jw. Qiw,ciw Real-valued coefficient matrices of quadratic 2 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. function derived from Jiw. qi1,2,3, ci1,2Elements of Qiand ci, respectively. A,bReal-valued coefficient matrices of constraint of optimizing problems. E1,2Important tangent points in orthogonal coordinate (vxu, vxl)for solving the optimization problems. vxy_E1,2Coordinate of E1,2. I. INTRODUCTION THE modular multilevel converter (MMC) has attracted great interest from power electronics academia and industry in recent years since 2003, when its concept was first proposed [1]. The structure of a three-phase MMC is illustrated in Fig. 1. Each phase leg consists of two arms and each arm is composed of a fixed number (denoted by N) of identical submodules (SMs) connected in series and a buffer inductor. The SM usually adopts the half-bridge topology as shown in Fig. 1. The modular structure of the MMC makes its output voltage and power highly scalable. Low voltage rating and low switching frequency of individual power devices and high-quality waveforms are also among the benefits from its topology. Owing to these features, the MMC has been extensively employed in high voltage/power applications [2]– [6]. However, the unique structure of the MMC also presents several challenges. To begin with, the SM capacitor voltages have unavoidable ripples, which then give rise to the circulating currents. Both detrimental effects lower the efficiency and power quality of the MMC system, even posing threat to the stability of the system if left uncontrolled. Therefore, effective modulation and control techniques are essential for the MMC operation in order to achieve high-quality output current (and therefore active/reactive power), i.e., the SMs need to be switched properly according to the operating conditions. Main control objectives of an MMC generally include the regulation of phase currents, circulating currents and SMs capacitor voltages. As primary control objective, the phasecurrent control can be carried out in abc,αβ or dq frame, aiming at a well tracking of the corresponding references given according to specific applications [3]. The circulatingcurrent controller, normally based on resonant controllers, is designed in most cases to eliminate the AC component for loss reduction, or to track DC-plus-AC references for SM- capacitor-voltage regulation [2]. The SM-capacitor voltages are normally controlled in two stages, namely the arm-total- capacitor-voltage control (or arm-energy control) which regulates the DC component of SM capacitor voltages to the nominal value [7], and the capacitor-voltage balancing among the SMs within the same arm which is achieved depending on which modulation scheme is applied once the arm-voltage references are determined by the above-mentioned controllers. An extra individual capacitor-voltage-balancing controller is required for each SM if the gate signals are determined directly by the conventional pulse-width modulation (PWM) strategy [8], [9]. However, in high-power applications, the MMC usually contains several dozens or even hundreds of SMs per arm, thus making it complex to equip every SM with a carrier and an individual capacitor-voltage controller. From this perspective, it is preferable to determine the gate signals indirectly employing pseudomodulation techniques [10]: the number of SMs to be switched on (also called insertion index) of each arm is obtained first, followed by the sortingbased SM selection technique seeking for the balance of SM capacitor voltages. Among this stream of methods [7], [11]– [19], the nearest level control (NLC) [12], [13] is extensively adopted for its simple implementation, which is especially advantageous for high-power applications such as high-voltage direct current (HVDC) transmission [20]. Several finite control set-model predictive control (FCSMPC) schemes belong to this class [14], [15], where the insertion indexes of arms are determined employing predictive techniques, avoiding the huge amount of computation effort brought by evaluating all combinations of individual SM switching states. Normally, for each phase, different combinations of insertion indexes of the upper and lower arms are compared in order to select the optimal one, leading to (N+ 1)2candidates to be evaluated [15]. Several research works have dedicated effort to refine the candidates according to the operation [16]–[19]. However, the resulting size of control set is still, at least, proportional to N, and moreover, with possibility of losing optimality. As an alternative of the FCS-MPC method, the continuous control set-MPC (CCS-MPC) scheme derives the output voltage reference analytically and then applies the modulation stage to generate the switching signals [21], [22]. In the context of the MMC, several CCS-MPC techniques have been derived during the last years [23]–[25], which normally integrate the PWM scheme to generate the derived arm voltage reference, leading to a better regulated harmonic spectrum. In this paper, the FCS-MPC and the CCS-MPC techniques 3 are both considered to develop a joint control method to operate the MMC. The core step is to determine the arm voltage references of the two arms of each phase according to the cost function which includes various control objectives of the system in terms of the regulation of output current, circulating current, and arm energy. In order to achieve this task, a quadratic programming problem is structured considering the range of each arm voltage as a constraint. After obtaining the arm voltage references, the variable rounding level control (VRLC) method developed in [26] is employed to select the optimal insertion indexes of both arms of each phase among four candidates determined by the arm voltage references. The main contributions of this paper can be summarized as follows: •The arm-voltage references which correspond to the minimum cost-function value under the specified feasible set are directly obtained employing quadratic programming technique, instead of evaluating many combinations of arm-voltage levels as required by conventional FCSMPC schemes, leading to optimal performance defined by the cost function in terms of phase-current tracking, circulating-current tracking and arm-energy regulation. •Desired arm-voltage levels can be obtained with much reduced computational burden since only 4 evaluations of arm-voltage-level combinations per phase per control period are required, regardless of the number of SMs of the MMC, leading to similar (almost identical) performance with the conventional FCS-MPC method [15] that evaluates all the (N+1)2arm-voltage-level combinations. •Both rigorous and simplified procedures are provided for the generation of arm-voltage references, which can be integrated into different MMC-based applications in various operating conditions by simply implementing the proposed method as the inner control loop and applying the conventional outer control loop for the determination of phase/circulating-current references. •Further removing the barriers for the prospective industrial applications of the MPC techniques to the MMCs, especially those with large number of SMs in high voltage applications, allowing for an implementation based on microprocessors with reduced processing capability, leading to potential economic gains. The rest of the paper is arranged as follows. Section II derives the constrained optimization problems from the system model and the cost function. The solutions are provided in section III. Section IV addresses the determination of arm voltage levels and gate signals. Then, section V and VI provide the experimental and simulation validation with comparative analysis based on the results. At last, section VII concludes the paper. II. FORMULATION OF OPTIMIZATION PROBLEM A. System Model From the MMC topology shown in Fig. 1, the mathematical models of the output current and circulating current (of phase x=a, b, c) can be obtained as (Larm +2L)dix dt = (−Rarm −2R)ix−2vSx −vxu +vxl (1) (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. 2Larm dicirc,x dt =−2Rarmicirc,x +Vdc −vxu −vxl.(2) The AC side of the MMC is connected through an RL filter to a grid with phase voltage of vSx and the common-mode voltage of its neutral point relative to that of the DC side is neglected since it is reasonable in a balanced operation. The model is flexible because in case of off-grid connection, vSx is equal to zero and R,Ldenote the corresponding load parameters. To aid comprehension, the per-phase circuit of the investigated MMC-based system and the corresponding decoupled phase-current and circulating-current models are provided in Fig. 2. The total energy of all SMs of the upper/lower arm is denoted by Wxy (y=u, l). In the same phase x, the difference and sum of the total energy of both arms are defined as Wx∆=Wxu −Wxl (3) WxΣ=Wxu +Wxl (4) respectively. Their dynamics are dWx∆ dt =ixuvxu −ixlvxl (5) dWxΣ dt =ixuvxu +ixlvxl.(6) Applying the forward Euler approximation to (1), (2), (5) and (6) during time step kto k+ 1 (with sampling time of Ts), the corresponding discrete models can be derived as ˆ ix(k+ 1) = ix(k) + Ts Larm + 2L[(−Rarm −2R)ix(k) −2vSx(k)−vxu(k) + vxl(k)] (7) ˆ icirc,x(k+ 1) = icirc,x(k) + Ts 2Larm [−2Rarmicirc,x(k) +Vdc −vxu(k)−vxl(k)] (8) ˆ Wx∆(k+ 1) = Wx∆(k) + Ts[ixu(k)vxu(k)−ixl(k)vxl(k)] (9) ˆ WxΣ(k+ 1) = WxΣ(k) + Ts[ixu(k)vxu(k) + ixl(k)vxl(k)] (10) 4 where the total arm energy is given as Wxy(k) = C 2 N X z=1 vCxyz (k)2≈Cvdcxy(k)2 2N(11) neglecting the difference between different SM capacitor voltages of the same arm. vdcxy is defined as the sum of all the SM capacitor voltages of the corresponding arm, also serving as the maximum of the arm voltage as 0≤vxy(k)≤vdcxy(k) = N X z=1 vCxyz (k).(12) B. Cost Function 1) Considering only current tracking: In order to employ the MPC technique, the cost function is introduced as Ji=p1(i∗ x−ˆ ix)2+p2(i∗ circ,x −ˆ icirc,x)2(13) taking into account the output current and circulating current which are the most important control objectives of an MMC. p1,2>0denote the weighting factors and the variables with ∗ denote the corresponding references. The current references are determined by the outer control loop according to the applications and/or operating conditions (in most of the cases implemented by the conventional field-oriented control (FOC) and voltage-oriented control (VOC) for motor drives and grid connected applications respectively), which are considered here as given information and the investigation will be focused on the inner current control loop implemented with an MPC. Applying (7) and (8) into (13), the cost function can be expressed as a function of vxu(k)and vxl(k)as Ji(vxu(k), vxl(k)) = p1g2 1[vxu(k)−vxl(k) + f1(k)]2 +p2g2 2[vxu(k) + vxl(k) + f2(k)]2(14) where g1=Ts Larm + 2L, g2=Ts 2Larm (15) and f1,2(k)are defined as f1(k)=2vSx(k)+(Rarm + 2R)ix(k) + i∗ x(k+ 1) −ix(k) g1 (16) f2(k)=2Rarmicirc,x(k)−Vdc +i∗ circ,x(k+ 1) −icirc,x(k) g2 (17) For the sake of simplicity, setting λ1=p1g2 1, λ2=p2g2 2(18) then (14) can be reformed as Ji(vxu(k), vxl(k)) = λ1[vxu(k)−vxl(k) + f1(k)]2+ λ2[vxu(k) + vxl(k) + f2(k)]2.(19) (a) (b) Fig. 3. Geometrical interpretation. (a) The elliptic paraboloid in orthogonal coordinates. (b) 9 regions where the unconstrained solutions fall. 2) Considering arm energy regulation: To regulate the energy stored in the SMs, the cost function can be defined as follows Jw=p3(W∗ x∆−ˆ Wx∆)2+p4(W∗ xΣ−ˆ WxΣ)2.(20) Similar to Ji,Jwcan be arranged into the following form Jw(vxu(k), vxl(k)) = λ3[ixu(k)vxu(k)−ixl(k)vxl(k) +f3(k)]2+λ4[ixu(k)vxu(k) + ixl(k)vxl(k) + f4(k)]2(21) where λ3=p3T2 s, λ4=p4T2 s(22) f3(k) = Wx∆(k)−W∗ x∆(k+ 1) Ts (23) f4(k) = WxΣ(k)−W∗ xΣ(k+ 1) Ts .(24) Normally, Jwis not applied alone, but together with Jias Jiw =Ji+Jw.(25) C. Optimization Problem Within the frame of the MPC, the cost function (either Ji or Jiw) has to be minimized to achieve optimal performance. In the previously proposed MPC schemes, the minimization of the cost function is realized by evaluating different voltage levels of vxy(k)in an exhaustive manner, leading to a considerable computational cost. As an alternative proposed in this paper, the cost function can be minimized employing numerical optimization techniques. Thus, optimization problems need to be formulated. 1) Case of Ji:Considering optimizing Ji, it can be inferred from (19) that the cost function reaches the minimum (zero) when and only when the following equations are satisfied vxu(k)−vxl(k) + f1(k) = 0 (26) vxu(k) + vxl(k) + f2(k) = 0 (27) leading to the global solution as vxu_0(k) = −f1(k) + f2(k) 2, vxl_0(k) = f1(k)−f2(k) 2. (28) To gain a further insight into the cost function, (19) is altered into the following matrix form as follows. For simplicity, the 5 time instant kis not explicitly indicated. Throughout the paper, all variables denote those of time instant kunless otherwise stated. Ji(vxy) = vT xyQivxy + 2cT ivxy +di(29) where the related variables are defined as follows Qi=qi1qi2 qi2qi3=λ1+λ2−λ1+λ2 −λ1+λ2λ1+λ2(30a) ci=ci1 ci2=λ1f1+λ2f2 −λ1f1+λ2f2,vxy =vxu vxl .(30b) Since det(Qi)=4λ1λ2>0,(31) Qiis positive definite. Therefore, (29) is strictly convex with at most one minimum point [27], and (29) represents an elliptic paraboloid in orthogonal coordinates (vxu, vxl, Ji(vxu, vxl)) as shown in Fig. 3(a). The contour lines of Jiare a series of ellipses with the same center point (vxu_0, vxl_0), the same eccentricity, and the same major/minor axis with equation of (26)/(27). As the weighting factor λ2increases relative to λ1, the elliptic paraboloid is stretched along the axis of (27). However, the arm voltage is bounded as (12). Thus, the unconstrained solution (28) cannot be adopted directly. Since the constrains are linear, the optimization problem to be solved is a typical quadratic programming problem, which can be formulated as min vxy Ji(vxy),subject to Avxy b(32) where A=1 0 −1 0 0 1 0 −1T (33a) b=vdcxu vdcxl 0 0T.(33b) 2) Case of Jiw:Similar to (29) and (30), Jwcan be rearranged into the matrix form as Jw(vxy) = vT xyQwvxy + 2cT wvxy +dw(34) where Qw=qw1qw2 qw2qw3 =i2 xu(λ3+λ4)ixuixl(−λ3+λ4) ixuixl(−λ3+λ4)i2 xl(λ3+λ4)(35a) cw=cw1 cw2=ixu(λ3f3+λ4f4) ixl(−λ3f3+λ4f4)(35b) From (25), (29) and (34), Jiw can be derived as Jiw(vxy) = vT xyQiwvxy + 2cT iwvxy +diw (36) where Qiw =qiw1qiw2 qiw2qiw3=Qi+Qw(37a) ciw =ciw1 ciw2=ci+cw(37b) The problem to be optimized is min vxy Jiw(vxy),subject to Avxy b.(38) B B1 A1 Case 1 Case 2 Case 3 E1 A (a) 1 1 1 2 2 1 (b) Fig. 4. Solution for Regions II and III. Since det(Qiw)=4λ1λ2+ 4i2 xui2 xlλ3λ4+i2 x(λ1λ3+λ2λ4) +4i2 circ,x(λ1λ4+λ2λ3)>0(39) The optimization problem (38) is a positive-definite quadratic programming problem as (32). III. SOLVING FOR ARM VOLTAGE REFERENCES USING QUADRATIC PROGRAMMING A. Rigorous Solution 1) Solution of Ji:As mentioned above, the global solution of the unconstrained version of (32) can be obtained as (vxu_0, vxl_0)determined from (28). As an equivalent measure, (vxu_0, vxl_0)can be obtained from ∇Ji(vxy_0) = 2(Qivxy_0+ci) = 0.(40) Considering the constraints, several techniques are available such as the active-set, interior-point or first-order methods, which normally require multiple steps of iterations and/or solutions of a linear system of equations [28]. Since the feasible set of (32) is only two-dimensional, the solution can be derived explicitly from geometrical interpretation. According to the constraints of (32), the Euclidean plane of (vxu, vxl)can be divided into 9 regions as shown in Fig. 3(b). Apparently (vxu_0, vxl_0)is not the solution of (32) when it falls outside region I (the feasible region). In that case, the optimization problem (32) is equivalent to looking for the first point in region I that touches the elliptical contour lines expanding from the center (vxu_0, vxl_0). To solve (32), the 9 regions shown in Fig. 3(b) can be divided into three types: I, II and III. For each group, the procedures of obtaining the solution are explained as follows: 1) (vxu_0, vxl_0)falls into region I: In this case, the solution of (32) is (28), as shown in Fig. 3(b). 2) (vxu_0, vxl_0)falls into region II: Region II-1 is studied here as an example because the other three regions (II-2,3,4) are similar. In this situation, the optimum of (32) lies on segment AB and can be divided into three cases as illustrated in Fig. 4(a). Firstly, the point E1on the segment A1B1that first touches the 6 expanding contours (i.e., the tangent point) is obtained by solving ∂Ji ∂vxu vxl_E1=vdcxl = 2(qi1vxu_E1+qi2vdcxl +ci1)=0 (41) leading to vxu_E1=−qi2Vdcxl +ci1 qi1 (42) which specifies the coordinate of E1as (vxu_E1, vdcxl). Then, the solution of (32) can be achieved as follows: a) Case 1: if E1is on AB (0≤vxu_E1≤vdcxu), the solution is E1. b) Case 2: if E1is on AA1(vxu_E1<0), the solution is A. c) Case 3: if E1is on BB1(vxu_E1> vdcxu), the solution is B. 3) (vxu_0, vxl_0)falls into region III: Region III-1 is studied here as an example because the other three regions (III-2,3,4) are similar. In this situation, the optimum of (32) lies on segment AB or AC and can be divided into five cases as illustrated in Fig. 4(b). Firstly, the tangent points on the line A1B1and A2C1 that first touch the expanding contour lines are obtained as E1and E2in a similar manner with the case of region II. Then, the solution of (32) can be achieved as follows: a) Case 1: if E1is on AA1and E2is on AA2, the solution is A. b) Case 2: if E1is on AB, the solution is E1. c) Case 3: if E2is on AC, the solution is E2. d) Case 4: if E1is on BB1, the solution is B. e) Case 5: if E2is on CC1, the solution is C. The complete solution for all the regions are provided in the appendix. 2) Solution of Jiw:The solution of (38) can be obtained in a similar manner since it presents an objective function with the same constraints and structure introduced in (32). Similar to (40), the global optimal solution (without constraints) should satisfy ∇Jiw(vxy_0) = 2(Qiwvxy_0+ciw) = 0(43) leading to vxy_0=vxu_0vxl_0T=−Q−1 iwciw.(44) Then, the procedures described above for solving (32) can be repeated for (38). In terms of implementation, the above-described scheme of region assignation and solution determination can be well designed to minimize the involved calculation cost. The corresponding flowchart with the least possible calculations is outlined in Fig. 5. B. Simplified Solution Although the rigorous procedures do not involve complex calculations, (32) and (38) can be solved in a simpler manner to accelerate the implementation. During normal operation of Level 0 Level N Level 0 Level N Level N Level N (a) (b) Fig. 6. Determination of optimal arm voltage levels of (a) a normal case. (b) an abnormal case. the MMC, the sum of arm voltages of each phase should always be kept close to the DC-voltage as vxu +vxl ≈Vdc (45) in order to avoid generating high circulating current. Thus, all (vxu_0, vxl_0)should fall close to the diagonal line shown in Fig. 3(b) (the green dotted line) even if they are outside region I. In addition, the contour lines of the cost function are normally not extremely long-and-narrow. Considering the above factors, a simplified solution of (32) and (38) can be obtained by simply passing the unconstrained solution through a saturation block as v∗ xy =     0vxy_0<0 vxy_00≤vxy_0≤vdcxy vdcxy vxy_0> vdcxy (46) which is equivalent to the rigorous solution with λ1=λ2and λ3=λ4= 0, forcing the elliptic contour lines mentioned above to evolve into circular contour lines. IV. DETERMINATION OF ARM VOLTAGE LEVEL AND GATE SIGNALS From the references of arm voltages (denoted as v∗ xy) obtained from the proposed method, the SM gate signals are determined applying a specific modulation method, such as the phase-shift/level-shift PWM or the NLC. Since the objective of this paper is to determine the optimal arm voltage level of the MMC, the variable rounding level control (VRLC) pseudomodulation scheme proposed in [26] is adopted. The basic principle of VRLC is to select ceiling() or floor() rounding function after normalizing the arm voltage reference as N∗ xy = ceiling/floor v∗ xy vdcxy/N (47) and the decision is made by the FCS-MPC technique. Further details of the VRLC method can be found in [26]. It is worth noting that if Jiis employed without arm-energy regulating terms, vdcxy =Vdc has to be adopted as indirect capacitorvoltage regulation for stability [16]. To facilitate the understanding, the above concept is illustrated in Fig. 6. The whole feasible region is divided by (N+ 1)2candidates into N2units. In the case of Fig. 6(a), (47) adopts vdcxy =Vdc for normalization and thus each unit 7 Calculating Calculating Calculating Y N Y N N N Y Y Y Y Y N N Y N N N N Y Y Calculating Calculating Calculating Y N Y N N N Y Y Y Y Y N N Y N N N N Y Y Fig. 5. Flowchart of region-search and solution-determination scheme. is a square. In addition, Jiis adopted as the cost function leading to an angle of π/4between the major/minor axis of the contour lines and the vxy axis as inferred from (26) and (27). Therefore, it can be deduced that in this case, the final solution of N∗ xy can be obtained by just evaluating the four vertices of the unit where the unconstrained solution (the center of the elliptical contour lines) falls. The optimal candidate is marked by the red-star icon. It is important to mention that the optimal candidate obtained by the proposed technique is identical to that determined by the conventional MPC scheme [15], which however has to evaluate all the (N+ 1)2possible candidates. In the case of Fig. 6(b), the SM capacitor voltages are used for (47), and the N2units are thus rectangular but not necessarily square. In addition, Jiw is adopted as the cost function (with λ3,46= 0), making the major/minor axis of the contour lines form an angle different from π/4with the vxy axis. In this case, the reduced four-candidate-evaluation scheme adopted by the VRLC technique may not be applicable, and the solution may be outside the unit as shown in Fig. 6(b). In a real MMC implementation, however, vdcxu ≈vdcxl ≈ Vdc since the capacitor voltage ripples should be much lower than Vdc. Additionally, the incorporation of Jwfor arm energy regulation is auxiliary and should not change the slope of the major/axis (of the contour lines) to a large extent. Therefore, the assumption of Fig. 6(a) can be made and the VRLC technique can be applied without sacrificing performance compared with the original method evaluating all (N+ 1)2 candidates [15]. The last step is to select the individual switched-on SMs within each arm. The conventional sorting scheme [1] is employed to undertake the SM-capacitor-voltage-balancer (SMCVB) task as follows 1) if the corresponding arm current is positive, the SMs with the lowest capacitor voltages will be switched on. 2) if the corresponding arm current is negative, the SMs with the highest capacitor voltages will be switched on. 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 * Optimizer of (32)/(38) VRLC SM-CVB SM-CVB Phase-x of MMC Fig. 7. Block diagram of the entire proposed method (Z={1,2, ..., N}). The entire control scheme of phase-xis outlined in Fig. 7, which starts from a set of given references since their determination is not the focus of this paper. For a specified application, the proposed method can be easily integrated by simply introducing the corresponding outer-loop controllers that lead to the phase-current references i∗ x, such as the speed/torque and flux controllers in motor drive applications, active- and reactive-power controllers in grid-connected applications, etc. A detailed review of outer-loop-controller design can be found in [20]. In terms of the setting of circulating-current reference i∗ circ,x,idc/3or its DC component (determined based on the power balance of the MMC between its AC side and DC side) is commonly adopted [15], [18]. In addition, some harmonics can be included for various purposes [29]. For further enhancing the regulation of SM-capacitor voltages, i∗ circ,x can be given by extra arm-voltage controllers [25]. The proposed technique can also be incorporated into abnormal operations. As an example, in case of unbalanced grid conditions, negative-sequence component needs to be included in i∗ xto maintain a constant active- and reactive-power transfer and i∗ circ,x is recommended to be set as a pure DC value (instead of idc/3which contains ripples) for the regulation of zero-sequence and unbalanced circulating current [14], [23]. In addition, another issue in this scenario is that the commonmode voltage of the AC-side neutral point has to be considered in the phase-current model (1). 8 PHASE A PHASE B PHASE C Fig. 8. The experimental setup (left/right cabinet: upper/lower arms). TABLE I PARAMETERS OF THE MMC-BASED SYSTEMS UNDER STUDY. Parameter Value in Experiment Simulation Number of SMs per arm, N4 200 SM capacitance, C1 mF 20 mF Arm inductance, Larm 15 mH 40 mH Arm resistance, Rarm 0.5 Ω0.1 Ω DC-link voltage, Vdc 180 V 128 kV Grid voltage, peak value of vSx 0 57 kV Load/grid inductance, L15 mH 5 mH Load/grid resistance, R20.5 Ω0.1 Ω AC frequency, f50 Hz 50 Hz Sampling/control time, Ts0.2 ms 0.2 ms Remark: Since only four candidates are evaluated per phase per sampling period (regardless of N), the proposed method presents a significant advantage in terms of computational burden compared with the conventional FCS-MPC schemes which evaluate a great number of (at least proportional to N) candidates [14]–[18]. Taking into account the calculation of vxy_0, the worst-case implementation of Fig. 5 (both vxu_E1 and vxl_E2are calculated) encompasses operations of 5 additions/subtractions, 8 multiplications, 4 divisions and 7 value comparisons. Additionally, every single evaluation of the cost function encompasses operations of 5 additions/subtractions, 8 multiplications and 1 value comparison. Thus, the proposed approach achieves much lower computational cost compared with the exhaustive MPC method. This advantage is achieved even if the MMC has a reduced number of SMs and becomes a critical issue when the number of SMs is large. V. EXPERIMENTAL ANALYSIS To evaluate the performance of the proposed method, a three-phase MMC setup with N= 4 SMs per arm (shown in Fig. 8) is employed to conduct the experimental tests. Important system parameters are summarized in Table I. The proposed control scheme is implemented on the dSPACE 1007 platform. The one-sampling-period delay is compensated with the technique presented in [30], which applies an extra step of prediction before employing the proposed method in the second-step prediction. A. Steady-State Performance The steady-state tests are evaluated first. i∗ xis assigned a balanced three-phase sinusoidal waveform with peak value of 3.7 A. i∗ circ,x is set as a constant value of 0.86 A, which is determined manually (ideal DC reference [14] plus a slight adjustment accounting for the converter losses) instead of using a total-arm-energy controller [7] in order to observe the current tracking performance. W∗ x∆is set as zero and W∗ xΣis set as CN(Vdc/N −0.8)2considering the influence of IGBT voltage drop as 0.8 V. The performance of the proposed techniques including both the rigorous and simplified solutions is shown in Fig. 9 and Fig. 10, while the emphasis is given to phase-a. For comparative purposes, the conventional FCS-MPC method [15] is also tested as an optimal-performance reference since it evaluates all (N+ 1)2insertion-index combinations in each phase. All tests in Fig. 9 evaluate the cost function Jiwith λ1= 1, λ2= 0.25 while adopting vdcxy =Vdc for stability (though a perfect regulation of arm energy is not guaranteed as shown in the unbalanced capacitor voltage waveforms), and those in Fig. 10 evaluate the cost function Jiw with λ1= 1,λ2= 0.25, λ3= 1 ×10−3,λ4= 1 ×10−4. It can be observed that in both groups of tests, the proposed methods (both the rigorous and simplified solutions) exhibit similar performance with the conventional FCS-MPC method in terms of the phase current quality, circulating current ripple, and the waveforms of phase voltage and capacitor voltages. For a quantitative comparison, the THD (%, up to the 50th harmonic) of the phase current, the RMS value (A) of the circulating current ripple, and the average switching frequency (Hz) of all the SMs of phase-aapplying the different methods are summarized in Table II. More tests with different weighting factors are added for a more complete analysis. In all cases, both proposed solutions present almost identical results to the conventional exhaustive FCS-MPC method. B. Dynamic Response The dynamic performance of the above-mentioned methods is summarized in Fig. 11 (Ji) and Fig. 12 (Jiw). In the transient, the amplitude of i∗ xis reduced to half (1.85 A) and i∗ circ,x is reduced to 0.215 A correspondingly. Fast dynamics in terms of phase currents and circulating current (reflected by the DC-link current idc) are achieved in all tests. Similar to the steady-state tests, both proposed solutions present similar results with the conventional exhaustive control scheme. C. Computational Burden Analysis It is important to highlight that in terms of complexity, the proposed solutions involve much less computational burden than the conventional FCS-MPC scheme. The execution time (µs) of the determination of insertion indexes (neglecting the 9 (a) Conventional method (b) Conventional method (c) Rigorous solution (d) Rigorous solution (e) Simplified solution (f) Simplified solution Fig. 9. Steady-state performance of different methods using Ji(with vdcxy =Vdc): (a)(c)(e) show ia(green, 2.00 A/), icirc,a (pink, 2.00 A/), iau (blue, 2.00 A/), ial (red, 2.00 A/); (b)(d)(f) show vCau2(yellow, 10.0 V/), vCal2(green, 10.0 V/), va(blue, 50.0 V/); time (10.00 ms/). (a) Conventional method (b) Conventional method (c) Rigorous solution (d) Rigorous solution (e) Simplified solution (f) Simplified solution Fig. 10. Steady-state performance of different methods using Jiw: (a)(c)(e) show ia(green, 2.00 A/), icirc,a (pink, 2.00 A/), iau (blue, 2.00 A/), ial (red, 2.00 A/); (b)(d)(f) show vCau2(yellow, 10.0 V/), vCal2(green, 10.0 V/), va(blue, 50.0 V/); time (10.00 ms/). TABLE II SUMMARY OF STEADY-STATE PERFORMANCE OF DIFFERENT METHODS IN EXPERIMENTAL TESTS. Cost function Weighting factors Conventional method Rigorous solution Simplified solution THD RMS fsw tex THD RMS fsw tex THD RMS fsw tex Jiλ1= 1 λ2= 0.25 1.81 0.190 1196 4.32 1.81 0.187 1197 1.49 1.80 0.192 1199 1.45 Jiw λ1= 1 λ3= 1 ×10−3 λ4= 1 ×10−4 λ2= 0.25 1.57 0.177 1070 4.34 1.55 0.175 1069 1.50 1.53 0.173 1069 1.44 λ2= 1 2.02 0.131 1081 4.35 2.02 0.130 1082 1.49 2.06 0.126 1080 1.45 λ2= 4 2.87 0.126 1179 4.36 2.92 0.123 1180 1.49 2.88 0.126 1178 1.45 common part of the three methods such as calculating the matrix of cost function, delay compensation, etc.) of different tests are summarized in Table II. Both proposed solutions present much lower computational cost than the conventional exhaustive technique, which coincides with the analysis of calculation time in section IV. In addition, it is important to notice that the proposed rigorous solution presents only slightly higher calculation than the simplified solution. In any case, the potential reduction of the computational burden of both proposed methods becomes significant considering the real MMC applications with much higher number of SMs. This would permit to increase the sampling frequency of the MMC leading to improved output waveform quality. Also, it would permit to define more complex cost functions including extra control terms to improve the MMC performance. D. Advantage of Rigorous Solution In all the tests above, both the rigorous and simplified solutions show a similar behavior. However, in certain conditions where the transient of circulating current is dramatic, the advantages of the rigorous solution can be observed. An example of harmonic injection of circulating current is investigated. Fig. 13 shows the performances of different methods with Jiw (λ1= 1,λ2= 4,λ3= 5×10−3,λ4= 1×10−3). A negativesequence second harmonic is injected in the circulating current, which is eliminated from i∗ circ,x during the transient. In steady state, all the three methods exhibit similar quality of circulating current injection. In the transient, all cases show fast dynamic, while a slightly higher distortion of iacan be observed in Fig. 13(a) and Fig. 13(b). To further investigate the