Model predictive control: a review of its applications in power electronics

1 Model Predictive Control: A Review of Its Applications in Power Electronics Sergio Vazquez, Jose I. Leon, Leopoldo G. Franquelo, Jose Rodriguez, Hector A. Young, Abraham Marquez and Pericle Zanchetta Abstract— Model predictive control (MPC) for power converters and drives is a control technique that has gained attention into the research community. The main reason is that although MPC presents high computational burden it can handle multivariable case and system constrains and nonlinearities easily in a very intuitive way. Taking advantage of that, MPC has been successfully used for different applications such as active front end, power converters connected to RL loads, uninterruptible power supplies and high performance drive for induction machines among others. This paper provides a review of the application of MPC in the power electronics area. I. INTRODUCTION MODEL-BASED predictive control (MPC) presents a dramatic advance in the theory of modern automatic control [1]. Originally, MPC was studied and applied in the process industry, where it has been in use for decades [2]. Now, predictive control is being considered in other areas, such as power electronics and drives [3]–[6]. The reason for the growing interest in the use of MPC in this field is the existence of very good mathematical models to predict the behavior of the variables under control in electrical and mechanical systems. In addition, today’s powerful microprocessors can perform the large amount of calculations needed in MPC at high speed and reduced cost. The analysis of the research works published in the period form 2007 to 2012 in IEEE xplorer rperforming a search using the keywords ”predictive” and ”power converters” generate more than 200 papers of MPC applied to PWM power converters published in conferences and journals [7]. The applications covered by these research works can be categorized in four main groups: Grid connected converters, inverters with RL output load, inverters with output LC filters and high performance drives. Fig. 1 shows how these research works are distributed among these four groups. It is also interesting to study how these categories have attracted the attention of the research community along the last years. Fig. 2 and Fig. 3 present information about this issue. Fig. 2 shows that grid connected converters and high performance drives are the application where researchers have paid more attention being a current focus of interest. Fig. 3 addresses that research community attention has not decrease in this period and it is still increasing. It should be noticed the for all categories the cumulative line trends are positives. S. Vazquez, J.I Leon, L.G. Franquelo and A. Marquez are with the Electronic Engineering Department, Universidad de Sevilla (Spain), (e-mail: [email protected]). Grid conected Motor drives Inverter with output LC filter Inverter with RL load Fig. 1. Research works of MPC for PWM power converters published in IEEE conferences and journals from 2007 to 2012: Distribution regarding applications. 0 2 4 6 8 10 12 14 16 18 20 2007 2008 2009 2010 2011 2012 Grid conected Motor drives Inverter with output LC filter Inverter with RL load Fig. 2. Research works of MPC for PWM power converters published in IEEE conferences and journals from 2007 to 2012: Distribution regarding applications and year of publication. 0 10 20 30 40 50 60 70 80 90 2007 2008 2009 2010 2011 2012 Grid conected Motor drives Inverter with output LC filter Inverter with RL load Fig. 3. Research works of MPC for PWM power converters published in IEEE conferences and journals from 2007 to 2012: Cumulative analysis for each application category. This paper presents the use of MPC for the four main categories of applications for PWM power converters that can be found in the literature. This includes a variety of applications like: Grid connected converters; Inverters with RL output load, Inverters with output LC filters and High performance drives. Basic issues of well established MPC algorithms are presented for these applications and new challenges for MPC control for 2 power converters and drives are also addressed. II. THE MPC CONTROL STRATEGY Predictive control is understood as a wide class of controllers which main characteristic is the use of the model of the system for the prediction of the future behavior of the controlled variables over a prediction horizon N. This information is used by the MPC control strategy to provide the control action sequence for the system by optimizing a user defined cost function [8]. It should be noticed that the algorithm is executed again every sampling period and only the first value of the optimal sequence is applied to the system at instant k. The cost function can have any form but usually it is defined as g=X i λi(x∗ i−xp i)2.(1) Where x∗ iis the reference command and xp ithe predicted value for variable xi,λiis a weighting factor and index istands for the number of variables to be controlled. In this simple way it is possible to include several control objectives (multivariable case), constrains and nonlinearities. The predicted values xp i are calculated by means of the model of the system to be controlled. A. MPC for power converters Application of MPC for power converters has increased due to the improvement of digital microcontrollers [3], [9]. This control technique requires a non negligible amount of calculations during small sampling times when applied for controlling power converters and drives. There are several approaches to face the computational burden problem. In some cases, it is possible to solve the optimization problem offline by multi-parametric programming, thus the implementation is reduced to some calculations and a look-up-table [10]. Another way consists of using predictive techniques as generalized predictive control (GPC). GPC provides an online solution to the optimization problem and can be used for long prediction horizons without increasing significantly the computational cost [8], [11]. It should be noticed that GPC does not take into account switching of power semiconductors when it is applied for power electronics and drives. Therefore GPC only gives an exact solution to an approximated optimization problem. This approach can be followed when an explicit solution to the problem can be found. Usually this requires an unconstrained problem, but it calculates the output voltage reference to the inverter. This voltage should be generated by a PWM-SVM modulator. Thus GPC technique can take advantage of well established knowledge about PWM-SVM to optimize some aspects of the power converter systems [12]. Finally, the discrete nature of power converters can be considered for implementing MPC control strategies. In this way, finding the solution to the optimization problem can be reduced to evaluate the cost function only for the prediction of the system behavior for the power converter possible switching states. As a finite number of control actions are evaluated, this Converter Load Predictive model Optimization Measurement and estimation J Fig. 4. FCS-MPC block diagram. Optimize Measure Apply Store Evaluate Predict Fig. 5. Time diagram of the execution of the FS-MPC algorithm. approach is called finite control set MPC (FCS-MPC). This technique has been extensively used for power converters due to the finite number of switching states they present [6]. B. FCS-MPC control principle Fig. 4 shows the block diagram of FCS-MPC, where a generic converter is used to feed a generic load. The converter presents Jdifferent switching states. The control objective pursuits that variable xhas to follow the reference x∗. The FCS-MPC algorithm has the following basic steps: 1) Measure and/or estimate the controlled variables. 3 n Load Fig. 6. Power circuit of the AFE. 2) Apply the optimal switching state (computed in the previous sampling period). 3) For every switching state of the converter, predict (using the mathematical model) the behavior of variable xin the next sampling interval xp. 4) Evaluate the cost function, or error, for each prediction as, for instance: g=|x∗−xp| 5) Select the switching state that minimizes the cost function, Sopt and store it for being applied to the converter at the next sampling period. As discussed in [13] it is convenient to perform the prediction two time steps ahead, in order to reduce the effects of the delay introduced by the implementation of FCS-MPC in a digital platform. Another possibility to avoid the effect of the computation delay is to use a control strategy that only requires a small computation time. In this way, the optimal switching state is applied to the converter with this small delay, and before the following sampling instant [14]. A time diagram of the execution of the FCS-MPC algorithm is presented in Fig.5. III. MPC FOR GRID CONNECTED CONVERTERS Several applications use grid connected converters as one of their main components. This application includes active front end (AFE) for high performance drives, rectifiers, grid integration of renewable energies like wind or PV, energy storage systems, and are also used in FACTS devices as STATCOM, active power filter, or as a part of an UPFC or UPQC [15]–[17]. A. Control of an Active Front End The power circuit of a grid connected converter through a smoothing inductor Lis presented in Fig. 6. As shown, the main system variables are the grid current iL,abc, grid voltage vS,abc and the output capacitor dc-link voltage vdc. The load connected to the dc-link represents any generic load connected to an AFE. Thus it can be a resistor for a rectifier, a PV panel or a converter to control the torque and/or speed of a wind turbine for grid integration of renewable energies, etc. The main objective of the control strategy is to calculate the output inverter voltage vI,abc in order to regulate the output J Predictive model Minimization of cost function Sopt n Load J PI Fig. 7. Block diagram of the FCS-MPC control strategy for the AFE. Fig. 8. Experimental results of the FCS-MPC-DPC for a three-phase twolevel AFE. dc-link capacitor voltage to a reference v∗ dc for any connected load and inject into the grid any reactive power command reference q∗. There are several alternatives for designing the control algorithm for an AFE. In general, a cascade control structure is used. An external control loop is employed to regulate the dc-link voltage. On the other hand an internal control loop is adopted to track the grid currents or the instantaneous active and reactive power references regarding the states variables used to develop this controller [18], [19]. MPC has been mainly used as control strategy for the inner control loop. Although some works developing grid current controllers can be found in the literature the main approach has been the direct power control (DPC) for tracking the commands for the instantaneous active and reactive powers, Pand Q. Application of FCS-MPC-DPC and predictive DPC 4 n Load J Predictive model P-DPC J PI PWM-SVM Sabc vI;abc Fig. 9. Block diagram of the P-DPC control strategy for the AFE. with SVM modulation strategy (P-DPC) can be considered as well established [14], [20]–[22]. The block diagram of the FCS-MPC-DPC strategy is presented in Fig. 7. In this case, the model of the system is used to predict values of the instantaneous active and reactive power over a prediction horizon N= 1,Pp(k+ 2),Qp(k+ 2). In [14], [20] a three-phase two-level AFE was controlled adopting this strategy. The algorithm was developed in the αβ frame. Therefore, only the seven possible output vectors were considered to perform the prediction, thus the number of switching states is J= 7. Once the seven output voltage predictions are calculated the cost function g= (P∗(k+2) −Pp(k+2))2+ (Q∗(k+2) −Qp(k+2))2(2) is minimized in order to find the inverter output vector that should be applied in the next sampling period. Fig. 8 presents experimental results obtained using this strategy [14]. It should be noticed that predictions in instants (k+ 2) are used in order to compensate for the control action delay of the digital implementation of the control strategy. Another way to perform the predictive controller for the AFE is the P-DPC. The block diagram of the P-DPC strategy is presented in Fig. 9. Like GPC, P-DPC strategy does not take into account switching of power semiconductors therefore it just provides an exact solution to an approximated optimization problem. Besides, P-DPC considers an unconstrained MPC problem. Thus an explicit solution can be obtained providing the control action to be applied once the cost function (2) is minimized. Therefore, an optimal switching vector sequence can be calculated. The control strategy provides the switching vectors and the switching times thus a PWM-SVM modulation strategy is necessary to generate the firing pulses. Compared with FCS-MPC, the P-DPC algorithm uses an external modulator thus constant switching frequency is obtained. This can be considered as an advantage specially in the AFE application because for grid connected converters exist high demanding codes that impose strict limits to the low order harmonics that can be injected into the grid. FCSMPC presents variable switching frequency thus grid current has a widespread harmonic spectrum. On the other hand P- DPC provides constant switching frequency thus grid current harmonic spectrum is concentrated around the switching frequency decreasing the cost of the output L filter. Fig. 10 shows experimental results obtained using the P-DPC strategy for a STATCOM application when an instantaneous reactive power command step is imposed [22]. 10 0.2 0.4 0.5 0.6 0.8 0 100 200 300 400 500 600 700 800 Time (s) vdc [V] vdc v? dc (a) 0.45 0.47 0.49 0.51 0.53 0.55 −10 −5 0 5 10 Time (s) q [kVAr], p [kW] q q? p (b) 0.45 0.47 0.49 0.51 0.53 0.55 −400 −300 −200 −100 0 100 200 300 400 Time (s) vSa [V], iLa x10 [A] vSa iLa x10 (c) Fig. 10. Experimental results of the P-DPC for a three-phase two-level AFE for a reactive power command step from 10 to -10 kVAr. (a) dc-link voltage; (b) Instantaneous active and reactive power; (c) Grid voltage and output current for phase a. It should be noticed that the outer control loop to regulate the dc-link capacitor voltage is usually solved using a conventional PI controller. However, there exist some solutions that replace the PI control for a MPC strategy [14]. B. Control of an Active Filter In its classical configuration, an active power filter (APF) basically consists of a voltage source inverter (VSI) whose dc side is connected to a capacitors bank whereas its ac side is connected to the mains through a suited filter, usually formed by a set of series inductors as shown in Fig. 11 (referring to the most common 3-wires configuration without neutral). In such configuration, ideally the APF is able to operate as a controllable current generator, drawing from the mains any set of current waveforms having a null sum. Therefore, an APF is ideally able to compensate the unbalanced, reactive and harmonic components of the currents drawn by any load, in such a way that the global equivalent load, as seen from the grid, resembles a resistive balanced load drawing about the 5 Non-linear load ilb ila ilc iSa iSb iSc Fig. 11. Power circuit of a 3-wires APF. J Predictive model Minimization of cost function Sopt JNon-linear load ilb ila ilc iSa iSb iSc Reference design Sopt(k) vS,abc(k) iL,abc(k) il,abc(k) J Fig. 12. Block diagram of the FCS-MPC control strategy for the APF. same active power. In fact, since under steady-state conditions the voltage of the dc bus is intended to remain about constant and close to the design level to permit an indefinitely long operation then in practice the currents drawn by the APF must give rise to a small net average power flow to exactly balance its internal losses. The active filter modeling procedure and the reference quantities calculations are reported in [23], while the control block scheme is shown in Fig. 12. The MPC uses the prediction model and the reference derivation to select the switching functions S(k+ 1) which minimize a cost function as: g=λ1 vdcR (ev∗ dc(k+ 2) −vp dc(k+ 2))2(3) +λ2 PSR (P∗ S(k+ 2) −Pp S(k+ 2))2 +λ3 PSR (Q∗ S(k+ 2) −Qp S(k+ 2))2 Where v∗ dc,P∗ Sand Q∗ Sare the reference values of dclink voltage, active and reactive power respectively; vp dc,Pp S and Qp Sare predicted values of dc-link voltage, active and reactive power respectively; PSR, and vdcR are respectively (a) (b) (c) Fig. 13. Experimental results of the FCS-MPC for a 3-wired three-phase two-level APF. (a) Supply current before compensation; (b) Supply current after compensation and supply voltage. (c) Spectrum of currents in (a) and (b). rated values of active power and dc link voltage, λ1,λ2and λ3are weighting factors that allow a proper balance among deviations in voltage, active and reactive power, ev∗ dc(k+ 2) represents a filtered voltage reference with a certain prediction horizon [23]. According to the scheme of Fig. 11, the APF experimental prototype chosen to verify the effectiveness of the FCS-MPC control includes a standard 3-leg IGBT based VSI inverter. The implemented control algorithm operates at fixed sampling frequency fs= 50 kHz. To reproduce a distorted current in the 6 grid, a non-linear load constituted by a 3-phase diode bridge rectifier supplying a resistor having a rated power PSR = 5 kW was considered [23]. At full power the load draws the distorted current in Fig. 13a where the vertical axis measures 5A/div while the horizontal one 10ms/div; after APF compensation the mains currents waveform is shown in Fig. 13b (5A/div) together with the supply voltage (100V/div). The compensation action results in a unity power factor operation and quasi sinusoidal current with a superimposed high-frequency ripple due to inverter commutation and the nature of the FCS-MPC control action itself. The achieved benefits and therefore the effectiveness of the control action were also confirmed in spectral terms by comparing the mains current spectrum and the load current spectrum in Fig. 13c, resulting in a reduction of major loworder harmonics, which permits to achieve a THD<5% starting from a THD>29%, where the THD is calculated including up to the 50th harmonic. IV. MPC FOR INVERTERS WITH RL-LOAD A. Control of a matrix converter The direct matrix converter (DMC) is a type of static power converter in which the load is directly connected to the mains through a set of bidirectional switches. The power circuit of a DMC is shown in Fig. 14. This topology does not require a dc-link stage with energy storage as most power converters; this reduces the weight and size of the converter, making it suitable for applications that require high power density, such as aerospace. Control approaches based on FCS-MPC have been extensively tested for the DMC [24], [25], showing the effectiveness and relative simplicity of the predictive methods over the traditional ones such as space-vector modulation [26]. The block diagram of the predictive control of load current and input power factor is presented in Fig. 15. The predictive controller relies on mathematical models for the prediction of both the input reactive power and the output current. Each prediction block depicted in Fig. 15 yields 27 predictions for the controlled variables, one for each of the different valid switching states of the DMC. A further stage of the algorithm then minimizes a cost function in order to determine the optimal combination of gating signals to be applied to the converter at the next sampling period. The cost function for the simultaneous control of input reactive power Qand output current iois the following: g=|i∗ o−ip o|+λ|Q∗−Qp|,(4) where i∗ o,Q∗,ip oand Qpare the reference and the predicted values of the output current and the input reactive power, respectively; λis a weighting factor used to adjust the relative importance of both control objectives within the cost function. Results of the predictive control of the DMC are shown in Fig. 16. The output current tracks its reference accurately, as can be seen in Fig. 16a. The input reactive power is controlled starting from time t= 0.4 [s], in Fig. 16c. It can be observed that from that instant on, after a short transient, the input current becomes sinusoidal and in phase with the line voltage. Load Direct Matrix CoverterInput filter Fig. 14. Power circuit of a three-phase direct matrix converter. Input Filter DMC Minimization of cost function 9 3 Reactive power prediction 27 27 Output current prediction -Load Fig. 15. Block diagram of the predictive current and reactive power control of a direct matrix converter. B. SHE and SHM for power converters 1) Selective harmonic elimination technique: The selective harmonic elimination (SHE) strategy is specially well suited for high-power applications where the power losses must be kept below strict limits. The well-known SHE technique is based on offline calculations and the generation of a preprogrammed voltage waveforms with very low number of commutations and eliminating some low order harmonics [27]. Using the predictive control to implement the SHE method, the MPC-SHE control objective is to follow the voltage reference, to eliminate low order harmonics and to reduce switching losses [28]. These three objectives are included in the cost function g=SDFTf1{|v∗ s−vp s|} (5) +λf M X i SDFTfi {|v∗ s−vp s|} (6) +λswxp sw,(7) i= 0,2,3,4, . . . , M. In this cost function, the sliding discrete fourier transform (SDFT) is used. The SDFT is a recursive implementation of 7 00.02 0.04 0.06 0.08 0.10 0 10 5 -10 -5 ia [A] (a) 00.02 0.04 0.06 0.08 0.10 0 400 200 -400 -200 va [V] (b) Time [s] is vs 00.02 0.04 0.06 0.08 0.10 0 10 5 -10 -5 is [A] 0 400 200 -400 -200 vs [V] (c) Fig. 16. Predictive current control of a direct matrix converter. (a) Output current. (b) Output voltage. (c) Input current and grid voltage (phase a). the discrete Fourier Transform algorithm used to calculate a finite number of single frequency spectral components with very low computational cost [29]. The first term (5) evaluates the error between the reference and the predicted output voltage vector tuned to the fundamental frequency f1. The second term (6) is the sum of all those frequencies (up to Mth order) that need to be eliminated. The weighting factor λfis used to control the importance of this term in the cost function, in this way the frequency elimination can be relaxed or strengthened in comparison to the fundamental frequency tracking, depending on design considerations. Finally, the third term (7) is used to reduce the number of commutations introducing weighting factor λsw to keep the power losses below acceptable limits. 2) Selective harmonic mitigation technique: An evolution of SHE is the selective harmonic mitigation (SHM) technique which is based on pre-programmed waveforms non eliminating Predictive Model Minimization of cost function SDFT f0 SDFT fK . J Weighting Factors Tuning executed each 20ms MPC-SHM technique executed each Ts Power Converter Sopt λi J J + - . . . . v s * v s v s p Fig. 17. Block diagram of the proposed MPC-SHM technique. Vdc - + H-Bridge2 a H-Bridge3 a H-Bridge2 b H-Bridge3 bH-Bridge3 c H-Bridge2 c H-Bridgek j - + - + - + - + - + - + - + H-Bridge1 aH-Bridge1 bH-Bridge1 c - + - + - + S1S2 S1S2 N La Ra van Lb Rb vbn Lc Rc vcn n a b c iaibic Fig. 18. Power circuit of a H-bridge multilevel converter (three H-bridges per phase) connected to a RL load. the low order harmonics but reducing the distortion below the limits imposed by a grid code [30], [31]. The control objective of the MPC-SHM is to follow the voltage reference, to control the harmonic distortion keeping it below the limits imposed in the grid code and to reduce the switching losses as much as possible [32]. These three objectives are included in a cost function which is similar to that introduced for the MPC-SHE method. In this case, the second term (6) has to be modified being the sum of 8 1.79 1.795 1.8 1.805 1.81 1.815 1.82 1.825 1.83 1.835 1.84 200 300 400 Phase Voltage v aN [V] 1.79 1.795 1.8 1.805 1.81 1.815 1.82 1.825 1.83 1.835 1.84 −400 −200 0 200 400 1.79 1.795 1.8 1.805 1.81 1.815 1.82 1.825 1.83 1.835 1.84 −300 −200 −100 0 100 200 300 1.79 1.795 1.8 1.805 1.81 1.815 1.82 1.825 1.83 1.835 1.84 −15 −10 −5 0 5 10 15 0 500 1000 0 0.2 0.4 0.6 0.8 1 0 500 1000 0 0.2 0.4 0.6 0.8 1 0 500 1000 0 0.2 0.4 0.6 0.8 1 0 500 1000 0 0.2 0.4 0.6 0.8 1 Line Voltage v ab [V] Load Voltage v an [V] Phase Currents iabc [A] Time [s] Frequncy [Hz] 0 100 Fig. 19. Predictive harmonic mitigation phase output voltage vaN , line voltage vab, load voltage van and load currents ia,ib,icfor a 7-level CHB. the distortion of those harmonics that have to be kept below acceptable limits imposed by the grid code from the 2nd to the Kth harmonic. In fact, the mathematical expression of the MPC-SHM cost function is similar to the MPC-SHE one but considering harmonics up to Kth order where Kis higher than M. Another difference between both cost functions is that a weighting factor λiis associated to each ith order harmonic distortion that has to be reduced. This fact gives the chance to relax the condition of eliminating the distortion of those harmonics. The proposed MPC-SHM technique can be summarized using the block diagram represented in Fig. 17 where a weighting factors tuning block has been added to the previously introduced MPC-SHE method. For each harmonic, the weighting factors tuning block relaxes the weighting factor λiif the ith harmonic distortion is below the limit of the grid code or make it higher in the opposite case. This dynamic weighting factors adjustment is made at fundamental frequency. The MPC-SHM method has been tested in a three-phase cascaded H-bridge multilevel converter (three H-bridges per phase) connected to a RL load as depicted in Fig. 18. On the other hand, Fig. 19 shows the converter phase output voltage VaN , the line-line voltage Vab, the load voltage Van and the load currents ia,iband ic. In addition, the respective harmonic spectra are shown next to each waveform. A deep analysis of the data shows that the distortion of harmonics considered in the cost function (up to harmonic 10th) is always below the limit imposed. In addition, the average switching frequency of the MPC-SHM method is lower to that obtained with the MPC-SHE technique dealing with the same number of harmonics. This phenomenon appears because the SHM relaxes the conditions of the harmonic distortions compared with the SHE method and this fact makes easier to find better solutions leading to lower power losses. C. Control of multilevel inverters The FCS-MPC method has been applied to multilevel converters for multiple applications. Among the multilevel converter topologies, the neutral-point-clamped converter (NPC), the flying-capacitor converter (FC) and the cascaded H-bridge converter (CHB) are the ones with vast industrial success. These topologies are normally used for medium-voltage highpower applications at the expense of a large number of power semiconductors and more complex control and modulation algorithms. The FCS-MPC method for multilevel inverters has to take into account the usual control objectives present in other converter topologies and applications but extra control targets have to be included such as the balance of the floating dc voltages (if needed) and the reduction of the switching losses (required because for high-power applications the effective switching frequency and consequently the power losses have to be limited). Some examples are here addressed: 1) NPC inverter topology: The three-level NPC converter has the dc-link bus divided in two parts that should be balanced. So, this fact has to be included in the cost function. The dc voltage balance is achieved by the FCS-MPC method but at the expense of changing the switching state almost each sampling time. So, the result is not satisfactory because it leads to high switching losses. So, a limitation in the switching frequency has to be included in the cost function as well. Thus a possible cost function could be g=|i∗ α−ip α|+|i∗ β−ip β|+λdc |vp C1−vp C2|+λnnc.(8) In the cost function, the first term is focused on the current tracking which is the application of this FCS-MPC method. The second term is proportional to the absolute difference between the voltage predictions of both capacitors, so a switching state that generates smaller differences will be preferred leading to a voltage balance situation. Finally, the third term is proportional to the number of commutations to get to the next switching state nc, so a switching state that implies fewer commutations of the power semiconductors will be preferred. The weighting factors λdc and λnhandle the relation between terms dedicated to current reference tracking, voltage balance and reduction of switching frequency [33]. 2) CHB inverter topology: The FCS-MPC method can be also applied to achieve current tracking control in a three-phase cascaded H-bridge converter with two H-bridges per phase [34]. In this case, the cost function only takes into account the current tracking: g=|i∗ α−ip α|+|i∗ β−ip β|.(9) In the CHB inverter case, there are a high number of possible switching combinations (125 for this topology where the dc voltages of the H-bridges are fixed) so the computational cost of the FCS-MPC method can become excessively high. In [34], this is solved eliminating the redundant switching states with higher common-mode voltages. In addition, only the last 9 v dc Capacitor Voltages [V] 100 200 300 400 Inverter Output Voltage [V] Output Currents [A] 100 200 300 400 0 1 2 3 4 5 -1 -2 -3 -4 -5 THD =1.5% c (a) v c a1 v c a2 10 20 05 1 5 Time [ms](c) (b) ratio 3:2:1 van i a i c i b 0 0 Fig. 20. Experimental results for the FCS-MPC applied to the FC with ratio 3:2:1: Capacitor voltages of phase a, inverter output voltage of phase a and output currents (taken from [35]). applied switching state and the six states surrounding it in the space vector diagram are taken into account. This fact reduces to seven the number of possible combinations leading to the same computational cost of a conventional three-phase twolevel converter at the expense of solving a suboptimal problem and loosing dynamic response. 3) FC inverter topology: A similar FCS-MPC strategy can be applied to the FC converter. In [35], it is presented a cost function to achieve current tracking and floating voltages control of a three-phase FC converter with two floating capacitors per phase. In this case, the cost function includes the current tracking term and the floating voltages control term as follows: g=ga+gb+gc(10) where, ga,gband gcfollow the next expression (x∈a, b, c) gx= (i∗ x−ip x)2+λdc1(v∗ c1x−vp c1x)2+λdc2(v∗ c2x−vp c2x)2. (11) An interesting point to be highlighted is that, with the three-phase FC converter with two floating capacitors per phase, there are 512 possible switching combinations so the Load LC Filter Fig. 21. Power circuit of a three-phase inverter with output LC filter. Load LC Filter Predictive model Minimization of cost function Sopt J Fig. 22. Block diagram of the FCS-MPC control strategy for a three-phase inverter with output LC filter. 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 −300 −200 −100 0 100 200 300 vC;abc [V] 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 −20 −10 0 10 20 Time [s] iO;abc [A] Fig. 23. Experimental results for the FCS-MPC: output voltages and currents in steady state for a non-linear load and a reference amplitude of 200V. computational cost of the FCS-MPC method can become excessively high. In [35], this is solved ignoring the interaction through the load neutral point in the prediction step. This reduces the possible switching combinations to 24 leading to a high reduction in the computational cost at the expense of limiting a control degree. In order to illustrate the good performance of FCS-MPC method for multilevel converters, the results for the FC inverter are represented in Fig. 20. As can be observed, the control objectives, current tracking and control of the floating voltages, are achieved. V. MPC FOR INVERTERS WITH OUTPUT LC FILTER Inverters with output LC filter are used when it is necessary to obtain a sinusoidal output voltage with very low harmonic content. This is the case of sensitive loads, or drives for