Evolutionary Gaps Stator Current Control of Multi-phase Drives Balancing Harmonic Content

Abstract—Multi-phase machines are increasingly used as Voltage Vector (VV). The control action is obtained miniin research and industry applications due to their inherent mizing a cost function (CF) having several objectives balanced advantages. Stator current control is a common strategy by Weighting Factors (WF). Previous works have considered face for this type of systems. The most important issue it must is regulation of currents in the torque producing plane parametric tuning of the CF [2]–[4], as well as the design and the harmonic plane. For this task, Finite Control Set of parameter-free CFs [5], [6]. However, it has been shown Model Predictive Control (FCS-MPC) constitutes an interthat trade-offs between figures of merit are unavoidable by esting alternative to methods using modulation. However, simply adjusting the CF [7], [8]. Researchers have considered the implementation of FCS-MPC is characterized by a high reducing the sampling period to improve current tracking. This computational demand, limiting the sampling frequency. work proposes a predictive algorithm that needs less In addition, wide-band converters achieve high commutation computation time. As a result, the sampling period can frequencies, needing high sampling frequencies [9]. However, be reduced while producing predictive control. This brings FCS-MPC is intensive in computation. As a result, a limit about several benefits resulting from improved current on the sampling frequency is found for any given computing tracking. The proposed method avoids the combinatorial hardware. Fast computation of the control action has become time-consuming optimization phase of standard FCS-MPC which is the most part. The algorithm is based on physical an interesting research topic in recent years. The case of threeinsights obtained from the application of FCS-MPC to multiphase converters is easier to study as there are no x−y currents phase drives leading to the concept of evolutionary gaps and the dynamics of the load is simpler [10]. The case of multiregions. The experimental results for a five-phase motor phase drives is more demanding. One alley of research has demonstrate improved performance. Moreover, the method been restraining the combinatorial space. For instance, in [11] tween is flexible enough to balance the trade-off appearing bethe torque producing plane and the harmonic plane. the Voltage Vector (VV) search is restricted to an area within 60 degrees of the voltage space diagram. Similarly, in [12], an angular region is pre-selected to reduce the size of control phase Index Terms—AC motor drives, Digital control, Multiset to 7 candidates for bipolar charging stations. Similarly, systems, Predictive control, Table lookup. in [13], only the outermost maximum vector are considered FCS-MPC is a direct digital control scheme where the referred to as Synthetic Voltage Vector (SVV). The method control moves are the Voltage Source Inverter (VSI) configuraaims at reducing the harmonic plane (x−y) content by issuing tions. Each configuration produces a voltage that is referred to the VV in pairs [15]. The optimization is simplified as just the torque-producing plane is used. Also, the number of SVV is usually less than that of the full control set of VVs. For Manuscript received Month xx, 2xxx; revised Month xx, xxxx; acexample, ten SVV are applied in [16] for a 5-phase VSI, 12 e cepted Month x, xxxx. This work was supported by Ministerio de Ciencia Innovacion ´ of Spain, Agencia Estatal de Investigacion, ´ NextGenerain [17] and 18 in [18]. Other proposals, related to SVV, use tionEU, grant TED2021-129558B-C22, and by Ministerio de Ciencia e the deadbeat concept [19]. It must be also remarked that, in Innovacion ´ of Spain, Agencia Estatal de Investigacion, ´ grant PID2021- all SVV approaches x − y currents are actually regulated in for the optimization. However, these approaches do not fully utilize the VV set that the converter can produce. Avoiding F ibility INITE has to been incorporate Control proposed Set dif Model for ferent electrical Predicti objectiv systems es. ve Control This due is ( particularly FCS-MPC to its flex- ) this algorithm the problem, in [14], a Sphere Decoding reduction can be Optimization compute (SDO) is proposed to reduce control action. However, the the time needed to improved useful in multi-phase Induction Machines (IM), where several since SDO does not completely eliminate iterations. Another research line is that of Virtual Voltage Vector also IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Evolutionary Gaps Stator Current Control of Multi-phase Drives Balancing Harmonic Content Manuel R. Arahal, Federico Barrero, Manuel G. Satu´ e, Cristina Mart´ ın and Mario Berm´ udez I. INTRODUCTION planes (not all torque producing) must be dealt with [1]. 125189OB-I00. Manuel R. Arahal and Manuel G. Satu´ e are with the Systems Engineering and Automation Department of the University of Seville, Seville, 41092, Spain (e-mail: [email protected]; [email protected]). Federico Barrero is with the Electronic Engineering Department of the University of Seville, Seville, 41092, Spain (e-mail: [email protected]). Cristina Mart´ ın and Mario Berm´ udez are with the Electrical Engineering Department of the University of Seville, Seville, 41092, Spain (e-mail: cmar[email protected]; [email protected]). (Corresponding author Manuel G. Satu´ e, phone: +34 954487353; fax: +34 954487353; e-mail: [email protected]). open loop, which is a major issue [20]. General methods for faster computation of the control action have also been used, for instance, Explicit Model Predictive Control (EMPC). EMPC is a technique where a quadratic programming problem is split into multiple local sub-problems each valid on a certain region [21]. Real-time implementation of the controller involves only the evaluation of the actual region to retrieve the appropriate control law. An example, is [22], where the current control of a three-level neutral point clamped inverter is tackled. The current error and the voltage balancing of the capacitor are considered, dividing the space vector set into sectors. Although the method is viable, it has been reported that reducing the set of available VVs reduces flexibility. In [23] the EMPC technique is considered for a permanent magnet synchronous motor. Another EMPC algorithm is presented in [24], where pre-computed control laws represented by parameterized gains are provided. In both cases, instead of FCS-MPC a modulation block is used. Also, the region-dividing methodology is applied systematically, without realizing that a better partition is possible considering physical insights. In this regard, in [25] a new division that allows for faster computation is presented. However, the tradeoff between α − β and x − y planes is not tackled. A fast method for the FCS-MPC optimization phase for stator current control of multi-phase IMs is presented. The justification for such proposal is the realization that a sampling period reduction benefits the drive operation [7]. The proposal explicitly considers the torque-producing plane (α−β) and the harmonic plane (x − y). The method avoids the combinatorial optimization of FCS-MPC. As a result, the computational burden is lower than that of FCS-MPC. This allows for a reduction in sampling time which improves tracking. The proposal does not rely on SVV for the regulation of x − y currents, instead they are considered in closed loop. Moreover, the proposed method is flexible enough to balance the trade-off The novelty derives from the application of the Evolutionary Gap concept, which has been recently introduced by the authors. Such concept is based on insights obtained from the application of FCS-MPC to multi-phase IMs. It allows converting the combinatorial optimization into a region determination problem. Furthermore, the derivation of the regions is based The rest of the paper is organized as follows. Section II presents the basics of FCS-MPC for controlling the stator current of a 5pIM. The proposed method is introduced in Section III, where an algorithm is given for its real-time implementation and is then evaluated using a laboratory setup II. FCS-MPC FOR THE STATOR CURRENT CONTROL OF 5PIM In FCS-MPC, a predictive model provides predictions of stator currents for time k + 2. The VSI state providing the lowest value for the CF is issued for the whole k + 1 period. The predictive model is obtained discretizing the continuous- IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS A. Novelty and Contributions appearing between α−βand x−yplanes. The proposed method is related to EMPC but instead of providing control laws, it provides control actions. In this way, the method needs no modulation block. This allows for an increase in the control bandwidth compared to the case where modulation techniques (such as PWM) are used [26]. on physical insights. in Section IV. The paper ends with a Conclusions section. time IM equations as: TABLE I NOMENCLATURE Electrical Variables iCurrent vV oltage ωAngular speed TLTorque load Control Variables eControl error ERoot mean squared error gControl Gap HRotor current term NNumber of points for root mean squared computation QQuadrant rModulus of gap vector ρActive modulus τActive ratio σAuxiliary variable for look-up table access Control Parameters λWeighting factor of the CF TsSampling period Wxy Trade-off parameter Drive Parameters LInductance RResistance PPole pairs VDC Direct-Current voltage Derived Coefficients c1=LsLr−L2 Mc2=Lr/c1 c3=1/Lls c4=LM/c1 a2=−Rsc2a3=−Rsc3 a4=−LMc4ωrγc h= cos hϑ γs h= sin hϑ ϑ = 2π/5 δDegradation ξDetuning factor Subscripts and superscripts α−βTorque subspace x−yHarmonic subspace cCorrect dDetuned hPhase i,j Matrix indices ∗Reference ˆPrediction ˆ i(k+ 1) = Φ(ω)i(k)+Ψu(k)(1) where icontains stator currents on the α,β,xand yaxes, ˆ i is the prediction, ωis the angular speed and uis the control signal, defined as u= (K1, K2,· · · , K5)⊤, where the values Khindicate the state of the corresponding VSI switch for phases h∈ {1,2,3,4,5}. Matrices Φand Ψare obtained from modeling and applying time discretization with sampling time Ts. This gives Φ=(I+ATs), and Ψ = TsVDC BTM, where the following matrices are used. A=     a2−a40 0 a4a20 0 0 0 a30 000a3     , B =     c20 0 0 0c20 0 0 0 c30 0 0 0 c3     (2) M=2 5       1γc 1γc 2γc 3γc 4 0γs 1γs 2γs 3γs 4 1γc 2γc 4γc 1γc 3 0γs 2γs 4γs 1γs 3 1/2 1/2 1/2 1/2 1/2       ,(3) The components of Tare Tii = 4/5for the diagonal terms and Tij =−1/5for the off-diagonal terms. The DC link voltage (VDC ) and other coefficients are described in Table I. The two-step ahead predictions are found as ˆ i(k+ 2) = Φ(ω)ˆ i(k+ 1) + Ψu(k+ 1) + H(k)(4) IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS −0.05 0 0.05 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 gα (A) gβ (A) −0.05 0 0.05 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 gx (A) gy (A) 30 gα (A) gβ (A) on deviations of α−β and x−y currents from their references to the previous definitions one gets g(k+1) = TsVDC BT Mu, where u takes any of the 32 VSI states as is the common practice in FCS-MPC. This produces the 32 points shown on where eˆ = . i∗ − ˆ i denotes the predicted current error. During the optimal VV depends just on gx−y(k+1). Furthermore, the operation, the FCS-MPC tries to make the α − β components region in the subspace gx−y where a particular VV is selected is a sinusoidal wave set by the speed/torque controller. III. PROPOSAL The minimization of J in FCS-MPC is done by combinatodefine the actual table from which the control action is selected Fig. 1. Regions in gap space, projected to the gα−gβplane (left graph), to the gx−gyplane (middle graph) and example of a region (right graph). The regions are identified by the index of the VV that is optimal for each one. where Haccounts for the effect of the rotor currents, which are usually unmeasured variables and can be estimated using a backtracking procedure (see [7]). The CF in imposes penalties using WF denoted as λαβ and λxy. This results in J(k+ 2) = λαβ∥ˆeαβ(k+ 2)∥2+λxy∥ˆexy(k+ 2)∥2(5) of stator currents follow a reference signal i∗(k). The reference rial search. This is a computationally expensive task, especially for multi-phase systems. The proposal in this paper is based on the distribution of optimal control actions with respect to a quantity termed gap that is defined in what follows. According to equation (4), the prediction for k+ 2 is a function of the state predicted for k+ 1 and u(k+ 1). Now, if one chooses to apply a null VV, then the evolution of stator currents is given solely by their own dynamics. The gap in k+1 is defined as the prediction of the control error for k+2 when a null VV is applied. g(k+ 1) . =i∗(k+ 2) −Aˆ i(k+ 1|k) + H(k)(6) where g(k+ 1) is termed the evolutionary gap. If the control action applied does not correspond to a null VV, then Equation (4) allows one to write the predicted control error as ˆe(k+ 2) = g(k+ 1) −Ψu(k+ 1) (7) The theoretical justification of the proposal is based on the observation that the minimization of the CF given of (5) is related to the gap. Suppose that λαβ = 0 and λxy = 0, then the optimal VV depends just on gα−β(k+ 1). Also the region in the subspace gα−βwhere a particular VV is selected happens to be a convex region, more precisely a trapeze, as illustrated by Fig. 1. This is easily shown as the lines defining the trapezes are perpendicular bisectors of adjacent VVs. They are determined by equation (7) considering that the objective is achieving ˆe(k+ 2) = 0. Recall that the available control moves are the VV that the VSI can produce. Then, according each plane in Fig. 1. For the case where λαβ = 0 and λxy = 0 it is found that is again a trapeze. The more general case for λαβ = 0 and λxy = 0 can be treated by introducing a trade-off parameter Wxy to provide weighted values for the gaps. The subspace (α−βor x−y) that holds the largest weighted gap is used to using the following algorithm. In short, the proposal is supported by equations (6)-(7) and their connections with the control model of (4) and the cost function of (5). The available control moves can close the gap that might exist at any given sampling period. With this idea in mind, the concept is simple: closing the gap is equivalent to minimizing J. A. Algorithm According to the previous paragraphs, the control action can be decided by a table. The region where the gap is located must first be identified. This can be done following the steps of the flowchart of Fig. 2. In the flowchart (step 5), the computation of the quadrant of vector gais indicated as Γ(ga). Said vector is computed from g(k+ 1) (steps 1 to 4) according to the relative values of |(gα, gβ)|and Wxy|(gx, gy)|, where Wxy is a trade-off parameter. Parameters U1and U2are based on geometric considerations given the ten-fold symmetry of the regions, resulting in U1= 1.3764,U2= 0.3249. Parameters GL,GM, and GSare based on the modulus of gaps. This set of parameters is machine dependent and can be computed from the equations of the model (1)-(4). They correspond to the midpoints between different VV coronas: Large (L), Medium (M) and Small (S). For example, GLis the midpoint between L and M. For the case analyzed, the values for σ= 1 are IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS TABLE II VV SELECTION TABLE Cases Quadrant σ= 1 Quadrant σ= 0 στσρ1 2 3 4 1 2 3 4 A 1 29 13 4 20 27 11 6 22 A 2 9 31 24 2 3 28 30 5 A 3 21 14 12 19 25 15 8 18 A 4 1 1 1 1 1 1 1 1 B 1 25 15 8 18 19 12 14 21 B 2 30 5 3 28 31 9 2 24 B 3 27 11 6 22 20 4 13 29 B 4 1 1 1 1 1 1 1 1 C 1 26 7 7 26 23 10 10 23 C 2 17 16 16 17 17 16 16 17 C 3 10 23 23 10 7 26 26 7 C 4 1 1 1 1 1 1 1 1 Gα−β S= 0.0089 (A), Gα−β M= 0.0234 (A), Gα−β L= 0.0340 (A). Similarly, for σ= 0 the values are Gx−y S= 0.0122 (A), Gx−y M= 0.0318 (A), and Gx−y L= 0.0515 (A). Begin Compute g(k+1) eq. (6) rαβ = |(gα, gβ)|, rxy = |(gx, gy)| 1 2 3 4 Q = Γ(ga), τ = |ga2 /ga1|, ρ= |ga| τ>U1?τ<U2? στ= A στ= C στ= B End 5 6 7 8 9 10 Wxy rxy < rαβ ? σ= 1, ga = (gα, gβ)| σ= 0, ga= |(gx , gy)| ρ>GL?ρ>GM? σρ= 1 σρ= 2 ρ>GS? σρ= 3 σρ= 4 Select VV from Table II yes yes yes yes yes yes The reduction in computational effort attained by the proposal, in comparison with FCS-MPC, is due to the following. • Step 1) involves matrix multiplication and 2 additions of 4-element vectors. Compared to FCS-MPC this operation Fig. 2. Flowchart of the proposed algorithm. B. Computational Cost TABLE III COMPARISON OF PROCESSOR TIMES (TMS320F28335). FCS-MPC Proposal Task A/D conversion 2.14 (µs) 2.14 (µs) PI + delay compensation 1.80 (µs) 1.80 (µs) Optimization 25.80 (µs)3.61 (µs) Data logging and other processes 7.15 (µs) 7.15 (µs) In contrast, the FCS-MPC algorithm needs 32 times the following: a 4 × 4 matrix multiplication and 2 additions of 4-element vectors (for the predictions), two vector modulus, a multiplication and an addition (for the computation of the CF). The processor times needed in both cases for a Texas Instruments digital signal processor (TMS320F28335) are presented in Table III. It can be seen that the proposal substantially reduces the requirements for the optimization IV. EXPERIMENTAL RESULTS AND COMPARISON has exactly the same complexity as one two-step ahead prediction computation using (4). •Steps 2-4) need 2 2D-vector modulus operations for computing the rvalues, a scalar multiplication and a comparison to find the value of σ; then a 2D-vector assignation is needed to compute ga. •Step 5) requires two comparisons to find the quadrant Q, a division and scalar modulus to compute τ, and another 2D-vector modulus fo find ρ. •Steps 6-7 need two comparisons to determine the case (A, B or C) and an assignation to determine στ. •Steps 8-9 require three comparisons to find σρ. •Step 10 needs a table search (based on σ,στ,Qand σρ) to obtain the VV index. phase. The experiments are carried out in a laboratory setup including: a 5pIM (with parameters given in Table IV), a VSI made by two SEMIKRON SKS 22F modules, a DC power supply, and a TMS320F28335 digital signal processor. Speed feedback is provided by a GHM510296R/2500 encoder. A DC motor is used to generate a torque load (TL) for the different tests. The assessment of the controller is made using the current tracking error for Nsample, as Eαβ =v u u t 1 N N X k=1 e2 αβ(k), Exy =v u u t 1 A series of tests have been performed to assess the proposal. Table V summarizes the results obtained comparing the performance of the proposed method with FCS-MPC (columns marked Std.) and with a method with less computational burden employing the technique of [13], where the number of VVs is restrained (columns marked Red). Columns marked G1 and G2 in Table V correspond to the proposed method N N X k=1 e2 xy(k).(8) A. Steady State Analysis IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS TABLE IV PARAMETERS OF THE UTILIZED 5PIM Parameter Value Unit Stator resistance, Rs12.85 Ω Rotor resistance, Rr4.80 Ω Stator leakage inductance, Lls 79.93 mH Rotor leakage inductance, Llr 79.93 mH Mutual inductance, LM681.7 mH Number of pairs of poles, P3 - 0.16 0.18 0.2 0.22 0.24 0.26 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Currents (A) iαsiαs *ixs iys 0.16 0.18 0.2 0.22 0.24 0.26 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Currents (A) iαsiαs *ixs iys 0.16 0.18 0.2 0.22 0.24 0.26 0.28 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Currents (A) iαsiαs *ixs iys TABLE V COMPARISON WITH STANDARD AND REDUCED BURDEN FCS-MPC Fig. 3. Comparison of conventional FCS-MPC technique (top plot), a FCS-MPC with reduced computational burden (middle plot), and evolutionary gaps (bottom figure). with Wxy = 0.95 and Wxy = 0.5 respectively. In said Table V, the tracking error is provided in (mA) to facilitate the reading. The sampling period for standard FCS-MPC is 50 (µs), 40 (µs) for FCS-MPC with reduced burden and 33 (µs) for the proposal.Three different operating points are considered, where different load torques are applied by an independent DC machine. For this, values of 10 %, 50 % and 100 % the nominal torque are used. The best result for each figure of merit is highlighted in bold for each test. The better TLG1 G2 Std. Red. (%) Eαβ Exy Eαβ Exy Eαβ Exy Eαβ Exy 10 99 74 75 74 113 108 99 112 50 135 88 98 88 142 109 125 114 99 190 107 167 133 221 113 181 115 5 10 15 20 25 30 35 40 10 20 30 Sample VV index Proposal Std Fig. 4. Indices of voltage vectors selected by standard FCS-MPC and the proposal. TABLE VI RESULTS IN STEADY STATE FOR DIFFERENT VALUES OF Wxy Wxy Eαβ Exy Wxy Eαβ Exy 0.95 190 107 0.50 167 133 0.70 183 116 0.40 166 271 0.60 171 129 0.30 165 278 performance of the proposal is clearly visible. In addition, Fig. 3 depicts stator currents for α and x axes. The top graphs correspond to Std FCS-MPC and the bottom ones to the proposal (G2 case). A reference speed ωm ∗ = 500 (rpm) is used. The sampling period for the FCS-MPC is 50 (µs) and 33 (µs) for the proposal. Current tracking is more accurate for the proposal, with less oscillations in α axis and A proof of the region’s index of the VV in experimental results is provided by Fig. 4. The graph shows the VV selected by the standard FCS-MPC algorithm with combinatorial optimization (blue) and by the proposal (red). The agreement between the region based method and the combinatorial search method is remarkable. The small differences arise from the different way in which the trade-off between α − β vs. x − y less content in xaxis. tracking is performed. The harmonic distribution of the stator currents is shown in Fig. 5. The performance differences can be appreciated, where the proposed method shifts the harmonic content to higher frequencies while producing less Total Harmonic Distortion (THD). The design parameter Wxy can be used to balance the trade-off between α−βtracking and x−yregulation. Its effect on performance can be analyzed with the experimental hodographs of Fig. 6. On each graph, points laying on the outer corona are (isα, isβ)pairs. The corresponding (i∗ sα, i∗ sβ) values are presented with a dashed line. The points in the inner corona are (isx, isy)pairs. The left graph corresponds to a value Wxy = 70 and the right one to Wxy = 35. The IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 5 10 15 20 25 30 35 0 0.02 0.04 0.06 0.08 0.1 Harmonic number Amplitude (A) THD = 10.20 % I3 = 0.90 % 5 10 15 20 25 30 35 0 0.02 0.04 0.06 0.08 0.1 Harmonic number Amplitude (A) THD = 8.10 % I3 = 0.23 % Fig. 5. Harmonic distribution for FCS-MPC (top) and the proposal (bottom). Values of THD and third harmonic amplitude (I3) are provided. −1 0 1 −1 −0.5 0 0.5 1 iα, ix (A) iβ , iy (A) −1 0 1 −1 −0.5 0 0.5 1 iα, ix (A) iβ , iy (A) Fig. 6. Hodographs of stator currents in α−βplane (blue) and x−y plane (cyan) for two values of the balancing parameter Wxy. TABLE VII EFFECT OF SAMPLING PERIOD Ts(µs) 80 60 40 20 Eαβ (mA) 162.54 122.70 82.32 41.61 Exy (mA) 222.68 167.73 114.78 60.54 The effect of sampling period in the results can be checked with the help of Table VII, where the figures of merit are provided for several values usually found in the literature. It can be seen that a reduction in sampling time provides tighter control of α − β and x − y stator currents. As an example consider the time trajectories of stator currents shown in Fig. 7. The sampling period for the proposal has been artificially increased to compare the results with standard FCS-MPC not having the computational burden reduction. In this case the proposal is run with Ts = 40 (µs) It can be seen that the better tracking of the proposal is hindered in this case, proving that the benefits of the method arise from the reduction in sampling period and not from the different treatment of the α − β and trade-off between α−βtracking versus x−yregulation is clearly visible. B. Effect of the Sampling Period x−ysubspaces. 0.16 0.18 0.2 0.22 0.24 0.26 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Currents (A) iαsiαs *ixs iys 0.16 0.18 0.2 0.22 0.24 0.26 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Currents (A) iαsiαs *ixs iys Parametric uncertainty is an important issue for model based controllers. For FCS-MPC, some papers have analyzed the effect of detuned models in performance [27], [28]. To avoid redundancies the present analysis is focused on those parameters that might have a more appreciable influence in the proposed method. In the following it is assumed that nominal parameters are those shown in Table IV, corresponding to the real system. The predictive model, however, will use other values computed as R ˆs = ξ1Rs, R ˆr = ξ2Rr, L ˆls = ξ3Lls, L ˆlr = ξ4Llr. The different values of ξi provide the means to introduce various detuning situations. Please note that a value of ξ = 0.5 means that the model is using a parameter A reversal test is applied using a reference mechanical speed of ±500 (rpm). During this test, the reference speed is abruptly changed from +500 (rpm) to −500 (rpm). The actual mechanical speed is monitored and included in the upper part of Fig. 8. In this way the proposed method is compared with a conventional FCS-MPC technique. Note that the outer speed control loops use the same PI tuning. The sampling period for the FCS-MPC is 50 (µs) and 33 (µs) for the proposal. In the lower part of said figures, the stator current reference i∗ sq computed by the PI is shown for both approaches. It can be seen that the proposed controller offers a speed regulation similar to that of the conventional FCS-MPC method, reducing the ripple of the torque-producing stator current, isq. In both Fig. 7. Comparison of between FCS-MPC technique (top plot) and the proposal (bottom figure) for a different sampling period. C. Transient Response Test cases the speed reversal is attained as expected. D. Parametric Uncertainty Analysis IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −600 −400 −200 0 200 400 600 ωm (rpm) ωm Proposal ωm Std. FCSMPC ωm * 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −3 −2 −1 0 1 Time (s) is q (A) Isq Proposal Isq Std. FCSMPC Isq * Fig. 8. Reversal test using the proposal (red) and FCS-MPC (blue). TABLE VIII EFFECT OF PARAMETRIC UNCERTAINTY. Change Conventional Proposed δα−βδx−yδα−βδx−y ξ1= 0.51.09 1.02 1.02 1.01 ξ1= 2.01.01 1.01 1.00 1.01 ξ2= 0.51.46 1.14 1.21 1.13 ξ2= 2.01.41 1.05 1.23 1.03 ξ3= 0.51.18 1.03 1.17 1.01 ξ3= 2.01.31 1.04 1.21 1.02 ξ4= 0.51.00 1.01 1.01 1.00 ξ4= 2.01.01 1.01 1.01 1.00 • The proposal outperforms the conventional FCS-MPC technique in terms of the stator currents’ indicators. This reduction in the figures of merit considered was anticipated in [7], and is a consequence of the reduction in the computational cost of the proposal. The experimental results show that the reduction in tracking error results from the reduction in sampling period and not from value half the correct value, ξ= 1 indicates no detuning and ξ= 2 implies that the model uses a parameter value double the correct one. In Table VIII the effect of the detuning is indicated as a degradation factor δdefined as δ=Ed/Ec, where Eis a figure of merit (either Eαβ or Exy). The superscript dindicates detuned parameter and the superscript cindicates correct value (i.e. ξi= 1 for all i). It can be seen that the uncertainty effect is less noticeable in the proposal due to the reduction in Ts. Finally, Fig. 9 shows the results for a robust performance test. The stator current tracking can be compared with that of Fig. 3. Despite the use of a detuned model, the method is able to provide adequate control of stator current with little effect on performance. E. Discussion and remarks The reduction in sampling time provides results that are highlighted and analyzed in what follows: changes in the CF. •The proposal is flexible in dealing with the trade-off between α−βtracking and x−yregulation. This is 0.16 0.18 0.2 0.22 0.24 0.26 −1 −0.5 0 0.5 1 Time (s) Currents (A) iαsiαs *ixs iys Fig. 9. Results for a robust performance test. • The proposal competes with standard FCS-MPC in terms of transient response. However, in this case, the outer PI tuning has a more appreciable effect. To ensure a fair comparison such tuning has been kept the same for both It has been shown that a faster computation of the control signal in FCS-MPC allows for a reduction in the sampling period. This in turn, produces better tracking of stator currents. As a result, other measures such as harmonic content are improved. As a side-effect, the proposal is better suited for A 5pIM has been used as a case example of the proposal. For higher number of phases the positive effects should be more noticeable as the combinatorial search of FCS-MPC clearly visible in the experimental results as the best values for the α−βtracking are always found for G2, while the best values for the regulation of the stator current components x−yare found using G1. approaches. •The problems associated with parameter uncertainty are less acute in the proposal. This is another positive effect of the reduction in sampling time. This is so because the model based controller receives a quicker feedback from the system. V. CONCLUSION its use with wide-band-gap semiconductor power devices. would need more computations. ACKNOWLEDGMENT The research work has been funded by the Ministerio de Ciencia e Innovaci´ on of Spain, Agencia Estatal de Investigaci´ on, NextGenerationEU, grant TED2021-129558B-C22, and by the Ministerio de Ciencia e Innovaci´ on of Spain, Agencia Estatal de Investigaci´ on, grant PID2021-125189OB- I00. REFERENCES [1] A. Tenconi, S. Rubino, and R. Bojoi, “Model predictive control for multiphase motor drives–a technology status review,” in 2018 International Power Electronics Conference (IPEC-Niigata 2018-ECCE Asia), pp. 732–739. IEEE, 2018. [2] M. R. Arahal, C. Mart´ ın, A. Kowal, M. Castilla, and F. Barrero, “Cost function optimization for predictive control of a five-phase IM drive,” Optimal Control Applications and Methods, vol. 41, no. 1, pp. 84–93, 2020. [3] S. R. Mohapatra and V. Agarwal, “Model predictive controller with reduced complexity for grid-tied multilevel inverters,” IEEE Transactions on Industrial Electronics, vol. 66, no. 11, pp. 8851–8855, 2018. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS [4] X. Liu, L. Qiu, W. Wu, J. Ma, Y. Fang, Z. Peng, and D. Wang, “Neural predictor-based low switching frequency FCS-MPC for MMC with online weighting factors tuning,” IEEE Transactions on Power Electronics, vol. 37, no. 4, pp. 4065–4079, 2021. [5] K. Eshwar and V. K. Thippiripati, “Weighting-factorless predictive torque control scheme for dual inverter fed open-end-winding PMSM with single DC source,” IEEE Transactions on Power Electronics, vol. 36, no. 11, pp. 12 968–12 978, 2021. [6] V. Jayan and A. M. Ghias, “Weighting factor free model predictive control for a flying capacitor converter in a DC microgrid,” IEEE Transactions on Energy Conversion, vol. 37, no. 2, pp. 1030–1041, 2021. [7] M. R. Arahal, F. Barrero, M. J. Duran, M. G. Ortega, and C. Martin, “Trade-offs analysis in predictive current control of multi-phase induction machines,” Control Engineering Practice, vol. 81, pp. 105–113, 2018. [8] H. Fretes, J. Rodas, J. Doval-Gandoy, V. Gomez, N. Gomez, M. Novak, J. Rodriguez, and T. Dragiˇ cevi´ c, “Pareto optimal weighting factor design of predictive current controller of a six-phase induction machine based on particle swarm optimization algorithm,” IEEE Journal of Emerging and Selected Topics in Power Electronics, 2021. [9] A. Gonz´ alez-Prieto, I. Gonz´ alez-Prieto, M. Dur´ an, and J. Aciego, “A memory-based model predictive control for multiphase electric drives using SiC switches,” in 2022 International Conference on Electrical Machines (ICEM), pp. 136–142. IEEE, 2022. [10] X. Liu, D. Wang, and Z. Peng, “A computationally efficient FCS-MPC method without weighting factors for NNPCs with optimal duty cycle control,” IEEE/ASME Transactions on Mechatronics, vol. 23, no. 5, pp. 2503–2514, 2018. [11] I. M. Alsofyani and L. M. Halabi, “Unidirectional finite control setpredictive torque control of ipmsm fed by three-level npc inverter with simplified voltage-vector lookup table,” Electronics, vol. 12, no. 1, p. 252, 2023. [12] H. T. Nguyen, K. Al Hosani, A. S. Al-Sumaiti, J. Y. Alsawalhi, K. A. Al Jaafari, T. H. Nguyen, and M. S. El Moursi, “Precise voltage regulation of unbalanced bipolar dc-grid based charging stations: A new multi-objective predictive control approach,” IEEE Transactions on Power Electronics, 2023. [13] H. Zhou, H. Li, X. Xiang, B. Yuan, T. Zhou, and W. Li, “Model predictive control algorithm of dual three-phase motor considering global single vector,” in 2022 25th International Conference on Electrical Machines and Systems (ICEMS), pp. 1–6. IEEE, 2022. [14] T. Dorfling, H. du Toit Mouton, T. Geyer, and P. Karamanakos, “Longhorizon finite-control-set model predictive control with nonrecursive sphere decoding on an FPGA,” IEEE Transactions on Power Electronics, vol. 35, no. 7, pp. 7520–7531, 2019. [15] M. G. Satue, M. R. Arahal, and D. R. Ramirez, “Estimation of rotor currents in polyphase machines for predictive control,” Revista Iberoamericana de Autom´ atica e Inform´ atica Industrial, vol. 20, no. 1, pp. 25–31, 2023. [16] B. Yu, W. Song, J. Li, B. Li, and M. S. Saeed, “Improved finite control set model predictive current control for five-phase VSIs,” IEEE Transactions on Power Electronics, vol. 36, no. 6, pp. 7038–7048, 2020. [17] S. Liu and C. Liu, “Virtual-vector-based robust predictive current control for dual three-phase PMSM,” IEEE Transactions on Industrial Electronics, vol. 68, no. 3, pp. 2048–2058, 2020. [18] P. Garcia-Entrambasaguas, I. Zoric, I. Gonzalez-Prieto, M. J. Duran, and E. Levi, “Direct torque and predictive control strategies in ninephase electric drives using virtual voltage vectors,” IEEE Transactions on Power Electronics, vol. 34, no. 12, pp. 12 106–12 119, 2019. [19] M. S. Saeed, W. Song, B. Yu, and X. Wu, “Low-complexity deadbeat model predictive current control with duty ratio for five-phase PMSM drives,” IEEE Transactions on Power Electronics, vol. 35, no. 11, pp. 12 085–12 099, 2020. [20] C. Xue, W. Song, X. Wu, and X. Feng, “A constant switching frequency finite-control-set predictive current control scheme of a five-phase inverter with duty-ratio optimization,” IEEE Transactions on Power Electronics, vol. 33, no. 4, pp. 3583–3594, 2017. [21] A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear model predictive control, pp. 345–369. Springer, 2009. [22] M.-V. Doi, B.-X. Nguyen, and N.-V. Nguyen, “A finite set model predictive current control for three-level npc inverter with reducing switching state combination,” in 2019 IEEE 4th International Future Energy Electronics Conference (IFEEC), pp. 1–9. IEEE, 2019. [23] M. Zhao, Y. Cao, Y. Yan, Z. Zhang, T. Shi, and C. Xia, “Weighting factors tuning method in explicit model predictive direct speed control of permanent magnet synchronous motor,” in 2021 24th International Conference on Electrical Machines and Systems (ICEMS), pp. 653–658. IEEE, 2021. [24] K. Belda and P. P´ ıˇ sa, “Explicit model predictive control of PMSM drives,” in 2021 IEEE 30th International Symposium on Industrial Electronics (ISIE), pp. 01–06. IEEE, 2021. [25] M. R. Arahal, F. Barrero, M. G. Satu´ e, and M. Berm´ udez, “Fast finitestate predictive current control of electric drives,” IEEE Access, 11, 12821-12828., 2023. [26] C.-S. Lim, E. Levi, M. Jones, N. A. Rahim, and W.-P. Hew, “A comparative study of synchronous current control schemes based on FCS-MPC and PI-PWM for a two-motor three-phase drive,” IEEE Transactions on Industrial Electronics, vol. 61, no. 8, pp. 3867–3878, 2013. [27] C. Martin, M. Berm´ udez, F. Barrero, M. R. Arahal, X. Kestelyn, and M. J. Dur´ an, “Sensitivity of predictive controllers to parameter variation in five-phase induction motor drives,” Control Engineering Practice, vol. 68, pp. 23–31, 2017. [28] A. Altan and R. Hacıo˘ glu, “Model predictive control of three-axis gimbal system mounted on UAV for real-time target tracking under external disturbances,” Mechanical Systems and Signal Processing, vol. 138, p. 106548, 2020. Manuel R. Arahal was born in Dos Hermanas, Seville, Spain, in 1966. He received a master’s degree and Ph.D. in Industrial Engineering from the University de Seville, in 1991 and 1996, respectively. He is currently a Full Professor with the Ingenier´ ıa de Sistemas y Autom´ atica Department, U. Seville. Prof. Arahal has received the Best Paper Awards from the IEEE Trans. on Ind. Electronics in 2009 and from the IET Electric Power Applications in 2010/2011. Federico Barrero received the M.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Seville, in 1992 and 1998, respectively. In 1992, he joined the Electronic Engineering Department, University of Seville, where he is currently a Full Professor. Prof. Barrero was the recipient of the Best Paper Awards from the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 2009 and the IET Electric Power Applications in 2010/2011. Manuel G. Satu´ ewas born in Seville, Spain. He received the master’s degree in industrial engineering and the Ph.D. degree in industrial engineering from the University of Seville, Seville, Spain in 2008 and 2021, respectively. His research interests include optimization for energy systems and predictive control. He is currently a Teaching Assistant with the with the Ingenier´ ıa de Sistemas y Autom´ atica Department, University of Seville. Cristina Mart´ın was born in Seville, Spain, in 1989. She received the Industrial Engineer degree from the University of M´ alaga, Spain, in 2014, and the Ph.D. degree in electrical/electronic engineering from the University of Seville, Spain, in 2020. She is currently a substitute professor in the Department of Electrical Engineering, University of Seville. Mario Berm´udez was born in M´ alaga, Spain, in 1987. He received the B.Eng. degree in industrial engineering from the University of M´ alaga, M´ alaga, Spain, in 2014, and the Ph.D. degree in electrical/electronic engineering jointly from Arts et M´ etiers ParisTech, Lille, France, and from the University of Seville, Seville, Spain, in 2018. He is currently an Assistant Professor in the Department of Electrical Engineering, University of Seville.