Voronoi Multi-phase Predictive Current Control with Variable Application Times

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 1 Voronoi Multi-phase Predictive Current Control with Variable Application Times Manuel R. Arahal, Federico Barrero, Manuel G. Satu´ e and Francisco Colodro Abstract—Predictive Stator Current Control (PSCC) is a flexible technique for drives of different types. For the multiphase case, PSCC must deal with an increased number of available control options (the voltage vectors) and cope with the different current spaces: torque producing (α−β) and harmonic planes (x−y). In this paper, a Voronoi region based scheme is designed to cope with both issues while maintaining an affordable commutation frequency. The proposal is enhanced by using variable application times with fine resolution. The method is compared with state of the art dead-beat multi-vector approach. The comparison is made using a real, laboratory setup designed for experimentation based on a five-phase induction machine. The comparison shows enhanced control results together with a reduction in harmonic content, without compromising the switching frequency limits of the power converters and maintaining flexibility due to the cost function. Index Terms—Multi-phase drives, Predictive control, Variable application time, Voronoi regions. I. INTRODUCTION VARIABLE speed drives require precise dynamic regulation that can be achieved through adequate control of stator currents. Control methods considering continuous voltage must use some modulation [1]. Predictive Stator Current Control (PSCC), however, avoids the modulation stage and is able to deal with various stator current planes; α−β, and x−y[2]. PSCC is characterized by its high flexibility to accommodate different objectives in a cost function. For instance, managing the conflicting criteria of torque production vs. x−yplane content. In addition it features built-in fault tolerance [3]. The number of reported applications of PSCC is large, encompassing different types of drives based on Induction Machines (IM) [4], [5], permanent magnet machines [6], [7], and others. However, several problems have been reported requiring new solutions. For instance, the computational cost associated with the minimization of the cost function [8]. Several schemes have been put forward to diminish the computational load, such as: reduction of the control set [9] (i.e. the set of allowed Voltage Vectors (VV)), within sample This work is part of project I+D+i / PID2021-125189OB-I00, funded by MCIN/AEI/10.13039/501100011033/FEDER, UE ”ERDF A way of making Europe”. 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 and Francisco Colodro are with the Electronic Engineering Department of the University of Seville, Seville, 41092, Spain (e-mail: fbar- [email protected]; [email protected]). (Corresponding author Federico Barrero, phone: +34 954481304; fax: +34 954487372; e-mail: [email protected]). Manuscript received Month xx, 2xxx; revised Month xx, xxxx; accepted Month x, xxxx. modulation (or multi-vector approach) [10], simplification of the combinatorial search [11], etc. The works directly related to the proposal are critically reviewed in what follows. A. Literature review Multi-vector approaches use a dead-beat concept aiming at computing a desired voltage V∗that must be synthesized using different VV during one sampling period [12], [13]. This approach, referred to as DB-PSCC in what follows, is fast because computing V∗does not need iterations. DB-PSCC has been extensively reported for the three-phase case [14], and recently for multi-phase drives [15]–[17]. The method, however, must switch the Voltage Source Inverter (VSI) configuration more often than single vector approaches. This is easy to understand since the vectors in the multi-vector scheme have different VSI configurations that are accessible only through commutation. This is a problem that has not been widely reported because fast methods for single-vector PSCC did not exist until recently. Another problem of DB-PSCC is that the dead-beat objective requires the exact cancellation of control error in one shot. This cannot be achieved in practice due to: unmodelled dynamics, parameter excursions and the fact that V∗must be synthesized with various VV, so that an average effect is obtained. This has been criticized in [18] as the ”average deception problem.” In short, this occurs because the optimal voltage is not acting during the whole sampling period but rather is an average value resulting from the successive application of basic VV. This is specially important in methods that consider x−ycomponents in open loop, relying on an average null excitation of the x−yplane, but without feedback. In relation with this last aspect, the trade-offs between α−βtracking and x−yregulation are lost in DB-PSCC methods. These trade-offs have been first reported in [19]. Their importance has been recognized in [20] and [21], where the possibility of balancing the trade-off is seen as an extra degree of freedom. This is important for energy efficiency considerations as x−ycurrents produce losses. Another criticism that DB-PSCC methods face is that of DC bus utilization. This problem arises as combinations of basic VV do not grant the same voltage span as the basic VV themselves as pointed out in [18]. Finally, trigonometric functions are used by DB-PSCC. These functions are known to be time-consuming and might be not available in low-end hardware. 0000–0000/00$00.00 © 2021 IEEE This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 2 On the other hand, existing single-vector predictive methods (SV-PSCC) have a number of problems of their own. Previous works have been hindered by the lack of fast methods for VV computation. The problem is specially acute in multi-phase drives since the number of VV grows exponentially with the number of phases [8]. The set of attempts to diminish the computational load include: reduction of the control set [22], simplification of the combinatorial search [11], use of specialized hardware [23], and the use of model-free approaches [24]. None of these achieve the levels of computational speed of DB-PSCC. As a result they must resort to higher sampling periods, degrading the control loop (unless special hardware is used). More recently, a region-based method for multi-phase drives has been presented [25]. This constitutes an extension to the multi-phase variable-speed case of previous works [26]–[31]. With this technique it is possible to compete with multi-vector approaches. Another problem of SV-PSCC methods in general, is that the application time of the VV (Ts) always covers the whole sampling period (Ts). This produces a harmonic distortion in stator currents that is more noticeable that those found in traditional approaches using modulation. The idea of variable sampling time has been proposed in [32] for PSCC. The method divides the sampling period into sub-intervals. This allows to optimize both, the VV to be applied, and its application time. However, combinatorial search is in both cases. This entails a large computational burden. This burden prevents the method from using fine resolution of the application times. The idea of variable sampling times has been applied to multiphase systems before with the help of current observers in [33]. However, the method still uses exhaustive search for the VV computation, being unable to compete with DB-PSCC in terms of sampling time reduction. B. Contributions and novelty As opposed to previous methods, the proposal uses reduced and non uniform sampling times for the application of a single basic VV. The proposal achieves a number of goals that are relevant (as the previous analysis of the state of the art shows). The contributions of the paper can be summarized as follows. 1) Fast Voronoi based computation of a single VV (i.e. providing less commutations per sampling period). 2) Closed loop treatment of x−ycurrents considering the whole x−yplane, with flexibility to balance α−β tracking and x−yregulation. 3) Reduced control period that reduces the reaction lag. 4) No reduction in DC bus voltage utilization. The rest of the paper is organized as follows. Section II revisits the PSCC control structure using a five phase induction machine for the presentation and experimental results. The proposal is detailed in Section III. Experimental results on a real induction machine are provided to compare the proposal with previous approaches. The last section presents the main conclusions of the work. * iq ωm * * id ωm PI θa PSCC * iαβ u ii D ωm ωm VSC 5 VSC 1 Fig. 1. Diagram of PSCC for a five phase IM drive. II. MULTI-PHASE PREDICTIVE CURRENT CONTROL The Predictive Stator Current Control PSCC uses the Clarke transformation to α−βand xj−yjplanes. For a five phase IM, just one x−yplane is produced. Currents in α−βspace must follow a reference that is obtained from the speed control loop. The scheme is presented in Fig. 1, where flux and torque are independently regulated. The flux set point is provided by i∗ dwhereas reference i∗ qis used for the electrical torque. These references are projected to the α−βspace using the Park transformation, obtaining a reference for stator current in α−βplane as I∗ α−β=Di∗ d, i∗ q⊺, where matrix Dis given by D=cos θasin θa −sin θacos θa(1) The flux angle θais obtained as θa=Rωedt. As a result, the set point for stator current tracking i∗(k)has an amplitude I∗=qi∗2 d+i∗2 q. Finally, the α−βreferences can be expressed as i∗ α(t) = I∗sin ωet,i∗ β(t) = I∗cos ωet, i∗ x(t) = 0,i∗ y(t) = 0. In single-vector PSCC, the control action is the state of the VSI u= (Ka, Kb,· · · , Ke)⊤, where the values Kj indicate the state of the corresponding VSI switch for each phase. For a five phase IM There are 32 possible VSI configurations each producing a VV as shown in Fig. 2. The stator voltages provided by each VSI state can be found as V(k) = VDC TMu(k), where VDC is the DC-link voltage and T=1 5       4−1−1−1−1 −1 4 −1−1−1 −1−1 4 −1−1 −1−1−1 4 −1 −1−1−1−1 4       ,(2) M=2 5       1γc 1γc 2γc 3γc 4 0γs 1γs 2γs 3γs 4 1γc 2γc 4cϑγc 3 0γs 2γs 4γs 1γs 3 1/2 1/2 1/2 1/2 1/2       .(3) where γc h= cos hϑ,γs h= sin hϑ,ϑ= 2π/5. The PSCC uses a predictive model to link future stator currents to actual voltages. This model is often a set of discrete-time state-space equations as follows: ˆ i(k+ 1) = (C(ω) + G)i(k) + BV (k)(4) This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 3 31 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 0 β α 13 18 Fig. 2. Distribution of the 32 basic VV of a five phase VSI in α−βplane. where icontains the α,β,xand ystator currents, ˆ iis the prediction of iand ωis the angular speed. Matrices Cand B are obtained from first principles applying time discretization with sampling time Ts. This gives C= (I+AcTs), and B= TsBc, where Ac=    a2a40 0 a4a20 0 0 0 a30 000a3     , Bc=    c20 0 0 0c20 0 0 0 c30 0 0 0 c3     (5) In the previous expressions the coefficients used are: c1= LsLr−L2 M,c2=Lr/c1,c3= 1/Lls,c4=LM/c1, a2=−Rsc2,a3=−Rsc3,a4=LMc4ωr,a= 4/5, b=−1/5,c=−b,d= 2/5,ar4=Rrc4,al4=Lrc4ωr. The measured values of the electrical parameters corresponding to the machine used in the experiments are shown in Table II. To compensate for the time needed to compute the control action, a delay compensation scheme is used. This is the standard practice for both single vector and multi vector PSCC methods. This amounts to let go a whole sampling period and optimize for k+ 2 during that time. Then, the two-step ahead prediction of the stator currents must be considered. From equation (4) the prediction is found to be ˆ i(k+ 2) = A(ω)ˆ i(k+ 1) + BV (k+ 1).(6) yielding the predicted control error as ˆe(k+ 2) = i∗(k+ 2) −A(ω)ˆ i(k+ 1) + BV (k+ 1),(7) this prediction is used to derive the control action as will be shown in what follows. Voltage vector determination Control action computation in PSCC can be performed in a variety of ways. For this paper it is relevant to consider a) the case of multi-vector approximation to the dead-beat solution and b) the case of single-vector minimization of a cost function. In case a), and using equation (7), one can derive the voltage that, applied during the k+ 1 interval, would drive the error to zero at k+ 2. Denoting it by V∗it turns out to be 2h v 3h v 1h v Fig. 3. Regions resulting from three adjacent VV determined by three planes. V∗=B−1i∗(k+ 2) −A(ω)ˆ i(k+ 1).(8) Voltage V∗might not be one of the 32 VV that the VSI can produce. Then it must be synthesized using various basic VV applied at different times and for different sub-periods during the whole sampling period. The basic VV that need be applied can be found using trinonometric rules and look-up tables. To further simplify matters, these algorithms suppose open-loop x−ycurrent elimination, so they can work with just the α−β plane. In case b), just a single basic VV is applied. The dead-beat concept cannot be applied. The usual way to find the control signal is by minimization of the following cost function J(k+ 2) = ∥ˆe(k+ 2)∥2 w,(9) where ∥.∥wrepresents a weighted norm defined as ∥e∥2 w=e2 α+e2 β+λxye2 x+λxye2 y,(10) where λxy is the Weighting Factor (WF) of the CF that allows balancing α−βtracking and x−yregulation. III. PSCC WITH VARIABLE APPLICATION TIMES The proposal computes the solution that minimizes the CF defined in (9). The single basic VV is then applied during a sampling time of variable duration. This is the core idea of the method. The expected benefits have been already commented in the introduction section. In what follows, these two ingredients are explained with technical detail. A. Voronoi region of optimal voltage vector To understand the proposal it is important to notice that, minimization of Jamounts to minimizing ∥V∗−Vh∥2 w, where index hgoes from 1 to the number of available VV. This can be checked noting that ˆe=i∗−ˆ iwith ˆ i=A(ω)ˆ i(k+1)+BV hfor some basic VV of index h. As a result the following identity is found B−1ˆe=B−1i∗−A(ω)ˆ i(k+ 1)−Vh,(11) This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 4 where the right hand side is exactly V∗−Vhby definition. So, the term ˆecan be replaced by B(V∗−Vh)in the expression of the cost function. Finally, since Bdoes not depend on the VV selected, minimizing Jand minimizing ∥V∗−Vh∥2 wyields the same result. Another important step in the derivation of the proposal is understanding that minimization of ∥V∗−Vh∥wcan be made using the idea of Voronoi regions. This will allow the avoidance of the cumbersome exhaustive exploration. Voronoi regions appear because ∥V∗−Vh∥wdefines a metric in the mathematical sense [34]. Then, to compute the VV minimizing the cost function one just needs to find in which Voronoi region is V∗placed. The Voronoi region containing V∗is easily found since the regions are bounded by planes as shown in Fig. 3. The planes in question are the perpendicular bisectors of each pair of adjacent Vh. The planes are completely defined by their normal vectors, found as nij =Vj−Vi, where indices i and jrepresent two adjacent regions, corresponding to voltage vectors Viand Vj. It is well known that, given a plane Πand a point V, their relative position can be found as the scalar product p=V⊺·Π,(12) if p > 0then point Vlies in a subspace with relation to Π corresponding to region i. However, if p < 0then point Vlies in the other subspace, corresponding to region j. The case in which p= 0 corresponds to Vbeing part of plane Π. In this case the two adjacent VV produce the same value for the CF. Given the fact that a large number of basic VV are provided by a multi-phase VSI, many scalar products and comparison would be needed to establish the region for V∗. However, if the quadrant of V∗is known, then just three comparisons are needed as shown in Fig. 3. This is interesting because the quadrant Q∗can be found easily either using some nested if-else statements or in the following way Q∗= 1 + S∗·(8,4,2,1)⊺,(13) where S∗is a one-by-four vector of signs computed as S∗= sgn(V∗), and sgn() is the sign function. To support the method consider Table I where the indices of the basic VV lying on each quadrant Qare shown. Just three VVs are found for each quadrant and identified by their indices h1,h2, and h3. Please notice that VV lying on the frontiers of quadrants are counted as belonging to both adjacent quadrants; in this way, they are not left out of the pertinent comparisons. With this in mind, the determination of the region containing V∗is done considering just three products: TABLE I INDICES hOF NON-ZERO VV BELONGING TO EACH QUADRANT Q. Q h1h2h3Q h1h2h3 01 16 24 26 09 06 22 30 02 16 20 28 10 04 06 22 03 08 09 25 11 10 14 15 04 09 25 29 12 12 13 15 05 16 18 19 13 02 06 22 06 16 17 21 14 06 22 23 07 09 25 27 15 03 11 15 08 01 09 25 16 05 07 15 POWER ELECTRONIC CONVERTERS DSP DC MOTOR 5-PHASE IM Phase Currents Switching Signals Position Encoder a b c d e Fig. 4. Diagram and photographs of the laboratory setup used in the experiments. p1=V∗⊺·nh1,h2(14) p2=V∗⊺·nh1,h3(15) p3=V∗⊺·nh2,h3,(16) where h1,h2,h3are retrieved from Table I from the row indicated by Q∗. Finally, the index of the region containing V∗is found as h∗=h1if p1<0,p2<0,h∗=h2if p1>0and p3<0, h∗=h3if p2>0,p2=p3>0. It is quite apparent that the computational burden is small, requiring a few scalar products and comparisons. This allows the computation of the optimal single VV without iterations and considering the original cost function. In this way the x−y plane is considered properly and in full. Also, the weighting factor can retained in the cost function. This constitutes a degree of freedom to balance α−βtracking with x−ycontent. B. Non uniform sampling time The sampling time can be modified from one period to another in order to make the selected VV have an optimal duration. In this way, the application time Tsbecomes a degree of freedom that can be used to obtain better results. The idea This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 5 of the non uniform sampling time is as follows. Instead of the typical read-compute-hold-wait scheme of a digital control system, the variable sampling time uses an additional step in which the sampling time is computed. In this way, the waiting part of the digital control method has a variable duration. With this idea in mind, one would like to use the selected VV and select the sampling time (Ts) to minimize the deviation Fdefined as F=i∗(t1+Ts)−ˆ ih∗(t1+Ts),(17) where t1is the time corresponding to the previous sampling period, h∗is the index to the optimal VV computed as explained in the preceding section and Tsis the application time being optimized. Using the continuous-time model of the IM it is possible to write the time derivative of ˆ ih∗as D∗=Acˆ i(t1) + BcVh∗,(18) where ˆ i(t1) = ˆ i(k+ 1) is the one-step ahead prediction and Vh∗is the basic VV selected in the optimization phase. The value D∗of equation (18) represents the direction of a line (in current space) that is transversed as time progresses from t1to t1+Ts. The point in the line that is closest to the reference value is found when Fis perpendicular to said line. This provides the following equation F⊺·Acˆ i(t1) + BcVh∗= 0.(19) The only unknown is Ts, appearing only in the Ffactor. Thus, the solution can be found as Ts=i∗(t1+Ts)−ˆ i(t1)⊺ ·D∗/∥D∗∥2,(20) this can be checked substituting (20) into (19). This solution has fine resolution, unlike previous methods that relied on discretization of the application time. IV. EXPERIMENTAL RESULTS The proposal is put to test using a test-bench that is described below. The performance indicators used for the comparisons are presented later on. The comparison is drawn between the proposed regionbased PSCC with variable application times and a state of the art multi-vector approach presented in [13], referred to as DB-PSCC. Said contender method uses Virtual Voltage Vectors (VVV) obtained without combinatorial search, thus providing very low computation time. The applied voltage is further optimized by combining the optimal VVV with a zero configuration for a non-fixed time topt. Furthermore, to avoid variable switching frequency the multi-vector method considers a modulation process in which the basic VV are produced in a particular sequence (see Fig. 7 in [13]). In said DB-PSCC, the harmonic plane content is considered in openloop, since the VVV are designed to provide (on average) zero content in the third-harmonic. The x−yplane is thus TABLE II PARAMETERS OF THE EXPERIMENTAL FIVE PHASE IM 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 Rotational inertia, Jm0.02 kg m2 Number of pairs of poles, P3 - TABLE III SUMMARY OF RESULTS FOR DB MULTI-VECTOR (TOP)AND THE PROPOSAL (BOTTOM). Case Eα−βEx−yASF T HD (mA) (mA) (kHz) (%) A) 73.27 81.87 9.50 9.3 B) 84.25 100.1 9.50 3.9 C) 107.7 145.0 9.50 4.1 D) 101.2 136.3 9.50 6.5 E) 81.96 101.0 9.50 5.5 Avg. 89.70 112.9 9.50 5.5 Case Eα−βEx−yASF T HD (mA) (mA) (kHz) (%) A) 69.06 80.43 9.46 7.2 B) 74.22 85.16 7.30 3.4 C) 102.6 109.5 6.34 4.1 D) 99.69 106.5 8.48 5.8 E) 79.40 88.27 7.62 5.4 Avg. 85.00 94.01 7.84 4.6 not considered in full. Finally, the method makes use of a trigonometric function (see Eq. (28) in [13]) to derive the sector of the reference voltage V∗. The method is here adapted for the five phase IM used in the experiments. A. Test facility A test bench is used for the experiments to appraise the proposal. Fig. 4 shows photographs of its main lements: a five phase IM, a power converter with SEMIKRON SKS 22F modules and a 300V DC power supply. For the control, the following elements are used: a MSK28335 board including a TMS320F28335 digital signal processor, a GHM510296R/2500 incremental position encoder, and Hall effect sensors (LH25-NP) to measure the stator phase currents. Finally, a DC motor provides an opposing torque for the tests. Table II presents the parameters of the system. B. Figures of Merit Several figures of merit are considered: speed ripple, Total Harmonic Distortion (THD), stator current error in the torque This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 6 producing plane, x−ycurrent content, and switching frequency at the VSI. The switching frequency for most PSCC methods is not constant. However its value is important for hardware selection (e.g., standard IGBTs or SiC-based power switches) and for efficiency considerations (VSI commutation losses). Thus, an average value, referred to as Average Switching Frequency (ASF), is usually considered. Having this in mind, the following efficiency factors are considered: Eα−β=v u u t 1 N k1+N X k=k1 e2 αβ(k)(21) Ex−y=v u u t 1 N k1+N X k=k1 e2 xy(k)(22) ASF =1/(5 ·2) t(k1+N)−t(k1) k1+N X k=k1 ∆S(k)(23) THD =100 I1 v u u t ∞ X i=2 I2 i(24) where ∆S(k) = P5 i=1 |ui(k+ 1) −ui(k)|is the number of switch changes produced at the VSI when configuration u(k) is changed to u(k+ 1), and Iiis the amplitude of the i-th harmonic component of stator currents. These quantities are defined over a temporal horizon of Nsamples, defined by the discrete-time index k1to k1+N. Please notice that, in the case of the proposal, and due to the variable sampling, times t(k1) and t(k1+N)are not, in general, exact multiples of a base period Ts. In other approaches on simply gets t(k) = k·Ts as Tsis a constant. C. Steady State Analysis The feasibility of the proposal is first tested on the experimental benchmark for rated conditions. The results can be observed in Fig. 5, where the α−βstator currents, and phase currents are shown as waveforms. Tracking of sinusoidal α−β reference can be clearly seen. The figures of merit for this test are: Eα−β= 0.1(A), Ex−y= 0.08 (A), ASF = 8240 (Hz), and THD = 4.1(%). These results improve previous schemes as will be shown later. It is interesting to note that the ASF is low for the converter used that can handle up to 15 (kHz). This means that the proposal is competitive without resorting to high commutation rates. A comparison with DB-PSCC is presented in the next section for different operating regimes of the variable-speed drive. D. Operating Regime Analysis A series of steady state tests on different operating points have been performed to compare the performance of the proposal with DB-PSCC. Both controllers use the same hardware and other resources. In particular, the sampling periods are adjusted so that both methods use similar switching frequency. 0 0.01 0.02 0.03 0.04 0.05 −2 −1 0 1 2 Time (s) Currents (A) 0 0.01 0.02 0.03 0.04 0.05 −2 −1 0 1 2 Time (s) Fig. 5. Steady state results for the proposal in rated conditions. Top plot is for α−βstator currents, and bottom plot for phase currents. Different operating points are used for the comparison. The operating points feature a percentage of rated mechanical speed and opposing load provided by the DC machine. The pairs (speed, opposing load) for operating points A) to F) are: A) (20 %, 10 %), B) (80 %, 50 %), C) (100 %, 100 %), D) (20 %, 100 %), and E) (50 %, 30 %). This covers many cases where the behavior of the IM is different. This coverage allows for a thorough evaluation on differerent operating regimes of the variable-speed drive. Table III summarizes the results obtained. Please notice that, in order to make the comparison fair both approaches use approximately the same switching frequency. In fact, the proposal features a lower ASF, so the contender is in a better position. It can be seen, in said Table III, that the proposal obtains the best results for all figures of merits and for all cases considered. In particular the Ex−yof the DB-PSCC are much larger. This is a reflection of the fact that the x−y components have not been considered, just the third harmonic. This has some relevance as x−ycurrents produce losses. It is also interesting to see that, the lower control errors of the proposal are obtained with slightly less ASF than the DBPSCC approach. The variable switching frequency is thus the only parameter that remains sub-par in SV-PSCC with respect to other approaches. The proposal also has an interesting feature that cannot be achieved with the DB approaches: the x−ycontent can be traded with Eα−βto suit the particular application at hand (see [19]). For instance, if one is concerned more with x−yrelated losses and a bit less with tracking, then one can increase the value of λxy. To demonstrate this, consider again the results for the proposal in Table III (where λxy = 0.05 is used) and compare it with Table IV (where λxy = 0.08 is used). It can be seen that in the latter case, the x−ycontent has been diminished, yet the average values for the other figures of merit still remain lower than those for the contender method This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 7 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 3 Currents (A) i* , iα , iβ , ix , iy 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 3 Time (s) Currents (A) Fig. 6. Steady state results for the proposal in operating point A for the proposal (top plot) and for DB-PSCC (bottom plot). DB-PSCC. The more important bit, however, is that the single vector approach has flexibility whereas DB-PSCC has not. TABLE IV EXPERIMENTAL RESULTS FOR THE PROPOSAL WITH REDUCED x−y CONTENT Operating Eα−βEx−yASF T HD Point (mA) (mA) (kHz) (%) A) 70.90 58.80 10.41 7.6 B) 75.68 62.25 7.88 3.8 C) 104.4 80.11 8.24 4.1 D) 100.5 77.88 9.07 6.2 E) 79.71 64.52 8.76 6.5 Avg. 86.25 68.71 8.87 5.2 In addition to the previous tables, a series of graphs are used to present the experimental trajectories of stator currents in α−βplane and x−yplane for the proposal and for the DB-PSCC method. These are shown in Figs. 6 to 9. The different operating points are indicated in the caption for each figure. The correct behavior of the proposal is clearly seen. According to [19], in the single-vector approach, the x−y content is traded off with the other figures of merit. In this way the method can be tuned for the specific application unlike the multi-vector approach. Also, due to the use of all 32 basic VV the proposal achieves better results. Recall that the DB-PSCC can only use the specific VVV, this restricts the possibilities to cope with different situations. 0 0.005 0.01 0.015 0.02 0.025 −2 −1 0 1 2 3 Currents (A) i* , iα , iβ , ix , iy 0 0.005 0.01 0.015 0.02 0.025 −2 −1 0 1 2 3 Time (s) Currents (A) Fig. 7. Steady state results for the proposal in operating point B for the proposal (top plot) and for DB-PSCC (bottom plot). 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 3 Currents (A) i* , iα , iβ , ix , iy 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 3 Time (s) Currents (A) Fig. 8. Steady state results for the proposal in operating point C for the proposal (top plot) and for DB-PSCC (bottom plot). This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 8 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 3 Currents (A) i* , iα , iβ , ix , iy 0 0.005 0.01 0.015 0.02 0.025 0.03 −2 −1 0 1 2 3 Time (s) Currents (A) Fig. 9. Steady state results for the proposal in operating point D for the proposal (top plot) and for DB-PSCC (bottom plot). E. Transient Response Tests In addition to the steady state results, the capabilities of the proposal have been tested for transient conditions. A speed reversal experiment has been performed where the desired speed goes from 500 (rpm) to -500 (rpm). The results can be seen in Fig. 10, where the trajectories of the measured speed are compared for the proposal and for the multi-vector method (DB-PSCC). Please notice that the results are similar. This is due to the fact that the same PI is used for both methods. This PI is largely responsible for the performance in mechanical terms. Please notice that this is not a negative result since the same mechanical performance has been obtained for the proposal with better electrical indicators, thus, in a more energy efficient way (i.e. less commutations and less harmonic content). F. Robustness and stability One problem of model-based approaches is that mismatched parameters can cause degradation in control results. This is the case of PSCC because the predictive model contains parameters that can differ from the real ones. This situation can appear as a result of the identification procedure or from parameter excursions. Excursions can appear for instance due to temperature variations. Robust behavior is achieved when the influence of parameter detuning is acceptable. In the case of PSCC, parameters Rs, Rr,Lls,Llr, and LMplay a role in the production of predictions that are later used to derive the control action. Analysis of PSCC robustness have been carried out in the literature considering detuned parameters. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −600 −400 −200 0 200 400 600 ω (rpm) ω* , Proposal, DB−PSCC 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −3.0 −2.0 −1.0 0.0 1.0 Time (s) is q (A) Fig. 10. Results obtained in a reversal test using the proposal and the multivector DB approach. For the analysis it is assumed that the parameters in Table II correspond to the tuned system. A series of test is then performed using a detuned predictive model. The parameters used by the model will be: ˆ Rs=ϕ1Rs,ˆ Rr=ϕ2Rr,ˆ Lls =ϕ3Lls, ˆ Llr =ϕ4Llr, and ˆ LM=ϕ5LM. The ϕcoefficients allow to consider various situations with mismatched parameters. Please note that for ϕ= 0.5the model is using a parameter with a value half the correct one. Similarly, ϕ= 1 indicates no detuning and ϕ= 2 implies that the model uses a parameter value double of the correct one. In Table V, the effect caused by the mismatched parameters is indicated as a degradation factor δ, defined as δ=Ed/Et. The degradation is considered for a performance indicator E. The superscript dindicates detuning and the superscript tindicates perfect tuning (i.e. ϕi= 1 for all i). In said Table V the indicators are Eα−βand Ex−y. TABLE V ROBUSTNESS ANALYSIS FOR DETUNED MODELS. Detuning Conventional Proposal Factor δα−βδx−yδα−βδx−y ϕ1= 0.51.09 1.02 1.08 1.02 ϕ1= 2.01.01 1.01 1.01 1.01 ϕ2= 0.51.46 1.14 1.45 1.14 ϕ2= 2.01.41 1.05 1.41 1.06 ϕ3= 0.51.18 1.03 1.18 1.03 ϕ3= 2.01.31 1.04 1.30 1.03 ϕ4= 0.51.00 1.01 1.01 1.00 ϕ4= 2.01.01 1.01 1.01 1.01 ϕ5= 0.51.00 1.00 1.00 1.00 ϕ5= 2.01.95 1.22 1.90 1.22 From the results of the analysis, it is clear that the proposed method has a similar robustness as that of the conventional PSCC. Similar results are obtained regarding stability. Please note that stability proof for PSCC consider Lyapunov methods that use the cost function as stepping stone. In Section III-A it This article has been accepted for publication in IEEE Transactions on Energy Conversion. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TEC.2024.3497212 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 9 has been shown that the minimization due to Voronoi region identification is equivalent to minimization of J. As a result the stability proof would for the proposal would follow the same steps as existing proofs for PSCC and are, thus, omitted. V. CONCLUSION Model based predictive methods have become an important research area in modern electromechanical drives, providing a flexible solution for multivariable-constrained control problems. One of its main drawbacks, is the computational burden that limits its applicability in low-end and medium DSP solutions. This results in high application times producing high THD values. Both problems are solved with the proposal by delivering a fast, Voronoi-based computation method and variable application times. The experimental results show that the proposal retains the ability of PSCC to deal with the trade-off between torque producing tracking and x−ycontent. Compared with a state of the art multi-vector solution, the proposal improves all figures of merit even using less switching frequency. The proposal, thus, puts back single-vector approaches in the frontier of research making it comparable with multi-vector approaches. 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