Multi-Phase Stator Current Tracking with Gradual Penalization of Commutations

Citation: Arahal, M.R.; Satué, M.G.; Martínez-Heredia, J.M. Multi-Phase Stator Current Tracking with Gradual Penalization of Commutations. Appl. Sci. 2024,14, 6005. https://doi.org/ 10.3390/app14146005 Academic Editor: Roberto Zivieri Received: 17 June 2024 Revised: 3 July 2024 Accepted: 6 July 2024 Published: 10 July 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Multi-Phase Stator Current Tracking with Gradual Penalization of Commutations Manuel R. Arahal 1,†,‡ , Manuel G. Satué 1,*,†,‡ and Juana M. Martínez-Heredia 2,†,‡ 1Systems Engineering and Automation Department, University of Seville, 41092 Seville, Spain 2Electronic Engineering Department, University of Seville, 41092 Seville, Spain; [email protected] *Correspondence: [email protected] †These authors contributed equally to this work. ‡ Current address: Escuela Técnica Superior de Ingeniería, Campus Cartuja, Avda. de los Descubrimientos s/n, 41092 Sevilla, Spain. Abstract: Energy efficiency in drives is an important issue. In converter-supplied variable-speed drives, switching losses can amount to a significant portion of all losses. This has been considered in Predictive Stator Current Control (PSCC), considering commutations at the power converter. However, in multi-phase drives, the computational burden limits the application of said techniques. Recent fast predictive algorithms have enabled shorter application times with enhanced tracking results. However, the switching frequency becomes larger with diminishing sampling periods. This paper presents a method that retains the fast computation of recent methods while reducing the switching frequency. The proposal revolves around a modification of the cost function to penalize commutations in a nonlinear way. For this task, a novel, gradual penalization is introduced. The method is experimentally applied to a five-phase induction motor. Experimental results show a significant reduction in switching frequency without compromising other control objectives. This results in an enhanced PSCC with a small sampling period and reduced switching losses. Keywords: induction machines; multi-phase systems; predictive control 1. Introduction Model predictive control (MPC) is a flexible method for electrical systems using power converters [ 1 , 2 ]. In particular, Predictive Stator Current Control (abbreviated PSCC) has been explored in the specialized literature. PSCC relies on the sample-wise optimization of a cost function (CF), penalizing deviations from control objectives. In particular, for multi-phase drives, reference tracking in α−β (torque producing) and x−y (harmonic) subspaces constitutes an important control objective [ 3 ]. In addition, many papers expanded the CF by adding a term penalizing commutations in the legs of the power converter. This is achieved to keep the average switching frequency (ASF) below acceptable limits. Limiting the maximum ASF has a relevant impact on the drive and enables the selection of the hardware (e.g., standard IGBTs or wideband switches). Also, ASF affects the energetic efficiency of the drive, as commutation losses are an important portion of all losses [4]. Average switching frequency reduction constitutes a challenge since the switching frequency in PSCC is not constant [ 5 ]. Also, the ASF shows a nonlinear behavior with speed and load [ 6 , 7 ]. Previous papers dealing with ASF in predictive control can be classified into two categories: (a) those that include a term in the CF and (b) those that use physical insight to reduce the available set of voltage vectors (VVs). These two categories apply to different predictive methods, including those using a single-vector per sampling period and those relying on the multi-vector concept. This latter lot includes the virtual VV method (also known as synthetic VV method) [ 8 , 9 ] and methods derived from the deadbeat (DB) concept [10]. Appl. Sci. 2024,14, 6005. https://doi.org/10.3390/app14146005 https://www.mdpi.com/journal/applsci Appl. Sci. 2024,14, 6005 2 of 14 For single-vector PSCC based on the CF, the work of [ 11 ] presented a neural solution for online weighting factor tuning. A previous work used scheduled tuning with no adaptation [ 6 ]. In [ 12 ], VV pre-selection was used mainly to reduce the computational burden. Similar ideas were used in [ 13 ]. A memory stage was proposed in [ 14 ] to reduce the number of commutations considering the past. Analytical normalization was proposed in [ 15 ] to eliminate the weighting factor, thus reducing the complexity of the problem. Quadratic current prediction and boundary constraint design were used in [ 16 ] to enable a CF that reduces the ASF with some increase in the computational cost. A further step along these lines consists of the elimination of the average switching frequency term by reducing the set of available voltage vectors on the fly, such as in [ 17 ]. Intermediate solutions do use a CF but avoid exploring all possible switching sequences, as in [18]. Regarding the multi-vector approach, several avenues have been investigated. In [ 19 ], four candidate voltage vectors were considered during each sampling period. A look-up table was used to derive the optimal VV. The table allowed just one commutation, thus reducing the ASF. In [ 20 ], a deadbeat method with a hybrid exploration state was used to reduce the computational burden for a given average switching frequency. A similar approach was used in [ 21 ] in conjunction with an asymmetric space vector pulse width modulation method. In [ 22 ], a multi-vector solution for a dual three-phase permanent magnet motor was presented. Just two VVs were applied, where switching commands were changed in two phases at most. In [ 23 ], the DB concept was used in a multi-variable setup to control all system variables in a single predictive loop and using modulation techniques. Similar concepts were proposed in [24]. Multi-vector approaches consider an optimal (or reference) voltage V∗ to be synthesized using basic VVs. This is important for multi-phase systems, where torque-producing currents (i.e., currents in α−β plane) and non-torque-producing currents (i.e., currents in x−y plane) must be dealt with. The deadbeat concept has been exploited to (1) eliminate the combinatorial search of the optimization phase of the predictive method and (2) somehow get rid of the problem posed by x−y currents. It has been pointed out that the elimination of x−y voltages is not perfect in practice since it operates in an open loop. On the other hand, single-vector approaches have been hindered by the lack of fast methods for VV computation. Several methods have been proposed to diminish the computational load: the reduction of the control set [ 25 ], the simplification of the combinatorial search [ 26 ], and the use of a region identification algorithm to derive the control action [ 27 ]. Recently, a region-based method for multi-phase systems was presented [ 28 ]. This constituted an extension of previous works, using geometrical insights for VV determination [ 29 , 30 ]. With this technique, it is possible to compete with multi-vector approaches. From the state of the art presented above, it is clear that single-vector predictive solutions need some advancement. The goal is to compete with fast multi-vector solutions while retaining the advantages of single-vector schemes. Among these advantages, the most important are as follows: 1. The possibility of more complex cost functions; 2. The appropriate consideration of the whole x−yplane in a closed loop; 3. Attaining a low switching frequency despite the sampling period used. Having this in mind, the novelty of the paper lies in that it solves the above problem. This is achieved by means of a fast procedure for the optimization phase of MPC. The method employs a weighted metric to consider the x−y components entirely. In this way, x−y content is considered in a closed loop. Also, the allowable x−y content can be decided using a weighting factor in the CF (unlike multi-vector approaches). Furthermore, the proposal constitutes a novel method for ASF reduction. The method considers the switch changes required by the VV selection. The cost of using this selection is compared with the cost of not commuting. A trade-off solution is found with the flexibility to be adapted to the requirements of the actual VSI used. The proposal is tested in an experimental laboratory setup using a five-phase induction machine (5pIM). Appl. Sci. 2024,14, 6005 3 of 14 The rest of the paper is organized as follows. Section 2details the PSCC control structure considered for the five-phase induction machine. Later, in Section 3, the proposal is introduced, presenting some background on the fast optimization method and detailing the new control block for ASF reduction. Experimental results are provided to assess the proposal and compare it with previous approaches. In the last section, the main conclusions drawn are presented. 2. Multi-Phase Predictive Stator Current Control The Predictive Stator Current Control (PSCC) for multi-phase drives is applied considering the Clarke projection of stator currents into the α−β and xj−yj planes, as shown in Figure 1. For a 5pIM, just one x−y plane is produced. The torque-producing currents must follow a reference set by the torque/speed control loop. The x−y current references are set to zero since they contribute to losses. The scheme is presented in Figure 1, where flux and torque are independently regulated. The flux set point is provided by i∗ d , whereas reference i∗ q is used for the electrical torque. These references are projected to the α−β space using the Park transformation, obtaining a reference for stator current in the α−β plane 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 θa is obtained as θa=Rωedt . The reference signal ( r ) is set by the speed/torque controller, where the α−β plane receives a sinusoidal wave with amplitude I∗ and frequency fe and the x−y plane is set to zero. As a result, the reference vector for the α−β−x−yaxes is r(k) = i∗ s(k)defined by r=(I∗sin 2πfekTs,I∗cos 2πfekTs, 0, 0)⊺. (2) 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 a to e. For a 5pIM, there are 32 possible VSI configurations, each producing a VV [ 28 ]. The stator voltages provided by each VSI state can be found as V(k) = VDCTMu(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       , (3) 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       (4) where γc h=cos hϑ,γs h=sin hϑ, and ϑ=2π/5. * iqωm * * id ωm PI θa PSCC ωm VSC 5 VSC 1 * iαβ u ii D ωm Figure 1. Diagram of PSCC for a five-phase induction motor. Appl. Sci. 2024,14, 6005 4 of 14 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)(5) where i contains the α , β , x , and y stator currents; ˆ i is the prediction of i ; and ω is the angular speed. Matrices C and B are obtained from first principles applying time discretization with sampling time Ts. This gives C= (I+AcTs)and B=TsBc, where Ac=    a2−a40 0 a4a20 0 0 0 a30 0 0 0 a3     , (6) Bc=    c2000 0c20 0 0 0 c30 000c3     . (7) The term G accounts for the effect of the rotor currents, which are usually unmeasured variables. The technique of [ 31 ] is used to directly estimate the effect of rotor currents on the evolution of stator currents, then Gcan be written as Gα−β(k) = I+Tsar4al4 −al4ar4·cos δ−sin δ sin δcos δ, (8) with Gx−y(k) = 0 since rotor currents do not induce x−y components in the stator. In the previous expressions (6)–(8), the following coefficients are used: 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 , κ= 0.92, and δ= 0.007 rad. The measured values of the electrical parameters corresponding to the machine used in the experiments are shown in Table 1. From Equation (5), the two-step ahead prediction of the stator currents can be found as ˆ i(k+2) = A(ω)ˆ i(k+1) + BV(k+1)(9) where A=C+G is introduced for better readability. The predicted control error is thus found as ˆ e(k+2) = i∗(k+2)−A(ω)ˆ i(k+1) + BV(k+1)(10) Table 1. 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 Rotational inertia Jm0.02 kg m2 Number of pairs of poles P3 - Control Action Computation in Single-Vector and Multi-Vector Cases A unified approach is now delivered to cope with control action computation both in (a) the case of multi-vector approximation to the deadbeat solution and (b) the case of single-vector minimization of a cost function. Appl. Sci. 2024,14, 6005 5 of 14 In case (a) and using Equation (10), one can derive that the voltage applied during the k+1 interval would drive the error to zero at k+2. Denoting it by V∗, it turns out to be V∗=B−1i∗(k+2)−A(ω)ˆ i(k+1). (11) Now, voltage V∗ may not be one of the 32 VVs that the VSI can produce. Then, it must be synthesized using various basic VVs applied at different times and for different sub-periods during the whole sampling period. These algorithms suppose open-loop x−y current elimination, so they can work with just the α−β plane. Simple trigonometric rules and look-up tables can be used to identify the region with very low computing requirements. In case (b), just a single basic VV is applied. The deadbeat concept cannot be applied, so the commanded voltage is found to minimize a cost function, such as J(k+2) = ∥ˆ eαβ(k+2)∥2+λxy∥ˆ exy(k+2)∥2+λscSC (12) The VSI configuration minimizing J is applied and held for the k+ 1 interval. Parameter λxy is a weighting factor that allows balancing α−β tracking and x−y regulation. The parameter λsc is another weighting factor used to penalize changes at the VSI, and SC stands for switch changes and is computed considering the state Kj of each VSI phase when u(k)is changed to u(k+1). In mathematical form, SC =∑ j |Kj(k+1)−Kj(k)|. (13) Weighting factors have been proposed for elimination in several works, including [ 32 , 33 ]; however, it was shown in [ 34 , 35 ] that this parameter allows balancing the solution depending on the application. 3. Proposal The optimization phase of FCSMPC is accomplished by computing the value of the CF for each candidate control move. This entails a high computational load when the number of possible control moves is high (e.g., in multi-phase and/or multi-level systems). In [ 28 ], a method is provided for avoiding the exhaustive minimization of the CF. The method relies on a vector quantity referred to as “gap”. Knowing the gap vector g , it is possible to determine the control move that reduces the predicted error the most. However, the method considers just one plane, either α−β or x−y , at a time. In order to provide current tracking in both planes, the method must shift objectives in different sampling periods. This is achieved by computing which plane ( α−β or x−y ) will develop a larger weighted error in k+ 2. Also, the commutation frequency of the VSI is not explicitly considered. The proposal, which is put forward in the following, avoids such change in objectives and is able to consider penalization on commutation frequency. 3.1. Preliminaries In the following, some preliminary material is presented to clarify the proposal. First, the idea of gap is defined. Given a measurement vector y and a reference vector r (both in α−β−x−y space), the gap is the predicted two-step ahead control error if no control action is issued (i.e., if a null VV is selected as control action). In this paper, a weighted metric is used to account for the effect of parameter λxy of the CF. As a result, the gap is defined as gαβxy =r(k+2)−ˆ yF(k+2), (14) where ˆ yF(k+2)is the free-response weighted two-step ahead prediction computed as ˆ yF W(k+2) = Aˆ yα−β(k+1) pλxy Aˆ yx−y(k+1). (15) Notice that the weighted metric needs the introduction of factor pλxy affecting the x−ycomponents of vector ˆ yF W(k+2). Appl. Sci. 2024,14, 6005 6 of 14 With this metric, the regions of gap space corresponding to a particular VV can be defined by hyper-planes. This is very convenient because, given a hyper-plane and a point, it is very easy to compute their relative position. Every pair of VVs defines a plane, which, in turn, produces two subspaces. The VVs are marked as VVi and VVj . The plane ( Π ) passes through the mid-point of the segment that links VVi and VVj . The plane is defined using the orthogonal vector n . It is easy to see that any point g in one subspace produces a scalar product g⊺n> 0. Conversely, any point g in the other subspace produces a scalar product g⊺n< 0. Thus, for any given point, its associated subspace is determined by taking the sign function of the scalar product by n. In a general case, many hyper-planes can be needed to determine the region where a particular control action is optimal. One of the findings of this work is that just three hyper-planes are needed for each hyper-quadrant. So, after computing the actual hyperquadrant ( Q∈ { 1, 2, . . . , 16 } ), the three hyper-planes produce just eight regions. On each one, one VV provides the optimal solution in terms of the α−β vs. x−y tradeoff. So, the optimal VV is found by means of a region determination procedure without exhaustive optimization. The issue of commutation frequency is handled by the proposal by adding an extra layer where the switch changes required by the VV (obtained by the region-based selection) are compared with the cost of not commuting, i.e., the value J0=J(k+ 2 ) when u(k+1) = u(k). This cost is easily computed as J0=∥ˆ eαβ(k+2)∥2+λxy∥ˆ exy(k+2)∥2(16) In this way, commutations on the VSI are issued only when the cost of not doing so ( J0 ) is high enough. The balance between tracking quality and commutation frequency is treated with flexibility by means of parameter λsc , as the allowed cost can be defined in terms of λsc and SC . Since SC can take just six values (corresponding to SC = 0 commutations to SC =5 commutations), a simple table can provide the permissible cost. 3.2. Algorithm According to the previous ideas, in the proposal, the control action is determined by the region that contains a gap vector with a modified metric. The region is determined by hyper-planes, making it easy to compute online. After this, an extra layer is used to decide whether to implement the computed optimal VV or the previous one in order to offset commutation frequency with tracking accuracy. The following algorithm is a step-by-step description of the method. 1. Compute the r(k+2)in α−β−x−yaxes using (2). 2. The one-step ahead prediction for stator currents is computed using (5). 3. The weighted gap for the desired reference r(k+2)is computed using (14). 4. The hyper-quadrant Q, where gαβxy lies, is computed as Q=1+sαβxy ·(8, 4, 2, 1)⊺, (17) where sαβxy is a one-by-four vector of signs computed as sαβxy =sgn(gα), sgn(gβ), sgn(gx), sgn(gy), (18) where sgn stands for the sign function. 5. Retrieve from Table 2the three hyper-planes ( Π1 , Π2 and Π3 ) that are active for Q . Compute the vector connecting the VVs as n=BVi−Vj, (19) where iand jare the indices or the VV that define the hyper-plane. 6. Compute the position of gαβxy with respect to the active hyper-planes by multiplying the normal vector by gαβxy and applying the sign function as Appl. Sci. 2024,14, 6005 7 of 14 sj=sgn(g⊺ αβxy ·nj), (20) where njwith j=1, 2, 3 are the vectors orthogonal to planes Πj. 7. Obtain the selected VV from the triple sΠ= (s1,s2,s3)as IVV =1+sΠ·(4, 2, 1)⊺, (21) The selected VV is computed as VVsel =TVV(Q , IVV ) , with the values of TVV provided in Table 3. 8. Compute SC(k+1)associated with VVsel, as indicated by Equation (13). 9. Compute J0 according to (16) and the permissible cost Ja=J0/Jap , where Jap is provided by Figure 2and J0 represents the mean value of J0 . Now, if J0<Ja , then the previous control action is maintained VVapp(k+ 1 ) = VVapp(k) , else the computed VV is applied: VVapp(k+1) = VVsel. 10. Issue VVapp(k+1); wait for the next sampling period and go back to 1. Table 2. Active hyper-planes for each hyper-quadrant Q. QΠ1Π2Π3 01 05-10 05-11 09-10 02 07-11 07-10 09-11 03 05-10 05-11 09-10 04 05-10 05-11 09-10 05 07-10 07-08 09-10 06 07-10 07-08 10-11 07 02-10 02-08 07-10 08 02-10 02-08 07-10 09 03-11 03-06 05-11 10 03-11 03-06 05-11 11 03-06 02-03 05-06 12 03-06 03-04 05-06 13 02-04 02-08 03-04 14 02-04 02-08 03-04 15 02-06 02-04 03-06 16 02-04 02-08 03-04 Table 3. Voltage vectors for each quadrant Qand IVV. IVV Q1 2 3 4 5 6 7 8 01 05 05 11 11 09 10 09 10 02 07 07 10 10 09 11 09 11 03 05 05 11 11 09 10 09 10 04 05 05 11 11 09 10 09 10 05 07 07 08 08 09 10 09 10 06 07 07 08 08 10 11 10 11 07 02 02 08 08 07 10 07 10 08 02 02 08 08 07 10 07 10 09 03 03 06 06 05 11 05 11 10 03 03 06 06 05 11 05 11 11 02 02 03 03 05 06 05 06 12 03 03 04 04 05 06 05 06 13 02 02 08 08 03 04 03 04 14 02 02 08 08 03 04 03 04 15 02 02 04 04 03 06 03 06 16 02 02 08 08 03 04 03 04 Appl. Sci. 2024,14, 6005 8 of 14 SC AZ 0 1 2 34 5 Jap 10 30 70 120 200 (%) 0 J Figure 2. Allowed zone (AZ) for commuting as a function of the future number of switch changes (SCs). Figure 3summarises the described algorithm in a flowchart. Begin Compute r(k+2) Equation (2) 1 3 End 10 Apply VVapp Compute gαβxy for r(k+2) Equation (14) Compute i(k+1) Equation (5) ෡ 2 Compute hyper-quadrant Q Equation (17) Compute normal nj, j=1,2,3 Equation (19) Compute sΠ=(s1,s2,s3) Equation (20) Obtain VVsel from eq. (21) and Table 2 Compute SC(k+1) Equation (13) Compute J0eq. (16) and Ja= J0 / Jap J0< Ja? VVapp (k+1) = VVapp (k) VVapp (k+1) = VVsel 4 5 6 7 8 9yes Figure 3. Flowchart of the proposal. The numbers on the left correspond to the steps in the algorithm. Appl. Sci. 2024,14, 6005 9 of 14 4. Experimental Results A five-phase IM is considered for the presentation of the method and its assessment (see Figure 4). The five-phase induction motor has the parameters shown in Table 1. The experimental setup included two three-phase inverters SKS-22F (SEMIKRON DANFOSS, Nürnberg, Germany), of which just five power legs were connected to the machine (phases a, b, c, d, and e). The DC-link voltage was provided by an appropriate DC power supply. A MSK28335 control board (Technosoft, Blue Ash, OH, USA) and a TI TMS320F28335 digital signal processor (Texas Instruments, Dallas, TX, USA) were used to implement the control algorithms. For the assessment of the proposal, the root mean squared (RMS) control error in the α−β and x−y planes and the average switching frequency (ASF) are used. These are defined mathematically in what follows: Eα−β=v u u t 1 (k2−k1+1) k2 ∑ k=k1 e2 αβ(k)(22) Ex−y=v u u t 1 (k2−k1+1) k2 ∑ k=k1 e2 xy(k)(23) ASF =1/(2·5) Ts(k2−k1+1) k2 ∑ k=k1 SC(k)(24) where eαβ is the actual (not predicted) tracking error for α−β plane, exy is the actual tracking error for x−y plane, and Ii is the amplitude of the i -th harmonic component of stator currents. These quantities are defined over a temporal horizon defined by the discrete-time indices k1 , k2 , which should be large enough to provide robust estimations (some electrical cycles). POWER ELECTRONIC CONVERTERS DSP DC MOTOR 5-PHASE IM i Switching Signals Position Encoder Figure 4. Diagram of the laboratory setup used in the experiments. Signal i corresponds to stator currents for phases a to d obtained with Hall sensors. 4.1. Steady State Results In order to assess the proposal, the operating range of the 5pIM is discretized considering the electrical frequency fe and the amplitude of the stator current reference in α−β plane (I∗ s). For the comparisons, the following controllers are considered.