Model Predictive Base Direct Speed Control of Induction Motor Drive—Continuous and Finite Set Approaches

Article Model Predictive Base Direct Speed Control of Induction Motor Drive—Continuous and Finite Set Approaches

Karol Wróbel * , Piotr Serkies and Krzysztof Szabat

Department of Electrical Drives and Measurements, Wrocław University of Science and Technology, 50-370 Wrocław, Poland; [email protected] (P.S.); [email protected] (K.S.) * Correspondence: [email protected]; Tel.: +48-71-320-3359

Received: 28 January 2020; Accepted: 3 March 2020; Published: 5 March 2020

Abstract: In the paper a comparative study of the two control structures based on MPC (Model Predictive Control) for an electrical drive system with an induction motor are presented. As opposed to the classical approach, in which DFOC (Direct Field Oriented Control) with four controllers is considered, in the current study only one MPC controller is utilized. The proposed control structures have a cascade free structure that consists of a vector of electromagnetic (torque, flux) and mechanical (speed) states of the system. The first investigated framework is based on the finite-set MPC. A short horizon predictive window is selected. The continuous set MPC is used in the second framework. In this case the predictive horizon contains several samples. The computational complexity of the algorithm is reduced by applying its explicit version. Different implementation aspects of both MPC structures, for instance the model used in prediction, complexity of the control algorithms, and their properties together with the noise level are analyzed. The effectiveness of the proposed approach is validated by some experimental tests.

Keywords: model predictive control; continuous set; ﬁnite set; induction motor drive

1. Introduction Electrical drives play a very important role in modern industries. They transform electrical energy into a required one, e.g., mechanical. Different types of electrical machines are used in industries. One of the most popular is an induction motor (IM). This is because of multiple reasons [1–5]. They are cheap in manufacturing and services, and IMs are very robust under different operational and environmental factors. However, the nonlinear mechanical characteristic as well as difficulty in speed regulation are the classical drawbacks of IMs. Therefore, IMs were used in classical industries where there is no speed regulation. Due to the progress of power electronics and microprocessor technique, different control concepts have been proposed to regulate the speed of IMs. Nowadays, vector control methods are the industrial standard for IMs. The most popular ones are DFOC (direct field-oriented control) and DTC (direct torque control). The DFOC algorithm was first proposed in the late of seventies and has become an industrial standard. These structures have two major control loops, namely flux and torque. For the separate control of those variables, transformation blocks and a decoupling circuit are required. Although there are unquestionable advantages to the DFOC structure, it has some drawbacks, such as: complicated control structure, limited robustness to the parameter changes, and limited control performance. In the DFOC there are four controllers, which are tuned separately for an assumed linear point of the work. When working conditions change, the performances of the drive worsen. Due to the industrial demands for efficient regulation, new control algorithms are sought after.

Energies 2020, 13, 1193; doi:10.3390/en13051193 www.mdpi.com/journal/energies Energies 2020, 13, 1193 2 of 15

The model predictive control (MPC), being a sophisticated control strategy, has received much interest from a variety of industries. The MPC has been introduced in petrochemical engineering in the 1970s [6]. From then on, the MPC gradually has become popular in other branches, including those of power electronics and electrical drives. Firstly, it functions better than the classical PID system. The prediction of the future behavior of the plant allows to shorten the transition times. A crucial feature of the MPC is its capability of taking into account the constraints of the control signal and state variables. Still, a main drawback of MPC is its requirement for large computational power [7–16]. Due to the progress of microprocessor technology it is no longer a critical factor. Nowadays, the MPC-based algorithm is applied to the different industrial plants with high sampling rate, including modern electrical drives. In the MPC algorithm the mathematical model of the plant is utilized so as to indicate the future states of the object. The optimization problem is solved on the basis of the specified cost function in order to determine the best sequence of the control signal. Different MPC concepts can be found in the literature [5,9–11,16–32]. The algorithm can be divided, taking into account the prediction horizon length. The first framework is called the long-horizon MPC, in which the prediction windows is several samples ahead [9,10,14,15,23,33,34]. This allows to obtain a better control performance of the plant, especially with respect to state constraints. The second approach is referred to as the short-horizon MPC, in which only a few samples are predicted ahead (usually one or two) [10,23,35]. This allows to minimize the complexity of the calculations in the algorithm, yet at the same time the dynamics of the plant could be worse, especially in the presence of the constraints. However, according to the review of the literature, this approach is commonly used in the domain of power electronics and drive control, particularly of current and torque control [5,8–12,18,23,28–34,36–41]. The MPC can also be divided, with reference to the character of the control signal, into CS-MPC (Continuous Set Model Predictive Control) and FS-MPC (Finite Set Model Predictive Control) [9–11,33, 34,36–38]. In the first framework the MPC controller generates the signal, which takes any value within the specified range. It means that in the control structure for IM the modulator is evident. This approach is obviously quite demanding in realization as far as computation is concerned. The second framework based on the fact that the number of the power switches is limited (six in a standard 2-level inverter). This means that the possible values of the control signal are also reduced, which in turn decreases the computational effort of the algorithm. Hence, it is not necessary to include a modulator in the control structure. The MPC algorithm can also be divided into linear or non-linear models of the plant. Obviously, the non-linear approach ensures the performances of the drive but at the same time drastically increases the computational effort, which is especially visible in the long-horizon approach [9,10,19]. Therefore, the short horizon FS MPC is preferable, especially for a practical implementation. The linear MPC is far less numerically demanding, thus the long horizon CS is preferable. In a significant number of papers, the FS-MPC [14,15,18,20,21,37,38] is used to control the electromagnetic part of the induction motor. Usually, based on the similarity to the DFOC structure, the rotor torque and the component of the stator current are regulated. The speed is controlled in the outer loop of the drive with the help of PI controller. Additionally, in the CS-MPC, the direct regulation of all variables (electromagnetic and mechanical) of IM is a rare approach. However, due to the progress in processor technique, the computational power is no longer a limitation factor. Therefor the control structure which directly regulate the electromagnetic and mechanic variables seem to be attractive for future applications. In the literature there is a limited number of papers in which the comparative studies of the application of different MPC strategies for power converters and drives are presented. In [31] applications of the FS- and CS-MPC for a DC-DC converter were demonstrated. The authors described the abovementioned algorithms in detail. Then, the performance of MPC approaches in steady-stay and dynamic responses were evaluated. The theoretical considerations and simulation results were supported by experimental tests. The results of application of two methodologies (FS- and CS-MPC) to Energies 2020, 13, 1193 3 of 15 control of a grid-connected two-level inverter were shown in [32]. The authors tested these two MPC approaches and then proposed the so-called hybrid algorithm, which takes advantage of both of them. The hybrid solution allows to obtain better performances for switching losses and low current ripple. The features of the system have been specified through simulation studies. The application of the two MPC approaches (FS- and CS-MPC) to the speed and torque control of an IM drive is presented in [40]. The authors have proposed a cascade system, in which two major loops are evident, i.e., torque and speed. The speed MPC based loop generates a control signal for the torque loop. In both loops a different cost function is specified. The investigated control strategies are implemented and tested with the help of an experimental ring with a 3.7 kW IM. The general conclusion formulated in the paper is that both strategies ensure similar dynamic performances, yet the CS- possess a smaller torque ripple. In [41] comparative studies discussing the application of FS- and CS-MPC to a power converter and PMSM are presented. The system performances in terms of current ripple and switching frequency are evaluated. Both systems are characterized by similar dynamic responses, although the CS-MPC is preferred. The control of PMSM with the help of the FS- and CS- is presented in [42]. The authors have proposed the optimal current control taking into account the voltage of the inverter. Additionally, the parameter variation is considered during tests. The effectiveness of the proposed algorithm is verified through experimental tests. Contrary to the above-discussed paper, in the current research a cascadeless control structure is proposed. It requires to define only one cost function in which the performance of speed and torque control are specified. The cascadeless structure allows to react to the changes in the system faster. The main goal of the work is to presents the comparative analysis of two MPC algorithms applied for the direct speed control of the IM drive. The first approach is based on the short horizon non-linear FS MPC. The length of the horizon is set to one sample ahead and the non-linear model of the IM motor is used here. There is no modulator in the control structure; the power electronics switches are directly controlled by algorithm. The second approach relies on long horizon linear CS MPC. The prediction windows are composed with 10 samples in this case. The model of IM is linearized with the use of decoupling circuits. Moreover, the off-line version was used, which further reduced the complexity of the algorithm. These two algorithms have been tested in the laboratory stands. The influence of the different design parameters on the dynamic performance is shown. On the basis of the obtained results a critical comparison was carried out.

2. Model Predictive Control In the MPC algorithm, the plant’s explicit model is used to predict the reaction of the process output to the changes of control signal [43,44]. Usually, a discrete-time, linear state-space model is applied in MPC as presented below:

x(k + 1) = Ax(k) + Bu(k) (1) y(k) = Cx(k) where x(t) n, u(t) m, y(t) p are the system vectors (state, input, and output). A n n, B ∈< ∈< ∈< ∈< × ∈< n m, C p n are matrices of the object, and k is a discrete constant. × ∈< × The problem of optimization is computed at each time step k.

 N Nu 1  min X T X−   re f re f + T  T T  y + yk+i k Q y + yk+i k uk+i kRuk+i k (2a) u , ... , u  k i k − k i k − |  0 Nc 1 i= | | | | i= |  − 0 0 Energies 2020, 13, 1193 4 of 15

umin uk+i k umax i = 0, 1, , Nu 1, ≤ | ≤ ··· − ymin yk+i k ymax i = 0, 1, , N, ≤ | ≤ ··· (2b) xk+i+1 k = Axk+i k + Buk+i, i 0 | | ≥ yk+i k = Cxk+i k i 0, xk k = x(k) | | ≥ | where Q 0 and R > 0 mean the matrices with weights, Nu and N mean the control and prediction ≥ horizon, and the system’s input and output constraints are umin, umax, ymin, and ymax. The following inequality held in the system: Nu N. ≤ With the help of quadratic programming, formula (2a) can be presented [43,44]:

~ ~ J(U, x(k)) = XTQX + UTRU (3) where X N and U Nu are predictive vectors of state variables and controls: ∈ < ∈ <      x(k + N1 k)   u(k k)   |   |   .   .  X(k) =  . , U (k) =  .  (4)  .   .      x(k + N k) u(k + Nu 1 k) | − | ~ ~ The matrices Q NxN and R NuxNu have the following form: ∈ < ∈ <      Q 0 0   R 0 0  ~   ~   =  ..  =  ..  Q  0 . 0 ;R  0 . 0  (5)      0 0 Q   0 0 R 

Finally, the problem of optimal control using quadratic programming can be formulated:

T 1 V(x(k)) = x(k) Yx(k) + min U0HU + x0(t)FU U 2 (6) subject to GU W + Ex(k) ≤ where H, F, and Y are deﬁned in the following way:

~ T ~ ~ ~ ~ T ~ ~ ~ T ~ ~ H = B QB + R; F = A QB; Y = A QA (7)   0 0  I       ···     B 0   A   ···     . . .  ~   ~  . .. .  A =  , B =   (8)  ···Nu   Nu   A   A B     ···     . . .     . . .   ···N    A  N 1 PN Nu i   A B − A B  − ··· i=0 There are two main implementing frameworks of the MPC algorithm, taking into account the way of solving the optimization problem (2). Originally, the optimization procedure is calculated on-line in a receding-horizon fashion for a provided x(k). Thus at the current time k, only the first element of the control signal uk is supplied to the plant. The remaining part of the control sequence is discarded. Next, the described procedure is repeated in the step (k+1) for the output y(k+1). The presented strategy could be too computationally demanding for the system with a very small time constant. The optimization problem can be also solved off-line for all the possible combinations of the state vector Xf with the use of multi-parametric programming [45,46]. If the state vector x(k) is treated as a parameter vector, it could be proven that the parameter space Xf can be further divided into specific Energies 2020, 13, 1193 5 of 15 regions. Then, the optimizer can be presented as an explicit function of its parameters. The following formula describes the output of the controller:

U(x) = Krx + gr, x Pr (9) ∀ ∈ where Pr is represented as:

n Pr = x Hrx dr , r = 1, ... Nr (10) ∈ < ≤ and Nr indicatesEnergies 2019 the, 12 number, x FOR PEER of REVIEW polyhedral regions [45,46]. 5 of 15 Energies 2019, 12, x FOR PEER REVIEW 5 of 15

3. Mathematical163 and Nr indicates Model the of IMnumber and of MPC-Based polyhedral regions Control [45,46]. Structures 163 and Nr indicates the number of polyhedral regions [45,46]. 164Di fferent3. Mathematical mathematical Model models of IM and of theMPC-Based controlled Control plant Structures can be selected as a starting point of the 164 analysis. In3. Mathematical control engineering Model of the IM modeland MPC-Based shouldbe Control accurate Structures enough to detect the different physical 165 Different mathematical models of the controlled plant can be selected as a starting point of the 165 Different mathematical models of the controlled plant can be selected as a starting point of the phenomena166 analysis. but it In should control engineering not be too the computationally model should be accurate demanding, enough toto detect allow the real different time calculations.physical 166 analysis. In control engineering the model should be accurate enough to detect the different physical Taking167 thesephenomena factors intobut it account, should not the be present too computationally article is based demanding, on the IMto allow model real with time thecalculations. space vector 167 phenomena but it should not be too computationally demanding, to allow real time calculations. orientated168 Taking on the these rotor factors flux. Theinto account, values arethe expressedpresent article in peris based unit on system. the IM model The equations with the space below vector describe 168 Taking these factors into account, the present article is based on the IM model with the space vector 169 orientated on the rotor flux. The values are expressed in per unit system. The equations below the169 IM, includingorientated the on acknowledgedthe rotor flux. The simplifications values are expressed [47]: in per unit system. The equations below 170 describe the IM, including the acknowledged simplifications [47]: 170 describe the IM, including the acknowledged simplifications [47]: ddΨΨr rrrxxM rrr 171 TTN dΨr == rr xM iisx− rr ΨΨr (11) (11) 171 TN dtdtr = rxxMr isx −−xrxrΨr (11) N dt xr sx xr r r x r r rrx M 0 == r M isy − ωωrΨΨr (12) 172 0 rr xM i − (12) 172 0 = r isy − ω r Ψr (12) xr sy r r xr xM me = Ψrisy (13) = xxMr ()Ψ 173 m xM i (13) 173 me = ()Ψrisy (13) The motion equations can be presentede as: xr r sy xr 174 The motion equations can be presented as: 174 The motion equations can be presentedd as:1 ω = (m m ) (14) dtd T1 e L 175 d ω = 11 ()m −−m (14) 175 ω = ()me − m L (14) dt T1 e L where is, ir are the vectors of stator anddt rotor currents,T1 Ψr is the vector of rotor flux, ω, ωr are the 176 where is, ir are the vectors of stator and rotor currents, Ψr is the vector of rotor flux, ω, ωr are the angular176 shaftwhere and is, slipir arepulsation, the vectors Tof1 ,TstatorN are and the rotor mechanical currents, Ψ andr is referencethe vector timeof rotor constant, flux, ω, ωrr areis the the rotor 177 angular shaft and slip pulsation, T1, TN are the mechanical and reference time constant, rr is the rotor resistance,177 angularxr, xM shaftare theand reactanceslip pulsation, of rotorT1, TN are and the magnetizing, mechanical and and referenceme, eL timeare theconstant, electromagnetic rr is the rotor and 178 resistance, xr, xM are the reactance of rotor and magnetizing, and me, eL are the electromagnetic and 178 resistance, xr, xM are the reactance of rotor and magnetizing, and me, eL are the electromagnetic and load179 torque. load torque. 179 load torque. 180The additionalThe additional components components are calculated are calculated in in order order to to perform perform the the decoupling decoupling of the of theflux fluxand and 180 The additional components are calculated in order to perform the decoupling of the flux and torque181 controltorque loops control and loops inserted and inserted to the to system the system (15)–(20) (15)-(20) [47 [47]:]: 181 torque control loops and inserted to the system (15)-(20) [47]: d 182 u = r i +T ddΨ + jω Ψ sk== s sk ++ N sk ++ωk sk (15) 182 uskusk rrssiisksk TTNN dt ΨΨsksk j jωk ΨkΨsk sk (15) (15) dtdt = + 183 Ψ sk = xsi sk + xM i rk (16) 183 Ψsksk = xxssiisksk + xMiirk (16) (16) 184 Ψ = x i + x i Ψ rk==x ri rk + xM ski (17) 184 Ψrkrk xrrirkrk xMMisksk (17) (17)

ddiiss xMxM dΨrdΨr xM xM 185 u =r i +T x σ di + jω ψ x σ i + x T dΨ + jω ψ x Ψ (18) 185 us =urs i=s r+sisT+NTxNsxσsσ s ++ jjωωssψxxsσsσiiss++M TN TN r + jω+s jωMsΨψ r Ψr (18) s s s N s dtdt sψ s s xr xrN dt dt sψ xr rx r (18) dt xr dt xr Ψ disx xM d r 186 u = r i + T x σ di −ω ψ x σ i + x T dΨ (19) 186 usx = rsisx + TN xsσ sx −ωs xsσ isy + M TN r (19) sx s sxN s dt sψ s sy xr N dt (19) dt  xr dt  fx  ex  fx ex

disy xM 187 u = r i +T x σ disy +ω ψ x σ i +ω ψ x Ψ (20) 187 usy = rsisy +TN xsσ +ωs xsσ isx +ωs M Ψr (20) sy s syN s dt sψ s sx sψ xr r dt  xr  (20)  f y  ey  f y ey 188 where σ is the total engine scattering factor, ωsΨ is the field pulsation, usk, isk, Ψsk, and Ψrk are the 188 where σ is the total engine scattering factor, ωsΨ is the field pulsation, usk, isk, Ψsk, and Ψrk are the 189 vectors of stator/rotor voltage, current, and flux rotating with speed ωk, in this case ωk=ωsΨ, and xs is 189 vectors of stator/rotor voltage, current, and flux rotating with speed ωk, in this case ωk=ωsΨ, and xs is 190 the reactance of stator. 190 the reactance of stator.

Energies 2019, 12, x FOR PEER REVIEW 6 of 15

191 The CS-MPC is depicted in Figure 1. Unlike the DFOC, it comprises one controller instead of 192 four. Drawing on the system states, two control signals for the speed and torque paths are generated 193 by the MPC, with the sampling frequency of the modulator set to 10 kHz. 194 Due to the fact that the four traditional PI controllers are replaced with one predictive controller, 195 the number of controller parameters is reduced, and hence, the system tuning process is less 196 troublesome. Moreover, the use of a single predictive controller allows for simple and effective 197 entering of the constraints of individual variables. Traditional structures equipped with PI regulators 198 involve anti-windup systems, which can saturate quickly in transient states. As a consequence, such 199 a structure may serve as an open loop system.   r  s 0 0 0 0 0 0 T T x   1   N s  0 0 0 0 0 0 x r  r T x   M r r 0 0 0 0 0 B   N s  , Energies 2020, 13, 1193  1 6 of 15 xrTN xrTN  0 0 0 0 0 0   r   T x   0 0 s 0 0 0 0  N s  200 T x 0 1 0 0 0 1 0  (21) whereAσis the total engine scatteringN s factor, ωs,Ψ is theC  ﬁeld pulsation, usk, isk, Ψ,sk, and Ψrk are the  nom    vectors of stator/rotor voltage, r x current,M 1 and ﬂux rotating with0 0 speed0 1ωk,0 in this0 case1 ωk = ωsΨ, and xs is  0 0 0 0 0 the reactance of stator. x T T ref ref T  r 1 1  x  isx  r isy  mL  r   , The CS-MPC 0 is depicted0 0 in Figure0 10. Unlike0 0 the DFOC, it comprises one controller instead of four.  f x  Drawing on the0 system0 states,0 two control0 0 signals0 0 for the speed andu   torque paths are generated by the    f y  MPC, with the0 sampling0 frequency0 0 of the0 modulator0 0 set to 10 kHz.

u d ref ud S fy a Predictive controller Sa r - Predictive controller s x y S N + V Sb S N u ey ref Dynamic optimisation b J  yTQy  uT Ru M  k k  p p fx S Power s Power k  N 1 p  0 c min g(k) Sc rs - modul modul Usα, +ex Decoupling  Usβ Calculation circuit s Prediction of: of voltages i (k+n) (k+n) (k+n) (k+n) isy  -ß sa Ψ , I , T , Usα,Usβ x-y is s s e isb

-ß 

isx s is abc  ωsΨ Estimator Estimation of: Isa   -ß r of Ψ (k), Ψ (k) mL Ψ and m s r I ,I Isb r L sα sβ abc 1 ref ref   r ud u d 1

m ref ref L Control of T1 TL Control of T1 load load

201 Figure 1. The CS-MPC (a) and FS-MPC (b) based control structure. 202 Figure 1. The CS-MPC (a) and FS-MPC (b) based control structure. Due to the fact that the four traditional PI controllers are replaced with one predictive controller, 203 the numberElectromagnetic of controller (11) parameters–(13) and ismotion reduced, (14) andequations hence, determine the system the tuning discrete process model is of less the troublesome. plant 204 in MPC. By virtue of the decoupling terms (19)–(20), the system is treated as linear and an explicit Moreover, the use of a single predictive controller allows for simple and eﬀective entering of the 205 MPC algorithm can be applied. In the state vector reference variables for rotor flux and speed are constraints of individual variables. Traditional structures equipped with PI regulators involve 206 introduced. The matrices describing the system are submitted in Equation (21). 207 anti-windupThe c systems,ontrol signal which and can system saturate states quickly can be in limited transient by the states. MPC As algorithm. a consequence, In the suchfollowing a structure 208 maystudies serve the as anlimit openations loop of stator system. current in x and y axis are specified. The cost function utilized in the 209 paper is described by Equation (22):  rs   1 T T− x 0 0 0 0 0 0  0 0 0 0 0 0   σ NN s   σTNNu xs1   xMrr rr ref 2  refB = 2 ref 1 ref   , min q − k 0k 0 q 0 0k 0   k   0r u 0 p  r 0u 0p 0 0   xrTN 11xrTr N  r   22  12        1 x σTNxs 2  y   ux rs  " # 210 u k01 0 − 0 0 0 0  p0  (22) y σTNxs  0 1 0 0 0 1 0  nom  C = − ,  ψr xM max 1 max  max max A =  0 0 usx  usx0 ; usy 0usy 0 ;,isx  isx ; isy 0 i 0sy 0 1 0 0 1 (21)  xrT1 T1  −   h re f iT  0 0 0 0 0 0 0  x = i i m re f ,   sx ψr sy ω L ψr ω   " #  0 0 0 0 0 0 0  fx   u = 0 0 0 0 0 0 0 fy

Electromagnetic (11)–(13) and motion (14) equations determine the discrete model of the plant in MPC. By virtue of the decoupling terms (19)–(20), the system is treated as linear and an explicit MPC algorithm can be applied. In the state vector reference variables for rotor ﬂux and speed are introduced. The matrices describing the system are submitted in Equation (21). The control signal and system states can be limited by the MPC algorithm. In the following studies the limitations of stator current in x and y axis are speciﬁed. The cost function utilized in the paper is described by Equation (22):    N 2 Nu 1   P re f re f 2 P− h re f re f i min  q11 ψr(k) ψr (k) + q22 ω12(k) ω (k) + r1 ux (p) + r2 uy (p)  k=1 − − p=0  ∆ux (22) ∆u y max max max max usx u ; usy u ; isx i ; isy i | | ≤ sx ≤ sy | | ≤ sx ≤ sy Energies 2020, 13, 1193 7 of 15

where N and Nu means length of prediction and control horizon, respectively, q11, q22 are weights for output vectors, and r1, r2 are weights for controls. The block diagram of the control structure based on the FS-MPC is demonstrated in Figure1b, where the following elements can be distinguished: the FS-MPC speed controller, drive system, supplied converter, and measuring devices. State variables can be measured or in some cases estimated. Drawing on the present value of the state vector and control signals, the future process states are predicted using the mathematical model of the drive (23)–(27). The reference values are compared in an objective function with the computed future system states. The conditions for meeting the restrictions are also checked. The control signal that minimizes the objective function and meets the restriction conditions is the output signal of the regulator. In the proposed controller the model of the motor in α-β coordinates is used. The control signal is generated in the sequence as follows: estimation of rotor ﬂux (23) and the prediction of: stator ﬂux (24), stator current (25), electromagnetic torque (26), and speed (27): Lr LrLs Ψr(k) = Ψs(k) + Is(k) Lm (23) Lm − Lm

Ψs(k + 1) = Ψs(k) + TsVs(k) TsRsIs(k) (24) − ( " ! #) Ts Ts 1 kr Is(k + 1) = 1 + Is(k) + kr jΩ Ψr(k) + Vs(k + 1) (25) τσ τσ + Ts Rσ τr − 3 n o T (k + 1) = pIm Ψ (k + 1)I (k + 1) (26) e 2 s s

Ts Ω(k + 1) = (Te(k + 1) TL) + Ω(k) (27) J1 − 2 2 where Ts is the sample time, τσ = σLs/Rσ, Rσ = Rs+kr Rr, kr = Lm/Lr, τr = Lr/Rr, σ = 1-Lm /(LrLs), k is the sampling instant, Rs,Rr are the stator and rotor resistance, Ls,Lr,Lm are the inductances: stator, rotor, and magnetizing, Vs is the vector of stator voltages, Ω is the speed, Te,TL are the electromagnetic and load torque, p is the number of poles pair, and J is the moment of inertia of the drive. The sequence given above is reproduced for each of the assumed steps of prediction. Estimating the value of the objective function is the final step, in which the speed and flux errors are minimized. The cost function is also equipped with penalty coefficients for going beyond limits and switching the keys of the converter. A direct influence on the system’s performance is exerted by the weighing coefficients, which at the same time have an effect of individual components and hence on the final value of this function. The objective function is characterized by Equation (28). The optimal control vector is selected owing to an assessment of the objective function value:

XN XN XN re f p re f p g = µn Ω Ω (k + n) + λn Ψ Ψ (k + n) + (αn fn + βnhn) (28) − s − s n=1 n=1 n=1

ref p ref p where Ω , Ω are the reference and predicted speeds, Ψs , Ψs are the reference and predicted stator ﬂux:, hn, fn are the penalties coeﬃcients for limits exceeding and switching frequency of the converter’s keys, µ, λ, α, β are the weighting factors, and N is the prediction horizon.

4. Results The following section is devoted to the presentation of the experimental results related to both control structures. Figure2 depicts the block diagram of the experimental set-up, including an induction motor of nominal power 1.1 kW driven by a power converter. The motor and the load machine were connected with a shaft. Incremental encoders of 36,000 pulses provided measurement of the speed and position of the drive. A digital signal processor using the dSPACE 1103 calculated the Energies 2020, 13, 1193 8 of 15

Energies 2019, 12, x FOR PEER REVIEW 8 of 15 control algorithms. The sampling time of the predictive controller was set to 0.2 ms. A simulation Energies 2019, 12, x FOR PEER REVIEW 8 of 15 250 currentthe control model algorithms. was utilized The to sampling estimate time the rotorof the ﬂux predictive [47]. controller was set to 0.2 ms. A simulation 251 current model was utilized to estimate the rotor flux [47]. 250 the control algorithms. The sampling time of the predictive controller was set to 0.2 ms. A simulation 251 current model was utilizedVoltage to estimate the rotor flux [47]. DS1103 inverter with extended box measurementVoltage DS1103 invertersystem with extended boxPC+Matlab measurement system PC+MatlabdSPACE Encoder Conector dSPACEpanel Encoder Conector panel

Drive motor Load motor 252 Drive motor Load motor 252 253 FigureFigure 2. 2.The The blockblock diagramdiagram ofof thethe laboratorylaboratory set-up.set-up. 253 Figure 2. The block diagram of the laboratory set-up. To begin with, both systems were tested under a reverse condition and for a low value of the 254 To begin with, both systems were tested under a reverse condition and for a low value of the reference speed. The drive worked under the following conditions. At time t = 1.9 s, the value of the 255254 referenceTo beginspeed with. The, bothdrive systems worked were under tested the underfollowing a reverse conditions. condition At andtime for t=1.9 a low s, the value value of theof the reference signal changed from -0.2 to 0.2. The switching on and oﬀ of the load torque was performed 256255 referencereference signal speed changed. The drive from worked -0.2 to under 0.2. The the switchingfollowing conditions.on and off Atof thetime load t=1.9 torque s, the wasvalue performed of the 256 at a time of 2.2 s and 2.5 s, respectively (it is marked in picture with arrow and description). At a time 257 atreference a time of signal 2.2 s and changed 2.5 s, fromrespectively -0.2 to 0.2. (it Theis marked switching in picture on and offwith of arrowthe load and torque description) was performed. At a time 257 ofat 2.9 a stime the of reference 2.2 s and signal 2.5 s, respectively of the speed (it changed is marked to in -0.02. picture Then, with the arrow process and ofdescription) loading of. At the a systemtime 258 of 2.9 s the reference signal of the speed changed to -.02. Then, the process of loading of the system 258 wasof repeated.2.9 s the reference At a time signal of 2.9 of s thethe negativespeed changed set value to - was.02. Then reversed., the process Some transients of loading of of the the systems system are 259 was repeated. At a time of 2.9 s the negative set value was reversed. Some transients of the systems 259 shownwas repeated. in Figure A3.t a time of 2.9 s the negative set value was reversed. Some transients of the systems 260 are shown in Figure 3. 260 are shown in Figure 3.

261261 262 FigureFigure 3. 3.Selected Selected transients transients ofof thethe CS-MPCCS-MPC ( a,ba,b ) and FC FC-MPC-MPC (c,d (c,d) )structures: structures: speed speed transients transients (a,c ()a ,c) 262 Figure 3. Selected transients of the CS-MPC (a,b ) and FC-MPC (c,d) structures: speed transients (a,c) 263 andand fragment fragment of of the the stator stator currents currents ((bb,d,d)) forfor thethe value of the the reference reference signal signal equal equal ωref=0,2ωref =. 0.2. 263 and fragment of the stator currents (b,d) for the value of the reference signal equal ωref=0,2. 264 InI Figuren Figure3 the3 the transients transients of of mechanical mechanical velocitiesvelocities (Figure(Figure 33a,c)a,c) and fragments of of the the currents currents 264 In Figure 3 the transients of mechanical velocities (Figure 3a,c) and fragments of the currents 265 (Figure(Figure3b,d) 3b,d) are are demonstrated demonstrated in CS CS-MPC-MPC (a,b) (a,b) and and FS-MPC FS-MPC control control structures. structures. From the From above the- 265 (Figure 3b,d) are demonstrated in CS-MPC (a,b) and FS-MPC control structures. From the above- 266 above-presentedpresented transients transients the following the following remarks remarks can be formulated. can be formulated. Both control Both structures control were structures working were 266 presented transients the following remarks can be formulated. Both control structures were working 267 workingproperly properly and follow anded followed the reference the reference signal accurately. signal accurately. The raising The time raising depended timedepended on the set driving on the set 267 properly and followed the reference signal accurately. The raising time depended on the set driving 268 drivingtorque torque limitation limitation in the system. in the system.Applying Applying the load torque the load resulted torque in resulted the small in disruption the small in disruption the rotor in 268 torque limitation in the system. Applying the load torque resulted in the small disruption in the rotor 269 thespeed, rotor speed, yet the yet CS the-MPC CS-MPC structure structure possesses possesses smaller smaller speed speed fall. fall. The The difference diﬀerence between between the thetwo two 269 speed, yet the CS-MPC structure possesses smaller speed fall. The difference between the two 270 presentedpresented approaches approaches is is clearly clearly visible visible in in the the shape shape of of the the stator stator currants currants (Figure (Figure3b,d). 3b,d). In In CS-MPC CS-MPC the 270 presented approaches is clearly visible in the shape of the stator currants (Figure 3b,d). In CS-MPC 271 currentsthe currents contain contain much much fewer fewer high-frequency high-frequency noises. noises. It hasIt has an an almost almost sinusoidal sinusoidal shape. shape. ContraryContrary to 271272 theto currentsthat, in FS contain-MPC the much current fewer possesses high-frequency a lot of ha noises.rmonics It, hasstemming an almost from sinusoidal the lack of shape. a modulator Contrary 272273 toin that, the systemin FS-MPC as well the as current the short possesses prediction a lot horizon. of harmonics , stemming from the lack of a modulator 273 in the system as well as the short prediction horizon.

Energies 2020, 13, 1193 9 of 15

that, in FS-MPC the current possesses a lot of harmonics, stemming from the lack of a modulator in the system as well as the short prediction horizon. EnergiesEnergies 2019 2019, ,12 12, ,x x FOR FOR PEER PEER REVIEW REVIEW 99 of of 15 15 Next, the eﬀect of the limitation of the stator current was investigated. The FS-MPC was analyzed 274274 ﬁrst. InNextNext Figure, , thethe4 transients effect effect of of the ofthe the limitation limitation states are of of presented. the the stator stator current current waswas investigated.investigated. The The FS FS-MPC-MPC waswas 275275 analyzedanalyzed first. first. In In Figure Figure 4 4 transients transients of of the the states states are are presented. presented.

276276 277277 FigureFigureFigure 4. 4. 4.Selected Selected Selected transientstransients transients of of the the FS FS-MPC:FS-MPC:-MPC: speed speed ( a( aa),),), electromagnetic electromagnetic electromagnetic torque torque torque ( b(b) (,)b ,and ),and and currents currents currents ( c,d,e(c,d,e (c,)d ) ,e) 278278 forforfor di different ﬀdifferenterent value value value of of of the the the stator stator stator currentcurrent current limitation:limitation: limitation: abs(imax)=1, abs(imax)abs(imax)=1,= 1.5 1,1.5 1.5,, ,and and and 2 2 and and 2 and ωref=0,2 ωref=0,2ωref.= . 0.2.

279279 InInIn Figure Figure Figure4 transients4 4 transients transients of of of the the the speeds speeds speeds (a),(a) (a), , driving driving torque torquetorque (b) (b),(b), ,and and stator stator stator current current current ( c,d,e(c,d,e (c,d,e)) )for for for different different di fferent max 280280 valuesvaluevalues ofs of of the the the statorstator stator limitation limitation ( (abs( (abs(abs(iiimaxmax)=1,)=1,) =1.5 1.51, and and 1.5 2 and2) )are are 2)presente presente are presented.d.d. As As can can be Asbe concluded concluded can be concluded from from Figure Figure from 281281 Figure44, ,the the4 system, system the system worked worked worked correctly. correctly. correctly. A A significant significant Depending limitation limitation on the, ,the the assumed bigger bigger torque signaltorque, , limits,was was generated generated di fferent, ,and and torque shorter shorter levels 282282 andrisingrising speed time time rising of of the the times speed speed were was was obtained.obtained obtained. .Depending Depending Despite correct on on the the work assumed assumed of the signal signal control limits, limits, structure, different different some torque torque oscillations levels levels 283283 wereandand evident speed speed rising rising into torquetimes times were were transients. obtained. obtained. Those Despite Despite oscillations correct correct work causedwork of of the the control shapecontrol ofstructure structure the speed, ,some some transients oscillation oscillation duringss 284284 thewerewere raising evidentevident time, into into which torque torque are transients. transients. not straight. Those Those The oscillations oscillations current possessed cause causedd thethe high shape shape frequency of of the the speed speed noises, transients transients which are 285285 characteristicduringduring the the raising raising for FS-MPC. time time, ,which which are are not not straight. straight. The The current current possessed possessed high high frequency frequency noises noises, ,which which 286 are characteristic for FS-MPC. 286 areThen, characteristic the CS-MPC for FS was-MPC. investigated. Different values of the stator current component isy were set 287 Then, the CS-MPC was investigated. Different values of the stator current component isy were 287 duringThen tests., Thethe CS obtained-MPC was transients investigat areed. presented Different in value Figures of5 .the stator current component isy were 288288 setset during during tests. tests. The The obtained obtained transients transients are are presented presented in in Figure Figure 5 5. .

289289 Figure 5. Selected transients of the CS-MPC: speed (a), driving torque (b), and currents (c,d,e) for 290290 FigureFigure 5. 5. Selected Selected transients transients of of the the CS CS-MPC:-MPC: speed speed ( a(a),), driving driving torque torque ( b(b),) ,and and currents currents ( c,d,e(c,d,e) )for for diﬀerent value of the stator current limitation: isy = 1, 1.5, and 2 and value of the reference signal different value of the stator current limitation: isy=1, 1.5, and 2 and value of the reference signal ωref=0,2. 291291 different value of the stator current limitation: isy=1, 1.5, and 2 and value of the reference signal ωref=0,2. ωref = 0.2. 292292 TheThe transients transients of of the the speeds, speeds, driving driving torque torque, ,and and stator stator current currentss a arere presented presented in in Figure Figure 5 5 for for CS CS-- 293293 MPCMPC for for different different value valuess of of the the stator stator component component limitation. limitation. In In this this case case the the electromagnetic electromagnetic torque torque 294294 transientstransients had had no no oscillations. oscillations. This This resulted resulted in in the the straight straight shape shape of of the the speed speed transients transients during during start start--

Energies 2020, 13, 1193 10 of 15

The transients of the speeds, driving torque, and stator currents are presented in Figure5 for Energies 2019, 12, x FOR PEER REVIEW 10 of 15 CS-MPCEnergiesfor 2019 di, 12ﬀ,erent x FORvalues PEER REVIEW of the stator component limitation. In this case the electromagnetic10 torque of 15 transients had no oscillations. This resulted in the straight shape of the speed transients during start-up. 295 up. Additionally, the overshoots in speed transients are much smaller than previously. The bigger 295 up. Additionally, the overshoots in speed transients are much smaller than previously. The bigger 296 Additionally,the value of the limitation, overshoots the in smaller speed transientsthe obtained are settling much time smaller of the than speed. previously. The current The transients bigger the 296 the value of the limitation, the smaller the obtained settling time of the speed. The current transients 297 valuewere of much the limitation, smoother theas compare smaller thed to obtained the FS-MPC. settling The time set limitation of the speed. value The of currentthe current transients was not were 297 were much smoother as compared to the FS-MPC. The set limitation value of the current was not 298 muchviolated smoother. as compared to the FS-MPC. The set limitation value of the current was not violated. 298 violated. 299 TheThe robustness robustness of of both both controlcontrol algorithmsalgorithms for for changes changes of of the the selected selected parameters parameters of ofthe the IM IM was was 299 The robustness of both control algorithms for changes of the selected parameters of the IM was 300 investigated.investigated. Firstly, Firstly, the the FC-MPC FC-MPC toto thethe variation of of the the stator stator resistance resistance and and inertia inertia of ofthe the rotor rotor was was 300 investigated. Firstly, the FC-MPC to the variation of the stator resistance and inertia of the rotor was 301 tested.tested The. The value value of of the the parametersparameters was changed changed into into the the model model of of the the object object use usedd in inthe the algorithm. algorithm. 301 tested. The value of the parameters was changed into the model of the object used in the algorithm. 302 SomeSome of of the the obtained obtained transients transients areare shownshown in FiguresFigure 66 and and 77.. 302 Some of the obtained transients are shown in Figure 6 and 7.

303 303 304 Figure 6. Selected transients of the FS-MPC: speeds (a), fragment of speeds (b), stator flux (c), and 304 FigureFigure 6. Selected6. Selected transients transients of of the the FS-MPC: FS-MPC: speedsspeeds ((aa),), fragmentfragment of of speeds speeds ( (bb),), stator stator flux ﬂux (c ()c, ),and and 305 driving torques (d) for various value of the stator resistance and value of the reference signal ωref=0,5. 305 drivingdriving torques torques (d) ( ford) for various various value value of of the the stator stator resistance resistance and and value value of of the the reference reference signalsignal ωωrefref=0,5= 0.5..

306 306 Figure 7. Selected transients of the FS-MPC: speeds (a), electromagnetic torques (b), and stator flux for 307 Figure 7. Selected transients of the FS-MPC: speeds (a), electromagnetic torques (b), and stator flux 307 variousFigure value 7. Selected of the motortransients inertia: of the Jn FS(c);-MPC: 0.5Jn (speedsd); 0.25J (an), (electromagnetice); 2Jn (f). torques (b), and stator flux 308 for various value of the motor inertia: Jn (c); 0.5Jn (d); 0.25Jn (e); 2Jn (f). 308 for various value of the motor inertia: Jn (c); 0.5Jn (d); 0.25Jn (e); 2Jn (f). 309 InIn Figure Figure6, the6, the speed, speed, stator stator flux, flux and, torqueand torque transients transients for di ffforerent different values value of thes statorof the resistance stator 309 In Figure 6, the speed, stator flux, and torque transients for different values of the stator 310 areresistance presented. are Thepresented. reference The valuereference of the value speed of the was speed set towas the set half to the of thehalf nominalof the nominal one. Theone. systemThe 310 resistance are presented. The reference value of the speed was set to the half of the nominal one. The 311 wassystem working was working under reverse under conditions.reverse conditions. It should It should be noted be noted that in that this in casethis thecase shapethe shape of the of speedthe 311 system was working under reverse conditions. It should be noted that in this case the shape of the 312 duringspeed start-up during start was- straighterup was straighter than for than smaller for smaller value of value the reference of the reference signal. Thesignal. decrease The decrease of the valueof 312 speed during start-up was straighter than for smaller value of the reference signal. The decrease of 313 ofthe the value stator of resistance the stator influencedresistance influence the systemd the properties system properties significantly. significantly. The settling The settling time of time the speed of 313 the value of the stator resistance influenced the system properties significantly. The settling time of 314 the speed was bigger than for the other cases. Additionally, oscillations of the speed during transients 314 the speed was bigger than for the other cases. Additionally, oscillations of the speed during transients 315 were bigger (Figure 6b). The increase of the stator resistance value had a small effect on the system 315 were bigger (Figure 6b). The increase of the stator resistance value had a small effect on the system

Energies 2020, 13, 1193 11 of 15

was bigger than for the other cases. Additionally, oscillations of the speed during transients were biggerEnergies (Figure2019, 12,6 xb). FOR The PEER increase REVIEW of the stator resistance value had a small e ffect on the system speed.11 of 15 The settling time of the speed was almost identical as for the nominal parameters. The stator flux was 316 controlledspeed. The correctly. settling time of the speed was almost identical as for the nominal parameters. The stator 317 flux Then,was controlled the effect ofcorrectly. the changing of the inertia was analyzed using transients presented in Figure7. 318 Then, the effect of the changing of the inertia was analyzed using transients presented in Figure The followingThen, the cases effect were of the considered: changing of inertia the inertia equal was to nominal analyzed value using (J transients= Jn), decrease presented (J = 0.5J in Figuren and 319 7. The following cases were considered: inertia equal to nominal value (J=Jn), decrease (J=0.5Jn and J7=. The0.25J followingn), and increase cases (Jwere= 2J nconsidered:) of the inertia. inertia The equal analyzed to nominal system workedvalue (J=J under), decrease a reverse (J=0.5J condition; and 320 theJ=0.25J referencen), and speedincrease changed (J=2Jn) fromof the 0.2 inertia to .0.2. TheThe analyzed transient system of the worked speed, under torque, a reverse and flux condition for the; − 321 nominalthe reference system speed parameter changed is presented from 0.2 withto -0.2. the The blue transient line. The of increase the speed, of the torque inertia, and had aflux relatively for the 322 smallnominal effect system on the parameter transients is ofpresented the system. with The the shapesblue line. of The the systemincrease variables of the inertia were had similar a relatively to the 323 systemsmall effect with nominalon the transients parameters. of the Contrary system. to The this, shape the decreases of the of system the inertia variables can be were seen similar more clearly to the 324 insystem the transients. with nominal The settlingparameters. time Contrary of the speed to this increased,, the decrease and the of hodographthe inertia can of the be seen flux wasmore noisier. clearly 325 in theThen, transients. the performance The settling of time the of CS-MPC the speed was increased tested for, and the the variation hodograph of the of systemthe flux parameters.was noisier. 326 The transientsThen, the of performance the plant for of changes the CS-MPC of rotor was resistance tested for and the inertiavariation are of shown the system in Figures parameters.8 and9. The 327 transients of the plant for changes of rotor resistance and inertia are shown in Figure 8 and Figure 9.

328

329 FigureFigure 8.8. SelectedSelected transients transients of of the the CS CS-MPC:-MPC: speeds speeds (a), (a ),fragment fragment of ofspeeds speeds (b), ( bstator), stator flux ﬂux (c), and (c), 330 anddriving driving torques torques (d) (d for) for different diﬀerent value valuess of of the the stator stator resistance resistance and and of the reference reference signal signal valuevalue ω = 0.5. 331 ωrefref=0,5.

332

333 FigureFigure 9. 9.Selected Selected transients transients of of the the CS-MPC: CS-MPC: speeds speeds (a ),(a electromagnetic), electromagnetic torques torques (b), ( andb), and stator stator ﬂux flux for 334 diforﬀerent different values value of thes of motor the motor inertia: inertia Tn (:c T);n 0.5T(c); 0.5Tn (d);n (d); 0.25T 0.25Tn (e);n (e) 2T;n 2T(f).n (f).

335 In Figure 8, transients of mechanical speeds (Figure 8a) and fragments of speeds (Figure 8b), 336 rotor torques (Figure 8c), and electromagnetic torques (Figure 8d) are demonstrated for different 337 values of the stator resistance. After analyzing the above-presented transients, the following 338 conclusions can be reached. The changes of the stator resistance had a much smaller effect to the

Energies 2020, 13, 1193 12 of 15

In Figure8, transients of mechanical speeds (Figure8a) and fragments of speeds (Figure8b), rotor torques (Figure8c), and electromagnetic torques (Figure8d) are demonstrated for di fferent values of the stator resistance. After analyzing the above-presented transients, the following conclusions can be reached. The changes of the stator resistance had a much smaller effect to the transients of the drive than in FS-MPC. The speed transients were similar. For the decreasing of the stator resistance the settling time of the speed was slightly bigger. The oscillation evident for speed transients was relatively small (Figure8b). It resulted from the smooth shape of the torque transients (Figure8d). Small fluctuations were evident in the rotor flux transients in all cases (Figure8c). Then, the effect of the inertia changes on the properties of the system was investigated, similar to previous work, where the increase (J = 2Jn) and decrease (J = 0.5Jn and J = 0.25Jn) of inertia was considered. The transients of the speed (Figure9a), torques (Figure9b), and the rotor flux (Figure9c) are presented. The quoted figures may lead to the following remarks. The changes of the inertia had limited impact on the transients of the drive in CS-MPC. The shape of all variables was similar for all cases. The differences between states were smaller than in the FC-MPC.

5. Conclusions The core of this paper is a detailed investigation of two MPC-based control structures for speed control of an IM drive. The first control algorithm relies on FS-MPC approach, while the second one works with the use of the CS-MPC. Theoretical considerations and experimental validations may be summarized as follows. When referring to the classical field-oriented control structure of the IM drive, the MPC approach seems to be an easier case at first glance. Instead of four classical PI controllers evident in the DFOC control structure, only one controller regulating all variables is visible. Practical implementation of the long horizon CS-MPC is rather difficult due to the fact that this algorithm is computationally complex. Hence, the following steps should be taken. First, an IM model is made linear by means of inserting a decoupling circuit, i.e., the transformation blocks evident in DFOC, has to be applied. As a next step, the off-line MPC version is utilized in order to simplify the algorithm. This allows to decrease the complexity of traditional long-horizon MPC and run real time tests. The mathematic model of IM does not include nonlinearities evident in the real system, such as the non-linear magnetizing characteristic, nonlinear friction. Those factors will reduce the performance in the drive especially in specific condition (e.g., in ultra- speed region). A similar problem arises for the change of the system parameters. Thus, the on-line version of MPC can be used, but the complexity of the algorithm increases as a result. For the short horizon FS-MPC the optimization problem is solved on-line. Non-linearities may be included in the mathematical model of the plant in a natural way, which is why it is especially recommended for the low speed operation range. Moreover, when the parameters of the plant change, the changes can be easily incorporated in the algorithm. In the control structure of CS-MPC the modulator is evident. It determines the constant value of the switching frequency. Contrary to this case, the modulator is not required in the FS-MPC. In this case the switching frequency is directly determinate as a function of the switches, which means that the modulation frequency is changeable. In order to hold it constant, the additional algorithm have to be applied. In the FS-MPC, the phase currents possess many more components with high frequency. Those noises generate oscillations into electromagnetic torque. On the contrary, the CS-MPC results in the sinusoidal currents and smooth driving torque. Both control algorithms allow for directly including the limitation (the control signal, state variables). Energies 2020, 13, 1193 13 of 15

The proposed algorithms worked correctly in experimental tests. The reference signal as followed quickly by the system and changes of the load torque were removed swiftly. The constraints formulated in the control problem were not violated during the tests. The CS-MPC is less sensitive to parameter changes of the plant, which results from the length of the horizon. In the CS-MPC it has a value of 10 samples, contrary to this approach, in FS-MPC only one sample is calculated ahead. The attempt to increase the length of horizon in FS-MPC has been done. However, real-time implementation was not possible due to the computational requirements of the experimental processor. Thus, both implemented structures CS- and FS- need similar computational power.

Author Contributions: All authors contributed equally to this paper, in particular: conceptualization, K.S., P.S. and K.W.; methodology, P.S. and K.W.; software, K.W. and P.S.; validation, K.S.; formal analysis, K.S., P.S. and K.W.; investigation, K.W.; resources, K.S.; data curation, K.W.; writing—original draft preparation, K.S. and K.W.; writing—review and editing, K.S., K.W.; visualization, K.W.; supervision, K.S.; project administration, K.S.; funding acquisition, K.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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