FixedSwitching Frequency FiniteState Model Predictive Thrust and Primary Flux Linkage Control for LIM
Mahmoud F. Elmorshedy ^{1,2,†}, Abualkasim Bakeer ^{3,†,*}, Dhafer Almakhles ^{1}

Renewable Energy Lab., College of Engineering, Prince Sultan University, Riyadh, Saudi Arabia

Electrical Power and Machines Engineering Department, Faculty of Engineering, Tanta University, Tanta, Egypt

Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan 81542, Egypt

† These authors contributed equally to this work.
* Correspondence: Abualkasim Bakeer
Academic Editor: Mohammad Jafari
Special Issue: Optimal Energy Management and Control of Renewable Energy Systems
Received: February 14, 2023  Accepted: April 27, 2023  Published: May 09, 2023
Journal of Energy and Power Technology 2023, Volume 5, Issue 2, doi:10.21926/jept.2302017
Recommended citation: Elmorshedy MF, Bakeer A, Almakhles D. FixedSwitching Frequency FiniteState Model Predictive Thrust and Primary Flux Linkage Control for LIM. Journal of Energy and Power Technology 2023; 5(2): 017; doi:10.21926/jept.2302017.
© 2023 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.
Abstract
The special design of linear induction machines (LIMs) leads to adverse effects caused by the longitudinal and end effects. These effects make the thrust control of the LIMs most attractive because its value decreases sharply with the speed increase. Thus, finitestate model predictive control (FSMPC) is developed to increase the performance of the LIMs. However, the variable switching frequency is the main drawback of this control. Consequently, the main objectives of this paper are to propose FSMPC with a constant switching frequency, directly control the linear speed, and overcome the problems resulting from the longitudinal and end effects. Therefore, the proposed FSMPC is based on the thrust and primary flux linkage (TF) control concept. In addition, the end effect is considered during the modeling of the proposed control method. The proposed FSMPTFC method has been tested under different working cases using MATLAB/Simulink to check its validity. Parameters of a 3 kW arc induction machine have been used during the simulation results.
Keywords
Linear induction motor; finitestate model predictive control; thrust control; flux control; speed control
1. Introduction
In general, linear induction machines (LIMs) are created by cutting apart the rotor and stator of rotary induction machines (RIMs) and flattening them. The RIMs have been widely used in applications such as drive systems like lifts, electric vehicles, renewable energy, and so on [1,2,3,4,5,6]. LIMs have many advantages and hence can be used in different applications. One of the most widespread applications is the employment of linear metro due to their substantial benefits of direct linear motion without any transformation gears, which can benefit from powerful acceleration or deceleration, outstanding hillclimbing ability, low noise, and so on [7,8,9]. The HSST in the TobuKyuryoLine, the Guangzhou Subway Line 4, the airport rapid transport line in China, the Kennedy Airline in America, the Vancouver light train in Canada, and others are just a few of the more than 30 commercial lines that have been built to date [10,11]. However, the mutual inductance of LIM fluctuates with operation speed with significant nonlinear features, which is why it is thought that the primary problem with these machines is their end effects, which would negatively affect the drive performance of the entire system [12].
Up until now, the LIM and drive have primarily used fieldoriented control (FOC) and direct thrust control (DTC) [13,14]. The FOC generally has issues with changing parameters, transformation matrices, and delayed response, whereas the DTC has issues with high thrust ripple and variable switching frequency. To solve the issues with classical control, model predictive control (MPC) methodologies are proposed [15,16,17]. MPC is typically divided into two categories: continuousstate MPC (CSMPC) and finitestate MPC (FSMPC) [18]. In this work, the FSMPC is used due to its straightforward implementation, quick dynamic reaction, and other factors after rigorous comparison [19,20,21,22]. In a nutshell, there are two forms of FSMPC: FSMPCC, which is based on predictive current control, and FSMPTFC, which is based on predictive thrust and flux control [8,23]. The long computation times caused by the Clark transformation are often the most significant FSMPCC issues.
Therefore, the FSMPTC has been developed and discussed widely in recent research work such as [24,25,26,27,28,29,30,31]. In [24] and [25], the speed estimation algorithm for the LIMs is proposed based on a model reference adaptive system (MRAS), while the FSMPDTC is used as a controller. The FSMPDTC with the speed estimation is used to improve the LIM drive system performance, where low cost, low thrust and flux ripple, and fast response can be achieved. In [25], fuzzy logic control (FLC) is adopted instead of the proportional integral control for both the outer speed control loop and the adaptation mechanism of the MRAS with a minimum number of membership functions to decrease the computation time.
In [26] and [27], the FSMPC is improved for the LIMs to remove the weighting factor from the cost function and reduce the calculation burden. These proposed methods are called finitestate model predictive voltage control (FSMPVC) and finitestate model predictive flux control (FSMPFC). In [28], a comparison between the FSMPTC and the FSMPFC is presented to illustrate the capability of each control method. In [28] and [29], the sliding mode control is used in the outer control loop of the FSMPTC for the LIMs and compared with the PI control loop. The target of combining the SMC with the FSMPTC is higher tracking accuracy and faster error convergence. In [23], the FSMPTC is improved to increase the efficiency of the LIMs by achieving the maximum thrust per ampere, where the optimum flux linkage is calculated according to the electromagnetic thrust and set as a reference in the cost function. In [30], the FSMPTC is combined with the DTC concept to reduce the number of predicting voltage vectors and hence reduce the computation time.
Although extensive research works have been done to achieve and increase the performance of the LIMs drive system based on the FSMPTC concept, all of these FSMPTC are based on the variable switching frequency concept, which is not preferred for the threephase voltage inverter as the inverter can be damaged if the switching frequency is increased above the allowable rating values.
As a result, this research suggests using a fixedswitching frequency in a finite state model predictive thrust and flux control to eliminate the problem of the variable switching frequency. The following are the main key points of the paper:
 Design a fixedswitching frequency finite state model predictive thrust and flux control for VSIdriven LIM.
 Define optimally the dwell times associated with the vectors of the selected sector.
 Provide high power quality for the VSIdriven LIM by fixing the switching frequency.
 Validate the proposed fixedswitching FSMPTFC using MATLAB/Simulink software.
The structure of this essay is as follows: The mathematical model of the LIM is described in Section II. The conventional FSMPTFC approach is presented in Section III. Section IV details the proposed approach of the fixedswitching frequency for FSMPTFC. In Section V, simulation results are discussed to demonstrate that the proposed method can fix the switching frequency and improve poor performance brought on by variable switching frequency. Finally, section VI reports the conclusions of the paper.
2. LIM Mathematical Model
Researchers were concerned about the LIM's dynamic model due to the endeffect activities that cause the airgap flux linkage to wander [10,31]. Based on Duncan's equivalent circuit, the LIM's dynamic model is presented [31]. In [8], the entire dynamic model, including the endeffect, is displayed in αβaxis coordinates. The following relations describe the modeling of the LIM in the stationary reference frame.
\[ \begin{matrix}u_{\alpha1}=R_{\alpha1}i_{\alpha1}+\frac{\text{d}\lambda_{\alpha1}}{dt}\\ u_{\beta1}=R_{\beta1}i_{\beta1}+\frac{\text{d}\lambda_{\beta1}}{dt} \end{matrix} \Biggr\rangle \tag{1} \]
\[ \begin{aligned}0&=R_{\alpha2}i_{\alpha2}+\frac{\text{d}\lambda_{\alpha2}}{dt}+(\unicode[Times]{x03C9} _1\unicode[Times]{x03C9}_2)\lambda_{\beta2}\\ 0&=R_{\beta2}i_{\beta2}+\frac{\text{d}\lambda_{\beta2}}{dt}+(\unicode[Times]{x03C9}_1\unicode[Times]{x03C9}_2)\lambda_{\alpha2}\end{aligned} \Biggr\rangle \tag{2} \]
Where u_{α}, u_{β} are the αβaxis voltages, i_{α}, i_{β} the αβaxis currents, and λ_{α}, λ_{β} the αβaxis fluxlinkages. R is the resistance and L stands for selfinductance. Meanwhile ω_{1} and ω_{2} are the primary and the secondary linear speed, respectively. In the meantime, subscripts 1 and 2 refer to the primary and the secondary.
The αβaxes of the primary and secondary fluxlinkages are calculated from
\[ \begin{matrix}\lambda_{\alpha1}&=L_1i_{\alpha1}+L_{meq}i_{\alpha2}\\ \lambda_{\beta1}&=L_1i_{\beta1}+L_{meq}i_{\beta2}\end{matrix} \Biggr\rangle \tag{3} \]
\[ \begin{aligned}\lambda_{\alpha2}&=L_2i_{\alpha2}+L_{meq}i_{\alpha1}\\ \lambda_{\beta2}&=L_2i_{\beta2}+L_{meq}i_{\beta1}\end{aligned} \Biggr\rangle \tag{4} \]
In addition, L_{meq} is calculated using the mutual inductance after endeffect modification and is computed by
\[ L_{meq} = \bigl(1f(Q)\bigr)L_m \tag{5} \]
where f(Q) is a coefficient introduced by the dynamic end effect, and L_{m} is the mutual inductance at static. f(Q) is the dynamic endeffect and it is calculated from
\[ f(Q)=\frac{[1\exp(Q)]}{Q}\text{where} \ Q=\frac{D_sR_2}{(v_2[L_{l2}+L_m])} \tag{6} \]
The motion relation for the LIM is given by
\[ F_e=F_l+M\frac{dv_2}{dt}+Bv_2 \tag{7} \]
Meanwhile, the electromagnetic thrust can be calculated from
\[ F_e=\frac{3\pi}{2\tau}\left(\vec{\lambda}_1^*\otimes\vec{l}_1\right) \tag{8} \]
3. Conventional FiniteState Model Predictive Thrust and Flux Control (FSMPTFC)
FSMPTFC is offered for the LIM to obtain a quicker response, reduced thrust ripples, and the lowest primary fluxlinkage ripples. The FSMPTFC operates on the same principles as the traditional DTC, except for using an already established switching table. However, the switching vector that provides a minimal value for the cost function is chosen by the FSMPTFC. This control method can be broken down into three critical steps to maximize efficiency. The most important stage in selecting the best vector is parameter estimate, followed by the prediction step, and cost function optimization step. These key points are outlined below.
 The estimation step for both primary and secondary flux linkages is determined through the following relations:
\[ \vec{\lambda}_1(k)=\vec{\lambda}_1(k1)+Ts\big(\vec{v}_1(k)R_1\vec{l}_1(k)\big) \tag{9} \]
\[ \vec{\lambda}_2(k)=\frac{L_2}{L_{meq}}\vec{\lambda}_1(k)+\left(L_{meq}\frac{L_2L_1}{L_{meq}}\right)\vec{\iota}_1(k) \tag{10} \]
 With the use of the firstorder Euler technique, predictions for the primary fluxlinkage, λ_{1}(k + 1), primary current, i_{1}(k + 1), and electromagnetic thrust, F_{e}(k + 1) are determined by
\[ \lambda_{\alpha1,i}(k+1)=\lambda_{\alpha1}(k)+T_s\left(u_{\alpha1,i}(k)R_1i_{\alpha1}(k)\right) \tag{11} \]
\[ \lambda_{\beta_{1,i}}(k+1)=\lambda_{\beta_{1}}(k)+T_{s}\left(u_{\beta_{1,i}}(k)R_{1}i_{\beta_{1}}(k)\right) \tag{12} \]
\[ \begin{matrix} i_{\alpha1,k}(k+1)=[i_{\alpha1}(k)]\times\left[\left(\frac{T_S}{Z}\right)\left(R_1+\frac{R_2}{\tau_l^2}\right)+1\right] \\ +\left(\frac{T_S}{Z}\right)\times\left(u_{\alpha1,k}(k)+\left(\frac{1}{\tau_{_r}\tau_{_l}}\frac{\omega_{_2}}{\tau_{_l}}\right)\lambda_{\beta_2}(k)\right) \end{matrix} \tag{13} \]
\[ \begin{matrix} i_{\beta1,k}(k+1)=\left[\left(\frac{T_S}{Z}\right)\left(R_1+\frac{R_2}{\tau_l^2}\right)+1\right]\times\left[i_{\beta1}(k)\right] \\ +\left(\frac{T_S}{Z}\right)\times\left(u_{\beta1,k}(k)+\left(\frac{1}{\tau_{_r}\tau_{_l}}\frac{\omega_{_2}}{\tau_{_l}}\right)\lambda_{\alpha_2}(k)\right) \end{matrix} \tag{14} \]
\[ F_e(k+1)=\frac{3\pi}{2\tau}\begin{pmatrix}\lambda_{\alpha1}(k+1)*i_{\beta1}(k+1)\\ +\lambda_{\beta1}(k+1)*i_{\alpha1}(k+1)\end{pmatrix} \tag{15} \]
whereas $\mit\unicode[Times]{x03C4} _r=\frac{L_2}{R_2}$, $Y=\frac{(T_S)}{[L_2+R_2T_S]}$ , $Z=(L_1\frac{L^2_{meq}}{L_2})$, $\mit\unicode[Times]{x03C4} _{l}=\frac{L_{2}}{L_{meq}}$ u_{α,} _{k}(k) and u_{β,} _{k}(k) are the αβaxis voltage vectors. $i_{\alpha_1}(k)$ and $i_{\beta_1}(k)$ are the αβaxis measured currents.
The proposed cost function, g_{T}, is designed as follows:
\[ g_T=\leftF_e^*F_{e,i}(k+1)\right+K_1\left\lambda_1^*\lambda_{1,i}(k+1)\right \tag{16} \]
where K_{1} stands for the weighting factors. A single PI controller controls the linear speed, and the output of this PI is subsequently employed as a reference thrust in the cost function. The complete block diagram of the conventional FSMPTFC is illustrated in Figure 1.
Figure 1 FSMDTFC for the LIM drive system.
4. Proposed FixedSwitching Frequency FSMPTFC for LIM Drive System
As a result of space vector modulation, it is possible to precisely position each vector within the vector space information in the (αβ) plane, as shown in Figure 2. The (αβ) plane can be divided into six sectors, each corresponding to a certain direction. According to the proposed technique, the two active vectors that comprise each sector are calculated from the predicted values of the two active vectors (S_{n} where n ∈ [1,6]) at every sampling time (T_{s}) and evaluates the total cost function.
Figure 2 Space vectors of the output voltage at the 2LVSI terminals.
The cost function measures the difference between the actual and predicted vector output. Next, the fixedswitching frequency FSMPTFC uses this information to adjust the active vectors so that the predicted vector output matches the actual vector output. In the last step, the cost function for each sector is evaluated, and predictions are made, where duty cycles are calculated for active vectors and zero vectors using the following equation:
\[ d_x=\frac{\delta}{J_x} \tag{17} \]
where δ denotes the proportionality constant, the subscript x refers to the adjacent vectors, in the current case (x = 1; 2) for the active vectors in the sector, while x = 0 corresponds to the zero vector.
The sum of the dutycycle for the two active vectors and the zero vector is always equal to one; see Eq. (16) in which d_{1} is the dutycycle for the first active vector in the sector, d_{2} is the dutycycle of the second active vector in the sector, and d_{0} is the dutycycle of the zero vector. The value of the duty cycle for each voltage vector can be found by solving Eq. (15) and Eq. (16), yielding Eq. (17).
\[ d_1+d_2+d_0=1 \tag{18} \]
\[ \begin{cases}d_1=\frac{\sigma J_2J_0}{J_1J_2+\sigma J_1J_0+\sigma J_2J_0}\\ d_2=\frac{\sigma J_1J_0}{J_1J_2+\sigma J_1J_0+\sigma J_2J_0}\\ d_0=\frac{ J_1J_2}{J_1J_2+\sigma J_1J_0+\sigma J_2J_0}\\ \end{cases} \tag{19} \]
A tuning parameter σ is associated with the cost function during the zerovoltage vector (i.e., J_{0}) [32]. Adjusting σ affects the zerovector time and therefore affects the performance of the 2LVSI on the LIM. In the current work, the value of the σ parameter is set employing trial and error until achieving the desired performance. At every time step t, the following cost function is evaluated to determine the optimal sector selection as
\[ g(k+1)=d_1J_1+d_2J_2 \tag{20} \]
where J_{1} and J_{2} are the cost functions associated with the tested sector's first and second voltage vectors, respectively.
The two vectors that minimize the cost function are chosen and applied in the next sampling interval. To determine how long each vector will be applied for in one sampling period, we need to find the corresponding time for each vector, which denotes the dwell time. This can be obtained by using the obtained duty cycle of each voltage vector in Eq. (17) and the value of the sampling time as:
\[ \begin{cases}T_1=d_1T_s\\ T_2=d_2T_s\\ T_0=d_0T_s\end{cases} \tag{21} \]
After defining the optimal sector S_{n} and dwell time for each vector, the next step is distributing these vectors within one sampling interval. This is important so that the vector distribution is consistent over time. For example, when the optimal sector is odd (n = 1, 2, or 5), the switching sequence in Figure 3(a) should be followed. At the even optimal sector (that is n = 2, 4, or 6), the switching sequence in Figure 3(b) should be followed.
Figure 3 Switching pattern of the 2LVSI vectors in the case of the optimal sector is a) odd and (b) even.
The complete flowchart of the proposed fixedswitching frequency FSMPTFC for 2LVSI loading with LIM is depicted in Figure 4. The proposed fixedswitching frequency FSMPTFC begins by measuring the necessary measurements for predicting the control objectives. These measurements are filtered from the noise to improve accuracy. Next, the fixedswitching frequency FSMPTFC stage generates an optimal switching sector and dwells time pattern, which is then fed to the SVPWM stage to generate the required switching pattern. As a result of the computational burdens of realtime implementation, applying the chosen switching state after the next sample instant is a simple solution to the delay [33,34].
Figure 4 Flowchart of the proposed fixedswitching frequency FSMDTFC for LIM driven by 2LVSI.
5. Simulation Results
Dynamic analyses are used to prove that the proposed FSMPTFC with fixedswitching frequency is viable. The essential arc induction machine (AIM) prototype characteristics are used to analyze the simulation results produced by the MATLAB/Simulink model. The data and parameters related to this AIM are listed in Table 1. This control strategy is tested under different reference speeds and sample load intervals to guarantee validity.
Table 1 LIM Parameters.
5.1 Speed Change Condition
The LIM drive system is tested using the suggested fixedswitching frequency FSMPTFC with variable speed. Speed increases from 6 m/s to 8 m/s while the load remains constant at 50 N. As seen in Figure 5, the electromagnetic thrust created ensures that the appropriate thrust load is followed. At the same time, the actual speed tracks the reference value. Figure 6 depicts the dynamic response of the electromagnetic and load thrust. As seen from the axis in Figure 7, the principal flux linkage is fixed at the reference value in the interim. Figure 8 displays the threephase primary current's dynamic response. In addition, Figure 9 shows the threephase voltage corresponding to the ideal switching vector with fixedswitching frequency. Finally, Figure 10 depicts the secondary flux linkage corresponding to the speed change.
Figure 5 Dynamic response of reference and measured speed of the LIM.
Figure 6 Dynamic response of the electromagnetic thrust during speed change of the LIM.
Figure 7 Dynamic response of the primary flux linkage during speed change of the LIM.
Figure 8 Dynamic response of the primary current during speed change of the LIM.
Figure 9 Output voltage corresponding to the optimum switching vectors during speed change of the LIM.
Figure 10 Dynamic response of the secondary flux linkage during speed change of the LIM.
5.2 Dynamic Performance of the Drive System under Load Change
In this scenario, the thrust load rise from 60 N up to 150 N while the reference speed remians constant at 7 m/s. Figure 11 and Figure 12 demonstrate the electromagnetic thrust and the speed response. The electromagnetic thrust is seen to match the necessary thrust load. The actual speed also follows the reference value. In addition, Figure 13 and Figure 14 display the threephase current and voltages that correlate to the load variation. Finally, Figure 15 and Figure 16 show the primary and secondary flow linkages, respectively.
Figure 11 Dynamic response of the electromagnetic thrust during load thrust change of the LIM.
Figure 12 Dynamic response of the actual speed during the LIM load thrust change.
Figure 13 Dynamic response of the primary during load thrust change of the LIM.
Figure 14 Output voltage corresponding to the optimum switching vectors during load thrust change of the LIM.
Figure 15 Dynamic response of the primary flux linkage during load thrust change of the LIM.
Figure 16 Dynamic response of the secondary flux linkage during load thrust change of the LIM.
5.3 Comparison between the Proposed FixedSwitching FSMPTFC and Conventional FSMPTFC
The switching frequency characteristics in Figure 17a show that the VSI operates on a fixed frequency of 10 kHz with the proposed FSMPTFC. Therefore, the harmonics spectrum appears when at multiplications of the switching frequency. On the other hand, the conventional FSMPTFC has a wider harmonics spectrum, as shown in Figure 17b.
Figure 17 THD evaluation of the primary current: a) proposed fixedswitching FSMPTFC, and b) conventional FSMPTFC.
6. Conclusions
This research article proposes an improved finitestate model predictive thrust and primary flux linkage control (FSMPTFC) for the linear induction machine used in the linear metro. The proposed FSMPTFC solved the problem of the variable switching frequency existing in the conventional FSMPTC control. The proposed FSMPTFC is based on the fixedswitching frequency to protect the inverter from damage when a high switching frequency is generated and reduce the thrust and primary flux linkage ripples, increasing the linear metro drive system performance. In addition, the linear speed of the LIM is directly controlled by adding an external PI controller to the FSMPTFC, where the output of this PI controller modifies the reference thrust to fasttrack the changeable speed. Utilizing MATLAB/Simulink, the proposed control mechanism's validity has been examined, and the outcomes demonstrated the potential of the suggested control strategy to deliver the required performance.
Author Contributions
The authors contributed equally to this work.
Competing Interests
The authors have declared that no competing interests exist.
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