Fixed-Switching Frequency Finite-State Model Predictive Thrust and Primary Flux Linkage Control for LIM

Mahmoud F. Elmorshedy ^1,2,†, Abualkasim Bakeer ^3,†,*, Dhafer Almakhles ¹

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

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

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

4. † 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. Fixed-Switching Frequency Finite-State 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.

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 hill-climbing ability, low noise, and so on [7,8,9]. The HSST in the Tobu-Kyuryo-Line, 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 field-oriented 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: continuous-state MPC (CS-MPC) and finite-state MPC (FS-MPC) [18]. In this work, the FS-MPC 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 FS-MPC: FS-MPCC, which is based on predictive current control, and FS-MPTFC, 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 FS-MPCC issues.

Therefore, the FS-MPTC 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 FS-MPDTC is used as a controller. The FS-MPDTC 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 FS-MPC is improved for the LIMs to remove the weighting factor from the cost function and reduce the calculation burden. These proposed methods are called finite-state model predictive voltage control (FS-MPVC) and finite-state model predictive flux control (FS-MPFC). In [28], a comparison between the FS-MPTC and the FS-MPFC 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 FS-MPTC for the LIMs and compared with the PI control loop. The target of combining the SMC with the FS-MPTC is higher tracking accuracy and faster error convergence. In [23], the FS-MPTC 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 FS-MPTC 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 FS-MPTC concept, all of these FS-MPTC are based on the variable switching frequency concept, which is not preferred for the three-phase 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 fixed-switching 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 fixed-switching frequency finite state model predictive thrust and flux control for VSI-driven LIM.
• Define optimally the dwell times associated with the vectors of the selected sector.
• Provide high power quality for the VSI-driven LIM by fixing the switching frequency. 
• Validate the proposed fixed-switching FS-MPTFC 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 FS-MPTFC approach is presented in Section III. Section IV details the proposed approach of the fixed-switching frequency for FS-MPTFC. 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 end-effect activities that cause the air-gap 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 end-effect, is displayed in αβ-axis 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 αβ-axis voltages, i_α, i_β the αβ-axis currents, and λ_α, λ_β the αβ-axis flux-linkages. R is the resistance and L stands for self-inductance. Meanwhile ω₁ and ω₂ are the primary and the secondary linear speed, respectively. In the meantime, subscripts 1 and 2 refer to the primary and the secondary.

The αβ-axes of the primary and secondary flux-linkages 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 end-effect modification and is computed by

\[ L_{meq} = \bigl(1-f(Q)\bigr)L_m \tag{5} \]

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 end-effect 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 Finite-State Model Predictive Thrust and Flux Control (FS-MPTFC)

FS-MPTFC is offered for the LIM to obtain a quicker response, reduced thrust ripples, and the lowest primary flux-linkage ripples. The FS-MPTFC 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 FS-MPTFC. 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:

\[ \vec{\lambda}_1(k)=\vec{\lambda}_1(k-1)+Ts\big(\vec{v}_1(k)-R_1\vec{l}_1(k)\big) \tag{9} \]

\[ \vec{\lambda}_2(k)=\frac{L_2}{L_{meq}}\vec{\lambda}_1(k)+\left(L_{meq}-\frac{L_2L_1}{L_{meq}}\right)\vec{\iota}_1(k) \tag{10} \]

• With the use of the first-order Euler technique, predictions for the primary flux-linkage, λ₁(k + 1), primary current, i₁(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 αβ-axis voltage vectors. $i_{\alpha_1}(k)$ and $i_{\beta_1}(k)$ are the αβ-axis measured currents.

The proposed cost function, g_T, is designed as follows:

\[ g_T=\left|F_e^*-F_{e,i}(k+1)\right|+K_1\left|\lambda_1^*-\lambda_{1,i}(k+1)\right| \tag{16} \]

where K₁ 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 FS-MPTFC is illustrated in Figure 1.

Figure 1 FS-MDTFC for the LIM drive system.

4. Proposed Fixed-Switching Frequency FS-MPTFC 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 2L-VSI terminals.

The cost function measures the difference between the actual and predicted vector output. Next, the fixed-switching frequency FS-MPTFC 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 duty-cycle for the two active vectors and the zero vector is always equal to one; see Eq. (16) in which d₁ is the duty-cycle for the first active vector in the sector, d₂ is the duty-cycle of the second active vector in the sector, and d₀ is the duty-cycle 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 zero-voltage vector (i.e., J₀) [32]. Adjusting σ affects the zero-vector time and therefore affects the performance of the 2L-VSI 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 J₁ and J₂ 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 2L-VSI vectors in the case of the optimal sector is a) odd and (b) even.

The complete flowchart of the proposed fixed-switching frequency FS-MPTFC for 2L-VSI loading with LIM is depicted in Figure 4. The proposed fixed-switching frequency FS-MPTFC begins by measuring the necessary measurements for predicting the control objectives. These measurements are filtered from the noise to improve accuracy. Next, the fixed-switching frequency FS-MPTFC 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 real-time 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 fixed-switching frequency FS-MDTFC for LIM driven by 2L-VSI.

5. Simulation Results

Dynamic analyses are used to prove that the proposed FS-MPTFC with fixed-switching 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 fixed-switching frequency FS-MPTFC 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 three-phase primary current's dynamic response. In addition, Figure 9 shows the three-phase voltage corresponding to the ideal switching vector with fixed-switching 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 three-phase 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 Fixed-Switching FS-MPTFC and Conventional FS-MPTFC

The switching frequency characteristics in Figure 17a show that the VSI operates on a fixed frequency of 10 kHz with the proposed FS-MPTFC. Therefore, the harmonics spectrum appears when at multiplications of the switching frequency. On the other hand, the conventional FS-MPTFC has a wider harmonics spectrum, as shown in Figure 17b.

Figure 17 THD evaluation of the primary current: a) proposed fixed-switching FS-MPTFC, and b) conventional FS-MPTFC.

6. Conclusions

This research article proposes an improved finite-state model predictive thrust and primary flux linkage control (FS-MPTFC) for the linear induction machine used in the linear metro. The proposed FS-MPTFC solved the problem of the variable switching frequency existing in the conventional FS-MPTC control. The proposed FS-MPTFC is based on the fixed-switching 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 FS-MPTFC, where the output of this PI controller modifies the reference thrust to fast-track 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.