Improved performance of linear induction motors based on optimal duty cycle finite-set model predictive thrust control

Abstract

Linear induction machines (LIMs) find widespread adoption in various applications, owing to their inherent advantages such as low noise, compact turning radius, and excellent climbing capability. LIMs are extensively utilized in linear metro applications. However, in practical operations, the output thrust shrinks with the increase in speed, which is attributed to end effects. This phenomenon leads to a reduction in efficiency. In addition, fluctuations in the normal force impact system stability, posing disturbances to the suspension system. To address these challenges, this paper suggests a finite-set model predictive thrust control (FS-MPTC) for a drive system employing a linear induction motor (LIM). The FS-MPTC optimizes the duty cycle for the active voltage vector and allocates the remaining period to one of the zero voltage vectors. The selected zero voltage vector reduces the switching transition between it and the active voltage vectors. The duty cycle is calculated for the six voltage vectors by incorporating the deadbeat concept and the derivative value for the electromagnetic thrust. In the prediction stage of the FS-MPTC, the computed duty cycle and the corresponding voltage vector are used simultaneously and repeated for the six voltage vectors. The cost function comprises two terms: the error between the reference thrust and predicted thrust value as the first term and the error between the rated primary flux linkage and its predicted value as the second term. The reference thrust is generated from the outer speed control loop. To validate the effectiveness of the proposed control approach, Japanese 12000 linear induction machine parameters are employed for verification. Comparative analysis of the performance between the suggested control method with the optimal duty cycle and the same process with a fixed duty cycle demonstrates superior performance when utilizing the optimal duty cycle. Finally, the proposed FS-MPTC with the optimal duty cycle offers a promising solution to enhance the operational efficiency of the LIM-based drive systems.

Abbreviations and Nomenclatures

Abbreviations
DC	Duty cycle	MPC	Model predictive control
DTC	Direct thrust control	MPCC	Model predictive current control
FS	Finite set	MPFC	Model predictive flux control
FOC	Field-oriented control	MPTC	model predictive thrust control
FDC	Fixed duty cycle	ODC	Optimum duty cycle
LIM	Linear induction motor	PWM	Pulse width modulation
LIMs	Linear induction machines	VSI	Voltage source inverter
RMS	root mean square	FSMPTC	Finite-set model predictive thrust control
Nomenclatures
Vll	Line voltage	Q	End effect coefficient
Lmo	Mutual inductance at standstill	Ll2	Secondary leakage inductance
p	Number of poles	i2	Secondary current
Pr	Rated power	i1	Primary current
R1	Primary resistance	λ1	Primary flux linkage
Ll1	Primary leakage inductance	λ2	Secondary flux linkage
D	Primary length	Fe	Electromagnetic thrust
τ	Pole pitch	Fl	Load thrust
R2	Secondary resistance	M	Mass
V2	Speed	B	viscous friction
K1	Weighting factor	Ts	Sample time
Ta	Active duration time	Tn	Null duration time
ui	voltage vectors of VSI	g	Cost function
LLs	Primary length	$X_n$	Value of thrust resulting from the zero-voltage vector
$X_a$	Thrust variation attributed to the active voltage vector	ω	Secondary angular velocity

1. Introduction

1.1. Motivation

In recent years, linear induction machines (LIMs) have garnered substantial attention from academics and industry due to their versatile applications and potential for innovation. Moreover, these machines offer inherent advantages, including a diminished turning radius and expanded operational speed range [1,2]. Not only that but this growing interest is also driven by several factors, including the unique operational benefits of LIMs, such as their capability for direct linear motion without the need for rotary-to-linear conversion mechanisms. This results in simpler mechanical designs, reduced maintenance requirements, and enhanced overall system efficiency. Moreover, LIMs offer robust performance in environments where traditional rotary motors may face challenges, such as in dusty, wet, or high-temperature conditions [[3], [4], [5]]. Despite the inherent advantages, LIMs are not without noteworthy limitations, marked notably by increased air gap dimensions and consequential end effects. These drawbacks exert a discernible impact on the dynamics of thrust and the mutual inductance, manifesting in continuous variations in their values with alterations in operational speed [6,7]. This is owing to the LIM's unique structure, in which the primary is like that of rotary induction machines, but it is sliced open and rolled flat. In addition, the secondary comprises a conductor sheet with a return channel of iron for the magnetic flux. The end effect exerts a discernible influence on the traction characteristics of the LIM, giving rise to two prominent outcomes: a) a non-uniform distribution of magnetic flux density within the air gap, and b) a non-uniform distribution of eddy currents in the secondary. Consequently, the mutual inductance undergoes variations attributable to this end effect, and its magnitude is contingent upon the operational speed of the system.

Nevertheless, the LIM finds diverse applications across various categories, serving as a traction motor, facilitating sliding door operations, driving metallic belt conveyors, contributing to transportation systems encompassing both medium and high-speed vehicles, and operating in electromagnetic pump applications, among others [[8], [9], [10], [11], [12], [13], [14]]. Particularly noteworthy is the extensive utilization of LIMs in subway systems, as depicted in Fig. 1, with substantial implementations observed in numerous countries globally, including but not limited to China, Japan, and the United States [15].

Fig. 1.

Linear metro based on linear induction machine.

1.2. Literature survey

Over the past decade, several investigations have been presented to explore, assess, and deliberate on diverse control methodologies applied to conventional rotary machines. These investigative efforts have subsequently been extended to encompass the domain of linear induction machines, thereby broadening the scope of control techniques under scrutiny. The drive system performance of LIMs depends significantly on the selection of the control strategies. Various control techniques are developed for the LIMs to improve their performance. These encompass field-oriented control (FOC) [[15], [16], [17], [18]], conventional direct thrust control (DTC) [19,20], DTC-based space vector modulation [21,22], combined field-oriented control with direct thrust control [23], and model predictive current control [5,24], model predictive thrust control [25], and model predictive flux control [26]. It is noteworthy that the driving performance of the LIM based on DTC and FOC may yield suboptimal outcomes due to potential side effects [25]. Therefore, a recommended control methodology, as proposed in Ref. [23], is developed to combine the advantages of both FOC and DTC for the LIM. However, the complexity associated with this method needs to be improved to become easier and achieve better performance.

Consequently, contemporary control methodologies have been advanced across diverse applications to garner heightened efficacy and surmount challenges. Prominent among these techniques are model predictive and sliding mode control, as elucidated in Refs. [[27], [28], [29], [30]]. Therefore, the utilization of model predictive control (MPC) is suggested and embraced for optimizing the performance of the LIM drive system [[24], [25], [26],31,32]. The nomenclature of the MPC is generally derived from the terms employed in the cost functions. In Ref. [5], the cost function depends on the current; hence, the method is known as the model predictive current control (MPCC). While MPCC offers the advantage of avoiding the need for the weighting factor, its dynamic response is slow, and it is unsuitable for metro applications, which demand a fast dynamic response.

Furthermore, an alternative approach, model predictive flux control (MPFC), is proposed in Ref. [26] in order to achieve better performance compared to the MPCC. Despite its lack of a weighting factor, it requires two Proportional-Integral (PI) controllers. As a result, in Ref. [33], the finite-set model predictive thrust control (FSMPTC) is introduced to address the need for expedited dynamic responses where thrust and flux linkage errors are included in the cost function. Owing to the discrete nature of two-level voltage source inverters (VSI), the finite-set model predictive torque control (FSMPTC) is used as an alternative to direct thrust control, overcoming the drawbacks of higher thrust and primary flux ripples. The discrepancy between the required torque and the predicted torque value, along with the error between the required flux linkage and its predicted value, make up the goal function in the FSMPTC approaches.

Nevertheless, it is essential to note that this method necessitates the incorporation of a weighting factor, the best value of which can be selected through empirical insights and expertise. Notably, these control methods are based on the utilization of one voltage vector within the sampling period despite the availability of eight distinct voltage vectors in the context of the three-phase two-level voltage source inverter (VSI). For a two-level three-phase inverter, there are typically eight possible voltage vectors. These vectors are formed by the various combinations of the ON and OFF states of the six switches (two for each phase). Each switch can be either ON or OFF, resulting in 2^6 = 64 possible combinations. However, due to the constraints of the inverter circuit, not all combinations are valid. Therefore, after eliminating invalid combinations, eight distinct voltage vectors remain as illustrated in Fig. 2. It is worth highlighting that a control strategy outlined in Ref. [34], introduces the integration of two voltage vectors within one sampling period, but the control method is very complex.

Fig. 2.

The available voltage vectors of the three-phase two-level VSI.

1.3. Novelty and contributions

In the reviewed literature outlined in subsection 1.2, a prevalent trend is identified, where studies on the LIM based on the MPC predominantly focused on employing either one voltage vector throughout the entire sample time or two voltage vectors without emphasizing the calculation of the optimal duration. Consequently, the current work strategically concentrates on utilizing two voltage vectors – one active and one zero – within a single sample period based on the MPTC. The objective is to achieve a faster dynamic response while simultaneously reducing thrust ripples, flux ripples, and distortion in the primary current. To enhance performance, the proposed methodology involves determining the duty cycle for all voltage vectors, which is then seamlessly incorporated during the prediction stage along with the corresponding voltage vector. After that, the cost function evaluates the six voltage vectors and determines the optimal voltage vector with its duty cycle. Then, the zero-voltage vector, which achieves a lower switching transition regarding the preselected active voltage vector, is chosen for the remaining sample period. This integrated approach not only ensures a faster dynamic response but also contributes to diminishing thrust ripples, flux ripples, primary current distortion, and lower switching loss, ultimately extending the lifetime of the driving system.

Furthermore, this work suggests two distinct methods for determining the duty cycle of the active and zero voltage vectors. The first method maintains a constant duty cycle for both vectors, while the second method calculates the duty cycle for the six active voltage vectors and strategically selects the optimal one, aligning it with the corresponding voltage vector. This dual-pronged approach provides flexibility and optimization in managing the duty cycles, contributing to the overall efficiency and longevity of the LIM-based system.

1.4. Organization of the manuscript

The manuscript is structured as follows: Section 1 provides an introduction overview, elucidating the fundamental context. Section 2 describes a comprehensive depiction of the LIM dynamic representation. In Section 3, the FSMPTC is expounded upon, detailing the proposed approach. Subsequently, Section 4 delves into an in-depth discussion of the simulation work conducted, encompassing diverse operational scenarios. Lastly, Section 5 encapsulates the study's outcomes, presenting a succinct summary of the findings derived from the research endeavor.

2. Representation of the LIM dynamics

Several investigations have delved into the mathematical characterization of the LIMs, specifically addressing the ramifications of end effects. These studies have scrutinized the impact of end effects on distinct components of the dynamic model such as the mutual inductance only, the secondary resistance only, and both mutual inductance and secondary resistance. Moreover, certain scholarly works have incorporated these effects exclusively within the framework of the dq-axis or d-axis alone [7,19,23]. In this work, the dynamics of the LIM are adapted based on the model expounded in Refs. [6,7], and elucidated in Fig. 3. The ensuing relationships proffer a comprehensive representation of the dynamic characteristics of the LIM under investigation.

Fig. 3.

Linear induction machine equivalent circuit.

2.1. Voltage relations

Within this subsection, the primary and secondary voltage are delineated within a stationary reference frame, elucidated in a complex form denoted by (1) and (2).

$${\overrightarrow{u}}_1=u_{\alpha 1}+j{u}_{\beta 1}=R_1{\overrightarrow{i}}_1+\frac{d{\overrightarrow{\lambda }}_1}{dt}$$ (1)

$${\overrightarrow{u}}_2=u_{\alpha 2}+j{u}_{\beta 2}=R_2{\overrightarrow{i}}_2+\frac{d{\overrightarrow{\lambda }}_2}{dt}-j\omega {\overrightarrow{\lambda }}_2=0$$ (2)

The subscripts 1 and 2 refer to the primary and secondary components of the LIM, respectively. The symbols u, R, i, and λ denote voltage, resistance, current, and flux linkage, respectively. Additionally, ω signifies the secondary angular velocity.

2.2. Flux linkage relations

The flux linkages are contingent upon currents, leakage inductances, and mutual inductance. The main influence of the end effect is on the mutual inductance. The ensuing relations (3) and (4) delineate these effects.

$${\overrightarrow{\lambda }}_1=L_{l1}{\overrightarrow{i}}_1+L_{meq}\left({\overrightarrow{i}}_1+{\overrightarrow{i}}_2\right)$$ (3)

$${\overrightarrow{\lambda }}_2=L_{l2}{\overrightarrow{i}}_2+L_{meq}\left({\overrightarrow{i}}_1+{\overrightarrow{i}}_2\right)$$ (4)

where L_l1, L_l2, and L_meq denote the primary leakage inductance, secondary leakage inductance, and mutual inductance, respectively. In this adapted model, the end effect exerts its influence on the mutual inductance through the coefficient f(Q) as articulated in (5).

$$L_{meq}=\left[1-f\left(Q\right)\right]L_m$$ (5)

This f(Q) coefficient derives its basis from the Q factor which in turn is tied to the linear speed, v₂, and secondary resistance, R₂, and the primary length, L_Ls. Calculating both these factors is given in (6) and (7).

$$f\left(Q\right)=\frac{\left(1-\mathit{exp}\left(-Q\right)\right)}{Q}$$ (6)

$$Q=\frac{L_{Ls}{R}_2}{v_1\left(L_{l2}+L_m\right)}$$ (7)

2.3. Electromagnetic relations

There are various mathematical expressions available for determining electromagnetic thrust in linear induction motor. Among these formulations, one pivotal relation given by (8) depends on the interplay between the primary current and primary flux linkage.

$$F_e=\frac{3}{2}\frac{\pi }{\tau }\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}•{\overrightarrow{i}}_1\right)$$ (8)

where τ is the pole pitch.

2.4. Motion equation

The motion equation delineates the interrelation among the electromagnetic thrust, load thrust, and other effective terms including mass and viscous friction. This mathematical representation of motion equation is given by (9).

$$F_e=F_l+M\frac{d{v}_2}{dt}+B{v}_1$$ (9)

where F_l and v₁ are the thrust load and the primary linear speed. M and B are the mass and the viscous friction respectively.

3. Proposed model predictive thrust control

This section serves as a platform to elucidate the distinctions inherent in two distinct variants of FS-MPTC methods: the optimal duty cycle (ODC) -based approach and its fixed duty cycle (FDC) counterpart. The comprehensive structure of FS-MPTC consists of three pivotal stages: the estimating stage, the prediction stage, and the cost function stage. The determination of states utilized in the estimation and prediction stages is contingent upon the chosen cost function, elucidating the interdependence of these stages. The ensuing subsections delve into characterizing each of the two control methodologies, offering a detailed comparative analysis of their operational frameworks and implications.

3.1. FS-MPTC based on fixed duty cycle

In this particular methodology, the sample time is structured to accommodate two voltage vectors: one characterized as active and the other as zero. Notably, the duty cycle of the active voltage vector is fixed and equals 90 % of the sample time, while the remaining 10 % is used for the zero-voltage vector. This distinctive feature limits the prediction stage exclusively to the active six voltage vectors, deviating from the conventional FS-MPTC paradigm, which typically incorporates all eight voltage vectors during this stage. After identifying the optimal active voltage vectors, the process involves their transmission to the Pulse Width Modulation (PWM) generation stage, where one of the zero voltage vectors is selected. This zero-voltage vector is strategically selected to minimize transitions during switching, thereby optimizing the system's efficiency. The following stages delve into the constituent terms used in the cost function, offering a meticulous examination of the parameters influencing the control strategy. Then, a thorough exploration of the states necessitating estimation and prediction can be determined.

3.1.1. Cost function stage

The formulated cost function comprises two distinct terms. The initial term encapsulates the difference between the required electromagnetic thrust and its corresponding predicted value. Meanwhile, the second term encapsulates the deviation between the required primary flux linkage amplitude and its predicted amplitude value. The ensuing relation (10) explicates the designed cost function [2].

$$g_1=\left|F_e^{ref}-F_e\left(k+1\right)\right|+K_1\left|{\lambda }_1^{ref}-\left|{\lambda }_1\left(k+1\right)\right|\right|$$ (10)

where the determination value of the weighting factor, K₁ is accomplished through an iterative trial and error methodology. Concurrently, the predicted values of the electromagnetic trust, F_e (k+1), and the primary flux linkage, λ₁ (k+1) are essential prerequisites for the computation of the formulated cost functions.

3.1.2. Prediction stage

In addition to predicting both the electromagnetic thrust and the primary flux linkage, as mentioned earlier, it is imperative to predict the primary current for accurate electromagnetic thrust prediction. The calculation of predicted states relies on the utilization of (11), (12), and (13), employing the first-order Euler method [21]:

$${\overrightarrow{\lambda }}_1\left(k+1\right)={\overrightarrow{\lambda }}_1\left(k\right)+T_s\left({\overrightarrow{u}}_1\left(k\right)-R_1{\overrightarrow{i}}_1\left(k\right)\right)$$ (11)

$${\overrightarrow{i}}_1\left(k+1\right)=\left[1-\frac{T_s}{\sigma }\left(R_1+\frac{R_2}{{\tau }_m^2}\right)\right]{\overrightarrow{i}}_1\left(k\right)+\frac{T_s}{\sigma }\left({\overrightarrow{u}}_1\left(k\right)+\frac{1}{{\tau }_2{\tau }_m}{\overrightarrow{\lambda }}_2\left(k\right)-j\frac{\omega }{{\tau }_m}{\overrightarrow{\lambda }}_2\left(k\right)\right)$$ (12)

$$F_e\left(k+1\right)=\frac{3}{2}\frac{\pi }{\tau }\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}\left(k+1\right)•{\overrightarrow{i}}_1\left(k+1\right)\right)$$ (13)

where ${\overrightarrow{\lambda }}_1\left(k\right)$ represents the primary flux linkage requiring estimation, T_s denotes the sample time, ${\overrightarrow{i}}_1\left(k\right)$ signifies the measured primary current and ${\overrightarrow{u}}_1\left(k\right)$ signifies the voltage vectors (0, 1, …7). The remaining variables are derived from the following interrelated relations $\sigma =\left(\frac{L_1{L}_2-L_{meq}^2}{L_2}\right)$, ${\tau }_2=L_2/R_2$, ${\tau }_m=L_2/L_{meq}$.

3.1.3. Estimation stage

In this stage, determining the secondary flux linkage is crucial for the primary flux linkage estimation. Therefore, both the primary value and secondary value of flux linkages have to be accurately estimated. These estimated values are derived from the LIM mathematical model, as illustrated by (14) and (15) [31].

$${\overrightarrow{\lambda }}_2\left(k\right)=\frac{R_2{T}_s{L}_{meq}{\overrightarrow{i}}_1\left(k\right)+\left(L_{l2}+L_{meq}\right){\overrightarrow{\lambda }}_2\left(k-1\right)}{\left[\left(L_{l2}+L_{meq}\right)+R_2{T}_s\right]}-j\frac{T_s\left(L_{l2}+L_{meq}\right)\omega {\overrightarrow{\lambda }}_2\left(k\right)}{\left[\left(L_{l2}+L_{meq}\right)+R_2{T}_s\right]}$$ (14)

$${\overrightarrow{\lambda }}_1\left(k\right)=\frac{L_{meq}{\overrightarrow{\lambda }}_2\left(k\right)}{\left(L_{l2}+L_{meq}\right)}-\frac{L_{meq}\left(L_{meq}-\frac{\left(L_{l1}+L_{meq}\right)\left(L_{l2}+L_{meq}\right)}{L_{meq}}\right){\overrightarrow{i}}_1\left(k\right)}{\left(L_{l2}+L_{meq}\right)}$$ (15)

The symbols λ₂, λ₁, R₂, T_s, i₁, L_l2, L_meq, and ω denote secondary flux linkage, primary flux linkage, secondary resistance, sample time, primary current, secondary leakage inductance, mutual inductance, and secondary angular velocity, respectively.

3.2. FS-MPTC based on optimal duty cycle with the corresponding voltage vector

This subsection discusses the proposed optimal duty cycle for the FS-MPTC. This control paradigm employs two voltage vectors within the sample time - one active and the other zero vector - like the previous method. The key distinction lies in the dynamic nature of the duty cycle, where its value is optimized rather than kept as a fixed parameter. In this methodological approach, the first step involves the calculation of optimal duty cycles corresponding to the six active voltage vectors. Then, these optimal duty cycles, along with their corresponding voltage vector, are incorporated into the prediction stage. The cost function selects the best voltage vector. The resultant output from the FS-MPTC comprises the optimal voltage vector and its corresponding duty cycle, serving as the inputs to the PWM generation unit. The active voltage vector is applied for its corresponding optimal duration, while the remaining period of the sample time is used for the null voltage vector. The deadbeat principle is used to determine the appropriate duration for each of the six active voltage vectors. The thrust is the only variable considered to determine this duration; hence, the process of calculating the optimal duration starts from the relation of electromagnetic thrust and the value of its derivative as calculated from (16) and (17).

$$F_e=\frac{3\pi }{2\tau }\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}•{\overrightarrow{i}}_1\right)=C\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}•{\overrightarrow{i}}_1\right)$$ (16)

$$\frac{d{F}_e}{dt}=C\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}\frac{d{\overrightarrow{i}}_1}{dt}+{\overrightarrow{i}}_1\frac{d{\overrightarrow{\lambda }}_1^{*}}{dt}\right)$$ (17)

where $C=\frac{3\pi }{2\tau }$.

The derivative values of the primary flux linkage and the primary current are computed through the following relations (18) and (19) [22].

$$\frac{d{\overrightarrow{\lambda }}_1^{*}}{dt}={\overrightarrow{u}}_1^{*}-R_1{\overrightarrow{i}}_1^{*}$$ (18)

$$\frac{d{\overrightarrow{i}}_1}{dt}=-\frac{1}{\sigma }\left(R_1+\frac{R_2}{{\tau }_m^2}\right){\overrightarrow{i}}_1+\frac{1}{\sigma }\left({\overrightarrow{u}}_1+\frac{1}{{\tau }_2{\tau }_m}{\overrightarrow{\lambda }}_2-j\frac{\omega }{{\tau }_m}{\overrightarrow{\lambda }}_2\right)$$ (19)

where ${\overrightarrow{u}}_1$ and R₁ signify the voltage vectors and primary resistance, respectively. The remaining variables are determined from the following interrelated relations $\sigma =\left(\frac{L_1{L}_1-L_{meq}^2}{L_2}\right)$, ${\tau }_2=\frac{L_2}{R_2}$, ${\tau }_m=\frac{L_2}{L_{meq}}$.

By substituting the expressions derived in (18) and (19) into (17), the derivative of thrust can be succinctly expressed by (20):

$$\frac{d{F}_e}{dt}=C\frac{1}{\sigma }\mathfrak{I}\mathrm{m}\left(-\left(R_1+\frac{R_2}{{\tau }_m^2}\right){\overrightarrow{\lambda }}_1^{*}{\overrightarrow{i}}_1+\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{u}}_1+\frac{1}{{\tau }_2{\tau }_m}{\overrightarrow{\lambda }}_1^{*}{\overrightarrow{\lambda }}_2\right)\right)+C\mathfrak{I}\mathrm{m}\left({\overrightarrow{i}}_1{\overrightarrow{u}}_1^{*}\right)-C\frac{\omega }{\sigma {\tau }_m}\mathfrak{R}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{\lambda }}_2\right)$$ (20)

The symbols $\mathfrak{I}\mathrm{m}$ and $\mathfrak{R}$ represent the imaginary and real parts.

By reformulating (20) into (21), it can be noticed that by using the zero-voltage vector, the variation of the electromagnetic thrust is negative. In contrast, the active voltage vector introduces the possibility of positive or negative variation in thrust.

$$\frac{d{F}_e}{dt}=-C\frac{1}{\sigma }\left(R_1+\frac{R_2}{{\tau }_m^2}\right)\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{i}}_1\right)-C\frac{1}{\sigma {\tau }_2{\tau }_m}\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_2^{*}{\overrightarrow{\lambda }}_1\right)-C\frac{\omega }{\sigma {\tau }_m}\mathfrak{R}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{\lambda }}_2\right)+C\frac{1}{\sigma }\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{u}}_1\right)+C\mathfrak{I}\mathrm{m}\left({\overrightarrow{i}}_1{\overrightarrow{u}}_1^{*}\right)$$ (21)

Based on (21), the change in thrust corresponding to both the zero and active voltage vectors is delineated by (22) and (23).

$$X_n=-C\frac{1}{\sigma }\left(R_1+\frac{R_2}{{\tau }_m^2}\right)\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{i}}_1\right)-C\frac{1}{\sigma {\tau }_2{\tau }_m}\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_2^{*}{\overrightarrow{\lambda }}_1\right)-C\frac{\omega }{\sigma {\tau }_m}\mathfrak{R}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{\lambda }}_2\right)$$ (22)

$$X_a=X_n+C\frac{1}{\sigma }\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}{\overrightarrow{u}}_1\right)+C\mathfrak{I}\mathrm{m}\left({\overrightarrow{i}}_1{\overrightarrow{u}}_1^{*}\right)$$ (23)

where $X_n$ represents the value of thrust resulting from the zero-voltage vector while $X_a$ signifies the thrust variation attributed to the active voltage vector.

As a result of using two voltage vectors within one sample time, T_s the predicted electromagnetic thrust can be calculated by (24).

$$F_e\left(k+1\right)=F_e\left(k\right)+T_a{X}_a+T_n{X}_n$$ (24)

where T_a and T_n represent the respective duration times of the active and zero voltage vectors, respectively.

The application of deadbeat theory, as elucidated in Ref. [35] and expressed by (25), is used to calculate the optimal duration time, T_a of the active voltage vector expressed by (26).

$$T_a=\frac{\left[F_e^{*}-F_e\left(k\right)-X_n{T}_s\right]}{\left[X_a-X_n\right]}$$ (25)

$$F_e\left(k+1\right)=F_e^{*}$$ (26)

As previously mentioned, the optimal duration value is calculated for each of the six active voltage vectors. Subsequently, the prediction stage is updated by incorporating the optimal duration along with the corresponding active voltage vector. Hence, the modified states within the prediction stage are determined from (27), (28), and (29).

$${\overrightarrow{\lambda }}_1\left(k+1\right)={\overrightarrow{\lambda }}_1\left(k\right)+T_a\left({\overrightarrow{u}}_1\left(k\right)-R_1{\overrightarrow{i}}_1\left(k\right)\right)$$ (27)

$${\overrightarrow{i}}_1\left(k+1\right)=\left[1-\frac{T_a}{\sigma }\left(R_1+\frac{R_2}{{\tau }_m^2}\right)\right]{\overrightarrow{i}}_1\left(k\right)+\frac{T_a}{\sigma }\left({\overrightarrow{u}}_1\left(k\right)+\frac{1}{{\tau }_2{\tau }_m}{\overrightarrow{\lambda }}_2\left(k\right)-j\frac{\omega }{{\tau }_m}{\overrightarrow{\lambda }}_2\left(k\right)\right)$$ (28)

$$F_e\left(k+1\right)=\frac{3}{2}\frac{\pi }{\tau }\mathfrak{I}\mathrm{m}\left({\overrightarrow{\lambda }}_1^{*}\left(k+1\right)•{\overrightarrow{i}}_1\left(k+1\right)\right)$$ (29)

The schematic representation of the proposed speed drive system using optimal duty cycle FS-MPTC is shown in Fig. 4.

Fig. 4.

Schematic illustration of the proposed system using FS-MPTC.

This methodological refinement exemplifies a meticulous consideration of system dynamics, leveraging the deadbeat control principle to optimize the durations of active voltage vectors based on thrust dynamics. The proposed optimal duty cycle FS-MPTC introduces a sophisticated approach, significantly enhancing the precision and adaptability of thrust control in LIM drive systems. By dynamically adjusting the duty cycle, the FS-MPTC method ensures optimal performance, accommodating varying operational conditions and improving overall system efficiency. This advanced control strategy not only refines the thrust control but also minimizes ripples and fluctuations, leading to smoother and more reliable operation of LIM drive systems. The precise calculation of the optimal duration for each of the six active voltage vectors allows for a finely tuned response, aligning the system's performance closely with desired outcomes. The schematic representation of the proposed speed drive system utilizing the optimal duty cycle FS-MPTC is illustrated in Fig. 4. This figure provides a visual overview of the system configuration, highlighting the integration of the FS-MPTC method and its role in optimizing drive performance. By incorporating this innovative control strategy, the LIM drive system achieves superior dynamic response, reduced energy losses, and enhanced operational stability, marking a significant advancement in the field of motor drive technology.

4. Simulation results

This section rigorously evaluates the efficacy of the proposed optimal duty cycle FS-MPTC (ODC-FS-MPTC) for the LIM drive system. The performance assessment utilizes the parameters of the Japanese 12000 linear induction motor, as outlined in Table 1 [22], to comprehensively model the whole drive system performance. The MATLAB/Simulink software program is adopted for the meticulously validations. The validation process involves a comparative analysis between the proposed ODC-FS-MPTC and the conventional FDC-FS-MPTC. To ensure a fair evaluation, identical operating conditions, including sample time, speed, and thrust load, are maintained. The assessment unfolds with the application of FDC-FS-MPTC for the initial 8 s, followed by the seamless transition to ODC-FS-MPTC for the remaining duration, both operating under the same conditions. This study encompasses two distinct tests, each accompanied by a detailed explanation and analysis, providing a comprehensive understanding of the performance characteristics and advantages offered by the proposed ODC-FS-MPTC in comparison to the traditional FDC-FS-MPTC.

Table 1.

Utilized parameters of the Japanese 12000 LIM [22].

Parameter	Symbol	Value	Unit
Line voltage	Vll	1100	V
Mutual inductance at standstill	Lmo	26.477	mH
Number of poles	p	10
Power	Pr	120	kW
Primary resistance	R1	0.138	Ω
Primary leakage inductance	Ll1	6.7	mH
Primary length	D	2.476	m
Pole pitch	τ	0.2808	m
Secondary resistance	R2	0.576	Ω
Speed	v1	40	km/h

4.1. Effect of FS-MPTC based on fixed and optimal duty cycle under constant linear speed

In this test condition, a reference speed of 21.6 km/h, which is equivalent to 6 m/s, is selected alongside a thrust load of 5000 N, which represents 42 % of the rated thrust load. Aligning with the machine parameters of the Japanese 12000 LIM, the rated value of the primary flux linkage is established at 6.25 Wb. In order to confirm and validate the response of the two control methods (FDC-FS-MPDTC and ODC-FS-MPDTC), the ensuing results depicted in Fig. 5 through Fig. 9 meticulously delineate the comparative analysis between the two control methodologies. To achieve a more accurate comparison, both control methods are tested under identical conditions, with a timer used to alternate between them. During the first 8 s, the FDC-FS-MPDTC is activated. After this period, the FDC-FS-MPDTC is deactivated, and the ODC-FS-MPDTC is activated for the remainder period of the test.

Fig. 5.

Effect of FS-MPTC based on fixed and optimal duty cycle on the actual linear speed.

Fig. 9.

Effect of FS-MPTC based on fixed and optimal duty cycle on the instantaneous αβ-axis flux linkage.

The response of the actual linear speed is illustrated in Fig. 5, which unequivocally demonstrates that both control methods adeptly track the reference linear speed.

For detailed analysis and discussion, the dynamic response of the electromagnetic thrust, primary flux linkage, and the three-phase primary current are presented. Fig. 6, Fig. 7 illustrate a comprehensive insight into the electromagnetic thrust and the amplitude of the primary flux linkage, showcasing a remarkable alignment with their respective reference values. Notably, during the steady state, the ODC-FS-MPTC exhibits lower ripples than the FDC-FS-MPTC, as evidenced by the meticulous analysis of both control methods. This will enhance the overall performance of the drive system and improve the waveform of the primary current. Furthermore, Fig. 8 offers a glimpse into waveforms of the primary three-phase instantaneous current. It is noteworthy that the activation of ODC-FS-MPTC results in lower fluctuations and, hence, lower total harmonic distortion, underscoring the efficacy of this control method in achieving enhanced performance and reduced fluctuations when compared to the FDC-FS-MPTC.

Fig. 6.

Effect of FS-MPTC based on fixed and optimal duty cycle on the electromagnetic thrust.

Fig. 7.

Effect of FS-MPTC based on fixed and optimal duty cycle on the actual amplitude flux linkage.

Fig. 8.

Effect of FS-MPTC based on fixed and optimal duty cycle on the instantaneous three-phase primary current.

Finally, the αβ-axis primary flux linkage, represented by the amplitude in Fig. 7, is further presented in Fig. 9 to illustrate the effect of the optimal duty on the ripples for the instantaneous waveforms.

4.2. Effect of FS-MPTC based on optimal duty cycle under variable linear speed

This case study is a compelling illustration of the dynamic response exhibited by the LIM when subjected to ODC-FS-MPTC under variable linear speed conditions. Also, the FDC-FS-MPTC is applied for the first 8 s, and the change in the linear speed happens during the operation of the ODC-FS-MPTC period. In this case, the thrust load is increased twofold compared to the previous case and reaches 10,000 N, which is close to the rated thrust load, as illustrated in Fig. 10. It can be seen that the electromagnetic thrust developed from the LIM equals the load thrust and exactly tracks the reference value generated from the controller. Further, an observation is a significant reduction in oscillations of the electromagnetic thrust when the ODC-FS-MPTC is employed, highlighting its effectiveness in reducing the thrust ripples and mitigating fluctuations. The reduction of the thrust ripples reaches 500 N when using the ODC-FS-MPTC compared to the FDC-FS-MPTC. Simultaneously, the linear speed undergoes a considerable increment, surging from 5 m/s up to 8 m/s (i.e. 18 km/h up to 28.8 km/h). Fig. 11 portrays the remarkable proficiency of ODC-FS-MPTC in accurately tracking the dynamic changes in linear speed, all while maintaining comparable oscillation levels in the electromagnetic thrust. It can be observed that the controller generates the maximum allowable reference thrust when the linear speed increases from 5 m/s to 8 m/s. Consequently, the Linear Induction Motor (LIM) produces its maximum thrust to track the new reference speed quickly, as illustrated in Fig. 10.

Fig. 10.

Effect of FS-MPTC based on optimal duty cycle on the electromagnetic thrust under variable linear speed.

Fig. 11.

Effect of FS-MPTC based on optimal duty cycle on the variable reference and actual linear speed.

In addition, Fig. 12, Fig. 13 present a detailed analysis of the amplitude and the αβ-axis of the primary flux linkage. Notably, the oscillation in the amplitude of the primary flux linkage is clearly reduced with the proposed control method, emphasizing its efficacy in enhancing stability. Finally, the comprehensive evaluation extends to different waveforms of the primary current, as showcased in Fig. 14, Fig. 15, Fig. 16. These figures include the responses of the dq-axis of the primary current, RMS value, and instantaneous three-phase. Notably, applying the suggested ODC-FS-MPTC results in a notable decrease in oscillations across these three waveforms. This observation underscores the ability of ODC-FS-MPTC to contribute to a more consistent and stable primary current profile. Further, this substantiates the robustness and adaptability of ODC-FS-MPTC, showcasing its ability to provide stable performance even under challenging conditions with increased thrust loads and dynamic changes in linear speed.

Fig. 12.

Effect of FS-MPTC based on optimal duty cycle on the actual amplitude flux linkage under variable linear speed.

Fig. 13.

Effect of FS-MPTC based on optimal duty cycle on the αβ-axis flux linkage under variable linear speed.

Fig. 14.

Effect of FS-MPTC based on optimal duty cycle on the actual dq-axis primary current under variable linear speed.

Fig. 15.

Effect of FS-MPTC based on optimal duty cycle on the RMS value of the primary current under variable linear speed.

Fig. 16.

Effect of FS-MPTC based on optimal duty cycle on the Instantaneous three-phase primary current under variable linear speed.

4.3. Effect of sample time and switching frequency on the ripples

This part of the investigation explores the impact of sample time and switching frequency on system ripple behavior, focusing on the thrust ripples and primary flux linkage ripples, which are most important and affect the whole drive performance. Therefore, three distinct scenarios, maintaining identical loading and reference speed, are meticulously examined and delineated in Fig. 17, Fig. 18, Fig. 19. These testing conditions are performed at a target speed set of 5 m/s, with an accompanying thrust load of 10 kN.

Fig. 17.

Effect of the 5 μs sample time and 10 kHz switching frequency on (a) linear speed, (b) Electromagnetic thrust, (c) RMS primary current, and (d) Primary flux linkage.

Fig. 18.

Effect of the 10 μs sample time and 10 kHz switching frequency on (a) linear speed, (b) Electromagnetic thrust, (c) RMS primary current, and (d) Primary flux linkage.

Fig. 19.

Effect of the 10 μs sample time and 20 kHz switching frequency on (a) linear speed, (b) Electromagnetic thrust, (c) RMS primary current, and (d) Primary flux linkage.

In the first scenario, a sample time of 5 μs and a switching frequency of 10 kHz are utilized. Meanwhile, in the second scenario, the sample time is extended to 10 μs, with the switching frequency remaining at 10 kHz. In the third scenario, the sample time remains at 10 μs, but the switching frequency is doubled to 20 kHz.

Each case is comprehensively illustrated through four figures, which depict linear speed, electromagnetic thrust, RMS primary current, and the amplitude of the primary flux linkage, respectively.

Notably, with a fixed duty cycle (FDC-FS-MPDTC), all three methods with different sample times and switching frequencies yielded identical thrust and primary flux linkage ripples. Conversely, under the optimal duty cycle conditions (ODC-FS-MPDTC), the amplitude of the ripples for both thrust and primary flux linkage are greatly reduced, as observed in Fig. 17, Fig. 18, Fig. 19. In terms of comparison of the three cases with only ODC-FS-MPDTC, it can be seen that altering the sample time while maintaining the switching frequency showed no impact on system performance, as demonstrated in cases one and two Fig. 17(a–d) and 18 (a - d). Meanwhile, a significant reduction in ripple is achieved by increasing the switching frequency, as validated in Fig. 19(a–d). This confirms that the switching frequency has much effect in reducing the ripples compared to the sample time. For further illustration, the results for the three cases are provided in Fig. 20, Fig. 21, Fig. 22, enabling a comprehensive understanding of the impact of sample time and switching frequency variations on the system's dynamic response.

Fig. 20.

Effect of sample time and switching frequency on electromagnetic thrust.

Fig. 21.

Effect of sample time and switching frequency on RMS primary current.

Fig. 22.

Effect of sample time and switching frequency on primary flux linkage.

5. Conclusion

This paper proposed two methodologies within the finite-set model predictive thrust control (FS-MPTC) framework, both designed to optimize the utilization of voltage vectors within a single sample time. The first method is called fixed duty cycle FS-MPTC (FDC-FS-MPTC) and has been used to determine the optimal voltage vector while ensuring a constant duty cycle for the active and zero voltage vectors. Conversely, the second method is named optimal duty cycle FS-MPTC (ODC-FS-MPTC). It has been adopted to calculate the duty cycle for different voltage vectors and select the optimal one along with its duty cycle. Comparative analyses of these methods have been carried out for two distinct study cases, revealing significant advantages of the second approach. It has the advantage of notably reducing thrust ripples, minimizing oscillations of the primary flux linkage, and attenuating distortion of the primary current. These enhancements collectively contribute to an extended operational lifespan of the drive system. Furthermore, both methods successfully maintain speed, thrust, and flux linkage at their respective reference values, underscoring their effectiveness in achieving desired control objectives. It has to be mentioned that the FDC-FS-MPTC method can be suggested for applications that operate at a fixed operating condition, like production lines and so on, wherein those applications, a low processor controller with a lower price and build the control algorithm on it can be used. In addition, in the fixed duty cycle FS-MPTC approach, the heavy computation of the optimal duty will be avoided rather than the ODC-FS-MPTC. However, the selection percentages between active and zero voltage vectors are based on trial and error, which needs the best selection for a better response.

Funding

This article is derived from a research grant funded by the Research, Development, and Innovation Authority (RDIA), Saudi Arabia, with grant number (13,354-psu-2023-PSNU-R-3-1-EI).

Data availability

All the data is available in this study.

CRediT authorship contribution statement

Mahmoud F. Elmorshedy: Writing – original draft, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Dhafer Almakhles: Writing – review & editing, Supervision, Funding acquisition. Said M. Allam: Writing – original draft, Visualization, Supervision, Resources, Project administration, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.