Substantiation of a Rational Model of an Induction Motor in a Predictive Energy-Efficient Control System

Abstract

The development and implementation of scientifically substantiated solutions for the improvement and modernization of electromechanical devices, systems, and complexes, including electric drives, is an urgent theoretical and applied task for energetics, industry, transport, and other key areas, both in global and national contexts. The aim of this paper is to identify a rational model of an induction motor that balances computational simplicity and control system performance based on predictive approaches while ensuring maximum energy efficiency and reference tracking during the operation in dynamic modes. Five main mathematical models of an induction machine with different levels of detail have been selected. Three predictive control models have been implemented using GRAMPC (v 2.2), Matlab MPC Toolbox (v 24.1), and fmincon (R2024a) (from Matlab Optimization Toolbox). It has been established that in the dynamic mode of operation, the equivalent induction motor circuit with parameters $R_{fe} =const$, $L_{\mu }=f\left(I_{1d}\right)$, and $T_F=f\left({\omega }_{Rm}\right)$ is the most appropriate in terms of the following criteria: accuracy of control action generation, computation speed, and calculation of energy consumption.

1. Introduction

The sustainable development and modernization of electromechanical devices, systems, and complexes, including electric drives, constitute a pressing and multifaceted scientific and practical challenge [1]. This issue is particularly relevant in key sectors such as energy, industrial manufacturing, transportation, and automation, where reliable and efficient electromechanical solutions are fundamental to performance and innovation. Addressing this challenge is essential for enhancing operational efficiency, reducing environmental impact, and ensuring long-term technological competitiveness on both national and global scales [2].

Enhancing existing technologies and developing new approaches for electric drive control significantly contributes to the overall performance and reliability of technological equipment and production systems [3]. Advanced control methods allow for a more accurate adjustment of motor operation parameters in real time, which leads to improved stability and reduced energy losses [4]. These innovations facilitate adaptive responses to variable load conditions and operating environments, enabling more efficient and flexible system behavior [5]. As a result, enterprises benefit from increased energy efficiency, the extended service life of devices, and enhanced operational reliability [6]. Such improvements support greater precision and responsiveness in process control, which is vital for high-performance industrial applications. In addition, the adaptability of equipment is improved to meet the evolving requirements of automation systems and to comply with stringent environmental and energy-efficiency standards [7].

Today, the practical implementation of scientific progress in the field of electromechanical system control is evolving along several complementary directions [8]. These include the development of intelligent control algorithms, predictive and adaptive control strategies, the integration of digital twin technologies, and the application of advanced modeling techniques for real-time diagnostics and optimization [1,4,5,6,7,8,9]:

• Optimization of energy consumption;
• Integration of digitalization and intellectualization methods and tools;
• Increasing the share of renewable energy sources;
• Reducing the negative impact on the environment;
• Reduction in weight and dimensions;
• Increasing adaptability and modularity.

Details of global trends in the optimization of operating modes and improvement of electromechanical devices and systems with the expected applied effect are shown in the form of a structure in Figure 1.

The research on substantiating a rational model of an induction motor within a predictive energy-efficient control system aligns closely with ongoing developments in mathematical modeling and advanced control strategies for electromechanical systems in the mining and energy sectors [10]. This study contributes to a growing body of work focused on improving operational efficiency, control accuracy, and energy sustainability in high-demand industrial environments [10,11]. While previous research explored wave processes in three-winding transformers with magnetic flux considerations, the present study addresses the dynamic behavior of electrical machines by integrating predictive algorithms [12]. Both efforts reflect a broader aim: achieving better synchronization between theoretical modeling and real-world operational constraints in energy-intensive sectors such as mining [13].

This research is complemented by studies investigating the dynamic modes of asynchronous motors under voltage asymmetry [14]. While those works primarily focus on irregular supply conditions, the current study emphasizes predictive modeling and rational control strategies to enhance energy efficiency and operational reliability [15]. Similarly, research examining the influence of voltage reserves on power compensators in mining highlights the importance of effective energy management under load variations—an issue also addressed by other models through predictive adjustment and optimization techniques [16]. These interconnected research directions underscore the necessity of flexible, intelligent control systems capable of adapting to diverse operational and electrical environments [17].

This study resonates with broader interdisciplinary perspectives on energy consumption culture and the impact of route topology on electric vehicle energy demand [17,18]. Although these works target different applications, they share a common objective: promoting sustainability and efficient resource utilization [19]. The modeling of neuro-fuzzy control modes in bioheat generators further parallels the approach of a previous study, as both rely on intelligent control frameworks for optimizing energy performance [20]. Therefore, this research not only reinforces the technical foundation for the predictive control of induction motors in industrial contexts but also contributes meaningfully to the wider discourse on sustainable and intelligent energy management across various sectors.

Therefore, the conducted analysis of global trends in the creation and sustainable development of software and technological support for the energy-efficient control of electromechanical devices confirms the relevance and urgency of practice-oriented research in this field [21]. It highlights the growing need to develop and validate advanced methods, mathematical models, and technical solutions aimed at optimizing electric drive-control systems based on the criterion of energy efficiency [22]. This includes the integration of intelligent algorithms, predictive control strategies, and real-time data processing to ensure adaptability and performance in dynamic industrial environments [23]. The application of modern digital and information technologies in these control systems opens new opportunities for enhancing operational sustainability, reliability, and resource efficiency across various sectors.

The current trend towards increased attention to the problems of exhaustibility of natural resources [10,19], their rational use, and environmental protection in all industries [21,22,23,24] has led to an increase in the share of electric vehicles [25] in the transport industry (about 25.9 million electric vehicles in use worldwide [26]), and, therefore, one of the most topical issues is to improve the energy efficiency of electric drive systems as a component of electric mobility. Applications related to electric mobility [27] require electric machines to operate in a dynamic mode (short-time operation, intermittent operation, and others), where torque and speed change frequently and current and voltage can reach limit values [28]. In this regard, induction machines are widely used in industrial applications and electric mobility because of their straightforward design and operation when compared to other types of machines. This simplicity enables a more cost-effective and resource-efficient manufacturing process [29]. Numerous techniques for achieving steady-state optimization at a constant speed and torque for induction machines are already well-established. These methods are discussed, for example, in [29,30].

Control methods based on the power loss model [31] are one of the most developed approaches to improving the energy efficiency of electric machines [32]. Given the above, dynamic operating modes require an adequate mathematical description of electric motors that considers key physical processes, energy losses, and efficiency criteria [32,33]. One of the most important aspects is the level of detail of the machine model [34], which affects the accuracy of the generation of control signals, computation speed, and calculation of energy consumption.

This study presents a rational model of an induction motor designed for integration within a predictive energy-efficient control system. This approach effectively addresses the challenges posed by dynamic operating conditions in predictive control by striking a balance between model accuracy and computational feasibility [35]. Unlike highly detailed motor models, which provide precise representations of physical phenomena but impose significant computational burdens, the proposed rational model reduces complexity without compromising essential dynamic behavior [9,36]. Such a compromise is vital for enabling real-time implementation in fast control loops, where computational delays can adversely affect system performance [37]. Furthermore, the integration of deep neural networks in the control architecture facilitates the intelligent management of energy losses by learning complex nonlinear relationships and adapting to varying operating conditions [38]. This data-driven component enhances the predictive capability of the system, allowing for more efficient energy usage and improved overall motor performance. Consequently, the combined model offers a promising solution for optimizing induction motor control in industrial applications requiring both precision and real-time responsiveness.

However, as with many predictive control strategies, the trade-off between model accuracy and computational demand remains a central issue [10,12,39]. While the simplified rational model improves real-time performance due to its reduced computational complexity, there is an inherent risk of losing fidelity in capturing certain nonlinear or transient dynamics of the induction motor [40]. Consequently, the model’s effectiveness largely depends on how well it approximates critical behaviors that impact control accuracy. Incorporating non-destructive diagnostics of power equipment insulation into the control system can further enhance reliability by detecting insulation degradation without interrupting operation [41]. This study contributes to the ongoing discussion by proposing a model that carefully balances complexity and accuracy, thereby enabling energy-efficient control suitable for practical applications [42]. Ultimately, this approach supports the development of smarter, more resilient motor control systems capable of meeting industrial demands.

Unlike most existing studies that focus either on developing new mathematical models of induction motors or on improving individual control algorithms, this research proposes a comprehensive approach to selecting a rational model through a systematic comparison of five mathematical models with different levels of detail, combined with three predictive control algorithms (GRAMPC, fmincon, and MPC Toolbox) under identical dynamic conditions. This approach makes it possible to establish clear model selection criteria based on the trade-off between accuracy, computational speed, and energy efficiency, thereby providing a direct foundation for integrating the obtained results into industrial predictive energy-efficient control systems.

Accordingly, to determine a rational motor model in the development and implementation of energy-efficient control algorithms and systems, it is necessary to assess how the level of model detail affects the above indicators. Thus, the scientific task here is to identify a rational motor model that balances computational simplicity and control system performance while ensuring maximum energy efficiency and reference tracking during the operation of asynchronous drive in transient modes.

2. Materials and Methods

To solve the stated scientific and practical problem, generally accepted research methods are used, which are based on the fundamental principles of the theory of electric machines, the theory of electric drive, the theory of predictive control of technical objects, and mathematical and computer modeling using Matlab R2024a. To implement energy-efficient control, GRAMPC based on the augmented Lagrange approach using a gradient search algorithm and a built-in nonlinear model of the control object [35], Matlab MPC Toolbox [43], and the fmincon function of Matlab from the Optimization Toolbox, which is designed to find the minimum of a constrained nonlinear multidimensional function [44], are used. An induction motor with the parameters shown in

Table 1. Motor data of the 370 W induction motor.

PN	370 W	TN	2.59 Nm
Zp	2	JM	22·10−4 kg·m2
nN	1370 rpm
R1	27.8 Ω	Lσ	0.6 H
R2	17.24 Ω	Lμ	0.142 H
RFe	2300 Ω	tR	35 ms
ωRm,1	52.4 rad/s ≅ 500 rpm	ωRm,2	157 rad/s ≅ 1500 rpm
TL	0.64 Nm

, which are typical for 370 W models, is used as the object of predictive control.

The following notations are used in Table 1: P_N—nominal power; Z_p—the number of pole pairs; n_N—nominal speed; T_N—nominal torque; J_M—the total moment of inertia of the induction machine and the load machine, as well as the coupling between the two machines and the torque sensor; R₁—stator resistance; R₂—equivalent rotor resistance; R_Fe—iron resistance; L_σ—dissipation inductance; L_μ—main inductance; t_R—rotor time constant; ω_Rm,1—initial value of the rotor angular speed; ω_Rm,2—final value of the rotor angular speed; and T_L—load torque.

In analyzing mathematical models of induction motors of various degrees of detail, it has been established that the most valuable in practice are those that provide high accuracy of electromagnetic processes in both stationary and transient modes of motor operation and allow the accurate calculation of actual state and control vectors depending on the operating conditions and tasks [45]. This is especially important when integrating into automated control systems, where models can be implemented as embedded software when monitoring and optimizing motor operation modes. Based on the results of the analysis of scientific sources [31,34,46,47,48], five main mathematical models of an induction machine with different levels of detail were identified, as shown below.

First is the Type I detailed model $L_{\mu }\ne const, R_{Fe}\left({\omega }_1\right), T_F\left({\omega }_{Rm}\right)$, which is shown in Figure 2, where the components are as follows: ${\underset{\_}{U}}_1$—stator voltage phasor; $R_1$—stator resistance; $R_2$—equivalent rotor resistance; $L_{\sigma }$—dissipation inductance; $R_{Fe}$—iron resistance; $L_{\mu }$—magnetization inductance (main inductance); ${\underset{\_}{I}}_{\mu }$—current phasor of the main inductance; ${\underset{\_}{I}}_{R_{Fe}}$—current phasor of the iron; ${\underset{\_}{I}}_1, {\underset{\_}{I}}_2$—stator current phasor and the phasor of the equivalent rotor current; and ${\omega }_1$—angular speed of rotation of the stator magnetic field.

Model $L_{\mu }\ne const, R_{Fe}\left({\omega }_1\right), T_F\left({\omega }_{Rm}\right)$ state equations:

$${\displaystyle \frac{d{I}_{1d}}{dt}}=Z_p{\omega }_{Rm}{I}_{1q}+{\displaystyle \frac{R_2{I}_{1q}^2}{{\Psi }_{2d}}}-{\displaystyle \frac{R_1{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{R_{Fe}\left({\omega }_1\right)}{R_{Fe}\left({\omega }_1\right)+R_2}}\left({\displaystyle \frac{R_2{I}_{1d}}{L_{\sigma }}}-{\displaystyle \frac{{\Psi }_{2d}}{T_2{L}_{\sigma }}}\right)+{\displaystyle \frac{U_{1d}}{L_{\sigma }}},$$

(1)

$${\displaystyle \frac{d{I}_{1q}}{dt}}=-{\displaystyle \frac{R_2{I}_{1q}{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{Z_p{\omega }_{Rm}{\Psi }_{2d}}{L_{\sigma }}}-{\displaystyle \frac{\left(R_1+R_2\right)I_{1q}}{L_{\sigma }}}-Z_p{\omega }_{Rm}{I}_{1d}+{\displaystyle \frac{U_{1q}}{L_{\sigma }}},$$

(2)

$${\displaystyle \frac{d{\Psi }_{2d}}{dt}}={\displaystyle \frac{R_{Fe}\left({\omega }_1\right)}{R_{Fe}\left({\omega }_1\right)+R_2}}\left(R_2{I}_{1d}-{\displaystyle \frac{{\Psi }_{2d}}{T_2}}\right),$$

(3)

$${\displaystyle \frac{d{\omega }_{Rm}}{dt}}={\displaystyle \frac{T_e-T_L-T_F\left({\omega }_{Rm}\right)}{J_M+J_L}},$$

(4)

where the components are as follows: $Z_p$—the number of pole pairs; ${\omega }_{Rm}$—angular speed of the rotor, equal to ${\omega }_{Rm}=\left({\omega }_1-{\omega }_2\right)/Z_p$; ${\omega }_2$—slip speed; $T_2$—rotor time constant, determined by the formula $T_2=L_{\mu }\left(I_{1d}\right)/R_2$; $J_M$  i  $J_L$—moments of inertia of the motor and the load; $T_e$—motor torque, which is equal to $T_e={\displaystyle \frac{3}{2}}{Z}_p{\Psi }_{2d}{I}_{1q}$; $T_L$—load torque; $T_F\left({\omega }_{Rm}\right)$—a function that models mechanical losses depending on the motor speed and load; dependence of the main inductance $L_{\mu }$ on the stator $d$-current $I_{1d}$: $L_{\mu }\left(I_{1d}\right)=a_1{I}_{1d}^5+\cdots +a_5{I}_{1d}+a_6$; friction torque $T_F\left({\omega }_{Rm}\right)=b_1{\omega }_{Rm}^7+\cdots +b_7{\omega }_{Rm}+b_8$; dependence of magnetic circuit (iron) resistance on angular speed $R_{Fe}\left({\omega }_1\right)={\displaystyle \frac{R_{Fe, nom}-R_{Fe0}}{{\omega }_{1N}}}{\omega }_1+R_{Fe0}$; $R_{Fe, nom}$ and $R_{Fe0}$—iron resistance at the nominal speed and at 0, respectively; and ${\omega }_{1N}$—nominal angular speed.

The power loss function corresponding to the Type I model is as follows:

$$P_{Loss}=P_{CuLoss}+P_{FeLoss},$$

(5)

$$P_{CuLoss}={\displaystyle \frac{3}{2}}\left(R_1+R_2\right)\left(I_{1d}^2+I_{1q}^2\right)+{{\displaystyle \frac{3}{2}}R}_2\left({\displaystyle \frac{{\Psi }_{2d}^2}{L_{\mu }^2\left(I_{1d}\right)}}-{\displaystyle \frac{2{\Psi }_{2d}{I}_{1d}}{L_{\mu }\left(I_{1d}\right)}}\right),$$

(6)

$$P_{FeLoss}={\displaystyle \frac{3}{2}}{R}_{Fe}\left({\omega }_1\right)\left({\left({\displaystyle \frac{\dot{{\Psi }_{2d}}}{R_{Fe}\left({\omega }_1\right)}}\right)}^2+{\left({\displaystyle \frac{{\omega }_1{\Psi }_{2d}}{R_{Fe}\left({\omega }_1\right)}}\right)}^2\right).$$

(7)

Type II: the substitution scheme is shown in Figure 2 with model parameters $L_{\mu }\ne const, R_{Fe}=const, T_F\left({\omega }_{Rm}\right)$. State equations of the model under the conditions $L_{\mu }\ne const, R_{Fe}=const, T_F\left({\omega }_{Rm}\right)$ are as follows:

$${\displaystyle \frac{d{I}_{1d}}{dt}}=Z_p{\omega }_{Rm}{I}_{1q}+{\displaystyle \frac{R_2{I}_{1q}^2}{{\Psi }_{2d}}}-{\displaystyle \frac{R_1{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{R_{Fe}}{R_{Fe}+R_2}}\left({\displaystyle \frac{R_2{I}_{1d}}{L_{\sigma }}}-{\displaystyle \frac{{\Psi }_{2d}}{T_2{L}_{\sigma }}}\right)+{\displaystyle \frac{U_{1d}}{L_{\sigma }}},$$

(8)

$${\displaystyle \frac{d{I}_{1q}}{dt}}=-{\displaystyle \frac{R_2{I}_{1q}{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{Z_p{\omega }_{Rm}{\Psi }_{2d}}{L_{\sigma }}}-{\displaystyle \frac{\left(R_1+R_2\right)I_{1q}}{L_{\sigma }}}-Z_p{\omega }_{Rm}{I}_{1d}+{\displaystyle \frac{U_{1q}}{L_{\sigma }}},$$

(9)

$${\displaystyle \frac{d{\Psi }_{2d}}{dt}}={\displaystyle \frac{R_{Fe}}{R_{Fe}+R_2}}\left(R_2{I}_{1d}-{\displaystyle \frac{{\Psi }_{2d}}{T_2}}\right),$$

(10)

$${\displaystyle \frac{d{\omega }_{Rm}}{dt}}={\displaystyle \frac{T_e-T_L-T_F\left({\omega }_{Rm}\right)}{J_M+J_L}}.$$

(11)

The power loss function corresponding to the Type II model is as follows:

$$P_{Loss}=P_{CuLoss}+P_{FeLoss},$$

(12)

$$P_{CuLoss}={\displaystyle \frac{3}{2}}\left(R_1+R_2\right)\left(I_{1d}^2+I_{1q}^2\right)+{{\displaystyle \frac{3}{2}}R}_2\left({\displaystyle \frac{{\Psi }_{2d}^2}{L_{\mu }^2\left(I_{1d}\right)}}-{\displaystyle \frac{2{\Psi }_{2d}{I}_{1d}}{L_{\mu }\left(I_{1d}\right)}}\right),$$

(13)

$$P_{FeLoss}={\displaystyle \frac{3}{2}}{R}_{Fe}\left({\left({\displaystyle \frac{\dot{{\Psi }_{2d}}}{R_{Fe}}}\right)}^2+{\left({\displaystyle \frac{{\omega }_1{\Psi }_{2d}}{R_{Fe}}}\right)}^2\right).$$

(14)

Type III: the substitution scheme is shown in Figure 3 with model parameters $L_{\mu }\ne const, R_{Fe}=\infty , T_F\left({\omega }_{Rm}\right)$.

The state equations of the model under the conditions $L_{\mu }\ne const, R_{Fe}=\infty , T_F\left({\omega }_{Rm}\right)$ are as follows:

$${\displaystyle \frac{d{I}_{1d}}{dt}}=Z_p{\omega }_{Rm}{I}_{1q}+{\displaystyle \frac{R_2{I}_{1q}^2}{{\Psi }_{2d}}}-{\displaystyle \frac{\left(R_1+R_2\right)}{L_{\sigma }}}{I}_{1d}+{\displaystyle \frac{{\Psi }_{2d}}{T_2{L}_{\sigma }}}+{\displaystyle \frac{U_{1d}}{L_{\sigma }}},$$

(15)

$${\displaystyle \frac{d{I}_{1q}}{dt}}=-{\displaystyle \frac{R_2{I}_{1q}{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{Z_p{\omega }_{Rm}{\Psi }_{2d}}{L_{\sigma }}}-{\displaystyle \frac{\left(R_1+R_2\right)I_{1q}}{L_{\sigma }}}-Z_p{\omega }_{Rm}{I}_{1d}+{\displaystyle \frac{U_{1q}}{L_{\sigma }}},$$

(16)

$${\displaystyle \frac{d{\Psi }_{2d}}{dt}}=R_2{I}_{1d}-{\displaystyle \frac{{\Psi }_{2d}}{T_2}},$$

(17)

$${\displaystyle \frac{d{\omega }_{Rm}}{dt}}={\displaystyle \frac{T_e-T_L-T_F\left({\omega }_{Rm}\right)}{J_M+J_L}}.$$

(18)

The power loss function corresponding to the Type III model is as follows:

$$P_{CuLoss}={\displaystyle \frac{3}{2}}\left(R_1+R_2\right)\left(I_{1d}^2+I_{1q}^2\right)+{{\displaystyle \frac{3}{2}}R}_2\left({\displaystyle \frac{{\Psi }_{2d}^2}{L_{\mu }^2\left(I_{1d}\right)}}-{\displaystyle \frac{2{\Psi }_{2d}{I}_{1d}}{L_{\mu }\left(I_{1d}\right)}}\right).$$

(19)

Type IV: the substitution scheme is shown in Figure 3 with model parameters $L_{\mu }=const, R_{Fe}=\infty , T_F\left({\omega }_{Rm}\right)$.

The state equations of the model under the conditions $L_{\mu }=const, R_{Fe}=\infty , T_F\left({\omega }_{Rm}\right)$ are as follows:

$${\displaystyle \frac{d{I}_{1d}}{dt}}=Z_p{\omega }_{Rm}{I}_{1q}+{\displaystyle \frac{R_2{I}_{1q}^2}{{\Psi }_{2d}}}-{\displaystyle \frac{\left(R_1+R_2\right)}{L_{\sigma }}}{I}_{1d}+{\displaystyle \frac{{\Psi }_{2d}}{T_2{L}_{\sigma }}}+{\displaystyle \frac{U_{1d}}{L_{\sigma }}},$$

(20)

$${\displaystyle \frac{d{I}_{1q}}{dt}}=-{\displaystyle \frac{R_2{I}_{1q}{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{Z_p{\omega }_{Rm}{\Psi }_{2d}}{L_{\sigma }}}-{\displaystyle \frac{\left(R_1+R_2\right)I_{1q}}{L_{\sigma }}}-Z_p{\omega }_{Rm}{I}_{1d}+{\displaystyle \frac{U_{1q}}{L_{\sigma }}},$$

(21)

$${\displaystyle \frac{d{\Psi }_{2d}}{dt}}=R_2{I}_{1d}-{\displaystyle \frac{{\Psi }_{2d}}{T_2}},$$

(22)

$${\displaystyle \frac{d{\omega }_{Rm}}{dt}}={\displaystyle \frac{T_e-T_L-T_F\left({\omega }_{Rm}\right)}{J_M+J_L}},$$

(23)

where the rotor time constant $T_2=L_{\mu }/R_2$.

The power loss function corresponding to the Type IV model is as follows:

$$P_{CuLoss}={\displaystyle \frac{3}{2}}\left(R_1+R_2\right)\left(I_{1d}^2+I_{1q}^2\right)+{{\displaystyle \frac{3}{2}}R}_2\left({\displaystyle \frac{{\Psi }_{2d}^2}{L_{\mu }^2}}-{\displaystyle \frac{2{\Psi }_{2d}{I}_{1d}}{L_{\mu }}}\right).$$

(24)

The Type V substitution scheme is shown in Figure 3 with model parameters $L_{\mu }=const, R_{Fe}=\infty , T_F\left({\omega }_{Rm}\right)=0$. The state equations of the model under the conditions $L_{\mu }=const, R_{Fe}=\infty , T_F\left({\omega }_{Rm}\right)=0$ are as follows:

$${\displaystyle \frac{d{I}_{1d}}{dt}}=Z_p{\omega }_{Rm}{I}_{1q}+{\displaystyle \frac{R_2{I}_{1q}^2}{{\Psi }_{2d}}}-{\displaystyle \frac{\left(R_1+R_2\right)}{L_{\sigma }}}{I}_{1d}+{\displaystyle \frac{{\Psi }_{2d}}{T_2{L}_{\sigma }}}+{\displaystyle \frac{U_{1d}}{L_{\sigma }}},$$

(25)

$${\displaystyle \frac{d{I}_{1q}}{dt}}=-{\displaystyle \frac{R_2{I}_{1q}{I}_{1d}}{{\Psi }_{2d}}}-{\displaystyle \frac{Z_p{\omega }_{Rm}{\Psi }_{2d}}{L_{\sigma }}}-{\displaystyle \frac{\left(R_1+R_2\right)I_{1q}}{L_{\sigma }}}-Z_p{\omega }_{Rm}{I}_{1d}+{\displaystyle \frac{U_{1q}}{L_{\sigma }}},$$

(26)

$${\displaystyle \frac{d{\Psi }_{2d}}{dt}}=R_2{I}_{1d}-{\displaystyle \frac{{\Psi }_{2d}}{T_2}},$$

(27)

$${\displaystyle \frac{d{\omega }_{Rm}}{dt}}={\displaystyle \frac{T_e-T_L}{J_M+J_L}}.$$

(28)

The power loss function corresponding to the Type V model is as follows:

$$P_{CuLoss}={\displaystyle \frac{3}{2}}\left(R_1+R_2\right)\left(I_{1d}^2+I_{1q}^2\right)+{{\displaystyle \frac{3}{2}}R}_2\left({\displaystyle \frac{{\Psi }_{2d}^2}{L_{\mu }^2}}-{\displaystyle \frac{2{\Psi }_{2d}{I}_{1d}}{L_{\mu }}}\right).$$

(29)

The problem of optimal control according to the criterion of minimum energy losses and maximum accuracy of maintaining the rotor angular velocity in terms of a formal description can be defined in the form of Equations (11)–(17).

$$\begin{array}{c}min\\ u\end{array}=J\left(u;x_k\right)={\int }_{t_0}^{t_f}\left[P_{loss}\left(\tau \right)+q{\left(x_4\left(\tau \right)-{\omega }_{Rm,ref}\left(\tau \right)\right)}^2\right]d\tau$$

(30)

$$s.t. \dot{x\left(\tau \right)}=f\left(x\left(\tau \right),u\left(\tau \right)\right),x\left(t_0\right)=x_k$$

(31)

$$h_1\left(u\left(\tau \right)\right)={u_1\left(\tau \right)}^2+{u_2\left(\tau \right)}^2-U_{1,max}^2\le 0,$$

(32)

$$h_2\left(x\left(\tau \right)\right)={x_1\left(\tau \right)}^2+{x_2\left(\tau \right)}^2-I_{1,max}^2\le 0,$$

(33)

$$h_3\left(x\left(\tau \right)\right)={x_1\left(\tau \right)}^2+{x_2\left(\tau \right)}^2-I_{1,max}^2\le 0,$$

(34)

$$h_4\left(x\left(\tau \right)\right)=-x_1\left(\tau \right)-I_{1d,max}\le 0,$$

(35)

$$u\left(\tau \right)\in \left[u_{min},u_{max}\right].$$

(36)

where the cost function is J, which depends on $x\in {\mathbb{R}}^{N_x}$ (where $x_1$, $x_2$, $x_3$ and $x_4$ are the state variables $I_{1d}$, $I_{1q}$, ${\Psi }_{2d}$ and ${\omega }_{Rm}$, respectively) and $u\in {\mathbb{R}}^{N_u}$ (where $u_1$ and $u_2$ are the control variables $U_{1d}$ and $U_{1q}$, respectively). The conditions imposed on the cost function (30) are the dynamics of the system (31), which is given by the system of differential Equations (1)–(4), (8)–(11), (15)–(18), (20)–(23), or (25)–(28) and constraints (32)–(36). The constraints on the control variables are $u_{min}={\left[{-U}_{max},{-U}_{max}\right]}^T$ and $u_{max}={\left[U_{max},U_{max}\right]}^T$, used to improve the overall stability of the search algorithm. The four constraints (32)–(36) have different orders of magnitude, i.e., $I_{1,max}^2$, $U_{1,max}^2$, and $I_{1d,max}$. Therefore, the given constraints must be scaled by the maximum value:

$$h_1={\displaystyle \frac{U_{1d}^2+U_{1q}^2}{U_{1,max}^2}}-1\le 0,$$

(37)

$$h_2={\displaystyle \frac{I_{1d}^2+I_{1q}^2}{I_{1,max}^2}}-1\le 0,$$

(38)

$$h_3={\displaystyle \frac{-I_{1d}}{I_{1d,max}}}\le 0,$$

(39)

$$h_4={\displaystyle \frac{I_{1d}}{I_{1d,max}}}-1\le 0.$$

(40)

Consequently, the induction motor models described above with varying degrees of detail, a formal description of the optimization problem, and modeling conditions from [49] form the functional basis for solving the scientific and applied problem of substantiating a rational model in a predictive control system.

3. Results and Discussion

During the study, three predictive control models were implemented based on the following: (1) GRAMPC (v 2.2) is a software framework for predictive control (MPC) based on the augmented Lagrangian approach using a gradient search algorithm and a built-in nonlinear model of the control object; (2) fmincon is a Matlab function from the Optimization Toolbox designed to find the minimum of a constrained nonlinear multidimensional function; (3) MPC Toolbox (v 24.1) is a nonlinear predictive controller that calculates optimal control actions over a forecast horizon using a nonlinear predictive model, a nonlinear cost function, and nonlinear constraints and fmincon.

The sampling time Δt = 250 µs is selected as a compromise between the stability of GRAMPC (v 2.2) and the deviation in the optimal trajectories calculated by GRAMPC (v 2.2) from the stationary optima. At a given time, t = 0 s, the induction machine operates at the speed n = 500 rpm and the optimal flux linkage Ψ_2d. The load torque is 25% of the nominal value in Table 1. The load torque is not constant; friction torque components are added to it. At t = 0.4 s, a command is given to change the speed from 500 rpm to 1500 rpm for 0.4 s. At n = 1500 rpm, the load torque is 25% + 7% of the nominal torque. The change in load torque from the initial to the final value is assumed to be linear, like the acceleration curve.

Figure 4, Figure 5, Figure 6 and Figure 7 show, as a representative example, the transient responses of the Type I motor model under the three predictive control algorithms. Figure 4 shows that GRAMPC achieves a smooth acceleration profile from 500 rpm to 1500 rpm within the prescribed transition time, with minimal overshoot in rotor speed. The stator currents exhibit a coordinated increase in the d and q components, which is characteristic of GRAMPC’s augmented Lagrangian approach optimizing both torque production and flux control. Figure 5 visualizes the same scenario with explicit constraint trajectories, demonstrating that GRAMPC maintains all variables within the defined operational limits throughout the transient.

Figure 6 presents results for the fmincon-based optimization, where the flux linkage increases in advance of the speed command. This anticipatory adjustment reduces simultaneous peaks in stator current components, thereby lowering instantaneous power losses during acceleration. Figure 7 shows the MPC Toolbox performance, which similarly applies anticipatory flux changes but with slightly different current profiles due to its nonlinear optimization strategy.

The similarity in dynamic behavior across models of different levels of detail is why only Type I results are shown in Figure 4, Figure 5, Figure 6 and Figure 7. The optimization results show that fmincon and MPC Toolbox increase the flux linkage Ψ_2d before the speed command. This way, the I_1d and I_1q currents do not increase simultaneously, which reduces power losses. Such a trajectory of Ψ_2d indicates that changing this indicator in an anticipatory manner in relation to changes in the speed and/or load torque command is important for minimizing power losses in dynamic operation.

The generalized result of simulating the models in MATLAB (v 24.1) (five levels of detail of the mathematical model of an induction machine), based on the comparison of the obtained characteristics, allowed us to determine the most rational motor model according to the following criteria: accuracy of control action generation, computation speed, and calculation of energy consumption. In particular, the models have been tested based on Matlab MPC Toolbox. Models with a low level of detail showed a high computational speed, while models with a high level of detail were more accurate, and the computational time was inversely related. Quantitative differences between models are captured in

Table 2. The results of determining the most rational model.

Model Type	Level of Detail	Elapsed Time, s	Control Accuracy (rad/s)
I	$R_{fe}=f\left({\omega }_1\right), L_{\mu }=f\left(I_{1d}\right), T_F=f\left({\omega }_{Rm}\right)$	44.12	0.15
II	$R_{fe}=const,L_{\mu }=f\left(I_{1d}\right), T_F=f\left({\omega }_{Rm}\right)$	24.71	0.22
III	$R_{fe}=\infty ,L_{\mu }=f\left(I_{1d}\right), T_F=f\left({\omega }_{Rm}\right)$	25.56	0.25
IV	$R_{fe}=\infty ,L_{\mu }=const, T_F=f\left({\omega }_{Rm}\right)$	27.68	0.20
V	$R_{fe}=\infty ,L_{\mu }=const, T_F=0$	26.86	0.23

, where accuracy and computation time are directly compared. In addition, Figure 8 presents the quantitative differences between the five induction motor models in terms of copper losses (E_cu) and iron losses (E_fe) in joules. The values shown are averaged over all three predictive control methods, providing a consolidated comparison of the energy performance of all models.

The comparative analysis conducted in this study differs from previous works in that it simultaneously evaluates accuracy, computational performance, and the influence of model detail level on energy consumption under transient operating conditions. Most publications address only one of these aspects, which does not allow for a comprehensive assessment of a model’s suitability for use in predictive control systems. The proposed methodology provides engineers and developers with a clearly structured set of criteria for model selection, taking into account computational resource constraints and energy-saving requirements.

The results obtained have direct practical relevance for industry, as they enable the rapid integration of optimally selected models into digital management platforms such as enterprise resource planning (ERP) systems or manufacturing execution systems (MESs). This, in turn, allows for greater flexibility in technological processes, timely response to load changes, and improved energy efficiency in accordance with Industry 4.0 principles. Therefore, this study forms a scientifically substantiated and practically oriented basis for deploying intelligent predictive energy-efficient control systems in industrial environments.

4. The Significance of the Research Findings for Operational and Strategic Management in Industry

The results obtained in this study, including the identification of a rational mathematical model of the induction motor and the application of three predictive control algorithms (GRAMPC, fmincon, and the MPC Toolbox), can serve as key components supporting energy, technology, and operational process management. The adopted approach enables the accurate prediction of dynamic load variations and energy demand in drive systems. Consequently, not only does the local optimization of drive performance become feasible but also strategic energy consumption planning at the level of entire production facilities. This translates into more informed management of operating costs and the effective implementation of sustainable development policies within energy management systems. The discussed models can also be successfully integrated with digital enterprise management platforms, such as enterprise resource planning systems and manufacturing execution systems. The implementation of predictive control based on GRAMPC or the MPC Toolbox leads to the more efficient use of operational data and its transformation into managerial knowledge. As a result, it becomes possible to dynamically adjust machine operation schedules, improve responsiveness to load changes, and enhance process flexibility in accordance with the principles of Industry 4.0. The diversity of the analyzed mathematical models also enables well-founded technological decisions, depending on the level of detail required in each application. Models with higher computational resolution offer greater precision in forecasting energy losses, which is essential for the development of efficiency monitoring systems, environmental reporting, and life cycle cost analyses.

The experimental results also confirm the potential for forecasting and minimizing operational risks. The ability to detect overloads or incorrect torque selection at an early stage forms the basis for reducing unplanned downtimes and improving maintenance processes. This approach contributes to increased production reliability and the improved economic performance of enterprises.

Identifying the optimal motor model in terms of accuracy, computational speed, and energy efficiency creates a foundation for standardizing the implementation of control technologies across different industrial environments. Such standardization facilitates benchmarking processes, supports the transfer of technical knowledge, and strengthens strategic technology portfolio management within the organization.

Another important aspect is the ability to translate technical data (e.g., power losses, response time, and dynamic parameters) into measurable business indicators. Transforming these data into performance metrics, return on investment, or equipment effectiveness facilitates better understanding of the value of technological implementations by management and supports decision making based on sound engineering principles.

5. Conclusions

The research results presented in this article constitute a significant contribution to the development of modern, intelligent, and sustainable industrial management systems. They highlight the necessity of further strengthening collaboration between technical sciences and management, particularly in the context of industrial digitalization and energy transition. In this paper, a topical scientific and applied problem has been addressed: the identification of a rational motor model that balances computational simplicity with control system performance while ensuring maximum energy efficiency and reference tracking during the operation of an asynchronous drive in transient modes. The main findings of the research are as follows:

• Three approaches to the implementation of predictive control models based on GRAMPC, fmincon, and the MPC Toolbox have been investigated. This enabled the examination of how the level of detail in induction motor models affects the efficiency of their control.
• The models were simulated in MATLAB, and the relationships between electric energy consumption indicators and the accuracy of maintaining motor speed were established in terms of their impact on the overall energy efficiency of electric drives under dynamic operating conditions. This makes it possible to optimize control strategies.
• Based on the results of computer experiments using predictive methods (GRAMPC, fmincon, and the MPC Toolbox) in MATLAB (v 24.1) and Simulink, it has been determined that under dynamic operating conditions, the most suitable model for controlling induction motors is an equivalent circuit with the parameters $R_{fe}=const$, $L_u=f\left(I_{1d}\right)$ and $T_F=f\left({\omega }_{Rm}\right)$. This allows for improved control approaches for induction motors.
• Promising directions for further research have been substantiated. These include, on the one hand, unifying the representation of processes across various electric machines with a rotating magnetic field to support the development of standardized energy-efficient control methods, and on the other hand, enhancing the functionality of the GRAMPC algorithm to accommodate complex, time-varying trajectories of desired speed/load torque in real-time optimization of the cost function.