Research on the Coordinated Control of Mining Multi-PMSM Systems Based on an Improved Active Disturbance Rejection Controller

Abstract

This study focuses on the problems of poor control performance, synchronization performance and stability in multi-motor permanent magnet drive systems in mining belt conveyors when a Proportional Integral Derivative (PID) controller is used to control the multi-motor. In this paper, a system model for three-motor synchronous control of a mine belt conveyor is established. On this basis, an Enhanced first-order Active Disturbance Rejection Controller (Efal_ADRC) is designed based on an optimized nonlinear function. Additionally, a weighted arithmetic mean is used to enhance the compensator of the ring coupling control structure. Finally, the system model is evaluated and simulated using various algorithms. Results show that synchronous control of a multi-Permanent Magnet Synchronous Motor (multi-PMSM) drive system based on the Efal_ADRC ring coupling control structure has better anti-interference ability, control accuracy and synchronization, which is conducive to the stable and efficient safe operation of the belt conveyor.

1. Introduction

The continuous development of modern industrial intelligence has rendered the control of a single motor incapable of meeting the high-performance requirements for equipment in modern industries. Consequently, multi-motor coordinated control systems have become a focal point of research in the field of modern industry. Among these, the belt conveyor, which is the principal equipment used for underground coal transportation in coal mines, is undergoing a process of intensification and automation. In the contemporary era, belt conveyors are predominantly operated by asynchronous induction motors in conjunction with intermediate transmission devices, such as reducers and hydraulic couplings. The belt conveyor drive system is constrained by a number of limitations, including low efficiency, unstable starting and difficulty in starting with heavy loads [1]. In comparison, the Permanent Magnet Synchronous Motor (PMSM) has a working principle that is distinct from an asynchronous motor. The PMSM exhibits the characteristics of low speed, large torque and high power. It does not require an intermediate transmission device and can directly drive the belt conveyor, thus improving the transmission efficiency and reliability of the system.

The multi-motor drive system of the belt conveyor allows for convenient intelligent control. However, when multiple motors operate simultaneously, it is possible for asynchronous operation between the motors to occur, which could potentially lead to damage to the motors. In response to this issue, researchers in both domestic and foreign research institutions have proposed a range of advanced control methods and control strategies. Deng et al. [2] proposed an Active Disturbance Rejection Controller (ADRC) parameter tuning method based on a particle swarm algorithm, which simplified parameter setting, but could not realize online parameter adjustment. Shi et al. [3] proposed an ADRC parameter tuning method based on a hybrid optimization method, which proved effective but involved the integration of two algorithms other than ADRC. Zhou et al. [4] proposed a novel harmonic separation scheme that has the capacity to directly extract the voltage distortion caused by the nonlinearity of the inverter in situ, obviating the necessity for offline testing. This method boasts lower computational costs, independent parameters and better adaptability. However, in the case of a saturated region and large torque, it will produce high-order harmonics and affect the steady-state error of the system. Liu et al. [5] proposed a nonlinear robust speed controller for double-non-identical parallel PMSM systems, which can effectively deal with motor parameter differences and external disturbances, and has certain innovations, but there are some limitations in model dependence and application range. Liu et al. [6] designed a perturbation adaptive control method that accurately compensates the nonlinearity and suppresses the external disturbances of the multi-motor system, but it is computationally expensive. Yu et al. [7] proposed an improved Linear Active Disturbance Rejection Controller (FTD-LADRC), which combines the advantages of FTD and LESO and not only maintains the simplicity of a LADRC, but also enhances disturbance immunity and dynamic response performance. However, this method is mainly aimed at magnetic suspension turbo-mechanical rotor systems, and whether it is suitable for other types of nonlinear systems needs further verification. Gao et al. [8,9] proposed a linear-nonlinear switching mechanism in the framework of active interference suppression control (ADRC), which can cope with possible interference in the system through the online selection of control modes and enhance the adaptability and robustness of the control system. Li et al. [10] proposed a fraction-order active interference suppression control (FADRC), which improved the traditional ADRC method by introducing fraction-order theory, thus improving the robustness and adaptability of the control system in dealing with complex nonlinear systems. However, there are some shortcomings such as its complicated implementation as well as its limited application range and parameter adjustment. Hu et al. [11] devised a control strategy based on an integral predictor, which mitigated the impact of high-frequency oscillation. The multi-motor system can still be controlled synchronously, but the controller design is complex and computationally intensive. Zhang et al. [12] designed optimized PID control based on the cross-coupling structure, which ensured control accuracy while reducing the amount of calculations. The speed was suboptimal, and there was a starting error. Ye [13] combined the advantages of cross-coupling and bias coupling to enhance the anti-interference performance of four motors. However, this approach has the disadvantages of error delay and a significant computational burden.

The majority of existing research focuses on high-speed and low-power PMSM, while there are relatively few studies on low-speed and high-power PMSM for mining. Although the combination of a PID algorithm with fuzzy control and neural network control algorithms can achieve the dynamic adjustment of PID parameters, the implementation process is usually cumbersome and requires a large amount of training to obtain precise parameter adjustment, making it difficult to achieve excellent control performance [14,15,16,17,18]. Similarly, although sliding mode variable structure control gets rid of the limitations of traditional PID control, there is a chattering problem when the control law is switched, which affects the control accuracy [19,20]. Consequently, the existing multi-PMSM control system for belt conveyance is deficient in several aspects, including an initial error of a substantial magnitude, a deficiency in precision in controlling the system, an inability to effectively mitigate the impact of external disturbances and a lack of clarity in the methodology employed for modifying the system’s parameters. In response to the shortcomings identified in the initial error, synchronization performance and disturbance rejection performance, this paper proposes a novel combination of the ADRC and speed regulation methodologies to enhance the system’s overall performance. An Enhanced first-order Active Disturbance Rejection Controller (Efal_ADRC) has been designed based on an optimized nonlinear function, with the objective of enhancing the disturbance immunity of the system. Furthermore, a weighted arithmetic mean is employed to reinforce the compensator of the ring coupling control structure. Through a comparative simulation analysis conducted in MATLAB(2019b)/Simulink, it is demonstrated that the strategy can effectively enhance the anti-interference capability, control precision and synchronization of the multi-motor drive system of the belt conveyor, which is conducive to the stable, efficient and safe operation of the belt conveyor.

2. Establishment of the Belt Conveyor Drive System Model

2.1. A Multi-Permanent Magnet Synchronous Motor Drive System Solution

This paper examines the use of a belt conveyor in an underground mine as its research object. In light of the limitations of a conventional multi-asynchronous motor drive system, an upgrade is proposed to a multi-permanent magnet direct drive system of equivalent power [21,22,23,24].

As illustrated in Figure 1, at the bottom of the belt conveyor, two permanent magnet synchronous motors jointly drive the same roller, and at the top, a PMSM drives the conveyor belt. This design increases the driving force at the bottom, so as to avoid the motor reversing phenomenon when starting with a large load, while also reducing the mechanical loss of the belt conveyor and improving the service life of the system. The fundamental characteristics of the enhanced belt conveyor are presented in

Table 1. Basic parameters of belt conveyor.

Physical Quantity	Parameter Size	Unit
Volume of traffic	2500	t/h
Transport distance	3000	m
Conveyor belt unit mass	67.2	kg/m
Unit mass of material	173.6	kg/m
Belt speed	4	m/s
Conveyor belt width	1400	mm
Rated power of motor	400	Kw

.

2.2. Mathematical Model of PMSM

In order to design the control strategy for the subsequent control system, it is first necessary to determine the model parameters of the PMSM. In order to simplify the analysis of the PMSM, it is necessary to take the PMSM as an idealized model, that is, one that meets the following conditions:

(1)

The core material is an ideal material, and there is no saturation;

(2)

The stator three-phase winding structure is symmetrical, and the current in the motor is a three-phase symmetric sine wave;

(3)

The above applies regardless of hysteresis loss and eddy current loss in the motor.

The structure diagram of the PMSM is shown in Figure 2. After coordinate transformation, the magnetic linkage and voltage equations of the PMSM are obtained:

$$\begin{array}{c}{\psi }_d=L_d{i}_d+{\psi }_f\\ {\psi }_q=L_d{i}_q\end{array}$$

(1)

$$\begin{array}{c}{u}_d=R{i}_d+L_d{\displaystyle \frac{d}{dt}}{i}_d-{\omega }_c{L}_q{i}_q\\ u_q=R{i}_q+L_q{\displaystyle \frac{d}{dt}}{i}_q+{\omega }_c(L_d{i}_d+{\psi }_f)\end{array}$$

(2)

The motor torque model is:

$$T_e=1.5P_n\left[{\psi }_f{i}_q+(L_d-L_q)i_q{i}_d\right]$$

(3)

In the formula, $T_e$ is the load torque; $L_d,L_q$ are each the inductance of the d-axis and q-axis, respectively; $P_n$ is the number of magnetic poles; $i_d,i_q$ are each the current of the d-axis and q-axis, respectively; and ${\psi }_f$ is the flux amplitude of the stator winding.

The research object of this article is a surface-mounted permanent magnet synchronous motor for mining [25], with $L_d=L_q$ and the rotational speed motion equation of the permanent magnet synchronous motor as follows:

$${\displaystyle \frac{dn}{dt}}={\displaystyle \frac{30}{\pi }}({\displaystyle \frac{-B_n}{J}}\omega +{\displaystyle \frac{-T_L}{J}}+{\displaystyle \frac{3P_n{\psi }_f{i}_q}{2J}})$$

(4)

In the formula, n is the motor speed; J is the load moment of inertia; $B_n$ is the damping coefficient of the motor; and $T_L$ is the load torque.

The motors under consideration in this paper are all of a particular brand of high-power PMSM, and the relevant parameters are presented in

Table 2. Parameters of permanent magnet synchronous motor.

Physical Quantity	Parameter Size	Unit
Number of pole-pairs Pn	16
Stator resistance R	0.0755	Ω
Stator d-, q-axis inductance Ld, Lq	6.53	mH
Rotor moment of inertia J	474	kg∙m2
Permanent magnet flux linkage φf	6.08	Wb
Damping coefficient B	0.002	N∙m∙s
Rated load torque TL	5 × 104	N∙m
Sampling time Ts	10−5	s
Dc voltage Vdc	1140	V
Rated power	400	Kw

.

3. Design of the First-Order Active Disturbance Rejection Controller

Given the inherent limitations of PID controllers, an ADRC can effectively compensate for their shortcomings. The theoretical basis of the ADRC is derived from both modern and classical control theories. The ADRC does not require the precise model parameters of the system, which greatly simplifies the construction of the system. Furthermore, the advantage of treating both internal and external disturbances as total disturbances for compensation can significantly enhance the anti-interference performance of the system [26].

3.1. Principle of the ADRC

The active disturbance rejection controller is comprised of three principal components: the tracking differentiator (TD), the extended state observer (ESO) and the nonlinear state error feedback (NLSEF) [27]. The most crucial component is the ESO, which is capable of treating system model errors resulting from changes in rotational inertia and external disturbances caused by alterations in load and tension as a unified disturbance for processing. In this article, an optimized first-order ADRC is employed to reduce parameters and enhance the synchronization performance of the system without compromising the control performance. The structure diagram of a typical first-order ADRC is depicted in Figure 3.

3.2. Optimization of Nonlinear Functions

The fundamental component of the ADRC is the ESO. A nonlinear function fal function has large gain and small error. When the system error is large, the fal function will reduce the gain to avoid drastic changes in the observer output due to excessive error, so that the observer can estimate the total disturbance of the system more smoothly. When the error is small, increasing the gain can estimate the disturbance more accurately, help to improve the observation accuracy of the system disturbance and then better compensate the disturbance and improve the anti-interference ability of the system. Moreover, fal function has better filtering performance than a first-order filter, and can suppress system buffeting to some extent. The greater the continuity of the nonlinear function, the more effective the controller performance. The most commonly utilized nonlinear function at present is $fal(e,\alpha ,\delta )$:

$$fal(e,a,\delta )\left\{\begin{array}{c}{\left|e\right|}^{a}sign(e)\\ {\displaystyle \frac{e}{{\delta }^{1-a}}}\end{array}\right.\begin{array}{ccc}& & \begin{array}{c}\left|e\right|>\delta \\ \left|e\right|\le \delta \end{array}\end{array}$$

(5)

In the formula, e is the observation error; $\alpha$ is a nonlinear factor; $\delta$ is the nonlinear interval; and $sign(·)$ is a symbolic function.

As illustrated in the preceding formula, the function’s curve at the piecewise point is not smooth and non-differentiable. Consequently, its control performance in the linear interval can be enhanced. To that end, a new nonlinear function, $Efal(e,\alpha ,\delta )$, is proposed in this paper for the interval $\left|e\right|\le \delta$, with the objective of improving the aforementioned control performance.

As illustrated by the preceding formula, the function’s curve at the segmentation point is not smooth and non-differentiable. Accordingly, in order to enhance the control efficacy of the controller, this paper proposes a novel nonlinear function to enhance its control performance.

Combined with the quadratic continuous function in the linear interval of the function, the new nonlinear function is designed as (6).

$$\left\{\begin{array}{c}Efal(e,\alpha ,\delta )=c_{1}e+c_2{e}^2+c_3\theta (e)\\ \theta (e)={\displaystyle \frac{e}{\left|e\right|+\epsilon }}\end{array}\right.$$

(6)

In the formula, $c_1,c_2,c_3$ are the term coefficients and $\epsilon$ is the normal number that tends to 0.

At this time, the $Efal(e,\alpha ,\delta )$ continuity and differentiability conditions must be met:

$$\left\{\begin{array}{l}Efal(e,\alpha ,\delta )={\delta }^{\alpha },e=\delta ;\\ Efa(e,\alpha ,\delta )=-{\delta }^{\alpha },e=-\delta ;\\ Efa{l}^{\prime }(e,\alpha ,\delta )=\alpha {\delta }^{\alpha -1},e=\pm \delta ;\\ Efa{l}^{\prime }(e,\alpha ,\delta )=1/{\delta }^{1-\alpha },e=0;\end{array}\right.$$

(7)

Solutions have to be as follows:

$$\left\{\begin{array}{l}{c}_1=(\alpha \delta +\alpha \epsilon -\epsilon ){\delta }^{\alpha -2}\\ c_2=0\\ c_3=(1-\alpha ){(\alpha +\epsilon )}^2{\delta }^{\alpha -2}\end{array}\right.$$

(8)

After a series of iterative optimization steps, the new nonlinear function has been obtained. The overall expression of $Efal(e,\alpha ,\delta )$ is as follows:

$$Efal(e,\alpha ,\delta )=\left\{\begin{array}{cc}{\left|e\right|}^{\alpha }sign(e)& \left|e\right|>\delta \\ (\alpha \epsilon +\alpha \delta -\epsilon ){\delta }^{\alpha -2}e+\\ {\displaystyle \frac{(1-\alpha ){(\delta -\epsilon )}^2{\delta }^{\alpha -2}e}{\left|e\right|+\epsilon }}& \left|e\right|\le \delta \end{array}\right.$$

(9)

In order to verify the control performance of the function, the parameters $\alpha =0.25$, $\delta =0.1$ and $\epsilon =0.05$ were taken in combination with the tuning experience of the ADRC. The comparison results were obtained by drawing the function curves of $fal(e,\alpha ,\delta )$ and $Efal(e,\alpha ,\delta )$ in MATLAB, as shown in Figure 4.

As illustrated in Figure 4, when compared with the traditional nonlinear function $fal(e,\alpha ,\delta )$, the $Efal(e,\alpha ,\delta )$ function is continuous and differentiable at the segmentation point, and has better control performance.

3.3. Design of the First-Order Active Disturbance Rejection Controller for Speed Loop

The initial design of the first-order ADRC of the system should be based on the aforementioned nonlinear function. In accordance with (3) and (4), the state-space expression of the rotational speed equation can be derived as follows:

$$\left\{\begin{array}{c}\dot{x}=a_v(t)+b_v{i}_q\\ y=x\end{array}\right.$$

(10)

In the formula, $\dot{x}$ is the system output signal $n$; $b_v$ is the adjustment coefficient of the controller $b_v={\displaystyle \frac{45P_n{\psi }_f}{J\pi }}$; and $a_v(t)$ is the comprehensive disturbance of the system $a_v(t)={\displaystyle \frac{30}{\pi }}({\displaystyle \frac{-B_n}{J}}\omega +{\displaystyle \frac{-T_L}{J}})$.

(1)

The second-order state observer ESO [28]:

$$\left\{\begin{array}{c}e=z_1-y\\ {\dot{z}}_1=z_2-{\beta }_{01}Efal(e,{\alpha }_1,{\delta }_1,\epsilon )\\ {\dot{z}}_2=-{\beta }_{02}Efal(e,{\alpha }_1,{\delta }_1,\epsilon )\end{array}\right.$$

(11)

In the formula, $z_1$ is the motor output speed n tracking signal; $z_2$ is the observed value of the system disturbance $a_v(t)$; and ${\beta }_{01}$, ${\beta }_{02}$ are each the disturbance factors and greater than 0.

(2)

First-order nonlinear error feedback control law NLSEF:

$$\left\{\begin{array}{l}{e}_1=v_1-z_1\\ u_0={\beta }_{1}Efal(e_1,{\alpha }_2,{\delta }_2,\epsilon )\\ u=u_0-z_2/b_v\end{array}\right.$$

(12)

In the formula, ${\beta }_1$ is the gain coefficient.

The overall structure diagram of a single PMSM speed regulation system based on the improved nonlinear function Efal_ADRC is presented in Figure 5.

4. Design of the New Compensator Based on Ring Coupling

4.1. Ring Coupling Control Principle

In a multi-motor collaborative control system, the ring coupling structure considers only the synchronization error between the real-time speed and the rated speed of each motor. Its feedback and compensation are performed only between two adjacent motors. Even if a large number of motors are involved, the control structure remains simple, making it more suitable for multi-motor collaborative control systems with multiple controlled motors [29,30]. The ring coupling control ensures that the system can follow a given signal consistently, and has strong synchronization and high-speed compensator operation. This ensures the following performance and anti-interference performance of the system starting process. It is currently an ideal multi-motor collaborative control strategy. The system control structure is shown in Figure 6.

As illustrated in Figure 6, the core of the multi-motor synchronous control system comprises three principal components: the motor drive system, the synchronous error compensator, and the controller. In response to the action of the synchronous error compensator, the detected error information is transmitted with precision to the corresponding motor. Concurrently, the multi-motor synchronous driver also receives this error signal in a synchronous manner. The compensation based on the error signal can enhance the synchronization and stability of the system, while reducing the disturbance of the motor speed caused by external forces. Figure 7 illustrates the three-motor ring coupling compensation structure [31].

It can be seen from its structure diagram that the tracking error of the i-th (i = 1,2,3) motor is:

$$e_i=n^{*}-n_i$$

(13)

The synchronization error between the i-th motor and the i + 1-th motor is defined as follows:

$${ee}_{i(i+1)}=n_i-n_{i+1}$$

(14)

The feedback through the ring-coupled compensator is:

$$e_{i(i+1)}=K_i{n}_i-n_{i+1}$$

(15)

In the formula, $n^{*}$ a is the system’s given speed; $n_i$ is the actual speed of the i-th motor; and $K_i$ is the compensation coefficient of the i-th motor.

4.2. Optimization of Compensator

The traditional ring coupling structure is susceptible to synchronization issues, which frequently result in delays in error feedback. Consequently, a design concept is proposed based on the original compensator, which adds dynamic factors to the speed compensator. This results in the design of a new compensator structure that meets the aforementioned requirements. The conventional compensator is only associated with the velocity of two proximate motors and possesses an error feedback delay. To mitigate this delay, the average velocity and dynamic error factors are introduced. The optimized structure diagram of the i-th motor compensator is depicted in Figure 8.

Figure 8 illustrates the operation of the optimized speed compensator. This device takes the average of the speeds of the two adjacent motors and the real-time speed of each motor, then calculates the difference between the speed of the i-th motor. The final step is to multiply the proportional coefficient in order to achieve real-time performance. This results in the optimized compensator, which is:

$$\left\{\begin{array}{l}{e}_{12}=K_1\left\{\left[{\omega }_1-{\displaystyle \frac{1}{2}}\left({\omega }_2+{\omega }_{ave}\right)\right]+\left[{\omega }_1-{\displaystyle \frac{1}{2}}\left({\omega }_3+{\omega }_{ave}\right)\right]\right\}\\ e_{23}=K_2\left\{\left[{\omega }_2-{\displaystyle \frac{1}{2}}\left({\omega }_3+{\omega }_{ave}\right)\right]+\left[{\omega }_2-{\displaystyle \frac{1}{2}}\left({\omega }_1+{\omega }_{ave}\right)\right]\right\}\\ e_{31}=K_3\left\{\left[{\omega }_3-{\displaystyle \frac{1}{2}}\left({\omega }_1+{\omega }_{ave}\right)\right]+\left[{\omega }_3-{\displaystyle \frac{1}{2}}\left({\omega }_2+{\omega }_{ave}\right)\right]\right\}\end{array}\right.$$

(16)

$$K_i={\displaystyle \frac{{\omega }_i}{2{\omega }_{ave}}}$$

(17)

In the formula, $e_1,e_2,e_3$ represent the speed compensation of each of the three motors, respectively; ${\omega }_{ave}$ is the average speed of the three motors; and $K_i$ is the speed compensation coefficient.

5. Simulation Experiments and Result Analysis

In order to verify the effectiveness of the proposed Efal_ADRC on the dynamic and static characteristics and anti-interference ability of the motor speed control system, this paper employs a three-motor system model. Figure 9 shows the single motor simulation model built on the basis of Figure 5. It includes the speed loop controller, current loop controller, display module, coordinate transformation module, decoupling module and so on. Figure 10 shows the simulation model diagram of the Efal_ADRC controller, built according to Figure 4 and Formulas (11) and (12).

On the basis of the above model, the simulation model of the three-motor control system based on the improved ring coupling control is built, as shown in Figure 11. Among the controllers employed, the PI controller is utilized in the current loop, while the traditional PI controller, ADRC, and Efal_ADRC are deployed in the speed loop. Simulation experiments were conducted to compare the master-slave structure, the deviated coupling structure, the ring coupling structure and the ring coupling structure of the improved compensator. The objective was to ascertain the superiority of Efal_ADRC-based ring coupling control in comparison to alternative control methodologies.

In order to reduce the difficulty of tuning parameters in the ADRC controller, the tuning of controller parameters is typically guided by certain rules. In the case of the second-order ESO, the undetermined parameters can be set as: ${\beta }_{01}=2{\omega }_0,{\beta }_{02}={{\omega }_0}^2$, while for the third-order ESO, the undetermined parameters can be set as: ${\beta }_{01}=3{\omega }_0$, ${\beta }_{02}=3{{\omega }_0}^2$, ${\beta }_{03}={\omega }_0^3$. Based on these settings, the ESO bandwidth in the speed loop is determined to be 280, and the controller parameters are shown in

Table 3. Efal_ADRC controller parameters.

r	h	${\mathit{\beta }}_1$	${\mathit{\beta }}_2$	${\mathit{\beta }}_{01}$	${\mathit{\beta }}_{02}$	${\mathit{\beta }}_{03}$
5000	0.001	20,000	1000	$3{\omega }_0$	$3{{\omega }_0}^2$	${\omega }_0^3$

. Concurrently, the parameters of the PI controller must be set as follows: Kp = 30 and Ki = 200 for the speed ring, and Kp = 20 and Ki = 150 for the d- and q-axis current loop.

5.1. Single Motor Speed Loop Control Simulation

The motor speed, n = 75 r/min, is set and the motor is started without a load. The simulation time is set to 15 s. After applying $T_L=5000$ $\mathrm{N}\cdot \mathrm{m}$ of load at 3 s, a $2000$ $\mathrm{N}\cdot \mathrm{m}$ load is subtracted at 5 s and a $3000$ $\mathrm{N}\cdot \mathrm{m}$ load is subtracted at 8 s. The simulation results for the Efal_ADRC, ADRC, and traditional PI controller are presented in Figure 12.

Figure 12 illustrates that, under no-load starting conditions, the motor based on the PI controller and the traditional ADRC controller exhibits a significant overshoot when starting, requiring a longer time to reach a stable rated speed. In contrast, the motor with the Efal_ADRC controller exhibits minimal overshoot, reaching a stable rated speed in 0.25 s. Furthermore, the motor output speed and speed error based on the Efal_ADRC controller exhibit smaller overall fluctuations. When the load is increased or decreased, the Efal_ADRC control exhibits smaller fluctuations than both ADRC control and PI control, demonstrates stronger anti-interference ability and has a shorter recovery time after being disturbed.

In conclusion, the control strategy based on the Efal_ADRC controller exhibits superior control performance. Consequently, this article will investigate the coordinated control strategy for multi-permanent magnet synchronous motors based on the Efal_ADRC controller.

5.2. Analysis of No-Load Operating Conditions

The no-load operation of the belt conveyor only needs to consider the quality of the conveyor belt. The load torque under the no-load condition is set to $T_L=5000$ $\mathrm{N}\cdot \mathrm{m}$. The resulting speed, torque, and speed difference for each motor are presented in Figure 13, Figure 14 and Figure 15.

As the system is not operating in real-time synchronization, it is necessary to acknowledge that the rotational speeds of the three motors will differ in actual operation. Figure 13, Figure 14 and Figure 15 illustrate that, under no-load conditions, the Efal_ADRC controller reaches stability within 0.25 s, while the ADRC controller reaches stability at 0.65 s. The maximum speed difference of the Efal_ADRC controller is less than 0.5 r/min, approaching 0, and its steady-state error fluctuation is minimal. It can thus be concluded that the three motors under the Efal_ADRC controller exhibit superior speed and stability under no-load conditions.

5.3. Analysis of Full-Load Operating Conditions

In order to study the operating characteristics of the control strategy designed in this article under full-load conditions, a simulation is conducted to model the heavy-load starting condition of the belt conveyor when it restarts after a shutdown accident due to a fault. It is assumed that the load torque at full-load starting is equal to the rated load torque $T_L=50,935$ $\mathrm{N}\cdot \mathrm{m}$. The output motor response curve under heavy-load starting conditions is presented in Figure 16, Figure 17 and Figure 18.

Figure 16 and Figure 17 demonstrate that, under full-load conditions, all three motors have reached a stable operating state. The ADRC controller exhibits a larger motor speed overshoot, yet its output electromagnetic torque fluctuation is smaller. In contrast, the Efal_ADRC controller’s motor speed experiences a slight overshoot, with a relatively fast time to reach stable operation and minimal output electromagnetic torque fluctuation during operation. As illustrated in Figure 18, after stabilization, the speed errors of the three motors under Efal_ADRC control are all within ±0.1 r/min, with the exception of the startup error, which was 1.7 r/min. This indicates that the stability performance is superior.

5.4. Analysis of Variable Load Operating Conditions

Given that the load on the belt conveyor is always in a state of flux, with considerable variation, this article sets the variable load to oscillate between no load, heavy load, and light load. The load torque is as follows:

Table 4. Load torque under variable load conditions.

Time	Operating Condition	Load Torque TL
0–1.3 s	no load	$5000$ $\mathrm{N}\cdot \mathrm{m}$
1.3–2 s	heavy load	$\mathrm{50,000}$ $\mathrm{N}\cdot \mathrm{m}$
2–2.5 s	light load	$\mathrm{41,000}$ $\mathrm{N}\cdot \mathrm{m}$
2.5–3 s	light load	$\mathrm{32,000}$ $\mathrm{N}\cdot \mathrm{m}$

. The motor response speed curves under variable output load operating conditions are obtained as shown in Figure 19, Figure 20 and Figure 21.

As illustrated in Figure 19 and Figure 20, the motor speed overshoot based on the ADRC controller is considerable, and the adjustment time is prolonged when subjected to load impact. The output electromagnetic torque of the motor exhibits a pronounced fluctuation during initial startup, but this diminishes significantly as the motor stabilizes. The motor, when operated under the Efal_ADRC controller, reaches the rated speed in a remarkably brief period with minimal overshoot. During operation, the output electromagnetic torque fluctuation is relatively stable, and the adjustment time is the shortest when subjected to load impact.

Figure 21 illustrates that when the system is subjected to significant disturbances, the maximum motor speed difference controlled by the Efal_ADRC is e = −3 r/min, which is maintained within 0.2 r/min at other times. In contrast, the speed difference controlled by the ADRC exhibits considerable fluctuations.

6. Conclusions

The present study considers three permanent magnet synchronous motors as the research objects. In order to enhance the synchronization performance and anti-interference capability of multiple motors, the optimized nonlinear function is employed to refine the ADRC. Concurrently, a novel circular coupling synchronous speed compensator is devised with the objective of reducing the synchronous speed error of the system. A synchronization control strategy for multi-permanent magnet synchronous motors in mining belt conveyors based on the Efal_ADRC is put forth, and a simulation model of a three-PMSM speed synchronous system is constructed in MATLAB. The simulation results demonstrate the following:

(1)

Based on the high-power PMSM ring coupling control strategy under improved Efal_ADRC control, smooth operation at low speed and high torque is achieved, and the motor has faster response speed, smaller overshoot, stronger anti-disturbance ability, and dynamic responsiveness. The motor’s ability to follow the reference speed is improved.

(2)

The new synchronous speed compensator exhibits superior synchronization performance in comparison to the traditional speed compensator, thereby reducing the synchronization error between motors, which in turn reduces motor losses and extends the operational lifespan.

In conclusion, the Efal_ADRC-based ring coupling control, which is the subject of this study, represents an improvement over the master-slave control commonly used in mining belt conveyors. It enhances the stable ore transport efficiency of the conveyor, enables it to operate smoothly over long distances and under heavy loads, reduces the incidence of transportation accidents and maintenance costs in mines and extends the service life of the conveyor. Nevertheless, the parameters of the controller remain difficult to adjust, and there is still a discrepancy between the simulation and the actual working conditions. In the future, the incorporation of intelligent algorithms may enhance the precision of the controller parameters and validate the actual operational conditions. In light of the above, the research presented in this paper is of significant value and importance with regard to the multi-motor drive system of the mine belt conveyor.