#### Abstract

Aiming at the problems existing in the method of using sensors to detect the rotor position of marine main propulsion motor at low speed, a sensorless rotor position identification method based on high-frequency pulse vibration signal is proposed. In this method, the low-pass filter is used to separate the high-frequency signal component, and the current component of d-axis to q-axis is demodulated. The control mode of PI action law is used to phase-locked processing of the demodulated signal, in which purpose is to obtain the rotor position signal. The design method is simulated and verified by MATLAB/Simulink software. The error between the estimated position and the actual position of the rotor in the simulation waveform is small, and the accuracy is high. The simulation results show that this method can achieve good results when applied to the ship electric propulsion system without position sensor, which provides a theoretical reference for the design of marine main propulsion motor control system.

#### 1. Introduction

Because the electric propulsion system has the advantages of energy saving, environmental protection, good maneuverability, high propulsion efficiency, and saving ship engine room space, and under the global “double carbon” background, more and more ships adopt electric propulsion system.

Ship electric propulsion system is mainly composed of main propulsion motor, transmission device, and propeller. The main propulsion motor drives the propeller to rotate and receives the reaction force of water; that is, it has thrust on the ship. The thrust mainly depends on the unit time flow of water; the greater the flow, the greater the thrust. Due to the ship's dead weight and heavy cargo, and the ship’s low speed when berthing and leaving the wharf and passing through narrow waterways, the main propulsion motor often uses the high-power synchronous motor as the prime mover to tow the propeller. In order to reduce the weight of the mechanical and electrical equipment on the ship and save the cabin space of the ship, the permanent magnet synchronous motor with small volume and light weight is mostly used as the marine main propulsion motor.

For the control mode of permanent magnet synchronous motor, vector control may not be suitable for applications requiring a very high dynamic response. DTC suffers from high torque and flux ripple and variable switching frequency. In the current study, various advanced methods are investigated to replace and improve traditional schemes, such as model predictive control (MPC). An improved MPC scheme is proposed, which combines a virtual vector expansion scheme and duty cycle control [1]. This scheme improves steady-state performance while having a faster dynamic response than VC. An MPC compensation method based on virtual voltage vectors is proposed for the single-phase open-circuit fault of six-phase permanent magnet synchronous in-wheel motors [2], which avoids the need for complex scheme reconstruction when using VC and DTC. One new method adopts a hybrid control strategy to achieve smooth switching from low speed to high speed [3]. The high-frequency signal injection method is used to effectively extract the back EMF signal at a low speed, and then, the FPS-PLL with high accuracy and robustness is used at high speed. The traditional vector control is replaced by the more advanced model predictive control [4]. The combination of MPC and FPS-PLL makes the controller structure simple and has better dynamic stability performance.

The traditional vector control needs to install encoder or Hall element on synchronous motor to detect the speed of synchronous motor rotor [5]. However, due to the high humidity of the ship sailing on the water, the detection elements are easy to be damaged by moisture [6], which will not only have a certain impact on the maintenance and stability of ship electric propulsion system but also increase the maintenance cost and use cost. Using the detection method without position sensor instead of position sensor on board can not only achieve the function of detecting motor speed but also avoid the risk of equipment failure. The IF control mode without position sensor mostly adopts current single closed loop [7], but the efficiency is low. The synovial observer method is a speed sensorless implementation method. However, due to the low back EMF at zero speed and low speed of the main propulsion motor [8], it is difficult to accurately estimate the rotor position by using the observer based on back EMF. The high-frequency signal injection method uses the nonideal characteristics of the motor to estimate the rotor position information [9, 10], and it does not involve physical quantities constrained by speed, such as back EMF, which also has good performance at zero speed and low speed. This method has been well applied in the low-speed operation of switched reluctance motor [11, 12], However, there are few reports on the low-speed control of high-power ship’s main propulsion motor when berthing and leaving the wharf, crossing narrow waterways and other special waters.

In order to solve the above problems, this paper uses the high-frequency pulse vibration injection method combined with the control mode without position sensor. A sensorless control mode of pulse high-frequency injection based on PLL rotor position observer is designed. Referring to the current sampling frequency and sampling accuracy commonly used in engineering, the amplitude and frequency of the injected voltage signal are reasonably set, which is to suppress the pulsation at low speed caused by nonideal factors such as dead time effect and current/voltage measurement error. Thus, the speed transition of the ship at low speed is smooth, and the comfort of the staff and passengers on board is improved.

#### 2. Mathematical Model of Marine Main Propulsion Motor

##### 2.1. Modeling Assumptions

In order to simplify the expression of the mathematical model, this paper idealizes the built-in salient pole permanent magnet main propulsion motor; that is, it is assumed that the marine main propulsion motor is an ideal motor and meets the following conditions:(1)The three-phase windings of the motor stator are symmetrical in space, with a mutual difference of 120 degrees.(2)The three-phase current of the input stator winding is sinusoidal AC, regardless of the influence of harmonics.(3)The magnetic saturation of the motor core should be ignored.(4)The damping effect of permanent magnet, and eddy current loss and hysteresis loss of motor are not considered.

##### 2.2. Mathematical Model

###### 2.2.1. Fundamental Stator Voltage Equation in Synchronous Rotating Coordinate System

In the formula (1), the letter *u*_{d} and *u*_{q}, respectively, represent the stator voltage components on the *d* and *q* axes in the fundamental wave state; the letter *R*_{s} represents stator resistance; the letter *i*_{d} and *i*_{q}, respectively, represent the stator current components on the *d* and *q* axes; the letter *L*_{d} and *L*_{q}, respectively, represent inductance components on the *d* and *q* axes; the letter *ω*_{e} represents the fundamental angular velocity; the letter *ψ*_{f} represents permanent magnet flux linkage; the letter *D* represents differential operator.

###### 2.2.2. Electromagnetic Torque Equation

In the formula (2), the letter *T*_{em} represents electromagnetic torque, and the letter *p* represents motor pole pairs.

###### 2.2.3. Equation of Motion

In the formula (3), the letter *T*_{L} represents load torque, and the letter *J* represents moment of inertia.

The stator voltage equation and flux linkage equation in *α-β* static coordinate system can be obtained from formula (1) and coordinate transformation method.

In the formula (5), *L* = (*L*_{q} + *L*_{d})/2; Δ*L* = (*L*_{d} − *L*_{q})/2, the letter *θ*_{r} represents the angle of rotor position.

#### 3. High-Frequency Pulse Voltage Injection

##### 3.1. Principle Description

If the high-frequency voltage or current signal is injected into the stator side of the motor, a magnetic field with corresponding frequency of this high-frequency signal inside the motor will be generated. The motor itself has the nonideal characteristics of rotor structure, such as salient polarity and rotor saturation, and this nonideal characteristic can modulate the high-frequency magnetic field. High-frequency voltage signals or current signals related to rotor speed and rotor position are generated on the stator side, and the position information of the motor rotor can be obtained by processing the voltage signal or current signal accordingly.

Pulse high-frequency voltage injection is to inject high-frequency sinusoidal voltage signal on the direct axis *d* of synchronous rotating coordinate system. The high-frequency signal will form a high-frequency pulse voltage signal in the static coordinate system. The signal frequency is fixed. The high-frequency current signal of axis *q* can be obtained through the band-pass filter [13–15]. After the signal is amplitude modulated, the information related to the rotor position can be obtained through the position estimator, so as to obtain the rotor speed. The principle is shown in Figure 1.

In Figure 1, the module “ABC/d^{/}-q^{/}” represents a coordinate conversion module, the three-phase current in the three-phase coordinate system is transformed into the current in the d-q coordinate system, and the current component on the d-axis can be obtained. The high-frequency response current signal represented by *i*_{qh} on the q-axis can be obtained by using the band-pass filter (BPF) to filter the carrier signal and the frequency selective output fundamental signal. The letter *i*_{qhe} represents the high-frequency response signal *i*_{qh} multiplied by the modulation signal sin(*ω*_{h}*t*). Then, the high-frequency component in the signal is filtered by the low-pass filter LPF, and the rotor position error signal *f*(*θ*_{e}) can be obtained. The position information represented by the letter *θ*_{est} in Figure 1 of the magnetic pole axis of the permanent magnet on the rotor can be estimated by phase-locked processing of the signal represented by the letter *f*(*θ*_{e}) in Figure 1. The letter *ω*_{est} represents the estimating fundamental angular frequency.

##### 3.2. Stator Voltage Equation under Pulse High-Frequency Voltage Injection

Generally, the frequency range of high-frequency voltage injection signal is 0.5 kHz∼2 kHz, which is much higher than the fundamental frequency. Under high-frequency signal, permanent magnet synchronous motor can be equivalent to a simple RL circuit, and the reactance value is much greater than the resistance value. Therefore, formula (1) can be rewritten as follows:

In formula (6), the letter *u*_{dh}, *u*_{qh}, *i*_{dh}, and *i*_{qh,} respectively, represent the stator voltage and current components on *d* and *q* axes under the condition of high-frequency signal injection. In this state, the expression of stator inductance can be expressed by the following formula:

According to the conversion relationship between two-phase rotating coordinates and two-phase stationary coordinates, formula (7) can be expressed as formula (8) in two-phase stationary coordinates.

In formula (8), the letter *θ*_{e} represents the rotor position angle.

In order to accurately estimate the rotor position, a new coordinate named d^{/}-q^{/} synchronous rotation coordinate system is established for estimating the rotor position. The relationship between the d^{/}-q^{/} synchronous rotation coordinate system and the actual d-q synchronous rotation coordinate system is to be analyzed.

##### 3.3. Current Response under High-Frequency Voltage Excitation of Pulse Vibration

The d^{/}-q^{/} synchronous rotation coordinate system and the actual d-q synchronous rotation coordinate system are shown as Figure 2.

In Figure 2, the d-q is the actual synchronous rotation coordinate system of the rotor, and d^{/}- q^{/} is the estimated synchronous rotation coordinate system of the rotor. *α*-*β* is two-phase stationary coordinate system. The letter *θ*_{e} represents the actual rotor position angle, and the letter *θ*_{est} represents the estimated position angle of the rotor. Then, the estimated angle error of rotor represented by the letter △*θ* can be expressed by the following formula:

From Figure 2 and formula (9), the relationship between voltage of high frequency and current of high frequency in d^{/}—*q*^{/} coordinate system can be expressed as following formula:

In formula (10), , the letters *i*_{d}^{/}, *i*_{q}^{/}, *u*_{d}^{/}, and *u*_{q}^{/}, respectively, represent the components of high-frequency current and high-frequency voltage on d^{/}-axis and q^{/}-axis in d^{/}—*q*^{/} coordinate system.

Pulse high-frequency voltage injection method is to inject high-frequency voltage signal into d^{/}-axis of the estimated synchronous rotation coordinate system.

In formula (11), the letter *V*_{h} represents the amplitude of the injected high-frequency voltage signal, and the letter *ω*_{h} represents the frequency of the injected high-frequency voltage signal. Bring formula (11) into formula (10), the current component of the estimated synchronous rotation coordinate system can be obtained and shown as the following formula:

It can be seen from formula (12) that as long as △L ≠ 0, that is, *L*_{d} ≠ *L*_{q}, the components of high-frequency current signal on d^{/}-axis and q^{/}-axis are related to the estimated angle error of rotor represented by the letter △*θ*. And if the estimated angular error is equal to zero (△*θ* = 0), the component of the high-frequency current signal on the q^{/}-axis is equal to zero (*i*_{q}^{/} = 0). Therefore, the estimated angle error represented by △*θ* can be obtained by processing the component *i*_{q}^{/} of the high-frequency current signal on the q^{/}-axis, which made the position of the rotor can be obtained.

#### 4. Rotor Pole Axis Position Estimation Based on Phase-Locked Loop

In order to realize the accurate estimation of the rotor pole axis position and achieve the purpose of zero static deviation, this paper uses PLL system combined with PI regulator to obtain the position angle of motor rotor. Its principle is shown in Figure 3.

In Figure 3, the method of obtaining the rotor position is to use the band-pass filter BPF to filter the frequency-selection basic signal and carrier signal. The d-axis current response signal *i*_{d} can be obtained by coordinate transformation, which is multiplied with the modulation signal represented by sin(*ω*_{h}t) and filtered the high-frequency component through the second-order low-pass filter LPF [16–18], and the rotor position error signal represented by *f*(△*θ*) can be obtained. Then, the rotor error signal can be processed by PLL to obtain the position of rotor magnetic pole axis.

The position error signal is expressed as the following formula:

The amplitude and phase of the current response signal will be changed after filtering, which will increase the position error. In order to reduce error, the low-pass filter LPF is set to be second order. And the transfer function of the second-order low-pass filter LPF in Figure 3 is expressed as the following formula:

In formula (14), the letter *ω*_{L} represents the natural oscillation frequency of low-pass filter LPF, and the frequency is set to 50 Hz. The letter *ξ* represents the relative damping coefficient, and the value is . Thus, the phase shift angle of the signal generated by the filter can be obtained through the phase-frequency characteristics, and the value is 1.63°.

The transfer function of PI regulator in Figure 3 is shown as the following formula:

In formula (15), the letter *k*_{p} and *k*_{i}, respectively, represent the magnification and integral gain of the regulator. In Figure 3, the transfer function of PI regulation closed loop is shown as the following formula:

In order to facilitate the tuning of the regulator parameters, three extreme points of formula (16) need to be arranged uniformly.

#### 5. Simulation

According to the fact that the synchronous motor has no self-starting ability, during the simulation, the given load torque is zero, and it is assumed that the resistance torque of no-load starting is also zero. The basic parameters in the simulation process are shown in Table 1.

In MATLAB/Simulink simulation environment, the Simulink simulation module diagram of rotor position tracking control system based on high-frequency pulse voltage injection method is shown in Figure 4.

In Figure 4, the letter *V*_{h} represents the amplitude of high-frequency voltage of pulse vibration and is set as 22V. The pulse frequency is set as 1100 Hz. The instantaneous expression is shown as formula.

The low-pass filter LPF in Figure 3 adopts Butterworth second-order filter, and the transfer function is shown as formula (14). The corresponding pass band edge angular frequency is set to 940 rad/s. The band-pass filter BPF adopts Butterworth second-order filter, and the corresponding low-pass band edge angular frequency is set to 6845 rad/s, and the high-pass band edge angular frequency is set to 6970 rad/s. The parameters of the three PI regulators in Figure 3 are shown in Table 2.

##### 5.1. Simulation of Constant Speed

###### 5.1.1. Current Simulation Waveform

The research object of this paper is salient pole synchronous motor with uneven air gap between stator and rotor. The simulation results of current component waveform of stator current in d-axis and q-axis are shown in Figures 5 and 6.

In this control system, the regulator with PI action law is adopted for the component of stator current on *d*-axis and *q*-axis, such as PI1 and PI2 in Figure 4. Both regulators adopt integral link to realize the function that the static deviation is equal to 0, so that the component of stator current in *d*-axis and *q*-axis can be basically stable. After stabilization, the amplitude of d-axis current component in Figure 5 is about 0.35 A, and the stable value of q-axis current component in Figure 6 is about 12.5 A.

###### 5.1.2. Speed Simulation and Analysis

In order to simulate the navigation environment of a real ship passing through a narrow waterway, berthing and leaving the wharf, the speed of the main propulsion motor is set as the operation state of low-speed working condition in this simulation process. In Figure 4, the given value of the speed of the main propulsion motor is set as 110 r/min. The simulation results are shown in Figure 7.

In the simulation results, in order to facilitate the analysis of the relationship between the actual speed value and the estimated value, the actual value waveform width is set to 5 and the estimated speed waveform width is set to 2. It can be seen from Figure 7 that the waveform of the estimated speed basically coincides with the waveform of the actual speed, reaches the minimum value at the same time, and tends to a stable state at the same time. Therefore, the design method in this paper is reasonable and effective from the waveform change trend of estimated speed and actual speed.

###### 5.1.3. Simulation and Analysis of Speed Estimation Error

The simulation waveform of speed estimation error is shown in Figure 8.

From the simulation waveform results of speed estimation error in Figure 8, it can be seen that the error value fluctuates greatly in the process of speed rise; when the speed tends to be stable, the estimation error becomes smaller and smaller, and gradually tends to be stable. It can be seen from Figure 7 that the stable speed estimation error is about 0.007 r/min, which is small enough to approach 0 and basically negligible. Therefore, for the speed estimation error, the simulation model built in this paper is correct, and the design method is reasonable and effective.

###### 5.1.4. Simulation and Analysis of Rotor Position Estimation

The comparison between the estimated value of rotor position and the actual value of rotor position is shown in Figure 9 from the simulation module. As can be seen from Figure 9, the estimated value of rotor position is basically consistent with the actual value of rotor position.

The variation curve of rotor position estimation error is shown in Figure 10. It can be seen from Figure 10 that the error value is very small, in the order of ten thousandth of an order of magnitude, and is about equal to 0, which can be ignored.

Figures 5–10 show the simulation waveform corresponding to the constant speed of 110 r/min. The simulation waveform is reasonable, and the change trend is in line with the theoretical analysis. The simulation results reflect the correctness of the mathematical model. The following is the simulation analysis of the sudden change process of speed.

##### 5.2. Simulation of Sudden Change of Speed

Sudden change of speed is a frequent process when a ship is maneuvering navigation. In Figure 4, the speed setting changes from 110 r/min to 130 r/min at 0.05 s. The corresponding simulation results of the system are shown in Figures 11–14.

###### 5.2.1. Current Simulation Waveform

The variation waveform of stator current component on *d*-axis and *q*-axis during sudden change of speed is shown in Figures 11 and 12.

It can be seen from the d-axis current waveform in Figure 11 that the d-axis current increases slightly after the sudden increase of speed, and the amplitude of the d-axis current increases from 0.35 A before the sudden change to 0.4 A after the sudden change. The q-axis current waveform in Figure 12 can be read out. After the sudden increase of speed, the q-axis current increases slightly after stabilizing, and the amplitude increases from 12.5 A before the sudden change to 14.5 A after the sudden change.

###### 5.2.2. Speed Simulation Analysis

It can be seen from Figure 13 that during the sudden change of speed, the waveform of the actual value of speed and the estimated value of speed basically coincides, reaches the minimum value at the same time, and tends to a stable state at the same time.

It can be seen from Figure 14 that the error increases slightly when the speed changes suddenly. When the speed in Figure 13 gradually stabilizes, the error waveform gradually stabilizes. The error size is basically the same as that in Figure 8 before the sudden change of speed. It can be seen from the figure that the estimation error will not change with the sudden change of speed, which further proves that the design of this paper is reasonable and effective.

###### 5.2.3. Simulation and Analysis of Rotor Position Estimation

It can be seen from Figure 15 that the waveform of the actual position of the rotor and the estimated position of the rotor during the sudden change of speed is basically consistent with that in the constant speed state. The waveform reflected in Figure 16 is consistent with the waveform trend in Figure 15. It can be seen from Figure 16 that under the condition of sudden change of speed, the estimation error of rotor position is basically consistent with the estimation error of constant speed in Figure 10. That is, the estimated position of the rotor and the estimation error of the rotor position do not change due to the sudden change of the speed, which are consistent with the change waveform when the speed is constant. Thus, the rationality and effectiveness of the mathematical model and control method are proved.

From the simulation results of the above two different working states, it can be seen that under the two different working conditions of constant speed and sudden change of speed, the estimated waveform of rotor speed and the estimated waveform of rotor position tend to coincide under the two different working conditions. Under two different working conditions, the waveform of rotor speed estimation error and the waveform of rotor position estimation error are basically the same. The simulation conclusion is that the sudden change of speed has little effect on the speed estimation error and rotor position estimation error, which are in the order of one thousandth, and can be almost ignored. This proves the correctness and effectiveness of the system design in this paper.

#### 6. Conclusion

In order to avoid the influence of installing position sensor on motor design and system stability, this paper applies the high-frequency pulse vibration injection method to the rotor position estimation and calculation of ship main propulsion motor under low-speed navigation conditions. In this paper, the principle of estimating rotor position by high-frequency pulse vibration is analyzed, and PI controller is used. Combined with the current sampling accuracy and frequency in engineering practice, the specific parameters of high-frequency pulse vibration method are set. The corresponding simulation module is built by using MATLAB/Simulink, and the corresponding parameters are simulated and calculated. The simulation results show that the design method in this paper is reasonable and effective, has fast response speed and good stability, and can be better used for the control of marine main propulsion motor at low speed.

#### Data Availability

All data generated or analyzed during this study are included in this article, and the data that support the findings of this study are available from the corresponding author upon reasonable request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The study was supported by the National Natural Science Foundation of China.