Volume 16, Issue 16 p. 2724-2738
ORIGINAL RESEARCH
Open Access

If closed-loop starting strategy of high-speed PMSM based on current vector adaptive regulation

Guxuan Xu

Corresponding Author

Guxuan Xu

Institute of Electrical Engineering Chinese Academy of Sciences, Beijing, People's Republic of China

University of Chinese Academy of Sciences, Beijing, People's Republic of China

Correspondence

Guxuan Xu, Institute of Electrical Engineering Chinese Academy of Sciences Beijing 100190, People's Republic of China.

Email: [email protected]

Contribution: Conceptualization, Data curation, Formal analysis, ​Investigation, Methodology, Software, Writing - original draft, Writing - review & editing

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Feng Zhao

Feng Zhao

Institute of Electrical Engineering Chinese Academy of Sciences, Beijing, People's Republic of China

Contribution: Funding acquisition, Validation

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Tao Liu

Tao Liu

Tiangong University, Tianjin, People's Republic of China

Contribution: Supervision, Validation, Writing - review & editing

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First published: 23 October 2023

Abstract

Conventional I–f startup control algorithms often lead to significant speed fluctuations, extended convergence times, and sluggish dynamic responses when the motor is working at the acceleration process. To address these challenges, the article constructs a discrete kinetic model of the motor, extracts and summarizes the change law of the motor torque angle fluctuation amplitude and frequency with the motor parameters and the reference current vector during I–f starting process based on the Runge–Kutta method, reveals the fundamental mechanism of the pole slipping of the I–f control. On this basis, the article proposes a closed-loop I–f starting control strategy for high-speed permanent magnet motors based on current vector adaptive regulation, The strategy notably improves speed tracking performance and current dynamic response to I–f startup control. Experimental results validate the efficacy of the proposed algorithm.

1 INTRODUCTION

As the central component of the fuel cell air system, its performance directly influences the energy conversion efficiency of the cell system and the overall performance of hydrogen fuel cell vehicles. The development trend is toward a high flow-to-pressure ratio, high dynamic performance, and significant dependability [1-3]. The ultra-high-speed permanent magnet motor has become the standard drive motor for centrifugal air compressors because of its high power density, compact design, superior transmission efficiency, prolonged durability, and rapid dynamic response [1, 4, 5]. Most centrifugal compressors operate at speeds exceeding 150,000 rpm, which complicates the use of position sensors with frequency ranges up to 2500 Hz. Therefore, sensorless control has become a necessity for high-speed permanent magnet motor drives [6]. Due to the characteristics of high-speed motors, such as the absence of a saliency effect and the small amplitude of the back EMF at low speeds [7, 8], there is a pressing need to transition quickly to an idle speed to elevate compressor pressure, especially for centrifugal air compressor applications. Hence, devising a method for a rapid and stable start of high-speed motors without sensors remains a significant research challenge [9, 10].

In recent years, research on low-speed startup technology for permanent magnet synchronous motors (PMSM) without position sensors has deepened, and I–f startup control is increasingly adopted for ultra-high-speed PMSM with surface-mounted structures [11]. The application of the I–f startup method to PMSM was first introduced by Professor Marius Fatu of Aalborg University [12]. The fundamental control strategy supplies a rotating reference current vector within the synchronous rotating coordinate system, where the current vector amplitude remains constant, and its rotating speed aligns with the I–f control reference speed [13]. However, the conventional open-loop I–f starting control strategy has a limitation: the current amplitude and frequency cannot be automatically adjusted, resulting in large fluctuations in output torque, acceleration, and speed during the dynamic process [14]. Especially for centrifugal air compressor applications, once the motor starts, it is necessary to rapidly accelerate to the idling state, and the angle fluctuations can induce pole slipping. Pole slipping denotes an instability condition when the generated torque is insufficient to maintain the rotor in synchronism with the rotating stator magnetic field. This leads to a pulsating torque on the rotor, resulting in severe acceleration and deceleration. Finally, the motor halts. Unsuitable reference current vectors and accelerations can induce pole slipping during open-loop If startup control.

To mitigate acceleration fluctuations in open-loop I–f startup, a closed-loop I–f control is proposed to prevent pole slipping and reduce speed oscillations. Frede Blaabjerg provided an upper limit for the rotational speed selection of the current vector for stable I–f control based on the principle of constant average torque in a single control cycle [15]. In [16], the prerequisites that must be met by the reference current vector and motor parameters for stable operation during I–f starting process are discussed. In [17], a mathematical derivation and stability analysis are conducted for the open-loop control system of PMSM, considering the constraints set by the vector and motor parameters. In addition, research indicates that real-time speed fluctuations can be reduced by making closed-loop adjustments to the current vector amplitude and frequency during I–f startup control. In [18], acceleration is regulated through reactive power, using the voltage-current vector angle to indirectly measure torque angle changes to adjust the current amplitude. In [19], the feedback of the disturbance component of the motor activation active power is employed to increase the system damping torque. In [20], the high-frequency component of the active power is used to increase the damping torque of the system by feeding back the disturbance. It also compensates the reference rotational speed of the current vector by assessing the load variations, aiming to achieve stable operation under sudden loads in a steady state. In [21], an offline self-learning method was adopted to obtain a no-load torque angle curve for online position angle compensation, which can reduce the risk of the pole slipping under shock loads.

The aforementioned studies identify the primary causes for pole slipping during I–f startup control: mostly owing to a mismatch between the reference acceleration and current parameters. However, the fundamental mechanism behind the pole slipping in I–f startup control has not been addressed. Therefore, the dynamic performance of the I–f starting strategy cannot be fundamentally improved. These elements have not received sufficient attention:
  1. Most enhanced algorithms cater solely to constant acceleration modes, neglecting validation for alternative acceleration modes, such as centrifugal air compressor applications, whose acceleration continues rising during the starting process.

  2. The reasons for high-frequency torque angle fluctuations and the fluctuation pattern have not been systematically investigated, resulting in unclear mechanisms of pole slipping during I–f startup control.

In this article, the motor I–f starting strategy in variable acceleration mode is investigated. Based on the motor dynamics model, the numerical Runge–Kutta solution method is used to extract the fluctuation laws of the actual speed, acceleration, torque angle, and motor parameters. The influence relationship between the fluctuation frequency and amplitude of torque angle with the reference current vector and motor parameters is analysed. This provides a theoretical basis for selecting adaptive filter parameters to extract the high-frequency component of the instantaneous power, implements adaptive adjustments to the I–f closed-loop control parameters according to the reference acceleration and speed during the acceleration interval, and enhances motor speed convergence rate and system stability effectively. Experimental findings demonstrate that the control strategy has good dynamic performance in both fixed and variable acceleration mode, as well as robustness against unexpected acceleration process.

2 MATHEMATICAL MODEL OF PMSM UNDER I–F CONTROL

The paper introduces a virtual dv-qv current vector coordinate system with rotor flux orientation to investigate current vector characteristics of the high-speed SPMSM during I–f startup control. As shown in Figure 1, the dv axis is the direct axis in the virtual coordinate system, the qv axis is the quadrature axis, the d axis is the direct axis in the actual coordinate system, the q axis is the quadrature axis, and the directions of qv axis and current vector is remain identical. The torque angle θ ̂ e ${\hat \theta _e}$ defines the angle between the qv axis and the d axis. Both id and iq represent the projection of is from the dvqv to the d–q coordinate system.

Details are in the caption following the image
Before and after I–f startup space vector diagram in the dual coordinate system.

As shown in Figure 1a, the torque angle θ ̂ e ${\hat \theta _e}$ is 0 at the instant of I–f startup control. The actual dq coordinate system is 90° ahead of the virtual dvqv current vector coordinate system, and the reference current vector is acts on the d-axis of the actual coordinate system. Therefore, no torque is generated initially. After a reference current vector command is issued, as in Figure 1b, acceleration in the dvqv coordinate system aligns it with the actual d-q coordinate system, and θ ̂ e ${\hat \theta _e}$ will increase. The reference current vector operates on the actual q-axis, leading to a proportional increase of the current issin θ ̂ e ${\hat \theta _e}$ with torque angle θ ̂ e ${\hat \theta _e}$ . The rotor commences its rotation when the electromagnetic torque generated by the actual q-axis current exceeds load torque.

According to Figure 1, the stator voltage equation in the dvqv virtual coordinate system can be expressed as:
u s d v = n p ω i L s i s n p ω r ψ r cos θ ̂ e u s q v = R s i s + n p ω r ψ r sin θ ̂ e + L s d i s d t $$\begin{eqnarray} {u_{s{d^v}}} &=& - {n_p}{\omega _i}{L_s}{i_s} - {n_p}{\omega _r}{\psi _r}\cos {{\hat \theta }_e}\nonumber\\ {u_{s{q^v}}} &=& {R_s}{i_s} + {n_p}{\omega _r}{\psi _r}\sin {{\hat \theta }_e} + {L_s}\frac{{d{i_s}}}{{dt}} \end{eqnarray}$$ (1)
where usdv and usqv are the dv-axis and qv-axis voltages in the virtual coordinate system, Rs is the stator resistance, Ls is the stator inductance, ψr is the permanent magnet flux, ωi is the rotating mechanical angular velocity of the current vector, ωr is the actual rotating mechanical angular velocity of the rotor, and np is the number of rotor pole pairs.
According to the electromechanical principles, the torque equation of SPMSM under I–f control can be expressed as:
T e = 3 2 n p ψ r i s sin θ ̂ e $$\begin{equation}{T_e} = \frac{3}{2}{n_p}{\psi _r}{i_s}\sin {\hat \theta _e}\end{equation}$$ (2)
where Te is the electromagnetic torque of the motor.
J d ω r d t = T e B ω r T L = 3 2 n p ψ r i s sin θ ̂ e B ω r T L $$\begin{equation}J\frac{{d{\omega _r}}}{{dt}} = {T_e} - B{\omega _r} - {T_L} = \frac{3}{2}{n_p}{\psi _r}{i_s}\sin {\hat \theta _e} - B{\omega _r} - {T_L}\end{equation}$$ (3)
where J is the rotational inertia of the motor rotor, B is the viscous friction factor of the motor, and TL is the load torque.
According to Figure 1, the torque angle θ ̂ e ${\hat \theta _e}$ can be determined in relation to the rotating mechanical angular velocity ωi of the current vector and the actual rotating mechanical angular velocity ωr of the rotor as follows:
d θ ̂ e d t = Δ ω = n p ω i ω r $$\begin{equation}\frac{{d{{\hat \theta }_e}}}{{dt}} = \Delta \omega = {n_p}\left( {{\omega _i} - {\omega _r}} \right)\end{equation}$$ (4)

3 THEORETICAL ANALYSIS OF OPEN-LOOP I–f CONTROL METHOD

3.1 General open-loop I–f control scheme

The overall control block diagram of the open-loop I–f control system is depicted in Figure 2. Combining its space vector diagram and mathematical model shows that specific offline settings for the current vector amplitude is and rotational angular velocity ωi must be based on specific motor parameters for practical applications. The “torque self-balancing” function is employed to minimize the likelihood of pole slipping during the motor starting process.

Details are in the caption following the image
Surface-mounted permanent magnet synchronous motors (SPMSM) open-loop I–f control block diagram.

3.2 Analysis of the open-loop I–f control

According to Equations (3) and (4), the SPMSM should satisfy the following 2D differential equations under I–f control, regardless of whether it is in initial acceleration or steady-state operation phase:
d ω r d t = 3 n p ψ r i s sin θ ̂ e 2 J T L J B J ω r d θ ̂ e d t = n p ω i ω r $$\begin{equation} \left\{ { \def\eqcellsep{&}\begin{array}{rcl} \frac{{d{\omega _r}}}{{dt}} &=& \frac{{3{n_p}{\psi _r}{i_s}\sin {{\hat \theta }_e}}}{{2J}} - \frac{{{T_L}}}{J} - \frac{B}{J}{\omega _r}\\[3pt] \frac{{d{{\hat \theta }_e}}}{{dt}} &=& {n_p}\left( {{\omega _i} - {\omega _r}} \right) \end{array} } \right.\end{equation}$$ (5)

According to the theory of differential equation solution, it is challenging to derive a precise solution for the torque angle θ ̂ e ${\hat \theta _e}$ . Therefore, in order to investigate the variation characteristics of each parameter during the I–f control operation, the numerical solution method is utilized to analyse the torque angle fluctuation law here.

Various numerical techniques exist for solving second-order differential equations have their strengths and limitations. Considering the nonlinear trigonometric functions characteristics in the open-loop I–f control differential equation for torque angle θ ̂ e ${\hat \theta _e}$ , the fourth order Runge-Kutta method is chosen. This method operates on its fundamental principle: the selection of distinct points proximate to various time instances, the linear combination of function values at these points, and the utilization of the resultant combined value as a substitute for the derivative in the Taylor series expansion. The Runge–Kutta approach provides a numerical solution to the differential equations in (5). Matlab is used to chart the behaviour of torque angle θ ̂ e ${\hat \theta _e}$ , actual rotational mechanical angular velocity ωr, actual rotational mechanical angular acceleration Kar, reference current vector is, and motor parameters.

For simplifying the numerical analysis, the load torque is set to 0 under open-loop I–f control, the reference current vector is constant, the reference acceleration by the current vector is constant, and the initial torque angle θ ̂ 0 ${\hat \theta _0}$ is 0°. As illustrated in Figure 3, the numerical solution derived from the initial value of the reference current vector reveals fluctuating patterns with a certain regularity for torque angle θ ̂ e ${\hat \theta _e}$ , actual rotor angular velocity ωr, and actual rotor angular acceleration Kar.

Details are in the caption following the image
Open-loop I–f startup Runge–Kutta method numerical solution result.

Then, by selecting a fluctuation period (t1t4) in Figure 3, we can analyse the fluctuation law of torque angle, actual acceleration, and motor actual speed in conjunction with the If control vector diagram in dual coordinates.

As depicted in Figure 4, the I–f startup acceleration process can be categorized into three predominant stages:
  • Phase 1: During t1-t2, In Figure 4a, the torque angle shows a rising trend as the virtual coordinate system edges closer to the actual coordinate system. Figure 4b shows that the actual rotor acceleration gradually increases from 0 to the constant acceleration provided by open-loop I–f control at t2. In Figure 4c, the actual motor speed is always less than the open-loop reference speed, with the error speed peaking at t2.
  • Phase 2: During t2- t3, the torque angle continues increasing in Figure 4a. Figure 4b demonstrates that the actual rotor acceleration continues to rise with both the torque angle and the actual rotor acceleration reaching respective maxima at t3. In Figure 4c, the actual speed remains below reference speed but with the error decreasing to 0 at t3.
  • Phase 3: During t3-t4, the torque angle in Figure 4a is decreasing, as is the actual rotor acceleration in Figure 4b, and the torque angle at t4 is the minimal value close to 0. The actual speed is equal to the open-loop reference speed, and the actual speed error is currently 0. At this point, the actual speed acceleration reaches a value near to 0, while the acceleration error reaches a positive maximum. In Figure 4c, the actual speed exceeds the open-loop reference speed, and the error speed is always more than 0.
Details are in the caption following the image
Space vector diagram of I–f control for starting acceleration under dual coordinate system.

Assuming current vector is constant, both the numerical solution results and the vector diagram of open-loop I–f control indicate that the torque angle and actual speed always fluctuate periodically during motor operation at the same frequency. When the fluctuation amplitude of the torque angle surpasses 90° between the virtual and actual coordinate systems, it heightens the risk of pole slipping. Therefore, theoretical research into the high frequency fluctuation mechanism and influence factors is required.

4 I–f CURRENT VECTOR CLOSED-LOOP ADAPTIVE REGULATION ALGORITHM

4.1 Mechanism analysis of high frequency oscillations under I–f control

According to Equation (5), the function f ( θ ̂ e ${\hat \theta _e}$ , ωi) about the torque angle θ ̂ e ${\hat \theta _e}$ and the rotational mechanical angular velocity ωi of the current vector can be expressed as:
J d 2 θ ̂ e d t 2 + B d θ ̂ e d t + 3 2 n p ψ r i s sin θ ̂ e = J d ω i d t + B ω i + T L $$\begin{equation}J\frac{{{d^2}{{\hat \theta }_e}}}{{d{t^2}}} + B\frac{{d{{\hat \theta }_e}}}{{dt}} + \frac{3}{2}{n_{\mathrm{p}}}{\psi _{\mathrm{r}}}{i_s}\sin {\hat \theta _e} = J\frac{{d{\omega _i}}}{{dt}} + B{\omega _i} + {T_{\mathrm{L}}}\end{equation}$$ (6)
When we insert Equation (4) into Equation (6) and extract its high-frequency and fundamental frequency components, we can derive the following result:
J d 2 θ ̂ e d t 2 + B d θ ̂ e d t + 3 2 n p ψ r i s sin θ ̂ e = B ( ω i ω r ) + J d ω i d t d ω r d t + 3 2 n p ψ r i sin ( ω i ω r ) d t $$\begin{eqnarray} J\frac{{{d^2}{{\hat \theta }_e}}}{{d{t^2}}} &+& B\frac{{d{{\hat \theta }_e}}}{{dt}} + \frac{3}{2}{n_{\mathrm{p}}}{\psi _{\mathrm{r}}}{i_s}\sin {{\hat \theta }_e} = B({\omega _i} - {\omega _r})\nonumber\\ &+& J \left(\frac{{d{\omega _i}}}{{dt}} - \frac{{d{\omega _r}}}{{dt}}\right) + \frac{3}{2}{n_{\mathrm{p}}}{\psi _r}i\sin \left(\int {({\omega _i} - {\omega _r})dt} \right)\quad \end{eqnarray}$$ (7)
Under the open-loop I–f control, Equation (7) shows that ωi is the fixed acceleration profile (Kait) provided by open-loop, where Kai is the fixed acceleration provided by the current vector. Therefore, the term associated with ωi can be regarded as the fundamental component, and its first derivative also being the constant Kai, which is fixed. According to the right side of Equations (7) and (6), the high-frequency component of the actual rotating mechanical angular velocity ωr of the rotor satisfies the following equation:
J ω r B ω r + 3 2 n p ψ r i s sin 1 2 K a i t 2 ω r t T L = 0 $$\begin{equation} - J\omega _r^\prime - B{\omega _r} + \frac{3}{2}{n_{\mathrm{p}}}{\psi _r}{i_s}\sin \left(\frac{1}{2}{K_{ai}}{t^2} - {\omega _r}t\right) - {T_L} = 0\end{equation}$$ (8)

According to Equation (8), the actual torque angle of the motor governed by open-loop I–f control is influenced not only by the motor mechanical parameters BJTL and the reference current vector is, but also by the actual speed of the rotor ωr. Unsuitable control parameters, load disturbances etc. can easily cause θ ̂ e ${\hat \theta _e}$ to deviate from the stable range, resulting in pole slipping. The high frequency fluctuation of ωr is the fundamental cause of pole slipping during the I–f starting process. Therefore, this paper will analyse the law of high frequency fluctuation and find suitable methods to reduce the speed fluctuation.

4.2 Analysis of numerical solutions for the change law of fluctuating frequency under I–f control

As shown in Figure 3, the torque angle fluctuation frequency aligns with the actual motor speed fluctuation frequency, Thus, this study correlates the characteristics of torque angle fluctuations with actual speed oscillations. To quantify the high frequency fluctuation of the torque angle, we define two parameters: fluctuation frequency and fluctuation amplitude. where fluctuation frequency reflects the torque angle fluctuation cycle and fluctuation amplitude reflects the range within a single cycle. The study employs four parameters to investigate the fluctuation law of torque angle: the initial value of torque angle θ ̂ 0 ${\hat \theta _0}$ , the reference current vector acceleration Kai, the moment of inertia J, and viscous damping coefficient B of the motor.

Figure 5 displays numerical solution results for the torque angle with varying B, J, Kai and ωr, where Jpu and Bpu are the original moment of inertia and viscosity coefficient of the motor. The reference value of Kapu is 600 r/min/s, and speed reference value is 10,000 r/min.

Details are in the caption following the image
Relationship between different B, J, Kai, θ ̂ 0 ${\hat \theta _0}$ and torque angle θ ̂ e ${\hat \theta _e}{\mathrm{\;}}$ under I–f control.

The variation curves for torque angle θ ̂ e ${\hat \theta _e}$ fluctuation frequency and amplitude under distinct B, J, Kai and ωr are obtained in Figure 6 based on the numerical solution results in Figure 5.

Details are in the caption following the image
Different B, J, Kai, ωr and torque angle fluctuation frequency and amplitude curves.

Figure 6a depicts that the fluctuation frequency of the torque angle under various B, J, Kai and ωr conditions. The frequency exhibits an approximately linear as reference acceleration Kai and viscosity coefficient B increase from the reference value 1, and decreases further as speed ωr rises, with the slope continuously increasing.

Figure 6b presents the variation curve of torque angle fluctuation amplitude across different B, J, and Kai values. It indicates that the amplitude remains largely unchanged with variations in the actual speed ωr. With an increase in the moment of inertia and acceleration, the fluctuation amplitude of the torque angle increases linearly, but diminishes with an increase in B. From the numerical solution, the relationship is described by Equation (9), where fwave(∆ θ ̂ e ${\hat \theta _e}$ ) represents the frequency of torque angle fluctuation and Max(∆θ) represents its fluctuation amplitude.
f w a v e ( Δ θ ̂ e ) J , B , K a i , ω r M a x ( Δ θ ̂ e ) J , K a i , 1 / B $$\begin{eqnarray} &&{f_{wave}}(\Delta {{\hat \theta }_e}) \propto J,B,{K_{ai}},{\omega _r}\nonumber\\ &&Max(\Delta {{\hat \theta }_e}) \propto J,{K_{ai}},1/B \end{eqnarray}$$ (9)

Equation (9) shows the torque angle fluctuation frequency varies continuously with speed and reference acceleration, and the torque angle fluctuation amplitude is consistently influenced by the reference acceleration. To suppress the high-frequency fluctuations of the rotational speed effectively, we can devise a closed-loop algorithm based on the fluctuation frequency variation rule of the torque angle and the influencing factors of fluctuation amplitude.

4.3 I–f closed-loop current vector adaptive regulation algorithm overall scheme

An I–f closed-loop control strategy is proposed here to adapt to the abrupt acceleration interval and reduce speed oscillations during I–f starting process, based on instantaneous active power feedback and adaptive current vector adjustment, the block diagram of the overall control structure is depicted in Figure 7.

Details are in the caption following the image
Diagram of the I–f startup current vector adaptive control structure block.

The main idea of the control algorithm: Calculating the instantaneous active power P of the SPMSM in real time based on the voltage and current in the dvqv virtual coordinate system. Combining the fluctuation law of Equation (9) with the real-time calculation of the adaptive filter cut-off frequency based on the reference current vector and estimated rotational speed, a feedback adjustment algorithm is constructed based on the high frequency fluctuation, and the current vector is adjusted in real time by adding the virtual damping torque component to the reference rotational speed ωi. It enables the motor to converge to the synchronized state rapidly between the virtual and actual coordinate system during the I–f starting process. It can also automatically adjust control parameters to accelerate the convergence of torque angle and improve the stability and adaptability of system when faced with abrupt changes in reference acceleration.

4.4 Algorithm for current vector adaptive regulation based on instantaneous power

According to the electromagnetic torque Equation (2), the variation of the torque angle θ ̂ e ${\hat \theta _e}$ in the vicinity of the steady-state angle θ0 over ∆t is derived as
Δ T e = 3 2 n P ψ r i s cos θ 0 Δ θ ̂ e $$\begin{equation}\Delta {T_{\mathrm{e}}} = \frac{3}{2}{n_{\mathrm{P}}}{\psi _r}{i_s}\cos {\theta _0}\Delta {\hat \theta _e}\end{equation}$$ (10)
Using Equations (10) and (4), we can derive:
Δ T e = k e ω i ω r t $$\begin{equation}\Delta {T_{\mathrm{e}}} = {k_e}\left( {{\omega _i} - {\omega _r}} \right)t\end{equation}$$ (11)
In the equation:
k e = 3 2 n p 2 ψ r i s cos θ 0 , θ 0 = T e t t = t 0 $$\begin{equation}{k_e} = \frac{3}{2}n_p^2{\psi _r}{i_s}\cos {\theta _0},{\theta _0} = \frac{{\partial {T_e}}}{{\partial t}}\left| {_{t = {t_0}}} \right.\end{equation}$$ (12)
To implement the closed-loop control between the actual speed and current vector reference speed, Equations (10), (11) shows ∆ θ ̂ e ${\hat \theta _e}$ in ∆Te is related to ∆ωi. An increment ∆ωi is considered to superimpose on the reference speed of the current vector, the electromagnetic torque increment ∆ T e $T_e^*$ is generated by the increment ∆ωi as follows:
Δ T e = 3 2 n P ψ f i s cos θ 0 · Δ ω i d t $$\begin{equation}\Delta T_e^ * = \frac{3}{2}{n_{\mathrm{P}}}{\psi _{\mathrm{f}}}{i_s}\cos {\theta _0} \cdot \int \Delta {\omega _i}dt\end{equation}$$ (13)
It can be seen from Equation (13) that the integral term of increment ∆ωi is proportional to ∆Te. Therefore, in order to adjust ∆Te, it is necessary to adjust the integral term of increment ∆ωi, defined as:
Δ ω i = k ω · d Δ ω r d t $$\begin{equation}\Delta {\omega _i} = - {k_\omega } \cdot \frac{{d\Delta {\omega _{\mathrm{r}}}}}{{dt}}\end{equation}$$ (14)
Applying Equation (14) to Equation (13) yields:
Δ T e = 3 2 n P ψ f i s cos θ 0 · k ω Δ ω r $$\begin{equation}\Delta T_e^ * = - \frac{3}{2}{n_{\mathrm{P}}}{\psi _{\mathrm{f}}}{i_s}\cos {\theta _0} \cdot {k_\omega }\Delta {\omega _r}\end{equation}$$ (15)

In Equation (14), ∆ωr represents the fluctuation of rotor speed at the steady-state operating point, and kω is a positive quantity that can be adjusted. In Equation (15), ∆ T e $T_{\mathrm{e}}^{\mathrm{*}}$ is proportional to ∆ωr, exhibiting virtual damping torque characteristics. When the speed fluctuates, real-time compensation of the reference speed is achieved by determining the fluctuation amount. Consequently, the electromagnetic torque acquires a virtual damping component, enabling adaptive damping for rapid actual speed tracking of the reference current vector speed.

The closed-loop current vector adaptive regulation algorithm is depicted in Figure 8. According to the equation for calculating active power, it can be expressed as:
P = U I s cos φ $$\begin{equation}P = U{I_s}\cos \varphi \end{equation}$$ (16)
where P is the active power of the SPMSM, U is the voltage vector amplitude in the real coordinate system, Is is the reference current vector amplitude is, and φ is the angle between the motor voltage in the real coordinate system and the reference current vector is in the virtual coordinate system, which is determined as follows from the vector diagram:
U cos φ = u s q v = R s i s + n p ω r ψ r sin θ ̂ e + L s d i s d t $$\begin{equation}U\cos \varphi = {u_{s{q^v}}} = {R_s}{i_s} + {n_p}{\omega _r}{\psi _r}\sin {\hat \theta _e} + {L_s}\frac{{d{i_s}}}{{dt}}\end{equation}$$ (17)
Details are in the caption following the image
Block diagram of the I–f current vector adaptive control algorithm.
When the reference current is constant, ∆ωr can be reflected by the instantaneous change in voltage Ucosφ. Therefore, under I–f control, ∆ωr and P/is can be regarded as proportional approximatively, which can be expressed as:
Δ ω r Δ P i s = Δ U cos φ $$\begin{equation}\Delta {\omega _{\mathrm{r}}} \propto \frac{{\Delta P}}{{{i_s}}} = \Delta U\cos \varphi \end{equation}$$ (18)
Under normal operating conditions, motor active power increases with speed, and the quantity of active power fluctuation can be achieved in real time by an adaptive high-pass filter, so according to the voltage equation in the virtual dv-qv coordinate system, the following can be derived:
Δ ω r Δ P i s = G HPF u d v i d v + u q v i q v i s $$\begin{equation}\Delta {\omega _{\mathrm{r}}} \propto \frac{{\Delta P}}{{{i_s}}} = {G_{{\mathrm{HPF}}}}\left( {\frac{{u_d^vi_d^v + u_q^vi_q^v}}{{{i_s}}}} \right)\end{equation}$$ (19)
According to Equations (14) and (19), the closed-loop adjustment increment ∆ωi can be calculated as follows:
Δ ω i = k ω u d v i d v + u q v i q v i s T P s T P s + 1 $$\begin{equation}\Delta {\omega _i} = - {k_\omega }\frac{{u_d^vi_d^v + u_q^vi_q^v}}{{{i_s}}}\frac{{{T_{\mathrm{P}}}s}}{{{T_{\mathrm{P}}}s + 1}}\end{equation}$$ (20)
where Tp represents the time constant associated with the adaptive high-pass filter and kω is the feedback compensation adaptive adjustment coefficient determined by the numerical solution analysis of Max(∆θ).
Here, the time constant of the filter is designed based on the numerical solution results presented in Equation (9), Combining the reference current vector amplitude, reference acceleration, and motor parameters, determined as follows:
T p = 1 f w a v e ( Δ θ ̂ e ) $$\begin{equation}{T_p} = \frac{1}{{{f_{wave}}(\Delta {{\hat \theta }_e})}}\end{equation}$$ (21)

4.5 Analysis of the stability of the vector adaptive regulation algorithm

According to the relationship between motor power, torque, and speed, the instantaneous active power of SPMSM can also be expressed:
P = T e ω r $$\begin{equation}P = {T_e}{\omega _r}\end{equation}$$ (22)
Substituting the torque Equation (2) and the motion Equation (3) into Equation (22), we obtain:
P = J d ω r d t ω r + B ω r 2 T L ω r $$\begin{equation}P = J\frac{{d{\omega _r}}}{{dt}}{\omega _r} + B\omega _r^2 - {T_L}{\omega _r}\end{equation}$$ (23)
Taking the derivative of the Equation (23), the fluctuation in active power caused by fluctuations in speed can be expressed as:
Δ P = J ω 0 d Δ ω r d t + 2 B ω 0 Δ ω r + Δ ω r T L 0 $$\begin{equation}\Delta P = J{\omega _0}\frac{{d\Delta {\omega _r}}}{{dt}} + 2B{\omega _0}\Delta {\omega _r} + \Delta {\omega _r}{T_{L0}}\end{equation}$$ (24)
where ω0TL0 are the motor speed and load torque in steady state, which can be derived from Equation (18) and (19):
Δ ω i = k p Δ P $$\begin{equation}\Delta {\omega _i} = - {k_p}\Delta P\end{equation}$$ (25)
where kp equals kω/is, by substituting Equations (24), (25) into Equation (10), the transfer function of the torque variation generated by ∆ωr is derived as:
Δ T e ( s ) = ( J k e k ω ω 0 + k e 1 + 2 k ω B ω 0 + k ω T L 0 s ) Δ ω r $$\begin{equation}\Delta {T_{\mathrm{e}}}(s) = - (J{k_e}{k_\omega }{\omega _0} + \frac{{{k_e}\left( {1 + 2{k_\omega }B{\omega _0} + {k_\omega }{T_{L0}}} \right)}}{s})\Delta {\omega _r}\end{equation}$$ (26)
Taking the derivative of the Equation (3), the equation of motion can be derived as:
J d Δ ω r d t = Δ T e B Δ ω r Δ T L $$\begin{equation}J\frac{{d\Delta {\omega _r}}}{{dt}} = \Delta {T_e} - B\Delta {\omega _r} - \Delta {T_L}\end{equation}$$ (27)
According to Equation (27), the torque component produced by ∆ωr can be expressed by a transfer function block diagram. F(s) can be deduced from Equation (26):
F ( s ) = J k e k ω ω 0 + k e 1 + 2 k ω B ω 0 + k ω T L 0 s $$\begin{equation}F(s) = J{k_e}{k_\omega }{\omega _0} + \frac{{{k_e}\left( {1 + 2{k_\omega }B{\omega _0} + {k_\omega }{T_{L0}}} \right)}}{s}\end{equation}$$ (28)
As shown in Figure 9, the open-loop transfer function of torque under closed-loop control can be expressed as:
G T ( s ) = B J k e k p ω 0 s + B k e 1 + 2 k ω B ω 0 + k ω T L 0 B s 2 + J s $$\begin{equation}{G_T}(s) = \frac{{BJ{k_e}{k_p}{\omega _0}s + B{k_e}\left( {1 + 2{k_\omega }B{\omega _0} + {k_\omega }{T_{L0}}} \right)}}{{B{s^2} + Js}}\end{equation}$$ (29)
Details are in the caption following the image
Block diagram of the transfer function of torque disturbance component generated by ∆ωr.
The defining equation of the open-loop transfer function is:
s 2 + k 1 s + k 2 = 0 $$\begin{equation}{s^2} + {k_1}s + {k_2} = 0\end{equation}$$ (30)
In the equation:
k 1 = J k e k p ω 0 + J B , k 2 = k e 1 + 2 k p B ω 0 + k p T L 0 J $$\begin{equation}{k_1} = J{k_e}{k_p}{\omega _0} + \frac{J}{B},\;\;\;{\kern 1pt} {k_2} = \frac{{{k_e}\left( {1 + 2{k_p}B{\omega _0} + {k_p}{T_{L0}}} \right)}}{J}\end{equation}$$ (31)
The specific solution root of the equation is:
x 1 , 2 = k 1 ± k 1 2 4 k 2 2 $$\begin{equation}{x_{1,2}} = \frac{{ - \left( {{k_1}} \right) \pm \sqrt {{{\left( {{k_1}} \right)}^2} - 4{k_2}} }}{2}\end{equation}$$ (32)

Based on Equation (31), it is inferred that k1 > 0, which implies that the real portion of the characteristic root is negative. According to the principle of automatic control theory, a sufficient necessary condition for the stability of a linear system is that the characteristic equation closed-loop system roots have negative real components. Therefore, the closed-loop system is stable.

When identification errors exist in the motor parameters B and J, we analyse the dynamic performance and stability based on the characteristic roots. Consequently, we introduced error terms for B and J as ∆B and ∆J, respectively. With these errors in mind, the specific solution root of the equation becomes:
x 1 , 2 = k 1 ± k 1 2 4 k 2 2 $$\begin{equation}x_{1,2}^* = \frac{{ - \left( {k_1^*} \right) \pm \sqrt {{{\left( {k_1^*} \right)}^2} - 4k_2^*} }}{2}\end{equation}$$ (33)
In the equation:
k 1 = ( J + Δ J ) k e k p ω 0 + ( J + Δ J ) ( B + Δ B ) , k 2 = k e 1 + 2 k p ( B + Δ B ) ω 0 + k p T L 0 ( J + Δ J ) $$\begin{eqnarray} k_1^* &=& (J + \Delta J){k_e}{k_p}{\omega _0} + \frac{{(J + \Delta J)}}{{(B + \Delta B)}},\nonumber\\ k_2^* &=& \frac{{{k_e}\left( {1 + 2{k_p}(B + \Delta B){\omega _0} + {k_p}{T_{L0}}} \right)}}{{(J + \Delta J)}} \end{eqnarray}$$ (34)
Based on control theory principles, the real part of the characteristic root of system affects the transient response. The closer the real part is to zero, the slower the decay, which implies that the system takes a longer time to stabilize. On the other hand, the imaginary part determines the oscillation frequency and response speed. From the above analysis, we deduce the subsequent conclusions:
  1. For ∆B > 0 (B is overestimated): As the characteristic roots are closer to the imaginary axis, the system responds more quickly, and the oscillation frequency may increase.

  2. For ∆B < 0 (B is underestimated): As the characteristic roots are further from the imaginary axis, the response speed may decrease, and the oscillation frequency may decrease.

  3. For ∆J > 0 (J is overestimated): As the characteristic roots are further from the imaginary axis, the response speed may decrease, and the oscillation frequency may reduce.

  4. For ∆J < 0 (J is underestimated): As the characteristic roots are closer to the imaginary axis, the system responds more quickly, and the oscillation frequency may increase.

5 ANALYSIS OF EXPERIMENTAL RESULTS

In order to experimentally validate the proposed control strategy, the experimental platform in Figure 10 was built, and the motor parameters are shown in Table. 1. The motor controller consists of a chip of type TMS320F28377S and an inverter with SiC components. The switching frequency and control period are both 50 kHz, the experimental phase current waveform is measured in Figure 10b, and other experimental results were transmitted in real time through the CAN communication of the motor controller and the human-machine interface based on Labview software development, data transmission period is 10 ms.

Details are in the caption following the image
SPMSM and its control system experiment platform.
TABLE 1. Parameters of SPMSM model.
Parameters Numerical values
Stator phase inductance/H 0.000827
Stator phase resistance/Ω 0.047
Permanent magnet flux / Wb 0.14
Inertia/(kg⋅m)2 0.063
Viscous friction factor/(N⋅m⋅s) 0.0054
Number of pole pairs p 1
Rated speed nr /(r/min) 20,000
Rated voltage Udc /(V) 540
Rated current I/(Arms) 54
Rated torque Te /(N⋅m) 11

It should be emphasized that the aim of the research in this paper is to improve the stability of I–f starting process of high-speed PMSM and the robustness under different accelerations, so the next experiments were conducted to verify the I–f closed-loop algorithm only at the idle speed (20% of rated speed).

5.1 Experimental verification and analysis under fixed acceleration of I–f startup conditions

Figure 11 shows experimental acceleration phase waveforms under open-loop I–f control and closed-loop adaptive regulation of the current vector at 720 r/min/s constant acceleration.

Details are in the caption following the image
Experimental waveforms of various control algorithms for I–f startup control fixed acceleration.
Details are in the caption following the image
FIGURE 11 (continued)
Experimental waveforms of various control algorithms for I–f startup control fixed acceleration.

Figure 11a illustrates the experimental waveforms of the reference current vector tracking, speed tracking, and torque angle under open-loop I–f control. It can be seen that maximum speed fluctuation tracking error exceeds 100 r/min, the maximum torque angle fluctuation is 90°, the reference current vector tracking error is about 10A, and the tracking error increases larger and larger during the whole I–f starting process. Figure 11b presents the supplied current, speed, and torque angle tracking waveforms based on closed-loop I–f startup control. The experimental results indicate that the torque angle fluctuation only occurs during the early stage of starting, peaking at 20° and stabilizing within 0.3 s. Throughout the whole process, the speed fluctuation tracking error peaks at approximately 5 r/min, and the reference current vector always tracks around the specified current vector amplitude of 15 A.

Figure 12 shows experimental torque angle fluctuations during If startup acceleration, comparing the algorithm proposed in this study with conventional If closed-loop control methods. Under the adaptive filter closed-loop control algorithm proposed in this paper, the torque angle converges more rapidly, and the torque angle fluctuations have the least amplitude. If the filter frequency is chosen incorrectly, the convergence speed of torque angle is clearly affected. In comparison to the conventional closed-loop methods with fixed filter cut-off frequency, the vector adaptive regulation method suggested in this study has a shorter convergence time of at least 0.5 s and more flexibility for variable acceleration of starting conditions.

Details are in the caption following the image
Experimental waveform of torque angle fluctuation for various I–f closed-loop control methods.

5.2 Experimental verification and analysis under variable acceleration of I–f startup control conditions

In order to verify the robustness of the current vector closed-loop adaptive control algorithm, experiments were conducted under specific acceleration conditions during I–f starting process. The experimental results are compared and analysed from four angles: the characteristics of torque angle convergence, the tracking performance of the reference current vector, speed tracking features, and the rate of phase current convergence.

Figures 13 and 14 are the experimental waveforms of the conventional I–f closed-loop control and the current vector adaptive algorithm under the reference acceleration mutation condition. Reference acceleration is fixed at 500 r/min/s during the 0–3 s period, increased to 1000 r/min/s at t = 3 s.

Details are in the caption following the image
Experimental waveforms of the conventional closed-loop control method under abrupt acceleration.
Details are in the caption following the image
FIGURE 13 (continued)
Experimental waveforms of the conventional closed-loop control method under abrupt acceleration.
Details are in the caption following the image
Experimental waveforms of the current vector adaptive control algorithm for abrupt acceleration.
Details are in the caption following the image
FIGURE 14 (continued)
Experimental waveforms of the current vector adaptive control algorithm for abrupt acceleration.

In Figure 13, which depicts the conventional current vector closed-loop control, the torque angle fluctuates up to 40° and takes approximately 1 s to converge, the reference current vector has a tracking error of around 2 A, and the tracking error between the reference speed and the actual speed is about 20 r/min. On the contrary, under the closed-loop I–f startup control strategy based on current vector adaptive regulation in Figure 14, the maximum fluctuation of torque angle is 25° and converges within 0.4 s, current tracking error is around 1 A and speed tracking error remains under 5 r/min.

When motor speed reaches 1500 r/min, the acceleration is raised from 500 to 1000 r/min/s. Figure 13a shows that the torque angle undergoes an abrupt change before it converges slowly to a stable value with conventional closed-loop I–f control algorithm. The torque angle fluctuates ±35° after a sudden change in acceleration, requiring 1.2 s to converge. The tracking error of the reference current amplitude is about 5 A, which is 30% of the 15 A fundamental value. Within the grey highlighted region, the motor speed experiences a continuous non-tracking phenomenon for 0.5 s, leading to a tracking error of up to 50 r/min. From the motor phase current waveform presented in Figure 14b, it is evident that the current of motor phase A oscillates between 10 and 25 A, reaching relative stability after 1.2 s during the abrupt acceleration.

Under the adaptive closed-loop control of the current vector in this paper, Figure 14a demonstrates that after the sudden change of acceleration, the torque angle fluctuates just ±10°, the torque angle quickly converges to the new stable value within 0.3 s, and the actual speed tracks the reference speed of current vector consistently throughout the entire process, the speed tracking error is only about 10 r/min even during changes in acceleration, the maximum reference current tracking error is about 1 A. Figure 14b shows the motor phase A current stabilizes within 0.4 s after acceleration changes, 0.8 s faster than the conventional I–f closed-loop control.

The experimental results in Table. 2 and 3 indicate that the current vector adaptive regulation closed-loop algorithm has better robustness during I–f startup under fixed and abrupt accelerations. It provides smaller torque angle fluctuations, faster motor convergence, and improved current tracking compared to conventional I–f control.

TABLE 2. Experimental comparison of I–f closed-loop control algorithms with fixed acceleration.
Fixed acceleration Conventional control Adaptive control
Torque angle fluctuation amplitude/(°) 40 25
Rotational speed tracking error/(r/min) 20 5
Torque angle convergence time/(s) 1 0.4
TABLE 3. Comparison of I–f closed-loop control scheme experimental results under the abrupt acceleration.
Abrupt acceleration Conventional control Adaptive control
Torque angle fluctuation amplitude/(°) 35 10
Rotational speed tracking error/(r/min) 50 10
Tracking error for a reference current (A) ±5 ±1
Convergence time after mutation/(s) 1.2 0.4

6 CONCLUSION

This study proposes a closed-loop I–f control strategy with adaptive current vector regulation based on active power feedback to address the slow dynamic response and long convergence time of the classic I–f control for high-speed SPMSMs. Compared to conventional I–f methods, it notably diminishes torque angle fluctuations, improves speed tracking performance, and enhances current tracking proficiency. The following conclusions were reached:
  1. Motor parameters, reference current vector, and actual speed determine the frequency and magnitude of speed and torque angle variations under I–f control. The high-frequency fluctuation of rotational speed is the underlying cause of the pole slipping under I–f control. The variation laws of torque angle fluctuation frequency and amplitude provide the theoretical basis for devising the I–f closed-loop control algorithms.

  2. Adjusting the current vector reference speed based on motor instantaneous active power feedback can minimize torque angle fluctuation magnitude, reduce torque fluctuation, prevent pole slipping and reduce speed oscillations.

  3. Experimental results demonstrate this proposed control method provides good performance for fixed and varying I–f acceleration, increasing convergence speed 50% and reducing torque angle fluctuation 30% under acceleration changes versus conventional regulation. Which provides an important algorithmic basis to enhance startup stability for ultra-high speed centrifugal air compressor applications.

Finally, it is worth noting that while significant identification errors in motor parameters can affect the performance of the closed-loop algorithm, they do not compromise the stability of system. Specifically, the response, oscillation frequency, and stabilization time of system can vary based on the overestimation or underestimation of B and J.

AUTHOR CONTRIBUTIONS

Guxuan Xu: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Writing—original draft; Writing—review and editing. Feng Zhao: Funding acquisition; Validation. Tao Liu: Supervision; Validation; Writing—review and editing

ACKNOWLEDGEMENTS

This work was supported in part by the National Natural Science Foundation of China under grant number 51807139.

    FUNDING INFORMATION

    This work was supported in part by the National Natural Science Foundation of China under grant number 51807139.

    CONFLICT OF INTEREST STATEMENT

    The authors declare no conflicts of interest.

    DATA AVAILABILITY STATEMENT

    The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.