Phase Current Reconstruction of PMSG-Based Three-Phase PWM Rectifiers Using Linear Extended State Observer

Abstract

As a core power supply component of the more electric aircraft (MEA), the reliability of the permanent magnet synchronous generator (PMSG) is of paramount importance. Phase current reconstruction technology can enhance the redundancy of current sensors, thereby improving system reliability. However, owing to the generally high engine speeds in MEAs, the employment of traditional d-axis current–zero control not only induces DC-link voltage fluctuations but also leads to inaccurate DC-link sampling points and distortion in the reconstructed current. In this paper, a lead-angle flux-weakening control strategy is introduced into the PMSG rectification system. This approach guarantees the normal operation of the current loop when the rotational speed exceeds the rated speed of the PMSG, ensuring the accuracy of the sampling points for phase current reconstruction. To further enhance the reconstruction accuracy, a phase current reconstruction technology based on a linear extended state observer (LESO) is proposed. The LESO not only filters the reconstructed current but also ensures that the observer performance remains robust against PMSG parameter perturbations. Finally, the effectiveness of the proposed method is validated through Hardware-in-the-Loop results.

1. Introduction

Possessing advantages such as high power density, high efficiency, and a wide speed regulation range, the permanent magnet synchronous generator (PMSG) serves as a core power supply component in transportation electrification systems, including the more electric aircraft (MEA) and all-electric aircraft [1,2,3]. In the pulse width modulation (PWM) rectification system of a PMSG, at minimum, two-phase current sensors are required to acquire phase current. However, these sensors are susceptible to failure due to environmental corrosion, which degrades measurement accuracy and may lead to machine shutdown, thereby severely compromising the reliability of the MEA [4]. The DC-link current sensor not only provides overcurrent protection functions but also facilitates phase current reconstruction, owing to the specific relationship between the DC-link current and the three-phase currents under active voltage vectors. Consequently, phase currents can be reconstructed based on the DC-link current to enhance the redundancy of PMSG current sensors, thereby improving the reliability of the MEA power supply system [5,6].

The technique utilizing a DC-link current sensor to reconstruct three-phase currents is referred to as phase current reconstruction technology, or single DC-link current sensor control. Phase current reconstruction necessitates the retrieval of two-phase currents during two active voltage vectors within a single control cycle. However, when the duration of the active voltage vectors is insufficient for accurate DC-link current measurement, current reconstruction dead zones emerge in the low modulation and sector boundary regions, thereby degrading reconstruction accuracy [7]. Numerous studies have investigated this issue, categorizing solutions into modified PWM methods, multi-branch sampling methods, and observer-based methods. In [8], measurement vectors with durations exceeding the minimum sampling time were injected into each PWM cycle. However, this method is inapplicable in the high modulation region and increases switching losses. The switching-state phase shift (SSPS) method was proposed and could suppress current reconstruction dead zones by phase shifting the switching control signals of either one phase (when the voltage vector is located in the sector boundary region) or two phases (when located in the low modulation region) [9]. Lu et al. replaced the zero vector at the end of the PWM cycle with an active voltage vector to enable phase current reconstruction. However, SSPS remains necessary when the duration of the active voltage vector is excessively short [10]. In [11], all zero vectors within the PWM cycle were replaced by active vectors, and the sectors of space vector pulse width modulation (SVPWM) were redefined. However, the absence of zero vectors in this approach results in increased DC-link current fluctuation. Furthermore, modified PWM methods induce asymmetry in the PWM waveforms, inevitably increasing current harmonics.

Multi-branch sampling methods achieve phase current reconstruction using a single current sensor that simultaneously samples the sum of currents from multiple branches. This allows for the acquisition of valid phase current information even during zero vectors, thereby shifting the dead zones from the low modulation and sector boundary regions to the high modulation region [12,13,14]. However, this approach is limited to utilizing Hall-effect current sensors for multi-point sampling and requires modifications to the rectifier topology. Consequently, it cannot be applied to intelligent power modules (IPMs), significantly constraining its scope of application. The utilization of state observers allows for the suppression of current reconstruction dead zones without altering the three-phase PWM waveforms. When the current can be accurately reconstructed, the reconstructed current serves as feedback for the observer. Conversely, when the system operates within a current reconstruction dead zone, the observer functions in an open-loop state [15,16]. Saritha et al. proposed a parameter-independent observer based on asymptotic curves. However, this method demonstrates superior performance only when the phase current is sinusoidal [16]. In [17], three independent adaptive current observers were constructed to compensate for three-phase current asymmetry. Furthermore, the difference between the reconstructed current and the observed current was utilized as an adaptive correction term to enhance reconstruction accuracy. In [18,19], Luenberger observers were employed to filter the reconstructed current, similarly improving reconstruction accuracy. However, the observation performance is susceptible to parameter perturbations. Zhu et al. proposed a robust current observer that improves current reconstruction accuracy and current loop bandwidth while ensuring the observer remains unaffected by parameter variations [20]. However, the parameter tuning of this method is too complex. Extended state observers are commonly employed to address uncertainties in parameters and disturbances in advanced control schemes, such as model predictive control, in power electronic systems [21,22].

Research on phase current reconstruction technologies primarily focuses on three-phase inverters and motors. Studies in [15,23,24] have investigated single DC-link current sensor control strategies for three-phase rectifiers. However, the impact of excessively high generator speed on phase current reconstruction has not been addressed. To enhance the reliability of the MEA, this paper investigates phase current reconstruction for the PMSG three-phase rectification system. The main contributions of this paper are summarized as follows:

(1)

Unlike previous studies that address flux-weakening and current reconstruction separately, this paper analyzes their coupling relationship and proposes an integrated solution that ensures both stable DC-link voltage and accurate reconstruction sampling points at high speeds.

(2)

Compared to existing robust current observers that require optimization of multiple parameters through iterative algorithms, the proposed LESO achieves comparable robustness with only a single tuning parameter, i.e., observer bandwidth, significantly reducing implementation complexity.

The remainder of this paper is organized as follows: Section 2 introduces the basic principle of phase current reconstruction. Section 3 proposes the LESO-based phase current reconstruction technology considering flux-weakening control. Experimental verification through a Hardware-in-the-Loop (HIL) test rig is provided in Section 4. Finally, Section 5 concludes this paper.

2. Principle of Phase Current Reconstruction Technology

In PMSG systems, closed-loop current control requires the detection of three-phase currents. Employing a DC-link current sensor for phase current reconstruction increases the redundancy of the current sensors, thereby enhancing system reliability. Phase current reconstruction is achieved by determining the current flow path under various voltage vectors. This path establishes the correspondence between the sampled DC-link current and the phase currents. Subsequently, the DC-link current is sampled twice within one switching cycle to obtain the information for two-phase currents, and the third phase current is determined using Kirchhoff’s current law.

Defining the positive directions for the three-phase currents and the DC-link current as shown in the topology of the three-phase two-level voltage source rectifier in Figure 1, the relationship between the DC-link current i_dc and the three-phase currents i_a, i_b, and i_c is given by

i_dc = S_ai_a + S_bi_b + S_ci_c,

(1)

where S_x(x = a, b, c) is the switching function for phase x. S_x = 1 represents the turn-on of the upper bridge switch or its anti-parallel diode and the turn-off of the lower switch, while S_x = 0 represents the turn-on of the lower bridge switch or its anti-parallel diode and the turn-off of the upper switch.

The six switches can generate eight switching states and the corresponding voltage vectors V₀–V₇, which include six active voltage vectors V₁–V₆ and two zero vectors V₀ and V₇. The correspondence between the DC-link current and the three-phase currents under the eight voltage vectors is presented in

Table 1. The relationship between the DC-link current and three-phase currents.

V	V0	V1	V2	V3	V4	V5	V6	V7
SaSbSc	000	100	110	010	011	001	101	111
idc	0	+ia	−ic	+ib	−ia	+ic	−ib	0

. Each SVPWM switching cycle includes two active voltage vectors, which allow for the separate reconstruction of two-phase currents. Assuming the zero-sequence current of the PMSG is zero, the third phase current is obtained from the relationship: i_a + i_b + i_c = 0, thus completing the phase current reconstruction for that switching cycle.

Owing to the existence of the minimum sampling time T_min, current reconstruction dead zones exist in the low modulation region and the sector boundary regions of the SVPWM hexagon, as illustrated in Figure 2. T_min is primarily composed of three components: the reserved dead time (to prevent shoot-through of the upper and lower bridge arms of the same phase), the switch stabilization time, and the analog-to-digital conversion (ADC) time [25]. When the voltage vector is located within the dead zone, meaning the duration of the active voltage vector is less than T_min, the reconstructed current becomes distorted. To resolve this issue, the SSPS method is commonly used to actively extend the duration of the active voltage vector. The principle of SSPS is shown in Figure 3 [9]. In this figure, t_a and t_c represent the sampling instants for reconstructing phase a and phase c currents, respectively, while i_aS and i_cS denote the reconstructed currents for phase a and phase c. To ensure that the duration of the active voltage vectors V₁ and V₂ exceeds the minimum sampling time, the switching signal S_a (corresponding to the highest duty cycle) is phase-shifted to the left (advanced), and S_c (corresponding to the lowest duty cycle) is phase-shifted to the right (delayed). This guarantees reliable DC-link current sampling, enabling the reconstruction of phase a and phase c currents.

3. Phase Current Reconstruction Technology Based on Linear Extended State Observer Considering PMSG Flux-Weakening Control

As discussed in the previous section, the SSPS method is required to mitigate the current reconstruction dead zone issue. Nevertheless, SSPS introduces asymmetry in the PWM waveform, leading to increased phase current harmonics and total harmonic distortion (THD), and consequently impacting the current loop bandwidth. To address these limitations, this paper proposes a PMSG phase current reconstruction technology based on a linear extended state observer (LESO). This section begins with the modeling of the PMSG PWM rectification system. Subsequently, the lead-angle flux-weakening control strategy is introduced to ensure the stability of the rectifier output DC-link voltage and protect the DC-link current sampling points from the influence of excessively high speeds. Finally, the construction of the LESO is detailed.

3.1. Description of PMSG System and Flux-Weakening Control

While an MEA incorporates multiple generation systems comprising PMSGs, the fundamental configuration of each system is identical. Therefore, this paper describes the modeling process using a single PMSG system as a representative example. The voltage equations in the dq rotor flux reference frame for the PMSG, are given by [26]

$$\left\{\begin{array}{l}{u}_d=R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_e{L}_q{i}_q\\ u_q=R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e{L}_d{i}_d+{\omega }_e{\psi }_f\end{array}\right.$$

(2)

where u_s is the stator voltage vector, and u_d, u_q are its orthogonal components in the rotor flux-oriented dq reference frame; i_s is the stator current vector, and i_d, i_q are the d-axis and q-axis stator currents, respectively; L_d and L_q are the d-axis and q-axis stator inductances, respectively; R_s is the stator resistance; ψ_f is the permanent magnet flux linkage; and ω_e is the PMSG electrical angular frequency, that coincides with the rotor electrical speed.

In the three-phase two-level voltage source rectifier shown in Figure 1, neglecting the power loss of the PWM rectifier, the power balance between the DC side and the AC side can be expressed as

u_dci_dc = 1.5 (u_di_d + u_qi_q)

(3)

where u_dc denotes the rectifier output DC-link voltage. According to the MIL-STD-704F standard, u_dc = 270 V is the rated voltage in the MEA; thus, the reference DC-link voltage $u_{\mathrm{dc}}^{\ast }$ is set to 270 V in this paper [3].

The closed-loop controllers utilized in this paper are all proportional–integral (PI) controllers. The parameters of the current control loop are tuned using internal model control, as detailed in Equation (4) [27]. The reference and feedback for the voltage loop controller are $\frac{1}{2}{u_{\mathrm{dc}}^{\ast }}^2$ and $\frac{1}{2}{u}_{\mathrm{dc}}^2$, respectively [28].

$$\left\{\begin{array}{l}{K}_{pd}=2\pi f_c{L}_d, K_{id}=2\pi f_c{R}_s\\ K_{pq}=2\pi f_c{L}_q, K_{iq}=2\pi f_c{R}_s\end{array}\right.$$

(4)

where K_pd and K_pq are the proportional coefficients for the d-axis and q-axis, respectively; K_id and K_iq are the integral coefficients for the d-axis and q-axis, respectively; and f_c is the current loop bandwidth.

Owing to the relatively high engine speed of the MEA, flux-weakening control is necessary for the PMSG to ensure the PWM rectifier can stably output a 270 V DC-link voltage even when operating above the rated speed. Additionally, flux-weakening control ensures that the three-phase PWM operates as intended, thereby guaranteeing the accuracy of the DC-link current sampling points during phase current reconstruction.

When the PMSG operates at high speed, the other terms in the voltage equations are negligible compared to the PMSG back electromotive force, allowing Equation (2) to be simplified as

$$\left\{\begin{array}{l}{u}_d=-{\omega }_e{L}_q{i}_q\\ u_q={\omega }_e{L}_d{i}_d+{\omega }_e{\psi }_f\end{array}\right.$$

(5)

Based on Equation (5), the PMSG stator voltage u_s can be expressed as

$$u_s=\sqrt{u_d^2+u_q^2}={\omega }_e\sqrt{{\left(L_q{i}_q\right)}^2+{\left({\psi }_f+L_d{i}_d\right)}^2}\le u_{s\max }$$

(6)

where u_smax is the stator voltage limit, with u_smax = $u_{\mathrm{dc}}/\sqrt{3}$ for SVPWM. As can be seen from Equation (6), when the rectifier output limit is reached, if the PMSG speed continues to increase, reducing i_q and increasing the negative i_d are the only ways to maintain voltage balance and prevent DC-link voltage overmodulation.

The lead angle β is defined as the angle between the stator current vector i_s and the q-axis current i_q, as shown in Equation (7). The basic principle of the lead-angle flux-weakening control is illustrated in Figure 4, where the superscript * denotes the reference value. Compared to conventional dual closed-loop control, an additional lead angle β control loop is introduced, where a PI controller regulates the magnitude of β. The angle β must be limited within the range of −π/2 ≤ β < 0 to achieve the objective of negatively increasing i_d (i.e., demagnetization).

$$\left\{\begin{array}{c}{i}_d=i_{\mathrm{s}}\sin \beta \\ i_q=i_{\mathrm{s}}\cos \beta \end{array}\right.$$

(7)

The PMSG lead-angle flux-weakening control is primarily divided into two operating stages:

(1)

During 0 ≤ ω_e ≤ ω_e1, where ω_e1 is the transition speed given by Equation (8), the stator voltage u_s is less than the voltage limit u_smax. The lead-angle controller is in forward saturation, yielding β = 0 and i_d = 0. In this stage, the PMSG operates in the constant torque region below base speed.

$${\omega }_{e1}={\displaystyle \frac{u_{s\max }}{\sqrt{{\left(L_q{i}_q\right)}^2+{\psi }_f^2}}}$$

(8)

(2)

During ω_e ≥ ω_e1, the stator voltage u_s reaches the voltage limit u_smax. The lead-angle controller input becomes negative, exiting saturation, and generating a negative β (where −π/2 ≤ β < 0). This simultaneously produces a negative d-axis current $i_d^{\ast }$, and the PMSG enters the flux-weakening state. However, $i_d^{\ast }$ is physically limited by i_lim and i_dmax = ψ_f/L_d, resulting in

$$i_d^{\ast }⩽\min (i_{d\max }, i_{\lim })$$

(9)

where $i_d^{\ast }$ represents the d-axis current reference; i_dmax represents the maximum demagnetizing current and the upper bound to prevent irreversible demagnetization determined by ψ_f/L_d; i_lim represents rated current limit and the maximum allowable continuous current determined by the thermal constraints of the PMSG windings and rectifier power semiconductors. When β = −π/2 and i_q = 0, the d-axis current i_d reaches its maximum value $i_{d\max }^{\ast }$. The maximum electrical angular frequency ω_emax of the PMSG is then given by

$${\omega }_{e\max }={\displaystyle \frac{u_{s\max }}{{\psi }_f-L_d{i}_{d\max }^{\ast }}}$$

(10)

3.2. LESO-Based Phase Current Reconstruction Technology

The phase shifting performed by SSPS introduces asymmetry in the three-phase PWM waveform, leading to increased current harmonics. While filtering the reconstructed current using an observer can improve the steady-state performance of current reconstruction, the observer’s performance is susceptible to degradation due to PMSG parameter perturbations. To address this issue, this paper proposes a phase current reconstruction technology based on the linear extended state observer (LESO) to enhance reconstruction accuracy while simultaneously achieving parameter robustness.

During actual PMSG operation, parameters such as the stator resistance R_s, inductances L_d and L_q, and flux linkage ψ_f are subject to variations influenced by temperature and stator current. These variations introduce dq-axis disturbances d_d and d_q into the PMSG dynamics in Equation (2) as follows:

$$\left\{\begin{array}{l}{u}_d=R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_e{L}_q{i}_q+d_d\\ u_q=R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e{L}_d{i}_d+{\omega }_e{\psi }_f+d_q\end{array}\right.$$

(11)

The lumped disturbances are defined as follows:

$$\left\{\begin{array}{l}{d}_d^L\triangleq -L_d^{-1}\left(R_s{i}_d-L_q{\omega }_e{i}_q+d_d\right)\\ d_q^L\triangleq -L_q^{-1}\left(R_s{i}_q+L_d{\omega }_e{i}_d+{\psi }_f{\omega }_e+d_q\right)\end{array}\right.$$

(12)

The PMSG current dynamics can thus be derived as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dt}}{i}_d=L_d^{-1}{u}_d+d_d^L\\ {\displaystyle \frac{d}{dt}}{i}_q=L_q^{-1}{u}_q+d_q^L\end{array}\right.$$

(13)

As can be seen from Equation (13), the PMSG d-axis and q-axis current dynamic equations exhibit the same form after the lumped disturbances are defined. Taking the state variables of the LESO for the y-axis (y = d, q) as i_y and $d_y^L$, the LESO is constructed as follows:

$$\left\{\begin{array}{lll}s{\hat{i}}_y& =& L_y^{-1}{u}_y+{\hat{d}}_y^L+h_1{e}_y\\ s{\hat{d}}_y^L& =& h_2{e}_y\\ e_y& =& i_y-{\hat{i}}_y\end{array}\right.$$

(14)

where ${\hat{i}}_y$ and ${\hat{d}}_y^L$ are the y-axis estimated current and the estimated lumped disturbance, respectively. e_y represents the error between the actual and estimated y-axis current. h₁ and h₂ are the LESO parameters related to the LESO bandwidth, which are set as $h_1=2{\omega }_o$ and $h_2={\omega }_o^2$, with ω_o being the observer bandwidth [29]. Based on Equation (14), the disturbance estimation transfer function $G_{DE}(s)$ of the LESO can be derived as

$$G_{DE}(s)={\displaystyle \frac{{\hat{d}}_y^L(s)}{d_y^L(s)}}={\displaystyle \frac{h_2}{s^2+h_{1}s+h_2}}={\displaystyle \frac{{\omega }_o^2}{{\left(s+{\omega }_o\right)}^2}}$$

(15)

And the current estimation transfer function $G_{CE}(s)$ can be derived as

$$G_{CE}(s)={\displaystyle \frac{{\hat{i}}_y(s)}{i_y(s)}}={\displaystyle \frac{h_{1}s+h_2}{s^2+h_{1}s+h_2}}={\displaystyle \frac{2{\omega }_{o}s+{\omega }_o^2}{{\left(s+{\omega }_o\right)}^2}}$$

(16)

Based on Equations (15) and (16), s² + h₁s + h₂ is the characteristic equation of the LESO. By setting h₁ and h₂ to $h_1=2{\omega }_o$ and $h_2={\omega }_o^2$, respectively, the two roots of the characteristic equation are both placed at −ω_o, ensuring the LESO is stable. Applying the forward Euler method to discretize the LESO yields

$$\left\{\begin{array}{l}{\hat{i}}_y(z)={\displaystyle \frac{T_s}{z-1}}\left[L_y^{-1}{u}_y(z)+{\hat{d}}_y^L(z)+h_1{e}_y(z)\right]\\ {\hat{d}}_y^L(z)={\displaystyle \frac{T_s}{z-1}}{h}_2{e}_y(z)\\ e_y(z)=i_y(z)-{\hat{i}}_y(z)\end{array}\right.$$

(17)

where T_s is the sampling period. The pole plots for the discrete transfer functions G_DE and G_CE after LESO discretization are shown in Figure 5. It is observed that when the parameters are set to $h_1=2{\omega }_o$ and $h_2={\omega }_o^2$, the poles are both located inside the unit circle and lie on the real axis, which indicates system stability and a non-oscillatory system response. Furthermore, a larger LESO bandwidth ω_o moves the poles further away from the unit circle, resulting in a faster transient response.

To facilitate implementation in a practical embedded system, converting Equation (17) into a difference equation yields

$$\left\{\begin{array}{lll}\hfill e_y(k)& =& i_y(k)-{\hat{i}}_y(k)\\ \hfill {\hat{i}}_y(k+1)& =& {\hat{i}}_y(k)+T_s\left[L_y^{-1}{u}_y(k)+{\hat{d}}_y^L(k)+h_1{e}_y(k)\right]\\ \hfill {\hat{d}}_y^L(k+1)& =& {\hat{d}}_y^L(k)+T_s{h}_2{e}_y\end{array}\right.$$

(18)

By subtracting the estimated lumped disturbance ${\hat{d}}_y^L$ from the output of the current controller, the effects introduced by PMSG parameter perturbations can be effectively compensated for

$$\left\{\begin{array}{l}{u}_d^{\ast }=\left(K_{pd}+{\displaystyle \frac{K_{id}}{s}}\right)\left(i_d^{\ast }-{\hat{i}}_d\right)-L_d{\hat{d}}_y^L\\ u_q^{\ast }=\left(K_{pq}+{\displaystyle \frac{K_{iq}}{s}}\right)\left(i_q^{\ast }-{\hat{i}}_q\right)-L_q{\hat{d}}_q^L\end{array}\right.$$

(19)

where the reference values $i_d^{\ast }$ and $i_q^{\ast }$ for the dq-axis current controller are obtained through the lead-angle flux-weakening control detailed in Section 3.1.

The control block diagram illustrating the LESO and the proposed LESO-based phase current reconstruction technology is presented in Figure 4. In this work, the lead-angle flux-weakening control is employed to prevent DC-link voltage overmodulation caused by excessive MEA engine speed and to ensure accurate sampling for phase current reconstruction. Concurrently, the LESO-based phase current reconstruction technology enhances current sensor reliability, as the LESO improves the accuracy of phase current reconstruction while also accounting for the effects of parameter perturbations during PMSG operation.

3.3. Comparison Between LESO and Robust Current Observer

To clarify the contribution of the proposed LESO relative to existing observer-based phase current reconstruction methods, particularly the RCO proposed in [20], the key differences are summarized as follows:

(1)

Observer structure: The RCO employs a traditional Luenberger-type state observer combined with an external integral-form disturbance compensator. The disturbance compensation requires additional dq-axis decoupling terms to handle cross-coupling effects. In contrast, the proposed LESO treats the lumped disturbance as an extended state variable within a unified observer framework. The d-axis and q-axis LESOs are completely independent, eliminating the need for cross-coupling compensation.

(2)

Parameter tuning complexity: The RCO requires tuning six parameters through a gradient-based iterative optimization algorithm that minimizes the condition number of the eigenvector matrix. The proposed LESO requires only one parameter, i.e., the observer bandwidth ω_o, with gains directly calculated as $h_1=2{\omega }_o$ and $h_2=2{\omega }_o^2$. This bandwidth-parameterized approach enables intuitive tuning: a larger ω_o increases tracking speed but amplifies measurement noise, resulting in straightforward tradeoffs.

(3)

Frequency-domain characteristics: The LESO provides explicit transfer functions for disturbance estimation G_DE and current estimation G_CE. These transfer functions reveal that the LESO acts as a second-order low-pass filter for disturbance estimation with unity DC gain, ensuring accurate steady-state disturbance rejection. The RCO does not have such explicit transfer function representations due to its external compensation structure.

(4)

Stability guarantee: According to Equations (15) and (16), the two poles of the LESO are located at s = −ω_o. This repeated real pole configuration guarantees asymptotic stability and critically damped response without oscillation, regardless of the operating speed. The RCO stability depends on satisfying the Lipschitz condition, which requires careful verification through iterative computation.

In summary, compared with the RCO, the LESO represents a fundamentally different observer architecture that offers simpler tuning, transparent frequency-domain design, and guaranteed stability characteristics.

4. Experimental Verification

To validate the proposed method, the RT-Box (OPAL-RT Technologies Inc., Montréal, QC, Canada)-based Hardware-in-the-Loop (HIL) experimental platform shown in Figure 6 was established. The controller utilized is the 32-bit DSP TMS320F28379D (Texas Instruments Inc., Dallas, TX, USA). The HIL platform and power modules specifications are summarized in

Table 2. The HIL platform and power modules specifications.

Parameter	Specification
Real-time simulator	PLECS RT-Box 1
Discretization step size	2.5 µs
Configuration of power modules	Sub-cycle average
Semiconductor symbol	MOSFET
Switch model	Ideal

. In the HIL PMSG model, iron losses, cross-saturation, magnetic saturation, spatial harmonics, and temperature effects are neglected. The parameters of the PMSG rectification system are detailed in

Table 3. PMSG rectification system parameters.

Quantity	Symbol	Value
Stator resistance	Rs	1.058 mΩ
d-axis inductance	Ld	99 µH
q-axis inductance	Lq	99 µH
Permanent magnet flux linkage	ψf	0.03644 Wb
Pole pairs	Pn	3
Rated power	Prate	15 kW
Rated speed	nrate	8000 r/min
DC-link capacitor	C	1.2 mF
Load resistance	RL	5 Ω

, and the engine operates at a constant speed. The PMSG rotor position is detected by a 2500-line incremental optical encoder. Experimental waveforms are output through the DSP’s PWM-to-DAC conversion module and subsequently captured using an oscilloscope. The switching frequency of the PWM rectifier is set to 10 kHz, the dead time is 1 µs, and the minimum sampling time T_min is 5 µs. The control parameters of the voltage loop are as follows: K_pv = 2.0, K_iv = 0.025, and the parameters of the lead angle β control loop are as follows: K_pa = 1.0, K_ia = 0.002. The current loop bandwidth f_c and the ESO bandwidth are both set to 500 Hz.

Initially, the necessity of PMSG flux-weakening control is verified. The engine speed is set to 12,000 r/min. The experimental comparison of the DC-link voltage and the actual dq-axis currents with and without flux-weakening control (i_d = 0) is shown in Figure 7. In the figure, n represents the actual PMSG speed, and i_dq is the actual PMSG dq-axis current vector. As can be observed from Figure 7a, without flux-weakening control, the DC-link voltage fluctuation, namely the standard deviation (STD), reaches 21.94 V and the average value is much higher than 270 V, and the STDs of d-axis and q-axis current are 67.13 A and 17.36 A, respectively. This demonstrates that the PWM rectifier is unable to maintain the desired DC-link voltage control. Upon the introduction of the flux-weakening control, as shown in Figure 7b, the DC-link voltage fluctuation decreases significantly to 11.08 V, and the STDs of d-axis and q-axis currents are reduced to 10.6 A and 9.07 A, respectively. This confirms that the DC-link voltage fluctuation is contained within a reasonable level and the DC-link voltage converges to 270 V due to the implemented flux-weakening control.

To verify the necessity of flux-weakening control for PMSG phase current reconstruction, Figure 8 shows the three-phase PWM waveforms at 12,000 r/min without and with flux-weakening control. As seen from Figure 8a, since the speed is above the rated speed, i_d = 0 control leads to saturation of the current loop output. Consequently, the PWM duty cycles are not output normally according to the expected SVPWM, causing the DC-link current sampling points to deviate from the ideal sampling instants (active voltage vectors). However, when flux-weakening control is employed, the current loop exits saturation, the three-phase PWM duty cycles and DC-link current sampling points return to normal, allowing for proper phase current reconstruction. Figure 9 shows the actual three-phase currents and the currents reconstructed via SSPS without flux-weakening control. In the figure, i_{abc_A} denotes the actual three-phase currents obtained using two-phase current sensors, i_{abc_SSPS} denotes the three-phase currents reconstructed using SSPS; e_{a_SSPS} is the error between the SSPS phase a reconstructed current and the phase a actual current. As observed from Figure 9a, even the actual currents sampled by the two-phase current sensors contain significant current harmonics. In Figure 9b, the current reconstruction error is substantial due to the abnormal DC-link current sampling without flux-weakening control, with the STD of e_{a_SSPS} reaching 59.95 A. The actual three-phase current and SSPS reconstructed current waveforms after incorporating flux-weakening control are shown in Figure 10a,b, respectively. Through comparison, it is evident that the sinusoidal quality of both the actual and reconstructed phase currents is significantly enhanced by utilizing the lead-angle flux-weakening control. Furthermore, due to the restored integrity of the DC-link current sampling, the STD of e_{a_SSPS} is significantly reduced compared to the case without flux-weakening control. These experiments collectively prove the necessity of introducing flux-weakening control during phase current reconstruction. However, by comparing Figure 9b and Figure 10b, it can be observed that SSPS still exhibits a large current reconstruction error magnitude caused by the inherent current reconstruction dead zone.

The proposed LESO-based phase current reconstruction technology is compared with the robust current observer (RCO)-based phase current reconstruction technology proposed in [20]. Their experimental results at 12,000 r/min with flux-weakening control are shown in Figure 10c,d, where i_{abc_RCO} and i_{abc_LESO} denote the three-phase reconstructed current waveforms obtained using the RCO and LESO, respectively, while e_{a_RCO} and e_{a_LESO} denote the corresponding phase a current reconstruction errors. By comparing the three experimental results in Figure 10, the STDs of current reconstruction errors for SSPS, RCO, and LESO are found to be 30.72 A, 24.08 A, and 17.72 A, respectively. The current reconstruction accuracy of the LESO is improved by 42.32% and 26.41% compared to SSPS and RCO, respectively, demonstrating the highest current reconstruction accuracy. The STD of the reconstruction error and its proportion relative to the rated current for the various methods are summarized in

Table 4. Standard deviation at various speeds and the proportion of STD relative to rated current for three methods.

Methods	12,000 r/min		8000 r/min
	STD	Proportion	STD	Proportion
SSPS	30.72 A	12.02%	18.27 A	12.1%
RCO	24.08 A	9.42%	14.75 A	9.77%
LESO	17.72 A	6.93%	11.99 A	7.94%

. It can be observed that the proportion of the STD to the rated current for all three methods remains small, with the proposed method exhibiting the smallest proportion.

To further compare the steady-state performance of the proposed method, Figure 11 presents the three-phase current experimental results at 8000 r/min, and Figure 12 shows the actual DC-link voltage, speed, and dq-axis currents experimental results. Similar to the experimental results in Figure 10, the proposed method exhibits the highest phase current reconstruction accuracy in Figure 11 and Table 4. Compared to SSPS and RCO, the STDs of phase a current reconstruction errors using the proposed method is reduced by 34.37% and 18.71%, respectively. In Table 4, the proposed method exhibits the smallest proportion. Benefiting from the superior current reconstruction accuracy of the proposed method, the DC-link voltage ripple shown in Figure 12 for the LESO is the smallest among the three methods, and its magnitude is comparable to the DC-link voltage ripple achieved using two-phase current sensor sampling. It is worth noting that the dq-axis current ripple obtained using the proposed phase current reconstruction method in Figure 12d is smaller than the dq-axis current ripple sampled directly by the two-phase current sensors in Figure 12a. This is because sampling errors, primarily caused by DC offset and gain mismatch between the two-phase current sensors, reduce the accuracy of the current loop control.

To investigate the effect of the LESO bandwidth ω_o on phase current reconstruction accuracy at 8000 r/min, Figure 13a,b present the three-phase reconstructed currents and their error waveforms for ω_o = 3ω_c and ω_o = 5ω_c, respectively, with ω_c = 2πf_c. Compared with Figure 11d, i.e., ω_o = ω_c, it can be observed that a larger LESO bandwidth results in a larger STD of the reconstructed current error and thus poorer phase current reconstruction accuracy. The settling time of LESO can be derived from Equation (16). For the ±5% criterion, the settling time of LESO satisfies t_s ≈ 4/ω_o. A reduction in the LESO bandwidth increases the observer settling time and slows down the transient response. Therefore, a tradeoff between reconstruction accuracy and transient performance is required according to the practical system. For single-DC-link current sensor control, reconstructed current accuracy is the primary concern. Otherwise, the current sampling accuracy required by the PMSG current control loop cannot be satisfied, making closed-loop operation infeasible. Therefore, it is recommended to select the LESO bandwidth to be one to two times the current loop bandwidth.

To verify the robustness of the proposed method, it is compared with the Luenberger observer (LO)-based phase current reconstruction technology proposed in [19]. Figure 14 and Figure 15 respectively show the DC-link voltage and the estimated dq-axis currents of the two methods at 8000 r/min under variations in dq-axis inductances and variations in stator resistance and permanent magnet flux linkage. In the figures, i_{dq_LO} and i_{dq_LESO} denote the dq-axis currents obtained by the Luenberger observer and the proposed method, respectively. It can be observed that the Luenberger observer is unable to withstand PMSG parameter perturbations; its current estimation accuracy is significantly affected by variations in dq-axis inductances, stator resistance, and permanent magnet flux linkage, particularly the q-axis inductance. In contrast, the DC-link voltage and dq-axis currents of the proposed method remain largely unaffected by PMSG parameter variations, demonstrating strong robustness.

Table 5. Execution time of various algorithms on the TMS320F28379D.

	Flux-Weakening Control	SSPS	RCO	LESO
Time	0.48 µs	1.84 µs	3.64 µs	1.43 µs

presents the execution time of the various algorithms on the TMS320F28379D. It can be observed that although the lead-angle flux-weakening control incorporates a PI controller and two trigonometric function operations, its execution time only consumes 0.48 μs. Compared to the robust current observer, the LESO reduces the execution time by 2.21 μs, requiring fewer computational resources while still achieving improved phase current reconstruction accuracy.

5. Conclusions

To enhance the reliability of the PMSG rectification system, this paper introduces phase current reconstruction technology to provide current sensor redundancy. Given the high engine speed characteristics of MEA, lead-angle flux-weakening control is integrated into the system to guarantee stable DC-link voltage output and accurate sampling for phase current reconstruction. Furthermore, addressing the current harmonics and three-phase PWM waveform asymmetry caused by the conventional SSPS method, this paper proposes a novel phase current reconstruction method based on the LESO. Compared to the RCO, the LESO requires only a single tuning parameter, eliminating the need for iterative optimization. By utilizing the LESO to filter the reconstructed current with inherent robustness, current harmonics are suppressed, and the overall reconstruction accuracy is significantly improved.

HIL results confirmed that the integration of lead-angle flux-weakening control significantly reduces DC-link voltage ripple by 49.5% and improves phase current reconstruction accuracy by 48.76% in the PMSG system. Furthermore, the proposed LESO-based phase current reconstruction method demonstrates strong immunity to parameter perturbations. Under rated operating conditions, the current reconstruction accuracy is improved by 34.37% compared to SSPS and by 18.71% compared to the RCO. Future work will focus on developing observer-only solutions that eliminate the dependence on SSPS, and experimental validation on physical PMSG test benches. The proposed method is not only applicable to power generation systems but also to motor drive applications such as servo control. Its minimal computational resource requirement in embedded systems underscores its high practical application value.