A novel control strategy for PWM rectifier based on underactuated characteristics

Abstract

In this paper, a double closed loop controller based on equivalent input disturbances (EID) error perturbation is proposed to solve the problem of abnormal operation of three-phase pulse width modulation (PWM) rectifier caused by load and external conditions. By analysing the buffeting problem and response time problem caused by traditional sliding mode in the current control process, a new current control method of terminal fuzzy sliding mode is designed. The system error is estimated by EID error estimation, and fuzzy control is introduced to improve the accuracy of error estimation, which solves the current buffeting and response problem. In order to solve the problem of response rate and harmonics in the voltage controller design, the extended state observer (ESO) in active disturbance rejection control (ADRC) control is improved, EID error estimation is introduced, the response rate and anti-disturbance ability of the system are further improved, and the harmonics of the system are suppressed. The experimental results show the effectiveness and superiority of the proposed method.

Keywords

Pulse width modulation rectifier underactuated characteristic terminal fuzzy sliding mode control equivalent input disturbance active disturbance rejection control

Introduction

Nowadays, green low-carbon environmental protection has become a common development trend.^1,2 Electricity is the most widely used energy source in both life and industrial development, so there are high requirements on the quality of electricity.^3,4 For the purpose of getting high quality electricity and reduce energy losses, the electricity needs to be controlled and transformed in its production, transmission, distribution and use.^5,6 The current intelligentisation of electricity grids puts forward much more high demands to performance of power electronic appliances, and the stabilisation and reliable operation of the converter, as a key device in power conversion, also need to have strict requirements.^7,8 A three-phase PWM rectifier extensively applied to microgrid,⁹ superconducting energy storage,¹⁰ motor drive speed regulation,¹¹ and other fields with the advantages of high power factor and bi-directional energy transfer, and its stability and reliability research value is high.^12,13

This paper considers a PWM rectifier as a target, analysing rectifiers underactuated properties,¹⁴ aiming to promote power utilisation in grids and to ensure that the PWM rectifier can operate normally under load transformations and external disturbances. To these two ends, the PI control methods were designed but cannot meet the control demand well, so many advanced control methods are generated.^15,16 For example, Zou et al.¹⁷ presented a generalised Clark’s transform and enhanced dual-circuit control system. Interleaved single-phase PWM monostage resonant rectifiers featuring high voltage isolation was created by literature.¹⁸ Interleaved Single-Stage resonant rectifiers for three-phase power PWM combined with high voltage isolation was created by literature.¹⁹ A quasi-resonant prolonged phase observer-based predicted current regulation method was presented by literature.²⁰ For the single phase PWM rectifiers, the grid-voltage sensorless modelling for forecasting control was presented by literature.²¹ Ashry et al.²² development of a simplicity mould for UHF rectifiers to enable the optimisation in rectifier performances. Additionally, other nonlinear control methods have been used, such as direct power control,^23,24 fuzzy self-tuning,^25,26 active disturbance rejection control strategy^27,28 and model predictive control,^29,30 etc. For PWM rectifier, high power factor operation and fast response are essential, and to assure an effective system response to external perturbations and has high stability in the face of parameter variations, for example, voltage surges or load transformations.

In this paper, we analysed PWM rectifier’s characteristics and use its characteristics to divide the PWM rectifier system into a dual closed-loop control system with actuated subsystem and underactuated subsystem for reducing the energy usage. The actuated subsystem is a current-loop control system, which uses terminal fuzzy sliding mode control based on EID error estimation to reduce the current jitter and negative sequence current caused by conventional sliding mode, and optimises the nonlinear function in sliding mode control to improve suppression effect. Underactuated subsystem is a voltage loop control system, and the EID-based active disturbance rejection control (ADRC) strategy is designed to improve the system immunity while suppressing the harmonics in DC-side voltage. The purpose of adding EID is to diminish the current square wave jitter under ADRC control strategy. Finally, a simulation experiment platform is built on MATLAB/Simulink in order to verified the efficiency for proposed strategies this research suggests. Then simulation results shows an achievement of control objectives of PWM rectifier: (1) unit power operation of the system; (2) achieving current sinusoidalisation. Experiments validate the accuracy and superiority of the controlling method. In summary, there are three key contributions from this paper.

(1) The underactuated characteristics of PWM rectifier are analysed, and the feasibility is explained for dual closed-loop control based on its underactuated characteristics.

(2) Aiming at the current loop control method, we analyse the reasons of leading to the systems large buffeting in the traditional sliding control and the advantages and disadvantages of its control, and proposed a novel sliding control method which can reduces the current buffeting as well as improves the control effect.

(3) For the voltage loop control method, we propose an ADRC control method based on EID error estimation, which effectively solves the voltage tuning problem, and also improves the dynamics and stability properties for systems.

A structural framework of content of the paper has been organisation as follows. In Section ‘Modeling and analysis’, we explain the modeling, and carry out its characteristic analysis and feasibility analysis. In Section ‘Current loop controller design’, we propose a new controlling strategy to target the electric circuit control network. In Section ‘Voltage loop controller design’, we propose an ADRC control method based on EID error estimation for voltage loops. In Section ‘Simulation experiments’, we carry out experimental verification for different working scenarios of PWM rectifier. Finally, the work of this paper is summarised.

Modeling and analysis

Mathematical model

The principle diagram of the circuit structure of the 3-phase voltage PWM rectifier as shown in Figure 1.

Figure 1.

PWM rectifier circuit structure topology diagram.

Define the switch function as shown in equation (1).

\begin{matrix} S_{K (K = a, b, c)} = {\begin{matrix} 0, Upper bridge arm conduction \\ 1, Lower bridge arm conduction \end{matrix} \end{matrix}

(1)

As shown in equation (2), the PWM rectifier is mathematically modelled using Kirkhoff’s voltage and current laws.³¹

{\begin{matrix} L \frac{d i_{a}}{d t} + R i_{a} = e_{a} - u_{0} S_{a} - \frac{1}{3} (S_{a} + S_{b} + S_{c}) u_{0} \\ L \frac{d i_{b}}{d t} + R i_{b} = e_{b} - u_{0} S_{b} - \frac{1}{3} (S_{a} + S_{b} + S_{c}) u_{0} \\ L \frac{d i_{c}}{d t} + R i_{c} = e_{c} - u_{0} S_{c} - \frac{1}{3} (S_{a} + S_{b} + S_{c}) u_{0} \\ C \frac{d u_{0}}{d t} + \frac{u_{0}}{R_{L}} = i_{a} S_{a} + i_{b} S_{b} + i_{c} S_{c} \end{matrix}

(2)

where, $e_{a}, e_{b}, e_{c}$ are the AC voltage; $i_{a}, i_{b}, i_{c}$ are the AC currents; $L and R$ respectively are the AC side inductor and resistance; $C$ is the DC side capacitor; $R_{L}$ is the DC side resistance; $u_{0}$ is the DC voltage.

We substitute the above equation into the dq synchronous rotating coordinate system and use the equal power coordinate transform method to get the simplified mathematics mode of the PWM rectifier as shown in equation (3).

{\begin{matrix} L \frac{d i_{d}}{d t} = e_{d} - R i_{d} + ω L i_{q} - S_{d} u_{0} \\ L \frac{d i_{q}}{d t} = e_{q} - ω L i_{d} - R i_{q} - S_{q} u_{0} \\ C \frac{d u_{0}}{d t} = S_{d} i_{d} + S_{q} i_{q} - \frac{u_{0}}{R_{L}} \end{matrix}

(3)

Characteristic analysis

By using equation (2), a state space expression is obtained as shown in equation (4).

[\begin{matrix} {\overset{\cdot}{i}}_{a} \\ {\overset{\cdot}{i}}_{b} \\ {\overset{\cdot}{u}}_{0} \end{matrix}] = A [\begin{matrix} i_{a} \\ i_{b} \\ u_{0} \end{matrix}] + B [\begin{matrix} S_{a} \\ S_{b} \\ S_{c} \end{matrix}] + C [\begin{matrix} e_{a} \\ e_{b} \\ 0 \end{matrix}]

(4)

Where

\begin{matrix} A = [\begin{matrix} - \frac{R}{L} & 0 & 0 \\ 0 & - \frac{R}{L} & 0 \\ 0 & 0 & - \frac{1}{R_{L} C} \end{matrix}], B = [\begin{matrix} - \frac{2 u_{0}}{3 L} & \frac{u_{0}}{3 L} & \frac{u_{0}}{3 L} \\ \frac{u_{0}}{3 L} & - \frac{2 u_{0}}{3 L} & \frac{u_{0}}{3 L} \\ \frac{i_{a}}{C} & \frac{i_{b}}{C} & \frac{- i_{a} - i_{b}}{C} \end{matrix}] . \end{matrix}

According to the state space expression described in equation (4), the coefficient matrices $rank (B) < 3$ and $rank (A) = 3$ can be derived, which determines that the number of control inputs to the system is less than the number of degrees of freedom it has. Based on the definition of an underactuated system, it can be determined that PWM rectifier system is a electrical and electronic system with underactuated characteristics. Uses the underactuated characteristic, the system is divided into a drive subsystem and an underactuated subsystem, and its control strategy is analysed and designed from an underactuated perspective.

Feasibility analysis

The final control strategy designed is given in Figure 2. Following this, a feasibility analysis of the above options is performed using mathematical methods. As can be seen from equation (3), $u_{0}$ itself, as the state variable of the system, has a coupling relationship with the other two state variables $i_{d} and i_{q}$ , and the state variables $i_{d}$ and $i_{q}$ reflect the active power and reactive power of the system itself, which means that as long as the stability of $i_{d}$ and $i_{q}$ is controlled, the power stability of the system can be achieved, thereby indirectly achieving the stability of the DC voltage. Then the feasibility analysis is carried out.

Figure 2.

Control strategies structure diagram.

The instantaneous power balance for PWM rectifiers is expressed as equation (5) by means of an iso-power coordinate transformation.

e_{d} i_{d} = L (i_{d} \frac{d i_{d}}{d t} + i_{q} \frac{d i_{q}}{d t}) + R (i_{d}^{2} + i_{q}^{2}) + C u_{0} \frac{d u_{0}}{d t} + \frac{u_{0}^{2}}{R_{L}}

(5)

In equation (5), in order for the current loop to achieve precise control, the current needs to converge to a given value, define $i_{d} = i_{dr}$ , $i_{q} = 0$ , illustrated in equation (6).

\frac{d u_{0}}{d t} = \frac{1}{C u_{0}} (e_{d} i_{dr} - {Ri}_{dr}^{2} - \frac{u_{0}^{2}}{R_{L}}) = \frac{1}{R_{L} C u_{0}} (u_{r}^{2} - u_{0}^{2})

(6)

From equation (6), so that $u_{0}$ converges to $u_{r}$ , that is, $\frac{d u_{0}}{d t} = 0$ , illustrated in equation (7).

\frac{d u_{0}}{d t} = \frac{1}{C u_{0}} (e_{d} i_{dr} - {Ri}_{dr}^{2} - \frac{u_{0}^{2}}{R_{L}}) = 0

(7)

According to equation (2), the transfer function equation of $u_{0}$ and $i_{d}$ can be obtained as

u_{0} (s) = \frac{R_{L} (e_{d} - 2 R i_{dr} - sL i_{dr})}{(2 + s R_{L} C) u_{r}} i_{d} (s)

(8)

Therefore, it is theoretically only necessary that $i_{d}$ converge to given value $i_{dr}$ and $i_{q}$ converge to 0, which indirectly allows $u_{0}$ to converge to the expected value of $u_{r}$ . This leads to the feasibility of the scheme where $u_{0}$ is the underactuated variable.

Current loop controller design

Aiming at the design of current loop controller, the chattering problem of traditional sliding mode in the process of current control is analysed first, which is also an important problem of sliding mode control, and the reason of chattering in current control is analysed that the sliding mode gain is too large. To solve this large gain problem, we propose a terminal fuzzy sliding mode control method based on EID error estimation. The EID error estimation is used to predict the system interference, and the fuzzy control is used to improve the prediction accuracy and ensure that the designed sliding mode gain is close to the system interference, so as to reduce the current buffeting.

Conventional sliding mode control

Current loop system of control as illustrated by equation (9).

{\begin{matrix} L \frac{d i_{d}}{d t} = - R i_{d} + ω L i_{q} - s_{d} u_{0} + e_{d} \\ L \frac{d i_{q}}{d t} = - ω L i_{d} - R i_{q} - s_{q} u_{0} + e_{q} \end{matrix}

(9)

In order to better reflect the active power and reactive power of the system, two sliding mode surfaces are selected for control, and $i_{d}$ and $i_{q}$ are selected as control variables.

Definition of error is:

{\begin{matrix} e_{1} = i_{dr} - i_{d} \\ e_{2} = i_{qr} - i_{q} \end{matrix}

(10)

Design the sliding surface as shown in equation (11).

{\begin{matrix} S_{1} = e_{1} \\ S_{2} = e_{2} \end{matrix}

(11)

The isokinetic convergence rate is:

{\begin{matrix} {\overset{\cdot}{S}}_{1} = - K_{1} sgn (S_{1}) K_{1} > 0 \\ {\overset{\cdot}{S}}_{2} = - K_{2} sgn (S_{2}) K_{2} > 0 \end{matrix}

(12)

Combining equations (9) and (12) yields

{\begin{matrix} s_{d} = \frac{- R i_{d} + ω L i_{q} + e_{d} - L K_{1} sgn (S_{1})}{u_{0}} \\ s_{q} = \frac{- R i_{q} - ω L i_{d} + e_{q} - L K_{2} sgn (S_{2})}{u_{0}} \end{matrix}

(13)

When there is parameter uptake or external disturbance in the system, the current control system description changes as equation (14).

{\begin{matrix} L \frac{d i_{d}}{d t} = - R i_{d} + ω L i_{q} - s_{d} u_{0} + e_{d} + d_{1} (t) \\ L \frac{d i_{q}}{d t} = - ω L i_{d} - R i_{q} - s_{q} u_{0} + e_{q} + d_{2} (t) \end{matrix}

(14)

where $d_{1} (t)$ and $d_{2} (t)$ are the disturbance signal caused by external interference.

Bringing the control equation of equations (13) into (14), we get

{\begin{matrix} {\overset{\cdot}{S}}_{1} = d_{1} (t) - K_{1} sgn (S_{1}) \\ {\overset{\cdot}{S}}_{2} = d_{2} (t) - K_{2} sgn (S_{2}) \end{matrix}

(15)

To make sure that the system is stable under disturbed states, it is necessary to make the sliding mode have a stable condition, that is,

{\begin{matrix} S_{1} {\overset{\cdot}{S}}_{1} < 0 \\ S_{2} {\overset{\cdot}{S}}_{2} < 0 \end{matrix}

(16)

which in turn leads to

{\begin{matrix} K_{1} > | d_{1} (t) | \\ K_{2} > | d_{2} (t) | \end{matrix}

(17)

In practice, the disturbance is often unknown, so the upper bound $K_{1}$ and $K_{2}$ of the disturbance are usually set at a large values to ensure that the system can be stable under the disturbance. But the larger gain will make the systems acceleration converging to sliding mode surface larger, which leads to a large velocity when reaching the sliding-mode surface and will cause jitter by crossing the sliding-mode surface frequently. In this paper, a global fast terminal fuzzy sliding mode control method is used to ameliorate the effect of disturbances and improve the current loop control performance.

Terminal fuzzy sliding mode control based on EID error estimation

Design slip surface:

{\begin{matrix} s_{1} = e_{1} + β_{1} e^{p / q} \\ s_{2} = e_{2} + β_{2} e^{p / q} \end{matrix}

(18)

where, the errors $e_{1}$ and $e_{2}$ are shown in equation (10), $β_{1}$ and $β_{2}$ are the positive odd. Because the current control system is a first-order system, there is no need to consider the singularity problem.

Derivation of equation (18) yields:

{\begin{matrix} {\overset{\cdot}{s}}_{1} = \frac{R i_{d} - ω L i_{q} + s_{d} u_{0} - e_{d}}{L} + β_{1} \frac{p}{q} e_{1}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{1} \\ {\overset{\cdot}{s}}_{2} = \frac{ω L i_{d} + R i_{q} + s_{q} u_{0} - e_{q}}{L} + β_{2} \frac{p}{q} e_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{2} \end{matrix}

(19)

Define the Lyapunov function

{\begin{matrix} V_{1} = \frac{1}{2} s_{1}^{2} \\ V_{2} = \frac{1}{2} s_{2}^{2} \end{matrix}

(20)

The derivative of equation (20) yields:

{\begin{matrix} {\overset{\cdot}{V}}_{1} = s_{1} (\frac{R i_{d} - ω L i_{q} + s_{d} u_{0} - e_{d}}{L} + β_{1} \frac{p}{q} e_{1}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{1}) \\ {\overset{\cdot}{V}}_{2} = s_{2} (\frac{ω L i_{d} + R i_{q} + s_{q} u_{0} - e_{q}}{L} + β_{2} \frac{p}{q} e_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{2}) \end{matrix}

(21)

According to the system stability principle, $\dot{V_{1}} \leq 0$ and $\dot{V_{2}} \leq 0$ , the design control rate is

{\begin{matrix} s_{d} = \frac{- R i_{d} + ω L i_{d} + e_{d} - L k_{1} (t) sgn (s_{1}) - L β_{1} \frac{p}{q} e_{1}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{1}}{u_{0}} \\ s_{q} = \frac{- R i_{q} - ω L i_{d} + e_{d} - L k_{2} (t) sgn (s_{2}) - L β_{2} \frac{p}{q} e_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{2}}{u_{0}} \end{matrix}

(22)

where, $k_{1} (t) = max | d_{1} (t) | + η_{1}$ , $k_{2} (t) = max | d_{2} (t) | + η_{2}$ , $η_{1}$ and $η_{2}$ are small normal numbers, $sgn (\cdot)$ is Symbolic function.

Substituting equations (22) into (14) for a current system with external disturbances yields

{\begin{matrix} {\overset{\cdot}{V}}_{1} = s_{1} {\overset{\cdot}{s}}_{1} = - k_{1} (t) | s_{1} | - s_{1} d_{1} (t) \leq - η_{1} | s_{1} | \\ {\overset{\cdot}{V}}_{2} = s_{2} {\overset{\cdot}{s}}_{2} = - k_{2} (t) | s_{2} | - s_{2} d_{2} (t) \leq - η_{2} | s_{2} | \end{matrix}

(23)

When $\dot{V_{1}} \equiv 0$ and $\dot{V_{2}} \equiv 0$ , $\overset{\cdot}{s} \equiv 0$ . The system is asymptotically stable according to LaSalle’s invariance principle.³²

In order to ensure that the system can be better stabilised under disturbances, the fuzzy control theory is used to optimize the problem that the upper bounds of disturbances $k_{1} (t)$ and $k_{2} (t)$ are set too large. In the terminal sliding mode control rate equation (22), we make the disturbance upper bounds $k_{1} (t)$ and $k_{2} (t)$ vary with the disturbance $d_{1} (t)$ and $d_{2} (t)$ and satisfy the $K_{1, 2}$ at every moment everywhere slightly larger than the disturbance.

In order to make $k_{1, 2} (t)$ as close as possible to the real $d_{1, 2} (t)$ , we first set up the EID strategy to estimate the system error, thereby obtaining the current loop error observations ${\hat{d}}_{1} (t)$ and ${\hat{d}}_{2} (t)$ . From the above analysis can be obtained the switching gain $k_{1, 2} (t)$ needs to be slightly greater than the error $d_{1, 2} (t)$ , so this paper added a very small normal number $η$ in the above design of the switching gain. But if the error is too large, the error measurement of the observer plus a very small $η$ cannot guarantee that the switching gain $k_{1, 2} (t)$ at this time to maintain good results. So this paper uses the fuzzy control to design an adaptive $η$ to solve the problem.

The expression of the switching gain is shown in equation (24).

{\begin{matrix} {\tilde{k}}_{1} (t) = {\hat{d}}_{1} (t) + H_{1} \int_{0}^{t} Δ η_{1} dt \\ {\tilde{k}}_{2} (t) = {\hat{d}}_{2} (t) + H_{2} \int_{0}^{t} Δ η_{2} dt \end{matrix}

(24)

where, $H_{1}$ and $H_{2}$ are empirical coefficients.

Firstly, EID error estimation is used to estimate the current system errors $d_{1} (t)$ and $d_{2} (t)$ . We can write the system control equation as shown in equation (25).

{\begin{matrix} \frac{d i_{d}}{d t} = \frac{- R i_{d} + ω L i_{q} - s_{d} u_{0} + e_{d}}{L} + v_{1} \\ \frac{d i_{q}}{d t} = \frac{- ω L i_{d} - R i_{q} - s_{q} u_{0} + e_{q}}{L} + v_{2} \end{matrix}

(25)

To better describe the system, let $x = {[\begin{matrix} x_{1} & x_{2} \end{matrix}]}^{T} = {[\begin{matrix} i_{d} & i_{q} \end{matrix}]}^{T}$ , $u = {[\begin{matrix} u_{1} & u_{2} \end{matrix}]}^{T} = {[\begin{matrix} s_{d} & s_{q} \end{matrix}]}^{T}$ , define the external perturbation as $v = {[\begin{matrix} v_{1} & v_{2} \end{matrix}]}^{T}$ , and set the grid voltage $e = {[\begin{matrix} e_{d} & e_{q} \end{matrix}]}^{T}$ with external disturbance $v$ as the total disturbances $d = {[\begin{matrix} d_{1} & d_{2} \end{matrix}]}^{T}$ .

Expressing equation (25) as

{\begin{matrix} \overset{\cdot}{x} = Ax + Bu + d \\ y = Cx \end{matrix}

(26)

where, $A = [\begin{matrix} - \frac{R}{L} & ω \\ - ω & - \frac{R}{L} \end{matrix}]$ , $B = [\begin{matrix} - \frac{u_{0}}{L} & 0 \\ 0 & - \frac{u_{0}}{L} \end{matrix}]$ , $C = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ .

Design observers as

\overset{\cdot}{\hat{x}} = A \hat{x} + B u_{f} + L [y - C \hat{x}]

(27)

where, $L = {[\begin{matrix} L_{1} & L_{2} \end{matrix}]}^{T}$ are the observer gains.

According to the principle of EID error estimation described in the literature,³³ the interference estimate is obtained

\hat{d} = B^{+} LC (x - \hat{x}) + u_{f} - u

(28)

Where $B^{+} = (B^{T} B)^{- 1} B^{T}$ .

Second, the following fuzzy system is designed using $η$ , $s \overset{\cdot}{s}$ for system inputs, and $Δ η_{1}$ , $Δ η_{2}$ for system outputs. The fuzzy set is set as

s \overset{\cdot}{s} = {NB NM NS ZO PS PM PB}

(29)

Δ η_{1, 2} = {NB NM NS ZO PS PM PB}

(30)

When $s \overset{\cdot}{s}$ is $PM$ and $PB$ , to restore the system to a steady state quickly, the switching gain needs to be increased, that is $Δ η$ needs to be increased, then the fuzzy value of $Δ η$ can be $PB$ ; when $s \overset{\cdot}{s}$ is $PS$ , the switching gain does not need to be too large, then $Δ η$ needs to be fine-tuned, then the fuzzy value of $Δ η$ can be $PM$ . similarly, when $s \overset{\cdot}{s}$ is $NM$ and $NB$ , in order to let the system quickly eliminate jitter, the switching gain needs to be quickly reduced, then the fuzzy value of $Δ η$ can be $PB$ , Then we can let the fuzzy value of $Δ η$ be $NB$ , when $s \overset{\cdot}{s}$ is $NS$ , the switching gain at this time does not need to be too large, then we need to fine-tune $Δ η$ , then we can let the fuzzy value of $Δ η$ be $NM$ , when $s \overset{\cdot}{s}$ is $ZO$ , $Δ η$ is $ZO$ , based on the above analysis, the fuzzy rule design table of this paper is shown in the Table 1, membership function are shown in Figures 3 and 4.

Table 1.

Fuzzy inference rule table.

$s \overset{\cdot}{s}$	NB	NM	NS	ZO	PS	PM	PB
$Δ η_{1, 2}$	NB	NB	NM	ZO	PM	PB	PB

Figure 3.

Membership function corresponding to fuzzy input $s \overset{\cdot}{s}$ : (a) $s_{1} {\overset{\cdot}{s}}_{1}$ function and (b) $s_{2} {\overset{\cdot}{s}}_{2}$ Function.

Figure 4.

Membership function corresponding to fuzzy input $Δ η_{1, 2}$ : (a) $Δ η_{1}$ function and (b) $Δ η_{2}$ function.

The main part of the current return controller has been designed. Meanwhile, to solving the problem that sign functions are not continuous in zero point or not smooth at transition points, We use hyperbolic tangent function instead of sign function. The mathematical expression is shown in equation (31).

\tanh (x) = \frac{e^{2 ax} - 1}{e^{2 ax} + 1}

(31)

where, $a$ denotes positive real number.

The controller finally gets as shown in equation (32)

{\begin{matrix} s_{d} = \frac{- R i_{d} + ω L i_{d} - L {\tilde{k}}_{1} (t) \tanh (s_{1}) - L β_{1} \frac{p}{q} e_{1}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{1}}{u_{0}} \\ s_{q} = \frac{- R i_{q} - ω L i_{d} - L {\tilde{k}}_{2} (t) \tanh (s_{2}) - L β_{2} \frac{p}{q} e_{2}^{\frac{p}{q} - 1} {\overset{\cdot}{e}}_{2}}{u_{0}} \end{matrix}

(32)

Voltage loop controller design

For the design of voltage loop controller, the problems that may occur in the process of voltage control by the active disturbance rejection control method are analysed, that is, the response rate of the system voltage and the harmonics of the system. To solve this problem, an active disturbance rejection control method based on EID error estimation is proposed. Based on the traditional active disturbance rejection algorithm, ESO is optimised and EID error estimation is introduced. Improve the response rate of the system and anti-harmonic problems.

Conventional ADRC

ADRC has been proposed as one of the nonlinear control methods for nonlinear uncertain systems, as shown in Figure 5.

Figure 5.

Structure diagram of ADRC.

TD can arrange a reasonable transition process according to the bearing capacity of the controlled object, which can reduce the overshoot while ensuring the rapidity. The unmodeled dynamic and unknown internal and external disturbances are summarized into total disturbances by the ESO, which is estimated and compensated, so that the key to the design the controller does not consist in deriving an accurate mathematical model of the object or disturbance. Meanwhile, a compensation effect can realise the system without static adjustment. NLSEF control law uses nonlinear structure to suppress system error and improve system control performance.

A general nonlinear uncertainty system could denoted as

\overset{\cdot\cdot}{x} = f (x, \overset{\cdot}{x}, t) + d (t) + bu

(33)

where $x$ is system state, $f (x, \overset{\cdot}{x}, t)$ is system unknown state function, $d (t)$ is disturbance, $u$ is control input and $b$ is control input coefficient.

Based on the nonlinear uncertainty system, the following ADRC controller is designed.

(1) tracking differentiator

{\begin{matrix} {\overset{\cdot}{v}}_{1} = v_{2} \\ {\overset{\cdot}{v}}_{2} = - M sgn (v_{1} - r + \frac{v_{2} | v_{2} |}{2 M}) \end{matrix}

(34)

(2) extended state observer

{\begin{matrix} e_{0} = z_{1} - y \\ {\overset{\cdot}{z}}_{1} = z_{2} - β_{1} fal (e_{0}, α_{1}, δ_{1}) \\ {\overset{\cdot}{z}}_{2} = z_{3} - β_{2} fal (e_{0}, α_{2}, δ_{2}) + bu \\ {\overset{\cdot}{z}}_{3} = - β_{3} fal (e_{0}, α_{3}, δ_{3}) \end{matrix}

(35)

(3) nonlinear state error feedback

{\begin{matrix} e_{1} = v_{1} - z_{1} \\ e_{2} = v_{2} - z_{2} \\ u_{0} = k_{1} fal (e_{1}, α_{4}, δ_{4}) + k_{2} fal (e_{2}, α_{5}, δ_{5}) \\ u = \frac{u_{0} - z_{3}}{b} \end{matrix}

(36)

where, $r$ is input the given value; $v_{1}$ is the tracking value of $r$ ; $M$ is adjustable parameter; $sgn (.)$ is Symbolic function; $y$ is system output; $z_{1}$ , $z_{2}$ and $z_{3}$ indicate respectively as estimates of $v_{1}$ , $v_{2}$ and total disturbance; $β_{1}$ , $β_{2}$ and $β_{3}$ indicate adjustable parameters; $k_{1}$ and $k_{2}$ are the nonlinear state-error feedback control mothed for gain coefficient; $fal (.)$ is nonlinear function.

Let $ξ = u_{0}^{2}$ , $ξ^{*} = u_{r}^{2}$ , according to equation (5), the instantaneous power relations for system as

\overset{\cdot}{ξ} = - \frac{2 ξ}{R_{L} C} + \frac{2 e_{d} i_{dr} - 2 R {(i_{dr})}^{2}}{C}

(37)

According to equation (33), setting $f (ξ, d_{e} (t))$ as a sum of internal kinetics of system, uncertain disturbances, and unmodeled system dynamics, then equation (37) is given as

\overset{\cdot}{ξ} = f (ξ, d_{e} (t)) + \frac{2 e_{d}}{C} i_{dr}

(38)

Design ADRC controller according to equation (38) as

{\begin{matrix} {\overset{\cdot}{u}}_{01} = u_{02} \\ {\overset{\cdot}{u}}_{02} = - k_{1} sign (u_{01} - r + \frac{u_{02} | u_{02} |}{2 k_{1}}) \\ e_{0} = z_{1} - ξ \\ {\overset{\cdot}{z}}_{1} = z_{2} - β_{1} fal (e_{0}, α_{1}, δ_{1}) + b i_{d} \\ {\overset{\cdot}{z}}_{2} = - β_{1} fal (e_{0}, α_{2}, δ_{2}) \\ e_{1} = ξ^{*} - z_{1} \\ i_{d 0} = k_{1} fal (e_{1}, α_{3}, δ_{3}) \\ i_{dr} = i_{d 0} - \frac{z_{2}}{b} \end{matrix}

(39)

However, under this ADRC action, $i_{dr}$ as the voltage outer loop controller output, there is a certain amplitude of square wave-like jitter ( $\pm 0.5 A$ ), as shown in Figure 6. This small jitter may have a small impact on the general system. However, for PWM rectifiers, the $i_{dr}$ , as the active current of the system and also as a set value for current loop, will directly increase harmonic content of AC current and generate square wave pulses on the DC voltage, thus affecting the system performance. Therefore, the conventional ADRC needs to be improved to reduce such problems.

Figure 6.

$i_{d, q}$ waveform with ADRC.

EID/ADRC control

In response to the analysis discussed earlier, an ADRC/EID error estimation is proposed to optimise the fluctuation of the current $i_{d}$ in this paper, and to solve the ADRC control tuning problem. EID/ADRC controller structure is shown in Figure 7.

Figure 7.

EID/ADRC controller.

According to equation (38), an error $d_{e} (t)$ is required to forecast before the calibration. Thus, in this paper, EID error estimation is used in order to compensate the errors generated by the systems.

Based on transfer function relations among $u_{0}$ and $i_{d}$ in equation (8), a state space expression can be obtained as

{\begin{matrix} \overset{\cdot}{x} (t) = A_{1} x (t) + A_{2} u_{0} (t) + d_{e} (t) \\ i_{d} (t) = A_{3} x (t) + A_{4} u_{0} (t) \end{matrix}

(40)

where, $u_{0} (t)$ is system input, $i_{d} (t)$ is system output, $x = i_{d} + \frac{R_{L} C u_{r}}{L i_{dr}} u_{0}$ , $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ are observer parameters, $A_{1} = \frac{R_{L} (e_{d} - 2 R i_{dr})}{L i_{dr}}$ , $A_{2} = - \frac{R_{L}^{2} C u_{r} + 2 u_{r} L i_{dr}}{L^{2} i_{dr}^{2}}$ , $A_{3} = 1$ , $A_{4} = - \frac{R_{L} C u_{r}}{L i_{dr}}$ .

Design of state observer for equation (40) is

{\begin{matrix} \hat{\overset{\cdot}{x}} (t) = A_{1} \hat{x} (t) + A_{2} u_{f} (t) + M [i_{d} (t) - {\hat{i}}_{d} (t)] \\ {\hat{i}}_{d} (t) = A_{3} \hat{x} (t) + A_{4} u_{f} (t) \end{matrix}

(41)

where $\hat{x} (t)$ is the estimated value of $x (t)$ , ${\hat{i}}_{d} (t)$ is observer output, $M$ is observer gain, $u_{f}$ is nonlinear error function output.

A definitive estimation for $d_{e} (t)$ is

{\hat{d}}_{e} (t) = A_{2}^{+} M A_{3} [x (t) - \hat{x} (t)] + u_{f} (t) - i_{dr} (t)

(42)

where, $A_{2}^{+} = (A_{2}^{T} A_{2})^{- 1} A_{2}^{T}$ .

For the first-order voltage control system, the EID/ADRC controller based on equations (42) and (39) is designed as shown in equation (43).

{\begin{matrix} {\overset{\cdot}{u}}_{01} = u_{02} \\ {\overset{\cdot}{u}}_{02} = - 1.73 r u_{02} - r^{2} (u_{01} - u_{r}^{2}) \\ e_{0} = z_{1} - ξ \\ {\overset{\cdot}{z}}_{1} = {\hat{d}}_{e} (t) - β_{1} fal (e_{0}, α_{1}, δ_{0}) + i_{dr} \\ e_{1} = ξ^{*} - z_{1} \\ u_{f} = k_{2} fal (e_{1}, {α'}_{1}, δ_{1}) \\ i_{dr} = u_{f} - {\hat{d}}_{e} (t) \end{matrix}

(43)

Simulation experiments

Through the relevant information the interval of the commonly used operating parameters of the PWM rectifier is obtained literatures,^34,35 and the system parameters for the simulation of this paper are determined as shown in Table 2. Three-phase PWM rectifier simplification modelling created in MATLAB software for experimentation.

Table 2.

The system parameters.

Name	Symbols	Value	Unit
Peak grid phase voltage	$U_{N}$	60	V
AC voltage frequency	$f_{N}$	50	Hz
AC side inductor	L	3	mH
DC side capacitor	$C$	3000	$μ F$
Equivalent resistance	$R$	0.1	$Ω$
Load resistance	$R_{L}$	20	$Ω$
Switching frequency	$f_{s}$	7000	Hz
DC side voltage reference	$u_{r}$	200	V

Steady-state operation simulation

As can be seen from the curves in the result plots of Figure 8, the strategy that we propose in this paper possesses following advantages of shorter response time and smoother voltage profile compared with the conventional ADRC method.

Figure 8.

$u_{0}$ waveforms (steady-state operation).

Figures 9 and 10 respectively illustrate rectifier phase A voltages and currents waveforms under the two control methods. From the waveforms, it is observed with proposed control method that the sinusoidalisation of the same phase voltage and current is achieved after 0.1 s, while the traditional ADRC control method can also sinusize the current at 0.1 s, but within 0.1 s at the beginning, the current can be sinusoidal. The phase is inconsistent with the voltage, the current in the subsequent results will spike at the peak, and the buffeting is more obvious. However, the phase of our method is basically consistent with the voltage within 0.1 s, and the current curve is smooth without obvious spikes.

Figure 9.

$e_{a}$ and $i_{a}$ waveforms with this paper (steady-state operation).

Figure 10.

$e_{a}$ and $i_{a}$ waveforms with ADRC (steady-state operation).

Figures 11 and 12 reflect the $i_{d, q}$ waveforms of the active and reactive currents in systems under the rotation axes d and q. It is evident that the proposed method is effective in suppressing current buffering as can be obtained from the figure. The current jitter for proposed control method is within a range from 0.2 A, while that of the traditional ADRC method is within the range of 0.4 A, and the reactive current fluctuation in proposed method is small, while in ADRC method it is up to 10A.

Figure 11.

$i_{d, q}$ waveforms with this paper (steady-state operation)

Figure 12.

$i_{d, q}$ waveforms with ADRC (steady-state operation)

Figures 13 and 14 reflect total harmonics distortion (THD) value and harmonic frequency of the system. The THD value directly reflects system’s harmonic content. Lower values are better for the power quality in a system. Observations and conclusions from the figure, harmonics of a system are generally concentrated on odd harmonics, and the THD value of the proposed control method is 0.44%, which is 1.85% points lower than that of the traditional method of 2.29%, indicating that the proposed method has a good control effect under steady state operation.

Figure 13.

THD value with this paper.

Figure 14.

THD value with ADRC.

Voltage surge simulation

Through the previous tests, we verified both availability and excellence in the method in steady-state situations. Next, we carried out theoretical experiments by simulating the sudden voltage change caused by the external environment. The steady state voltage of 200 V was set to change to 250 V at 0.1 s, and the results of the experiment are analysed as follows.

The simulations in Figure 15 lead to these conclusions: the voltage starts to change at 0 s and reaches the given value, when the voltage changes abruptly, the DC voltage can be stabilized near the new DC voltage reference value after a short time under the two control strategies. The voltage will not be distorted because of the sudden change of the AC voltage, both of them can achieve a good following, and the dynamic response speed is faster. The response time difference between the proposed method and the ADRC method is not much, and both delays are smaller, but proposed method has much shorter regulation period. A control method proposed in this paper has some advantages over the ADRC control method.

Figure 15.

$u_{0}$ waveforms (voltage surge).

Figures 16 and 17 respectively illustrate rectifier phase A voltages and currents waveforms with the two control methods loaded into the controller. From the waveforms, We could find the two control methods could sinusoidalise in-phase voltages and currents within 0.01 s when a sudden voltage surge occurs. The phase of this paper’s method is basically consistent with the phase of the reference voltage at 0.01 s, and the current curve is smooth without obvious spikes, it is capable to achieve fast following of voltage and current.

Figure 16.

$e_{a}$ and $i_{a}$ waveforms with this paper (voltage surge).

Figure 17.

$e_{a}$ and $i_{a}$ waveforms with ADRC (voltage surge).

Waveforms for the active and reactive currents in a system under the rotation axes d and q are reflected in Figures 18 and 19. From the figures, it can be observed the proposed method is effective in suppressing the current jitter generated. When a sudden voltage change occurs in 0.1 s, the current using the proposed method in this paper can quickly reach stabilisation in 0.02 s, and the reactive current fluctuation is small.

Figure 18.

$i_{d, q}$ waveforms with this paper (voltage surge).

Figure 19.

$i_{d, q}$ waveforms with ADRC (voltage surge).

From the figures, it is observed that the proposed methods have good dynamic and stability effectiveness even under sudden voltage surge, and has obvious advantages compared with the traditional method in voltage stability time and current sinusoidal waveform.

Load transformation simulation

Next, we conducted theoretical experiments by simulating the load transformation. The steady state voltage of 200 V was set to change to 250 V at 0.1 s, and the results of the experiment are analysed as follows.

From the simulated outcomes in Figure 20, we can conclude that compared to conventional ADRC control method, which proposed control methods have some advantages in response time, the voltage profile fluctuates is less when it is affected by the load transformation, it recovers quickly and continues to follow the reference voltage, and the voltage profile is also smoother. However, the conventional ADRC control method is significantly weaker than our proposed method in terms of response time.

Figure 20.

$u_{0}$ waveforms (load Transformation).

The waveforms for voltage and current in phase A in the rectifier under load transformation with the two control methods as shown in Figures 21 and 22. From the waveforms, it is observed that a control method of the paper achieves in-phase voltage and current and achieves sinusoidal current after 0.01 s, while the conventional ADRC control method also achieves sinusoidal current at 0.01 s, but the current can be sinusoidal in the beginning of the 0.01 s. The phase is not consistent with the voltage, the current in the subsequent results will be spiked at the peak, and the buffering phenomenon is more obvious. In contrast, our method is basically consistent with the phase and voltage within 0.1 s, the current curve is smooth without obvious spikes, and when a load transformation occurs in 0.1 s, our proposed method can respond quickly and the following curve is smoother.

Figure 21.

$i_{d, q}$ waveforms with this paper (load transformation).

Figure 22.

$i_{d, q}$ waveforms with ADRC (load transformation).

The data from the experimental results were collated and compared, as detailed in Table 3.

Table 3.

Comparison of the experimental results.

Performance indicators	ADRC	Methodology of this paper
Response time (steady state operation)	0.025 s	0.015 s
Response time (voltage mutation)	0.02 s	0.01 s
Response time (load transformation)	0.015 s	0.01 s
Grid current THD	2.29%	0.44%
Power factor	0.995	0.999
$i_{d, q}$ steady-state time (steady state operation)	0.025 s	0.01 s
$i_{d, q}$ steady-state time (voltage mutation)	0.02 s	0.01 s
$i_{d, q}$ steady-state time (load transformation)	0.02 s	0.01 s

Conclusion

This paper uses an underactuated characteristic of PWM rectifiers system to verify the feasibility of its double closed-loop control structure, and conducts controller theory analysis and design, and system performance analysis. The terminal fuzzy sliding mode control based on EID error estimation is designed to achieve current stabilization in a shorter period of time, which can effectively solve the problem that the unknown system sliding mode gain design is too large, reduce the system current jitter and vibration, and improve the system power efficiency. For the ADRC control method in the face of the problem of the DC voltage square wave pulses are too large, the input disturbance elimination feature of EID error estimation is utilized to reduce the voltage square wave fluctuation interval and enhance the system working performance. Experimental tests demonstrate the proposed algorithm achieves the fast DC voltage response and unit grids powers operating for PWM rectifier. It improves the response rate of DC side voltage in steady state operation, has a low THD value, reduce the harmonic content of the control system, commend to better stability, dynamic properties indexes and improves energy qualities under sudden voltage surges and load transformations, as well as improving system perturbations resistance.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Nature Science Foundation of Hubei Province (No. 2023AFB380), the Graduate Innovative Fund of Wuhan Institute of Technology (No. CX2023550, No. CX2023578), the Hubei Key Laboratory of Digital Textile Equipment (Wuhan Textile University) (No. KDTL2022003), the Hubei Key Laboratory of Intelligent Robot (Wuhan Institute of Technology) (No. HBIRL202301), and the Open Research Fund of Vehicle Measurement Control and Safety Key Laboratory of Sichuan Province of Xihua University (No. QCCK2024-0011).

ORCID iD

Zixin Huang

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

Tseng

Huang

Cheng

, et al. A single-switch converter with high step-up gain and low diode voltage stress suitable for green power-source conversion. IEEE Journal of Emerging and Selected Topics in Power Electronics 2016; 4(2): 363–372.

Aman

Solangi

Hossain

, et al. A review of safety, health and environmental (SHE) issues of solar energy system. Renewable and Sustainable Energy Reviews 2015; 41: 1190–1204.

Han

Liu

Catalão

. Optimization of distribution network and mobile network with interactive balance of flexibility and power. IEEE Transactions on Power Systems 2023; 38(3): 2512–2524.

Nejati

Amjady

Zareipour

, et al. A new multi-resolution closed-loop wind power forecasting method. IEEE Transactions on Sustainable Energy 2023; 14(4): 2079–2091.

Sun

Liu

, et al. Multiple-modular high-frequency DC transformer with parallel clamping switched capacitor for flexible MVDC and HVDC system applications. IEEE Journal of Emerging and Selected Topics in Power Electronics 2020; 8(4): 4130–4143.

Kaur

Kaddoum

Zeadally

. Blockchain-based cyber-physical security for electrical vehicle aided smart grid ecosystem. IEEE Transactions on Intelligent Transportation Systems 2021; 22(8): 5178–5189.

Pan

Wang

Zheng

, et al. Fractional-order linear active disturbance rejection control strategy for grid-side current of PWM rectifiers. IEEE Journal of Emerging and Selected Topics in Power Electronics 2023; 11(4): 3827–3838.

Liu

Wang

, et al. Energy sharing management for microgrids with PV prosumers: A stackelberg game approach. IEEE Transactions on Industrial Informatics 2017; 13(3): 1088–1098.

Tushar

Zeineddine

Assi

, et al. Demand-side management by regulating charging and discharging of the EV, ESS, and utilizing renewable energy. IEEE Transactions on Industrial Informatics 2018; 14(1): 117–126.

10.

Chen

Zhang

Jiang

, et al. An SMES-based current-fed transformerless series voltage restorer for DC-Load protection. IEEE Transactions on Power Electronics 2021; 36(9): 9698–9703.

11.

Chen

, et al. Real-time fault diagnosis of pulse rectifier in traction system based on structural model. IEEE Transactions on Intelligent Transportation Systems 2022; 23(3): 2130–2143.

12.

Costa

Ewerling

Font

, et al. Unidirectional three-phase voltage-doubler SEPIC PFC rectifier. IEEE Transactions on Power Electronics 2021; 36(6): 6761–6773.

13.

Beres

Wang

Liserre

, et al. A review of passive power filters for three-phase grid-connected voltage-source converters. IEEE Journal of Emerging and Selected Topics in Power Electronics 2016; 4(1): 54–69.

14.

Wang

Huang

Wei

, et al. A novel ADRC control strategy for PWM rectifier. IFAC-PapersOnLine 2023; 56(2): 422–427.

15.

Sun

, et al. Linear active disturbance rejection control for three-phase voltage-source PWM rectifier. IEEE Access 2020; 8: 45050–45060.

16.

Geng

Wang

, et al. Performance analysis and improvement of PI-type current controller in digital average current mode controlled three-phase six-switch boost PFC rectifier. IEEE Transactions on Power Electronics 2022; 37(7): 7871–7882.

17.

Zou

Zhang

Xing

, et al. Generalized clarke transformation and enhanced dual-loop control scheme for three-phase PWM converters under the unbalanced utility grid. IEEE Transactions on Power Electronics 2022; 37(8): 8935–8947.

18.

Ruan

Wang

, et al. An interleaved three-phase PWM single-stage resonant rectifier with high-frequency isolation. IEEE Transactions on Industrial Electronics 2020; 67(8): 6572–6582.

19.

Song

Zhang

Gao

, et al. Power model free voltage ripple suppression method of three-phase PWM rectifier under unbalanced grid. IEEE Transactions on Power Electronics 2022; 37(11): 13799–13807.

20.

Yang

, et al. A quasi-resonant extended state observer-based predictive current control strategy for three-phase PWM rectifier. IEEE Transactions on Industrial Electronics 2022; 69(12): 13910–13917.

21.

Zhang

Wang

Jiao

, et al. Grid-voltage sensorless model predictive control of three-phase PWM rectifier under unbalanced and distorted grid voltages. IEEE Transactions on Power Electronics 2020; 35(8): 8663–8672.

22.

Ashry

Sharaf

Ibrahim

, et al. A simple and accurate model for RFID rectifier. IEEE Systems Journal 2008; 2(4): 520–524.

23.

Yan

Yang

Hui

, et al. A review on direct power control of pulsewidth modulation converters. IEEE Transactions on Power Electronics 2021; 36(10): 11984–12007.

24.

Xie

Wang

, et al. A family of asymmetrical SVPWM schemes with normal and open-circuit fault-tolerant operation capability for a three-phase isolated AC-DC matrix converter. IEEE Transactions on Power Electronics 2022; 37(2): 1788–1803.

25.

Zheng

Yuan

Cai

, et al. Large-Signal stability analysis of DC side of VSC-HVDC system based on estimation of domain of attraction. IEEE Transactions on Power Systems 2022; 37(5): 3630–3641.

26.

Hang

Liu

Yan

, et al. An improved deadbeat scheme with fuzzy controller for the grid-side three-phase PWM boost rectifier. IEEE Transactions on Power Electronics 2011; 26(4): 1184–1191.

27.

Zhang

Liu

Yao

, et al. Suppression of low-frequency oscillation in traction network of high-speed railway based on auto-disturbance rejection control. IEEE Transactions on Transportation Electrification 2016; 2(2): 244–255.

28.

Xie

, et al. An active disturbance rejection control-based voltage control strategy of single-phase cascaded h-bridge rectifiers. IEEE Transactions on Industry Applications 2020; 56(5): 5182–5193.

29.

Wei

Yuan

Lei

, et al. Direct-Current predictive control strategy for inhibiting commutation failure in HVDC converter. IEEE Transactions on Power Systems 2014; 29(5): 2409–2417.

30.

Mehreganfar

Saeedinia

Davari

, et al. Sensorless predictive control of AFE rectifier with robust adaptive inductance estimation. IEEE Transactions on Industrial Informatics 2019; 15(6): 3420–3431.

31.

Gui

, et al. A voltage modulated DPC approach for three-phase PWM rectifier. IEEE Transactions on Industrial Electronics 2018; 65(10): 7612–7619.

32.

Lasalle

. Stability theory for ordinary differential equations. Journal of Differential Equations 1968; 4(1): 57–65.

33.

She

Fang

Ohyama

, et al. Improving disturbance-rejection performance based on an equivalent-input-disturbance approach. IEEE Transactions on Industrial Electronics 2008; 55(1): 380–389.

34.

Chelladurai

Sundarabalan

Santhanam

, et al. Interval type-2 fuzzy logic controlled shunt converter coupled novel high-quality charging scheme for electric vehicles. IEEE Transactions on Industrial Informatics 2021; 17(9): 6084–6093.

35.

Pang

Liu

Wei

, et al. Online diode fault detection in rotating rectifier of the brushless synchronous starter generator. IEEE Transactions on Industrial Informatics 2020; 16(11): 6943–6951.