Research on the Nonlinear Control Strategy of Three-Phase Bridgeless Rectifier under Unbalanced Grids

Abstract

The three-phase Y-connected bridgeless rectifier is essentially a nonlinear system, and it is difficult to obtain superior dynamic performance under the action of traditional linear controller. Under the condition of unbalanced power grids, this paper has established a mathematical model based on Euler–Lagrange (EL) equations with line voltage and line current as state variables. Furthermore, it then designed a passivity-based controller in inner current loop based on the mathematical model. The hybrid nonlinear control strategy consisting of active disturbance rejection controller (ADRC) in the outer voltage loop and passivity-based controller (PBC) in the inner current loop is adopted to control the system, which does not need to consider the positive and negative sequence components. The control structure is simple and can improve the steady-state accuracy, dynamic performance and anti-interference ability. The feasibility of the proposed control strategy is verified by computer simulation, which has a guiding significance for the application of three-phase bridgeless rectifier in practical engineering.

1. Introduction

Compared with the traditional two or three level rectifiers, multilevel rectifiers are more and more used in the field of AC/DC-DC/AC power conversion because of its advantages such as lower current distortion rate, lower switching frequency, lower switching device voltage stress and higher power quality [1,2,3]. The cascaded H-bridge rectifier can be connected with the power grid without the multi-winding line-frequency transformers, which is currently receiving considerable attention [4,5]. Furthermore, the H-bridge module in the cascaded H-bridge rectifier can be replaced by the bridgeless module, which can further reduce the number of fully controlled devices in the multilevel rectifiers as well as the cost and control complexity of the system [6,7].

Nevertheless, the analysis and discussion of the control strategies of the cascaded H-bridge or bridgeless are mostly conducted with the assumptions of ideal power grids, which possesses balanced input voltages and good stability [7,8,9]. Beyond that, an unbalanced grid does exist because of the asymmetry of power grid parameters, the access of nonlinear loads and power grid faults. At the same time, a large number of harmonics will be injected into the power grids [10]. If the control strategy under the ideal power grid is continued, the rectifier will not work normally and even damage the devices in serious cases, resulting in the power failure [11,12]. According to the published literature, there is little research or analysis of control strategies for the cascaded bridgeless rectifier under unbalanced power grids. A linear controller was designed for the cascaded bridgeless rectifier with the unbalanced power grids to control its positive and negative sequence components respectively in [13,14]. Obviously, the control structure is relatively complex. Further, the traditional dual closed loop linear control strategy was proposed by [13,14]. While the cascaded H-bridge or bridgeless multilevel rectifier is essentially a nonlinear system, and the adoption of linear control strategy requires the establishment of an accurate small signal model for a specific working point, the design of linear controller parameters is difficult to meet the stability of the system in a large scale [15,16].

Passivity-based control (PBC) theory was proposed by R. Ortega and M. Spong in 1989. The passivity-based controller is essentially a nonlinear controller, can be stable globally in a nonlinear system and obtains the advantages of simple design and easy implementation. When ensuring excellent steady-state performance of the system, the passivity-based controller can also improve the dynamic performance of the system and the anti-interference ability of the system [17,18,19]. Active disturbance rejection control (ADRC) technology was formally proposed by Professor Jingqing Han in 1998. It is a nonlinear control technology which developed from the traditional PID technology. It does not require the accurate mathematical model of the controlled object and meanwhile has the advantages of easy implementation, shorter adjustment time, higher precision, stronger anti-interference ability and so forth [20,21,22].

In order to overcome the drawback of existing control strategies for the three-phase cascaded bridgeless rectifier, taking one bridgeless model in each phase of three-phase cascaded bridgeless multilevel converter as an example, this paper designed the current inner loop PBC controller and outer loop ADRC controller with the operating principle and mathematical model of three-phase cascaded bridgeless rectifier under unbalanced power grids. In this proposed control strategy, using line voltages and line currents as state variables, the mathematical model of rectifier has been established without considering the positive and negative sequence components of the above variables. The overall control scheme is easy to be realized in practical engineering. Finally, the simulation model of three-phase bridgeless rectifier is built through Matlab/Simulink platform. According to the analysis of the simulation results, the proposed control strategy and the inner current loop PBC controller and outer voltage loop ADRC controller have improved the steady-state accuracy and dynamic performance of the system. At the same time, the research on the control strategy of three-phase bridgeless rectifier in this paper is also applicable to three-phase cascaded multilevel rectifier with multiple bridgeless or H-bridge modules in each phase.

This paper is organized as follows. In Section 2, the topology and operating principle are introduced and analyzed thoroughly. In Section 3, the mathematical model based on EL equation is established. In Section 4, the hybrid nonlinear controller is designed, and the overall control scheme is proposed. In Section 5, the simulation results and discussion are presented. Finally, Section 6 presents the discussion.

2. Topology and the Operating Principle

Figure 1 shows the topology of three-phase bridgeless rectifier. According to Figure 1, the rectifier is composed of three bridgeless modules, each phase consisting of one module. The three modules adopt Y-connected method as Figure 1. u_a, u_b, u_c and i_a, i_b, i_c are three-phase input voltages and currents respectively. L is the AC side filter inductance; D_j(1,2), (j = a, b, c) and T_j(1,2), (j = a, b, c) are the diodes and power switches IGBTs in the three-phase respectively; C₁, C₂, C₃ and R₁, R₂, R₃ are the DC-link capacitors and resistors respectively. In this paper, the two power switches in each phase are driven complementarily. S_j(1,2), (j = a, b, c) is the switching function of the IGBT. When S_j(1,2), (j = a, b, c) is equal to 1, it represents that the IGBT is turned on, and when S_j(1,2), (j = a, b, c) is equal to 0, it represents that the power switch is turned off.

The operating principle of three-phase bridgeless rectifier is analyzed through taking phase-a as an example. When the current flowing through phase-a i_a is greater than 0, if S_a1 = 1 and S_a2 = 0, the i_a flowed into T_a1 through the AC side inductance L and then flowed into the neutral point O through T_a2. The inductance L is releasing energy at this time and the synthetic voltage on the AC side of phase-a is equal to 0; If S_a1 = 0 and S_a2 = 1, the i_a flowed into the diode D_a1 through the AC side inductance L and then flowed into the neutral point O through T_a2. The inductance L is storing energy at this time, and the synthetic voltage on the AC side of phase-a is equal to u_dc.

When the current flowing through phase-a i_a is less than 0, if S_a1 = 1 and S_a2 = 0, the i_a flowed from the neutral point O into the diode D_a2 and then flowed into the AC side inductance L through T_a1, the inductance L is storing energy at this time, and the synthetic voltage on the AC side of phase-a is equal to −u_dc; If S_a1 = 0 and S_a2 = 1, the i_a flowed from the neutral point O into T_a2 and then flew into the AC side inductance L through T_a1. The inductance L is releasing energy at this time, and the synthetic voltage on the AC side of phase-a is equal to 0.

3. Mathematical Model of the Rectifier

In order to facilitate the establishment of the mathematical model of the rectifier, it is assumed that the inductor and capacitor in the three-phase bridgeless rectifier are all ideal components, and the power switches have no loss and delay. Under the condition of unbalanced power grids, the three-phase input voltages and currents are Δu = u_a + u_b + u_c ≠ 0 and i_a + i_b + i_c = 0 respectively. In addition, the switching function can be obtained as ΔS = S_a + S_b + S_c ≠ 0, and the S_j1 = S_j2 (j = a, b, c) because of the IGBTs are driven synchronously. In order to facilitate the establishment of mathematical model, S_j (j = a, b, c) is used to represent the switching function of each phase IGBTs. A new switching function is defined as follows:

$$d_j=\{\begin{array}{l}1-S_j\begin{array}{cc},& \end{array}{i}_j>0\\ S_j-1\begin{array}{cc},& \end{array}{i}_j<0\end{array}$$

(1)

Considering the condition of DC-link resistors in three phases are equal, namely R₁ = R₂ = R₃ = R, the DC-link voltages during the steady-state of the rectifier are also equal, which means u_dc1 = u_dc2 = u_dc3 = u_dc. The mathematical model in abc coordinates of the three-phase bridgeless rectifier can be obtained according to Figure 1:

$$\{\begin{array}{l}L\frac{\mathrm{d}{i}_{\mathrm{a}}}{\mathrm{d}t}=u_{\mathrm{a}}-d_{\mathrm{a}}{u}_{\mathrm{dc}}+\frac{1}{3}\Delta d{u}_{\mathrm{dc}}-\frac{1}{3}\Delta u\\ L\frac{\mathrm{d}{i}_{\mathrm{b}}}{\mathrm{d}t}=u_{\mathrm{b}}-d_{\mathrm{b}}{u}_{\mathrm{dc}}+\frac{1}{3}\Delta d{u}_{\mathrm{dc}}-\frac{1}{3}\Delta u\\ L\frac{\mathrm{d}{i}_{\mathrm{c}}}{\mathrm{d}t}=u_{\mathrm{c}}-d_{\mathrm{c}}{u}_{\mathrm{dc}}+\frac{1}{3}\Delta d{u}_{\mathrm{dc}}-\frac{1}{3}\Delta u\\ 3C\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}=d_{\mathrm{a}}{i}_{\mathrm{a}}+d_{\mathrm{b}}{i}_{\mathrm{b}}+d_{\mathrm{c}}{i}_{\mathrm{c}}-3\frac{u_{\mathrm{dc}}}{R}\end{array}$$

(2)

where Δd = d_a + d_b + d_c.

For the sake of realizing the components of 0-axis is equal to zero in dq0 coordinates, the mathematical model of the three-phase bridgeless rectifier can be established as Formula (3) by taking line voltages, line switching functions, phase currents and DC-link voltage as variables. These variables are following: line voltages u_ab = u_a − u_b, u_bc = u_b − u_c, u_ca = u_c − u_a, line switching functions d_ab = d_a − d_b, d_bc = d_b − d_c, d_ca = d_c − d_a, phase currents i_a, i_b, i_c and DC-link voltage u_dc.

$$\{\begin{array}{l}L\frac{\mathrm{d}{i}_{\mathrm{a}}}{\mathrm{d}t}=(u_{\mathrm{ab}}-u_{\mathrm{bc}})-(d_{\mathrm{ab}}-d_{\mathrm{bc}})u_{\mathrm{dc}}\\ L\frac{\mathrm{d}{i}_{\mathrm{b}}}{\mathrm{d}t}=(u_{\mathrm{bc}}-u_{\mathrm{ab}})-(d_{\mathrm{bc}}-d_{\mathrm{ab}})u_{\mathrm{dc}}\\ L\frac{\mathrm{d}{i}_{\mathrm{c}}}{\mathrm{d}t}=(u_{\mathrm{ca}}-u_{\mathrm{bc}})-(d_{\mathrm{ca}}-d_{\mathrm{bc}})u_{\mathrm{dc}}\\ C\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}=\frac{1}{3}[(d_{\mathrm{ab}}-d_{\mathrm{bc}})i_{\mathrm{a}}+(d_{\mathrm{bc}}-d_{\mathrm{ab}})i_{\mathrm{b}}+(d_{\mathrm{ca}}-d_{\mathrm{bc}})i_{\mathrm{c}}]-\frac{u_{\mathrm{dc}}}{R}\end{array}$$

(3)

For the three-phase input currents i_a, i_b, i_c in the mathematical model of three-phase bridgeless rectifier, the matrix M_abc/dq0 is used for equal rotation transformation, and the matrix M_abc/ldq0 is used for equal rotation transformation of line voltages u_ab, u_bc, u_ca and line switching functions d_ab, d_bc, d_ca. The equivalent rotation matrices M_abc/dq0 and M_abc/ldq0 are:

$${\mathit{M}}_{\mathrm{abc}/\mathrm{dq}0}=\frac{2}{3}\left(\begin{array}{ccc}\cos \omega t& \cos (\omega t-2\pi /3)& \cos (\omega t+2\pi /3)\\ \sin \omega t& \sin (\omega t-2\pi /3)& \sin (\omega t+2\pi /3)\\ 1/2& 1/2& 1/2\end{array}\right)$$

(4)

$${\mathit{M}}_{\mathrm{abc}/\mathrm{dq}0}=\frac{2}{3}\left(\begin{array}{ccc}\cos (\omega t+\pi /6)& \cos (\omega t-\pi /2)& \cos (\omega t+5\pi /6)\\ \sin (\omega t+\pi /6)& \sin (\omega t-\pi /2)& \sin (\omega t+5\pi /6)\\ 1/2& 1/2& 1/2\end{array}\right)$$

(5)

After the above rotation transformation, the mathematical model of three-phase bridgeless rectifier, that is, the Formula (3) has changed into the following forms:

$$\{\begin{array}{l}L\frac{\mathrm{d}{i}_{\mathrm{d}}}{\mathrm{d}t}=\frac{\sqrt{3}}{3}{u}_{\mathrm{ld}}+\omega L{i}_{\mathrm{q}}-\frac{\sqrt{3}}{3}{d}_{\mathrm{ld}}{u}_{\mathrm{dc}}\\ L\frac{\mathrm{d}{i}_{\mathrm{q}}}{\mathrm{d}t}=\frac{\sqrt{3}}{3}{u}_{\mathrm{lq}}-\omega L{i}_{\mathrm{d}}-\frac{\sqrt{3}}{3}{d}_{\mathrm{lq}}{u}_{\mathrm{dc}}\\ \frac{2}{3}C\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}=\frac{\sqrt{3}}{3}(d_{\mathrm{ld}}{i}_{\mathrm{d}}+d_{\mathrm{lq}}{i}_{\mathrm{q}})-\frac{2}{3}\frac{u_{\mathrm{dc}}}{R}\end{array}$$

(6)

where, u_lq ≠ 0 due to the unbalanced three-phase input voltages as well as d_ld and d_lq represent the corresponding components of the line switching functions in the dq0 coordinates.

To facilitate the design of PBC for the three-phase bridgeless rectifier, organizing the Formula (6) into the form of Euler–Lagrange Equation [23,24,25] as follows:

$$\mathit{M}\stackrel{•}{\mathit{x}}+\mathit{J}\mathit{x}+\mathit{R}\mathit{x}=\mathit{u}$$

(7)

where:

$$\begin{gathered} \mathit{M}=\left(\begin{array}{ccc}L& 0& 0\\ 0& L& 0\\ 0& 0& 2C/3\end{array}\right),\mathit{x}=\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left(\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\\ u_{\mathrm{dc}}\end{array}\right), \\ \mathit{J}=\left(\begin{array}{ccc}0& -\omega L& \sqrt{3}{d}_{\mathrm{ld}}/3\\ \omega L& 0& \sqrt{3}{d}_{\mathrm{lq}}/3\\ -\sqrt{3}{d}_{\mathrm{ld}}/3& -\sqrt{3}{d}_{\mathrm{lq}}/3& 0\end{array}\right),\mathit{R}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 2/3R\end{array}\right),\mathit{u}=\left(\begin{array}{c}\frac{\sqrt{3}}{3}{u}_{\mathrm{ld}}\\ \frac{\sqrt{3}}{3}{u}_{\mathrm{lq}}\\ 0\end{array}\right) \end{gathered}$$

And the matrix J satisfies the property of an antisymmetric matrix, that is, J = −J^T, x^TJx = 0. According to the reference [25], we can know that Euler–Lagrange equation has the characteristics of non-power and passivity. For the non-power characteristic, it is reflected in that matrix J is an antisymmetric matrix and does no work to the system. Therefore, it can be seen that the above formula has the properties of the Euler–Lagrange equation.

4. Design of Hybrid Nonlinear Controller

In this paper, a hybrid nonlinear control strategy for three-phase bridgeless rectifier is proposed. The outer loop voltage ADRC controller and the inner loop current PBC controller are designed respectively. The function of this hybrid nonlinear controller is to ensure the sinusoidal currents input, unity power factor, quick response of DC-link voltage as well as the strong anti-interference ability of the rectifier. These control objectives are reflected in the mathematical model when the components of three-phase input currents in the dq0 coordinate i_d, i_q can be stable at the desired value i^*_d, i^*_q under the condition of unbalanced three-phase input voltages and existing external interference. In addition, the DC-link voltage of each phase can be stable at the desired value u^*_dc, namely x^* = [i^*_d i^*_q u^*_dc]^T = [i^*_d 0 u^*_dc]^T. According to the judgment of passivity about three-phase voltage source PWM rectifier, the three-phase bridgeless rectifier is strict passive [23], and a PBC controller can be designed for it.

Traditionally, it is very often to design a PI controller to regulate the outer loop voltage of the rectifier, and then achieve the desired value i^*_d of inner loop current. Nevertheless, the three-phase bridgeless rectifier is a truly complex nonlinear system. As for the traditional PI controller, it is a linear controller based on error elimination, and the actual output and desired output are mainly considered without the influence between the other factors of the system. In this way, the controller can only ensure the system stable at a static working point, so it is sometimes difficult to ensure the stability of the system in a large scale, and the anti-interference ability is weak [24]. In order to overcome the weakness of the traditional linear PI controller, a ADRC controller is designed to regulate the outer loop voltage of the three-phase bridgeless rectifier. ADRC is an advanced nonlinear control technology. It can treat all the uncertain factors of the system, namely the internal and external disturbance, as an unknown disturbance. It can then estimate the disturbance in real time. Above all, the ADRC controller can make the system obtain the better ability of dynamics and anti-interference.

4.1. Design of ADRC Controller

The ADRC controller is a nonlinear controller which is improved by traditional linear PI controller. It is completely independent of the specific mathematical model of the controlled object and only needs a general description of the controlled object. The ADRC controller mainly includes three parts: the tracking differentiator (TD), the extended state observer (ESO), the nonlinear state error feedback (NLSEF). In order to simplify the design of the controller, the first-order ADRC is adopted in this paper. The detailed design process of each part in the controller is as follows:

• Design of TD

First-order TD is as follows:

$${\stackrel{•}{v}}_1=-m{\sin }_{\mathrm{sgn}}(v_1-k{u}_{\mathrm{dc}}^{*},n)$$

(8)

The tracking differentiator is composed of nonlinear saturation function sin_sgn [25], where:

$${\sin }_{\mathrm{sgn}}(A,n)=\{\begin{array}{l}1,\begin{array}{cc}& \begin{array}{cc}& \end{array}A>n\end{array}\\ \sin \frac{\pi A}{2n},\begin{array}{cc}& \left|A\right|\le n\end{array}\\ -1,\begin{array}{ccc}& & A<-n\end{array}\end{array}$$

(9)

The tracking signal of DC-link voltage can be obtained by the above first-order tracking differentiator. The parameter m in Formula (8) determines the tracking accuracy of the tracking differentiator.

2.

Design of ESO

Second-order ESO is as follows:

$$\{\begin{array}{l}e=z_1-u_{\mathrm{dc}}\\ {\stackrel{•}{z}}_1=z_2-{\beta }_{1}fal(e,{\alpha }_1,{\delta }_1)\\ {\stackrel{•}{z}}_2=-{\beta }_{2}fal(e,{\alpha }_2,{\delta }_2)\end{array}+b{i}_{\mathrm{d}}^{*}$$

(10)

where the fal function [25] is as follows:

$$fal(e,\alpha ,\delta )=\{\begin{array}{ll}{\left|e\right|}^{\alpha }\mathrm{sign}(e),& \left|e\right|>b\\ \frac{e}{b^{\alpha }},& \left|e\right|\le b\end{array}$$

(11)

The tracking signal of system’s output voltage and estimation of total disturbance z₂ can be obtained by the second-order ESO. In Formula (10), the β₁, β₂ are the parameters which are selected properly for the observer. Through regulating the parameters α, δ in the nonlinear fal function, the filtering effectiveness of ESO on the system as well as the adaptability to the uncertainty and disturbance of the system’s mathematical model can be enhanced.

3.

Design of NLSEF

NLSEF is as follows:

$$\{\begin{array}{l}{e}_1=v_1-z_1\\ u_{\mathrm{o}}={\beta }_{3}fal(e,{\alpha }_3,{\delta }_3)\\ i_{\mathrm{d}}^{*}=u_{\mathrm{o}}-z_2/b\end{array}$$

(12)

The NLSEF calculates the control quantity u_o of the ADRC controller according to the e₁ between the tracking differentiator and the output of ESO, and then obtained the final desired value i^*_d of inner loop currents through the comprehensive of total disturbance compensation quantity z₂/b.

The ADRC controller replaces the PI controller to maintain the DC-link voltage stability and provide the desired value i^*_d of inner loop currents. The outer loop voltage control scheme based on ADRC is shown in Figure 2.

4.2. Design of PBC Controller

According to the EL mathematical model of the three-phase bridgeless rectifier above, the inner loop current PBC controller is designed. Let the energy storage function H(x) and the error energy function H_e(x). The purpose of the PBC controller is to make the system converge to the desired equilibrium point as soon as possible, that is, x^* = [i_d^* i_q^* u_dc^*]^T = [i_d^* 0 u_dc^*]^T. If the derivate of the error energy function ${\mathit{H}}_{\mathrm{e}}={\mathit{x}}_{\mathrm{e}}^{\mathrm{T}}\mathit{M}{\mathit{x}}_{\mathrm{e}}$ holds, then the convergence of at the desired equilibrium point x^* of system can be realized, that is, x→x^*. In order to accelerate the convergence of error energy function, it is necessary to inject virtual damping R_a into the system. The virtual damping term after injecting virtual damping into the system is R_d = (R_a + R). R_d is the system damping matrix after injecting virtual damping. R is the system inherent damping matrix. R_a is the injected virtual damping matrix as follows:

$${\mathit{R}}_{\mathrm{a}}=\left(\begin{array}{ccc}{r}_{\mathrm{a}1}& 0& 0\\ 0& r_{\mathrm{a}2}& 0\\ 0& 0& r_{\mathrm{a}3}\end{array}\right)$$

The EL mathematical model of the system after injecting virtual damping is as follows:

$$\mathit{M}\stackrel{•}{{\mathit{x}}_{\mathit{e}}}+\mathit{J}{\mathit{x}}_{\mathit{e}}+{\mathit{R}}_{\mathit{d}}{\mathit{x}}_{\mathit{e}}=\mathit{u}-(\mathit{M}\stackrel{•}{{\mathit{x}}^{*}}+\mathit{J}{\mathit{x}}^{*}+\mathit{R}{\mathit{x}}^{*}-{\mathit{R}}_{\mathit{a}}{\mathit{x}}_{\mathit{e}})$$

(13)

For avoiding coupling in the process of dynamic response, the above formula is further sorted out:

$$\mathit{M}\stackrel{•}{{\mathit{x}}_{\mathit{e}}}+{\mathit{R}}_{\mathit{d}}{\mathit{x}}_{\mathit{e}}=\mathit{u}-[\mathit{M}\stackrel{•}{{\mathit{x}}^{*}}+\mathit{J}({\mathit{x}}^{*}+{\mathit{x}}_{\mathit{e}})+\mathit{R}{\mathit{x}}^{*}-{\mathit{R}}_{\mathit{a}}{\mathit{x}}_{\mathit{e}})]$$

(14)

The PBC controller can be obtained from the Formula (14):

$$\mathit{u}=\mathit{M}\stackrel{•}{{\mathit{x}}^{*}}+\mathit{J}\mathit{x}+\mathit{R}{\mathit{x}}^{*}-{\mathit{R}}_{\mathit{a}}{\mathit{x}}_{\mathit{e}}$$

(15)

Under the action of PBC controller, the error energy function $\stackrel{•}{{\mathit{H}}_{\mathit{e}}}(\mathit{x})=-{\mathit{x}}_{\mathit{e}}^{\mathit{T}}{\mathit{R}}_{\mathit{d}}{\mathit{x}}_{\mathit{e}}$, which is always less than 0. Thus, the system can converge to the desired equilibrium point x^* = [i_d^* i_q^* u_dc^*]^T = [i_d^* 0 u_dc^*]^T quickly. Obviously, the convergence speed of the error energy function depends on the injected virtual damping R_a.

In addition, the $\mathit{M}\stackrel{•}{{\mathit{x}}^{*}}$ term is always equal to 0, so the PBC controller can be further simplified as follows:

$$\mathit{u}=\mathit{J}\mathit{x}+\mathit{R}{\mathit{x}}^{*}-{\mathit{R}}_{\mathit{a}}{\mathit{x}}_{\mathit{e}}$$

(16)

The line switching functions d_ld and d_lq can be obtained as follows:

$$\{\begin{array}{l}{d}_{\mathrm{ld}}=\frac{\sqrt{3}{u}_{\mathrm{ld}}-3L{i}_{\mathrm{d}}^{*}+3\omega L{i}_{\mathrm{q}}-3r_{\mathrm{a}1}(i_{\mathrm{d}}-i_{\mathrm{d}}^{*})}{\sqrt{3}{u}_{\mathrm{dc}}}\\ d_{\mathrm{lq}}=\frac{\sqrt{3}{u}_{\mathrm{lq}}-3\omega L{i}_{\mathrm{d}}-3r_{\mathrm{a}2}{i}_{\mathrm{q}}}{\sqrt{3}{u}_{\mathrm{dc}}}\end{array}$$

(17)

Then, the above line switching functions are rotated from the synchronous rotation transformation inverse matrix M_ldq0/abc to obtain the switching functions: d_ab, d_bc, d_ca. Where the rotation matrix is as follows:

$${\mathit{M}}_{\mathrm{ldq}0/\mathrm{abc}}=\left(\begin{array}{ccc}\cos (\omega t+\pi /6)& \sin (\omega t+\pi /6)& 1\\ \cos (\omega t-\pi /2)& \sin (\omega t-\pi /2)& 1\\ \cos (\omega t+5\pi /6)& \sin (\omega t+5\pi /6)& 1\end{array}\right)$$

(18)

The relationship between d_ab, d_bc, d_ca and d_a, d_b, d_c is

$$\{\begin{array}{l}{d}_{\mathrm{a}}=\frac{1}{3}(d_{\mathrm{ab}}-d_{\mathrm{ca}}+\Delta d)\\ d_{\mathrm{b}}=\frac{1}{3}(d_{\mathrm{bc}}-d_{\mathrm{ab}}+\Delta d)\\ d_{\mathrm{c}}=\frac{1}{3}(d_{\mathrm{ca}}-d_{\mathrm{bc}}+\Delta d)\end{array}$$

(19)

The AC side voltage equations of three-phase bridgeless rectifier under the condition of grid imbalance are

$$\{\begin{array}{l}{u}_{\mathrm{a}}=L\frac{\mathrm{d}{i}_{\mathrm{a}}}{\mathrm{d}t}+d_{\mathrm{a}}{u}_{\mathrm{dc}}+\frac{1}{3}(\Delta u-\Delta d{u}_{\mathrm{dc}})\\ u_{\mathrm{b}}=L\frac{\mathrm{d}{i}_{\mathrm{b}}}{\mathrm{d}t}+d_{\mathrm{b}}{u}_{\mathrm{dc}}+\frac{1}{3}(\Delta u-\Delta d{u}_{\mathrm{dc}})\\ u_{\mathrm{c}}=L\frac{\mathrm{d}{i}_{\mathrm{c}}}{\mathrm{d}t}+d_{\mathrm{c}}{u}_{\mathrm{dc}}+\frac{1}{3}(\Delta u-\Delta d{u}_{\mathrm{dc}})\end{array}$$

(20)

By observing Formula (20), it can be seen that in order to overcome the influence of unbalanced voltage Δu on the rectifier only needs to meet the equation Δd = Δu/u_dc, then the above Formula (20) becomes

$$\{\begin{array}{l}{u}_{\mathrm{a}}=L\frac{\mathrm{d}{i}_{\mathrm{a}}}{\mathrm{d}t}+d_{\mathrm{a}}{u}_{\mathrm{dc}}\\ u_{\mathrm{b}}=L\frac{\mathrm{d}{i}_{\mathrm{b}}}{\mathrm{d}t}+d_{\mathrm{b}}{u}_{\mathrm{dc}}\\ u_{\mathrm{c}}=L\frac{\mathrm{d}{i}_{\mathrm{c}}}{\mathrm{d}t}+d_{\mathrm{c}}{u}_{\mathrm{dc}}\end{array}$$

(21)

Formula (21) shows that in order to ensure the normal operation of the three-phase bridgeless rectifier under the condition of the grid imbalance, a set of phase-switching functions that can compensate the grid voltages imbalance must be obtained as follows:

$$\{\begin{array}{l}{d}_{\mathrm{a}}=\frac{1}{3}(d_{\mathrm{ab}}-d_{\mathrm{ca}}+\frac{\Delta u}{u_{\mathrm{dc}}})\\ d_{\mathrm{b}}=\frac{1}{3}(d_{\mathrm{bc}}-d_{\mathrm{ab}}+\frac{\Delta u}{u_{\mathrm{dc}}})\\ d_{\mathrm{c}}=\frac{1}{3}(d_{\mathrm{ca}}-d_{\mathrm{bc}}+\frac{\Delta u}{u_{\mathrm{dc}}})\end{array}$$

(22)

Then, according to the relationship between the switch functions S_j (j = a, b, c) and d_j (j = a, b, c), the driving signal of power switches in each phase is obtained.

4.3. Input Current Zero-Crossing Distortion Elimination

Obviously, the unidirectional rectifier has the problem of input current zero-crossing distortion in essence. In order to eliminate the zero-crossing distortion, this paper adopted the method, through controlling the voltage u_ON between the neutral point of the rectifier input terminal and three-phase power supply to eliminate the zero-crossing distortion, which was described in detail in [13]. This paper will briefly discuss this method as follows:

Step 1: Assuming the zero-crossing distortion is not existing, following the changing law of voltage u_aN, u_bN, u_cN and then obtain the desired value u_aN^*, u_bN^*, u_cN^*.

Step 2: Judge whether the input current in Phase x (x = a, b, c) is flowing into the zero-crossing distortion region or not. The judging criterion is defined as the Formula (23).

$$i_x^{\ast }\times u_{xN}^{\ast }\le 0$$

(23)

Step 3: If the input current in Phase x (x = a, b, c) is out of the zero-crossing distortion region, let the desired voltage u_aN^*, u_bN^*, u_cN^* are equal to the desired voltage u_aO^*, u_bO^*, u_cO^*.

Step 4: If the input current in Phase x (x = a, b, c) is flowing into the the zero-crossing distortion region, control the corresponding voltage u_xO of input terminal to 0 and control the other voltages u_xO of input terminals to make their fundamental component follow the changing law of u_xN^*, so as to achieve the elimination of input current zero-crossing distortion. The whole adjustment process can be expressed by the following formula:

$$\{\begin{array}{l}{u}_{a\mathrm{O}}^{\ast }=u_{a\mathrm{N}}^{\ast }-u_{x\mathrm{N}}^{\ast }\\ u_{b\mathrm{O}}^{\ast }=u_{b\mathrm{N}}^{\ast }-u_{x\mathrm{N}}^{\ast }\\ u_{c\mathrm{O}}^{\ast }=u_{c\mathrm{N}}^{\ast }-u_{x\mathrm{N}}^{\ast }\end{array}$$

(24)

Above all, the overall control scheme of the three-phase bridgeless rectifier is shown in Figure 3 as follows:

5. Simulation Verification and Discussion

Main Parameters of System

To verify the effectiveness of the hybrid nonlinear control strategy proposed and the designed nonlinear controller in this paper, a three-phase bridgeless rectifier simulation model is built by using the power electronics module in Matlab/Simulink for verification. The main circuit parameters of the prototype are set as shown in the

Table 1. Main circuit parameters of prototype.

Parameters	Value
Rated Phase-a voltage/V	220
Rated Phase-b voltage/V	180
Rated Phase-c voltage/V	260
Grid frequency/Hz	50
AC inductance L/mH	5
Switching frequency/kHz	15
DC-link capacitor/μF	500
DC-link resistor/Ω	50
Rated DC-link voltage/V	500

. At the same time, it is compared with the traditional linear control strategies, such as PI + proportional resonance (PR).

In the hybrid nonlinear control strategy, the parameter settings of PBC controller and ADRC controller are shown in the

Table 2. Control coefficients of prototype.

Parameters	Value
ADRC k	1
ADRC m	8 × 106
ADRC n	3 × 10−6
ADRC α1	0.5
ADRC α2	0.5
ADRC α3	0.7
ADRC β1	100
ADRC β2	50
ADRC β3	30
ADRC δ1	0.01
ADRC δ2	0.01
ADRC δ3	0.02
Damping coefficient ra1	100
Damping coefficient ra2	50
Damping coefficient ra3	50

.

Set the main circuit parameters and control coefficients of prototype according to the Table 1 and Table 2. Simulated the prototype under the condition of grid voltages imbalance. The Figure 4 shows the AC side input voltages and currents under the control strategy proposed in this paper. Moreover, in order to verify the sensitivity of the control strategy to the phase difference of three-phase grid voltage, the phase angle of phase-b is −100 degrees and the phase of phase-c is 130 degrees. The simulation results are shown in Figure 5. It can be seen from Figure 4 and Figure 5 that the proposed strategy can ensure the AC side three-phase input currents sinusoidal and realize the unity power factor under the condition of grid voltages imbalance.

To better verify that the control strategy in this paper can cope with worse grid voltage quality, it is proposed to inject 10% of the 3rd, 5th and 7th harmonic voltages into Phase a and Phase b, respectively, on the basis of the above three-phase unbalanced grid voltages. The grid voltages and three-phase input currents on the AC side after harmonic injection are shown in Figure 6.

The response curves of DC-link output voltages u_dc1, u_dc2 and u_dc3 during steady-state operation of three-phase Y-connected bridgeless rectifier are shown in Figure 7. It can be seen from Figure 7 that the DC-link voltages can track the desired value well. At the same time, the DC-link voltages response curves of the three-phase basically coincide, which proves that the balance between the three phases can be realized under the hybrid PBC and ADRC.

To verify the advantages of hybrid nonlinear control of PBC and ADRC, the Figure 8 is a comparison with the DC-link output voltage response curves under the traditional PI + PR control strategy. Under the PI + PR control strategy, the voltage responses between three phases are not balanced well. Beyond that, voltage fluctuations under PI + PR control are greater than the PBC + ADRC control. Although traditional PI + PR control can stabilize the DC-link voltages at the desired value, under the hybrid PBC and ADRC, the DC-link voltage response is closer to the desired value u_dc^*, and the voltage difference between the three phases is smaller. The comparison of DC-link voltage response curves in Figure 8. Figure 9 compares the three-phase input currents of AC side under two different control strategies. The three-phase input currents of hybrid PBC and ADRC, PI + PR control strategy are 1.34% and 4.61%, respectively. Proving that the hybrid PBC and ADRC control strategy makes the rectifier have lower current harmonics. In addition, it can be seen that through adding the modified rules, the input current zero-crossing distortion is eliminated very well as shown in Figure 9.

To test the anti-interference ability of the rectifier under the hybrid PBC and ADRC control strategy, load disturbance is injected into the rectifier at the time of 0.3 s, that is, the DC-link load R_L changes from 50 Ω to 25 Ω, the disturbance lasts for 0.4 s, the disturbance disappears at the time of 0.7 s, and the DC-link load R_L returns to 50 Ω. The Figure 10 is the response curves of the DC-link voltage u_dc1 of the rectifier when the load changes suddenly. It can be seen from the response curve that the rectifier can quickly sense the load disturbance and stabilize the DC-link voltage at the desired value. At the same time, it also compares the DC-link voltage response under load disturbance of PI + PR control strategy. As can be seen from Figure 10, under the PI + PR control strategy, the DC-link voltage drop is greater when the load changes, and the voltage recovery time is relatively long. It is further proved that the rectifier has better anti-interference performance under the control strategy of hybrid PBC and ADRC.

6. Discussion

This paper has studied the hybrid nonlinear control strategy of three-phase Y-connected bridgeless rectifier under the condition of power grid imbalance. The working principle of the three-phase Y-connected bridgeless rectifier is analyzed and the EL mathematical model is established. A hybrid nonlinear control strategy using voltage outer loop ADRC controller and current inner loop PBC controller are designed. With the control strategy, there is no requirement to consider the positive and negative sequence components of the voltages and the currents. The detection and processing of voltages and currents are simplified. Only the real-time values of grid voltages and AC side input currents can ensure that three-phase Y-connected bridgeless rectifier can maintain excellent steady-state and dynamic characteristics under the condition of power grid imbalance. A simulation model is constructed by Matlab/Simulink power electronics module to verify the proposed control strategy. At the same time, compared with the PI + PR controller, it also proves that the superiority of the control strategy proposed in this paper. It has important guiding significance for the application of three-phase Y-connected bridgeless rectifier in engineering. In order to further study the nonlinear control of the rectifier, some other nonlinear control methods, such as model predictive control and sliding mode control, can be considered in the future work. In addition, their advantages and disadvantages can be compared and summarized. Moreover, the parameter tuning of nonlinear controller and the power density of the system need to be further studied and optimized.

7. Conclusions

In this paper, a nonlinear control strategy for three-phase Y-connected bridgeless rectifier is proposed. The control strategy considers the nonlinear essence of rectifier, so as to improve the steady-state accuracy and dynamic performance of system under the complex working conditions of power grid imbalance. In addition, it can fundamentally solve the problem of large-scale asymptotic stability of nonlinear system. At the same time, the control strategy has simple control structure, can be applied to the various of power electronic converters, and has important popularization and application value.