Impedance Modeling and Stability Factor Assessment of Grid-connected Converters Based on Linear Active Disturbance Rejection Control

Hailiang Xu; Fei Nie; Zhongxing Wang; Shinan Wang; Jiabing Hu

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Impedance Modeling and Stability Factor Assessment of Grid-connected Converters Based on Linear Active Disturbance Rejection Control PDF

- ORCID：
Hailiang Xu (Member, IEEE)
✉
- ORCID：
Fei Nie
✉
- ORCID：
Zhongxing Wang
✉
- ORCID：
Shinan Wang
✉
- ORCID：
Jiabing Hu (Senior Member, IEEE)
✉

China University of Petroleum (East China), Qingdao, China； Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Updated：2021-11-23

DOI：10.35833/MPCE.2021.000280

OUTLINE

Abstract

With the increase of converter-based renewable energy generation connected into the power grid, the interaction between renewable energy and grid impedance has introduced lots of new issues, among which the sub- and super-synchronous oscillation phenomenon makes a big concern. The linear active disturbance rejection control (LADRC) is a potential way to improve the damping characteristics of the grid-connected system, but the key factors and influencing mechanism on system stability are unknown. This paper establishes the equivalent impedance and coupling admittance models of a typical three-phase grid-connected converter. Then, the influence of the key factors such as the bandwidth of the LADRC and grid impedance on the stability and frequency coupling effect is assessed in detail. Finally, the theoretical analysis results are verified by simulations and experiments.

Keywords

Grid-connected converter; linear active disturbance rejection control (LADRC); sub- and super-synchronous oscillation; stability

I. Introduction

NOWADAYS, more and more renewable energy generations are being connected into the power grid, where the grid-connected converters play an irreplaceable role between the renewable energy and the power grid [

1]. It should be noted that large-scale renewable energy generations such as wind and photovoltaic farms are usually located in remote areas. Hence, the long distance transmission and multistage transformers make the grid impedance in-negligible in stability analysis [2], [3]. Moreover, with the increase of grid impedance, the grid short-circuit ratio (SCR) decreases gradually, which will impair the current quality of grid-connected renewable energy [4].

The interaction between the renewable energy generation and weak grid may lead to serious stability issues. For instance, in Hami wind farm located in Xinjiang Uygur Autonomous Region of China, a sub-synchronous oscillation accident caused by the interaction between the wind converters and the weak grid occurred in 2015 [

5]. Note that such oscillation phenomenon has a common feature, i.e., a pair of interdependent sub- and super-synchronous oscillation components will appear synchronously. It is usually called the frequency coupling (FC) [6], or mirror frequency (MF) effect [7].

It is essential to point out that this kind of oscillation is quite different from that in the conventional single input and single output (SISO) system [

8], [9]. Due to the asymmetrical control links such as phase-locked loop (PLL) [10], and the considerable grid impedance [11], the current response will contain a disturbance component with its frequency at 2f₁-f_p or f_p-2f₁ in addition to the original frequency at f_p. Thus, the system becomes a single input and double output (SIDO) system, rather than the SISO one [12].

Consequently, the original stability criterion for the SISO system is no longer applicable. Instead, a generalized Nyquist criterion (GNC) can be utilized [

13], [14]. Due to the complex analysis of the GNC [15], the parameters of the system cannot be conveniently modified according to the stability margin. In [7], an improved sequence impedance modeling method is proposed, which can greatly simplify the modeling process. The method is also valid when the frequency coupling effect is considered [16], [17]. In the dq frame, a method to transform the impedance model from a multi-input multi-output (MIMO) system into an SISO one is proposed in [9]. This transformation can greatly simplify the stability analysis [18].

Various methods have been presented to suppress the oscillation components [

8], [19], [20]. In [19], an asymmetrical current control method is proposed to make up for the asymmetrical characteristics introduced by PLL. In [8], a flexible symmetrical control structure is designed by introducing the concept of complex phase angle. Theoretical analysis and experimental results demonstrate that the symmetrical PLL can significantly suppress the frequency coupling effect. Nevertheless, this method inevitably needs to alter the internal structure of PLL [21], and makes the stability analysis process more complicated.

As an alternative, the linear active disturbance rejection control (LADRC) is applied into the PLL in [

22], which can suppress the effect of FC caused by the grid impedance and unsymmetrical control loops. The original concept of the LADRC can be traced back to the ADRC in [23]. And LADRC is proposed in [24] by linearizing the controller and extended state observer (ESO), which makes the algorithm simple and easy to implement. Nowadays, the LADRC has been widely used in engineering because of its unique disturbance-rejection characteristic. Especially, in the grid-connected converter system, LADRC shows its superior control performance. In [25], the LADRC is introduced into the DC-link voltage controller to replace the conventional proportional-integral (PI) regulator for variable speed hydro-electric plants, and the transient performance of the LADRC-based system with a stiff grid is satisfactory. In [26] and [27], the LADRC is applied to the DC-link voltage control loop, which enhances the robustness of the grid-connected system. In addition, an improved LESO is proposed to form a new generalized integrator-ESO (GI-ESO) in PLL [28]. This new type of LADRC has an excellent performance in suppressing the background harmonics of the grid voltage.

Although the LADRC has been applied in converter-based grid-connected systems, the stability issue has not yet been sufficiently investigated, especially when the grid impedance is considerable. In addition, the LADRC is a potential way to improve the damping characteristics of grid-connected systems, but the key factors and the influencing mechanism on stability are unknown.

To address this issue, this paper presents a comprehensive stability assessment of LADRC-based converter control system with weak grid connection. And the single-parameter LADRC is adopted in the grid-connected system, considering that it cannot only effectively reduce the number of parameters to be tuned, but also has a ideal phase margin, compared with the traditional PI control [

29], [30]. As a specific contribution, this paper establishes the equivalent impedance and coupling admittance models of a typical three-phase pulse width modulation (PWM) converter under weak grid condition, where the LADRC is utilized into the DC-link voltage control. Furthermore, the system stability is analyzed according to frequency characteristic curves and Nyquist curves. Moreover, the influence of the key factors on system stability and frequency coupling effect is assessed in detail. Simulations and experiments are finally carried out to validate the correctness of the theoretical analysis.

II. LADRC-based DC-link Voltage Control of Grid-connected Converter

A. Configuration of Three-phase Grid-connected Converter

Figure 1 shows the main circuit and control structure of a three-phase grid-connected converter. Note that the LADRC is implemented in the DC-link voltage control loop, while the d- and q-axis current control and PLL still use the traditional PI regulator. In Fig. 1, $u_{d}$ and $u_{q}$ are the dq-axis components of PCC voltage; $i_{d r e f}$ and $i_{q r e f}$ are the reference currents of dq-axis components, respectively; m_i is the PWM signal; $m_{d}$ and $m_{q}$ are the dq-axis components of m_i; Q_1-6 is the output signal of space vector pulse width modulation (SVPWM); u_PCC is the point of common coupling (PCC) voltage; i_g is the grid-connected current; U_dc and U_dcref are the DC-link voltage and its reference, respectively; θ_PLL is the output angle of PLL; $H_{P L L}$ and $H_{i}$ are the PI controller of PLL and inner current loop, respectively; C_dc is the DC-link capacitor; R_load is the equivalent resistance of the DC-link load; and L_f and L_g are the equivalent inductances of filter and the power grid, respectively. For simplification, the grid impedance is expressed as $Z_{g} = j ω_{0} L_{g}$ , and $ω_{0}$ is the fundamental frequency.

Fig. 1 Main circuit and control structure of three-phase grid-connected converter with LADRC.

As for the DC-link circuit, it can be given as:

C_{d c} \frac{d U_{d c}}{d t} = i_{d c} - i_{l o a d}

(1)

where i_dc and i_load are the total DC-link current and the equivalent load current, respectively.

In order to obtain the parameters in LADRC, the relation between U_dc and i_d_ref needs to be determined firstly. Note that U_dc is a dynamic quantity, while i_d_ref is a steady-state one. Thus, it is necessary to make the following assumptions: ① the power losses on the converters and the internal resistor of the inductive filter can be neglected, i.e., P_AC≈P_DC; ② the DC-link voltage and grid-connected current are well controlled, i.e., U_dc≈U_dcref and i_d≈i_d_ref [

31].

\{\begin{array}{l} P_{A C} = \frac{3}{2} U_{1} i_{d} \approx \frac{3}{2} U_{1} i_{d r e f} \\ P_{D C} = U_{d c} i_{d c} \approx U_{d c r e f} i_{d c} \end{array} \Rightarrow i_{d c} \approx \frac{3 U_{1}}{2 U_{d c r e f}} i_{d r e f}

(2)

where U₁ is the fundamental magnitude of grid voltage.

Based on (1) and (2), the relation between i_d_ref and U_dc can be expressed by:

\frac{d U_{d c}}{d t} = \frac{3 U_{1}}{2 C_{d c} U_{d c r e f}} i_{d r e f} - \frac{U_{d c}}{C_{d c} R_{l o a d}}

(3)

It is notable that this is a transitional expression that can be used to derive the control parameters of LADRC.

B. Design of Second-order LADRC

Suppose that a second-order LADRC is applied in the DC-link voltage loop. Then, its model can be represented by:

\ddot{y} = \underset{f_{t o t a l}}{\underset{︸}{- a_{1} \dot{y} - a_{2} y + w + (b - b_{0}) u}} + b_{0} u

(4)

where u, y, and w are the input, output and unknown external disturbances, respectively; a₁ and a₂ are the parameters of the unknown system; b is the unknown input gain, while b₀ is a known value; and f_total is the total disturbance, containing both the internal and external uncertainties.

Assume that $y = U_{d c}$ and $u = i_{d r e f}$ . By taking the derivation of (3), the expression of b₀ can be obtained, as show in (5).

\frac{d^{2} U_{d c}}{d t^{2}} = - \underset{b_{0}}{\underset{︸}{\frac{3 U_{1}}{2 R_{l o a d} C_{d c}^{2} U_{d c r e f}}}} i_{d r e f} + \underset{f_{t o t a l}}{\underset{︸}{\frac{U_{d c}}{(C_{d c} R_{l o a d})^{2}}}}

(5)

Introduce state variables x₁, x₂, and extended state variable x₃, with $x_{1} = y$ , $x_{2} = \dot{y}$ , and $x_{3} = f_{t o t a l}$ . Then, the LESO can be organized as:

\{\begin{array}{l} {\dot{z}}_{1} = z_{2} + k_{1} (y - z_{1}) \\ {\dot{z}}_{2} = z_{3} + k_{2} (y - z_{1}) + b_{0} u \\ {\dot{z}}_{3} = k_{3} (y - z_{1}) \end{array}

(6)

where z₁, z₂, and z₃ are the estimations of x₁, x₂, and f_total, respectively; and k₁, k₂ and k₃ are the observer gains, respectively.

If the estimation error between z₃and f_total can be neglected, i.e., $z_{3} \approx x_{3} = f_{t o t a l}$ , the original nonlinear control system becomes a linear integrated series control system. Therefore, the linear state error feedback (LESF) law can be designed as:

u = \frac{1}{b_{0}} [l_{2} (r - z_{1}) - l_{1} z_{2} - z_{3}]

(7)

where r is the input reference.

Finally, the expression of LESO (6) and the expression of LSEF law (7) make up the whole LADRC system.

Note that there are five parameters to be confirmed, i.e., l₁, l₂, k₁, k₂, and k₃. In [

29], it reveals that the single-parameter structure enables the system to obtain a fast step-free response without overshoot, and it is superior to the traditional PI controller in suppressing low-frequency external and high-frequency internal disturbances. Hence, a parameter

ω_{L}

called bandwidth of the LADRC is introduced. Consequently, l₁, l₂, k₁, k₂, and k₃ can then be expressed by

ω_{L}

as:

\{\begin{array}{l} l_{1} = 2 ω_{L} \\ l_{2} = ω_{L}^{2} \\ k_{1} = 3 ω_{L} \\ k_{2} = 3 ω_{L}^{2} \\ k_{3} = ω_{L}^{3} \end{array}

(8)

It is worth noting that $ω_{L}$ corresponds to the control bandwidth of the traditional PI controller, and is different from the controller bandwidth $ω_{c}$ , which is usually 3-10 times the observer bandwidth $ω_{o}$ in the traditional LADRC. In other words, $ω_{L} = ω_{c} = ω_{o}$ . Consequently, this design can reduce the number of control parameters, which further simplifies the tuning process of parameters. In addition, it can also make the system have a satisfactory phase margin [

30].

C. Analysis of Tracking and Anti-interference Characteristics

Taking the design of d-axis current controller as an example, and considering the signal-sampling delay of the inner current loop, the relationship between i_d_ref and U_dc is shown in Fig. 2, where k_pi and k_ii are the control parameters of inner current loop, respectively.

Fig. 2 Control structure of inner current loop.

According to Fig. 2, the transfer function between i_d_ref and U_dc can be deduced as:

G_{p} (s) = \frac{3 U_{1} R_{l o a d}}{2 U_{d c r e f} (R_{l o a d} C_{d c} s + 1) (4 T_{s} s + 1)}

(9)

where T_s is the sampling time.

Figure 3 shows the DC-link voltage control loop, where the LADRC is adopted. Assume $U (s)$ , $Y (s)$ , and $R (s)$ are the Laplace transforms of u, y, and r, respectively. For a clear description, define C₁(s) and C(s) to be the respective feedforward and feedback controllers, with their transfer functions shown in (10). Consequently, Fig. 3 can be simplified to the standard structure of LADRC, as shown in Fig. 4.

Fig. 3 DC-link voltage control loop.

Fig. 4 Simplified control structure of LADRC.

\{\begin{array}{l} C_{1} (s) = \frac{ω_{L}^{2} (s + ω_{L})^{3}}{ω_{L}^{3} (10 s^{2} + 5 ω_{L} s + ω_{L}^{2})} \\ \begin{array}{l} C (s) = \frac{ω_{L}^{3} (10 s^{2} + 5 ω_{L} s + ω_{L}^{2})}{b_{0} s (s^{2} + 5 ω_{L} s + 10 ω_{L}^{2})} = \\ \underset{C_{L A D R C_P I D} (s)}{\underset{︸}{(\frac{5 ω_{L}}{b_{0}} + \frac{ω_{L}^{2}}{b_{0} s} + \frac{10}{b_{0}} s)}} \underset{C_{L A D R C_L P F} (s)}{\underset{︸}{\frac{1}{s^{2} + 5 ω_{L} s + 10 ω_{L}^{2}}}} \end{array} \end{array}

(10)

where C_{LADRC_PID}(s) and C_{LADRC_LPF}(s) are the equivalent proportional integral differential (PID) and second-order filter parts in C(s), respectively.

The frequency response characteristics of the open-loop transfer function are shown in Fig. 5(a). It can be observed that the two parts of the LADRC, i.e., C_{LADRC_PID}(s) and C_{LADRC_LPF}(s), are indivisible. It means that only when the two parts get combined, we can obtain the satisfactory bandwidth and phase margin.

Fig. 5 Frequency characteristic curves of L(s). (a) $L_{L A D R C} (s) = C (s) G_{p} (s)$ , $L_{L A D R C_P I D} (s) = C_{L A D R C_P I D} (s) G_{p} (s), a n d$ $L_{L A D R C_L P F} (s) = C_{L A D R C_L P F} (s) G_{p} (s)$ . (b) $L_{L A D R C} (s) = C (s) G_{p} (s), L_{P I} (s) = H_{P I} (s) G_{p} (s), a n d L_{L P F} (s) = H_{L P F} (s) G_{p} (s)$ .

To make it more clear, Fig. 5(b) compares bode diagrams of L(s). Note that the bandwidths of these three controllers are set to be the same. With the same control bandwidth, the system phase margin with LADRC is much higher than that of the PI or LPF, implying that the system stability can be evidently enhanced when the LADRC is adopted.

Then, according to Fig. 4, the close-loop transfer function of the LADRC-based system can be derived as:

y = \frac{C_{1} (s) C (s) G_{P} (s)}{1 + C (s) G_{P} (s)} r + \frac{G_{P} (s)}{1 + C (s) G_{P} (s)} f_{t o t a l}

(11)

Figure 6 depicts the frequency characteristic curves with different $ω_{L}$ . As shown in Fig. 6(a), when $ω_{L}$ increases from 300 rad/s to 700 rad/s, the control bandwidth of the tracking signal gets enlarged. However, the attenuation ability of the high frequency signal will be weakened. To ensure enough tracking and anti-interference performance, $ω_{L}$ should be 5-10 times of the bandwidth of traditional PI controller [

25].

Fig. 6 Frequency characteristic curves with different $ω_{L}$ . (a) Tracking characteristic. (b) Anti-interference characteristic.

Figure 6(b) shows the anti-interference characteristics of the LADRC as $ω_{L}$ increases. It is notable that with the increase of $ω_{L}$ , the anti-interference characteristic on the middle and low frequency band will be influenced, while the high frequency band is not effected significantly. However, because of the high frequency roll-off, the high-frequency internal disturbance will be suppressed as well.

III. Impedance Modeling of Three-phase Grid-connected Converter with LADRC

A. Equivalent Impedance Modeling of System with LADRC

When the grid impedance L_g is considered, the coupling current disturbance $I_{ω_{p} - 2 ω_{0}}$ is superimposed with the original one, which intensifies the frequency coupling effect of the system. This disturbance transfer process can be represented as the signal flow diagram shown in Fig. 7.

Fig. 7 Disturbance transfer process considering L_g.

In Fig. 7, $Y_{p p} (s)$ and $Y_{p n} (s)$ represent the equivalent admittance from current disturbance $I_{ω_{p}}$ and coupling current disturbance $I_{ω_{p} - 2 ω_{0}}$ to voltage disturbance $U_{ω_{p}}$ , and $Y_{p p} (s)$ and $Y_{p n} (s)$ can be deduced as (12) and (13).

Y_{p p} (s) = \frac{|\begin{matrix} M_{1} + H_{2, ω_{p}} I_{1}^{*} + H_{2, ω_{p} - 2 ω_{0}} I_{1} & - [(s - j ω_{0}) C_{d c} + 1 / R_{l o a d}] / (3 k_{P W M}) + H_{4, ω_{p}} I_{1}^{*} + H_{4, ω_{p} - 2 ω_{0}} I_{1} & - H_{3, ω_{p}} I_{1}^{*} - H_{3, ω_{p} - 2 ω_{0}} I_{1} \\ - k_{P W M} U_{d c r e f} H_{2, ω_{p}} & - k_{P W M} U_{d c r e f} H_{4, ω_{p}} - k_{P W M} M_{1} & - 1 + k_{P W M} U_{d c r e f} H_{3, ω_{p}} \\ - k_{P W M} U_{d c r e f} H_{2, ω_{p} - 2 ω_{0}} - (s - j 2 ω_{0}) L_{f} & - k_{P W M} U_{d c r e f} H_{4, ω_{p} - 2 ω_{0}} - k_{P W M} M_{1}^{*} & k_{P W M} U_{d c r e f} H_{3, ω_{p} - 2 ω_{0}} \end{matrix}|}{|\begin{matrix} M_{1}^{*} + H_{1, ω_{p}} I_{1}^{*} + H_{1, ω_{p} - 2 ω_{0}} I_{1} & M_{1} + H_{2, ω_{p}} I_{1}^{*} + H_{2, ω_{p} - 2 ω_{0}} I_{1} & - [(s - j ω_{0}) C_{d c} + 1 / R_{l o a d}] / (3 k_{P W M}) + H_{4, ω_{p}} I_{1}^{*} + H_{4, ω_{p} - 2 ω_{0}} I_{1} \\ - k_{P W M} U_{d c r e f} H_{1, ω_{p}} - s L_{f} & - k_{P W M} U_{d c r e f} H_{2, ω_{p}} & - k_{P W M} U_{d c r e f} H_{4, ω_{p}} - k_{P W M} M_{1} \\ - k_{P W M} U_{d c r e f} H_{1, ω_{p} - 2 ω_{0}} & - k_{P W M} U_{d c r e f} H_{2, ω_{p} - 2 ω_{0}} - (s - j 2 ω_{0}) L_{f} & - k_{P W M} U_{d c r e f} H_{4, ω_{p} - 2 ω_{0}} - k_{P W M} M_{1}^{*} \end{matrix}|}

(12)

Y_{p n} (s) = \frac{|\begin{matrix} M_{1}^{*} + H_{1, ω_{p}} I_{1}^{*} + H_{1, ω_{p} - 2 ω_{0}} I_{1} & - [(s - j ω_{0}) C_{d c} + 1 / R_{l o a d}] / (3 k_{P W M}) + H_{4, ω_{p}} I_{1}^{*} + H_{4, ω_{p} - 2 ω_{0}} I_{1} & - H_{3, ω_{p}} I_{1}^{*} - H_{3, ω_{p} - 2 ω_{0}} I_{1} \\ - k_{P W M} U_{d c r e f} H_{1, ω_{p}} - s L_{f} & - k_{P W M} U_{d c r e f} H_{4, ω_{p}} - k_{P W M} M_{1} & - 1 + k_{P W M} U_{d c r e f} H_{3, ω_{p}} \\ - k_{P W M} U_{d c r e f} H_{1, ω_{p} - 2 ω_{0}} & - k_{P W M} U_{d c r e f} H_{4, ω_{p} - 2 ω_{0}} - k_{P W M} M_{1}^{*} & k_{P W M} U_{d c r e f} H_{3, ω_{p} - 2 ω_{0}} \end{matrix}|}{|\begin{matrix} M_{1}^{*} + H_{1, ω_{p}} I_{1}^{*} + H_{1, ω_{p} - 2 ω_{0}} I_{1} & M_{1} + H_{2, ω_{p}} I_{1}^{*} + H_{2, ω_{p} - 2 ω_{0}} I_{1} & - [(s - j ω_{0}) C_{d c} + 1 / R_{l o a d}] / (3 k_{P W M}) + H_{4, ω_{p}} I_{1}^{*} + H_{4, ω_{p} - 2 ω_{0}} I_{1} \\ - k_{P W M} U_{d c r e f} H_{1, ω_{p}} - s L_{f} & - k_{P W M} U_{d c r e f} H_{2, ω_{p}} & - k_{P W M} U_{d c r e f} H_{4, ω_{p}} - k_{P W M} M_{1} \\ - k_{P W M} U_{d c r e f} H_{1, ω_{p} - 2 ω_{0}} & - k_{P W M} U_{d c r e f} H_{2, ω_{p} - 2 ω_{0}} - (s - j 2 ω_{0}) L_{f} & - k_{P W M} U_{d c r e f} H_{4, ω_{p} - 2 ω_{0}} - k_{P W M} M_{1}^{*} \end{matrix}|}

(13)

where $H_{i, ω_{p}}$ and $H_{i, ω_{p} - 2 ω_{0}}$ $(i = 1, 2, 3, 4)$ are the influence coefficients of each disturbance component on the PWM waveforms, respectively; $M_{1} = (U_{1} - s L_{f} I_{1}) / (k_{P W M} U_{d c r e f})$ is the fundamental component of PWM waveform; U₁ and I₁ are the fundamental voltage and current, respectively; $U_{ω_{p} - 2 ω_{0}}$ is the coupling voltage disturbance; ‘*’ represents a conjugation of the variable.

Note that $Y_{n n} (s)$ and $Y_{n p} (s)$ can be deduced by $Y_{p p} (s)$ and $Y_{p n} (s)$ [

32], [33], i.e.,

\{\begin{matrix} Y_{n n} (s) = Y_{p p}^{*} (s - j 2 ω_{0}) \\ Y_{n p} (s) = Y_{p n}^{*} (s - j 2 ω_{0}) \end{matrix}

(14)

Therefore, the equivalent impedance of the grid-side converter considering FC under the weak grid condition can be written as:

Z_{a c e q} (s) = \frac{U_{ω_{p}}}{I_{ω_{p}}} = \frac{1}{Y_{p p} (s) \underset{Y_{c} (s)}{\underset{︸}{- \frac{Y_{p n} (s) Y_{n p} (s) Z_{g} (s - j 2 ω_{0})}{1 + Y_{n n} (s) Z_{g} (s - j 2 ω_{0})}}}}

(15)

where $Y_{c} (s)$ is the equivalent admittance of reactive frequency-coupling effect in Fig. 7.

B. Equivalent Impedance Modeling of Control Links

The disturbance component of DC-link voltage in frequency domain can be expressed as:

U_{d c, ω_{p} - ω_{0}} = \frac{3 k_{P W M} (M_{ω_{p}} I_{1}^{*} + M_{ω_{p} - 2 ω_{0}} I_{1} + M_{1}^{*} I_{ω_{p}} + M_{1} I_{ω_{p} - 2 ω_{0}})}{(s - j ω_{0}) C_{d c} + 1 / R_{l o a d}}

(16)

M_{a p} (f) = \{\begin{array}{l} M_{ω_{p}} = H_{1, ω_{p}} (s) I_{ω_{p}} + H_{2, ω_{p}} (s) I_{ω_{p} - 2 ω_{0}} + H_{3, ω_{p}} (s) U_{ω_{p}} + H_{4, ω_{p}} (s) U_{d c, ω_{p}} & f = \pm f_{p} \\ M_{ω_{p} - 2 ω_{0}} = H_{1, ω_{p} - 2 ω_{0}} (s) I_{ω_{p}} + H_{2, ω_{p} - 2 ω_{0}} (s) I_{ω_{p} - 2 ω_{0}} + H_{3, ω_{p} - 2 ω_{0}} (s) U_{ω_{p}} + H_{4, ω_{p} - 2 ω_{0}} (s) U_{d c, ω_{p}} & f = \pm (f_{p} - 2 f_{1}) \end{array}

(17)

where $M_{ω_{p}}$ and $M_{ω_{p} - 2 ω_{0}}$ are the disturbances of PWM waveform at frequency of $f_{p}$ and $f_{p} - 2 f_{1}$ , respectively.

As a result, a disturbance component with frequency $f_{p} - f_{0}$ will be contained in the reference of inner current loop or the DC-link output of the voltage loop controller, i.e.,

\begin{matrix} I_{d r e f, ω_{p} - ω_{0}} (f) = U_{d c, ω_{p} - ω_{0}} (f) C (s \mp j ω_{0}) & f = \pm f_{p} \end{matrix}

(18)

In addition, θ_PLL will be affected by the existence of voltage disturbance component. Further, the d- and q-axis components of the grid-connected current will be influenced by the angle disturbance in the output of PLL. Considering the coordinate transformation links, the disturbance of output angle $Δ θ_{P L L}$ [

16] can be expressed as:

\begin{matrix} Δ θ_{P L L} (f) = \mp j T_{P L L} (s) U_{ω_{p}} & f = \pm (f_{p} - f_{1}) \end{matrix}

(19)

T_{P L L} (s) = \frac{H_{P L L} (s)}{1 + U_{1} H_{P L L} (s)}

(20)

where $H_{P L L} (s) = (k_{p p} + k_{i p} / s) / s$ is the forward channel transfer function of PLL, and k_pp and k_ip are the control parameters of PLL.

Then, the d- and q-axis components of the grid-connected current affected by the disturbance of output angle can be deduced by:

\{\begin{array}{l} I_{d} (f) = I_{ω_{p}} + I_{ω_{p} - 2 ω_{0}} \mp I_{1} (s i n φ_{i 1}) Δ θ_{P L L} & f = \pm (f_{p} - f_{1}) \\ I_{q} (f) = \mp j I_{ω_{p}} \pm j I_{ω_{p} - 2 ω_{0}} \pm I_{1} (c o s φ_{i 1}) Δ θ_{P L L} & f = \pm (f_{p} - f_{1}) \end{array}

(21)

where $φ_{i 1}$ is the angle of fundamental current.

According to Fig. 1, the d- and q-axis components of the PWM modulated signal can be expressed as:

\{\begin{array}{l} m_{d} (s) = [(U_{d c r e f} - U_{d c}) C (s) - i_{d}] H_{i} (s) - ω_{0} L_{f} i_{q} \\ m_{q} (s) = (i_{q r e f} - i_{q}) H_{i} (s) + ω_{0} L_{f} i_{d} \end{array}

(22)

where H_i(s)=k_pi+k_ii/s is the transfer function of current PI controller.

And the PWM waveform of phase a can be expressed as:

m_{a} (s) = c o s (θ_{P L L} (s)) m_{d} (s) - s i n (θ_{P L L} (s)) m_{q} (s)

(23)

By considering (16)-(23), the coefficients in (12) and (13) can be solved and obtained as:

\{\begin{array}{l} H_{1, ω_{p}} (s) = H_{i} (s - j ω_{0}) - j ω_{0} L_{f} \\ H_{1, ω_{p} - 2 ω_{0}} (s) = 0 \end{array}

(24)

\{\begin{array}{l} H_{2, ω_{p}} (s) = 0 \\ H_{2, ω_{p} - 2 ω_{0}} (s) = H_{i} (s + j ω_{0}) + j ω_{0} L_{f} \end{array}

(25)

\{\begin{array}{l} \begin{array}{l} H_{3, ω_{p}} (s) = j \frac{1}{2} T_{P L L} (s - j ω_{0}) H_{i} (s - j ω_{0}) C (s - j ω_{0}) + \\ T_{P L L} (s - j ω_{0}) {[j ω_{0} L_{f} - H_{i} (s - j ω_{0})] I_{1} + M_{1}} \end{array} \\ \begin{array}{l} H_{3, ω_{p} - 2 ω_{0}} (s) = j \frac{1}{2} T_{P L L} (s + j ω_{0}) H_{i} (s + j ω_{0}) C (s + j ω_{0}) + \\ T_{P L L} (s + j ω_{0}) {[j ω_{0} L_{f} + H_{i} (s + j ω_{0})] I_{1}^{*} - M_{1}^{*}} \end{array} \end{array}

(26)

\{\begin{array}{l} H_{4, ω_{p}} (s) = \frac{1}{2} C (s - j ω_{0}) H_{i} (s - j ω_{0}) \\ H_{4, ω_{p} - 2 ω_{0}} (s) = \frac{1}{2} C (s + j ω_{0}) H_{i} (s + j ω_{0}) \end{array}

(27)

IV. Key Factor Assessment on System Stability

According to the impedance-based stability criterion, if the ratio between the grid impedance Z_g and the inverter output impedance Z_eq satisfies the Nyquist criterion, the system would be stable [

32]. This section will evaluate the influence of LADRC on the system stability.

A. Stability Mechanism Analysis

Figure 8 displays the bode diagrams of Z_eq(s), Z_g(s), and Y_couple(s) based on PI controller with different SCRs of the system. The damping characteristics of the system get deteriorated when the SCR decreases. As a consequence, the magnitude-frequency characteristic curves of Z_eq(s) and Z_g(s) intersect at several points, which are not conducive to the stability of the system. For instance, when $S C R = 2$ , the two curves intersect at four frequencies, i.e., 72 Hz, 80 Hz, 171 Hz, and 175 Hz, with the phase differences being 13°, 227°, 215°, and 95°, respectively. Note that, if the phase difference is more than 180°, the system will fall into an oscillation risk area.

Fig. 8 Frequency characteristic curves with PI controller ( $S C R = 2, 4,$ and 8). (a) Z_eq(s) and Z_g(s). (b) Y_couple(s).

In addition, as demonstrated in Fig. 8(b), the decrease of SCR increases the magnitude-frequency characteristic curve of Y_couple(s). It indicates that the intensity of the FC effect is negatively correlated with the SCR. Figure 9 presents the Nyquist curves with PI controller. With the decrease of SCR, the corresponding Nyquist curves gradually encircles (-1, j0). And when $S C R = 2$ , the system becomes unstable.

Fig. 9 Nyquist curves with PI controller. (a) $S C R = 2, 4,$ and 8. (b) $f_{v} = 20, 30$ and 40 Hz.

In [

10], it has revealed that reducing the bandwidth of the control loop can enhance the stability and depress the FC effect of the system under weak grid, which is verified by the Nyquist curve shown in Fig. 9(b). Furthermore, the stability and FC degree of the system with low SCR are compared when the traditional PI controller, the bandwidth reduced PI (BR-PI) controller, and LADRC work, and comparison results are shown in Fig. 10.

Fig. 10 Frequency characteristic curves with various kinds of control strategy. (a) Z_eq(s) and Z_g(s). (b) Y_couple(s).

From Fig. 10, it can be observed that when LADRC is introduced, both the magnitude-frequency characteristic curves of Z_eq(s) and Y_couple(s) show better performance than the other two controllers. Though reducing the bandwidth of PI controller can improve the stability of the system and reduce the degree of frequency coupling, it will worsen the dynamic performance of the system. Figure 10 also indicates that after adopting LADRC, the system stability can be enhanced with weak grid by improving the damping characteristics of the system.

B. Effect of Key Factors on Stability

In order to study the influence of LADRC parameters and grid impedance on the stability of the system, the frequency characteristic curves and Nyquist curves of $Z_{e q} (s)$ and $Y_{c o u p l e} (s)$ are drawn when the values of the $ω_{L}$ and $L_{g}$ are changed.

Figures 11 and 12 show the frequency characteristic curves with different L_g and ω_L, while Fig. 13(a) and (b) presents the corresponding Nyquist curves. As shown in Fig. 11(a), there is almost no intersection between Z_eq(s) and Z_g(s). Even if at the narrow intersection area (when the SCR is further reduced to 1.5), the phase difference is still less than 180°. Similar to Fig. 8(b), it can be observed from Fig. 11(b) that the magnitude of the coupling admittance increases a little as the SCR decreases. However, the Nyquist curves do not surround (-1, j0) in these cases, as shown in Fig. 13(a). It indicates that after the introduction of the LADRC, the system can be stable with a fairly low SCR.

Fig. 11 Frequency characteristic curves with LADRC when $S C R = 1.5, 1.75,$ and 2. (a) Z_eq(s) and Z_g(s). (b) Y_couple(s).

Fig. 12 Frequency characteristic curves with LADRC when $ω_{L} = 100$ , 500, 700, and 1000 rad/s. (a) Z_eq(s) and Z_g(s). (b) Y_couple(s).

Fig. 13 Nyquist curves with LADRC. (a) $S C R = 1.5, 1.75, a n d 2$ . (b) $ω_{L} = 100, 500, 700, a n d 1000 H z$ .

As shown in Fig. 12(a), the magnitude-frequency characteristic curve of Z_eq(s) at low frequency band gets large as $ω_{L}$ increases, which means that the damping of the system can be enhanced.

Meanwhile, as shown in Fig. 12(b), the magnitude of the coupling admittance decreases when $ω_{L}$ increases, indicating that the FC effect is significantly suppressed. Furthermore, although the Nyquist curves in Fig. 13(b) gradually approximate (-1, j0) with the increase of $ω_{L}$ , the adjustable range is much larger than that of the traditional PI controller. Besides, compared with the results shown in Figs. 8 and 9, the grid-connected system can remain stable and have a satisfactory controllable bandwidth range with the LADRC under weak grid.

In summary, it can be concluded that the LADRC is able to enhance the stability and robustness of the system under weak grid. Furthermore, it can effectively suppress the FC effect caused by the grid impedance and asymmetrical control links. But the bandwidth of LADRC should not be too large, otherwise the contrary results may be obtained.

V. Simulation and Experimental Results

In this section, simulation and experiments are carried out to verify the stability analysis of the grid-connected converter system with LADRC. A 2 kW converter prototype is built in the lab, as shown in Fig. 14. Simulation and experimental parameters are shown in Table I and Table II.

Fig. 14 Hardware of experimental platform.

TABLE I Simulation System Parameters

Parameter	Value
U_dcref	650 V
U₁	311 V
L_g	6.3, 3.2, 1.6 mH (SCR = 2, 4, 8)
L_f	3.5 mH
C_dc	4400 $μ F$
R_load	20 Ω
k_pp	0.367
k_ip	21.036
k_pv	0.50, 0.75, 1.007
k_iv	28.78, 64.76, 115.15
k_pi	4.003
k_ii	2289
ω_L	100, 300, 500, 700 rad/s

TABLE II Expertimental System Parameters

Parameter	Value
U_dcref	440 V
U₁	179.59 V
f_switch, f_sample	20, 10 kHz
L_g	18, 21, 24 mH ( $S C R = 2.5, 2.1, a n d 1.9$ )
L_f	1.5 mH
C_dc	1230 μF
R_load	100 Ω
k_pp	0.26
k_ip	2.3
k_pv	0.04
k_iv	22
k_pi	27
k_ii	15
ω_L	100, 300, 500, 700 rad/s

A. Stability Validation of LADRC

The simulation results with different control strategies are shown in Fig. 15(a). It demonstrates that when SCR decreases from 8 to 2, the oscillation of grid-connected current and DC-link voltage gradually intensifies. When SCR is reduced to 2, a pair of sub- and super-synchronous oscillation appears at 20 Hz and 80 Hz, respectively. This result corresponds to the theoretical analysis result shown in Fig. 8. Moreover, it can be observed from Fig. 15(b) that when LADRC is introduced, the sub- and super-synchronous oscillation can be suppressed well when $S C R = 2$ , and the total harmonic distortion (THD) value also reduces from 18.23% to 4.94% evidently.

Fig. 15 Simulation results with different control strategies. (a) $U_{d c}$ , $I_{g}$ , and THD with PI controller. (b) $U_{d c}$ , $I_{g}$ , and THD with LADRC. (c) Magnitude with PI controller. (d) Magnitude with LADRC.

Figure 16 gives the experimental results with different control strategies, it can be observed that the LADRC has a better performance than traditional PI controller in suppressing the sub- and super-synchronous oscillation in the grid-connected current. Furthermore, the system can remain stable when SCR is reduced to 1.9.

Fig. 16 Experimental results with different control strategies. (a) U_dc and I_a with PI controller when $S C R = 2.5$ . (b) Fast Fourier transform (FFT) result of I_a with PI controller when $S C R = 2.5$ . (c) U_dc and I_a with LADRC controller when $S C R = 2.5$ . (d) FFT result of I_a with LADRC controller when $S C R = 2.5$ . (e) U_dc and I_a with PI controller when $S C R = 2.1$ . (f) FFT result of I_a with PI controller when $S C R = 2.1$ . (g) U_dc and I_a with LADRC controller when $S C R = 1.9$ . (h) FFT result of I_a with LADRC controller when $S C R = 1.9$ .

B. Parameter Robustness and Dynamic Performance of LADRC

As mentioned above, changing the bandwidth of LADRC will have a certain impact on the system stability. As shown in Fig. 17, with the increase of $ω_{L}$ , the stability of the system will be enhanced to some extent. However, when $ω_{L}$ increases to 700 rad/s, high frequency ripples will emerge, which is consistent with the analysis in Section II. The ability to attenuate the high frequency disturbance will weaken with a too high bandwidth of $ω_{L}$ . Nevertheless, $ω_{L}$ can still change in a large range to obtain better robustness.

Fig. 17 Experimental results with different $ω_{L}$ . (a) $ω_{L} = 100 r a d / s$ . (b) $ω_{L}$ = $300 r a d / s$ . (c) $ω_{L} = 500 r a d / s .$ (d) $ω_{L} = 700 r a d / s .$

To test the dynamic performance of the system after the introduction of the LADRC, the value of the $U_{d c r e f}$ is suddenly changed from 440 V to 450 V when $S C R = 2.1$ . Figure 18 shows the experimental results with different $ω_{L}$ when $U_{d c r e f}$ increases from 440 V to 450 V. A bigger $ω_{L}$ can decrease the response time and there is no overshoot.

Fig. 18 Experimental results with different ω_L when U_dcref increases from 440 V to 450 V. (a) $ω_{L} = 100 r a d / s$ . (b) $ω_{L} = 300 r a d / s$ . (c) $ω_{L} = 500 r a d / s$ . (d) $ω_{L} = 700 r a d / s$ .

Figure 19 shows experimental results with different $ω_{L}$ when $I_{q r e f}$ increases from 0 to 3 A. Similarly, the system remains stable when $ω_{L}$ changes in a wide range. It is worth noting that the transient response process is shortened obviously, and there is no overshoot, either. This is consistent with the situation when U_dcref changes.

Fig. 19 Experimental results with different ω_L when I_q_ref increases from 0 to 3 A. (a) $ω_{L} = 100 r a d / s$ . (b) $ω_{L} = 300 r a d / s$ . (c) $ω_{L} = 500 r a d / s$ . (d) $ω_{L} = 700 r a d / s$ .

In summary, the system stability is greatly improved by LADRC. Although the bandwidth of LADRC can be adjusted in a large range, attention should be paid to avoid the influence of system stability due to more ripple.

VI. Conclusion

This paper establishes the equivalent impedance and the coupling admittance models considering the influence of LADRC, inner current loop controller, PLL, and the grid impedance. Based on these models, the influences of several key factors are studied. And the specific conclusions can be summarized as follows.

1) Based on the analysis of the established impedance model, the damping characteristic of system at low SCR is significantly enhanced with the introduction of LADRC, and the FC effect can be greatly suppressed.

2) Only $ω_{L}$ needs to be adjusted during the tuning process of single-parameter structure LADRC. Moreover, the system can keep robust when $ω_{L}$ is changed in a wide range.

3) Though $ω_{L}$ can be tuned in a wide range, it is necessary to set the $ω_{L}$ selection at 2-5 times of the bandwidth of the traditional PI controller to guarantee a good dynamic performance. This can be used as a guide in the tuning of LADRC parameters.

4) Considering the satisfactory dynamic and static performance of the LADRC, it can also be used in other control loops of the grid-connected system, e.g., the PLL and the current control. Similarly, the stability issue needs to be evaluated carefully when adopting the LADRC.

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