Linear Active Disturbance Rejection Control and Stability Analysis for Modular Multilevel Converters Under Weak Grid

Hailiang Xu; Mingkun Gao; Pingjuan Ge; Jiabing Hu

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Linear Active Disturbance Rejection Control and Stability Analysis for Modular Multilevel Converters Under Weak Grid PDF

- ORCID：
Hailiang Xu
✉
- ORCID：
Mingkun Gao
✉
- ORCID：
Pingjuan Ge
✉
- ORCID：
Jiabing Hu
✉

the School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Updated：2023-11-15

DOI：10.35833/MPCE.2022.000654

OUTLINE

Abstract

The modular multilevel converters (MMCs) are popularly used in high-voltage direct current (HVDC) transmission systems. However, for the direct modulation based MMC, its complex internal dynamics and the interaction with the grid impedance would induce the frequency coupling effect, which may lead to instability issues， especially in the case of weak grid. To effectively suppress the sub- and super-synchronous oscillations, this paper proposes a linear active disturbance rejection control (LADRC) based MMC control strategy. The LADRC mainly consists of the linear extended state observer (LESO) and the linear state error feedback (LSEF). And it is a potential method to enhance the system stability margin, attributing to its high anti-interference capability and good tracking performance. Thereupon, the system small-signal impedance model considering frequency coupling is established. And the effect of the introduction of the LADRC on the system stability is further investigated using the Nyquist criterion. Particularly, the influences of key control parameters on the stability are discussed in detail. Meanwhile, the impact of LADRC on the transient performance is explored through closed-loop zero poles. Finally, the correctness of the theoretical analysis and the effectiveness of the proposed control strategy are verified via electromagnetic simulations.

Keywords

Modular multilevel converter (MMC); linear active disturbance rejection control (LADRC); sub- and super-synchronous oscillation; stability; weak grid

I. Introduction

IN recent years, the high-voltage direct current (HVDC) transmission technology has been developed rapidly. Compared with the AC transmission systems, the HVDC systems have the advantages of flexible controllability and high efficiency [

1]. Particularly, the modular multilevel converter (MMC) based HVDC has the merits of high scalability, active and reactive power decoupling control, and low harmonics [2]-[4], which is widely used for the large-scale long-distance transmission of renewable energy sources, asynchronous grid interconnection [5], [6], etc.

Depending on the reference value used in obtaining the insertion indexes, the modulations of MMC can be categorized into two types, i.e., compensated modulation (CM) [

7] and direct modulation (DM) [8]. Since the capacitor voltage ripples are compensated, the CM-based MMC can ignore the internal dynamics such as the circulating current. However, the CM-based MMC requires additional energy-based controller to ensure the capacitor voltage stability [7]. Furthermore, since the measurement systems inevitably exhibit distortion and delay, the CM is difficult to be implemented in practice [9]. In contrast, the DM is widely used in real projects because it is easy to realize [10]. But in the DM-based MMC, its internal dynamics have to be considered and the circulating current suppression control (CCSC) needs to be incorporated. More critically, the complex internal dynamics may have detrimental effects on the stable operation of the MMC-based grid-connected system. Especially, for the power electronics converters connected to the weak grid, their interaction with grid impedance may aggravate the stability issue [4].

It is worth noting that the instability accidents in the power electronics-based grid-connected renewable energy systems no longer manifest as oscillations of a single frequency [

11], [12]. They behaves as oscillations consisting of a pair of sub- and super-synchronous frequency components, which are interdependent. This coupled oscillation phenomenon is also defined as frequency coupling [13], which is characterized as: when a voltage perturbation at frequency

f_{p}

is injected into the point of common coupling (PCC), a coupling perturbation component at frequency

f_{p} - 2 f_{0}

will exist in the grid-connected current [14] in addition to the perturbation at the original frequency

f_{p}

, where

f_{0}

is the fundamental frequency. It has been shown that the cause of frequency coupling in the power electronics-based grid-connected systems is the asymmetric control, such as phase-locked loops (PLLs) and DC voltage controllers [15]. In addition, the presence of grid impedance would aggravate the frequency coupling [16]. The voltage perturbation at frequency

f_{p} - 2 f_{0}

is generated when the current perturbation at frequency

f_{p} - 2 f_{0}

flows through the grid impedance, which in turn will cause reverse coupling to generate the current perturbation at frequency f_p. And the frequency coupling oscillation issues are particularly serious under the weak grid.

For the MMC, its complex internal dynamics are also contributing to the frequency coupling. The voltage perturb-ation at frequency f_p interacts with the steady-state harmonics of the circulating currents and the sub-module (SM) capacitor voltages, which results in the voltage perturbation at frequency $f_{p} - 2 f_{0}$ . The complicated coupling characteristic of oscillation components will change the original single-input single-output (SISO) characteristic of the MMC system and thus seriously threaten the system stability. Therefore, the internal dynamics of MMC and the frequency coupling effect cannot be neglected in stability analysis. To analyze the system stability of the grid-connected MMC, an efficient method is the impedance-based analytical approach [

17]. The multi-input multi-output (MIMO) impedance modeling methods of the MMC, with the internal dynamics considered, have been reported, such as the harmonic state-space [17], [18] and the multi-harmonic linearization [19]-[21]. However, the models in [19] and [20] are usually decoupled into SISO models, focusing on the input and output relationship at the same frequency, which makes it difficult to fully consider the effect of frequency coupling. Furthermore, most of these studies lack the detailed mechanistic analysis for the frequency coupling phenomenon caused by the internal dynamics of the MMC.

So far, many methods for suppressing the frequency coupling oscillations have been proposed [

22]-[25], but most focus on 2-level grid-connected converters. A symmetric control strategy for the DC voltage controller and PLL is proposed in [22] to suppress the frequency coupling. In [23] and [24], two different structures of symmetric PLLs are proposed to make up for the asymmetrical control characteristics. These methods can inhibit the frequency coupling caused by the asymmetry control to some extent, but the suppression of the frequency coupling caused by the internal dynamics of the MMC may not be effective. And, it inevitably needs to alter the internal structure of the PLL or the voltage controller, which makes the small-signal modeling process complicated.

In [

25], the linear active disturbance rejection control (LADRC) is applied to the PLL of the 2-level converter, which can increase the damping ratio of the PLL. The LADRC is composed of the linear extended state observer (LESO) and the linear state error feedback (LSEF) [26]. It considers all factors that cause the system output to deviate from the reference value as generalized disturbances, and estimates disturbances by means of the LESO, which is then compensated by the feedforward. Thus, the LADRC has a strong anti-interference capability [27]. Furthermore, it has the advantages of good tracking performance, simple algorithm, and convenient implementation [28]. Due to the good control performance, the LADRC has been applied to the power electronics systems such as virtual synchronous generators [29] and permanent magnet synchronous motors [30].

Although the LADRC has been applied to power electronics, most studies have not investigated the effect of LADRC on the system stability or only discussed individual control link. Moreover, because of the complex internal dynamics, there are more complex control structures and more control requirements in the MMC compared with the 2-level converter. And the influence mechanism with the introduction of LADRC on the stability of the MMC system are not clear, especially in the case of weak grid. Also, there is no comprehensive comparison between the LADRC and proportional-integral (PI) controller for system stability and transient performance.

To cope with such issues, and as specific contributions, ① this paper proposes the LADRC-based MMC control strategy, which replaces the conventional PI controllers with the LADRC in CCSC, current inner-loop, DC voltage outer-loop, and PLL; ② the induced mechanism of the MMC internal dynamics on the frequency coupling effect is revealed, and the LADRC-based small-signal impedance model of MMC accounting for the frequency coupling effect is established; ③ the stability of the LADRC-based system under low short-circuit ratio (SCR) power grid is further analyzed, with the influence of key control parameters on the stability being explored in detail, which provides a reference for parameter tunings; ④ the transient performance of the LADRC is also investigated through closed-loop zero poles; ⑤ finally, the time-domain simulations are carried out to verify the correctness of theoretical analysis.

II. LADRC-based MMC

A. Configuration and Averaged Model of MMC

The circuit topology of the MMC connected into AC grid and the control structure of the MMC with LADRC are presented in Fig. 1. $V_{d c}$ and $i_{d c}$ are the DC-link voltage and current, respectively. $I_{d c}$ is the output current of the DC source. $v = [v_{g j}]$ is the voltage at the PCC, and the subscript j represents three phases $(j = a, b, c)$ . $i_{g} = [i_{g j}]$ is the grid-connected current. $i_{j u}$ and $i_{j l}$ are the currents in the upper and lower arms, respectively. The DM is used in the MMC, and its internal circulating current is expressed as $i_{c j} = (i_{j u} + i_{j l}) / 2$ , and $i_{c} = [i_{c j}]$ . Note that the subscripts d and q denote the d- and q-axis components of the corresponding variables, and the subscript ref denotes the given reference value of the corresponding variables. θ is the grid voltage tracking phase of PLL. $m_{v} = [m_{v j}]$ and $m_{c} = [m_{c j}]$ are the three-phase modulation indexes in the double-closed-loop control and the CCSC, respectively. $L_{0}$ and $R_{0}$ are the arm inductance and resistance, respectively. $L_{f}$ and $R_{f}$ are the AC-link equivalent inductance and resistance, respectively. C is the equivalent capacitance of the arms. For simplification, the grid impedance is expressed as $Z_{g} = j ω_{0} L_{g}$ , $ω_{0}$ is the grid angulan frequency, and $L_{g}$ is the grid inductance.

Fig. 1 Circuit topology and control structure of MMC-based grid-connected system. (a) Circuit topology. (b) Control structure of MMC with LADRC.

Taking phase a for example and omitting the subscript a, the averaged model of MMC can be obtained as:

\{\begin{array}{l} L_{0} \frac{d i_{u}}{d t} + R_{0} i_{u} = \frac{V_{d c}}{2} - L_{f} \frac{d i_{g}}{d t} - R_{f} i_{g} - m_{u} v_{u} - v \\ L_{0} \frac{d i_{l}}{d t} + R_{0} i_{l} = \frac{V_{d c}}{2} + L_{f} \frac{d i_{g}}{d t} + R_{f} i_{g} - m_{l} v_{l} + v \\ \frac{C_{s m}}{N} \frac{d v_{u}}{d t} = m_{u} i_{u} \\ \frac{C_{s m}}{N} \frac{d v_{l}}{d t} = m_{l} i_{l} \end{array}

(1)

where $N$ is the number of SMs in each arm; $C_{s m}$ is the SM capacitance; $v_{u}$ and $v_{l}$ are the sum of the SM capacitance voltages in the upper and lower arms, respectively; and $m_{u}$ and $m_{l}$ are the modulation indexes in the upper and lower arms, respectively, which can be expressed as:

\{\begin{array}{l} m_{u} = 0.5 - m_{v} + m_{c} \\ m_{l} = 0.5 + m_{v} + m_{c} \end{array}

(2)

From (1) and (2), the following expression can be obtained, i.e.,

L \frac{d i_{g}}{d t} + R i_{g} = - v - (\frac{1}{2} + m_{c}) v_{d i f} + m_{v} v_{c o m}

(3a)

L_{0} \frac{d i_{c}}{d t} + R_{0} i_{c} = \frac{V_{d c}}{2} - (\frac{1}{2} + m_{c}) v_{c o m} + m_{v} v_{d i f}

(3b)

C \frac{d v_{c o m}}{d t} = (\frac{1}{2} + m_{c}) i_{c} - \frac{1}{2} m_{v} i_{g}

(3c)

C \frac{d v_{d i f}}{d t} = \frac{1}{2} (\frac{1}{2} + m_{c}) i_{g} - m_{v} i_{c}

(3d)

where $L = L_{f} + L_{0} / 2$ ; $R = R_{f} + R_{0} / 2$ ; $C = C_{s m} / N$ ; $v_{c o m} = (v_{u} + v_{l}) / 2$ ; and $v_{d i f} = (v_{u} - v_{l}) / 2$ . For the steady-state values, it can be considered that $v_{u} = v_{l} = V_{d c} = V_{d c r e f}$ [

2].

B. Design Principles of First-order LADRC

As for a first-order plant, it can be represented as:

\dot{y} = \underset{f_{d}}{\underset{︸}{- a_{1} y + w + (b - b_{0}) u}} + b_{0} u

(4)

where u, y, and w are the input, output, and unknown external disturbance, respectively; a₁ represents the parameter of unknown system; b represents the unknown input gain, while b₀ represents the known value; and f_d is the generalized disturbance, containing both the internal and external disturbances.

Figure 2 illustrates the control structure of the typical first-order LADRC, which includes the LESO, LSEF, and the control object G_p.

Fig. 2 Control structure of first-order LADRC.

The LESO can realize real-time observations of the actual system variables $y$ and $f_{d}$ , which can be designed as:

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

(5)

where z₁ and z₂ are the estimations of y and $f_{d}$ , respectively; and $β_{1}$ and $β_{2}$ are the observer gains.

The LSEF can amplify the feedback control quantity through proportional control, improving the system transient response. The LSEF can be organized as:

\{\begin{array}{l} u_{0} = k_{p} (r - z_{1}) \\ u = \frac{1}{b_{0}} (u_{0} - z_{2}) \end{array}

(6)

where k_p, u₀, and r are the error feedback coefficient, LSEF output, and control reference, respectively.

To facilitate the parameter tuning, the single-parameter LADRC is utilized [

26], which introduces a parameter

ω_{L}

and defines it as the LADRC bandwidth. And

k_{p}

β_{1}

, and

β_{2}

can then be expressed by

ω_{L}

as:

\{\begin{array}{l} k_{p} = ω_{L} \\ β_{1} = 2 ω_{L} \\ β_{2} = ω_{L}^{2} \end{array}

(7)

Therefore, the single-parameter LADRC only needs to tune the bandwidth $ω_{L}$ . What is more, the bandwidth of LADRC $ω_{L}$ corresponds to that of the traditional PI controller. Thus, such structure can reduce the number of parameters, which substantially simplifies the tuning process.

According to Fig. 1 and (7), the following transfer relationships of the LADRC can be obtained, i.e.,

\{\begin{array}{l} \frac{U (s)}{Y (s)} = - \frac{(β_{2} + β_{1} k_{p}) s + β_{2} k_{p}}{b_{0} s^{2} + (b_{0} β_{1} + b_{0} k_{p}) s} = - \frac{ω_{L}^{2} (3 s + ω_{L})}{b_{0} (s^{2} + 3 ω_{L} s)} \\ \frac{U (s)}{R (s)} = \frac{k_{p} s^{2} + β_{1} k_{p} s + β_{2} k_{p}}{b_{0} s^{2} + (b_{0} β_{1} + b_{0} k_{p}) s} = \frac{ω_{L} (s^{2} + 2 ω_{L} s + ω_{L}^{2})}{b_{0} (s^{2} + 3 ω_{L} s)} \end{array}

(8)

where $U (s)$ , $Y (s)$ , and $R (s)$ represent the Laplace transforms of u, y, and r, respectively. Based on Fig. 1 and (8), the LADRC can be simplified to a two degree-of-freedom (2DOF) control system. Its simplified structure is shown in Fig. 3, in which $C (s)$ and $C_{f} (s)$ can be written as (9).

Fig. 3 Simplified structure of LADRC.

\{\begin{array}{l} C (s) = - \frac{U (s)}{Y (s)} = \frac{ω_{L}^{2} (3 s + ω_{L})}{b_{0} (s^{2} + 3 ω_{L} s)} \\ C_{f} (s) = \frac{U (s)}{R (s) C (s)} = \frac{s^{2} + 2 ω_{L} s + ω_{L}^{2}}{ω_{L} (3 s + ω_{L})} \end{array}

(9)

In the conventional feedback control, $C (s)$ plays a role similar to the PI controller, which can be regarded approximately as a PI controller in series with a low-pass filter. The low-pass characteristics of LADRC enables the controller to filter out disturbance components with a higher frequency than the cut-off frequency. It prevents the controller to amplify disturbances in this frequency band [

27].

C_{f} (s)

acts as a softening starter and can also be regarded as playing a pre-filtering role, for the purpose of reducing overshoot at the step response by attenuating the signal from the cut-off frequency.

For the discrete-time first-order LADRC, it is mainly implemented by the state-space-based method [

31], the Euler-discretization-based method [32], the transfer-function-based method [33], and the improved transfer-function-based method [34]. The comparison of their computational costs is given in Table I. It indicates that the computational efforts and storage requirements of the LADRC are not high, especially for the improved method. And it can be easily implemented on the real-time microcontroller, even in low-cost embedded systems [34].

TABLE I Comparison of Computational Costs with Different Methods

Method	Addition	Multiplication	Variables
State-space-based	11	10	2
Euler-discretization-based	7	6	8
Transfer-function-based	7	6	4
Improved transfer-function-based	7	6	2

C. LADRC-based MMC Control Strategy

Based on design principles of the first-order LADRC, the CCSC, current inner-loop, DC voltage outer-loop, and PLL of the MMC can be implemented as follows. It is necessary to point out that in the LADRC-based CCSC and current inner-loop, the coupling effect between d- and q-axis can be considered as internal disturbance of the system [

35], which is contained in

f_{d}

. Thus, the LADRC enables decoupling control of d- and q-axis currents. Considering the symmetric property, only the d-axis control is explained as follows.

1)　LADRC-based CCSC

Based on (3a) and (3b) and Park transformation, the mathematical model of MMC in the dq-axis can be derived as:

\frac{d i_{c d}}{d t} = - \frac{V_{d c r e f}}{L_{0}} m_{c d} - \frac{R_{0}}{L_{0}} i_{c d} - 2 ω i_{c q}

(10)

\frac{d i_{g d}}{d t} = \frac{V_{d c r e f}}{L} m_{v d} - \frac{R}{L} i_{g d} - \frac{1}{L} v_{d} + ω i_{g q}

(11)

where $ω$ is the angular frequency of grid voltage.

According to (4)-(6) and (10), the CCSC can be designed as:

\{\begin{array}{l} y_{c} = i_{c d} \\ u_{c} = m_{c d} \\ b_{0, c} = - \frac{V_{d c r e f}}{L_{0}} \\ f_{d, c} = - \frac{R_{0}}{L_{0}} i_{c d} - 2 ω i_{c q} \end{array}

(12)

Note that, in this paper, the subscripts c, i, v, and pll represent the CCSC, current inner-loop, voltage outer-loop, and PLL, respectively. And, the design principles of other control loops will be described later.

2)　LADRC-based Current Inner-loop

From (4)-(6) and (11), the current inner-loop can be designed as:

\{\begin{array}{l} y_{i} = i_{g d} \\ u_{i} = m_{v d} \\ b_{0, i} = \frac{V_{d c r e f}}{L} \\ f_{d, i} = - \frac{R}{L} i_{g d} - \frac{1}{L} v_{d} + ω i_{g q} \end{array}

(13)

3)　LADRC-based DC Voltage Outer-loop

Based on Fig. 1(a), the AC- and DC-link power of MMC, i.e., $P_{a c}$ and $P_{d c}$ , respectively, can be expressed as:

\{\begin{array}{l} P_{a c} = \frac{3}{2} (v_{g d} i_{g d} + v_{g q} i_{g q}) = \frac{3}{2} V_{1} i_{g d} \\ P_{d c} = V_{d c} i_{d c} = V_{d c} (I_{d c} - C_{d c} \frac{d v_{d c}}{d t}) \end{array}

(14)

where V₁ is the fundamental magnitude of grid voltage; and $C_{d c}$ is the MMC DC-side capacitor. In order to design the LADRC-based DC voltage outer-loop, the relationship between $V_{d c}$ and $i_{g d r e f}$ needs to be determined. Thus, it is necessary to assume that: ① the power loss of MMC can be ignored, i.e., $P_{a c} \approx P_{d c}$ ; ② the DC-link voltage and grid-connected current can be controlled well, i.e., $V_{d c} \approx V_{d c r e f}$ and $i_{g d} \approx i_{g d r e f}$ [

36]. Consequently, according to (14), the following relationship can be obtained as:

\frac{d v_{d c}}{d t} = - \frac{3 V_{1}}{2 C_{d c} V_{d c r e f}} i_{g d r e f} + \frac{I_{d c}}{C_{d c}}

(15)

From (4)-(6) and (15), the voltage outer-loop can be designed as:

\{\begin{array}{l} y_{v} = v_{d c} \\ u_{v} = i_{g d r e f} \\ b_{0, v} = - \frac{3 V_{1}}{2 C_{d c} V_{d c r e f}} \\ f_{d, v} = \frac{I_{d c}}{C_{d c}} \end{array}

(16)

4)　LADRC-based PLL

Referring to [

25], the differential equation of PLL can be obtained as:

\frac{d v_{q}}{d t} = - V_{1} (ω_{p l l} - \frac{d θ}{d t})

(17)

where $ω_{p l l}$ is the tracked angular velocity of PLL.

Based on (4)-(6) and (17), the PLL can be designed as:

\{\begin{array}{l} y_{p l l} = v_{q} \\ u_{p l l} = ω_{p l l} \\ b_{0, p l l} = - V_{1} \\ f_{d, p l l} = V_{1} \frac{d θ}{d t} \end{array}

(18)

The open-loop Bode diagrams for transfer function of each control loop with LADRC ( $ω_{L}$ ) and PI controller ( $ω_{P I}$ ) are shown in Fig. 4. For each control loop, the bandwidths of the LADRC and PI controller are the same, i.e., the cut-off frequencies are equal in open-loop Bode diagrams. In Fig. 4, the bandwidth of the CCSC $ω_{P I, c} = ω_{L, c} = 1256 r a d / s$ ; the bandwidth of the current inner-loop $ω_{P I, i} = ω_{L, i} = 502 r a d / s$ ; the bandwidth of the voltage outer-loop $ω_{P I, v} = ω_{L, v} = 9 r a d / s$ ; and the bandwidth of the PLL $ω_{P I, p l l} = ω_{L, p l l} = 251 r a d / s$ . Compared with the PI controller, the LADRC has the larger phase margin at the cut-off frequency. As the phase margin is positively correlated with stability, the introduction of LADRC will be beneficial to improve the system stability.

Fig. 4 Open-loop Bode diagrams for transfer function of each control loop. (a) CCSC ( $ω_{P I, c} = ω_{L, c} = 1256 r a d / s$ ). (b) Current inner-loop ( $ω_{P I, i} = ω_{L, i} = 502 r a d / s$ ). (c) DC voltage outer-loop ( $ω_{P I, v} = ω_{L, v} = 9 r a d / s$ ). (d) PLL ( $ω_{P I, p l l} = ω_{L, p l l} = 251 r a d / s$ ).

III. Impedance Modeling of LADRC-based MMC

Considering the frequency coupling effect, the small-signal impedance model of the LADRC-based MMC is derived by multi-harmonic linearization method in this section.

A. Expressions of Frequency Coupling Effect

For the MMC system, its complex internal dynamics are responsible for the frequency coupling effect. Figure 5 reveals the induced mechanism of the frequency coupling in the MMC. If there is voltage perturbation of frequency $f_{p}$ $\hat{v} (f_{p})$ in the MMC, it will interact with the harmonic perturbations of the modulation indexes, circulating current, and SM capacitance voltage, which eventually generates the voltage perturbation at frequency $f_{p} - 2 f_{0} \hat{v} (f_{p} - 2 f_{0})$ . It is notable that the amplitude of harmonic disturbances is inversely proportional to its order, and the 4^th order and higher harmonics can be ignored, without affecting the accuracy of the small-signal model [

3]. In Fig. 5, the symbol ^ represents the perturbation variables.

Fig. 5 Schematic diagram of frequency coupling in MMC.

In addition, the current perturbations ${\hat{i}}_{g} (f_{p})$ and ${\hat{i}}_{g} (f_{p} - 2 f_{0})$ flowing through the grid impedance will generate voltage perturbations $\hat{v} (f_{p})$ and $\hat{v} (f_{p} - 2 f_{0})$ at the corresponding frequencies, which will exacerbate the frequency coupling effect. The frequency coupling process considering the grid impedance is depicted in Fig. 6. Thus, to ensure the correctness of the stability analysis, the internal dynamics of the MMC and the frequency coupling need to be fully considered in the small-signal model.

Fig. 6 Frequency coupling diagram considering grid impedance.

Since the SISO model cannot describe the frequency coupling effect, the 2×2 MIMO model is adopted, i.e.,

[\begin{matrix} {\hat{i}}_{g} (f_{p}) \\ {\hat{i}}_{g} (f_{p} - 2 f_{0}) \end{matrix}] = \underset{Y}{\underset{︸}{[\begin{matrix} Y_{p p} (f_{p}) & Y_{n p} (f_{p} - 2 f_{0}) \\ Y_{p n} (f_{p}) & Y_{n n} (f_{p} - 2 f_{0}) \end{matrix}]}} [\begin{matrix} \hat{v} (f_{p}) \\ \hat{v} (f_{p} - 2 f_{0}) \end{matrix}]

(19)

where $Y_{n p} (f_{p} - 2 f_{0}) = Y_{p n}^{*} (2 f_{0} - f_{p})$ ; $Y_{n n} (f_{p} - 2 f_{0}) = Y_{p p}^{*} (2 f_{0} - f_{p})$ ; and * is the conjugate symbol.

From Fig. 6, the equivalent impedance of MMC with frequency coupling $Z_{m m c} (f_{p})$ can be derived as:

\{\begin{array}{l} Z_{m m c} (f_{p}) = - \frac{\hat{v} (f_{p})}{{\hat{i}}_{g} (f_{p})} = \frac{1}{Y_{p p} (f_{p}) + Y_{c} (f_{p})} \\ Y_{c} (f_{p}) = - \frac{Y_{p n} (f_{p}) Y_{n p} (f_{p} - 2 f_{0}) Z_{g} (f_{p} - 2 f_{0})}{1 + Y_{n n} (f_{p} - 2 f_{0}) Z_{g} (f_{p} - 2 f_{0})} \end{array}

(20)

where $Y_{c} (f_{p})$ is the equivalent admittance of frequency coupling in the blue dashed area of Fig. 6.

B. Frequency-domain Model of MMC

The MMC presents complex internal harmonic characteristics. For analysis convenience, the auxiliary functions $F_{1, k}$ and $F_{2, k}$ are introduced to describe the phase sequence and output characteristic of the perturbation at frequency $f_{p} + k f_{0}$ , where k is the harmonic order and $| k | \leq 3$ is taken in this paper. The expressions for $F_{1, k}$ and $F_{2, k}$ can then be expressed as:

F_{1, k} = \{\begin{array}{l} 1 (p o s i t i v e - s e q u e n c e) k = 3 i \\ - 1 (n e g a t i v e - s e q u e n c e) k = 3 i + 1 \\ 0 (z e r o - s e q u e n c e) k = 3 i - 1 \end{array}

(21)

F_{2, k} = \{\begin{array}{l} 1 (o u t p u t c o m p o n e n t) k = 2 i \\ 0 (c i r c u l a t i o n c o m p o n e n t) k = 2 i + 1 \end{array}

(22)

The model of the MMC in (3) is time-periodic, which needs to be converted into the frequency domain. To this end, the perturbation quantity of an arbitrary variable x needs to be presented in terms of Fourier coefficients, i.e.,

\hat{x} = [x_{p - k} \dots x_{p - 1} x_{p} x_{p + 1} \dots x_{p + k}]^{T}

(23)

where $x_{p \pm k}$ is the Fourier coefficient at frequency $f_{p} \pm k f_{0}$ . And the steady-state variables X of MMC should be transformed to the Toeplitz matrices [

20]. Consequently, the frequency-domain model of MMC can be obtained as:

\{\begin{array}{l} {\hat{i}}_{g} = Y (- \hat{v} - \frac{1}{2} {\hat{v}}_{d i f} - M_{c} {\hat{v}}_{d i f} - V_{d i f} {\hat{m}}_{c} + M_{v} {\hat{v}}_{c o m} + V_{c o m} {\hat{m}}_{v}) \\ {\hat{i}}_{c} = Y_{0} (\frac{1}{2} {\hat{v}}_{d c} - \frac{1}{2} {\hat{v}}_{c o m} - M_{c} {\hat{v}}_{c o m} - V_{c o m} {\hat{m}}_{c} + M_{v} {\hat{v}}_{d i f} + V_{d i f} {\hat{m}}_{v}) \\ {\hat{v}}_{c o m} = Z_{c} (\frac{1}{2} {\hat{i}}_{c} + M_{c} {\hat{i}}_{c} + I_{c} {\hat{m}}_{c} - \frac{1}{2} M_{v} {\hat{i}}_{g} - \frac{1}{2} I_{g} {\hat{m}}_{v}) \\ {\hat{v}}_{d i f} = Z_{c} (\frac{1}{4} {\hat{i}}_{g} - M_{v} {\hat{i}}_{c} - I_{c} {\hat{m}}_{v} + \frac{1}{2} M_{c} {\hat{i}}_{g} + \frac{1}{2} I_{g} {\hat{m}}_{c}) \end{array}

(24)

where Y, Y₀, and Z_c are the diagonal matrices representing the equivalent admittance, the arm inductor admittance, and the capacitor impedance, respectively. The expressions can be described as:

\{\begin{array}{l} Y = d i a g [\frac{1}{j 2 π (f_{p} + k f_{0}) L + R}] \\ Y_{0} = d i a g [\frac{1}{j 2 π (f_{p} + k f_{0}) L_{0} + R_{0}}] \\ Z_{c} = d i a g [\frac{1}{j 2 π (f_{p} + k f_{0}) C}] \end{array}

(25)

C. Modeling of Control System

1)　CCSC Modeling

According to the CCSC block diagram in Fig. 1(b), the relationship between ${\hat{m}}_{c}$ and ${\hat{i}}_{c}$ can be deduced as:

{\hat{m}}_{c} = T_{c} {\hat{i}}_{c}

(26)

\{\begin{array}{l} T_{c} = d i a g [|F_{1, k}| 1 - |F_{2, k}| C_{c, k}] \\ C_{c, k} = \frac{j 6 π ω_{L, c}^{2} [f_{p} + (k + 2 F_{1, k}) f_{0}] + ω_{L, c}^{3}}{- 4 π^{2} b_{0, c} [f_{p} + (k + 2 F_{1, k}) f_{0}]^{2} + j 6 π ω_{L, c}^{} [f_{p} + (k + 2 F_{1, k}) f_{0}]} \end{array}

(27)

2)　Current Inner-loop Modeling

Similarly, the influence of the current inner-loop on ${\hat{m}}_{v}$ can be described as:

{\hat{m}}_{v}^{'} = T_{i} {\hat{i}}_{g}

(28)

\{\begin{array}{l} T_{i} = d i a g [|F_{1, k}| |F_{2, k}| C_{i, k}] \\ C_{i, k} = \frac{j 6 π ω_{L, i}^{2} [f_{p} + (k - F_{1, k}) f_{0}] + ω_{L, i}^{3}}{- 4 π^{2} b_{0, i} [f_{p} + (k - F_{1, k}) f_{0}]^{2} + j 6 π ω_{L, i}^{} [f_{p} + (k - F_{1, k}) f_{0}]} \end{array}

(29)

3)　PLL Modeling

Likewise, the impact of the PLL on ${\hat{m}}_{v}$ can be written as:

{\hat{m}}_{v}^{‴} = T_{p l l} \hat{v}

(30)

\{\begin{array}{l} T_{p l l} (k + 1, k + 1) = G_{p l l, - 1} (C_{i, 0} I_{u, 1} + M_{u, 1}) \\ T_{p l l} (k - 1, k + 1) = - G_{p l l, - 1} (C_{i, - 2} I_{u, - 1} + M_{u, - 1}) \\ G_{p l l, k} = \frac{C_{p l l, k}}{1 + V_{1} C_{p l l, k}} \\ C_{p l l, k} = \frac{j 6 π ω_{L, p l l}^{2} (f_{p} + k f_{0}) + ω_{L, p l l}^{3}}{- j 8 π^{3} b_{0, p l l} (f_{p} + k f_{0})^{3} - 12 π^{2} ω_{L, c}^{} (f_{p} + k f_{0})^{2}} \end{array}

(31)

where the coefficient matrix $T_{p l l}^{}$ consists of the non-zero elements $T_{p l l} (k + 1, k + 1)$ and $T_{p l l} (k - 1, k + 1)$ .

4)　Voltage Outer-loop Modeling

From Fig. 1, the relationship between ${\hat{m}}_{v}$ and ${\hat{v}}_{d c}$ can be deduced as:

{\hat{m}}_{v}^{″} = T_{v} {\hat{v}}_{d c} = T_{i}^{'} T_{v}^{'} {\hat{v}}_{d c}

(32)

\{\begin{array}{l} T_{i}^{'} = d i a g [|F_{1, k}| |F_{2, k}| C_{f i, k} C_{i, k}] \\ C_{f i, k} = \\ \frac{- 4 π^{2} [f_{p} + (k - F_{1, k}) f_{0}]^{2} + j 4 π ω_{L, i}^{} [f_{p} + (k - F_{1, k}) f_{0}] + ω_{L, i}^{2}}{j 6 π ω_{L, i}^{} [f_{p} + (k - F_{1, k}) f_{0}] + ω_{L, i}^{2}} \\ T_{v}^{'} (k + 1, k) = T_{v}^{'} (k - 1, k) = \frac{1}{2} C_{v, - 1} \\ C_{v, k} = \frac{j 6 π ω_{L, v}^{2} (f_{p} + k f_{0}) + ω_{L, v}^{3}}{- 4 π^{2} b_{0, v} (f_{p} + k f_{0})^{2} + j 6 π ω_{L, v}^{2} (f_{p} + k f_{0})} \\ {\hat{v}}_{d c} = - 3 Z_{d c} {\hat{i}}_{c} \\ Z_{d c} = d i a g [(1 - |F_{1, k}|) (1 - |F_{2, k}|) Z_{d c, k}] \end{array}

(33)

where the coefficient matrix $T_{v}^{'}$ is composed of non-zero elements $T_{v}^{'} (k + 1, k)$ and $T_{v}^{'} (k - 1, k)$ .

D. AC-side Equivalent Impedance Model of MMC

According to (24)-(33), the relationship between current perturbation ${\hat{i}}_{g}$ and voltage perturbation $\hat{v}$ can be obtained as:

{\hat{i}}_{g} = A \hat{v} = \frac{A_{4} - A_{6} A_{1}^{- 1} A_{3}}{A_{5} + A_{6} A_{1}^{- 1} A_{2}} \hat{v}

(34)

where $A (k - 1, k + 1) =$ $Y_{p p}$ ; $Y_{p n} =$ $A (k + 1, k + 1)$ ; and A₁-A₆ are expressed as (35).

\{\begin{array}{l} A_{1} = E + Y_{0} [\frac{1}{4} Z_{c} + \frac{1}{2} Z_{c} M_{c} + \frac{1}{2} Z_{c} I_{c} T_{c} + \frac{1}{2} M_{c} Z_{c} + M_{c} Z_{c} M_{c} + M_{c} Z_{c} I_{c} T_{c} + V_{c o m} T_{c} + M_{v} Z_{c} M_{v} - \frac{1}{2} M_{v} Z_{c} I_{g} T_{c} + \\ (\frac{3}{2} E + \frac{3}{4} Z_{c} I_{g} T_{v} + \frac{3}{2} M_{c} Z_{c} I_{g} T_{v} - 3 M_{v} Z_{c} I_{c} T_{v} + 3 V_{d i f} T_{v}) Z_{d c}] \\ A_{2} = Y_{0} (\frac{1}{4} Z_{c} M_{v} + \frac{1}{4} Z_{c} I_{g} T_{i} + \frac{1}{2} M_{c} Z_{c} M_{v} + \frac{1}{2} M_{c} Z_{c} I_{g} T_{i} + \frac{1}{4} M_{v} Z_{c} - M_{v} Z_{c} I_{c} T_{i} + \frac{1}{2} M_{v} Z_{c} M_{c} + V_{d i f} T_{i}) \\ A_{3} = Y_{0} (\frac{1}{4} Z_{c} I_{g} T_{p l l} + \frac{1}{2} M_{c} Z_{c} I_{g} T_{p l l} - M_{v} Z_{c} I_{c} T_{p l l} + V_{d i f} T_{p l l}) \\ A_{4} = Y (E - \frac{1}{2} Z_{c} I_{c} T_{p l l} - M_{c} Z_{c} I_{c} T_{p l l} + \frac{1}{2} M_{v} Z_{c} I_{g} T_{p l l} - V_{c o m} T_{p l l}) \\ A_{5} = Y (- \frac{1}{8} Z_{c} + \frac{1}{2} Z_{c} I_{c} T_{i} - \frac{1}{4} Z_{c} M_{c} - \frac{1}{4} M_{c} Z_{c} + M_{c} Z_{c} I_{c} T_{i} - \frac{1}{2} M_{c} Z_{c} M_{c} - \frac{1}{2} M_{v} Z_{c} M_{v} - \frac{1}{2} M_{v} Z_{c} I_{g} T_{i} + V_{c o m} T_{i}) - E \\ A_{6} = Y [\frac{1}{2} Z_{c} M_{v} - \frac{1}{4} Z_{c} I_{g} T_{c} + M_{c} Z_{c} M_{v} - \frac{1}{2} M_{c} Z_{c} I_{g} T_{c} - V_{d i f} T_{c} + \frac{1}{2} M_{v} Z_{c} + M_{v} Z_{c} M_{c} + M_{v} Z_{c} I_{c} T_{c} + \\ (- \frac{3}{2} Z_{c} I_{c} T_{v} - 3 M_{c} Z_{c} I_{c} T_{v} + \frac{3}{2} M_{v} Z_{c} I_{g} T_{v} - 3 V_{c o m} T_{v}) Z_{d c}] \end{array}

(35)

where $E$ is the unit matrix.

IV. Key Factor Assessment on System Stability and Transient Performance Analysis

A. Stability Analysis Under Weak Grid

Once the small-signal impedance model of MMC considering the frequency coupling effect is derived, the Nyquist criterion can be applied to analyze the interaction stability between the MMC and the grid [

2]. It can be measured by the Nyquist curves of impedance ratio

L = Z_{g} / Z_{m m c}

. If the curve encloses the point

(- 1, j 0)

, the system is unstable; otherwise, the system is stable.

Figure 7 shows the Nyquist curves of system impedance ratio with different SCRs. With the reduction of SCR, both the Nyquist curves gradually approach the $(- 1, j 0)$ point, i.e., the system stability becomes weak. On the contrary, the Nyquist curve of PI controller based system encircles $(- 1, j 0)$ when the SCR drops to 1.5, while the LADRC-based system has a large stability margin. It indicates the application of LADRC effectively enhances the system stability under weak grid.

Fig. 7 Nyquist curves of system impedance ratio. (a) PI controller based MMC system. (b) LADRC-based MMC system.

B. Influence of Key Factors on System Stability

In addition to the grid condition, the unsuitable control parameters may also incur system instability [

4]. In this subsection, the effects of key control parameters

ω_{L, v}, ω_{L, p l l}, ω_{L, i}

, and

ω_{L, c}

on the system stability are discussed in detail, and the parameter robustness of LADRC is assessed as well. It can provide a reference on the rational adjustment of parameters.

1)　Influence of Voltage Outer-loop Parameters

The gain b_0,v and bandwidth $ω_{L, v}$ are the main parameters of the LADRC-based DC voltage outer-loop, in which b_0,v can be calculated by (16) and is usually considered as a known value. The impact of $ω_{L, v}$ on the system stability when $S C R = 1.5$ and $b_{0, v} = - 1.2$ is shown in Fig. 8. As $ω_{L, v}$ increases from 9 to 57 rad/s, the Nyquist curve surrounds $(- 1, j 0)$ , i.e., the system is destabilized. It shows the increase of $ω_{L, v}$ weakens the system stability.

Fig. 8 Nyquist curves with various ω_L,v when SCR=1.5 and $b_{0, v} = - 1.2$ .

Figure 9 exhibits the system stability domain with different SCRs, b_0,v, and $ω_{L, v}$ . It shows that the system stability gets weakened with the lower SCR, which is consistent with the previous analysis. The effect of b_0,v on stability is little, even though it is 10-15 times larger than the calculated value. Only in the low SCR case does it have slight influence on system stability. In contrast, the influence of $ω_{L, v}$ on system stability is obvious: increasing $ω_{L, v}$ diminishes the stability margin, and the influence degree aggravates with the reduction of the SCR. In consequence, the bandwidth $ω_{L, v}$ is not permitted to be set too large under weak grid.

Fig. 9 Stability domain with different SCRs, b_0,v, and $ω_{L, v}$ .

Figure 10 shows the effects of different voltage controllers on system stability when $S C R = 2$ . As shown in Fig. 10(a), the system is destabilized when the bandwidth of PI controller based voltage outer-loop $ω_{P I, v}$ drops to 57 rad/s. Indeed, it is almost in a critical state when $ω_{P I, v} = 28$ rad/s. Conversely, Fig. 10(b) shows that the Nyquist curves of the LADRC-based voltage outer-loop are consistently distant from $(- 1, j 0)$ when $ω_{L, v}$ is varied.

Fig. 10 Nyquist curves with various ω_PI,v and ω_L,v when $S C R = 2$ . (a) PI controller based voltage outer-loop. (b) LADRC-based voltage outer-loop.

Figure 11 illustrates the system stability domains with different voltage controllers. It is obvious that the LADRC-based one has a larger stability domain, compared with the PI controller based one. This proves that the LADRC-based voltage outer-loop has stronger parameter robustness, which is beneficial to improve the stability.

Fig. 11 Stability domains with different voltage controllers.

2)　Influence of PLL Parameters

Similarly, the PLL gain $b_{0, p l l}$ can be calculated from (18), and its effect on stability can be estimated. The Nyquist curves with various $ω_{L, p l l}$ when SCR=1.5 is indicated in Fig. 12. As can be observed, with decreasing $ω_{L, p l l}$ , the Nyquist curves gradually move away from $(- 1, j 0)$ . It indicates that decreasing $ω_{L, p l l}$ can improve the system stability and $ω_{L, p l l}$ is allowed to vary in a large range without incurring the system instability concern.

Fig. 12 Nyquist curves with various ω_L,pll when $S C R = 1.5$ .

3)　Influence of Current Inner-loop Parameters

Figure 13 illustrates the system Nyquist curves when the bandwidth of current inner-loop $ω_{L, i}$ changes and SCR=1.5. The Nyquist curves are slightly close to $(- 1, j 0)$ as $ω_{L, i}$ decreases from 628 to 251 rad/s, which indicates that increasing $ω_{L, i}$ could slightly enhance the system stability and the stability margin is large when $ω_{L, i}$ varies over a relatively wide range.

Fig. 13 Nyquist curves with various $ω_{L, i}$ when $S C R = 1.5$ .

4)　Influence of CCSC Parameters

Figure 14 shows the Nyquist curves with different bandwidths of CCSC $ω_{L, c}$ when SCR=1.5. As can be observed, it has almost no effect on the Nyquist curves when $ω_{L, c}$ increases from 628 to 2513 rad/s, which demonstrates that the effect of $ω_{L, c}$ on stability is little and its value can be adjusted substantially. However, it is worth noting that the Nyquist curve of the system without CCSC encloses $(- 1, j 0)$ , indicating that the system is unstable. This is because the internal circulation current of the MMC causes resonant peak in the sub- and super-synchronous frequency band, and the resonant peak will induce oscillation due to interactions with the grid impedance or the load converter [

3]. The CCSC can provide internal damping to the MMC system, which is beneficial to enhance the stability.

Fig. 14 Nyquist curves with different ω_L,c when $S C R = 1.5$ .

As a summary, the most critical factor is the bandwidth of voltage outer-loop $ω_{L, v}$ , which has an important influence on system stability. To ensure the system stability, its value should be set not too large. Moreover, decreasing the bandwidth of PLL $ω_{L, p l l}$ and increasing the bandwidth of current inner-loop $ω_{L, i}$ can enhance the system stability effectively. The CCSC parameters have almost no effect on the system stability, but the CCSC is indispensable. More importantly, the LADRC allows the bandwidth $ω_{L}$ to vary within a relatively large range, but not to induce the system instability issue, which is quite different from the PI controller. It confirms that the LADRC has better control parameter robustness.

C. Analysis of Transient Performance

In order to analyze the effect of LADRC on the transient performance, this subsection discusses and compares the LADRC and PI controller by closed-loop zero-pole maps.

1)　CCSC

The closed-loop control structure of the CCSC is illustrated in Fig. 15. According to its control structure, the closed-loop zero-pole maps with different R₀ can be obtained, as shown in Fig. 16. The transient performance of the system is dominated by its poles. Note that except for poles A, B, C and a, b, the effect of remaining poles on the transient performance is offset by the zeros in their vicinity. Therefore, it is sufficient to focus on the poles A, B, C and a, b.

Fig. 15 Closed-loop control structure of CCSC.

Fig. 16 Zero-pole maps of CCSC with increasing R₀.

The damping ratio (0.3, 0.5, 0.68, 0.81, 0.89, 0.945, 0.976, 0.994) is equal to the cosine of the angle between the line of pole-origin and the real axis. With larger damping ratio, the overshoot is smaller. Figure 16 shows that the damping ratios at poles A, B, and C are greater than those of a and b. Thus, the response oscillations in the LADRC-based CCSC are less than the PI controller based one. In addition, the response time is inversely proportional to the distance of the pole from the imaginary axis. It can be concluded from Fig. 16 that the response speed of the LADRC-based CCSC is faster than that of the PI controller based one.

2)　Current Inner-loop

As shown in Fig. 17, the closed-loop control structure of the current inner-loop is similar to that of the CCSC. Likewise, it can be concluded from Fig. 18 that the closed-loop poles A, B, and C of the LADRC-based current iner-loop have a longer distance from the imaginary axis and have a larger damping ratio than the PI-based one. It means that the LADRC-based current inner-loop has better transient performance.

Fig. 17 Closed-loop control structure of current inner-loop.

Fig. 18 Zero-pole maps of current inner-loop with increasing R.

3)　Voltage Outer-loop

Figure 19 illustrates the closed-loop control structure of the voltage, where $G_{i} (s)$ is the current closed-loop transfer function of current. And Fig. 20 gives the zero-pole maps of the voltage outer-loop with increasing $V_{d c r e f}$ . As can be observed, the poles A, B, C, and D of the LADRC-based voltage outer loop have a larger damping ratio and farther distance from the imaginary axis than that of the poles a and b of the PI controller based one. It indicates that the LADRC-based control has excellent dynamic control performance.

Fig. 19 Closed-loop control structure of voltage.

Fig. 20 Zero-pole maps of voltage outer-loop with increasing V_dcref.

4)　PLL

The closed-loop control structure and zero-pole maps of PLL are illustrated in Figs. 21 and 22, respectively. Similarly, the LADRC-based PLL has the satisfactory damping ratio and response speed compared with the PI controller based one.

Fig. 21 Closed-loop control structure of PLL.

Fig. 22 Zero-pole maps of PLL with increasing V₁.

In conclusion, the LADRC has preferable damping ratio and dynamic responsiveness, compared with the PI controller.

V. Simulation Verification

In order to verify the correctness of theoretical analysis, electromagnetic transient simulations are carried out in MATLAB/Simulink based on the structure of MMC grid-connected system in Fig. 1. The simulation parameters are shown in Table II.

TABLE II Simulation Parameters

Parameter	Value	Parameter	Value
N	250	ω_L,c (rad/s)	628, 1257, 2513
f₀ (Hz)	50	ω_L,i (rad/s)	251, 502, 628
P (MW)	400	ω_L,v (rad/s)	9, 28, 57
V_dc (kV)	500	ω_L,pll (rad/s)	126, 251, 377
I_dc (A)	800	b_0,c	-12.5
V₁ (kV)	200	b_0,i	19.2
C_sm (mF)	8	b_0,v	-1.2
R₀ (Ω)	0.1	b_0,pll	-20000
L₀ (mH)	80	k_p,c, k_i,c	50, 10892
R_f (Ω)	12	k_p,i, k_i,i	17, 8967
L_f (mH)	6	k_p,v, k_i,v	2.1, 129
L_g (mH)	106.1, 159.2, 212.2	k_p,pll, k_i,pll	0.011, 0.145

A. System Stability Comparison

The simulation results with different SCRs are exhibited in Fig. 23. It demonstrates that the grid-connected current and DC-side voltage of the PI controller based MMC system get to oscillate and gradually diverge when SCR decreases from 3 to 1.5, as shown in Fig. 23(a). What is more, the fast Fourier transform (FFT) results reveal that a pair of sub- and super-synchronous oscillation components appears at 37 Hz and 63 Hz when SCR decreases to 1.5. In contrast, the LADRC-based MMC system reaches the steady state rapidly even though SCR is reduced to 1.5, as can be observed in Fig. 23(b). Moreover, the sub- and super-synchronous oscillation can be suppressed well and the total harmonic distortion (THD) value also gets reduced from 17% to 1.2%. The simulation results correspond to the theoretical analysis in Fig. 7.

Fig. 23 Simulation results with SCR reduction. (a) PI controller based MMC system. (b) LADRC-based MMC system.

B. Influence of LADRC Parameters on System Stability and Control Robustness

Figure 24 illustrates the simulation results when $ω_{L, i}$ decreases from 628 to 251 rad/s when $S C R = 2$ . The results show that the system always remains stable as $ω_{I, i}$ decreases from 638 to 251 rad/s. Figure 25 displays the simulation results of different voltage control methods with varying the bandwidth of voltage loop when $S C R = 2$ . As shown in Fig. 25(a), the system gets oscillated when $ω_{P I, v}$ increases from 9 to 57 rad/s and its dominant oscillation frequencies are 18 Hz and 82 Hz, respectively. On the contrary, Fig. 25(b) shows that the system is consistently stable, even though $ω_{I, v}$ gets increased from 9 to 57 rad/s. In addition, the sub- and super-synchronous oscillations can be well suppressed and the THD is reduced from 31.4% to 1.3%. The simulation results prove that the LADRC has advantageous parameter robustness compared with the PI controller. It also verifies the analytical results in Section IV.

Fig. 24 Simulation results with different ω_I,i when SCR=2.

Fig. 25 Simulation results with different $ω_{P I, v}$ and $ω_{I, v}$ when SCR=2. (a) PI controller based MMC system. (b) LADRC-based MMC system.

C. Transient Performance

In order to study the transient performance of the LADRC, simulation tests are performed with the q-axis current reference changed directly from 0 to 600 A, as demonstrated in Fig. 26. Compared with the PI controller based MMC system, the LADRC-based one has lower overshoot and shorter response time when $I_{g q r e f}$ abrupt changes.

Fig. 26 Simulation results when i_g_q_ref changes. (a) PI controller based MMC system. (b) LADRC-based MMC system.

Further, the simulation results of grid voltage dips with different control strategies are described in Figs. 27 and 28. From Fig. 27(a), it can be observed that as for the PI controller, when the amplitude of the grid voltage suddenly drops 10%, the overshoot and response time become too large, which may trigger the protection mechanism. And the system is unstable when the grid voltage drops 14%, as shown in Fig. 27(b). However, the LADRC-based MMC system can recover the steady state rapidly with the minimum response oscillation even if the amplitude of the grid voltage suddenly drops 14%, as illustrated in Fig. 28.

Fig. 27 Simulation results of PI controller based MMC system with grid voltage drops. (a) 10% drop in grid voltage. (b) 14% drop in grid voltage.

Fig. 28 Simulation results of LADRC-based MMC system with 14% drop in grid voltage.

The above results verify that the LADRC has excellent damping ratio and transient responsiveness compared to the PI controller, which is consistent with the analysis in Section IV.

D. Control Performance During AC Fault

The simulation results with different control strategies during the AC fault in case of $S C R = 3$ are illustrated in Fig. 29. At 5.5 s, the two-phase short-circuit fault (phases a and b) occurs; and at 6.5 s, the AC fault is eliminated. It can be observed from Fig. 29 that when the fault happens, the currents in phases a and b increase sharply, and due to the grid impedance, there is a difference in amplitude, while the phases are approximately opposite. It is noteworthy that when the AC fault arises, the DC-side voltage of the PI controller based MMC system oscillates and tends to diverge. Contrastively, the LADRC-based one has good control performance during the fault, and the fluctuations of the DC-side voltage are smaller. In addition, the response overshoot is not so significant before and after the grid fault. The simulation results prove that the LADRC has satisfactory control performance during grid fault condition.

Fig. 29 Simulation results during AC fault in case of $S C R = 3$ . (a) PI controller based MMC system. (b) LADRC-based MMC system.

VI. Conclusion

This paper proposes an LADRC-based MMC control strategy and its equivalent impedance model is established. Furthermore, the system stability and transient characteristics are analyzed and the influences of key factors are discussed in detail. The conclusions can be drawn as follows.

1) The LADRC has excellent traceability and anti-disturbance capability. Especially, the mutual coupling between d- and q-axis can be regarded as internal disturbances in the LADRC-based CCSC and current inner-loop, which achieves decoupling control.

2) The damping characteristic of system is improved by the application of LADRC, which enhances the system stability in the sub- and super-synchronous frequency bands and effectively suppresses the frequency coupling effect under weak grid.

3) The bandwidth of voltage loop $ω_{I, v}$ behaves as the most critical factor of system stability. It should not be too large, especially under weak grid. It is found that reducing the bandwidth of PLL $ω_{I, p l l}$ and increasing the bandwidth of current inner-loop $ω_{I, i}$ can increase the system stability margin. Although the bandwidth of CCSC $ω_{I, c}$ has little effect on the stability, the CCSC is indispensable because it can enhance the internal damping of the MMC system. Furthermore, the LADRC-based MMC has outstanding control robustness compared with the PI controller based one.

4) In comparison with the PI controller based MMC, the LADRC-based one has larger damping ratio and faster dynamic response. This might provide an idea for the enhancement of the system transient stability. However, its mechanism and vital factors need further analysis and discussion, which will be done in the near future.

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