Adaptive Switching Control of Voltage Source Converters in Renewable Energy Station Based on Operating Short Circuit Ratio

Abstract

By integrating the grid-following control and grid-forming control, the adaptability of grid-connected converters to the grid impedance fluctuation can be effectively improved, and a stable operation in a wide short circuit ratio range can be realized. The existing fusion control schemes focus on the influence of the short circuit ratio on the stability of the converter, ignoring the influence of the active power fluctuation of the renewable energy in the design of the fusion scheme. In order to improve this shortcoming, an adaptive switching control of voltage source converters in the renewable energy station is proposed in this paper. Based on the oscillation characteristics of the grid-following converter and the grid-forming converter, this method selects the operating short circuit ratio as the switching index of the grid-following mode and the grid-forming mode. Compared with the current switching schemes based on the short circuit ratio, the operating short circuit ratio replaces the rated capacity of the station with the active output of the station, so it can more reasonably reflect changes in stability caused by active power fluctuations and then give the appropriate switching command, which means that unnecessary switching can be reduced and the control mode can be correctly converted to enhance stability when the system state changes.

1. Introduction

The international community is actively promoting the development of renewable energy in order to cope with climate change and environmental pollution. Traditional fossil energy is gradually being replaced by renewable energy such as wind and solar power [1]. In the future power system, the increasing penetration of renewable energy sources is an inevitable trend, and the utilization of renewable energy has emerged as a pivotal area of emphasis in contemporary research and development [2]. The existing renewable energy generation devices are generally connected to the grid through grid-following (GFL) voltage source converters (VSCs) [3]. A large number of GFL VSCs connecting to the grid have significantly changed the characteristics of traditional power systems [4,5,6]. As synchronous generators are gradually replaced by renewable energy sources, the voltage support capability of the grid decreases, resulting in a weaker grid strength [7]. Existing research indicates that the grid strength is directly related to the stability margin of GFL converter-based systems [8,9,10], and the coupling between renewable energy converters and the grid becomes stronger under weak grid conditions. This leads to oscillation issues under low short circuit ratio (SCR) conditions [11], posing a threat to the safe and stable operation of power systems and limiting the integration capacity of renewable energy sources. To cope with these oscillation problems, early approaches mainly involved optimizing the parameters of various control loops in the converter [12] or adding additional damping control [13]. However, these methods cannot change the inherent reliance of GFL converters on following the grid voltage, thereby still posing stability risks under extremely weak grid conditions.

Grid-forming (GFM) control enables converters to exhibit voltage source characteristics, allowing them to provide voltage support similar to synchronous generators [14,15]. Therefore, GFM converters are suitable for areas lacking synchronous machines’ support, such as a renewable energy base in a desert area. Different from GFL converters, GFM converters can significantly reduce the risk of subsynchronous oscillation under the condition that the power grid is weak. In [16], the impedance characteristics of GFL and GFM converters are compared. It is found that the phase of the GFM converter is inductive, so it can avoid forming a negative damping oscillation circuit with a weak grid. It has been proven that GFM converters can still ensure small-signal stability under the condition that the grid is extremely weak with an SCR value close to one through small-signal analysis and experimental verification [17]. However, converters adopting GFM control exhibit poor stability under strong grid conditions [18]. Additionally, due to the p-f (active power–frequency) droop characteristic, grid-forming control has to reserve capacity to provide active power support when the grid frequency drops below the fundamental frequency, and it is hard to achieve maximum power point tracking (MPPT) like grid-following control [19]. In the long term, reserving capacity leads to a certain loss of energy generation and poor economic performance.

Some experts and scholars have carried out research on hybrid GFL and GFM control in order to comprehensively utilize the advantages of the two types of control. In [20,21], the time domain simulation method is used to analyze the influence of GFM units’ capacity on the stability of the regional power grid in the United Kingdom and southeastern Australia. References [7,22] employ the generalized short circuit ratio (gSCR) to assess the small-signal stability of power systems and further propose formulating the configuration of GFM converters as an optimization problem with the gSCR as the objective function. However, the aforementioned configuration schemes are designed for specific operating points and neglect scenarios with significant operational condition fluctuations.

Some studies have explored how to solve the adaptability of the converter’s control scheme to large fluctuation conditions. In [23], phase-locked loop and power synchronization control are incorporated to achieve hybrid synchronization control, permitting a stable and well-performing operation irrespective of the grid strength. In [24], the modulation signals from GFL control and GFM control are weighed to obtain the inverter modulation signal. Based on small-signal analysis, the optimal weighting factor is obtained, which makes the short circuit ratio range of stable operation the widest. Adaptive control dynamically adjusts control strategies based on real-time system states and can also be employed to enhance the adaptability of the converter’s control schemes to fluctuating operating conditions [25]. In [26], a mode-switching scheme is proposed, where the SCR is taken as a criterion to determine the switching boundary between GFL mode and GFM mode. In [27], a phase-locked loop and power synchronization loop are mixed in a variable proportion. The proportional coefficient varies with the SCR, and the variation law imitates the hysteresis curve. Existing research focuses on improving the adaptability of control schemes to SCR variations, ignoring the active power fluctuations, which has a significant impact on the small-signal stability [28].

To address the limitations of existing research that neglects active power fluctuations, this paper proposes an adaptive switching control for converters in the renewable energy station based on the operating short circuit ratio (OSCR), which can comprehensively reflect the stability variation law of GFL converters and GFM converters under the condition of grid impedance and active power output fluctuations. The rest of this paper is organized as follows: Section 2 establishes a mathematical model for the mode-switchable VSC in the renewable energy station. Section 3 analyzes the effect of the SCR, active power and the capacity of GFM VSCs on the oscillation characteristic of the station by employing an eigenvalue analysis. Section 4 proposes an adaptive switching control scheme for the mode-switchable VSCs based on the operating short circuit ratio (OSCR), including the grid impedance identification, the capacity configuration of mode-switchable converters and the setting of the switching boundary. Section 5 is simulation verification. Section 6 concludes this article.

2. The Mathematical Model of the Mode-Switchable Voltage Source Converter

The control block diagram of the VSC is shown in Figure 1, where the inner loop for both GFL and GFM control modes is identical. The expressions of the proportional integral loop H_m(s) (m = 1, 2, …, 8) and the inertia loop T(s) are as follows:

$$\left\{\begin{array}{c}{\mathrm{H}}_m(s)={\mathrm{K}}_{\mathrm{p}m}+{\displaystyle \frac{{\mathrm{K}}_{\mathrm{i}m}}{s}}, m=1,2,\dots ,8\hfill \\ T(s)={\displaystyle \frac{1}{1+s{T}_{\mathrm{F}}}}\hfill \end{array}\right.,$$

(1)

where K_pm is the proportionality coefficient, K_im is the integral coefficient and T_F is the inertia time constant.

This paper primarily focuses on the characteristics of the inverter side of the VSC. Hence, the DC side is modeled as an ideal DC voltage source. The system parameters are listed in

Table 1. The parameters of the system.

Parameter	Value
T(s)	1/(1 + 0.002s)
H1(s), H8(s)	0.5 + 40/s
H2(s)	0.01 + 0.1/s
H3(s), H4(s)	0.2 + 10/s
H6(s), H7(s)	2.5 + 5/s
H5(s)	3.2 + 2000/s
LF	0.5 mH
CF	30 μF
Rg	0.1 ohm
Lg	calculate based on SCR
ω0	314.15926 rad/s
J	0.002
KD	0.1

. When the VSC works in one mode, the other mode is idle. If the control loop of idle mode does not follow the state of the system, when the control mode is switched, the system will be greatly disturbed due to the sudden change in the control loop state. Since the two modes share the inner loop, the key to realizing smooth switching between GFL mode and GFM mode is to ensure that the reference values of the inner loop will not mutate before and after the mode switching so as to avoid the sudden change in the voltage modulation wave. When the control mode is switched, the state of the system at the current time is sampled, which is assigned to the corresponding integrators as their initial values by the reset pulses shown in Figure 1.

For the proportional integral loops in the system, their outputs are taken as state variables. The state variable of the proportional integral loop m is named z_m (m = 1, 2, …, 8). As for the inertia loops in the inner loop, their outputs are taken as state variables as well. The variable in the d-axis is named x_d, while the one in the q-axis is named x_q. There is an integrator related to the inertia coefficient J in the power synchronization loop, and its output is taken as a state variable, which is recorded as x_J. The superscript c represents the value of the variable in the GFL or GFM controller coordinate, and the superscript s represents the value of the variable in the global coordinate, the d-axis of which coincides with the AC grid voltage vector.

2.1. The Mathematical Model of GFL Mode

According to Kirchhoff’s voltage law and Kirchhoff’s current law, the dynamic equations of the AC circuit can be obtained as (2)

$$\left\{\begin{array}{c}(L_{\mathrm{T}}+L_{\mathrm{g}}){\displaystyle \frac{d{i}_{\mathrm{d}}^{\mathrm{c}}}{dt}}=u_{\mathrm{td}}^{\mathrm{c}}-u_{\mathrm{sd}}^{\mathrm{c}}-R_{\mathrm{g}}{i}_{\mathrm{d}}^{\mathrm{c}}+{\omega }_0(L_{\mathrm{T}}+L_{\mathrm{g}})i_{\mathrm{q}}^{\mathrm{c}}\hfill \\ (L_{\mathrm{T}}+L_{\mathrm{g}}){\displaystyle \frac{d{i}_{\mathrm{q}}^{\mathrm{c}}}{dt}}={\mathrm{u}}_{\mathrm{tq}}^{\mathrm{c}}-u_{\mathrm{sq}}^{\mathrm{c}}-R_{\mathrm{g}}{i}_{\mathrm{q}}^{\mathrm{c}}-{\omega }_0(L_{\mathrm{T}}+L_{\mathrm{g}})i_{\mathrm{d}}^{\mathrm{c}}\hfill \\ C_{\mathrm{F}}{\displaystyle \frac{d{u}_{\mathrm{td}}^{\mathrm{c}}}{dt}}=i_{\mathrm{vd}}^{\mathrm{c}}-i_{\mathrm{d}}^{\mathrm{c}}+{\omega }_0{C}_{\mathrm{F}}{\mathrm{u}}_{\mathrm{tq}}^{\mathrm{c}}\hfill \\ C_{\mathrm{F}}{\displaystyle \frac{d{u}_{\mathrm{tq}}^{\mathrm{c}}}{dt}}=i_{\mathrm{vq}}^{\mathrm{c}}-i_{\mathrm{q}}^{\mathrm{c}}-{\omega }_0{C}_{\mathrm{F}}{u}_{\mathrm{td}}^{\mathrm{c}}\hfill \\ L_{\mathrm{F}}{\displaystyle \frac{d{i}_{\mathrm{vd}}^{\mathrm{c}}}{dt}}=u_{\mathrm{vd}}^{\mathrm{c}}-u_{\mathrm{td}}^{\mathrm{c}}+{\omega }_0{L}_{\mathrm{F}}{i}_{\mathrm{vq}}^{\mathrm{c}}\hfill \\ L_{\mathrm{F}}{\displaystyle \frac{d{i}_{\mathrm{vq}}^{\mathrm{c}}}{dt}}=u_{\mathrm{vq}}^{\mathrm{c}}-u_{\mathrm{tq}}^{\mathrm{c}}-{\omega }_0{L}_{\mathrm{F}}{i}_{\mathrm{vd}}^{\mathrm{c}}\hfill \end{array}\right.,$$

(2)

where ω₀ is the fundamental angular frequency. $i_{\mathrm{d}}^{\mathrm{c}}$, $i_{\mathrm{q}}^{\mathrm{c}}$, $u_{\mathrm{td}}^{\mathrm{c}}$, $u_{\mathrm{tq}}^{\mathrm{c}}$, $i_{\mathrm{vd}}^{\mathrm{c}}$ and $i_{\mathrm{vq}}^{\mathrm{c}}$ are the state variables as well as the output variables of the circuit part. $u_{\mathrm{sd}}^{\mathrm{c}}$, $u_{\mathrm{sq}}^{\mathrm{c}}$, $u_{\mathrm{vd}}^{\mathrm{c}}$ and $u_{\mathrm{vq}}^{\mathrm{c}}$ are the input variables.

The dynamic equations of the inner loop can be represented as (3). In (3), $z_1$, $z_8$, $x_{\mathrm{d}}$ and $x_{\mathrm{q}}$ are the state variables of the inner loop. $u_{\mathrm{vd}}^{\mathrm{c}}$ and $u_{\mathrm{vq}}^{\mathrm{c}}$ are the output variables. $i_{\mathrm{vdrefGFL}}^{\mathrm{c}}$, $i_{\mathrm{vqrefGFL}}^{\mathrm{c}}$, $i_{\mathrm{vd}}^{\mathrm{c}}$, $i_{\mathrm{vq}}^{\mathrm{c}}$, $u_{\mathrm{td}}^{\mathrm{c}}$, $u_{\mathrm{tq}}^{\mathrm{c}}$ and ω_GFL are the input variables. The dynamic equations of the GFL outer loop can be represented as (4). In (4), the state variables are z₃ and z₄. The output variables are $i_{\mathrm{vdrefGFL}}^{\mathrm{c}}$ and $i_{\mathrm{vqrefGFL}}^{\mathrm{c}}$. The input variables are $u_{\mathrm{td}}^{\mathrm{c}}$, $i_{\mathrm{d}}^{\mathrm{c}}$, $u_{\mathrm{tq}}^{\mathrm{c}}$ and $i_{\mathrm{q}}^{\mathrm{c}}$. The dynamic equations of the phase-locked loop (PLL) can be represented as (5). In (5), the state variables of the PLL are z₅ and θ_GFL. The input variable is $u_{\mathrm{tq}}^{\mathrm{c}}$. The output variables are ω_GFL and θ_GFL:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{z}_1}{dt}}=i_{\mathrm{vdrefGFL}}^{\mathrm{c}}-i_{\mathrm{vd}}^{\mathrm{c}}\hfill \\ {\displaystyle \frac{d{z}_8}{dt}}=i_{\mathrm{vqrefGFL}}^{\mathrm{c}}-i_{\mathrm{vq}}^{\mathrm{c}}\hfill \\ {\displaystyle \frac{d{x}_{\mathrm{d}}}{dt}}={\displaystyle \frac{-x_{\mathrm{d}}+u_{\mathrm{td}}^{\mathrm{c}}}{T_{\mathrm{F}}}}\hfill \\ {\displaystyle \frac{d{x}_{\mathrm{q}}}{dt}}={\displaystyle \frac{-x_{\mathrm{q}}+u_{\mathrm{tq}}^{\mathrm{c}}}{T_{\mathrm{F}}}}\hfill \\ u_{\mathrm{vd}}^{\mathrm{c}}=U_{\mathrm{vdref}}^{\mathrm{c}}=x_{\mathrm{d}}+K_{\mathrm{p}1}(i_{\mathrm{vdrefGFL}}^{\mathrm{c}}-i_{\mathrm{vd}}^{\mathrm{c}})+\hfill \\ \hfill K_{\mathrm{i}1}{z}_1-{\omega }_{\mathrm{GFL}}{L}_{\mathrm{F}}{i}_{\mathrm{vq}}^{\mathrm{c}}\hfill \\ u_{\mathrm{vq}}^{\mathrm{c}}=U_{\mathrm{vqref}}^{\mathrm{c}}=x_{\mathrm{q}}+K_{\mathrm{p}8}(i_{\mathrm{vqrefGFL}}^{\mathrm{c}}-i_{\mathrm{vq}}^{\mathrm{c}})+\hfill \\ \hfill K_{\mathrm{i}8}{z}_8+{\omega }_{\mathrm{GFL}}{L}_{\mathrm{F}}{i}_{\mathrm{vd}}^{\mathrm{c}}\hfill \end{array}\right.$$

(3)

$$\left\{\begin{array}{c}{\displaystyle \frac{d{z}_3}{dt}}=P_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}+u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})\hfill \\ {\displaystyle \frac{d{z}_4}{dt}}=Q_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}-u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})\hfill \\ i_{\mathrm{vdrefGFL}}^{\mathrm{c}}=K_{\mathrm{p}3}[P_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}+u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})]+K_{\mathrm{i}3}{z}_3\hfill \\ i_{\mathrm{vqrefGFL}}^{\mathrm{c}}=K_{\mathrm{p}4}[Q_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}-u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})]+K_{\mathrm{i}4}{z}_4\hfill \end{array}\right.$$

(4)

$$\left\{\begin{array}{c}{\displaystyle \frac{d{z}_5}{dt}}=u_{\mathrm{tq}}^{\mathrm{c}}\hfill \\ {\displaystyle \frac{d{\theta }_{\mathrm{GFL}}}{dt}}=K_{\mathrm{p}5}{u}_{\mathrm{tq}}^{\mathrm{c}}+K_{\mathrm{i}5}{z}_5={\omega }_{\mathrm{GFL}}\hfill \end{array}\right.$$

(5)

The equations of coordinate transformation between the controller coordinate and the global coordinate are as follows:

$$\left\{\begin{array}{c}{u}_{\mathrm{sd}}^{\mathrm{s}}=u_{sd}^c\cos \alpha -u_{sd}^c\sin \alpha \hfill \\ u_{\mathrm{sq}}^{\mathrm{s}}=u_{\mathrm{sq}}^{\mathrm{c}}\sin \alpha +u_{\mathrm{sq}}^{\mathrm{c}}\cos \alpha \hfill \\ u_{\mathrm{sd}}^{\mathrm{c}}=u_{\mathrm{sd}}^{\mathrm{s}}\cos \alpha +u_{\mathrm{sd}}^{\mathrm{s}}\sin \alpha \hfill \\ u_{\mathrm{sq}}^{\mathrm{c}}=-u_{\mathrm{sq}}^{\mathrm{s}}\sin \alpha +u_{\mathrm{sq}}^{\mathrm{s}}\cos \alpha \hfill \\ {\displaystyle \frac{d\alpha }{dt}}={\omega }_{\mathrm{GFL}}-{\omega }_0\hfill \end{array}\right.,$$

(6)

where α is the angle of the controller coordinate leading the global coordinate. ω_GFL is the input variable. u_sd and u_sq before coordinate transformation are input variables, while u_sd and u_sq after coordinate transformation are output variables.

The above dynamic equations can be linearized at the working point to obtain their own small-signal models. Based on the input and output variables of each part’s small-signal model, the connection between them can be determined, and then the overall small-signal model of the GFL converter can be obtained. The mathematical equations of the GFL converter’s small-signal model are as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{d\Delta {\mathit{X}}_{\mathrm{GFL}}}{dt}}={\mathit{A}}_1\Delta {\mathit{X}}_{\mathrm{GFL}}+{\mathit{B}}_1\Delta \mathit{U}\hfill \\ \Delta {\mathit{Y}}_{\mathrm{GFL}}={\mathit{C}}_1\Delta {\mathit{X}}_{\mathrm{GFL}}+{\mathit{D}}_1\Delta \mathit{U}\hfill \end{array}\right.,$$

(7)

where X_GFL = [$i_{\mathrm{d}}^{\mathrm{c}}$, $i_{\mathrm{q}}^{\mathrm{c}}$, $u_{\mathrm{td}}^{\mathrm{c}}$, $u_{\mathrm{t}\mathrm{q}}^{\mathrm{c}}$, $i_{\mathrm{v}\mathrm{d}}^{\mathrm{c}}$, $i_{\mathrm{vq}}^{\mathrm{c}}$, θ_GFL, z₅, z₁, z₈, x_d, x_q, z₃, z₄]^T is the state vector, A₁ is the state matrix and B₁ is the input matrix. U = [$u_{\mathrm{s}\mathrm{d}}^{\mathrm{s}}$, $u_{\mathrm{sq}}^{\mathrm{s}}$] is the input vector. C₁ is the output matrix and D₁ is the direct transfer matrix. Y_GFL = [$i_{\mathrm{d}}^{\mathrm{s}}$, $i_{\mathrm{q}}^{\mathrm{s}}$]^T is the output vector.

2.2. The Mathematical Model of GFM Mode

The dynamic equations of the inner loop can be represented as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{z}_1}{dt}}=i_{\mathrm{vdrefGFM}}^{\mathrm{c}}-i_{\mathrm{vd}}^{\mathrm{c}}\hfill \\ {\displaystyle \frac{d{z}_8}{dt}}=i_{\mathrm{vqrefGFM}}^{\mathrm{c}}-i_{\mathrm{vq}}^{\mathrm{c}}\hfill \\ {\displaystyle \frac{d{x}_{\mathrm{d}}}{dt}}={\displaystyle \frac{-x_{\mathrm{d}}+u_{\mathrm{td}}^{\mathrm{c}}}{T_{\mathrm{F}}}}\hfill \\ {\displaystyle \frac{d{x}_{\mathrm{q}}}{dt}}={\displaystyle \frac{-x_{\mathrm{q}}+u_{\mathrm{tq}}^{\mathrm{c}}}{T_{\mathrm{F}}}}\hfill \\ u_{\mathrm{vd}}^{\mathrm{c}}=U_{\mathrm{vdref}}^{\mathrm{c}}=x_{\mathrm{d}}+K_{\mathrm{p}1}(i_{\mathrm{vdrefGFM}}^{\mathrm{c}}-i_{\mathrm{vd}}^{\mathrm{c}})+\hfill \\ \hfill K_{\mathrm{i}1}{z}_1-{\omega }_{\mathrm{GFM}}{L}_{\mathrm{F}}{i}_{\mathrm{vq}}^{\mathrm{c}}\hfill \\ u_{\mathrm{vq}}^{\mathrm{c}}=U_{\mathrm{vqref}}^{\mathrm{c}}=x_{\mathrm{q}}+K_{\mathrm{p}8}(i_{\mathrm{vqrefGFM}}^{\mathrm{c}}-i_{\mathrm{vq}}^{\mathrm{c}})+\hfill \\ \hfill K_{\mathrm{i}8}{z}_8+{\omega }_{\mathrm{GFM}}{L}_{\mathrm{F}}{i}_{\mathrm{vd}}^{\mathrm{c}}\hfill \end{array}\right..$$

(8)

In (8), $z_1$, $z_8$, $x_{\mathrm{d}}$ and $x_{\mathrm{q}}$ are the state variables of the inner loop. $u_{\mathrm{vd}}^{\mathrm{c}}$ and $u_{\mathrm{vq}}^{\mathrm{c}}$ are the output variables. $i_{\mathrm{vdrefGFM}}^{\mathrm{c}}$, $i_{\mathrm{vqrefGFM}}^{\mathrm{c}}$, $i_{\mathrm{vd}}^{\mathrm{c}}$, $i_{\mathrm{vq}}^{\mathrm{c}}$, $u_{\mathrm{td}}^{\mathrm{c}}$, $u_{\mathrm{tq}}^{\mathrm{c}}$ and ω_GFM are the input variables.

The dynamic equations of the GFM outer loop can be represented as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{z}_2}{dt}}=Q_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}-u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})\hfill \\ {\displaystyle \frac{d{z}_6}{dt}}=K_{\mathrm{p}2}[Q_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}-u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})]+\hfill \\ K_{\mathrm{i}2}{z}_2+U_0-u_{\mathrm{td}}^{\mathrm{c}}\hfill \\ {\displaystyle \frac{d{z}_7}{dt}}=-u_{\mathrm{tq}}^{\mathrm{c}}\hfill \\ i_{\mathrm{vdrefGFM}}^{\mathrm{c}}=K_{\mathrm{p}6}{K}_{\mathrm{p}2}[Q_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}-u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})]+\hfill \\ \hfill K_{\mathrm{p}6}{K}_{\mathrm{i}2}{z}_3+K_{\mathrm{p}6}{U}_0-K_{\mathrm{p}6}{u}_{\mathrm{td}}^{\mathrm{c}}+K_{\mathrm{i}6}{z}_6\hfill \\ i_{\mathrm{vqrefGFM}}^{\mathrm{c}}=-K_{\mathrm{p}7}{u}_{\mathrm{tq}}^{\mathrm{c}}-K_{\mathrm{i}7}{z}_7\hfill \end{array}\right..$$

(9)

In (9), the state variables of the GFM outer loop are z₂, z₆ and z₇. The output variables are $i_{\mathrm{vdrefGFM}}^{\mathrm{c}}$ and $i_{\mathrm{vqrefGFM}}^{\mathrm{c}}$. The input variables are $u_{\mathrm{td}}^{\mathrm{c}}$, $i_{\mathrm{d}}^{\mathrm{c}}$, $u_{\mathrm{tq}}^{\mathrm{c}}$ and $i_{\mathrm{q}}^{\mathrm{c}}$.

The dynamic equations of the power synchronization loop can be represented as follows:

$$\left\{\begin{array}{c}J{\omega }_0{\displaystyle \frac{d{x}_J}{dt}}=-K_{\mathrm{D}}{\omega }_0{x}_J+P_{\mathrm{ref}}-{\displaystyle \frac{3}{2}}(u_{\mathrm{td}}^{\mathrm{c}}{i}_{\mathrm{d}}^{\mathrm{c}}+u_{\mathrm{tq}}^{\mathrm{c}}{i}_{\mathrm{q}}^{\mathrm{c}})\hfill \\ {\displaystyle \frac{d{\theta }_{\mathrm{GFM}}}{dt}}={\omega }_{\mathrm{GFM}}=x_J+{\omega }_0\hfill \end{array}\right..$$

(10)

In (10), the state variables are x_J and θ_GFM. The input variables are $u_{\mathrm{td}}^{\mathrm{c}}$, $i_{\mathrm{d}}^{\mathrm{c}}$, $u_{\mathrm{tq}}^{\mathrm{c}}$ and $i_{\mathrm{q}}^{\mathrm{c}}$. The output variables are ω_GFM and θ_GFM.

Similarly, the mathematical equations of the GFM converter’s small-signal model are as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{d\Delta {\mathit{X}}_{\mathrm{GFM}}}{dt}}={\mathit{A}}_2\Delta {\mathit{X}}_{\mathrm{GFM}}+{\mathit{B}}_2\Delta \mathit{U}\hfill \\ \Delta {\mathit{Y}}_{\mathrm{GFM}}={\mathit{C}}_2\Delta {\mathit{X}}_{\mathrm{GFM}}+{\mathit{D}}_2\Delta \mathit{U}\hfill \end{array}\right.,$$

(11)

where X_GFM = [$i_{\mathrm{d}}^{\mathrm{c}}$, $i_{\mathrm{q}}^{\mathrm{c}}$, $u_{\mathrm{td}}^{\mathrm{c}}$, $u_{\mathrm{t}\mathrm{q}}^{\mathrm{c}}$, $i_{\mathrm{v}\mathrm{d}}^{\mathrm{c}}$, $i_{\mathrm{vq}}^{\mathrm{c}}$, θ_GFM, x_J, z₁, z₈, x_d, x_q, z₆, z₇]^T is the state vector, A₂ is the state matrix and B₂ is the input matrix. U = [$u_{\mathrm{s}\mathrm{d}}^{\mathrm{s}}$, $u_{\mathrm{sq}}^{\mathrm{s}}$] is the input vector. C₂ is the output matrix, and D₂ is the direct transfer matrix. Y_GFM = [$i_{\mathrm{d}}^{\mathrm{s}}$, $i_{\mathrm{q}}^{\mathrm{s}}$]^T is the output vector.

2.3. The Verification of Small-Signal Model

The accuracy of the small-signal model for mode-switchable converters in both grid-following (GFL) and grid-forming (GFM) modes is validated under SCR values of 2, 3 and 5. The active power from the VSC is 1 p.u. and the reactive power from the VSC is 0.5 p.u. The disturbance is the step of the active power reference value from 1 p.u. to 0.98 p.u. Figure S1 in the Supplementary File shows the dynamic response of the small-signal model and the electromagnetic transient simulation model in PSCAD/EMTDC when the VSC works in GFL mode. Figure S2 in the Supplementary File shows the dynamic response of the small-signal model and the electromagnetic transient simulation model in PSCAD/EMTDC when the VSC works in GFM mode. The dynamic response of the small-signal model and the electromagnetic transient simulation model in PSCAD/EMTDC is basically the same, which verifies the correctness of the small-signal model.

3. The Oscillation Characteristics of Renewable Energy Station

Based on the mathematical model of the mode-switchable VSC, the small-signal model of the renewable energy station with 50 VSCs is obtained by using single-unit multiplication. All GFL VSCs are aggregated into an equivalent GFL VSC, while all GFM VSCs are aggregated into an equivalent GFM VSC. The impact of the SCR, actual active power of renewable energy and the capacity of GFM VSCs on the station’s oscillation characteristic is analyzed in this section. The control loop parameters of the mode-switchable VSC are as shown in Table 1. In different control groups, the values of active power dispatched by the station are set to 1 p.u., 0.75 p.u and 0.5 p.u., while the values of the SCR are set to 1, 3 and 5. With varying GFM VSCs’ capacity in the station, the dominant eigenvalues corresponding to different combinations of the SCR and active power are calculated for each capacity.

The dominant eigenvalues and the frequencies of the condition where all the VSCs work in GFL mode are listed in

Table 2. The dominant eigenvalues and their frequencies in the station with all VSCs working in GFL mode.

Active Power P (p.u.)	SCR = 1	SCR = 3	SCR = 5
0.5	−0.775	−1.17	−1.3
	±46.2j	±45.6j	±45.4j
	(7.36 Hz)	(7.28 Hz)	(7.23 Hz)
0.75	−0.554	−1.04	−1.21
	±45.8j	±45.6j	±45.4j
	(7.29 Hz)	(7.28 Hz)	(7.25 Hz)
1.0	−0.459	−0.924	−1.12
	±44.9j	±45.6j	±45.5j
	(7.15 Hz)	(7.26 Hz)	(7.25 Hz)

. With an increase in the SCR, the real part of the dominant eigenvalues shows a decreasing trend, whereas it increases as the active power increases. The subsynchronous stability of the station is positively correlated with the SCR and negatively correlated with active power, while the oscillation frequency shows little dependence on either parameter.

The dominant eigenvalues and their frequencies of the condition that all VSCs work in GFM mode are listed in

Table 3. The dominant eigenvalues and their frequencies in the station with all VSCs working in GFM mode.

Active Power P (p.u.)	SCR = 1	SCR = 3	SCR = 5
0.5	−2.01	−1.9	−1.74
	±0.591j	±0.998j	±1.35j
	(0.33 Hz)	(0.34 Hz)	(0.35 Hz)
0.75	−2.13	−2.06	−1.91
	±0.374j	±0.897j	±1.32j
	(0.34 Hz)	(0.36 Hz)	(0.37 Hz)
1.0	−2.96	−2.34	−2.22
	±0.00j	±0.592j	±1.23j
	(0.36 Hz)	(0.38 Hz)	(0.40 Hz)

. With an increase in the SCR, the real part of the dominant eigenvalues shows an increasing trend, while it increases as the active power decreases. The low-frequency stability of the station is positively correlated with active power and negatively correlated with the SCR, while the oscillation frequency shows little dependence on either parameter.

The previous analysis indicates that when the value of the SCR is large or the active power is low, the stability of the station with all the VSCs working in GFL mode is good enough so that there is no need for mode switching. When the value of the SCR is small and the active power is high, there is a risk of instability, so part of the VSCs should work in GFM mode under such conditions. However, the specific capacity of the VSCs to be converted from GFL mode to GFM mode has not been determined yet. Several operating points with relatively poor stability of subsynchronous oscillation mode are selected to investigate the impact of the GFM VSCs’ capacity on the stability of the station. The dominant eigenvalues of different GFM VSCs’ capacity proportion in the subsynchronous frequency band are listed in

Table 4. The dominant eigenvalues of different GFM VSCs’ capacity proportion in the subsynchronous frequency band.

(SCR, Active Power)	GFM VSCs’ Capacity Proportion in the Station
	40%	50%	70%
(2, 1 p.u.)	−1.3	−1.44	−1.49
	±45.3j	±45.4j	±45.9j
(2, 0.5 p.u.)	−1.35	−1.45	−1.52
	±45.9j	±46.1j	±46.4j
(2, 0.3 p.u.)	−1.4	−1.48	−1.55
	±46.1j	±46.2j	±46.6j
(3, 1 p.u.)	−1.32	−1.50	−1.57
	±45.5j	±45.6j	±45.8j
(3, 0.5 p.u.)	−1.40	−1.52	−1.59
	±45.7j	±45.8j	±46.0j

. The dominant eigenvalues of different GFM VSCs’ capacity proportion in the low-frequency band are listed in

Table 5. The dominant eigenvalues of different GFM VSCs’ capacity proportion in the low-frequency band.

(SCR, Active Power)	GFM VSCs’ Capacity Proportion in the Station
	40%	50%	70%
(2, 1 p.u.)	−2.08	−2.30	−2.39
	±1.57j	±1.20j	±1.00j
(2, 0.5 p.u.)	−1.59	−1.74	−1.89
	±1.56j	±1.30j	±1.00j
(2, 0.3 p.u.)	−1.50	−1.65	−1.82
	±1.60j	±1.40j	±1.10j
(3, 1 p.u.)	−1.59	−1.90	−2.25
	±2.00j	±1.70j	±1.20j
(3, 0.5 p.u.)	−1.29	−1.49	−1.75
	±1.80j	±1.70j	±1.30j

. For the five operating points listed in Table 4 and Table 5, the stability of both the low-frequency dominant oscillation mode and the subsynchronous dominant oscillation mode becomes stronger as the capacity of the GFM VSCs increases. Comparing Table 4 and Table 5 when the SCR and active power are equal to three and 0.5 p.u., the low-frequency eigenvalue replaces the subsynchronous frequency eigenvalue as the dominant eigenvalue of the station. Both oscillation modes may become the dominant mode, and thus the capacity of the GFM VSCs should satisfy the constraint of ensuring stability in both subsynchronous and low-frequency oscillation modes.

According to the analysis of the oscillation characteristics of the renewable energy station, the OSCR can be selected as the index to reflect the stability of the subsynchronous oscillation mode and the low-frequency oscillation mode of the station. The OSCR is defined as follows [29]:

$$\mathrm{OSCR}={\displaystyle \frac{S_{\mathrm{ac}}}{P}}={\displaystyle \frac{S_{\mathrm{ac}}}{S_{\mathrm{N}}}}\times {\displaystyle \frac{S_{\mathrm{N}}}{P}}={\displaystyle \frac{1}{\left|Z^{*}\right|\times P^{*}}},$$

(12)

where S_ac is the short-circuit capacity of the grid, S_N is the rated capacity of the station, P is the actual active power of the station, Z^* is the per unit quantity of grid impedance and P^* is the per unit quantity of active power. When the value of the SCR is larger and the active power of the station is lower, the OSCR value is larger. In this condition, the stability of the subsynchronous oscillation mode of the station is better, and the stability of the low-frequency oscillation mode is worse. It is reasonable to set the switching boundary based on the OSCR.

4. The Adaptive Switching Control Scheme Based on Operating Short Circuit Ratio

4.1. The Identification of Grid Impedance

To calculate the OSCR, it is necessary to monitor the active power of the station and the grid impedance. The formula for estimating grid impedance in the αβ coordinate axis by injecting non-characteristic harmonics into the grid can be represented as follows [30]:

$$U_{\mathrm{g}}(h_x)=U_{\mathrm{g}\mathrm{\alpha }}^{hx}+j{U}_{\mathrm{g}\mathrm{\beta }}^{hx},$$

(13)

$$I_{\mathrm{g}}(h_x)=I_{\mathrm{g}\mathrm{\alpha }}^{hx}+j{I}_{\mathrm{g}\mathrm{\beta }}^{hx},$$

(14)

$$R_{\mathrm{g}}={\displaystyle \frac{U_{\mathrm{g}\mathrm{\alpha }}^{hx}{I}_{\mathrm{g}\mathrm{\alpha }}^{hx}+U_{\mathrm{g}\mathrm{\beta }}^{hx}{I}_{\mathrm{g}\mathrm{\beta }}^{hx}}{I_{\mathrm{g}\mathrm{\alpha }}^{hx}{I}_{\mathrm{g}\mathrm{\alpha }}^{hx}+I_{\mathrm{g}\mathrm{\beta }}^{hx}{I}_{\mathrm{g}\mathrm{\beta }}^{hx}}},$$

(15)

$$L_{\mathrm{g}}={\displaystyle \frac{U_{\mathrm{g}\mathrm{\beta }}^{hx}{I}_{\mathrm{g}\mathrm{\alpha }}^{hx}-U_{\mathrm{g}\mathrm{\alpha }}^{hx}{I}_{\mathrm{g}\mathrm{\beta }}^{hx}}{{\omega }_{hx}(I_{\mathrm{g}\mathrm{\alpha }}^{hx}{I}_{\mathrm{g}\mathrm{\alpha }}^{hx}+I_{\mathrm{g}\mathrm{\beta }}^{hx}{I}_{\mathrm{g}\mathrm{\beta }}^{hx})}},$$

(16)

where h_x is the non-characteristic harmonic frequency, U_g is the voltage at the point of common coupling (PCC), I_g is the current injected into the grid from the PCC, R_g is the equivalent resistance, L_g is the equivalent inductance and ω_hx is the harmonic angular frequency. The fundamental impedance of the grid can be calculated as follows:

$$Z_{\mathrm{g}}=\sqrt{{R_{\mathrm{g}}}^2+{({\omega }_0{L}_{\mathrm{g}})}^2}.$$

(17)

In this section, a complex filter is used to extract harmonic voltage and current signals. Figure 2 shows the block diagram of a complex filter [30]. The superscripts + and − represent positive-sequence and negative-sequence components of the fundamental frequency, respectively. ω_hc is the cut-off angular frequency of the harmonic filter, and ω_c is the cut-off angular frequency of the fundamental filter. Figure 3 shows the amplitude–frequency characteristic of the complex filter for non-characteristic harmonics. The complex filter can effectively suppress the influence of positive-sequence and negative-sequence components of the fundamental frequency on the measurement of non-characteristic harmonics. Figure 4 shows the process of harmonic injection and extraction.

4.2. The Setting of the Switching Boundary

The switching logic of the mode-switchable VSC in the renewable energy station is shown in Figure 5. When the OSCR crosses the boundary value C₁ from large to small, the stability of the station with all the VSCs working in GFL mode is weakened, and there is a risk of subsynchronous oscillation. Part of the VSCs should change their control mode from GFL mode to GFM mode. In order to avoid the repeated switching of the VSCs’ control mode caused by the fluctuation in renewable energy in a short period of time, a hysteresis switching strategy is adopted. When the OSCR changes from small to large over the boundary value C₂, all the VSCs’ control methods should be restored to GFL mode.

The damping ratio required to restore stability at different oscillation frequencies is different, which can be calculated as follows:

$${\zeta }_{cri}={\displaystyle \frac{1}{\sqrt{a·{f_i}^2+1}}},$$

(18)

$$a={\displaystyle \frac{1-{\zeta }_{std}^2}{{\zeta }_{std}^2{f}_{std}^2}},$$

(19)

where ζ_cri is the damping ratio threshold of oscillating mode i, f_i is the frequency of oscillating mode i and a is the per unit coefficient. The calculation of a is based on a low-frequency mode, where the damping ratio threshold ζ_std is 0.05 and the frequency f_std is 2.5 Hz.

The setting of the switching boundary is based on the variation in station stability with the OSCR. Due to the worse MPPT ability of the GFM converters discussed above, when the system can remain stable, all converters should adopt GFL control. When all converters adopt GFL control, the stability of the system in the subsynchronous frequency band is positively correlated with the OSCR. When the OSCR is reduced to a critical value, the system stability margin is insufficient, and mode-switchable converters need to change their control modes from GFL mode to GFM mode. The corresponding critical value is C₁. The small-signal model of the system with all converters adopting GFL control is established, and then the specific value of C₁ is determined by an eigenvalue analysis.

Considering that the calculation of the OSCR involves grid impedance and the active power of the station, even if the OSCR values are the same, the corresponding grid impedance and active power of the station are not unique, and the stability of the station is not unique. Therefore, when setting the value of C₁, the active power of the station is fixed at the maximum value, and the value of the SCR is gradually reduced. When the damping ratio of the subsynchronous dominant mode is below the threshold value, the critical OSCR value is marked as C₁₁. Then, the SCR is fixed to the minimum value, and the active power of the station gradually increases. When the damping ratio of the subsynchronous dominant mode is below the threshold value, the critical OSCR value is marked as C₁₂. The larger one in C₁₁ and C₁₂ is taken as the value of C₁. By setting C₁ in the above way, it can ensure that when the OSCR value is greater than C₁, the subsynchronous dominant modal damping ratio of the station satisfies the requirement of stability, and it also considers the situation that the control mode of the VSCs does not need to convert to GFM mode when the active power is low enough, even when they are connected to weak grid. The maximum active power is set to 1 p.u. According to the rated capacity of the renewable energy station and the short-circuit capacity of the power grid provided by the grid operator, the minimum SCR value that may occur is determined. It is worth noting that since the simulation case in this paper is not derived from the actual project, a reasonable assumption can be made for the minimum SCR value. In this paper, the minimum SCR value is assumed to be one so as to test the adaptability of the proposed scheme to the extremely weak power grid.

When the OSCR increases from small to large over C₁, the converters adopting GFM control can be restored to GFL converters. If the OSCR fluctuates around the critical value C₁, converters will change control mode repeatedly. In order to avoid the frequent switching of the control mode, this paper applies the idea of hysteresis control, which is shown in Figure 5. When the OSCR crosses the switching boundary C₁ from large to small, mode-switchable converters change their control mode from GFL mode to GFM mode. When the OSCR crosses the switching boundary C₁ from small to large, mode-switchable converters keep adopting GFM control. When the OSCR is further increased to a larger value C₂, mode-switchable converters change their control mode from GFM mode to GFL mode. The switching boundary C₂ can be calculated with an incremental ΔC on the basis of C₁. There is no mandatory requirement for the value of ΔC, which is set to one in this paper. The example in Section 5.3 sets the switching boundaries C₁ and C₂ according to the above ideas.

4.3. The Capacity Configuration Scheme of the Mode-Switchable Voltage Source Converters

If the station has the ability to remain stable in the case of large-scale fluctuations in active power and the SCR, the capacity configuration scheme of the mode-switchable VSCs must satisfy the stability requirement of the station under extreme conditions with the worst stability. Firstly, considering the extreme condition with the worst stability of the subsynchronous mode with all the VSCs working in GFL mode, the configured capacity of the mode-switchable VSCs needs to ensure that the subsynchronous oscillation mode dominated by the VSCs working in GFL mode in the station is stable under this condition. The extreme condition in this paper is that the SCR is set to be the minimum value, and the active power of the station is set to be the maximum value. At the same time, the VSCs working in GFM mode introduce a low-frequency oscillation mode dominated by themselves. The capacity of the VSCs working in GFM mode is only a part of the station capacity, so it may be much smaller than the short-circuit capacity of the grid, even though the value of the SCR is low at this time. As a result, the GFM VSCs in the station are equivalent to operating under strong grid conditions. Therefore, under the condition that the OSCR is set to the minimum value, the capacity of the GFM VSCs gradually increases from small to large until the synchronous and low-frequency modal damping ratio of the system is greater than the corresponding threshold. The number of GFM VSCs at this time is recorded as N₁. Although there is no risk of instability in the subsynchronous frequency band under weak grid conditions due to the support of GFM VSCs, whether the low-frequency oscillation mode introduced by GFM VSCs under strong grid conditions will be stable or not is still unknown. On the switching boundary where the OSCR is equal to C₂, the capacity of GFM VSCs in the station also increases from small to large until the low-frequency modal damping ratio is greater than the threshold. The number of GFM VSCs at this time is recorded as N₂. The number of mode-switchable VSCs to be configured in the station takes the larger value of N₁ and N₂.

5. Simulation Verification

5.1. The Verification of Mode Smooth Switching Strategy

A station containing 50 VSCs is taken as an example to verify the effectiveness of the smooth switching strategy. The parameters involved in this section are shown in Table 1. Considering the time delay of signal transmission, mode switching occurs after 20 ms of parameter variation. The control delay of the converter control system is not considered, which mainly causes high-frequency oscillation. It is not in the range of low-frequency and subsynchronous frequency bands studied in this paper. The simulation results are shown in Figures S3–S7 in the Supplementary File. Figure S3 shows the steady-state switching performance. In Figure S3, the VSCs switch from GFM mode to GFL mode at 10 s and switch from GFL mode to GFM mode at 20 s. Figures S4–S7 show the dynamic switching performance. The change in the active power of the station and the change in the equivalent impedance of the power grid, that is, the change in the SCR, will trigger the mode switching. When the active power value crosses over 0.75 p.u. from large to small at 20 s, the VSCs switch from GFM mode to GFL mode. When the active power value crosses over 0.75 p.u. from small to large at 24 s, the VSCs switch from GFL mode to GFM mode. When the value of the SCR crosses over 2.5 from small to large at 40 s, the VSCs switch from GFM mode to GFL mode. When the value of the SCR crosses over 2.5 from large to small at 44 s, the VSCs switch from GFL mode to GFM mode. Regardless of steady-state switching or dynamic switching, the simulation results show that the smooth switching strategy of GFL mode and GFM mode can significantly reduce the maximum fluctuation amplitude of the station’s active power output and effectively avoid the sudden change in voltage and current waveform at the PCC caused by the mode switching.

The mechanism of smooth switching can be explained by circuit principles. The equivalent circuit of the grid-connected converter system is illustrated in Figure 6. From the perspective of the circuit, U_vabc and U_sabc are the excitation sources, and the remaining voltage and current variables in the circuit are the response of the excitation sources. In the short process of mode switching, the change in the grid voltage is ignored. If the mutation of U_vabc can be limited during the mode-switching process, the remaining voltage and current variables in the circuit will not mutate. This is the basic principle of the smooth switching strategy proposed in this paper. The implementation of this basic principle depends on the smooth switching strategy shown in Figure 1, which makes sure that the modulation wave signal of the converter will not mutate during the switching process. According to Figures S3d,h and S6 in the Supplementary File, whether it is static switching or dynamic switching, the smooth switching strategy proposed in this paper can effectively suppress the mutation of U_vabc, thus effectively suppressing the disturbance in the remaining voltage and current variables in the circuit.

5.2. The Grid Impedance Identification

In this paper, the frequency of the disturbance signal is selected as −75 Hz to avoid the influence of the background harmonics of the power grid on the impedance identification results. The amplitude–frequency characteristic of the complex filter is used to extract the voltage component and current component of −75 Hz for impedance identification. The effectiveness of the impedance identification scheme to eliminate the background harmonic interference of the power grid is further verified by a PSCAD simulation. The fifth harmonic component and the seventh harmonic component are added to the grid voltage, and the amplitude of the two harmonic components is 5% of the fundamental frequency component. In order to test the sensitivity of the proposed method to noise, we also add white noise to the grid voltage, with a maximum amplitude of 5% of the fundamental frequency component. The spectrum analysis of grid voltage is shown in Figure 7. Figure 8 shows that the impedance identification method described in this paper still has good identification ability in the presence of grid background harmonics and white noise.

It should be noted that identifying grid impedance by injecting harmonics will inevitably affect the power quality. On the one hand, the impact on the power quality can be reduced by limiting the amplitude of the injected harmonic components. In this study, the amplitude of the injected harmonic components is set at 1% of the fundamental frequency component. On the other hand, the duration of the harmonic injection can be reduced by improving the impedance identification method. For example, passive methods have been proposed [31,32]. This category of techniques eliminates the requirement for impedance parameter estimation through external excitation injection from inverters. Instead, it relies on extracting impedance characteristics by analyzing inherent information embedded in grid voltage/current signals. Consequently, these passive methods achieve impedance parameter identification while maintaining power quality without additional harmonic injection. However, the disadvantage of passive methods is that the identification accuracy is insufficient. Subsequent research can consider combining the characteristics of harmonic injection and passive methods for impedance identification. The passive methods are used to monitor whether the grid impedance changes. Once the grid impedance changes, harmonics are injected to obtain the exact value of the impedance. This improvement approach can be proposed as a potential direction for future research, although it has not been thoroughly explored in this paper.

5.3. The Switching Boundary Setting and Capacity Configuration of Mode-Switchable VSCs

According to Figure 5, the station containing 50 VSCs mentioned in Section 5.1 is taken as an example to illustrate the process of the switching boundary setting and capacity configuration of mode-switchable VSCs. The switching boundary is set by using the small-signal model of the station with all the VSCs working in GFL mode. In this paper, the maximum active power is set to its rated value (1 p.u.). The minimum SCR that may occur can be obtained from power grid operators. It should be noted that in order to facilitate the implementation process of the proposed method, the example in this paper assumes that the minimum SCR is one, which is not a general value method. The minimum SCR of different systems may be different, which needs to be obtained according to the actual situation of the power grid.

Firstly, the reference value of active power is fixed at 1 p.u, and the value of the SCR is changed. The curve of the dominant eigenvalue damping ratio of the subsynchronous frequency band with the OSCR is shown in Figure 9. The frequency of the dominant eigenvalue corresponding to each working point is also marked in the figure. According to (18) and (19), the damping ratio threshold can be calculated by combining the frequency information of the dominant eigenvalues in the subsynchronous frequency band. When the value of the OSCR is less than 2.4, the dominant eigenvalue damping ratio is below the threshold. Therefore, C₁₁ can be set to 2.4. Similarly, the curve of the dominant eigenvalue damping ratio with the OSCR is shown in Figure 10 when the value of the SCR is fixed to one and the active power is changed. When the value of the OSCR is less than 2.22, the dominant eigenvalue damping ratio is below the threshold. Therefore, C₁₂ can be set to 2.22. In summary, C₁ can be set to 2.4, and C₂ can be set to 3.4.

The variation trend in the dominant eigenvalue damping ratio with the GFM VSCs’ capacity in the subsynchronous frequency band and low-frequency band is shown in Table S1 in the Supplementary File. According to the minimum SCR value and the minimum active power value, the minimum value of the OSCR is one. When the OSCR is set to its minimum value, placing nine GFM VSCs can ensure that the damping ratio of the dominant eigenvalue in the subsynchronous frequency band exceeds the corresponding threshold. At the same time, the stability of the low-frequency dominant oscillation mode is good enough. The low-frequency damping ratios all exceed the corresponding thresholds. Therefore, N₁ can be set to nine. When the OSCR value is 3.4 (the SCR value is 3.4 and the active power value is 1 p.u.), the number of GFM VSCs needs to reach 14 to satisfy the damping ratio value requirement in the low-frequency band.

The small-signal stability of the station on the switching boundary C₂ is further tested. Under the condition that the OSCR value is 3.4, the dominant eigenvalue damping ratios corresponding to 36 GFL VSCs and 14 GFM VSCs in the station under different SCR and active power combinations are as shown in Table S1. The result shows that the configuration of 14 GFM VSCs is sufficient to ensure that the oscillation mode of the low-frequency band and the subsynchronous frequency band are stable at the same time. Therefore, N₂ can be set to 14. In summary, the number of mode-switchable VSCs configured in the station can be finally set to 14.

5.4. The Comparison Between the Switching Scheme Based on SCR and the Switching Scheme Based on OSCR

To demonstrate the superiority of the proposed scheme in this paper, a comparative analysis is conducted in this section between the switching scheme based on the OSCR proposed in this work and the switching scheme based on the SCR presented in [26]. This paper mainly studies the improvement in the switching index. The capacity configuration of the mode-switchable converters and the setting idea of the switching boundary in the two switching schemes are consistent.

In terms of the setting of the switching boundary, the switching scheme based on the SCR also adopts a hysteresis strategy, which is shown in Figure 11. The SCR reflects the stability of the system under the condition that the active power output by the converter is the rated value. Therefore, the active power is fixed to 1 p.u., and the SCR value is gradually reduced. When the active power is fixed to 1 p.u., the SCR is equal to the OSCR. The variation trend in the dominant eigenvalue damping ratio with the SCR can be referred to in Figure 9 in the manuscript. When the SCR is less than 2.4, the stability margin of the system does not meet the requirements. Therefore, A₁ is set to 2.4. Correspondingly, A₂ is set to 3.4.

In terms of the capacity configuration of the mode-switchable converters, both switching schemes adopt a unified configuration principle focused on the operating point with the worst subsynchronous stability. When the OSCR is used to measure the subsynchronous stability of the system, the active power value of the worst operating point is exactly 1 p.u. At this time, the SCR is equal to the OSCR. For the two switching schemes, the operating point with the worst synchronization stability is actually the same. Consequently, both switching schemes demonstrate identical capacity configurations of mode-switchable converters; that is, there are 14 mode-switchable converters in the station, which totally contains 50 converters.

Figure S8 in the Supplementary File presents the simulation results of a comparative analysis between the two switching schemes. The simulation condition is set to the fluctuations in the SCR and active power reference value of the station, as shown in Figure S8a,b. The identification values of the SCR and OSCR are computed online through impedance identification. Whether the identification values cross the switching boundary is used as the basis for the VSC’s mode switching, as shown in Figure S8d,e.

As evidenced in Figure S8f,g, the scheme based on the OSCR demonstrates smoother active power profiles compared to the scheme based on the SCR. The performance divergence between the two schemes primarily manifests during five distinct simulation time intervals, as explicitly annotated in Figure S8f,g. Combined with Figure S8f–h, it can be seen that if the SCR is used as the switching index, the mode-switchable converters work in GFM mode in the five simulation time intervals. The underutilized active power output leads to bad low-frequency modal stability dominated by GFM converters, resulting in low-frequency oscillations when subjected to disturbances. The OSCR effectively captures the deterioration of low-frequency modal stability due to the reduced active power. Consequently, the mode-switchable converters maintain GFL mode throughout all five intervals, successfully avoiding low-frequency oscillation.

From Figure S8h, it can be seen that the mode-switching number of the scheme based on the OSCR is significantly less than that based on the SCR. The SCR identification value crosses the switching boundary many times, whereas the OSCR values, incorporating actual active power measurements, maintain generally higher magnitudes with fewer boundary crossings. Under low SCR conditions, the OSCR properly reflects the subsynchronous modal stability when the active power is low, thereby preventing unnecessary switching from GFL mode to GFM mode. In conclusion, the OSCR provides a more accurate representation of system small-signal stability characteristics, establishing it as a more theoretically sound basis for switching boundary determination.

Furthermore, the adaptability of the proposed scheme based on the OSCR to SCR and power fluctuations is verified by simulation. In the case that the SCR and the active power fluctuations lead to the continuous migration of the equilibrium point, the method proposed in this paper can make the system maintain a sufficient stability margin, and quickly attenuate the oscillation at different equilibrium points, which is verified by Figure S9 in the Supplementary File. It should be noted that the analysis of the SCR and active power on small-signal stability is equivalent to comparing the small-signal stability of multiple different equilibrium points. This paper does not study the large-signal stability under the SCR and active power step conditions, but rather the differences in small-signal stability between different equilibrium points.

6. Conclusions

This article proposes an adaptive switching control scheme of voltage source converters based on the OSCR in the renewable energy station. Firstly, the oscillation characteristics of the renewable energy station are analyzed based on the small-signal model of the station. The characteristic indicates that the OSCR can reflect the low-frequency stability and subsynchronous stability of the station more reasonably, so it is chosen as the switching index. Secondly, the specific steps of the adaptive switching control scheme are given, including the grid impedance identification method, the switching boundary setting of the hysteresis switching strategy and the capacity configuration method of the mode-switchable VSCs. Finally, the superiority of the switching scheme based on the OSCR over the switching scheme based on the SCR is analyzed by simulation. The OSCR considers the influence of grid strength and active power fluctuations on the stability of the station comprehensively. The adaptive switching control scheme proposed in this paper can realize stable operation under the condition of large fluctuations in grid impedance and active power.

It is important to note that this research has several limitations that could be explored as directions for future studies. First, regarding the impedance identification method, this study considered the scenario of RL (resistive–inductive) lines but did not account for series-compensated lines. Further investigation is required to address the parameter identification challenges in series-compensated line applications. Second, grid impedance identification through harmonic injection may adversely affect the power quality. Subsequent research can consider combining the characteristics of harmonic injection and passive methods for impedance identification. The passive methods are used to monitor whether the grid impedance changes. Once the grid impedance changes, harmonics are injected to obtain the exact value of the impedance. Third, in terms of converter modeling, future studies should address high-frequency oscillation issues potentially induced by control delays. Finally, this paper simplifies the circuit part other than converters to the equivalent impedance, and further research can be carried out on more detailed modeling, such as considering the influence of nonlinear loads.