A Seamless Transition Control Strategy Based on Adaptive Fusion Between Grid-Following and Grid-Forming Inverters for Wide-Ranging Grid-Strength Fluctuations

Abstract

To tackle the degraded stability and non-smooth mode transitions under wide-range grid-strength variations with high renewable penetration, an adaptive fusion and disturbance-free switching control strategy is proposed, where parameter stability regions are analyzed using the D-partition method, thereby improving robustness over single-mode grid-following/grid-forming operation and reducing transients from conventional switching. First, a unified frequency-domain characteristic equation that incorporates the fusion weight is derived based on the sequence-impedance stability criterion, providing a consistent theoretical basis for stability modeling and assessment across operating conditions. Next, under wide-range grid-strength conditions, the controller-parameter stability region is computed subject to multiple constraints, including phase margin, gain margin, and short-circuit ratio, and the resulting robust feasible set is geometrically characterized on the parameter plane. Furthermore, to suppress transient disturbances induced by variations of the fusion weight with grid strength near the switching threshold, a D-zone-based multi-partition, stage-by-stage smoothing adaptive fusion strategy is developed. A nonlinear weight mapping yields a continuous transition trajectory, enabling seamless, disturbance-free transitions from weak to strong grids. Finally, simulation and experimental results quantitatively validate the superiority of the proposed method. Under severe weak-grid conditions with a short-circuit ratio of 1, the fusion strategy enlarges the parameter-stability feasible region by approximately 11.5% compared to single-mode operations. Moreover, the proposed D-zone strategy achieves a peak fusion advantage ratio of 43.11%, ensuring robust and disturbance-free switching across a wide range of grid-strength scenarios where the short-circuit ratio varies from 1 to 30.

1. Introduction

With the continuously increasing installed capacity of wind and photovoltaic generation, power systems are being accelerated toward a power-electronics-dominated paradigm [1]. In typical scenarios such as the integration of offshore wind farms via high-voltage direct current (HVDC) systems [2], the grid is characterized by wide-range variations in short-circuit ratio (SCR), a standard indicator of grid strength, low inertia, and insufficient damping [3]. As two representative grid-interfacing technologies, grid-forming (GFM) and grid-following (GFL) converters critically determine overall system stability through their control performance. However, a single control mode cannot satisfy all operating conditions. Specifically, GFL control relies on a phase-locked loop (PLL) to track the grid voltage, and in weak grids, low-frequency synchronization oscillations may be excited due to coupling between the PLL bandwidth and grid impedance [4]. By contrast, although GFM control can emulate synchronous-machine behavior and provide active support [5,6], under strong-grid conditions, high-frequency resonances may be triggered by interactions between the converter’s equivalent output impedance and the grid impedance, and overcurrent risks may arise under large disturbances [7,8]. Therefore, developing a GFL–GFM hybrid control architecture that achieves wide-range grid-strength adaptability and disturbance-free transitions is essential for overcoming stability bottlenecks in high-renewable-penetration power systems.

To address the coordinated operation of GFM and GFL converters, systematic investigations have been conducted on hybrid-system modeling, stability analysis, and control design. In [4,9], a frequency-domain impedance model of a parallel GFL–GFM system was established, and the impact of converter capacity ratios on damping characteristics was analyzed, indicating that appropriate sizing can effectively suppress oscillations. Ref. [10] systematically examined sizing rules for mixed GFM–GFL grid integration from the perspective of key influencing factors, providing guidance for capacity allocation and stability-oriented design. A duality theory between GFL and GFM control was proposed in [11], revealing their intrinsic relationship at the control-architecture level. In [12], a sixth-order transient model of the hybrid system was developed; however, due to the complexity of the high-order state matrix, closed-form stability conditions are difficult to obtain. The dynamic coupling mechanisms across multiple time scales were investigated in [13] using a state-space framework, and an impedance-based generalized stability criterion was presented in [14]. Collectively, these studies provide an important theoretical foundation for understanding steady-state characteristics and parameter tuning in hybrid systems. However, in the practical operation of modern power systems, the SCR at the point of interconnection (POI) is no longer a static constant but instead exhibits pronounced wide-range time variability. As reported in [15], due to intermittency and weather uncertainty, stochastic renewable-power fluctuations can directly cause dynamic drift in the system’s effective grid strength. Ref. [16] further revealed an explicit coupling between multi-plant SCR and nodal injected power, implying that dispatch changes can correspond to substantial variations in grid strength. It was explicitly noted in [17] that conventional stability assessment based on static grid strength cannot meet real-time operational requirements. To enhance wide-range adaptability to grid-strength variations, ref. [18] analyzed transient stability during multi-mode switching, and a hybrid control framework based on virtual synchronous machine concepts or modified outer-loop characteristics was proposed in [19] to combine GFL power-tracking capability with GFM voltage-support advantages. The effectiveness of GFM solutions in improving transient synchronization stability and voltage support in weak grids was validated in [20].

Although substantial progress has been made in steady-state modeling and equilibrium-point control, GFL–GFM coordinated operation still faces major challenges when grid strength varies continuously over a wide range. Specifically, most existing coordination strategies adopt fixed ratios or logic-based mode switching to combine GFL and GFM control [21]. Near the critical grid-strength boundary, this discontinuous regulation can cause abrupt changes in control weighting, leading to transient voltage/current impacts and potentially inducing unintended frequent switching under measurement noise [22]. More importantly, the coupling mechanisms over the full parameter space have not been systematically elucidated, and how parameter sensitivity evolves with SCR remains unclear [13,14]. As SCR changes, dominant poles and stability boundaries shift significantly, making conventional fixed-point analysis insufficient for quantifying the stability-feasible region and the hybrid-control advantage region [23]. The lack of visualization based on global parameter boundaries makes it difficult to implement unified scheduling that simultaneously ensures steady-state performance and transient smoothness during cross-regime operation.

The D-partition (D-decomposition) method can construct stability regions and track their boundary evolution in the parameter plane, thereby providing quantitative guidance for converter tuning [24,25]. Accordingly, the D-partition method is introduced to quantitatively characterize the stability-advantage space of hybrid control from the viewpoint of geometric evolution in the parameter plane. On this basis, a smooth switching function is designed to ensure that the fusion weight varies continuously with SCR and transitions smoothly across regions, thereby enabling disturbance-free switching and improving robustness and stability preservation from weak to strong grids.

The main contributions of this paper are summarized as follows:

First, the topology of a three-phase grid-connected converter with an LCL filter is described, and a unified GFL–GFM control model incorporating the fusion path is established.

Second, a unified GFL–GFM frequency-domain characteristic equation that includes the fusion weight is established based on the D-partition method and a sequence-impedance stability criterion. The design is evaluated under a set of multi-dimensional specifications, namely gain margin, phase margin, and bandwidth. The constrained stability region is computed and visualized in the parameter space to quantify the enlarged advantage region and tuning benefits over single-mode control.

Third, a hierarchical smooth fusion strategy across multiple D-zones is proposed to address discontinuous weighting and transient impacts near the SCR threshold. A continuously differentiable switching curve is constructed to realize adaptive fusion and disturbance-free transition, while additional constraints are introduced in strong-grid regions to mitigate oscillation risks associated with excessively large fusion coefficients.

Finally, simulations and experiments are conducted to validate the proposed strategy in enlarging the weak-grid stability region, suppressing switching-induced transient oscillations, and improving wide-range operating adaptability.

2. Unified Frequency-Domain Modeling of an Integrated GFL/GFM System

2.1. GFL/GFM Topology

The topology of the investigated three-phase grid-connected converter with an LCL filter is shown in Figure 1. The system is supplied by a DC source $V_{dc}$. The LCL filter is composed of the converter-side inductor $L_1$, the shunt capacitor $C_f$, and the grid-side inductor $L_2$; the parasitic/line resistances are denoted by $R_1$ and $R_2$, and a series damping resistor $R_{cd}$ is included with the filter capacitor. After linearization around the rated operating point, the converter AC side is represented as a controlled voltage source $u_{inv}$, whereas the grid side is modeled by a Thevenin equivalent consisting of a voltage source $u_g$ in series with an equivalent inductance $L_g$. This representation establishes the transfer relationship between the converter output voltage $u_{inv}$ and the injected grid current $i_g$, thereby providing the basis for subsequent frequency-domain modeling and stability analysis.

2.1.1. GFM Control Structure

In the GFM scheme, a droop-based power loop is employed, together with inner voltage and current loops, to form a GFM control architecture, as shown in Figure 2. In the outer power loop, the active power $P$ and reactive power $Q$ are used to regulate the output frequency and voltage magnitude via the droop coefficients $m_p$ and $m_q$, respectively. The corresponding control laws can be expressed as follows:

$$\begin{array}{l}{\omega }_{GFM}={\omega }_{ref}+m_p(P_{ref}-P)\\ U_{GFM}=U_{ref}+m_q(Q_{ref}-Q)\end{array}$$

(1)

In these equations, ${\omega }_{GFM}$ and $U_{GFM}$ denote the output angular frequency and voltage magnitude of the GFM converter, respectively; ${\omega }_{ref}$ and $U_{ref}$ denote the nominal angular frequency and nominal voltage magnitude, respectively; $P_{ref}$ and $Q_{ref}$ are the reference values of active and reactive power; $P$ and $Q$ are the measured converter output powers; and $m_p$ and $m_q$ are the active- and reactive-power droop coefficients, respectively.

The converter regulates its output frequency and voltage magnitude based on the instantaneous active and reactive powers $P$ and $Q$. Specifically, the output angular frequency ${\omega }_{GFM}$ is adjusted from ${\omega }_{ref}$ via the active-power droop coefficient $m_P$, and the d-axis voltage reference $u_d^{GFM}$ is adjusted from $U_{\mathrm{ref}}$ via the reactive-power droop coefficient $m_q$. The inner current loop tracks the dq-axis current references $i_{dref}$ and $i_{qref}$ and generates the modulation signal $c_{dq}^{GFM}$. Meanwhile, an internal phase angle ${\theta }_{GFM}$ is obtained by integrating ${\omega }_{GFM}$, which is then used to construct the internal synchronous reference frame and enable self-synchronized operation.

Under the above control structure, the GFM converter does not require a PLL to actively establish voltage-magnitude and frequency references and thus exhibits electrical characteristics similar to those of a synchronous source.

2.1.2. GFL Control Structure

The control structure of the GFL converter is shown in Figure 3. A PLL is used to track the POI voltage phase ${\theta }_{PLL}$, and synchronization is achieved by relying on the external AC grid. The GFL controller is composed of an outer power/voltage loop, an inner current loop, and a PLL, with the active-power and voltage commands $P_{ref}$ and $V_{ref}$ serving as external setpoints. The three-phase converter output voltage is denoted as $u_{abc}$, and its fundamental magnitude is denoted as $V_1$. In the synchronous rotating frame defined by the PLL angle ${\theta }_{GFL}$, the output current is expressed as $i_d$ and $i_q$. The outer power loop generates the d-axis current reference $i_d^{ref}$ from $P_{ref}$, whereas the voltage loop generates the q-axis current reference $i_q^{ref}$ from $V_{ref}$. Based on these references, the converter computes the dq-axis voltage commands $u_d^{GFL}$ and $u_q^{GFL}$. Subsequently, the dq-frame modulation signal $c_{dq}^{GFL}$ is obtained by the modulator, and three-phase gating signals $e_{mabc}$ are generated through coordinate transformation and PWM mapping.

Unlike GFM control, GFL control depends on the grid to provide the frequency and phase references; therefore, its dynamics and stability are strongly influenced by grid strength, and synchronization and oscillation issues are likely to occur under weak-grid conditions.

To characterize the equivalent external behavior of grid-connected converters under different control modes, the output impedance is modeled based on the GFL and GFM control structures. Based on existing small-signal modeling methods [4], the equivalent output impedance of the conventional GFM converter is given as follows:

$$Z_{o\_GFM}(s)={\displaystyle \frac{Z_{num}(s)}{Z_{den}(s)}}={\displaystyle \frac{Z_1{Z}_2+Z_c(Z_1+Z_2)+G_i(s)G_d(s)(Z_c+Z_2)}{Z_1+Z_c+G_i(s)G_d(s)\left[1+G_v(s)Z_c\right]+G_{droop}(s)H_Q(s)}}$$

(2)

In this expression, $s$ denotes the Laplace operator. $Z_{\mathrm{num}}\left(s\right)$ is the numerator polynomial of the impedance model, characterizing the open-loop dynamics of the passive LCL network and the inner current loop, whereas $Z_{\mathrm{den}}\left(s\right)$ is the denominator polynomial, reflecting the damping and stiffness support introduced by the outer voltage loop. The complex impedances of the converter-side and grid-side filter inductors are $Z_1=s{L}_1+R_1$ and $Z_2=s{L}_2+R_2$, respectively, and the capacitor-branch impedance is $Z_{\mathrm{c}}=1/s{C}_{\mathrm{f}}+R_{\mathrm{cd}}$. $G_v\left(s\right)$ and $G_i\left(s\right)$ denote the transfer functions of the PI controllers in the AC voltage loop and inner current loop, respectively. The digital control and sampling delay are modeled as $G_{\mathrm{d}}\left(s\right)=e^{-1.5T_{\mathrm{s}}s}$, where $T_{\mathrm{s}}$ is the sampling period. $G_{\mathrm{droop}}\left(s\right)$ denotes the reactive-power–voltage droop gain and is typically approximated as $G_{\mathrm{droop}}\left(s\right)\approx m_q$.

Similarly, considering the impact of PLL dynamics as derived in established literature [26], the equivalent output impedance of the GFL converter is given as follows:

$$Z_{o-GFL}(s)={\displaystyle \frac{Z_{num}(s)}{Z_{den}(s)+G_{pwr}(s)H_{\mathrm{Q}}(s)K_{\mathrm{v}}(s)+Y_{PLL}(s)Z_{num}(s)}}$$

(3)

In this expression, $Z_{num}\left(s\right)$ and $Z_{den}\left(s\right)$ are defined as in the previous section, and $G_{pwr}\left(s\right)$ denotes the transfer function of the PI controller in the reactive-power loop. Specifically, $G_{pwr}\left(s\right)$ is given by $G_{\mathrm{pwr}}\left(s\right)=k_{p\mathrm{Q}}+k_{i\mathrm{Q}}/S$, where $k_{p\mathrm{Q}}$ and $k_{i\mathrm{Q}}$ are the proportional and integral gains of the reactive-power loop, respectively. $H_{\mathrm{Q}}\left(s\right)$ denotes the transfer function of the reactive-power calculation and low-pass filtering (LPF) block and is typically selected as $H_{\mathrm{Q}}\left(s\right)=1/1+T_{\mathrm{s}}s$, where $T_{\mathrm{s}}$ is the sampling period of the converter. $K_{\mathrm{v}}\left(s\right)$ represents an approximation of the closed-loop gain of the voltage loop, i.e., the ratio that maps the command generated by the power loop to the resulting voltage response. Finally, $Y_{\mathrm{PLL}}\left(s\right)$ denotes the small-signal equivalent shunt admittance introduced by the PLL dynamics.

The expression for the equivalent admittance $Y_{\mathrm{PLL}}\left(s\right)$ is

$$Y_{PLL}(s)={\displaystyle \frac{I_{1d}}{U_{1d}}}\cdot {\displaystyle \frac{U_{1d}\left(k_{pPLL}+\frac{k_{iPLL}}{s}\right)}{s+U_{1d}\left(k_{pPLL}+\frac{k_{iPLL}}{s}\right)}}$$

(4)

In this expression, $U_{1d}$ and $I_{1d}$ denote the d-axis voltage and current components at the steady-state operating point, respectively, and $k_{p\mathrm{PLL}}$ and $k_{i\mathrm{PLL}}$ are the proportional and integral gains of the PLL controller.

In particular, the SCR [14] is adopted in this study to quantify grid strength, and its relationship with the equivalent grid inductance $L_g$ is given as follows:

$$L_g(\mathrm{SCR})={\displaystyle \frac{U_n^2}{S_n{\omega }_0\mathrm{SCR}}}$$

(5)

In this expression, SCR denotes the short-circuit ratio, which is used to quantify grid strength; $L_g$ is the equivalent grid inductance; $U_n$ is the rated rms voltage at the POI; $S_n$ is the rated apparent power of the grid-connected converter; and ${\omega }_0=2\pi f_0$ is the fundamental grid angular frequency.

To account for the sampling delay of the digital control system and the PWM transport delay, a unified delay term is introduced as $G_d\left(s\right)=e^{-1.5T_{s}s}$. For frequency-domain analysis, the delay term is linearized using a first-order Padé approximation, yielding $G_{\mathrm{d}}\left(\mathrm{S}\right)=\left(1-0.75T_s\right)/\left(1+0.75T_s\right)$. The resulting generalized plant model is defined as follows:

$$G_p(s,\mathrm{SCR})=G_{LCL}(s)\cdot G_d(s)$$

(6)

In this expression, $G_{LCL}\left(S\right)$ denotes the transfer function of the LCL filter from the inverter output voltage to the grid current.

2.2. Fusion Synchronization Control

If both synchronization mechanisms are enabled, the converter may exhibit both GFL and GFM characteristics. To model and analyze the stability of the two control schemes within a unified framework, a weighted-output fusion form is adopted, and a fusion ratio coefficient $\lambda \in [0, 1]$ is introduced to construct a convex combination of the two control outputs [27]. The corresponding control block diagram is shown in Figure 4.

Here, $\lambda =0$ corresponds to single GFL control, whereas $\lambda =1$ corresponds to single GFM control. This fusion formulation unifies the representation of the control mode and grid-strength parameter, thereby establishing a consistent theoretical basis for (i) deriving a characteristic equation that explicitly includes $\lambda$ and SCR, (ii) obtaining an analytical mapping of the stability region using the D-partition method, and (iii) designing a continuous switching strategy over a wide SCR range.

The converter’s final output angular frequency is obtained by proportionally weighting the outputs of the two control modes, i.e.,

$${\omega }_{cmd}(s)=\left[\lambda {\omega }_{ref}^{GFM}(s)+(1-\lambda ){\omega }_{ref}^{GFL}(s)\right]$$

(7)

where $\lambda$ is the fusion coefficient ($\lambda \in (0,1)$), and ${\omega }_{ref}^{GFM}$ and ${\omega }_{ref}^{GFL}$ denote the POI angular frequencies produced by the GFM and GFL controllers, respectively, and ${\omega }_{cmd}$ denotes the commanded POI angular frequency.

Based on the previously developed GFL, GFM, and fusion models, time-domain simulations were conducted for the grid-connected system. Following the modeling and switching procedure in [13], the SCR was specified piecewise, and step changes in SCR were applied at 0.5 s intervals during the simulation to emulate a continuously varying grid-strength condition.

As shown in Figure 5a, under progressively weakening grid conditions where the SCR drops stepwise from 5 to 2 and ultimately to 1, the conventional GFL control exhibits noticeable power oscillations when the SCR decreases to 2 and becomes completely unstable at 1. From a physical mechanism perspective, this instability arises because the PLL severely interacts with the high grid-side impedance in a weak grid environment. In contrast, the fusion control reduces its reliance on the PLL by incorporating a GFM synchronization mechanism, thereby operating smoothly throughout the entire grid-weakening process.

As shown in Figure 5b, under progressively strengthening grid conditions where the SCR increases stepwise from 5 to 10 and then to 30, the single GFM control generates transient phase differences due to its rigid characteristics, resulting in significant power fluctuations and slow recovery. While retaining robust voltage support, the fusion control effectively overcomes this inherent drawback by utilizing the flexible damping of the GFL control path. Simulation waveforms demonstrate that during abrupt changes in grid impedance, the fusion control significantly suppresses transient fluctuations and converges more rapidly to the new steady-state operating point. This proves that the fusion strategy adopted in this study achieves an ideal balance between system synchronization stability and transient dynamic performance, laying a solid foundation for the subsequent unified global stability analysis.

2.3. Unified Open-Loop Model and Characteristic Equation Under Coupled Fusion Ratio and SCR

Within the unified frequency-domain modeling framework, it is assumed that the GFL and GFM control modes operate near the steady state in the same synchronously rotating reference frame. Based on the aforementioned fusion control structure, the corresponding equivalent control block $G_{eq}\left(s,\lambda \right)$ can be expressed as follows:

$$G_{eq}(s,\lambda )=\lambda G_c^{\mathrm{GFM}}(s)+(1-\lambda )G_c^{\mathrm{GFL}}(s)$$

(8)

In this expression, $G_c^{GFL}\left(s\right)$ and $G_c^{GFL}\left(s\right)$ denote the equivalent transfer functions of the GFM and GFL synchronization mechanisms, respectively, mapping $\epsilon \left(s\right)$ to the output angular–frequency deviation. Furthermore, by incorporating the equivalent plant $G_p\left(s,\mathrm{SCR}\right)$ of the converter and its inner control loops, as well as the grid-coupling block $G_{grid}\left(s,\mathrm{SCR}\right)$, the overall open-loop transfer function $L\left(s,\lambda ,\mathrm{SCR}\right)$ can be written in a unified form as

$$L(s,\lambda ,\mathrm{SCR})=G_{eq}(s,\lambda )G_p(s,\mathrm{SCR})G_{grid}(s,\mathrm{SCR})$$

(9)

Accordingly, the closed-loop characteristic equation $D\left(s\right)$ is obtained as

$$D(s)=1+L(s,\lambda ,\mathrm{SCR})=0$$

(10)

This model explicitly captures the coupling between the control mode (via $\lambda$) and SCR, thereby providing a unified mathematical basis for the subsequent global stability analysis using the D-partition method.

3. Converter Modeling and Stability Analysis Based on the D-Partition Method

3.1. Sequence-Impedance Modeling

To reveal the frequency-domain impedance-coupling characteristics between the converter and the grid and to support parameter-domain mapping, positive- and negative-sequence impedance models are established on the basis of the unified open-loop model using the harmonic linearization method [27]. Model validity is then verified by frequency-sweep comparisons. Since the stability boundary under the considered operating conditions is dominated by the positive-sequence impedance, the loop gain is formulated using the positive-sequence model, and parameter-domain mapping analysis is performed accordingly. According to impedance-based stability theory, the positive-sequence output impedance of the fusion-controlled system, $Z_{out}^{+}\left(s,\lambda \right)$, is defined around the balanced operating point as

$$Z_{out}^{+}(s;\lambda )={\displaystyle \frac{\Delta V^{+}(s)}{\Delta I^{+}(s)}}$$

(11)

where $\Delta V^{+}\left(s\right)$ and $\Delta I^{+}\left(s\right)$ denote the small-signal perturbations of the positive-sequence voltage and current, respectively. Considering the weighted-output property of the fusion converter, the positive-sequence output impedance can be expressed as a weighted combination of the impedances under GFM and GFL control, i.e.,

$$Z_{out}^{+}(s;\lambda )=\lambda Z_{out,GFM}^{+}(s)+(1-\lambda )Z_{out,GFL}^{+}(s)$$

(12)

where $Z_{out,GFM}^{+}\left(s\right)$ and $Z_{out,GFL}^{+}\left(s\right)$ are the converter positive-sequence output impedances corresponding to the GFM and GFL control modes, respectively, evaluated under identical plant parameters and the same linearized operating point.

The grid positive-sequence equivalent impedance depends on SCR: a smaller SCR indicates a weaker grid and corresponds to a larger magnitude of the equivalent grid impedance. For stability analysis, the positive-sequence loop gain $L^{+}\left(s,\lambda ,\mathrm{SCR}\right)$ is defined within the positive-sequence impedance framework as

$$L^{+}(s;\lambda ,\mathrm{SCR})={\displaystyle \frac{Z_{out}^{+}(s;\lambda )}{Z_g^{+}(s;\mathrm{SCR})}}$$

(13)

In this expression, $Z_g^{+}\left(s;\mathrm{SCR}\right)$ denotes the grid positive-sequence equivalent impedance parameterized by SCR.

The corresponding positive-sequence stability characteristic equation is given as follows:

$$1+L^{+}(s;\lambda ,\mathrm{SCR})=0$$

(14)

With the loop gain defined in (13), the positive-sequence closed-loop system is stable when the Nyquist plot of the loop gain in (14) satisfies the Nyquist stability criterion with respect to the critical point $\left(-1,j0\right)$.

To validate the positive-sequence impedance model and the associated stability criterion established in (11)–(14), a grid-connected model including the complete control loops was implemented in an electromagnetic transient simulation platform. Under $P_{\mathrm{r}\mathrm{e}\mathrm{f}}=1.0$ pu and $Q_{\mathrm{r}\mathrm{e}\mathrm{f}}=0$, small-signal frequency sweeps from 100 to 3000 Hz were applied for different SCR values. As shown in Figure 6, the theoretically calculated frequency response of the positive-sequence equivalent impedance agrees well with the sweep-identified results in both magnitude and phase, thereby validating the accuracy of the proposed unified frequency-domain model.

Furthermore, the frequency range in which the output-impedance phase is below $-90°$ is defined as the capacitive negative damping (CND) band. The analysis indicates that when SCR is <2, the GFL-mode output impedance is prone to coupling with the grid impedance in the CND band, thereby triggering instability.

3.2. Principle of the D-Partition Method and Its Equivalence to the Sequence-Impedance Approach

Although sequence-impedance modeling can determine system stability under specific operating conditions, it provides limited visualization for controller-parameter tuning. To enable a quantitative parameter-domain representation of stability constraints, the D-partition method is employed to analytically map frequency-domain stability conditions onto the controller-parameter plane. Moreover, the equivalence between the D-partition method and the sequence-impedance criterion is established in terms of mathematical form, and a unified characteristic equation with clear physical interpretation is formulated, thereby providing a consistent framework for subsequent feasible-region analysis and tuning-boundary determination.

3.2.1. Impedance-Characteristic Boundary Under the Sequence-Impedance Approach

By examining the characteristic equation formed by the converter positive-sequence output impedance and the grid impedance and recasting it over a common denominator and rearranging it with respect to the controller parameters, the mapping relationship to the D-partition characteristic equation can be directly revealed.

According to (13)–(14), the positive-sequence stability boundary satisfies

$$1+{\displaystyle \frac{Z_{out}^{+}(s)}{Z_g^{+}(s)}}=0$$

(15)

Considering that the converter’s closed-loop output impedance $Z_{out}^{+}\left(s\right)$ is determined by the open-loop impedance $Z_{open}\left(s\right)$ and the loop gain $T\left(s\right)$. With $T\left(s\right)=G_{PI}(s)H_{plant}(s)$, $Z_{out}^{+}(s)$ is given by

$$Z_{out}^{+}(s)={\displaystyle \frac{Z_{open}(s)}{1+G_{PI}(s)H_{plant}(s)}}$$

(16)

where $Z_{open}\left(s\right)$ is the open-loop positive-sequence output impedance under the same plant boundary conditions as in (16), $H_{plant}\left(s\right)$ denotes the generalized plant, and $G_{PI}\left(s\right)=K_p+K_i/s$ denotes the PI controller to be tuned.

By substituting (16) into the impedance criterion $Z_{out}^{+}\left(s\right)+Z_g^{+}\left(s\right)=0$ and rewriting the resulting expression over a common denominator, the following form is obtained:

$$Z_{open}(s)+Z_g^{+}(s)\left[1+\left(K_p+{\displaystyle \frac{K_i}{s}}\right)H_{plant}(s)\right]=0$$

(17)

To eliminate the integral term $s$ in the denominator, both sides of (17) are multiplied by $s$, and the expression is rearranged and grouped with respect to the controller parameters $K_p$ and $K_i$, yielding a unified characteristic equation in terms of the control parameters:

$$\begin{array}{c}D(s;\lambda ,\mathrm{SCR},K_p,K_i)=P(s)+K_p{Q}_p(s)+K_i{Q}_i(s)\\ D(s;\lambda ,\mathrm{SCR},K_p,K_i)=0\end{array}$$

(18)

In these expressions, $Q_i\left(s\right)$ and $Q_p(s)$ are the coefficient terms associated with the integral gain $K_i$ and proportional gain $K_p$ respectively, where $P(s)$, $Q_p(s)$, and $Q_i(s)$ are the polynomial coefficients of the characteristic equation rearranged for $K_p$ and $K_i$, whereas $P\left(s\right)$ is a constant term independent of the controller parameters. Moreover, $P\left(s\right)$, $Q_p(s)$, and $Q_i\left(s\right)$ are uniquely determined by the system impedance characteristics:

$$\left\{\begin{array}{l}{Q}_i(s)=Z_g^{+}(s)H_{plant}(s)\\ Q_p(s)=s{Z}_g^{+}(s)H_{plant}(s)\\ P(s)=s[Z_{open}(s)+Z_g^{+}(s)]\end{array}\right.$$

(19)

3.2.2. Analytical Determination of the Parameter-Stability Boundary Using the D-Partition Method

According to the D-partition principle [24], the crossing of the imaginary axis by the roots of the characteristic equation corresponds to a critical stability condition. For a given fusion coefficient $\lambda$ and grid strength (SCR), the stability boundary in the parameter plane consists of the zero-root boundary and the purely imaginary-root boundary.

Let the characteristic root in (18) be $s=j\omega \left(\omega \in \left(0,+\infty \right)\right)$. By substituting this into the unified characteristic equation and separating the real and imaginary parts, the following expressions are obtained:

$$\left\{\begin{array}{l}\mathrm{Re}[D(j\omega )]=P_R(\omega )+K_p{Q}_{pR}(\omega )+K_i{Q}_{iR}(\omega )=0\hfill \\ \mathrm{Im}[D(j\omega )]=P_I(\omega )+K_p{Q}_{pI}(\omega )+K_i{Q}_{iI}(\omega )=0\hfill \end{array}\right.$$

(20)

where $j$ is the imaginary unit, and $\omega$ is the angular frequency. $D\left(j\omega \right)$ is defined in (18). $P_R\left(\omega \right)$ and $P_I\left(\omega \right)$ denote the real and imaginary parts of $P\left(j\omega \right)$, respectively, i.e., $P\left(j\omega \right)=P_R\left(\omega \right)+j{P}_I\left(\omega \right)$. Similarly, $Q_p\left(j\omega \right)=Q_{pR}\left(\omega \right)+j{Q}_{pI}\left(\omega \right)$ and $Q_i\left(j\omega \right)=Q_{iR}\left(\omega \right)+j{Q}_{iI}\left(\omega \right)$. Here, $K_p$ and $K_i$ are the PI gains to be tuned.

By applying Cramer’s rule, analytical expressions of the parameter boundary can be obtained, where $K_p\left(\omega \right)$ and $K_i\left(\omega \right)$ denote the boundary values of the proportional and integral gains at frequency $\omega$, respectively.

$$K_p(\omega )={\displaystyle \frac{P_I(\omega )Q_{iR}(\omega )-P_R(\omega )Q_{iI}(\omega )}{Q_{pR}(\omega )Q_{iI}(\omega )-Q_{pI}(\omega )Q_{iR}(\omega )}}$$

(21)

$$K_i(\omega )={\displaystyle \frac{P_R(\omega )Q_{pI}(\omega )-P_I(\omega )Q_{pR}(\omega )}{Q_{pR}(\omega )Q_{iI}(\omega )-Q_{pI}(\omega )Q_{iR}(\omega )}}$$

(22)

The scanning trajectories defined by (21)–(22) are mapped onto the $K_p-K_i$ plane to trace the complex-domain locus $D_2$ corresponding to the characteristic roots crossing the imaginary axis. Together with the static-gain boundary $D_0$ determined by $s=0$, the parameter-stability region $\mathrm{\Omega }\left(\mathrm{SCR},\lambda \right)$ in which all closed-loop characteristic roots lie in the left-half plane can be identified using the Boundary Crossing Rule. Accordingly, the following expression is obtained:

$$\left\{\begin{array}{l}{D}_0: \underset{\omega \to 0^{+}}{\lim }D(j\omega ;\lambda ,\mathrm{SCR},K_p,K_i)=0\hfill \\ D_{\infty }: \underset{\omega \to +\infty }{\lim }{\displaystyle \frac{D(j\omega ;\lambda ,\mathrm{SCR},K_p,K_i)}{{(j\omega )}^n}}=0\hfill \\ D_{\omega }: D(j\omega ;\lambda ,\mathrm{SCR},K_p,K_i)=0, \omega \in (0,+\infty )\hfill \end{array}\right.$$

(23)

where $D_0$ denotes the zero-frequency boundary, $D_{\infty }$ denotes the infinite-frequency boundary, $n$ is the highest order of $D(s)$, and $D_{\omega }$ denotes the boundary generated by purely imaginary-root crossings.

By comparison with (18), the sequence-impedance analysis and the D-partition method are mathematically equivalent. The former characterizes converter–grid interaction stability through the frequency-domain impedance ratio, whereas the latter parameterizes the same stability condition and maps it onto the $K_p$–$K_i$ plane. Consequently, a stability-feasible parameter region can be constructed from the boundary curves, providing a geometric interpretation that facilitates controller tuning.

3.3. Converter Stability Analysis Using the D-Partition Method

Based on the analytical boundary expressions derived using the D-partition method, the stability regions of the single-mode controllers are analyzed under different SCR conditions. To ensure satisfactory dynamic performance, additional constraints on gain margin, phase margin, and control bandwidth are imposed on the stability regions: $G_m\ge 3 \mathrm{d}\mathrm{B}$, $30°\le P_m\le 45°$, and $400 \mathrm{H}\mathrm{z}\le B_w\le 750 \mathrm{H}\mathrm{z}$. The feasible region satisfying the comprehensive requirements is obtained as the intersection of these constraints.

As shown in Figure 7, the evolution of the feasible regions for the GFL controller parameters $\left(K_p,K_i\right)$ and the GFM droop parameters $\left(m_p,m_q\right)$ is compared. If the PI parameters are selected within the stable region, satisfactory dynamic response and steady-state performance can be simultaneously ensured under ideal grid conditions.

The blue curves represent the D-partition stability boundaries, and the shaded regions indicate the feasible intersections that satisfy the performance constraints. As SCR decreases, the feasible regions of both modes shrink. In comparison, the GFL mode is more sensitive to weak grids and exhibits a more pronounced reduction in the feasible intersection, whereas the GFM mode retains a non-negligible feasible range in the weak-grid regime, although it is likewise constrained by the performance requirements. These results provide the basis for subsequent weight scheduling of the fusion strategy over the full SCR range.

3.4. Fusion Advantage Region

The analytical model of parameter stability boundaries established using the D-partition method is intended to quantitatively evaluate the robustness gain of the fusion control strategy under wide-range grid-strength variations. By constructing feasible-region sets for the single-mode controllers (GFL and GFM) and the fusion mode, a fusion-advantage metric is defined, and its distribution in the parameter space and its variation with SCR are characterized.

3.4.1. Set-Theoretic Definition and Mapping of Stability Regions

For a given grid strength (SCR), the parameter-stability-region sets corresponding to the GFL and GFM modes are defined as

$$\left\{\begin{array}{l}{\mathrm{\Omega }}_{GFL}(\mathrm{SCR})\subset {\mathbb{R}}_{(K_p,K_i)}^2 \\ {\mathrm{\Omega }}_{GFM}(\mathrm{SCR})\subset {\mathbb{R}}_{(m_p,m_q)}^2\end{array}\right.$$

(24)

In these expressions, ${\mathrm{\Omega }}_{GFL}\left(\mathrm{SCR}\right)$ denotes the feasible region of the GFL mode in the $(K_p,K_i)$ parameter plane under a given SCR, satisfying both stability and performance constraints; ${\mathrm{\Omega }}_{GFM}\left(\mathrm{SCR}\right)$ denotes the feasible region of the GFM mode in the $(m_p,m_q)$ parameter plane under the same SCR; and ${\mathbb{R}}_{(K_p,K_i)}^2$ and ${\mathbb{R}}_{(m_p,m_q)}^2$ denote the corresponding two-dimensional real spaces of the parameter coordinates.

For the fusion control strategy, a stability region $\mathrm{\Omega }\left(\mathrm{SCR},\lambda \right)$ can be obtained for $\lambda \in [0,1]$. Accordingly, the global joint feasible region of the fusion controller under a given SCR, denoted as ${\mathrm{\Omega }}_{Fusion}\left(\mathrm{SCR}\right)$ [28], is defined as

$$\begin{array}{l}{\mathrm{\Omega }}_{con}^{Fusion}(\mathrm{SCR};m_p,m_q)={\mathrm{\Omega }}_{sta}^{Fusion}(\mathrm{SCR};m_p,m_q)\cap {\mathrm{\Omega }}_{per}^{Fusion}(\mathrm{SCR};m_p,m_q)\\ {\mathrm{\Omega }}_{Fusion}(\mathrm{SCR};m_p,m_q)\triangleq \left\{(K_p,K_i)\in \Xi | (K_p,K_i)\in {\mathrm{\Omega }}_{con}^{Fusion}(\mathrm{SCR};m_p,m_q)\right\}\end{array}$$

(25)

where ${\mathrm{\Omega }}_{Fusion}\left(\mathrm{SCR};m_p,m_q\right)$ denotes the stability–performance feasible region of the fusion controller in the $(K_p,K_i)$ plane for a given SCR and droop-parameter pair $(m_p,m_q)$; ${\mathrm{\Omega }}_{con}^{Fusion}\left(\mathrm{SCR};m_p,m_q\right)$ denotes the corresponding comprehensive constraint set; $\mathsf{\Xi }$ denotes the PI-parameter search domain; ${\mathrm{\Omega }}_{sta}\left(\mathrm{SCR};m_p,m_q\right)$ denotes the stability-constraint set determined by the D-partition criterion; and ${\mathrm{\Omega }}_{per}\left(\mathrm{SCR};m_p,m_q\right)$ denotes the dynamic-performance constraint set.

3.4.2. Fusion Advantage Region and Area Gain

To quantify the gain of the fusion strategy relative to the single-mode schemes, a fusion-advantage-region metric is introduced. The baseline feasible region for the single-mode schemes is defined as follows:

$${\mathrm{\Omega }}_{base}(\mathrm{SCR};m_p,m_q)={\mathrm{\Omega }}_{GFL}(\mathrm{SCR};m_p,m_q)\cup {\mathrm{\Omega }}_{GFM}(\mathrm{SCR};m_p,m_q)$$

(26)

where ${\mathrm{\Omega }}_{base}\left(\mathrm{SCR};m_p,m_q\right)$ denotes the combined feasible region formed by the two endpoint modes, i.e., single GFL and single GFM.

Both ${\mathrm{\Omega }}_{base}\left(\mathrm{SCR};m_p,m_q\right)$ and ${\mathrm{\Omega }}_{Fusion}\left(\mathrm{SCR};m_p,m_q\right)$ are obtained as the intersection of the stability-constraint set ${\mathrm{\Omega }}_{sta}\left(\mathrm{SCR};m_p,m_q\right)$ and the performance-constraint set ${\mathrm{\Omega }}_{per}\left(\mathrm{SCR};m_p,m_q\right)$. Accordingly, the fusion advantage region of the fusion strategy relative to the endpoint modes, $\Delta \mathrm{\Omega }\left(\mathrm{SCR}\right)$, is defined as

$$\Delta \mathrm{\Omega }(\mathrm{SCR})={\displaystyle \frac{{\mathrm{\Omega }}_{Fusion}(\mathrm{SCR};m_p,m_q)}{{\mathrm{\Omega }}_{base}(\mathrm{SCR};m_p,m_q)}}$$

(27)

The corresponding area gain ratio ${\eta }_{adv}(SCR)$ [29] is defined as follows:

$${\eta }_{adv}(\mathrm{SCR})={\displaystyle \frac{\mathrm{Area}\left(\Delta \mathrm{\Omega }(\mathrm{SCR})\right)}{\mathrm{Area}\left({\mathrm{\Omega }}_{base}(\mathrm{SCR};m_p,m_q)\right)}}\times 100\%$$

(28)

To provide an intuitive representation of the change in feasible solutions under different SCR conditions, the number of feasible points satisfying both stability and performance constraints, $N\left(\mathrm{S}\mathrm{C}\mathrm{R}\right)$, is counted. Figure 8 compares $N\left(\mathrm{S}\mathrm{C}\mathrm{R}\right)$ for the three control schemes. It can be observed that the feasible-point counts of the single GFL and single GFM schemes decrease markedly as SCR varies, whereas the fusion scheme maintains a higher and smoother $N\left(\mathrm{S}\mathrm{C}\mathrm{R}\right)$ over the entire SCR scan range. This indicates that fusion control can continuously provide a larger stability–performance feasible set under grid-strength fluctuations, thereby significantly improving robustness and tunability in weak-grid conditions.

To further characterize the stability margin in the parameter space, a stability-region area maximization principle is proposed, and the stability-region area index under a given $(\lambda ,\mathrm{SCR})$ is defined as follows:

$$S_{area}(\lambda ,\mathrm{SCR})={\displaystyle {\iint }_{(K_p,K_i)\in \mathrm{\Omega }(\lambda ,\mathrm{SCR})}\mathrm{d}}{K}_p\mathrm{d}{K}_i$$

(29)

where $S_{\mathrm{area}}\left(\lambda ,\mathrm{SCR}\right)$ denotes the area of the feasible region in the $(K_p,K_i)$ plane for the given SCR and fusion ratio $\lambda$, and $\mathrm{d}{K}_p\mathrm{d}{K}_i$ is the differential area element.

Here, $\mathrm{\Omega }\left(\lambda ,\mathrm{SCR}\right)$ is jointly determined by the D-partition stability boundary and the performance constraints. The same performance constraints as those defined above are adopted here.

As shown in Figure 9, under the weak-grid condition of SCR = 1, the feasible-region area of the fusion control strategy is increased by approximately 11.5% relative to ${\mathrm{\Omega }}_{\mathrm{GFL}}\cup {\mathrm{\Omega }}_{\mathrm{GFM}}$, indicating that the fusion strategy enlarges the tunable parameter set under combined stability and performance constraints.

However, if the fusion ratio is selected as a fixed value or via simple piecewise switching, weight jumps may occur at partition boundaries when SCR fluctuates, thereby inducing transient shocks and reducing the stability margin. To address the inherent limitations of such discontinuous switching strategies, a D-zone-based fusion-weight curve construction method is developed for the full SCR range, such that the fusion weight varies continuously with grid strength, enabling adaptive disturbance-free transitions between GFM and GFL characteristics.

4. Adaptive Disturbance-Free Switching Control Strategy Based on the D-Partition Method

To satisfy both stability and dynamic-performance requirements under weak- and strong-grid conditions, the SCR axis is partitioned into four typical regions, and a nominal fusion ratio ${\lambda }_j(j=1,\dots ,4)$ is assigned to each region to reflect the required weighting between GFM and GFL control across grid strengths. The region boundaries and the corresponding piecewise specification are given later in this section. The allowable range of the fusion coefficient is specified as follows:

$$\begin{array}{l}\lambda (\mathrm{SCR})\in \left[{\lambda }_{\min },{\lambda }_{\max }\right]\\ {\lambda }_{\min }=0.05, {\lambda }_{\max }=0.95\end{array}$$

(30)

where $\lambda \left(\mathrm{SCR}\right)$ denotes the fusion ratio scheduled as a function of SCR, and ${\lambda }_{\min }$ and ${\lambda }_{\max }$ denote the lower and upper bounds, respectively.

If a piecewise-constant value (e.g., ${\lambda }_j$) is directly adopted, discontinuous jumps arise at region boundaries and may lead to control-weight chattering. To suppress boundary discontinuities and improve practical robustness, the piecewise curve should be made continuous and smoothed.

4.1. Adaptive Disturbance-Free Switching Strategy

A hyperbolic-tangent-based smoothing weight $\sigma$ is defined as

$$\sigma (\mathrm{SCR};{\mathrm{SCR}}_0,d)={\displaystyle \frac{1}{2}}\left[1+\tanh \left({\displaystyle \frac{\mathrm{SCR}-{\mathrm{SCR}}_0}{d}}\right)\right]$$

(31)

where $\sigma$ is the smoothing-weight function with $\sigma \in (0,1)$, $SC{R}_0$ is the center point of the boundary between adjacent regions, and $d>0$ is the smoothing factor that determines the transition width.

Taking the nominal values ${\lambda }_a$ and ${\lambda }_b$ of two adjacent regions as an example, a smoothing weight $\mu \left(\mathrm{SCR}\right)=\sigma \left(\mathrm{SCR};{\mathrm{SCR}}_0,d\right)$ is introduced, and the fusion ratio $\lambda \left(SCR\right)$ is continuousized near the boundary as follows:

$$\lambda (\mathrm{SCR})=(1-\mu (\mathrm{SCR})){\lambda }_a+\mu (\mathrm{SCR}){\lambda }_b$$

(32)

In this expression, ${\lambda }_a$ and ${\lambda }_b$ denote the preselected nominal fusion ratios for two adjacent SCR intervals, respectively.

Furthermore, the modulation voltage reference of the fusion controller can be expressed as follows:

$${\omega }_{ref}^{Fusion}(s,\mathrm{SCR})=\lambda (\mathrm{SCR}){\omega }_{ref}^{GFM}(s)+[1-\lambda (\mathrm{SCR})]{\omega }_{ref}^{GFL}(s)$$

(33)

where ${\omega }_{ref}^{Fusion}\left(s;\mathrm{SCR}\right)$ denotes the reference frequency produced by the fusion controller, and ${\omega }_{ref}^{GFM}\left(s\right)$ and ${\omega }_{ref}^{GFL}\left(s\right)$ denote the reference frequencies under the single GFM and single GFL modes, respectively.

The corresponding control block diagram is shown in Figure 10.

This recursive structure enables smooth transitions in the multi-dimensional parameter space, thereby achieving disturbance-free switching. For any SCR operating point, the first- and second-order derivatives of the control gain, ${\displaystyle \frac{dK}{d{S}_{cr}}}$ and ${\displaystyle \frac{d^{2}K}{d{S}_{cr}^2}}$, remain continuous, effectively suppressing gain discontinuities caused by hard switching and the resulting control non-smoothness.

4.2. D-Zone Fusion-Coefficient Curve

To achieve seamless interconnection among multiple regions, a cascaded recursive form is further constructed. The SCR range is partitioned into four typical regions with switching thresholds $S_{\mathrm{t}\mathrm{h},1}$, $S_{\mathrm{t}\mathrm{h},2}$, and $S_{\mathrm{t}\mathrm{h},3}$, and the corresponding smoothing factors are ${\tau }_1$, ${\tau }_2$, and ${\tau }_3$. The nominal fusion coefficients are denoted by ${\lambda }_j\left(j=1,\dots ,4\right)$, equivalently written as ${\lambda }_1$–${\lambda }_4$. The boundary weight at the $i$-th interface is defined as follows:

$${\sigma }_i(\mathrm{SCR})={\displaystyle \frac{1}{2}}\left[1+\tanh \left({\displaystyle \frac{\mathrm{SCR}-S_{th,i}}{{\tau }_i}}\right)\right],\hspace{1em}i=1,2,3$$

(34)

where ${\sigma }_i\left(\mathrm{SCR}\right)$ denotes the smoothing weight at the $i$-th region boundary, $S_{\mathrm{th},i}$ denotes the corresponding SCR threshold, and ${\tau }_i>0$ is the associated smoothing factor.

The nominal values for each region are specified as follows:

$${\lambda }_{base}(\mathrm{SCR})=\left\{\begin{array}{l}{\lambda }_1,\hspace{1em}\mathrm{SCR}<S_{th,1}\hfill \\ {\lambda }_2,\hspace{1em}{S}_{th,1}\le \mathrm{SCR}<S_{th,2}\hfill \\ {\lambda }_3,\hspace{1em}{S}_{th,2}\le \mathrm{SCR}<S_{th,3}\hfill \\ {\lambda }_4,\hspace{1em}\mathrm{SCR}\ge S_{th,3}\hfill \end{array}\right.$$

(35)

where ${\lambda }_{base}\left(\mathrm{SCR}\right)$ denotes the piecewise nominal fusion ratio, and ${\lambda }_1\sim {\lambda }_4$ denote the preselected nominal fusion ratios for the corresponding SCR intervals.

To mitigate the risk of high-frequency resonance caused by an excessively large gain $K$ in the strong-grid regime, the nominal value in the strong-grid region can be further modified using a linear constraint as follows:

$${\lambda }_4(\mathrm{SCR})=\max ({\lambda }_{\min },{\lambda }_{ref}-k_s(\mathrm{SCR}-{\mathrm{SCR}}_0))$$

(36)

Here, ${\lambda }_4\left(\mathrm{SCR}\right)$ denotes the baseline fusion proportion in the high-SCR region after correction via a linear constraint; ${\lambda }_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the reference fusion proportion for this region; and $k_s\ge 0$ is the slope coefficient of the linear constraint.

On this basis, a hierarchical smoothing and bumpless switching scheme was adopted to enable seamless inter-region coordination, thereby yielding the final fusion coefficient:

$${\lambda }^{(1)}(\mathrm{SCR})=(1-{\sigma }_1(\mathrm{SCR})){\lambda }_1(\mathrm{SCR})+{\sigma }_1(\mathrm{SCR}){\lambda }_2(\mathrm{SCR})$$

(37)

$${\lambda }^{(2)}(\mathrm{SCR})=(1-{\sigma }_2(\mathrm{SCR})){\lambda }^{(1)}(\mathrm{SCR})+{\sigma }_2(\mathrm{SCR}){\lambda }_3(\mathrm{SCR})$$

(38)

$${\lambda }_{final}(\mathrm{SCR})=(1-{\sigma }_3(\mathrm{SCR})){\lambda }^{(2)}(\mathrm{SCR})+{\sigma }_3(\mathrm{SCR}){\lambda }_4(\mathrm{SCR})$$

(39)

In Equation (37), ${\lambda }^{\left(1\right)}\left(\mathrm{SCR}\right)$ denotes the intermediate fusion proportion obtained by smoothly transitioning from ${\lambda }_1$ to ${\lambda }_2$ in the vicinity of the threshold $S_{\mathrm{th},1}$. In Equation (38), ${\lambda }^{\left(2\right)}\left(\mathrm{SCR}\right)$ denotes the intermediate fusion proportion after further smoothing near the threshold $S_{\mathrm{th},2}$. In Equation (39), ${\lambda }_{final}\left(\mathrm{SCR}\right)$ is the final grid-strength-adaptive smooth switching function over a wide SCR range. Equations (37)–(39) ensure that ${\lambda }_{final}\left(\mathrm{SCR}\right)$ varies continuously over the entire SCR range and effectively suppresses abrupt weight changes introduced at region boundaries by piecewise switching.

In practical implementations, real-time SCR estimation is inherently susceptible to measurement noise and delays. To guarantee robustness against such uncertainties, the raw estimated SCR is first processed through an LPF before evaluating Equation (34). Furthermore, the hyperbolic tangent functions physically act as buffers via the smoothing factors (${\tau }_i$). This dual mechanism ensures that even if the filtered SCR fluctuates around the thresholds, the final fusion weight ${\lambda }_{final}$ updates continuously without inducing oscillatory mode-jumping.

A comparison of the active-power dynamic responses under different SCR conditions is presented in Figure 11 for the fixed-fusion and D-zone strategies. The results indicate that the D-zone response remains smoother as SCR transitions across regions, with no pronounced spikes or high-frequency oscillations, demonstrating that the bumpless transition of ${\lambda }_{final}\left(\mathrm{SCR}\right)$ effectively mitigates transient disturbances caused by abrupt weight variations.

To further investigate the multi-variable dynamic characteristics during grid-strength transitions, a comprehensive time-domain simulation was conducted, as shown in Figure 12. By evaluating the point of common coupling (PCC) voltage, grid current, active power, and reactive power, this analysis establishes a direct comparative baseline for the subsequent experimental verification.

As illustrated in Figure 12, conventional switching strategies induce severe transient voltage distortions and massive current spikes at the instants of grid-strength transitions. These simulated transients theoretically reveal the physical roots of the instability tendencies. In contrast, the proposed D-zone strategy maintains highly stable and smooth waveforms across all physical variables throughout the wide-range SCR reduction, demonstrating excellent transient suppression capabilities.

To validate the superiority of the proposed D-zone bumpless switching strategy, comparisons were conducted over the full SCR range against hysteresis switching [26], traditional piecewise switching [30], and hard switching [31]; the gain-evolution trajectories of the four strategies are shown in Figure 13.

As shown in Figure 13a, hard switching exhibits discontinuous jumps, traditional piecewise switching still tends to trigger chattering at region boundaries, and linear switching is continuous but struggles to simultaneously satisfy stability-margin and performance constraints in weak grids. In contrast, the D-zone strategy achieves bumpless transitions at partition boundaries while constraining $K$ in the strong-grid region. Figure 13b further shows that the D-zone strategy maintains a higher and smoother fusion advantage ratio over a wider SCR range, reaching 43.11%, which indicates that the benefits of fusion control can be released more consistently across a broad range of grid strengths.

To explicitly demonstrate the practical replicability of the proposed method, the complete execution framework is illustrated in Figure 14. The control system is structurally decoupled into an offline analysis phase and an online execution loop. The complex stability boundary mapping via the D-partition method is strictly executed offline to determine the switching thresholds alongside the optimal baseline parameters, specifically $K_p$, $K_i$, $m_p$, and $m_q$. Consequently, the real-time digital signal processor (DSP) execution is computationally highly efficient. This process requires only grid strength estimation, LPF for noise attenuation, and the arithmetic calculation of the continuous fusion weight ${\lambda }_{\mathrm{final}}$ to synthesize the final pulse-width modulation (PWM) commands.

5. Experimental Validation

5.1. RT-LAB Hardware-in-the-Loop (HIL) Real-Time Testing Platform and Parameters

The RT-LAB real-time HIL simulation platform (RT-LAB, Version 2020.2.2.82, OPAL-RT Technologies, Montréal, QC, Canada) was established in this study, as shown in Figure 15. Specifically, it should be clarified that this HIL system functions as a semi-physical experimental setup, which effectively bridges purely software-based simulations and full-scale physical experiments. The physical controller board was composed of a TMS320F28335 main control chip and associated peripheral circuits. RT-LAB was used to emulate the DC source, the converter, and an ideal AC grid with a precise simulation step size of 0.2 microseconds, while the host computer was used for code deployment and data acquisition, and a digital oscilloscope (SIGLENT Technologies Co., Ltd., Shenzhen, China) was used to capture actual hardware waveform data. Furthermore, regarding computational efficiency, the proposed strategy is fundamentally designed with an offline–online decoupled architecture. Because the computationally intensive stability boundary mapping is executed strictly offline, the online execution on the physical TMS320F28335 DSP involves only lightweight tasks. Since the online control loop strictly avoids any iterative algorithms or complex-domain matrix inversions, its computational footprint is inherently minimal.

The hardware specifications and controller parameters of the experimental platform are summarized in

Table 1. Experimental parameters.

Parameter	Value	Parameter	Value
Grid line-to-line RMS voltage, $V_g$	380 V	LPF cutoff angular frequency, ${\omega }_{\mathrm{L}\mathrm{P}\mathrm{F}}$	200 rad/s
Grid frequency, $f_g$	50 Hz	Inverter-side inductor, $L_1$	2.4 mH
Rated active power, $P_n$	30 kW	LCL filter capacitor, $C_f$	40 μF
Rated reactive power, $Q_n$	0 var	Grid-side inductor, $L_2$	1 mH
DC input voltage, $V_{dc}$	800 V	Grid-side inductor (baseline), $L_g$	5 mH
Switching frequency, $f_{sw}$	10 kHz	Grid-side resistor (baseline), $R_g$	48 mΩ

.

5.2. Stability-Region Verification Under Typical Operating Conditions

After the converter was connected to the grid, the rated active power was regulated and maintained at 30 kW. Stability tests were then conducted under different SCRs for the piecewise switching strategy, the hysteresis switching strategy, and the proposed D-zone bumpless switching strategy. For each strategy, the grid current, the voltage at the PCC, and the active/reactive power waveforms were recorded.

The experimental waveforms obtained with the piecewise function scheduling strategy at SCR = 20, 5, and 2 are presented in Figure 16, including the PCC voltage and grid current. The experiment was conducted as follows: The system was first operated with a 0.25 mH series inductor (SCR = 20), under which stable operation was maintained. The 0.25 mH inductor was then removed and replaced with a 1 mH inductor (SCR = 5), and the grid strength was further reduced to SCR = 2. As shown in Figure 14, when SCR was reduced from 20 to 5 and then to 2, pronounced transient disturbances were introduced at the region boundaries by piecewise switching, and the voltage and current oscillations were intensified, indicating a tendency toward instability. These observations indicate that the piecewise switching strategy provides an insufficient stability margin under weak-grid conditions.

The experimental waveforms obtained with the hysteresis switching strategy at SCR = 20, 8, and 2 are presented in Figure 17, including the PCC voltage, grid current, and a zoomed-in view of the switching interval. The system was first operated with a 0.25 mH series inductor (SCR = 20), where stable operation was preserved. The equivalent grid inductance was then increased to reduce SCR to 8 and subsequently to 2. As shown in Figure 17, when SCR entered the hysteresis band near the threshold, the control mode repeatedly toggled across the boundary, causing pronounced transient disturbances at the switching instants and exacerbating voltage and current oscillations. Under the weaker-grid condition of SCR = 2, the oscillation amplitude was further amplified, and an unstable tendency was observed. These results indicate that hysteresis switching tends to introduce weight chattering and transient disturbances under SCR fluctuations.

Figure 18 illustrates the experimental waveforms of the D-zone strategy under different SCR conditions, and the experimental procedure is described below. In the test setup, the inductor values corresponding to different SCRs were 0.167 mH (SCR = 30), 0.25 mH (SCR = 20), 0.5 mH (SCR = 10), 1 mH (SCR = 5), and 5 mH (SCR = 1). In stark contrast to the severe weight chattering and transient voltage spikes observed in Figure 16 and Figure 17 under the hysteresis method, the proposed D-zone strategy leverages continuous hyperbolic tangent functions to ensure a smooth and monotonic evolution of the fusion weight. By adopting this continuous parameter adaptation mechanism, the fusion ratio adapts seamlessly to grid strength variations. This not only effectively suppresses chattering at the partition boundaries but also guarantees bumpless transitions, thereby effectively mitigating any transient shocks or electrical stresses on the converter hardware. Consequently, highly stable voltage, current, and power are maintained across the entire wide-range operating spectrum. Even at severely low SCRs, the proposed strategy continuously optimizes the stability margin to yield high-quality sinusoidal current injection, which is perfectly consistent with the theoretical stability analysis.

6. Conclusions

In this paper, a D-zone-based smooth scheduling mechanism is proposed for adaptive GFL/GFM fusion control with bumpless switching. Based on D-partitioning stability-domain analysis in the parameter space, robust parameter tuning and smooth, bumpless transitions of the converter were achieved under wide-range grid-strength variations. The main conclusions are summarized as follows:

(1)

By applying the D-partition method to quantitatively characterize the parameter space, the expanded fusion-advantage region and the tuning gain achieved by fusion control relative to a single mode were explicitly revealed. Specifically, under severe weak-grid conditions with an SCR of 1, the fusion strategy enlarges the parameter stability feasible region by approximately 11.5% compared to single-mode operation. Theoretical analysis further indicates that fusion control provides a wider, robust stability boundary under cross-region grid conditions.

(2)

The proposed hierarchical, adaptive D-zone bumpless switching strategy alleviates switching transients under wide-SCR conditions. Quantitative evaluation demonstrates that the proposed D-zone strategy achieves a peak fusion advantage ratio of 43.11%. By constructing a nonlinear and continuous mapping between the fusion weights and grid strength, step-like discontinuities during SCR variations were eliminated, thereby reducing the risk of transient oscillations and mitigating electrical stresses on power semiconductor devices to enhance converter reliability.

(3)

The simulation results agree well with the RT-LAB HIL experimental results, indicating that the proposed method can effectively suppress transient oscillations and achieve smooth, bumpless transitions from weak to strong grids over an SCR range of 1 to 30.

It should be noted that the present study primarily focuses on the small-signal stability and control design of a single grid-connected inverter under grid impedance variations. Future work will extend the proposed strategy to multi-inverter parallel systems to investigate their interaction mechanisms. Additionally, evaluating the large-signal transient stability and fault ride-through (FRT) capability of the fusion strategy under severe grid faults (e.g., short circuits) remains a key focus for our subsequent research.