Large-Signal Stability Enhancement for FIS: Criterion-Based Parameter Optimization and Power Differentiation Feedforward Control

Abstract

Flexible interconnection systems (FISs) improve distribution flexibility, yet they remain vulnerable to pronounced nonlinear instability and potentially severe DC-link voltage collapse during large disturbances such as constant power load (CPL) surges. Conventional linear control methods are often unable to prevent deep transient voltage dips under these conditions. To address this issue, this paper proposes a novel large-signal stability criterion based on mixed potential function (MPF) theory. Unlike conventional Lyapunov-based approaches, the proposed formulation explicitly incorporates the dynamics of the DC capacitor, thereby enabling the derivation of a closed-form stability boundary. On this basis, the proportional gains of the outer voltage loop are first optimized to guarantee an adequate static stability margin. Subsequently, a power differentiation feedforward control strategy is developed. Rather than passively counteracting transients, the proposed method dynamically adjusts the DC voltage reference according to the rate of change in power, thereby actively reshaping the transient trajectory. In this way, the simple PI control framework is preserved while avoiding the heavy computational burden associated with advanced methods such as model predictive control. Simulation results show that the proposed strategy increases the permissible CPL step power by 8.7%, from 92 kW to 100 kW. Moreover, under severe load surges and weak grid conditions, the method prevents voltage collapse and maintains the transient trajectory above the practical 600 V safe-operation threshold. This computationally efficient strategy significantly improves the robustness and continuity of operation of practical FISs.

1. Introduction

With the ongoing global energy transition and the increasing penetration of renewable energy sources such as wind and solar power, modern power systems are experiencing pronounced stochastic fluctuations on the load side and a continuously rising share of power electronic interfaces [1,2]. Flexible interconnection systems (FISs), which employ power electronic converters to realize flexible interconnection and dynamic energy dispatch among multiple regional grids, have, therefore, become key equipment for enhancing distribution network flexibility and renewable energy accommodation [3,4]. However, under large disturbances such as sudden load surges, FISs are prone to instability and even voltage collapse. Traditional small-signal stability analysis, which relies on linearized models, is inherently limited by local linearization and, therefore, cannot accurately capture the nonlinear dynamic behavior or instability mechanisms of the system under large disturbances [5,6]. Consequently, rigorous large-signal stability analysis of FISs, together with corresponding parameter optimization and stability-oriented control design, is essential for ensuring the secure and stable operation of modern power systems.

Unlike small-signal analysis, large-signal stability analysis must explicitly account for the strong nonlinearity of the system. Existing large-signal stability analysis methods for power-electron-dominated systems mainly include the Lyapunov direct method, the Takagi–Sugeno (T-S) fuzzy model approach, and mixed potential function (MPF) theory [7,8,9]. The Lyapunov direct method is theoretically rigorous; however, constructing an appropriate Lyapunov function often depends heavily on experience, and it is difficult to generalize to high-order complex systems [10]. Although T-S fuzzy model-based methods can effectively approximate nonlinear systems, their computational burden increases rapidly with model order because of the “curse of dimensionality” [11]. By contrast, the MPF framework provides a systematic, topology-oriented approach for constructing the system energy function by explicitly combining the stored energy of dynamic components with the dissipative content of resistive elements. Because the resulting MPF is closely aligned with the physical characteristics of the circuit, it can generally provide a less conservative and more physically interpretable stability criterion. Rather than relying on the geometrically complex estimation of the region of attraction required by many classical Lyapunov approaches, MPF theory can yield explicit algebraic parameter boundaries. This feature is particularly attractive for controller parameter design because it offers a tighter and more practical characterization of the safe operating region [12,13,14].

Building on these methodological advantages, many studies have applied MPF theory to specific power electronic systems. In [15], the voltage source converter (VSC) was simplified as a DC circuit with a controlled current source, and an MPF-based large-signal stability criterion was derived. In [16], the load converter was modeled as a constant power load (CPL), the MPF of the cascaded system was established, and the influence of control parameters on stability was analyzed. To improve model fidelity, [17] further represented the load side as an equivalent controlled source. In [18], the cascaded system formulation was combined with the VSC model to obtain a stability criterion for VSC-HVDC systems, whereas [19] derived the MPF for modular multilevel converters (MMCs) and introduced a conservative correction factor to reduce over-conservatism. Nevertheless, existing criteria are often derived from overly simplified models, which would compromise their accuracy and limit their ability to reveal stability relationships over a broad parameter range.

With regard to large-signal stability enhancement, existing studies mainly focus on advanced nonlinear control algorithms, hardware modification, and refinements of conventional control schemes. Advanced nonlinear methods—such as model predictive control (MPC), sliding mode control (SMC) [20], and backstepping [21]—can deliver excellent dynamic performance during severe transients. However, their large-scale industrial deployment is still constrained by the need for accurate mathematical models and substantial real-time computational resources. Alternatively, adding auxiliary hardware, such as active capacitors [22], can significantly improve stability but at the expense of increased system volume and cost. Improvements within the classical control framework also face strict physical limitations. For example, transient path optimization [23] has difficulty coping with stochastic disturbances in real time, while virtual capacitor strategies [24] and complex feedforward loops [25] are fundamentally limited by the converter overcurrent margin. Likewise, improved droop control methods [26,27] depend strongly on the availability of sufficient power reserve at neighboring terminals in networked systems.

Therefore, bridging theoretical stability analysis and practical engineering implementation without relying on computationally intensive algorithms or costly additional hardware remains an important challenge. Translating stability criteria into actionable controller design is especially important in this context. Parameter optimization based on the MPF stability boundary offers a systematic route for selecting proportional–integral (PI) gains that improve system robustness. In addition, by introducing a power differentiation feedforward control scheme that dynamically reshapes the DC voltage reference according to the measured rate of change in power, the active disturbance rejection capability of the system can be further strengthened.

Accordingly, this paper proposes an economical, practical, and margin-independent stability enhancement strategy for FISs under severe large-signal disturbances. The main contributions and novelties of this study are summarized as follows:

• A comprehensive nonlinear model of the FIS is established, and an improved MPF-based stability criterion is derived by explicitly incorporating the dynamic characteristics of the DC-side capacitor. The resulting criterion provides a more accurate stability margin than conventional MPF formulations.
• A systematic parameter optimization method is developed to connect the abstract stability criterion with practical PI controller tuning, thereby improving large-signal stability without additional hardware cost.
• A novel power differentiation feedforward control strategy is proposed. By sensing the power rate of change in real time, the strategy dynamically reshapes the voltage reference during transient events, effectively suppressing voltage fluctuations and enhancing large-signal stability under severe grid and load disturbances.
• Extensive simulations verify the effectiveness of the proposed strategy and demonstrate superior performance over conventional methods in both stability range and transient recovery.

The remainder of this paper is organized as follows. Section 2 describes the system topology and establishes the basic mathematical model of the two-terminal flexible interconnection system. Section 3 constructs the mixed potential function, derives the large-signal stability criterion, and presents the parameter optimization method. Section 4 details the proposed power differentiation feedforward control strategy. Section 5 presents comprehensive simulation results to validate the effectiveness of the proposed method. Section 6 discusses the applicability of the proposed strategy to multi-terminal systems with droop control. Finally, Section 7 concludes the paper.

2. System Description and Mathematical Modeling

The topology of a typical two-terminal VSC-based flexible interconnection system (FIS) used in distribution networks is shown in Figure 1. The FIS consists of two voltage source converters (VSCs) sharing a common DC-link capacitor. The rectifier is located at the sending end and adopts constant DC voltage control to regulate the DC bus voltage, whereas the inverter is located at the receiving end and operates under constant power control to inject power into the AC grid.

In Figure 1, the main variables and parameters are defined as follows: u_sabc denotes the grid voltage on the rectifier side, and i_abc represents the rectifier grid current; L_r and R_r are the equivalent inductance and resistance of the AC-side line, respectively; L_s and R_s denote the filter inductance and its equivalent series resistance, respectively; P_rec denotes the active power absorbed by the rectifier from the grid; u_abc is the AC-side voltage of the rectifier; u_dc is the DC bus voltage; and C_dc is the DC-link capacitance. In addition, P_inv and u_{sabc_inv} denote the inverter output power and the inverter-side grid voltage, respectively.

According to Figure 1, the Kirchhoff voltage law (KVL) for the AC side of the FIS can be written as

$${\mathit{u}}_{\mathit{s}\mathit{a}\mathit{b}\mathit{c}}-{\mathit{u}}_{\mathit{a}\mathit{b}\mathit{c}}=L_f{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathit{a}\mathit{b}\mathit{c}}}{\mathrm{d}t}}+R_f{\mathit{i}}_{\mathit{a}\mathit{b}\mathit{c}}$$

(1)

where L_f = L_r + L_s and R_f = R_r + R_s are the lumped equivalent inductance and resistance on the AC side, respectively. By applying the amplitude-invariant transformation to the variables in (1), the model can be transformed into the synchronous dq0 rotating reference frame. In this paper, the d-axis is aligned with the grid voltage vector, and the d-axis leads the q-axis by 90°. Accordingly, u_sd ≈ U_m and u_sq ≈ 0, and the mathematical model in the dq frame can be obtained as

$$\left\{\begin{array}{l}{u}_{sd}-u_d=L_f{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}+R_f{i}_d-\omega L_f{i}_q\hfill \\ u_{sq}-u_q=L_f{\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}+R_f{i}_q+\omega L_f{i}_d\hfill \end{array}\right.$$

(2)

where u_sd, u_d, and i_d denote the d-axis components of u_sabc, u_abc, and i_abc, respectively, whereas u_sq, u_q, and i_q denote the corresponding q-axis components. ω represents the grid angular frequency.

The active power input to the rectifier is given by

$$P_{rec}={\displaystyle \frac{3}{2}}\left(u_d{i}_d+u_q{i}_q\right)$$

(3)

The control block diagram of the rectifier in the FIS is shown in Figure 2. The rectifier adopts constant DC voltage control to support the DC bus voltage. In Figure 2, u_dcref denotes the DC voltage reference, and u_dc is the measured DC bus voltage; i_dref and i_qref are the reference values of the AC-side current components i_d and i_q, respectively; u_kds and u_kqs represent the reference values of the converter output voltage components u_d and u_q, respectively.

As shown in Figure 2, the outer voltage loop regulates the DC voltage to its nominal value and generates the d-axis current reference for the inner loop. The corresponding control law is expressed as

$$\left\{\begin{array}{l}{i}_{dref}=k_{pv}\left(u_{dcref}-u_{dc}\right)+k_{iv}{\displaystyle \int \left(u_{dcref}-u_{dc}\right)}\mathrm{d}t\hfill \\ i_{qref}=0\hfill \end{array}\right.$$

(4)

Similarly, the current inner-loop controller can be expressed as

$$\left\{\begin{array}{l}{u}_{kds}=u_{sd}-k_{pi}\left(i_{dref}-i_d\right)-k_{ii}{\displaystyle \int \left(i_{qref}-i_q\right)}\mathrm{d}t+\omega L_f{i}_q\hfill \\ u_{kqs}=u_{sq}-k_{pi}\left(i_{qref}-i_q\right)-k_{ii}{\displaystyle \int \left(i_{qref}-i_q\right)}\mathrm{d}t-\omega L_f{i}_d\hfill \end{array}\right.$$

(5)

where u_kds and u_kqs are then processed by PWM modulation to generate the control signals for the rectifier.

3. Large-Signal Stability Analysis Based on MPF

3.1. Mixed Potential Function Theory

The mixed potential function (MPF) theory, proposed by R.K. Brayton and J.K. Moser in 1964, can be regarded as a generalized extension of the Lyapunov function method. It has been widely applied to the large-signal stability analysis of AC/DC systems containing constant power loads (CPLs). Brayton and Moser showed that, for nonlinear RLC circuits with different topologies, although the corresponding characteristic equations may vary, they can be represented in a unified mathematical form, expressed as

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial P(\mathit{i},\mathit{v})}{\partial \mathit{i}}}=\mathit{L}{\displaystyle \frac{\mathrm{d}\mathit{i}}{\mathrm{d}t}}\hfill \\ {\displaystyle \frac{\partial P(\mathit{i},\mathit{v})}{\partial \mathit{v}}}=-\mathit{C}{\displaystyle \frac{\mathrm{d}\mathit{v}}{\mathrm{d}t}}\hfill \end{array}\right.$$

(6)

where P denotes the mixed potential function (MPF) of the circuit; L represents the inductance in the circuit, and i is the corresponding inductor current; C denotes the capacitance, and v is the corresponding capacitor voltage. For a circuit with known parameters, the MPF can be employed to construct a Lyapunov-type energy function for assessing circuit stability. The MPF consists of a voltage potential function and a current potential function, and its standard form is given by

$$P(\mathit{i},\mathit{v})=-A(\mathit{i})+B(\mathit{v})+(\mathit{i},\mathit{\gamma }\mathit{v}-\mathit{\alpha })$$

(7)

where A(i) is the current potential function of the non-energy storage elements; B(v) is the voltage potential function of the non-energy storage elements; and the term (i,γv-α) is determined by the circuit topology. Once the MPF has been established, (6) can be used to verify its correctness. The construction procedure of the MPF can be summarized as follows:

• Determine the potential functions of the non-energy storage elements in the circuit;
• Calculate the energy associated with the capacitor elements;
• Sum the values obtained in the previous two steps and obtain a standard form.

Based on the MPF model, the theory provides a theorem for large-signal stability analysis. Let A_ii(i) denote the second-order partial derivative of the current potential function with respect to current and let B_vv(v) denote the second-order partial derivative of the voltage potential function with respect to voltage. Furthermore, let μ₁ be the minimum eigenvalue of L^−1/2A_ii(i)L^−1/2 and let μ₂ be the minimum eigenvalue of C^−1/2B_vv(v)C^−1/2. If the condition

$${\mu }_1+{\mu }_2⩾\delta \hspace{1em}\delta >0$$

(8)

is satisfied and, simultaneously, the auxiliary function P*(i,v) satisfies the condition

$$P^{*}(i,v)={\displaystyle \frac{{\mu }_1-{\mu }_2}{2}}P(i,v)+{\displaystyle \frac{1}{2}}\left(P_i,L^{-1}{P}_i\right)+{\displaystyle \frac{1}{2}}\left(P_v,C^{-1}{P}_v\right)\to \infty$$

(9)

as the state variables approach infinity, where P_i and P_v are the partial derivatives of the MPF P(i,v) with respect to current and voltage, respectively, then all system trajectories converge to the steady-state operating point, thereby ensuring stable operation of the system.

3.2. Mixed Potential Function of the Flexible Interconnection System

Based on the flexible interconnection system shown in Figure 1, a nonlinear average model is established, as shown in Figure 3. Since the inverter adopts constant-power control and its dynamic response is typically much faster than that of the rectifier, the inverter side can be simplified as a constant power load (CPL). Although practical inverters may reach current limits or experience control saturation during severe transients, thereby deviating from ideal CPL behavior, modeling the inverter as an ideal CPL represents the worst-case operating condition associated with the strongest negative impedance effect. Therefore, the stability boundary obtained under this assumption is inherently conservative and robust [19]. In Figure 3, i_o represents the DC output current, i_C denotes the capacitor current, and P_CPL is the inverter-side load power.

In Figure 3, i_C denotes the current flowing through the DC-link capacitor C_dc. According to Figure 3, the potential function P_dc of the non-energy storage elements on the DC side of the FIS can be written as

$$\begin{array}{l}{P}_{dc}=u_{sd}{i}_d-{\displaystyle \frac{1}{2}}{R}_f{i}_d^2-{\displaystyle {\int }_{\Gamma }{u}_d}\mathrm{d}{i}_d+{\displaystyle {\int }_{\Gamma }\omega L_f{i}_q}\mathrm{d}{i}_d+u_{sq}{i}_q-{\displaystyle \frac{1}{2}}{R}_f{i}_q^2\\ -{\displaystyle {\int }_{\Gamma }{u}_q}\mathrm{d}{i}_q-{\displaystyle {\int }_{\Gamma }\omega L_f{i}_d}\mathrm{d}{i}_q+{\displaystyle {\int }_{\Gamma }{u}_{dc}}\mathrm{d}{i}_o-{\displaystyle {\int }_{\Gamma }{u}_{dc}}\mathrm{d}{\displaystyle \frac{P_{CPL}}{u_{dc}}}\end{array}$$

(10)

where Γ denotes the integration path. This path starts from the equilibrium point and extends from the equilibrium DC voltage u_dce to the state variable u_dc [28]. The term P_C associated with the capacitor in the circuit is expressed as

$$P_C=-i_o{u}_{dc}+P_{CPL}$$

(11)

By combining (10) and (11), the MPF P(i,v) of the DC side of the system is obtained as

$$\begin{array}{l}P(i,v)=u_{sd}{i}_d-{\displaystyle \frac{1}{2}}{R}_f{i}_d^2-{\displaystyle {\int }_{\Gamma }{u}_d}\mathrm{d}{i}_d+{\displaystyle {\int }_{\Gamma }\omega L_f{i}_q}\mathrm{d}{i}_d+u_{sq}{i}_q-{\displaystyle \frac{1}{2}}{R}_f{i}_q^2\\ -{\displaystyle {\int }_{\Gamma }{u}_q}\mathrm{d}{i}_q-{\displaystyle {\int }_{\Gamma }\omega L_f{i}_d}\mathrm{d}{i}_q+{\displaystyle {\int }_{\Gamma }\frac{P_{CPL}}{u_{dc}}}\mathrm{d}{u}_{dc}-{\displaystyle {\int }_{\Gamma }{i}_o}d{u}_{dc}\end{array}$$

(12)

Taking the partial derivative of the MPF in (12) with respect to voltage yields

$$\left\{\begin{array}{l}\begin{array}{l}{\displaystyle \frac{\partial P(i,v)}{\partial i_d}}=u_{sd}-R_f{i}_d+\omega L_f{i}_q-u_d=L_f{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}\\ {\displaystyle \frac{\partial P(i,v)}{\partial i_d}}=u_{sq}-R_f{i}_q+\omega L_f{i}_d-u_q=L_f{\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}\\ {\displaystyle \frac{\partial P(i,v)}{\partial u_{dc}}}={\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o=-C_{dc}{\displaystyle \frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}}\end{array}\hfill \end{array}\right.$$

(13)

It is evident that the derived potential function satisfies the condition in (6). Therefore, (12) represents the MPF of the system shown in Figure 1. Expressed in the standard form of the MPF, it yields

$$\left\{\begin{array}{l}\begin{array}{l}A(i)=-u_{sd}{i}_d+{\displaystyle \frac{1}{2}}{R}_f{i}_d^2+{\displaystyle {\int }_{\Gamma }{u}_d}\mathrm{d}{i}_d-{\displaystyle {\int }_{\Gamma }\omega L_f{i}_q}\mathrm{d}{i}_q\\ -u_{sq}{i}_q+{\displaystyle \frac{1}{2}}{R}_f{i}_q^2+{\displaystyle {\int }_{\Gamma }{u}_q}{i}_q+{\displaystyle {\int }_{\Gamma }\omega L_f{i}_d}\mathrm{d}{i}_q\\ B(u_{dc})=-{\displaystyle {\int }_{\Gamma }{i}_o}d{u}_{dc}+{\displaystyle {\int }_{\Gamma }\frac{P_{CPL}}{u_{dc}}}d{u}_{dc}\end{array}\hfill \end{array}\right.$$

(14)

Taking the second-order partial derivative of the voltage potential function in (14) yields

$$\left\{\begin{array}{l}\begin{array}{l}{A}_{ii}(i)=\left[\begin{array}{cc}{R}_f+{\displaystyle \frac{\partial u_d}{\partial i_d}}& 0\\ 0& R_f+{\displaystyle \frac{\partial u_q}{\partial i_q}}\end{array}\right]\\ B_{vv}(u_{dc})=-{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}-{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\end{array}\hfill \end{array}\right.$$

(15)

3.3. Derivation of the Stability Criterion Based on MPF

Since the response speed of the current loop is significantly faster than that of the voltage loop, the dynamics of the inner current loop can be neglected, provided that the PI parameters are properly tuned. It is, therefore, assumed that the inner current loop tracks the reference generated by the outer voltage loop with negligible delay. Hence,

$$\left\{\begin{array}{l}\begin{array}{l}{i}_d\approx i_{dref}\\ i_q\approx 0\end{array}\hfill \end{array}\right.$$

(16)

According to (15), evaluating the second-order partial derivative of the voltage potential function requires determining the partial derivative of the DC current i_o with respect to the DC voltage udc:

$${\displaystyle \frac{\partial i_o}{\partial u_{dc}}}={\displaystyle \frac{1}{u_{dc}}}{\displaystyle \frac{\partial \left(u_{dc}{i}_o\right)}{\partial u_{dc}}}-{\displaystyle \frac{i_o}{u_{dc}}}$$

(17)

Neglecting the internal losses of power electronic devices and invoking the principle of power conservation, we obtain

$$u_{dc}{i}_o={\displaystyle \frac{3}{2}}{u}_d{i}_d$$

(18)

Substituting the d-axis voltage equation in (2) into (18), the product u_dci_o can be expressed as

$$u_{dc}{i}_o={\displaystyle \frac{3}{2}}(u_{sd}{i}_d-R_f{i}_d^2-L_f{i}_d{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}})$$

(19)

Differentiating both sides of (19) with respect to u_dc yields

$${\displaystyle \frac{\partial \left(u_{dc}{i}_o\right)}{\partial u_{dc}}}={\displaystyle \frac{3}{2}}\left(u_{sd}{\displaystyle \frac{\partial i_d}{\partial u_{dc}}}-2R_f{i}_d{\displaystyle \frac{\partial i_d}{\partial u_{dc}}}-L_f{\displaystyle \frac{\partial i_d}{\partial u_{dc}}}{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}-L_f{i}_d{\displaystyle \frac{\partial \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}{\partial u_{dc}}}\right)$$

(20)

Substituting (20) into (17), we obtain

$${\displaystyle \frac{\partial i_o}{\partial u_{dc}}}={\displaystyle \frac{3}{2u_{dc}}}\left(u_{sd}{\displaystyle \frac{\partial i_d}{\partial u_{dc}}}-2R_f{i}_d{\displaystyle \frac{\partial i_d}{\partial u_{dc}}}-L_f{\displaystyle \frac{\partial i_d}{\partial u_{dc}}}{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}-L_f{i}_d{\displaystyle \frac{\partial \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}{\partial u_{dc}}}\right)-{\displaystyle \frac{i_o}{u_{dc}}}$$

(21)

According to (21), the partial derivative of i_o with respect to u_dc depends on ∂i_d/∂u_dc, di_d/dt, ∂(di_d/dt)/∂u_dc. Therefore, the explicit expression can be obtained by determining these three terms separately.

Based on the control law in (4), the time derivative of the current i_d is

$${\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}=-k_{pv}{\displaystyle \frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}}+k_{iv}\left(u_{dcref}-u_{dc}\right)$$

(22)

According to the constitutive relation of the DC-link capacitor, the capacitor current i_C can be written as

$$i_C=C_{dc}{\displaystyle \frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}}$$

(23)

Combining this relation with Kirchhoff’s current law (KCL), the time derivative of the DC voltage u_dc is obtained as

$${\displaystyle \frac{\mathrm{d}{u}_{dc}}{\mathrm{d}t}}={\displaystyle \frac{1}{C_{dc}}}\left(i_o-{\displaystyle \frac{P_{CPL}}{u_{dc}}}\right)$$

(24)

Substituting (24) into (22) yields

$${\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}={\displaystyle \frac{k_{pv}}{C_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right)+k_{iv}\left(u_{dcref}-u_{dc}\right)$$

(25)

Differentiating both sides of (25) with respect to u_dc, we obtain

$${\displaystyle \frac{\partial \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}{\partial u_{dc}}}=-{\displaystyle \frac{k_{pv}}{C_{dc}}}{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}-{\displaystyle \frac{k_{pv}}{C_{dc}}}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}-k_{iv}$$

(26)

Differentiating the control equation in (4) with respect to u_dc, the partial derivative of the d-axis current i_d with respect to u_dc is given by

$${\displaystyle \frac{\partial i_d}{\partial u_{dc}}}=-k_{pv}$$

(27)

Substituting (25), (26), and (27) into (21), the partial derivative of i_o with respect to u_dc is obtained as

$$\begin{array}{l}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}=-{\displaystyle \frac{3}{2u_{dc}}}\left(u_{sd}{k}_{pv}-2R_f{i}_d{k}_{pv}-L_f{k}_{pv}\left({\displaystyle \frac{k_{pv}}{C_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right)+k_{iv}\left(u_{dcref}-u_{dc}\right)\right)\right)\\ -{\displaystyle \frac{3}{2u_{dc}}}{L}_f{i}_d\left(-{\displaystyle \frac{k_{pv}}{C_{dc}}}{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}-{\displaystyle \frac{k_{pv}}{C_{dc}}}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}-k_{iv}\right)-{\displaystyle \frac{i_o}{u_{dc}}}\end{array}$$

(28)

Collecting like terms yields

$$\begin{array}{l}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}=-{\displaystyle \frac{3k_{pv}}{2u_{dc}}}\left(u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)\right)\\ +{\displaystyle \frac{3L_f{k}_{pv}^2}{2u_{dc}{C}_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right)+{\displaystyle \frac{3L_{\mathrm{f}`}{i}_{\mathrm{d}}{k}_{pv}}{2C_{dc}{u}_{dc}}}\left({\displaystyle \frac{\partial i_o}{\partial u_{dc}}}+{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)+{\displaystyle \frac{3L_f{i}_d{k}_{iv}}{2u_{dc}}}-{\displaystyle \frac{i_o}{u_{dc}}}\end{array}$$

(29)

Substituting (15) into (29), we obtain

$$\begin{array}{l}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}=-{\displaystyle \frac{3k_{pv}}{2u_{dc}}}\left(u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)\right)\\ +{\displaystyle \frac{3L_f{k}_{pv}^2}{2u_{dc}{C}_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right)-{\displaystyle \frac{3L_{\mathrm{f}`}{i}_{\mathrm{d}}{k}_{pv}}{2C_{dc}{u}_{dc}}}{B}_{vv}(u_{dc})+{\displaystyle \frac{3L_f{i}_d{k}_{iv}}{2u_{dc}}}-{\displaystyle \frac{i_o}{u_{dc}}}\end{array}$$

(30)

Substituting (30) back into (15) gives

$$\begin{array}{l}{B}_{vv}(u_{dc})={\displaystyle \frac{3k_{pv}}{2u_{dc}}}\left(u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)\right)\\ -{\displaystyle \frac{3L_f{k}_{pv}^2}{2u_{dc}{C}_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right)+{\displaystyle \frac{3L_{\mathrm{f}`}{i}_{\mathrm{d}}{k}_{pv}}{2C_{dc}{u}_{dc}}}{B}_{vv}(u_{dc})-{\displaystyle \frac{3L_f{i}_d{k}_{iv}}{2u_{dc}}}+{\displaystyle \frac{i_o}{u_{dc}}}-{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\end{array}$$

(31)

Rearranging the equation to isolate B_vv(u_dc) on the left-hand side yields

$$\begin{array}{l}\left(1-{\displaystyle \frac{3L_{\mathrm{f}`}{i}_{\mathrm{d}}{k}_{pv}}{2C_{dc}{u}_{dc}}}\right)B_{vv}(u_{dc})={\displaystyle \frac{3}{2u_{dc}}}{k}_{pv}\left(u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)\right)\\ -{\displaystyle \frac{3L_f{k}_{pv}^2}{2u_{dc}{C}_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right)-{\displaystyle \frac{3L_f{i}_d{k}_{iv}}{2u_{dc}}}+{\displaystyle \frac{i_o}{u_{dc}}}-{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\end{array}$$

(32)

Simplifying (32), the expression for B_vv(u_dc) is derived as

$$B_{vv}(u_{dc})={\displaystyle \frac{1}{1-M_1}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_1-\left(1+{\displaystyle \frac{3L_{\mathrm{f}}{k}_{pv}^2}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)$$

(33)

where M₁ and N₁ are intermediate variables introduced to simplify the mathematical expression, and they are defined as follows:

$$\left\{\begin{array}{l}\begin{array}{l}{M}_1={\displaystyle \frac{3L_{\mathrm{f}`}{i}_{\mathrm{d}}{k}_{pv}}{2C_{dc}{u}_{dc}}}\\ N_1=k_{pv}\left(u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)\right)+{\displaystyle \frac{L_f{k}_{pv}^2{i}_o}{C_{dc}}}-L_f{i}_d{k}_{iv}+{\displaystyle \frac{2i_o}{3}}\end{array}\hfill \end{array}\right.$$

(34)

By substituting (5) and (16) into (15), the second-order partial derivatives of the current potential function are obtained as

$$A_{ii}(i)=\left[\begin{array}{cc}{R}_f& 0\\ 0& R_f\end{array}\right]$$

(35)

In summary, the minimum eigenvalues μ₁ and μ₂ of L^−1/2A_ii(i)L^−1/2 and C^−1/2B_vv(v)C^−1/2, respectively, are obtained as follows:

$$\left\{\begin{array}{l}\begin{array}{l}{\mu }_1={\displaystyle \frac{R_f}{L_f}}\\ {\mu }_2={\displaystyle \frac{1}{C_{dc}\left(1-M_1\right)}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_1-\left(1+{\displaystyle \frac{3L_f{k}_{pv}^2}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)\end{array}\hfill \end{array}\right.$$

(36)

Based on (36), μ₁ + μ₂ is given by

$${\mu }_1+{\mu }_2={\displaystyle \frac{1}{C_{dc}\left(1-M_1\right)}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_1-\left(1+{\displaystyle \frac{3L_f{k}_{pv}^2}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)+{\displaystyle \frac{R_f}{L_f}}>0$$

(37)

According to the third theorem of MPF theory, imposing the condition μ₁ + μ₂ > 0 yields the following system stability criterion:

$${\displaystyle \frac{1}{C_{dc}\left(1-M_1\right)}}\left(1+{\displaystyle \frac{3L_f{k}_{pv}^2}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}<{\displaystyle \frac{1}{C_{dc}\left(1-M_1\right)}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_1\right)+{\displaystyle \frac{R_f}{L_f}}$$

(38)

3.4. Stability Analysis and Parameter Design

The stability criterion in (38), derived from MPF theory, determines the maximum constant power load (CPL) that can be connected to the DC bus of the flexible interconnection system (FIS), thereby providing a theoretical basis for parameter design. It should be noted that, although incorporating the fast dynamics of the inner current loop could yield a more precise stability boundary, stability analysis based on a simplified model that assumes time-scale separation and neglects these inner-loop dynamics generally produces conservative results [8]. However, this study mainly focuses on the operational stability of converter-based FISs under severe large-signal disturbances for grid restoration during extreme events. Therefore, the sufficient but not necessary stability condition derived from the simplified MPF model can strictly guarantee safe system operation. Moreover, the conservativeness introduced by neglecting the inner-loop dynamics provides an additional safety margin, thereby enhancing the operational resilience of the system under extreme conditions. A detailed analysis of criterion (38) is presented below.

The system parameters are listed in

Table 1. Main circuit parameters.

Parameters	Value
Nominal DC bus voltage udc	750 V
Grid voltage us	380 V
Initial load power PCPL1	50 kW
AC-side filter resistance Rs	0.025 Ω
AC-side filter inductance Ls	2.6 mH
AC-side line resistance Rr	0.01 Ω
AC-side line inductance Lr	1 mH
DC capacitance Cdc	1000 μF
Controller parameter kiv	200
Controller parameter kpv	0.7

. Let P_CPL1 and P_CPL2 denote two different constant power loads. The transient stability problem associated with a load step from P_CPL1 to P_CPL2 is then analyzed. When evaluating (38), the post-disturbance equilibrium point should be used as the integration reference. To assess system stability after the load step, P_CPL in criterion (38) is set to P_CPL2. According to the continuity of state variables, i.e., the switching theorem of capacitor voltage, inductor current and other state variables at the pre-disturbance equilibrium point are substituted into the calculation. By substituting the parameters in Table 1 into (37), the relationship among the stability index μ₁ + μ₂, the constant power load P_CPL, and the proportional gain k_pv of the outer voltage loop can be obtained, as shown in Figure 4.

In Figure 4, the intersection curve between the surface and the plane μ = 0 represents the large-signal stability boundary of the system. As can be seen, the effects of the constant power load P_CPL and the proportional gain k_pv on system stability are not linearly related. The stable operating region first expands and then contracts as k_pv increases.

Reference [29] presents a stability criterion for the flexible interconnection system. By setting the DC voltage derivative du_dc/dt = 0 and following the derivation procedure described above, the conventional stability criterion of the flexible interconnection system can be obtained as

$$P_{CPL}<{\displaystyle \frac{3u_{dc}}{2}}\left(u_{sd}{k}_{pv}+{\displaystyle \frac{2}{3}}{i}_o-2R_f{i}_d{k}_{pv}-L_f{i}_d{k}_{iv}\right)$$

(39)

According to (39), the proportional gain k_pv in the conventional criterion exhibits a linear relationship with the maximum load power P_CPL that the system can support. To compare the differences between criterion (38) and criterion (39) more intuitively, the influence of k_pv on system stability under the two criteria is shown in Figure 5. Here, k_pvmax denotes the extremum point of the curve corresponding to (38).

Figure 5 shows that, with the other parameters fixed, the value of μ₁ + μ₂ in the criterion (39) is positively correlated with k_pv; that is, a larger k_pv implies a higher maximum stable load power and stronger stability. By contrast, for the proposed criterion (38), when k_pv < k_pvmax, system stability improves as k_pv increases, whereas when k_pv > k_pvmax, system stability deteriorates as k_pv continues to increase.

Under large disturbances, greater changes in the operating condition lead to more pronounced fluctuations in the DC voltage u_dc during the transient process. Since criterion (39) neglects the derivative of the DC bus voltage, a significant discrepancy arises between (38) and (39) when P_CPL is large. As shown by the stable operating regions in Figure 5, a larger load power step causes larger fluctuations in u_dc and capacitor current i_C; therefore, leading to a more pronounced difference between the two criteria.

According to the relationship between k_pv and system stability shown in Figure 5, with the other parameters held constant, there exists an optimal value k_pvmax that maximizes the loadable power of the system. Based on stability criterion (38), k_pvmax satisfies

$$k_{pv\max }={\displaystyle \frac{L_f{i}_d{k}_{iv}+\sqrt{{\left(L_{f`}{i}_d{k}_{iv}\right)}^2+\frac{2C{\left(u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)\right)}^2}{3L_f}}}{u_{sd}-2R_f{i}_d-L_f{k}_{iv}\left(u_{dcref}-u_{dc}\right)}}$$

(40)

According to the criterion equation, when the denominator 1-A < 0, μ₁ + μ₂ cannot remain positive during the transition of the AC current i_d from i_d1 to i_d2 after the disturbance. Only when 1-A > 0 can the validity of the criterion be guaranteed throughout the entire transient process. Therefore, to ensure system stability, k_pv must satisfy

$$k_{pv}<{\displaystyle \frac{2C_{dc}{u}_{dc}}{3L_f{i}_d}}$$

(41)

Based on criterion (38), and by substituting the parameters in Table 1, the stable operating ranges of the system under different DC capacitance values C_dc are shown in Figure 6.

Figure 6 shows that system stability is positively correlated with DC capacitance C_dc. Under the same set of other parameters, the stable operating region expands as C_dc increases. Comparing the relationship between k_pv and stability for different capacitance values, a larger capacitance causes the stable operating boundary to approach the curve predicted by (39) in Figure 5 more closely. This indicates that only when the DC capacitance C_dc is sufficiently large does k_pv exhibit an approximately linear relationship with the maximum permissible load P_CPL. Under such a condition, criterion (38) becomes approximately equivalent to criterion (39).

To further examine the influence of key parameters on the large-signal stability boundary, a sensitivity analysis is carried out by substituting the system parameters into criterion (38). Figure 7 illustrates the stable operating boundaries of the system under different equivalent resistance values R_f.

As illustrated in Figure 7, as the equivalent series resistance $R$ increases from 0.035 Ω to 0.235 Ω, the maximum allowable CPL step range decreases, indicating a deterioration in large-signal stability. From a physical viewpoint, a larger series resistance causes a more severe transient voltage drop across the line and filters elements during load surges. Owing to the constant-power characteristic of the CPL, the reduced DC bus voltage forces the load to draw a larger transient current to maintain power balance. This positive feedback effect aggravates the negative-impedance characteristic of the CPL and consequently reduces the large-signal stability region of the system. In addition, Figure 7 shows that the overall influence of R_f on system stability is relatively limited. For example, at k_pv = 0.7, P_CPL decreases only from 92 kW to 87 kW. Therefore, although practical parasitic losses in power electronic devices are unavoidable, the inherent conservativeness of MPF theory provides a sufficient stability margin to justify neglecting these minor losses in the theoretical model.

Similarly, based on criterion (38), Figure 8 shows the influence of different AC-side equivalent inductance values L_f on the stable operating range of the system.

As illustrated in Figure 8, the stable operating boundary of the system shrinks significantly as the AC-side equivalent inductance L_f increases. From a physical perspective, a larger front-end line inductance severely limits the slew rate (di/dt) of the grid-side current. When the system is subjected to a severe CPL step disturbance, excessive line inductance prevents the front-end rectifier from drawing energy from the AC grid in time to support the DC bus voltage. In particular, under weak grid conditions with a short-circuit ratio (SCR) below 3, where the grid-side impedance is very large, the large-signal stability of the system faces the most severe challenge.

4. Converter Stability Control Strategy Based on Power Differentiation Feedforward

According to the stability criterion in (38), system stability is correlated with the DC voltage reference u_dcref. The stable operating range of the system under different values of u_dcref is shown in Figure 9.

As shown in Figure 9, with all other parameters held constant, a smaller DC voltage reference u_dcref corresponds to a larger maximum load power and stronger system stability. This indicates that moderately reducing u_dcref after a disturbance can effectively improve the large-signal stability of the system.

Although directly modifying the DC voltage reference u_dcref can improve system stability, load variations are often difficult to predict, making it challenging to adjust u_dcref in a timely manner after a disturbance. In addition, permanently changing the reference value may shift the operating point of the DC voltage away from its nominal range and potentially trigger undesired operating conditions or even faults. Therefore, this paper proposes a power differentiation feedforward control strategy that adjusts the voltage reference according to power variations. The corresponding control block diagram is shown in Figure 10.

As shown in Figure 10, the corresponding control law can be written as

$$i_{dref}=k_{pv}\left(u_{dcref}-u_{dc}-\lambda {\displaystyle \frac{\mathrm{d}{P}_{rec}}{\mathrm{d}t}}\right)+k_{iv}{\displaystyle \int \left(u_{dcref}-u_{dc}-\lambda \frac{\mathrm{d}{P}_{rec}}{\mathrm{d}t}\right)}\mathrm{d}t$$

(42)

where λ is the control coefficient to be designed. According to (42), the DC bus voltage is related to the rate of change in the rectifier power. When a load power step occurs, the rectifier output power changes rapidly toward a new operating point during the initial stage of the disturbance, resulting in a large derivative term. Consequently, the equivalent DC voltage reference decreases. During the later stage of the transient, as the system gradually approaches steady state and P_rec becomes constant, the DC voltage u_dc returns to and remains at its nominal value. Compared with directly modifying the DC voltage reference, the proposed strategy introduces power feedforward to achieve adaptive voltage regulation while preventing the DC voltage from permanently deviating from its nominal operating point.

Next, the influence of the proposed strategy on system stability is analyzed on the basis of mixed potential function (MPF) theory.

For the flexible interconnection system shown in Figure 1, when the power differentiation feedforward control strategy is adopted, the expressions for the voltage potential function B(u_dc) and the DC current i_o still satisfy (14) and (19), respectively. Therefore, in the derivation of the stability criterion, it is only necessary to calculate the three partial derivatives in (21) and substitute them back into the criterion. Differentiating (42) with respect to time t and voltage u_dc, respectively, yields

$$\left\{\begin{array}{l}\begin{array}{l}{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}={\displaystyle \frac{k_{pv}+\lambda k_{iv}{i}_o}{C_{dc}}}{\displaystyle \frac{P_{CPL}-u_{dc}{i}_o}{u_{dc}}}+{\displaystyle \frac{\lambda k_{iv}{P}_{CPL}-\lambda k_{iv}{u}_{dc}{i}_o}{C_{dc}}}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\hfill \end{array}\\ {\displaystyle \frac{\partial \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}{\partial u_{dc}}}=-{\displaystyle \frac{k_{pv}+\lambda k_{iv}{i}_o}{C_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}^2}}+{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\right)-k_{iv}+{\displaystyle \frac{\lambda k_{iv}}{C_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right){\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\\ {\displaystyle \frac{\partial i_d}{\partial u_{dc}}}=-k_{pv}-\lambda k_{iv}{i}_o\end{array}\right.$$

(43)

Following a derivation similar to that in Section 3.3, substituting (43) into (21) yields the partial derivative of the DC current with respect to voltage:

$$\begin{array}{l}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}=-{\displaystyle \frac{3}{2u_{dc}}}\left(u_{sd}-2R_f{i}_d\right)\left(k_{pv}+\lambda k_{iv}{i}_o\right)\\ +{\displaystyle \frac{3L_f}{2u_{dc}}}\left({\displaystyle \frac{\left(k_{pv}+\lambda k_{iv}{i}_o\right)\left(P_{CPL}-u_{dc}{i}_o\right)}{C_{dc}{u}_{dc}}}+{\displaystyle \frac{\lambda k_{iv}{P}_{CPL}-\lambda k_{iv}{u}_{dc}{i}_o}{C_{dc}}}{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\right)\left(k_{pv}+\lambda k_{iv}{i}_o\right)\\ -{\displaystyle \frac{3}{2u_{dc}}}{L}_f{i}_d\left(-{\displaystyle \frac{k_{pv}+\lambda k_{iv}{i}_o}{C_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}^2}}+{\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\right)-k_{iv}+{\displaystyle \frac{\lambda k_{iv}}{C_{dc}}}\left({\displaystyle \frac{P_{CPL}}{u_{dc}}}-i_o\right){\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\right)-{\displaystyle \frac{i_o}{u_{dc}}}\end{array}$$

(44)

Collecting like terms and isolating the current partial derivative term on the left-hand side yield

$$\begin{array}{l}\left(1+{\displaystyle \frac{3\lambda L_f{k}_{iv}}{2C_{dc}{u}_{dc}^2}}\left(P_{CPL}-u_{dc}{i}_o\right)\left(i_d-\left(k_{pv}+\lambda k_{iv}{i}_o\right)u_{dc}\right)\right){\displaystyle \frac{\partial i_o}{\partial u_{dc}}}\\ =-{\displaystyle \frac{3}{2u_{dc}}}\left(u_{sd}-2R_f{i}_d\right)\left(k_{pv}+\lambda k_{iv}{i}_o\right)+{\displaystyle \frac{3L_f{i}_d{k}_{iv}}{2u_{dc}}}-{\displaystyle \frac{i_o}{u_{dc}}}\\ +{\displaystyle \frac{3L_f}{2u_{dc}}}{\displaystyle \frac{{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2\left(P_{CPL}-u_{dc}{i}_o\right)}{C_{dc}{u}_{dc}}}-{\displaystyle \frac{3L_f{i}_d}{2u_{dc}}}{\displaystyle \frac{k_{pv}+\lambda k_{iv}{i}_o}{C_{dc}}}{B}_{vv}(u_{dc})\end{array}$$

(45)

After simplification, the partial derivative of the current can be obtained as

$${\displaystyle \frac{\partial i_o}{\partial u_{dc}}}=-{\displaystyle \frac{1}{1+\alpha }}\left({\displaystyle \frac{3}{2u_{dc}}}\left(N_2-{\displaystyle \frac{{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2{L}_f{P}_{CPL}}{C_{dc}{u}_{dc}}}\right)+M_2{B}_{vv}(u_{dc})\right)$$

(46)

where α, M₂, and N₂ are intermediate variables introduced to simplify the mathematical expression, and they are defined as follows:

$$\left\{\begin{array}{l}\begin{array}{l}\alpha ={\displaystyle \frac{3\lambda L_f{k}_{iv}}{2C_{dc}{u}_{dc}^2}}\left(P_{CPL}-u_{dc}{i}_o\right)\left(i_d-\left(k_{pv}+\lambda k_{iv}{i}_o\right)u_{dc}\right)\\ M_2={\displaystyle \frac{3L_f{i}_d}{2C_{dc}{u}_{dc}}}\left(k_{pv}+\lambda k_{iv}{i}_o\right)\\ N_2=\left(k_{pv}+\lambda k_{iv}{i}_o\right)\left(u_{sd}-2R_f{i}_d\right)-L_f{i}_d{k}_{iv}+{\displaystyle \frac{2i_o}{3}}+{\displaystyle \frac{{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2{L}_f{i}_o}{C_{dc}}}\end{array}\hfill \end{array}\right.$$

(47)

Substituting (46) into (16), the second-order partial derivative of the voltage potential function is obtained as

$$B_{vv}(u_{dc})={\displaystyle \frac{1}{1+\alpha }}\left({\displaystyle \frac{3}{2u_{dc}}}\left(N_2-{\displaystyle \frac{{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2{L}_f{P}_{CPL}}{C_{dc}{u}_{dc}}}\right)+M_2{B}_{vv}(u_{dc})\right)-{\displaystyle \frac{P_{CPL}}{u_{dc}^2}}$$

(48)

Collecting like terms yields

$$B_{vv}(u_{dc})={\displaystyle \frac{1}{1+\alpha -M_2}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_2-\left(1+\alpha +{\displaystyle \frac{3{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2{L}_f}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)$$

(49)

Based on (49), the minimum eigenvalues μ₁ and μ₂ of L^−1/2A_ii(i)L^−1/2 and C^−1/2B_vv(v)C^−1/2, respectively, can be calculated as follows:

$$\left\{\begin{array}{l}\begin{array}{l}{\mu }_1={\displaystyle \frac{R_f}{L_f}}\\ {\mu }_2={\displaystyle \frac{1}{C_{dc}\left(1+\alpha -M_2\right)}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_2-\left(1+\alpha +{\displaystyle \frac{3{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2{L}_f}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)\end{array}\hfill \end{array}\right.$$

(50)

Accordingly, μ₁ + μ₂ is given by

$$\begin{array}{l}{\mu }_1+{\mu }_2={\displaystyle \frac{1}{C_{dc}\left(1+\alpha -M_2\right)}}\left({\displaystyle \frac{3}{2u_{dc}}}{N}_2-\left(1+\alpha +{\displaystyle \frac{3{\left(k_{pv}+\lambda k_{iv}{i}_o\right)}^2{L}_f}{2C_{dc}}}\right){\displaystyle \frac{P_{CPL}}{u_{dc}^2}}\right)\\ +{\displaystyle \frac{R_f}{L_f}}>0\end{array}$$

(51)

Compared with the stability criterion under constant DC voltage control, the criterion under the proposed strategy becomes more complex because of the additional power differentiation feedforward term. When λ = 0, the proposed power differentiation feedforward control degenerates into conventional constant DC voltage control. Under this condition, the derived criterion reduces to the original one. By substituting the parameters in Table 1 into the resulting criterion, the stable operating ranges under different control coefficients λ can be obtained, as shown in Figure 11.

As shown in Figure 11, compared with conventional constant DC voltage control (λ = 0), the stable operating range of the system expands after the introduction of power differentiation feedforward. With the other parameters unchanged, a larger λ, within a certain range, leads to stronger system stability. Meanwhile, increasing λ also enlarges the DC voltage dip following a disturbance. Similarly to the analysis in Section 3.4, to ensure system stability, the corresponding criterion must satisfy 1+α-M₂ > 0, which implies that λ must satisfy

$$\begin{array}{l}2C_{dc}{u}_{dc}^2+3\lambda L_f{k}_{iv}\left(P_{CPL}-u_{dc}{i}_o\right)\left(i_d-\left(k_{pv}+\lambda k_{iv}{i}_o\right)u_{dc}\right)\\ -3L_f{i}_d{u}_{dc}\left(k_{pv}+\lambda k_{iv}{i}_o\right)>0\end{array}$$

(52)

Clearly, (52) is a quadratic inequality with respect to λ. To identify the maximum stable operating range of the system, the stable operating regions under different values of λ are obtained by combining the corresponding stability criterion with (52) and substituting the relevant parameters from Table 1, as illustrated in Figure 12.

As shown in Figure 12, for the flexible interconnection system with the parameters listed in Table 1, the maximum stable operating range is achieved when λ = 8 × 10⁻⁶. Compared with conventional constant DC voltage control, the maximum allowable load step increases from 92 kW to 100 kW, corresponding to an improvement of 8.7%.

Beyond the theoretical expansion of the stability boundary, the practical engineering value of the proposed strategy also warrants attention. It should be noted that, although advanced nonlinear control methods such as model predictive control (MPC) and sliding mode control (SMC) can provide excellent dynamic performance during severe transients, their widespread industrial application remains constrained by several practical limitations. For example, MPC requires an accurate mathematical model and considerable real-time computational resources for rolling optimization, often necessitating high-performance and costly digital controllers.

By contrast, the feedforward control strategy proposed in this paper is highly attractive from an engineering perspective because of its simplicity. It improves the large-signal stability margin without abandoning the mature and widely adopted proportional–integral (PI) control framework. Moreover, the proposed method introduces only a negligible computational burden, making it a cost-effective and easily deployable solution for upgrading existing DC microgrid systems.

5. Simulation Verification

A two-terminal flexible interconnection system (FIS), as shown in Figure 1, is built in MATLAB/Simulink R2022b to verify the derived large-signal stability criterion and the proposed power differentiation feedforward control strategy. The system parameters are listed in Table 1. The rectifier adopts a dual-loop control structure consisting of an AC-side inner current loop and a DC-side outer voltage loop, whereas the inverter operates under PQ control to maintain constant output power. In the simulations, large disturbances are introduced through step changes in the active power reference of the inverter.

5.1. Verification of the Validity of the Proposed Criterion

Based on the system parameters listed in Table 1, substituting the corresponding values into the large-signal stability criterion (38) yields the maximum constant power load (CPL) that the system can withstand:

$$P_{CPL}\le 92\mathrm{kW}$$

(53)

According to (53), with an initial power of 50 kW, the maximum allowable total CPL power is less than 92 kW.

Table 2. Simulation cases under large disturbances.

Group	CPL Power Step	Prediction by (38)
I	50 kW→92 kW	Stable
II	50 kW→100 kW	Unstable

lists two simulation cases with different load steps. Group I satisfies criterion (38), whereas Group II does not.

Case I: The inverter-side load power steps from 50 kW to 92 kW using the parameters of Group I. The simulation results are shown in Figure 13.

As shown in Figure 13, the CPL steps from 50 kW to 92 kW at t = 0.3 s. After an oscillatory transient lasting approximately 0.15 s, the DC bus voltage recovers to 750 V, and the AC-side three-phase currents gradually return to normal. This response indicates that, although the flexible interconnection system (FIS) remains stable, it is already operating close to the critical stability boundary. This simulation result is fully consistent with the theoretical prediction of criterion (38).

Case II: The inverter-side CPL power P_CPL steps from 50 kW to 100 kW. The simulation results are shown in Figure 14.

As shown in Figure 14, when the load steps to 100 kW at t = 0.3 s, the h can no longer be maintained at 750 V, and the AC-side three-phase currents become unstable. This confirms that the flexible interconnection system cannot operate stably when the load step violates criterion (38).

In summary, the system becomes unstable when the power step exceeds the range predicted by the criterion. This verifies the accuracy of the large-signal stability criterion (38) derived in this paper.

Next, the influence of the proportional gain k_pv and the DC capacitance C_dc on large-signal stability is further examined.

Figure 5 compares the proposed stability criterion (38) with the conventional criterion (39). With the other parameters held constant, criterion (39) suggests that the stable operating range increases linearly with k_pv. However, according to the proposed criterion (38), stability becomes negatively correlated with k_pv when k_pv > 0.7.

When k_pv = 1.2, criterion (38) indicates that the allowable inverter-side load power satisfies

$$P_{CPL}\le 86\mathrm{kW}$$

(54)

According to (54), if k_pv in Group I is increased from 0.7 to 1.2 while all other parameters remain unchanged, the maximum load power limit decreases from 92 kW to approximately 86 kW. Consequently, a load step to 92 kW exceeds the allowable limit, and the system should become unstable.

Table 3. Simulation cases under different k_pv values.

Group	CPL Power Step	kpv	Prediction by (38)	Prediction by (39)
III	50 kW→92 kW	1.2	Unstable	Stable
IV	50 kW→92 kW	0.4	Unstable	Unstable

summarizes the simulation cases under different k_pv values.

Case III: Under the condition k_pv = 1.2, the load power P_CPL steps from 50 kW to 92 kW. The simulation results are shown in Figure 15.

Case IV: Under the condition k_pv = 0.4, the load power P_CPL steps from 50 kW to 92 kW. The simulation results are shown in Figure 16.

Comparing Cases I, III, and IV under the same load step (50 kW→92 kW), Figure 13, Figure 15, and Figure 16 show that when k_pv = 0.4 or k_pv = 1.2, the DC bus voltage drops and oscillates after the disturbance and fails to recover to the rated value, while the AC-side currents become unstable. The system remains stable only when k_pv = 0.7.

More importantly, Case III (k_pv = 1.2) exhibits instability in the simulation. This result contradicts the prediction of the conventional criterion (39), which suggests improved stability, but agrees with the proposed criterion (38), which predicts a reduced stability margin (P_CPL < 86 kW). This demonstrates that the proposed criterion (38) predicts the stable operating range under large disturbances more accurately than criterion (39).

Case V: With C_dc = 2 mF, the load power P_CPL steps from 50 kW to 100 kW. The simulation results are shown in Figure 17.

According to criterion (38), increasing C_dc raises the maximum allowable power limit. When C_dc is increased to 2 mF, with all other parameters unchanged, the system is subjected to the same disturbance as in Case II (50 kW→100 kW). As shown in Figure 17, after the load step at t = 0.3 s, the DC bus voltage returns to 750 V, and the AC-side currents remain stable. This behavior differs from that in Case II (C_dc = 1 mF), confirming that increasing the DC capacitance improves system stability, which is consistent with the analysis in Figure 6.

5.2. Verification of the Proposed Control Strategy

In practical industrial applications, the dynamic reconstruction of the DC voltage induced by the power differentiation feedforward term must be coordinated with the over-voltage and under-voltage protection thresholds of the converter hardware. To prevent the reconstructed voltage reference from triggering under-voltage lockout during transients, an additional saturation limit is imposed on the DC voltage reference command. In this paper, the reference variation is constrained within a safe operating range of 650 V to 800 V throughout the transient process.

According to the stability criterion derived for the proposed feedforward strategy, adding power differentiation feedforward to the dual-loop control system increases the allowable power step limit and enables the system to remain stable under conditions that would otherwise lead to instability. As shown in Figure 12, when the power differentiation feedforward control is adopted with λ = 8 × 10⁻⁶, the maximum load step satisfies

$$P_{CPL}\le 100\mathrm{kW}$$

(55)

According to (55), if the system in Group II, which previously became unstable, adopts the proposed control strategy, it should remain stable. To verify this, the rectifier control is modified according to Figure 10, with λ = 8 × 10⁻⁶ and all other parameters unchanged from Table 1. The specific load-step settings in the two new simulation cases are summarized in

Table 4. Simulation cases for verifying the proposed control strategy.

Group	CPL Power Step	Prediction by the Proposed Criterion
VI	50 kW→92 kW	Stable
VII	50 kW→100 kW	Stable

.

5.2.1. Verification Under Severe CPL Surges

Case VI: Under power differentiation feedforward control, the load power P_CPL steps from 50 kW to 92 kW. The simulation results are shown in Figure 18.

As shown in Figure 18, the system remains stable under the proposed feedforward control. Compared with Figure 13, the voltage nadir is improved from 627 V to 638 V. When a load surge occurs (dP_rec/dt > 0), the feedforward term proactively reduces the voltage reference u_dcref to track the inevitable transient decline in the actual DC voltage u_dc. This dynamic tracking limits the PI controller input error e_u = u_dcref − u_dc, thereby suppressing integral windup and preventing excessively aggressive instantaneous current commands. Consequently, the system draws energy from the grid at a smooth and controlled di_d/dt rate. As the disturbance decays, u_dcref gradually returns to 750 V without overshoot. By providing a transient buffering effect, the proposed strategy effectively suppresses DC voltage fluctuations and expands the large-signal stability boundary.

Case VII: Under power differentiation feedforward control, the load power P_CPL steps from 50kW to 100kW. The simulation results are shown in Figure 19.

Figure 19 shows the simulation results under the proposed control. At t = 0.3 s, the load steps from 50 kW to 100 kW. Under the action of the feedforward control, the DC bus voltage stabilizes at 750 V after the disturbance, and the AC-side currents remain stable. Compared with the corresponding case without feedforward control, the proposed strategy exhibits enhanced disturbance rejection capability and maintains stable operation. These simulation results are consistent with the theoretical analysis of Figure 11 and verify the effectiveness of the proposed stability control strategy.

5.2.2. Performance Under Weak Grid Conditions

To further evaluate the robustness of the proposed control strategy under severe grid conditions, the simulation scenario is extended to a weak grid environment. In practical flexible interconnection systems, a weak AC grid usually exhibits a low short-circuit ratio (SCR) and a large equivalent grid impedance, which significantly reduces system damping and challenges large-signal stability during transients.

In Case VIII, an extremely weak grid condition with an SCR of 1.5 is considered. Based on the rated power and rated voltage of the system, the grid-side line impedance parameters are calculated as R_r = 0.2 Ω and L_r = 3 mH, assuming an X/R ratio of 5. At t = 0.3 s, a large-signal disturbance is applied, in which the constant power load steps from 50 kW to 65 kW.

Figure 20 compares the dynamic responses under the two control strategies. As shown in Figure 20a, under conventional PI control, the severe weak grid condition leads to rapid voltage collapse. After the disturbance, the DC bus voltage drops to approximately 400 V, indicating a complete loss of large-signal stability. By contrast, the proposed feedforward control shown in Figure 20b successfully maintains system stability. Despite the very high grid impedance, the DC voltage dip is effectively limited to about 700 V before rapidly recovering to the nominal value of 750 V, and the AC-side currents transition smoothly to the new steady state. This result clearly demonstrates the robustness of the proposed strategy in preventing voltage collapse under severe weak grid conditions.

5.2.3. Robustness Against AC Voltage Sags

Although the stability criterion in (38) is derived specifically for severe nonlinear instability caused by CPLs, practical DC microgrids are also subject to external AC-grid disturbances. To further verify the overall robustness of the proposed control strategy, an additional simulation is conducted under a symmetrical AC-grid voltage sag.

Figure 21 compares the dynamic responses of the system under the two control strategies during a 31% AC voltage sag. As shown in Figure 21a, under conventional PI control, the transient power delivery interruption caused by the grid fault induces severe and divergent oscillations in the DC bus voltage, ultimately leading to instability. By contrast, as illustrated in Figure 21b, the proposed feedforward control exhibits strong disturbance rejection capability. It rapidly mitigates the transient power mismatch, effectively suppresses DC voltage fluctuations, and maintains the DC voltage around its nominal value of 750 V, while the AC-side currents smoothly return to steady state. This further confirms that the proposed strategy not only handles large-signal load step disturbances effectively but also maintains satisfactory stability performance under severe grid-side voltage sags.

6. Discussion

In practical distributed DC microgrids and flexible interconnection systems (FISs), advanced droop control techniques have been extensively studied. For example, dynamic decentralized droop control has been proposed to facilitate high-efficiency renewable energy integration in microgrid systems [30]. Although such decentralized strategies can effectively achieve steady-state power sharing, they inherently introduce steady-state voltage deviations, which may continuously reduce the large-signal stability margin in systems containing constant power loads (CPLs). By contrast, the power differentiation feedforward strategy proposed in this paper acts strictly as a transient compensator. Since it only modifies the local voltage reference on the basis of local measurements, it is inherently a decentralized control method. Therefore, it can be naturally integrated into individual source nodes in complex multi-terminal systems. In this way, critical local nodes can be reinforced against severe transient CPL surges while the global power-sharing capability of the network and the efficient integration of renewable energy are preserved.

Although the proposed control strategy is physically scalable and easy to implement, the analytical scope of this study should also be clearly stated. The rigorous large-signal stability criterion and the corresponding parameter optimization results presented in this paper are mathematically derived from a simplified two-terminal source-load model. This fundamental framework provides useful physical insight and guarantees the local stability of individual converter nodes. However, it may not fully capture the dynamic complexity of modern multi-terminal or mesh-connected FISs. In such networked systems, dynamic interactions, power flow redistribution, and cross-coupling through line impedances become increasingly important. Therefore, extending the current mixed potential function (MPF) methodology to construct a global energy function for mesh networks, and thereby enabling network-level large-signal stability assessment, remains an important and challenging direction for future research.

7. Conclusions

Focusing on a typical two-terminal flexible interconnection system, this paper establishes a system model based on the mixed potential function (MPF) method. By incorporating the dynamic characteristics of the DC-side capacitor, a novel large-signal stability criterion is derived. Furthermore, an improved control strategy for enhancing the disturbance rejection capability of the system is proposed and verified through MATLAB/Simulink simulations. The main conclusions are summarized as follows:

1. The proposed stability criterion reveals the constraint relationships among key circuit and control parameters, including the constant power load (CPL), the PI parameters of the rectifier voltage loop, and the DC-side shunt capacitance. Unlike conventional large-signal stability analyses, the proposed criterion explicitly accounts for the effect of the DC voltage derivative, or equivalently the capacitor current, on system stability. In addition, it enables PI parameter optimization under stability constraints, thereby providing useful theoretical guidance for controller design.

2. The proposed power differentiation feedforward control strategy dynamically reshapes the DC voltage reference during transient events, thereby suppressing voltage fluctuations and improving the disturbance rejection capability of the system. By actively mitigating transient power mismatch, the strategy enhances large-signal stability without relying heavily on additional hardware buffering devices. As a result, the method shows strong practical potential for maintaining continuous operation under severe CPL power steps and grid-side voltage sags.

Future work will focus on integrating the proposed strategy with advanced nonlinear control methods, such as model predictive control (MPC), and experimentally validating its practical robustness through hardware-in-the-loop (HIL) testing.