Transient Stability Analysis and Enhancement Strategies for AC Side of Hydro-Wind-PV VSC-HVDC Transmission System

Abstract

To analyze and enhance the transient stability of a hydro-wind-PV VSC-HVDC transmission system, this paper establishes a transient stability analytical model and proposes strategies for stability improvement. Based on the dynamic interaction mechanisms of multiple types of power sources, an analytical model integrating GFM converters, GFL converters, and SGs is first developed. The EAC is employed to investigate how the factors such as current-limiting thresholds and fault locations influence transient stability. Subsequently, a parameter tuning method based on optimal phase angle calculation and delayed control of current-limiting modes is proposed. Theoretical analysis and PSCAD simulations demonstrate that various factors affect transient stability by influencing the PLL of converters and the electromagnetic power of synchronous machines. The energy transfer path during transient processes is related to fault locations, parameter settings of current-limiting modes in converters, and the operational states of equipment. The proposed strategy significantly improves the transient synchronization stability of multi-source coupled systems. The research findings reveal the transient stability mechanisms of hydro-wind-PV VSC-HVDC transmission systems, and the proposed stability enhancement method combines theoretical innovation with engineering practicality, providing valuable insights for the planning and design of such scenarios.

1. Introduction

Currently, the energy structure worldwide is continuously being optimized and upgraded, with the share of installed clean energy power generation capacity steadily increasing [1,2,3,4]. As of August 2024, China’s total installed capacity of wind and solar power nationwide has reached 1.22 billion kilowatts [5], achieving ahead of schedule the 2030 development target of exceeding 1.2 billion kilowatts for combined wind and solar capacity. During the phase of rapid energy transition, regions such as Western China, the Mongolian Western Region, and the Baziro site will continue to support it by their exceptional advantages in renewable energy resources and development conditions [6,7]. Tibet alone possesses approximately 180 million kilowatts of technically exploitable wind energy and 200 million kilowatts of average annual hydropower potential [8,9,10], with its theoretical hydropower reserves accounting for about 30% of the nation’s total. To achieve flexible regulation and optimal allocation of renewable energy and hydropower, China has planned and constructed multiple hydro-wind-PV energy bases in western regions, including the Kela photovoltaic power station [11]. Furthermore, countries such as Egypt are also planning or constructing hybrid hydro-wind-PV power systems in regions including Ataka and Suez [12]. Due to the reverse distribution pattern between China’s energy resources and economic development [13], the long distance between western hydro-wind-PV complementary energy bases and eastern coastal load centers necessitates HVDC transmission systems for long-distance power delivery [14]. VSC-HVDC technology demonstrates significant advantages in long-distance, high-voltage, large-capacity power transmission scenarios [15], showing enormous potential for future renewable energy delivery applications. The bundled hydro-wind-PV hybrid transmission system via MMC-HVDC has attracted widespread attention [16,17,18], with ongoing projects like the Gansu-Zhejiang and Tibet-Yunnan HVDC links adopting this technical approach.

Based on synchronization control methods, grid-connected converters for renewable energy generation can be categorized into two main types: GFM converters and GFL converters. Among them, GFM converters actively establish grid synchronization references through self-synchronization mechanisms, and are more suitable for weak grid conditions, with transient instability manifesting as virtual power-angle oscillations [19,20,21]. However, GFL converters synchronize with external grids via PLL demonstrating better stability in strong grids, where PLL-weak grid interaction dominates instability factors [22,23]. In hydro-wind-PV complementary energy bases employing VSC-HVDC transmission, the sending-end AC system involves interactions among multiple devices including GFM converters, GFL converters, and SGs. Compared with conventional transmission scenarios, the combined effects of heterogeneous device dynamics, control strategy coupling, and oscillation interactions in weak grid conditions make the synchronization stability mechanism under large disturbances significantly more complex.

Regarding transient stability issues in AC systems involving different types of power source equipment, some scholars have conducted corresponding research and made progress. Reference [21] proposed measures to improve the transient stability of grid-connected systems with GFM and GFL converters but did not mention their interaction characteristics. Reference [24] analyzed the mutual influence between GFM and GFL converters in terms of transient stability but did not consider the influence of SG integration into the system. Reference [25] focused on hybrid power systems where GFL/GFM renewable generation units and SGs are connected together. It used space vector diagrams and phase plane methods to analyze the influence of power electronic device control strategies and controller parameters on the transient stability of various generation units in the system. However, the study focused on scenarios where multiple types of power sources are directly connected to the main grid rather than independently forming a sending-end grid, and it did not consider the influence of the GFM converter limiting current during faults. Reference [26] quantitatively evaluated the power transmission capacity of a hydro-wind-PV VSC-HVDC transmission system, but the analysis was limited to steady-state electrical parameters and small-signal stability without involving large-disturbance stability analysis. Therefore, there is still limited research on the synchronization stability of the hydro-wind-PV VSC-HVDC transmission system with GFM converters, GFL converters, and SG, particularly when considering the current-limiting control of converters.

This paper focuses on the sending-end AC system of a hydro-wind-PV VSC-HVDC transmission base, investigating the transient stability of power systems incorporating different types of converters and SGs while considering converter current-limiting control. The study establishes transient stability analytical models for GFM converters, GFL converters, and SGs. Based on the power-angle synchronization mechanisms of different converter types, it elaborates the transient stability evaluation method for hydro-wind-PV hybrid systems and the mechanism of various influencing factors on system transient stability. Furthermore, the paper proposes transient stability enhancement strategies according to the revealed influence mechanisms.

2. The Sending-End AC System of a Hydro-Wind-PV VSC-HVDC Transmission System

This study takes the sending-end AC system of the hydro-wind-PV VSC-HVDC transmission system as the research object. In this system, the renewable energy base connects to the converter station’s AC bus via AC Line 1, the hydropower station connects to the converter station’s AC bus via AC Line 2, and the valve-side outlet AC bus of the sending-end converter station is designated as PCC1. For the investigated transient stability problem, this paper adopts the following premises and assumptions:

• The DC voltage of the sending-end converter station is kept constant by energy storage or other converter stations, with its dynamic variations neglected in the study;
• The inner-loop current controllers of converters, including both VSC-HVDC converters and renewable energy converters, respond significantly faster than outer-loop controllers [27], allowing the current inner-loop dynamics to be disregarded in transient stability analysis;
• The SGs in this study employ the classical second-order model, maintaining constant q-axis transient EMF E′_q during transient processes;
• The resistive components in the equivalent impedances of transformers and SGs are relatively small, thus their influence is neglected;
• Limited by overcurrent capability, converters will switch to current-limiting mode upon detecting a voltage drop below the threshold during severe AC-side short-circuit faults.

Considering the simplified conditions, the sending-end system is represented by the equivalent circuit in Figure 1. In this equivalent circuit, Converter 1 representing the sending-end converter station adopts a GFM strategy with constant AC voltage and frequency control (i.e., V/f control), while Converter 2 representing the renewable energy base adopts a GFL strategy with constant power control.

The terms in Figure 1 are defined as follows:

• The AC bus node of Converter 2 is denoted as PCC2;
• R_Line1 and X_Line1 represent the resistance and reactance of Line 1, respectively;
• R_Line2 and X_Line2 represent the resistance and reactance of Line 2, respectively;
• X_Ts1 and X_Ts2 represent the connection reactance of Converter 1 and Converter 2, respectively;
• X_TG represents the leakage reactance of the SG’s step-up transformer;
• X′_d represents the transient reactance of the SG;
• Z₁ and Z₂ represent the equivalent impedances between PCC2 and PCC1, and between the E′_q node and PCC1, respectively;
• U₀ and I₁ represent the voltage and injected current at PCC1, respectively;
• U_s, I₂, P_s and Q_s represent the voltage, injected current, active power output and reactive power output at PCC2, respectively;
• U_g, I_g, P_g and Q_g represent the terminal voltage, injected current, active power output and reactive power output of the SG, respectively.

Based on the equivalent circuit, this paper will establish a mathematical model for transient stability analysis of the sending-end system in the hydro-wind-PV VSC-HVDC transmission system, aiming to quantitatively analyze the interaction mechanisms within the hybrid hydro-wind-PV system during transient evolution processes.

3. Generic Model for Transient Stability Analysis of Sending-End AC Systems

The research objects in this study include Converter 1, Converter 2, and the SG. To distinguish them, this paper denotes the dq rotating reference frames corresponding to the synchronization unit output phase angle of Converter 1, the PLL output phase angle of Converter 2, and the q-axis transient electromotive force E′_q of the SG as the dq frame, dsqs frame, and dgqg frame, respectively. The vector transformation relationships between these coordinate systems are provided in Equation (A1) in Appendix A.1.

3.1. Generic Equivalent Circuit Model

Based on the topology presented in Figure 1, this paper normalizes the system equivalent circuits for both normal and AC fault conditions into a unified generic circuit model as shown in Figure 2. The equivalent representation of system circuits under both normal and fault conditions can be effectively achieved by adjusting the grounding resistance values. In this model, the following are defined:

• U_f1 and U_f2 represent the voltage phasors at fault points f₁ and f₂, respectively;
• Z_f1 and Z_f2 denote the equivalent impedances of faults occurring on Line 1 and Line 2, respectively, which are determined by the fault type;
• Z₁₁ and Z₁₂ represent the impedances between fault point f₁ and the PCC2, and between f₁ and the PCC of Converter 1, respectively;
• Z₂₁ and Z₂₂ represent the impedances between fault point f₂ and the transient EMF equivalent point of the SG, and between f₂ and the PCC1, respectively;
• The ratio of Z_i2 to Z_i is defined as K_i (i = 1 or 2), which characterizes the relative distance between fault point f_i and Converter 1.

Let I_1max and I_2max be the current phasors of Converter 1 and Converter 2 in the current-limiting state, respectively, with φ₁ and φ₂ representing the angles between I_1max and the d-axis, and between I_2max and the ds-axis, respectively. According to Equations (A2)–(A4) in the Appendix A.2, U_f1, U_f2, and I_g are all coupled with I_1max, I_2max, and E′_q. Considering that power systems are more prone to transient instability during three-phase short-circuit faults [21], this study primarily focuses on the more severe three-phase metallic grounding faults. Based on the equivalent circuit shown in Figure 2, the voltages at the fault points of Line 1 and Line 2 are given by Equations (1) and (2), respectively:

$$\left\{\begin{array}{c}{\dot{U}}_{\mathrm{f}1}=0\hfill \\ {\dot{U}}_{\mathrm{f}2}={\displaystyle \frac{(1-K_2)K_1{\dot{Z}}_1{\dot{Z}}_2}{K_1{\dot{Z}}_1+{\dot{Z}}_2}}\cdot {\dot{I}}_{1\max }+{\displaystyle \frac{K_1{\dot{Z}}_1+K_2{\dot{Z}}_2}{K_1{\dot{Z}}_1+{\dot{Z}}_2}}\cdot {\dot{E}}_{\mathrm{q}}^{\prime }\end{array}\right.$$

(1)

$$\left\{\begin{array}{c}{\dot{U}}_{\mathrm{f}1}=K_2{\dot{Z}}_2\cdot {\dot{I}}_{1\max }+(K_1{\dot{Z}}_1+K_2{\dot{Z}}_2)\cdot {\dot{I}}_{2\max }\\ {\dot{U}}_{\mathrm{f}2}=0\hfill \end{array}\right.$$

(2)

From the above analysis, when a metallic fault occurs on Line 1, the output current I₂ₘₐₓ of Converter 2 does not affect the SG or the output voltage of Converter 1. And when a metallic fault occurs on Line 2, E′_q does not influence the output voltages of either Converter 1 or Converter 2.

3.2. Transient Stability Analysis Model for GFM Converters

The equivalent circuit shown in Figure 2 employs V/f control for Converter 1. After specifying the initial phase θ_initial, its transient stability analysis model refers to the synchronization signal θ₀ generation strategy of Converter 1 at t:

$${\theta }_0={\displaystyle \underset{0}{\overset{t}{\int }}{\omega }_{0}dt}+{\theta }_{\mathrm{initial}}$$

(3)

For Converter 1 adopting V/f control, the influence of its inner current control loop can be neglected in transient stability analysis [28]. Since the V/f control directly provides the synchronization signal, Converter 1 does not face synchronization stability issues. Therefore, this paper focuses specifically on analyzing the transient stability of Converter 2 and the SG.

3.3. Transient Stability Analysis Model for GFL Converters

The PLL output signal θ_PLL of GFL converters is a critical research focus for synchronization stability analysis [29]. This paper therefore concentrates on investigating the PLL’s dynamic characteristics. When Converter 2 is connected to an infinite bus, its dynamic model in the dsqs coordinate system exhibits similar characteristics to the second-order SG equations [30]. Based on the equivalent circuit shown in Figure 2, the mathematical model for transient stability analysis of Converter 2 in hydro-wind-PV VSC-HVDC transmission system can be derived as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\delta }_{\mathrm{PLL}}}{dt}}={\omega }_{\mathrm{PLL}}-{\omega }_0\\ J_{\mathrm{s}i}{\displaystyle \frac{d^2{\delta }_{\mathrm{PLL}}}{dt}}=P_{\mathrm{ms}i}-P_{\mathrm{es}i}-D_{\mathrm{s}i}{\displaystyle \frac{d{\delta }_{\mathrm{PLL}}}{dt}}\end{array}\right.$$

(4)

In the equations, the following apply:

• δ_PLL represents the angle between the ds-axis and d-axis;
• J_si, P_msi, P_esi, and D_si denote Converter 2’s equivalent inertia coefficient, equivalent mechanical power, equivalent electromagnetic power, and equivalent damping coefficient, respectively;
• i indicates the system operating state, where i = 0 corresponds to normal operation and i = 1 or 2 indicates a fault on Line 1 or Line 2, respectively. The same meaning applies hereafter.

For different system operating conditions, including normal operation, Line 1 fault, or Line 2 fault, Equation (5) can be expressed as Equations (6), (7) and (8), respectively:

$$\left\{\begin{array}{l}{J}_{\mathrm{s}0}={\displaystyle \frac{1-L_1{I}_{\mathrm{sd}}}{k_{\mathrm{i}}}}, P_{\mathrm{ms}0}=R_1{I}_{\mathrm{sq}}+X_1{I}_{\mathrm{sd}}\\ P_{\mathrm{es}0}=U_0\sin {\delta }_{\mathrm{PLL}}, D_{\mathrm{s}0}={\displaystyle \frac{k_{\mathrm{p}}}{k_{\mathrm{i}}}}{U}_0\cos {\delta }_{\mathrm{PLL}}-L_1{I}_{\mathrm{sd}}\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}{J}_{\mathrm{s}1}={\displaystyle \frac{1-L_1{I}_{2\max }(1-K_1)\cos {\phi }_2}{k_i}}, P_{\mathrm{ms}1}=I_{2\max }(1-K_1)(R_1\sin {\phi }_2+X_1\cos {\phi }_2)\\ P_{\mathrm{es}1}=0, D_{\mathrm{s}1}=-L_1{I}_{2\max }(1-K_1)\cos {\phi }_2\end{array}\right.$$

(6)

$$\left\{\begin{array}{c}{J}_{\mathrm{s}2}={\displaystyle \frac{1-(L_1+K_2{L}_2)I_{2\max }\cos {\phi }_2}{k_{\mathrm{i}}}}\hfill \\ P_{\mathrm{ms}2}=(R_1+K_2{R}_2)I_{2\max }\sin {\phi }_2+(X_1+K_2{X}_2)I_{2\max }\cos {\phi }_2\hfill \\ P_{\mathrm{es}2}=K_2{Z}_2{I}_{1\max }\sin ({\delta }_{\mathrm{PLL}}-{\phi }_1-{\theta }_2)\hfill \\ D_{\mathrm{s}2}={\displaystyle \frac{k_{\mathrm{p}}}{k_{\mathrm{i}}}}{K}_2{Z}_2{I}_{1\max }\cos ({\delta }_{\mathrm{PLL}}-{\phi }_1-{\theta }_2)-(L_1+K_2{L}_2)I_{2\max }\cos {\phi }_2\end{array}\right.$$

(7)

In the equations, the following apply:

• L₁ and R₁ represent the inductance and resistance values of impedance Z₁, respectively;
• L₂ and R₂ denote the inductance and resistance values of impedance Z₂, respectively;
• I_sd and I_sq are the ds-axis and qs-axis components of the current phasor I_s.

3.4. Transient Stability Analysis Model for SGs

This paper analyzes the transient stability of SGs based on the second-order rotor swing equation, with the corresponding transient stability model expressed as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d\delta }{dt}}={\omega }_{\delta }-{\omega }_0\\ J_{\mathrm{g}}{\displaystyle \frac{d^2\delta }{dt}}=P_{\mathrm{mg}}-P_{\mathrm{eg}i}-D_{\mathrm{g}}{\displaystyle \frac{d\delta }{dt}}\end{array}\right.$$

(8)

In the equation, P_egi represents the electromagnetic power of the SG.

Considering that the resistance components in high-voltage AC transmission lines are significantly smaller than the reactance components, the resistance can be neglected when analyzing the transient stability of SGs. Through derivations, the expression for the electromagnetic power P_egi is obtained as follows:

$$P_{\mathrm{eg}0}={\displaystyle \frac{E_{\mathrm{q}}^{\prime }{U}_0}{X_2}}\sin \delta$$

(9)

$$P_{\mathrm{eg}1}=E_{\mathrm{q}}^{\prime }{I}_{1\max }\cdot {\displaystyle \frac{K_1{X}_1}{K_1{X}_1+X_2}}\cdot \cos (\delta -{\phi }_1-\pi )=P_{\mathrm{eg}1\max }\cos (\delta -{\phi }_1-\pi )$$

(10)

$$P_{\mathrm{eg}2}=0$$

(11)

Based on the generic transient stability analysis model for the sending-end AC system of the hydro-wind-PV VSC-HVDC transmission system developed above, this paper will subsequently employ the EAC to analyze the transient stability of the coupled converter-SG system [31].

4. Transient Stability Analysis of the Sending-End AC System

4.1. Transient Stability Assessment

This paper employs the EAC to analyze the transient stability of the sending-end AC system in a hydro-wind-PV VSC-HVDC transmission system, with quantitative evaluation based on the CCT of system faults [32].

As shown in Figure 3a, when using EAC to assess the transient stability of a SG, the process is divided into the following key stages:

Stage 1: During steady-state operation, the SG maintains equilibrium at the SEP, where the mechanical power P_mg equals the electromagnetic power P_eg0. The SEP is determined by the pre-fault load level of the system, and its corresponding rotor angle δ_SEP is given by the following:

$${\delta }_{\mathrm{SEP}}=\arcsin ({\displaystyle \frac{P_{\mathrm{mg}}}{P_{\mathrm{eg}0\max }}})=\arcsin ({\displaystyle \frac{P_{\mathrm{mg}}{X}_2}{E_{\mathrm{q}}^{\prime }{U}_0}})$$

(12)

In the equation, P_eg0max represents the maximum output power of the SG under steady-state operation.

In addition to the SEP, the condition P_mg = P_eg0 yields another intersection point, namely the UEP, with its corresponding rotor angle δ_UEP given by δ_UEP = π − δ_SEP.

Stage 2: At the instant of fault occurrence, the operating point shifts to point b. The mechanical torque T_mg exceeds the electromagnetic torque T_eg, causing the rotor to accelerate toward increasing δ. The operating point moves toward point c, where P_eg1max represents the maximum electromagnetic power of the SG under Line 1 fault conditions. When the fault is cleared at angle δ_T, the rotor accelerating area S₊ is expressed as follows:

$$S_{+}={\displaystyle \underset{{\delta }_{\mathrm{SEP}}}{\overset{{\delta }_T}{\int }}(P_{\mathrm{mg}}-P_{\mathrm{eg}1})d\delta }$$

(13)

Stage 3: After fault clearance, the operating point shifts to point d, initiating rotor deceleration. If the fault is cleared promptly (i.e., δ_T < δ_cr), the maximum rotor angle swing δ_max will not exceed δ_UEP. The decelerating area S₋ is then given by the following:

$$S_{-}={\displaystyle \underset{{\delta }_T}{\overset{{\delta }_{\max }}{\int }}(P_{\mathrm{eg}\max }\sin \delta -P_{\mathrm{mg}})d\delta }$$

(14)

When δ_max = δ_UEP, S₋ is the maximum decelerating area S_−max for the given δ_T.

Stage 4: If the fault is cleared promptly (i.e., S_−max > S₊), the rotor speed will decelerate to the rated synchronous speed before the operating point reaches the UEP, then continue to decelerate and eventually stabilize back to the SEP under damping effects. Conversely, if the fault clearance is delayed (i.e., S_−max < S₊), the rotor angle δ will exceed δ_UEP, resulting in transient instability of the SG. The clearing angle δ_CCA corresponding to S_−max = S₊ is defined as the CCA, and its associated fault duration is the CCT. Here, δ_CCA is determined by solving the following equation:

$${\displaystyle \underset{{\delta }_{\mathrm{SEP}}}{\overset{{\delta }_{\mathrm{CCA}}}{\int }}(P_{\mathrm{mg}}-P_{\mathrm{eg}1})d\delta }={\displaystyle \underset{{\delta }_{\mathrm{CCA}}}{\overset{{\delta }_{\mathrm{UEP}}}{\int }}(P_{\mathrm{eg}\max }\sin \delta -P_{\mathrm{mg}})d\delta }$$

(15)

In the equivalent power curve of Converter 2, the equivalent power angle θ_SEP corresponding to SEP is:

$${\theta }_{\mathrm{SEP}}=\arcsin ({\displaystyle \frac{P_{\mathrm{ms}}}{P_{\mathrm{es}\max }}})$$

(16)

The δ_PLL of Converter 2 exhibits dynamic characteristics similar to the power angle δ of a SG. When assessing its transient stability using EAC, the principle is consistent with that of SGs, which will not be reiterated here. The derived equivalent power-angle characteristic curve is shown in Figure 4.

4.2. Factors Affecting Transient Stability

4.2.1. Limiting Current in Converter Current-Limiting Mode

(1) Influence of Limiting Current on δ_PLL During Line 2 Fault

It can be seen from Equation (7) that when Line 2 fails, the amplitude and phase of the P_es2 curve of Converter 2 are affected by |I_1max| and φ₁, respectively, while the magnitude of P_ms2 is influenced by |I_2max| and φ₂. During the Line 2 fault, δ_PLL is jointly determined by I_1max and I_2max. Based on the vector analysis method, the influence mechanism of the current limit on δ_PLL is analyzed.

The PLL of Converter 2 adopts PI control, and the dynamic characteristics of δ_PLL are determined by U_sq:

$$U_{\mathrm{sq}}(k_{\mathrm{p}}+{\displaystyle \frac{k_{\mathrm{i}}}{s}})={\displaystyle \frac{d{\delta }_{\mathrm{PLL}}}{dt}}$$

(17)

When K₁ in Equation (2) is set to 1, U_f1 equals U_s. After coordinate transformation, the expressions for U_s and U_sq in the dsqs coordinate system can be derived as follows:

$$\left\{\begin{array}{l}{\dot{U}}_{\mathrm{sq}}={\dot{U}}_{\mathrm{sq}-\mathrm{I}1}+{\dot{U}}_{\mathrm{sq}-\mathrm{I}2}\\ {\dot{U}}_{\mathrm{sq}-\mathrm{I}1}=\left[\left|Z_{\mathrm{eq}-\mathrm{I}1}{I}_{1\max }\right|\sin ({\theta }_{\mathrm{eq}-\mathrm{I}1}+{\phi }_1-{\delta }_{\mathrm{PLL}})\right]\cdot e^{\mathrm{j}{\displaystyle \frac{\pi }{2}}}\\ {\dot{U}}_{\mathrm{sq}-\mathrm{I}2}=\left[\left|Z_{\mathrm{eq}-\mathrm{I}2}{I}_{2\max }\right|\sin ({\theta }_{\mathrm{eq}-\mathrm{I}2}+{\phi }_2)\right]\cdot e^{\mathrm{j}{\displaystyle \frac{\pi }{2}}}\\ {\dot{Z}}_{\mathrm{eq}-\mathrm{I}1}=K_2{\dot{Z}}_2=Z_{\mathrm{eq}-\mathrm{I}1}\angle {\theta }_{\mathrm{eq}-\mathrm{I}1}\\ {\dot{Z}}_{\mathrm{eq}-\mathrm{I}2}={\dot{Z}}_1+K_2{\dot{Z}}_2=Z_{\mathrm{eq}-\mathrm{I}2}\angle {\theta }_{\mathrm{eq}-\mathrm{I}2}\end{array}\right.$$

(18)

According to Equations (17) and (18), the dynamic characteristics of δ_PLL are determined by U_sq, which in turn depends on I_1max and I_2max. Therefore, during a Line 2 fault, δ_PLL is jointly governed by I_1max and I_2max.

Equation (18) indicates that the U_sq component U_sq-I1 determined by I_1max varies with δ_PLL, leading to changes in both magnitude and phase of U_sq, whereas the U_sq component U_sq-I2 determined by I_2max remains unaffected by δ_PLL. According to Equation (17), the effect of U_sq on δ_PLL is analogous to the effect of net torque on the power angle of a SG. Therefore, I_1max and I_2max, respectively, correspond to electromagnetic power and mechanical power of a SG. Among them, the magnitude of the former varies with δ_PLL or power angle, while the latter remains constant.

In Figure 5: U_s-I2-1 and U_s-I2-2 represent the voltages generated by I_2max at the PCC of Converter 1 And Converter 2, respectively, while ΔU_s-I1 denotes the increment in U_s caused by the increase in I_1max.

As shown in Figure 5, The U_sq component that is in phase (or anti-phase) with the qs-axis will drive the dsqs coordinate system to rotate counterclockwise (or clockwise) in the dq coordinate system. Therefore, within the dsqs coordinate system, I_1max and U_s-I1 exhibit a tendency to rotate clockwise (or counterclockwise), causing U_sq-I1 to vary in magnitude and polarity within the range of |U_s-I1|. In contrast, I_2max and U_sq-I2 remain stationary relative to the dsqs coordinate system. Based on the above analysis, when I_2max is fixed and |I_1max| increases within a certain range, the dynamic behavior of δ_PLL during the fault undergoes the following changes:

• If the initial value of U_sq (U_sq0) is nonzero and |I_1max| results in |U_sq-I1| < |U_sq-I2|, δ_PLL will monotonically increase (if U_sq0 > 0) or decrease (if U_sq0 < 0).
• As |I_1max| increases, δ_PLL may either stabilize due to damping or exhibit oscillatory growth/decay, depending on the phase of I_1max.
• If the phase of I_1max remains unchanged, further increasing |I_1max| will cause δ_PLL that was stabilizing due to damping to continue stabilizing, while δ_PLL that was oscillating will either oscillate more prominently or transition to a stable state.

(2) Influence of Limiting Current on δ_PLL During Line 1 Fault

From Equation (6), it can be seen that when Line 1 fails, Converter 2 is only affected by I_2max, where |I_2max| only influences the magnitude of P_ms1 while φ₂ affects both the magnitude and sign of P_ms1. Through vector analysis, the following is found:

$$\left\{\begin{array}{l}{\dot{U}}_{\mathrm{sq}-\mathrm{I}1}=0\\ {\dot{U}}_{\mathrm{sq}-\mathrm{I}2}=\left[\left|(1-K_1)Z_1{I}_{2\max }\right|\sin ({\theta }_1+{\phi }_2)\right]\cdot e^{\mathrm{j}{\displaystyle \frac{\pi }{2}}}\end{array}\right..$$

(19)

As shown in Equation (19), |I_2max| only affects the magnitude of U_sq-I2, while φ₂ influences both the magnitude and phase of U_sq-I2. Consequently, different φ₂ values may cause δ_PLL to exhibit varying directional changes. If φ₂ remains constant, an increase in |I_2max| will amplify |U_sq-I2|, accelerating the variation of δ_PLL during the fault. Therefore, with a fixed fault clearing time, as |I_2max| increases, the post-fault δ_PLL will transition from stable to unstable, and larger |I_2max| values increasingly undermine the transient stability of δ_PLL.

(3) Influence of Limiting Current on δ During Line 1 Fault

From Equation (10), it can be seen that the δ of the SG is only affected by the current limit during a fault on Line 1 and is solely influenced by I_1max, where |I_1max| and φ₁ determine the amplitude and phase of its electromagnetic power curve, respectively. Therefore, for δ, φ₁ determines the increase or decrease in S₊, while |I_1max| affects the magnitude of this increase or decrease. Through vector analysis, the following is found:

U₀ is determined by E′_q and I_1max. Since the U₀ component U_0-E′q determined by E′_q remains constant and its phase nearly aligns with U₀ when the parameters of the two lines are similar, changes in the magnitude and phase of U₀ are caused by the U₀ component U_0-I1 determined by I_1max. As shown in Figure 6, when |I_1max| is fixed, adjusting φ₁ can alter the phase and amplitude of the PCC voltage U₀ of Converter 1. If φ₁ is set such that the angle between E′_q and U_0-I1 lies within 0~180°, according to the superposition theorem, increasing |I_1max| will raise P_es1, thereby enhancing the synchronous stability of the generator during the fault. However, the current-carrying capacity of the equipment |I_1max| limits the magnitude of U_0-I1, resulting in a smaller amplitude of the I_1max -coupled component in P_eg1. Consequently, the influence of changes in I_1max on the transient stability of the SG is limited.

4.2.2. Fault Location

(1) A fault occurs on the outgoing line of Converter 2

For Converter 2, under Fault 1, P_es1 = 0. Combined with Equation (6), it can be seen that the closer the fault point f₁ is to Converter 2, the smaller P_ms1 becomes, making Converter 2 less prone to equivalent power-angle instability. In principle, the following apply:

The larger K₁ is, the smaller the impedance from PCC2 to the fault ground becomes. Limited by |I_2max|, both |U_s| and the corresponding |U_sq| decrease, slowing the change in δ_PLL and increasing the CCT, which enhances the transient stability of Converter 2. In this case, if φ₂ ∈ (−θ₁, π−θ₁), U_sq will remain in a positive phase, and the increase in δ_PLL will slow further as K₁ increases; for other values of φ₂, the decrease will also slow.

Additionally, Equation (5) shows that the smaller the line impedance of Z₁, the smaller P_ms0 of Converter 2 under normal conditions, which improves its transient stability. This aligns with the conclusion that a higher short-circuit ratio leads to better PLL tracking performance.

For the SG, P_eg1 is coupled with I_1max. According to Equation (10), P_eg1max depends on K₁Z₁ and the magnitude of I_1max. When |I_1max| is fixed, the distance of f₁ from Converter 1 is monotonically related to CCT, and the value of φ₁ determines whether this correlation is positive or negative. In principle, the larger the impedance of K₁Z₁ from Converter 1 to the fault ground, the greater the current injected by Converter 1 (acting as a current source) into the SG. If φ₁ ensures the injected current promotes an increase in P_eg1, the resulting P_eg1 enhancement due to I_1max coupling will improve the transient stability of the SG. Conversely, if φ₁ opposes this effect, it will degrade transient stability.

(2) A fault occurs on the outgoing line of the SG

In high-voltage transmission systems, the reactance of line impedance significantly exceeds the resistance, with impedance angles consistently approaching 90°. Therefore, the following assumptions are made:

$${\theta }_1\approx {\theta }_2$$

(20)

For Converter 2, both P_ms2 and P_es2 are directly related to the line parameters. The difference between P_ms2 and P_es2 acting on δ_PLL, denoted as ΔP_s2, reflects the rate of change in ωPLL. The expression for ΔP_s2 is as follows:

$$\mathsf{\Delta }{P}_{s2}=Z_1{I}_{2\max }\sin ({\theta }_1+{\phi }_2)+K_2{Z}_2\mathsf{\Delta }{P}_{s2-2}$$

(21)

$$\mathsf{\Delta }{P}_{s2-2}=I_{2\max }\sin ({\theta }_1+{\phi }_2)-I_{1\max }\sin ({\delta }_{PLL}-{\phi }_1-{\theta }_1)]$$

(22)

The variable K₂ characterizing the location of fault point f₂ only affects ΔP_s2-2. When K₂ is small, the sign of ΔP_s2 is determined by the first term. If φ₂ ≠ −θ₁, δ_PLL will exhibit monotonic variation during the fault, with the direction of change determined by φ₂. As K₂ increases, the influence of ΔP_s2 becomes more pronounced—both the direction and rate of δ_PLL variation are affected by the magnitude and phase of ΔP_s2. Under parameter-matched conditions, δ_PLL may reach an equilibrium point during sustained faults.

For the SG, when a fault occurs on Line 2, its active power output becomes zero, and its transient stability becomes independent of the fault location.

5. Parameter Tuning Methodology for Transient Stability Enhancement

5.1. Parameter Tuning Methodology for GFL Converter Outgoing Line Faults

5.1.1. Parameter Tuning Methodology for SG

The magnitudes of |I_1max| and |I_2max| are constrained by the withstand capability of power equipment, making the controllable variables in the system φ₁ and φ₂. During Line 2 faults, the SG becomes decoupled from the system. Therefore, when designing parameter tuning methods to enhance the transient stability of SGs, only Line 1 fault conditions need to be considered.

After fault occurrence, the following occurs if the value of φ₁ is set to simultaneously satisfy the subsequent conditions:

• For a given δ_T, S₊ reaches the minimum value among all possible φ₁ settings;
• δ_T corresponds to the δ_CCA under this specific φ₁.

At this δ_T, any other φ₁ values would lead to an increase in S₊, causing the maximum rotor angle swing to exceed δ_UEP and resulting in transient instability of the SG. This specific φ₁ is therefore considered the optimal value for the given clearing angle δ_T. The corresponding power-angle characteristic curve under this condition is shown in Figure 7.

Through systematic derivation, the solution for φ₁ satisfying the aforementioned conditions is obtained as:

$$A\cos [2{\phi }_1-{\delta }_{\mathrm{eq}}]-{\displaystyle \frac{B}{1+\frac{C}{K_1}}}\sin [{\phi }_1-{\delta }_{\mathrm{eq}}]=E$$

(23)

$$\left\{\begin{array}{l}A={\displaystyle \frac{E_{\mathrm{q}}^{\prime }{U}_0}{X_2}}, B=2E_{\mathrm{q}}^{\prime }{I}_{1\max }, C={\displaystyle \frac{X_2}{X_1}}, D=P_{\mathrm{mg}}(\pi -2{\delta }_{\mathrm{eq}})\\ {\delta }_{\mathrm{eq}}={\delta }_{\mathrm{SEP}}=\arcsin ({\displaystyle \frac{P_{\mathrm{mg}}{X}_2}{E_{\mathrm{q}}^{\prime }{U}_0}}),E=D-A\cos {\delta }_{\mathrm{eq}}\end{array}\right.$$

(24)

In the equation: A, B, C, D, E, and δ_eq are all constants determined by the system topology parameters and operating conditions.

Under the aforementioned operating conditions, S₊ is given by the following:

$$S_{+}=2P_{\mathrm{mg}}({\phi }_1-{\delta }_{\mathrm{SEP}}-\pi )+2P_{\mathrm{eg}1\max }\sin ({\phi }_1-{\delta }_{\mathrm{SEP}})$$

(25)

$$P_{\mathrm{eg}1\max }={\displaystyle \frac{E_{\mathrm{q}}^{\prime }{I}_{1\max }}{1+\frac{X_2}{X_1{K}_1}}}$$

(26)

Subject to operational constraints, the relevant variables must satisfy the following requirements:

$$\left\{\begin{array}{l}{P}_{\mathrm{mg}}\hspace{0.17em}>\hspace{0.17em}\hspace{0.17em}0, K_{1\hspace{0.17em}\hspace{0.17em}}>\hspace{0.17em}\hspace{0.17em}0\\ {\phi }_1={\displaystyle \frac{{\delta }_{\mathrm{SEP}}+{\delta }_T}{2}}+\pi \in ({\delta }_{\mathrm{SEP}}+\pi ,{\displaystyle \frac{\pi }{2}}+\pi )\end{array}\right.$$

(27)

Consequently, the following can be derived:

$$\left\{\begin{array}{l}2P_{\mathrm{mg}}({\phi }_1-{\delta }_{\mathrm{SEP}}-\pi )\in (0,\pi P_{\mathrm{mg}})\\ 2P_{\mathrm{eg}1\max }\sin ({\phi }_1-{\delta }_{\mathrm{SEP}})\in (-2P_{\mathrm{eg}1\max },0)\end{array}\right.$$

(28)

The closer fault f₁ is to Converter 1, the smaller P_eg1max becomes and the larger S₊ grows, making it more difficult for the SG to achieve transient stability. As shown in Figure 8, when the aforementioned two conditions are satisfied, faults occurring at PCC1 represent the most severe scenario for the SG across different fault locations. Compared to other fault points, this location yields the smallest δ_CCA, denoted as δ_CCA-min. When δ_T = δ_CCA-min, the φ₁ value satisfying both conditions ensures the following:

• For any fault location along Line 1, δ_CCA remains greater than δ_CCA-min;
• This φ₁ value maximally increases δ_CCA in the vicinity of δ_CCA-min.

Thus, from the perspective of overall transient stability enhancement, this φ₁ setting provides improvement.

When fault f₁ approaches Converter 1, the previously derived power-angle analytical equation can be simplified to the following:

$$A\cos (2{\phi }_1-{\delta }_{\mathrm{eq}})=E$$

(29)

Incorporating the operational constraints, the solution yields the following:

$${\phi }_1={\displaystyle \frac{\arccos \gamma +{\delta }_{\mathrm{SEP}}}{2}}$$

(30)

$$\left\{\begin{array}{l}\gamma =(\pi -2{\delta }_{\mathrm{SEP}})\cdot \sin {\delta }_{\mathrm{SEP}}-\cos {\delta }_{\mathrm{SEP}}\\ \arccos \gamma \in (2\pi ,3\pi )\end{array}\right.$$

(31)

Thus, the value of φ₁ can be determined solely based on the pre-fault steady-state power angle δ_SEP. In actual operation and maintenance, the control system can monitor δ_SEP in real time, automatically calculate the φ₁ value, and dynamically adjust the current phase angle of current-limiting mode.

5.1.2. Parameter Tuning Methodology for GFL Converter

When a fault occurs on Line 1, Converter 2 becomes decoupled from the system. Under this condition, P_es1 = 0 and P_ms1 depends solely on I_2max and Line 1 parameters. Therefore, minimizing P_ms1 is essential to reduce the rate of change of δ_PLL. Based on Equation (6), to achieve the smallest possible P_ms1, the optimal φ₂ is given by [33]:

$${\phi }_2=-{\theta }_1$$

(32)

5.2. Parameter Tuning Methodology for SG Outgoing Line Faults

During a Line 1 fault, the optimal phase angle for I_2max is set to φ₂ = −θ₁. According to Equation (7), this value simultaneously ensures P_ms2 = 0 when Line 2 faults occur, thereby improving transient stability. However, this φ₂ will cause P_ms2 to be less than P_es2 at the instant of fault inception, resulting in reverse swinging of δ_PLL. Under sustained fault conditions, the system will ultimately stabilize at θ_SEP-New, with the corresponding equivalent power-angle characteristics shown in Figure 9. Based on the discussion of φ₁ range limits in Equation (27), the power-angle curve offset angle of Converter 2’s equivalent electromagnetic power P_es2 must satisfy the following:

$${\phi }_1+{\theta }_2\in ({\displaystyle \frac{3\pi }{2}},2\pi )$$

(33)

Thus, the range of θ_SEP-New is determined to be the following:

$${\theta }_{\mathrm{SEP}-\mathrm{New}}\in (-{\displaystyle \frac{\pi }{2}},0)$$

(34)

If Converter 2 immediately resumes normal operation after fault clearance, P_ms2 will rapidly recover to P_ms0, corresponding to the acceleration area shown by the blue shaded region in Figure 9. Given the substantial acceleration area, Converter 2 is highly susceptible to transient instability during the recovery phase. In this scenario, prolonging its current-limiting mode duration can maintain the equivalent mechanical power near zero, thereby reducing the original acceleration area during recovery. This allows δ_PLL to stabilize via oscillations within the neighborhood of a 0° phase angle before the normal control is restored, ultimately returning δ_PLL to its pre-fault equilibrium point. This strategy can be transformed into a standard operation and maintenance procedure, involving three sequential steps: triggering the current-limiting mode through fault detection, maintaining the current-limiting duration, and progressively restoring the normal control mode.

6. Results

6.1. Case Overview

To validate the correctness of the research content and the effectiveness of the proposed parameter tuning method, this study establishes a corresponding simulation test system in PSCAD based on Figure 2, where both Converter 1 and Converter 2 employ MMC topology, and the SG adopts a 6th-order model. In the test system, the active power outputs of the SG and Converter 2 are 400 MW and 250 MW, respectively, with Converter 2’s reactive power set to 0 MVar. All faults are assumed to occur at 7.0 s, with the main circuit parameters listed in

Table 1. System main circuit parameters.

Equipment	Parameter	Value
Converter 1	Rated Capacity/MVA	1000
	Rated DC Voltage/kV	400
	AC-side Voltage/kV	230
Converter 2	Rated Capacity/MVA	500
	Rated DC Voltage/kV	400
	AC-side Voltage/kV	230
SG	Rated Capacity/MVA	500
	Rated Voltage/kV	10
Line 1	Line Impedance/p.u.	0.046
	Impedance Angle/°	84.2904
Line 2	Line Impedance/p.u.	0.185
	Impedance Angle/°	84.2904

. For the base case (unless otherwise specified), the parameters are configured as follows: fault duration t_f = 0.1 s, faults occurring only on a single line with K₁ = K₂ = 0.5, |I_1max| = |I_2max| = 1.2 p.u., φ₁ = 230°, and φ₂ = 0°. In this case study, Converter 2 is considered transiently unstable if the steady-state value of δ_PLL exceeds ±180°.

6.2. Simulation of Transient Stability Influencing Factors

In studying the influence of the phase angle and magnitude of I_1max on Converter 2, a fault was applied to Line 2. The simulation results in Figure 10a demonstrate that when φ₁ = −20°, δ_PLL exhibits forward swing, and as |I_1max| increases, the rate of forward swing decreases, thereby enhancing stability; when φ₁ = 160°, δ_PLL exhibits reverse swing, and as |I_1max| increases, the rate of reverse swing increases, consequently weakening stability. The mechanism of I_1max’s influence on the dynamic characteristics of δ_PLL is confirmed by the simulation results.

When investigating the influence of the phase angle and magnitude of I_1max on the SG, a fault was applied to Line 1. The simulation results in Figure 10b show that when φ₁ = 0°, I_1max promotes the increase in δ; when φ₁ = 180°, I_1max suppresses the increase in δ. The larger |I_1max| is, the more pronounced its effect on either promoting or suppressing the increase in δ becomes. However, overall, the influence of increasing |I_1max| on the transient stability of the SG remains limited, which aligns with the analytical conclusions.

When investigating the influence of the phase angle and magnitude of I_2max on Converter 2, a fault was applied to Line 1 with a duration of t_f = 0.15 s. The simulation results in Figure 11a,b demonstrate that the phase of I_2max determines the direction of change in the δ_PLL output angle; once the phase of I_2max is determined, an increase in |I_2max| accelerates the change in δ_PLL and weakens the stability of Converter 2, which is consistent with the analytical conclusions presented earlier.

To verify the influence of fault location on transient stability, simulations were conducted on the test system with f₁ and f₂ positioned at different locations.

Simulations with f₁ at different locations in Figure 12a,b reveal that when f₁ is closer to Converter 2, the U_sq of Converter 2 decreases, leading to a slower rate of change in δ_PLL and relatively better synchronization stability; meanwhile, the P_es1 of the SG increases, resulting in a slower rate of change in δ and relatively better power-angle stability.

Simulations of the test system with f₂ at different locations in Figure 13 demonstrate that when f₂ is closer to the SG, the variation trends of δ_PLL and δ become more pronounced. These simulation results are consistent with the analytical conclusions.

Based on the proposed parameter tuning method, the optimal φ₁ and φ₂ values for this test system are determined to be 237.9827° and −84.2904°, respectively. To validate the effectiveness of the proposed approach, simulations of CCT under different φ₁ values were conducted, with the results shown in Figure 14. In this system, since Converter 2 exhibits significantly better stability than the SG during Fault 1, Figure 11b obtained from simulations is used here to verify and explain the optimal φ₂ value rather than CCT.

As shown in Figure 14, the analytical CCT results are conservative due to the omission of system damping and other factors in the analytical model. However, simulation results indicate that the transient stability under the proposed parameter tuning method surpasses that achieved with other values of φ₁ and φ₂, thus verifying the effectiveness of the proposed tuning strategy for φ₁ and φ₂ values.

This paper proposes enhancing transient stability during Line 2 faults by extending the current-limiting mode duration. To validate the effectiveness of the delay-time tuning method, simulations results of CCT under different delay durations during Line 2 faults were conducted in

Table 2. Validation of the effectiveness of the delayed exit duration setting method.

Delay Time/s	CCT/ms	Dominant Unstable Equipment
0.000	30	Converter 2
0.005	102	Converter 2
0.010	124	SG
0.015	124	SG

. The simulations demonstrate that the proposed method significantly increases CCT, primarily improving the transient stability of Converter 2. As the delay duration extends, CCT increases within a certain range, after which the main unstable component shifts from Converter 2 to the SG, and CCT becomes determined by the SG. Beyond this point, further extending the delay duration no longer significantly enhances the transient stability of Converter 2. Therefore, selecting an appropriate exit time for the current-limiting mode to improve system transient stability is feasible.

7. Conclusions

This paper investigates the transient stability of the AC side in a hydro-wind-PV VSC-HVDC transmission system. An analytical model for transient stability analysis is established, revealing the dynamic interactions among various power sources. Based on influencing factors, a parameter tuning method is proposed to enhance the transient stability of the sending-end system. The main research findings are as follows:

• In the context of hydro-wind-PV VSC-HVDC transmission, multiple types of equipment are mutually coupled. Various factors influence its transient stability by affecting the phase and magnitude of the PLL input phasor U_sq and the electromagnetic power of synchronous machines.
• The energy transfer paths and control interactions during transient processes are related to the fault location, parameter settings of the current-limiting mode in converter stations, and the operational states of the equipment.
• This paper derives an optimal phase angle calculation formula for the current-limiting mode of converters and proposes a tuning method for extending the current-limiting mode duration during the recovery stage. These measures aim to enhance system transient stability when transmission line faults occur. The correctness of the analysis and the effectiveness of the proposed method are verified by PSCAD.

This paper focuses on the transient stability of the AC side in hydro-wind-PV VSC-HVDC transmission system, providing a reliable technical approach for the planning and design of multi-energy complementary systems and the optimization of system parameters for multi-source cooperative stability. The research outcomes offer guiding significance for enhancing renewable energy integration levels and ensuring the secure and stable operation of new power systems.