Open AccessArticle Optimal Configuration Strategy for Flexible DC Control Parameters Considering System Operational Constraints by Qiang Guo Qiang Guo SciProfiles Scilit Preprints.org 1, Nan Feng Nan Feng SciProfiles Scilit Preprints.org 1, Yuyao Feng Yuyao Feng SciProfiles Scilit Preprints.org 1, Aiqiang Pan Aiqiang Pan SciProfiles Scilit Preprints.org 1 and Tao Niu Tao Niu SciProfiles Scilit Preprints.org Tao Niu received his Ph.D. degree in Electrical Engineering from Tsinghua University in 2019. He the [...] Read more 2,* 1 State Grid Shanghai Electric Power Research Institute, Shanghai 200437, China 2 School of Electrical Engineering, Chongqing University, Chongqing 400044, China * Author to whom correspondence should be addressed. Processes 2026, 14(12), 1849; https://doi.org/10.3390/pr14121849 (registering DOI) Submission received: 1 April 2026 / Revised: 28 May 2026 / Accepted: 28 May 2026 / Published: 7 June 2026 Abstract With the large-scale integration of renewable energy sources, the stability and control of flexible DC (VSC-HVDC) grid-connected systems have become critical issues. This paper proposes an optimal configuration strategy for the control parameters of grid-forming VSC-HVDC systems considering multiple operational constraints. First, a state-space model of the grid-forming VSC-HVDC system connected to a wind farm is established, and the effects of key control parameters on the small-signal stability are analyzed using eigenvalue and participation factor methods. Then, based on the stability analysis, an optimization model is constructed with the objectives of minimizing the steady-state DC operating voltage under operational constraints and maximizing system damping. To solve the optimization problem, the NSGA-II genetic algorithm is employed. Case studies in MATLAB/Simulink demonstrate that the proposed method effectively enhances the small-signal stability of the system across various operating points, reduces overshoot and settling time during power step changes, and ensures stable operation under the maximum transferable power limit. The results verify the robustness and effectiveness of the proposed parameter configuration strategy, providing a practical approach for the design and tuning of grid-forming VSC-HVDC systems in renewable energy integration applications. Keywords: flexible DC transmission; grid-forming control; parameter optimization; small-signal stability 1. Introduction The global energy landscape is undergoing a profound transformation, characterized by the rapid and large-scale integration of renewable energy sources (RES), primarily wind and solar photovoltaic power [ 1]. This transition is essential for achieving climate goals but introduces formidable technical challenges to the stability, reliability, and controllability of modern power systems. A critical challenge lies in the efficient and reliable transmission of bulk renewable power, often generated in remote, resource-rich areas, to distant load centers [ 2]. Suboptimal parameter selection can degrade dynamic response, cause poorly damped oscillations, or even lead to small-signal instability following minor disturbances [ 13]. This necessitates a systematic approach to parameter design. Small-signal modeling and eigenvalue analysis are fundamental tools for assessing and predicting the stability of such converter-dominated systems [ 14]. Prior research has extensively applied these tools to investigate the stability of VSC-HVDC systems and the impact of various parameters. Studies have analyzed the influence of PLL bandwidth [ 15], current controller gains [ 16], DC voltage control dynamics [ 17], and the interaction between multiple converters [ 18]. Furthermore, the specific stability challenges and parameter design considerations for GFM converters, such as the trade-offs among inertia, damping, and bandwidth, have been explored in [ 19, 20]. However, significant gaps remain when considering the global parameter optimization for a complete grid-forming VSC-HVDC system considering real-world operational constraints. (1) Many existing studies on VSC-HVDC parameter optimization focus on single objectives or a single type of controller, without holistically considering the coupled effects of all major control loops and main circuit parameters [ 27]. (2) A comprehensive set of hard operational constraints are seldom integrated simultaneously into the optimization framework [ 28]. Neglecting these constraints can yield parameter sets that are theoretically stable but practically unfeasible or detrimental to equipment health. (3) While NSGA-II has been applied in related domains, its application to the co-optimization of both control and key main circuit parameters for a GFM-VSC-HVDC link, with stability and economic objectives under full constraints, remains underexplored [ 29, 30]. To bridge these gaps, this paper proposes a comprehensive, constraint-aware, multi-objective optimal configuration strategy for the control parameters of a grid-forming VSC-HVDC system. The main contributions of this work are fourfold: (1) A linearized state-space model of GFM-VSC-HVDC transmission link is developed. A systematic eigenvalue-based analysis is conducted to elucidate the individual and coupled effects of key GFM control parameters on the system’s small-signal stability. (2) The NSGA-II algorithm is employed to solve this complex, high-dimensional, and constrained optimization problem, efficiently searching the parameter space to identify a Pareto-optimal set of system configurations that offer the best trade-offs between the competing objectives. 2. State-Space Model of Grid-Forming VSC-HVDC Grid-Connected System Wind power transmission technology generally employs two approaches: HVAC and HVDC. VSC-HVDC can avoid the impact of line-to-ground capacitance and meets the transmission requirements for large-capacity, long-distance offshore wind power. It does not suffer from commutation failure, possesses the ability to independently regulate active and reactive power, offers flexible operation and control, and can achieve fault isolation between the wind farm and the power grid. 2.1. Topology and Operating Principle of VSC-HVDC Grid Connection The fundamental operational principle of VSC-HVDC grid integration is characterized by a two-stage power conversion process. Initially, the alternating current (AC) power harvested and consolidated at the busbar of direct-drive wind farms is rectified into direct current (DC) power by the rectifier station. This DC power is then transmitted across the high-voltage DC link to the inverter station at the receiving end. Finally, the inverter station performs an inversion process to convert the DC power back into AC power, which is subsequently fed into the receiving-end AC grid. The topological structure of the overall renewable energy integration system via VSC-HVDC is illustrated in Figure 1. Comprising a wind farm, a sending-end voltage source converter (SE-VSC), a receiving-end voltage source converter (RE-VSC), DC transmission lines, and the host AC grid, the entire system constitutes a typical multi-terminal power transmission architecture. In this paper, the large-scale wind farm is modeled using a single-machine equivalent aggregation method. The dynamic characteristics of active and reactive power output at the grid-connected port are retained, while the subtle internal differences among individual wind turbines are neglected. This modeling approach is suitable for small-signal stability analysis and control interaction research of the system. The instantaneous active and reactive power output at the point of common coupling (PCC) of the wind farm are expressed as: P w = U s w I w d Q w = U s w I w q (1) where U s w is the AC voltage amplitude at the grid-connected point of the wind farm; I w d , I w q are the d-axis and q-axis components of the wind farm output current, respectively; P w , Q w denote the equivalent active and reactive power output of the wind farm. Considering the power regulation inertia of wind turbine converters, a first-order inertial link is adopted to characterize the power dynamic response of the wind farm: T w d P w d t = P w r e f − P w T w d Q w d t = Q w r e f − Q w (2) where T w is the equivalent response time constant of the wind farm; P w r e f and Q w r e f are the reference values of active and reactive power of the wind farm, respectively. The output current dynamics of the wind farm satisfy the grid-connected voltage balance relationship: L w d I w d d t = U w d r e f − U s w + ω L w I w q L w d I w q d t = U w q r e f − ω L w I w d (3) where L w is the equivalent grid-connected filter inductance of the wind farm; ω is the system synchronous angular frequency; U w d r e f and U w q r e f are the modulation voltage reference values of the wind turbine grid-side converter. Wind speed fluctuation and operating condition switching of the wind farm induce random disturbances in P w and Q w , which directly cause voltage and power flow deviation at the sending-end grid-connected point, and further disturb the power reference commands and AC-side operating point of the sending-end VSC. The disturbance signals are transmitted step by step through the inner and outer loop control of the converter and the charging–discharging dynamics of the DC link capacitor, altering the system damping configuration and oscillation modes, and ultimately affecting the small-signal stability and transient disturbance recovery performance of the entire VSC-HVDC system. Therefore, incorporating the equivalent dynamic model of the wind farm is an essential prerequisite for guaranteeing research integrity and analysis accuracy. This paper mainly focuses on the converter control of VSC-HVDC, DC link dynamics, and system low-frequency small-signal stability, with the research frequency band concentrated in the low-frequency oscillation range. Accordingly, reasonable simplifications are adopted for the wind farm as follows: (a) The mechanical transients of the wind turbine shaft system and the high-frequency dynamics of pitch angle control are neglected. The modal frequencies of these components are much higher than the frequency band of interest in this paper and have negligible influence on the dominant oscillation modes of the system; (b) A clustered equivalent model is used to replace the detailed modeling of individual wind turbines. While maintaining the external characteristics of voltage, power, and current at the grid-connected port unchanged, the system order is greatly reduced, which facilitates state-space modeling and eigenvalue analysis; (c) The first-order inertial dynamics of wind farm power and grid-connected current control dynamics are retained, which can fully characterize the disturbance effect of wind speed fluctuation and output variation on the sending-end VSC-HVDC system. 2.2. Modeling of the Sending-End and Receiving-End Converter Station Employed for large-scale wind power integration, the sending-end converter station is usually separated from load centers and primarily acts as a collector and consolidator of power produced by wind farms. This station generally adopts the constant voltage/frequency ( v- f) control strategy to keep the output voltage and frequency at their preset reference values. In detail, the fixed AC voltage and frequency are regulated by its outer loop, while the inner loop adopts a dual loop control structure that incorporates voltage and current regulation. Figure 2 illustrates the corresponding control block diagram of the sending-end converter station. In Figure 2, for the v- f control, ω f r e f is the per-unit reference value of the given angular frequency, ω f is the actual value of the given angular frequency, and θ f is the output phase angle. U g v r e f is the reference value for the outer-loop AC voltage control, U g v m is the measured voltage at the collection bus, u g d v r e f is the d-axis voltage output from the outer-loop AC voltage control, and the q-axis voltage reference is set as u g q v r e f = 0 . Further, i e d r e f and i e q r e f are the d- and q-axis current reference quantities output from the inner-loop voltage control, respectively, while iedm and ieam are the measured d- and q-axis currents of the SE-VSC AC line, respectively. Additionally, i g d v m and i g q v m are the measured d- and q-axis currents of the AC transmission line, respectively, and u g d v m and u g q v m are the measured d- and q-axis voltages at the collection bus, respectively. The specific expressions for the d- and q-axis voltage/current measurement links and the collection bus voltage measurement are as follows: d i e d m d t = i e d − i e d m T i e d , d i e q m d t = i e q − i e q m T i e q d i g d v m d t = i g d v − i g d v m T i g d v , d i g q v m d t = i g q v − i g q v m T i g q v d u g d v m d t = u g d v − u g d v m T u g d v , d u g q v m d t = u g q v − u g q v m T u g q v U g m = ( u g d v m ) 2 + ( u g q v m ) 2 (4) where i e d and i e q are the d- and q-axis current components of i e ; i g d and i g q are the d- and q-axis current components of i g ; u g d and u g q are the d- and q-axis voltage components of u g . T i e d , T i e q , T i g d , T i g q , T u g d , and T u g q are the time constants of the inertial measurement links for the d- and q-axis voltages and currents, respectively. Defining the integral of the error between the reference and actual values of the outer-loop AC voltage PI controller as x u g v , the mathematical model of the outer-loop fixed-frequency and constant AC voltage control is as follows: d θ f d t = ω f = ω B a s e θ f r e f d x u g v d t = U g v r e f − U g v m , u g d v r e f = K p u g v d x u g v d t + K i g v g x u g v (5) where K p u v and K i u v are the proportional and integral parameters of the outer-loop AC voltage control, respectively. Defining the integrals of the error between the reference and actual values for the PI controllers in the dual voltage-current closed loops as x u g v d , x u g v a , x i e d , and x i e q , the mathematical model of the decoupling control for the voltage inner loop can be obtained as follows: d x u g r d d t = u g d r f − u g d r m i e d r f = i g d r m − K p g n d d x u g r d d t + K i u g r d x u g r d + ω f C 2 u g p m d x u g r v d t = u g q v r − u g q m i e q r f = i g q m − K p q n g d x u g r v d t + K i q n g x u g r v − ω f C 2 u g d r m d x i e d d t = i e d r f − i e d m u e d = u g d r m − K p i e d d x i e d d t + K i i e d x i e d + ω f L 2 i e q m d x i e q d t = i e q r f − i e q m u e q = u g q m − K p i e q d x i e q d t + K i i e q x i e q − ω f L 2 i e d m (6) where K p u g v d and K i g v g d are the proportional and integral parameters of the inner-loop d -axis voltage control, respectively; K p u g v q and K i g v q are the proportional and integral parameters of the inner-loop q -axis voltage control, respectively. K p i d and K i i d are the proportional and integral parameters of the inner-loop d -axis current control, respectively; K p i e q and K i e q are the proportional and integral parameters of the inner-loop q -axis current control, respectively. Finally, u e d and u e q are the reference voltages on the AC side of the SE-VSC output by the inner-loop current control. The differential equations for the dq-axis voltage components ( u g d v , u g q v ) of the wind farm collection bus voltage u g and the dq-axis current components ( i e d , i e q ) of the current i e flowing through the SE-VSC AC line are as follows: d u g d r d t = 1 C 2 ( i g d r − i e d + ω f C 2 u g q r ) d u g q r d t = 1 C 3 ( i g q r − i e q − ω f C 2 u g d r ) d i e d d t = 1 L 2 ( u g d r − u e d + ω f L 2 i e q ) d i e q d t = 1 L 3 ( u g q r − u e q − ω f L 2 i e d ) (7) The DC link capacitor is a critical energy buffering unit of the VSC-HVDC system, which undertakes the functions of AC-DC power balance, DC voltage support, filtering, and voltage regulation. Its dynamic characteristics directly restrict system stability and transient response performance. The dynamic equation of DC link voltage can be expressed as: C d c d U d c d t = P a c − P d c (8) where C d c is the equivalent capacitance of the DC link; U d c denotes the DC bus voltage; P a c represents the active power input at the AC side of the converter; P d c is the active power delivered by the DC line. When load disturbance, grid fault, or operating condition switching occurs in the system, an instantaneous AC-DC power imbalance arises, which triggers continuous charging and discharging of the DC capacitor and further causes dynamic fluctuation of the DC bus voltage. The DC voltage fluctuation signal is fed back to the outer-loop DC voltage control and inner-loop current control of the converter, changing the control reference commands and modulation outputs, and forming strong coupling with the electrical dynamics of the AC network. Ultimately, it affects the system damping characteristics, low-frequency oscillation modes, and transient recovery process. The capacitance value of the DC link has a remarkable impact on system dynamic characteristics. A small capacitance enables a fast DC voltage response but weak anti-disturbance capability, which easily induces voltage oscillation during operating condition switching. A large capacitance can effectively suppress DC voltage fluctuations and improve system voltage inertia and stability, yet it reduces the rapidity of system dynamic regulation. Therefore, fully considering the inherent dynamics of the DC link and its interaction with the hierarchical control system is an essential prerequisite for analyzing the small-signal stability and transient performance of VSC-HVDC systems. The strength of the receiving-end AC grid tends to weaken progressively with the increasing integration of renewable energy generation, DC transmission systems, and power electronic devices. To proactively provide voltage and frequency support to the receiving-end AC grid while ensuring DC voltage stability, a grid-forming control strategy is adopted for the RE-VSC in this paper. The distinct naming of variables and control parameters for the receiving-end converter station serves solely to differentiate them from those associated with the grid-forming direct-drive wind turbine side converter. The control block diagram of the receiving-end converter station is shown in Figure 3. After completing the modeling of the sending-end and receiving-end converter stations, we supplement and present the unified nonlinear mathematical model of the system and the standardized small-signal state-space expression after linearization. The composition form and physical significance of the state matrix A, input matrix B, output matrix C, and direct transmission matrix D are fully given. Specifically, all nonlinear differential equations and algebraic equations of the system are combined to establish the unified system model: x ˙ = f ( x , u ) , y = g ( x , u ) (9) Then, the standardized small-signal model is derived by linearization at the steady-state operating point: Δ x ˙ = A Δ x + B Δ u , Δ y = C Δ x + D Δ u (10) 3. Stability Analysis and Optimal Configuration of Control Parameters for the System The practical operational capability of a VSC-HVDC grid-connected system is affected by its small-signal stability. Whether the system can achieve its physical limits during actual operation or maintain good operational stability under extreme conditions depends not only on the selection of main circuit parameters and grid characteristics but also critically on the choice of control parameters. The quality of control parameter selection directly influences the system’s small-signal stability. Improper parameter selection can degrade the system’s dynamic performance and increase the risk of oscillatory instability under small disturbances. Therefore, tuning and optimizing the control parameters of the grid-forming VSC-HVDC grid-connected system is beneficial for improving the system’s dynamic response characteristics and ensuring good small-signal stability even under severe operating conditions. 3.1. Analysis of the Impact of Control System Parameters on Small-Signal Stability (1) Impact of inertia time constant on system stability When the inertia time constant T H of the phase-side generation link is gradually increased from 0.02 to 0.22, the resulting partial trajectory of the system eigenvalues near the imaginary axis is shown in Figure 4. As shown in Figure 4, as the inertia time constant T H gradually increases, the eigenvalue corresponding to the system’s dominant mode λ 34 is a negative real root on the real axis when T H 3 U N 2 γ 2 + 9 U d c 2 1 4 ω 36 ω 2 C 0 2 U c o 2 − I d c 2 (15) ω res = 1 2 N L arm C 0 < 1.55 ω (16) Traditional optimization methods are often limited when addressing such high-complexity optimization problems, as they cannot adequately handle the coupling relationships among multiple variables and the challenges posed by multiple constraints. To overcome the issues encountered in global parameter optimization, this paper employs the NSGA-II genetic algorithm as the optimization algorithm for the global parameter optimization design of the grid-forming VSC-HVDC grid-connected system, aiming to achieve better parameter optimization results. Genetic algorithms, based on the principle of “survival of the fittest and elimination of the inferior” from Darwin’s theory of evolution, simulate natural evolutionary processes. The fundamental idea is to transform the problem to be solved into individuals (which may be in binary, integer, real number, or other forms) and then evaluate the fitness of each individual using a fitness function. Based on this evaluation, genetic operations such as crossover and mutation are performed to generate new offspring. This process repeats for multiple generations until Pareto-optimal solutions are found, i.e., a set of solution vectors in the solution space where no solution vector dominates another. Due to their ability to automatically adjust and optimize the search direction and their good global optimization capability, genetic algorithms have been widely applied to various complex multi-objective optimization scenarios in recent years. However, NSGA suffers from issues such as inefficient non-dominated sorting, difficulty in maintaining population diversity, and challenges in parameter selection, which severely hinder its application. In the planning and design of the VSC-HVDC system using this method, the first step is to clarify the maximum power transfer capability of the converter, i.e., the maximum power output on the AC side. The next step is to determine the maximum power transmission capacity of the converter. Therefore, based on the active power transmission limit constraints of the converter station, the maximum transmissible power limit for the AC-side converter in the area can be calculated and used as the setpoint for the optimization algorithm. Subsequently, multi-objective optimization is performed for the selection of various system parameters. Combining the aforementioned analysis, the optimization objectives of the system can be expressed as f 1 = min U d c f 2 = min max Re ( λ i ) (17) The optimization objective of “minimization of DC voltage level” adopted in this paper does not refer to reducing voltage deviation, suppressing overvoltage, minimizing voltage ripple, or changing the rated operating point. In the flexible DC transmission system, the DC voltage is still regulated in a closed-loop manner around its rated value. The actual implication of this objective is: under the premise of satisfying all operational constraints including power transmission capability, voltage modulation ratio, capacitor voltage fluctuation, and equipment insulation stress, the steady-state operating point of DC voltage is reduced to the lowest feasible value within the allowable range. The underlying purpose is to reduce the electrical stress and thermal stress of key equipment such as IGBTs, DC capacitors and DC transmission lines, lower the requirements for insulation design, and improve the operational safety and economic efficiency of the system. Meanwhile, the optimal DC voltage operating point satisfying all constraints is determined while guaranteeing the maximum transmittable power. We have fully supplemented the above explanations in Section 3.2: Optimization Objective Configuration of the revised manuscript, and revised the relevant statements in the Abstract and Conclusions. This revision clarifies the definition, physical significance, and engineering value of the optimization objective, and thoroughly resolves the ambiguity of the objective interpretation in the original version. A systematic methodology for the global parameter optimization of the grid-forming VSC-HVDC grid-connected system is established in this paper. The design procedure commences with characterizing the AC grid at the renewable energy-sending side, specifically by defining its equivalent short-circuit ratio (SCR) and impedance profiles. Taking the maximization of transmissible active power as the core design objective for the converter station, the maximum deliverable power capacity is first derived based on stringent active power transmission limitations. Subsequently, the feasible parameter ranges for the primary circuit components are delineated. Concurrently, the constraint boundaries for key electrical variables, which are essential for the subsequent multi-objective optimization framework, are explicitly formulated. Based on the preliminary conclusions from small-signal stability analysis under varying parameters, the approximate ranges for control parameters that can maintain system small-signal stability are provided. Using the NSGA-II multi-objective optimization algorithm, with (11) as the objective function and (5)–(10) as the constraint function, several sets of main circuit and control parameter values are solved that achieve the lowest DC voltage level and the best system small-signal stability. An optimal set is then selected as the final design parameters. The corresponding flowchart is shown in Figure 10. It should be noted that as a metaheuristic evolutionary algorithm, NSGA-II does not provide a strict theoretical guarantee of global optimality. However, for the high-dimensional, non-convex, multi-objective, and strongly constrained optimization problem in this paper, traditional convex optimization methods are difficult to apply. The NSGA-II algorithm with fast non-dominated sorting and elitist preservation can effectively search the Pareto-optimal front, avoid local optima, and obtain solutions that meet the engineering optimality requirements of grid-forming VSC-HVDC systems. Based on the above design process, a complete set of global parameter optimal design methods for the grid-forming VSC-HVDC grid-connected system that meets the expected power delivery level of the converter station can be obtained. This method remains applicable even when AC system characteristics change or component specifications are modified. 4. Case Analysis To verify the effectiveness of the proposed parameter configuration method, a grid-forming VSC-HVDC model with large-scale wind power integration was built in the MATLAB R2023b environment. The selected IGBT device parameters are 3.5 kV/1 kA. According to the IGBT device selection rules, a voltage withstand margin of 1.5 times should be reserved. Therefore, the submodule voltage limit is known to be within 2.3 kV. The submodule capacitance range is selected as 1~5 mF. The basic parameters of the NSGA-II genetic algorithm are shown in Table 1. The parameters of the NSGA-II algorithm in this study are set to a population size of 200, and the maximum number of iterations at 100. The detailed computing environment and time are listed below: the optimization calculation is conducted on a computer equipped with an Intel Core i7-12700K @ 3.60 GHz processor and 32 GB RAM. Under the above hardware configuration, the total computational time for the entire parameter optimization process is approximately 12.42 s. To ensure the reproducibility of optimization results, all key parameters of NSGA-II are fixed in this paper, including population size of 200, maximum iterations of 100, crossover probability of 0.75, and mutation probability of 0.08. A fixed random seed is adopted in the program. Ten independent optimization runs are carried out under the same conditions, and the results show that the Pareto front distributions are highly consistent, with the parameter deviation less than 1%. The final optimal parameters are selected from the stable central region of the Pareto front, which further ensures the robustness and repeatability of the proposed strategy. To further validate the rationality of the selected objective functions in Equation (11), comparative evaluations with alternative objective combinations were conducted during the parameter design phase. A commonly utilized alternative is the dynamic performance-driven objective, such as minimizing the integral time absolute error (ITAE) of the power step response, or minimizing the transient overshoot and settling time. Simulations indicate that while these dynamic error-driven objectives are indeed highly effective in accelerating system response and achieving precise waveform tracking, they tend to yield overly aggressive control parameters (e.g., excessively high proportional gains in PI controllers). Consequently, the system’s small-signal stability margin is severely compromised, resulting in a lower damping ratio for the dominant modes. This makes the system more susceptible to oscillatory instability during severe operating point drifts or grid faults. Furthermore, optimizing purely for control dynamics without incorporating steady-state electrical levels as an objective often leaves the system operating at an unnecessarily high steady-state DC voltage. As analyzed previously, this imposes continuous elevated electrical and thermal stresses on critical main circuit components such as IGBTs and DC-link capacitors, which ultimately increases insulation requirements and reduces operational economy. Therefore, the combination of maximizing system damping and minimizing the steady-state DC operating voltage, as formulated in Equation (11), provides a superior and more practical Pareto-optimal solution. It successfully strikes a balance between securing robust small-signal stability (safety) and alleviating physical stress on hardware (economics), strictly adhering to the fundamental engineering requirements of VSC-HVDC systems. The main circuit and control system parameters of the grid-forming VSC-HVDC grid-connected system, obtained through the global parameter optimization design method proposed in this paper, are presented in Table 2. A comparison of the eigenvalue distributions of the system before and after optimization under the same power transmission level is shown in Figure 11. It can be seen that before optimization, the dominant eigenvalue of the system was a single negative eigenvalue on the real axis at −1.872. After optimization, the dominant eigenvalues become a pair of conjugate eigenvalues at −4.831 and −2.056. The dominant eigenvalues have moved leftward, away from the imaginary axis. Furthermore, it can be observed that most of the other eigenvalues of the optimized system have shifted leftward compared to those before optimization, indicating an improvement in the system’s small-signal stability. Although the dominant mode exhibits an underdamped oscillatory characteristic, its decay coefficient ensures rapid energy dissipation after a disturbance. Its equivalent damping ratio is approximately 0.91, close to critical damping, achieving a good balance between rapidity and smoothness. The significant increase in the absolute value of the real part of the eigenvalues directly corresponds to a faster decay rate of the system’s state variables. This indicates that the optimized parameters effectively enhance the bandwidth and response speed of each control loop. The overall leftward shift of the remaining eigenvalues suggests that not only the dominant mode but also the stability of higher-frequency modes related to circulating current suppression and delay links has been enhanced. This comprehensively reduces the risk of system resonance or oscillation across different frequency bands, further validating the rationality of the system parameters selected by the optimization algorithm proposed in this paper. Based on the optimized system, the active power transmission level is gradually increased from 0.32 p.u. to 0.76 p.u. The eigenvalue loci of the grid-forming VSC-HVDC grid-connected system are shown in Figure 12. This indicates that the optimized parameter set exhibits good consistency across different operating points, and changes in the system load linearization points do not lead to alterations in the dominant instability modes. Although the real parts of some modes shift slightly to the right as the power increases, they always maintain a sufficient distance from the imaginary axis. This verifies that the optimized design is not only targeted at a specific operating condition but also considers the operating point drift caused by power fluctuations, ensuring global stability across the expected operational range. As shown in Figure 12b, for the optimized grid-forming VSC-HVDC grid-connected system at different power transmission levels, all system eigenvalues are located in the left half of the complex plane, confirming that the system maintains small-signal stability at each power level. The systems before and after optimization are simulated and verified using an electromagnetic transient model in Simulink in Matlab R2023b. The initial operating condition is set with the system transmitting 2.5 MW active power in rectifier mode. At 5 s, the transmitted active power steps from 2.5 MW to 2.7 MW. The dynamic response processes of various electrical quantities under this condition for the systems before and after optimization are compared in Figure 13. From the comparison of the dynamic responses of the electrical quantities before and after optimization in Figure 13, it can be observed that the significant reduction in the active power response overshoot directly corresponds to the optimized adjustment of the damping coefficient D in the phase generation link and the retuning of the current inner-loop PI parameters, which effectively suppresses the adverse interaction between the power loop and the current loop. The reduction in settling time reflects an increase in the overall closed-loop system bandwidth. This benefits from the retuning of the DC voltage control loop PI parameters and the coordinated optimization of main circuit parameters such as the arm inductance and submodule capacitance, which collectively improve the dynamic energy balancing speed from the DC side to the AC side. The more stable responses of the d- and q-axis voltages and currents at the PCC indicate that the optimized dual-loop controller parameters have enhanced the decoupling control performance, reduced the dynamic coupling between active–reactive power and d-q axes, resulting in less impact on the AC grid during power transients. Under the same power step condition, the optimized system shows significant improvement in both overshoot and system damping compared to the system before optimization, which also verifies the effectiveness of the parameter optimization method proposed in this section. Scenario A in Figure 14a: Active power step from 0.5 p.u. to 0.55 p.u. We have retained and refined the small-signal power step test occurring at 5 s, as suggested. The comparison results clearly reproduce the original conclusion: the optimized system (red solid line) exhibits smaller overshoot and faster convergence speed compared with the unoptimized one (blue solid line). Scenario B in Figure 14b: New severe disturbance case. To verify the robustness against large disturbances, a temporary three-phase short-circuit fault lasting 0.1 s at the point of common coupling (PCC) is simulated at 6 s. The comparative results demonstrate that, in the face of severe voltage sag and subsequent large-current transients, the proposed optimization method can effectively suppress violent oscillations during the fault and achieve a remarkably faster stability recovery after fault ride-through. Scenario C in Figure 14c: New case with different loading levels. To validate robustness under different initial operating states, the system performance is evaluated under a high initial loading level of 0.8 p.u. An identical power step change of 0.03 p.u. (rising to 0.83 p.u.) is applied at 5 s. The comparison confirms that even under near-full-load operating conditions, the proposed control method can still maintain excellent dynamic damping performance, and the decoupling control between active and reactive power remains stable and robust. To further validate the effectiveness of the proposed NSGA-II-based optimization strategy, comparative studies are carried out with the empirical tuning method and particle swarm optimization (PSO). Three typical indexes are compared: dominant eigenvalue real part (stability margin), power response overshoot, and settling time. The comparison results are listed in Table 3. As can be seen from Table 3, the proposed NSGA-II method exhibits obvious advantages over the empirical tuning method and MOPSO algorithm in all key indexes. In terms of small-signal stability, the real part of the dominant eigenvalue obtained by NSGA-II is −4.831, which is much farther away from the imaginary axis than those of empirical tuning (−1.872) and MOPSO (−2.945), indicating a significant improvement in stability margin. For dynamic response performance, the active power overshoot is controlled within 2.15%, and the settling time is only 0.13 s, which are obviously superior to the other two methods. The above quantitative comparison fully demonstrates that the parameter configuration obtained by NSGA-II can better balance the requirements of stability, rapidity, and damping, thus verifying the effectiveness and advancement of the proposed strategy. 5. Conclusions This paper presents a systematic method for optimally configuring the control parameters of a grid-forming VSC-HVDC system, considering both small-signal stability and practical operational constraints. A detailed state-space model was established, and eigenvalue analysis revealed the critical impact of virtual inertia, damping, and DC voltage control parameters on system stability. A multi-objective optimization model was then formulated and solved using the NSGA-II algorithm, successfully balancing the trade-off between maximizing system damping and minimizing steady-state DC operating voltage under full operational constraints while satisfying all operational limits. Case studies demonstrated the effectiveness of the proposed strategy. The optimized parameters significantly enhanced the system’s small-signal stability margin, as evidenced by a leftward shift of the eigenvalue spectrum. Time-domain simulations confirmed superior dynamic performance, with substantially reduced overshoot and shorter settling time during power transients. Importantly, the system maintained stable operation across the entire power range, including at the transmission limit. In summary, this work provides a practical, constraint-aware framework for the design of stable and efficient grid-forming VSC-HVDC systems. The proposed methodology offers a valuable tool for ensuring the reliable integration of large-scale renewable energy into modern power grids. Author Contributions Conceptualization, Q.G. and Y.F.; methodology, Q.G., Y.F. and A.P.; software, Q.G. and Y.F.; validation, Q.G.; formal analysis, Q.G.; investigation, N.F.; resources, N.F. and Y.F.; data curation, N.F., A.P. and T.N.; writing—original draft, Q.G., N.F. and Y.F.; writing—review and editing, Q.G., N.F., Y.F. and T.N. All authors have read and agreed to the published version of the manuscript. Funding This research was supported by the Science and Technology Foundation of the State Grid Corporation of China (Grant No. 52094025002U). Data Availability Statement The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Conflicts of Interest This study received funding from the State Grid Corporation of China (SGCC). The authors of the paper, Qiang Guo, Nan Feng, Yuyao Feng, and Aiqiang Pan, are employees of Shanghai Electric Power Research Institute, which is affiliated with SGCC. They participated in the
