Transient Stability Analysis for the Wind Power Grid-Connected System: A Manifold Topology Perspective on the Global Stability Domain
Abstract
1. Introduction
- Equivalent Model of DFIG: By analyzing the dynamic behavior of DFIGs during faults and their impact on the power grid, a simplified equivalent model for grid-connected DFIGs is established.
- Precise BSD Characterization via Manifold–Singularity Synergy: Through analyzing the stable/unstable manifold topological structures near system equilibrium points and incorporating SAIs to characterize nonlinear behaviors under extreme conditions, high-precision BSD computation is achieved, providing new insights for stability analysis of wind power grid-connected systems.
- Quantitative Mechanism of DFIG’s Impact on Stability Domain: By studying the BSD topology, the relationship between DFIG penetration rate and stability margins is revealed, elucidating the dynamic mechanism of stability domain contraction under high wind power penetration.
- Transient Stability Enhancement via Principle of Singularity Invariance (PSI): A transient stability principle based on preserving the topological characteristics of SAIs is proposed, demonstrating stability margin improvement through parameter regulation in the practical IEEE 39-bus system. The comparative study of transient stability margin enhancement in this paper is through a comparison based on the specific physical mechanisms of singularity invariance and energy function invariance. This is significantly different from the current mainstream cutting-edge methods (e.g., data-driven transient stability rapid assessment based on transient stability [13,14], AI-assisted optimal control strategies [15,16], etc.) in terms of theoretical foundation, modeling assumptions, optimization goals, and application focus. While the existing frontier methods mainly focus on rapid identification of destabilization modes or data-based prediction of control boundaries, the core of this study is to reveal the intrinsic constraints on stability margins imposed by a specific physical mechanism and to derive a direct principle. Therefore, at this stage, directly comparing the quantitative performance of the results of this study with these methods with different theoretical foundations under a unified standard may make it difficult to fairly evaluate their respective advantageous features. As it is well known that extensive comparisons for verifying innovativeness and generalizability are important, these will be the key directions for the future in-depth development of this study.
2. Modeling of the Wind Power Grid-Connected System and Its Global BSD Topology Characterization
2.1. Dynamic Characteristics of DFIG
- Transient reactance:
- Internal voltage:
2.2. Multi-Machine System Dimensionality Reduction
2.3. The Local Manifold Topology
- For the SEP , the sets of trajectories converging to and diverging from the SEP are, respectively, defined as:
- For the UEP , the corresponding manifolds are:
2.4. The Global Manifold Topology
- The tangent plane at ;
- The spherical surface at .
2.5. The Principle of Singularity Invariance
2.6. Restrictive Conditions
- The damping ratio shows positive correlation with damping ()—increasing enhances and accelerates oscillation decay.
- Negative correlation exists with inertia ()—larger J reduces , slowing system response and potentially degrading stability.
- : faster response but higher overshoot.
- : critically damped with significantly slower response.
3. Simulation Analysis
3.1. Validation of the Proposed Method
3.2. Validation of the Superiority of the Proposed Method
3.2.1. Selection of Different J
3.2.2. Selection of Different
3.2.3. Overcoming the Conservativeness of the EF Method
3.3. Sensitivity Analysis
4. Discussion
- (1)
- Addressing the randomness of new energy output: Current research is based on deterministic conditions, while actual wind power output exhibits significant randomness and intermittency. Future research can integrate stochastic dynamic system theory, introduce a Weibull distribution wind speed model, and analyze the dynamic perturbation effects of power fluctuations on the stability domain. Additionally, through stochastic differential equations and Monte Carlo simulations, the stability probability of the system under uncertainty can be quantified, thereby enhancing the robustness and practical applicability of the method.
- (2)
- High-dimensional system expansion: In large test cases with thousands of variables, the proposed method demonstrates good scalability and computational efficiency by combining the manifold topology theory and SAI analysis. Firstly, the high-dimensional multi-machine system is reduced to a low-dimensional nonlinear differential equation by the equivalent model of the DFIG and the dimensionality reduction technique (e.g., the SIME method), and the specific dimensionality reduction process can be referred to in Section 2.2. So as long as the equations of a large test system with thousands of variables are derived, its transient power angle stability can be analyzed using the method of this paper. At this stage, it is not possible to study the performance of the method of this paper in real large-scale test system optimization problems, and we will continue to study it in depth in the future.
- (3)
- Extension to other methods: In existing studies, there are various methods for the transient stability analysis of wind power grid-connected systems, but each of them has certain limitations. In order to more comprehensively assess the advantages and applicability of the method proposed in this paper, Table 3 systematically compares this proposed method with the existing reference methods in terms of the core method, research focus, etc.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Abbreviations | |
DFIG | Doubly fed induction generator |
BSD | Boundary of stability domain |
SEP | Stable equilibrium point |
UEP | Unstable equilibrium point |
SAIs | Singularities at infinity |
OMIB | One-machine infinite bus |
PSI | Principle of singularity invariance |
CCT | Critical clearing time |
EF | Energy function |
SIME | Single-machine equivalent |
SG | Synchronous generator |
3D | Three-dimensional |
IDRS% | Evaluation index of dynamic response speed |
ICCT% | Evaluation index of critical clearing time |
VSG | Virtual synchronous generator |
DC | Direct current |
CPS | Cyber–physical systems |
Symbols | |
Stator voltage | |
Stator current | |
Rotor current | |
Stator flux linkage | |
Rotor flux linkage | |
Stator resistance | |
Rotor resistance | |
Synchronous angular velocity | |
Stator self inductance | |
Rotor self inductance | |
Mutual inductance | |
Transient reactance | |
Internal voltage | |
Voltage phase angle at the point of DFIG grid connection | |
s | Slip ratio |
Active power | |
Reactive power | |
DFIG equivalent impedance | |
Impedance between generator G1 and reference node | |
Impedance between infinite power source and reference node | |
Self impedance | |
Self-impedance angle | |
Mutual impedance | |
Mutual impedance angle | |
Equivalent resistance | |
Equivalent reactance | |
Excitation voltage | |
System voltage | |
Central angle of S group | |
Central angle of A group | |
Inertia of S group | |
Inertia of A group | |
Self conductance | |
Mutual susceptance | |
J | Inertia |
Damping | |
The power angle of the i-th generator | |
The angular velocity of the i-th generator | |
Relative rotor angle | |
Relative angular velocity | |
Total inertia | |
Equivalent inertia | |
Equivalent damping | |
Mechanical power | |
Electromagnetic power | |
Reference power | |
Damping patio | |
Overshoot | |
Settling time | |
Synchronous angular frequency | |
Natural angular frequency | |
Critical clearing time | |
Stable manifold | |
Unstable manifold | |
Auxiliary coordinates | |
Coordinates in the original system | |
Projected coordinates | |
State space radius | |
Projected space radius | |
Trajectory function | |
Phase shift angle | |
SAI coordinate parameters | |
SAI position functions |
References
- Kim, J.; Bialek, S.; Unel, B.; Dvorkin, Y. Strategic Policymaking for Implementing Renewable Portfolio Standards: A Tri-Level Optimization Approach. IEEE Trans. Power Syst. 2021, 36, 4915–4927. [Google Scholar] [CrossRef]
- Alsmadi, Y.M.; Xu, L.; Blaabjerg, F.; Ortega, A.J.P.; Abdelaziz, A.Y.; Wang, A.; Albataineh, Z. Detailed Investigation and Performance Improvement of the Dynamic Behavior of Grid-Connected DFIG-Based Wind Turbines Under LVRT Conditions. IEEE Trans. Ind. Appl. 2018, 54, 4795–4812. [Google Scholar] [CrossRef]
- Mohammadpour, H.A.; Santi, E. SSR Damping Controller Design and Optimal Placement in Rotor-Side and Grid-Side Converters of Series-Compensated DFIG-Based Wind Farm. IEEE Trans. Sustain. Energy 2015, 6, 388–399. [Google Scholar] [CrossRef]
- Oraa, J.; Samanes, J.; Lopez, J.; Gubia, E. Single-Loop Droop Control Strategy for a Grid-Connected DFIG Wind Turbine. IEEE Trans. Ind. Electron. 2024, 71, 8819–8830. [Google Scholar] [CrossRef]
- Ghosh, S.; Bakhshizadeh, M.K.; Yang, G.; Kocewiak, Ł. An extended nonlinear stability assessment methodology for type-4 wind turbines via time reversal trajectory. CSEE J. Power Energy Syst. 2025, 1–10. [Google Scholar]
- Faried, S.O.; Billinton, R.; Aboreshaid, S. Probabilistic Evaluation of Transient Stability of a Wind Farm. IEEE Trans. Energy Convers. 2009, 24, 733–739. [Google Scholar] [CrossRef]
- Muljadi, E.; Butterfield, C.P.; Parsons, B.; Ellis, A. Effect of Variable Speed Wind Turbine Generator on Stability of a Weak Grid. IEEE Trans. Energy Convers. 2007, 22, 29–36. [Google Scholar] [CrossRef]
- Zhang, X.; Huang, Y.; Fu, Y. Oscillation Energy Transfer and Integrated Stability Control of Grid-Forming Wind Turbines. IEEE Trans. Sustain. Energy 2025, 16, 826–839. [Google Scholar] [CrossRef]
- Bakhshizadeh, M.K.; Ghosh, S.; Yang, G.; Kocewiak, Ł. Transient Stability Analysis of Grid-Connected Converters in Wind Turbine Systems Based on Linear Lyapunov Function and Reverse-Time Trajectory. J. Mod. Power Syst. Clean Energy 2024, 12, 782–790. [Google Scholar] [CrossRef]
- Odun-Ayo, T.; Crow, M.L. Structure-Preserved Power System Transient Stability Using Stochastic Energy Functions. IEEE Trans. Power Syst. 2012, 27, 1450–1458. [Google Scholar] [CrossRef]
- Wang, X.; Chiang, H.-D.; Wang, J.; Liu, H.; Wang, T. Long-Term Stability Analysis of Power Systems with Wind Power Based on Stochastic Differential Equations: Model Development and Foundations. IEEE Trans. Sustain. Energy 2015, 6, 1534–1542. [Google Scholar] [CrossRef]
- Ma, X.; Wan, Y.; Wang, Y.; Dong, X.; Shi, S.; Liang, J.; Zhao, Y.; Mi, H. Multi-Parameter Practical Stability Region Analysis of Wind Power System Based on Limit Cycle Amplitude Tracing. IEEE Trans. Energy Convers. 2023, 38, 2571–2583. [Google Scholar] [CrossRef]
- Wang, T.; Chiang, H.-D. On the Number of System Separations in Electric Power Systems. IEEE Trans. Circuits Syst. I Regul. Pap. 2016, 63, 661–670. [Google Scholar] [CrossRef]
- Qiu, G.; Liu, J.; Liu, Y.; Liu, T.; Mu, G. Ensemble Learning for Power Systems TTC Prediction with Wind Farms. IEEE Access 2019, 7, 16572–16583. [Google Scholar] [CrossRef]
- Jafarzadeh, S.; Genc, I.; Nehorai, A. Real-time transient stability prediction and coherency identification in power systems using Koopman mode analysis. Electr. Power Syst. Res. 2021, 201, 107565. [Google Scholar] [CrossRef]
- Ali, B.M. Deep Reinforcement Learning for Wind-Power: An Overview. In Proceedings of the 2023 6th International Conference on Engineering Technology and its Applications (IICETA), Al-Najaf, Iraq, 15–16 July 2023; pp. 245–251. [Google Scholar]
- Sun, K.; Yao, W.; Fang, J.; Ai, X.; Wen, J.; Cheng, S. Impedance Modeling and Stability Analysis of Grid-Connected DFIG-Based Wind Farm with a VSC-HVDC. IEEE J. Emerg. Sel. Top. Power Electron. 2020, 8, 1375–1390. [Google Scholar] [CrossRef]
- Karaagac, U.; Mahseredjian, J.; Jensen, S.; Gagnon, R.; Fecteau, M.; Kocar, I. Safe Operation of DFIG-Based Wind Parks in Series-Compensated Systems. IEEE Trans. Power Deliv. 2018, 33, 709–718. [Google Scholar] [CrossRef]
- Hu, B.; Nian, H.; Li, M.; Xu, Y.; Liao, Y.; Yang, J. Impedance-Based Analysis and Stability Improvement of DFIG System Within PLL Bandwidth. IEEE Trans. Ind. Electron. 2022, 69, 5803–5814. [Google Scholar] [CrossRef]
- Xiao, Y.T.; Li, G.H.; Wang, W.S.; He, G.Q.; Zhen, N. Oscillation Mechanism Analysis and Suppression Strategy of Renewable Energy Base Connected into LCC-HVDC (Part II): Impedance Characteristics and Oscillation Mechanism Analysis. Proc. CSEE 2024, 44, 4245–4260. [Google Scholar]
- Chen, S.; Yu, S.S.; Zhang, G.; Zhang, Y. Novel VSG Control to Enhance DFIG Oscillatory Stability in Weak Grids via Admittance Modeling: Analysis and Experimental Validation. IEEE Trans. Energy Convers. 2025, 37, 1–13. [Google Scholar] [CrossRef]
- Xing, F.; Xu, Z. Investigation on Negative-Resistance Effect of Doubly-Fed Induction Generator. Acta Energiae Solaris Sin. 2022, 43, 324–332. [Google Scholar]
- Pavella, M.; Ernst, D.; Ruiz-Vega, D. Transient Stability of Power Systems: A Unified Approach to Assessment and Control; Kluwer: Norwell, MA, USA, 2000. [Google Scholar]
- Kuznetsov, Y.A. Elements of Applied Bifurcation Theory, 2nd ed.; Springer: New York, NY, USA, 2004; pp. 40–50. [Google Scholar]
- Si, Y.; Wang, Z.; Liu, L.; Anvari-Moghaddam, A. Impacts of Uncertain Geomagnetic Disturbances on Transient Power Angle Stability of DFIG Integrated Power System. IEEE Trans. Ind. Appl. 2023, 59, 2615–2625. [Google Scholar] [CrossRef]
- Amaral, F.M.; Alberto, F.C. Stability Region Bifurcations of Nonlinear Autonomous Dynamical Systems: Type-Zero Saddle-Node Bifurcations. Int. J. Robust Nonlinear Control 2011, 21, 591–612. [Google Scholar] [CrossRef]
- Meiss, J.D. Differential Dynamical Systems; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2007; pp. 165–186. [Google Scholar]
- Chen, X.; Ma, M.; Yan, Z. An Improved Parameter Boundary Calculation Method for Virtual Synchronous Generator with Capacity Constraints of Energy Storage System. Electr. Power Syst. Res. 2025, 241, 111391. [Google Scholar] [CrossRef]
- Zhuo, Q.; Liu, W.; Xu, Z.; Zhang, H.; Zhai, Y.; Yang, L.; Xu, F. Active Power-Frequency Oscillation Suppression Strategy for Parallel VSG Grid-Connected Power System. In Proceedings of the 2024 10th International Conference on Power Electronics Systems and Applications (PESA), Hong Kong, 5–7 June 2024; pp. 1–6. [Google Scholar]
- Pouresmaeil, M.; Sangrody, R.; Taheri, S.; Montesinos-Miracle, D.; Pouresmaeil, E. Enhancing Fault Ride Through Capability of Grid-Forming Virtual Synchronous Generators Using Model Predictive Control. IEEE J. Emerg. Sel. Top. Ind. Electron. 2024, 5, 1192–1203. [Google Scholar] [CrossRef]
- Li, Y.; Sahoo, S.; Ou, S.; Leng, M.; Vazquez, S.; Dragičević, T.; Blaabjerg, F. Flexible Transient Design-Oriented Model Predictive Control for Power Converters. IEEE Trans. Ind. Electron. 2024, 71, 11377–11387. [Google Scholar] [CrossRef]
- Jongudomkarn, J.; Liu, J.; Ise, T. Virtual Synchronous Generator Control with Reliable Fault Ride-Through Ability: A Solution Based on Finite-Set Model Predictive Control. IEEE J. Emerg. Sel. Top. Power Electron. 2020, 8, 3811–3824. [Google Scholar] [CrossRef]
- Shen, J.; Weng, P.; Shen, Y.; Chen, B.; Yu, L. Distributed Fusion Estimation for Nonlinear Cyber-Physical Systems with Attacked Control Signals. IEEE Syst. J. 2022, 17, 1216–1223. [Google Scholar] [CrossRef]
- Gupta, A.; Sekhar, P.C. Dynamic Droop Control for Optimal Power Sharing in Renewable Rich Hybrid Islanded Microgrids. In Proceedings of the 2024 IEEE International Conference on Power Electronics, Drives and Energy Systems (PEDES), Mangalore, India, 18–21 December 2024; pp. 1–6. [Google Scholar]
- Li, C.; Yang, Y.; Li, Y.; Liu, Y.; Blaabjerg, F. Modeling for Oscillation Propagation With Frequency-Voltage Coupling Effect in Grid-Connected Virtual Synchronous Generator. IEEE Trans. Power Electron. 2025, 40, 82–86. [Google Scholar] [CrossRef]
Conditions | J | ||||
---|---|---|---|---|---|
Original parameters | 0.90 | 1.00 | 0.75 | 8.39 | 3.10 |
Adjusted parameters | 1.00 | 1.11 | 0.84 | 7.21 | 3.40 |
Case | Method | Select J | (s) | (s) | IDRS% | ICCT% | ||
---|---|---|---|---|---|---|---|---|
1 | Proposed Method | 0.95 | 1.06 | 0.779 | 7.54 | 3.26 | −6.22 | 5.16 |
EF Method | 0.95 | 1.03 | 0.757 | 7.76 | 3.19 | −3.48 | 2.90 | |
2 | Proposed Method | 1.00 | 1.11 | 0.837 | 7.21 | 3.40 | −10.32 | 9.68 |
EF Method | 1.00 | 1.05 | 0.792 | 7.62 | 3.28 | −5.22 | 5.81 | |
3 | Proposed Method | 1.05 | 1.17 | 0.904 | 6.84 | 3.55 | −14.93 | 14.52 |
EF Method | 1.05 | 1.08 | 0.835 | 7.40 | 3.35 | −7.96 | 8.06 | |
Case | Method | Select | (s) | (s) | IDRS% | ICCT% | ||
4 | Proposed Method | 0.99 | 1.10 | 0.825 | 7.26 | 3.35 | −9.70 | 8.06 |
EF Method | 1.19 | 1.10 | 0.866 | 7.27 | 3.33 | −9.58 | 7.42 | |
5 | Proposed Method | 1.04 | 1.15 | 0.885 | 6.94 | 3.51 | −13.68 | 13.23 |
EF Method | 1.19 | 1.15 | 0.946 | 6.95 | 3.50 | −13.56 | 12.90 | |
6 | Proposed method | 1.08 | 1.20 | 0.940 | 6.67 | 3.66 | −17.04 | 18.06 |
EF Method | 1.30 | 1.20 | 1.030 | 6.68 | 3.64 | −16.92 | 17.42 |
Comparison | Methodology of This Paper | Methodology in the References |
---|---|---|
Core Methods | Precise characterization of the BSD based on manifold topology and SAIs, combined with PSI parameter optimization. | Diverse (Lyapunov theory, energy functions, etc.), mostly involving local or linearized analysis. |
Research Focus | Transient stability domain contraction mechanism and PSI regulation in DFIG grid-connected systems. | Parameter optimization, stability domain partitioning (Reference [8]), etc., mostly focusing on single issues. |
Advantages | High-precision BSD calculation; linear parameter optimization ( conservation); balancing CCT and response speed. | Reference [29] (communication-less control) and Reference [30] (multi-fault adaptation) each have their own advantages. |
Disadvantages | Dimensionality reduction of high-dimensional systems may ignore high-frequency dynamics; randomness is not explicitly considered. | Reference [29] (only VSG), Reference [31] (only DC systems), and Reference [32] (high computational complexity) have obvious limitations. |
Applicable Scenarios | High-proportion renewable energy systems (especially DFIG) require global stability domain analysis and parameter optimization. | For example, microgrid oscillations (Reference [29]), attacked CPS (Reference [33]), and DC systems (Reference [32]) are scenario-specific. |
Comparison of Advantages | Global nonlinearity, theoretical universality, and simple parameter optimization; dimensionality reduction verification is required. | The advantages of the references are mostly local performance (such as the economic efficiency in reference [34] and the oscillation threshold in reference [35]), but they lack generalizability. |
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Yuan, J.; Ma, M.; Jia, Y. Transient Stability Analysis for the Wind Power Grid-Connected System: A Manifold Topology Perspective on the Global Stability Domain. Electricity 2025, 6, 44. https://doi.org/10.3390/electricity6030044
Yuan J, Ma M, Jia Y. Transient Stability Analysis for the Wind Power Grid-Connected System: A Manifold Topology Perspective on the Global Stability Domain. Electricity. 2025; 6(3):44. https://doi.org/10.3390/electricity6030044
Chicago/Turabian StyleYuan, Jinhao, Meiling Ma, and Yanbing Jia. 2025. "Transient Stability Analysis for the Wind Power Grid-Connected System: A Manifold Topology Perspective on the Global Stability Domain" Electricity 6, no. 3: 44. https://doi.org/10.3390/electricity6030044
APA StyleYuan, J., Ma, M., & Jia, Y. (2025). Transient Stability Analysis for the Wind Power Grid-Connected System: A Manifold Topology Perspective on the Global Stability Domain. Electricity, 6(3), 44. https://doi.org/10.3390/electricity6030044