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Stability and multi-frequency dynamic characteristics of nonlinear grid-connected pumped storage-wind power interconnection system

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Abstract

This paper researches the stability and multi-frequency dynamic characteristics of nonlinear grid-connected pumped storage-wind power interconnection system (PS-WPIS). Firstly, a nonlinear model of grid-connected PS-WPIS is established. Then, the system stability and multi-frequency characteristics are revealed through stable domain and dynamic response analysis. Furthermore, the coupling mechanism of grid-connected PS-WPIS is explained, and the effect of capacity ratio on system stability is studied. Finally, the effect of hydraulic, mechanical and electrical parameters on grid-connected PS-WPIS is revealed. The results show that the stable domain of grid-connected PS-WPIS consists of two horizontal bifurcation lines and one curving bifurcation line. The former is related to wind power subsystem, and the latter is related to pumped storage subsystem. The grid-connected PS-WPIS contains the phenomenon of multi-frequency oscillations. The multi-frequency oscillations are generated by the coupling effect of pumped storage subsystem and wind power subsystem. The capacity increase of pumped storage or wind power worsens the stability and dynamic response of grid-connected PS-WPIS. The regulation performance of grid-connected PS-WPIS can be significantly improved by selecting smaller values of flow inertia time constant of penstock and time constant of wind turbine shafting.

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The data are available from the corresponding author upon reasonable request.

References

  1. Li, W.Y., Ji, Z.S., Dong, F.G.: Global renewable energy power generation efficiency evaluation and influencing factors analysis. Sustain Prod Consum 33, 438–453 (2022)

    Google Scholar 

  2. Chang, C.H., Lo, S.F.: Impact analysis of a national and corporate carbon emission reduction target on renewable electricity use: a review. Energies 15(5), 1794 (2022)

    Google Scholar 

  3. Wang, Y., Guo, C.H., Du, C., Chen, X.J., Jia, L.Q., Guo, X.N., Chen, R.S., Zhang, M.S., Chen, Z.Y., Wang, H.D.: Carbon peak and carbon neutrality in China: goals, implementation path and prospects. China Geology 4, 720–746 (2021)

    Google Scholar 

  4. IRENA Renewable Capacity Statistics 2022 (available at: https://www.irena.org/publications/2022/Apr/Renewable-Capacity-Statistics-2022)

  5. Zhao, J., He, Y.Q., Fang, Y.D., Weng, Y.X., Ma, W.Z., Xiao, S.Y., Liang, Y.L.: Multi-source optimal dispatch considering ancillary service cost of pumped storage power station based on cooperative game. Energy Rep. 7, 173–186 (2021)

    Google Scholar 

  6. Xu, Y.H., Zheng, Y., Du, Y., Yang, W., Peng, X.Y., Li, C.S.: Adaptive condition predictive-fuzzy PID optimal control of start-up process for pumped storage unit at low head area. Energy Convers. Manage. 177, 592–604 (2018)

    Google Scholar 

  7. Guo, W.C., Wu, F.L.: Hydraulic-mechanical coupling vibration performance of pumped storage power station with two turbine units sharing one tunnel. J Energy Storage 53, 105082 (2022)

    Google Scholar 

  8. Chai, Z.S., Li, H., Xie, X.J., Abdeen, M., Yang, T., Wang, K.: Output impedance modeling and grid-connected stability study of virtual synchronous control-based doubly-fed induction generator wind turbines in weak grids. Int. J. Electr. Power Energy Syst. 126, 106601 (2021)

    Google Scholar 

  9. Li, R., Zhao, S.Q., Gao, B.F., Zhang, R.X., Hu, Y.T.: Sub synchronous torsional interaction of steam turbine under wind power oscillation in wind-thermal power bundled transmission system. Int. J. Electr. Power Energy Syst. 108, 445–455 (2019)

    Google Scholar 

  10. Ma, N.N., Xie, X.R., He, J.B., Wang, H.: Review of wide-band oscillation in renewable and power electronics highly integrated power systems. Proc CSEE 40(15), 4720–4732 (2020)

    Google Scholar 

  11. Yang, W.J., Yang, J.D.: Advantage of variable-speed pumped storage plants for mitigating wind power variations: Integrated modelling and performance assessment. Appl. Energy 237, 720–732 (2019)

    Google Scholar 

  12. Deng, Y.W., Wang, P.F., Morabito, A., Feng, W.F., Mahmud, A., Chen, D.Y., Hendrick, P.: Dynamic analysis of variable-speed pumped storage plants for mitigating effects of excess wind power generation. Int. J. Electr. Power Energy Syst. 135, 107453 (2022)

    Google Scholar 

  13. Zhang, T.Y., Zhou, J.Z., Lai, X.J., Huang, Y., Li, M.Y.: Nonlinear stability and dynamic characteristics of grid-connected hydropower station with surge tank of a long lateral pipe. Int. J. Electr. Power Energy Syst. 136, 107654 (2022)

    Google Scholar 

  14. Liu, J., Yao, W., Wen, J.Y., Fang, J.K., Jiang, L., He, H.B., Cheng, S.J.: Impact of power grid strength and PLL parameters on stability of grid-connected DFIG wind farm. IEEE Trans Sustain Energy 11(1), 545–557 (2020)

    Google Scholar 

  15. Yang, L.H., Xu, Z., Østergaard, J., Dong, Z.Y., Wong, K.P., Ma, X.K.: Oscillatory stability and eigenvalue sensitivity analysis of a DFIG wind turbine system. IEEE Trans. Energy Convers. 26(1), 328–339 (2011)

    Google Scholar 

  16. Li, J.N., Huang, H.Q., Chen, X.N., Peng, L.X., Wang, L., Luo, P.: The small-signal stability of offshore wind power transmission inspired by particle swarm optimization. Complexity 13, 1–13 (2020)

    Google Scholar 

  17. Zhang, S.S., Yang, W.J., Li, X.D., Zhao, Z.G., Wang, R., Li, Y.L.: Economic evaluation of wind-PV-pumped storage hybrid system considering carbon emissions. Energy Rep. 8, 1249–1258 (2022)

    Google Scholar 

  18. Zhang, J.J., Mahmud, A., Govaerts, W., Chen, D.Y., Xu, B.B., Xiong, H.L.: Sensitivity analysis and low frequency oscillations for bifurcation scenarios in a hydraulic generating system. Renewable Energy 162, 334–344 (2020)

    Google Scholar 

  19. Zhu, Z.W., Tan, X.Q., Lu, X.D., Liu, D., Li, C.S.: Hopf bifurcation and parameter sensitivity analysis of a doubly-fed variable-speed pumped storage unit. Energies 15(1), 204 (2022)

    Google Scholar 

  20. Gao, C.Y., Yu, X.Y., Nan, H.P., Men, C.S., Zhao, P.Y., Cai, Q.S., Fu, J.N.: Stability and dynamic analysis of doubly-fed variable speed pump turbine governing system based on Hopf bifurcation theory. Renewable Energy 175, 568–579 (2021)

    Google Scholar 

  21. Khosravi, S., Zamanifar, M., Derakhshan-Barjoei, P.: Analysis of bifurcations in a wind turbine system based on DFIG. Emerg Sci J 2(1), 39–52 (2018)

    Google Scholar 

  22. Li J, Zhou XS, Dong GL. Hopf bifurcation analysis on dynamic voltage stability of wind power systems with SVC. In Proceedings of 2014 IEEE International Conference on Mechatronics and Automation; Tianjin, China, 2014.

  23. Yang, L.H., Ma, X.K., Dai, D.: Hopf bifurcation in doubly fed induction generator under vector control. Chaos Solitons Fractals 41(5), 2741–2749 (2009)

    Google Scholar 

  24. Sonawane, A., Umarikar, A.: Small-signal stability analysis of PV-based synchronverter including PV operating modes and DC-link voltage controller. IEEE Trans. Industr. Electron. 69(8), 8028–8039 (2022)

    Google Scholar 

  25. Wang, F.Z., Tan, T.Y., Liu, K.P., Zhu, S., Yang, J., Li, Y.X., Hu, C.G., Qin, L.: Small-signal stability analysis of combined operation system of variable-speed pumped storage unit and direct-drive wind turbine unit. Electric Power Auto Equip 41(7), 65–72 (2021)

    Google Scholar 

  26. Yue, L., Xue, A.C., Li, Z.Q., Meng, H.M., Yan, J.F., Li, B.Q., Bi, T.S., Yang, Q.X.: Effects on extra-low frequency oscillation caused by hydro generator governor system and model suitability analysis. Proc CSEE 39(227–235), 337 (2019)

    Google Scholar 

  27. Wang, P., Liu, T.Q., Wang, S.L., Jiang, Q., Yang, L., Li, B.: Small-signal model and influencing factors analysis of ultra-low frequency oscillation in hydropower unit. Proc CSEE 40(18), 5942–5955 (2020)

    Google Scholar 

  28. Zhu, Y.M., Sun, H.S., Qin, S.Y., Han, Y.S., Huang, B.Y., Mao, Y.J., Wang, D.Z.: Sub-synchronous oscillation in DFIG system connected to weak grid. Power Syst Technol 45, 1641–1648 (2021)

    Google Scholar 

  29. Du, X.B., Guo, C.Y., Zhao, C.Y., Xu, L.Q., Zhang, J.Q.: Modeling and multi-frequency oscillation mode analysis of hydropower unit integration through HVDC system. Auto Electric Power Syst 46, 75–83 (2022)

    Google Scholar 

  30. Zhao, K.J., Xu, Y.H., Guo, P.C., Qian, Z.D., Zhang, Y.C., Liu, W.: Multi-scale oscillation characteristics and stability analysis of pumped-storage unit under primary frequency regulation condition with low water head grid-connected. Renewable Energy 189, 1102–1119 (2022)

    Google Scholar 

  31. Lai, X.J., Li, C.S., Guo, W.C., Xu, Y.H., Li, Y.G.: Stability and dynamic characteristics of the nonlinear coupling system of hydropower station and power grid. Commun. Nonlinear Sci. Numer. Simul. 79, 104919 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Liu, Y., Guo, W.C.: Multi-frequency dynamic performance of hydropower plant under coupling effect of power grid and turbine regulating system with surge tank. Renewable Energy 171, 557–581 (2021)

    Google Scholar 

  33. D’Annibale, F.: Piezoelectric control of the Hopf bifurcation of Ziegler’s column with nonlinear damping. Nonlinear Dyn. 86, 2179–2192 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Luongo, A., D’Annibale, F.: Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads. Int. J. Non-Linear Mech. 55, 128–139 (2013)

    Google Scholar 

  35. Luongo, A., D’annibale, F.: Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces. Int. J. Mod. Phys. B 26(25), 1246015 (2012)

    Google Scholar 

  36. Casalotti, A., D’Annibale, F.: Improving the linear stability of the visco-elastic Beck’s beam via piezoelectric controllers. J Appl Comput Mech 7, 1098–1109 (2021)

    Google Scholar 

  37. Casalotti, A., Lacarbonara, W.: Nonlinear vibration absorber optimal design via asymptotic approach. Procedia IUTAM 19, 65–74 (2016)

    Google Scholar 

  38. Casalotti, A., Zulli, D., Luongo, A.: Nonlinear dynamics of a tubular beam considering distortion of the cross sections and internal resonances. Nonlinear Dyn. 111(8), 6961–6983 (2023)

    Google Scholar 

  39. Taig, G., Ranzi, G., D’Annibale, F.: An unconstrained dynamic approach for the generalised beam theory. Continuum Mech. Thermodyn. 27, 879–904 (2015)

    MathSciNet  MATH  Google Scholar 

  40. D’Annibale, F., Luongo, A.: A damage constitutive model for sliding friction coupled to wear. Continuum Mech. Thermodyn. 25, 503–522 (2013)

    MATH  Google Scholar 

  41. D’Annibale, F., Ferretti, M., Luongo, A.: Shear-shear-torsional homogenous beam models for nonlinear periodic beam-like structures. Eng. Struct. 184, 115–133 (2019)

    Google Scholar 

  42. Chen, Z.Q.: Stability and control of the wind power and pumped storage system. Nanjing University of Science and Technology, Jiangsu (2012)

    Google Scholar 

  43. Chang, J.S.: Transient process of hydraulic machinery. Higher Education Press, Beijing (2005)

    Google Scholar 

  44. Guo, W.C., Zhu, D.Y.: Critical stable sectional area of downstream surge tank of hydropower plant with sloping ceiling tailrace tunnel. Energy Sci Eng 9(8), 1090–1102 (2021)

    Google Scholar 

  45. Popescu, M., Arsenie, D., Vlase, P.: Applied hydraulic transients: for hydro-power plants and pumping stations. CRC Press, Boca Raton (2003)

    Google Scholar 

  46. Guo, W.C., Peng, Z.Y.: Hydropower system operation stability considering the coupling effect of water potential energy in surge tank and power grid. Renewable Energy 134, 846–861 (2019)

    Google Scholar 

  47. Zheng, Y.: Study on model predictive control of the regulating system of hydropower unit. China; Huazhong University of Science and Technology, Wuhan (2018)

    Google Scholar 

  48. Guo, W.C., Yang, J., Yang, J.B.: Effect of throttling orifice head loss on dynamic behavior of hydro-turbine governing system with air cushion surge chamber. IOP Conf Ser Earth Environ Sci 240, 052018 (2019)

    Google Scholar 

  49. Liu, S.B., Wang, D.L., Ma, N.N., Deng, W., Zhou, X., Wu, S.J., He, P.: Study on characteristics and suppressing countermeasures of ultra-low frequency oscillation caused by hydropower units. Proceedings of the CSEE 39(18), 5354-5362,5582 (2019)

    Google Scholar 

  50. Feijóo, A., Cidrás, J., Carrillo, C.: A third order model for the doubly-fed induction machine. Electric Power Systems Research 56(2), 121–121 (2000)

    Google Scholar 

  51. Ekanayake, J.B., Holdsworth, L., Jenkins, N.: Comparison of 5th order and 3rd order machine models for doubly fed induction generator (DFIG) wind turbines. Electric Power Systems Research 67(3), 207–215 (2003)

    Google Scholar 

  52. Wu, F., Zhang, X.P., Godfrey, K., Ju, P.: Small signal stability analysis and optimal control of a wind turbine with doubly fed induction generator. IET Gener. Transm. Distrib. 1(5), 751–760 (2007)

    Google Scholar 

  53. Du, W.J., Wang, H.F., Bu, S.Q.: Small-signal stability analysis of power systems integrated with variable speed wind generators. Springer, Switzerland (2018)

    Google Scholar 

  54. Chatterjee, A., Ghoshal, S.P., Mukherjee, V.: Transient performance improvement of grid connected hydro system using distributed generation and capacitive energy storage unit. Int. J. Electr. Power Energy Syst. 43(1), 210–221 (2012)

    Google Scholar 

  55. Guo, W.C., Zhu, D.Y.: Nonlinear modeling and operation stability of variable speed pumped storage power station. Energy Sci Eng 9(10), 1703–1718 (2021)

    Google Scholar 

  56. Shi, Y.S., Zhou, J.Z.: Stability and sensitivity analyses and multi-objective optimization control of the hydro-turbine generator unit. Nonlinear Dyn. 107(3), 2245–2273 (2022)

    Google Scholar 

  57. Shevitz, D., Paden, B.: Lyapunov stability theory of nonsmooth systems. IEEE Trans. Autom. Control 39(9), 1910–1914 (1994)

    MathSciNet  MATH  Google Scholar 

  58. Wei, Y.: Lyapunov stability theory for nonlinear nabla fractional order systems. IEEE Trans. Circuits Syst. II Express Briefs 68(10), 3246–3250 (2021)

    Google Scholar 

  59. Jana, A.K.: Differential geometry-based adaptive nonlinear control law: application to an industrial refinery process. IEEE Trans. Industr. Inf. 9(4), 2014–2022 (2013)

    Google Scholar 

  60. Ivancevic, V.G., Ivancevic, T.T.: Applied differential geometry: a modern introduction. World Scientific, Singapore (2007)

    MATH  Google Scholar 

  61. Lee, S.Y., Cho, H.S.: A fuzzy controller for an aeroload simulator using phase plane method. IEEE Trans. Control Syst. Technol. 9(6), 791–801 (2001)

    Google Scholar 

  62. Lee, S.Y., Cho, H.S.: A fuzzy controller for an electro-hydraulic fin actuator using phase plane method. Control. Eng. Pract. 11(6), 697–708 (2003)

    Google Scholar 

  63. Wu, D., Chen, K.: Frequency-domain analysis of nonlinear active disturbance rejection control via the describing function method. IEEE Trans. Industr. Electron. 60(9), 3906–3914 (2012)

    Google Scholar 

  64. Kim, E., Lee, H., Park, M.: Limit-cycle prediction of a fuzzy control system based on describing function method. IEEE Trans. Fuzzy Syst. 8(1), 11–22 (2000)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project No. 51909097).

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  1. School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wencheng Guo & Jiening Li

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  1. Wencheng Guo
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Appendix A

Appendix A

$$ A = \left[ {\begin{array}{*{20}c} {a_{1,1} } & 0 & {a_{1,3} } & 0 & {a_{1,5} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {a_{2,3} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {a_{3,1} } & 0 & {a_{3,3} } & 0 & {a_{3,5} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {a_{4,4} } & 0 & 0 & 0 & 0 & 0 & 0 \\ {a_{5,1} } & 0 & {a_{5,3} } & 0 & {a_{5,5} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {a_{6,6} } & {a_{6,7} } & {a_{6,8} } & {a_{6,9} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {a_{7,6} } & {a_{7,7} } & {a_{7,8} } & {a_{7,9} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {a_{8,6} } & {a_{8,7} } & 0 & {a_{8,9} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {a_{10,9} } & 0 \\ \end{array} } \right],\;B = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {b_{3,1} } & {b_{3,2} } & {b_{3,3} } & {b_{3,4} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {b_{4,3} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {b_{5,1} } & {b_{5,2} } & {b_{5,3} } & {b_{5,4} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {b_{6,5} } & 0 & 0 & 0 & 0 & 0 & 0 & {b_{6,12} } \\ 0 & 0 & 0 & 0 & {b_{7,5} } & 0 & 0 & 0 & 0 & 0 & {b_{7,11} } & 0 \\ 0 & 0 & 0 & 0 & {b_{8,5} } & 0 & 0 & 0 & 0 & 0 & {b_{8,11} } & {b_{8,12} } \\ 0 & 0 & 0 & 0 & {b_{9,5} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {b_{10,5} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], $$
$$ C = \left[ {\begin{array}{*{20}c} 0 & {c_{1,2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{1,10} } \\ 0 & {c_{2,2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{2,10} } \\ 0 & 0 & {c_{3,4} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {c_{5,2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{5,10} } \\ 0 & {c_{6,2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{6,10} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{7,10} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{8,10} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{9,10} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {c_{10,10} } \\ 0 & 0 & 0 & 0 & 0 & 0 & {c_{11,7} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {c_{12,6} } & 0 & 0 & 0 & 0 \\ \end{array} } \right],\;D = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {d_{1,4} } & 0 & 0 & {d_{1,7} } & {d_{1,8} } & 0 & 0 & 0 & 0 \\ 0 & 0 & {d_{2,3} } & 0 & 0 & 0 & {d_{2,7} } & {d_{2,8} } & 0 & 0 & 0 & 0 \\ 0 & {d_{3,2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {d_{4,1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {d_{5,3} } & {d_{5,4} } & 0 & 0 & 0 & {d_{5,8} } & 0 & 0 & 0 & 0 \\ 0 & 0 & {d_{6,3} } & {d_{6,4} } & 0 & 0 & {d_{6,7} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {d_{7,11} } & {d_{7,12} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {d_{8,11} } & {d_{8,12} } \\ 0 & 0 & 0 & 0 & {d_{9,5} } & {d_{9,6} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {d_{10,5} } & {d_{10,6} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {d_{11,10} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {d_{12,9} } & 0 & 0 & 0 \\ \end{array} } \right], $$

Note that: \(a_{1,1} = - \frac{1}{{T_{wt0} }}\left( {\frac{1}{{e_{qh} }} + \frac{{2h_{t0} }}{{H_{0} }}} \right),\;a_{1,3} = \frac{{e_{qx} }}{{T_{wt0} e_{qh} }},\;a_{1,5} = \frac{{e_{qy} }}{{T_{wt0} e_{qh} }},\;a_{2,3} = 2\pi f_{b} ,\;a_{3,1} = \frac{{e_{h} }}{{T_{J} e_{qh} }},\)

$$ a_{3,3} = \frac{1}{{T_{J} }}\left( {\frac{{e_{x} e_{qh} - e_{qx} e_{h} }}{{e_{qh} }} - D} \right),\;a_{3,5} = \frac{1}{{T_{J} }}\frac{{e_{y} e_{qh} - e_{qy} e_{h} }}{{e_{qh} }},\;b_{3,1} = - \frac{{I_{dg} }}{{T_{J} }},\;b_{3,2} = - \frac{{I_{qg} }}{{T_{J} }},\;b_{3,3} = - \frac{{V_{dg} }}{{T_{J} }},\;b_{3,4} = - \frac{{V_{qg} }}{{T_{J} }}, $$

\(a_{4,4} = - \frac{1}{{T^{\prime}_{d0} }},\;b_{4,3} = - \frac{{X_{d} - X^{\prime}_{d} }}{{T^{\prime}_{d0} }},\;a_{5,1} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}a_{3,1} ,\;a_{5,3} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}a_{3,3} - \frac{{K_{w} K_{I0} }}{{1 + b_{p} K_{P0} }},\)

$$ a_{5,5} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}a_{3,5} - \frac{{b_{p} K_{I0} }}{{1 + b_{p} K_{P0} }},\;b_{5,1} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}b_{3,1} ,\;b_{5,2} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}b_{3,2} ,\;b_{5,3} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}b_{3,3} , $$

\(b_{5,4} = - \frac{{K_{w} K_{P0} }}{{1 + b_{p} K_{P0} }}b_{3,4} ,\;a_{6,6} = - \frac{{\omega_{b} R_{r} }}{{L_{rr} }},\;a_{6,7} = \omega_{b} \left( {\omega_{pll} - \omega_{r} } \right),\;a_{6,8} = - \omega_{b} E_{q} ,\;a_{6,9} = \omega_{b} K_{Ipll} \left( {E_{q} - \frac{{R_{r} L_{m}^{2} }}{{L_{rr}^{2} }}I_{qs} - \frac{{L_{m} }}{{L_{rr} }}V_{qr} } \right),\)

$$ b_{6,5} = - \omega_{b} K_{Ppll} \left( {E_{q} - \frac{{R_{r} L_{m}^{2} }}{{L_{rr}^{2} }}I_{qs} - \frac{{L_{m} }}{{L_{rr} }}V_{qr} } \right),\;b_{6,12} = - \omega_{b} \omega_{pll} \frac{{R_{r} L_{m}^{2} }}{{L_{rr}^{2} }},\;a_{7,6} = - \omega_{b} \left( {\omega_{pll} - \omega_{r} } \right),\;a_{7,7} = - \frac{{\omega_{b} R_{r} }}{{L_{rr} }}, $$

\(a_{7,8} = \omega_{b} E_{d} ,\;a_{7,9} = \omega_{b} K_{Ipll} \left( { - E_{d} + \frac{{R_{r} L_{m}^{2} }}{{L_{rr}^{2} }}I_{ds} + \frac{{L_{m} }}{{L_{rr} }}V_{dr} } \right),\;b_{7,5} = - \omega_{b} K_{Ppll} \left( { - E_{d} + \frac{{R_{r} L_{m}^{2} }}{{L_{rr}^{2} }}I_{ds} + \frac{{L_{m} }}{{L_{rr} }}V_{dr} } \right),\)

$$ b_{7,11} = \omega_{b} \omega_{pll} \frac{{R_{r} L_{m}^{2} }}{{L_{rr}^{2} }},\;a_{8,6} = \frac{{I_{ds} }}{{T_{j} }},\;a_{8,7} = \frac{{I_{qs} }}{{T_{j} }},\;a_{8,9} = - \frac{{K_{Ipll} }}{{T_{j} }}\left( {E_{d} I_{ds} + E_{q} I_{qs} } \right),\;b_{8,5} = \frac{{K_{Ppll} }}{{T_{j} }}\left( {E_{d} I_{ds} + E_{q} I_{qs} } \right),\;b_{8,11} = \frac{{E_{d} }}{{T_{j} }}, $$

\(b_{8,12} = \frac{{E_{q} }}{{T_{j} }},\;b_{9,5} = - 1,\;a_{10,9} = K_{Ipll} ,\;b_{10,5} = - K_{Ppll} ,\;c_{1,2} = \cos \delta - X_{TL43} \left[ {\cos \left( {\delta - \delta_{pll} } \right)I_{ds} + \sin \left( {\delta - \delta_{pll} } \right)I_{qs} } \right],\)

$$ c_{1,10} = X_{TL43} \left[ {\cos \left( {\delta - \delta_{pll} } \right)I_{ds} + \sin \left( {\delta - \delta_{pll} } \right)I_{qs} } \right],\;d_{1,4} = - \left( {X_{TL41} + X_{TL43} } \right),\;d_{1,7} = - X_{TL43} \sin \left( {\delta - \delta_{pll} } \right), $$

\(d_{1,8} = X_{TL43} \cos \left( {\delta - \delta_{pll} } \right),\;c_{2,2} = - \sin \delta - X_{TL43} \left[ { - \sin \left( {\delta - \delta_{pll} } \right)I_{ds} + \cos \left( {\delta - \delta_{pll} } \right)I_{qs} } \right],\)

$$ c_{2,10} = - X_{TL43} \left[ {\sin \left( {\delta - \delta_{pll} } \right)I_{ds} - \cos \left( {\delta - \delta_{pll} } \right)I_{qs} } \right],\;d_{2,3} = X_{TL41} + X_{TL43} ,\;d_{2,7} = - X_{TL43} \cos \left( {\delta - \delta_{pll} } \right), $$

\(d_{2,8} = - X_{TL43} \sin \left( {\delta - \delta_{pll} } \right),\;c_{3,4} = \frac{1}{{X^{\prime}_{d} }},\;d_{3,2} = - \frac{1}{{X^{\prime}_{d} }},\;d_{4,1} = \frac{1}{{X_{q} }},\;c_{11,7} = - \frac{1}{{X_{1} }},\;d_{11,10} = \frac{1}{{X_{1} }},\)

$$ c_{5,2} = X_{TL43} \left[ {\sin \left( {\delta - \delta_{pll} } \right)I_{qg} - \cos \left( {\delta - \delta_{pll} } \right)I_{dg} } \right],\;c_{6,2} = - X_{TL43} \left[ {\cos \left( {\delta - \delta_{pll} } \right)I_{qg} + \sin \left( {\delta - \delta_{pll} } \right)I_{dg} } \right], $$

\(c_{5,10} = \cos \delta \cos \left( {\delta - \delta_{pll} } \right) + \sin \delta \sin \left( {\delta - \delta_{pll} } \right) - X_{TL43} \left[ {\sin \left( {\delta - \delta_{pll} } \right)I_{qg} - \cos \left( {\delta - \delta_{pll} } \right)I_{dg} } \right],\)

\(d_{5,3} = - X_{TL43} \sin \left( {\delta - \delta_{pll} } \right),\;d_{5,4} = - X_{TL43} \cos \left( {\delta - \delta_{pll} } \right),\;d_{5,8} = X_{TL42} + X_{TL43} ,\)

\(c_{6,10} = - \sin \delta \cos \left( {\delta - \delta_{pll} } \right) + \cos \delta \sin \left( {\delta - \delta_{pll} } \right) + X_{TL43} \left[ {\cos \left( {\delta - \delta_{pll} } \right)I_{qg} + \sin \left( {\delta - \delta_{pll} } \right)I_{dg} } \right]\),

$$ d_{6,3} = X_{TL43} \cos \left( {\delta - \delta_{pll} } \right),\;d_{6,4} = - X_{TL43} \sin \left( {\delta - \delta_{pll} } \right),\;d_{6,7} = - \left( {X_{TL42} + X_{TL43} } \right), $$

\(c_{7,10} = - \sin \left( {\delta - \delta_{pll} } \right)I_{ds} + \cos \left( {\delta - \delta_{pll} } \right)I_{qs} ,\;d_{7,11} = \cos \left( {\delta - \delta_{pll} } \right),\;d_{7,12} = \sin \left( {\delta - \delta_{pll} } \right),\)

$$ c_{8,10} = - \cos \left( {\delta - \delta_{pll} } \right)I_{ds} - \sin \left( {\delta - \delta_{pll} } \right)I_{qs} ,\;d_{8,11} = - \sin \left( {\delta - \delta_{pll} } \right),\;d_{8,12} = \cos \left( {\delta - \delta_{pll} } \right), $$

\(c_{9,10} = - \sin \left( {\delta - \delta_{pll} } \right)V_{ds} - \cos \left( {\delta - \delta_{pll} } \right)V_{qs} ,\;d_{9,5} = \cos \left( {\delta - \delta_{pll} } \right),\;d_{9,6} = - \sin \left( {\delta - \delta_{pll} } \right),\)

$$ c_{10,10} = \cos \left( {\delta - \delta_{pll} } \right)V_{ds} - \sin \left( {\delta - \delta_{pll} } \right)V_{qs} ,\;d_{10,5} = \sin \left( {\delta - \delta_{pll} } \right),\;d_{10,6} = \cos \left( {\delta - \delta_{pll} } \right),\;c_{12,6} = \frac{1}{{X_{1} }},\;d_{12,9} = - \frac{1}{{X_{1} }}. $$

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Guo, W., Li, J. Stability and multi-frequency dynamic characteristics of nonlinear grid-connected pumped storage-wind power interconnection system. Nonlinear Dyn 111, 20929–20958 (2023). https://doi.org/10.1007/s11071-023-08942-5

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