| The broadband oscillations appearing in modular multilevel converter(MMC)bring challenges to the safe and stable operation of the flexible DC interconnected grid.The development of a broadband dynamic model for MMC is an urgent engineering problem.Harmonic interaction is an essential characteristic of MMC,and the high-order harmonic components in the system have an important impact on the dynamic modeling of MMC.The traditional single-harmonic dynamic modeling method ignores the role of higher harmonics,and therefore cannot accurately describe the broadband dynamic characteristics of MMC.At present,the dynamic model of MMC is still limited to specific scenarios,and there is a lack of systematic modeling and analysis method for different control strategies and broadband stability problems.To address the broadband oscillation of MMC in real projects,this paper focuses on the dynamic modeling and small-signal stability analysis of MMC.The broadband dynamic models of MMC considering harmonic interaction and detailed control dynamics are developed.The mechanism of harmonic coupling in MMC is revealed,and the influence of higher harmonics on MMC dynamic modeling and stability analysis is clarified.The key influencing factors for the oscillation of flexible DC interconnected system are identified.Finally,the corresponding stability enhancement strategies are proposed.First of all,A nonlinear time-periodic system model of MMC is established based on the arm averaged model.Then,the linear time-periodic system model of MMC is obtained by linearizing the above model around the periodic orbit.The multiplication operation rules of dynamic phasor in the generalized case are derived.The dynamic phasor model of the MMC power stage is established and combined with the controller dynamic equations to obtain the nonlinear time-invariant equations representing the closed-loop control dynamics.The three types of established time-domain models all take into account the harmonic interaction characteristics and detailed control dynamics,where the controller delay terms are converted into rational transfer functions for analysis by the Pade approximation.Besides,the proposed models are verified by electromagnetic transient simulations based on the aggregated model.Secondly,a d-q frame impedance model containing multiple rotational coordinates is derived based on the dynamic phasor model of MMC and the transfer function matrix modeling method,and the modular modeling of AC and DC side impedance is realized.In the extended harmonic domain containing multiple harmonic components,the harmonic impedance model of MMC is developed based on the harmonic state-space method.Modular programming of the harmonic impedance model is realized,and the model can be extended to consider arbitrary high-order harmonics.In addition,the relation and difference between dynamic phasor and harmonic state-space modeling are discussed.The equivalence relation of the impedance models between the multi-d-q domain and the multi-harmonic domain is verified.The delay models in d-q domain are derived based on the equivalence relation,and the d-q axis coupling mechanism caused by control delay is revealed.Then,the influence of high-order harmonics on the frequency response of MMC under different control strategies is analyzed.The harmonic coupling mechanism of power stage and controller is studied,and it is found that different frequency components as well as the different sequence(or d-q axis)components of the system are involved in the harmonic interaction.The order of the dynamic model is selected by the comparison of frequency response.The effects of sideband effects and controller dynamics on the highfrequency characteristics of the MMC are studied.A reduced-order model based on the singular-perturbation theory is established for the analysis of low-frequency oscillations of power systems.Through the eigenvalue analysis and the decoupling of time scale based on the calculation of participation factor,the dominant dynamics of the system is identified and the analytical expressions of the reduced-order model are obtained.The effectiveness of the reduced-order model is verified by embedding the MMC to a four-machine two-area system.Finally,the sub-synchronous interaction between MMC and variable speed pumped storage-wind turbine joint operation system is investigated.An MMC sub-synchronous oscillation suppression method based on total capacitor energy control is proposed and compared with active damping control and virtual arm resistance control to illustrate its control performance.In addition,the mechanism of the proposed controller to enhance the system stability is explained from the impedance perspective.The effects of controller delay,grid impedance and voltage feed-forward link on the high-frequency stability of the MMC are analyzed,and the high-frequency characteristics of the MMC are compared with the two-level converter.The high-frequency resonant waveform in the Yu-E flexible DC project is reproduced.Moreover,the applicability of Floquet analysis and generalized Nyquist criterion are discussed,the stability analysis results in harmonic domain and d-q domain are compared,and the influence of high-order harmonic dynamics on the stability analysis of MMC is studied.The results of the theoretical analysis in this paper are verified by electromagnetic transient simulation and are close to the field experience of flexible DC projects such as Xia-Men and Yu-E,which fully illustrate the correctness of modeling.The harmonicinteraction characteristics also exist in other topologies of power electronics,so this paper can provide inspiration for the broadband modeling of other power electronics.It also provides a reference for the stability analysis and control strategy design of complex power electronic systems. |
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