Quantitative Study For Power Angle Dynamics Of Grid-forming Converter With Damping Effects

In the context of achieving carbon peaking and carbon neutrality goals,the development of renewable energy generations accelerates power electronization process of power system.The compatibility between converter capability and system demand is crucial to its security,stabil-ity and economy,and the bidirectional adaptive iteration becomes an inevitable trend for power electronics based power system,which requires quantitative study for converter transient char-acteristics.Moreover,grid forming converter is commonly regarded as a promising equipment due to its superior frequency and voltage supporting ability,and therefore its transient charac-teristics quantitative study is the theoretical basis for power system transient analysis.However,the state-of-art theories guide its over-current design with extensive empirical criteria,and eval-uate transient stability with qualitative method,which seriously restricts the optimal utilization of converter transient capability and brings severe challenges to the safe,stable and economic operation of grid.This paper focuses on the power angle dynamics of grid forming converter with damping effects.In response to the dual demands of trajectory analytical description and stability boundary quantitative depiction,the parameter-perturbed analytical calculation method and transient damping energy quantification method have been developed.In this manner,the quantitative theoretical framework for power angle dynamics of grid forming converter with damping effects is formed,which lays a good foundation for the bidirectional adaptive iteration between converter and grid.For power angle trajectory description,an averaging idea based analytical method is first proposed.The traditional methods are developed from nonlinear mechanical vibration with weak-damping and simple perturbation function,which are difficult to be applied to grid form-ing converter with large damping effects and complex perturbation function directly.This paper introduces Krylov-Bogoliubov(KB)averaging method and reconstructs its solvable motion,which reduces the perturbation parameter from damping inertia ratio D_p/H to reciprocal of in-ertia 1/H and decreases the ratio of perturbation function.Furthermore,the new idea of using linear solution as an average trajectory to calculate amplitude and phase differentials is also pro-posed,which preserves derivation simplicity and considers more nonlinear features,and then the explicit analytical expression can be obtained.The proposed method solves the quantitative description problem of the relationship between parameters and transient trajectory,which may provide a good foundation for the transient over-current design of grid forming converter.To solve the accuracy limitation of averaging method,an asymptotic idea based analyti-cal method is further proposed.The accuracy of averaging method is limited by its low order nature,and the quantitative relationship is still hidden behind the complex summation,which brings difficulty to its intuitive understanding and direct application.In this paper,the Krylov-Bogoliubov-Mitropolsk(KBM)asymptotic method is further adopted,which uses the high or-der asymptotic expansion to approximate its real trajectory based on linear solution and reduces the truncation error from O(ε)to O(ε~m).Furthermore,the differential simplification method and the tailored integral method are proposed to solve the anti-derivative problem caused by complex perturbation function,and the explicit trajectory expression with simple form,clear relationship and accurate calculation is then obtained.In this manner,the proposed method clarifies the dominated relationship between control or grid parameters and the main character-istics of transient trajectories,such as oscillation frequency and decaying rate,which effectively improves its practicability in the transient over-current design of grid forming converter.For transient stability boundary characterization,a transient damping energy visualiza-tion and approximation method is first proposed.The traditional energy function simplifies the weak-damping swing motion as a quasi-steady-state periodic process,which is hard to be ap-plied to the decaying process with strong damping dissipation.In this paper,the new concept of transient damping area is proposed,which visualizes the transient damping energy to a tai-lored phase portrait and gives it clear physical meaning.In this manner,the classical power angle diagram could be combined to realize the intuitive analysis of various transient energy.Furthermore,the geometry area approximation is utilized to describe transient damping energy accurately and linearly,which evolves the transient stability boundary characterization from numerical iteration into algebraic calculation.The proposed method solves the visual presenta-tion and accurate characterization problem of the quantitative relationship between acceleration and deceleration energy in grid forming converter,which can provide a theoretical basis for the transient stability quantitative evaluation of grid forming converter.To solve the dynamic interaction problem of two-machines,a transient damping energy visualization and approximation method for two-machines paralleled system,which is the pro-totype and foundation of large system analysis,is further proposed.Nonetheless,the failure of system infinite equivalence leads to the center frequency of inertia shift and the non-uniformity of damping coefficients destroys the formal consistency of power angle equation in different coordinate system,which brings great challenges to system transient stability analysis.In this paper,the new concept of uniform and non-uniform damping energy is proposed,and their effects on system transient stability are analyzed,which reveals the non-conservative risk of the classical energy function under negative non-uniform damping energy and breaks through the inherent cognitive limitations that damping dissipation results in stabilization.Moreover,the tailored uniform and non-uniform damping phase portraits is used to map transient damp-ing energy and the geometry area approximation is extended to non-uniform damping energy analysis.The proposed method achieves the quantitative analysis of two-machines paralleled system considering damping effects,which provides theoretical support for the online transient stability analysis and the optimal resources configuration of modern power systems.This paper verifies the validity and accuracy of the theoretical research results through simulation and experiments,which provides a theoretical and scientific basis for the quantitative iteration of bidirectional adaptation of converter and system.