Equivalent Multiple $dq$-Frame Model of the MMC Using Dynamic Phasor Theory in the $\alpha \beta z$-Frame

Small-Signal Modeling of AC Power Electronic Systems: Critical Review and Unified Modeling

“…The initial phase impact in the GDQ modeling was not discussed in previous works [40], [41], since the modeling was considered for a single converter system. When the system is linearized around the local steady-state dq frames of the converter system, the modeling can be simplified by selecting a time reference properly, such that the initial phase is assumed to be zero.…”

“…Their relationship is further revealed in [39], pointing out that the DP model can be transformed into the HSS model through the Laplace transformation. The equivalence between the GDQ model and the DP model was reported in [40] with the proofs in the complex space. The equivalence of the GDQ model and HSS model was also studied in [41] based on the principle of harmonic balance.…”

“…However, the relationships between the linearization and transformation have not been fully understood for these modeling methods. • The impacts of the initial phases for the GDQ modeling of ac systems are not considered in the existing studies [40], [41]. The existing studies on the initial-phase effect [8], [42] are merely for modeling balanced ac systems in a single dq frame.…”

Mathematical Relationships of Linearized Modeling Methods of AC Power Electronic Systems

“…In [37], the equivalence of two methods was proved by the eigenvalue analysis of the state-space matrices, through a comparative study on a converter system. For the GDQ modeling, its equivalence to applying the DP modeling in the αβ frame was studied in [38] by proofs in the complex space. The equivalence of the GDQ model and HSS model was studied in [39] based on the principle of harmonic balance using Fourier series, and their relationships were explained by the complex transformation.…”

“…While the equivalences of these models have been proved by the eigenvalue analysis [37], [38] and been explained based on the harmonic balance [33], [39], there is still lack of a discussion on the common mathematical basis that leads to the model equivalence. Furthermore, the explicit mathematical relationships among the principles of these methods are still missing.…”