Output Impedance Modeling of Single-Phase Grid-Tied Inverters With Capturing the Frequency-Coupling Effect of PLL

The output impedance model for the precise stability analysis on the single-phase grid-tied inverters considering the frequency-coupling effect of the phase-locked loop (PLL) is studied. A multifrequency admittance matrix (MAM) is developed to predict the inverter-grid system stability by applying the generalized Nyquist criterion. Based on the MAM, a novel output impedance model characterized by clear physical meaning is proposed, which reveals that the inverter output impedance <inline-formula><tex-math notation="LaTeX">$(Z_{op}|_{(Z\text{pcc}, {Im})})$</tex-math></inline-formula> varies along with the grid impedance <inline-formula><tex-math notation="LaTeX">$Z_{{\text{pcc}}}$</tex-math></inline-formula> and the current reference <inline-formula><tex-math notation="LaTeX">$I_{m}$</tex-math></inline-formula>. The resonance and the stability problems can be identified by applying the impedance-based stability criterion to the impedance ratio <inline-formula><tex-math notation="LaTeX">$Z_{op}| _{(Z\text{pcc},\,{Im})}/ Z_{\text{pcc}}$</tex-math></inline-formula>. The proposed impedance model is further extended to the <italic>n</italic>-parallel inverter system. It is found that the output impedances of two <italic>n</italic>-parallel homogeneous-inverter systems are identical if the number <italic>n</italic> and the total injected current <italic>i</italic>_tot are the same, and the system stability is indicated by the impedance ratio <inline-formula><tex-math notation="LaTeX">$\frac{1}{\boldsymbol{n}}$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">$Z_{op}| _{(n\cdot Z{\text{pcc}},\,{I}{{\rm{avg)}}}}/ Z_{{\text{pcc}}}$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$Z_{op}| _{(n\cdot Z{\text{pcc}},\,{I{\text{avg)}}}}$</tex-math></inline-formula> is the output impedance of one special inverter with the average current <italic>i</italic>_tot/<italic>n</italic> injected to the grid with <inline-formula><tex-math notation="LaTeX">$n\cdot Z_{{\text{pcc}}}$</tex-math></inline-formula>. Further on, the design procedure of the PLL bandwidth <inline-formula><tex-math notation="LaTeX">$f_{b}$</tex-math></inline-formula> is elaborated, which guides the selection of <inline-formula><tex-math notation="LaTeX">$f_{b}$</tex-math></inline-formula> with the anticipated phase margin. Compared to the conventional impedance model, the developed output impedance is more qualified to precisely identify the harmonic stability problem in the weak grid. The selection of <inline-formula><tex-math notation="LaTeX">$f_{b}$</tex-math></inline-formula> and the correctness of the proposed impedance model are experimentally verified by the measured output impedance and the waveforms of the elaborated cases.