## 4 Harmonic Sources and Simulations  
As illustrated in Figure 5c,d, when the PV inverters were running but no \"resonances\" existed, the spectra of PV PCC waveforms contained fundamental voltage and current components only. Figure 5c reveals no background harmonics in the micro-grid. To determine the position of harmonic source, the \(dq\) conversion was implemented to the three-phase voltage at PCC and obtained sinusoidal ripples on the \(dq\) axis of the inverter output (DC component removed). The results are depicted in Figure 11.  
Figure 11: _Cont._  
Figure 10: The harmonic varieties vs. the quantity of PV inverters in parallel at PCC: (**a**) voltage RMS of PCC harmonics vs. time; (**b**) current RMS of PCC harmonics vs. time; and (**c**) frequencies of PCC harmonics vs. time. RMS: root mean square.  
As illustrated in Figure 11, there were 252.90 Hz ripples with high amplitudes on \(d\)-axis and \(q\)-axis of PV inverters. The inter-harmonics (202.90 Hz and 302.90 Hz) with significant amplitudes at PCC were generated by this sinusoidal alternating component. This sinusoidal alternating component was generated by nonlinear self-oscillation of PV inverters and micro-grid systems.  
To verify the opinions of nonlinear self-oscillation to generate harmonics at PV PCC, a Simulink (Ver. R2016a, Mathworks, Natick, MA. 01760-2098, USA, 2016) model of \"PV inverter and micro-grid\" is implemented based on the actual parameters of PV inverters and line impedances of the micro-grid in field, as shown in Figure 12. The PV inverter works in current source mode; the inputs of PV inverter are connected to an energy storage system with constant DC voltage as the PV plant; PLL is based on PCC voltages to produce the real-time phase information; PI control algorithm is adopted by inverter controller in \(dq\) domain; and a special self-oscillation source is used to generate steady-state oscillation signals to equivalent to all nonlinear factors of the \"grid-tied inverter and weak grid\" system.  
\(V_{\rm DC}\) is an ideal DC source with 400 V voltage; the phase voltage of micro-grid is 220 V AC; and the P and I parameters of controller are 0.03 and 0.01, respectively. The equivalent inductance and resistance of line impedances are 0.15 mH and 0.15 \(\Omega\), respectively, which were calculated according to the harmonic voltages and currents measured at PV PCC in field. The output filter and isolation  
Figure 11: Harmonic voltage at \(d\)-axis and \(q\)-axis: (**a**) harmonic voltages and spectra at \(d\)-axis; and (**b**) harmonic voltages and spectra at \(q\)-axis.  
Figure 12: Simulation of nonlinear self-oscillations of PV grid-tied inverters.  
transformer have the same parameters, as shown in Figure 8. The \"oscillation source\" in Figure 12 can generate a sinusoidal signal with frequency of 252.90 Hz which is employed to the \(d\)-axis and \(q\)-axis of the inverter, and the inter-harmonics with frequencies of 202.90 Hz and 302.90 Hz appear at the PV PCC. The waveforms and spectra of these inter-harmonics are presented in Figure 13.  
As illustrated in Figure 13, significant harmonics appear at PCC and the current THD exceeds 24.50%. The simulation shows that the pulsation signals in PV inverters are the key source of the steady inter-harmonics of PV inverter outputs. The discussion below presents that the pulsation signal is generated by nonlinear self-oscillation of the \"inverter-weak grid\" system.