# Dynamic Interactions between Parallel Grid-Forming Inverters in a Microgrid
## 1 Introduction
For example, when there is a step change in voltage, each will try to control and set the voltage to its reference value. However, since each converter control has different gain parameters and control speed, this probably could cause voltage oscillation. Several real world issues have been reported in [11; 12] regarding multi-parallel inverters. In [11], resonant issues were observed in a Dutch network with high penetration of PV inverters. Several unstable operation records of PV plants are presented in [12]. One case is a PV plant connected to utilities and containing 30 inverters operating in parallel. No local load is supplied. Based on the inverter specifications, the total harmonic distortion is below 2% at rated power. As it can be observed from the event, the current becomes highly distorted. Because of the instability, the PV inverters restart and restore normal operation. FFT analysis shows the 7 th harmonic current magnitude at 20% of its rated current. Another case is described in [12] where PV plants shut down their inverters partially because of oscillations. In both cases, there was no reported outage or fault in the system that could be the cause of such incidents. In West China, an SSO event occurred due to dynamic interaction between DFIG wind and STATCOM [13]. The event occurred in the Guohua area between the DFIG-based wind and STATCOM. From the recorded voltage waveforms during the event, three stages were experienced. After a small disturbance, 37 Hz voltage oscillations were observed in the first stage. At the second stage, due to voltage oscillation, harmonic reactive power from STATCOMs decreased. Then, oscillation increased and led to a trip of the wind farms. The existing stability analysis regarding the parallel inverters primarily focuses on evaluating the small-signal behavior of the droop control scheme adopted [18; 19; 20; 21]. In these studies, the interaction analysis of the current and the voltage controls are ignored or simplified since the bandwidth of the droop control loop and the voltage-current control loops are well separated [22; 23]. Therefore, the dynamic interaction between the outer voltage loop, inner control loop, and the remaining system dynamics is overlooked. Such interaction may lead to instability issues at a higher frequency, as shown in [23]. Several techniques for analyzing stability have been proposed in the literature. The traditional state-space method, which is ideal for linear time-invariant structures, can be used [24]. The stability of the electric network is judged based on the system matrix,notated as \(A\). The system will be stable if all of \(A\)'s eigenvalues are found in the left half-plane (LHP). Building a state-space LTI model is not as straightforward as building an \(s\)-domain model. Therefore, in the current literature, building \(s\)-domain impedance models and evaluating stability via the \(s\)-domain models is another viable path. In [25], the \(s\)-domain nodal admittance matrix (\(Y(s)\)) was proposed to be used for eigenvalue analysis in 1999. The method is further developed in [26; 27; 28]. The method is based on the fact that the system stability could be judged by the zeros of the determinant of \(Y(s)\), also known as the system eigenvalues. Furthermore, using the admittance matrix, we may identify the influencing factors using the branch current or the nodal voltage participation factor in [29]. This paper aims to analyze the dynamic interactions of two parallel grid-forming voltage-source converters and identify potential unstable oscillation modes and influencing factors. The study utilizes admittance-based eigenvalue and mode shape analysis, examining the effects of circuit impedance, virtual impedance unit, and voltage feed-forward unit. The accuracy of the findings is validated through electromagnetic transient (EMT) simulations. The analysis method utilized in this study takes full advantage of the derived modular impedance blocks by formulating a five-node system admittance network for eigenvalue and mode shape analysis.