## 1 Introduction
However, if the ratio \(m=f_{s}/f_{g}\), known as the _modulation ratio_, is enough high which is the case in many applications, _quasi-static approximation_ can be used. Specifically, when the system dynamics at the switching scale are sufficiently fast, then the quasi-static approximation leads to enough accurate expressions for predicting fast-scale instability in AC-DC rectifiers [25] and in DC-AC inverters [26]. This approximation is a commonly used technique for analyzing DC-AC inverters [26; 27] and AC-DC rectifiers [25; 27]. This kind of systems is also featured with multi-time scale operation and their dynamical behavior has received and still receiving a great attention from researchers all over the world. The fast-scale operation in DC-AC inverters is usually averaged and the switching details have been conventionally overlooked. Although some past works dealing with nonlinear dynamical behavior of DC-AC inverters exist, the PV-fed grid-tied DC-AC boost inverter for AC module applications has not been studied from the perspective of nonlinear dynamics and its slope compensation has not been addressed. Fast-scale instability in such AC modules can cause the power generation system to be subject to faults or unfavorable conditions and the system reliability may be degraded which can negatively impacts the system performance. This is because fast-scale instability could increase the ripples of the electrical variables which results in higher overall losses and current stress on the switching devices. The power quality can also be compromised by this fast-scale instability since it can increase the Total Harmonic Distortion (THD). That being the case, accurate modeling of the dynamical behavior of the modules and their fast-scale stability analysis are needed. To have a reliable functioning, the instability must be avoided for the full range of system parameter values. For that, the right compensating must be used. In general, it can be claimed that the methods of analysis and modeling applied to power electronics systems are based, to a large extent, on the averaging approach. DC-DC converters have been also studied by using accurate discrete-time approaches. However, to date little has been done to study the qualitative dynamical aspects of DC-ACinverters based on accurate modeling and analysis approaches. The aim of this study is to faithfully determine the outbreak of fast-scale instability in a grid-tied differential boost AC module by combining numerical and analytical procedures and to obtain the required compensation ramp signal slope ensuring stability of the module for the full range of the operating duty cycle. The rest of this paper is structured as follows. In Section 2, the PV-fed DC-AC grid-tied AC module under a three-loop differential Current Mode Control (CMC) is briefly described. In Section 3, the dynamics of the AC module is explored disclosing that its behavior depends on parameters of different nature but all affecting the operating quasi-static operating duty cycle. In particular, the AC module is shown to manifest fast-scale instability over specific intervals within the grid period causing bubbling phenomenon in some of the state variables waveforms. Thereafter, in Section 4, approximate analysis and prediction of fast-scale instability is preformed obtaining the required compensating slope for achieving stability for all operating duty cycle values and for an optimum fast-scale _deadbeat_ response of the AC module using an adaptive compensation scheme. Finally, in Section 5 the concluding remarks of this study are outlined.