# Stability Analysis of Buck Converter Based on Passivity-Based Stability Criterion
## 1 Introduction  
The flexible microgrid structure has arisen during a historic moment to integrate new energy sources, such as distributed generation (DG) systems and energy storage systems (ESSs), in a coordinated manner. Unlike the AC grid, the DC microgrid does not need to track the phase and frequency of the voltage or consider the eddy current loss in the distribution line and the reactive energy absorbed by the line [1]. Therefore, the DC microgrid has the advantages of simple control, high efficiency, flexibility, low construction cost [2], and broader application prospects in electric vehicles, residential load power supply and other fields [3]. In DC microgrids, the bus voltage becomes the only standard by which to measure the power balance of system, since the system does not need to consider reactive power [4]. In the DC microgrid, new energy sources such as DG and ESSs are usually connected to the DC bus through the source converters. At the same time, the loads are usually connected to the DC bus through closed-loop controlled load converters, and they present constant power load characteristics from the input terminal, which are regarded as constant power loads (CPLs). However, the CPLs exhibit negative impedance characteristics at their inputs, which will amplify the disturbance, including positive feedback, and destabilize microgrid systems [5]. The interaction between multi-converters can also lead to oscillation and even instability of DC microgrids. Therefore, it is of great significance to study the small-signal stability of DC microgrids to ensure the stable operation of DC microgrids. At present, impedance-based methods are usually used to analyze the small-signal stability of DC microgrids. This approach based on impedance allows direct stability criteria to be defined for each individual subsystem through convenient impedance specifications [6]. Initially, the Middlebrook impedance ratio criterion was proposed for stability analysis. On this basis, a series of stability criteria were derived, including Gain Margin and Phase Margin (GMPM), Opposing Argument (OA) criterion, energy source analysis consortiumcriterion (ESACC), root exponential stability criterion (RESC), maximum peak criterion (MPC), Modified Phase Margin and Opposing Argument (MPMOA) [7; 8; 9] and so on. These criteria divide the system into source and load subsystems and set different restricted areas for the Nyquist curve of the impedance ratio between source and load [10]. This makes them potentially assume a unidirectional power flow, but it is not applicable to the systems which contain ESSs. In order to solve the above problems, the system is no longer divided into source and load subsystems. The converters in the system are classified as bus voltage-controlled converters (BVCCs) and bus current controlled converters (BCCs); meanwhile, the system stability is judged by the general impedance stability criterion listed in [11]. However, this method needs to meet the premise that each converter is stable when it works alone. Secondly, when the power flow direction of the system changes, the converters in the system need to be reclassified and the input or output impedance needs to be recalculated. Therefore, the stability analysis is complex. Traditional stability is a condition that implies boundedness for dynamic closed system. If an autonomous closed-loop system is interconnected with another, its stability may be affected [12]. The Lyapunov direct method is the analytical method used to analyze the disturbance stability of the system. However, the acquisition of the Lyapunov function does not have an established method, and the criterion conditions may be conservative [13]. In addition, the artificial intelligence algorithm [14], singular perturbation theory [15], the dynamic phasor [16] method and other methods can also be used for isolated DC microgrid stability analyses.