## 2 Problem Formulation  
### Dynamic Model of Inverter-Based Distributed Generations (DGs)  
Figure 1 presents the block diagram of an islanded AC microgrid containing several DGs. Three-phase inverter-based \(DG_{i}\) (\(i=1,\,\ldots,N\)) is connected to the microgrid through a DC/AC inverter, an LC filter, and an output connector. Moreover, \(L_{i}^{s}\), \(R_{i}^{s}\) and \(C_{i}^{s}\) represent the inductance, resistance, and capacitance of the LC filter, respectively. \(L_{i}^{c}\), \(R_{i}^{c}\) represent the inductance and resistance of the output connector, respectively. The local control of \(DG_{i}\) is made up of the droop-based power controller, PI voltage controller and PI current controller in a cascade configuration. The power controller provides the reference for the voltage controller and the reference angular frequency for the whole control system. For \(DG_{i}\), droop technique which is used in the power controller describes the desired relationship between the frequency \(\omega_{i}\) and active power \(P_{i}\), and the relationship between the reference of output voltage magnitude \(v^{*}_{o,magi}\) and reactive power \(Q_{i}\), which are given by  
\[\left\{ \begin{array}{l}v^{dref}_{i}=V^{*}_{i}-n^{Q}_{i}Q_{i}\\ v^{dref}_{i}=0\\ \omega_{i}=\omega^{*}_{i}-m^{p}_{i}P_{i}\end{array} \right. \tag{1}\]  
where \(v^{dref}_{i}\) and \(v^{qref}_{i}\) are the \(d\)-axis and \(q\)-axis components of \(v^{*}_{o,magi}\), and output voltage \(v^{ad}_{i}\) can approach the reference value \(v^{*}_{o,magi}\) in steady-state. \(m^{p}_{i}\) and \(n^{Q}_{i}\) are the frequency and voltage droop coefficients. \(P_{i}\) and \(Q_{i}\) are fundamental components of active and reactive power which can be gained through a low-pass filter. \(\omega^{*}_{i}\) and \(V^{*}_{i}\) are the primary control reference, and \(V^{*}_{i}\) is typically set equal to \(V_{norm}\), where \(V_{norm}\) is the nominal voltage. According to the references by the power controller, \(v^{dref}_{i}\) and \(v^{qref}_{i}\), the differential algebraic equations of the voltage controller are written as  
\[\left\{ \begin{array}{l}t^{dref}_{i}=\left(K_{PVi}+\frac{K_{IVi}}{s}\right) \left(v^{dref}_{i}-v^{ad}_{i}\right)-\omega_{b}C^{s}_{i}v^{qq}_{i}+F_{i}t^{ad} _{i}\\ \dot{t}^{qref}_{i}=\left(K_{PVi}+\frac{K_{IVi}}{s}\right)\left(\dot{v}^{qref}_ {i}-v^{qq}_{i}\right)+\omega_{b}C^{s}_{i}v^{d}_{i}+F_{i}t^{aq}_{i}\end{array} \right. \tag{2}\]  
where \(K_{PVi}\) and \(K_{IVi}\) are the proportional and integral coefficients of the PI voltage control loop respectively, \(F_{i}\) is the feedforward gain, and \(\omega_{b}\) is the nominal angular frequency. By introducing feedforward quantities \(F_{i}t^{ad}_{i}\) and \(F_{i}t^{aq}_{i}\), the dynamic performance of voltage controllers under disturbance can be improved. With adoption decoupling items \(\omega_{b}C^{s}_{i}v^{qq}_{i}\) and \(\omega_{b}C^{s}_{i}v^{ad}_{i}\), the current in the \(d\)-\(q\) axis can be completely decoupled, which ensures that \(v^{ad}_{i}\) and \(v^{ad}_{i}\) are able to be controlled separately. Figure 1: Block diagram of an inverter-based islanded microgrid. The current controller is responsible to make inductive currents \(i_{i}^{dd}\) and \(i_{i}^{dd}\) to track their references, \(t_{i}^{dref}\) and \(i_{i}^{dref}\), provided by the voltage controller. The output of the current controller can be expressed as  
\[\left\{ \begin{array}{l}v_{i}^{dPPM}=\left(K_{PCI}+\frac{K_{ICI}}{s}\right) \left(i_{i}^{dref}-i_{i}^{dd}\right)-\omega_{b}{\rm{I}}_{i}^{s}i_{i}^{dq}\\ v_{i}^{dPPM}=\left(K_{PCI}+\frac{K_{ICI}}{s}\right)\left(i_{i}^{dref}-i_{i}^{dq }\right)+\omega_{b}{\rm{I}}_{i}^{s}i_{i}^{dd}\end{array} \right. \tag{3}\]  
where \(K_{PCI}\) and \(K_{ICI}\) denote the proportional and integral coefficients of the PI current control loop respectively. The output of current controller, \(v_{i}^{dPPM}\) and \(v_{i}^{dPPM}\), are used as references for the PWM generator.