## 3 Evaluation the Impacts of Virtual Inertia and VMPE on the Proposed Dynamic Model  
By use of the achieved active and reactive power-based dynamic model (2) along with the proposed controller-based time domain dynamic Equation (10), an evaluation related to the virtual inertia and VMPE effects is executed in this section. All power and angular frequency variations are firstly defined as:  
\[\Delta x=x^{*}-x\qquad x\in\{P,Q,P_{m},\omega\} \tag{12}\]  
The time-based dynamic model of (10) can be achieved as:  
\[\left(\omega^{*2}J\right)\frac{d\omega}{dt}+(P_{m}^{*}-P^{*})\omega+(\omega^{* })P-(\omega^{*})P_{m}=\left[\omega^{*2}J\dot{\omega}^{*}+(P_{m}^{*}-P^{*}) \omega^{*}-\omega^{*}P_{m}^{*}+\omega^{*}P^{*}\right] \tag{13}\]  
\[ \begin{array}{c}\left(\omega^{*2}P^{*}\right)\frac{d\omega}{dt}+(P^{*}(P_{ m}^{*}-P^{*}))\omega-(\omega^{*}Q^{*})Q-(\omega^{*}P^{*})P_{m}=\\ \left[\omega^{*2}P^{*}J\dot{\omega}^{*}+P^{*}(P_{m}^{*}-P^{*})\omega^{*}- \omega^{*}P^{*}P_{m}^{*}-\omega^{*}Q^{*2}\right]\end{array}  \tag{14}\]  
As can be understood from (13) and (14), the zero dynamic of angular frequency i.e., \(d\omega/dt\) is dependent on the active power, reactive power, virtual inertia, VMP, the angular frequency, and also their reference values. Considering the desired value for \(d\omega/dt\) in (10) and (11) that is \(d\omega/dt=0\), the relationships (13) and (14) can be rewritten based on linear relationships for both active and reactive power as,  
\[P=\left[\omega^{*}J\dot{\omega}^{*}+(P_{m}^{*}-P^{*})-P_{m}^{*}+P^{*}\right]- \frac{(P_{m}^{*}-P^{*})}{(\omega^{*})}\omega+P_{m} \tag{15}\]  
\[Q=-\frac{\left[\omega^{*}P^{*}\dot{\omega}^{*}+P^{*}(P_{m}^{*}-P^{*})-P^{*}P_{ m}^{*}-Q^{*2}\right]}{Q^{*}}+\frac{P^{*}(P_{m}^{*}-P^{*})}{\omega^{*}Q^{*}} \omega-\frac{P^{*}}{Q^{*}}P_{m} \tag{16}\]  
The diagrams related to (15) and (16) are drawn in Figure 2(a),b, respectively. As can be seen from Figure 2(a), a constant ramp is achieved for the active power diagram. The reference value increment and decrement of the VMPE can change the initial value of the active power diagram according to Figure 2(a). On the other hand, as demonstrated in Figure 2(b), a changeable ramp generated by varying the reactive power reference value can be observed for the reactive power diagram. Moreover, the initial value of the reactive power diagram can be altered by \(P_{m}^{*}\) and \(\Delta\omega\). These diagrams illustrate that how improved control of \(P_{m}\) and \(\omega\) can help both active and reactive power approach to desired values. Figure 3: Diagrams of (**a**) the active power based on the virtual mechanical power (VMP) with changing \(P_{m}^{*}\) and \(\Delta\omega\) and (**b**) the reactive power based on VMP with changing \(P_{m}^{*}\), \(Q^{*}\) and \(\Delta\omega\). In order to reach an accurate zero dynamic-based analysis associated with the proposed controller-based model, the second order derivative of the angular frequency should be driven by use of (13) and (14) as:  
\[\left(\omega^{*2}J\right)\frac{d^{2}\omega}{dt^{2}}+(P_{m}^{*}-P^{*})\frac{d \omega}{dt}+(\omega^{*})\frac{dP}{dt}=\frac{d\left[ \begin{array}{c}\omega^{* 2}J\dot{\omega}^{*}+(P_{m}^{*}-P^{*})\omega^{*}\\ -\omega^{*}P_{m}^{*}+\omega^{*}P^{*}+(\omega^{*})P_{m}\end{array} \right]}{dt} \tag{17}\]  
\[\left(\omega^{*2}P^{*}J\right)\frac{d^{2}\omega}{dt^{2}}+(P^{*}(P_{m}^{*}-P^{*} ))\frac{d\omega}{dt}-(\omega^{*}Q^{*})\frac{dQ}{dt}=\frac{d\left[ \begin{array}[] {c}\omega^{*2}P^{*}J\dot{\omega}^{*}+P^{*}(P_{m}^{*}-P^{*})\omega^{*}\\ -\omega^{*}P^{*}P_{m}^{*}-\omega^{*}Q^{*2}+(\omega^{*}P^{*})P_{m}\end{array}  \right]}{dt} \tag{18}\]  
The zero dynamic of the VMP is found in (17) and (18). Moreover, it can be realized from (17) and (18) that the zero dynamics of the active and reactive power can highly impact on the angular frequency variations.