# On Transient Stability Problems in DC Microgrids
## 3 Analysis of Control on Transient Stability
### Analysis of Droop-Controlled DC Generators  
The droop controller is another typical power controller \(G_{p(s)}\)[3; 19], which allows decentralized power sharing among multiple DC generators. Normally, the droop controller can take two forms. The first droop controller excludes the low-pass filter (LPF), expressed as  
\[v_{\text{c}}=v_{\text{c\_ref}}+k_{\text{droop}}(p_{\text{c\_ref}}-p_{\text{c}}), \tag{7}\]  
where \(k_{\text{droop}}\) denotes the droop coefficient. The second droop controller employs an LPF, which can not only attenuate high-frequency ripples but also introduce inertia to DC grids [15]  
\[v_{\text{c}}=v_{\text{c\_ref}}+k_{\text{droop}}\underbrace{\frac{\omega_{\text {c}}}{s+\omega_{\text{c}}}}_{\text{LPF}}(p_{\text{c\_ref}}-p_{\text{c}}), \tag{8}\]  
where \(\omega_{\text{c}}\) represents the cut-off angular frequency of the LPF. In AC microgrids, the important role inertia plays in frequency regulation stability has been publicly acknowledged [20]. However, the effect of inertia on the transient stability of DC microgrids remains to be explored. Figure 5: _Cont._  
First, we consider the first droop controller. Substitution of (2) into (7), the latter yields the two equilibrium points of \(v_{c}\) for droop-controlled DC generators:  
\[v_{\text{cl,2\_ droop}}=\frac{(v_{\text{g}}k_{\text{ droop}}-R_{\text{c}})\pm\sqrt{(R_{\text{c}}-v_{\text{g}}k_{\text{ droop}})^{2}+4k_{\text{ droop}}(R_{\text{c}}v_{\text{c\_ref}}+R_{\text{c}}k_{\text{ droop}}p_{\text{c\_ref}})}}{2k_{\text{ droop}}}. \tag{9}\]  
Remember that the existence of equilibrium points in (9) necessitates a positive term in the bracket of (9), leading to  
\[p_{\text{c\_ref}}\geq\frac{-(R_{\text{c}}-v_{\text{g}}k_{\text{ droop}})^{2}-4k_{\text{ droop}}R_{\text{c}}v_{\text{c\_ref}}}{4k_{\text{ droop}}^{2}R_{\text{c}}}. \tag{10}\]  
It is noted that (10) determines the limitation of power absorption for droop-controlled DC generators. If (10) satisfies, the corresponding powers are  
\[p_{\text{cl,2\_ droop}}=\frac{v_{\text{c\_ref}}-v_{\text{cl,2\_ droop}}}{k_{\text{ droop}}}+p_{\text{c\_ref}}, \tag{11}\]  
As (7) related to the first droop controller does not involve any system dynamics, both the two equilibrium points in (9) can be reached immediately. However, this will not happen in practice due to filtering and control dynamics. Thus, we will investigate the second droop controller, which includes the LPF, to analyze the transient stability of droop-controlled generators. Likewise, we substitute (2) into (8) and derive the expression of \(\text{d}v_{\text{c}}/\text{d}t\) as  
\[\frac{\text{d}v_{\text{c}}}{\text{d}t}=\omega_{\text{c}}(v_{\text{c \_ref}}-v_{\text{c}})+k_{\text{ droop}}\omega_{\text{c}}\bigg{[}p_{\text{c\_ref}}-\frac{v_{\text{c}}(v_{\text{c}}-v_{ \text{g}})}{R_{\text{c}}}\bigg{]}. \tag{12}\]  
Let (12) equal zero. The two equilibrium points of \(v_{\text{c}}\) for droop-controlled DC generators remain unchanged as in (9). As such, the corresponding limitation of power absorption and powers follow (10) and (11), respectively. Figure 6 illustrates the curve of \(\text{d}v_{\text{c}}/\text{d}t\) as a function of \(v_{\text{c}}\) with droop control. By comparing Figures 5 and 6, it is clear that the undesirable equilibrium point \(v_{\text{c\_droop}}\)  
Figure 5: \(\text{d}v_{\text{c}}/\text{d}t\) as a function of \(v_{\text{c}}\) with different PI control parameters. becomes negative in Figure 6, which is an important feature. According to (9), this statement holds valid when  
\[\
u_{\text{c\_ref}}+k_{\text{droop}}p_{\text{c\_ref}}>0, \tag{13}\]  
which is true for normal operating points. In addition, different from (4), (12) always contains a parabolic equation in the right-hand side, therefore translating into only one stable equilibrium point \(v_{\text{c1\_droop}}\) and one unstable \(v_{\text{c2\_droop}}\). Therefore, as compared to PI power-controlled DC generators, droop-controlled DC generators feature an enlarged stable region down to \(v_{\text{c}}<0\) V (or \(v_{\text{c}}>v_{\text{c2\_droop}}\)).