# On Transient Stability Problems in DC Microgrids
## 3 Analysis of Control on Transient Stability
### Analysis of PI Power-Controlled DC Generators  
Figure 4 shows the schematic and control block diagram of DC generators in DC microgrids. Without loss of generality, a buck converter serves as an exemplified DC generator interface. It consists of a pair of semiconductor switches \(S_{c1}\) and \(S_{c2}\), which operate complimentarily and can step down the input voltage \(v_{in}\) to the output voltage \(v_{c}\) via an LC filter.  
The control architecture involves three cascaded control loops, comprising the inner current controller \(G_{i(s)}\), middle voltage controller \(G_{v(s)}\), and outer power controller \(G_{p(s)}\), which are designed to be well separated in time scales. For transient stability analysis, we normally ignore voltage and current controllers due to their fast dynamics and assume them to have unity gains [16; 17]. However, such an assumption may affect the small-signal stability of grid-forming converters in the case of fast power control [18]. Henceforth, we only consider the power control loop, which determines the converter output voltage \(v_{c}\)[7]. One typical power controller \(G_{p(s)}\) is a PI power controller [1], resulting in a converter voltage of  
\[v_{c}=v_{c\_ref}+(k_{\text{pp}}+\frac{k_{\text{pi}}}{s})(p_{c\_ref}-p_{c}), \tag{3}\]  
where \(v_{c\_ref}\) and \(p_{c\_ref}\) stand for the converter voltage and power references, respectively. \(k_{\text{pp}}\) and \(k_{\text{pi}}\) are the proportional and integral gains of the PI controller, respectively. \(s\) refers to the Laplacian complex frequency variable. Substitution of (2) into (3), the latter can be reorganized in the time domain as  
\[\frac{\text{d}v_{c}}{\text{d}t}=\frac{k_{\text{pi}}p_{c\_ref}R_{c}-k_{\text{ pi}}v_{c}(v_{c}-v_{\text{g}})}{(R_{c}+2k_{\text{pp}}v_{c}-k_{\text{pp}}v_{ \text{g}})}. \tag{4}\]  
Let (4) equal zero. We can derive the two equilibrium points as  
\[v_{c1,2\_PI}=\frac{v_{\text{g}}\pm\sqrt{v_{\text{g}}^{2}+4p_{c\_ref}R_{c}}}{2}. \tag{5}\]  
Notably, both equilibrium points are positive for PI control when \(p_{c\_ref}<0\). In addition, the two equilibrium points only exist when the term inside the brackets is positive or  
Figure 4: Schematic and control block diagram of DC generators and DC microgrids.  
\[p_{\text{c\_ref}}\geq p_{\text{c\_min}}=-\frac{\
u_{\text{g}}^{2}}{4R_{\text{c}}}. \tag{6}\]  
Otherwise, no equilibrium point exists, and transient stability problems appear. Therefore, transient instabilities will occur in the following two cases. First, the active power reference \(p_{\text{c\_ref}}\) drops below \(p_{\text{c\_min}}\). Second, the active power pc passes through the undesirable equilibrium point.  
Figure 5 plots two curves of \(\text{d}v_{\text{c}}/\text{d}t\) as a function of \(v_{\text{c}}\) with different control parameters \(k_{\text{pp}}=0.1\) and \(k_{\text{pp}}=0.3\). It can be inferred that the signs of \(\text{d}v_{\text{c}}/\text{d}t\) around the equilibrium points determine their stability. Referring to Figure 5a, we find that the equilibrium point \(v_{\text{c1 PI}}\) is stable as its nearby voltages converge to it. In contrast, \(v_{\text{c2 PI}}\) is unstable. In Figure 5b, both \(v_{\text{c1 PI}}\) and \(v_{\text{c2 PI}}\) are stable. Regardless of stability, \(v_{\text{c2 PI}}\) is undesirable and should be avoided in practice due to excessive currents. As such, the stable region of the PI power-controlled generator is (at most) as low as \(v_{\text{c2 PI}}\).