## 2 Structure and Modeling of Island DC Microgrid
### Dimension Reduction Modeling of Multi-Voltage Control Unit under Droop Control  
On the basis of the work done above, the multi-branch combination model of DC microgrids with multiple voltage control units was obtained, as illustrated in Figure 3.  
Figure 1: Typical configuration of an island direct current (DC) micro grid.  
Figure 2: Distributed Resources model. (**a**) Maximum Power Point Tracking (MPPT) control. (**b**) Droop control.  
\(V_{i}\) is the source voltage and \(v_{bus}\) is the voltage of the dc bus. \(R_{di}\) is the droop coefficient of the \(i\)th droop controller, \(R_{ti}\) and \(L_{ti}\) are the line resistance and inductance from \(i\)th energy storage unit to the bus, respectively. \(C\) is the equivalent capacitance on the bus [25]. The droop resistance and line resistance in series of the \(i\)th control unit can be represented by:  
\[R_{i}{=R_{di}}{+}R_{ti}, \tag{2}\]  
Assume that the value of droop resistance is much greater than the line resistance, Equation (2) can be rewritten as:  
\[R_{i}\approx R_{di}, \tag{3}\]  
As a complex multi-dimensional system, the DC microgrid power system is not conducive to analysis [26]. It is necessary to simplify the \(n\)-dimensional equation of representative DC microgrid system. Each distributed unit can be modeled as a differential equation in (4).  
\[\begin{array}{l}\frac{di_{1}}{dt}=\frac{1}{L_{ti}}(V_{1}{-}v_{bus})-\frac{R_ {di}}{L_{ti}}i_{1},\\ \frac{di_{2}}{dt}=\frac{1}{L_{2}}(V_{2}-v_{bus})-\frac{R_{di}}{L_{2}}i_{2},\\ \vdots\\ \frac{di_{3}}{dt}=\frac{1}{L_{ni}}(V_{n}-v_{bus})-\frac{R_{di}}{L_{ni}}i_{n}.\end{array} \tag{4}\]  
Furthermore, the total current \(i_{s}\) provided by the resources can be denoted by  
\[i_{s}{=i_{1}}{+}i_{2}+\cdots{+}i_{n}, \tag{5}\]  
Therefore, the sum of the \(n\) differential equations in (4) can be simplified to  
\[\frac{di_{s}}{dt}=\left(\sum_{i=1}^{n}\frac{1}{L_{ti}}\right)(V_{i}-v_{bus})- \frac{R_{di}}{L_{ti}}i_{s}, \tag{6}\]  
In reality, multiple identical energy storage units often exist in the form of clusters [27], and the overall power supply capacity is evenly divided into individual battery unit. Assuming that the specifications of each energy storage unit and the line length to the bus are the same. Thus, the following equation is obtained  
\[\begin{array}{l}V_{ref}{=}V_{1}{=}V_{2}=\cdots=V_{n},\\ R_{d1}{=}R_{d2}=\cdots=R_{dn},\\ L_{t1}{=}L_{t2}=\cdots=L_{tn}.\end{array} \tag{7}\]  
Figure 3: Multi-branch combination model of \(n\) droop control resources.  
Multiplying both sides of (6) by  
\[L_{t}=\frac{1}{\sum_{i=1}^{n}\frac{1}{L_{tn}}}, \tag{8}\]  
The system is simplified from multiple differential equations to a simple differential equation in (9). Thus, the conversion from multi-dimensional to a low-dimensional is realized  
\[L_{t}\frac{di_{s}}{dt}=\left(V_{ref}-v_{bus}\right)-L_{t}\frac{R_{dn}}{L_{tn}}i _{s}, \tag{9}\]  
And under the condition:  
\[R_{d}=L_{t}\frac{R_{dn}}{L_{tn}}, \tag{10}\]  
From (9), the equivalent steady-state model of the DC microgrid system can be obtained, as illustrated in Figure 4.  
where \(V_{ref}\) is the reference voltage of the bus. Resistance \(R_{d}\) is the droop control coefficient. \(P\), \(R_{dc}\) correspond to the equivalent power of CPLs and the resistive load, respectively. The corresponding whole system parameters are listed in Table 1.