## 2 Fundamentals of the DC Microgrid
### Problem Definition and Control Objectives  
Conventionally, a droop controller is employed to realize proportional power sharing in an islanded DC microgrid where multiple DERs connect to a common bus. The primary control loop of the DC-DC converter, associated with each DER, incorporates the droop control, which operates in a decentralized manner alongside PI-based voltage and current controllers. In addition, the droop control function can be utilized to generate the voltage reference signal, enabling current sharing among multiple sources. Mathematically, this function can be represented as:  
\[U_{ref_{i}}=U^{*}-k_{d_{i}}I_{o_{i}} \tag{5}\]  
where \(k_{d}\) denotes the virtual droop resistance and \(U^{*}\) represents the rated DC bus voltage of the converter operating with the \(i\)th DER. Conventionally, the DER DC voltage output, \(U\), must track the voltage reference as:  
\[U=U_{ref_{i}} \tag{6}\]  
Thus, combining (2) to (6), we derive:  
\[U_{dc}=U^{*}-\big{(}R_{line_{i}}+k_{d_{i}}\big{)}I_{o_{i}} \tag{7}\]  
Figure 2: Communication network of multiple DERs. (**a**) Communication graph. (**b**) Communication matrix. Based on (7), we can deduce that the voltage at the common DC bus, \(U_{dc}\), will consistently differ from the rated DC voltage, \(U^{*}\), as long as the current output, \(I_{o}\), remains non-zero. Similarly, the extent of the voltage deviation is contingent upon the transmission line resistance, \(R_{line}\), and droop resistance value, \(k_{d}\). Higher values of \(R_{line}\) will amplify the deviations in the DC bus voltage, while the magnitude of \(R_{line}\) has an impact on the bus voltage. This study focuses on an islanded DC microgrid comprising \(M\) DERs, aiming to actualize average power sharing to attain the load requirements. It can be derived from (7) that:  
\[\big{(}R_{line_{i}}+k_{d_{i}}\big{)}I_{o_{i}}=\Big{(}R_{line_{j}}+k_{d_{j}} \Big{)}I_{o_{j}}\qquad\forall\,i,j=1,\dots,M \tag{8}\]  
Consequently, the current sharing ratio between the converters is determined by the combined effect of \(k_{d}\) and \(R_{line}\) of each DER in the microgrid:  
\[\frac{I_{o_{i}}}{I_{o_{j}}}=\frac{R_{line_{j}}+k_{d_{j}}}{R_{line_{i}}+k_{d_{i}}} \tag{9}\]  
In previous studies, the line resistance has often been overlooked for simplicity during analysis. However, in real-world DC microgrid systems, the line resistance cannot be neglected due to the varying locations of DERs in relation to the common bus. While the converters typically have the same nominal output voltage, it may not be assumed that the line resistances and externally added droop resistances are the same, especially when they are located at different positions. In the occurrence of a slight voltage variation, a circulating current proportional to the voltage error initiates its flow. The formulation and analysis of the proposed method consider the inclusion of line resistance. To achieve coordinated current distribution among the DERs and sustain the bus voltage at the nominal DC voltage, the formulated control objectives for the direct current MG are two-fold:  
1. To ensure voltage recovery on the DC bus, the voltage variation needs to be eliminated entirely. This objective is global, meaning that a single DC voltage needs to be upheld in the MG, and it can be formulated as: \[ \begin{split}\Delta U_{dc}=U_{dc}-U^{*},\\ \lim_{t\rightarrow\infty}\Delta U_{dc}(t)=0.\end{split} \] (10)
2. To achieve equal current distribution among the DERs actively operating within the microgrid, minimization of the current distribution error in each DER is necessary and ideally reduced to zero. This can be formulated as follows: \[ \begin{split}\Delta I_{o_{i}}(t)=-\sum_{j\in M,j\
eq i}k_{ij} \left(\frac{I_{o_{i}}(t)}{P_{i}}-\frac{1}{M-1}\frac{I_{o_{j}}(t)}{P_{j}} \right),\\ \lim_{t\rightarrow\infty}\Delta I_{o_{i}}(t)=0.\end{split} \] (11) where \(P\) denotes the current distribution ratio of DERs.