*2.2. DC Distribution Network Power Flow Model*

According to the branch flow model of the AC distribution network, the branch flow model of the DC distribution network can be derived as follows:

$$P\_{\rm dc,jk} = P\_{\rm dc,ij} - \frac{r\_{\rm dc,ij} \left(P\_{\rm dc,ij}\right)^2}{\left(V\_{\rm dc,i}\right)^2} - P\_{\rm dc,j} \tag{7}$$

$$(V\_{\rm dc,j})^2 = (V\_{\rm dc,i})^2 - \frac{r\_{\rm dc,ij}P\_{\rm dc,ij}}{V\_{\rm dc,i}} \tag{8}$$

$$P\_{\rm dc,j} = P\_{\rm dc,j,L} - P\_{\rm dc,j,DG} \tag{9}$$

where *V*dc,*<sup>i</sup>* is the voltage amplitude of the DC node *i*; *r*dc,*ij* is the resistance between the branches *ij*; *P*dc,*ij* is the active power between the branches *ij*; *P*dc,*j*,*<sup>L</sup>* and *P*dc,*j*,DG are the load active power and

DG active power at the node *j*, respectively. The VSC model is shown in Figure 1. Where *V*∠*θ<sup>V</sup>* is the voltage at the junction of the converter and the AC distribution system, *R*VSC and *X*VSC are the equivalent resistance and reactance inside the converter, *V*VSC∠*θ*VSC is the phase voltage of the input converter, *P*VSC + *jQ*VSC is the power of the input converter, *P*dc and *V*dc are the output power and voltage of the converter. *Appl. Sci.* **2019**, *9*, x FOR PEER REVIEW 4 of 16

**Figure 1.** Voltage source converter (VSC) model. **Figure 1.** Voltage source converter (VSC) model.

where *V* ∠θ *<sup>V</sup>* is the voltage at the junction of the converter and the AC distribution system, *R*VSC and *X*VSC are the equivalent resistance and reactance inside the converter, *V*VSC VSC ∠θ is the phase voltage of the input converter, *P jQ* VSC VSC + is the power of the input converter, *P*dc and *V*dc are the output The branch flow model of *<sup>R</sup>*VSC <sup>+</sup> *jX*VSC is as shown in Equation (10). The AC three-phase activepower of the input VSC is equal to the DC power of the VSC output. The input voltage and the output voltage satisfy the following relationship:

$$V\_{\rm VSC} = \frac{\sqrt{3}}{3} \mu V\_{\rm dc} \tag{10}$$

(10)

power of the input VSC is equal to the DC power of the VSC output. The input voltage and the output voltage satisfy the following relationship: where *µ* is the DC voltage utilization under SPWM modulation, which is 0.866.
