## 3 A Hybrid AC/DC Topology
### Unified VSC Model  
Core to the MVDC interface model is the unified fundamental frequency model for the VSC which has already been introduced by us in [9; 27; 28]. The AC and DC sides of the VSC are coupled together through the following simple voltage relationship \(V_{cr}=km_{a}E_{dc}\), essentially modelling the VSC in a fundamental frequency as an ideal transformer with a continuous complex tap ratio (\(m_{a}\)) signifying the pulse width modulation (PWM) control of a real VSC. The voltage relationship allows fo ther coupling of the AC and DC sides together for developing unified formulations of power flow equations for both the AC and DC sides of the converter and by extension for the hybrid network [9; 27]. Consequently, the VSC control actions (i.e., AC power control and DC voltage control) may be modelled for power flows as well as optimal power flows on one unified frame of reference and solved simultaneously. The VSC model equivalent circuit as introduced in [9] is given in Figure 4.  
From Figure 4 the set of nodal current injections for the AC and DC sides of the converter are written as such:  
\[\left[\begin{array}{c}\overline{I}_{g}\\ \overline{I}_{0}\end{array}\right]=\left[\begin{array}{cc}\overline{Y}_{cr}&- km_{a}\overline{Y}_{cr}e^{\mathrm{j}\phi}\\ -km_{a}\overline{Y}_{cr}e^{-\mathrm{j}\phi}&G_{sw}+k^{2}m_{a}^{2}(\overline{Y}_ {cr}+\mathrm{j}B_{sw})\end{array}\right]\times\left[\begin{array}{c} \overline{V}_{g}\\ E_{dc}\end{array}\right] \tag{2}\]  
From Equation (2) we can then calculate the set of nodal power injections on both sides:  
\[\left[\begin{array}{c}\overline{S}_{g}\\ \overline{S}_{0}\end{array}\right]=\left[\begin{array}{cc}\overline{V}_{g}&0 \\ 0&E_{dc}\end{array}\right]\left[\begin{array}{c}\overline{I}_{g}\\ \overline{I}_{0}\end{array}\right]^{*} \tag{3}\]  
The nodal power injections at either side of the VSC, i.e., the bus \(g\) (AC side) and bus 0 (DC side) are calculated by expanding Equation (3) and knowing that the VSC is interfaced to the AC side by an equivalent phase reactor with admittance, \(\overline{Y}_{cr}=\mathrm{j}/x_{cr}=\mathrm{j}B_{cr}\):  
\[P_{g}=-B_{cr}km_{a}E_{dc}V_{g}\sin(\theta-\phi) \tag{4}\] \[Q_{g}=-B_{cr}V_{g}^{2}+B_{cr}km_{a}E_{dc}V_{g}\cos(\theta-\phi)\] (5) \[P_{0}=G_{sw}E_{dc}^{2}-B_{cr}km_{a}E_{dc}V_{g}\sin(\phi-\theta)\] (6) \[Q_{0}=-(B_{cr}+B_{sw})k^{2}m_{a}^{2}E_{dc}^{2}+B_{cr}km_{a}E_{dc }V_{g}\cos(\phi-\theta) \tag{7}\]  
An additional constraint is imposed to ensure that reactive power injections into the DC buses are kept at zero:  
\[-(B_{cr}+B_{sw})k^{2}m_{a}^{2}E_{dc}^{2}+B_{cr}km_{a}E_{dc}V_{g}\cos(\phi- \theta)=0 \tag{8}\]  
where \(B_{sw}\) is an equivalent shunt susceptance at the AC side that models the reactive power injection/absorption of the converter.  
Figure 4: The equivalent VSC (voltage source converter) fundamental frequency model.  
Moreover, the shunt resistance of the VSC, \(G_{sw}\) is used to model the switching losses inside the converter. This term increases with the switching frequency of the PWM. When the VSC operates at a nominal current and rated DC voltage, the switching losses are represented with a constant conductance, \(G_{0}\). Even though the DC voltage remains constant, the current injection changes according to the active and reactive power injections of the converter into the network. Therefore, the \(G_{sw}\) can be derived from the constant conductance and corrected by the quadratic of the operating current, \(I_{act}\) to the nominal current, \(I_{nom}\) as expressed in Equation (9) [27]:  
\[G_{sw}=G_{0}\left(\frac{I_{act}}{I_{nom}}\right)^{2} \tag{9}\]