## 3 A Hybrid AC/DC Topology
### Unified Hybrid AC/DC Network Model  
This section derives a unified model for steady-state analysis (conventional and optimal power flow calculation). The complete model consists of the VSCs and the AC/DC network. Figure 5 shows a feeder in the hybrid AC/DC network with the MVDC interface. The VSCs are represented using the unified modelling methodology presented in Section 3.1, while the network and the derivation of the complete model will be described in detail now.  
The entire hybrid AC/DC network will be modelled as a notionally only-AC system, as proposed in [29; 30; 31]. Specifically, the DC parts of the network are represented as notional only-AC parts, with resistive AC lines (zero reactance). This is possible due the equivalence of the power-flow equations of resistive AC lines in AC systems and the power-flow equations, as discussed in [28]. With this modelling approach, bus voltages and currents of the DC systems will be complex numbers, as those in AC systems, but with a zero imaginary part. Hence, the entire hybrid network, represented as a notionally only-AC equivalent system, will have an admittance matrix. Therefore, the nodal power-flow injections can be written with:  
\[\mathbf{S}=diag(\mathbf{V})\times(\mathbf{Y}^{*}_{acdc}\mathbf{V}^{*}) \tag{10}\]  
where the vector \(\mathbf{V}\) contains all nodal voltages (of the AC and DC systems) belonging to the set \(N\) and \(\mathbf{Y}_{acdc}\) is the admittance matrix of the complete AC/DC system. The entire network can be modelled using Equation (10), as a notionally only-AC system by assuming that there is no reactance in the DC side. Meanwhile, the nodal power injections of the AC feeders can be written as (see Figure 5 and Equation (10)):  
\[\overline{S}_{g\{l\}}=\overline{Y}_{ac\{l\}}[V_{g}^{2}-\overline{V}_{g} \overline{V}_{h}^{*}] \tag{11}\]  
Figure 5: One feeder in the hybrid AC/DC network with the MVDC interface.  
The power balance equations of every AC feeder can be written as:  
\[P_{g\{c\}}+\sum_{l=1}^{nl}P_{g\{l\}}+P_{d\{g\}}-P_{w\{g\}}=0 \tag{12}\] \[Q_{g\{c\}}+\sum_{l=1}^{nl}Q_{g\{l\}}+Q_{d\{g\}}=0 \tag{13}\]  
where \(nl\) is the total number of lines connected to bus \(g\). The subscripts \(d\) and \(w\) refer to the loads and wind-based DGs at their respective buses.  
Following Equation (10), the active and reactive power injections in the DC side can be written as:  
\[P_{0\{l\}} = G_{dc\{l\}}[E_{dc\{0\}}^{2}-E_{dc\{0\}}E_{dc\{n\}}\cos(\theta_{0} -\theta_{n})]=G_{dc\{l\}}E_{dc\{0\}}\triangle E_{dc} \tag{14}\] \[Q_{0\{l\}} = -E_{dc\{0\}}E_{dc\{n\}}G_{dc\{l\}}\sin(\theta_{0}-\theta_{n})]=0 \tag{15}\]  
In Equation (14) and (15), \(\triangle E_{dc}=E_{dc\{0\}}-E_{dc\{n\}}\) and \(Y_{dc\{l\}}=G_{dc\{l\}}\). By imposing the zero-reactive power injection condition, as expressed in Equation (8) and (15), the angles of the voltages of the DC buses will be zero \(\theta_{n}=\theta_{0}=0\) (initial angles of zero are assumed). Therefore, the total nodal active power injection at bus 0 of the the DC side is given by:  
\[\sum_{l=1}^{nl}P_{0\{l\}}=\sum_{l=1}^{nl}G_{dc\{l\}}E_{dc\{0\}}\triangle E_{dc} \tag{16}\]  
And the power balance at bus 0 of the DC side reads:  
\[P_{0\{c\}}+\sum_{l=1}^{nl}P_{0\{l\}}-P_{w\{0\}}=0 \tag{17}\]  
It follows from Equation (10) and from the assumptions on the DC side, that in the hybrid AC/DC network for any node, \(b\), the total calculated power injections can be calculated as below, considering the nodal power injections for the converter and the MVDC feeder derived in Equation (4)-(7), Equation (11), as well as Equation (14)-(15), respectively:  
\[\sum_{c\in\mathbb{C}}P_{b\{c\}}+\sum_{l\in L}P_{b\{l\}}+P_{d\{b\} }-P_{w\{b\}}=0\qquad\forall b\in N \tag{18}\] \[\sum_{c\in\mathbb{C}}Q_{b\{c\}}+\sum_{l\in L}Q_{b\{l\}}+Q_{d\{b\} }=0\qquad\forall b\in N \tag{19}\]  
This unified model can now be implemented in the proposed security ANM framework described in the earlier section.