## 2 Converter Model  
A schematic of the converter is shown in Figure 3. The converter is built with two T-structures (\(T_{1}\) and \(T_{2}\)) with three arms each. In turn, each arm is built with an inductor \(L\) and N cascaded H-bridges named _series_, _derivation_, and _output_ stacks. The use of H-bridges in the stacks allows the possibility to operate with bidirectional power flow. Moreover, the inductors are required to couple the ac voltage components produced by each stack for current control purposes. The resistor \(R\) represents the equivalent resistance of the inductor and each T-structure is grounded through a _derivation_ stack.  
Figure 4 shows the average model of the converter in terms of the voltages and currents in each arm of the topology. Time-varying voltages \(v_{se_{1,2}}(t)\), \(v_{de_{1,2}}(t)\), and \(v_{o_{1,2}}(t)\) are produced by stacks _series_, _derivation_, and _output_, respectively. These time-varying voltages are dependent on the arm currents \(i_{se_{1,2}}(t)\), \(i_{de_{1,2}}(t)\), and \(i_{o_{1,2}}(t)\), respectively. \(V_{i}\) is the dc input voltage. \(V_{o}\) is the converter output voltage which has a dc component and harmonics multiple of the switching frequency. In an MT-HVdc grid, \(V_{o}\) is another terminal and can deliver power to \(V_{i}\).  
Figure 4: Average model of the double-T topology.  
Figure 3: Scheme of the double-T topology.  
Applying voltage and current laws to the meshes, then the main equations defining the average model can be derived:  
\[V_{i}=L\frac{di_{se1}(t)}{dt}-L\frac{di_{de1}(t)}{dt}+R(i_{se1}(t)-i_{de1}(t))+v_ {se1}(t)+v_{de1}(t) \tag{1}\]  
\[V_{i}=L\frac{di_{se2}(t)}{dt}-L\frac{di_{de2}(t)}{dt}+R(i_{se2}(t)-i_{de2}(t))+v_ {s2}(t)+v_{d2}(t) \tag{2}\]  
\[0=L\frac{di_{de1}(t)}{dt}+L\frac{di_{01}(t)}{dt}+R(i_{de1}(t)+i_{01}(t))-v_{de1 }(t)+v_{o1}(t)+V_{o} \tag{3}\]  
\[0=L\frac{di_{de2}(t)}{dt}+L\frac{di_{io2}(t)}{dt}+R(i_{de2}(t)+i_{02}(t))-v_{ de2}(t)+v_{o2}(t)+V_{o} \tag{4}\]  
\[i_{de1}(t)=i_{01}(t)-i_{se1}(t) \tag{5}\]  
\[i_{de2}(t)=i_{02}(t)-i_{so2}(t) \tag{6}\]  
\[-i_{i}(t)-i_{de1}(t)-i_{de2}(t)+i_{o}(t)=0 \tag{7}\]  
The voltages \(v_{se1,2}(t)\), \(v_{de1,2}(t)\) and \(v_{o1,2}(t)\) are defined in (8)-(10) where \(S_{i,se1,2}(t)\), \(S_{i,de1,2}(t)\) and \(S_{i,o1,2}(t)\) are the switching functions of each sub-module (H-bridge) of each stack of the topology. The value of the switching functions can be either \(-1\), \(0\), or \(1\). Finally, \(C_{i}\) are the capacitors of each sub-module.  
\[v_{se_{1,2}}(t)=\sum_{j=1}^{2}\sum_{i=1}^{N}\frac{1}{C_{i}}\int\Bigl{(}i_{se_{ j}}(t)\!\cdot\!S_{i,se_{j}}(t)\Bigr{)}\:dt \tag{8}\]  
\[v_{de_{1,2}}(t)=\sum_{j=1}^{2}\sum_{i=1}^{N}\frac{1}{C_{i}}\int\Bigl{(}i_{de_ {j}}(t)\!\cdot\!S_{i,de_{j}}(t)\Bigr{)}\:dt \tag{9}\]  
\[v_{o_{1,2}}(t)=\sum_{j=1}^{2}\sum_{i=1}^{N}\frac{1}{C_{i}}\int\Bigl{(}i_{o_{j }}(t)\!\cdot\!S_{i,o_{j}}(t)\Bigr{)}\:dt \tag{10}\]  
From (1)-(7) it is possible to obtain the expressions for the instantaneous currents. Let \(x(t)=[i_{o1}(t)\:i_{o2}(t)\:i_{se1}(t)\:i_{s2}(t)]^{T}\) be the state vector of independent currents and \(u(t)=[v_{se1}(t)\:v_{s2}(t)\:v_{de1}(t)\:v_{de2}(t)\:v_{o1}(t)\:v_{o2}(t)]^{T}\) the control variables vector, the state-space model is given by:  
\[\frac{d}{dt}\begin{bmatrix}i_{o1}(t)\\ i_{o2}(t)\\ i_{s2}(t)\end{bmatrix}=-\frac{R}{L}\begin{bmatrix}i_{o1}(t)\\ i_{o2}(t)\\ i_{s1}(t)\\ i_{s2}(t)\end{bmatrix}+\frac{1}{3L}\begin{bmatrix}-1&0&1&0&-2&0\\ 0&-1&0&1&0&-2\\ -2&0&-1&0&-1&0\\ 0&-2&0&-1&0&-1\end{bmatrix}\begin{bmatrix}v_{se1}(t)\\ v_{s2}(t)\\ v_{d2}(t)\\ v_{o1}(t)\\ v_{o2}(t)\end{bmatrix}+\begin{bmatrix}V_{i}-2V_{o}\\ V_{i}-2V_{o}\\ V_{i}-V_{o}\\ 2V_{i}-V_{o}\\ \end{bmatrix} \tag{11}\]