## 3 Discrete Adjoint Model of DC Grid  
The characteristic of discrete adjoint model is that the lumped parameter element is converted into a parallel connection of an admittance and a current source according to the state of its history voltage and current. Thus, a circuit network originally described by differential-algebraic equations could be transformed into a network that could only be described by algebraic equations [4].  
The discrete adjoint model of the series-connected resistor-inductor branch is shown in Figure 3, and the corresponding voltage and current characteristics are listed in Equation (10).  
\[\left\{\begin{array}{l}\dot{t}_{ij}^{k+1}=G_{\text{RL}}\Big{(}V_{i}^{k+1}-V_{ j}^{k+1}\Big{)}+J_{\text{RL}}^{k}\\ \zeta_{\text{RL}}=h/(L+hR)\\ \dot{J}_{\text{RL}}^{k}=L\dot{t}_{ij}^{k}/(L+hR)\end{array}\right. \tag{10}\]  
where, \(h\) is the time step. \(L\) is the inductance of the branch; \(R\) is the resistance of the branch. The superscript \(k\) indicates the current step, and the superscript \((k+1)\) indicates the next step. \(i\) is the branch current. \(V\) is the node voltage. \(G_{\text{RL}}\) is the equivalent admittance, and \(J_{\text{RL}}\) is the equivalent current source.  
The discrete adjoint model of the capacitor branch is shown in Figure 4, and the corresponding voltage and current characteristics are shown in Equation (11).  
\[\left\{\begin{array}{l}\dot{t}_{ij}^{k+1}=G_{\text{C}}\Big{(}V_{i}^{k+1}-V_{ j}^{k+1}\Big{)}+J_{\text{C}}^{k}\\ G_{\text{C}}=C/h\\ J_{\text{C}}^{k}=-C\Big{(}V_{i}^{k}-V_{j}^{k}\Big{)}/h\end{array}\right. \tag{11}\]  
where, \(G_{\text{C}}\) is the equivalent admittance and \(J_{\text{C}}\) is the equivalent current source. The other variables have been defined as above.  
By replacing the MMC equivalent circuit with the discrete adjoint model of the basic components, the discrete adjoint model of an MMC could be obtained, as shown in Figure 5.  
Combing Equations (10) and (11) with the KVL and KCL laws:  
\[\begin{array}{l}\dot{t}_{mi}^{k+1}=G_{\text{mi}}u_{\text{dc}}^{k+1}+J_{\text {mi}}^{m}\\ G_{\text{mi}}=1/\Big{(}\frac{L_{\text{dc}}^{k}}{L_{\text{eq}}}+R_{\text{eq}}+ \frac{L_{\text{eq}}}{L_{\text{eq}}}\Big{)}\\ J_{mi}^{m}=\Big{(}u_{\text{c}}^{n}-\frac{L_{\text{eq}}}{h}\dot{t}_{\text{dc}}^{ n}-\frac{L_{\text{eq}}}{L_{\text{eq}}}\dot{t}_{g}^{n+1}\Big{)}/\Big{(}\frac{L_{ \text{eq}}}{h}+R_{\text{eq}}+\frac{h}{L_{\text{eq}}}\Big{)}\end{array} \tag{12}\]  
Figure 4: Discrete adjoint model of capacitor branch: (**a**) original circuit; (**b**) discrete adjoint model.  
Figure 3: Discrete adjoint model of series-connected resistance-inductor branch: (**a**) original circuit; (**b**) discrete adjoint model.  
In a large-scale DC grid, the overhead transmission lines are used to deliver bulk power over a long distance. The DC overhead lines are represented as a RL series circuit. The smoothing reactor on both sides of the MMC could be involved into the equivalent model of the lines. As shown in Figure 2, \(R\) represents the equivalent resistance of the DC line and \(L\) represents the DC reactor.  
All MMC converters can be equivalent to RLC series circuits parallel with the controlled current sources, and DC lines can be equivalent to RL series circuits. On this basis, it is easy to obtain the discrete adjoint model of the DC grid.