# Minimum Frequency and Voltage Stability Constrained Unit Commitment for AC/DC Transmission Systems
## 4 Model Decomposition and Solution
### Security Constraints Sub-Model  
This sub-model is used to validate whether the optimization result of the UC master model satisfies the operating constraints in the hybrid AC/DC power system. In essence, the process is equal to the calculation of the power flow with HVDC operational constraints, generator reactive power constraint, bus voltage, and transmission power limits. This paper solves such problem by the traditional Newton-Raphson method which is transformed into successive linear programming. The advantage is that the Benders cut could be formed conveniently, as shown in Formulas (30)-(39). Among these variables, \(MP_{1}\) (\(MP_{2}\)), \(MQ_{1}\) (\(MQ_{2}\)), \(MPL_{1}\) (\(MPL_{2}\)), and \(MPD_{1}\) (\(MPD_{2}\)) are non-negatively and used to relax equations of the AC/DC hybrid power flow calculation. Among formulas, Formula (31) is the linear equation of buses active and reactive power; Formula (32) is the linear equation of the active and reactive power of the HVDC transmission system; Formula (33) is the linear equation of line active power; Formulas (34)-(37) are the upper and lower limits of generator active and reactive power, transmission power and buses voltage; Formulas (38) and (39) are the upper and lower limits of the relevant AC/DC variables in the HVDC transmission system. We use \(H,N,J,L,O,D,E,F,Z,S\) as the first derivative coefficient matrix of the corresponding equation to independent variables. In addition to the equilibrium node, the active increment of other nodes should meet Formula (34). \(\Lambda,\underline{\psi}\) and \(\overline{\psi}\) are the corresponding simplex multiplier of the constraints. \[Min\ w^{t} =\sum\left(MP_{1}+MP_{2}\right)+\sum\left(MQ_{1}+MQ_{2}\right)\] \[+\sum\left(MPL_{1}+MPL_{2}\right)+\sum\left(MPD_{1}+MPD_{2}\right) \tag{30}\]  
\[\left[ \begin{array}{c}\Delta P_{G}\\ \Delta Q_{G}\end{array} \right]+\left[ \begin{array}{cc}H&N\\ J&L\end{array} \right]\left[ \begin{array}{c}\Delta\theta\\ \Delta V\end{array} \right]+\left[ \begin{array}{c}\Delta P_{dc}\\ \Delta Q_{dc}\end{array} \right]+\left[ \begin{array}{c}MP_{1}\\ MQ_{1}\end{array} \right]-\left[ \begin{array}{c}MP_{2}\\ MQ_{2}\end{array} \right]=\left[ \begin{array}{c}dP\\ dQ\end{array} \right] \tag{31}\]  
\[\left[ \begin{array}{c}\Delta P_{dc}\\ \Delta Q_{dc}\end{array} \right]+\left[ \begin{array}{cc}O&D\\ E&F\end{array} \right]\left[ \begin{array}{c}\Delta X_{ac}\\ \Delta X_{dc}\end{array} \right]+\left[ \begin{array}{c}MP_{1}D\\ MQD_{1}\end{array} \right]-\left[ \begin{array}{c}MPD_{2}\\ MQD_{2}\end{array} \right]=\left[ \begin{array}{c}dP_{dc}\\ dQ_{dc}\end{array} \right] \tag{32}\]  
\[\Delta PL+\left[ \begin{array}{cc}Z&S\end{array} \right]\left[ \begin{array}{c} \Delta\theta\\ \Delta V\end{array} \right]+MPL_{1}-MPL_{2}=dPL \tag{33}\]  
\[\Delta P_{G} =0\ \ \ \ \lambda \tag{34}\]  
\[\Delta Q_{\min} \leq\Delta Q\leq\Delta Q_{\max}\ \ \ \underline{\psi},\overline{\psi} \tag{35}\]  
\[\Delta PL_{\min} \leq\Delta PL\leq\Delta PL_{\max} \tag{36}\]  
\[\Delta V_{\min} \leq\Delta V\leq\Delta V_{\max} \tag{37}\]  
\[\Delta X_{ac,\min} \leq\Delta X_{ac}\leq\Delta X_{ac,\max} \tag{38}\] \[\Delta X_{dc,\min} \leq\Delta X_{dc}\leq\Delta X_{dc,\max} \tag{39}\]  
The process of solving the sub-model is as follows [17]:* Set up the maximum permissible error \(\varepsilon\) and the maximum iteration number \(IT_{\max}\).