# Power Stability Analysis and Evaluation Criteria of Dual-Infeed HVDC with LCC-HVDC and VSC-HVDC
## 2 Mathematical Model of Hybrid Dual-Infeed HVDC Systems
### Model Solution  
For the above-mentioned hybrid dual-infeed HVDC system, the solution of the mathematical model can be divided into two cases: rated operating condition and non-rated operating condition.  
Under the rated operating condition, all variables of the AC/DC system are the nominal values, and the reactive power consumed by LCC-HVDC is completely provided by the filters and compensators connected at the converter bus. It satisfies:  
\[\left\{\begin{array}{l}\gamma_{1}=\gamma_{\text{N}1}\\ V_{1}=V_{\text{N}1}\\ V_{\text{d}1}=V_{\text{d}\text{N}1}\\ I_{\text{d}1}=I_{\text{d}\text{N}1}\\ Q_{\text{c}1}=Q_{\text{d}1}\end{array}\right. \tag{16}\]  
where \(\gamma_{\text{N}1}\), \(V_{\text{N}1}\), \(V_{\text{d}\text{N}1}\), and \(I_{\text{d}\text{N}1}\) are the extinction angle, AC bus voltage, DC voltage, and DC current of LCC-HVDC under the rated state, respectively.  
For VSC-HVDC, its rated state has nothing to do with the control mode, and it satisfies:  
\[\left\{\begin{array}{l}V_{2}=V_{\text{N}2}\\ P_{\text{d}2}=P_{\text{d}\text{N}2}\\ Q_{\text{d}2}=Q_{\text{d}\text{N}2}\end{array}\right. \tag{17}\]where \(V_{\text{N2}}\), \(P_{\text{dN2}}\), and \(Q_{\text{dN2}}\) are the AC bus voltage, active power, and reactive power of VSC-HVDC under the rated state, respectively. Normally, the VSC-HVDC does not deliver reactive power to the AC system under the rated state--that is, \(Q_{\text{dN2}}=0\).  
In addition, when the whole system is in the rated state, the active power exchange \(P_{12}\) between the AC system 1 and 2 is set to a certain planned value \(P_{\text{tie0}}\) (usually 0). It satisfies:  
\[P_{12}=P_{\text{tie0}}=0 \tag{18}\]  
If we assume that the parameters \(Z_{i}\angle\varphi_{i}\), \(Z_{ij}\angle\varphi_{ij}\) and \(X_{\text{t1}}\) are known and let \(\delta_{1}=0\), then, according to Equations (1)-(4) and (8)-(18), the value of parameters \(E_{i}\angle\delta_{i}\), \(\tau_{1}\) and \(B_{c1}\) under the rated state can be obtained by applying the Newton-Raphson method.  
Under the non-rated operating condition, keeping the parameters \(E_{i}\angle\delta_{i}\), \(\tau_{1}\) and \(B_{c1}\) constant, there are 21 unknown operating state variables in the whole system: \(\mu_{1}\), \(\gamma_{1}\), \(I_{\text{d1}}\), \(V_{\text{d1}}\), \(\theta_{i}\), \(P_{\text{d1}}\), \(Q_{\text{d1}}\), \(P_{\text{i}}\), \(Q_{\text{d1}}\), \(P_{\text{i}}\), \(Q_{\text{d1}}\), \(P_{\text{i}}\), \(Q_{\text{d1}}\), \(P_{\text{i}}\), \(Q_{\text{d1}}\) (\(i\), \(j\) = 1, 2 and \(i\
eq j\)). According to Equations (1)-(15), it can be seen that the whole system contains 19 constraint equations. Therefore, only two operating state variables are independent. Select \(\gamma_{1}\) and \(I_{\text{d1}}\) as the independent variables of the system. Once the independent variables are determined, the remaining operating state variables will also be determined. It should be noted that when the VSC-HVDC operates in constant AC voltage control mode, the maximum allowable power limit \(S_{\text{2 max}}\) of the VSC converter should be considered. Under the premise that the connected AC bus voltage is \(V_{\text{2 ref}}\), whether the actual reactive power \(Q_{\text{d2 act}}\) provided by VSC-HVDC exceeds its reactive power limit \(Q_{\text{2 limit}}\) should be calculated. When \(Q_{\text{d2 act}}\) exceeds the limit, its control mode is changed from constant AC voltage control to constant reactive power control, and its reactive power reference value is:  
\[Q_{\text{d2 ref}}=\left\{\begin{array}{cc}\sqrt{S_{\text{2 max}}^{2}-P_{\text {d2 ref}}^{2}}&Q_{\text{d2 act}}\geq Q_{\text{2 limit}}\\ -\sqrt{S_{\text{2 max}}^{2}-P_{\text{d2 ref}}^{2}}&Q_{\text{d2 act}}\leq-Q_{\text {2 limit}}\end{array}\right. \tag{19}\]