# 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  
### Description of the Mathematical Model  
As illustrated in Figure 1, a simplified hybrid dual-infeed HVDC transmission system consists of one LCC-HVDC link and one VSC-HVDC link. The connected AC system is represented by the Thevenin equivalent circuit.  
For the LCC-HVDC link, the quasi-steady-state DC model [12] is adopted, which can be expressed by the following algebraic equations:  
\[P_{\rm d1}=V_{\rm d1}I_{\rm d1} \tag{1}\]  
\[Q_{\rm d1}=P_{\rm d1}\frac{2\mu_{1}+\sin 2\gamma_{1}-\sin 2(\gamma_{1}+\mu_{1})}{ \cos 2\gamma_{1}-\cos 2(\gamma_{1}+\mu_{1})} \tag{2}\]  
\[V_{\rm d1}=K_{\rm b}\left(\frac{3\sqrt{2}V_{1}}{\pi\tau_{1}}\cos\gamma_{1}- \frac{3X_{\rm t1}}{\pi}I_{\rm d1}\right) \tag{3}\]  
\[\mu_{1}=\cos^{-1}\left(\cos\gamma_{1}-\frac{\sqrt{2}X_{\rm t1}\tau_{1}I_{\rm d 1}}{V_{1}}\right)-\gamma_{1} \tag{4}\]  
For the VSC-HVDC link, it can realize flexible control of active and reactive power [19]. When the VSC converter operates in constant active power control mode, its characteristic equation can be described as:  
\[P_{\rm d2}=P_{\rm d2 ref} \tag{5}\]  
Two kinds of control modes are considered for reactive power control. When the VSC converter operates in constant reactive power control mode, there is:  
\[Q_{\rm d2}=Q_{\rm d2 ref} \tag{6}\]  
When the VSC converter operates in constant AC voltage control mode, there is:  
\[V_{2}=V_{\rm 2 ref} \tag{7}\]  
For the AC network, the power flow equations can be described as follows:  
\[P_{i}=\frac{V_{i}^{2}\cos\varphi_{i}-E_{i}V_{i}\cos(\theta_{i}+\varphi_{i}- \delta_{i})}{Z_{i}} \tag{8}\]  
Figure 1: Hybrid dual-infeed HVDC system model.  
\[Q_{i}=\frac{V_{i}^{2}\sin\varphi_{i}-E_{i}V_{i}\sin(\theta_{i}+\varphi_{i}-\delta_ {i})}{Z_{i}} \tag{9}\]  
\[P_{ij}=\frac{V_{i}^{2}\cos\varphi_{ij}-V_{i}V_{j}\cos(\theta_{i}-\theta_{j}+ \varphi_{ij})}{Z_{ij}} \tag{10}\]  
\[Q_{ij}=\frac{V_{i}^{2}\sin\varphi_{ij}-V_{i}V_{j}\sin(\theta_{i}-\theta_{j}+ \varphi_{ij})}{Z_{ij}} \tag{11}\]  
At the converter bus, the following balance equations can be obtained:  
\[P_{i}+P_{ij}-P_{\text{d}i}=0 \tag{12}\]  
\[Q_{1}+Q_{12}+Q_{\text{d}1}-Q_{\text{c}1}=0 \tag{13}\]  
\[Q_{2}+Q_{21}-Q_{\text{d}2}=0 \tag{14}\]  
where:  
\[Q_{\text{c}1}=V_{1}^{2}B_{\text{c}1} \tag{15}\]  
In Equations (1)-(15), \(P_{\text{d}1}\), \(-Q_{\text{d}1}\), \(P_{\text{d}2}\), \(Q_{\text{d}2}\) are the active power and reactive power injected into the AC system by LCC-HVDC and VSC-HVDC, respectively; \(V_{\text{d}1}\), \(I_{\text{d}1}\) are the DC voltage and DC current of LCC-HVDC; \(\mu_{1}\), \(\gamma_{1}\) are the commutation overlap angle and extinction angle of LCC-HVDC; \(\tau_{1}\), \(X_{\text{t}1}\) are the transformer turns ratio and leakage reactance of LCC-HVDC; \(K_{\text{b}}\) is the number of six-pulse bridges of LCC-HVDC; \(P_{\text{d}2\text{ref}}\), \(Q_{\text{d}2\text{ref}}\) and \(V_{\text{2 ref}}\) are the reference values of the VSC-HVDC constant active power control, constant reactive power control and constant AC voltage control, respectively; \(V_{i}\angle\theta_{i}\) is the converter bus voltage; \(E_{i}\angle\delta_{i}\) is the equivalent source voltage of the connected AC system; \(P_{i}\) and \(Q_{i}\) are the active power and reactive power of the AC system \(i\); \(P_{ij}\) and \(Q_{ij}\) are the active power and reactive power flowing from the AC system \(i\) to \(j\); \(Z_{i}\angle\varphi_{i}\) and \(Z_{ij}\angle\varphi_{ij}\) are the equivalent impedance of the AC system \(i\) and the coupling impedance between the AC system \(i\) and \(j\), respectively; \(B_{\text{c}1}\), \(Q_{\text{c}1}\) are the equivalent admittance and reactive power of the filters and compensators for LCC-HVDC; where \(i,j=1\), \(2\) and \(i\
eq j\).