# Power Stability Analysis and Evaluation Criteria of Dual-Infeed HVDC with LCC-HVDC and VSC-HVDC
## 3 Power Stability Analysis of Hybrid Dual-Infeed HVDC Systems
### Power Stability Margin  
The power stability of LCC-HVDC is the main factor restricting the normal and stable operation of HVDC systems [12; 21]. For any operating point on the \(P_{\rm d1}\)-\(I_{\rm d1}\) curve, its slope is defined as the power stability index (PSI) of the HVDC system at this point [22]:  
\[\rm PSI=\frac{d\textit{P}_{\rm d1}}{d\textit{I}_{\rm d1}} \tag{23}\]  
According to Figure 2, when the HVDC system operates on the left side of the maximum available power point M, \(P_{\rm d1}\) increases with the increase of \(I_{\rm d1}\)--that is, PSI > 0; When the system operates on the right side of point M, \(P_{\rm d1}\) decreases with the increase of \(I_{\rm d1}\)--that is, PSI < 0. Here, the small disturbance method can be used to test the power stability of the left and right sides of point M. For the left area of point M, it is assumed that there is a positive small disturbance \(\Delta I_{\rm d1}\). According to the maximum power curve, the increase of \(I_{\rm d1}\) will lead to the increase of \(P_{\rm d1}\). Under the action of the constant power controller of the HVDC system, the command value of \(I_{\rm d1}\) will be reduced, so that the current will return to the initial point. Therefore, when PSI > 0, the HVDC system can maintain power stability. On the other hand, for the right area of point M, the existence of \(\Delta I_{\rm d1}\) will lead to the decrease of \(P_{\rm d1}\). Under the action of the constant power controller, the command value of \(I_{\rm d1}\) will be increased, so that the DC power \(P_{\rm d1}\) will further drop. The system cannot return to the initial equilibrium point. Therefore, when PSI < 0, the HVDC system is difficult to operate stably. Power stability is essentially a stability problem under small disturbances. Next, the main influencing factors of the PSI value are further analyzed.  
When the inverter side of LCC-HVDC is in constant extinction angle control mode (\(\gamma_{1}\) = \(\gamma_{1\rm N}\)), according to Equations (21) and (22), \(P_{\rm d1}\) is a function of \(V_{1}\) and \(I_{\rm d1}\). Then, we can obtain:  
\[\rm PSI=\frac{d\textit{P}_{\rm d1}}{d\textit{I}_{\rm d1}}=K_{b}\left(\frac{3 \sqrt{2}V_{1}}{\pi\tau_{1}}\cos\gamma_{1\rm N}+\frac{3\sqrt{2}I_{\rm d1}}{\pi \tau_{1}}\cos\gamma_{1\rm N}\frac{\textit{d}V_{1}}{d\textit{I}_{\rm d1}}- \frac{6X_{\rm t1}}{\pi}I_{\rm d1}\right) \tag{24}\]  
Equation (24) is the detailed analytical expression of PSI. For any operating point, as long as \(V_{1}\), \(I_{\rm d1}\), and \(\textit{d}V_{1}/\textit{d}I_{\rm d1}\) are known, the PSI value can be calculated. At the rated operating point, \(V_{1}=1\), \(I_{\rm d1}=1\), the value of PSI is only related to \(\textit{d}V_{1}/\textit{d}I_{\rm d1}\). The power stability of the HVDC system is related to the voltage support strength of the receiving end system. The greater the voltage support strength of the receiving end system, the smaller the decrease of \(V_{1}\) with the same increment of \(I_{\rm d1}\), resulting in the greater value  
Figure 2: Curves of \(V_{1}\), \(V_{\rm d1}\) and \(P_{\rm d1}\) versus \(I_{\rm d1}\) of LCC-HVDC.  
of \(\mathrm{d}V_{1}/\mathrm{d}I_{\mathrm{d}1}\). Therefore, the value of PSI is greater, and the power stability of the HVDC system is better.