## 3 Voltage Stability of a Multi-Infeed LCC-HVDC System
### Voltage Stability Factor (VSF)  
The voltage-stability-factor-based approach was initially proposed in [18]. It computes the ratio between changes in voltage to the varying reactive power injection. This ratio is named voltage stability factor (VSF). A system is said to be stable if the VSF is small and positive. The increasing value of the VSF indicates that the system is losing stability. The system moves to an unstable region when the VSF becomes infinite. On the other hand, if the VSF is negative, then the system is also unstable. The correlation between MAP- and VSF-based approaches is first reported in [19]. It is investigated that at the maximum of the MPC, the VSF becomes infinite for inverters operating at a constant extinction angle (CEA). In ref. [20], it is reported that the power flow Jacobian has great influence on the voltage stability of a power network. If the Jacobian of the power network is becoming singular, the power system is losing its stability and operates with less stability margin. This opens new directions in the field of power system stability. In ref. [21], the voltage stability analysis for multi-infeed HVDC is discussed using the eigenvalue-based approach. The eigenvalues of a reduced Jacobian matrix are inverse of the VSF. In ref. [22], a method is presented to estimate the voltage stability by decomposing the Jacobian matrix into eigenvalues at steady-state conditions. The bus angle is decoupled by assuming \(\Delta P=0\), which means no active power injection [23]. The Jacobian matrix turns to the reduced Jacobian matrix by considering \(\Delta P=0\), which only shows the influence of reactive power on the stability of the AC/DC power system. The reduced Jacobian matrix can easily be decomposed into eigenvalues by using a modal analysis [22]. The reactive power injection (\(\Delta Q\)) and change in bus voltage (\(\Delta V\)) are related by eigenvalues. For voltage stability, all eigenvalues must be positive. The relationship between Q and V is in-phase if the eigenvalues are positive and antiphase if the eigenvalues are negative. The in-phase relation corresponds to a stable operation, as the bus voltage increases with an increase in reactive power. The antiphase relation indicates an unstable operation; the bus voltage decreases as the reactive power is injected. Similarly, when eigenvalues become zero, the reduced Jacobian becomes singular, and a tiny change in reactive power leads to an infinite change in bus voltage. This also corresponds to an unstable operation. The HVDC system is modelled as described in [24]. The converters in a multi-infeed scenario are connected via coupling impedance. Each converter is occupied with reactive power support by installed filters and static capacitors. All converters in multi-infeed are normalized on a common base. The Thevenin impedance of each AC system to which the HVDC is connected and the coupling impedance greatly affect the system stability. An equivalent single infeed model of a multi-infeed HVDC system is proposed in [25]. The voltage stability using equivalent nodal analysis is discussed in [26]. The method considers N node, L load, and M generator buses to compute an equivalent model. The impact of a control system on voltage stability is investigated in [27]. The linearized equations are used to model the modal analysis of a multi-infeed HVDC system. It is also reported that both active and reactive powers influence the voltage stability, but in order to simplify the equations, only the reactive power injection is considered. The modal analysis for voltage stability considering the point of collapse is reported in [28]. It is discussed that the modal analysis can be applied to a power system to evaluate the vulnerability of voltage stability. Modal analysis results in an appropriate location for installing reactive power support. The modal analysis can judge the system behavior for stability only in case of incremental change. A computer program for estimating the small signal stability of a power system is described in [29].