# Analyzing Six Indices for Online Short-Term Voltage Stability Monitoring in Power Systems
## 2 Mathematical Preliminaries: SVS Indices
### Voltage Stability Risk Index (\(Vspi)\)  
Voltage stability index (VSRI) has been used in [12] as a dynamic voltage stability criterion to assess voltage stability for determining a suitable location for the load curtailment. The index has been considered in [22] as a long-term voltage stability index to identify the vulnerability of the system to the voltage instability at each bus following a contingency. It utilizes the time series data of the bus voltages and does not require any input for other buses.  
This index can be computed from time series of the voltages RMS values measured by PMU at systems' buses. For calculating \(VSRI\), let PMU measurements for the voltage of each bus are specified as \(y(t)=[y_{1},y_{2},\ldots,y_{M}]^{T}\). The following steps are used for \(VSRI\) calculation [12; 22]:  
1. The moving average value of the bus voltage at \(j\)th instant with taking \(N\) initial samples is described by (19). \[\left\{\begin{array}{cc}v_{j}=\sum\limits_{k=1}^{j}y_{k}/_{j}&,\quad j=1,2, \ldots,N\\ v_{j}=\sum\limits_{k=j-N+1}^{j}y_{k}/_{N}&,\quad j=N+1,\ldots,M\end{array}\right.\] (19) According to (19), the average \(v_{j}\) is computed by using \(j\) samples existing in the data window before filling the window by \(N\) initial samples (\(j<=N\)). When the window is filled by \(N\) samples, according to (19), the average \(v_{j}\) is computed on a moving window including \(N\) samples from \(j-N+1\) till \(j\) for \(j>N\). The voltage stability is monitored by \(M\) measured samples.
2. The percentage diversity, \(d_{j}\), at \(j\)th instant between \(y_{j}\) and \(v_{j}\) is computed as \[d_{j}=(y_{j}-v_{j})\times\
icefrac{{100}}{{v_{j}}}\quad,\quad j=1,2,\ldots,M\] (20) In (20), \(d_{j}\) indicates the normalized deviation of the current sample, \(y_{j}\) with respect to the moving average based on \(v_{j}\). Thus, it can reflect the current situation of buses voltage with respect to the average of \(N\) past samples. Small (big) \(d_{j}\) means a small (big) deviation respect to the average of past samples.
3. By calculating integral of \(d_{j}\) curve and averaging over the \(N\), VSRI is calculated as follows: \[VSRI_{j}=\sum\limits_{k=1}^{j}\frac{(d_{k}+d_{k-1})}{2j} \quad j\leq N\] (21) \[VSRI_{j}=\sum\limits_{k=j-N+1}^{j}\frac{(d_{k}+d_{k-1})}{2N} \quad N+1\leq j\leq M\]  
The area under \(d_{j}\) curve indicates the summation of the deviation of the bus voltage with respect to the dynamic average value of it during an interval from the fault clearing time to the final study time. Thus, VSRI models the dynamic behavior of the buses voltage and it can be used to evaluate short-term voltage stability.  
After these steps, \(VSRI\) for individual buses can be determined. If the buses voltage are stable after disturbance, VSRI of all buses converges to zero. The convergence time depends on load and system characteristics. The buses with the smallest negative VSRI are identified as the most vulnerability to the voltage instability.  
For computing the system's \(VSRI\), in the second step, \(D_{j}\) is used instead of \(d_{j}\) in Equation (21). \(D_{j}\) comes from Equation (22) in which \(d_{j,i}\) refers to percentage diversity at \(j\)th instant (\(d_{j}\)) in bus \(i\).  
\[D_{j}=\frac{1}{n}\sum_{i=1}^{n}d_{j,i} \tag{22}\]  
However, to use \(VSRI\) in short-term voltage stability monitoring, an algorithm is proposed in Section 3.2.6.