# Analyzing Six Indices for Online Short-Term Voltage Stability Monitoring in Power Systems
## 2 Mathematical Preliminaries: SVS Indices
### Lyapunov Exponents  
One of the introduced indices for SVS monitoring is a Lyapunov exponent, which is adapted from ergodic theory of dynamic systems. LEs in nonlinear systems can be considered as eigenvalues in linear systems. The purpose of using LEs in the SVS monitoring is voltage stability/instability prediction by investigation of converging/diverging voltage trajectories. If maximum LEs of the system are positive/negative, it implies convergence/divergence of nearby system trajectories [7].  
The following algorithm is used for calculating system LE using measurement data from PMU.  
1. Consider \(V_{t}=[v_{1}(t),v_{2}(t),\dots,v_{n}(t)]\) as voltage time series vector measured by PMU or obtained from simulations, at \(n\) buses of system in times \(t=0,\Delta t,2\Delta t,\dots,M\Delta t\). \(\Delta t\) is the sampling period of PMU measurement.  
Choose N initial value of the time series that are satisfied in the following relation:  
\[\varepsilon_{1}<\left\|V_{m\Delta t}-V_{(m-1)\Delta t}\right\|<\varepsilon_{2}\ \,m=1,2, \ldots,N \tag{1}\]  
\(\varepsilon_{1},\varepsilon_{2}\) are two small fixed numbers and \(0<\varepsilon_{1}<\varepsilon_{2}\).  
For \(k=1,2,\ldots,M\) the maximum Lyapunov exponent at time \(k\Delta t\) is defined as follows:  
\[\begin{array}{l}\Lambda(k\Delta t)=\frac{1}{Nk\Delta t}\times\sum\limits_{m =1}^{N}log\frac{\left\|V(k+m)\Delta t-V(k+m-1)\Delta t\right\|}{\left\|V(m) \Delta t-V(m-1)\Delta t\right\|}\\ k>N\end{array} \tag{2}\]  
In Equation (2), by taking \(N\) initial condition from measurement data and studying its evolution over time, the system's LE would be calculated. LE in time \(t\) is calculated by normalizing distance between \(N\) initial conditions at time \(t\) to this distance at time zero and averaged over the number of initial conditions. With n PMU in the system that measures n buses voltage and by taking measured quantities after fault clearing time, there are n voltage time series. In [7], it has been proven that system LE is more than maximum individual buses LE. Therefore, it is sufficient to use system LE for instability detection.  
For the purpose of online implementation, the Lyapunov exponent needs to be computed over a finite time window [7]. The appropriate size of this window will be crucial for reliable, accurate, and timely stability monitoring depending on considered stability study. In particular, a smaller size window will lead to inaccurate stability prediction, while a larger size window will lead to accurate, but untimely prediction. There exists an optimal window that provides reliable and timely stability predictions. The optimal window depends on fault and system characteristics [7]. Thus, the basic LE algorithm can be improved to avoid oscillation issues by selecting the optimal or suitable parameters and computation window through systematic or empirical methods such as those have been in [13; 20]. For example, the LE is estimated by a recursive least squares based method in [13; 20], and two critical parameters, the computation window and the LE estimation initial time step, are carefully chosen or, in [13; 20], a phase rectification method has been proposed to eliminate the negative influence of oscillation of LE. These methods improve the LE performance in the cost of increasing method complexity and serving a considerable time for setting parameters for a specific system and a specific stability study.