# A Predictive Model Using Long Short-Time Memory (LSTM) Technique for Power System Voltage Stability
## 2 The Proposed System Methodology  
In this system, a constant voltage source represents an infinite bus, and the line and load impedances are denoted by the symbols \(Z\) and \(Z_{x}\), respectively. When the maximum power transmission is reached, more power can be delivered to the load by lowering the impedance \(Z_{x}\). The voltage drop will then grow more noticeable as \(Z_{x}\) is decreased, which, in turn, increases the power demand. This increased power demand will continue to rise, eventually resulting in less power being supplied to the load. The PV curve represents the process illustrated in Figure 2. At this moment, the load is receiving active power, denoted as \(P_{0}\). The \(V_{x}^{\prime}\) and \(V_{y}^{\prime}\) are voltages at the receiving and sending ends; \(P_{x}^{\prime}\) and \(P_{y}^{\prime}\) are active power values at the receiving and sending ends; \(\wp_{x}\) and \(\wp_{y}\) are apparent powers at the receiving and sending ends; \(Q_{x}^{\prime}\) and \(Q_{y}^{\prime}\) are reactive powers at the receiving and sending ends of the system, respectively; \(Y\) is the admittance; \(R\) is resistance; \(X\) is reactance; \(Z\) is impedance; \(\theta\) is the impedance angle of the lines and I is the current passing through the line. Figure 2: The P-V curve [16]. \[\Re_{x}=P_{x}^{ir}+jQ_{x}^{ir}=V_{x}^{\prime}I \tag{1}\]  
\[I=\frac{V_{x}^{\prime}\angle\wp_{x}-V_{y}^{\prime}\angle\wp_{y}}{Z\angle\theta} \tag{2}\]  
\[\Re_{x}=\frac{V_{x}^{\prime}V_{y}^{\prime}\angle\left(-\wp_{x}+\wp_{y}\right)-V _{y}^{\prime 2}\angle 0}{Z\angle\left(-\theta\right)} \tag{3}\]  
\[\Re_{x}=\frac{V_{x}^{\prime}V_{y}^{\prime}}{Z}\angle\left(\theta-\wp_{x}+\wp_ {y}\right)-\frac{-V_{y}^{\prime 2}}{Z}\angle\theta \tag{4}\]  
If \(\wp=\wp_{x}-\wp_{y}\), it can be expressed at the receiving end for real and active power:  
\[P_{x}^{ir}=\frac{V_{x}^{\prime}V_{y}^{\prime}}{Z}\cos(\theta-\wp)-\frac{V_{x }^{\prime 2}}{Z}\cos\theta \tag{5}\]  
\[Q_{x}^{ir}=\frac{V_{x}^{\prime}V_{y}^{\prime}}{Z}\sin(\theta-\wp)-\frac{V_{x }^{\prime 2}}{Z}\sin\theta \tag{6}\]  
\[V_{s}^{\prime}=\frac{P_{\max}^{\prime}-P_{0}}{P_{\max}^{\prime}} \tag{7}\]  
A classical load system in Equation (2) is included in the scheme model of Equation (3) with the continuation power flow (CPF) approach [16]. \[\left\{ \begin{array}{l}P_{x}^{ir}={P_{x0}^{i}}^{\prime}+\lambda_{0}P_{xd}^{i }\\ \mathcal{Q}_{x}^{i}=Q_{x0}^{i}+\lambda_{0}Q_{xd}^{i}\end{array} \right. \tag{8}\]  
where \(P_{x}^{ir}\) and \(Q_{x}^{ir}\) represent powers provided to the load bus \(i^{th}\); the limitation \(\lambda_{0}\) is an integer that defines the system's loading level; \(P_{xd}^{i^{\prime}}\) and \(Q_{xd}^{i^{\prime}}\) represent the velocities at which the load bus\(i^{th}\) active and reactive powers change, respectively. \[\left\{ \begin{array}{l}P_{i}^{\prime}={V_{i}^{\prime 2}}L_{ii}+V_{i}^{ \prime}\sum\limits_{i\
eq c}V_{c}^{\prime}(L_{ic}\cos\theta_{ic}+M_{ic}\sin \theta_{ic})\\ Q_{i}^{\prime}=-{V_{i}^{\prime 2}}M_{ii}+V_{i}^{\prime}\sum\limits_{i\
eq c}V_{c}^{ \prime}(L_{ic}\cos\theta_{ic}-M_{ic}\sin\theta_{ic})\end{array} \right. \tag{9}\]  
whereas the voltage consequence at bus \(i^{th}\) is represented as \(V_{i}^{\prime}\) and the angle voltage variation involving nodes \(i^{th}\) with \(V_{i}^{\prime}\) is denoted as L0; the actual and fictional sections of the \(ic^{th}\) part of the admittance conditions of the structure are represented by \(\theta_{ic}\); \(L_{ic}\); and \(M_{ic}\). Variations in active and reactive power as the constraint \(\lambda_{0}\) changes establish the value of revolution in active and reactive power. The CPF approach raises the \(\lambda_{0}\) value gradually until the maximum load limit is reached, increasing the system load. The \(\lambda_{0}\), \(P_{x}^{\prime}\), and \(Q_{x}^{\prime}\) are equivalent at the bifurcation node to their maximum values, \(\lambda_{\max}^{\prime}\), \(P_{\max}^{\prime}\), and \(Q_{\max}^{\prime}\), respectively.