## 2 Materials and Methods
### Modelling of SMIB Power System Installed with a TCSC  
The SMIB power system with a TCSC, which is employed for stability purposes, is depicted in Figure 1. A synchronous generator \(G\) is shown in Figure 1 with an excitation system, terminal voltage \(V_{t}\), and a TCSC damping controller connected to an infinite bus with voltage \(V_{b}\) through transmission lines of impedance \(x_{t}\), where \(R_{E}\) and \(X_{E}\) are real and imaginary parts, respectively, and \(x_{tcsc}\) stands for TCSC variable reactance. With the use of this figure, TCSC damping controllers are designed [2]. The simplified model of a single-machine infinite-bus system installed with TCSC is as follows:  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]  
\[\dot{\dot{E}}_{q}=[-E_{q}+E_{fd}]/\hat{T}_{do} \tag{5}\]  
\[\dot{E}_{fd}=-\frac{E_{fd}}{T_{A}}+\frac{K_{A}}{T_{A}}[V_{trg}-V_{t}] \tag{6}\]  
where  
\[P_{t}=\frac{\dot{E}_{q}V_{b}}{(\dot{x}_{d}+x_{t}-x_{tesc})}\sin\delta-\frac{V_ {b}^{2}}{2}(x_{q}-\dot{x}_{d})\frac{\sin 2\delta}{(\dot{x}_{d}+x_{t}-x_{tesc})(x_{q}+x_ {L}-x_{tesc})} \tag{7}\]  
\[E_{q}=\frac{\dot{E}_{q}(x_{d}+x_{L}-x_{tesc})}{(\dot{x}_{d}+x_{t}-x_{tesc})}- \frac{(x_{d}-\dot{x}_{d})V_{b}\cos\delta}{(\dot{x}_{d}+x_{t}-x_{tesc})} \tag{8}\]  
\[E_{fd}=E_{fdo}+E_{fd} \tag{9}\]  
\[v_{td}=\frac{x_{q}V_{b}\sin\delta}{(x_{q}+x_{L}-x_{tesc})} \tag{10}\]  
\[v_{tq}=\frac{\dot{x}_{d}V_{b}\cos\delta}{(\dot{x}_{d}+x_{t}-x_{tesc})}+\frac{ \dot{E}_{q}x_{t}}{(\dot{x}_{d}+x_{t}-x_{tesc})} \tag{11}\]  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]0  
By linearising Equations (3)-(12), we have  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]1  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]2  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]3  
where  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]4  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]5  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]6  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]7  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]8  
\[\dot{\delta}=\omega_{o}(\omega-1) \tag{3}\]9  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]0 Single-machine infinite-bus power system installed with a TCSC. \[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]1  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]2  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]3  
The values of Equations (13)-(15) are placed in Equations (3)-(6). Hence, the Equations (25)-(28) are the extended Phillips-Heffron model of the single-machine infinite-bus power system installed with the TCSC stabiliser, which is shown in \[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]4  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]5  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]6 \[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]7 (27) \[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]8  
The system in Figure 2 expressed by a linearised state space mathematical model is represented as follows:  
\[\dot{\omega}=[P_{m}-P_{t}-\mathrm{D}(\omega-1)]/\mathrm{M} \tag{4}\]9  
\[\dot{\dot{E}}_{q}=[-E_{q}+E_{fd}]/\hat{T}_{do} \tag{5}\]0  
\[\dot{\dot{E}}_{q}=[-E_{q}+E_{fd}]/\hat{T}_{do} \tag{5}\]1  
\[A_{tcsc}= \begin{bmatrix}0&\omega_{o}&0&0\\ -\frac{K_{1}}{M}&-\frac{D}{M}&-\frac{K_{2}}{M}&0\\ -\frac{K_{4}}{T_{A}}&0&-\frac{K_{3}}{T_{A}}&\frac{1}{T_{d_{0}}}\\ -\frac{K_{4}K_{5}}{T_{A}}&0&-\frac{K_{4}K_{5}}{T_{A}}&-\frac{1}{T_{A}}\end{bmatrix}  \tag{32}\]  
Figure 2: Extended Phillips–Heffron model of the single-machine infinite-bus power system installed with a TCSC stabiliser. \[B_{tcc}= \begin{bmatrix}0\\ -\frac{K_{p}}{T_{d}}\\ -\frac{K_{d}}{T_{d}}\\ -\frac{K_{d}K_{p}}{T_{A}}\end{bmatrix}  \tag{33}\]  
\[U= \begin{bmatrix}\Delta x_{tcc}\end{bmatrix}