## 4 Design of Adaptive Global Fast Terminal Sliding Mode Control  
To prove that the system based on AGFTSMC is stable, the following assumption is made.  
**Assumption 1**.: _The unknown load disturbances are bounded, setting \(\|\omega(t)\|\leq Q_{1}\)._  
To guarantee the stable operation of each area in an interconnected power system and mitigate the impact of unknown load disturbances, adaptive laws are designed to estimate the unknown upper bound:  
\[\dot{\hat{Q}}_{1}=\bigg{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\bigg{)}s(t) \tag{17}\]  
\[\dot{\hat{Q}}_{2}=\bigg{(}\frac{\alpha}{M_{i}^{2}}+\frac{1}{M_{i}}\bigg{)}s(t) \tag{18}\]  
The AGFTSMC controller can be designed as follows:  
\[\begin{array}{ll}u(t)&=-\alpha_{1}s(t)-\alpha_{2}\mathrm{sgn}(s(t))-\alpha_ {3}|s(t)\,\mathrm{l}^{v}\mathrm{sgn}(s(t))+Lx(t)\\ &+\bigg{[}\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{1}+\bigg{(} \frac{\alpha}{M_{i}^{2}}+\frac{1}{M_{i}}\bigg{)}\hat{Q}_{2}\bigg{]}\mathrm{sgn }(s(t))\\ &+k\lambda\mathrm{e}^{-kt}\end{array} \tag{19}\]  
where \(\hat{Q}_{1}\) and \(\hat{Q}_{2}\) are the estimate of \(Q_{1}\) and \(Q_{2}\), respectively, and \(\widetilde{Q}_{1}=Q_{1}-\hat{Q}_{1}\) and \(\widetilde{Q}_{2}=Q_{2}-\hat{Q}_{2}\) are estimation errors.  
The Lyapunov function is developed as follows:  
\[V(t)=\frac{1}{2}s^{2}(t)+\frac{1}{2}\widetilde{Q}_{1}^{2}+\frac{1}{2}\widetilde{Q }_{2}^{2} \tag{20}\]  
By deriving Equation (20), the following is obtained:  
\[\dot{V}(t) =s(t)[-\alpha_{1}s(t)-\alpha_{2}\text{sgn}(s(t))-\alpha_{3}|s(t) \,|^{\upsilon}\text{sgn}(s(t))\] \[+k\lambda\text{e}^{-kt}+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i }}\Big{)}\omega(t)-\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\dot{ \omega}(t)\] \[+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{1} \text{sgn}(s(t))-\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{2 }\text{sgn}(s(t))\] \[-k\lambda\text{e}^{-kt}]-\tilde{Q}_{1}\dot{\tilde{Q}}_{1}-\tilde{ Q}_{2}\dot{\tilde{Q}}_{2}\] \[\leq-\alpha_{1}s^{2}(t)-\alpha_{2}s^{2}(t)-\alpha_{3}|s(t)\,|^{1+\
u} \tag{21}\] \[+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\Big{(}\tilde{ Q}_{1}+\hat{Q}_{1}\Big{)}s(t)-\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)} \hat{Q}_{1}|\,s(t)|\] \[+\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\Big{(} \widetilde{Q}_{2}+\hat{Q}_{2}\Big{)}s(t)-\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1 }{M_{i}}\Big{)}\hat{Q}_{2}|\,s(t)|\] \[-\widetilde{Q}_{1}\dot{\tilde{Q}}_{1}-\widetilde{Q}_{2}\dot{ \tilde{Q}}_{2}\] \[\leq-(\alpha_{1}+\alpha_{2})s^{2}(t)-\alpha_{3}\Big{|}s(t)\,|^{1+\
u}\] \[+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\widetilde{Q}_ {1}s(t)+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{1}|\,s(t)|\] \[-\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{1}|\,s (t)|+\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\widetilde{Q}_{2}s(t)\] \[+\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{2}|\, s(t)\,|-\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{M_{i}}\Big{)}\hat{Q}_{2}|\,s(t)|\] \[-\widetilde{Q}_{1}\dot{\tilde{Q}}_{1}-\widetilde{Q}_{2}\dot{ \tilde{Q}}_{2}\] \[\leq-(\alpha_{1}+\alpha_{2})s^{2}(t)-\alpha_{3}\Big{|}s(t)\,|^{1+\
u}\] \[+\Big{[}-\dot{\hat{M}}_{1}+\Big{(}\frac{D_{i}}{M_{i}}+\frac{1}{M _{i}}\Big{)}s(t)\Big{]}\widetilde{Q}_{1}\] \[+\Big{[}-\dot{\hat{M}}_{2}+\Big{(}\frac{\pi}{M_{i}^{2}}+\frac{1}{ M_{i}}\Big{)}s(t)\Big{]}\widetilde{Q}_{2}\]  
By integrating Equations (17) and (18), Equation (22) can be obtained:  
\[\dot{V}(t)\leq-(\alpha_{1}+\alpha_{2})s^{2}(t)-\alpha_{3}\Big{|}s(t)\,|^{1+\
u}<0 \tag{22}\]  
Therefore, the preceding analysis demonstrates that the system can remain stable under the action of the proposed controller.