## 3 Design of an Observer for Detecting Tie-Line Power Considering Uncertainties
\begin{bmatrix}\Gamma_{11}&0&\Gamma_{13}&0&0&\Gamma_{16}&\Gamma_{17}\\ *&\Gamma_{22}&0&\Gamma_{24}&\Gamma_{25}&0&0\\ *&*&\Gamma_{33}&0&0&0&0\\ *&*&*&\Gamma_{44}&0&0&0\\ *&*&*&*&\Gamma_{55}&0&0\\ *&*&*&*&*&\Gamma_{66}&0\\ *&*&*&*&*&\Gamma_{77}\end{bmatrix} <0 \tag{21}\]  
_where_  
\[ \begin{split}\Gamma_{11}=& PA_{i}+A_{i}^{T}P+2\delta_{u} \mathbf{\Psi}_{a}^{T}\mathbf{\Psi}_{a}+\delta_{a}^{-1}P\Phi\Phi^{T}P\\ &+\delta_{b}^{-1}P\Phi\Phi^{T}P\\ \Gamma_{22}=& Q(\[PERSON]_{i}A_{i}-Y_{i1}\mathbb{C}_{i})+(\Xi_{i}A_{i}-Y_{i1} \mathbb{C}_{i})^{T}Q+I\\ &+\delta_{a}^{-1}Q(\[PERSON]_{i}\[PERSON])(\[PERSON]_{i}\[PERSON])^{T}Q+\delta_{b}^{-1}Q(\[PERSON]_{i} \[PERSON])(\[PERSON]_{i}\[PERSON])^{T}Q\\ &\Gamma_{33}=-\gamma^{2}I+2\delta_{b}\mathbf{\Psi}_{b}^{T}\mathbf{ \[PERSON]}_{b}\\ &\Gamma_{44}=\Gamma_{55}=\Gamma_{66}=\Gamma_{77}=-\gamma^{2}I\\ &\Gamma_{13}=PB_{i},\ \ \Gamma_{16}=PE_{ij},\ \ \Gamma_{17}=PF_{i}\\ &\Gamma_{24}=Q\dot{Y}_{i1}G_{i},\ \Gamma_{25}=QN_{i}G_{i},\ \delta_{a}>0,\ \delta_{b}>0\end{split} \]  
Proof. : Taking the following Lyapunov function:  
\[V(t)=x_{i}^{T}(t)Px_{i}(t)+e^{T}(t)Qe(t) \tag{22}\]  
Using (1)-(4), (19) and (22), we can derive  
\[ \begin{split}\dot{V}=& x_{i}^{T}(PA_{i}+A_{i}^{T}P) x_{i}+2x_{i}^{T}PB_{i}u_{i}\\ &+2x_{i}^{T}PE_{ij}\Delta f_{j}+2x_{i}^{T}PF_{i}\Delta P_{i}^{d}-2 e^{T}QY_{i1}G_{i}\omega_{i}\\ &-2e^{T}QN_{i}G_{i}\dot{\omega}_{i}+e^{T}(Q(\Xi_{i}A_{i}-Y_{i1} \mathbb{C}_{i})\\ &+(\Xi_{i}A_{i}-Y_{i1}\mathbb{C}_{i})^{T}Q)e+2x_{i}^{T}P\Phi\Phi \sigma\Psi_{a}u_{i}\\ &+2x_{i}^{T}P\Phi\sigma\Psi_{b}u_{i}+2e^{T}Q\Xi_{i}\Phi\sigma \Psi_{a}x_{i}\\ &+2e^{T}Q\Xi_{i}\Phi\sigma\Psi_{b}u_{i}\\ \leq& x_{i}^{T}(PA_{i}+A_{i}^{T}P)x_{i}+2x_{i}^{T}PB_{i}u_{i}\\ &+2x_{i}^{T}PE_{ij}\Delta f_{j}+2x_{i}^{T}PF_{i}\Delta P_{i}^{d}-2 e^{T}QY_{i1}G_{i}\omega_{i}\\ &-2e^{T}QN_{i}G_{i}\dot{\omega}_{i}+e^{T}(Q(\Xi_{i}A_{i}-Y_{i1} \mathbb{C}_{i})\\ &+(\Xi_{i}A_{i}-Y_{i1}\mathbb{C}_{i})^{T}Q)e+\delta_{a}^{-1}x_{i} ^{T}P\[PERSON]\Phi^{T}Px_{i}\\ &+\delta_{a}^{-1}x_{i}^{T}\[PERSON]_{b}^{T}\[PERSON]_{a}x_{i}+\delta_{b}^{-1 }x_{i}^{T}P\Phi\Phi^{T}Px_{i}\\ &+\delta_{b}^{-1}u_{i}^{T}\Psi_{b}^{T}\Psi_{b}u_{i}+\delta_{a}^{-1 }e^{T}Q(\[PERSON])(\[PERSON])^{T}Qe\\ &+\delta_{a}^{-1}x_{i}^{T}\Psi_{b}^{T}\Psi_{a}x_{i}+\delta_{b}^{-1 }e^{T}Q(\[PERSON])(\[PERSON])^{T}Qe\\ &+\delta_{b}^{-1}u_{i}^{T}\Psi_{b}^{T}\Psi_{b}u_{i}\end{split}  \tag{23}\]  
where \(\delta_{a}\) and \(\delta_{b}\) are positive scalars. Letting \(\beta=[x_{i}\ \ e\ \pi^{T}]^{T}\), we can derive:  
\[\dot{V}\leq\beta^{T}\Gamma\beta+\gamma^{2}\alpha^{T}\alpha-e^{T}e \tag{24}\]where  
\[\Gamma= \begin{bmatrix}\Gamma_{11}&0&\Gamma_{13}&0&0&\Gamma_{16}&\Gamma_{17}\\ *&\Gamma_{22}&0&\Gamma_{24}&\Gamma_{25}&0&0\\ *&*&\Gamma_{33}&0&0&0&0\\ *&*&*&\Gamma_{44}&0&0&0\\ *&*&*&*&\Gamma_{55}&0&0\\ *&*&*&*&*&\Gamma_{66}&0\\ *&*&*&*&*&*&\Gamma_{77}\end{bmatrix}  \tag{25}\]  
If the aforementioned two requirements are satisfied, the system satisfies the Lyapunov stability criteria. The proof is completed. Based on the proposed observer, we can derive the observation value \(\Delta p_{i}^{to}\) of the tie-line power \(\Delta p_{i}^{t}\). Considering uncertain communication delay and noise in the process of transmission to the defense center, there is deviation between the measured data and the observed data received by the defender, which can be written as  
\[ \begin{split} diff(t)=&\Delta p_{i}^{tm}(t)-\Delta p_{i}^{td}(t) \\ =&\Delta p_{i}^{tm}(t)-(\Delta p_{i}^{lo}(t-\tau)+N_{d}(t-\tau)) \end{split}  \tag{26}\]  
where \(diff(t)\) represents the deviation between the measured data and the observed data; \(\Delta p_{i}^{td}(t)\) is the observed data received by defender; \(\tau\) is the communication delay; and \(N_{d}(t)\) is the communication noise. This deviation also exists when the system is not compromised. Defenders cannot judge if this deviation is caused by the observer or the FDI attacks. Therefore, the similarity between the observed and real data should be further considered before applying the observed data for attack detection.