## 5 Conclusions
\[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]5 \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]6 \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]7 \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]8 \[\tilde{\mathbf{W}}_{a,i}(k){=}\mathbf{W}_{a,i}(k)-\mathbf{W}_{a,i}^{*}(k), \tilde{\mathbf{W}}_{c,i}(k){=}\mathbf{W}_{c,i}(k)-\mathbf{W}_{c,i}^{*}(k) \tag{12}\]9 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]0 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]1 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]2 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]3 (A9)  
The second term \(V_{i,2}(k)\) is taken from Equation (A5), and then Equation (A8) is substituted into \(\Delta V_{i,2}(k)\). Thereby, the following equation is obtained. \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]4 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]5 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]6 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]7 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]8 \[V_{i}(k)=\frac{1}{2}\left\|\mathbf{x}_{i}(k)\right\|^{2}+\frac{1}{2}\left\| \tilde{\mathbf{W}}_{a,i}(k)\right\|_{F}^{2}+\frac{1}{2}\left\|\tilde{\mathbf{ W}}_{c,i}(k)\right\|_{F}^{2} \tag{13}\]9 \[\Delta V_{i}(k)=V_{i}(k+1)-V_{i}(k)=\frac{1}{2}\left\|\tilde{ \mathbf{W}}_{a,i}(k+1)\right\|_{F}^{2}-\frac{1}{2}\left\|\tilde{\mathbf{W}}_{ a,i}(k)\right\|_{F}^{2}\]0 (A10)  
According to the model (9) of aggregated distributed energy grids under DoS attacks, \(V_{i,3}(k)\) can be expressed as Equation (A11).