# Fourier Singular Values-Based False Data Injection Attack Detection in AC Smart-Grids
## 4 Case Studies  
### Cyber-Physical Model  
As can be seen from Figure 2, an Islanded Micro-grid (IMG) consists of m-th DG units that have been linked in parallel style. In this IMG, some agents are in the mode of controlling voltage and frequency, and consequently, they are in duty of fixing the ISMG voltage. Other DG units are in the state of sharing the loads between units and current control mode [49]. The power circuit of a typical single-phase inverter is illustrated in Figure 3 which connected to the ISMG. The output LC filter is recruited to lessen the output voltage signal's harmonics which are produced via the Pulse Width Modulation (PWM) that is used for inverter switching. The state equations of the assumed inverter displayed in Figure 3 are presented in Equations (14) and (15), respectively:  
\[\mathrm{L}\frac{\mathrm{d}\mathrm{I}_{\mathrm{L}}}{\mathrm{dt}}+\mathrm{V}_{ \mathrm{o}}=\mathrm{V}_{\mathrm{INV}} \tag{14}\]  
\[\mathrm{I}_{\mathrm{L}}=\mathrm{I}_{\mathrm{c}}+\mathrm{I}_{\mathrm{o}}\ \ \ \ \ \,\ \ \ \ \ \ \ \ \mathrm{I}_{\mathrm{c}}=\mathrm{C}\frac{\mathrm{dV}_{\mathrm{o}}}{\mathrm{dt}} \tag{15}\]  
Figure 3: Typical layout of a single-phase distributed generation unit in front of a cyber-attack. Figure 2: Typical CPS sample of AC-ISMG. Here, \(\mathrm{V_{INN}}=\mathrm{uV_{dc}}\) defines the inverter's output voltage, and \(\mathrm{u}\) expresses the controller's input signal. By combining Equations (14) and (15), we have Equation (16):  
\[\frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c}\mathrm{V_{o}}\\ \mathrm{i_{L}}\end{array} \right]=\left[ \begin{array}{cc}0&\frac{1}{C}\\ -\frac{1}{C}&0\end{array} \right]\left[ \begin{array}{c}\mathrm{V_{o}}\\ \mathrm{I_{L}}\end{array} \right]+\left[ \begin{array}{c}0\\ \frac{\mathrm{v_{dc}}}{\mathrm{I_{L}}}\end{array} \right]\mathrm{u}+\left[ \begin{array}{c}-\frac{\mathrm{I_{o}}}{C}\\ 0\end{array} \right] \tag{16}\]  
Here, the inductive current (\(\mathrm{I_{L}}\)) and the capacitor voltage (\(\mathrm{V_{o}}\)) have been chosen as the state variables. \(\mathrm{V_{dc}}\), \(\mathrm{I_{c}}\) and \(\mathrm{I_{o}}\) represent the DC link voltage, the output currents of capacitor and filter, respectively. Based on the sliding mode control method, the FDI attack detection's indexes in voltage/current parameters are defined as follows:  
\[\mathrm{\sigma_{V}}=\widetilde{\widetilde{\vartheta}}-\lambda\widetilde{ \vartheta},\ \widetilde{\vartheta}=\vartheta-\vartheta_{\mathrm{base}} \tag{17}\]  
where \(\mathrm{\sigma_{V}}\) (voltage's switching surface) is the voltage index to detect FDIA that is used as an input of FFT; \(\lambda\) defines a positive value, \(\vartheta\) is the ISMG voltage, and \(\vartheta_{\mathrm{base}}\) is the base voltage of ISMG which owns a firm frequency and amplitude. \[\mathrm{\sigma_{I}}=\varphi-\varphi_{\mathrm{base}} \tag{18}\]  
where \(\mathrm{\sigma_{I}}\) (current's switching surface) is the current index to detect FDIA that is used as an input of Fourier transform; \(\varphi\) is the ISMG current, which is produced by distributed generation units, and \(\varphi_{\mathrm{base}}\) is the base current of ISMG loads that is measurable. As can be seen, the independent AC-ISMG suggested in this study is depicted in Figure 2. DC resources linked by DC/AC converters are inter-connected through lines, consequently forming the ISMG physical layer. Any DC/AC converter acts according to the reference values to maintain the output voltage. An undirected cyber figure of the communication network is illustrated in this study that transmits/receives data through other units where located in their neighborhood. Further, the output converter of each agent is connected to loads and other units. Each source is considered as a different agent; the communication figure is displayed as a digraph through links/edges through a vicinity matrix \(\mathrm{A}=\left[\mathrm{a_{ij}}\right]\ e\ \mathrm{R^{M+N}}\), which suggests the communication weights that are given in Equation (19):  
\[\mathrm{a_{ij}}=\left\{