## 2 Problem Description
### A Multi-Attack Mode
#### 2.2.1 FDIA Construction Based on the Partial Grid Information Known  
This paper takes the perspective of the attacker into account during the formulation of attack vectors. Given the attacker's limited comprehension of the electric power CPS network structure, we postulate that only partial network information is accessible to the attacker. Based on the varying degrees of information available, diverse attack vectors are formulated [23]. To enhance the comprehension of the security issues inherent to the target power grid, the grid is divided into two critical regions: the attack region \(\alpha\) and the unattacked region \(\beta\), as depicted in Figure 1. In Figure 1, the loads and transformers in the two regions are represented by L and T, respectively, and a partial depiction of the network structure is illustrated in Figure 1.  
The attackers possess a restricted understanding of the network information affiliated with the attack region, and their aim is to perpetrate the attack by manipulation of measurement data within that area. To effectively evade detection by the BDD detection module, the attackers must guarantee that their manipulating actions do not result in an elevation of the measurement residuals, thereby evading the system's suspicion. The relationship between the target grid's state variables and the measurement data can be formulated as follows:  
\[z= \begin{bmatrix}z_{\alpha}\\ z_{\beta}\end{bmatrix} = \begin{bmatrix}H_{\alpha\alpha}&H_{\alpha\beta}\\ 0&H_{\beta\beta}\end{bmatrix}\begin{bmatrix}\hat{\mathbf{x}}_{\alpha}\\ \hat{\mathbf{x}}_{\beta}\end{bmatrix} + \begin{bmatrix}\mathbf{e}_{\alpha}\\ \mathbf{e}_{\beta}\end{bmatrix} , \tag{3}\]  
where \(z\), \(z_{\alpha}\), and \(z_{\beta}\) represent the number of system measurements for the total system, the attacked area, and the unattacked area; \(\hat{\mathbf{x}}_{\alpha}\) and \(\hat{\mathbf{x}}_{\beta}\) are the estimated values of the state variables for the corresponding areas, respectively; \(\mathbf{e}_{\alpha}\) and \(\mathbf{e}_{\beta}\) are the corresponding measurement errors; and \(H_{\alpha\alpha}\), \(H_{\alpha\beta}\), and \(H_{\beta\beta}\) are the Jacobi matrices associated with the state variables. Expanding the aforementioned equation in accordance with the provided expression results in:  
\[z_{\alpha}=H_{\alpha\alpha}\hat{x}_{\alpha}+H_{\alpha\beta}\hat{x}_{\beta}+ \mathbf{e}_{\alpha}, \tag{4}\]  
\[z_{\beta}=H_{\beta}\hat{x}_{\beta}+\mathbf{e}_{\beta}, \tag{5}\]  
The attacker constructs the following false data based on the knowledge of the network in area \(\alpha\):  
\[z^{\prime}_{\alpha}=H_{\alpha\alpha}x_{\alpha}+H_{\alpha\beta}\hat{x}_{\beta}, \tag{6}\]  
At this point, the measurement vector of the system becomes \(z^{\prime}=[z^{\prime}_{\alpha}z_{\beta}]^{T}\), and the state vector becomes \(\hat{x}^{\prime}=[x_{\alpha}\hat{x}_{\beta}]^{T}\). The 2-norm residual \(r^{\prime}\) test of the measurement data satisfies the following derived relationship:  
\[r^{\prime}=\left\|z^{\prime}-H(\hat{x}^{\prime})\hat{x}^{\prime}\right\|_{2}= \left\| \begin{array}{c}z^{\prime}_{\alpha}-(H_{\alpha\alpha}x_{\alpha}+H_{ \alpha\beta}\hat{x}_{\beta})\\ z_{\beta}-H_{\beta\beta}\hat{x}_{\beta}\end{array} \right\|_{2}=\left\|\begin {array}[]{c}0\\ \mathbf{e}_{\beta}\end{array}\right\|_{2}=\left\| \begin{array}{c}\mathbf{e}_{\beta }\end{array} \right\|_{2}<r=\left\| \begin{array}{c}\mathbf{e}_{\alpha}\\ \mathbf{e}_{\beta}\end{array} \right\|_{2}, \tag{7}\]  
Based on the information available to the attackers concerning the network within the attack area, the formulation of false data becomes a straightforward process, as depicted in  
Figure 1: Diagram of the division of the attacked and unattacked areas. Equation (6). Once these false data points infiltrate the system, as described by Equation (7), the resultant measurement residuals of the system exhibit reduced magnitudes compared to the residuals corresponding to scenarios without the presence of false data.