# Enhancing Cybersecurity in Smart Grids: False Data Injection and Its Mitigation
## 3 Classifications of Cyber-Attacks in SGs
### False Data Injection Attacks (FDIA)  
False Data Injection Attacks aim to inject malicious measurements and modify the results. FDIA could violate data integrity in various regions as transmission, communication, generation, control, etc. It can be seen in a different part of the SG that contains data. In this section, the FDIA will be evaluated in the grid without categorizing. It will be examined with the same approach for all regions. Figure 7 depicts the diagram of the following formulated system [100]. In terms of common features in systems, basic principles of FDIA can be formulated as follows [102]:  
Let \(z_{d}\) represent the measurements vector which contains false and malicious data, and \(z_{d}\) can be formulated as; \(z_{d}=z+d\), where \(z\) is the original vector measurements \(z=(z_{1},z_{2},\ldots,z_{m})^{T}\), and \(d\) is false or malicious data \(d=(d_{1},d_{2},\ldots,d_{m})^{T}\) which added to  
Figure 7: FDIA general block diagram. the original measurements. It is referred to d as a false data attack vector. The element of \(d_{i}\) means non-zero, and the attacker conciliates the \(i_{th}\) meter and then displaces its original mensuration \(z_{i}\) with a false extent \(z_{i}+d_{i}\). Let \(\hat{X}_{false}\) and \(\hat{x}\) specify the forecasts of \(x\) using the false mensuration's \(z_{d}\) and the original values \(z\), respectively. \(\hat{X}_{false}\) may be written as \(\hat{x}+f\), where \(f\) is a non-zero \(n\)-dimensional vector. It is worth noting that \(f\) states the attacker's calculation error. The intruder, on the other hand, should choose f as a linear combination of \(H\)'s column vectors (i.e., \(d=H\rho\)). Therefore, FDIA with complete information is following and \(z_{d}\) can pass the detector as long as \(z\) is able to pass it [102]. It is assumed that an attacker accesses the \(H\) matrix and injects false measurements in [103], \(m\) meters provide \(m\) measurements \(z_{1},\ldots,z_{m}\), and also, it is assumed that there are \(n\) state variables \(x_{1},\ldots,z_{n}\). The \(m\times n\) matrix \(H\) can be characterized by a relationship between \(m\) meter measurements and \(n\) state variables. The measurement noise [104] is formulated with \(W\) diagonal matrix as  
\[\hat{x}=\left(H^{\mathsf{T}}WH\right)^{-1}\!H^{\mathsf{T}}W_{Z} \tag{1}\]  
In [102], an attack in which the attack vector d equals \(H_{f}\), where \(f\) is an arbitrary non-zero vector, is a false data injection attack. Seeing that \(z\) can pass the detection and where \(\tau\) is the threshold, \(\|z-H\hat{X}\|\leq\tau\) is had; then, the vector of estimated state variables acquired from \(z_{d}\) can be demonstrated as \(\hat{x}+f\). Described previously if \(d=H_{f}\), the resulting measurement follows:  
\[\|z-H\hat{X}_{false}\|=\|z+d-H(\hat{x}+f)\|\]  
\[=\|z-H\hat{x}+(d-Hf)\|\]  
\[=\|z-H\hat{x}\|\leq\tau \tag{2}\]  
Attackers generate malicious measurements based on the \(H\) matrix and then inject it by starting FDIAs; they can manage the injected false data to overcome the bad measurement detection and represent random errors into the state estimation (SE) output. On the contrary, if attackers have no complete information about \(H\) matrix, if \(d\
eq\ H_{f}\), \(\varphi\) error matrix is created, then the solution of the state estimation follows:  
\[\hat{x}_{d}=\left(H^{\mathsf{T}}WH\right)^{-1}\!H^{\mathsf{T}}W_{Zd}\]  
\[=\left(H^{\mathsf{T}}WH\right)^{-1}\!H^{\mathsf{T}}W(z+d)\]  
\[=\left(H^{\mathsf{T}}WH\right)^{-1}\!H^{\mathsf{T}}W\!\left(z+H_{f}+\varphi_{ f}\right)\]  
\[=\hat{x}+f+\left(H^{\mathsf{T}}WH\right)^{-1}\!H^{\mathsf{T}}W\varphi_{f}\]  
\[=\hat{x}+\overline{f} \tag{3}\]  
\[\hat{x}=\left(H^{\mathsf{T}}WH\right)^{-1}\!H^{\mathsf{T}}W_{Z} \tag{1}\]0  
The attacker aims to hack the multiple sensors and phasor measurement units (PMUs) readings to mislead the smart grid's decision-making process in FDIA [103,104]. False Data Injection is one of the most dangerous types of attack among cyber-attacks. Therefore, it should be examined most carefully.