## 2 Background Information
### Unbalanced Parameter FDI Attack Correction Model  
In (1), the possibility of errors in the parameter data is not considered. Instead, if one considers \(z=h(x,p)+e\), where \(p\) is the parameter in error, this function can be expanded into a Taylor Series [19]:  
\[z_{i}=h_{i,0}+\frac{\partial h_{i}(x,p)}{\partial p}\bigtriangleup p, \tag{5}\]  
where \(\bigtriangleup p\) denotes the parameter error. From (5), the parameter error can be calculated to be as follow:  
\[\bigtriangleup p=\frac{z_{i}-h_{i,0}}{H_{p,0}}, \tag{6}\]  
where \(H_{p,0}\) denotes the Jacobian of the parameter. All the quantities are known, so the parameter error can be calculated by (6), which is called the relaxed model here, since it considers the measurement without error. With this model, parameter error can be corrected by using measurement value of reactive power flow corresponding to the line where the parameter attack happened through iterations. However, system net parameter values includes three components, which are series conductance \(g\), series susceptance \(b\) and shunt susceptance \(b^{sh}\). In the meantime, the weights of these three components are decided by network parameter database, so one can only correct the parameter error through this model when these components have the exactly same percentage attack, lets say, 10% on \(g\), 10% on \(b\), and 10% on \(b^{sh}\). Thus unbalanced FDI in parameter values are not considered by this model, for example, 30% on g, 20% on b,and 10% on \(b^{sh}\). To address this issue, an unbalanced correction model is presented as follows [31]:  
\[\begin{pmatrix}\Delta g_{\text{bm}}\\ \Delta b_{\text{bm}}\\ \Delta b^{sh}_{\text{bm}}\end{pmatrix}=\tau_{n}^{-1}\begin{pmatrix}Z_{R-m( loss)}-H_{R-m( loss)}^{n}\\ Z_{R-n}-H_{R-m}^{n}\\ Z_{Q_{k-n}}-u^{n}_{Q_{k-m}}\end{pmatrix}, \tag{7}\]  
where the parameter correction Jacobian matrix \(\tau\) is defined as:  
\[\tau=\begin{pmatrix}|E_{k}-E_{m}|^{2}&0&0\\ V_{k}^{2}-V_{k}V_{m}cos\theta_{km}&-V_{k}V_{m}sin\theta_{km}&0\\ -V_{k}V_{m}sin\theta_{km}&-V_{k}^{2}+V_{k}V_{m}cos\theta_{km}&-V_{k}^{2}\end{pmatrix}. \tag{8}\]  
In (7), \(n\) denotes the iteration index; \(Z_{P_{k-m(loss)}},Z_{P_{k-m}},Z_{Q_{k-m}}\) are recorded measurement values of real power loss, real power flow and reactive power flow for FDI attacked line from bus \(k\) to \(m\); \(h^{n}_{P_{k-m(\text{feas})}}\), \(h^{n}_{P_{k-m}}\), \(h^{n}_{Q_{k-m}}\) denote the continuous nonlinear differentiable function of above three quantities at \(n_{th}\) iteration. In (8), parameter correction Jacobian matrix \(\tau\) uses the magnitude of the voltage drop \(|E_{k}-E_{m}|^{2}\), voltage injection \(V\) and phase difference \(\theta\) for bus \(k\) and \(m\) to perform parameter correction. One can see from (7) and (8), parameter errors of conductance \(\Delta g_{km}\), series susceptance \(\Delta b_{km}\) and shunt susceptance \(\Delta b_{km}^{sh}\) are corrected by using 3 measurements. However, two issues may yield a failure of this process: (1) There are incomplete measurements dataset needed by this model; (2) conductance error \(\Delta g_{km}\) can only be estimated from corresponding real power loss measurement which is relatively small in Transmission Line (TL), so an incorrect estimated \(\Delta g_{km}\) will cause a wrong estimation of series susceptance error \(\Delta b_{km}\) and shunt susceptance error \(\Delta b_{km}^{sh}\). To address this issue, a synthetic measurement enhanced parameter correction model is presented in this paper.