## 2 Background  
### State Estimation  
AC State estimation aims solving a non-linear algebraic differentiable set of equations that have the following form [27]:  
\[\mathbf{z}=\mathbf{h}(\mathbf{x})+\mathbf{e}. \tag{1}\]  
where \(\mathbf{z}\in\mathbb{R}^{m}\) is the measurement vector, \(\mathbf{x}\in\mathbb{R}^{N}\) is the state variables vector (typically voltage magnitudes \(V\) and voltage angles \(\theta\)), \(\mathbf{h}(\mathbf{x})\):\(\mathbb{R}^{m}\rightarrow\mathbb{R}^{N},(m>N)\) is a non-linear differentiable function that relates the states to the measurements, \(\mathbf{e}\) is the measurement error vector assumed with zero mean, standard deviation \(\sigma\) and having Gaussian probability distribution, and \(N=2n-1\) is the number of unknown state variables and \(n\) is the number of buses in the system. Hence, in classical Weight Least Square State Estimation (WLS SE), the approach consists of solving the following minimization problem:  
\[\min_{\mathbf{x}}\quad J(\mathbf{x})=[\mathbf{z}-\mathbf{h}(\mathbf{x})]^{T} \mathbf{W}[\mathbf{z}-\mathbf{h}(\mathbf{x})]. \tag{2}\]  
where \(\mathbf{W}\) is a diagonal weight matrix composed by the inverse of the squared values of measurement standard deviations (\(\sigma\)): \(\mathbf{W}=diag([\sigma_{1}^{-2},\ldots,\sigma_{m}^{-2}]^{T})\). \(J(x)\) index is a norm in the measurements vector space. The measurement model in (1) relies on two data sets: measurements set and grid graph, i.e, connectivity and system parameters. If corrupted data is used, then the obtained solution will mislead the operators who monitor the grid. Corrupted data could be attributed to measurement(s) and/or system parameters (database). Given the non-linear relationship, it would be a difficult task to distinguish the source of bad data when there is a simultaneous attack [25]. Hence, in this work, the way is paved for the model to be able to clearly distinguish the error source in the measurement model seen in (1), i.e., is the FDI on the measurement set, system model parameters, or both, and how to correct this? The database in this context is the model representation of different components that compose the physical power grid. For instance, a typical model of a long transmission system line is represented by the \(\pi\)-model. Hence, in SE, this model contributes to the bus admittance matrix, i.e., \(Y_{bus}\) through its parameters such as line conductance \(g_{km}\), line susceptance \(b_{km}\) and shunt admittance \(b_{km}^{sh}\). Depending on the system under study, a combination of those parameters might be considered. For instance, in short and medium transmission lines, \(b_{km}^{sh}\) has a negligible effect on the voltage. Hence, it could be excluded from the model. For long transmission lines, however, \(b_{km}^{sh}\) is important for estimating the voltage. The challenging scenario is when all parameters are included. Therefore, with any perturbation in these parameters, the state estimator might lead to a solution that does not depict the true underlying physical system. The task would be even more challenging when both measurements and parameters have contributed to estimate an untrue states, i.e., \(V\) and \(\theta\), how one could identify the source of erroneous with confidence? The classical WLS model in (2) minimizes the residual. The work in [28] proved, however, that the error in (1) has a unique decomposition; detectable and undetectable components. The error can be written as follows  
\[\left\|e\right\|^{2}=\left\|e_{D}\right\|^{2}+\left\|e_{II}\right\|^{2}. \tag{3}\]  
where \(e_{D}\) is the detectable error while \(e_{U}\) is the undetectable error. Hence, the Innovation concept, i.e., \(II\), is used to quantify the undetectable part as follows:  
\[II_{i}=\frac{\left\|e_{D}^{i}\right\|}{\left\|e_{U}^{i}\right\|}=\frac{\sqrt{1- P_{ii}}}{\sqrt{P_{ii}}}. \tag{4}\]where \(P_{ii}\) is the \(i\)th entry in the projection matrix \(P\).