## 2 Background  
### State Estimation with the Innovation Concept  
The power system with \(n\) buses and \(m\) measurements is modeled as a set of non-linear equations as follows [19]:  
\[z=h(x)+e \tag{1}\]  
where \(z\in\mathbb{R}^{m}\) is the measurement vector, \(x\in\mathbb{R}^{N}\) is the state variables vector, \(h(x)\):\(\mathbb{R}^{m}\rightarrow\mathbb{R}^{N},(m>N)\) is a non-linear differentiable function that relates the states to the measurements, \(e\) 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. Weighted Least Square (WLS) is a classical state estimator that searches for the best estimates of the states \(x\) of the well-known problem that minimizes the cost function as follows:  
\[J(x)=\left\|z-h(x)\right\|_{\mathbb{R}^{-1}}^{2}=[z-h(x)]^{T}R^{-1}[z-h(x)] \tag{2}\]  
where \(R\) is the measurement covariance matrix. \(J(x)\) index is a norm in the measurements vector space. Let \(\hat{x}\) be the solution of the aforementioned minimization problem. Then, the estimated measurement vector is \(\hat{z}=h(\hat{x})\). The residual is defined as the difference between \(\hat{z}\) and \(z\): \(r=z-\hat{z}\). Linearizing (1) at a certain operating point \(x^{*}\) yields:\[\bigtriangleup z=H\bigtriangleup x+e \tag{3}\]  
where \(H=\frac{\partial u}{\partial x}\) is the Jacobian matrix of \(h\) calculated at the point \(x^{*}\). \(\bigtriangleup z=z-h(x^{*})=z-z^{*}\) and \(\bigtriangleup x=x-x^{*}\) are the correction of measurement and state vector, respectively. Under an observable condition, i.e., \(\text{rank}(H)\geq N\), the vector space of measurements can be decomposed into two sub-spaces that are orthogonal to each other. Let \(P\) be a linear operator such that \(\bigtriangleup\hat{z}=P\bigtriangleup z\) and the residual vector \(r\) be \(\bigtriangleup z-\bigtriangleup\hat{z}\). Then, the vector \(\bigtriangleup\hat{z}=H\bigtriangleup\hat{x}\) is orthogonal to the residual vector \(r\), since \(P\) projects the measurement vector mismatch \(\bigtriangleup z\) orthogonality in the range space of \(H\). Equivalently, in mathematical form one can write the following:  
\[<\bigtriangleup\hat{x},r>=(H\bigtriangleup x)^{T}R^{-1}(\bigtriangleup z-H \bigtriangleup\hat{x})=0 \tag{4}\]  
Solving for \(\bigtriangleup\hat{x}\), one can obtain the following:  
\[\bigtriangleup\hat{x}=(H^{T}R^{-1}H)^{-1}H^{T}R^{-1}\bigtriangleup z \tag{5}\]  
The projection matrix \(P\) is found to be the idempotent matrix that can be derived as follow: using the solution in (5), the estimated increment in measurement is:  
\[\bigtriangleup\hat{z}=H\bigtriangleup\hat{x}. \tag{6}\]  
Therefore, by substituting (5) into (6), the projection matrix P can be calculated using the following expression:  
\[P=H(H^{T}R^{-1}H)^{-1}H^{T}R^{-1}. \tag{7}\]  
It is possible to decompose the measurement error vector into two components. The component \(e_{D}\) is the detectable error which is the residual in the classical WLS model while the component \(e_{U}\) is the undetectable error. \(e_{D}\) is in the orthogonal space to the range space of Jacobian whereas \(e_{U}\) is hidden in the Jacobian space. Hence, the error can be written as follows:  
\[\|e\|^{2}=\|e_{D}\|^{2}+\|e_{U}\|^{2}. \tag{8}\]  
The error vector in (8) is called Composed Measurement Error (\(CME\)). In order to find the masked error and compose it, the Innovation Index (II) introduced in [20] is used to quantify the undetectable error as presented in the following:  
\[II_{i}=\frac{\left\|e_{D}^{i}\right\|}{\left\|e_{U}^{i}\right\|}=\frac{\sqrt{1 -P_{ii}}}{\sqrt{P_{ii}}}. \tag{9}\]  
A low Innovation Index, i.e., II, indicates there is a large component of error which is not reflected in the residual. Therefore, the residual will be very small even if there is a gross error.