## 2 Background Information  
### State Estimation Augmented with Synthetic Measurements  
State estimation aims at solving a set of non-linear algebraic differentiable equations that have the following form [33]:  
\[\mathbf{z}=\mathbf{h}(\mathbf{x})+\mathbf{e} \tag{1}\]  
where \(\mathbf{z}\in\mathbb{R}^{m}\) is the measurement vector, \(\mathbf{x}\in\mathbb{R}^{N}\) is the state variables vector, \(\mathbf{h}(\mathbf{x}):\mathbb{R}^{m}\rightarrow\mathbb{R}^{N},(m>N)\) 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. Hence, in the weighted least square state estimation (WLS SE), the approach consists of solving the following minimization problem:  
\[\min_{\mathbf{x}}\quad J(\mathbf{x})=[\mathbf{z}-\mathbf{h}(\mathbf{x})]^{T} \mathbf{W}[\mathbf{z}-\mathbf{h}(\mathbf{x})]\enspace, \tag{2}\]  
where \(\mathbf{W}\) is a diagonal weight matrix composed by the inverse of the squared values of measurement standard deviations (\(\sigma\)): \(\mathbf{W}=diag([\sigma_{1}^{-2},\dots,\sigma_{m}^{-2}]^{T})\). \(J(x)\) index is a norm in the measurements vector space.  
The measurement model in (1) relies on the characteristics of the grid, i.e., connectivity and system parameters. If corrupted data is used, then the obtained solution will be physically incorrect, and could potentially mislead the operators who monitor the grid.  
In the view of the minimization problem in (2), WLS SE considers minimizing the error as described in (1), which assumes that the residual tends to follow a normal distribution. From the Central Limit Theorem [33], adding large number of independent random variables that follow any distribution with bounded variance, their properly normalized sum tends to approximate a normal distribution. Therefore, for detecting errors using classical WLS, the hypothesis test solely relies on the distribution of \(\chi^{2}\). If \(\chi^{2}\) distribution does not follow a normal distribution, then the hypotheses test will fail. In ordered to have \(\chi^{2}\) distribution to follow a normal distribution the measurement model degrees of freedom needs to be increased. Increasing degree of freedom comes with the financial cost of increasing the measurement set through additional meters.  
Ideally, one would want to have a measurement reading in every bus and line section. However, this is not realistic, considering the inherent financial cost. Instead, measurements can be created artificially at locations where no real-life measurement or historical data exist. These measurements, named synthetic measurements, were modeled in [32]. One should not confuse synthetic measurements with pseudo measurements, which are created considering available historical data [33]. The main idea of creating synthetic measurements is to approximate the residual of the measurement model to a normal distribution. In doing so, not only parameter FDI correction is enhanced, as will be presented, but also the global redundancy level of the system increases, which enhances gross error detection and identification [32].