# A Comprehensive Survey on Enabling Techniques in Secure and Resilient Smart Grids
## 4 Security and Resilient Architecture
### Power System State Estimation (PSSE)
#### 4.2.3 Static State Estimation (SSE)  
Most current power monitoring systems and energy management systems (EMSs) are based on the quasi-steady state system model and static state estimation model, which rely on slow scan rates and no timestamps measurements from SCADA systems [55; 51]. State estimation is a knowledge-based approach to detecting malicious attacks. In a common steady DC power system model, the system state can be formulated using a linear regression model [21]  
\[z=h(x)+\epsilon \tag{1}\]  
where \(z\) is the measurement vector usually includes bus voltage magnitude, power injections, and power line flow, \(x\) is the state vector including voltage angle and voltage magnitude, \(\epsilon\) is the Gaussian measurement white noise with zero mean and covariance matrix, and \(h\) function is the Jacobian matrix related to power system topology [56; 9; 57]. The measurement residual is usually constructed with the help of weighted least-square observers and then compared with a predetermined threshold [18]. The measurement noise in PSSE usually uses the Gaussian distribution model; however, according to Pacific Northwest National Laboratory's study, the measurement noise follows a \"thick tail\" non-Gaussian distribution model [58; 59]. Weighted least square (WLS) is one of the commonly used traditional static state estimation approaches, which is based on the non-linear mathematical relations between the actual measurements and the state estimations. In WLS, the residuals are usually defined as a column vector, according to the regression model. \[ \begin{split} r&=z-h(x)\\ &=[z_{1}-h(x_{1}),z_{2}-h(x_{2}),\ldots,z_{n}-h(x_{n})]^{-1}\end{split}  \tag{2}\]  
The weighing matrix \(\mathsf{W}\) can be represented as the inverse of the covariance matrix \(\mathsf{R}\) in the WLS solution, which reflects the precision or reliability of the observations. Because the measurement error is the zero-mean, this means all measurement noise is independent and unrelated to each other. The variance (\(\sigma^{2}\)) of the measurement errors can be used as an estimate of the uncertainty associated with each measurement. \[W=R^{-1}=diag(\frac{1}{\sigma^{2}}) \tag{3}\]The WLS objective function is used to minimise the sum of the squares of the weighted residuals, which can be formulated below  
\[J(x)=\sum_{i=1}^{m}Wr^{2}=\sum_{i=1}^{m}r^{T}Wr=[z-h(x)]^{T}W[z-h(x)] \tag{4}\]  
To minimise the objective function, the derivation \(J(x)\) needs to be calculated equal to 0, and \(X\) is the point, where \(H(X)\) is the Jacobian matrix, which refers to the partial derivatives of the model function \(h(X)\) with respect to the parameters \(X\). \[g(X)=\frac{\partial(r^{T}Wr)}{\partial X}=-H^{T}(X)Wr=0 \tag{5}\]  
Non-linear WLS estimation is usually solved by the Gauss-Newton iterative algorithm, where is the \(k\)-th iteration of \(x\), which has the Gain matrix calculated as  
\[G(x_{k})=\frac{\partial g(x_{k})}{\partial x}=H^{T}(x_{k})WH(x_{k}) \tag{6}\]  
The weights allow the WLS model to assign more weight to observations that are more reliable or have less error variance and less weight to observations that are less reliable or have more error variance, which provides more accurate and stable estimates of the regression coefficients for WLS. The WLS solution's disadvantages are very obvious: it is not very robust and vulnerable to non-Gaussian processes, complex and highly non-linear systems, and cannot capture history records, even though it can process very fast [58; 60]. Some SSE studies are listed in Table 2: