## 3 Distribution System State Estimation  
SE calculation is the function of the Energy Management System (EMS) in a power grid. SE calculations provide a way to monitor real-time power grid operation and develop more power grid analysis, including power flow calculations, power dispatching, observability analysis, bad-data detection, reliability assessment, etc. Transmission state estimation techniques have been established, but these techniques cannot be applied directly for DNs due to the differences between transmission and distribution grids. New DSSE strategies should be designed according to the needs and requirements of DNs. The first DSSE technique was established when the SCADA system was introduced in 1990 [66]. Traditional SE techniques are commonly performed based on conventional measurements, which are commonly obtained from SCADA systems [67]. However, since SCADA measurements are not enough to acquire SE calculations, pseudo-measurements are considered along with traditional measurements to make a system observable [3]. In order to calculate state estimation variables, the WLS method is generally applied to both TNs and DNs [66]. By using WLS, the state vector can be defined based on node voltage [68,69] or branch current variables [70,71]. When state variables are calculated, other network parameters can also be determined. In node voltage definition, bus voltage phase angle, as well as voltage magnitude, are considered as a state vector [72]: \(x=[\theta,|V|]\). When only conventional measurements are available, and no measurement devices are placed at the slack bus, the state vector is defined in polar coordination as follows:  
\[x=[\theta_{2},\cdots,\theta_{N},V_{1},\cdots,V_{N}], \tag{1}\]  
where, \(N\) is the number of buses. The slack bus is considered as a reference: \(\theta_{1}=0\). When a PMU is located at the slack bus, the phase angle at the slack bus is not zero, and it should be involved in the state vector. The state vector is represented as:  
\[x=[\theta_{1},\cdots,\theta_{N},V_{1},\cdots,V_{N}]. \tag{2}\]  
The relationship between the state variables and the measurement data is called the measurement function, which is represented as:  
\[z=h(x)+e, \tag{3}\]  
Figure 1: (**a**) SCADA and PMU measurements in a power network; (**b**) SCADA and PMU measurement refresh rates. where \(z=[z_{1},\cdots,z_{M}]^{T}\) is the measurement vector, and \(M\) is the number of measurements. Bus voltage magnitudes, phase angles, branch current magnitudes, and phases, as well as active and reactive power flows and injections, can be involved in the measurement vector. For completeness of the paper, we include the equations for standard active (\(P_{i}\)) and reactive (\(Q_{i}\)) power injections at bus \(i\):  
\[P_{i}=V_{i}{\sum}_{j=0}^{N}\ V_{j}\big{(}G_{ij}cos\theta_{ij}+B_{ij}sin\theta_ {ij}\big{)} \tag{4}\]  
\[Q_{i}=V_{i}{\sum}_{j=0}^{N}\ V_{j}\big{(}G_{ij}sin\theta_{ij}-B_{ij}cos\theta_ {ij}\big{)}. \tag{5}\]  
\(G_{ij}\) and \(B_{ij}\) are the real and imaginary part, respectively, in a nodal admittance matrix element \(Y_{ij}\), and \(\theta_{ij}=\theta_{i}-\theta_{j}\) is the standing phase angle difference between buses \(i\) and \(j\). The active (\(P_{ij}\)) and reactive (\(Q_{ij}\)) power flows from bus \(i\) to bus \(j\) are expressed as:  
\[P_{ij}=V_{i}V_{j}\big{(}G_{ij}cos\theta_{ij}+B_{ij}sin\theta_{ij}\big{)}-G_{ij }V_{i}^{2} \tag{6}\]  
\[Q_{ij}=V_{i}V_{j}\big{(}G_{ij}sin\theta_{ij}+B_{ij}cos\theta_{ij}\big{)}+B_{ij }V_{i}^{2}. \tag{7}\]  
The relationship between the measurement vector and the state variable \(x=[x_{1},\cdots,x_{N}]^{T}\) (\(N\) is the number of state variables) is shown by \(h=[h_{1},\cdots,h_{M}]^{T}\). \(h\) is the list of measurement functions, which is commonly nonlinear.