## 4 D-PMU Placement Order Determination Based on Spatial Electrical Distance
### Spatial Electrical Distance  
Traditional electrical distance characterizes the closeness of electrical connections between two buses. Spatial electrical distance considers the influence of various buses in the system.  
There are many ways to define electrical distances, most of which are defined as the sensitivity of reactive power variations to the voltage amplitude magnitude. However, this kind of method is defined as applied to voltage control. Voltage amplitude can be measured directly, and the phase angle needs to be measured indirectly in a traditional power grid. Therefore, state estimation can pay more attention to phase angle than to amplitude. In addition, compared with SCADA, the D-PMU can measure phase angle data. In state estimation with D-PMU data, the D-PMU can significantly improve the accuracy of the phase angle data.  
Therefore, this paper uses the formula characterizing the effect of active power variations on the voltage phase angle to calculate the spatial electrical distance. This method can improve the phase angle estimation accuracy as rapidly as possible during the D-PMU placement stages.  
Based on the traditional power flow equation, keeping the reactive power constant, the sensitivity of the active power to the voltage phase angle can be expressed as:  
\[\Delta\delta=S\Delta\textbf{P}=\left[\textbf{{H}}-\textbf{{N}}\textbf{{L}}^{-1} \textbf{{J}}\right]^{-1}\Delta\textbf{P}, \tag{29}\]  
where \(\Delta\delta\) is the variation of the bus voltage phase angle and \(\textbf{{S}}=\left[\textbf{{H}}-\textbf{{N}}\textbf{{L}}^{-1}\textbf{{J}} \right]^{-1}\) is the sensitivity matrix, which represents the sensitivity of active power variation to voltage phase angle variation. \(\Delta\textbf{P}\) is the bus active power variation vector, and \(\textbf{{H}},\textbf{{N}},\textbf{{J}}\) and \(\textbf{{L}}\) are Jacobian submatrices, of which coefficients are real numbers.  
According to the sensitivity matrix \(\textbf{{S}}\), the sensitivity between buses can be calculated as:  
\[k_{ij}=\left|\text{lg}\left(\frac{S_{jj}}{S_{ij}}\right)\right|, \tag{30}\]  
where \(k_{ij}\) represents the voltage phase angle variation ratio between bus \(i\) and \(j\) when bus \(j\) active power changes. The larger that \(k_{ij}\) is, the smaller that the effect of bus \(j\) is on bus \(i\) voltage phase angle.  
According to \(k_{ij}\), the associated sensitivity matrix \(\textbf{{K}}\) is calculated as:  
\[\textbf{{K}}=\left[k_{ij}\right]_{n\times n}=\left[\begin{array}{cccc}k_{1,1 }&k_{1,2}&\cdots&k_{1,n-1}&k_{1,n}\\ k_{2,1}&k_{2,2}&\cdots&k_{2,n-1}&k_{2,n}\\ \vdots&\vdots&\cdots&\vdots&\vdots\\ k_{n-1,1}&k_{n-1,2}&\cdots&k_{n-1,n-1}&k_{n-1,n}\\ k_{n,1}&k_{n,2}&\cdots&k_{n,n-1}&k_{n,n}\\ \end{array}\right], \tag{31}\]  
where \(n\) is the number of buses in the network.  
According to the matrix \(\textbf{{K}}\), the Euclidean distance between buses \(i\) and \(j\) is calculated as:  
\[\begin{array}{ll}d_{ij}&=\left\|\textbf{{k}}_{i}-\textbf{{k}}_{j}\right\|_{ 2}\\ &=\left[\left(k_{i1}-k_{j1}\right)^{2}+\left(k_{i2}-k_{j2}\right)^{2}+\cdots+ \left(k_{in}-k_{jn}\right)^{2}\right]^{1/2}\end{array} \tag{32}\]  
where \(d_{ij}\) is the spatial electrical distance between buses \(i\) and \(j\). The smaller that \(d_{ij}\) is, the greater that the influence of bus \(j\) is on bus \(i\) voltage phase angle, and the closer that the electrical connection is. In contrast, the higher that \(d_{ij}\) is, the greater that the electrical coupling degree is between buses \(i\) and \(j\).  
According to spatial electrical distance \(d_{ij}\), the associated weight between buses \(i\) and \(j\) is defined as \(w_{ij}\):  
\[w_{ij}={d_{ij}}^{-1}. \tag{33}\]  
If topology variation is considered, the above formula is modified as:  
\[w_{ij}=\sum_{c=1}^{R}a_{c}{d_{cij}}^{-1}. \tag{34}\]  
where \(d_{cij}\) is the spatial electrical distance between bus \(i\) and \(j\) under the \(c\)-th topology.