## 4 Multi Objective Model of Optimal PMU Placement
### Updated Constraint Considering SCADA Measurement  
The installation cost can be high if only PMU measurements are applied to meet the constraints. In the process of OPP, the existing SCADA measurements are applied to meet the full observability.  
The full observability indicates that the voltage phasors of all buses in the network are known. In this paper, power flow measurements and injection measurements are applied to the OPP model by updating the constraints and equivalent.  
The relationship between power flow and bus voltage phasor can be expressed as Equation (25):  
\[S_{ij}=\dot{U_{i}}\Big{(}\dot{\mathbf{U_{i}}}-\dot{\mathbf{U_{j}}}\Big{)}\dot{\mathbf{y_{i }}}_{ij}, \tag{25}\]  
where \(S_{ij}\) is the power flow of branch \(i-j\); \(\dot{U_{i}}\) represent the voltage phasor of bus \(i\); \(\dot{\mathbf{U_{i}}}\) and \(\dot{\mathbf{U_{j}}}\) represent the conjugations of the voltage phasor of bus \(i\) and bus \(j\) respectively; \(\dot{\mathbf{y_{ij}}}\) represents the conjugation of branch admittance.  
According to Equation (25), if the power flow and the voltage phasor of the bus at one end of the branch are known, the voltage phasor of the bus at another end can be calculated.  
If the power flow of branch \(i-j\) is measured, the constraints corresponding to bus \(i\) and bus \(j\) can be modified to Equation (26):  
\[\begin{array}{ccc}m_{i}=\mathbf{A_{i}}\mathbf{X}+o_{i}&\Rightarrow&m_{i(new)}=\bm {A_{i}}\mathbf{X}+o_{i}+(1-k_{i})k_{j}\\ m_{j}=\mathbf{A_{j}}\mathbf{X}+o_{j}&m_{j(new)}=\mathbf{A_{j}}\mathbf{X}+o_{j}+\big{(}1-k_{j} \big{)}k_{i}\end{array}, \tag{26}\]  
where \(\mathbf{A_{i}}\) and \(\mathbf{A_{j}}\) are respectively the \(i-\)th and \(j-\)th row of \(\mathbf{A}\); \(k_{i}\) and \(k_{j}\) are 0-1 variables, which can be calculated as Equation (27):  
\[k_{i}=\begin{cases}1&m_{i}\geq 1\\ 0&m_{i}=0\end{cases}k_{j}=\begin{cases}1&m_{j}\geq 1\\ 0&m_{j}=0\end{cases}. \tag{27}\]  
If \(k_{i}=1\), the bus \(i\) is observable. If \(k_{i}=0\), the bus \(i\) is unobservable.  
The relationship between the power injection and the bus voltage phasor can be expressed as Equation (28):  
\[S_{i}=\overset{\cdot}{U_{i}}\overset{\cdot}{l_{i}}=\overset{\cdot}{U_{i}}\sum \limits_{j\in i}\overset{\cdot}{Y_{ij}}\overset{\cdot}{U_{j}}, \tag{28}\]  
where \(S_{i}\) represents the power injection at bus \(i\); \(\overset{\cdot}{l_{i}}\) represents the conjugation of the current phasor of the bus \(i\); \(j\in i\) represents bus \(j\) is connected to bus \(i\) in the network; \(\overset{\cdot}{Y_{ij}}\) represents the conjugation of the admittance matrix element.  
According to Equation (28), if the power injection at the bus \(i\) is known, and the number of observable buses (bus whose voltage phasor is known) among bus \(i\) and its \(k\) neighbors is \(k\), the voltage phasor of bus \((k+1)\) can be calculated by KCLs. The observation property of the bus whose injection is known is the same as that of ZIB.  
Therefore, if the power injection at the bus is known, this bus is equivalent to ZIB, which is named as \"lesser ZIB\" in this paper. If the \"lesser ZIB\" has two neighbors, the \"lesser ZIB\" is merged with one of its neighbors and the network is compressed again. If the \"lesser ZIB\" has more than two neighbors, the modeling process is shown in Equations (22)-(24).  
The reason why this kind of bus is named as \"lesser ZIB\" is that ZIB measurement is perfect while the power injection measurement has the errors. Although the bus with known power injection has some similar characteristics of ZIB, the weights should be different when these measurements are applied to state estimation.