## 3 Multi-Objective OPP Model
### Multi-Objective OPP Model Considering Topology Changes  
The frequent economic reconfiguration of a distribution network can change the network's topology, which means that D-PMUs may not be able to monitor adjacent nodes for a long time. Thus, the impact of economic reconfiguration on observability should be considered. The optimal reconfiguration of distribution systems with DGs helps attain multiple objectives, including load balancing, voltage profile improvements and loss reduction [22; 23; 24; 25]. Equation (22) considers current profile improvements, voltage profile improvements and loss reduction [23; 24; 25]. The three parts of (22) are respectively derived from [22; 23; 24; 25]. The objective function is formulated as follows:  
\[F=u_{load}\cdot\sum_{i}^{N_{br}}\left(\frac{I_{i}-I_{ave}}{I_{ave}}\right)^{2}+ u_{voltage}\cdot\sum_{i}^{N}\left(\frac{V_{i}-V_{Ni}}{V_{Ni}}\right)^{2}+u_{ \text{loss}}\ \cdot\left(\sum_{i}^{N_{br}}\left|I_{i}^{2}\right|\cdot r_{i}/P_{\text{loss 0}} \right)\,, \tag{22}\]  
where \(N_{br}\) is the number of branches; \(I_{i}\) is the current of the \(i\)-th branch, \(I_{\text{ave}}\) is the average current of all branches; \(V_{i}\) and \(V_{Ni}\) are the true and reference voltage of the \(i\)-th bus; \(r_{i}\) is the resistance of the \(i\)-th branch; \(P_{\text{loss0}}\) is the power loss of the normal network's topology; and \(u_{\text{load}}\), \(u_{\text{voltage}}\) and \(u_{\text{loss}}\) are the weights of different optimal objectives.  
If the load distribution and network parameters are known in each reconfiguration period, the optimal reconfiguration scheme can be calculated according to Equation (22). Ref. [26] provides a load forecasting method that can predict the probability of load distribution in multiple time scales. The scene set of distributed generation outputs and load distributions can be produced by the Monte Carlo simulation combined with distributed generation output forecasting, which is considered a reconfigured topology set in the proposed OPP model. Then, the topology weight can be calculated according to the frequency of each topology during the simulation. Each topology weight is: \(a_{1}\), \(a_{2}\),..., \(a_{R}\).  
Therefore, each objective function can be changed to the average value under the main topologies to ensure good performance of the placement scheme under multiple topologies. The OPP model can be updated to:\[\begin{split}&\min F_{1}(\mathbf{X})=N_{\rm D-PMU}=\sum\limits_{i=1}^{N}x_{ i}\\ &\max F_{2}(\mathbf{X})=NOBCS=\sum\limits_{c=1}^{K}\sum\limits_{k=1}^{N \rm MU}\alpha_{c}n_{\alpha,k^{\prime}}/\left(N_{\rm D-PMU}\cdot R\right)\\ &\max F_{3}(\mathbf{X})=NMR=\sum\limits_{c=1}^{R}\sum\limits_{l=1}^{N }\alpha_{c}m_{ci}/R\\ & s.t.\,\mathbf{M}_{c}\geq\mathbf{b}\,c=1,\,2,\,\cdots,\,R\end{split} \tag{23}\]  
where \(R\) is the number of topologies considered in the model; \(c\) is the topology number;\(n_{\alpha,k}\) is the value of \(n_{\alpha,k}\) under the \(c\)-th topology; \(m_{ci}\) is the value of \(m_{i}\) under the \(c\)-th topology; and \(\mathbf{M}_{c}\) is the measurement redundancy vector \(\mathbf{M}\) under the \(c\)-th topology. The three objective functions in Equation (23) correspond to the weighted average of the three objective functions in Sections 3.1-3.3 under multi-topology.  
If there are special requirements for the accuracy of state estimation, the influence of inherent accuracy and environmental noise on measurement placement should be considered. In Ref. [10], we proposed a state estimation error calculation method using mixed measurements. The state estimation error calculated by this method can be used as a constraint to meet the needs of state estimation. Limited by the length of the paper, we do not discuss this method in detail.