# A Data-Driven State Estimation Based on Sample Migration for Low-Observable Distribution Networks
## 3 State Estimation Model Based on Sample Migration  
### Fundamentals of Distribution Network State Estimation  
Distribution network state estimation iteratively determines the current operating state by solving the measurement equations as follows:  
\[\mathbf{z}=h(\mathbf{x})+\mathbf{v}\]where \(z\) and \(x\) represent the measurements (\(m\) dimension) and state variables (\(n\) dimension), respectively, and \(v\) signifies a measurement error vector. \(h(x)\) denotes measurement functions between measurements and states, which are nonlinear.  
The measurement configuration for distribution network state estimation in this paper includes node voltage amplitude \(V_{i}\), node injection power \(P_{i}\), \(Q_{i}\), and branch power flow \(P_{ij}\), \(Q_{ij}\), which serve as the inputs to the state estimation model, with node voltage as the output. First, based on the historical operational modes of distribution networks, the network topology under different operating conditions is obtained. By applying the corresponding network topology data, measurement equations of the distribution networks can be established. Please refer to [26] for the specific expression of the measurement equations.  
Traditional model-driven SE algorithms include WLS and WLAV. The objective function of WLS is to minimize the weighted sum of squared residuals, while WLAV is to minimize the weighted sum of the absolute value of the residuals. The specific formulas of WLS and WLAV are as follows:  
\[\text{min}J_{\text{WLS}}(x)=\sum_{i=1}^{m}w_{i}r_{i}^{2} \tag{3}\]  
\[\text{min}J_{\text{WLAV}}(x)=\sum_{i=1}^{m}w_{i}|r_{i}| \tag{4}\]  
where \(w_{i}\) represents the weight of associated measurement, and \(r_{i}\) represents residual, \(r_{i}=z_{i}-h_{i}(x)\).  
The Newton formulation is generally used to iteratively solve WLS [26], while the interior point method is often used to solve WLAV [27].  
Due to the large scale of measurements in the distribution networks, overfitting may occur during CNN training. To eliminate the impact of data dimensionality on model training, this paper normalizes the data of the model. The specific formula is as follows:  
\[o_{i}^{\prime}=\frac{o_{i}-\text{min}o_{i}}{\text{max}o_{i}-\text{min}o_{i}} \tag{5}\]  
where \(o_{i}\) and \(o_{i}^{\prime}\)  
where \(\mathbf{U}\mathbf{U}^{\mathrm{T}}=\mathbf{I}_{m}\), \(\mathbf{W}\mathbf{W}^{\mathrm{T}}=\mathbf{I}_{n}\), and \(\mathbf{I}_{m}\) and \(\mathbf{I}_{n}\) represent \(m\)-order and \(n\)-order identity matrices. \(\Sigma\) is a diagonal matrix, and its diagonal elements are composed of singular value sequences of the original matrix \(\mathbf{M}\).  
In actual distribution networks, the singular value sequences of high-observable distribution networks can be obtained using the aforementioned principles. Clustering algorithms, such as \(K\)-means clustering, can then be employed to group these singular value sequences into several typical topology sets. In these typical topology sets, the measurement configurations of distribution networks are rich, and historical measurement data are abundant. Therefore, these typical distribution network topologies can serve as the source domain in this paper. Similarly, a topology model for low-observable distribution networks can be established. Using the singular value sequence as a topology similarity evaluation index, the corresponding source domain model connected to the low-observable distribution networks can be identified from the typical topology sets. Subsequently, sample migration techniques can be applied to transfer and supplement high-observable distribution network samples to low-observable distribution networks.