# Distribution System State Estimation Using Model-Optimized Neural Networks
## 1 Introduction  
Reliable electric power transmission and distribution has ever been at the forefront of power system studies. Power system reliability, as defined by [1], encompasses both system adequacy and system security. The former aims to satisfy the balance between energy supply and energy demand within the system, while the latter aims to establish the system's ability to bounce back from disturbances. These days, more and newer devices such as renewable energy sources and converter interfaced generation are introduced to the network. These induce uncertainties as shown by [2] and harmonic disturbances in the voltage signal as shown by [3]. Further, integration of DC sources and loads into hybrid AC/DC distribution systems are also increasing [4]. To help maintain reliability, reference [5] mentions that operators need full system monitoring to maximize renewable energy production and to make a more versatile power system. However, system monitoring implements such as SCADA, AMI, and PMUs are expensive; thus, optimal use is normally required. As proposed in [6], while fully covering the system with measurement devices ensures reliable services, they experience diminishing returns on investment. Due to that, optimal placement papers such as [6, 7, 8] exist to attempt to find a configuration which makes the system observable and cost effective. This then leads to the problem ofcompleting the missing measurements of the current system state given limited information; thus, power system state estimation is developed. State estimation works by pooling the limited measurement data available and finding a suitable system state that will approximate the given measurements with the least amount of error. This is largely analogous to regression analyses wherein a best fit function's parameters are changed in order to reconcile the data and the function mapping. However, instead of the usual forecasting paradigm, state estimation aims to deduce current values given current and historical input data. The key difficulty here is establishing an effective model to relate the measurement values to the system state, that is, completely modeling the power system including the non-linear components, the noise acquired during data acquisition, and the statistical uncertainties of the renewable energy sources. There are few references that consider hybrid distribution system implementations, some of which include [9; 10] wherein the conventional state estimation method has been decentralized to evaluate the AC and DC subgrids separately. The work in [9] capitalized on the duality principle while the work in [10] introduced an intermediate non-linear variable in their three-stage approach. Since the conventional state estimation method is a snapshot in time, the distributed generation and renewable energy sources can be summarily modeled as current injections in the DC subgrid. For references [9; 10] too, however, the conventional process is computationally intensive and is open to failures in assumption as [11] mentions. Due to the nature of this problem and its analog, regression analysis, there exists a method that excels in summarily modeling unseen and complex relationships between data: machine learning, specifically, neural networks. There have already been other allied works that have used neural networks in some form or another for power system state estimation. Works such as [12; 13] have used neural networks simply to prepare and clean measurement data to aid in state estimation. One work [14], far back in 1996, has already explored the use of neural networks for topology processing and static state estimation. A more recent work [15] adopts Bayesian state estimation using machine learning to power system state estimation so that estimation of unobservable parts of the system is made possible. Some other works try to include the original components of state estimation into the training process such as [16].