# Distribution System State Estimation Using Model-Optimized Neural Networks
## 2 Power System State Estimation  
The intuition behind state estimation (SE) for power systems is to solve for a system's state given a set of measurement data (making the system observable) while also considering possible data acquisition error. The basic formulation shown in Equation (1), and also shown in most state estimation works such as in [19; 20; 21; 22; 23], where \(z\) is the measurement data, \(h\) is a function that translates the true system states, \(x\), to measurement values, and \(\epsilon\) is the modeled acquisition error. For every state estimation problem, there exists required conditions so that the system state can be accurately and validly estimated. First is confirmation of network topology, achieved through confirmation and/or estimation of breaker statuses. Second is observability, which intuitively means, the state of all the nodes of the system can be solved given a set of state measurements in the form shown in Equation (1) [24]. \[z=h(x)+\epsilon \tag{1}\]  
\[\min J=\sum_{i=1}^{n}\frac{\left(z_{i}-h_{i}(x)\right)^{2}}{\sigma_{i}^{2}} \rightarrow\left[z-h(x)\right]^{T}\cdot W\cdot\left[z-h(x)\right] \tag{2}\]  
Due to the strong dependency of this process on measured data, any estimate of the system state will only be as good as the measurement data it was fed [23; 25]. Thus, the third requirement is bad data processing, which is why papers that deal with bad data itself [26], detection [27], and suppression [28] exist. Once data have been cleaned and prepared, all of this information is inserted into the weighted least squares (WLS) formulation as seen in Equation (2) with the aim of minimizing \(\epsilon\), thus called WLS-SE. \(\sigma_{i}\) represents the standard deviation of the measurements--or, loosely put, error--depending on the device and the quantity measured. Examples of this are \(\sigma_{i}=1\%\) to \(3\%\) for voltage-based measurements, \(5\%\) for current-based measurements, and \(15\%\) for power-based measurements. The general flow of WLS-SE is shown in Figure 1 which is closely similar to [19]. One of the more important relationships to be established is the relationship between the measurements and the different system states, i.e., the relationship of \(z_{i}\) with \(x_{i}\), which is shown by \(h(x)\). At this point, it is important to be clear about what parameters will  
Figure 1: The general WLS-SE flowchart. be the system state and which parameters will be measured. Based on traditional power system analysis methods, knowledge of system-wide bus voltages and angles are sufficient in order to fully establish current flows and phase differences. Thus, the system state, \(x_{i}\), becomes a \(2n\times 1\) vector consisting of voltage magnitude and angle quantities for all \(n\) buses; the measurements largely become a combination of either a subset of the bus voltage vector or a subset of the known real and reactive power injections. With that in mind, and depending on the measurement data used, \(h(x)\) can take its forms from traditional power flow analysis, with power measurements derived from the system-wide power balance equation shown in Equation (3) and is related to the system state via nodal analysis as shown in Equation (4), as presented in textbooks such as [29]. \[P_{gen}=P_{load}+P_{loss} \tag{3}\]  
\[\mathbf{Y_{bus}\cdot V}=\mathbf{I} \tag{4}\]  
The solved state estimates will only be as precise as the model and data used, which is why papers that deal with parameter estimation also exist [30]. Other ways to improve the state estimate also include transforming the measurements into more indicative quantities such as branch current measurements [20]. The details about these steps have been neatly laid out by [21] from network topology modeling down to bad data processing. While this paper focuses on static state estimation, other forms also exist such as dynamic state estimation where [31] uses the Extended Kalman Filter to effectively consider unknown inputs of synchronous machines (generators).