# WAMs Based Eigenvalue Space Model for High Impedance Fault Detection
## 1 Introduction
Despite encouraging results produced by Frequency Domain Estimators, several limitations of this approach must be noted: First, there is a lack of a generalized solution, particularly in regards to the system's topology, and secondly, the estimation of system parameters, including the fault itself, has not been standardized. Some of the solutions based on frequency domain estimators attempt to streamline the process by modeling and estimating a reduced number of parameters, which leads to marginal observability and near singular Jacobian matrices. Both of these lead to reduced redundancy and a possible magnification of errors [1; 14]. Another important limitation of WSL estimators to be considered is that the vast majority of them only consider the residual of the error, which does not always represent the total error [15; 16]. An approach based on neural networks is used in [17], which derives estimates of the parameters of a feeder during fault conditions. The estimation is carried out in the time domain, and utilizes polynomial approximation techniques. This approach is hindered by assumptions related to the time-frame when measurements are taken and the location of the load [18]. Recently, in [19], the harmonic distortion generated during the fault is analyzed in the spectral domain. This solution detects HIFs by identifying parameter errors in the fundamental and the third harmonic through a WLS estimator. This technique yields a high rate of detection and an accurate location of the fault, however, implementation presents serious challenges as some of the parameters required to solve the estimator must be calculated manually. Finally, solutions based on WAMs technology were utilized by studies in [20; 21; 22]. In [20], a Fourier Series based approach is proposed in which parameters are estimated from harmonics measured by PMUs. This approach is similar to other frequency domain approaches, and although the idea seems promising, implementation could be an issue due to short windows of time when measurements must be made, and the use of specialized measurements and parameters that could lead to singular matrices. It must also be noted that [20] was not evaluated in a recognized environment such as an \(IEEE\) Test Feeder. In [21], a combination of measurements and simulations are used to locate faults. As mentioned before, the need to simulate parameters leads to solutions that are not general and can be difficult to implement. In [22], measurements from multiple sections of a line are utilized to derive a model based on symmetrical components with emphasis on line capacitance to detect and locate faults. The results were interesting; however, it was noted that the performance of the algorithm is affected by the length of the line, and the solution was not tested with resistances beyond 100 \(\Omega\). Moreover, this solution appears to be geared towards systems with underground cables. This review of the literature has lead to the following conclusions:  
1. Most acceptable solutions are based on the Apparent Impedance framework. 2. The Time Domain vs. Frequency Domain debate is still open. 3. Most solutions focus on modeling and identifying specific aspects of HIFs. 4. Most solutions suffer from a lack of generality, meaning that manual adjustments have to be made each time the solution is deployed. In view of the limitations of existing solutions and the current state of smart grid technology, this work presents a unified, dynamic hybrid data-driven and analytical-based model, which advances the state-of-the-art of HIFs detection in the following ways:  
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state space simplifies the process of generalization as all systems of the same type and order will have the same State Space structure. Many techniques have been developed for dynamic state state estimation, most of them are based on the Kalman Filter [23; 24; 25]. Subspace estimation, used in this work, is now introduced.