# WAMs Based Eigenvalue Space Model for High Impedance Fault Detection
## 1 Introduction  
High impedance faults in electric power systems (EPS) represent a liability in regards to safety and reliability. High impedance faults (HIFs) arise when a connection is made between an energized conductor and a surface with high resistance that is not part of the EPS. Common surfaces that can lead to HIFs include trees, asphalt, concrete, sand, and grass [1]. Significant damage to equipment, property, and even fatalities have been attributed to this phenomenon [2]. Despite these significant implications, the problem of HIFs remains unsolved [3]. One of the main challenges in this area is presented by the unique nature of HIFs [4]: The nonlinear relationship between currents and voltages, the presence of electrical arcs, and most importantly the low current magnitudes. These characteristics make it difficult to detect HIF by the traditional means of over-current protection. A protection device on a heavily loaded feeder will most likely not operate in the presence of HIFs. On the other hand, choosing to set a protective device too close to the expected loading level can lead to nuisance tripping. To compound the problem even further, the dynamic behavior of both the fault and the load must be taken into account. As a result of the clear limitations of over-current protection in the detection of HIFs, efforts have been made to find alternate solutions to this problem. Several techniques have been proposed in recent years, but a unified and general solution is yet to be accepted by the industry [5]. Most techniques can be divided into the following groups: knowledge-based techniques, network topology based techniques, and apparent impedance-based techniques. Knowledge-based techniques leverage data acquired by meters and utilize machine learning (ML) approaches to identify outliers [6]. One of the drawbacks of these techniques is the reliance on meters, and the infrastructure necessary to process large amounts of data. Methods based on network topology are somewhat similar to knowledge-based solutions in that they rely on metering [7]. Meter sensors in key areas are deployed to create a network capable of detecting HIFs. This approach requires a significant investment in terms of equipment, installation and maintenance of meters. Moreover, this type of approach suffers from a lack of generality, which means that significant engineering is needed in real-life applications. Transmission protection techniques have inspired researchers to develop solutions for HIFs based on methods such as Apparent Impedance. These techniques are appealing due to the relatively low investment cost and the familiarity of protection engineers with the underlying principle of impedance (distance) relaying. A significant obstacle these techniques face is the stochastic behavior of HIFs. Several of these solutions make the assumption that the impedance at the fault point is purely resistive, which would lead to a fault signature that is fairly linear and constant [8]. These techniques were advanced by taking into account the non-linear behavior of an electrical arc. These models face two challenges: Being able to detect the change in impedance, and doing so in small time windows. Still, despite having multiple unresolved issues, impedance-based techniques have gone on to become accepted as the framework for the development of new HIF detection and location solutions [9]. Solutions based on state estimators, such Weighted Least Squares (WLS) [10; 11], have been proposed more recently. These solutions aim to detect HIFs by identifying statistical outliers during the estimation process. Estimation-based methods such as [12] have been proposed in the time domain, as well as in the frequency domain as presented in [8; 13]. In regards to state estimators for HIFs, frequency domain techniques appear to have greater momentum compared to time domain approaches due to reported advantages such as noise suppression and relative ease of implementation.