Measurement equipment located in electrical distribution grids, as power-quality measurement devices, substation meters, or customer smart meters do not provide phasor measurements due to the lack of high resolution time synchronization. Instead such measurement devices only measure magnitudes of voltages and currents and the local phase angle between these. In addition, these measurements are subject to measurement errors of up to few percent of the measurand. In order to utilize measurements for grid monitoring, this paper presents and assesses a stochastic grid calculation approach that derives confidence regions for the resulting current and voltage phasors. Two different metering models are introduced: a PMU model, which is used to validate theoretical properties of the estimator, and an Electric Meter model for which a Gaussian approximation is introduced. The estimator results are compared for the two metering models and case study results for a real Danish distribution grid are presented.

Anomaly detection in measurements within electrical distribution grids is essential for maintaining accurate state estimation and ensuring efficient grid management. This paper presents a novel anomaly detection framework that extends a linear state estimation model with statistical F-tests to identify deviations in distributed voltage or current phasor measurements. Assuming knowledge of the grid topology and correct phasor measurements, but subject to zero-mean complex Gaussian noise with known standard deviations, the anomaly detection approach checks whether deviations in measured data exceed normal fluctuations. The proposed algorithm flags potentially anomalous phasor measurements when the corresponding p-value from an F-test falls below a predefined significance level. Extensive simulations on a medium-voltage grid demonstrate the framework's ability to detect subtle magnitude and angle anomalies across a wide range of operating scenarios, maintaining robust performance even under reduced measurement coverage. The approach is scalable, interpretable, and well-suited for deployment in resource-constrained monitoring and data quality assurance within electrical distribution grids. Code available at https://github.com/DauselJensen/F-testcode

There is a lack of point process models on linear networks. For an arbitrary linear network, we consider new models for a Cox process with an isotropic pair correlation function obtained in various ways by transforming an isotropic Gaussian process which is used for driving the random intensity function of the Cox process. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

The electrical state of the power distribution grid can be estimated based on noisy measurements of a subset of voltages and currents in the grid. The accuracy and cost of such state estimation depends strongly on the selection of measurands. This paper addresses the selection of measurands, i.e. choice of which voltage and current phasors to measure, for the purpose of grid state estimation. The investigation is based on calculation of so-called leverages known from mathematical statistics. Based on the leverages a greedy measurand selection algorithm is prosed. The performance of the algorithm is studied by simulation in a small and a larger example grid.

We develop parametric classes of covariance functions on linear networks and their extension to graphs with Euclidean edges, i.e., graphs with edges viewed as line segments or more general sets with a coordinate system allowing us to consider points on the graph which are vertices or points on an edge. Our covariance functions are defined on the vertices and edge points of these graphs and are isotropic in the sense that they depend only on the geodesic distance or on a new metric called the resistance metric (which extends the classical resistance metric developed in electrical network theory on the vertices of a graph to the continuum of edge points). We discuss the advantages of using the resistance metric in comparison with the geodesic metric as well as the restrictions these metrics impose on the investigated covariance functions. In particular, many of the commonly used isotropic covariance functions in the spatial statistics literature (the power exponential, Mat{e}rn, generalized Cauchy, and Dagum classes) are shown to be valid with respect to the resistance metric for any graph with Euclidean edges, whilst they are only valid with respect to the geodesic metric in more special cases.

In this paper we consider point processes specified on directed linear networks, i.e. linear networks with associated directions. We adapt the so-called conditional intensity function used for specifying point processes on the time line to the setting of directed linear networks. For models specified by such a conditional intensity function, we derive an explicit expression for the likelihood function, specify two simulation algorithms (the inverse method and Ogata's modified thinning algorithm), and consider methods for model checking through the use of residuals. We also extend the results and methods to the case of a marked point process on a directed linear network. Furthermore, we consider specific classes of point process models on directed linear networks (Poisson processes, Hawkes processes, non-linear Hawkes processes, self-correcting processes, and marked Hawkes processes), all adapted from well-known models in the temporal setting. Finally, we apply the results and methods to analyse simulated and neurological data.

The purpose of the present study is to introduce a point process model for characterizing the pattern of atrial fibrillation (AF) episodes. A variant of the bivariate Hawkes process is proposed, accounting for clustered episodes. The model parameters are inferred by the maximum likelihood method. The goodness-of-fit analysis show that model fits the data in most of the recordings (27 out of 32). The information provided by this approach is complementary to AF burden.

Objective: The present study proposes a model-based, statistical approach to characterizing episode patterns in paroxysmal atrial fibrillation (AF). Thanks to the rapid advancement of noninvasive monitoring technology, the proposed approach should become increasingly relevant in clinical practice. Methods: History-dependent point process modeling is employed to characterize AF episode patterns, using a novel alternating, bivariate Hawkes self-exciting model. In addition, a modified version of a recently proposed statistical model to simulate AF progression throughout a lifetime is considered, involving non-Markovian rhythm switching and survival functions. For each model, the maximum likelihood estimator is derived and used to find the model parameters from observed data. Results: Using three databases with a total of 59 long-term ECG recordings, the goodness-of-fit analysis demonstrates that the proposed alternating, bivariate Hawkes model fits SR-to-AF transitions in 40 recordings and AF-to-SR transitions in 51; the corresponding numbers for the AF model with non-Markovian rhythm switching are 40 and 11, respectively. Moreover, the results indicate that the model parameters related to AF episode clustering, i.e., aggregation of temporal AF episodes, provide information complementary to the well-known clinical parameter AF burden. Conclusion: Point process modeling provides a detailed characterization of the occurrence pattern of AF episodes that may improve the understanding of arrhythmia progression.

Knowledge of currents in individual Low Voltage feeders of a secondary substation is interesting for distribution system operators for a variety of purposes. Deploying measurement devices at each feeder in each substation, however, can be costly. Due to the increasing deployment of Smart Meters, the knowledge about currents at each connected customer is in principle available. This paper proposes and evaluates an approach to determine the feeder currents taking into account the impact of measurement errors of Smart Meter measurements. The developed approach makes a rigorous derivation of confidence intervals for the calculated voltage and current values utilizing a subset of measured voltages and currents as input. The approach is applied to two realistic low voltage grids and the impact of measurement errors and missing smart meter measurements is quantitatively analyzed.

In recent decades, Norway spruce (Picea abies L. Karst.) forests of the High Tatra Mountains have suffered unprecedented tree mortality caused by European spruce bark beetle (Ips typographus L.). Analysis of the spatiotemporal pattern of bark beetle outbreaks across the landscape in consecutive years can provide new insights into the population dynamics of tree-killing insects. A bark beetle outbreak occurred in the High Tatra Mountains after a storm damaged more than 10,000 ha of forests in 2004. We combined yearly Landsat-derived bark beetle infestation spots from 2006 to 2014 and meteorological data to identify the susceptibility of forest stands to beetle infestation. We found that digital elevation model (DEM)-derived potential radiation loads predicted beetle infestation, especially in the peak phase of beetle epidemic. Moreover, spots attacked at the beginning of our study period had higher values of received solar radiation than spots at the end of the study period, indicating that bark beetles prefer sites with higher insolation during outbreak. We conclude that solar radiation, easily determined from the DEM, better identified beetle infestations than commonly used meteorological variables. We recommend including potential solar radiation in beetle infestation prediction models.

Methods for unsupervised clustering is an important part of the statistical toolbox in numerous scientific disciplines. Tewari, Giering, and Raghunathan (2011) proposed to use so-called Gaussian Mixture Copula Models (GMCM) for general unsupervised clustering. Li, Brown, Huang, and Bickel (2011) independently discussed a special case of these GMCMs as a novel approach to meta-analysis in high-dimensional settings. GMCMs have attractive properties which make them highly flexible and therefore interesting alternatives to well-established methods. However, parameter estimation is hard because of intrinsic identifiability issues and intractable likelihood functions. Both aforementioned papers discuss similar expectation-maximization-like (EM) algorithms as their pseudo maximum likelihood estimation procedure. We present and discuss an improved implementation in R of both classes of GMCMs along with various alternative optimization routines to the EM algorithm. The software is freely available through the accompanying R package GMCM. The implementation is fast, general, and optimized for very large numbers of observations. We demonstrate the use of GMCM through different applications.

Power or energy losses are an important metric to describe efficiency of distribution grids. They can also be relevant input metrics for fault and theft detection approaches. When considering low-voltage grids, power losses have to be obtained from distributed low-cost measurement devices, which leads to measurement inaccuracies. When calculating power losses, these measurement inaccuracies need to be taken into account. This paper presents first steps for obtaining unbiased estimators and confidence intervals of loss calculation taking into account statistical errors on input measurands. Smart meter measurements from a real low-voltage grid are used to show the validity of approximations that are useful for efficient and effective confidence interval calculation.

These short lecture notes contain a not too technical introduction to point processes on the time line. The focus lies on defining these processes using the conditional intensity function. Furthermore, likelihood inference, methods of simulation and residual analysis for temporal point processes specified by a conditional intensity function are considered.

Control of assets in electricity distribution grids is becoming an increasingly interesting solution to address challenges arising in the grid with an increasing penetration of distributed renewable energy generation. This paper looks at a voltage control scenario in low voltage (LV) distribution grids, which targets a reduction of over- and under voltage level situations by adjusting reactive power production of selected low voltage grid assets. The paper models different information access schemes between remote assets and controller, which is activated only when certain voltage thresholds have been crossed at any measurement point in the grid. We assess the information access methods on information reliability and how this affects control performance. We focus on two different information quality metrics; (1) information age and (2) mismatch probability, which are expressed via stochastic models. We investigate in this paper the suitability for using these two metrics for optimization in a voltage grid control scenario. We conclude that, while the mismatch probability is very useful compared to the simpler information age metric from a network designers and operators point of view in setting quality of service requirements, it is not as helpful for control engineers.

Networked control is challenged by stochastic delays that are caused by the communication networks as well as by the approach taken to exchange information about system state and set-points. Combined with stochastic changing information, there is a probability that information at the controller is not matching the true system observation, which we call mismatch probability (mmPr). The hypothesis is that the optimization of certain parameters of networked control systems targeting mmPr is equivalent to the optimization targeting control performance, while the former is practically much easier to conduct. This is first analyzed in simulation models for the example system of a wind-farm controller. As simulation analysis is subject to stochastic variability and requires large computational effort, the paper develops a Markov model of a simplified networked control system and uses numerical results from the Markov model analysis to demonstrate that mmPr based optimization can improve control performance.

Analyzing point patterns with linear structures has recently been of interest in e.g. neuroscience and geography. To detect anisotropy in such cases, we introduce a functional summary statistic, called the cylindrical $K$-function, since it is a directional $K$-function whose structuring element is a cylinder. Further we introduce a class of anisotropic Cox point processes, called Poisson line cluster point processes. The points of such a process are random displacements of Poisson point processes defined on the lines of a Poisson line process. Parameter estimation based on moment methods or Bayesian inference for this model is discussed when the underlying Poisson line process and the cluster memberships are treated as hidden processes. To illustrate the methodologies, we analyze a two and a three-dimensional point pattern data set. The 3D data set is of particular interest as it relates to the minicolumn hypothesis in neuroscience, claiming that pyramidal and other brain cells have a columnar arrangement perpendicular to the pial surface of the brain.

Methods for clustering in unsupervised learning are an important part of the statistical toolbox in numerous scientific disciplines. Tewari, Giering, and Raghunathan (2011) proposed to use so-called Gaussian mixture copula models (GMCM) for general unsupervised learning based on clustering. Li, Brown, Huang, and Bickel (2011) independently discussed a special case of these GMCMs as a novel approach to meta-analysis in highdimensional settings. GMCMs have attractive properties which make them highly flexible and therefore interesting alternatives to other well-established methods. However, parameter estimation is hard because of intrinsic identifiability issues and intractable likelihood functions. Both aforementioned papers discuss similar expectation-maximization-like algorithms as their pseudo maximum likelihood estimation procedure. We present and discuss an improved implementation in R of both classes of GMCMs along with various alternative optimization routines to the EM algorithm. The software is freely available in the R package GMCM. The implementation is fast, general, and optimized for very large numbers of observations. We demonstrate the use of package GMCM through different applications.

We consider the ability of a mobile sensor to locate its own geographical location, the so-called self-localization prob- lem. The need to locate people and objects has inspired the development of many systems for automatic localiza- tion. Most systems are based on location information and measured radio propagation characteristics for received sig- nals from sensors in the proximity of the mobile sensor. It is of fundamental importance that such systems also works in critical situations such as loss of observability or the presence of multipath. The present paper suggest a framework to as- sess the performance of localization algorithms in mobile and critical situations. This is done by exploring the per- formance of various filtering techniques for self-localization of a mobile sensor in a field of sensors. More specifically, we model the mobility of the sensor such that the veloc- ity varies according to an autoregressive model. Measure- ment uncertainty is assumed to follow a Gaussian distribu- tion and the probability for detecting a distance to a given sensor is assumed to fall off exponentially with squared dis- tance. The combined model is formulated as a nonlinear state space model and Bayesian inference is performed with the extended Kalman filter (EKF) and a particle filtering method. Precision of the position estimate is evaluated by the root mean square error (RMSE). A lower bound on the RMSE of the estimate is derived, thus providing important information on the best achievable precision for any algo- rithm. We report a number of simulation experiments which validate our proposed algorithms and theoretical results. We conclude that the performance of the EKF and particle fil- tering methods are comparable and that the derived lower bound is a useful lower limit on the RMSE.