arXiv subjects

Matthias Wächter

Publications and source records attributed to Matthias Wächter.

At least 19 recordsLinked to original sources

arXiv2024

Extension of the Langevin power curve analysis by separation per operational state

In the last few years, the dynamical characterization of the power output of a wind turbine by means of a Langevin equation has been well established. For this approach, temporally highly resolved measurements of wind speed and power output are used to obtain the drift and diffusion coefficients of the energy conversion process. These coefficients fully determine a Langevin stochastic differential equation with Gaussian white noise. The drift term specifies the deterministic behavior of the system whereas the diffusion term describes the stochastic behavior of the system. A precise estimation of these coefficients is essential to understand the dynamics of the power conversion process of the wind turbine. We show that the dynamics of the power output of a wind turbine have a hidden dependency on turbine's different operational states. Here, we use an approach based on clustering Pearson correlation matrices for different observables on a moving time window to identify different operational states. We have identified five operational states in total, for example the state of rated power. Those different operational states distinguish non-stationary behavior in the mutual dependencies and represent different turbine control settings. As a next step, we condition our Langevin analysis on these different states to reveal distinctly different behaviors of the power conversion process for each operational state. Moreover, in our new representation hysteresis effects which have typically appeared in the Langevin dynamics of wind turbines seem to be resolved. We assign these typically observed hysteresis effects clearly to the change of the wind energy system between our estimated different operational states. In this contribution, we discuss further consequences for the meaning of hysteric switching and detection of malbehaviors in wind turbines.

physics.data-an↗

arXiv2024

Local statistical moments to capture Kramers-Moyal coefficients

This study introduces an innovative local statistical moment approach for estimating Kramers-Moyal coefficients, effectively bridging the gap between nonparametric and parametric methodologies. These coefficients play a crucial role in characterizing stochastic processes. Our proposed approach provides a versatile framework for localized coefficient estimation, combining the flexibility of nonparametric methods with the interpretability of global parametric approaches. We showcase the efficacy of our approach through use cases involving both stationary and non-stationary time series analysis. Additionally, we demonstrate its applicability to real-world complex systems, specifically in the energy conversion process analysis of a wind turbine.

stat.ME↗

arXiv2024

Dynamics of wind turbine operational states

Modern wind turbines gather a wealth of data with Supervisory Control And Data Acquisition (SCADA) systems. We study the short-term mutual dependencies of a variety of observables by evaluating Pearson correlation matrices on a moving time window. Using clustering on these matrices, we identify multiple stable operational states, which characterize the non-stationarity of mutual dependencies at a single turbine. They represent different turbine operational settings. Moreover, we combine the clustering analysis with a construction of a stochastic process to study the switching dynamics of those states in more detail. Calculating the distances between correlation matrices we obtain a time series that describes the behavior of the complex system in a collective way. Assuming this time series to be governed by a Langevin equation, we estimate the deterministic (drift) and stochastic (diffusion) components of the dynamics to understand the underlying non-stationarity. After adapting our method to specific features of our data, we are able to study the dynamics of operational states and their transitions as well as to resolve hysteresis effects.

physics.flu-dyn↗

arXiv2023

Assessing the Suitability of the Langevin Equation for Analyzing Measured Data Through Downsampling

The measured time series from complex systems are renowned for their intricate stochastic behavior, characterized by random fluctuations stemming from external influences and nonlinear interactions. These fluctuations take diverse forms, ranging from continuous trajectories reminiscent of Brownian motion to noncontinuous trajectories featuring jump events. The Langevin equation serves as a powerful tool for generating stochasticity and capturing the complex behavior of measured data with continuous stochastic characteristics. However, the traditional modeling framework of the Langevin equation falls short when it comes to capturing the presence of abrupt changes, particularly jumps, in trajectories that exhibit non-continuity. Such non-continuous changes pose a significant challenge for general processes and have profound implications for risk management. Moreover, the discrete nature of observed physical phenomena, measured with a finite sample rate, adds another layer of complexity. In such cases, data points often appear as a series of discontinuous jumps, even when the underlying trajectory is continuous. In this study, we present an analytical framework that goes beyond the limitations of the Langevin equation. Our approach effectively distinguishes between diffusive or Brownian-type trajectories and trajectories with jumps. By employing downsampling techniques, where we artificially lower the sample rate, we derive a set of measures and criteria to analyze the data and differentiate between diffusive and non-diffusive behaviors. To further demonstrate its versatility and practical applicability, we have applied our proposed method to real-world data in various scientific fields, turbulence, optical tweezers for trapped particles, neuroscience, renewable energy, and market price analysis.

cond-mat.stat-mech↗

arXiv2022

An open source MATLAB package to perform basic and advanced statistical analysis of turbulence data and other complex systems

We present a user-friendly open-source MATLAB\textsuperscript{\textregistered} package developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg. Firstly, this package helps the user to perform a very basic statistical analysis of a given turbulent data set which we believe to be useful to the entire turbulence community. It can be used to estimate the statistical quantities of turbulence such as the spectrum density, turbulent intensity, integral length scale, Taylor microscale, Kolmogorov scale and dissipation rate. Different well-known methods available in the literature were selected so that they can be compared. Secondly, this package also performs an advanced analysis which includes the scale-dependent statistical description of turbulent cascade using the Fokker-Planck equation which consequently leads to the assessment of integral fluctuation theorem. This is utilized to estimate velocity increments, structure functions and their scaling exponents, drift and diffusion coefficients of the Fokker-Planck equation and consequently the total entropy production of the turbulent cascade. As a precondition for the stochastic process approach, Markovian properties of the turbulent cascade in scale are tested. The knowledge of a Fokker-Planck equation allows to determine for each independent cascade trajectories a total entropy production. The estimation of total entropy production allows to verify a rigorous law of non-equilibrium stochastic thermodynamics, namely the integral fluctuation theorem, which must be valid if Markov properties hold and the Fokker-Planck equation is correct. This approach to the turbulent cascade process has the potential for a new way to link the statistical description of turbulence, non-equilibrium stochastic thermodynamics and local turbulent flow structures.

physics.flu-dyn↗

arXiv2022

Instantons and the path to intermittency in turbulent flows

Processes leading to anomalous fluctuations in turbulent flows, referred to as intermittency, are still challenging. We consider cascade trajectories through scales as realizations of a stochastic Langevin process for which multiplicative noise is an intrinsic feature of the turbulent state. The trajectories are conditioned on their entropy exchange. Such selected trajectories concentrate around an optimal path, called instanton, which is the minimum of an effective action. The action is derived from the Langevin equation, estimated from measured data. In particular instantons with negative entropy pinpoint the trajectories responsible for the emergence of non-Gaussian statistics at small-scales.

physics.flu-dyn↗

arXiv2018

The Fokker-Planck Approach to Complex Spatio-Temporal Disordered Systems

When the complete understanding of a complex system is not available, as, e.g., for systems considered in the real-world, we need a top-down approach to complexity. In this approach one may start with the desire to understand general multi-point statistics. Here such a general approach is presented and discussed based on examples from turbulence and sea waves. Our main idea is based on the cascade picture of turbulence, entangling fluctuations from large to small scales. Inspired by this cascade picture, we express the general multi-point statistics by the statistics of scale-dependent fluctuations of variables and relate it to a scale-dependent process, which finally is a stochastic cascade process. We show how to extract from empirical data a Fokker-Planck equation for this cascade process, which allows to generate surrogate data to forecast extreme events as well as to develop a non-equilibrium thermodynamics for the complex systems. For each cascade events an entropy production can be determined. These entropies fulfil accurately a rigorous law, namely the integral fluctuations theorem.

cond-mat.stat-mech↗

arXiv2017

Normal behaviour models for wind turbine vibrations: An alternative approach

The identification of abnormal behaviour in mechanical systems is key to anticipate and avoid their potential failure. Thus wind turbine health is commonly assessed monitoring series of $10$-minute SCADA and high frequency data from sensors. To monitor wind turbine vibrations, normal behaviour models are built to predict tower top accelerations and drive-train vibrations. Signal deviations from model prediction are labelled as anomalies and are further investigated. More efficient models are expected to help enhancing the identification of abnormal behaviour. In this paper we assess a stochastic approach to reconstruct the $1$ Hz tower top acceleration signal, which was measured in a wind turbine located at the wind farm Alpha Ventus in the German North Sea. We compare the resulting data reconstruction with that of a model based on a neural network, which has been previously reported as a data-mining algorithm suitable for reconstructing this signal. In order to focus the discussion on the similarities and differences of both approaches, we limit our evaluation to a single input-output system; in doing so, we avoid other differences, i.e. loading type, pre-processing or model complexity. Our results present evidence that the stochastic approach outperforms the neural network in the high frequency domain ($1$ Hz). Although neural network retrieve accurate step-forward predictions, with low mean square errors, the stochastic approach predictions better preserve the statistics and the frequency components of the original signal, remaining high accuracy levels. The implementation of our stochastic approach is available as open source code and can easily be adapted for other situations involving stochastic data reconstruction. Based on our findings we argue that such an approach could be implemented in signal reconstruction for monitoring purposes or for abnormal behaviour detection.

physics.data-an↗

arXiv2016

The Langevin Approach: An R Package for Modeling Markov Processes

We describe an R package developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg, which extracts the (stochastic) evolution equation underlying a set of data or measurements. The method can be directly applied to data sets with one or two stochastic variables. Examples for the one-dimensional and two-dimensional cases are provided. This framework is valid under a small set of conditions which are explicitly presented and which imply simple preliminary test procedures to the data. For Markovian processes involving Gaussian white noise, a stochastic differential equation is derived straightforwardly from the time series and captures the full dynamical properties of the underlying process. Still, even in the case such conditions are not fulfilled, there are alternative versions of this method which we discuss briefly and provide the user with the necessary bibliography.

physics.data-an↗

arXiv2015

Capturing rogue waves by multi-point statistics

As an example for complex systems with extreme events we investigate ocean wave states exhibiting rogue waves. We present a statistical method of data analysis based on multi-point statistics which for the first time allows grasping extreme rogue wave events in a statistically highly satisfactory manner. The key to the success of the approach is mapping the complexity of multi-point data onto the statistics of hierarchically ordered height increments for different time scales for which we can show that a stochastic cascade process with Markov properties is governed by a Fokker-Planck equation. Conditional probabilities as well as the Fokker-Planck equation itself can be estimated directly from the available observational data. With this stochastic description surrogate data sets can in turn be generated allowing to work out arbitrary statistical features of the complex sea state in general and extreme rogue wave events in particular. The results also open up new perspectives for forecasting the occurrence probability of extreme rogue wave events, and even for forecasting the occurrence of individual rogue waves based on precursory dynamics.

physics.data-an↗

arXiv2015

Langevin power curve analysis for numerical WEC models with new insights on high frequency power performance

Based on the Langevin equation it has been proposed to obtain power curves for wind turbines from high frequency data of wind speed measurements u(t) and power output P (t). The two parts of the Langevin approach, power curve and drift field, give a comprehensive description of the conversion dynamic over the whole operating range of the wind turbine. The method deals with high frequent data instead of 10 min means. It is therefore possible to gain a reliable power curve already from a small amount of data per wind speed. Furthermore, the method is able to visualize multiple fixed points, which is e.g. characteristic for the transition from partial to full load or in case the conversion process deviates from the standard procedures. In order to gain a deeper knowledge it is essential that the method works not only for measured data but also for numerical wind turbine models and synthetic wind fields. Here, we characterize the dynamics of a detailed numerical wind turbine model and calculate the Langevin power curve for different data samplings. We show, how to get reliable results from synthetic data and verify the applicability of the method for field measurements with ultra-sonic, cup and Lidar measurements. The independence of the fixed points on site specific turbulence effects is also confirmed with the numerical model. Furthermore, we demonstrate the potential of the Langevin approach to detect failures in the conversion process and thus show the potential of the Langevin approach for a condition monitoring system.

physics.data-an↗

arXiv2015

Parameter-free resolution of the superposition of stochastic signals

This paper presents a direct method to obtain the deterministic and stochastic contribution of the sum of two independent sets of stochastic processes, one of which is composed by Ornstein-Uhlenbeck processes and the other being a general (non-linear) Langevin process. The method is able to distinguish between all stochastic process, retrieving their corresponding stochastic evolution equations. This framework is based on a recent approach for the analysis of multidimensional Langevin-type stochastic processes in the presence of strong measurement (or observational) noise, which is here extended to impose neither constraints nor parameters and extract all coefficients directly from the empirical data sets. Using synthetic data, it is shown that the method yields satisfactory results.

physics.data-an↗

arXiv2015

Stochastic modeling and performance monitoring of wind farm power production

We present a new stochastic approach to describe and remodel the conversion process of a wind farm at a sampling frequency of 1Hz. When conditioning on various wind direction sectors, the dynamics of the conversion process appear as a fluctuating trajectory around an average IEC-like power curve, see section II. Our approach is to consider the wind farm as a dynamical system that can be described as a stochastic drift/diffusion model, where a drift coefficient describes the attraction towards the power curve and a diffusion coefficient quantifies additional turbulent fluctuations. These stochastic coefficients are inserted into a Langevin equation that, once properly adapted to our particular system, models a synthetic signal of power output for any given wind speed/direction signals, see section III. When combined with a pre-model for turbulent wind fluctuations, the stochastic approach models the power output of the wind farm at a sampling frequency of 1Hz using only ten-minute average values of wind speed and directions. The stochastic signals generated are compared to the measured signal, and show a good statistical agreement, including a proper reproduction of the intermittent, gusty features measured. In parallel, a second application for performance monitoring is introduced in section IV. The drift coefficient can be used as a sensitive measure of the global wind farm performance. When monitoring the wind farm as a whole, the drift coefficient registers some significant deviation from normal operation if one of twelve wind turbines is shut down during less than 4% of the time. Also, intermittent anomalies can be detected more rapidly than when using ten-minute averaging methods. Finally, a probabilistic description of the conversion process is proposed and modeled in appendix A, that can in turn be used to further improve the estimation of the stochastic coefficients.

physics.data-an↗

arXiv2015

Stochastic model for aerodynamic force dynamics on wind turbine blades in unsteady wind inflow

The paper presents a stochastic approach to estimate the aerodynamic forces with local dynamics on wind turbine blades in unsteady wind inflow. This is done by integrating a stochastic model of lift and drag dynamics for an airfoil into the aerodynamic simulation software AeroDyn. The model is added as an alternative to the static table lookup approach in blade element momentum (BEM) wake model used by AeroDyn. The stochastic forces are obtained for a rotor blade element using full field turbulence simulated wind data input and compared with the classical BEM and dynamic stall models for identical conditions. The comparison shows that the stochastic model generates additional extended dynamic response in terms of local force fluctuations. Further, the comparison of statistics between the classical BEM, dynamic stall and stochastic models' results in terms of their increment probability density functions gives consistent results.

physics.flu-dyn↗

arXiv2015

Towards a Simplified Dynamic Wake Model using POD Analysis

We apply the proper orthogonal decomposition (POD) to large eddy simulation data of a wind turbine wake in a turbulent atmospheric boundary layer. The turbine is modeled as an actuator disk. Our analyis mainly focuses on the question whether POD could be a useful tool to develop a simplified dynamic wake model. The extracted POD modes are used to obtain approximate descriptions of the velocity field. To assess the quality of these POD reconstructions, we define simple measures which are believed to be relevant for a sequential turbine in the wake such as the energy flux through a disk in the wake. It is shown that only a few modes are necessary to capture basic dynamical aspects of these measures even though only a small part of the turbulent kinetic energy is restored. Furthermore, we show that the importance of the individual modes depends on the measure chosen. Therefore, the optimal choice of modes for a possible model could in principle depend on the application of interest. We additionally present a possible interpretation of the POD modes relating them to specific properties of the wake. For example the first mode is related to the horizontal large scale movement. Besides yielding a deeper understanding, this also enables us to view our results in comparison to existing dynamic wake models.

physics.flu-dyn↗

arXiv2015

The Langevin Approach: a simple stochastic method for complex phenomena

We describe a simple stochastic method, so-called Langevin approach, which enables one to extract evolution equations of stochastic variables from a set of measurements. Our method is parameter-free and it is based on the nonlinear Langevin equation. Moreover, it can be applied not only to processes in time, but also to processes in scale, given that the data available shows ergodicity. This chapter introduces the mathematical foundations of the Langevin approach and describes how to implement it numerically. A specific application of the method is presented, namely to a turbulent velocity field measured in the laboratory, retrieving the corresponding energy cascade and comparing with the results from a computational simulation of that experiment. In addition, we describe a physical interpretation bridging between processes in time and in scale. Finally, we describe extensions of the method for time series reconstruction and applications to other fields such as finance, medicine, geophysics and renewable energies.

physics.data-an↗

arXiv2014

Fatigue Loads Estimation Through a Simple Stochastic Model

We propose a procedure to estimate the fatigue loads on wind turbines, based in a recent framework used for reconstructing data series of stochastic properties measured at wind turbines. Through a standard fatigue analysis, we show that it is possible to accurately estimate fatigue loads in any wind turbine within one wind park, using only the load measurements at one single turbine and the set of wind speed measurements. Our framework consists of deriving a stochastic differential equation that describes the evolution of the torque at one wind turbine driven by the wind speed. The stochastic equation is derived directly from the measurements and is afterwards used for predicting the fatigue loads at neighboring turbines. Such a framework could be used to mitigate the financial efforts usually necessary for placing measurement devices in all wind turbines within one wind farm. Finally, we also discuss the limitations and possible improvements of the proposed procedure.

physics.data-an↗

arXiv2014

Reconstructing the intermittent dynamics of the torque in wind turbines

We apply a framework introduced in the late nineties to analyze load measurements in off-shore wind energy converters (WEC). The framework is borrowed from statistical physics and properly adapted to the analysis of multivariate data comprising wind velocity, power production and torque measurements, taken at one single WEC. In particular, we assume that wind statistics drives the fluctuations of the torque produced in the wind turbine and show how to extract an evolution equation of the Langevin type for the torque driven by the wind velocity. It is known that the intermittent nature of the atmosphere, i.e. of the wind field, is transferred to the power production of a wind energy converter and consequently to the shaft torque. We show that the derived stochastic differential equation quantifies the dynamical coupling of the measured fluctuating properties as well as it reproduces the intermittency observed in the data. Finally, we discuss our approach in the light of turbine monitoring, a particular important issue in off-shore wind farms.

physics.data-an↗