Stochastic timeseries analysis in electric power systems and paleo-climate data

In this thesis a data science study of elementary stochastic processes is laid, aided with the development of two numerical software programmes, applied to power-grid frequency studies and Dansgaard--Oeschger events in paleo-climate data. Power-grid frequency is a key measure in power grid studies. It comprises the balance of power in a power grid at any instance. In this thesis an elementary Markovian Langevin-like stochastic process is employed, extending from existent literature, to show the basic elements of power-grid frequency dynamics can be modelled in such manner. Through a data science study of power-grid frequency data, it is shown that fluctuations scale in an inverse square-root relation with their size, alike any other stochastic process, confirming previous theoretical results. A simple Ornstein--Uhlenbeck is offered as a surrogate model for power-grid frequency dynamics, with a versatile input of driving deterministic functions, showing not surprisingly that driven stochastic processes with Gaussian noise do not necessarily show a Gaussian distribution. A study of the correlations between recordings of power-grid frequency in the same power-grid system reveals they are correlated, but a theoretical understanding is yet to be developed. A super-diffusive relaxation of amplitude synchronisation is shown to exist in space in coupled power-grid systems, whereas a linear relation is evidenced for the emergence of phase synchronisation. Two Python software packages are designed, offering the possibility to extract conditional moments for Markovian stochastic processes of any dimension, with a particular application for Markovian jump-diffusion processes for one-dimensional timeseries. Lastly, a study of Dansgaard--Oeschger events in recordings of paleoclimate data under the purview of bivariate Markovian jump-diffusion processes is proposed, augmented by a semi-theoretical study of bivariate stochastic processes, offering an explanation for the discontinuous transitions in these events and showing the existence of deterministic couplings between the recordings of the dust concentration and a proxy for the atmospheric temperature

The stochastic nature of power-grid frequency in South Africa

In this work, we explore two mechanisms that explain non-Gaussian behaviour of power-grid frequency recordings in the South African grid. We make use of a Fokker–Planck approach to power-grid frequency that yields a direct relation between common model parameters such as inertia, damping, and noise amplitude and non-parametric estimations of the same directly from power-grid frequency recordings. We propose two explanations for the non-Gaussian leptokurtic distributions in South Africa: the first based on multiplicative noise in power-grid frequency recordings, which we observe in South Africa; the second based on the well-known scheduled and unscheduled load shedding and rolling blackouts that beset South Africa. For the first we derive an analytic expression of the effects of multiplicative noise that permits the estimation of all statistical moments—and discuss drawbacks in comparison with the data; for the second we employ a simple numerical analysis with a modular power grid of South Africa. Both options help understand the statistics of power-grid frequency in South Africa—particularly the presence of heavy tails.The stochastic nature of power-grid frequency in South AfricapublishedVersio

Stability Bounds of Droop-Controlled Inverters in Power Grid Networks

The energy mix of future power systems will include high shares of electricity generation by wind turbines and solar photovoltaics. These generation facilities are generally connected via power-electronic inverters. While conventional generation responds dynamically to the state of the electric power system, inverters are power-electronic hardware and need to be programmed to react to the state of the system. Choosing an appropriate control scheme and the corresponding parameters is necessary to guarantee that the system operates safely. A prominent control scheme for inverters is droop control, which mimics the response of conventional generation. In this work, we investigate the stability of coupled systems of droop-controlled inverters in arbitrary network topologies. Employing linear stability analysis, we derive effective local stability criteria that consider both the overall network topology as well as its interplay with the inverters’ intrinsic parameters. First, we explore the stability of an inverter coupled to an infinite grid and uncover stability and instability regions. Second, we extend the analysis to a generic topology of inverters and provide mathematical criteria for the stability and instability of the system. Last, we showcase the usefulness of the criteria by examining two model systems using numerical simulations. The developed criteria show which parameters might lead to an unstable operating state.Stability Bounds of Droop-Controlled Inverters in Power Grid NetworkspublishedVersio

jumpdiff: A Python Library for Statistical Inference of Jump-Diffusion Processes in Observational or Experimental Data Sets

We introduce a Python library, called jumpdiff, which includes all necessary functions to assess jump-diffusion processes. This library includes functions which compute a set of non-parametric estimators of all contributions composing a jump-diffusion process, namely the drift, the diffusion, and the stochastic jump strengths. Having a set of measurements from a jump-diffusion process, jumpdiff is able to retrieve the evolution equation producing data series statistically equivalent to the series of measurements. The back-end calculations are based on second-order corrections of the conditional moments expressed from the series of Kramers-Moyal coefficients. Additionally, the library is also able to test if stochastic jump contributions are present in the dynamics underlying a set of measurements. Finally, we introduce a simple iterative method for deriving secondorder corrections of any Kramers-Moyal coefficient

Anticipating critical transitions in multi-dimensional systems driven by time- and state-dependent noise

The anticipation of bifurcation-induced transitions in dynamical systems has gained relevance in various fields of the natural, social, and economic sciences. When approaching a co-dimension 1 bifurcation, the feedbacks that stabilise the initial state weaken and eventually vanish; a process referred to as critical slowing down (CSD). This motivates the use of variance and lag-1 autocorrelation as indicators of CSD. Both indicators rely on linearising the system's restoring rate. Additionally, the use of variance is limited to time- and state-independent driving noise, strongly constraining the generality of CSD. Here, we propose a data-driven approach based on deriving a Langevin equation to detect local stability changes and anticipate bifurcation-induced transitions in systems with generally time- and state-dependent noise. Our approach substantially generalizes the conditions underlying existing early warning indicators, which we showcase in different examples. Changes in deterministic dynamics can be clearly discriminated from changes in the driving noise. This reduces the risk of false and missed alarms of conventional CSD indicators significantly in settings with time-dependent or multiplicative noise. In multi-dimensional systems, our method can greatly advance the understanding of the coupling between system components and can avoid risks of missing CSD due to dimension reduction, which existing approaches suffer from

Complexity and irreducibility of dynamics on networks of networks

We study numerically the dynamics of a network of all-to-all-coupled, identical sub-networks consisting of diffusively coupled, non-identical FitzHugh--Nagumo oscillators. For a large range of within- and between-network couplings, the network exhibits a variety of dynamical behaviors, previously described for single, uncoupled networks. We identify a region in parameter space in which the interplay of within- and between-network couplings allows for a richer dynamical behavior than can be observed for a single sub-network. Adjoining this atypical region, our network of networks exhibits transitions to multistability. We elucidate bifurcations governing the transitions between the various dynamics when crossing this region and discuss how varying the couplings affects the effective structure of our network of networks. Our findings indicate that reducing a network of networks to a single (but bigger) network might be not accurate enough to properly understand the complexity of its dynamics.Comment: 8 figure

Understanding electricity prices beyond the merit order principle using explainable AI

Electricity prices in liberalized markets are determined by the supply and demand for electric power, which are in turn driven by various external influences that vary strongly in time. In perfect competition, the merit order principle describes that dispatchable power plants enter the market in the order of their marginal costs to meet the residual load, i.e. the difference of load and renewable generation. Many market models implement this principle to predict electricity prices but typically require certain assumptions and simplifications. In this article, we present an explainable machine learning model for the prices on the German day-ahead market, which substantially outperforms a benchmark model based on the merit order principle. Our model is designed for the ex-post analysis of prices and thus builds on various external features. Using Shapley Additive exPlanation (SHAP) values, we can disentangle the role of the different features and quantify their importance from empiric data. Load, wind and solar generation are most important, as expected, but wind power appears to affect prices stronger than solar power does. Fuel prices also rank highly and show nontrivial dependencies, including strong interactions with other features revealed by a SHAP interaction analysis. Large generation ramps are correlated with high prices, again with strong feature interactions, due to the limited flexibility of nuclear and lignite plants. Our results further contribute to model development by providing quantitative insights directly from data.Comment: 13 pages, 6 figure

Stable stadial and interstadial states of the last glacial's climate identified in a combined stable water isotope and dust record from Greenland

During the last glacial interval, the Northern Hemisphere climate was punctuated by a series of abrupt changes between two characteristic climate regimes. The existence of stadial (cold) and interstadial (milder) periods is typically attributed to a hypothesised bistability in the glacial North Atlantic climate system, allowing for rapid transitions from the stadial to the interstadial state – the so-called Dansgaard–Oeschger (DO) events – and more gradual yet still fairly abrupt reverse shifts. The physical mechanisms driving these regime transitions remain debated. DO events are characterised by substantial warming over Greenland and a reorganisation of the Northern Hemisphere atmospheric circulation, which are evident from concomitant shifts in the δ18O ratios and dust concentration records from Greenland ice cores. Treating the combined δ18O and dust record obtained by the North Greenland Ice Core Project (NGRIP) as a realisation of a two-dimensional, time-homogeneous, and Markovian stochastic process, we present a reconstruction of its underlying deterministic drift based on the leading-order terms of the Kramers–Moyal equation. The analysis reveals two basins of attraction in the two-dimensional state space that can be identified with the stadial and interstadial regimes. The drift term of the dust exhibits a double-fold bifurcation structure, while – in contrast to prevailing assumptions – the δ18O component of the drift is clearly mono-stable. This suggests that the last glacial's Greenland temperatures should not be regarded as an intrinsically bistable climate variable. Instead, the two-regime nature of the δ18O record is apparently inherited from a coupling to another bistable climate process. In contrast, the bistability evidenced in the dust drift points to the presence of two stable circulation regimes of the last glacial's Northern Hemisphere atmosphere.Stable stadial and interstadial states of the last glacial's climate identified in a combined stable water isotope and dust record from GreenlandpublishedVersio

Non-linear, bivariate stochastic modelling of power-grid frequency applied to islands

Mitigating climate change requires a transition away from fossil fuels towards renewable energy. As a result, power generation becomes more volatile and options for microgrids and islanded power-grid operation are being broadly discussed. Therefore, studying the power grids of physical islands, as a model for islanded microgrids, is of particular interest when it comes to enhancing our understanding of power-grid stability. In the present paper, we investigate the statistical properties of the power-grid frequency of three island systems: Iceland, Ireland, and the Balearic Islands. We utilise a Fokker-Planck approach to construct stochastic differential equations that describe market activities, control, and noise acting on power-grid dynamics. Using the obtained parameters we create synthetic time series of the frequency dynamics. Our main contribution is to propose two extensions of stochastic power-grid frequency models and showcase the applicability of these new models to non-Gaussian statistics, as encountered in islands