Stochastic Model for Power Grid Dynamics

We introduce a stochastic model that describes the quasi-static dynamics of an electric transmission network under perturbations introduced by random load fluctuations, random removing of system components from service, random repair times for the failed components, and random response times to implement optimal system corrections for removing line overloads in a damaged or stressed transmission network. We use a linear approximation to the network flow equations and apply linear programming techniques that optimize the dispatching of generators and loads in order to eliminate the network overloads associated with a damaged system. We also provide a simple model for the operator's response to various contingency events that is not always optimal due to either failure of the state estimation system or due to the incorrect subjective assessment of the severity associated with these events. This further allows us to use a game theoretic framework for casting the optimization of the operator's response into the choice of the optimal strategy which minimizes the operating cost. We use a simple strategy space which is the degree of tolerance to line overloads and which is an automatic control (optimization) parameter that can be adjusted to trade off automatic load shed without propagating cascades versus reduced load shed and an increased risk of propagating cascades. The tolerance parameter is chosen to describes a smooth transition from a risk averse to a risk taken strategy...Comment: framework for a system-level analysis of the power grid from the viewpoint of complex network

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

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

Machine learning of power grid frequency dynamics and control: prediction, explanation and stochastic modelling

A reliable supply of electric power is not a matter of course. Power grids enable the transport of power from generators to consumers, but their stable operation constantly requires corrective measures and a careful supervision. In particular, power generation and demand have to be balanced at all times. A large power imbalance threatens the reliability of the power supply and can, in extreme cases, lead to a large-scale blackout. Therefore, the power imbalance is constantly corrected through distinct control schemes. The power grid frequency measures the balance of power generation and demand. To guarantee frequency stability, and thereby a balance of generation and demand, load-frequency control constantly counteracts large frequency deviations. However, the transition of the energy system to renewable energy sources challenges frequency stability and control. Wind and solar power do not provide intrinsic inertia, which leads to increasingly fast frequency dynamics. Different economic sectors become strongly coupled to the power system, as, for example, the adoption of electric vehicles will interconnect the transport sector and the power system. Finally, wind and solar power are weather-dependent, which increases the variability of power generation. All in all, this gives rise to diverse, interdependent and stochastic impact factors, that drive the balance of power demand and generation, and thus the grid frequency. How can we predict, explain and model frequency dynamics given its strong non-autonomous and stochastic character? In this thesis, I use machine learning to disentangle the effects of external drivers on grid frequency dynamics and control. First, I propose a prediction model that only uses historic frequency data, but fails in representing external impacts. Therefore, I include time series of techno-economic drivers and model their impact on grid frequency data using explainable machine learning methods. These methods reveal the dependencies between external drivers and frequency deviations, such as the important impact of forecast errors in the Scandinavian grid or the varying effects of different generation types. Finally, I integrate these drivers into a stochastic dynamical model of the grid frequency, which both represents short-term dynamics and long-term trends due to techno-economic impacts. My work complements traditional simulation-based approaches through validation and modelling inspiration. It offers flexible modelling and prediction tools for power system dynamics, which are profitable for systems with diverse impact factors but noisy and insufficient data

Control of Synchronization in two-layer power grids

In this work we suggest to model the dynamics of power grids in terms of a two-layer network, and use the Italian high voltage power grid as a proof-of-principle example. The first layer in our model represents the power grid consisting of generators and consumers, while the second layer represents a dynamic communication network that serves as a controller of the first layer. In particular, the dynamics of the power grid is modelled by the Kuramoto model with inertia, while the communication layer provides a control signal $P_i$ for each generator to improve frequency synchronization within the power grid. We propose different realizations of the communication layer topology and different ways to calculate the control signal. Then we conduct a systematic survey of the two-layer system against a multitude of different realistic perturbation scenarios, such as disconnecting generators, increasing demand of consumers, or generators with stochastic power output. When using a control topology that allows all generators to exchange information, we find that a control scheme aimed to minimize the frequency difference between adjacent nodes operates very efficiently even against the worst scenarios with the strongest perturbations

Self-Organized Dynamics of Power Grids: Smart Grids, Fluctuations and Cascades

Climate change is one of the most pressing issues of our time and mitigating it requires a reduction of CO2 emissions. A big step towards achieving this goal is increasing the share of renewable energy sources, as the energy sector currently contributes 35% to all greenhouse gas emissions. However, integrating these renewable energy sources challenges the current power system in two major ways. Firstly, renewable generation consists of more spatially distributed and smaller power plants than conventional generation by nuclear or coal plants, questioning the established hierarchical structures and demanding a new grid design. Restructuring becomes necessary because wind and solar plants have to be placed at favorable sites, e.g., close to coasts in the case of wind. Secondly, renewables do not provide a deterministic and controllable power output but introduce power fluctuations that have to be controlled adequately. Many solutions to these challenges are build on the concept of smart grids, which require an extensive information technology (IT) infrastructure communicating between consumers and generators to coordinate efficient actions. However, an intertwined power and IT system raises great privacy and security concerns. Is it possible to forgo a large IT infrastructure in future power grids and instead operate them purely based on local information? How would such a decentrally organized system work? What is the impact of fluctuation on short time scales on the dynamical stability? Which grid topologies are robust against random failures or targeted attacks? This thesis aims to establish a framework of such a self-organized dynamics of a power grid, analyzing its benefits and limitations with respect to fluctuations and discrete events. Instead of a centrally monitored and controlled smart grid, we propose the concept of Decentral Smart Grid Control, translating local power grid frequency information into actions to stabilize the grid. This is not limited to power generators but applies equally to consumers, naturally introducing a demand response. We analyze the dynamical stability properties of this framework using linear stability methods as well as applying numerical simulations to determine the size of the basin of attraction. To do so, we investigate general stability effects and sample network motifs to find that this self-organized grid dynamics is stable for large parameter regimes. However, when the actors of the power grid react to a frequency signal, this reaction has to be sufficiently fast since reaction delays are shown to destabilize the grid. We derive expressions for a maximum delay, which always desynchronizes the system based on a rebound effect, and for destabilizing delays based on resonance effects. These resonance instabilities are cured when the frequency signal is averaged over a few seconds (low-pass filter). Overall, we propose an alternative smart grid model without any IT infrastructure and analyze its stable operating space. Furthermore, we analyze the impact of fluctuations on the power grid. First, we determine the escape time of the grid, i.e., the time until the grid desynchronizes when subject to stochastic perturbations. We simulate these events and derive an analytical expression using Kramer's method, obtaining the scaling of the escape time as a function of the grid inertia, transmitted power, damping etc. Thereby, we identify weak links in networks, which have to be enhanced to guarantee a stable operation. Second, we collect power grid frequency measurements from different regions across the world and evaluate their statistical properties. Distributions are found to be heavy-tailed so that large disturbances are more common than predicted by Gaussian statistics. We model the grid dynamics using a stochastic differential equation to derive the scaling of the fluctuations based on power grid parameters, identifying effective damping as essential in reducing fluctuation risks. This damping may be provided by increased demand control as proposed by Decentral Smart Grid Control. Finally, we investigate discrete events, in particular the failure of a single transmission line, as a complementary form of disturbances. An initial failure of a transmission line leads to additional load on other lines, potentially overloading them and thereby causing secondary outages. Hence, a cascade of failures is induced that propagated through the network, resulting in a large-scale blackout. We investigate these cascades in a combined dynamical and event-driven framework, which includes transient dynamics, in contrast to the often used steady state analysis that only solves static flows in the grid while neglecting any dynamics. Concluding, we identify critical lines, prone to cause cascades when failing, and observe a nearly constant speed of the propagation of the cascade in an appropriate metric. Overall, we investigate the self-organized dynamics of power grids, demonstrating its benefits and limitations. We provide tools to improve current grid operation and outline a smart grid solution that is not reliant on IT. Thereby, we support establishing a 100% renewable energy system

Electromechanical Dynamics of High Photovoltaic Power Grids

This dissertation study focuses on the impact of high PV penetration on power grid electromechanical dynamics. Several major aspects of power grid electromechanical dynamics are studied under high PV penetration, including frequency response and control, inter-area oscillations, transient rotor angle stability and electromechanical wave propagation.To obtain dynamic models that can reasonably represent future power systems, Chapter One studies the co-optimization of generation and transmission with large-scale wind and solar. The stochastic nature of renewables is considered in the formulation of mixed-integer programming model. Chapter Two presents the development procedures of high PV model and investigates the impact of high PV penetration on frequency responses. Chapter Three studies the impact of PV penetration on inter-area oscillations of the U.S. Eastern Interconnection system. Chapter Four presents the impacts of high PV on other electromechanical dynamic issues, including transient rotor angle stability and electromechanical wave propagation. Chapter Five investigates the frequency response enhancement by conventional resources. Chapter Six explores system frequency response improvement through real power control of wind and PV. For improving situation awareness and frequency control, Chapter Seven studies disturbance location determination based on electromechanical wave propagation. In addition, a new method is developed to generate the electromechanical wave propagation speed map, which is useful to detect system inertia distribution change. Chapter Eight provides a review on power grid data architectures for monitoring and controlling power grids. Challenges and essential elements of data architecture are analyzed to identify various requirements for operating high-renewable power grids and a conceptual data architecture is proposed. Conclusions of this dissertation study are given in Chapter Nine

Distributed Machine Learning Approach to Fast Frequency Response-based Inertia Estimation in Low Inertia Grids

Recent updates to the IEEE 1547-2018 standard allow active participation of distributed energy resources (DERs) in power grid services with the goal of increased grid reliability and resiliency. With the rapid growth of DERs towards a low inertia converter-dominated grid, the DERs can provide fast frequency response (FFR) services that can quickly counteract the change in system frequency through inertial support. However, in low voltage grids, frequency and voltage face dynamics coupling due to a high resistance to reactance ratio and cannot be controlled separately as in the bulk electric grid. Due to the coupling effect, the control of one parameter also affects the dynamics of the other parameter. A part of this work highlights the role of DERs to provide grid ancillary services underscoring the challenges of combined voltage and frequency control in low voltage grids. Increasing penetration of renewable energy sources (RES) also decreases the power system inertia, there by affecting the stability of bulk grid. The stochastic nature of RES makes the power system inertia a time-varying quantity. Furthermore, converter-dominated grids have different dynamics compared to conventional grids and therefore estimates of the inertia constant using existing dynamic power system models are unsuitable. This work proposes a novel inertia estimation technique based on convolutional neural networks (CNN) that use local frequency measurements. The model uses a non-intrusive excitation signal to perturb the system and measure frequency using a phase-locked loop. The estimated inertia constants, have significant accuracy for the training, validation, and testing sets. Additionally, the proposed approach can be applied over traditional inertia estimation methods that do not incorporate the dynamic impact of renewable energy sources. The frequency response of power systems changes drastically when multi-area power systems with interconnected tie-lines are considered. Furthermore, higher penetration of RES increases the stochasticity in interconnected power systems. Hence, it is important to estimate the multi-area parameters ensuring communication and coordination between each of the areas. A robust and secure client-server-based distributed machine learning framework is used to estimate power system inertia in a two-area system. The proposed approach can be efficiently optimized to increase the training performance. It is important to analyze the performance of a trained machine learning model in a real-world scenario with unknown dynamics. A pre-trained CNN is tested on a system with model predictive controller (MPC)-based virtual inertia (VI) unit. Results show that the frequency and inertial response of conventional synchronous generators-based system differs drastically as compared to the system with non-synchronous generator-based VI support