arXiv subjects

Leonardo Rydin Gorjão

Publications and source records attributed to Leonardo Rydin Gorjão.

At least 19 recordsLinked to original sources

arXiv2026

Modelling the Nordic power system expansion with PyPSA-Eur: a comparison with the Nordic Grid Development Perspective 2025

We compare the Nordic Grid Development Perspective 2025 (NGDP) with a least-cost transition pathway for the Nordic power system obtained with PyPSA-Eur for the period 2025-2050 under a net-zero CO$_2$ constraint. The model co-optimises generation, storage, and transmission expansion with hourly system operation while constraining the Nordic generation mix to remain consistent with NGDP milestone ranges. Overall, the optimised capacity pathways reproduce the broad NGDP expansion of wind and solar power, with Denmark emerging as the main locus of offshore wind and solar deployment, Finland as a major site for onshore wind expansion, and Norway and Sweden relying more strongly on existing hydropower and nuclear capacity. The main discrepancy between the two pathways does not lie in the broad build-out of variable renewables, but in the timing and role of hydrogen. Whereas the NGDP treats hydrogen as an early and steadily increasing component of industrial demand and system development already from 2030, PyPSA-Eur delays substantial hydrogen deployment until 2045-2050, when it becomes cost-effective primarily as a flexibility and seasonal storage option in a strongly variable-renewable-energy-dominated system. This contrast suggests that early hydrogen build-out in the Nordics is not a robust outcome of cost optimisation alone, but depends strongly on assumptions regarding industrial demand growth, sector coupling, and anticipatory infrastructure planning. We therefore argue that hydrogen remains the principal strategic uncertainty in the Nordic transition, with the NGDP representing a more anticipatory planning perspective and PyPSA-Eur a more conservative least-cost timing of deployment.

eess.SY↗

arXiv2026

A graph theoretic view on small signal stability of inverter-based power grids

Dynamic grid stability is traditionally ensured with synchronous generators. Modern grids rely substantially more on inverter-based resources, which require grid-forming control to guarantee adequate system-wide synchronization and stability. Small-signal stability has granted various centralized and decentralized stability certificates - but these have primarily been limited to sufficient criteria only. In this work, we construct a necessary and sufficient small-signal stability criterion for lossless inverter-based power grids with arbitrary topology. We show that asymptotic stability is equivalent to the positive definiteness of a single matrix that combines network topology, operating point, and effective droop gains. We derive graph-theoretic stability criteria based on an augmented cone graph and show that the contribution of graph cycles is typically small, as illustrated for three IEEE test cases. The resulting framework yields decentralized stability criteria, quantifies the conservatism introduced by decentralization, and may support the development of future grid codes.

eess.SY↗

arXiv2026

PowerSINDy: Identifying Nonlinear Time-Dependent Dynamics in Power Grid Frequency

System identification plays a crucial role in physics and machine learning for discovering governing equations directly from data. A powerful approach is the Sparse Identification of Nonlinear Dynamics (SINDy) method, which assumes that only a few dominant terms drive the essential behavior of a nonlinear dynamical system. While SINDy methods have shown excellent results, they are most often illustrated on synthetic or simulated systems, leaving open the question of how well they perform on complex, noisy, real-world data. Power grid frequency dynamics provide a highly relevant and challenging environment for advancing system identification methods. In this work, we propose PowerSINDy as a framework for empirical power system data. We apply this framework to empirical frequency data from the Continental Europe (CE) and South Korea (SK) synchronous grids, two major power systems with distinct dynamical characteristics. PowerSINDy, which also includes time-dependent terms, can identify the dynamics of these complex real-world systems. Furthermore, we benchmark three sparsity-promoting regression strategies: Sequentially Thresholded Least Squares (STLSQ), Least Absolute Shrinkage and Selection Operator (LASSO), and Sparse Relaxed Regularized Regression (SR3) to evaluate trade-offs between accuracy, sparsity, and robustness. Results show that LASSO consistently achieves the lowest stable RMSEs, reaching 0.0101 for the CE, while STLSQ provides the best balance between accuracy and stability. SR3 exhibits higher variability and sensitivity to regularization, with L0 and L1 producing nearly indistinguishable outcomes.

eess.SY↗

arXiv2026

Understanding the complexity of frequency and phase angle fluctuations in power grids

Power grids must modernize to meet climate goals while maintaining reliable and stable operating conditions. Yet progress is hindered by a limited understanding of the stochastic processes underlying grid frequency and phase-angle fluctuations, which are induced by the growing penetration of renewable generation, consumer demand fluctuations, and market trading. This issue is particularly acute in Africa, where grids often face weak investment. Here, we present results from a newly collected, large-scale, high-resolution dataset of grid frequency and phase angles for the United Kingdom and South Africa, comprising close to one billion data points. Using superstatistical modeling, we treat market-driven power fluctuations as a slowly varying parameter driving grid dynamics and incorporate nonlinear frequency control. As a result, we derive an analytical model that reproduces multimodal frequency distributions previously obtained from numerical simulations, as well as heavy-tailed fluctuations and double-exponential frequency autocorrelation decays, all in excellent agreement with experimental measurements. Beyond frequency, we also address the so far largely overlooked problem of characterizing spatial phase-angle fluctuations. By comparing our predictions with measurement data, we demonstrate that a low-dimensional effective grid model accurately fits South African data despite the grid's complexity. We also highlight significant differences between the grids of South Africa and the United Kingdom. Our results clarify how energy markets and control policies shape grid dynamics across countries with contrasting infrastructure maturity.

physics.soc-ph↗

arXiv2026

Jump-diffusion models of parametric volume-price distributions

We present a data-driven framework to model the stochastic evolution of volume-price distribution from the New York Stock Exchange (NYSE) equities. The empirical distributions are sampled every 10 minutes over 976 trading days, and fitted to different models, namely Gamma, Inverse Gamma, Weibull, and Log-Normal distributions. Each of these models is parameterized by a shape parameter, $phi$, and a scale parameter, $θ$, which are detrended from their daily average behavior. The time series of the detrended parameters is analyzed using adaptive binning and regression-based extraction of the Kramers-Moyal (KM) coefficients, up to their sixth order, enabling to classification of its intrinsic dynamics. We show that (i) $ϕ$ is well described as a pure diffusion with a linear mean regression for the Gamma, Inverse Gamma, and Weibull models, while $θ$ shows dominant jump-diffusion dynamics, with an elevated fourth- and sixth-order moment contributions; (ii) the log-normal model shows however the opposite: $θ$ is predominantly diffusive, with $ϕ$ showing weak jump signatures; (iii) global moment inversion yields jump rates and amplitudes that account for a large share of total variance for $θ$, confirming that rare discontinuities dominate volatility.

cs.NE↗

arXiv2025

Discontinuous stochastic forcing in Greenland ice core data

Paleoclimate proxy records from Greenland ice cores, archiving e.g. $δ^{18}$O as a proxy for surface temperature, show that sudden climatic shifts called Dansgaard-Oeschger events (DO) occurred repeatedly during the last glacial interval. They comprised substantial warming of the Arctic region from cold to milder conditions. Concomitant abrupt changes in the dust concentrations of the same ice cores suggest that sudden reorganisations of the hemispheric-scale atmospheric circulation have accompanied the warming events. Genuine bistability of the North Atlantic climate system is commonly hypothesised to explain the existence of stadial (cold) and interstadial (milder) periods in Greenland. However, the physical mechanisms that drove abrupt transitions from the stadial to the interstadial state, and more gradual yet still abrupt reverse transitions, remain debated. Here, we conduct a one-dimensional data-driven analysis of the Greenland temperature and atmospheric circulation proxies under the purview of stochastic processes. We take the Kramers-Moyal equation to estimate each proxy's drift and diffusion terms within a Markovian model framework. We then assess noise contributions beyond Gaussian white noise. The resulting stochastic differential equation (SDE) models feature a monostable drift for the Greenland temperature proxy and a bistable one for the atmospheric circulation proxy. Indicators of discontinuity in stochastic processes suggest to include higher-order terms of the Kramers-Moyal equation when modelling the Greenland temperature proxy's evolution. This constitutes a qualitative difference in the characteristics of the two time series, which should be further investigated from the standpoint of climate dynamics.

physics.ao-ph↗

arXiv2024

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

Anticipating bifurcation-induced transitions in dynamical systems has gained relevance in various fields of the natural, social, and economic sciences. Before the annihilation of a system's equilibrium point by means of a bifurcation, the system's internal feedbacks that stabilize the initial state weaken and eventually vanish, a process referred to as critical slowing down (CSD). In one-dimensional systems, this motivates the use of variance and lag-1 autocorrelation as indicators of CSD. However, the applicability of variance is limited to time- and state-independent driving noise, strongly constraining the generality of this CSD indicator. In multidimensional systems, the use of these indicators is often preceded by a dimension reduction in order to obtain a one-dimensional time series. Many common techniques for such an extraction of a one-dimensional time series generally incur the risk of missing CSD in practice. Here, we propose a data-driven approach based on estimating a multidimensional 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 under which CSD can reliably be detected, as demonstrated in a suite of examples. In contrast to existing approaches, changes in deterministic dynamics can be clearly discriminated from changes in the driving noise using our method. This substantially reduces the risk of false or missed alarms of conventional CSD indicators in settings with time-dependent or multiplicative noise. In multidimensional 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.

math.DS↗

arXiv2023

Probabilistic Forecasting of Day-Ahead Electricity Prices and their Volatility with LSTMs

Accurate forecasts of electricity prices are crucial for the management of electric power systems and the development of smart applications. European electricity prices have risen substantially and became highly volatile after the Russian invasion of Ukraine, challenging established forecasting methods. Here, we present a Long Short-Term Memory (LSTM) model for the German-Luxembourg day-ahead electricity prices addressing these challenges. The recurrent structure of the LSTM allows the model to adapt to trends, while the joint prediction of both mean and standard deviation enables a probabilistic prediction. Using a physics-inspired approach - superstatistics - to derive an explanation for the statistics of prices, we show that the LSTM model faithfully reproduces both prices and their volatility.

cs.LG↗

arXiv2023

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.

physics.data-an↗

arXiv2023

Stability bounds of droop-controlled inverters in power grid networks

The energy mix of future power systems will include high shares of wind power and solar PV. 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 in an analytic fashion and uncover stability and instability regions. Secondly, we extend the analysis to a generic topology of inverters and provide mathematical criteria for 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.

eess.SY↗

arXiv2022

Initial analysis of the impact of the Ukrainian power grid synchronization with Continental Europe

When Russia invaded Ukraine on the 24\textsuperscript{th} of February 2022, this led to many acts of solidarity with Ukraine, including support for its electricity system. Just 20 days after the invasion started, the Ukrainian and Moldovan power grids were synchronized to the Continental European power grid to provide stability to these grids. Here, we present an initial analysis of how this synchronization affected the statistics of the power grid frequency and cross-border flows of electric power within Continental Europe. We observe faster inter-area oscillations, an increase in fluctuations and changes in the cross-border flows in and out of Ukraine and surrounding countries as an effect of the synchronization with Continental Europe. Overall these changes are small such that the now connected system can be considered as stable as before the synchronization.

physics.soc-ph↗

arXiv2022

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.

cs.LG↗

arXiv2022

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.

stat.AP↗

arXiv2022

Microscopic fluctuations in power-grid frequency recordings at the sub-second scale

Complex systems, such as the power grid, are essential for our daily lives. Many complex systems display (multi-)fractal behavior, correlated fluctuations and power laws. Whether the power-grid frequency, an indicator about the balance on supply and demand in the electricity grid, also displays such complexity remains a mostly open question. Within the present article, we utilize highly resolved measurements to quantify the properties of the power-grid frequency. We show that below 1 second, the dynamics may fundamentally change, including a suddenly increasing power spectral density, emergence of multifractality and a change of correlation behavior. We provide a simplified stochastic model involving positively correlated noise to reproduce the observed dynamics, possibly linked to frequency dependent loads. Finally, we stress the need for high-quality measurements and discuss how we obtained the data analyzed here.

physics.soc-ph↗

arXiv2022

Collective effects and synchronization of demand in real-time demand response

Sustainable energy systems require flexible elements to balance the variability of renewable energy sources. Demand response aims to adapt the demand to the variable generation, in particular by shifting the load in time. In this article, we provide a detailed statistic analysis of the collective operation of many demand response units. We establish and simulate a model for load shifting in response to real-time electricity pricing using local storage systems. We show that demand response drives load shifting as desired, but also induces strong collective effects that may threaten system stability. The load of individual households synchronizes, leading to extreme demand peaks. We provide a detailed statistical analysis of the grid load, and quantify both the likeliness and extent of extreme demand peaks.

physics.soc-ph↗

arXiv2021

Validation Methods for Energy Time Series Scenarios from Deep Generative Models

The design and operation of modern energy systems are heavily influenced by time-dependent and uncertain parameters, e.g., renewable electricity generation, load-demand, and electricity prices. These are typically represented by a set of discrete realizations known as scenarios. A popular scenario generation approach uses deep generative models (DGM) that allow scenario generation without prior assumptions about the data distribution. However, the validation of generated scenarios is difficult, and a comprehensive discussion about appropriate validation methods is currently lacking. To start this discussion, we provide a critical assessment of the currently used validation methods in the energy scenario generation literature. In particular, we assess validation methods based on probability density, auto-correlation, and power spectral density. Furthermore, we propose using the multifractal detrended fluctuation analysis (MFDFA) as an additional validation method for non-trivial features like peaks, bursts, and plateaus. As representative examples, we train generative adversarial networks (GANs), Wasserstein GANs (WGANs), and variational autoencoders (VAEs) on two renewable power generation time series (photovoltaic and wind from Germany in 2013 to 2015) and an intra-day electricity price time series form the European Energy Exchange in 2017 to 2019. We apply the four validation methods to both the historical and the generated data and discuss the interpretation of validation results as well as common mistakes, pitfalls, and limitations of the validation methods. Our assessment shows that no single method sufficiently characterizes a scenario but ideally validation should include multiple methods and be interpreted carefully in the context of scenarios over short time periods.

cs.LG↗

arXiv2021

Change of persistence in European electricity spot prices

The European Power Exchange has introduced day-ahead auctions and continuous trading spot markets to facilitate the insertion of renewable electricity. These markets are designed to balance excess or lack of power in short time periods, which leads to a large stochastic variability of the electricity prices. Furthermore, the different markets show different stochastic memory in their electricity price time series, which seem to be the cause for the large volatility. In particular, we show the antithetical temporal correlation in the intraday 15 minutes spot markets in comparison to the day-ahead hourly market. We contrast the results from Detrended Fluctuation Analysis (DFA) to a new method based on the Kramers--Moyal equation in scale. For very short term ($<12$ hours), all price time series show positive temporal correlations (Hurst exponent $H>0.5$) except for the intraday 15 minute market, which shows strong negative correlations ($H<0.5$). For longer term periods covering up to two days, all price time series are anti-correlated ($H<0.5$).

q-fin.ST↗

arXiv2021

Dynamic stability of electric power grids: Tracking the interplay of the network structure, transmission losses and voltage dynamics

Dynamic stability is imperative for the operation of the electric power system. This article provides analytical results and effective stability criteria focusing on the interplay of network structures and the local dynamics of synchronous machines. The results are based on an extensive linear stability analysis of the third-order model for synchronous machines, comprising the classical power-swing equations and the voltage dynamics. The article explicitly covers the impact of Ohmic losses in a linear approximation in power grids, which are often neglected in analytical studies. Necessary and sufficient stability conditions are formulated, and different routes to instability are analysed, yielding concrete mathematical criteria applicable to all scales of power grids, from transmission to distribution grids, as well as microgrids. A subsequent numerical study of the criteria is presented, without and with resistive terms, to test how tight the derived analytical results are.

nlin.AO↗