# Recherche

# Delabays Robin

### Professeur-e HES Assistant-e

### Professeur-e HES Assistant-e

Bureau: ENP.23.N401

Rue de l'Industrie 23, 1950 Sion, CH

**Domaine**

Technique et IT

**Filière principale**

Systèmes industriels

En cours

**Rôle: **Requérant(e) principal(e)

**Financement: **
FNS

**Description du projet : **
The safe and reliable operation of power grids rely crucially, among other factors, on an accurate estimation
of their current and future states. Whereas our society has been operating larger and larger power grids over
the last 150 years, our fundamental understanding thereof is still partial and incomplete.
The past decades have seen a tremendous body of work dedicated to deciphering the impact of network
structures and parameters on the behavior of voltages and currents. The power flow equations, relating the
balance of active and reactive power to the complex voltages, are a fundamental tool for the operation and
planning of power grids, as well as for their theoretical analysis (see, e.g., [1, Sec. 6.4] or [2, Sec. 3.5]). Up
to this day, there is little analytical understanding of the relation between the systems characteristics (parameters,
coupling network) and the properties of the power flow solutions (existence, uniqueness, stability).
Events such as large scale loop flows around geographic obstacles, as Lake Erie [3] in 2007, are far from being
totally understood. It is however clear that their occurrence is highly related to the network structure of the
grid.

**Equipe de recherche au sein de la HES-SO:**
Délitroz Jim
, Delabays Robin

**Partenaires académiques: **VS - Institut Energie et environnement

**Durée du projet:**
01.10.2023 - 30.09.2027

**Montant global du projet: **248'964 CHF

**Statut: ** En cours

2024

**Résumé:**

There is growing evidence of systematic attempts to influence democratic elections by controlled and digitally organized dissemination of fake news. This raises the question of the intrinsic robustness of democratic electoral processes against external influences. Particularly interesting is to identify the social characteristics of a voter population that renders it more resilient against opinion manipulation. Equally important is to determine which of the existing democratic electoral systems is more robust to external influences. Here we construct a mathematical electoral model to address these two questions. We find that, not unexpectedly, biased electorates with clear-cut elections are overall quite resilient against opinion manipulations, because inverting the election outcome requires to change the opinion of many voters. More interesting are unbiased or weakly biased electorates with close elections. We find that such populations are more resilient against opinion manipulations (i) if they are less polarized and (ii) when voters interact more with each other, regardless of their opinion differences, and that (iii) electoral systems based on proportional representation are generally the most robust. Our model qualitatively captures the volatility of the US House of Representatives elections. We take this as a solid validation of our approach.

2023

**Résumé:**

This letter studies the celebrated Kuramoto-Sakaguchi model of coupled oscillators adopting two recent concepts. First, we consider appropriately-defined subsets of the n -torus called winding cells. Second, we analyze the semicontractivity of the model, i.e., the property that the distance between trajectories decreases when measured according to a seminorm. This letter establishes the local semicontractivity of the Kuramoto-Sakaguchi model, which is equivalent to the local contractivity for the reduced model. The reduced model is defined modulo the rotational symmetry. The domains where the system is semicontracting are convex phase-cohesive subsets of winding cells. Our sufficient conditions and estimates of the semicontracting domains are less conservative and more explicit than in previous works. Based on semicontraction on phase-cohesive subsets, we establish the at most uniqueness of synchronous states within these domains, thereby characterizing the multistability of this model.

**Résumé:**

A forced oscillation event in power grids refers to a state where malfunctioning or abnormally operating equipment causes persisting periodic disturbances in the system. While power grids are designed to damp most perturbations during standard operations, some of them can excite normal modes of the system and cause significant energy transfers across the system, creating large oscillations thousands of miles away from the source. Localization of the source of such disturbances remains an outstanding challenge due to a limited knowledge of the system parameters outside of the zone of responsibility of system operators. Here, we propose a new method for locating the source of forced oscillations that addresses this challenge by performing a simultaneous dynamic model identification using a principled maximum likelihood approach. We illustrate the validity of the algorithm on a variety of examples where forcing leads to resonance conditions in the system dynamics. Our results establish that an accurate knowledge of system parameters is not required for a successful inference of the source and frequency of a forced oscillation. We anticipate that our method will find a broader application in general dynamical systems that can be well described by their linearized dynamics over short periods of time.

*Chaos, solitons fractals: the interdisciplinary journal of nonlinear science, and nonequilibrium and complex phenomena*,
2023, vol. 168, article no. 113166

**Résumé:**

Networked systems have been used to model and investigate the dynamical behavior of a variety of systems. For these systems, different levels of complexity can be considered in the modeling procedure. On one hand, this can offer a more realistic and rich modeling option. On the other hand, it can lead to intrinsic difficulty in analyzing the system. Here, we present an approach to investigate the dynamics of Kuramoto oscillators on networks with different levels of connections: a network of networks. To do so, we utilize a construction in network theory known as the join of networks, which represents “intra-area” and “inter-area” connections. This approach provides a reduced representation of the original, multi-level system, where both systems have equivalent dynamics. Then, we can find solutions for the reduced system and broadcast them to the original network of networks. Moreover, using the same idea we can investigate the stability of these states, where we can obtain information on the Jacobian of the multi-level system by analyzing the reduced one. This approach is general for arbitrary connection schemes between nodes within the same area. Finally, our work opens the possibility of studying the dynamics of networked systems using a simpler representation, thus leading to a better understanding of the dynamical behavior of these systems.

2021

*Chaos: An Interdisciplinary Journal of Nonlinear Science*,
2021, vol. 31, no. 10, article no. 103117

**Résumé:**

The dynamics of systems of interacting agents is determined by the structure of their coupling network. The knowledge of the latter is, therefore, highly desirable, for instance, to develop efficient control schemes, to accurately predict the dynamics, or to better understand inter-agent processes. In many important and interesting situations, the network structure is not known, however, and previous investigations have shown how it may be inferred from complete measurement time series on each and every agent. These methods implicitly presuppose that, even though the network is not known, all its nodes are. Here, we investigate the different problem of inferring network structures within the observed/measured agents. For symmetrically coupled dynamical systems close to a stable equilibrium, we establish analytically and illustrate numerically that velocity signal correlators encode not only direct couplings, but also geodesic distances in the coupling network within the subset of measurable agents. When dynamical data are accessible for all agents, our method is furthermore algorithmically more efficient than the traditional ones because it does not rely on matrix inversion.

2020

**Résumé:**

Inspired by the Deffuant and Hegselmann-Krause models of opinion dynamics, we extend the Kuramoto model to account for confidence bounds, i.e., vanishing interactions between pairs of oscillators when their phases differ by more than a certain value. We focus on Kuramoto oscillators with peaked, bimodal distribution of natural frequencies. We show that, in this case, the fixed-points for the extended model are made of certain numbers of independent clusters of oscillators, depending on the length of the confidence bound -- the interaction range -- and the distance between the two peaks of the bimodal distribution of natural frequencies. This allows us to construct the phase diagram of attractive fixed-points for the bimodal Kuramoto model with bounded confidence and to analytically explain clusterization in dynamical systems with bounded confidence.

2019

*Chaos: An Interdisciplinary Journal of Nonlinear Science*,
2019, no. 29, article no. 113129

**Résumé:**

The Kuramoto model with high-order coupling has recently attracted some attention in the field of coupled oscillators in order, for instance, to describe clustering phenomena in sets of coupled agents. Instead of considering interactions given directly by the sine of oscillators’ angle differences, the interaction is given by the sum of sines of integer multiples of these angle differences. This can be interpreted as a Fourier decomposition of a general 2π-periodic interaction function. We show that in the case where only one multiple of the angle differences is considered, which we refer to as the “Kuramoto model with simple qth-order coupling,” the system is dynamically equivalent to the original Kuramoto model. In other words, any property of the Kuramoto model with simple higher-order coupling can be recovered from the standard Kuramoto model.

*Chaos: An Interdisciplinary Journal of Nonlinear Science*,
2019, vol. 29, no. 10, article no. 103130

**Résumé:**

In modern electric power networks with fast evolving operational conditions, assessing the impact of contingencies is becoming more and more crucial. Contingencies of interest can be roughly classified into nodal power disturbances and line faults. Despite their higher relevance, line contingencies have been significantly less investigated analytically than nodal disturbances. The main reason for this is that nodal power disturbances are additive perturbations, while line contingencies are multiplicative perturbations, which modify the interaction graph of the network. They are, therefore, significantly more challenging to tackle analytically. Here, we assess the direct impact of a line loss by means of the maximal Rate of Change of Frequency (RoCoF) incurred by the system. We show that the RoCoF depends on the initial power flow on the removed line and on the inertia of the bus where it is measured. We further derive analytical expressions for the expectation and variance of the maximal RoCoF, in terms of the expectations and variances of the power profile in the case of power systems with power uncertainties. This gives analytical tools to identify the most critical lines in an electric power grid.

**Résumé:**

Complex physical systems are unavoidably subjected to external environments not accounted for in the set of differential equations that models them. The resulting perturbations are standardly represented by noise terms. We derive conditions under which such noise terms perturb the dynamics strongly enough that they lead to stochastic escape from the initial basin of attraction of an initial stable equilibrium state of the unperturbed system. Focusing on Kuramoto-like models we find in particular that, quite counterintuitively, systems with inertia leave their initial basin faster than or at the same time as systems without inertia, except for strong white-noise perturbations.

**Résumé:**

Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have an identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted, and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely, acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted, and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted, and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.

2017

**Résumé:**

In dynamical systems, the full stability of fixed point solutions is determined by their basins of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [Wiley et al., Chaos 16, 015103 (2006) and Menck et al. Nat. Phys. 9, 89 (2013)]. Here, we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [Chaos 16, 015103 (2006)] that inspired the title of the present manuscript and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number q and the number n of oscillators. We find that the basin volumes scale as (1−4𝑞/𝑛)𝑛, contrasting with the Gaussian behavior postulated in the study by Wiley et al.. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks.

**Résumé:**

We investigate the scaling properties of the order parameter and the largest nonvanishing Lyapunov exponent for the fully locked state in the Kuramoto model with a finite number N of oscillators. We show that, for any finite value of N, both quantities scale as (K−KL)1/2 with the coupling strength K sufficiently close to the locking threshold KL. We confirm numerically these predictions for oscillator frequencies evenly spaced in the interval [−1,1] and additionally find that the coupling range δK over which this scaling is valid shrinks like δK∼N−α with α≈1.5 as N→∞. Away from this interval, the order parameter exhibits the infinite-N behavior r−rL∼(K−KL)2/3 proposed by Pazó [Phys. Rev. E 72, 046211 (2005)]. We argue that the crossover between the two behaviors occurs because at the locking threshold, the upper bound of the continuous part of the spectrum of the fully locked state approaches zero as N increases. Our results clarify the convergence to the N→∞ limit in the Kuramoto model.

**Résumé:**

The number 𝒩 of stable fixed points of locally coupled Kuramoto models depends on the topology of the network on which the model is defined. It has been shown that cycles in meshed networks play a crucial role in determining 𝒩 because any two different stable fixed points differ by a collection of loop flows on those cycles. Since the number of different loop flows increases with the length of the cycle that carries them, one expects 𝒩 to be larger in meshed networks with longer cycles. Simultaneously, the existence of more cycles in a network means more freedom to choose the location of loop flows differentiating between two stable fixed points. Therefore, 𝒩 should also be larger in networks with more cycles. We derive an algebraic upper bound for the number of stable fixed points of the Kuramoto model with identical frequencies, under the assumption that angle differences between connected nodes do not exceed 𝜋/2. We obtain 𝒩≤∏𝑐𝑘=1[2⋅Int(𝑛𝑘/4)+1], which depends both on the number c of cycles and on the spectrum of their lengths {nk}. We further identify network topologies carrying stable fixed points with angle differences larger than 𝜋/2, which leads us to conjecture an upper bound for the number of stable fixed points for Kuramoto models on any planar network. Compared to earlier approaches that give exponential upper bounds in the total number of vertices, our bounds are much lower and therefore much closer to the true number of stable fixed points.

2016

**Résumé:**

Geographical features such as mountain ranges or big lakes and inland seas often result in large closed loops in high voltage AC power grids. Sizable circulating power flows have been recorded around such loops, which take up transmission line capacity and dissipate but do not deliver electric power. Power flows in high voltage AC transmission grids are dominantly governed by voltage angle differences between connected buses, much in the same way as Josephson currents depend on phase differences between tunnel-coupled superconductors. From this previously overlooked similarity we argue here that circulating power flows in AC power grids are analogous to supercurrents flowing in superconducting rings and in rings of Josephson junctions. We investigate how circulating power flows can be created and how they behave in the presence of ohmic dissipation. We show how changing operating conditions may generate them, how significantly more power is ohmically dissipated in their presence and how they are topologically protected, even in the presence of dissipation, so that they persist when operating conditions are returned to their original values. We identify three mechanisms for creating circulating power flows, (i) by loss of stability of the equilibrium state carrying no circulating loop flow, (ii) by tripping of a line traversing a large loop in the network and (iii) by reclosing a loop that tripped or was open earlier. Because voltages are uniquely defined, circulating power flows can take on only discrete values, much in the same way as circulation around vortices is quantized in superfluids.

**Résumé:**

Determining the number of stable phase-locked solutions for locally coupled Kuramoto models is a long-standing mathematical problem with important implications in biology, condensed matter physics, and electrical engineering among others. We investigate Kuramoto models on networks with various topologies and show that different phase-locked solutions are related to one another by loop currents. The latter take only discrete values, as they are characterized by topological winding numbers. This result is generically valid for any network and also applies beyond the Kuramoto model, as long as the coupling between oscillators is antisymmetric in the oscillators’ coordinates. Motivated by these results, we further investigate loop currents in Kuramoto-like models. We consider loop currents in nonoriented n-node cycle networks with nearest-neighbor coupling. Amplifying on earlier works, we give an algebraic upper bound 𝒩≤2 Int[𝑛/4]+1 for the number 𝒩 of different, linearly stable phase-locked solutions. We show that the number of different stable solutions monotonically decreases as the coupling strength is decreased. Furthermore stable solutions with a single angle difference exceeding π/2 emerge as the coupling constant K is reduced, as smooth continuations of solutions with all angle differences smaller than π/2 at higher K. In a cycle network with nearest-neighbor coupling, we further show that phase-locked solutions with two or more angle differences larger than π/2 are all linearly unstable. We point out similarities between loop currents and vortices in superfluids and superconductors as well as persistent currents in superconducting rings and two-dimensional Josephson junction arrays.

2016

*Proceedings of 2016 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), 9-12 October 2016, Ljubljana, Slovenia*

**Résumé:**

Demand side management (DSM) is known for generating synchronized behaviors of aggregated loads that can lead to large power fluctuations. In contrast to this well-studied occurrence, we report here on the emergence of novel synchronized behaviors of thermostatically-controlled electric heating systems in buildings with good thermal insulation and important solar radiation gains without DSM. To suppress the resulting large load fluctuations on the distribution grid we propose a centralized DSM algorithm that smoothens the total load curve - including electric heating and all other domestic appliances - of the cluster of dwellings it pilots. Setting up the baseline load is based on weather forecasts for a receding time-horizon covering the next 24 hours, while control actions are based on a priority list which is constructed from the current status of the dwellings. We show numerically that our DSM control scheme can be generically used to modify load curves of domestic households to achieve diverse goals such as minimizing electricity costs, peak shaving and valley filling.

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