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# Jacquod Philippe

### Professeur-e HES Ordinaire

#### Main skills

### Professeur-e HES Ordinaire

Desktop: ENP.23.N401

Rue de l'Industrie 23, 1950 Sion, CH

**Faculty**

Technique et IT

**Main Degree Programme**

Systèmes industriels

2021

**Summary:**

We investigate coherent oscillations in large scale transmission power grids, where large groups of generators respond in unison to a distant disturbance. Such long wavelength coherent phenomena are known as inter-area oscillations. Their existence in networks of weakly connected areas is well captured by singular perturbation theory. However, they are also observed in strongly connected networks without time-scale separation, where applying singular perturbation theory is not justified. We show that the occurrence of these oscillations is actually generic. Applying matrix perturbation theory, we show that, because these modes lie at the edge of the system's spectrum of eigenvalues, they are only moderately sensitive to increased network connectivity between well chosen, initially weakly connected areas, and that their general structure remains the same, regardless of the strength of the inter-area coupling. This is qualitatively understood by bringing together the standard singular perturbation theory and Courant's nodal domain theorem.

Pagnier Laurent, Chertkov Michael, Fritzsch Julien, Jacquod Philippe

*arXiv.org*,

**Summary:**

This manuscript is the first step towards building a robust and efficient model reduction methodology to capture transient dynamics in a transmission level electric power system. Such dynamics is normally modeled on seconds-to-tens-of-seconds time scales by the so-called swing equations, which are ordinary differential equations defined on a spatially discrete model of the power grid. We suggest, following Seymlyen (1974) and Thorpe, Seyler and Phadke (1999), to map the swing equations onto a linear, inhomogeneous Partial Differential Equation (PDE) of parabolic type in two space and one time dimensions with time-independent coefficients and properly defined boundary conditions. The continuous two-dimensional spatial domain is defined by a geographical map of the area served by the power grid, and associated with the PDE coefficients derived from smoothed graph-Laplacian of susceptances, machine inertia and damping. Inhomogeneous source terms represent spatially distributed injection/consumption of power. We illustrate our method on PanTaGruEl (Pan-European Transmission Grid and ELectricity generation model). We show that, when properly coarse-grained, i.e. with the PDE coefficients and source terms extracted from a spatial convolution procedure of the respective discrete coefficients in the swing equations, the resulting PDE reproduces faithfully and efficiently the original swing dynamics. We finally discuss future extensions of this work, where the presented PDE-based reduced modeling will initialize a physics-informed machine learning approach for real-time modeling, n−1 feasibility assessment and transient stability analysis of power systems.

Tyloo Melvyn, Delabays Robin, Jacquod Philippe

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

**Summary:**

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.

**Summary:**

Interconnecting power systems has a number of advantages such as better electric power quality, increased reliability of power supply, economies of scales through production and reserve pooling and so forth. Simultaneously, it may jeopardize the overall system stability with the emergence of so-called inter-area oscillations, which are coherent oscillations involving groups of rotating machines separated by large distances up to thousands of kilometers. These often weakly damped modes may have harmful consequences for grid operation, yet despite decades of investigations, the mechanisms that generate them are still poorly understood, and the existing theories are based on assumptions that are not satisfied in real power grids where such modes are observed. Here we construct a matrix perturbation theory of large interconnected power systems that clarifies the origin and the conditions for the emergence of inter-area oscillations. We show that coherent inter-area oscillations emerge from the zero-modes of a multi-area network Laplacian matrix, which hybridize only weakly with other modes, even under significant capacity of the inter-area tie-lines, i.e. even when the standard assumption of area partitioning is not satisfied. The general theory is illustrated on a two-area system, and numerically applied to the well-connected PanTaGruEl model of the synchronous grid of continental Europe.

2020

**Summary:**

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.

Tyloo Melvyn, Jacquod Philippe

*IEEE Control Systems Letters*,
2021, vol. 5, no. 3, pp. 929-934

**Summary:**

Many recent works in control of electric power systems have investigated their synchronization through global performance metrics under external disturbances. The approach is motivated by fundamental changes in the operation of power grids, in particular by the substitution of conventional power plants with new renewable sources of electrical energy. This substitution will simultaneously increase fluctuations in power generation and reduce the available mechanical inertia. It is crucial to understand how strongly these two evolutions will impact grid stability. With very few, mostly numerical exceptions, earlier works on performance metrics had to rely on unrealistic assumptions of grid homogeneity. Here we show that a modified spectral decomposition can tackle that issue in inhomogeneous power grids in cases where disturbances occur on time scales that are long compared to the intrinsic time scales of the grid. We find in particular that the magnitude of the transient excursion generated by disturbances with long characteristic times does not depend on inertia. For continental-size, high-voltage power grids, this corresponds to power fluctuations that are correlated on time scales of few seconds or more. We conclude that power fluctuations arising from new renewables will not require per se the deployment of additional rotational inertia. We numerically illustrate our results on the IEEE 118-Bus test case and a model of the synchronous grid of continental Europe.

Jacquod Philippe

*Physical Review E*,
2020, vol. 101, no. 6, article no. 062141

**Summary:**

We investigate eigenstate thermalization from the point of view of vanishing particle and heat currents between a few-body fermionic Hamiltonian prepared in one of its eigenstates and an external, weakly coupled Fermi-Dirac gas. The latter acts as a thermometric probe, with its temperature and chemical potential set so that there is neither particle nor heat current between the two subsystems. We argue that the probe temperature can be attributed to the few-fermion eigenstate in the sense that (i) it varies smoothly with energy from eigenstate to eigenstate, (ii) it is equal to the temperature obtained from a thermodynamic relation in a wide energy range, (iii) it is independent of details of the coupling between the two systems in a finite parameter range, (iv) it satisfies the transitivity condition underlying the zeroth law of thermodynamics, and (v) it is consistent with Carnot's theorem. For the spinless fermion model considered here, these conditions are essentially independent of the interaction strength. When the latter is weak, however, orbital occupancies in the few-fermion system differ from the Fermi-Dirac distribution so that partial currents from or to the probe will eventually change its state. We find that (vi) above a certain critical interaction strength, orbital occupancies become close to the Fermi-Dirac distribution, leading to a true equilibrium between the few-fermion system and the probe. In that case, the coupling between the Fermi-Dirac gas and few-fermion system does not modify the state of the latter, which justifies our approach a posteriori. From these results, we conjecture that for few-body systems with sufficiently strong interaction, the eigenstate thermalization hypothesis is complemented by ensemble equivalence: individual many-body eigenstates define a microcanonical ensemble that is equivalent to a canonical ensemble with grand canonical orbital occupancies.

2019

Tyloo Melvyn Sandy, Pagnier Laurent, Jacquod Philippe

*Science Advances*,
2019, vol. 5, no. 11, article no. eaaw8359

**Summary:**

Identifying key players in coupled individual systems is a fundamental problem in network theory. We investigate synchronizable network-coupled dynamical systems such as high-voltage electric power grids and coupled oscillators on complex networks. We define key players as nodes that, once perturbed, generate the largest excursion away from synchrony. A spectral decomposition of the coupling matrix gives an elegant solution to this identification problem. We show that, when the coupling matrix is Laplacian, key players are peripheral in the sense of a centrality measure defined from effective resistance distances. For linearly coupled systems, the ranking is efficiently obtained through a single Laplacian matrix inversion, regardless of the operational synchronous state. The resulting ranking index is termed LRank. When nonlinearities are present, a weighted Laplacian matrix inversion gives another ranking index, WLRank. LRank provides a faithful ranking even for well-developed nonlinearities, corresponding to oscillator angle differences up to approximately Δθ ≲ 40°.

Jacquod Philippe

*IEEE Access*,
2019, vol. 7, pp. 145889-145900

**Summary:**

The increasing penetration of new renewable sources of electrical energy reduces the overall mechanical inertia available in power grids. This raises a number of issues regarding grid stability over short to medium time scales. A number of approaches have been proposed to compensate for this inertia reduction by deploying substitution inertia in the form of synchronous condensers, flywheels or power-electronic-based synthetic inertia. These resources are limited and expensive; therefore, a key issue is to determine how to optimally place them in the power grid, for instance, to mitigate voltage angle and frequency disturbances following an abrupt power loss. Performance measures in the form of H 2 - norms have recently been introduced to evaluate the overall magnitude of such disturbances. However, despite the mathematical convenience of these measures, analytical results can only be obtained under rather unrealistic assumptions of a uniform damping-to-inertia ratio or a homogeneous distribution of the inertia and/or primary control. Here, we introduce and apply matrix perturbation theory to obtain analytical results for an optimal inertia and primary control placement in the case where both are heterogeneous. This powerful method allows us to construct two simple algorithms that independently optimize the geographical distribution of the inertia and primary control. The algorithms are then implemented for a numerical model of the synchronous transmission grid of continental Europe with different initial configurations. We find that an inertia redistribution has little effect on the grid performance but that the primary control should be redistributed on the slow modes of the network, where the intrinsic grid dynamic requires more time to damp frequency disturbances. For a budget-constraint optimization, we show that increasing the amount of primary control in the periphery of the grid, without changing the inertia distribution, achieves 90 % or more of the maximal possible optimization, already for relatively moderate budgets.

Delabays Robin, Tyloo Melvyn, Jacquod Philippe

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

**Summary:**

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.

Tyloo Melvyn Sandy, Jacquod Philippe

*Physical Review E*,
2019, vol. 100, no. 3, article no. 032303

**Summary:**

In complex network-coupled dynamical systems, two questions of central importance are how to identify the most vulnerable components and how to devise a network making the overall system more robust to external perturbations. To address these two questions, we investigate the response of complex networks of coupled oscillators to local perturbations. We quantify the magnitude of the resulting excursion away from the unperturbed synchronous state through quadratic performance measures in the angle or frequency deviations. We find that the most fragile oscillators in a given network are identified by centralities constructed from network resistance distances. Further defining the global robustness of the system from the average response over ensembles of homogeneously distributed perturbations, we find that it is given by a family of topological indices known as generalized Kirchhoff indices. Both resistance centralities and Kirchhoff indices are obtained from a spectral decomposition of the stability matrix of the unperturbed dynamics and can be expressed in terms of resistance distances. We investigate the properties of these topological indices in small-world and regular networks. In the case of oscillators with homogeneous inertia and damping coefficients, we find that inertia only has small effects on robustness of coupled oscillators. Numerical results illustrate the validity of the theory.

Hindes Jason, Jacquod Philippe, Schwartz Ira B.

*Physical Review E*,
2019, vol. 100, no. 5, article no. 052314

**Summary:**

Many networks must maintain synchrony despite the fact that they operate in noisy environments. Important examples are stochastic inertial oscillators, which are known to exhibit fluctuations with broad tails in many applications, including electric power networks with renewable energy sources. Such non-Gaussian fluctuations can result in rare network desynchronization. Here we build a general theory for inertial oscillator network desynchronization by non-Gaussian noise. We compute the rate of desynchronization and show that higher moments of noise enter at specific powers of coupling: either speeding up or slowing down the rate exponentially depending on how noise statistics match the statistics of a network's slowest mode. Finally, we use our theory to introduce a technique that drastically reduces the effective description of network desynchronization. Most interestingly, when instability is associated with a single edge, the reduction is to one stochastic oscillator.

Tyloo Melvin Sandy, Delabays Robin, Jacquod Philippe

*Physical Review E*,
2019, vol. 99, no. 6, article no. 062213

**Summary:**

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.

Coletta Tomaso, Jacquod Philippe

*IEEE Transactions on Control of Network Systems*,
2020, vol. 7, no. 1, pp. 221-231

**Summary:**

Classes of performance measures expressed in terms of H 2 -norms have been recently introduced to quantify the response of coupled dynamical systems to external perturbations. So far, investigations of these performance measures have been restricted to nodal perturbations. Here, we go beyond these earlier works and consider the equally important, but so far neglected case of line perturbations. We consider a network-reduced power system, where a Kron reduction has eliminated passive buses. Identifying the effect that a line fault in the physical network has on the Kron-reduced network, we find that performance measures depend on whether the faulted line connects two passive, two active buses, or one active to one passive bus. In all cases, performance measures depend quadratically on the original load on the faulted line times a topology-dependent factor. Our theoretical formalism being restricted to Dirac-δ perturbations, we investigate numerically the validity of our results for finite-time line faults. For uniform damping over inertia ratios, we find good agreement with theoretical predictions for longer fault durations in systems with more inertia, for which eigen modes of the network are harder to excite.

**Summary:**

Conventional generators in power grids are steadily substituted with new renewable sources of electric power. The latter are connected to the grid via inverters and as such have little, if any rotational inertia. The resulting reduction of total inertia raises important issues of power grid stability, especially over short-time scales. With the motivation in mind to investigate how inertia reduction influences the transient dynamics following a fault in a large-scale electric power grid, we have constructed a model of the high voltage synchronous grid of continental Europe. To assess grid stability and resilience against disturbance, we numerically investigate frequency deviations as well as rates of change of frequency (RoCoF) following abrupt power losses. The magnitude of RoCoF’s and frequency deviations strongly depend on the fault location, and we find the largest effects for faults located on the support of the slowest mode—the Fiedler mode—of the network Laplacian matrix. This mode essentially vanishes over Belgium, Eastern France, Western Germany, northern Italy and Switzerland. Buses inside these regions are only weakly affected by faults occuring outside. Conversely, faults inside these regions have only a local effect and disturb only weakly outside buses. Following this observation, we reduce rotational inertia through three different procedures by either (i) reducing inertia on the Fiedler mode, (ii) reducing inertia homogeneously and (iii) reducing inertia outside the Fiedler mode. We find that procedure (iii) has little effect on disturbance propagation, while procedure (i) leads to the strongest increase of RoCoF and frequency deviations. This shows that, beyond absorbing frequency disturbances following nearby faults, inertia also mitigates frequency disturbances from distant power losses, provided both the fault and the inertia are located on the support of the slowest modes of the grid Laplacian. These results for our model of the European transmission grid are corroborated by numerical investigations on the ERCOT transmission grid.

Dörfler Florian, Delabays Robin, Jacquod Philippe

*SIAM Journal on Applied Dynamical Systems*,
2019, vol. 18, no.1 pp. 458-480

**Summary:**

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.

2018

**Summary:**

Conventional generators in power grids are steadily substituted with new renewable sources of electric power. The latter are connected to the grid via inverters and as such have little, if any rotational inertia. The resulting reduction of total inertia raises important issues of power grid stability, especially over short-time scales. We have constructed a model of the synchronous grid of continental Europe with which we numerically investigate frequency deviations as well as rates of change of frequency (RoCoF) following abrupt power losses. The magnitude of RoCoF's and frequency deviations strongly depend on the fault location, and we find the largest effects for faults located on the slowest mode - the Fiedler mode - of the network Laplacian matrix. This mode essentially vanishes over Belgium, Eastern France, Western Germany, northern Italy and Switzerland. Buses inside these regions are only weakly affected by faults occuring outside. Conversely, faults inside these regions have only a local effect and disturb only weakly outside buses. Following this observation, we reduce rotational inertia through three different procedures by either (i) reducing inertia on the Fiedler mode, (ii) reducing inertia homogeneously and (iii) reducing inertia outside the Fiedler mode. We find that procedure (iii) has little effect on disturbance propagation, while procedure (i) leads to the strongest increase of RoCoF and frequency deviations. These results for our model of the European transmission grid are corroborated by numerical investigations on the ERCOT transmission grid.

Bonvin Michel, Jacquod Philippe

*Eau énergie air*,
2018, 110, 1, pp. 13-17

**Summary:**

Le réchauffement climatique impacte de plus en plus fortement les conditions des régions alpines. Avec l'augmentation des températures, on assiste à une fonte accrue des volumes glaciaires, à une fonte précoce des neiges annuelles et à des modifications de régimes saisonniers de précipitation. La présente étude se penche sur l'impact combiné de ces changements climatiques et de la transition énergétique sur les modes de fonctionnement des installations hydroélectriques de notre pays. Alors que la production hydro électrique annuelle sera réduite à cause du réchauffement climatique d'ici à 2050, son fonctionnement relatif sera accru en hiver et réduit en été. Ce nouveau régime de production permettra de compenser plus aisément les fluctuations saisonnières de production d'électricité dues aux nouvelles sources d’énergie renouvelable qui seront introduites par la stratégie énergétique 2050 de la Confédération.

Pagnier Laurent, Jacquod Philippe

*Energy*,
2018, 157, pp. 550-560

**Summary:**

The paradox of the energy transition is that the low marginal costs of new renewable energy sources(RES) drag electricity prices down and discourage investments in flexible productions that are needed to compensate for the lack of dispatchability of the new RES. The energy transition thus discourages the investments that are required for its own harmonious expansion. To investigate how this paradox can be overcome, we argue that, under certain assumptions, future electricity prices are rather accurately modeled from the residual load obtained by subtracting non-flexible productions from the load. Armed with the resulting economic indicator, we investigate future revenues for European power plants with various degree of flexibility. We find that, if neither carbon taxes nor fuel prices change, flexible productions would be financially rewarded better and sooner if the energy transition proceeds faster but at more or less constant total production, i.e. by reducing the production of thermal power plants at the same rate as the RES production increases. Less flexible productions, on the other hand, would see the irrevenue grow more moderately. Our results indicate that a faster energy transition with a quicker withdrawal of thermal power plants would reward flexible productions faster.

Tyloo Melvyn, Coletta Tommaso, Jacquod Philippe

*Physical review letters*,
2018, 120, 8

**Summary:**

In network theory, a question of prime importance is how to assess network vulnerability in a fast and reliable manner. With this issue in mind, we investigate the response to external perturbations of coupled dynamical systems on complex networks. We find that for specific, nonaveraged perturbations, the response of synchronous states depends on the eigenvalues of the stability matrix of the unperturbed dynamics, as well as on its eigenmodes via their overlap with the perturbation vector. Once averaged over properly defined ensembles of perturbations, the response is given by new graph topological indices, which we introduce as generalized Kirchhoff indices. These findings allow for a fast and reliable method for assessing the specific or average vulnerability of a network against changing operational conditions, faults, or external attacks.

2017

Delabays Robin, Tyloo Melvyn, Jacquod Philippe

*Chaos: An Interdisciplinary Journal of Nonlinear Science*,

**Summary:**

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.

**Summary:**

We investigate the dependence of transmission losses on the choice of a slack bus in high voltage AC transmission networks. We formulate a transmission loss minimization problem in terms of slack variables representing the additional power injection that each generator provides to compensate the transmission losses. We show analytically that for transmission lines having small, homogeneous resistance over reactance ratios r/x≪1, transmission losses are generically minimal in the case of a unique \textit{slack bus} instead of a distributed slack bus. For the unique slack bus scenario, to lowest order in r/x, transmission losses depend linearly on a resistance distance based indicator measuring the separation of the slack bus candidate from the rest of the network. We confirm these results numerically for several IEEE and Pegase testcases, and show that our predictions qualitatively hold also in the case of lines having inhomogeneous r/x ratios, with optimal slack bus choices reducing transmission losses by 10% typically.

**Summary:**

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.

Delabays Robin, Coletta Tommaso, Jacquod Philippe

*Journal of Mathematical Physics*,
2017, vol. 58, article no. 032703

**Summary:**

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

Coletta Tommaso, Delabays Robin, Adagidedi Inanc, Jacquod Philippe

*New Journal of Physics*,
2016, vol. 18, article no, 103042

**Summary:**

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.

Delabays Robin, Coletta Tommaso, Jacquod Philippe

*Journal of Mathematical Physics*,
2016, vol. 57, no. 3, article no. 032701

**Summary:**

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.

Coletta Tommaso, Jacquod Philippe

*Physical Review*,
2016, vol. 93, article no. 032222

**Summary:**

We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled-oscillator networks. Using a simple model, we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric power grids where a new line can lead to four different scenarios corresponding to enhanced or reduced grid stability as well as increased or decreased power flows. Our analysis shows that the Braess paradox may occur in any complex coupled system, where the synchronous state may be weakened and sometimes even destroyed by additional couplings.

2014

Nichele Fabrizio, Hennel Szymon, Pietsch Patrick, Wegscheider Werner, Stano Peter, Jacquod Philippe, Ihn Thomas, Ensslin Klaus

*Physical Review Letters*,
2014, vol. 114, no. 20, article no. 206601

**Summary:**

Storing, transmitting, and manipulating information using the electron spin resides at the heart of spintronics. Fundamental for future spintronics applications is the ability to control spin currents in solid state systems. Among the different platforms proposed so far, semiconductors with strong spin-orbit interaction are especially attractive as they promise fast and scalable spin control with all-electrical protocols. Here we demonstrate both the generation and measurement of pure spin currents in semiconductor nanostructures. Generation is purely electrical and mediated by the spin dynamics in materials with a strong spin-orbit field. Measurement is accomplished using a spin-to-charge conversion technique, based on the magnetic field symmetry of easily measurable electrical quantities. Calibrating the spin-to-charge conversion via the conductance of a quantum point contact, we quantitatively measure the mesoscopic spin Hall effect in a multiterminal GaAs dot. We report spin currents of 174 pA, corresponding to a spin Hall angle of 34%.

2013

Jacquod Philippe, Büttiker Markus

*Physical Review*,
2013, vol. 88, article no. 241409(R)

**Summary:**

We investigate persistent currents in metallic rings interrupted by a Coulomb-blockaded topological superconducting segment. We show that the presence of Majorana bound states in the superconductor is reflected in the emergence of an h / e harmonic in the persistent current. The Majorana bound states further render the current finite at zero flux, with a sign that is determined by the fermion parity of the superconductor. The resulting peculiar symmetry of the persistent current is compatible with a free energy that is even in time-reversal symmetry-breaking fields. These unique features of the persistent currents are robust against disorder and provide unambiguous signatures of the presence of Majorana fermions.

2018

*Proceedings of 2018 IEEE Conference on Decision and Control (CDC), 17-19 December 2018, Miami, FL, USA*

**Summary:**

New classes of performance measures have been recently introduced to quantify the transient response to external disturbances of coupled dynamical systems on complex networks. These performance measures are time-integrated quadratic forms in the system's coordinates or their time derivative. So far, investigations of these performance measures have been restricted to Dirac- -δ impulse disturbances, in which case they can be alternatively interpreted as giving the long time output variances for stochastic white noise power demand/generation fluctuations. Strictly speaking, the approach is therefore restricted to power fluctuating on time scales shorter than the shortest time scales in the swing equations. To account for power productions from new renewable energy sources, we extend these earlier works to the relevant case of colored noise power fluctuations, with a finite correlation time . We calculate a closed-form expression for generic quadratic performance measures. Applied to specific cases, this leads to a spectral representation of performance measures as a sum over the non-zero modes of the network Laplacian. Our results emphasize the competition between inertia, damping and the Laplacian modes, whose balance is determined to a large extent by the noise correlation time scale τ.

2017

*Proceedings of 2017 IEEE Manchester PowerTech, 18-22 June 2017, Manchester, UK*

**Summary:**

The energy transition is well underway in most European countries. It has a growing impact on electric power systems as it dramatically modifies the way electricity is produced. To ensure a safe and smooth transition towards a pan-European electricity production dominated by renewable sources, it is of paramount importance to anticipate how production dispatches will evolve, to understand how increased fluctuations in power generations can be absorbed at the pan-European level and to evaluate where the resulting changes in power flows will require significant grid upgrades. To address these issues, we construct an aggregated model of the pan-European transmission network which we couple to an optimized, few-parameter dispatch algorithm to obtain time- and geographically-resolved production profiles. We demonstrate the validity of our dispatch algorithm by reproducing historical production time series for all power productions in fifteen different European countries. Having calibrated our model in this way, we investigate future production profiles at later stages of the energy transition - determined by planned future production capacities - and the resulting interregional power flows. We find that large power fluctuations from increasing penetrations of renewable sources can be absorbed at the pan-European level via significantly increased electricity exchanges between different countries. We identify where these increased exchanges will require additional power transfer capacities. We finally introduce a physically-based economic indicator which allows to predict future financial conditions in the electricity market. We anticipate new economic opportunities for dam hydroelectricity and pumped-storage plants.

2016

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

**Summary:**

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.

Achievements