Transient chaos enforces uncertainty in the British power grid
TTransient chaos enforces uncertainty in the Britishpower grid
Lukas Halekotte , Anna Vanselow and Ulrike Feudel Institute for Chemistry and Biology of the Marine Environment, Carl vonOssietzky University Oldenburg, Oldenburg, GermanyE-mail: [email protected]
February 2021
Abstract.
Multistability is a common phenomenon which naturally occurs incomplex networks. If coexisting attractors are numerous and their basins of attractionare complexly interwoven, the long-term response to a perturbation can be highlyuncertain. We examine the uncertainty in the outcome of perturbations to thesynchronous state in a Kuramoto-like representation of the British power grid.Based on local basin landscapes which correspond to single-node perturbations, wedemonstrate that the uncertainty shows strong spatial variability. While perturbationsat many nodes only allow for a few outcomes, other local landscapes show extremecomplexity with more than a hundred basins. Particularly complex domains inthe latter can be related to unstable invariant chaotic sets of saddle type. Mostimportantly, we show that the characteristic dynamics on these chaotic saddles canbe associated with certain topological structures of the network. We find that oneparticular tree-like substructure allows for the chaotic response to perturbations atnodes in the north of Great Britain. The interplay with other peripheral motifsincreases the uncertainty in the system response even further.
Keywords : transient chaos, uncertainty, multistability, chaotic saddle, complexnetworks, power grid, Kuramoto model with inertiaPrepared using the IOP Publishing L A TEX 2 ε preprint class file a r X i v : . [ n li n . AO ] F e b ransient chaos enforces uncertainty in the British power grid
1. Introduction
The Kuramoto model with inertia (KM + ) has been the subject of tremendous researchefforts within the last decade [1]. Certainly, one of the main reasons is that it hasbecome common practice to use networks of nonlinear oscillators – such as the KM + –as coarse-scale representations of real power grids [2, 3, 4]. In addition, the interest inconceptual models of power grids has been fueled by the necessary transformation oftodays electrical distribution grids. It is in this context that the KM + has been appliedto tackle some of the major challenges which accompany the decarbonization of powersupply, like increasing frequency fluctuations [5, 6, 7], the loss of inertia within the grid[8, 9] or the progressive decentralization of power generation [10, 11].A main theme in studies modelling power grids as networks of nonlinear oscillatorsis maintaining a stable operation and thus the avoidance of failures. In particular,the KM + is suited to unravel the relation between network topology and grid stability.Approaches to this issue are manifold and include various stability concepts based, on theone hand, on different perturbation scenarios such as arbitrary small perturbations [12],specific [2, 10] or random [13] large perturbations or stochastic fluctuations [14, 15] and,on the other hand, on the considered system response, like transient excursion [16, 17],asymptotic behavior [13] or both [18] or the severity of cascading failures [19, 20].The most fundamental approaches use properties which are inherent in the systemto quantify its stability. For instance, if small perturbations are considered, eigenvaluesof the operating state can be consulted, whereas properties of its basin of attractionare more suitable if perturbations under consideration are large. The latter explicitlytakes into account that power grids – and thus reasonable parametrizations of the KM + – exhibit multistability. In fact, the phase space can be populated by multiple desiredsynchronized states [21, 22] representing different operating modes as well as severalundesired non-synchronized states representing power outages [18, 23, 24, 25]. In arecent study [26], we contributed to the field of multistability-based stability analysesby determining the minimal fatal shock in a Kuramoto-like representation of the Britishpower grid (figure 1(a)). The minimal fatal shock complements the most commonlyapplied basin stability approach [13, 24, 27, 28, 29] as it considers another property ofthe basin landscape (see e.g. [30, 31, 32] for elaborated explanations on this subject):While the basin stability quantifies the stability of the desired state based on the shareof volume its basin takes within a predefined frame [33, 34], the minimal fatal shock usesthe shortest distance between the desired state and its basin boundary as a criterion ofstability.When determining the minimal fatal shock, the only criterion concerning thelong-term behavior was that the corresponding perturbation should be fatal. In theframework of the KM + , the attribute fatal simply means that a perturbation inducesthe desynchronization of at least one oscillator or, in other words, that it pushes thesystem into any basin of attraction but the one of the fully synchronized state. Whilewe were able to calculate the direction and the magnitude with reasonable precision, ransient chaos enforces uncertainty in the British power grid Figure 1. (a) Topology of the British power grid [35]. (b,c) Different outcomesfollowing the ’same’ perturbation. The perturbation vector applied in (c) is only1 . − times longer than the vector in (b) and points in the same direction. Thenodes which are desynchronized in the end are colored in red and yellow, respectively. the precise outcome of the minimal fatal shock was uncertain. In fact, the slightestvariation – either in the perturbation vector or the integration precision – could lead toseverely different final states, characterized by distinct sets of desynchronized oscillators(figure 1(b),(c)). A high sensitivity towards small variations in the perturbation is anindicator for complexly interwoven basins of attraction possessing fractal boundaries[36, 37]. This impression appears to be confirmed by the erratic – seemingly chaotic –transient dynamics following the fatal shock (figure 1(b),(c)). If the dynamics are indeedtemporarily chaotic, they showcase the influence of a non-attracting chaotic invariantset – a so-called chaotic saddle – embedded within the basin boundary [38, 39, 40].Concerning the outcome of specific perturbations, the fractal dimension of basinboundaries is a sufficient determinant of its predictability. If basin boundaries are indeedfractal, small uncertainties in the initial conditions transfer into large uncertainties inthe final state which is the classic case of final state sensitivity [41, 36]. If howeverperturbations are only vaguely defined which means that already the initial conditionshold some level of uncertainty, not only the dimension of basin boundaries, but furtheraspects like the extension of the fractal domains and the number of distinct basinswithin the – now larger – considered region of the basin landscape are decisive [42, 43].In network applications, a typical constraint regards the number of elements, e.g. nodes ransient chaos enforces uncertainty in the British power grid + as a representation of a power grid did not elaborate on the outcome of aharmful perturbation. In fact, due to the focus being on maintaining the synchronousstate, it is often only distinguished between safe and unsafe perturbations (e.g. [26]).However, this rough classification is systemically justified as the KM + , which neglectsvoltage dynamics [48], ohmic losses [25] and control mechanisms [20], provides a validapproximation of the synchronization dynamics in real power grids only on short timescales [49]. Nevertheless, we believe that an analysis of the complexity and uncertaintyin the basin landscape of the KM + will provide valuable insights, especially concerningthe interplay between topology and uncertainty. For instance, the impact of chaoticinvariant sets might be significant even on short time scales and might determine theseverity of failures within power grids.The whole work is based on the British power grid [35] which has been establishedas a benchmark for power grid analyses [10, 12, 16, 44, 45, 50, 51, 52]. To address theissue of uncertainty within this grid, we proceed as follows: We start by introducing themodel equations and the chosen parametrization (section 2). In the following (section3), we concentrate on cross sections of the basin landscape obtained by perturbing thestate variables of single oscillators. Based on these local basin landscapes, we analyzethe distribution of uncertainties throughout the grid. We find that uncertainties varystrongly and that some topological features in the northernmost part of the grid can berelated to particularly complex basin structures. Therefore, we continue by examiningsome of the local landscapes in the northernmost area in more detail. In this context,special emphasis is given to the invariant sets of saddle-type found in two cross sectionswhich are responsible for the high complexity in the corresponding landscapes and thehigh diversity of accessible attractors. Ultimately, we provide a short parameter studyand conclude with a discussion of our results (section 4).
2. The British power grid
We consider the Kuramoto model with inertia (KM + ) as a model which captures thedesynchronization dynamics within power grids [2, 10]. In this framework, a power grid ransient chaos enforces uncertainty in the British power grid φ i d t = ω i d ω i d t = P i − αω i + N − (cid:88) j =0 K ji sin( φ j − φ i ) , (1)where φ i and ω i denote the phase and frequency deviation of oscillator i from the grid’srated frequency (hereinafter φ i and ω i will simply be called phase and frequency). Inaccordance with the interpretation as a power grid, the parameters α and P i are thegrid’s damping constant and the net power input/output of oscillator i , respectively.The capacities of the transmission lines and therefore also the topology of the grid iscontained in the matrix K , with K ji = K ij > i and j are connected and K ij = 0 otherwise.Throughout this work, we consider only one network which represents the high-voltage transmission grid of the United Kingdom [35] (figure 1(a)). The grid consists of120 nodes and 165 transmission lines and the parametrization is chosen in accordancewith our former work [26]: We assume one half of the oscillators to be generators( P i = + P ) and one half to be consumers ( P i = − P ). The distribution of generatorsand consumers within the grid has been drawn randomly. Furthermore, the maximumcapacity is the same for all transmission lines and thus either K ij = K or K ij = 0.With α = 0 . P = 1 . K = 5 .
0, the model parameters are chosen in order toensure the coexistence of (at least) one desired synchronous state, with constant phases φ ∗ i and frequencies ω ∗ i = 0 for all i ∈ [0 , ω i (cid:54) = 0 for at least one node i . We obtain thesynchronous state by setting the initial phases and frequencies to φ i = 0 and ω i = 0 forall nodes and integrating the system until it reaches a stable state. The perturbationswhich are considered in this work are always applied to this stable state and thus willbe specified by their phase ∆ φ i and frequency difference ∆ ω i = ω i in the following. Aside from the synchronous state, the KM + has been shown to hold a variety of differentattractors. For instance, the network of oscillators can be split into multiple clusterswhich are decoupled from each other but internally synchronized [23]. Accordingly,emerging clusters can be characterized by the common time-average of their oscillator’sfrequencies (cid:104) ω i (cid:105) .Another class of non-synchronous attractors which is expected to be found quietoften – especially for single-node perturbations – are solitary states [25] in which one(1-solitary) or more ( n -solitary) nodes are effectively decoupled from the rest of the gridand swing around their natural frequency ˜ ω i (here ˜ ω i = P i /α = ± P /α = ±
10) whilethe rest of the network forms a synchronized cluster. In the course of this work, we willrefer to corresponding nodes as solitary detached nodes . ransient chaos enforces uncertainty in the British power grid et al [18] reported another, less common class of attractorsin which the detachment of a solitary node is seemingly incomplete or weak. The weakdetachment of a node manifests in a mean frequency lower than its natural frequencyand in a higher amplitude of frequency fluctuations. In a way, the weakly detached node still feels the impact of the opposing cluster.A network of the size of the British grid naturally allows for a large number ofcoexisting solitary states and cluster states. Moreover, diverse composites which containmultiple clusters and solitary detached nodes are possible. We therefore assume theKM + -representation of the British power grid to be highly multistable. In this work, topological characteristics which allow for an easy detachment of subpartsof the grid are of special interest. For the sake of clarity, we will give a short definitionof the most important concepts which we use throughout this work.The first concept of relevance is a tree . A tree is defined as a connected graphwhich contains no loops [53]. Clearly, the British grid is no tree. However, it involvessubgraphs which are tree-shaped [18] and some which resemble a tree-shaped part. Atree-shaped part is a subgraph which, if separated from the rest of the network, due tothe removal of a single node, fulfills the definition of a tree. The simplest tree-shapedpart is a dead end which is simply defined as a node with a single link (degree 1).The significance of dead ends in KM + -representations of power grids has already beenstressed [13, 18]. It should be noted that dead ends and trees are strongly linked to thenetwork theoretical concepts of articulation points and bridges which denote nodes andedges whose removal would cut a network into multiple subgraphs.
3. Uncertainty in the British power grid
The scope of this work can be roughly summarized as examining the relation betweenthe location of a perturbation and the uncertainty of its outcome. By the outcome of aperturbation we mean the asymptotic long-term behavior or attractor which the systemapproaches after being perturbed. In order to ensure that their location is well-defined,we focus on perturbations which, in each case, only affect the phase φ i and frequency ω i of a single node i . Since we additionally assume that the system is situated in the fullysynchronized state prior to a disturbance, all perturbations are basically drawn fromtwo-dimensional cross sections of the high-dimensional phase space. In the following,we will use properties of these cross sections to assess the distribution of uncertainty inthe KM + -representation of the British power grid.We start our analysis by explicitly inspecting each single-node cross section in theBritish grid. To this end, we assign a set of 750 ×
750 initial conditions to each node,wherein phase and frequency deviations at node i are equally distributed in a frame ransient chaos enforces uncertainty in the British power grid φ i ∈ [ − π, π ] and ∆ ω i = ω i ∈ [ − , φ j = 0 and ω j = ω ∗ j = 0 ∀ j (cid:54) = i .Using numerical integration [54], we then determine the attractors to which the initialconditions converge. Attractors are thereby differentiated on the basis of the temporalmean and the fluctuation magnitude of nodal frequencies [18, 23]: We assume that twotrajectories belong to the same attractor if, after a sufficiently long transient (maximumintegration time t max = 5 · ), the differences between their temporal mean, minimumand maximum frequency are smaller than predefined thresholds (0 .
2, 0 .
7, 0 .
7) for each ω j with j ∈ [0 , local basin landscape for each node. With respect to Daza et al [42] and to theusual presentation of basins, we also refer to the relation between initial condition andattractor as the color of that initial condition.The visual inspection of four exemplary local basin landscapes (figure 2(a)-(d)),each corresponding to a generator in the network, shows that their coarse structure bearssome similarity, but that their complexity differs immensely. We find that each locallandscape includes two, more or less dominant, characteristic basins, corresponding tothe fully synchronized state (blue) and a non-synchronous state in which the perturbednode itself is solitary detached (orange). However, while the smooth boundary betweenthese two basins represents the only boundary in the local landscape of N58 (figure2(d)), the landscapes of N5, N14 and N76 (figure 2(a)-(c)) contain domains of multiple,highly intertwined basins. The fine structure and extent of these domains again variesstrongly ranging from rather thin stripes (figure 2(c)) to extended areas (figure 2(b)).Differences in the structure and complexity of local basin landscapes implydifferences in the uncertainty of the outcome of single-node perturbations. In thefollowing, we will address the distribution of uncertainties within the British grid basedon two attributes which capture appropriate characteristics of the local basin landscapes:the coloring and the basin entropy . As the coloring of a node we simply denote thenumber of colors found in the corresponding landscape. Since every color representsan attractor, the coloring is a seemingly natural indicator of the number of possibleoutcomes due to perturbations at the specific node. It should, however, be noted that –due to the finiteness of the set of initial conditions, the uneven distribution of complexdomains and the potential length of some transients – the coloring itself holds a certaindegree of uncertainty. In fact, depending on the distribution of basins within a locallandscape, the real number of accessible attractors can be severely underestimated (seeAppendix A). Nevertheless, we assume that the coloring is instructive since it allows usto gain some insight into the diversity of accessible attractors.The coloring is, however, not suitable as an exclusive measure of uncertainty sinceit neglects essential aspects like the frequency of colors or the mingling of correspondingbasins. Suitable measures which could complement the coloring in this regard are thebasin stability [33] and the uncertainty exponent [41], both of which have been appliedin similar contexts to capture one of these aspects [46, 47]. We choose neither of thetwo but follow a more holistic approach which has been introduced by Daza et al [42]. ransient chaos enforces uncertainty in the British power grid Figure 2.
Uncertainty in the British power grid. (a-d) Local basin landscapes of fourexemplary nodes: (a) N5, (b) N14, (c) N76, (d) N58. Each color depicts a differentbasin of attraction. While the darker blue always corresponds to the basin of the fullysynchronized state, the remaining basins are sorted according to their relative size inthe landscape. The seven most common basins receive a specific color while additionalbasins are colored in recurring shades of gray. (Middle column) Basin entropy (innercolor) and coloring (edge color) of all nodes. Circles portray generators and squaresconsumers.
Their measure – called basin entropy – has been shown to be sensitive to the fractalityand size of basin boundaries as well as to the number of colors. It thus incorporatesmultiple aspects of uncertainty within a single index. In order to obtain a nodewiseuncertainty index, we apply a single node-version of the basin entropy which is basedon the local basin landscapes. The basin entropy S b is defined as S b = 1 N b N b (cid:88) k =1 (cid:32) − m k (cid:88) l =1 p kl log p kl (cid:33) . (2)The procedure to obtain S b can be outlined as follows: Divide a region in phase space– in our case, one of the cross sections – into N b boxes, calculate the Gibbs entropy foreach box k (term in brackets in (2)) and average over all boxes. The calculation of theGibbs entropy is thereby based on the distribution of colors in the corresponding box.In this context, m k denotes the number of colors found in box k and p kl the probability ransient chaos enforces uncertainty in the British power grid l (approximatedby the number of initial conditions with color l divided by the total number of initialconditions in box k ). We calculate the basin entropy for each node (figure 2(e)) basedon the set of 750 ×
750 initial conditions which we divide into 150 ×
150 equally sizedboxes, each holding 25 initial conditions.The distribution of the coloring and basin entropy over the British grid showsthat the uncertainty in the outcome of a perturbation exhibits strong variations, bothglobally and locally (figure 2(e)). We find that large parts of the grid – especiallyin the center – possess minimal uncertainties including a coloring of 2 (light yellowcoloring). On the contrary, we find several peripheral areas – e.g. in the southeast, in thewest and in the north in particular – in which the outcomes of localized perturbationsare highly uncertain (darker colors). It is notable that the proximity of nodes withextraordinary high values of basin entropy and coloring always includes other nodeswhose basins exhibit high complexities as well. An observation which might suggesta relation between uncertainty and specific larger topological structures. Nevertheless,we also find that within areas of high uncertainty, basin entropy and coloring varystrongly among nodes. For instance, nodes which are easily detached – like dead ends– exhibit lower uncertainties than adjacent nodes which show relatively high values ofbasin entropy and coloring.
Particularly high indices of uncertainty are found in the northernmost or ’Scottish’ partof the grid (figure 3(b) and figure 2(e)). Especially in the northeast, the high basinentropies of several adjacent nodes are striking. Importantly, this region of pronounceduncertainty coincides with an exceptional topological structure which resembles a deadend in the sense that it also holds a single connection (bridge originating at N14; figure3(b)) to the rest of the grid. We refer to the corresponding structure as tree-like (yellowshaded region in figure 3(b)) since it contains multiple bridges and articulation pointswhich are characteristic for trees but one more edge than an actual tree.Despite the accumulation of nodes with high uncertainty around the tree-like, thehighest coloring (1851) is found elsewhere, rather distant at N5 (still in Scotland!). Infact, not only the most but also the second (696) and third (458) most colorful node islocated in this area. Compared to the highest coloring found at the tree-like (coloring of101 at N14), all three allow for a massive number of possible outcomes. Topologically,the close proximity of three associated dead ends in this region is striking: Each of thethree most colorful nodes is linked to a dead end (N0, N1 and N6 ; figure 3(b)).In order to uncover the origin of the high uncertainties in these two distinct regions,we will examine two exemplary cross sections in more detail. We choose the node N5 asit exhibits by far the highest coloring as well as the highest basin entropy apart from thetree-like and its single neighbor N14. As a representative of the tree-like, we considerN14 as it represents, in a way, the topological equivalent to N5: Both are articulation ransient chaos enforces uncertainty in the British power grid Figure 3.
Spatial resolution of the Gibbs entropy – term in brackets in (2) – for thenodes N5 (a) and N14 (c). Minimum and maximum Gibbs entropies are 0 (black)and − log (1 / ≈ .
22 (yellow), respectively. Minimum/maximum: All 25 initialconditions in one box approach the same/different attractor(s). (b) Enlarged displayof the northern part of the British grid. The nodes N5 (a) and N14 (c) are framedin magenta. The yellow shaded region highlights the tree-like substructure. As areference for upcoming results, the dead ends N0, N1 and N6 are marked in cyan.Circles represent generators and squares consumers. points which link a peripheral motif – dead end and tree-like, respectively – to the restof the grid.The basin boundaries in the complex domains of both local basin landscapes areindeed fractal (uncertainty exponent γ ≈ .
02 for N5, γ ≈ .
03 for N14; see AppendixB). Furthermore, according to the basin entropy (2), the uncertainty of the two nodesis pretty much the same (N5: S b ≈ . S b ≈ . − log (1 /
25) which means that from a set of 5 × In the following, we expand our analysis on the two exemplary local basin landscapes,starting with the node which shows the lower coloring, N14. Despite its high complexity,the local basin landscape of N14 can be divided into three distinct regions. The first twocorrespond to the basins of the two most commonly approached attractors, characteristic ransient chaos enforces uncertainty in the British power grid
Figure 4. (d) Division of the local basin landscape of N14 into three regions – blue,red and grayscale – owing to three distinct transient behaviors. The different shadesof gray denote the lifetime of the tree-like cluster. (a-c) Exemplary trajectories, eachcorresponding to one of the three regions (corresponding colored stripes). Thin graylines depict the actual frequencies and thick colored lines the moving average of thefrequencies of N14 (cyan), the nodes in the tree-like (yellow) and in the rest of Scotland(magenta).
Differences between the three regions (blue, orange, mixed) become apparentespecially in the transient dynamics. While the convergence of initial conditions in theblue and orange region is straightforward and usually fast (figure 4(a),(b)), trajectoriesinitiated in the mixed region often pass rather long transient stages before settling onone of several accessible attractors (figure 4(c)). Importantly, most of these transientsshow a characteristic behavior which involves strong, erratic frequency fluctuations. ransient chaos enforces uncertainty in the British power grid
We have seen that the complex basin landscape at N14 is due to transients whichtemporary follow an unstable chaotic invariant set. As this chaotic saddle is visiblyassociated with the tree-like, the interrelationship between network topology and basincomplexity becomes apparent. However, the origin of the extraordinary high uncertaintyat the rather distant node N5 is still unclear (figure 2(a),(e)).To obtain a first hint at the source of its colorfulness, we take a look at some of theattractors which are accessible due to perturbations at N5. The three most commonattractors which are the fully synchronized state (not shown), a 2-solitary state in whichN5 and its dead end neighbor N6 are detached (figure 5(a)) and a 1-solitary state inwhich only N6 is detached (figure 5(b)) are not surprising. Furthermore, finding thetopological counterpart of the latter in which N6 is only weakly detached is hardlysurprising, although it is comparatively rare (figure 5(e)). ransient chaos enforces uncertainty in the British power grid Figure 5.
Six exemplary attractors approached after a perturbation at node N5.Shown are the mean frequencies (circles and squares) and the range of oscillationsbounded by the minimum and maximum frequency value (vertical dark gray lines) forthe upper 27 oscillators within the British grid. The color saturation of the markersdepict the mean frequencies (blue - negative, red - positive). (a-c) The three mostcommon attractors – besides the fully synchronized state ( ω i = 0 ∀ i ) – which areapproached by 93529 (a), 75314 (b) and 23640 (c) initial conditions. (d-f) Less commonattractors which are approached by 963 (d), 142 (e) and 5 (f) initial conditions. Aside from the few expectable outcomes, the response to perturbations at N5 canbe exceptionally complex which shows, for instance, in severe outcomes involving thedesynchronization of large parts of the grid (figure 5(f)). However, also the accessibilityof some of the rather common outcomes is surprising. An illustrative example is a1-solitary state whose detached node lies within the tree-like and thus far from theoriginally perturbed node (figure 5(c)). Another example resembles the chaotic saddlein the sense that it involves the same weakly detached cluster but one additionallydetached dead end, N0 (figure 5(d)). Interestingly, both attractors are related to thetree-like which might indicate its impact on the complexity of the local basin landscapeof N5.In order to test this, we again take a look at the transient dynamics. We findthat trajectories initiated within the mixed areas of the cross section of N5 exhibit a ransient chaos enforces uncertainty in the British power grid
Figure 6.
Chaotic transients originating from the local basin landscape of N5. (a)The grayscale displays the lifetime of the tree-like cluster. Magenta colored initialconditions approach an attractor in which the cluster persists. The cluster is not formedfor initial conditions situated in white areas. (b) Depiction of the nodes (brownishyellow, cyan, dark blue, black) which are detached during the existence of the cluster.Gray points mark the additional desynchronization of no other nodes and magentapoints the additional detachment of N0 and any combination of further nodes.
These long lifetimes come as no surprise since we have already noticed thatperturbations at N5 can lead to an attractor which involves the weakly detached tree-likecluster (figure 5(d)). The obvious difference between the chaotic transient approachedfor perturbations at N14 and this attractor is that the node N0 is solitary detached inthe latter (figure 5(d)). Furthermore, we find multiple additional transient dynamicswhich involve the weakly detached tree-like cluster but differ with regard to the nodeswhich are additionally detached, especially N1 and N6. Each combination of additionallydetached nodes accounts for a particular chaotic saddle (or attractor) which ultimatelyexplains the extraordinary coloring of N5 – each saddle paves the way to a distinct setof attractors. A spatial depiction of the temporarily approached saddles shows that ransient chaos enforces uncertainty in the British power grid
We have seen that the existence and accessibility of chaotic saddles is a decisive factordetermining the uncertainty in the outcome of single-node perturbations. However, sofar, our examinations have been entirely based on one particular parametrization ofthe British grid. In the following, we aim at a brief insight into the variability of theuncertainty depending on varying parameters. As an exemplary parameter we choose K which determines the maximum capacity of all transmission lines. Since we need thefully synchronized state as a reference for the positioning of our cross sections, we vary K within a range which ensures the existence and stability of this state. For each valueof K , we then calculate the coloring and basin entropy in accordance with our earlierexplanations. For the sake of simplicity, we concentrate on one particular node, N14.The uncertainty in the outcome of perturbations is particularly high for a ratherwide range of low transmission capacities which coincide with a less voluminous basinof the fully synchronized state (figure 7(a)). In this respect, the decrease in uncertaintywith increasing K is non-surprising as the basin of the fully synchronized state takesincreasing amounts of the tested cross section while, at the same time, the number ofaccessible attractors diminishes. However, in contrast to the coloring, the progressionof the basin entropy is not at all monotonous but does indeed possess pronounced localminima and maxima (especially b and c in figure 7(a)).The resulting topography – valley: minima at b in figure 7, peak: maxima at c infigure 7 – emphasizes once again that particular unstable invariant sets determine thecomplexity of the local basin landscape. To recognize this, a closer look at the dominantregions of the local landscape for some exemplary values of K is worthwhile (figure 2(b)and figure 7(b)-(d)): For K = 5 .
0, the local landscape is populated by three dominantregions of which one can be characterized on account of the transient dynamics whichshow the impact of a chaotic saddle. As K is increased, this chaotic invariant setbecomes attractive. Accordingly, the formerly mixed region of initial conditions whichwas determined by the attraction and subsequent repulsion of the saddle does nowapproach this attractor (orange domain in figure 7(b)). Since now the local landscapeis mainly filled by three basins, the basin entropy exhibits comparatively low values (bin figure 7(a)). The subsequent increase in the basin entropy is then established due tothe solitary state losing its stability and thereby making room for a ’new’ mixed region(figure 7(c)). Interestingly, at this point, the two attractors whose basins populatemost of the local landscape – aside from the fully synchronized state – both include theweakly detached tree-like cluster and only differ slightly in the mean frequency withinthe cluster. Subsequently, the ’new’ mixed region is again taken over by the basin of asolitary state in which N14 is detached. At the same time the expansion of other basinsdiminishes which ultimately leads to basin entropies at constantly low levels. ransient chaos enforces uncertainty in the British power grid Figure 7.
Uncertainty in the local basin landscape of N14 depending on thetransmission capacity K . (a) Basin entropy and coloring. Magenta arrows referto the local basin landscapes shown in figure 2(b) and in the subfigures (b), (c), (d).(b)-(d) Local basin landscapes for K = 6 . K = 6 . K = 8 .
4. Discussion
Inspired by findings in an earlier work [26] (figure 1), we examined the uncertainty in theoutcome of localized perturbations within a Kuramoto-like representation of the Britishpower grid. Using local basin landscapes corresponding to perturbations at single nodes,we demonstrated that the basin complexity and thus the uncertainty in the outcomeof a perturbation is highly variable with low uncertainty predominantly in the core ofthe network and high uncertainty concentrated close to peripheral regions (figure 2).Particularly complex domains in the basin landscape which resemble riddled structurescan be related to unstable invariant chaotic sets which are accessible by initial conditionswithin these areas (figure 4 and figure 6). However, most importantly, we showed thatthe chaotic dynamics on the saddles are related to certain topological structures within ransient chaos enforces uncertainty in the British power grid ransient chaos enforces uncertainty in the British power grid
Acknowledgments
The simulations were performed at the HPC Cluster CARL, located at the Universityof Oldenburg (Germany) and funded by the DFG through its Major ResearchInstrumentation Programme (INST 184/157-1 FUGG) and the Ministry of Science andCulture (MWK) of the Lower Saxony State.
Appendix A. Some comments on the coloring
A short reminder: In our framework, the coloring simply denotes the number of basins ofattraction found in one local basin landscape. This is, however, not necessarily the sameas the number of basins which are actually located in the landscape, since each landscapeis only approximated by a finite set of initial conditions. In order to demonstrate therelation between the number of tested initial conditions and the coloring being detected,we determine the coloring for random subsets of the complete set of 750 ×
750 initialconditions and associated basins (see Section 3.1). To this end, we randomly select N ini initial conditions from the complete set and count the number of basins whichare approached from this subset. By varying the size of the subset N ini , we obtain arelation between the coloring and the number of initial conditions (figure A1(b)). Inconsideration of the random selection, we calculate the coloring for each value of N ini as the mean of 100 realizations.We find that for many nodes, the increase in coloring depending on the number ofconsidered initial conditions is far from being saturated. Interestingly, for sufficientlylarge numbers of initial conditions, this relation follows approximately a power law(figure A1(b)). The power law behavior is useful as it allows us to visualize therelation between coloring and the number of initial conditions for the whole networkby approximating the power k for each node (figure A1(a)). The distribution of powersshows that the ’true’ coloring could be particularly underestimated for nodes whichalready hold a high coloring, such as N5. Furthermore, the powers show some spatialcorrelation (see e.g. the neighborhood of N5 or N14). This might indicate that the localbasin landscapes of certain adjacent nodes hold very similar sets of accessible attractors ransient chaos enforces uncertainty in the British power grid Figure A1.
Dependency of the coloring on the number of tested initial conditions.(a) Power k which describes the increase in coloring with increasing number of initialconditions in a log-log plot. (b) Relation between the coloring and the number of testedinitial conditions for four exemplary nodes. Initial conditions are drawn randomly fromthe complete set of initial conditions assigned to each node. Each small point is basedon 100 realizations. Big points denote the coloring for the complete set of 750 × which are, however, embedded in complex domains of different size. Appendix B. Fractal basin boundaries at N5 and N14
We have seen that the local basin landscapes of the nodes N5 and N14 contain complexregions of mixed basins or colors, respectively (figure 2(a),(b) and figure 3(a),(c)). Inorder to verify that the contained basin boundaries are indeed fractal, we calculatethe uncertainty exponent γ [41]. To this end, we compute the fraction f ( (cid:15) ) of initialconditions which are uncertain with respect to an initial error (cid:15) . In this framework,an initial condition is called uncertain with respect to (cid:15) if it holds a different colorthan an associated initial condition which has a distance of (cid:15) (they converge to differentattractors). The scaling relation between f ( (cid:15) ) and (cid:15) is expected to be f ( (cid:15) ) ∼ (cid:15) γ , (B.1)with an uncertainty exponent γ < ransient chaos enforces uncertainty in the British power grid Figure B1.
Final state sensitivity in the mixed regions of the local basin landscapesof N5 and N14. (a,c) Boundary boxes (areas containing basin boundaries) are coloredblack. (b) Relation between fraction of uncertain initial conditions and initial error (cid:15) within boundary boxes. The slope of the yellow and blue line gives the uncertaintyexponent γ for N5 and N14, respectively. conditions in the local basin landscapes (see figure 2(a),(b)) to establish a space-fillingset of boxes which are either defined as boundary boxes or no-boundary boxes . Eachbox covers the phase space between 2 × (cid:15) in the ω i -direction (uncertaintywith respect to the frequency of node i ). In accordance with the description in [38],we proceed until we obtain 5000 pairs which denote an uncertain initial condition andthus obtain f ( (cid:15) ) ≈ /N , with N being the number of selected pairs. By varyingthe error (cid:15) within the interval (cid:15) ∈ [10 − , − ], we obtain a relation between f ( (cid:15) ) and (cid:15) which fulfills (B.1) for lower values of (cid:15) (figure B1(b)). Appendix C. Trajectories on the chaotic invariant sets
Using the stagger-and-step method [55], we create pseudo-trajectories of the non-attracting invariant sets which are approached from initial conditions in the mixed/grayregions of the local basin landscapes of N14 (figure 4(d)) and N5 (figure 6). In figure C1,segments of two of these pseudo-trajectories are shown. The first is the transient chaotictrajectory which we found for perturbations at N14 (figure C1(a)). In this ’standard’case, the weakly detached tree-like cluster is established while all remaining nodes forman opposing giant cluster. The second segment shows the transient chaotic trajectorywhere, simultaneously to the formation of the weakly detached tree-like cluster, the nodeN1 is solitary detached (figure C1(b)). Furthermore, the attracting chaotic invariant setis shown (figure C1(c)). ransient chaos enforces uncertainty in the British power grid | r ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N − (cid:88) j =0 e i φ j ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (C.1)where N is the number of oscillators in the network and | r | ∈ [0 ,
1] with | r | = 1 indicatingtotal synchronization.We see that if the tree-like cluster is formed and the node N0 is desychronized, thechaotic dynamics are stabilized (at least they persist for very long times; see figure 6).This stabilization goes along with the disappearance of characteristic drops in the timeseries (figure C1(c)) which show up in the mean frequencies of the tree-like and thephase coherence of the pseudo-trajectories (figure C1(a),(b)). Interestingly, we find thatthe detachment of N1 already affects the characteristics of these drops (figure C1(b)).The drops are lower and occur less frequently than for the ’standard’ case in which noadditional node is detached (figure C1(a)). Since we also observe that the escape fromthe chaotic transients takes place during these drops (figure 4(c)), the stabilization ofthe chaotic dynamics can be ascribed to the demise of these escape drops.Finally, using the corresponding method in the JiTCODE module [54] which isbased on the approach by Benettin et al [67], we compute the largest Lyapunov exponent λ for each of the three exemplary time series (figure A1). We find that all three exhibita Lyapunov exponent λ > λ ≈ .
07 for the pseudo-trajectory in figure C1(a), λ ≈ .
06 for figure C1(b) and λ ≈ .
003 for figure C1(c). Accordingly, the dynamicsof all three time series are chaotic, although, the chaotic signal of the dynamics whichallow for the persistence of the weakly detached tree-like cluster is rather weak. ransient chaos enforces uncertainty in the British power grid Figure C1.
Chaotic time series. Upper subfigures show the moving average over 25time units of the frequencies of the 27 nodes within Scotland, including the tree-like.Lower subfigures depict the absolute value of the order parameter in dark gray andits moving average in magenta. (a) Pseudo-trajectory of the ’standard’ chaotic saddle.(b) Pseudo-trajectory of the chaotic saddle in which the node N6 is solitary detached.(c) Trajectory of the chaotic attractor in which the node N0 is solitary detached. ransient chaos enforces uncertainty in the British power grid References [1] Francisco A Rodrigues, Thomas K DM Peron, Peng Ji, and J¨urgen Kurths. The Kuramoto modelin complex networks.
Physics Reports , 610:1–98, 2016.[2] Giovanni Filatrella, Arne Hejde Nielsen, and Niels Falsig Pedersen. Analysis of a power grid usinga Kuramoto-like model.
The European Physical Journal B-Condensed Matter and ComplexSystems , 61(4):485–491, 2008.[3] Takashi Nishikawa and Adilson E Motter. Comparative analysis of existing models for power-gridsynchronization.
New Journal of Physics , 17(1):015012, 2015.[4] Mehrnaz Anvari, Frank Hellmann, and Xiaozhu Zhang. Introduction to focus issue: Dynamics ofmodern power grids.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 30(6):063140,2020.[5] Katrin Schmietendorf, Joachim Peinke, and Oliver Kamps. The impact of turbulent renewableenergy production on power grid stability and quality.
The European Physical Journal B ,90(11):1–6, 2017.[6] Hauke Haehne, Jannik Schottler, Matthias Waechter, Joachim Peinke, and Oliver Kamps. Thefootprint of atmospheric turbulence in power grid frequency measurements.
EPL (EurophysicsLetters) , 121(3):30001, 2018.[7] Hauke Haehne, Katrin Schmietendorf, Samyak Tamrakar, Joachim Peinke, and Stefan Kettemann.Propagation of wind-power-induced fluctuations in power grids.
Physical Review E ,99(5):050301, 2019.[8] Bala Kameshwar Poolla, Saverio Bolognani, and Florian D¨orfler. Optimal placement of virtualinertia in power grids.
IEEE Transactions on Automatic Control , 62(12):6209–6220, 2017.[9] Laurent Pagnier and Philippe Jacquod. Inertia location and slow network modes determinedisturbance propagation in large-scale power grids.
PloS one , 14(3):e0213550, 2019.[10] Martin Rohden, Andreas Sorge, Marc Timme, and Dirk Witthaut. Self-organized synchronizationin decentralized power grids.
Physical Review Letters , 109(6):064101, 2012.[11] Martin Rohden, Andreas Sorge, Dirk Witthaut, and Marc Timme. Impact of network topology onsynchrony of oscillatory power grids.
Chaos: An Interdisciplinary Journal of Nonlinear Science ,24(1):013123, 2014.[12] Tommaso Coletta and Philippe Jacquod. Linear stability and the Braess paradox in coupled-oscillator networks and electric power grids.
Physical Review E , 93(3):032222, 2016.[13] Peter J Menck, Jobst Heitzig, J¨urgen Kurths, and Hans Joachim Schellnhuber. How dead endsundermine power grid stability.
Nature communications , 5(1):1–8, 2014.[14] Benjamin Sch¨afer, Moritz Matthiae, Xiaozhu Zhang, Martin Rohden, Marc Timme, and DirkWitthaut. Escape routes, weak links, and desynchronization in fluctuation-driven networks.
Physical Review E , 95(6):060203, 2017.[15] Melvyn Tyloo, Robin Delabays, and Philippe Jacquod. Noise-induced desynchronization andstochastic escape from equilibrium in complex networks.
Physical Review E , 99(6):062213, 2019.[16] Frank Hellmann, Paul Schultz, Carsten Grabow, Jobst Heitzig, and J¨urgen Kurths. Survivabilityof deterministic dynamical systems.
Scientific reports , 6:29654, 2016.[17] Melvyn Tyloo, Laurent Pagnier, and Philippe Jacquod. The key player problem in complexoscillator networks and electric power grids: Resistance centralities identify local vulnerabilities.
Science advances , 5(11):eaaw8359, 2019.[18] Jan Nitzbon, Paul Schultz, Jobst Heitzig, J¨urgen Kurths, and Frank Hellmann. Decipheringthe imprint of topology on nonlinear dynamical network stability.
New Journal of Physics ,19(3):033029, 2017.[19] Martin Rohden, Daniel Jung, Samyak Tamrakar, and Stefan Kettemann. Cascading failures in acelectricity grids.
Physical Review E , 94(3):032209, 2016.[20] Benjamin Sch¨afer, Dirk Witthaut, Marc Timme, and Vito Latora. Dynamically induced cascadingfailures in power grids.
Nature communications , 9(1):1–13, 2018. ransient chaos enforces uncertainty in the British power grid [21] T Coletta, R Delabays, I Adagideli, and Ph Jacquod. Topologically protected loop flows in highvoltage ac power grids. New Journal of Physics , 18(10):103042, 2016.[22] Debsankha Manik, Marc Timme, and Dirk Witthaut. Cycle flows and multistability in oscillatorynetworks.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 27(8):083123, 2017.[23] Simona Olmi, Adrian Navas, Stefano Boccaletti, and Alessandro Torcini. Hysteretic transitionsin the Kuramoto model with inertia.
Physical Review E , 90(4):042905, 2014.[24] Heetae Kim, Sang Hoon Lee, J¨orn Davidsen, and Seung-Woo Son. Multistability and variationsin basin of attraction in power-grid systems.
New Journal of Physics , 20(11):113006, 2018.[25] Frank Hellmann, Paul Schultz, Patrycja Jaros, Roman Levchenko, Tomasz Kapitaniak, J¨urgenKurths, and Yuri Maistrenko. Network-induced multistability through lossy coupling and exoticsolitary states.
Nature Communications , 11(1):1–9, 2020.[26] Lukas Halekotte and Ulrike Feudel. Minimal fatal shocks in multistable complex networks.
Scientific reports , 10(1):1–13, 2020.[27] Heetae Kim, Sang Hoon Lee, and Petter Holme. Building blocks of the basin stability of powergrids.
Physical Review E , 93(6):062318, 2016.[28] Yannick Feld and Alexander K Hartmann. Large-deviations of the basin stability of power grids.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 29(11):113103, 2019.[29] Cristian Camilo Galindo-Gonz´alez, David Angulo-Garc´ıa, and Gustavo Osorio. Decreasedresilience in power grids under dynamically induced vulnerabilities.
New Journal of Physics ,22(10):103033, 2020.[30] MS Soliman and JMT Thompson. Integrity measures quantifying the erosion of smooth and fractalbasins of attraction.
Journal of Sound and Vibration , 135(3):453–475, 1989.[31] Brian Walker, Crawford S Holling, Stephen R Carpenter, and Ann Kinzig. Resilience, adaptabilityand transformability in social–ecological systems.
Ecology and society , 9(2), 2004.[32] Chiranjit Mitra, J¨urgen Kurths, and Reik V Donner. An integrative quantifier of multistabilityin complex systems based on ecological resilience.
Scientific reports , 5(1):1–10, 2015.[33] Peter J Menck, Jobst Heitzig, Norbert Marwan, and J¨urgen Kurths. How basin stabilitycomplements the linear-stability paradigm.
Nature physics , 9(2):89–92, 2013.[34] Ulrike Feudel, Celso Grebogi, Brian R Hunt, and James A Yorke. Map with more than 100coexisting low-period periodic attractors.
Physical Review E , 54(1):71, 1996.[35] Ingve Simonsen, Lubos Buzna, Karsten Peters, Stefan Bornholdt, and Dirk Helbing. Transientdynamics increasing network vulnerability to cascading failures.
Physical Review Letters ,100(21):218701, 2008.[36] Steven W McDonald, Celso Grebogi, Edward Ott, and James A Yorke. Fractal basin boundaries.
Physica D: Nonlinear Phenomena , 17(2):125–153, 1985.[37] Ulrike Feudel. Complex dynamics in multistable systems.
International Journal of Bifurcationand Chaos , 18(06):1607–1626, 2008.[38] Ying-Cheng Lai and Tam´as T´el.
Transient chaos: complex dynamics on finite time scales , volume173. Springer Science & Business Media, 2011.[39] Celso Grebogi, Edward Ott, and James A Yorke. Metamorphoses of basin boundaries in nonlineardynamical systems.
Physical Review Letters , 56(10):1011, 1986.[40] Helena E Nusse and James A Yorke. A procedure for finding numerical trajectories on chaoticsaddles.
Physica D: Nonlinear Phenomena , 36(1-2):137–156, 1989.[41] Celso Grebogi, Steven W McDonald, Edward Ott, and James A Yorke. Final state sensitivity: anobstruction to predictability.
Physics Letters A , 99(9):415–418, 1983.[42] Alvar Daza, Alexandre Wagemakers, Bertrand Georgeot, David Gu´ery-Odelin, and Miguel AFSanju´an. Basin entropy: a new tool to analyze uncertainty in dynamical systems.
Scientificreports , 6(1):1–10, 2016.[43] Alexandre R Nieto, Euaggelos E Zotos, Jes´us M Seoane, and Miguel AF Sanju´an. Measuringthe transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems.
Nonlinear Dynamics , 99(4):3029–3039, 2020. ransient chaos enforces uncertainty in the British power grid [44] Dirk Witthaut, Martin Rohden, Xiaozhu Zhang, Sarah Hallerberg, and Marc Timme. Critical linksand nonlocal rerouting in complex supply networks. Physical Review Letters , 116(13):138701,2016.[45] Chiranjit Mitra, Anshul Choudhary, Sudeshna Sinha, J¨urgen Kurths, and Reik V Donner.Multiple-node basin stability in complex dynamical networks.
Physical Review E , 95(3):032317,2017.[46] Everton S Medeiros, Rene O Medrano-T, Iberˆe L Caldas, and Ulrike Feudel. Boundaries ofsynchronization in oscillator networks.
Physical Review E , 98(3):030201, 2018.[47] Vagner Dos Santos, Fernando S Borges, Kelly C Iarosz, Iberˆe L Caldas, JD Szezech, Ricardo LViana, Murilo S Baptista, and Antonio M Batista. Basin of attraction for chimera states ina network of R¨ossler oscillators.
Chaos: An Interdisciplinary Journal of Nonlinear Science ,30(8):083115, 2020.[48] Katrin Schmietendorf, Joachim Peinke, Rudolf Friedrich, and Oliver Kamps. Self-organizedsynchronization and voltage stability in networks of synchronous machines.
The EuropeanPhysical Journal Special Topics , 223(12):2577–2592, 2014.[49] Jan Machowski, Zbigniew Lubosny, Janusz W Bialek, and James R Bumby.
Power systemdynamics: stability and control . John Wiley & Sons, 2020.[50] Dirk Witthaut and Marc Timme. Braess’s paradox in oscillator networks, desynchronization andpower outage.
New journal of physics , 14(8):083036, 2012.[51] Debsankha Manik, Martin Rohden, Henrik Ronellenfitsch, Xiaozhu Zhang, Sarah Hallerberg, DirkWitthaut, and Marc Timme. Network susceptibilities: Theory and applications.
PhysicalReview E , 95(1):012319, 2017.[52] Robin Delabays, Melvyn Tyloo, and Philippe Jacquod. The size of the sync basin revisited.
Chaos:An Interdisciplinary Journal of Nonlinear Science , 27(10):103109, 2017.[53] Mark Newman.
Networks . Oxford university press, 2018.[54] Gerrit Ansmann. Efficiently and easily integrating differential equations with jitcode, jitcdde, andjitcsde.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 28(4):043116, 2018.[55] David Sweet, Helena E Nusse, and James A Yorke. Stagger-and-step method: Detecting andcomputing chaotic saddles in higher dimensions.
Physical Review Letters , 86(11):2261, 2001.[56] Everton S Medeiros, Rene O Medrano-T, Iberˆe L Caldas, Tam´as T´el, and Ulrike Feudel. State-dependent vulnerability of synchronization.
Physical Review E , 100(5):052201, 2019.[57] Suso Kraut and Ulrike Feudel. Multistability, noise, and attractor hopping: The crucial role ofchaotic saddles.
Physical Review E , 66(1):015207, 2002.[58] Tilman Weckesser, Hj¨ortur J´ohannsson, and Jacob Østergaard. Impact of model detail ofsynchronous machines on real-time transient stability assessment. In , pages 1–9. IEEE, 2013.[59] Sabine Auer, Kirsten Kleis, Paul Schultz, J¨urgen Kurths, and Frank Hellmann. The impact ofmodel detail on power grid resilience measures.
The European Physical Journal Special Topics ,225(3):609–625, 2016.[60] Thomas Lilienkamp and Ulrich Parlitz. Terminating transient chaos in spatially extended systems.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 30(5):051108, 2020.[61] Rub´en Cape´ans, Juan Sabuco, Miguel AF Sanju´an, and James A Yorke. Partially controllingtransient chaos in the lorenz equations.
Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences , 375(2088):20160211, 2017.[62] Mukeshwar Dhamala and Ying-Cheng Lai. Controlling transient chaos in deterministic flows withapplications to electrical power systems and ecology.
Physical Review E , 59(2):1646, 1999.[63] Alex Arenas, Albert D´ıaz-Guilera, Jurgen Kurths, Yamir Moreno, and Changsong Zhou.Synchronization in complex networks.
Physics reports , 469(3):93–153, 2008.[64] Arkady Pikovsky and Michael Rosenblum. Dynamics of globally coupled oscillators: Progress andperspectives.
Chaos: An Interdisciplinary Journal of Nonlinear Science , 25(9):097616, 2015. ransient chaos enforces uncertainty in the British power grid [65] Arthur T Winfree. The geometry of biological time , volume 12. Springer Science & BusinessMedia, 2001.[66] Yoshiki Kuramoto.
Chemical oscillations, waves, and turbulence . Courier Corporation, 2003.[67] Giancarlo Benettin, Luigi Galgani, Antonio Giorgilli, and Jean-Marie Strelcyn. Lyapunovcharacteristic exponents for smooth dynamical systems and for Hamiltonian systems; a methodfor computing all of them. part 1: Theory.