Abstract
Behavioral homogeneity is often critical for the functioning of network systems of interacting entities. In power grids, whose stable operation requires generator frequencies to be synchronized—and thus homogeneous—across the network, previous work suggests that the stability of synchronous states can be improved by making the generators homogeneous. Here, we show that a substantial additional improvement is possible by instead making the generators suitably heterogeneous. We develop a general method for attributing this counterintuitive effect to converse symmetry breaking, a recently established phenomenon in which the system must be asymmetric to maintain a stable symmetric state. These findings constitute the first demonstration of converse symmetry breaking in realworld systems, and our method promises to enable identification of this phenomenon in other networks whose functions rely on behavioral homogeneity.
Introduction
In an alternating current power grid, the generators provide electrical power that oscillates in time as sinusoidal waves. As these waves are superimposed before reaching the consumers, they need to be synchronized to the same frequency; otherwise, timedependent cancellation between these waves would cause the delivered power to fluctuate, which can lead to equipment malfunction and damage^{1}. Maintaining frequency synchronization is challenging because the system is complex in various ways, with every generator responding differently to the continual influence of disturbances and varying conditions^{2}. Adding to the challenge is the increase in perturbations resulting from the ongoing integration of energy from intermittent sources^{3}, the emergence of gridconnected microgrids^{4}, and the expansion of an increasingly open electricity market^{5}. Furthermore, the inherent heterogeneities in the parameters of system components and in the structure of the interaction network are perceived as obstacles to achieving synchronization. Consistent with the view that heterogeneities may generally inhibit frequency homogeneity, an earlier study showed that homogenizing the (otherwise heterogeneous) values of generator parameters can lead to stronger stability of synchronous states than in the original system^{6}. An outstanding question remains, however, as to whether there is a heterogeneous parameter assignment (different from the nominal one) that would enable even stronger stability for synchronous states than the best homogeneous parameter assignment. Though motivated by its significance for power grids, this question is broadly relevant for improving the stability of homogeneous dynamics in complex network systems in general, including consensus dynamics in networks of human or robotic agents^{7,8}, coordinated spiking of neurons in the brain^{9,10}, and synchronization in communication networks^{11,12}.
To gain insights into the potential role of heterogeneity in enhancing stability, it is instructive to first consider the case of damped harmonic oscillators. For a single oscillator, the optimal stability corresponds to the fastest convergence to the stable equilibrium and is achieved when the oscillator is critically damped: underdamping would lead to lingering oscillations around the equilibrium, and overdamping would lead to slowed convergence due to excess dragging. This optimization is exploited in door closers (devices that passively close doors in a controlled manner), which are designed to be critically damped for the door to close fast without slamming. When multiple damped oscillators are coupled, the damping giving rise to optimal stability will be influenced by the network interactions. More important, we can show that the optimal stability in such a network requires different oscillators to have different damping (even when their other parameters are all identical and they are positioned identically in the network), as illustrated in Fig. 1.
In this paper, we first demonstrate that an analogous effect occurs in powergrid networks: heterogeneity in generator parameters can robustly enhance both the linear and the nonlinear stability of synchronous states in power grids from North America and Europe. Since these systems have heterogeneity in the network structure in addition to the tunable generator parameters, one possibility is that the effect arises entirely from compensation: stability reduction due to one heterogeneity is compensated by another heterogeneity, leading to a stability enhancement when the latter heterogeneity is added. An alternative, which we validate here, involves the recently established phenomenon of converse symmetry breaking (CSB)^{13}, in which the stability of a symmetric state requires the system’s symmetry to be broken. Owing to its counterintuitive nature, this phenomenon had not been recognized until it was recently predicted and experimentally confirmed^{13,14} for synchronization in oscillator networks (a class of network dynamics widely studied in the literature^{15,16,17}). Despite its conceptual generality and potential to underlie symmetric states of many systems, this phenomenon has not yet been observed outside laboratory settings. The symmetry relevant here is nodepermutation symmetry, since in a synchronized state the states of different nodes are equal and can be permuted without altering the state of the system. For power grids, CSB would translate to a stability enhancement mechanism in which maintaining the stability of synchronous (and thus symmetric) states requires the generator parameters to be heterogeneous (thus making the system asymmetric). By systematically removing all the other system heterogeneities and isolating the effect of the generator heterogeneity, we establish that CSB is responsible for a significant portion of the stability improvement observed in the power grids we consider. This offers insights into mechanisms underlying the parameter heterogeneity that arises when the generators are tuned to damp oscillations^{18,19} (e.g., by adjusting devices called power system stabilizers). Our results are of particular relevance given that CSB has thus far not been observed in any realworld system outside controlled laboratory conditions, let alone powergrid networks.
Results
Powergrid dynamics and stability
To describe the dynamics of n generators in a powergrid network, we represent each generator node as a constant voltage source behind a reactance (the socalled classical model) and their interactions through intermediate nongenerator nodes as effective impedances (a process known as Kron reduction)^{20}. We assume that the system is operating near a synchronous state in which the voltage frequencies of the n generators are all equal to a constant reference frequency ω_{s}, and we examine whether the homogeneous state is stable against dynamical perturbations. Such perturbations, whether they are small or large, may come for instance from sudden changes in generation and/or demand due to shifting weather condition at wind or solar farms, variations in power consumption, switching on/off connections to microgrids, etc. The shortterm dynamics (of the order of one second or less) are then governed by the socalled swing equation^{20,21}:
where δ_{i} is the phase angle variable for generator i (representing the generator’s internal electrical angle, relative to a reference frame rotating at the reference frequency ω_{s}); \(\beta_{i} \equiv D_{i}/{\!}(2H_{i})\) is an effective damping parameter (corresponding to b_{i} in the massspring system of Fig. 1), with constant D_{i} capturing both mechanical and electrical damping and constant H_{i} representing the generator’s inertia; a_{i} is a parameter representing the net power driving the generator (i.e., the mechanical power provided to the generator, minus the power demanded by the network, including loss due to damping); and c_{ik} and γ_{ik} are respectively the coupling strength and phase shift characterizing the electrical interactions between the generators. The parameters in Eq. (1) for a given system are determined by computing the active and reactive power flows between network nodes from system data and using them to calculate the complexvalued effective interaction (and thus its magnitude c_{ik} and angle γ_{ik}) between every pair of generators. In real power grids, stable system operation is ensured by a hierarchy of controllers that adjust generator power outputs and thus the parameters in Eq. (1). Here, however, these parameters can be regarded as constants, since the lowest level of control (known as the primary control) is modeled as a dampinglike effect captured by the β_{i} term in Eq. (1), while the upperlevel controls (known as the secondary and tertiary controls) act on time scales much longer than that of the shortterm generator dynamics described by the model. In addition, fluctuations in power generation and demand on the time scales of minutes or longer (which can come, e.g., from renewable energy sources) do not affect the shortterm dynamics. Equation (1) has recently been studied extensively in the network dynamics community^{3,6,22,23,24,25}.
We first analyze the stability of the synchronous state against small perturbations. The synchronous state corresponds to a fixed point of Eq. (1) given by \({\delta }_{i}={\delta }_{i}^{* }\) and \({\dot{\delta }}_{i}=0\), which represents frequency synchronization because \({\dot{\delta }}_{i}\) is the frequency relative to the reference ω_{s}. The Jacobian matrix of Eq. (1) at this point can be written as
where O and I denote the n × n null and identity matrices, respectively; P = (P_{ik}) is the n × n matrix defined by
which expresses the effective interactions between the generators; and B is the n × n diagonal matrix with β_{i} as its diagonal elements. We note that, while the form of the Jacobian matrix for coupled damped harmonic oscillators is the same as in Eq. (2), power grids are different in that they can have P ≠ P^{T} because c_{ik} ≠ c_{ki} in general and because γ_{ik} appears in Eq. (3). The stability under noiseless conditions is determined by the Lyapunov exponent defined as \({\lambda }^{\max }\equiv {\max }_{i\ge 2}{\rm{Re}}({\lambda }_{i})\), where λ_{i} are the eigenvalues of J. The identically zero eigenvalue, which comes from the zero rowsum property of P and is denoted here by λ_{1}, is excluded because it is associated with the invariance of the equation under uniform shift of phases. If \({\lambda }^{\max }\,<\,0\), then the synchronous state is asymptotically stable, and smaller \({\lambda }^{\max }\) implies stronger stability (this is known as smallsignal stability analysis in power system engineering). Since real powergrid dynamics are noisy due to power generation/demand fluctuations and various other disturbances occurring on short time scales, \({\lambda }^{\max }\) needs to be sufficiently negative to keep the system close to the synchronous state. Indeed, a previous study^{14} showed that, for broad classes of noise dynamics, there is a (negative) threshold value of \({\lambda }^{\max }\) for such stability: the system is stable if and only if \({\lambda }^{\max }\) is below the threshold. This stability threshold depends on the noise intensity level. For impulselike disturbances, the intensity level corresponds to the maximum deviation of δ_{i} that can be induced by a single disturbance, such as a sudden loss of a generator or a spike in power demand. For continual disturbances, the intensity level can be quantified by the variances of the fluctuating power generation and demand, which can be modeled by adding a randomly varying term to the parameter a_{i}. Since the stability threshold is generally lower for higher noise levels, the lower the value of \({\lambda }^{\max }\) for a given power grid, the more intense disturbances and fluctuations the system can endure without losing stability. Incidentally, the optimal damping in the massspring system of Fig. 1 is given precisely by minimizing \({\lambda }^{\max }\) for that system.
Enhancing stability with generator heterogeneity
We now study \({\lambda }^{\max }={\lambda }^{\max }({\mathbf{\beta}})\) as a function of \({\mathbf{\beta}} {\,}\equiv{\,} (\beta_{1},\ldots , \beta_{n})\) for a selection of power grids whose dynamics can be described by Eq. (1) with the parameter values based on data. Using the same model, it was previously shown^{6} that, under the constraint that all β_{i}’s have the same value, \({\lambda }^{\max }\) is minimized when β = β_{=}, where \({\mathbf{\beta}}_{=}\equiv (\beta_{=}, \ldots , \beta_{=})\) and \({\beta }_{ = }\equiv 2\sqrt{{\alpha }_{2}}\), with α_{2} denoting the smallest nonidentically zero eigenvalue of matrix P. The eigenvalue α_{2} is associated with the least stable eigenmode, and we assume that it is real and positive (as confirmed in all systems we consider). It was further shown that, at this homogeneous optimal point β_{=}, the function \({\lambda }^{\max }({\mathbf{\beta}})\) is nondifferentiable (which precludes the use of a standard derivative test), but its onesided derivative along any given straightline direction is positive, i.e., the directional derivative \({D}_{\mathbf{\upbeta}^{\prime} }{\lambda }^{{\mathrm{max}}}({\mathbf{\beta}}_{ = })\) is positive in the direction of any ndimensional vector \({\mathbf{\beta}}^{\prime}\). Thus, moving away from β_{=} along any straight line would necessarily increase \({\lambda }^{\max }\) from the local minimum value \({\lambda }^{\max }({\mathbf{\beta}}_{ = })=\sqrt{{\alpha }_{2}}\) and hence only reduce the stability of the synchronous state.
Despite the apparent impossibility of improving on \({\lambda }^{\max }({\mathbf{\beta}}_{ = })\) locally, we first show that there can be curved paths starting at β_{=} along which \({\lambda }^{\max }\) can be further minimized with heterogeneous β_{i}. Indeed, Fig. 2a illustrates using a 3generator system that such curved paths exist and can connect β_{=} to the (unique) global minimum, which we denote by β_{≠} as its components are all different. The corresponding optimal \({\lambda }^{\max }({\mathbf{\beta}}_{\ne })\approx 9.41\) represents more than 8% improvement over \({\lambda }^{\max }({\mathbf{\beta}}_{ = })\approx 8.69\). In general, if a curved path starts at β_{=}, and if \({\lambda }^{\max }\) decreases monotonically along that path, then it cannot be oriented in an arbitrary direction in the βspace. We show that it needs to be tangent to a systemspecific plane (or hyperplane of codimension one for n > 3), denoted here by L and defined by the equation \(\mathop{\sum }\nolimits_{i = 1}^{n}{u}_{i}{v}_{i}{\beta }_{i}=0\), where u_{i} and v_{i} are the ith component of the left and right eigenvectors, respectively, associated with the eigenvalue α_{2}. This result, illustrated by the three example paths in Fig. 2a, follows from the derivation of a formula for \({\lambda }^{\max }\) and the full analytical characterization of the stability landscape near β_{=} (both presented in Supplementary Note 1).
The curved paths of decreasing \({\lambda }^{\max }\) are part of the complex structure of the stability landscape. These paths generally lie on a cusp surface, defined by the property that, at any point on the surface, \({\lambda }^{\max }\) is nondifferentiable and locally minimum along any direction transverse to the surface. The three paths shown in Fig. 2a all lie on the same cusp surface, which contains both β_{=} and β_{≠}. The intersection between this cusp surface and the plane M (the one perpendicular to L) is the green path of monotonically decreasing \({\lambda }^{\max }\) shown in Fig. 2. In fact, there are infinitely many different paths of decreasing \({\lambda }^{\max }\) on this cusp surface. Of these paths, the red and blue paths shown in Fig. 2a share the additional property of being an intersection between pairs of cusp surfaces. Each of these paths consists of at least two parts that are intersections between different pairs of cusp surfaces, which explains the kinks observed in Fig. 2 as points at which the curve switches from one intersecting surface to another. In larger systems, we find that their higherdimensional βspaces are sectioned by many entangled cusp hypersurfaces associated with spectral degeneracies (as illustrated in Supplementary Fig. 2 using the four larger systems we will introduce below). Their intersections, which themselves form cusp hypersurfaces of lower dimensions, are expected to contain curved paths of monotonically decreasing \({\lambda }^{\max }\). The existence of kinks and cusp surfaces in the stability landscape, which makes numerical search for global optima challenging, is not unique to power grids nor phase oscillator networks. It is a consequence of a much more general mathematical observation that the largest real part of the eigenvalues of a matrix (known as the spectral abscissa), such as \({\lambda }^{\max }\) we consider here, is a nonsmooth, nonconvex, and nonLipschitz function of the matrix elements^{26}.
Stabilizing heterogeneity in real power grids
Having established that heterogeneous β_{≠} can improve stability over the homogeneous β_{=} for a small example system, we now show that this result extends to much larger, realworld power grids. Specifically, we study the 48generator NPCC portion of the North American power grid and the 69generator German portion of the European power grid. Assessing the stability against small perturbations based on Eqs. (1)–(3) has the advantage of reducing the complexity of these systems to a single matrix P and its eigenvalues. For each system, we identify a local minimum β_{≠} that has heterogeneous β_{i} (and thus is distinct from β_{=}) and achieves the lowest \({\lambda }^{\max }\) over 200 independent runs of simulated annealing. We find simulated annealing to be more effective than other methods in locating a minimum on a nondifferentiable landscape^{27}. The resulting stability improvement over β_{=} is substantial: \({\lambda }^{\max }({\mathbf{\beta}}_{ = }){\lambda }^{\max }({\mathbf{\beta}}_{\ne })=0.66\) for the NPCC network and \({\lambda }^{\max }({\mathbf{\beta}}_{ = }){\lambda }^{\max }({\mathbf{\beta}}_{\ne })=0.42\) for the German network. The optimized β_{i} assignment in β_{≠} exhibits substantial heterogeneity across each network and also across the corresponding geographical area (Fig. 3).
To validate the prevalence of such stabilityenhancing heterogeneity, we also analyze \({\lambda }^{\max }\) as a function of system stress level (to be precisely defined below) for four different systems, including the two used in Fig. 3 (see “Methods” for detailed descriptions of the systems and data sources). To increase or decrease the level of stress in a given system, we scale the power output of all generators and the power demand at all nodes by a common constant factor. We then recompute the power flows across the entire network and the parameters of Eq. (1). The system stress level is then defined to be the common scaling factor used in this procedure. Thus, a stress level of 1 for a given system corresponds to the original demand level in the corresponding dataset. For each stress level, we estimate \({\lambda }^{\max }\) at β = β_{≠} from 200 independent simulated annealing runs. Over the entire range of stress levels considered, we consistently observe a smaller \({\lambda }^{\max }\) for β_{≠} compared to β_{=}, the optimal homogeneous β_{i} assignment, and to β_{0}, the original β_{i} assignment in the dataset (Fig. 4a).
To test the robustness of the identified optimal \({\lambda }^{\max }\) against uncertainties in the β_{i} values, we study how \({\lambda }^{\max }\) changes under perturbations along random directions in the βspace in the vicinity of β_{≠} and (for comparison) in the vicinity of β_{=}. For the stress level of 1 and for each random direction, we compute \({\lambda }^{\max }\) as a function of the perturbation size ε, measured in 2norm. The resulting statistics from 1000 random directions indicate that, for each system, there is a sizable neighborhood of the optimum β_{≠} in which \({\lambda }^{\max }\) is significantly lower than at β_{=}, representing a stability improvement against small perturbations (Fig. 4b).
To show that the improvement is also observed for stability against large perturbations, we define a generalized notion of attraction basin as a set of initial conditions whose corresponding trajectories satisfy a criterion for convergence to synchronous states (a variation of the socalled basin stability^{28}). Here, the convergence criterion we use is that the instantaneous frequency enters into a narrow band around ω_{s} (within ± 0.3 Hz) and remains inside the band until \({t}_{\max }=10\) seconds. This criterion is similar to what is typically used for transient stability analysis in power system engineering. It also captures a variety of synchronous states, including not only those corresponding to fixed points of Eq. (1) (with constant phase angle differences), but also those corresponding to timedependent solutions of Eq. (1). To account for large perturbations, we consider initial conditions with arbitrary phase angles and frequencies within 1 Hz of the nominal frequency (60 Hz for the New England and NPCC systems; 50 Hz for the U.K. and German systems). Each initial condition can be regarded as resulting from a large impulselike disturbance, such as a disconnection of a significant portion of the grid or a systemwide demand surge. The size of the basin can then be quantified using the fraction f of the corresponding trajectories that converge before a given cutoff time t_{c}, i.e., the fraction of those that satisfy \( {\dot{\delta }}_{i}(t) /(2\pi )\le 0.3\) Hz for all \(t\in [{t}_{c},{t}_{\max }]\). For each t_{c}, the fraction f is estimated using 1000 initial conditions sampled randomly and uniformly from all states satisfying the criteria described above. As shown in Fig. 4c, we find that the estimated f is significantly larger for β_{≠} than for β_{=} (which in turn is much larger than for β_{0}). This indicates that the likelihood for the system to return to stable operation after a large disturbance is higher for the heterogeneous optimal β_{i} than for the homogeneous optimal ones. We also observe that larger systems tend to exhibit larger increase in the size of the asymptotic basins (i.e., in the value of f for t_{c} → ∞).
Isolating converse symmetry breaking
Since real power systems generally have heterogeneity in a_{i}, c_{ik}, and γ_{ik}, the stability improvement enabled by the β_{i} heterogeneity (and the associated system asymmetry) could in principle be a compensation for heterogeneity in the network structure, power demand and generation, or other component parameters (and the associated system asymmetries). To illustrate that no such compensation is needed and that CSB can be responsible for stability improvement, we use an example system consisting of four generators connected to each other and to one load (see Supplementary Fig. 1 for a system diagram). This system is symmetric with respect to the permutation of generators 2 and 3 if β_{2} = β_{3}, and this symmetry is reflected in the property that P_{2j} = P_{3j} for all j in the corresponding interaction matrix P (Fig. 5a). The minimum \({\lambda }^{\max }\) possible for this symmetric system is \({\lambda }^{\max }\approx 2.40\), which can be decreased further by >20% to \({\lambda }^{\max }\approx 2.97\) if the β_{2} = β_{3} constraint is lifted (Fig. 5b–d). This demonstrates CSB for this system under a range of noise levels: breaking the system’s symmetry under the permutation of generators 2 and 3 is required for \({\lambda }^{\max }\) to cross the stability threshold and make the (symmetric) synchronous state stable. We note that the observation of CSB depends on the system’s symmetry. While CSB is observed in this 4generator system (with a twogenerator permutation symmetry), we do not observe CSB in a variant of the system with the fourgenerator permutation symmetry. We also note that, while the optimal β_{i} assignment does not share the twogenerator permutation of the system, the twodimensional stability landscape does, and it features a pair of equally optimal assignments related to each other through the symmetry (Fig. 5b). It is instructive to compare this result with the massspring system in Fig. 1, where similar breaking of a permutation symmetry (between masses 1 and 3) for a symmetric landscape (where optimal b_{1} and b_{3} are necessarily different but can be swapped) is shown to underlie optimal damping.
Having established that β_{i} heterogeneity alone can enhance stability through CSB, we now introduce a systematic method to separate CSB from other mechanisms that involve interplays between multiple heterogeneities. For this purpose, we transform matrix P for each system used in Fig. 4, which does not have a pairwise node permutation symmetry, to a slightly different matrix \({\bf{P}}^{\prime}\) that does have the symmetry. More precisely, for a given pair of nodes i_{1} and i_{2}, we define this symmetrized matrix \({\bf{P}}^{\prime}\) by \({P}_{{i}_{1}j}^{\prime}={P}_{{i}_{2}j}^{\prime}\equiv ({P}_{{i}_{1}j}+{P}_{{i}_{2}j})/2\) for all j ≠ i_{1} nor i_{2}, making it symmetric under the permutation of nodes i_{1} and i_{2}. To elucidate CSB for each system, we choose a node pair that simultaneously minimizes the difference between P and \({\bf{P}}^{\prime}\) and maximizes the amount of stability improvement observed for \({\bf{P}}^{\prime}\). For the four systems in Fig. 4, the stability of the symmetrized system can clearly be enhanced by allowing \({\beta }_{{i}_{1}}\ne\, {\beta }_{{i}_{2}}\), and much of the enhancement is maintained as one interpolates from the symmetrized system back to the original system (Fig. 6). This indicates that a significant portion of the stability enhancement for the original system can be attributed to CSB.
Discussion
Our demonstration that heterogeneity of generators can enhance the stability of synchronous states in a range of power grids suggests that there is large previously underexplored potential for tuning and upgrading current systems for better stability. Since larger conventional generators have larger inertia and thus larger impact on the stability of other generators, tuning of their parameters may be particularly beneficial. While we focused on the heterogeneity of a specific generator parameter here, further stability enhancement is likely to be possible by exploiting heterogeneity in other generator parameters and in the parameters of other network components as well as in the network topology. We suggest that such stability enhancement opportunities exist beyond power systems and may extend to any network whose function benefits from homogeneous dynamics and whose stability depends on tunable system parameters. For example, the results presented here suggest that CSB can potentially be observed for coupled oscillatory flows in microfluidic networks and for networks of coupled chemical reactors whose oscillatory node dynamics is close to a Hopf bifurcation. It is known^{13} that such systems can be parameterized so that their Jacobian matrices take a form that generalizes Eq. (2) and thus is conducive to the emergence of CSB. The approach we developed here to isolate CSB is versatile and can be applied broadly to systems for which different heterogeneities cooccur. Determining how prevalent CSB is and how it depends on the properties of the system (e.g., the network size, link distribution, and node dynamics) are important questions for future research.
It is instructive to interpret our results and contrast them with past approaches in network optimization. In seeking the best approach, one may form two complementary hypotheses. One hypothesis, invoked in the past, was that the stability of the desired homogeneous states would be optimal when the system is homogeneous; the approach would thus be to limit the optimization search to the lowdimensional parameter subspace corresponding to networks with identical parameter values for all nodes. The other hypothesis, validated here, is that optimal stability of the desired homogeneous states is generally obtained with heterogeneous parameter assignment, which implies that the search for this optimum requires exploring the highdimensional parameter space without making a priori assumptions on how the parameters of different nodes are related. Recognizing this can lead to new control approaches designed to manipulate these parameters for further optimization of stability. We suggest that the fresh opportunities for network optimization and control revealed in this study apply to network systems in general and thus have the potential to inspire new discoveries in many different disciplines.
Methods
Powergrid datasets
Here, we describe the sources of data for the six powergrid networks considered (the 3generator system in Fig. 2; the New England, NPCC, U.K., and German systems in Figs. 3, 4, and 6; and the 4generator system in Fig. 5). For each system, the data provide the net injected real power at all generator nodes, the power demand at all nongenerator nodes, and the parameters of all power lines and transformers. These parameters are sufficient to determine all active and reactive power flows in the system using a standard power flow calculation. The data also provide the generators’ dynamic parameters H_{i}, D_{i}, and x_{int,i} used in our stability calculations. The parameters H_{i} and D_{i} are the inertia and damping constants, respectively, that define the effective damping parameter through the relation β_{i} = D_{i}/(2H_{i}). The parameter x_{int,i} represents the internal reactance of generator i and is used in the calculation of the parameters a_{i}, c_{ij}, and γ_{ij}. In each system, nodes are indexed as in the original data source (except for the German power grid; see below).

3generator test system (3gen). For this IEEE 3generator, 9node test system, which appeared in ref. ^{20}, we used the data file (data3m9b.m) available in the PST toolbox^{29}. This system represents the Western System Coordinating Council (WSCC), which was part of the region now called the Western Electricity Coordinating Council (WECC) in the North American power grid. The data file provides all necessary dynamical parameters for each generator.

New England test system (10gen). For the IEEE 10generator, 39node test system, as described in refs. ^{30} and ^{31}, we used the data file (case39.m) available in the MATPOWER toolbox^{32}, with dynamic parameters added manually from ref. ^{30}. This is a reduced model representing the New England portion of the Eastern Interconnection in the North American power grid, with one generator representing the connection to the rest of the grid.

NPCC power grid (48gen). For the 48generator, 140node NPCC power grid^{33}, we used the data file (data48em.m) available in the PST toolbox^{29}. The system represents the former NPCC region of the Eastern Interconnection in the North American power grid and includes an equivalent generator/load node representing the rest of the Interconnection. The data file provides H_{i} and x_{int,i} for all generators (while it assumes D_{i} = 0). We generated D_{i} randomly by sampling from the uniform distribution on the interval [1, 3] (in per unit on the system base, as specified by the data file). The geographic coordinates of the nodes used in Fig. 3a were extracted from ref. ^{34}, and the coastline and boundary data used to draw the map were obtained from Natural Earth^{35}.

U.K. power grid (66gen). For the 66generator, 29node U.K. power grid, we used the data file (GBreducednetwork.m) available from ref. ^{36}. The system represents a reduced model for the power grid of Great Britain. The dynamical parameters, H_{i}, D_{i}, and x_{int,i}, were generated randomly by sampling from the uniform distribution on the intervals, [1, 5], [1, 3], and [0.001, 0.101], respectively. The generated parameters values for each generator are in per unit on its own machine base, i.e., normalized by the reference values computed from the power base for the generator (chosen to be 1.5 times the maximum real power generation provided in the data file). For stability calculations, we converted these values to the corresponding values in per unit on a common system base.

German power grid (69gen). For the 69generator, 228node German power grid, we created the data from the ENTSOE 2009 Winter model^{37}. The ENTSOE model is a DC power flow model of the continental Europe and contains 1,486 nodes and 565 generators. We first created a dynamical model for the entire ENTSOE network by solving the DC power flow and converting it to an AC power flow solution (assuming a 0.95 power factor at each node), and then generating dynamical parameters using the same method as for the U.K. grid. For any node with multiple generators attached, the net reactive power injection was distributed among these generators in proportion to their real power generation. From this full ENTSOE model, we extracted the German portion by eliminating (using Kron reduction) all the nodes outside Germany (identified using the country label “D” representing Germany in the dataset). We reindexed the extracted nodes consecutively, preserving the original ordering. The geographic coordinates of the nodes used in Fig. 3b were extracted from the PowerWorld data files available from ref. ^{37}, and the coastline and boundary data used to draw the map were obtained from Natural Earth^{35}.

4generator example system. For the 4generator, 5node example system used in Fig. 5, we show a full system diagram in Supplementary Fig. 1, indicating the main parameters of the components. When the damping parameters of generators 2 and 3 are equal (i.e., β_{2} = β_{3}), the system is symmetric under the permutation of these generators. MATLAB code for running simulations on this system, which includes the full set of parameters and uses the MATPOWER toolbox^{32}, is available from our GitHub repository^{38}.
Aggregation of generators and effective damping parameter β _{i}
If a subset of generators are synchronized in the sense that δ_{i} − δ_{j} is constant in time for any two generators i and j in the subset, then they can be represented by a single equivalent generator using a Zhukovbased aggregation method similar to that described in ref. ^{33}. In this method, the equivalent generator has inertia constant ∑_{i}H_{i} and damping constant ∑_{i}D_{i}, where the sums are taken over the generators i in the subset. The effective damping parameter of the equivalent generator is then \({\sum }_{i}{D}_{i}/(2{\sum }_{i}{H}_{i})=\bar{D}/(2\bar{H})\), where \(\bar{D}\) and \(\bar{H}\) are respectively the average of the inertia and damping constants of the generators in the subset. Thus, the aggregation does not introduce any artifactual heterogeneity.
Data availability
Data on all six systems we consider (described in “Methods”) and detailed data of the core results presented in the figures are available from our GitHub repository^{38}.
Code availability
Essential code for reproducing the core results in all figures, as well as scripts for generating plain versions of the figures, is available from the GitHub repository^{38}.
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Acknowledgements
The authors thank Alex Mercanti and Yuanzhao Zhang for insightful discussions. This research was supported by Northwestern University’s Finite Earth Initiative (supported by Leslie and Mac McQuown) and ARPAE Award No. DEAR0000702 and benefited from logistical support provided by Northwestern University’s Institute for Sustainability and Energy.
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F.M., T.N., and A.E.M. designed the research, analyzed the results, and wrote the paper. F.M. performed the simulations. All authors approved the final manuscript.
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Molnar, F., Nishikawa, T. & Motter, A.E. Asymmetry underlies stability in power grids. Nat Commun 12, 1457 (2021). https://doi.org/10.1038/s41467021212905
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