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1 DOI: 1.138/NPHYS535 Spontaneous synchrony in power-grid networks Adilson E. Motter, Seth A. Myers, Marian Anghel and Takashi Nishikawa Supplementary Sections S1. Power-grid data. The data required for power flow calculations were obtained as follows. For the 1-generator system, known as the New England test system, the parameters were taken from ref. 47. For the 3- and 5-generator systems, the parameters were taken from refs. 3 and 48, respectively. The data for the Guatemala and Northern Italy systems were provided by F. Milano (University of Castilla La Mancha), and the data for the Poland system were provided as part of the MATPOWER software package 49. The dynamic data required to calculate the synchronous state, determine its stability, and simulate equation () in the main text, are not all available for the real power grids, and were obtained as follows. For all systems, we assumed that before any optimization the damping coefficient and droop parameter satisfy D i +1/R i = 5 per unit for all generators. The parameters x d,i and H i are available for the three test systems from the respective references mentioned above. For the Guatemala, Northern Italy, and Poland systems, we estimated x d,i and H i using the strong correlation observed in the test systems between each of these parameters and the power P i injected by generator i into the network, as shown in Supplementary Fig. S1. The estimated values are x d,i in per unit, and H i is in seconds. 9.8P 1.3 i and H i.4p i, where P i is in megawatts, x d,i is S. Derivation of the swing equation. Equation () in the main text can be derived by setting the rate of change of the angular momentum of the rotor equal to the net torque acting on the rotor: J d δ i dt = T mi T ei, (S1) where J is the moment of inertia in kg m, T mi is the mechanical torque in N m accelerating the rotor, and T ei is the typically decelerating torque in N m due to electrical load in the network. Multiplying both sides by ω i and using the fact that the torque in N m multiplied by the angular velocity in radians per second gives the power in watts, the equation can be written in terms of power: d δ i Jω i dt = P mi P ei. (S) To make P mi and P ei per unit quantities, we divide both sides of the equation by the rated power P R (used as a reference). The factor Jω i then becomes H i /ω i, where we defined the inertia constant 1 NATURE PHYSICS 1

2 DOI: 1.138/NPHYS535 H i = W i /P R (in seconds) and the kinetic energy of the rotor W i = 1 Jω i (in joules). Noting that ω i is approximately equal to the reference frequency ω R in systems close to synchronization, we obtain equation () in the main text. A more detailed description of this derivation can be found in ref. 3 (second edition, pp ). S3. Relation to the master stability formalism. Equations (4) and (5) in the main text, when written individually for each node i, can be expressed as d δ i dt = 1 δ i n + δ j P ij. (S3) β j=1 i 1 δ i δ i This is in the same general form as the variational equation for the class of coupled oscillators considered in ref. 34, which for a synchronous state s = s(t) is δ j d dt x i = DF(s) x i + σ j G ij DH(s) x j, (S4) where ẋ i = F(x i ) describes the node dynamics, H is the coupling function, and σg ij represents the strength of coupling from node j to i. The factors DF(s) and DH(s) in equation (S4) are both constant matrices in equation (S3), and P ij in equation (S3) corresponds to σg ij in equation (S4), which are relations that can be used to derive Λ β (α) from equations (4) and (5) based on the results in ref. 34. Therefore, while the formalism in ref. 34 cannot be directly applied to equation () in the main text, it can be applied near the synchronization manifold, which is a procedure previously proposed for a broad class of coupled oscillators 35. S4. Local analysis of λ max (β opt + εβ ) along all directions β. For a fixed direction β = (β 1,...,β n) T in the space of all β i values and β opt =(β opt,...,β opt ) T, we consider β = β opt + εβ and study the behavior of the maximum Lyapunov exponent λ max as a function of ε in the limit of small ε. The same transformation that leads to equation (7) in the main text, now applied to the general case of nonidentical β i values, results in Ż = MZ, where 1 M =, Z = Z 1, J B and B = Q 1 BQ. Here and 1 denote the null and the identity matrix, respectively, and we recall that J, Z 1, Z, B, and Q are defined in the main text. We denote the eigenvalues of M as Z NATURE PHYSICS

3 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION ν j± = ν j± (ε), j =1,...,n, and index them so that, as ε, each ν j± approaches continuously the corresponding Lyapunov exponent given by equation (8) in the main text for the case β i = β opt, i.e., ν j± (ε) ε ν j± () = λ j± (α j,β opt )= β opt ± 1 βopt 4α j = α ± α α j. Here, as in the main text, the eigenvalues of the matrix P are denoted α 1,...,α n, among which α 1 =and α is the smallest nonzero eigenvalue. Also note that ν 1± (S5) are not relevant for determining the stability of synchronous states, since these eigenvalues are associated with perturbation modes that do not affect synchronization. Thus, the Lyapunov exponent λ max determining the stability is the largest real component among the remaining eigenvalues: λ max (ε) = max j n max {±} Re ( ν j± (ε) ). (S6) For ε =, which corresponds to β i = β opt, it follows from equation (S5) and α α j, j, that λ max () = β opt = α λ max opt. (S7) Our goal here is to show that this is a local minimum of λ max (ε) for any given direction β. To help with our analysis, we introduce auxiliary variables by writing the characteristic polynomial of M as (ν + c 1 ν + d 1 )(ν + c ν + d ) (ν + c n ν + d n ) (S8) and indexing the coefficients c j and d j so that the jth quadratic factor corresponds to the pair ν j±, i.e., ν + c j ν + d j =(ν ν j+ )(ν ν j ). The coefficients c j (ε) and d j (ε) are thus related to ν j± (ε) by c j = (ν j+ + ν j ) and d j = ν j+ ν j, (S9) or equivalently, ν j± = c j ± 1 c j 4d j. (S1) Equations (S5) and (S9) can be used to show that for ε =, we have c j () = β opt and d j () = α j. From equations (S6) and (S1), it follows that the Lyapunov exponent λ max (ε) is bounded from below by a function of c and d : λ max (ε) Re ν + = Re ( c + 1 ) c 4d f(c,d ). (S11) Since f(c (ε),d (ε)) = λ max opt when ε =, it suffices to show that f(c (ε),d (ε)) >λ max opt for all sufficiently small ε = in order to establish that ε =is a local minimum of λ max (ε). The lower 3 NATURE PHYSICS 3

4 DOI: 1.138/NPHYS535 bound function f(c,d ), whose functional form is system independent, is shown in Supplementary Fig. S. Note that the set of points at which f(c,d ) is not smooth form the curve c =4d, and at these points the associated eigenvalues are degenerate, i.e., ν + = ν. It is evident from the figure (and also from equation (S11)) that the region in the (c,d )-plane characterized by f(c,d ) >λ max opt can be expressed as { (c,d ): c <β opt and/or d α < } α. (S1) c β opt We thus aim to show that the point (c (ε),d (ε)) remains is this region for small ε = for any given direction β. We now expand the functions c j (ε) and d j (ε) in terms of ε around the points c j () = β opt and d j () = α j, respectively: c j = β opt + c (1) j ε + c () j ε + O(ε 3 ), d j = α j + d (1) j ε + d () j ε + O(ε 3 ). (S13) To derive equations satisfied by the linear coefficients c (1) j and d (1) j, we first write the characteristic polynomial as det(m ν1), set it equal to expression (S8), and compare the coefficients of ν k to obtain an equation relating c j and d j to the components of the matrix B for each k. We then substitute β i = β opt + εβ i and expansions (S13) into this equation and equate the first-order terms in ε to obtain a system of linear equations. In matrix form, this system reads Cc (1) + Dd (1) = Cb, (S14) where c (1) =(c (1) 1,...,c (1) n ) T, d (1) =(d (1) 1,...,d (1) n ) T, b =(b 1,...,b n ) T, and b j = i u jiv ji β i. Here C and D are constant matrices that depend only on α j (recall that α j is the jth eigenvalue of the matrix P), and u ji and v ji are the ith component of the left and right eigenvectors of P corresponding to the eigenvalue α j, respectively. We see that a solution of equation (S14) is given by c (1) j = b j and d (1) j =for all j, which corresponds to c j = β opt + b j ε + O(ε ), d j = α j + O(ε ). (S15) This means that to first order in ε, the function c j (ε) is linear (b j ) or constant (b j =), and d j (ε) is constant. In particular, the behavior of the point (c,d ) as ε varies, which determines whether it remains in region (S1), depends on the value of b. 4 4 NATURE PHYSICS

5 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION If b, this leads to a first-order approximation formula for f(c,d ): f(c,d ) λ max opt b ε/, εb, λ max opt + β opt b ε/, εb >. (S16) The function f attains the minimum value λ max opt at ε =, and the approximation is valid for all directions β = (β 1,...,β n) T outside the (n 1)-dimensional hyperplane L defined by b = i u iv i β i =in the space of all β i values. Thus, along such a direction, the Lyapunov exponent λ max (ε) attains a local minimum λ max opt at ε =. This establishes that the parameter choice β i = β opt locally minimizes λ max in an arbitrary power-grid network along almost all directions in the β -space (i.e., all directions, except those corresponding to b =). Supplementary Fig. S3 (black dashed curves) shows the behavior of c (ε) and d (ε) in equation (S15), as well as of the corresponding λ max (ε), along a randomly chosen direction in the β i -space of the 3-generator test system in Table 1 of the main text. The same conclusion holds true even if c (1) b, as long as c (1) and d (1) =. In this general case, we define the hyperplane L by c (1) = i c(1) i β i =, where c (1) i are the linear coefficients that arise in the full n-dimensional expansion of c and d : c (εβ )=β opt + d (εβ )=α + n c (1) i=1 n d (1) i=1 i εβ i + i k i εβ i + i k c () ik (εβ i)(εβ k)+o ( (ε β ) 3), d () ik (εβ i)(εβ k)+o ( (ε β ) 3). (S17) Since we can write d (1) = i d(1) i β i using the linear coefficients of d in equation (S17), the local minimum of λ max at β opt is established for all directions outside L (for which c (1) ) if d (1) i = for all i. This condition is numerically validated below. If b =(or c (1) =in the more general case), which corresponds to the hyperplane L, the first-order approximation is not sufficient to describe the local behavior of the Lyapunov exponent. We thus use the second-order terms in ε to show that λ max is locally minimum at β opt along all directions β L. The condition for (c,d ) to stay in region (S1) for small ε can be expressed in terms of the second-order coefficients as d () c () < α, (S18) 5 NATURE PHYSICS 5

6 DOI: 1.138/NPHYS535 since d α = d() ε + O(ε 3 ) c β opt c () ε + O(ε 3 ) d () as ε. c () In order to verify condition (S18) for the power-grid systems in Table 1, we rewrite it in terms of β i as d () = < α c (), (S19) i k d() ik β iβ k i k c() ik β i β k where c () ik and d() ik are the quadratic coefficients in equation (S17). We have thus reduced the problem of establishing the local optimality of λ max (β opt + εβ ) along the directions with b = to the problem of computing the n(n + 1)/ quadratic coefficients in equation (S17) and using the computed coefficients to verify condition (S19) for all β L. As a first step toward verifying condition (S19), we determine the linear and quadratic coefficients in equation (S17) for a neighborhood of size ε (ε = 1 for the Guatemala and Italy networks and ε = 1 4 for all other networks) from the values of c and d computed at n(n+3)/ sample points in the β -space (the minimum number of points required to determine the n linear coefficients and n(n + 1)/ quadratic coefficients of each function). For a given ε, we place two sample points on each β i-axis (β i = ε or ε and β k =for all k =i) and one for each pair of coordinate axes (β i = ε and β k = ε for a pair i and k =i, and β l =for all other l). The values of c and d are computed from the estimated ν + and ν through relations (S9). The resulting quadratic approximations for c (ε) and d (ε), as well as for the maximum Lyapunov exponent λ max (ε), are shown in Supplementary Fig. S3 (black solid curves) for a typical direction β outside the hyperplane L. Before proceeding to verify condition (S19) using the computed coefficients, we note that for all systems we consider, the linear coefficients satisfy d (1) i =within numerical precision; this condition was used above to study the local behavior of λ max (β opt + εβ ) within the hyperplane L. Due to the large condition numbers of the matrices involved in the computation, the observed numerical errors are larger than the machine precision. We thus further validate d (1) i = by quadratically fitting many values of d (taken to be 1 in our simulations) computed along each β i -axis to estimate d (1) i more accurately. Assuming that numerical errors in computing d are Gaussian, such estimates of d (1) i follow a distribution with a standard deviation σ. Using the observed residuals in the quadratic fit to estimate σ, we find that for each system and for each i the standard deviation σ is at most of order 1 9 and zero is within σ of our estimate of d (1) i. This provides further validation of the local behavior of λ max established above for all power grids. 6 6 NATURE PHYSICS

7 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION Given the values of the quadratic coefficients c () ik and d() ik, along with eigenvector components u i and v i, we now seek to verify condition (S19) for all β in the hyperplane L. To this end, we first formulate an auxiliary constrained optimization problem: Find β =(β 1,...,β n) T that maximizes subject to i k i k d () ik β iβ k c () ik β iβ k = K and b = i u i v i β i = (S) for a given arbitrary constant K. If condition (S19) is satisfied for all solutions of this optimization problem for all K (i.e., the maximum value found is less than K α ), then the condition is also satisfied for all directions within the hyperplane L. Using the method of Lagrange multipliers, this optimization problem can be reduced to the generalized eigenvalue problem Dη = µ Cη, where the matrices D and C are defined by D = R T ˆDR, C = R T ĈR, ˆDik = d () ik + d() ii δ ik, and Ĉik = c () ik + c() ii δ ik. Here R is an n (n 1) matrix of orthonormal basis vectors for the hyperplane L. Specifically, a solution β of the optimization problem must satisfy β = Rη k, where η k is a generalized eigenvector satisfying Dη k = µ k Cηk. Note that the eigenvectors do not depend on the constant K except via the normalization factors, since the matrices D and C do not depend on K. Since the r.h.s. of inequality (S19) does not change upon scaling of β, the condition can be verified using normalized eigenvectors η k without loss of generality. It can also be shown that the verification of condition (S19) is independent of the choice of the matrix R. Thus, for a given system, the local minimum of λ max (β opt + εβ ) can be established for all directions β by computing the generalized eigenvectors η k with an arbitrary choice of K and R, and verifying condition (S19) for each β = Rη k. To proceed with the verification of condition (S19), we implement an additional numerical step in the procedure. This is necessary because the variations of c and d for ε ε ε along the direction corresponding to a generalized eigenvector η k can be smaller than the numerical error involved in the computation if (as in the procedure above) the sample points are not taken along this direction, and this could prevent the estimates of c () and d () from being sufficiently accurate. We thus compute c () and d () in equations (S13) directly through a quadratic fit using a number points much larger than 3 sampled along this particular direction on an interval of ε much wider than ε (to capture sufficiently large variations in c and d ). Carrying this out for β corresponding to each generalized eigenvector η k, we find that condition (S19) is indeed satisfied 7 NATURE PHYSICS 7

8 DOI: 1.138/NPHYS535 for each system with a significant margin (see Supplementary Table S4 for the smallest margins we found). This shows that the rate of increase in λ max (ε) is strictly positive, thus showing that ε = is locally a minimum, for all directions β in the hyperplane L. Together with the result from the first-order approximation outside the hyperplane L, this establishes the local optimality of λ max at β i = β opt along all directions β (within and outside L). Supplementary Fig. S4 illustrates the behavior of c (ε), d (ε), and λ max (ε) for the 3-generator test system in Table 1 along the direction corresponding to the maximum of d () /c (), which is also the direction β of least increase in λ max (ε). S5. Further implications. The optimization in equation (1) of the main text can be effective even when modeling load dynamics 5 (Supplementary Fig. S5). In that context, our approach for enhancing synchronization stability is complementary to an approach proposed recently 51 to mitigate saddle-node instability by adjusting power scheduling or line impedances (Supplementary Fig. S6). Conversely, the adjustment of power scheduling and/or line impedances can also be effective within the reduced network model used in our study (Supplementary Fig. S7). Supplementary References 47. Pai, M. Energy function analysis for power system stability (Kluwer Academic Publishers, Norwell, 1989). 48. Vittal, V. Transient stability test systems for direct stability methods. IEEE T. Power Syst. 7, (199). 49. Zimmerman, R. D., Murillo-Sánchez, C. E. & Thomas, R. J. MATPOWER: Steady-state operations, planning and analysis tools for power systems research and education. IEEE T. Power Syst. 6, 1 19 (11). 5. Bergen, A. R. & Hill, D. J. A structure preserving model for power system stability analysis. IEEE T. Power Ap. Syst. PAS-1, 5 35 (1981). 51. Mallada, E. & Tang, A. Improving damping of power networks: Power scheduling and impedance adaptation. IEEE Conference on Decision and Control and European Control Conference (11). 8 NATURE PHYSICS

9 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION Supplementary Figures Supplementary Fig. S1: Relationship between the power produced by individual generators and their parameters in the three test systems of Table 1. a, The transient reactance x d,i versus the power P i injected into the network. b, The inertia constant H i versus P i. We used the approximate functional relations revealed by these data to estimate the transient reactance and the inertia constant for each generator in the Guatemala, Northern Italy, and Poland systems. NATURE PHYSICS 9

10 DOI: 1.138/NPHYS535 Slope Supplementary Fig. S: Lower bound function f(c,d ) for the Lyapunov exponent λ max. Region (S1), in which f(c,d ) >λ max opt holds true, is shown by the shades of blue, while the other region is shown by the shades of green. The whole color scheme is used to indicate the values of f(c,d ). In Supplementary Section S4, we show that, given a direction β in the β i -space, the point (c (ε),d (ε)) remains in the blue region for any sufficiently small ε =, thus establishing λ max (ε) >λ max opt. 1 1 NATURE PHYSICS

11 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION 1 a b c d Supplementary Fig. S3: Synchronization stability near β i = β opt along a typical direction β. For a fixed direction β =(β 1,β,β 3) T in the β i -space, which we chose randomly and normalized to unit length, we consider the deviation β i = β opt + εβ i with ε for the 3-generator test system in Table 1. In all panels, the blue dots represent values computed directly from the eigenvalues of the matrix M defined in Supplementary Section S4, and the black curves represent the linear (dashed) and quadratic (solid) approximation from expansion (S13) in the same section. a, b, The characteristic polynomial coefficients c (panel a) and d (panel b) as a function of ε, relative to their nominal values at ε =(which are β opt and α, respectively). c, The path followed by the point (c,d ) as ε varies in the interval [, ]. The red curve indicates the points where the Lyapunov exponents are degenerate (ν + = ν ) and f(c,d ) is not smooth as a function of c and d (Supplementary Fig. S). It is evident that no blue dots (and hence no eigenvalues of M) fall in the gray shaded region, in which f(c,d ) <λ max opt = α. d, The maximum Lyapunov exponent λ max as a function of ε. NATURE PHYSICS 11

12 DOI: 1.138/NPHYS535.1 a b c.6 d.4 Slope. 1 1 Supplementary Fig. S4: Synchronization stability near β i = β opt along the direction β of least increase in λ max. The information presented is the same as in Supplementary Fig. S3 except that β is chosen to be the direction that gives the maximum value of d () /c() (which corresponds to the approximate slope of the curves followed by the blue dots in panel c). 1 NATURE PHYSICS

13 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION Improvement factor Supplementary Fig. S5: Enhancement of synchronization stability in power-grid model incorporating load dynamics. Colors indicate the factor by which the stability improves after adjusting all β i to the value β opt, where the darkest red corresponds to any factor > 7 and black to any factor < 1. The stability of a synchronous state is measured by the largest nonzero Lyapunov exponent λ max computed within the Bergen-Hill model 5. In this model, instead of eliminating non-generator nodes by the Kron reduction, load dynamics is modeled by assuming that the real power is a linear function of the voltage frequency for all power-consuming nodes, while the voltage magnitude is assumed to be constant for all nodes. For illustration, we use the 3-generator system studied in ref. 51, with the power injection modified to P i =5, 6, 7 per unit. The improvement factor is shown as a function of the parameters D gen and D load, where D gen is a coefficient given by (D i +1/R i )/ω R (assumed be the same for all generator nodes) and D load is the frequency coefficient (assumed to be the same for all the other nodes). In most cases, our method yields significant stability improvement, demonstrating its effectiveness beyond the reduced network model. NATURE PHYSICS 13

14 DOI: 1.138/NPHYS535 Supplementary Fig. S6: Combination of complementary approaches to mitigate instabilities associated with saddle-node bifurcations. As an example, we use the same 3-generator system described in the caption of Supplementary Fig. S5 with power injection P i =7.994, 3.6, 7., simulated using the Bergen-Hill model 5. For D gen = D load =1per unit, the system is near a saddle-node bifurcation, and thus the Lyapunov exponent λ max is negative real and close to zero. At each iteration of the gradient descent-like method described in ref. 51, which changes the power injected by the generators (independently of the values of β i and thus of D i ), we compute λ max both with and without adjusting β i to β opt by tuning the damping coefficients D i. The improvement factor, measured by the ratio between the smallest λ max obtained through the iterative process with and without the β i -adjustment, is shown as a function of D gen for different values of D load. In most cases, significant additional improvement results from the adjustment of β i, illustrating that near instabilities the two methods can be combined to achieve enhancement not possible by either method alone. This result is robust against heterogeneity in the network parameters, as illustrated in the inset histogram for the system with D gen =.5and D load =1(blue dot in the main plot), where each of these coefficients is independently perturbed according to the Gaussian distribution with mean zero and standard deviation NATURE PHYSICS

15 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION Iterations Iterations Supplementary Fig. S7: Enhancement of synchronization stability via tuning of power injection within the reduced network model. We use the same 3-generator system as in Supplementary Figs. S5 and S6, where all system parameters are the same as in ref. 51, except for the power injection and consumption, which are initially set to 6% of the values used in that reference. This makes the system unstable, as indicated by the positivity of the Lyapunov exponent λ max for the synchronous state. The power injections were then adjusted iteratively according to the gradient descent-like scheme of ref. 51 and λ max is plotted against the number of iterations for several values of the coefficient D gen. In all cases, the power adjustment stabilizes the synchronous state, which is a consequence of increasing α across zero, as shown in the inset. Note that α is independent of D gen because the synchronous state in the reduced network model is unaffected by changes in D gen. 15 NATURE PHYSICS 15

16 DOI: 1.138/NPHYS535 Supplementary Tables Supplementary Table S1: Phase difference in synchronous states and the real versus imaginary parts of the admittances in the physical and effective networks. Physical network Effective network System Mean δij /π Mean G ij Mean B ij Mean G ij Mean B ij 3-generator system generator system generator system Guatemala Northern Italy: original x d,i adjusted Poland: original x d,i adjusted The real and imaginary components of the admittance Y ij are denoted G ij and B ij, respectively. Supplementary Table S: -norm of the symmetric and antisymmetric parts of the matrix P. System Symmetric Antisymmetric 3-generator system generator system generator system Guatemala Northern Italy: original.45.3 x d,i adjusted Poland: original x d,i adjusted NATURE PHYSICS

17 DOI: 1.138/NPHYS535 SUPPLEMENTARY INFORMATION Supplementary Table S3: Fraction of positive off-diagonal elements in the matrices (B ij ), (cos δij ), and (B ij cos δij ). System B ij cos δij B ij cos δij 3-generator system generator system generator system Guatemala Northern Italy: original x d,i adjusted Poland: original x d,i adjusted Supplementary Table S4: Property of the estimated quadratic coefficients of c and d. ( System min β α d () ) /c() 3-generator system generator system generator system.87 Guatemala.9 Northern Italy (x d,i adjusted) 1.85 Poland (x d,i adjusted).61 The minimum is taken over all directions β within the hyperplane L. 17 NATURE PHYSICS 17