Asymmetry underlies stability in power grids

TL;DR

This work develops 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.

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 real-world systems, and our method promises to enable identification of this phenomenon in other networks whose functions rely on behavioral homogeneity.

Figures (8)

• Figure 1: Stabilizing effect of heterogeneity in a mass-spring system. a System consisting of a linear chain of three unit masses connected by two identical springs. The masses are constrained to move horizontally, and their dynamics are governed by the equation shown, where $x_i$ is the displacement of mass $i$ relative to its equilibrium and $b_i$ is its damping coefficient. b Total potential energy of the springs vs. time for three different damping scenarios. The optimal damping (red), corresponding to the fastest energy decay, is achieved for $b_1 \approx 2.47$, $b_2 \approx 3.17$, $b_3 \approx 1.47$ (or equivalently, $b_1 \approx 1.47$, $b_2 \approx 3.17$, $b_3 \approx 2.47$), despite the fact that masses $1$ and $3$ are otherwise identical and identically coupled. Overdamping leads to a slower monotonic decay, while underdamping results in a slower oscillatory decay, as shown in blue by varying $b_1$ and $b_3$ by a factor of $5$. In all cases, the initial conditions are $(x_1,x_2,x_3)=(1,0,-1)$ and $(\dot{x}_1,\dot{x}_2,\dot{x}_3)=(0,0,0)$.
• Figure 1: System diagram for the 4-generator example system in Fig. \ref{['fig_small_example']}. The generators at nodes 1--4 produce active and reactive power $P_g$ and $Q_g$ (in MW and MVAR, respectively). The load at node 5 consumes active and reactive power $P_{\ell}$ and $Q_{\ell}$. The other generator parameters are identical among nodes 1--4, except for the tunable damping parameters $\beta_i$, while the power line parameters are identical among all six lines connecting the nodes.
• Figure 2: Enhancing the stability of the 3-generator system along curved paths.a Examples of curved paths (red, green, and blue) in the $\boldsymbol{\upbeta}$-space from $\boldsymbol{\upbeta}_{=}$ to $\boldsymbol{\upbeta}_{\neq}$ along which the Lyapunov exponent $\lambda^{\max}$ decreases monotonically. The numerically generated curves confirm our theoretical prediction that these paths are all tangent to the plane $L$ at the point $\boldsymbol{\upbeta}_{=}$. The droplines indicate that these curves are outside $L$. The contour levels of $\lambda^{\max}$ are shown on $L$. Also shown is the plane $M$, which is defined as the plane perpendicular to $L$ that contains $\boldsymbol{\upbeta}_{=}$ and $\boldsymbol{\upbeta}_{\neq}$. b Contour levels of $\lambda^{\max}$ on the plane $M$, which contains the green path from a. The orange lines in a and b indicate the intersection between planes $L$ and $M$. The red dashed curves (and the green path) trace cusp surfaces associated with degeneracies of the real parts of the eigenvalues of $\mathbf{J}$ that determine $\lambda^{\max}$. Three types of degeneracy are indicated: a real eigenvalue equal to the real parts of a pair of complex conjugate eigenvalues ($a$, $a \pm bi$), two different complex conjugate eigenvalues with equal real parts ($a \pm b_1i$, $a \pm b_2i$), and two equal real eigenvalues ($a$, $a$). For details on the system, see Methods.
• Figure 2: Cusp hypersurfaces in the stability landscape of larger networks.a--d Contour levels of $\lambda^{\max}$ on the plane $M$, defined as the hyperplane perpendicular to $L$ that contains $\boldsymbol{\upbeta}_{=}$ and $\boldsymbol{\upbeta}_{\neq}$ (as in Fig. \ref{['fig4']}), for the four systems used in Fig. \ref{['fig_2node']}. The horizontal coordinate $\xi_1$ represents the Euclidean distance from the point $\boldsymbol{\upbeta}_{=}$ along the line connecting $\boldsymbol{\upbeta}_{=}$ to $\boldsymbol{\upbeta}_{\neq}$, and $\xi_2$ is the distance along the line orthogonal to the $\xi_1$-axis. A white curve indicates the (one-dimensional) cross section of a codimension-one cusp hypersurface and corresponds to single degeneracy of the real parts of the eigenvalues of the Jacobian $\mathbf{J}$, such as two identical real eigenvalues $\{a,a\}$, two pairs of complex conjugate eigenvalues with matching real parts $\{a \pm b_1 i,\ a \pm b_2 i\}$, and one real eigenvalue matching the real parts of a conjugate eigenvalue pair $\{a,\ a \pm b i\}$. The green dot and star indicate $\boldsymbol{\upbeta}_{=}$ and $\boldsymbol{\upbeta}_{\neq}$, respectively. These two points and the white dots correspond to the cross sections of cusp hypersurfaces of higher codimensions, each representing specific double or higher degeneracy of the real parts of the eigenvalues.
• Figure 3: Heterogeneity of optimized generator parameters $\boldsymbol{\upbeta_i}$ for two power grids.a Portion of the North American power grid corresponding to the former Northeast Power Coordinating Council (NPCC) region. b German portion of the European power grid. In both panels, a color-coded circle represents a generator or an aggregate of generators (see Methods for the aggregation procedure used), with the color indicating the corresponding optimal $\beta_i$ in the vector $\boldsymbol{\upbeta}_{\neq}$. The arrows above the color bars indicate $\beta_{=}$, the optimal uniform value of $\beta_i$. The $\lambda^{\max}$ values for the uniform and non-uniform optimal $\beta_i$ (in the vectors $\boldsymbol{\upbeta}_{=}$ and $\boldsymbol{\upbeta}_{\neq}$, respectively) are indicated at the bottom of each panel. The radius of the circle is proportional to the real power output of the generator in megawatts (MW). Small green dots indicate non-generator nodes. Details on these systems, including data sources, can be found in Methods.