Enhancing power grid synchronization and stability through time-delayed feedback control

We study the synchronization and stability of power grids within the Kuramoto phase oscillator model with inertia with a bimodal natural frequency distribution representing the generators and the loads. The Kuramoto model describes the dynamics of the ac voltage phase and allows for a comprehensive understanding of fundamental network properties capturing the essential dynamical features of a power grid on coarse scales. We identify critical nodes through solitary frequency deviations and Lyapunov vectors corresponding to unstable Lyapunov exponents. To cure dangerous deviations from synchronization we propose time-delayed feedback control, which is an efficient control concept in nonlinear dynamic systems. Different control strategies are tested and compared with respect to the minimum number of controlled nodes required to achieve synchronization and Lyapunov stability. As a proof of principle, this fast-acting control method is demonstrated for different networks (the German and the Italian power transmission grid), operating points, configurations, and models. In particular, an extended version of the Kuramoto model with inertia is considered that includes the voltage dynamics, thus taking into account the interplay of amplitude and phase typical of the electrodynamical behavior of a machine.

Figure 1

(a) Map of the German ultra-high-voltage power grid, consisting of 95 net generators (green squares) and 343 net consumers (red dots) connected by 662 transmission lines (black lines). [(b) and (c)] Histograms showing distributions of net power: (b) Realization of an artificial bimodal Gaussian; $P_0=105$ MW, $\sigma =P_0/2$. (c) Based on the German power grid. The light green (dark red) bars correspond to generators (consumers). Panels (a) and (c) are constructed from elmod-de data reported in Ref. [38].

Figure 2

Spatial distribution of nodal power demands constructed from elmod-de data reported in Ref. [38]. The size of the circles indicate the value of $p_i{P}_{\text{Total}}$.

Figure 3

Spatial distribution of nodal generation capacities constructed from elmod-de data reported in Ref. [38]. The size of the circles indicate the value of $C_i$.

Figure 4

Power characteristics of time-delayed feedback using the extended model and the German grid with $P^R$: (a) Frequencies ${\omega }_i\left(t\right)$ and (b) control powers $c_i$ of controlled solitary nodes versus time $t$. The vertical line indicates the onset of control. The set of controlled nodes ($N_c=10$) is chosen according to the strategy shown in Fig. 14. Delay time $\tau =4$ s. The feedback gain $g_i$ is set to $g_i=1.3$ for controlled nodes and $g_i=0$ for other nodes. Parameters: $m_v=1$, $E_{f,i}=1$, $X_i=1$, $K=1307$ MW, $\alpha =2$ ${\mathrm{s}}^{-1}$, $I_i=I=40\times {10}^3\text{kg}{\text{m}}^2$, ${\omega }_G=2\pi 50$ Hz.

Figure 5

German power grid with bimodal Gaussian distribution $P^{\mathrm{G}}$: (a) Average frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ and (b) largest Lyapunov exponent ${\lambda }_1$ versus coupling strength $K$. The solid (dashed) lines correspond to the adiabatic upsweep (downsweep) of $K$. [(c)–(e)] Average frequencies ${〈{\omega }_i〉}_t$ and [(f)–(h)] Lyapunov vector components ${\xi }_i$ versus node index $i$ for the $K$ values marked by black circles in the top panels from left to right. Parameters: $0\le K\le 3142$ MW in steps of $\mathrm{\Delta }K\approx 21$ MW with $\alpha =5/6$ ${\mathrm{s}}^{-1}$, $I_i=I=40\times {10}^3\text{kg}{\text{m}}^2$, ${\omega }_G=2\pi 50$ Hz. Averages taken over 100 s after discarding a transient time of 400 s. Lyapunov exponents and vectors calculated for a duration of $4\times {10}^5$ s. Lyapunov exponents are expressed in units of $\mathrm{\Delta }{t}^{-1}=5$ ${\mathrm{s}}^{-1}$.

Figure 6

German power grid with real-world distribution $P^R$: Same as in Fig. 5. Parameters: $0\le K\le 4500$ MW in steps of $\mathrm{\Delta }K\approx 25$ MW with $\alpha =2$ ${\mathrm{s}}^{-1}$. Other parameters as in Fig. 5.

Figure 7

Time-averaged standard frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ (blue dots) and maximum Lyapunov exponent ${\lambda }_1$ (orange line) versus coupling strength $K$ for the bimodal Gaussian $P^G$ (a) and for the real-world distribution $P^R$ (b), calculated for the upsweep protocol. The dashed black lines highlight the operating $K$ points. Time-averaged phase velocity profile ${〈{\omega }_i〉}_t$ (dark blue dots) and Lyapunov vector components ${\xi }_i$ (light orange triangles) versus node index $i$ for the bimodal Gaussian at $K\approx 817$ MW (c) and for the real-world distribution at $K\approx 729$ MW (d). Data are ordered in descending order of ${\xi }_i$. The insets show a zoom for small $i$. Large circles (triangles) mark ${〈{\omega }_i〉}_t$ (${\xi }_i$) of solitary nodes. For $P^G$ $0\le K\le 3142$ MW in steps of $\mathrm{\Delta }K\approx 21$ MW with $\alpha =5/6$ ${\mathrm{s}}^{-1}$. For $P^R$ $0\le K\le 4500$ MW in steps of $\mathrm{\Delta }K\approx 25$ MW with $\alpha =2$ ${\mathrm{s}}^{-1}$ [50]. Other parameters as in Fig. 5.

Figure 8

Time-delayed feedback control: Standard deviation of frequencies $\mathrm{\Delta }\omega \left(t\right)$ versus time $t$. The insets show the frequencies ${\omega }_i\left(t\right)$ versus time $t$ of five arbitrary solitary nodes. Panel (a) corresponds to the distribution $P^G$ with $P^{\mathrm{Max}}\approx 817$ MW and (b) to $P^R$ with $P^{\mathrm{Max}}\approx 729$ MW. The dashed vertical lines indicate the onset of control; delay time $\tau =4$ s. The feedback gain $g_i$ is set to $g_i=1$ for solitary nodes and $g_i=0$ for other nodes; other parameters as in Fig. 5 for panel (a) and Fig. 6 for panel (b).

Figure 9

Efficiency of time-delayed feedback control: Time-averaged frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ (blue dots) and maximum Lyapunov exponent ${\lambda }_1$ (orange triangles) vs number of controlled nodes $N_c$ following different control strategies: [(a) and (d)] Solitary nodes sorted in descending order of ${\xi }_i$. [(b) and (e)] Solitary nodes sorted in descending order of $\left|{〈{\omega }_i〉}_t\right|$. [(c) and (f)] Randomly picked nodes. At each step of each control strategy, one more node is controlled, picked from one of the three mentioned lists, and both the level of synchronization and the instability are recalculated via ${〈\mathrm{\Delta }\omega 〉}_t$ and ${\lambda }_1$. Panels (a)–(c) correspond to the distribution $P^G$ and (d)–(f) to $P^R$. The dashed line marks ${〈\mathrm{\Delta }\omega 〉}_t=0$, the dotted line denotes ${\lambda }_1=0$. Control acts for a duration of 40 s and is then turned off; delay time $\tau =4$ s, feedback gain $g=1$, time averages over 80 s, and operating points $K\approx 817$ ($K\approx 729$) for $P^G$ ($P^R$). Other parameters as in Fig. 5 for (a)–(c) and Fig. 6 for (d)–(f).

Figure 10

Bimodal Gaussian natural frequency distribution, control strategy (i): Lyapunov exponents ${\lambda }_n$ versus $n$ for $K=817$ MW. For simplicity only the first 13 exponents of the spectrum are plotted. Panels (a) to (j) are arranged according to the number of controlled nodes $N_c$ increasing by one from $N_c=0$ to $N_c=9$. Lyapunov exponents are expressed in units of $\mathrm{\Delta }{t}^{-1}=5$ ${\mathrm{s}}^{-1}$.

Figure 11

Real-world natural frequency distribution, control strategy (i): Lyapunov exponents ${\lambda }_n$ versus $n$ for $K=729$ MW. For simplicity only the first 13 exponents of the spectrum are plotted. Panels (a) to (j) are arranged according to the number of controlled nodes $N_c$ increasing by one from $N_c=0$ to $N_c=11$. Lyapunov exponents are expressed in units of $\mathrm{\Delta }{t}^{-1}=5$ ${\mathrm{s}}^{-1}$.

Figure 12

Source of solitary nodes: Lyapunov vector components ${\xi }_i$ versus maximum neighborhood degree $D_i$ for (a) $P^G$ and (b) $P^R$. Only solitary nodes are shown, and filled circles identify nodes which belong to dead trees. [(c) and (d)] Absolute time-averaged frequency $|{〈{\omega }_i〉}_t|$ versus node index $i$ for $P^G$, where $4\mathrm{\Delta }P$ is added to the inherent frequency of an arbitrary nonsolitary node $k$ (green diamond). In (c) dead-tree nodes (filled circles) adjacent to $k$ are controlled and in (d) $k$ is controlled. The black line indicates the synchronized cluster. The instantaneous frequencies ${\omega }_i\left(t\right)$ of node $k$ (light green) and dead-tree nodes adjacent to $k$ (dark red) versus time are shown in the insets. Vertical dashed lines mark activation and deactivation of control. Parameters as in Figs. 4 and 7, time averages over 80 s.

Figure 13

German power grid with real-world distribution $P^R$ using the extended model: (a) Average frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ and (b) largest Lyapunov exponent ${\lambda }_1$ versus coupling strength $K$. The solid lines (dashed lines) correspond to the adiabatic upsweep (downsweep) of $K$. [(c)–(e)] Average frequencies ${〈{\omega }_i〉}_t$ and [(f)–(h)] Lyapunov vector components ${\xi }_i$ versus node index $i$ for the $K$ values marked by black circles in the top panels from left to right. Parameters: $m_v=1$, $E_{f,i}=1$, $X_i=1$. Lyapunov exponents and vectors calculated for a duration of $8\times {10}^3$ s. Lyapunov exponents are expressed in units of $\mathrm{\Delta }{t}^{-1}=5$ ${\mathrm{s}}^{-1}$. Other parameters as in Fig. 6.

Figure 14

Efficiency of time-delayed feedback control for the German power grid with real-world distribution $P^R$ using the extended model: Time-averaged frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ (blue dots) and maximum Lyapunov exponent ${\lambda }_1$ (orange triangles) vs number of controlled nodes $N_c$ following different control strategies: (a) Solitary nodes sorted in descending order of ${\xi }_i$; (b) solitary nodes sorted in descending order of $\left|{〈{\omega }_i〉}_t\right|$. The dashed line marks ${〈\mathrm{\Delta }\omega 〉}_t=0$ and the dotted line denotes ${\lambda }_1=0$. Control acts for a duration of 40 s and is then turned off; delay time $\tau =4$ s and feedback gain $g=1.3$, $K\approx 1307$ MW [middle point of Fig. 13]. Other parameters as in Fig. 13.

Figure 15

Efficiency of time-delayed feedback control for a different operating point: Time-averaged frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ (blue dots) and maximum Lyapunov exponent ${\lambda }_1$ (orange triangles) vs number of controlled nodes $N_c$ following different control strategies: [(a) and (c)] Solitary nodes sorted in descending order of ${\xi }_i$; [(b) and (d)] solitary nodes sorted in descending order of $\left|{〈{\omega }_i〉}_t\right|$. Panels (a) and (b) correspond to the distribution $P^G$ and (c) and (d) to $P^R$. The dashed line marks ${〈\mathrm{\Delta }\omega 〉}_t=0$ and the dotted line denotes ${\lambda }_1=0$. Control acts for a duration of 40 s and is then turned off; delay time $\tau =4$ s and feedback gain $g=1.5(g=1)$ for $P^G\left(P^R\right)$. Other parameters as in Fig. 5 for the top panels (Fig. 6 for the bottom panels). In comparison to Fig. 9, the coupling strengths $K$ are smaller ($K\approx 565$ MW and $K\approx 578$ MW for $P^G$ and $P^R$, respectively) and the initial sets of solitary nodes are bigger, thus being more difficult to control.

Figure 16

(a) Map of the Italian ultra-high-voltage power grid, consisting of 127 nodes connected by 171 transmission lines (red lines) [54]. (b) Histogram shows a realization of an artificial bimodal Gaussian distribution of net power with $N=127$, $P_0=105$ MW, $\sigma =P_0/2$. The light green (dark red) bars correspond to generators (consumers).

Figure 17

Italian power grid with Gaussian distribution $P^{\mathrm{G}}$: (a) Average frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ and (b) largest Lyapunov exponent ${\lambda }_1$ versus coupling strength $K$. The solid lines (dashed lines) correspond to the adiabatic upsweep (downsweep) of $K$. [(c)–(e)] Average frequencies ${〈{\omega }_i〉}_t$ and [(f)–(h)] Lyapunov vector components ${\xi }_i$ versus node index $i$ for $K$ values marked by the black circles in the top panels from left to right. Lyapunov exponents and vectors calculated for a duration of $2\times {10}^4$. Other parameters as in Fig. 5.

Figure 18

Efficiency of time-delayed feedback control for the Italian grid: Time-averaged frequency deviation ${〈\mathrm{\Delta }\omega 〉}_t$ (blue dots) and maximum Lyapunov exponent ${\lambda }_1$ (orange triangles) vs number of controlled nodes $N_c$ following different control strategies. (a) Solitary nodes sorted in descending order of ${\xi }_i$. (b) Solitary nodes sorted in descending order of $\left|{〈{\omega }_i〉}_t\right|$. The dashed lines mark ${〈\mathrm{\Delta }\omega 〉}_t=0$ and the dotted lines denote ${\lambda }_1=0$. Control acts for a duration of 40 s and is then turned off; delay time $\tau =4$ s and feedback gain $g=1$, $K\approx 461$ MW (middle point of Fig. 17). Other parameters as in Fig. 17.