The effect of HVDC lines in power-grids via Kuramoto modelling

TL;DR

Using the EU2016 European HV grid as a testbed, the authors study synchronization and cascade dynamics under adaptive second-order Kuramoto dynamics with inertia. They augment the AC grid model by incorporating HVDC links through frequency-difference-based activation functions, comparing adaptive and static schemes. The work links observed synchronization and cascade behavior to finite-size scaling on graphs with spectral dimension $d_s<4$ and discusses Braess-like paradoxes when altering net transmission. The findings show that adaptive HVDC can improve steady-state synchronization at the cost of longer relaxation times, and that HVDC segmentation can reduce cascade sizes for certain activation choices, offering guidance for grid design under high-renewable conditions.

Abstract

We present a numerical study on the synchronization and cascade failure behaviour by solving the adaptive second-order Kuramoto model on a large high voltage (HV) European power-grid. This non-perturbative analysis takes into account non-linear effects, which occur even when phase differences are large, when the system is away from the steady state, and even during a blackout cascade. Our dynamical simulations show that improvements in the phase synchronziation stabilization as well as the in the cascade sizes can be related to the finite size scaling behaviour of the second order Kuramoto on graphs with $d_s<4$ spectral dimensions. On the other hand drawbacks in the frequency spread and Braess effects also occur by varying the total transmitted power at large and small global couplings, presumably when the fluctuations are small, causing a freezing in the dynamics. We compare simulations of the fully AC model with those of static or adaptive High Voltage Direct Current (HVDC) line replacements. The adaptive (local frequency difference-based) HVDC lines are more efficient in the steady state, at the expense of very long relaxation times.

Figures (6)

• Figure 1: Different types of activation functions used in the adaptive HVDC modelling scenarios.
• Figure 2: HVDC cables in the continental European grid based on the 2016 data in GridKit. While the Baltic area is quite well documented, the Southern-Europe part is less so. In this work only the Baltic seabed cables were turned from AC to DC. (c.f. https://www.hvdcworld.com/hvdc-map?authReq=true&lat=45.266983&lng=6.752306&zoom=8.00&id=250).
• Figure 3: Example timeseries for HVDC modelling with $softsign(x)$ function as activation. The top row displays the thermalization phase, and the bottom one corresponds to the cascade part of the simulations. We are showing results for our main synchronization metrics, $R(t_k)$, $\Omega(t_k)$ and $R_{uni}(t_k)$, for various global coupling values, marked by different colors and symbols. Saturation to the steady state values requires long times, typically $10^4$ iterations and happens non-monotonically as we start from fully phase ordered initial states. The $R_{uni}(t_k)$ curves overlap above coupling $K=500$, being a hallmark of good local synchronisation .
• Figure 4: Comparison of the steady state results after the thermalization of the fully AC and different HVDC approaches as the function of global coupling $K$ in case of the EU2016 network. The inset shows the standard deviation of these quantities. Notice the fact that even static power input/output at DC buses helps the synchronisation to improve. On the other hand, the frequency spread is increasing, which can be understood, since DC connections do not transfer phase information.
• Figure 5: Comparison of the dynamical cascade simulation results of the fully AC and different HVDC approaches as the function of global coupling $K$ in case of the EU2016 network. The inset shows the standard deviation of these quantities. Even during cascade simulation, a higher synchronisation is achieved when we account for the HVDC connections; however, the different approaches separate less than in the steady state (cf. Fig. \ref{['fig:therm_metrics_eu16']}). In terms of frequency spread the full AC model yields better results. This can be understood by the aforementioned fact that DC connections do not carry phase or frequency information.