Lunar Power Grid: Network Structure and Spontaneous Synchronization

TL;DR

This work addresses stable synchronization in NASA's proposed lunar kilohertz-range hub-and-spokes power grid by formulating a reduced swing-equation model for a central hub with many small generators. It establishes a concise, necessary-and-sufficient coupling bound $|D_i \Delta \omega_{sync}-A_i|<K_{1i}$ that guarantees a unique locally stable synchronized mode and proves stability with a Lyapunov energy function, enabling passive stabilization without active feedback. The analysis further reveals an infinite-period bifurcation at $|\mu_i|=1$ (where $\mu_i=\frac{D_i \Delta \omega_{sync}-A_i}{K_{1i}}$) that leads to large-amplitude oscillations and loss of synchronization, highlighting operational bounds for safe grid operation. The results generalize to strongly heterogeneous hub-and-spoke networks and have implications for both lunar and terrestrial inverter-based grids, informing design choices, protection strategies, and modular expansion plans under high-frequency operation. The work contributes a rigorous stability framework for high-frequency, topology-driven synchronization in complex networks, with broad relevance to resilient power delivery in extreme environments.

Abstract

Achieving stable synchronized operation in an alternating current power network is critical to the continuity and reliability of energy delivery. In this paper, we study a dynamic model for synchronization in the proposed power network which the National Aeronautic and Space Agency plans to build on the lunar surface to support continuous human presence on the Moon and a lunar economy. This network is quite remarkable in the sense that it is expected to be the first power network to operate at the unprecedented operating frequency in the $\text{kHz}$ range. The particular structure of this network allows us to derive the necessary and sufficient conditions guaranteeing the existence of a unique locally stable synchronized mode, which will provide a passive control mechanism for the system. Furthermore, we study the bifurcation process leading to the loss of synchronization when the system parameters fall outside of the stable regime. Our results have broader implications for many complex networks with parametric heterogeneity to enhance stability and resilience.

Figures (5)

• Figure 1: Conceptual depiction of the lunar power grid. The design features a large fission nuclear generation unit that is coupled to a number of relatively smaller solid state solar and battery technologies csank2022power. The expected loads are the human habitations - whether on the surface or in lunar pits carrer2024radar - rover charging for surface exploration and scientific purposes, mining inside lunar craters, and space vehicle landing and take-off sites. The lunar power grid will have comparatively limited generation capacity, due to the cost of deployment and the limitations on availability of resources (because of long lunar nights), requiring source and load power levels to be closely matched thomas2023lunar. These features make the lunar grid especially susceptible to instability carbone2019bifurcationcarbone2023voltage. Additionally, the sensitive and critical nature of loads, and the need to serve loads in permanently shadowed regions of the Moon, warrant a high degree of robustness to prevent interruptions.
• Figure 2: Network topology for the lunar power grid.(\ref{['fig:crater']}) Shackleton Crater that lies at the lunar South Pole - the potential site of the first lunar power grid - and approximate location of generators and loads. This graphic is produced using publicly available lunar images by NASA. The location of components are estimated according to the NASA's trade studies thomas2023lunar. (\ref{['fig:graph_toplogy']}) General topology of a Hub-and-spoke network which is NASA has found to be the mass-optimal design for the lunar power network thomas2023lunar, and the focus of this study. A large fission nuclear generation unit in the middle as the hub provides a robust inertial dominance through its centrality. It is coupled with relatively smaller solid state solar and battery technologies and loads as its spokes. Each load will feature its own local small power generation and energy storage. Loads include a human habitat, in situ resource utilization (ISRU) mining operations, and an ISRU production facility which will process mined materials into usable materials. Note that although this proposed design is suggested by the trade studies thomas2023lunarthomas2022establishing, this design does not imply or reflect an official endorsement from NASA. Image Credit: LROC (Lunar Reconnaissance Orbiter) and ShadowCam teams with images provided by NASA/KARI/ASU nasa_image.
• Figure 3: The log of the energy function $E$ is depicted on the vertical axis. The synchronized state is at $(0,0)$ on the horizontal plane. $\rho$ quantifies deviation from the synchronized state in phase $\delta_2$, and $\sigma$ quantifies deviation in frequency $\Delta \omega_2$. Note that in all cases, the synchronized fixed point is at the bottom of an energy well, indicating that it is stable. (\ref{['fig:Completely_Homogeneous_Contours']}) Homogeneous Inertia - Base Coupling. This base case shows the energy function when the generators are identical, and the coupling constants $K_{ij}$ are at a constant baseline value. Note the stable fixed point at $(0,0)$ and the unstable fixed points near $\rho = \pm \pi$. (\ref{['fig:H_hetero_Contours']}) Heterogeneous Inertia - Base Coupling. This case illustrates a heterogeneous grid - generator $1$ is taken to be a large, high inertia generator, while the remaining generators are taken to be small solid state generators with very low inertia, $100\times$ smaller than baseline. Note that the lower inertia decreases the overall energy, but has no effect on the stability of $(0,0)$. (\ref{['fig:Homogeneous_Large_Coupling_Contours']}) Homogeneous Inertia - Large Coupling - The grid is again taken to be homogeneous, but with coupling $K_{ij}$ increased by a factor of $10$ relative to baseline. Increased coupling increases the overall energy. Notice that the unstable fixed points have moved to almost exactly $\rho = \pm \pi$. For this reason, large coupling increases the radius of the basin of attraction of the stable fixed point. (\ref{['fig:H_hetero_Large_Coupling_Contours']}) Heterogeneous Inertia - Large Coupling - Again, a heterogeneous grid, similar to figure (b), but with coupling increased by a factor of $10$ over baseline. Compared to figure (c), the lower inertia decreases the overall energy, but the synchronized mode remains stable.
• Figure 4: Depiction of the infinite-period bifurcation that destroys the stable synchronized fixed point. The system is $2\pi$ periodic - if the plots above were extended to show the range $[-2\pi,2\pi],$ one would see the same patterns repeated every $2\pi$. Trajectories are therefore plotted on a torus - the top of each phase plane at $2\pi$ is identified with (or "glued to") the bottom at 0, and the left side is identified with the right. When trajectories wrap around torus, it is denoted with the thin dotted line. The plots were created by simulating a 6 generator system for various parameter values. Hollow circles depict unstable fixed points, and filled circles depict the stable fixed point. In the rightmost graphic, there are no fixed points - instead initial conditions are marked with a small grey dot. Differently colored lines correspond to different initial conditions. (\ref{['fig:Phase_Plot_Base']}) In the base case, all trajectories lead to stable fixed point in the bottom left of the phase plane. There are 4 fix points pictured, only one of which is stable. (\ref{['fig:Phase_Plot_Weaker']}) Coupling $K_{ij}$ is decreased. Synchronized fixed point is still stable. Notice that fixed points move closer to each other. (\ref{['fig:Phase_Plot_Weakest']}) Coupling $K_{ij}$ is decreased still further. Synchronized state is still stable. Note fixed points are extremely close. (\ref{['fig:Phase_Plot_Drift']}) Coupling decrease below critical value. Fixed points have pairwise annihilated in saddle-node bifurcations. Trajectories now wind around the torus in large periodic motions, shown as the vertical curve in the left part of the plane.
• Figure 5: The synchronization boundary for lunar power grid. The central graphic in the $A_n$-$D_n$ plane shows the region of parameter space satisfying Inequality (\ref{['eq:unique_solution_precise_main']}). The data in this figure was generated by simulating equations (1) of Supplementary Note 2, using $n=10$ generators, whose parameters are given in Supplementary Note 9. The three graphs show the result of numerical simulation of the swing equations (\ref{['eq:Swing_Equation-simplified']}) with $10$ generators. In each subfigure, the damping value of just a single generator was changed to give rise to an under-damped, adequately-damped, and over-damped scenario, respectively. Notice that both the under-damped and over-damped scenarios cause generator $10$ to enter into wild oscillations as a result of the infinite-period bifurcation. In the adequately damped case, notice that the solid state generators quickly synchronize with the central fission reactor, and the whole system then steadily increases towards the desired synchronization frequency of $1\text{kHz}\approx 6,283 \text{ rad}/\text{sec}$. For the given choice of parameters, the synchronization condition $|A_n - D_n \Delta\omega_{sync}|<K_{n1}$ yields the boundary lines $-12.7+247.08 D_n < A_n < 12.7+776.25 D_n$. We can see a single generator having damping outside of the allowable range can cause undesirable effects throughout the rest of the grid. In the over-damped case, the rogue generator's frequency is less than the rest of the generators, and in the under-damped scenario, the rogue generator's frequency is higher than the remaining generators. In both cases, because it continues to influence the rest of the grid, the rogue generator pulls the remaining generators out of equilibrium.