Energy-as-a-Service for RF-Powered IoE Networks: A Percolation Theory Approach

TL;DR

The paper addresses scalable connectivity for RF-powered IoE under Energy-as-a-Service by modeling ESs and IoE devices as independent PPPs and constructing a WET-aware connectivity graph (WC-RGG). Using continuum percolation, it proves the existence of a finite critical ES density $\\lambda_f^c$ that enables large-scale D2D connectivity, and derives approximate expressions via inner-city and Gilbert disk models to estimate CAPEX. The proposed approximations capture the transition behavior and provide practical guidance for ES deployment, highlighting significant CAPEX savings over full coverage while acknowledging spatial outages and non-idealities. Overall, the work offers a rigorous, geometry-based toolkit to plan energy provisioning for massive RF-powered IoE deployments with tangible real-world deployment implications.

Abstract

Due to the involved massive number of devices, radio frequency (RF) energy harvesting is indispensable to realize the foreseen Internet-of-Everything (IoE) within 6G networks. Analogous to the cellular networks concept, shared energy stations (ESs) are foreseen to supply energy-as-a-service (EaaS) in order to recharge devices that belong to different IoE operators who are offering diverse use cases. Considering the capital expenditure (CAPEX) for ES deployment along with their finite wireless energy transfer (WET) zones, spatial energy gaps are plausible. Furthermore, the ESs deployment cannot cover 100% of the energy-harvesting devices of all coexisting IoE use cases. In this context, we utilize percolation theory to characterize the feasibility of large-scale device-to-device (D2D) connectivity of IoE networks operating under EaaS platforms. Assuming that ESs and IoE devices follow independent Poisson point processes (PPPs), we construct a connectivity graph for the IoE devices that are within the WET zones of ESs. Continuum percolation on the construct graph is utilized to derive necessary and sufficient conditions for large-scale RF-powered D2D connectivity in terms of the required IoE device density and communication range along with the required ESs density and WET zone size. Fixing the IoE network parameters along with the size of WET zones, we obtain the approximate critical value of the ES density that ensures large-scale connectivity using the inner-city and Gilbert disk models. By imitating the bounds and combining the approximations, we construct an approximate expression for the critical ES density function, which is necessary to minimize the EaaS CAPEX under the IoE connectivity constraint.

Figures (8)

• Figure 1: Illustration for RF-powered IoE operation under the EaaS platform. The IoE devices inside the WET zones of ESs can be activated (marked in red), while other IoE devices are inactive (marked in blue). A successful connection between two active IoE devices can be achieved when the distance between them is smaller than the maximum connection range.
• Figure 2: Illustration for the inactive faces, outer envelope, and inactive circuit. As shown in Fig.\ref{['fig:ClosedFace']}, a hexagon without any active IoE devices is considered an inactive face. Fig.\ref{['fig:outerenvelope']} is an extreme case we use to derive the lower bound. If there is no ES inside the outer envelope, all IoE devices in the hexagon are inactive. Fig.\ref{['fig:ClosedCircuit']} shows that an inactive circuit appears when $\mathbb{P}\{{\mathcal{H}}_L \rm{\ is\ inactive}\}>\frac{1}{2}$. Because $r_r=a_L$, the IoE device inside the circuit can not be connected with those outside the circuit.
• Figure 3: Illustration for the active faces, inner envelope, and face percolation. As shown in Fig.\ref{['fig:OpenFace']}, a hexagon with at least one active IoE device is considered an active face. Fig.\ref{['fig:innerenvelope']} is an extreme case we use to derive the upper bound. If there is an ES in the inner envelope, all IoE devices in the hexagon are activated. Fig.\ref{['fig:FacePercolation']} shows that face percolation happens when $\mathbb{P}\{{\mathcal{H}}_U \rm{\ is\ active}\}>\frac{1}{2}$. Because $r_r=\sqrt{13}a_U$, the active IoE devices in two adjacent hexagons can communicate with each other, which helps the IoE network percolate at the same time.
• Figure 4: Illustration for the approximation based on a simple Gilbert disk model. When the density of IoE devices is very large the necessary condition for connecting IoE devices in two adjacent WET areas is to make the distance between these two adjacent ESs (the centers of WET zones) at most $2r_f+r_r$, where the distance between the two points at the critical position (marked in blue) is exactly the maximum communication range $r_r$.
• Figure 5: Comparison between curves of bounds, approximations, and simulation results of the critical ES density. As the density of IoE devices increases, the bounds, approximations and simulated critical ES density do not increase.

Theorems & Definitions (13)

• Definition 1: Inactive face in sub-critical regime
• Lemma 1: Lower bound of critical ES density
• Remark 1
• Definition 2: Active face in super-critical regime
• Lemma 2: Upper bound of critical ES density
• Remark 2
• Theorem 1: Phase transition
• Lemma 3: Critical ES density function
• Lemma 4: Approximation of critical ES density using the inner-city model and Gilbert disk model
• Remark 3