# Optimal Power Procurement for Green Cellular Wireless Networks Under Uncertainty and Chance Constraints

## TL;DR

This paper proposes a new method for optimizing energy procurement in cellular networks using renewable energy while ensuring service quality.

## Contribution

A novel time-continuous Lagrangian relaxation approach is introduced to handle probabilistic QoS constraints in energy procurement.

## Key findings

- The proposed method efficiently solves the stochastic optimal control problem with probabilistic constraints.
- Numerical results show the approach is computationally efficient and provides near-optimal solutions in practical timeframes.

## Abstract

Given the increasing global emphasis on sustainable energy usage and the rising energy demands of cellular wireless networks, this work seeks an optimal short-term, continuous-time power-procurement schedule to minimize operating expenditure and the carbon footprint of cellular wireless networks equipped with energy-storage capacity, and hybrid energy systems comprising uncertain renewable energy sources. Despite the stochastic nature of wireless fading channels, the network operator must ensure a certain quality-of-service (QoS) constraint with high probability. This probabilistic constraint prevents using the dynamic programming principle to solve the stochastic optimal control problem. This work introduces a novel time-continuous Lagrangian relaxation approach tailored for real-time, near-optimal energy procurement in cellular networks, overcoming tractability problems associated with the probabilistic QoS constraint. The numerical solution procedure includes an efficient upwind finite-difference solver for the Hamilton–Jacobi–Bellman equation corresponding to the relaxed problem, and an effective combination of the limited memory bundle method (LMBM) for handling nonsmooth optimization and the stochastic subgradient method (SSM) to navigate the stochasticity of the dual problem. Numerical results, based on the German power system and daily cellular traffic data, demonstrate the computational efficiency of the proposed numerical approach, providing a near-optimal policy in a practical timeframe.