Critical Infrastructure: The Key Role of Mathematical Algorithms

In a world where the efficiency and security of critical infrastructure are essential, mathematical algorithms provide innovative solutions to address complex challenges.

In particular, three key areas where they prove to be indispensable: the reduction of fare evasion in public transport, the optimization of self-consumption in energy communities, and the protection of computer networks against malicious targeted attacks.

By: Bernard Fortz

Reducing Fraud in Public Transport

Fare evasion in public transport, especially in systems without physical barriers where checks are random, is a major economic issue for operators. Losses caused by fraudulent passengers can be considerable, making it necessary to optimize control strategies.

A recent study proposes using the Stackelberg game to model the interaction between transport operators, who plan the checks, and opportunistic passengers who try to avoid them. This approach allows for more effective inspection strategies by planning patrols unpredictably, which complicates the task of fraudsters.

According to this study, the method improves the management of checks by integrating passenger behavior into the planning. The model uses mixed strategies to establish inspection probabilities, and heuristics have been developed to improve the quality of solutions, even if optimality is not always guaranteed. This methodology can be applied to different transport systems (metro, bus, tram, train...), allowing for a significant reduction in fare evasion.

Optimization of Self-Consumption in Energy Communities

Energy communities, where members share the energy they produce, store, and consume, represent a promising path toward a more sustainable energy transition. These communities remain connected to the public grid but aim to maximize their collective self-consumption rate, thereby reducing their dependence on the grid and their energy costs.

A pilot project, called Smart Lou Quila, was developed by the Montpellier-based start-up Beoga. It allows the seven members of the community to share 100% renewable and local electricity, produced by photovoltaic panels installed on the roofs of houses and on the roof of the municipal stadium.

In collaboration with this pilot project, a mixed-integer linear programming model was developed to optimize the self-consumption of an energy community. A mixed-integer linear model is a mathematical optimization method where the objective is to maximize or minimize a linear function, subject to linear constraints. What distinguishes this model is that some or all decision variables must take integer (and not continuous) values. It is widely used to solve problems involving discrete choices, such as planning, allocation, or scheduling, but it is often more complex to solve than a classic linear model due to the integrality constraint.

The model we developed manages the use of electrical devices, energy storage, and exchanges between community members. Thanks to this model, it has been shown that joining an energy community can reduce energy consumption from the public grid by at least 15%, with substantial savings on bills. The use of operational research tools for better management of energy communities was also the research focus of the European project SEC-OREA (Supporting Energy Communities – Operational Research and Energy Analytics) in which I participated.

Protection of Networks Against Targeted Attacks

Service networks, whether communication, logistics, or critical infrastructures, are particularly vulnerable to cyberattacks. These attacks aim to disable specific nodes of the network, causing severe malfunctions and disrupting the service.

To counter these threats, a new approach based on game theory has been developed. Game theory is a branch of mathematics that studies strategic interactions between rational actors, called “players.” It models situations where each player’s decisions influence the outcomes of the others, whether in cooperation or competition.

Our approach models the interaction between a network operator, who tries to protect their infrastructure, and an attacker seeking to maximize damage.

The researchers present mathematical formulations to optimize the protection of critical network nodes. The new mathematical formulations proposed outperform existing methods, offering better anticipation of attacks and a reduction in potential damages. These models are applicable not only to communication networks but also to other critical sectors such as public infrastructure and medical systems.

The challenges faced by transport operators, energy communities, and network managers are varied, but mathematical algorithms offer effective and adaptable solutions. Whether it’s about maximizing public transport revenue, optimizing energy self-consumption, or protecting networks against cyberattacks, advances in mathematical modeling and optimization play a central role in securing and optimizing our critical infrastructures, strengthening the resilience and sustainability of our systems in the face of current and future challenges.

This article was written in collaboration with Dr. Arnaud Stiepen, expert in scientific communication.

Translated from the original French version published in The Conversation.