The Role of Energy in Urban Expansion Models: An Approach Based on Electrical Systems Analogy

The search for an optimal urban expansion model, balancing densification with horizontal sprawl, remains a central challenge in sustainable planning. This paper proposes an innovative analytical framework that applies electrical distribution system principles to urban growth dynamics. Premised on the logic that urban infrastructure and services (the 'Attractor Poles') function similarly to electrical conductors, we mathematically demonstrate that horizontal expansion necessitates a polynomial replication of infrastructure, following the sequence 1:7:19... under the worst-case assumption of conditions of isolated, non-adaptive planning. To bridge this theoretical upper bound with empirical urban scaling laws, we introduce a Network Coupling Efficiency Factor η, which quantifies the operational gap between pure geometric sprawl and the sublinear sharing of infrastructure (β ≈ 0.85) demonstrated by Bettencourt et al. (2007). We argue that the spatial transition from circular stranded cable geometries to space-filling urban hexagons is a direct morphological consequence of this sublinear optimization. Moving beyond the electrical analogy, the article establishes the 'Right to Sunlight' as an illustrative boundary condition to prevent predatory verticalisation, public health degradation, and the loss of distributed solar generation potential. Finally, a multidimensional optimization framework—the "Umbrella Algorithm"—is proposed as a non-linear quantitative tool for homeostatic urban governance.