Constraint-Embedded Fractal Self-Similar Generation of Synthetic Topologies for Receiving-End Power Grids with Rich Renewable Energy: A Hybrid Metaheuristic Approach

Fractal geometry offers a natural language for the self-similarity and scale invariance of power networks, yet future receiving-end grids shaped by high renewable penetration, high load density and large HVDC infeed have no historical counterpart. This paper formalizes the synthesis of such grids as a Constrained Fractal Iteration System (CFIS), in which synthetic topology is generated by the scale-progressive composition of three fractal operators, each coupling a spatial clustering/topology operator with an engineering constraint-projection operator. A hybrid metaheuristic realizes the operators through dynamic-weighted K-means (DWK-Means), social-learning particle swarm optimization (SL-PSO) and constraint-embedded 2-opt (CE-2-opt). On a provincial receiving-end grid at 162, 234, and 363 GW, the generated topologies are scale-invariant over a well-defined box-size range: the box-counting dimension rises from 1.675 to 1.800 with load and shows a clear voltage-layered hierarchy; the constrained dimension remains within 1.5% of the unconstrained value, showing that engineering feasibility costs almost no fractal quality. The 500 kV station set attains a correlation dimension of 1.81 versus 1.69 for the real existing grid, consistent with the interpretation of the synthetic grid as an idealized fractal upper bound. Short-circuit levels are brought within the 63 kA limit after correction, and renewable curtailment and load shedding remain zero; N-1 screening of all elements shows that the worst post-contingency loading falls from 98.5% at 162 GW to 91.4% at 363 GW, satisfying the thermal limits under single contingencies. Against classical topology generators built on the same node set, the proposed backbone attains full degree-two redundancy and a comparable fractal dimension with about 24% less line length, and an ablation of the redundancy bound shows that it, rather than the short-circuit limit, governs steady-state security at the low load scale.