In this extended study, the focus is on advancing the generation of synthetic distribution grids (SDGs) through the introduction of a new algorithm based on the Barabási-Albert Random Graph model. The initial use of the Erdős model to create SDGs revealed limitations in size and structural adjustability beyond the number of vertices. To address these limitations and push the research forward, the new algorithm utilizes the Barabási-Albert model to provide more control over the structural features of the generated Graphs through the introduction of a novel tuning parameter known as the “richness index”. The effectiveness of both algorithms in producing SDGs of various sizes is demonstrated by generating SDGs with different sizes, confirming their ability to mimic synthetic radial distribution grids successfully. Additionally, a detailed examination of degree-based parameters and Pearson coefficients for SDGs of sizes from 20 to 1000 uncovers significant patterns. Furthermore, the proposed algorithm is examined in the terms of the variation of richness index in branching rate and μ-PMU placement, confirming the scale-free characteristic of the method. A comparison of the Erdős and Barabási-Albert models shows variations in maximum degree values, branching rates, and mixing patterns. The original Barabási-Albert model tends to have nodes with higher degrees and increased branching rates, which can be adjusted by the richness index. These findings emphasize the ability of the Barabási-Albert model to generate scale-free SDGs with diverse structures by fine-tuning the richness index.