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In this paper, we propose an optimization model that considers two pathways for injecting renewable content into natural gas pipeline networks. The pathways include (1) power-to-hydrogen or PtH, where off-peak electricity is converted to hydrogen via electrolysis, and (2) power-to-methane, or PtM, where carbon dioxide from different source locations is converted into renewable methane (also known as synthetic natural gas, SNG). The above pathways result in green hydrogen and methane, which can be injected into an existing natural gas pipeline network. Based on these pathways, a multi-period network optimization model that integrates the design and operation of hydrogen from PtH and renewable methane is proposed. The multi-period model is a mixed-integer non-linear programming (MINLP) model that determines (1) the optimal concentration of hydrogen and carbon dioxide in the natural gas pipelines, (2) the optimal location of PtH and carbon dioxide units, while minimizing the overall system cost. We show, using a case study in Ontario, the optimal network structure for injecting renewable hydrogen and methane within an integrated natural gas network system provides a $12M cost reduction. The optimal concentration of hydrogen ranges from 0.2 vol % to a maximum limit of 15.1 vol % across the network, while reaching a 2.5 vol % at the distribution point. This is well below the maximum limit of 5 vol % specification. Furthermore, the optimizer realized a CO2 concentration ranging from 0.2 vol % to 0.7 vol %. This is well below the target of 1% specified in the model. The study is essential to understanding the practical implication of hydrogen penetration in natural gas systems in terms of constraints on hydrogen concentration and network system costs. [ABSTRACT FROM AUTHOR]