As a potential energy utilization paradigm, power and natural gas coupled system (PGCS) with gas turbines and power to gas (P2G) provides an effective approach to improve the energy efficiency with an environmentally friendly and reliable energy supply. To fully exploit complementary potentialities of PGCS and accommodate wind power with P2G devices under system uncertainties, a novel robust multi objective power and gas flow (RM-PGF) incorporating a two-stage interval robust dispatch is proposed in this paper. Besides, to achieve a complementary energy utilization with an information privacy and a decentralized autonomy for these two sub-systems, this paper establishes a novel two-layer convex decoupled-synergetic framework for PGCS. In the outer optimization, the original nonconvex issue is convexified into an iterative second-order cone programming (SCOP) through a penalty convex concave procedure (PCCP). While in the inner optimization, the convexified RM-PGF is solved with three decoupled-synergetic strategies, including alternating direction method of multipliers (ADMM), auxiliary problem principle (APP) and generalized Benders decomposition (GBD) based strategies. Finally, numerical simulations validate the superiority of the proposed RM-PGF with P2Gs for wind power accommodation, as well as the computational efficiency and robustness of these convex decoupledsynergetic strategies PCCP-ADMM, PCCP-APP and PCCP-GBD. (C) 2018 Elsevier Ltd. All rights reserved.