Real-time electricity markets are the main transaction platforms for providing necessary balancing services, where the market clearing (nodal or zonal prices depending on markets) is very close to real time operations of power systems. We present single and multiple time period decentralized market clearing models based on the DC power flow model. The electricity market we study consists of a set of Generation Companies (GenCos) and a set of Distribution System Operators (DSOs). The Independent System Operator (ISO) determines the market clearing generation and demand levels by coordinating with the market participants (GenCos and DSOs). We exploit the problem structure to obtain a decomposition of the market-clearing problem where the GenCos and DSOs are decoupled. We propose a novel semismooth Newton algorithm to compute the competitive equilibrium. Numerical experiments demonstrate that the algorithm can obtain several orders of magnitude speedup over a typical subgradient algorithm with no modification to the existing communication protocol between the ISO and market participants.