A direct and rigid algorithm approach based on Equivalent Current Injection (ECI) for large-scale distribution power flow analysis is proposed in this dissertation. This algorithm used two primary matrices: BI and ZV-BC. Two matrices, which are built from the topological characteristics of distribution networks, are used to achieve the power flow solutions. BI matrix is the bus injection to branch current matrix and the ZV-BC matrix describes the relationship between the bus voltage mismatches and the branch current. The building algorithm is easily programmable and can be accomplished by a simple search technique with the two proposed matrices. Four connected cases are considered in this dissertation. The proposed algorithm is robust and accurate. Test results demonstrate the potential and validity of the proposed algorithm in distribution applications. Secondly, this thesis also presents a fault analysis with hybrid compensation for unbalanced distribution systems is proposed. The method employs the unbalanced three-phase model to analyze faults. BI and ZV-BC matrices containing information of the topological characteristics of distribution networks were built along with the proposed hybrid compensation method for analysis. Appropriate boundary conditions can be obtained for a fault to solve various types of single or simultaneous faults. The time-consuming LU decompositions, the Jacobian matrix, or the Y admittance matrix, required in the traditional algorithms, are not needed in the new development. Test results show that the proposed method is efficient, easy to program, also with advantages of high speed, robustness, improved accuracy, and lower memory requirements. This thesis also presents a hybrid programming (HP) technique to solve the reconfiguration problem for loss reduction and service restoration in Smart Grid application.