The ever-increasing popularity of distributed generation resources and modernization of power system devices is making the power grid operations more complex. Fur- thermore, the non-convexity of the optimal power flow (OPF) challenges the solver to reach the global optimal solution and affects the overall accuracy of the solution. To overcome this problem, the convex relaxation methods are increasingly adopted to improve the computational efficiency of the solution and reach the global optimal point. New convex optimization methods for Optimal Power Flow and Unit Com- mitment applications are introduced in this research work. First, a multi-objective OPF formulation for transmission networks is proposed. The objective function in- cludes total generation cost and voltage stability margin. The effect of the weighting factor of the objective functions on the solution has been observed. The formula- tion is then tested on IEEE 14 bus and IEEE 118 bus systems, and the results are analyzed. Second, a combined UC-OPF formulation is presented based on the mixed- integer semidefinite programming. The UC and OPF consist of separate operating constraints for power system operation. Thus combining the constraints can cover the entire range of power system operations constraints. Since UC is mixed-integer linear programming (MILP) problem and OPF is a convex optimization problem, thus combining the two becomes a MISDP problem. The algorithm is developed and tested using IEEE 14 bus, and IEEE 118 bus systems, and the results were validated. Also, a branch-and-bound (BnB) approach is formulated for the combined UC-OPF problem, and the solutions from the BnB approach are compared and validated with a two-staged MISDP approach. Third, an SDP relaxed OPF formulation is presentediii iv for multi-phase unbalanced distribution networks. The approach is based on the branch flow model of the network. The formulation includes detailed modeling of voltage regulators, mutual coupling of the branch impedance matrix, and network switches. The formulation is tested on IEEE 123 bus system and modified 650 bus system, a part of the IEEE 8500 bus network. The solutions were compared with the non-linear power flow solution using the same operating scenario. Due to the increase of power system equipment and distributed generation resources in the distribution network, along with the massive size of the network, the distributed approach to solve the OPF problem has become a significant field of research. An alternating direction method of multipliers(ADMM) based OPF formulation is discussed to solve a large- scale network. For this, the power grid is divided into multiple sub-networks based on geographical position or the location of the regulators then OPF sub-problems are solved for each of the sub-networks iteratively until the global convergence is achieved. The solution is compared with the centralized approach and validated. The algorithm is tested on IEEE 123 bus system and 2500 bus system. The accuracy of the solution in all the cases. It was found that all the methods are accurate, computationally efficient, and provide global optimal solutions.