Fractional programming-based energy efficiency optimization under nonlinear operating constraints in sustainable power networks

Abstract

Generically, the efficiency of a power network can take a fractional form, which is the ratio between the useful delivered power and the total cost and the losses of the system. However, the maximization of such an efficiency is ambiguous due to the non-convexity of the involved functions resulting from the nonlinear power flow Equations in which the voltage magnitudes and angles are coupled. The application of standard NLP solvers on this objective function may lead to local optimality because of the same reason. This paper presents a Dinkelbach-type parametric reformulation where the initial non-convex fractional efficiency maximization problem is transformed into a set of zero-root parametric subproblems that are solved via a minimax saddle-point formulation over a semidefinite relaxation of the nonlinear constraints obtained by lifting voltage products into a rank-relaxed Hermitian variable. The existence of a saddle point is verified through the application of Slater’s condition based on the minimax duality concept; besides, it has been shown that a two-layer iterative procedure based on the combination of the inner primal-dual interior point solver and the outer Dinkelbach method converges to the unique optimal value. The numerical verification on the IEEE 30-bus and 118-bus systems demonstrates the theoretical convergence of the proposed approach, that is, the outer iteration converges in 5 and 7 optimizing the efficiency ratio of 4.821 and 8.194 according to the 30-bus and 118-bus systems respectively and surpassing 17.4% and 16.7% gains of the SQP approach. These results demonstrate that the proposed fractional programming and minimax duality framework provides a mathematically rigorous and computationally efficient route to globally optimal energy efficiency operation in nonlinearly constrained power networks.