Power systems are undergoing unprecedented transformations with the incorporation of larger amounts of renewable energy sources, distributed generation and demand response. All these changes, while potentially making power grids more responsive, efficient and resilient, also pose significant implementation challenges. In particular, operating the new power grid will require new tools and algorithms capable of predicting if the current state of the system is operationally safe. In this paper we study and generalize the so-called energy function as a tool to design algorithms to test if a high-voltage power transmission system is within the allowed operational limits. In the past the energy function technique was utilized primarily to access the power system transient stability. In this manuscript, we take a new look at energy functions and focus on an aspect that has previously received little attention: Convexity. We characterize the domain of voltage magnitudes and phases within which the energy function is convex. We show that the domain of the energy function convexity is sufficiently large to include most operationally relevant and practically interesting cases. We show how the energy function convexity can be used to analyze power flow equations, e.g. to certify solution uniqueness or non-existence within the domain of convexity. This and other useful features of the generalized energy function are described and illustrated on IEEE 14 and 118 bus models. Power systems are experiencing revolutionary changes. Integration of renewable generation, distributed generation, smart metering, direct or price-based load-control capabilities have contributed to the revolution, but are also posing significant operational challenges by making the power system inherently stochastic and inhomogeneous. With these changes, the system operators will no longer have the luxury of large positive and negative reserves. Moreover, operating the future power grid will require developing new computational tools that can assess the system state and its operational margins more accurately and efficiently than current approaches. Specifically, these new techniques need to go beyond linearized methods of analysis and ensure that the power system is safe even in the presence of large disturbances and uncertainty. In this paper, we study the energy function approach and argue that this classic tool can be generalized to help solve a number of existing and emerging problems in operating modern power systems.
The optimization problem in REF was formulated in polar coordinates, i.e. in terms of the voltage magnitude and phase at each bus.
Convexity of Energy-Like Functions: Theoretical Results and Applications to Power System Operations
arXiv: Optimization and Control
Mathematics