Employing phase trajectory length concept as performance index in linear power oscillation damping controllers

In this paper, phase trajectory length concept is employed to introduce a performance index in order to investigate the oscillation of linear power systems. At first, Phase Trajectory Length (PTL) concept in the state space of a Multi Input Multi Output (MIMO) linear system is defined as the traversed distance from a certain point in the state-space to the equilibrium point of the system. Moreover, lower and upper bounds of the PTL are computed. In order to evaluate and compare the oscillatory nature of the power system, oscillation number is defined as the ratio of the PTL to the radial distance of a certain point to the equilibrium. Based on this criterion, it is demonstrated that shortening the PTL leads to effective oscillation damping for all variables which are linear combinations of the main states of the intended system. It is proven that considering the aforementioned index in controller design leads to satisfy a certain finite boundary for Integral Absolute Error (IAE) of system states. Predicated on expressed features of the PTL, corresponding Hamilton Jacobi Bellman (HJB) equations with Minimum Length Controller (MLC) is represented to design damping controller. In order to reduce the settling time in oscillation damping, the desired time weight is augmented in the calculation of the considered objective function. The proposed index can be employed to tune the controller parameters. In this regard, a numerical algorithm is suggested to design a full state feedback controller as MLC. At the end, the given linear power examples, both in simulation and experimental results, show the benefits of this approach for analyzing and designing the oscillation damping controller in linear power systems.