A Closed-form Formulation of Eigenvalue Sensitivity Based on Matrix Calculus for Small-signal Stability Analysis in Power System

With the rapid development of power-electronicsenabled power systems,the new converter-based generators are deteriorating the small-signal stability of the power system.Although the numerical differentiation method has been widely used for approximately calculating the eigenvalue sensitivities,its accuracy has not been carefully investigated.Besides,the element-based formulation for computing closed-form eigenvalue sensitivities has not been used in any commercial software due to the average efficiency,complicated formulation,and errorprone characteristics.Based on the matrix calculus,this paper proposes an easily manipulated formulation of the closed-form eigenvalue sensitivities with respect to the power generation.The distinguishing feature of the formulation is that all the formulas consist of vector and matrix operations,which can be performed by developed numerical algorithms to take full advantages of architectural features of the modern computer.The tests on WSCC 3-machine 9-bus system,New England 10-machine 39-bus system,and IEEE 54-machine 118-bus system show that the accuracy of the proposed formulation is superior to the numerical differentiation method and the efficiency is also greatly improved compared to the element-based closed-form formulation.The proposed formulation will be helpful to perform a more accurate and faster stability analysis of a power grid with converter-based devices.