Estimation Of Power System Stability Domain By Sum Of Square Polynomials Programming

Stability is essential to safe operation of the power system.Considering the nonlinearity of power system,determination of the domain of attraction(DA)is an important issue.The main methods in transient stability analysis,i.e.the time domain simulation method and the energy function method,are mutually complementary.The energy function method merits in the rapid computation,and is potential in online application.But it is insufficient in the adaption to system model and the accuracy.Sum of square polynomial programming(SOSP)method approximates the DA via optimization of the energy function.Since no CUEP calculation is required,the method can overcome the main difficulties of the prevailing energy function method,i.e.construction of the energy function and CUEP calculation.This paper studies the effectiveness of SOSP in DA estimation of the power system.First,the principle of SOS programming is clarified.The SOS polynomial and its matrix representation are introduced.The SOS polynomial is checked by linear matrix inequality(LMI)feasibility test.The SOS index is introduced and then the LMI feasibility test is transformed into a semi-definite program(SDP).The lower bound of the polynomial is estimated by SOS polynomials.Without the restraints,the generalized SOS index is used in determining the positivity of the polynomial and rational polynomial.When the restraints are considered,the determination of the positivity of the polynomial by using the Positivestellensatz(abbreviated as P-satz)is discussed.That is to determine the lower bound of the polynomial by solving a SDP problem.Second,the procedure of DA estimation is studied.The power system is described by polynomial model with equality constraints.According to the P-satz,the problem of determining the coefficients in the polynomial in the energy function is transformed into a SDP problem,which can be solved by LMI feasibility test or one parameter optimization.The iteration procedure in energy function modification is clarified.Finally,case studies are performed on a single machine system and a three machine system.Starting from the initial energy function,the DA of the power systems are estimated by SOSP.The estimation is modified by iteration.The results show that,the DA estimation obtained by SOSP is conservative.No offensive result is exhibited.The estimated DA can be enlarged by iteration and thus the conservativeness is reduced.