Partial Spectral Discretization-Based Wide-Area Damping Control Of Large-Scale Delayed Cyber-Physical Power System

The large-scale development trend of modern power system intensifies the inter-area low frequency oscillations,which restrict the transmission capacity and stability of the system.The wide-area measurement system(WAMS)based on the phasor mea-surement unit(PMU)provides a new information platform for situational awareness,wide-area protection and coordination control of large-scale interconnection power sys-tem.By introducing the remote signals which can effectively reflect the inter-area low frequency oscillations,the wide-area damping control can significantly improve the damping of the weakly damped electromechanical oscillations.However,measuring,routing,processing and transmitting wide-area signals in WAMS introduce inevitable communication delays,which can deteriorate the damping characteristic of wide-area damping controllers(WADCs)and even bring the risk for the stability operation of power grid.When the effects of communication delays are considered,the large-scale power system evolves into large-scale delayed cyber-physical power system(DCPPS).Therefore,it is necessary to construct corresponding method system including model-ing,analysis and control of DCPPS.Aim at solving the delay problem involved in wide-area damping control,the spec-tral operator discretization-based eigen-analysis methods in the fields of computational mathematics and numerical analysis are imported into the power domain in this disser-tation.Based on the idea of partial spectral discretization,the dissertation concentrates on the small signal stability analysis and wide-area damping control of DCPPS.The research includes the following two aspects.The critical eigenvalues of large-scale DCPPS are accurately and efficiently computed based on the partial discretization of two spectral operators,i.e.,infinitesimal generator and solution operator.And,the pa-rameters of WADCs are optimally tuned based on eigenvalue optimization.The major contributions and innovations are summarized as follows.(1)Theoretical framework of partial spectral discretization-based eigenvalue com-putation methods is given for small signal stability analysis of large-scale DCPPS.It in-cludes partition of system state variables,spectral mapping,partial spectral discretiza-tion,spectral transformation,spectral estimation and spectral correction.The core of spectral discretization methods is firstly to build the transfer function of DCPPS by us-ing two spectral operators,which transforms the delayed differential equation(DDE)describing the system dynamics into functional ordinary differential equation(ODE).Subsequently,the eigenvalues of DCPPS are turned into the spectra of infinitesimal generator and solution operator,which can avoid the direct solution of the exponential terms involved in the characteristic equation of DCPPS.Then,the partial spectral dis-cretizaiton of infinitesimal generator and solution operator is implemented by discretiz-ing the state variables in the past time which are related to the states in current time or current time interval.Accordingly,the spectra with infinite dimension are transformed into eigenvalues of finite-dimensional discretized matrices.Finally,the least damped or the rightmost eigenvalues of DCPPS are derived from the critical eigenvalues of discretized matrices,which can be used for stability analysis of DCPPS.(2)The partial explicit infinitesimal generator discretization(PEIGD)method is pro-posed for eigenvalue computation of large-scale DCPPS.Firstly,the transformation of DDEs into functional ODE yields the spectra problem of infinitesimal generator in Banach space.Secondly,the pseudo-spectral discretization scheme is adopted to discrete the infinitesimal generator to obtain the partial discretization matrix.Then,the shift-invert transformation is executed to transform the rightmost eigenvalues of DCPPS into eigenvalues with the largest moduli of preconditioned partial discretiza-tion matrix.Subsequently,the implicitly restarted Arnoldi(IRA)algorithm is adopted to compute these critical eigenvalues with priority by fully using the sparsity of dis-cretization matrix and system state matrices.Finally,the accurate eigenvalues of D-CPPS are corrected by the Newton method.The PEIGD method is validated on the 1 6-generator 68-bus test system,the Shandong power grid and the ultra high voltage North China-Central China interconnected power grid,respectively.Simulations show that the PEIGD method can accurately capture the critical eigenvalues of large-scale D-CPPS.Compared with the EIGD method,the PEIGD method can dramatical ly enhance the computation efficiency while ensuring the accuracy.Especially for large-scale sys-tem,the efficiency can be improved by about 10 times,which is comparable to that of the traditional eigenvalue computation for system without any time delay.(3)The partial pseudo-spectral collocation discretization method of solution oper-ator(PSOD-PS)is proposed for eigenvalue computation of large-scale DCPPS.First-ly,the eigenvalues of DCPPS are transformed into the spectra of solution operator in Banach space based on the transformation of DDEs into functional transfer function.Secondly,applying the pseudo-spectral discretization scheme to discrete the solution operator yields the pseudo-spectral discretization matrix.Based on the idea of par-tial spectral discretization,the partial discretization matrix can be derived by remov-ing the discretization of plenty of past states which have no contribution to the states of current time interval.Through two implementations of rotation-and-multiplication preconditioning technique,the preconditioned discretized matrix is generated,whose eigenvalues with largest moduli are computed with priority by IRA.In IRA algorithm,the sparsity of discretization matrix and system state matrices is fully utilized.Fip nally,the Newton method is applied to capture accurate eigenvalues of DCPPS.The PSOD-PS method features obtaining the eigenvalues of DCPPS with damping ratios smaller than a given value by one calculation.The accuracy,efficiency and scalability for large-scale system of the PSOD-PS method are studied on the 16-generator 68-bus test system and two real-life systems.The PSOD-PS method can obviously improve the computation efficiency with the identical computational accuracy to the SOD-PS method.In the analysis of large-scale systems,the PSOD-PS method can save as much as 57%computation time.(4)The eigenvalue optimization-based method is proposed to optimally tune WAD-C parameters.Firstly,the objective function is proposed to maximize the minimum damping ratio of the targeted modes,i.e.,weakly damped interarea oscillation modes,under multiple operating conditions.The presented objective can effectively describe the characteristic of WADCs and is immune to the potential "mode masking" problems,which essentially ensure the optimal damping features of controllers.An eigenvalue tracking method is proposed to reliably trace the targeted modes based on matrix per-turbation theory.Then the constrained optimization problem is rewritten by the penal-ty function method.Subsequently,the Broyden-Fletcher-Goldfarb-Shanno(BFGS)method combined with the gradient sampling technique and the weak Wolfe criteri-on is employed to efficiently solve the presented nonsmooth,nonconvex and nonlinear eigenvalue optimization-based problem.Specifically,the steepest descent directions at differentiable and non-differentiable points are searched by the BFGS method and the gradient sampling technique,respectively,which avoids the nonoptimal stagnates.The step-length is searched by the weak Wolfe criterion.The effectiveness,optimality and robustness of the proposed optimal tuning method are extensively studied on the two-area four-machine test system and the Shandong power grid.Test results show that the optimally tuned WADCs can efficiently improve the damping of DCPPS.The feature of the critical eigenvalue computation methods for large-scale DCPP-S is summarized as follow.The application of these methods enables that DCPPS can be analyzed by using the perfect theoretical framework and theoretical achieve-ments of eigenvalue-based small signal stability analysis for traditional power system.Moreover,these methods facilitate establishing the foundation of revealing the effec-t mechanism of time delays on wide-area damping control and designing the optimal WADCs for DCPPS.The optimal tuning of WADCs based on eigenvalue optimization develops the new idea and method for wide-area damping control of modern pow-er system,thereby promoting the development and application of wide-area damping control.This work has the realistic theoretical significance and application value in solving the low frequency oscillation problem in Chinese interconnection power grid,and then guaranteeing the safe and stable operation of power grids.