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The existing spectral discretization-based methods (SDMs) are capable of accurately computing critical eigenvalues of large power systems when time delays in wide-area damping control loops are considered. However, SDMs suffer from huge dimension of the discretized matrices of spectral operators, which is usually dozens of times of actual system states. To resolve the problem, an idea of partial spectral discretization (PSD) is proposed in this paper where only the retarded state variables instead of all system states are discretized. Following the PSD idea, a partial and explicit infinitesimal generator discretization (PEIGD) method is presented for highly efficient eigen-analysis of large closed-loop delayed cyber-physical power system (DCPPS). In contrast to the original EIGD method, the order of the resultant discretized matrix of the infinitesimal generator is greatly reduced and close to the number of actual state variables. The computational burden of PEIGD can be an order of magnitude less than that of EIGD and nearly the same as eigen-analysis of a system without time delay. Moreover, PEIGD is endowed with exactly the same accuracy as EIGD in capturing critical oscillation modes. Both theoretical analyses and intensive tests on the 16-generator 68-bus test system as well as a 516-bus and a 33028-bus real-life large power systems verify the accuracy and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

Highlights • Matrix perturbation theory is applied to stability analysis and control of DCPPS. • A perturbation-based eigenvalue tracking method is offered to track targeted modes. • It is a building block for optimally tuning WADCs to damp interarea oscillations. • Delay approximation effect on SOD-based eigenvalue method's accuracy is analyzed. • Identified key factors guide to select suitable parameters to precondition DCPPS. Abstract In this paper, the matrix perturbation theory is introduced to delayed cyber-physical power system (DCPPS) for analyzing the variations of eigenvalues and eigenvectors when small changes are imposed on system parameters and time delays. On one hand, a perturbation-based eigenvalue tracking method is presented to reliably track the targeted interarea oscillation modes so that wide-area damping controllers (WADCs) can be optimally designed to damp them. On the other hand, key factors affecting the accuracy of the solution operator discretization (SOD)-based eigenvalue computation methods for DCPPS are identified by matrix perturbation analysis, which provide a guidance for selecting suitable parameters to precondition the DCPPS. The study results on the 16-generator 68-bus test system validate the effectiveness of the two applications of matrix perturbation theory. [ABSTRACT FROM AUTHOR]

This paper presents a nonsmooth optimization-based method to optimally tune wide-area damping controllers in large delayed cyber-physical power system (DCPPS). The objective is to maximize the damping ratios of the weakly damped interarea oscillation modes under multiple operating conditions. An eigenvalue-perturbation and receding-tracking method is then proposed to reliably trace the targeted interarea oscillation modes during the optimization process. The Broyden–Fletcher–Goldfarb–Shanno method combined with the gradient sampling technique is employed to efficiently solve the presented nonsmooth, nonconvex, and nonlinear eigenvalue optimization-based problem. The effectiveness of the proposed method is validated by the significantly improved damping of the interarea oscillation modes in the two-area four-machine test system and a real-life large DCPPS. [ABSTRACT FROM AUTHOR]

In this paper, solution operator discretization methods with linear multistep and implicit Runge–Kutta (SOD-LMS/IRK) are presented for efficient eigen-analysis of large delayed cyber-physical power system (DCPPS). First, the time integration-based discretization generates highly structured approximate matrices to the solution operator. Exploitation of the structure ensures low computational burden and high efficiency in solving the matrix-inversion-vector product involved in eigenvalue computation. Second, the implicitly restarted Arnoldi algorithm is employed to compute critical eigenvalue from the solution operator's discretized matrices. SOD-LMS/IRK are endowed with scalability in analyzing very large DCPPS by fully utilizing the inherent sparsity in the augmented system state matrices. The main contribution of the presented SOD-LMS/IRK is the improved efficiency and scalability of existing SOD-PS (pseudo-spectral collocation) and SOD-LMS/IRK. Numerical results on the 16-generator 68-bus test system, a 516-bus, and a 33028-bus real-life large transmission systems validate the effectiveness of the proposed SOD-LMS/IRK. [ABSTRACT FROM AUTHOR]

In eigenanalysis of large delayed cyber-physical power system (DCPPS), power engineers are interested in critical electromechanical oscillation modes with damping ratios less than a specified threshold. To efficiently compute these modes, a method based on pseudo-spectral discretization of the solution operator of DCPPS (SOD-PS) is presented in this paper. First, the unique spectral mapping properties of solution operator are analyzed. The largest eigenvalues in moduli of the operator correspond to the ones of DCPPS with the largest real parts. Second, a rotation-and-multiplication preconditioning technique is presented to enhance the dispersion among eigenvalues of the solution operator's discretized matrix. Third, critical electromechanical oscillation modes of DCPPS with the least damping ratios are captured with priority and an accelerated convergence rate by the implicitly restarted Arnoldi algorithm. Subsequently, the small signal stability of DCPPS can be readily and reliably determined. In SOD-PS, the unique property of Kronecker product and the inherent sparsity in augmented system matrices are fully exploited to guarantee efficiency and scalability. The accuracy and efficiency of the presented method are intensively studied and thoroughly validated on the 16-generator 68-bus test system and a real-life large transmission grid. [ABSTRACT FROM PUBLISHER]

This letter presents a computationally efficient approach to implement the spectral discretization-based methods for eigenanalysis of large delayed cyber-physical power system by intensively exploiting the unique property of Kronecker product. Theoretical analyses and numerical tests demonstrate the novelty and effectiveness of the efficiently implemented explicit infinitesimal generator discretization method. [ABSTRACT FROM PUBLISHER]

• The idea of partial spectral discretization is applied to the SOD-PS method. • Only the retarded system states are discretized, leading to the PSOD-PS method. • The PSOD-PS method can largely improve the availability and efficiency of SOD-PS. • Two implementation ways of rotation-multiplication preconditioning are derived. To analyze the small signal stability of large-scale time-delay power systems (TDPSs), critical oscillation modes are required to be reliably computed. The solution operator discretization with pseudo-spectral collocation (SOD-PS)-based method has been proposed to indirectly calculate critical eigenvalues from the solution operator's discretization matrix. However, dimension of the discretization matrix is dozens of times the number of system state variables, and thus limits the efficiency of SOD-PS. To resolve this, SOD-PS is evolved into partial SOD-PS (PSOD-PS) in this paper by applying the idea of partial spectral discretization (PSD). Specifically, only the discretization of retarded states is reserved instead of discretizing all system states in delay interval. The resultant partial discretization matrix exhibits very low order, which is close to the number of system state variables. In addition, the rotation-and-multiplication preconditioning technique is studied to efficiently capture oscillation modes with damping ratios less than the given threshold. Specifically, the preconditioning technique is implemented by two ways and their consistency is derived in detail. Both theoretical analyses and intensive tests on the 16-machine 68-bus test system, a real-life 516-bus provincial transmission system of China and the 33028-bus ultra-high-voltage (UHV) interconnected system of China verify the accuracy and efficiency of the presented PSOD-PS. [ABSTRACT FROM AUTHOR]

Modeling and formulation of a delayed cyber-physical power system (DCPPS) are the basis of small-signal stability analysis and control synthesis of the system. First, a logic describing the interconnections between various systems of delayed differential algebraic equations (DDAEs) and delayed differential equations (DDEs) is established. It clearly shows the transformation of the most general DDAEs of the non-index-1 Hessenberg form to the widely used DDEs with finite terms of actual delays. Second, the DDAE model of the index-1 Hessenberg form is formulated in detail for DCPPS, which features in preserving the inherent sparsity in system's augmented state matrix. Exploiting the sparsity guarantees the scalability in stability analysis and controller design of a large DCPPS. Third, three delay merge properties are presented with complete proofs, which indicate that the feedback and control delays in one control loop can be treated as one single time delay. The properties facilitate modeling and formulating DCPPS. Last, numerical studies are carried out on a real-life large DCPPS with wide-area damping controllers embedded. Test results validate the correctness of the presented theoretical backgrounds. [ABSTRACT FROM AUTHOR]

To efficiently analyze the stability of large delayed cyber-physical power systems (DCPPS) incorporating wide-area damping controllers, an iterative infinitesimal generator discretization-based method (IIGD) for computing critical eigenvalues of the system is presented. IIGD contains three core techniques to guarantee efficiency and scalability. First, the sparsity of the infinitesimal generator's discretized matrix, which possesses identical spectrum to DCPPS, is explored by reformulating its blocks into Kronecker products. Especially, the dominant block is factorized as sum of Kronecker products of constant Lagrange vectors and system state matrices, which lays the basis of further utilizing the sparsities in the augmented state matrices of DCPPS. Second, the shift-invert preconditioning technique is applied to transform the required eigenvalues into those dominated in moduli. Third, the inverse iteration of the discretized matrix involved in sparse eigenvalue computation is iteratively achieved by utilizing the induced dimension reduction method (IDR( s )). Subsequently, the discretized matrix–vector product embedded in the method is efficiently implemented by exploiting the unique property of Kronecker product and the inherent sparsities in augmented system state matrices. The correctness, accuracy, efficiency and scalability of IIGD are extensively studied and thoroughly validated on the two-area four-machine test system and a real-life large transmission grid. [ABSTRACT FROM AUTHOR]