Improved Sparsity Techniques for Solving Network Equations in Transient Stability Simulations

A fully parallel nested bordered block diagonal form (BBDF) method for power system transient stability simulations is proposed in this paper. The fully parallel nested BBDF (FNBBDF) method is developed by combining the fully parallel BBDF method with the nested BBDF method based on subnet-core mapping and mixed programming of MPI and OpenMP. The FNBBDF method is compared with existing methods from the perspectives of computational burden, parallel overhead, and subnet-core mapping. Detailed tests of the FNBBDF method are carried out in the 2383wp system and two practical power systems (13490-node and 24886-node). The test results prove the efficiency of the proposed method. The FNBBDF method can increase the speedup ratio to more than 21x with 64 CPU cores in the 24886-node test power system.

This efficient application of any numerical integration method depends on accurate estimation of the local truncation error to govern step size control. In this paper, the local truncation error of the multirate method using the forward Euler integration is derived leading to the calculation of the optimal step ratio. The techniques of applying multirate method in power systems are discussed. By implementing the multirate method in a practical power system example, the multirate strategies are proved to be efficient compared with traditional simulation methods.

A new simultaneous solution method using module bi-directional iteration is proposed for power system transient stability simulation. In this method, power network is partitioned into a tree hierarchy; computation modules are established for decomposed power networks and various power system components respectively. Through representing every computation module by a computation node, a computation tree is constructed by connecting the nodes together according to their electrical relations in power systems. A tree-traversing procedure called forward reduction and backward evaluation is performed to calculate correction factors of the variables in Newton iterations. This high-efficiency simulation method is feasible to be applied in parallel computation for large interconnected systems. Simulation tests are conducted on the New England 10-generator test power system and the North China-Northeast interconnected system, and the results are compared with those of the commercial software BPA to validate the effectiveness and correctness of this method.

This study presents a graphical presentation of the harmonic operating principles in voltage source converter (VSC) with sparse vector methods and provides a visualisation method for analysing the harmonic operations of VSC. Based on the dynamic phasor sequence components model of the VSC, the operation analysis of VSC has been transformed into a solution of an algebraic equations. With the sparse characteristics of the vectors and coefficient matrix, the sparse vector method is applied into solving equations. The coupling from the ac‐side to the dc‐side can be represented by the forward substitution operations and the coupling from the dc‐side to the ac‐side can be represented by the back solution operations in the sparse vector method correspondingly. Therefore, the factorisation paths which describe the forward substitution and the back solution operations can describe the harmonic operating principles of VSC too. When the set of factorisation paths are presented as the weighted directed graphs based on graph theory, the harmonic operating principles of VSC can be displayed as the graphs, which is direct and clear. Finally, the simulation results testify the correctness of the studies.

This paper proposes the application of the symplectic Runge-Kutta algorithm combined with the flexible GMRES method to solve the differential algebraic equations encountered in the power system transient stability simulation. In the proposed method, the s-stage 2s-order symplectic Runge-Kutta is used to convert the differential algebraic system in to a set of nonlinear algebraic equations, and then the large algebraic system is solved using a flexible inner-outer preconditioned GMRES method. The proposed method is of inherently parallel. The numerical simulation results obtained on IEEE 145-bus test system show that, the proposed method can achieve significant improvement in speed as compared to the conventional approach.

This paper presents a new asymmetrical model of synchronous generators for power system transient stability simulations. The symmetrical component method has been used traditionally in the analysis of unbalanced faults in power systems. This is mainly because the synchronous generators, which play an important role in the power system, can be easily treated by the method. However, when analyzing a system that has elements with asymmetrical impedance, it may not be easy to use the symmetrical component method. In such cases, the simulations can be easily executed by using the phase coordinate method. However, there are few papers that derive a phase coordinate model of synchronous machines. This paper analyzes synchronous generators under asymmetrical conditions in detail by using the symmetrical component method, then derives a model of synchronous generators for the phase coordinate method. © 1997 Scripta Technica, Inc. Electr Eng Jpn 119(2): 46–59, 1997

This paper proposes a new approach for power system transient stability simulation, which is based on a semianalytical solution (SAS) of power system differential-algebraic equations. In this paper, an SAS is derived using the Adomian decomposition method as a closed-form explicit function of symbolic variables such as time, the initial state and other variables on system conditions, and hence it can directly give a power system's dynamic trajectory being accurate for a certain time window. Unlike a traditional numerical integration based simulation approach, the proposed new approach offline derives an SAS and online evaluates the SAS by plugging in values of symbolic variables for a series of time windows making up the desired simulation period. This paper further studies the maximum length of the time window for an SAS being accurate and proposes a divergence indicator for simulation using adaptive time windows. Implementation of this new approach on parallel computers is also studied. The new approach is validated through contingency simulation of the IEEE 10-generator 39-bus system with detailed generator models.

Electrical impedance tomography (EIT) is an inverse imaging technique used to reconstruct resistive distributions. Some EIT inverse methods, such as the Gauss-Newton method, use the finite element method in the forward problem to solve for the Jacobian matrix. The authors detail methods to improve the solution time in the forward problem and accurate Jacobian calculation. The first suggestion is to keep the same node grounded for all projection angles by floating the current source. Details of an optimised Jacobian formation are also presented including full exploitation by sparse vector methods. The application of EIT to interface pressure distribution measurements makes use of a pressure sensitive resistive mat that can be cut to any shape or size. This flexibility can be taken advantage of and a square or rectangular domain comprising square bilinear elements is chosen. The sparse matrix nested dissection node renumbering algorithm then gives further speed advantages and also enables best optimisation with sparse vector techniques.

Evaluation study for the integration of electromagnetic transients simulator and transient stability simulator

This paper considers the use of hybrid simulation combining electromagnetic transients (EMT) simulation and transient stability (TSA) simulation to model large networks. By comparing with a full detailed EMT simulation, the paper shows that the hybrid approach gives more accurate results than TSA simulation alone, when power electronic equipment is included in the EMT part of the model. The paper also considers line switching and faults in the TSA part of the model which were ignored in earlier approaches, and shows that the accuracy is a function of the electrical distance of the fault to the interface bus bars. A new method to improve this accuracy with the use of switchable frequency dependent network equivalents (FDNE) is also introduced.

It is common knowledge that the power flow can have multiple solutions; one is the normal (desired) solution while the others are alternative solutions, which are sometimes characterized by low voltages. Usually these alternative solutions represent erroneous or unstable operation conditions. The paper presents a method for quickly assessing whether the power flow has likely converged to an alternative solution by considering, on a bus by bus basis, the sensitivity of the bus voltage magnitude to a change in the bus's reactive power injection, coupled with whether the bus is closely connected with negative reactance branches. Sparse vector methods are used to calculate these sensitivities quickly even for large systems with 10,000+ buses. The paper shows that these solutions can be particularly problematic when assessing voltage stability impacts of geomagnetic disturbances on the grid, and uses the method to identify problematic areas of a system under a GMD event.

In transient stability simulators such as PSS/E, the bus frequency is estimated using a window of (positive-sequence) phasor angle measurements. A digital filter is often used to account for the filtering effect in actual measurement devices and to eliminate sudden changes during frequency computations. Although transient stability simulators label the filtered angle derivative as the “frequency,” the frequency provided by such programs does not match actual measurements reported by phasor measurement units (PMUs), which makes it difficult to gauge the validity of studies (e.g., wide-area event detection and control applications) that are based on such simulations. In this paper, we implement a frequency computation method directly using positive phasor angles provided by simulators. The proposed method is tested on two systems extensively and validated against the measurements of an actual PMU. The improved frequency measurements closely match PMU responses. Cross-validation results also suggest the method may be used for other systems without conducting full time-domain simulations.

Most transient stability programs can solve very large systems but the representation of individual components of the system, in particular power electronic devices and control systems associated with FACTS and HVDC transmission systems cannot be represented in detail. Electromagnetic transients simulators, on the other hand, can represent system components and control systems in great detail, but are impracticable for the simulation of very large systems, and dynamic equivalents are required. It is apparent that the future of simulation lies in the development of hybrid simulators, which would allow practicable simulation of large-scale power systems with embedded modeling on power electronic devices in devices level. In this paper, an attempt has been made to incorporate an electromagnetic transient model of a static VAr compensator (SVC) into the traditional transient stability simulation program.

The sensitivity of constrained linear inversions to the selection of the Lagrange multiplier is demonstrated for the can of inferring columnar aerosol size distributions from spectral aerosol optical depth measurements. Since negative values of the aerosol size distribution constitute an unphysical solution, the Lagrange multiplier is varied within a restricted range until a range of values is reached for which all elements of the solution vector are positive. In addition to the constraint that the solution vector be positive, it is necessary for the final solution to be a smooth function and to satisfy the original integral equation to within the noise level of the measurements. An iterative method is presented whereby an initial estimate of the size distribution is modified until the final solution satisfies both the positivity constraint and the requirement that the regression fit to the data using the inverted size distribution he consistent with the measurement errors. A formula for calculating the variances and covariances in the inversion solution is derived and applied to optical depth measurements obtained at the University of Arizona and at Goddard Space Flight Center. In the former case an estimate of the measurement errors is available and thus the inversion formula and error analysis explicitly includes the magnitudes of the measurement variances. In the latter case the measurement errors are not known and the analysis assumes the errors in the measurements are equal and uncorrelated. Results of the error analysis show that the variances in the solution vector are large for radii where the information content of the measurements is small.

This paper demonstrates the design and implementation of smart on-line fast load shedding to maintain system stability and operation when a major generating unit is suddenly tripped. The design stage included load flow and transient stability study cases to determine system stability limits, transient frequency variation curves and transient voltage profiles. Numerous outage scenarios that cover all possible operating conditions are simulated and key system stability parameters are identified. A fast load shedding priority list has been provided by system operations and has been included as needed into transient stability simulation cases to maintain system stability and generation/load balance. Fast load shedding logic diagrams are developed with triggering parameters including generating unit breaker status and on-line frequency measurements. On-line measurements of system generation, tie line flow and loads are continuously monitored and fed into a PLC system that processes information and issues load shedding trip signals to maintain balanced generation/load and stability. This fast load shedding scheme has been tested in real time outages several times, and proved to be accurate and in agreement with transient stability simulation results. Since its implementation, this load shedding scheme has successfully saved the oil field power system from total shutdown and maintained system operation after sudden tripping of major generating units