Power System Coherency and Model Reduction

This introductory chapter gives a brief overview of power system coherency and model reduction literature. This survey focuses on both the early results and some more recent developments, and organizes power system model reduction techniques into two broad categories. One category of methods is to use coherency and aggregation methods to obtain reduced models in the form of nonlinear power system models. The other category is to treat the external system or the less relevant part of the system as an input–output model and obtain a lower order linear or nonlinear model based on the input–output properties. This chapter also provides a synopsis of the remaining chapters in this monograph.

There has been a continuing need over the past several decades to model larger and larger interconnection wide models. Models of the complete interconnections with up to 50,000 buses are regularly used for system planning studies. These models typically go down to 115 and 69 kV levels, but ignore underlying 35 kV sub-transmission networks. With the growing deployment of plug-in vehicles, distributed generation and smart load controls, along with the need to perform realistic system restoration drills there is a need to model interconnections down to the feeder breaker level. Restoration drills also require modeling of power plant auxiliaries and emergency generator systems, especially for nuclear units. It is conceivable that the size of interconnection wide models could grow by another order of magnitude. The EPRI DYNRED (Dynamic Reduction) computer program reduces a large-scale system model into a smaller equivalent model for use in transient stability studies. The program has been used since the 1970s to build equivalent models of the Eastern U.S. and Western U.S. interconnected power systems. The DYNRED program accepts a normal transient stability database as input, and develops an equivalent that is a fraction of the size of the full power system representation, while adequately retaining the dynamic characteristics of the full system. The reduction process requires only a fraction of the time needed for a transient stability simulation.

This chapter presents the theory and analysis of slow coherency and aggregation. The main idea is that slow coherency arises from interarea modes, that is, groups of machines swinging together against each other at oscillatory frequencies slower than the local modes of the machines in the same coherent group swinging against each other. We show analytically that this phenomenon can be attributed to the coherent areas being weakly coupled, either because of higher impedance transmission lines, heavily loaded transmission lines, or fewer connections between the coherent areas compared to the connections within a coherent area. These system properties allow the use of singular perturbations to display the time-scale separation of the interarea modes and local modes, resulting in eigenvector-based algorithms to identify coherent machines. In addition, the singular perturbations technique has the capability to provide correction terms for improving reduced-order models in capturing the slow coherent dynamics.

Constructing a dynamic equivalent for a power system involves several steps: the partition of the system into coherent areas, the aggregation of coherent generator buses, and the aggregation of the coherent generators and their control devices. These steps have been discussed in previous chapters, including a method in Chap. 2 to aggregate the exciter models using frequency response. In this chapter, we investigate a trajectory sensitivity method to tune the aggregate exciter parameters of the equivalenced model. The optimal parameters of the aggregated exciter yielding the least error are used to evaluate the sensitivity technique against the aggregation method in the DYNRED program and a weighted MVA-based method. A three-machine system with one coherent area satisfying the theoretical coherency conditions is used to investigate the impact of the variations of the individual generator, network, and exciter parameters on the aggregate exciter model parameters. The technique is then applied to the exciter aggregation of a larger NPCC 48-machine system.

In this chapter, a hybrid dynamic equivalent consisting of both a coherency-based conventional equivalent and an artificial neural network (ANN)-based equivalent is developed and analyzed. The ANN-based equivalent complements the coherency-based equivalent at all the boundary buses of the retained area. It is designed to compensate for the discrepancy between the full system model and the reduced equivalent developed using any commercial software package, such as the dynamic reduction program (DYNRED), by providing appropriate power injections at all the boundary buses. These injections are provided by the ANN-based equivalent which is trained using the outputs from a trajectory sensitivity simulation of the system responses to a candidate set of disturbances. The proposed approach is tested on a system representing a portion of the Western Electricity Coordinating Council (WECC) system. The case study shows that the hybrid dynamic equivalent can enhance the accuracy of the coherency-based dynamic equivalent without significantly increasing the computational effort.

In this chapter, we discuss two mathematical approaches for model reduction of power systems which do not use coherency information. The advantages of these approaches lie in the ability to handle large systems. The Krylov and balanced truncation methods take into account system reachability and observability in obtaining reduced-order models of the external system, and thus would perform better than methods based simply on eigenvalues. In the case of balanced truncation approach, a sensitivity analysis is carried out. Through the eigenvalue information, we also establish a connection of these methods with the coherency- based approaches.

This chapter illustrates, by a practical case study, an application of the dynamic model reduction for a large power system model. This example concerns the creation of a reduced-order model from a full WECC model, to be used by BC Hydro for on-line dynamic security assessment (DSA). This model reduction was conducted using the software DYNRED. The focus of this case study is on the objective of the model reduction, the approach and procedure, the results, and the benchmarking work to ensure the suitability of the reduced models for the required applications. It is shown that, following the procedure used in the case study, a reduced model can be obtained for use with a wide range of system conditions with acceptable results and computation time saving.

Selective Modal Analysis (SMA) is a comprehensive methodology for the modeling, analysis, and control of selected parts of the dynamics of systems described by large linear time-invariant (LTI) models. The main components of SMA are sensitivity tools and algorithms for reduced-order eigenanalysis. Sensitivity tools developed within the SMA framework include participation factors, which measure the participation of the state variables in the eigenmodes and vice versa. Participation factors play a central role in SMA developments. This chapter shows the role of participation factors in the identification of dynamic patterns, design of damping controllers, and reduced-order eigenanalysis.

Interarea oscillations are predominantly governed by the slower electromechanical modes which, in turn, are determined by the coherent machine rotor angles and speeds. The issue is that, although these rotor angles and speeds provide the best visibility of such modes, currently they are not available from phasor measurement units (PMU). As such, the aim of this chapter is to demonstrate that interarea oscillations are observable in the network variables, such as voltages and line currents, which are measured by PMU. By analyzing the electromechanical modes in the network variables, we can trace how electromechanical oscillations are spread through the power network following a disturbance. Applying eigenvalue and sensitivity analysis, we provide an analytical framework to understand the nature of these network oscillations through a relationship termed network modeshapes. Using this relationship, a novel concept, “dominant interarea oscillation paths,” is developed to identify the passageways where the interarea modes of concern travel the most. We demonstrate the concept of the dominant path with an equivalent two-area system. We propose an algorithm for identification of the dominant paths and illustrate with a reduced model of a large-scale network. Finally, we end this chapter with an important application of the concept: feedback input signal selection for damping controller design.