This paper records the development of a state variable model of the megawatt-frequency control problem of multiarea electric energy systems. The model is in a mathematical form necessary for application of theorems of modem optimal control theory.

The Megawatt-Frequency Control Problem: A New Approach Via Optimal Control Theory

This paper deals with the problem of the estimation of regions of asymptotic stability for continuous, autonomous, nonlinear systems. The first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields. In the second part certain topological considerations are first developed and the "trajectory reversing method" is then presented together with a theorem on which it is based. In the final part, several examples of application are reported, showing the efficiency of the proposed technique for low-order (second and third) systems.

On the estimation of asymptotic stability regions: State of the art and new proposals

Paper: Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems

Selective Modal Analysis (or SMA) is a physically motivated framework for understanding, simplifying and analyzing complicated linear time-invariant (or LTI) models of dynamic systems [1,2,3]. SMA allows one to focus on any prespecified dynamic pattern of intrest in the model. In particular, One can efficiently and accurately compute the eigene values and eigenvectors of the natural modes of interest and their sensitivities, and also determine physically meaningful reduced order models containing these natural modes. SMA is particularly suitable for dealing with composite models, i.e. models consisting of several dynamic subsystems interrelated by static constraints. An introduction to the basic concepts of SMA pertinent to the applications being considered here is presented in the companion paper [3]. This paper concerns the application of SMA to the Dynamic Stability problem in electric power systems; it is shown how SMA is well suited to meet the demanding requirements of Dynamic Stability analysis. This is illustrated in [3] with examples, including a 60-machine model of a dynamic instability occurrence in an actual power system.

/pdf/selective-modal-analysis-with-applications-to-electric-power-4rzqn9kfrx.pdf

Selective Modal Analysis With Applications to Electric Power Systems, Part II: The Dynamic Stability Problem

With the increasing use of voltage source converters (VSCs) in power electronics dominated power systems, oscillation phenomena in DC-link voltage control (DVC) timescale (around 10 Hz) among multiple VSCs have occurred. Several studies have tried to analyze these oscillation problems, but all associated with the single VSC situation. To consider the dynamic interactions between VSCs in DVC timescale, especially in the weak grid condition, this paper presents a small-signal model to understand VSC external characteristics based on motion equation concept also featured in synchronous generator (SG). Comparisons of time-domain simulation responses and eigenvalues show that the proposed model can hold the main behaviors of concern. The form of the model is very similar to the rotor motion equation in SG, with which power engineers have been very familiar. In addition, by establishing the relationship between the unbalanced powers and state variables of internal voltage (viz., VSC output voltage), the modeling idea introduced in this paper can be applied to other power electronic devices.

Modeling of Grid-Connected VSCs for Power System Small-Signal Stability Analysis in DC-Link Voltage Control Timescale

In recent years important research has been done in the area of system optimization by control engineers. Many theoretical results have been published but application examples have mainly been on low-order systems. An attempt is made to apply a certain class of optimal control theory, known as the state regulator problem, to obtain an optimal controller to improve the dynamic response of a power system. The system differential equations are written in the first-order state variable form. A cost functional is then chosen, and the matrix Riccati equation is solved. Puri's and Gruver's method is applied for the numerical computation, and the system is made initially stable by shifting the system eigenvalues.

Application of an Optimal Control Theory to a Power System

For an optimal linear regulator design a performance function of the quadratic form must be chosen. The question arises of how to decide the weighting matrix Q of the performance function. A new method is developed in this paper to determine Q in conjunction with a left shift of the dominant eigenvalues as far as the practical controllers permit. The method is then applied to the optimal control design of a typical power system. Three cases are investigated, the first with an optimal excitation control uE, the second with optimal governor controls uG and uG, with and without the dash-pot, and the third with uE plus uG control. The stabilizing signals thus obtained are given nonlinear tests on the same power system. It is found from the results that the optimal controls are more effective than conventional excitation control, that the optimal governor control without dash-pot is just as good as the optimal excitation control, and that the optimal uE plus uG control is the best way to stabilize a power system.

Optimal Power System Stabilization Through Excitation and/or Governor Control

The stabilization of power systems with excitation control using speed or power signals has been very well developed,1-6 while the application of optimal control to power system stability study is relatively new.7 Both are based on linear models. In this paper, a comparison is made of these two techniques. Signals derived from a lower order linear model are tested on a high order non-linear system modelled as a single machine-infinite bus as well as multi-machine systems.

Stabilization and Optimal Control Signals for a Power System

The optimal linear regulator design technique of determining the weighting matrix Q with dominant eigenvalue shift, developed in a companion paper[6], is applied to the optimal stabilization of a multi- machine system. Two cases are studied, the first is with one optimal controller uE1 on one machine of the m-machine system and the other with m optimal controllers uEM for the m-machine system. For comparison arison a one optimal controller uE on one machine with the rest approximated as an infinite system and m optimal controllers of individual idual design are also investigated. Nonlinear tests are given. It is found that uEI control is better than uE control, uEM control is better still than UEI control, and individual uE controllers are not good for the overall system.

Optimal Stabilization of a Multi-Machine System

In this paper the dynamic interactions between mechanical and reactive modes of machines in a multi-machine power system are investigated. The dynamic interactions are expressed in terms of machine angles, transient voltages, and coefficients which are functions of system parameters and operation conditions. The effect of the dynamic interaction can be examined by the electric torque loci on the A6-Aw phase plane. It is found that the interaction between the mechanical modes of machines has significant effect on stability characterized by the natural mechanical oscillation frequency and damping of individual machines.

Yao-nan Yu

Papers

Application of an Optimal Control Theory to a Power System

Optimal Power System Stabilization Through Excitation and/or Governor Control

Stabilization and Optimal Control Signals for a Power System

Optimal Stabilization of a Multi-Machine System

Dynamic Interaction of Multi-Machine Power System and Excitation Control