Investigation of the structure of dynamic stabilization problems of power pool N. N. Lizalek, A. N. Ladnova, M. V. Danilov Institute of Power System Automation Siberian Branch of JSC R&D Center for Power Engineering

I NTRODUCTION In this article the prerequisites of the structural analysis power systems stability are considered as a result of grand disturbance. On an example of short circuits the approach to structural investigations of the stability, founded on the analysis of oscillatory structures of system and an estimation of their limit excitation up to stability is stated. A complete stability analysis of an electric power system necessitates answering not only the question "Will there be a violation of stability at this or that disturbance?", but also the question: "Along which section will be a loss of stability and how the spatial position of this section depends on the disturbance?" Structural researches of an electrical power system stability we will base on the analysis of electromechanical transient oscillatory structure. Basic introduced concepts: An oscillatory / wave structure; An oscillatory degrees of freedom for a power system; A structured motion ; A potential and an actual trajectories. We will relate the processes leading to the loss of stability with some stability-ultimate (critical) disturbance of at least one of the oscillatory degrees of freedom. As a quantitative characteristics of the ultimate excitation intensity, we can use, for instance, the stability-ultimate oscillation energy of this oscillatory degree of freedom under some disturbing action. 2

An oscillatory structure – it is the form of the idea of wave motion. It describes such partition of system into the subsystems, in which in every two adjacent of them (at the given instant) the displacement of all vectors emf of synchronous machines relative to the coordinate of the center of inertia of system has opposite directions. An oscillatory degrees of freedom for a power system – is a the structural mapping of laws governing the free oscillatory motions in the power system. A potential trajectory – is a trajectory on which are carried out the equations of balance and the law of momentum conservation. A wave structure – this is the constant oscillatory structure of a certain free ocsillatory motion in the linearized system with the "switched off" damping Structural model of system (structure of system) – is the partition of system into the subsystems and their uniting intersystem links. A quantity of subsystems, entering the structure, gives its dimensionality R (S). The motion of system, described as a hierarchical system of relative processes, we will call structural organized on S. We can apply the notion of oscillatory degrees of freedom to the development of algorithms for evaluating the stability conditions for power systems based on the consideration of power system motion in structured forms. 3 B ASIC CONCEPTS

In the region of low frequencies the oscillatory structures have the chained structure ( Hz). On the measure increases in the frequency of the vibrations ( Hz - more than 1.1 Hz) of structure are re-formed and acquire the branched out chained structures, are formed rings and star-shaped structures. For the fluctuations with frequencies of about 1 Hz a quantity of subsystems in maximally developed directions of wave structures does not exceed Examples of the oscillatory structures The chained structure The branched out chained structure The star-shaped structure Hz Hz More than 1.1 Hz Hz Hz More than 1.1 Hz

5 Estimation of the critical on the stability excitation of oscillatory degrees of freedom for a power system. Estimation of the time of propagation of the traveling waves in the EES of Russia As a quantitative characteristics of the ultimate excitation intensity, we can use, for instance, the stability-ultimate oscillation energy of this oscillatory degree of freedom under some disturbing action. The nature of the developing instability in this case will be determined by the comparative rates of the processes of the racing of the objects of oscillatory degrees of freedom for a power system to the maximum on the stability deviations from the position of equilibrium. Calculations of stable transients in the UES of Russia spent on nonlinear model with "switched off" damping for the purpose of definition of running wave passage time for various frequencies, yield rather close results to the received estimations. It confirms the made assumption of possibility of revealing of directions and an estimation of speeds of wave propagation not only "small", but also nonlinear waves on the basis of wave structures for "small" standing waves.

Equations of relative motions We represent the speed of rotation of the j-th synchronous machine as the sum of a time independent component (speed of rotation in the initial regime) and three relative processes:  гj (t)=  0 +   гjs (t)+   s0 (t)+   0 (t), (1) here,   0 (t)=  0 (t)-  0 ;   s0 (t)=  s0 (t)-  0 (t);   гjs (t)=  гj (t)-  s0 (t); Irrespective of the manner used to divide the whole system into subsystems, there hold the following equalities (momentum conservation law): (2) here, - is the total moment of inertia of the i-th subsystem. Conventional signsName  0 (t) the travel velocity of the center of inertia of the system  s0 (t) the travel velocity of the center of inertia of the subsystem to which a given synchronous machine belongs i и i s the sets of the synchronous machines in the entire system and in the s-th subsystem of the system JiJi the moments of inertia   0 (t)the value by which the travel velocity of the center of inertia of the system deflects from the stationary frequency  0   s0 (t) the synchronous motion of the subsystem, presenting a regional process of a synchronous machine   гjs (t) the individual motion, presenting the local motion of a synchronous machine 6

The angular-velocity components satisfy the following equations of relative motion: (3) The components of the incremental kinetic energies are due to the works done over incremental displacements : (4) Conventional signsName the total moment of inertia of the system MэMэ the total excess torque moment at machine shafts in the system MsMs the total excess torque moment of the s-th subsystem M s0 =  M s -  M э J s /J э is the excess torque moment acting on the s-th subsystem as all subsystems exercise their relative motion about the center of inertia of the system M is =  M i -  M s J i /J s the excess moment that acts upon a synchronous machine as the machine exercises its relative motion about the center of inertia of the subsystem to which this machine belongs 7 Equations of relative motions

The angular-velocity increments The system level The regional level The local level Distribution of the added kinetic energy The system level The regional level The local level The response of the system at equilibrium to an impact (pulsed) disturbance acting on the system during a short time interval  t : 8 Estimation of maximum impulse disturbances for the objects of one oscillatory degree of freedom Provided that the energy of the object at the time t 0 is equal to, or greater than, the estimated margin in terms of deceleration work, it can be expected that the stability will be violated under an angular displacement equal to the critical one in a time interval: (5) Here, is the critical energy of the object at the time t 0, and is the deceleration work for the object as a function of its deflection from the position at time t 0.

9 The “Area method”

The identification algorithm for instability structures of a power system at finite-duration shunt faults Selection of the certain k-th oscillatory degree of freedom Estimate of the ultimate pulsed disturbance for selected oscillatory degree of freedom Step-by-step calculation of the accumulating acceleration area Determination of a total variation of the kinetic energy of the system and the time Determination of the times required for the subsystem to come to the ultimate displacement 10 The calculated data for several oscillatory degrees of freedom can be conveniently represented as an energy-time diagram

The energy-time diagram 11

12 The energy-time diagram

Development of identification methods for the structure of dynamic stability problems implies a wider formulation of these problems resulting from additional consideration of points concerning the spatial structure of unstable motion. Power system is characterized by the natural three-dimensional structural organization of the oscillatory motions by its reflected oscillating structure. A study of oscillatory processes in the system can be most conveniently conducted by invoking the notion of oscillatory degrees of freedom and their wave and oscillatory structures used for obtaining structured motions. A structured motion involves the motion of the system as a whole, regional motions of subsystems, and local motions inside subsystems, i.e. presents a hierarchy of motions. C ONCLUSIONS 13

An analysis of energy relations for the various hierarchical-level objects of structured motion enables evaluation of stability of these objects with respect to environment. The structural analysis of stability of electromechanical transient processes in electric power systems uses energy relations for structured motion. The main tool here is the formulation of the stability analysis problem in the form of a set of mutually complementing stability problems for variously structured forms of the motion of interest. The use of structural analysis algorithms for stability allows identification of the structure of stabilization problems with quantitative estimates of their main characteristics, such as ultimate disturbances, position of asynchronous- operation sections, time margins for exerting the control, etc. The verification of the structural analysis algorithms for stability under closed-circuit faults based on the comparison of the predictions yielded by these algorithms with calculated data for transient processes has proven these algorithms to be applicable to the study of main developmental features of unstable motion in complex power systems. C ONCLUSIONS 14