Phần 21 KHÓA ĐÀO TẠO TÍNH TOÁN ỔN ĐỊNH VÀ ỨNG DỤNG TRÊN PHẦN MỀM PSSE CHO KỸ SƯ HỆ THỐNG ĐIỆN (Ổn định kích thích nhỏ và ứng dụng trên Phần mềm PSSE)

Ổn định kích thích nhỏ và ứng dụng trên Phần mềm PSSE.NỘI DUNG CHÍNH PHẦN 21 (small signal stability and application of small signal stability): 1. Transient Stability: a. Time Domain Analysis. b. Step wise Integration of Differential Equations. 2. SmallSignal Stability. a. Frequency Domain Analysis. b. Eigen ValueVector Analysis using Linearized Differential Equations Differential Equations

A Division of Global Power

POWER SYSTEM STABILITY CALCULATION TRAINING

D 10 S ll Si l St bilit Day 10 - Small-Signal Stability

July 17, 2013 Prepared by: Peter Anderson

SMALL SIGNAL STABILITY

SMALL-SIGNAL STABILITY

Transient Stability:

Small-Signal Stability

 Frequency Domain Analysis

 Eigen Value/Vector Analysis using Linearized

Differential Equations

APPLICATIONS APPLICATIONS

Power System Size

 I i th h G th i I t ti

 Increasing through Growth in Interconnections

 Driven by Potential Cost Savings (Economies of

Scale, Use of Lowest-cost Generating Units)

 Focus on Generation-Not on Transmission

Disadvantages

 Increased Vulnerability y

 Inter-Area Oscillations

 System Disintegration/Widespread Blackouts

EIGEN-VALUE ANALYSIS EIGEN VALUE ANALYSIS

Applied to a Linearized Model of the Power

 Sub synchronous Torsional Interactions

 Sub-synchronous Torsional Interactions

 Electro-mechanical Performance in the

Low Frequency Range (0 1 to 3Hz)

COMPARISON OF APPROACHES COMPARISON OF APPROACHES

Non‐linearities represented in  detail

Weakly Damped Modes may not 

be Excited or Observed

Frequencies/Damping are Mixed Evaluation of Results‐Difficult Frequency 

Domain

Reveals Rules behind System  Dynamics

Non‐linearities not well  represented

can be Difficult

Siting and Tuning of Damping  g g p g Controllers

instability

SWING MODES SWING MODES

CONTROLLER MODES CONTROLLER MODES

APPLICATION OF THE APPROACH

APPLICATION OF THE APPROACH

A Division of Global Power

POWER SYSTEM STABILITY CALCULATION TRAINING

D 10 A li ti f S ll Si l St bilit Day 10 - Application of Small-Signal Stability

July 17, 2013 Prepared by: Mohamed El Chehaly

OUTLINE OUTLINE

• Small-Signal Stability

SMALL-SIGNAL STABILITY eBook for You

Modal Analysis

Modal Analysis

 Exclusively suitable for small signal

 Exclusively suitable for small signal

stability studies

 Also know as Eigenvalue analysis

 Analysis of linear systems

 Linearization of non-linear systems at a

 Linearization of non linear systems at a

specified operating point (steady-state

load flow condition))

 Typical applications include inter area

oscillations, sub synchronous torsional

interactions, voltage stability…

Modal Analysis

Modal Analysis

Modal Analysis

Modal Analysis

 Simulation method in the time domain:

 Disturbances are applied

 System responses are calculatedy p

 Dynamics are observed through plotted curves

 Model analysis

 Not necessary to apply any disturbances

 Inherent properties of a studied dynamic system

are revealed by Eigenvalues and Eigenvectors

Modal Analysis

Modal Analysis

 Modal analysis provides the following

 Modal analysis provides the following

information

 Frequencies and damping

 Mode observability and controllability

 Controller location and tuning

Modal Analysis

Modal Analysis

 Example of a multi – machine system

Simulation Method Advantages

Simulation Method - Advantages

 Wide application fields

 Nonlinearities represented in detail

 No modeling limitations

 Time domain results in curves show a

representation of the real system

representation of the real system

behaviour

 Programs for time domain simulation are

 Programs for time domain simulation are

well established and available worldwide

Simulation Method Disadvantages

Simulation Method - Disadvantages

 Trial-and-Error approach by applying

 Trial and Error approach by applying

disturbances and observing responses

 Different disturbances have to be applied

 For each load flow, new cases are required

 Certain weakly damped and unstable

 Certain weakly damped and unstable

modes may not be observed

 Modes of different frequencies and

 Modes of different frequencies and

damping are mixed

effective damping controllers

Modal Analysis Advantages

Modal Analysis - Advantages

 Systematic approach which reveals rules

 Systematic approach which reveals rules

behind complicated phenomena

 No need to apply disturbances

 For each load flow one modal calculation

is sufficient

 Weakly damped and unstable modes are

picked out and analyzed in detail

 Individual modes are analyzed

Modal Analysis Disadvantages

Modal Analysis - Disadvantages

 Only suitable for small-signal stability

 Nonlinearities are not well reflected

 Linearization of some elements is difficult

 Frequency domain modal results are not

familiar to many people

 Requires a lot of memory for large

systems

 System modeling and Eigenvalue

algorithms are sophisticated g p

Eigenvalue

 State space representation of a linear

 State space representation of a linear

dynamic system

 Transfer function

 Eigenvalues (Modes) are the solution of

 Eigenvalues (Modes) are the solution of

the characteristic equation

Eigenvalue

Eigenvalue: mathematical term

Mode: engineering term

Eigenvalue

Mode Overview

Mode Overview

 To ensure a stable power system all the

 To ensure a stable power system, all the

modes must be located on the left side of

the complex s-plane p p

 Three indices for damping

 Absolute damping: real part of a mode

Mode Overview

Mode Overview

Mode Overview

Mode Overview

Mode Overview

Mode Overview

Mode Overview

Mode Overview

Observability

 A right eigenvector determines the relative

activity of its eigenvalue on components

of system variables (generator rotor angle,

real power and reactive power)

 Different outputs can be observed

concerning network behaviour (voltage

deviation of buses and power deviations

deviation of buses and power deviations

of transmission lines

Controllability

 A left eigenvector (with its initial state)

determines the dominance of its mode

 With the mode observability, the system

can display which generators swing

against other generators as well as how

significant a role each generator plays

Controller Siting

Controller Siting

 Time domain simulation offers no

systematic information regarding the

optimal location for POD

 Early methods used right eigenvectors

which considered only observability

 In practice, the priorities for installing PSS

are on large generators

 Improved method uses the mode shape of

rotor speed deviation weighted by the

generator size

Controller Siting

Controller Siting

 Further improvement of the method uses

mode participation factors as a siting

index where both mode observability and

controllability are contained

 Methods using participation factors have

shortcomings because of dynamic

shortcomings because of dynamic

properties of excitation systems through

which PSS operates are not considered

 Better methods are developed by the

evaluation of transfer function residues

Controller Siting

Controller Siting

 Transfer function residues

Nature of Modes

Nature of Modes

generators located in the same area

 Regional: a group of generators at a power station

swing against the rest of the system

 Frequency: relatively high

 Damping: relatively strong

 POD if necessary: PSS

 Heavy power transfer between weakly

 Heavy power transfer between weakly

interconnected areas and poor system damping

 Frequency: relatively low

 Damping: relatively weak

 POD if necessary: PSS, FACTS, TCSC…

Nature of Modes

Nature of Modes

 Controller modes

 Associated with controllers of generator voltage

regulators, generator speed governing systems

FACTS controllers

FACTS controllers,…

 Most are monotonous modes with strong damping

associated with first order delay elements with y

small time constants

 Some monotonous modes near the origin which

have low damping associated with first order delay

have low damping associated with first order delay

elements with large time constants (reheaters of

steam turbines) These are not dangerous

Nature of Modes

Nature of Modes

 Inter area oscillations

Dominant Mode

Dominant Mode

 Very significant mode

NEVA – PSS NETOMAC eBook for You

Introduction

 Program used for modal analysis

 Offers a comprehensive tool box of

 Transfer function residues

 Controller siting indices

 Frequency response plots

 Linear impulse and step response

Solve Eigenvalues

Solve Eigenvalues

 Solve Eingenvalues using NEVA

Calculation of Eigenvalue

Calculation of Eigenvalue

 Complete Eigenvalue solution

 Whenever possible, it is always desired to obtain

complete Eigenvalues for a studied linear

complete Eigenvalues for a studied linear dynamic system

 Guaranteed that no unstable modes are missed

 Possible to calculate frequency response and

linear time response

 Use of QR algorithm which is based on similarity

 Use of QR algorithm which is based on similarity

transformation of a state matrix

 Advantage of QR transformation is its superior g p

numeric stability and accuracy

Calculation of Eigenvalue

Calculation of Eigenvalue

 Complete Eigenvalue solution

 Disadvantage of QR transformation is that its

memory requirement and computation time rises

memory requirement and computation time rises quickly as the order of a state matrix grows

 Such a disadvantage makes QR impractical to

use in high order systems such as large scale power systems

Calculation of Eigenvalue

Calculation of Eigenvalue

 Partial Eigenvalue solution

 It is difficult or impossible to solve all Eigenvalues

of a large scale system

 Even if all Eigenvalues are solvable, it is

sometimes unnecessary to calculate all of them

 Usually, those Eignevalues are weakly damped

and unstable modes or of particular physical meaning

meaning

 When system parameters are changed,

Eigenvalues have to be recalculated It is often sufficient to update results for a few Eigenvalues

Calculation of Eigenvalue

Calculation of Eigenvalue

Calculation of Eigenvalue

Calculation of Eigenvalue

 Mode distribution on the complex s-plane

Calculation of Eigenvalue

Calculation of Eigenvalue

 Mode waveforms (step unit)

Calculation of Eigenvalue

Calculation of Eigenvalue

 Maximum participated state for a mode

with respect to one another to a particular mode

 Units oscillating together have the bars in the

same direction

 Units oscillating against one another have the

bars in opposite directions

 Units with no bars means these machines do not

contribute in the oscillations

 The unit with the highest contribution to a

problematic mode or high participation factor is a good candidate for POD siting

Calculation of Eigenvalue

Calculation of Eigenvalue

 Residue of transfer function

QUESTIONS?

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