Stability Analysis of Time Delayed Generator Excitation Control System using Resultant Theory

IJSTE - International Journal of Science Technology & Engineering | Volume 4 | Issue 5 | November 2017 ISSN (online): 2349-784X

Stability Analysis of Time Delayed Generator Excitation Control System using Resultant Theory T. Jesintha Mary Assistant Professor R.M.D Engineering College, Kavaraipettai

J. Sumithra Associate Professor R.M.D Engineering College, Kavaraipettai

Abstract Time Delays have become unavoidable in Power System due to the usage of measurement devices and communication links for transferring the control signals. This paper investigates the effect of time delays on the stability of a generator excitation control system with a stabilizing transformer. An analytical method based on frequency domain approach is proposed to compute the delay margin for stability. First, the transcendental characteristic equation is transformed into a polynomial using Rekasius substitution.

Then with the help of Resultant theory the lower bound of crossing frequency is determined. After determining the crossing frequency, the delay margin is determined using back transformation formula. The theoretical delay margin values are computed for different sets of controller gains .The theoretical results indicate that the addition of a stabilizing transformer to the AVR system increases the delay margin and improves the damping performance of the system. The theoretical delay margin results are validated using MATLAB/SIMULINK. Keywords: Delay margin, generator excitation control system, Rekasius’s substitution, Resultant theory, Time delays ________________________________________________________________________________________________________ I.

INTRODUCTION

In Power systems, Automatic Voltage Regulator (AVR) equipment is installed for each generator unit to maintain the generator output voltage magnitude within acceptable limits when there is any change in load [1]. This paper analyzes the stability of the generator excitation control system considering the effect of time delays. The schematic diagram of generator excitation control system is shown in figure.1. It comprises of an exciter, a generator, a voltage transducer, a PMU, a rectifier, a stabilizing transformer and a regulator. The exciter excites the generator field winding. The voltage transducer senses the output voltage magnitude of generator and drives the PMU. The PMU converts the voltage sensed by the transducer to phasor information. The rectifier rectifies and filters the generator terminal voltage to DC quantity. The regulator has a PI controller and an amplifier. The stabilizing transformer gives additional input signal to the PI controller to damp out the oscillations [2].Because of the use of Phasor Measurement Unit for transferring the measured signals the time delays have become a major problem in Power System Operation and Control. The time delays can be in the range of 0.5-1.0 second [3-5]. The PMU introduces two delays i) Voltage transducer delay ii) Processing Delay. These time delays reduce the damping performance of power system and may even lead to loss in synchronous stability. Therefore the effect of time delays on the stability of power system has to be investigated. Particularly, it is very important to determine the maximum tolerable value of time delay until which the system remains stable. There are various ways to determine the delay margin [6-14]. In previous literature, various computational methods for deriving delay margin have been addressed in Load Frequency Control [15, 16]. However less attention has been shown on analyzing the effect of time delay on generator excitation control system. The effect of time delay can be determined by computing delay margin of the system.

Generally the stability of any time delayed system can be analyzed by two methods i) time domain approach ii) Frequency domain approach. This paper applies frequency domain approach based on Rekasius’s substitution and Resultant theory to compute the delay margin of the system. First, Rekasius’s substitution is done to convert the transcendental characteristic equation in to algebraic polynomial. Then Resultant theory is applied in the algebraic polynomial equation to determine the lower bound of crossing frequency .With the availability of crossing frequency, the delay margin is derived using back transformation formula. In this way, the delay margin values are determined for various PI controller values. The theoretical delay margin results are validated in Matlab/Simulink. From the simulation results, it is observed that the system is stable for all time delays less than delay margin.

The delay margin results of AVR with stabilizing transformer are compared with those of AVR without stabilizing transformer.

From the comparison, it is proved that the stability margin of the system is enhanced after connecting stabilizing transformer.

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Stability Analysis of Time Delayed Generator Excitation Control System using Resultant Theory (IJSTE/ Volume 4 / Issue 5 / 036)

Fig. 1: Block Diagram of Generator Excitation Control System

II. DYNAMIC MODEL OF AVR WITH TIME DELAY The dynamics of the excitation control systems are commonly analyzed by linear models. The block diagram of an AVR with time delay is shown in Figure 2 as presented in [20].

Fig. 2: Block diagram of the excitation control system with a stabilizing transformer and time delay

In Figure 2, the exponential term in the feedback part represents total measurement and communication delays. The first-order transfer function of amplifier, exciter, generator and sensor is given below:

K

K

K

K

T

T

T

T

G and R are the gains of exciter, amplifier, generator and sensor, respectively, and E , A G and R are Where E , A their corresponding time constants. A stabilizing transformer is included in the system by adding a derivative feedback to improve the dynamic performance.

Transfer function of stabilizing transformer is described as

where K F and T F are the gain and time constant of stabilizing transformer .The stabilizing transformer increases the relative stability of the closed-loop system by adding a zero to the open-loop transfer function of AVR. Transfer function of PI controller is given below.

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Stability Analysis of Time Delayed Generator Excitation Control System using Resultant Theory (IJSTE/ Volume 4 / Issue 5 / 036)

K

p

And K i are the proportional and integral gains respectively. The PI controller is used to shape the response of the generator

excitation system and achieve the desired performance. The measurement and communication delay is represented as an exponential block the delay is assumed as constant.

III. DELAY-DEPENDENT STABILITY ANALYSIS The characteristic equation of the above system can be written as ď „ ( s, ď ´ )  P ( s )  Q ( s )e

 sď ´

0

(1)

Where P(s) = p4 s 4 + p3 s 3 + p2 s 2 + p1 s Q(s) = q1 s + q0 P(s) and Q(s) Are polynomials of s having real coefficients? The coefficients of these polynomials are presented in AppendixI. The system is said to be stable if and only if all the roots of the characteristic equation lie in the left half of the complex plane.

Because of the exponential term in equation (1), the characteristic equation has infinite number of roots. It is very difficult to find all the roots. For analyzing the stability of AVR system, it is sufficient to investigate the variation of the roots with respect to time delay.

The main idea of the proposed method is to examine the location of the roots of the characteristic equation and to compute the delay margin using Rekasius’s substitution [17, 18] and Resultant theory [19]. The first step is to apply Rekasius’s substitution [18], [19] 1−sT e−sđ?œ? = T ≼ 0, (2) 1+sT Which is defined only on the imaginary axis in the complex plane, s = jω. However, equation (2) is exact if and only if 2 đ?‘˜đ?œ‹ đ?œ? = đ?‘Ąđ?‘Žđ?‘›âˆ’1 (ωT) + (3) ω 2 k = 0, 1, 2, .After this step the transcendental characteristic equation (1) is transformed in to an algebraic polynomial equation as given below.

1−jωT h(jω, T) = (h(jω, đ?œ? )|e−sđ?œ? = , T ≼ 0) (1 + jωT) (4) 1+jωT

This polynomial equation (4) has real and imaginary parts in terms of T and ω and it can be rewritten as h(jω, T) = hR (jω, T) + jhI (jω, T) (5) Where â„Žđ?‘… = ďƒ‚ ( h ) ;â„Žđ??ź = ďƒ ( h ) .Then the resultant of hR and hI can be obtained using Resultant theory [19], eliminating T. The resultant polynomial Z(ω) in terms of ω is.

Z(ω) = R t (hR , hI , T) = 0 (6) Then the crossing frequency ω0 is obtained by finding the minimum positive real roots of Z (ω). The relation between đ?œ?đ?‘˜ and T is given by (3).Then the time delay margin (đ?œ?đ?‘˜ ) is computed using the equation 2 đ?œ?đ?‘˜ = tan−1 (ω0 T) (7) ω0

Fig. 3: Movement of the roots with respect to time delay

Figure.3 shows the movement of the roots of the characteristic equation when the time delay đ?œ? varies. It is assumed that the system is stable for time delay đ?œ?= 0. When the time delay increases, the roots of the characteristic equation move from LHP to All rights reserved by www.ijste.org

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Stability Analysis of Time Delayed Generator Excitation Control System using Resultant Theory (IJSTE/ Volume 4 / Issue 5 / 036)

RHP. The time delay margin is defined as the time delay at which the roots move from LHP to RHP and cross the imaginary axis.

Simulation is performed in the next section for different values of time delay (đ?œ?) to validate the theoretical delay margin.

IV. RESULTS AND DISCUSSION Theoretical Results The system parameters used in the analysis are tabulated in Table.1. The stability of the system is investigated and the Delay margin (đ?œ?đ?‘˜ ) at which the roots of characteristic equation cross jω axis is computed using the proposed method. The results of the delay margin of AVR system with and without stabilizing transformer for different sets of PI controller parameters are tabulated in Table.2 and Table.3. Theoretical delay margin results are validated using Matlab/simulink.

Table - 1 Parameters of AVR system Parameters values

KE

K

A

KG

1.0

5.0

1.0

KR

KF

TE

TA

TG

TR

1.0 2.0 0.4 0.1 1.0 0.05 Table - 2 Delay margin for different sets of and (with stabilizing transformer) ď ´ d (s) KI

K P  0 .1

TF

0.04

K P  0 .7

Simulation Theoretical Simulation 4.06 3.65 3.645 3.24 3.256 3.252 2.97 3.0169 3.016 2.84 2.8929 2.89 Table - 3 Delay margin for different sets of and (without stabilizing transformer) ď ´ d (s) 0.2 0.4 0.6 0.8

KI

0.2 0.4 0.6 0.8

Theoretical 4.055 3.243 2.9782 2.8469

K P  0 .1

Theoretical 0.9746 0.4866 0.2464 0.1076

Simulation 0.974 0.486 0.246 0.1074

K P  0 .7

Theoretical 0.3385 0.2752 0.2132 0.1574

Simulation 0.3387 0.2756 0.2137 0.1577

From Tables.2 and 3 it is clear that the stabilizing transformer in AVR system increases the delay margin and makes the system more stable. For example, when K P  0 . 1 and K I  0 . 4 , the delay margin is found to be đ?œ?=3.243 s when a stabilizing transformer is not included while it is đ?œ?=0.4866s when a stabilizing transformer is connected. Thus the stability performance of the system is significantly improved by introducing a stabilizing transformer in AVR system.

Validation of Theoretical Delay Margin Results Simulation is performed to assess the accuracy of the theoretical value of delay margin obtained using proposed method. This is implemented by increasing the time delay step by step from 0 secs in simulink model until the system becomes unstable. The time delay at which the system is marginally stable is defined as Delay margin. The system response for PI controller gain for KP= 0.7 and KI=0.8 is shown in Figure.4.

Fig. 4: Time response of the system with stabilizing transformer for KP= 0.7 and KI=0.8

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Stability Analysis of Time Delayed Generator Excitation Control System using Resultant Theory (IJSTE/ Volume 4 / Issue 5 / 036)

The dashed dotted lines in Figure.4 show the growing oscillations for time delay Ď„ = 3.2đ?‘ . The dotted lines in Figure.4 show decreasing oscillations for time delay Ď„ = 2.4đ?‘ .Thus the system loses its stability somewhere between 2.4 s and 3.2s and delay margin should lie within the range [2.4s, 3.2s]. The solid lines in Figure.4 show the sustained oscillations for delay margin Ď„k = 2.89s which indicates the system is marginally stable. This delay margin value is closer to the theoretically calculated delay margin 2.8929s given in Table.2.Simulation results prove that the proposed method is highly accurate in finding delay margin. It also reveals that the system is stable for all time delays less than delay margin and it becomes unstable when the time delay exceeds delay margin.

Discussion The effect of the PI controller gains on the delay margin has been examined. The theoretical delay margin results indicate that the delay margin increases as the integral gain decreases for a fixed proportional gain. Such an increase in delay margin implies a more stable AVR system. Furthermore, it has been observed that the AVR system with a stabilizing transformer increases the delay margin of the system .Thus AVR system can be made more stable by introducing a stabilizing transformer in the model. These observations can be used as a reference while selecting the PI controller gains so that the AVR system is made more stable and a desired damping performance can be achieved even if certain amount of time delays prevails in the system.

V. CONCLUSION An alternative approach based on frequency domain approach is proposed for the delay dependent stability analysis of AVR system with stabilizing transformer. The proposed method uses two concepts namely Rekasius substitution and Resultant Theory to determine the stability delay margin. Using Rekasius substitution the transcendental characteristic equation of AVR system is transformed to polynomial equation. Then Resultant theory is applied to find the exact lower bound of crossing frequency. After determining the crossing frequency, the maximal time delay đ?œ?đ?‘˜ is derived using back transformation formula. Simulation is carried out to show that the system is asymptotically stable for all time delays less than delay margin. The simulation result also proves that the proposed method is highly accurate and provides better stability region, less computation time.

VI. APPENDIX-I The coefficients of polynomials P (đ?‘ ) and Q (đ?‘ ) presented in equation (1) in terms of time constants and gains of the AVR system: p 6  T A T E T F TG T R p 5  T A T E T F T G  T R (T A T E T F  T A T E T G  T A T F T G  T E T F T G )

p 4  T A T E T F  T A T E T G  T A T F T G  T E T F T G  T R (T A T E  T E T F  T A T F  T A T G  T E T G  T F T G  K A K E K F K P T G ) p 3  T AT E  T E T F  T AT F  T ATG  T E TG  T F TG  K A K E K F K P TG  T R (T A  T E  T F  TG  TG  K A K E K F K P  K A K E K F K I TG )

p 2  T A  T E  T F  T G  K A K E K F K I T G  K A K E K F K P  T R ( K A K E K F K I  1) p1  K A K E K F K I  1 q 2  K A K E K P K G K RTF q1  K A K E K G K R ( K P  K I T F ) q0  K A K E K G K R K I

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