stability of multi-machine power system by used lqg controller€¦ · · 2014-04-11stability of...

Stability of Multi-Machine Power System by used LQGController

AHMED SAID OSHABA1,2

e-mail: [email protected],

1- Power Electronics & Energy Conversion Dep., Electronics Research Institute (ERI), Egypt,2- Electric Engineering Dept., Faculty of Engineering, Jazan University, KSA

Abstract:- This paper has been designed optimize feed-back controller for dynamic response of the power systems.

The power system consists of the infinite bus through a transmission line supplied by a synchronous machine and

also multi machine power system. The effect of two control signals fed to the voltage regulator and the mechanical

system is investigated. Robust Linear quadratic Gawssian (LQG) control technique based power system stabilizer

is developed for excitation system control and the mechanical system control. The proposed robust LQG-PSS is

simple, effective, and can ensure that the system is asymptotically stable for all admissible uncertainties and

abnormal operating conditions. To validate the effectiveness of the proposed power system stabilizer, a sample

power system consists of multi machine and single machine are simulated and subject to different disturbance and

parameter variations. Kalman Filter is used for compound with LQR to get robust LQG control. The results prove

the robustness and powerful of proposed LQG controller stabilizer than LQR controller in terms of fast damping

response and less settling time of power system states responses.

Key-word:- LQR controller , LQG controller, power system stabilizer and multi-machine power system

1 Introduction

Many papers have been published on thesynthesis of the power system stabilizer (PSS) controlsystem. Some approach it by complex frequencymethods using the concept of synchronizing anddamping torques [1, 2], some by optimal controlmethods and also, by using pole placement methods[3-5]. In control system designed a satisfactorycontroller cannot be obtained by considering theinternal stability objective alone. The interconnectedpower system can be achieved by conventionalcontroller as[1, 3]. A brief overview of the theoreticalfoundation of H synthesis is introduced in [7]. The

H formulation and solution procedures areexplained, and guidelines on how to choose properweighting functions that reflect the robustness andperformance goals are given in [8,9,10 ]. Hsynthesis is carried out in two stages. First, in what iscalled the H formulation procedure, robustness tomodeling errors and weighting the appropriate input-output transfer functions usually reflects performancerequirements. The weights and the dynamic model ofthe power system are then augmented into an

H standard plant [9]. Second, in what is called the

H solution procedure, the standard plat isprogrammed into a computer aided design software,

WSEAS TRANSACTIONS on POWER SYSTEMS Ahmed Said Oshaba

E-ISSN: 2224-350X 143 Volume 9, 2014

such as MATLAB[11], and the weights are iterativelymodified until an optimal controller that satisfies the

H optimization problem is found. Time responsesimulations are used to validate the results obtainedand to illustrate the dynamic system response to statedisturbances. The effectiveness of such controllers isexamined at different extreme operating conditions.Using the linear quadratic regulator (LQR) forcomparison with the proposed robust H controller.The present paper used the LQR approach andKalman filter to design a robust LQG power systemstabilizer for stabilization the dynamic responses atdifferent operating conditions.

2 Power System Model

Two power system models are studying in this

research as follow:

2.1. Single machine model

A synchronous machine connected to infinite busthrough transmission line is obtained in a aninterconnected power system between automaticvoltage regulation and load frequency control asshown in a block diagram of Fig.1. Where :

ω = the mechanical speed.∆ω = speed deviationR = regulation constant.μs = damping factorKA = exciter constant.TA = exciter time constant.

∆δ = change in torque angle.∆Efd = change equivalent excitation voltage∆E'

q = change internal voltage behind transientreaction

K1 to K6 = constant of linear Zed model ofsynchronic machine

The state space formulation can be obtained asfollows :

Steady-state Representation

w.

(1)

dm PMTM

qEMKwMDMKw

)/1()/1(

')/()/()/( 21

.

(2)

fd

q

EdoT

qEdoTKdoTKE

)'/1(

')'/1()'/( 34

'.

(3)

gtmtm PTTTT )/1()/1(.

(4)

2

.

)/1()/1()/1( UTPTwRTP ggggg (5)

16

5

.

)/(')/(

)/()/1(

UTKqeTKK

TKKETE

AAAA

AAfdAfd

(6)

In a matrix form as follows:

dPUBXAX .

(7)

where;

tfdgmq EPTEX '

Fig.1: The block diagram of single machine power system.


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AA

A

A

A

gg

tt

dododo

TT

KK

T

KKTRT

TT

TTKT

KMM

K

M

D

M

Kwo

A

1000

01

001

0

011

000

100

)(

10

001

00000

65

''3

'4

21

t

g

A

At

T

T

K

B

BB

01

0000

00000

2

1 ,

tUUU 21 t

M

0000

10

2.2. Multi-machine model

The power system in this model consists of threesynchronous machines connected to infinite bus andits dynamic performance is represented in the statevariables form. The single line diagram model for thesystem is shown in the Fig.2 and is based upon thefollowing assumptions

1- Saturation is neglected,2- Armature transformer voltage is

neglected,3- Damper winding effect is neglected.

.Once the A, B and C matrices are determined,applying the Linear Quadratic Gaussian LQGcontroller on it. The multi machine power systemdata and load flow are displayed in tables 1, 2 [2, 12].

Fig.2: Three machine-infinite bus system.

Table 1: The Multi-machine Power System Data

Table 2 : The Multi-machine load flow data.

M/C Machine data

Xd Xq Xj Tdo H KA TA Base quantities

1 1.68 1.66 0.32 4.0 2.31 40.0 0.05 360 MVA, 13.8KV2 0.88 0.53 0.33 8.0 3.40 45.0 0.05 503 MVA, 13.8KV3 1.02 0.57 0.20 7.76 4.63 50.0 0.05 1673 MVA, 13.8KV

Bus Power flow Qo MVA Vto pu. o. degreesPo, MW

1 26.5 37.0 1.3 102 518 -31.5 1.025 32.523 1582 -69.9 1.3 45.824 410.0 49.1 1.6 20.69


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Each plant is represented by a 4th -order generator equipped with a static exciter. The state equation of this systemis given by

BUAXX .

(8)Where

X = [W1, W2, W3, 1, 2, 3, e َ◌q1, e َ◌q2, e َ◌q3, eFD1, eFD2, eFD3]T

U = [u1 u2 u3]T

A = Matrix systemB = input matrixIs the input vector .The system A and B are given as follows

3 Control Design Strategy

3.1 Optimal LQR control design

The object of the optimal control design isdetermining the optimal control law u(t,x) which cantransfer system from its initial state to the final statesuch that given quadratic performance index isminimized.

[KLQR ,S,E]=lqr(A,B,Q,R,N) (9)

Where: Q is positive semi definite matrix and R isreal symmetrical matrix. The problem is to find thevector feedback K of control law, by choosing matrix

Q and R to minimize the quadratic performance indexJ is described by :

0

1 )( dtuRuxQxJ tt

The optimal control law is written as

Δ u(t)= K Δ x (t)

KLQR = - R-1 Bt P (10)

The matrix P is positive definite, symmetricsolution to the matrix Riccati equation, which haswritten as:

200.00.04.541.278.61.707.11.1049.89337.4693.33

0.0200.041.1167.2155.1299.1609.10648.1891.17111.51647.64

0.00.020194.10501.3943.60051.62599.241.30167599.9114.309

0.10.00.0197.0011.0002.02.122.0083.0131.10.00.0

0.010.006.021.0021.046.06.1121.0754.0885.1131.1

0.00.01072.0024.0922.025.0087.0266.0131.1754.0393.3

0.00.00.00.00.00.00.00.00.03770.00.0

0.00.00.00.00.00.00.00.00.00.03770.0

0.00.00.00.00.00.00.00.00.00.00.0377

0.00.00.0009.00.0003.0056.0017.0001.0017.001.0017.0

0.00.00.00.0008.000645.0079.0149.0004.0028.0032.0034.0

0.00.00.0003.00013.0046.0022.0147.002.0004.0039.0

A

T

B

00800000000000

09000000000000

100000000000000


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P A + At P + Q - P B R-1 Bt P = 0 (11)

3.2 optimal compensator LQG control

We have introduced the Kalman filter, which is anoptimal observer for multi-output plants in thepresence of process and measurement noise, modeledas white noises. The optimal compensator LinearQuadratic Gaussian (LQG) consists of combinebetween optimal LQR control and Kalman filter asshown in Figs.3,4. In short, the optimal compensatorLQG design process is the following:

1- Design an optimal regulator LQR for a linearplant assuming full-state feedback (i.e.assuming all the state variables are availablefor measurement) and a quadratic objectivefunction.

2- Design a Kalman filter for the plant assuminga known control input, u(t), a measuredoutput, y(t), and white noises, v(t) and z(t).The Kalman filter is designed to improve anoptimal estimate of the state-vector.

3- Combine the separately designed optimalregulator LQR and Kalman filter into anoptimal compensator LQG.

4- The optimal regulator feedback gain matrix,K, and the Kalman filter gain matrix, L, areused to complete closed compensator systemLQG as follows:

From Eq. 10 get optimal regulator gain matrix KLQR.Calculate the Kalman filter gain as follows. Let thesystem as

.x = Ax + Bu + Gw {State equation} (12)y = Cx + Du + v {Measurements}

with unbiased process noise w and measurementnoise v with covariance’s

E{ww'} = Q, E{vv'} = R, E{wv'} = N ,

[L,P,E] = LQE(A,G,C,Q,R,N) (13)

Returns the observer gain matrix L such that thestationary Kalman filter.

x_e = Ax_e + Bu + L(y - Cx_e - Du)

Produces an optimal state estimate x_e of x using thesensor measurements y. The resulting Kalmanestimator can be formed with estimator. The noisecross-correlation N is set to zero when omitted. Alsoreturned are the solution P of the associated Riccatiequation.

AP+PA'-(PC'+G*N)R(CP+N'*G')+G*Q*G'=0 (14)

and the estimator poles E = EIG(A-L*C).

Using MATLAB function readymade command regto construct a state-space model of the optimalcompensator LQG, given a state-space model of theplant, sysp, the optimal regulator feedback gainmatrix K, and the Kalman filter gain matrix L. Thiscommand is used as follows:

),,(_ LLQRKsyspregclosedsys (15)

Where;sys_closed is the state-space model of the LQGcompensator. The final, get the system overallfeedback sysCL as:

)_,( closedsyssyspfeedbacksysCL (16)

Where, sysCL is the state-space of LQG plus state-space of system with open-loop

Fig. 3: Linear Quadratic Gaussian (LQG) control system.


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Fig.4 : The LQG synthesis.Where: A = system matrixB = control matrixC = output matrixS =laplace operator

3.3 State estimation using Kalman filter

Alternatively, it is possible to estimate thewhole state vector by using a Kalman filter. TheKalman filter optimally filters noise in the measuredvariables and allows the estimation of unmeasuredstates. The Kalman filter uses a model of the systemto find a state estimate ˆx(t) by integrating thefollowing state observer equation:

(17)

where my is the measured output and Kf is the

Kalman filter gain. The Kalman filter assumes thatthe measurements obey the following model:

(18)

Where G is a noise distribution matrix, w and v arewhite noise processes. v is themeasurement noise and is assumed to have acovariance matrix Rf. w is known as process noiseand is assumed to have a covariance matrix Qf . TheKalman filter gain Kf is found as follows:

where P is the solution of the algebraic Ricattiequation:

)

If we use the control law given in Equation 17 with astate estimate obtained using a Kalman filter, then weare using the LQG (Linear Quadratic Gaussian)control law:

(20)

4 Digital Simulation Results

4.1 Simulation of single machine model

From LQR control (Eqn. 10), the feedback gain andsolution of Reccati equation are :

0.0032-0.04130.0015-0.1779-1.73020.0214-

0.08310.1071-0.00400.44933.9708-0.0655LQRK

0.00000.0000-0.00000.00010.0010-0.0000

0.0000-0.00030.0000-0.0014-0.01380.0002-

0.00000.0000-0.00000.0001-0.0005-0.0000

0.00010.0014-0.0001-0.02930.1226-0.0005-

0.0010-0.01380.0005-0.1226-1.65240.0025

0.00000.0002-0.00000.0005-0.00250.0004

*1000S

From LQG and Kalman filter control (Eqn. 13), theobserver gain matrix L and solution of reccatiequation P are :

0.0043-0.1305-

0.00060.0390

0.05348.7476

0.0007-0.1471-

0.01560.2762

0.276214.4339

L

0.01150.0025-0.0708-0.00100.0065-0.1971-

0.0025-0.03260.03800.0006-0.00090.0589

0.0708-0.038017.40020.7356-0.080613.2088

0.00100.0006-0.7356-0.05590.0010-0.2221-

0.0065-0.00090.08060.0010-0.02350.4171

0.1971-0.058913.20880.2221-0.417121.7952

P

(19)


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Table 3: Eigen values calculation with and without controllers of single machine power system.

Operatingpoint

Withoutcontrol

LQR-Control WithKalman

LQG+FeedbackControl

P=1, Q=0.25pu.

Lag p.f load

-0.0367 +6.9961i-0.0367 - 6.9961i-14.2953-12.4821-2.7625-3.7201

-1.1161 + 7.2542i-1.1161 - 7.2542i

-43.3537-14.2787-5.6556-2.9596

-7.24 +10.0732i-7.24 -10.0732i-14.3023-12.4881-3.7076-2.8026

-7.2411 +10.0732i-7.2411 -10.0732i-1.1161 + 7.2542i-1.1161 - 7.2542i

-43.3537-14.3023-14.2787-12.4881-5.6556-3.7076-2.8026-2.9596

P=1, Q= -0.25puLead p.f

0.1033 + 6.3047i0.1033 - 6.3047i

-14.9008-12.4804-2.4303-3.7285

-1.2812 + 6.6267i-1.2812 - 6.6267i-6.1062-1.5921

-43.3498-14.8640

-2.28 + 6.6889i-2.28 - 6.6889i-3.7220-2.3389

-14.9022-12.4809

-1.2812 + 6.6267i-1.2812 - 6.6267i-2.2857 + 6.6889i-2.2857 - 6.6889i-14.9022-14.8640-12.4809-6.1062-1.5921-2.3389-3.7220

-43.3498

Figure 5 shows the rotor angle deviation responsedue to 0.1 load disturbance with and without LQGand LQR controllers at lag power factor load (P=1,Q=0.25 pu). Fig.6 depicts the rotor speed deviationresponse due to 0.1 load disturbance with and withoutLQG and LQR controllers at lag power factor load(P=1, Q=0.25 pu). Fig. 7 shows the rotor speeddeviation response due to 0.1 load disturbance withLQG compared with LQR controllers at lag powerfactor load (P=1, Q=0.25 pu). Fig. 8 displays therotor speed deviation response due to 0.1 loaddisturbance with LQG compared with LQRcontrollers at lead power factor load (P=1, Q= - 0.25pu). Fig. 9 shows the rotor speed deviation responsedue to 0.1 load disturbance with and without LQGand LQR controllers at lead power factor load (P=1,Q= -0.25 pu) .Moreover, Table 3 displays theEignvalues calculation with and without controllersfor single machine power system. Also, Table 4shows the Settling time for single machine modelwith and without controllers

0 1 2 3 4 5 6 7 8 9 10-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

time in Sec.

roto

r ang

le d

ev. i

n pu

.

rotor angle dev. response( P=1,Q= 0.25pu.)

W/O -CONTROLLQR-ControlLQG-Control

Fig.5: Rotor angle dev. Response due to 0.1 loaddisturbance with and without LQG and LQR controllers at

lag power factor load (P=1, Q=0.25 pu).


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0 1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

1

2

3

4x 10-3

time in Sec.

roto

r spe

ed d

ev. i

n pu

.

rotor speed dev. response( P=1,Q= 0.25pu.)

W/O -CONTROLLQR-ControlLQG-Control

Fig.6: Rotor speed dev. Response due to 0.1 loaddisturbance with and without LQG and LQR controllers at


0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1x 10-3

time in Sec.

roto

r spe

ed d

ev. i

n pu

.

rotor speed dev. response( P=1,Q= 0.25pu.)

LQR-ControlLQG-Control

Fig. 7: Rotor speed dev. Response due to 0.1 loaddisturbance with LQG compared with LQR controller at


0 1 2 3 4 5 6 7 8 9 10-5

-4

-3

-2

-1

0

1

2x 10-3

time in Sec.

Roto

r spe

ed d

ev. i

n pu

.

Rotor speed dev. response at(P=1,Q= -0.25pu.)

LQG -Control LQR- Control

Fig. 8: Rotor speed dev. Response due to 0.1 loaddisturbance with LQG compared with LQR controller at

lead power factor load (P=1, Q= - 0.25 pu).

0 1 2 3 4 5 6 7 8 9 10-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

time in Sec.

roto

r spe

ed d

ev. i

n pu

.

rotor speed dev. (P=1,Q= -0.25pu.)

W/O -CONTROLLQR-controlLQG - CONTROL

Fig. 9: Rotor speed dev. Response due to 0.1 loaddisturbance with and without LQG and LQR controllers at

lead power factor load (P=1, Q= -0.25 pu).

Table 4: Settling time for single machine model with andwithout controllers

States WithoutControl

LQR-Control

LQG-Control

P=1,Q=0.25pu.

RotorSpeed

> 10 Sec. 7 Sec. 4 Sec.

RotorAngle

>10 Sec. 5 Sec. 2.5 Sec.

P=1, Q=-0.25 pu.

RotorSpeed

3.5 Sec. 2 Sec.

RotorAngle

2 Sec. 0.5 Sec.

4.2 Simulation results of multi-machinesystem

From LQR control (Eqn. 10), the feedback gain is:

0.00000.00000.00000.00010.00000.00000.00020.00010.0000-0.0310-0.0060-0.0014-

0.00050.01710.00040.00900.02320.00510.0105-0.21180.0314-5.3269-0.3101-0.1203-

0.00330.00010.00040.02220.00230.00770.11170.0034-0.00076.1716-0.9033-0.3427-

LQRK

From LQG and Kalman filter control (Eqn. 13), theobserver gain matrix L and solution of reccatiequation P for multimachine power system arecalculated as:


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P = 1.0e+005 *

3.05022.33442.50290.29800.47960.32971.19090.52190.30370.0002-0.00410.0026

2.33447.25262.87730.39480.68620.48970.98171.50010.35540.0015-0.0041-0.0027

2.50292.87732.41410.26070.39700.30250.92880.61160.11570.0016-0.00300.0011

0.29800.39480.26070.13330.23600.13100.16880.12380.08340.0017-0.0005-0.0000-

0.47960.68620.39700.23600.53760.23720.30270.21320.14930.0027-0.0023-0.0001-

0.32970.48970.30250.13100.23720.14090.17300.13310.05560.0017-0.0007-0.0005

1.19090.98170.92880.16880.30270.17300.52620.24110.17320.00020.00100.0009

0.52191.50010.61160.12380.21320.13310.24110.35360.12480.0010-0.00010.0004

0.30370.35540.11570.08340.14930.05560.17320.12480.28840.0009-0.0003-0.0001

0.0002-0.0015-0.0016-0.0017-0.0027-0.0017-0.00020.0010-0.0009-0.00010.00000.0000

0.00410.0041-0.00300.0005-0.0023-0.0007-0.00100.00010.0003-0.00000.00010.0000

0.00260.00270.00110.0000-0.0001-0.00050.00090.00040.00010.00000.00000.0001

16.0888-272.6171169.9381

97.1338-269.7776-180.2242

103.3462-197.853374.3507

114.5776-33.9607-2.9718-

175.6031-153.5171-6.6915-

109.9142-45.7872-31.0354

10.630167.908957.6099

64.0363-6.603029.3742

61.8858-18.7586-5.5492

5.15330.74040.5642

0.74046.29190.5023

0.56420.50235.6175

L

Figure 10 depicts the rotor speed deviation responsedue to 0.1 load disturbance with and without LQGcontrol of M/C-1. Fig.11 shows the rotor speeddeviation response due to 0.1 load disturbance withand without LQG control of M/C-2. Also, Fig.12shows the rotor speed deviation response due to 0.1load disturbance with LQG compared with LQRcontrollers of M/C-2. Fig. 13 depicts the rotor speeddeviation response due to 0.1 load disturbance withand without LQG control of M/C-3. Fig.14 shows therotor speed deviation response due to 0.1 loaddisturbance with and without LQG and LQR controlof M/C-3. Moreover, Fig. 15 depicts the rotor speeddeviation response due to 0.1 pu load disturbancewith LQR control for three machines. Also, Fig.16displays the rotor speed deviation response due to0.1 pu load disturbance with LQG control for threemachines. Table 5 displays the Settling time formulti-machine model with and without controllers.Also, table 6 shows the Eignvalues calculation withand without controllers of multi- machine model

0 1 2 3 4 5 6 7 8 9 10-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time in Sec.

roto

r spe

ed d

ev. i

n pu

.

rotor speed dev. in M/C-1

W/O -CONTROLLQG-controL

Fig. 10: Rotor speed dev. Response due to 0.1 loaddisturbance with and without LQG control of M/C-1.

0 1 2 3 4 5 6 7 8 9 10-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time in Sec.

roto

r sp

eed

dev.

in p

u. rotor speed dev. in M/C-2


Fig.11: Rotor speed dev. Response due to 0.1 loaddisturbance with and without LQG control of M/C-2.

0 1 2 3 4 5 6 7 8 9 10-7

-6

-5

-4

-3

-2

-1

0

1

2

3x 10-3

time in Sec.

roto

r spe

ed d

ev. i

n pu

.


LQR- ControlLQG - ControL

Fig.12: Rotor speed dev. Response due to 0.1 loaddisturbance with LQG and LQR controllers of M/C-2.


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0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4x 10-3

time in Sec.

Rot

or S

peed

dev

. in

pu.

Rotor Speed dev. with LQG control in pu.

M/C-1M/C-2M/C-3

Table 5: Settling time for multi-machine model with andwithout controllers

OperatingPoint

Rotorspeedof

Withoutcontroller

WithLQR-Control

WithLQG –Control

P=1,Q=0.25pu

M/C-1

12 Sec. 7 Sec.

M/C-2

9 Sec. 5.5 Sec.

M/c-3 5 Sec. 3 Sec

5 Discussion

From Table 6, the eignvalues with the LinearQuadratic Gaussian controller(LQG) is the best thanLinear Quadratic Regulator (LQR) controller.Kalman Filter using with the regulator LQR toproduced the LQG. Also, Table 5 displays thedecreasing in the settling time for three machineswith using LQG controller than other controller.Moreover, the three machines on multi machinesystem are unstable without control but withproposed LQG controller all machine become stablewith less settling time.

Fig. 16: Rotor speed dev. Response due to 0.1 pu loaddisturbance with LQG control for three machines.

0 1 2 3 4 5 6 7 8 9 10-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

time in Sec.

roto

r spe

ed d

ev. i

n pu

.



Fig. 13: Rotor speed dev. Response due to 0.1 loaddisturbance with and without LQG control of M/C-3.

Fig.14: Rotor speed dev. Response due to 0.1 load disturbancewith and without LQG and LQR control of M/C-3.

Fig. 15: Rotor speed dev. Response due to 0.1 pu loaddisturbance with LQR control for three machines.

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10-3

time in Sec.

Rot

or S

peed

dev

. in

pu.

Rotor Speed dev. with LQR control in pu.

M/C-1M/C-2M/C-3

0 1 2 3 4 5 6 7 8 9 10-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

time in Sec.

Rot

or S

peed

dev

. in

pu.

Rotor Speed dev. in M/C-3 in pu.

W/O -CONTROLLQR-controllqg- CONTROL


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Table 6: Eigen values calculation with and without controllers of multi- machine model

Without control With LQR With Kalman With LQG With LQG +Feedback control

CertainOperatingPoint

-18.8713-15.1893-17.05190.0953 + 7.8364i0.0953 - 7.8364i-0.0627 + 7.3692i-0.0627 - 7.3692i0.2637 + 4.0915i0.2637 - 4.0915i-5.8914-3.4305-1.5112

-34.1322-19.5804-15.7307-0.0721 + 7.8603i-0.0721 - 7.8603i-0.0776 + 7.3715i-0.0776 - 7.3715i-0.2581 + 4.0793i-0.2581 - 4.0793i-5.5632-3.0683-1.1913

-18.8684-17.0507-15.1612-2.7296 + 7.4647i-2.7296 - 7.4647i-2.8210 + 7.0795i-2.8210 - 7.0795i-6.4027-2.4204 + 2.8257i-2.4204 - 2.8257i-3.4688-1.5217

-34.1388-19.6481-15.7647-2.9275 + 7.5560i-2.9275 - 7.5560i-2.8056 + 7.1482i-2.8056 - 7.1482i-4.1097 + 3.2205i-4.1097 - 3.2205i-4.7248-2.0980-1.0843

-34.1250-19.5046-18.8652-17.0493-15.6930-15.1210-0.2888 + 7.9098i-0.2888 - 7.9098i-0.0909 + 7.3805i-0.0909 - 7.3805i-2.4990 + 7.3546i-2.4990 - 7.3546i-2.8479 + 6.9797i-2.8479 - 6.9797i-0.9751 + 4.7816i-0.9751 - 4.7816i-7.5320-0.8193 + 2.4713i-0.8193 - 2.4713i-5.7117-3.6632-3.3731-1.2828-1.5345

6 ConclusionThe present paper introduces an application of a

robust linear quadratic Gaussian LQG controller todesign a power system stabilizer. The LQG optimalcontrol has been developed to be included in powersystem in order to improve the dynamic response andgive the optimal performance at any loadingcondition. The LQG controller design gives goodperformance in terms of fast damping dynamicoscillation. The LQG is better than LQR controller interms of small settling time and less overshoot andunder shoot. The digital results show that theproposed PSS based upon the LQG can achieve goodperformance over a wide range of operatingconditions.

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[8] Francis, "State-space solutions tostandard H2 and Hinf control problems,"

IEEE Trans. Automat. Contr., AC-34, no.8, PP. 831-847, Aug. 1989.

[9] Ali M. Yousef, and Amer Abdel-Fatah " Arobust H synthesis controller for power

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[13] Ali M. Yousef and Ahmed M. Kassem“ Optimal Power System StabilizerEnhancement of Synchrony Zing andDamping Torque Coefficient” wseas,issue 2, Vol. 7, PP. 70-79, 7 April 2012.


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