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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,
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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
lag power factor load (P=1, Q=0.25 pu).
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
lag power factor load (P=1, Q=0.25 pu).
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
W/O -CONTROLLQG-controL
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
.
rotor speed dev. in M/C-2
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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E-ISSN: 2224-350X 151 Volume 9, 2014
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
.
rotor speed dev. in M/C-3
W/O -CONTROLLQG-controL
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
WSEAS TRANSACTIONS on POWER SYSTEMS Ahmed Said Oshaba
E-ISSN: 2224-350X 152 Volume 9, 2014
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.
7 References[1] M. K. El-Sherbiny and D. Mehta “ Dynamic
stability, part I, investigation of the effect ofdifferent loading and excitation system”IEEE Trans. Power Appar. Sys., PAS-92(1973), PP 1538-1546.
[2] Chern-Lin Chen and Yuan Yih Hsu “Coordinated synthesis of multi machineHpower system stabilizer using an efficientdecentralized modal control algorithm” IEEETransaction on power system, vol. PWRS-2,No. 3, August 1987.
[3] Lan Petersen, “ Minimax LQG control”,School of information technology andelectrical engineering, university of Newsouth Wales, Australia, Vol 16, No. 3, PP.209:323, 2006.
[4] M. K. El-Sherbiny, A.I.Saleh and A.A.M.El-Gaafary " Optimal Design of AnOverall Controller of Saturated SynchronousMachine Under Different Loading", EEETransaction of power Apparatus and System,Vol.PAS-102,No.6,June 1983.
[5] M.K. El-Sherbiny , G.El-Saady and Ali M.Yousef "Robust controller for power systemstabilization" IEEE MEPCON'2001, HelwanUniversity, Cairo, Egypt, December 29-31,2001, PP. 287-291.
[6] M.K. El-Sherbiny , M. M. Hassan, G.El-Saady and Ali M. Yousef " Optimal poleshifting for power system stabilization",Electric Power Systems Research Journal,No.66 PP. 253-258. 2003.
[7] S. C. Tripathy, R. Balasubramanian andZahid Jamil " Interaction of Voltage andFrequency Control Loops and Optimization
of Parameters", IE(I) Journal-EL, Vol. 62,paper No. UDC 621-316-2, October 1981.J. Doyle, K. Glover, P. Khargonekar, and B.
WSEAS TRANSACTIONS on POWER SYSTEMS Ahmed Said Oshaba
E-ISSN: 2224-350X 153 Volume 9, 2014
[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
system stabilization " International journal ofautomatic control and system engineeringVol.2, pp. 1-6, Feb. 2005, Germany.
[10] P. P. Khargonekar, I. R. Petersen, and M. A.Rotea, "Hinf optimal control with
state feedback," IEEE Trans. Automat.Contr., AC-33, pp. 783-786, 1988.
[11] "MATLAB Math Library User's Guide", bythe Math Works. Inc., 2007.
[12] Ali M. Yousef, Labib Daloub, and M. El-Rezaghi “ Power System Stabilizer DesignBased on Robustic Control for Multi-Machine System” Al-Azhar EngineeringEight International Conference, Vol. 2,No. 4,,PP. 631:640, Egypt, Apr. 2007.
[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.
WSEAS TRANSACTIONS on POWER SYSTEMS Ahmed Said Oshaba
E-ISSN: 2224-350X 154 Volume 9, 2014