Lqr Tuning Of Power System Stabilizer For Damping Oscillations

> Nuraddeen Magaji and Dr. Mohd Wazir bn Mustapha

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

LQR Tuning of Power System Stabilizer for Damping Oscillations N. Magaji, and M, Wazir, Mustafa. Faculty, of Electrical Engineering University Teknologi Malaysia

Abstract— Power System Stabilizers (PSS) are added to excitation systems to enhance the damping

during low frequency oscillations. In this paper, the design of PSS for single machine connected to an infinite bus using optimal control techniques is considered. A method for shifting the unstable openloop poles to optimum positions is presented. In each step of this approach, I solve a first-order or a second-order linear matrix Lyapunov equation for shifting unstable pole. This presented method yields a solution, which is optimal with respect to a quadratic performance index. The attractive feature of this method is that it enables solutions to complex problem to be easily found without solving any non-linear algebraic Riccati equation. The gain feedback is calculated one time only and it works over wide range of operating conditions. A comparison between the effect of the PSS based on conventional approach, and the proposed Linear Quadratic Regulator (LQR ) is reported using MATLAB/SIMULINK for simulation.

Index Terms— K-constants, LQR, SMIB, PSS and Power system oscillations. I. INTRODUCTION Power systems experience low-frequency oscillations due to disturbances. These low frequency oscillations are related to the small signal stability of a power system. The phenomenon of stability of synchronous machine under small perturbations is explored by examining the case of a single machine connected to an infinite bus system (SMIB). The analysis of SMIB [1] gives physical insight into the problem of low frequency oscillations. These low frequency oscillations are classified into local mode, inter area mode and torsional mode of oscillations. The SMIB system is predominant in local mode low frequency oscillations. These oscillations may sustain and grow to cause system separation if no adequate damping is available. In recent years, modern control theory has been applied to PSS design problems. These include optimal

> Nuraddeen Magaji and Dr. Mohd Wazir bn Mustapha

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control, adaptive control, variable structure control, and intelligent control. Despite the potential of modern control techniques with different structures, power system utilities still prefer the conventional lead-lag PSS structure. The reasons behind that might be the ease of on-line tuning and the lack of assurance of the stability related to some adaptive or variable structure techniques. The main objective of this paper is to evaluate a control technique, to design a damping controller for power system stabilizer (PSS). This paper uses LQR control approach to design a PSS [2]. An expression for synchronizing and damping torque coefficients with optimal controller is established.

II. SYSTEM INVESTIGATED A single machine-infinite bus (SMIB) system is considered for the present investigations. A machine connected to a large system through a transmission line may be reduced to a SMIB system, by using Thevenin’s equivalent of the transmission network external to the machine. Because of the relative size of the system to which the machine is supplying power, the dynamics associated with machine will cause virtually no change in the voltage and frequency of the Thevenin’s voltage (infinite bus voltage). The Thevenin equivalent impedance shall henceforth be referred to as equivalent impedance (i.e. Re+jXe). The synchronous machine is described as the fourth order model. The two-axis synchronous machine representation with a field circuit in the direct axis but with out damper windings is considered for the analysis. The equations describing the steady state operation of a synchronous generator connected to an infinite bus through an external reactance can be Linearized about any particular operating point as follows(eq:1-4):

Tm Te 2H

d 2  dt 2

(1)

Te K 1K 2 E q'

(2)

K3 K K E 'q  E fd  3 ' 4  ' 1 sTd 0K 3 1 sT d 0K 3

(3)

Vt K 5K 6 E q'

(4)

The K-constants are given in appendix. The interaction between the speed and voltage control equations of the machine is expressed in terms of six constants k 1-k6.[3] These constants with the exception of k 3, which is only a function of the ratio of impedance, are dependent upon the actual real and reactive power loading as well as the excitation levels in the machine.

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Conventional PSS comprising cascade connected lead networks with generator angular speed deviation (Δw) as input signal has been considered. Fig.1 shows the Linearized model of the single machine infinite bus (SMIB) connected to large system which is linearized around the operating points. From the transfer function block diagram, the following state variables are chosen for single machine system. The Linearized differential equations can be written in the form state space form as X (t ) Ax(t ) Bu(t )

(5)

Where X  t   E q ’ E fd   T

0   D M A=   K 4   K A TA  K5  

B=

B K 1 M 0 0

0     0        0 , x  Eq     K A   E FD      T A 

y []T

(6) 0  K 2M

1 K 3 T ' do

 K A TA  K6

  0  ' 1 T do   -1 T A   0

(7)

(8)

(9)

System state matrix A is a function of the system parameters, which depend on operating conditions. Control matrix B depends on system parameters only. Control signal u is the PSS output. From the operating conditions and the corresponding parameters of the system considered, the system eigenvalues are obtained. III.

CONVENTIONAL PSS AND DESIGN CONSIDERATIONS

The exciter considered here is only having the gain of K A and the time constant of TA. The typical PSS consists of a washout function, a phase compensator (lead/lag functions), and a gain. It is well known that the performance of the PSS is mostly affected by the phase compensator and the gain. Therefore, these are the main focus of the tuning process. Two first order phase compensation blocks are considered. If the degree of compensation required is small, a single first-order block may be used. Generally small under compensation is preferable so that the PSS does not contribute to the negative synchronizing torque component. Washout function (Tw) has the value of anywhere in the range of 1 to 20 seconds [4, 10]. The main considerations are that it should be long enough to pass stabilizing signals at the frequencies of interest

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relatively unchanged, but not so long that it leads to undesirable generator voltage excursions as a result of stabilizer action during system- islanding conditions. For local mode of oscillations in the range of 0.8 to 2 Hz, a wash out of 1.5 sec is satisfactory. From the view point of low-frequency interarea oscillations, a wash out time constant of 10 seconds or higher is desirable, since low- time constants result in significant phase lead at low frequencies [5]. The stabilizer gain K has an important effect on damping of rotor oscillations. The value of the gain is chosen by examining the effect for a wide range of values. The damping increases with an increase in stabilizer gain up to a certain point beyond which further increase in gain results in a decrease in damping. Ideally, the stabilizer gain should be set at a value corresponding to maximum damping. However the gain is often limited by other considerations. The transfer function model of the SMIB system with the PSS is given in Fig.2. The transfer function of PSS is given by

H1 ( s) K*

1 sT1 1 sT3 10s *( )( ) (1 10s) 1 T2 s 1 T4 s

(10)

Where, K—PSS gain T w—washout time constant T 1-T4---phase lead time constants A low value of T2=T4=0.05 second is chosen from the consideration of physical realization. Tw=10 sec is chosen in order to ensure that the phase shift and gain contributed by the wash out block for the range of oscillation frequencies normally encountered is negligible. The wash out time constant (Tw) is to prevent steady state voltage off sets as system frequency changes. By considering two identical cascades connected lead-lag networks for the PSS H1(s) where T 1=T 3, the problem now reduces to the tuning of gain (K) and T1 only. The parameters of the PSS obtained for the damping ratio of 0.3. The oscillation frequency is generally about 0.8-2 Hz for the local mode of oscillations. In this SMIB system only local mode of oscillations are considered for the tuning of PSS. The local mode of oscillation occurs when a machine supplies power to a load center over long, weak transmission lines.

A. Conventional PSS design The eigenvalues of the above A matrix are obtained using Matlab function eig.. It is evident from the open loop eigenvalues, the system without PSS is unstable and therefore it is necessary to stabilize the system by shifting these eigenvalues to the LHP and far off from the imaginary axis. The location of the desired

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eigenvalues is calculated by choosing a damping factor ζfor the dominant root. The real part is -ζ wn and the imaginary part is wn  1 2 Where wn is the undamped natural frequency of the corresponding roots [6]. For the determination of PSS parameters a damping factor of ζ =0.3 is chosen (maximum damping). Corresponding to this damping factor the desired eigenvalues are obtained as.

1 n n 1 2

(11)

2 n n 1 2

(12)

It is to be noted here that the some of the eigenvalues need not be shifted since they are already in left hand S-plane and satisfied the required damping. If any electromechanical modes of oscillations are present then PSS needs to be added to enhance the dynamic stability of the system. By using Decentralized modal control (DMC) algorithm the parameters of the conventional PSS are found [7, 11]. IV. PROPOSED LQR PSS ALGORITHM The LQR controller generates the parameters of the gain and the phase lead time constant by minimizing the error criteria in eqn (14). Consider a linear system characterized by eqn. (5) where (A, B) is stabilizable. We define the cost index then determine the matrix K of the LQR vector[9] u Kx

(13)

So in order to minimize the performance index  J ( x, u, Q, R) ( xT Qx uT Ru) dt, Q 0,R>0 0

(14)

Where Q and R are the positive-definite Hermitian or real symmetric matrix. Note that the second term on the right side account for the expenditure of the energy on the control efforts. The matrix Q and R determine the relative importance of the error and the expenditure of this energy. From the above equations we get   J ( xT Qx xT KT RKx) dt  xT ( QxKT RK ) xdt 0 0

(15)

where (A,Q1/2 ) is detectable and ( A – BK ) is stable. The linear quadratic regulation problem is to find a control u Kx law such that and J is minimized, the solution is given by K R 1B T P

(16)

The controller K can be derived using parameter-optimization techniques, we set d xT (Q K T RK) x  ( xT Px) dt

(17)

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Then we obtain T T T T x (Q K RK ) x x Px x Px x T [( A BK )T P P ( A BK )]x

(18)

Comparing both sides of the above equation and note that this equation must hold true for any x, we require that ( A BK )T P P ( A BK )] (Q K T RK )

(19)

R T T

(20)

T

Where T is a nonsingular matrix, and T T 1 T T A P PA [TK (T ) B P]  [TK (TT )1 BT P ] PBR1 BT Q 0

(21)

The minimization J of with respect to K requires the minimization of xT [TK (T T )1 BT P]T [TK (T T ) 1 BT P] x

(22)

Which this equation is nonnegative, the minimum occurs when it is zero, or when TK (T T )1 BT P

(23)

Hence K T 1(T T )1BT P R 1BT P

(24)

Thus we get a control law as

1BT Px (t ) u (t ) Kx (t ) R 

(25)

In which P must satisfy the reduced Riccati equation: 1 TP Q PA A TP  PBR  B  0

(26) A. Weight Matrix Selection

The LQR design selects the weight matrix Q and R such that the performances of the closed loop system can satisfy the desired requirements mentioned earlier. The selection of Q and R is weakly connected to the performance specifications, and a certain amount of trial and error is required with an interactive computer simulation before a satisfactory design results. Now given these linear models we can use LQR or pole placement techniques to design full state feedback controllers, u=-Kx. The lqr function allows you to choose two parameters, R and Q, which will balance the relative importance of the input and state in the cost function that you are trying to optimize. The simplest case is to assume

> Nuraddeen Magaji and Dr. Mohd Wazir bn Mustapha

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R=1, and Q=C'*C. Essentially, the lqr method allows for the control of all outputs. In this case, it is pretty easy to do. The controller can be tuned by changing the nonzero elements in the Q matrix to get a desirable response the Matlab function lqr can be used to derive optimal control gains for a continuous controller. B. Numerical example Consider the mathematical Linearized state space model which represents the power system in equation (5)(9) Choosing the machine parameters and nominal operating point as;

Machine (p.u):

x d = 1.70 , xd' = 0.245 xq = 1.64 , T d' 0 5 .9 s, D= 0.0, M=2H = 4.74 and f=60Hz Transmission line (p.u): re = 0.02, xe = 0.4 Exciter : KA = 400 T A= 0.05 s Operating point: Vt0 Eb0 1.0 , KE=-0.17,TE=0.95 P0 = 1.0, Q0 = 0.1 and δ0 = 53.74 0 The A matrix given in Eq. (7) is evaluated as 377 0 0   0   0 0.3 0.1 0   A 0.8 0 19.2  0.2    1361.5 0  2871.5  20 

Consider the mechanical torque is constant and the vector B is obtained as follows: B  0 0 0 8,000  T

The open-loop poles are:

1,2  3.9054 j4.8082

 =-41.52and  =-5.792 3 4 The dominant poles 3 .9 0 5 4  j4 .8 0 8 2 can be rewritten as: 1,2  n  jn 1 2 Where ζ : = damping coefficient; ωn = frequency. The real part of dominant pole from Eq. (13) is 3.905 and imaginary part is j4.8082 which implies 0.6305 and n 4.8082rad / s  f=0.765Hz .Therefore this electromechnical mode characterized by pair of eigenvalues 3.9054 ±j4.8082 is negatively damped. The desired value of the damping coefficient to damp the oscillation of speed is obtained by optimum controller

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after shifting the unstable poles from 3.6 ±j25.9 to optimum position that is -5.726±9.498. Note; these values -5.726±9.498 are obtained by using Matlab function lqr by taking R=1 and Q=C’*C. The dominant closed loop poles are now specified at ζ =0.756 and ωn=9.497rad/s The new system is now stable with LQR Controller in closed loop: V. COMPARISON OF VARIOUS DESIGN TECHNIQUES The Linearized incremental state space model for a single machine system with its voltage regulator with four state variables has been developed. The single machine system without PSS is found unstable with roots in RHP. The system dynamic response without PSS is simulated using Simulink for 0.05 p.u disturbance in mechanical torque. MATLAB coding is used for conventional PSS design techniques, the final values of gain (K), and phase lead time constant (T) as 16 and 0.2287 respectively are taking from [8] and given to the Simulink block. While LQR PSS, design techniques obtain a vector of K from Matlab function lqr are also given to the Simulink block. The dynamic response curves for the variables Δω, Δδ and ΔEfd are taken from the Simulink. The system responses curves of open loop system, the conventional PSS (CPSS), LQR based PSS are compared. Shaft speed deviation is taken as the input to the all the PSSs. So the PSS is also called as delta-omega PSS. The system dynamic response with PSS is simulated using these Simulink diagrams for 0.05 p.u step change in mechanical torque ΔTm. The dynamic response curves for the variables change in speed deviation (Δω), change in rotor angle deviation (Δδ ) and change in Exciter voltage deviation (ΔEfd) of the single machine system with PSS are plotted for three different types of PSSs are shown in Figs. 3– 11. It is observed that the oscillations in the system output variables with PSS are well suppressed. Table 1-3 shows the eigenvalues of different load condition while Table 4-5 shows conventional PSS parameters and LQR parameters for different load condition VI. S IMULATION RESULTS

Performance of fixed-gain CPSS is better for particular operating conditions. It may not yield satisfactory results when there is a drastic change in the operating point. The proposed PSS has robustness control property with power system operating points change and its parameters variation and uncertainty Dynamic response shows that the LQR based PSS has optimum response and the response is smooth and it has less over shoot and settling time as compared to the open loop response and the traditional Lead-lag PSS.

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VII. CONCLUSION In this study, an optimal control algorithm is proposed to the PSS design problem. The method is illustrated by applying it to PSS controller design for damping oscillations. The procedure is developed identifying the most effective optimum controller based on state feedback. A numerical example illustrates the effectiveness of LQR damping performance by shifting all the negative damping poles to positive damping poles. The potential of the proposed design approach has been demonstrated by comparing the response curves of various PSS (PSS) design techniques. In addition, the simulation results show that the proposed LQR can work effectively and robustly over wide range of loading conditions and system configurations

APPENDIX Derivation of K-constants

All the variables with subscript 0 are values of variables evaluated at their pre -disturbance steady-state operating point from the known values of P 0 , Q 0 and Vt0.

iq 0 =

P0V to (P0 x q ) 2 (Vt 20 Q0 x q ) 2

(27)

vd 0 = iq 0 x q

(28)

vqo = Vt 20 v t20

(29)

Q0 xq iq20 id 0 = vq 0

(30)

E q 0 = vq 0 id 0 xq

(31)

E 0 = (vd 0 xeiq 0 )2 ( vq 0 xe id 0 )2

(32)

1

0 = tan

(vd 0 xeiq 0 ) (v q 0 xe id 0 )

xq xd' E E cos 0 K1 = i E sin 0  q 0 0 ' q0 0 xe xd xe xq K2 =

E0 sin 0 ' x e xd

(33)

(34)

(35)

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xd' xe K3 = xd xe

xq xd'

K4 =

xe x d'

(36)

E 0 sin 0

(37)

xq vd0 xd' vq0 K5  E0 cos E0 sin 0 0 xe xq Vt 0 xe  xd' Vt 0 K6 =

(38)

xe vq 0 xe x 'd Vt 0

(39)

Nomenclature All quantities are per unit on machine base. D

Damping Torque Coefficient

M

Inertia constant

ω

Angular speed

δ

Rotor angle

Id, Iq

Direct and quadrature components of armature current

Xd , X q Synchronous reactance in d and q axis Xd ’, X q’ Direct axis and Quadrature axis transient reactance EFD

Equivalent excitation voltage

KA

Exciter gain

TA

Exciter time constant

Tm,T e

Mechanical and Electrical torque

Tdo’

Field open circuit time constant.

Vd ,Vq Direct and quadrature components of terminal voltage K1

Change in T e for a change in δwith constant flux linkages in the d axis

K2

Change in Te for a change in d axis flux linkages with constant δ

K3

Impedance factor

K4

Demagnetising effect of a change in rotor angle

K5

Change in Vt with change in rotor angle for constant E q’

K6

Change in Vt with change in E q’ constant rotor angle

REFERENCES

[1] Y.L.Abdel-Maidand M.M. Dawoud, Tuning of Power system stabilizers using genetic algorithms, Electric Power Systems Research,Vol. 39, Jul.(1996), pp. 137-143.

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[2] L. Fan and A. Feliachi, Robust TCSC Control Design for Damping Inter-Area Oscillations, Power Engineering Society Summer Meeting, 2001. IEEE Vol. 2, 15-19 July (2001), pp.784 - 789. [3] J. Chow, J.Sanchez-Gasca, H. Ren and Sh. Wang, Power System damping Controller Design Using Multiple Input Signals, IEEE Control Systems Magazine, Volume 20, August( 2000) ,82-90. [4] M.Klein, G.J.Rogers, S.Moorty, P.Kundur, Analytical investigation of factors influencing Power system stabilizers performance, IEEE trans. On energy conversions, vol.7, Sep.(1992),No.3, pp. 382-390. [5] Yao-nan, Qing-hua Li , Pole placement power system stabilizers design of an unstable nine- machine system, IEEE transactions on power systems, Vol 5, (1990),No.2, pp.353-358. [6] Bikash Pal , Balarko Chaudhuri Robust Control in power System power electronics and Power series Editors. New York USA Springer 2005. [7] M.E. Aboul-Ela, A.A. Salam, J.D. McCalley and A.A. Fouad, Damping Controller Design for Power System Oscillations Using Global Signals, IEEE Transactions on Power Systems, Vol. 11, No2, May (1996), pp. 767-773. [8] K,R Padiyer Power System Dynamics Stability and Control page 279, Anshan Limited UK 2nd Edition 1994 [9] Liqun Xing A Comparison of Pole Assignment & LQR Design Methods for Multivariable Control for STATCOM Master Theses , Department of Mechanical Engineering, College Of Engineering, Florida State University .2003. [10] Joe H.Chow, George E.Boukarim and Alexander Murdoch, Power system stabilizers as undergraduate control design projects, IEEE trans. on power systems, vol 19, Feb.(2004). Pp.144-151. [11] A. Kazemi, and M. V. Sohforouzani Power System Damping Using Fuzzy Controlled FACTS Devices International Conference on Power System Technology - POWERCON 2004 Singapore, 21-24 November 2004

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K1 Tm

B s

1 2Hs



K4 K2

K5 K3 1 sTdoK3

KE 1 sT E

K6

Fig 1 Open loop model of SIMB using Heffron-Philips Constants

Vref

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Light load DW response

0.015

s p e e d d e v ia t io n

0.01

0.005

0

-0.005

Conventional PSS NO PSS LQR PSS

-0.01

-0.015 0

10

20

30

40

50

60

70

80

90

100

Time in sampling of 0.1 Fig.3 ΔωVs time for Light load condition of the SMIB system.

Normal load DW esponse 0.025 0.02

p ed d ev ia tio n

0.015 0.01 0.005 0 -0.005 -0.01 -0.015

N0 PSS Conventional PSS LQR PSS

-0.02 -0.025 0

20

40

60

80

Time in 0.1 sampling

Fig. 4 ΔωVs time for normal load condition of the SMIB system

100

120

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Heavy Load Speed deviation DW response 0.04 NO PSS Conventional PSS LQR PSS

0.03

sp ee d d ev ia tion

0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 0

10

20

30

40

50

60

70

80

90

100

Time in 0.1 sampling

Fig. 5 ΔωVs time for Heavy load condition of the SMIB system Light load DW response

0.015

speed deviation

0.01

0.005

0

-0.005

Conventional PSS NO PSS LQR PSS

-0.01

-0.015 0

10

20

30

40

50

60

70

80

90

Time in sampling of 0.1

Fig.6 ΔδVs time for Light load condition of the SMIB system

100

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Normal load angle deviation 4 No PSS Conventional PSS LQR PSS

A n g le d e viat io n

3 2 1 0

-1 -2 -3 0

20

40

60

80

100

120

Time 0.1 sampling Fig.7 ΔδVs time for Normal load condition of the SMIB system Heavy load Field voltage response 1.4 1.2

deviation in E'q voltage

1 0.8 0.6

0.4 CPSS

0.2

No PSS LQR PSS

0 -0.2 -0.4 0

20

40

60

80

100

Time (s)

Fig. 8 ΔE’ q Vs time for Heavy load condition of the SMIB system

120

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Normal load field voltage response 1.2

deviation in E'q voltage

1

0.8

0.6

0.4

CPSS No PSS LQr PSS

0.2

0

-0.2 0

10

20

30

40

50

60

70

80

90

100

90

100

Tim(s)

Fig. 9 ΔE’ q Vs time for Normal load condition o f the SMIB system

Light load field voltage response 1.2

deviation in E'q voltage

1

0.8

0.6

CPSS No PSS LQr PSS

0.4

0.2

0

-0.2 0

10

20

30

40

50

Tim(s)

60

70

Fig. 10 ΔE’ q Vs time for light load condition of the SMIB system

80

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Normal load Excitor voltage responseTime(s) 100

Devition of Efd voltage

80

CPSS No PSS LQR PSS

60

40

20

0

-20

-40 0

20

40

60

80

100

Time (s)

Fig. 11 ΔEfd Vs time for Normal load condition of the SMIB system

120

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TABLE I

OPERATING CONDITIONS P(pu) Q(pu)

Case 1 Normal

1.0

0.1

2 Heavy

1.2

0.2

3 Light

0.2

0.05

TABLE 2 0PEN LOOP EIGENVALUE Heavy Light

Normal

3.9095±j4.815

3.89±j5.039

-1.676±j7.04

-41.529

-41.137

-48.199

-5.802

-6.177

12.202

TABLE 3 C LOSED LOOP EIGENVALUE Heavy

Normal

Light

-5.726 ±j9.498

-5.681 ±j9.439

-4.27±8.83

-41.529

-41.129

-48.197

-11.549

-11. 484

-13.23

TABLE 4 CONVENTIONAL PSS CONSTANTS CCONVENTIONAL PSS K = 16 T2 =T4=0.05 T1=T3= 0.2287 TABLE 5 LQR C ONSTANTS K Normal Heavy Light K1

0.8673

0.8472

1.7525 -79.5827

K2

-75.669

76.1599 -

-0.7972 0.0038

K3

-0.5995

0.5805

K4

0.0031

0.0031

Address Nuraddeen Magaji U3B 603 Kolege Perdana, 81310 UTM Skudai

Jahor, Malaysia Emal [email protected], +60177739011, +607536745