LQR Tuning of Power System Stabilizer for Damping Oscillations

> Nuraddeen Magaji and Dr. Mohd Wazir bn Mustapha 1

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 open-

loop 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

LQR Tuning of Power System Stabilizer forDamping Oscillations

N. Magaji, and M, Wazir, Mustafa. Faculty, of Electrical EngineeringUniversity Teknologi Malaysia


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):

2

22

m e

dT T H

dt(1)

'1 2 qTe K K E (2)

' 3 3 4' '0 3 0 31 1q fd

d d

K K KE E

sT K sT K

(3)

'5 6t qV K K E (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 k1-k6.[3] These constants with the exception of k3,

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.


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

’ T

q fdX t E E (6)

B

1 2' '

4 3 do do

A A 5 A A 6 A

0 0 0D M K M K M 0

A=K 0 1 K T 1 T

K T K 0 K T K -1 T

(7)

00

,0

A

A

x EqK

EFDT

B=

(8)

[ ] Ty (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 KA 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


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

3

2 4

( * ) )1 1

1 1 sT1 sT10s

H s) K* ( (1 (1 10s) T s T s (10)

Where,

K—PSS gain

Tw—washout time constant

T1-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 T1=T3, 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


eigenvalues is calculated by choosing a damping factor ζfor the dominant root. The real part is -ζwn and the

imaginary part is wn21 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.

211 n n (11)

22 1n n (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

( , , , ) ( ) ,0

T TJ x u Q R x Qx u Ru dt Q 0,R>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

( ) ( )0 0

T T T T TJ x Qx x K RKx dt x Qx K RK xdt

(15)

where 1/2 )(A,Q 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

1 TK R B P (16)

The controller K can be derived using parameter-optimization techniques, we set

( ) ( )dT T Tx Q K RK x x Pxdt

(17)


Then we obtain

( )

[( ) ( )]

T T T Tx Q K RK x x Px x PxT Tx A BK 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

( ) ( )] ( )T TA BK P P A BK Q K RK (19)

TR T T (20)

Where T is a nonsingular matrix, and

1[ ( ) ]1 1[ ( ) ] 0

T T T TA P PA TK T B PT T TTK T B P PBR B Q

(21)

The minimization J of with respect to K requires the minimization of

1 1[ ( ) ] [ ( ) ]T T T T T Tx TK T B P TK T B P x (22)

Which this equation is nonnegative, the minimum occurs when it is zero, or when

1( )T TTK T B P (23)

Hence

1 1 1( )T T TK T T B P R B P (24)

Thus we get a control law as

1( ) ( ) ( )Tu t Kx t R B Px t (25)

In which P must satisfy the reduced Riccati equation:

1 0T TPA A P PBR B P Q (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


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):

dx = 1.70 , 'dx = 0.245 qx = 1.64 , '

0 5.9dT s, D= 0.0, M=2H = 4.74 and f=60Hz

Transmission line (p.u):

re = 0.02, xe = 0.4

Exciter :

KA = 400 TA= 0.05 s Operating point: 0 0 .0 t bV E 1 , 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

0 377 0 0

0 0.3 0.1 0

0.8 0 19.2 0.21361.5 0 2871.5 20

A

Consider the mechanical torque is constant and the vector B is obtained as follows:

0 0 0 8,000 TB

The open-loop poles are:

1,2 3.9054 j4.8082

3 4=-41.52and =-5.792

The dominant poles 3 .9 0 5 4 j4 .8 0 8 2 can be rewritten as: 21,2 1n nj

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 4 8082 n. and . rad / 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


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. SIMULATION 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.


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 P0 , Q0 and Vt0.

0qi =2

020

20

0

)()( qtq

to

xQVxP

VP

(27)

0dv = qq xi 0 (28)

qov = 20

20 tt vV (29)

0di =0

200

q

qq

vixQ

(30)

0qE = qdq xiv 00 (31)

0E = 200

200 )()( deqqed ixvixv (32)

0 =)(

)(tan

00

001

deq

qed

ixv

ixv

(33)

K1 =qe

qq

de

dq

xxEE

Eixxxx

000

000'

' cossin

(34)

K2 = '00 sin

de xxE

(35)


K3 =ed

ed

xxxx

'

(36)

K4 = 00'

'

sinExx

xx

de

dq

(37)

'00

0 0 0 0'0 0

cos sinq

q qd5

e t e d t

x vv xdK E E

x x V x x V

(38)

K6 =0

0'

t

q

de

e

V

v

xxx

(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, Xq Synchronous reactance in d and q axis

Xd’, Xq

’ Direct axis and Quadrature axis transient reactance

EFD Equivalent excitation voltage

KA Exciter gain

TA Exciter time constant

Tm,Te Mechanical and Electrical torque

Tdo’ Field open circuit time constant.

Vd,Vq Direct and quadrature components of terminal voltage

K1 Change in Te 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 Eq’

K6 Change in Vt with change in Eq’ constant rotor angle

REFERENCES

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


[2] L. Fan and A. Feliachi, Robust TCSC Control Design for Damping Inter-Area Oscillations, PowerEngineering 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 UsingMultiple 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 systemstabilizers 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- machinesystem, 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 seriesEditors. New York USA Springer 2005.

[7] M.E. Aboul-Ela, A.A. Salam, J.D. McCalley and A.A. Fouad, Damping Controller Design for PowerSystem 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 Edition1994

[9] Liqun Xing A Comparison of Pole Assignment & LQR Design Methods for Multivariable Control forSTATCOM Master Theses , Department of Mechanical Engineering, College Of Engineering, FloridaState University .2003.

[10] Joe H.Chow, George E.Boukarim and Alexander Murdoch, Power system stabilizers as undergraduatecontrol 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 DevicesInternational Conference on Power System Technology - POWERCON 2004 Singapore, 21-24 November2004


1K

2K5K

6K

4K

12 H s

B

s

3

31 do

KsT K 1

E

E

KsT

mT

refV

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


0 10 20 30 40 50 60 70 80 90 100-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time in sampling of 0.1

spee

dd

evi

atio

n

Light load DW response

Conventional PSSNO PSSLQR PSS

Fig.3 Δω Vs time for Light load condition of the SMIB system.

0 20 40 60 80 100 120-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Time in 0.1 sampling

ped

dev

iati

on

Normal load DW esponse

N0 PSSConventional PSSLQR PSS

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


0 10 20 30 40 50 60 70 80 90 100-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Time in 0.1 sampling

spee

dd

evia

tion

Heavy Load Speed deviation DW response

NO PSSConventional PSSLQR PSS

Fig. 5 Δω Vs time for Heavy load condition of the SMIB system

0 10 20 30 40 50 60 70 80 90 100-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time in sampling of 0.1

spee

dd

evia

tion

Light load DW response

Conventional PSSNO PSSLQR PSS

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


0 20 40 60 80 100 120-3

-2

-1

0

1

2

3

4Normal load angle deviation

Time 0.1 sampling

Ang

lede

viat

ion

No PSSConventional PSSLQR PSS

Fig.7 Δδ Vs time for Normal load condition of the SMIB system

0 20 40 60 80 100 120-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Heavy load Field voltage response

Time(s)

devi

atio

nin

E'q

volta

ge

CPSSNo PSSLQR PSS

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


0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Tim(s)

devi

atio

nin

E'q

voltage

Normal load field voltage response

CPSSNo PSSLQr PSS

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

0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Tim(s)

devi

atio

nin

E'q

volt

age

Light load field voltage response

CPSSNo PSSLQr PSS

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


0 20 40 60 80 100 120-40

-20

0

20

40

60

80

100

Time(s)

Dev

itio

nof

Efd

volta

geNormal load Excitor voltage responseTime(s)

CPSSNo PSSLQR PSS

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

>
Nuraddeen Magaji and Dr. Mohd Wazir bn Mustapha 18
Address

Nuraddeen MagajiU3B 603 Kolege Perdana, 81310 UTM SkudaiJahor, MalaysiaEmal [email protected],+60177739011, +607536745

TABLE IOPERATING CONDITIONS

Case P(pu) Q(pu)

1 Normal

2 Heavy

3 Light

1.0

1.2

0.2

0.1

0.2

0.05

TABLE 20PEN LOOP EIGENVALUE

Normal Heavy Light

3.9095±j4.815

-41.529

-5.802

3.89±j5.039

-41.137

-6.177

-1.676±j7.04

-48.199

12.202

TABLE 3CLOSED LOOP EIGENVALUE

Normal Heavy Light

-5.726 ±j9.498

-41.529

-11.549

-5.681 ±j9.439

-41.129

-11. 484

-4.27±8.83

-48.197

-13.23

TABLE 4CONVENTIONAL PSS CONSTANTS

CCONVENTIONAL PSS K = 16 T2=T4=0.05T1=T3= 0.2287

TABLE 5LQR CONSTANTS

K Normal Heavy Light

K1

K2

K3

K4

0.8673

-75.669

-0.5995

0.0031

0.8472

76.1599 -

0.5805

0.0031

1.7525 -79.5827

-0.7972 0.0038

mailto:[email protected]

Documents

LQR Tuning of Power System Stabilizer for Damping Oscillations