Robust Decentralized Power System Stabilizer for ...

12th IEEE International Workshop on Variable Structure Systems, VSS’12, January 12-14, Mumbai, 2012

Robust Decentralized Power System Stabilizer for Multi-machine Power System Using Reduced Order Observer designed by the Duality to Sliding Surface Design A J Mehta and H A Mehta

Abstract— This paper presents design of robust decentralized Power System Stabilizer (PSS) for multi-machine power system using reduced order observer designed by the duality to sliding surface design. Firstly, the discrete-time sliding surface is designed and then by the duality principle the discretetime reduced order observer for the PSS of multi-machines is configured. The novelty of the method is that the observer and the controller both are designed simultaneously using the same sliding surface. The performance of the proposed robust decentralized PSSs are investigated under different loading conditions and disturbances. The simulation results are presented to show the efficacy of the proposed method.

I. I NTRODUCTION Power System Stabilizer (PSS) have long been regarded as an effective way to enhance the damping of electromechanical oscillations in power system [1]. In other words, the PSSs extends the angular stability limits of a power system by providing supplemental damping to the oscillation of synchronous machine rotors through the generator excitation [2]. This damping is provided by an electric torque applied to the rotor that is in phase with the speed variation. Once the oscillations are damped out, the thermal limits of the tie-lines in the system may then be approached. This supplementary signal is very useful during large power transfers and line outages [3]. Over the past few decades, various control strategies have been proposed for PSS design to improve overall system performance. Among these, conventional PSS of the lead-lag compensation type [4], [5] have been adopted by most utility companies because of their simple structure, flexibility and ease of implementation. However, the performance of these stabilizers can be considerably degraded with the changes in the operating condition during normal operation. Since power systems are highly nonlinear, conventional fixed-parameter PSSs cannot cope up with great changes in the operating conditions. There are two main approaches are proposed in the literature for stabilizing a power system over a wide range of operating conditions, namely adaptive control and robust control [6]. Adaptive control is based on the idea of continuously updating the controller parameters according A J Mehta is with Department of Electrical Engineering, Gujarat Power Engineering and Research Institute (GPERI), Mehsana, Gujarat, INDIA

Email:[email protected] H A Mehta is with Department of Electrical Engineering, Gujarat Power Engineering and Research Institute (GPERI), Mehsana, Gujarat, INDIA

Email:[email protected]

to recent measurements. However, adaptive controllers have generally poor performance during the learning phase, unless they are properly initialized. Successful operation of adaptive controllers requires the measurements to satisfy strict persistent excitation conditions otherwise the adjustment of the controller’s parameters fails. On the other hand, Robust control strategy provides an effective approach to dealing with uncertainties introduced by variations of operating conditions. Among many robust control techniques available in the control literature, H∞ and variable structure have received considerable attention in the design of PSSs. The H∞ approach is applied by Chen [6] to PSS design for a single machine infinite bus system. The basic idea is to carry out a search over all possible operating points to obtain a frequency bound on the system transfer function. Then a controller is designed so that the worst-case frequency response of the closed loop system lies within pre-specified frequency bounds. It is to be noted that the H∞ , design requires an exhaustive search and results in a high order controller. On the other hand, the variable structure control is designed to drive the system to a sliding surface on which the error decays to zero [8]–[10]. The Perfect performance of the controller is achieved even if parameter uncertainties are present. However, such performance is obtained at the cost of high control activities (chattering) [11]. One of the requirement while applying the robust controller is that all the states variables must be available for measurement. But in actual practice, it is seldom available and if available it may be very costly. So the solution would be to estimate the state variables from the output and if the design of estimator can be done simultaneously along with the controller design then the efforts required for the observer design gets reduced. A J Mehta and B Bandyopadhyay [12], [13] has proposed a method for discrete-time reduced order observer design based on duality to sliding surface design and also proposed a reduced order observer based PSS design for single machine infinite bus (SMIB) system in [13]. In this paper, we propose a robust decentralized PSSs design for multi-machine power system (4-machine, 10-bus system) using reduced order observer designed by the duality principal. The duality between discrete-time reduced order observer and sliding surface design is reviewed. Using the same sliding surface and the estimated states, the sliding mode controller for PSSs is designed. The novelty is that the observer and the controllers both are designed simultaneously using the same sliding surface. The performance

142


of the proposed robust decentralized PSSs are investigated under different disturbances and loading conditions. The paper is organized as follows. Section II presents basics of power system stabilizer and power system analysis. Section III presents the review on duality between reduced order observer and the sliding surface design. Section IV presents the decentralized output feedback based PSS design for multi-machines (4-machine, 10-bus) power system using the duality principal. The simulation studies for various operating conditions and disturbances are presented in Section V. Conclusions are drawn in Section VI. II. P OWER S YSTEM S TABILIZER It is well established that fast acting exciters with high gain AVR can contribute to oscillatory instability in power systems. This type of instability is characterized by low frequency(0.2 to 3.0 Hz) oscillations which can persist (or even grow in magnitude) for no apparent reasons [2]. The major factors that contribute the instability are (a) loading of the generator or tie line (b) power transfer capability of transmission lines (c) power factor of the generator(leading power factor operation is more problematic than lagging power factor operation) (d) AVR gain. A cost efficient and satisfactory solution to the problem of oscillatory instability is to provide damping for generator rotor oscillations. This is conveniently done by providing Power System Stabilizer (PSS) which are supplementary controllers in the excitation systems. The signal Vs in Fig. 1 is the output from PSS which has input signal derived from rotor speed, frequency, electrical power or a combination of these variables. The objective of designing PSS is to provide additional damping torque without affecting the synchronizing torque at critical frequencies [3]. A. Basic Concept The basic function of a PSS is to extend the angular stability of a power system. This is done by providing supplemental damping to the oscillation of synchronous machine rotors through the generator excitation. This damping is provided by a electric torque applied to the rotor that is in phase with the speed variations. The oscillations of concern typically occur in the frequency range of 0.2 to 3.0 Hz, and insufficient damping of these oscillations may limit ability to transmit power. In practical system, the various modes (of oscillation) can be grouped into three broad categories [3]. • A. Intra-plant modes (generator G1 swings against G2 ) in which only the generators within a power plant participate. The oscillation frequencies are generally high in the range of 1.5 to 3.0 Hz. • B. Local modes in which several generators (G1 and G2 swing together against G3 ) in an area participate. The frequencies of oscillations are in the range of 0.8 to 1.8 Hz. • C. Inter area modes in which generators (generators G1 to G3 swing against G4 ) over an extensive area participate. The oscillation frequencies are low and in the range of 0.2 to 0.5 Hz.

Fig. 1.

Single line diagram of 4 machine 10 Bus System

The above categorization can be illustrated with the help of a system consisting of two areas connected by weak AC tie as shown in Fig. 1. The area 1 contains 3 generators G1 , G2 and G3 while the area 2 is represented by a single generator G4 . The distinction between local modes and inter area modes applies mainly for those systems which can be divided into distinct areas which are separated by long distances. For systems in which the generating stations are distributed uniformly over a geographic area, it would be difficult to distinguish between local and inter area modes from physical considerations. However, a common observation is that the inter area modes have the lowest frequency and participation from most of the generators in the system spread over a wide geographic area [7]. The PSSs are designed mainly to stabilize local and inter area modes. B. Performance objectives The main objective of providing PSS is to increase the power transfer in the network, which would otherwise be limited by oscillatory instability. The PSS also must function properly when system is subjected to large disturbances. PSS can extend power transfer stability limits which are characterized by lightly damped or spontaneously growing oscillations in the 0.2 to 3.0 Hz frequency range. This is accomplished via excitation control to contribute damping to the system modes of oscillations. Consequently, it is the stabilizers ability to enhance damping under the least stable conditions is important. Additional damping is primarily required under the conditions of weak transmission and heavy load which may occur, while attempting to transmit power over long transmission lines from the remote generating plants or relatively weak tie between systems. Contingencies, such as line outage, often precipitate such conditions. Hence system normally have adequate damping can often benefit from stabilizers during such conditions. C. State Space Model of Multi-machine Power System The dynamics of the machine rotor circuits, excitation systems, prime mover and other devices are represented by differential equations. The result is that the complete system

143


model consists of large number of ordinary differential and algebraic equations [3]. Model 1.0 is assumed for synchronous machine by neglecting the damper windings. In addition, the following assumptions are made for simplicity [2]. 1) The loads are represented by constant impedances. 2) Transient saliency is ignored by considering xq = xB 3) The mechanical power input to the generator is constant. The single line diagram of a 4 machine 10 bus system is shown in Fig. (1). The state space model of a multi-machine power system can be represented using machine data, line data and load flow data as follows, x˙ = y =

Ax + B(ΔVref + ΔVs ) Cx,

(1) (2)

where z(k) is (n−m) and xˆ(k) is an estimate of x(k). The estimate x ˆ(k) converges to x(k) if the coefficient matrices D, E, F, P, V satisfy the condition as given by T Φτ − DT = EC, P T + V C = In ,

III. R EVIEW OF D UALITY BETWEEN R EDUCED O RDER O BSERVER AND S LIDING S URFACE D ESIGN The design of discrete-time reduced order observer using the duality to discrete-time sliding surface design is proposed in [12]. Consider the discrete-time linear time invariant system x(k + 1) = Φτ x(k) + Γτ u(k),

(3)

y(k) = Cx(k),

(4)

where x ∈ is the state variable, Φτ ∈ , Γτ ∈ n×m is full rank, u ∈ m is the control input, C ∈ p×n such that CΓτ is nonsingular and y ∈ p is the output. We assume that (Φτ , Γτ ) is completely controllable and m < n. The (n − p)th - order reduced order state observer to estimate the state variable x(k) from measurement y(k) and input u(k) is given by [14]–[16] n

z(k + 1) = xˆ(k) =

n×n

Dz(k) + Ey(k) + F u(k),

(5)

P z(k) + V y(k),

(6)

(7)

where T is (n − m) × n matrix. The observer design is to find the coefficient matrices satisfying the above conditions (7). A detailed design scheme for a discrete-time reduced order state observer from the discrete-time sliding surface design is proposed in [12] and the duality of coefficients are obtained as shown in Table I. TABLE I D UALITY TABLE

where x = [x1 , x2 , ..., x10 ]T , and y = [y 1 , y 2 , ..., y 10 ]T . xj (j = 1, ..., 10) denotes the states of j th machine, and y j (j = 1, ..., 10) denotes the output of the j th machine. x denotes the state variables of each machine and are given as, ´q ]T , x = [Sm , δ, Ef d , E where, • Sm is machine slip. • δ is machine shaft angular displacement in degrees. • Ef d is generator field voltage in pu. ´q is voltage proportional to field flux linkages of • E machine in p.u. Further, • Vs is the correction voltage(PSS Output). • Vref is the reference voltage. • A is the system (state) matrix. • B is the input matrix. • C is the output matrix.

F = T Γτ , D is stable,

Observer Coefficients

Sliding Surface Design Coefficients

Φτ

ΦT τ

C

ΓT

D

JT

E

LT

P

UT

V

CsT

T

WT

IV. D ECENTRALIZED O UTPUT F EEDBACK BASED PSS D ESIGN FOR M ULTI - MACHINES (4 -MACHINE, 10BUS) P OWER S YSTEM U SING THE D UALITY The single line diagram of the 4-machine 10-bus system is shown in Fig. 1. The nonlinear differential equations governing the behavior of the system is linearized about an operating point to obtain a linear model, which represents the small signal oscillatory response of the power system. The AVR data, machine data and line data are given in the appendix [3]. The linearized model is discretized at a sampling rate of 0.02 sec. The slip of the each machine is taken as available output measurement for the observer. The reduced order observer design procedure proposed by the same author in [12] is used for designing the PSS. Firstly, the desired locations of the poles are assigned for the = 0.85 and the matrix Ps is obtained from the Riccati equation where and Ps are defined as in [12]. The matrix T0 is considered as given in appendix. The reduced order observer co-efficients D, E, F , P and V are obtained as given in the appendix and the control law is derived using the Gao’s reaching law [17] and the same sliding surface co-efficients is obtained as u(k) = −(Cs Γτ )−1 [Cs Φτ x(k) − (1 − qτ )s(k) +ρτ sgn(s(k))]

(8)

The simulation with the nonlinear model is carried out as mentioned in the next section.

144


V. S IMULATION WITH NONLINEAR MODEL The slip of the each machine is taken as output to estimate the state variables by the proposed reduced order observer. The control signal obtained by the state feedback control law 8 is used to damp out the small signal disturbances via modulating the generator excitation along with Vref signal. A limiter is also added to Vref signal that limits the PSS output at ±0.1. The disturbance considered here is a self clearing fault which is cleared after 0.1 second. The nonlinear simulation results of different generators with the classical PSSs and with the proposed PSSs for various models (i.e. different operating conditions) are shown in Fig. 2 to Fig. 7. The results are compared with the classical PI controller. As shown in the simulation results, the proposed PSS is able to damp out the oscillations in 1 second after clearing the fault for the range of active power with external line inductance variation of xe = 0.25pu to xe = 0.6pu.

[13] A.J. Mehta, B. Bandyopadhyay and A. Inoue, “Design of DiscreteTime Reduced Order Observer Using the Duality to the Discrete-Time Sliding Surface Design: An Application To Power System Stabilizer”, Proc. of 33rd Annual Conference of the IEEE Industrial Electronics Society (IECON 2007), Taipei, Taiwan, pp. 908 - 914, 5-8th Nov. 2007. [14] D. G. Luenberger, Observer for Multivariable Systems, IEEE Trans. on Automatic Control, vol. 11, no. 2, pp. 190-197, Apr. 1966. [15] D. G. Luenberger, An Introduction to Observers, IEEE Trans. on Automatic Control, vol. 16, no. 6, pp. 596-602, Dec. 1971. [16] D. G. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications, Wiley, 1979. [17] W. Gao, Y. Wang, and A. Homaifa, Discrete-time variable structure control systems, IEEE transactions on Industrial Electronics, vol. 42, no. 2, pp. 117122, April 1995.

A PPENDIX

TABLE II

VI. C ONCLUSION

LINE DATA: 4 MACHINE AND 10 BUS SYSTEM

In this paper, a design scheme of the Power System Stabilizers for multi-machine power system using reduced order observer is proposed. The efficacy of the proposed design method is substantiated by the simulation results for various operating conditions. The slip signal is taken as output for the output feedback sliding mode control that is applied at an appropriate sampling rate. It is found that designed controller provides good damping enhancement of each machine for multi-machine power system as compared with classical PSSs. It is also found that single control structure is enough to damp out all oscillations generated because of various models. R EFERENCES [1] F. DeMello and C.Concordia, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. on Power Apparatus and Systems, vol. PAS-88, pp. 316329, 1969. [2] P. Kundur, Power System Stability and Control, McGraw-Hill, Inc. Newyork, 1993. [3] K.R.Padiyar, Power System Dynamics Stability and Control, Interline publishing private Ltd. Bangalore, 1996. [4] E.V.Larsen and D.A.Swann, Applying power system stabilizers parti: General concepts, IEEE Trans. on Power Apparatus and Systems, vol. PAS-100, no. 6, pp. 30173024, June 1981. [5] E.V.Larsen and D.A.Swann, Applying power system stabilizers partiii: General concepts, IEEE Trans. on Power Apparatus and Systems, vol. PAS-100, no. 6, pp. 30343046, June 1981. [6] S.Chen and O.P.Malik, ”H∞ optimization- based power system stabilizer design”, IEE proceedings Part-C, vol. 142, no. 2, pp. 179-184, March 1995. [7] A. S. R. Ramamurthy, S. Parameswaran, and K. Ramar, Design of decentralized variable structure stabilizers for multimachine power systems, Electrical Power and Energy Systems, vol. 18, no. 8, pp. 535546, 1996. [8] S. V. Drakunov and V. I. Utkin, Sliding Mode in dynamic systems, Int. Journal of Control vol. 55, pp. 1029-1037,1990. [9] V.I. Utkin, Sliding Modes in Control Optimization, Springer-Verlag, New York, 1992. [10] G Bartolini, A. Ferrara, V. I. Utkin, ”Adaptive Sliding mode Control in Discrete-time Systems”, Automatica, vol. 31, no. 5, pp. 769-7733, 1995. [11] C. Edward and S.K. Spurgeon, Sliding Mode Control: Theory and Application, Springer-Verlag, New York, 1992. [12] A.J. Mehta, B. Bandyopadhyay and A. Inoue, “Reduced Order Observer Design for Servo System Using Duality to Sliding Surface Design”, IEEE Transaction on Industrial Electronics, Vol. 57, No. 11, pp. 3793-3800, 2010.

145

From

To

Series

Series

Shunt

Bus

Bus

Resistance Rs

Reactance Xs

Reactance Bs

1

6

0.01

0.012

0

2

5

0.01

0.012

0

9

10

0.022

0.22

0.33

9

10

0.022

0.22

0.33

9

10

0.022

0.22

0.33

9

5

0.002

0.02

0.03

9

5

0.002

0.02

0.03

3

8

0.001

0.012

0

4

7

0.001

0.012

0

10

7

0.002

0.02

0.03

10

7

0.002

0.02

0.03

6

5

0.005

0.05

0.075

6

5

0.005

0.05

0.075

8

7

0.005

0.05

0.075

8

7

0.005

0.05

0.075

TABLE III LOAD FLOW DATA: 4 MACHINE AND 10 BUS SYSTEM Bus

V

θ

PG

QG

PL

QL

Bl

No.

(pu)

(deg.)

(pu)

(pu)

(pu)

(pu)

(pu)

1

1.03

8.22

7

1.34

0

0

0

2

1.01

-1.5

7

1.6

0

0

0

3

1.03

0

7.21

1.45

0

0

0

4

1.01

-10.21

7

1.81

0

0

0

5

1.01

3.66

0

0

0

0

0

6

0.99

-6.24

0

0

0

0

0

7

1.01

-4.7

0

0

0

0

0

8

0.99

-14.94

0

0

0

0

0

9

0.98

-14.42

0

0

11.59

2.12

3

10

0.97

-23.3

0

0

15.75

2.88

4


TABLE IV MACHINE DATA: 4 MACHINE AND 10 BUS SYSTEM Ra

xd

x´d

xq

x´q

H

(pu)

(pu)

(pu)

(pu)

(pu)

(pu)

1

.00028

0.2

.033

0.19

.061

54

2

.00028

0.2

.033

0.19

.061

54

3

.00028

0.2

.033

0.19

.061

63

4

.00028

0.2

.033

0.19

.061

63

Gen

TABLE V MACHINE DATA: 4 MACHINE AND 10 BUS SYSTEM T´d0

T´q0

Xl

D

KE

TE

(pu)

(pu)

(pu)

(pu)

(pu)

(pu)

1

8

0.4

.022

0

200

0.05

2

8

0.4

.022

0

200

0.05

3

8

0.4

.022

0

200

0.05

4

8

0.4

.022

0

200

0.05

Gen

⎡

⎤

⎡

7.4926

⎢ ⎥ ⎥ =⎢ ⎣ 8.9934 ⎦ ; F −0.0793 ⎡ ⎤ 0 0 0 ⎢ ⎥ ⎢ 1 0 0 ⎥ ⎥ =⎢ ⎢ 0 −1 1 ⎥ ; V ⎣ ⎦ 0 0 1

E

P

⎡

T0

D

=

=

⎤

−0.0007

⎢ ⎥ ⎥ =⎢ ⎣ −61.3719 ⎦ ; 0.1701 ⎡ ⎤ 1.0000 ⎢ ⎥ ⎢ −0.0019 ⎥ ⎥ =⎢ ⎢ −0.0000 ⎥ ⎣ ⎦ −0.0020 ⎤

0

0

1

1

0

1

0

1

1

0

1

1

1

1

1

0

1

1

0

1

0

1

0

1

0

1

1

0

1

0

1

0

1

1

0

1

0

1

1

1

1

0

1

1

0

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

0

1

1

1

1

0

1

1

0

1

1

0

0

0

1

1

1

1

0

1

0

1

0

1

0

1

1

1

1

0

1

1

0

1

1

1

0

1

1

1

0

1

0

1

0

1

1

0

1

1

0

1

1

1

0

1

1

1

1

0

1

1

1

1

0

1

1

1

0

1

1

0

1

1

1

1

1

0

1

1

0

1

0

1

1

0

1

1

0

1

1

0

0

1

1

0

1

1

1

0

1

1

1

0

1

1

1

1

⎡

0

0

1

1

0

1

1

0

1

1

0

1

1

0

1

0

114.8431

76.3761

148.6503

−183.9561

113.8125

31.4439

−1.0739

74.6321

−77.4586

−296.9832

77.8488

−77.8532

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

113.8911

77.3350

148.6510

−183.8877

113.7828

31.4317

−1.0660

74.5923

−77.5089

−296.9367

77.8307

−77.8321

104.9568

87.7307

237.9629

−279.3462

127.4199

30.3793

−12.0736

162.5707

−73.7262

−385.6140

86.3614

−86.3448

231.7851

107.6629

191.8935

−311.4963

234.8560

118.6321

−9.9035

91.8548

−105.0942

−549.9342

101.2574

−101.2627

−12.0697

−86.3439

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

104.9593

87.7309

236.9679

−279.3442

128.4133

30.3793

162.5695

−73.7305

−385.6156

86.3605

240.4849

96.2690

103.5245

−216.9057

221.0544

120.5485

1.1047

3.8990

−108.7863

−460.9107

92.7082

−92.7345

104.9763

87.7612

237.1173

−279.4280

127.4154

30.3301

−11.0761

162.6593

−73.7934

−385.7020

86.4021

−86.3841 −92.7572

240.6819

96.2948

103.4464

−216.9536

221.2216

119.6934

1.1111

4.7808

−108.8339

−461.1603

92.7309

105.0180

87.6954

236.9708

−279.2862

127.4005

30.3752

−12.0669

162.5276

−72.7844

−385.5912

86.3471

−86.3275

113.8421

76.3795

148.6565

−183.9539

113.8070

31.4387

−1.0735

74.6359

−77.4618

−295.9844

77.8478

−77.8523

253.2477

67.0163

93.3647

−212.6645

223.5412

143.5431

−6.8443

5.2615

−97.4163

−469.0863

64.1134

−63.8173

126.6005

47.1820

138.5232

−179.6528

116.2564

55.3401

−8.9956

75.9558

−66.1708

−305.0704

48.3276

−48.0096

146

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦


−3

1

−3

Generator 1

x 10

3 Classical PSS Proposed PSS Relative slip

Relative slip

0

1 0

−0.5

−1

0

1

2

3

4

−3

1

5 6 Time in second

7

8

9

−2

10

0

1

2

3

4

−3

Generator 2

x 10

1.5 Classical PSS Proposed PSS

5 6 Time in second

7

8

Relative slip

0

9

10

Generator 4

x 10

Classical PSS Proposed PSS

1

0.5 Relative slip


2

0.5

−1

Generator 3

x 10

0.5 0 −0.5 −1

−0.5

−1.5 −1

0

1

2

3

4

5 6 Time in second

7

8

9

−2

10

0

1

2

3

4

Generator 1

8

9

10

Generator 3

0.24


0.22


0.05

0.2

Relative delta

Relative delta

7

Slip responses of 4-machine, 10-Bus system with Classical (with PI controller) PSS and Proposed PSS (Model 1 P go = [7, 7, 7.2172, 7]pu)

Fig. 2.

0.18 0.16 0.14

0 −0.05 −0.1

0.12 0.1

5 6 Time in second

0

1

2

3

4

5 6 Time in second

7

8

9

−0.15

10

0

1

2

3

4

Generator 2

5 6 Time in second

7

8

9

10

Generator 4

0.06

−0.05 Classical PSS Proposed PSS

0.04

Classical PSS Proposed PSS Relative delta

Relative delta

−0.1 0.02 0 −0.02

−0.15

−0.2 −0.04 −0.06

0

1

2

3

4

7

8

9

−0.25

10

−3

3

Relative slip

Relative slip

0

−0.5

3

4

5 6 Time in second

7

8

9

10

Generator 3

x 10


1 0 −1

0

1

2

3

4

−3

5 6 Time in second

7

8

9

−2

10

0

1

2

3

4

−3

Generator 2

x 10


5 6 Time in second

7

8

Relative slip

0

9

10

Generator 4

x 10


1

0.5 Relative slip

2

2

0.5

1

1

−3

Generator 1

x 10


−1

0

Delta responses of 4-machine, 10-Bus system with Classical (with PI controller) PSS and Proposed PSS (Model 1 P go = [7, 7, 7.2172, 7]pu)

Fig. 3.

1

5 6 Time in second

0.5 0 −0.5 −1

−0.5

−1.5 −1

Fig. 4.

0

1

2

3

4

5 6 Time in second

7

8

9

−2

10

0

1

2

3

4

5 6 Time in second

7

8

9

10

Slip responses of 4-machine, 10-Bus system with Classical (with PI controller) PSS and Proposed PSS (Model 2 P go = [7.05, 7, 7.2172, 7]pu)

147


Generator 1

Generator 3

0.24


0.2 0.18 0.16 0.14

0 −0.05 −0.1

0.12 0.1


0.05 Relative delta

Relative delta

0.22

0

1

2

3

4

5 6 Time in second

7

8

9

−0.15

10

0

1

2

3

4

Generator 2

5 6 Time in second

7

8

9

10

Generator 4

0.06


0.04


Relative delta

−0.1 0.02 0 −0.02

−0.15

−0.2 −0.04 −0.06

0

1

2

3

4

5 6 Time in second

7

8

9

−0.25

10

0

1

2

3

4

5 6 Time in second

7

8

9

10

Fig. 5. Delta responses of 4-machine, 10-Bus system with Classical (with PI controller) PSS and Proposed PSS (Model 2 P go = [7.05, 7, 7.2172, 7]pu)

−3

1

−3

Generator 1

x 10

3 Classical PSS Proposed PSS Relative slip

Relative slip

0

−0.5

0

1

2

3

4

5 6 Time in second

7

8

9

0

−2

10

0

1

2

3

4

−3

Generator 2

x 10


5 6 Time in second

7

8

Relative slip

0

9

10

Generator 4

x 10


1

0.5 Relative slip

1

−1

−3

1


2

0.5

−1

Generator 3

x 10

0.5 0 −0.5 −1

−0.5

−1.5 −1

0

1

2

3

4

5 6 Time in second

7

8

9

−2

10

0

1

2

3

4

7

8

9

10

Slip responses of 4-machine, 10-Bus system with Classical (with PI controller) PSS and Proposed PSS (Model 3 P go = [7, 7.05, 7.2172, 7]pu)

Fig. 6.

Generator 1

Generator 3

0.24


0.22 0.2 0.18 0.16 0.14

0 −0.05 −0.1

0.12 0.1


0.05 Relative delta

Relative delta

5 6 Time in second

0

1

2

3

4

5 6 Time in second

7

8

9

−0.15

10

0

1

2

3

4

Generator 2

5 6 Time in second

7

8

9

10

Generator 4

0.06


0.04


Relative delta

−0.1 0.02 0 −0.02

−0.15

−0.2 −0.04 −0.06

0

1

2

3

4

5 6 Time in second

7

8

9

−0.25

10

0

1

2

3

4

5 6 Time in second

7

8

9

10

Fig. 7. Delta responses of 4-machine, 10-Bus system with Classical (with PI controller) PSS and Proposed PSS (Model 3 P go = [7, 7.05, 7.2172, 7]pu)

148