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Design of power system stabilizer for single machine system using robust fast output sampling feedback technique
Electric Power Systems Research 65 (2003) 247 /257 www.elsevier.com/locate/epsr
Design of power system stabilizer for single machine system using robust fast output sampling feedback technique Rajeev Gupta a,*, B. Bandyopadhyay a, A.M. Kulkarni b a
b
Systems and Control Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Received 5 February 2002; received in revised form 2 December 2002; accepted 11 December 2002
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 fast output sampling feedback is proposed. The nonlinear model of a machine is linearized at different operating points and 16 linear plant models are obtained. For all of these plants a single stabilizing state feedback gain, F is obtained. A robust fast output sampling feedback gain which realizes this state feedback gain is obtained using linear matrix inequalities approach. This method does not require state of the system for feedback and is easily implementable. This robust fast output sampling control is applied to non-linear plant model of one machine at different operating (equilibrium) points. This method gives very good results for the design of PSS. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Power system stabilizer; Fast output sampling; Robust controller; Single machine with infinite bus
1. Introduction In the late 1950s and early 1960s most of the new generating units added to the electric utility systems were equipped with continuously acting voltage regulators. As these units became a larger percentage of generating capacity, it became apparent that the voltage regulator action had a detrimental impact upon the dynamical stability (or perhaps more properly steadystate stability) of the power system. Oscillations of the small magnitude and low frequency often persists for the long periods of time and in some case it can cause limitation on power transfer capability. Power system stabilizers (PSS) were developed to aid in damping these oscillation via modulation of the generator excitation. The art and science of applying PSS has been developed over the past 30/35 years since the first widespread application to the Western systems of the US. The development has evolved the use of various tuning techniques and input signals and learning to deal with turbine-generator shaft torsional mode of vibrations [1]. * Corresponding author. Fax: /91-22-2572-3480.
PSS are added to excitation system to enhance the damping of electric power system during low frequency oscillations. Several methods are used in the design of PSSs. Tuning of supplementary excitation controls for stabilizing system modes of oscillation has been the subject of much research during the past 30/35 years. Two basic tuning techniques have been successfully utilized with system stabilizer applications: phase compensation method and the root locus method. Phase compensation consists of adjusting the stabilizer to compensate for the phase lags through the generator, excitation system and power system such that the stabilizer path provides torque changes which are in phase with the speed changes [2 /8]. This is the most straightforward approach, easily understood and implemented in the field and most widely used. Synthesis by root locus involves shifting the eigenvalues associated with power system modes of oscillation by adjusting the stabilizer pole and zero locations in the s-plane [9,10]. This approach gives additional insight to performance by working directly with the closed-loop characteristics of the systems, as opposed to the open-loop nature of the phase compensation technique, but is more complicated to apply, particularly in the field. However, the
0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00017-8
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performance of these stabilizers can be considerably degraded with the changes in the operating condition during normal operation and can only deal with static effects. Recently, modern control methods have been used by several researchers to take advantage of optimal control techniques. These methods utilize a state space representation of power system model and calculate a gain matrix which when applied as a state feedback control will minimize a prescribed objective function. Successful application of the optimal control to PSS requires that the constraints imposed by power system non-linearities be used effectively and that a limited number of feedback signals is included [11]. In recent years, there have been several attempts at designing PSS using H based robust control techniques [12]. In this approach, the uncertainty in the chosen system is modeled in terms of bounds on frequency response. A H optimal controller is then synthesized which guarantees robust stability of the closed-loop system. However, this will lead to dynamic output feedback, which may be feasible but leads to a higher feedback system [13]. In practice, not all of the states are available for measurement. In the state feedback case the optimal control law requires the design of a state observer. This increases the implementation cost and reduces the reliability of control system. Another disadvantage of the observer based control system is that even slight variations of the model parameters from their nominal values may result into significant degradation of the closed-loop performance. Hence it is desirable to go for an output feedback design. However, the static output feedback problem is one of the most investigated problems in control theory. The complete pole assignment and guaranteed closed-loop stability is not obtained by using static output feedback. Another approach to pole placement problem is to consider the potential of time-varying fast output sampling feedback. With fast output sampling approach proposed by Werner and Furuta [14], it is generically possible to simultaneously realize a given state feedback gain for a family of linear, observable models. This approach requires to increase the low rank of the measurement matrix of an associated discretized system, which can be achieved by sampling the output several times during one input sampling interval, and constructing the control signal from these output samples. Since the feedback gains are piecewise constant, their method could be easily implemented and indicated a new possibility. Such a control law can stabilize a much larger class of systems than the static output feedback. This paper proposes the design of a robust PSS for single machine system using fast output sampling feedback. The brief outline of the paper is as follows: Section 2 presents basics of PSS whereas Section 3 contains the modeling of single machine system. Section 4 presents a
brief review on fast output sampling feedback control method. Section 5 contains the simulations of single machine at different operating points with the proposed controller followed by the concluding section.
2. PSS 2.1. Performance objectives PSS can extend power transfer stability limits which are characterized by lightly damped or spontaneously growing oscillations in the 0.2 /2.5 Hz frequency range. This is accomplished via excitation control to contribute damping to the system modes of oscillation. Consequently, it is the stabilizer’s ability to enhance damping under the least stable conditions, i.e., the ‘performance conditions’, which is important. Additional damping is primarily required under the conditions of weak transmission and heavy load as occurs, for example, when 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 systems which normally have adequate damping can often benefit from stabilizers during such conditions. It is important to realize that the stabilizer is intended to provide damping for small excursions about a steadystate operating point, and not to enhance transient stability, i.e., the ability to recover from a severe disturbance. In fact, the stabilizer will often have deleterious effect on transient stability by attempting to pull the generator field out of ceiling too early in response to a fault. The stabilizer output is generally limited to prevent serious impact on transient stability, but stabilizer tuning also has a significant impact upon the system performance following a large disturbance [2]. 2.2. Classical stabilizer implementation procedure Implementation of a PSS implies adjustment of its frequency characteristic and gain to produce the desired damping of the system oscillations in the frequency range of 0.2 /3.0 Hz. The transfer function of a generic PSS may be expressed as PSS Ks
Tw s(1 sT1 )(1 sT3 ) (1 Tw s)(1 sT2 )(1 sT4 )
FILT(s)
(1)
where Ks represents stabilizer gain and FILT(s) represents combined transfer function of torsional filter and input signal transducer. FILT(s)
v2n s2 2zvn s v2n
(2)
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The stabilizer frequency characteristic is adjusted by varying the time constant Tw , T1, T2, T3 and T4. It will be noted that the stabilizer transfer function includes the effect of both the input signal transducer and filtering required to attenuate the stabilizer gain at turbinegenerator shaft torsional frequencies. These effects, dictated by other consideration, must be considered in addition to the ‘plant’. The torsional filter in the PSS is essentially a band rejection filter to attenuate the first torsional modes frequency. The maximum possible change in damping (z ) of any torsional mode is less than some fraction of the inherent torsional damping. The phase lag of the filter in the frequency (vn ) range of 1 /3 Hz is minimized. This filter may not be needed in case torsional modes are well damped. A PSS can be made more effective if it is designed and applied with the knowledge of associated power characteristics. PSS must provide adequate damping for the range of frequencies of the power system oscillation modes. To begin with, simple analytical models, such as that of a single machine connected to an infinite bus, can be useful in determining the frequencies of local mode oscillations. PSS should also be designed to provide stable operation for the weak power system conditions and associated loading. Designed stabilizer must ensure for the robust performance and satisfactory operation with an external system reactance ranging from 20 to 80% on the unit rating [15].
3. State space model of single machine system From the block diagram shown in Fig. 1, the following state space equations for the entire system can be derived using Heffron /Phillip’s model: [16 /18]
x˙ [A]x[B](DVref DVs );
249
(3)
where xT [Dd DSm 2 0 6 K1 6 6 6 2H 6 K [A] 6 6 4 6 T? 6 do 6 K K 4 E 5 TE "
(5)
[B]T 0
(6)
[C] [0
DE?q DEfd ]; 3 0 0 vB 7 D K 7 2 0 7 2H 2H 7 7 1 1 7; 0 7 T?do K3 T?do 7 7 KE K6 1 7 5 0 TE TE # KE 0 ; TE
0 1
0
0]:
(4)
(7)
The damping term D is included in the swing equation. The eigenvalues of the matrix should lie in LHP in the s-plane for the system to be stable. The effect of various parameters (for example, KE and TE ) can be examined from eigenvalue analysis. It is to be noted that the elements of [A ] are dependent on the operating condition.
4. Review on fast output sampling feedback The problem of simultaneous stabilization has received considerable attention. In the given literature, a family of plants in state space representation (Fi , Gi ), i /1,. . ., M , find a linear state feedback gain F such that (Fi /Gi F ) is stable for i/1,. . ., M , or determine that no such F exists. But, the method is of use only in the case where whole state information is available.
Fig. 1. Block diagram of single machine.
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One way of approaching this problem with incomplete state information is to use observer based control laws, i.e., dynamic compensators. The problem here is that the state feedback and state estimation cannot be separated in face of the uncertainty represented by a family of systems. Assuming that a simultaneously stabilizing F has been found, it is possible to search for a simultaneously stabilizing full order observer gain, but this search is dependent on the F previously obtained. If no stabilizing observer for this state feedback exists, nothing can be said because there may exist stabilizing observers for different feedback gains. With the fast output sampling approach proposed in Werner and Furuta [14], it is generically possible to simultaneously realize a given state feedback gain for a family of linear, observable models. For fast output sampling gain L to realize the effect of state feedback gain F , find the L such that (Fi /Gi LC ) is stable for i/ 1,. . ., M, if there exist a set of F ’s, there should exist a common L for given family of plants. One of the problems with this approach is that large feedback gains tend to render the system very noise-sensitive. To overcome this problem the design problem can be posed as a multiobjective optimization problem in an linear matrix inequalities (LMI) formulation [14]. Consider a plant described by a linear model x ˙ AxBu; y Cx;
(8)
with (A , B ) controllable and (C , A ) observable. Assume the plant is to be controlled by a digital controller, with sampling time t and zero order hold, and that a sampled data state feedback design has been carried out to find a state feedback gain F such that the closed-loop system x(ktt)(Ft Gt F )x(kt);
(9)
has desired properties. Hence Ft /eA t and Gt t f0 eAs ds B: Instead of using a state observer, the following sampled data control can be used to realize the effect of the state feedback gain F by output feedback. Let D /t/N , and consider 2 3 y(ktt) 6y(kttD)7 7 Lyk ; u(t) [L0 L1 LN1 ]6 (10) 4n 5 y(ktD) for kt 0/t B/(k/1)t , where the matrix blocks Lj represent output feedback gains, and the notation Lyk has been introduced for convenience. Note that 1/t is the rate at which the loop is closed, whereas output samples are taken at the N -times faster rate 1/D. This control law is illustrated in Fig. 2. To show how a fast output sampling controller Eq. (10) can be designed to realize the given sampled data
Fig. 2. Fast output sampling feedback method.
state feedback gain, we construct a fictitious, lifted system for which Eq. (10) can be interpreted as static output feedback. Let (F,G,C ) denote the system Eq. (8) at the rate 1/D. Consider the discrete-time system having at time t /kt the input uk /u (kt), state xk /x (kt) and output yk as xk1 Ft xk Gt uk ; yk1 C0 xk D0 uk ;
(11)
where 2
2 3 3 0 C 6 7 6CF 7 7: 7; UD0 6CG C0 6 4n 5 4n 5 P N2 j N1 C j0 F G CF Assume that the state feedback gain F has been designed that (Ft /Gt F ) has no eigenvalues at the origin. Then, assuming that in intervals kt 0/t B/kt/t u(t)Fx(kt);
(12)
one can define the fictitious measurement matrix C(F ; N)(C0 D0 F )(Ft Gt F )1 ;
(13)
which satisfies the fictitious measurement equation yk / Cxk . For L to realize the effect of F , it must satisfy LC F :
(14)
Let n denote the observability index of (F,G). It can be shown that for N E/n , generically C has full column rank, so that any state feedback gain can be realized by a fast output sampling gain L . If the initial state is unknown, there will be an error Duk /uk /Fxk in constructing the control signal under state feedback. One can verify that the closed-loop dynamics are governed by
Xk1 F Gt F Gt xk t : (15) Duk1 0 LD0 F Gt Duk To see this, apply the coordinate transformation
R. Gupta et al. / Electric Power Systems Research 65 (2003) 247 /257
T
I F
0 ; I
(16)
to the equation
Xk1 Gt F xk t ; uk1 LC0 LD0 Duk
(17)
and use Eq. (13). Thus, one can say that the eigenvalues of the closed-loop system under a fast output sampling control law Eq. (10) are those of Ft /Gt F together with those of LD0/F Gt . One feature of fast output sampling control that makes it attractive for robust controller design, is the fact that a result similar to the above can be shown to hold when the same state feedback is applied simultaneously to a family of models representing different operating conditions of the plant.
x ˙ Ai xBi u; (18)
By sampling at the rate of 1/D, we get a family of discrete-time systems {Fi ,Gi ,Ci }. Now consider the augmented system defined below. 2 3 0 F1 0 60 F2 n 7 7; F˜ 6 :: 4n n : n 5 0 FM 2 3 G1 0 0 60 G2 n 7 7; G˜ 6 :: 4n n : n 5 (19) 0 GM C˜ [C1
C2
...
CM ]
Consider the family of discrete-time systems Eq. (18) having at time t/kt the input uk /u (kt), state xk / x (kt ) and output yk as xk1 Fti xk Gti uk ; yk1 C0i xk D0i uk ;
u(t)Fx(kt);
(21)
one can define the fictitious measurement matrix Ci (F ; N)(C0i D0i F )(Fti Gti F )1 ;
(22)
which satisfies the fictitious measurement equation yk / Ci xk . For robust fast output sampling gain L to realize the effect of F , it may satisfy i 1; . . . ; M:
(23)
The Eq. (23) can be written as
For multimodel representation of a plant, it is necessary to design controller which will robustly stabilize the multimodel system. Multimodel representation of plants can arise in several ways. When a nonlinear system has to be stabilized at different operating points, linear models are sought to be obtained at those operating points. Even for parametric uncertain linear systems, different linear models can be obtained for extreme points of the parameters. The models are used for stabilization of the uncertain system. Now consider a family of plant S /{Ai ,Bi ,Ci }, defined by i 1; . . . ; M:
2 3 0 6Ci Gi 7 7: D0 6 4n 5 PN2 j Ci j0 Fi Gi
Assume that (Fti ,Gi ) are controllable. Then we can find a robust state feedback gains F such that (Fti / Gi F ) has no eigenvalues at the origin. Then, assuming that in intervals kt 0/t B/kt/t
LCi F :
5. Multimodel synthesis
y Ci x
where 2 3 Ci 6Ci Fi 7 7; C0i 6 4n 5 Ci FN1 i
251
(20)
LC˜ F˜;
(24)
where C˜ [C1 F˜ [F
C2 F
CM ]; ...
F ]:
˜ G; ˜ C): ˜ It Let n˜ denote the observability index of (F; ˜ ˜ generically C has full can be shown that for N E/n; column rank, so that robust state feedback gain can be realized by a fast output sampling gain L . When this idea is realized in practice, i.e., fast output sampling gain L have been obtained by realising the state feedback gain F , two problems are required to be addressed.The first one is apparent from Eq. (15). With this type of controller, the unknown states are estimated implicitly, using the measured output samples and assuming that initial control is generated by state feedback. If initial state causes an estimation error, then decay of this error will be determined by the eigenvalues of the matrix (LD0i /F Gti ) which depends on L and whose dimension equals the number of control input. For stability these eigenvalues have to be inside the unit disc, and for fast decay they should be as close to the origin as possible. This problem must be taken into account while designing L . The second problem is that the gain matrix L may have elements with large magnitude. Because these values are only weights in linear combination of output samples, large magnitudes do not necessarily imply large control signal, and in theory and noise free simulation they pose no problem. But in practice they amplify measurement noise, and it is desirable to keep these values low. This objective can be expressed by an upper bound r on the norm of the gain matrix L .
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When trying to deal with these problem, it is better not to insist on an exact solution to the design Eq. (23): one can allow a small deviation and use an approximation LCi :/F , which hardly affects the desired closedloop dynamics, but may have considerable effect on the two problems described above. Instead of looking for an exact solution to the equalities, the following ineqaulities are solved kLkBr1 ; kLD0i F Gti kBr2i ; kLCi F kBr3i :
i 1; . . . ; M; (25)
Here three objectives have been expressed by upper bounds on matrix norms, and each should be as small as possible. The r1 small means low noise sensitivity, r2 small means fast decay of estimation error, most important */r3-small means that fast output sampling controller with gain L is a good approximation of the originally designed state feedback controller. If r3 /0 then L is exact solution. Using the Schur complement, it is straight forward to bring these conditions in the form of LMI.
2 r1 I L B0; LT I
2 (LD0i F Gti ) r2i I B 0; (LD0i F Gti )T I
2 (LCi F ) r3i I B0: (26) (LCi F )T I In this form, the function mincx() of the LMI control tool box for MATLAB can be used immediately to minimize a linear combination of r1, r2, r3. The following approach turned out to be useful. If the actual measurement noise is known, the magnitude of L is fixed accordingly. Likewise eigenvalues of (LD0i /F Gti ) less than 0.05 cause no problem. So we can fix r1, r2 and only r3 is minimized subject to these constraints. This is illustrated in the following section. In this form, the LMI tool box of MATLAB can be used for synthesis [19]. The fast output sampling feedback controller obtained by the above method requires only constant gains and hence is easier to implement. The example of single machine system dynamics with infinite bus is used to demonstrate the method.
6. Case study The non-linear differential equations governing the behavior of a power system can be linearized about a particular operating point to obtain a linear model which represents the small signal oscillatory response of a power system. Variations in the operating condition of
the system result in the variations in the parameters of the small signal model. A given range of variations in the operating conditions of a particular system thus generates a set of linear models each corresponding to one particular operating condition. Since, at any given instant, the actual plant could correspond to any model in this set, a robust controller would have to import adequate damping to each one of this entire set of linear models. The 16 plants of single machine system connected to an infinite bus are considered for designing robust fast output sampling feedback controller using LMI approach of MATLAB software. Then controller gains are applied to simulate a non-linear plant of single machine system connected to an infinite bus at different operating points. 6.1. Example The discrete models of different plants with different power (Pg 0) and different external line inductance(xe) are obtained by sampling time t/0.5 s and is given in Appendix A. The output matrix is given as C [0
1
0 0]:
(27)
Using the method discussed in Section 5 common stabilizing output injection gain matrix F is obtained for all 16 plants as given below. FT [0:4101
19:3152 0:4879 0:0016]:
(28)
Using LMI approach, Eqs. (25) and (26) are solved using different values of r1, r2, and r3 to find the gain matrix L . The robust fast output sampling feedback gain L is obtained as L [L1
L2 ];
(29)
where L1 [18:7233 14:0543 L2 [48:7778 34:0972
7:6172 31:0312
46:6674];
0:9996 51:0546 121:8813]:
The eigen values of (Fti /Gti LCi ) are found to be within the unit circle and no eigenvalues at the origin. 6.2. Simulation with non-linear plant at different operating points A SIMULINK based block diagram including all the non-linear blocks is generated [16]. The slip of the machine is taken as output and the intial value of slip is taken as zero. The output slip signal with gain L and a limiter is added to Vref signal which is used to provide additional damping. This is used to damp out the small signal disturbances via modulating the generator excitation. The output must be limited to prevent the PSS acting to counter action of AVR. Different operating
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253
Fig. 3. Open- and closed-loop responses of non-linear system for various operating conditions.
points are taken as the different plants. The disturbances considered is self clearing fault at generator terminal cleared after 0.1 s. The limits of PSS output are taken as 9/0.25 and exciter output limits (Efd ) are taken 9/6 pu. The conventional PSS is designed as compensator with washout circuit [16]. The transfer function of PSS
compensator is selected as PSS(s)
16(1 0:08s) (1 0:027s)
and washout circuit constant is selected as 2.0 s.
Fig. 4. Open- and closed-loop responses of non-linear system for various operating conditions.
(30)
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R. Gupta et al. / Electric Power Systems Research 65 (2003) 247 /257
Fig. 5. Open- and closed-loop responses of non-linear system for various operating conditions.
Simulation results at different operating points (Pg 0) and different external line inductance (xe) are shown in Figs. 3 /10 without controller and with the proposed controller. The control input as modulating voltage is also shown in these figures. The responses are compared with the conventional PSS. The simulataion results show that conventional PSS is able to damp out the oscillations upto active power, Pg 0 /1.5 with external line inductance, xe /0.2 and upto active power, Pg 0 /1.0
with external line inductance, xe /0.4, where as the proposed fast output sampling feedback controller is more robust and a single controller is able to damp out the oscillations for all cases. As shown in plots, the proposed controller is able to damp out the oscillations in 2/3 s after clearing the fault upto active power, Pg 0 /1.0 with external line inductance from xe /0.2 to 0.8 and 5 /15 s upto active power, Pg 0 /2.0 with external line inductance, xe /0.2.
Fig. 6. Open- and closed-loop responses of non-linear system for various operating conditions.
R. Gupta et al. / Electric Power Systems Research 65 (2003) 247 /257
255
Fig. 7. Open- and closed-loop responses of non-linear system for various operating conditions.
7. Conclusion In this paper, a design scheme of robust PSS for single machine connected to an infinite bus using fast output sampling feedback has been developed. The slip signal is
taken as output and the fast output sampling feedback is applied at appropriate sampling rate. The method is more general than static output feedback and the control input for these plants are required of small magnitudes. It is found that designed robust controller
Fig. 8. Open- and closed-loop responses of non-linear system for various operating conditions.
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Fig. 9. Open- and closed-loop responses of non-linear system for various operating conditions.
provides good damping enhanchement for various operating points of single machine system connected to an infinite bus. This method is also compared with conventional PSS. The proposed method results better response behaviour to damp out the oscillations. The conventional PSS is dynamic in nature and required to
tune according to power characteristics. Where as, the proposed controller gains are static in nature and one controller structure is able to damp out the oscillations for all plants. The method described has been extended to the multi-machine system and will be presented else where.
Fig. 10. Open- and closed-loop responses of non-linear system for various operating conditions.
R. Gupta et al. / Electric Power Systems Research 65 (2003) 247 /257
Appendix A: Plant parameters The following parameters are used for simulation of 16 plants H/5, Tdo /6 s, D /0.0, KE /100, TE /0.02 s, xe /0.2 pu
Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pg 0
Qg 0
K1
K2
K3
K4
K5
K6
xe
0.4 0.5 0.6 0.75 1.0 1.2 1.4 1.5 1.6 1.7 1.8 2.0 1.0 1.0 1.0 1.0
0.0160 0.0251 0.0361 0.0566 0.1010 0.1461 0.2000 0.2303 0.2629 0.2979 0.3352 0.4174 0.1535 0.2087 0.3333 0.5000
1.0170 1.1624 1.4336 1.1624 1.5772 1.6342 1.6263 1.6117 1.6263 1.6117 1.5911 1.4977 1.3197 1.3288 0.7430 0.3903
1.1239 1.2983 1.5855 1.2983 1.7402 1.8588 1.8752 1.8883 1.8752 1.8883 1.8987 1.9174 1.5042 1.7402 1.0776 0.8904
0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.3263 0.3600 0.4180 0.4667
1.4385 1.6618 2.0294 1.6618 2.2274 2.3792 2.4003 2.4170 2.4003 2.4170 2.4303 2.4543 1.9254 1.7009 1.3794 1.1397
0.0643 0.0577 0.0194 0.0577 /0.0309 /0.1130 /0.1328 /0.1524 /0.1328 /0.1524 /0.1717 /0.1878 /0.0641 /0.1002 /0.1739 /0.2468
0.3291 0.3083 0.2628 0.3308 0.2300 0.1988 0.1939 0.1898 0.1939 0.1898 0.1866 0.1805 0.3020 0.3608 0.4561 0.5380
xe xe xe xe xe xe xe xe xe xe xe xe 1.5xe 2xe 3xe 4xe
References [1] E.V. Larsen, D.A. Swann, Applying power system stabilizers Part-I: general concepts, IEEE Trans. Power Apparatus Syst. PAS-0100 (6) (1981) 3017 /30242. [2] E.V. Larsen, D.A. Swann, Applying power system stabilizers Part-II: performance objective and tuning concepts, IEEE Trans. Power Apparatus Syst. PAS-100 (6) (1981) 3025 /3033. [3] F.R. Schlief, J.H. White, Damping for the North-west /southwest Tie line oscillations */a analog study, IEEE Trans. Power Apparatus Syst. PAS-85 (1966) 1239 /1246. [4] F.R. Schlief, R.R. Angell, Damping of system oscillations with a hydrogenerating unit, IEEE Trans. Power Apparatus Syst. PAS86 (1967) 438 /442. [5] C. Concordia, F.P. de Mello, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Apparatus Syst. PAS-88 (1969) 316 /329.
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[6] F.R. Schlief, H.D. Hunkins, E.E. Hattan, W.B. Gish, Control of rotating exciters for power system damping: pilot applications and experience, IEEE Trans. Power Apparatus Syst. PAS-88 (1969) 1259 /1266. [7] E.J. Warchol, F.R. Schlief, W.B. Gish, J.R. Church, Alignment and modeling of hanford excitation control for system damping, IEEE Trans. Power Apparatus Syst. PAS-90 (1971) 714 /724. [8] A.D. Gerhart, T. Hillesland, Jr., J.F. Luini, M.L. Rockfield, Power system stabilizer field testing and digital simulation, IEEE Trans. Power Apparatus Syst. PAS-90 (1971) 2095 /2100. [9] K.E. Bollinger, A. Laha, R. Hamilton, T. Harras, Power system stabilizer design using root locus methods, IEEE Trans. Power Apparatus Syst. PAS-94 (1975) 1484 /1488. [10] P. Kundur, D.C. Lee, H.M. Zein El-din, Power system stabilizers for thermal units: analytical techniques and on-site validation, Paper F80-227-9 presented IEEE PES Winter Meeting, New York, February (1980). [11] F.A. Saleh, M.S. Mahmoud, Design of power system stabilizer using reduced-order methods, Electric Power Syst. Res. 33 (1995) 219 /226. [12] S. Chen, O.P. Malik, H based power system stabilizer design, IEE Proc. Part C 142 (2) (1995) 179 /184. [13] P. Shrikant Rao, I. Sen, Robust pole placememt stabilizer design using linear matrix inequalities, IEEE Trans. Power Syst. 15 (1) (2000) 3035 /3046. [14] H. Werner, K. Furuta, Simultaneous stabilisation based on output measurement, Kybernetika 31 (1995) 395 /411. [15] E.V. Larsen, D.A. Swann, Applying power system stabilizers Part-III: practical consideration, IEEE Trans. Power Apparatus Syst. PAS-100 (6) (1981) 3035 /3046. [16] K.R. Padiyar, Power System Dynamics Stability and Control, Interline publishing private Ltd, Bangalore, 1996. [17] W.G. Heffron, R.A. Phillips, Effect of a modern amplidyne voltage regulator on under excited operation of large turbine generators, AIEE Trans. 71 (Part 3) (1952) 692 /697. [18] F.D. DeMello, C. Concordia, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Apparatus Syst. PAS-88 (1969) 316 /329. [19] P. Gahenet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Tool Box with for MATLAB, The Math works Inc, Natick MA, 1995.
Design of power system stabilizer for single machine system using robust fast output sampling feedback technique Rajeev Gupta a,*, B. Bandyopadhyay a, A.M. Kulkarni b a
b
Systems and Control Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Received 5 February 2002; received in revised form 2 December 2002; accepted 11 December 2002
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 fast output sampling feedback is proposed. The nonlinear model of a machine is linearized at different operating points and 16 linear plant models are obtained. For all of these plants a single stabilizing state feedback gain, F is obtained. A robust fast output sampling feedback gain which realizes this state feedback gain is obtained using linear matrix inequalities approach. This method does not require state of the system for feedback and is easily implementable. This robust fast output sampling control is applied to non-linear plant model of one machine at different operating (equilibrium) points. This method gives very good results for the design of PSS. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Power system stabilizer; Fast output sampling; Robust controller; Single machine with infinite bus
1. Introduction In the late 1950s and early 1960s most of the new generating units added to the electric utility systems were equipped with continuously acting voltage regulators. As these units became a larger percentage of generating capacity, it became apparent that the voltage regulator action had a detrimental impact upon the dynamical stability (or perhaps more properly steadystate stability) of the power system. Oscillations of the small magnitude and low frequency often persists for the long periods of time and in some case it can cause limitation on power transfer capability. Power system stabilizers (PSS) were developed to aid in damping these oscillation via modulation of the generator excitation. The art and science of applying PSS has been developed over the past 30/35 years since the first widespread application to the Western systems of the US. The development has evolved the use of various tuning techniques and input signals and learning to deal with turbine-generator shaft torsional mode of vibrations [1]. * Corresponding author. Fax: /91-22-2572-3480.
PSS are added to excitation system to enhance the damping of electric power system during low frequency oscillations. Several methods are used in the design of PSSs. Tuning of supplementary excitation controls for stabilizing system modes of oscillation has been the subject of much research during the past 30/35 years. Two basic tuning techniques have been successfully utilized with system stabilizer applications: phase compensation method and the root locus method. Phase compensation consists of adjusting the stabilizer to compensate for the phase lags through the generator, excitation system and power system such that the stabilizer path provides torque changes which are in phase with the speed changes [2 /8]. This is the most straightforward approach, easily understood and implemented in the field and most widely used. Synthesis by root locus involves shifting the eigenvalues associated with power system modes of oscillation by adjusting the stabilizer pole and zero locations in the s-plane [9,10]. This approach gives additional insight to performance by working directly with the closed-loop characteristics of the systems, as opposed to the open-loop nature of the phase compensation technique, but is more complicated to apply, particularly in the field. However, the
0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00017-8
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performance of these stabilizers can be considerably degraded with the changes in the operating condition during normal operation and can only deal with static effects. Recently, modern control methods have been used by several researchers to take advantage of optimal control techniques. These methods utilize a state space representation of power system model and calculate a gain matrix which when applied as a state feedback control will minimize a prescribed objective function. Successful application of the optimal control to PSS requires that the constraints imposed by power system non-linearities be used effectively and that a limited number of feedback signals is included [11]. In recent years, there have been several attempts at designing PSS using H based robust control techniques [12]. In this approach, the uncertainty in the chosen system is modeled in terms of bounds on frequency response. A H optimal controller is then synthesized which guarantees robust stability of the closed-loop system. However, this will lead to dynamic output feedback, which may be feasible but leads to a higher feedback system [13]. In practice, not all of the states are available for measurement. In the state feedback case the optimal control law requires the design of a state observer. This increases the implementation cost and reduces the reliability of control system. Another disadvantage of the observer based control system is that even slight variations of the model parameters from their nominal values may result into significant degradation of the closed-loop performance. Hence it is desirable to go for an output feedback design. However, the static output feedback problem is one of the most investigated problems in control theory. The complete pole assignment and guaranteed closed-loop stability is not obtained by using static output feedback. Another approach to pole placement problem is to consider the potential of time-varying fast output sampling feedback. With fast output sampling approach proposed by Werner and Furuta [14], it is generically possible to simultaneously realize a given state feedback gain for a family of linear, observable models. This approach requires to increase the low rank of the measurement matrix of an associated discretized system, which can be achieved by sampling the output several times during one input sampling interval, and constructing the control signal from these output samples. Since the feedback gains are piecewise constant, their method could be easily implemented and indicated a new possibility. Such a control law can stabilize a much larger class of systems than the static output feedback. This paper proposes the design of a robust PSS for single machine system using fast output sampling feedback. The brief outline of the paper is as follows: Section 2 presents basics of PSS whereas Section 3 contains the modeling of single machine system. Section 4 presents a
brief review on fast output sampling feedback control method. Section 5 contains the simulations of single machine at different operating points with the proposed controller followed by the concluding section.
2. PSS 2.1. Performance objectives PSS can extend power transfer stability limits which are characterized by lightly damped or spontaneously growing oscillations in the 0.2 /2.5 Hz frequency range. This is accomplished via excitation control to contribute damping to the system modes of oscillation. Consequently, it is the stabilizer’s ability to enhance damping under the least stable conditions, i.e., the ‘performance conditions’, which is important. Additional damping is primarily required under the conditions of weak transmission and heavy load as occurs, for example, when 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 systems which normally have adequate damping can often benefit from stabilizers during such conditions. It is important to realize that the stabilizer is intended to provide damping for small excursions about a steadystate operating point, and not to enhance transient stability, i.e., the ability to recover from a severe disturbance. In fact, the stabilizer will often have deleterious effect on transient stability by attempting to pull the generator field out of ceiling too early in response to a fault. The stabilizer output is generally limited to prevent serious impact on transient stability, but stabilizer tuning also has a significant impact upon the system performance following a large disturbance [2]. 2.2. Classical stabilizer implementation procedure Implementation of a PSS implies adjustment of its frequency characteristic and gain to produce the desired damping of the system oscillations in the frequency range of 0.2 /3.0 Hz. The transfer function of a generic PSS may be expressed as PSS Ks
Tw s(1 sT1 )(1 sT3 ) (1 Tw s)(1 sT2 )(1 sT4 )
FILT(s)
(1)
where Ks represents stabilizer gain and FILT(s) represents combined transfer function of torsional filter and input signal transducer. FILT(s)
v2n s2 2zvn s v2n
(2)
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The stabilizer frequency characteristic is adjusted by varying the time constant Tw , T1, T2, T3 and T4. It will be noted that the stabilizer transfer function includes the effect of both the input signal transducer and filtering required to attenuate the stabilizer gain at turbinegenerator shaft torsional frequencies. These effects, dictated by other consideration, must be considered in addition to the ‘plant’. The torsional filter in the PSS is essentially a band rejection filter to attenuate the first torsional modes frequency. The maximum possible change in damping (z ) of any torsional mode is less than some fraction of the inherent torsional damping. The phase lag of the filter in the frequency (vn ) range of 1 /3 Hz is minimized. This filter may not be needed in case torsional modes are well damped. A PSS can be made more effective if it is designed and applied with the knowledge of associated power characteristics. PSS must provide adequate damping for the range of frequencies of the power system oscillation modes. To begin with, simple analytical models, such as that of a single machine connected to an infinite bus, can be useful in determining the frequencies of local mode oscillations. PSS should also be designed to provide stable operation for the weak power system conditions and associated loading. Designed stabilizer must ensure for the robust performance and satisfactory operation with an external system reactance ranging from 20 to 80% on the unit rating [15].
3. State space model of single machine system From the block diagram shown in Fig. 1, the following state space equations for the entire system can be derived using Heffron /Phillip’s model: [16 /18]
x˙ [A]x[B](DVref DVs );
249
(3)
where xT [Dd DSm 2 0 6 K1 6 6 6 2H 6 K [A] 6 6 4 6 T? 6 do 6 K K 4 E 5 TE "
(5)
[B]T 0
(6)
[C] [0
DE?q DEfd ]; 3 0 0 vB 7 D K 7 2 0 7 2H 2H 7 7 1 1 7; 0 7 T?do K3 T?do 7 7 KE K6 1 7 5 0 TE TE # KE 0 ; TE
0 1
0
0]:
(4)
(7)
The damping term D is included in the swing equation. The eigenvalues of the matrix should lie in LHP in the s-plane for the system to be stable. The effect of various parameters (for example, KE and TE ) can be examined from eigenvalue analysis. It is to be noted that the elements of [A ] are dependent on the operating condition.
4. Review on fast output sampling feedback The problem of simultaneous stabilization has received considerable attention. In the given literature, a family of plants in state space representation (Fi , Gi ), i /1,. . ., M , find a linear state feedback gain F such that (Fi /Gi F ) is stable for i/1,. . ., M , or determine that no such F exists. But, the method is of use only in the case where whole state information is available.
Fig. 1. Block diagram of single machine.
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One way of approaching this problem with incomplete state information is to use observer based control laws, i.e., dynamic compensators. The problem here is that the state feedback and state estimation cannot be separated in face of the uncertainty represented by a family of systems. Assuming that a simultaneously stabilizing F has been found, it is possible to search for a simultaneously stabilizing full order observer gain, but this search is dependent on the F previously obtained. If no stabilizing observer for this state feedback exists, nothing can be said because there may exist stabilizing observers for different feedback gains. With the fast output sampling approach proposed in Werner and Furuta [14], it is generically possible to simultaneously realize a given state feedback gain for a family of linear, observable models. For fast output sampling gain L to realize the effect of state feedback gain F , find the L such that (Fi /Gi LC ) is stable for i/ 1,. . ., M, if there exist a set of F ’s, there should exist a common L for given family of plants. One of the problems with this approach is that large feedback gains tend to render the system very noise-sensitive. To overcome this problem the design problem can be posed as a multiobjective optimization problem in an linear matrix inequalities (LMI) formulation [14]. Consider a plant described by a linear model x ˙ AxBu; y Cx;
(8)
with (A , B ) controllable and (C , A ) observable. Assume the plant is to be controlled by a digital controller, with sampling time t and zero order hold, and that a sampled data state feedback design has been carried out to find a state feedback gain F such that the closed-loop system x(ktt)(Ft Gt F )x(kt);
(9)
has desired properties. Hence Ft /eA t and Gt t f0 eAs ds B: Instead of using a state observer, the following sampled data control can be used to realize the effect of the state feedback gain F by output feedback. Let D /t/N , and consider 2 3 y(ktt) 6y(kttD)7 7 Lyk ; u(t) [L0 L1 LN1 ]6 (10) 4n 5 y(ktD) for kt 0/t B/(k/1)t , where the matrix blocks Lj represent output feedback gains, and the notation Lyk has been introduced for convenience. Note that 1/t is the rate at which the loop is closed, whereas output samples are taken at the N -times faster rate 1/D. This control law is illustrated in Fig. 2. To show how a fast output sampling controller Eq. (10) can be designed to realize the given sampled data
Fig. 2. Fast output sampling feedback method.
state feedback gain, we construct a fictitious, lifted system for which Eq. (10) can be interpreted as static output feedback. Let (F,G,C ) denote the system Eq. (8) at the rate 1/D. Consider the discrete-time system having at time t /kt the input uk /u (kt), state xk /x (kt) and output yk as xk1 Ft xk Gt uk ; yk1 C0 xk D0 uk ;
(11)
where 2
2 3 3 0 C 6 7 6CF 7 7: 7; UD0 6CG C0 6 4n 5 4n 5 P N2 j N1 C j0 F G CF Assume that the state feedback gain F has been designed that (Ft /Gt F ) has no eigenvalues at the origin. Then, assuming that in intervals kt 0/t B/kt/t u(t)Fx(kt);
(12)
one can define the fictitious measurement matrix C(F ; N)(C0 D0 F )(Ft Gt F )1 ;
(13)
which satisfies the fictitious measurement equation yk / Cxk . For L to realize the effect of F , it must satisfy LC F :
(14)
Let n denote the observability index of (F,G). It can be shown that for N E/n , generically C has full column rank, so that any state feedback gain can be realized by a fast output sampling gain L . If the initial state is unknown, there will be an error Duk /uk /Fxk in constructing the control signal under state feedback. One can verify that the closed-loop dynamics are governed by
Xk1 F Gt F Gt xk t : (15) Duk1 0 LD0 F Gt Duk To see this, apply the coordinate transformation
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T
I F
0 ; I
(16)
to the equation
Xk1 Gt F xk t ; uk1 LC0 LD0 Duk
(17)
and use Eq. (13). Thus, one can say that the eigenvalues of the closed-loop system under a fast output sampling control law Eq. (10) are those of Ft /Gt F together with those of LD0/F Gt . One feature of fast output sampling control that makes it attractive for robust controller design, is the fact that a result similar to the above can be shown to hold when the same state feedback is applied simultaneously to a family of models representing different operating conditions of the plant.
x ˙ Ai xBi u; (18)
By sampling at the rate of 1/D, we get a family of discrete-time systems {Fi ,Gi ,Ci }. Now consider the augmented system defined below. 2 3 0 F1 0 60 F2 n 7 7; F˜ 6 :: 4n n : n 5 0 FM 2 3 G1 0 0 60 G2 n 7 7; G˜ 6 :: 4n n : n 5 (19) 0 GM C˜ [C1
C2
...
CM ]
Consider the family of discrete-time systems Eq. (18) having at time t/kt the input uk /u (kt), state xk / x (kt ) and output yk as xk1 Fti xk Gti uk ; yk1 C0i xk D0i uk ;
u(t)Fx(kt);
(21)
one can define the fictitious measurement matrix Ci (F ; N)(C0i D0i F )(Fti Gti F )1 ;
(22)
which satisfies the fictitious measurement equation yk / Ci xk . For robust fast output sampling gain L to realize the effect of F , it may satisfy i 1; . . . ; M:
(23)
The Eq. (23) can be written as
For multimodel representation of a plant, it is necessary to design controller which will robustly stabilize the multimodel system. Multimodel representation of plants can arise in several ways. When a nonlinear system has to be stabilized at different operating points, linear models are sought to be obtained at those operating points. Even for parametric uncertain linear systems, different linear models can be obtained for extreme points of the parameters. The models are used for stabilization of the uncertain system. Now consider a family of plant S /{Ai ,Bi ,Ci }, defined by i 1; . . . ; M:
2 3 0 6Ci Gi 7 7: D0 6 4n 5 PN2 j Ci j0 Fi Gi
Assume that (Fti ,Gi ) are controllable. Then we can find a robust state feedback gains F such that (Fti / Gi F ) has no eigenvalues at the origin. Then, assuming that in intervals kt 0/t B/kt/t
LCi F :
5. Multimodel synthesis
y Ci x
where 2 3 Ci 6Ci Fi 7 7; C0i 6 4n 5 Ci FN1 i
251
(20)
LC˜ F˜;
(24)
where C˜ [C1 F˜ [F
C2 F
CM ]; ...
F ]:
˜ G; ˜ C): ˜ It Let n˜ denote the observability index of (F; ˜ ˜ generically C has full can be shown that for N E/n; column rank, so that robust state feedback gain can be realized by a fast output sampling gain L . When this idea is realized in practice, i.e., fast output sampling gain L have been obtained by realising the state feedback gain F , two problems are required to be addressed.The first one is apparent from Eq. (15). With this type of controller, the unknown states are estimated implicitly, using the measured output samples and assuming that initial control is generated by state feedback. If initial state causes an estimation error, then decay of this error will be determined by the eigenvalues of the matrix (LD0i /F Gti ) which depends on L and whose dimension equals the number of control input. For stability these eigenvalues have to be inside the unit disc, and for fast decay they should be as close to the origin as possible. This problem must be taken into account while designing L . The second problem is that the gain matrix L may have elements with large magnitude. Because these values are only weights in linear combination of output samples, large magnitudes do not necessarily imply large control signal, and in theory and noise free simulation they pose no problem. But in practice they amplify measurement noise, and it is desirable to keep these values low. This objective can be expressed by an upper bound r on the norm of the gain matrix L .
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When trying to deal with these problem, it is better not to insist on an exact solution to the design Eq. (23): one can allow a small deviation and use an approximation LCi :/F , which hardly affects the desired closedloop dynamics, but may have considerable effect on the two problems described above. Instead of looking for an exact solution to the equalities, the following ineqaulities are solved kLkBr1 ; kLD0i F Gti kBr2i ; kLCi F kBr3i :
i 1; . . . ; M; (25)
Here three objectives have been expressed by upper bounds on matrix norms, and each should be as small as possible. The r1 small means low noise sensitivity, r2 small means fast decay of estimation error, most important */r3-small means that fast output sampling controller with gain L is a good approximation of the originally designed state feedback controller. If r3 /0 then L is exact solution. Using the Schur complement, it is straight forward to bring these conditions in the form of LMI.
2 r1 I L B0; LT I
2 (LD0i F Gti ) r2i I B 0; (LD0i F Gti )T I
2 (LCi F ) r3i I B0: (26) (LCi F )T I In this form, the function mincx() of the LMI control tool box for MATLAB can be used immediately to minimize a linear combination of r1, r2, r3. The following approach turned out to be useful. If the actual measurement noise is known, the magnitude of L is fixed accordingly. Likewise eigenvalues of (LD0i /F Gti ) less than 0.05 cause no problem. So we can fix r1, r2 and only r3 is minimized subject to these constraints. This is illustrated in the following section. In this form, the LMI tool box of MATLAB can be used for synthesis [19]. The fast output sampling feedback controller obtained by the above method requires only constant gains and hence is easier to implement. The example of single machine system dynamics with infinite bus is used to demonstrate the method.
6. Case study The non-linear differential equations governing the behavior of a power system can be linearized about a particular operating point to obtain a linear model which represents the small signal oscillatory response of a power system. Variations in the operating condition of
the system result in the variations in the parameters of the small signal model. A given range of variations in the operating conditions of a particular system thus generates a set of linear models each corresponding to one particular operating condition. Since, at any given instant, the actual plant could correspond to any model in this set, a robust controller would have to import adequate damping to each one of this entire set of linear models. The 16 plants of single machine system connected to an infinite bus are considered for designing robust fast output sampling feedback controller using LMI approach of MATLAB software. Then controller gains are applied to simulate a non-linear plant of single machine system connected to an infinite bus at different operating points. 6.1. Example The discrete models of different plants with different power (Pg 0) and different external line inductance(xe) are obtained by sampling time t/0.5 s and is given in Appendix A. The output matrix is given as C [0
1
0 0]:
(27)
Using the method discussed in Section 5 common stabilizing output injection gain matrix F is obtained for all 16 plants as given below. FT [0:4101
19:3152 0:4879 0:0016]:
(28)
Using LMI approach, Eqs. (25) and (26) are solved using different values of r1, r2, and r3 to find the gain matrix L . The robust fast output sampling feedback gain L is obtained as L [L1
L2 ];
(29)
where L1 [18:7233 14:0543 L2 [48:7778 34:0972
7:6172 31:0312
46:6674];
0:9996 51:0546 121:8813]:
The eigen values of (Fti /Gti LCi ) are found to be within the unit circle and no eigenvalues at the origin. 6.2. Simulation with non-linear plant at different operating points A SIMULINK based block diagram including all the non-linear blocks is generated [16]. The slip of the machine is taken as output and the intial value of slip is taken as zero. The output slip signal with gain L and a limiter is added to Vref signal which is used to provide additional damping. This is used to damp out the small signal disturbances via modulating the generator excitation. The output must be limited to prevent the PSS acting to counter action of AVR. Different operating
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253
Fig. 3. Open- and closed-loop responses of non-linear system for various operating conditions.
points are taken as the different plants. The disturbances considered is self clearing fault at generator terminal cleared after 0.1 s. The limits of PSS output are taken as 9/0.25 and exciter output limits (Efd ) are taken 9/6 pu. The conventional PSS is designed as compensator with washout circuit [16]. The transfer function of PSS
compensator is selected as PSS(s)
16(1 0:08s) (1 0:027s)
and washout circuit constant is selected as 2.0 s.
Fig. 4. Open- and closed-loop responses of non-linear system for various operating conditions.
(30)
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Fig. 5. Open- and closed-loop responses of non-linear system for various operating conditions.
Simulation results at different operating points (Pg 0) and different external line inductance (xe) are shown in Figs. 3 /10 without controller and with the proposed controller. The control input as modulating voltage is also shown in these figures. The responses are compared with the conventional PSS. The simulataion results show that conventional PSS is able to damp out the oscillations upto active power, Pg 0 /1.5 with external line inductance, xe /0.2 and upto active power, Pg 0 /1.0
with external line inductance, xe /0.4, where as the proposed fast output sampling feedback controller is more robust and a single controller is able to damp out the oscillations for all cases. As shown in plots, the proposed controller is able to damp out the oscillations in 2/3 s after clearing the fault upto active power, Pg 0 /1.0 with external line inductance from xe /0.2 to 0.8 and 5 /15 s upto active power, Pg 0 /2.0 with external line inductance, xe /0.2.
Fig. 6. Open- and closed-loop responses of non-linear system for various operating conditions.
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255
Fig. 7. Open- and closed-loop responses of non-linear system for various operating conditions.
7. Conclusion In this paper, a design scheme of robust PSS for single machine connected to an infinite bus using fast output sampling feedback has been developed. The slip signal is
taken as output and the fast output sampling feedback is applied at appropriate sampling rate. The method is more general than static output feedback and the control input for these plants are required of small magnitudes. It is found that designed robust controller
Fig. 8. Open- and closed-loop responses of non-linear system for various operating conditions.
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Fig. 9. Open- and closed-loop responses of non-linear system for various operating conditions.
provides good damping enhanchement for various operating points of single machine system connected to an infinite bus. This method is also compared with conventional PSS. The proposed method results better response behaviour to damp out the oscillations. The conventional PSS is dynamic in nature and required to
tune according to power characteristics. Where as, the proposed controller gains are static in nature and one controller structure is able to damp out the oscillations for all plants. The method described has been extended to the multi-machine system and will be presented else where.
Fig. 10. Open- and closed-loop responses of non-linear system for various operating conditions.
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Appendix A: Plant parameters The following parameters are used for simulation of 16 plants H/5, Tdo /6 s, D /0.0, KE /100, TE /0.02 s, xe /0.2 pu
Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant Plant
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Pg 0
Qg 0
K1
K2
K3
K4
K5
K6
xe
0.4 0.5 0.6 0.75 1.0 1.2 1.4 1.5 1.6 1.7 1.8 2.0 1.0 1.0 1.0 1.0
0.0160 0.0251 0.0361 0.0566 0.1010 0.1461 0.2000 0.2303 0.2629 0.2979 0.3352 0.4174 0.1535 0.2087 0.3333 0.5000
1.0170 1.1624 1.4336 1.1624 1.5772 1.6342 1.6263 1.6117 1.6263 1.6117 1.5911 1.4977 1.3197 1.3288 0.7430 0.3903
1.1239 1.2983 1.5855 1.2983 1.7402 1.8588 1.8752 1.8883 1.8752 1.8883 1.8987 1.9174 1.5042 1.7402 1.0776 0.8904
0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.2889 0.3263 0.3600 0.4180 0.4667
1.4385 1.6618 2.0294 1.6618 2.2274 2.3792 2.4003 2.4170 2.4003 2.4170 2.4303 2.4543 1.9254 1.7009 1.3794 1.1397
0.0643 0.0577 0.0194 0.0577 /0.0309 /0.1130 /0.1328 /0.1524 /0.1328 /0.1524 /0.1717 /0.1878 /0.0641 /0.1002 /0.1739 /0.2468
0.3291 0.3083 0.2628 0.3308 0.2300 0.1988 0.1939 0.1898 0.1939 0.1898 0.1866 0.1805 0.3020 0.3608 0.4561 0.5380
xe xe xe xe xe xe xe xe xe xe xe xe 1.5xe 2xe 3xe 4xe
References [1] E.V. Larsen, D.A. Swann, Applying power system stabilizers Part-I: general concepts, IEEE Trans. Power Apparatus Syst. PAS-0100 (6) (1981) 3017 /30242. [2] E.V. Larsen, D.A. Swann, Applying power system stabilizers Part-II: performance objective and tuning concepts, IEEE Trans. Power Apparatus Syst. PAS-100 (6) (1981) 3025 /3033. [3] F.R. Schlief, J.H. White, Damping for the North-west /southwest Tie line oscillations */a analog study, IEEE Trans. Power Apparatus Syst. PAS-85 (1966) 1239 /1246. [4] F.R. Schlief, R.R. Angell, Damping of system oscillations with a hydrogenerating unit, IEEE Trans. Power Apparatus Syst. PAS86 (1967) 438 /442. [5] C. Concordia, F.P. de Mello, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Apparatus Syst. PAS-88 (1969) 316 /329.
257
[6] F.R. Schlief, H.D. Hunkins, E.E. Hattan, W.B. Gish, Control of rotating exciters for power system damping: pilot applications and experience, IEEE Trans. Power Apparatus Syst. PAS-88 (1969) 1259 /1266. [7] E.J. Warchol, F.R. Schlief, W.B. Gish, J.R. Church, Alignment and modeling of hanford excitation control for system damping, IEEE Trans. Power Apparatus Syst. PAS-90 (1971) 714 /724. [8] A.D. Gerhart, T. Hillesland, Jr., J.F. Luini, M.L. Rockfield, Power system stabilizer field testing and digital simulation, IEEE Trans. Power Apparatus Syst. PAS-90 (1971) 2095 /2100. [9] K.E. Bollinger, A. Laha, R. Hamilton, T. Harras, Power system stabilizer design using root locus methods, IEEE Trans. Power Apparatus Syst. PAS-94 (1975) 1484 /1488. [10] P. Kundur, D.C. Lee, H.M. Zein El-din, Power system stabilizers for thermal units: analytical techniques and on-site validation, Paper F80-227-9 presented IEEE PES Winter Meeting, New York, February (1980). [11] F.A. Saleh, M.S. Mahmoud, Design of power system stabilizer using reduced-order methods, Electric Power Syst. Res. 33 (1995) 219 /226. [12] S. Chen, O.P. Malik, H based power system stabilizer design, IEE Proc. Part C 142 (2) (1995) 179 /184. [13] P. Shrikant Rao, I. Sen, Robust pole placememt stabilizer design using linear matrix inequalities, IEEE Trans. Power Syst. 15 (1) (2000) 3035 /3046. [14] H. Werner, K. Furuta, Simultaneous stabilisation based on output measurement, Kybernetika 31 (1995) 395 /411. [15] E.V. Larsen, D.A. Swann, Applying power system stabilizers Part-III: practical consideration, IEEE Trans. Power Apparatus Syst. PAS-100 (6) (1981) 3035 /3046. [16] K.R. Padiyar, Power System Dynamics Stability and Control, Interline publishing private Ltd, Bangalore, 1996. [17] W.G. Heffron, R.A. Phillips, Effect of a modern amplidyne voltage regulator on under excited operation of large turbine generators, AIEE Trans. 71 (Part 3) (1952) 692 /697. [18] F.D. DeMello, C. Concordia, Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Apparatus Syst. PAS-88 (1969) 316 /329. [19] P. Gahenet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Tool Box with for MATLAB, The Math works Inc, Natick MA, 1995.