Development of the Discrete-Time Adaptive Sliding Mode Power System Stabilizer

Copyright © IFAC Control of Power Plants and Power Systems, Cancun, Mexico, 1995

Development of the Discrete-time Adaptive Sliding Mode Power System Stabilizer Young-Moon Park and Wook Kim* * Department of Electrical Engineering, Seoul National University, Shinlim-dong, Gwanak-ku, Seoul 151-742, Korea, E-mail: s_wookCcd4680.snu.ac.kr

:Abstract .. A n~wly developed discrete-time adaptive sliding mode power system stabilizer( PSS ) IS proposed In thIs paper. Because the proposed PSS is developed in the pure discrete-time domain, it is able to maintain the stability with the rather slower sampling frequency compared to discretized conventional continuous-time sliding mode PSS. The proposed PSS has the three main superiorities to the conventional PSS. First, because the proposed PSS utilizes the sliding mode control method, it has the strong robustness to the variations of the system parameters. Hence, it can overcome the minor disturbances such as identification errors, modeling errors, etc. Second, the proposed PSS needs only input/output measurements as feedback signals . Hence, it does not need the measurements of the state variables as the conventional sliding mode PSS or PSS based on the linear quadratic control method does. Finally, because the proposed PSS have the adaptivity property, it is able to overcome the various power system faults, such as line-to-line faults, line-to-ground faults, etc. To verify the performance of the developed PSS simulations with the sample power system with single-machine and infinity-bus have been performed. Key Words. Power system stabilizer; discrete-time sliding mode control; parameter identification; adaptive control

gineering problems, after the invariance property of the sliding mode method having been reported(Drazenovic, 1969}.

1. INTRODUCTION Stabilization of power system is one of the most typical control problems which need the strong robustness of the controllers. There exist several kind of disturbances which hinder the power system to be stabilized, for example, the mathematical model errors, interactions between the generators, variations of the parameters of the generators and tie-lines, etc. The performance and even the stability of the power system may be influenced by these disturbances including the measurement noise if the stabilizer do not have any corresponding facilities. Hence, the robustness of the stabilizer is strongly needed to maintain or improve the dynamical stability of the power system in the presence of the various disturbances and measurement noises. During the last several decades this robustness property of the controller has been extensively studied. And as the results, various robust control algorithms are introduced and reported to be applied successfully. HOC, LQG/LTR, variable structure control, etc., are the most significant examples.

The most attractive feature of sliding mode control is that the control system which is invariant to the parametric uncertainties and external disturbances can be achieved only with very simple structure ofthe switched feedback gains. And this simple structure makes VSC possible to be hybridized with the other useful control algorithms such as model following, model reaching, adaptive and optimal control, and state observation, etc. Such hybrid approaches can make the control algorithm to have significantly new desirable characteristics. Meanwhile there exist a lot of research results which apply the sliding mode control to the power system stabilization problem. See the references (Bengiamin and Chan, 1982; Chan and Hsu, 1983; Fleming and Sun, 1992; Hsu and Chan, 1983; Matthews et al., 1986; Sivaramakrishnan et al., 1984) . Almost of all these results are based on the conventional continuous-time sliding mode control theory and the simulation results given in the papers are rather satisfactory.

Although the sliding mode control was first introduced in the 1950's in the Soviet Union by Emelyanov and several coresearchers, It did not attract much attentions at first. But, today, researchs and developments continue to apply sliding mode control to a wide variety of en-

However, when the continuous-time sliding mode control is implemented with the digital computers, there is two respects which cannot be disre-

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garded. First, sliding mode control systems can only undergo quasi-sliding modes(Milosavljevic, 1985), i.e., the state of the system can approach the switching surface but cannot stay on it, in general. This is due to the fact that control actions can only be activated at sampling instants and the control effort is constant over each sampling period. Second, when the state does reach the switching surface, the subsequent discrete-time switching cannot generate the equivalent control to keep the state on the surface. Hence, just simple discretization can make the performance of the closed-loop system rather deteriorative or even unstable.

-

V...J +

>--

AVR& Exciter

~eFD

~v

Sync. Machine ~co

u

PSS

y

Fig. 1. Schematic diagram of the power generation system

and even some of these state variables can not be defined if the structure of the power generation system varies significantly. So we do not use such internal state signals for designing PSS, instead , we design PSS using only the sampled supplimentary excitation input signals u and the sampled angular velocity deviation output signals ~w .

So the power system stabilizer based on sliding mode control theory in the pure discrete-time domain is considered in this paper. The sliding sector which is the corresponding concept of the sliding surface in the continuous-time sliding mode control theory is adopted and the pure discretetime control law to make the system states to remain in this sliding sector is proposed.

With the measured sequences of the inputs and outputs of the power system to be controlled, the power system can be represented as the following ARMA model.

The main superiority of the proposed PSS is its adaptation ability. The proposed PSS gathers the input and output data from the controlled power plant and makes the mathematical model from the data. Hence, the PSS does not need to measure all the state variables, one of the most significant shortcomings of the conventional sliding mode PSS. Furthermore, because it adjusts the parameters of the model whenever new data are obtained , the proposed PSS can cope with the abrupt changes of the system parameters caused by the power system faults, such as line-to-line faults, line-to-ground faults, and changes of the machine constants, etc.

y(k + 1) = aly(k) + ... + any(k - n + 1) +b 1 u(k) + ... + bnu(k - n + 1)

(1)

where u(k) and y(k) are assumed to be the supplimentary excitation control signals and the frequency deviation output signals D-w, respectively. The famouse recursive least square identification method with the forgetting factor (Astrom and Wittenmark, 1989) is used to identify the parameters of the above equation, i.e ., al, ... , an, and bl ,· · . , bn .

In section 2, we formulate the adaptive power system stabilization problem in discrete-time domain . In section 4 and 5, the structure of adaptive variable structure power system stabilizer is proposed. And section 6 will show the simulation results to verify the performance and robustness of the proposed stabilizer.

From the parameters in Eq. (1) the following statespace forms can be achieved.

x(k + 1) = Ax(k) y(k) = Cx(k)

+ Bu(k)

(2)

where x(k) is n x 1 vector, and u(k) and y(k) are scalar. A,B and C are given as the following controllable canonical form

2. PROBLEM FORMULATION The overall schematic diagram of the power generation system is presented in Fig. 1.

o o

1 0

o o

o

0

1

A=

The objective of PSS described in this paper is to give an auxiliary control input of the exciter to enhance the stability of the controlled power system . In the figure, D-w, D-PG, D-XG, D-O, etc., can be considered as the internal state variables. The conventinal sliding mode PSS uses these state variables as the feedback signals, but sometimes this state vectors can not be measured exactly,

(3)

Instead of obtaining the above controllable canonical form directly from Eq. (1) , the intermediate step is necessary for the ease of estimating state estimations. The detailed procedures to get

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Eq.(3) is explained in section 3.

4. DISCRETE-TIME SLIDING MODE CONTROL

Because all the variables of the power system described in Fig. (1) are usually defined as the deviations from the norminal values, the objective of the power system stabilizer is to regulate the output variables to zeros. Hence the control sequence u(k) should stabilize the output of the system y(k) in (1) to zeros as time goes infinity. This is equivalent to stabilize all the states defined in (2) to zeros, because the output is the linear combinations of the defined state variables.

In designing the conventional continuous-time sliding mode controller, a sliding surface are defined as

S = {x Is = O}. but, for discrete-time systems , the sliding mode cannot occur exactly on this sliding surface because of the finite switching rate . So we adopt the sliding sector instead of the sliding surface as follows(Pan and Furuta, 1994) : ~

3. STATE SPACE REALIZATIONS

~1

s(k)

c1xdk) Cx(k)

. [

(4)

o

C=[10

~

c5(x(k))}

(5)

where c5(x(k)) is a function of the defined state x (k) . s( k) is the linear function of the state variables defined as

To obtain Eq.(3) the state space realization of the following observable canonical form is considered .

A=

= {x(k) Ils(k)1

+ C2X2(k) + ... + cnxn(k) (6)

=

=

where C [Cl ... cn ] and we assume Cn 1 for simplicity without loss of generality. Here the following discrete-time variable structure control law is proposed .

0]

The advantage of the above form is that the state variables can be defined as the combinations of the history of the input and output signals as follows ,

u(k) =

_(CB)-l{CAx(k) - (Js(k) -Fx(k)}

y(k) y(k - 1) + (J2u(k - 1)

(7)

The control schemes described in this paper are originated from those of Pan and Furuta's paper (Pan and Furuta, 1994) and we slightly modify the control parameters to enhance the robustness of the controller. In equation (7), the elements of matrix F = [flh .. ·Inl are defined as

y(k - n + 1) + (J2u(k - n + 1) + ··· +{Jn u(k-1) Comparing the last row in Eq.(3) and Eq.(l) the relationships between a1 ,"' , an and {J1 ,"', (In in Eq. (1) and the parameters a1 ," ', an and b1 , .. . , bn can be obtained as follows,

Is(k)1 ~ c5k(X(k)) -1{Jlcma:t'sign({Js(k)x;(k)) ,

_ { 0,

I; -

Is(k)1 > c5k (x(k)) where c5k (x(k)) is defined as n

and

c5k (x(k))

= Cma:t' L

IXj(k)1

j=l

where Cma:t'

= max {Icjl} 1$3$n

and {J is a constant such that

I{JI < 1,{J # 0 It is easy to transform Eq.(4) into Eq.(3) (Astrom and Wittenmark, 1990) . Although Eq.(3) can be obtained directly from the parameters of Eq .(l) , the intermediate step of Eq.(4) provides the easy ways to estimate the state variables.

The proof of the stability of the closed-loop system can found in (Pan and Furuta, 1994) with the slight modifications. As a result, the control law 7 guarantees the following condition of the existence of the sliding sector(Sarpturk et al., 1987).

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and the gain matrix K is given by

Is(k

+ 1)1 < Is(k)1

(8)

(13) where P is a symmetric positive definite matrix that satisfies the following discrete-time algebraic matrix Riccati equation :

5. DESIGN OF THE OPTIMAL SLIDING SECTOR

P=Q+ATpA-ATpBx (I+B T pB)-lB T pA (14)

The inequality 8 means that once the system states enter the sliding sector, they will remain in the sector . Hence the dynamics of the sliding sector plays the most important role in determining the dynamics of the system when the system is in the sliding sector. And because the dynamics of the sliding sector is determined by the coefficient matrices of Eq. (6), determining the dynamics of the sliding sector is equivalent to design the coefficient matrices of the sliding sector.

The control equation (12) is substituted into the system (2) which yields

x(k + 1) = Aax(k)

(15)

where

Aa = A - BK or equivalently,

Inside the sliding sector, i.e.,

Is(k)1 ::; c5(x(k)) the closed-loop system has the form,

x(k + 1) = Acx(k)

Hence, the corresponding characteristic equation

(9)

IS

qn + (an + kn)qn-l + ... +(a2 + k2)q + (al + k l ) = 0

where

Ac = A - B(CB)-l(CA - (3C)

(16)

equivalently, equivalently, 1 where Zl is one of the real zeros of the characteristic equation (16) . Comparing (10) and (17), the following results is concluded, So the consequent characteristic equation of (9) is given by

qn

+ (Cn-l

- {3c n )qn-l + ... +(Cl - {3C2)q - {3cl = 0

(3

Zl

Ci

hi,

i=1,2,·· · ,n-1

Hence, we can calculate the coefficient matrices of the optimal sliding sector.

which yields

(q - (3)(qn-l

+ cn_lqn-2 + ...

6. SIMULATIONS

(10)

+C2q + Cl) = 0

We demonstate the results of the simulations with the single-machine and infinite-bus sample power system. The important parameters of the power system are shown in Fig . 2. The constants in the synchronous machine are given in the following tables . The above sample system is a linearized model with unstable poles and zeros. The simulations are done with the assumption that the initial frequency deviation, f).c5 is perturbed with the small amounts, i.e., 0.1 p.u. The conventional lead/lag compensator PSS(Yu, 1983) is used for the comparison . The structure of the conventional

To this equation in mind, the following quadratic performance index is chosen, 00

J =

L

{x(kf Qx(k) + U(k)2}

(11)

k=O

From the theory of the optimal control method, the optimal solution which minimizes (11) for the system (2) has the form,

u(k)

= -Kx(k)

(12)

46

~

~...__1+_ST_ST_...H. ._K_Cl_~_:_~_2T._l) ~

Vo Z = -0.034 + jO.077

....

Fig. 3. Structure of the conventional PSS

Table 3

Classical PSS Data

Y = 0.249 + jO.262 3.0

0.685

0.1

7.09

Fig. 2. Single-machine and infinite-bus power system

8. REFERENCES Table 1 Generator Data M 9.26

7.76

o

Astrom, K.J. and B. Wittenmark (1989) . Adaptive Control. Addison-Wesley. NY. Astrom , K.J . and B. Wittenmark (1990). Computer Controlled Systems, theory and design. second ed .. Prentice-Hall, Inc .. NJ . Bengiamin, N.N. and W .C. Chan (1982) . Variable structure control of electric power generation . IEEE Transactions on Power Apparatus and Systems. Chan, W.C. and Y.Y. Hsu (1983) . Optimal control of electric power generation using variable structure controllers. Electric Power Systems Research. Drazenovic, B. (1969) . The invariance conditions in variable structure systems. A utomatica. Fleming, R.J . and J . Sun (1992) . An optimal variable structure stabilizer for a synchronous generator. International Journal of Power and Energy Systems. Hsu, Y.Y. and W .C. Chan (1983). Stabilization of power systems using a variable structure stabilizer. Electric Power Systems Research. Matthews, G .P., R .A. DeCarlo, P. Hawley and S. Lefebvre (1986) . Toward a feasible variable structure control design for a synchronous machine connected to an infinite bus. IEEE Transactions on A utomatic Control. Milosavljevic, D . (1985) . General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable struc-

XI

D

d

0.973

0.19

0.55

PSS is shown in Fig. 3 and the parameters are given in Table 3. The results of the computer simulations are shown in Fig. 4 and Fig . 5. Fig . 4 shows the resulting frequency deviations from the nominal values by the conventional PSS by Yu(Yu, 1983) and by the proposed discrete-time variable structure adaptive PSS . Here the sampling time is assumed to be 0.05[sec] for both cases. Fig. 5 shows the graph for the control amounts to acquire the result for Fig 4. The comparisons of both the conventional and the proposed PSS show that the performance of the proposed PSS is very desirable.

7. CONCLUSIONS To date we can find the various research results which applied the variable structure control to the stabilization of the power system. But , because almost all the results were based on the control laws based on the continuous-time domain, it is very hard to implement the research results with the digital computer. In this paper of designing the power system stabilizer, we developed the new variable structure control law derived in the discrete-time domain . In addition to this, by adopting the adaptive mechanism using the recursive parameter identification method, we could develop the power system stabilizer which is considerably robust to the variations of the system parameters and , furthermore, various power system faults . The simulation results to the one-machine and infinite-bus sample power system showed the preferable performance of the proposed power system stabilizer . Table 2

.a

S. ~

"> co

o

."

0. 0. .0.

............... ~

l;e:

er ~

u..

Exciter Data 2

50

0 . 05

3 Time [sec]

4

Fig. 4. Simulation Result ( Frequency Deviation )

47

6

u~--,,-.------------------------~

-0.01

-0.02 .

2

3 TIme [sec]

4

5

6

Fig. 5. Simulation Result ( Control Amount)

ture systems. Automations and Remote Control. Pan, Y. and K. Furuta (1994). Vss controller design for discrete-time systems. Control- Theory and Advanced Technology, Part 1. Sarpturk, S.Z., Y. Istefanopulos and O. Kaynak (1987) . On the stability of discrete-time sliding mode control systems. IEEE Transactions on A utomatic Control. Sivaramakrishnan, A.Y., M.V. Hariharan and M.C . Srisailam (1984). Design ofvariable structure load frequency controller using pole assignment technique . International Journal of Control. Yu, Y.N . (1983). Electric Power System Dynamics. Academic Press. NY.

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