Hamiltonian modelling and nonlinear disturbance attenuation control of TCSC for im roving power system stability

Y.Z. Sun, Q.J. Liu, Y.H. Song and T.L. Shen

Abstract: To tackle the obstacle of applying passivity-based control (PBC) into power systems, an affine non-linear system widely existing in power systems is formulated as a standard Hamiltonian system using a pre-feedback method. The port controlled Hamiltonian with dissipation (PCHD) model of TCSC is then established corresponding with a revised Hamiltonian function. Furthermore, employing the modified Hamiltonian function directly as the storage function, a non-linear adaptive L2 gain control method is proposed to solve the problem of L2 gain disturbance attenuation for this Hamiltonian system with parametric perturbations. Finally, simulation results are presented to verify the validity of the proposed controller.

1 Introduction

The restructuring of the electricity industry has increased the need for optimum system use. Increasing attention has been devoted to enhancing the stability of powcr systems, which are chardcterised by high non-linearity and influ- enced by external disturbances. Many non-linear control methods have been introduced into power systems. They include [ I ] : (i) feedback linearisation, such as the differ- ential geometry approach and direct feedback methods based on exact models with no uncertainties, and (ii) robust control based on robust models including non- linear H , control and direct feedback linearisation compensating control, etc. However, the former method cannot guarantee its final strict optimality and robustness to the plant system, while thc latter relies on solving the Hamilton-Jacobi-Issacs inequality, whose general analytic solution is still unobtainable [2].

A powerful design technique for the stabilisation of non- linear systems is PBC [3-61. The excellent features of PBC make it an excellent method for designing robust controllers for physical systems described by Euler-Lagrange equa- tions of motion, which includes mechanical, electrical and electromechanical applications [6]. Two stages are required in the PBC design process [7]: an energy shaping stage to let the new potential energy function have a strict local mini- mum in the desired equilibrium, and a damping injection stage to ensure asymptotic stability.

Recently, the PBC design methodology has been extended to a broader class of systems described by PCHD models, since they can capture directly the physical

Q IEE, 2002 IEE Pmceedings online no. 20020399 DOI: 10. 1049lip-cta:20020399 Paper firs1 received 6th July 2001 and in revised Form 8th March 2002 Y.Z. Sun and Q.J. Liu are with Tsinghua University, Beijing 100084. China Y.H. Song is with Brunel University, London, UK T.L. She" i s with Sophia University, Tokyo, Japan

constraints of the system and reveal the structural obstacles for energy shaping and damping injection [S-lo]. Indeed, the Hamiltonian function in PCHD systems is the total energy of the plant device, and can play the role of the Lyapunov function, making it promising in power systems where energy generation, transmission and distribution play the main role. The authors of [9-11] have proved that the excitation control problem can he formulated as a PCHD system with successhl application.

In this paper we consider the practical application of a TCSC using an energy-based control method; a pre- feedback method is also proposed for the transformation of FACTS devices such as a thyristor controlled serial compensator (TCSC) into a standard Hamiltonian struc- ture. Based on this, a non-linear disturbance attenuation control technique that does not require feedback linearisa- tion is designed to render the system the L2-gain from disturbance to a penalty signal less than a given level. Furthermore, its asymptotical stability can also be guaran- teed. On the other hand, control of uncertain non-linear dynamic systems using compensation of estimated perm- bations has recently been a topic of considerable interest in the control research area. In this passivity-based synthesis framework, an adaptivc mechanism is also adopted to compensate for perturbation of the parameters and render the closed-loop system passive. Compared with the recur- sive design procedure [ I ] , the Hamiltonian function denotes the total energy of the physical system and is adopted directly as a storage function in this proccss, which not only gives a high level of precision in mathe- matical form, hut also has explicit physical meaning.

2 PCHD modelling of TCSC using pre-feedback

Consider the PCHD system shown in Fig. 1 [lo]

x = [J(x) ~

278

resistive elements

environment energy storing power consewing

element interconnection

Fig. 1 Port-contrulled Hamiltonian .qmem with di.wipatiun

where x E M (M is a manifold), and u’, y E R”’ represent the input and output, respectively; J(x) is a skew-symmetric matrix, i.e. J(x) = -J‘(x), which defines a generalised Poisson bracket on M. R(x) is a non-negative symmetric matrix and defines a symmetric bracket on the state manifold M. In general, the Hamiltonian function H(x) represents the total stored energy. If the Lyapunov stability criterion is satisfied then the function can act as a Lyapunov function, which is one of the characteristics of (1). From Fig. 1 , we can see that the PCHD model makes a clear distinction between the internal interconnection struc- ture and the interconnection with the environment, i.e. the control action. Hence the derivative of the Hamiltonian function will include two items: the first term represents the cnergy-dissipation due to the resistive elements in the system, while the second term is the power externally supplied to the system [IO].

The Hamiltonian system is intrinsically a class of non- linear systems with a well-defined structural form and a prerequisite Hamiltonian function. In fact, there is no general method to construct such a Hamiltonian function, which creates more difficulties for this type of transform. It has been shown in [3-5] that an affine non-linear system under certain matching conditions can be transformed into a passive system using a feedback equivalence method, and more techniques are designed to find the proper Hamiltonian function. However, those general mathemati- cally driven approaches neglect the physical structural characteristics of practical systems and can barely cope with non-linearity and uncertainty in practical engineering problems [6]. In this paper, we will show that these problems can he well solved in power systems using pre- feedback techniques based on definite physical implica- tion, especially for plant devices in a single machine system, such as excitation, steam valve, FACTS, etc. As an example of FACTS, consider the following TCSC device.

If we view the TCSC as a controllahle model with respect to the impedance and consider the system shown in Fig. 2, if the dynamic behaviour of TCSC can be described as a first-order differential equation, then we can formulate the whole system dynamics as follows [ 131:

s = wow,

T I

T<k

1 W, = - (f, - Dw, - E’Usydc sin 6)

Vd, = - ( :Vdc + Y,,, + UdC) (2) 1

where 6 is the rotor angle in rad/s, u Y = w - I is the deviation of the angular velocity of the rotor in units, wa=2rlfo; U, denotes the voltage of the infinite-bus in

Fig. 2 wilh TCSC

Eqquivulent circuit of single-machin~-infinit~-h~,s system

units; P, is the constant mechanical input power in units; the parameters T, and D are the inertia constant in seconds and the damping coefficient in units, respectively; T,, is the time constant in seconds; x$ is the direct axis transient reactance of the generator, .rT being the reactance of the transformer, xL the transmission line reactance, yd, = I/(& - x,,,) is the total inductance of the system while ydcs denotes the unit value at the stable operation point, and x&=.rj+xT+.rL, udc is the control variable in units.

Since the power system shown in Fig. 2 does not satisfy the structural form of a Hamiltonian system with a classical single-machine energy function, an effective way is to transform the original plant device into a system with a basic Hamiltonian structure by introducing suitable pre-feedback. Drawing inspiration from the theoretical framework and practical application, a reactive power feedback is adopted on the control input udc for the TCSC system shown in Fig. 3 as follows:

where Qe and Qe.T denote the reactive power at the opera- tion and stable equilibrium point, respectively; KQ is its control gain, while the practical measurable output z relates to the Hamiltonian form and will be defined later.

Decomposition on the reactive power expression gives

Qe - Q, = (E’U,Y,, cos 6 - Urzydc)

= E’U,vydc(cos6 - cos 6J - (E‘U,,JJ~, cos 6, - U?ydC)

(4)

We can then formulate the following closed loop system with TCSC

into a standard Hamiltonian system as follows:

Fig. 3 Hamiltonian system modelled hy pre-feedback cvnhol

279

where

and

aH -(x) = ax

~

~

g(x) = [ o 0 I ] '

aH 1

ay,,l E'Uydc sin S - P,, 1

At the same time we can also obtain a concise Hamiltonian function

1 2

H ( I ) = -T,cD& - [P,n(d ~ 8,) + E'li,y,

1 x (cos 6 - cos 6,)] + - (ydi ~ y;-,)2 (7)

2% where the Hamiltonian denotes the overall physical energy of the plant device, which comprises the classical single- machine energy function and revised TCSC item, and the proposed pre-feedback also has an explicit physical expla- nation, which mcans that we can obtain a full imagc for the resistive element and energy exchangeable channel of TCSC, which will simplify our final control strategies. In general, S E (0, x).

It can he seen from this deduction process that this pre- feedback method is different from those set-point regula- tions shown in [S-II], because the physical energy of TCSC is included in the classical single-machine energy function to form the final Hamiltonian function. This energy shaping process formulates the closed-loop system with the structural form of a controlled Hamilto- nian system, while the revised Hamiltonian function can he proven to have a minimum at the desired operating point. The same technique can he easily extended into PCHD modelling for other FACTS devices and steam valve control.

3 Disturbance attenuation control for PCHD

Consider a PCHD system with input disturbances

aH i = [J(x) - Wx)l ax (x) + &)(u' t W ) (8)

where w t R m is a strict bounded unknown disturbance. Since PCHD systems define passive operators with a

280

storage function, the total energy function H(x), i t can he easily seen that the unforced Hamiltonian system is stable at the equilibrium point xn when w is set to zero.

The practical output function z must he selected from measurable variables of the plant device. In this Section, we therefore discuss the problem of non-linear L, gain disturbance attenuation for a PCHD system, which can he described as follows: given a positive disturbance attenua- tion level y , a penalty signal z=y(x) and a desired equilibrium point xncR" , our aim is to find a feedback control law u'=p(x) and a positive storage function VIX) such that the dissipation inequality [3]:

(9) 1 + Q(x) 5 ~ [ ~ 2 1 1 ~ ~ 1 1 2 - l l ~ l l ~ } , VW

holds along all trajectories of the closed-loop systsrm comprising (8) and the feedback law, where Q(x) (Q(x)#O, Vx#xu) is a given positive semi-definite func- tion.

3.7 L, gain disturbance attenuation control In this Section, a general Lz gain disturbance attenuation control law will he designed by constructing a suitable storage function and damping injection for the PCHD system. Suppose that there exists pre-feedback u = a ( x ) + u' such that the closed-loop system has the Hamil- tonian stntcture form, and the Hamiltonian function has a strict minimum at the desired equilibrium point xn.

For a PCHD system with input disturbance, we dcfne the penalty signal as follows:

(10) 3H

2 = h(x)g7(x) (4

where h(x) is a weight matrix. By injecting proper damping into the Hamiltonian

system (S), i.e. employing the feedback control law u'=/?(x) in this case, the Lz gain disturbance attenuation problem can he easily solved. Along any trajectory of this system, a straightforward calculation using complete square gives

dH(x) -~ "H(x)R(X)aH(x) ~- - dt ax a K

let

then we have

Thus we obtain the following L2 gain disturbance attenua- tion control law:

.(x) = 4 4 + P(x)

Therefore, if we inject additional damping

R(x) + R(.v) g'(x) (15) I into the closed loop system, we can render the closed-loop system finite Lz gain. Another interesting property is that the Hamiltonian function serves directly as the storage function for the closed-loop system in the design procc- dure. For practical systcms the damping injection is limited by the physical size and capacity of the system, hence the disturbance attenuation level must be optimally selected according to the control performance.

3.2 Adaptive case In this Section we will consider the case where the Hamiltonian system involvcs parametric perturbations. We will show that if the perturbations satisfy a certain matching condition, an adaptation mechanism can be introduced into the feedback control law to estimate the controller parameters corresponding to the parameter perturbations.

Assume that parameter perturbations exist in the PCHD system, which can be expressed by the vector p whose nominal value is zero, i.e.

8H 1 . z [ J ( ~ . P ) - R ( ~ , P ) I ~ ~ ( x . P ) + ~ ( x ) ( u ' + , ~ ) (16)

and the perturbed parameters vectorp can be broken down as follows:

R(x,P) R(x) + A R ( ~ , P ) J 1 X . P ) = /(XI + d,(x,p) (17)

where A@, p ) satisfies A @ , O ) = O ( i=H, R, J), which denotes the broken down functions corresponding to the parametric uncertainties.

Proposition I : For system (16) , if there exists a function Y(x) E R Y x n ' to satisfy the following parameter evaluation condition for all x, i.e.

[ J ( ~ , P ) - R(x.p)ld~,(x,p) = g(x)pr(x)G (18)

Then the ?-dissipation problem of the pcrturbed closed- loop system can be solved by

where r=diag(p,p: . . . p y } , p i > 0 ( i = j , 2,. . . , y) is the parameter adaptation gain matrix, and 0 E R" is the esti- mate of the unknown parameter vector 0. Proof.; Define the following storage function

1 2

v ( ~ , i) = N ( ~ ) + - ( e - 6)rr -1(e - 6) (20)

/.?E Pm.-Conrml Theon App., YVI. 149. No 4. Jd.v 2UU2

Considering ( I 1) and substituting (17), (IS), (19) into (16), we obtain the derivative of V(x) as follnws:

1 I '+Q(x) ~ ~ ( ~ ~ 1 1 ~ l l ~ - l l ~ l l ~ ] (21)

Thus an adaptive L2 gain disturbance attenuation control is obtained for the perturbed controlled Hamiltonian system, and only linear parametrisation is assumed in the process. It should be pointed out that 0 and Y(x) can be directly obtained from the concrete form of the unknown parameters in the controller, thus (18) can be finally held. Furthermore, the presence of parameter uncertainty will induce a shift in the achieved equilibrium point, which disappears as the estimate error converges to zero.

4 Application with TCSC

In the following Section, we will present the procedure for designing TCSC controllers using the aforementioned method.

4.1 TCSC controller design The characteristic of TCSC is usually defined by the overall reactance of the device versus the firing angle of the TCR that is connected in parallel with a fixed capacitor. It can changc its apparent reactance smoothly and rapidly, which meets the demands of modern power systems, which must operate flexibly and react quickly. Under suitable control strategies, TCSC is able to directly schedule power flow along desired paths and allow the system to operate closer to the line limits. Furthermore, it can also improve transient stability and attenuate sub-synchronous reso- nance (R) in power systems [ 1 3 ] . Most of the control methods employed in practice use a static TCSC model which cannot reflect its practical effect in power systems [14, 151.

In this Section, we will adopt the above third-order dynamic model, while an adaptive non-linear controller is proposed to analyse its control characteristics.

Step (i): Consider the penalty signal with the weighting function h ( x ) = [ q l x l y2x2 yzx3]: y i>O ( i = 1, 2, 3) and define the penalty function

z = h(x)g'(x)-(x) aH ax

= [ :11] 43x3

Step (ii): If we neglect the effect of parameter perturbation, we obtain the following TCSC control strategy from (14):

1

( K Q

x - (ydc -ydcs) -E'Us(cos6 - cusfi,)

Step (iii): Taking the unknown parameters into account, especially for those changes in the novel structure, the possible shift of the operation point stemming from the

281

fault occurring in the power system, and the parameter perturbation existing in the TCSC itself, we can select the perturbedparameterydc.s+yyd,+pI, cosfi,-tcos6,+~12, K Q i KQ+p3, where pi ( i = l , 2, 3 ) is the bounded parameter perturbation.

It is then easy to check that the structure matrices J(x,p), R(w, p ) can be broken down by

where

d,=[O 0 o;]

-51 ' 5 10 0

ttme, s

~1 1 0 5 10

time, s

b Fig. 4 Simulution results tinder dfferent y of TCSC a Rotor angle b Reactive power

282

If we define the parameter evaluation vector it= [i, &,Ir and let the matrix function YT(x) = [ I (y<,? - ydiJ], such that the following matching condition can be satisfied:

r -

( 2 5 )

where e = $ , 621'' is the evaluation to the parameter perturbation and can be presented by

lime. s

a

0 5 10

time. 5

b

Fig. 5 Paramrtev estimation under di0i.wnt y-dissipatioion Ia~els of TCSC

b 0, 0 8 1

IEE Pmc.-Coniml T h e q Appl.. Yol. 14Y. No. 4 , July ZOO2

we can obtain the final L2 gain adaptive disturbance attenuation controller of TCSC from (19)

I 1 1.Q x -(ydc -ydcs) - E'UJcos6 - cos6,)

-ydC.,) - E'(/,(cosb - cosd,)

4.2 Simulation results In order to test the control performance of the proposed FACTS controllers presented above, simulation studies have been carried out on the power system shown in Fig. 2 with the parameters given in the Appendix, where

PID control . . . . . . . .

' 0

3.5,

5

time. s

a

0 5 10

time. s

c

Fig. 7 a Rotor angle h Speed c output power dContral output of TCSC

ILE Pruc.-Conwl T h m q Appl.. Yo/. 149, No. 4, JuB. 2002

Simirlurion resulrr of TCSC compared with PID

Fig. 6 Block diagrum of PID conmlfor TCSC a Rator anglc h Speed

the generator is only installed with AVR. This simulation is implemented in the PSASP package, which is a profes- sional testing system for power systems designed by the Chinese Electric Power Research Institute.

In the following studies, a three phase to ground fault occurs with a fault duration of 0.1 s before the fault is cleared and the normal operation condition for the power system with TCSC is J0 = 0.2298 rad, E' = 1.06679 p.u. The fault point is located at a distance of 50% of the total length of the transmission line from the infinite bus, and the compensation level of TCSC is given as 70%

4.2.1 Disturbance attenuation and adaptive mechanism: Fig. 4 shows the dynamic response of the rotor angle under different disturbance attenuation levels y between 0.1-10 in the power system with TCSC.

10

0

lime. s

b

5 lime, s

d

10

283

The parameter estimation of the adaptive controller is shown in Fig. 5.

From Fig. 40, there is no distinct difference between these curvcs, the reason being that the pre-feedback has sufficiently strong passivity to stabilise the closed loop PCHD system, which shows the robustness of the system to exogenous disturbance from the input channel. Thus we can allow for more flexibility in the modulation ofy. Fig. 4b shows that the pre-feedback also takes part in the controller response for the sudden disturbance, since it is a feedback passivity equivalence controller.

From Fig. 5 , it can be seen that when the structure parameters change, the adaptation mechanism can provide accurate estimates of the system states and perturhation, with a fast track rate without any phase delay. Thus the proposed controller can work well for different system topologies, parameters and operating conditions.

4.2.2 Comparison with PID: For comparison, the PID controller shown in Fig. 6 was also employed for TCSC control (the details for this model can be found in 1131). Simulation results are given in Fig. 7.

Fig. 7 shows the comparison results of the proposed controller with PID undcr disturbance attenuate level y=O.I. From the Figures above, we can easily observe that the passivc controllers can dampen any oscillation of the rotor angle. Hence_ the designed controller is able to deal with the uncertainties of parameters, non-linear systems dynamics and external disturbances. Furthermore, we can also say that the proposed control law is effective over a broad range of operation points of the power system.

5 Conclusion

An energy-based control method has been applied to TCSC. Firstly, the port controlled Hamiltonian model of TCSC was presented using a pre-feedback method. Secondly, to deal with the problem of L2 gain disturbance attenuation for this Hamiltonian system with parametric perturbations, an adaptive L2 gain control law was designed using the Hamiltonian function as the storage function, which guarantees robustness to parametric uncer- tainties and also attenuates bounded exogenous distur- bances to the system in the form of L2 gain. The proposed TCSC controller not only preserves the non- linearity of physical models, but also has a clear physical meaning in precise mathematical terms. Finally, the simu- lation result shows its validity in assuring asymptotical stability, and attenuating disturbance effectively.

6 Acknowledgments

The authors would like to acknowledge the support of the National Natural Sciencc Foundation of China for National

Outstanding Young Investigators (No. 59825 104) and National Natural Science Foundation of China for Overseas Young Scientists (No. 59928705).

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(2).pp. 1 3 0 ~ 1 3 1 1

8 Appendix

System data: TJ= 12.922 s, y,= I.Op.u., x:i=0.1045 p.u., xr=0.0292 D.u.. oh = 314 rad/s. D= I D.u.. . , , , XL = 0.0266 p.u.

284 IEE Pmc.-Cunwl The"? Appi , YOI. 14Y. No 4, Juiv 2002