Output-feedback IDA control design via structural properties: Application to Thyristor Controlled Series Capacitors

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

WeCIn2.3

Output–feedback IDA Control Design via Structural Properties: Application to Thyristor Controlled Series Capacitors Gerardo Espinosa–P´erez, Paul Maya–Ortiz, Arnau D`oria–Cerezo and Jaime A. Moreno Abstract— In this paper the properties exhibited by the IDA Passivity–based Controller (PBC) design methodology for developing Output–Feedback Controllers (OFC) are illustrated by means of the particular system composed by a synchronous generator connected to an infinite bus via a Thyristor Controlled Series Capacitors (TCSC). Two OFC are presented, one that does not involve the unmeasurable state and other that, although including this state, present some Input–to–State Stability (ISS) properties that allow for establishing a sort of separation principle concerning an observer–based structure for the closed–loop system.

I. I NTRODUCTION It is well known that controller design for dynamical systems must primarily deal with fundamental properties that involve topics like stability, performance and robustness. However, another issue that must be also incorporated in this design in order to completely solve a given problem, specially for practically motivated problems, is to provide the controller structure with the necessary features that allow it for dealing with constraints found for its implementation, e.g. those related with the impossibility for measuring all the state variables. Output Feedback Control (OFC) is the branch of control theory that deals with the problem of designing control schemes involving only available for measurement information. Roughly speaking, this task can be carried out following two general approaches, namely: One that synthesize the control law using only the available for measurement states (pure OFC) and other that substitutes the unavailable state for a corresponding estimate obtained from a dynamical observer (observer–based OFC). Although the available possibilities in the literature for solving a given nonlinear OFC problem are very varied and is difficult to assert about the advantages and drawbacks of each of the existing solutions, a claim that is widely accepted is that the attractiveness of a design technique is in direct correlation with the possibility for carrying its application out in a systematic way. G. Espinosa–P´erez is with DEPFI – UNAM, A.P. 70-256, 04510 M´exico D.F., MEXICO. [email protected]. He is on sabbatical leave at LSS–SUPELEC, France. Part of his work and stay were supported by DGAPA–UNAM and CONACYT, grants IN103306 and 51050, and SUPELEC. P. Maya–Ortiz is with Facultad de Ingenier´ıa – UNAM, A.P. 70-256, 04510 M´exico D.F., MEXICO. [email protected] A. D`oria-Cerezo is with the Department of Electrical Engineering and the Institute of Industrial and Control Engineering, Universitat Politcnica de Catalunya, 08800 Vilanova i la Geltr, Spain [email protected] J. A. Moreno is with Instituto de Ingeniera – UNAM, A.P. 70-472, 04510 M´exico D.F., MEXICO. [email protected]

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

The aim of this paper is to contribute towards the establishment of a novel alternative for approaching in a systematic way the OFC problem design for a class of nonlinear systems. Specifically, it is considered the kind of systems defined by those that can be stabilized by the application of the so–called Interconnection and Damping Assignment (IDA) Passivity–based Control (PBC) methodology design [1]. The interest on developing output feedback controllers under this perspective, comes from a feature of this methodology design that has been repeatedly reported under several scenarios, e.g. in [1] regarding the control of an electromechanical system or in [3] concerning the induction motor control, but that has not been systematically developed, namely, the flexibility offered for solving the Matching Equation (ME), a key step in the procedure for regulating the behavior of a non–linear system by assigning it a desired Hamiltonian structure, without using unavailable states. The main contribution of this paper is to illustrate, by means of the practically important problem of improving the transient stability properties of Power Systems (PS), how tailoring the ME of the IDA design it is possible to propose both a pure OFC and, after a further exploitation of the ME structural properties, an observer–based OFC with the particular feature that the dependency of the control law with respect to the unmeasurable state can be decided during the controller design. In particular, it is shown that the structure of the control law can be chosen in such a way that the closed–loop stability analysis is simplified. Special attention is given to the possibility of finding Input–to–State Stability (ISS) properties with the aim of establishing a sort of separation principle. Regarding the case study approached in this paper, it must be pointed out that it is important by itself, since PS are experiencing oscillations due to different reasons, e.g. existence of disturbances, that if are not damped could lead to unstable operation [4]. Usually these undesirable behaviors are compensated considering the synchronous generator Power System Stabilizer (PSS) and the Automatic Voltage Regulator (AVR) as the mechanisms to enhance the system damping, although sometimes this solution does not completely solve the problem. Thus, in this paper it is assumed that the action of the PSS is complemented by introducing in the network a flexible AC transmission system (FACT), due to their proved capability for improving transient stability properties beside to their primary functions, such as voltage and power flow control [4]. Actually, the considered system is composed by the swing equation of a synchronous generator connected to

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WeCIn2.3 an infinite bus, i.e. a Single Machine Infinite Bus (SMIB) system, via a Thyristor Controlled Series Capacitor (TCSC). The importance of this device has been widely recognized in the literature and it is possible to find several schemes for controlling it developed under different perspectives, e.g. state–feedback pole placement [5], immersion and invariance [6], standard passivity–based [7] and state–feedback IDA techniques [8]. The remaining of this paper is organized as follows. Section II is devoted to formulate, in a general fashion, the problem of finding an OFC by exploiting the structural properties of the ME. The presentation of the model considered in this paper for the SMIB–TCSC system is introduced in Section III together with the proposed controllers and the corresponding stability analysis. Section IV contains a numerical evaluation of the proposed schemes while Section V is dedicated to the presentation of some concluding remarks. II. P ROBLEM FORMULATION The problem of stabilizing an equilibrium point of nonlinear systems of the form x˙ = f (x, t) + g(x)u

(1)

where x ∈ Rn is the state vector, u ∈ Rm (m < n) is the control action and g(x) is assumed full rank, is approached from the IDA–PBC perspective by splitting the controller design into two stages, namely, the Energy–shaping and Damping injection steps. For the first step the objective is to find some functions Jd : Rn × R → Rn×n ,

Hd : Rn → R,

satisfying the skew–symmetry condition for the interconnection matrix Jd (x, t) + JdT (x, t) = 0, (2) and the equilibrium assignment condition for the desired total stored energy x⋆ = arg min Hd (x) n

with x⋆ ∈ R the equilibrium to be stabilized, such that the closed–loop defines a port–controlled Hamiltonian (PCH) system of the form

x˙ y

= [Jd (x, t) − Rd (x)]∇Hd + g(x)v = g ⊤ (x)∇Hd .

(3)

where the damping matrix R d (x) = Rd⊤ (x) ≥ 0 is defined by Rd (x) = g(x)Kdi g ⊤ (x), and v is a signal introduced to define the port variables. The required stability properties of the last closed–loop system can be proved by noting that with v = 0 it satisfies H˙ d = −y ⊤ Kdi y. Then it is easy to prove (e.g., Lemma 3.2.8 of [2]) that the equilibrium x ⋆ will be asymptotically stable if it is detectable from y, i.e. if the implication y(t) ≡ 0 ⇒ limt→∞ x(t) = x⋆ is true. In spite of its clear formulation, the major problem for finding the controller above comes for the necessity of solving the so–called Matching Equation (ME) 1 g ⊥ (x)f (x, t) = g ⊥ (x)Jd (x, t)∇Hd ⊥

(4)

(n−m)×n

where g (x) ∈ R is a full–rank left–annihilator of g(x), that is, g ⊥ (x)g(x) = 0 and rank g ⊥ (x) = n − m, in order to obtain the functions J d (x, t) and Hd (x) that are compatible with the dynamic behavior of system (1) given in terms of f (x, t). In this sense, a recently reported alternative for solving this equation that simplify at some extent this problem and, at the same time, enlarge the set of systems that can be stabilized under the IDA–PBC approach, is a variation known as Simultaneous IDA–PBC (SIDA– PBC) which intents for finding in one step both the skew– symmetric Jd (x, t) and the damping R d (x) matrices [3]. This objective is achieved by solving the alternative representation for the matching equation given by g ⊥ (x)f (x, t) = g ⊥ (x)Fd (x, t)∇Hd

(5)

where, instead of the skew–symmetric constraint (2), matrix Fd (x, t) must satisfy the weaker condition Fd (x, t) + FdT (x, t) ≤ 0

(6)

leading to the control law

x˙ = Jd (x, t)∇Hd + g(x)udi y = g ⊤ (x)∇Hd .

ues = [g T (x)g(x)]−1 g T (x){Fd (x, t)∇Hd − f (x, t)} (7)

The rational behind of choosing the Hamiltonian structure is that the conservative structure exhibited by this kind of systems, concluded from the fact that H˙ d = 0, allows for guaranteeing the stability of x ⋆ if Hd (x) is considered as Lyapunov function. On the other hand, the purpose of the damping injection step is to improve, from stable to asymptotically stable, the properties of the equilibrium point x ⋆ by adding dissipation to the closed–loop system. This task is carried out by feeding back the passive output y. In a general context the controller that achieves the aforementioned conditions is given by u = u es + udi with ues = [g T (x)g(x)]−1 g T (x){Jd (x, t)∇Hd − f (x, t)}

⊤ > 0, since the closed–loop and udi = −Kdi y, Kdi = Kdi leads to the PCH system with dissipation

Being the solution of (4) or (5) a fundamental element for implementing the (S)IDA–PBC, a lot of research has been devoted to this topic and currently it is possible to identify several techniques to solve this equation (the interested reader is refereed to [1] for a detailed discussion on different methods of solution of the matching equations). However, one property that (up to the authors knowledge) has not been analyzed or exploited in a systematic way, although has been recognized in several particular applications, is the related with the possibility for solving this equation in an output 1 All vectors in the paper are column vectors, even the gradient of a scalar ∂ . When clear from the context the subindex function denoted ∇(·) = ∂(·) will be omitted.

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WeCIn2.3 feedback way. Hence, the main purpose of this paper is to contribute towards this study by giving an answer, for the particular case of the control design for the TCSC, to the following problem Consider the system (1) with state vector given by  T x = xTm , xTnm

where xm and xnm stand for the measurable and unmeasurable states, respectively. Design a control law u = u es + udi by finding a solution of the ME (4) or (5) and an structure of the damping injection mechanism in such a way that one of the next two conditions is satisfied: • The control law does not depend on the unmeasurable states xnm , or • The dependency of the proposed controller on the unmeasurable states allows for designing an observerbased control scheme in a simple as possible way. III. C ASE STUDY: T HYRISTOR C ONTROLLED S ERIES C APACITOR Introduced at the late 80’s as a rapid adjustment method for network impedance, TCSC is currently a widely accepted alternative for both controlling the power flow in a line and improving the transient stability properties [4] of a given power system by canceling a portion of the reactive line impedance. The basic SMIB–TCSC system, shown in Fig. 1, consists of a series compensating capacitor (X C ) shunted by a thyristor–controller reactor (X L ). In this figure E is the voltage of the generation unit, V the infinite bus voltage while XΣ es the line reactance.

T the rotor inertia, D the damping coefficient and T dc the time constant included to model the dynamic response of the TCSC. In order to formulate the control problem related with this system, it is necessary to notice that its equilibria are given by the two solutions of x∗2 Pm x3

= 0 = a2 x∗3 sin(x∗1 ) =

(9)

x∗3

and that its practical operation region is delimited by 0 ≤ x1 ≤ π2 . Moreover it must be also considered that the total inductance of the system x 3 is not available for measurement and that the equilibrium point that is located into the aforementioned region, given by x ∗1 = sin−1 (Pm /a2 x∗3 ), is already stable. Under these conditions, the control objective is to further enhance the stability properties (the region of attraction) of the equilibrium point x ∗ = (x∗1 , x∗2 , x∗3 ) = (sin−1 (Pm /a2 x∗3 ), 0, x∗3 ). In order to carry the controller design out, notice that the ME for this particular system can be written as f (x) + gu = Fd

∂Hd (x) ∂x

where    x2 0 f (x) =  Pm − a1 x2 − a2 x3 sin(x1 )  ; g =  0  −b1 (x3 − x∗3 ) b1 

while the matrix Fd is assumed to have the following form   F11 F12 F13 Fd =  F21 F22 F23  F31 F32 F33

with each entry Fij of appropriate dimension. With the definitions above, solving the imposed control problem reduces to find suitable functions that satisfy the following equalities x2 = F11 Fig. 1.

Single Machine Infinity Bus System with TCSC.

=

x2

x˙ 2 x˙ 3

= =

Pm − a1 x2 − a2 x3 sin(x1 ) b1 (−x3 + x∗3 + u)

(8)

where x1 = δ is the power angle, x 2 = ω is the angular velocity of the rotor, x 3 is the total admittance of the system while Pm = P/T , a1 = D/T , a2 = E ′ V and b1 = 1/Tdc are positive constants which depend on E ′ the transient voltage of the generator, P m the constant mechanical power,

(10)

Pm − a1 x2 − a2 x3 sin(x1 ) = F21 ∂H∂xd (x) + F22 ∂H∂xd (x) 1 2 +F23 ∂H∂xd (x) 3 (11)

If it is considered that the generator is provided with a PSS–AVR, it is possible to consider only the mechanical dynamic of the Synchronous Generator, i.e. the so-called swing equation, for describing the behavior of the considered system resulting in a model given by x˙ 1

∂Hd (x) ∂Hd (x) ∂Hd (x) + F12 + F13 ∂x1 ∂x2 ∂x3

−b1 (x3 − x∗3 ) + b1 u = F31 ∂H∂xd (x) + F32 ∂H∂xd (x) 1 2 +F33 ∂H∂xd (x) 3

(12)

Due to the underactuated nature of the system, it is important to notice that equations (10) and (11) does not depend on the control input u leading to the fact that they must be solved before dealing with the control dependent equation (12). In this sense, from (10) and considering

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Hd (x) =

k1 2 e + Hd1 (x1 , x3 ); k1 > 0 2 2

(13)

WeCIn2.3 with e2 = x2 − x∗2 leads to F11 = F13 = 0, F12 = k11 which in its turn defines F 21 = − k11 , F22 = − ka11 implying, considering (11) and (13), that ∂Hd (x) 1 ∂Hd (x) + F23 = −a2 x3 sin(x1 ) + Pm − k1 ∂x1 ∂x3 Taking into account, from (9), the equilibrium value of Pm , then it is possible to define Hd1 (x1 , x3 ) = k1 a2 x3 [cos(x∗1 ) − cos(x1 )] −k1 a2 x∗3 sin(x∗1 )e1 + Hd2 (x3 )

(14)

where e1 = x1 − x∗1 , leading to  ∂Hd2 (x3 ) ∗ F23 k1 a2 [cos(x1 ) − cos(x1 )] + =0 ∂x3

forcing to F23 = 0. Once the fixed part of matrix F d has been defined, the rest of the design is related with choosing the free part of matrix Fd involved with equation (12). In this sense, notice that under the definition of the desired energy function (13-14), equation (12) takes the form F31 k1 [−a2 x∗3 sin(x∗1 ) + a2 x3 sin(x1 )] −b1 e3 + b1 u =

+F32 k1 e2 + F33 k1 a2 [cos(x∗1 ) − cos(x1 )] +

∂Hd2 (x3 ) ∂x3

(15) where e3 = x3 − x∗3 . As can be seen, there exist several possibilities for solving equation (15). Among them, the designer must look for those that while satisfying condition (6) at the same time hold with the contraint argmin{Hd (x)} = x∗ with x∗ the equilibrium point to be stabilized. In the rest of this section two different solutions are presented, the first gives as a result a control law that does not require the unmeasurable state x 3 while the second, although depending on this variable, exhibits some properties that simplify the design of an observer– based control. A. Pure Output Feedback Control If in (15) it is defined H d2 (x3 ) = this equation reduces to

γ 2 2 e3

and F33 = − bγ1 ,

b1 u = F31 [−k1 a2 x∗3 sin(x∗1 ) + k1 a2 x3 sin(x1 )] + F32 k1 e2 − b1 kγ1 a2 [cos(x∗1 ) − cos(x1 )] (16) Then, F31 = 0 and F32 = k produces the pure output feedback control law kk1 k1 a2 u=− [cos(x∗1 ) − cos(x1 )] x2 − (17) b1 γ with

0 Fd =  − k11 0 

1 k1 − ak11

k

0 0  − bγ1 

and applying standard arguments for achieving asymptotic convergence of the trajectories to the equilibrium point. B. Observer–based Output Feedback Control In contrast to the pure OFC design, the observer–based controller design requires to cover several steps in order to achieve the posed stabilization objective. Specifically, in addition to the (in this case state–feedback) controller design, a dynamic observer must be proposed and the stability of the whole system must be guaranteed. In this section these three topics are approached. Concerning the state–feedback design, the key point that must be taken into account is the complete control that the designer has about how the unmeasurable state appears in the control law defined by (15), since as long as F 31 is different from zero x3 will appear in the control law in a nonlinear fashion while definition of F 33 will determine if this state appears in a linear way (due to the definition of H d2 ). Hence, given that a linear dependency leads to a more treatable structure, nonlinear dependency will be avoided. 2 Thus, considering that H d2 (x3 ) is as above, F33 = − b1 +k γ , F31 = 0 and F32 = k produce that the control law (15) takes the form (b1 + k2 )k1 a2 kk1 [cos(x∗1 ) − cos(x1 )] − k2 e3 x2 − u=− b1 b1 γ (18) with   1 0 0 k1 a 1  0 Fd =  − k1 − k11 2 0 k − b1 +k γ The stability properties are assured in a similar way than in the pure OFC design but in this case with the control gains satisfying 4a1 (b1 + k2 ) k1 a2 sin2 (x∗1 ) ; >k γ> x∗3 cos(x∗1 ) k1 γ

Regarding the dynamic observer design, it is useful to recognize that model in (8), x 3 appears in a linear way with respect to the measurable states x 1 , x2 , as can be noticed if this expression is equivalently written as σ˙ x˙ 3

(19)

= −b1 x3 + B(u)

(20)

T

where B(u) = b1 (x∗3 + u), σ = [x1 , x2 ] and



 0 x2 ; ψ0 (σ) = ψ1 (σ) = −a1 x2 − Pm −a1 sin(x1 )

With this alternative representation, defining the variable y = σ˙ − ψ0 (σ) the system reads as x˙ 3 y

Hence, the stability properties of the closed–loop scheme are proved by noting that F d + FdT ≤ 0 if and only if k1 a2 sin2 (x∗1 ) 4a1 b1 γ> ; >k x∗3 cos(x∗1 ) k1 γ

= ψ1 (σ)x3 + ψ0 (σ)

= −b1 x3 + B(u) = ψ1 (σ)x3

exhibiting a structure that allows for proposing, using classical arguments, an observer of the form

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x ˆ˙ 3

=

yˆ =

−b1 x ˆ3 + B(u) + K(σ)(y − yˆ)

(21)

ψ1 (σ)ˆ x3

(22)

WeCIn2.3 with K(σ), a time-varying gain that depends on the measurable state, to be determined below. Under these conditions the dynamic of the estimation error ˆ3 is given by x ˜3 = x3 − x x ˜˙ 3 = − [b1 + K(σ)ψ1 (σ)] x ˜3 where, to guarantee convergence, the gain K(σ) must be chosen such that b 1 + K(σ)ψ1 (σ) > 0 for all time. Unfortunately, observer (21)–(22) is not implementable since the variable y = σ˙ − φ(σ, u) depends on the time derivative of σ which in its turn depends on the unmeasurable state x3 . In order to avoid this problem consider the following alternative representation of the observer x ˆ˙ 3 − K(σ)σ˙ = − [b1 + K(σ)ψ1 (σ)] xˆ3 +B(u) − K(σ)ψ0 (σ) x3 yˆ = ψ1 (σ)ˆ Defining the available for measuring variable s = x ˆ 3 − β(σ) where β(σ) is a function of the measurable state, the last observer representation takes the implementable form s˙ = − [b1 + K(σ)ψ1 (σ)] (s + β(σ)) + B(u) − K(σ)ψ0 (σ) x ˆ3 = s + β(σ)

˜3 with The first point to be noticed is that u o = u + k2 x u the original state feedback controller (18). Under these conditions, the closed–loop system takes the form ∂Hd (x) + gk2 x ˜3 ∂x due to the fact that the equilibrium point is constant. If x ˜3 = 0 then x∗ is locally asymptotically stable. If x˜ 3 = 0 then the time derivative of H d (x) along the trajectories of the closed–loop system reads as  T ∂Hd (x) ¯ d ∂Hd (x) + ∂Hd (x) k2 x R H˙ d (x) = − ˜3 ∂x ∂x ∂x3 e˙ = Fd

¯d = R ¯ T ≥ 0 is the non-zero part of matrix 1 (Fd + where R d 2 FdT ). From this last expression it is possible to show that T  ∂Hd (x) ¯ d ∂Hd (x) ˙ R Hd (x) ≤ −(1 − θ) ∂x23 ∂x23 with ∂Hd (x) = ∂x23

, 0 1 ≤ γ

∂Hd (x) ¯d} ∂x θλmin {R

∂x3

(23)

if k3 > 0. The final step in the observer–based control design is related with the stability proof of the system composed by the plant, the state–feedback control and the observer. In this sense, the advantage of developing the controller design as above, lies in the fact that guaranteeing the stability properties of the closed–loop system can be achieved in a (relatively) simple way. For instance, as reported in [10], it is possible to attain this objective by proving that the map from the observation ˆ3 , with xˆ3 the estimate of x3 , to the control error x ˜3 = x3 − x error e = x − x∗ exhibits some Input to State Stability (ISS) properties [11]. The motivation for guaranteeing this kind of property comes from the fact that under ISS, for any bounded observation error the control error will be also bounded and that if the observation error tends to zero so does the control error. In order to state the ISS properties of controller (18), consider the system (8) in closed loop with the output– feedback version of this controller given by

∂Hd (x) ∂x2 ∂Hd (x) ∂x3

whenever the following constraint holds

provided ∂β(σ) K(σ) = ∂σ



 k1 a2 x∗3 [sin(x1 ) − sin(x∗1 )]  0 = ∗ )] −k a [cos(x ) − cos(x 1 2 1 1   k1 a2 sin(x1 )(x3 − x∗3 )  (x2 − x∗2 ) + (x3 − x∗3 )

d (x) leading to the that ∂H∂x

≤(γ1 + γ2 ) x − x∗ with  fact γ1 > k1 a2 (x∗3 )2 + 1 and γ2 > k12 a21 + 1, which in its turn gives as a result that

c1 ˜ x3 ≤ x − x∗ ; c1 =

|k2 | ¯ d} γ(γ1 + γ2 )θλmin {R

i.e. the trajectories of the system will tend to a ball, centered at x = x∗ and radius depending on ˜ x 3 , showing that if this last term tends to zero, then the control error will also tend to zero. IV. S IMULATION RESULTS The usefulness of the proposed output feedback controllers is illustrated in this section via some numerical simulations. For this purpose, the parameters of the SMIB model (given in pu) were taken from [7] as P m = 16, a1 = 1, a2 = 21.3358 and b1 = 20. The considered experiment was to generate a three phase short circuit at the generator bus that started at 0.5s with a critical clearing time of t cl = 182ms. Regarding the controller gains, for the pure output feedback control they were set at k1 = 0.01, γ = 90.2977 and k = 9.4125, while

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WeCIn2.3 0.04

2.2

2

1.8

x1

1.6

1.4

1.2

1

0.8

0.6 0

1

2

3

4

5

6

7

8

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10

time

Fig. 2.

Phase angle.

0.03

0.02

0.01

u

for the observer–based control the considered values were k1 = 0.01, k2 = 0.00001, γ = 90.2977, k = 9.4125 with the observer gain given by k 3 = 0.001. Figure 2 and Figure 3 show the phase angle and the angular velocity of the system, respectively 2. The first feature to be noticed is that with both controllers the stabilization objective is achieved despite of the large clearing time, as predicted from the theoretical analysis. Moreover, as can be seen from the aforementioned figures, the performance accomplished with both controllers is very similar, showing the pure OFC only a slight improvement with respect to the observer–based control. However, as depicted in Figure 4 and Figure 5, the advantage of the former is more relevant. In these figures it is shown how the total admittance and the control effort required by the OFC are significatively lower with respect to the other scheme.

0

−0.01

−0.02

−0.03 0

1

2

3

4

5

6

7

8

9

10

time

Fig. 5.

Control effort.

PBC design methodology can be used to generate Output– feedback controllers. This illustration was carried out by considering the practically important problem of improving the transient stability properties of a power system composed by a synchronous generator connected to an infinite bus via a TCSC. For this system both a pure and an observer–based OFC were reported. In addition it was also illustrated how the aforementioned flexibility in tailoring the ME can be further exploited with the aim of simplifying the design of the observer–based control. Specifically, it was shown that deciding the structure of the state–feedback control could allow for the use of some tools, like ISS, for establishing a sort of Separation Principle which in its turn gives to the designer some freedom for approaching the observer design problem in an independent way with respect to the controller proposition. R EFERENCES

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2.5

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1.5

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−1.5

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Fig. 3.

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0.93

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Fig. 4.

Total admittance.

V. C ONCLUDING REMARKS In this paper it has been illustrated how the flexibility offered for solving the ME in the application of the IDA–

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2 Dashed lines correspond to the observer–based scheme while continuous lines correspond to the pure OFC.

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