Generalized Control Energy Function forControllable TCSC DevicesDaniel S. Siqueira, Luis F. C. Alberto and Newton G. Bretas

EESC - University of Sao PauloDepartament of Electrical Engineering

Sao Paulo, Sao Carlos, BrazilEmail: danielsiqueira@usp.br, lfcalberto@usp.br and ngbretas@sc.usp.br

Abstract—In this paper the technique of non-linear controlbased on the Generalized Energy Function is employed to designcontrollers for TCSC devices. This technique, recently developedin [1], extends the ideas of Control Lyapunov Function for alarger class of problems. Besides allowing the controller design,the technique provides estimates of the stability region of thesystem and therefore can support the systematic evaluation ofthe controller contribution to the system transient stability.Index Terms—Nonlinear Control, Generalized Energy Func-

tion, Generalized Energy Function of Control, FACTS Devices,TCSC Devices.

I. INTRODUCTIONThe Thyristor Controlled Series Capacitor is one of the

most effective dynamic compensators used in power systems.It offers a flexible, quick, and reliable adjustment enabling theapplication of advanced theories to the design of control. Thesedevices can control the flow of power in specific lines allowingthe optimal use of the network and thus bringing manybenefits, such as: (i) damping of oscillations, (ii) improvementin transient stability limits.Due to the benefits of dynamic compensation [2], [3],

studies have been developed in order to synthesize control lawsfor FACTS devices and in particular for TCSC devices [4],[5]. However, in general, these studies are based on classicalcontrol techniques and design control laws for linearized mod-els. Their efficiency is verified a posteriori through nonlinearsimulations of the corresponding nonlinear model. However,these analyses cannot ensure the performance of the controllerwhen the disturbances lead the trajectories to points distantfrom those used for the controller design.Aiming to better representing the physical phenomena oc-

curring in the real power systems, nonlinear models havebeen used, and controller design techniques such as feedbacklinearization and Control Lyapunov Functions (CLF) havebeen increasingly developed [6], [7], [8]. However, the dis-advantages of some of these techniques are that the controllaws obtained by them are functions of variables of difficultsynthesis, and based on simplified system models.The control technique based on the CLF employed in [8]

has shown to be very interesting for the design of nonlinear

This work supported by Fundacao de Amparo a Pesquisa do Estado de SaoPaulo (FAPESP), under grant number 2011/23345-8

controller for the TSCC. The control laws obtained by thismethod are independent of the network topology and thedisturbance location. Moreover they use local feedback signalswhich increase the stability region of the equilibrium point. Incontrast, the CLF method is inappropriate to work with moredetailed system models, since it is not a trivial task to find anassociated Lyapunov function.The concept of a Generalized Control Energy Function

(GCEF), based on the extension of the LaSalle’s invarianceprinciple [9], has been recently developed in [1]. The GCEFallows its derivative to be positive in regions of the statespace. Generalized Energy Functions are usually more easilyobtained as compared to Energy Functions and therefore canbe applied to a large class of problems, including detailedmodels of power systems [10]. Another advantage of theFEGC technique is that it offers an estimation of the stabilityregion [1] which cannot be generally obtained with CLFs.In this paper, we will employ the technique of nonlinear

control based on the Generalized Control Energy Function todesign control laws for stabilizing TCSC devices consideringlosses in the system. Besides allowing the controller design,the technique provides estimates of the system stability region.That can be used to include the impact of the controller in thetransient stability.

II. TCSC DEVICESThe use of series compensators is an effective and econom-

ical solution for controlling the energy flow in transmissionlines (TL). The TCSC device is a variable reactance controlledby thyristors with the function of varying the degree ofcompensation quickly and continuously.The TCSC offers several benefits to the system: (i) rapid

and continuous control of power flow in TLs, (ii) dampinginter-area oscillations, (iii) improving the limits of stability,among others [11]. Figure 1 shows the basic configuration ofthe TCSC [12].

Fig. 1. Typical Configuration of a TCSC.

A. Network ModelThe TCSC may be modeled in the network as a capacitive

reactance in series with the transmission line. Figure 2 showsthe transmission line with a TCSC connected at one of the TLend.

�� ��

� �

�������������������

���

Fig. 2. Transmission line with TCSC installed.

In figure 2, Ei and Ej are the phasor voltages at buses i andj respectively, Zij = rij + jxij is the line impedance, and S′ij, Iij are respectively the complex power and the TL current.The current flowing from bus i to bus j is given by:

Iij =Ei −Ej

Zij − jxC

=Eie

jθi − Ejejθj

rij + j(xij − xC)(1)

We can express the voltage in TCSC as follows:

VC = −jxCIij (2)

The TCSC can be modeled as a voltage source in series oras a current source in parallel with the line, figure 3.

��

��

� �

��

��

� �

���������� �����������

�����������

�

Fig. 3. Transformation of the TCSC voltage source to the current source.

Modeling the TCSC as a current source we define Sij as thecomplex power transmitted on the line without the effect of theTCSC, and SiC as the complex power injected by the effectof the TCSC. The total transmitted power with the TCSC willbe:

S′ij = Sij + Sic (3)

The complex power in the line without the effect of theTCSC is calculated as follows:

Sij = Pij + jQij (4)

Pij = GijE2i −Dijcos(θij)− Cijsen(θij) (5)

Qij = −BijE2i −Dijsen(θij) + Cijcos(θij) (6)

where,Dij = GijEiEj Cij = BijEiEj (7)

The complex power injected on the line by the TCSC iscalculated as follows:

SiC = Ei · (IC)∗ (8)

SiC =jxC

rij − j(xij − xC)· (Pij + jQij) (9)

defining,

⎧⎪⎪⎪⎨⎪⎪⎪⎩

g′ij =rij

r2ij + (xij − xC)2

b′ij = −xij − xc

r2ij + (xij − xC)2

(10)

Equation (9) can be rewritten as follows:

SiC = xC · [(−b′ijPij − g′ijQij)

+ j(g′ijPij − b′ijQij)](11)

The total transmitted power is the sum of equations (4) and(11).

S′ij = [Pij · (1 + u)] + j[Qij · (1 + v)] (12)

where, ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

u = xC ·

(−b′ij − g′ij

Qij

Pij

)

v = xC ·

(−b′ij + g′ij

Pij

Qij

) (13)

III. GENERALIZED ENERGY FUNCTION

The Lyapunov Function theory can be explored to designor provide guidelines for the choice of feedback signals. Forthis choice, we impose constraints on the construction ofthe feedback law to ensure the derivative of the LyapunovFunction candidate to be negative. This idea is not new, but hasbeen recently developed with the introduction of the ControlLyapunov function. However, the major obstacle of the CLFis to find a Lyapunov function or an energy function for thesystem in question.With the intention to overcome the difficulties imposed by

the Control Lyapunov Function, the concept of GeneralizedControl Energy Function was proposed in [1] with the goal ofdesigning stabilizing control laws for nonlinear systems, evenwhen the derivative of the energy function is positive in somebounded regions of the state space.

A. Generalized Control Energy FunctionThe GEF is a generalization of the energy function proposed

in [13]. The requirements on the scalar function V are relaxedin [1], allowing the treatment of a larger class of nonlinearsystems.Consider the following autonomous system:

x = f(x) (14)where x ∈ R

n is the vector of state variables and f : Rn −→R

n is a C1 - function.

Definition 1 (Generalized Energy Function). [1] A scalar C1

function V : Rn → R is a Generalized Energy Function if:

(i) any compact set intersects a finite number of isolatedbounded connected components Ci of the set C:= {x ∈ R

n :V (x) ≥ 0}(ii) supt≥0 | V(ϕ(t, x0)) |< ∞ implies ϕ(t, x0) is boundedfor t ≥ 0The concept of Generalized Energy Function allows the

derivative of the function along the trajectories to be positivein bounded sets Ci′s [14].Now consider the following controlled autonomous system:

x(t) = F (x, u) (15)

where x ∈ Rn is a vector of state variables, u ∈ R

m is theinput control and F : Rn × R

m −→ Rn is a C1 function.

Suppose that the origin is an equilibrium point of the openloop system (u = 0), ie F (0, 0) = 0. Our goal is to obtaina feedback law u = h(x), such that the origin of the system(16) is asymptotically stable.

x(t) = F (x, h(x)) (16)

With that in mind, we introduce the concept of a General-ized Control Energy Function.

Definition 2 (Generalized Control Energy Function). [1] A C1

function V : Rn → R is a Generalized Control Energy Func-tion of (15), if there exists a feedback control law u = h(x),of class C1, such that W (x) = V (x, h(x)) is a GeneralizedEnergy Function of the closed loop system.

Next theorem explore the existence of a GCEF to offersufficient conditions to guarantee the existence of an asymp-totically stable set and an estimate of its stability region.

Theorem 1 (Stability and Stability Region of the Closed LoopSystem). [1] Suppose system (15) admits a GCEF V . LetL ∈ R be a real number such that the connected componentSc(L) of {x ∈ R

n : W (x) < L} is bounded. Suppose thatsupx∈C∩Sc(L)W (x) := l < L. Then Sc(l) := {x ∈ Sc(L) :W (x) ≥ l} contains an invariant and asymptotically stableset H and Sc(L) is an estimate of the stability region ofH . Moreover, H has a nonempty intersection with the setC ∩ Sc(l).

Further details about the contents of this section can befound in [1] and [15].

B. Application in Power SystemConsider the system of figure 4 composed of a generator

connected to a large system via a transmission line compen-sated with a TCSC device.

� �� ��

�����

Fig. 4. Single Machine - Infinite Bus System.

Considering the classical model for the generator, thesystem can be modeled by the following set of differentialequations:

δ = ω, (17)

ω =1

M(Pm − Pger − Tω).

According to section II.A, the active power generated canbe expressed as:

Pger = (1 + u) · Pe, (18)

where Pe is the flow on the transmission line without the effectof the TCSC, given by:

Pe = GE2 − C cos(δ)−D sin(δ). (19)

Now, consider the following scalar function as a candidatefor the Generalized Energy Function:

W (δ, ω) =1

2Mω2 − Pδ + C cos(δ)− βωPl + κ, (20)

where P = Pm−GE2, Pl = Pm−Pe, β is a parameter to bedetermined and κ is an arbitrary constant. The candidate fora GEF W was inspired in the ideas developed in [16]. In thispaper, an extension of the invariance principle [9] and a scalarfunction, similar to W was employed to study stability of apower system model that does not admit a general Lyapunovfunction.The goal is to show that the scalar function (20) satisfies the

requirements imposed by the GEF for the open loop system(u = 0). Differentiating the scalar function (20) one has:

W (δ, ω) = −

[Pl

ω

]TA

[Pl

ω

]+D cos(δ)ω, (21)

where, A is a symmetric matrix,

A =

⎡⎣

βM

− βT2M

− βT2M T−β

(C cos(δ)+D sin(δ)

)⎤⎦ (22)

The parameter β has to be chosen to make the quadraticfunction (21) positive definite. Applying the Silvester’s crite-ria, it is concluded that,

β <T

C +D + T 2

4M

. (23)

With this choice of β, the quadratic form is positive definiteand only the term ’D cos(δ)ω’ is responsible for generatingregions where the derivative ofW (δ, ω) is positive. Therefore,the scalar function (20) is a Generalized Energy Function forsystem (17) and will be used for the synthesis of control lawsfor the TCSC.Consider the system (17) with the control input u. The

derivative of W (δ, ω) can be expressed by:

W (δ, ω) = −

[Pl

ω

]TA

[Pl

ω

]+D cos(δ)ω

−

(Peω −

β

MPlPe

)h(δ, ω),

(24)

where A and β are determined according to (22) e (23) respec-tively. The objective is to choose a control law u = h(δ, ω)that satisfies the requirements of a GCEF. For this, considerthe following control law:

h(δ, ω) = KPetω, (CL)

where, K is the controller gain, ω is the angular speed andPet is the line power flow given by:

Pet = G′E2 − C ′ cos(δ)−D′ sin(δ), (25)

where

G′ =r

r2 + (x− xC)2, B′ = −

x− xc

r2 + (x− xC)2, (26)

D′ = EV G′, C ′ = EV B′. (27)

Thus we have the following expression for W (δ, ω):

W (δ, ω) = −

[Pl

ω

]TA

[Pl

ω

]+D cos(δ)ω

−

(Peω −

β

MPlPe

)KPetω,

W (δ, ω) = −

[Pl(δ)ω

]TA

[Pl(δ)ω

]+D cos(δ)ω

−

(KPetPeω

2 −β

MKPetPl(δ)Peω︸ ︷︷ ︸

Generates pos. deriv.

),

(28)

Even with the inclusion of the control law (CL), W (δ, ω)has regions where the derivative is positive. It can be easilychecked that these regions are always close to equilibriumpoints. Thus for D and β sufficiently small, we can guaranteethat W is a GCEF for the system (17).

IV. EXAMPLEConsider the system shown in Figure 5 where a generator is

connected to an infinite bus by two parallel transmission lines.One of them has a TCSC device installed on it. The generatoris supplying an active power to the system of 1 [pu] and atthe instant t a short circuit occurs in the middle of the linewhere the TCSC is not installed. The fault is eliminated byopening the line after an interval of time tcl.

� ���

����� ���

���� ����

��

� �Fig. 5. System Configuration.

The dynamic performance of this system will be analyzedfor the following parameters: Generator data: Pm = 1, M =0.00318, x′d = 0.2, ra = 0.002, E′q = 1.11 and T = 0.014.Lines data: rL1 = 0.04, xL1 = 1, rL2 = 0.04 e xL2 = 0.8.TCSC data: x0

C = 0.2, xminC = 0.1 and xmax

C = 0.5.This system has a pre-fault equilibrium point with an angle

of 32.18o (0.5617 [rad]) and the post-fault equilibrium pointwith an angle of 61.41o (1.0719 [rad]). The graph in Figure 6shows the dynamic behavior of the system.

0 5 10 150

0.5

1

1.5

2

2.5 (a) Angle (δ)

δ

0 5 10 15

−2

−1

0

1

2

3

(b) Speed (ω)

ω

[s]

No controlControl

No ControlControl

Fig. 6. Dynamic behavior of the system: (a) speed, (b) angular variation.

The critical clearing time (CCT) of the fault with a staticTCSC is 297 [ms] with a damping ratio of 4.83%. Applyingthe control law (CL) with the gain K = 0, 03, we achievea damping ratio of ξ = 14, 94%. The CCT of this systemwith control law was 384 [ms]. The graph in Figure 7 showsthe dynamic behavior of the TCSC. Clearly, the CL improvesthe transient stability margin as well as the small signal stateperformance.

0 5 10 150.1

0.15

0.2

0.25

0.3

(a) Reactance of TCSC

XTC

SC

0 5 10 150

0.5

1

1.5

(b) Injection of Power / Reactance of TCSC

Pe

/ X

TCSC

[s]

XTCSCInjection of Power

Fig. 7. (a) Variation of the TCSC reactance versus time, (b) comparison ofthe variation of the injection power and the reactance of the TCSC a functionof time.

Figure 8 shows the estimate of the stability region for thecontrol law (CL). The constant (κ = 1.815) was chosen so thatthe function equals zero at the post-fault stable equilibriumpoint and β = 7.9617 10−3.

0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

2

3

4

δ

ω

Contours of WEstimation of the stability regionRegions of derived positivePos −fault equilibrium pointPre−fault equilibrium pointTrajectory of the system

C

Sc(L)

Fig. 8. Estimation of the stability region for the post-fault system with theinclusion of control.

According to theorem 1, set Sc(L) is bounded for L = 2.3.Within Sc(L) there is a bounded region where the derivative ofW is positive (set C). The set C is contained in the set Sc(l)and, moreover, Sc(l) contains the largest and asymptoticallystable invariant set H . Set H has non empty intersection withC. Thus, Sc(L) is an estimate of the stability region.

V. CONCLUSIONSIn this paper a control law for the TCSC device using

the technique of Generalized Control Energy Function wasdeveloped. This technique has been recently developed andextends the ideas of Control Lyapunov Function for a largerclass of problems. With the GCEF technique a control law fora machine versus infinite bus system, considering the lossesin the transmission line is derived. This law leads to goodresponse of the system to large disturbances, and increasesthe critical clearing time. Moreover, the synthesized controllaw is composed of signals that are easy to obtain. It wasalso possible to adjust the gain of the controller so as to have

a desired damping ratio near the post-fault equilibrium. Thetechnique allowed us to estimate the stability region.As prospects for future work, we intend to extend the

presented ideas for multi-machine systems.

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