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Enhancement of Power System Transient Stability Using a Novel Adaptive Backstepping control law Aritra Mitra

Mriganka Mukherjee

Kishankumar Naik

Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur- 208016, India Email: [email protected]

Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur- 208016, India Email: [email protected]

Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur- 208016, India Email: [email protected]

Abstract—In this paper a novel non linear feedback excitation controller has been proposed for improvement of transient stability of power systems. The design of the controller is based on the adaptive backstepping control principle. The controller takes into consideration system uncertainties. A significant merit of this non linear controller in comparison with existing non linear excitation control schemes is its ability to maintain transient stability even under the occurrence of large disturbances, such as a 3-phase fault at the generator terminals. Another advantage is that the proposed technique does not require the existence of a solution of a designed algebraic Riccati equation unlike the controllers based on the robust nonlinear Direct Feedback Linearization (DFL) technique. Simulation results show the rapid convergence of the power angle and the rotor speed to their equilibrium values following the occurrence of a large sudden fault. Keywords—Adaptive control, Backstepping control, Transient stability, Nonlinear Systems, Synchronous generators.

I.

I NTRODUCTION

Today, the problem of maintaining a high degree of reliability and ensuring stability becomes even more pertinent with large interconnected electric power systems operating more and more closely to their stability limits. To achieve these goals, advanced non-linear excitation control techniques have gained much prominence over the years. These controllers must be designed to prevent an electric power system from losing synchronism following the occurrence of a 3-phase fault. The active electric power collapses in the event of such large disturbances. This in turn leads to acceleration of the rotors of the electric generators due to imbalance between the mechanical and electrical torques acting on the shafts. Enhancement of the post fault region of stability of a power system necessitates the need for auxiliary control action in addition to the rapid fault detection equipment and fast acting circuit breakers [5].

Feedback Linearization (DFL) is one such technique which has been proposed in [12]-[14]. However the DFL based control law design requires exact knowledge of the system parameters. Also, it causes cancellation of several useful nonlinearities. Other non linear control schemes involve the Sliding mode control which has been employed in [17].[16] employs a Particle Swarm Optimization (PSO) technique to search for optimal settings of PSS parameters. Non linear control actions based on the Backstepping design approach have been used in [5]-[8]. In our paper, we propose a novel adaptive backstepping control [19] technique for enhancement of transient stability. One of the merits of the proposed controller includes its ability to model system uncertainties. In this paper, uncertainty in the damping coefficient has been considered. Further, the control action does not depend on the existence of a solution of a designed algebraic Riccati equation, unlike the robust nonlinear DFL based controller in [15]. Although the technique based on DFL and an adaption approach in [18] considers the transient stability and voltage regulation problem of power systems with some uncertainties, they fail to maintain transient stability if a fault occurs at a distance lesser than or equal to one-tenth of the transmission line from the generator. A significant merit of our designed controller is that it manages to ensure transient stability even under the occurrence of a fault at the generator terminals. This will be evident from the simulation results presented in a later section of the paper. The organization of the paper is as follows. Section II describes the dynamical model of the system under consideration. Section III deals with the formulation of the proposed adaptive backstepping control law. In Section IV, the simulation results have been presented. The conclusions are considered in Section V.

Techniques based on linear control theory [1]-[3] prove to be effective only if the post fault region experiences small oscillations. In such cases linearization of the system around the operating point would be a feasible option. However simplified linear models as in [4] can be employed only for small disturbances around a specific operating point and may not guarantee stability if a large disturbance occurs [9]-[11].

For analyzing the effectiveness of the proposed controller, a simplified model of a power system comprising of a single synchronous generator connected to a very large network approximated by an infinite bus, is taken. The model is shown in Fig. 1.

Over the last few years, several non linear excitation control techniques have been proposed for improving transient stability of power systems experiencing large disturbances. Direct

The classical third order synchronous machine model [20] is used in this paper. The equations governing the machine dynamics and generator electric dynamics are

978-1-4799-4445-3/15/$31.00 ©2015 IEEE

II.

DYNAMIC M ODEL

space model where the states represent deviations from an equilibrium point. We define x1 = δ(t) − δ0 , x2 = ω(t) − ω0 , x3 = Eq (t) − eq 0 . where δ0

the operating point of the power angle

ω0

the synchronous speed of the generator

eq 0

the operating point of the quadrature axis EMF of the generator

Fig. 1.

Based on the states defined, the state space model in strictfeedback form is as follows

Simulink model of the power system

˙ = ω(t) − ω0 δ(t) ω(t) ˙ =−

x˙ 1 x˙ 2 x˙ 3

(1)

ω0 Vs Eq (t) D sin δ(t)) (2) (ω(t) − ω0 ) + (Pm − 2H 2H xd

′ Eq (t) Vs Ef (t) E˙ q (t) = − + ′ (xd − xd ) sin δ(t)ω(t) + (3) ′ ′ x Td Td d

where

= x2 = θx2 + f1 + g1 x3 = f2 + g2 u

(4) (5) (6)

D represents the uncertainty in the model due where θ = − 2H to the damping coefficient D,

f1 = g1 =

Vs eq 0 ω0 sin(x1 + δ0 )], (7) [Pm − 2H xd ω0 Vs − sin(x1 + δ0 ), (8) 2Hxd ′ Vs 1 − ′ (eq 0 + x3 ) + ′ (xd − xd ) sin(x1 + δ0 )x2 , x Td d (9) 1 (10) ′ , Td Ef (t) (11)

δ(t)

Power angle (in rad);

ω(t)

Rotor speed (in rad/s);

ω0

Synchronous speed (in rad/s);

D

Damping constant (in p.u);

H

Inertia constant(in secs);

Pm

Mechanical power input to the generator (in p.u);

Vs

Infinite bus voltage (in p.u);

Eq (t)

EMF in the quadrature axis (in p.u);

Ef (t)

Equivalent EMF in the field (in p.u);

xd

d axis reactance (in p.u);

Now, α1 = −c1 x1 is the stabilizing feedback for the first subsystem, where c1 is a positive constant. Let us define z2 = x2 − α1 . From (4) and the definitions of z1 , α1 , and z2 , we have z˙1 = x˙ 1 = x2 = (z2 + α1 ) = (z2 − c1 z1 ) (13)

d axis transient reactance (in p.u);

Let V1 = 12 z1 2 be the Lyapunov function for the first subsystem. Using (13), the time derivative of V1 is

f2 =

g2 = u =

Step 1. Design of the virtual control α1 for the first subsystem. Let z1 = x1 . So,

xd Td

′

′

III.

d axis transient short circuit time constant (in secs);

D ESIGN OF THE P ROPOSED A DAPTIVE BACKSTEPPING C ONTROLLER

In this section we formulate the non linear control law based on the backstepping principle taking into account the uncertainties in the damping coefficient D. Accordingly, estimates of D have been used to design an adaptive controller. In order to apply the control action we need to define a state

z˙1 = x˙ 1

V˙ 1 = z1 (z2 − c1 z1 ) = z1 z2 − c1 z1 2

(12)

(14)

Step 2. Design of the virtual control α2 for the second subsystem. From the definition of z2 , z˙2 = x˙ 2 − α˙1

(15)

From (5), and the definition of α1 , we get z˙2 = θx2 + f1 + g1 x3 + c1 x˙ 1

(16)

From (13), and the definition of z2 , we get z˙2 = θ(z2 + α1 ) + f1 + g1 x3 + c1 (z2 − c1 z1 )

(17)

Let α2 be the stabilizing function for the second sub-system and z3 be defined as follows

From (17), and the definition of z3 , we get z˙2 = θ(z2 + α1 ) + f1 + g1 (z3 + α2 ) + c1 (z2 − c1 z1 ) (18) Let the virtual control for the second sub-system be defined as 1 [−f1 − (v1 + c1 )(z2 − c1 z1 ) − c2 z2 ] g1

where v1 is an estimate of θ and c2 is a positive constant. Replacing the expressions for α1 and α2 in (18), we have z˙2 = (θ − v1 )(z2 − c1 z1 ) + g1 z3 − c2 z2

(19)

Let us define the Lyapunov function V2 as follows 1 1 2 (θ − v1 ) V2 = V1 + z2 2 + 2 2γ where γ represents the adaption gain. Taking the derivative of V2 with respect to time, we have 1 V˙ 2 = V˙ 1 + z2 z˙2 − (θ − v1 )v˙ 1 γ

From (23), (25) and (26), we have ∂α2 x2 V˙ 3 = z1 z2 − c1 z1 2 − c2 z2 2 + z3 [g1 z2 + f2 + g2 u − ∂x1

z3 = x3 − α2

α2 =

From the definition of V3 , its derivative with respect to time is 1 V˙ 3 = V˙ 2 + z3 z˙3 − (θ − v2 )v˙ 2 (26) γ

∂α2 1 ∂α2 (θx2 + f1 + g1 x3 ) − γz2 (z2 − c1 z1 )] − (θ − v2 )v˙ 2 ∂x2 ∂v1 γ (27) Let us define the actual control law u as follows ∂α2 ∂α2 1 x2 + (v2 x2 + f1 + g1 x3 ) u = [−g1 z2 − f2 + g2 ∂x1 ∂x2 ∂α2 γz2 (z2 − c1 z1 ) − c3 z3 ] + ∂v1 (28) −

Replacing u in (27), we get ∂α2 1 V˙ 3 = z1 z2 −c1 z1 2 −c2 z2 2 −c3 z3 2 −z3 (θ−v2 )x2 − (θ−v2 )v˙ 2 ∂x2 γ (29) Simplifying (29), we get ∂α2 1 V˙ 3 = z1 z2 − c1 z1 2 − c2 z2 2 − c3 z3 2 − (θ − v2 )(z3 x2 + v˙ 2 ) ∂x2 γ

(20)

From (14), (19) and (20), we have

(30) Let us define the parameter update law for v2 as follows

1 ∂α2 V˙ 2 = z1 z2 −c1 z1 2 +z2 [(θ−v1 )(z2 −c1 z1 )+g1 z3 −c2 z2 ]− (θ−v1 )v˙ 1 v˙ 2 = −γz3 x2 γ ∂x2 (21) From the definition of v˙ 2 and (30), we have Simplifying (21), we get V˙ 3 = z1 z2 − c1 z1 2 − c2 z2 2 − c3 z3 2 (31) 1 V˙ 2 = z1 z2 −c1 z1 2 −c2 z2 2 +g1 z3 z2 +(θ−v1 )[z2 (z2 −c1 z1 )− v˙ 1 ] 2 1 2 1 2 γ Now, (z1 − z2 ) ≥ 0∀z1 , z2 . Thus, z1 z2 − 2 z1 − 2 z2 ≤ 0∀z1 , z2 . Let c1 , c2 be chosen such that c1 > 12 and c2 > 21 . (22) Then, as c3 is a positive constant, z1 z2 < c1 z1 2 +c2 z2 2 +c3 z3 2 Let v˙ 1 = γz2 (z2 − c1 z1 ) be the parameter update law for v1 . for non-zero values of z1 , z2 and z3 . Hence, we can conclude Using this parameter update law and (21), the derivative for that V˙ 3 < 0 for non-zero values of z1 , z2 and z3 and V˙ 3 = 0 V2 takes the form only if z1 = z2 = z3 = 0. 2 2 ˙ V2 = z1 z2 − c1 z1 − c2 z2 + g1 z3 z2 (23) Thus, V˙ 3 is negative definite. As V˙ 3 is negative definite, the T equilibrium state z = [z1 , z2 , z3 ] is globally asymptotically Step 3. Design of the actual control law u stable. Thus, the fictitious stabilizing functions α1 = −c1 z1 and α2 = g11 [−f1 − (v1 + c1 )(z2 − c1 z1 ) − c2 z2 ], along rd For the 3 sub-system, let us define the Lyapunov function with the control law u given by (28) guarantee that as V3 as follows t → ∞, z1 , z2 , z3 → 0. From the definitions of z1 , z2 and 1 1 z3 , it is clear that z1 , z2 , z3 → 0 implies x1 , x2 , x3 → 0. Thus, V3 = V2 + z3 2 + (θ − v2 )2 2 2γ the controller causes δ to reach δ0 and ω to reach ω0 . Here v2 is another estimate of θ. From (6), and the definitions of z3 and α2 , we have z˙3 = x˙ 3 − α˙ 2 = f2 + g2 u −

∂α2 ∂α2 ∂α2 x˙ 1 − x˙ 2 − v˙ 1 (24) ∂x1 ∂x2 ∂v1

IV.

S IMULATION R ESULTS

For simulation, the power system model consisting of a synchronous machine connected to an infinite bus, shown in Fig. 1 is used. This model is developed in Simulink platform of MATLAB. The system parameters are given below:

From (4), (5), (24) and the parameter update law for v1 , we have δ0 = 21.6860, N0 = 1800 RPM (nominal rotor speed), ∂α2 ∂α2 ∂α2 z˙3 = f2 +g2 u− x2 − (θx2 +f1 +g1 x3 )− γz2 (z2 −c1 z1 ) e = 0.9707 p.u, H = 4 secs, x = 1.078 p.u, x ′ = .1505 d d q0 ∂x1 ∂x2 ∂v1 p.u, Pm = .4568 p.u, Vs = 1 p.u. (25)

Fig. 2a.

Power angle in absence of controller

Fig. 3b.

Rotor speed in presence of controller

For the system shown in Fig. 1, a 3-phase symmetric fault occurs at the generator terminals at t = 1 sec and is cleared at t = 1.1 sec. The transient stability of the system in terms of power angle and rotor speed deviations from their nominal values is studied. Fig. 2(a) shows the power angle variations in absence of the control action. Fig. 2(b) shows the rotor speed variations in absence of the control action. Figures 3(a) and 3(b) show the stabilizing action of the proposed controller on the power angle and rotor speed deviations respectively. It is seen that the controller minimizes rotor speed oscillations and simultaneously causes rapid convergence of the power angle and rotor speed to their nominal values. Thus, from these figures, the improvement in transient stability is evident even under the sudden occurrence of the 3-phase fault at the generator terminals. Fig. 2b.

V.

Rotor speed in absence of controller

C ONCLUSION

In this paper, an adaptive backstepping principle based non linear excitation controller has been designed and implemented with the purpose of enhancing transient stability of power systems. The design procedure is recursive in nature and basically involves the determination of virtual control laws for each subsystem based on the Lyapunov stability criterion. A major advantage of backstepping is that it has the flexibility to avoid cancellations of useful nonlinearities and pursue the objectives of stabilization and tracking, rather than that of linearization. The effectiveness of the proposed controller is reflected in the simulation results. Unlike existing non linear excitation controllers, the proposed control action guarantees transient stability even for 3- phase faults at the generator terminals. The control law takes into account the uncertainties in system damping and ensures rapid convergence of the system states to their equilibrium values. R EFERENCES [1] Fig. 3a.

Power angle in presence of controller

Y.N. Yu and G. Siggers, “Stabilization and optimal control signals for a power system”, IEEE Trans PAS, 91:1469-1480, 1971. [2] A.V. Machias, E.N. Dialynas, and C.D. Papageorgiou, “Application of an optimal control reduction algorithm to a power system”, Found.Contro1 Eng, 12(4):181-194, 1987.

[3]

S . Ammari, Y. Besanger, N. Hadj-Said, and D. Georges, “Robust solutions for the interaction phenomena between dynamic loads and facts controllers”, IEEE Power Engineering Society Summer Meeting 2000, 1:401-406, 2000. [4] F.P. deMello and C.Concordia, “Concepts of synchronous machine stability as affected by excitation control”, IEEE Trans. Power Appr.syst., vol.88,pp.316-329, 1969. [5] R.Yan, Z.Y.Dong, T.K.Saha and R.Majumder, “Power system transient stability enhancement with an Adaptive Control Scheme Using Backstepping Design”, Power Engineering Society General Meeting, IEEE Digital Object Identifier, 2007. [6] Jun Fu and Jun Zhao, “Robust Nonlinear Excitation Control Based on A Novel Adaptive Back-stepping Design for Power Systems”, American Control Conference, Proceedings of the 2005 Digital Object Identifier, vol.4,pp.2715-2720, 2005. [7] Ali Reza Roosta, Didier Georges and Nouredine Hadj-Said, “Nonlinear Control for Power Systems based on a Backstepping Method”, Proceedings of the 40th IEEE Conference on Decsion and Control, vol.4, pp.30373042, 2001. [8] Ming-Shan Kung, Yu-Feng Kuo and Chih-Chiang Cheng, “Design of the Robust Backstepping Tracking Controllers for Synchronous Generators”, Proceedings of SICE Annual Conference , pp.300-306, 2010. [9] M. Jalili and M. J. Yazdanpanah, “Transient stability enhancement of power systems via optimal nonlinear state feedback control”, Electrical Engineering, vol. 89, pp. 149-156, 2006. [10] M. Galaz, R. Ortega, A. S. Bazanella, and A. M. Stankovic, “An energy shaping approach to the design of excitation control of synchronous generators”, Automatica, vol. 39, pp. 111-119, 2003. [11] A. G. Loukianov, J. M. Canedo, V. I. Utkin, and J. Cabrera-Vazquez, “Discontinuous controller for power systems: sliding-mode block control approach”, IEEE Transactions on Industrial Electronics, vol. 51, no. 2,pp. 340-353, April, 2004. [12] Youyi Wang, David J. Hill, Richard H. Middleton, and Long Gao, “Transient stability enhancement and voltage regulation of power systems”, IEEE Transactions on Power Systems, vol. 8, no. 2, pp. 620627, May, 1993. [13] L. Gao, L. Chen, Y. Fan, and H. Ma, “A nonlinear control design for power systems”, Automatica, vol. 28, no. 5, pp. 975979, Sep, 1992. [14] J. W. Chapman and M. D. Ilic, “Stabilizing a multimachine power system via decentralized feedback linearizing excitation control”, IEEE Transactions on Power Systems, vol. 8, pp. 830839, 1993. [15] Youyi Wang and David J. Hill, “Robust nonlinear coordinated control of power systems”, Automatica, vol. 32, no. 4, pp. 611618, 1996. [16] M. A. Abido, “Optimal design of power-system stabilizers using particle swarm optimization”, IEEE Transactions Power Systems, vol. 17, no. 3, pp. 406413, 2002. [17] Y.Cao, L.Jiang and S.Cheng et al., “A nonlinear variable structure stabilizer for power system stability”, IEEE Trans. Energy Conversion, vol. 9 (3),1994, pp 489-495. [18] Youyi Wang, David J. Hill, Rick H. Middleton, and Long Gao, “Transient stabilization of power systems with an adaptive control law”, Automatica, vol. 30, no. 9, pp. 14091413, 1994. [19] M. Krsti´c, I. Kanellakopoulos, and P. V. Kokotovi´c, Nonlinear and Adaptive Control Design, New York: Wiley, 1995. [20] P. Kundur, Power Systems Stability and Control, The EPRI Power System Engineering Series. McGraw-Hill, New York, 1994.

Lihat lebih banyak...
Mriganka Mukherjee

Kishankumar Naik

Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur- 208016, India Email: [email protected]

Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur- 208016, India Email: [email protected]

Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur- 208016, India Email: [email protected]

Abstract—In this paper a novel non linear feedback excitation controller has been proposed for improvement of transient stability of power systems. The design of the controller is based on the adaptive backstepping control principle. The controller takes into consideration system uncertainties. A significant merit of this non linear controller in comparison with existing non linear excitation control schemes is its ability to maintain transient stability even under the occurrence of large disturbances, such as a 3-phase fault at the generator terminals. Another advantage is that the proposed technique does not require the existence of a solution of a designed algebraic Riccati equation unlike the controllers based on the robust nonlinear Direct Feedback Linearization (DFL) technique. Simulation results show the rapid convergence of the power angle and the rotor speed to their equilibrium values following the occurrence of a large sudden fault. Keywords—Adaptive control, Backstepping control, Transient stability, Nonlinear Systems, Synchronous generators.

I.

I NTRODUCTION

Today, the problem of maintaining a high degree of reliability and ensuring stability becomes even more pertinent with large interconnected electric power systems operating more and more closely to their stability limits. To achieve these goals, advanced non-linear excitation control techniques have gained much prominence over the years. These controllers must be designed to prevent an electric power system from losing synchronism following the occurrence of a 3-phase fault. The active electric power collapses in the event of such large disturbances. This in turn leads to acceleration of the rotors of the electric generators due to imbalance between the mechanical and electrical torques acting on the shafts. Enhancement of the post fault region of stability of a power system necessitates the need for auxiliary control action in addition to the rapid fault detection equipment and fast acting circuit breakers [5].

Feedback Linearization (DFL) is one such technique which has been proposed in [12]-[14]. However the DFL based control law design requires exact knowledge of the system parameters. Also, it causes cancellation of several useful nonlinearities. Other non linear control schemes involve the Sliding mode control which has been employed in [17].[16] employs a Particle Swarm Optimization (PSO) technique to search for optimal settings of PSS parameters. Non linear control actions based on the Backstepping design approach have been used in [5]-[8]. In our paper, we propose a novel adaptive backstepping control [19] technique for enhancement of transient stability. One of the merits of the proposed controller includes its ability to model system uncertainties. In this paper, uncertainty in the damping coefficient has been considered. Further, the control action does not depend on the existence of a solution of a designed algebraic Riccati equation, unlike the robust nonlinear DFL based controller in [15]. Although the technique based on DFL and an adaption approach in [18] considers the transient stability and voltage regulation problem of power systems with some uncertainties, they fail to maintain transient stability if a fault occurs at a distance lesser than or equal to one-tenth of the transmission line from the generator. A significant merit of our designed controller is that it manages to ensure transient stability even under the occurrence of a fault at the generator terminals. This will be evident from the simulation results presented in a later section of the paper. The organization of the paper is as follows. Section II describes the dynamical model of the system under consideration. Section III deals with the formulation of the proposed adaptive backstepping control law. In Section IV, the simulation results have been presented. The conclusions are considered in Section V.

Techniques based on linear control theory [1]-[3] prove to be effective only if the post fault region experiences small oscillations. In such cases linearization of the system around the operating point would be a feasible option. However simplified linear models as in [4] can be employed only for small disturbances around a specific operating point and may not guarantee stability if a large disturbance occurs [9]-[11].

For analyzing the effectiveness of the proposed controller, a simplified model of a power system comprising of a single synchronous generator connected to a very large network approximated by an infinite bus, is taken. The model is shown in Fig. 1.

Over the last few years, several non linear excitation control techniques have been proposed for improving transient stability of power systems experiencing large disturbances. Direct

The classical third order synchronous machine model [20] is used in this paper. The equations governing the machine dynamics and generator electric dynamics are

978-1-4799-4445-3/15/$31.00 ©2015 IEEE

II.

DYNAMIC M ODEL

space model where the states represent deviations from an equilibrium point. We define x1 = δ(t) − δ0 , x2 = ω(t) − ω0 , x3 = Eq (t) − eq 0 . where δ0

the operating point of the power angle

ω0

the synchronous speed of the generator

eq 0

the operating point of the quadrature axis EMF of the generator

Fig. 1.

Based on the states defined, the state space model in strictfeedback form is as follows

Simulink model of the power system

˙ = ω(t) − ω0 δ(t) ω(t) ˙ =−

x˙ 1 x˙ 2 x˙ 3

(1)

ω0 Vs Eq (t) D sin δ(t)) (2) (ω(t) − ω0 ) + (Pm − 2H 2H xd

′ Eq (t) Vs Ef (t) E˙ q (t) = − + ′ (xd − xd ) sin δ(t)ω(t) + (3) ′ ′ x Td Td d

where

= x2 = θx2 + f1 + g1 x3 = f2 + g2 u

(4) (5) (6)

D represents the uncertainty in the model due where θ = − 2H to the damping coefficient D,

f1 = g1 =

Vs eq 0 ω0 sin(x1 + δ0 )], (7) [Pm − 2H xd ω0 Vs − sin(x1 + δ0 ), (8) 2Hxd ′ Vs 1 − ′ (eq 0 + x3 ) + ′ (xd − xd ) sin(x1 + δ0 )x2 , x Td d (9) 1 (10) ′ , Td Ef (t) (11)

δ(t)

Power angle (in rad);

ω(t)

Rotor speed (in rad/s);

ω0

Synchronous speed (in rad/s);

D

Damping constant (in p.u);

H

Inertia constant(in secs);

Pm

Mechanical power input to the generator (in p.u);

Vs

Infinite bus voltage (in p.u);

Eq (t)

EMF in the quadrature axis (in p.u);

Ef (t)

Equivalent EMF in the field (in p.u);

xd

d axis reactance (in p.u);

Now, α1 = −c1 x1 is the stabilizing feedback for the first subsystem, where c1 is a positive constant. Let us define z2 = x2 − α1 . From (4) and the definitions of z1 , α1 , and z2 , we have z˙1 = x˙ 1 = x2 = (z2 + α1 ) = (z2 − c1 z1 ) (13)

d axis transient reactance (in p.u);

Let V1 = 12 z1 2 be the Lyapunov function for the first subsystem. Using (13), the time derivative of V1 is

f2 =

g2 = u =

Step 1. Design of the virtual control α1 for the first subsystem. Let z1 = x1 . So,

xd Td

′

′

III.

d axis transient short circuit time constant (in secs);

D ESIGN OF THE P ROPOSED A DAPTIVE BACKSTEPPING C ONTROLLER

In this section we formulate the non linear control law based on the backstepping principle taking into account the uncertainties in the damping coefficient D. Accordingly, estimates of D have been used to design an adaptive controller. In order to apply the control action we need to define a state

z˙1 = x˙ 1

V˙ 1 = z1 (z2 − c1 z1 ) = z1 z2 − c1 z1 2

(12)

(14)

Step 2. Design of the virtual control α2 for the second subsystem. From the definition of z2 , z˙2 = x˙ 2 − α˙1

(15)

From (5), and the definition of α1 , we get z˙2 = θx2 + f1 + g1 x3 + c1 x˙ 1

(16)

From (13), and the definition of z2 , we get z˙2 = θ(z2 + α1 ) + f1 + g1 x3 + c1 (z2 − c1 z1 )

(17)

Let α2 be the stabilizing function for the second sub-system and z3 be defined as follows

From (17), and the definition of z3 , we get z˙2 = θ(z2 + α1 ) + f1 + g1 (z3 + α2 ) + c1 (z2 − c1 z1 ) (18) Let the virtual control for the second sub-system be defined as 1 [−f1 − (v1 + c1 )(z2 − c1 z1 ) − c2 z2 ] g1

where v1 is an estimate of θ and c2 is a positive constant. Replacing the expressions for α1 and α2 in (18), we have z˙2 = (θ − v1 )(z2 − c1 z1 ) + g1 z3 − c2 z2

(19)

Let us define the Lyapunov function V2 as follows 1 1 2 (θ − v1 ) V2 = V1 + z2 2 + 2 2γ where γ represents the adaption gain. Taking the derivative of V2 with respect to time, we have 1 V˙ 2 = V˙ 1 + z2 z˙2 − (θ − v1 )v˙ 1 γ

From (23), (25) and (26), we have ∂α2 x2 V˙ 3 = z1 z2 − c1 z1 2 − c2 z2 2 + z3 [g1 z2 + f2 + g2 u − ∂x1

z3 = x3 − α2

α2 =

From the definition of V3 , its derivative with respect to time is 1 V˙ 3 = V˙ 2 + z3 z˙3 − (θ − v2 )v˙ 2 (26) γ

∂α2 1 ∂α2 (θx2 + f1 + g1 x3 ) − γz2 (z2 − c1 z1 )] − (θ − v2 )v˙ 2 ∂x2 ∂v1 γ (27) Let us define the actual control law u as follows ∂α2 ∂α2 1 x2 + (v2 x2 + f1 + g1 x3 ) u = [−g1 z2 − f2 + g2 ∂x1 ∂x2 ∂α2 γz2 (z2 − c1 z1 ) − c3 z3 ] + ∂v1 (28) −

Replacing u in (27), we get ∂α2 1 V˙ 3 = z1 z2 −c1 z1 2 −c2 z2 2 −c3 z3 2 −z3 (θ−v2 )x2 − (θ−v2 )v˙ 2 ∂x2 γ (29) Simplifying (29), we get ∂α2 1 V˙ 3 = z1 z2 − c1 z1 2 − c2 z2 2 − c3 z3 2 − (θ − v2 )(z3 x2 + v˙ 2 ) ∂x2 γ

(20)

From (14), (19) and (20), we have

(30) Let us define the parameter update law for v2 as follows

1 ∂α2 V˙ 2 = z1 z2 −c1 z1 2 +z2 [(θ−v1 )(z2 −c1 z1 )+g1 z3 −c2 z2 ]− (θ−v1 )v˙ 1 v˙ 2 = −γz3 x2 γ ∂x2 (21) From the definition of v˙ 2 and (30), we have Simplifying (21), we get V˙ 3 = z1 z2 − c1 z1 2 − c2 z2 2 − c3 z3 2 (31) 1 V˙ 2 = z1 z2 −c1 z1 2 −c2 z2 2 +g1 z3 z2 +(θ−v1 )[z2 (z2 −c1 z1 )− v˙ 1 ] 2 1 2 1 2 γ Now, (z1 − z2 ) ≥ 0∀z1 , z2 . Thus, z1 z2 − 2 z1 − 2 z2 ≤ 0∀z1 , z2 . Let c1 , c2 be chosen such that c1 > 12 and c2 > 21 . (22) Then, as c3 is a positive constant, z1 z2 < c1 z1 2 +c2 z2 2 +c3 z3 2 Let v˙ 1 = γz2 (z2 − c1 z1 ) be the parameter update law for v1 . for non-zero values of z1 , z2 and z3 . Hence, we can conclude Using this parameter update law and (21), the derivative for that V˙ 3 < 0 for non-zero values of z1 , z2 and z3 and V˙ 3 = 0 V2 takes the form only if z1 = z2 = z3 = 0. 2 2 ˙ V2 = z1 z2 − c1 z1 − c2 z2 + g1 z3 z2 (23) Thus, V˙ 3 is negative definite. As V˙ 3 is negative definite, the T equilibrium state z = [z1 , z2 , z3 ] is globally asymptotically Step 3. Design of the actual control law u stable. Thus, the fictitious stabilizing functions α1 = −c1 z1 and α2 = g11 [−f1 − (v1 + c1 )(z2 − c1 z1 ) − c2 z2 ], along rd For the 3 sub-system, let us define the Lyapunov function with the control law u given by (28) guarantee that as V3 as follows t → ∞, z1 , z2 , z3 → 0. From the definitions of z1 , z2 and 1 1 z3 , it is clear that z1 , z2 , z3 → 0 implies x1 , x2 , x3 → 0. Thus, V3 = V2 + z3 2 + (θ − v2 )2 2 2γ the controller causes δ to reach δ0 and ω to reach ω0 . Here v2 is another estimate of θ. From (6), and the definitions of z3 and α2 , we have z˙3 = x˙ 3 − α˙ 2 = f2 + g2 u −

∂α2 ∂α2 ∂α2 x˙ 1 − x˙ 2 − v˙ 1 (24) ∂x1 ∂x2 ∂v1

IV.

S IMULATION R ESULTS

For simulation, the power system model consisting of a synchronous machine connected to an infinite bus, shown in Fig. 1 is used. This model is developed in Simulink platform of MATLAB. The system parameters are given below:

From (4), (5), (24) and the parameter update law for v1 , we have δ0 = 21.6860, N0 = 1800 RPM (nominal rotor speed), ∂α2 ∂α2 ∂α2 z˙3 = f2 +g2 u− x2 − (θx2 +f1 +g1 x3 )− γz2 (z2 −c1 z1 ) e = 0.9707 p.u, H = 4 secs, x = 1.078 p.u, x ′ = .1505 d d q0 ∂x1 ∂x2 ∂v1 p.u, Pm = .4568 p.u, Vs = 1 p.u. (25)

Fig. 2a.

Power angle in absence of controller

Fig. 3b.

Rotor speed in presence of controller

For the system shown in Fig. 1, a 3-phase symmetric fault occurs at the generator terminals at t = 1 sec and is cleared at t = 1.1 sec. The transient stability of the system in terms of power angle and rotor speed deviations from their nominal values is studied. Fig. 2(a) shows the power angle variations in absence of the control action. Fig. 2(b) shows the rotor speed variations in absence of the control action. Figures 3(a) and 3(b) show the stabilizing action of the proposed controller on the power angle and rotor speed deviations respectively. It is seen that the controller minimizes rotor speed oscillations and simultaneously causes rapid convergence of the power angle and rotor speed to their nominal values. Thus, from these figures, the improvement in transient stability is evident even under the sudden occurrence of the 3-phase fault at the generator terminals. Fig. 2b.

V.

Rotor speed in absence of controller

C ONCLUSION

In this paper, an adaptive backstepping principle based non linear excitation controller has been designed and implemented with the purpose of enhancing transient stability of power systems. The design procedure is recursive in nature and basically involves the determination of virtual control laws for each subsystem based on the Lyapunov stability criterion. A major advantage of backstepping is that it has the flexibility to avoid cancellations of useful nonlinearities and pursue the objectives of stabilization and tracking, rather than that of linearization. The effectiveness of the proposed controller is reflected in the simulation results. Unlike existing non linear excitation controllers, the proposed control action guarantees transient stability even for 3- phase faults at the generator terminals. The control law takes into account the uncertainties in system damping and ensures rapid convergence of the system states to their equilibrium values. R EFERENCES [1] Fig. 3a.

Power angle in presence of controller

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