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Robust nonlinear control of transient stability of power systems

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Proceedings of the 42nd IEEE
Conference on Decision and Control
Maui, Hawaii USA, December 2003 TuA09-4

Robust Nonlinear Control of Transient Stability

of Power Systems

Brahim Behar, Françoise Lamnabhi-Lagarrigue Tarek Ahmed-Ali

Laboratoire des Signaux et systèmes Ecole Nationale Supérieure des Etudes
L2S-Supélec-CNRS et Techniques d’Armement
3, Rue Joliot Curie 2-Rue Francois Verny
91192 Gif sur Yvette, France. 29806 Brest cedex 09, France
{behar,lamnabhi}@lss.supelec.fr Ahmedali@ensieta.fr

Abstract— In This paper we present a robust nonlinear exci- Infinite Bus

tation controller of synchronous generator connected through Vt
one transmission line to an infinite bus (SMIB) to enhance tran-
sient stability. Simulation results show that transient stability Generator Transformer
of a Single Machine Infinite Bus power system under a large
sudden fault can be improved.
G
I. INTRODUCTION
In this paper we present a new solution to stabilization of
a single machine connected through one transmission line to Transmission line
infinite bus (SMIB), as it is shown in Figure 1. The direct Fig. 1. A single machine infinite bus power system (SMIB)
current motor or turbine transmits the mechanic power Pm
with the generator, and this synchronous generator transmits
the electric power Pe through transmission line with infinite
bus. control loops be written as follow [4][9]
In practical, this system (SMIB) is subject to various distur-
bances, as transmission line xL , variation of desired terminal Mechanical equations:
voltage Vs , load conditions and network interconnection. δ̇(t) = w(t) (1)
Various stabilizing control results based on nonlinear power
D w0
system models have been obtained [3,9,11] for single ma- ẇ(t) = − w(t) + [Pm (t) − P e(t)] (2)
chine systems, this controllers which can only deal with local 2H 2H
stability arround an operating point, the controlled system where
can endure large disturbances and retain a steady post-fault δ(t)[rad] is the power angle of the generator relative to the
condition, on the other hand, these controlllers do not ensure angle of the infinite bus rotating at synchronous speed ws
the regulation of the terminal voltage. w(t) = wg (t) − ws (t)[rad/s]
In this paper, we design a robust nonlinear control to maintain wg is the generator angular speed
the transient stability and voltage regulation of a power H(s) is the per unit inertia constant
system when subjected to a severe disturbances. D is the per unit damping constant
The paper is organized as follows. In the section II a Pm is the input mechanical power in per unit
description of mathematical algebraic model of a single Pe is the active electric power delivered by the generator to
machine infinite bus in the to axis d and q is given. In the infinite bus.
the section III we present and test the existing controller 0

performances as Direct Feedback Linearization and Voltage The quadrature Eq and the transient quadrature Eq are
Controller from on different fault sequences. In the section given by
IV, a robust nonlinear excitation controller will be presented 0 1
and tested. And we finished with results simulations and Ėq = [Ef (t) − Eq (t)] 0 (3)
Td0
conclusions.
Ef (t) = Kc uf (t)
II. DESCRIPTION OF MATHEMATICAL POWER
SYSTEM MODEL Electrical equations
The quadrature’s EMF Eq (t) is given by
A simple dynamical model of a single machine infinite 0
bus power system is considered in this paper. xds 0 xd − x d
Eq = 0 Eq − 0 Vs cos δ = xad If (4)
A model for the generator with both excitation and power xds xds

0-7803-7924-1/03/$17.00 ©2003 IEEE 294

The active power Pe is given by [7] A. Feed-Back linearization nonlinear control
Vs Eq (t) The system (1), (10), (11) is can represented in the forme
Pe (t) = sinδ(t) (5) as (-see[9] (DFL Transformation) ):
xds
Vs is the voltage infinite bus, xds is the total reactance : δ̇(t) = w(t) (12)
0
xds = xT + 21 xL + xd and xds is the total transient reactance D w0
0 0
: xds = xT + 21 xL + xd . ẇ(t) = − w(t) − ∆Pe (t) (13)
2H 2H
The quadrature axis current Iq is given by
˙ e (t) = − 10 ∆Pe − 10 vf
∆P (14)
Vs Pe T T
Iq (t) = sin δ(t) = (6) where
xds xad If
∆Pe = Pe − Pm0 ,
1 2 2 1
Vt (t) = [x E (t) + Vs2 x2d + 2xs xd xds Pe (t) cot δ(t)] 2 0
xds 0
xds s q 0
T = T ,
(7) xds d0
Substituting (5) into (2) and (4) into (3), we obtain the state Vs 0 0 Vs
space model: vf = sin δ[Kc uf + Td0 (xd − xd ) sin δw]
xds xds
δ̇(t) = w(t) 0 Vs
+T Eq cos δw − Pm0 . (15)
D w0 Vs 0 xds
ẇ(t) = − w(t) + [Pm (t) − 0 Eq (t) sin δ(t) The feed-back linearization compensating control law is
2H 2H Xds
obtained as :
0
xd − x d 2 vf = −(αδ δ + αw w + α∆Pe ∆Pe ) (16)
+ V sinδ(t) cos δ(t)] (8)
xds xds s
0
Where αδ , αw and α∆Pe are the linear gains obtained from
the solutions of Algebraic Riccati Equation (ARE) [9]
0
0 xds 0 xd − x d The real excitation control is given by [9]:
Ėq = [Kc uf − 0 E (t) + Vs cosδ(t)] (9)
xds q
0
xds xds 0 Vs
uf = [vf − T Eq cos δ(t)w(t) + Pm0 ]−
0
Kc Vs sin δ(t) xds
in which (δ, w, Eq ) is the states and uf is the control 0 0 Vs
0
input. since Pe is measurable while Eq is not, it is convenient −Td0 (xd − xd ) sin δ(t)w(t) (17)
Kc xds
to express the state space model using (δ, w, Pe ) as states
B. Voltage Controller
[1].
Differentiating (5), and using (4) and (9), we obtain The voltage regulation is an important issue particularly in
the post-transient period. Its basic to regulate the voltage to
reach its nominal value. Voltage controllers have been given
δ̇(t) = w(t)
in [3] using LQ-optimal techniques and in [11] using a linear
robust control technique. Both of them have the problem that
D w0
ẇ(t) = − w(t) − [Pe (t) − P m(t)] (10) they deteriorate transient stability over the whole operating
2H 2H region. For example, as proposed in [11], differentiating
1 Vs equation (15) gives
Ṗe = − 0 {Pe − [ sin δ(t)(Kc uf +
Td0 xds f2 (t) f2 (t)
∆V̇t = f1 (t)w + 0 ∆Pe + 0 vf (18)
Td0 Td0
0 Vs 0 0
Td0 (xd − xd ) 0 w sin δ(t)) + Td0 Pe wcosδ(t)} (11) where f1 (t) and f2 (t) are highly nonlinear functions of δ, Pe
xds
and Vt (see [11] for details). A new linearized system which
0 x
0 is represented by the vector [∆, Vt , w, Pe ] can be developed.
Where Td0 = xd0ds
Td0 is the direct axis transient short circuit Robust linear control techniques can be applied to obtain
time constant.
vf = −kv ∆Vt − kw w − kp ∆Pe (19)
III. TRANSIENT STABILITY CONTROL PROBLEM
where kv , kw , kP are linear gains dependent on the bounds
Several controllers performances has used and tested on of f1 (t) and f2 (t).
different publication for the studies of the transient stability However, the voltage controller achieves the voltage reg-
[5] [9]. ulation, but is only valid locally as shown in simulations.

295
C. simulation of existing controllers IV. ROBUST NONLINEAR CONTROLLER
The parameters of the SMIB power system which is In this section, we present a new robust nonlinear
shown in Figure 1 are as follow: controller which ensures asymptotic stability in presence of
0
xd = 1.863, xd = 0.257, xT = 0.127, unknowns disturbances bounded by an unknown bound.
0
Td0 = 6.9, xL = 0.4853, H = 4, D = 5,
Kc = 1, xad = 1.712, w0 = 314.159. Theorem
Let us consider the following class of uncertain nonlinear
The operating point of the power system used in the systems
simulations is :
ẋ = f (x) + g(x)ξ + η(x, ξ, t) (22)
δ0 = 72deg, Pm0 = 0.9p.u., Vt0 = 1.0p.u..
ξ˙ = u (23)
The faults with its sequences described as
where x ∈ R, ξ ∈ R and u ∈ Rn . We suppose that the
Case 1.
following assumptions holds :
Stage 1: The system is in a pre-fault steady state;
A1: The uncertainties η(x, ξ, t) are bounded by unknown
Stage 2: At t = t0 to t = t1 the voltage decrease has 0.1p.u.;
positive constant µ.
Stage 3: At t = t2 to t = t3 the mechanical power increase
A2 :There exist a positive Lyapunov function V which
has 30%;
satistfies
Stage 4: The system is in a post-fault state.
Case 2. c1 |x|2 ≤ V (x) ≤ c2 |x|2
Stage 1: The system is in a pre-fault steady state; ∂V
Stage 2: At t = t0 to t = t1 the voltage decrease has 0.1p.u.; | | ≤ c3 |x|
∂x
Stage 3: At t = t2 to t = t3 the mechanical power increase (24)
has 30%;
Stage 4: At t = t4 to t = t5 the voltage decrease has 0.1p.u.; and a smooth function α(x) such that
Stage 5: The system is in a post-fault state. ∂V
Case 3. [f (x) + g(x)α(x)] ≤ −β0 |x|2 . (25)
∂x
Stage 1: The system is in a pre-fault steady state;
Stage 2: At t = t0 to t = t1 the voltage decrease has 0.1p.u.; where α(0) = 0 and β0 , c1 , c2 , c3 are positive constants.
Stage 3: At t = t2 the mechanical power increase has 30%; Then there exist a controller U (x, ξ, t) such that the system
Stage 4: At t = t4 to t = t5 the voltage decrease has 0.1p.u.; is asymptotically stable in closed loop.
Stage 5: The system is in a post-fault state. Proof
In order to derive the controller U (x, ξ, t), let us consider
We choose in the simulations t0 = 0.1s, t1 = 0.25s, the sub-system
t2 = 1s, t3 = 1.4s, t4 = 1.5s, and t5 = 1.65s ẋ = f (x) + g(x)ξ + η(x, ξ, t) (26)

The controllers employed in the simulations are [10] [11]: and the following Lyapunov function
1
DFL nonlinear controller: W = V + (µ − µ̂)2 (27)
2
vf = 22.36δ + 12.81w − 82.45∆Pe (20) then it’s first derivative will be
∂V ∂V ˙ − µ̂)
Voltage controller: Ẇ = [f (x) + g(x)ξ] + η(x, ξ, t) − µ̂(µ
∂x ∂x
vf = −40.14∆Vt + 10.11w − 30.81∆Pe (21) Now, let us consider ξ as input and suppose that

ξ = ξ¯ = α(x) + α1 (x, t) (28)

From the simulations results (Fig.2, Fig.3 and Fig.4) it can
be observed that the transient nonlinear controller (DFL then we can write
controller) stabilize the disturbed system only in the Case
∂V ∂V ˙
1 but not in the Case 2 and 3. The voltage controller can not Ẇ = [f (x)+g(x)α(x)+α1 (x, t)]+ η(x, ξ, t)−µ̂(µ−µ̂)
∂x ∂x
stabilize the disturbed system in the Case 3.
In the next section, we will propose a new robust nonlinear this means that
excitation control able to ensure the stability of the disturbed ∂V ∂V
Ẇ ≤ −β0 |x|2 + g(x)α1 (x, t) + | ˙ − µ̂)
|µ − µ̂(µ
system in the three cases. ∂x ∂x

296
 150 5

140
4

Speed deviation (Rad/s)

130
3

Angle (Degrees)
120

2
110

100 1

90 0

80
−1

70
−2
60

−3
50

40 −4
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Tiem (sec)

1.2 2.5

2
Terminal voltage (p.u.)

Electrical Power (p.u.)

1

1.5

0.8

0.6

0.5

0.4
0

0.2 −0.5
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

Fig. 2. Power system responses for Case 1 : (- -) DFL nonlinear controller, (-) Voltage controller

150 6

140 5

Speed deviation (Rad/s)

130
4
Angle (Degrees)

120
3

110
2
100
1
90
0
80

−1
70

−2
60

50 −3

40 −4
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

1.4 5

1.2
Terminal Voltage (p.u.)

Electrical Power (p.u.)

1
3

0.8

0.6

1
0.4

0
0.2

0 −1
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

Fig. 3. Power system responses for Case 2 : (- -) DFL nonlinear controller, (-) Voltage controller

150 10

140
8
Speed deviation (Rad/s)

130
Angle (Degrees)

120 6

110

4
100

90
2

80

70 0

60
−2
50

40 −4
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

1.4 5

1.2
4
Terminal voltage (p.u.)

Electrical power (p.u.)

1
3

0.8

0.6

1
0.4

0
0.2

0 −1
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

Fig. 4. Power system responses for Case 3 : (- -) DFL nonlinear controller, (-) Voltage controller

Now, if we use the following controller By using Barbalat lemma, we deduce that x → 0 asymptot-
ically.
1 µ̂2 ∂V
∂x
In order to derive the global controller U (x, ξ, t), we consider
α1 (x, t) = − the following augmented Lyapunov function
g(x) |µ̂ ∂V
∂x | + e
−λt

µ̂˙ = |x| 1 ¯ 2 + 1 (µ̂1 − µ)2 .

W1 = W + (ξ − ξ) (29)
2 2
we can say that Then it’s first derivative is

Ẇ ≤ −β0 ||x||2 + e−λt . Ẇ1 = Ẇ + (ξ − ξ)(u ˙¯ + µˆ˙ (µ̂ − µ)

¯ − ξ) 1 1

297
then where a, b are positive contants. By replacing ∆Pe by it’s
¯ value, we obtain
Ẇ1 ≤ −β0 ||x||2 + e−λt + (ξ − ξ)[u ¯ − ∂ ξ (f (x) + g(x)ξ)
∂x V̇ ≤ −aS 2 + e−bt .
or
∂ ξ¯ ∂ ξ¯ ∂V Note that, it is not difficult to see that there exits a constant
η(x, ξ, t) − + g(x)] + µˆ˙1 (µ̂1 − µ). Γ such that
∂x ∂t ∂x
Now, let us choose the following controller |P̄˙ e | ≤ Γ ∀(δ, w, Pe ) ∈ Ω

∂ ξ¯ ∂ ξ¯ ∂V Now, in order to derive the control u, we consider this

u = u1 + (f (x) + g(x)ξ) + − g(x) (30) augmented Lyapunov function
∂x ∂t ∂x
we have 1 1
W = V + (P̄e − ∆Pe )2 + (Γ̂ − Γ)2
¯ 2 2
Ẇ1 ≤ −β0 |x|2 +e−λt +(ξ−ξ)[u ¯ 1 − ∂ ξ η(x, ξ, t)]+µˆ˙1 (µˆ1 −µ) Then
∂x
˙
if we choose u1 as follows Ẇ ≤ V̇ + (P̄˙ − ∆P
e
˙ )(P̄ − ∆P ) + Γ̂(
e e Γ̂ − Γ)
e

µˆ1 2 | ∂x
∂ ξ̄ 2 ¯
| (ξ − ξ) or
u1 = ¯ −
−β1 (ξ − ξ) w0 ˙
∂ ξ̄
|µˆ1 ∂x (ξ − ξ)|¯ + e−λt Ẇ ≤ −aS 2 + e−bt + (P̄e − ∆Pe )(P̄˙ e − Ṗe + ) + Γ̂(Γ̂ − Γ)
2H
∂ ξ¯ Let us set
uˆ˙1 = ¯
|(ξ − ξ) | (31)
∂x xds Vs
we have u = [vf − T 0 Eq cos(δ)w + Pm0 ]
Kc Vs sin(δ) xds
¯ 2 + 2e−λt .
Ẇ1 ≤ −β0 ||x||2 − β1 ||ξ − ξ|| 0 Vs
− Td0 (xd − x0ds ) sin(δ)w (36)
Kc xds
From this it is not difficult to see that the systems will be
then, we have
asymptotically stable.
1 w0 ˙
A. Application to power system Ẇ ≤ −aS 2 +e−bt +(P̄e −∆Pe )(P̄˙ e − 0 vf + )+Γ̂(Γ̂−Γ)
T 2H
In this section, we apply the above controller to power by choosing
system. We suppose that the machine evoluates in a closed w0
set Ω ∈ R3 . Let us consider the sub-system defined by vf = T 0 [a(P̄e − ∆Pe ) + +
2H
δ̇ = w (32) Γ̂2 (P̄e − ∆Pe )2
+ ] (37)
D w0 |Γ̂(P̄e − ∆Pe )| + e−bt
ẇ = − w+ (Pm − ∆Pe ) (33)
2H 2H ˙
Γ̂ = |P̄e − ∆Pe | (38)
where ∆Pe is considered as input and Pm as disturbance
bounded by an unknown disturbance Pmax . Let us consider we have
the following manifold Ẇ ≤ −aS 2 − a(P̄e − ∆Pe )2 + 2e−bt
S = (δ − δ̄) + λ0 w By using Barbalat lemma, we deduce that S → 0 and
∆Pe → ∆P ¯ e asymptotically. Then δ → δ0 , w → 0 and
where λ0 > 0 and the following Lyapunov function
Pe → Pm0 asymptotically.
1 2 1 The results of simulations is shown in following fogures
V = S + (P̂max − Pmax )2
2 2 (Fig.5, Fig.6, Fig.7).
then, we have
V. CONCLUSIONS
D λ0 w0 ˙
V̇ = S[w−λ0 w+λ0 (Pm −Pe )]+P̂ max (P̂max −Pmax ) In this paper, a new robust nonlinear excitation controller
2H 2H is proposed to improve the transient stability and to achieve
from this and by using the above theorem, we deduce the the voltage regulation of power systems.
control The advantage of this controller is of ensure the asymptotic
2H D global stability and voltage regulation of power system under
∆Pe = P̄e = [w − w − aS −
w0 2H a large sudden faults as it is shown in the simulations (Fig.5,
2
P̂max S Fig.6 and Fig.7) .
− ] (34) The robust nonlinear excitation controller proposed here can
P̂max |S| + e−bt
easily implemented in Real-Time and be extended to the case
˙
P̂ max = |S| (35) of multi-machine power systems.

298
 90 2

88 1.8

Terminal voltage (p.u)

86 1.6

Angle (degrees)
84 1.4

82 1.2

80 1

78 0.8

76 0.6

74 0.4

72 0.2

70 0
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

100 2

1.8
95

Terminal voltage (p.u.)

1.6
90
Angle (degrees)

1.4

85
1.2

80 1

0.8
75

0.6
70
0.4

65
0.2

60 0
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Time (sec) Time (sec)

Fig. 5. Power system responses for Case 2 (Upper) and Case 3 (Lower) : (-) Robust nonlinear controller, (...) Voltage controller, (- -) DFL Controller.

VI. REFERENCES [10] Y.Wang, D.J. Hill, R.H. Middleton, and L. Gao ”Tran-
sient stability enhancement and voltage regulation of
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299

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