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On Transient Stabilization of Multi–Machine
Power Systems: A “Globally” Convergent
Controller for Structure–Preserving Models
Wissam Dib∗ , Romeo Ortega∗ , Andrey Barabanov† and Françoise Lamnabhi-Lagarrigue∗
∗
Laboratoire des Signaux et Systèmes, CNRS-SUPELEC, 91192 Gif-sur-Yvette, France
(Email: name@lss.supelec.fr)
† Department
of Mathematics and Mechanics, Saint-Petersburg University
198904 St.-Petersburg, Russia
(Email: Andrey.Barabanov@pobox.spbu.ru)
Abstract
The design of excitation controllers to improve transient stabilization of power systems is a topic of renewed
interest in the control community. Existence of a state–feedback stabilizing law for multi–machine aggregated reduced
network models has recently been established. In this paper we extend this result in two directions: first, in contrast
with aggregated models, we consider the more natural and widely popular structure–preserving models that preserve
the identity of the network components (generators, loads and lines) and allow for a more realistic treatment of the
loads. Second, we explicitly compute a control law that, under a detectability assumption, ensures that all trajectories
converge to the desired equilibrium point, provided that they start and remain in the region where the model makes
physical sense.
Index Terms
Global Nonlinear control, nonlinear differential and algebraic system, nonlinear loads, structure preserving power
system, stability.
I. I NTRODUCTION
C
LASSICAL research on transient stabilization of power systems has relied on the use of aggregated reduced
network models that represent the system as an n–port described by a set of ordinary differential equations.
Several excitation controllers that establish Lyapunov stability of the desired equilibrium (with a Lyapunov function
and a well–defined domain of attraction) of these models have been reported. The nonlinear controller design
This work has been done in the context of the European Network of Excellence HYCON.
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techniques that have been considered include feedback linearization [14], damping injection [7], [15], [16], as well
as, the more general, interconnection and damping assignment passivity–based control, see [11], [10] and [9]. In
[9] the existence of a nonlinear static state feedback law that ensures stability of the operating point for a general
n–machine system including transfer conductances is proven. Unfortunately, due to computational complexity, an
explicit expression of the controller is given only for the case n ≤ 3. Moreover, due to the use of aggregated
models, the identity of the network components is erased and an unrealistic treatment of the loads is imposed.
In this paper, we abandon the aggregated n–port view of the network and consider the more natural structure–
preserving models, first proposed in [2]. Since these models consist of differential algebraic equations (DAE) they
require the development of some suitably tailored tools for controller synthesis and stability analysis. Another
original feature of the present work is that we do not aim at Lyapunov stability, but establish instead a “global”
convergence result.1
In [6] structure–preserving models were used to identify—in terms of feasibility of a linear matrix inequality—a
class of power systems with nonlinear (so-called ZIP) loads and leaky lines for which a linear time–invariant
controller renders the overall linearized system dissipative with a (locally) positive definite storage function, thus
ensuring stability of the desired equilibrium for the nonlinear system. Unfortunately, a full–fledged nonlinear analysis
of the problem was not possible due to the difficulty in handling the complicated interdependence of the variables
appearing in the algebraic constraints of the DAEs. The Lyapunov function in that paper is obtained by adding
a quadratic term in the rotor angle to the classical energy function of [13]. This quadratic term is needed to
compensate for a linear term (in rotor angle) appearing in the energy function of [13] and render the new storage
function positive definite. To obtain our “global” convergence result we observe that removing the linear term from
the energy function of [13] and increasing the quadratic term in bus voltages yields a function whose time derivative
can be arbitrarily assigned with a “globally” defined static state feedback. Furthermore, although this new function
is not positive definite, it is bounded from below and has some suitable radial unboundedness properties—features
that are essential to establish boundedness of trajectories. We then select a control law that renders “globally”
attractive the level set of this function that contains the desired equilibrium point. If, furthermore, the function
defines a detectable output, then all trajectories will asymptotically converge to the equilibrium. The only critical
assumption required to establish this result is that the loads are constant impedances—a condition that is implicitly
assumed in all controllers derived for aggregated models.
The structure of the paper is as follows. Section II presents the mathematical model of the various elements
comprising the power system. Then, we formulate the control problem in Section III and give a key preliminary
lemma. Section IV contains our main “global” convergence result that relies on the aforementioned detectability
assumption. In Section V we prove that the system is indeed detectable in the single–machine case and simulations
are given in Section VI. We wrap up the paper with concluding remarks on future research in Section VII. Proofs
1 The
precise meaning of the qualifier “global” will be given in the sequel. It essentially boils down to restricting to the trajectories that remain
in the region where the model makes physical sense.
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of some of the Lemmas are presented in the appendices. Note that a preliminary version of this paper has been
submitted to [3].
Notation All vectors in the paper are column vectors, even the gradient of a scalar function: ∇ x =
function f : Rn → R, we define ∇zj f (z) :=
∂f
∂zj (z),
∂
∂x .
For any
and for vector functions g : Rn → Rn , we define the Jacobian
∇g(z) := [∇g1 (z), . . . , ∇gn (z)]> ∈ Rn×n . To simplify notation we introduce the sets
Mn := Sn × Rn × Rn>0 × Sn × Rn>0 , n ∈ n̄ := {1, .., n},
where Rn>0 := {x ∈ Rn | xi > 0}.
II. S TRUCTURE –P RESERVING M ODELLING
In this section we recall the well–known structure–preserving model for n–machine power systems comprised
by synchronous machines and loads interconnected by transmission lines [12].
To simplify the presentation of our results we will assume a simplified network topology where attached to
each bus there is a machine and a load.2 Each bus, and their corresponding machine and load, have an associated
identifier j ∈ n̄. Buses are interconnected through transmission lines that are identified by the double subindex
jk ∈ Ω ⊂ n̄ × n̄, indicating that the line jk connects the bus j ∈ n̄ with the bus k ∈ n̄; the set avoids obvious
repetitions, e.g., if jk ∈ Ω then kj ∈
/ Ω. We also define the set Ωj := {k ∈ n̄ | ∃ jk ∈ Ω}, that is, the set of buses
that are linked to the bus j through some transmission line. Obviously, j 6∈ Ω j .
All elements share as port variables the angle θj and the magnitude Vj of the bus voltage phasor yj = col(θj , Vj ) ∈
S×R>0 . Associated to each bus are the active and reactive powers entering the machine, the load or the transmission
lines, that will be denoted


PjM
QM
j
 
,
 
PjL
,
QL
j
Pjk
Qjk

 ∈ R2 ,
(1)
respectively. Following standard convention, we take active and reactive powers as positive when entering their
corresponding component.
A. Synchronous machines model
Each synchronous machine is described by a set of third order DAE’s, [13]:
δ̇j =ωj
Mj ω̇j =Pmj − Dj ωj + PjM
xd
τj Ėj =− x0 j
d
j
PjM = −
E j Vj
x0d
Ej +
xdj −x0d
x0d
j
(2)
Vj cos(δj − θj ) + EFj ,
j
sin(δj − θj ) − Y2j Vj2 sin(2(δj − θj ))
j
2
QM
j = (YVj − Y2j cos(2(δj − θj )))Vj −
E j Vj
x0d
(3)
cos(δj − θj ),
j
2 As
will become clear below the derivations are also applicable for other network topologies—at the expense of a more cluttered notation.
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where, to simplify notation, we defined the constants
Y2j :=
x0dj − xqj
2xqj x0dj
, YVj :=
x0dj + xqj
2xqj x0dj
.
The state variables xj := col(δj , ωj , Ej ) ∈ S × R × R>0 denote the rotor angle, the rotor speed and the quadrature
axis internal e.m.f., respectively, and EFj is the field voltage. The latter is split in two terms, EF? j + vj , the first is
constant and fixes the equilibrium value, while the second one is the control action. The parameters are denoted
as in [13], and are fairly standard. We will make the physically reasonable assumptions D j > 0, xdj − x0dj > 0.
B. Loads model
Loads are described by the standard ZIP model, see [5],
PjL
= PZj Vj2 + PIj Vj + P0j
QL
j
= QZj Vj2 + QIj Vj + Q0j ,
(4)
which explicitly represent the contribution of each type of load (constant impedance, current or power). As will
become clear below, to state our main result we must consider a simplified model for the loads. Namely, we will
assume only constant impedance loads:
PjL
= PZj Vj2
QL
j
= QZj Vj2
(5)
This simplification, which is necessary to obtain the lumped parameter model used in must transient stability
controller design studies, allows us to transform the algebraic constraints into a set of linear equations for which
we can give conditions for solvability.
C. Transmission lines model
The transmission lines are modeled with the standard lumped Π circuit, see [1],
Pjk = Gjk Vj2 + Bjk Vj Vk sin(θj − θk ) − Gjk Vj Vk cos(θj − θk )
c
Qjk = (Bjk − Bjk
)Vj2 − Bjk Vj Vk cos(θj − θk ) − Gjk Vj Vk sin(θj − θk )
(6)
where jk ∈ Ω, while Gjk , Bjk and Gcjk denote the lines conductance, series and shunt susceptance, respectively.
The active and reactive power entering at node k, Pkj and Qkj can be obtained by a simple change of indexes.
Remark 1: In comparison with previous works on transient stabilization, for generality we consider lines with
capacitive effects, a parameter that is usually small, hence reasonable to neglect.
Remark 2: In contrast with reduced network models Gjk here is the effective line conductance and not the
transfer conductance that lumps the effects of the line conductance and the load impedances. While G jk may,
sometimes, be neglected it is impermissible to neglect the transfer conductances [9]. We are interested in this paper
in the more realistic case of leaky lines.
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D. Bus equations
From Kirchhoff’s laws, at each bus we have
0 =
0 =
P
P
k∈Ωj
Pjk + PjM + PjL
k∈Ωj
L
Qjk + QM
j + Qj
(7)
where we recall that Ωj is the set of buses that are linked to the bus j through some transmission line.
Remark 3: We bring to the readers attention the fact that Vj , being a magnitude of a phasor, is non–negative.
Similarly, due to physical considerations, Ej > 0. These fundamental physical constraints of the model will be
assumed for our derivations.
III. C ONTROL P ROBLEM
AND A
K EY L EMMA
To obtain the overall model we group all the algebraic constraints and write the system equations in the compact

 ẋ = f (x, y) + L v
v
 0 = g(x, y),
form
(8)
where (x := col(xj ), y := col(yj )) ∈ Mn , v := col(vj ) ∈ Rn , the matrix Lv := diag{col(0, 0, τ1j )} ∈ R3n×n , and
the functions f : Mn → R3n , and g : Mn → R2n are defined by (2), and the replacement of (3), (5) and (6) into
(7), respectively.
A. Problem formulation
Assumption A1. There exists an isolated asymptotically stable open loop equilibrium (x ? , y ? ) of the system (8).
Assumption A2. The matrix ∇y g(x, y) is invertible for all (x, y) ∈ Mn .
Asymptotic Convergence Problem. Consider the system (8) satisfying Assumptions A1 and A2. Find a control
law v = v̂(x, y) such that:
(x(t), y(t)) ∈ Mn , ∀t ≥ 0 ⇒ lim (x(t), y(t)) = (x? , y ? ).
t→∞
?
?
Consequently, (x , y ) is an attractive equilibrium of the closed–loop provided trajectories start, and remain, in
Mn —the set where the model is physically valid.
Remark 4: Assumption A1 is standard in transient stability studies where v is included to enlarge the domain
of attraction of the operating point. Assumption A2 is needed to compute the control law. In all practical situations
∇y g(x∗ , y ∗ ) is non–singular ensuring, via the Implicit Function Theorem, that ∇ y g(x, y) is (locally) invertible. We
have assumed that this is true throughout Mn to avoid cluttering notation in the main result.
Remark 5: Notice that we do not aim at proving that trajectories starting in M n actually remain there, but we
only assume it. In spite of that, and with obvious abuse of notation, we will say that a controller satisfying the
implication above ensures “global” convergence.
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B. Proposed solution strategy
The solution to the problem stated above that we propose in the paper proceeds along the following steps:
1) Representation of the system dynamics as a perturbed port–Hamiltonian system using a Hamiltonian function
with desired characteristics, e.g., lower bounded and proper.
2) Construction of a control signal that, assigning the derivative of the Hamiltonian function, ensures that
trajectories will converge to the level set of the Hamiltonian that contains the equilibrium point. Trajectories
will then converge to the equilibrium if the Hamiltonian function defines a detectable output.
3) Prove that the resulting controller is well defined and convergence is guaranteed—provided the trajectories
remain in Mn .
The first two steps can be carried out for the model with the general ZIP loads (4). Invoking the existence of an
isolated local minimum of Assumption A1, using some continuity arguments and assuming detectability we can,
therefore, conclude that the proposed controller renders the equilibrium locally attractive. This kind of local results
are easily obtained using linearization, and known in the power systems community as small–signal stability. In
this paper we are interested in the nonlinear transient stability phenomenon, i.e., the large–signal stability problem,
therefore, the third step is indispensable. To complete this third step the explicit solution of the algebraic equations
is essential—unfortunately, this imposes the restrictive requirement of constant impedance loads (5).
C. Solution of g(x, y) = 0
In this subsection we present an explicit solution to the algebraic constraints g(x, y) = 0, a result which is of
interest on its own. To simplify the presentation we define, for j ∈ n̄, the complex variables
Vj := Vj eiθj ∈ C,
iδ
Ej := Ej e j ∈ C,
(9)
and
V
:= col(Vj )j∈n̄ ∈ Cn ,
n
E := col(Ej )j∈n̄ ∈ C
E
:= col(Ej )j∈n̄ ∈ Rn ,
δ := diag{δj }j∈n̄ ∈ Rn×n .
Notice that, for notational convenience, we have defined δ as a (real) diagonal matrix while E is a (real) vector.
Lemma 1: Consider the algebraic equations g(x, y) = 0 of the power systems model (8) defined by (3), (5), (6)
and (7). If
X
1
c
+ Q Zj >
Bjk
,
0
xd j
j ∈ n̄,
(10)
k∈Ωj
g(x, y) = 0 has a “globally” defined solution. That is, there exists a function ŷ : S n × Rn>0 → Sn × Rn>0 such that
g(x, ŷ(x)) = 0.3 Furthermore, this function can be written in the form
V = W (δ)E,
3 Notice
ŷ(x).
(11)
that g(x, y) is a function only of (δj , Ej ), therefore the domain of ŷ is as defined. However, to simplify notation, we have written
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where W : Rn×n → Cn×n is bounded and invertible, with elements, given in (33) in Appendix A, rational functions
of cos(δj ) and sin(δj ).
The proof is given in Appendix A.
Remark 6: As indicated in Remark 1, (10) is always verified in (standard) transient stability studies, where it is
c
assumed that Bjk
= 0. Also, it is clear that the construction of ŷ directly follows from (9) and (11), and is omitted
for brevity.
IV. M AIN R ESULTS
This section contains our main “global” convergence result, which is derived proceeding along the steps delineated
in Subsection III-B.
A. Perturbed port–Hamiltonian representation
The j-th synchronous machine model dynamics (2) can be written as a perturbed port–Hamiltonian system
ẋj = (Jj − Rj )∇xj Hj (xj , yj ) + Lvj vj + ξj
(12)
with the Hamiltonian functions Hj : M1 → R,
Hj :=
1
1
Y2j
E j Vj
1
Mj ωj2 + YEj Ej2 + [∆j + YVj ]Vj2 − YFj EF? j Ej −
cos 2(θj − δj )Vj2 − 0 cos(θj − δj ) (13)
2
2
2
2
xd j
and we defined the matrices

0


1
Jj := 
−
Mj
0
Lvj := [0, 0, τ1j ]> , ξj := [0,
Pmj
Mj

0 0
0
1
0
D
j

Mj 
0
0
 = −Jj> , Rj :=  Mj2
0 0

1
0 0
τ j YF j
0 0
, 0]> , and the constants
YEj :=
xd j
0
xdj (xdj −
x0dj )
,
YFj :=



,

1
,
xdj − x0dj
where Rj ≥ 0 and ∆j ≥ 0 is a key design parameter.
One important property of the Hamiltonian Hj is that it is quadratic in Zj := col(ωj , Ej , Vj ) and, furthermore,
bounded from below. (Consequently, if Hj is non–increasing, we can conclude that all signals are bounded—because
Zj will be bounded and θj and δj live in compact sets.) To prove this fact, let us write the function in the form
Hj =
where we have defined

Mj

 0
Tj := 


0
1 >
Z Tj (θj − δj )Zj + Zj> bj
2 j
0
Y Ej
0
− cos(θj − δj )
x0dj
− cos(θj − δj )
∆j + YVj − Y2j cos 2(θj − δj )
x0dj
(14)






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and bj := col(0, −YFj EF? j , 0). Using the fact that (for all ∆j ≥ 0)
∆j + YVj > Y2j ,
it is possible to show that, uniformly in θj − δj , there exists j > 0 such that Tj ≥ j I. Consequently, after some
basic bounding, we can prove that
Hj ≥ −
(YFj EF? j )2
2j
.
(15)
Remark 7: The functions Hj defined in (13) should be contrasted with the energy functions used in [12], see
also [6]. On one hand, the latter includes an additional term −P mj δj .4 On the other hand, we have included a term
∆j Vj2 that, as will come clear below, will prove essential for the construction of the control law.
Remark 8: To handle the linear term −Pmj δj in a Lyapunov–like analysis we must take care of some delicate
theoretical issues that have, unfortunately, been overlooked in the literature and we discuss in detail here—see also
discussion in [9] and [15]. Since this function is not defined in S, but in R, if we look at the system as evolving
in Mn it will be a discontinuous function and (standard) Lyapunov arguments will not hold true. To avoid this
difficulty, we should consider that δj evolves in R, instead of S. In this case, the function Hj is not lower bounded
anymore, styming the establishment of the property of trajectory boundedness needed for LaSalle–based arguments. 5
Remark 9: Due to the presence of the term ξj in (12) it is clear that the set of open–loop equilibria and the set
of minima of Hj are disjoint. Therefore, the new Hamiltonian cannot qualify as a Lyapunov function candidate
(for the desired equilibrium).
B. Evaluating the time derivative of the Hamiltonian
Besides being lower bounded and quadratic (in Zj ) we will prove in the paper another fundamental property
P
of the function Hj , namely, that the derivative of the function H(x, y) :=
j∈n̄ Hj (xj , yj ) can be arbitrarily
assigned with a suitable selection of the control v.
To prove this fact we will invoke Lemma 1, which requires the differentiation of a complex valued function, and
is done in Subsection IV-C. To motivate the subsequent calculations we first compute Ḣ in this subsection using
standard—real domain—derivations. Towards this end, let us define
X
H(x, y) :=
Hj (xj , yj )
(16)
j∈n̄
and compute
>
−1
˜
Ḣ = −∇>
v + ∇>
x HR∇x H + ξ(x, y) + L (x, y)τ
y H ẏ
4 Removing
this term induces the appearance of the constant term ξ j in (12), but we will show in the sequel that its effect can be compensated
with a suitable selection of the control law.
5 This
unfortunate mistake is made in many papers. For instance, in Proposition 1 of [4], where the interesting idea of damping injection for
structure preserving models is proposed, boundedness of trajectories is never established—nor assumed.
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where
R := diag{Rj }j∈n̄ ∈ R3n×3n , ξ˜ :=
X
∇>
x j Hj ξ j =
j∈n̄
and τ := diag{τj }j∈n̄ ∈ R
n×n
X
n
ωj Pmj ∈ R, L := col(∇>
xj Hj Lvj )j∈n̄ = ∇E H ∈ R ,
j∈n̄
. To evaluate ẏ we differentiate the algebraic constraints g(x, y) = 0 yielding
>
∇>
x g ẋ + ∇y g ẏ = 0.
Invoking Assumption A2 we obtain
ẏ = M (x, y)ẋ,
(17)
where
>
2n×3n
.
M := −∇−>
y g∇x g ∈ R
Replacing (12) in (17) we have that ẏ = F (x, y) + G(x, y)v, where F := M col((J j − Rj )∇xj Hj + ξj )j∈n̄ ∈ R2n
and G := M diag{Lvj }j∈n̄ ∈ R2n×n . Therefore,
>
−1
˜
Ḣ = −∇>
v
x HR∇x H + ξ0 (x, y) + L̃ (x, y)τ
(18)
where
ξe0
:=
ξ˜ + ∇>
y HF ∈ R,
e := L + τ G> ∇y H ∈ Rn .
L
e is bounded away from zero, we
Let us take a brief respite to analyze (18). It is clear that, wherever the vector L
e y) 6= 0,
can easily select a control law v that assigns an arbitrary function to Ḣ. Instead of trying to prove that L(x,
for all (x, y) ∈ Mn , we show in the next subsection that, if we restrict the analysis to the set g(x, y) = 0 (using
Lemma 1), we can establish a property that allows us to assign arbitrarily Ḣ(x, ŷ(x)).
C. “Global” assignment of Ḣ(x, ŷ(x))
The following lemma, whose proof is given in Appendix B, is instrumental to compute Ḣ restricted to the set
g(x, y) = 0.
Lemma 2: Consider the quadratic function f : R × R → R,
f (µ1 , µ2 ) = c1 µ21 + 2c3 µ1 µ2 + c2 µ22 ,
with ci ∈ R and µ1 , µ2 : R → R. Define z = µ1 + iµ2 ∈ C and the function fC : C → R such that
fC (z) = f (µ1 , µ2 ).
Then,
f˙ = Re
where (·)H denotes complex conjugate transpose.
(
∂fC
∂z
H )
ż ,
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Proposition 1: Consider the power systems model (8) with Assumptions A1 and A2 and the Hamiltonian function
(13). There exists ∆min
> 0 such that, for all ∆j ≥ ∆min
we have
j
j
e > (x, ŷ(x))E > 0
L
for all x ∈ Sn × Rn × Rn>0 ,
where L̃ is given in (19). Therefore, for any function α : Mn → R, the “globally” defined control law
v=
ensures Ḣ = α.
1
e
[α(x, y) + ∇>
x HR∇x H − ξ0 ]τ E
L̃> E
(19)
Proof: To establish the proof we will compute Ḣ(x, ŷ(x)) invoking Lemma 1. Since the lemma uses complex
notation, we express the Hamiltonian function (13), (16) in terms of the complex variables V defined in (9). To
this end, notice that
Vj2 cos 2(θj − δj ) = Re{e−2iδj Vj2 }.
Therefore, we define
HC (δ, w, E, V) :=
1
1 >
w M w + E > YE E + VH (YV + ∆) V − Re{V> Y2 e−2iδ V} − Re{EH X V} − E > YF EF
2
2
where EF := col(EF? j )j∈n̄ , YV , Y2 , X are defined in Appendix A and
M := diag{Mj }j∈n̄ ∈ Rn×n , YE := diag{YEj }j∈n̄ ∈ Rn×n ,
YF := diag{YFj }j∈n̄ ∈ Rn×n , ∆ := diag{∆j }j∈n̄ ∈ Rn×n .
To compute the time derivative of HC we invoke Lemma 2 above, which indicates that we need to compute the
n
o
term Re (∇V HC )H V̇ . It is easy to see that
∇V H = (YV + ∆) V −Y2 e2iδ V∗ −X E
where (·)∗ denotes complex conjugation of the elements of (·). We recall now the identity (30) established in
Appendix A that, for ease of reference, we recall here
A0 V∗ −X E∗ −Y2 e−2iδ V = 0.
Substituting the complex conjugated of the latter in ∇V H above we get
∇V H = (YV + ∆ − A∗0 ) V,
and by definition of A0 , given in (29), we get ∇V H = D V, where
D := ∆ − QZ − B d + B + i(−PZ − Gd + G) ∈ Cn×n .
We recall that the matrices ∆, QZ , PZ , B d and Gd are diagonal. All these matrices are defined in Appendix A.
Let us now compute V̇. In Lemma 1 it is shown that
V = W E, where
W = eiδ Y0 (Aδ A∗δ − I)−1 (Aδ + I)C
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was derived in (33) of Appendix A. Therefore,
V̇ = Ẇ E + W Ė
The function Ẇ depends on δ and ω, but is independent of v, while Ė will bring along terms on v.
We now come back to Ḣ, that takes the form6
n
o
H
−1
>
˜
Ḣ = −∇>
v + Re (∇V H) V̇ .
x HR∇x H + ξ + ∇E Hτ
That, replacing the computations above, can be compactly written as
>
−1
v
Ḣ = −∇>
x HR∇x H + Ξ(δ, w, E, V ) + L (δ, w, E, V )τ
(20)
where we defined the (real valued) functions
n
h
io
Ξ := ξ˜ + Re (∇V H)H Ẇ E −W (τ YF )−1 ∇E H
n
oo
n
H
.
L> :=
∇>
E H + Re (∇V H) W
Comparing (18) with (20) we identify the terms Ξ = ξ˜0 (x, ŷ(x)) and L = L̃(x, ŷ(x)), and we concentrate our
attention on the latter critical term.
It is necessary to prove that the control law (19) is “globally” well defined. By definition,
L> E = Re (∇V H)H W (δ)E + ∇>
E HE.
(21)
Let us consider the first term. Since V = W (δ)E and ∇V H = D V we have
D + D∗
Re (∇V H)H W E = Re VH D V = VH
V
2
The matrix D is symmetric (not Hermitian self conjugate). Therefore,
D + D∗
= ∆ − QZ − B d + B,
2
where QZ , B d and B are constant matrices defined in Appendix A. The quadratic form above can then be made
arbitrarily large by choosing a large ∆ > 0.
Using again V = W E and invertibility of W we see that the second term in (21) is also a quadratic function of
V, that can be written in the form
H
H
?
∇>
E HE = V S(δ) V + Re(V s(δ)EF ),
for some suitable matrices S, s : Rn×n → Cn×n . From boundedness of W −1 we have that S and s are also
bounded and we can conclude that, throughout Mn , the first term in (21) can be made strictly greater than the
second. Therefore, the denominator in (19) is always larger than zero, completing the claim.
6 Obviously,
since HC (δ, w, E, V) = H(x, y), their time derivatives are equal.
LSS INTERNAL REPORT
12
D. A “globally” convergent controller
In this subsection we propose to select the function α such that, under a detectability assumption, trajectories
converge to (x? , y ? ).
Proposition 2: Consider the power systems model (8) with Assumptions A1 and A2 in closed–loop with the
control (19) with
α(x, y) = −λ[H(x, y) − H ? ],
(22)
where H ? := H(x? , y ? ), λ > 0, ∆j ≥ ∆min
, and ∆min
is as in Proposition 1.
j
j
(i) Assume (x(t), y(t)) ∈ Mn , ∀t ≥ 0. Then, trajectories are bounded.
(ii) If, additionally,
Assumption A3. The function H(x, y) − H ? defines a detectable output for the closed–loop system.
Then, limt→∞ (x(t), y(t)) = (x? , y ? ).
Proof: First, note that
d
[H(x, y) − H ? ] = −λ[H(x, y) − H ? ].
dt
Hence H is bounded, ensuring boundedness of trajectories. Furthermore, we have that H(x(t), y(t)) → H ? . The
proof is completed invoking LaSalle’s Invariance Principle and the definition of detectability.
Remark 10: The controller of Proposition 1 drives the trajectories towards the level set {(x, y) ∈ M n | H(x, y) =
H ? }. The analysis of the dynamics restricted to this set is quite involved and is currently under investigation—hence
the need of the detectability assumption. However, we prove in the next subsection that the assumption is verified
for the classical single machine infinite bus (SMIB) system.
Remark 11: A practically interesting property of the control law (19) is that it is “almost” decentralized. Indeed,
it is of the form vi = β(x, y)Ei , where the scalar function β : Mn → R is the only information that needs to be
transferred among the generators.
Remark 12: We recall that the minima of H are not equilibria of the system—hence, it is not a Lyapunov function
candidate and the property Ḣ ≤ 0 is not sufficient to guarantee some stability/convergence properties.
LSS INTERNAL REPORT
13
V. SINGLE MACHINE SYSTEM
For the elementary case of a SMIB system neglecting the generator saliency, i.e., n = 1 and Y 2 = 0, the model
(2) reduces to
δ̇
= ω
M ω̇
= Pm − Dω + P M
τ Ė
= − xx0d E +
d
PM
=
QM
=
xd −x0d
x0d V
cos(δ − θ) + EF? + v,
− x10 EV sin(δ − θ)
d
x0d +xq 2
1
2xq x0d V − x0d EV cos(δ
(23)
− θ).
The algebraic constraints imposed by the bus equations (7), assuming for simplicity G = B c = 0, are
2
− EV
x0 sin(δ − θ) + BV sin(θ) + PZ V = 0
d
(YV + B + QZ )V 2 −
EV
x0d
(24)
cos(δ − θ) − BV cos(θ) = 0,
where, following standard convention, the magnitude and the angle of the voltage phasor at the infinite bus are
taken equal to 1 and 0, respectively.
To set up the notation used in the sequel, and for the sake of completeness, we give now a simplified version of
Lemma 1.
Lemma 3: Assume the voltage V (t) > 0 for all t ≥ 0. The algebraic constraints (24) are equivalent to
Re{A0 }
E
0}
V cos(δ − θ) = Im{A
B
sin(δ)
+
+
B
cos(δ)
,
0
|A0 |2
|A0 |2
xd
Im{A0 }
E
0}
V sin(δ − θ) = Re{A
|A0 |2 B sin(δ) − |A0 |2
x0 + B cos(δ) ,
(25)
(26)
d
where
A0 := YV + QZ + B − iPZ ∈ C.
Proof: Writing (24) in complex notation we obtain
E
iV (YV + QZ + B − iPZ ) V − 0 ei(θ−δ) − Beiθ = 0
xd
Since V ∈ R>0 , the term in brackets should be zero. Multiplying this term by ei(δ−θ) it can be written as
1
A∗0
iδ
i(δ−θ)
E
+
Be
Ve
=
x0d
|A0 |2
(27)
The proof is completed taking the real and the imaginary part of (27).
We will now check the detectability condition (Assumption A3) for the SMIB model (23), (24) in closed–loop
with the control (19), (22). Towards this end, we introduce the coordinate transformation (δ, ω, E) ! η, where
e
η1 = δ, η2 = ω, η3 = H,
e := H − H ? and H reduces to
we defined H
H=
M 2 YE 2 1
EV
ω +
E + [∆ + YV ]V 2 − YF EF? E − 0 cos(θ − δ).
2
2
2
xd
(28)
LSS INTERNAL REPORT
14
Using (25) and (26) the inverse transformation for the third coordinate is obtained as E = Φ(η 1 , η2 , η3 ) where
i
p
1 h
Φ :=
−b(η1 ) + b2 (η1 ) − 4ac(η2 , η3 )
2a
with
2a := YE +
b :=
1
x0d |A0 |2
1
[∆ − YV − 2(B + QZ )],
(x0d )2 |A0 |2
(∆ − B − QZ ) B cos(η1 ) − YF EF? +
BPZ
0
xd |A0 |2
sin(η1 )
and
c :=
M η22
(YV + ∆)B 2
+
− H ? − η3 .
2
2|A0 |2
The closed–loop system, in the new coordinates, takes the form
η˙1
M η˙2
η˙3
= η2
= Pm − Dη2 −
1
x0d Φ(η1 , η2 , η3 )Ψ(η1 , Φ)
= −λη3 ,
where
Ψ(η1 , Φ) :=
Im{A0 } Φ
Re{A0 }
B sin(η1 ) −
(
+ B cos(η1 )).
2
|A0 |
|A0 |2 x0d
Establishing detectability with respect to η3 is tantamount to proving that the equilibrium (δ ? , 0) of the two–
dimensional system
η̃˙ 1
M η̃˙ 2
= η̃2
= Pm − Dη̃2 −
1
x0d Φ(η̃1 , η̃2 , 0)Ψ(η̃1 , Φ)
is asymptotically stable. For, we recall that in Proposition 2 we have already established boundedness of trajectories.
Hence, recalling that trajectories in plane systems can only diverge, converge or go to a limit cycle, it suffices to
prove that the latter will not occur. From Poincare–Bendixson’s Theorem we know that a necessary and sufficient
condition for non–existence of limit cycles in a planar system η̃˙ = f (η̃) is
∇η̃1 f1 + ∇η̃2 f2 6= 0.
Computing this expression yields
−
D
+ ∇Φ f2 ∇η̃2 Φ(η̃1 , η̃2 , 0) 6= 0.
M
We have numerically evaluated this function for the classical example used in the next subsection with P Z = QZ =
0.8 and B = 6.2112, for which the condition above is satisfied for all η̃ 1 (resp., δ) and for η̃2 ∈ (−5.5, 5.5) (resp.,
ω)—an interval far beyond the normal range of operation of the SMIB system.
LSS INTERNAL REPORT
15
TABLE I
PARAMETERS OF THE SMIB MODEL
xd
0.8958
x0d
0.8645
τ
6
M
0.0407
D
0.795 × 10−3
w0
314.1593
Pm
1.63
δ(rad)
1.5
1
0.5
0
0
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
E(pu)
3
2.5
Control input
2
0
10
0
−10
0
6
Time (sec)
Fig. 1.
SMIB system, with G = 0 and tcl =0.9s, in closed loop. Behavior of load angle, interval voltage and control input.
VI. SIMULATIONS
In this section, we present numerical simulations of the proposed controller for the SMIB with and without line
losses. The parameters of the SMIB taken from [1] are listed in Table I. The derivation of the equilibrium point is
done with the software package PSAT [8].
We analyze the response of (23), (24) (system without losses) to a short circuit which consists of the temporary
connection of a small impedance between the machine’s terminal and the ground. The fault is introduced at t = 1s
and removed after a certain time (called the clearing time, and denoted t cl ), after which the system is back to its
pre–disturbance topology. During the fault the trajectories make away from the equilibrium, the largest time interval
“before instability”,7 called the critical clearing time (tcr ), is determined via simulation.
7 This
is practically detected observing the evolution of the signals that should remain within physically reasonable values, e.g, E > 0.
LSS INTERNAL REPORT
16
2.5
δ(rad)
2
1.5
1
0.5
0
0
1
2
3
4
5
6
E
E(pu)
4
3
2
1
0
1
2
3
4
5
6
1
2
3
4
5
Time(sec)
6
Control input
10
5
0
−5
−10
0
Fig. 2.
SMIB system, with G = 1.1876 S and tcl =1.05s, in closed loop. Behavior of load angle, interval voltage and control input.
The SMIB has a critical clearing time tcr = 0.44s in open loop. With the proposed controller, taking λ = 0.001
and ∆ = 2000, this time could be increased to 19.4s—a value that is far beyond the time scale of interest in this
e to
problem. Notice that, we can increase this critical clearing time by decreasing λ but then the convergence of H
zero will be slower.
To tune the controller there is a compromise between the choices of ∆ that, as indicated in Proposition 1, should
be big enough to ensure that the denominator of the controller will stay away from zero, and λ that determines the
e as ∆V 2 where V represents, in some way,
speed of convergence to the desired level set. Indeed, ∆ appears in H
e will be in the transient phase, and we have to decrease λ to
the perturbation. Then, the bigger ∆ is, the bigger H
eliminate impulsive responses in the controller during the perturbation.
We then consider the effect of the losses in the transmission lines setting G = 1.1876 S. Similarly to the lossless
case, the proposed controller increases the critical clearing time from 0.36s to 7.2s. Figs.1 and 2 present the transient
behavior of the system with and without losses.
VII. C ONCLUSIONS
We have presented in this paper an excitation controller to improve the transient stability properties of multi–
machine power systems described by structure–preserving models with leaky lines including capacitive effects. Our
main contribution is the explicit computation of a control law that ensures “global” asymptotic convergence to the
desired equilibrium point of all trajectories starting and remaining in the physical domain of the system—provided a
LSS INTERNAL REPORT
17
detectability assumption is satisfied. To the best of our knowledge, no equivalent result is available in the literature
at this level of generality. Numerical simulations were presented for the standard SMIB system, for which the
detectability assumption was numerically verified for a classical example.
Similarly to most developments reported by the control theory community on the transient stability problem, it
is clear that the complexity of the proposed controller—as well as its sensitivity to the system parameters and the
assumption of full state measurement—stymies the practical application of this result. This kind of work pertains,
however, to the realm of fundamental research where basic issues like existence of stabilizing controllers are
addressed. In [9] we proved the existence of an asymptotically stabilizing controller (with a suitable Lyapunov
function) for aggregated models—alas, we could only give a constructive solution for n ≤ 3. The present paper
proves that, under a detectability assumption, a solution to the “global” convergence problem for the more natural
structure preserving models can indeed be explicitly constructed. Current research is under way to further investigate
the implications of this assumption. Moreover, we are working on the development of a realistic simulation example
for the three-machines problem. The outcome of this research will be reported in the near future.
ACKNOWLEDGMENT
The third author thanks Alessandro Astolfi for pointing out an error in a previous version of this paper and
suggesting to stabilize the desired level set.
R EFERENCES
[1] P. M. Anderson and A. A. Fouad, Power Systems Control and Stability, The Iowa State University Press, 1977.
[2] A. Bergen and D. Hill, A structure preserving model for power system stability analysis, IEEE Trans Power App. Syst., volume. PAS-100,
No. 1, pages 25–35, 1981.
[3] W. Dib, A. Barabanov, R. Ortega and F. Lamnabhi-Lagarrigue, On transient stabilization of multi–machine power systems: a “globally”
convergent controller for structure–preserving models, Submitted to International Federation of Automatic Control, 2007.
[4] J. Hao, C. Chen, L. Shi and J. Wang, Nonlinear decentralized disturbance attenuation excitation control for power systems with nonlinear
loads based on the hamiltonian theory, IEEE Trans on Energy Conversion, volume 22, No. 2, pages 316–324, June 2007.
[5] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994.
[6] A. Giusto, R. Ortega and A. Stankovic, On transient stabilization of power systems: a power–shaping solution for structure–preserving
models, 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 2006.
[7] Y-H. Moon, B-K. Choi and T-H. Roh, Estimating the domain of attraction for power systems via a group of damping-reflected energy
functions, Automatica, volume 36, pages 419–425, 2000.
[8] F. Milano, An open source Power System Analysis Toolbox, IEEE Trans on Power Systems, volume 20, No. 3 pages 1199–1206, August
2005.
[9] R. Ortega, M. Galaz, A. Astolfi, Y. Sun and T. Shen, Transient stabilization of multimachine power systems with nontrivial transfer
conductances, IEEE Trans on Automatic Control, volume 50, No. 1, pages 60–75, January 2005.
[10] T. Shen, R. Ortega, Q. Lu, S. Mei and K. Tamura, Adaptive disturbance attenuation of Hamiltonian systems with parametric perturbations
and application to power systems, Asian Journal of Control, volume 5, No. 1, 2003.
[11] Y. Z. Sun, Y. H. Song and X. Li, Novel energy-based Lyapunov function for controlled power systems, IEEE Power Engineering Review,
pages 55–57, 2000.
[12] N. A. Tsolas, A. Arapostathis and P. P. Varaiya, A structure preserving energy function for power system transient stability analysis, IEEE
Trans on Circuits and Systems, volume 32, pages 1041–1049, 1985.
LSS INTERNAL REPORT
18
[13] P. Varaiya, F. Wu and R. Chen, Direct methods for transient stability analysis of power systems: Recent results Proc IEEE, volume 73,
No. 12, pages 1703–1714, December 1985.
[14] Y. Wang, D. J. Hill, R. H. Middleton and L. Gao, Transient stability enhancement and voltage regulation of power system, IEEE Trans
on Power Systems, volume 8, pages 620–627, 1993.
[15] Y. Wang, D. Cheng, C. Li and Y. Ge, Dissipative hamiltonian realization and energy-based L 2 disturbance attenuation control of
multimachine power systems, IEEE Trans on Automatic Control, volume 48, No. 8, pages 1428–1433, 2003.
[16] Y. Wang, G. Feng, D. Cheng and Y. Liu, Adaptative L 2 disturbance attenuation control of multimachine power systems with SMES units,
Automatica, volume 42, pages 1121–1132, 2006.
A PPENDIX A
P ROOF
OF
L EMMA 1
The following lemma, whose proof is given in Appendix B, will be instrumental to establish the proof of Lemma
1.
Lemma 4: Let A, L ∈ Rn×n , A = A> , L = L> and A diagonal dominant, that is,
ajj >
X
|ajk |,
j ∈ n̄.
k6=j
Then, for any diagonal matrix Y0 > 0 we have
σmin (I + Y0 AY0 + iL) > 1,
where σmin is the minimal singular value and I is the n × n identity matrix.
The following basic identities concerning the phasors (9) will be used in the sequel:
−i E∗j Vj
−i(Vj )2 e−2iδj
−i Vj Vk∗
= −Ej Vj [sin(δj − θj ) + i cos(δj − θj )],
= −Vj2 [sin(2(δj − θj )) + i cos(2(δj − θj ))],
= −Vj Vk [− sin(θj − θk ) + i cos(θj − θk )],
where (·)∗ denotes complex conjugation. Using these identities in (3), (5), (6), and after some simple calculations,
we obtain
PjM + iQM
j
PjL + iQL
j
Pjk + iQjk
= iYVj | Vj |2 −
i Vj ∗
2 −2iδj
,
E −iY2j (Vj ) e
x0dj j
= (PZj + iQZj )| Vj |2 ,
c
= [−Gjk − iBjk ] Vj Vk∗ +[Gjk + i(Bjk − Bjk
)]| Vj |2 .
Replacing the expressions above in (7) we see that the j-th bus equation takes the form

i V j  Y Vj
i
X
k∈Ωj


X
1
∗
∗
−2iδj
c
[(Bjk − Bjk
) − iGjk ] Vj∗ +
Vj − 0 Ej −Y2j e
Vj + QZj − iPZj +
xdj
k∈Ωj

(iBjk + Gjk ) Vk∗  = 0.
LSS INTERNAL REPORT
19
Since the voltage Vj ∈ R>0 , the term in brackets should be zero leading to the following linear equation


X
X
1
c
YVj + QZj − iPZj +
[(Bjk − Bjk
) − iGjk ] Vj∗ − 0 E∗j −Y2j e−2iδj Vj +
(−Bjk + iGjk ) Vk∗ = 0.
xdj
k∈Ωj
k∈Ωj
Grouping all bus equations (j ∈ n̄) leads to a square linear system for the complex vector V that can be written
in the form
(YV + QZ − iPZ + B d − iGd − B + iG) V∗ −X E∗ −Y2 e−2iδ V = 0,
where we have defined the n × n real matrices
YV := diag{YVj }, Y2 := diag{Y2j }, B := {Bjk }, PZ := diag{PZj }, QZ := diag{QZj }, G := {Gjk },
Gd := diag{Gdj }, X := diag{
1
}, B d := diag{Bjd },
x0dj
where j, k ∈ n̄, Bjk = Gjk = 0 if k 6∈ Ωj and we introduced the scalars
Bjd :=
X
c
(Bjk − Bjk
),
Gdj :=
k∈Ωj
X
Gjk .
k∈Ωj
A more compact expression is obtained defining the matrix
A0 := YV + QZ − iPZ + B d − iGd − B + iG ∈ Cn×n ,
(29)
A0 V∗ −X E∗ −Y2 e−2iδ V = 0.
(30)
which yields
To solve the algebraic equation above we must eliminate the term V∗ . Define the diagonal matrix
Y0−1 := diag{
so that Y2 = Y0−2 . Define also an auxiliary vector
p
Y2j }j∈n̄ ∈ Rn×n
−iδ −1
n
Vδ := e Y0 V ∈ C .
(31)
Multiply (30) from the left by eiδ Y0 . Then, it can be written in the form
∗
Vδ = Aδ Vδ −CE,
where we recall that E = col(Ej )j∈n̄ = eiδ E∗ ∈ Rn and we defined
p −1
iδ
−iδ
0
Aδ := e Y0 A0 Y0 e ,
C := diag
xdj Y2j
By complex conjugation of (32) we obtain
∗
∗
Vδ = Aδ Vδ −CE
After substitution of the latter back in (32) and grouping terms we get
(Aδ A∗δ − I) Vδ = (Aδ + I)CE
(32)
.
j∈n̄
LSS INTERNAL REPORT
20
If the matrix (Aδ A∗δ − I) is nonsingular we can express Vδ as a function of E and, plugging this back into (31),
obtain the desired relation (11) with
W (δ) = eiδ Y0 (Aδ A∗δ − I)−1 (Aδ + I)C
(33)
We will prove now that, under condition (10), this matrix is indeed invertible. Towards this end, notice that the
matrix A0 is symmetric by definition since Bjk = Bkj and Gjk = Gkj and all other matrices are diagonal.
∗
H
Therefore, AH
is the conjugate transpose matrix. For this reason
0 = A0 where (·)
(Aδ A∗δ − I) = (Aδ AH
δ − I)
This matrix is invertible iff kA∗δ zk > kzk for any nonzero complex vector z ∈ Cn . Define w = eiδ z, note that
kwk = kzk. Then,
kA∗δ zk = ke−iδ Y0 A∗0 Y0 eiδ zk = kY0 A∗0 Y0 wk,
and it is sufficient to prove that for any nonzero complex vector w ∈ C n
kY0 A∗0 Y0 wk > kwk
This can be expressed also as σmin (Y0 A∗0 Y0 ) > 1. It follows from definition of A0 that
Y0 A∗0 Y0 = I + Y0 AY0 + iL
where
A
= YV − Y2 + QZ + B d − B,
L
= Y0 (PZ + Gd − G)Y0
Both matrices A and L are symmetric. The diagonal entries of the matrix A are
ajj =
X
1
c
(Bjk − Bjk
),
+ Q Zj +
0
xdj
k6=j
where we have used the fact that YVj − Y2j =
1
x0d
. The non-diagonal entries of A are equal to ajk = −Bjk , j 6= k.
j
The proof of the claim follows using condition (10) and invoking Lemma 4.
A PPENDIX B
P ROOF
OF
L EMMATA
Proof of Lemma 2. By definition fC can be written in the following form:
fC (z) =
c1 + c 2 2 c1 − c 2
|z| +
Re{z 2 } − Re{ic3 z 2 }
2
2
Where Re{ic3 z 2 } = −2c3 µ1 µ2 . Then
∂fC
∂z
= (c1 + c2 )z + (c1 − c2 )z ∗ + 2ic3 z ∗
= 2 [(c1 µ1 + c3 µ2 ) + i(c2 µ2 + c3 µ1 )]
LSS INTERNAL REPORT
21
Hence,
˙ 1 , µ2 ) = 2(c1 µ1 + c3 µ2 )µ˙1 + 2(c2 µ2 + c3 µ1 )µ˙2
f(µ
∗ = 2 Re [(c1 µ1 + c3 µ2 ) + i(c2 µ2 + c3 µ1 )] ż
Proof of Lemma 4. It is required to prove that for any complex vector x ∈ C n , x 6= 0
k(I + Y AY + iL)xk > kxk.
Make the transformation:
k(I + Y AY + iL)xk2 − kxk2
= kx + (Y AY + iL)xk2 − kxk2
≥ 2xH Y AY x
because (iL)H = −iL and AH = A. Let x = col(xj )j∈n̄ and define yj = Yj xj . Then
H
x Y AY x =
n
X
2
ajj |yj | +
j=1
≥
n
X
=
n
X
ajk 2 Re(yjH yk )
j=1 k=j+1
ajj |yj |2 −
j=1
n−1
X
n−1
X
n−1
X
n
X
|ajk |2|yj | |yk |
j=1 k=j+1
n
X
j=1 k=j+1
|ajk |(|yj | − |yk |)2 +
n
X
j=1

ajj −
n
X
k6=j

|ajk | |yj |2 > 0,
where we have used the condition of diagonal dominance to obtain the last bound.