Stability reserve in stochastic linear s
Stability Reserve in Stochastic Linear Systems with
Applications to Power Systems
Humberto Verdejo
Luis Vargas
Wolfgang Kliemann
Department of Electrical Engineering
University of Chile
Department of Electrical Engineering
University of Chile
Department of Mathematics
Iowa State University
Abstract—This paper studies linear systems under sustained
random perturbations with the purpose of defining a stochastic
stability reserve, i.e., of computing for a given size of the
pertubation the values of the system parameters for which
the system shows the best stability behavior. The stochastic
perturbation model is given by a bounded Markov diffusion
process. The Lyapunov exponent is used for computing the
stability reserve. This paper presents a short description of four
numerical methods for the computation of the Lyapunov exponent
and the methodology is applied to linear oscillator in dimension
2 and to a one machine - infinite bus electric power system.
I. I NTRODUCTION
In their operation, electric power systems are subjected to
a variety of random perturbations, which affect all aspects of
system behavior, including small signal analysis as well as
contingency studies and transient stability analysis. Basically
all probabilistic analysis of power system behavior addresses
contingency and security / reliability analyses, see e.g. [10].
Notable exceptions that deal with small signal analysis under
random perturbations include, e.g., [8], [9], and [13]. These
papers consider systems under white noise excitation and
discuss stability criteria ([8] and [9]) or noise-induced chaos
([13]).
In the context of small signal analysis of linearized systems,
Lyapunov exponents provide information about the almost sure
(exponential) stability of systems under random perturbations,
generalizing the real parts of the eigenvalues of deterministic linear systems. In [12] we presented the theoretical
development for computing the Lyapunov exponents of linear
stochastic systems. In this paper, we consider perturbations of
varying size, and study their impact on the resulting Lyapunov
exponents. This allows us to study the stability reserve in such
systems.
This paper is organized as follows: In Section II we briefly
introduce the model and discuss almost sure Lyapunov exponents. Section III presents the model of the perturbation and reviews numerical methods for computing Lyapunov exponents.
Section IV describes the two examples for which we present
stochastic stability results in Section V.
II. M ETHODOLOGY
A. Lyapunov Exponents and Stability of Stochastic Linear
Systems
Consider the system
ẋ = A(ξt )x
in Rd ,
(1)
where ξt is a perturbation process of the following type: Let
dηt = Y0 (ηt )dt +
l
X
Yi (ηt ) ◦ dWt
in M
(2)
i=1
be a stochastic differential equation on a compact
C ∞ −manifold M with C ∞ −vector fields Y0 , ..., Yl . Here
”◦” denotes the Stratonovic stochastic differential. We assume
that (2) has a unique stationary and ergodic solution with
invariant distribution ν on M , which is guaranteed, e.g. by
the weak nondegeneracy condition
dim LA {Yi , i = 1, ..., l} (y) = dim M
for all y ∈ M ,
(3)
where LA denotes the Lie algebra generated by the vector
fields Yi . The system perturbation ξt in (1) is modeled as a
function of the background noise ηt in the form
ξt = f (ηt )
f : M → U ⊂ Rm
(4)
with f a C ∞ function that is onto a compact set U ⊂ Rm
with 0 ∈ intU , the interior of U . This set-up gives us great
flexibility when modeling a (bounded, stationary, Markovian)
perturbation of the system.
We denote the solutions of (1) at time t ≥ 0 with initial
value x ∈ Rd by ϕ(t, x, ξt ). Their exponential growth behavior
is given the Lyapunov exponents
1
λ(x, ω) = lim sup log(kϕ(t, x, ξt (ω))k),
t→∞ t
(5)
where ω is an element of the probability space on which the
stochastic differential equation (2) is defined. For background
material on Lyapunov exponents of stochastic systems we refer
the reader to [1], [4] and [14].
In general, the stochastic system (1) with ergodic perturbation ξt can have up to d different Lyapunov exponents. Under
a mild nondegeneracy condition on the invariant directions of
A(ξt ), however, one has a unique exponent with probability 1.
This condition is expressed in terms of the induced system on
the sphere in Rd : Since the system (1) is linear, its projection
onto the sphere Sd−1 ⊂ Rd is given by the random differential
equation
ṡ = h(A(ξt ), s)
s ∈ Sd−1
(6)
T
h(A, s) = (A − s As · I)s
PMAPS 2010
where ”·T ” denotes the transpose of a vector, and I is the
d × d identity matrix. We assume the following ’richness’
condition on the background noise (2) and A(η) projected onto
the sphere
P
Y0 + wi Yi
η
l
dim LA
= dim M + d − 1
,w∈R
s
h(A(f (·)), s)
(7)
for all ηs ∈ M × Sd−1 . With these preparations we have the
following result:
Theorem 2.1: Consider the linear system (1) with stochastic
perturbation (2,4) under the assumptions (3,7). Then the system
has a unique Lyapunov exponent
λ ≡ λ(x, ω) = lim
1
t→∞ t
log(kϕ(t, x, ξt (ω))k)
(8)
for all x ∈ Rd \ {0}, almost surely.
This theorem was first proved in [3], the version presented
here follows the set-up of [4]. The Lyapunov exponent from
Theorem 2.1 determines the stability behavior of the solutions
of (1) in the following way:
Corollary 2.2: Under the conditions of Theorem 2.1, the
zero solution ϕ(t, 0, ξt ) ≡ 0 of the stochastic linear system
ẋ(t) = A(ξt )x(t) is almost surely exponentially stable if and
only if λ < 0.
Theorem 2.1 and Corollary 2.2 are the basis for numerical
methods for Lyapunov exponents, which we address in the next
section. The setup presented here is quite flexible in the sense
that the background noise ηt in (2) allows the modeling of
bounded, stationary perturbations with any rational spectrum
(e.g. as projection of an Ornstein-Uhlenbeck process onto a
sphere), and hence the system perturbation ξt = f (ηt ) can
model stochastic processes with a wide variety of statistical
characteristics. Of course, in applications to actual power
systems both ηt and f need to be estimated from actual data,
such as variations in generation or loads.
So far we have considered systems with a fixed random
perturbation ξt , t ≥ 0. To judge the actual stability reserve
in a system, we study the system response to perturbations of
varying size. Keeping the background noise in (2) fixed, we
introduce a parameterized family of functions
f ρ : M → U ρ ⊂ Rm ,
ρ
f (η) := ρ · f (η),
U ρ := ρU
ρ ≥ 0.
Recall the requirement that 0 ∈ intU , and hence this setup
models varying perturbation ranges. For ρ = 0 we recover
the unperturbed system, and for ρ = 1 we obtain the system
(1). The assumptions (3) and (7) are valid for all ρ > 0 if
they hold for ρ = 1. Hence Theorem 2.1 and Corollary 2.2
guarantee the existence of a unique Lyapunov exponent λρ for
all ρ > 0. For the sake of comparison with the unperturbed
(deterministic) system we define λ0 to be the maximal real
part of the eigenvalues of the system matrix A := A(ξt = 0).
III. N UMERICAL M ETHODS
A. Perturbation model
For the case studies in Section IV we use as background
noise an Ornstein-Uhlenbeck process
dηt = −ηt dt + dWt
in R1 .
(9)
The system noise is given by
ξt = ρ · sin(ηt ),
ρ≥0
(10)
resulting on an Ornstein-Uhlenbeck process on the circle S1
with magnitude 1.
The stochastic differential equation (9) is solved numerically using the Euler scheme, compare [6]. The resulting
linear equation (1) and the projected equation (6) are solved
numerically using an explicit 4-th order Runge-Kutta scheme.
B. Method 1: Computing the Lyapunov exponent from the
linear system
For general background information on the numerical computation of Lyapunov exponents we refer the reader to [5],
[11], and [14].
We fix a time interval [0, T ], T ∈ N, and a step size
h = k1 > 0, k ∈ N, for the simulation of the background
process ηt , resulting in β time series ηt (i), i = 1, ..., β
of length T k. We pick α initial conditions xj0 ∈ Sd−1 ,
j = 1, ..., α. For each initial condition the linear system (1)
is solved on the time interval [0, 1], resulting in β time series
xj(n) (i) with n = 0, ..., k. We define xj1 (i) := xj(k) (i) and use
sj1 (i) := xj(k) (i)/
xj(k) (i)
∈ Sd−1 to be the initial condition
for the time interval [1, 2], and we continue in this way over the
time interval [0, T ]. For each trajectory ηt (i) of the background
noise and each initial value xj0 we obtain an approximation of
the Lyapunov exponent via
λj (i) =
T
1 X
log
xjn (i)
.
T n=1
(11)
Averaging expression (11) over the β realizations of the
background noise results in an estimate
λj =
β
1X j
λ (i)
β i=1
(12)
of the time-T Lyapunov exponent from the initial value xj0 ∈
Sd−1 , and further averaging over the initial conditions gives
the estimate
α
1X j
λ=
λ
(13)
α j=1
for the time-T Lyapunov exponent of the linear stochastic
system (1).
Since the Lyapunov exponent of a linear stochastic system is
obtained as the limit of the time-T exponents for T → ∞, one
often obtains more consistent simulation results when using a
’burn-in’ period, i.e. one starts the actual computation of the
exponent after time T1 > 0. For T1 ∈ N with T1 < T , one
obtains instead of (11) the following formula
λj (i) =
T
X
1
log
xjn (i)
.
T − T1
(14)
n=T1
The other averages are then computed as in (12) and (13).
C. Method 2: Computing the Lyapunov exponent from the
projection of the linear system
As in Section III-B we fix a time interval [0, T ], T ∈ N, a
step size h, and we pick α initial conditions xj0 ∈ Sd−1 , j =
1, ..., α. The time series of the background noise are computed
as in Method 1. In this approach, for each initial condition the
linear system (1) is solved for one time step h, and the resulting
point in Rd is projected onto the sphere Sd−1 . Continuing
in
this way, one obtains a time series sjn (i) := xjn (i)/
xjn (i)
for
n = 0, ..., T k. This time series is a numerical approximation
of the solution of the projected system (6). Using Formula (13)
in [12] we obtain
1
λ (i) =
T
j
ZT
sjt (i)T A(ξt )sjt (i)dt
as the time-T approximation of the Lyapunov exponent for
the initial condition xj0 ∈ Sd−1 and the background trajectory ηt (i). For the results in Section IV we have used the
trapezoidal rule to evaluate the integral in (15). Averages over
the background trajectories and the initial conditions are now
computed as in (12) and (13).
As for Method 1, one often obtains more consistent results
when using a burn-in period T1 > 0, i.e. by considering
1
λ (i) =
T − T1
ZT
sjt (i)T A(ξt )sjt (i)dt.
Pd−1
For lower dimensional systems, Formula (17) often provides
a convenient way to compute the a.s. Lyapunov exponent of
a linear system, compare the discussion in [14]. For higher
dimensional systems, such as power systems, the reliable
computation of the density of κ is prohibitive in terms computation time and data storage. Therefore, we do not pursue
this approach in the current study.
(15)
0
j
E. Method 4: Invariant distribution on the sphere
Under the conditions of Theorem 2.1 the pair process
Ps(t, s0 , ξt )
has a unique stationary distribution µ on
ηt
d−1
P
× M with marginal ν on M , compare, e.g., [2]. Here
Pd−1 denotes the projective space of Rd , and this space needs
to be considered since ϕ(t, αx0 , ξt (ω)) = αϕ(t, x0 , ξt (ω)) for
all α ∈ R, α 6= 0. The map P : Sd−1 → Pd−1 is the projection,
identifying opposite points on the sphere. On Sd−1 × M we
either have one stationary distribution (iff the support of µ is
all of Pd−1 × M ), or two that are reflections at the origin of
each other. Let κ be the marginal of µ on Pd−1 (or one of its
copies on Sd−1 ), then the ergodic theorem implies that
Z
sT As dκ almost surely.
(17)
λ=
(16)
T1
For this method, such burn-in periods are often essential since
the expression sT As in (15) can take, for some systems such
as realistic power systems, very large values when s ∈ Sd−1
is close to one of the axes in Rd .
D. Method 3: Computing the Lyapunov exponent from the
projected system
This method is the same as Method 2, except that the
trajectories sjn (i) for n = 0, ..., T k on the sphere Sd−1 are
computed directly using the nonlinear differential equation (6)
of the projected system. We have used an explicit RungeKutta scheme of order 4 for the case studies in Section IV.
Within this setup, Formulas (15) and (16) hold without change.
When using this method, burn-in periods are essential since the
expression sT As also appears in the vector field on the right
hand side of (6).
We have tested all four methods for a variety of systems,
compare e.g. the results in [12], and we have chosen Method
1 for this paper because of its accuracy and speed.
IV. E XAMPLES
A. Two-dimensional linear oscillator
ẋ + 2bẏ + (1 + ξt )y = 0
(18)
or in the state space form
0 0
0
1
x.
ẋ =
+ ξt
−1 0
−1 −2b
The key parameter of the linear oscillator is the damping
b, for which we chose values b ∈ [0, 3]. For the range of the
random perturbation we used ρ ∈ [0, 4], in increments of 0.2.
The results in Section V were produced with a number β = 60
of realizations, and a number α = 41 of initial conditions,
spaced uniformly.
B. One machine-infinite bus power system
In the area of power systems, we present the classical
example of a one machine-infinite bus system, compare [7].
LT
G
HT
Xt
j0.15
Et
P→
Q→
Xl
j0.5
Xl
j0.5
EB
Inf. Bus
Fig. 1: Machine to infinite bus
V. R ESULTS
We use the following parameters
P = 0.9(p.u) Q = 0.3(p.u)
A. Example 1: The linear oscillator
(overexcited)
Ėt = (1.0∠36o ) E˙B = (0.995∠0o ).
The state vector for the linear system ∆ẋ = A∆x, is
∆x = [∆ω, ∆δ, ∆Ψf d , v1 , v2 , v3 ]
(19)
where v1 , v2 , v3 are variables of the PSS, using expressions
defined in [7]. The matrix A has the structure
a11 a12 a13 0
0
0
a21 0
0
0
0
0
0 a32 a33 a34 0 a36
A=
0
0 a42 a43 a44 0
a51 a52 a53 0 a55 0
a61 a62 a63 0 a65 a66
and the differential equation for the field circuit is:
K3
(∆Ef d − K4 ∆δ) ,
1 + sT3
∆Ψf d =
(a) Lyapunov Exponents
(20)
with excitation system
∆Ef d = −KA ∆v1 ,
(21)
where v1 is the output of the voltage transductor. The perturbation has been introduced as affecting the reference signal.
This situation is described by changing the element a34 in the
matrix A to
∆Ef d = −KA (1 + ξt )∆v1 .
(22)
In this section we consider the (linearized) one machine infinite bus system ẋ = Ax with system matrix
0
−0.11 −0.12
0
0
0
377
0
0
0
0
0
0
−0.19
−0.42
−27.4
0
27.4
,
A=
0
−7.3
20.8
−50
0
0
0
−1
−1.1
0
−0.71
0
0
−4.8 −5.4
0
26.9 −30.3
where a3,4 = a3,4 · (1 + ξt ) is a stochastic perturbation as
described above. The stochastic perturbation is ξt = ρ · sin(ηt )
with ηt an Ornstein-Uhlenbeck process as in (9).
The key parameter in this system is the gain of the PSS,
KP SS , whose nominal values was chosen as in [7]. To study
the stability reserve of this system, we used gain values Kγ :=
γ ·KP SS , with γ = 0.5, 1, 5, 10, 15, 20, 25. For the range of the
random perturbation we used ρ ∈ [0, 4], with a step size of 0.2.
The results in Section V were produced with a number β = 30
of realizations, and a number α = 54 of initial conditions,
spaced uniformly on a regular grid in (∆δ, ∆ω)− space (angle,
angular velocity).
The time of simulation for the results in Section V is T =
30sec. We tested the system also with T = 60sec and T =
90sec, and obtained similar results.
(b) Level curves
Fig. 2: Two-dimensional linear oscillator
Figure 2a shows the almost sure Lyapunov exponent λρ (b)
depending on two parameters, the perturbation size ρ and the
damping b. Figure 2b contains the different levels curves of
the surface in Figure 2a. These figures show two important
results:
1) For a fixed value b0 ≤ 1 of the damping, the a.s.
Lyapunov exponent λρ (b0 ) grows in a monotone fashion
with the perturbation size ρ, while for b0 > 1 the
exponent λρ (b0 ) attains a minimum for some ρ(b0 ) > 0.
E.g. for b0 = 1.25, the minimal value λρ (1.25) = −0.65
is attained at ρ(1.25) = 1.5. Note that b = 1.0 is
the optimal damping value for the unperturbed system
with ρ = 0. In other words, damping beyond the
optimal deterministic value causes an increase in stability
reserve for small perturbations up to the range ρ(b0 ), and
diminishing stability reserve for further increases of the
perturbation range.
2) For the unperturbed system (with ρ = 0) the Lyapunov
exponent is at its minimum for b = 1, with λ0 (1) = −1.
For a fixed value ρ0 > 0 of the perturbation range, the
behavior is similar, except that the value b(ρ0 ) of the
damping, at which the minimal Lyapunov exponent is
realized, increases with increasing perturbation range ρ.
These two observations allow us to determine, for each
perturbation size ρ ≥ 0, the damping value b(ρ) at which the
stability reserve of the system is maximal, i.e. at which the
Lyapunov exponent λρ (b) is minimal.
This example shows basically the same qualitative behavior
as Example V-A. The only difference is that for small perturbation ranges ρ the optimal PSS gain γ(ρ) is somewhat smaller
than the optimal deterministic value. But for larger ρ we see
the same behavior as before: The optimal stability reserve of
the system is attained for gain factors γ(ρ) that are larger than
the optimal deterministic value γ0 = 10.
C. Example 3: One machine - infinite bus system, Part II
B. Example 2: One machine - infinite bus system, Part I
(a) Lyapunov Exponents
(a) Lyapunov Exponents
(b) Level curves
(b) Level curves
Fig. 3: One machine-infinite bus power system with huge
values
Figure 3a shows the almost sure Lyapunov exponent λρ (γ)
depending on two parameters, the perturbation size ρ and the
factor γ determining the PSS gain Kγ as described in Section
IV-B. Figure 3b contains the different levels curves of the
surface in Figure 3a. Note that in this example the range of
the PSS gain is very large, up to 25 times of the nominal value
KP SS .
Fig. 4: One machine-infinite bus power system with tipical
values
In this example we have considered the one machine infinite bus system from Example V-B for PSS gains that are
close to the nominal value KP SS , i.e. for values of γ in the
range γ ∈ [0, 1.4]. Figures 4a and 4b show a more detailed
picture of lower ranges in Figures 3a and 3b. As expected,
for fixed ρ > 0 the Lyapunov exponent λρ (γ) is basically
flat, independent of the value of γ in this range. This shows
that for the nominal one machine - infinite bus system from
Section IV-B, tuning of the PSS gain within the common range
of gain factors hardly affects the stochastic stability reserve of
the system.
VI. C ONCLUSION
In this paper we have analyzed the stability reserve of a linear stochastic system using almost sure Lyapunov exponents.
The idea is to compute the exponent λρ (p) for various values
of the perturbation range ρ ≥ 0 and of the tunable system
parameter(s) p. The computed surface of λρ (p) allows us, for
each noise range, to obtain the optimal tuning value p(ρ) for
the parameter, resulting in maximal stability reserve for this
noise level. We studied the case of the linear oscillator (with
damping coefficient b as the tuning parameter) and of the one
machine - infinite bus power system (with PSS gain constant as
the tuning parameter). In both cases the optimal tuning value
under random perturbations is larger than the optimal value of
the unperturbed (deterministic) system. However, if we restrict
the PSS gain to a small neighborhood of its nominal value,
the λρ (p) level curves are basically flat, i.e. the PSS gain in
this range does not have much of an effect on the stochastic
stability reserve of the system.
ACKNOWLEDGMENT
This work was supported by The Institute of Complex
Engineering System, by Mecesup (Project FSM0601) and
BECAS-CHILE program.
R EFERENCES
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Applications to Power Systems
Humberto Verdejo
Luis Vargas
Wolfgang Kliemann
Department of Electrical Engineering
University of Chile
Department of Electrical Engineering
University of Chile
Department of Mathematics
Iowa State University
Abstract—This paper studies linear systems under sustained
random perturbations with the purpose of defining a stochastic
stability reserve, i.e., of computing for a given size of the
pertubation the values of the system parameters for which
the system shows the best stability behavior. The stochastic
perturbation model is given by a bounded Markov diffusion
process. The Lyapunov exponent is used for computing the
stability reserve. This paper presents a short description of four
numerical methods for the computation of the Lyapunov exponent
and the methodology is applied to linear oscillator in dimension
2 and to a one machine - infinite bus electric power system.
I. I NTRODUCTION
In their operation, electric power systems are subjected to
a variety of random perturbations, which affect all aspects of
system behavior, including small signal analysis as well as
contingency studies and transient stability analysis. Basically
all probabilistic analysis of power system behavior addresses
contingency and security / reliability analyses, see e.g. [10].
Notable exceptions that deal with small signal analysis under
random perturbations include, e.g., [8], [9], and [13]. These
papers consider systems under white noise excitation and
discuss stability criteria ([8] and [9]) or noise-induced chaos
([13]).
In the context of small signal analysis of linearized systems,
Lyapunov exponents provide information about the almost sure
(exponential) stability of systems under random perturbations,
generalizing the real parts of the eigenvalues of deterministic linear systems. In [12] we presented the theoretical
development for computing the Lyapunov exponents of linear
stochastic systems. In this paper, we consider perturbations of
varying size, and study their impact on the resulting Lyapunov
exponents. This allows us to study the stability reserve in such
systems.
This paper is organized as follows: In Section II we briefly
introduce the model and discuss almost sure Lyapunov exponents. Section III presents the model of the perturbation and reviews numerical methods for computing Lyapunov exponents.
Section IV describes the two examples for which we present
stochastic stability results in Section V.
II. M ETHODOLOGY
A. Lyapunov Exponents and Stability of Stochastic Linear
Systems
Consider the system
ẋ = A(ξt )x
in Rd ,
(1)
where ξt is a perturbation process of the following type: Let
dηt = Y0 (ηt )dt +
l
X
Yi (ηt ) ◦ dWt
in M
(2)
i=1
be a stochastic differential equation on a compact
C ∞ −manifold M with C ∞ −vector fields Y0 , ..., Yl . Here
”◦” denotes the Stratonovic stochastic differential. We assume
that (2) has a unique stationary and ergodic solution with
invariant distribution ν on M , which is guaranteed, e.g. by
the weak nondegeneracy condition
dim LA {Yi , i = 1, ..., l} (y) = dim M
for all y ∈ M ,
(3)
where LA denotes the Lie algebra generated by the vector
fields Yi . The system perturbation ξt in (1) is modeled as a
function of the background noise ηt in the form
ξt = f (ηt )
f : M → U ⊂ Rm
(4)
with f a C ∞ function that is onto a compact set U ⊂ Rm
with 0 ∈ intU , the interior of U . This set-up gives us great
flexibility when modeling a (bounded, stationary, Markovian)
perturbation of the system.
We denote the solutions of (1) at time t ≥ 0 with initial
value x ∈ Rd by ϕ(t, x, ξt ). Their exponential growth behavior
is given the Lyapunov exponents
1
λ(x, ω) = lim sup log(kϕ(t, x, ξt (ω))k),
t→∞ t
(5)
where ω is an element of the probability space on which the
stochastic differential equation (2) is defined. For background
material on Lyapunov exponents of stochastic systems we refer
the reader to [1], [4] and [14].
In general, the stochastic system (1) with ergodic perturbation ξt can have up to d different Lyapunov exponents. Under
a mild nondegeneracy condition on the invariant directions of
A(ξt ), however, one has a unique exponent with probability 1.
This condition is expressed in terms of the induced system on
the sphere in Rd : Since the system (1) is linear, its projection
onto the sphere Sd−1 ⊂ Rd is given by the random differential
equation
ṡ = h(A(ξt ), s)
s ∈ Sd−1
(6)
T
h(A, s) = (A − s As · I)s
PMAPS 2010
where ”·T ” denotes the transpose of a vector, and I is the
d × d identity matrix. We assume the following ’richness’
condition on the background noise (2) and A(η) projected onto
the sphere
P
Y0 + wi Yi
η
l
dim LA
= dim M + d − 1
,w∈R
s
h(A(f (·)), s)
(7)
for all ηs ∈ M × Sd−1 . With these preparations we have the
following result:
Theorem 2.1: Consider the linear system (1) with stochastic
perturbation (2,4) under the assumptions (3,7). Then the system
has a unique Lyapunov exponent
λ ≡ λ(x, ω) = lim
1
t→∞ t
log(kϕ(t, x, ξt (ω))k)
(8)
for all x ∈ Rd \ {0}, almost surely.
This theorem was first proved in [3], the version presented
here follows the set-up of [4]. The Lyapunov exponent from
Theorem 2.1 determines the stability behavior of the solutions
of (1) in the following way:
Corollary 2.2: Under the conditions of Theorem 2.1, the
zero solution ϕ(t, 0, ξt ) ≡ 0 of the stochastic linear system
ẋ(t) = A(ξt )x(t) is almost surely exponentially stable if and
only if λ < 0.
Theorem 2.1 and Corollary 2.2 are the basis for numerical
methods for Lyapunov exponents, which we address in the next
section. The setup presented here is quite flexible in the sense
that the background noise ηt in (2) allows the modeling of
bounded, stationary perturbations with any rational spectrum
(e.g. as projection of an Ornstein-Uhlenbeck process onto a
sphere), and hence the system perturbation ξt = f (ηt ) can
model stochastic processes with a wide variety of statistical
characteristics. Of course, in applications to actual power
systems both ηt and f need to be estimated from actual data,
such as variations in generation or loads.
So far we have considered systems with a fixed random
perturbation ξt , t ≥ 0. To judge the actual stability reserve
in a system, we study the system response to perturbations of
varying size. Keeping the background noise in (2) fixed, we
introduce a parameterized family of functions
f ρ : M → U ρ ⊂ Rm ,
ρ
f (η) := ρ · f (η),
U ρ := ρU
ρ ≥ 0.
Recall the requirement that 0 ∈ intU , and hence this setup
models varying perturbation ranges. For ρ = 0 we recover
the unperturbed system, and for ρ = 1 we obtain the system
(1). The assumptions (3) and (7) are valid for all ρ > 0 if
they hold for ρ = 1. Hence Theorem 2.1 and Corollary 2.2
guarantee the existence of a unique Lyapunov exponent λρ for
all ρ > 0. For the sake of comparison with the unperturbed
(deterministic) system we define λ0 to be the maximal real
part of the eigenvalues of the system matrix A := A(ξt = 0).
III. N UMERICAL M ETHODS
A. Perturbation model
For the case studies in Section IV we use as background
noise an Ornstein-Uhlenbeck process
dηt = −ηt dt + dWt
in R1 .
(9)
The system noise is given by
ξt = ρ · sin(ηt ),
ρ≥0
(10)
resulting on an Ornstein-Uhlenbeck process on the circle S1
with magnitude 1.
The stochastic differential equation (9) is solved numerically using the Euler scheme, compare [6]. The resulting
linear equation (1) and the projected equation (6) are solved
numerically using an explicit 4-th order Runge-Kutta scheme.
B. Method 1: Computing the Lyapunov exponent from the
linear system
For general background information on the numerical computation of Lyapunov exponents we refer the reader to [5],
[11], and [14].
We fix a time interval [0, T ], T ∈ N, and a step size
h = k1 > 0, k ∈ N, for the simulation of the background
process ηt , resulting in β time series ηt (i), i = 1, ..., β
of length T k. We pick α initial conditions xj0 ∈ Sd−1 ,
j = 1, ..., α. For each initial condition the linear system (1)
is solved on the time interval [0, 1], resulting in β time series
xj(n) (i) with n = 0, ..., k. We define xj1 (i) := xj(k) (i) and use
sj1 (i) := xj(k) (i)/
xj(k) (i)
∈ Sd−1 to be the initial condition
for the time interval [1, 2], and we continue in this way over the
time interval [0, T ]. For each trajectory ηt (i) of the background
noise and each initial value xj0 we obtain an approximation of
the Lyapunov exponent via
λj (i) =
T
1 X
log
xjn (i)
.
T n=1
(11)
Averaging expression (11) over the β realizations of the
background noise results in an estimate
λj =
β
1X j
λ (i)
β i=1
(12)
of the time-T Lyapunov exponent from the initial value xj0 ∈
Sd−1 , and further averaging over the initial conditions gives
the estimate
α
1X j
λ=
λ
(13)
α j=1
for the time-T Lyapunov exponent of the linear stochastic
system (1).
Since the Lyapunov exponent of a linear stochastic system is
obtained as the limit of the time-T exponents for T → ∞, one
often obtains more consistent simulation results when using a
’burn-in’ period, i.e. one starts the actual computation of the
exponent after time T1 > 0. For T1 ∈ N with T1 < T , one
obtains instead of (11) the following formula
λj (i) =
T
X
1
log
xjn (i)
.
T − T1
(14)
n=T1
The other averages are then computed as in (12) and (13).
C. Method 2: Computing the Lyapunov exponent from the
projection of the linear system
As in Section III-B we fix a time interval [0, T ], T ∈ N, a
step size h, and we pick α initial conditions xj0 ∈ Sd−1 , j =
1, ..., α. The time series of the background noise are computed
as in Method 1. In this approach, for each initial condition the
linear system (1) is solved for one time step h, and the resulting
point in Rd is projected onto the sphere Sd−1 . Continuing
in
this way, one obtains a time series sjn (i) := xjn (i)/
xjn (i)
for
n = 0, ..., T k. This time series is a numerical approximation
of the solution of the projected system (6). Using Formula (13)
in [12] we obtain
1
λ (i) =
T
j
ZT
sjt (i)T A(ξt )sjt (i)dt
as the time-T approximation of the Lyapunov exponent for
the initial condition xj0 ∈ Sd−1 and the background trajectory ηt (i). For the results in Section IV we have used the
trapezoidal rule to evaluate the integral in (15). Averages over
the background trajectories and the initial conditions are now
computed as in (12) and (13).
As for Method 1, one often obtains more consistent results
when using a burn-in period T1 > 0, i.e. by considering
1
λ (i) =
T − T1
ZT
sjt (i)T A(ξt )sjt (i)dt.
Pd−1
For lower dimensional systems, Formula (17) often provides
a convenient way to compute the a.s. Lyapunov exponent of
a linear system, compare the discussion in [14]. For higher
dimensional systems, such as power systems, the reliable
computation of the density of κ is prohibitive in terms computation time and data storage. Therefore, we do not pursue
this approach in the current study.
(15)
0
j
E. Method 4: Invariant distribution on the sphere
Under the conditions of Theorem 2.1 the pair process
Ps(t, s0 , ξt )
has a unique stationary distribution µ on
ηt
d−1
P
× M with marginal ν on M , compare, e.g., [2]. Here
Pd−1 denotes the projective space of Rd , and this space needs
to be considered since ϕ(t, αx0 , ξt (ω)) = αϕ(t, x0 , ξt (ω)) for
all α ∈ R, α 6= 0. The map P : Sd−1 → Pd−1 is the projection,
identifying opposite points on the sphere. On Sd−1 × M we
either have one stationary distribution (iff the support of µ is
all of Pd−1 × M ), or two that are reflections at the origin of
each other. Let κ be the marginal of µ on Pd−1 (or one of its
copies on Sd−1 ), then the ergodic theorem implies that
Z
sT As dκ almost surely.
(17)
λ=
(16)
T1
For this method, such burn-in periods are often essential since
the expression sT As in (15) can take, for some systems such
as realistic power systems, very large values when s ∈ Sd−1
is close to one of the axes in Rd .
D. Method 3: Computing the Lyapunov exponent from the
projected system
This method is the same as Method 2, except that the
trajectories sjn (i) for n = 0, ..., T k on the sphere Sd−1 are
computed directly using the nonlinear differential equation (6)
of the projected system. We have used an explicit RungeKutta scheme of order 4 for the case studies in Section IV.
Within this setup, Formulas (15) and (16) hold without change.
When using this method, burn-in periods are essential since the
expression sT As also appears in the vector field on the right
hand side of (6).
We have tested all four methods for a variety of systems,
compare e.g. the results in [12], and we have chosen Method
1 for this paper because of its accuracy and speed.
IV. E XAMPLES
A. Two-dimensional linear oscillator
ẋ + 2bẏ + (1 + ξt )y = 0
(18)
or in the state space form
0 0
0
1
x.
ẋ =
+ ξt
−1 0
−1 −2b
The key parameter of the linear oscillator is the damping
b, for which we chose values b ∈ [0, 3]. For the range of the
random perturbation we used ρ ∈ [0, 4], in increments of 0.2.
The results in Section V were produced with a number β = 60
of realizations, and a number α = 41 of initial conditions,
spaced uniformly.
B. One machine-infinite bus power system
In the area of power systems, we present the classical
example of a one machine-infinite bus system, compare [7].
LT
G
HT
Xt
j0.15
Et
P→
Q→
Xl
j0.5
Xl
j0.5
EB
Inf. Bus
Fig. 1: Machine to infinite bus
V. R ESULTS
We use the following parameters
P = 0.9(p.u) Q = 0.3(p.u)
A. Example 1: The linear oscillator
(overexcited)
Ėt = (1.0∠36o ) E˙B = (0.995∠0o ).
The state vector for the linear system ∆ẋ = A∆x, is
∆x = [∆ω, ∆δ, ∆Ψf d , v1 , v2 , v3 ]
(19)
where v1 , v2 , v3 are variables of the PSS, using expressions
defined in [7]. The matrix A has the structure
a11 a12 a13 0
0
0
a21 0
0
0
0
0
0 a32 a33 a34 0 a36
A=
0
0 a42 a43 a44 0
a51 a52 a53 0 a55 0
a61 a62 a63 0 a65 a66
and the differential equation for the field circuit is:
K3
(∆Ef d − K4 ∆δ) ,
1 + sT3
∆Ψf d =
(a) Lyapunov Exponents
(20)
with excitation system
∆Ef d = −KA ∆v1 ,
(21)
where v1 is the output of the voltage transductor. The perturbation has been introduced as affecting the reference signal.
This situation is described by changing the element a34 in the
matrix A to
∆Ef d = −KA (1 + ξt )∆v1 .
(22)
In this section we consider the (linearized) one machine infinite bus system ẋ = Ax with system matrix
0
−0.11 −0.12
0
0
0
377
0
0
0
0
0
0
−0.19
−0.42
−27.4
0
27.4
,
A=
0
−7.3
20.8
−50
0
0
0
−1
−1.1
0
−0.71
0
0
−4.8 −5.4
0
26.9 −30.3
where a3,4 = a3,4 · (1 + ξt ) is a stochastic perturbation as
described above. The stochastic perturbation is ξt = ρ · sin(ηt )
with ηt an Ornstein-Uhlenbeck process as in (9).
The key parameter in this system is the gain of the PSS,
KP SS , whose nominal values was chosen as in [7]. To study
the stability reserve of this system, we used gain values Kγ :=
γ ·KP SS , with γ = 0.5, 1, 5, 10, 15, 20, 25. For the range of the
random perturbation we used ρ ∈ [0, 4], with a step size of 0.2.
The results in Section V were produced with a number β = 30
of realizations, and a number α = 54 of initial conditions,
spaced uniformly on a regular grid in (∆δ, ∆ω)− space (angle,
angular velocity).
The time of simulation for the results in Section V is T =
30sec. We tested the system also with T = 60sec and T =
90sec, and obtained similar results.
(b) Level curves
Fig. 2: Two-dimensional linear oscillator
Figure 2a shows the almost sure Lyapunov exponent λρ (b)
depending on two parameters, the perturbation size ρ and the
damping b. Figure 2b contains the different levels curves of
the surface in Figure 2a. These figures show two important
results:
1) For a fixed value b0 ≤ 1 of the damping, the a.s.
Lyapunov exponent λρ (b0 ) grows in a monotone fashion
with the perturbation size ρ, while for b0 > 1 the
exponent λρ (b0 ) attains a minimum for some ρ(b0 ) > 0.
E.g. for b0 = 1.25, the minimal value λρ (1.25) = −0.65
is attained at ρ(1.25) = 1.5. Note that b = 1.0 is
the optimal damping value for the unperturbed system
with ρ = 0. In other words, damping beyond the
optimal deterministic value causes an increase in stability
reserve for small perturbations up to the range ρ(b0 ), and
diminishing stability reserve for further increases of the
perturbation range.
2) For the unperturbed system (with ρ = 0) the Lyapunov
exponent is at its minimum for b = 1, with λ0 (1) = −1.
For a fixed value ρ0 > 0 of the perturbation range, the
behavior is similar, except that the value b(ρ0 ) of the
damping, at which the minimal Lyapunov exponent is
realized, increases with increasing perturbation range ρ.
These two observations allow us to determine, for each
perturbation size ρ ≥ 0, the damping value b(ρ) at which the
stability reserve of the system is maximal, i.e. at which the
Lyapunov exponent λρ (b) is minimal.
This example shows basically the same qualitative behavior
as Example V-A. The only difference is that for small perturbation ranges ρ the optimal PSS gain γ(ρ) is somewhat smaller
than the optimal deterministic value. But for larger ρ we see
the same behavior as before: The optimal stability reserve of
the system is attained for gain factors γ(ρ) that are larger than
the optimal deterministic value γ0 = 10.
C. Example 3: One machine - infinite bus system, Part II
B. Example 2: One machine - infinite bus system, Part I
(a) Lyapunov Exponents
(a) Lyapunov Exponents
(b) Level curves
(b) Level curves
Fig. 3: One machine-infinite bus power system with huge
values
Figure 3a shows the almost sure Lyapunov exponent λρ (γ)
depending on two parameters, the perturbation size ρ and the
factor γ determining the PSS gain Kγ as described in Section
IV-B. Figure 3b contains the different levels curves of the
surface in Figure 3a. Note that in this example the range of
the PSS gain is very large, up to 25 times of the nominal value
KP SS .
Fig. 4: One machine-infinite bus power system with tipical
values
In this example we have considered the one machine infinite bus system from Example V-B for PSS gains that are
close to the nominal value KP SS , i.e. for values of γ in the
range γ ∈ [0, 1.4]. Figures 4a and 4b show a more detailed
picture of lower ranges in Figures 3a and 3b. As expected,
for fixed ρ > 0 the Lyapunov exponent λρ (γ) is basically
flat, independent of the value of γ in this range. This shows
that for the nominal one machine - infinite bus system from
Section IV-B, tuning of the PSS gain within the common range
of gain factors hardly affects the stochastic stability reserve of
the system.
VI. C ONCLUSION
In this paper we have analyzed the stability reserve of a linear stochastic system using almost sure Lyapunov exponents.
The idea is to compute the exponent λρ (p) for various values
of the perturbation range ρ ≥ 0 and of the tunable system
parameter(s) p. The computed surface of λρ (p) allows us, for
each noise range, to obtain the optimal tuning value p(ρ) for
the parameter, resulting in maximal stability reserve for this
noise level. We studied the case of the linear oscillator (with
damping coefficient b as the tuning parameter) and of the one
machine - infinite bus power system (with PSS gain constant as
the tuning parameter). In both cases the optimal tuning value
under random perturbations is larger than the optimal value of
the unperturbed (deterministic) system. However, if we restrict
the PSS gain to a small neighborhood of its nominal value,
the λρ (p) level curves are basically flat, i.e. the PSS gain in
this range does not have much of an effect on the stochastic
stability reserve of the system.
ACKNOWLEDGMENT
This work was supported by The Institute of Complex
Engineering System, by Mecesup (Project FSM0601) and
BECAS-CHILE program.
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