Electric Power Systems Research 96 (2013) 246–254
Contents lists available at SciVerse ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier.com/locate/epsr
Computation of multi-parameter sensitivities of equilibrium points in electric
power systems
Enrique A. Zamora-Cárdenas a , Claudio R. Fuerte-Esquivel b,∗
a
b
Universidad de Guanajuato, Electrical Engineering Department, Salamanca, Guanajuato, Mexico
Universidad Michoacana de San Nicolás de Hidalgo, Faculty of Electrical Engineering, Morelia, Michoacán, Mexico
a r t i c l e
i n f o
Article history:
Received 15 March 2012
Received in revised form
21 November 2012
Accepted 23 November 2012
Available online 23 December 2012
Keywords:
Small signal stability
Parameter sensitivities
Equilibrium points
Hopf bifurcation
Selective modal analysis
a b s t r a c t
This paper proposes to use the trajectory sensitivity theory to assess the influence of the system’s
parameters on the power system equilibrium. The proposed parameter sensitivity approach provides
complementary information to that given by selective modal analysis which can help determine how the
state variables linked with the critical eigenvalues are affected by the system’s parameters. The parameter sensitivities also provide a way of judging how the system’s parameters affect the oscillatory behavior
of a power system. The proposed method is demonstrated on the 3-machine, 9-bus WSCC system and a
46-machine, 190-bus equivalent Mexican system.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
A power system normally operates at a stable Equilibrium Point
(EP) that depends on a very large number of parameters such as
generation, load and network condition. When small disturbances
occur in some of these parameters, the system adjusts its current
EP to match the parameter change; however, as the stable EP is
moving in the parameter space, it is prone to instability. As parameters change, the EP can lose its stability in such a way that a pair
of complex conjugate eigenvalues of the linearized model crosses
the imaginary axis of the complex plane, from the left-half plane to
the right-half plane, such that the system may start oscillating with
small amplitude [1,2]. This mechanism of loss of stability is associated with a Hopf Bifurcation (HB) [3] where a periodic oscillation
emerges from a stable EP, and another small perturbation on system
parameters provokes the onset of growing oscillations. In this context, there are two types of Hopf bifurcations [3]: supercritical and
subcritical. In the former, the EP becomes unstable with increasing
oscillations which are eventually attracted to a stable limit cycle. In
the latter, the EP becomes unstable, but the system oscillates with
growing amplitude, such that an unstable limit cycle occurs.
From a perspective of power system studies, the HB can be
detected by using a Small Signal Stability (SSS) analysis [4]: when
the reduced Jacobian of the linearized system has a simple pair of
∗ Corresponding author. Tel.: +52 443 3 27 97 28; fax: +52 443 3 27 97 28.
E-mail address: [email protected] (C.R. Fuerte-Esquivel).
0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.epsr.2012.11.013
purely imaginary eigenvalues, referred to as critical eigenvalues,
and no other eigenvalue with zero real parts, a HB occurs. Furthermore, control strategies are adopted in the state variables that are
involved with the critical eigenvalues to maintain a certain oscillatory stability margin and to improve the system’s damping [4].
These associated state variables are identified through the Selective
Modal Analysis (SMA) [5,6] by means of participation factors that
measure the interaction between oscillation modes and the state
variables of a linear system, such that the most influential generator’s states in the EP’s oscillatory instability can be determined.
Although the SMA determines how a given eigenvalue is affected
by the system’s state variables, it is not possible to directly assess
how the parameters of electric components composing the power
system affect the state variables linked to the critical eigenvalues,
and hence the EP’s stability, which is of paramount importance to
design control strategies. A perturbation approach has been proposed to compute a parameter sensitivity that permits quantifying
the effect of a specified parameter on the EP’s stability by analyzing changes of the eigenvalues of the system state matrix [4].
From a base case, the nominal value of a parameter is perturbed,
and a SSS analysis is performed at the new equilibrium point. The
difference between the resulting eigenvalues associated with the
base and perturbed cases provide the effect of such a parameter
on the stability. In order to predict the EP’s stability under different parameter settings, the eigenvalues must be recomputed at
each set of the parameter selection, which could be impractical in
large-scale power systems because of the multiplicity of parameter
variation possibilities. Furthermore, when multiple parameters are
E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
simultaneously perturbed, this technique does not allow the determination of how the EP’s stability is affected by each parameter.
System eigenvalues are, in general, functions of system parameters, such that first and second-order eigenvalue sensitivities can
be used to estimate how the EP’s stability changes in response
to a parameter variation, if this parameter appears explicitly in
the formulation of the system state matrix [7,8]. Eigenvalue firstorder sensitivities have been applied for locations of controllers
and their parameter optimization [9,10], for voltage contingency
ranking [11], as well as for predicting and detecting oscillatory
instabilities [12–14]. When a critical eigenvalue is a nonlinear function of changes in system parameters, the inclusion of second-order
sensitivity terms is necessary to guarantee accuracy for a wide
range of parameter variations [15,16]. Despite the reduction of
the computational cost of employing eigenvalues sensitivities over
repeated eigenvalue computation to assess parameter sensitivities, numerical problems for the computation of derivatives occur
in some numerical schemes if the eigenvectors are nearly linearly
dependent [17]. Additionally, the derivatives of the eigenvalues and
eigenvectors vanish at parameter values where there are multiple
eigenvalues [13].
In order to circumvent the above-mentioned shortcomings, this
paper proposes a numerical scheme to determine how the state
variables linked to the critical eigenvalues are affected by the
system’s parameters. The proposed approach is based on the Trajectory Sensitivity (TS) theory [18,19], and complements the SSS
analysis of EPs: assessing which systems’ parameters have a major
effect on the possible occurrence of an oscillatory instability is possible by computing the parameter sensitivities of associated state
variables, instead of computing the parameter sensitivity of critical
eigenvalues. The TS theory is revisited in this paper and applied in
the context of small signal stability to obtain parameter sensitivities at the equilibrium, which to the best of the authors’ knowledge,
has not been reported yet in the literature. In this case, the contributions of the paper are as follows: (i) parameter sensitivities
of those state variables associated with the critical eigenvalues are
directly computed through a time-domain simulation w.r.t. all system’s parameters of interest, thus providing valuable insights into
the influence of these parameters on the associated states and on
the dynamic behavior of the system in the context of small signal stability and (ii) sensitivities are computed w.r.t. the power
demanded by loads in order to determine critical loading directions
that could steer the system to an oscillatory instability.
The proposed approach is detailed in the rest of the paper which
is structured as follows: Section 2 describes the application of
the Trajectory Sensitivity theory to Differential-Algebraic Equation
(DAE) systems. Section 3 then describes the proposed approach to
quantify the parameters’ effect on the state variables in the context of small-signal stability. Section 4 discusses the results and
validates the proposed approach by using two test systems of 9bus, 3-generators and of 190-bus, 46-generators. Finally, the main
results and contributions are summarized and highlighted in Section 5.
2. Trajectory sensitivity theory for a Differential-Algebraic
Equation model
Power system dynamics can be described by a set of DifferentialAlgebraic Equations (DAEs), as given by (1), where x is a
n-dimensional vector of dynamic state variables with initial conditions x(t0 ) = x0 , y is a m-dimensional vector of instantaneous state
(algebraic) variables (usually the real and imaginary parts or the
magnitudes and phase angles of the complex node voltages) with
initial conditions y(t0 ) = y0 , and ˇ is a set of time-invariant parameters of the system. The dynamics of the equipment, e.g. generators
247
and controls, are explicitly modeled by the set of differential equations through the function f(·). The set of algebraic equations 0 = g(·)
represents the stator algebraic equations and mismatch power flow
equations at each node:
ẋ = f (x, y, ˇ) f : n+m+p → n
0 = g(x, y, ˇ) g : n+m+p → m
x ∈ X ⊂ n
y ∈ Y ⊂ m
(1)
ˇ ∈ ˇ ⊂ p .
2.1. Analytical formulation
Let ˇ0 be the nominal values of ˇ, and assume that the nominal
set of DAEs ẋ = f (x, y, ˇ0 ), 0 = g(x, y, ˇ0 ) has a unique nominal trajectory solution x(t, x0 , y0 , ˇ0 ) and y(t, x0 , y0 , ˇ0 ) over t ∈ [t0 , tend ],
where t0 and tend are the initial and final times, respectively, of the
study time period. Thus, for all ˇ sufficiently close to ˇ0 , the set
of DAEs (1) has a unique perturbed trajectory solution x(t, x0 , y0 ,
ˇ) and y(t, x0 , y0 , ˇ) over t ∈ [t0 , tend ] that is close to the nominal
trajectory solution. This perturbed solution is given by [18]
tend
x(·) = x0 +
f x(·), y(·), ˇ ds
t0
(2)
0 = g x(·), y(·), ˇ .
(3)
The sensitivities of the dynamic and algebraic state vectors w.r.t.
a chosen system’s parameter, xˇ = ∂x(·)/∂ˇ and yˇ = ∂y(·)/∂ˇ, at a
time t along the trajectory are obtained from the partial derivative
of (2) and (3) w.r.t. ˇ:
∂x(·)
=
∂ˇ
0=
tend
t0
∂f (·) ∂x
∂f (·) ∂y
∂f (·)
+
+
∂x ∂ˇ
∂y ∂ˇ
∂ˇ
ds
(4)
∂g(·) ∂y
∂g(·)
∂g(·) ∂x
+
+
.
∂y ∂ˇ
∂ˇ
∂x ∂ˇ
(5)
Lastly, the smooth evolution of the sensitivities along the trajectory is obtained by differentiating (4) and (5) w.r.t. t:
ẋˇ =
∂f (·) ∂y
∂f (·)
∂f (·) ∂x
+
+
≡ fx xˇ + fy yˇ + fˇ ; xˇ (t0 ) = 0
∂x ∂ˇ
∂y ∂ˇ
∂ˇ
(6)
and
0=
∂g(·) ∂x
∂g(·) ∂y
∂g(·)
+
+
≡ gx xˇ + gy yˇ + gˇ ; yˇ (t0 ) = 0 (7)
∂x ∂ˇ
∂y ∂ˇ
∂ˇ
where fx , fy , fˇ , gx , gy and gˇ , are time-varying matrices computed
along the system trajectories. An expanded version of these equations is reported in the Appendix A.
2.2. Trajectory sensitivity computation
Computing the TS with respect to any parameter requires the
solution of the set of nonlinear DAEs (1) and the set of linear
time-varying DAEs (6) and (7), whose differential equations are
algebraized by using the trapezoidal rule to obtain the following
set of algebraic-difference equations:
F1 (·) = xk+1 − xk −
t k+1
f
+fk = 0
2
F2 (·) = g k+1 = 0
F3 (·) = xˇk+1 − xˇk −
t
2
(8)
(9)
k+1
k+1
fxk+1 xˇk+1 + fyk+1 yˇ
+ fˇ
k +fk
+fxk xˇk + fyk yˇ
ˇ
k+1
F4 (·) = gxk+1 xˇk+1 + gyk+1 yˇ
+ gˇk+1 = 0
=0
(10)
(11)
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E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
where t is the integration time-step, and the superscript k is an
index for the time instant tk at which
and functions are
variables
evaluated, e.g. xk = x(tk ) and f k = f xk , yk .
The Newton–Raphson (NR) algorithm provides an approximate solution to the nonlinear equations (8) and (9) by solving
T
for xk yk
in the linear problem Ji ·Xi = −F(X)i , given in
expanded form by (12), where J is known as the Jacobian matrix
and the index i to the NR iteration step:
⎡
⎣
gxk+1
⎤i
t k+1
f
2 x
I−
−
i
i
t k+1 xk
F1 (·)
f
2 y ⎦
=−
.
F2 (·)
yk
g k+1
y
For given values
guess
X i
Ji
x0k+1 = xk
xk
yk
y0k+1 = yk
T
T
(12)
⎣
gxk+1
and updates the solution at each
T
k
⎢
xˇk +
t
S
2
k
fxk xˇk + fyk yˇ
+ fˇk + fˇk+1
⎤
−gˇk+1
⎥
⎦.
(13)
B
The coefficient matrix on the left side of (13) corresponds to
the Jacobian matrix used in the final NR iteration to solve for xk+1
and yk+1 in (12). Based on this observation, the computational burden for the calculation of TS is substantially reduced because the
coefficient matrix is already factored, and only a forward/backward
k+1
at each
substitution is required for the solution of xˇk+1 and yˇ
discrete time tk+1 of the integration period [19].
2.3. Multi-parameter sensitivity
Computing the sensitivities with respect to more than one
parameter requires multiple solutions of (13) at each time of the
whole integration period to compute all sensitivities. In this case,
the solution approach described in Section 2.2 can be directly
extended to compute multi-parameter trajectory sensitivities associated with Np parameters of the system by expressing (13) as (14),
k+1
k+1
and yˇi
∀i = 1,
which is solved Np times for the calculation of xˇi
. . ., Np
J Sˇ1
Sˇ2
···
SˇNp
S
= Bˇ1
Bˇ2
···
BˇNp .
(14)
B
2.4. Sensitivity index
The theoretical background supporting the interpretation of
time evolution of sensitivities in the context of power systems
is reported in [18], based on the fact that the state trajectories become more sensitive to variations of parameters when
the system becomes more stressed. Furthermore, as numerically
demonstrated in [18,20], when the system approaches an unstable operation condition, the sensitivities of state trajectories have
∂ˇ
∂ˇ
∀ = 1, . . . , Np
, the method starts from an initial
t k+1 k+1 x
f
2 y ⎦ ˇ
k+1
yˇ
gyk+1
J
⎡
=⎣
−
∂ˇ
m=1
⎤
t k+1
f
2 x
I−
ng 2 2 (t
)
∂ı
∂ı
(t
)
∂ω
(t
)
j
k
m
m
k
k
SN (tk ) = −
+
F(·)i
iteration i, i.e. xk+1 = xk + xk yk+1 = yk + y
, until a convergence criterion is satisfied.
Once the states have been computed for a new time-step, the TS
are calculated from (13), which is derived from (10) and (11):
⎡
more rapid changes in magnitudes and larger excursions than the
state trajectories. Taking advantage of these properties, an Euclidian norm of the trajectory sensitivities vector, referred to as a
sensitivity norm, is proposed in [20] as a measure of proximity to
instability. This sensitivity norm also permits the computation of
the critical parameters whose variations steer the system much
faster to an unstable operation condition
The time-varying index of proximity to oscillatory instability
proposed in [20] is based on the sensitivity norm for a ng -generator
system defined at each integration time-step t, and is given by
(15)
where j denotes the reference machine. Regarding the interpretation of the dynamic evolution of sensitivities, if the dynamic
sensitivities are attracted to a stationary equilibrium point, this
implies that the system is small-signal transiently stable; otherwise, the system is prone to an oscillatory behavior. Hence, the time
evolution of sensitivities is necessary to quantify which system
parameters have the most significant effect on the possible occurrence of an oscillatory instability by evaluating the time-varying
sensitivity index given by (15) [20]. In this context, the critical
parameter is the one that has the largest value of sensitivity index
within the integration period, not the largest final value.
Note that the application of this sensitivity norm has offered a
suitable approach to explore the critical clearing time of a faulted
system [20], the best possible location of FACTS controllers for transient stability enhancement [21,22] and the TCSC control design to
enhance transient stability [23].
3. Parameter sensitivities of state variables around
equilibrium
A mathematical approach to directly compute parameter sensitivities of state variables at the equilibrium is proposed in this
section based on the TS theory described in Section 2. Details of the
proposal are given below.
The equilibrium values of state and algebraic variables satisfy
0 = f(xe , ye , ˇ) and 0 = g(xe , ye , ˇ); from the TS theory viewpoint this
implies that the sensitivities of these variables w.r.t. system parameters are also constants: xˇ = k1 and yˇ = k2 where k1 and k2 are
n-dimensional and m-dimensional vectors, respectively, and the
state variables’ sensitivity changes w.r.t. time are null, ẋˇ = 0. Based
on these facts, the sensitivities at the equilibrium can be computed
by solving the linearized set of DAEs (13) until the steady-state has
been achieved.
In this solution process, the state and algebraic variables are
set at their equilibrium values during the integration process, such
that the matrices gx , gy , gˇ and fx , fy , fˇ are time-invariant in the
linear sensitivity model (13). Therefore, the set of equations (13) is
reduced to
⎡
⎣
t
fx
2
I−
gx
⎡
=⎣
⎤
−
gy
(xe ,ye ,ˇ) J
xˇk +
t
fy 2
⎦
xˇk+1
k+1
yˇ
S
⎤
t k
k
fx xˇ + fy yˇ
+ 2fˇ
2
⎦,
−gˇ
B
(16)
E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
249
and sensitivities are simply computed by a forward/backward substitution performed at each discrete time tk+1 of the integration
period in order to obtain the sensitivities equilibrium values at
t∞ = t→ ∞. The solution process starts from initial conditions xˇ0 =
0 = 0. In addition to the computation of sensitivities at the equilibyˇ
rium, all sensitivities computed at each time of the solution process
are used to compute the sensitivity index (15), which is employed to
quantify how the associated state variables are affected by changes
in system parameters.
The proposed procedure to assess the effect of a set of system’s
parameters on the associated state variables at the equilibrium
point is summarized as follows:
Step 1. – For an arbitrary set of fixed parameters ˇ, the system’s
equilibrium is computed by solving the set of nonlinear algebraic
equations (1) for x and y considering ẋ = 0. The NR approach is
applied to obtain this solution given by the values xe and ye that
satisfy 0 = f(xe , ye , ˇ) and 0 = g(xe , ye , ˇ).
Step 2. – Based on the Schur and the implicit function
theorems
[2,24],
compute the reduced Jacobian matrix JR =
fx − fy gy−1 gx that has the same dynamic and algebraic
(xe ,ye ,ˇ)
properties of the system’s Jacobian matrix.
Step 3. – Determine the critical eigenvalues of JR , and perform a
SMA to identify the associate state variables.
Step 4. – Compute the sensitivities of associated state variables
w.r.t. the selected system’s parameters at the equilibrium point,
t→∞ , by solving (16). The integration process is started
xˇt→∞ and yˇ
0 = 0 for the parameter sensitivities,
with initial conditions xˇ0 = yˇ
while the state and algebraic variables are set at their equilibrium values during the solution process. The time evolution of
sensitivities is computed under the assumption that a very small
perturbation is carried out in the system such that the state
and algebraic variables are infinitesimally perturbed, their values, therefore, can be considered constant during the computation
of the sensitivity index. These assumptions permit us to directly
relate our proposal to the theory of trajectory sensitivities.
Step 5. – Quantify the interaction between the system parameters
and the associated state variables by using the sensitivity index
(15). Since this index is a function of the sensitivities of those state
variables directly associated with the oscillatory modes and the
critical eigenvalues, it can be used to quantify the effect of the ith parameter on these variables. In this case, the highest values
of the sensitivity norms indicate the most sensitive parameters.
Furthermore, the sensitivity index value increases as the system
is approaching an oscillatory stability problem.
The eigenvalue and selective modal analyses are carried out in
step 3 just to obtain the associated state variables and the equilibrium point stability. Once these variables have been determined,
assessing how these associated variables are affected by changes
in system’s parameters is possible. This information is important
in order to perform a corrective control action to damp out system
oscillations and improve the system stability.
Note that even though the oscillatory behavior of sensitivities is implicitly related to the critical eigenvalue that determines
the damping and frequency oscillation of associated variables,
directly obtaining the associated state variables from the trajectory
sensitivity analysis is not possible because sensitivities uniquely
associate changes of the state variables with respect to variations
of system parameters.
One way to determine the associated variables directly from
the TS approach is by obtaining sensitivities of state variables with
respect to changes in eigenvalues, but it requires formulating the
Fig. 1. WSCC 9-buses, 3-generators.
set of DAEs representing the power system as an explicit function
of the eigenvalues, which is out of the scope of this paper.
4. Study cases
The suitability of the proposed method is tested on the 3machine, 9-bus WSCC system and the 46-machine, 190-bus model
of the Mexican power system. For the purpose of the studies presented in this section, the system generators are represented by
means of the two-axis model with a simple fast exciter loop containing max/min ceiling limits. The system loads are represented
by means of the constant power load model. The design of the study
cases is given below.
4.1. 3-Machine, 9-bus WSCC system
The WSCC system shown in Fig. 1 is considered in this section to
describe the application and validation of the proposed approach.
The active power demanded at node 5 is varied in order to assess
the stability of the new EP and to determine the parameter sensitivity of associated states w.r.t. this demanded power. The demand
varies from 4.2 pu to 4.5 pu with load changes of 0.1 pu on a base
of 100 MVA. The critical eigenvalues, associated states and participation factors computed at each demand scenario are reported
in columns 2, 3 and 4, respectively, of Table 1. Examination of the
eigenvalues analysis indicates that the system becomes unstable at
PL5 = 4.5 pu, while the SMA points out that the mode of instability is
because of the electromechanical variables ı and ω of generator 2.
The point at which the eigenvalues cross over to the right-half plane
corresponds to a subcritical HB point responsible for the power system’s unstable oscillatory behavior. This statement is numerically
Table 1
Selective modal analysis.
PL5 (pu)
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Critical eigenvalues
−0.50
−0.53
−0.03
0.70
1.51
2.66
5.37
±
±
±
±
±
±
±
7.30i
6.85i
6.14i
5.89i
5.71i
5.50i
4.58i
Associated states
Participation factors
ω2 , ı2 , Ed2
ω2 , ı2 , Ed2
ı2 , ω2 , Ed2
Eq1
, ı2 , ω2
Eq1
, ı2 , ω2
Eq1
, ı2 , ω2
Eq1
, ı2 , ω2
1.0, 0.99, 0.18
1.0, 0.99, 0.28
1.0, 0.99, 0.46
1.0, 0.85, 0.85
1.0, 0.81, 0.80
1.0, 0.79, 0.78
1.0, 0.77, 0.75
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E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
Fig. 3. Parameter sensitivities w.r.t. PL5 = 4.3 pu, crit = −0.5395 ± 6.8512i.
Fig. 2. Time evolution of ı2 and ω2 from a demand scenario of PL5 = 4.5 pu.
demonstrated by computing the dynamics of ı2 and ω2 from the
unstable equilibrium state corresponding to a demand scenario of
PL5 = 4.5 pu. The profile of these variables as a function of time is
shown in Fig. 2, where their growing oscillatory behavior, without
being attracted to a stable limit cycle, is observed.
The proposed approach is applied to compute the parameter sensitivities of electromechanical variables of generators 2
and 3 w.r.t. PL5 at the equilibrium points defined by PL5 = 4.3 pu,
PL5 = 4.4 pu, and PL5 = 4.41 pu, respectively. Note that there are no
sensitivities for generator 1 because its angle is used as the reference angle for these simulations. Fig. 3 shows the evolution of
parameter sensitivities w.r.t. PL5 = 4.3 pu, which converges to a fixed
value, i.e. the sensitivity oscillations are damped out. This result
agrees with the information provided by the critical eigenvalue
crit = −0.5395 ± 6.8512i in the sense that the EP is stable. A comparative analysis of the waveforms of sensitivities shows that the
maximum peak values correspond to the sensitivities of generator
Fig. 4. Parameter sensitivities w.r.t. PL5 = 4.4 pu, crit = −0.0305 ± 6.1462i.
2, indicating that the critical variables are ı2 and ω2 . This result also
agrees with that obtained by using the SMA.
The sensitivities of ı2 , ω2 , ı3 and ω3 w.r.t. PL5 = 4.4 pu and
PL5 = 4.41 pu are shown in Figs. 4 and 5, respectively. The
Fig. 5. Parameter sensitivities w.r.t. PL5 = 4.41 pu, crit = 0.0462 ± 6.1105i.
E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
sensitivities present a sustained oscillatory behavior at PL5 = 4.4 pu,
indicating that the system is operating close enough to the imaginary axis in the complex plane; hence, the system is operating in
the vicinity of a HB point, as confirmed by the critical eigenvalue
crit = −0.0305 ± 6.1462i related to this operating condition. Therefore, a very small increment in the active power demanded at bus 5
steers the system to an oscillatory instability. This statement is confirmed by the sensitivities shown in Fig. 5 for a PL5 = 4.41 pu. Finally,
the sensitivities correctly predict the most sensitive state variables
w.r.t. PL5 in all cases, and they correspond to the associated state
variables computed by SMA.
The modeling of the load has an important effect on the system’s dynamic behavior. The results reported above have been
obtained considering a constant power (PQ) load model, which
creates an artificial oscillatory instability that easily destabilizes
the system as far as network loadability is concerned. However,
this oscillatory behavior does not necessarily occur for the same
level of loadability when other types of load models are considered
in the simulations. The proposed approach has also been applied
for computing the trajectory sensitivities of rotors angles w.r.t.
PL5 , considering an operating point of PL5 = 4.5 pu and QL5 = 0.5 pu,
as well as both constant current and constant impedance load
models [4]. The results are reported in Fig. 6 for the three types
of loads. Note that the sensitivities (system) become(s) unstable
when all loads are modeled as constant PQ, whereas, the trajectory
sensitivities (system dynamics) are attracted to an asymptotically
stable equilibrium for the other two types of loads. The peak values
of these sensitivities also indicate that the system is more stable when the constant impedance model is adopted than when
the constant current model is considered. This last observation
is demonstrated by repeating the study considering PL5 = 4.6 pu
and QL5 = 0.5 pu, as well as a high gain of generator exciters set at
KA = 175 [4]. The trajectory sensitivities are graphically reported in
Fig. 7 for the three types of load models. In this case, an oscillatory instability occurs when both constant PQ and constant current
load models are used, while the state variable sensitivities indicate that the system remains stable with a constant impedance
model.
The effect of the time-step t in the computational burden
related to the calculation of the parameter sensitivities is analyzed
by computing the sensitivities of ı2 , ω2 , ı3 and ω3 w.r.t. PL5 = 4.4 pu,
but by using three different values of t: 0.001 s, 0.01 s, and 0.1 s.
The number of forward/backward substitutions required to obtain
the sensitivities’ evolution for a period of 14 s were 14,000, 1400
Fig. 6. Parameter sensitivities w.r.t. PL5 = 4.5 pu with three types of load models.
251
Fig. 7. Parameter sensitivities w.r.t. PL5 = 4.6 pu with three types of load models.
Fig. 8. Effect of the integration time-step on the evolution of parameter sensitivities.
and 140 for each case, respectively. Acknowledging the simulation with t = 0.001 s as a base case, the increase of t implies
a reduction of 90% and 99% of the number of sensitivity solutions,
respectively, without degrading the sensitivities’ profile w.r.t. time,
as shown in Fig. 8 for the rotor angle sensitivity of generator 2.
In real operative scenarios, the system loading is constantly
varying in NP - loading directions; therefore, assessing the most critical load whose variations could steer the system to an unstable
operating condition is very important. The computation of multiparameter sensitivities w.r.t. Np parameters is carried out by using
(14), which permits the identification of the critical generators
(states) and determining the most critical loading directions. Fig. 9
shows the sensitivity norm (SN) evolution w.r.t. the active power
demanded at load nodes when all loads are modeled as constant
PQ. A comparison of these indices shows that the EP is very sensitive to changes in load embedded at node 8 (PL8 ), while being less
affected by changes in loads connected at nodes 6 and 5, respectively. In order to validate these results, three parameter studies
have been performed by loading the system through the increment
of the active power demanded by the three loads, one at a time, until
a HB is located. The results are reported in Table 2 as follows: the
first column indicates the load nodes, and columns 2 and 3 present
the maximum value of the SN computed by the proposed approach
and the active powers demanded by each load at the base operating point. In addition, the fourth column provides the active power
demanded by each load at the HB point and, lastly, the fifth column
252
E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
3
1.8
1.6
SN w.r.t. PL6
1.2
SN w.r.t. PL5
2.5
node: 147
SN: 2.73
Sensitivity norm SNi
Sensitivity Norm SN
SN w.r.t. PL8
1.4
1
0.8
0.6
node: 152
SN: 2.93
2
1.5
node: 120
SN: 1.78
1
0.4
0.2
0
0.5
0
1
2
3
4
5
6
7
Seconds
0
40
70
100
130
160
190
190
Load nodes
Fig. 9. Loads’ effect on the EP’s stability (WSCC system).
Fig. 11. Loads’ effect on the equilibrium point stability (Mexican system).
indicates the active power increment from the base case to the
HB point for each loading direction. Note that the increment PL8
steers the system to a HB point faster than PL6 , and an increase
in PL6 provokes an instable operation condition faster than PL5 .
This agrees with the SNs reported in Fig. 9 and the sensitivity indices
in Table 2.
Table 2
Sensitivity norm and Hopf bifurcation (WSCC system).
Node
SN
Pbase (MW)
PHB (MW)
PHB (MW)
8
6
5
1.723
0.997
0.739
100.0
90.0
125.0
399
404
441
299
314
316
4.2. Mexican 190-bus, 46-generators system
The proposed approach has been applied to a reduced model of
the Mexican Interconnected System including the northern, northeastern, western, central and southeastern areas, as shown in Fig. 10
[25]. This equivalent consists of 190 buses, 46 generators, 90 loads,
23 shunt compensators, 180 transmission lines and 83 transformers operating at voltage levels ranging from 400 kV to 115 kV. The
constant PQ model is adopted for all loads.
The parameter sensitivities of all electromechanical variables ı
and ω are computed w.r.t. the active power demanded at each node
of the power system in order to attain the sensitivity norms (15).
Fig. 10. Mexican power system.
E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
253
Table 3
Sensitivity norm and Hopf bifurcation (Mexican system).
Node
Pbase (MW)
P60 (60 MW)
152
150
151
147
153
145
120
172.64
188.24
18.72
104.00
78.00
83.20
308.88
232.64
248.64
78.72
164.00
138.00
143.20
368.88
The maximum values of SNs are reported in Fig. 11 indicating that
the power system equilibrium is more sensitive to changes in loads
embedded at nodes 152, 150 and 151. Therefore, increments in the
active power demanded by these loads will steer the system to
an oscillatory behavior (HB point) faster than increments in other
loads. The sensitivities were obtained by solving (16) for a period
of 29 s. The integration time-steps considered in our analysis were
t = 0.001 s, t = 0.01 s and t = 0.1 s, which took 29,000, 2900, and
290 forward/backward substitutions, respectively. This last case
represents only the 1% of the computational burden associated with
the first case without degrading the sensitivities’ profile.
Fig. 12 shows the evolution of rotor angle sensitivities w.r.t. the
active power demanded at bus 152, PL152 , which was identified as
the most critical load in the sensitivity norm assessment. From
these results the rotor angles of generators 32 and 33 are clearly
the most sensitive state variables to changes in PL152 . The damped
oscillations in the sensitivities’ evolution also indicate that the EP
is stable, which agrees with the stability qualitative information
inferred from the critical eigenvalues crit = −0.05 ± 7.85i of the EP.
In order to validate the critical loads’ screening via the sensitivity norm, a SSS analysis has been performed to show how the
increments in the most sensitive loading directions influence the
EP’s stability. The results obtained from this analysis are reported
in Table 3 as follows: Column 1 indicates the most sensitive loads
resulting from the proposed approach shown in Fig. 11; Column 2
shows the value of active power demanded by these loads at the
base operating point Pbase at which the parameter sensitivities were
computed; Columns 3 and 4 represent the increment of 60 MW in
each loading direction (P60 ), one at the time, as well the critical
eigenvalues (crit ) for the new EP resulting from such an increment,
respectively. Lastly, columns 5 and 6 indicate the value of active
power demanded by each load (PHB ) and the increment of demand
from the base case (PHB ) at the HB point for each loading direction.
From these results one observes that the most critical load
clearly requires the lowest increment of active power to steer the
crit
0.021
0.018
0.015
0.012
−0.011
−0.015
−0.050
±
±
±
±
±
±
±
5.02i
5.03i
5.03i
5.05i
8.83i
5.13i
7.85i
PHB (MW)
PHB (MW)
225.64
242.24
72.72
160.00
141.00
155.20
497.88
53.0
54.0
54.0
56.0
63.0
72.0
189.0
system to a sustained oscillatory behavior. We reiterate that the
loads’ effect in the power system equilibrium is not only dependent on the magnitude of changes in loads, but also on the topologic
location of loads. By way of example, the power demand embedded
at bus 120 is 17 times larger than the load at bus 151; however, the
load at 151 has a bigger effect on the EP’s stability than the load
embedded at bus 120, as shown in Table 3. Finally, these observations agree with those results obtained by applying the proposed
approach.
5. Conclusions
This paper proposes an approach to compute multi-parameter
sensitivities to quantify the influence of the system’s parameters
on the power system equilibrium. Since parameter sensitivities are
obtained at the equilibrium point, their computation only involves
the solution of a set of linear algebraic equations by forward and
backward substitutions. Based on this proposal, the direct assessment of the existing interaction between the system’s parameters
and the state variables related to the oscillatory behavior of a power
system has been possible. Simulation results were presented showing the effectiveness of the proposed method and its applicability
to efficiently analyze a real-life power system.
Appendix A.
The mathematical expressions of all elements composing the
sensitivity matrices (7) are obtained analytically and their numerical values are calculated based on the state and algebraic variables
computed at the equilibrium point. In order to clarify this issue, the
sensitivity equations (6) and (7) are reported in an expanded form
in this Appendix.
The dynamic and algebraic state variables are given by
E
E ]
x = [ıi ωi Edi
qi fdi
y = [Idi Iqi k Vk ]
T
T
∀
i ∈ Ng
(17)
k ∈ Nb
while the function vectors are
⎡
ωi − ω0
⎤
⎥
⎢ 1 ⎢ M PMi − Eqi
Iqi + Edi
Idi + (Xqi
− Xdi
)Idi Iqi −Di (ωi − ω0 )) ⎥
⎥
⎢ i
⎥
⎢
⎥
⎢
1 ⎥
⎢
−E
+
(X
−
X
)I
qi
qi
di
qi
ẋ = f (x, y, ˇ) = ⎢
Tq0i
⎥
⎥
⎢
⎥
⎢
1 −E
−
(X
−
X
)I
+
E
⎥
⎢
di
di
fdi
di
qi
Td0i
⎥
⎢
⎦
⎣
1
TAi
Fig. 12. Evolution of parameter sensitivities w.r.t. PL152 , crit = −0.05 ± 7.85i.
KAi (Vrefi − Vi ) − Efdi
(18)
254
E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel / Electric Power Systems Research 96 (2013) 246–254
⎡ E
di
− Vi sin(ıi − i ) + Xqi
Iqi
⎤
⎢ Eqi
− Vi cos(ıi − i ) − Xdi
Idi ⎥
⎥
0 = g(x, y, ˇ) = ⎢
⎣ PGi − PDk − PCk
⎦
QGi − QDk − QCk
(19)
(2Ng+2Nb)x(2Ng+2Nb)
The sensitivities of dynamic and algebraic state variables with
respect to a system’s parameter are given by
Eqi,ˇ
Efdi,ˇ ]
xi,ˇ = [ıi,ˇ ωi,ˇ Edi,ˇ
yk,ˇ = [k,ˇ Vk,ˇ Idi,ˇ Iqi,ˇ ]
T
T
∀
i ∈ Ng
k ∈ Nb
,
(20)
and the changes of the sensitivities of dynamic state variables w.r.t.
time are
T
ẋi,ˇ = [ı̇i,ˇ ω̇i,ˇ Ėdi,ˇ Ėqi,ˇ Ėfdi,ˇ ] .
⎡
(21)
Lastly, the set of trajectory sensitivities’ equations are given by
⎤
⎢
⎢ ω̇i,ˇ
⎢
⎢ Ė
⎢ di,ˇ
⎢
⎢ Ė
⎣ qi,ˇ
⎡ ı
i,ˇ
⎥
⎢ω
⎥ ⎢ i,ˇ
⎥
∂f (x, y, ˇ) ⎢
⎥
⎢ E
=
⎥
⎢ di,ˇ
∂x
⎥
⎢ ⎥
⎣ Eqi,ˇ
⎦
Efdi,ˇ
Efdi,ˇ
ı̇i,ˇ
+
⎥
⎥ ⎥
⎥ + ∂f (x, y, ˇ)
⎥
∂y
⎥
⎦
⎡
Idi,ˇ
Vk,ˇ
(22)
Efdi,ˇ
∂g(x, y, ˇ)
.
∂ˇ
⎤
⎥
⎢
⎢ Iqi,ˇ ⎥
⎥
⎢
⎢ ⎥
⎣ k,ˇ ⎦
∂f (x, y, ˇ)
∂ˇ
⎡ ı
i,ˇ
⎢ω
⎢ i,ˇ
∂g(x, y, ˇ) ⎢
⎢ E
[0] =
⎢ di,ˇ
∂x
⎢ ⎣ Eqi,ˇ
+
⎤
⎤
⎥
⎥ ⎥
⎥ + ∂g(x, y, ˇ)
⎥
∂y
⎥
⎦
⎡
Idi,ˇ
⎤
⎥
⎢
⎢ Iqi,ˇ ⎥
⎥
⎢
⎢ ⎥
⎣ k,ˇ ⎦
Vk,ˇ
(23)
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