Computation of Singular and Singularity
Induced Bifurcation Points of DifferentialAlgebraic Power System Model
Saffet Ayasun, Student Member, IEEE, Chika O. Nwankpa1, Member, IEEE, Harry G. Kwatny, Fellow, IEEE
Abstract-In this paper, we propose an efficient algorithm to compute singular points and singularity induced bifurcation
points of the differential-algebraic equations for a multimachine power system model. Power systems are often modeled as a
set of differential-algebraic equations (DAE) with a vector of β parameters. The algebraic part of DAE brings singularity
issues into dynamic stability assessment of power systems. Roughly speaking, the singular points are points that satisfy the
algebraic equations, but at which the vector field is not defined. In terms of power system dynamics, around singular points,
the generator angles (the natural states variables) are not defined as a graph of the load bus variables (the algebraic
variables). Thus, the causal requirement of the DAE model breaks down and it cannot predict system behavior. Singular
points constitute important organizing elements of power system DAE models. This paper proposes an iterative method to
compute singular points at any given parameter value. With a theorem presented in this paper, we are also able to locate
singularity induced bifurcation points upon identifying the singular points. The proposed method is implemented into
Voltage Stability Toolbox and simulations results are presented for 3- and IEEE 118-bus systems.
Index Terms-Power systems, differential-algebraic equation, bifurcation, singular points, singularity induced bifurcation and
stability enhancement
2
I.
INTRODUCTION
With deregulation of the electric utility industry, there is a trend to operate power systems in ways that utilize the
transmission network quite differently from its original intent. As a consequence, power systems now operate closer to theirs
stability limits. These new modes of utilization of the system make it necessary to investigate mechanisms of power system
instability, including voltage collapse, that are associated with stressed transmission systems. It is widely accepted that
practical analyses of power system stability and voltage collapse require the consideration of bifurcation phenomena. This
view has become increasingly relevant, as the operating environment has forced systems to function deeper into nonlinear
regions. A comprehensive overview of bifurcation concepts and methods in relation to power system stability may be found
in [1].
Bifurcation analysis is the study of behavioral variations within families of nonlinear systems - commonly defined as
parameter dependent sets of ordinary differential equations (ODE) or a mixture of differential and algebraic equations
(DAE). Bifurcation is a term broadly used to describe significant qualitative changes that occur in the orbit structure of a
dynamical system, as parameters on which the dynamical system depends are varied [2,3]. Local bifurcations are readily
evident in power systems as important elements of voltage instability. Local bifurcation analysis often results in three
different bifurcation types, namely saddle node (SN), Hopf and singularity induced (SI) bifurcations.
Saddle node bifurcation has become a widely accepted paradigm for one important form of voltage instability and linked
to voltage collapse. It is the common view in literature that the saddle node bifurcation happens with a bifurcation of
equilibrium points during slow changes of parameters (real and reactive power injections in power systems). Specifically,
one stable equilibrium point (SEP) meets a type-1 unstable equilibrium point (UEP) at the saddle node bifurcation point and
the load flow Jacobian matrix has an eigenvalue at the origin. The saddle node bifurcation in the voltage collapse frame has
been investigated in subsequent papers such as [4-12]. The importance of Hopf bifurcation has been increasingly
recognized, as it became clear that stability of the equilibrium could be lost by this mechanism well before reaching the
point of collapse. Hopf bifurcation occurs when a pair of complex conjugate eigenvalues moves from the left to right half of
the complex plane crossing the imaginary axis at points other than the origin. There is substantial literature available on
Hopf bifurcation in power systems and its relation to instability [13-18].
The other significant qualitative change in the system equilibria happens when the system equilibrium is at the singularity
of the algebraic part of the DAE model. This bifurcation is known as singularity induced (SI) bifurcation [19-23]. At the
singularity induced bifurcation the equilibria undergo stability exchanges and one of the eigenvalues of the system Jacobian
3
matrix becomes unbounded. Significant research conducted recently illustrates the growing interest in the SI bifurcation and
the algebraic singularity of the DAE model. One important implication of the occurrence of the SI bifurcation is the
existence of a singular set (impasse surface) in the state space containing singular points at each parameter value. In power
systems, such points were encountered by DeMarco and Bergen [24] in connection with transient stability studies (impasse
points), by Kwatny et al. [4] in connection with static bifurcation analysis (noncausal points). Hiskens and Hill [25,26] have
extended the concept of voltage causal points to a voltage causal region within which load bus variables follow generator
angle behavior2 and the DAE model can be reduced to a locally equivalent set of ordinary differential equations (ODE). The
constraint manifold is decomposed into voltage causal regions separated by the singular surface (impasse surface in [25]).
Their studies reveal the fact that there is a strong relationship between voltage collapse and loss of voltage causality.
Specifically, trajectories passing near a singular point may bifurcate and the voltage magnitudes may settle to a physically
infeasible operating point (i.e. low voltage profile). Based on the feasibility region concept introduced by
Venkatasubramanian et al. [19-23], Praprost and Loparo [28,29] also consider the effects of the singular surface (boundary
of solution “sheets”) on dynamic stability evaluation of power systems. They have shown that trajectories that are tangent to
the singular surface form a stability boundary for the DAE model. At the singular points, the DAE model breaks down and
load bus variables cannot be defined as functions of generator angles, which implies the loss of voltage causality. A strong
relationship between the loss of voltage causality and short-term voltage collapse has also been established in [25,26]. An
underlying issue is that at the singular surface, the DAE model cannot predict the voltage behavior and the validity of the
model, as a characterization of the power system, is questionable. It is likely that uncertainties, neglected in the model, now
become central to the local power system behavior.
Recently, there is a trend in the literature to use singular points to obtain a more accurate estimate of critical energy for
dynamic stability analysis. The energy function approach [30] has been considered to investigate connections between
voltage collapse and the algebraic singularity of the DAE model for transient conditions [25-29]. However, none of them
have presented a rigorous method to identify singular points at any given parameter value. The identification of singular
points, as parameter(s) vary (load changes) is a significant part of the over-all problem. However, this problem is a complex
one for larger size power systems and a systematic method is not available in the literature yet.
The aim of this paper is to propose a systematic method to locate and identify the algebraic singularity of the DAE model
as parameters vary. In particular, we modify and implement known methods such as the Newton-Raphson (NR) and
Newton-Raphson-Seydel (NRS) [31] methods, which are used to identify voltage collapse points [32-34]. We use generator
angles to parameterize the algebraic part of the DAE model and identify the singular points as being a saddle node
4
bifurcation point of the algebraic part. The reason we use the generator angles as the parameters is that we can control them
by controlling the mechanical input power to the generators. As an interesting aside, it is likely that in the future operation
environment of power systems, one will be able to control load bus variables by implementing Flexible AC Transmission
System (FACTS) [43] devices such as the thyristor-controlled series capacitors (TCSC), static var compensators (SVC). We
propose a search method that implements a combined NR/NRS search method in the state space to locate the algebraic
singularity at a fixed parameter value (i.e., load demand is constant), which is the general assumption in direct stability
assessment methods. This paper contributes to the analysis of the DAE power system models in two ways:
•
We provide a static method to identify singular points that are on the boundary segment of the dynamic stability
region. Our method does not require any time-domain integration over predetermined trajectories as in other
“controlling UEP” methods [30].
•
By a theorem presented in Section II, we show that any singular point at a given set of generator3 bus injections is
also an equilibrium point at another set of bus injections. Thus, it is a singularity induced bifurcation point.
Our approach allows us to combine static and dynamic information on the current operating point. In the daily operation
of the power system, operators need to have a quantitative measure to assess the stability of the current operating point. The
saddle node bifurcation point is also known as maximum loading point or point of collapse. In the appropriate parameter
space, it provides the operator with information on how far the current operating point is away from the collapse point. That
static stability margin could be measured in terms of actual physical quantities such MW transfer or in terms of energy [3536]. As stated in [25-29], voltage collapse may occur during a transient too if the sustained fault trajectory encounters the
singular surface. Therefore, one needs to bring this information into the big picture of the stability assessment of the system.
The dynamic stability margin could be computed as the energy difference between the current operating point and the
singular point. That energy value provides a dynamic security index to the operator. By doing so, one provides both static
and dynamic stability margins for the current operating point. The traditional nose curves are usually used to indicate the
stability margins in the parameter space. We bring the singular point information into the nose curve and illustrate both
system equilibria and singular points as the load demand increases. This way of bringing the information gathered from state
space to parameter space gives a visual representation of the static and dynamic boundaries together in the same picture.
It is worth mentioning here that stability analysis based on the DAE model is a more realistic characterization of power
system operations than ODE models obtained with the assumption of constant impedance loads. The ODE representation of
power system dynamics that follows from this assumption is tantamount to the assumption of global voltage causality. Such
models ignore the actual behavior of load buses and lead to the erroneous conclusion that all stability issues relate to the
5
generator side of the network. Moreover, eliminating load bus variables and injections in order to get a reduced order
network model does not fit into today’s deregulated power market in which generation, transmission and distribution parts
of the network may be owned and operated by different utility companies.
We have updated our Voltage Stability Toolbox (VST)4 to include singularity analysis of the DAE model of power
systems. The VST is a MATLAB based software package developed by us to analyze bifurcation phenomena in power
systems [38-41]. The VST is a portable software package that combines computational and analytical capabilities of
bifurcation theory, visualization and symbolic computation capabilities of MATLAB. We have encountered singularity
induced bifurcations in almost all of the IEEE standard systems. We have also observed that the location and identification
of the SI bifurcation points are method and tool dependent. In addition, we have found out, not surprisingly, that step-sizes
used in Newton-Raphson (NR) and Newton-Raphson-Seydel (NRS) [31] iteration methods are among the key issues in the
identification of SI bifurcation points [40].
In Section II we describe the classical power system model used in our analysis and review some basic definitions and
results regarding the DAE model of power systems and singularities. Section II also includes two simple examples of the
DAE model (one of which is a power system model) to motivate and illustrate the key concept of the paper. In Section III
we describe methods to identify equilibrium and singular points at any given parameter value. In Section IV we present the
simulation results for the IEEE 118-bus system using the Voltage Stability Toolbox. In Section V we discuss the possible
practical implications of the algorithm and results from a power system operating point of view and possible future
enhancement of the algorithm. In Section VI we conclude the paper and provide some future work.
II. DIFFERENTIAL-ALGEBRAIC POWER SYSTEM MODEL AND SINGULARITIES
A. Classical Power System
We consider that the network consists of n generators and n + m + l buses where buses i = 1,..., n are the internal buses
of n generators, buses i = n + 1,..., n + m are m voltage controlled (PV) load buses and buses i = n + m + 1,..., n + m + l are l
PQ load buses [4]. After elimination of constant admittance and network internal buses, the system equation could be
written as follows:
M i δ&&i + Di δ&i + f i = Pmi ,
i = 1,..., n
(2.1)
f i = Pdi ,
i = n + 1,..., n + m
(2.2)
f i = Pdi , g i = Q di ,
i = n + m + 1,..., n + m + l
(2.3)
where f i and g i are the load flow equations at each bus and defined as follows:
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n + m+l
f i = ∑ [ ViV j Bij sin( δ i − δ j ) + ViV j G ij cos( δ i − δ j )],
i = 1,..., n + m + l
(2.4)
i = n + m + 1..., n + m + l
(2.5)
j =1
n + m +l
g i = ∑ [ Vi V j G ij sin( δ i − δ j ) − ViV j Bij cos( δ i − δ j )],
j =1
where,
M i : The rotor inertia of the ith generator
Di : The damping coefficient of the ith generator
Vi : The ith bus voltage magnitude
δ i : The angle associated with each of the n + m + l buses
Bij : The transfer susceptance
G ij : The transfer conductance
Pmi : The turbine mechanical power injection at ith generator
Pdi , Q di : The load real and reactive power components at the ith bus
In order to obtain a vector form of (2.1) - (2.3), define the following vector form of variables;
δ g = [ δ 1 ,...,δ n ] T
(2.6)
δ l = [ δ n +1 ,...,δ n + m +l ] T
(2.7)
V = [ V n + m +1 ,...,V n + m +l ] T
(2.8)
y = [ δ lT ,V T ] T
(2.9)
and the functions
f g ( δ g , y , β ) = [ f 1 − Pm1 ,..., f n − Pmn ] T
(2.10)
f l ( δ g , y , β ) = [ f n +1 − Pd ( n +1) ,...., f n + m + l − Pd ( n + m + l ) ] T
(2.11)
g l ( δ g , y , β ) = [ g n + m +1 − Qd ( n + m +1 ) ,..., g n + m + l − Qd ( n + m + l ) ] T
(2.12)
g ( δ g , y , β ) = [ f lT , g lT ] T
(2.13)
Then, we have the following compact form of (2.1)-(2.3):
Mδ&& + Dδ& + f g ( δ g , y , β ) = 0
(2.14a)
fl ( δ g , y, β ) = 0
(2.14b)
g l ( δ g , y, β ) = 0
(2.14c)
where M denotes the diagonal matrix of generator rotor inertias and D is the damping matrix.
M = diag( M 1 ,..., M n )
(2.15a)
D = diag ( D1 ,..., D n )
(2.15b)
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It is clear that the classical power system model defined (2-14a)-(2.14c) is a parameter-dependent differential algebraic type
whose parameters are the bus real and reactive power injections and the transmission line admittances. It is convenient to fix
the transmission line admittances and treat the bus injections as parameters that can be varied. These parameters can be
isolated in the equations as follows:
f g ( δ g , y , β ) = [ f 1 ,..., f n ] T − Pm
(2.16a)
f l ( δ g , y , β ) = [ f n +1 ,...., f n + m + l ] T − Pd
(2.16b)
g l ( δ g , y , β ) = [ g n + m +1 ,..., g n + m + l ] T − Qd
(2.16c)
where,
Pm = [ Pm1 ,..., Pmn ] T : Vector containing the mechanical input power to the generators.
Pd = [ Pd ( n + 1 ) ,..., Pd ( n + m + l ) ] T : Vector containing the real power demands at m PV and l PQ buses.
Q d = [ Q d ( n + m +1) ,..., Q d ( n + m + l ) ] T : Vector containing the reactive power demands at l PQ buses.
Let β g = Pm , β lP = Pd and β lQ = Q d . Then we can rewrite (2.14) as a set of first order differential equations in terms of
β ’s:
δ& = ω
D
1
ω−
f g ( δ g ,δ l ,V ) − β g
M
M
f l ( δ g ,δ l ,V ) − β lP = 0
ω& = −
(2.17)
g l ( δ g ,δ l ,V ) − β lQ = 0
[
Let x = δ g
ω ]T and y = [ δ lT ,V T ] T and rewrite (2.17) in terms of x and y.
x& = f ( x , y ) − β G
0 = g( x , y ) − β L
[
where β G = 0
(2. 18)
β g ]T and β L = [β lP
x& = f ( x , y , β )
0 = g( x , y , β )
β lQ ]T or in a much more compact form we have
(2.19)
From this point forward we assume that x ∈ ℜ n , y ∈ ℜ m and α ∈ ℜ k where k = n + m + 2l . Since det [ M ] ≠ 0 (2.17) is
clearly of the type (2.19). The DAE model of (2.18) or (2.19) admits various enhancements to the classical model, including
excitation systems, tap changing transformers, nonlinear and dynamic loads, and more elaborate generator models. In our
analysis, the classical model with the constant power load model is implemented in Voltage Stability Toolbox [38]. The
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above model has two essential features: 1) the explicit parameter dependence, 2) the differential-algebraic structure. In the
following, we study the implications of these two features.
B. Singularities of the DAE Power System Model
The parameter dependence implies changes in the system behavior as a consequence of parameter variations.
Specifically, it is essential to study the changes in the solution structure of the static equations of (2.19) (system equilibria).
For a given set of parameter, an equilibrium point satisfies two sets of algebraic equations. The set of all equilibrium points
is called the equilibrium surface. It is defined as follows:
{
}
E = ( x , y , β ) ∈ ℜ n+ m + k f ( x , y , β ) = 0 , g( x , y , β ) = 0
(2.20)
The sign of the real parts of the eigenvalues of system Jacobian matrix (if it exists)
Asys = [ D x f ] − [ D y f ][ D y g ] −1 [ D x g ]
determines the stability properties of any equilibrium point.
The algebraic part of (2.19) requires that any motion be constrained to the set:
{
M ( β ) = ( x , y ) ∈ ℜ n + m g( x , y , β ) = 0 , β = constant
}
(2.21)
Typically, we expect M to be composed of one or more disconnected (differentiable) manifolds [37] called components. In
general, when we refer to M, we will mean a particular one of these components called the principle component. M is a
regular manifold of dimension n if
 ∂g ∂g 
rank 
 = m on M
 ∂x ∂y 
(2.22)
The structure of M depends, of course, on the parameter β . Even for very simple power system models, (2.22) may not be
satisfied for some values of β . The manifold M is the state space for the dynamical system defined by (2.19) which induces
a vector field on M. The vector field may not be well defined at all points of M. At any point ( x , y ) ∈ M we have
x& = f ( x , y , β ). If det[ D y g( x , y )] ≠ 0 , then y& is uniquely defined by
−1
 ∂g   ∂g 
y& = −     f ( x , y , β )
 ∂y   ∂x 
(2.23)
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If [ D y g ( x , y )] is singular at a point ( x , y ) ∈ M then vector field is not well defined at that point. Typically, such singular
points lie on codimension 1 submanifolds of M.
Definition [4]: Suppose M is a regular manifold for all β near β ∗ , and that det[ D y g( x , y )] ≠ 0 at a point β = β ∗ ,
( x , y ) = ( x ∗ , y ∗ ) ∈ M . Then ( x ∗ , y ∗ , β ∗ ) is said to be causal. Otherwise it is noncausal. <
Hiskens and Hill have extended the causality of a point to define the causality of a region [25]. Within any voltage causal
region load bus voltages and angles follow generator angles’ behavior. At any causal point ( x ∗ , y ∗ , β ∗ ) in the region, say
C k , the implicit function theorem ensures that there exists a function ψ ( x , β ) defined on a neighborhood of ( x ∗ , β ∗ )
with y ∗ = ψ ( x ∗ , β ∗ ) and that satisfies g( x ,ψ ( x , β ), β ) = 0 . It follows that within a voltage causal region, trajectories of
the DAE are locally defined by the ordinary differential equation:
x& = φ ( x , β ) := f ( x ,ψ ( x , β ), β )
(2.24)
The boundary of a voltage causal region where local equivalence (to an ODE) is not possible is referred as a singular
surface [19-26]:
{
S ( β ) = ( x , y ) ∈ ℜ n + m g( x , y ) = 0 , det[ D y g( x , y )] = 0
}
(2.25)
In order to predict the system behavior around the singular surface, Praprost et al. (much earlier, Arapostathis et al. [45])
has proposed a singularly perturbed differential equation (SPDE) as the power system model [28] and their simulation
results indicate that rapid decline in bus voltage magnitudes may occur if trajectories pass close to the singular surface.
It is observed in power systems that the singular surface defined by (2.25) and the equilibrium surface given by (2.20) may
intersect at a certain parameter value. This results in what is known as the singularity induced bifurcation [19-23].

f ( x , y , β ) = 0 , g( x , y , β ) = 0 
SI = ( x , y , β ) ∈ ℜ n + m + k

det[ D y g( x , y , β )] = 0


(2.26)
We have found that identification and location of SI bifurcation points are method and tool dependent [40]. For example,
the step-size in the NRS algorithm is one of the key issues in the identification of SI bifurcation points. We have observed
that in order to capture the SI bifurcation we ought to choose a smaller step-size. Observe that a SI bifurcation point belongs
to both the equilibrium surface (2.20) and singular surface (2.25). A natural question that arises : Is it possible to identify a
SI bifurcation point ( x SIN , y SIN ) ∈ SI if we are given a singular point ( x S , y S ) ∈ S ( β ) at a fixed parameter value β . The
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following theorem shows that this is indeed possible due to the decoupled structure of the power system model of our
interest (2.18).
Theorem : A singular point of (2.18) ( x S , y S ) ∈ S ( β S ) at a given parameter value β S is also an equilibrium point, hence,
a singularity induced bifurcation point, at another paramater value β NEW .
Proof:
Suppose that ( x S , y S ) is a singular point of the decoupled DAE of (2.18) at the parameter value
β S = [β SG
β SL ]T such that
x& = f ( x S , y S ) − β SG ≠ 0
0 = g( x S , y S ) − β SL
(2.27)
det[ D y ( g( x S , y S ) − β SL )] = 0
Observe that since ( x S , y S ) is not (in general) an equilibrium point, we have a non-zero mismatch at the generator buses.
mismatch
≠ 0 . In order to force a zero mismatch at the generator buses, we can
Let this mismatch be x& = f ( x S , y S ) − β SG = β SG
mismatch
at the generator buses such that:
always define a new set of injections β GNEW = β SG + β SG
mismatch
x& = f ( x S , y S ) − ( β SG + β SG
) = f ( x S , y S ) − β GNEW = 0
0 = g SL ( x S , y S ) − β SL
(2.28)
det[ D y ( g SL ( x S , y S ) − β SL )] = 0
Therefore, a singular point ( xS , yS ) ∈ S ( β S ) = S ( β NEW ) at the parameter β S = [β SG
β SL ]T is an equilibrium point,
indeed it is a singularity induced bufurcation point at the new parameter
β NEW = β GNEW
[
β SL
]
T
where
mismatch
β GNEW = β SG + β SG
. <
This theorem enables us to identify the singularity induced bifurcation points once singular points and their corresponding
mismatch values at the singular points are available. The subscript “SL” in (2.28) indicates the fact that the injections at the
load buses remain the same. In order to make a singular point a SI bifurcation point we need to adjust only the injections at
the generator buses, which are the mechanical input to generators. This theorem assumes that mechanical input to the
generators are controllable, which is realistic. This assumption also indicates that we can control the generator angles δ g ,
which leads us to propose an iterative method to identify singular points and thus singularity induced bifurcation points by
the previous theorem.
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C. Two Illustrative Examples
Before we present an algorithm to identify singular points and thus singularity induced bifurcation points we provide two
illustrative examples of the DAE model (one of which is a power system model) in order to exhibit the types of bifurcations
in the solution structure of the system equilibria that may occur and the application of the theorem for a relatively simple
power system. The following example is adapted from [27]:
Example I:
Consider the following parameter dependent DAE:
x& = − x + y + c 1 = f ( x , y , c 1 )
0 = x 2 + y 2 − c 22 = g( x , y , c 2 )
(2.29)
where x is the dynamic variable, y is the algebraic variable and, c1 and c 2 are the parameters.
Note that the DAE of (2.29) is in the form of (2.18) where parameters c1 and c 2 are decoupled from the rest of the
equations. We assume that the parameter c 1 is controllable and we analyze the qualitative changes, known as bifurcations in
system equilibria as the parameter c1 varies. These qualitative changes include changes in the number of equilibria and
most importantly, coincidence of the equilibria with the singularity of the algebraic part of the DAE.
Note that for any given parameters the equilibrium points are the intersection of the two curves: i) y = x − c 1 , a line, and
ii) x 2 + y 2 = c 22
, a circle centered at the origin with a radius c 2 , as shown in Fig. 1. Fig. 1 depicts the constraint manifold
(i.e., x 2 + y 2 = c 22 ) and two-isolated equilibrium points for the case c1 < 0 . Two equilibrium points are labeled as EQ1 and
EQ2, one being stable while other unstable. Observe that the Jacobian matrix of the algebraic equation of (2.29) (i.e.
D y g( x , y , c 2 ) = 2 y ) is singular along the x-axis and intersection of the constraint manifold with the x-axis gives two
singular points of the DAE model ( −c 2 ,0 ) and ( c 2 ,0 ) , which are labeled as S1 and S2. In the following, we seek for
conditions on parameters, specifically c 1 to study the effects of changes in c 1 on the system equilibria. Analytical
computations yield the following three conditions on c 1 and c 2 in terms of number of system equilibria:
2c 22 − c 12 > 0 ⇒ two equilibrium points, EQ1 and EQ2 in Fig. 1
•
If
•
If 2c 22 − c 12 = 0 ⇒ EQ1 and EQ2 meet at ( x , y ) = ( c 1 / 2 ,−c 1 / 2 ) known as saddle node bifurcation shown in Fig. 2
for c 1 = ± 2 c 2
•
If 2c 22 − c 12 < 0 ⇒ loss of equilibria
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Next, we study conditions for which one of the equilibria will be at the singularity of the algebraic part of (2.29), which is
known as the singularity induced bifurcation. Singular points are in the set defined by
{
S = ( x , y ) x 2 + y 2 − c 22 = 0 and det[D y g ( x , y , c 2 )] = 2 y = 0
}
(2.30)
As depicted in Fig. 3, there are two singular points ( −c 2 ,0 ) and ( c 2 ,0 ) that satisfy the conditions given by (2.30).
Specifically, if one is able to control the parameter c1 , then it is possible that one of the system equilibria coincides with a
singular point. Fig. 3 indicates two different values of c1 such that singularity induced bifurcation occurs:
•
For c 1 = −c 2 < 0 , the equilibrium point EQ2 is at the singular point ( −c 2 ,0 )
•
For c 1 = c 2 > 0 , the equilibrium point EQ2 is at the singular point ( c 2 ,0 )
Observe that parameter c 2 is kept constant and we adjust the parameter c1 to either c 1 = −c 2 < 0 or c 1 = c 2 > 0 such that
one of two equilibrium points of (2.29) is at the singular point, which is a singularity induced bifurcation point, as claimed
in the theorem. The constraint manifold is composed of two regions that are connected through the singular points ( −c 2 ,0 )
and ( c 2 ,0 ) . These two regions that are the half circles defined analytically as follows:
{
= {( x , y ) − c
}
−x }
C 0 = ( x , y ) − c 2 < x < c 2 and y = c 22 − x 2
C1
2
< x < c 2 and y = − c 22
(2.31)
2
(2.32)
The induced vector fields in each region can be given as:
x& = − x + y + c 1
y& = −
x
( − x + y + c1 )
y
(2.33)
As can be seen from (2.33) the vector field y& is defined everywhere on the constraint manifold except at the two singular
points ( −c 2 ,0 ) and ( c 2 ,0 ) . The local ODEs in each region are:
x& = − x + c 22 − x 2 + c 1 on C 0
x& = − x − c 22 − x 2 + c 1 on C 1
(2.34)
This example illustrates that even a simple DAE model may exhibit bifurcations and loss of causality. Specifically, it makes
the fact clear that if we are able to control any of the system parameters, we can intentionally cause the equilibria to
bifurcate. The occurrence of saddle node and singularity induced bifurcations with the qualitative changes in the number of
system equilibria are summarized in Fig. 4 when the parameter c1 is assumed to be controllable. This example intuitively
proves that any singular point at a given parameter value could easily be an equilibrium point, indeed a singularity induced
13
bifurcation point at another parameter value. In the next example we study bifurcations and singularities of a DAE power
system model.
Example II:
We now present a 3-bus power system example that exhibits different bifurcations. This 3-bus system, whose one-line
diagram is shown in Fig. 5, has 2 generator buses and one constant PQ load bus connected to each other radially, and is
taken from [28]. Similar 3-bus systems have been studied extensively by Kwatny et al. [4] and by Tavora and Smith [44].
The data describing the system is as follows:
•
x d = 0.1 pu and includes the transient reactance of generator plus the reactances of the transformer and transmission
line, and is the same for each generator-line connection.
•
Pd + jQ d = 10 + j 3 pu is the real and reactive power at the load bus, bus 3.
•
V1 = E 1 = V 2 = E 2 = 1.2 pu are the internal generator voltage magnitudes
•
Pm1 = Pm 2 = 5 pu are the generator mechanical power inputs
•
D =[0.3 0.3] are the generators’ damping coefficients
•
M = [1 1] are the generators’ inertia moments
•
Generator 1 is chosen as the swing bus with a zero rotor angle; load bus voltage angle (bus 3) and generator 2 rotor
angle (bus 2) are measured relative to generator 1 (bus 1)
We use the Voltage Stability Toolbox (VST) [38-41] to analyze bifurcations and singularities of the 3-bus system. In order
to investigate the dynamic behavior of the DAE model, we first determine the equilibrium points and their stability features
as one of the system parameters varies (the method for identification of the system equilibria will be briefly summarized in
Section III). In the VST, the concerned parameters are bus injections. Specifically, loads can be varied according to
following equations:
P = P0 + alpha * directionP
Q = Q0 + alpha * directionQ
(2.35)
where P0 , Q0 and P , Q are vectors of initial and actual real and reactive power injections, respectively; directionP and
directionQ are vectors of searching directions which can be set to be positive, zero or negative for each bus; alpha (α) is a
scalar adjustable parameter. Fig. 6 illustrates how load bus voltage magnitude at bus 3, V3, for various equilibria changes
when the real power and reactive power demand at bus 3 varies according to (2.35) while power factor is kept constant.
Note that for any given parameter value alpha, equilibrium points are dynamically stable. The portion of the equilibrium
14
surface around the nose point is the only unstable section. As power demand at bus 3 increases both high and low voltage
solutions are dynamically stable and stability exchanges (stable→unstable) along the lower part of the nose curve occur at
parameter value close to the saddle node (SN) bifurcation due to a singularity induced (SI) bifurcation. Then, one stable and
one type-1 unstable equilibrium point meet at the SN bifurcation. This can be seen clearly with an appropriate step size in
the search method used to locate equilibria [40]. Fig. 6 also depicts the singular points at different parameter values with a
nose curve in a 2-dimensional space for load bus voltage magnitude V3. Each of the singular points depicted by (x) belongs
to the singular set defined by (2.25). At a fixed parameter value, the singular point lies on the boundary between voltage
causal regions and changes smoothly as the parameter varies. In Section III we will present the method to identify and locate
these singular points at any given parameter value β .
In order to locate the SI bifurcation, we choose a small step-size in the search algorithm for the system equilibria by
specifying the number of steps in Newton-Raphson (NR) and Newton-Raphson-Seydel (NRS) [31] algorithms. Both search
algorithms in VST are set up in such a way that the step-size is inversely proportional to the number of steps. Therefore, we
use NR_steps = 500 and NRS_steps = 500. Fig. 7 shows how the eigenvalues of the system matrix
Asys = [ D x f ] − [ D y f ][ D y g ] −1 [ D x g ] move as we increase the parameter α and illustrates the occurrence of the SI
bifurcation. The dashed-arrows indicate the direction of increase in parameter α. Just before the SI bifurcation; the system
has all its eigenvalues in the left half plane of the complex plane, which implies stability. The eigenvalues that are labeled
as stable seem to be on the imaginary axis due to the large scaling of x-axis even though there exists small damping in the
system. At the SI bifurcation point, one complex eigenvalue moves (in a jump fashion) to the right half plane with a large
real number, which implies instability of the system. Therefore, the stability status of the equilibria undergoes an
instantaneous change from stable to unstable with exactly one eigenvalue. Note that the other complex eigenvalue stays in
the left half plane but becomes a large negative real eigenvalue. Theoretically, these real eigenvalues become unbounded at
the SI bifurcation [19-23]. We can obtain a clearer picture of the SI bifurcation occurrence with a much larger eigenvalue at
the expense of simulation time [40]. Further increase in the parameter causes a stable equilibrium point and a type-1
unstable equilibrium point to meet at the SN bifurcation. As seen in Fig. 6, the SN and SI bifurcations look like the same
point in terms of parameter α, but they are not. The SI bifurcation happens at α = 0.68943 while the SN α = 0.68960 and
they are two qualitatively different operating points.
Recall that the algebraic part of (2.19) requires any motion to take place on the constraint manifold. Thus, decomposition
of the constraint manifold into voltage causal regions and singular surfaces is essential to investigate the dynamic behavior
15
of the system. Fig. 8 and 9 depict 2-dimensional (2-D) projections of the constraint manifold onto the (δ2,V3) and (δ2,δ3)spaces for α = 0.06. Observe that the constraint manifold consists of two voltage causal regions (C1 and C2) separated by the
singular surface. The singular surfaces are defined by the sets:
S = {( δ 2 ,V 3 ,δ 3 ) = ( − 0 .49931 , 0 .72467 ,−0 .88932 ), ( 0 .49931 , 0 .72467 , − 0 .39001 )}
(2.36)
The two voltage causal regions each contain dynamically stable equilibrium points labeled as ( S .E .P )h and ( S .E .P )l ,
respectively. These stable equilibrium points correspond to high and low voltage solutions for both parameter values shown
in the nose curve, Fig. 6. In these figures, the singular points are labeled as S1 and S2 and they indicate bifurcation points of
the algebraic equations when generator angle δ2 is considered as the parameter. Note that S1 is the same singular point at
α = 0.06 shown in Fig. 6. Table 1 summarizes all the significant points (i.e., equilibria and singular) for two different
parameter values α = 0.06 and 0.42.
We apply the theorem for the 3-bus example. Note that the dynamic variables are x = [δ 2
variables are y = [V3
ω 2 ]T and the algebraic
δ 3 ]T . We have presented two singular points S1 and S2 for α = 0.06. For this parameter value
β G and β L consist of net power injection at the buses. Specifically, β G = [0 P2 ]T and β L = [P3
Q3 ] where P2 , P3 and
T
Q3 are the net real power injection at bus 2 and the net real and reactive power injections at bus 3, respectively. At the
singular points S1 and S2, β SG = [0 5] pu and β SL = [− 10.06
− 3.018 ] pu. If we follow the theorem for the singular
point S1, we obtain the following results: Recall that S1 is [δ 2
V3
T
T
δ 3 ]T = [− 0.49931 0.72467 − 0.88932]T and the
mismatch
= [0 − 1.69378 ] pu. Based on (2.28), we can adjust the net injection at bus 2
corresponding mismatch at bus 2 is β SG
T
such that the mismatch will be zero at S1. Then, the new injection would be β GNEW = [0 3.30622] pu while the injections at
T
bus 3 remains unchanged, β SL = [− 10.06
this
new
injection.
EQ 2 = [− 0.42355 0.80676
− 3.018 ] pu. One can run a single load flow to obtain two equilibrium points at
T
Those
would
be
EQ1 = [− 0.49931 0.72467 − 0.88932] and
T
− 0.77207 ] . Observe that EQ1 is the same point as singular point S1 given in Table 1, which
T
also verify the claim of the Theorem: That is EQ1 is a singularity induced bifurcation point. We can make similar analysis
for the other singular point S2. It is worth to mention here that the mismatch at the swing bus (generator 1) for the singular
points S1 and S2 are 6.75378 pu and 3.30622 pu, respectively as expected since P1 + P2 + P3 = 0 for a lossless power
16
network model. Moreover, the swing bus injection is not affected by the adjustment of mechanical input power to the
generator 2.
Praprost and Loparo have reported similar results for the 3-bus system in [28] in the context of transient stability analysis
of power systems. Hiskens and Hill [25,26] have considered the effects of impasse surfaces in the dynamic stability
evaluation of power system. However, none of them have presented a rigorous method to identify singular points at any
given parameter value. But, they are among the first who have recognized the importance of singular surfaces in energy
function evaluation of power system dynamics and voltage collapse. Their simulation results indicate that parts of the
stability boundary are formed by trajectories that are tangent to the singular surface and if any disturbance causes
trajectories to pass by the singular surface, the system may settle down to a different stable equilibrium point. However, one
of the equilibrium points (i.e., ( S .E .P )l in C2 in Fig. 8) is an infeasible operating point due to its low voltage profile, which
may cause a further disturbance in the system leading to a voltage collapse problem.
III. IDENTIFICATION OF EQUILIBRIUM AND SINGULAR POINTS
A. Identification of System Equilibria
The equilibrium points of the DAE model of (2.19) satisfy the following algebraic equations:
f ( x, y , β ) = 0
g( x , y , β ) = 0
(3.1)
For the rest of the paper, we refer to equilibrium equations (3.1) as the load flow equations. In the following, we summarize
methods and algorithms implemented in Voltage Stability Toolbox (VST) to identify the system equilibria as the parameter
varies [34,38,39]. The starting point for the static analysis of the power system model (2.19) is the identification of system
equilibria. For a given set of parameters β , an equilibrium point ( x ∗ , y ∗ ,β ∗ ) satisfies two algebraic equations given by
(3.1) and is said to be on the equilibrium surface defined by (2.20). Load flow analysis is basically the identification of the
equilibrium set given by (2.20). The VST implements load flow calculations that function up to the point of collapse (saddle
node bifurcation point). Conventional numerical methods for computing equilibria, such as the Newton-Raphson (NR)
method, must be modified in order to obtain reliable results near bifurcation points. Variants of two basic methods have
been applied to power system analysis: the continuation (or homotopy) [42] method and the direct (or point of collapse)
method [31] . We describe our implementation of the Seydel method, which is a direct method. For convenience, let
[
z = xT
yT
] , F ( z , β ) = [f ( x, y , β )
T
g( x , y , β )] and N = n + m . Then, we have
T
17
F( z, β ) = 0
(3.2)
First recall that a load flow calculation consists of solving a set of algebraic equations of the form (3.2) for the dependent
variables z ∈ ℜ N , given the (independent) parameters β ∈ ℜ k . The standard Newton-Raphson method applied to (3.2) is
[D z F ( z i , β )]∆z = − F ( z i , β ),
 ∂f
∂
F
   ∂x
where, [D z F ] =   = 
 ∂z   ∂g
 ∂x
z i +1 = z i + ∆z
(3.3)
∂f 
∂y 
 is the load flow Jacobian matrix.
∂g 
∂y 
However, the standard Newton-Raphson method breaks down near (static) bifurcation points, i.e. when [D z F ] is singular
( rank[ D z F ] < N ). In generic one-parameter families the dimension of Ker [D z F ] at a bifurcation point is precisely one,
i.e. rank [ D z F ] = N − 1 . Thus, to locate such a point we seek values for z ∈ ℜ N , β ∈ ℜ , and nontrivial v or w ∈ ℜ N
which satisfy
F( z, β ) = 0
(3.4a)
[D z F ( z i , β )]v = 0 ,
or w T [D z F ( z i , β )] = 0
(3.4b)
The requirement for nontriviality of v, w may be stated by
v = 1, or w = 1
(3.4c)
One basic approach to finding bifurcation points is to apply the Newton-Raphson method to (3.4). This is the NewtonRaphson-Seydel (NRS) method. Data that satisfies (3.4) will be denoted z b , β b , v b , wb and we designate the Jacobian
[D z F ]:= D z F ( z b , β b ) .
Note that the vectors
vb , wb have special significance. They are, respectively, the right and left
eigenvectors corresponding to the zero eigenvalue of [D z F ] . The eigenvector
vb spans the kernel of [D z F ] and wb spans
the kernel of [D z F ] . Once a bifurcation point is located, it is feasible to modify the above method to compute points
T
around the fold (nose) of the equilibrium surface:
F( z , β ) = 0
[D z F ( z , β ) − λI ]v = 0
for values of λ ∈ [−ε 1
(3.5)
ε 2 ] with ε 1 , ε 2 > 0 .
This allows computation of equilibrium points close to the bifurcation point where conventional Newton-Raphson
calculations would fail. Of course λ = 0 corresponds to the saddle node bifurcation point. Note that in order to implement
the NRS method, second order derivatives are required. In the VST, governing equations of the classical model and
18
Jacobian matrix including the second derivatives have been constructed symbolically. In the NRS method, the load flow
equations and the Jacobian matrix are evaluated numerically through the following callable MEX function in the VST.
[f
J ] = function( data , z, param, v)
(3.6)
The inputs are: data- a data set for load flow calculations associated with the system model that is automatically loaded into
the MATLAB workspace when the system to be investigated is identified, z- the state vector, param- the parameter vector
(real and reactive power injections), and v- at the bifurcation point v is the null space spanning vector. The outputs are: f- a
vector containing the values of load flow equations, and the matrix J = [J 1
J 2 ] is the Jacobian matrix containing first and
second derivatives. The explicit expressions of J 1 and J 2 are stated below.
 ∂f
 ∂F   ∂x
J1 =   = 
 ∂z   ∂g
 ∂x
∂f 
∂y  (3.7a) J = ∂ [J v ] = ∂  ∂F

2
1
∂z
∂z  ∂z
∂g 
∂y 
 ∂f

 ∂  ∂x
v =

 ∂z  ∂g

 ∂x

∂f 
∂f
 ∂f

vx +
vy



v
∂y  x   ∂ ∂x
∂y   J 11


=
=
∂g  v y   ∂z  ∂g
∂g   J 21
 
v +
v
 ∂x x ∂y y 
∂y 

J 12 
(3.7b)
J 22 
where each block entries in J 2 can be given as follows:
J 11 =


∂f
∂f
∂g 
∂g 
∂  ∂f
∂  ∂f
∂  ∂g
∂  ∂g
v y  , J 12 =
v y  , J 21 =
v y  , J 22 =
vy 
 vx +
 vx +
 vx +
 vx +
∂x  ∂x
∂y 
∂y  ∂x
∂y 
∂x  ∂x
∂y 
∂y  ∂x
∂y 
[
Note that the eigenvector v is partitioned into two components as v = v x
vy
] where
T
(3.7c)
v x corresponds to the dynamic
variables (i.e., generator angles) and v y corresponds to the algebraic variables (i.e., load bus voltage magnitudes and
angles).
The equilibrium and singular surfaces are multi-dimensional surfaces even for a relatively small sized power system
making it difficult to visualize these surfaces in a multi-dimensional space. Therefore, 2-dimensional (2-D) depictions of the
surfaces, for example nose curves, are usually used to illustrate the equilibrium and singular points of (2.19) upon parameter
variations. Bifurcation points indicating different bifurcations and stability characteristics of the equilibrium points are
illustrated schematically in Fig. 10. Fig. 10 indicates the occurrence of SI and SN bifurcations and the singular surface. Note
that when the equilibria and singular surface meet at α = α SI SI bifurcation occurs. As can be seen from Fig. 10, a large
portion of the lower part of the nose curve is dynamically stable. Therefore, a trajectory moving from a particular stable
equilibrium point (upper part of the nose curve) in many cases may reach the corresponding singular point that should be
used as the fictitious u.e.p.’s and critical energy should be calculated based on these fictitious u.e.p.’s, shown in Fig. 10.
19
For a larger-scale power system, it is impossible to visualize nose curves and singular surfaces in a multi-dimensional
space. Therefore, we have use a three-dimensional (3-D) space to depict the singular surface. Fig. 11 illustrates a 3-D
equilibrium surface with a 2-D singular surface cutting it. This 2-D singular surface (shown as planar) is really an
approximation of a non-planar surface that will actually cut the 3-D equilibrium surface. Note that two nose curves are
plotted with each one representing a specific search direction (load increase pattern) in the parameter space (denoted as
curves ➀ and ➁). These nose curves represent the equilibrium surface given by (2.20) and dashed surface represents the
singular surface given by (2.25). When the nose curves cross the singular surface, singularity induced bifurcation occurs.
Two singularity induced bifurcation points on the surface are labeled by the symbol ⊗. In Fig. 11, there are two equilibrium
points of the nose curve ➁ at a given parameter value α. The upper and lower voltage solutions are labeled as ( V h ,δ h ,α )
and ( Vl ,δ l ,α ) , respectively. We believe that the illustration of a singular set (or family of singular points) and nose curves
in a 3-D (or 2-D) space consisting of bus voltage magnitude (V), and angle (δ) at a given bus and load parameter (α) is
much more useful and informative than a state-space depiction from the power engineering point of view. This is further
justified by the fact that voltage instability has often been viewed as a local phenomenon. Using these pictures, Fig. 10 and
11, one can determine how far a current operating point in the parameter space is away from the stability boundary and
appreciate the effects of bifurcations on the stability of the system. It is worth to state that there is no reason to expect the
singular surface to form a smooth surface as shown in Fig. 11. However, we do expect it to be a set (not necessarily)
connected with a boundary.
B. Identification of Singular Points
In the previous section, studies on the equilibrium structure of the model (2.19) provide us with information in the
parameter space such as maximum loading point (i.e., saddle node bifurcation) and help us identify static constraints. In
order to complete the picture, we fix the parameter and analyze the structure of constraint manifold (i.e., g ( x , y ) = 0 ),
shown in Fig. 12. Recall that constraint manifold is decomposed into voltage causal regions separated by a singular surface.
Within a voltage causal region, the DAE model of power system could be reduced to a set of locally equivalent ordinary
differential equations (ODE). On the other hand, the equivalence is not possible at any singular point that imposes a
dynamic constraint on the system. Thus, location of singular points in the constraint manifold is valuable information to
assess the stability of the system.
20
Fig. 12 depicts a hypothetical constraint manifold composed of two voltage causal regions at a fixed parameter value. We
intentionally divide the constraint manifold into two different voltage causal regions, C1 and C2. Observe that each region
has a dynamically stable equilibrium point. In order to identify the singular point, we want to implement a search method
initiated at one region, say C1 and end up at another point in region C2, crossing the singular surface in between. We use the
generator angles to parameterize the algebraic equation g ( x , y ) = 0 at a fixed parameter β assuming that we are able to
control the mechanical input to the generators. This assumption also allow us to identify singularity induced bifurcation
point by the theorem in Section II and we have demonstrated the case for the 3-bus power system example in the previous
section. In Fig. 12 two different search paths are shown to indicate the fact that there may be more than one singular point
at a given parameter value. Actually, the total number of possible search directions is ( 2 n −1 − 1 ) because the dimension of
the dynamic variables (generator angles) is n. Each path (search direction) is closely related to which of the dynamic
variables are parameterized and may result in a different singular point. The geometry of the singular surface and the left
eigenvector corresponding to the zero eigenvalue of [ D y g ] may provide useful information on computing an optimum
search direction in the state space to locate the closest singular point to the current operating point [12]. At this point, we do
not intend to find an optimum search direction, so we parameterize all the generator angles and implement it by the
following expression:
[
δ g = T ( 1 − µ )δ gu + µδ gl
]
(3.8)
where δ u and δ l are (n-1)-dimensional vectors representing the generator angles at the upper and lower equilibrium points
at a given parameter value β , respectively, and µ is a new scalar parameter used to locate the singular points. Note that
swing bus angle is not included in the parameterization since it is assumed to be zero. T ∈ ℜ ( n −1 )×( n −1 ) is the direction matrix
used to specify the generator angles to be parameterized. The matrix T contains entries either zero or one. Because all the
generator angles are identically parameterized, the direction matrix T is a unity matrix. It is clear that different scaling of
generator angles results in different direction matrix T and thus, different singular points.
In order to compute singular points, we modify the NR and NRS algorithms explained in the previous section as follows:
Recall that we seek points ( x , y ) in the constraint manifold such that g ( x , y ) = 0 and rank [ D y g( x , y )] = m − 1 (i.e.,
Ker [ D y g ( x , y )] is exactly one). To compute such a singular point, we seek values for
21
y ∈ ℜ m , µ ∈ ℜ (thus,
x ∈ ℜ n through equation (3.8)), and nontrivial v y ∈ ℜ m (a nontrivial v y assures the singularity of [ D y g( x , y )] ) that
satisfy:
g ( x( µ ), y ) = 0
[D y g( x( µ ), y )] v y = 0
(3.9)
vy −1 = 0
Once, a singular point is located, it is easy to modify equation (3.9) to obtain points around the fold of the constraint
manifold:
h1 = g( x( µ ), y ) = 0
h2 = [D y g( x( µ ), y ) − λI ] v y = 0
(3.10)
h3 = v y - 1
The iterative process would be:
[Dẑ H ( ẑi )]∆ẑ = − H ( ẑi ),
H = [h1
ẑi +1 = ẑi + ∆ẑ
(3.11)
[
]
h3 ] = g( x( µ ), y ) ( D y g( x( µ ), y ) − λI )
T
h2
[
where, ẑ = y v y
µ]
T
 J 11
and [D ẑ H ] =  J 21
 J 31
J 12
J 22
J 32
vy −1
T
(3.12)
J 13 
J 23  is the corresponding extended Jacobian matrix whose each block
J 33 
entry is defined below:
J 11 =
∂g
∂g ∂δ g  ∂g
∂g
∂g
, J 12 =
= 0 , J 13 =
=
=
∂y
∂v y
∂µ ∂δ g ∂µ
 ∂δ g
J 21 =
∂
∂
∂  ∂g 
( D y g − λI )v y =
( D y g − λI )v y = ( D y g − λI )
 v y  J 22 =
∂v y
∂y
∂y  ∂y 
(3.14)
J 23 =
 ∂  ∂g   ∂δ g  ∂  ∂g  
∂
u
l
( D y g − λI )v y = 
=
 v y  − δ g + δ g
 v y 
δ
y
µ
δ
y
∂
∂
∂
∂µ
∂
∂
 g 
 g 
 
 
(3.15)
J 31 =
v Ty
∂
∂
∂
v y − 1 = 0 , J 32 =
vy −1 =
, J 33 =
vy −1 = 0
∂y
∂v y
∂µ
vy
[
]
[
[

u
l
 −δ g +δ g

[
[
[
[
(3.13)
]
]
]
]
]
[
]
]
(3.16)
The iterative process provides a singular point ( x , y ) such that [ D y g( x , y )] evaluated at ( x , y ) has a zero eigenvalue
( λ = 0 ) with its corresponding eigenvector v y . It is worth to mention here, Jacobian matrices containing second order
22
derivatives are already available from equilibria and bifurcation analysis in the VST. The initial condition for the method
above is a stable equilibrium point, which is a normal operation point and again available at any parameter (injection) value
from equilibria and bifurcation analysis in parameter space. Our implementation of these calculations is as follows:
•
The user specifies an initial value of the state (i.e., upper solution at given parameter value, β ) and chooses
dynamic variables (generator angles) to parameterize (i.e., search direction in the state space). We choose the
upper equilibrium point as the initial value because this is a feasible operating point satisfying voltage magnitude
constraints.
•
Newton-Raphson (NR) computations proceed starting at the upper equilibrium point along the search direction
until it fails to converge as shown in Fig. 13.
•
The last successful NR data point is used to implement an inverse-iteration computation to obtain an estimate of
the eigenvalue of [ D y g ] nearest 0, and its eigenvector, v y
•
This data is then used to initiate a Newton-Raphson-Seydel (NRS) procedure using (3.10) to compute around the
singular point. The value λ = 0 is always included and data at the singular point is thereby obtained.
The above algorithm is implemented in the VST. In the next section, we illustrate simulation results for the IEEE 118-bus
system.
IV. SIMULATION RESULTS
In order to show that the VST and specifically the identification method for singular points work for larger systems, we
present results on singularity induced bifurcation and singular points for the IEEE 118 bus test system. We have the
following two cases:
Case I: The real and reactive power at bus 75 have been increased according to (2.35). In order to capture singularity
induced bifurcation we use NR_steps = 300 and NRS_steps = 300 for the step sizes in the NR and NRS algorithms [40].
Fig. 14 illustrates a 2-D equilibrium surface defined by (2.20). The voltage magnitude at bus 75 is selected to show how
equilibria and their stability properties change with parameter variations. As can be seen, two kinds of bifurcations are
identified, namely saddle node (SN) and singularity induced (SI) bifurcations. As the parameter α increases both upper and
lower parts of the nose curve are dynamically stable. At α = 7.92034 , the system undergoes a stability exchange associated
with the SI bifurcation and the stability feature of the lower equilibrium points changes qualitatively, from stable to
unstable. As stated in Section II, when SI bifurcation occurs, [ D y g( x , y )] becomes singular and the point ( x , y ) belongs
23
to both the singular surface defined by (2.25) and the equilibrium surface given by (2.20). As the parameter further
increases, one stable (upper part) and one type-1 unstable (lower part) equilibrium point meet at the SN bifurcation point
α = 8.69104 , which is the tip of the nose curve. Beyond the SN bifurcation point, there is no feasible solution to the load
flow equations. The stability properties of the lower part of the nose curve are certainly model dependent. A constant power
model is assumed in VST with a classical generator model. Fig. 14 also depicts singular points at different parameter values.
The parameter value α = 7.92034 is especially important in the sense that it enables us to check whether the singular point
search method gives correct results. Recall that at α = α SI = 7.92034 singularity induced bifurcation occurs and all the state
information at this parameter value is available to us from equilibria and bifurcation analysis. The SI bifurcation point also
belongs to the singular set defined by (2.25). Thus we expect that the proposed singular point search method should give the
same result as that of the bifurcation analysis at α = α SI = 7.92034 . As seen from Fig. 14, this is indeed the case. Observe
that for the voltage magnitude at bus 75 singular points at each parameter value lies between the higher and lower voltage
solutions until the SI occurs. Note that the lower part of the nose curve may not be practically feasible operating points due
to the low voltage profile at bus 75. However, this is not the general case as shown in Fig. 15. Fig. 15 depicts the voltage
magnitude at bus 63 for the same load increase pattern. The lower part of the nose curve and singular points including the SI
points lie above 0.95 pu, which is usually considered to be the low voltage thresh-hold value for a normal operation of the
power systems.
Fig. 16 shows how the eigenvalues of ASYS move as we increase the parameter and indicates the occurrence of the SI
bifurcation clearly. The dashed-arrows indicate the direction of increase in the parameter value. Just before the SI
bifurcation, the system has all its eigenvalues in the left half plane of the complex plane, which implies stability. The
eigenvalues that are labeled as stable seem to be on the imaginary axis due to the large scaling of x-axis even though there
exists small damping in the system. At the SI bifurcation point, one complex eigenvalue ( λ = −0.15 + i175.832 ) moves to
the right half plane and a large real number ( λ = 29.712 ), which implies the instability of the system. Therefore, the
stability status of the equilibria undergoes an instantaneous change from stable to unstable with exactly one eigenvalue.
Note that the other complex eigenvalue ( λ = −0.15 − i175.832 ) stays in the left half plane but becomes a large negative real
eigenvalue ( λ = −30.012 ). Theoretically, these real eigenvalues become unbounded at the SI bifurcation [19-23]. It is
possible to obtain a clear picture of the SI bifurcation with a much larger eigenvalue at the expense of simulation time [40]
if we use a smaller step-size in the NR/NRS algorithms.
24
Case II: The real and reactive powers at bus 76 have been increased according to (2.35) while power factor is kept constant
and initial load demand at bus 76 is: P0 = 0.27 pu and Q0 = 0.11 pu on a 100 MVA base. Fig. 17 illustrates the voltage
magnitude at bus 76 and the singular points at different parameter values. Observe that this time almost the whole part of the
nose curve is stable. The SI bifurcation happens at a parameter value α = 6.2422 so close to the SN bifurcation point
α = 6.2702 . That is the main reason why we present this case. The SI and SN bifurcations are very close to each other in
terms of parameter. One may mistakenly consider that these two qualitatively different points are the same. So, the question
to be answered is: Under which condition(s) can one identify and locate these two distinguished operating points? We
interpret this problem as a numerical issue in which the minimum step-size in the NR/NRS methods is located to provide
both a quantitative and qualitative depiction of SI bifurcation point. We have reported in [40] that the dynamics of SI
bifurcation are so fast that one should choose an appropriate step-size to capture it. The SN and SI bifurcation points can be
distinguished from each other by evaluating the eigenvalues of ASYS , which are not presented for this case.
As stated earlier, the proposed singular point search method uses generator angles to parameterize the algebraic part of
the DAE model (2.19) and this parameterization has been achieved through (3.8). The core of the algorithm is that it
identifies the saddle node bifurcations of algebraic variables y in the state space at a fixed parameter. As seen in (3.10), we
seek for ( x , y ) such that the Jacobian matrix [ D y g ] evaluated at ( x , y ) has an eigenvalue at the origin (i.e., [ D y g ] is
singular) with its corresponding nontrivial eigenvector v y . Therefore, we expect that singular point search algorithm will
give us a nose curve like evolution of the algebraic variables y (i.e., load bus voltage magnitudes and angles) as the
generator angles change. In order to clearly illustrate the saddle node bifurcation idea in the state space, we fix the
parameter α at α = 5.0 and run the search algorithm for case I. Fig. 18 shows the evaluation of voltage magnitudes at bus 75
as the generator angles change through parameter µ in the state space at α = 5.0 while Fig. 20 shows the case for
α = 7.92034 . Observe that in these figures we have a nose curve like evolution of the voltage magnitudes in the state space.
The tips of the curves are the saddle node bifurcation point of the algebraic part of the DAE model, which is also the
singular point. Recall that the smallest real eigenvalue of [ D y g ] λ is used as a parameter in (3.10) as to go around the
singular point. Fig. 19 and 21 illustrate how the smallest real eigenvalue of [ D y g ] moves in the NRS algorithm (3.10) as
the generator angles change in the state space for α = 5.0 and α = 7.92034 , respectively. The point ( x , y ) at which λ = 0
is the singular point at the corresponding parameter.
25
V. DISCUSSION
One of the most important issues in the method for singular point identification is the selection of the generator angles to
parameterize the algebraic part of the DAE model at a given set of bus injections. We choose generator angles as parameters
because we can control them by controlling the mechanical input to generators. The results presented in the paper are based
on the assumption that we have the ability to control the mechanical input power to all generators. If this is not case, i.e.,
only some of the generator injections are controllable, then one can choose the angles of the controllable generators as
parameters and the method would work efficiently. Of course, in the future operation environment of the power system,
Flexible AC Transmission System (FACTs) devices will make it possible to control load bus variables (i.e. voltage
magnitudes and angles) at load buses locally. Then, it would be possible to include these controllable load bus variables in
parameters to be chosen. The states (i.e. generator angles) need not to also be parameterized at the same rate. In that case,
the entries of the matrix T in (3.8) could be scaled based on the right eigenvector and sensitivity information.
We use both the NR and NRS algorithms in our method. Because, like any other Newton-iterative method, the NRS
algorithm needs a good initial condition, that is, a point close enough to the singular point along with the smallest real
eigenvalue of [ D y g( x , y )] and the corresponding right eigenvector. Otherwise, we may experience convergence problems.
The NR method plus an inverse iteration method for estimating the eigenvalue-eigenvector pair provides a good initial
condition for the NRS algorithm.
We claim in a theorem that any singular point at a given set of injections is a singularity induced bifurcation point at
another injection set. This is true, if we can adjust the input mechanical power to generators; so as to have a zero mismatch
at the generator at the singular point, implying that it becomes a singularity induced bifurcation point. As a by-product of the
singular point search algorithm and by this theorem, we can easily identify as many singularity induced bifurcations as
required once the singular points are available. We can distribute the total mismatch among all the generators that are
controllable based on their limits and sensitivity information obtain from the right eigenvector corresponding the zero
eigenvalue [ D y g( x , y )] of evaluated at the singular point. Note that the swing bus injection is not affected by this
adjustment, which is desired since utilities do not want their swing bus to be affected by normal operating power transfers.
VI. CONCLUSION AND FUTURE WORK
In this paper, we have presented an iterative method to locate and identify singular points of the DAE model of power
systems at any given parameter value. In the method, we use generator angles to parameterize the algebraic part of the DAE
model in the state space at a given set of bus injections and we identify the singular points as being the saddle node
26
bifurcation of the algebraic part of the DAE model. The corresponding Jacobian matrix evaluated at the singular points has
an eigenvalue at the origin. We have shown by a theorem that any singular point at a given set of bus injections is a
singularity induced bifurcation point at another set of bus injections, which enables us to identify singularity induced
bifurcation points. This new set of injections could be reached by adjusting the mechanical power input to the generators.
We have combined static information from the saddle node bifurcation and dynamic information from the singular point
together in order to provide a comprehensive picture of the system stability. We have updated the Voltage Stability Toolbox
to include singular point computations. Simulation results on a 3-bus system and the IEEE 118-bus system have been
presented. We have illustrated singular points with the traditional nose curve for different load change scenarios. Finally, we
have discussed some numerical issues and the possible practical implications of the simulation results from a power systemoperating point of view and possible future enhancement of the algorithm.
As future work, an energy function approach should be implemented in order to provide a dynamic security index that
considers the singular points. Specifically, the dynamic stability margin of a given operating point could be computed as the
energy difference between the current operating point and the singular point. This scalar energy value would be the dynamic
security index. Moreover, in order to complete the picture a static security index as being the energy difference between the
current operating point and the point of collapse should be combined with the dynamic security index.
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29
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30
VIII. ILLUSTRATIONS
y
y
( 0 ,c2 )
EQ1
EQ1
Constraint manifold
x +y =c
2
2
2
2
( 0 ,c2 )
Constraint manifold
x 2 + y 2 = c 22
Singularity
induced bifurcation
det[ D y g( x , y )] = 0
singular point
( −c2 ,0 )
S1
det[ D y g ( x , y )] = 0
( c2 ,0 )
S2
( 0 ,0 )
( −c2 ,0 )
EQ2
x
( c2 ,0 )
EQ2
x
( 0 ,0 )
singular point
Singularity
induced bifurcation
EQ2
c1 = − c2 < 0
( 0 ,−c 2 )
c1 < 0
( 0 ,−c2 )
EQ1
c1 = 0
c1 = c2 > 0
Fig. 3. The occurrence of singularity induced bifurcations for
two different values of c 1
Fig. 1. Constraint manifold, system equilibria and
singularities
Saddle node bifurcations
y
( 0 ,c2 )
Saddle node
bifurcation
two equilibrium points
loss of equilibria
EQ1
− 2c 2
Constraint manifold
loss of equilibria
c2
0
2 c2
c1
x 2 + y 2 = c22
( c1 / 2 ,−c1 / 2 )
det[ D y g( x , y )] = 0
singular point
Singularity induced bifurcations
singular point
( −c2 ,0 )
( 0 ,0 )
( c 2 ,0 )
x
Fig. 4. The saddle node and singularity induced bifurcations
with qualitative changes in the number of the system
equilibrium points
c1 = − 2 c2 < 0
( c1 / 2 ,−c1 / 2 )
EQ2
c1 = 0
− c2
( 0 ,−c 2 )
Saddle node
bifurcation
Gen 2
c1 = 2 c 2 > 0
Bus 2
Bus 3
jxd
Bus 1
Gen 1
jxd
Fig. 2. The occurrence of saddle node bifurcations for two
different values of c 1
Pd, Qd
Fig. 5. A 3-bus system with 2 generators and 1 load buses
from [28]
31
C1
C1
(S.E.P)h
singular points
S1
S2
(S.E.P)l
C2
Fig. 8. Load bus voltage magnitude vs. generator 2 rotor
angle for Pd= 10.06 pu and Qd = 3.018 pu (alpha = 0.06)
Fig. 6. Load bus voltage magnitude V3 vs. parameter with
singular points
stable
C2
C1
SI bifurcation
S2
α increases
α increases
(S.E.P)h
unstable
C1
C2
α increases
α increases
(S.E.P)l
S1
C2
SI bifurcation
Fig. 9. Load bus voltage phase angle vs. generator 2 rotor
angle for Pd= 10.06 pu and Qd = 3.018 pu (alpha = 0.06)
Fig. 7. Eigenvalues of the system matrix ASYS as the
parameter α varies
States
Singular Point I S1
Upper Equilibrium
Lower Equilibrium
Singular Point II S2
TABLE 1: SUMMARY OF THE STATES
Case I (α = 0.06)
Case II (α = 0.42)
V3 (pu)
V3 (pu)
δ2 (rad)
δ3 (rad)
δ3 (rad)
δ2 (rad)
-0.49931
-0.006387
-0.010031
0.49931
0.72467
0.88791
0.59021
0.72467
-0.88932
-0.49485
-0.79381
-0.39001
32
-0.33286
-0.04849
-0.07164
0.33286
0.73752
0.84237
0.64500
0.73752
-0.80610
-0.56591
-0.77396
-0.47324
Statically constrained
x,or y
V
( xu,yu )
Vh
(xSIN,ySIN)
Dynamically constrained
stable
unstable
singular surface
Singularity Induced
Bifurcation Points
(Vh,δh,α)
1
2
( xSN,ySNαSN)
x
Vl
Saddle node (SN)
bifurcation
(Vl ,δl ,α)
α
x
α
(xl,yl)
( xSI,ySIαSI)
Singular surface
where Dy g is singular
Singularity induced (SI)
bifurcation
0
αSI αSN
αs
α
δ
Fig. 10. Illustration of static and dynamic constraints
imposed by saddle node bifurcation and singular points
δl
δh
α, V, δ - surface for a fixed
search direction in α - space
Singular surface as seen
from within given, α, V, δ surface
Fig. 11. A 3-dimesional depiction of equilibrium surface
with a singular set
y
Switch from NR to NRS
u u
(x ,y )
C1
(xSIN , ySIN )
Singular Point
Singular points
(xl,yl)
C2
µ
Fig. 13. Tracing process for singular points in the state space
Fig. 12. Constraint manifold with voltage causal regions and
search directions
33
α increases
α increases
SI bifurcation
α increases
α increases
Fig. 14. Voltage magnitude at bus 75 vs. parameter with
singular points when the real and reactive power at bus 75
increase
Fig. 16. Eigenvalues of the system matrix ASYS as the
Fig. 15. Voltage magnitude at bus 63 vs. parameter with
singular points when the real and reactive power at bus 75
increase
Fig. 17. Voltage magnitude at bus 76 vs. parameters with
singular points when the real and reactive power at bus 76
increase
parameter α varies
34
Fig. 18. Saddle node bifurcation in the state space
for α = 5.0 : Voltage magnitude at bus 75, V75
Fig. 20. Saddle node bifurcation in the state space for
Fig. 19. Evolution of the smallest real eigenvalue of
D y g( x , y ) in the NRS search process for α = 5.0
Fig. 21. Evolution of the smallest real eigenvalue of
D y g( x , y ) in the NRS search process for α = 7.92034
α = 7.92034 : Voltage magnitude at bus 75, V75
35
IX. BIOGRAPHIES
Saffet Ayasun (MS’97) is currently pursuing his Ph.D. degree in Electrical and Computer Engineering Department at Drexel
University, Philadelphia, PA. He received his MS degree from Drexel University in 1997. His research interests include
stability of the large-scale nonlinear dynamical system, applied mathematics, nonlinear control theory, power systems, and
bifurcation theory.
Chika O. Nwankpa received the Magistr Diploma in electric power systems from Leningrad Polytechnical Institute, USSR, in
1986, and the Ph.D. degree in the electrical and computer engineering from the Illinois Institute of Technology, Chicago, in
1990. He is currently an Associate Professor of Electrical and Computer Engineering at Drexel University, Philadelphia, PA.
His research interests are in the areas of power systems and power electronics. Dr. Nwankpa is a recipient of the 1994
Presidential Faculty Fellow Award and the 1991 NSF Engineering Research Initiation Award.
Professor Harry G. Kwatny joined the Mechanical Engineering & Mechanics Department as an Assistant Professor in 1966, and
where he is currently the S. Herbert Raynes Professor of Mechanical Engineering. Prior to joining the University he received a BSME
degree from Drexel Institute of Technology, a SM in Aeronautics & Astronautics from the Massachusetts Institute of Technology and
a Ph.D. (EE) from the University of Pennsylvania. He is a Fellow of the IEEE. Professor Kwatny’s research interests include
modeling, analysis and control of nonlinear, parameter-dependent systems with specific applications to electric power systems and
power plants, aircraft, spacecraft and ground vehicles. In recent years he has focused on symbolic computing as a vehicle for bringing advances in
nonlinear system theory into engineering practice. He has published over 100 papers in these areas. He is the coauthor with Professor Gilmer Blankenship
(U. of Maryland) of the forthcoming book “Nonlinear Control and Analytical Mechanics: a computational approach” to be published by Birkhauser in the
spring of 2000. He is also a coauthor of the software package TSi ProPac, a Mathematica package for nonlinear control system design and multibody
mechanical system modeling.
36
Manuscript received ___________________________.
Saffet Ayasun and Chika O. Nwankpa are with Department of Electrical and Computer Engineering, Drexel University,
3141 Chestnut St. Philadelphia, PA 19104 USA.
Harry G. Kwatny is with Department of Mechanical Engineering & Mechanics, Drexel University, 3141 Chestnut St.
Philadelphia, PA 19104 USA
Footnotes:
1.
The corresponding author. Phone: (215) 895 2218, Fax: (215) 895 1695,
e-mail: chika@nwankpa.ece.drexel.edu
2.
The algebraic equations are locally solvable in terms of the generator angles.
3.
Actually, PV bus injections, not necessarily generator bus injections.
4.
VST may be obtained free of charge from power.ece.drexel.edu
Acknowledgment:
This research was supported by the National Science Foundation under Project Number ECS-9453407.
37
Figure Captions:
Fig. 1. Constraint manifold, system equilibria and singularities.
Fig. 2. The occurrence of saddle node bifurcations for two different values of c 1 .
Fig. 3. The occurrence of singularity induced bifurcations for two different values of c 1 .
Fig. 4. The saddle node and singularity induced bifurcations with qualitative changes in the number of the system
equilibrium points.
Fig. 5. A 3-bus system with 2 generators and 1 load buses from [28].
Fig. 6. Load bus voltage magnitude V3 vs. parameter with singular points.
Fig. 7. Eigenvalues of the system matrix ASYS as the parameter α varies.
Fig. 8. Load bus voltage magnitude vs. generator 2 rotor angle for Pd= 10.06 pu and Qd = 3.018 pu (alpha = 0.06).
Fig. 9. Load bus voltage phase angle vs. generator 2 rotor angle for Pd= 10.06 pu and Qd = 3.018 pu (alpha = 0.06).
Fig. 10. Illustration of static and dynamic constraints imposed by saddle node bifurcation and singular points.
Fig. 11. A 3-dimensional depiction of equilibrium surface with a singular set
Fig. 12. Constraint manifold with voltage causal regions and search directions.
Fig. 13. Tracing process for singular points in the state space.
Fig. 14. Voltage magnitude at bus 75 vs. parameter with singular points when the real and reactive power at bus 75 increase.
Fig. 15. Voltage magnitude at bus 63 vs. parameter with singular points when the real and reactive power at bus 75 increase.
Fig. 16. Eigenvalues of the system matrix ASYS as the parameter α varies.
Fig. 17. Voltage magnitude at bus 76 vs. parameter with singular points when the real and reactive power at bus 76 increase.
Fig. 18. Saddle node bifurcation in the state space for α = 5.0 : Voltage magnitude at bus 75, V75.
Fig. 19. Evolution of the smallest real eigenvalue of D y g( x , y ) in the NRS search process for α = 5.0 .
Fig. 20. Saddle node bifurcation in the state space for α = 7.92034 : Voltage magnitude at bus 75, V75.
Fig. 21. Evolution of the smallest real eigenvalue of D y g( x , y ) in the NRS search process for α = 7.92034 .
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