Grigsby, L.L., Hanson, A.P., Schlueter, R.A., Alemadi, N. “Power Systems”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
63
Power Systems
63.1 Power System Analysis
Introduction ? Types of Power System Analyses ? The Power Flow
Problem ? Formulation of the Bus Admittance Matrix ? Example
Formulation of the Power Flow Equations ? P-V Buses ? Bus
Classifications ? Generalized Power Flow Development ? Solution
Methods ? Component Power Flows
63.2 Voltage Instability
Voltage Stability Overview ? Voltage Stability Models and
Simulation Tools ? Kinds, Classes, and Agents of Voltage
Instability ? Proximity to Voltage Instability ? Future Research
63.1 Power System Analysis
Introduction
The equivalent circuit parameters of many power system components are described in Chapters 61, 64, and
66. The interconnection of the different elements allows development of an overall power system model. The
system model provides the basis for computational simulation of the system performance under a wide variety
of projected operating conditions. Additionally, “post mortem” studies, performed after system disturbances
or equipment failures, often provide valuable insight into contributing system conditions. The different types
of power system analyses are discussed below; the type of analysis performed depends on the conditions to be
assessed.
Types of Power System Analyses
Power Flow Analysis
Power systems typically operate under slowly changing conditions which can be analyzed using steady state
analysis. Further, transmission systems operate under balanced or near balanced conditions allowing per phase
analysis to be used with a high degree of confidence in the solution. Power flow analysis provides the starting
point for many other analyses. For example, the small signal and transient stability effects of a given disturbance
are dramatically affected by the “pre-disturbance” operating conditions of the power system. (A disturbance
resulting in instability under heavily loaded system conditions may not have any adverse effects under lightly
loaded conditions.) Additionally, fault analysis and transient analysis can also be impacted by the “pre-distur-
bance” operating point of a power system (although, they are usually affected much less than transient stability
and small signal stability analysis).
Fault Analysis
Fault analysis refers to power system analysis under severely unbalanced conditions. (Such conditions include
downed or open conductors.) Fault analysis assesses the system behavior under the high current and/or severely
L.L. Grigsby and
A.P. Hanson
Auburn University
R.A. Schlueter and
N. Alemadi
Michigan State University
? 2000 by CRC Press LLC
unbalanced conditions typical during faults. The results of fault analyses are used to size and apply system
protective devices (breakers, relays, etc.) Fault analysis is discussed in more detail in Section 61.5.
Transient Stability Analysis
Transient stability analysis, unlike the analyses previously discussed, assesses the system’s performance over a
period of time. The system model for transient stability analysis typically includes not only the transmission
network parameters, but also the dynamics data for the generators. Transient stability analysis is most often
used to determine if individual generating units will maintain synchronism with the power system following
a disturbance (typically a fault).
Extended Stability Analysis
Extended stability analysis deals with system stability beyond the generating units’ “first swing.” In addition to
the generator data required for transient stability analysis, extended stability analysis requires excitation system,
speed governor, and prime mover dynamic data. Often, extended stability analysis will also include dynamics
data for control devices such as tap changing transformers, switched capacitors, and relays. The addition of
these elements to the system model complicates the analysis, but provides comprehensive simulation of nearly
all major system components and controls. Extended stability analyses complement small signal stability
analyses by verifying the existence of persistent oscillations and establishing the magnitudes of power and/or
voltage oscillations.
Small Signal Stability Analysis
Small signal stability assesses the stability of the power system when subjected to “small” perturbations. Small
signal stability uses a linearized model of the power system which includes generator, prime mover, and control
device dynamics data. The system of nonlinear equations describing the system are linearized about a specific
operating point and eigenvalues and eigenvectors of the linearized system found. The imaginary part of each
eigenvalue indicates the frequency of the oscillations associated with the eigenvalue; the real part indicates
damping of the oscillation. Usually, small signal stability analysis attempts to find disturbances and/or system
conditions that can lead to sustained oscillations (indicated by small damping factors) in the power system.
Small signal stability analysis does not provide oscillation magnitude information because the eigenvalues only
indicate oscillation frequency and damping. Additionally, the controllability matrices (based on the linearized
system) and the eigenvectors can be used to identify candidate generating units for application of new or
improved controls (i.e., power system stabilizers and new or improved excitation systems).
Transient Analysis
Transient analysis involves the analysis of the system (or at least several components of the system) when
subjected to “fast” transients (i.e., lightning and switching transients). Transient analysis requires detailed
component information which is often not readily available. Typically only system components in the immediate
vicinity of the area of interest are modeled in transient analyses. Specialized software packages (most notably
EMTP) are used to perform transient analyses.
Operational Analyses
Several additional analyses used in the day-to-day operation of power systems are based on the results of the
analyses described above. Economic dispatch analyses determine the most economic real power output for each
generating unit based on cost of generation for each unit and the system losses. Security or contingency analyses
assess the system’s ability to withstand the sudden loss of one or more major elements without overloading the
remaining system. State estimation determines the “best” estimate of the real-time system state based on a
redundant set of system measurements.
The Power Flow Problem
Power flow analysis is fundamental to the study of power systems. In fact, power flow forms the core of power
system analysis. A power flow study is valuable for many reasons. For example, power flow analyses play a key
role in the planning of additions or expansions to transmission and generation facilities. A power flow solution
? 2000 by CRC Press LLC
is often the starting point for many other types of power system analyses. In addition, power flow analysis and
many of its extensions are an essential ingredient of the studies performed in power system operations. In this
latter case, it is at the heart of contingency analysis and the implementation of real-time monitoring systems.
The power flow problem (popularly known as the load flow problem) can be stated as follows:
For a given power network, with known complex power loads and some set of specifications or restrictions on
power generations and voltages, solve for any unknown bus voltages and unspecified generation and finally for
the complex power flow in the network components.
Additionally, the losses in individual components and the total network as a whole are usually calculated.
Furthermore, the system is often checked for component overloads and voltages outside allowable tolerances.
Balanced operation is assumed for most power flow studies and will be assumed in this chapter. Consequently,
the positive sequence network is used for the analysis. In the solution of the power flow problem, the network
element values are almost always taken to be in per unit. Likewise, the calculations within the power flow
analysis are typically in per unit. However, the solution is usually expressed in a mixed format. Solution voltages
are usually expressed in per unit; powers are most often given in kVA or MVA.
The “given network” may be in the form of a system map and accompanying data tables for the network
components. More often, however, the network structure is given in the form of a one-line diagram (such as
shown in Fig. 63.1).
Regardless of the form of the given network and how the network data are given, the steps to be followed
in a power flow study can be summarized as follows:
1. Determine element values for passive network components.
2. Determine locations and values of all complex power loads.
3. Determine generation specifications and constraints.
4. Develop a mathematical model describing power flow in the network.
5. Solve for the voltage profile of the network.
FIGURE 63.1 The one line diagram of a power system.
? 2000 by CRC Press LLC
6. Solve for the power flows and losses in the network.
7. Check for constraint violations.
Formulation of the Bus Admittance Matrix
The first step in developing the mathematical model describing the power flow in the network is the formulation
of the bus admittance matrix. The bus admittance matrix is an n×n matrix (where n is the number of buses
in the system) constructed from the admittances of the equivalent circuit elements of the segments making up
the power system. Most system segments are represented by a combination of shunt elements (connected
between a bus and the reference node) and series elements (connected between two system buses). Formulation
of the bus admittance matrix follows two simple rules:
1. The admittance of elements connected between node k and reference is added to the (k, k) entry of the
admittance matrix.
2. The admittance of elements connected between nodes j and k is added to the ( j, j) and (k, k) entries of
the admittance matrix. The negative of the admittance is added to the ( j, k) and (k, j) entries of the
admittance matrix.
Off nominal transformers (transformers with transformation ratios different from the system voltage bases
at the terminals) present some special difficulties. Figure 63.2 shows a representation of an off nominal turns
ratio transformer.
The admittance matrix mathematical model of an isolated off nominal transformer is:
(63.1)
where
–
Y
e
is the equivalent series admittance (referred to node j)
–
c is the complex (off nominal) turns ratio
–
I
j
is the current injected at node j
–
V
j
is the voltage at node j (with respect to reference)
Off nominal transformers are added to the bus admittance matrix by adding the corresponding entry of the
isolated off nominal transformer admittance matrix to the system bus admittance matrix.
Example Formulation of the Power Flow Equations
Considerable insight into the power flow problem and its properties and characteristics can be obtained by
consideration of a simple example before proceeding to a general formulation of the problem. This simple case
will also serve to establish some notation.
FIGURE 63.2 Off nominal turns ratio transformer.
I
I
YcY
-c*Y c Y
V
V
j
k
ee
ee
j
k
?
?
?
?
?
?
=
??
?
?
?
?
?
?
?
?
?
?
?
?
?2
? 2000 by CRC Press LLC
A conceptual representation of a one-line diagram for a four-bus power system is shown in Fig. 63.3. For
generality, a generator and a load are shown connected to each bus. The following notation applies:
–
S
G1
= Complex complex power flow into bus 1 from the generator
–
S
D1
= Complex complex power flow into the load from bus 1
Comparable quantities for the complex power generations and loads are obvious for each of the three other
buses.
The positive sequence network for the power system represented by the one-line diagram of Fig. 63.3 is
shown in Fig. 63.4. The boxes symbolize the combination of generation and load. Network texts refer to this
network as a five-node network. (The balanced nature of the system allows analysis using only the positive
sequence network; reducing each three-phase bus to a single node. The reference or ground represents the fifth
node.) However, in power systems literature it is usually referred to as a four-bus network or power system.
For the network of Fig. 63.4, we define the following additional notation:
–
S
1
=
–
S
G1
–
–
S
D1
Net complex power injected at bus 1
–
I
1
= Net positive sequence phasor current injected at bus 1
–
V
1
= Positive sequence phasor voltage at bus 1
The standard node voltage equations for the network can be written in terms of the quantities at bus 1
(defined above) and comparable quantities at the other buses.
FIGURE 63.3 Conceptual one-line diagram of a four-bus power system.
? 2000 by CRC Press LLC
(63.2)
(63.3)
(63.4)
(63.5)
The admittances in Eqs. (63.2) through (63.5),
–
Y
ij
, are the ijth entries of the bus admittance matrix for the
power system. The unknown voltages could be found using linear algebra if the four currents
–
I
1
…
–
I
4
were
known. However, these currents are not known. Rather, something is known about the complex power and
voltage at each bus. The complex power injected into bus k of the power system is defined by the relationship
between complex power, voltage, and current given by Eq. (63.6).
(63.6)
Therefore,
(63.7)
By substituting this result into the nodal equations and rearranging, the basic power flow equations for the
four-bus system are given as Eqs. (63.8) through (63.11)
(63.8)
FIGURE 63.4 Positive sequence network for the system of Fig. 63.3.
I YVYVYVYV
1 11 1 12 2 13 3 14 4
=+++
I YVYVYVYV
2 21 1 22 2 23 3 24 4
=+++
I YVYVYVYV
3 31 1 32 2 33 3 34 4
=+++
I YVYVYVYV
4 41 1 42 2 43 3 44 4
=+++
SVI
kkk
*
=
I
S
V
SS
V
k
k
*
k
*
Gk
*
Dk
*
k
*
==
?
S – S V YV YV YV YV
G1
*
D1
*
1
*
11 1 12 2 13 3 14 4
=+++
[]
? 2000 by CRC Press LLC
(63.9)
(63.10)
(63.11)
Examination of Eqs. (63.8) through (63.11) reveals that, except for the trivial case where the generation
equals the load at every bus, the complex power outputs of the generators cannot be arbitrarily selected. In
fact, the complex power output of at least one of the generators must be calculated last because it must take
up the unknown “slack” due to the, as yet, uncalculated network losses. Further, losses cannot be calculated
until the voltages are known. These observations are a result of the principle of conservation of complex power
(i.e., the sum of the injected complex powers at the four system buses is equal to the system complex power
losses).
Further examination of Eqs. (63.8) through (63.11) indicates that it is not possible to solve these equations
for the absolute phase angles of the phasor voltages. This simply means that the problem can only be solved
to some arbitrary phase angle reference.
In order to alleviate the dilemma outlined above, suppose
–
S
G4
is arbitrarily allowed to float or swing (in
order to take up the necessary slack caused by the losses) and that
–
S
G1
,
–
S
G2
, and
–
S
G3
are specified (other cases
will be considered shortly). Now, with the loads known, Eqs. (63.7) through (63.10) are seen as four simulta-
neous nonlinear equations with complex coefficients in five unknowns
–
V
1
,
–
V
2
,
–
V
3
,
–
V
4
, and
–
S
G4
.
The problem of too many unknowns (which would result in an infinite number of solutions) is solved by
specifying another variable. Designating bus 4 as the slack bus and specifying the voltage
–
V
4
reduces the problem
to four equations in four unknowns. The slack bus is chosen as the phase reference for all phasor calculations,
its magnitude is constrained, and the complex power generation at this bus is free to take up the slack necessary
in order to account for the system real and reactive power losses.
The specification of the voltage
–
V
4
decouples Eq. (63.11) from Eqs. (63.8) through (63.10), allowing calcu-
lation of the slack bus complex power after solving the remaining equations. (This property carries over to
larger systems with any number of buses.) The example problem is reduced to solving only three equations
simultaneously for the unknowns
–
V
1
,
–
V
2
, and
–
V
3
. Similarly, for the case of n buses, it is necessary to solve n-1
simultaneous, complex coefficient, nonlinear equations.
Systems of nonlinear equations, such as Eqs. (63.8) through (63.10), cannot (except in rare cases) be solved
by closed-form techniques. Direct simulation was used extensively for many years; however, essentially all power
flow analyses today are performed using iterative techniques on digital computers.
P-V Buses
In all realistic cases, the voltage magnitude is specified at generator buses to take advantage of the generator’s
reactive power capability. Specifying the voltage magnitude at a generator bus requires a variable specified in
the simple analysis discussed earlier to become an unknown (in order to bring the number of unknowns back
into correspondence with the number of equations). Normally, the reactive power injected by the generator
becomes a variable, leaving the real power and voltage magnitude as the specified quantities at the generator bus.
It was noted earlier that Eq. (63.11) is decoupled and only Eqs. (63.8) through (63.10) need be solved
simultaneously. Although not immediately apparent, specifying the voltage magnitude at a bus and treating
the bus reactive power injection as a variable results in retention of, effectively, the same number of complex
unknowns. For example, if the voltage magnitude of bus 1 of the earlier four bus system is specified and the
reactive power injection at bus 1 becomes a variable, Eqs. (63.8) through (63.10) again effectively have three
complex unknowns. (The phasor voltages
–
V
2
and
–
V
3
at buses 2 and 3 are two complex unknowns and the
angle δ
1
of the voltage at bus 1 plus the reactive power generation Q
G1
at bus 1 result in the equivalent of a
third complex unknown.)
S – SVYVYVYVYV
G2
*
D2
*
2
*
21 1 22 2 23 3 24 4
=+++
[]
S – S VYV YV YV YV
G3
*
D3
*
3
*
31 1 32 2 33 3 34 4
=+++
S – S V YV YV YV YV
G4
*
D4
*
4
*
41 1 42 2 43 3 44 4
=+++
[]
? 2000 by CRC Press LLC
Bus 1 is called a voltage controlled bus because it is apparent that the reactive power generation at bus 1 is
being used to control the voltage magnitude. This type of bus is also referred to as a P-V bus because of the
specified quantities. Typically, all generator buses are treated as voltage controlled buses.
Bus Classifications
There are four quantities of interest associated with each bus:
1. real power, P
2. reactive power, Q
3. voltage magnitude, V
4. voltage angle, δ
At every bus of the system two of these four quantities will be specified and the remaining two will be
unknowns. Each of the system buses may be classified in accordance with the two quantities specified. The
following classifications are typical:
? Slack bus—The slack bus for the system is a single bus for which the voltage magnitude and angle are
specified. The real and reactive power are unknowns. The bus selected as the slack bus must have a source
of both real and reactive power, because the injected power at this bus must “swing” to take up the
“slack” in the solution. The best choice for the slack bus (since, in most power systems, many buses have
real and reactive power sources) requires experience with the particular system under study. The behavior
of the solution is often influenced by the bus chosen. (In the earlier discussion, the last bus was selected
as the slack bus for convenience.)
? Load bus (P-Q bus)—A load bus is defined as any bus of the system for which the real and reactive power
are specified. Load buses may contain generators with specified real and reactive power outputs; however,
it is often convenient to designate any bus with specified injected complex power as a load bus.
? Voltage controlled bus (P-V bus)—Any bus for which the voltage magnitude and the injected real power
are specified is classified as a voltage controlled (or P-V) bus. The injected reactive power is a variable (with
specified upper and lower bounds) in the power flow analysis. (A P-V bus must have a variable source
of reactive power such as a generator or a capacitor bank.)
Generalized Power Flow Development
The more general (n bus) case is developed by extending the results of the simple four-bus example. Consider
the case of an n-bus system and the corresponding n+1 node positive sequence network. Assume that the buses
are numbered such that the slack bus is numbered last. Direct extension of the earlier equations (writing the
node voltage equations and making the same substitutions as in the four-bus case) yields the basic power flow
equations in the general form.
The Basic Power Flow Equations (PFE)
(63.12)
and
(63.13)
S P jQ V Y V
for k = 1, 2, 3, , n – 1
k
*
kkk
*
ki i
n
i=1
=? =
∑
…
PjQ VYV
nnn
*
ni i
n
i=1
?=
∑
? 2000 by CRC Press LLC
Equation (63.13) is the equation for the slack bus. Equation (63.12) represents n-1 simultaneous equations in
n-1 complex unknowns if all buses (other than the slack bus) are classified as load buses. Thus, given a set of
specified loads, the problem is to solve Eq. (63.12) for the n-1 complex phasor voltages at the remaining buses.
Once the bus voltages are known, Eq. (63.13) can be used to calculate the slack bus power.
Bus j is normally treated as a P-V bus if it has a directly connected generator. The unknowns at bus j are
then the reactive generation Q
Gj
and δ
j
because the voltage magnitude, V
j
, and the real power generation, P
gj
,
have been specified.
The next step in the analysis is to solve Eq. (63.12) for the bus voltages using some iterative method. Once
the bus voltages have been found, the complex power flows and complex power losses in all of the network
components are calculated.
Solution Methods
The solution of the simultaneous nonlinear power flow equations requires the use of iterative techniques for
even the simplest power systems. Although there are many methods for solving nonlinear equations, only two
methods are discussed here.
The Newton-Raphson Method
The Newton-Raphson algorithm has been applied in the solution of nonlinear equations in many fields. The
algorithm will be developed using a general set of two equations (for simplicity). The results are easily extended
to an arbitrary number of equations.
A set of two nonlinear equations are shown in Eqs. (63.14) and (63.15).
f
1
(x
1
, x
2
) = k
1
(63.14)
f
2
(x
1
, x
2
) = k
2
(63.15)
Now, if x
1
(0)
and x
2
(0)
are inexact solution estimates and ?x
1
(0)
and ?x
2
(0)
are the corrections to the estimates
to achieve an exact solution, Eqs. (63.14) and (63.15) can be rewritten as:
f
1
(x
1
+ ?x
1
(0)
, x
2
+ ?x
2
(0)
) = k
1
(63.16)
f
2
(x
1
+ ?x
1
(0)
, x
2
+ ?x
2
(0)
) = k
2
(63.17)
Expanding Eqs. (63.16) and (63.17) in a Taylor series about the estimate yields:
(63.18)
(63.19)
where the superscript, (0), on the partial derivatives indicates evaluation of the partial derivatives at the initial
estimate and h.o.t. indicates the higher order terms.
Neglecting the higher order terms (an acceptable approximation if ?x
1
(0)
and ?x
2
(0)
are small), Eqs. (63.18)
and (63.19) can be rearranged and written in matrix form.
fx ,x
f
x
x
f
x
x h.o.t. k
11
(0)
2
(0) 1
1
(0)
1
(0) 1
2
(0)
2
(0)
1
( )
+++=
?
?
?
?
??
fx ,x
f
x
x
f
x
x h.o.t. k
11
(0)
2
(0) 2
1
(0)
1
(0) 2
2
(0)
2
(0)
2
( )
+++=
?
?
?
?
??
? 2000 by CRC Press LLC
(63.20)
The matrix of partial derivatives in Eq. (63.20) is known as the Jacobian matrix and is evaluated at the initial
estimate. Multiplying each side of Eq. (63.20) by the inverse of the Jacobian yields an approximation of the
required correction to the estimated solution. Since the higher order terms were neglected, addition of the
correction terms to the original estimate will not yield an exact solution, but will provide an improved estimate.
The procedure may be repeated, obtaining sucessively better estimates until the estimated solution reaches a
desired tolerance. Summarizing, correction terms for the H5129th iterate are given in Eq. (63.21) and the solution
estimate is updated according to Eq. (63.22).
(63.21)
x
(H5129+1)
= x
(H5129)
+ ?x
(H5129)
(63.22)
The solution of the original set of nonlinear equations has been converted to a repeated solution of a system
of linear equations. This solution requires evaluation of the Jacobian matrix (at the current solution estimate)
in each iteration.
The power flow equations can be placed into the Newton-Raphson framework by separating the power flow
equations into their real and imaginary parts and taking the voltage magnitudes and phase angles as the
unknowns. Writing Eq. (63.21) specifically for the power flow problem:
(63.23)
The underscored variables in Eq. (63.23) indicate vectors (extending the two equation Newton-Raphson
development to the general power flow case). The (sched) notation indicates the scheduled real and reactive
powers injected into the system. P
(H5129)
and Q
(H5129)
represent the calculated real and reactive power injections based
on the system model and the H5129th voltage phase angle and voltage magnitude estimates. The bus voltage phase
angle and bus voltage magnitude estimates are updated, the Jacobian re-evaluated and the mismatch between
the scheduled and calculated real and reactive powers evaluated in each iteration of the Newton-Raphson
algorithm. Iterations are performed until the estimated solution reaches an acceptable tolerance or a maximum
number of allowable iterations is exceeded. Once a solution (within an acceptble tolerance) is reached, P-V bus
reactive power injections and the slack bus complex power injection may be evaluated.
Fast Decoupled Power Flow Solution
The fast decoupled power flow algorithm simplifies the procedure presented for the Newton-Raphson algorithm
by exploiting the strong coupling between real po wer and bus voltage phase angles and reactive power and bus
?
?
?
?
?
?
?
?
f
x
f
x
f
x
f
x
x
x
kfx,x
kfx,x
1
1
(0)
1
2
(0)
2
1
(0)
2
2
(0)
1
(0)
2
(0)
111
(0)
2
(0)
221
(0)
2
(0)
?
?
?
?
?
?
?
?
?
?
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≈
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( )
?
( )
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?
?
?
?
x
x
f
x
f
x
f
x
f
x
kfx,x
kfx,x
1
()
2
()
1
1
()
1
2
()
2
1
()
2
2
()
111
()
2
()
221
()
2
()
l
l
ll
ll
ll
ll
?
?
?
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≈
?
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( )
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( )
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?
?
?
δ
?
?δ
?
?
?
?δ
?
?
()
()
() ()
() ()
()
()
sched
sched
l
l
ll
ll
l
l
V
PP
V
QQ
V
PP
QQ
?
?
?
?
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≈
?
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( )
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? 2000 by CRC Press LLC
voltage magnitudes commonly seen in power systems. The Jacobian matrix is simplified by approximating the
partial derivatives of the real power equations with respect to the bus voltage magnitudes as zero. Similarly, the
partial derivatives of the reactive power equations with respect to the bus voltage phase angles are approximated
as zero. Further, the remaining partial derivatives are often approximated using only the imaginary portion of
the bus admittance matrix. These approximations yield the following correction equations:
(63.24)
(63.25)
where B′ is an approximation of the matrix of partial derviatives of the real power flow equations with respect
to the bus voltage phase angles and B″ is an approximation of the matrix of partial derivatives of the reactive
power flow equations with respect to the bus voltage magnitudes. B′ and B″ are typically held constant during
iterative process, eliminating the necessity of updating the Jacobian matrix (required in the Newton-Raphson
solution) in each iteration.
The fast decoupled algorithm has good convergence properties despite the many approximations used during
its development. The fast decoupled power flow algorithm has found widespread use since it is less computa-
tionally intensive (requires fewer computational operations) than the Newton-Raphson method.
Component Power Flows
The positive sequence network for components of interest (connected between buses i and j) will be of the
form shown in Fig. 63.5.
An admittance description is usually available from earlier construction of the nodal admittance matrix. Thus,
(63.26)
Therefore, the complex power flows and the component loss are:
(63.27)
(63.28)
(63.29)
FIGURE 63.5 Typical power system component.
?δ
() ()ll
= ′
[]( )
?B schedPP
?VQQB sched
() ()
= ′′
[]( )
?
I
I
YY
YY
V
V
i
j
ab
cd
i
j
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
SVIVYVYV
ij i i
*
iai bj
== +
[]
*
SVIVYVYV
ji j j
*
ici dj
== +
[]
*
SSS
loss ij ji
=+
? 2000 by CRC Press LLC
The calculated component flows combined with the bus voltage magnitudes and phase angles provide extensive
information about the operating point of the power systems. The bus voltage magnitudes may be checked to
ensure operation within a prescribed range. The segment power flows can be examined to ensure no equipment
ratings are exceeded. Additionally, the power flow solution may used as the starting point for other analyses.
An elementary discussion of the power flow problem and its solution is presented in this chapter. The power
flow problem can be complicated by the addition of further constraints such as generator real and reactive power
limits. However, discussion of such complications is beyond the scope of this chapter. The references provide
detailed development of power flow formulation and solution under additional constraints. The references also
provide some background in the other types of power system analysis discussed at the begining of the chapter.
Defining Terms
Bus admittance matrix: The nodal admittance matrix for an electric network resulting from a power system.
Newton-Raphson algorithm: An iterative technique for solving a system of nonlinear algebraic equations.
Fast-decoupled algorithm: An extension of the Newton-Raphson iterative technique for solving power flow
equations for a power system.
The power flow problem: A model of a power system, the solution of which provides the system voltage
profile and power flows from the sources to the loads.
Related Topics
3.2 Node and Mesh Analysis ? 3.5 Three-Phase Circuits
References
A. R. Bergen, Power Systems Analysis, Englewood Cliffs, N.J.: Prentice-Hall.
O. I. Elgerd, Electric Energy Systems Theory - An Introduction, 2nd ed., New York: McGraw-Hill.
J. D. Glover and M. Sarma, Power System Analysis and Design, 2nd ed., Boston, Mass.: PWS Publishing.
C. A. Gross, Power System Analysis, 2nd ed., New York: John Wiley & Sons.
W. D. Stevenson, Elements of Power System Analysis, 4th ed., New York: McGraw-Hill.
Further Information
The references provide clear introductions to the analysis of power systems. An excellent review of many issues
involving the use of computers for power system analysis is provided in Proceedings of the IEEE (Special Issue
on Computers in the Power Industry), July 1974. The quarterly journal IEEE Transactions on Power Systems
provides excellent documentation of more recent research in power system analysis.
63.2 Voltage Instability
Robert A. Schlueter and Nassar Alemadi
Voltage Stability Overview
Retention of voltage stability and viability is the ability of a power system to preserve the voltage at an operating
equilibrium under normal condition and to maintain an acceptable voltage at all buses after being subjected to a
disturbance. A loss of voltage viability occurs when voltage declines below acceptable levels but does not decline to
progressively lower values. A system loses voltage stability when a disturbance, changes in system operating condi-
tion, or increase in load demand causes a progressive and spreading drop in voltage [Kundur, 1994]. The incapability
of the power system to meet the reactive power demand is the main cause of voltage instability. The drop in voltage
results in increased network reactive losses due to (1) reduced shunt capacitive reactive supply and (2) increased
magnetic field due to increased current flow. The increased network losses results in (1) reduced reactive power
flow to the region that needs the most reactive supply and (2) exhaustion of the reactive reserves on generators,
? 2000 by CRC Press LLC
synchronous condensers, or SVCs causing loss of voltage control. This loss of voltage control can cause further
voltage drop and further increase in network reactive losses that produce a voltage collapse [Schlueter, 1998d].
Voltage instability has become the principal constraint on power system operation for many utilities. Thermal
(overheating) constraints or transient stability constraints were the principal limitations on power system
operation just 15 years ago. Low or high voltage limit violation (voltage viability) constraints and voltage
instability constraints have become the major operational limitation on many utilities. Voltage instability is a
unique problem because it can produce an uncontrollable, cascading instability that results in blackout for a
large region or an entire country. Major blackouts have affected the Pennsylvania, New Jersey, Maryland
Interconnection, the Western System Coordinating Council (WSCC) system, Florida, France, Sweden, and
Japan. A more complete list of voltage instabilities incidents is contained in [Taylor, 1994; NERC 1991]. Thermal,
overload voltage limit violation transient angle instabilities do not have the potential to cause the uncontrollable
cascading instability that affects such large regions as does voltage instability.
The modeling required and simulation tools used to accurately assess retention and loss of voltage stability
is reviewed in the next section. Voltage instability has been studied using both a loadflow (algebraic) and a
differential algebraic model, and both are discussed. The kinds, classes, and agents of voltage instability that
can develop are discussed in in the third section. A bifurcation is a sudden change in response, usually stable
to unstable, for a smooth, continuous, slow change in load or operating condition. Saddle node, Hopf, singu-
larity induced, and algebraic bifurcation are the kinds of bifurcation that have been observed on a power system
differential algebraic model, and all these different bifurcation have been associated with voltage instability.
Clogging and loss of control voltage instability are the two kinds of bifurcation in a loadflow (algebraic) model.
Methods for assessing proximity to voltage instability in the loadflow model are reviewed. A Voltage Stability
Security Assessment and Diagnosis Method [Schlueter, 1998d] is discussed that can answer the voltage instability
diagnostic questions of where, when, why, and proximity and cure for each equipment outage, transaction
combination, or both.
Proximity to voltage instability has also been studied in a differential algebraic model. It has been shown
that bifurcation sequences occur in a differential algebraic model that can include saddle node, Hopf, singularity
induced, or algebraic bifurcation [Zaborsky, 1993; Guo et al., 1994]. Instability in the dynamics can occur
before the bifurcation affects the algebraic model. It has been shown that saddle node bifurcation in a differential
algebraic model at equilibrium is a bifurcation in the loadflow model that includes both the algebraic submodel
and differential submodel at equilibrium. [Schlueter, 1998e; Liu, 1998]. In other cases, the bifurcation is solely
in the algebraic model and has no effects on generator dynamics (algebraic bifurcation) or is in the algebraic
model that produces very rapid changes in generator dynamics (singularity-induced bifurcation). The bifur-
cation in the algebraic equations is almost always associated with the ultimate blackout, even when saddle node
or Hopf bifurcation initiates the instability that results in blackout.
Voltage Stability Models and Simulation Tools
A differential algebraic model for a power system can be written as [Schlueter et al., 1994].
(63.30)
(63.31)
(63.32)
where:
·
x
1
= state of generator, automatic voltage regulator (AVR), power system stabilizer, field current
limiter, and armature current limiter on each generator in the system
p
1
= overexcitation limiter relay limits that disable the AVR and trip the generator, armature current
relay limits that trip generators
˙
,,xfxxp
11131
=
( )
˙
,,xfxxp
22232
=
( )
ofxxxp=
( )
31233
,,,
? 2000 by CRC Press LLC
·
x
2
= state of large induction motors, large thermostatic loads where temperature control is per-
formed; generic load that represents action of under load tap changers, switchable shunt
capacitors, aggregate of small induction motor load, and aggregate of small thermostatic load
models
p
2
= parameters of the load model
x
3
= voltage and phase of network power balance equations
p
3
= parameters of network under load tap changers and switchable shunt capacitors
f
1
( ) = model of generators, synchronous condensers, and FACTS devices and their controls
f
2
( ) = model of large induction motors; large thermostatic loads where temperature is controlled;
and generic load models that represent aggregated action of under load tap changes, switchable
shunt capacitors, smaller induction motors, and smaller thermostatic controlled load and their
controls
f
3
( ) = network algebraic model that can include load where temperature, energy, or voltage control
dynamics have no effect, under load tap changers, and switchable shunt capacitors
The loadflow model is:
(63.33)
or
(63.34)
where
Voltage instability is most accurately assessed by a differential algebraic model (Eq. (63.30-63.32), but can
generally be accurately assessed using the loadflow model (Eq. (63.34)). The loadflow model simulation is used
to screen for equipment outages, and for accurate assessment of retention or loss of voltage stability for those
instability problems identified via loadflow. The size and complexity of the loadflow model required to accu-
rately determine the steady-state equilibrium x
o
after one or more equipment outages, transfer and wheeling
transaction combinations, or both have grown from less than 1000 buses to above 15,000 buses for utilities
with voltage instability problems. The model has grown to include distribution system buses in the study system,
the transmission and subtransmission network electrically distant from the study system, as well as virtually
all reactive generation devices in the interconnection. The exact sequence of actions on (1) field current limiters
reduction of field current, (2) tap position changes of under load tap changers, and (3) switchable shunt
capacitors insertions have been shown to be quite important in accurately obtaining the equilibrium produced
by the simulation of the differential algebraic model. A quasi steady-state (QSS) approximation is an improve-
ment on loadflow, that incorporates the effects of the control delays on devices (1–3) without incorporating
effects of faster generator and load dynamics. QSS is being implemented by several software vendors as a means
fxxp
fxxp
fxxxp
1131
2232
31233
0
0
0
,,
,,
,,,
( )
=
( )
=
( )
=
fxp,
( )
= 0
x
x
x
x
p
p
p
p
fxp
f
f
f
=
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
( )
=
( )
( )
( )
?
?
?
?
?
?
?
?
?
?
?
?
1
2
3
1
2
3
1
2
3
,
? 2000 by CRC Press LLC
of accurately obtaining an equilibrium without a full time simulation. The size and complexity of mid-term
and long-term stability simulations that simulate the trajectory and determine the equilibrium also increased
in order to accurately simulate the response of a system to one or more equipment outages, transfer and
wheeling transaction combinations, or both. Timing of control actions must obviously be modeled in mid-
term and long-term simulations in order to accurately simulate the trajectory and determine if it converges
asymptotically or does not converge to an equilibrium or limit cycle. The actual tap position limits on under-
load tap changers, field current limits on field current limiters, and switchable shunt capacitor capacity limits
can have dramatic effects on loadflow, QSS, or mid-term and long-term simulation results. This is due to the
fact that loss of voltage control on these devices (1) eliminates actions that force reactive supply to areas with
reactive need or (2) choke off available reactive supply due to dramatic increases in I
2
x losses and shunt capacitive
supply withdrawal that occur with voltage drop that results without proper voltage control.
Kinds, Classes, and Agents of Voltage Instability [Schlueter, 1994]
Voltage instability can be classified in terms of the time frame required for voltage instability to develop. Long-
term voltage instability is defined as developing over a time frame of 1 to 20 min [Taylor, 1994] and transient
voltage instability is defined as developing over a time frame of 1 to 10 s [Taylor, 1994]. Another method of
classifying voltage instability as short term or long term is to determine the dynamics that play the central role
in the development of voltage instability [Van Cutsem et al., 1998]. Short-term voltage instability [Van Cutsem
et al., 1998] occurs due to (1) motor stalling, (2) motor stalling after a short circuit occurs and is cleared,
(3) instability in generator flux decay dynamics when the AVR is disabled by an over-excitation limiter relay
[Schlueter, 1998e], and (4) oscillations in generator dynamics, [Sauer, 1992] or between groups of generators
or between generators and induction motors [Van Cutsem et al., 1998]. All of these short-term stability problems
can be accurately captured in mid-term or long-term simulation but cannot be captured in a loadflow or QSS
simulation [Van Cutsem, 1998]. Action of tap changers, switchable shunt capacitors, generic load change,
thermostatic load change, generator field current limiters and protection are long-term dynamics because their
response can take 1 to 20 min or longer [Van Cutsem, 1998]. In many cases, these long-term dynamics result
in instability in short-term dynamics that ultimately produce lack of solution or instability in the algebraic
equations. Loadflow and QSS models can capture most long-term voltage instability problems. The lack of a
loadflow solution can indicate that there exists no steady-state equilibrium to the differential algebraic model
after some equipment outage, transaction combination, or both. Diagnostics can be applied to determine if
voltage instability is the cause of the lack of an equilibrium solution [Schlueter, 1998d]. Voltage instability is
known to have occurred if a cure can be found based on these diagnostics that corrects the lack of solution.
The system matrix
(63.35)
can be used to test for stability of the equilibrium if one can be computed by the loadflow. Thus, obtaining a
loadflow solution does not guarantee stability of the dynamics at equilibrium. Bifurcation is a discontinuous
change in the qualitative behavior of the dynamics as some parameter changes slowly, continuously, and
smoothly and generally implies a change from a stable to an unstable response. There are many different kinds
of bifurcation that have been observed to occur in a power system model in both the long-term and short-
term dynamics, as noted in the next paragraph. Different kinds of bifurcation can occur in the same dynamics
in the same subsystems or in different subsystems in a power system model. Bifurcation subsystem analysis
[Ben-Kilani, 1997] is being used to identify all of the different subsystems (classes) experiencing each kind of
bifurcation. Classes for each kind of bifurcation are the specific short-term (inertial dynamics of induction
motor or generators, flux decay and excitation control dynamics of generators or induction motors) or long-
term dynamics (tap changer, switchable shunt capacitor, thermostatic load control dynamics, field current
limiter dynamics) that experience or produce the particular kind of bifurcation. The subsystem experiencing
A
ff
ff
f
f
fff
xx
xx
x
x
xxx
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
[][ ]
?
11
22
1
2
3
1
33
12
12
3
3
312
? 2000 by CRC Press LLC
the particular kind of bifurcation indicates whether it is short-term or long-term instability as noted in [Van
Cutsem, et al. 1998]. It may be that the long-term dynamic changes bring about the particular bifurcation in a
short-term dynamics subsystem, as noted above. It is necessary to specify the time frame (long term or short
term) for instability to develop, the kind of bifurcation (saddle node, Hopf, singularity induced), the class of
bifurcation, and the agent of that particular kind and class to describe a particular voltage instability event.
Agents that experience bifurcation are the specific devices in a class that are in specific locations that experience
or produce the particular kind of bifurcation. Bifurcation subsystem analysis is identifying the agents for each
kind and class of bifurcation. Only the generic kinds of bifurcations that have been studied extensively will now
be discussed, although a more extensive list of bifurcation kinds observed in power systems is given in [Schlueter,
1994].
Saddle node bifurcation occurs if both J and A have a zero eigenvalue as p changes toward bifurcation value p*
and certain transversality and genericity conditions hold [Zaborsky, 1994]. Saddle node bifurcation occurs
when the region of attraction, where the trajectory converges to the stable equilibrium, becomes a null set and
the dynamics associated with the bifurcation eigenvalue of A evaluated at the equilibrium become infinitely
slow and then unstable. Saddle node bifurcation occurs when a loadflow experiences bifurcation [Canizares
et al., 1992]. Hopf bifurcation occurs if the real part of a complex pair of eigenvalues cross the jω axis as p
changes toward bifurcation values p* if certain transversality and genericity conditions hold [Zaborsky, 1994].
The Hopf bifurcation occurs when a stable or unstable limit cycle (oscillation) is formed. Singularity induced
bifurcation occurs when f
3x
3
has an eigenvalue that approaches zero and the eigenvalue of A approaches infinity,
becomes negative infinity, and then approaches zero [Zaborsky, 1994]. Algebraic bifurcation occurs when f
3x
3
has an eigenvalue approaching zero and A has no eigenvalue experiencing bifurcation.
There is often a sequence of bifurcations associated with a particular voltage instability event. One such
sequence is a Hopf followed by singularity-induced bifurcation [Sauer et al., 1992] as shown in Figs. 63.6 and
63.7. Figure 63.6 shows the PV curve stress test and Fig. 63.7 shows the pair of bifurcating eigenvalues that
experience the sequence of bifurcations. Another bifurcation sequence is saddle node followed by singularity-
induced bifurcation, as shown in Figs. 63.8 and 63.9. Figure 63.8 shows the Q-V curve stress test and Fig. 63.9
shows the sequence of bifurcations. It is proven that after the saddle node bifurcation, the unstable internal
generator voltage decline can produce a dynamic Q-V stress test that inevitably leads to the singularity induced
biburcation and blackout [Schlueter, 1998e]. Motor stalling is a saddle node bifurcation that can also produce
FIGURE 63.6 PV curve on an example system.
? 2000 by CRC Press LLC
a dynamically administered stress test that leads to singularity-induced bifurcation and blackout [Van Cutsem,
1998].
FIGURE 63.7 Bifurcation sequence of Hopf (A), node-focus (B), singularity-induced (C) bifurcation produced on an
example system by the PV curve in Fig. 63.6.
FIGURE 63.8 Q-V curve produced on an example system.
? 2000 by CRC Press LLC
Proximity to Voltage Instability
This proximity to voltage instability is initially assessed using a loadflow model (Eq. (63.34)). If a particular
contingency, transfer, or wheeling combination, or both is found to experience voltage instability based on a
loadflow model, it will be confirmed using a mid-term or long-term simulation. Using a loadflow has been
found to be quite a satisfactory model for accurate assessment of proximity to long-term voltage instability.
The loadflow model is f(x,p) = 0, where x is the n dimension state of the model and is of the same dimension
as f(x,p) and p is an m vector of parameters that can change and produce bifurcation or instability if p changes
smoothly and continuously. The implicit function theorem can indicate when a solution exists and the solution
is unique.
Theorem (Implicit function theorem) [Apostol, 1974]
Let f = (f
1
,…f
n
) be a vector valued defined on an open set S in R
n+m
with values in R
n
. Suppose f∈C￠ or that
f is continuously differentiable on S. Let (x
0
;p
0
) be a point in S for which f(x
0
;p
0
) = 0 and for which the
determinant of the n × n jacobian det[f
x
(x
0
;p
0
)] ≠ 0. Then there exists an m-dimensional open set P
0
containing
p
0
and one, and only one, vector-valued function g, defined on P
0
and having values in R
n
, such that
(1) g∈C′on P
0
(2) g(p
0
) = x
0
(3) f(g(p);p) = 0 for every p in P
0
.
When the jacobian is nonsingular at a point in S, the implicit function theorem indicates there exist solutions
that are unique for all p∈P
0
. When a solution exists, the system is stable when all eigenvalues of f
x
(x
0
, p
0
) are
positive or unstable, depending on whether there are non-positive eigenvalues of the jacobian f
x
(x
0
, p
0
). When
no solution exists at p
0
, the system is considered unstable. The vector p can be changed, usually reduced, from
p
0
until solutions exist. Singularity of the loadflow jacobian can be used to detect the point of voltage instability
p* < p
0
where instability is initiated. When the det[f
x
(x*, p*)] is zero (or the jacobian f
x
(x*, p*) is singular),
FIGURE 63.9 Bifurcation sequence of saddle node (A), followed by singularity-induced bifurcation (B) produced by the
Q-V curve in Fig. 63.8.
? 2000 by CRC Press LLC
the implicit function theorem does not provide any information but it may imply no solution x* = x(p*) exists
at p* or there are multiple solutions x*
i
(p*).
Proximity Indices
A number of indices have been developed to test for loadflow bifurcation when p is changed smoothly and
continuously in some direction n in R
m
:
p = p
0
+ kn (63.36)
via change in k until det[f
x
(x*,p*)] = 0. The indices come from tracking the miminum eigenvalue λ*
i
[IEEE,
1993] using
(63.37)
where λ *
i
(k) = min
i
[λ
i
(k)] and u
i
(k) is a right eigenvector or alternatively the minimum singular value σ*
i
(k)
obtained [IEEE, 1993] using
(63.38)
where Σ(k) = diag[σ
1
(k), σ
2
(k), L, σ
n
(k)] and σ*
i
(k) =
min
i
[σ
i
(k)]. The singular values σ
i
(k) are the eigenvalues
of
(63.39)
and satisfy
(63.40)
(63.41)
where v
i
(k) and w
i
(k) are the right and the left singular vectors of σ
i
(k) and are columns of matrices V
σ
(k) and
W
σ
(k) above.
The Q-V and P-V curves [IEEE, 1993] are particular scalar (m = 1) proximity measures where:
1. For a Q-V curve, the direction n is a unit vector in Eq. (63.36) where the voltage at a bus i is the only
nonzero element, k is the real valued negative number that starts at zero and decreases, and –Q
i
(V
i
) is
the reactive load that is added at bus i for each value of V
i
. The curve Q
i
(V
i
), shown in Fig. 63.8 is the
reactive injection at bus i obtained by (1) changing bus type from a load (PQ) bus to a generator bus
(PV) and (2) reducing the voltage V
i
until Q
i
(V
i
) reaches a minimum at V
i
min
with maximum added load
–Q
i
(V
i
min
) = –Q
i
min
≥ 0. The value of (V
i
min
, Q
i
min
) defines the minimum of the Q-V curve when ?Q
i
/?V
i
=
0, that corresponds to the bifurcation point (x*, p*).
2. The P-V curve, shown in Fig. 63.6, can add active power load at a bus i or at several load buses
simultaneously
p = p
0
+ kn
load
(63.42)
fxkpkuk k k
x i ii
() () ()
=
() ()
, λ u
fxkpk fxkpk Wk kV k
x
T
x
T
( ) ( )( ) ( ) ( )( )
=
( ) ( ) ( )
,,
σσ
Σ
fxkpkf xkpk
xx
T
( ) ( )( ) ( ) ( )( )
,,
fxkpkvk kwk
xiii
( ) ( )( ) ( )
=
( ) ( )
, σ
wkfxkpk kvk
i
T
xii
T
( ) ( ) ( )( )
=
( ) ( )
, σ
? 2000 by CRC Press LLC
and pick up that power at several generators
g = g
0
+ kn
gen
(63.43)
where n is made up of n
gen
and n
load
and both are participation vectors where
Σ
?
i
n
i
= 1. A P-V curve can
also be computed for transfer power from one set of generators with generation g* to another set of
generators with generation
?
g.
(63.44)
(63.45)
Note n
gen
, n
load
, n*, and
?
n are unit vectors where one or more elements are nonzero and
Σ
?
i
n
i
= 1. The
P-V curve plots voltage at some bus i for change in k = P
system
where it represents system power load
change or k = P
transfer
represents the total power transfer change.
3. Optimization-based methods have been used to calculate Q-V and P-V curves [Reppen et al., 1991].
These scalar optimization-based methods optimize performance index Q
i
to produce a Q-V curve with
loadflow equality constraints
F(x, p, u) = 0 (63.46)
and inequality constraints on voltage controls u, states, x, and parameter p. These controls can include
under load tap changer tap position, switchable shunt capacitor susceptance, and possibly generator
excitation voltage control setpoints. The P-V curve computed by loadflow for varying k = P
system
or P
transfer
has all or most of these controls fixed rather than optimizing their values. The P-V curve computed via
an optimal power flow program would optimize k for a particular transfer or wheeling transaction with
the same loadflow model, same controls u, and inequality constraints on controls u, states, and param-
eters used in computing the Q-V curve. The particular transfer or wheeling transaction is defined via
specification of n
gen
, n
load,
n*, and
?
n. In [Van Cutsem, 1991], a scalar optimization-based method was
used to maximize the reactive power stress when n in Eq. (63.36) was a unit vector with several nonzero
elements. This optimal power flow generalized Q-V curve allows added reactive load at several buses in
the participation factor normal direction, rather than just one as in a typical loadflow-based Q-V curve
calculation. The approach used in [Van Cutsem, 1991] eliminates the active power and phase angle
relationship, using active power generation as control, and imposes reactive power limits on the gener-
ators. [Dobson et al., 1991] was first to develop a vector optimization-based method that optimizes the
normal direction vector n and the loading factor k in Eq. (63.36). The vector n is in the right eigenvector
direction w
i
of the bifurcating eigenvalue if the loadflow is continuously differentiable. This method and
extensions [IEEE, 1993] computes the smallest change in the active or reactive load power and thus flow
that produces saddle node bifurcation. The proximity measure |p*–p
0
| to saddle node bifurcation, where
p
0
and p* represent the current load power and the bifurcation value of load power, respectively, was
first noted by Dobson [1991].
Voltage Stability Security Assessment and Diagnosis
All of these methods used [IEEE, 1993; Reppen, 1991; Dobson, 1991; and Van Cutsem, 1991] assess bifurcation
in a single mode due to continuous, smooth, scalar, or vector parameter variation. The methods are not practical
for assessing voltage instability or stability because the loadflow most often has no solution when voltage
instability occurs, and all these methods require loadflow solutions to make any assessment of whether voltage
instability occurs or has not occurred. This is true because these methods cannot determine whether voltage
instability, algorithmic convergence difficulties, or round-off error is the reason for the lack of solution. These
methods [IEEE, 1993; Reppen, 1991; Dobson, 1991; and Van Cutsem, 1991] can be viewed as based on implicit
function theorem as long as the model is continuously differentiable. Implicit function theory and bifurcation
gg kn** *=+
0
?? ?
gg kn=?
0
? 2000 by CRC Press LLC
theory assumptions are both violated for the case when a loadflow does solve after discontinuous parameter
change because the parameter variation is not continuous and smooth and the power system model may not
be continuously differentiable at the point (x
0
, p
0
). The P-V curve, or Q-V curve, or eigenvalues and eigenvectors
could be computed and used to assess proximity to voltage instability after each equipment outage or discon-
tinuous parameter change when a loadflow solution exists to establish whether the solutions is stable or unstable
at values of p above p
0
. The computation of the P-V curve, Q-V curve, or eigenvalues and eigenvectors requires
significant computation and is not practical for screening thousands of contingencies for voltage instability or
for assessing proximity to instability although they are used to assess stability and proximity to instability after
a few selected contingencies. These methods also do not explicitly take into account the many discontinuities
in the model and eigenvalues that occur for continuous parameter and discontinuous parameter changes. In
many cases, the eigenvalue changes due to discontinuities is virtually all the change that occurs in an eigenvalue
that approaches instability [IEEE, 1993] and the above methods have particular difficulty in such cases. The
above methods cannot assess the agents that lose voltage instability for a particular event and cannot diagnose
a cure when the loadflow has no solution for an equipment outage, wheeling or transaction combination, or
both. These methods can provide a cure when a loadflow solution exists but its capabilities have not been
compared to the f Security Assessment and Diagnosis proposed cure. The Voltage Stability
Security Assessment and Diagnosis (VSSAD) [Schlueter, 1998d] overcomes the above difficulties because:
1. It determines the number of discontinuities in any eigenvalue that have already occurred due to generator
PV to load PQ bus type changes that are associated with an eigenvalue compared to the total number
that are needed to produce voltage instability when the eigenvalue becomes negative. The eigenvalue is
associated with a coherent bus group (voltage control area) [Schlueter, 1998a; f]. The subset of generators
that experience PV-PQ bus type changes (reactive reserve basin) for computing a Q-V curve at any bus
in that bus group are proven to capture the number of discontinuities in that eigenvalue [Schlueter,
1998a; f]. An eigenvalue approximation for the agent, composed of the test voltage control area where
the Q-V current is computed and its reactive reserve basin, is used to theoretically justify the definitions
of a voltage control area and the reactive reserve basin of an agent. The VSSAD agents are thus proven
to capture eigenvalue structure of the loadflow jacobian evaluated at any operating point (x
0
, p
0
). The
reactive reserve on generators in each voltage control area of a reactive reserve basin is proven to measure
proximity to each of the remaining discontinuities in the eigenvalue required for bifurcation.
2. It can handle strictly discontinuous (equipment outage or large transfer or wheeling transaction changes)
or continuous model or parameter change (load increase, transfer increases, and wheeling increases)
whereas the above methods are restricted to continuous changes to assess stability or instability at a
point p
0
.
3. It can simultaneously and quickly assess proximity to voltage instability for all agents where each has a
bifurcating eigenvalue. Proximity to instability of any agent is measured by assessing (1) the percentage
of voltage control areas containing generators in a reactive reserve basin with non-zero reserves, and
(2) the percentage of base case reactive reserves remaining on reactive reserve basin voltage control areas
that have not yet exhausted reserves [Schlueter, 1998b; f].
4. It can assess the cure for instability for contingencies that do not have a solution. The cure can be either
(1) adding needed reactive reserve on specific generators to obtain a solution that is voltage stable,
(2) adding reactive supply resources needed in one or more agents, or (3) the reduction in generation
and load in one or more agents or between one or more agents to obtain a solution and assure that it
is a stable solution. These cures can be obtained in an automated fashion [Schlueter, 1998b; f]. The
diagnosis can also indicate if the lack of a solution is due to convergence difficulties or round-off error
if the diagnosis indicates the contingency combination does not produce sufficient network reactive
losses to cause instability or any agent.
5. It can provide operating constraints or security constraints on each agent’s reactive reserve basin reserves
that prevent voltage instability in an agent in a manner identical to how thermal constraints prevent
thermal overload on a branch and voltage constraints prevent bus voltage limit violation at a bus
[Schlueter, 1998c; f].
? 2000 by CRC Press LLC
6. The reactive reserve basin operating constraints allow optimization that assures that correcting one
voltage instability problem due to instability in one or more agents will not produce other voltage stability
problems in the rest of the system [Schlueter, 1998c; f].
7. The reactive reserve basin constraints after an equipment outage and operating change combination
allows optimization of transmission capacity that specifically corrects that particular equipment outage
and transaction change induced voltage instability with minimum control change [Schlueter, 1998c; f].
8. It requires very little computation per contingency and can find multiple contingencies that cause voltage
instability by simulating only a small percentage of the possible multiple contingencies [Schlueter, 1998d].
Kinds of Loadflow Instability
Two kinds of voltage instability have been associated with a loadflow model: loss of control voltage instability
and clogging voltage instability [Schlueter, 1998d]. Loss of control voltage instability is caused by exhaustion
of reactive power supply that produces loss of voltage control on some of the generators or synchronous
condensers. Loss of voltage control on these reactive supply devices implies both lack of any further reactive
supply from these devices and loss of control of voltage that will increase network reactive losses that absorb
a portion of the flow of reactive power supply and prevent it from reaching the subregion needing that reactive
supply. Loss of voltage control develops because of equipment outages (generator, transmission line, and
transformer), operating condition changes (wheeling, interchange, and transfer transactions), and load/gener-
ation pattern changes. Loss of control voltage instability occurs in the subtransmission and transmission system
[Schlueter, 1998d]. It produces either saddle node or singularity-induced bifurcation in a differential algebraic
model. On the other hand, clogging develops because of increasing reactive power losses, and switching shunt
capacitors and tap changers reaching their limits. These network reactive losses, due to increasing magnetic
field and shunt capacitive supply withdrawal, can completely block reactive power supply from reaching the
subregion with need [Schlueter, 1998d]. Clogging voltage instability can produce algebraic bifurcation in a
differential algebraic model. The VSSAD method can diagnose whether the voltage instability occurs due to
clogging or loss of control voltage instability for each equipment outage, transaction combination, or both that
have no solution.
Theoretical Justification of the Diagnosis in VSSAD
A bifurcation subsystem analysis has been developed that theoretically justifies the diagnosis performed by
[Schlueter, 1997; 1998a; b; d; f]. This bifurcation subsystem analysis for a loadflow model attempts to break
the loadflow model into a subsystem model and external model
(63.47)
and to break the state x into two components where x
s
is the dimension of f
s
(x
s
, x
e
, p) = O
n
i
.
The bifurcation occurs at p* = p
o
+ μ*
o
n when
(63.48)
fx x p
fxxp
fxxp
O
O
se
sse
ese
n
n
,,
,,
,,
( )
=
( )
( )
?
?
?
?
?
?
?
?
=
?
?
?
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?
?
?
?
1
2
x
x
x
s
e
=
?
?
?
?
?
?
?
?
()
?
?
()
?
?
()
?
?
()
?
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()
()
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=
()
(
(
?
?
?
?
f
x
x*,p *
f
x
x*,p *
f
x
x*,p *
f
x
x*,p *
up*
wp*
p*
u p
w p
s
s
s
e
e
s
e
e
i
i
i
i
i
λ
? 2000 by CRC Press LLC
The vector is the right eigenvector of eigenvalue λ
i
(p*) = 0 at bifurcation point p*. A bifurcation
subsystem exists if two conditions hold:
(63.49)
(63.50)
The first condition is called the bifurcation subsystem condition and the second is called the geometric
decoupling condition. Finding a bifurcation subsystem for any bifurcation of the full system model requires
finding the combination of correct dimension, correct subset of equations, and correct subset of variables such
that the subsystem experiences the bifurcation (Eq. (63.49)) of the full system model (Eq. (63.48)) but also
produces that bifurcation since the external model is completely uncoupled from the bifurcation subsystem in
the direction of the right eigenvector (Eq. (63.50)). The right eigenvector is an approximation of the center
manifold at bifurcation, and the center manifold is the subsystem that actually experiences the bifurcation and
is obtained via a nonlinear transformation of the model. The expectation of finding a bifurcation subsystem
for any loadflow bifurcation, noting the above requirements for identifying such a bifurcation subsystem, is
that the difficulty in finding a bifurcation subsystem would be great even though one may exist for some
bifurcations. The results in [Schlueter, 1998b; f] prove that one cannot only describe the bifurcation subsystem
(where) for every clogging voltage instability and for every loss of control voltage instability, but also can
theoretically establish diagnostic information on when, proximity, and cure for a specific bifurcation in a specific
bifurcation subsystem for clogging or for loss of control voltage instability [Schlueter, 1998b; f].
The analysis establishes that:
1. The real power balance equations are a bifurcation subsystem for angle instability when the loadflow
model is decoupled ( and are assumed null) [Schlueter, 1998b; f].
2. The reactive power balance equations are a bifurcation subsystem for voltage instability when the
loadflow model is assumed decoupled [Schlueter, 1998b; f].
3. A voltage control area is the bifurcation subsystem (agent) for clogging voltage instability. The agent is
vulnerable to voltage instability for loss of generation in the agent, line outage in the agent boundary,
or increased real and reactive flow across the agent boundary based on analysis of the lower bound
approximation of the eigenvalue associated with that agent. The cure for clogging voltage instability in
this agent is to reduce the real and reactive flow across the boundary of the agent [Schlueter, 1998b; f].
4. A voltage control area and its associated reactive reserve basin are the bifurcation subsystem (agent) for
loss of control voltage instability. The agent is vulnerable to voltage instability for loss of generation in
the agent, line outages, transfer or wheeling transactions that reduce reactive reserve basin reserves based
on analysis of the lower bound approximation of the eigenvalue associated with that agent. The cure for
voltage instability in the agent is to add reactive reserves on the reactive reserve basin via capacitor
insertion, generator voltage setpoint changes on reactive reserve basin generators, or reverse tap position
changes on underload tap changers [Schlueter, 1998b; f].
5. The percentage of reserves unexhausted in the reactive reserve basin is theoretically justified as a proximity
measure for clogging instability in any clogging voltage instability agent. The percentage of voltage control
areas in a reactive reserve basin with unexhausted reactive reserve is theoretically justified as a proximity
measure for each loss of control voltage instability agent [Schlueter, 1998b; f].
6. Exhaustion of reactive reserves in a particular locally most vulnerable agent’s reactive reserve basin causes
cascading exhaustion of reactive reserves and loss of control voltage instability in agents with successively
up
wp
i
i
*
*
( )
( )
?
?
?
?
?
?
?
?
( ) ( )
=
f
x
xp up
s
s
i
*, * * 0
?
?
?
?
?
?
( )
=
?
f
x
f
x
f
x
up
s
e
e
e
e
s
i
1
0*
dP
dV
dQ
dθ
? 2000 by CRC Press LLC
larger reactive reserve basins. This partially explains why voltage collapse occurs [Schlueter, 1998a; d; f]
which is a cascading loss of stability in several agents.
The automated diagnostic procedures in VSSAD are thus theoretically justified via this bifurcation subsystem
analysis.
Future Research
Research is needed to:
1. Develop improved nonlinear dynamic load models that are valid at any particular instant and that are
valid when voltage decline is severe. The lack of accurate load models makes it difficult to accurately
simulate the time behavior and/or assess the cause of the voltage instability. The lack of knowledge of
what constitutes an accurate load model makes accurate postmortem simulation of a particular blackout
a process of making trial and error assumptions on the load model structure to obtain as accurate a
simulation as possible that conforms with time records of the event. Accurate predictive simulation of
events that have not occurred is very difficult [Taylor, et al. 1998].
2. Explain (a) why each specific cascading sequence of bifurcations inevitably occurs in a differential
algebraic model, and (b) the dynamic signature associated with each bifurcation sequence. Work is
underway to explain why instability in generator and load dynamics can inevitably cause a singularity-
induced bifurcation to occur. The time signature for singularity-induced bifurcation changes dependence
on why it occurs is discussed in [Schlueter, 1998e; Liu, 1998].
3. Extend bifurcation subsystem analysis to the differential algebraic model and link the bifurcation sub-
system in a differential algebraic model, to those obtained in the loadflow model. The bifurcation
subsystems for different Hopf and saddle node bifurcations can explain why the subsystem experiences
instability, as well as how to prevent instability as has been possible for bifurcation subsystems in the
algebraic model. Knowledge of bifurcation subsystems in the algebraic model may assist in identifying
bifurcation subsystems in the differential algebraic model.
4. Develop a protective or corrective control for voltage instability. A protective control would use con-
straints on the current operating condition for contingencies predicted to cause voltage instability if they
occurred. These constraints on the current operation would prevent voltage instability if and when the
contingency occurred. A corrective control would develop a control that correct the instability in the
bifurcation subsystems experiencing instability only after the equipment outages or operating changes
predicted to produce voltage instability have occurred. The implementation of the corrective control
requires a regional 5-s updated data acquisition system and control implementation similar to that used
in Electricité de France and elsewhere in Europe.
Defining Terms
Power system stability: The property of a power system that enables it to remain in a state of operating
equilibrium under normal operating conditions and to converge to another acceptable state of equilib-
rium after being subjected to a disturbance. Instability occurs when the above is not true or when the
system loses synchronism between generators and between generators and loads.
Small signal stability: The ability of the power system to maintain synchronism under small disturbances
[Kundur, 1994].
Transient stability: The ability of a power system to maintain synchronism for a severe transient disturbance
[Kundur, 1994].
Rotor angle stability: The ability of the generators in a power system to remain in synchronism after a severe
transient disturbance [Kundur, 1994].
Voltage viability: The ability of a power system to maintain acceptable voltages at all buses in the system
after being subjected to a disturbance. Loss of viability can occur if voltage at some bus or buses are
below acceptable levels [Kundur, 1994]. Loss of viability is not voltage instability.
? 2000 by CRC Press LLC
Voltage stability: The ability of the combined generation and transmission system to supply load after a
disturbance, increased load, or change in system conditions without an uncontrollable and progressive
decrease in voltage [Kundur, 1994]. Loss of voltage instability may stem from the attempt of load dynamics
to restore power consumption beyond the capability of the combined transmission and generation system.
Both small signal and transient voltage instability can occur.
Voltage collapse: An instability that produces a cascading (1) loss of stability in subsystems, and/or (2) outage
of equipment due to relaying actions.
Bifurcation: A sudden change in system response from a smooth, continuous, slow change in parameters p.
References
T.M. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley Publishing, 1974.
K. Ben-Kilani, Bifurcation Subsystem Method and its Application to Diagnosis of Power System Bifurcations
Produced by Discontinuities, Ph.D. Dissertation, Michigan State University, August 1997.
C.A. Canizares, F.L. Alvarado, C.L. DeMarco, I. Dobson, and W.F. Long, Point of collapse methods applied to
AC/DC power system, IEEE Trans. on Power System, 7, 673–683, 1992.
I. Dobson and Liming Lu, Using an iterative method to compute a closest saddle node bifurcation in the load
power parameter space of an electric power system, in Proceedings of the Bulk Power System Voltage
Phenomena. II. Voltage Stability and Security, Deep Creek Lake, MD, 1991.
T.Y. Guo and R.A. Schlueter, Identification of generic bifurcation and stability problems in a power system
differential algebraic model, IEEE Trans. on Power Systems, 9, 1032–1044, 1994.
IEEE Working Group on Voltage Stability, Suggested Techniques for Voltage Stability Analysis, IEEE Power
Engineering Society Report, 93TH0620-5PWR, 1993.
P. Kundur, Power System Stability and Control, Power System Engineering Series, McGraw-Hill, 1994.
S. Liu, Bifureation Dynamics as a Cause of Recent Voltage Collapse Problems on the WSCC System, Ph.D.
Dissertation, Michigan State University, East Lansing, MI, 1998.
N.D. Reppen and R.R. Austria, Application of the optimal power flow to analysis of voltage collapse limited
power transfer, in Bulb Power System Voltage Phenomena. II. Voltage Stability and Security, August 1991,
Deep Creek Lake, MD.
Survey of Voltage Collapse Phenomena: Summary of Interconnection Dynamics Task Force’s Survey on Voltage
Collapse Phenomena, Section III Incidents, North American Reliability Council Report, August, 1991.
P.W. Sauer, C. Rajagopalan, B. Lesieutre, and M.A. Pai, Dynamic Aspects of voltage/power characteristics, IEEE
Trans. on Power Systems, 7, 990–1000, 1992.
R.A. Schlueter, K. Ben-Kilani, and U. Ahn, Impact of modeling accuracy on type, kind, and class of stability
problems in a power system model, Proceedings of the ECC & NSF International Workshop on Bulk Power
System Voltage Stability, Security and Control Phenomena-III, pp. 117–156. August 1994.
R.A. Schlueter, A structure based hierarchy for intelligent voltage stability control in planning, scheduling, and
stabilizing power systems, Proceedings of the EPRI Conference on Future of Power Delivery in the 21st
Century, La Jolla, CA, November 1997.
R.A. Schlueter and S. Liu, Justification of the voltage stability security assessment as an improved modal analysis
procedure, Proceedings of the Large Engineering System Conference on Power System Engineering,
pp. 273–279, June 1998.
R.A. Schlueter, K. Ben-Kilani, and S. Liu, Justification of the voltage security assessment method using the
bifurcation subsystem method, Proceedings of the Large Engineering System Conference on Power Systems,
pp. 266-272, June 1998.
R.A. Schlueter and S. Liu, A structure based hierarchy for intelligent voltage stability control in operation
planning, scheduling, and dispatching power systems, Proceedings of the Large Engineering System Con-
ference on Power System Engineering, pp. 280–285, June 1998.
R.A. Schlueter, A voltage stability security assessment method, IEEE Trans. on Power Systems, 13, 1423-1438,
1998.
? 2000 by CRC Press LLC
R.A. Schlueter, S. Liu, K. Ben-Kilani, and I.-P. Hu, Static voltage instability in generator flux decay dynamics
as a cause of voltage collapse, accepted for publication in the Journal on Electric Power System Research,
July 1998.
R. Schlueter, S. Liu, and N. Alemadi, Intelligent Voltage Stability Assessment Diagnosis, and Control of Power
Systems Using a Modal Structure, Division of Engineering Research Technical Report, December 1998
and distributed to attendees of Bulk Power System Dynamics and Control IV; Restructuring, August 24–28,
1998, Santorini, Greece.
C. Taylor, Power System Voltage Stability, Power System Engineering Series, McGraw-Hill, New York, 1994.
C. Taylor, D. Kostorev, and W. Mittlestadt, Model validation for the August 10, 1996 WSCC outage, IEEE Winter
Meeting, paper PE-226-PWRS-0-12-1997.
T. Van Cutsem, A method to compute reactive power margins with respect to voltage collapse, in IEEE Trans.
on Power Systems, 6, 145–156, 1991.
T. Van Cutsem and C. Vournas, Voltage stability of electric power systems, Power Electronic and Power System
Series, Kluwer Academic Publisher, Boston, MA, 1998.
V. Venkatasubramanian, X. Jiang, H. Schattler, and J. Zaborszky, Current status of the taxonomy theory of large
power system dynamics, DAE systems with hard limits, Proceedings of the Bulk Power System Voltage;
Phenomena-III Stability, Security and Control, pp. 15–103, August 1994.
Further Reading
There are several good books that discuss voltage stability. Kundur [1994] is the most complete in describing
the modeling required to perform voltage stability as well as some of the algebraic model-based methods for
assessing proximity to voltage instability. Van Cutsem and Vournas’ book [1998] provides the only dynamical
systems discussion of voltage instability and provides a picture of the various dynamics that play a role in
producing voltage instability. Methods for analysis and simulation of the voltage instability dynamics are
presented. This analysis and simulation is motivated by a thorough discussion of the network, generator, and
load dynamics models and their impacts on voltage instability. Taylor [1994] provides a tutorial review of
voltage stability, the modeling needed, and simulation tools required and how they can be used to perform a
planning study on a particular utility or system.
The IEEE Transactions on Power Systems is a reference for the most recent papers on voltage viability and
voltage instability problems. The Journal of Electric Power Systems Research and Journal on Electric Machines
and Power Systems also contain excellent papers on voltage instability.
? 2000 by CRC Press LLC
