Power System Analysis
K. Tomsovic
V. Venkatasubramanian
School of Electrical Engineering and Computer Science
Washington State University
Pullman, WA
1. Introduction
The interconnected power system is often referred to as the largest and most complex
machine ever built by humankind. This may be hyperbole, but it does emphasize an
inherent truth: the complex interdependency of different parts of the system. That is,
events in geographically distant parts of the system may interact strongly and in
unexpected ways. Power system analysis is concerned with understanding the operation
of the system as a whole. Generally, the system is analyzed either under steady-state
operating conditions or under dynamic conditions during disturbances.
Electric power is primarily transmitted as a three phase signal. That is, three AC current
currents are sent that are out of phase by 120o but of equal magnitude. Such balanced
currents sum to zero and thus, obviate the need for a return line. If the voltages are
balanced as well, the total power transmitted will be constant in time, which is a more
efficient use of equipment capacity. For large scale systems analysis, the assumption is
usually made that the system is balanced. Each phase can be then analyzed independently
greatly simplifying computations. In the following, the implicit assumption is that three
phase systems are being used.
2. Steady-State Analysis
In steady state analysis, any transients from disturbances are assumed to have settled
down and the system state is unchanging. Specifically, system load, including
transmission system losses, are precisely matched with power generation so that the
system frequency is constant, e.g., 60 Hz in North America. Perhaps, the foremost
concern during steady-state is economic operation of the system but reliability is also
important as the system must be operated to avoid outages should disturbances occur.
The primary analysis tool for steady-state operation is the so-called power flow analysis,
where the voltages and power flow through the system is determined. This analysis is
used for both operation and planning studies and throughout the system at both the high
transmission voltages and the lower distribution system voltages.
The power system can be roughly separated into three subcomponents: generation,
transmission and distribution, and load. The transmission and distribution network
consists of power transformers, transmission lines, capacitors, reactors and protection
devices. The vast majority of generation is produced by synchronous generators. Loads
consist of a large number of, and a diverse assortment, of devices, from home appliances
and lighting to heavy industrial equipment to sophisticated electronics. As such,
modeling the aggregate effect is a challenging problem in power system analysis. In the
following, the appropriate models for these components in the steady-state are
introduced.
2.a Modeling
2.a.1 Transformers
A transformer is a device used to convert voltage levels in an AC circuit. They have
numerous uses in power systems. To begin, it is more efficient to transmit power at high
voltages and low current than low voltage and high current. Conversely, lower voltages
are safer and more economic for end use. Thus, transformers are used to step-up voltages
from the generators and then used to step-down the voltage for end use. Another wide use
of transformers is for instrumentation so that sensitive equipment can be isolated from the
high voltages and currents of the transmission system. Transformers may also be used as
means of controlling real power flow by phase-shifting.
Transformers function by the linkage of magnetic flux through a core of ferromagnetic
material. Figure 2.1a illustrates a magnetic core with a single winding. When a current I
is supplied to the first set of windings, called the primary windings, a magnetic field, H,
will develop and magnetic flux, φ, will flow in the core. Ampère’s Law relates the
enclosed current to the magnetic field encountered on a closed path. If we assume that H
is constant throughout the path then
Hl = NI
(2.1)
where l is the path length through the core and N is the number of windings on the core
so that NI is the enclosed current by the path referred to as the magnetomotive force
(mmf).
I
φ
I
φ
Figure 2.1 a) flux flows through core from first winding, b) flux is linked to a second set
of windings
The magnetic field is related to the magnetic flux by the properties of the material,
specifically, the permeability. If we assume a linear relationship, i.e., neglecting
hysteresis and saturation effects, then the flux density B or the flux φ is
B = µH = µ
NI
NI
or φ = µA
l
l
(2.2)
where A is the cross-sectional area of the core. This relationship between the flux flow in
the core and the mmf is called the reluctance, R, of the core so that
Rφ = NI
(2.3)
Now, if a second set of windings, the secondary windings, is wrapped around the core as
shown in Figure 2.1b, the two currents will be linked by magnetic induction. Assuming
that no flux flows outside the core, then the two windings will be see the exact same flux,
φ. Since, the two windings also see the same core reluctance, the two mmf’s are identical,
i.e.,
N1 I 1 = N 2 I 2
(2.4)
If the flux φ is changing in time, or equivalently the current I, then according to Faraday’s
Law, a voltage will be induced. Assuming this ideal transformer has no losses, the power
input will be the same as the power output so
V1 I 1 = V2 I 2
(2.5)
where V1 and V2 are the primary and secondary voltages, respectively. Substituting (2.4)
and rearranging shows
V2 N 2
=
V1 N 1
(2.6)
Thus, the voltage gain in an ideal transformer is simply the ratio of the number of
primary and secondary windings. A practical transformer experiences several non-ideal
effects. Specifically, these include non-zero winding resistance, finite permeability of the
core, eddy currents that flow within the core, hysteresis (the effect arising from the
energy required to reorient the magnetic dipoles as the magnetic polarity changes), and
magnetic saturation. For steady-state studies of the large system, we desire linear circuit
models. These non-ideal effects are typically modeled as a combination of series and
parallel impedances in the following way:
•
•
Series impedances – Since the transformer core has a finite permeability, some of
the magnetic flux flows outside the core. This leakage flux will not link the
primary and secondary windings. Thus, the voltage at the input sees not only the
voltage that links the primary and secondary windings, but also a voltage drop
caused by this leakage inductance. Similarly, the finite winding resistance causes
an additional voltage drop to be seen at the terminals.
Shunt impedances - Finite permeability implies non-zero core reluctance and so
requires current to magnetize the core (i.e., a non-zero mmf). This difference
between the primary and secondary mmf’s can be modeled as a shunt inductance.
Hysteresis and eddy currents lead to energy losses in the core that can be
approximately modeled by a shunt resistor. Saturation is an important non-linear
effect that results in additional losses and the creation of odd order harmonics in
the current and voltage signals. Since in the steady-state system analysis, only the
60 Hz component of the currents and voltages are considered, saturation effects
are typically ignored.
An equivalent circuit for the transformer model described above is shown in Figure 2.2.
Figure 2.2 Transformer circuit model
The main difficulty with this model as it now stands is the ideal transformer component.
Carrying this component around in the calculations creates unnecessary complexity.
Further from engineering point of view, the voltages and currents in the system are most
easily seen relative to their rated values. Thus, most system analysis is done on a
normalization called the per unit system. In the per unit system, a system power base is
established and the rated voltages at each point in the network are determined. All system
variables are then given relative to this value. These base quantities for the currents can
be found as
SB
VB
(2.7)
VB VB2
ZB =
=
I B SB
(2.8)
IB =
and for impedances
This normalization has the great added advantage of reducing the need to represent the
ideal transformer in the circuit. One must simply keep track of the nominal base voltage
in each part of the network.In this way, the equivalent transformer model is as given in
Figure 2.3. Note, phase-shifting and off-nominal transformer ratios result in asymmetric
circuits and require some additional manipulation in the per unit framework. Those
details are omitted for brevity.
Figure 2.3 Simplified transformer circuit model under per unit system
2.a.2 Transmission line parameters
As mentioned previously, electric power is transmitted in three phases. This accounts for
the common site of three lines, or for dual circuits six lines, seen strung between
transmission towers. Typically, a high voltage transmission line has several feet of
spacing between the three conductors. The conductors themselves are stranded wire for
improved mechanical properties, as well as electrical properties. If the currents will be
large, several conductors may be strung per phase. This improves cooling compared to
using one large conductor. This geometry is important as it impacts the electrical
properties of the line.
To begin, as current flows in each conductor, a magnetic field develops. Adjacent lines
then may induce voltages in nearby conductors through mutual induction (as we saw for
transformers only now the coupling is not as tight). This interaction largely determines
the inductance seen by the respective phase currents. To understand this phenomenon,
consider a single line of radius r and infinite length with some current flow, I, as sketched
in Figure 2.4. Similar to the transformer development, we will apply Ampère’s Law to
characterize the magnetic field. The magnetic field at some distance x from the line can
be found by assuming that the field is constant at all points equal distance from the line.
Then the closed path is a circle with circumference 2πx, which gives
I
x
Figure 2.4 Infinite transmission line
2πxH = I or H =
I
2πx
(2.9)
If x is less than the line radius, the closed path will not link all of the current. Assuming
an equal distribution of current throughout the wire, then
2πxH = I
x2
Ix
or H =
2
r
2πr 2
(2.10)
Now, then once again the flux density is determined by the permeability of the material,
in this case, either the conductor itself if x<r or that of free space for x>r. Then the flux
relationship is
dφ = µ
Ix
dx
2πr 2
(2.11)
For x>r, the flux linked up to some radial distance R per unit of length is simply
R
λexternal = ∫ µ 0
r
I
2πx
dx =
µ0 I R
ln
r
2π
(2.12)
For x<r, only the enclosed current will be linked and continuing with the even
distribution of current assumption, the flux linked is
r
λ internal = ∫ µ c
0
Ix 3
2πr 4
dx = µ c
I
8π
(2.13)
For simplicity assume the permeability of the conductor is simply that of free space, then
the total flux linkage is
λ=
µ0 I µ0 I R µ0 I
R
+
ln =
ln
8π
2π
r
2π
re
−1
4
(2.14)
Typically, re 4 is written as r ′ . Now consider a three phase transmission line of phase
currents Ia, Ib, and Ic, with each line spaced equally by the distance D. Flux from each of
the currents will link with each of the other conductors. The flux linkage for phase a out
to some point R a far distance away from the conductors is approximately
−1
λa =
µ0 
R
R
R
 I a ln + I b ln + I c ln 
2π 
r′
D
D
(2.15)
with balanced currents, i.e., I a + I b + I c = 0 , and recalling that inductance is simply the
ratio of flux linkage to current, the series inductance in phase a per unit length of line will
be
µ
D
~
La = 0 ln
2π r ′
(2.16)
D
D
D
Figure 2.5 End view of equally spaced phase conductors
In practice, the phase conductors may not be equally spaced. This results in unbalanced
conditions due to the imbalance in mutual inductance. High voltages transmission lines
with such a layout can be transposed so that on average the distance between phases is
equal canceling out the imbalance. The equivalent distance of separation between phases
can then be found as the geometric mean of this spacing. Similarly, if several conductors
are used per phase an equivalent conductor radius can be found as the geometric mean.
Transmission lines also exhibit capacitive effects. That is, whenever a voltage is applied
to a pair of conductors separated by a non-conducting medium, charge accumulates
leading to capacitance. Similar to the previous development for inductance, we can
determine the capacitance based on Gauss’s Law. For a point P at a distance x from a
conductor with charge q, the electric flux density D is
D=
q
2πx
(2.17)
Assuming a homogeneous medium, the electric field density E is related to D by the
permitivity ε of the dielectric, which in this case will be assumed to be that of free space.
E=
q
(2.17)
2πε 0 x
Integrating E over some path (a radial path is chosen for simplicity) yields the voltage
difference between the two end points.
V12 =
R2
q
∫ 2πε
R1
0
x
dx =
q
2πε 0
ln
R1
R2
(2.18)
Now consider a three phase transmission line again with each line spaced equally by the
distance D. Superposition holds so that the voltage arising from each of the charges can
be added. To find the voltage from phase to ground arising from each of the conductors,
assume first, a balanced system with q a + qb + q c = 0 , and two, a neutral located at some
far distance R from phase a so
Van =
1 
1
R
R
R
D
q a ln
 q a ln + qb ln + q c ln  =
r
D
D  2πε 0
r
2πε 0 
(2.19)
Now recalling that capacitance is simply the ratio of charge to voltage, the capacitance
from phase a to ground per unit length of line will be
q
2πε 0
~
C an = a =
D
Van
ln
r
(2.20)
Again if the conductors are not evenly spaced, transposition results in an equivalent
geometric mean distance and using bundled conductors per phase can also be
accommodated by using a geometric mean.
Finally, conductors have finite resistances that depend upon the temperature, the
frequency of the current, the conductor material, and so on. For most systems analysis
problems, these can be based on values provided by manufacturers or compiled into
tables for commonly used conductors and typical ambient conditions.
2.a.3 Transmission line circuit models
Transmission lines may be classified based on their total length. If the line is around 50
miles or less, a so-called “short line,” capacitance can be neglected and the series
inductance and resistance can be modeled as lumped parameters. Figure 2.6 depicts the
short line model per phase. The series resistance and inductance is simply the per unit
distance parameters times the line length so at 60 Hz for line length l the line impedance
is
~
~
Z = R l + j120πL l = R + jX
(2.22)
Figure 2.6 Short line model
For lines longer than 50 miles, up to around 150 miles, capacitance can no longer be
neglected. A reasonable circuit model is to simply split the total capacitance evenly with
each half represented as a shunt capacitor at each end of the line. This is depicted as the
π-circuit model in Figure 2.7. Again, the total capacitance is simply the per unit distance
capacitance times the line length so
~
Y = j120πC an l = jB
(2.23)
Figure 2.7 Medium line model
For line lengths longer than 150 miles, the lumped parameter model may not provide
sufficient accuracy. To see this, note that at 60 Hz for a low loss line, the wavelength is
around 3000 miles. Thus, a 150 mile line begins to cover a significant portion of the wave
and the well-know wave equations must be used. The relationship between voltage and
current at a point x, i.e., distance along the line, to the receiving end voltage, Vr, and
current, Ir, is
V ( x ) = Vr cosh γx + I r Z c sinh γx
I ( x ) = I r cosh γx +
Vr
sinh γx
Zc
(2.24)
(2.25)
where Z c = z y is the characteristic impedance of the line and γ = zy is the
propagation constant. It would be useful, and it turns out to be possible, to continue to
use the π-model for the transmission line and simply modify the circuit parameters to
represent the distributed parameter effects. The relationship between the sending end
voltage, Vs, and current, Is, to the receiving end voltage and current in a π-model can be
found to be
 YZ 
V s = Vr  1 +
 + IrZ
2 

 YZ 
 YZ 
I s = VrY 1 +
 + I r 1 +

4 
2 


(2.26)
(2.27)
Now equating (2.24) - (2.25) for a line of length l to (2.26)-(2.27) and solving shows the
equivalent shunt admittance and series impedance for a long line to be
Z ′ = Z c sinh γl = Z
sinh γl
γl
Y′ 1
γl Y tanh γl 2
tanh =
=
2 Zc
2 2 γl 2
(2.28)
(2.29)
where the prime indicates the modified circuit values arising from a long line.
2.a.4 Generators
Three phase synchronous generators produce the overwhelming majority of electricity in
the modern power systems. Synchronous machines operate by applying a DC excitation
to a rotor that, when mechanically rotated, induces a voltage in the armature windings
due to changing flux linkage. The per phase flux for a balanced connection can be written
as
λ = K f I f sin θ m
(2.30)
where If is the field current, θm is the angle of the rotor relative to the armature and Kf is a
constant that depends on the number of windings and the physical properties of the
machine. The machine may have several poles so that the armature will “see” multiple
rotations for each turn of the rotor. So for example, a four pole machine appears
electrically to be rotating twice as fast as two pole machine. For a machine rotating at ωm
rad/s with p poles, the electric frequency is
ωs = ωm
p
2
(2.31)
with ω s the desired synchronous frequency. If the machine is rotated at a constant speed
Faraday’s Law tells us the induced voltage will be
V =
dλ
= K f I f ω s sin (ω s t + θ 0 )
dt
(2.32)
If a load is applied to the armature windings, then current will flow and the armature flux
will link with the field. This effectively puts a mechanical load on the rotor and so power
input must be matched to this load in order to maintain the desired constant frequency.
Some of the armature flux “leaks” and does not link with the field. In addition, there are
winding resistive losses but those are commonly neglected. The circuit model shown in
Figure 2.8 is a good representation for the synchronous generator in the steady state. Note
that most generators are operated at some fixed terminal voltage with a constant power
output. Thus, for steady state studies the generator is often referred to as a PV bus since
the terminal node has fixed power P and voltage V.
Figure 2.8 Simple synchronous generator model
2.a.5 Loads
Modeling power system loads remains a difficult problem. The large number of different
devices that could be connected to the network at any given time renders precise
modeling intractable. Broadly speaking, loads may vary with voltage and frequency. In
the steady-state frequency is constant so one only needs to be concerned with the voltage.
For most steady-state analysis, a fixed, i.e., constant over an allowable voltage range,
power consumption model can be used. Still, some analysis requires consideration of
voltage effects to be useful and then the traditional exponential model can be used to
represent real power consumption P and reactive power consumption Q
P = P0V a
(2.33)
Q = Q0V b
(2.34)
where the voltage V is normalized to some rated voltage. The exponents a and b can be 0,
1 or 2 where they could represent constant power, current or impedance loads,
respectively. Alternatively, they can represent composite loads with a generally ranging
between 0.5 and 1.8 and b ranging between 1.5 and 6.
2.b Power flow analysis
Power flow equations represent the fundamental balancing of power as it flows from the
generators to the loads through the transmission network. Both real and reactive power
flows play equally important roles in determining the power flow properties of the
system. Power flow studies are amongst the most significant computational studies
carried out in power system planning and operations in the industry. Power flow
equations allow us to compute the bus voltage magnitudes and their phase angles, as well
as the transmission line current magnitudes. In actual system operation, both the voltage
and current magnitudes need to be maintained within strict tolerances, for meeting
consumer power quality requirements and for preventing overheating of the transmission
lines, respectively. The difficulty in computing the power-flow solutions arises from the
fact that the equations are inherently nonlinear arising from the balancing of power
quantities. Moreover, the large size of the power network implies that the power-flow
studies involve solving a very large number of simultaneous nonlinear equations.
Fortunately, the sparse interconnected nature of the power network reflects itself in the
computational process, facilitating the computational algorithms.
We first study a simple power flow problem in Section 2.b.1 to gain insight into the
nonlinear nature of the power flow equations. We then formulate the power flow problem
for the large power system in Section 2.b.2. In Section 2.b.3, a classical power flow
solution method based on the Gauss-Seidel algorithm is studied. In Section 2.b.4, the
popular Newton-Raphson algorithm, which is the most commonly used power flow
method in the industry today, is introduced. In Section 2.b.5, we will briefly consider the
fast decoupled power flow algorithm, which is a heuristic method that has proved quite
effective for quick power-flow computations. Finally, in Section 2.b.6, we will learn
about the DC power-flow solution that is a highly simplified algorithm for computing
approximate linear solutions of the power-flow problem, and becoming widely used for
electricity market calculations.
2.b.1 Simple example of a power flow problem:
1
0
V
δ
jx
P+jQ
Figure 2.9 A simple power system
Let us consider a single generator delivering the load P + j Q through the transmission
line with the reactance x. The generator bus voltage is assumed to be at the rated voltage
and thus it is at 1 pu. The generator bus angle is defined as the phasor reference and
hence the generator bus voltage phase angle is set to be zero. The load bus voltage has
magnitude V and phase angle δ. Since the line has been assumed to be lossless, note that
the generator real power output must be equal to the real power load P. However, the
reactive power output of the generator will be the sum of the reactive load Q and the
reactive power “consumed” by the transmission line reactance x.
Let us write down the power-flow equations for this problem. Given a loading condition
P + j Q, we want to solve for the unknown variables, namely, the bus voltage magnitude
V and the phase angle δ. For simplicity, we will assume that the load is at unity power
factor, that is, Q = 0. The line current phasor I from the generator bus to the load bus is
easily calculated as
1∠0 − V∠δ
jx
=
I
(2.35)
Next, the complex power S delivered to the load bus can be calculated as
S
= V I*
=
V∠(δ + π 2 ) V 2 ∠(π 2)
−
x
x
(2.36)
Therefore, we get the real and reactive power balance equations to be
P =
− V sin δ
,
x
Q =
− V 2 +V cos δ
x
(2.37)
After setting Q=0 in (2.37), we can simplify the two equations in (2.37) into a quadratic
equation in V2 as follows
V 4 − V 2 + x2 P2 = 0
(2.38)
Therefore, given any real power load P, the corresponding power-flow solution for the
bus voltage V can be solved from (2.38). We note that for nominal load values, there are
two solutions for the bus voltage V and they are the positive roots of V2 in the next
equation
V2
=
1 ± 1 − 4x 2 P 2
2
(2.39)
Equation (2.39) implies that there exist two power-flow solutions for load values P < Pmax
where Pmax = 1/(2x), and there exist no power-flow solutions for P > Pmax. A qualitative
plot of the power-flow solutions for the bus voltage V in terms of different real power
loads P is shown in Figure 2.10.
From the plot and from the analysis thus far, we can make the following observations:
1. The dependence of the bus voltage V on the load P is very much nonlinear. It has
been possible for us to compute the power-flow solutions analytically for this
simple system. In the large power system with hundreds of generators delivering
power to thousands of loads, we have to solve for thousands of bus voltages and
their phase angles from large coupled sets of nonlinear power-flow equations, and
the computation is a nontrivial task.
2. Multiple power-flow solutions can exist for a specified loading condition. In
Figure 2.10, there exist two solutions for any load P < Pmax . Among the two
solutions, the solution on the upper locus with voltage V near 1 pu is considered
the nominal solution. For the solutions on the lower locus, the bus voltage V may
be unacceptably low for normal operation. The lower voltage solution also
requires higher line current in order to deliver the specified load P, and the line
current values can become unacceptably high. In general, for any specified
loading condition, we would like to locate the power-flow solution that has the
most acceptable values of voltages and currents among the multiple power-flow
solutions. In this example of a single generator delivering power to a single load,
there exist two power-flow solutions. In a large power system, there may exist a
very large number of possible power-flow solutions.
3. Once the bus voltage V has been computed from (2.39), the bus voltage phase
angle δ can be computed from (2.37). Then, the line current phasor I can be
solved from (2.35). Specifically, we would like to ensure that the magnitude of
the line current I stays below the thermal limit of the transmission line for
preventing potential damage to the expensive transmission line.
4. Power-flow solutions may fail to exist at high loading conditions, for instance,
when P > Pmax in Figure 2.10. The loading value Pmax beyond which power-flow
solutions do not exist is called the static limit in the power literature. Since the
power-flow solutions denote the steady state operating conditions in our
formulation, lack of power-flow solutions implies that it is not possible to transfer
power from the generator to the load in a steady state fashion, and the dynamic
interactions of the generators and the loads become significant. Operating the
power system at loading conditions beyond the static limit may lead to
catastrophic failure of the system.
V
1
P
0
Pmax
Figure 2.10 Qualitative plot of the power-flow solutions
2.b.2 Power-flow problem formulation
In this subsection, we will construct the power-flow equations in a structured manner
using the admittance matrix Ybus representation of the transmission network. The
admittance matrix Ybus is assumed to be known for the system under consideration. Let us
first look at the complex power-balance at any bus, say bus i, in the network.
SG i
Ii
Si
SL i
Vi
Figure 2.11 Complex power-balance at Bus i
The power balance equation is given by
= Vi I i
Si
*
=
(2.40)
S Gi − S Li
Let us denote the vector of bus voltages as Vbus and the vector of bus injection currents as
Ibus . By definition, the admittance matrix Ybus provides the relationship Ibus = Ybus Vbus.
Suppose the (i,j)-th entry Yij of the Ybus matrix has the magnitude Yij and the phase γij.
Then, we can simplify the current injection Ii as
Ii
=
∑Y V
ij
j
∑ Y V ∠(δ
=
ij
j
j
j
+ γ ij )
(2.41)
j
Then, combining (2.40) and (2.41), we get the complex power-balance equations for the
network as
Si
=
S Gi − S Li
=
∑ Y V V ∠(δ
ij
i
j
i
− δ j − γ ij )
(2.42)
j
Taking the real and imaginary parts of the complex equation (2.42) gives us the real and
reactive power-flow equations for the network.
Pi
Qi
=
PGi − PLi
= QGi − Q Li
=
∑Y V V
ij
i
j
cos (δ i − δ j − γ ij )
(2.43)
j
=
∑Y V V
ij
i
j
sin (δ i − δ j − γ ij )
(2.44)
j
Generally speaking, our objective in this section is to solve for the bus voltage
magnitudes Vi and the phase angles δi when the power generations and loads are
specified. For a power system with N buses, there are 2N number of power-flow
equations. At each bus, there are six variables PGi, QGi, PLi, QLi, Vi, and δi. Depending
on the nature of the bus, four of these variables will be specified at each bus, leaving two
unknown variables at each bus. We will end up with 2N unknown variables related by 2N
equations (2.43) and (2.44), and our aim in the rest of this section is to develop
algorithms for solving this problem.
Let us consider a purely load bus first, that is, with PGi = QGi = 0. In this case, the loads
PLi and QLi are assumed to be known either from measurements or from load estimates,
and the bus voltage variables Vi, and δi are the unknown variables. Purely load buses with
no generation support are called “PQ buses” in the power-flow studies since both realpower injection Pi and reactive power Qi have been specified at these buses.
Typically, every generator in the system consists of two types of internal controls, one for
maintaining the real power output of the generator, and the other for regulating the bus
voltage magnitude. In power-flow studies, we usually assume that both these control
mechanisms are operating perfectly and so the real power output PGi and Vi are
maintained at their specified values. Again, the load variables PLi and QLi are also
assumed to be known. This leaves the generator reactive output QGi and the voltage phase
angle δi as the two unknown variables for the bus. In terms of injections, the real power
injection Pi and the bus voltage Vi are then the specified variables, and thus the generator
buses are normally denoted “PV buses” in power-flow studies.
In reality, the generator voltage control for keeping the bus voltage magnitude at a
specified value becomes inactive when the control is pushed to the extremes, say when
the reactive output of the generator becomes either too high or too low. This voltage
control limitation of the generator can be represented in the power-flow studies by
keeping track of the reactive output QGi. When the reactive generation QGi becomes
larger than a prespecified maximum value say QGi,max or goes lower than a prespecified
minimum value QGi,min, the reactive output is assumed to be fixed at the limiting value
QGi,max or QGi,min respectively, and the voltage control is disabled in the formulation. That
is, the reactive power QGi becomes a known variable, either at QGi,max or QGi,min and the
voltage Vi then becomes the unknown variable for bus i. In power-flow terminology, we
say that the generator at bus i has reacted its reactive limits and hence, bus i has changed
from a PV bus to a PQ bus. Owing to space limitations, we will not discuss generator
reactive limits in any more detail in this section.
In addition to PQ buses and PV buses, we also need to introduce the notion of a “slack
bus” in the power-flow formulation. Note that power conservation demands that the real
power generated from all the generators in the network must equal the sum of the total
real power loads and the line losses on the transmission network.
∑P
i
Gi
=
∑P
i
Li
+ ∑∑ Plossesij
i
(2.45)
j
The line losses associated with any transmission line in turn depends on the line
resistance and the line current magnitude. As stated earlier, one of the main objectives of
the power-flow studies to compute the line currents, and as such, the line current values
are not known at the beginning of a power-flow computation. Therefore, we do not know
the actual values for the line losses in the transmission network. Looking at (2.45), we
need to assume that at least of one of the variables PGi or PLi should be a free variable for
satisfying the real power conservation. Traditionally, we assume that one of the
generations is a “slack” variable and such a generator bus is denoted the “slack bus”. At
the slack bus, we specify both the voltage Vi and the angle δi. The power injections Pi and
Qi are the unknown variables. Again, by tradition, we set the voltage at slack bus to be
the rated voltage or at 1 pu, and the phase angle to be at zero.
Like in standard text books, slack bus is defined in this section to be the first bus in the
network with V1=1 and the angle δ1=0. Assuming the number of generators to be NG, the
buses 2 through NG+1 are set to be the PV buses. The remaining buses NG+2 through N
are then the PQ buses.
2.b.3. Gauss-Seidel algorithm:
Let us consider a set of simultaneous linear equations of the form A x = b, where A is an
n X n matrix, and, x and b are n X 1 vectors. Clearly, there exists a unique solution to the
problem when the matrix A is invertible and the solution is given by x = A-1 b. When the
matrix size is very large, it may not be possible to compute the inverse of the matrix A for
finding the solution and there exist other numerical techniques. Gauss-Seidel algorithm is
one such classical algorithm that tries to arrive at the solution x = A-1 b iteratively by
starting from an approximate initial condition say x0. The iteration for the solution xk+1
from the previous iterate xk proceeds as follows.
xi
k +1
=
1
aii


 bi − ∑ aij x j k +1 − ∑ aij x j k  for i = 1,2,..., n


j <i
j >i


(2.46)
Here aij denotes the (i,j)-the entry of the matrix A as usual. It can be shown that the
iterative solution xk converges to the exact solution A-1 b for any initial condition x0,
provided the matrix A satisfies certain “diagonal dominance” properties. The details are
limited here to save space and they can be seen in standard numerical analysis text-books.
In the previous section, we have formulated the power-flow problem as a set of
simultaneous nonlinear equations (2.42) and as such, it is not obvious how the GaussSeidel algorithm can be applied for solving these equations. The trick here is to visualize
the power-balance equations to be arising from the network admittance equations Ybus
Vbus = Ibus. The matrix Ybus takes over the role of the matrix A in the linear equations. We
will be solving for the bus voltage vector Vbus. The current injections Ibus are not known
per se. The current injections are in fact dependent on the bus voltages. As we see next,
they can also be computed iteratively from the power injections Si by using the
relationship Ii = Si* / Vi*. For a PQ bus, the injection Si is a specified variable and hence is
known. For PV buses, only the real power injection Pi is known while the reactive
injection Qi is evaluated first using the latest estimate of bus voltages Vbus.
An outline of the Gauss-Seidel algorithm for solving the power-flow equations (2.42) is
presented next. Let us start with an initial condition for the bus voltages Vbus0 and we
would like to compute the iterate Vbusk+1 from the previous iterate Vbusk. Recall that bus 1
k
is a slack bus and hence V1 = 1∠0 for all iterations. Also, buses 2 through NG+1 are PV
buses, and hence, we need to keep Vik at their specified values Vi, specified = Vi0 for all
iterations for the PV buses.
Gauss-Seidel iterations:
k +1
a. Slack bus (i=1):
V 1 = 1∠0
b. PV buses (i=2, …, NG+1):
(1) Compute the reactive power generation at bus i.
k +1
k
k +1
k
k +1
k
k
k
k
Qi = ∑ YijVi V j sin δ i − δ j − γ ij + ∑ YijVi V j sin δ i − δ j − γ ij
j <i
(
)
j ≥i
(2) Update the bus voltage phasor Vik+1.
k +1

1  Pi − jQi
k +1
k 
k +1
Vi =
Y
V
Y
V
−
−
∑
∑
ij
j
ij
j

Yii  V k *
j <i
j >i
i


(3) Normalize the magnitude Vik+1 to be Vi0.
k +1
Vi
k +1
0
Vi
=
V
k +1 i
Vi
c. PQ buses (i=NG+2, …, N):
(4) Update the bus voltage phasor Vik+1.

1  Pi − jQi
k +1
k 
k +1
Vi =
Y
V
Y
V
−
−
∑
∑
ij
j
ij
j

Yii  V k *
j <i
j >i
i


(
)
(2.47)
(2.48)
(2.49)
(2.50)
At the end of the (k+1)-th iteration, we have the updated values of the bus voltages
Vbusk+1. The values of Vbusk+1 can be compared with the previous set of values Vbusk to
check whether the solution has converged or not.
For the power-flow problems, Gauss-Seidel algorithm has been known to converge even
from poor initial conditions, which is one of its main strengths. The algorithm is typically
used for “flat starts” when all the initial voltage magnitudes are set to be at their rated
values (say Vi0 = 1 pu for all the PV buses) and the bus voltage angles are set to be zero
(that is, δi0=0 for all buses). The relative simplicity of the computations involved in each
iteration step in (2.47) to (2.50) implies that the algorithm is very fast to implement. On
the other hand, the error convergence rate is typically linear in the sense that the ratios of
the error norm from one iteration to the previous iteration tends to be a constant.
Therefore, while the Gauss-Seidel algorithm can converge to the proximity of the actual
solution in tens of iterations, it typically takes large number of iterations to get to an
accurate solution estimate.
2.b.4 Newton-Raphson algorithm:
Unlike the Gauss-Seidel algorithm, which was originally developed for solving
simultaneous linear equations, Newton-Raphson (NR) algorithm is specifically designed
for solving nonlinear equations. The algorithm proceeds iteratively by linearizing the
nonlinear equations into linear equations at each step, and by solving the linearized
equations exactly.
Suppose we want to solve the nonlinear equations F(x) = 0 where x is a n X 1 vector, and
F : ℜ n → ℜ n is a smooth nonlinear function. We have been given an initial condition x0.
Then, for computing the estimate xk+1 from xk, we first linearize the functions F(x) at xk as
follows
k
0 = F ( x) ≈ F ( x ) +
∂F
∂x
(x − x )
k
(2.51)
xk
The solution to the linearized equations (2.b.1.17) is defined as the iteration estimate xk+1.
x
k +1
=
 ∂F
x −
 ∂x

−1
k
x
k

 F xk


( )
(2.52)
Note that the linearization (2.51) will be a good approximation if the estimate xk is close
to the true solution say x* since F(x*)=0. The NR algorithm stated in (2.52) can be proved
to converge to the true solution x*, when the initial condition x0 is sufficiently close to x*.
On the other hand, for initial conditions away from x*, the approximation (2.51) becomes
poorly justified, and the iterations can quickly diverge away from x*. When the iterations
converge, owing to the linearized nature of the algorithm, the norm of the error decreases
to zero in a “quadratic” fashion. Roughly speaking, the ratios of the error norm from one
iteration to the square of the error norm in the previous iteration tends to be a constant.
An example would be that the error norms decrease from 0.1 in one iteration, to 0.01 in
the next iteration, to 0.0001 in the following iteration. Therefore, given good initial
conditions, the NR algorithm can typically get to an accurate solution estimate within a
few iterations.
Let us apply the NR algorithm for solving the power-flow equations (2.43) and (2.44).
We will solve for the unknown variables among the bus voltage magnitudes Vi and angles
δi first. That is, we define the vector x as consisting of all the PV and PQ bus angles, and
all the PQ bus voltages. PV bus voltages are known and hence, they are not included in x.
x =
(δ
2
, ..., δ N G +1 , δ N G + 2 , ..., δ N , V N G + 2 , ...,V N
The corresponding power-flow equations are as follows
)
T
(2.53)


P2 − ∑ Y2, j V2V j cos(δ 2 − δ j − γ 2, j )


j

 P2 − p 2 ( x)  
.............
 


.............
  PN +1 − ∑ YN +1, jV N +1V j cos δ N +1 − δ j − γ N +1, j 

G
G
G
  G
P

j
N G +1 − p N G +1 ( x )
 P

− Y N G + 2, jV N + 2V j cos δ N G + 2 − δ j − γ N G + 2, j 
 PN G + 2 − p N G + 2 ( x)   N G + 2 ∑

j




............
F ( x) = 
............
=

PN − ∑ Y N , j V N V j cos(δ N − δ j − γ N , j )
 PN − p N ( x)  

j
 


 Q N G + 2 − q N G + 2 ( x)   Q N + 2 − ∑ Y N + 2, j V N + 2V j sin δ N + 2 − δ j − γ N + 2, j 
G
G
G
  G


j
............
 


............
 Q N − q N ( x)  

Q N − ∑ Y N , j V N V j sin (δ N − δ j − γ N , j )


j


(2.54)
(
(
)
)
(
)
The entries of the F function in (2.54) are the differences between the specified power
injections and the computed power injections from the current power-flow solutions, and
these are usually denoted as the real and reactive power mismatches at the different
buses. In the power-flow problem, then we want to find a solution that makes the power
mismatches in (2.54) to be zero.
Suppose an initial condition x0 has been specified. Then, the NR algorithm for solving the
power-flow equations (2.54) proceeds iteratively as follows:
Newton-Raphson iterations:
(1) Compute the power mismatches F(xk) for step k from (2.54). If the mismatches
are within desired tolerance values, the iterations stop.
∂F
k
. Owing to the nice structure of the
(2) Compute the power-flow Jacobian J =
∂ x xk
equations in (2.54), explicit formulas can be derived for the entries of the
Jacobian matrix, and the Jacobian for step k can be evaluated by substituting the
current values of xk into these formulas.
(3) Compute the correction factors ∆xk from (2.52) by solving a set of simultaneous
linear equations
k
J ∆x
k
( )
= −F x
k
(2.55)
The Jacobian matrix Jk is extremely sparse even for very large power systems,
which facilitates the solution of ∆xk in (2.55).
(4) Evaluate xk+1 from xk by adding the correction factors ∆xk.
x
k +1
=
k
x + ∆x
k
(2.56)
As compared with the Gauss-Seidel algorithm, each iteration step in the NR algorithm is
computationally much more intensive because of (i) evaluating the Jacobian and (ii)
solving the linear equations in (2.55). On the other hand, the error convergence rate of the
NR algorithm is spectacularly faster, and hence, the NR algorithm requires much fewer
iterations to reach comparable solution accuracies. In usual practice, Gauss-Seidel
algorithm is used only for flat starts with poorly known initial conditions. In most other
situations, NR algorithm is the preferred choice.
2.b.5. Fast decoupled power-flow algorithm:
Both the Gauss-Seidel algorithm and the Newton-Raphson algorithm are general methods
for solving linear and nonlinear equations respectively, and they were tailored towards
solving the power-flow problem. On the other hand, we study a power system specific
method in this section called the fast decoupled power-flow algorithm, which is a
heuristic method that is derived by exploiting power system specific properties.
The fast decoupled power-flow algorithm is essentially a highly simplified and
approximated version of the Newton-Raphson algorithm of the previous subsection. We
recall that the NR iteration steps are computation intensive because of evaluating Jk and
solving the Jacobian equation (2.55). In this subsection, we will proceed to simplify this
by replacing the iteration specific Jk with a constant matrix.
It is a well-known property of power systems that variations in the bus voltage
magnitudes mostly affect the reactive power injections under nominal operating
conditions. Similarly, the variations in the bus voltage phase angles mostly influence the
real power injections. By idealizing this property, we assume that all the Jacobian entries
∂pi
∂qi
and
are all identically zero. Next, if we assume that the bus voltage
of the form
∂V j
∂δ j
magnitudes are all close to one pu, and the voltage phase angle differences on the two
ends of any transmission line are all close to zero, it can be shown that the NR algorithm
greatly simplifies to the fast decoupled algorithm stated below.
Let us split the power-flow state vector x in (2.53) into the voltage magnitude V and angle
δ counterparts.
δ
=
(δ
2
, ..., δ N G +1 , δ N G + 2 , ..., δ N
),
T
V
=
(V
NG + 2
, ..., V N
Similarly, we separate the real and reactive power mismatches in (2.54).
)
T
(2.57)
 P2 − p 2 ( x) 


.............


P
− p N G +1 ( x) 
,
∆ P =  N G +1
 PN G + 2 − p N G + 2 ( x) 


............


 P − p ( x) 
N
N


∆Q =
 Q NG + 2 − q N G + 2 ( x) 


............


 Q − q ( x) 
N
N


(2.58)
Suppose initial conditions for the voltage magnitude vector V0 and angle vector δ0 have
been specified. Then, the fast decoupled algorithm proceeds iteratively as follows.
Fast decoupled iterations:
(1) Compute the real and reactive power mismatches ∆P(xk) and ∆Q(xk). If the
mismatches are within desirable tolerance, the iterations end.
(2) Normalize the mismatches by dividing each entry by its respective bus voltage
magnitude.
~k
∆P

 ∆P k V k
2
2




.............


∆P k N G +1 V k N G +1 

=
,
 ∆P k N + 2 V k N + 2 
G
G


............


 ∆P k V k 
N
N


~k
∆Q
k
k
 ∆Q

 NG + 2 VNG + 2 


............


k
k
 ∆Q N V N



=
(2.59)
(3) Solve for the voltage magnitude and angle correction factors ∆Vk and ∆δk by
using the constant matrices B and B which are extracted from the bus admittance
matrix Ybus.
~k
~k
k
k
B ∆δ = ∆ P ,
B ∆V = ∆Q
(2.60)
Define Bij=imag(Yij). The matrices B and B are constructed as follows.
 B2 , 2

B = −  ...
 B N , 2
... B2, N 

...
... ,
... B N , N 
 B N G + 2, N G + 2

...
B=−
 BN , N +2
G

... B N G + 2, N 

...
... 
... B N , N 
(2.61)
(4) Update the voltage magnitude and angle vectors.
δ k +1 = δ k + ∆δ k ,
V
k +1
k
= V + ∆V
k
(2.62)
By using the constant matrices B and B in (2.60), the time taken for evaluating one
iteration for the fast decoupled algorithm is considerably lesser than that of the NewtonRaphson algorithm. In repeated power-flow runs of the same power system, the inverses
of the matrices B and B can directly be stored, which enables the implementation to be
very fast. However, the convergence speed of the fast decoupled algorithm is not
quadratic, and it takes considerably more iterations to converge to an accurate powerflow solution. Fast decoupled algorithm is used in applications where quick approximate
estimates of power-flow solutions are required.
2.b.6. DC power-flow algorithm:
A further simplification of the fast decoupled algorithm is the highly approximate DC
power-flow algorithm, which completely transforms the nonlinear power-flow equations
into linear equations by using drastic assumptions. In addition to the assumptions used in
deriving the fast decoupled method, we also assume that all the voltage magnitudes are at
1 pu, and all the transmission lines are lossless.
With these assumptions, the voltage correction factors ∆Vk become irrelevant in (2.60).
Moreover, the angle variables can be solved explicitly by the linear equations
Bδ = P
(2.63)
where P is the vector of bus power injections. The resulting solution for the bus voltage
phase angles is called the DC power-flow model. It gives approximate values for the
phase angles across the power system. The phase angles can be used to approximate the
real power-flow on any transmission line by dividing the phase angle difference between
the two ends of the transmission lines by the line reactance. The advantage of the DC
power-flow model is its extreme simplicity in finding a power-flow solution. However,
the limitations of the solution need to be kept in mind, in light of the drastic assumptions
that were used to simplify the nonlinear power-flow equations into linear equations.
3. Dynamic Analysis
The power system in practice is constantly undergoing changes either due to changing
loads, planned outages of equipment for maintenance or other disturbances, such as,
equipment failures, line faults, lightning strikes or any number of other events that cause
outages. During disturbances, the precise balance between generation and load is not
maintained. These disturbances may lead to oscillations and the system must be able to
dampen these and reach a viable steady-state operating condition. Extremely fast
electromagnetic transients, such as those that arise from lightning strikes or switching
actions, are not considered from a system point-of-view and so the network models
introduced in the previous are still valid. Still dynamic models for generator units models
must be introduced in order to understand system response. Load dynamics are also
important, particularly from large induction motors, but such details are beyond the scope
here. This section focuses on the transient response of the system over fractions of a
second to oscillations over several seconds and then up to slow dynamics that occur with
voltage problems over several minutes.
3.a Modeling
Electric Generators
To understand modeling generators for dynamic analysis, some more details on the
physical construction are needed. Most generators are three phase synchronous machines,
which means they are designed to operate at a constant frequency. The machine rotor is
driven mechanically by the prime mover governing system to control power output and
speed. Synchronous machine rotors can be classified as either cylindrical or salient pole.
Cylindrical rotors have an even air gap at all points between the stator, i.e., the stationary
part of the machine, and the rotor. This construction is used for machines that rotate at
high speed, typically steam-driven generators. Steam generators generally are two-pole or
four-pole machines and so rotate, in North America, at 1800 RPM or 3600 RPM to
produce the desired 60 Hz signal. Hydro generators, conversely, may have numerous pole
pairs, as it is more efficient to drive them at lower speeds. These generators have a salient
pole construction that leads to a variable air gap between the stator and the rotor.
For modeling purposes, the pole construction is important not only to represent the
mechanical speed of rotations but also since the transfer of power from the rotor to the
stator through mutual inductance depends on the size of the air gap (i.e., the reluctance of
the air gap and so the effective coupling). During disturbances, these variations are most
evident and the effective circuit inductances must be modeled accurately. In modern
power system modeling for dynamic analysis, a rotating frame of reference is chosen to
represent these effects. The circuit equations are then written in terms of direct and
quadrature axes. For simplicity, the more involved rotating frame of reference models are
not developed here but instead a simple model approximating the variable inductances is
presented. To begin, there are primarily two sets of windings of concern:
Armature windings – The windings on the stator are referred to as the armature windings.
The armature is the source from which the power generated will be drawn. A voltage is
induced in the armature from the rotation of the field generated by currents on the rotor.
For purposes of modeling, the self inductance, including leakage, and mutual inductance
between phases of the armature windings describe the circuit performance as current
flows from the terminals. The armature windings are distributed in slots around the stator
so as to produce a high quality sinusoidal signal. Winding resistance may also be
included but is small and often neglected in simple studies.
Field windings – These windings reside on the rotor and provide the primary excitation
for the machine. The windings are supplied with a DC current so that as they rotate
induce a voltage in the armature windings. The current is controlled by the exciter in
order to provide the desired voltage across the armature windings. This current must be
constrained to avoid overheating of the windings and the modeling of these limits along
with excitation control circuit is critical for analysis.
In addition, in salient pole machines, there are solid conducting bars in the pole faces
called damper windings, or amortisseurs windings, that influence the effective inductanc.
These windings serve to damp out higher frequency oscillations. Similar effects are also
seen in the cylindrical case through eddy current flows in the rotor even though damper
windings may not be present. These details are not pursued further here.
The armature will be modeled as a voltage controlled by the rotor current behind a simple
transient reactance as illustrated in Figure 3.1. The induced voltage E ' ∠θ , referred to as
the voltage behind the reactance, is connected to the generator terminal through the
transient reactance xd’ , where the subscript indicates a direct axis quantity and the
superscript a transient quantity.
j xd’
E’ θ
Figure 3.1 A simple model of a synchronous generator armature for dynamic studies
To control the terminal voltage, the field current is controlled by the exciter which in turn
varies the voltage behind the reactance. A simple high gain exciter with limits can be
modeled as in Figure 3.2. For the field circuit, there is a time constant associated with the
winding inductance and resistance. Together, these describe the basic electromagnetic
time constants for a simple generator model. There is often a supplementary stabilization
control on large units referred to as a PSS (Power System Stabilizer). The interested
reader is referred to the literature.
Figure 3.2 Basic components of simple voltage regulation and excitation system
Mechanically, the generator is similar to a mass spring system with some frictional
damping. If there is a net imbalance of torque acting on the rotor, then, neglecting
damping, the machine will accelerate according to the well-known swing equations
dω m
1
= ω& m = (Tm − Te )
dt
J
(3.1)
where J is the rotational inertia, Tm is the mechanical torque input and Te the
electromagnetic torque that arises from producing an electrical power output. Recall that
power equals the torque times angular velocity so multiplying both sides of (3.1) by ω m
yields
ω& mω m =
1
( Pm − Pe )
J
(3.2)
Typically, engineers normalize the machine inertia based on the power machine rating
and use the per unit inertia constant H as
1 J (ω mo ) 2
H=
2 SB
(3.3)
with ω m0 the synchronous mechanical speed. If in addition, the speed and the real powers
are expressed on a per unit basis, then substituting (3.3) into (3.2) gives
ω& =
ωs
2H
( Pm − Pe ) or ω& =
πf s
H
( Pm − Pe )
(3.4)
where ω& is now the acceleration relative to synchronous speed. The electrical power
output is a function of the rotor angle θ and so we need to write the simple differential
equation that relates rotor angle to speed
θ& = ω − ω s
(3.5)
Thus, the mechanical equations can be expressed as a second order dynamic system. In
the next section, the analysis of this in a connected system is discussed.
3.b Power system stability analysis
The power system is a nonlinear dynamical system consisting of generators, loads and
control devices, which are coupled together by the transmission network. The dynamic
equations of the generator have been discussed in the previous section. The interactions
of the generators with the loads and the control devices can result in diverse nonlinear
phenomena. Specifically, we are interested in understanding whether the system response
can return to an acceptable operating condition following disturbances. Normally, we
distinguish between two types of disturbances in the power system context.
Minor disturbances such as normal random load fluctuations are denoted “smalldisturbances”, and the ability of the power system to damp out such small disturbances is
called “small-signal stability”. In Section 3.b.1, we will learn that small-signal stability
can be understood by computing the eigenvalues of the system Jacobian matrix that is
evaluated at an equilibrium point.
Major disturbances such as transmission line trippings and generator outages are denoted
“large disturbances” and the capability of the power system to return to acceptable
operating conditions following a large disturbance is called the “transient stability”. In
Section 3.b.2, we will study techniques for verifying the transient stability of a power
system following a specific large disturbance.
3.b.1 Small-signal stability:
Consider a nonlinear system described by the following ordinary differential equation
dx
= f ( x ), x ∈ ℜ n ,
dt
f : ℜn → ℜn ,
f is smooth
(3.6)
Suppose that x* is an equilibrium point for the system. That is, f(x*)=0. We define the
∂f
Jacobian matrix A for the equilibrium to be the matrix of partial derivatives
∂x
*
evaluated at x . Then, classical analysis in nonlinear dynamical system theory tells us
that the equilibrium x* is locally stable if all the eigenvalues of the matrix A have negative
real parts. Recall that the eigenvalues of a matrix A are defined as the solutions λi of the
polynomial matrix characteristic equation det( λ In – A ) = 0 where In denotes the n X n
identity matrix. Also, the equilibrium will be locally unstable if any one of the
eigenvalues has positive real part.
In the power system context, the concept of local stability is known as the small-signal
stability or the small-disturbance stability. We will look at a simple power system
example next for studying the concept in more detail.
A single generator that is connected to an ideal generator bus through a transmission line
is shown in Figure 3.3. The ideal generator maintains its bus voltage at 1 pu irrespective
of the external dynamics and is referred to as an infinite bus. It also defines the reference
angle for this power system and the angle is set at zero. The other generator is
represented by the classical machine model with a voltage source of E ' ∠θ connected to
the generator terminal through the transient reactance xd’. In the classical machine model,
the internal induced voltage E’ is assumed to remain constant, and the rotor angle θ
follows the second order swing equations as in (3.4) but here including a damping
component Pd
2 H &&
θ = Pm − Pe − Pd
(3.7)
ωs
This can be rewritten into the standard form of
ω −ωs


 θ& 

  =  ωs
 ω& 


(
)
P
P
P
−
−
m
e
d 
 
 2H

(3.8)
j xd’
jx
E’
θ
1 0
Figure 3.3. A simple power system
For the power system in Figure 3.3, the electrical power output Pe is easily calculated as
Pe
=
E ' sin θ
xd ' + x
= Pmax sin θ
(3.9)
where Pmax = E’/(xd’+x) is a constant. The remaining term, the damping power Pd, is
usually defined as Pd = D (ω – ωs ) where D is known as the damping constant for the
generator, which is a positive constant. Substituting the entries for Pe and Pd into (3.8),
we get the dynamic equations for the generator as
ω −ωs


 θ& 

  =  ωs
 ω& 


(
(
))
P
P
sin
θ
D
ω
ω
−
−
−
m
max
s 
 
 2H

(3.10)
The equilibrium points for (3.10) can be easily solved by setting the derivatives to be
zero. (θ*, ωs)T is an equilibrium point for this system if Pm = Pmax sin θ*. The equilibrium
points can be identified visually by plotting Pe and Pm as shown in Figure 3.4. The
intersection of the constant line Pm with the curve Pmax sin θ highlighted by the black
marks depict the two possible equilibrium points when Pm < Pmax. Let us denote the
equilibrium point between 0 and π/2 by θs and the other equilibrium between π/2 and π by
θu.
Pmax sin θ
θs
θu
Pm
θ
Figure 3.4 Plot of Pe and Pm
We will show that the equilibrium (θs, ωs)T is small-signal stable, while the other
equilibrium (θu, ωs)T is small-signal unstable. For assessing the small-signal stability, we
need to evaluate the Jacobian matrix A at the respective equilibrium point. The Jacobian
is first derived as
∂f
∂x
0
1


ωs

=  ωs
 2 H (− Pmax cosθ ) 2 H (− D )
(3.11)
The eigenvalues of the system matrix A can easily be computed at the equilibrium point
(θs, ωs)T as the solutions of the second order polynomial
λ2 +
ωs
2H
Dλ +
ωs
2H
Pmax cosθ s
= 0
(3.12)
Since θs by definition lies between 0 and π/2, cos θs is positive. Therefore, the two roots
of the characteristic equation (3.12) have negative real parts and hence the equilibrium
(θs, ωs)T is small-signal stable. Therefore, the system will return to the equilibrium
condition following any small perturbations away from this equilibrium point. On the
other hand, when the equilibrium point (θu, ωs)T is considered, note that cos θu is negative
since θu lies between π/2 and π. Then, the characteristic equation (3.12) will have one
positive real eigenvalue and one negative real eigenvalue, when θs is replaced by θu.
Therefore, the equilibrium (θu, ωs)T is small-signal unstable.
Normally, the power system is operated only at the stable equilibrium point (θs, ωs)T. The
unstable equilibrium point (θu, ωs)T also plays an important role in determining the large
disturbance response of the system that we study in the next subsection. Unlike this
simple system, assessing the small-signal stability properties of a realistic large power
system is an extremely challenging task computationally.
3.b.2 Transient stability:
Assume that the power system is operating at a small-signal stable equilibrium point.
Suppose the system is suddenly subject to a large disturbance such as a line fault. Then,
the power system protective relays will detect the fault and they will isolate the faulty
portion of the network by possibly tripping some lines. The occurrence of fault and the
subsequent line trippings will cause the system response to move away from the
equilibrium condition. After the fault has been cleared, the ability of the system to return
to nominal equilibrium condition is called the transient stability for that fault scenario.
We will study a transient stability example for the simple system in Figure 3.5.
jx
j xd’
jx
E’
θ
1 0
Figure 3.5 Pre-fault power system
Suppose we want to study the occurrence of a solid three-phase-to-ground fault at the
middle point of the lower transmission line in Figure 3.5. We usually assume that the
system is operating at a nominal equilibrium point before the fault occurs. For this prefault system, the effective transmission line reactance is x/2, since there are two
transmission lines in parallel each with reactance x. The dynamic equations for the prefault system are then given by
ω −ωs


 θ& 


  =
ω
pre
 ω& 
 s (Pm − Pmax

(
)
)
sin
θ
−
D
ω
−
ω
s 
 
 2H

(3.13)
E'
. The equilibrium points can be solved by setting the derivatives to
x d '+ x
2
zero in (3.13) and let us denote the stable equilibrium by (θspre, ωs)T where θspre is the
equilibrium solution of the rotor angle between 0 and π/2.
pre
where Pmax
=
Next, let us say that the fault occurs at time t=0. When the solid fault is present at the
middle of the lower transmission line, the system in Fig. 3.3 changes to the configuration
shown in Fig. 3.6.
jx
j xd’
j x/2
E’ θ
1 0
Figure 3.6 Fault-on power system
The computation of the generator electrical power output Pe for the fault-on system in
Fig. 3.4 requires a little more work. Looking for the generator terminal bus, the effective
x/2
1
Thevenin voltage is
∠0 which is ∠0 . Next, the effective Thevenin reactance
x+ x/2
3
is the parallel equivalent of reactance x and the reactance x/2, which is x/3. Therefore, the
electrical power Pe during the fault-on period is given by
Pe fault
fault
sin θ
= Pmax
=
E' 3
sin θ
x d '+ x 3
(3.14)
The dynamic equations for the fault-on system are then given by
ω −ωs


 θ& 

  =  ωs
fault
 ω& 


(
(
)
)
P
P
sin
θ
D
ω
ω
−
−
−
m
max
s 
 
 2H

(3.15)
Let us assume that the relays clear the fault at time t=tc by opening the lower transmission
line at both the sending and receiving end. Then, the system configuration becomes as
shown in Figure 3.3, and the dynamic equations for the post-fault system are given by
ω −ωs


 θ& 

  =  ωs
post
 ω& 

(Pm − Pmax sin θ − D(ω − ω s ))
 
 2H
(3.16)
E'
. If the system settles down to the stable equilibrium of the postx d '+ x
fault system (3.16) after the fault is cleared, we say that the system is transient stable for
the fault under study. In order to verify this numerically, we proceed as follows.
post
where Pmax
=
Transient stability study by numerical integration:
(1) The system is at (θspre, ωs)T before the fault occurs. Therefore, we start the
simulation with (θ, ω)T = (θspre, ωs)T at time t=0.
(2) During the fault-on period, the system dynamics is governed by (3.15) and the
fault-on period is from time t=0 through time t=tc. Therefore, we integrate the
equations (3.15) starting from the initial condition (θspre, ωs)T at time t=0 for a
time period from t=0 to t=tc. Let us denote the system state at the end of faulton period numerical integration at time t=tc by the clearing state (θc, ωc)T.
(3) At time t=tc, the system equations change to the post-fault equations (3.16).
Therefore, we integrate the equations (3.16) starting from the clearing state
(θc, ωc)T at time t=tc for a period of several seconds to assess whether the
system response converges to the stable equilibrium (θspost, ωs)T or not. If the
post-fault response settles down to the nominal equilibrium, we can determine
the system to be transient stable for the disturbance. If the post-fault system
response diverges away, the system is not transient stable.
The steps involved in the numerical integration procedure for a realistic large system are
similar to the outline above for the simple system. The models are of very large
dimensions for real-size power systems. Hence, the formulation of the pre-fault, fault-on
and post-fault models becomes a nontrivial task, and the numerical integration becomes
highly time-consuming.
Equal area criterion:
In order to verify the transient stability, it is possible to derive analytical conditions for
the power system in Figure 3.4. For simplicity, we assume that the damping constant
D=0, and that the fault clearing is instantaneous (tc=0). We recall that the dynamics of the
pre-fault system are described by
2 H &&
pre
θ = Pm − Pmax
sin θ
ωs
pre
where Pmax
=
(3.17)
E'
. Similarly, the equations for the post-fault system are described by
x d '+ x
2
2 H &&
post
θ = Pm − Pmax
sin θ
ωs
(3.18)
E'
pre
post
. Clearly, it follows that Pmax
> Pmax
since x/2 < x. Intuitively, more
x d '+ x
power can be transferred in the pre-fault configuration with two parallel lines as
compared to one transmission line present in the post-fault configuration.
post
where Pmax
=
As stated earlier, the system is operating at the pre-fault equilibrium (θspre, ωs)T before the
Pm
fault occurs where sin θ spre = pre
, and θspre lies between 0 and π/2. When the fault is
Pmax
cleared instantaneously at time t=0, we would like to know whether the transient starting
from (θspre, ωs)T will settle to the post-fault system equilibrium (θspost, ωs)T or whether it
will diverge away. Convergence to (θspost, ωs)T or divergence implies transient stability or
instability, respectively.
pre
post
Let us start with the analysis. First, we note that θspre < θspost since Pmax
> Pmax
. For
visualization, let us plot Pe and Pm for the post-fault system as shown in Fig. 3.7.
Pmaxpost sin θ
Pm
θspost
θupost
θspre
θ
Figure 3.7 Power-angle curve for the post-fault system
The dynamics of the transient starting from (θspre, ωs)T is governed by the second order
equation,
2 H &&
θ =
ωs
2H
ωs
post
ω& = Pm − Pmax
sin θ
(3.19)
post
Therefore, the sign of the term Pm − Pmax
sin θ determines whether the speed derivative
ω& is positive or negative. Inspecting the plot Figure 3.7, we conclude that the rotor
frequency ω increases whenever the rotor angle θ is below the Pm line since then
post
Pm − Pmax
sin θ will be positive. Similarly, the rotor speed decreases in value when the
post
angle θ is above the Pm line since then Pm − Pmax
sin θ will be negative.
Let us recall that the post-fault system response starts at (θspre, ωs)T at time t=0. As noted
earlier, the initial rotor angle θspre lies beneath the Pm line in Figure 3.7, and hence, the
speed ω increases from the initial value ω=ωs at time t=0, as soon as the fault is cleared.
When ω increases above ωs, the rotor angle starts to increase from θspre since
θ& = ϖ − ϖ s for the machine dynamics. Therefore, the rotor angle moves up on the powerangle curve in Figure 3.7 and the rotor speed keeps increasing until the rotor angle
reaches the value θspost at the intersection point with the Pm line in Figure 3.7. Let us say
that the angle θ takes time t1 seconds to increase from the initial value θspre at time t=0, to
the value θspost. During this time period from t=0 to t=t1, the speed ω has increased from
ωs to some higher value say ω1. The dynamic state of the post-fault system at time t=t1 is
then given by (θspost, ω1)T. Note that even though the rotor angle θ equals θspost at time
t=t1, the system is not in the equilibrium condition, since the speed ω equals ω1 at time t1,
and ω1 is greater than the equilibrium speed value ωs.
By construction, the speed value ω1 is defined as follows by the dynamics (3.19).
ω1 − ω s
=
t =t1
∫ ω& dt =
t =0
θ spost
ωs
∫ 2 H (P
m
post
sin θ )dθ
− Pmax
θ spre
=
ωs
2H
Aa
(3.20)
where Aa is the shaded area shown in Figure 3.8. Since the rotor acceleration θ&& has been
positive during this time period from t=0 to t=t1, the angle has been accelerating in the
area shown, and hence, the area Aa is called the “acceleration area”.
Pmaxpost sin θ
Pm
θspost
θupost
Aa
θspre
θ
Figure 3.8 Acceleration area for the post-fault system
When the transient reaches (θspost, ω1)T at time t=t1, the rotor angle keeps on increasing
since θ& = ϖ 1 − ϖ s > 0 at time t1. However, for time t>t1, the rotor angle moves above the
Pm line in Figure 3.7, and hence, the derivative of speed becomes negative. That is, the
speed ω starts to decrease from the value ω1 as the time increases from t=t1. Only after
the speed ω has decreased below the synchronous speed ωs can the rotor angle start to
decrease. Till then, the rotor angle will keep increasing and the rotor speed keeps
decreasing as time increases from t=t1.
Looking at the power-angle plot in Figure 3.8, the rotor angle stays above the Pm line
only up to the unstable equilibrium value θupost. If the rotor angle were to increase above
θspost, then the speed derivative ω& becomes positive again, and the speed will start to
increase.In this case, there is no scope for speed ω to decrease to the synchronous speed
ωs, and transient instability results. Therefore, for any chance of transient stability, and
for response to settle down around (θspost, ωs)T, we need the speed ω to decrease below ωs
before the rotor angle reaches the critical value θupost. Graphically, this implies that the
maximum deceleration area Admax shown in Figure 3.9 needs to be larger than the
acceleration area Aa shown in Figure 3.8.
Pmaxpost sin θ
Pm
θspost
θupost
Admax
θspre
θ
Figure 3.9 Maximum deceleration area for the post-fault system
When Admax > Aa, as the rotor angle decelerates past θspost from time t=t1, the deceleration
area will become equal to the accelaration area at some intermediate value of θ between
θspost and θupost at some time say t=t2. For time t>t2, the speed ω falls below ωs, and the
rotor angle θ starts to decrease back towards θspost. The alternating scenarios of rotor
angle acceleration and deceleration will continue before the angle swings are damped out
eventually by the rotor damping effects which have been ignored thus far. Therefore, we
say that the system is transient stable whenever Admax > Aa.
On the contrary, when Admax < Aa, the rotor speed stays above ωs, when the rotor angle
reaches θupost. Then, the rotor speed starts to increase away from ωs monotonously. In this
case, the rotor speed never recovers below ωs, and the rotor angle continuously keeps
increasing. The transient diverges away thus resulting in transient instability.
The analytical criterion presented in this section for the simple system can be extended to
multimachine models using Lyapunov theory and based on the concepts of energy
functions. However, development of analytical criteria for checking the transient stability
of large representative dynamic models remains a research area. Numerical integration
procedures outlined in the previous section are commonly used by the power industry for
studying the transient stability properties of large power systems.
4. Summary Remarks
This chapter has introduced the readers to the basic concepts in power system analysis,
namely modeling issues, power flow studies, and dynamic stability analysis. The
concepts have been illustrated on simple power system representations. In real power
systems, power flow studies and system stability studies are routinely carried out for
enduring the reliability and security of the electric grid separation. While the basic
concepts here have been summarized in this chapter on simple examples, the real power
systems are large-scale, nonlinear systems. The large interconnected nature of electric
networks makes the computation aspects challenging. We have highlighted some of these
issues in this section, and the readers are encouraged to refer to advanced power system
analysis textbooks for additional details.
References
A. Bergen and V. Vittal, Power Systems Analysis, Prentice Hall, 2nd Ed., New Jersey,
2000.
S. Chapman, Electric Machinery and Power System Fundamentals, McGraw-Hill, New
York, 2002.
J. Glover and M. Sarma, Power System Analysis and Design, 2nd Ed., PWS Publishing,
Boston, 1994.
J.J. Grainger and W.D. Stephenson, Power System Analysis, McGraw-Hill, New York,
1994.
P. Kundur, Power System Stability and Control, McGraw-Hill, New York, 1994.
H. Saadat, Power System Analysis, McGraw-Hill, 2nd Ed., New York, 2002.