Summary of Papers

1
Summary of Papers
1. P. Sauer and M. Pai, Power System
Steady-State Stability and the Load Flow
Jacobian, IEEE Transactions on Power
Systems, Vol. 5, No. 4, Nov. 1990 2. V. Ajjarapu
and C. Christy, The Continuation Power Flow A
Tool for Steady-State Voltage Stability
Analysis, IEEE Transactions on Power Systems,
Vol. 7, No. 1, Feb., 1992. 3. S. Greene, I.
Dobson, and F. Alvarado, Sensitivity of the
Loading Margin to Voltage Collapse with
Respect to Arbitrary Parameters, IEEE
Transactions on Power Systems, Vol. 12, No. 1,
Feb. 1997, pp. 232-240. 4. S. Greene, I. Dobson,
and F. Alvarado, Contingency Ranking for Voltage
Collapse via Sensitivities from a Single Nose
Curve, IEEE Transactions on Power Systems, Vol.
14, No. 1, Feb. 1999, pp. 262-272.
2
Voltage Security

• Voltage security is the ability of the system to
  maintain
• adequate and controllable voltage levels at all
  system load buses.
• The main concern is that voltage levels outside
  of a specified
• range can affect the operation of the customers
  loads.
• Voltage security may be divided into two main
  problems
• 1. Low voltage voltage level is outside of
  pre-defined range.
• 2. Voltage instability an uncontrolled voltage
  decline.
• You should know that
• low voltage does not necessarily imply voltage
  instability
• no low voltage does not necessarily imply
  voltage stability
• voltage instability does necessarily imply low
  voltage

3
Resources

• There have been several individuals that have
  significantly
• progressed the field of voltage security. These
  include
• Ajjarapu from ISU
• Van Cutsem See the book by Van Cutsem and
  Vournas.
• Alvarado, Dobson, Canizares, Greene

• There are a couple other texts that provide good
  treatments of
• the subject
• Carson Taylor Power System Voltage Stability
• Prabha Kundur Power System Stability Control

4
Our treatment of voltage security will proceed as
follows

• Voltage instability in a simple system
• Voltage instability in a large system
• Brief treatment of bifurcation analysis
• Continuation power flow (path following) methods
• Sensitivity methods

5
Voltage instability in a simple system
Consider the per-phase equivalent of a very
simple three phase power system given below
V1
V2
ZRjX
Node 1
Node 2
I


V2
V1
_
_
S12
SD-S12
6
Note Bgt0
Let G0. Then.
7

• Now we can get SDPDjQD-(P21jQ21) by
• - exchanging the 1 and 2 subscripts in the
  previous equations.
• - negating

Define ?12 ?1- ?2
8
Define ? is the power factor angle of the load,
i.e.,
Then we can also express SD as
Note that phi, and therefore beta, is positive
for lagging, negative for leading.
Define ßtan?. Then
9
So we have developed the following equations.
Equating the expressions for PD and for QD, we
have
Square both equations and add them to get..
10
Manipulation yields
Note that this is a quadratic in V22. As such,
it has the solution
11
Lets assume that the sending end voltage is
V11.0 pu and B2 pu. Then our previous
equation becomes
pf 0.97 lagging beta0.25 pdn0 0.1 0.2 0.3
0.4 0.5 0.6 0.7 0.78 v2nsqrt((1-beta.pdn -
sqrt(1-pdn.(pdn2beta)))/2) pdp0.78 0.7 0.6
0.5 0.4 0.3 0.2 0.1 0 v2psqrt((1-beta.pdp
sqrt(1-pdp.(pdp2beta)))/2) pd1pdn
pdp v21v2n v2p pf 1.0 beta0 pdn0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99 v2nsqrt((1-beta.pdn - sqrt(1-pdn.(pdn2
beta)))/2) pdp0.99 0.9 0.7 0.6 0.5 0.4 0.3 0.2
0.1 0 v2psqrt((1-beta.pdp
sqrt(1-pdp.(pdp2beta)))/2) pd2pdn
pdp v22v2n v2p pf .97
leading beta-0.25 pdn0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8 0.9 1.0 1.1 1.2 1.3 v2nsqrt((1-beta.pd
n - sqrt(1-pdn.(pdn2beta)))/2) pdp1.3 1.2
1.1 1.0 0.9 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 v2psqrt((1-beta.pdp sqrt(1-pdp.(pdp2bet
a)))/2) pd3pdn pdp v23v2n
v2p plot(pd1,v21,pd2,v22,pd3,v23)
You can make the P-V plot using the
following matlab code.
12
Plots of the previous equation for different
power factors
V2
Real power loading, PD
13

• Some comments regarding the PV curves
• 1. Each curve has a maximum load. This value is
  typically called the maximum system load or the
  system loadability.
• 2. If the load is increased beyond the
  loadability, the voltages will
• decline uncontrollably.
• 3. For a value of load below the loadability,
  there are two
• voltage solutions. The upper one corresponds to
  one that can be
• reached in practice. The lower one is correct
  mathematically, but I
• do not know of a way to reach these points in
  practice.
• 4. In the lagging or unity power factor
  condition, it is clear that the
• voltage decreases as the load power increases
  until the loadability.
• In this case, the voltage instability phenomena
  is detectable, i.e.,
• operator will be aware that voltages are
  declining before the
• loadability is exceeded.
• 5. In the leading case, one observes that the
  voltage is flat, or perhaps
• even increasing a little, until just before
  the loadability. Thus, in
• the leading condition, voltage instability is
  not very detectable.
• The leading condition occurs during high
  transfer conditions when the
• load is light or when the load is highly
  compensated.

14
QV Curves
We consider our simple (lossless) system again,
with the equations
Now, again assume that V11.0, and for a given
value of PD and V2, compute ?12 from the first
equation, and then Q from the second equation.
Repeat for various values of V2 to obtain a
QV curve for the specified real load PD.
You can make the P-V plot using the following
matlab code.
v11.0 b1.0 pd10.1 v21.1,1.05,1.0,.95,.90,.
85,.80,.75,.70,.65,.60,.55,.50,.45,.40,.35,.30,.25
,.20,.15 sinthetapd1./(bv1.v2) thetaasin(si
ntheta) qd1-v2.2bv1bv2.cos(theta)
plot(qd1,v2)
The curve on the next page illustrates.
15
Q-V Curve
V2
QD
16
Homework 1. Draw the PV-curve for the following
cases, and for each, determine the
loadability. a. B2, V11.0, pf0.97
lagging b. B2, V11.0, pf0.95 lagging c.
B2, V11.06, pf0.97 lagging d. B10,
V11.0, pf0.97 lagging Identify the
effect on loadability of power factor,
sending-end voltage, and line reactance. 2. Draw
the QV-curves for the following cases, and for
each, determine the maximum QD. a. B1,
V11.0, PD0.1 b. B1, V11.0, PD0.2 c.
B1, V11.06, PD0.1 d. B2, V11.0, PD0.1
Identify the effect on maximum QD of real
power demand, sending-end voltage, and line
reactance.
17
Some comments regarding the QV Curves

• In practice, these curves may be drawn with a
  power flow program
• by
• 1. modeling at the target bus a synchronous
  condenser (a
• generator with P0) having very wide reactive
  limits
• 2. Setting V to a desired value
• 3. Solving the power flow.
• 4. Reading the Q of the generator.
• 5. Repeat 2-4 for a range of voltages.
• QV curves have one advantage over PV curves
• ?They are easier to obtain if you only have a
  power flow (standard
• power flows will not solve near or below
  the nose of PV curves
• but they will solve completely around the
  nose of QV curves.)

18

• Voltage instability in a large system
• Influential factors
• Load modeling
• Reactive power limits on generators
• Loss of a circuit
• Availability of switchable shunt devices

Two important ideas on which understanding of the
above influences rest

• Voltage instability occurs when the reactive
  power supply cannot meet the reactive power
  demand of the network.
• Transmission line loading is too high
• Reactive sources (generators) are too far from
  load centers
• Generator terminal voltages are too low.
• Insufficient load reactive compensation
• 2. Reactive power cannot be moved very far in a
  network
• (vars do not travel), since I2X is large.

Implication The SYSTEM can have a var surplus
but experience voltage instability if a local
area has a var deficiency.
19
Load modeling
In analyzing voltage instability, it is necessary
to consider the network under various voltage
profiles. Voltage stability depends on the
level of current drawn by the loads. The level
of current drawn by the loads can depend on the
voltage seen by the loads. Therefore, voltage
instability analysis requires a model of how
the load responds to load variations. Thus, load
modeling is very influential in voltage
instability analysis.
20
Exponential load model
A typical load model for a load at a bus is the
exponential model
where the subscript 0 indicates the initial
operating conditions.
The exponents ? and ? are specific to the type of
load, e.g., ? ? Incandescent
lamps 1.54 - Room air conditioner 0.50 2.5
Furnace fan 0.08 1.6 Battery
charger 2.59 4.06 Electronic compact
florescent 1.0 0.40 Conventional
florescent 2.07 3.21
21
Polynomial load model
The ZIP or polynomial model is a special case of
the more general exponential model, given by a
sum of 3 exponential models with specified
subscripts
where again the subscript 0 indicates the initial
operating conditions.
Usually, values p2 and q2 are the largest.

• So this model is composed of three components
• constant impedance component (p1, q1) - lighting
• constant current component (p2, q2)
  motor/lighting
• constant power component (p3,, q3) loads
  served by LTCs

22
Effect of Load modeling
Understanding the effect of each component on
voltage instability depends on understanding two
ideas 1. Voltage instability is alleviated when
the demand reduces. This is because I
reduces and I2X reactive losses in the circuits
reduce. 2. Since voltage instability causes
voltage decline, alleviation of voltage
instability results if demand reduces with
voltage decline. This gives the key to
understanding the effect of load modeling.

• constant impedance load (p1) is GOOD since
  demand
• reduces with square of voltage.
• constant current load (p2) is OK since demand
  reduces
• with voltage.
• Constant power load (p3) is BAD since demand
  does
• not change as voltage declines.

23
Some considerations in load modeling

• The effects of voltage variation on loads, and
  thus of loads on voltage instability, cannot be
  fully captured using exponential or
• polynomial load models because of the following
  three aspects.
• Thermostatic load recovery
• Induction motor stalling/tripping
• Load tap changers

24
Thermostatic load recovery
Heating load is the most common type of
thermostatic load, and it is one for which we are
all quite familiar. Although much heating is done
with natural gas as the primary fuel, some
heating is done electrically, and even gas
heating systems always contain some electric
components as well, e.g., the fans. Other
thermostatic loads include space heaters/coolers,
water heaters, and refrigerators. When voltage
drops, thermostatic loads initially decrease in
power consumption. But after voltages remain low
for a few minutes, the load regulation devices
(thermostats) will start the loads or will
maintain them for longer periods so that more of
them are on at the same time. This is referred to
as thermostatic load recovery, and it tends to
exacerbate voltage problems at the high voltage
level.
25
Induction motor stalling/tripping
Three phase induction motors comprise a
significant portion of the total load and so its
response to voltage variation is
important, especially since it has a rather
unique response. Consider the steady-state
induction motor per-phase equivalent model.
I2
ZaR1jX1
X2
Zb Rc//jXm
V1
R2R2(1-s)/s R2 / s
26
Induction motor stalling/tripping
The (referred to stator) rotor current is given
by
where
and
Under normal conditions, the slip s is typically
very small, less than 0.05 (5). In this case,
R2/s gtgt R2, and I2 is small. But as voltage
V1 decreases, the electromagnetic torque
developed decreases as well, the motor slows
down. Ultimately, the motor may stall. In this
case, s1, causing R2/s R2. Thus, one sees
that the current I2 is much larger for stalled
conditions than for normal conditions. Because of
X1 and X2 of the induction motor, the large
stall current represents a large reactive
load. Large motors have undervoltage tripping to
guard against this, but smaller motors
(refrigerators/air conditioners) may not.
27
Tap changers
Load tap changers (LTC, OLTC, ULTC, TCUL) are
transformers that connect the transmission or
subtransmission systems to the distribution system
s. They are typically equipped with regulation
capability that allow them to control the voltage
on the low side so that voltage deviation on the
high side is not seen on the low side.
t1
V1 and t are given in pu.
HV side
V1/t
LV side
V1

• In per unit, we say that the tap is t1, where
• t may range from 0.85-1.15 pu
• a single step may be about 0.005 pu
  (5/80.00625 is very common)
• a change of one step typically requires about 5
  seconds.
• there is a deadband of 2-3 times the tap step to
  prevent excessive tap change.

Under low voltage conditions at the high side,
the LTC will decrease t in order to try and
increase V1/t.
28
Tap changers
Thus, as long as the LTC is regulating (not at a
limit), a voltage decline on the high side does
not result in voltage decline at the load, in
the steady-state, so that even if the load is
constant Z, it appears to the high side as if it
is constant power. So a simple load model for
voltage instability analysis, for systems using
LTC, is constant power! There are 2
qualifications to using such a simple model
(constant power) 1. Fast voltage dips are seen
at the low side (since LTC action typically
requires minutes), and if the dip is low enough,
induction motors may trip, resulting in an
immediate decrease in load power. 2. Once the
LTC hits its limit (minimum t), then the low
side voltage begins to decline, and it becomes
necessary to model the load voltage sensitivity.
29
Generator capability curve
Field current limit due to field
heating, enforced by overexcitation limiter on If.
Q
Qmax
Armature current limit due to armature heating,
enforced by operator control of P and If.
Typical approximation used in power flow
programs.
P
Qmin
Limit due to steady-state instability (small
internal voltage E gives small EVBsin?), and
due to stator end-region heating from induced
eddy currents, enforced by underexcitation
limiter (UEL).
30
Effect of generator reactive power limits
1. Voltage instability is typically preceded by
generators hitting their upper reactive
limit, so modeling Qmax is very important to
analysis of voltage instability. 2. Most power
flow programs represent generator Qmax as fixed.
However, this is an approximation, and one
that should be recognized. In reality, Qmax is
not fixed. The reactive capability diagram
shows quite clearly that Qmax is a function of P
and becomes more restrictive as P increases.
A first-order improvement to fixed Qmax is to
model Qmax as a function of P. 3. Qmax is set
according to the Over-eXcitation Limiter (OXL).
The field circuit has a rated steady-state
field current If-max, set by field circuit
heating limitations. Since heating is
proportional to , we see that smaller
overloads can be tolerated for longer times.
Therefore, most modern
OXLs are set with a time-inverse
characteristic 4. As soon as the OXL acts to
limit If, then no further increase in
reactive power is possible. When drawing PV
or QV curves, the action of a generator
hitting Qmax, will manifest itself as a
sharp discontinuity in the curve.
2.0
OXL characteristic
If Irated
120
1.0
10
Overload time (sec) ?
31
Effect of OXL action on PV curve
One generator hits reactive limit
V
?
No reactive limits modeled
o
P
(demand)
Note Georgia Power Co. models its loadability
limit at point x, not point o.
32
Loss of a circuit
Compare reactive losses with and without second
circuit
Assume both circuits have reactance of X.
I/2
I
X
X
I/2
P
P
Qloss(I/2)2X (I/2)2XI2X/2
QlossI2X
Implication Loss of a circuit will always
increase reactive losses in
the network. This effect is compounded by the
fact that losing a circuit
also means losing its line charging
capacitance.
33
Kundur, on pp. 979-990, has an excellent example
which illustrates many of the aforementioned
effects. The illustration was done using a
long-term time domain simulation program
(Eurostag).
34
Influence of switched shunt capacitors
I
I
P
P
V
With capacitor
Without capacitor
P
(demand)
35

• But, shunt compensation has some drawbacks
• It produces reactive power in proportion to the
  square of the
• the voltage, therefore when voltages drop, so
  does the reactive
• power supplied by the capacitor.
• It has a maximum compensation level beyond which
  stable
• operation is not possible (See pg. 972 of
  Kundur, and next slide).

(A synchronous condenser and an SVC do not have
these 2 drawbacks)

• It results in a flatter PV curve and therefore
  makes voltage
• instability less detectable. Therefore, as the
  load grows in areas
• lacking generation, more and more shunt
  compensation is used to
• keep voltages in normal operating ranges. By so
  doing, normal
• operating points progressively approach
  loadability.

36
V11.0
Each QV curve/Capacitor characteristic
intersection shows the operating point. Note that
for the first three operating points, a small
increase in Q-comp (indicated by arrows) results
in voltage increase, but for the last operating
point (950), more Q-comp (say 960) results in a
voltage decrease.
V2
PL QL0
SV22BSbase with V21.0
V2
675 Mvar
450 Mvar
300 Mvar
1.2
950 Mvar
PL1300 mw
1.0
PL1900 mw
0.8
QV-curves drawn using synchronous condensor
approach.
PL1700 mw
PL1500 mw
0.6
400
600
800
1000
1200
1400
1600
200
Capacitive Mvars
37
Bifurcation analysis (ref A. Gaponov-Grekhov,
Nonlinearities in action and also Van Cutsem
Vournas, Voltage stability of electric power
systems.)
A bifurcation, for a dynamic system, is an
acquisition of a new quality by the motion the
dynamic system, caused by small changes in its
parameters. A power system that has experienced
a bifurcation will generally have corresponding
motion that is undesirable.
Consider representing the dynamics of the power
system as
Eqts. 1
A differential-algebraic system (DAS) Here x
represents state variables of the system (e.g.,
rotor angles, rotor speed, etc), y represents the
algebraic variables (bus voltage magnitudes
voltage angles), and p represents the real and
reactive power injections at each bus. The
function F represents the differential equations
for the generators, and the function G
represents the power flow equations.
38
Types of bifurcations

• There are at least two types of bifurcation
• Hopf two eigenvalues become purely imaginary
• a birth of oscillatory or periodic motion.
• Saddle node a disappearance of an equilibrium
  state.
• The stable operating equilibrium coalesces with
  an unstable
• equilibrium and disappears. The dynamic
  consequence of a
• generic saddle node bifurcation is
• a monotonic decline in system variables.

So we think it is the saddle node bifurcation
that causes voltage instability.
39
The unreduced Jacobian
The Jacobian matrix of eqts. 1 is
and it is referred to as the unreduced Jacobian
of the DAS, where
Eqt. 2
40
The reduced Jacobian
We may reduce eq. 2 by eliminating the variable
??y
This means we need to force the top right hand
submatrix to 0, which we can do by multiplying
the bottom row by -FYGY-1 and then adding to the
top row.
This results in
So that the reduced Jacobian matrix is a Schurs
complement
41
Stability

• Fact 1 The conditions for a saddle node
  bifurcation are
• Equilibrium
• Singularity of the unreduced Jacobian
• ? det(J)0 (a 0 eigenvalue, J noninvertible) .

Implication 1 The stability of an equilibrium
point of the DAS depends on the eigenvalues of
the unreduced Jacobian J. The system will
experience a SNB as parameter p increases when J
has a zero eigenvalue.
Fact 2 The determinant of a Schurs complement
times the determinant of GY gives the determinant
of the original matrix det(J)det(A)det(GY) i
f GY is nonsingular.

• Implications 2
• If GY is nonsingular, then singularity of A
  implies singularity of J so that we may analyze
  eigenvalues of A to ascertain stability.
• The fact that GY may be nonsingular, yet A
  singular, means that load flow convergence is not
  a sufficient condition for voltage stability.

42
Singularity of load flow Jacobian

• Implications 2
• If GY is nonsingular, then singularity of A
  implies singularity of J so that we may analyze
  eigenvalues of A to ascertain stability.
• The fact that GY may be nonsingular, yet A
  singular, means that load flow convergence is not
  a sufficient condition for voltage stability.

Singular (unstable)
Singular
Singular
Nonsingular (stable)
Nonsingular
Nonsingular
A
GY
J
43
Singularity of load flow Jacobian
So voltage instability analysis using only a load
flow Jacobian may yield optimistic results when
compared to results from analysis of A, that is,
stable points (based on Gy) may not be really
stable. gt However, I believe it is true that
points identified as unstable using the load flow
Jacobian will be really unstable (Schurs
complement does not support that singularity of
GY implies singularity of J, however, because it
is only valid if GY is nonsingular). Note
Sauer and Pai, 1990, provide an in-depth analysis
of the relation between singularity of GY and
singularity of J, and show some special cases for
which singularity of GY implies singularity of J.
Singular (unstable)
Singular (unstable)
Singular (unstable)
Nonsingular (stable)
Nonsingular (stable)
Nonsingular (stable)
A
GY
J
44
Singularity of load flow Jacobian
So, we assume that load flow Jacobian analysis
provides an upper bound on stability. Fact The
bifurcation (zero eigenvalue of GY) of the load
flow Jacobian corresponds to the turn-around
point (i.e., the nose point) of a P-V or Q-V
curve drawn using a power flow program.
This can be proven using an optimization
approach. See pp. 218-220 of the text by Van
Cutsem and Vournas.
We have previously denoted the power flow
equations as G(x,y,p)0, but now we denote them
as G(y,p)0, without the dependence on the state
variables x (which relate to the machine modeling
and include, minimally, ? and ? of each machine).
45

• So we turn our effort to identifying the saddle
  node bifurcation
• (SNB) for the power flow Jacobian matrix.
• The Jacobian can reach a SNB in many ways. For
  example,
• increase the impedance in a key tie line
• increase the generation level at a generator
  with weak transmission, while
• decreasing generation at all other generators.
• increase the load at a single bus
• increase the load at all buses.
• In all cases, we are looking for the nose point
  of the
• V-? curve, where ? is the parameter that is being
  increased.)
• Most applications focus on the last method
  (increase load at all buses).
• Key questions here are
• direction of increase are bus loads increased
  proportionally, or in some other way?
• dispatch policy how do the generators pick up
  the load increase ?
• We will assume proportional load increase with
  governor load flow
• (generators pick up in proportion to their rating)

V
?
?
46
Define critical point - the operating
conditions, characterized by a
certain value of ?, beyond which operation is
not acceptable.
Question 1 What can cause the critical point to
differ from the SNB point ?
V
?
?
Question 2 How can knowledge of the critical
point provide a security measure?
Question 3 Does the P-V curve provide a forecast
of the system trajectory ?
47
Solution approaches to finding ?, the value of ?
corresponding to SNB.
Approach 1 Search for ? using some iterative
search procedure.
1. i1 2. Using ?(i), solve power flow using
Newton-Raphson. Here, we iteratively solve
G(y,p)0. At each step, we must solve for ?y
in the eqt GY ?y ?p 3. If solved, ?(i1)
?(i)? ?. ii1 go to 2 else if not solved,
? ?(i1) endif 4. End
But big problem as ? gets close to ?, GY
becomes ill-conditioned (close to singular).
This means that at some point before the
critical point, step 2 will no longer be feasible.
48
Approach 2 Use the continuation power flow (CPF).
Predictor step
Corrector step
Pass ? ?
No.
Select continuation parameter
Yes.
Stop
49
The predictor step
The power flow equations are functions of the bus
voltages and bus angles and the bus injections
Augment the power flow equations so that they are
functions of ? (dependence on p is carried
through the dependence on ?).
p??p0 ?
Now recognize that
so that

• If we want to compute the change in the power
  flow equations dG
• due to small changes in the variables ?, V, and
  ?,
• that move us closer to the loadability point
• as we move from one solution i to another
  close solution i1, then
• dG G(?(i),V(i),?(i))- G(?(i1),V(i1),?(i1))
  0 0 0

50
Here, each set of partial derivatives are
evaluated at the operating conditions
corresponding to the old solution. If the power
flow equations are linear with the 3 sets of
variables in the region between the old solution
and the (close) new one, the following is
satisfied
Eq. 3
BUT, we have added one unknown, ?, to the power
flow problem without adding a corresponding
equation, i.e., in G(?,V,?)0, there are are N
equations but N1 variables, so that in eq. 3,
the matrix G? GV, G?, has N rows (the number of
eqts being differentiated) and N1 columns (the
number of variables for which each eqt is
differentiated). So we need another equation in
order to solve this. What to do ?
51
The answer to this can be found by identifying
how we will be using using the solution to eqt.
3. Note the solution corresponding to the new
point is
Here the p indicates that this is the
predicted point.
If we define ? to be the step size, then we can
rewrite this as
where
52
We call the update vector (with the
differentials) the tangent vector, denoted by
t.
This vector provides the direction to move in
order to find a new solution (i1,p) from the
old one (i). We can think of this in terms of
the following picture..
53
Tangent vector
V
?
54
Note In specifying a direction using an
n-dimensional vector, only n-1 of the elements
are constrained - one element can be chosen to be
any value we like.
For example, consider a 2-dimensional vector.
x2x1tan(30) so - the direction is specified
by selecting x11, x20.5774, - the direction
is specified by selecting x10.5, x20.2246.
x2
Direction 30o
x1
So we can set one of the tangent vector elements
to any value we like, then compute the other
elements. This provides us with our other
equation.
55
Suppose that we set the k-th parameter in the
tangent vector to be ?1.0. Then our equation
given as eq. 3 can be augmented to become
where


0
...
0
1
0
...
0
0

e
k





k


To select ?, we would have
Which would force d?1.


1
...
0
0
0
...
0
0

e
k
56
The parameter for which we select k is called the
continuation parameter, and it can be any load
level (or group of load levels), or it can be a
voltage magnitude. Initially, when the solution
is far from the nose, the continuation parameter
is typically ?.
The parameter ? is called the step size, and it
can be selected using various techniques. The
simplest of these is to just set it to a
constant. Lets try this on our simple problem
formulated at the beginning of these slides.
57
HOMEWORK 2, Due Monday, Jan 26. 1. Using the
equations at the bottom of slide 7, with the
left-hand side (PD and QD) and also V1 given
by the problem statement, we know everything
except V2 and theta. 2. Now, just bring the right
hand side of these 2 equations over to the
left-hand side, and you have the 2 equations that
correspond to G(y,p)0. 3. Solve these
equations to get the corresponding power flow
solution (but you do not need Newton-Raphson
to do this you can just use the equation at
the bottom of slide 10). 4. Now you need to
replace the value specified in the equations for
PD (which is 0.4 according to the problem
statement) with 0.4lambda. This gives you
the equations in the form of slide 49
0G(theta,V,lambda). Note, however, that G is
really two equations G1 and G2. 5. Now you need
to formulate the equations on the slide 55. This
is a matter of taking derivatives and then
evaluating those derivatives at the solution
that you obtained above. Note, however, the each
element in the matrix of slide 55 actually
represents 2 elements. That is
dG1/dtheta dG1/dV dG1/dlambda
dG2/dtheta dG2/dV dG2/dlambda 0 0
1 6. Evaluate each of the above
matrix elements at the solution obtained in
step 3. 7. Then solve these equations for the
tangent vector. You can do this by inverting
the above matrix (use matlab or a calculator to
do this) and then multiply the
right-hand-side by this inverted matrix. 8. Then
take a step using an appropriately chosen step
size per the equation on slide 56. 9.
Beginning from your predicted point that you
identified in step 8 of 2a, develop equations
for approach a, solve them, and identify the
resulting corrected point in terms of voltage and
power. 10. Repeat 9 except implement approach
b.
9 and 10 will be explained in next few slides.
58
Corrector step
Note, however, that the predicted point will
satisfy the power flow equations only if the
power flow equations are linear, which they are
not. So our point needs correction. This leads
to the corrector step. There are two different
approaches for performing the corrector step.
Approach a Perpendicular intersection method.
Approach b Parameterization method
59
Approach a perpendicular intersection Here, we
find the intersection between the power
flow equations (the PV curve) and a plane that is
perpendicular to the tangent vector.
V
? t
y(i)
Solve simultaneously, for y(i1)
y(i1,p)
y(i1)
The last equation says the inner (dot) product
of 2 ? vectors is zero.
?
Use Newton-Raphson to solve the above (requires
only 1-3 iterations since we have good starting
point). If no convergence, cut step size (?) by
half and repeat.
60

• Approach b Parameterization
• The corrector step is performed by
• identifying a continuation parameter (see slide
  62) can be ?
• fixing it at the value found in the predictor
  step
• then solving the power flow equations.

V
? t
y(i)
Solve simultaneously, for y(i1)
y(i1,p)
Vertical corrections correspond to a
fixed load-continuation parameter,
horizontal corrections to a fixed voltage-continua
tion parameter.
y(i1)
?
Here, yk(i1) is the continuation parameter it
is the variable yk(i1) that corresponds to the
k-th element dyk(i1) in the tangent vector and
is usually ? at first but often becomes something
else as the nose point is neared. The parameter
? is the value to which yk is set, which would
be the value found in the predictor step. As in
approach a, we can solve this using
Newton-Raphson. If no convergence, cut step size
(?) by half and repeat.
61
Detection of critical point
We will know that we have surpassed the critical
point when the sign of d? in the tangent vector
becomes negative, because it is at this point
where the loading reaches a maximum point and
begins to decrease.
? increasing
V
x
? decreasing
?
62
Selection of continuation parameter

• The continuation parameter is selected from among
  ?
• and the state variables in y according to the one
  that is
• changing the most with ?. This will be the
  parameter that
• has the largest element in the tangent vector.
• relatively unstressed conditions (far from
  nose) generally ?
• relatively stressed conditions (close to nose)
  generally the
• voltage magnitude of the weakest bus, as it
  changes a great
• deal as ? is changed, when we are close to ?.

The one changing the most with ? is most
sensitive and represents a variable that we
want to be careful with as we look for another
solution, so it makes sense to keep it
constant.
Typically, yk is going to be one of these.
63
Selection of continuation parameter (unstressed
condition)

• The continuation parameter is selected from among
  ?
• and the state variables in y according to the one
  that is
• changing the most with ?. This will be the
  parameter that
• has the largest element in the tangent vector.
• relatively unstressed conditions (far from
  nose) generally ?.
• gt This looks like below.

V
y(i)
y(i1,p)
y(i1)
Here, ? is fixed.
?
64
Selection of continuation parameter (stressed
condition)

• relatively stressed conditions (close to nose)
  generally the
• voltage magnitude of the weakest bus. Here, the
  voltage being
• plotted is chosen as the continuation parameter.

V
y(i)
y(i1,p)
y(i1)
Here, V is fixed.
?
Essentially, a variable is fixed as a parameter
(the voltage), and the parameter (?) is treated
as a variable. This process of selecting a
variable to fix is sometimes called the
parameterization step. -Scott Greene, Ph.D.
dissertation, 1998.
65
A central question How does the continuation
technique alleviate the ill- conditioning problem
experienced by a regular power flow ?
66
The Jacobian of the power flow equations is just
Gy, but the Jacobian of the equations in the two
corrector approaches will have an extra row and
column.
Here, C is the additional equation, and xk is the
selected continuation parameter.
This addition of a row and column to the Jacobian
has the effect of improving the conditioning so
that the previously singular points can in fact
be obtained. In other words, the additional row
and column provides that this Jacobian is
nonsingular at ? where the standard Jacobian is
singular.
67
Known codes for continuation methods

• Claudio Canizarres at University of Waterloo
  C-code
• See http//www.power.uwaterloo.ca/claudio/claudi
  o.html
• UWPFLOW is a research tool that has been
  designed to calculate local bifurcations related
  to system
• limits or singularities in the system Jacobian.
  The program also generates a series of output
  files that
• allow further analyses, such as tangent vectors,
  left and right eigenvectors at a singular
  bifurcation
• point, Jacobians, power flow solutions at
  different loading levels, voltage stability
  indices, etc
• I have Matlab code that does it from Scott
  Greene.
• Venkataramana Ajjarapu (ISU) Fortran code
• Powertech has a program

68
Calculation of sensitivities for voltage
instability analysis
What is a sensitivity ? It is the derivative of
an equation with respect to a variable. It shows
how parameter 1 changes with parameter 2. It
is exact when parameter 2 depends linearly on
parameter 1. It is approximate when parameter 2
depends nonlinearly on parameter 1, but it
is quite accurate if it is only used close to
where it is calculated.
69
Consider the system characterized by G(y).
Then is the sensitivity of the equation G with
respect to y, evaluated at y.
G(y)
Slope is ?G/?y evaluated at y.
?y
y
y
y
Its usefulness is that once it is calculated, it
can be used to QUICKLY evaluate f(y) from
G(y)?G(y) (?G/?yy)?y, BUT ONLY AS LONG AS y
IS CLOSE TO y.
70

• Consider parameter p we desire to obtain the
  sensitivity of
• G(y,p) to p. Typical parameters p would be a bus
  load, a bus
• power factor, or a generation level.
• Very important to distinguish between
• voltage sensitivities
• voltage instability sensitivities

• What is the difference between them in terms of
• what they mean ?
• how to compute them ?

71
Sensitivities for bus voltage
These we compute at the current operating
condition. For a given continuation parameter,
they can be obtained from the first predictor
step in the continuation power flow.
V
Recall that this provides us with the tangent
vector, given by
The tangent vector is the vector of
sensitivities with respect to a small change in
?, so the portion of the vector designated as dV
is exactly the voltage sensitivities.
?
Current operating point
72
Sensitivities for voltage instability
Here, it is important to realize that the measure
of voltage instability, the loading margin,
depends on an operating condition different from
the present operating condition.
The implication is that we must look at
sensitivities of the loading margin, not of the
voltage.
V
So we want the sensitivities evaluated at this
point, i.e., the SNB point.
Loading margin
?
Current operating point
73
Derivation of loading margin sensitivities at SNB
point.
Let S be the vector of real and reactive load
powers, and k be the direction of load increase.
Also, define L as the loading margin (a scalar),
so that the load powers resulting in the SNB
point are given by
We desire to find the sensitivity of the loading
margin L to a change in the parameter p. We
denote this sensitivity by Lp.
74
Consider the system characterized by G(y,S,
p) 0 Assumption the system has a SNB at
(y,S, p), i.e., 1. G(y,S, p) 0 (an
equilibrium point) 2. Gy(y,S, p) is singular
(zero eigenvalue), and w is a left
eigenvector of Gy(y,S, p), corresponding
to the zero eigenvalue so that (by definition of
the left eigenvector) wT Gy(y,S, p)
0 wT0 Note that Gy(y,S, p), being
singular, cannot be inverted, but we can
compute it (that is, Gy (y,S, p)), and its
eigenvectors. 3. wT GS(y,S, p) ? 0
We want the sensitivity of the loading margin to
p.
75
The points (y,S, p) satisfying numbers 1 and 2
correspond to SNB points, and we can obtain a
curve of such points by varying p about its
nominal value p. Linearization of this curve
about the SNB point results in
where the notation indicates the derivatives
are evaluated at the SNB point.
Pre-multiplication by the left eigenvector w
results in
By 2 on the previous slide, the first term in
the above is zero. So...
76
Eqt.
Now recall the relation of the load powers to the
loading margin.
Substituting this expression for the load powers
into eqt. ,
And the loading margin sensitivity to parameter p
is
So p may be, for example, real power load at a
bus (to detect the most effective load shedding)
or reactive power at a bus (to determine where
to site a shunt cap).
77
Some comments about computing Lp

• The left eigenvector w must be computed for the
• Jacobian Gy evaluated at the SNB point.
• You only need to compute w and GS once,
  independent of
• how many sensitivities you need. Methods to
  compute the left eigenvector
• w include QR or inverse iteration.
• The vector of derivatives with respect to the
  parameter p, which is Gp, is
• typically sparse. For example, if you want to
  compute the sensitivity to a
• bus power, then there would be only 1 non-zero
  entry in Gp.
• The matrix of derivatives with respect to the
  load powers, GS, using constant
• power load models, is a diagonal matrix with
  ones in the rows corresponding
• to load buses. This is because a particular
  load variable would ONLY occur
• in the equation corresponding to the bus where
  it is located, and for these
• equations, these variables appear linearly
  with 1 as coefficient.

78
Some comments about extensions

• Multiple sensitivities may be computed using Gp
  (a matrix) instead of Gp (a vector).
• In this case, the result is a vector.

• Getting multiple sensitivities can be especially
  attractive when we want to find
• the sensitivity to several simultaneous
  changes. One good example is to find the
• sensitivity to changes in multiple loads.
• A special case of this is to find the
  sensitivity to changes at ALL loads, which is
• very typical, given a particular loading
  direction k . Then

• A sensitivity to a line outage may be obtained
  by letting p contain elements
• corresponding to the outaged line parameters.

79
Some comments about extensions

• A sensitivity to a line outage may be obtained
  by letting p contain elements
• corresponding to the outaged line parameters
  R (series conductance), X (series
• reactance), and B (line charging). Then use
  the multiple parameter approach.

• Here, ?p R X BT.
• Note that ?p is NOT SMALL ! Therefore ?L may
  have considerable error.
• For that reason, this one needs to be careful
  about using this approach to
• compute the actual loading margins following
  contingencies.
• However, it certainly can be used for RANKING
  contingencies. One might
• consider having a quick approximation and a
  long exact risk calculation.

80
Some comments about alternatives

• Greene, et al., also propose a quadratic
  sensitivity which requires calculation of a
  second order term Lpp . This is used together
  with the linear sensitivity according to

It requires significantly more computation but
can provide greater accuracy over a larger range
of ?p.

• Invariant Subspace Parametic Sensitivity (ISPS)
  by Ajjarapu.
• Advantages
• based on differential-algebraic model
• provides sensitivities at ANY point on the P-V
  curve