Mrinal K Pal. Power system stability
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VOLTAGE STABILITY
10-63
be a more efficient alternative to the conventional power flow program. In situations where
voltage instability is a real possibility these programs are not recommended.
A Typical Voltage Collapse Scenario
In a modern interconnected power system, a total system collapse is a relatively rare occurrence.
A study of the world’s major voltage collapse incidents reveals certain similarities in the system
operating conditions existing prior to the collapse, and the subsequent sequence of events that led
to the collapse. The initiating event may be a large disturbance, e.g., the loss of a heavily loaded
line and/or a large generating unit. This kind of contingency is considered in system planning
and in most cases this would not cause serious problem. Occasionally, however, the system may
already be overloaded beyond planning limit, and/or another line or equipment is outaged, either
inadvertently or due to protective action. This drives the system beyond its loading limit and
results in depressed voltages at the load centers. If the load is predominantly dynamic, e.g.,
motor load with fast recovery characteristics, voltage instability and collapse would follow
quickly.
A more likely scenario might, however, be the following: In a typical utility system the
composite load characteristic being voltage sensitive and/or slow recovery type, an immediate
voltage instability and collapse does not occur. Due to the depressed voltage there is
considerable load relief initially. In parts of the system some load might trip by undervoltage
protection due to excessively low voltage. At this time the system’s additional reactive demand
(caused by excessive reactive losses due to system overload) is being met by the temporary
reactive support from the remaining generators utilizing their short-term overload capability.
Due to the load relief caused by depressed voltage there is now excess generation and one or
more generators trip from overspeed. As reactive supports from these generators are lost the
voltages are depressed further resulting in further overloading of transmission lines, some of
which might start to trip by protective action. By this time some of the loads attempt to restore to
their nominal level.
By now the remaining generators’ short-term (overload) reactive capability is exhausted and the
overexcitation limiters start to bring the excitation down to the steady-state limit. There is now
serious reactive imbalance in the system. This causes widespread voltage decline and frequency
swing, resulting in tripping of more generators and loads. Soon the entire system collapses. The
collapse takes place over a period of several to many minutes. Note that speed governors, AGC
and other power plant controls which would normally respond to generation load imbalance
could not be very effective due to the severity of the problem, and, in fact, might have
aggravated the problem.
Notice that in the scenario presented there was no rotor angle instability or voltage instability in
the conventional sense. In fact, if a simulation of the initiating events were carried out using a
conventional transient stability program, everything would probably have looked normal in the
simulated time span of a few seconds. There was no transient or oscillatory instability. Voltage
did not collapse because of voltage instability -- as the events progressed, voltages went down in
steps following each event and stayed there until the next one. The system collapsed because of
the continued imbalance in generation and load, accompanied by the depletion of reactive
reserve. This was a case of overall system instability and collapse.
VOLTAGE STABILITY
10-64
If the final outcome were anticipated at the initial stage, from the overall system condition
prevailing at the beginning of the disturbance, well before the onset of the collapse, the system
might have been saved by operator action, by promptly initiating appropriate amount of load
shedding at the appropriate locations. The initial stage may not, however, indicate the need for
such actions.
References
1. IEEE VAR Management Working Group Report, “Bibliography on Reactive Power and Voltage Control,”
IEEE Trans., PWRS - 2, (2), pp.361-370, May 1987.
2. T.J.E. Miller, Reactive Power Control in Electric Power Systems, Wiley, N.Y. 1982.
3. N. Flatabo, R. Ognedal and T. Carlsen, “Voltage Stability Condition in a Power Transmission System
Calculated by Sensitivity Methods,” IEEE Trans., PWRS - 5, (4), pp1286-1293, November 1990.
4. M.K. Pal, discussion of “An Investigation of Voltage Instability Problems,” by N. Yorino, H. Sasaki, Y.
Masuda, Y. Tamura, M. Kitagawa and A. Ohshimo, IEEE Trans., PWRS - 7, (2), pp.608-609, May 1992.
5. M.K. Pal, discussion of “Fast Calculation of a Voltage Stability Index,” by P.A. Lof, T. Smed, G. Anderson
and D.J. Hill, IEEE Trans., PWRS – 7, (1), pp.61-62, February 1992.
6. M.K. Pal, discussion of “Point of Collapse Methods Applied to AC/CD Power Systems,” by C.A. Canizares,
F.L. Alvarado, C.L. DeMarco, I. Dobson and W.F. Long, IEEE Trans. PWRS - 7, (2), p.681, May 1992.
7. M.K. Pal, discussion of “Voltage Stability Evaluation Using Modal Analysis,” by B. Gao, G.K. Morison and
P. Kundur, IEEE Trans., PWRS - 7, (4), p.1537, November 1992.
8. M.K. Pal, discussion of “Point of Collapse and Continuation Methods for Large AC/DC Systems,” by C.A.
Canizares and F.L. Alvarado, IEEE Trans., PWRS - 8 (1), p.8, February 1993.
9. M.K. Pal, discussion of “Voltage Stability Analysis Using Static and Dynamic Approaches,” by G.K.
Morison, B. Gao and P. Kundur, IEEE Trans., PWRS - 8, (3), p.1166, August 1993.
10. M.K. Pal, discussion of “Identification of Generic Bifurcation and Stability Problems in Power System
Differential-Algebraic Model,” by T.Y. Guo and R.A. Schlueter, IEEE Trans., PWRS - 9, (2), p.1041, May
1994.
11. M.K. Pal, discussion of “Different Types of Voltage Instability,” by W.R. Lachs and D. Sutanto, IEEE Trans.,
PWRS - 9, (2), pp.1132-1133, May 1994.
12. M.K. Pal, discussion of “Computation of Closest Bifurcations in Power Systems,” by F. Alvarado, I. Dobson
and Y. Hu, IEEE Trans., PWRS - 9, (2), p.925, May 1994.
13. M.K. Pal, discussion of “Voltage Stability and Controllability Indices for Multimachine Power Systems,” by
C.D. Vournas, IEEE Trans., PWRS - 10, (3), p.1191, August 1995.
14. M.K. Pal, discussion of “On Bifurcations, Voltage Collapse and Load Modeling,” by C.A. Canizares, IEEE
Trans., PWRS - 10, (1), p.520, February 1995.
15. M.K. Pal, discussion of “Voltage Stability Analysis in Transient and Mid-term Time Scales,” by T. Van
Cutsem and C.D. Vournas, IEEE Trans., PWRS - 11, (1), p.153, February 1996.
16. M.K. Pal, discussion of “A Piecewise Global Small-Disturbance Voltage Stability Analysis of Structure
Preserving Power System Models,” by B. Lee and V. Ajjarapu, IEEE Trans., PWRS - 10, (4), p.1969,
November 1995.
17. M.K. Pal, discussion of “Estimating the Loading Limit Margin Taking into Account Voltage Collapse Areas,”
by J. Barquin, T. Gomes and F. Luis Pagola, IEEE Trans., PWRS - 10, (4), p.1959, November 1995.
18. M.K. Pal, “Voltage Stability Conditions Considering Load Characteristics,” IEEE Trans., PWRS - 7, (1),
pp.243-249, February 1992.
VOLTAGE STABILITY
10-65
19. M.K. Pal, “Voltage Stability: Analysis Needs, Modelling Requirements, and Modelling Adequacy,” IEE Proc.
C - 140 (4), pp.279-286, July 1993.
20. C. Rajagopalan, B. Lesieutre, P.W. Sauer and M.A. Pai, “Dynamic Aspects of Voltage/Power
Characteristics,” IEEE Trans., PWRS - 7, (3), pp.990-1000, August 1992.
21. M.K. Pal, discussion of [20].
22. J. Deuse and M. Stubbe, “Dynamic Simulation of Voltage Collapse,” IEEE Trans., PWRS - 8, (3), pp.894-
904, August 1993.
23. M.K. Pal, discussion of [22].
24. M.K. Pal, “Assessment of Corrective Measures for Voltage Stability Considering Load Dynamics,” Electrical
Power and Energy Systems, Vol.17, (5), pp.325-334, 1995.
25. C.W. Taylor, “Concepts of Undervoltage Load Shedding for Voltage Stability,” IEEE Trans., PWRD - 7, (2),
pp.480-488, 1992.
26. M.K. Pal, discussion of “Emergency Load Shedding to Avoid Risks of Voltage Instability Using Indicators,”
by T. Quoc Tuan, J. Fandino, N. Hadjsaid, J.C. Sabonnadiere and H. Vu, IEEE Trans., PWRS - 9, (1), pp.349-
350, February 1994.
27. N.W. Miller, R. D’Aquila, K.M. Jimma, M.T. Sheehan and G.L. Comegys, “Voltage Stability of the Puget
Sound System Under Abnormally Cold Weather Conditions,” IEEE Trans., PWRS - 8, (3), pp.1133-1142,
August 1993.
28. M.K. Pal, discussion of [27].
29. F. Bourgin, G. Testud, B. Heilbronn and J. Verseille, “Present Practices and Trends on the French Power
system to Prevent Voltage Collapse,” IEEE Trans., PWRS - 8, (3), pp.778-788, August 1993.
30. M.K. Pal, discussion of [29].
31. S. Abe, Y. Fukunaga, A. Isono and B. Kondo, “Power System Voltage Stability,” IEEE Trans., PAS - 101,
pp.3830-3840, 1982.
32. H. Ohtsuki, A. Yokoyama and Y. Sekine, ”Reverse Action of On-Load Tap Changer in Association with
Voltage Collapse,” IEEE Trans., PWRS - 6, (2), pp.300-306, May 1991.
33. A.E. Fitzgerald and C. Kingsley, Jr., Electric Machinery, McGraw-Hill, N.Y. 1961.
A-1
APPENDIX A
REVIEW OF MATRICES
The set of linear equations
mnmnmm
nn
nn
yxaxaxa
yxaxaxa
yxaxaxa
=+++
=+++
=
+
+
+
LL
LLLL
LLLL
LL
LL
2211
22222121
11212111
(A.1)
constitutes a set of relationships between the variables ,,,,
21 n
xxx L and the variables
.,,,
21 n
yyy L This relationship, or linear transformation of the x variables into the y variables is
completely characterized by the ordered array of the coefficients a
ij
. If this ordered array is
denoted by
A, and written as
=
mnmm
n
n
aaa
aaa
aaa
L
MOMM
L
L
21
22221
11211
A
(A.2)
then the set of linear equations can be written as
Ax = y, by a suitable definition of the product
Ax.
A matrix is a rectangular array such as shown in equation (A.2) that obeys certain rules of
addition, subtraction, multiplication, and equality.
The elements of the matrix, ,,,,
1211 ij
aaa L are written with a double subscript notation. The
first subscript indicates the row where the element appears in the array, and the second subscript
indicates the column. The elements may be real or complex, or functions of specified variables.
A matrix with m rows and n columns is called an m × n (m by n) matrix or is said to be of order
(m, n). For a square matrix (m = n) the matrix is of order n. It is common to use bold-faced
letters to denote matrices.
Types of matrices
Column matrix: An m × 1 matrix is called a column matrix or a column vector, such as
=
m
a
a
a
M
2
1
a
Row matrix: A matrix containing a single row of elements, such as a 1 × n matrix, is called a row
matrix or a row vector.
REVIEW OF MATRICES
A-2
Diagonal matrix: The principal diagonal of a square matrix consists of the elements a
ii
. A
diagonal matrix is a square matrix whose off-diagonal elements are zero. Denoting the diagonal
matrix by
D,
=
nn
d
d
d
O
22
11
D
Unit matrix: A unit matrix is a diagonal matrix whose principal diagonal elements are equal to
unity. It is often denoted by
I.
=
1
1
1
O
I
Null matrix: A matrix which has all of its elements identically equal to zero is called a zero or
null matrix.
Transpose of a matrix: Transpose of a matrix
A is obtained by interchanging the rows and
columns of
A. The transpose of a matrix A is denoted by A′ (or A
T
). If A is an m × n matrix, then
A′ is an n × m matrix.
Symmetric matrix: A matrix is said to be symmetric if it is equal to its transpose, i.e., if
A = A′.
Skew-symmetric matrix: A matrix is said to be skew-symmetric if
A = −A′. This, of course,
implies that the elements on the principal diagonal are identically zero.
Conjugate matrix: If the elements of the matrix
A are complex, then the conjugate matrix B has
elements which are complex conjugates of the elements of
A. This is written as B = A
*
.
Addition of matrices
If two matrices A and B are both of order (m, n), then the sum of these two matrices is the matrix
C, C = A + B, where the elements of C are defined by
ijijij
bac
+
=
(A.3)
Addition of matrices is commutative and associative, i.e.,
A + B = B + A (commutative)
A + (B + C) = (A + B) + C (associative)
Subtraction of matrices
The difference of two matrices A and B, both of order (m, n), is the matrix D, D = A - B, where
the elements of
D are defined by
ijijij
bad
−
=
(A.4)
REVIEW OF MATRICES
A-3
Equality of matrices
Two matrices A and B, which are of equal order, are equal if their corresponding elements are
equal. Thus
A = B, if and only if a
ij
= b
ij
.
Multiplication of matrices
Consider the set of equations
2221212
2121111
xaxay
xaxay
+=
+
=
(A.5)
Equation (A.5) can be visualized as the matrix
A transforming the vector x into the vector y. This
transformation can be written in the form
y = Ax (A.6)
where
=
=
=
2
1
2221
1211
2
1
,,
x
x
aa
aa
y
y
xAy
Assume now that the vector
x is formed by the linear transformation
2221212
2121111
zbzbx
zbzbx
+=
+
=
(A.7)
which can be written in the form
x = Bz (A.8)
The transformation from the column vector
z to the column vector y can be written in matrix
notation as
y = ABz (A.9)
The product
AB can be viewed as the matrix C where
C = AB (A.10)
A typical element c
ij
of C is given by
∑
=
=
2
1
k
kjikij
bac
(A.11)
Problem
Verify equation (A.11) by obtaining the relationship between the vectors y and z by substituting
(A.7) into (A.5)
In order for the product to be defined, the number of columns of
A must equal the number of
rows of
B. Proceeding to the general case, the product of two matrices, A (m × n) and B (n × p),
is defined in terms of the typical element of the product
C as
∑
=
=
n
k
kjikij
bac
1
(A.12)
Clearly, the product
C is an m × p matrix. In general, multiplication is not commutative, i.e.,
AB ≠ BA
REVIEW OF MATRICES
A-4
Matrix multiplication is associative and distributive.
A(BC) = (AB)C (associative)
A(B + C) = AB + AC
(distributive)
(
A + B)C = AC + BC
Scalar multiplication
Pre-multiplication or post-multiplication of a matrix by a scalar multiplier k multiplies each
element of the matrix by k.
Multiplication of transpose matrices
The product of two transpose matrices, B′ and A′, is equal to the transpose of the product of the
original two matrices
B and A, taken in reverse order, i.e.,
B′A′ = (AB)′ (A.13)
Problem
Verify equation (A.13).
Multiplication by a diagonal matrix
Post-multiplication of a matrix A by the diagonal matrix D is equivalent to an operation on the
columns of
A. Pre-multiplication of a matrix A by the diagonal matrix D is an operation on the
rows of
A.
Problem
Verify the above two statements.
Obviously, pre-multiplication or post-multiplication by the unit matrix
I leaves the matrix
unchanged, i.e.,
IA = AI = A.
Partitioning of matrices
Sometimes it is convenient to partition matrices. For example, by drawing the vertical and
horizontal dotted lines in the 3 × 3 matrix
A, a 2 × 2 matrix can be written using the submatrices
A
1
, A
2
, A
3
, and A
4
.
=
=
43
21
333231
232221
131211
AA
AA
A
aaa
aaa
aaa
Assuming that the matrix
B, which is also of order (3, 3) is partitioned in the same manner,
yielding
=
=
43
21
333231
232221
131211
BB
BB
B
bbb
bbb
bbb
REVIEW OF MATRICES
A-5
++
++
=+
4433
2211
BABA
BABA
BA
and
()
(
)
()( )
++
++
=
44233413
42213211
BABABABA
BABABABA
BA
In general, the product of two matrices can be expressed in terms of the submatrices, only if the
partitioning produces submatrices which are conformable. The grouping of columns in
A must
be equal to the grouping of rows in
B.
Differentiation of a matrix
Let A(t) be an m × n matrix whose elements a
ij
(t) are differentiable functions of the scalar
variable t. The derivative of
A(t) with respect to the variable t is defined as
[]
==
)()()(
)()()(
)()()(
)()(
21
22221
11211
ta
dt
d
ta
dt
d
ta
dt
d
ta
dt
d
ta
dt
d
ta
dt
d
ta
dt
d
ta
dt
d
ta
dt
d
tt
dt
d
mnmm
n
n
L
MOMM
L
L
&
AA (A.14)
From above, it is evident that
[]
)()()()( tttt
dt
d
BABA
&
&
+=+ (A.15)
The derivative of a matrix product is formed in the same manner as the derivative of a scalar
product, with the exception that the order of the product must be preserved. Examples:
[]
)()()()()()( tttttt
dt
d
BABABA
&
&
+= (A.16)
[]
)()()()()()()()(
223
tttttttt
dt
d
AAAAAAAA
&&&
++= (A.17)
Integration of a matrix
The integral of a matrix is defined in the same way as the derivative of a matrix. Thus
∫
∫∫∫
∫∫∫
∫∫∫
=
dttadttadtta
dttadttadtta
dttadttadtta
dtt
mnmm
n
n
)()()(
)()()(
)()()(
)(
21
22221
11211
L
MMMM
L
L
A
(A.18)
Determinant
The determinant of a square matrix A is denoted by enclosing the elements of the matrix A
within vertical bars. For example, for a 2 × 2 matrix
A,
REVIEW OF MATRICES
A-6
2221
1211
det
aa
aa
=
A
(A.19)
If the determinant of
A is equal to zero, then the matrix is said to be singular. The value of a
determinant is determined by obtaining the minors and cofactors of the determinants. The minor
of an element a
ij
of a determinant of order n is a determinant of order n −1, obtained by removing
the row i and the column j of the original determinant. The cofactor of a given element of a
determinant is the minor of the element with either a plus or minus sign attached as shown
below:
cofactor of
ij
ji
ijij
Ma
+
−== )1(
α
(A.20)
where M
ij
is the minor of a
ij
. For example, the cofactor of the element a
23
of
3231
1211
23
5
23
333231
232221
131211
)1(
det
aa
aa
M
aaa
aaa
aaa
−=−=
=
α
A
The value of a determinant of second order (2 × 2) is
()
21122211
2221
1211
aaaa
aa
aa
−=
(A.21)
The general nth order determinant has a value given by
∑
=
=
n
j
ijij
a
1
det
α
A , with i chosen for one row,
or (A.22)
∑
=
=
n
i
ijij
a
1
det
α
A , with j chosen for one column.
That is, the elements a
ij
are chosen for a specific row (or column) and that the entire row (or
column) is expanded according to equation (A.22).
The adjoint matrix of a square matrix
A is formed by replacing each element a
ij
by the cofactor
α
ij
and then transposing the matrix. Therefore
=
nnnn
n
n
ααα
ααα
ααα
L
MMMM
L
L
21
22212
12111
adjoint A
(A.23)