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Electromechanical distance measure for decomposition of power systems
Electromechanical distance measure for decomposition of power systems M A Pal University of Illinois, Urbana, IL 61801, USA
R P Adgaonkar Goa Engineering College, Farmagudi, India
A concept ofelectromechanical distance measure which reflects the interaction among the machines during a transient is presented. This measure is then used as a criterion for decomposing the power system into regions, each requiring different levels o f model complexity. The complexity o f models for regions decreases progressively away from the point o f fault. Grouping o f coherent generators within each region can also be done using the results o f this analysis. The analysis takes into account the fault location, magnitude o f disturbance, and changes in boundary between the study area and the external system. Keywords: modelling o f energy systems, electric power system disturbances, transient stability
I. I n t r o d u c t i o n In forming dynamic equivalents, the boundary between the study system and the external system is drawn on the basis of geographical or ownership considerations. The generators in the external area are then made equivalent mathematically by modal analysis 1 or by grouping coherent machines z-s. It is quite possible that, in certain instances, machines in the external system close to the boundary may have a significant influence on the dynamics of the study system, and consequently making them equivalent may lead to errors in simulation. Quite independently, it may be necessary in a large interconnected system to decrease the model complexity gradually as one moves away from the study area. Thus, in different regions one may have different models of the generating units. These generating units may consist of aggregated units and generators that are not coherent with any other generators. Hence, the problems of decomposition and coherency analysis become concurrent problems. Earlier work in this area by Lee and Schweppe 6 proposed admittance and reflection distances to divide a power system into three or more regions. This is followed by a pattern recognition algorithm to group machines in each region separately.
Received: 1 September 1982, Revised: 1 March 1984
Vol 6 No 4 October 1984
In this paper, the concept of electromechanicat distance measure (EMD) for decomposition into regions around the point of fault is introduced. The complexity of generator models decreases away from the point of fault. The method is validated on a 31-generator system considered in Reference 6. A procedure to accommodate changes in boundary between the study area and the external system is also indicated. Using results of this analysis, coherency grouping within each region can also be performed.
II. Power system model I1.1 Assumptions In the formulation of the system model, linearized swing equations and classical representation of synchronous machines are assumed. The electrical disturbances, such as faults and loss of generation, are approximated by considering the unfaulted network and increasing the mechanical input power to each generator by an amount equal to the accelerating power of the machine at t = 0 ÷. For electrical faults, the period of application of this input is limited to the clearing time of the fault only. Such modelling of electrical disturbances has been proposed by Podmore 2 and is valid for typical fault clearing times in modern day power systems. For generator or load shedding, the duration of the input is valid for t ~>0 ÷. The case of electrical faults is considered only with no line switching. 11.2 State space model Considering a power system with n synchronous generators and under the above hypotheses, the linearized swing equation of the ith machine for t > 0 is given by:
Mi
d2(A6i) dt 2
d( A6i) +Di
dt
n
- - A P m i - - Z EiEi(--Gii sin$° i=1 + Bii cos6~.) A6ii
i=1,2 ..... n
t>O
(1)
where the parameters are defined as follows.
8i
Di
APmi
Ei
coefficient of inertia of ith machine rotor angle of ith machine with respect to a synchronously rotating frame, electrical radians coefficient of damping, s step input to generator, i.e. accelerating power at t = 0 ÷, p.u. magnitude of the internal voltage
0142-0615/84/040249-06 $03.00 © 1984 Butterworth & Co (Publishers) Ltd
249
real and imaginary parts of i]th element of prefault reduced admittance nratrix 6 i - 61, superscript 'o' indicates the prefault value
G#, B 6 8ij
The faulted period t e is generally of the order of a few cycles, and during that period APmi (i = 1 , 2 . . . . . n) is assumed to be a step input equal to the accelerating powers at t = 0 +. For t > t e (i.e. after the fault is cleared with no line switching), APmi ==-O. This enables equation (1) to be used in the state space form during the fault period 0 < t < t e and the post-fault period t > t e as follows (considering the nth generator from the study system as reference generator).
very small compared with the transient period of interest, Ihe angular deviations between the nrachines during the faulted period C0 < I < [ e) will also be small. Therefore. only the postfault system equation (2b) is considered for obtaining the expression for EMD. However, the system equation (2a) in the faulted period is integrated to oblain X e, which forms the initial conditions for equation (2hi. Assuming that matrix [A] has all distinct eigenvalues. tile response of equation (2b) is expressed as a linear combination of the decoupled modes as: 2/7-1
X=
k
= [A] X + BU
0 < t<
(2a)
te
) ~ : [A]X; t>~te; X(te)=Xe
(2b)
where t e is the fault clearing time, X e is the state of the system at t e and X = [A61n , A~2n .....
A6n_l,
n " A~a)l,
n -- 1
Ao)
2 ....
, AOon]
(3)
E kiXieXit i= 1
where )k i is the eigenvalue of [A], X i is the eigenvector corresponding to Xi, and k i is the scalar depending upon the initial conditions X e and the reciprocal basis vector V i corresponding to X i.
T
Consider that [A] has m pairs of complex and ( 2 n - 2 m - - l ) real eigenvalues*. Equation (3) is simplified to:
n 1
0
...
0
--1
0
1
...
0
--1
m
n--I
[Ol
x=y
[pi(e air cos~it) + qi(e air sin/3it)]
i 1 0
[A] =
0
...
I wCl i i
I [Ao] I
o •
EC
2
1
--1
° . .
0
0
. . .
0
0
0
2n-I E si eait i--2m+l
(4)
where:
1 1
+
0
. . .
0
--C n
Pi = ( krXr + k~X~) _
n
[ [0l ] .-1,
i
r
r
i
qi -- ( k i X i -- k i X i ) Si : k f X f
B = t[Diag M,r~] k r : (Vri, X e ) Ci = D i / M i U = [APm, , . . . . Apron ] T ai,/3i = real and imaginary parts of )k i
The elements o f [Ao] are obtained as: X[., X~I = real and imaginary parts of X i Ei~ aii
:
--
~"
-~i
(--Gi]
sinSg. + Bii cos6~.)
V~, V ii = real and imaginary parts of V i
j--I Eit:) ai] : - ~ i (--Gii sin6~ + Bq cos8}})
All the eigenvectors and reciprocal basis vectors are normalized and satisfy the usual relationships 6. The expression for A6in , i.e. the rotor angle deviation between machine i and the reference machine n, is written from equation (4) as: m
III.
Electromechanical distance
A logical measure of the electromechanical interaction between a pair of machines during the transient period could be the synchronizing power flow or the maximum angular excursion• The EMD between two machines is defined here as a measure of the maximum angular excursion between the machines during the transient period• The expression for EMD is derived from the transient response of the power system models given by equation (2). Since the typical fault clearing times are
250
[Pi~ ( ea~ t cosfl~ t) + qiQ ( e~Q t sin/3~t)]
A~ in = Z Q=I
+
2n- 1 E si~eC~t ~=2m+l
(5)
* U n d e r the assumptions of negligible damping and Gij < Bij , it can be shown 7 that [A] has (n --1 ) pairs of complex eJgenvalues with negative real parts and one negative real eigenvalue. This implies t h a t m = n i 1 in t h e subsequent theoretical development.
Electrical Power & Energy Systems
where Pi~, qi~ and si~ are the elements of the vectors Pi, qi and si, respectively, corresponding to the variable ASin. The.suffix £ represents the index of the mode. The EMD between the generator i and the reference generator n is defined as ~-rn
din =
2
2n-1
2
(Pi~ + qi~) + I~ 1
2
1/2 2 ]
Sir2
(6)
~=2m+l
The physical meaning of the EMD is as follows. The elements Pin, qi~ and si~ represent the amplitudes of the complex modes and the real modes, respectively, in A~in. A measure (the Euclidean norm) of such amplitudes for all the modes is taken and defined as the EMD. Thus the EMD din reflects the maximum angular excursion between the generator i and the reference generator n. Similarly, the EMD between generators i and j (j 4: n) is derived from the expression for ( A 6 i n - - A ~ / n ) and is given by:
di i =
numbers that determine the boundaries between the regions. The decomposition is based on the EMDs with respect to the generator n which is in the study system. Since the transient behaviour of the generators in a study system subjected to the local disturbances is to be studied in detail, the level of complexity in representing the generating units in region 1 should be the highest. The generators in region 4, being far away (in the sense of EMD) from the study system, can be represented by the classical models. The degree of detail in modelling the generators in regions 2 and 3 can be suitably graded in between these two. V. C o h e r e n c y analysis Since the EMD dii reflects the maximum angular excursion between the generators i and j during the transient period, it can be used as a measure to determine the coherency. Two generators i a n d j are classified as coherent* if:
[(Pi~ _ p]~)2 + (qi~ _ q]~)2]
dii< e
1 2n- 1 + y. ~=2m+l
] 1/2
(s,~ -
Sl02]
(7)
The EMD dii gives a measure of the maximum angular excursion between the generators i and j. In the special case when all eigenvalues of [A] are on the imaginary axis, cti = 0 (i = 1, . . . , 2 n - - l ) and si = 0. It can be verified that equation (7) reduces to the RMS coherency measure 4. It is important to note that since the EMD is a measure of the magnitude of angular excursion and does not consider the sign (positive or negative) of the angular deviation, dij will not be equal to Idin -- din I. III. 1 Normalized EMD Since it is convenient to refer to the angular deviations between the machines as a percentage of the maximum angular excursion in the system, the din and dii are normalized. Max (din), i.e. the maximum of all dins , is taken as a unit deviation. In the subsequent analysis, din and dij always refer to the normalized EMDs.
(8)
where e is the tolerance. In this technique, a distinct departure in the approach to the formation of coherent groups in the external system is adopted. In the existing algorithms 2-4, the decomposition of power system and the formation of coherent groups are not interfaced. As a result, the comparisons for formation of groups have to be extended to all the generators in the external system. Here, the power system as explained in Section IV is decomposed separately and the coherent groups in each region of the external system formed separately. For such an approach, the obvious question about the coherency among the generators belonging to the different regions has to be answered. Two generators i and j belonging to different regions will be coherent only if •
they belong to the adjacent regions,
• din ~ din, i.e. they are in the vicinity of the boundary separating the regions,
• di] satisfies the coherency criterion in equation (8).
IV. Decomposition of power system In order to determine the level of complexity in the modelling of generators, a power system is divided into different regions based on the concept of EMD. This decomposition of a power system also forms the primary step in the coherency analysis described in Section V. The proposed criterion for the decomposition of a power system is as follows: •
region 1: study system
•
region 2: bl ~din<~b2
•
region 3 : b 2 ~ din <- b3
•
region4: b a < d i n <, 1
The study system is defined by geographical, ownership or other considerations, h i , b2 and ba are the known
Vol 6 No 4 October 1984
If such cases are detected, then the boundaries between the regions may be adjusted by changing the values of bis marginally. V.1 Formation of groups in a region A comparison generator is selected for each group, and all other eligible generators are compared against this generator in order to determine whether they should be included in the same group. To start with, all the generators in a region belonging to the external system are marked as eligible and a comparison generator, say i, is selected for the first group. The distances diis are computed for alljs belonging to the eligible generators. The first group is determined by the criterion in equation (8). Then all the generators in the first group are deleted and the remaining generators are marked * Since we are dealing w i t h [ A ] of equation (2b), it involves the disturbance in s o m e s e n s e . Hence, it may be termed as disturbancebased coherency 8.
251
as eligible for forming the second group. The subsequent groups are determined in a similar manner. This is the transitive algorithm of Reference 2.
V I . Change in study system Each time the study system is changed extensively, the decomposition of system and coherent groups must be redefined. In such cases, recomputation of eigenvectors and reciprocal basis vectors is avoided by using the concept of similarity transformation. When the study system is moved, the reference generator of the system matrix is changed, because it has to be selected from the study system under consideration. For a given prefault state of the system, this amounts to a similarity transformation of the [A] matrix or the rotation of the axis of the [A] matrix. Therefore, the eigenvectors and reciprocal basis vectors corresponding to the new reference generator are obtained by using the transformation matrix. The transformation matrix for changing the reference of [A] from ith t o / t h generator is given by :
Xi
=
Table 1. Electromechanical distances din Generato~
din
I 2 3 6 8 9 10 11 12 13 14 15 16
0.233 1.000 0.566 0.223 0.130 0.224 0.132 0.149 0.154 0.240 0.245 0.297 0.280
1 0 5 4 1 4 7 9 2 5 4 7 8
Generator
din
17 21 22 23 24 25 26 27 28 29 30 31
0.656 0.129 0.133 0.184 0.270 0.230 0.206 0.197 0.129 0.216 0.146 0.135
5 0 l 0 3 6 0 3 0 5 9 2
TXj
where X i and Xj are the state vectors with respect to the generators i and j, respectively. The matrix [T] has elements 0, 1 or --1 and can be written by inspecting the vectors X i and Xj 3.
VII. Example The above theory for decomposition and coherency analysis is now illustrated with a 31-generator model of a sample power system previously considered in Reference 6. A single line diagram of the power system is given in Figure 1. •
study system: generators 4, 5, 7, 18, 19, 20
•
reference: generator 5
•
fault: a three-phase fault on the terminals of generator 5 cleared in 0.1 s is considered.
V II. 1 Decomposition of the power system The electromechanical distances din are given in Table 1. The different regions corresponding to the proposed criterion given above are:
Figure 2 Decomposition and coherent grouping of the power system
•
region 1 : study system
•
region 2: generators 8, 10, 11, 12, 21,22, 23, 26, 27, 28, 30, 31
(0 < din ~ 0.2) •
region 3: generators 1,6, 9, 13, 14, 15, 16, 24, 25, 29 (0.2 <
•
din ~ 0.5)
region 4: generators 2, 3, 17
The above decomposition is shown in Figure 2.
Studysys?em
Figure 1 Single line diagram of the 31-machine system6
252
V 11.2 Groups of coherent generators The electromechanical distances dij for different generators in region 2 and region 3 are given in Tables 2 and 3, respectively. Each Table is written in accordance with the pro-
Electrical Power & Energy Systems
Table 2. F o r m a t i o n o f coherent groups in region 2
•
Generators in the region
The groups of coherent generators as identified above are verified by comparing the swing curves obtained from the base case simulation.
8 10 11 12 21 22 23 26 27 28 30 31
Comparison generator of group 30
31
26
23
dij
dij
dij
dii
0.096 3* 0.067 8* 0.1746 0.1864 0.096 4* 0.094 0* 0.216 2 0.224 9 0.197 7 0.112 5* 0.000 0* 0.151 6
0.101 4* 0.1135" 0.189 4 0.185 5 0.173 9 0.000 0*
0.234 9 0.000 0* 0.068 2* -
0.0" -
Table 3. Formation o f coherent groups in region 3
Generators in the region
1 6 9 13 14 15 16 24 25 29
Comparison generator of group 25
24
16, 15, 1 4 , 9 , 6 , 1
dij
dii
dqt
0.360 0.241 0.345 0.420 0.316 0.429 0.666 0.458 0.000 0.120
2 3 3 1 9 7 0 6 0* 0*
0.304 0.453 0.355 0.105 0.310 0.543 0.679 0.000 -
8 2 3 8* 0 1 0 0*
region 4: no coherent group
V I I I . Discussion In the above numerical example, it is interesting to note that so far as coherency is concerned, only geographically close generators are coherent. The results agree with those in Reference 3. Generators that are not coherent with other generators exhibit widely differing behaviour as evidenced by the indices dins. Hence, two neighbouring noncoherent generators can be modelled differently depending on which region they belong to, e.g. generators 9 and 23. Similarly, the noncoherent generators having the simplest model may be geographically apart, e.g. generators 2, 3 and 17. The gradation of model complexity for different regions will improve the overall dynamic equivalence of the external system.
IX. Conclusion An approach for the decomposition of a power system based on the concept of electromechanical distance in the transient period is proposed and has been illustrated on a practical system. The decomposition of the power system into different regions provides a logical basis for determining the degree of detail in the modelling of the generating units. The formation of coherent groups is confined to each region separately, thereby reducing the number of comparisons. The suitability of the technique has been demonstrated for the cases when the study system is moved around. The procedure for grouping can be improved upon if the commutative grouping algorithm 4 is used instead of the transitive grouping algorithm 2. Other areas to be investigated relate to using the slow modes only instead of the fast modes s. It would be also interesting to establish a connection between the method proposed here and the selective modal analysis approach 9, as well as the frequency response method using the same mathematical model 1°.
t These distancesare not given here for the sake of brevity, as no more coherent groups could be determined
cedure of grouping the generators as described in Section V. The groups of coherent generators are determined with tolerance 0.15 corresponding to the deviation of 10 ° in the base case studies. The first generator in each group is the reference generator used in the transitive algorithm 2. •
region 2 o o © o
•
group group group group
1: generators 30, 28, 22, 21, 10, 8 2: generators 31, 12, 11 3 : generators 26, 27 4: generator 23 alone
region 3 o group 4: generators 29, 25 © group 5: generators 24, 13 The other generators in this group are not coherent with any other generator.
Vol 6 No 4 October 1984
X. Acknowledgement The authors would like to thank the Department of Science and Technology, Government of India, for supporting the research. M A Pai also acknowledges the support of the University of Illinois Power Affiliates Program.
XI. References 1 Undrill, J M, Casazza, J A, Gulachenski, E M and Kirschmeyer, L K 'Electromechanical equivalents for use in power system stability" IEEE Trans. Power Appar. Syst. Vol PAS-90 (September 1971 ) pp 20602071 Podmore, R 'Identification of coherent generators for dynamic equivalents' IEEE Trans. Power Appar. Syst. Vol PAS-97 (July/August 1978) pp 1344-1354 Pai, M A and Adgaonkar, R P ' Identification of coherent generators using weighted eigenvectors' Proc. IEEE PES Winter Meeting New York, USA A79 022-5 (February 1979)
253
Lawler, J S and Schlueter, R A 'Computational algorithms for constructing modal-coherent dynamic equivalents' IEEE Trans. Power Appar. & Syst. Vol PAS-101 No 5 (May 1982) pp 1070-1080
Dicaprio, U I 'Theoretical and practical dynamic equivalents in multimachine power systems' Int. J. E/ectr. Power & Energ. Syst. Part I (October 1982) Part II (January 1983)
Winkelman, J R, Chow, J H, Bowler, B, Avramovic, B and Kokotovic, P V 'An analysis of inter-area dynamics
Perez-Arriaga, I J, Schweppe, F C and Verghese, G C
of multi-machine power systems" IEEE Trans. Power Appar. & Syst. Vol PAS-100 (February 1981) pp 754-763 Lee, S T Y and Schweppe, F C 'Distance measures and coherency recognition for transient stability equivalents' IEEE Trans. Power Appar. & Syst. Vol PAS-92 (September 1973) pp 1550-1557
Chow, J H (Ed.) 'Time scale modeling of dynamic networks with applications to power systems' Lecture notes in control and information sciences Vol 46, Springer-Verlag, USA (1982)
254
'Selective model analysis with applications to power systems' IEEE Trans. Power Appar. & Syst. Parts I and II, Vol PAS-101 (September 1982) 10 Hiyama, T ' Identification of coherent generators using frequency response' Proc. lEE Vol 128 Part C No 5 (September 1981 )
Xl. Bibliography Desoer, C A 'Modes in linear circuits'/RE Trans. Circuit Theory Vol CT-7 (September 1960) pp 211-224
Electrical Power & Energy Systems
R P Adgaonkar Goa Engineering College, Farmagudi, India
A concept ofelectromechanical distance measure which reflects the interaction among the machines during a transient is presented. This measure is then used as a criterion for decomposing the power system into regions, each requiring different levels o f model complexity. The complexity o f models for regions decreases progressively away from the point o f fault. Grouping o f coherent generators within each region can also be done using the results o f this analysis. The analysis takes into account the fault location, magnitude o f disturbance, and changes in boundary between the study area and the external system. Keywords: modelling o f energy systems, electric power system disturbances, transient stability
I. I n t r o d u c t i o n In forming dynamic equivalents, the boundary between the study system and the external system is drawn on the basis of geographical or ownership considerations. The generators in the external area are then made equivalent mathematically by modal analysis 1 or by grouping coherent machines z-s. It is quite possible that, in certain instances, machines in the external system close to the boundary may have a significant influence on the dynamics of the study system, and consequently making them equivalent may lead to errors in simulation. Quite independently, it may be necessary in a large interconnected system to decrease the model complexity gradually as one moves away from the study area. Thus, in different regions one may have different models of the generating units. These generating units may consist of aggregated units and generators that are not coherent with any other generators. Hence, the problems of decomposition and coherency analysis become concurrent problems. Earlier work in this area by Lee and Schweppe 6 proposed admittance and reflection distances to divide a power system into three or more regions. This is followed by a pattern recognition algorithm to group machines in each region separately.
Received: 1 September 1982, Revised: 1 March 1984
Vol 6 No 4 October 1984
In this paper, the concept of electromechanicat distance measure (EMD) for decomposition into regions around the point of fault is introduced. The complexity of generator models decreases away from the point of fault. The method is validated on a 31-generator system considered in Reference 6. A procedure to accommodate changes in boundary between the study area and the external system is also indicated. Using results of this analysis, coherency grouping within each region can also be performed.
II. Power system model I1.1 Assumptions In the formulation of the system model, linearized swing equations and classical representation of synchronous machines are assumed. The electrical disturbances, such as faults and loss of generation, are approximated by considering the unfaulted network and increasing the mechanical input power to each generator by an amount equal to the accelerating power of the machine at t = 0 ÷. For electrical faults, the period of application of this input is limited to the clearing time of the fault only. Such modelling of electrical disturbances has been proposed by Podmore 2 and is valid for typical fault clearing times in modern day power systems. For generator or load shedding, the duration of the input is valid for t ~>0 ÷. The case of electrical faults is considered only with no line switching. 11.2 State space model Considering a power system with n synchronous generators and under the above hypotheses, the linearized swing equation of the ith machine for t > 0 is given by:
Mi
d2(A6i) dt 2
d( A6i) +Di
dt
n
- - A P m i - - Z EiEi(--Gii sin$° i=1 + Bii cos6~.) A6ii
i=1,2 ..... n
t>O
(1)
where the parameters are defined as follows.
8i
Di
APmi
Ei
coefficient of inertia of ith machine rotor angle of ith machine with respect to a synchronously rotating frame, electrical radians coefficient of damping, s step input to generator, i.e. accelerating power at t = 0 ÷, p.u. magnitude of the internal voltage
0142-0615/84/040249-06 $03.00 © 1984 Butterworth & Co (Publishers) Ltd
249
real and imaginary parts of i]th element of prefault reduced admittance nratrix 6 i - 61, superscript 'o' indicates the prefault value
G#, B 6 8ij
The faulted period t e is generally of the order of a few cycles, and during that period APmi (i = 1 , 2 . . . . . n) is assumed to be a step input equal to the accelerating powers at t = 0 +. For t > t e (i.e. after the fault is cleared with no line switching), APmi ==-O. This enables equation (1) to be used in the state space form during the fault period 0 < t < t e and the post-fault period t > t e as follows (considering the nth generator from the study system as reference generator).
very small compared with the transient period of interest, Ihe angular deviations between the nrachines during the faulted period C0 < I < [ e) will also be small. Therefore. only the postfault system equation (2b) is considered for obtaining the expression for EMD. However, the system equation (2a) in the faulted period is integrated to oblain X e, which forms the initial conditions for equation (2hi. Assuming that matrix [A] has all distinct eigenvalues. tile response of equation (2b) is expressed as a linear combination of the decoupled modes as: 2/7-1
X=
k
= [A] X + BU
0 < t<
(2a)
te
) ~ : [A]X; t>~te; X(te)=Xe
(2b)
where t e is the fault clearing time, X e is the state of the system at t e and X = [A61n , A~2n .....
A6n_l,
n " A~a)l,
n -- 1
Ao)
2 ....
, AOon]
(3)
E kiXieXit i= 1
where )k i is the eigenvalue of [A], X i is the eigenvector corresponding to Xi, and k i is the scalar depending upon the initial conditions X e and the reciprocal basis vector V i corresponding to X i.
T
Consider that [A] has m pairs of complex and ( 2 n - 2 m - - l ) real eigenvalues*. Equation (3) is simplified to:
n 1
0
...
0
--1
0
1
...
0
--1
m
n--I
[Ol
x=y
[pi(e air cos~it) + qi(e air sin/3it)]
i 1 0
[A] =
0
...
I wCl i i
I [Ao] I
o •
EC
2
1
--1
° . .
0
0
. . .
0
0
0
2n-I E si eait i--2m+l
(4)
where:
1 1
+
0
. . .
0
--C n
Pi = ( krXr + k~X~) _
n
[ [0l ] .-1,
i
r
r
i
qi -- ( k i X i -- k i X i ) Si : k f X f
B = t[Diag M,r~] k r : (Vri, X e ) Ci = D i / M i U = [APm, , . . . . Apron ] T ai,/3i = real and imaginary parts of )k i
The elements o f [Ao] are obtained as: X[., X~I = real and imaginary parts of X i Ei~ aii
:
--
~"
-~i
(--Gi]
sinSg. + Bii cos6~.)
V~, V ii = real and imaginary parts of V i
j--I Eit:) ai] : - ~ i (--Gii sin6~ + Bq cos8}})
All the eigenvectors and reciprocal basis vectors are normalized and satisfy the usual relationships 6. The expression for A6in , i.e. the rotor angle deviation between machine i and the reference machine n, is written from equation (4) as: m
III.
Electromechanical distance
A logical measure of the electromechanical interaction between a pair of machines during the transient period could be the synchronizing power flow or the maximum angular excursion• The EMD between two machines is defined here as a measure of the maximum angular excursion between the machines during the transient period• The expression for EMD is derived from the transient response of the power system models given by equation (2). Since the typical fault clearing times are
250
[Pi~ ( ea~ t cosfl~ t) + qiQ ( e~Q t sin/3~t)]
A~ in = Z Q=I
+
2n- 1 E si~eC~t ~=2m+l
(5)
* U n d e r the assumptions of negligible damping and Gij < Bij , it can be shown 7 that [A] has (n --1 ) pairs of complex eJgenvalues with negative real parts and one negative real eigenvalue. This implies t h a t m = n i 1 in t h e subsequent theoretical development.
Electrical Power & Energy Systems
where Pi~, qi~ and si~ are the elements of the vectors Pi, qi and si, respectively, corresponding to the variable ASin. The.suffix £ represents the index of the mode. The EMD between the generator i and the reference generator n is defined as ~-rn
din =
2
2n-1
2
(Pi~ + qi~) + I~ 1
2
1/2 2 ]
Sir2
(6)
~=2m+l
The physical meaning of the EMD is as follows. The elements Pin, qi~ and si~ represent the amplitudes of the complex modes and the real modes, respectively, in A~in. A measure (the Euclidean norm) of such amplitudes for all the modes is taken and defined as the EMD. Thus the EMD din reflects the maximum angular excursion between the generator i and the reference generator n. Similarly, the EMD between generators i and j (j 4: n) is derived from the expression for ( A 6 i n - - A ~ / n ) and is given by:
di i =
numbers that determine the boundaries between the regions. The decomposition is based on the EMDs with respect to the generator n which is in the study system. Since the transient behaviour of the generators in a study system subjected to the local disturbances is to be studied in detail, the level of complexity in representing the generating units in region 1 should be the highest. The generators in region 4, being far away (in the sense of EMD) from the study system, can be represented by the classical models. The degree of detail in modelling the generators in regions 2 and 3 can be suitably graded in between these two. V. C o h e r e n c y analysis Since the EMD dii reflects the maximum angular excursion between the generators i and j during the transient period, it can be used as a measure to determine the coherency. Two generators i a n d j are classified as coherent* if:
[(Pi~ _ p]~)2 + (qi~ _ q]~)2]
dii< e
1 2n- 1 + y. ~=2m+l
] 1/2
(s,~ -
Sl02]
(7)
The EMD dii gives a measure of the maximum angular excursion between the generators i and j. In the special case when all eigenvalues of [A] are on the imaginary axis, cti = 0 (i = 1, . . . , 2 n - - l ) and si = 0. It can be verified that equation (7) reduces to the RMS coherency measure 4. It is important to note that since the EMD is a measure of the magnitude of angular excursion and does not consider the sign (positive or negative) of the angular deviation, dij will not be equal to Idin -- din I. III. 1 Normalized EMD Since it is convenient to refer to the angular deviations between the machines as a percentage of the maximum angular excursion in the system, the din and dii are normalized. Max (din), i.e. the maximum of all dins , is taken as a unit deviation. In the subsequent analysis, din and dij always refer to the normalized EMDs.
(8)
where e is the tolerance. In this technique, a distinct departure in the approach to the formation of coherent groups in the external system is adopted. In the existing algorithms 2-4, the decomposition of power system and the formation of coherent groups are not interfaced. As a result, the comparisons for formation of groups have to be extended to all the generators in the external system. Here, the power system as explained in Section IV is decomposed separately and the coherent groups in each region of the external system formed separately. For such an approach, the obvious question about the coherency among the generators belonging to the different regions has to be answered. Two generators i and j belonging to different regions will be coherent only if •
they belong to the adjacent regions,
• din ~ din, i.e. they are in the vicinity of the boundary separating the regions,
• di] satisfies the coherency criterion in equation (8).
IV. Decomposition of power system In order to determine the level of complexity in the modelling of generators, a power system is divided into different regions based on the concept of EMD. This decomposition of a power system also forms the primary step in the coherency analysis described in Section V. The proposed criterion for the decomposition of a power system is as follows: •
region 1: study system
•
region 2: bl ~din<~b2
•
region 3 : b 2 ~ din <- b3
•
region4: b a < d i n <, 1
The study system is defined by geographical, ownership or other considerations, h i , b2 and ba are the known
Vol 6 No 4 October 1984
If such cases are detected, then the boundaries between the regions may be adjusted by changing the values of bis marginally. V.1 Formation of groups in a region A comparison generator is selected for each group, and all other eligible generators are compared against this generator in order to determine whether they should be included in the same group. To start with, all the generators in a region belonging to the external system are marked as eligible and a comparison generator, say i, is selected for the first group. The distances diis are computed for alljs belonging to the eligible generators. The first group is determined by the criterion in equation (8). Then all the generators in the first group are deleted and the remaining generators are marked * Since we are dealing w i t h [ A ] of equation (2b), it involves the disturbance in s o m e s e n s e . Hence, it may be termed as disturbancebased coherency 8.
251
as eligible for forming the second group. The subsequent groups are determined in a similar manner. This is the transitive algorithm of Reference 2.
V I . Change in study system Each time the study system is changed extensively, the decomposition of system and coherent groups must be redefined. In such cases, recomputation of eigenvectors and reciprocal basis vectors is avoided by using the concept of similarity transformation. When the study system is moved, the reference generator of the system matrix is changed, because it has to be selected from the study system under consideration. For a given prefault state of the system, this amounts to a similarity transformation of the [A] matrix or the rotation of the axis of the [A] matrix. Therefore, the eigenvectors and reciprocal basis vectors corresponding to the new reference generator are obtained by using the transformation matrix. The transformation matrix for changing the reference of [A] from ith t o / t h generator is given by :
Xi
=
Table 1. Electromechanical distances din Generato~
din
I 2 3 6 8 9 10 11 12 13 14 15 16
0.233 1.000 0.566 0.223 0.130 0.224 0.132 0.149 0.154 0.240 0.245 0.297 0.280
1 0 5 4 1 4 7 9 2 5 4 7 8
Generator
din
17 21 22 23 24 25 26 27 28 29 30 31
0.656 0.129 0.133 0.184 0.270 0.230 0.206 0.197 0.129 0.216 0.146 0.135
5 0 l 0 3 6 0 3 0 5 9 2
TXj
where X i and Xj are the state vectors with respect to the generators i and j, respectively. The matrix [T] has elements 0, 1 or --1 and can be written by inspecting the vectors X i and Xj 3.
VII. Example The above theory for decomposition and coherency analysis is now illustrated with a 31-generator model of a sample power system previously considered in Reference 6. A single line diagram of the power system is given in Figure 1. •
study system: generators 4, 5, 7, 18, 19, 20
•
reference: generator 5
•
fault: a three-phase fault on the terminals of generator 5 cleared in 0.1 s is considered.
V II. 1 Decomposition of the power system The electromechanical distances din are given in Table 1. The different regions corresponding to the proposed criterion given above are:
Figure 2 Decomposition and coherent grouping of the power system
•
region 1 : study system
•
region 2: generators 8, 10, 11, 12, 21,22, 23, 26, 27, 28, 30, 31
(0 < din ~ 0.2) •
region 3: generators 1,6, 9, 13, 14, 15, 16, 24, 25, 29 (0.2 <
•
din ~ 0.5)
region 4: generators 2, 3, 17
The above decomposition is shown in Figure 2.
Studysys?em
Figure 1 Single line diagram of the 31-machine system6
252
V 11.2 Groups of coherent generators The electromechanical distances dij for different generators in region 2 and region 3 are given in Tables 2 and 3, respectively. Each Table is written in accordance with the pro-
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Table 2. F o r m a t i o n o f coherent groups in region 2
•
Generators in the region
The groups of coherent generators as identified above are verified by comparing the swing curves obtained from the base case simulation.
8 10 11 12 21 22 23 26 27 28 30 31
Comparison generator of group 30
31
26
23
dij
dij
dij
dii
0.096 3* 0.067 8* 0.1746 0.1864 0.096 4* 0.094 0* 0.216 2 0.224 9 0.197 7 0.112 5* 0.000 0* 0.151 6
0.101 4* 0.1135" 0.189 4 0.185 5 0.173 9 0.000 0*
0.234 9 0.000 0* 0.068 2* -
0.0" -
Table 3. Formation o f coherent groups in region 3
Generators in the region
1 6 9 13 14 15 16 24 25 29
Comparison generator of group 25
24
16, 15, 1 4 , 9 , 6 , 1
dij
dii
dqt
0.360 0.241 0.345 0.420 0.316 0.429 0.666 0.458 0.000 0.120
2 3 3 1 9 7 0 6 0* 0*
0.304 0.453 0.355 0.105 0.310 0.543 0.679 0.000 -
8 2 3 8* 0 1 0 0*
region 4: no coherent group
V I I I . Discussion In the above numerical example, it is interesting to note that so far as coherency is concerned, only geographically close generators are coherent. The results agree with those in Reference 3. Generators that are not coherent with other generators exhibit widely differing behaviour as evidenced by the indices dins. Hence, two neighbouring noncoherent generators can be modelled differently depending on which region they belong to, e.g. generators 9 and 23. Similarly, the noncoherent generators having the simplest model may be geographically apart, e.g. generators 2, 3 and 17. The gradation of model complexity for different regions will improve the overall dynamic equivalence of the external system.
IX. Conclusion An approach for the decomposition of a power system based on the concept of electromechanical distance in the transient period is proposed and has been illustrated on a practical system. The decomposition of the power system into different regions provides a logical basis for determining the degree of detail in the modelling of the generating units. The formation of coherent groups is confined to each region separately, thereby reducing the number of comparisons. The suitability of the technique has been demonstrated for the cases when the study system is moved around. The procedure for grouping can be improved upon if the commutative grouping algorithm 4 is used instead of the transitive grouping algorithm 2. Other areas to be investigated relate to using the slow modes only instead of the fast modes s. It would be also interesting to establish a connection between the method proposed here and the selective modal analysis approach 9, as well as the frequency response method using the same mathematical model 1°.
t These distancesare not given here for the sake of brevity, as no more coherent groups could be determined
cedure of grouping the generators as described in Section V. The groups of coherent generators are determined with tolerance 0.15 corresponding to the deviation of 10 ° in the base case studies. The first generator in each group is the reference generator used in the transitive algorithm 2. •
region 2 o o © o
•
group group group group
1: generators 30, 28, 22, 21, 10, 8 2: generators 31, 12, 11 3 : generators 26, 27 4: generator 23 alone
region 3 o group 4: generators 29, 25 © group 5: generators 24, 13 The other generators in this group are not coherent with any other generator.
Vol 6 No 4 October 1984
X. Acknowledgement The authors would like to thank the Department of Science and Technology, Government of India, for supporting the research. M A Pai also acknowledges the support of the University of Illinois Power Affiliates Program.
XI. References 1 Undrill, J M, Casazza, J A, Gulachenski, E M and Kirschmeyer, L K 'Electromechanical equivalents for use in power system stability" IEEE Trans. Power Appar. Syst. Vol PAS-90 (September 1971 ) pp 20602071 Podmore, R 'Identification of coherent generators for dynamic equivalents' IEEE Trans. Power Appar. Syst. Vol PAS-97 (July/August 1978) pp 1344-1354 Pai, M A and Adgaonkar, R P ' Identification of coherent generators using weighted eigenvectors' Proc. IEEE PES Winter Meeting New York, USA A79 022-5 (February 1979)
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Lawler, J S and Schlueter, R A 'Computational algorithms for constructing modal-coherent dynamic equivalents' IEEE Trans. Power Appar. & Syst. Vol PAS-101 No 5 (May 1982) pp 1070-1080
Dicaprio, U I 'Theoretical and practical dynamic equivalents in multimachine power systems' Int. J. E/ectr. Power & Energ. Syst. Part I (October 1982) Part II (January 1983)
Winkelman, J R, Chow, J H, Bowler, B, Avramovic, B and Kokotovic, P V 'An analysis of inter-area dynamics
Perez-Arriaga, I J, Schweppe, F C and Verghese, G C
of multi-machine power systems" IEEE Trans. Power Appar. & Syst. Vol PAS-100 (February 1981) pp 754-763 Lee, S T Y and Schweppe, F C 'Distance measures and coherency recognition for transient stability equivalents' IEEE Trans. Power Appar. & Syst. Vol PAS-92 (September 1973) pp 1550-1557
Chow, J H (Ed.) 'Time scale modeling of dynamic networks with applications to power systems' Lecture notes in control and information sciences Vol 46, Springer-Verlag, USA (1982)
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'Selective model analysis with applications to power systems' IEEE Trans. Power Appar. & Syst. Parts I and II, Vol PAS-101 (September 1982) 10 Hiyama, T ' Identification of coherent generators using frequency response' Proc. lEE Vol 128 Part C No 5 (September 1981 )
Xl. Bibliography Desoer, C A 'Modes in linear circuits'/RE Trans. Circuit Theory Vol CT-7 (September 1960) pp 211-224
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