- Email: [email protected]
Transient stability index for online stability assessment and contingency evaluation
Transient stability index for online stability assessment and
contingency evaluation M Ribbens-Pavella, P G Murthy and J L Horward Department of Electrical Engineering, University of Liege, Sart-Tilman, B-4000 Liege, Belgium
J L Carpentier Electricit~ de France, 2 Rue Louis Murat, 75008 Paris, France
An online methodology is proposed for assessing the robustness o f a power system from the point o f view o f transient stability, and a scalar expression, the transient stability index, is accordingly derived. The reliability and sensitivity o f this index are tested by means o f simulations for a number o f power system cases. The index is shown to be appropriate for online stability assessment, contingency evaluation and preventive control Keywords: electric power systems, network analysis, transient stability, security assessment
I. I n t r o d u c t i o n Stable operation of synchronous generators connected together and running in parallel has long been considered one of the foremost requirements for reliable service. It has also been recognized that the study of stability forms an integral part of power-system planning and design. Conventional transient stability evaluation is normally done by simulating severe disturbances (generally one at a time) and determining the corresponding critical clearing times. To reduce the computational burden as far as possible, direct methods employing Lyapunov's stability theory have been developed since the 1960s. They have been proved to be very encouraging and promising, at least in the initial stages of system design 1' 2. Further improvements to counteract the conservative character of the Lyapunov method have recently been proposed (see, for instance, those reported in References 3 - 7 ) . Note that despite all the improvements achieved in this field, conventional transient stability analysis, by its very nature, remains essentially an offline procedure.
Contrary to an abundance of literature available on the conventional approach, the online assessment of power system stability while in operation is a fairly recent development and is still in its infancy. Nevertheless, this Received: 27 April 1981
Vol 4 No 2 April 1982
stability aspect is at least as important as the conventional offline one. Indeed, a power system is never in a steady state, in the true sense, during operation. There are always haphazard changes in the load levels with associated adjustments in generation. But, much more important than these random changes are the large network disturbances associated with possible disconnection of certain important transmission lines in the event of unsuccessful reclosure leading to quasi-permanent changes in the network topology. Other modifications during operation may also arise, leading to important changes in the network configuration. A knowledge of how safe or critical the system is at any given instant with respect to stability is therefore essential in order for the operating personnel to take appropriate decisions to maintain the security of the system. In accordance with these motivations, the problem of power-system stability assessment, suitable for online implementation, has recently been considered. Some of the significant work in this direction includes an approach based on probabilistic considerations 8, methods for fast computation of transient stability 9, lo and the concept of a deterministic transient stability index n. The deterministic index aims at assessing the stability of a system with a given configuration in terms of the value that it assumes for this configuration. Obviously, the evaluation of the (scalar) expression of this index should be computationally quite straightforward in order to constitute a true online stability measure. The present work is based on the same concept as above. However, significant improvements have been included to take care of certain inconsistencies encountered while testing the index of Reference 11. This has led the authors of the present paper to reexamine the physical foundations and to define a new transient stability index. The developments related to this new index, the derivation of its expression and some of its potential practical uses are reported in Section III. Section IV introduces the notions of reliability and sensitivity and explores them thoroughly. The investigations are based upon simulations performed
0142-0615/82/020091-09 $03.00 © 1982 Butterworth & Co (Publishers) Ltd
91
on six different appropriately selected power systems; the results obtained are tabulated, commented on and discussed in detail. It is found that the index behaves remarkably well. It is also shown that this index introduces a substantial improvement compared with the index of Reference 11. Finally, Section V suggests another potential use of the index concerning transient preventive control.
II. Some preliminaries In the light of the present analysis, it is proposed to introduce some terminology and a few definitions for a better appreciation of the principle underlying the concept. In a power system it is always possible to have different critical clearing times (CCTs) depending upon the severity of the fault, its location, load-generation levels and the network configuration prevalent at the instant of fault inception. Further, there is no loss of generality in assuming that the severest fault is a 3-phase shortcircuit located at a generator high-voltage bus and adversely affecting the stability of the system. The 'minimum critical clearing time' (MeeT) in a power system under known load-generation balance and network topology is given by the smallest value of the critical clearing times evaluated for an assumed 3-phase shortcircuit at each of the generator high-voltage buses in the system. Of course, this M e e T varies with changes in generation, loads and network topology. We are then justified in accepting that the M e e T is a positive indication of how safe or critical the system is, and can be considered as a measure of relative stability of the system in a given 'configuration'. 'Configuration' is used to denote a given load flow and topology of the system. The 'transient stability index' (TSI) is a figure that is obtained by considering the maximum relative acceleration experienced by the system when referred to a chosen reference machine immediately after the initiation of a 3-phase shortcircuit at each of the generator high-voltage buses in the system. The 'average acceleration' of the system is the mean value of the accelerations of all the machines in the system at to+ when a 3-phase shortcircuit has taken place at a generator bus. Obviously, the average acceleration is different for faults at different generator buses. The 'reference machine' is the one having the maximum nominal power available at the busbar in the system. If more than one machine possesses the same maximum nominal power, any one of them can equally well serve as the reference machine.
III. Transient stability index Ilia Concept of TSI The main consideration involved in evolving the TSI is its relationship to the most dangerous disturbance, i.e. the one capable of causing the severest instability in a given system. In other words, the main aim is to derive an expression whose value indicates the severity of the most dangerous disturbance in a power system with a given configuration, without identifying this disturbance and without evaluating its corresponding CCT. In fact, the aim
is to totally replace the computation of the M e e T (which, as is well known, is quite time consuming since it implies simulating many disturbances and finding their corresponding CCTs) by using an appropriate scalar expression (the TSI) which is independent of any transient stability run and whose computation is quite straightforward. In deriving such an expression, the authors were guided by the acceleration approach used to assess the practical stability domain estimate of the Lyapunov-like criterion and by the following reasoning 7' n. Experience shows that the severest disturbance that a power system may be subjected to is a 3-phase shortcircuit at a machine's busbar. According to the Lyapunov-like criterion, the value of the CCT of a given disturbance depends on the unstable point relative to the machine possessing the 'largest initial acceleration' during the transients; hence the most dangerous disturbance will be related to the machine possessing the largest initial acceleration when a 3-phase shortcircuit acts at the busbar of that or another machine. In fact, in a power system consisting of several synchronous generators running in parallel, it is quite possible, when a shortcircuit occurs, that some of the machines accelerate and some decelerate. Consequently, the machine having maximum absolute acceleration need not necessarily fall out of step. In other words, the maximum initial acceleration of any machine when viewed in isolation may not always provide the true picture of the system's response. What is more appropriate under these conditions, i.e. what decides the stability behaviour of the system, is the maximum initial relative acceleration. Following the above reasoning, the severity of the most dangerous disturbance may be assessed by means of the maximum difference between this largest initial relative acceleration and the mean acceleration of the overall system's machines. The definition of the TSI now follows.
111.2 TSI derivation
Initially, a 3-phase shortcircuit is created at the ith generator bus at to. Let 7k denote the acceleration of the kth machine at to +; clearly, 7k
P m k - Pek
Mk
k = 1,2 . . . . . n ; k ~ i (1)
Pmi Mi where Mk, Pink and Pek denote the inertia constant, the "Yi-
mechanical input power and the electrical output power of the kth machine respectively. This latter power may be expressed by: Pek = Re Ek
iYk/Ej
(2)
/=1 where E k is the internal voltage of generator k (behind its transient reactance) and where iYk! is the kith element of the admittance matrix reduced to the generator nodes
when a 3-phase shortcircuit acts at the ith generator node. Then, let "7[ denote the absolute value of the maximum relative acceleration with respect to the reference machine R, as:
T~ A
Irk -
max k=l,2,...,n--
T~
I
(3)
I
Moreover, let the absolute value of the average acceleration of the system be designated by "7i under these conditions. Thus:
(4)
Since the main concern is to get Peg at to + different numerical techniques can be used to reduce the number of iterations and hence the computational time for solving the algebraic equations. One alternative is to eliminate completely the reduction step and simulate the entire system for the solution of the network performance equations.
(5)
If the network-reduction step is avoided, it is possible to represent loads according to their respective load characteristics; it will not, therefore, be necessary to assume a constant impedance representation for the loads.
Now let A i be the difference between the absolute value of the maximum relative acceleration and that of the average acceleration of the system, as:
Ai & 1 7 f - ' ~ i l Finally, the above three-step procedure is repeated for all the generator buses in the system, and the values of the Ais are calculated. The transient stability index, r~, for a specified load flow and network configuration is now given by: n =a
max
Ai
Remarks
The only time-consuming computation is that of the reduction of the system admittance matrix to the generator nodes when each of the generators is subjected to a 3-phase shortcircuit, as can be seen in equation (2). Consequently, there are, in principle, a maximum ofn such matrix reductions to be calculated. However, owing to the superposition technique used in Reference 11, these n reductions may be replaced by only one with a few additional computations of some simple analytical expressions. This allows for a substantial saving in computing time.
A sound knowledge of the power system avoids a series of computations and therefore saves computing time when evaluating the TSI. Indeed, in a given power system, there are generally a number of 'uninteresting' generators in that the A corresponding to a 3-phase shortcircuit at (6)
i=l,2,...,n
Calculate initial load flow for specified load- generation balance and network configuration
Remarks Formally, the difference between the expression of the above index and the one suggested earlier (r/m in Reference 11) is that ~ in equation (6) proposed here is onthe basis of the maximum relative acceleration rather than the maximum absolute acceleration acquired by a generator.
t Read in data of machines for translent
stability studies with reference generator indicated Compute initial state
Rather than considering the average acceleration 3; expressed by equation (4), the acceleration of the centre of inertia
T Set bus count = 0
",/g t/
Create fault at ith generator terminal bus at time to and set terminal voltage to zero
ZM; ; j=l -'/g - - g/
Solve algebraic equations for reduced network to obtain electrical power outputs (Pet) of all generators at to+
EMi
j-I
could be taken and another stability index derived accordingly. However, this modification has not been found worthwhile. Although derived from the Lyapunov criterion, the TSI is free from the simplifying assumptions that are usually made when constructing Lyapunov functions for power system transient stability analysis.
I
Compute accelerations of all machines from equation (I) Calculate relative accelerations with respect to reference machine and obtain 7/" from equation (3], ~. from equation (4) and Z~i from equation (5)
Determine
111.3 TSI computation The flowchart given in Figure 1 indicates the various steps of the general procedure involved in the computation of the TSI. It may be noted that calculating 7?implies simulating a 3-phase shortcircuit at the busbar of each of the system's machines successively.
Vol 4 No 2 April 1982
[ Print results I
Figure 1. Flowchart for calculatingTS1
93
Table 1. Reliability tests: A versus CCT
Version
Fault location (gen. bus no.)
3-machine
3 2 1 7-machine 7 l 5 3 2 4 6 9-machine 3 2 1 6 9 5 7 4 14-machine 34 0 84 89 76 77 59 21 19 1 17 33 64 15-machine K1 N1 C1 GI Z 40-machine AWlR 4 BRES 4 VILG 4 FARC 4 LANB 3 SCHE 3A TRIV 3
CCT, s
A m of Reference 1, rad,s -2
A of equation (5), rad, x-z
0.18 0.32 0.39 0.33 0.35 0.35 0.39 0.41 0.50 0.52 0.26 0.44 0.45 0.47 0.48 0.49 0.59 0.60 0.30 0.32 0.39 0.43 0.47 0.48 0.49 0.53 0.53 0.57 0.59 0.64 0.70 0.36 0.40 0.41 0.42 0.50 0.23 0.24 0.25 0.27 0.28 0.30 0.32
27.160 18.343 11.218 25.363 28.132 28.238 23.916 22.508 14.020 14.510 44.684 17.183 13.926 14.746 13.364 12.264 10.360 16.418 33.347 21.862 23.231 19.669 17.032 14.905 16.683 15.868 12.233 21.723 10.694 11.861 10.223 25.333 20.431 20.962 20.819 16.662 40.738 41.093 37.659 35.033 32.591 30.132 28.468
30.595 19.073 1.311 24.242 23.228 16.952 16.612 18.495 8.203 5.525 41.995 17.643 7.795 4.150 11.877 9.583 3.109 10.947 30.632 27.433 20.679 18.553 16.589 14.810 12.054 11.509 13.330 17.977 12.069 10.953 7.774 21.077 14.500 9.246 8.8t3 6.581 40.718 41.079 37.662 35.035 32.589 30.087 28.464
111.4.1 Online stability assessment Based on a sound knowledge of the power system, it is possible to specify a 'safe' configuration that may be used as a basis for assessing the margin of safety with respect to stability. The term 'safe' here can be attributed to the system's ability to remain in synchronism under such conditions as the occurrence of a 3-phase shortcircuit at the most vulnerable location in the system and its successful removal by primary protection while the system is under normal operation. The safe configuration can as well be based on an 'allowable' or 'safe' MCCT for the system. Let ~s denote the corresponding value of the TSI. If ~? is the TSI calculated for any new configuration, then the margin of safety for stability r is given by: 77s --
(7)
rls This margin of safety will furnish the necessary information regarding the stability behaviour of the system in its present configuration.
111.4.2 Contingency analysis Since the TSI as defined in the present paper is found to be effectively sensitive for changes in system conditions (see Section IV), it can also be used for performing contingency analysis. For this purpose, the TSI for the present configuration is first computed; then, assuming a set of other contingencies, the r7 is calculated for each contingency. The set of rTs thus obtained can identify whether the system is secure or alert.
IV.
Reliability and sensitivity tests
In order to explore the main features of the TSI and to assess its behaviour from a practical viewpoint, two sets of simulations wilt be conducted, related to the reliability and the sensitivity as defined and investigated below. It must be stressed at this point that the tests described below are totally unrelated to using the TSI for stability evaluation, and are performed merely to prove the effectiveness Of the index. The simulations are performed on several test systems and practical networks selected to represent a wide variety of power system characteristics by their number of generators and by their particular distribution-concentration of the power generation-consumption. These systems are: •
their busbars is a priori known to be significantly smaller than the max A = r/(see Table 1 in Section IV).
77
3-machine test system,
• 7-machine CIGRE test system,
111.4 Practical use o f TSI For a given power system, a sensitivity analysis should first be carried out offline by considering several typical configurations and evaluating the TSI for each of them; the sensitivity curve, i.e. ~ against the MCCT can thus be obtained. This sensitivity curve may be used for the two following purposes:
•
9-machine system,
•
14-machine Greek system,
•
15-machine test system,
•
40-machine Belgian HV equivalent power system.
• online transient stability assessment,
The data and the single-line diagrams of these systems, with the exception of the Greek one, are given in Reference 11. The data for the latter can be found in Reference 12.
• contingency analysis.
nA
Electrical Power & Enerqv Svstems
IV.1 Reliability tests The TSI will be considered as reliable if in a power system and for a given configuration it detects effectively the most critical situation. Hence, for establishing the TSI's reliability, many simulations have been performed to find out whether for a system in a given configuration, the MCCT corresponds closely to the maximum A i (i.e. to r~); subsequently, although not very importantly, the variation of A with CCT has also been shown. The simulations concern 3-phase shortcircuits, generally at the busbars of the a priori most critical generators, further assumed to be cleared at the CCT, so that the system's final configuration coincides with the initial, prefaulted, one. For each of these disturbances, the following quantities have been computed: the CCT as provided by a standard numerical integration program, where the generators are represented by a classical simplified model of constant EMF behind transient reactance, i.e. the same model as used for evaluating the TSI; it may be noted that the reported CCT values correspond to the stable case, •
the corresponding Am of Reference 11,
•
the corresponding A as proposed in the present paper.
Table 1 shows the results obtained with the systems listed above considered in their 'base case' configuration. The reference machines for these systems are chosen according to the considerations in Section II and are respectively generators 1, 4, 2, 0, Z and ROMK 1. One can see that in the case of the 7-machine CIGRE test system, the MCCT did not correspond to ?./m computed on the basis of maximum absolute acceleration in the system 11. Obviously, this discrepancy has been solved with the TSI. In the case of the 40-machine system, it can be observed that the MCCT is 0.23 s whereas the maximum A corresponds to a CCT of 0.24 s. This minor inconsistency of less than 1% in A is attributed to the integration technique used as well as to the static equivalent of the large original network computed in evaluating the CCTs. A more thorough study has been performed for the 7-machine CIGRE system and even more for the 14-machine Greek power system for which many configurations (four and 12 in addition to the base case, respectively)have been built up and investigated. To obtain these configurations the following changes with respect to the base case were made (a detailed description of the various changes are reported in Reference 12): • generation levels at the generators being within their rated power outputs, tripping of major loads without changing the generation schedule (the load-generation balance being taken care of by the slack-bus), •
tripping of a group of transmission lines at one time.
Changes in generation are found to be more effective, as they would in general, in causing a considerable shift in the MCCT in the system.
Vol 4 No 2 April 1982
Table 2. Reliability tests on CIGRE system; A versus CCT
Version 1
2
3
4
Fault location (gen. bus no.) 7 5 2 1 7 5 3 2 7 5 3 2 7 5 3 2
CCT,
Am of Reference 1,
s
r a d , s -2
A of equation (5), rad, g-2
0.28 0.35 0.41 0.41 0.23 0.35 0.39 0.41 0.20 0.35 0.38 0.41 0.1_7 0.35 0.38 0.41
31.368 28.164 22.540 22.425 37.373 28.156 23.960 22.539 42.523 28.127 23.929 22.520 48.535 28.063 23.852 22.477
30.994 17.991 18.479 16.778 37.753 16.978 16.219 18.423 43.554 16.924 15.955 18.337 50.331 16.803 15.564 18.190
Tables 2 and 3 show the various versions of the CIGRE and Greek systems studied with the CCTs computed at some of the most interesting generator buses. IV.1.1 Remarks It is only for the sake of economy that the a priori, less interesting disturbances have not been evaluated. However, a good number of checks have been performed in order to make sure that only the smallest values of A are not accurately evaluated and reported in the Tables. The 15-machine system has a symmetrical construction; hence, the complete set of reliability tests would consist of eight rather than 15 3-phase shortcircuits. IV.2 Sensitivity tests Once the TSI is obtained for specified conditions in a given power system, the main concern is then to determine its use in solving the transient preventive control problem 'a. If the index can be considered as a measure of a system's relative stability via the concept of the MCCT, it is possible to predict the stability of the system for any assumed contingency and thus the objective can be achieved. Alternatively, it is essential to assess how sensitive the TSI is for changes in the MCCT, i.e. whether there exists a well defined functional relationship between rt and the MCCT. This aspect of the problem is investigated by conducting appropriate simulations on the five versions of the 7-machine CIGRE test system and on the 13 versions of the 14-machine Greek power network. Tables 4 and 5 list the values of the corresponding 7/s with the MCCTs for the different versions taken from Tables 1-3. It may be noted that the reference machines for these two systems coincide with those of their corresponding base case. The resulting sensitivity curves are shown in Figures 2 and 3. Figures 4 and 5 show the sensitivity curves provided by ~?m of Reference 11. One can see that these curves are less satisfactory than the corresponding curves of Figures 2 and 3.
95
Table 3. Reliability test on Greek system; A versus CCT
Version 1
2
3
4
5
6
7 8
9 10
na
Fault location (gen. bus no.) 34 84 89 76 1 59 77 ~I = 19 33 0 34 84 89 1 76 77 34 1 84 89 0 76 77 0 21 59 84 89 34 76 0 34 84 89 59 76 34 84 l 89 0 34 84 89 34 64 76 34 0 76 84 89 1
CCT, s
Am of Reference 1, rad, s - z
A of equation (5), tad, s-z
0.29 0.38 0.42 0.46 0.47 0.48 0.48 0.50 0.52 0.62 0.62 0.27 0.38 0.42 0.45 0.46 0.48 0.36 0.38 0.39 0.43 0.44 0.47 0.49 0.14 0.31 0.35 0.38 0.43 0.43 0.47 0.20 0.38 0.39 0.43 0.46 0.47 0.25 0.39 0.40 0.43 0.27 0.34 0.26 0.30 0.32 0.32 0.35 0.40 0.41 0.42 0.43 0.43 0.44
34.633 23.648 20.058 17.496 22.515 16.764 15.118 15.320 11.963 12.296 9.545 35.698 23.545 19.985 22.427 17.471 15.100 26.453 24.680 23.251 19.678 15.530 17.098 14.928 38.132 16.676 16.740 23.015 19.467 18.672 16.854 31.063 24.563 23.128 19.560 16.727 16.928 41.425 23.277 27.562 19.741 25.520 29.695 37.531 30.726 29.777 28.514 25.691 23.509 16.861 20.133 20.627 19.592 20.870
34.306 25.298 21.169 17.894 22.472 17.687 15.515 16.311 15.752 12.111 9.923 35.535 25.151 21.084 22.409 17.865 15.493 24.748 22.805 22.771 19.642 17.087 17.055 15.071 58.639 22.860 13.749 15.377 15.675 13.474 15.485 44.496 20.314 17.626 16.885 9.775 15.939 40.152 23.204 25.904 19.939 34.050 26.333 40.562 32.483 29.360 27.603 25.738 21.525 18.649 20.041 19.596 19.292 18.614
Table 3. continued
Version 11
12
Fault location (gen. bus no.) 0 21 84 34 0 21 59 84 89 34 76 77
CCT, s
Am of Reference 1, rad, s -a
A of equation (5), rad,s -2
0.17 0.38 0.39 0.40 0.12 0.27 0.30 0.38 0.43 0.46 0.47 0.48
34.480 16.488 23.077 21.622 40.555 16.803 16.740 22.969 19.433 16.823 16.831 14.834
51.237 19.677 16.527 16.887 63.273 25.051 14.890 14.631 15.278 11.351 15.342 14.082
Table 4. Sensitivity test on CIGRE system
Version 4 3 1 0
MCCT, s
r/m of r/of Reference equation 1, (6), rad, s -2 rad, s -2
Fault location corresponding to MCCT (bus no. )
0.17 0.20 0.23 0.28 0.33
48.535 42.523 37.373 31.368 25.365
7 7 7 7 7
50.331 43.554 37.753 30.994 24.242
Table 5. Sensitivity test on Greek system
Version
MCCT, s
r/m of r/of Reference equation 1, (6), rad, s-2 rad, s-2
12 4 11 5 6 3 7 2 1 0 9 3 10
0.12 0.14 0.17 0.20 0.25 0.26 0.27 0.27 0.29 0.30 0.32 0.36 0.40
40.555 38.132 34.480 31.063 41.425 37.531 25.520 35.698 34.633 33.347 28.514 26.453 23.509
63.273 58.639 51.237 44.496 40.152 40.562 34.050 35.535 34.306 30.632 27.603 24.748 21.525
Fault location corresponding to MCCT (bus no.) 0 0 0 0 34 84 0 34 34 34 64 34 34
gl~etrical P o w e r & Enerav Systems
50
50-m
4O
m
40--
15
30--
30
i
20--
O/
I
I
OI5
02
I 025
I
I I I I I
~8
I
015
O 33
fcm
r~ MCCT
I
1 025
I
1 035
tcm
Figure 2. Sensitivity test on C l G R E 7-machine system; versus
Figure 4. Sensitivity test on Greek 14-machine system; r/versus MCCT
i
40 °o
i
11
20~-
I
o~
o2
>5
o3 o4 tom Figure 3. Sensitivity behaviour of r/m of Reference 11 in
w o u l d throw sufficient light on the stability behaviour of p o w e r systems. From the various results presented in Tables 1-5 and Figures 2 - 5 , the f o l l o w i n g observations can be
C lG R E system case
made.
Vol 4 No 2 April 1982
97
Comparisons of Am and ~rn with A and 77, respectively, from the point of view of reliability and sensitivity show that the location of the fault providing the MCCT influences A m and r?m. However, this problem does not exist if the faults providing the MCCTs are located at the same busbar while effecting changes in the system configuration. The MCCT of a power system for a given load flow and network configuration corresponds to ~ without exception. In other words, r~ can be considered as a mapping of the MCCT and, hence, a one-to-one correspondence between these two always exists for a given power system. The sensitivity curves show that the minimum of the MCCTs corresponds well to the maximum r/. The index has a remarkable sensitivity, especially for lower values of the MCCT, i.e. for the most interesting part of the curve. The sensitivity curve gives a true picture of the criticality of the system at any given instant during operation. Therefore, the TSI can undoubtedly be considered as one measure of relative stability. Also, since the index mirrors all conceivable changes that can take place in a power system, one may conclude that it is indeed possible to specify a measure of relative stability that is valid for all situations arising in a power system. The index can also indicate with sufficient accuracy the first few pairs of machines (with a reference machine implied) that are likely to create problems. IV.4 Supplementary remarks Although the index is deterministic, the approach should partially be based on pattern-recognition techniques in order to determine the 'threshold' r for a given power system, i.e. for fixing a 'safe configuration' (base-case configuration) and a critical value of r? above for which the system should be considered to be in an alert state. In order to obtain a dimensionless stability index for a given power system, one can use r/' = r~/rls instead oft/, where r/s corresponds to the base case configuration or any other configuration considered to be the 'safe' one, as shown in Figures 2 and 3. The simulations carried out up to this point have been concerned with 3-phase shortcircuits at the generator buses only; this is because, as observed above, the severest disturbance has always been found to be of this kind. It may be, however, that the fault lies elsewhere - for inStance at a point subjected to adverse weather conditions - and there is no difficulty in assessing the severity of any other kind of disturbance by evaluating its corresponding A. For illustrative purposes, simulations have been performed of 3-phase shortcircuits located at other points than the generator buses in the Greek power system. Table 6 lists the values of the As found and compares them with their corresponding CCTs. One can see that the consistency in reliability is preserved when inserting these values into the corresponding versions.
98
Table 6. Reliability tests on Greek system: A versus CCT (fault location different from generator buses)
Version
Fault location (bus no.) 37 72 91 37 72 91
CCT, s
A m of Reference 1, rad, s-2
A of equation (5), rad, s-2
0.26 0.27 0.32 0.33 0.35 0.40
39.534 37.654 32.465 27.930 26.219 23.097
38.110 36.127 30.666 27.474 25.746 23.083
Similarly, the occurrence of simultaneous faults has not been considered in the present work; if necessary, this could be included in the calculations. It is possible that more than one generator (or group of generators) in a power system has the same maximum nominal power available at its busbar, as in the case of the 15 machine test system studied. In such cases, any one of these machines (or groups of machines) can be taken as the reference machine. While evaluating the A for a 3-phase shortcircuit at these generator buses, however, it is suggested that the reference machine is chosen to correspond to the generator under investigation.
V. Practical use of TSI for online preventive control The TSI may be used to perform not only transient stability assessment but also transient preventive control, for instance when associated with an optimal power-flow program. Indeed, the TSI has the advantage of being a function of the static control variables defining the state of the system before perturbation as the real generated powers P and the voltage magnitudes V at the generator buses. In these conditions, the inequalities ~(P, V) ~< ~s may be added to the statement of an optimal power 17ow used for preventive control. Without the above inequality, the usual optimal power flows determine the values a l P and V that meet economy and security requirements considered from a static point of view only. When adding these inequalities, the optimal power flows will determine the values a l P and V that not only meet economy and static security but also prevent the system from being unstable, i.e. they will perform a 'transient preventive control'. From a practical point of view, handling this kind of constraint could be easily embedded in 'compact' optimal power-flow methods, where the computation is performed in two steps used iteratively. In the first step, a 'reduced model' is built where the useful constraints to the system are addressed in terms of the static control variables before perturbation, limited to a linear or quadratic development. Then, in the second step, this model is optimized, i.e. the constraints are met while minimizing the operating cost. In these conditions, in order to handle the new constraints ~(P, V) ~< ~s it is sufficient, when building the reduced model, to compute B and the first partial derivatives o f t
Electrical Power & Energy Systems
with respect to P and V, which may be realized by using sensitivity techniques with the transposed Jacobian method. Such developments are now expected to include transient preventive control in the differential injections method 14 used for optimal power flows.
Ribbens-Pavella, M 'Critical survey of transient stability studies of multimachine power systems by Lyapunov's direct-method' Proc. 9th Allerton Conf. on Circuits andSystem Theory University of Illinois, USA (1971) pp 751-767 3 Sastry, V R and Murthy, P G 'Derivation of completely controllable and completely observable state models for multimachine power system stability studies' Int. J. Control Vol /6 No 4 (October 1972) pp 777-788
V I . Conclusions A transient stability index has been proposed for online stability analysis of power systems. Conceptually, it is intended to provide a measure of the robustness of the system in a given configuration without any actual conventional disturbance simulation and subsequent search for the corresponding critical clearing time.
Although initially based upon considerations derived from the Lyapunov criterion applied to conventional transient stability analysis, the TSI eventually yields a simple expression that is physically meaningful and computationally very fast and straightforward. Moreover, this expression is free from the simplifying assumptions usually required for the construction of Lyapunov functions for power systems. And, what is even more interesting, the flexibility of the TSI expression allows for the accounting of various models of the system's components (generators, loads, etc.). The suitability of the TSI has been illustrated by means of numerous simulations performed on six appropriate power systems. The simulations have been mainly aimed at exploring two essential characteristics of the index, namely its reliability and its sensitivity. They have proved that the index behaves very well: it has a remarkable sensitivity, especially on the most interesting part of the sensitivity curve, and it reflects properly the stability behaviour of a power system.
Ribbens-Pavella, M and Lemal, B 'Fast determination of stability regions for online transient power system studies' Proc. Inst. Electr. Eng. Vol 123 No 7 (July 1976) pp 689-696 Athay, T et al. 'Transient energy stability analysis' presented at Engineering Foundation Conference US Department of Energy Davos, Switzerland (30 September-5 October 1979) Kakimoto, N, Ohsawa, Y and Hayashi, M 'Transient stability analysis of multimachine power systems in the field flux decays via Lyapunov's direct method IEEE PES Winter Meeting (3-8 February 1980) Paper No A 80 231 Ribbens-Pavella, M et al. 'Direct methods for offline and online transient stability assessment of power systems' Proc. 18th AIlerton Conf. (1980)pp 735-744
Billington, A and Kuruganty, P R S 'A probabilistic index for transient stability' IEEE Trans. Power Appar. & Syst. Vol PAS-99 No 1 (January/February 1980) pp 195-206
Baratella, Pet al. 'Fast computation of the transient overloadings in multimachine systems' Proc. 6th PSCC Darmstadt (August 1978) pp 953-956
Three different uses of the TSI have been suggested: •
online stability assessment,
•
contingency analysis,
•
online preventive control.
The first two uses have been thoroughly investigated in the present paper. In conclusion, the authenticity of the TSI allows for predicting its effective implementation in online stability assessment and control.
V l I. References 1 Fouad, A A 'Stability-theory criteria for transient stability' Proc. Eng. Foundation Conf. on Systems Engineering for Power," Status and Prospects Henniker, USA (August 1975) pp 421-450
Vol 4 No 2 April 1982
10 Chamorro, R S, Anderson, M D and Richards, E F 'Fast transient contingency evaluation in power systems' presented at IEEE PES Summer Meeting Minneapolis, USA (July 1980) Paper No 80 SM 589-2 11 Ribbens-Pavella, M, Calvaer, A and Gheury, J 'Transient stability index for online evaluation' presented at IEEE PES Winter Meeting New York, USA (3-8 February 1980) Paper No A 80 013-3 12 Ribbens-Pavella, M, Murthy, P G and Horward, J L 'Transient stability index: an online tool for stability assessment and contingency evaluation' Int. Rep. No LML/2 University of Liege, Belgium (December 1980) 13 Carpentier, J 'Prospects for security control in electric power systems' presented at 8th IFA C World Congress Japan (24-28 August 1981)
14 Carpentier, J L 'Differential injections method, a general method for secure and optimal load flow' Proc. PICA (1973) pp 255-262
99
contingency evaluation M Ribbens-Pavella, P G Murthy and J L Horward Department of Electrical Engineering, University of Liege, Sart-Tilman, B-4000 Liege, Belgium
J L Carpentier Electricit~ de France, 2 Rue Louis Murat, 75008 Paris, France
An online methodology is proposed for assessing the robustness o f a power system from the point o f view o f transient stability, and a scalar expression, the transient stability index, is accordingly derived. The reliability and sensitivity o f this index are tested by means o f simulations for a number o f power system cases. The index is shown to be appropriate for online stability assessment, contingency evaluation and preventive control Keywords: electric power systems, network analysis, transient stability, security assessment
I. I n t r o d u c t i o n Stable operation of synchronous generators connected together and running in parallel has long been considered one of the foremost requirements for reliable service. It has also been recognized that the study of stability forms an integral part of power-system planning and design. Conventional transient stability evaluation is normally done by simulating severe disturbances (generally one at a time) and determining the corresponding critical clearing times. To reduce the computational burden as far as possible, direct methods employing Lyapunov's stability theory have been developed since the 1960s. They have been proved to be very encouraging and promising, at least in the initial stages of system design 1' 2. Further improvements to counteract the conservative character of the Lyapunov method have recently been proposed (see, for instance, those reported in References 3 - 7 ) . Note that despite all the improvements achieved in this field, conventional transient stability analysis, by its very nature, remains essentially an offline procedure.
Contrary to an abundance of literature available on the conventional approach, the online assessment of power system stability while in operation is a fairly recent development and is still in its infancy. Nevertheless, this Received: 27 April 1981
Vol 4 No 2 April 1982
stability aspect is at least as important as the conventional offline one. Indeed, a power system is never in a steady state, in the true sense, during operation. There are always haphazard changes in the load levels with associated adjustments in generation. But, much more important than these random changes are the large network disturbances associated with possible disconnection of certain important transmission lines in the event of unsuccessful reclosure leading to quasi-permanent changes in the network topology. Other modifications during operation may also arise, leading to important changes in the network configuration. A knowledge of how safe or critical the system is at any given instant with respect to stability is therefore essential in order for the operating personnel to take appropriate decisions to maintain the security of the system. In accordance with these motivations, the problem of power-system stability assessment, suitable for online implementation, has recently been considered. Some of the significant work in this direction includes an approach based on probabilistic considerations 8, methods for fast computation of transient stability 9, lo and the concept of a deterministic transient stability index n. The deterministic index aims at assessing the stability of a system with a given configuration in terms of the value that it assumes for this configuration. Obviously, the evaluation of the (scalar) expression of this index should be computationally quite straightforward in order to constitute a true online stability measure. The present work is based on the same concept as above. However, significant improvements have been included to take care of certain inconsistencies encountered while testing the index of Reference 11. This has led the authors of the present paper to reexamine the physical foundations and to define a new transient stability index. The developments related to this new index, the derivation of its expression and some of its potential practical uses are reported in Section III. Section IV introduces the notions of reliability and sensitivity and explores them thoroughly. The investigations are based upon simulations performed
0142-0615/82/020091-09 $03.00 © 1982 Butterworth & Co (Publishers) Ltd
91
on six different appropriately selected power systems; the results obtained are tabulated, commented on and discussed in detail. It is found that the index behaves remarkably well. It is also shown that this index introduces a substantial improvement compared with the index of Reference 11. Finally, Section V suggests another potential use of the index concerning transient preventive control.
II. Some preliminaries In the light of the present analysis, it is proposed to introduce some terminology and a few definitions for a better appreciation of the principle underlying the concept. In a power system it is always possible to have different critical clearing times (CCTs) depending upon the severity of the fault, its location, load-generation levels and the network configuration prevalent at the instant of fault inception. Further, there is no loss of generality in assuming that the severest fault is a 3-phase shortcircuit located at a generator high-voltage bus and adversely affecting the stability of the system. The 'minimum critical clearing time' (MeeT) in a power system under known load-generation balance and network topology is given by the smallest value of the critical clearing times evaluated for an assumed 3-phase shortcircuit at each of the generator high-voltage buses in the system. Of course, this M e e T varies with changes in generation, loads and network topology. We are then justified in accepting that the M e e T is a positive indication of how safe or critical the system is, and can be considered as a measure of relative stability of the system in a given 'configuration'. 'Configuration' is used to denote a given load flow and topology of the system. The 'transient stability index' (TSI) is a figure that is obtained by considering the maximum relative acceleration experienced by the system when referred to a chosen reference machine immediately after the initiation of a 3-phase shortcircuit at each of the generator high-voltage buses in the system. The 'average acceleration' of the system is the mean value of the accelerations of all the machines in the system at to+ when a 3-phase shortcircuit has taken place at a generator bus. Obviously, the average acceleration is different for faults at different generator buses. The 'reference machine' is the one having the maximum nominal power available at the busbar in the system. If more than one machine possesses the same maximum nominal power, any one of them can equally well serve as the reference machine.
III. Transient stability index Ilia Concept of TSI The main consideration involved in evolving the TSI is its relationship to the most dangerous disturbance, i.e. the one capable of causing the severest instability in a given system. In other words, the main aim is to derive an expression whose value indicates the severity of the most dangerous disturbance in a power system with a given configuration, without identifying this disturbance and without evaluating its corresponding CCT. In fact, the aim
is to totally replace the computation of the M e e T (which, as is well known, is quite time consuming since it implies simulating many disturbances and finding their corresponding CCTs) by using an appropriate scalar expression (the TSI) which is independent of any transient stability run and whose computation is quite straightforward. In deriving such an expression, the authors were guided by the acceleration approach used to assess the practical stability domain estimate of the Lyapunov-like criterion and by the following reasoning 7' n. Experience shows that the severest disturbance that a power system may be subjected to is a 3-phase shortcircuit at a machine's busbar. According to the Lyapunov-like criterion, the value of the CCT of a given disturbance depends on the unstable point relative to the machine possessing the 'largest initial acceleration' during the transients; hence the most dangerous disturbance will be related to the machine possessing the largest initial acceleration when a 3-phase shortcircuit acts at the busbar of that or another machine. In fact, in a power system consisting of several synchronous generators running in parallel, it is quite possible, when a shortcircuit occurs, that some of the machines accelerate and some decelerate. Consequently, the machine having maximum absolute acceleration need not necessarily fall out of step. In other words, the maximum initial acceleration of any machine when viewed in isolation may not always provide the true picture of the system's response. What is more appropriate under these conditions, i.e. what decides the stability behaviour of the system, is the maximum initial relative acceleration. Following the above reasoning, the severity of the most dangerous disturbance may be assessed by means of the maximum difference between this largest initial relative acceleration and the mean acceleration of the overall system's machines. The definition of the TSI now follows.
111.2 TSI derivation
Initially, a 3-phase shortcircuit is created at the ith generator bus at to. Let 7k denote the acceleration of the kth machine at to +; clearly, 7k
P m k - Pek
Mk
k = 1,2 . . . . . n ; k ~ i (1)
Pmi Mi where Mk, Pink and Pek denote the inertia constant, the "Yi-
mechanical input power and the electrical output power of the kth machine respectively. This latter power may be expressed by: Pek = Re Ek
iYk/Ej
(2)
/=1 where E k is the internal voltage of generator k (behind its transient reactance) and where iYk! is the kith element of the admittance matrix reduced to the generator nodes
when a 3-phase shortcircuit acts at the ith generator node. Then, let "7[ denote the absolute value of the maximum relative acceleration with respect to the reference machine R, as:
T~ A
Irk -
max k=l,2,...,n--
T~
I
(3)
I
Moreover, let the absolute value of the average acceleration of the system be designated by "7i under these conditions. Thus:
(4)
Since the main concern is to get Peg at to + different numerical techniques can be used to reduce the number of iterations and hence the computational time for solving the algebraic equations. One alternative is to eliminate completely the reduction step and simulate the entire system for the solution of the network performance equations.
(5)
If the network-reduction step is avoided, it is possible to represent loads according to their respective load characteristics; it will not, therefore, be necessary to assume a constant impedance representation for the loads.
Now let A i be the difference between the absolute value of the maximum relative acceleration and that of the average acceleration of the system, as:
Ai & 1 7 f - ' ~ i l Finally, the above three-step procedure is repeated for all the generator buses in the system, and the values of the Ais are calculated. The transient stability index, r~, for a specified load flow and network configuration is now given by: n =a
max
Ai
Remarks
The only time-consuming computation is that of the reduction of the system admittance matrix to the generator nodes when each of the generators is subjected to a 3-phase shortcircuit, as can be seen in equation (2). Consequently, there are, in principle, a maximum ofn such matrix reductions to be calculated. However, owing to the superposition technique used in Reference 11, these n reductions may be replaced by only one with a few additional computations of some simple analytical expressions. This allows for a substantial saving in computing time.
A sound knowledge of the power system avoids a series of computations and therefore saves computing time when evaluating the TSI. Indeed, in a given power system, there are generally a number of 'uninteresting' generators in that the A corresponding to a 3-phase shortcircuit at (6)
i=l,2,...,n
Calculate initial load flow for specified load- generation balance and network configuration
Remarks Formally, the difference between the expression of the above index and the one suggested earlier (r/m in Reference 11) is that ~ in equation (6) proposed here is onthe basis of the maximum relative acceleration rather than the maximum absolute acceleration acquired by a generator.
t Read in data of machines for translent
stability studies with reference generator indicated Compute initial state
Rather than considering the average acceleration 3; expressed by equation (4), the acceleration of the centre of inertia
T Set bus count = 0
",/g t/
Create fault at ith generator terminal bus at time to and set terminal voltage to zero
ZM; ; j=l -'/g - - g/
Solve algebraic equations for reduced network to obtain electrical power outputs (Pet) of all generators at to+
EMi
j-I
could be taken and another stability index derived accordingly. However, this modification has not been found worthwhile. Although derived from the Lyapunov criterion, the TSI is free from the simplifying assumptions that are usually made when constructing Lyapunov functions for power system transient stability analysis.
I
Compute accelerations of all machines from equation (I) Calculate relative accelerations with respect to reference machine and obtain 7/" from equation (3], ~. from equation (4) and Z~i from equation (5)
Determine
111.3 TSI computation The flowchart given in Figure 1 indicates the various steps of the general procedure involved in the computation of the TSI. It may be noted that calculating 7?implies simulating a 3-phase shortcircuit at the busbar of each of the system's machines successively.
Vol 4 No 2 April 1982
[ Print results I
Figure 1. Flowchart for calculatingTS1
93
Table 1. Reliability tests: A versus CCT
Version
Fault location (gen. bus no.)
3-machine
3 2 1 7-machine 7 l 5 3 2 4 6 9-machine 3 2 1 6 9 5 7 4 14-machine 34 0 84 89 76 77 59 21 19 1 17 33 64 15-machine K1 N1 C1 GI Z 40-machine AWlR 4 BRES 4 VILG 4 FARC 4 LANB 3 SCHE 3A TRIV 3
CCT, s
A m of Reference 1, rad,s -2
A of equation (5), rad, x-z
0.18 0.32 0.39 0.33 0.35 0.35 0.39 0.41 0.50 0.52 0.26 0.44 0.45 0.47 0.48 0.49 0.59 0.60 0.30 0.32 0.39 0.43 0.47 0.48 0.49 0.53 0.53 0.57 0.59 0.64 0.70 0.36 0.40 0.41 0.42 0.50 0.23 0.24 0.25 0.27 0.28 0.30 0.32
27.160 18.343 11.218 25.363 28.132 28.238 23.916 22.508 14.020 14.510 44.684 17.183 13.926 14.746 13.364 12.264 10.360 16.418 33.347 21.862 23.231 19.669 17.032 14.905 16.683 15.868 12.233 21.723 10.694 11.861 10.223 25.333 20.431 20.962 20.819 16.662 40.738 41.093 37.659 35.033 32.591 30.132 28.468
30.595 19.073 1.311 24.242 23.228 16.952 16.612 18.495 8.203 5.525 41.995 17.643 7.795 4.150 11.877 9.583 3.109 10.947 30.632 27.433 20.679 18.553 16.589 14.810 12.054 11.509 13.330 17.977 12.069 10.953 7.774 21.077 14.500 9.246 8.8t3 6.581 40.718 41.079 37.662 35.035 32.589 30.087 28.464
111.4.1 Online stability assessment Based on a sound knowledge of the power system, it is possible to specify a 'safe' configuration that may be used as a basis for assessing the margin of safety with respect to stability. The term 'safe' here can be attributed to the system's ability to remain in synchronism under such conditions as the occurrence of a 3-phase shortcircuit at the most vulnerable location in the system and its successful removal by primary protection while the system is under normal operation. The safe configuration can as well be based on an 'allowable' or 'safe' MCCT for the system. Let ~s denote the corresponding value of the TSI. If ~? is the TSI calculated for any new configuration, then the margin of safety for stability r is given by: 77s --
(7)
rls This margin of safety will furnish the necessary information regarding the stability behaviour of the system in its present configuration.
111.4.2 Contingency analysis Since the TSI as defined in the present paper is found to be effectively sensitive for changes in system conditions (see Section IV), it can also be used for performing contingency analysis. For this purpose, the TSI for the present configuration is first computed; then, assuming a set of other contingencies, the r7 is calculated for each contingency. The set of rTs thus obtained can identify whether the system is secure or alert.
IV.
Reliability and sensitivity tests
In order to explore the main features of the TSI and to assess its behaviour from a practical viewpoint, two sets of simulations wilt be conducted, related to the reliability and the sensitivity as defined and investigated below. It must be stressed at this point that the tests described below are totally unrelated to using the TSI for stability evaluation, and are performed merely to prove the effectiveness Of the index. The simulations are performed on several test systems and practical networks selected to represent a wide variety of power system characteristics by their number of generators and by their particular distribution-concentration of the power generation-consumption. These systems are: •
their busbars is a priori known to be significantly smaller than the max A = r/(see Table 1 in Section IV).
77
3-machine test system,
• 7-machine CIGRE test system,
111.4 Practical use o f TSI For a given power system, a sensitivity analysis should first be carried out offline by considering several typical configurations and evaluating the TSI for each of them; the sensitivity curve, i.e. ~ against the MCCT can thus be obtained. This sensitivity curve may be used for the two following purposes:
•
9-machine system,
•
14-machine Greek system,
•
15-machine test system,
•
40-machine Belgian HV equivalent power system.
• online transient stability assessment,
The data and the single-line diagrams of these systems, with the exception of the Greek one, are given in Reference 11. The data for the latter can be found in Reference 12.
• contingency analysis.
nA
Electrical Power & Enerqv Svstems
IV.1 Reliability tests The TSI will be considered as reliable if in a power system and for a given configuration it detects effectively the most critical situation. Hence, for establishing the TSI's reliability, many simulations have been performed to find out whether for a system in a given configuration, the MCCT corresponds closely to the maximum A i (i.e. to r~); subsequently, although not very importantly, the variation of A with CCT has also been shown. The simulations concern 3-phase shortcircuits, generally at the busbars of the a priori most critical generators, further assumed to be cleared at the CCT, so that the system's final configuration coincides with the initial, prefaulted, one. For each of these disturbances, the following quantities have been computed: the CCT as provided by a standard numerical integration program, where the generators are represented by a classical simplified model of constant EMF behind transient reactance, i.e. the same model as used for evaluating the TSI; it may be noted that the reported CCT values correspond to the stable case, •
the corresponding Am of Reference 11,
•
the corresponding A as proposed in the present paper.
Table 1 shows the results obtained with the systems listed above considered in their 'base case' configuration. The reference machines for these systems are chosen according to the considerations in Section II and are respectively generators 1, 4, 2, 0, Z and ROMK 1. One can see that in the case of the 7-machine CIGRE test system, the MCCT did not correspond to ?./m computed on the basis of maximum absolute acceleration in the system 11. Obviously, this discrepancy has been solved with the TSI. In the case of the 40-machine system, it can be observed that the MCCT is 0.23 s whereas the maximum A corresponds to a CCT of 0.24 s. This minor inconsistency of less than 1% in A is attributed to the integration technique used as well as to the static equivalent of the large original network computed in evaluating the CCTs. A more thorough study has been performed for the 7-machine CIGRE system and even more for the 14-machine Greek power system for which many configurations (four and 12 in addition to the base case, respectively)have been built up and investigated. To obtain these configurations the following changes with respect to the base case were made (a detailed description of the various changes are reported in Reference 12): • generation levels at the generators being within their rated power outputs, tripping of major loads without changing the generation schedule (the load-generation balance being taken care of by the slack-bus), •
tripping of a group of transmission lines at one time.
Changes in generation are found to be more effective, as they would in general, in causing a considerable shift in the MCCT in the system.
Vol 4 No 2 April 1982
Table 2. Reliability tests on CIGRE system; A versus CCT
Version 1
2
3
4
Fault location (gen. bus no.) 7 5 2 1 7 5 3 2 7 5 3 2 7 5 3 2
CCT,
Am of Reference 1,
s
r a d , s -2
A of equation (5), rad, g-2
0.28 0.35 0.41 0.41 0.23 0.35 0.39 0.41 0.20 0.35 0.38 0.41 0.1_7 0.35 0.38 0.41
31.368 28.164 22.540 22.425 37.373 28.156 23.960 22.539 42.523 28.127 23.929 22.520 48.535 28.063 23.852 22.477
30.994 17.991 18.479 16.778 37.753 16.978 16.219 18.423 43.554 16.924 15.955 18.337 50.331 16.803 15.564 18.190
Tables 2 and 3 show the various versions of the CIGRE and Greek systems studied with the CCTs computed at some of the most interesting generator buses. IV.1.1 Remarks It is only for the sake of economy that the a priori, less interesting disturbances have not been evaluated. However, a good number of checks have been performed in order to make sure that only the smallest values of A are not accurately evaluated and reported in the Tables. The 15-machine system has a symmetrical construction; hence, the complete set of reliability tests would consist of eight rather than 15 3-phase shortcircuits. IV.2 Sensitivity tests Once the TSI is obtained for specified conditions in a given power system, the main concern is then to determine its use in solving the transient preventive control problem 'a. If the index can be considered as a measure of a system's relative stability via the concept of the MCCT, it is possible to predict the stability of the system for any assumed contingency and thus the objective can be achieved. Alternatively, it is essential to assess how sensitive the TSI is for changes in the MCCT, i.e. whether there exists a well defined functional relationship between rt and the MCCT. This aspect of the problem is investigated by conducting appropriate simulations on the five versions of the 7-machine CIGRE test system and on the 13 versions of the 14-machine Greek power network. Tables 4 and 5 list the values of the corresponding 7/s with the MCCTs for the different versions taken from Tables 1-3. It may be noted that the reference machines for these two systems coincide with those of their corresponding base case. The resulting sensitivity curves are shown in Figures 2 and 3. Figures 4 and 5 show the sensitivity curves provided by ~?m of Reference 11. One can see that these curves are less satisfactory than the corresponding curves of Figures 2 and 3.
95
Table 3. Reliability test on Greek system; A versus CCT
Version 1
2
3
4
5
6
7 8
9 10
na
Fault location (gen. bus no.) 34 84 89 76 1 59 77 ~I = 19 33 0 34 84 89 1 76 77 34 1 84 89 0 76 77 0 21 59 84 89 34 76 0 34 84 89 59 76 34 84 l 89 0 34 84 89 34 64 76 34 0 76 84 89 1
CCT, s
Am of Reference 1, rad, s - z
A of equation (5), tad, s-z
0.29 0.38 0.42 0.46 0.47 0.48 0.48 0.50 0.52 0.62 0.62 0.27 0.38 0.42 0.45 0.46 0.48 0.36 0.38 0.39 0.43 0.44 0.47 0.49 0.14 0.31 0.35 0.38 0.43 0.43 0.47 0.20 0.38 0.39 0.43 0.46 0.47 0.25 0.39 0.40 0.43 0.27 0.34 0.26 0.30 0.32 0.32 0.35 0.40 0.41 0.42 0.43 0.43 0.44
34.633 23.648 20.058 17.496 22.515 16.764 15.118 15.320 11.963 12.296 9.545 35.698 23.545 19.985 22.427 17.471 15.100 26.453 24.680 23.251 19.678 15.530 17.098 14.928 38.132 16.676 16.740 23.015 19.467 18.672 16.854 31.063 24.563 23.128 19.560 16.727 16.928 41.425 23.277 27.562 19.741 25.520 29.695 37.531 30.726 29.777 28.514 25.691 23.509 16.861 20.133 20.627 19.592 20.870
34.306 25.298 21.169 17.894 22.472 17.687 15.515 16.311 15.752 12.111 9.923 35.535 25.151 21.084 22.409 17.865 15.493 24.748 22.805 22.771 19.642 17.087 17.055 15.071 58.639 22.860 13.749 15.377 15.675 13.474 15.485 44.496 20.314 17.626 16.885 9.775 15.939 40.152 23.204 25.904 19.939 34.050 26.333 40.562 32.483 29.360 27.603 25.738 21.525 18.649 20.041 19.596 19.292 18.614
Table 3. continued
Version 11
12
Fault location (gen. bus no.) 0 21 84 34 0 21 59 84 89 34 76 77
CCT, s
Am of Reference 1, rad, s -a
A of equation (5), rad,s -2
0.17 0.38 0.39 0.40 0.12 0.27 0.30 0.38 0.43 0.46 0.47 0.48
34.480 16.488 23.077 21.622 40.555 16.803 16.740 22.969 19.433 16.823 16.831 14.834
51.237 19.677 16.527 16.887 63.273 25.051 14.890 14.631 15.278 11.351 15.342 14.082
Table 4. Sensitivity test on CIGRE system
Version 4 3 1 0
MCCT, s
r/m of r/of Reference equation 1, (6), rad, s -2 rad, s -2
Fault location corresponding to MCCT (bus no. )
0.17 0.20 0.23 0.28 0.33
48.535 42.523 37.373 31.368 25.365
7 7 7 7 7
50.331 43.554 37.753 30.994 24.242
Table 5. Sensitivity test on Greek system
Version
MCCT, s
r/m of r/of Reference equation 1, (6), rad, s-2 rad, s-2
12 4 11 5 6 3 7 2 1 0 9 3 10
0.12 0.14 0.17 0.20 0.25 0.26 0.27 0.27 0.29 0.30 0.32 0.36 0.40
40.555 38.132 34.480 31.063 41.425 37.531 25.520 35.698 34.633 33.347 28.514 26.453 23.509
63.273 58.639 51.237 44.496 40.152 40.562 34.050 35.535 34.306 30.632 27.603 24.748 21.525
Fault location corresponding to MCCT (bus no.) 0 0 0 0 34 84 0 34 34 34 64 34 34
gl~etrical P o w e r & Enerav Systems
50
50-m
4O
m
40--
15
30--
30
i
20--
O/
I
I
OI5
02
I 025
I
I I I I I
~8
I
015
O 33
fcm
r~ MCCT
I
1 025
I
1 035
tcm
Figure 2. Sensitivity test on C l G R E 7-machine system; versus
Figure 4. Sensitivity test on Greek 14-machine system; r/versus MCCT
i
40 °o
i
11
20~-
I
o~
o2
>5
o3 o4 tom Figure 3. Sensitivity behaviour of r/m of Reference 11 in
w o u l d throw sufficient light on the stability behaviour of p o w e r systems. From the various results presented in Tables 1-5 and Figures 2 - 5 , the f o l l o w i n g observations can be
C lG R E system case
made.
Vol 4 No 2 April 1982
97
Comparisons of Am and ~rn with A and 77, respectively, from the point of view of reliability and sensitivity show that the location of the fault providing the MCCT influences A m and r?m. However, this problem does not exist if the faults providing the MCCTs are located at the same busbar while effecting changes in the system configuration. The MCCT of a power system for a given load flow and network configuration corresponds to ~ without exception. In other words, r~ can be considered as a mapping of the MCCT and, hence, a one-to-one correspondence between these two always exists for a given power system. The sensitivity curves show that the minimum of the MCCTs corresponds well to the maximum r/. The index has a remarkable sensitivity, especially for lower values of the MCCT, i.e. for the most interesting part of the curve. The sensitivity curve gives a true picture of the criticality of the system at any given instant during operation. Therefore, the TSI can undoubtedly be considered as one measure of relative stability. Also, since the index mirrors all conceivable changes that can take place in a power system, one may conclude that it is indeed possible to specify a measure of relative stability that is valid for all situations arising in a power system. The index can also indicate with sufficient accuracy the first few pairs of machines (with a reference machine implied) that are likely to create problems. IV.4 Supplementary remarks Although the index is deterministic, the approach should partially be based on pattern-recognition techniques in order to determine the 'threshold' r for a given power system, i.e. for fixing a 'safe configuration' (base-case configuration) and a critical value of r? above for which the system should be considered to be in an alert state. In order to obtain a dimensionless stability index for a given power system, one can use r/' = r~/rls instead oft/, where r/s corresponds to the base case configuration or any other configuration considered to be the 'safe' one, as shown in Figures 2 and 3. The simulations carried out up to this point have been concerned with 3-phase shortcircuits at the generator buses only; this is because, as observed above, the severest disturbance has always been found to be of this kind. It may be, however, that the fault lies elsewhere - for inStance at a point subjected to adverse weather conditions - and there is no difficulty in assessing the severity of any other kind of disturbance by evaluating its corresponding A. For illustrative purposes, simulations have been performed of 3-phase shortcircuits located at other points than the generator buses in the Greek power system. Table 6 lists the values of the As found and compares them with their corresponding CCTs. One can see that the consistency in reliability is preserved when inserting these values into the corresponding versions.
98
Table 6. Reliability tests on Greek system: A versus CCT (fault location different from generator buses)
Version
Fault location (bus no.) 37 72 91 37 72 91
CCT, s
A m of Reference 1, rad, s-2
A of equation (5), rad, s-2
0.26 0.27 0.32 0.33 0.35 0.40
39.534 37.654 32.465 27.930 26.219 23.097
38.110 36.127 30.666 27.474 25.746 23.083
Similarly, the occurrence of simultaneous faults has not been considered in the present work; if necessary, this could be included in the calculations. It is possible that more than one generator (or group of generators) in a power system has the same maximum nominal power available at its busbar, as in the case of the 15 machine test system studied. In such cases, any one of these machines (or groups of machines) can be taken as the reference machine. While evaluating the A for a 3-phase shortcircuit at these generator buses, however, it is suggested that the reference machine is chosen to correspond to the generator under investigation.
V. Practical use of TSI for online preventive control The TSI may be used to perform not only transient stability assessment but also transient preventive control, for instance when associated with an optimal power-flow program. Indeed, the TSI has the advantage of being a function of the static control variables defining the state of the system before perturbation as the real generated powers P and the voltage magnitudes V at the generator buses. In these conditions, the inequalities ~(P, V) ~< ~s may be added to the statement of an optimal power 17ow used for preventive control. Without the above inequality, the usual optimal power flows determine the values a l P and V that meet economy and security requirements considered from a static point of view only. When adding these inequalities, the optimal power flows will determine the values a l P and V that not only meet economy and static security but also prevent the system from being unstable, i.e. they will perform a 'transient preventive control'. From a practical point of view, handling this kind of constraint could be easily embedded in 'compact' optimal power-flow methods, where the computation is performed in two steps used iteratively. In the first step, a 'reduced model' is built where the useful constraints to the system are addressed in terms of the static control variables before perturbation, limited to a linear or quadratic development. Then, in the second step, this model is optimized, i.e. the constraints are met while minimizing the operating cost. In these conditions, in order to handle the new constraints ~(P, V) ~< ~s it is sufficient, when building the reduced model, to compute B and the first partial derivatives o f t
Electrical Power & Energy Systems
with respect to P and V, which may be realized by using sensitivity techniques with the transposed Jacobian method. Such developments are now expected to include transient preventive control in the differential injections method 14 used for optimal power flows.
Ribbens-Pavella, M 'Critical survey of transient stability studies of multimachine power systems by Lyapunov's direct-method' Proc. 9th Allerton Conf. on Circuits andSystem Theory University of Illinois, USA (1971) pp 751-767 3 Sastry, V R and Murthy, P G 'Derivation of completely controllable and completely observable state models for multimachine power system stability studies' Int. J. Control Vol /6 No 4 (October 1972) pp 777-788
V I . Conclusions A transient stability index has been proposed for online stability analysis of power systems. Conceptually, it is intended to provide a measure of the robustness of the system in a given configuration without any actual conventional disturbance simulation and subsequent search for the corresponding critical clearing time.
Although initially based upon considerations derived from the Lyapunov criterion applied to conventional transient stability analysis, the TSI eventually yields a simple expression that is physically meaningful and computationally very fast and straightforward. Moreover, this expression is free from the simplifying assumptions usually required for the construction of Lyapunov functions for power systems. And, what is even more interesting, the flexibility of the TSI expression allows for the accounting of various models of the system's components (generators, loads, etc.). The suitability of the TSI has been illustrated by means of numerous simulations performed on six appropriate power systems. The simulations have been mainly aimed at exploring two essential characteristics of the index, namely its reliability and its sensitivity. They have proved that the index behaves very well: it has a remarkable sensitivity, especially on the most interesting part of the sensitivity curve, and it reflects properly the stability behaviour of a power system.
Ribbens-Pavella, M and Lemal, B 'Fast determination of stability regions for online transient power system studies' Proc. Inst. Electr. Eng. Vol 123 No 7 (July 1976) pp 689-696 Athay, T et al. 'Transient energy stability analysis' presented at Engineering Foundation Conference US Department of Energy Davos, Switzerland (30 September-5 October 1979) Kakimoto, N, Ohsawa, Y and Hayashi, M 'Transient stability analysis of multimachine power systems in the field flux decays via Lyapunov's direct method IEEE PES Winter Meeting (3-8 February 1980) Paper No A 80 231 Ribbens-Pavella, M et al. 'Direct methods for offline and online transient stability assessment of power systems' Proc. 18th AIlerton Conf. (1980)pp 735-744
Billington, A and Kuruganty, P R S 'A probabilistic index for transient stability' IEEE Trans. Power Appar. & Syst. Vol PAS-99 No 1 (January/February 1980) pp 195-206
Baratella, Pet al. 'Fast computation of the transient overloadings in multimachine systems' Proc. 6th PSCC Darmstadt (August 1978) pp 953-956
Three different uses of the TSI have been suggested: •
online stability assessment,
•
contingency analysis,
•
online preventive control.
The first two uses have been thoroughly investigated in the present paper. In conclusion, the authenticity of the TSI allows for predicting its effective implementation in online stability assessment and control.
V l I. References 1 Fouad, A A 'Stability-theory criteria for transient stability' Proc. Eng. Foundation Conf. on Systems Engineering for Power," Status and Prospects Henniker, USA (August 1975) pp 421-450
Vol 4 No 2 April 1982
10 Chamorro, R S, Anderson, M D and Richards, E F 'Fast transient contingency evaluation in power systems' presented at IEEE PES Summer Meeting Minneapolis, USA (July 1980) Paper No 80 SM 589-2 11 Ribbens-Pavella, M, Calvaer, A and Gheury, J 'Transient stability index for online evaluation' presented at IEEE PES Winter Meeting New York, USA (3-8 February 1980) Paper No A 80 013-3 12 Ribbens-Pavella, M, Murthy, P G and Horward, J L 'Transient stability index: an online tool for stability assessment and contingency evaluation' Int. Rep. No LML/2 University of Liege, Belgium (December 1980) 13 Carpentier, J 'Prospects for security control in electric power systems' presented at 8th IFA C World Congress Japan (24-28 August 1981)
14 Carpentier, J L 'Differential injections method, a general method for secure and optimal load flow' Proc. PICA (1973) pp 255-262
99