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On-line steady-state security assessment of power systems
Electric Power Systems Research 30 (1994) 123-134
ELSEVIER
ELBgTRIO POWER 8W$TEM8 FIEBEflROH
On-line steady-state security assessment of power systems A.S. Farag a, 1, E.Z. Abdel-Aziz a, T.C. Cheng b,*, S.M. E1-Sobki c, M.A. E1-Shibini c aKing Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia bDepartment of Electrical Engineering, University of Southern California, Los Angeles, CA 90089-2560, USA cCairo University, Cairo, Egypt
Accepted 22 January 1994
Abstract
Steady-state security assessment primarily addresses the question of the adequacy of generating and transmission capacity to meet load demand. In security assessment, there are two aspects of system failure that should be considered, namely, load curtailment and system collapse. The study of steady-state security has become essential, specially in modern large interconnected power systems. In this paper, a comprehensive conceptual framework and technique for on-line steady-state security assessment is suggested. As security of a power system is characterized in terms of power injections, naniely, load demand and generation, the technique mainly depends on the idea of the partial solution of the electric power system. It assumes a subsystem built from three concentric tiers of buses around the faulted component while the rest of the power system outside the third tier is represented by the active and reactive power injections whose values are those before the contingency occurrence. The basic idea of the conceptual framework and technique for the unified approach to steady-state security assessment assumes that the changes are concentrated only inside the subsystem and those changes outside the third tier are small and can be neglected. The accuracy of the technique is compared with the exact solution, the full AC and DC solutions, of the selected IEEE 30-bus power system model with the same contingency cases and is found satisfactory. Keywords: Security assessment; Contingency analysis
I. Introduction
In the past, informal methods of security analysis and control were performed by system operators, based upon their experience and knowledge, as the power systems were smaller and less complicated [1]. M o d e m power systems are quite large and more extensively interconnected, making the task of security analysis and control difficult for operators. During system difficulties or abnormal conditions, system operators are often so occupied that they m a y not be able to properly interpret worsening conditions or the development of a new problem. Steady-state security assessment of an electric power system is used to identify which contingency cases cause limit violations and the severity of such violations. Violations are c o m m o n l y presented when branch flow
*Author to whom correspondence should be addressed. ~Currently Visiting Professor at the University of Southern California. 0378-7796/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0378-7796(94) 00845 -U
limits, bus voltage limits, generator var production limits, or a combination of them, are exceeded [1-3]. On-line contingency analysis and steady-state security assessment are gaining attention as new methods and techniques are developed and applied. The on-line steady-state security assessment of power systems requires the evaluation of all possible contingency cases of the system. A direct approach to study the problem is to perform a full AC load flow solution for every contingency case, followed by checking for limit violations. Such an approach can be extremely time consuming for large systems, in opposition to the requirements of security [ 1-4]. Several techniques h a v e been suggested, but none has given a perfect solution to the problem. A set of methods, called ranking methods, rank the contingency cases in approximate order of security in a list called the contingency list. A C load flow is used to check the contingency cases, starting with the most severe case and proceeding down to the cases where no violations exist. At this stage, there is no need to continue checking since all remaining contingency cases are less severe.
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A.S. Farag et al./Electric Power Systems Research 30 (1994) 123 134
part where the contingency occurs. This relies on the fact that the part very close to the contingency is the most affected, while the rest of the system can be completely eliminated and substituted by constant active and reactive power injections through the buses representing the third tier of buses around the contingency location. The proposed technique assumes that all changes in the system conditions (bus voltages and power flows) are in the area around the faulted section, while these conditions for the rest of the system can be assumed to be the same as those before the contingency occurs. The accuracy of the technique is compared with the exact solution for the selected IEEE 30-bus power system model with the same contingency cases and is found to be satisfactory [4].
This is called the 'stopping-point criterion'. In practice, due to inaccuracies in the ranking process, the analysis is performed for some cases after the stopping point. Ranking methods should have two main properties: (i) the accuracy of the ranking process should be reasonable so that no severe cases are overlooked (a measuring tool called a capture ratio compares the ranking obtained by the ranking method used and the actual ranking obtained by a full AC analysis [5-11]); and (ii) the selection process and subsequent AC analysis time must be less than that required for a full AC analysis. Problems may arise from the inconsistent and unreliable ranking due to the following considerations: (a) magnitude of violation, (b) number of violations occurring, (c) relative importance of each violation, and (d) nearness to security limits [6-8]. Screening methods deal directly with the system to identify the contingency cases causing limit violations. The popular DC load flow, which is a fast straightforward technique is considered to be one of the most powerful methods for on-line steady-state security analysis. MW-flow (PQ) contingency analysis has been performed satisfactorily for a number of years using distribution factors. The voltage ( Q V ) contingency analysis problem is a more complicated task due to its highly nonlinear nature and presents a heavier computational burden than P Q contingency analysis [9-11]. In this paper, a technique related to the screening methods suitable for the security study of power systems under steady-state conditions is suggested. The proposed technique depends on the full solution of the AC load flow, but only for a portion of the network around the
2. The proposed technique The proposed technique utilizes the idea of treating the part of the system near the contingency and assumes that the conditions of the rest of the system, which is far away from the contingency, will not change. As an example, consider the power system in Fig. 1 with NB buses and NL lines. The circuit k, which ties buses il and j~, will be assumed to be dropped to simulate a single-contingency case. The concept of the concentric tiers can be applied as follows: (i) the two buses il andjl are considered as thefirst tier; (ii) buses i21, i22, i23, /24, J21, J22, J23, and J24 are considered as the second tier;
i31
L14
j31
L
i22 ~.~ 2
~
/
i33
L5
S0
/
i35
,_
/
j22
"23
i34
Fig. 1. A single-line diagram for a power system of Na buses and N L lines. Only the connections for the subsystem are shown.
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
i31 ~
i22
125
j31
"~L2
i33
L15 L4
i217-L1
Fig. 2. The connections of the third tier of the subsystem with the outside system.
(iii) buses i31, i32, /33, i34, i35, J31, J32, J33, a n d J34 a r e considered as the third tier. The system consisting of these three concentric tiers of buses will be called the 'subsystem'. Lines L] . . . . . L2~ are the lines of the subsystem. The rest of the power system will be called the 'Outside system', which has the property that the electrical conditions, bus voltage magnitudes and power line flows, will not be changed from those of the precontingency conditions. In Fig. 2, the buses of the third tier are connected to the outside system by certain lines denoted by a31, b31 ~ c31 ~ a32 , . . . , etc. The new injections for the third tier of buses are to be recalculated from the precontingency conditions and the standard AC load flow analysis is applied to the subsystem inside the third tier of buses. As an example, consider the new injections for bus i3~. The lines a3i , b31, and c3~ are the lines which connect bus i3~ with the outside system. If the precontingency active power flows in these lines were Pa31, Pb3~ and P,3~ and the injection at that bus was P ~ b in the directions shown in Fig. 3, the new active power injection P~3~ is given as Pi31 = --ei31b
-
-
Pa3t -- Pb31 -[- Pc3]
.
-
Pb31
Fig. 3. Precontingency active power conditions at bus i3~.
'k 131 Fig. 4. Precontingency reactive power conditions at bus i31.
Similarly, the same concept will be applied for the new reactive power injection Qi3~ at bus i3~, shown in Fig. 4:
Qi3, =
- Q i 3 ~ b -- Qa3, -
Qb3, + Q,~I
After calculating these new injections for the third-tier buses, the subsystem can be redrawn taking into consideration these new injections as shown in Fig. 5. Now it is possible to consider the subsystem as an integrated system which can be solved by the standard method of AC load flow analysis to get the line flows and/or the bus voltage magnitudes. The outside system is considered to be unchanged and the results for the power line flows and voltage magnitudes obtained under the precontingency conditions are assumed to be the postcontingency values. It is also possible to start the AC load flow analysis for the subsystem with the precontingency bus voltage magnitudes and angles as the initial conditions of the analysis. Using this starting point, the number of iterations is effectively reduced, since the results of this subsystem will not be very far from the precontingency conditions for a large number of buses inside the subsystem. A flowchart for the proposed method is shown in Fig. 6.
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A.S. Farag et al./ Electric Power Systems Research 30 (1994) 123-134
J31
131 t13
132
L
L II 2!
121
LI4
J22
"~122 L8
J3z
~{LI5 L3
L9
123 1 34
lz3 LI9
124 rio
LIT
/
124
LIB
135
J34
Fig. 5. Subsystem with the new third-tier bus injections.
3. AC applications of the tested power system
Full AC analysis for the precontlngency l to get bus voltages and line flows J
[ Jcbotlngency ~elect ]
[ I Construct the subsystem formation of Y bus
J
l for the third tier buses
I Solve the subsystem by AC 1o.d flow for bus voltages and line flows starting by the base case value as initial conditions
[ ohao
violatio.s ,
1.o
yes
J select t , J another J" contingencyJ
Fig. 6. Flowchart for the proposed method.
[~rite output]
The objective is to test the accuracy of the subsystem concept when the sample contingencies are selected. The subsystem for each contingency is built from the first three concentric tiers of buses of the whole system. Next, an AC load flow analysis is carried out to test this subsystem, and the results of the subsystem analysis are compared with those of the AC analysis for the whole system. The comparison includes the bus voltage magnitudes and power line flows. The selected power system is the IEEE 30-bus power system model with some modifications in the original bus loadings. These modifications were made in order to have a reasonable number of out-of-limit cases, line overloads and/or bus voltage problems for the studied line outages. This was done by increasing the bus loadings by 25% and neglecting two reactive power compensating units. The modified system is shown in Fig. 7. Line data, bus loadings, and reactive power limits for the voltage controlled buses are given in Tables 1, 2, and 3, respectively. The complete system was solved using the AC load flow technique and the Gauss iterative method to compare the results with those obtained from the proposed technique. In this application, two line outages were considered, line nos. 1 and 2. Each case was studied individually. The first step of the solution was the determination of the first-, the second-, and the third-tier boundaries. These are shown in Figs. 8 and 9 for the line outage nos. 1 and 2, respectively.
A.S. Farag et al./Electric Power Systems Research 30 (/994) /23-134
127
Jill
7'
13
2
18 17
7"T-:5 24
21
12
=,.T"
~
~
2
fO
28 41
31
28
24
~
40
26
~
38
27
29
3g
Fig. 7. IEEE 30-bus power system model. The second step of the solution was the determination o f the active and reactive power injections at the buses of the third tier, which can be calculated from the
base-case results. As an example, the calculations are shown in Fig. 10 for the third-tier bus no. 11 for the case of line outage no. 1. F r o m the values indicated
2-1 j I0~
1
First Tier Buses -Second Tier Buses
Fig. 8. Subsystem for line outage no. l.
--
Third Tier Buses Dropped Line
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
128
Table 1 Line data for the IEEE 30-bus power system (all values are in p.u. on a base of 100 MVA) Line
SB
EB
PL
Shunt admittance
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
1 1 2 7 2 2 8 3 9 4 9 9 5 11 8 6 13 13 13 14 16 15 18 19 12 12 12 12 21 15 22 23 24 25 25 28 27 27 29 4 9
2 7 8 8 3 9 9 10 10 9 11 12 1l 12 13 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 27 29 30 30 28 28
1.6000 1.3000 0.6500 1.3000 1.3000 0.6500 0.9000 0.7000 1.3000 0.6500 0.6500 0.3200 0.6500 0.6500 0.6500 0.6500 0.3200 0.3200 0.3200 0.1600 0.1600 0.1600 0.1600 0.3200 0.3200 0.3200 0.3200 0.3200 0.3200 0.1600 0.1600 0.1600 0.1600 0.1600 0.1600 0.6500 0.1600 0.1600 0.1600 0.3200 0.3200
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Series impedance 0.0264 0.0204 0.0184 0.0042 0.0209 0.0187 0.0045 0.0102 0.0085 0.0045 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0214 0.0065
0.0192 0.0452 0.0570 0.0132 0.0472 0.0581 0.0119 0.0460 0.0267 0.0120 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1231 0.0662 0.0945 0.2210 0.0824 0.1070 0.0639 0.0340 0.0936 0.0324 0.0348 0.0727 0.0116 0.1000 0.1150 0.1320 0.1885 02544 0.1093 0.0000 0.2198 0.3202 0.2399 0.0636 0.0169
0.0575 0.1852 0.1737 0.0379 0.1983 0.1763 0.0414 0.1160 0.0820 0.0420 0.2080 0.5560 0.2080 0.1100 0.2560 0.1400 0.2559 0.1304 0.1987 0.1997 0.1932 0.2185 0.1292 0.0680 0.2090 0.0845 0.0749 0.1499 0.0236 0.2020 0.1790 0.2700 0.3292 0.3800 0.2087 0.3960 0.4153 0.6027 0.4533 0.2000 0.0599
SB = sending bus; EB = end bus; PL = maximum power limit.
in the Figure, the new power injections for bus no. 11 are PI~ = 0.0 + 0.1522 - 0.3774 = - 0 . 2 2 5 2 Q~ ~ = 0.0 + 0.1690 - 0.2494 = - 0.0804 Similarly, the third-tier bus injections can be calculated from the base-case injections and line flows. The new injections for both line outages are shown in Table 4. After calculating the third-tier bus injections, a full A C load flow program was applied on the whole system to evaluate both bus voltage magnitudes and
power line flows. The AC load flow program was also applied on the whole system for every contingency case separately. The output results of these two programs, full solution and subsystem solution, with the base-case solution are shown in Tables 5 and 6 for comparison of the voltage magnitudes and line flows (apparent power), respectively, for line outage no. 1, while the results of line outage no. 2 are given in Tables 7 and 8. Graphic comparison can show the accuracy of the subsystem algorithm. Such comparisons are shown in Figs. 11 and 12 for line outage nos. 1 and 2, respectively.
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
129
Table 2 Bus injection of the IEEE 30-bus power system (all values are in p.u. on a base of 100 MVA) Bus
Generation P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Slack bus 0.6105 0.2689 0.2769 0.1518 0.1500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Load Q
P
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.2712 1.1775 0.3750 0.0000 0.0000 0.0300 0.0900 0.0000 0.2850 0.0000 0.0725 0.1400 0.0775 0.1025 0.0438 0.1125 0.0400 0.1188 0.0275 0.2188 0.0000 0.0400 0.1088 0.0000 0.0438 0.0000 0.0000 0.0300 0.1325
Net injection Q
P
Q
0.0150 0.0200 0.0000 0.1363 0.0000 0.0250 0.0938 0.0200 0.0313 0.0225 0.0725 0.0113 0.0425 0.0088 0.1400 0.0000 0.0200 0.0838 0.0000 0.0288 0.0000 0.0000 0.0113 0.0238
0.3393 -0.9086 -0.0981 0.1518 0.1500 -0.0300 -0.0950 0.0000 -0.2850 0.0000 -0.0775 -0.1400 -0.0775 -0.1025 -0.0438 -0.1125 -0.0400 -0.1188 -0.0275 -0.2188 0.0000 -0.0400 -0.1088 0.0000 -0.0428 0.0000 0.0000 -0.0300 -0.1325
-0.0150 -0.0200 0.0000 -0.1363 0.0000 -0.0250 -0.0938 -0.0200 -0.0313 -0.0225 -0.0725 -0.0113 -0.0425 -0.0088 -0.1400 0.0000 -0.0200 -0.0838 0.0000 -0.0288 0.0000 0.0000 -0.0113 -0.0238
Table 3 Reactive power limits of the voltage controlled buses of the IEEE 30-bus power system (all values are in p.u. on a base of 100 MVA) Bus
1 2 3 4 5 6
Voltage magnitude
1.00 1.00 1.00 1.00 1.00 1.00
Reactive power limit Minimum
Maximum
Slack bus -0.20 -0.15 - 0.15 -0.10 -0.15
Slack bus 0.80 0.75 0.60 0.50 0.60
First Tier Buses Second Tier Buses --
Third Tier Buses - - Dropped Line
Fig. 9. Subsystem of line outage no. 2.
4. Results and discussion
From the comparisons of the results in Tables 5 and 6 for the voltage magnitudes and line flows in the three
cases, namely, the base case, the full solution, and the reduced system (subsystem), for the same contingency, it can be seen that: (i) the maximum voltage deviation between the base-case solution and the full AC contingency case solution outside the third tier is at bus no. 30 with a percentage error of 2.49; (ii) the maximum deviation between the bus voltage magnitudes in the case of the full AC solution and the
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
130
Table 4 The new third-tier bus injections for line outage no. 1 and no. 2 Line outage no. 1 Pll = -0.2252 P12 = -0.1607 P13 = -0.4010 P28 = -0.1915 P4 = -0.1317
Q11 = Q12 = Q13 = Q28 = Q4a
Line outage no. 2 -0.0804 -0.0684 -0.0370 -0.0159
P9 = - 1.1251 P13 = -0.4010 P3 = - 0 . 7 8 0 7
Q9 = +0.3690 Q13 = -0.0370 Q3a
aQ is not defined (voltage controlled bus).
subsystem solution is at bus no. 13 with a percentage error of 3.0; (iii) the m a x i m u m deviation in the line flow between the base-case solution and the full AC contingency case solution outside the third tier is in line no. 21 with a percentage error of 31.5. (iv) the m a x i m u m deviation in the line flow between the full AC load flow solution and the subsystem solution is in the line no. 11 with a percentage error of 15.4.
Table 5 Voltage magnitude results for line outage no. 1 Bus
1" 2a 3a 4a 5 6 78 8a 9a 10a 11a 12~ 138 14 15
Voltage magnitude
Bus
Base-case solution
Full AC solution
Subsystem solution
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9812 0.9774 0.9815 0.9783 0.9630 0.9355 0.9645 0.9396 0.9316
1.0000 1.0000 1.0000 0.9830 1.0000 1.0000 0.9337 0.9426 0.9602 0.9650 0.9499 0.9197 0.9489 0.9261 0.9147
1.0000 1.0000 1.0000 0.9810
0.9288 0.9374 0.9569 0.9630 0.9377 0.9100 0.9204
Voltage magnitude Base-case solution
Full AC solution
16 17 18 19 20 21 22 23 24 25 26 27 28 ~ 29 30
0.9405 0.9295 0.9158 0.9108 0.9159 0.9179 0.9187 0.9139 0.9022 0.9122 0.8873 0.9307 0.9774 0.9026 0.8864
0.9261 0.9140 0.9007 0.8954 0.9003 0.9018 0.9025 0.8985 0.8853 0.8926 0.8671 0.9098 0.9565 0.8800 0.8643
Line
Apparent power flow (% loading)
Subsystem solution
0.9526
8Buses which appear in the subsystem. Table 6 Line apparent power flow (percentage loading) for line outage no. 1 Line
1 2a 3" 4a 5 6 7a 8a 9a 10a 11a 12a 13 14 15a 16 17 18 19 20 21
Apparent power flow (% loading) Base-case solution
Full AC solution
97.10 56.26 64.47 53.07 63.29 85.79 71.53 38.34 33.13 75.89 37.26 56.42 36.33 69.58 62.69 48.43 33.06 77.70 33.53 15.34 35.31
Out 197.49 77.76 182.66 45.57 32.05 165.60 80.59 53.86 79.90 31.59 50.48 44.16 68.22 69.28 61.01 34.52 83.72 39.36 17.75 46.44
aLines which appear in the subsystem.
Subsystem solution
199.89 82.92 183.64 45.85 34.84 173.55 82.16 54.06 94.39 37.48 57.34
63.05
Base-case solution 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
50.50 24.30 27.13 36.75 24.38 72.92 34.70 9.05 50.66 51.47 22.96 14.87 33.66 47.69 39.60 50.78 57.81 29.54 30.50 59.72
Full AC solution 57.14 30.57 23.90 33.40 19.63 72.69 34.56 9.17 57.17 50.94 28.40 11.68 33.68 24.15 37.93 50.85 57.91 29.56 31.59 56.73
Subsystem solution
59.76
A.S. Farag et al./ Electric Power Systems Research 30 (1994) 123-134
131
Table 7 Voltage magnitude results for line outage no. 2 Bus
Ia 2a 3a 4 5 6 7a 8a 9a 10 11 12 13a 14 15
Voltage magnitude
Bus
Base-case solution
Full AC solution
Subsystem solution
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9812 0.9774 0.9815 0.9783 0.9630 0.9355 0.9645 0.9396 0.9316
1.0000 1.0000 1.0000 0.9964 1.0000 1.0000 0.9694 0.9703 0.9767 0.9753 0.9604 0.9329 0.9585 0.9362 0.9287
1.0000 1.0000 1.0000
0.9592 0.9601 0.9657
0.9439
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Voltage magnitude Base-case solution
Full AC solution
0.9405 0.9295 0.9158 0.9108 0.9159 0.9179 0.9187 0.9139 0.9022 0.9122 0.8873 0.9307 0.9774 0.9026 0.8864
0.9378 0.9270 0.9130 0.9081 0.9132 0.9153 0.9161 0.9110 0.8992 0.9088 0.8838 0.9271 0.9728 0.8989 0.8826
Subsystem solution
aBuses which appear in the subsystem.
0.0
O }~.0.1522
O. 1 6 9 0
--y--- ,l
~"
~0. 3774
0.2494
Fig. 10. Base-case condition at bus no. 11.
The absolute percentage tolerance in the case of circuit loading is not completely accurate because the circuit might be already slightly loaded and any change in its loading level may result in a large percentage tolerance, as in circuit no. 21 in Table 6. However, the graphic comparisons in Figs. 11 and 12 show the accuracy of the subsystem results more conveniently in the case of circuit overloading. To investigate the accuracy
Table 8 Line apparent power flow (percentage loading) for line outage no. 2 Line
1a 2 3a 4a 5a 6a 7a 8 9 10 11 12 13 14 15a 16 17 18 19 20 21
Apparent power flow (% loading)
Line
Base-case solution
Full AC solution
Subsystem solution
97.10 56.26 64.47 53.07 63.29 85.79 71.53 38.34 33.13 75.89 37.26 56.42 36.33 69.58 62.69 48.43 33.06 77.70 33.53 15.34 35.31
148.36 Out 118.71 2.49 72.90 127.00 36.04 30.55 24.02 80.08 38.27 57.52 37.79 71.02 58.89 51.38 32.54 75.52 31.51 14.69 32.07
149.26
aLines which appear in the subsystem.
128.50 2.45 63.32 137.73 38.54
62.96
Apparent power flow (% loading) Base-case solution
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
50.50 24.30 27.13 36.75 24.38 72.92 34.70 9.05 50.66 51.47 22.96 14.87 33.66 47.69 39.60 50.78 57.81 29.54 30.50 59.72
Full AC solution 47.89 22.12 28.54 38.13 26.61 73.22 34.92 8.68 49.06 52.94 22.40 14.53 33.65 47.85 39.79 50.77 57.81 29.54 31.77 60.22
Subsystem solution
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123 134
132
Table 10 Comparison between the AC and DC full solutions and the subsystem solution for lines inside the third tier in the case of line outage no. 2
Table 9 Comparison between the AC and DC full solutions and the subsystem solution for lines inside the third tier in the case of line outage no. 1
Line Line
Full solution AC
DC
AC
DC
197.49 77.76 182.66 45.57 56.73 165.66 80.59 53.86 79.90 31.59 50.48 69.28 56.73
160.35 38.60 158.05 41.29 58.17 142.26 53.13 50.53 19.56 32.65 47.32 65.90 58.17
199.59 82.90 183.64 45.85 59.76 173.55 82.16 54.06 94.39 37.48 57.34 63.05 59.76
160.40 39.44 158.09 41.42 60.39 146.62 52.89 50.40 19.78 36.28 51.23 59.12 60.39
2 3 4 5 6 7 8 9 10 11 12 15 41
2 3 4 5 6 7 15
ll.S
Subsystem solution
AC
DC
AC
DC
148.38 118.71 2.49 72.90 127.00 36.04 58.89
130.29 115.61 2.31 66.92 123.46 29.37 55.72
149.26 128.50 2.45 63.32 137.73 38.54 62.96
126.29 118.18 2.31 58.42 128.05 35.71 49.50
is considered to be a fast straightforward technique, for the whole system, as shown in Tables 9 and 10 for line outage nos. 1 and 2, respectively. It is very clear from the above remarks that the voltage magnitude and line flows outside the third tier are slightly affected by the contingency inside the tier itself. It is also clear that the accuracy of the results for the line flows and voltage magnitudes inside the third tier is reasonable and within an acceptable range for system security assessment studies.
:~f the subsystem concept, its results were compared with lready existing standard techniques for the purpose of ecurity assessment. The results of the subsystem soluion can be compared with the full DC solution, which p
Full solution
Subsystem solution
o
|.0.
0
Exact
1,4
d
solution
Subsystem
solution.
Base
solution
case
I.I
l.l
I.I.
1.0.
o-ll
A
o-
o-
A A
o
O.
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A.S. Farag et al./ Electric Power Systems Research 30 (1994) 123-134 p
133
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The main problem in on-line contingency analysis is the time constraint. In AC load flow analysis, the time is mainly consumed in the iterative part. The iteration
[ 2
4f S ~
25
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~
. c' ' s ' so
Fig. 13. Subsystem for line outage no. 36.
27
time depends on the number of equations to be solved and the number of iterations for specified accuracy. For the same degree of accuracy, the proposed technique
~
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134
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
can reduce the consumed time considerably. The time will be reduced mainly because the number of buses in the subsystem is much less than that in the whole system, hence the number of iterations will be reduced, and consequently the time per iteration, which depends on the number of buses, will be effectively reduced. The proposed technique can yield quick results with acceptable accuracy for on-line applications to estimate the voltage changes and line overloads as a result of a circuit contingency. It should be noted that the subsystem built by considering three concentric tiers of buses as presented could not be applied in some contingency cases without splitting the produced subsystem. As an example of such a contingency, line outage no. 36, shown in Fig. 13, causes subsystem splitting. To overcome this problem, the number of tiers has to be increased or elements considered which would be suitable to build a connected subsystem.
5. Conclusions
The object of this work is to overcome the time constraint of on-line security assessment. A comprehensive conceptual framework is used to reduce the time of assessment by reducing the size of the power system to be analyzed. The subsystem concept can reduce the size of the power system effectively depending on the number of tiers chosen. Comparative Tables show that the results outside the third tier do not change much from those of the base case, while the changes in the first three concentric tiers are considerable. It is clear that three tiers are enough to obtain results with an acceptable degree of accuracy. It should be noted that, whatever the size of the power system, the size of the subsystem does not depend on the size of the whole system but only on the number of tiers. Therefore, even for large power systems, it is always possible to
construct a subsystem with a very limited number of buses. The analysis presented in this paper is limited to single-contingency cases. The same concept may be applied to multiple-contingency cases. Finally, the subsystem method presented meets both the speed and accuracy conditions for detecting system violations in the case of a contingency which are required for on-line steady-state security assessment.
References [1] M.G. Lauby, T.M. Mikolinnas and N.D. Reppen, Contingency selection of branch outages causing voltage problems, IEEE Trans. Power Appar. Syst., PAS-102 (1983) 3899-3904. [2] F. Albuyeh, A. Bose and B. Heath, Reactive power considerations in automatic contingency selection, IEEE Trans. Power Appar. Syst., PAS-101 (1982) 107-112. [3] J. Zaborszky, K.W. Whang and K. Prasad, Fast contingency evaluation using concentric relaxation, IEEE Trans. Power Appar. Syst., PAS-99 (1980) 28-36. [4] O. Alsac and B. Stott, Fast decoupled load flow, IEEE Trans. Power Appar. Syst., PAS-93 (1974) 859-869. [5] G.C. Ejebe and B.F. Wollenberg, Automatic contingency selection, IEEE Trans. Power Appar. Syst., PAS-98 (1979)97-109. [6] G. Irisarri and D. Levner, Automatic contingency selection for on-line analysis--real-time tests, IEEE Trans. Power Appar. Syst., PAS-98 (1979) 1552 1559. [7] T.A. Mikolinnas and B.F. Wollenberg, An advanced contingency selection algorithm, IEEE Trans. Power Appar. Syst., PAS-IO0 (1981) 608-617. [8] G.D. Irisarri and A.M. Sasson, An automatic contingency selection method for on-line security analysis, IEEE Trans. Power Appar. Syst., PAS-IO0 (1981) 1838-1844. [9] R.G. Wasley and M. Daneshdoost, Identification and ranking of critical contingencies in dependent variable space, IEEE Trans. Power Appar. Syst., PAS-102 (1983) 881-892. [10] K. Nara et al., On-line contingency selection algorithm for voltage security analysis, IEEE Trans. Power Appar. Syst., PAS104 (1985) 847 856. [11] F. Aboytes and G. Arroyo, Security assessment in the operation of longitudinal power systems, IEEE Trans. Power Syst., PWRS-I (1986) 225-232.
ELSEVIER
ELBgTRIO POWER 8W$TEM8 FIEBEflROH
On-line steady-state security assessment of power systems A.S. Farag a, 1, E.Z. Abdel-Aziz a, T.C. Cheng b,*, S.M. E1-Sobki c, M.A. E1-Shibini c aKing Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia bDepartment of Electrical Engineering, University of Southern California, Los Angeles, CA 90089-2560, USA cCairo University, Cairo, Egypt
Accepted 22 January 1994
Abstract
Steady-state security assessment primarily addresses the question of the adequacy of generating and transmission capacity to meet load demand. In security assessment, there are two aspects of system failure that should be considered, namely, load curtailment and system collapse. The study of steady-state security has become essential, specially in modern large interconnected power systems. In this paper, a comprehensive conceptual framework and technique for on-line steady-state security assessment is suggested. As security of a power system is characterized in terms of power injections, naniely, load demand and generation, the technique mainly depends on the idea of the partial solution of the electric power system. It assumes a subsystem built from three concentric tiers of buses around the faulted component while the rest of the power system outside the third tier is represented by the active and reactive power injections whose values are those before the contingency occurrence. The basic idea of the conceptual framework and technique for the unified approach to steady-state security assessment assumes that the changes are concentrated only inside the subsystem and those changes outside the third tier are small and can be neglected. The accuracy of the technique is compared with the exact solution, the full AC and DC solutions, of the selected IEEE 30-bus power system model with the same contingency cases and is found satisfactory. Keywords: Security assessment; Contingency analysis
I. Introduction
In the past, informal methods of security analysis and control were performed by system operators, based upon their experience and knowledge, as the power systems were smaller and less complicated [1]. M o d e m power systems are quite large and more extensively interconnected, making the task of security analysis and control difficult for operators. During system difficulties or abnormal conditions, system operators are often so occupied that they m a y not be able to properly interpret worsening conditions or the development of a new problem. Steady-state security assessment of an electric power system is used to identify which contingency cases cause limit violations and the severity of such violations. Violations are c o m m o n l y presented when branch flow
*Author to whom correspondence should be addressed. ~Currently Visiting Professor at the University of Southern California. 0378-7796/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0378-7796(94) 00845 -U
limits, bus voltage limits, generator var production limits, or a combination of them, are exceeded [1-3]. On-line contingency analysis and steady-state security assessment are gaining attention as new methods and techniques are developed and applied. The on-line steady-state security assessment of power systems requires the evaluation of all possible contingency cases of the system. A direct approach to study the problem is to perform a full AC load flow solution for every contingency case, followed by checking for limit violations. Such an approach can be extremely time consuming for large systems, in opposition to the requirements of security [ 1-4]. Several techniques h a v e been suggested, but none has given a perfect solution to the problem. A set of methods, called ranking methods, rank the contingency cases in approximate order of security in a list called the contingency list. A C load flow is used to check the contingency cases, starting with the most severe case and proceeding down to the cases where no violations exist. At this stage, there is no need to continue checking since all remaining contingency cases are less severe.
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A.S. Farag et al./Electric Power Systems Research 30 (1994) 123 134
part where the contingency occurs. This relies on the fact that the part very close to the contingency is the most affected, while the rest of the system can be completely eliminated and substituted by constant active and reactive power injections through the buses representing the third tier of buses around the contingency location. The proposed technique assumes that all changes in the system conditions (bus voltages and power flows) are in the area around the faulted section, while these conditions for the rest of the system can be assumed to be the same as those before the contingency occurs. The accuracy of the technique is compared with the exact solution for the selected IEEE 30-bus power system model with the same contingency cases and is found to be satisfactory [4].
This is called the 'stopping-point criterion'. In practice, due to inaccuracies in the ranking process, the analysis is performed for some cases after the stopping point. Ranking methods should have two main properties: (i) the accuracy of the ranking process should be reasonable so that no severe cases are overlooked (a measuring tool called a capture ratio compares the ranking obtained by the ranking method used and the actual ranking obtained by a full AC analysis [5-11]); and (ii) the selection process and subsequent AC analysis time must be less than that required for a full AC analysis. Problems may arise from the inconsistent and unreliable ranking due to the following considerations: (a) magnitude of violation, (b) number of violations occurring, (c) relative importance of each violation, and (d) nearness to security limits [6-8]. Screening methods deal directly with the system to identify the contingency cases causing limit violations. The popular DC load flow, which is a fast straightforward technique is considered to be one of the most powerful methods for on-line steady-state security analysis. MW-flow (PQ) contingency analysis has been performed satisfactorily for a number of years using distribution factors. The voltage ( Q V ) contingency analysis problem is a more complicated task due to its highly nonlinear nature and presents a heavier computational burden than P Q contingency analysis [9-11]. In this paper, a technique related to the screening methods suitable for the security study of power systems under steady-state conditions is suggested. The proposed technique depends on the full solution of the AC load flow, but only for a portion of the network around the
2. The proposed technique The proposed technique utilizes the idea of treating the part of the system near the contingency and assumes that the conditions of the rest of the system, which is far away from the contingency, will not change. As an example, consider the power system in Fig. 1 with NB buses and NL lines. The circuit k, which ties buses il and j~, will be assumed to be dropped to simulate a single-contingency case. The concept of the concentric tiers can be applied as follows: (i) the two buses il andjl are considered as thefirst tier; (ii) buses i21, i22, i23, /24, J21, J22, J23, and J24 are considered as the second tier;
i31
L14
j31
L
i22 ~.~ 2
~
/
i33
L5
S0
/
i35
,_
/
j22
"23
i34
Fig. 1. A single-line diagram for a power system of Na buses and N L lines. Only the connections for the subsystem are shown.
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
i31 ~
i22
125
j31
"~L2
i33
L15 L4
i217-L1
Fig. 2. The connections of the third tier of the subsystem with the outside system.
(iii) buses i31, i32, /33, i34, i35, J31, J32, J33, a n d J34 a r e considered as the third tier. The system consisting of these three concentric tiers of buses will be called the 'subsystem'. Lines L] . . . . . L2~ are the lines of the subsystem. The rest of the power system will be called the 'Outside system', which has the property that the electrical conditions, bus voltage magnitudes and power line flows, will not be changed from those of the precontingency conditions. In Fig. 2, the buses of the third tier are connected to the outside system by certain lines denoted by a31, b31 ~ c31 ~ a32 , . . . , etc. The new injections for the third tier of buses are to be recalculated from the precontingency conditions and the standard AC load flow analysis is applied to the subsystem inside the third tier of buses. As an example, consider the new injections for bus i3~. The lines a3i , b31, and c3~ are the lines which connect bus i3~ with the outside system. If the precontingency active power flows in these lines were Pa31, Pb3~ and P,3~ and the injection at that bus was P ~ b in the directions shown in Fig. 3, the new active power injection P~3~ is given as Pi31 = --ei31b
-
-
Pa3t -- Pb31 -[- Pc3]
.
-
Pb31
Fig. 3. Precontingency active power conditions at bus i3~.
'k 131 Fig. 4. Precontingency reactive power conditions at bus i31.
Similarly, the same concept will be applied for the new reactive power injection Qi3~ at bus i3~, shown in Fig. 4:
Qi3, =
- Q i 3 ~ b -- Qa3, -
Qb3, + Q,~I
After calculating these new injections for the third-tier buses, the subsystem can be redrawn taking into consideration these new injections as shown in Fig. 5. Now it is possible to consider the subsystem as an integrated system which can be solved by the standard method of AC load flow analysis to get the line flows and/or the bus voltage magnitudes. The outside system is considered to be unchanged and the results for the power line flows and voltage magnitudes obtained under the precontingency conditions are assumed to be the postcontingency values. It is also possible to start the AC load flow analysis for the subsystem with the precontingency bus voltage magnitudes and angles as the initial conditions of the analysis. Using this starting point, the number of iterations is effectively reduced, since the results of this subsystem will not be very far from the precontingency conditions for a large number of buses inside the subsystem. A flowchart for the proposed method is shown in Fig. 6.
126
A.S. Farag et al./ Electric Power Systems Research 30 (1994) 123-134
J31
131 t13
132
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121
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124 rio
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3. AC applications of the tested power system
Full AC analysis for the precontlngency l to get bus voltages and line flows J
[ Jcbotlngency ~elect ]
[ I Construct the subsystem formation of Y bus
J
l for the third tier buses
I Solve the subsystem by AC 1o.d flow for bus voltages and line flows starting by the base case value as initial conditions
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Fig. 6. Flowchart for the proposed method.
[~rite output]
The objective is to test the accuracy of the subsystem concept when the sample contingencies are selected. The subsystem for each contingency is built from the first three concentric tiers of buses of the whole system. Next, an AC load flow analysis is carried out to test this subsystem, and the results of the subsystem analysis are compared with those of the AC analysis for the whole system. The comparison includes the bus voltage magnitudes and power line flows. The selected power system is the IEEE 30-bus power system model with some modifications in the original bus loadings. These modifications were made in order to have a reasonable number of out-of-limit cases, line overloads and/or bus voltage problems for the studied line outages. This was done by increasing the bus loadings by 25% and neglecting two reactive power compensating units. The modified system is shown in Fig. 7. Line data, bus loadings, and reactive power limits for the voltage controlled buses are given in Tables 1, 2, and 3, respectively. The complete system was solved using the AC load flow technique and the Gauss iterative method to compare the results with those obtained from the proposed technique. In this application, two line outages were considered, line nos. 1 and 2. Each case was studied individually. The first step of the solution was the determination of the first-, the second-, and the third-tier boundaries. These are shown in Figs. 8 and 9 for the line outage nos. 1 and 2, respectively.
A.S. Farag et al./Electric Power Systems Research 30 (/994) /23-134
127
Jill
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24
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Fig. 7. IEEE 30-bus power system model. The second step of the solution was the determination o f the active and reactive power injections at the buses of the third tier, which can be calculated from the
base-case results. As an example, the calculations are shown in Fig. 10 for the third-tier bus no. 11 for the case of line outage no. 1. F r o m the values indicated
2-1 j I0~
1
First Tier Buses -Second Tier Buses
Fig. 8. Subsystem for line outage no. l.
--
Third Tier Buses Dropped Line
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
128
Table 1 Line data for the IEEE 30-bus power system (all values are in p.u. on a base of 100 MVA) Line
SB
EB
PL
Shunt admittance
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
1 1 2 7 2 2 8 3 9 4 9 9 5 11 8 6 13 13 13 14 16 15 18 19 12 12 12 12 21 15 22 23 24 25 25 28 27 27 29 4 9
2 7 8 8 3 9 9 10 10 9 11 12 1l 12 13 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 27 29 30 30 28 28
1.6000 1.3000 0.6500 1.3000 1.3000 0.6500 0.9000 0.7000 1.3000 0.6500 0.6500 0.3200 0.6500 0.6500 0.6500 0.6500 0.3200 0.3200 0.3200 0.1600 0.1600 0.1600 0.1600 0.3200 0.3200 0.3200 0.3200 0.3200 0.3200 0.1600 0.1600 0.1600 0.1600 0.1600 0.1600 0.6500 0.1600 0.1600 0.1600 0.3200 0.3200
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Series impedance 0.0264 0.0204 0.0184 0.0042 0.0209 0.0187 0.0045 0.0102 0.0085 0.0045 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0214 0.0065
0.0192 0.0452 0.0570 0.0132 0.0472 0.0581 0.0119 0.0460 0.0267 0.0120 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1231 0.0662 0.0945 0.2210 0.0824 0.1070 0.0639 0.0340 0.0936 0.0324 0.0348 0.0727 0.0116 0.1000 0.1150 0.1320 0.1885 02544 0.1093 0.0000 0.2198 0.3202 0.2399 0.0636 0.0169
0.0575 0.1852 0.1737 0.0379 0.1983 0.1763 0.0414 0.1160 0.0820 0.0420 0.2080 0.5560 0.2080 0.1100 0.2560 0.1400 0.2559 0.1304 0.1987 0.1997 0.1932 0.2185 0.1292 0.0680 0.2090 0.0845 0.0749 0.1499 0.0236 0.2020 0.1790 0.2700 0.3292 0.3800 0.2087 0.3960 0.4153 0.6027 0.4533 0.2000 0.0599
SB = sending bus; EB = end bus; PL = maximum power limit.
in the Figure, the new power injections for bus no. 11 are PI~ = 0.0 + 0.1522 - 0.3774 = - 0 . 2 2 5 2 Q~ ~ = 0.0 + 0.1690 - 0.2494 = - 0.0804 Similarly, the third-tier bus injections can be calculated from the base-case injections and line flows. The new injections for both line outages are shown in Table 4. After calculating the third-tier bus injections, a full A C load flow program was applied on the whole system to evaluate both bus voltage magnitudes and
power line flows. The AC load flow program was also applied on the whole system for every contingency case separately. The output results of these two programs, full solution and subsystem solution, with the base-case solution are shown in Tables 5 and 6 for comparison of the voltage magnitudes and line flows (apparent power), respectively, for line outage no. 1, while the results of line outage no. 2 are given in Tables 7 and 8. Graphic comparison can show the accuracy of the subsystem algorithm. Such comparisons are shown in Figs. 11 and 12 for line outage nos. 1 and 2, respectively.
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
129
Table 2 Bus injection of the IEEE 30-bus power system (all values are in p.u. on a base of 100 MVA) Bus
Generation P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Slack bus 0.6105 0.2689 0.2769 0.1518 0.1500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Load Q
P
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.2712 1.1775 0.3750 0.0000 0.0000 0.0300 0.0900 0.0000 0.2850 0.0000 0.0725 0.1400 0.0775 0.1025 0.0438 0.1125 0.0400 0.1188 0.0275 0.2188 0.0000 0.0400 0.1088 0.0000 0.0438 0.0000 0.0000 0.0300 0.1325
Net injection Q
P
Q
0.0150 0.0200 0.0000 0.1363 0.0000 0.0250 0.0938 0.0200 0.0313 0.0225 0.0725 0.0113 0.0425 0.0088 0.1400 0.0000 0.0200 0.0838 0.0000 0.0288 0.0000 0.0000 0.0113 0.0238
0.3393 -0.9086 -0.0981 0.1518 0.1500 -0.0300 -0.0950 0.0000 -0.2850 0.0000 -0.0775 -0.1400 -0.0775 -0.1025 -0.0438 -0.1125 -0.0400 -0.1188 -0.0275 -0.2188 0.0000 -0.0400 -0.1088 0.0000 -0.0428 0.0000 0.0000 -0.0300 -0.1325
-0.0150 -0.0200 0.0000 -0.1363 0.0000 -0.0250 -0.0938 -0.0200 -0.0313 -0.0225 -0.0725 -0.0113 -0.0425 -0.0088 -0.1400 0.0000 -0.0200 -0.0838 0.0000 -0.0288 0.0000 0.0000 -0.0113 -0.0238
Table 3 Reactive power limits of the voltage controlled buses of the IEEE 30-bus power system (all values are in p.u. on a base of 100 MVA) Bus
1 2 3 4 5 6
Voltage magnitude
1.00 1.00 1.00 1.00 1.00 1.00
Reactive power limit Minimum
Maximum
Slack bus -0.20 -0.15 - 0.15 -0.10 -0.15
Slack bus 0.80 0.75 0.60 0.50 0.60
First Tier Buses Second Tier Buses --
Third Tier Buses - - Dropped Line
Fig. 9. Subsystem of line outage no. 2.
4. Results and discussion
From the comparisons of the results in Tables 5 and 6 for the voltage magnitudes and line flows in the three
cases, namely, the base case, the full solution, and the reduced system (subsystem), for the same contingency, it can be seen that: (i) the maximum voltage deviation between the base-case solution and the full AC contingency case solution outside the third tier is at bus no. 30 with a percentage error of 2.49; (ii) the maximum deviation between the bus voltage magnitudes in the case of the full AC solution and the
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
130
Table 4 The new third-tier bus injections for line outage no. 1 and no. 2 Line outage no. 1 Pll = -0.2252 P12 = -0.1607 P13 = -0.4010 P28 = -0.1915 P4 = -0.1317
Q11 = Q12 = Q13 = Q28 = Q4a
Line outage no. 2 -0.0804 -0.0684 -0.0370 -0.0159
P9 = - 1.1251 P13 = -0.4010 P3 = - 0 . 7 8 0 7
Q9 = +0.3690 Q13 = -0.0370 Q3a
aQ is not defined (voltage controlled bus).
subsystem solution is at bus no. 13 with a percentage error of 3.0; (iii) the m a x i m u m deviation in the line flow between the base-case solution and the full AC contingency case solution outside the third tier is in line no. 21 with a percentage error of 31.5. (iv) the m a x i m u m deviation in the line flow between the full AC load flow solution and the subsystem solution is in the line no. 11 with a percentage error of 15.4.
Table 5 Voltage magnitude results for line outage no. 1 Bus
1" 2a 3a 4a 5 6 78 8a 9a 10a 11a 12~ 138 14 15
Voltage magnitude
Bus
Base-case solution
Full AC solution
Subsystem solution
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9812 0.9774 0.9815 0.9783 0.9630 0.9355 0.9645 0.9396 0.9316
1.0000 1.0000 1.0000 0.9830 1.0000 1.0000 0.9337 0.9426 0.9602 0.9650 0.9499 0.9197 0.9489 0.9261 0.9147
1.0000 1.0000 1.0000 0.9810
0.9288 0.9374 0.9569 0.9630 0.9377 0.9100 0.9204
Voltage magnitude Base-case solution
Full AC solution
16 17 18 19 20 21 22 23 24 25 26 27 28 ~ 29 30
0.9405 0.9295 0.9158 0.9108 0.9159 0.9179 0.9187 0.9139 0.9022 0.9122 0.8873 0.9307 0.9774 0.9026 0.8864
0.9261 0.9140 0.9007 0.8954 0.9003 0.9018 0.9025 0.8985 0.8853 0.8926 0.8671 0.9098 0.9565 0.8800 0.8643
Line
Apparent power flow (% loading)
Subsystem solution
0.9526
8Buses which appear in the subsystem. Table 6 Line apparent power flow (percentage loading) for line outage no. 1 Line
1 2a 3" 4a 5 6 7a 8a 9a 10a 11a 12a 13 14 15a 16 17 18 19 20 21
Apparent power flow (% loading) Base-case solution
Full AC solution
97.10 56.26 64.47 53.07 63.29 85.79 71.53 38.34 33.13 75.89 37.26 56.42 36.33 69.58 62.69 48.43 33.06 77.70 33.53 15.34 35.31
Out 197.49 77.76 182.66 45.57 32.05 165.60 80.59 53.86 79.90 31.59 50.48 44.16 68.22 69.28 61.01 34.52 83.72 39.36 17.75 46.44
aLines which appear in the subsystem.
Subsystem solution
199.89 82.92 183.64 45.85 34.84 173.55 82.16 54.06 94.39 37.48 57.34
63.05
Base-case solution 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
50.50 24.30 27.13 36.75 24.38 72.92 34.70 9.05 50.66 51.47 22.96 14.87 33.66 47.69 39.60 50.78 57.81 29.54 30.50 59.72
Full AC solution 57.14 30.57 23.90 33.40 19.63 72.69 34.56 9.17 57.17 50.94 28.40 11.68 33.68 24.15 37.93 50.85 57.91 29.56 31.59 56.73
Subsystem solution
59.76
A.S. Farag et al./ Electric Power Systems Research 30 (1994) 123-134
131
Table 7 Voltage magnitude results for line outage no. 2 Bus
Ia 2a 3a 4 5 6 7a 8a 9a 10 11 12 13a 14 15
Voltage magnitude
Bus
Base-case solution
Full AC solution
Subsystem solution
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9812 0.9774 0.9815 0.9783 0.9630 0.9355 0.9645 0.9396 0.9316
1.0000 1.0000 1.0000 0.9964 1.0000 1.0000 0.9694 0.9703 0.9767 0.9753 0.9604 0.9329 0.9585 0.9362 0.9287
1.0000 1.0000 1.0000
0.9592 0.9601 0.9657
0.9439
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Voltage magnitude Base-case solution
Full AC solution
0.9405 0.9295 0.9158 0.9108 0.9159 0.9179 0.9187 0.9139 0.9022 0.9122 0.8873 0.9307 0.9774 0.9026 0.8864
0.9378 0.9270 0.9130 0.9081 0.9132 0.9153 0.9161 0.9110 0.8992 0.9088 0.8838 0.9271 0.9728 0.8989 0.8826
Subsystem solution
aBuses which appear in the subsystem.
0.0
O }~.0.1522
O. 1 6 9 0
--y--- ,l
~"
~0. 3774
0.2494
Fig. 10. Base-case condition at bus no. 11.
The absolute percentage tolerance in the case of circuit loading is not completely accurate because the circuit might be already slightly loaded and any change in its loading level may result in a large percentage tolerance, as in circuit no. 21 in Table 6. However, the graphic comparisons in Figs. 11 and 12 show the accuracy of the subsystem results more conveniently in the case of circuit overloading. To investigate the accuracy
Table 8 Line apparent power flow (percentage loading) for line outage no. 2 Line
1a 2 3a 4a 5a 6a 7a 8 9 10 11 12 13 14 15a 16 17 18 19 20 21
Apparent power flow (% loading)
Line
Base-case solution
Full AC solution
Subsystem solution
97.10 56.26 64.47 53.07 63.29 85.79 71.53 38.34 33.13 75.89 37.26 56.42 36.33 69.58 62.69 48.43 33.06 77.70 33.53 15.34 35.31
148.36 Out 118.71 2.49 72.90 127.00 36.04 30.55 24.02 80.08 38.27 57.52 37.79 71.02 58.89 51.38 32.54 75.52 31.51 14.69 32.07
149.26
aLines which appear in the subsystem.
128.50 2.45 63.32 137.73 38.54
62.96
Apparent power flow (% loading) Base-case solution
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
50.50 24.30 27.13 36.75 24.38 72.92 34.70 9.05 50.66 51.47 22.96 14.87 33.66 47.69 39.60 50.78 57.81 29.54 30.50 59.72
Full AC solution 47.89 22.12 28.54 38.13 26.61 73.22 34.92 8.68 49.06 52.94 22.40 14.53 33.65 47.85 39.79 50.77 57.81 29.54 31.77 60.22
Subsystem solution
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123 134
132
Table 10 Comparison between the AC and DC full solutions and the subsystem solution for lines inside the third tier in the case of line outage no. 2
Table 9 Comparison between the AC and DC full solutions and the subsystem solution for lines inside the third tier in the case of line outage no. 1
Line Line
Full solution AC
DC
AC
DC
197.49 77.76 182.66 45.57 56.73 165.66 80.59 53.86 79.90 31.59 50.48 69.28 56.73
160.35 38.60 158.05 41.29 58.17 142.26 53.13 50.53 19.56 32.65 47.32 65.90 58.17
199.59 82.90 183.64 45.85 59.76 173.55 82.16 54.06 94.39 37.48 57.34 63.05 59.76
160.40 39.44 158.09 41.42 60.39 146.62 52.89 50.40 19.78 36.28 51.23 59.12 60.39
2 3 4 5 6 7 8 9 10 11 12 15 41
2 3 4 5 6 7 15
ll.S
Subsystem solution
AC
DC
AC
DC
148.38 118.71 2.49 72.90 127.00 36.04 58.89
130.29 115.61 2.31 66.92 123.46 29.37 55.72
149.26 128.50 2.45 63.32 137.73 38.54 62.96
126.29 118.18 2.31 58.42 128.05 35.71 49.50
is considered to be a fast straightforward technique, for the whole system, as shown in Tables 9 and 10 for line outage nos. 1 and 2, respectively. It is very clear from the above remarks that the voltage magnitude and line flows outside the third tier are slightly affected by the contingency inside the tier itself. It is also clear that the accuracy of the results for the line flows and voltage magnitudes inside the third tier is reasonable and within an acceptable range for system security assessment studies.
:~f the subsystem concept, its results were compared with lready existing standard techniques for the purpose of ecurity assessment. The results of the subsystem soluion can be compared with the full DC solution, which p
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A.S. Farag et al./ Electric Power Systems Research 30 (1994) 123-134 p
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Fig. 12. Graphic comparison between the subsystem solution and the full AC solutions for line active power flow in the case of line outage no. 2.
The main problem in on-line contingency analysis is the time constraint. In AC load flow analysis, the time is mainly consumed in the iterative part. The iteration
[ 2
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Fig. 13. Subsystem for line outage no. 36.
27
time depends on the number of equations to be solved and the number of iterations for specified accuracy. For the same degree of accuracy, the proposed technique
~
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134
A.S. Farag et al./Electric Power Systems Research 30 (1994) 123-134
can reduce the consumed time considerably. The time will be reduced mainly because the number of buses in the subsystem is much less than that in the whole system, hence the number of iterations will be reduced, and consequently the time per iteration, which depends on the number of buses, will be effectively reduced. The proposed technique can yield quick results with acceptable accuracy for on-line applications to estimate the voltage changes and line overloads as a result of a circuit contingency. It should be noted that the subsystem built by considering three concentric tiers of buses as presented could not be applied in some contingency cases without splitting the produced subsystem. As an example of such a contingency, line outage no. 36, shown in Fig. 13, causes subsystem splitting. To overcome this problem, the number of tiers has to be increased or elements considered which would be suitable to build a connected subsystem.
5. Conclusions
The object of this work is to overcome the time constraint of on-line security assessment. A comprehensive conceptual framework is used to reduce the time of assessment by reducing the size of the power system to be analyzed. The subsystem concept can reduce the size of the power system effectively depending on the number of tiers chosen. Comparative Tables show that the results outside the third tier do not change much from those of the base case, while the changes in the first three concentric tiers are considerable. It is clear that three tiers are enough to obtain results with an acceptable degree of accuracy. It should be noted that, whatever the size of the power system, the size of the subsystem does not depend on the size of the whole system but only on the number of tiers. Therefore, even for large power systems, it is always possible to
construct a subsystem with a very limited number of buses. The analysis presented in this paper is limited to single-contingency cases. The same concept may be applied to multiple-contingency cases. Finally, the subsystem method presented meets both the speed and accuracy conditions for detecting system violations in the case of a contingency which are required for on-line steady-state security assessment.
References [1] M.G. Lauby, T.M. Mikolinnas and N.D. Reppen, Contingency selection of branch outages causing voltage problems, IEEE Trans. Power Appar. Syst., PAS-102 (1983) 3899-3904. [2] F. Albuyeh, A. Bose and B. Heath, Reactive power considerations in automatic contingency selection, IEEE Trans. Power Appar. Syst., PAS-101 (1982) 107-112. [3] J. Zaborszky, K.W. Whang and K. Prasad, Fast contingency evaluation using concentric relaxation, IEEE Trans. Power Appar. Syst., PAS-99 (1980) 28-36. [4] O. Alsac and B. Stott, Fast decoupled load flow, IEEE Trans. Power Appar. Syst., PAS-93 (1974) 859-869. [5] G.C. Ejebe and B.F. Wollenberg, Automatic contingency selection, IEEE Trans. Power Appar. Syst., PAS-98 (1979)97-109. [6] G. Irisarri and D. Levner, Automatic contingency selection for on-line analysis--real-time tests, IEEE Trans. Power Appar. Syst., PAS-98 (1979) 1552 1559. [7] T.A. Mikolinnas and B.F. Wollenberg, An advanced contingency selection algorithm, IEEE Trans. Power Appar. Syst., PAS-IO0 (1981) 608-617. [8] G.D. Irisarri and A.M. Sasson, An automatic contingency selection method for on-line security analysis, IEEE Trans. Power Appar. Syst., PAS-IO0 (1981) 1838-1844. [9] R.G. Wasley and M. Daneshdoost, Identification and ranking of critical contingencies in dependent variable space, IEEE Trans. Power Appar. Syst., PAS-102 (1983) 881-892. [10] K. Nara et al., On-line contingency selection algorithm for voltage security analysis, IEEE Trans. Power Appar. Syst., PAS104 (1985) 847 856. [11] F. Aboytes and G. Arroyo, Security assessment in the operation of longitudinal power systems, IEEE Trans. Power Syst., PWRS-I (1986) 225-232.