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A computerized technique for reliability evaluation in power system transmission planning
Electric Power Systems Research, 5 (1982) 245 - 252
245
A C o m p u t e r i z e d T e c h n i q u e f o r R e l i a b i l i t y E v a l u a t i o n in P o w e r S y s t e m Transmission Planning T. A. M. S H A R A F and G. J. BERG
Department of Electrical Engineering, The University of Calgary, Calgary, T2N IN4 (Canada) (Received March 16, 1982)
SUMMARY
The paper discusses two computer programs to calculate the composite system reliability indices, the loss o f load probability and the expected demand not served, for use in long-range power system transmission expansion planning. The programs are based on the maximum-flow-minimum.cut technique and a decomposition principle which transforms the network capacity state space into nonoverlapping subsets to calculate the reliability indices. A modified version is also discussed in which the respective individual load reliability indices at each bus are determined. The programs have been applied to different test systems and typical results are presented in the paper.
1. INTRODUCTION
In long-range system transmission expansion planning, the object is to select the most desirable transmission network, given a generation expansion pattern and projected demand. Economics and reliability considerations are involved in the appraisal of alternative solutions to the transmission planning problem and acceptable schemes must satisfy technical requirements as well as other constraints. A basic problem in transmission planning is the determination of adequate capacity with forced outage of various system components. The utilization of a quantitative reliability criterion provides a consistent approach in this respect, i.e., investment in transmission facilities may be made at the most desired locations in the system while maintaining 0378"7796/82/0000-0000/$02.75
acceptable risk levels at the respective load points [ 1]. The components of service quality considered are (a) continuity of supply, (b) constant bus voltage levels, and (c) constant frequency. The last item, constant frequency, is dependent upon the ability of the system to handle overloads and forced outages, which are included in dealing with the first component, and also on the load-frequency control used. Frequency control is not included in the longrange planning study. The levels of voltage and voltage variations are checked after establishing viable transmission alternatives. Optimization techniques for reactive power compensation can be used to ensure that desired voltage levels are maintained. In this paper, the impact of steady state system disturbance on the supply is considered. The object is to develop a suitable technique for determining whether the state of a system is acceptable or not after a disturbance has reached steady state. An acceptable state implies that no system component is overloaded and that all demands are met. Sullivan [2] has discussed the application of the maximum-flow-minimum-cut technique [3], together with the decomposition principle [4], which allows the transformation of the network capacity state space into nonoverlapping subsets to calculate the composite system reliability indices. The reliability indices used are the system loss of load probability (LOLP) and the system expected demand not served (EDNS). The composite system reliability index is a measure of system performance to which all system parts contribute. Although the overall © Elsevier Sequoia/Printed in The Netherlands
246
system index may be satisfactory, the performance of particular system parts could be well below acceptable levels, or indeed performance could be above a justifiable level. Evaluation of reliability at specific buses therefore seems just as important as the determination of the overall system index. In this paper the reliability algorithm presented in ref. 2 has been extended to allow determination of the individual load reliability indices (LOaF(i) and EDNS(i)) at the respective buses in the system. Calculation of the reliability indices for individual buses as well as for the composite system is outlined. For purposes of determining the respective indices, two computer programs have been developed. The first program uses only the maximum-flow-minimum-cut technique, while in the second program the decomposition principle is also applied. The two programs have been applied to two systems: the IEEE five-bus test system and the southern portion of the Saskatchewan Power Corporation (SPC) system. In Section 2, the problem is formulated in terms of the probabilistic indices. Rules for classifying the state space into acceptable and unacceptable states are presented. The Section ends with the subset decompositions for system LOLP and EDNS calculations. The two computer programs are described in Sections 3 and 4. Application of the programs to the two systems considered and the results are discussed at the end of the paper.
Let Pr[X = x] be the probability associated with a point x E X. Then, m
Pr[X = x] = l l P r [ v i ]
(1)
j=l
where
Pr[vj]=pj if vj = 1.0 =qj if vi = 0 . 0 Also Pr[X = x] = Pr[X] = 1.0
(2)
x@X
2.2. Statement of the problem The problem is to decompose the entire state space X into states that are acceptable, A, and states that are unacceptable, B [2]. Unacceptable states are system capacity states for which the load, L, cannot be satisfied, either because of insufficient generation capacity or because of insufficient transmission capacity. The system LOLP is defined by LOaF~
~ Pr[X=x]
(3)
x~B
where B is the subset of unacceptable states. For each unacceptable state, the a m o u n t of load served is F < L, and the EDNS may be expressed, EDNS ~ ~ Pr[X = x] (L -- F)
(4)
x~B
2. B A S I C
THEORY
AND
DEFINITIONS
2.1. General Let $1 be a discrete random variable (j = 1, ..., m) where m is the number of elements in the system. The m random variables Si are assumed to be independent. A random variable S 1 assumes either the 0 state, with probability qj, in which it has no capacity and is out-of-service, or the 1 state, with probability pj, in which it has capacity c; and is in-service. A state point x will be denoted as x = (vl, v2..... v~..... vm), where vj is either 1 or 0. The entire state space is denoted by X. The system will have 2 m distinct capacity states. The state points x(1, 1 . . . . . 1) and x(0, 0, ..., 0) are called limiting state points for X. Upper and lower limiting capacity states are denoted by and _x respectively.
The subset B representing unacceptable states may be decomposed into overlapping individual subsets, Bind(i ) where (i) refers to the bus number. Individual load unacceptable subset, Bind(i), contains system capacity states for which the load L(i) cannot be satisfied. The individual load LOLP(i) is defined by LOaF{i) =
~
Pr[X = x]
(5)
x E Bind(i)
For each individual load unacceptable state x, the a m o u n t of individual load served is F(i) < L(i). Therefore the individual load EDNS(i) will be the sum of the products of individual demands not served and the probability that the associated state occurred, i.e., EDNS(i) =
~ x E Bind(i)
Pr[X = x] [L(i) -- F(i)] (6)
247
It appears computationally inefficient to solve the problem by considering each point of the state space one after the other. In the suggested method, using the decomposition principle, sequences of nonoverlapping subsets are constructed in which all state points are acceptable, and nonoverlapping subsets in which all state points are unacceptable.
(9)
Pr[B~] = qi I ] Pr[vj >/vi~] i< i
where Pr[vj 1> vj~] = 1.0 if vj~ = 0 Pr[vi/> vj~] =pj if v~ = 1.0
if
2.5. Subset decomposition for system EDNS calculation As for calculating system LOLP, use of (4) to calculate EDNS would require excessive computations. To reduce the requirements of this kind, unacceptable states may be decomposed into subsets with the same maximum flow F (i.e., the same minimum-cut elements). An upper critical state xU(Bi) exists [2] that defines the boundary between states in B~ with the same minimum cut as the upper limiting state of Bi. xU(Bi) is defined by establishing a maximum-flow pattern through the network corresponding to the upper limiting state of Bi. Therefore, for minimum-cut elements j E M,
Vn~ <~ Vn ~ Vnu
viU(Bi) = vi~(Bi) of the lower limiting state,
and
and f o r / ~ M,
vi>~Vju for j < n
viU(B~) = vj~(Bi) if fj = 0
Each U, is considered as the new entire state space and the process is repeated. For determination of the upper and lower critical states, see ref. 2.
vjU(S~) = 1.0 if fj > 0
2.3. Classification rules Initially all the states are unclassified. We define upper and lower critical states and denote them by Xu and x~, such that any system state x t> Xu is acceptable and any state x < xQ is unacceptable [2]. The states x~ ~< x < x~ will be defined as unclassified, meaning that Xu and x~ are insufficient to ascertain acceptability of these states. The set of unclassified states, U, may be decomposed into m subsets, U , , as follows: x ~ U,
(7)
2.4. Subset decomposition for system LOLP calculation The calculation of LOLP using (3) requires extensive calculations. Decomposing the unacceptable states, B, into nonoverlapping subsets greatly reduces the computational requirements. The decomposition algorithm [4] may be described briefly as follows:
(10)
In other words, any state x whose capacity equals or exceeds that defined by x~(Bi) will have the same minimum cut M: x E B.
(11)
if
vj >1 v~U(Bi)
j = 1, 2, ..., m
if
As a result of decomposing set B~, we shall have a portion of the subset that does not satisfy the condition of Bn. The remaining states must be further decomposed using the associated limiting capacity states, until all the states are classified. Starting with (4), it can be shown [2] that
Vi < v~
EDNS=L(1--Pr[A])--
x E Bi
(8)
and
~ ~ ~Pr[S.] B. JeM Pr[vj(B.)]
(12)
vi>lvi~ for a l l j < i In words, the decomposition rules group in a subset all those states that are unacceptable because of the outage of a given element vi. The second condition ensures that x is n o t unacceptable because of any element j < i. Then,
~=
~, cl(vi)Pr[vi] oi=o_i
--- the expected value of the capacity of the minimum-cut element j
248
Pr[vj(B,)] = ~
Pr[vj(B,)]
I
W=vi
= the probability that the minimumcut element resides in a state vi E B,
I I
I
[] Determinethe limitingI states ,I
Pr[B,] = I l P r [ v i t> v~U(Bi)] i
I
[] Determinethe criticall states J
3. COMPUTER PROGRAM F O R EVALUATION OF RELIABILITY INDICES
3.2. First version The block diagram is shown in Fig. 1. 1. Read input data. Network topology, element capacities, forced outage rates (FORs), and individual loads are fed to the computer.
I
state space
and
3.1. General As indicated in the last section, use of the decomposition principle appears to reduce the computational requirements. The decomposition of the unacceptable set, B, into nonoverlapping subsets to calculate the system LOLP and EDNS has been discussed in subsections 2.4 and 2.5, respectively. In order to calculate the individual load reliability indices, the unacceptable set has to be decomposed into overlapping subsets. This requires consideration of every point in the unacceptable set; hence in this case there is no need to perform the decompositions discussed in subsections 2.4 and 2.5. Two versions of a computer program have been developed to calculate the desired indices. In the first version, the maximumf l o w - m i n i m u m - c u t technique is used to assist in classifying the state space into acceptable, unacceptable, and unclassified sets as described in subsection 2.3. The unacceptable set is decomposed into overlapping subsets. Systems LOLP and EDNS, and individual loads LOLP(i) and EDNS(i), are calculated using (3), (4), (5) and (6), respectively. In the second version, the state space is classified as before. Then, the unacceptable set is decomposed into the t w o groups of nonoverlapping subsets as described in subsections 2.4 and 2.5. The systems LOLP and EDNS are calculated using (9) and (12). Individual load reliability indices are not calculated in this version.
I ioput I
I 1 cl sifYaccopt blo. I unacceptable, and unclassified states
I a b i l i t y indices
I
[ ] Deconrpose unclassified states I
[ [] Print the results l
Fig. 1. The block diagram of version I of the computer program.
2. Determine the initial state space. With the two-state model, the system can reside in any of 2 m different capacity states. Create these states. 3. Determine the limiting states. For the state space considered, determine the upper and lower limiting states. 4. Determine the critical states. Using the procedure described in ref. 2, determine the upper and lower critical capacity states. 5. Classify acceptable, unacceptable, and unclassified states. Classify the states in the state space being considered according to the classification rules in subsection 2.3. 6. Calculate the reliability indices. Use (1) to calculate the probability that the system resides in an acceptable capacity state, Pr[A]. Calculate also the probability that the system resides in an unacceptable state, P r [ B ] , using (3). The probability that the system resides in an unclassified state is then given by
249
~r[U]
=
(13)
1 -- Pr[A] --Pr[B]
Using (4), calculate the system EDNS. Calculate the individual loads LOLP(i) and EDNS(i) using (5) and (6). 7. If Pr[U] is less than a small value 7, Which is too small to be considered in the planning process, go to step 9; if not, go to step 8. 8. Decompose unclassified states. The probabilities Pr[A], Pr[B], EDNS, LOLP(i) and EDNS(i) are intermediate values because the unclassified set is not empty. Decompose the unclassified states into m subsets according to (5). Return to step 3 and repeat steps 3 to 7 for each subset until constraint 7 is fulfilled. 9. Print the results. The probabilities Pr[A], Pr[B], EDNS, LOLP(i) and EDNS(i) are final. System LOLP equals Pr[B].
Im
[
I
input data t
I ['2l Determine t h e i n i t i a l
I
state space
I I[] Determine stathe tes criticall ,
I ['5] Classify acceptable, unacceptable, and unclassified states
I
I
Decomposethe unacceptable states. Calculate Pr[A], Pr[B],
and Pr[U]
[] Decompose unclassified
states
1'[]
I
°LP
one.
9. Final LOLP. The probabilities Pr[A] and Pr[B] are final. The final LOLP equals Pr[B]. 10. Decompose unacceptable subset Bi. Decompose subset Bi into subsets with the same minimum-cut elements using (11) and the procedure described in subsection 2.5. 11. Calculate system EDNS(BD. Use (12) to calculate system EDNS(Bi), which is the contribution of subset B~ to system EDNS. 12. Augment i to i + 1 and add EDNS(Bi) to the system EDNS. 13. If i < m, return to step 10; otherwise terminate.
l
I I[] Determine stathe tes limitingI
Fg]
3.3. Second version The block diagram for the second version of the program is shown in Fig. 2. Steps 1 to 5 are the same as in version one. 6. Decompose the unacceptable states. Decompose the unacceptable states into m subsets, Bi, according to (8). Then, use (9) to calculate Pr[B]. Calculate Pr[A] from (1), then use (13) to calculate Pr[U]. Steps 7 and 8 are the same as in version
4. EXAMPLES METHOD
OF
APPLICATION
I
)
L3
subsetBi
L4 J
I
I [] L2
Fig. 2. The block diagram of version 2 of the computer program.
THE
4.1. General The method described has been applied to evaluate the reliability indices of two systems: the IEEE five-bus test system (see Fig. 3), and the southern portion of the SPC system (see Fig. 5) [1]. Peak loads have been assumed for purposes of illustrating the evaluation of LOLP and EDNS indices. A more detailed load model could be used, depending on the purposes of the study. In planning, peak load conditions are often appropriate. Before applying the algorithm (version one or version two) to any network, the network
Decompose unacceptable
]
OF
r
,)
N,
L5
Fig. 3. The IEEE five-bus test system.
250 G /-
[]
_-
OLd,/,
~~.~.
~d'-~.
1~. / I.. 2
Fig. 4. The modified five-bus system.
should first be modified. The capacity of the generators and the transmission elements should be integral values; in ot her words, a suitable base MVA is chosen so t hat any fraction can safely be neglected. Then, all generator nodes are c o n n e c t e d to a c o m m o n new node using elements t ha t have the same capacity and forced outage rates as t he generating units. The last step is to c o n n e c t all nodes having loads to a n o t h e r c o m m o n new node with elements that are one h u n d r e d per cent reliable and have capacities equal to the corresponding loads. Figures 4 and 6 show the t w o networks o f Figs. 3 and 5, respectively, after being modified.
4.2. System 1 Table 1 indicates the element capacities and forced outage rates [5] for the IEEE fivebus test system. T he last column in Table 1 shows how to convert the element capacities into integral values before applying the algorithm. The system reliability indices are calculated using the two versions. The stopping criterion, 7, is based on the fact t hat we can neglect the states which have probability less than a certain specified percentage (for planning studies a value of 5% is reasonable). TABLE 1 I n p u t d a t a for s y s t e m 1
Element 1
2 3 4 5 6 7 8 9 10 11 12 13
Buses
Capacity c (p.u.)*
FOR q
c' = 4 ×c
G G 1 1 2 2 2 3 4 L2 L3 L4 L5
1.50 0.50 1.00 0.50 0.25 0.50 0.75 O.25 0.25 0.25 0.50 0.50 0.75
0.02 0.03 0.22 0.32 0.29 0.29 0.24 0.19 0.32 0.00 0.00 0.00 0.00
6 2 4 2 1 2 3 1 1 1 2 2 3
2 3 3 4 5 4 5
*Base value = 100 MVA.
[]
Fig. 5. The southern portion of the SPC. I
L4
L°.
I
g ' ~ , " q [] Fig. 6. The modified southern portion of the SPC.
Table 3 shows the n u m b e r of states, the n u m b e r o f acceptable, unacceptable, and unclassified states, along with their associated probabilities. Also given are the system LOLP and EDNS. For this system the execut i on time required by version t w o is 30% of that required by version one. This is at the expense of increased c o m p l e x i t y o f the algorithm and increased m e m o r y requirements. The individual load reliability indices are calculated using version one with less than 1% increase in e x e c u t i o n time. Table 3 shows the individual load LOLP(i) and EDNS(i) for loads at buses 2, 3, 4 and 5. Notice the differences in the reliability indices for the individual loads from each ot her and from those for the total system. While the loss of load probability for the total system is 0.7279, it is 0.0071 for the load at bus 2. Notice also the great difference in LOLP between the load at bus 2 and t hat
251
at bus 4 where it is 0.4697. It should be noted that EDNS for different loads should not be compared in absolute values, b u t as a percentage of the corresponding load. Thus, multilevel reliability can be considered quantitatively in the planning process, since the individual load reliability indices can easily be calculated and compared with each other and with the system indices.
4.3. System 2 Table 2 gives the required input data. The stopping criterion, ~, is 5% as before. The system reliability indices are calculated using the t w o versions. The relative reduction in execution time and the increase in memory requirements are almost the same as for system 1. The total number of states, the number of acceptable, unacceptable, and unclassified states and their associated probabilities are shown in Table 3. The individual loads LOLP(i) and EDNS(i) for loads at buses 1, 2, 3, 4 and 5 are shown in Table 3. Base values (MVA) for the two systems are completely different; this should be noted when comparing their respective EDNS.
TABLE 3 Calculated reliability indices Quantity
System 1
System 2
Total No. o f states No. o f a c c e p t a b l e states A No. o f u n a c c e p t a b l e states B No. o f unclassified states U Pr[A] Pr[B] Pr[U ] System LOLP System EDNS* L I * * LOLP(1) L 1 EDNS(1) L 2 LOLP(2) L 2 EDNS(2) L 3 LOLP(3) L 3 EDNS(3) L 4 LOLP(4) L 4 EDNS(4) L 5 LOLP(5) L 5 EDNS(5)
512 8 504 0 0.2721 0.7279 0.0000 0.7279 2.4487 --0.0071 0.0071 0.3336 0.5841 0.4697 0.8100 0.4191 1.0488
2048 65 1940 43 0.0568 0.9129 0.0303 0.9129 14.2323 0.2037 1.2220 0.4433 2.3798 0.3582 1.7909 0.7302 5.8415 0.7495 2.9981
* N o t i c e t h e d i f f e r e n c e in base values for t h e t w o systems. * * B u s No. 1 in s y s t e m 1 does n o t have load.
5. C O N C L U S I O N S
TABLE 2 I n p u t data for s y s t e m 2 Element
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Buses
G1 G2 1 -2 1 -3 2 -3 1 -5 1 -4 4 -5 6 -7 1 -6 7 -2 L1 L2 L3 L4 L5
Capacity c
FOR
(p.u.)*
q
5.55 1.36 1.75 1.75 1.75 1.75 1.75 1.75 5.00 4.00 4.00 0.58 1.84 0.48 0.77 0.38
0.021 0.029 0.704 0.485 0.475 0.886 0.724 0.423 0.613 0.020 0.020 0.000 0.000 0.000 0.000 0.000
*Base value = 100 MVA.
c ° = 10.3 × c
57 14 18 18 18 18 18 18 51 41 41 6 19 5 8 4
For purposes of long-range transmission system expansion planning the composite system reliability indices, the loss of load probability and the expected demand not served, can be determined using the m a x i m u m - f l o w minimum-cut technique. A program has been developed for this purpose, and for calculating individual reliability indices, the loss of load probability, and the expected demand n o t served at each load bus. The approach and method discussed permits multilevel reliability to be considered quantitatively in the planning process. The use of the decomposition principle to transform the network capacity state space into nonoverlapping subsets to calculate the system reliability indices is very effective in reducing overall computational requirements.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the financial assistance granted by NSERC in support of this project.
252 REFERENCES 1 M. P. Bhavraju and R. Billinton, Transmission planning using a reliability criterion. Part II. Transmission planning, IEEE Trans., PAS-90 (1971) 70 - 78. 2 R. L. Sullivan, Power System Planning, McGrawHill, New York, 1977.
3 L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, NJ, 1962. 4 P. V. Moeseke, Mathematical Programs for Activity Analysis, North Holland, Amsterdam, and Elsevier, New York, 1974, Chap. 9. 5 IEEE reliability test system, IEEE Trans., PAS-98 (1979) 2047 - 2054.
245
A C o m p u t e r i z e d T e c h n i q u e f o r R e l i a b i l i t y E v a l u a t i o n in P o w e r S y s t e m Transmission Planning T. A. M. S H A R A F and G. J. BERG
Department of Electrical Engineering, The University of Calgary, Calgary, T2N IN4 (Canada) (Received March 16, 1982)
SUMMARY
The paper discusses two computer programs to calculate the composite system reliability indices, the loss o f load probability and the expected demand not served, for use in long-range power system transmission expansion planning. The programs are based on the maximum-flow-minimum.cut technique and a decomposition principle which transforms the network capacity state space into nonoverlapping subsets to calculate the reliability indices. A modified version is also discussed in which the respective individual load reliability indices at each bus are determined. The programs have been applied to different test systems and typical results are presented in the paper.
1. INTRODUCTION
In long-range system transmission expansion planning, the object is to select the most desirable transmission network, given a generation expansion pattern and projected demand. Economics and reliability considerations are involved in the appraisal of alternative solutions to the transmission planning problem and acceptable schemes must satisfy technical requirements as well as other constraints. A basic problem in transmission planning is the determination of adequate capacity with forced outage of various system components. The utilization of a quantitative reliability criterion provides a consistent approach in this respect, i.e., investment in transmission facilities may be made at the most desired locations in the system while maintaining 0378"7796/82/0000-0000/$02.75
acceptable risk levels at the respective load points [ 1]. The components of service quality considered are (a) continuity of supply, (b) constant bus voltage levels, and (c) constant frequency. The last item, constant frequency, is dependent upon the ability of the system to handle overloads and forced outages, which are included in dealing with the first component, and also on the load-frequency control used. Frequency control is not included in the longrange planning study. The levels of voltage and voltage variations are checked after establishing viable transmission alternatives. Optimization techniques for reactive power compensation can be used to ensure that desired voltage levels are maintained. In this paper, the impact of steady state system disturbance on the supply is considered. The object is to develop a suitable technique for determining whether the state of a system is acceptable or not after a disturbance has reached steady state. An acceptable state implies that no system component is overloaded and that all demands are met. Sullivan [2] has discussed the application of the maximum-flow-minimum-cut technique [3], together with the decomposition principle [4], which allows the transformation of the network capacity state space into nonoverlapping subsets to calculate the composite system reliability indices. The reliability indices used are the system loss of load probability (LOLP) and the system expected demand not served (EDNS). The composite system reliability index is a measure of system performance to which all system parts contribute. Although the overall © Elsevier Sequoia/Printed in The Netherlands
246
system index may be satisfactory, the performance of particular system parts could be well below acceptable levels, or indeed performance could be above a justifiable level. Evaluation of reliability at specific buses therefore seems just as important as the determination of the overall system index. In this paper the reliability algorithm presented in ref. 2 has been extended to allow determination of the individual load reliability indices (LOaF(i) and EDNS(i)) at the respective buses in the system. Calculation of the reliability indices for individual buses as well as for the composite system is outlined. For purposes of determining the respective indices, two computer programs have been developed. The first program uses only the maximum-flow-minimum-cut technique, while in the second program the decomposition principle is also applied. The two programs have been applied to two systems: the IEEE five-bus test system and the southern portion of the Saskatchewan Power Corporation (SPC) system. In Section 2, the problem is formulated in terms of the probabilistic indices. Rules for classifying the state space into acceptable and unacceptable states are presented. The Section ends with the subset decompositions for system LOLP and EDNS calculations. The two computer programs are described in Sections 3 and 4. Application of the programs to the two systems considered and the results are discussed at the end of the paper.
Let Pr[X = x] be the probability associated with a point x E X. Then, m
Pr[X = x] = l l P r [ v i ]
(1)
j=l
where
Pr[vj]=pj if vj = 1.0 =qj if vi = 0 . 0 Also Pr[X = x] = Pr[X] = 1.0
(2)
x@X
2.2. Statement of the problem The problem is to decompose the entire state space X into states that are acceptable, A, and states that are unacceptable, B [2]. Unacceptable states are system capacity states for which the load, L, cannot be satisfied, either because of insufficient generation capacity or because of insufficient transmission capacity. The system LOLP is defined by LOaF~
~ Pr[X=x]
(3)
x~B
where B is the subset of unacceptable states. For each unacceptable state, the a m o u n t of load served is F < L, and the EDNS may be expressed, EDNS ~ ~ Pr[X = x] (L -- F)
(4)
x~B
2. B A S I C
THEORY
AND
DEFINITIONS
2.1. General Let $1 be a discrete random variable (j = 1, ..., m) where m is the number of elements in the system. The m random variables Si are assumed to be independent. A random variable S 1 assumes either the 0 state, with probability qj, in which it has no capacity and is out-of-service, or the 1 state, with probability pj, in which it has capacity c; and is in-service. A state point x will be denoted as x = (vl, v2..... v~..... vm), where vj is either 1 or 0. The entire state space is denoted by X. The system will have 2 m distinct capacity states. The state points x(1, 1 . . . . . 1) and x(0, 0, ..., 0) are called limiting state points for X. Upper and lower limiting capacity states are denoted by and _x respectively.
The subset B representing unacceptable states may be decomposed into overlapping individual subsets, Bind(i ) where (i) refers to the bus number. Individual load unacceptable subset, Bind(i), contains system capacity states for which the load L(i) cannot be satisfied. The individual load LOLP(i) is defined by LOaF{i) =
~
Pr[X = x]
(5)
x E Bind(i)
For each individual load unacceptable state x, the a m o u n t of individual load served is F(i) < L(i). Therefore the individual load EDNS(i) will be the sum of the products of individual demands not served and the probability that the associated state occurred, i.e., EDNS(i) =
~ x E Bind(i)
Pr[X = x] [L(i) -- F(i)] (6)
247
It appears computationally inefficient to solve the problem by considering each point of the state space one after the other. In the suggested method, using the decomposition principle, sequences of nonoverlapping subsets are constructed in which all state points are acceptable, and nonoverlapping subsets in which all state points are unacceptable.
(9)
Pr[B~] = qi I ] Pr[vj >/vi~] i< i
where Pr[vj 1> vj~] = 1.0 if vj~ = 0 Pr[vi/> vj~] =pj if v~ = 1.0
if
2.5. Subset decomposition for system EDNS calculation As for calculating system LOLP, use of (4) to calculate EDNS would require excessive computations. To reduce the requirements of this kind, unacceptable states may be decomposed into subsets with the same maximum flow F (i.e., the same minimum-cut elements). An upper critical state xU(Bi) exists [2] that defines the boundary between states in B~ with the same minimum cut as the upper limiting state of Bi. xU(Bi) is defined by establishing a maximum-flow pattern through the network corresponding to the upper limiting state of Bi. Therefore, for minimum-cut elements j E M,
Vn~ <~ Vn ~ Vnu
viU(Bi) = vi~(Bi) of the lower limiting state,
and
and f o r / ~ M,
vi>~Vju for j < n
viU(B~) = vj~(Bi) if fj = 0
Each U, is considered as the new entire state space and the process is repeated. For determination of the upper and lower critical states, see ref. 2.
vjU(S~) = 1.0 if fj > 0
2.3. Classification rules Initially all the states are unclassified. We define upper and lower critical states and denote them by Xu and x~, such that any system state x t> Xu is acceptable and any state x < xQ is unacceptable [2]. The states x~ ~< x < x~ will be defined as unclassified, meaning that Xu and x~ are insufficient to ascertain acceptability of these states. The set of unclassified states, U, may be decomposed into m subsets, U , , as follows: x ~ U,
(7)
2.4. Subset decomposition for system LOLP calculation The calculation of LOLP using (3) requires extensive calculations. Decomposing the unacceptable states, B, into nonoverlapping subsets greatly reduces the computational requirements. The decomposition algorithm [4] may be described briefly as follows:
(10)
In other words, any state x whose capacity equals or exceeds that defined by x~(Bi) will have the same minimum cut M: x E B.
(11)
if
vj >1 v~U(Bi)
j = 1, 2, ..., m
if
As a result of decomposing set B~, we shall have a portion of the subset that does not satisfy the condition of Bn. The remaining states must be further decomposed using the associated limiting capacity states, until all the states are classified. Starting with (4), it can be shown [2] that
Vi < v~
EDNS=L(1--Pr[A])--
x E Bi
(8)
and
~ ~ ~Pr[S.] B. JeM Pr[vj(B.)]
(12)
vi>lvi~ for a l l j < i In words, the decomposition rules group in a subset all those states that are unacceptable because of the outage of a given element vi. The second condition ensures that x is n o t unacceptable because of any element j < i. Then,
~=
~, cl(vi)Pr[vi] oi=o_i
--- the expected value of the capacity of the minimum-cut element j
248
Pr[vj(B,)] = ~
Pr[vj(B,)]
I
W=vi
= the probability that the minimumcut element resides in a state vi E B,
I I
I
[] Determinethe limitingI states ,I
Pr[B,] = I l P r [ v i t> v~U(Bi)] i
I
[] Determinethe criticall states J
3. COMPUTER PROGRAM F O R EVALUATION OF RELIABILITY INDICES
3.2. First version The block diagram is shown in Fig. 1. 1. Read input data. Network topology, element capacities, forced outage rates (FORs), and individual loads are fed to the computer.
I
state space
and
3.1. General As indicated in the last section, use of the decomposition principle appears to reduce the computational requirements. The decomposition of the unacceptable set, B, into nonoverlapping subsets to calculate the system LOLP and EDNS has been discussed in subsections 2.4 and 2.5, respectively. In order to calculate the individual load reliability indices, the unacceptable set has to be decomposed into overlapping subsets. This requires consideration of every point in the unacceptable set; hence in this case there is no need to perform the decompositions discussed in subsections 2.4 and 2.5. Two versions of a computer program have been developed to calculate the desired indices. In the first version, the maximumf l o w - m i n i m u m - c u t technique is used to assist in classifying the state space into acceptable, unacceptable, and unclassified sets as described in subsection 2.3. The unacceptable set is decomposed into overlapping subsets. Systems LOLP and EDNS, and individual loads LOLP(i) and EDNS(i), are calculated using (3), (4), (5) and (6), respectively. In the second version, the state space is classified as before. Then, the unacceptable set is decomposed into the t w o groups of nonoverlapping subsets as described in subsections 2.4 and 2.5. The systems LOLP and EDNS are calculated using (9) and (12). Individual load reliability indices are not calculated in this version.
I ioput I
I 1 cl sifYaccopt blo. I unacceptable, and unclassified states
I a b i l i t y indices
I
[ ] Deconrpose unclassified states I
[ [] Print the results l
Fig. 1. The block diagram of version I of the computer program.
2. Determine the initial state space. With the two-state model, the system can reside in any of 2 m different capacity states. Create these states. 3. Determine the limiting states. For the state space considered, determine the upper and lower limiting states. 4. Determine the critical states. Using the procedure described in ref. 2, determine the upper and lower critical capacity states. 5. Classify acceptable, unacceptable, and unclassified states. Classify the states in the state space being considered according to the classification rules in subsection 2.3. 6. Calculate the reliability indices. Use (1) to calculate the probability that the system resides in an acceptable capacity state, Pr[A]. Calculate also the probability that the system resides in an unacceptable state, P r [ B ] , using (3). The probability that the system resides in an unclassified state is then given by
249
~r[U]
=
(13)
1 -- Pr[A] --Pr[B]
Using (4), calculate the system EDNS. Calculate the individual loads LOLP(i) and EDNS(i) using (5) and (6). 7. If Pr[U] is less than a small value 7, Which is too small to be considered in the planning process, go to step 9; if not, go to step 8. 8. Decompose unclassified states. The probabilities Pr[A], Pr[B], EDNS, LOLP(i) and EDNS(i) are intermediate values because the unclassified set is not empty. Decompose the unclassified states into m subsets according to (5). Return to step 3 and repeat steps 3 to 7 for each subset until constraint 7 is fulfilled. 9. Print the results. The probabilities Pr[A], Pr[B], EDNS, LOLP(i) and EDNS(i) are final. System LOLP equals Pr[B].
Im
[
I
input data t
I ['2l Determine t h e i n i t i a l
I
state space
I I[] Determine stathe tes criticall ,
I ['5] Classify acceptable, unacceptable, and unclassified states
I
I
Decomposethe unacceptable states. Calculate Pr[A], Pr[B],
and Pr[U]
[] Decompose unclassified
states
1'[]
I
°LP
one.
9. Final LOLP. The probabilities Pr[A] and Pr[B] are final. The final LOLP equals Pr[B]. 10. Decompose unacceptable subset Bi. Decompose subset Bi into subsets with the same minimum-cut elements using (11) and the procedure described in subsection 2.5. 11. Calculate system EDNS(BD. Use (12) to calculate system EDNS(Bi), which is the contribution of subset B~ to system EDNS. 12. Augment i to i + 1 and add EDNS(Bi) to the system EDNS. 13. If i < m, return to step 10; otherwise terminate.
l
I I[] Determine stathe tes limitingI
Fg]
3.3. Second version The block diagram for the second version of the program is shown in Fig. 2. Steps 1 to 5 are the same as in version one. 6. Decompose the unacceptable states. Decompose the unacceptable states into m subsets, Bi, according to (8). Then, use (9) to calculate Pr[B]. Calculate Pr[A] from (1), then use (13) to calculate Pr[U]. Steps 7 and 8 are the same as in version
4. EXAMPLES METHOD
OF
APPLICATION
I
)
L3
subsetBi
L4 J
I
I [] L2
Fig. 2. The block diagram of version 2 of the computer program.
THE
4.1. General The method described has been applied to evaluate the reliability indices of two systems: the IEEE five-bus test system (see Fig. 3), and the southern portion of the SPC system (see Fig. 5) [1]. Peak loads have been assumed for purposes of illustrating the evaluation of LOLP and EDNS indices. A more detailed load model could be used, depending on the purposes of the study. In planning, peak load conditions are often appropriate. Before applying the algorithm (version one or version two) to any network, the network
Decompose unacceptable
]
OF
r
,)
N,
L5
Fig. 3. The IEEE five-bus test system.
250 G /-
[]
_-
OLd,/,
~~.~.
~d'-~.
1~. / I.. 2
Fig. 4. The modified five-bus system.
should first be modified. The capacity of the generators and the transmission elements should be integral values; in ot her words, a suitable base MVA is chosen so t hat any fraction can safely be neglected. Then, all generator nodes are c o n n e c t e d to a c o m m o n new node using elements t ha t have the same capacity and forced outage rates as t he generating units. The last step is to c o n n e c t all nodes having loads to a n o t h e r c o m m o n new node with elements that are one h u n d r e d per cent reliable and have capacities equal to the corresponding loads. Figures 4 and 6 show the t w o networks o f Figs. 3 and 5, respectively, after being modified.
4.2. System 1 Table 1 indicates the element capacities and forced outage rates [5] for the IEEE fivebus test system. T he last column in Table 1 shows how to convert the element capacities into integral values before applying the algorithm. The system reliability indices are calculated using the two versions. The stopping criterion, 7, is based on the fact t hat we can neglect the states which have probability less than a certain specified percentage (for planning studies a value of 5% is reasonable). TABLE 1 I n p u t d a t a for s y s t e m 1
Element 1
2 3 4 5 6 7 8 9 10 11 12 13
Buses
Capacity c (p.u.)*
FOR q
c' = 4 ×c
G G 1 1 2 2 2 3 4 L2 L3 L4 L5
1.50 0.50 1.00 0.50 0.25 0.50 0.75 O.25 0.25 0.25 0.50 0.50 0.75
0.02 0.03 0.22 0.32 0.29 0.29 0.24 0.19 0.32 0.00 0.00 0.00 0.00
6 2 4 2 1 2 3 1 1 1 2 2 3
2 3 3 4 5 4 5
*Base value = 100 MVA.
[]
Fig. 5. The southern portion of the SPC. I
L4
L°.
I
g ' ~ , " q [] Fig. 6. The modified southern portion of the SPC.
Table 3 shows the n u m b e r of states, the n u m b e r o f acceptable, unacceptable, and unclassified states, along with their associated probabilities. Also given are the system LOLP and EDNS. For this system the execut i on time required by version t w o is 30% of that required by version one. This is at the expense of increased c o m p l e x i t y o f the algorithm and increased m e m o r y requirements. The individual load reliability indices are calculated using version one with less than 1% increase in e x e c u t i o n time. Table 3 shows the individual load LOLP(i) and EDNS(i) for loads at buses 2, 3, 4 and 5. Notice the differences in the reliability indices for the individual loads from each ot her and from those for the total system. While the loss of load probability for the total system is 0.7279, it is 0.0071 for the load at bus 2. Notice also the great difference in LOLP between the load at bus 2 and t hat
251
at bus 4 where it is 0.4697. It should be noted that EDNS for different loads should not be compared in absolute values, b u t as a percentage of the corresponding load. Thus, multilevel reliability can be considered quantitatively in the planning process, since the individual load reliability indices can easily be calculated and compared with each other and with the system indices.
4.3. System 2 Table 2 gives the required input data. The stopping criterion, ~, is 5% as before. The system reliability indices are calculated using the t w o versions. The relative reduction in execution time and the increase in memory requirements are almost the same as for system 1. The total number of states, the number of acceptable, unacceptable, and unclassified states and their associated probabilities are shown in Table 3. The individual loads LOLP(i) and EDNS(i) for loads at buses 1, 2, 3, 4 and 5 are shown in Table 3. Base values (MVA) for the two systems are completely different; this should be noted when comparing their respective EDNS.
TABLE 3 Calculated reliability indices Quantity
System 1
System 2
Total No. o f states No. o f a c c e p t a b l e states A No. o f u n a c c e p t a b l e states B No. o f unclassified states U Pr[A] Pr[B] Pr[U ] System LOLP System EDNS* L I * * LOLP(1) L 1 EDNS(1) L 2 LOLP(2) L 2 EDNS(2) L 3 LOLP(3) L 3 EDNS(3) L 4 LOLP(4) L 4 EDNS(4) L 5 LOLP(5) L 5 EDNS(5)
512 8 504 0 0.2721 0.7279 0.0000 0.7279 2.4487 --0.0071 0.0071 0.3336 0.5841 0.4697 0.8100 0.4191 1.0488
2048 65 1940 43 0.0568 0.9129 0.0303 0.9129 14.2323 0.2037 1.2220 0.4433 2.3798 0.3582 1.7909 0.7302 5.8415 0.7495 2.9981
* N o t i c e t h e d i f f e r e n c e in base values for t h e t w o systems. * * B u s No. 1 in s y s t e m 1 does n o t have load.
5. C O N C L U S I O N S
TABLE 2 I n p u t data for s y s t e m 2 Element
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Buses
G1 G2 1 -2 1 -3 2 -3 1 -5 1 -4 4 -5 6 -7 1 -6 7 -2 L1 L2 L3 L4 L5
Capacity c
FOR
(p.u.)*
q
5.55 1.36 1.75 1.75 1.75 1.75 1.75 1.75 5.00 4.00 4.00 0.58 1.84 0.48 0.77 0.38
0.021 0.029 0.704 0.485 0.475 0.886 0.724 0.423 0.613 0.020 0.020 0.000 0.000 0.000 0.000 0.000
*Base value = 100 MVA.
c ° = 10.3 × c
57 14 18 18 18 18 18 18 51 41 41 6 19 5 8 4
For purposes of long-range transmission system expansion planning the composite system reliability indices, the loss of load probability and the expected demand not served, can be determined using the m a x i m u m - f l o w minimum-cut technique. A program has been developed for this purpose, and for calculating individual reliability indices, the loss of load probability, and the expected demand n o t served at each load bus. The approach and method discussed permits multilevel reliability to be considered quantitatively in the planning process. The use of the decomposition principle to transform the network capacity state space into nonoverlapping subsets to calculate the system reliability indices is very effective in reducing overall computational requirements.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the financial assistance granted by NSERC in support of this project.
252 REFERENCES 1 M. P. Bhavraju and R. Billinton, Transmission planning using a reliability criterion. Part II. Transmission planning, IEEE Trans., PAS-90 (1971) 70 - 78. 2 R. L. Sullivan, Power System Planning, McGrawHill, New York, 1977.
3 L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, NJ, 1962. 4 P. V. Moeseke, Mathematical Programs for Activity Analysis, North Holland, Amsterdam, and Elsevier, New York, 1974, Chap. 9. 5 IEEE reliability test system, IEEE Trans., PAS-98 (1979) 2047 - 2054.