- Email: [email protected]
Probability measure of adequacy assessment using a fuzzy approach
[tEOTRIO
POt,IAn
ELSEVIER
Electric Power Systems Research 33 (1995) 7 15
Probability measure of adequacy assessment using a fuzzy approach Jeeng-Min Ling, Chin-E. Lin*, Ching-Lien Huang Department ~[ Electrical Engineering, Institute o[' Aeronautics and Astronautics, National Cheng Kung University, Tainan. Taiwan Received 10 October 1994
Abstract
An approach based on fuzzy set theory is developed for defining the failure levels in assessing the adequacy of facilities during the stages of operational and long-term planning in a power system. After analyzing the stochastic uncertainties of load variation and generation nonavailability by the probability load flow algorithm, the 'probability measure of fuzzy events' is proposed to combine these uncertainties and the linguistic failure inexactness to evaluate the adequacy indices of both transfer capacity and spinning reserve. The impact of load uncertainty on the adequacy assessment is investigated in nine different cases, and the resulting adequacy indices of each case are summarized. The correlations of the demand characteristics in the relatively short term are also examined for an actual system. Results show that the proposed fuzzy adequacy indices provide more information and a better realization than the deterministic adequacy index. Furthermore, some potential risk to facilities can easily be detected by the fuzzy failure level, whereas the deterministic specified-value level will fail to do so. The proposed technique is implemented on the IEEE 25-bus system to demonstrate its feasibility.
Keywords: Fuzzy set theory: Adequacy assessment; Probability analysis; Power system planning
1. Introduction
Uncertainty has recently been recognized as an inevitable factor when dealing with the inherently random nature of an actual electric power system, so the characteristics of systems can therefore be considered as random variables. Well-investigated stochastic uncertainties include reading errors and random errors related to analog measurements in online computer applications [1], nonlinear relationships between load and weather changes, inaccuracies in weather forecasts for short-term load forecasting, etc. Many papers have successfully applied probability methodology to account for these stochastic uncertainties in power system applications [2]. This research has already attained a significant level of theoretical maturity, and can be considered to be currently in a stage of emerging development. In this paper, we analyze the impacts of uncertainties on adequacy assessment during the stages of operational and long-term planning. System adequacy relates to the existence of sufficient facilities to satisfy system demand [3,4]. For a composite (generation and transmission) power system, this means that the generating facilities are adequate to generate sufficient energy, and the *Corresponding author. 0378-7796/95'$09.50 ~t'~ 1995 Elsevier Science S.A. All rights reserved S S D I 0378-7796(95)00921-4
associated transmission facilities are adequate to transfer the energy to actual load points. In general, adequacy is associated with static conditions which do not include system disturbances. Techniques for adequacy assessment have been realized by probabilistic analyses considering various uncertainties [2,3,5,6]. The basic probabilistic technique uses the probabilistic load flow (PLF) approach [6]. Both stochastic load variation and generation unavailability are recognized and analyzed in the PLF algorithm. The results of the P L F assessment give a complete picture of the system and assess the likelihood of occurrence of system problems that cannot be obtained from a conventional deterministic analysis [7]. Compared with another similar probabilistic technique, composite reliability evaluation, P L F analyses reflect system problems before the remedial action evaluation algorithm is used [6]. Uncertainty involves unpredictable factors which have a major influence on a utility, and are not under control or cannot be predicted with certainty. In this sense, a 'risk' assessment should be conferred to indicate the hazards to which a utility is exposed [5,8]. Risk exposure involves facilities which significantly affect the adequacy or the reliability of a system. If we focus on the assessment of the facilities' adequacy under uncertainty, defining the 'failure level' is an essential step.
8
J.M. Ling et al.
EA,ctric Power Systems Research 33 (1995) 7 15
The risk exposure of two competitive alternatives for the Brazilian system has been evaluated by defining failure levels by specific numbers [6]. These numbers were assigned based on the capacity limits of facilities or the steady-state stability limits (operating constraints). However, in adequacy assessment, an event with most of its elements close to but not exceeding the failure level will be recognized as no risk, which is an unreasonable judgement. A deterministic number cannot properly represent the entire characteristics of a failure level, nor consistently define the true risk to the system [9]. Intuitively, some past experiences and expected future events can be integrated to define a 'soft' failure level. These assignments are particularly promising and useful in the fields of adequacy assessment and system planning when human perception and the process of decision making are inevitable. From the viewpoint of experience interpretation, a 'soft' bound for the failure level is more flexible. In this paper, an approach based on fuzzy set theory is proposed for the interpretation of the characteristics of failure [10,I1]. After analyzing the results of the PLF technique, the 'probability measure of fuzzy events' is used to assess the adequacy indices of facilities. To test the adequacy assessment in an actual system during the planning stage, some significant considerations about load, namely, load correlation, load growth, and load uncertainty, are investigated using the IEEE 25-bus test system. Results show that a more reasonable adequacy assessment can be achieved by the proposed probability measure.
2. Analytical approach to uncertainty Probability and fuzzy analytical approaches used to deal with stochastic and linguistic uncertainties are introduced in this section. Theoretically, probability theory deals with inherent stochastic uncertainty due to the occurrence of random events, while fuzzy set theory deals with uncertainty in the use of human experience in linguistic terms. The details of analytical concepts are described, and a feasible framework is proposed to integrate the two approaches. 2.1. Stochastic uncertainty In the field of adequacy assessment, stochastic uncertainties have been measured by probabilistic analyses [2,3,5]. These uncertainties include two main types: load forecast uncertainty and generation unavailability. Load forecast uncertainty is an extremely important parameter, combining the financial, social, environmental, and other uncertainties which electric power utilities must face [4]. Past research suggests that such uncertainty can be reasonably approximated by a normal
distribution. In composite reliability evaluation [4], load forecast uncertainty has been included in the risk computations by dividing the normal probability distribution into class intervals which are dependent on the desired accuracy. Similar discretization techniques can be used to model load forecast uncertainty in probabilistic analysis. Another relevant sampling method uses a tabulating technique [12]. The basic generating unit parameter used in static capacity evaluation is the probability of finding the unit on forced outage at some distant time in the future [4]. This probability is known in systems engineering as the unit unavailability, and historically in power system applications as the unit forced outage rate (FOR). In contrast to the multistate model used in generating capacity reliability evaluation, the probabilistic algorithm has been associated with a simple two-state model and defined accurately by a binomial distribution or a recursive technique [4,13,14]. The impact of generating unit unavailability and load uncertainty can be measured using the PLF algorithm to describe the steady-state performance of a given system. The results of the PLF assessment also reveal the range of all possible conditions that might be encountered as a result of expected uncertainties in inputs. Based on such implementation, the future system performance in an uncertain environment can therefore be predicted in a more consistent way. 2.2. Linguistic inexactness Another type of uncertainty, linguistic inexactness, associated with the specification of the failure level should be considered in the adequacy assessment. Occurrences of deficiency (nonadequacy) rarely occur in an actual system. This fact leads to the representation of the characteristics of a failure level not being properly described by either the probability technique, because it violates the law of large numbers, or by a specified deterministic number. Qualitative observation is one of the essential attributes of adequacy assessment, decision making and other fields involving human judgement. Human experts tend to use linguistic terms to describe their experience and heuristic rules. In this sense, high-level reasoning requires vagueness in order to manage problems without getting bogged down in details. Fuzzy set theory is proposed to capture the psychological reality of experience because of its linguistic elasticity [11]. The concept of a fuzzy set can be defined as follows. Definition 1. Fuzzy set Let U be the universe of discourse and x be the universal elements. Then, a fuzzy set A is an ordered pair:
J.M. Ling et al./Electric Power Systems Research 33 (1995) 7 15
9
Deterministic Failure 0.06
Level -1.20
-
0.05
/
~
/
0.04
/ ~0,03
s
/
• 1.00
/
4
.a~'v/-
"
[ /
.~ ~ 0.02
0.01
kJ III
°'~s.o
'I
so.o
"~'"
ss.o
I * I I | I * I | J * | L4'|
r~.o
¢~.o
~o.o
* I I*
?s.o
I I
"d. Jl
oo.o
.0.40
'0.20
~"
~.o
90.o
' l I I I l J
I I I I I I I I
g ~ . o ~oo.o Io5~o . o . o
P o w e r J~o~ i~ l~,~e 5 - I ? a n d ff~ membership .f~net~on of overload Fig. 1. The probabilistic real power flow and the failure level.
w h e r e / ~ is the membership function, and the grade of membership typically falls within [0, 1]. A fuzzy number is one of the most common forms of fuzzy set application [16]. A convex and normalized fuzzy set defined by a set of real numbers is called a fuzzy number. One typical form of representation, termed the trapezoidal fuzzy number, which is characterized by a quadruplet, is adopted in this study.
2.3. A framework Jor combining stochastic uncertainty and linguistic inexactness Probability theory and fuzzy set theory are both used to study imprecision. It is interesting to consider a general framework which can integrate both the stochastic and fuzzy uncertainties to solve the following problem: if the probability distribution of the power flow is derived from the PLF analysis, what is the probability that a line is 'overloaded'? Here, 'overloaded' is a vague linguistic term. One way to deal with this problem is to fuzzify the concept of a random variable (RV) [10]. From classical probability theory, if the sample space contains all 'outcomes of a random experiment', then a function from the sample space into the set of real numbers is called an RV. An 'event' is a precisely specified collection of points in the sample space. However, in everyday experience one frequently encounters situations in which an 'event' is an ill-defined rather than a sharply defined collection of points. For example, "it is a warm day" and " x is approximately equal to 5" are fuzzy events. A fuzzy event is a fuzzy subset of the sample
space. Therefore, we can define a fuzzy event in Euclidean n-space, R', to fuzzify the concept of an RV by the following definition.
Definition 2. Probability measure of a Juzzy event Let (R', ~, P) be a probability space in which ~ is the a-field of Borel sets in R" and P is a probability measure over R". Then a fuzzy event in R" is a fuzzy set A in R" whose membership function /~a (/~A: R " ~ [0, 1]) is Borel measurable. The probability measure of a fuzzy event A is defined as P(A)
=
~ fla(X) Q/ Rn
dP = E ( f l A )
(2)
The probability measure of a fuzzy event is interpreted as the expectation of its membership function. In this study, the expectation is proposed to identify the adequacy indices in the adequacy assessment studies. To clarify the concept of Eq. (2), a simple but significant diagram is exemplified in Fig. 1 to answer the question posed at the beginning of this section. Fig. 1 depicts the probabilistic distribution of real power flow in line 5-17 (P5 17) and the membership function of 'overload' for transfer capacity. The probability of P5 ~7 being overloaded can be evaluated from Eq. (2) and is equal to 36.2%.
3. The fuzzy approach to adequacy assessment The adequacy indices of two significant attributes arc described in this section, including the probability
l0
J.M. Ling et al.
Electric Power Systems Research 33 (1995) 7 15
of any particular power flow being greater than the corresponding equipment's thermal rating, and the probability of active generation exceeding acceptable levels. The process of adequacy assessment is concerned with two basic steps. Step I. Summarize the heuristic rules identified through the intuition of experienced operators, and construct the membership functions of the failure levels for significant attributes using the fuzzy set approach. Step 2. Use the proposed framework described by Eq. (2) to evaluate the desired adequacy indices based on the results of the PLF analysis and the heuristic membership functions of the related levels. Details of the specifications of two particular levels, namely, the 'state' level used for assessing the state of the facilities, and the 'failure' level used for assessing the risk index of the facilities, for the attributes of transfer capacity and generation reserve will be discussed in the following.
1.~0
1.25
Heavy
io~
~dium
/
1.00
I I
O.~tO
Medium
I I
l
O.Z5
,,,,,,,, t,,, /,,,/
0.00
.....
. 3 0 t . 5 0 t . . ?Q~ 80%
percentage o ~ ' ~ i e r
Three categories, light, medium, and heavy, are used to assess the 'state' levels of transfer capacity. The heuristic quadruplets of various fuzzy members in terms of the 'state' and 'failure' levels are depicted in Figs. 2 and 3. All quadruplets are expressed as a percentage of
A simple measure for the adequacy aspect of system reliability is the transmission system's limitation. Theoretically, the maximum power P ..... otherwise known as the steady-state stability limit, that can be transferred from generation to load is related to the bus voltages and line impedance. Under actual system operating conditions, it is natural that the failure level of transfer capacity cannot be considered to be a crisply defined value, like P . . . . since operators tend to use linguistic representations to assess the 'failure' level of line flow, such as
3.2. Assessment of spinning reserve Another significant measure of the adequacy aspect of system reliability is the spinning reserve. Variations in the consumption of electricity must be met by an equivalent modification of the generation output. This implies that some of the capacity in the system must be saved for 'reserve' purposes. In the PLF analysis, the reserve can be measured by evaluating the probability density curve of the overall balance of power in the system. In the meantime, the probability of deficiency in the system can also be evaluated. However, what percentage of spinning reserve should be kept in the operation? The answer to this question involves the operational condition of the system and the strategy of the utilities. Since forced outage is possible and unreliable load forecasts exist, there should be 'enough' reserve to meet all likely load
Overload
Underload
~( Risk )
"The line flow over a certain line is 'heavy'" How 'heavy' is heavy? How do we handle the characteristics of such a linguistic term in the adequacy assessment? Since the 'imprecision' described above is difficult to manage by conventional probability theory, a fuzzy set concept is proposed.
¢¢m..¢
Fig. 2. The state levels of transfer capacity.
Pmax •
The term 'overloaded' is imprecise. How many real power flows are identified by the 'overloaded' state? The operator may confirm that those lines that are loaded 110% (in terms of Pmax) or more will cause immediate security problems, and it is also certain that a lighter loading of 70% or less will cause a robust state. It is less certain whether a line loading of 70% 110% will cause risk. Other operators may place different assignments on the 'overloaded' and 'underloaded' levels because of different considerations. For the 'state' level of adequacy assessment, an operator may judge that
lltt,,,,,,,,,,,
i .....
- 7 0 t -50% ~30%
-80:
3.1. Assessment of transfer capacity
"The line flow over a certain line causes it to be 'overloaded'"
/
Heavy
Overload
(Robus ness)
( Risk )
A ,
.....
,V o,
:
/ ....
p c r c e . J ~ e of t r o ~ J e r Z*,'~J
Fig. 3. The failure levels of transfer capacity.
. . . . . . . . .
'+I
J.M. Ling et al./ Electric Power Systems Research 33 (1995) 7+ 15
12.5
Deficient
Surplus
Abundant
t.00
o+I 0.25
. . . . . . . .
,,
ltl
15%///// " ~ 2 0 % ,,,+ ,/1,t,11,1,% ,1,
Pe~'ee~taxd~ of S;4~.~ 4
iii
~1 li
Reser'ue
Fig. 4. The state levels of spinning reserve.
variations. How much is 'enough'? Is 10% enough? From the practical point of view, a specified value cannot properly define the generation reserve in the field of risk assessment. It is more flexible and instructive to consider a gradual transition rather than a crisply defined bound, like 10% or 15%. After taking significant aspects of the operational system and operator experience into account, the heuristic fuzzy concept for the 'state' and 'failure' levels of spinning reserve is related. Figs. 4 and 5 depict the shape of the proposed trapezoidal fuzzy numbers. All quadruplets are expressed as a percentage of peak demand.
4. Applications of the proposed technique 4.1. Test system and probabilistic technique In our test case the proposed 'probability measure of a fuzzy event' was applied to the IEEE 25-bus system shown in Fig. 6. The deterministic base-case data and line data are depicted in Fig. 6. The probabilistic nodal test data were modified from the deterministic base-case data and the same expected values maintained. It is
1.25
Deficient
Enough
O,,?.5
5+ 0~040
, I a , , I h , ,
lo+ i , , , , , , i i i i i i i i i i i i I
Pcrce~Jo@e oi S p ~ m { ~ R e ~ e
Fig. 5. The failure levels of spinning reserve.
II
noted that the data listed in Table I relating to the characteristics of consumer demands are not completely independent, i.e. there exists some linear dependence between various groups of nodal powers. The adequacy assessment studies began with the evaluation of the PLF analysis. Probability input data were mathematically convoluted to derive the output probability density curve for each line by using the Laplace transform technique [14]. During the process of convolution, the probabilities of discontinuous nodal powers, i.e. binomial and discrete, were truncated to a cumulative probability of 10 +'. A considerable number of states were deleted using this truncation technique. Another weighted-average technique [14] was used to reduce storage and computation time by grouping convoluted discontinuous points with the predetermined points. For a continuous normal nodal power, nine discretization points within three standard deviations from the mean were sampled. Based on the results of PLF analyses, the adequacy indices were evaluated using the proposed measure implemented by Eq. (2). These indices are represented by the percentage probability in this study.
4.2. Demonstration and discussion (~1 short-term studies Various reasons for correlations exist between nodal powers, especially in relatively short-term planning exercises [13,15]. The partial correlation probabilistic data used in the present study are listed in Table 1, and contain the following assumptions. Positive linear dependence between nodal powers exists at buses 3, 11, 17, and 22 (group 1), 8, 12, and 16 (group 2), 9, 14, and 18 (group 3), and 15 and 25 (group 4). All other nodal powers are assumed to be independent of each other and of the above dependent groups. The state and risk indices for assessing the adequacy of transfer capacity in each line of the test system were evaluated by the proposed fuzzy approach and by the deterministic approach. The maximum transfer capacity in the test system was assumed to be 100 MW. The five lines with most severe conditions are listed in Table 2. Results show that the risk index of some critical lines can be evaluated successfully with the proposed approach, but may fail (zero probability) to be evaluated in the deterministic analysis. The proposed fuzzy approach can be verified easily from the probability density curves of real power in line 5 17 depicted in Fig. 1. It is obvious that zero probability evaluated by the deterministic technique does not confirm the transfer capacity of line 5 17 to be in a robust state, because most power distributions bordering on the capacity limit will affect the identification of some potential critical conditions. Using the proposed approach, the operator may pay more attention to line 5 17 in the operational planning stage.
J.M. Ling et al. Electric Power 5)'.slem.~ Research 33 (1995) 7 15
12
30MW
25MW
15MW
20MW
@r,-~ ~ T ~ - ~ 2 ~ 3 0 3o o764{ I
• '~l
tio.145B
L--- ~
l
I I
-~ ......
.
I I T 1 I ] | { [ ]0"3897
.]0 1379 "
5(~1o~
l'
/
jo.3~93
T, "~
25MW
2595~[~22 30 1007~r,-21 j0.1613 "
[|['
20MW
, 154,~w_jo.2s47
T
501~t
~ /9
{I {
2001~/
(~)
jo.287~
I
.
1117;o.2119
I ltlo 25MW
O. 2935
[jO. 1736
O. 2372
0.2372
5MW
I (9
30.27°3
15MW
.I0 15MW
15MW
25MW
IOOMW
Fig. 6. The IEEE 25-bus test system.
Table 2 The five lines with the most severe conditions in the operational planning (see Fig. 1)
Table 1 Probabilistic bus data for the 25-bus test system Binomial distribution
Line
Fuzzy adequacy indices (5)
Deterministic index (%)
Bus no. No. of units Unit rating (MW) Forced outage rate (%) l 2 3 4 5
3 2 3 1 3
90 52 52 52 70
5.926 3.846 3.846 3.846 4.762
From
To
Low
Medium
Heavy
Risk
5 4 5 I 3
17 19 19 16 14
1.54 36.28 10.87 51.16 38.16
16.26 58.04 87.07 48.82 61.09
82.20 5.68 2.06 0.03 0.00
36.22 1.58 0.51 0.00 0.00
0.0 0.0 0.0 0.0 0.0
Normal distribution Bus no.
Mean (MW)
¢r (%)
Bus no.
Mean (MW)
13 14 16 1.7 18 19 20 21 22 23 24
25 -20 30 - 60 - 15 -15 25 -20 20 -15 15
a (%)
Discrete distribution
T h e o v e r a l l b a l a n c e o f real p o w e r in t h e s y s t e m c a n be measured by the adequacy index of the spinning r e s e r v e . F o r t h e s t a t e i n d i c e s , t h r e e levels h a v e b e e n derived: 4.12% (deficient), 16.46% (surplus), and 79.42% ( a b u n d a n t ) . T h e risk i n d e x w h i c h is e q u a l t o 4 . 1 2 % ( t h e r o b u s t n e s s i n d e x is 9 5 . 8 8 % ) i n d i c a t e s t h a t t h e r e is a 4.12% probability of the system being deficient and unable to meet the demand. Comparing the index used b y t h e d e t e r m i n i s t i c a n a l y s i s t a k i n g 5 % as t h e f a i l u r e level, a n d t h e f u z z y risk i n d e x e q u a l t o 3.9%, t h e l a t t e r risk i n d e x is a c o n s e r v a t i v e ( p e s s i m i s t i c ) e v a l u a t i o n b e c a u s e o f its ' s o f t ' b o u n d a s s i g n m e n t , w h i c h i n c o r p o r a t e s all t h e i n f o r m a t i o n ' c l o s e t o ' t h e f a i l u r e level.
Bus no. Value (MW) Prob. (%) Bus no. Value (MW) Prob. (%)
4.3. Demonstration and discussion o f long-term studies
15 15 15 15
For long-term planning studies, the forecasting methods used to predict the expected peak loads are mainly based on demographic and economic factors which can be considered the most common reasons for correlation.
1
2 3 4 5 6 7 8 9 10 11
12
50 -10 - 50 30 -25 -15 - 15 25 -15 15
5.6 4.0 5.0 4.7 3.2 3.2 2.8 3.8 4.2 2.5 5.0 3.8
-
-
5
-- 10
25.7 - 30.2 - 34.9 27.9
15 30 25 30
25 25 25 25
3.7 4.2 3.8 5.0 4.2 4.5 3.2 3.9 5.0 3.2 3.2
-
-20.5 -23.7 -28.3 -25.3
15 30 25 30
13
J.M. Ling et al./Electric Power Systems Research 33 (1995) 7 15
However, due to forecasting uncertainties, the demand random variables are generally independent, and therefore demands are reasonably accepted as being independent [7,13,15]. To simulate the effects of load uncertainties on the long-term adequacy assessment studies, nine comparison cases in terms of different peak loads and standard deviations were investigated. The mean value of the peak load for the load buses was evaluated by the load growth law,
Table 3 The nine cases used in the comparison
L,, = (1 + r)"Lo
the fuzzy indices than by the deterministic index. Furthermore, potential exposure can easily be detected by the fuzzy failure level, but not by a specified deterministic level. The impacts of load growths and uneertainties on the adequacy of transfer capacity can be examined by comparing the resulting fuzzy risk/adequacy indices in these nine cases to obtain some important information. The larger load growth rate (8%) does not guarantee a larger risk/severity probability, as demonstrated by the cases of power flow in lines 5 19 and 5-17 shown in Tables 4 and 5. It can be seen that cases C, F, and I cause more severe conditions that the other cases. These trends interpret the fact that the sensitivity coefficient of the power flow plays an important role in affecting or weighting the power flow distribution. On the other hand, the larger load uncertainty level increases the risk/severity degree for assessing adequacy. Such a trend agrees with other approaches [4,12], which use the composite reliability evaluation technique. The resulting adequacy indices of the spinning reserve are summarized in Table 7. According to the results of these nine cases studies, it is found that the risk of adequacy of the spinning reserve is affected significantly by the load growths, but is less influenced by the load uncertainty levels, In this situation, the fuzzy risk indices are always bigger than the deterministic indices. This is due to the fact that some potential exposures bordering on the failure level have been incorporated, which reflects the conservative characteristics of fuzzy indices.
(3)
where r is the annual load growth rate, and n is the time basis. In the present study, a five-year basis with three growth rates, 8%, 5%, and Y'/o, was considered. The growth of the peak load reflects the correlation features of economic and demographic growths. The load uncertainty levels were weighted by various values of the standard deviation measured as a percentage of the expected values. Also, three different grades, 10%, 8%, and 6%, were assigned to reflect the independent forecasting errors. In comparison, a standard deviation of approximately 3'70-5% for the active power has been derived for the short-term load behavior [6]. For purposes of comparison, Table 3 lists the different load growth rates and uncertainty levels for the nine cases. After analyzing all the probabilistic line flows in the test system, the adequacy of the transfer capacity is assessed by the fuzzy and deterministic indices. Three main categories for evaluation, listed in Tables 4 6, are summarized. Referring to the first category, illustrated in Table 4, both the fuzzy index and the deterministic index indicate zero probability for risk exposure, i.e. the transmission states are in a robust condition. However, how robust is the transmission state in each case? This information can be obtained by ranking the states according to the probability of the fuzzy 'heavy' state. No obvious information is gained from the deterministic analysis, nor from the analytical probability approach, which evaluates the expected value and standard deviation and the complete density function of the power flows. Most transmission lines in the test system belong to this category. The feasibility achieved by the fuzzy approach is demonstrated by results for the second category, listed in Table 5 for the power flow in line 5-17. Some immediate benefits were outlined in the above short-term studies. For the final category, both the fuzzy index and the deterministic index indicate the risk probability, as shown by the results listed in Table 6, while the fuzzy indices contain ample clear information for a linguistic state description, which is very useful in heuristic applications, such as expert systems. From the above demonstrations, it is clear that more information and better realization can be supplied by
Load growth rate
Uncertainty level c~
10% 8% 6%
8%
5'~/,,
3%,
A D G
B E H
C F I
Table 4 Adequacy indices of transfer capacity for line 5 19 P5 ~,~
Fuzzyadequacy indices (%)
Cases
Low
Medium Heavy
Risk
A B C D E F G H l
20.59 16.21 13.83 20.22 16.23 13.80 19.97 16.21 13.78
79.41 83.70 85.66 79.78 83.73 85.77 80.03 83.78 85.88
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.00 0.09 0.51 0.00 0.04 0.42 0.00 0.01 0.34
Deterministic index (%) 0.0 0.0 0.0 0,0 0,0 0,0 0.0 0.0 0.0
14
J.M. Ling et al./ Electric Power Systems Research 33 (1995) 7 15
Table 5 Adequacy indices of transfer capacity for line 5-17 P5- 17
Fuzzy adequacy indices (%,)
Deterministic index (%)
Cases
Low
Medium
Heavy
Risk
A B C D E F G H I
5.153 2.991 2.102 5.114 2.892 1.993 5.092 2.791 1.893
22.13 17.39 16.91 20.79 17.17 16.94 19.50 17.16 17.81
72.73 79.62 80.99 74.10 79.94 81.06 75.41 80.05 81.10
4.18 10.29 14.83 3.71 10.15 14.80 3.25 10.12 14.79
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Table 6 Adequacy indices of transfer capacity for line 1-16 PI 16
Fuzzy adequacy indices (%,)
Cases
Low
Medium
Heavy
Risk
A B C D E F G H I
0.00 0.07 6.02 0.00 0.01 4.69 0.00 0.00 3.39
15.35 89.27 92.78 12.22 90.22 94.32 9.02 90.75 95.73
84.20 10.65 1.21 87.50 9.77 1.00 90.70 9.25 0.88
16.39 0.64 0.05 15.24 0.53 0.04 14.11 0.44 0.03
Deterministic index (%) 3.213 0.041 0.001 2.829 0.030 0.001 2.354 0.021 0.000
ing the c o n c e p t o f the ' p r o b a b i l i t y m e a s u r e o f fuzzy events' is p r o p o s e d to i n c o r p o r a t e two basic uncertainties, the stochastic u n c e r t a i n t y a n d the linguistic inexactness o f failure, a n d to evaluate the a d e q u a c y indices o f transfer c a p a c i t y a n d spinning reserve. The p r o p o s e d fuzzy a d e q u a c y indices exhibit m a n y advantages. M o r e i n f o r m a t i o n a n d better realization can be s u p p o r t e d by the fuzzy indices than by the deterministic index. Potential risk to facilities can be detected a n d p r e v e n t e d by k n o w i n g the fuzzy failure level. It is c o n c l u d e d that the i m p a c t o f l o a d uncert a i n t y on a d e q u a c y e v a l u a t i o n will increase the risk p r o b a b i l i t y o f the spinning reserve, b u t c a n n o t be t a k e n as a general conclusion when assessing the transfer c a p a c i t y because the latter involves c o n s i d e r a t i o n o f the value a n d sign o f the sensitivity coefficients. T h e results o f the fuzzy state indices a n d fuzzy risk indices p r o v i d e an engineer with a c o m p l e t e picture o f the system's capabilities which c o u l d not be achieved deterministically. S o m e significant results have been attained. H o w e v e r , in a m u c h wider sense, the features o f qualitative assessment p e r m i t m a n y p o t e n t i a l applic a t i o n s involving heuristic concepts, e.g. the use o f expert systems, the process o f decision m a k i n g , etc. T h e results o f the present study m a y p r o v e p r o m i s i n g in further applications.
References Table 7 Adequacy indices of spinning reserve in long-term planning Spinning reserve
Fuzzy adequacy indices (%) Deficient Surplus Abundant Risk
A B C D E F G H l
100.00 80.74 36.40 100.00 80,98 36.37 100.00 81.61 36.35
0.00 19.26 39.50 0.00 19.02 39.55 0,00 18.39 39.15
0.00 0.00 24.10 0.00 0.00 24.08 0.00 0.00 24.50
Deterministic index (%) at 7%
100.00 100.00 80.74 70.81 36.40 34.38 100.00 100.00 80.98 70.81 36.37 33.41 100.00 100.00 81.61 70.81 36.35 33.96
5. Conclusions A n a p p r o a c h b a s e d on fuzzy set t h e o r y is d e v e l o p e d for defining the failure levels when assessing the adeq u a c y o f facilities d u r i n g the stages o f o p e r a t i o n a l a n d l o n g - t e r m planning. A d e q u a c y risk m a i n l y results f r o m the c o n s i d e r a t i o n o f uncertainties. H o w to fully integrate significant uncertainties b e c o m e s a very i m p o r t a n t p r o c e d u r e in successfully i n t e r p r e t i n g the true c h a r a c teristics o f a d e q u a c y indices. A flexible f r a m e w o r k us-
[1] M.M. Adibi and J.P. Stovall, On estimation of uncertainties in analogy measurements, IEEE Trans. Power Syst., 5 (1990) 1222 .k 1230. [2] M.Th. Schilling, A.M. Leite da Silva, R. Billinton and M.A. El-Kady, Bibliography on power system probabilistic analysis, IEEE Trans. Power Syst., 5 (1990) 1-10. [3] R.N. Allan and R. Billinton, Concepts of power system reliability evaluation, Electr. Power Energy Syst., 10 (1988) 139 140. [4] R. Billinton and R.N. Allan, Reliability Evaluation ~d Power Systems, Plenum, New York, 1984. [5] H.M. Merrill and A.J. Wood, Risk and uncertainty in power system planning, Electr. Power Energy' Syst., 13 (1991) 81 90. [6] A.M. Leite da Silva, S.M.P. Ribeiro, V.L. Arienti, R.N. Allan and M.B. Do Coutto Filho, Probabilistic load flow techniques applied to power system expansion planning, IEEE Trans. Power Syst., 5 (1990) 1047-1053. [7] R.N. Allan, B. Borkowska and C.H. Grigg, Probabilistic analysis of power flows, Proc. Inst. Electr. Eng., 121 (1974) 1551 1556. [8] E.O. Crousillat, P. Dorfner, R. Alvarado and H.M. Merrill, Conflicting objectives and risk in power system planning, 1EEE Trans. Power Syst., 8 (1993) 887 893. [9] Y.Y. Hsu and H.C. Kuo, Fuzzy-set based contingency ranking, IEEE Trans. Power Syst., 7 (1992) 1189-1195. [10] L.A. Zadeh, Probability measure of fuzzy events, J. Math. Anal. Appl., 10 (1968) 421-427. [11] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part I11, l n f Sci., 9 (1975) 43-80. [12] L. Wenyuan and R. Billinton, Effect on bus load uncertainty and correlation in composite system adequacy evaluation, 1EEE Trans. Power Syst., 6 0991) 1522 1529.
J.M. Ling et al./Electric Power Systems Research 33 (1995) 7 15
[13] R.N. Allan, C.H. Grigg, D.A. Newey and R.F. Simmons, Probabilistic power-flow techniques extended and applied to operational decision making, Proe. Inst. Electr. Eng., 123 (1976) 1317 1324. [14] R.N. Allan~ C.H. Grigg and M.R.G. AI-Shakarchi, Numerical techniques in probabilistic load flow problems, Int. J. Numer. Methods Eng., 10 (1976) 853 860.
15
[15] R.N. Allan and M.R.G. A1-Shakarchi, Linear dependence between nodal powers in probabilistic a.c. load flow, Proe. Inst. Electr. Eng., 124 (1977) 529 534. [16] A. Kaufmann and M.M. Gupta, Fuz=y Mathematical Models in Engineering and Management Science, North-Holland, Amsterdam, 1988.
POt,IAn
ELSEVIER
Electric Power Systems Research 33 (1995) 7 15
Probability measure of adequacy assessment using a fuzzy approach Jeeng-Min Ling, Chin-E. Lin*, Ching-Lien Huang Department ~[ Electrical Engineering, Institute o[' Aeronautics and Astronautics, National Cheng Kung University, Tainan. Taiwan Received 10 October 1994
Abstract
An approach based on fuzzy set theory is developed for defining the failure levels in assessing the adequacy of facilities during the stages of operational and long-term planning in a power system. After analyzing the stochastic uncertainties of load variation and generation nonavailability by the probability load flow algorithm, the 'probability measure of fuzzy events' is proposed to combine these uncertainties and the linguistic failure inexactness to evaluate the adequacy indices of both transfer capacity and spinning reserve. The impact of load uncertainty on the adequacy assessment is investigated in nine different cases, and the resulting adequacy indices of each case are summarized. The correlations of the demand characteristics in the relatively short term are also examined for an actual system. Results show that the proposed fuzzy adequacy indices provide more information and a better realization than the deterministic adequacy index. Furthermore, some potential risk to facilities can easily be detected by the fuzzy failure level, whereas the deterministic specified-value level will fail to do so. The proposed technique is implemented on the IEEE 25-bus system to demonstrate its feasibility.
Keywords: Fuzzy set theory: Adequacy assessment; Probability analysis; Power system planning
1. Introduction
Uncertainty has recently been recognized as an inevitable factor when dealing with the inherently random nature of an actual electric power system, so the characteristics of systems can therefore be considered as random variables. Well-investigated stochastic uncertainties include reading errors and random errors related to analog measurements in online computer applications [1], nonlinear relationships between load and weather changes, inaccuracies in weather forecasts for short-term load forecasting, etc. Many papers have successfully applied probability methodology to account for these stochastic uncertainties in power system applications [2]. This research has already attained a significant level of theoretical maturity, and can be considered to be currently in a stage of emerging development. In this paper, we analyze the impacts of uncertainties on adequacy assessment during the stages of operational and long-term planning. System adequacy relates to the existence of sufficient facilities to satisfy system demand [3,4]. For a composite (generation and transmission) power system, this means that the generating facilities are adequate to generate sufficient energy, and the *Corresponding author. 0378-7796/95'$09.50 ~t'~ 1995 Elsevier Science S.A. All rights reserved S S D I 0378-7796(95)00921-4
associated transmission facilities are adequate to transfer the energy to actual load points. In general, adequacy is associated with static conditions which do not include system disturbances. Techniques for adequacy assessment have been realized by probabilistic analyses considering various uncertainties [2,3,5,6]. The basic probabilistic technique uses the probabilistic load flow (PLF) approach [6]. Both stochastic load variation and generation unavailability are recognized and analyzed in the PLF algorithm. The results of the P L F assessment give a complete picture of the system and assess the likelihood of occurrence of system problems that cannot be obtained from a conventional deterministic analysis [7]. Compared with another similar probabilistic technique, composite reliability evaluation, P L F analyses reflect system problems before the remedial action evaluation algorithm is used [6]. Uncertainty involves unpredictable factors which have a major influence on a utility, and are not under control or cannot be predicted with certainty. In this sense, a 'risk' assessment should be conferred to indicate the hazards to which a utility is exposed [5,8]. Risk exposure involves facilities which significantly affect the adequacy or the reliability of a system. If we focus on the assessment of the facilities' adequacy under uncertainty, defining the 'failure level' is an essential step.
8
J.M. Ling et al.
EA,ctric Power Systems Research 33 (1995) 7 15
The risk exposure of two competitive alternatives for the Brazilian system has been evaluated by defining failure levels by specific numbers [6]. These numbers were assigned based on the capacity limits of facilities or the steady-state stability limits (operating constraints). However, in adequacy assessment, an event with most of its elements close to but not exceeding the failure level will be recognized as no risk, which is an unreasonable judgement. A deterministic number cannot properly represent the entire characteristics of a failure level, nor consistently define the true risk to the system [9]. Intuitively, some past experiences and expected future events can be integrated to define a 'soft' failure level. These assignments are particularly promising and useful in the fields of adequacy assessment and system planning when human perception and the process of decision making are inevitable. From the viewpoint of experience interpretation, a 'soft' bound for the failure level is more flexible. In this paper, an approach based on fuzzy set theory is proposed for the interpretation of the characteristics of failure [10,I1]. After analyzing the results of the PLF technique, the 'probability measure of fuzzy events' is used to assess the adequacy indices of facilities. To test the adequacy assessment in an actual system during the planning stage, some significant considerations about load, namely, load correlation, load growth, and load uncertainty, are investigated using the IEEE 25-bus test system. Results show that a more reasonable adequacy assessment can be achieved by the proposed probability measure.
2. Analytical approach to uncertainty Probability and fuzzy analytical approaches used to deal with stochastic and linguistic uncertainties are introduced in this section. Theoretically, probability theory deals with inherent stochastic uncertainty due to the occurrence of random events, while fuzzy set theory deals with uncertainty in the use of human experience in linguistic terms. The details of analytical concepts are described, and a feasible framework is proposed to integrate the two approaches. 2.1. Stochastic uncertainty In the field of adequacy assessment, stochastic uncertainties have been measured by probabilistic analyses [2,3,5]. These uncertainties include two main types: load forecast uncertainty and generation unavailability. Load forecast uncertainty is an extremely important parameter, combining the financial, social, environmental, and other uncertainties which electric power utilities must face [4]. Past research suggests that such uncertainty can be reasonably approximated by a normal
distribution. In composite reliability evaluation [4], load forecast uncertainty has been included in the risk computations by dividing the normal probability distribution into class intervals which are dependent on the desired accuracy. Similar discretization techniques can be used to model load forecast uncertainty in probabilistic analysis. Another relevant sampling method uses a tabulating technique [12]. The basic generating unit parameter used in static capacity evaluation is the probability of finding the unit on forced outage at some distant time in the future [4]. This probability is known in systems engineering as the unit unavailability, and historically in power system applications as the unit forced outage rate (FOR). In contrast to the multistate model used in generating capacity reliability evaluation, the probabilistic algorithm has been associated with a simple two-state model and defined accurately by a binomial distribution or a recursive technique [4,13,14]. The impact of generating unit unavailability and load uncertainty can be measured using the PLF algorithm to describe the steady-state performance of a given system. The results of the PLF assessment also reveal the range of all possible conditions that might be encountered as a result of expected uncertainties in inputs. Based on such implementation, the future system performance in an uncertain environment can therefore be predicted in a more consistent way. 2.2. Linguistic inexactness Another type of uncertainty, linguistic inexactness, associated with the specification of the failure level should be considered in the adequacy assessment. Occurrences of deficiency (nonadequacy) rarely occur in an actual system. This fact leads to the representation of the characteristics of a failure level not being properly described by either the probability technique, because it violates the law of large numbers, or by a specified deterministic number. Qualitative observation is one of the essential attributes of adequacy assessment, decision making and other fields involving human judgement. Human experts tend to use linguistic terms to describe their experience and heuristic rules. In this sense, high-level reasoning requires vagueness in order to manage problems without getting bogged down in details. Fuzzy set theory is proposed to capture the psychological reality of experience because of its linguistic elasticity [11]. The concept of a fuzzy set can be defined as follows. Definition 1. Fuzzy set Let U be the universe of discourse and x be the universal elements. Then, a fuzzy set A is an ordered pair:
J.M. Ling et al./Electric Power Systems Research 33 (1995) 7 15
9
Deterministic Failure 0.06
Level -1.20
-
0.05
/
~
/
0.04
/ ~0,03
s
/
• 1.00
/
4
.a~'v/-
"
[ /
.~ ~ 0.02
0.01
kJ III
°'~s.o
'I
so.o
"~'"
ss.o
I * I I | I * I | J * | L4'|
r~.o
¢~.o
~o.o
* I I*
?s.o
I I
"d. Jl
oo.o
.0.40
'0.20
~"
~.o
90.o
' l I I I l J
I I I I I I I I
g ~ . o ~oo.o Io5~o . o . o
P o w e r J~o~ i~ l~,~e 5 - I ? a n d ff~ membership .f~net~on of overload Fig. 1. The probabilistic real power flow and the failure level.
w h e r e / ~ is the membership function, and the grade of membership typically falls within [0, 1]. A fuzzy number is one of the most common forms of fuzzy set application [16]. A convex and normalized fuzzy set defined by a set of real numbers is called a fuzzy number. One typical form of representation, termed the trapezoidal fuzzy number, which is characterized by a quadruplet, is adopted in this study.
2.3. A framework Jor combining stochastic uncertainty and linguistic inexactness Probability theory and fuzzy set theory are both used to study imprecision. It is interesting to consider a general framework which can integrate both the stochastic and fuzzy uncertainties to solve the following problem: if the probability distribution of the power flow is derived from the PLF analysis, what is the probability that a line is 'overloaded'? Here, 'overloaded' is a vague linguistic term. One way to deal with this problem is to fuzzify the concept of a random variable (RV) [10]. From classical probability theory, if the sample space contains all 'outcomes of a random experiment', then a function from the sample space into the set of real numbers is called an RV. An 'event' is a precisely specified collection of points in the sample space. However, in everyday experience one frequently encounters situations in which an 'event' is an ill-defined rather than a sharply defined collection of points. For example, "it is a warm day" and " x is approximately equal to 5" are fuzzy events. A fuzzy event is a fuzzy subset of the sample
space. Therefore, we can define a fuzzy event in Euclidean n-space, R', to fuzzify the concept of an RV by the following definition.
Definition 2. Probability measure of a Juzzy event Let (R', ~, P) be a probability space in which ~ is the a-field of Borel sets in R" and P is a probability measure over R". Then a fuzzy event in R" is a fuzzy set A in R" whose membership function /~a (/~A: R " ~ [0, 1]) is Borel measurable. The probability measure of a fuzzy event A is defined as P(A)
=
~ fla(X) Q/ Rn
dP = E ( f l A )
(2)
The probability measure of a fuzzy event is interpreted as the expectation of its membership function. In this study, the expectation is proposed to identify the adequacy indices in the adequacy assessment studies. To clarify the concept of Eq. (2), a simple but significant diagram is exemplified in Fig. 1 to answer the question posed at the beginning of this section. Fig. 1 depicts the probabilistic distribution of real power flow in line 5-17 (P5 17) and the membership function of 'overload' for transfer capacity. The probability of P5 ~7 being overloaded can be evaluated from Eq. (2) and is equal to 36.2%.
3. The fuzzy approach to adequacy assessment The adequacy indices of two significant attributes arc described in this section, including the probability
l0
J.M. Ling et al.
Electric Power Systems Research 33 (1995) 7 15
of any particular power flow being greater than the corresponding equipment's thermal rating, and the probability of active generation exceeding acceptable levels. The process of adequacy assessment is concerned with two basic steps. Step I. Summarize the heuristic rules identified through the intuition of experienced operators, and construct the membership functions of the failure levels for significant attributes using the fuzzy set approach. Step 2. Use the proposed framework described by Eq. (2) to evaluate the desired adequacy indices based on the results of the PLF analysis and the heuristic membership functions of the related levels. Details of the specifications of two particular levels, namely, the 'state' level used for assessing the state of the facilities, and the 'failure' level used for assessing the risk index of the facilities, for the attributes of transfer capacity and generation reserve will be discussed in the following.
1.~0
1.25
Heavy
io~
~dium
/
1.00
I I
O.~tO
Medium
I I
l
O.Z5
,,,,,,,, t,,, /,,,/
0.00
.....
. 3 0 t . 5 0 t . . ?Q~ 80%
percentage o ~ ' ~ i e r
Three categories, light, medium, and heavy, are used to assess the 'state' levels of transfer capacity. The heuristic quadruplets of various fuzzy members in terms of the 'state' and 'failure' levels are depicted in Figs. 2 and 3. All quadruplets are expressed as a percentage of
A simple measure for the adequacy aspect of system reliability is the transmission system's limitation. Theoretically, the maximum power P ..... otherwise known as the steady-state stability limit, that can be transferred from generation to load is related to the bus voltages and line impedance. Under actual system operating conditions, it is natural that the failure level of transfer capacity cannot be considered to be a crisply defined value, like P . . . . since operators tend to use linguistic representations to assess the 'failure' level of line flow, such as
3.2. Assessment of spinning reserve Another significant measure of the adequacy aspect of system reliability is the spinning reserve. Variations in the consumption of electricity must be met by an equivalent modification of the generation output. This implies that some of the capacity in the system must be saved for 'reserve' purposes. In the PLF analysis, the reserve can be measured by evaluating the probability density curve of the overall balance of power in the system. In the meantime, the probability of deficiency in the system can also be evaluated. However, what percentage of spinning reserve should be kept in the operation? The answer to this question involves the operational condition of the system and the strategy of the utilities. Since forced outage is possible and unreliable load forecasts exist, there should be 'enough' reserve to meet all likely load
Overload
Underload
~( Risk )
"The line flow over a certain line is 'heavy'" How 'heavy' is heavy? How do we handle the characteristics of such a linguistic term in the adequacy assessment? Since the 'imprecision' described above is difficult to manage by conventional probability theory, a fuzzy set concept is proposed.
¢¢m..¢
Fig. 2. The state levels of transfer capacity.
Pmax •
The term 'overloaded' is imprecise. How many real power flows are identified by the 'overloaded' state? The operator may confirm that those lines that are loaded 110% (in terms of Pmax) or more will cause immediate security problems, and it is also certain that a lighter loading of 70% or less will cause a robust state. It is less certain whether a line loading of 70% 110% will cause risk. Other operators may place different assignments on the 'overloaded' and 'underloaded' levels because of different considerations. For the 'state' level of adequacy assessment, an operator may judge that
lltt,,,,,,,,,,,
i .....
- 7 0 t -50% ~30%
-80:
3.1. Assessment of transfer capacity
"The line flow over a certain line causes it to be 'overloaded'"
/
Heavy
Overload
(Robus ness)
( Risk )
A ,
.....
,V o,
:
/ ....
p c r c e . J ~ e of t r o ~ J e r Z*,'~J
Fig. 3. The failure levels of transfer capacity.
. . . . . . . . .
'+I
J.M. Ling et al./ Electric Power Systems Research 33 (1995) 7+ 15
12.5
Deficient
Surplus
Abundant
t.00
o+I 0.25
. . . . . . . .
,,
ltl
15%///// " ~ 2 0 % ,,,+ ,/1,t,11,1,% ,1,
Pe~'ee~taxd~ of S;4~.~ 4
iii
~1 li
Reser'ue
Fig. 4. The state levels of spinning reserve.
variations. How much is 'enough'? Is 10% enough? From the practical point of view, a specified value cannot properly define the generation reserve in the field of risk assessment. It is more flexible and instructive to consider a gradual transition rather than a crisply defined bound, like 10% or 15%. After taking significant aspects of the operational system and operator experience into account, the heuristic fuzzy concept for the 'state' and 'failure' levels of spinning reserve is related. Figs. 4 and 5 depict the shape of the proposed trapezoidal fuzzy numbers. All quadruplets are expressed as a percentage of peak demand.
4. Applications of the proposed technique 4.1. Test system and probabilistic technique In our test case the proposed 'probability measure of a fuzzy event' was applied to the IEEE 25-bus system shown in Fig. 6. The deterministic base-case data and line data are depicted in Fig. 6. The probabilistic nodal test data were modified from the deterministic base-case data and the same expected values maintained. It is
1.25
Deficient
Enough
O,,?.5
5+ 0~040
, I a , , I h , ,
lo+ i , , , , , , i i i i i i i i i i i i I
Pcrce~Jo@e oi S p ~ m { ~ R e ~ e
Fig. 5. The failure levels of spinning reserve.
II
noted that the data listed in Table I relating to the characteristics of consumer demands are not completely independent, i.e. there exists some linear dependence between various groups of nodal powers. The adequacy assessment studies began with the evaluation of the PLF analysis. Probability input data were mathematically convoluted to derive the output probability density curve for each line by using the Laplace transform technique [14]. During the process of convolution, the probabilities of discontinuous nodal powers, i.e. binomial and discrete, were truncated to a cumulative probability of 10 +'. A considerable number of states were deleted using this truncation technique. Another weighted-average technique [14] was used to reduce storage and computation time by grouping convoluted discontinuous points with the predetermined points. For a continuous normal nodal power, nine discretization points within three standard deviations from the mean were sampled. Based on the results of PLF analyses, the adequacy indices were evaluated using the proposed measure implemented by Eq. (2). These indices are represented by the percentage probability in this study.
4.2. Demonstration and discussion (~1 short-term studies Various reasons for correlations exist between nodal powers, especially in relatively short-term planning exercises [13,15]. The partial correlation probabilistic data used in the present study are listed in Table 1, and contain the following assumptions. Positive linear dependence between nodal powers exists at buses 3, 11, 17, and 22 (group 1), 8, 12, and 16 (group 2), 9, 14, and 18 (group 3), and 15 and 25 (group 4). All other nodal powers are assumed to be independent of each other and of the above dependent groups. The state and risk indices for assessing the adequacy of transfer capacity in each line of the test system were evaluated by the proposed fuzzy approach and by the deterministic approach. The maximum transfer capacity in the test system was assumed to be 100 MW. The five lines with most severe conditions are listed in Table 2. Results show that the risk index of some critical lines can be evaluated successfully with the proposed approach, but may fail (zero probability) to be evaluated in the deterministic analysis. The proposed fuzzy approach can be verified easily from the probability density curves of real power in line 5 17 depicted in Fig. 1. It is obvious that zero probability evaluated by the deterministic technique does not confirm the transfer capacity of line 5 17 to be in a robust state, because most power distributions bordering on the capacity limit will affect the identification of some potential critical conditions. Using the proposed approach, the operator may pay more attention to line 5 17 in the operational planning stage.
J.M. Ling et al. Electric Power 5)'.slem.~ Research 33 (1995) 7 15
12
30MW
25MW
15MW
20MW
@r,-~ ~ T ~ - ~ 2 ~ 3 0 3o o764{ I
• '~l
tio.145B
L--- ~
l
I I
-~ ......
.
I I T 1 I ] | { [ ]0"3897
.]0 1379 "
5(~1o~
l'
/
jo.3~93
T, "~
25MW
2595~[~22 30 1007~r,-21 j0.1613 "
[|['
20MW
, 154,~w_jo.2s47
T
501~t
~ /9
{I {
2001~/
(~)
jo.287~
I
.
1117;o.2119
I ltlo 25MW
O. 2935
[jO. 1736
O. 2372
0.2372
5MW
I (9
30.27°3
15MW
.I0 15MW
15MW
25MW
IOOMW
Fig. 6. The IEEE 25-bus test system.
Table 2 The five lines with the most severe conditions in the operational planning (see Fig. 1)
Table 1 Probabilistic bus data for the 25-bus test system Binomial distribution
Line
Fuzzy adequacy indices (5)
Deterministic index (%)
Bus no. No. of units Unit rating (MW) Forced outage rate (%) l 2 3 4 5
3 2 3 1 3
90 52 52 52 70
5.926 3.846 3.846 3.846 4.762
From
To
Low
Medium
Heavy
Risk
5 4 5 I 3
17 19 19 16 14
1.54 36.28 10.87 51.16 38.16
16.26 58.04 87.07 48.82 61.09
82.20 5.68 2.06 0.03 0.00
36.22 1.58 0.51 0.00 0.00
0.0 0.0 0.0 0.0 0.0
Normal distribution Bus no.
Mean (MW)
¢r (%)
Bus no.
Mean (MW)
13 14 16 1.7 18 19 20 21 22 23 24
25 -20 30 - 60 - 15 -15 25 -20 20 -15 15
a (%)
Discrete distribution
T h e o v e r a l l b a l a n c e o f real p o w e r in t h e s y s t e m c a n be measured by the adequacy index of the spinning r e s e r v e . F o r t h e s t a t e i n d i c e s , t h r e e levels h a v e b e e n derived: 4.12% (deficient), 16.46% (surplus), and 79.42% ( a b u n d a n t ) . T h e risk i n d e x w h i c h is e q u a l t o 4 . 1 2 % ( t h e r o b u s t n e s s i n d e x is 9 5 . 8 8 % ) i n d i c a t e s t h a t t h e r e is a 4.12% probability of the system being deficient and unable to meet the demand. Comparing the index used b y t h e d e t e r m i n i s t i c a n a l y s i s t a k i n g 5 % as t h e f a i l u r e level, a n d t h e f u z z y risk i n d e x e q u a l t o 3.9%, t h e l a t t e r risk i n d e x is a c o n s e r v a t i v e ( p e s s i m i s t i c ) e v a l u a t i o n b e c a u s e o f its ' s o f t ' b o u n d a s s i g n m e n t , w h i c h i n c o r p o r a t e s all t h e i n f o r m a t i o n ' c l o s e t o ' t h e f a i l u r e level.
Bus no. Value (MW) Prob. (%) Bus no. Value (MW) Prob. (%)
4.3. Demonstration and discussion o f long-term studies
15 15 15 15
For long-term planning studies, the forecasting methods used to predict the expected peak loads are mainly based on demographic and economic factors which can be considered the most common reasons for correlation.
1
2 3 4 5 6 7 8 9 10 11
12
50 -10 - 50 30 -25 -15 - 15 25 -15 15
5.6 4.0 5.0 4.7 3.2 3.2 2.8 3.8 4.2 2.5 5.0 3.8
-
-
5
-- 10
25.7 - 30.2 - 34.9 27.9
15 30 25 30
25 25 25 25
3.7 4.2 3.8 5.0 4.2 4.5 3.2 3.9 5.0 3.2 3.2
-
-20.5 -23.7 -28.3 -25.3
15 30 25 30
13
J.M. Ling et al./Electric Power Systems Research 33 (1995) 7 15
However, due to forecasting uncertainties, the demand random variables are generally independent, and therefore demands are reasonably accepted as being independent [7,13,15]. To simulate the effects of load uncertainties on the long-term adequacy assessment studies, nine comparison cases in terms of different peak loads and standard deviations were investigated. The mean value of the peak load for the load buses was evaluated by the load growth law,
Table 3 The nine cases used in the comparison
L,, = (1 + r)"Lo
the fuzzy indices than by the deterministic index. Furthermore, potential exposure can easily be detected by the fuzzy failure level, but not by a specified deterministic level. The impacts of load growths and uneertainties on the adequacy of transfer capacity can be examined by comparing the resulting fuzzy risk/adequacy indices in these nine cases to obtain some important information. The larger load growth rate (8%) does not guarantee a larger risk/severity probability, as demonstrated by the cases of power flow in lines 5 19 and 5-17 shown in Tables 4 and 5. It can be seen that cases C, F, and I cause more severe conditions that the other cases. These trends interpret the fact that the sensitivity coefficient of the power flow plays an important role in affecting or weighting the power flow distribution. On the other hand, the larger load uncertainty level increases the risk/severity degree for assessing adequacy. Such a trend agrees with other approaches [4,12], which use the composite reliability evaluation technique. The resulting adequacy indices of the spinning reserve are summarized in Table 7. According to the results of these nine cases studies, it is found that the risk of adequacy of the spinning reserve is affected significantly by the load growths, but is less influenced by the load uncertainty levels, In this situation, the fuzzy risk indices are always bigger than the deterministic indices. This is due to the fact that some potential exposures bordering on the failure level have been incorporated, which reflects the conservative characteristics of fuzzy indices.
(3)
where r is the annual load growth rate, and n is the time basis. In the present study, a five-year basis with three growth rates, 8%, 5%, and Y'/o, was considered. The growth of the peak load reflects the correlation features of economic and demographic growths. The load uncertainty levels were weighted by various values of the standard deviation measured as a percentage of the expected values. Also, three different grades, 10%, 8%, and 6%, were assigned to reflect the independent forecasting errors. In comparison, a standard deviation of approximately 3'70-5% for the active power has been derived for the short-term load behavior [6]. For purposes of comparison, Table 3 lists the different load growth rates and uncertainty levels for the nine cases. After analyzing all the probabilistic line flows in the test system, the adequacy of the transfer capacity is assessed by the fuzzy and deterministic indices. Three main categories for evaluation, listed in Tables 4 6, are summarized. Referring to the first category, illustrated in Table 4, both the fuzzy index and the deterministic index indicate zero probability for risk exposure, i.e. the transmission states are in a robust condition. However, how robust is the transmission state in each case? This information can be obtained by ranking the states according to the probability of the fuzzy 'heavy' state. No obvious information is gained from the deterministic analysis, nor from the analytical probability approach, which evaluates the expected value and standard deviation and the complete density function of the power flows. Most transmission lines in the test system belong to this category. The feasibility achieved by the fuzzy approach is demonstrated by results for the second category, listed in Table 5 for the power flow in line 5-17. Some immediate benefits were outlined in the above short-term studies. For the final category, both the fuzzy index and the deterministic index indicate the risk probability, as shown by the results listed in Table 6, while the fuzzy indices contain ample clear information for a linguistic state description, which is very useful in heuristic applications, such as expert systems. From the above demonstrations, it is clear that more information and better realization can be supplied by
Load growth rate
Uncertainty level c~
10% 8% 6%
8%
5'~/,,
3%,
A D G
B E H
C F I
Table 4 Adequacy indices of transfer capacity for line 5 19 P5 ~,~
Fuzzyadequacy indices (%)
Cases
Low
Medium Heavy
Risk
A B C D E F G H l
20.59 16.21 13.83 20.22 16.23 13.80 19.97 16.21 13.78
79.41 83.70 85.66 79.78 83.73 85.77 80.03 83.78 85.88
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.00 0.09 0.51 0.00 0.04 0.42 0.00 0.01 0.34
Deterministic index (%) 0.0 0.0 0.0 0,0 0,0 0,0 0.0 0.0 0.0
14
J.M. Ling et al./ Electric Power Systems Research 33 (1995) 7 15
Table 5 Adequacy indices of transfer capacity for line 5-17 P5- 17
Fuzzy adequacy indices (%,)
Deterministic index (%)
Cases
Low
Medium
Heavy
Risk
A B C D E F G H I
5.153 2.991 2.102 5.114 2.892 1.993 5.092 2.791 1.893
22.13 17.39 16.91 20.79 17.17 16.94 19.50 17.16 17.81
72.73 79.62 80.99 74.10 79.94 81.06 75.41 80.05 81.10
4.18 10.29 14.83 3.71 10.15 14.80 3.25 10.12 14.79
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Table 6 Adequacy indices of transfer capacity for line 1-16 PI 16
Fuzzy adequacy indices (%,)
Cases
Low
Medium
Heavy
Risk
A B C D E F G H I
0.00 0.07 6.02 0.00 0.01 4.69 0.00 0.00 3.39
15.35 89.27 92.78 12.22 90.22 94.32 9.02 90.75 95.73
84.20 10.65 1.21 87.50 9.77 1.00 90.70 9.25 0.88
16.39 0.64 0.05 15.24 0.53 0.04 14.11 0.44 0.03
Deterministic index (%) 3.213 0.041 0.001 2.829 0.030 0.001 2.354 0.021 0.000
ing the c o n c e p t o f the ' p r o b a b i l i t y m e a s u r e o f fuzzy events' is p r o p o s e d to i n c o r p o r a t e two basic uncertainties, the stochastic u n c e r t a i n t y a n d the linguistic inexactness o f failure, a n d to evaluate the a d e q u a c y indices o f transfer c a p a c i t y a n d spinning reserve. The p r o p o s e d fuzzy a d e q u a c y indices exhibit m a n y advantages. M o r e i n f o r m a t i o n a n d better realization can be s u p p o r t e d by the fuzzy indices than by the deterministic index. Potential risk to facilities can be detected a n d p r e v e n t e d by k n o w i n g the fuzzy failure level. It is c o n c l u d e d that the i m p a c t o f l o a d uncert a i n t y on a d e q u a c y e v a l u a t i o n will increase the risk p r o b a b i l i t y o f the spinning reserve, b u t c a n n o t be t a k e n as a general conclusion when assessing the transfer c a p a c i t y because the latter involves c o n s i d e r a t i o n o f the value a n d sign o f the sensitivity coefficients. T h e results o f the fuzzy state indices a n d fuzzy risk indices p r o v i d e an engineer with a c o m p l e t e picture o f the system's capabilities which c o u l d not be achieved deterministically. S o m e significant results have been attained. H o w e v e r , in a m u c h wider sense, the features o f qualitative assessment p e r m i t m a n y p o t e n t i a l applic a t i o n s involving heuristic concepts, e.g. the use o f expert systems, the process o f decision m a k i n g , etc. T h e results o f the present study m a y p r o v e p r o m i s i n g in further applications.
References Table 7 Adequacy indices of spinning reserve in long-term planning Spinning reserve
Fuzzy adequacy indices (%) Deficient Surplus Abundant Risk
A B C D E F G H l
100.00 80.74 36.40 100.00 80,98 36.37 100.00 81.61 36.35
0.00 19.26 39.50 0.00 19.02 39.55 0,00 18.39 39.15
0.00 0.00 24.10 0.00 0.00 24.08 0.00 0.00 24.50
Deterministic index (%) at 7%
100.00 100.00 80.74 70.81 36.40 34.38 100.00 100.00 80.98 70.81 36.37 33.41 100.00 100.00 81.61 70.81 36.35 33.96
5. Conclusions A n a p p r o a c h b a s e d on fuzzy set t h e o r y is d e v e l o p e d for defining the failure levels when assessing the adeq u a c y o f facilities d u r i n g the stages o f o p e r a t i o n a l a n d l o n g - t e r m planning. A d e q u a c y risk m a i n l y results f r o m the c o n s i d e r a t i o n o f uncertainties. H o w to fully integrate significant uncertainties b e c o m e s a very i m p o r t a n t p r o c e d u r e in successfully i n t e r p r e t i n g the true c h a r a c teristics o f a d e q u a c y indices. A flexible f r a m e w o r k us-
[1] M.M. Adibi and J.P. Stovall, On estimation of uncertainties in analogy measurements, IEEE Trans. Power Syst., 5 (1990) 1222 .k 1230. [2] M.Th. Schilling, A.M. Leite da Silva, R. Billinton and M.A. El-Kady, Bibliography on power system probabilistic analysis, IEEE Trans. Power Syst., 5 (1990) 1-10. [3] R.N. Allan and R. Billinton, Concepts of power system reliability evaluation, Electr. Power Energy Syst., 10 (1988) 139 140. [4] R. Billinton and R.N. Allan, Reliability Evaluation ~d Power Systems, Plenum, New York, 1984. [5] H.M. Merrill and A.J. Wood, Risk and uncertainty in power system planning, Electr. Power Energy' Syst., 13 (1991) 81 90. [6] A.M. Leite da Silva, S.M.P. Ribeiro, V.L. Arienti, R.N. Allan and M.B. Do Coutto Filho, Probabilistic load flow techniques applied to power system expansion planning, IEEE Trans. Power Syst., 5 (1990) 1047-1053. [7] R.N. Allan, B. Borkowska and C.H. Grigg, Probabilistic analysis of power flows, Proc. Inst. Electr. Eng., 121 (1974) 1551 1556. [8] E.O. Crousillat, P. Dorfner, R. Alvarado and H.M. Merrill, Conflicting objectives and risk in power system planning, 1EEE Trans. Power Syst., 8 (1993) 887 893. [9] Y.Y. Hsu and H.C. Kuo, Fuzzy-set based contingency ranking, IEEE Trans. Power Syst., 7 (1992) 1189-1195. [10] L.A. Zadeh, Probability measure of fuzzy events, J. Math. Anal. Appl., 10 (1968) 421-427. [11] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part I11, l n f Sci., 9 (1975) 43-80. [12] L. Wenyuan and R. Billinton, Effect on bus load uncertainty and correlation in composite system adequacy evaluation, 1EEE Trans. Power Syst., 6 0991) 1522 1529.
J.M. Ling et al./Electric Power Systems Research 33 (1995) 7 15
[13] R.N. Allan, C.H. Grigg, D.A. Newey and R.F. Simmons, Probabilistic power-flow techniques extended and applied to operational decision making, Proe. Inst. Electr. Eng., 123 (1976) 1317 1324. [14] R.N. Allan~ C.H. Grigg and M.R.G. AI-Shakarchi, Numerical techniques in probabilistic load flow problems, Int. J. Numer. Methods Eng., 10 (1976) 853 860.
15
[15] R.N. Allan and M.R.G. A1-Shakarchi, Linear dependence between nodal powers in probabilistic a.c. load flow, Proe. Inst. Electr. Eng., 124 (1977) 529 534. [16] A. Kaufmann and M.M. Gupta, Fuz=y Mathematical Models in Engineering and Management Science, North-Holland, Amsterdam, 1988.