Module PE.PAS.U19.5. U19.1 Introduction

Transcription

1 Module PE.PAS.U19.5 Generation adequacy evaluation 1 Module PE.PAS.U19.5 Generation adequacy evaluation U19.1 Introduction Probabilistic evaluation o generation adequacy is traditionally perormed or one o two classes o decision problems. The irst one is the generation capacity planning problem where one wants to determine the long-range generation needs o the system. The second one is the short-term operational problem where one wants to determine the unit commitment over the next ew days or wees. We may thin o the problem o generation adequacy evaluation in terms o Fig. U19.1. Fig. U19.1: Evaluation o Generation Adequacy In Fig. U19.1, we see that there are a number o generation units, and there is a single lumped load. Signiicantly, we also observe that all generation units are modeled as i they were connected directly to the load, i.e., transmission is not modeled. The implication o this is that, in generation adequacy evaluation, transmission is assumed to be perectly reliable. We begin our treatment by irst identiying the necessary modeling requirements in terms o, in Section U19.2, the generation side,

2 Module PE.PAS.U19.5 Generation adequacy evaluation 2 and, in Section U19.3, the load side. Section U19.4 describes a common computational approach associated with the generation capacity planning problem, and Section U19.5 illustrates how this approach is typically used or capacity planning. Section U19.6 provides an alternative way o computing generation capacity planning indices. Section U19.7 briely summarizes three important issues central to a more extended treatment o the topic. U19.2 Generator model In the basic capacity planning study, each individual generation unit is represented using the two-state Marov model illustrated in Fig. U19.2. Up =1/r λ=1/m Fig. U19.2: Two-State Marov Model This model was described in Sections U16.4-U16.5 and Section U o modules U16 & U18, respectively. Important relations or this model, in terms o long-run availability A and long-run unavailability U, are provided here again, or convenience, where m=mttf, r=mttr, µ and λ are transition rates (number o transitions per unit time) or repair ( to Up) and or ailure (Up to ), respectively; T is the mean cycle time, and is a requency which gives the expected number o direct transers between states per-unit time. U A m m r m T r FOR m r r T SH FOH SH FOH FOH SH (U19.1) (U19.2)

3 Module PE.PAS.U19.5 Generation adequacy evaluation 3 In (U19.2), the FOR is the orced outage rate. One should be careul to note that the FOR is not a rate at all but rather an estimator or a probability. The terms in the right-hand-expressions o (U19.1) and (U19.2) are deined as ollows: Forced outage hours (FOH) is the number o hours a unit was in an unplanned outage state; Service hours (SH) is the number o hours a unit was in the in-service state. It does not include reserve shutdown hours. Module U18 also describes how one can approximate the eects o derating (the unit is operating but at reduced capacity due to, or example, the outages o auxiliary equipment such as pulverizers, water pumps, ans, or environmental constraints) and, or peaing plants, o reserve shutdown (intentionally out o service on a requent basis, common or peaing units), by using the equivalent orced outage rate, EFOR, according to: EFOR orced equivalent orced outage hours derated hours equivalent reserve orced service shutdown orced outage hours hours derated hours EFOR FOH EFH FOH SH ERSFH (U19.3) The basis or (U19.3) is not simple, and so we will not address it here. But it is very well explained in Module U18. U Capacity outage table or identical units A capacity table is simply a probabilistic description o the possible capacity states o the system being evaluated. The simplest case is that o the 1 unit system, where there are two

4 Module PE.PAS.U19.5 Generation adequacy evaluation 4 possible capacity states: 0 and C, where C is the maximum capacity o the unit. Table U19.1 shows capacities and corresponding probabilities. Table U19.1: Capacity Table or 1 Unit System Capacity C Probability A 0 U We may also describe this system in terms o capacity outage states. Such a description is generally given via a capacity outage table, as in Table U19.2. Table U19.2: Capacity Outage Table or 1 Unit System Capacity Outage Probability 0 A C U Figure U19.3 provides the probability mass unction (pm) or this 1 unit system. A Probability U 0 C Capacity outaged 2C Fig. U19.3: pm or Capacity Outage o 1 Unit Example

5 Module PE.PAS.U19.5 Generation adequacy evaluation 5 Now consider a two unit system, with both units o capacity C. We can obtain the capacity outage table by basic reasoning, resulting in Table U19.3. Table U19.3: Capacity Outage Table or 2 Identical Units Capacity Outage Probability 0 A 2 C C AU UA 2C U 2 We may also more ormally obtain Table U19.3 by considering the act that it provides the pm o the sum o two random variables. eine X1 as the capacity outage random variable (RV) or unit 1 and X2 as the capacity outage RV or unit 2, with pms X1(x) and X2(x), each o which appear as in Fig. U19.3. We desire Y(y), the pm o Y, where Y=X1+X2. Recall rom Section U that we can obtain Y(y) by convolving X1(x) with X2(x), i.e., ( y) ( t) ( y t) dt Y X X 2 1 (U19.4) But, inspection o X1(x) and X2(x), as given by Fig. U19.3, indicates that, since X1 and X2 are discrete random variables, their pms are comprised o impulses. Convolution o any unction with an impulse unction simply shits and scales that unction. The shit moves the origin o the original unction to the location o the impulse, and the scale is by the value o the impulse. Fig. U19.4 illustrates this idea or the case at hand.

6 Module PE.PAS.U19.5 Generation adequacy evaluation 6 A A Probability U * Probability U 0 C 2C 0 C 2C Capacity outaged Capacity outaged Probability A 2 AU + Probability AU U 2 0 C 2C 0 C 2C Capacity outaged Capacity outaged Fig. U19.4: Convolution o Generator Outage Capacity pms Figure U19.5 shows the resultant pm or the capacity outage or 2 identical units each o capacity C. A 2 Probability 2AU U 2 0 C Capacity outaged 2C Fig. U19.5: pm or Capacity Outage o 2 Unit Example We note that Fig. U19.5 indicates there are only 3 states, but in Table U19.3, there are 4. One may reason rom Table U19.3 that there are two possible ways o seeing a capacity outage o C, either

7 Module PE.PAS.U19.5 Generation adequacy evaluation 7 unit 1 goes down or unit 2 goes down. Since these two states are the same, we may combine their probabilities, resulting in Table U19.4, which conorms to Fig. U19.5. Table U19.4: Capacity Outage Table or 2 Identical Units Capacity Outage Probability 0 A 2 C 2AU 2C U 2 In act, we saw this same ind o problem in Section U10.2 o module 10, where we showed that the probabilities can be handled using a binomial distribution, since each unit may be considered as a trial with only two possible outcomes (up or down). We may then write the probability o having r units out o service as: P X r n! r!( n r)! r ( nr ) Pr[ X r, n, U ] U ( A) (U19.5) where n is the number o units. It is interesting to note we may also thin about this problem via a state-space model, as shown in U19.6 where we have indicated the state o each unit together with the capacity outage level associated with each system state. Note that we are not representing the possibility o common mode or dependent ailures. 1u, 2u 0 out 1u, 2d C out 1d, 2u C out 1d, 2d 2C out Fig. U19.6: State Space Model or 2-Unit System

8 Module PE.PAS.U19.5 Generation adequacy evaluation 8 From Section U16.8 o module 16, since the two middle states o Fig. U19.3 satisy the merging condition (a group o (internal) states can be merged i the transition intensities to any external states are the same rom each internal state) and they satisy rule 3 (two states should be combined only i they are o the same state classiication in this case, the same capacity), we may combine them using rule 1 (when two (internal) states have transition rates that are identical to common external states, those two states can be merged into one; entry rates are added, exit rates remain the same.) Thereore, Fig. U19.6 becomes Fig. U u, 2u 0 out 2 1d, 2u 1u, 2d C out 2 1d, 2d 2C out Fig. U19.7: Reduced State Space Model or 2 Unit System The 2λ transition in Fig. U19.6 relects the act that the 0 out state may transition to the C out state because o unit 1 or because o unit 2, but it does not relect a common mode outage since the middle state is a state in which only 1 unit is ailed. Similarly, the 2 transition in Fig. U19.6 relects the act the 2C out state may transition to the C out state because o repair to unit 1 or repair to unit 2, but it does not relect a common mode repair since the middle state is a state in which only 1 unit is repaired. One may also compute requency and duration or each state in Fig. U19.7 according to (U16.32) and (U16.33), repeated here or convenience: p j j, j (U19.6) j

9 Module PE.PAS.U19.5 Generation adequacy evaluation 9 T j j 1 Table U19.5 tabulates all o the inormation. j (U19.7) Table U19.5: Capacity Outage Table or 2 Identical Units with Frequencies and urations Capacity Outage Probability Frequency uration 0 A 2 2λA 2 1/2λ C 2AU 2AUλ 1/λ 2C U 2 2U 2 1/2 U Capacity outage table or units having dierent capacities Reerence [1] provides a simple example or the more realistic case o having multiple units with dierent capacities, which we adapt and present here. Consider a system with two 3 MW units and one 5 MW unit, all o which have orced outage rates (FOR) o (The act that all units have the same FOR means that we could handle this using the binomial distribution, which would not be applicable i any unit had a dierent FOR). The pms o the two identical 3 MW units can be convolved as in Section U to give the pm o Fig. U19.8 and the corresponding capacity outage table o Table U19.6.

10 Module PE.PAS.U19.5 Generation adequacy evaluation Probability Capacity outaged Fig. U19.8: pm or Capacity Outage o Convolved 3 MW Units Table U19.6: Capacity Outage Table or Convolved 3 MW Units Capacity Outage Probability = (0.98)(0.02)= = Now we want to convolve in the 5 MW unit. The pm or this unit is given by Fig. U Probability Capacity outaged Fig. U19.9: pm or 5 MW Capacity Outage

11 Module PE.PAS.U19.5 Generation adequacy evaluation 11 Convolving the pm o Fig. U19.8 with the pm o Fig. U19.9 results in the pm illustrated in Fig. U19.10, with the corresponding capacity outage table given in Table U19.7. Table U19.7: Capacity Outage Table or Convolved 3 MW Units and 5 MW Unit Capacity Outage Probability = = = = = =

12 Module PE.PAS.U19.5 Generation adequacy evaluation Probability Unit 3 0 MW capacity outage convolved with two 3 MW units pm Capacity outaged Probability Unit 3 5 MW capacity outage convolved with two 3 MW units pm Capacity outaged Probability Resultant inal pm accounting or all three units Capacity outaged Fig. U19.10: Procedure or convolving Two 3 MW units with 5 MW Unit (top two plots) and inal 3 unit pm U Convolution algorithm The procedure illustrated above can be expressed algorithmically, which is advantageous in order to code it.

13 Module PE.PAS.U19.5 Generation adequacy evaluation 13 Two-state model: The algorithm is simplest i we assume that all units are represented using two-state models. Let denote the th unit to be convolved in, A and U its availability and FOR, respectively, and C its capacity. The composite capacity outage pm beore a convolution is denoted by Yold(y), and ater by Ynew(y), so that or unit, the capacity outage random variables are related by Ynew=Yold+X. We assume that there are N units to be convolved. The algorithm ollows. 1. Let =1. 2. Convolve in the next unit according to: y) (U19.8) Ynew ( A Yold ( y) U Yold ( y C ) or all values o y or which Yold(y)0 and/or Yold(y-C) I =N, stop, else =+1 and go to 2. Note that in (U19.8) the inluence o the argument in the last term Yold(y-C) is to shit the unction Yold(y) to the right by an amount equal to C. This corresponds to the shit inluence o the th unit pm impulse at X=C. U econvolution An interesting situation requently occurs, particularly in operations, but also in production costing programs, when the composite pm has been computed or a large number o units, and capacity outage probabilities are ully available. Then one o the units is decommitted, and the existing composite pm no longer applies. How to obtain a new one? One obvious approach is to simply start over and perorm the convolution or each and every unit. But this is time-consuming, and besides, there is a much better way! We have a better approach based on the ollowing act:

14 Module PE.PAS.U19.5 Generation adequacy evaluation 14 The computation o Ynew(y) is independent o the order in which the units are convolved. Consider, in (U19.8), the term Yold(y). This is the composite pm just beore the last unit was convolved in. Given we have Ynew(y), we assume that the last unit convolved in was the unit that we would lie to decommit. It may not have been the last unit, in actuality, but because the computation o Ynew(y) is independent o order, we can mae this assumption without loss o generality. In that case, we may convolve out the decommited unit. How to do that? Consider solving (U19.8) or Yold(y), resulting in: Yold ( y) ( y) U A Ynew Yold (U19.9) ( y C The problem with the above is that the unction we want to compute on the let-hand-side, Yold(y), is also on the right-handside, as Yold(y-C). There is a way out o this, however. It stems rom two acts. Fact 1: The probability o having capacity outage less than 0 is zero, i.e., the best that we can do is that we have no capacity outage, in which case the capacity outage is zero. Thereore any valid capacity outage pm must be zero to the let o the origin. Fact 2: Yold( ) is a valid capacity outage pm. Implication: For values o y such that 0<y<C, Yold(y-C) evaluates to the let o the origin and thereore, since Yold is a valid capacity outage pm, it MUST BE ZERO in this range. As a result, ( y) A Ynew ( y), 0 y C (U19.10) Yold But what about the case o C<y<IC, where IC is the total installed capacity? Here, we must use (U19.9). But let s assume that we )

15 Module PE.PAS.U19.5 Generation adequacy evaluation 15 have already computed Yold(y) or 0<y<C. Then the irst time we use (U19.9) is when y=c. Then we have: Yold ( C ) Ynew ( C ) U A Yold (0) But we already have computed Yold(0) rom (U19.10)! And we will be able to use the values o Yold(y), 0<y<C, in computing all values o Yold(y), C<y<2C. In act, we will be able to compute all o the remaining values o Yold(y) in this way! As an example, try deconvolving one o the 3 MW units rom the capacity outage table o Table U19.7 (which is also illustrated at the bottom o Fig. U19.10). In this case, C3=3, A3=0.98, U3=0.02. The computations are given in Table U19.8. Note that, since Yold(y-C)=0 or y<c, (U19.9) includes the case o (U19.10), and we can express the algorithm using (U19.9) only. The deconvolution algorithm is given below. There is just one step. We assume that we are deconvolving unit. 1. Compute: Yold ( y) Ynew ( y) U A Yold ( y C consecutively or y=0,.,ic such that Ynew(y)0 and/or Yold(y-C)0, where IC is the installed capacity o the system beore deconvolution. 2. Stop. )

16 Module PE.PAS.U19.5 Generation adequacy evaluation 16 Table U19.8: Computations or econvolution Example Capacity Outage y Ynew(y) Yold(y) Yold (0) (0) A Ynew Yold (3) Ynew (3) U 3 A (3 3) Yold (5) Ynew Yold (6) Ynew 3 3 (5) U 3 A (6) U 3 A Yold Yold (8) Ynew (8) U 3 A 3 Yold Yold (8 3) Yold (11) Ynew 3 Yold (11) U 3 A Yold (5 3) (6 3) (11 3)

17 Module PE.PAS.U19.5 Generation adequacy evaluation 17 U Multi-state models We have so ar addressed only the case where all units are represented by two-state models. It may be, however, that we would lie to account or derated units, in which case we need to address the multi-state model as well. This situation presents no additional conceptual diiculty relative to the two-state model, as the pm or each unit will still consist o only impulses, except now, each unit will have a pm consisting o as many impulses as it has states, instead o only two. We do, however, need to generalize the algorithms or convolution and deconvolution. Convolution algorithm or multi-state case: With N the total number o units: 1. Let =1. 2. Convolve in the next unit according to: Ynew n ( y) p j 1 j Yold ( y C j ) (U19.11) or all values o y or which Yold(y) or Yold(y-Cj) are non-zero. Here, n is the number o states or unit ; pj is the j th state probability or unit ; Cj is the j th capacity outage or unit. 3. I =N, stop, else =+1 and go to 2. Note that (U19.11) is the same as (U19.8) i n=2, with A=p1, and U=p2. econvolution algorithm or multi-state case: We again assume that we are deconvolving unit. To determine the deconvolution equation or the multi-state case, rewrite (U19.11) by extracting rom the summation the irst term, according to: Ynew ( y) p n ( y) p ( y C 1 Yold j Yold j j 2 )

18 Module PE.PAS.U19.5 Generation adequacy evaluation 18 where we have assumed that the irst capacity outage state or unit is zero, i.e., C1=0. Solving or Yold(y), we have: Yold ( y) Ynew n ( y) p j 2 p j 1 Yold ( y C We assume that we are deconvolving unit. The algorithm is: 1. Compute: Yold ( y) Ynew n ( y) p j 2 p j 1 Yold j ) ( y C j ) (U19.12) consecutively or y=0,.,ic, and y such that Ynew(y)0, Yold(y-Cj)0, where IC is the installed capacity o the system beore deconvolution. 2. Stop. U19.3 Load model Consider the plot o instantaneous demand as a unction o time, as given in Fig. U Load (MW) Time Fig. U19.11: Instantaneous load vs. time

19 Module PE.PAS.U19.5 Generation adequacy evaluation 19 Although this curve is only illustrated or seven days, one could easily imagine extending the curve to cover a ull year. From such a yearly curve, we may identiy the percent o time or which the demand exceeds a given value. I we assume that the curve is a orecasted curve or the next year, then this percentage is equivalent to the probability that the demand will exceed the given value in that year. The procedure or obtaining the percent o time or which the demand exceeds a given value is as ollows. 1. iscretize the curve into N equal time segments, so that the value o the discretized curve in each segment taes on the maximum value o continuous curve in that segment. 2. The percentage o time the demand exceeds a value d is obtained by counting the number o segments having a value greater than d and dividing by N. 3. Plot the demand d against the percent o time the demand exceeds a value d. A typical such plot is illustrated in Fig. U emand, d (MW) Percent o time 100 Fig. U19.12: Load duration curve Fig. U19.12 is oten generically reerred to as a load duration curve (LC). However, one should be aware that there is a

20 Module PE.PAS.U19.5 Generation adequacy evaluation 20 signiicant dierence between LCs based on hourly segments and LCs based on daily segments. Hourly: the load duration curve indicates the percentage o hours through the year that the hourly pea exceeds a value d. aily: the load duration curve indicates the percentage o days through the year that the daily pea exceeds a value d. Thus, one must realize that the load duration curve gives the percentage o time through the year that the load exceeds a value d, only under the assumption that Hourly: the load is constant throughout the hour at the hourly pea. aily: the load is constant throughout the day at the daily pea. Clearly, the smaller the segment, the better approximation is given by the LC to the actual percentage o time through the year that the load exceeds a value d. Nonetheless, both daily and hourlybased LCs are used in practice. The LC may also be drawn in another way that is convenient or computation. Consider irst normalizing the abscissa (xcoordinate) by dividing all values by 100, so that we obtain all abscissa values in the range o 0 to 1. The abscissa then represents the probability that the demand exceeds the corresponding value d. We denote this probability using the notation or a cumulative distribution unction (cd), F(d). However, one should realize that it is actually the complement o a true cd, i.e., F ( d) P( d) 1 P( d) Here, is a random variable and d are the values it may tae. Finally, we can switch the axes o the LC so that we plot F(d) as a unction o d. Figure U19.12 illustrates the curve, which we reer to as the load model or the given time period.

21 Module PE.PAS.U19.5 Generation adequacy evaluation 21 1 F(d) emand, d (MW) Fig. U19.12: Load shape Note that Chanan Singh in his notes on Load Modeling gives an algorithm or getting the load model rom a single scan o the hourly load data [12]. U19.4 Calculation by Capacity Outage Tables Module U17 identiies the loss o load probability (LOLP) and the loss o load expectation (LOLE) as two indices or characterizing generation adequacy ris. The LOLP is the probability o losing load throughout the time interval (year). LOLE is the number o time units (hours or days) per time interval (year) or which the load will exceed the demand. Fig. U19.13 illustrates a typical load-capacity relationship [1] where the load model is shown as a continuous curve or a period o 365 days. The capacity outage state, C, is shown so that one observes that load interruption only occurs under the condition that d>ic-c. The minimum demand or which this is the case is d=ic-c. Thus, the probability o having an outage o capacity C and o having the demand exceed d is given by the capacity outage pm and F(d), i.e., Y(C)F(d)= Y(C)F(IC-C).

22 Module PE.PAS.U19.5 Generation adequacy evaluation 22 t 365 F (d) 1 C 0 0 F (d ) or t d emand, d (MW) IC Fig. U19.13: Relationship between capacity outage, load model [1] The LOLP is then computed as: N LOLP C ) F ( IC C and the LOLE as: N 1 LOLE C ) F ( IC C 1 Y Y ( (U19.13) ( (U19.14) ) N )*365 1 Y ( C where N is the total number o capacity outage states. ) t

23 Module PE.PAS.U19.5 Generation adequacy evaluation 23 Example: Compute the LOLP and the LOLE or the capacity outage table o Table U19.7, or the load shape curve given by Fig. U Table U19.7 is repeated below or convenience. Table U19.7: Capacity Outage Table or Convolved 3 MW Units and 5 MW Unit Capacity Outage Probability = = = = = = t 365 F (d) d=3 d=5 d=6 d=8 IC=11 emand, d (MW) Fig. U19.14: Load shape curve or example

24 Module PE.PAS.U19.5 Generation adequacy evaluation 24 From (U19.13), we then have: N LOLP ( C ) F ( IC C 1 Y Y Y (0) F (6) F (5) / (11) Y year Y ) (3) F (8) F (8) (3) Y Y (5) F (11) F * * * * * *1 (6) (0) We could compute LOLE using (U19.14), but it is easier to just recognize that LOLE=LOLP*365= *365 = days/year. This means that we can expect to see complete days o load interruption each year, assuming that the pea load per day lasts all day. Another index oten cited is the years/day, in this case, 1/ = years/day. This is the number o years that must pass beore we see a ull day o load interruption. Two important qualiiers should be emphasized: This is the load outage time expected as a result o generation unavailability and does not include the eects o transmission or distribution system components unavailability. This amount o outage time would correspond to the long-run average o this system only i o all 3 units are always committed, i.e., no reserve shutdown, and there is no maintenance o the demand remains at its pea throughout the day These qualiiers are obviously pointing towards inaccuracies in the model and as a result, indicate that the indices computed should

25 Module PE.PAS.U19.5 Generation adequacy evaluation 25 not be perceived as accurate in an absolute sense. However, the indices should still serve well or comparative purposes. U19.5 A capacity planning example Reerence [1] provides an illustrative example showing how the generation adequacy calculation procedure in the previous section can be applied to the capacity planning problem. We adapt that example here. Consider a system containing ive 40 MW units each with a FOR=0.01, so that the installed capacity is 200 MW. The capacity outage table or this system is shown in Table U19.9, where capacity outage states having probabilities less than 10-6 have been neglected. Table U19.9: Capacity outage table or example [1] Capacity Outage Probability The next years s system load model is represented by the load shape curve o Fig. U19.15, which is a linear approximation o an actual load shape curve. Note that the orecasted annual system pea load is 120 MW.

26 Module PE.PAS.U19.5 Generation adequacy evaluation 26 t 365 F (d) 1 P pea emand, d (MW) Fig U19.15: Load shape curve or example [1] The procedure o the previous section was applied and the LOLE and years/day were computed as days/year and 498 years/day, respectively. Certainly this is a very reliable system! The reason or the high reliability is, o course, that the installed capacity is so much greater than the system annual pea. However, the load will grow in the uture, so it is o interest to see how these indices vary as pea load increases. Table U19.10 summarizes LOLE and years/day or the system pea beginning at 120 MW and increasing to 200 MW in units o 10 (this is to just illustrate the eect on the indices; the 10 MW increment should not be interpreted as an annual load growth).

27 Module PE.PAS.U19.5 Generation adequacy evaluation 27 Table U19.10: Variation in LOLE with System Annual Pea [1] System annual pea (MW) ays/year Indices Years/day The LOLE (days/year) is plotted on semi-log scale in Fig. U19.16.

28 Module PE.PAS.U19.5 Generation adequacy evaluation 28 Fig. U19.16: LOLE as a unction o system annual pea load [1] Obviously, we must add some capacity beore we reach an annual pea demand o 200 MW. But at what pea demand level should that be done? The answer to this question can be identiied i we select a threshold ris level beyond which we will not allow. This is basically a management decision, but o course, all management decisions can be acilitated by quantitative analysis. We will orego such analysis here and instead arbitrarily select 0.15 days/year as the threshold ris level. Assume: we have orecasted a 10% per year load growth, we have decided to add one 50 MW unit at a time, each with FOR=0.01, as the load grows, in order to ensure the system satisies the identiied threshold ris level.

29 Module PE.PAS.U19.5 Generation adequacy evaluation 29 The question is: when do we add the units? To answer this question, we will repeat the analysis o Table U19.10, except or our dierent installed capacities: 200 MW, 250 MW, 300 MW, and 350 MW, corresponding to additional units o 0, 1, 2, and 3, respectively. Table U19.11 summarizes the calculations. Fig. U19.17 illustrates the variation in LOLE with pea load or each case, together with vertical lines indicating the pea load value or each year. The unit additions would need to be made in years 2, 4, and 6. The dotted line tracs the year-by-year ris variation. This approach ensures that the stated reliability criteria are met; however, the other inluence to the decision-maing process is, as always, economic. Recall that we assumed that we would solve our capacity problem by adding capacity at increments o 50 MW at a time. It would be quite atypical i this were the only solution approach considered. For example, one might consider larger or smaller increments, or more or less reliable units (dierent FOR). ierent decisions would have dierent inluence on the system ris; they would also have dierent present worth values. The inluence on ris and present worth would need be weighed one against another in order to arrive a good decision. Question: Why would you want to perorm this ind o calculation or a system in which generators are built by electricity maret participants rather than a centralized vertically integrated utility company?

30 Module PE.PAS.U19.5 Generation adequacy evaluation 30 Table U19.11: LOLE Calculations or Example [1] System annual pea (MW) LOLE (days/year) 200 MW 250 MW 300 MW 350 MW

31 Module PE.PAS.U19.5 Generation adequacy evaluation 31 Fig. U19.17: Capacity planning example [1] U19.6 The eective load approach Most o what we have seen in sections U19.1-U19.5 characterize the view taen by [1]. We now provide another view, based on [2]. U Preliminary einitions Let s characterize the load shape curve with t=g(d), as illustrated in Fig. U It is important to note that the load shape curve

32 Module PE.PAS.U19.5 Generation adequacy evaluation 32 characterizes the (orecasted) uture time period and is thereore a probabilistic characterization o the demand. t T t=g(d) Here: d is the system load Fig. U19.18: Load shape t=g(d) d max emand, d (MW) t is the number o time units in the interval T or which the load is greater than d and is most typically given in hours or days t=g(d) expresses the unctional dependence o t on d T represents, most typically, a day, wee, month, or year The cumulative distribution unction (cd) introduced in Section U19.3 is given by t g( d) F ( d) P( d) T T (U19.15) One may also compute the total energy ET consumed in the period T as the area under the curve, i.e.,

33 Module PE.PAS.U19.5 Generation adequacy evaluation 33 The average demand in the period T is obtained rom d avg d d 1 E T T 1 T (U19.16) max max g( ) d F ( ) d (U19.17) 0 Now let s assume that the planned system generation capacity, i.e, the installed capacity, is CT, and that CT<dmax. This is an undesirable situation, since we will not be able to serve some demands, even when there is no capacity outage! Nonetheless, it serves well to understand the relation o the load duration curve to several useul indices. The situation is illustrated in Fig. U t E T d max 0 g ( ) dλ 0 T t=g(d) t C C T d max emand, d (MW) Fig. U19.19: Illustration o Unserved emand Then, under the assumption that the given capacity CT is perectly reliable, we may express three useul reliability indices: Loss o load expectation, LOLE: the number o time units that the load will exceed the capacity,

34 Module PE.PAS.U19.5 Generation adequacy evaluation 34 LOLE t T (U19.18) C g ( C T Loss o load probability, LOLP: the probability that the load will be interrupted during the time period T LOLP (U19.19) P( C ) F ( C ) T T One may thin that, given CT<dmax, then LOLP=1, i.e., the event load interruption during T is certain. The reason why it is not certain is because the load model is probabilistic. So LOLP is simply relecting the uncertainty associated with demand, i.e., the demand may or may not exceed CT, according to F(CT). Expected demand not served, ENS: I the average (or expected) demand is given by (U19.17), then it ollows that the expected demand not served would be: ENS d max F C T ( ) d ) (U19.20) which would be the same area as in U19.19 when the ordinate is normalized to provide F(d) instead o t. Reerence [2] provides a rigorous derivation or (U19.20). Expected energy not served, EENS: This is the total amount o time multiplied by the expected demand not served, i.e., EENS T d max F C T ( ) d which is the area shown in Fig. U U Eective load d (U19.21) max g( ) d The notion o eective load is used to account or the unreliability o the generation, and it is essential or understanding the view taen by [2]. The basic idea is that the total system capacity is always CT, and the eect o capacity outages are accounted or by changing the C T

35 Module PE.PAS.U19.5 Generation adequacy evaluation 35 load model in an appropriate ashion, and then the dierent indices are computed as given in (U19.18), (U19.19), and (U19.20). A capacity outage o Ci is thereore modeled as an increase in the demand, not as a decrease in capacity! We have already deined as the random variable characterizing the demand. Now we deine two more random variables: j is the random increase in load or outage o unit i. e is the random load accounting or outage o all units and represents the eective load. Thus, the random variables, e, and j are related according to: N e j j 1 (U19.21) It is important to realize that, whereas Cj represents the capacity o unit j and is a deterministic value, j represents the increase in load corresponding to outage o unit j and is a random variable. The probability mass unction (pm) or j is assumed to be as given in Fig. U19.20, i.e., a two-state model. We denote the pm or j as j(dj) j (d j ) A j U j 0 C j Outage load, d j Fig. U19.20: Two state generator outage model Recall rom module U13 that the pd o the sum o two random variables is the convolution o their individual pds. In addition, it is true that the cd o two random variables can be ound by

36 Module PE.PAS.U19.5 Generation adequacy evaluation 36 convolving the cd o one o them with the pd (or pm) o the other, that is, or random variables X and Y, with Z=X+Y, that F Z ( z) FX ( z ) Y ( ) d (U19.22) Let s consider the case or only one unit, i.e., rom (U19.21), (U19.23) Then, by (U19.22), we have that: F e ( d e j ) F e ( d ) j ( d (1) (0) ) e e ( j ) F ( ) (U19.24) where the notation indicates the cd ater the j th unit is convolved in. Under this notation, then, (U19.23) becomes ( j ) e and the general case or (U19.24) is: F ( d ) F ( d ( j 1) e ) ( j ) ( j1) ) e e e e j j ( d (U19.25) (U19.26) which expresses the equivalent load ater the j th unit is convolved in. Since j(dj) is discrete (i.e., a pm), we may rewrite (U19.26) as ( j ) ( j1) F ( d ) F ( d d ) ( d ) (U19.27) e e d j 0 e From an intuitive perspective, (U19.27) is providing the ( j 1) convolution o the load shape F ( ) with the set o impulse unctions comprising j(dj). When using a 2-state model or each generator, j(dj) is comprised o only 2 impulse unctions, one at 0 and one at Cj. Recalling that the convolution o a unction with an e j j j

37 Module PE.PAS.U19.5 Generation adequacy evaluation 37 impulse unction simply shits and scales that unction, (U19.27) can be expressed or the 2-state generator model as: F ( j ) e ( d e ) A F j ( j 1) e ( d e ) U j F ( j1) e ( d e C j ) (U19.28) So the cd or the eective load ollowing convolution with capacity outage pm o the j th unit, is the sum o the original cd, scaled by Aj and the original cd, scaled by Uj and right-shited by Cj. Example: Fig. U19.21 illustrates the convolution process or a single unit C1=4 MW supplying a system having pea demand dmax=4 MW, with demand cd given as in plot (a) based on a total time interval o T=1 year.

38 Module PE.PAS.U19.5 Generation adequacy evaluation * F r ( (0) d e ) (a) , j (d j ) (b) C 1 = d e (c) (d) d e d e = F r ( (1) d e ) (e) d e Fig. U19.21: Convolving in the irst unit Plots (c) and (d) represent the intermediate steps o the convolution (0) where the original cd ( ) was scaled by A1=0.8 and F e d e U1=0.2, respectively, and right-shited by 0 and C1=4, respectively. Note the eect o convolution is to spread the original cd. Plot (d) may raise some question since it appears that the constant part o the original cd has been extended too ar to the let. The reason or this apparent discrepancy is that all o the original cd, in plot (a), was not shown. The complete cd is illustrated in Fig.

39 Module PE.PAS.U19.5 Generation adequacy evaluation 39 (0) U19.22 below, which shows clearly that ( ) 1 relecting the act that P(e>de)=1 or de<0. F e d e or de<0, F r ( (0) d e ) d e Fig. U19.22: Complete cd including values or de<0 Let s consider that the irst unit we just convolved in is actually the only unit. I that unit were perectly reliable, then, because C1=4 and dmax=4, our system would never have loss o load. This would be the situation i we applied the ideas o Fig. U19.19 to Fig. U19.21, plot (a). However, Fig. U19.21, plot (e) tells a dierent story. Fig. U19.23 applies the ideas o Fig. U19.19 to Fig. U19.21, plot (e) to show how the cd on the equivalent load indicates that, or a total capacity o CT=4, we do in act have some chance o losing load F r ( (1) d e ) C T = d e Fig. U19.23: Illustration o loss o load region The desired indices are obtained rom (U19.18), (U19.19), and (U19.20) as:

40 Module PE.PAS.U19.5 Generation adequacy evaluation 40 LOLE t g ( C ) T F ( C 4) C e T T T A LOLE o 0.2 years is 73 days, a very poor reliability level that relects the act we have only a single unit with a high FOR=0.2. The LOLP is given by: LOLP P( C ) F ( C ) 0.2 and the ENS is given by: ENS d e T e, max F e ( ) d C T e which is just the shaded area in Fig. U19.23, most easily computed using the basic geometry o the igure, according to: The EENS is given by EENS T d e,max F C T 1 0.2(1) (3)(0.2) 0.5MW 2 e ( ) d d e,max g C T e r T ( ) d or TENS=1(0.5)=0.5MW-years, or 8760(0.5)=4380MWhrs. U19.7 Four additional issues A more extended treatment o generation adequacy evaluation would treat a number o additional issues. Here, we just point to these issues with a brie overview o each so that the interested reader may ollow up on them as desired. The main issues are model uncertainty (U16.7.1), maintenance (U16.7.2), convolution techniques (U16.7.3), and requency and duration approach (U16.7.4). U Model uncertainty We have modeled uncertainty in our analysis o generation adequacy. However, we have assumed that our uncertainty models years

41 Module PE.PAS.U19.5 Generation adequacy evaluation 41 are precise, i.e, the unit FORs and the load orecast used to obtain the load duration curves are both perectly accurate. The act o the matter is that the unit FORs and the load orecast are estimates o the true parameters, and they will always be estimates no matter how much data is collected! Thereore, it is o interest to model uncertainty in the model parameters and then identiy the inluence o these uncertainties on the resulting adequacy indices. One method o modeling parameter uncertainty is to represent each parameter with a numerical distribution. Then repeatedly draw values rom each distribution, and calculate the reliability indices using those values. I the parameter values are drawn as a unction o their probabilities, as indicated by the distribution, then the computed reliability indices will also orm a distribution, rom which we may compute their statistics, e.g., mean, variance, etc. For example, i the pea load is normally distributed, then the distribution may be discretized, and each interval o the distribution can be assigned to an interval on (0,1) in proportion to its area under the normal curve. Then a random draw on (0,1), which is then converted to the pea load value through the assignment, will relect the desired normal distribution. Figure U19.24 illustrates the process Fig. U19.24: Monte Carlo Simulation

42 Module PE.PAS.U19.5 Generation adequacy evaluation 42 This process is called Monte Carlo simulation (MCS) and is almost always an available option or computing reliability indices under parameter uncertainty. The advantage to MCS is that it is conceptually simple to implement. The disadvantage is that it is computationally intensive. Load orecast uncertainty: There are two basic methods. The irst, well articulated in [1], is the most computational but the easiest to understand. The approach is to model the pea load using a discretized normal distribution, as shown in Fig. U19.24, where the mean o the distribution corresponds to the orecasted load Standard deviations rom mean Fig. U19.25: Modeling o load uncertainty [1] The load shape curve is adjusted or each o the load values corresponding to the seven standard deviations rom the mean (-3, -2, -1, 0, 1, 2, 3), where 1 standard deviation is estimated based on the load orecasting program used and the amount o time over which the orecast is being done. A reasonable value could be 2%, or example. Then the indices are computed or each dierent load shape and composite indices are computed as a weighted unction o the individual indices, where the weights are the probabilities given in Fig. U

43 Module PE.PAS.U19.5 Generation adequacy evaluation 43 A second method is given in [1] but perhaps more thoroughly described in [2]. The basic idea is that a single cd is constructed that relects the uncertainty o the pea load orecast, using F (0) ( d) F (0) ( d ) ( ) (U19.29) Once this cd is obtained, the indices are computed using one o our standard approaches. It is important to realize that modeling o uncertainty in load orecast always results in indices relecting poorer reliability because the rate o increase o the indices is nonlinear with pea load, in that it is higher at higher load levels than at lower load levels. FOR uncertainty: Reerences [1, 4] address inclusion o FOR uncertainty using a covariance matrix corresponding to the capacity outage table. The method is based on [5]. One important conclusion rom this wor is that although FOR uncertainty certainly aects the distribution o the reliability indices, it does not aect their expected values. This is in contrast to load orecast uncertainty. U Maintenance The conceptually simplest method or including unit maintenance is through the capacity outage approach according to the ollowing: 1. Compute a ull capacity outage table. 2. ivide the year into Ny intervals and obtain a unique load shape cd Fp(d) or each period p. 3. For each interval p=1, Ny a. Identiy the units out on maintenance in this interval b. econvolve each outaged unit rom the capacity outage table to get a capacity outage table or period p, using the

44 Module PE.PAS.U19.5 Generation adequacy evaluation 44 algorithm o Section U enote the resulting capacity outage pm as Yp(y). c. Compute the LOLE or period p as (similar to (U19.14)): LOLE p N p ( C ) F ( IC C 1 Yp p )* N days N 1 Yp ( C ) t p (U19.30) where Np is the total number o capacity outage states or period p and Ndays are the number o days in period p. 4. The annual LOLE is then given as the sum o the LOLEp, i.e., LOLE N Y LOLE p p1 (U19.31) U Convolution techniques We have seen that convolution plays a major role in both the capacity table approach and the eective load approach. The convolution method illustrated or both approaches is called the recursive method. One drawbac o this method is that it is quite computationally intensive and can require signiicant computer resources when it is used or systems having a large number o units and/or units with a large number o derated states. As a result, there has been a great deal o research eort into developing aster convolution methods. This wor has resulted in, in addition to the recursive method, the ollowing methods [3]: Fourier transorm [6] Method o cumulants [7] Segmentation method [8, 9, 10] Energy unction method [3] O these, the method o cumulants is very ast, and the recursive method very is accurate. The segmentation method is said to achieve a good tradeo between speed and accuracy. Note that

45 Module PE.PAS.U19.5 Generation adequacy evaluation 45 Chanan Singh summarizes the method o cumulants in his notes [12]. U Frequency and duration approach The methods presented in this module so ar provide the ability to compute LOLP, LOLE, ENS, and EENS, but they do not provide the ability to compute Frequency o occurrence o an insuicient capacity condition The duration or which an insuicient capacity condition is liely to exist. A competing method which provides these latter quantities goes, quite naturally, under the name o the requency and duration (F&) approach. The F& approach is based on state space diagrams and Marov models. We touched on this at the end o Section U above by showing that we may represent a 2 generator system via a Marov model and then compute state probabilities, requencies, and durations or each o the states. The underlying steps or the F& approach, outlined in chapter 10 o [11], are: 1. evelop the Marov model and corresponding state transition matrix, A or the system. 2. Use the state transition matrix to solve or the long-run probabilities rom 0=pA and pj=1 (note that we have dropped the subscript or brevity, but it should be understood that all probabilities in this section are long-run probabilities). 3. Evaluate the requency o encountering the individual states rom (U16.31), repeated here or convenience: j j p j, p j, j (U19.32) j j which can be expressed as: j=pj,[total rate o departure rom state j]

46 Module PE.PAS.U19.5 Generation adequacy evaluation Evaluate the mean duration o each state, i.e., the mean time o residing in each state, rom (U16.33), repeated here or convenience: T j 1 p j, (U19.33) j j j (Note that [11] uses mj to denote the duration or state j and uses Tj to denote the cycle time or state j, which is the reciprocal o the state j requency j. One should careully distinguish between the cycle time and the mean duration. The cycle time is the mean time between entering a given state to next entering that same state. The duration is the mean time o remaining in a given state.) 5. Identiy the states corresponding to ailure, lumped into a cumulative state denoted as J. 6. Compute the cumulative probability o the ailure states pj as the sum o the individual state probabilities: p (U19.34) p J j jj 7. Compute the cumulative requency J o the ailure states (see section U16.8.2) as the total o the requencies leaving a ailure state j or an non-ailure state : J j (U19.35) J jj Because (see (U16.29)) j=λj pj,, (U19.35) can be expressed as

47 Module PE.PAS.U19.5 Generation adequacy evaluation 47 J J jj j p j, jj J jj p j p j, J j, 8. Compute the cumulative duration or the ailure states, as: p J j (U19.36) J T J (U19.37) The above approach is quite convenient or a system o just a very ew states, and it is important or our purposes because it lays out the underlying principles on which the F& is based. However, or a large system, the above approach is not very useul because o step 1 where we must develop the Marov model. This diiculty is circumvented by building the capacity outage table using recursive relations or the capacity outage (e.g. state) probabilities together with additional recursive relations or state transitions and state requencies [1, 2, 4, 11]. Reerences [1] R. Billinton and R. Allan, Reliability Evaluation o Power Systems, 2 nd edition, Plenum Press, [2] R. Sullivan, Power System Planning, McGraw Hill, [3] X. Wang and J. Mconald, Modern Power System Planning, McGraw Hill, [4] J. Endrenyi, Reliability modeling in electric power systems, Wiley and Sons, New Yor, [5] A. Patton and A. Stasinos, Variance and approximate conidence limits on LOLP or a single-area system, IEEE Trans. on Power Apparatus and Systems, Vol. 94. pp , July/August [6] N. Rau and K. Schen, Application o Fourier Methods or the Evaluation o Capacity Outage Probabilities, IEEE PES 1979 Winter Power Meeting, paper A [7] N. Rau, P. Toy, and K. Schen, Expected energy production costs by the method o moments, IEEE Trans. on Power Apparatus and Systems, vol. PAS-99, no. 5, pp , Sep/Oct., [8] K. F. Schen, R. B. Misra, S. Vassos and W. Wen, A New Method or the Evaluation o Expected Energy Generation and Loss o Load Probability, IEEE Transaction on Power Apparatus and Systems, Vol. PAS-103, No. 2, Feb [9] Y. ai, J. McCalley, and V. Vittal, Annual Ris Assessment or Thermal Overload, Proceedings o the 1998 American Power Conerence, Chicago, Illinios, April, [10] Y. ai,, Ph.. issertation, Iowa State University, [11] R. Billinton and R. Allan, Reliability evaluation o engineering systems, 2 nd edition, Plenum Press, New Yor, [12] C. Singh, Electric Power System Reliability Course Notes.