1052 IEEE Transactions on Power Systems, Vol. 11, No. 2, May 1996

APPLICATIONS OF CLUSTER ANALYSIS IN MONTE CARLO PRODUCTION SIMULATION

S. R. Huang Member, IEEE

Electrical Engineering Department Feng Chia University Taichung, Taiwan, R.O.C.

Y. W. Lin

Abstract - The paper describes a computer algorithm designed for Monte Carlo production simulation. The design seeks to enhance the precision of production cost estimation at a reduced computation time of the existing approach. The techniques included are cluster sampling and proportional estimate. The algorithm has been evaluated and compared numerically with the existing approach. The effectiveness on precision improvement is demonstrated in this paper.

Keyword: Production Cost Simulation, Monte Carlo Simulation, cluster Sampling, proportional Estimate

1. INTRODUCTION

Production simulation has the ability to estimate future energy production of power system generating units and their associated cost, fuel consumption and pollutants emission. Because of this ability, production simulation is an essential step to evaluate: (1) future generation expansion program [I, 21, (2) the maintenance schedule of system generating units [3], and (3) the cost/ benefit of load management program [4].

There are two approaches to production simulation: analytical and Monte Carlo simulations. The analytical approach [5-13] relies on a convolution process which convolves these random variables of outage capacity with the random variable of system load. The hourly load distribution is expressed into a probability density function. By this expression, the chronological information of system load is destroyed. Hence the analytical approach has difficulty in simulating the chronological constraints often imposed on generation scheduling. Without evaluating the chronological constraints, the simulation normally results in an underestimation of system production cost.

In the Monte Carlo simulation [14-16], a large population of (N) system states are specified by random draws to simulate the chronological random outage events of system generating units. Each state so specified represents one possible realization of the availability of system generating units over the study period. The system production cost estimated by applying the unit commitment, including economic dispatch, is applied to n (with n<<N) sample states within this population.

On basis of the n production costs calculated, the desired cost value, the population mean of the N unit commitment production costs, is estimated. By unit commitment, the chronology of load and power generation is thus preserved. However, sampling introduces imprecision to the estimation. To enhance the precision or to reduce the estimation variance, a Monte Carlo production simulation algorithm of applying the

95 SM 590-0 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the 1995 IEEE/PES Summer Meeting, July 23-27, 1995, Portland, OR. Manuscript submitted December 2, 1994; made available for printing April 28, 1995.

stratified and antithetic samplings as well as the linear regression estimation techniques has been developed by previous authors [17, 181. Their algorithms are generally effective, but have two limitations when applying to a real utility system (such as the Taipower system described in this paper). First, their stratification is based on engineering judgment, which has difficulty in applying in a complex system. Second, their estimation by linear regression requires preeevaluation of the relative correlation factor, which is computationally expensive, and the evaluatcd correlation factor could sometimes cause a high variance of estimation.

To overcome these limitations, a new algorithm, comprisin the following two advanced elements, is proposed: (i mathematically proved cluster for sampling, and (ii estimation of the population mean by proportional estimate.

In the following section, the structure of the algorithm is outlined. Then sections 2, 3, and 4 present the proposed techniques for cluster sampling [19 - 241, population mean estimation [19], and regression estimator. In section 5, numerical test results are examined.

'I

2. STRUCTURE OF MONTE CARLO PRODUCTION SIMULATION

The proposed algorithm is outlined in Fig. 1. As shown, first, a large number of combinations of system's available generating units are enumerated by random number generators to be described in Appendix A. For each combination or availability state, the load curve (LC) type of scheduler is applied to providing the hourly generation schedule during the evaluation period. Then cluster sampling techniques to be detailed in subsection 2.2 are used to sample n effective availability states. For each state, the conventional unit commitment including economic dispatch is applied to calculating the cost. On basis of the simulated results in blocks (C) and (D), a statistical estimation procedure to be described in sections 3 and 4 is applied to estimating true mean.

1 Formulate N Generation I (A) Avai lab i 1 i t y S t a t e s

. For Each States ,Evaluat Cost by LC Type

Sampling f o r n Effect ive Combinations

Evaluates Cost by Unit Commi tment . Including 1 (D) Economic Dispatch

I (E) I S t a t i s t i c a l Estimation f o r True Mean of Cost

Fig. 1 General s t ruc ture of proposed algorithm

0885-8950/96/$05.00 0 1995 JEEE

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step 1: Examine all pairs of adjacent groups and find those two which are closest together, the Euclidean distance between their nearest members.

step 2: Repeat step 2 until there is only one group.

An artificial example of the single-linkage method is shown in dendrogram form in Figure 2 for the data in Table 1. For instance, the nested tree structure of a dendrogram suggests that many different groups may be present in the data, and the obvious question is where to "cut" the tree so that the optimal number of groups is found. Assume L=2. Following the single linkage clustering method, which yields the clusters boundaries presented in Fig. 2.

(2) Number of Clusters

In our algorithm design, because at the cluster stage, the sample size n has not been decided yet, thus without sampling, L is selected in our algorithm by evaluating [19]:

2.1 Basic COI"t Of cluster %UDling

Following the cluster sampling theory, population Z is divided into L nonoverlapping subpopulations, called clusters. Let the population under consideration be divided into L clusters and a cluster simple random sample of size n be drawn from it. Then the sample size in the h-th clusi,ei- is nh so that

C nh=n. If z is the variate under study, an ui-tbiased estimate L

h = l of the population mean is given by

where w h is the proportion of units in the h-th cluster and Zh is the sample mean based on nh units drawn from the h-th cluster. The variance of the estimate & id found to be

To achieve the maximum precision, the cldster design that minimizes V(&) is desired. From Eq. (2), it is clear that the problem of cluster involves the simultaneous determinations of i) construction of L clusters, (ii) the numbei of clusters and iii) the sample allocation {nh}.

3.2 Prouosed Cluster Process

Our proposed cluster process comprises three major steps: 1 construction of the clusters L, 2 choice of the number of clusters L, and 3 choice of the sample size nh be taken from the h-th

cluster. 1 1 (1) Construction of Clusters

The first' step in using cluster analysis is to choose a measure of distance or similarity between the cases. For example, consider the Euclidean distance coefficient [21-241

(3)

where djk is the distance between cases j <ind k, n is the number of attributes, and xj is the v lue of the jth case. This

throughout this section. The following presents the step-by-step computational

procedure. Appendix B gives an artificial example to demonstrate each calculation step numerically.

measure seems to be most natural an i will be adopted for use

where fJh2 is the true variance of the population in cluster h. In this algorithm, VDRL is evaluated iteratively, with L=2 assumed in the first iteration and L=L+l in the subsequent iterations. Our piroposed algorithm suggests that number of clusters (L) be 13elected at the lowest decreasing rate of

CWgoh2 [19]. T , J e the same test problem demonstrated in Table 1 as an example, the number of clusters, L, is selected as 2 according to VDRLts in Table 2.

L

h = l

(3) Cluster After Selection of the Sample

Given a total sample size n, the proportional allocation yields the V(Y,t) [ 191:

(5)

where Wh=Nh/N as defined in Eq. (1). In our algorithm, Eq. (5) is applied to the selection of sample size n at a pre-desi nated estimation precision (i.e. V(yst) denoted by V in Eq. (2$), shown as follows:

(4) Optimum Sampling Allocation

In cluster sampling, the values of the sample sizes (nh) in the respective clusters are chosen by the sampler. The same as the existing approach to Monte Carlo production simulation, proportional allocation 1191 is adopted by our algorithm to decide the population sample size (n) at a pre-specified

Table 1 The data matrix Case

1 2 3 4 5 Attribute I 35 I 34 I 30 j 22 :1m

Table 2 Selected number of c lus te rs by c 1 usiter ing popu 1 a t ion variances in eiramp 1 e prob 1 em.

Variance Variance clusters(L1 decreasing

r a t e (VDR, )

Case Figure 2 Tree proposed by using the signle-linkage clustering

method on the artificial datamatrix Shown in Table 1.

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Table 4 Population mean estimation of a r t i f i c i a l '.'5 data matrix 1.90"

estimation precision (i.e., V(vst)). Then the proportional allocation [19] is followed for the choice of nh:

40 45 24 28 -- cluster2 09

O 2 18 22

W hbh L

h = I

nh=n.

Whbh (7)

where bh is the subpopulation mean of cluster h. Take the example problem in Table 1 for demonstration. With the V(yst) pre-specified at 4.5, a sample size n=3 is calculated by substituting both Nh and Uh of Table 3 into Eq. (6). Then sample sizes (nh) are calculated from Eq. (7), which yield nl='L, nz=l(see Table 4).

3. PROPOSED ESTIMATION BY PROPORTIONAL

In the proportional estimate method, an auxiliary variate Y correlated with Z is obtained for each unit in the sample. Following the proportional estimate, the estimate of the mean of population Z, denoted by zst [19], is expressed by

EST I M A T E

nhi where wh=nh/n is the weight Of cluster h, .&,=(l/nh)*.c Z h i

is the estimate of cluster mean of cluster h. For the same problem as the proceeding sections, the estimated cluster means and population mean are tabulated in Table 4.

1: 1

4. LINEAR REGRESSION ESTIMATOR

The linear regression estimator for production cost estimation proposed by [lo] is adopted. Before applying this estimator to estimating the expected unit commitment production cost of a year, coefficient /3 defined in [lo] and given in Eq. (9) below is evaluated for a typical week of each individual season. With /3 known, the expected unit commitment production cost of any specified simulation period within the year can be estimated by the mean of xss(p) defined as :

xss( P) =zss-P {Y ss-vss } (9)

In Eq. (9), zss and yss are the unit commitment and LC simulated production costs, respectively, both evaluated on the cluster sample states.

5. NUMERICAL TEST

The algorithm presented in Appendix A has been tested in the actual Taipower system. The effectiveness on variance reduction of our proposed cluster rule and state formulation by forced outage rate is examined. This section contains the test results obtained from a simplified Taipower system (presented in the Appendix C).

5.1 Test Cases

The following subalgorithms will be evaluated: (1) Cluster sampling (by our proposed rule) combined with

proportional estimate. (2) Stratified sampling (by baseload unit rule) combined with

linear regression estimator - subalgorithm (1) having our proposed cluster rule replaced with the conventional rule presented in [ZO].

5.2 Results DescnDtion

5.2.1 Cluster Sampling Combined with Proportional Estimate The algorithm is section 3 was applied to estimating the

production cost of the evaluation period. The computation results are summarized below:

stage 1: Monte Carlo simulation of units' outage combinations (with population size N-50).

stage 2: Approximate generation scheduling - The selected 50 production cost values are summarized in Fig. 3.

stage 3: Cluster sampling - With the left-side of Eq. (5) pre-specified at 8x1010, the sample size n as well as nk in the individual cluster is decided. The step-by-step results are given in Fig. 4 and Tables 5 - 7. As shown, the selected number of cluster (L) is 4, and sample size (n) is 10.

stage 4: Conventional unit commitment including economic dispatch.

stage 5: Population mean estimation - Referring to Table 8, the estimated mean (&) is 1.96309.107 US$. Take the population mean (1.9657~107 US$) as reference, the estimation error is 0.1323%.

With cluster sampling, the estimation variance decreases from the original population variance (7.6223~1011 US$') to 9.9238~109 US$2.

5.2.2 Stratified Sampling Combined with Linear Regression

The algorithm is section 4 was applied to estimating the production cost of the evaluation period. The computation results are summarized below:

Estimator

stage 1:

stage 2:

stage 3:

stage 4:

stage 5 .

stage

Monte Carlo simulation of units' outage combinations (with population size N=50).

Approximate generation scheduling - The selected 50 production cost values are summarized in Fig. 3.

Stratification sampling - As to the stratification by the conventional rule, the selected L=3(i.e., no outage of base-load units, one outage unit, two outage units, etc.). However, according to our computational experience on the test problem, the probability is extremely low for th: cases having more than two outage units. rherefore, by conventional stratification, L has been selected at L=3 in our numerical tests.

With the left-side of Eq. 5) pre-specified at 8x1010,

stratum is decided. The calculation results are given in Table 9. Conventional unit commitment including economic

the sample size n as we \ 1 as nk in the individual

dispatch. 6: Apply Eq. (9) to estimate the desired population

mean, assuming p in Eq. (9) has been calculated as stated in section 4. The estimation results are given in Table 9.

5.3 ComDarison of Two Production Cost Estima.tions

Referring to the test system in Table 11, units 1 through 6 are the base-load units. Due to the conventional stratification by the outage combination of base-load units, it seems adequate to select L=3(i.e., no outage of base-load, one outage unit, two outage units, etc.). As to the effectiveness on variance reduction, our proposed rule can always give better results than the conventional rule. Referring to Tables 8 and 10, the mean variance estimated by the proportional estimate

case

07 12 05 50 01 44 43 03 32 36 21 34 04 29 38 E _ _ 35 31 22 18 02 45 28 40 09 24 06 15

ii 14

26 IT 39 4 1 30

%P

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is less than that estimated by the linear regression ap roach. The minimum variance according to Eq. 6) is 1.72% lolo( US$2) by the conventional rule, and 9.9238.1 O9 (US$?) by our proposed rule. In theory, the variance of the regression estimate is smaller than that of the nonstratified estimate [19], which can be explained by Eq. (4). Referring to Eq. (8), a weight factor is given to each cluster, which is calculated from the proportional estimate. Among the random sample values drawn out of the same cluster, those deviating from the cluster mean will be weighted less than the sample values close to the cluster mean. In contrast, by linear regression, equal weights are given to all the random sample values drawn out oi the same stratum. The result shows that with the optimum choice the cluster proportional estimate has

a smaller population variance (i.e., C &,<<d, and h = l C U$,<<

U:). The estimator by proportional estimate can avoid the identification of regression model and save computation time. Besides, relying 011 engineering judgment, the stratification results by conventional approach are not so homogeneous as in Table 9. Table 9 explains the reason (ng=l) for selecting L=3 by applying the conventional rule to cost estimation.

L L

h = l

Table 5 Selecte number of c lus te rs by clustering oooulation variances

I Number of I Variance I Variance decreasing I c lus te rs (Gt, 1 0 ~ 1 0 Us$2) rate(VDRL)

4.9589 6.9535

0.6822 1.0021

0.5854 0.2754 2.1255

Table 6 Boundaries evaluated by the single linkage clustering method of t e s t proble

I Frequence (Nh) Interval I coy(h ) I (10E7 US$) 1.814305-1.855295 17 1.888155-1.931890 1.961730-2.019300 2.054485-2.094710

Table 7 Sample size allocation of the test problem

I I I , I I I ----t-- 0.05 0 . 1 0 0.15 0.20 d.25 0.30

Minimum distance between groups (10E7 US$)

Fig.4 Tree produced by applying single linkage clustering method to the data show i n Fig.3

t 0.35

I I I 1 I

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6. CONCLUSIONS

In this paper, a computer algorithm combined with variance reduction techniques to enhance the precision of Monte Carlo production simulation is presented. Numerid test results of the algorithm in Taipower system were examined. Speci-Eic conclusions arising from this work can be summarized as follows: (1) Our proposed stratificatian rule can always give better

variance reduction estimates than conventional stratification rule.

(2) Following our proposed proportional estimate for production cost estimation, users need not precalculate the correlation factor which is, however, needed in the linear regression estimation approach. This pre-calculation process requires calculatifig the unit commitment production cost for all the (N) states within the population, which thus is computationally expensive. Further, for the case with a low correlation, the linear regression approach would yield a high variance of estimation. With mutivariate consideration, there are easiness in applying simpling and estimate techniques.

Application of our algorithm to Tapower's fuel budgeting

(3)

is being conducted by the authors.

7. REFERENCES

[l] R.T. Jenkins and D.S. Joy, "WIEN Automatic System Planning Package (WASP) - An Electric Utility Optimal Generation Expansion Planning Computer Codes," Oak RidFe National Laboratory, Report ORNL-4945, 1974. [a] K.F. Schenk and S. Chan orporation and Impact of a Wind Energy Conv System in Generation Expansion Planning," presented at IEEE Power Engineerinc Societv Summer Meeting, 1981.

Table 9 Strutum boundaries evaluated by conventional u n i t outage rule on test problem in Fig.3

Remarks : (a) Mean of production cost evaluated by load currve : <,=l. 9333 10E7 US$

(b) ,B=0.9513 ( c ) Estimation error : 0.4436%

N.S. Rau, P. Toy, and K.F. Schenk

Sage Publications, 1984.

1993. 1241 B.S. Everitt, Cluster Analysis,

3 4

3-th 13di t ion, Wiley ,

- 1 - 1 - 1 - 8.1101 - I - I -

8. APPENDIX

A. STATE FORMULATION BY FORCED OlJTAGE RATE

Because the simulation period under evaluation always leads the present time, by months or even by years, the random number generator (RNG) used to decide units' up/down states at t = l , 2, 3, + . ., T. State formulations by forced outage rate (FOR) for the M weeks itre exactly the same. Let sjm denote the up/down status of unit j in week m.

step 1: Let j=1. step2:

step3:

Generate a random number and denote R within the interval (0, 1) by the uniforinly distributed RNG. If R<FOR, set sjpl=O; otherwise, sjm=l. Let j= j+ l . If j<J (unit numbers). go to step 2; otherwise, evaluate the desired production attribute by load curve simulation, and stop.

B. ARTIFICIAL EXAMPLE TO DEMONSTRATE THE COMPUTATIONAL PROCEDURE IN SINGLE LINKAGE METHOD

Table 1 shows the data of an artificial data matrix to be used to demonstrate our proposed clustering method computational procedure numerically.

step 1: Merge clusters 1 and 2, giving (12), 3, 4, and 5 at the value of d12=[(35-34)' 1/2= 1.0. Table B.l shows the value of initid Euclideah distance coefficient for each possible set of clusters, and dlz=l.O is the smallest. The results is shown in Table B .2.

Table B.l Initial Euclidean distance coefficient matrix 1 2 3 4 5

1 2 3 4 5

C. A SIMPLIFIED TAIPOWER SYSTEM

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step 2: Next we revise the initial Euclidean distance coefficient matrix by calculating d3 (12) , d4( 12) , and d5 (12) t,ranscribing the remainin initial Euclidean distance coefficients matrix from the initial Euclidean distance coefficients matrix to the second Euclidem distance coefficients matrix. We calculate the following:

5 6.00 2.00 - ( la) -1

step 3 Merge clusters (la), and 3, giving (45) and (113) at the value of d3(12)=4.00. The results is shown in Table E1.4.

Table B.3 Third Euclidean distance coefficient matrix

3

step 4 Merge clusters (45) and (123), giving (12345) at the value of d ( 45) 123) =6.00.

Table B.4 Fourth 13uclidean distance coefficient matrix

Table 10 Characteristics of Taipower's thermal generation system in July 1987

b " l i U e l Y (Cch/h) (Gca?/YYh) ( G c a l F b h ) (GcalA/IYah) $ij $pf ?:%/E%) A ~ % g e c ~ ~ : ' :::% ::?% "E::: up '"' ( h r . ) (hr ) type

mF. ' 2 u9a b 6080 15 5750 0 1 716 80 Coal 2.095 8 6228 1 8 16 80 Coal

6 4000 2 091

1 113.2 0.1938t-3 u.BYa1t-6 220 500 2 110.2 0 1936f-3 U 5Y a1116 220 500 5750 0 1

2 093 U620 0 Ob 71F 7YU 80 70 Goai L 8 6080 20 173 (16 BU Coal

'I 110 2 -0 1938L.3 0 5Y5lk-6 242 550 JlJV U 1

0 3T34t-2 6 be25 U uti 7Yu i D m a l

3 12u 1 1 3Ta U.3834L-2 - U 4 ~ 2 8 c 3 135 JUU

0 1936t-3 0 a 113 aYolt-6 242 560 ~6 14 0150 U 1 6 120.; - U 4 i m L . 5 133 sou 8.5445 20 ,2614 15

~

1006 0.08 790 74 Coal I 10 4312 0 oal

7 140 5 2 042 - 0 lTlOC3 0 .5269C6 80 200 12 0546 28.7086 10 U M Y 2 U 2 .110 0 5000E-6 0 .21VUG6 6 5 140 12 1050 2b ?gyg . O B 2716 144 C 9 67.22 2.265 6 12 b62j ti os 790 70 coal-

12.Y900 30 6 15 i u Coal 0 4500110 0 2 bOilk-6 131 3UU 12.YYO0 2Y 44 15

6.323 0.08 79U 1 G 4 u Z G 05 ~ 1 1 6 144 Coal

b 15 091

io 5 2 . l i 2.2m - @ 42aJL.3 o 7 7 B L 5 110 300 .6ola 11 99 65 2 0 6 3 - 0 4 2 i Z b I U .14IY116 57 80 12 1265 33

1 3 99 li 5206 3- I 'D U11 12 31.83 2 ,254 - 0 3077tr2 0 .1844114 68 140 15.3715 36 1zF5 Zti ia 0.06 r r l 6 144 L

1175 U u t l6Ul i a U 1 1 Y 8 i j C.07 1602 .65 2 Ok.5 - 0.42'211 4 0 14"YL - 6 120 500 I 3903

14 140.5 2 OFI .O J OUU l i .5988 3- a.= 7 575 0.0,; 9 i 2 49 Com

16 8Y.58 ,5 U 1 1 1 5 80.10 1 510 0.8100C3 - 0 6601f-6 100 580 17.5288 39 0448

U 4000G5 0 2 E 2 55 U11 l b 70 41 s l JY3i 0 05 9 , 2 4 Y F.?-

l i .3413 40 .lii& 1; 862. 0.0, 160 17.5113 4U rr;5

li 12U 0 2 1E 3 U O L - 6 12 3 00u - i a

0 d 5 0 0 G g 0 2JOOL 6 L)4 3 n 12 b 6 2 ~ 0.00 0 ut 1imI gS1 49 I O fl ULr 10 i z 44

13 1150 U 05 27 16 1 4 4 ti oai

11 641i 4U 3355 17.5413 4 G . K D C

7s 2 211 0.4aOUtr5 U .2aUUtr6 94 3

19 , r 6 dl 2 D f l

15 aZ,a 43.0000 2 1EZ -0 1 i l U t - 3 U 526YL.6 125 5011

21 10 16 2.616 -0.1412t-1 0.284Yt-3 26 42 6 0 0 Sa ti I.

34 4i50 6 2 S323 (r 1 72 L . I . a72 U U7 1uo2

10 35 Oa5O 81 7813

1 111 J.l u,,u 8 4 G40b 0 0 ( 0 L I . 373 0.UI 1 w 2

1 118 1 l l r

i l l 0 0.07 IUUS -- '

0 1 1 246 34 o m Is; gS13 a75 0 U8 7YU 7u L . 1 1 76 9 W21L-5 1 280 34 100 4098 an5 0 Cl i lUU2 1 76 - U 1a21t-5 1 390 34.2500 1 0 Y . T E E a i > (I 0- r) L t I 4" G I

G G ' l V t r 3 U - 2 m 6 12

2.110 0.5OGOtr6 U.210UL--6 125 500 17 5266 JY 3001 i n l a 0.07 1002

27716 144 Coal 1150 22 16.08 2.094 0 l r J M E - 1 U.304,t-3 24 42 13 5'173 43 m a

I r 3 b 24 U.2997b 2 U 1 ( 0 6 . 1 I U 38 24 0 29911;2 C' 1 - .

1058

5 . 6 5.4 5.2 5 . 0 4.8 4.G 4.4 4.2

(1.0 3.8 3.6 3.4 3.2 3.0 2.8

0 20 110 60 80 100 120 140 160 Hour

Flg 5 Load curve during 21-27 July, 1987

9. BIOGRAPHIES

Sp-Ruen Huan received his B.S. degree from Feng ChiagUruversity, his M.S. degree from National Tsing Hua University of Taiwan, and Ph.D degree from the same University, in 1988, 1988 and 1993, respectively. In 1993, he joined the faculty of Feng Chia University, Taiwan, where he is currently an associate professor. His interest is in power generation planning.

Ya-Whei Lin was born in 16, 1966. She received her E.E. Department of Feng Taiwan. She is currently a E.E. Department of Feng Taiwan. Her interest is in planning.

Taiwan, on Oct. B.S. degree from Chia University, M.S. student of

Chia University, power generation