Monte Carlo Analysis of Real-Time Electricity Pricing for Industrial Loads By Dr. Carl J. Spezia

Transcription

1 Volue 25, uber 3 - July 29 through Septeber 29 Monte Carlo Analysis of Real-Tie Electricity Pricing for Inustrial oas By Dr. Carl J. Spezia Peer-Referee Applie Papers Keywor Search Electricity Energy Research The Official Electronic Publication of The Association of Technology, Manageent, an Applie Engineering 29

2 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 Monte Carlo Analysis of Real-Tie Electricity Pricing for Inustrial oas By Dr. Carl J. Spezia Carl J. Spezia is an Assistant Professor in the Electrical Engineering Technology Progra in the Departent of Technology at Southern Illinois University Carbonale (SIUC). e joine the progra in998 as a Visiting Assistant Professor. e has eight years of utility experience as a power systes engineer an is a license professional engineer in Illinois. is inustrial assignents inclue power syste oeling, power systes protection, an substation esign. e receive his M.S. an Ph.D. fro SIUC in 99 an 22 respectively. e teaches courses in electric power an achinery, inustrial autoation, an electric circuits. e won outstaning epartental teaching awars two of the last five years. is research interests inclue power systes econoics, power arkets, an electric energy anageent. Abstract Maintaining a copetitive inustrial business eans close cost control. Free arket wholesale electricity suppliers offer a variety of purchase agreeents. Suppliers now offer real-tie pricing to all custoer classes with the proise of potential savings. Real-tie price tariffs irror the ynaic nature of the wholesale electricity arket. This arket exhibits high price volatility ue to constantly changing ean an lack of large-scale electricity storage technology. Businesses ay benefit fro these arket changes epening on their loa characteristics. This paper uses Monte Carlo analysis to eterine the potential econoic benefit of aopting a real-tie price rate uner ifferent loa paraeters. Sections of this paper introuce loa oels for electric ean, efine a benefit oel, evelop an expression for equivalent break-even fixe rates, exaine paraetric variation, an recor siulation results. Siulations show that businesses with a high loa factor have a greater probability of profiting fro real-tie price tariffs without aopting any loa control strategy. Businesses with low loa factors require higher equivalent break-even rates to benefit fro realtie price tariffs. Introuction A copetitive inustrial business ust pay attention to all prouction costs an take avantage of all process iproveents that provie potential savings. Free arket electricity increase the nuber of energy suppliers an create opportunities to reuce electricity costs. These suppliers offer a wie variety of service agreeents that ay reuce custoer costs when copare to fixe rate tariffs, but custoers ust assue loa curtailent risk an high price volatility (Borenstein, 26). Electricity price control is especially iportant in high consuption inustry sub-sectors. Feeral governent statistics show that the top five sub-sector electricity consuers are cheicals, priary etals, foo, paper an transportation equipent (Energy Inforation Ainistration, 29). These subsectors can realize significant savings by stuying consuption patterns an alternative tariffs. Suppliers now arket to all custoer classes a real-tie price (RTP) tariff as a cost saving alternative to the fixe rate tariff (Power Sart Pricing, 28). Over 7 U.S. utilities offer RTP tariffs on a peranent or pilot basis to various custoer classes (Barbose et al, 24). Progras exist in every region of the country an have varying levels of acceptance. The progras help itigate suppliers arket power an reuce price volatility by sening econoic signals to custoers that allow the to oify their power ean on arket price. The RTP tariff gives custoers access to the aily price variations of the wholesale electricity arket. These prices reflect the hourly supply an ean conitions on the gri an are highly volatile. Although this tariff ay have savings potential, custoers ust stuy their loa pattern an copare it to the RTP before aopting the tariff. oa patterns that peak coincient to RTP rates introuce higher electricity costs, especially for sustaine loa peaks. Custoers who require stea for inustrial processes can co-generate elec- 2

3 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 tricity to control their peak eans an reuce purchase electricity costs when RTP rates spike (Sariveis et al., 23; Coffy an Kutrowski, 25). In general, inustrial loa control woul require aitional investent in energy anageent systes an custoerowne generators to reuce consuption uring ties of high prices. Recovering the cost of this equipent epens on the potential savings, if any, (nt ) a realize fro aopting the RTP tariff. Shifting or curtailing inustrial operations is another etho of loa control but aybe too costly or ipractical to ipleent. Electrical oa Representation One etric for classifying inustrial electric loa patterns is loa factor (Turner, 2). oa factor copares average power ean to peak power ean for a efine perio such as a onth or year. Equation () efines loa factor atheatically. In this equation, E total is the (nt ) Equations (2) an (3) siulate high an low loa factor inustrial loas respectively. These ean functions generate ties series loa ata noralize to peak power ean. = sin(2πfnt ) + a 2 sin(2πf 2nT + φ 2 + ε φn ) + + ε bn E total F = () Ppeak (nt ) = a sin( 2πfnT + (3) φ ) + a 2 total perio electric energy usage in kilowatt-hours, (nt ) = a P peak is the axiu sin( 2πfnT + φ ) + a 2 sin( 2πf 2nT + φ 2 + ε φn ) + Previous researchers prouce eterinistic ynaic oels perio power ean in kilowatts (nt of ) = a inustrial sin( 2πfnT + φ ) + a 2 sin( 2πf 2nT + φ 2 + ε φn ) + + ε bn uring the perio, an is the perio custoer types using linear an nonlinear prograing to test the ipact loa factors present a nearly constant These equations represent inustrial length in hours. Custoers with high of RTP tariffs (Davi an ee, 989). power ean over tie to suppliers while custoers with a low loa changes. They inclue cycles for shift operations that exhibit aily perioic The oels generate optial operating scheules for plant prouction uner an factor have ore cyclic power ean work changes reflecting three eight RTP with prouction constraints. These patterns. ow loa factor inustrial hour work perios. The variable, T, theoretical oels are for evaluating operations exhibit cyclic electricity represents the ean eter totalizing supply an ean interactions, an consuption ue to aily prouction perio an n is the perio inex over have value in inustrial site planning an changes an shift operations. Inustrial the total stuy interval. The other process expansion. ughes an Bailey custoers that operate at full capacity paraeters are: evelope a scheuling ethoology continuously have higher loa factors. using iscounte cash flow oeling f = aily frequency = /T, T in hours, for optially allocating co-generating Electric ean eters totalize energy resources in a nylon plant using RTP consuption over perios varying fro f 2 = shift frequency = /T 2, T 2 in hours, inforation (ughes an Bailey, 24). five inutes to one hour. These interval values provie inustrial custo- a In this work, a Monte Carlo siulation = aily an shift coponent evaluate a nuber of fuel cost an ers with power ean inforation aplitues for high loa factor, electricity price scenarios to ai in the an energy consuption ata. Siple ecision process an risk analysis of calculations convert shorter interval a = aily an shift coponent optial co-generator scheules. ata into equivalent hourly values. aplitues for low loa factor, Graphing ean eter ata prouces This paper exaines ean functions to eterine what factors influ- consuption. A Fast Fourier Transfor nent for high loa factor, tie series plots of consuer electricity φ 2 = phase elay of shift loa copoence RTP tariff savings by introucing (FFT) ecoposition of power ean a benefits inex to calculate savings tie series values ientifies significant φ, φ 2 = phase elay of aily an shift potential. The noinal values of the cyclic loa coponents that can be loa coponents for low loa factor, ean functions erive fro analysis oele atheatically using sinusoial ters. The FFT ecoposition pro- of actual plant electricity consuption, = average power eans ata. The analysis uses a power ean vies aplitue, frequency an phase for high an low loa factors, easure, loa factor, to characterize inustrial loa types. The analysis 995). Rano loa variations occur in ε bn paraeters for the loa oel (Cohen, = stochastic average loa coponent, also copares the potential benefits of inustrial loas also. Stochastic phase aopting the RTP tariff to loa factors an aplitue paraeters represent this with high an low values. The analysis part of custoer loa. The su of the ε φn = stochastic phase elay coponent. assues inustrial custoers purchase perioic an stochastic loa coponents gives the total custoer power no aitional loa control equipent. ean function. (nt ) = a sin(2πf nt = a sin(2πfnt ) + a 2 sin(2πf 2nT + φ 2 + ε φn ) + a ) + 2 sin(2 (2) πf nt + ε bn 2 3

4 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 Per Unit Price an Dean Figure. Phase Relationship between Dean an RTP n Figure shows the phase elay relationship between the price an loa. Phase elay is the angular ifference between the real-tie price aily peak an loa series peak with price as reference. These paraeters relate the teporal ifference between the two tie series. The equations represent rano changes in operations with two stochastic variables: one for power ean an another for phase elay. Using loa oels on actual consuption ata allows energy anagers to correlate operation paraeters such a work start ties to the price series an exaine the ipact of paraeter changes quantitatively. Electricity Cost An Benefit Calculation Suppliers provie real-tie price inforation to custoers at regular intervals ranging fro five inutes to one hour. Soe arkets publish ay-ahea forecasts to help custoers anage their loas an reuce costs. These prices reflect the supply/ean relationship of the wholesale electricity arkets. The RTP eonstrates high volatility over a aily cycle reflecting the teporal nature of power ean. Weekly an seasonal price cycles appear in long-ter tie series, reflecting the changes in electric ean ue to 24 hours oralize Power Dean oralize RTP Tarriff reuce weeken activity an builing inoor environental control. Fixe rate tariffs reain constant over efine contract perios. Fixe rate tariffs account for utility fuel costs an capital investent in their systes, an are on average custoer consuption. Fixe rate tariffs inclue cross-subsiies between custoer classes such as inustrial, coercial, an resiential (Borenstein, 26). Most inustrial tariffs charge for both electric energy an power ean. Dean charges help suppliers recover costs associate with peak power generation, transission, an elivery. Electric syste owners ust construct facilities to hanle peaks that ay occur for only a single hour annually. The ean charge penalizes custoers with lower loa factors since their peak power ean is large relative to their total consuption over a billing cycle. The following analysis assues the sae peak ean charges occur for both RTP an fixe custoer tariffs. Diviing the coponents of the realtie price series, p n an the fixe rate, p f, by the axiu RTP price uring an interval gives a noralize price sequence an noralize fixe rate, f. Equations (4) an (5) copute noralize custoer cost for a fixe rate tariff over the interval [, ] using noralize power eans fro () an (2). The forulas assue a one hour etering interval, T =. The variables FEC an FEC are the fixe rate electricity cost inexes for high an low loa factor tie series respectively. FEC = f (n) FEC = f (n) (4) (5) Equations (6) an (7) copute noralize costs for RTP tariffs with REC an REC representing the custoer electricity cost inex for high an low loa factor consuption patterns. REC = n (n) REC = n (n) (6) (7) A benefit inex quantifies potential inustrial custoer savings fro aopting a RTP tariff. The inices shown in equations (8) an (9) are the ifferences between the fixe an RTP costs for both high an low loa factor power ean patterns. B = B = f n = f n = (n) (n) n = n = (n) (8) When the inex, B, is zero, aopting the RTP tariff prouces no aitional benefits over a fixe rate. If B is negative, then electricity costs uner the RTP tariff are larger than the fixe rate. Aopting the RTP rate is not cost effective for negative values of the benefits inex. If B is positive, then the cost of aopting the RTP rate structure is less n (n)(9) n 4

5 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 (n) + than the cost of the fixe rate, so a custoer woul realize savings by switching to the RTP tariff. Setting the benefit equations to zero fins the break-even noralize fixe rate for aopting a RTP tariff with a given loa factor, f. The RTP sequence can be written as the su of the average RTP over, ave, an the series resiuals, rn. Setting the benefits equation to zero an replacing the RTP series with the above su gives: n = n = f (n) = ave (n) + ) n = n = Solving this equation for f gives the break-even noralize fixe rate for any loa pattern an price tie series. r n (n) () f = ave + r n (n) (n) r n The secon ter in () quantifies the variation in cost ue to price changes about the ean. It accounts for the correlation between the price changes about the series ean an power ean series. ighly correlate prices an eans increase the agnitue of this ter. arge ean swings also increase it. Monte Carlo Experiental Design Monte Carlo siulation is a tool for analyzing systes with paraeter variation or incoplete knowlege of ata. This type of analysis is one way to quantify uncertainty in ata an oel paraeters. Monte Carlo siulation results in a probability istribution that escribes how uncertainty propagates (n through a syste. Statistical analysis of the resulting istribution escribes syste perforance (Wittwer, 24). Four statistical experients exaine the ipact of loa an price paraeter variations on custoer benefit functions, equations (7) an (8), for iffering loa factor ean patterns. Two statistical experients analyze the ipact of ean paraeter variations on the break-even fixe rate given by (). Table shows the fixe an rano variables for the six experients. The first four experients stuy how variation of average power ean, phase elay, an fixe rate prices ipact custoer benefits when copare to a real-tie price tariff. The reaining two experients exaine how loa function paraeter variations affect the break-even fixe rate. The first four experients capture loa factor effects by coparing the benefits fro the two ifferent ean patterns. For each experient, =648 hours. This nuber represents 27 consecutive ays of hourly usage. Each experient coputes results for 5 iterations of the benefit inices using the fixe an rano variables fro Table. The experients use historical real-tie price ata (Aeren, 29) noralize to the peak hourly price over a 27-ay interval in August, 27 for these coputations. All experients use this RTP tie series. The Aeren site archives an hourly RTP tie series beginning on Deceber 28, 26 an running to the present. This ata erives fro the Miwest regional wholesale electricity arket. Siilar ata is available for every region of the U.S. fro the Feeral Energy Regulatory Coission Website (Feeral Energy Regulatory Coission, 29). The Aeren site also posts ay-ahea prices that are next ay RTP forecasts. The site upates the ay-ahea prices with RTP prices at 4:3 p CST each ay. RTP tariff custoers use the ay-ahea prices as inicators of the actual RTP to ajust their loa profile. Table 2 lists the paraeter values an probability istributions for the power ean functions use in the first four experients. This analysis assues a unifor probability istribution for Experient Table. Custoer Benefits Experient Construction igh oa Factor Dean Function Paraeters ow oa Factor Dean Function Paraeters Fixe Rano Fixe Rano a, f, φ, φ 2, f 2 a, f φ, φ 2,, f 3 a, f, φ, f 4 a, f φ,, f 5 f a, φ, 6 f a, φ, φ 2, 5

6 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 phase elay, power ean, an fixe tariff paraeters. A noral istribution represents the rano phase elay an loa variations in the power ean functions. The analysis consiers a aily power ean perio of 24 hours an a prouction shift perio of eight hours. Figure 2 shows weekly high an low loa factor tie series copute using noinal paraeter values of φ 2 =φ 2 =π/4, φ =π/3 =.8 for high loa factor an =.5 for low loa factor. The high loa factor ean oel is on a case stuy of a blow-oling factory running continuously at full capacity proucing plastic soa bottles. The low loa factor ean oel is on a case stuy of a soa bottling facility operating a 5-ay week an a single shift. For price experients 5 an 6, a an a 2 are uniforly istribute noralize values. Table 3 lists the istributions use in experients 5 an 6 for the paraeters that iffer fro the benefits experients. The siulation coputes 5 iterations of the breakeven fixe rate, f, for both loa factor cases. Siulation Results An Discussion The Monte Carlo siulation results show how loa an fixe rate variations ipact potential custoer savings fro aopting a RTP tariff. These siulations assue that an inustrial custoer takes no other actions to control peak loa or overall electricity usage. Experient siulates custoer benefits using the low loa factor power ean function with uniforly istribute average power ean factors an fixe electricity rates. The aily an shift phase elay paraeters reain constant at φ =π/3 an φ 2 =3π/4. Experient 2 coputes custoer benefits for three uniforly istribute variables: average power ean, electricity rates, an phase elay. Experients 3 an 4 replicate Experients an 2 using the high loa factor power ean equation. Experients 3 an 4 only inclue a shift phase elay paraeter, φ 2 =3π/4. Table 2. Paraeter Values an Probability Distributions for oa Functions. Paraeter igh oa Factor () ow oa Factor () a.5.8 a φ (ra) /A P U (π/2,π) φ 2 (ra) P U (3π/2,3π) P U (3π/2,3π) P U (.3,.6) P U (.3,.6) T ( hrs) T 2 ( hrs) f ra/hr π/2 π/2 f 2 ra/hr π/4 π/4 ε φn P (M=, SD=2.5) P (Μ=, SD=2.5) ε bn P (Μ=.,SD=.25) P (Μ=.,SD=.25) f P U (.,.8) P U (.,.8) oralize Power Dean Figure 2. Power Dean Tie Series Showing Weekly Patterns of igh an ow oa Factor Inustrial Operations. igh F Dean ow F Dean Tie Inex (hours) Table 3. Paraeter Probability Distributions for Price Experients. Paraeter igh oa Factor () ow oa Factor () a P U (.5,.) P U (.8,.) a 2 P U (.8,. P U (.8,.) Table 4. Monte Carlo Siulation Results-Benefit Inex. oa Factor Benefits Inex Experient Mean Variance Mean Variance. ow loa factor, rano fixe rate an average loa paraeter ow loa factor, rano fixe rate, average loa, an phase elay paraeter igh loa factor, rano fixe rate an average loa paraeter igh loa factor, rano fixe rate, average loa, an phase elay paraeter

7 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 Table 4 suarizes siulation results for the first four experients. The low loa factor with rano average loas gives the lowest benefit inex value. In this case, the rano tie series paraeters are average power ean an fixe rate tariff. If power ean peaks when RTP peaks, the custoer experiences an increase in electricity costs copare to a fixe rate. Moving the power ean series with respect to the peak RTP series iproves the benefits inex by aking the peak ean an price coincie less. igh values of aily phase elay correspon to oving inustrial operations outsie the tie of peak electricity usage, which ay increase prouction costs ue to increase labor expenses. The high loa factor ean series prouces the highest benefit inex values. The inex oes not iprove significantly with changes in shift phase elay. The low variation in the aily loa pattern increases the savings potential of the RTP tariff since ost of the aily price cycle is lower than the fixe rate. The lower hourly rates prouce savings while the RTP is below the fixe rate. As long as price spikes are of short uration, an inustrial custoer will realize cost benefits fro using a RTP tariff if they have a high loa factor. Facilities that are operate aroun the clock an have alost constant process electricity eans exhibit a high loa factor. Figures 3 an 4 show the cuulative probability istributions for the four benefit siulations. The low loa factor siulation with no shift phase elay ajustent prouces the lowest probability of positive benefit. (P=.63, B>) Ajusting the phase elay along with the average ean paraeter iproves this probability to P=.72. The high loa factor cases prouce nearly ientical positive benefit inex probabilities with a value of P=.76. These results inicate that inustrial electrical loas that have high loa factors can prouce cost savings when a RTP tariff is applie. The savings are realize without instituting aitional energy anageent progras. ow loa factor eans can prouce cost savings on Figure 3. Cuulative Probability Distributions Coparing ow oa Factor Power Dean aving Rano Price, an Average Dean with a Case aving Rano Price, Average Dean, an Shift Phase Delay. Cuulative Probability Cuulative Probability Rano Price an Dean Rano Price,Dean an Phase an RTP rate, but ay require extensive loa control to achieve the benefit levels of a high loa factor eans. If the loa pattern is highly correlate with the price pattern, electricity cost will increase on a RTP tariff. Peak shaving an loa shifting on projecte prices iniize the risk of high cost consuption. owever, instituting these loa control progras coul be Benefit Inex Figure 4. Cuulative Probability Distributions Coparing igh oa Factor Power Dean aving Rano Price, an Average Dean with a Case aving Rano Price, Average Dean, an Shift Phase Delay Rano Price an Dean Rano Price,Dean an Phase Benefits Inex costly an itigate any potential savings fro a rate switch. Table 5 lists results fro ean equality tests of the benefits experients. The first two rows inicate that low loa factor ean benefits are statistically less than high loa factor ean benefits regarless of paraeter variation in the loa oel. Coparing ean benefits between low loa factor experients an 2 shows that elaying highly cyclic 7

8 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 loas with respect to the RTP tariff prouces a statistically significant iproveent in the benefit eans. igh loa factor experients show no statistically significant ifference between benefit eans when loa paraeters vary. This result inicates that ajusting shift operations in high loa factor eans will prouce no statistically significant benefits when copare to high loa factor eans without shift operation ajustent. Experients 5 an 6 exaine the ipact of power ean variations on the break-even noralize fixe rate. The histogras in Figures 5 an 6 show the siulation results. Experient Means Copare Table 5.Mean Equality Test Results. Z Statistically Different α= Yes Yes Yes o Figure 5. Break-Even Fixe Rate Distribution for ow oa Factor Power Dean M=.377 pu SD=.46 pu The fluctuations of the power ean an the RTP increase the variability of the break-even fixe rate for the low loa factor series. The break-even fixe rate varies.8 per unit (pu) over the range of siulation values. The loa variations a to the average RTP rate an relate to the loa changes. The high loa factor siulation results in a very sharp price istribution, which is consistent with the low ean variation about the ean in this oel. The break-even fixe rate is very near the average RTP for the analysis perio for high loa factor eans. Frequency oralize Break -Even Rate (ow F) The presente ethoology uses loa oels on Fourier ecoposition of actual loa ata with stochastic coponents to copute benefits. This technique ignores the ipact of outliers in the original ata series that ay affect stuy results. It also assues that operations reain cyclic over the stuy perio. Using actual noralize power ean ata to copute custoer benefits eliinates these liitations but ake it ore ifficult to ientify paraeters that relate to work processes. Conclusion Introucing real-tie price tariffs to all custoers creates opportunities for electricity cost saving in inustrial operations. The shape an tiing of the loa tie series relative to the price series ipacts the potential benefits a business realizes fro aopting a Figure 6. Break-Even Fixe Rate Distribution for igh oa Factor Power Dean. Frequency M=.29 pu SD=.3 pu oralize Break -Even Rate (igh F) 8

9 Journal of Inustrial Technology Volue 25, uber 3 July 29 through Septeber 29 real-tie price tariff. Results of Monte Carlo siulations using power ean atheatical oels with high an low loa factors inicate that higher loa factor tie series have a greater econoic benefit potential. The high loa factor power ean series exhibits a lower break-even fixe price when copare to the low loa factor series. Inustrial operations that operate aroun the clock an have a steay process loa can save electricity costs by aopting a real-tie rate without further investent in energy anageent or control systes. Using actual ata to copute benefits coul provie ore accurate case stuies but woul eliinate analysis paraeters. Future work will exten the analysis perio of the experients to cover an entire suer. References Aeren Corporation. (29). Retrieve January 2 29 fro www2.aeren.co/retailenergy/ rtpdownloa.aspx. Barbose, G, Golan, C, & eenan, B. (24). A survey of utility experience with real tie pricing, Ernest Orlano awrence Berkeley ation aboratory, B [Available Online] EMS/EMS_pubs.htl. Borenstein, S. (26, July). Custoer risk fro real-tie retail electricity pricing: bill volatility an hegability, University of California Energy Institute, working paper CSEM WP 55. [Available Online] ucei.org. Borenstein, S. (26, July). Wealth transfer aong large custoers fro ipleenting real-tie retail electricity pricing, University of California Energy Institute, working paper CSEM WP 56. [Available Online] Coffey, B. & Kutrowski, E. (25). Dean charge consierations in the optiization of cogeneration ispatch in a eregulate energy arket, International Journal of Energy Research, vol. 3, no. 7, pp Cohen,. (995). Tie frequency analysis st e. Englewoo Cliffs, J: Prentice all. Davi, A. K, & ee, Y. C. (989). Dynaic tariffs: theory of utilityconsuer interaction, IEEE Transactions of Power Systes, vol. 4, no. 3, pp Energy Inforation Ainistration. (29) Retrieve Jun, 29 fro preata/estiates.htl. Feeral Energy Regulatory Coission. (29) Retrieve June, 29 fro ughes, P. D. & Bailey, W. F. (24). Inustrial powerhouse optiization in the eregulate electricity arketplace, Energy Engineering, vol., no., pp Power Sart Pricing. (28). Retrieve February 2, 29 fro powersartpricing.org. Sariveis,. K. et al., (23). Optial energy anageent in pulp an paper ills, Energy Conversion an Manageent, vol. 44, no., pp Turner, W. C., (2). Energy anageent hanbook 4 th e. ilburn, GA: The Fairont Press, Inc. Wittwer, J. W. (24, June). Monte Carlo Siulation Basics, Vertex42. co. [Available Online] tex42.co/excelarticles/c/monte- CarloSiulation.htl. 9