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I Figure 2-3. Probabilifiy distribution aver quantity requiredforMonteCarloanalysis. Foreconomicanalysisembodiedinaspreadsheet,commercialadd-ins are available to carry out both Monte Carlo and sampling procedure^.'^ The end result of these procedures will be probabilistic descriptions of the chosen economic measures appliedtotheproject. Thesedescriptionscanbeexpressedinseveralways,thetwomostcommonbeing the probability density Eunction and the cumulative distribution, illustrated in Figures 2-1 and 2-2. The density function illustrates most directly which values of the economic value measure are more likely and how the relative likelihood changes over the range of possible values. The cumulative distribution, on the other hand, can be used directly to answer questions such as, "What is the probability that the value lies in a specified range?" Figure 2-4 illustrates the probability distribution over the project NPV resulting from the simple model described previously. This information was generated by a commercial software package (Crystal Ball) using Monte Carlo simulation. This analysis illustrates the importance of the probability density representation of the project's net present value. Looking at the project economics when dl the variables are set at nominal values indicates a substantiallypositiveNPV. However,thesensitivityanalysisrevealsthepossibilitythatuncertaintiesin price and quantity could lead to a negative NPV. The summary in Figure 2-4 is the most revealing. The most likely values of NPV, as it turns out, are the negative ones. The cumulative distribution curve of Figure 2-4 would reveal that the odds are lessthanfifty-fifty of achieving a positive NPV. Even in this drastically oversimplified model, it would have been difficult to foresee this conclusion from the basic information about input variables. lgCrystalBall, distributed by Decisioneering Software,Lakewood, Colorado, is one example of add-in software for performing uncertainty analysis. This package is an add-in for Excel and was used to develop the example shown in Figure 2-4. 0.7 T 0.6 0,5 0-4 02 082 0.1 ~OOOOO~oaooaumooao 2000000 250~000~ounooo Annual Sales,kWh 32PDF Image | A Manual for the Economic Evaluation of Energy Efficiency and Renewable Energy Technologies
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