EE-RM/STD-98-440 - J09q

. APPENDIX A: APPROACH FORUNCERTAINTY AND VARIABILITY
A.1 INTRODUCTION Analysis ofan energy efficiency standard involves calculations of impacts, for example the impact ofa standard on consumer life-cycle cost (LCC). In order to perform the calculation, the analyst must first: 1)specify the equation or model that will be used; 2)define the quantities in the equation; and 3) provide numerical values for each quantity. In the simplest case, the equation is unambiguous (contains all relevant quantities and no others), each quantity has a single numerical value, and the calculation results in a single value. However, unambiguity and precision are rarely the case. In almost all cases the model and/or the numerical values for each quantity in the model are not completely known (i.e, there is uncertainty) or the model and/or the numerical values for each quantity inthe model depend upon other conditions (i.e., there isvariability). Thorough analysis involves accounting for uncertainty and variability. While the simplest analysis involves a single numerical value for each quantity in a calculation, arguments can arise about what the appropriate value isfor each quantity. Explicit analysis ofuncertainty and variability isintendedto provide morecomplete information tothe decision process.
A.2 UNCERTAINTY When making observations of past events or speculating about the future, imperfect knowledge isthe rule rather than the exception. For example, the energy actually consumed by a particular appliance type (such as the average U.S. central air conditioner or heat pump) is not directly recorded, but rather estimated based upon available information. Even direct laboratory measurements have some margin oferror. When estimating numerical values expected for quantities atsome futuredate, the exact outcome israrelyknown inadvance.
A3 VARIABILITY Variability means that different applications or situations produce different numerical values when calculating a quantity. Specifying an exact value for a quantity may be difficult because the value depends on something else. For example, the number ofhours an air conditioner isoperated by a household depends upon the specific circumstances and behaviors of the occupants (e.g., number ofpersons, personal habits about how comfortable the person wants to be, etc.). Variability makes specifying an appropriate population value more difficult in as much as any one value may not be representative ofthe entire population. Surveys can be helpful here, and analysis ofsurveys can relate the variable ofinterest (e.g., hours ofuse) toother variables that are better known or easier to forecast (e.g., personsper household).

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### EE-RM/STD-98-440-_3 q a A.4 APPROACHES TOUNCERTAINTY AND VARIABILITY Two approachestoconsideruncertainty and variability aredescribed here: * scenario analysis and * probability analysis. Scenario analysis uses a single numerical value for each quantity in a calculation, then changes one (or more) ofthe numerical values and repeats the calculation. A number ofcalculations are done, which provide some indication of the extent to which the result depends upon the assumptions. For example, the life cycle cost ofan appliance could be calculated for energy rates of 2, 8,and 14cents perkWh. The advantages of scenario analysis are that each calculation is simple; a range ofestimates isused; and crossover points can be identified. (An example ofa crossover point is the energy rate above which the life cycle cost is reduced, holding all other inputs constant. That is,the crossover point is the energy rate at which the consumer achieves savings in operating expense that more than compensate for the increased purchase expense.) The disadvantage ofscenario analysis is that there isno information aboutthe likelihood ofeach scenario. Probability analysisconsiders the probabilities within arange ofvalues. For quantities with variability (e.g., electricity rates indifferent households), surveys can be used to generate a frequency distribution ofnumerical values (e.g., the number of households with electricity rates at particular levels) to estimate the probability of each value. For quantities with uncertainty, statistical or subjective measures can be usedto provide probabilities (e.g., manufacturing costto improve energy efficiencyto somelevel maybe estimated tobe $10± $3). The major disadvantage of the probability approach is that it requires more information, namely information about the shapes and magnitudes of the variability and uncertainty of each quantity. The advantage ofthe probability approach is that itprovides the greatest information about the outcome ofthe calculations, that is,the probability that the outcome will be in any particular range isprovided. Scenario and probability analysis provide some indication of the robustness of the policy given the uncertainties and variability. A policy is robust when the impacts are acceptable over a wide range ofpossible conditions.
A.5 PROBABILITY ANALYSIS AND THEUSE OF CRYSTAL BALL To quantify the uncertainty and variability that exist in inputs to the Engineering, Life-Cycle Cost (LCC), and Payback Period (PBP) analyses, Microsoft Excel spreadsheets combined with Crystal Ball (a commercially available add-in) were used to conduct probability analyses. The
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# q I probability analyses usedMonte Carlosimulationand probability distributions. Simulation refers to any analytical method meant to imitate a real-life system, especially when other analyses are too mathematically complex or too difficult to reproduce. Without the aid of simulation, a spreadsheet model will only reveal a single outcome, generally the most likely or average scenario. Spreadsheet risk analysis uses both a spreadsheet model and simulation to automatically analyze the effect of varying inputs on outputs ofthe modeled system. One type of spreadsheet simulation is Monte Carlo simulation, which randomly generates values for uncertain variables over andover to simulate a model. Monte Carlosimulationwas named for Monte Carlo, Monaco, where the primary attractions are casinos containing games ofchance. Games of chance such as roulette wheels, dice, and slot machines, exhibit random behavior. The random behavior in games ofchance is similar to how Monte Carlo simulation selects variable values at random to simulate amodel. Whenyou roll a die, you know that either a 1,2, 3,4, 5,or 6 willcome up,butyou don't know which for any particular roll. It's the same with the variables that have a known range of values but an uncertain value for any particular time or event (e.g. equipment lifetime, discount rate, and installationcost). For each uncertain variable (one that has a range of possible values), possible values are defined with aprobability distribution. The type ofdistribution selected isbased on the conditions surrounding thatvariable. Probability distribution types include:

NORMAL TRIANGULAR UNIFORM FigureA.la Normal Probability FigureA.lb TriangularProbability FigureA.lc Uniform Probability Distribution Distribution Distribution During a simulation, multiple scenarios of a model are calculated by repeatedly sampling values from the probabilitydistributionsfor the uncertain variables and using those values for the cell. Crystal Ball simulations can consist ofas many trials (or scenarios) as desired

- hundreds or even thousands. During a single trial, Crystal Ball randomly selects a value from the defined possibilities (the range and shape oftheprobability distribution)foreach uncertain variable andthen recalculatesthe spreadsheet.

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