8B -i APPEN DIX 8B. UN CERTAINTY AND VAR

IABILITY
TABLE OF CO NTENTS
8B.1 INTRODUCTI ON

.................................................... 8B-1 8B.2 UNCERTAINTY

..................................................... 8B-1 8B.3 VARIABI

LI TY

...................................................... 8B-1 8B.4 APPROACHES TO UNCERTAINTY AND VARI ABIL ITY

.................. 8B-2 8B.5 PR OBA BILITY ANA LYSIS AND THE US E OF CR YS TAL B ALL

............ 8B-3

LIST OF F IGURES Figure 8B.5.1 Normal Probability Distribution

................................... 8B-3 Figure 8B.5.2 Triangular Probability Distribution

................................. 8B-3 Figure 8B.5.3 Uniform Probability Distribution

................................... 8B-38B -1 APPEN DIX 8B. UN CERTAINTY AND VAR

IABILITY

8B.1 INTRODUC TION Analysis of an ene

rgy-efficiency

standard involves calculations of impacts—for example, the impact of a stan

dard on li fe-cy

cle cost (LCC) . To perf

orm the c alculati on, the an aly st must first: 1) specify the

equation or model that will be used; 2) define the quantities in the equation; and 3) pr ovide nume

rical va lues for each quantity . In the simplest case, the equati on is unambig uous (i.e ., contai

ns all rel evant qua

ntities an d no other

s), each quantity

has a sing

le numerical value, and the calculation results in a single va lue. However, unambiguity and precision are rarely

the case. In almost all case s, the model and/or the numerical values for each quantity in the model are not complete ly known (i.e., there is uncertainty) or

the model and/or the numeric al value s for ea

ch quanti ty in the model

depend on other c onditions ( i.e., the re is variability). Thorough analysis involves accounting for uncertainty and variability. While the simplest a naly sis involve s a singl e numeri cal value for ea ch quanti ty in a calcu lation, a rgumen ts can ari se about w

hat the a ppropria te value is for ea ch quanti ty. Explicit ana

lys is of unce

rtainty and variability is intended to provide more complete information to the decision-making process.

8B.2 UNCERTAI

NTY When making observations of past events or speculating a bout the future, imperfect kn ow le dg e i s t he ru le , r at he r t ha n t he ex ce pt io n. Fo r e xa mp le , t he en er gy a ct ua ll y co ns um ed by a particular piece of equipment (such as distribution transformer ) is not directly rec orded, but rather estimated based on available information. Even dire ct laboratory measurements have some margin of error. When estimating numeric al values expected for quantities at some future date, the exact outcome is rarely known in advance.

8B.3 VARIABILI TY Variability means that diff erent applications or situations produce different numerica l values w hen calc ulating a quantit y. Sp ecify

ing an exact value

for a qua ntity may be diffic ult because the value depends on somethi

ng else . For exa

mple, the number of

hours a tr

ansformer is run at a particular load depends on the specific circumstances and behaviors of the building owners or utilities (e.g., square footag e served, hours of operation). Variability makes specify

ing an appropriate population value more difficult, since any one value may

not be representative of the entir e populat ion. Surve

ys c an be he lpful her

e, and a naly sis of survey s can re late the variable of interest (e.g., hours of use) to other va riables that are better known or easier to f orecast (e.g., building square footage).8B -2

8B.4 APPR

OACHES TO UN CERTAINTY AND VAR

IABILITY This sect ion descr ibes two a

pproaches to unce

rtainty and vari ability :

C scenario analy sis, and

C probabil ity a

naly sis. Scenario analy sis uses a single numerical value for eac h quantity in a calcula

tion, then changes one (or more) of

the numerical values and repeats the ca lculation. A number of calculations are performed that provide some indica tion of the extent to which the result depends on the assumptions. For example, the LCC of a piece of e quipment could be calculated for energy rates of 2¢, 8¢, and 14¢ per kilowatt-hour (kWh). The advantages of scena

rio analysis are

that each calculation is simple, a range of estimates is used, and crossover points can be identified. (An example of a c rossover point is the energy rate above which the L CC is reduced, holding all other inputs constant. That is, the crossover point is the energy rate at which an owner achieve s savings in operating expense that more tha n compens ate for

the incr eased pu rchase expense.)

The disad

vantag e of sce

nario analysis is that there is no information about the likelihood of e ach scenario. Probability analy

sis considers the probabilities within a range of values. For quantities with variability (e.g ., electricity tar iffs in different regions), survey s can be used to generate a frequency distribution of numerical values (e.g., the number of buildings subjec t to particular tariffs) to estimate the probability of each value. For quantities with uncertainty , statistical or subjective measures can be used to provide probabilities (e.g ., manufacturing cost to improve energy efficiency

to some level may be e stimated to be $10 ± $3). The major disadvantage of the probability approach is that it requires more information, namely information about the sha pes and magnitudes of the variability and uncertainty

of each quantity. The advanta

ge of the probability

approach is that it provides greater infor mation about the outcome of the calculations, i.e., it provides the probability that the outcome will be in a particular range. Scenario and probability

analysis provide some indica tion of the robustness of a policy given the uncertainties and variability . A policy is robust when the impacts a re acceptable over a wide ran ge of p ossible c onditions. 8B.5 PROBABIL ITY ANALY SIS AND THE US E OF CRY STAL BALL To quantif

y the uncer

tainty and vari ability

that exist in

inputs to the

engine ering, LCC, and pay back pe riod ana lys es, the D epartme

nt used Mic

rosoft Exc el sprea

dsheets c ombined with Cry stal Bal l, a comme

rcially availab le add-i

n, to cond uct proba bility analy ses. The probabil ity analy ses used M onte Carl o simulation and pro bability distributi ons.8B -3

Fig ure 8B .5.1 Normal

Proba bility Distr ibut ion Fig ure 8B .5.2 Triangul ar Proba bility Distr ibut ion Fig ure 8B .5.3 Uniform Proba bility Distr ibut ion Simulation r

efers to any analy tical met hod meant

to imitate a

real-li fe system, esp ecially when other analy ses are too mathematically complex or too difficult to reproduce. Without the aid of simu

lation, a spreadsheet mode l will only reveal a single outcome, gener

ally the most likely or aver

age scenario. Spreadshee t risk analysis uses both a sprea dsheet model and simulation to automatically analy ze the effect of vary

ing inputs on outputs of the modeled sys tem. One t

ype of sprea dsheet si mulation is

Monte Car

lo simulation, which r

andomly gener ates val ues for u ncerta in varia bles ag ain and a

gain, to simulate a

model. Mon

te Carlo simulation was named for Monte Carlo, Monaco, where the primary attractions are casinos containing games of chanc e. Games of chance such as roulette w heels, dice, and slot machines exhibit ran

dom behav

ior. The r andom behavior in games of c hance is similar to h

ow Monte Carlo simulation selects variable values at random to simulate a model. When y ou roll a die, you know that either a 1, 2, 3, 4, 5, or 6 will come up, but y ou do not know which for any particular roll. It is the same with the variables that have a known range of values but an unce rtain value for any pa rticular time or event (e.g., equipment lifetime, discount rate , installation cost). For each uncertain variable (one that has a range of possible value s), possible values are defined with a probability distribution. The ty pe of distribution selected is based on the conditions surrounding that variable. Probability distribution types for normal, triang ular, and uniform are shown in Figures 8B.5.1, 8B .5.2, and 8B.5.3, respectively

.

During a simulation, multiple scenarios of a model are calc ulated by repe atedly sampling values from the probability distributions for the unc ertain variables and using those values for the cell. Crystal Ba

ll simulations can consist of as many trials (or sce narios) as desired—hundreds or even thousands. During a single trial, Cry stal Ball randomly se lects a value from the defined possibilities (the range and shape of the probability distribution) for each uncertain variable and then recalculates the spreadsheet.