A-1

APPENDIX A: APPROACH FOR UNCERTAINTY AND VARIABILITY

A.1 INTRODUCTION

Analysis of an energy efficiency standard involves calculations of impacts, for example the impact

of a 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 in the model depend upon other conditions (i.e., there

is variability).

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 is for each quantity. Explicit analysis of uncertainty and variability is intended to provide

more complete information to the decision process.

A.2 UNCERTAINTY

When making observations of past events or speculating about the future, imperfect knowledge is

the 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 of

error. When estimating numerical values expected for quantities at some future date, the exact outcome

is rarely known in advance.

A.3 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 of hours an air conditioner is operated by a household

depends upon the specific circumstances and behaviors of the occupants (e.g., number of persons, 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 represen tative of the entire

population. Surveys can be helpful here, and analysis of surveys can relate the variable of interest (e.g.,

hours of use) to other variables that are better known or easier to forecast (e.g., persons per household).A-2

A.4 APPROACHES TO UNCERTAINTY AND VARIABILITY

Two approaches to consider uncertainty and variability are described here:

C scenario analysis and

C probability analysis.

Scenario analysis uses a single numerical value for each quantity in a calculation, then changes one

(or more) of the numerical values and repeats the calculation. A number of calculations are done, which

provide some indication of the extent to which the result depends upon the assumptions. For example, the

life cycle cost of an appliance could be calculated for energy rates of 2, 8, and 14 cents per kWh.

The advantages of scenario analysis are that each calculation is simple; a range of estimates is used;

and crossover points can be identified. (An example of a 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 of scenario analysis is that there is no information about the likelihood

of each scenario.

Probability analys is considers the probabilities within a range of values. For quantities with

variability (e.g., electricity rates in different households), surveys can be used to generate a frequency

distribution of numerical 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 used to provide probabilities (e.g., manufacturing cost to improve energy efficiency to some level may

be estimated to be $10 ± $3).

The major disadvantage of the probability approach is that it requires more information, namely

information about the shapes and magnitud es of the variability and uncertainty of each quantity. The

advantage of the probability approach is that it provides the greatest information about the outcome of the

calculations, that is, the probability that the outcome will be in any particular range is provided.

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 of

possible conditions.

A.5 PROBABILITY ANALYSIS AND THE USE 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 probability analyses usedA-3 NORMAL

Figure A.1a Normal Probability

Distribution TRIANGULAR

Figure A.1b Triangular Probability

Distribution UNIFORM

Figure A.1c Uniform Probability

Distribution Monte Carlo simulation and 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

s preadsheet 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 of the modeled system. One type of spreadsheet simulation is Monte Carlo

simulation

, which randomly generates values for uncertain variables over and over to simulate a model.

Monte Carlo simulation was named for Monte Carlo, Monaco, where the primary attractions are casinos

containing games of chance. Games of chance such as roulette wheels, dice, and slot machines, exhibit

random behavior. The random behavior in games of chance is similar to how Monte Carlo simulation

selects variable values at random to simulate a model. When you roll a die, you know that either a 1, 2, 3,

4, 5, or 6 will come up, but you 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 installation cost).

For each uncertain variable (one that has a range of possible values), possible values are defined

with a probability distribution . The type of distribution selected is based on the conditions surrounding

that variable. Probability distribution types include:

During a simulation, multiple scenarios of a model are calculated by repeatedly sampling values

from the probability distributions for the uncertain variables and using those values for the cell. Crystal

Ball simulations can consist of as 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

of the probability distribution ) for each uncertain variable and then recalculates the spreadsheet.