Developing an Optimization Tool for Selecting Home Investment Strategies

A Senior Design Project

 

By: Brittany Hagedorn

Spring 2009

 

Overview

Company B works with homeowners to help them reduce their utility bills.  They do so through an energy audit, which involves an expert visiting the home and identifying opportunities for the owner to save energy.  Currently, the service does not include any energy estimates that are specific to the home’s existing condition.  To improve the energy audit, Company B would like to have a tool that they can use to assess which energy-saving opportunities should be top priority for the homeowner, as well as the detailed financial implications of those investments.  To do so, an optimization tool has been developed, using Excel Solver as a platform, to meet the homeowner’s goals (ROI, Savings, Payback Period), while staying within constraints (budget, possible of system combinations, etc).  The optimization uses client-specific input, combined with researched assumptions, to provide detailed estimates of savings, return on investment, and payback periods.  This information is used to prioritize investment strategies and results in a list of recommendations for the client.

 

Problem Background

An energy audit is a service for homeowners who are just beginning to consider energy efficiency and how it affects the utility bills they pay every month.  It consists of an energy expert coming to the home, discussing the homeowner’s priorities and any limitations they may have (such as budget), and then a home inspection to identify problem areas.  Throughout the visit, the energy expert is making suggestions of possible improvements, but these recommendations are not prioritized for the homeowner.  Currently, the homeowner does not end up with any personalized reports, simply the conversation and some basic information to get them started.

 

Development Process

There are several steps necessary for completion of a successful tool development process.  They included scope definition, identification of the alternatives to be included as investment options, choice of objective function parameters, data and assumption research, and model design and development.

1.  Scope Definition

The first step was to define the scope of the project, to ensure the resulting tool met the requirements discussed above.  The question was which aspects of an audit would be included in the optimization.  Since there are many components that contribute to how much energy a home consumes, I settled on four categories: HVAC systems, domestic hot water heaters, major appliances, and add-on features. 

2.  Alternative Identification

The next step was to identify the existing alternatives for each category being considered, and then narrow down the many options to those appropriate for consideration. 

3.  Objective Function Parameters

To ensure the accuracy and usefulness of the model, the objective function was carefully thought out.  There were several parameters considered as the objective for the entire optimization: maximization of total savings, maximization of return on investment, and minimization of payback period.

4.  Research

Given the many alternatives that were identified as relevant for inclusion in the tool, there was a lot of research into the assumptions were used for calculating the objective function parameters.  Unfortunately, most of the publicly available data is scattered in different sources and very little was originally in a form that fit neatly into the model as designed.

5.  Model design and development

The model itself was an optimization from the start, since I was developing the tool to make recommendations of which alternatives (decision variables) should be selected to meet a specific goal (objective function).

 

Note:  For information on steps 1-4, please see the report linked below.

 

Model Design and Development

To understand the way the model design developed, there is first a discussion of the basics: decision variables, constraints that were eventually included, and the platform used for development, in this case Excel Solver.  Following the initial design, there were several phases of design and redesign, resulting in the final design shown below.

Decision Variables

 

Category

Variable Designation

Heating  (H)

Binary – installed (1), not installed (0)

Cooling  (C)

Binary – installed (1), not installed (0)

Hot water  (W)

Binary – installed (1), not installed (0)

Refrigerator  (R)

Binary – installed (1), not installed (0)

Clothes Washer  (CW)

Binary – installed (1), not installed (0)

Insulation  (N)

Integer – number of additional inches installed

Add-ons  (A)

Integer – quantity purchased or in use

 

Constraints

There were four types of constraints that were eventually included.  They included single-system constraints, physical constraints, convenience constraints, and client-imposed constraints. 

Single-System

total number heating systems

1

=

1

total number cooling systems

1

=

1

total number hot water

1

=

1

total number refrigerators

1

=

1

total number stoves

1

=

1

total number clothes washers

1

=

1

Physical

can't have desuperheater wo/ geothermal HVAC

1

<=

1

can't add more fans than have space

4

<=

4

can't replace more incandescents than exist

10

<=

10

can't install set-back if already one there

1

<=

1

can't use power strips you don't own

2

<=

2

can't caulk if home already tight

0

<=

0

can't buy blanket if already own one

0

<=

0

can't add ducting insulation if already there

1

<=

1

Convenience

can't replace central air with window units

0

<=

0

can't add central air to buildings wo/ ducting

1

<=

2

can't add central heat to buildings wo/ ducting

1

<=

2

only one type of insulation

1

<=

1

Client-imposed

don't exceed budget!

$20,736

<=

$45,000

don't exceed payback period!

0.00

<=

4

 

Final Design

Decision variables: 

xijj  = decision variable for category i, alternative j

Objective function: 

Max:    EQ6.png

Subject to:   

                        SINGLE-SYSTEM constraints

PHYSICAL constraints

CONVENIENCE constraints

CLIENT-IMPOSED constraints

Equations:

            TOTAL NPV of all investments:

                        EQ1.png

TOTAL SAVINGS for heating, cooling, hot water, refrigerator, clothes washer:

 

 

TOTAL SAVINGS for insulation, add-ons:

EQ3.png

 

TOTAL INVESTMENT for all categories:

            EQ4.png

 

TOTAL DRC for heating, cooling, hot water, refrigerator, clothes washer:

            EQ5.png

 

Where

sij = yearly savings by selecting alternative ij

lij  = expected lifespan of alternative ij

pij = purchase price of new system ij

ocij = operating cost of system ij

                        yij = existing system (binary) for category i, alternative j

effi,input = system efficiency of alternative i, as input by user

effi,year = system efficiency of alternative i, by installation year

                        HES = new high-efficiency system

                        average = average new system

                        existing = system currently installed

 

Sample Run

To verify and demo the optimization program, an example home was used, in conjunction with reasonable expectations of budget and risk tolerance.  The existing system, recommendations, and results are listed below.

System

Existing

Recommended Investment

Heating

Electric furnace

Geothermal

Cooling

Central air

Geothermal

Domestic hot water

Electric storage

Desuperheater

Refrigerator

Top freezer

 

Clothes washer

Front load

 

Insulation

5” Fiberglass

Add 5” fiberglass

Add-on features

 

Install 4 ceiling fans

Replace all incandescent bulbs

Install set-back thermostat

Use existing power strips for efficiency

Purchase 1 additional power strip

 

Return on Investment = 198%

Total Savings = $35,468

Payback period = 13.4 years

 

LINKS to DOCUMENTS

Final Report

Poster

PowerPoint Presentation