Energy Cost Metrics

Learning Objectives

• Combine financial and energy quantities to create metrics usef for energy decisions.

Energy Metrics

• Payback period

• Cost of Conserved Energy

• Cost of Conserved Carbon

• Economic Thickness of Insulation

Economic Thickness of Insulation

If we consider both the annual cost of insulation and the annual cost of energy, there is an optimal thickness where the cost of both is minimized.

This is known as the economic thickness of insulation.

Note: Place example from class notes here.

We calculate

• The area of the cabin walls, floors, and ceiling

• The UA product of the cabin

• The yearly cost of the insulation

• The yearly cost of the natural gas

• The total cost of both

We observe that the total cost has a minimum at a given thickness of insulation.

Area

We have an ideal cabin shaped like a cube with a dimension of 15 feet. The area of the cube is $A = 6 \cdot (15\ \textrm{ft})^2$

area = 6 * (15 * u.feet)**2
area

1350 foot2

Insulation

We are getting our insulation from Home Depot.

• R-Tech 320821

• 10.48 for a 4 foot by 8 foot sheet

• R-value of 3.85 per sheet

We can calculate the price per square foot and then multiply by the area of the cabin.

$price = A * \frac{\textrm{price per sheet}}{\textrm{area of sheet}}$

number_sheets = 1
price_insulation = 10.48 * u.dollar / (32 * u.foot**2) * number_sheets
area * price_insulation

442.125 dollar

UA value

The UA value is the total area divided by the R value. This is the same as the total area multiplied by the U-value.

$UA = \frac{A}{R}$

r_value = 3.85 * u.hour * u.foot**2 * u.delta_degF / u.BTU * number_sheets
area / r_value

350.649350649 btu/(delta_degF hour)

Cost of Conserved Energy

Warning, error in this section

The cost of conserved energy is cost of the equivalent purchase of energy to a conservation measure.

This allows us to compare this metric to the cost of energy.

$CCE = \frac{\textrm{annual extra investment cost}}{\textrm{annual energy savings}}$

The key to this is a clear definition of the two scenarios you are comparing. The cost is the difference in cost and the energy is the difference in energy between the two scenarios.

Cost of conserved energy example

We can calculate the cost of conserved energy of adding an additional sheet of insulation. To do this, we compare two scenarios, for example, the energy of only one sheet of insulation and then the energy of two sheets and the cost of the additional sheet. Recall that our cost of energy in our previous calculation is 1 USD per therm.

Scenario 1

• One sheet of insulation

• 50.5 MMBTU

Scenario 2

• Two sheets of insulation

• 25.2 MMBTU

• 442 USD additional cost

• 44.20 USD yearly additional cost

$CCE = \frac{\textrm{annual extra investment cost}}{\textrm{annual energy avoided}}$ $CCE = 44.20 USD/25.3 MMBTU = 0.17 USD/therm$

Scenario 4

• Four sheets of insulation

• 12.6 MMBTU

Scenario 5

• Five sheets of insulation

• 10.1 MMBTU

• 44.20 USD yearly additional cost over scenario 4

We are comparing between scenario 5 and scenario 1

$CCE = 176.80 USD / 40.4 MMBTU = 0.44 USD/therm$

energy_price = 1e-5 * u.dollar / u.BTU
energy_price.to(u.dollar/u.therm)

1.0 dollar/thm

(44.20 * u.dollar / (2.5e6 * u.BTU)).to(u.dollar/u.therm)

1.768 dollar/thm

Cost of Conserved Carbon

This is a calculated example of the cost of conserved carbon for installing a heat pump water heater. It only takes into account the capital cost of the heater and neglects the energy savings.

from pint import UnitRegistry
u = UnitRegistry()
u.define('USD = []')
install_cost = 10e3 * u.USD
UA_value = 1500 * u.BTU / u.hour / u.delta_degF
degree_days = 6000 * u.delta_degF * u.day
HSPF = 17 * u.BTU / (u.watt * u.hour)
# https://www.eia.gov/tools/faqs/faq.php?id=73&t=11
natural_gas_carbon_intensity = 117 * u.pound / u.MBTU
carbon_intensity = 405 * u.pound / u.MWh
# crf of 1 is about a 5% 15 year loan
CRF = 0.1
cost_electricity = 0.10 * u.USD / u.kWh
cost_natural_gas = 1.0 * u.USD / u.therm

# fossil energy required scenario 1
energy = UA_value * degree_days
energy.to(u.BTU)

216000000.0 btu

# carbon emitted per year scenario 1
carbon_1 = (energy * natural_gas_carbon_intensity).to(u.kg)
carbon_1

11463.186374640001 kilogram

# cost of energy
cost_1 = (energy * cost_natural_gas).to(u.USD)
cost_1

2160.0000000000005 USD

# electricity required scenario 2
electricity = energy / HSPF
electricity.to(u.kWh)

12705.88235294118 kilowatt_hour

# electricity cost
cost_2 = electricity.to(u.kWh) * cost_electricity
cost_2

1270.588235294118 USD

# carbon emitted per year scenario 2
carbon_2 = (electricity * carbon_intensity).to(u.kg)
carbon_2

2334.1329722117653 kilogram

yearly_cost = install_cost * CRF
yearly_cost

1000.0 USD

# simple payback
install_cost / (cost_1 - cost_2)

11.243386243386242 dimensionless

ccc = (cost_1 - cost_2) / (carbon_1 - carbon_2)
ccc.to(u.USD / u.metric_ton)

97.42650475341813 USD/metric_ton