ECE 7800: Renewable Energy SystemsTopic 12: Economic Analysis of Renewable

Energy Systems

Spring 2010

© Pritpal Singh, 2010

Introduction to Energy Economics

When evaluating the economic viability of a renewable energy system, various factors must be taken into account including capital costs, operation and maintenance costs, fuel costs, escalation rates for the price of fuel, interest rates, etc.

Two approaches are commonly used to evaluate the economic viability of renewable energy systems:

1) Payback period analysis 2) Life cycle analysis

Net Present Value In order to calculate the payback

period of a renewable energy system, we need to incorporate the time value of money. Because of inflation, the value of money decreases over time. Also, money placed in a bank account or used to purchase bonds, can grow with interest over time. These factors must be accounted for in any economic analysis. A way of accomplishing this is through the net present value method.

Net Present Value (cont’d) In the net present value method, all future

costs are converted back to an equivalent present value (or worth) using a discount rate.

If we assume an annual interest rate of i, a principal amount P will be worth P(1+i) after 1 yr. After 2 yrs. it will be worth P(1+i)2. After n yrs. it will be worth P(1+i)n.

Thus future money, F, is worth F when (1+d)n

discounted back to the present at a discount rate, d.

Net Present Value (cont’d) Thus, $1,000 five years from now will be

worth $1,000 = $620.92 today

(1+0.1)5

assuming a 10% discount rate.

A renewable energy system will also deliver financial benefits year after year. To determine the financial benefits accumulated over several years, in present year dollars, we use the present value function (PVF). If the annual cash flow benefits are A, the present value of accumulated benefits, P = A.PVF(d,n) where d= discount rate and n=no. of yrs.

Net Present Value (cont’d)For a series of n annual amounts, starting from year 1, the total present value of the sum is given by:

The sum of this series is:

ndddndPVF

1

1...

1

1

1

1),( 2

n

n

dd

dndPVF

)1(

1)1(),(

Net Present Value (cont’d) The present value of all costs,

present and future, is called the life cycle cost of the system.

When choosing between two energy systems, their life cycle costs should be calculated and compared. The difference between these quantities is the net present value of the lower cost choice.

Net Present Value (cont’d)

Example 5.6

Net Present Value (cont’d) A simplified approach to calculating

the net present value is to compare the present value of future fuel savings (ΔA) to the extra first cost of the more efficient device (ΔP). Thus,

NPV = present value - added first

of annual savings cost

= {ΔA x PVF(d,n)} - ΔP

Internal Rate of Return (IRR) Another approach to calculating the

economics of a renewable energy system is to look at how the investment in a renewable energy system compares with an investment in a competing investment. This approach is termed the internal rate of return (IRR) method. IRR is the discount rate that makes the net present value of the energy investment equal to zero.

Internal Rate of Return (IRR) (cont’d)

From the earlier equation, we can write

NPV = {ΔA x PVF(IRR,n)} - ΔP = 0

Thus, PVF(IRR,n) = ΔP

ΔA

(simple payback period)

Tables of present value function can be helpful in solving this equation (see table on next slide). Also, spreadsheet programs can be used.

Internal Rate of Return (IRR) (cont’d)

Internal Rate of Return (IRR) (cont’d)

Another depiction can be provided as shown in the figure below:

Fuel EscalationThe NPV and IRR calculations should also take into account the rising cost of fuel over time. Of course, this has to be mitigated by the decrease in value of future dollars to pay for the fuel. Thus the present value function taking into account the fuel escalation rate, e, is given by:

n

n

d

e

d

e

d

enedPVF

1

1...

1

)1(

1

1),,( 2

2

Fuel Escalation (cont’d)An equivalent discount rate, d’, taking into account the fuel escalation rate, can be derived as follows:

Solving this equation gives:

d’ = (d-e)(1+e)

This equation can be used in all the previous equations for NPV and IRR calculations taking into account the rising cost of fuel over time.

)'1(

1

)1(

)1(

de

d

Fuel Escalation (cont’d)

Example 5.7

Annualizing the InvestmentAnother factor that must be taken into account in calculating the cost of energy investments is the cost of loans taken out to pay for the energy investments. The annual loan payments, A, is given by:

A = P x CRF (i,n)

where P is the principal borrowed and CRF(i,n) is the capital recovery factor given by:

PVFi

iiniCRF

n

n 1

1)1(

)1(),(

Annualizing the Investment (cont’d)

A table of capital recovery factors is shown below:

Annualizing the Investment (cont’d)If the loan payments are made on a monthly basis, the CRF can be re-written as:

per month.

1)12/(1

)12/(1)12/(),( 12

12

n

n

i

iiniCRF

Annualizing the Investment (cont’d)

Example 5.10

Levelized Bus-Bar Costs (cont’d)

In order to compare the cost per kWh of a renewable energy system vs. a fossil fuel system, the future price of fuel must be taken into account.

For a power plant, there are two main components to costs:

1) Up-front fixed cost to build the plant;

2) Recurring costs for fuel and operation

and maintenance (O&M) of the plant.

Levelized Bus-Bar Costs (cont’d)

A present value calculation is first

performed to convert the actual annual costs into levelized annual costs. If the annualized costs are A0 and a fuel escalation rate, e, is assumed, the levelized vs. actual costs are as shown:

Levelized Bus-Bar Costs (cont’d) The present value of the increasing

annual costs is given by:

Present value = A0.PVF(d’,n)

(annual costs)

The levelized annual costs are then

given by:

Levelized annual = A0[PVF(d’,n).CRF(d,n)]

costs

Levelized Bus-Bar Costs (cont’d)The quantity in brackets is called the levelizing factor and brings the future costs of fuel and O&M to a series of equal annual amounts. This levelizing factor is given by:

Note: For e=0, d’=d and LF=1.

1)1(

)1(.

)'1('

1)'1(n

n

n

n

d

dd

dd

dFactorLevelizing

Levelized Bus-Bar Costs (cont’d)

The impact of the levelizing factor can be high (see figure below):

Levelized Bus-Bar Costs (cont’d)

A fixed charge rate (FCR) should also be included that covers costs incurred even without the plant operating, e.g. depreciation, insurance, taxes, etc. The levelized fixed costs then become:

Levelized = capital costs x FCR

fixed costs 8760 h/yr x CF

Levelized Bus-Bar Costs (cont’d)

Example 5.11

Cash-Flow Analysis Commonly used to assess whether a

residential energy efficiency investment is economically sensible. See text pp. 254-256

Wind Turbine Economics Wind turbines are getting larger and some

economies of scale are leading to improved economics for such systems, particularly in high wind areas. The leading country in terms of per capita wind system installation is Denmark and the increasing size of wind systems in Denmark is shown below.

Wind Turbine Economics (cont’d) The average energy productivity per site has doubled (from 600kWh/yr. to about 1.2MWh/yr.) in the last 20 years because of more efficient machines, better wind sites, and increased tower heights.

The costs per kW has dropped as the wind turbines have gotten larger. The capital costs have dropped from about $1,400/kW in 1989 to approx. $800/kW.

Wind Turbine Economics (cont’d)

An example cost analysis for a 60MW wind farm is given below:

Wind Turbine Economics (cont’d)

To determine the levelized cost of energy delivered by a wind system, we need to divide the annualized costs by the annual energy delivered. Assuming a loan principal P and an interest rate, i, the annual loan payments are:

A= P. CRF (i,n) = P. i(1+i)n

(1+i)n-1

Wind Turbine Economics (cont’d)

Example 6.18

Wind Turbine Economics (cont’d)

See pp. 375-377 (including example 6.19) for example calculations for a wind farm.

Wind Turbine Economics (cont’d)

$0.00

$0.10

$0.20

$0.30

$0.40

1980 1984 1988 1991 1995 2000 2005

Levelized cost at excellent wind sites in nominal dollars, not including tax credit

38 cents/kWh

2.5-3.5 cents/kWh

Courtesy: American Wind Energy Association

Grid-Tied PV System Economics The cost of a grid-tied PV system

includes the costs of the PV array, inverter, and mounting hardware, wiring, circuit breakers, disconnect switches, etc. - generally referred to as balance-of-systems (BOS) costs – and installation costs. The economic viability of such a system is measured based on the cost of energy displaced, the cost of a loan to pay for the PV system, and any tax incentives/subsidies available.

Grid-Tied PV System Economics (cont’d) The detailed factors that come into

play are as follows:

• O&M costs;

• Future costs of utility electricity;

• Loan terms and tax situation of purchaser of PV power;

• System lifetime;

• Residual value when system is removed.

Grid-Tied PV System Economics (cont’d)

The industry uses $/Wpk as a cost estimator but this can be confusing. Does it mean cost of PV per peak output dc or ac (after the inverter)? Also, does it apply to a tracking system or fixed system? A way to include tracking systems is to incorporate a factor that takes account the additional power generated using tracking (see text pg. 544).

Grid-Tied PV System Economics (cont’d)

The installation costs of a residential system is shown below:

Grid-Tied PV System Economics (cont’d)

A simple way to estimate the cost of electricity generated by a grid-tied PV system is to assume that a loan is taken out by a residential customer to pay for the PV system and then using annual payments divided by annual kWh delivered to give the cost of electricity in ¢/kWh.

Grid-Tied PV System Economics (cont’d)The annual loan payments are given

by:

A = P. CRF(i,n)

Examples 9.11 and 9.12

Grid-Tied PV System Economics (cont’d)

The figure below shows how the cost of electricity varies as a function of average daily insolation throughout the year and installed system cost.