1 Levelized Cost Analysis 1 LEVELIZED COST ANALYSIS U.S.-Denmark Summer Workshop in Renewable Energy Levelized Cost Analysis 2 Economic Analysis and Optimization • Assess economic feasibility of energy systems • Identify anticipated cost of energy (COE) and other measures of economic performance using consistent methodologies • Compare alternatives over different time frames • Evaluate economies of scale and potential for optimization
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Levelized Cost Analysis 1
LEVELIZED COST ANALYSISU.S.-Denmark Summer Workshop in Renewable Energy
Levelized Cost Analysis 2
Economic Analysis and Optimization• Assess economic feasibility of energy
systems
• Identify anticipated cost of energy (COE) and other measures of economic performance using consistent methodologies
• Compare alternatives over different time frames
• Evaluate economies of scale and potential for optimization
2
Levelized Cost Analysis 3
Economic Analysis and Optimization• Review of economic analysis
Cash flow equivalence
Time value of money
Depreciation
Taxes
Escalation and inflation• Current and constant dollar analysis
Evaluation of alternatives• Present worth
• Levelized cost
• Rate of return
• Benefit/Cost
• Revenue Requirements method
• Sensitivity analysis
• Optimization
Levelized Cost Analysis 4
Cash Flow Equivalence
• Investment of money is intended to generate additional money ("return“ or “interest”)– Money placed in a bank account ("loaned" to the bank) is
anticipated to earn interest– If the interest earned is reinvested (e.g., left in the bank account),
then additional interest is to be earned on the interest already earned. This is known as compounding.
• Under compounding, cash flows of various types can be found to be equivalent
• A future sum of money resulting from the investment of a present sum of money is to be greater than the present sum (exclusive of inflation) due to the effects of interest
3
Levelized Cost Analysis 5
Future amount• A future sum, F ($) can be found from the
present sum, P ($), known as the present worth or present value, given an interest rate, i (period-1), and a number of compounding periods, n, as
• The factor (1+i)n is called the compound amount factor, F/P
F = P(1+i)n
time n1 2P
F
0
Levelized Cost Analysis 6
Cash Flow Diagram• By convention, an arrow pointing
downwards represents an investment, an arrow pointing upwards represents a return.
• Note that for future amounts, an end of period convention is utilized for a present worth occurring at time "zero"
time n1 2P
F
0
4
Levelized Cost Analysis 7
Present worth
• The present worth, P ($), of a future sum of money is
• The factor (1+i)-n is called the present worth factor, P/F
time n1 2P
F
0
ni
FP
)1(
Levelized Cost Analysis 8
Uniform (Level)Cash Flow
• An equivalence can be found between a present worth or a future worth and a uniform series of cash flows, designated A(for annuity, although the periods can be other than years)
• The uniform series of cash flows begins at the end of period 1 in the following analysis
timeP
F
n0
A
1 2
5
Levelized Cost Analysis 9
Uniform (Level)Cash Flow timeP
F
n0
A
1 2
1)1(
)1(n
n
i
iiPA
the term in brackets is known as the capital recovery factor, A/P, or sometimes, CRF
1)1( ni
iFA
the term in brackets is known as the sinking fund factor, A/F
Note that A/P = A/F + i
Note that if n = ∞, then A/P = i
Levelized Cost Analysis 10
Depreciation• Depreciation accounts for the loss in value of assets over time, due
to wear-out, obsolescence, or economic management. • Accounting for depreciation is necessary to ensure that replacement
assets can be purchased at the end of the useful life of the existing asset.
• Depreciation is also an important tax deduction. • Several depreciation methods may be used. For tax purposes the
tax code normally specifies the method to be used. • Accelerating the depreciation allowance beyond the constant rate of
straight line depreciation is beneficial for tax purposes if the business is profitable.
• Modified accelerated cost recovery systems (MACRS) are defined for U.S. federal tax purposes.
6
Levelized Cost Analysis 11
Depreciation: Straight Line
• Depreciation allowance to be taken in each year, Dn, by the straight line method can be found as
l
SBDn
B = Book value at the beginning of the yearS = Salvage value of assetl = remaining useful life at the beginning of the year
Levelized Cost Analysis 12
Depreciation: Straight Line
• The straight line method will depreciate the asset to its salvage value.
• The book value is the value of the asset, equal to the first cost less accumulated depreciation.
• Commonly, the value Dn is the same in each year and computed as
P = first cost (or purchase cost) of assetN = total useful or economic life of asset
(P - S)/N
7
Levelized Cost Analysis 13
Depreciation: Declining Balance
• Declining balance depreciation allowance taken in each year is
• Double (or 200%) declining balance is commonly used, in which case, = 2
• Only by coincidence will declining balance depreciate to the salvage value
= decline rate
BN
Dn
Levelized Cost Analysis 14
Depreciation: Declining Balance• The book value at the end of n years, (beginning
of n+1 years), is P(1 - /N)n, so that
• At some point, the depreciation allowance by declining balance may fall below that by straight line based on the remaining book value and economic life
• The depreciation method can be switched at that time to use straight line on the remaining book value while maintaining the highest depreciation allowance in each year
• Taxes are fees assessed on earnings and other transactions, generally determined as
T = (t)(It)
T = taxes ($)t = tax rate (decimal)It = taxable income ($)
9
Levelized Cost Analysis 17
Taxable income
• Taxable income is gross income adjusted by deductions
It = Ig - E - Dt - Di
Ig = gross income ($)E = expenses ($)Dt = depreciation for tax purposes ($)Di = interest on debt ($)
Levelized Cost Analysis 18
Depreciation for tax purposes• Tax depreciation procedures will normally
specify the schedule of depreciation, including how to handle salvage, although they are usually based on one or more of the previous methods e.g., double declining balance switching to
straight line
possibly with restrictions on the amount of depreciation allowed in the first year (e.g. MACRS half-year convention in first and last years)
10
Levelized Cost Analysis 19
Combined Tax Rate
• For federal purposes, state tax is a deduction, but federal tax is not normally considered a deduction for state tax
• A combined tax rate, t, can be computed:
t = tF (1 - tS) + tS
tF = federal tax rate (decimal)tS = state tax rate (decimal)
Levelized Cost Analysis 20
Escalation and Inflation
• Inflation results in an increase in the cost of a good or service over time and a decrease in purchasing power, and is generally modeled in a manner similar to interest rate
• Defining:• f = inflation rate (period-1)
• i = apparent (or quoted) interest rate
• i’= real interest rate (accounting for the effect of inflation)
11
Levelized Cost Analysis 21
Inflation
• Under inflation, a future amount of money, F, will not have the same purchasing power as if there had been no inflation.
• Expressed in the same value units as when the investment was made, that is, in time zero dollars, the year zero equivalent future value is
nof
FF
1
time n1 2
Fo
F
0
Levelized Cost Analysis 22
Inflation
• The present worth of Fo is
• P is related to F in the same way as before
time n1 2
F
0P
nnnno
i
F
fi
F
i
FP
)1()1('1'1
Fo
12
Levelized Cost Analysis 23
Inflation
• The apparent interest rate is
• The real interest rate is
fififii ''1)1)('1(
f
fi
f
ii
1
11
1'
Levelized Cost Analysis 24
Current and Constant Dollars
• A current dollar analysis includes the effect of inflation
• A constant dollar analysis attempts to adjust for the effect of inflation so that economic values may be compared on an equivalent basis typically expressed for some base year
for example, comparisons of the cost of alternative fuel resources in the future to the present cost of existing resources, exclusive of general inflation in the economy
13
Levelized Cost Analysis 25
Current and Constant Dollars• Example: consider that the cost of fuel, C,
over time will escalate at an apparent escalation rate, e, such that
• The present worth of Cn is
Cn = Co(1+e)n
Co = present cost of fuelCn = escalated cost of fuel in the future
non
n
onn
n kCi
eC
i
CP
)1(
)1(
)1(
k = (1+e)/(1+i)
Levelized Cost Analysis 26
Current and Constant Dollars• The total present worth of the fuel cost
over the entire analysis time is
• Factoring k
• Combining
nN
no kCP
1
nN
no
kkC
P 1
11
)1(
)1(
k
kkCP
N
o
14
Levelized Cost Analysis 27
Current Dollar LAC
• The current dollar levelized annual cost, LAC, is
1)1(
)1(
)1(
)1()/)((
N
NN
o i
ii
k
kkCPAPLAC
Levelized Cost Analysis 28
Constant Dollar LAC
• The constant dollar levelized annual cost adjusts for inflation by using the real escalation and interest rates
ki
ek
)'1(
)'1(' 1
)1(
)1('
f
ee 1
)1(
)1('
f
ii
1)'1(
)'1('
)'1(
)'1(''
N
NN
o i
ii
k
kCkLAC
15
Levelized Cost Analysis 29
Evaluation of Alternatives• Alternative energy technologies or strategies may be
compared economically using any of a number of different rational methods, all of which are equivalent and yield the same decision if employed properly.
• Other techniques, such as simple payback analysis, have different objectives and do not necessarily yield the same decision.
• Payback is useful when the rapid recovery of capital investment is required, however, it will not necessarily select the alternative that maximizes the amount of money made over the economic life (more precisely, that maximizes net present worth, NPW).
• Net Present Worth, Levelized Cost, and Benefit/Cost analysis assume an interest or discount rate.
• Rate of return analysis computes the interest rate earned on investment equal to that rate which makes NPW=0
Levelized Cost Analysis 30
Evaluation of AlternativesMethod Objective
Present Worth Compute net present worth (present worth of benefits less presentworth of costs) of alternatives.•Maximize Net Present Worth (NPW)
Levelized Cost Compute net level (typically annual) cash flow (level benefits less levelcosts) of alternatives.•Maximize Net Level Benefits•Minimize Net Level Costs
Rate of Return (ROR)
Find i = ROR such that NPW=0For feasible alternatives with ROR ≥ Minimum attractive rate of return(MARR), compute incremental ROR (∆ROR) between alternatives:•If ∆ROR≥MARR, choose higher cost alternative.•If ∆ROR<MARR, choose lower cost alternative.
Benefit-Cost Ratio (B/C)
Find feasible alternatives with ratio of present worth of benefits (PWB)to present woth of costs (PWC) greater than unity, i.e., PWB/PWC ≥ 1.Compute incremental B/C (∆B/C) between feasible alternatives.•If ∆B/C≥1, choose higher cost alternative.•If ∆B/C<1, choose lower cost alternative.
16
Levelized Cost Analysis 31
Evaluation of Alternatives
• Example: Choice of two equal-outcome alternatives in a production activityCategory: Alternative 1 Alternative 2
Capital Cost ($) 8,000 23,500
Operating Cost ($/y) 100 1,000
Energy Cost ($/y) 17,935 12,267
Gross Revenue ($/y) 19,815 19,815
Net Revenue ($/y) 1,780 6,548
Life (y) 10 10
Interest (Discount) Rate (%/y) 12 12
Levelized Cost Analysis 32
Evaluation of Alternatives: ExampleCategory: Alternative 1 Alternative 2
Capital Cost ($) 8,000 23,500
Operating Cost ($/y) 100 1,000
Energy Cost ($/y) 17,935 12,267
Gross Revenue ($/y) 19,815 19,815
Net Revenue ($/y) 1,780 6,548
Life (y) 10 10
Interest (Discount) Rate (%/y) 12 12
1. Present Worth Analysis—Objective: Max NPW
Alternative 1: NPW (Alt1) = -8,000 + 1,780(P/A, 12%, 10)
= -8,000 + 1,780(5.65) = +2,057
Alternative 2: NPW (Alt2) = -23,500 + 6,548(5.65) = +13,496
Decision: Max NPW--choose Alternative 2
17
Levelized Cost Analysis 33
Evaluation of Alternatives: Example
2. Levelized Cost—Objective: Max Net Level Benefits
Choose higher cost alternative, choose Alternative 2
Category Difference Alt 2 – Alt 1
Capital Cost ($) = -23,500 – (-8,000) = -15,500
Net Revenue ($/y) = 6,548 – 1,780 = 4,768
Levelized Cost Analysis 38
Revenue Requirements Method
• Approach is similar to the levelized cost method
• A revenue requirements analysis attempts to determine the necessary energy price to yield the stipulated rate of return
• The approach has commonly been used by regulated utilities to establish prices, but has general application
20
Levelized Cost Analysis 39
Revenue Requirements Method• The revenue requirements fall generally into four
categories: Capital repayment Return on investment Expenses Taxes
• Other categories may include tax credits and other subsidies
• Capital repayment recovers the capital cost of the project over the economic life. The return represents the interest earned on the investment over the economic life.
Levelized Cost Analysis 40
Revenue Requirements Method: Taxes• Because the revenue requirements method specifies the rate of return to be
earned, taxes, which are part of the revenue requirement, are computed in a special way, rather than directly as if the revenues were known a priori.
• The tax payment in each year of the analysis is
T = (t)(It)• The taxable income, It, is
It = Ig - E - Dt - Di
• The revenue requirements, Ig, are
Ig = Cr + Ir + T + E
Cr = capital (principal) repayment ($)Ir = return on investment ($)
21
Levelized Cost Analysis 41
Revenue Requirements Method: Taxes• The taxes are
T = t(Cr+ Ir + T + E - Dt -E - Di)= t(Cr + Ir + T - Dt - Di)
• Solve for T
• The term Ir - Di , the difference between the total return on the unrecovered investment and the interest on debt, is simply the return on equity.
• Defining the debt ratio, rD (decimal), as the fraction of the cost of the project covered by debt (as opposed to equity), and C as the unrecovered investment to date, the total return at rate of return, i, is
Ir = iC
)(1 irtr DIDC
t
tT
Levelized Cost Analysis 42
Revenue Requirements Method: Taxes• the interest or return on debt is
Di =iDrDC
• iD is the interest rate on debt
• Then
• Taxes are
Di
Ir
iDrD
i
i
riIDC
t
tT DD
rtr 11
22
Levelized Cost Analysis 43
• Power plant is assumed to burn biomass in a fluidized bed furnace, with a net electrical capacity of 25 MWe and an availability of 85% equal to the capacity factor
• The cost of the plant ($70M) is covered by both debt and equity funds 75% borrowed from the bank (debt), 25% equity supplied from the owners
• The rate of return on equity is assumed to be 15% • The debt interest rate is 5% • As part of the financing plan, the owner is required to place the equivalent of one year
of debt repayment in a savings account in case technical problems cause a suspension of plant operations and revenue is not generated. This "debt reserve" account earns simple interest, which is available to the plant as revenue.
• The plant also receives a payment for "capacity," when it can guarantee a certain minimum level of performance. This payment and the interest on the debt reserve, reduce the amount of revenue required to cover the cost of generating the electrical energy.
• A federal accelerated depreciation schedule (MACRS) based on double declining balance switching to straight line is used for tax purposes.
• Negative taxes are permitted under the assumption that the owner-company is large, and negative taxes merely reduce the tax burden of the company as a whole. If this is not the case, then negative taxes are not permitted. Carry-forward of taxes is not considered for this example. A production tax credit for open loop biomass is included
• Fuel costs and expenses are assumed to escalate at 2.1% annually, with inflation also running at 2.1%.
Revenue Requirements Method: Example Biomass Power Plant
Levelized Cost Analysis 44
• Model calculation in Excel
Revenue Requirements Method: Example Biomass Power Plant
23
Levelized Cost Analysis 45
Sensitivity Analysis
Levelized Cost Analysis 46
Optimization
• The opportunity often exists to optimize the energy system in some way, for example, the size or capacity of the system
• An optimization requires an objective function expressing explicitly how the optimum value is to be determined
• Tradeoffs among capital, operating, and fuel costs influenced by size or scale frequently lead to an optimum providing either minimum cost or maximum profit
24
Levelized Cost Analysis 47
Economy of Scale
• An economy of scale is said to occur when the cost per unit output of production varies depending on the size or scale of the facility. A common expression is:
Cp = capital cost of the plant ($) Co = capital cost of some base or reference size plant ($) Mp = size of the plant (e.g., kW) Mo = reference size (e.g., kW) s = scale factor (dimensionless)
Cp
Co
Mp
Mo
s
Levelized Cost Analysis 48
Economy of Scale
• For capital costs expressed per unit of capacity:
• K is the cost per unit of installed capacity (e.g., $ kW-1), K = C/M
Kp
Ko
Mp
Mo
s1
25
Levelized Cost Analysis 49
Objective Function
• Analytic function defining how the optimum condition is to be determined
• Example: Minimization of electricity production cost
• Note this does not address delivery into the final demand sector. Full optimization should consider this as well.
min[Cost of Electricity = Capital Cost + Operating Cost
+ Fuel Cost]
Levelized Cost Analysis 50
Optimization: Example
• Biomass power plant subject to economy of scale in both capital and non-fuel operating costs
• Option 1: s = constant
• Option 2: s = variable to account for greater risk at large scale
26
Levelized Cost Analysis 51
Capital Cost
• Annualized capital cost is Cp(A/P) where (A/P) is the capital recovery factor
• Annual energy production is Mph where his the number of hours per year the plant operates at its average capacity Mp
• Capital cost contribution to the levelized cost of energy, C ($ kWh-1):
Cp Kp Mp Ko
Mos 1
Mps
C (A / P)Cp
Mph
(A / P)Ko
h
Mp
Mo
s 1
Levelized Cost Analysis 52
Non-fuel O&M Costs
• Subject to economy of scale, operating and maintenance costs may scale as:
• the prime denotes that the costs are total annual costs (i.e., $ y-1)
• The value of s for O&M need not be the same as that for the capital cost
R
R o
Mp
Mo
s
27
Levelized Cost Analysis 53
Non-fuel O&M Costs
• Per unit energy output:
• R is the cost per unit energy ($ kWh-1)
R R
Mph
R Ro
Mp
Mo
s 1
Levelized Cost Analysis 54
Fuel Cost
• Biomass feedstock (fuel) cost derived for a rectangular road grid:
Davg 1
q1/ 2TQ1/ 2 2T 1/ 2
q = average biomass feedstock density in the area (t km-2) Q = annual powerplant feedstock demand (t) T = average truck payload (t)
28
Levelized Cost Analysis 55
Fuel Cost
• Total fuel cost (F', $ t-1) is composed of two components: acquisition cost (F'p, $ t-1) including the cost of
production, harvesting, storage, and processing, and
transportation cost (F't, $ t-1)
• Define F"t as the unit cost of transportation ($ t-1 km-1)
Levelized Cost Analysis 56
Fuel Cost
• Total fuel cost contribution to COE:
• Fuel cost contribution to the price of output electrical energy , F, $ kWh-1:
F F p F t F p FtDavgT
F 3.6 F
Qf Qf = heating value (or heat of combustion of the fuel), MJ t-1
= plant thermal efficiency (dimensionless)
29
Levelized Cost Analysis 57
Fuel Cost• Total fuel demand of the powerplant, Q, is
a function of the plant size:
• If Q>>T:
• E = Qf /3.6 (kWh t-1) is the reciprocal of the fuel rate
Q 3.6Mph
Qf
DavgT Q
q
1/ 2
Mph
qE
1/ 2
Levelized Cost Analysis 58
Fuel Cost• Fuel cost contribution to COE:
F F p
E
Mph
q
1/ 2F t
E 3/ 2
30
Levelized Cost Analysis 59
Objective Function
• Minimize COE:
min Pe C R F aMps 1 bMp
1/2 c
a Mo1 s (A / P)Koh
1 Ro b Ft
h1/ 2q1/ 2E3/ 2
c F p E1
Levelized Cost Analysis 60
Optimum Capacity
• The minimum, or optimal, value of Pe is found by taking the derivative of Pe with respect to Mp and setting equal to zero (dPe/dMp = 0)
• With F'p and F"t constant:
• where Mopt is the optimal size (kW) which minimizes Pe.
Mopt 2a(1 s)
b
1
1.5 s
31
Levelized Cost Analysis 61
Example: fixed s
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1 10 100 1,000 10,000
M (MW)
($/k
Wh
)
Capital + O&M Cost Feedstock Cost Total Cost
s = constant
Minimum COE
Mo = 25,000 kWKo = $2,000 kW-1
Ro = $0.02 kWh-1
h = 6,570 h y-1
75% capacity factorF"t = $0.1 t-1 km-1
q = 1000 t km-2
E = 1,389 kWh t-1
Qf = 20,000 MJ t-1
= 0.25i = 0.06l = 20 yearss = 0.9179*
Capital/O&M
Mopt = 1,253 MW
*derived from variable s model, see following
Note limited curvature in region of minimum
Levelized Cost Analysis 62
-25
-20
-15
-10
-5
0
5
0 0.2 0.4 0.6 0.8 1
s
dM
op
t/d
s (G
W)
0
500
1000
1500
2000
2500
Mo
pt
(MW
)
Influence of Scale Factor
on Optimal Capacity
and Sensitivity of Solution
(s = constant)
32
Levelized Cost Analysis 63
SensitivityFunctions
0
1
2
3
4
5
0 1 2 3 4 5
Relative z
Rel
ativ
e M
op
t
Ct
b
Mo
qRo
Ko
H
h
s
z=
F”=Ct
Y/X
Qf
z
Mopt
z
q 1
3 2sa4
1/ (1.5 s )q(s 1) /(1.5 s ); a4 2a1
a2˜ q 1/ 2
1 s Mo1s
Ko a5
1.5 s
a2
1/(s1.5) a5Ko a6 (s0.5)/ (1.5s);
a5 2f
h1 s Mo
1s, a6 2Ro 1 s Mo1s
Ro a7
1.5 s
a2
1/(s1.5) a7Ro a8 (s0.5)/(1.5s);
a7 2 1 s Mo1s, a8 2
f
hKo 1 s Mo
1s
H 1
12
3s
a91/(1. 5s )H (s) / (1.5 s) ; a9 2
a1
a2˜ H 3/ 2
1 s Mo1 s
1
12
3s
a101/(1.5 s) (s)/ (1.5 s ); a10 2
a1
a2˜ 3 /2
1 s Mo1 s
Ct 1
s 1.5a11
1/(1.5 s)Ct(2.5 s) / (s 1.5); a11 2
a1
a2˜ C t1
1 s Mo1s
1
s 1.5a12
1/ (1.5 s) (2. 5s )/(s 1.5); a12 2a1
a2˜ 1
1 s Mo1 s
f a13
1.5 s
a2
1/(s1.5) a13f a6 (s0.5)/ (1.5s);
a13 2Ko
h1 s Mo
1s
h 1
1.5 sa14h
3 /2 a15h1/ 2 (s 0.5) / (1.5 s)
3
2a14h
5/ 2 1
2a15h
3/ 2
;
a14 2
a2˜ h 1/ 2
fKo 1 s Mo1 s, a15
2
a2˜ h 1/ 2
Ro 1 s Mo1 s
Mo 1 s
1.5 sa16
1/(1.5s)Mo1/ (2s3); a16 2
a1
a21 s
b
1
1.5 sMopt
1 b
2b 1 b
s 1
1.5 sMopt
1
1.5 sln Mopt
1.5s 1
s1 ln Mo
˜ z indicates reference value to be inserted.
F”=Ct
Y/X=b
Qf=H
Levelized Cost Analysis 64
Variable s
• Use of fixed scaling factor, s, for all sizes ignores potential risks as the size becomes very large
• Large coal and nuclear plants (> 400 MW) trend toward s > 0.9
• Model with s = variable yields different optimal capacity depending on function used, s = f(M)