Levelized costs are calculated correctly

The Utility Dive recently published an opinion article that claimed that the conventional method of calculating the levelized cost of energy (LCOE) is incorrect. The UD article was derived from an article published in 2019 in the Electricity Journal by the same author, James Loewen. The article claimed that conventional method gave biased results against more capital intensive generation resources such as renewables compared to fossil fueled ones. I wrote a comment to the Electricity Journal showing the errors in Loewen’s reasoning and further reinforcing the rationale for the conventional LCOE calculation. (You have until August 9 to download my article for free.)

I was the managing consultant that assisted the California Energy Commission (CEC) in preparing one of the studies (CEC 2015) referenced in Loewen. I also led the preparation of three earlier studies that updated cost estimates. (CEC 2003, CEC 2007, CEC 2010) In developing these models, the consultants and staff discussed extensively this issue and came to the conclusion that the LCOE must be calculated by discounting both future cashflows and future energy production. Only in this way can a true comparison of discounted energy values be made.

The error in Loewen’s article arises from a misconception that money is somehow different and unique from all other goods and services. Money serves three roles in the economy: as a medium of exchange, as a unit of account, and as a store of value. At its core, money is a commodity used predominantly as an intermediary in the barter economy and as a store of value until needed later. (We can see this particularly when currency was generally backed by a specific commodity–gold.) Discounting derives from the opportunity cost of holding, and not using, that value until a future date. So discounting applies to all resources and services, not just to money.

Blanchard and Fischer (1989) at pp. 70-71, describe how “utility” (which is NOT measured in money) is discounted in economic analysis. Utility is gained by consumption of goods and services. Blanchard and Fischer has an extensive discussion of the marginal rate of substitution between two periods. Again, note there is no discussion of money in this economic analysis–only the consumption of goods and services in two different time periods. That means that goods and services are being discounted directly. The LCOE must be calculated in the same manner to be consistent with economic theory.

We should be able to recover the net present value of project cost by multiplying the LCOE by the generation over the economic life of the project. We only get the correct answer if we use the conventional LCOE.  I walk through the calculation demonstrating this result in the article.

15 thoughts on “Levelized costs are calculated correctly

    1. Richard McCann Post author

      We’ve already had an extended discussion about this via email. This is not about a fundamental mathematical question–it is about a fundamental conceptual question that needs to be answered before delving into the mathematical formula. Dr. Szymanski makes a fundamental error in overlooking that the discount rate applies to all resources, not just to money. Therefore both the numerator and the denominator in the levelized cost calculation must be discounted. His comment erroneously discounts only the numerator.

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      1. Adam Szymański

        In my article, I presented a method of introducing the LCOE definition from the costs-revenues equilibrium equation. It is a consistent method from an economic and mathematical point of view. I propose that Dr. MCann introduce a similar model. Then a constructive discussion will be possible.

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      2. Richard McCann Post author

        I already showed that the current conventional LCOE method is correctly established. The correct formulation calculates the net present value of both the monetary streams and the physical production streams and divides the monetary NPV by the physical units NPV. The mathematically equivalent calculation is to calculate the annualized payment and production amounts and divide the payment by the production. All of this has nothing to do with the principles of calculus. The method you propose is not economically consistent because it discounts the monetary stream but does not discount the physical production stream. Money is not a separate entity–it is a unit of barter between two types of physical units. Therefore the physical units also need to be discounted. I explained all of this in my comment on Loewen’s initial article.

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      3. Adam Szymański

        Dr. MCCann’s definition of the LCOE constant; LCOE = NPV (costs) / NPV (production) does not satisfy the cost-revenues equilibrium equation. The definition of LCOE should follow from fundamental economic principles, in this case from the equilibrium equation. NPV (production) has no rational economic justification. Discounting physical quantities is a pure nonsense. In my article, I showed a mathematically and economically consistent technique for defining LCOE. Production is technically degraded due to the depreciation of the quality of the equipment used for production, and this has nothing to do with the cash flow discounting function. This is clearly highlighted in my article. The degradation function is an equivalent of the discounting function for physical quantities. In the article, I showed where the formal assumption leads; discount function = degradation function. This is exactly what J. Loewen described. NPV (production) does not follow the economic definition of the discounting function.

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      4. Richard McCann Post author

        I showed in my initial comment that the method that you’re proposing leads to an underestimate of the total NPV for the entire project. life. Please read that section again where I demonstrate that mathematical fact.

        As for discounting physical quantities, I’ll pose the classic introductory economics example: Would you rather have a piece of cake today or one year from now? Would it take two pieces of cake a year from now to be equal to a piece of cake today? Of course you would prefer your cake now. That is by definition discounting of a physical product–there is no money involved. We know that five year olds show different levels of patience, which is the definition of discounting: https://www.vox.com/science-and-health/2018/6/6/17413000/marshmallow-test-replication-mischel-psychology.

        Discounting is NOT about a physical immutable process like gravity–it is entirely based in the human experience and how we perceive time and how that affects our preferences. It has nothing to do with degradation (that’s picked up in the depreciation rate, which is NOT the discount rate.) If you have an economic citation that shows how money is discounted while all other physical products and services have exactly the same experiential value in the future as today, then provide those citations. You need to provide much more than just your assertions that are based on a misunderstanding about discounting. I provided citations showing that products and services are discounted.

        We have nothing further to discuss if you can’t accept the principle that discounting extends beyond just money. Your method might be mathematically consistent only IF you separately treat money as a completely separate good with a unique set of properties. I discussed this issue in my comment and don’t yet have anything further to add on this. Loewen saw that his approach was incorrect based on this discussion which is why he withdrew his original article and submitted a different one that relied on the conventional method.

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      5. Adam Szymański

        Yes, my method is economically consistent because I treat money as a completely separate good with a unique set of properties. This results from the costs-revenues equilibrium equation, and this in turn is a mathematical description of what I wrote above. Unfortunately, your method relies on a verbal description of the problem and lacks a consistent mathematical approach, making it wrong. Of course, technical degradation has nothing to do with the discounting procedure. One could ask completely formally; what happens when the discount function equals the function of technical degradation? Then we get, once again, that your method does not make sense. This is also a mathematical proof. If you want to question it, you have to use purely mathematical methods, not verbal deliberations.

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      6. Richard McCann Post author

        First, money is not a stand alone “good”. Money is a means of transference of value amongst different goods and services. You cannot consume or directly use money to satisfy a human need or want–it is only a means of acquiring those goods and services. If you continue to assert this position, you need to provide a peer-reviewed citations that support this.

        Second, your method is not economically consistent. Let’s start with the definition of “levelized cost”. Levelized cost is the annual cashflow that arrives at the total net present value when summing the discounted stream of those cashflows. Mathematically, the equation is:

        NPV = sum(t)[CF(t)/(1+r)], with CF = levelized cost, where CF(t) = cost C(t) * output O(t) and sum(t) is the sum of the equation over time (t)

        That means that the annual CF is discounted over time and the CF is a function of the price or cost per unit times the physical output of the resource. So the equation can be expanded to

        NPV = sum(t)[(C(t) * O(t))/(1+r)] = sum(t)[C(t)/(1+r) * O(t)/(1+r)]

        Thus the physical output is also discounted in this equation as shown by the term O(t)/(1+r)

        We can see how this works in our every day life when examining load payments on a house or a car. The output of a house or car from this perspective is a period (month or year) of service Y, e.g., the year that we reside in a house. The loan payment can be calculated using this equation:

        Payment = Loan amount / sum(t)[Y(t) / (1+r)] where Y

        We can show an example with a loan of $500,000 over 30 years at a 3% mortgage rate. Using the Excel PMT function and adjusting to monthly, we arrive at a payment of $2,108 per month. Alternatively, we can calculate this by taking the present value of the months of housing service: sum(t)[1 month(t) / (1.0025)] That present value is 237.19 months (vs. a nominal total of 360 months):

        $500,000 / 237.19 months = $2,108 per month

        This is the levelized cost of homeownership.

        Given that the entire finance industry is premised on this equation, you are asserting that the entire structure of financial transactions are wrong.

        Before I approve another comment from you, you will need to provide the answers to these three questions. If you don’t answer these questions, I will assume that you have conceded the validity of my points and I will not publish your attempts to distract from those points or try to obscure it by using a mathematical method that is beyond what is required to demonstrate these points:

        (1) If given a choice when offered a piece of cake for dessert, most of the time would you prefer to eat that cake now or a week from today? If offered time off from work for a vacation, would you prefer that time off now or a year from now (generally)? If offered an additional benefit at work such as a new office or a new computer, would you prefer to receive the benefit today or a year from now?

        (2) Provide citations to peer reviewed journal articles that demonstrate that money is a separate good with unique properties whereas money is the only good or service that declines in value over time, and provide citations to peer reviewed articles that demonstrate that the value of physical goods and services hold the same intrinsic value to people regardless of how far into the future those goods and services will be consumed.

        (3) Provide a numerical example where the NPV of the stream of annual cashflows represented by your version of the LCOE provides sufficient amount to cover the entire initial capital cost of the resource. (You will find that it underrecovers.)

        By the way if technical degradation has nothing to do with discounting, why did you reference that relationship in your previous comment?

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      7. Richard McCann Post author

        An additional point on the home mortgage example: If we use your proposed LCOE method which implies dividing the house loan of $500,000 by the undiscounted total months (360), the monthly loan payment would be $1,389. The NPV of that stream of payments is only $329,430, far short of the loan amount.

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      8. Adam Szymański

        NPV = sum(t)[CF(t)/(1+r)], with CF = levelized cost, where CF(t) = cost C(t) * output O(t) and sum(t) is the sum of the equation over time (t).

        The correct definion is as follows,
        NPV = sum(t)[CF(t) x D(t)]
        where D(t) – non- dimensional discount function
        CF(t) – cash flow [USD]
        Hence,

        NPV [USD].

        Now,
        NPV = sum(t)[CFdis(t)],
        where CFdis(t) = CF(t) x D(t),
        NPV = sum(t)[(CFdis(t)/O(t)) x O(t)],
        where O(t) – pyhsical output [ appropriate unit].
        In this case NPV [USD]. The unit must be consistent !

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      9. Richard McCann Post author

        This discussion reminds me of the economist’s joke “Sure it works in practice, but does it work in theory???” You still haven’t answered my three questions, but I approved this comment so that I can show that your theoretical method does not work in practice.

        Let’s start with a fundamental of electricity power contracts which is directly tied to the LCOE. All power purchase agreements (PPA) in the U.S. are priced in $/MWH (as I know from working in the industry for more than 35 years.) The annual cashflow to the plant developer/owner equals $/MWH * annual MWH output. The developer/owner is planning on receiving over the term of the PPA at least the net present value of the revenues or cashflow equal initial capital cost of the plant. This is the identical principle of repaying a loan as I discussed in my previous response.

        So let’s compare the outcomes of the two LCOE methods based on this fundamental of PPA pricing. Assume the following parameters for a renewable power plant’s costs, annual output, cost of capital (=discount rate), and PPA term:

        Renewable plant size 100 MW
        Capital cost per MW $1,000,000 per MW
        Plant Cost (PV) $100,000,000

        Cost of capital/discount rate 7%
        Book life 30 years

        Renewable plant output 200 GWh

        The annual cashflow target is $8,058,060 over the 30 year term of the PPA. The total nominal undiscounted output over the 30 year term is 6,000 GWH.

        The conventional LCOE can be calculated in two ways from this information. The first, as shown in equation (1) of your comment, is the NPV of the annual cashflow targets, which equals the initial investment of $100,000,000, divided by the NPV of annual generation output. Over 30 years, that is 2,482 GWH. The LCOE equals $100,000,000 / 2.482 GWH = $40.29/MWH. The second method is to divide the annual revenue requirement by the annual generation. (The annual generation amount = annualized generation amount from the NPV of lifetime generation.) The LCOE equals $8,058,060 / 200 GWH = $40.29/MWH, the same amount as the first method. The annual cashflow is $40.29/MWH * 200 GWH = $8,058,060. The NPV over 30 years equals $100,000,000. The developer/owner will receive a satisfactory return on investment with this price.

        Equation 4 in your comment leads to the equation: Capital investment/Total lifetime output, or $100,000,000 / 6,000 GWH = $16.67 per MWH. The annual cashflow of $16.67/MWH * 200 GWH per year is $3,333,333. The NPV of that cashflow over 30 years is only $41,363,471, well short of the required NPV of $100,000,000. (I showed a similar example in my initial comment.) No plant developer would sign a PPA based on this price where they receive only 41.4% of the NPV of their initial investment.

        The LCOE is based on the methodology for determining price terms in PPAs and it must reflect those PPA terms to be useful for comparing resource costs. Any LCOE method that does not comply with this singular requirement fails this test.

        You will need to answer the other two questions that I posed before I will approve your response. (I’ve provided the real world example that I asked of you.)

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      10. Adam Szymański

        I don’t do calculations. I only check the consistency of the algorithms.
        As your field of interest is precisely practical calculations, as I noticed, I am enclosing a spreadsheet that is generally available in Poland and was implemented by Professor Mielczarski.
        The second sheet contains a fix that matches my article, we are discussing. I am convinced that it is the practical application of the flawed algorithm you are suggesting, and my correction that should resolve any doubts.
        https://drive.google.com/file/d/1bZX9yL3FseCvXGK0X6KvMPqnd3t1FMZW/view?usp=sharing
        https://drive.google.com/file/d/1jpfE6c0yt9lXHdTHV9WFo4hAyhllhi2w/view?usp=sharing

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      11. Richard McCann Post author

        That you don’t do calculations illustrates my economist’s joke. Economics is about observing and measuring reality–it isn’t solely theoretical mathematics. One must ALWAYS check the implied results from any economic construct. As I just demonstrated, the algorithm that you are proposing isn’t consistent because when you take a sum, divide it by a value and then multiply it back by a number, you don’t arrive back at the same sum. This violates the fundamental associative property.

        I will look at these files to see if they arrive at the required consistent result that the net present value of annual cashflows must sum to the initial investment amount. I urge others to also review these for consistency.

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