About Cost_Inv

[Ryo Ishida]

Dear all.

Please let me ask a basic question.
Could you explain Cost_Inv to me?

For example, when we input Capex 2000$ and 20 LIFETIME of one vehicle, Cost_Inv per year is 100$ (2000$ / 20yr) . isn't it?
I thought Cost_Inv / VAR_Cap = CAPEX / LIFETIME

But actually, I can say that expectations and reality do not seem to match.

Sorry for my poor English. Thank you for your advice in advance.

Ryo Ishida

[Antti-L]

Welcome to the Forum.

> Could you explain Cost_Inv to me?

I assume you are using VEDA-TIMES (other VEDA users may also use OseMoSys or TEMOA).

As described in the TIMES documentation, Cost_Inv is a reporting parameter representing "Annual undiscounted investment costs (caused by NCAP_COST) in period (t) spread over the economic lifetime (NCAP_ELIFE) of a process".  Admittedly, from that it is not quite clear what those "Annual undiscounted investment costs" exactly mean, but they refer to the annualized investment cost payments described in Part II, section 6.2.2. Consequently, the annual investment cost payments are equivalent to the lump-sum total investment costs when discounted back to the beginning of the commissioning year, using the technology-specific discount rate. Note also that by default these payments are spread over the economic lifetime, but when using the levelized annual cost option (SET ANNCOST LEV), they are spread over the technical lifetime (this option only affects reporting, not the objective function).

> I thought Cost_Inv / VAR_Cap = CAPEX / LIFETIME

Well, assuming that CAPEX=NCAP_COST, it would be so in the special case where the discount rate would be zero, but not in general, because with such annual payments the total investment cost would not be covered. Note also that if using NCAP_ILED, the payments would also include the effect of the interest during construction (IDC).

[Ryo Ishida]

Antti-san

Thank you very much for your reply.
Please let me additional questions for your reply.

>Cost_Inv is a reporting parameter representing "Annual undiscounted investment costs (caused by NCAP_COST) in period (t) spread over the economic lifetime (NCAP_ELIFE) of a process". Admittedly, from that it is not quite clear what those "Annual undiscounted investment costs" exactly mean, but they refer to the annualized investment cost payments described in Part II, section 6.2.2.
→I cannot understand "Annual undiscounted investment costs (caused by NCAP_COST) in period (t) spread over the economic lifetime (NCAP_ELIFE) of a process". If so, Cost_Inv / VAR_Cap = CAPEX / LIFETIME should be true? because with investment costs should not be discounted.

>because with such annual payments the total investment cost would not be covered.
→Could you be kind enough to explain the detail with something like formula...?

Thank you in advance.
Best regards

[Antti-L]

> Cost_Inv / VAR_Cap = CAPEX / LIFETIME should be true? because with investment costs should not be discounted.

Not true. Investment costs must be discounted just like any other costs. And your relation Cost_Inv / VAR_Cap = CAPEX / LIFETIME would be economically inconsistent with any positive discount rates.

> Could you be kind enough to explain the detail with something like formula...?

It is all explained in detail in the documentation, Part II, section 6.2 Objective function.  Specifically, section 6.2.1 introduces the Capital Recovery Factor (CRF), which is used to derive the annual investment payments (INVCOST(y)), and section 6.2.2 Investment costs: INVCOST(y), which gives the detailed formulas for deriving these annual payments (annual costs) from the full investment costs. They can be in simplified terms expressed for vintage v as shown below:

   INVCOST(v,y)  = CRF(v) × NCAP_COST(v) × VAR_NCAP(v),  for {y | v ≤ y ≤ v+NCAP_ELIFE(v)−1}

As you can see, from the expression above we get: INVCOST(v,y) / VAR_NCAP(v) = CRF(v) × NCAP_COST(v). That is quite different from CAPEX / LIFETIME, because the latter is indeed generally inconsistent and incorrect. And the reported Cost_Inv(v,y) values correspond to these INVCOST(v,y) values.

So, please just read the documentation: You can find it all explained there, with detailed formulas.

[Ryo Ishida]

Thank you for your reply!
I'll give it some thought...

[nukelucas]

Dear Antti,
I hope this message finds you well.
I have a question regarding the relationship between ELIFE and the OBJZ. I am currently working with a toy model (based on the Demo models) to test the behavior of nuclear plant lifetime extensions.
In our setup, we have two extension processes occurring in 2022 and 2025 — specifically, ELCTENUC02_EXT and ELCTENUC00_EXT.
In the LTO scenario, the extensions are set with a TLIFE of 32 and 24 years, respectively.
In the LTO_E scenario, the NCAP_TLIFE values remain the same, but we assign an ELIFE of 22 and 14 years — that is, 10 years shorter than the corresponding TLIFE values.
When comparing the results (as shown in the attached figure — only one plant extension is shown for simplicity, but the same pattern holds for both), the Cost_Inv behaves as expected:
higher and until '35 for LTO_E and lower and until '40 for LTO (because of the spreading). 




However, despite this difference, the objective function (OBJZ) is exactly the same in both scenarios.




Could you kindly help us understand why the model reports identical objective values when ELIFE is applied? Is the ELIFE somehow not influencing the objective function directly, even though it affects cost reporting?

Please note: The LTO scenario comprehends the SyS_Settings + the VT_REG_PRI_V03 file + the SubRes file + the ESC_BASE and the ESC_LTO. 
The LTO_E scenario has all the same files + the ESC_ELIFE file. 
Please let me know if anything else is needed for you to give us a hand with this. 

Thanks in advance. 

Kind regards, 
Lucas

[Antti-L]

Hmm... in order to get a more deep understanding of your question, can you elaborate in what way you would expect the economic life to affect the objective function in this case, and why that should be economically justified?  I assume that the investment costs should nonetheless be the same, in terms of a lump-sum overnight cost?

But anyway, the cost accounting is described in full detail in the documentation, Part II, and it has been designed by highly qualified authors (not me). Have you possibly identified some inconsistency with the documentation and the model results?  Or do you see some internal inconsistency in the cost methodology itself?  The economic life does affect the present value of the investment costs (and thereby the objective function) basically only through the technology-specific discount rate, if defined.  If not defined, there is no risk premium involved, and therefore the present value of the annualized investment cost payments will be equal to the lump-sum overnight cost plus interest during construction. As far as I can see, that would explain the equality in your case, and so I am curious to know the reasoning under which a difference in the objective function would be economically consistent there?

[nukelucas]

Thanks, Antti, for your response.

I agree with the points you raised, which is why I find it unclear why no difference appears in the Total Discounted System Cost (ObjZ). As shown in the Sub_Res file, we have defined a discount rate of 2.5% for the extensions, consistent with the other modelled technologies. Despite observing the expected variation in Cost_Inv between the LTO and LTO_E scenarios—driven by the reduced ELIFE in LTO_E—the ObjZ remains identical in both cases...

[Antti-L]

Ok, yes, I see now you have G_DRATE = 0.025 and NCAP_DRATE = 0.025, and so there is no risk premium. Your specification is thus equivalent to leaving NCAP_DRATE undefined.  Therefore, the explanation seems all clear now, would you agree? 

If you don't agree, could you answer my question: In what way you would expect the ObjZ to be different, and how should one derive that difference mathematically? I am sure your revised formulation could be implemented, if it is well justified.

[nukelucas]

Dear Antti,
Thank you for your detailed explanation.

In our case, shortening the economic life (ELIFE) of the extensions (from 24 years to 14 years) should result in higher annual payments, which leads to a higher present value and, consequently, a higher objective function. This was my initial expectation — that the shorter ELIFE would increase the objective function due to the accelerated amortization of the investment cost over a shorter period.
Answering your question, the objective function should be calculated by summing the total cost for each time period, amortized at the correct rate (Time_NPV).

[Antti-L]

> shortening the economic life (ELIFE) of the extensions (from 24 years to 14 years) should result in higher annual payments

Yes, I fully agree, and you can indeed see those higher annual payments, in accordance with the detailed TIMES cost accounting, in the Cost_Inv results you already showed above. And you even said yourself that "the Cost_Inv behaves as expected".

> which leads to a higher present value and, consequently, a higher objective function.

No, those higher annual payments DO NOT result in a higher present value, and thus neither to a higher value of the objective function.  And that is easy to verify:  When discounted back to the base year, or to the commissioning year, or to any common reference year, those annual payments have exactly the same present value for both ELIFE. That is a basic property of the method for deriving these payments in TIMES, which is the very standard, widely used annuity method for annualizing investment costs into a stream of annual payments, as fully described in the documentation.

Therefore, it seems that you would like the annual payments to be calculated in some way different from the standard annuity methods which are employed in TIMES (and many other energy system models). I would suggest that you provide the mathematical formulation (or a good reference) for your desired method for annualizing the costs, such that would result in different NPV values with different ELIFE. However, to me it looks quite possible that you may simply be confusing the use of higher rates of required return with the standard annualized payment calculation.