Cost calculation clarification needed

[smartie2076]

@remmyd wrote me a question of the cost calculation implemented in Offgridders. She found the following equations to describe calculation of the asset annuities:

She wonders:

Is there was a specific reason to calculte the annuity costs this way rather than multiplying the investment cost of each component by their CRF based on their respective lifetime?
Like so: sum(capex_i wacc(1+wacc)^t_i/((1+wacc)^t_i -1) + opex_i for i in Components)

[smartie2076]

So, I want to explain what is calculated above:

The costs for re-investment (c_last,i) and residual value (cr_res,i) are calculated. Together with the first investment in year 0, they result in the investment costs over the whole lifetime (capex_i).
For the optimization, we need the annuity of those investment costs, so we use the CRF. Additionally, there are annual operation and management costs (opex_i), that also need to be added to the annuity to properly size the assets.

Is there was a specific reason to calculte the annuity costs this way rather than multiplying the investment cost of each component by their CRF based on their respective lifetime?
Like so: sum(capex_i wacc(1+wacc)^t_i/((1+wacc)^t_i -1) + opex_i for i in Components)

I think you are then not accounting for the replacement costs of the asset, nor the residual value.
For an asset that has lifetime=project duration the results would be the same. But if lifetime<project duration, you need to invest into an replacement asset. Only part of that money you can get back due to residual costs (and the value is even less, as the time value of money result in a much higher present value of the replacement costs then the reimbursement). That adds costs on top of the annuities.
Your equation does not result in the same costs (see below).

Excel calculation: