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Hi Hasret,
To begin with, LEAP does not necessarily have a problem if supply is not sufficient to meet demand - it will simply import the balance of the fuels required, or let that demand go unmet, depending on how you've set up your model. So the least sophisticated strategy would be to interpret any imported or unmet electricity as a mix of imports or privately generated power, thereby not including any private electricity generation in your model at all. But this doesn't sound quite adequate for your purposes.
I think that I would begin by doing some spreadsheet analysis, looking at the fuels consumed by the electricity sector (both public and private) from your energy balance sheets. Assuming you know the historical production, as well as power plant characteristics for the public sector (most importantly the input fuels, historical availabilities, and efficiencies), you should be able to make a good guess at the remaining input fuels that are consumed by the private sector.
Using this information, as well as some educated guessing about the properties of power plants that exist in the private sector, you can let LEAP build power plants endogenously and dispatch them even during the Current Accounts period. You would do this by setting the 'First Simulation Year' variable for these plants equal to the base year for your model. I would guess that you will be able to replicate quite closely the additional electricity generation from the private sector.
You're certainly correct, though - often, actually using LEAP is only part of the battle. Data availability is something we grapple with on a regular basis! This is just my approach, but hopefully it helps to shed light on the problem.
Best,
Taylor