Saturday, June 27, 2009

So, you want to buy a solar plant, part 2.

I thought I'd extend my calculations from So, you want to buy a solar plant, part 1, to make them a bit more realistic. You'll find the basic logic for this post at that location. Here's a recap:

(1) Outgoing$Monthly = TotalPeakPower (kW) * Cost/Wp * C1

(2) Incoming$Monthly = TotalPeakPower (kW) * Rate ($/kWh) * C2

(3) C1 = i / (1 - (1+ i)^-n)

Note: i = .0042, n = 300. Previously, I had set n to 240 for a 20 year loan, but in this case, I'll be setting n to 300 for a 25 year payoff period.

(4) C2 = Insolation Ratio * 1year * 365days/year * 24hours/day / 12months

Note: Insolation Ratio = .1875. Previously I had used .20 for the Solar Plant's local Insolation Ratio. The particular location that's being considered is in the area of Bangalore. I see from this Insolation Map that that area seems to be recieving between 4kWh/day/m2 and 5kWh/day/m2 in Solar Energy. For this estimation's sake, I'll pick the middle, or 4.5kWh/day/m2. Using previous calculations, this works out to an Insolation Ratio of 18.75%.

(5) Setting Outgoing$Monthly = Incoming$Monthly gives the break-even long term average energy price per kWh with respect to the cost per Watt of the Installation:

(6) Cost/kWp ($/Wp) = Rate ($/kWh) * C2/C1

Or conversely:

(7) Cost/kWp ($/Wp) * C1/C2 = Rate ($/kWh)

I want to take this one step further, and take into account degradation of the system over time, as well as various losses like Inverter / Substation losses. I'll try to be aggressive on this correction, to err towards the worse case. So, over the 25 years over which I'm considering, I'll say that the modules lose 1% per year, and Inverter losses are 8%. Averaging the degradation over 25 years gives a loss of 12.5%, plus the 8% Inverter loss, gives a total of a 20.5% loss.

I'll reflect this in my equation by going back to another equation from Part 1 (one of the roots of the (6), above):

(8) TotalPeakPower (kW) * Cost/kWp * C1 = TotalPeakPower (kW) * Rate ($/kWh) * C2 (Hours/Year)

Just prior to (6), I had canceled TotalPeakPower out of my equation. Essentially, these equations should give a rough estimate of the Levelized Cost of Energy irrespective of the total size of the Installation. Of course, there are alot of variables involved when changing scale, and these aren't reflected here. Consider this to be more along the lines of an ideal solar farm. It costs the same per Watt to add a thousand Watts of capacity, or a hundred thousand Watts.

Remember that on the lefthand side of the equation is the initial cost per Watt of the system, while on the righthand side of (8) is the income per unit energy. So, essentially, the "TotalPeakPower" on the left is the ideal TotalPeakPower, while the TotalPeakPower on the right is the effective TotalPeakPower as reflected in the Energy produced over the lifetime of the system, by which income is derived.

So, I'll throw in some subscripts.

(9) TotalPeakPowerideal * Cost/kWp * C1 = TotalPeakPowereffective * Rate ($/kWh) * C2 (Hours/Year)

In the ideal case, TotalPeakPowereffective = TotalPeakPowerideal, but in this case, there's a 20.5% loss, so:

(10) TotalPeakPowereffective = .795 * TotalPeakPowerideal

Plugging (10) back into (9) gives:

(11) TotalPeakPowerideal * Cost/kWp * C1 = TotalPeakPowerideal * .795 * Rate ($/kWh) * C2 (Hours/Year)

So, I can now still cancel out the TotalPeakPowerideal, and I'm left with a correction to the income side of the equation, which reflects system degredation losses and Inverter losses.

The final result is:

(12) Cost/kWp * C1 = (Rate ($/kWh) * C2 (Hours/Year)) * .795

Taking a case, let's say your installation is going to cost $5.25/W, or $5250/kW.

Solving (12) for Rate gives:

(13) Rate ($/kWh) = Cost/kWp * C1/.795C2

C1 = i / (1 - (1+ i)^-n) = .0058 (i = .0042, n = 300 (25 years))

C2 = .1875 * 8760 Hours/Year / 12 Months/Year = 137 Hours/Month

C1/.795C2 = .00005325

Rate ($/kWh) = $5250/kWp * .00005325 = $.28/kWh

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