Tuesday, April 14, 2009

So, you want to buy a solar plant.

A Scenario.

Note: A follow-up scenario includes accounting for system degradation and inverter losses.

The cost of the install + Interest will equal some amount of money to be paid out per month. I'll call this Outgoing$Monthly.

Power generated per month will be sold on the market for some amount of money. I'll call this Incoming$Monthly.

Set Incoming$Monthly = Outgoing$Monthly.

This would be the point at which your investment broke even on a monthly basis (not including maintenance cost at the moment, this is just to include interest expense into the equation). It's not going to be quite right, because of seasonal variation, as mentioned below, but I'm not looking for anything exact, just a rough way to start gauging cost / benefits.

The end result will be a relationship between the Installation Cost per Watt, Interest Rate, and Required Sales Price of Energy produced in order to break even.

I'll skip to the chase, for those that don't want to read through the whole thing.

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

Note that the assumed interest rate (5%) for purposes of this post has been set and absorbed by C1, and the Insolation Ratio has been absorbed into C2.. Other assumptions are pointed out below.

To give an example of what this tries to point out, let's say you can sell the energy produced by the power plant for $.25/kWh (equal to the low range of this estimate of costs for future nuclear power plants).

Cost/kWp ($/Wp) = $.25/kWh * 146 Hours/Year / .0066 = $5,530/kWp, or $5.53/Wp.

So, if you can sell your power for $.25/kWh, then you break even (roughly) if you can complete the installation for $5.53/Wp or less. Note that the equations below DO NOT include the existing 30% Federal Tax Credit for Solar Installation. That's icing (of course, it also doesn't include lifetime performance degradation or inverter losses).

Fact: this is very much in the range of possibility in TODAY's market. Particularly in the case of mid-large scale installations.

The basis follows.

If there's one thing that I've learned being on the Internet this many years, it's that if you're wrong, somebody will point it out. Have at it with my thanks!

Here goes:

First, find the Monthly Payment required to make the loan payment for an installation of some total cost.

(1) Outgoing$Monthly = (Principle * i) / (1 - (1+ i)^-n) See http://en.wikipedia.org/wiki/Amortization_calculator.

This is the Monthly Payment on the loan for the power plant with the below assumptions.

Principle = Total Original Loan amount used to finance the entire plant = the Total Peak Power of the plant * the overall Cost per Watt of the system.
i = periodic interest rate (Monthly. Assume 5% APR, so i = .05 / 12 = .0042).
n = total number of payments (Months. Assume 20 Year Loan, so n = 240).

(2) Principle = TotalPeakPower * Cost/Wp

The Principle is the amount of the loan, where the total cost of the installation is given by the Total Peak Power * Cost per Watt. Substituting for "Principle," from (2) into (1) gives:

(3) Outgoing$Monthly = (TotalPeakPower * Cost/Wp * i) / (1 - (1+ i)^-n)

For simplicity, and ease of double-checking results, I'm going to treat n and i as constants (they are part of the assumptions above), and will pull a constant out of the above equation (3):

(4) Set C1 = i / (1 - (1+ i)^-n) and substitute into (3).

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

Now, to figure out what's coming in every month on the sale of the Energy.

This doesn't include seasonal variations. On thinking about it, though, in an Energy market where consumers are paying based on momentary supply and demand, wintertime prices could actually go up based on decreased supply, and so help to balance out the annual cycle for the energy supplier. Then, in the summer where supplies were higher, the prices to the consumer would decrease to offset some winter costs.

In any case, following similar logic to my note on Insolation, the Annual Energy output of the plant can be written as below.

(6) Annual Energy (kWh) = TotalPeakPower (kW) * 20% * 8760 Hours/Year * 1 Year

Start by writing down an equation to relate the Installation's Total Peak Power, to it's Annual Energy Output. I'm plugging in an assumption of a 20% Insolation Ratio, which would include a broad swath of non-sunbelt States. The Insolation Ratio Assumption for this post applies to such shady states as Tennessee, Missouri, and even North Dakota.

(7) Incoming$Yearly = Annual Energy (kWh) * Rate ($/kWh)

Multiplying the Annual Energy Output by the Rate at which it sells for, gives the Total Income for the year. Divide by 12 (below) and you have the Average Monthly Income.

(8) Incoming$Monthly = Incoming$Yearly / 12 Months

(9) Set C2 = .2 * 365 * 24 / 12

Once again, I'm going to pull all of the Constants out of the equation (6) to come up with C2.

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

Ok, so now we have the Monthly Outlay required for loan payments, and we have the Monthly Income from energy sales.

To break even - let's set them equal to each other.

(11) Set Outgoing$Monthly = Incoming$Monthly

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

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

Canceling out TotalPeakPower (kW) from both sides of the equation, gives a very simple equation relating the Rate at which the energy is sold, to the Cost/kWp of the initial plant installation.


Ok, so to an example and a factcheck.

First, Calculate out C1 and C2.

(14) C1 = i / (1 - (1+ i)^-n) = .0066 (i = .0042, n = 240)

(15) C2 = .20 * 8760 Hours/Year / 12 Months/Year = 146 Hours/Month

Then, pick a target Sale Price for the power that is produced by the Installation, and solve (13) for Cost/kWp. I'm using $.25 in this case, so:

(16) Cost/kWp = Rate * C2/C1 = $.25/kWh * 146 Hours/Month / .0066 = $5,530/kW

Now to check it, or at least check the Interest Calculations:

Since TotalPeakPower was canceled out of the above equation, I'll pick a value to use for the factcheck, say, 1000kW.

So, using (3), Outgoing$Monthly = (TotalPeakPower * Cost/Wp * i) / (1 - (1+ i)^-n) = 1000kW * $5,530/kW * .0042 / (1 - (1+ .0042)^-240) = $36,617/Month.

Then, using (6), Annual Energy (kWh) = TotalPeakPower (kW) * 20% * 8760 Hours/Year * 1 Year = 1,752,000kWh/Year and Dividing by 12 to get a monthly Energy Output, gives 146,000kWh/Month.

Multiplying this by $.25/kWh gives $36,500/Month

Pretty Close. Exponentials are subject to rounding errors. Another way to check would be to put the total cost, or Principle (in this case, $5,530,000) into any number of online mortgage calculators.


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