I have a tariff the electricity board pay on my solar panel. The tariff is 0.413 cents/kWh. It rises by q=2.4% each year. The system generates power P = 1785 kWh in the first year, going down by d= 0.5% each year (as its cells decay).

I want one formula that works out the total cumulative income after n years.

I've seen an incorrect answer on another board.

Ok, let x be the tariff at any time t years.

When t = 0 (initial tariff), x = 0.413
When t = 1, x = 0.413 + (0.413 * 2.4%)
When t = 2, x = 0.413 + (0.413 * 2.4% * 2.4%)
So, when t = n, x = 0.413 + 0.413 * 2.4%^(n)



or



Ok, second part. Let y be the electricity output, with the same time, t years.
When t = 0, y = 1785
When t = 1, y = 1785 - (1785 * 0.5%)
When t = 2, y = 1785 - (1785 * 0.5% * 0.5%)
So, when t = n, y = 1785 - (1785 * 0.5%^ n)



or



Net income is then given by: income - tariff.

You can do that. I did the most part of the problem.

Thanks. But I'm still a bit confused sorry. The second formula is right per year. That is what happens to the panel each year. The first formula is also a year by year one.

Below are the year by year figures I make by 0.413 c/kWh (going up at 2.4% each year) x 850 kWh x 2.1 = 1785 kWh per year going down by 0.5% each year) looks like. I want also a formula that sums these results up without having to use a spreadsheet. I want to input n and give me the cumulative total on that year. T would start a 1 rather than 0 by the way.

561.68
572.28
583.09
594.10
605.31
616.74
628.39
640.25
652.34
664.66
677.20
689.99
703.02
716.29
729.81
743.59
757.63
771.93
786.51
801.36
816.49
831.90
847.61
863.61
879.92

Sorry those are the wrong numbers. They are based on 1.6 x 850 = 1360 kWh - simply replace 1785 with this. Better still use P for power.

I used the zero because from the complete start, until 364 days, the tariff and income are being considered to be constant. As soon as that one year is passed, the tariff and income will change.

I think you want them in a single formula, but I gave it to you in two parts:





So, net income is income - tariff, or y - x



I'll try simplify it later. I'm running out of time; I need to go.

Doing this sum on a spreadsheet, 0.413 x 1785 = £737.21. That's the starting figure. Then the 2.4% and the 0.5% applies. The above formula doesn't work when I check it using a spreasheet. I don't know what its calculating?

Here are the results from a spreadsheet with a cumulative sum after 25 years - what I'm after for n years. If anyone can help please let me know?

£737.21
£751.12
£765.30
£779.75
£794.48
£809.48
£824.76
£840.33
£856.19
£872.36
£888.83
£905.61
£922.71
£940.13
£957.88
£975.96
£994.39
£1,013.16
£1,032.29
£1,051.78
£1,071.64
£1,091.87
£1,112.49
£1,133.49
£1,154.89
TOTAL: £23,278.11

Oops, sorry, I didn't notice that the tariff was connected to the income. I'll post an answer as soon as I can. I'm in a library right now, the internet at home is not working and I have limited time.

Ok, for that one, the first is given by:



Then, by:



Then,



Hence, the pattern:



Now, since it's only multiplication, we can safely 'mix' the two separate terms:



Furthering:





Ok, that's it for the formula for income for any particular year.

Now, sum of the years for a pattern of the form:



is given by:



So, the cumulative sum formula you're looking for is:



I hope it helped! Sorry for the delay, I had problems with the internet connection :o

Check for 25 years: 39047.13983(1.01888^25 -1) = 23,278.27 (with no lost decimals)

Correct! :)