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Math Help - Photovoltaic panels payback in Britain - formulas

  1. #1
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    Photovoltaic panels payback in Britain - formulas

    Recently the British govt decided to force energy companies to pay people who installed photovoltaic solar panels on their roof. The rates are as follows:

    Main Tariff: 41.3 p/kWh for everything you generate plus
    Export Tariff: 3 p/kWh for what they deem you export, e.g. they might say you use 50% and you export 50% of what the panels generate.

    Both tariffs rise by inflation rate r = 2.4% each year.

    A typical 1kW system in the UK generates 850 kWh per year. A 2.0 kW system will generate twice this and so on, whatever the size. Anything under 4 kW qualifies.

    Okay, so far, so good. I want to work out three short formulas that will predict my earnings from

    Main tariff (m)
    Export Tariff (e)
    for any given year 1-25
    rate rise = 2.4% (r)

    For example, year n=1,2,3,4....25 (the last year they'll pay).

    I can do this on a long complex spreadsheet, but I'm sure that there is a shorter formulaic way to do this.

    It's a good problem to solve actually, so I hope someone can provide the formulas.
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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by rintelen View Post
    Recently the British govt decided to force energy companies to pay people who installed photovoltaic solar panels on their roof. The rates are as follows:

    Main Tariff: 41.3 p/kWh for everything you generate plus
    Export Tariff: 3 p/kWh for what they deem you export, e.g. they might say you use 50% and you export 50% of what the panels generate.

    Both tariffs rise by inflation rate r = 2.4% each year.

    A typical 1kW system in the UK generates 850 kWh per year. A 2.0 kW system will generate twice this and so on, whatever the size. Anything under 4 kW qualifies.

    Okay, so far, so good. I want to work out three short formulas that will predict my earnings from

    Main tariff (m)
    Export Tariff (e)
    for any given year 1-25
    rate rise = 2.4% (r)

    For example, year n=1,2,3,4....25 (the last year they'll pay).

    I can do this on a long complex spreadsheet, but I'm sure that there is a shorter formulaic way to do this.

    It's a good problem to solve actually, so I hope someone can provide the formulas.
    P = power output
    M = Main tariff
    x_m = amount on main tariff
    x_e = 1-x_m = amount exported
    n = time (years)
    r = inflation (% value)



    Per kWh: 41.3x_m + 3(1-x_m)

    Annual amount = 850P \times (41.3x_m +3(1-x_m))

    Adjusting for inflation: \left(1+\frac{r}{100}\right)^n \times 850P \times (41.3x_m + 3(1-x_m))

    That's the way I'd do it anyway
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  3. #3
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    Another related question

    Hi

    That's works great for individual years. How would I sum them so I can work out say the cumulative income over say 5, 10 or 15 years? Or do I have to keep doing each formula and adding them up? I'm sure there's a trick I'm missing?

    Quote Originally Posted by e^(i*pi) View Post
    P = power output
    M = Main tariff
    x_m = amount on main tariff
    x_e = 1-x_m = amount exported
    n = time (years)
    r = inflation (% value)



    Per kWh: 41.3x_m + 3(1-x_m)

    Annual amount = 850P \times (41.3x_m +3(1-x_m))

    Adjusting for inflation: \left(1+\frac{r}{100}\right)^n \times 850P \times (41.3x_m + 3(1-x_m))

    That's the way I'd do it anyway
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  4. #4
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by rintelen View Post
    Hi

    That's works great for individual years. How would I sum them so I can work out say the cumulative income over say 5, 10 or 15 years? Or do I have to keep doing each formula and adding them up? I'm sure there's a trick I'm missing?
    Well it would stay constant apart from the inflation term assuming nothing changes

    If this is the case then it would increase by a factor of 1+r each year which would make it into a geometric series with common ratio 1+r

    First of all I'm going to denote inflation,  r as q in this next sum as not to confuse it with the common ratio which is denoted r. q = 0.024

    S_n = \frac{a(r-1)^n}{r-1}

    Applying your values: S_n = \frac{a(1+q-1)^n}{1+q} = \frac{aq^n}{1+q}

    If you add this to n \times 850P \times (41.3x_m +3(1-x_m))

    \sum_{i=0}^n \left[\left(1+\frac{r}{100}\right)^n \times 850P \times (41.3x_m + 3(1-x_m))\right] \: = \left(\frac{aq^n}{1+q} \times n \times 850P \times (41.3x_m +3(1-x_m))\right)

    Chances are my summation notation is wrong so ignore it if it makes no sense
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  5. #5
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    Thanks. Now to a new problem

    Thanks.

    However, how would I do the same if this was the problem instead:

    Let say that the rates themselves rose by 2.4% each year. And that the power of the PV unit diminished by 0.5%.

    Now I want the cumulative sum formula so that I just plug in n year from n=1 to 25 and I can work out the separate tariff cumulative incomes?

    It should provide growing numbers...
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