Doing the maths

According to Michael Boxwell's book, figuring out what we can achieve with solar panels, and what we'd need to install, requires taking lots of things into account.

Firstly, it depends on what kind of system we want. There are "grid tied" systems that essentially supply your house with electricity, and if you generate too much, the excess is sold to the power company, and if you generate too little, the power company sells you electricity (like they normally do).  It looks like that would be impractical here - there's no legislation to cover how it works, and although it does exist in Argentina, it's been installed in few enough places that all six are named in the government report on solar energy that came out a few years ago.

I suspect what we want is a "grid fallback" system - solar panels generate electricity which is stored in batteries.  The batteries, in turn, power our household appliances.  And unless we want to buy all new appliances that run on 12V or 24V DC, we need an "inverter" that converts low voltage DC power from the batteries to 220V AC, like what the grid supplies.

So far, so "off-grid"; the "grid fallback" part is that, if the batteries run out, then instead of sitting in the dark until sunrise, we would use power from the grid to tide us over until the sun charges the batteries again.

So to work out what I can reasonably expect from the measly roof space I have to work with, there are various factors:

• What we consume
• What direction the panels are pointing
• The amount of sun we get, across the whole year
• The efficiency of our batteries, and our inverter

Consumption

Contrary to my newly-awakened green principles, I'm pretty addicted to air conditioning at the height of the porteño summer. And my friolenta wife feels the cold in winter and likes to run a heater in the girls' room at night.  However, these are pretty power hungry contraptions, and I think for now I just have to accept that we use them at the expense of the future of humanity, and use the grid to power them in winter and summer months.

So we'd want 2.4kWh per day minimum, and ideally more like 10kWh per day to cover most of our consumption. 

Pointing the panels 

Solar panels work best when pointed directly at the sun, which annoyingly doesn't stay in the same place in the sky for very long.  It is possible, but complicated, expensive, and probably high-maintenance, to mount panels on moving 'trackers' that follow the sun across the sky during the day, and up and down in the sky across the seasons.  I don't think that's really practical for us.

So we'd be mounting them in a fixed position. It would be possible to put them horizontally flat on the roof (or vertically flat on a north-facing wall), but the best idea would be to angle them towards the north, to best catch the sun during the whole year.

What angle?  That depends on where you are on planet Earth.  The book tells you how to work it out from your latitude, but book's website has some handy calculators into which you can put your location and a bunch of other data, and it tells you (among other things) the angle you need.

For us, it would be 55° from vertical, if we wanted the the best average energy over the year, or 39° if we want to maximise generation in winter (but generate less in summer).

Insolation

It turns out that NASA has a whole bunch of data about how much sun every part of the globe gets. The aforementioned calculators on the website use this data to tell you how much energy you could theoretically get from the sun depending on your location.

Here, assuming panels angled at 55°, it's 6.12 kWh/m²/day in summer, 3.39 and kWh/m²/day in winter, or an average of 4.75 kWh/m²/day.

But how much we get in our own roof depends on other things like how much of the horizon we can see and where the nearby tall buildings are.

The northern half of our horizon

The book has a pretty easy-to-follow method for working that out with compass, protractor, etc., which gives you a single chart that covers the position of the sun all year (from its height in summer to the depths of winter).

However, there's also an app for that too, which was even easier to use.  It draws a nice chart of the sun's path against building shadows:

... on which you can see the white area between the green and blue lines is where panels would get direct sunlight, and the grey areas are shaded by buildings.

According to the app, building shadows mean we lose 19.2% of our sun because of the pesky neighbours.

There's one more thing about shade that I discovered from the book: solar panels essentially don't work if there is even a tiny bit of shadow on them. More accurately, they're made up of lots of solar cells, and the panel as a whole works at the capacity of the cell producing the lowest amount of energy.  This means that if one cell is shaded by a nearby TV antenna, then the whole panel will produce less electricity.

So I have to look not only at distant buildings, but also at the shadows cast by nearby objects, to figure out how much of the portería roof I can use.  So I went up to the roof every half an hour or so on a sunny day, and took photos:

Because of the shadows of the various chimneys sticking out of the roof, it looks like actually only the right half of the roof space is shadow free, and the water tank casts a shadow over the back half of that late in the day.  This is a little discouraging: it probably means I only have 6m² not even 10!

Inefficiencies

We would want to run our current domestic appliances - and if we want to use this for emergency water-pump supply, the pump works on 220V as well - so we'll need an inverter to convert battery-supplied DC to 220V AC.  That, according to the book, adds a 10% performance penalty on to the system - so we get 90% of the power we generate instead of all of it.

Also, batteries aren't perfectly either, and how efficient they are depends on the type of battery. There are apparently high-efficiency industrial batteries available, but they're lead-acid batteries, which contain acid, produce flammable gas, and require maintenance.  Instead, I'd prefer "gel" batteries that are maintenance and gas free, and much safer to have lying around the place. These are apparently 90% efficient, so I need to add a further 10% penalty to what we need to generate.

Putting it all together

With an open horizon, in the depths of winter, we get 3.39 kWh/m²/day insolation, but because of our pesky neighbours, we lose 19.2% of that, so we really get 2.74 kWh/m²/day.

How much energy you can get from a solar panel depends on its rating in Watts. This varies, but using as an example some solar panels I know I can get online, which are 1m × 0.66m in size (0.66m²), and rated at 100W (which makes the maths easy!).

To work out how many Watt-hours per day I can get out of one of these panels, I need to multiply the insolation by the panel rating:
2.74 kWh/m²/day × 100 W = 274 Wh per panel.

For the 'bare minimum' system, we need to generate 2.4kWh per day.  If we add to that what's lost by inverter and battery inefficiencies, we need 3 kWh per day. 

So with the 274 Wh panels above, 3000 Wh ÷ 274 Wh = 10.95 - so I'd need 11 panels to generate the minimum in winter. 11 panels takes up 7.3 m², and if I put them all side-by-side, on their short (0.66m) edge, they'd all be 7.3m long.

I've only got a strip 6m by 1m - 6m², and 6m fits only 9 of these panels if they're put on their short edge. If I could somehow arrange them in two rows, along their long (1m) edge, then I could do two rows of 6 panels, giving me 12 panels, one more than I need.

The panels won't be flat on the roof, they'll be angled at 55° from vertical, which is 35° from horizontal. If there are two rows, then the width of the strip of panels would be 2×0.66 = 1.32m. How much roof width would a 1.32m bank of panels angled at 35° take up?

There are tiny flags raising in my brain, about the "hypotenuse", "sine", something-or-other.  Can't really remember the details...

Yay for the internet:

So, c = 1.32, A is 35°, and we want to know b.

This works out to be 1.08m - 8cm more than we really have, unless the panels are mounted on a frame that's raised above the wee chimney in the middle of the portería roof.

Let's assume I got 12 panels instead of 11, to fill out 2 rows of 6.  Then the winter output would be (274Wh × 12 panels =) 3288Wh/day, a wee margin over the 3kW/day I need.

What would it generate in summer?  The summer insolation is 6.31kWh/m²/day, but my neighbours' shadows make it (6.31kWh/m²/day - 19.2% =) 5.1kWh/m²/day.  So each panel would generate (5.1kWh/m²/day × 100W =) 5100Wh/day - more than twice the 2.4kWh we would need in emergencies, and less than half the 13kWh/day we need for everything, if we're being austere with electricity (we need 10kWh/day, which is 13 after taking into account the battery/inverter inefficiencies).

So it looks like we can probably cover emergency supply with lots of panels squeezed into the only viable corner of the roof. Yay!

But it's a little disappointing that, on the sunniest day in summer, we couldn't even reach half what we generally use.  And that's not counting air conditioning, which we'd probably have cranked up that particular day.

Hmm.