What is the best tilt for solar PV?

We have a 2 kW solar PV system, and solar hot water. Our house is a net exporter of electricity (we don't use gas). This is by quite a margin: our average consumption is about 3 kWh/day and our average export is about 6.5 kWh/day.

Thus, we export more than double the power that we import from the grid.

However, at this time of year our import/export is about equal. We are using slightly more power than during the shoulder season (though we're yet to boost our solar hot water), but the production of our solar PV system has markedly decreased. There are two reasons for this:

1. The angle of the sun is much lower at this time of year, resulting in lower irradiance to the panels
2. Cloud cover

Clearly, there's nothing that can be done about cloud cover. But it is the sun's angle I want to talk about here.

PV tilt

There has been much discussion of the optimal tilt for PV panels. Back when there was a well-paying feed-in-tariff (FiT), people advocated installing the panels nearly flat. At Southern Australian latitudes, the annual production of clean panels is maximised when they are flat (note the word clean -- below ~10 degrees tilt, panels must be manually cleaned).

Now that there is no FiT, the equation has changed. For me, the FiT is less of a consideration -- my goal is to to live within my solar budget throughout the year. But why?

EDIT: It is worth mentioning that choosing the "best" tilt is important for all solar collectors: solar thermal, solar hot water, and solar PV. If anything, it is more important for solar thermal than solar PV, since solar PV collects diffuse light energy from the sky and not just the sun.

Your solar budget

Growing up in an age of cheap fossil fuels, we have been conditioned to the idea that gratification follows expectation. Want to heat your house to 35 C in the middle of winter? Sure. Want an outside spa in the snow? sure. This simply will not happen in the age of renewables. For a society powered on solar and wind, there will be times (sustained cloudy and calm periods) where power is significantly more expensive. By "significant", I imagine 10 times dearer or more.

This can be mitigated by installing battery storage, however batteries are expensive. To install sufficient storage to provide during prolonged periods of low production will be out of reach for many. I think in Australia a 3 kWh (usable) battery system would (currently) be affordable by the majority, and might cost $3000 including installation. Note that such a system would really only provide power for one day, and then only for the frugal.

This is a side issue, about which I will elaborate in another post, let's get back to the PV tilt.

It's all about winter production

If you want to live within your means, it is the winter production that is crucial. There is a solar bounty in summer, so there's no problem there. Therefore, we need to increase the tilt of the panels to increase winter PV production. Here comes some maths:

Let's imagine we have a solar panel that is directly facing the sun. We would say that its surface is normal or perpendicular to the sun. This maximises the production because it catches as much sun as is possible.
Now, let's imagine that we tilt the panel so that it is not directly facing the sun. Now it catches less sunlight (its shadow is smaller) and so it will produce less energy. If we keep turning it, eventually it will cast no shadow (it's sideways to the sun) and receive no direct sunlight [1] and produce little.

So, as we turn the panel from directly facing the sun, to being side-on to the sun, let's imagine a scaling factor that describes the production of the panel. This factor will be decreasing as we increase the angle away form the sun form 0 degrees (facing) to 90 degrees (side on). The curve that describes this is called the cosine curve.

The cosine curve, showing how production decreases as the PV panel is tilted away from direct sunlight. Remember, that this does not consider energy production from diffuse light (in practice a panel will still produce energy if oriented away form the sun, it will just produce a lot less). Note that for low angles (less than, say, 20 degrees) there is little effect on production, but by 40 degrees it is falling sharply. That is the characteristic of the cosine function.

Looking at the amount of power exported from my house since September 2014, this pattern is clearly visible as we approach winter.
These data start on December 20 2014 (summer solstice) and the decrease in production through Autumn 2015 is clear.

My panels are oriented at about 23 degrees, which means that at the winter solstice, my panels are oriented about 40 degrees from ideal, significantly effecting production.

Given that we have lots of available PV power in summer, I would happily sacrifice a little to gain extra winter production and I would do this by increasing the tilt of my panels. I think the optimum would be to increase their tilt to about 45 degrees, which would reduce the winter sun's angle to about 20 degrees and increase winter production (at the cost of some summer production).

Conclusion

Orienting PV panels to ensure optimal winter production is a strategy that will help maximise your self-consumption. This helps you get the most from your PV system during all seasons.

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[1] PV panels also produce energy from diffuse light form the sky, but it is a lot less than direct sunlight. Here, we consider only the direct sunlight.