If you are focused on the laptop, you can reduce DC/AC => AC/DC conversion losses with a DC/DC converter, such as
this.
For planning how much solar to use, follow the steps in
this guide I made for pandemic preparedness. For battery storage, determine how long you want to run the laptop on batteries. Deep cycle batteries are available in different voltages, most commonly 2v, 6v, and 12v. They are also rated for the number of amps they can provide in a given time frame, such as “amp-hour.” An important battery energy storage calculation is:
Volts x Amp-Hr = Watt-Hr
If you need roughly 20W for 2 hours per day, that's 40 Watt-Hrs per day. The same laptop run for 10 hours would use 200 Watt-Hrs. A 12v battery with 100A-Hr capacity has the equivalent of 1200 Watt-Hrs capacity.
Keep in mind that even deep-cycle batteries can be damaged if they are frequently discharged more than 50%. For this reason, provide a margin of error by never discharging a battery below 50% of its capacity. At all times then, you want to maintain the
battery reserve charge above the 50% Depth of Discharge (DoD) level. So multiply your daily energy needs by 2, which would bring our example up to 2500 x 2 = 5000 watt-hr (or 5 kW-hr).
We can stop here and use a rule of thumb that there would be another 40% losses and that you could run your laptop for about 18 hours with the above-mentioned battery starting out with a full charge. There are finer calculations if you want to be more certain.
Battery
charging losses are also a consideration during the charging phase, as the electrochemical processes within the battery are not 100% efficient, so subtract 25% for this loss. Most of this loss is due to heat and gases generated during charging.
Cloudy days decrease PV-panel energy production. To account for this, multiply the daily watt-hr requirement by the number of cloudy days you want to maintain operation.
As
batteries age, they gradually lose their capacity as their interior structure deteriorates and eventually becomes too weak to effectively store energy. To ensure that your system will operate correctly throughout a battery pack’s useful life, add 20% to your energy storage calculation.
Battery packs must match the system voltage, so if you have two 12v batteries for a 12-volt solar system, these would be connected in parallel, and you would add each of the amp-hours of the batteries together. If you had the two 12-volt batteries in a 24-volt system, you would connect them in series. For example, consider the following energy storage need and system voltage level calculation.
Total battery storage = Energy Storage Need x Battery Reserve x Charging Losses x Number of Cloudy Days x Battery Aging Factor
Using an example of a 12-volt device needing 1200 watt-hours daily with 3 days of cloudy day storage, we can determine the minimum battery energy storage requirements and the number of batteries needed.
Total battery storage = 1200 x 2 x 1.25 x 3 x 1.2 = 10368 watt-hours
If we use 12-volt AGM batteries that are rated at 108 amp-hr each, they produce:
12 volts x 108 amp-hrs = 1296 watt-hrs
10368/1296 = 8
So we would need at least 8 of these batteries in parallel in our pack to meet our storage needs for 3 consecutive days without the sun.