Renewable Energy Calculations

This is a playground for calculations and data relevant to renewable energy. I want to answer questions such as:

  • How many years would it take for wind or solar power to replace all fossil fuel usage at current rate of development?

  • How much storage will renewable energy require, and how expensive will it be? What will the cost and tradeoffs of different solutions be?

  • How much will renewable energy cost compared to other solutions such a nuclear power?

Basic Starting Facts

To get started it is useful to have some basic facts in place to play with, such as how much electric power or energy a normal person consumes. How much power a square meter of solar power outputs etc.

We can use the PVWatts calculator to get some ideas of how many kWh you can produce with 1 m² of solar cells. With standard settings a DC System size of 4 kW, implies about 25 m² of solar cells, so we got to divide result by 25.

I can test these settings for Barcelona in Spain. With a standard solar cell of efficiency 15% I get that 5194 kWh is produced by 25 m² of solar cells in one year.


We can try the same calculation by switching to premium cells with 19% efficiency and see if the results is the same as we could estimate. The calculator gives us 5230 kWh/year.

using Printf
estimate = 5194 * (19/15)
estimate_pr_sqmeter = estimate/25
@printf("Estimate is %.1f/25m² which is %.1f/m²", estimate, estimate_pr_sqmeter)

So our estimate is much higher than what the calculator gives us. Thus this should need some further investigation.

Calculating Solar Cell kWh from Scratch

Let us look at how we can calculate an estimate by ourselves from scratch rather than using an online calculator. The amount of sun that a house gets differs due to the seasons, clouds etc. Thus the number we use here is an annual average. It is what you get if you add up the sun hours every day for a year and then divide on the number of days in the year.

solar_hours = 5 # Number of hours of full solar intensity per day in USA
solar_flux = 1  # kW/m² of solar energy received per square meter
# Amount of energy received per m² from sun on the ground
received_energy = solar_hours * solar_flux
# Assume 15% efficiency from solar panel
efficiency = 0.15
# Energy captured by solar power per m²
captured_energy = received_energy * efficiency

So that comes out at 0.75 kWh/ per day. We could update our kWh produced per day to turn it into production per year for more easy comparison.

yearly_production = captured_energy * 365
println("$yearly_production/m² per year")

This is in the ballpark of what the PVWatts calculator gave us. Obviously it should have given a lower number as it includes inverter efficiency losses etc.

We can look at how much energy households around the world use to see how many square meters of solar panels we would need in different places. Here is data including the USA and Canada from 2010. Which suggests the US requires 11 879 kWh per dwelling per year, while e.g. Germany needs 3512 kWh per year.

Here is a more recent report covering various European nations in 2000 and 2017. According to this Denmark uses 3535 kWh per year per dwelling. My home country Norway is a major outlier at 16834 kWh. It is worth noting that Norway is a major anomaly due to a lot of power hungry industry and heating being done almost exclusively through electricity.

Let us calculate roof area required for some different different countries

using Printf
usa = 11879
denmark = 3535
italy = 2651
@printf("Household USA %.2f m²\n", usa/yearly_production)
@printf("Household Denmark %.2f m²\n", denmark/yearly_production)
@printf("Household Italy %.2f m²\n", italy/yearly_production)

That is actually not all that much area. I live in an apartment of about 80 m² in Oslo, in a building with 3 floors. A rough estimate is then that 80 m² has to be divided by 3 apartments. That is 27 m². Both at Danish and Italian electricity consumption level that would be totally doable.

Electricity Storage

But of course electricity production will fluctuate through the day and with the weather. The sun does not shine the whole day. We need to be able to store power. Then it is useful to know daily kWh needs.

@printf("USA %.2f kWh/day\n", usa/365)
@printf("Denmark %.2f kWh/day\n", denmark/365)
@printf("Italy %.2f kWh/day\n", italy/365)

A fairly affordable Tesla EV such as the Model 3 comes with battery capacity from 54 to 75 kWh. In Denmark that is enough battery capacity to last for 7 days.

This could be an option if cars could send current back to the grid. This is an option being developed. Until then the alternative for now would be the Tesla Powerwall 2 which has a capacity of 13.5 kWh which would be fine for a Danish or Italian house but not an American house. This costs $6500. The costs are a bit unclear another article mentions 10 kWh costing $3500.

Supplying households is in other words not that hard. But what about total electricity demand, including needs from industry?

Total Electricity Demand

On wikipedia we can find electric energy consumption for different countries per capita. This includes electric usage by industry, transport and commercial buildings which get excluded from the household numbers I showed earlier.

Let us pick from the table some of the countries we have already look at.

using Printf
total_usa = 14378 # kWh consumed per year per capita
total_denmark = 6912
total_italy = 6054
@printf("Total USA %0.2f kWh/day\n", total_usa/365)
@printf("Total Denmark %0.2f kWh/day\n", total_denmark/365)
@printf("Total Italy %0.2f kWh/day\n", total_italy/365)

It could be interesting to look at how many square meters of solar panels are needed to cover this electric demand.

using Printf
area_usa = total_usa/yearly_production
area_denmark = total_denmark/yearly_production
area_italy = total_italy/yearly_production
@printf("USA %0.2f m²\n", area_usa)
@printf("Denmark %0.2f m²\n", area_denmark)
@printf("Italy %0.2f m²\n", area_italy)

This is not too bad. This is the area of solar panels needed per capita for respective countries. In developed countries household size tends to be from 2.0 to 2.5. This is the most interesting case, because it is when countries reach this state of development they really need a lot of power. Households in poorer countries is much larger but these countries also don't use much power.

In Denmark it is around 2.1. That means that each household in Denmark needs to the following area to cover all the electric needs of Denmark.

@printf("Denmark %0.1f m²\n", area_denmark * 2.1)
@printf("100m² solar produce on avg %0.2f kWh/day\n", 100*yearly_production/365)

That is in fact not a too unreasonable amount of space needed on top of each household in a country.

Arable land per person in the world is about 0.2 hectars according to the World Bank. This translates into 2000 m² per person. 50 m² is just 2.5% of this land. Thus even if we had to utilize farmland for solar panels, they would not steal a lot of the land.


Let us look at the economics of different renewable solutions. Before reading any discussions of costs of various forms of power one ought to familiarize oneself with  levelized cost of energy  (LCOE) which is really just the total cost of some power plant: building it, interest rate on loans to build it, fuel costs, maintenance over its whole life time. Take all that and divide by all the energy the plant produced over its life time and you got the LCOE.

What is the point of that? It makes it possible to compare very different forms of power generation. E.g. how do you compare the cost of coal power and solar power? To produce power from a coal plant you need to buy coal and burn it each time. For a solar farm there is no cost like that. Cost is primarily in building the farm.

CarbonBrief has this new report from the British government estimating energy costs from different sources.

What I think is interesting here is that estimated nuclear costs in 2025 will be £102/MWh. While the cost of Gas power with Carbon Capture (CCS) will be £85/MWh. There is a lot of uncertainty with this number as CCS is not use at large scale.

Yet if this is realized, Nuclear power seems to have a problem. Offshore wind is expected to cost £57/MWh based on recent auctions and development. Why is this more significant than onshore wind or solar both both which are cheaper?

According to this article solar power prices are the following:

  • $20/MWh in best locations

  •  $30-60/MWh in Europe and the US

  •  $20-40/MWh in China and India

Here is a crazy thing from the same article. Solar PV with revenue support (don't know what that means) is cheaper than operating costs of coal and gas plants in China and India.

Capacity Factor

No power plant is ever producing power 24 hours a day 365 days per year at max. When we talk about installed capacity, that is usually the number quoted. The max possible production. In reality plants need to go down for maintenance, refueling etc. For wind and solar it is of course affected by the weather. Capacity factor is how much of max capacity a given power plant generates. Usually this is measured over time. Nuclear power e.g. is usually quite high at around 90%. But it can in some countries and areas be lower at around 60-70%. Coal plants are often around 50%.

At the bottom you got solar farms. This is highly dependent on country. In the US e.g. the capacity factor for solar farms is around 25%, but in the UK it is closer to 10%.

Wind is interesting. It is not always smart to look at averages because newer installed wind farms often have much improved capacity factor.

This is where offshore wind gets interesting. Onshore wind typically has capacity factor of 30%. But offshore we get more like 40-50%. This is because offshore wind is far more reliable. This is of course much better for the grid, to get stable supply.

However a wind turbine is actually producing power most of the time, just not at full capacity as pointed out by EWEA:

A modern wind turbine produces electricity 70-85% of the time, but it generatesdifferent outputs depending on the wind speed. Over the course of a year, it will typically generate about 24% of the theoretical maximum output (41% offshore). This is known as its capacity factor.

Nuclear Cost Problem

Anyway this is why Nuclear power has a cost problem. Offshore wind is at nearly half the price and produce power most of the time in a pretty reliable fashion. Wind power forecasting is actually quite good.

But of course it is not perfect. Reliable power generation has been the ace up the sleeve for nuclear power. However according to this estimate the kind of power generation which we already know pairs well with wind power, gas power plants can produce electricity with carbon capture at £85/MWh. That is still lower than nuclear and now you got 80% reduction of CO2 emissions which is already lowered when using gas.

The question of course is how much of the mix needs to be gas power. The current COIVD-19 crisis (2020) gives us some clues. From CarbonBrief we have this quote:

The chart below shows that the highest national shares were 65% in Denmark, up from 57% last year, followed by Germany at 45%, up from 34%, and Greece at 41% up from 28%. In the UK over the past 30 days, wind and solar supplied 32% of electricity, up from 24% last year.

We have not seen reports from these countries of grids collapsing or failing. Thus e.g. Denmark is pulling of having 65% of their power from intermittent (irregular output) sources.

This is a bit different from when wind and solar has an unusual high production. In this case demand has dropped.

Capacity Costs per MW

We can lookup various costs for wind turbines at wind industry.

The costs for a utility scale wind turbine range from about $1.3 million to $2.2 million per MW of nameplate capacity installed. Most of the commercial-scale turbines installed today are 2 MW in size and cost roughly $3-$4 million installed. 

# 1600 MW/8000 million euro (Olkiluoto 3)
println(1600/8e3, " MW/million € for Olkiluoto 3")
using Printf
@printf("%.2f MW/million € for Wind\n", 2/3.5)

Danish Analysis

Denmark is an interesting country to look at since, they have such a high degree of power generated from renewable energy.

Danish import of power in GWh per month

using Printf
danish_wind_capacity = 6128    # MW in 2019
danish_wind_produced = 16.15e3 # GWh in 2019
june_import = 1738
feb_import = 922
@printf("In June imports are %0.2f percent of Danish wind power produced per month\n", 100*june_import/(danish_wind_produced/12))
@printf("%0.2f percent in February\n", 100*feb_import/(danish_wind_produced/12))

Sources for wind production numbers are found here. Sources for imported power found here. NOTE: This analysis particular crap, and I have to review it. Need to find good numbers for electricity consumption, production in total.

This PDF has some good overview.

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