Solar insolation: by latitude, season, date, time and tilt?

Solar insolation varies from 600-2,500 kWh/m2/year at different locations on Earth, depending on their latitude, altitude, cloudiness, panel tilt and azimuth. This means the economics of solar can also vary by a factor of 4x. Seasonality is a key challenge at higher latitudes. Active strategies are emerging for orienting solar modules.

1,353 W/m2 of solar energy arrives at the top of the Earth’s atmosphere, based on the Planck Equation, equivalent to almost 12,000 kWh/m2/year. Amazingly, solar is changing the world, even though only c2-3% of this energy is ultimately getting harnessed today.

((The location of the losses in the chart above is also a reason for exploring solar in space, then beaming the power back to Earth)).

50% of all solar energy is inaccessible due to night time (chart below). Another 20-40% is inaccessible as it is absorbed by the atmosphere and clouds (depending on location). And of the insolation that does reach a solar module, only c20-25% is currently converted into useful electricity, in today best HJT modules.

Calculating the insolation ultimately available for solar modules depends on the mass of atmosphere that is traversed by incoming sunshine, which varies hour-by-hour, with the elevation of the sun in the sky (i.e., vertical height) and its azimuth (i.e., compass point bearing).

Calculating these numbers is quite complex, because the Earth is 23º declinated on its axis. Hence the sun’s elevation and azimuth vary hour-by-hour, day-by-day and by location. Nevertheless, the charts below plot elevation and azimuth at a 45-degree latitude, based on 8,760 calculations throughout the year (24 hours per day x 365 days). The latitude can be varied in the data-file, which also contains hour-by-hour granularity.

Insolation at ground level can thus be calculated, based on the mass of air that has been traversed (chart below left). However, fixed solar modules are not always pointed directly at the sun. This can sacrifice 30-60% of the maximum available insolation, simply due to misalignment (chart below right), which is also calculated hour by hour in the data-file.

For fixed modules, losses can be minimized by matching the tilt of the panels to the latitude at which they are situated (chart below). The losses can be reduced even further with solar trackers, which rotate the panels to follow the sun, although this does also add cost.

It is usually best to orient solar modules directly South (in the North Hemisphere). But efficiency may be sacrificed for economics! West-facing panels generate one-third less energy than South-facing ones. But the generation profile is 2-4 hours later, to smooth out the duck curve.

Insolation available to solar modules can realistically vary from 700 – 2,400 kWh/m2/year, depending on latitude and cloudiness. These numbers can be stress-tested in the data-file.

Depending on latitude, generation will also be 0-80% lower in the winter versus the summer. This is visible in the charts above, as high latitudes have short days in the winter, while even when the sun is up, it is only sitting at a low angle in the sky. This seasonality is extremely challenging to back up economically using batteries.

The full data-file allows you to calculate solar insolation, and resultant solar generation, hour-by-hour and then on a fully annualized basis; by stress-testing latitude, elevation, module tile, module azimuth, cloud cover, tracking efficiency and module efficiency. This is helpful for informing our solar economic models. The numbers match our findings from assessing real-world solar volatility. A fantastic resource that helped us with the equations is

Electromagnetic energy: Planck, Shockley-Queisser, power beaming?

Electromagnetic radiation is a form of energy, exemplified by light, infrared, ultraviolet, microwaves and radio waves. What is the energy content of light? How much of it can be captured in a solar module? And what implications? We answer these questions by modelling the Planck Equation and Shockley-Queisser limit from first principles.

Electromagnetic radiation is the synchronized, energy-carrying oscillation of electric and magnetic fields, which moves through a vacuum at the speed of light, which is 300,000 km per second.

Most familiar is visible light, with wavelengths of 400 nm (violet) to 700 nm (red), equating to frequencies of 430 (red) to 750 THz (violet).

At the center of the solar system, our sun happens to emit c40% of its energy in the visible spectrum, 50% as infra-red and c10% as ultraviolet, and very little else (e.g., X-rays, gamma rays at high frequency; microwaves and radio waves at high wavelength). But this is not a coincidence…

Planck’s Law: Spectral radiance as a function of temperature?

Planck’s Law quantifies the electromagnetic energy that will be radiated from a body of heat, across different electromagnetic frequencies, according to its temperature, the speed of light, Boltzmann’s constant (in J/ºK) and Planck’s constant (in J/Hz).

In the chart below, we have run Planck’s equation for radiating bodies at different temperatures from 3,000-8,000ºK, including the sun, whose surface is 5,772ºK. Then we have translated the units into kW per m2 of surface area and per nm of wavelength.

Hence by integration, the ‘area under the curve’ shows the total quantity of electromagnetic radiation per m2. If the surface of the sun were just 10% hotter, then it would emit c50% more electromagnetic radiation and 55% more visible light!

Charts like this also explain why the filament of an incandescent light bulb, super-heated to 2000-3000ºC is only going to release 2-10% of its energy as light. Most of the electromagnetic radiation is in the infra-red range here. And this is the reason for preferring LED lighting as a more efficient alternative. LEDs can reach 60-90% efficiency.

Planck’s Law and Solar Efficiency?

Planck’s Law also matters for the maximum efficiency of a solar module, and can be used to derive the famous Shockley-Queisser limit from first principles, which says that a single-junction solar cell can never be more than c30-33% efficient at harnessing the energy in sunlight.

Semiconductor material has a bandgap, which is the amount of energy needed to promote a single electron from its valence band into its conduction band: a higher energy state, from which electricity can be drawn out of a solar cell. For silicon, the bandgap is 1.1 eV.

What provides the energy is photons in light. The energy per photon can be calculated according to its wavelength. This involves multiplying Planck’s constant by the Speed of Light, dividing by the wavelength, and then converting from Joules to electronVolts. For a radiating body at 5,772ºK, the statistical distribution of photons and their energies is below.

So what bandgap semiconductor is best? If the bandgap is too high (e.g., 4eV), then most of the photons in light will not contain sufficient energy to promote valence band electrons into the conduction band, so they cannot be harnessed. Conversely, if the bandgap is too low (e.g., 0.5eV), then most of the energy in photons will be absorbed as heat not electricity (e.g., a photon with 2.0 eV would transfer 0.5eV into electron promotion, but the remaining 1.5 eV simply heats up the cell).

The mathematical answer is that a bandgap just above 1.3 eV maximizes the percent of incoming sunlight energy that can be transferred into promoting electrons within a solar cell from their valence bands to their conduction bands, at 43-44% (chart below).

If we run a sensitivity analysis on the bandgap, the next chart below shows that our 43-44% conversion limit holds for any semiconductors with a bandgap of 1.1-1.35eV, more of a plateau than a sharp peak.

The Shockley-Queisser limit is usually quoted at 30-34%, which is lower than the number above. In addition to the losses due to incomplete capture of photon energy, the maximum fill factor of a solar cell (balancing load, voltage and current) is around 77%, so only 77% x 44% = 34% of the incoming light energy could actually be harnessed as electrical energy. Moreover, in their original 1961 paper, Shockley and Queisser assumed an 87% efficiency limit for impedance matching relative to the 77%, which is why the number they originally quoted was around 30%.

Another issue is that the solar energy arriving at a given point on Earth has been depleted in certain wavelengths, as they are absorbed by the atmosphere. 1,362W/m2 of sunlight reaches the top of the Earth’s atmosphere. While on a clear day, only around 1,000W/m2 makes it to sea level at the equator. We know the atmosphere absorbs specific infrared wavelengths as heat, because this is the entire reason for worrying about the radiative forcing of CO2 or radiative forcing of methane.

Hence for an ultra-precise calculation of maximum solar efficiency, we should not take the Planck curve, but read-out the solar spectrum reaching a particular point on Earth, which will itself vary with weather!!

Multi-junction solar is inevitable?

The biggest limitation on the efficiency of single-junction solar cells is that they only contain a single junction. This follows from the discussion above. But what if we combine two semiconductors, with two bandgaps into a ‘tandem cell’. The top layer has a bandgap of 1.9eV (e.g. perovskite) and the second has a bandgap of 1.1eV (e.g., silicon). The same analysis now shows how the maximum efficiency can reach 44%.

Cells with multiple semiconductors are already being commercialized. For example, we wrote last year about heterojunction solar (HJT) and this year about the push towards perovskite tandems in solar patents from LONGi. It feels like the ultimate goal will be multi-junction cells that capture along the entire solar spectrum (chart below). It will simply take improvements in semiconductor manufacturing.

solar efficiency in record-breaking multi-junction cells

Power Beaming and Other

Elsewhere in the electromagnetic spectrum, this data-file also contains workings into the energy efficiency of microwave energy, transmitting it through space and converting it back to useful electricity via rectennas.

All of the numbers and calculations go back to first principles, in case you are looking to model the Planck Equation, Shockley-Queisser limit, multi-junction solar efficiency, lighting efficiency, or other calculations of electromagnetic radiation energy.

Vapor deposition: leading companies?

Leading vapor deposition companies by their revenue in 2023 and exposure to the PVD/CVD market.

This data-file is a screen of leading companies in vapor deposition, manufacturing the key equipment for making PV silicon, solar, AI chips and LED lighting solutions. The market for vapor deposition equipment is worth $50bn pa and growing at 8% per year. Who stands out?

Vapor deposition uses 250-1,250ºC temperatures and vacuums as low as 1 millionth of an atmosphere, to deposit nm-μm thick layers of ultra-pure materials onto semiconductor and solar substrates, to make PV silicon, solar modules, computer chips, AI chips, LEDs, plus for hardened metals, cutting tools, insulated glass and aluminized food packaging.

We figured that we needed to compile this screen after reviewing LONGi‘s patents in early-2024. The technology underpinning HJTs and TOPCON modules is very clever, but it is clear from the patents, that it all relies upon vapor deposition. Hence who are the crucial shovel-makers here?

Half of the $50bn pa market is dominated by five public companies with 25-50% exposure to vapor deposition and c30% EBIT margins, based on our screen of leading companies in vapor deposition.

In overall Semiconductor Production equipment, the world leader is Applied Materials, which is based in the US, produces vapor deposition for the solar industry plus for the ‘angstrom era’ of chips, and has $170bn of market cap, more than Schlumberger, Baker Hughes and Halliburton combined.

In chemical vapor deposition for the semiconductor industry, a large Japanese company stood out, claiming 43% market share, and also the only integrated product suite covering the four sequential processes of deposition, coating/developing, etching and cleaning.

In the $700M niche of Metal Organic CVD, as used to make 70% of LEDs globally, but also for wide-bandgap semiconductors, such as SiC and GaN, the market leader is a publicly listed German specialist, with 70% market share.

In laser annealing, which can modify chemical properties over 10-100nm within nanoseconds, for making AI chips, a US-listed specialist stood out as a leader, and it also has a well-regarded ion beam deposition line, seen as a successor to PVD as it achieves larger and uniformly deposited grains.

Our experience as energy analysts has been that companies in the semiconductor supply chain are now just as relevant to the future of global energy as those in the subsea supply chain. Hence over time we will add to this screen of leading companies in vapor deposition.

Global solar: absorption spectrum?

Historic and future solar capacity growth as percentage of total electricity demand growth for different regions

How much new solar can the world absorb in a given year? And are core markets such as the US now maturing? This 15-page note refines our solar forecasts using a new methodology. Annual solar adds will likely plateau at 50-100% of total electricity demand growth in most regions. What implications and adaptation strategies?

Silver pastes for solar contacts?

50 companies make conductive silver pastes to form electrical contacts in solar modules. This data-file tabulates the compositions of silver pastes based on patents, averaging 85% silver, 4% glass frit and 11% organic chemicals. Ten companies stood out, including a Korean small-cap.

Producing over 500GW pa of solar modules per year from 2024 onwards, each containing 15 g/kW-DC of silver electrical contacts, implies total silver consumption of 7.5kTpa, which is over 20% of the global silver market. Hence this work follows on from our screen of silver miners to look at silver pastes for solar module manufacturing.

There are 50 companies producing the silver pastes that are screen-printed onto the front of solar modules, and increasingly also the back too (in TOPCon and HJT cells), forming the electrical contacts with the underlying silicon substrate. Can any companies have an edge?

We reviewed 15 patents from 8 companies. Each has a subtly different formulation. But the average one is 85% silver, 4% glass frit, and 11% other organics. The silver particles average 1.3μm in diameter. These solid reagents are slurried in organic solvents, then screen-printed, then dried at 200-400ºC, then sintered at 800ºC.

Composition of silver pastes for solar panel contacts. On the average they contain 85% silver, 4% glass frit, and 11% other organics.

The key objectives in the patents are to improve efficiency (often by 0.1-0.2% overall, e.g., by ensuring electrical contacts and minimizing resistance), improve printability, improve adhesion, and replace lead from the glass frit. Some of these improvements will enable silver thrifting, perhaps getting the per surface silver intensity down to 10 g/kW-DC in future, per our model here.

The role of the glass frit is to carry silver particles through the passivation layer and into contact with the underlying substrate during sintering. PbO and Bi2O3 are often used. Poorly designed frits cost over 10% in the efficiency of a solar module. Further details in the data-file.

Other additives optimize the viscosity and prevent foaming. Solvents often include ethyl cellulose or butyl carbitols. There are dozens of variations, noted in the data-file.

Ten companies in particular are profiled in the Companies tab, especially based in Korea, China, the US and Japan. A Korean small-cap company stood out from the patents as the most focused on silver pastes, optimizing silver particle sizes, glass frit compositions, micro-porosity and low resistance after sintering. But the space is also highly competitive.

LONGi: technology review and solar innovations?

This data-file is our LONGi technology review, based on recent patent filings. The work helps us to de-risk increasingly efficient solar modules, a growing focus on perovskite-tandem cells, and low-cost solar modules, with simple manufacturing techniques that may ultimately displace bottlenecked silver from electrical contacts. Key conclusions within.

LONGi is the largest solar module producer in the world, on a trailing 5-year basis, producing 60GW of PV modules in 2023, founded in 2000, headquartered in Xi’an with shares publicly listed in Shanghai. The company features in our screen of solar module manufacturers.

LONGi aims to continue driving efficiency gains through the solar industry, especially via HJT cells and perovskite tandem cells. In November-2023, LONGi set a new world record of 33.9% cell-level efficiency for a silicon-perovskite tandem cell, which is the first ever cell to surpass the Shockley-Queisser (S-Q) theoretical efficiency limit.

Hence in this LONGi technology review, we have evaluated twenty recent patent families, mainly those filed from 2022 and 2023. Our conclusions, and key learnings from this exercise, are in the data-file.

Manufacturing details were the highlight. One patent covers the nineteen step process from silicon wafer to finished cell, step by step. What surprised us is the high reliance on simple processes (e.g., polymer adhesive tapes, lasering, vapor deposition) and away from more complex semiconductor manufacturing techniques.

Increasing efficiency was the underlying focus in 80% of LONGi’s patents (chart below). Increasing efficiency historically explains 40% of solar cost deflation and is very likely set to continue.

The breadth of options being explored strongly suggests that solar module efficiency will continue improving by at least 0.5%+ per year (absolute terms), and likely higher as perovskite/tandem cells reach commerciality (details in the data-file).

Silver bottlenecks in the solar industry have been a major feature in our recent research, and across our work into silver. 30% of the patents in our sample focused on ways to displace silver out of PV modules. Updated conclusions on silver are in the data-file.

Key challenges for perovskite/tandem solar cells are also described in LONGi’s patents, and summarized in our LONGi technology review. But how much can we de-risk the solutions intended to overcome these challenges, and how much running room lies ahead?

Solar module production by company?

The world produced over 400GW of solar modules in 2023, which is up 10x from a decade ago. This data-file breaks down solar module production by company and over time, comparing the companies by solar module selling prices ($/kW), margins (%), efficiency (%), transparency, and technology development.

Solar modules are produced when photovoltaic silicon (model here, company screen here) is sliced into wafers, then processed into cells using semiconductor manufacturing techniques, and then finally combined with front contacts, encapsulants, frames, reinforced glass, backsheets and wiring (cost build-up here).

Six Chinese companies (e.g., Longi, Trina) now produce two-thirds of the world’s solar modules, with 2023 output of 20-70GW each. Their growth has been enormous, ramping up by 7x in the past half-decade, and doubling their collective market share (chart below).

High levels of competition are shown by similar module selling prices across the companies in the screen ($/kW numbers in the data-file), and low EBIT margins (numbers also in the data-file by company).

The data also strongly imply that module shipments exceeded module sales in 2021-23, perhaps by as much as 5-10%, creating an overhang for the industry. The overhang was worst in 2021-22 and may have softened in 2023, as the excess was drawn down.

Hence 2023 was a terrible year for the solar industry, with many large PV module manufacturers seeing share price declines of 40-70%, due to interest rates and an overhang of modules, as the US and Europe imported 30-100% more modules than they deployed (chart below). Interestingly, companies with better reporting transparency were more resilient.

Solar module imports by country by month over 2022 and 2023. Imports peaked in late-2022/early-2023 and have fallen roughly 40% since.

Another trend is the shift from P-type towards N-type solar cells, such as TOPCons and HJTs, often to boost efficiency. Different numbers are noted for different companies in the data-file.

The full data-file aims to break down solar module production by company, annually, back to 2013, including useful metrics into their revenues per GW of module production, operating margins, capex intensity and labor intensity (charts below).

Data from the financial reports of solar module producing companies. Charts include: revenues per kW of modules produced, average operating margins, module manufacturing capex per GW produced, and employees per GW produced. Revenues, capex, and labor intensity have fallen over the past decade.

Companies can be differentiated by their technology focus and their geographic focus, with some prioritizing US expansions, and others retrenching to China.

Solar inverters: companies, products and costs?

This data-file tracks some of the leading solar inverter companies and inverter costs, efficiency and power electronic properties. As China now supplies 85% of all global inverters, at 30-50% lower $/W pricing than Western companies, a key question explored in the data-file is around price versus quality.

Solar inverters convert the DC output from solar modules in an AC waveform that can be transmitted across power grids or used in electronic devices. This is achieved via pulse width modulation (explained here) using IGBTs and MOSFETs (explained here).

This data-file covers solar inverter companies and the costs of solar inverters. Twenty companies account for about 90% of global inverter shipments, and the ‘top five’ account for two-thirds of inverters, of which four are Chinese companies, such as Huawei and Sungrow, while we have also explored electronics from SolarEdge.

Our utility-scale solar cost models assume $0.1/W inverter costs, and this is borne out by the data-file. Although costs per watt approximately double for every 10x reduction in inverter size.

Chinese manufacturers sell inverters for 30-50% less than Western companies, suggesting challenged margins and strong competition.

Decent inverters on the market in 2024 convert 98% of the incoming DC electricity into AC electricity, and have advanced power electronics. The ability to control reactive power with a +/- 0.8 leading/lagging power factor is typical. As is the ability to limit total harmonic distortion below 3% (charts below).

While Chinese-made inverters are 30-50% lower cost than Western-made inverters, a key question explored in the data-file is whether this also comes at the cost of lower power quality. Our views and their implications are summarized in the first tab of the data-file. The backup tabs contain the full data behind all of the other charts above.

Semiconductor physics: pièce de resistance?

Semiconductor physics

Semiconductors underpin solar panels, electric vehicles and electronics. Hence this 20-page note aims to explain semiconductor physics from first principles: their conductivity and resistance, their use in devices, plus implications for materials value chains and the energy transition itself?

Semiconductors: conductivity calculations?

Semiconductor conductivity

This datafile calculates the semiconductor conductivity and semiconductor resistivity from first principles, based on their bandgap, doping, electron and hole mobility, temperature, the Fermi-Dirac distribution and the Effective Density of States. Put in any inputs you like to compute the resistance of silicon, germanium or GaAs.

The purpose of this data-file is to break down the conductivity of semiconductors, in spreadsheet format, from first principles, as a useful reference file for our broader modelling of semiconductors, from solar panels to electric vehicles.

The basic formula for conductivity in metals is that S=neμ, where n is the number of charge carriers (usually an integer multiple of the number of atoms), e is the fundamental energy of an electron (1.6 x 10^-19 Joules per electron) and μ is the mobility of the electrons in the material (maybe around 40cm2/V-s as a good ballpark for copper). Resistivity in Ohm-cm is simply 1 ÷ conductivity in Siemens per centimeter.

The formula for conductivity in semiconductors is more complex than metals, because the charge carriers may be conduction band electrons, valence band holes, donor electrons from pentavalent dopants or acceptor holes from trivalent dopants. But generally semiconductor conductivity: σ = ne e μe + np e μp, and these values are in the data-file.

For intrinsic semiconductors, the number of conduction band electrons must be computed as a function of the bandgap, Boltzmann’s constant, temperature and the Effective Density of States (Nc). That final term, in turn, is calculated by multiplying the Fermi-Dirac distribution by the Density of States function and then using calculus to find the ‘area under the curve’. It is all in the data-file.

Semiconductor physics
How semiconductors work. Fermi-Dirac x Density of States yields Effective Density of States and number of Charge Carriers

Undoped silicon (aka intrinsic silicon) will likely have an intrinsic resistivity of around 250,000Ω-cm at room temperature, of which 90% comes from the conductivity of conduction band electrons (rather than holes). There might be 5 x 10^22 silicon atoms per cm3, yielding 1.6 x 10^10 conduction band electrons per cm3, with an average mobility of 1,425 cm2/V-S. These numbers can all be flexed in the data-file.

Doping makes semiconductors an order of magnitude more conductive, and enables their assembly into devices such as light emitting diodes, transistors and solar cells. Moderately doped silicon might contain 1 impurity per 10^7 silicon atoms yielding 5 x 10^14 charge carriers per cm3 (table below). Each 10x increase in doping concentration generally improves conductivity by 10x.

Semiconductor conductivity
Doping of semiconductors in terms of impurities per silicon atom and per cm3

Although even a heavily doped semiconductor will still be about 500-5,000x more resistive than a conductive metal such as copper. Please download the data-file to stress test semiconductor conductivity and resistivity. More of the theory is explained in our overview of semiconductors.

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