Wind and solar: curtailments over time?

Wind and solar curtailments average 5% across different grids that we have evaluated in this data-file, and have generally been rising over time, especially in the last half-decade. The key reason is grid bottlenecks. Grid expansions are crucial for wind and solar to continue expanding.

Curtailments occur when wind and solar are capable of generating electricity, but operators cannot dispatch that electricity into the grid.

This data-file tabulates curtailment rates in California, Australia, the UK, Germany, Spain, Chile and Ireland, averaging c5% in 2022.

The main reason for curtailment is bottlenecks in the grid — i.e., moving renewables from points of generation, to points of unmet demand — rather than renewables having saturated total grid demand.

In a recent research report into the ultimate share of renewables in power grids, we calculated that based on their statistical distributions, solar would only start meeting 100% of a grid’s total demand around 1% of the time when solar was providing c30% of the total grid, while wind would only start meeting 100% of a grid’s total demand around 1% of the time when wind was providing 40% of the total grid (see below). We are not at these levels yet, in the countries in this sample.

The bottleneck is power grids. You might have a 100MW grid, composed of 10 x 10MW inter-connected nodes, and the issue causing curtailment is trying to flow 20MW through a 10MW node.

This confirms that meeting the theoretical potential of renewables (per the note above) requires vast grid expansion, and indeed, countries that have seen YoY reductions in curtailment rates have often achieved this by building new interconnectors.

Second, it gives a new lens on energy storage. Battery deployments can absorb low-cost renewables and prevent curtailment, by circumventing grid bottlenecks, especially for renewables developers who fear bottlenecks in the grid will be persistent.

Commodity prices: metals, materials and chemicals?

Commodity prices

Annual commodity prices are tabulated in this database for 70 material commodities, as a useful reference file; covering steel prices, other metal prices, chemicals prices, polymer prices, with data going back to 2012, all compared in $/ton. 2022 was a record year for commodities. The average material commodity traded 25% above its 10-year average and 60% of all material commodities made ten-year highs.

Material commodity prices flow into the costs of producing substantively everything consumed by human civilization, and increasingly consumed as part of the energy transition. Hence this database of annual commodity prices is intended as a useful reference file. Note it only covers metals, materials and chemicals. Energy commodities and agricultural commodities are covered in other TSE data-files.

Source and methodology. The underlying source for this commodity price database is the UN’s Comtrade. This useful resource covers trade between all UN member countries, across thousands of categories, in both value terms ($) and mass terms (kg). Dividing values (in $) by masses (in kg) yields an effective price (in $/kg or $/ton). We have then aggregated, cleaned and averaged the data for 70 materials commodities.

The median commodity in the data-file costs $2,500/ton on an unweighted basis. Although this ranges from $20/ton for aggregates to $75M per ton for palladium metal.

2022 was a record year for material commodity prices. The average material commodity priced 25% above its 10-year average and 40 of the 70 commodities in the data-base made 10-year highs.

Steel prices reached ten-year highs in 2022, averaging $2,000/ton across the different steel grades that are assessed in the data-file. This matters as 2GTpa of steel form one of the most important underpinnings in all global construction. Our steel research is aggregated here.

Commodity prices
Steel Price by year by steel grade in $ per ton

Base metal prices averaged 40% above their ten year averages in 2022, as internationally traded prices rose sharply for nickel, rose modestly for aluminium and zinc, and remained high for copper (chart below).

Commodity prices
Base metal prices by year and over time for zinc aluminium copper and nickel in $ per ton

Battery metals and materials prices rose most explosively in 2022, due to bottlenecks in lithium, cobalt, nickel and graphite. This is motivating a shift in battery chemistries, both for vehicles and for energy storage. It also means that the average battery material in our datafile was higher priced than the average Rare Earth metal in the data-file (which is unusual, but not the first time).

Commodity prices
Battery material prices over time $ per ton for lithium cobalt manganese nickel LiPF6 and lithium carbonate in $ per ton

Commodity chemicals all rose in 2022, across every category tracked in our chart below. These chemicals matter as intermediates. On average sodium hydroxide prices reached $665/ton in 2022, sulphuric acid prices reached $140/ton and nitric acid prices reached $440/ton.

Commodity prices
Industrial Acids and Caustic Soda Prices over time NaH H2O2 HCl H2SO4 Sulfuric Acid HNO3 Nitric Acid H3PO4 Phosphoric Acid HCN and HF in $ per ton

500MTpa of global plastics and polymers demand is covered in our plastics demand database. Both finished polymer prices (first chart) and underlying olefins and aromatics (as produced by naphtha crackers, second chart) prices rose sharply in 2022. Our recent research has wondered whether terms of trade are likely to become particularly constructive for polyurethanes.

Commodity prices
Polymer prices by year LDPE HDPE PET EVA Polyurethanes Paints and Adhesives in $ per ton
Commodity prices
Olefins and Aromatics Prices over time

Silicon prices matter as they feed in to the costs of solar, and traded silicon prices also reached ten year highs in 2022, before correcting sharply in 2023. Silica prices surpassed $70/ton, silicon metal prices reached $4,000/ton and polysilicon prices surpassed $30/kg (charts below).

Commodity prices
Silica price, silicon price and polysilicon price in $ per ton

The full database captures 70 globally traded materials commodities and their annual prices over time in $/ton, year by year, from 2012-2022. These are: Acrylonitrile prices, Adhesives prices, Aggregates prices, Aluminium prices, Ammonia prices, Battery Graphite prices, Benzene prices, Butadiene prices, Carbon Fiber prices, Cement prices, Cobalt prices, Cobalt Oxide prices, Cold Rolled Steel prices, Concrete prices, Copper prices, Copper Wire prices, Cumene prices, Electric Motor and Generator prices, Electrical Transformer prices, Epoxide prices, Ethanol prices, Ethylene prices, Ethylene Oxide prices, EVA prices, Formaldehyde prices, Glass Fiber prices, Gold prices, Graphite Anode prices, Graphite paste prices, HCl prices, HDPE prices, HF prices, Hot Rolled Steel prices, Hydrogen Peroxide prices, Integrated Circuit prices, LDPE prices, LiPF6 prices, Lithium Carbonate prices, Lithium Metal prices, Manganese prices, Manganese Oxide prices, Methanol prices, NaCN prices, Nickel prices, Nitric Acid prices, Paint prices, Palladium prices, PET prices, Phosphoric Acid prices, Platinum prices, Polyethylene prices, Polysilicon prices, Polyurethane prices, Propylene prices, Propylene Oxide prices, PTFE prices, Rare Earth Magnet prices, Scandium & Yttrium prices, Silica prices, Silicon Metal prices, Silver prices, Sodium Hydroxide prices, Stainless Steel prices, Steel Alloy prices, Sulfuric Acid prices, Toluene prices, Tubular Steel prices, Urea prices, Vehicle prices, Xylene prices, Zinc prices.

Oscar Wilde noted that the cynic is the man who knows the price of everything, but the value of nothing. To avoid falling into this trap, we also have economic models for most of the commodities in this commodity price database too.

We will continue adding to this commodity price database amidst our ongoing research. You may find our template useful for running Comtrade queries of your own. Or alternatively, if you are a TSE subscription client and we can help you to use this useful resource, then please do email us any time.

California’s grid: wind and solar statistical distributions?

Wind and solar statistical distributions

This data-file aggregates wind and solar statistical distributions, plotting solar generation and wind generation, every 5-minutes, across California, for the entirety of 2022, in order to understand their volatility and curtailment rates. The data suggest that wind and solar will most likely peak at 50-55% of renewables-heavy grids.

How will ramping up wind and solar increase volatility of other generation sources, lower the utilization rates of the overall grid, and where will wind and solar naturally peak?

We can answer these questions by evaluating 100,000 x 5-minute intervals of power generation, from California’s power grid in 2022, then calculating the statistical distributions of wind, solar and total demand.

Our key findings are in our recent research note, assessing where will wind and solar peak? (see below).

As always, we publish the data behind our analysis, to give transparency on the methodology, and assumptions can be stress-tested.

Tabs ending _Averages plot the average load across different five-minute intervals, across the year, showing wind and solar statistical distributions, and how they vary throughout the day, in different months.

Tabs ending _DitsributionFlex plot the distribution of each time series, by percentile, across the entire year.

The analysis then asks: if the future distribution matches today’s distribution, how much of a hypothetical ‘constant’ grid load could wind, solar and wind+solar supply, at different curtailment thresholds.

If we assume that supply will stop growing when marginal capacity additions are likely to incur 30-60% curtailment rates, this suggests that solar would most likely peak out at 35-40% of a grid, wind would most likely peak out at 60-70% of a grid, while a mixture of wind and solar would most likely peak out at 50-55% of a grid.

Please see our broader research into power grids, into California’s grid and for similar statistical analysis of wind generation elsewhere, we have run similar analysis for the UK power grid.

California power generation over time?

California power generation

California’s power grid ranges from 15-61GW of demand. Utility scale solar has almost quadrupled in the past decade, rising from 5% to almost 20% of the grid. Yet it has not displaced thermal generation, which rose from 28% to 36% of the grid. We even wonder whether wind and solar are entrenching natural gas generators that can backstop their daily, weekly and even seasonal volatility.

This data-file aggregates descriptive statistics into California’s power grid, plotting California power generation over time, looking across 150MB of CAISO data into solar, wind, nuclear, hydro, imports and thermal generation, every five minutes, for almost a decade.

Over the past decade, California’s power grid has averaged 26GW of load, but the demand is highly variable, troughing at 15GW in April-2022 and peaking at over 61GW in July-2021. Summer demand is almost 50% higher than winter demand due to air conditioning.

California power generation
California Power Grid Descriptive Statistics Average Load, Minimum, Maximum and Quartiles

Utility-scale solar is the biggest change in the generation mix, rising from 5% of the grid to almost 20% of the grid over the past decade. Solar generation is volatile. The average load from solar was 4GW in the trailing twelve months, but c40% of the time, solar generation is zero, while 25% of the time it is above 10GW and 5% of the time it is above 13GW.

California power generation
California Utility-Scale Solar Generation Over Time in GW

Seasonality also matters for solar. Solar generates around 2x higher output during the summer months than the winter months. Similar charts for wind are in the data-file.

Thermal generation has not necessarily been displaced by the rise of solar and wind in California’s grid, but arguably, entrenched, as gas-fired power plants can rapidly ramp up production, after the sun sets, or in non-sunny months.

The increasing volatility of California’s power prices rewards natural gas generators, but is not high enough to bring longer-duration battery storage into the money.

Total thermal generation has actually risen from 28% of the grid in 2017 to 36% of the grid in 2023, albeit the picture is somewhat distorted by annual fluctuations in hydro output and the change in imports, tabulated in the data-file.

California power generation
California Thermal Generation Over Time in GW

Is California’s thermal generation base backstopping its renewables? There is also evidence of thermal generation plants being run more flexibly, to ‘backstop’ solar and wind in California’s grid, with rising interquartile ranges, absolute ranges and deviations between upper-decile and lower-decile generation.

California power generation
Thermal Generation Monthly Ranges

Full descriptive statistics are in the data-file, covering all of California’s main generation sources — solar, wind, nuclear, hydro, imports and thermal — and the proportionate share of each one. For each month, we plot each generation source’s minimum output, 10th percentile, lower quartile, median, mean, upper quartile, 90th percentile and maximum output.

Renewable grids: solar, wind and grid-scale battery sizing?

Grid-scale battery sizing

How much wind, solar and/or batteries are required to supply a stable power output, 24-hours per day, 7-days per week, or at even longer durations? This data-file stress-tests grid-scale battery sizing, with each 1MW of average load requiring at least 3.5MW of solar and 3.5MW of lithium ion batteries, for a total system cost of at least 18c/kWh.

Start by modelling a power demand curve. Then model how much wind or solar would need to be installed to provide this electricity demand across a comparable timeframe. Then model how big a battery is required to move the renewables to align with the timing of the power demand curve. This data-file works through the maths, for different batteries, including their round trip efficiencies, and their costs.

The minimum possible requirement for a fully solar-powered electricity grid is that each 1MW of load requires 3.5MW of solar modules and 3.5MW of lithium ion batteries with daily charging-discharging, in a location where every day is perfectly sunny, with no clouds, and no seasonality, for a total levelized cost (LCOTE) of 18c/kWh.

Introduce volatility into the weather pattern, and the requirement for a fully solar-powered grid is that each 1MW of average load requires 5MW of solar modules and 9MW of lithium ion batteries with full charging-discharging every 1.5 days on average, and a total levelized cost (LCOTE) of 35 c/kWh. For more detail, please see our data-file into the volatility of solar generation.

Grid-scale battery sizing
Sizing of a solar array and battery module to supply a 100MW load for 30-days including cloudy days

Introduce seasonality in the weather pattern, with 50% lower solar output in winter versus the summer, and the requirement for a fully solar-powered grid is that each 1MW of average load requires 6MW of solar modules and a somewhat insane 235MW of lithium ion batteries with full charging-discharging every 70-days on average, for a total levelized cost (LCOTE) of 800c/kWh. Which is also somewhat insane.

Grid-scale battery sizing
Sizing of a solar array and battery module to supply a 100MW load for 365-days including seasonality

Wind numbers are more demanding than solar numbers, all else equal, because the sun rises and sets daily (helping the utilization rate of the batteries), while wind can incur 2-3 windy days followed by 2-3 non-windy days (hurting the utilization rate of batteries). For more detail, please see our data into the volatility profile of wind generation.

Grid-scale battery sizing
Sizing of a wind farm and battery module to supply a 100MW load for 2-3 weeks including still days

Redox flow batteries are particularly helpful for integrating larger shares of renewables, and are modelled to result in total system costs that are c50% lower than using lithium ion batteries at grid scale. Please see our deep-dive research note into redox flow batteries.

This data-file provides underlying workings into renewable asset sizing, grid-scale battery sizing and total system costs for our recent research into renewables’ true levelized cost of electricity (LCOTE).

Density of gases: by pressure and temperature?

Density of gases

The density of gases matters in turbines, compressors, for energy transport and energy storage. Hence this data-file models the density of gases from first principles, using the Ideal Gas Equations and the Clausius-Clapeyron Equation. High energy density is shown for methane, less so for hydrogen and ammonia. CO2, nitrogen, argon and water are also captured.

The Ideal Gas Law states that PV = nRT, where P is pressure in Pascals, V is volume in m3, n is the number of mols, R is the Universal Gas Constant (in J/mol-K) and T is absolute temperature in Kelvin.

The Density of a Gas can be calculated as a function of pressure and temperature, simply by re-arranging the Ideal Gas Law, where Density ρ = P x Molecular Weight / RT. Our favored units are in kg/m3.

Density of methane in kg/m3 and kWh/m3

The Density of Methane (natural gas) can thus be derived from first principles in the chart below, using a molar mass of 16 g/mol, and then flexing the temperature and pressure. This shows how methane at 1 bar of pressure and 20ºC has a density of 0.67 kg/m3. LNG at -163ºC is 625x denser at 422 kg/m3. And CNG at 200-bar has a density of 180kg/m3.

Density of gases
Density of methane, LNG and CNG according to pressure and temperature

The Energy Density of Methane can thus be calculated by multiplying the density (in kg/m3) by the enthalpy of combustion in kJ/kg, and then juggling the energy units. A nice round number: the primary energy density of methane is 10 kWh/m3 at 1-bar and 20ºC, increasing with compression and liquefaction. CNG at 200-300 bar has around 30-60% of the energy density of gasoline, in kWh/m3 terms.

The energy density of methane is 10kWh/m3 as a nice rounded rule-of-thumb.

Clausius-Clapeyron: gas liquefaction?

Methane liquefies into LNG at -162ºC under 1-bar of pressure. The boiling points of other gases range from water at 100ºC, ammonia at -33ºC, CO2 at -78ºC, argon at -186ºC, nitrogen at -196ºC to hydrogen at -259ºC. This is all at 1-bar of pressure.

However, liquefaction temperatures rise with pressure, as can be derived from the Clausius-Clapeyron equation, and shown in the chart below. At 10-20 bar of pressure, you can liquefy methane into ‘pressurized LNG’ at just -105 – 123ºC, which can sometimes improve the efficiency of LNG transport. This can also help cryogenic air separation.

Density of gases
Boiling Points of Different Gases According to the Clausius Clapeyron Relationship

Density of CO2: strange properties?

The Density of CO2 is 1.87 kg/m3 at 20ºC and 1-bar of pressure, which is 45% denser than air (chart below). But CO2 is a strange gas. It cannot exist as a liquid below 5.2 bar of pressure, sublimating directly to a solid. CO2 can also be liquefied purely by compression, with a boiling point of 20-80ºC at 30-100 bar of pressure.

Density of Gas
Density of CO2 according to pressure and temperature in kg per m3

Hence often the disposal pipeline in a CCS or blue hydrogen value chain may often be pumping a liquid, rather than flowing a gas. And finally, these properties of CO2 open the door to surprisingly low cost CO2 transport by truck or in ships. This is all just physics.

Super-critical fluids: fourth state of matter?

There is also a fourth density state for all of the gases in the data-file. Above their critical temperature and critical pressure, fluids ‘become super-critical’. Sometimes this is described as ‘having properties like both a gas and liquid’. Mathematically, it means density starts rising more rapidly than would be predicted by the Ideal Gas Equations.

Super-critical gases behave unpredictably. Their thermodynamic parameters cannot be derived from simple formulae, but rather need to be read from data-tables and/or tested experimentally. This is why the engineering of supercritical systems tends to involve supercomputers.

Examples of super-critical gases? Steam becomes supercritical above 218-bar and 374ºC. CO2 becomes supercritical about 73-bar and 32ºC. Thus CO2 power cycles inevitably endure supercriticality.

Energy density of hydrogen lags other fuels?

The Density of Hydrogen is 0.08 kg/m3 at 20ºC and 1-bar of pressure, which is very low, mainly because of H2’s low molar mass of just 2g/mol. Methane, for example, is 8x denser. CO2 is 20x denser. In energy terms, gasoline is 3,000x denser per m3.

Hence hydrogen transportation and storage requires demanding compression or liquefaction. Tanks of a hydrogen vehicle might have a very high pressure of 700-bar, to reach a 40kg/m3 (the same density can be achieved by compressing methane to just 50-bar!). Liquefied hydrogen has a density around 70kg/m3 (LNG is 6x denser).

The density of hydrogen is just 0.08 kg/m3 at 20ºC of temperature and 1-bar of pressure

The energy density of hydrogen, in kWh/m3 also follows from these equations. At 1-bar and 20ºC, methane contains 3x more energy per m3 than hydrogen. Under cryogenic conditions, it contains 2x more energy. Under super-critical and ultra-compressed conditions, it contains 4x more.

The energy density of hydrogen is 50-75% lower than natural gas, even after compression/liquefaction

Data into the energy density of gases?

Similar energy density challenges constrain the use of ammonia as a fuel, as tabulated in the data-file, contrasted with other fuels, and discussed in our research note here.

This data-file allows density charts — in kg/m3 and in kWh/m3 — to be calculated for any gas, using the Ideal Gas Laws and the Clausius-Clapeyron equations. The data-file currently includes methane, CO2, nitrogen, ammonia, argon, water and hydrogen.

Storage tank costs: storing oil, energy, water and chemicals?

Storage tank costs are tabulated in this data-file, averaging $100-300/m3 for storage systems of 10-10,000 m3 capacity. Costs are 2-10x higher for corrosive chemicals, cryogenic storage, or very large/small storage facilities. Some rules of thumb are outlined below with underlying data available in the Excel.

This data-file tabulates 50 data-points into the costs of storage tanks for water, oil products, chemicals, LNG and hydrogen. In both $/m3 terms and $/ton terms.

This matters as storage tanks are used the downstream industry, materials value chains, and in several types of new energies such as redox flow batteries or pumped hydro.

We also think that some industrial facilities may be able to benefit from increasingly volatile power prices, amidst the build out of renewables, by demand shifting, which means timing their electrical loads to the times when renewables are generating. In some cases, this requires increasing the sizes of storage tanks to increase flexibility.

A good rule of thumb is that the storage tank costs for storing fluid commodities will average around $100-300/m3 of capacity, at capacities of 10m3 to 10,000 m3, for relatively simple and non-hazardous commodities such as water and fuel.

Generally tank costs fall (in $/m3 terms) as tank capacities rise. Bigger tanks benefit from economies of scale, and this is visible in the chart above for all categories.

Costs are typically 3-5x higher for corrosive chemicals that can require double tanks, stainless steel or specialized tank linings. Maybe $1,000/ton is fair here.

Costs are also typically 3-5x higher for storing cryogenic liquids, which can require specialized nickel steel and insulation.

Costs are also 2-3x higher for very small tanks (below 10m3, lacking economies of scale) or very large tanks (on the magnitude ot 100,000m3, so large that they need to be stick-built rather than simply purchased as finished modular units).

LNG storage tanks thus come in as some of the most expensive storage facilities in the data-file, because they are very large and cryogenic. Higher capex may be worthwhile to install higher grade tanks that minimize boil-off and improve energy efficiency.

Large-scale hydrogen storage would likely be higher cost than LNG storage, in our view. Please see our hydrogen storage model and broader hydrogen research.

Methodology. Mainly we have aimed to capture tank costs in the data-file, while excluding the costs of their foundations, pumps, valves and installation; but the lines get a little bit blurry, especially for some of the very large tanks.

Context matters. Some of the data-points are supplier quotes, some are estimates for technical papers, and some are disclosed data-points from specific projects. Please download the data-file for the additional context.

CCS absorbers: unit sizing and residence times?

CCS absorbers

Post-combustion CCS plants flow CO2 into an absorber unit, where it will react with a solvent, usually a cocktail of amines. This data-file quantifies operating parameters for CCS absorbers, such as their sizes, residency times, inlet temperatures, structural packings and the implications for retro-fitting CCS at pre-existing power plants.

Post-combustion CCS aims to capture the CO2 from pre-existing industrial facilities and power plants, by flowing exhaust gases upwards through an absorber unit, while a solvent simultaneously flows downwards and reacts with the CO2. Costs, energy penalties and leading solvent candidates are covered in our CCS research.

But how hard is it to find space for these absorber units at pre-existing industrial facilities? This data-file has compiled key parameters from various technical papers, most aiming for 90% capture rates.

Across a dozen CCS examples in the data-file, each m/s of inlet gas requires 7 m3 of absorber capacity. Hence the absorber units for a world-scale 500MW power plant can reach 3,000 – 10,000 m3 of volume, usually across 2-4 absorbers with 10-15m diameters and 15-25m heights.

For the ultimate space requirements of the CCS plant, multiply by 2-5x, for the desorbers, utilities, piping and balance of plant.

This model calculates the size of the absorber unit required, as a function of height, diameter, residency time, CO2 inlet concentration, CO2 capture rate, solvent properties and structural packing.

Generally larger absorber units are required at industrial facilities with higher CO2 inlet concentrations and lower target CO2 levels.

For example, removing 90% of the 4%-concentrated CO2 from our base case natural gas burner requires a 12m absorber. Absorbing 90% of the 12%-concentrated CO2 from a coal boiler requires a 20m absorber.

CCS absorbers
Larger absorber units are required for CCS plants that start with more CO2 and absorb more CO2

The average residency time within a CCS absorber is below 10-seconds. Although the number depends on the unit size, flow velocity, amine quality and temperature. These can all be flexed in the data-file.

CCS absorbers
Residence time for a CCS absorber is usually below 10 seconds. Hotter inlet gas and solvent allows for shorter residence times.

Smaller and less expensive absorbers are possible with faster-acting amines, shorter residency times and greater structural packing.

A listed mid-cap company based in Switzerland was often mentioned in technical papers, with a product range of packing materials that can achieve 200-1,200 m2/m3 of internal surface area to promote gas-liquid exchange and slim-line CCS absorbers.

Electrochemistry: battery voltage and the Nernst Equation?

battery voltage

What determines the Voltage of an electrochemical cell, such as a lithium ion battery, redox flow battery, a hydrogen fuel cell, an electrolyser or an electrowinning plant? This note explains electrochemical voltages, from first principles, starting with Standard Potentials and working through the Nernst Equation.

Redox Chemistry 101

Reducing agents. Some atoms “really want” to surrender unstable outer electrons and oxidize into positively charged ions. Lithium metal, for example, is so desperate to shed an outer electron that it will rip water apart, forming LiOH (Li+ and OH-) and H2 gas. This makes lithium metal a strong reducing agent. As it oxidizes, another atom must reduce.

Oxidizing agents. Other atoms “really want” to complete incomplete outer electron shells, gain electrons and thus “reduce” into negatively charged ions (or form covalent bonds). Oxygen, for example, is quite keen to accept electrons from metal atoms. So much so that ‘oxidation’ is named after it. As oxygen reduces, another atom must oxidize.

battery voltage
Standard Potential of Cations from Lithium to Gold in Volts relative to the Standard Hydrogen Half Cell

What is Standard Potential?

Standard Potential aims to quantify how much an atom wants to oxidize or reduce relative to the H2/H+ redox pair and measured in Volts. This sentence sounds confusing. But bear with us!

How should I think about standard potential? Imagine a magic electrochemical cell, with two electrodes, in two chambers, separated by a magical membrane. The key facet of this magic membrane is that ONLY H+ ions can cross the cell. On one side of the cell is a chamber of H2 gas. At any point in time, a very small portion of H2 gas is prone to dissociating into H+ and H- cations and anions. H2 <=> H+ + H- On the other side of the cell, we can add literally any reagent.

For example, if we add lithium to the other side of the cell, it will oxidize, surrendering an electron into an electrode, forming Li+. The Li+ will attract H- anions across the cell membrane to form LiH. This leaves an excess of H+ ions on the H-side of the cell. These H+ cations reduce, accepting an electron from a second electrode, and re-forming H2 gas. The chemical reaction is Li + H+ -> Li+ 0.5 H2. The thermodynamic result is that energy has been released by this reaction. The energy is imparted to the electrons that were ‘pushed out’ from the Li-side of the cell. Remember that 1 Volt simply means 1 Joule of energy per Coulomb of charge, where the elementary charge of 1 electron is 1.602×10^-19. Power = Voltage x Current. Hence our electrochemical cell is a power source, flowing a current (in Coulombs per second) with a voltage (in Joules/Coulomb). Lithium is one of the strongest reducing agents in chemistry with a Standard Potential of -3.04 Volts (relative to the H2/H+ redox pair).

Battery voltage
Schematic illustrating the Standard Potential for a hypothetical Lithium-Hydrogen battery including Standard Potentials

As another example, if we add Fluorine gas to the other side of the cell, it will reduce, pulling in an electron from the first electrode. The chemical reaction is F2 + 2e -> 2F-. The F- ions will attract H+ cations across the cell membrane to form HF. This leaves an excess of H- anions on the H-side of the cell. These H- anions oxidize, surrendering an electron from the second electrode, reforming H2 gas. The chemical reaction is F2 + H2 -> 2HF. The thermodynamic result is that energy has been released by this reaction. The energy is imparted to the electrons that were ‘pushed out’ from the H-side of the cell. Fluorine is one of the strongest oxidizing agents in chemistry with a Standard Potential of +2.86 Volts (relative to the H2/H+ redox pair).

Redox flow batteries are very similar to these idealized electrochemical cells. The example below uses Vanadium, in V2+, V3+, V4+ and V5+ oxidation states.

Batteries are Electrochemical Cells

All batteries work via this same redox chemistry principles that were illustrated above. And the standard potentials of each half-cell add together to yield the standard potential of the battery.

For example, imagine an electrochemical cell with Li metal on one side, F2 gas on the other, and a membrane that was ONLY permeable to Li+ ions. On the F-side of the cell, F will reduce to F-, with a Standard Potential of +2.86V. On the Li-side of the cell, Li will oxidize into Li+, with a Standard Potential of -3.04V. The resultant Li+ ions can travel across the membrane to the F-side, forming LiF salt. The Standard Potentials add, for a total voltage of 5.9V. Remember, this is the energy that is imparted per Coulomb of electrons flowing through this direct current circuit.

Lithium Fluoride would be one of the highest voltage electrochemical cells possible, with a Standard Potential of 5.9V, because lithium is one of the strongest reducing agents and fluorine is one of the strongest oxidizing agents. Wikipedia maintains a useful list of Standard Potentials.

Charging a battery is simply running this same circuit in reverse. Likewise, electrowinning is an electrical technology used to separate metals from mixtures, such as aluminium, zinc, copper, silver and gold. There are even attempts to electrowin steel. Green hydrogen is the electrochemical product of electrowinning water. For all of these systems, the required Voltage can be computed from the Standard Potentials of the redox pairs (our own attempt to do this is here).

The Nernst Equation for Open Circuit Voltage?

Actual voltage in an electrochemical cell differs from Standard Potential, according to reaction conditions, temperature and molar concentrations. These deviations are captured by the Nernst Equation.

The Nernst Equation approximates the energy potential (in Volts) of an electrochemical reaction, based on the reagents, their concentrations, their temperature and other constants. It was formulated by the German Chemist, Walther Nernst (1864-1941), who is also partly credited with formulating the third law of thermodynamics.

The Nernst Equation states that the Reaction Potential E = E0 – (RT/nF) lnQ. To unpack these variables:

Eo is the Standard Potential for the redox reaction between reagents.

R is the Universal Gas Constant of 8.314 (J/mol-K), in turn the product of the Boltzmann Constant (J/K) and Avogadro’s Constant (particles/mol).

T is the absolute temperature in Kelvin.

n is the number of electrons transferred in the redox reaction (e.g., 1 for Li -> Li+ + e-).

F is Faraday’s Constant (Coulombs/mol), in turn the product of the elementary charge of an electron (C) and Avogadro’s Constant (particles/mol).

Q is the ratio of concentrations between reagents in the reaction. Q is going to change over time, as reagents are consumed. Q is going to vary, in particular, in a battery, according to its state of charge.

How does lithium ion battery voltage vary with State of Charge?

We have used the Nernst Equation, in the chart above, to capture a lithium ion battery with a 3.7V Standard Potential.

Cell Voltage matches Standard Potential when the concentration of Li+ in solution matches the concentration of Li intercalated at the anode. Here [LiC6] = [LiMxOy]. Hence [LiMxOy]/[LiC6] = 1. Hence ln(1) = 0. Hence E = E0.

As the battery is discharged, most of the MxOy sites at the cathode have already been occupied at the cathode, and there is very little LiC6 left to dissociate at the anode. Hence the [LiMxOy]/[LiC6] term in the Nernst Equation becomes very large. Hence the Voltage of the Battery falls. Since Power = Voltage x Current, and Voltage is dropping, it would be necessary to flow faster currents to sustain the same power output. This is why users sometimes report batteries “running out quite suddenly”. 

The cutoff voltage for a lithium ion battery is around 3V. Battery degradation occurs when lithium ion batteries are over-discharged, such as dissolution of the copper current collector at the anode.

As the battery is charged, most of the LiC6 sites have already been occupied at the anode, and there is little LiMxOy left to dissociate at the cathode. Hence the [LiMxOy]/[LiC6] term in the Nernst Equation becomes very small. Note than when x is less than 1, ln(x) is negative. Hence the Voltage needed to continue charging the battery rises.

Higher voltages can chemically break down other materials in the battery and also contribute to battery degradation. For example, the standard potential for decomposition LiPF6 electrolyte is just over 5V. Hence there is an inherent trade-off, where cells with higher Voltage will tend to be more energy dense, but also more degradation-prone.

Why do batteries degrade faster at higher temperatures?

The Nernst Equation helps to explain why batteries degrade faster at higher temperatures. It features absolute temperature as an input variable.

With higher temperatures, voltage variations become more extreme, for both high charges and extreme discharges (chart below).

Other chemical side-reactions will also be promoted at higher temperatures via the Boltzmann Equations. Especially if there are hotspots in the battery during charging.

Does battery hysteresis impact efficiency?

When an electrochemical cell is charged, or discharged, the electrode materials become polarized. Polarizing the electrodes consumes a voltage. Resistance within the cell also causes a voltage drop. And there is usually an overpotential at the electrodes of any electrochemical cell. Hence the voltage needed to charge a cell is usually higher than the voltage measured when discharging it. These losses are relatively small, around 1-3%.

Other losses include Ohmic losses (from resistance), Coulombic losses (from chemical side-reactions and shunt currents) and System losses (e.g., from any battery management system, inverters, converters or other power electronics). But overall, the round-trip efficiency of a lithium ion battery should be around 85-90%.

Other real world issues?

The Nernst Equation gives a useful framework for understanding the voltage of electrochemical cells. Real world batteries have additional complexities, which can cause voltage curves to deviate. Electrodes are crystalline solids. The compositions, size and structure of these materials can impact Li/Li+ potentials. There can even be quite abrupt changes in Voltage, where the intercalation ‘reorganizes’ the crystal structure of cathode materials (e.g., from cubic spinel to tetragonal spinel).

The download linked below contains the maths and data behind the charts in the article. For other battery-related data-files, please see our broader research into batteries.

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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