Primary, final and useful energy

Framing

In 2023, the world's economies extracted, imported, harvested or otherwise put into productive use a quantity of energy that the IEA reports at approximately 625 exajoules. The same year, the world's consumers paid for energy at meters, pumps and counters totalling approximately 440 exajoules. The difference, around 185 exajoules, vanished in the transformation sector: in refineries, in power stations, in gas-processing plants, in cooling towers and grid losses. By the time the consumer's electricity reached the wall socket and the consumer's petrol reached the engine, almost a third of what nature had given up had already been spent as waste heat.

A third number is rarer in published statistics, because it is harder to measure and the methodology is contested. It is the useful energy: the share that actually does the work the consumer wanted done. The light that emerges from the bulb, not the heat that emerges with it. The kinetic energy of the moving car, not the heat in the tailpipe. The room warmth at twenty degrees, not the flame in the boiler. Estimates of global useful energy, made on different methodological bases, cluster around 100 to 130 exajoules. On the lower of those estimates, less than a fifth of the primary energy that humanity extracts actually does anything.

These three numbers — primary, final and useful — are not different things. They are the same physical quantity, measured at different accounting boundaries, with definite losses between them. The losses are not regrettable accidents. Most of them are thermodynamically necessary: heat engines cannot exceed the Carnot limit, and a flame that warms a room must also warm a chimney. But a sizeable fraction is not necessary. It reflects the technologies that happen to be installed, the order in which energy is converted along the chain, and the matching of energy quality to end use. The transition to a less wasteful energy system is, at its physical core, a transition that compresses the cascade.

This lesson teaches the three definitions, the principal points at which energy is lost between them, the accounting methodology disputes that change the apparent share of renewables and nuclear, and the worked-out implications of all this for the most-discussed end use of all: heating a house.

Key terms

1. Primary energy. Energy in the form in which it occurs in nature, at the point at which it enters the economic system. Examples: crude oil at the wellhead, natural gas at the gathering line, coal at the pit head, solar radiation on the panel, kinetic energy of wind at the rotor, hydraulic head behind a dam, fission heat in a reactor core.
2. Final energy. Energy in the form delivered to the consumer at the point of purchase. Examples: electricity at the wall socket, petrol or diesel at the pump, natural gas at the household meter, district heat at the radiator. Final energy equals primary energy minus the losses incurred in extraction, transformation, transmission and distribution.
3. Useful energy. Energy in the form that actually delivers the service the consumer wanted. Examples: visible light from a lamp, kinetic energy of a moving vehicle, room heat at twenty degrees, mechanical work of a lathe. Useful energy equals final energy minus the losses incurred at the device, at the point of end use.
4. Transformation losses. The energy dissipated in the conversion of primary into final. The dominant component is the heat rejected by thermal power stations to atmosphere or cooling water, but the category also includes refinery own-use, gas-processing losses, liquefaction and regasification losses in LNG, and pipeline and grid losses.
5. End-use efficiency. The ratio of useful energy to final energy at a specific device. Indicative values: an incandescent bulb converts about 5 per cent of its electricity into visible light, an LED bulb about 30 to 40 per cent; a petrol car engine converts about 25 per cent of its fuel into kinetic energy at the wheels, an electric motor about 90 per cent; a condensing gas boiler converts about 90 per cent of its gas into heat in the water circuit. Heat pumps are a special case (see term 6).
6. Coefficient of performance (COP). For a heat pump, the ratio of heat delivered to electricity consumed. A COP of three means one kilowatt-hour of electricity moves three kilowatt-hours of heat from outside to inside. The figure routinely exceeds one — and is often three or four — because the heat pump is not creating heat but moving it. The thermodynamic limit (the Carnot COP at the operating temperatures) is several times higher again. The seasonal COP, abbreviated SCOP, is the figure averaged over a heating season under real conditions.

The cascade in physical detail

The three tiers as accounting boundaries

It is helpful to think of primary, final and useful energy as the readings on three meters placed at three different points along the energy chain. The primary meter sits at the boundary of the producing well, the mine mouth, the dam, the photovoltaic panel, the wind turbine rotor, or the reactor core. The final meter sits at the consumer's point of purchase: the petrol pump, the household electricity meter, the gas meter, the radiator served by district heat. The useful meter sits inside the device, measuring what the device actually delivers as service.

National and international energy statistics report the first two of those readings comprehensively. The IEA's World Energy Balances database publishes, for every country and every year, both Total Energy Supply (the primary figure) and Total Final Consumption (the final figure), in tonnes of oil equivalent and in terajoules. Final consumption is then broken down by sector — industry, transport, residential, commercial, agriculture — and within each sector by fuel. The third reading, useful energy, is almost never reported in official datasets. It has to be estimated by combining final-energy statistics with end-use efficiency factors for every device in the economy, and the factors themselves are uncertain. The most ambitious global estimate, by Cullen and Allwood (2010), put global useful energy at about 11 per cent of primary on an exergy basis. More recent country-level primary-final-useful databases (Heun et al. 2024) suggest somewhat higher figures, in the range of 17 to 20 per cent, depending on the choice of efficiency factors and on whether the calculation is performed on an energy or exergy basis.

The conceptual point survives whichever method is used. The cascade is steep. Most of what is extracted is lost before it does any work. The interesting questions are where the losses occur, how much of each loss is thermodynamically necessary, and how the cascade changes when one set of technologies is substituted for another.

Where the losses occur

The cascade has five principal loss points, taken in order along the chain.

Extraction and gathering. The smallest of the losses in absolute terms, but not zero. Oil and gas operations consume between five and eight per cent of their gross production as own-use fuel and lose a further small fraction as fugitive emissions and flaring. Coal mining loses methane that escapes from the seam. The most important of these from a climate perspective is methane venting and flaring, because methane has a global warming potential roughly thirty times that of carbon dioxide over a hundred-year horizon. As pure energy loss, extraction is a few per cent.

Refining and gas processing. A modern oil refinery consumes about five to eight per cent of its crude oil input as own-use fuel: heat for the distillation columns, hydrogen for the hydrotreaters, steam for the cracking units, electricity for the pumps. The remainder leaves the refinery as a slate of products — gasoline, diesel, jet fuel, naphtha, fuel oil, LPG, bitumen — whose summed energy content is close to that of the input crude but never quite equal. Gas processing strips condensate, water, sulphur and carbon dioxide from raw natural gas; the energy cost is small in proportional terms but real in absolute. LNG liquefaction at minus 162 degrees consumes about eight to twelve per cent of the gas's energy as compressor work; regasification at the receiving terminal returns most of that as cold but loses a few more per cent.

Electricity generation. The single largest loss point in the global energy system. The fundamental cause is the Carnot limit introduced in Lesson 1.1: a heat engine running between a hot source and a cold sink can convert into work no more than the fraction (T_hot − T_cold) / T_hot, with the temperatures in kelvin. Practical thermal power stations work within reach of this limit but never at it. A subcritical coal plant converts about 33 per cent of its fuel into electricity. An ultra-supercritical coal plant reaches 45 per cent. A modern combined-cycle gas turbine, which combines a gas turbine with a steam turbine driven by its exhaust heat, reaches about 60 per cent. A pressurised-water nuclear reactor, constrained by the lower operating temperature its materials and chemistry allow, converts about 33 per cent of its fission heat into electricity. The remaining 40 to 67 per cent is dissipated as heat to atmosphere or cooling water. Hydropower, wind and solar photovoltaics do not face this loss at all: they produce electricity directly from a mechanical or quantum process and have no thermal stage.

Transmission and distribution. The electricity that leaves a generating station is shipped over a network of transformers and conductors that have resistance, and resistance dissipates a fraction of the power as heat. The global average loss between generation and consumer's meter is about eight per cent (IEA 2025), with national figures ranging from under five in countries with short, high-voltage networks to over fifteen in countries with old or overloaded distribution systems. Gas pipelines lose much less in proportional terms — typically one to two per cent of their throughput as compressor fuel.

End use. The single largest source of losses when totalled across the economy, although no individual device looms as large as the power-generation stage. An incandescent bulb dissipates 95 per cent of its electricity as heat and only 5 per cent as visible light. A petrol-engined passenger car dissipates about 75 per cent of its fuel as engine heat, exhaust heat and friction; the wheels see only the remaining 20 to 25 per cent. Even modern, efficient devices have material losses: an LED is roughly 30 to 40 per cent efficient at converting electricity to light; an electric motor 85 to 95 per cent efficient at converting electricity to shaft work; a condensing gas boiler about 90 per cent efficient at converting gas to delivered heat. The further down the cascade the device sits, and the lower the quality of the energy needed by the service it delivers, the more room exists in principle for substitution and improvement.

The cumulative effect is the figure quoted at the start of the framing. Roughly two-thirds of every joule that the world economy extracts is dissipated as waste before it does any useful work. The other third is the basis of modern civilisation.

Reading a Sankey diagram

The standard graphical representation of the cascade is the Sankey diagram, named after the Irish engineer Matthew Henry Phineas Riall Sankey, who used the form in 1898 to depict the steam-engine losses of a Royal Navy gunboat. The width of each band on a Sankey is proportional to the magnitude of the flow it represents. Bands divide at transformation points: the band of crude oil entering a refinery divides into bands of gasoline, diesel, jet fuel, fuel oil and own-use. Bands meet at consumption points: the band of electricity leaving a power station joins bands of gas, oil and coal at the boundary of the residential sector, and the combined incoming flow splits again into bands of heat, light, mechanical work and other services.

Two features of a competently drawn Sankey carry most of the information. The first is the relative width of the rejected-heat band: the band that leaves the system without doing any work. On a typical national Sankey, the rejected-heat band is the widest single outflow on the diagram, exceeding every individual sector of useful consumption. The second is the matching of energy quality to end use. A Sankey that traces useful energy reveals that high-quality electricity (capable of doing work) is sometimes used to deliver low-grade heat (which a lower-quality energy source could have provided), and that high-quality fuel (capable of high-temperature combustion) is sometimes used in low-temperature applications where its quality is wasted. The mismatch is one of the largest sources of avoidable loss in the modern economy.

The most widely cited national Sankey is the one published every year by the Lawrence Livermore National Laboratory for the United States. It is reproduced in textbooks, in the energy press and in IEA presentations, partly because it is clear and partly because the United States is the only country whose statistical agency publishes the underlying data in a Sankey-ready form. The IEA's World Energy Balance can be redrawn as a Sankey, and is in the annexes of the World Energy Outlook, but the country-by-country balances are normally read as tables.

The methodological dispute: how to count non-combustion sources

The cascade as described so far works straightforwardly for fuels that are burnt. A tonne of coal has a definite calorific value, measured in laboratory combustion, and the primary-energy content of the coal is that calorific value. A barrel of crude oil has a calorific value too: about 6.1 gigajoules on average. The same is true for natural gas, biomass and the chemical energy stored in any hydrocarbon.

For energy sources that are not burnt, the situation is more delicate. What is the primary-energy content of a kilowatt-hour of electricity generated by a wind turbine? The wind turbine has no fuel. The kinetic energy of the wind is not metered. There is no calorific value to multiply. The choice of how to count this electricity is not a question of physics but of convention, and three different conventions are in use.

The physical energy content method, used by the IEA, the OECD and Eurostat, counts the electricity at its first physical step. For hydro, wind, solar photovoltaics and geothermal flash plants, the primary-energy figure equals the electrical output. For nuclear, the convention is to count the heat output of the reactor — about three times the electrical output, reflecting the roughly 33 per cent thermal efficiency of a typical pressurised-water reactor. For solar thermal and concentrated solar power, the primary figure is the heat captured by the receiver.

The partial substitution method, used until 2024 by the BP Statistical Review of World Energy (now the Energy Institute Statistical Review), converts non-combustion electricity into a primary-energy equivalent by dividing by the average efficiency of the fossil generation it is presumed to displace, approximately 38 per cent. This inflates the apparent primary-energy figure of renewables and nuclear by a factor of about 2.6 relative to the IEA method. The rationale is that, in a hypothetical world without renewables and nuclear, the same electricity would have been generated by burning fossil fuels at 38 per cent efficiency, so the renewable contribution is in some sense "worth" the larger figure.

The direct equivalent method, used by the IPCC and by some academic analysts, counts every kilowatt-hour of electricity at its electrical output, regardless of generating technology. Under this method nuclear is counted at electrical output, not at heat output. The method has the cleanest physical interpretation: it asks, at the boundary of the electricity system, how much electricity was generated, and treats the answer as the primary figure for non-thermal sources. The cost is that it understates the heat losses in thermal generation, since the heat losses of a coal or nuclear plant disappear from the primary-energy total.

The numerical consequences of the choice are large. A wind sector producing 1,000 terawatt-hours per year contributes 3.6 exajoules under the physical energy content method (and the same under the IPCC direct equivalent), but 9.5 exajoules under the partial substitution method. The same physical quantity of wind generation appears at almost three times the share of global primary energy depending on which page of which annual report it is read from. The Energy Institute changed its methodology with the 2024 edition of its Statistical Review, replacing the partial substitution approach with a new "Total Energy Supply" measure aligned more closely with the IEA. The global primary-energy figure for 2023 changed by about 25 exajoules between the old and new methods, without any underlying physical quantity changing. The change is bookkeeping. It matters for trend statements ("renewables now account for X per cent of primary energy") but not for the physical reality of how much energy each technology produces.

There is no universally right choice among the three methods. Each captures a defensible question. The physical energy content method asks what was the energy at the first physical step of the chain; the partial substitution method asks what fossil energy was avoided; the direct equivalent method asks what energy emerged at the boundary of the electricity system. The discipline for the reader of energy statistics is to know which method a given publication uses, and to convert between them when comparing numbers from different sources.

Why this matters: the primary-energy story of electrification

The largest single primary-energy reduction available to the world over the next two decades is the substitution of direct-combustion end uses with electrified ones, where the electrification chain is more efficient than the combustion chain. The substitution does not change the final-energy service the consumer wants — the room is still at twenty degrees, the car still moves a tonne of metal at the same speed — but it changes the cascade behind that service. The largest gains arise in two places: heating, where heat pumps replace boilers, and ground transport, where electric drivetrains replace internal-combustion engines.

Both cases share a structural feature. The electrified chain has more transformation stages than the combustion chain (the fuel must be burnt to make electricity, the electricity must be transmitted, the electricity must drive the end-use device) but the final stage is many times more efficient than the combustion alternative (the electric motor at 90 per cent versus the petrol engine at 25 per cent; the heat pump at 300 per cent versus the boiler at 90 per cent). The total primary energy required for the same useful service falls, sometimes by a factor of two or three. As the electricity is decarbonised, the primary fuel input shifts from fossil to renewable, and the cascade compresses further: a wind-powered heat pump has no upstream fuel chain at all (under the physical energy content method).

This is the primary-energy logic that underlies the major net-zero scenarios. The IEA's Net Zero Emissions by 2050 scenario shows global final energy demand falling by roughly 10 per cent between 2022 and 2050, even as the world's population grows and its economy more than doubles. Primary energy in the same scenario falls by about 25 per cent over the same period (IEA 2024). The difference, 15 percentage points, is the compression of the cascade through electrification. It is invisible at the final-energy level and obvious at the primary-energy level. The reader who understands the distinction can read the IEA's tables fluently. The reader who does not, mistakes a cascade compression for a fall in service consumption.

Worked example: a UK household swaps gas for a heat pump

A typical British three- or four-bedroom dwelling consumes about 11 to 12 megawatt-hours of useful heat per year, after accounting for losses from the radiators into the room (DESNZ 2024). Take 12 MWh as the figure for this example. The household is considering replacing its existing condensing gas boiler with an air-source heat pump. The question we work through is how the primary-energy demand of the household for heating changes under each path.

Path 1. The condensing gas boiler

A modern condensing gas boiler delivers heat to the central-heating water at about 90 per cent of the calorific value of the gas it burns. To produce 12 MWh of useful heat, the boiler therefore consumes 12 / 0.90 = 13.3 MWh of gas as measured at the household meter. This is the final-energy figure. To trace it back to the wellhead, we add the upstream losses: gas processing at the field (one to two per cent), pipeline compression to the high-pressure transmission grid (one to two per cent), and a small further loss in the distribution network. Combining these gives a primary-to-final factor of about 0.95, so the primary-energy demand of the boiler is 13.3 / 0.95 = 14.0 MWh of gas at the wellhead. Per unit of useful heat delivered, the primary energy requirement is 1.17 — almost a fifth more primary energy is consumed at the well than ends up in the room.

Path 2. The air-source heat pump

The heat pump moves heat from the outdoor air into the central-heating water. Its seasonal coefficient of performance varies with the design and the climate. The Department for Energy Security and Net Zero's Electrification of Heat Demonstration Project, which monitored 427 UK installations, found a mean SCOP of 2.80 (DESNZ 2024). Recent independent monitoring of best-practice installations on the Heat Pump Monitor platform reports averages above 3.8 (Viessmann UK 2026). Take 2.80 as the conservative figure. The pump therefore consumes 12 / 2.80 = 4.29 MWh of electricity at the household meter. This is the final-energy figure for Path 2.

Tracing the electricity back to the primary stage requires three further steps. First, the grid transmission and distribution loss, about 8 per cent in the UK, raises the gross generation requirement to 4.29 / 0.92 = 4.66 MWh. Second, this 4.66 MWh is then attributed to the technologies in the UK generating mix. Take a representative recent year: approximately 33 per cent from gas-fired CCGTs at 50 per cent efficiency, 22 per cent from wind, 14 per cent from nuclear, 13 per cent from biomass, and the remainder from solar, hydro and imports. Third, the mix is converted into primary energy by whichever methodology one chooses.

Under the IEA's physical energy content method, the gas-fired component (33 per cent of 4.66 = 1.54 MWh of electricity) corresponds to 1.54 / 0.50 = 3.08 MWh of primary gas. The wind, solar and hydro components (31 per cent = 1.44 MWh of electricity) count at their electrical output: 1.44 MWh of primary energy. Nuclear (14 per cent = 0.65 MWh of electricity) counts at heat output: 0.65 / 0.33 = 1.97 MWh of primary energy. Biomass (13 per cent = 0.61 MWh of electricity, at about 35 per cent generating efficiency) corresponds to 1.74 MWh of primary biomass energy. Imports (9 per cent) are counted at their electrical output. Summing: roughly 8.7 MWh of primary energy delivers the 12 MWh of useful heat. Per unit of useful heat, the primary energy requirement is 0.72.

Under the partial substitution method, in which the non-combustion electricity components are inflated by dividing by an assumed fossil-displacement efficiency of 0.38, the total primary figure rises to roughly 11.0 MWh — an intensity of 0.92, and a saving versus the boiler of about 22 per cent. Under the IPCC direct equivalent method, in which non-combustion sources (including nuclear) are counted at electrical output, the total falls to roughly 7.3 MWh — an intensity of 0.61, and a saving of about 48 per cent. The heat pump uses less primary energy than the gas boiler under all three conventions. The saving ranges from about a fifth to about a half, depending on how the non-combustion components are counted.

What the example reveals

Two things. First, under the methodology most widely used in energy policy (the IEA's physical energy content method), the heat pump reduces the primary-energy demand of the household for heating by close to two-fifths. Second, the size of the gain is sensitive to two parameters that change over time: the SCOP of the heat pump and the carbon and methodology of the grid mix. A heat pump installed with a SCOP of 4.0 instead of 2.8 reduces final energy by a further third. A grid that decarbonises further reduces primary energy on a sliding scale. The boiler has no such trajectory: its efficiency is capped near 90 per cent and the upstream gas chain is essentially fixed.

The example also illustrates a feature that recurs throughout the energy literature. Final-energy statistics, which are the ones most commonly published in monthly bulletins and national accounts, would show the household's energy consumption falling from 13.3 MWh to 4.29 MWh — a 68 per cent reduction. The primary-energy reduction, under the physical energy content method, is from 14.0 MWh to 8.7 MWh — only 38 per cent. The two numbers measure different things and answer different questions. The reader of any energy publication should know which is being reported.

Retrieval and generative practice

Answer the following without referring to the text. Suggested answers are at the end of the document.

1. Define primary, final and useful energy. Give one residential-sector example of each.
2. Which single transformation contributes the largest absolute loss in the global energy cascade? Why is the loss thermodynamically necessary, and what physical principle sets its lower bound?
3. A wind farm and a nuclear plant produce the same 10 TWh of electricity in a year. How does their respective contribution to primary energy differ under (a) the IEA physical energy content method and (b) the IPCC direct equivalent method?
4. A condensing gas boiler is 90 per cent efficient. A heat pump has a SCOP of 3. The heat pump therefore appears to be more than 100 per cent efficient. Is that a violation of the first law of thermodynamics? Explain in two or three sentences.

Generative prompt. Choose one major industrial process you know something about — steelmaking, ammonia synthesis, cement, aluminium smelting, glass-making, ethylene cracking, pulp and paper. Trace its energy cascade from primary fuel (or electricity) to useful service. Identify the largest single loss point. Then sketch how an electrified or hydrogen-fed version of the same process would change the cascade for the same useful output. You will need this skeleton in Part VI of the course.

Further reading

• Vaclav Smil, Energy and Civilization: A History (Cambridge, MA: MIT Press, 2017). Chapter 4 traces the efficiency of human energy conversions across the millennia and sets the long-term frame for the cascade discussion in this lesson.
• Jonathan M. Cullen and Julian M. Allwood, "Theoretical efficiency limits for energy conversion devices," Energy 35(5): 2059–2069 (2010). The canonical paper estimating overall global energy conversion efficiency at 11 per cent on an exergy basis. The companion paper, "Reducing Energy Demand: What Are the Practical Limits?" (2011), brings the figure closer to a practical efficiency target.
• IEA, World Energy Balances (Paris: IEA, annual). The data source against which any claim about primary, final and transformation flows is checked. The accompanying database documentation explains the physical energy content method in full.
• Matthew Kuperus Heun and colleagues, country-level primary-final-useful (CL-PFU) database (IOP Energy, 2024). The most recent open-source effort to extend the IEA balances to a third tier of useful energy across 150 economies.
• Lawrence Livermore National Laboratory, Estimated U.S. Energy Consumption (annual Sankey, flowcharts.llnl.gov). The most widely cited single-page visualisation of a national energy cascade, redrawn each year as the data arrive.
• Eurostat, Energy Balance Guide (Luxembourg: Eurostat, current edition). The methodology document for European energy statistics, including the treatment of heat pumps as a primary-energy source under the EU Renewable Energy Directive.

References

• Cullen, Jonathan M. and Julian M. Allwood. 2010. "Theoretical Efficiency Limits for Energy Conversion Devices." Energy 35(5): 2059–2069.
• Cullen, Jonathan M., Julian M. Allwood and Edward H. Borgstein. 2011. "Reducing Energy Demand: What Are the Practical Limits?" Environmental Science and Technology 45(4): 1711–1718.
• DESNZ (Department for Energy Security and Net Zero). 2024. Electrification of Heat Demonstration Project: Heat Pump Performance Findings. London: DESNZ.
• Eurostat. 2024. Energy Balance Guide: Methodology Manual. Luxembourg: Publications Office of the European Union.
• Heun, Matthew Kuperus, et al. 2024. "A Country-Level Primary-Final-Useful (CL-PFU) Energy and Exergy Database: Overview of Its Construction and 1971–2020 World-Level Efficiency Results." Environmental Research: Energy 1(2): 025006.
• IEA. 2024. World Energy Outlook 2024. Paris: International Energy Agency.
• IEA. 2025. World Energy Balances Database, April 2025 Edition. Paris: International Energy Agency.
• IPCC. 2022. Climate Change 2022: Mitigation of Climate Change. Working Group III Contribution to the Sixth Assessment Report. Annex II: Methodology. Cambridge: Cambridge University Press.
• Smil, Vaclav. 2017. Energy and Civilization: A History. Cambridge, MA: MIT Press.
• Viessmann UK. 2026. Top of the SCOPs Competition Results, January 2026. Telford: Viessmann Climate Solutions UK.

Suggested answers to retrieval questions

1. Primary: the energy at the wellhead, mine mouth or first physical step of capture (e.g. the natural gas as it leaves the gathering line). Final: the energy at the consumer's point of purchase (e.g. the gas at the household meter). Useful: the energy that actually delivers the service (e.g. the heat in the room at twenty degrees). The losses between primary and final are transformation and distribution losses; the losses between final and useful are end-use device losses.
2. The largest single absolute loss in the global cascade is the rejected heat from thermal electricity generation. The lower bound is set by the Carnot limit: a heat engine running between a hot source and a cold sink cannot convert into work a fraction greater than (T_hot − T_cold) / T_hot, with the temperatures in kelvin. For typical power-plant operating temperatures, the limit is in the range of 60 to 70 per cent, and real plants reach 33 to 60 per cent depending on technology.
3. Under the IEA physical energy content method, the wind farm contributes 10 TWh of primary energy (counted at electrical output), while the nuclear plant contributes about 30 TWh of primary energy (counted at heat output, assuming 33 per cent thermal efficiency). Under the IPCC direct equivalent method, both contribute 10 TWh of primary energy. The choice changes nuclear's apparent share by a factor of three.
4. No, it is not a violation of the first law. A heat pump does not create heat; it moves heat from a cold reservoir (the outdoor air) to a warm one (the indoor water). The electricity input is the work needed to drive the compressor, and the heat output is the sum of that work and the heat moved. The number called efficiency in the boiler case is renamed coefficient of performance in the heat pump case, precisely because the device is not converting one form of energy into another but transferring an existing form against a temperature gradient. The thermodynamic limit on the COP is the Carnot COP, T_hot / (T_hot − T_cold), which for typical heating conditions is in the range 7 to 12.

Key insight. The global energy cascade loses roughly two-thirds of primary energy before any service is delivered. Most of the loss is the Carnot tax on thermal generation; some is end-use waste that can be compressed. Electrification compresses the cascade by replacing direct combustion with chains that end in motors and heat pumps — and the compression is invisible at the consumer's bill, but enormous at the primary level.