Calvin Digital Commons Calvin Digital Commons A physical supply-use table framework for energy analysis on the A physical supply-use table framework for energy analysis on the energy conversion chain (2) energy conversion chain (2)

(cid:129) Today, energy analysis addresses topics all along the energy conversion chain. (cid:129) The ﬁ eld of energy analysis would bene ﬁ t from a common analysis framework. (cid:129) In response, a physical supply-use table framework is presented. (cid:129) Real-world examples demonstrate the range of applicability of the framework. (cid:129) Bene ﬁ ts include data structure uniformity and methodological consistency. energy analysis framework through to energy services and application of physical supply-use tables to both energy and exergy analysis. The methodological advances enable the fol- lowing ﬁ rst-time contributions to the literature: (1) performing energy and exergy analyses on an energy conversion chain using physical supply-use table matrices comprised of disaggregated products in physical units when the last stage is any of ﬁ nal energy, useful energy, or energy services; (2) performing structural path analysis on an energy conversion chain; and (3) developing and utilizing a matrix approach to inhomogeneous units. The framework spans the entire energy conversion chain and is suitable for many sub- ﬁ elds of energy analysis, including net energy analysis, societal energy analysis, human needs and well-being, and structural path analysis, all of which are explored in this paper.


A recent history of energy analysis: expansion through revolution and evolution
The modern field of energy analysis is rooted in energy accounting, which emerged in the 1950s from Leontief's input-output (IO) methods [1] and Barnett's energy balance tables [2]. With studies of the U.S. economy by Schurr and Netschert [3] and Morrison and Readling [4], the field remained closely aligned to energy accounting methods through the 1960s (see Berndt [5] for an overview of the early history of energy analysis).
The 1970s oil crises caused a revolution in the field: its focus expanded from merely accounting for production and sale of primary and final energy carriers to many other aspects of energy in society and the economy. Reistad [6, p. 429] said, "In this period of concern for our energy resources and the environment, it is imperative to consider the manner in which our energy resources are consumed." The study of technical energy efficiency became prominent, illustrated by a 1973 conference presentation by Hatsopoulos [7] and the 1975 American Institute of Physics reports on second-law efficiency [8], automobiles [9], and industrial processes [10]. At an economy-wide level, studies of net energy [11], useful energy [6], and energy services [12] were conducted. Furthermore, new studies of interactions between energy and the economy appeared, covering topics such as the energy impact of consumption decisions [13], the entropic nature of economic processes [14], energy and "potential" GDP [15], and questioning the value of the concept of energy intensity [16]. In 1978, Roberts [17, p. 200] noted that the term "energy analysis" was now preferred to "energy accounting," the name change signifying that the revolution was underway.
Following the 1970s, evolution and specialization led to the creation of several energy analysis sub-fields. Net energy analysis evolved from the study of single fossil fuel sources (e.g., oil, coal, gas) [18] to renewables [19,20] and to the consideration of economy-wide issues such as the minimum energy return on (energy) invested (EROI) required for a functioning society [21], the implications of declining EROI [22], energy expenditure and economic growth [23], and input-output methods to determine national-level EROI [24]. World-wide issues also received attention, including detailed studies of oil and gas production [25], correlations between EROI and oil prices [26], and social implications [27]. The empirical study of energy efficiency and rebound [28] specialized into evaluation of direct [29], indirect [30], and sectoral and economy-wide rebound for energy in the UK [31] and for energy intensity as opposed to energy efficiency [32]. A new sub-field, societal exergy analysis, emerged. Building on the earlier work of Reistad [6], Wall [33], and Kümmel et al. [34], Ayres and co-authors made significant advances on the role of physical resources flows in endogenous growth models [35], the role of physical work in economic growth [36], efficiencies of specific energy and economic sectors [37], and the impact of natural resource consumption and technological change on economic growth [38]. Recent work has standardized allocation of final energy to useful exergy categories [39], improved estimates of exergetic efficiencies [40], and explored theoretical efficiency limits of end-use devices [41]. Another new sub-field (energy decomposition analysis) expanded greatly largely due to the efforts of Ang who developed log-mean divisia index (LMDI) methods [42], compared them against other decomposition approaches [43], applied them to monitoring energy intensity [44], and provided a practical guide for implementation [45]. Further specialization of energy analysis occurred as researchers considered the role of energy in economic growth in terms of energy constraints [46], primary energy sources [47], empirical evidence from many countries [48], and causality directions and substitution possibilities via time-series analysis [49]. The benefits [50] and limitations [51] of the metaphor "the economy is society's metabolism" were explored by several authors, and the magnitude of the industrial energetic and material metabolism has been estimated for the EU [52] and the world [53]. Others have explored the role that energy plays in satisfying human needs [54] across various nations [55], have studied how energy enables well-being [56], and have developed a sufficiency framework for decoupling human well-being from energy consumption [57]. Lastly, analysis of long-run energy transitions has received much recent attention, with researchers studying countries (the UK [58], the U.S. [59], and Sweden [60]), causes (energy cost share [61] and policy [62]), and policy needs for a transition away from oil [26] and toward a sustainable future [63].

The energy conversion chain (ECC)
A notable feature of this history is an expanding analysis boundary. In the 1960s, energy accountants were focused on primary energy sources and final energy carriers. Today, energy analysts also consider the consumption of useful energy produced by consumer-owned devices [39] to generate energy services [64,65] that satisfy human needs and enable human well-being and development [57]. The expanded boundary covers the entire energy conversion chain (ECC), a term (to our knowledge) introduced by Crowe [66, p. 3] to describe energy conversion processes in diesel generators and fuel cells. We find the phrase to be apt for all types of energy analysis, so we define it more broadly to be a set of energy carriers, energy transformation devices, and energy services within spatial and temporal boundaries of interest. In this paper, we focus on economy-spanning ECCs comprised of primary, final, and useful energy carriers as well as the energy services they enable. Fig. 1 shows an example ECC with two pathways: Natural gas (NG) to Residential end use and Crude to Transport end use. Activities in the Residential and Transport final demand sectors, made possible by the ECC, partially satisfy Human needs, some of which are shown. For simplicity, Fig. 1 ignores interactions between the two pathways (e.g., electricity to operate an oil refinery), self-consumption (e.g., of electricity by power plants), and distribution (of electricity and fuels by the grid and transport systems, respectively). Real-world examples in Section 3 incorporate these complexities.
The expanding analysis boundary was accompanied by an increase in the number of questions addressed by energy analysis. Note that emissions concerns trace upstream to Primary energy at the far left of Fig. 1, but satisfaction of human needs in the Residential and Transport sectors is downstream at the far right. And there is a growing realization that focusing on a single part of the ECC yields an incomplete analysis. Mayumi and Giampietro [67, p. 65] say "[w]e should not study in isolation either patterns of production or patterns of consumption of energy carriers. Any metabolic system works by integrating the two sides (production and consumption of energy carriers) in an organic whole capable of expressing a desirable set of functions".
Indeed, climate-altering emissions and the role of energy in human development are just two aspects of contemporary energy analysis. Four questions that represent important topics in energy analysis subfields today are: • Net energy analysis: What are the energy return ratios (ERRs) for energy production devices?
• Societal energy analysis: Where are the key energy saving opportunities in an economy?
• Human needs and well-being: How much primary energy is required to provide energy services?
• Structural path analysis: What are the key supply-chain paths through the ECC for delivering energy services?
These questions span the entire ECC from primary energy to energy services, they encompass issues relevant to many energy analysis subfields, and they require significant empirical data and interdisciplinary knowledge to address. We tackle these questions using a real-world ECC in Section 3.

The benefits and building blocks of an energy analysis framework for the ECC
In our opinion, efforts to address today's energy analysis questions would benefit from a data structure and associated analytical methods -an energy analysis framework-that (a) spans the entire ECC and (b) is suitable for many energy analysis sub-fields. Such a framework could organize and streamline questions to be asked, data to be gathered, analyses to be performed, and results to be reported.
We believe that an energy analysis framework with these benefits is possible, taking a physical supply-use table (PSUT) approach. In fact, several research communities have been developing techniques that provide the building blocks for such a framework. We identify five important developments below.
First, IO researchers have developed methods that employ supplyuse tables to overcome problems of co-production (one industry makes more than one product) [68], to deal with wastes [69], to perform decomposition analysis [70], to analyze environmental impacts [71,72], and to combine decomposition and impact analysis [73]. Importantly for this study, with supply-use tables a single energy conversion device (e.g., an Oil refinery) can produce multiple outputs (e.g., Petrol and Diesel).
Second, Pauliuk and co-authors have developed SUT-based techniques for accounting physical [74] resource flows [75,76], drawing on waste accounting frameworks [77] that employed physical IO tables [78]. These physical approaches have been employed to study wood and paper flows [79], among other commodities. Others investigate international flows of embodied energy using matrix-based [80] and network [81] methods. These advances demonstrate that physical flows (including, in our case, energy carriers and energy services) can be accommodated in an SUT analysis framework.
Third, life-cycle analysis practitioners have overcome methodological issues to demonstrate material balances [82] in physically-extended economic SUT frameworks [83]. This development gives confidence that an energy analysis framework that obeys the first and second laws of thermodynamics can be developed. Others have developed matrix-based methods for determining energy return ratios [84], giving confidence that matrix-based analysis of the entire ECC will be successful.
Fourth, Rocco [85], Guevara [86], and their respective co-authors have developed advanced, mixed-units, matrix-based SUT and IO techniques. These techniques have been applied to the broader economy for life cycle assessment of electricity production in waste-toenergy technology [87], for determining the primary exergy cost of goods and services [88], for understanding the energy metabolism of the world [89], for decomposition of primary energy use [90], and for decoupling of exergy use from economic growth [91]. Their work gives confidence that techniques developed over decades of economic IO and SUT research can be applied to energy flows and energy services in an ECC.
Finally, Chong et al. [92] obtain primary-to-final "energy quantity conversion factors" via Leontief inverse of an IO table comprised of aggregated physical quantities. To our knowledge, their work is the first example of obtaining ECC efficiencies via IO techniques, albeit in support of the narrow objective of performing LMDI decomposition analysis of final energy consumption in Guangdong Province, China. However, it shows that application of IO techniques with quantities expressed in purely physical, not monetary, units is both feasible and beneficial for energy analysis.

Aim, originality, and scope of paper
The aim of this paper, then, is to build upon these recent advances to develop and demonstrate a PSUT-based energy analysis framework (the "PSUT framework" for short) that spans the entire ECC and is pertinent to many energy analysis questions. Such a PSUT framework should have two important characteristics, namely (a) applicability to the entire ECC (i.e., primary energy to energy services) and (b) applicability to both energy and exergy analysis. The representative contemporary energy analysis questions posed in Section 1.2 provide a context for demonstrating the PSUT framework. Although answers to these energy analysis questions can inform policy debates, we consider policy to be beyond the scope of this paper. Table 1 provides a summary of differences between recent work and this study.
The paper proceeds as follows: Section 2 describes the PSUT framework. Section 3 gives real-world examples and answers the questions posed in Section 1.2. A discussion and conclusions (Sections 4 and 5) follow. Detailed appendices are provided for the interested reader.

Introduction to PSUT framework
Our energy analysis framework is a physical framework (the P in PSUT), because all values are quantified in physical units (e.g., ktoe, TJ, or passenger-km), not monetary units (e.g., $ or £). The framework accommodates industries with multiple inputs and multiple outputs, because it is based on supply-use table methods (the SUT in PSUT). The PSUT framework is applicable to analyses conducted in either energy or exergy terms, although we write simply "energy" where possible to avoid the awkward phrase "energy or exergy".
The structure of the PSUT framework comprises four matrices. The first three are typical of supply-use table (SUT) formulations of IO analyses, and we refer to them as the PSUT matrices: (an industry-by-product "supply" or "make" matrix), and × Y p s (a product-by-sector "final demand" matrix). A fourth matrix is an auxiliary product-by-unit summation matrix ( identifies the physical units in which products are measured. To indicate whether industries (i), products (p), final demand sectors (s), or units (u) appear in rows or columns of matrices, we adopt the notation shown in Table 2. When a matrix is introduced (and when needed for clarity thereafter), this notation is typeset beneath matrix symbols. We mostly follow the Eurostat nomenclature for matrix symbols and the categories of products, industries, and final demand sectors [93]. Table 3 provides a mapping between Eurostat categories and energy and services concepts in the PSUT framework. See Appendix A for a comprehensive table of nomenclature.
The U V , , and Y matrices can be arranged spatially as shown in Fig. 2. Entries in the use matrix (U) give the consumption of energy carriers and services (in rows) by energy transformation devices (in columns). Entries in the make matrix (V) indicate the production of energy carriers and services (in columns) by energy transformation devices (in rows). Entries in the final demand matrix (Y) specify consumption of energy carriers and services (in rows) by final demand sectors (in columns). (Non-energy uses of energy carriers or services appear in a column of Y.) Entries in the units summation matrix (S units ) specify the physical units (in columns) by which energy carriers and services (in rows) are measured. Note that not all final demand sectors in columns of Y correspond to final demand sectors in systems of national accounts. For example, Transport is an intermediate sector in systems of national accounts but a final demand sector here.

Building and manipulating the PSUT matrices
Building and manipulating the PSUT matrices involves deciding an analytical approach, constructing and verifying the PSUT matrices, formulating the IO structure of the ECC, and estimating the effect of changes in final demand on the ECC. Each activity is described in subsections below.

Analytical approach
Before constructing the PSUT matrices introduced in Section 2.1, an analytical approach must be decided, i.e. a set of decisions must be made about analysis choices that is sufficient to allow construction of the PSUT matrices. Analysis choices include, but are not limited to, (a) the country, device, or process of interest (spatial boundary); (b) the time period over which the analysis applies (temporal boundary); (c) the method of accounting for primary energy corresponding to renewable energy production (partial substitution method, physical content method, or resource content method [94]); (d) whether to include non-energy uses of energy carriers in PSUT matrices; (e) whether entries in PSUT matrices represent energy or exergy quantities; and (f) whether the last stage of analysis will be final energy, useful energy, or energy services.

PSUT matrix construction
The PSUT matrices are populated with energy and energy services data gathered from sources including, but not limited to, (a) the International Energy Agency (IEA) [95] (for primary and final energy data), (b) estimates of final-to-useful transformation device efficiencies [96,97,40,98] (for calculating useful energy), (c) exergy/energy ratios (ϕ) [99] (for calculating exergy content from energy values), and (d) national statistical datasets [100, Table 2] (for energy services data). Note that all primary-to-final, final-to-useful, and useful-to-services transformation devices are included as "industries" in the U V , , and Y matrices. All entries in the PSUT matrices should be non-negative numbers, and all energy entries must be in the same units, typically TJ/ year or ktoe/year for a large economy.

Thermodynamic verification
Regardless of analytical approach, PSUT matrices populated with energy carriers and energy services are verified by the first law of thermodynamics (see Appendix B for discussion of exergy and the second law of thermodynamics). Two fundamental input-output calculations are needed for first law verification: value added and aggregation. A value added matrix ( Aggregations are row, column, or matrix sums, and several are given in Table 4. With rare exception, the column sums of the value added matrix (i W T ) are positive in financial SUT analyses, because finished products are more valuable (in a monetary sense) than the raw materials from which they are made. (And industries with negative value added don't survive for long!). However, in the PSUT framework, column sums of the value added matrix (i W T ) are often negative, because energy transformation devices produce less useable energy than they consume i×i Industries in both rows and columns (e.g.,ĝ ) p×i Products in rows and industries in columns (e.g., U) i×p Industries in rows and products in columns (e.g., V) p×s Products in rows and final demand sectors in columns (e.g., Y) p×u Products in rows and units of products in columns (e.g., S units ) Table 3 Eurostat categories.
due to inefficiencies and wastes. For example, coal-fired power plants produce about 1/3 as much electrical energy as they consume in coal energy, the difference being waste heat. Indeed, i W T will contain positive entries for extractive industries (free gifts from nature) and negative entries for ECC transformation devices (due to wastes and waste heat).
Unless the PSUT matrices conform to the first law of thermodynamics, all further calculations will be wrong. With the aggregations of Table 4 and the value added matrix of Eq. (1) in hand, energy and services balances should be verified across products and across industries. To evaluate the first law across products, the following equation applies: Across industries, inputs must equal the sum of valuable products (outputs) and wastes. Thus, the first law can be expressed as Assuming homogeneous units in the U V , , and W matrices, total output by industry is g, waste by industry is −W i T (wastes are negative value added), and input by industry is U i T . Substituting into Eq. (3) yields For ECCs with inhomogeneous units in the U V , , and W matrices, we substitute V for outputs, −W T for wastes, and U T for inputs in Eq. (3) to obtain Note that Eqs. (4) and (5) are helpful identities for checking calculations. See Appendix D for a short proof of Eq. (4). See Appendix E for details of the shift from Eq. (4) to Eq. (5).

Input-output structure
After construction (Section 2.2.2) and verification (Section 2.2.3), the complete IO structure of the ECC can be formulated. The IO structure of the ECC is represented by the set of matrices shown in Table 5.
We employ Eurostat Model B (the industry technology assumption) wherein each industry has its own specific way of production, regardless of its product mix [93, p. 349]. This model is appropriate for analyzing ECCs, because each energy transformation device produces its products in its own way. For example, coal-fired and gas-fired power plants must be able to produce electricity, each with its own mix of energy inputs. Other Eurostat models, which employ different assumptions (in particular, Model A, which assumes that "each product is produced in its own specific way, irrespective of the industry where it is produced" [93, p. 347]), are inappropriate for the PSUT framework.  The calculations in Table 5 can be verified by Note that the description of IO structure in Table 5 requires that coproducts of any industry exhibit unit homogeneity. Specifically, rows of V must contain exactly one nonzero element. If an industry has coproducts with inhomogeneous units (e.g., Airlines make both Passenger transport [passenger-km/yr] and Freight transport [tonne-km/yr]), the industry should be split and inputs allocated as appropriate for each industry. (E.g., "Airlines" becomes "Passenger airlines" and "Freight airlines" with inputs to "Airlines" allocated between "Passenger airlines" and "Freight airlines.")

Effect of changes to final demand
After the IO structure of an ECC has been characterized by the matrices of Table 5, an important question may be answered with respect to final demand: "What would be the effect on the ECC of a change to final demand?" Calculations proceed as shown in Table 6 to perform an "upstream swim" from the adjusted final demand matrix ( ′ Y ) to resource extraction, thereby creating a second set of PSUT matrices (including ′ U and ′ V ) that describe an adjusted ECC associated with ′ Y . After the calculations in Table 6 are accomplished, the adjusted PSUT matrices ( ′ ′ U V , , and ′ Y ) can be (a) analyzed using Eq. (1) and Tables 4 and 5 and (b) verified using Eqs. (2) and (4)-(7).

Results: Demonstrating the PSUT framework for real applications
With the PSUT framework now established (Section 2), we provide results for one real-world example for each of the four contemporary energy questions in Section 1.2, thereby illustrating application across a range of energy analysis sub-fields, including net energy analysis (Section 3.1), societal energy analysis (Section 3.2), human needs and well-being (Section 3.3), and structural path analysis (Section 3.4). The real-world examples are at the economy-wide level, although the PSUT framework could be applied at any level: device, firm, sector, economywide, or global.
A real-world ECC (based on the two-path ECC of Fig. 1) illustrates the four numerical examples. The ECC is revealed sequentially as needed, the last stage extending from final energy (Section 3.1) to useful energy (Section 3.2) to energy services (Sections 3.3 and 3.4). All energy values are in ktoe/year, while energy services are expressed in differing physical units, e.g. passenger-km/year. All ECCs are constructed with energy quantification for energy carriers but could just as well have been constructed with exergy quantification by multiplying each energy flow by the appropriate exergy-to-energy ratio (ϕ, see Serrenho [99, Table 2]). See Appendix B for an ECC constructed from exergy flows. Data and calculations for all ECCs can be found in the data repository for this paper [101].
The real-world ECC is based on a portion of the UK's ECC in 2000, and energy and services data have been rounded to 1-2 significant figures. Thus, numerical results should be interpreted with caution. Data from any combination of country and year would suffice for this paper, because the real-world ECC is used for demonstration purposes only. Primary and final energy data come from IEA energy statistics [95]. Brockway et al. [40] provide useful energy. Energy services data have been obtained from several sources. Passenger and Freight transport data are from the UK Department for Transport, Tables TSGB0702 and TSGB0401, respectively [102]. Illumination data are from Fouquet and Pearson [103]. We estimate residential Space heating service for 25 million homes, each with representative 100 m 2 floor space, 3 m ceiling height, and average 10 K temperature difference between heated space and ambient.

Net energy analysis: What are the energy return ratios (ERRs) for energy production devices?
Within the sub-field of net energy analysis, an important question is What are the energy return ratios (ERRs) for energy production devices? A common ERR is energy return on (energy) invested (EROI), a metric first explored by Hall [104], and utilized extensively in subsequent years by Murphy and Hall [105,106], Heun and de Wit [26], Lambert et al. [107], Brand-Correa et al. [24], and many others.
Significance: Large ERRs indicate an effective energy-producing industry that provides a large rate of energy to society for small rate of energy investment.
We adopt the nomenclature of Brandt et al. [108] in which GER γ and NER γ indicate the gross and net energy return ratios, respectively, for an energy production device. The subscript γ denotes an ERR analysis boundary that accounts for multiple interacting energy pathways (e.g., Oil fields that consume Electricity). Inspired by Brandt [109], we include the net-to-gross energy ratio (r γ ) as well. Larger values of all ERRs indicate an energy system that is more effective at providing energy to society with less energy consumed (see Appendix F for derivations of relationships among the three ERRs).
In the context of the PSUT framework, all of GER NER , γ γ , and r γ become industry column vectors (ger ner , γ γ , and r γ ) given by Eqs.
To demonstrate, we calculate ERRs for each device of Fig. 3, the first version of our real-world ECC, wherein final energy is consumed by both (a) intermediate industries (Gas wells, Oil fields, Natural gas and Crude distribution, Power plants, and Oil refineries) and (b) final Table 6 Estimating the effect of changes to final demand ("upstream swim").

Equation
Note Table 5 Calculations for IO structure.  Fig. 6 shows ERR vectors for the ECC of Fig. 3. ERRs are most relevant for production stages of the ECC (Gas wells and Oil fields in this example), although (8)- (10) provide ERRs for all industries in the ECC. In Fig. 6, ERRs for Resources are ∞, because the energy to extract Resources is accounted in Gas wells and Oil fields. The ERRs for Elect grid are ∞, because there is no energy apart from Elect supplied to the Elect grid in Fig. 3.
Benefit of the PSUT framework: This real-world example shows that organizing ECC data in the PSUT framework allows computation of any ERR for all ECC devices with straightforward matrix mathematics.

Societal energy analysis:
Where are the key energy saving opportunities in an economy?
Within the sub-field of societal energy analysis, an important question is What are the device and sector energy efficiencies along an ECC?
Significance: Answers to this question identify key energy saving opportunities in an economy, which is important because "[t]he efficient provision of energy services not only reduces the required amounts of primary energy but in general also reduces adverse environmental impacts" [109, p. 421]. Fig. 7 extends the last stage of analysis in our real-world ECC from final energy to useful energy such that final demand includes Lowtemperature heat (LTH), Light, and Mechanical drive (MD). Some intermediate industries now also consume useful energy (e.g., the distribution industries consume MD-Truck engines, whereas in Fig. 3 they consumed Diesel, a final energy carrier). And for simplicity, self-consumption flows are internalized (e.g., self-consumption of 5000 ktoe of Diesel by Oil refineries in Fig. 3 is now internal to Oil refineries, thereby providing net Diesel output of 15,500 ktoe in Fig. 7). Given the ECC   shown in Fig. 7, PSUT matrices can be constructed as shown in Fig. 8.
A vector of ECC industry efficiencies (η E γ , ) can be calculated by Fig. 9 shows device efficiencies (η E γ , ) for the ECC shown in Figs. 7 and 8. Again, these are energy efficiency values for the γ system boundary, because they account for industry consumption of energy from other branches of the ECC (see Brandt et al. [108]). The η E γ , vector in Fig. 9 shows that Power plants, Car and Truck engines, and Light fixtures have much lower efficiencies than other devices.
Beyond the last energy stage, all energy transformations are accomplished within final demand sectors (in this ECC, Residential and Transport). The efficiency of a final demand sector can be evaluated by comparing two final demand matrices (Y). For example, Fig. 4 gives final demand by sector for final energy (Y f ) and Fig. 8 gives final demand by sector for useful energy (Y u ). A vector of efficiencies by which final demand sectors convert final energy to useful energy (η E fu , ) can be calculated by    3.3. Human needs and well-being: How much primary energy is required to provide energy services?
In general, purchasers of final energy are not interested in energy, per se. Rather, they are interested in the services that useful energy (when combined with infrastructure) provides. Indeed, energy services are desired because they contribute to human well-being by satisfying human needs such as subsistence, protection, participation, leisure, and freedom (see Fig. 1).
But final demand (whether expressed as final energy, useful energy, or energy services) contains "embodied" primary energy: the sum of all primary energy consumed and wasted throughout the ECC in the process of satisfying that final demand. And the ratio of final demand level to embodied primary energy is the consumption-based energy efficiency of meeting that final demand. When the ECC is extended through to energy services, this efficiency is important, because (a) primary-toservices efficiency is a factor in determining the primary energy requirements of providing energy services and (b) primary energy consumption is a proxy for environmental degradation and resource depletion. So an important question in sub-fields of human well-being and development is What is the consumption-based primary-to-services efficiency of providing energy services?
Significance: If we want to provide human well-being with minimal environmental impact, consumption-based primary-to-services energy efficiency is an important metric to monitor.
To illustrate the utility of the PSUT framework to comprehensively address these issues, we extend the ECC of Fig. 7 through to services, including final demand for Space heating, Illumination, Passenger transport, and Freight transport. Several intermediate industries now consume energy services rather than useful energy as they did in Fig. 7 (e.g., distribution industries consume Freight transport instead of MD-Truck engines). (Note that the ECC of Fig. 11 is reproduced in To assess the efficiency of providing an energy service, the embodied primary energy of that energy service is needed. The embodied primary energy of a final demand service is similar to a CO 2 [111] or material [112] "footprint." Just as a material footprint is the quantity of material consumed by all industries in the production chain to make a good (e.g., automobiles), so also embodied primary energy is the primary energy consumed by all industries in the ECC to provide an energy service (e.g., Passenger transport). And just as footprinting analysis is conducted with environmentally-extended input-output (EEIO) techniques developed for supply chains quantified in monetary units [113,114], calculation of embodied primary energy within the PSUT framework applies EEIO techniques to ECCs quantified in physical units.
In EEIO analysis, a diagonal matrix (ê ) is formed from a per-unitoutput vector of industry ancillary products ( × e i 1 ) and pre-multiplied into G to obtain a "footprint" matrix ( Extending EEIO analysis from supply chains in monetary units to ECCs in energy and energy services units in the context of the PSUT framework, we see that the choice of e determines the embodied product (energy carrier or service) obtained from Eq. (13). The starting point for forming any number of e vectors is the value added matrix (W), because its entries give the production (positive values) and consumption (negative values) of energy carriers and services by industry within the ECC. Matrix Thus, any product row P of E (e P T ) can serve as an appropriate e vector for Eq. (13): The matrix Q P contains positive and/or negative entries. Positive entries in Q P give the "footprint" of P embodied in the product of the j th column of Q P produced by the industry of the i th row of Q P . Negative entries in Q P show the consumption of P embodied in the product of the j th column of Q P by the industry of the i th row of Q P . Fig. 13 shows Q Crude and Q NG matrices for the ECC of Fig. 11. Using Q Crude as an example, we see that the embodied Crude oil in Passenger transport is 31,998 ktoe/year. Because the entry is in the top row of Q Crude , we know that the embodied Crude was produced by the Resources-Crude industry. Lesser, but still nonzero, amounts of Crude oil are embodied in Freight transport (17,736 ktoe/year), Illumination (102.2 ktoe/year), and Space heating (164.3 ktoe/year), due to interactions among sectors of the ECC. The amount of Crude oil embodied in all final demand products (the sum of all positive entries in Q Crude ) is 50,000 ktoe/year, the direct production of Crude by Resources-Crude (see Fig. 11).
Any product created in the ECC, not just primary energy carriers, can be analyzed like Crude oil and Natural gas above. Another interesting example for the ECC of Fig. 11 is Freight transport, for which Q Freight is shown in Fig. 14. There is some amount of Freight transport created by Trucks embodied in all energy services (bottom row of Fig. 14), again because of interactions among the industries in the ECC. Space heating, for example, embodies × 1 10 9 tonne-km/year of Freight transport that was produced by Trucks. The sum of the bottom row in Q Freight is × 1.5 10 11 tonne-km/year, the gross production of Freight transport.
With embodied primary energy in hand, the consumption-based primary-to-services efficiency of providing an energy service can be determined by dividing the magnitude of an energy service by the embodied primary energy for that energy service (summed across all primary energy carriers). For the example of Passenger transport, we obtain    11. A real-world ECC covering primary energy to energy services. All energy flows in units of ktoe/year; energy services in units shown. NG is Natural gas. LTH is Low-temperature heat. MD is Mechanical drive. "tes" is an abbreviation for metric tonnes. Line colors indicate products.
When each primary energy carrier is produced by a single Resources industry (as in Fig. 11), nonzero entries in the e P  and D matrices will be 1 (see Appendix G), and a vector of consumption-based primary-toservices energy efficiencies (η E ps , ) can be obtained directly with Fig. 15 shows consumption-based primary-to-services energy efficiencies (η E ps , ) for the real-world ECC in Fig. 11. We note that consumption-based primary-to-services exergetic efficiencies (as shown in Fig. 15) account for all direct and indirect primary energy demanded by each service (the embodied energy of the service). The provision of Passenger transport to final demand provides an illustrative example.
For a narrow analysis boundary around automobiles and the service they provide, the final-to-services energy efficiency of Passenger where the numerator is the service level provided by Cars (Passenger transport) and the denominator is the final energy consumed by Car engines (Petrol). However, the consumption-based primary-to-services energy efficiency of Passenger transport in Fig. 15 was obtained from an expanded analysis boundary (made possible by the PSUT framework), which accounts for all energy consumption in the ECC to provide Passenger transport by cars, including (a) Crude required to supply selfconsumption of Petrol and Diesel and (b) Natural gas required to make electricity. With the wider analysis boundary, we find the consumptionbased primary-to-services energy efficiency of providing Passenger transport to be 19% less: × 1.55 10 7 passenger-km/ktoe, as shown in Fig. 15.
The final-to-services energy efficiency of Passenger transport is, essentially, an expression of the fleet-average fuel efficiency of automobiles. We can cast the above results into familiar fuel economy units  Fig. 2a] shows fleet average fuel economy of about 10 litres/100 km or 23.5 miles/U.S. gallon, indicating that the rounded data in our real-world ECC are close to reality and that our estimate of 1.5 passenger-miles/car-mile is reasonable.) Accounting for all indirect energy consumption along the ECC (expressed in Petrol gallon equivalents), we obtain 20.2 car-miles/U.S. gallon for the expanded analysis boundary, again 19% less than the fuel economy obtained from the narrow boundary.
As discussed above, the ECCs of Figs. 3, 7, and 11 are meant to be representative, as they contain only two of the many energy pathways in the UK economy in 2000. Each additional energy pathway included in this analysis will further reduce the final-to-services energy efficiency of Passenger transport. Expanding the analysis boundary to include the  We showed that embodied primary energy of final demand can be determined within the PSUT framework in Section 3.3. But as we evaluate strategies for reducing embodied energy, an important question emerges: What are the critical supply chains involved in energy and services delivery to final demand?
Significance: One approach to reducing environmental impacts of economic activity is to minimize the embodied primary energy of final demand as energy moves through the ECC.
To address this question, one needs to trace the large number of pathways for delivering energy through an ECC. Structural path analysis (SPA) [116,117] is an established IO technique that uses the Taylor series expansion [118] to "unravel" the Leontief inverse ( × L p p ) and identify and quantify individual paths through a supply chain. SPA can be used within the PSUT framework to assess paths from resource extraction to final demand expressed in any form, including final energy (Fig. 3), useful energy (Fig. 7), or energy services (Fig. 11).
SPA provides two important results within the PSUT framework: (a) the lengths of paths from primary resources through the ECC to final demand and (b) the embodied primary energy of each path. The length of an ECC path is defined as the number of ECC industries through which energy or an energy service flows before reaching final demand. A zero length path is one where energy flows directly from resource extraction to final demand; a path of length 1 has a single industry between resources and final demand; etc. For simple supply chains, path lengths can be determined by inspection, but complex supply chains in real-world ECCs have far too many paths for each to be identified visually. Although the ECCs in this paper are increasingly complex (compare Figs. 3 and 11), it is obvious by inspection that there are no paths of length 0 or 1. For example, the shortest energy service delivery path in Fig. 11 has length 4, traversing from Natural gas through (1) Gas wells and processing to (2) Natural Gas distribution to (3) Furnaces to (4) Homes and, ultimately, to the Residential sector of final demand.
The magnitude of an ECC path is defined as the embodied primary energy of the service delivered by the path. For the real-world ECCs in this paper, ECC path magnitudes are measured in ktoe/year.
Calculations of path lengths rely on the Taylor series expansion of the Leontief where n is the number of terms retained for a finite approximation to the infinite sum. If the right side of Eq. To demonstrate SPA within the PSUT framework, we perform five separate analyses, one for each combination of primary energy resource (Crude and Natural Gas) and ECC (Figs. 7 and 11) and one for Freight transport in the ECC of Fig. 11. (Note that Figs. 7 and 11 involve different final demand matrices (Y): the final demand matrix of Fig. 7 is comprised of useful energy and the final demand matrix of Fig. 11 is comprised of energy services. SPA works with both types of final demand matrices within the PSUT framework. An SPA could also be performed with the ECC of Fig. 3, but we focus on Figs. 7 and 11 for simplicity.) All paths in all five analyses are evaluated for both length (the number of steps) and magnitude (embodied primary energy).
We first aggregate the magnitudes of all same-length paths originating at primary energy carriers to create Fig. 16, which shows aggregated magnitudes (as a fraction of total embodied primary energy) on the vertical axis and path lengths (from 0 to 9) on the horizontal axis. Nearly all (98%) of embodied Crude energy takes five steps to reach final demand expressed as useful energy (solid line in Fig. 16a). To reach final demand expressed as energy service, nearly all the embodied Crude energy takes six steps (dashed line in Fig. 16a). A review of Figs. 7 and 11 confirms that the energy service ECC (Fig. 11) has one additional stage compared to the useful energy ECC (Fig. 7). Indeed, inspection of Fig. 7 shows that the simplest path from Crude to final demand expressed as useful energy takes five steps: from Resources-Crude to (1) Oil fields to (2) Crude dist. to (3) Oil refineries to (4) Diesel or Petrol dist. to (5) Truck or Car engines to Transport. And inspection of Fig. 11 shows that the simplest path from Crude to final demand expressed as energy services takes six steps: from Resources-Crude to (1) Oil fields to (2) Crude dist. to (3) Oil refineries to (4) Diesel or Petrol dist. to (5) Truck or Car engines to (6) Trucks or Cars to Freight or Passenger Transport. Appendix G provides additional details of the path from Crude to Freight transport.
Paths from Natural gas extraction are slightly more complex. 61% of the embodied Natural gas takes three steps to reach final demand expressed as useful energy with a further 37% taking five steps (solid line in Fig. 16b). Again, the paths to final demand expressed as energy services (dashed line in Fig. 16b) are one step longer (four and six steps).
With the help of Figs. 7 and 11, it is possible to use Fig. 16 to interpret the primary paths through the real-world ECC. However, the embodied energy of more complex paths (e.g., paths which include  (19) is expanded again such that the path from industry i to industry j via industry k is described by matrix elements A ik and A kj . With the doublyexpanded form of the Leontief inverse, it is possible to identify and rank the most important (largest magnitude) paths through the ECC (see Appendix G for additional details).
To illustrate the capability of SPA to identify paths within an ECC, we show results for the most interesting combination of resource and ECC, namely Natural gas through to energy services in Fig. 11 (the dashed line in Fig. 16b). The 10 paths of largest magnitude are shown in Table 7, comprising 99.8% of all embodied Natural gas in the ECC of Fig. 11. The largest magnitude path has length 4: from Resources-NG to (1) Gas wells & proc. to (2) NG dist. to (3) Furnaces to (4) Homes to Residential final demand. The second-largest path is a six-step path and the shortest path that provides Illumination: from Resources-NG to (1) Gas wells & proc. to (2) NG dist. to (3) Power plants to (4) Elect. grid to (5) Light fixtures to (6) Rooms to Residential final demand. The results of Table 7 confirm that the embodied energy captured in 4 and 6 steps (shown by the dashed line in Fig. 16b) comprises two large-magnitude paths only.
However, there are several more-complex routes from Natural gas to final demand through the ECC of Fig. 11. For example, the routes of paths with size rank 2-10 in Table 7 go through Power plants and the Elect. grid to make Electricity available to other portions of the ECC, some of which flows to industries that serve end uses other than Illumination. It would be impossible to find all paths by inspection from Figs. 11 and 16b, and this detailed SPA method is likely the only way to identify the length and magnitude of paths longer than, say, 6 steps.
In addition to primary energy carriers created at the upstream end of an ECC, SPA can be performed on any product created anywhere in the ECC. In the ECC of Fig. 11, Freight transport is created by Trucks and delivered to Transport final demand as well as distribution industries within the ECC. Most (95%) embodied Freight transport reaches Transport final demand directly (0 steps), but some Freight transport is provided to the distribution industries within the ECC. The shortest indirect path through the distribution industries to Transport final demand has length 3: from Trucks to (1) Diesel dist. to (2) Truck engines to (3) Trucks to Transport final demand. (A similar path with length 3 goes through Petrol dist.) Exponentially-decreasing amounts of embodied Freight transport complete this cycle twice, then three times, then four times, etc., each much smaller in magnitude than the last. The semi-log plot in Fig. 17 is the Freight transport version of Fig. 16, and it shows exponentially-decreasing embodied Freight as a (nearly) loglinear descending function of path length.
Benefit of the PSUT framework: This extended example demonstrates that when ECC data are arranged in a PSUT format, quantification of the magnitude of embodied product flows through every route of an ECC can be accomplished using SPA techniques. Table 7 Top 10 largest magnitude paths from Natural gas to final demand expressed as energy services in Fig. 11.

Size rank
Step 0 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Path magnitude [ktoe  Fig. 17. Fraction of embodied Freight captured by paths of varying lengths in the energy services ECC of Fig. 11, expressed in log 10 such that, e.g., −2 on the vertical axis is 10 −2 or 1% of all embodied Freight transport.

Discussion
In this discussion, we briefly discuss the originality of this work (Section 4.1), explain limitations of the PSUT framework (Section 4.2), identify several additional applications (Section 4.3), and suggest future work (Section 4.4).

Originality
To our knowledge, the following elements of this paper are novel advances that appear in the literature for the first time:

Limitations
We suggest two limitations of the PSUT framework. The first arises from the fact that the accuracy and level of detail of analyses performed with the PSUT framework are a function of the accuracy and availability of ECC data. At the primary and final energy stages of an ECC, data are readily available from the IEA [95] and national energy agencies, but they must be applied correctly [119], are they not without measurement errors and inaccuracies [120].
On the other hand, data availability at the useful energy or energy services stages are a challenge. At the useful stage, energy flows must be calculated from (a) estimates of allocation of final energy to end-use devices and (b) estimates of final-to-useful end-use device efficiencies (η fu ). Many challenges arise when estimating allocations and device efficiencies. Progress is being made on allocation of IEA final energy data [97,40], and probabilistic models are under development to quantify effects of allocation uncertainty [121]. Estimating time series for final-to-useful device efficiencies (η fu ) is time consuming, because economy-wide efficiencies are a function of many factors, including diffusion rates of new technologies, statistical distributions of device vintage, maintenance schedules, etc. All of these factors must be evaluated per device for each economy when estimating time series of device efficiencies (η fu ). Fortunately, here, too, progress is being made, and many countries have been analyzed, including the U.S. [40,122], the UK [40,123], the EU-15 [99], China [98], Mexico [124], and Portugal [39].
When pushing through to energy services, some data are readily available (e.g., Freight and Passenger transport) while other data are less available (e.g., Lighting).
However, we note that data limitations are not unique to the PSUT framework: all energy analyses on the ECC face similar challenges. Indeed, the PSUT framework is not a means to obtain or generate final energy or energy services data. Rather, it is a way to organize available data and streamline analyses of that data.
The second limitation arises from the inherent linearity of IO and SUT analysis methods, which are often and rightly criticized for their inability to represent non-linear effects and dynamics related to changes in final demand (see Section 2.2.5). We note here that nonlinear effects exist in both (a) the physical realm (e.g., larger buildings are more efficient because heat loss scales with surface area but space heating service scales with volume) and (b) the economic realm (e.g., "economies of scale"). For the purposes of energy analysis, we believe that physical SUT techniques are less problematic than economic SUT techniques, because PSUT techniques avoid purely-economic non-linearities.
That said, we recognize that, at the economy-wide scale, the physical realm and the economic realm may interact in unexpected ways to produce non-linear effects. For example, if demand for electricity decreases, markets may prefer to mothball inefficient plants, thereby increasing the aggregate efficiency of electricity production. The methods of Section 2.2.5 would not predict such efficiency improvements and would instead assume that efficiency remains constant as final demand shifts. (To capture these non-linearities in a predictive sense, dynamic energy-economy models are needed.) However, when annual data for the entire ECC are available (e.g., IEA world energy statistics [95] as discussed in Appendix H), each year can be analyzed independently, and the PSUT framework will correctly observe and calculate year-toyear physical changes in an ECC (e.g., increasing efficiency of electricity production), regardless of their root cause (e.g., economic structural changes or technological efficiency changes). Section 4.3 discusses structural decomposition analysis (SDA) which can be applied to determine the dominant drivers of temporal trends.

Additional applications
There are many additional applications for the PSUT framework described in Section 2 and demonstrated in Section 3. Most of the additional applications are enabled by the supply-use table structure of the framework. For each additional application discussed below, we include questions that, taken together, illustrate the breadth of applicability of the PSUT framework. Due to space constraints, we do not provide real-world examples.
The PSUT framework could be used to study the question Which country can provide energy services most efficiently? To answer this question, a multi-regional PSUT (MR-PSUT) would need to be constructed. A MR-PSUT model would enable the calculation of embodied energy content of energy services consumed anywhere in the world, taking into account global supply chains that cover any energy conversion process in any country.
Further development of the MR-PSUT could involve producing annual tables, allowing the following question to be addressed: What are the most important drivers of difference of the embodied energy of final demand between (a) two countries at a given time? and (b) for a given country between two times? This question can be analyzed within the PSUT framework by the application of Structural Decomposition Analysis (SDA). SDA is an "analysis of …change by means of a set of comparative static changes in key parameters in an input-output table" [124, p. 3]. An SDA would be able to determine the importance of the following factors in contributing to country-by-country or year-by-year differences: (a) larger final demand for the energy service (b) the structure of the ECC involved in the delivery of the energy service, and (c) increasing or decreasing waste energy at various stages of the ECC.
Regarding energy services, an important question is Are energy services being provided more efficiently over time? Steps to answer this question using the PSUT framework would comprise: (a) gathering ECC time series data (through to energy services), (b) organizing time series data into PSUT matrices, with one set of U V , , and Y matrices for each time period (typically, one year) as shown in Appendix H, and (c) repeating the analysis of Section 3.3 for each year to obtain a time series of consumption-based primary-to-services energy efficiencies. The evolution of energy service efficiencies will then be obvious when graphed against time.
Turning to economics-related questions, one might want to know What are the consumption-based energy intensities (in GJ/$) of economic sectors as defined by the system of national accounts? (This question hearkens back forty years to the works of Bullard et al. [13], Costanza [126], and Roberts, whose 1978 definition of energy analysis was "a systematic way of tracing the flows of energy through an industrial system, resulting in the apportioning of a fraction of the primary energy inputs into the system to each of the outputs of that system" [17, p. 200].) To answer this question, the PSUT matrices should be embedded within a larger mixed units energy-economy SUT analysis that includes financial flows for non-energy sectors. Recent work by Guevara and coauthors [86,90,91] and Rocco and co-authors [85,87,88] has pursued this line of inquiry (see Table 1).
In addition, we speculate that analyses typically performed on individual energy conversion devices in the sub-field of exergoeconomics [127,128] could be applied to the economy-wide ECC boundary. A core question would be: What is the optimum design of an economy-wide ECC to minimize its costs or its exergy destruction? To answer this question, analysts would need to (a) obtain or generate efficiency vs. cost relationships for each device in an economy-wide ECC and (b) apply exergoeconomic techniques. Estimates of the cost of exergy destruction by each device in the ECC would be generated, and optimization of the ECC could be pursued for various objective functions, including minimizing exergy destruction or minimizing cost of energy service delivery. An optimal mix of exergy conversion devices could be determined for each objective.
Finally, we note that energy carriers and services change form through the ECC: all primary energy is completely consumed by transformation processes on the way to providing energy services. The PSUT framework is able to track all of these changes of form, even when the transformation is so complete that the quantities involved no longer exist as useable energy but rather as services only.
Of course, energy is not the only resource whose primary resources are "used up" to provide services measured in different units. Thus, we speculate that a version of the PSUT energy analysis framework could be applied to other service delivery networks involving other resource flows. For example, materials of all types (paper and wood [79], steel [129], water [130], the entire economy [112,131]) provide material services to society [132,133] and are (at least partially) "used up" in the process. A key question for a materials application of the PSUT framework would be What is the consumption-based efficiency of providing material services to society?

Future work
There are several areas available for future work on this topic. First, because the PSUT framework will be applied to real ECCs (see Hardt et al. [134] for application with LMDI decomposition analysis), efforts to improve the availability and accuracy of data along the ECC are needed, particularly regarding useful energy (see Section 4.2). If analyses are to reach energy services, robust data on services will be required. Therefore, additional work is encouraged on developing a common method to estimate useful energy and publishing databases of useful energy and energy services statistics. (These new directions should build on related efforts to develop consistent societal exergy accounting methods [94,135], to assess the effect of allocation uncertainties in societal exergy accounting [121], and to understand the basic driver of the energy system, end use [136].) Second, further development of the additional PSUT framework applications described in Section 4.3 should be undertaken and demonstrated via real-world examples, similar to those in Section 3.
Third, development of a generalized mathematical approach for unit inhomogeneity of sector co-products would advance the PSUT framework by removing the requirement of co-product unit homogeneity and the need to split industries whose co-products are unitinhomogeneous (see Section 2.2.4).

Conclusions
In this paper, we have built upon prior work in related fields to develop and demonstrate, via four real-world examples that address contemporary energy analysis questions, a new physical supply-use table energy analysis framework. The framework is applicable to all parts of the energy conversion chain and provides several important benefits to the field of energy analysis.
First, because physical supply-use table matrices can be asymmetric (i.e., non-square), the physical supply-use table framework allows analysis of energy conversion chains that include co-producing industries with disaggregated products. (In the energy conversion chain of Figs. 3, 7, and 11, Oil refineries co-produce Petrol and Diesel. Real refineries make dozens of products.) This characteristic overcomes a limitation of input-output-based methods that require symmetric (i.e., square) matrices, which, in turn, necessitate aggregations that discard energy conversion chain information.
Second, because the physical supply-use table framework allows analysis on the entire energy conversion chain, including co-producing industries, it can overcome communication challenges that may arise when different analysis techniques or different terms are used by different research communities who study different portions of the energy conversion chain.
Next, two advantages arise from units and product quantification. Because the physical supply-use table framework utilizes physical units exclusively, it overcomes a limitation of financial input-output and supply-use table methods in which monetary flows are proxies for physical flows, thereby introducing distortions into what otherwise should be purely physical (energy) analysis. Indeed, one of the main challenges with year-by-year economic, rather than physical, inputoutput or supply-use table energy decompositions is that effects of inflation must be removed before performing the analysis. If not, the importance of temporal changes in final demand is often exaggerated due to inflationary price increases. The physical supply-use table framework has a significant advantage over economic input-output and supply-use table analyses, because it uses data in physical units rather than economic spending information in monetary units. And because the physical supply-use table framework allows analyses in either energy or exergy quantifications of energy carriers, it can assist answering energy analysis questions posed in either energy or exergy terms.
Finally, we note that the physical supply-use table framework is useable by many sub-fields of energy analysis because it both (a) allows analysis anywhere along the energy conversion chain and (b) is flexible regarding energy quantification. For example, emissions footprinting is conducted with energy quantification and is concerned with extracted fossil fuels (primary energy) at the upstream end of the energy conversion chain, while societal exergy analysis is conducted with exergy quantification and is often concerned with useful exergy and exergy services at the downstream end of the energy conversion chain.
We believe that these advantages commend the physical supply-use table framework to the field of energy analysis. It can provide data structure uniformity and methodological consistency for many subfields. (For example, physical supply-use table matrices could complement the energy balance format currently employed by national and international energy agencies.) And, being a common framework, it could organize and streamline questions to be asked, data to be gathered, analyses to be performed, and results to be reported. None of the funding sources provided input to the research and/or preparation of the article; the design of the study; the collection, analysis, and interpretation of data; the writing of this article; or the decision to submit this article for publication.

Data Repository
A complete set of input and results datasets for this paper have been deposited at the University of Leeds Data Repository at https://doi.org/ 10.5518/393.

Appendix A. Nomenclature
We employ several symbol conventions in this paper. Boldface capital letters (e.g., U) represent matrices. Boldface lowercase letters (e.g., g) identify column vectors. (All vectors are assumed to be column vectors.) Symbols for PSUT matrices and vectors mostly follow Eurostat naming conventions [93, pp. 349-350]. Table A.1 lists the nomenclature for this paper.

Appendix B. Exergy quantification in the PSUT framework
As discussed in Section 1.1, the sub-field of societal exergy analysis has informed discussions of the role of energy in society and the economy in recent years [85,96,137]. Analyses within the sub-field of societal exergy analysis are conducted with exergy quantification for energy carriers. Exergy is an alternative quantification of energy that gives the maximum useful work that could be generated by bringing a system into equilibrium with its surroundings. The purpose of this appendix is to demonstrate that the analyses conducted with energy quantification in Section 3 can also be conducted with exergy quantification.
When exergy quantifications are used for energy carriers, the equations of Sections 2 and 3 are unchanged, but nonzero energy entries in the PSUT matrices are different by the exergy-to-energy ratio (ϕ, see Serrenho [99,Table 2]). We assume that wastes and waste heat from each industry represent exergy destroyed by the industry, accounted by the second law of thermodynamics. We re-present the key results of Sections 3.3 and 3.4 in exergy terms below. To begin, we convert all energy flows in Fig. 11 to exergy flows via multiplication by the exergy-to-energy ratio (ϕ), thereby obtaining Figs. B.1 and B.2.
Calculating the consumption-based primary-to-services exergetic efficiencies proceeds as discussed in Section 3.3 using the PSUT matrices of Fig.  B.2. Q Crude and Q NG are shown in Fig. B.3. The vector of consumption-based primary-to-services efficiencies (η X ps , ) is shown in Fig. B (B.1) slightly less than × 1.92 10 7 passenger-km/ktoe reported in Eq. (18) due to exergy quantification in the denominator of Eq. (B.1) and energy quantification in the denominator of Eq. (18). The denominators are different by the exergy-to-efficiency ratio for oil and oil products ( . Similarly, structural path analysis can be conducted using exergy quantification of energy carriers. Calculations proceed as discussed in Sections 3.4 and Appendix G using the PSUT matrices whose entries are now quantified as exergy (Fig. B.2). The fraction of embodied primary exergy captured by paths of varying lengths is shown in Fig. B.5, and largest magnitude paths for delivery of exergy services from natural gas are shown in Table B.1.
We note that Figs. 14 and 17 are unchanged for the exergy ECC of Fig. B.1, because the Freight transport quantities are unchanged between Figs. 11 and B.1.     Size rank Step 0 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Path magnitude [ktoe Third, Section 3 includes several terms of the form̂− b a 1 , which is the matrix algebra notation for an element-wise quotient of two same-length column vectors: Finally, we point out that Tables 5 and 6 contain several terms of the form̂− Mv 1 , which is equivalent to dividing each column of matrix M by the associated entry in column vector v . We begin with a restatement of Eq. (4).
Next, we substitute definitions for g and W from Table 4

Appendix E. Aggregation across products with inhomogeneous units
When inhomogeneous units are present along the product dimension of any of the U V W , , , or Y matrices, care must be taken to obtain appropriate row and column sums for energy and energy services balances. Inhomogeneous product units are likely when any of the U V W , , , or Y matrices contain energy services on their product dimensions. Under such circumstances, aggregation across products must be done in a unit-aware manner, as shown in Eq. (5).
A units summation matrix (S units ) facilitates such aggregations. S units is products × units and is formed by placing a "1" to indicate the units of any product. See Fig. E.1 for an example S units matrix for the ECC of Fig. 11. Note that if the U V W , , , and Y matrices are unit-homogeneous, S units simplifies to an identity vector that provides simple row sums (i) or column sums (i T ).
Post-multiplying V by S units or pre-multiplying U W , , or Y by S units T reduces the size of the product dimension from the number of products to the number of unique product units, aggregating all products of like units. We use an over-bar applied to a matrix symbol (e.g., V) to indicate that summation across products of the same units has occurred. Aggregation equations are given in Table E.1.
An example is instructive.   Appendix F. Relationships among energy return ratios This appendix demonstrates relationships among the three energy return ratios (ERRs) discussed in Section 3.1 (GER NER , , and r) and proves that any two can be expressed in terms of the third. We begin with the definitions of net energy (E net ), gross energy ratio (GER), net energy ratio (NER), and net-to-gross energy ratio (r).   Fig. 11. "tes" is an abbreviation for metric tonnes. showing that GER and NER can be expressed in terms of r and completing the proof that any ERR can be expressed in terms of the other two and that any two ERRs can be expressed in terms of the third. Table F.1 summarizes these results.

Appendix G. Details of structural path analysis
This appendix provides additional details of the calculations that identify the key supply-chain paths through the ECC. The technique, known as Structural Path Analysis (SPA), can be used to decompose the embodied primary energy associated with final demand to the sum of an infinite number of production chains, called paths. Wood and Lenzen [115, p. 371] describe this process as "unraveling the Leontief inverse using its series expansion." SPA was developed initially for use with symmetric input-output tables in monetary units [138,139] rather than the SUT format employed by the PSUT framework discussed in Section 2. We adapt SPA for the PSUT framework here.
SPA proceeds by substituting the right side of Eq.    In practice, Eq. (G.2) is implemented as the product of a series of entries in theê D Z , , , andŷ matrices for paths through the ECC found by a search algorithm (see Table 5 for definitions of vectors and matrices in the PSUT framework). The provision of Freight transport in the ECC of Fig. 11 provides an example. Fig. G.1 shows the calculation of the embodied Crude in Freight transport for a 6-step path. (There are other, longer, paths from Crude to Freight transport that are not captured by this calculation.) We start with the appropriate entries in theê and D matrices, followed by a series of 6 entries in the Z and D matrices, representing the 6 steps of the shortest path from Crude to Freight transport. Finally, the Freight transport entry inŷ is shown.
The product of all values in Fig. G.1 is 17,465 ktoe/year, the embodied primary energy of the 6-step path from Crude to Freight transport. Note that 17,465 ktoe/year is 98.5% of all embodied Crude in Freight transport (17,736 ktoe/year in Figs. 15 and 13a), the difference being Crude embodied in paths that take more than 6 steps to reach Freight transport.
In Fig. G.1, the product of the 12 values that comprise the A matrix (entries in Z and D at each step in the path) is × − 1.22204 10 7 ktoe/tonne-km. Fig. G.2 shows the A 6 matrix, which contains the sum of magnitudes of all length-6 paths from products in rows to products in columns of the ECC of Fig. 11. The only length-6 path from Crude to Freight transport appears in the Crude row and the Freight transport column of the A 6 matrix, and its value is the same as the product of all A entries in Fig. G.1.   Fig. G.1. The path from Resources-Crude to Freight transport through the A matrix for the ECC in Fig. 11. "tes" is an abbreviation for metric tonnes.   [95]. The sign of IEA entries affects the PSUT matrix in which data should be entered, as indicated by (+) and (−). Before placing negative numbers into the PSUT matrices, the absolute value should be taken, as indicated by "as +" below. The IEA energy statistics data contain Flow and Product columns whose entries become the names of rows or columns in the PSUT matrices, as indicated by (row×column) notation.