Balances

library(dplyr)
library(IEATools)
library(Recca)
library(tidyr)

Introduction

When working with material and energy conversion chains in the Recca package with matsindf data frames, it is helpful to calculate several types of balances. This vignette describes those balances and shows when you might want to use each of them. Balances include

inter-industry balances,
intra-industry balances, and
the $\mathbf{W}$ matrix.

Terminology

Recca assists with matrix-based input-output calculations when the data are organized into the $\mathbf{RUVY}$ matrices of the physical supply-use table (PSUT) framework. A generic processing stage is called an “industry.” The stuff that flows between industries is generically called a “product.” A “balance” is any calculation that compares inputs to outputs. Balances can be computed for all types of products, in particular materials (quantified in mass or exergy units) and energy (quantified as energy or exergy).

In the PSUT framework, the $\mathbf{R}$ matrix is an industry-by-product resource matrix. The $\mathbf{U}$ matrix is an product-by-industry use matrix. The $\mathbf{V}$ matrix is an industry-by-product make matrix. The $\mathbf{Y}$ matrix is an product-by-industry final demand matrix.

In this vignette, matrices are represented by boldface capital letters, such as $\mathbf{R}$ . Vectors are assumed to be column vectors and represented by boldface lowercase letters, such as the identity vector $\mathbf{i}$ .

Data

In the examples that follow, we use the energy conversion chain in UKEnergy2000mats from the Recca package.

ecc <- UKEnergy2000mats |> 
  tidyr::pivot_wider(names_from = matrix.name, 
                     values_from = matrix)
dplyr::glimpse(ecc)
#> Rows: 4
#> Columns: 12
#> $ Country    <chr> "GBR", "GBR", "GBR", "GBR"
#> $ Year       <dbl> 2000, 2000, 2000, 2000
#> $ EnergyType <chr> "E", "E", "E", "X"
#> $ LastStage  <chr> "Final", "Services", "Useful", "Services"
#> $ R          <list> <<matrix[2 x 2]>>, <<matrix[2 x 2]>>, <<matrix[2 x 2]>>, <<…
#> $ U          <list> <<matrix[12 x 9]>>, <<matrix[17 x 17]>>, <<matrix[13 x 13]>…
#> $ U_EIOU     <list> <<matrix[7 x 8]>>, <<matrix[3 x 8]>>, <<matrix[3 x 8]>>, <<…
#> $ U_feed     <list> <<matrix[9 x 9]>>, <<matrix[16 x 17]>>, <<matrix[12 x 13]>>…
#> $ V          <list> <<matrix[9 x 10]>>, <<matrix[17 x 18]>>, <<matrix[13 x 14]…
#> $ Y          <list> <<matrix[4 x 2]>>, <<matrix[4 x 2]>>, <<matrix[4 x 2]>>, <…
#> $ r_EIOU     <list> <<matrix[12 x 9]>>, <<matrix[17 x 17]>>, <<matrix[13 x 13]…
#> $ S_units    <list> <<matrix[12 x 1]>>, <<matrix[20 x 5]>>, <<matrix[16 x 1]>>…

The following subsections explain each type of balance and how to calculate it with the Recca package.

Inter-industry (between-industry) balances check for product consistency

In any conversion chain, products that leave one industry must enter another industry. These between-industry (or inter-industry) balances should be verified for every conversion chain to ensure consistency.

Inter-industry balances are obtained by the following computation.

$[(\mathbf{R} + \mathbf{V})^\mathrm{T} - (\mathbf{U} + \mathbf{Y})] \, \mathbf{i}$

If the conversion chain is consistent, the result should be the $\mathbf{0}$ vector with products in rows.

To compute inter-industry balances within the PSUT framework using the Recca package, use the following code.

inter_industry_balances <- ecc |> 
  calc_inter_industry_balance()

The result is added as a new column, named “SUTInterIndustryBalance” by default.

dplyr::glimpse(inter_industry_balances)
#> Rows: 4
#> Columns: 13
#> $ Country                 <chr> "GBR", "GBR", "GBR", "GBR"
#> $ Year                    <dbl> 2000, 2000, 2000, 2000
#> $ EnergyType              <chr> "E", "E", "E", "X"
#> $ LastStage               <chr> "Final", "Services", "Useful", "Services"
#> $ R                       <list> <<matrix[2 x 2]>>, <<matrix[2 x 2]>>, <<matrix…
#> $ U                       <list> <<matrix[12 x 9]>>, <<matrix[17 x 17]>>, <<mat…
#> $ U_EIOU                  <list> <<matrix[7 x 8]>>, <<matrix[3 x 8]>>, <<matrix…
#> $ U_feed                  <list> <<matrix[9 x 9]>>, <<matrix[16 x 17]>>, <<matr…
#> $ V                       <list> <<matrix[9 x 10]>>, <<matrix[17 x 18]>>, <<ma…
#> $ Y                       <list> <<matrix[4 x 2]>>, <<matrix[4 x 2]>>, <<matri…
#> $ r_EIOU                  <list> <<matrix[12 x 9]>>, <<matrix[17 x 17]>>, <<ma…
#> $ S_units                 <list> <<matrix[12 x 1]>>, <<matrix[20 x 5]>>, <<mat…
#> $ SUTInterIndustryBalance <list> <<matrix[12 x 1]>>, <<matrix[20 x 1]>>, <<mat…

The vectors in SUTInterIndustryBalance column have products in rows.

inter_industry_balances$SUTInterIndustryBalance[[1]]
#>                     Industry
#> Crude                      0
#> Crude [from Dist.]         0
#> Crude [from Fields]        0
#> Diesel                     0
#> Diesel [from Dist.]        0
#> Elect                      0
#> Elect [from Grid]          0
#> NG                         0
#> NG [from Dist.]            0
#> NG [from Wells]            0
#> Petrol                     0
#> Petrol [from Dist.]        0
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Industry"
inter_industry_balances$SUTInterIndustryBalance[[2]]
#>                                  Industry
#> Crude                        0.000000e+00
#> Crude [from Dist.]           0.000000e+00
#> Crude [from Fields]          0.000000e+00
#> Diesel                       0.000000e+00
#> Diesel [from Dist.]          0.000000e+00
#> Elect                        0.000000e+00
#> Elect [from Grid]            0.000000e+00
#> Freight [tonne-km/year]      3.051758e-05
#> Illumination [lumen-hrs/yr]  0.000000e+00
#> Light                        0.000000e+00
#> LTH                          0.000000e+00
#> MD [from Car engines]       -1.364242e-12
#> MD [from Truck engines]      4.547474e-13
#> NG                           0.000000e+00
#> NG [from Dist.]              0.000000e+00
#> NG [from Wells]              0.000000e+00
#> Passenger [passenger-km/yr]  0.000000e+00
#> Petrol                       0.000000e+00
#> Petrol [from Dist.]          0.000000e+00
#> Space heating [m3-K]         0.000000e+00
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Industry"
inter_industry_balances$SUTInterIndustryBalance[[3]]
#>                             Industry
#> Crude                   0.000000e+00
#> Crude [from Dist.]      0.000000e+00
#> Crude [from Fields]     0.000000e+00
#> Diesel                  0.000000e+00
#> Diesel [from Dist.]     0.000000e+00
#> Elect                   0.000000e+00
#> Elect [from Grid]       0.000000e+00
#> Light                   0.000000e+00
#> LTH                     0.000000e+00
#> MD [from Car engines]   0.000000e+00
#> MD [from Truck engines] 2.273737e-13
#> NG                      0.000000e+00
#> NG [from Dist.]         0.000000e+00
#> NG [from Wells]         0.000000e+00
#> Petrol                  0.000000e+00
#> Petrol [from Dist.]     0.000000e+00
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Industry"
inter_industry_balances$SUTInterIndustryBalance[[4]]
#>                                 Industry
#> Crude                       0.000000e+00
#> Crude [from Dist.]          0.000000e+00
#> Crude [from Fields]         0.000000e+00
#> Diesel                      0.000000e+00
#> Diesel [from Dist.]         0.000000e+00
#> Elect                       0.000000e+00
#> Elect [from Grid]           0.000000e+00
#> Freight [tonne-km/year]     3.051758e-05
#> Illumination [lumen-hrs/yr] 0.000000e+00
#> Light                       0.000000e+00
#> LTH                         0.000000e+00
#> MD [from Car engines]       0.000000e+00
#> MD [from Truck engines]     0.000000e+00
#> NG                          0.000000e+00
#> NG [from Dist.]             0.000000e+00
#> NG [from Wells]             0.000000e+00
#> Passenger [passenger-km/yr] 0.000000e+00
#> Petrol                      0.000000e+00
#> Petrol [from Dist.]         0.000000e+00
#> Space heating [m3-K]        0.000000e+00
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Industry"

As expected, these vectors are the $\mathbf{0}$ vector to within reasonable precision.

If you want to check the inter-industry balances for consistency, use verify_inter_industry_balance(). The tol argument gives the tolerance for non-zero product balances.

inter_industry_balances |> 
  verify_inter_industry_balance(tol = 1e-4) |> 
  dplyr::glimpse()
#> Rows: 4
#> Columns: 14
#> $ Country                  <chr> "GBR", "GBR", "GBR", "GBR"
#> $ Year                     <dbl> 2000, 2000, 2000, 2000
#> $ EnergyType               <chr> "E", "E", "E", "X"
#> $ LastStage                <chr> "Final", "Services", "Useful", "Services"
#> $ R                        <list> <<matrix[2 x 2]>>, <<matrix[2 x 2]>>, <<matri…
#> $ U                        <list> <<matrix[12 x 9]>>, <<matrix[17 x 17]>>, <<ma…
#> $ U_EIOU                   <list> <<matrix[7 x 8]>>, <<matrix[3 x 8]>>, <<matri…
#> $ U_feed                   <list> <<matrix[9 x 9]>>, <<matrix[16 x 17]>>, <<mat…
#> $ V                        <list> <<matrix[9 x 10]>>, <<matrix[17 x 18]>>, <<m…
#> $ Y                        <list> <<matrix[4 x 2]>>, <<matrix[4 x 2]>>, <<matr…
#> $ r_EIOU                   <list> <<matrix[12 x 9]>>, <<matrix[17 x 17]>>, <<m…
#> $ S_units                  <list> <<matrix[12 x 1]>>, <<matrix[20 x 5]>>, <<ma…
#> $ SUTInterIndustryBalance  <list> <<matrix[12 x 1]>>, <<matrix[20 x 1]>>, <<ma…
#> $ SUTInterIndustryBalanced <lgl> TRUE, TRUE, TRUE, TRUE

If a product is out of balance by more than tol (by default 1e-6), a warning occurs and FALSE appears in the new column, “SUTInterIndustryBalanced.”

Intra-industry (across-industry) balances

A second type of balance (the intra-industry balance) computes differences across industries. Intra-industry balances are obtained by the following computation.

$(\mathbf{U}^\mathrm{T} - \mathbf{V}) \, \mathbf{i}$

The result is a column vector with industries in rows.

The interpretation of the results depends upon the construction of the matrices.

In a PSUT description of conserved quantities (such as mass or energy), when all losses are accounted within the matrices themselves, the calculation of intra-industry balances should give the $\mathbf{0}$ vector. When losses are not accounted in the matrices, the calculation of intra-industry balances gives losses.

For PSUT matrices describing non-conserved quantities such as exergy (mass exergy, energy exergy, or both), when losses are accounted in the matrices, the intra-industry balance gives destruction of the non-conserved quantity. When losses are not accounted in the matrices, the intra-industry balance gives the sum of destruction and losses.

The following table summarizes the above points.

Matrices contain	Matrices include losses (wastes)	Matrices exclude losses (wastes)
Mass (MCC)	$\mathbf{0}$	$\mathbf{m}_L$
Energy (ECC)	$\mathbf{0}$	$\mathbf{e}_L$
Mass exergy (BCC)	$\mathbf{b}_D$	$\mathbf{b}_D + \mathbf{b}_L$
Energy exergy (XCC)	$\mathbf{x}_D$	$\mathbf{x}_D + \mathbf{x}_L$
Mass and energy exergy (BXCC)	$\mathbf{bx}_D$	$\mathbf{bx}_D + \mathbf{bx}_L$

MCC indicates a material conversion chain. ECC indicates an energy conversion chain. BCC indicates a material exergy conversion chain. XCC indicates an energy exergy conversion chain. BXCC indicates a combined material and energy exergy conversion chain. $\mathbf{0}$ indicates the result is the zero vector; $\mathbf{m}$ indicates the result is a mass vector; $\mathbf{e}$ indicates the result is an energy vector; $\mathbf{b}$ indicates the result is a mass exergy vector; $\mathbf{x}$ indicates the result is an energy exergy vector; and $\mathbf{bx}$ indicates the result is a the sum of mass and energy exergy vectors. Subscript $L$ indicates losses (or wastes), and subscript $D$ indicates exergy destroyed by the industry.

To compute intra-industry (across-industry) balances within the PSUT framework using the Recca package, use the following code.

intra_industry_balances <- ecc |> 
  dplyr::filter(LastStage %in% c("Final", "Useful")) |> 
  calc_intra_industry_balance()

The result is added as a new column, named “SUTIntraIndustryBalance” by default.

dplyr::glimpse(intra_industry_balances)
#> Rows: 2
#> Columns: 13
#> $ Country                 <chr> "GBR", "GBR"
#> $ Year                    <dbl> 2000, 2000
#> $ EnergyType              <chr> "E", "E"
#> $ LastStage               <chr> "Final", "Useful"
#> $ R                       <list> <<matrix[2 x 2]>>, <<matrix[2 x 2]>>
#> $ U                       <list> <<matrix[12 x 9]>>, <<matrix[13 x 13]>>
#> $ U_EIOU                  <list> <<matrix[7 x 8]>>, <<matrix[3 x 8]>>
#> $ U_feed                  <list> <<matrix[9 x 9]>>, <<matrix[12 x 13]>>
#> $ V                       <list> <<matrix[9 x 10]>>, <<matrix[13 x 14]>>
#> $ Y                       <list> <<matrix[4 x 2]>>, <<matrix[4 x 2]>>
#> $ r_EIOU                  <list> <<matrix[12 x 9]>>, <<matrix[13 x 13]>>
#> $ S_units                 <list> <<matrix[12 x 1]>>, <<matrix[16 x 1]>>
#> $ SUTIntraIndustryBalance <list> <<matrix[9 x 1]>>, <<matrix[13 x 1]>>

The vectors in the “SUTIntraIndustryBalance” column have industries in rows.

intra_industry_balances$SUTIntraIndustryBalance[[1]]
#>                   Product
#> Crude dist.           550
#> Diesel dist.          350
#> Elect. grid           125
#> Gas wells & proc.    2075
#> NG dist.               50
#> Oil fields           2575
#> Oil refineries       5075
#> Petrol dist.          750
#> Power plants         9700
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Product"
intra_industry_balances$SUTIntraIndustryBalance[[2]]
#>                      Product
#> Car engines       22999.6000
#> Crude dist.         545.0000
#> Diesel dist.        367.9998
#> Elect. grid         125.0000
#> Furnaces           5000.0000
#> Gas wells & proc.  2075.0000
#> Light fixtures     4800.0000
#> NG dist.             45.0000
#> Oil fields         2575.0000
#> Oil refineries     5075.0000
#> Petrol dist.        526.9997
#> Power plants       9700.0000
#> Truck engines     13250.0200
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Product"

Because the matrices in ecc do not contain losses, calc_intra_industry_balance() calculates energy losses (waste heat) ( $\mathbf{e}_L$ in the table above).

Endogenizing losses

The losses can be endogenized, i.e., added to the $\mathbf{RUVY}$ matrices of the PSUT framework. To do so, use the function endogenize_losses().

endogenized <- intra_industry_balances |> 
  endogenize_losses()

endogenize_losses() adds two new columns: by default V_prime and Y_prime. V_prime and Y_prime contain $\mathbf{V}$ and $\mathbf{Y}$ matrices with losses included directly in the matrices. The $\mathbf{V}$ matrices contain a new column (named “Waste heat” by default) that shows the heat losses from each industry.

endogenized$V_prime[[1]]
#>                   Crude [from Dist.] Crude [from Fields] Diesel
#> Crude dist.                    47500                   0      0
#> Diesel dist.                       0                   0      0
#> Elect. grid                        0                   0      0
#> Gas wells & proc.                  0                   0      0
#> NG dist.                           0                   0      0
#> Oil fields                         0               50000      0
#> Oil refineries                     0                   0  20500
#> Petrol dist.                       0                   0      0
#> Power plants                       0                   0      0
#>                   Diesel [from Dist.] Elect Elect [from Grid] NG [from Dist.]
#> Crude dist.                         0     0                 0               0
#> Diesel dist.                    15500     0                 0               0
#> Elect. grid                         0     0              6275               0
#> Gas wells & proc.                   0     0                 0               0
#> NG dist.                            0     0                 0           41000
#> Oil fields                          0     0                 0               0
#> Oil refineries                      0     0                 0               0
#> Petrol dist.                        0     0                 0               0
#> Power plants                        0  6400                 0               0
#>                   NG [from Wells] Petrol Petrol [from Dist.] Waste heat
#> Crude dist.                     0      0                   0        550
#> Diesel dist.                    0      0                   0        350
#> Elect. grid                     0      0                   0        125
#> Gas wells & proc.           43000      0                   0       2075
#> NG dist.                        0      0                   0         50
#> Oil fields                      0      0                   0       2575
#> Oil refineries                  0  26500                   0       5075
#> Petrol dist.                    0      0               26500        750
#> Power plants                    0      0                   0       9700
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Product"

The $\mathbf{Y}$ matrices also have a new column, by default named “Losses.”

endogenized$Y_prime[[1]]
#>                     Losses Residential Transport
#> Diesel [from Dist.]      0           0     14750
#> Elect [from Grid]        0        6000         0
#> NG [from Dist.]          0       25000         0
#> Petrol [from Dist.]      0           0     26000
#> Waste heat           21250           0         0
#> attr(,"rowtype")
#> [1] "Product"
#> attr(,"coltype")
#> [1] "Industry"

In addition, the $\mathbf{Y}$ matrices contain a new row (again named “Waste heat” by default) that shows the total heat loss in a new “Losses” column.

To accept the endogenized matrices, do the following:

endogenized |> 
  dplyr::mutate(
    V = V_prime,
    Y = Y_prime, 
    V_prime = NULL, 
    Y_prime = NULL, 
    SUTIntraIndustryBalance = NULL
  ) |> 
  dplyr::glimpse()
#> Rows: 2
#> Columns: 12
#> $ Country    <chr> "GBR", "GBR"
#> $ Year       <dbl> 2000, 2000
#> $ EnergyType <chr> "E", "E"
#> $ LastStage  <chr> "Final", "Useful"
#> $ R          <list> <<matrix[2 x 2]>>, <<matrix[2 x 2]>>
#> $ U          <list> <<matrix[12 x 9]>>, <<matrix[13 x 13]>>
#> $ U_EIOU     <list> <<matrix[7 x 8]>>, <<matrix[3 x 8]>>
#> $ U_feed     <list> <<matrix[9 x 9]>>, <<matrix[12 x 13]>>
#> $ V          <list> <<matrix[9 x 11]>>, <<matrix[13 x 15]>>
#> $ Y          <list> <<matrix[5 x 3]>>, <<matrix[5 x 3]>>
#> $ r_EIOU     <list> <<matrix[12 x 9]>>, <<matrix[13 x 13]>>
#> $ S_units    <list> <<matrix[12 x 1]>>, <<matrix[16 x 1]>>

The value added matrix ( $\mathbf{W}$ )

The value added matrix ( $\mathbf{W}$ ) obtained from calc_yqfgW() is similar to but different from the balances obtained from calc_inter_industry_balance() and calc_intra_industry_balance(). The following equations show the similarities and differences.

The $\mathbf{W}$ matrix is

$\mathbf{W} = \mathbf{V}^{\mathrm{T}} - \mathbf{U}$

Inter-industry balances are calculated by

$[(\mathbf{R} + \mathbf{V})^\mathrm{T} - (\mathbf{U} + \mathbf{Y})] \mathbf{i}$

and the intra-industry balances are calculated by

$(\mathbf{U}^\mathrm{T} - \mathbf{V}) \mathbf{i} \, .$

The $\mathbf{W}$ matrix is closest to the intra-industry balances, but with the opposite sense ( $\mathbf{W}$ has products in rows and the intra-industry balances have industries in rows), without the summation by the unit vector ( $\mathbf{i}$ ), and with the opposite sign. The following code shows the comparison between $\mathbf{W}$ and the intra-industry balances.

compare <- ecc |> 
  calc_yqfgW() |> 
  calc_intra_industry_balance()
compare$W[[1]] |> 
  # Calculate column sums and 
  # transpose to create a column vector 
  # for easier comparison
  matsbyname::colsums_byname() |> 
  matsbyname::transpose_byname()
#>                   Product
#> Crude dist.          -550
#> Diesel dist.         -350
#> Elect. grid          -125
#> Gas wells & proc.   -2075
#> NG dist.              -50
#> Oil fields          -2575
#> Oil refineries      -5075
#> Petrol dist.         -750
#> Power plants        -9700
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Product"
compare$SUTIntraIndustryBalance[[1]]
#>                   Product
#> Crude dist.           550
#> Diesel dist.          350
#> Elect. grid           125
#> Gas wells & proc.    2075
#> NG dist.               50
#> Oil fields           2575
#> Oil refineries       5075
#> Petrol dist.          750
#> Power plants         9700
#> attr(,"rowtype")
#> [1] "Industry"
#> attr(,"coltype")
#> [1] "Product"

Deprecated functions

Two related functions in the Recca package have been deprecated and will soon be removed.

`verify_SUT_energy_balance()`

verify_SUT_energy_balance() has been deprecated for two reasons.

verify_SUT_energy_balance() unfortunately combined calc_inter_industry_balance() and verify_inter_industry_balance() in a single step. We often need to know the balances independently from whether products are balanced. The combination of calc_inter_industry_balance() and verify_inter_industry_balance() provides the needed flexibility.
Using “energy” in the name of the function was an unfortunate choice, because we can use the same mathematics to balance masses or exergies. The new function names are independent of the type products in the $\mathbf{RUVY}$ matrices and are, therefore, an improvement.

For now, verify_SUT_energy_balance() remains in the Recca package but will likely be removed at a later date.

`verify_SUT_energy_balance_with_units()`

verify_SUT_energy_balance_with_units() seemed like a good idea at the time, but we have never used it. For now, verify_SUT_energy_balance_with_units() remains in the Recca package but will likely be removed at a later date.

Conclusion

There are several types of balances that are of interest to energy and material conversion chain analysts. The functions calc_inter_industry_balance(), verify_inter_industry_balance(), calc_intra_industry_balance(), verify_intra_industry_balance(), and endogenize_losses() assist with those analyses.

Balances

Introduction

Terminology

Data

Balances

Inter-industry (between-industry) balances check for product consistency

Intra-industry (across-industry) balances

Endogenizing losses

The value added matrix (𝐖\mathbf{W})

Deprecated functions

verify_SUT_energy_balance()

verify_SUT_energy_balance_with_units()

Conclusion

The value added matrix ( $\mathbf{W}$ )

`verify_SUT_energy_balance()`

`verify_SUT_energy_balance_with_units()`