Download Technical guide - Logarithmic Mean Divisia Index

Energy Analysis using Divisia Decomposition
Dr Jonathan Lermit
Methodology for Energy Efficiency Monitoring
This example considers the division of the energy consumption into structural and
intensity components. The change in energy use (usually from year to year) is split
into four components:



The Activity Effect measures changes in energy due to changes in activity of a
sector.
The Structural Effect which measures changes in energy resulting from
changes in relative activity between industries or transport modes.
The Fuel Switching Effect which measures changes in energy resulting from
the relative changes in fuel types. This accounts for changes in what fuel is
consumed in a sub-sector, taking into account the thermodynamic properties
of different energy sources. For example switching from a high quality fuel
(e.g. electricity) to a low quality fuel (e.g. coal) would result in more energy
being used overall to achieve the same outcome.. This is necessary as fuel use
is recorded independent of energy quality.
The remaining factor, the Energy Efficiency Effect measures the changes in energy
use after all the other explanatory effects have been taken into account.
All these effects can be separated out for each fuel type, thus giving additional insight
to changes in fuel use. Note that, when aggregated, the market share effects sum to
zero.
This analysis considers energy divided into three sectors:



Business, consisting of primary industries, manufacturing industries,
commercial and public services.
Transport, which is subdivided into Passenger and Freight
Residential.
The Business Sector
The economy is divided into N sub-sectors, which are taken as homogeneous. The
economic output and energy use are measured for each. These typically change over
time, as the economy develops, and processes change. The aim of this decomposition
technique is to separate out the effects of structural change from those of energy
intensity in individual sub-sectors.
E : Total Energy (  Ei )
i
Ei : Energy used in the i th sub-sector (= Eij )
i
th
Eij : Energy of the j fuel type used in the i th sub-sector
Ei* : The effective use of all fuels in the i th sub-sector (  Eij* )
j
*
ij
th
th
E : The effective use of the j fuel in the i sub-sector
 i : The quality ratio of fuel used in the i th sub-sector (= Ei / Ei* )
Y : Total economic output (GDP)
Yi : Economic output from the i th sub-sector
I : Energy:GDP ratio for the whole economy ( E / Y )
I i : Energy intensity of the i th sub-sector ( Ei / Yi )
I ij : Energy intensity of the i th fuel type in the i th sub-sector (= Eij /Yi )
I i* : Effective energy intensity of the i th sub-sector ( Ei* / Yi )
sij :Share of the i th fuel type of energy use in the i th sub-sector (= Eij /E i )
yi : Production share of the i th sub-sector ( Yi / Y )
wi : Energy share of the i th sub-sector ( Ei / E)
wij : Energy share of the j th fuel type in the i th sub-sector (= Eij /E)
All variables are continuous, but are typically observed only at certain points of time,
usually annually. Let t=0 and t=T be two such times.
The Divisia Index Approach
Energy use can be broken down by industry and by fuel type:
E   Ei
i

i
Ei Yi
Y
Yi Y
Ei Ei* Yi
 *
Y
i E Y Y
i
i
   i I i* yiY
i
We wish to calculate the changes in energy use due to changes in economic structure
(changes in yi ), and that due to changes in technical intensity ( I i* ). Following Ang
and Choi (1), we first consider the rate of change of (the logarithm of) intensity. The
reason for choosing the logarithm is that d ln E  dE / E , so that the derivative
represents the percentage change in the growth rate of intensity.
Taking logarithms and differentiating over time, it follows that:


 d ln yi d ln i d ln I i* dlnY 
d ln E


  wi 


i
dt 
dt
dt
dt
dt









 structure fuel switching efficiency activity 
where
wi 
Ei
, the energy share of the i th sub-sector.
E
Integrating this equation over the interval 0 to T yields:
T
d ln yi
ln(ET / E0 )   i  wi
dt
0
dt 


structural change
T
d ln I *i
  i  wi
dt
0
dt



fuel switching
T
d ln  i
  i  wi
dt
0
dt 


efficiency change


T
d lnY
   i wi
dt
0
dt


activity change
The change in energy use can therefore be calculated from:
where
is the Logarithmic Mean:
This change in energy can be split into its components as:
where:
These integrals can be evaluated approximately in terms of the end points to give the
following estimated values for the energy components:
where the weights {wi* } are (essentially) average values of the proportion of energy
used in each sector.
The two important points to note in this method are:


The use of logarithms ensures that the components are additive
The choice of weights ensures that the components sum exactly to the
total change, without any residual error.
The structural change measures changes in energy resulting from changes in
economic structure, for example an energy intensive industry, such as transport, may
increase as a proportion of total economic activity.
The fuel switching change reflects the relative changes in fuel types.
The activity change encompasses economic growth.
Derivation of the Weighting Function
Since the integrals used cannot be evaluated because only the values at the end points
are known, an estimation using only known data (usually annual) is required.
If wi were constant over time,
and similarly for
t = T. We therefore approximate
These involve only the end values at t = 0 and at
by:
where the wi* are currently undefined, but
w
i
*
i
 1. Then
These two must be identically equal; let  be the difference between them:
*
   k,T yk,T I k,T
k
  ln(Et / E0 )  ln(Yt / Y0 )  ln 
*
  k  k,0 yk,0 I k,0



Since yi  Yi / Y and I i  Ei /Yi , evaluated at either t  0 or t  T :
w
i
*
i
ln( i yi I i* )
 Ei Yi Ei* 
  w ln  *

i
 Ei Y Yi 
E
  wi* ln i
i
Y
*
  wi ln(wi,T / wi,0 )
*
i
i
Evaluating this expression at t  0 and t  T , and noting that the lnY cancel, the
value of  becomes:
   ln
i
 Ek,T
Ei,T
  wi* ln k
i
Ei,0
 Ek,0
k


Ei,T
Ei,0


  w ln
 ln
i
  Ek,T Ek,T
E
E
k k ,0 k,0 
k
*
i
  wi* ln(wi,T / wi,0 )
i
Defining the weights  wi*  by
ensures that
w
i
*
i
 1. The error term thus becomes:
Thus no remainder term is generated, regardless of the changes in economic structure
or technical changes within each sector.
The Transport Sector
Transport is a large consumer of energy; it is divided here into Passenger and Freight.
Each has a different ‘activity’ measure, Passenger-kilometres and Tonne-kilometres
respectively.
Passenger Transport
The energy use is divided into four sub-sectors:




Cars (light private vehicles)
Buses
Passenger rail
Domestic air
(Data for shipping, basically the Cook Strait ferries, are not available and fuel use is
included in freight).
Note that only internal travel is included, and not international travel. Since most
transport is oil based no fuel switching component is included.
Notation
E : Total Energy (  Ei )
i
Ei : Energy used in the i th sub-sector
Y : Total output (Passenger-km)
Yi : Passenger-km in the i th sub-sector
I : Energy:Pass.-km ratio for all passenger transport ( E / Y )
I i : Energy intensity of the i th sub-sector ( Ei / Yi )
yi : Transport share of the i th sub-sector ( Yi / Y )
wi : Energy share of the i th sub-sector ( Ei / E)
Proceeding as before, total change in energy use can be broken down into:
The  wi*  are defined as before.
Freight Transport
The energy use is divided into three sub-sectors:



Road (trucks)
Freight rail
Coastal shipping
Activity is measured in tonne-kilometres. As with passenger transport, only internal
transport is considered. The equations are similar to passenger transport.
Residential
Activity in this sector is simply population. Fuel switching is handled as in the
Business sector, and efficiency is measured as before. Because of the homogeneous
nature of households, no structural component is involved.
In this sector, in addition to population, size of households (the average number of
people living in each household), and the physical area of the house also have a
significant effect on energy consumption.
Notation
E : Total Energy
E j : Energy of the j th fuel type
E * : The effective use of all fuels (  E *j )
j
*
j
th
E : The effective use of the j fuel
P : Total population
H : Number of households
A : Total floor area
 : The quality ratio of fuel used (= E / E * )
 : The efficiency effect or effective energy per unit area ( E * / A)
 : The average household area ( A / H )
 : The average household occupancy ( H / P)
Energy use can then be decomposed as:
E E* A H
E  *    P
A H P
E
     P
As before, a change in energy use between two points in time t  0 and t  T can be
decomposed into:
where:
Reference 1. Ang, B.W. and Ki-Hong Choi Decomposition of Aggregate Energy and
Gas Emission Intensities for Industry: A Refined Divisia Index Method The Energy
Journal Vol. 18, No. 3.