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# Download Technical guide - Logarithmic Mean Divisia Index

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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.