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Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. WP1 Shared modelling framework and learnings D1.2 – Description of scientific methods Task 1.3 Methods for Life Cycle Impact Assessment Appendix – Mathematical model for calculating AGTP functions (Tier3) Lead Contractor: INSAT Author(s): Damien Arbault (INSAT) PROJECTS DETAILS Project title Aligning Life Cycle Assessment methods and bio-based sectors for improved environmental performance. Project acronym ALIGNED Start / Duration 01/10/2022 – 36 months Type of Action RIA Website www.alignedproject.eu This appendix details the mathematical model used to estimate AGTP functions. It is a simplified version from CCI Tool (Climate Change Impact tool for dynamic LCA). The complete version of CCI Tool is freely available online or for download at https://www.insa-toulouse.fr/cci-tool/. The model was simplified and adapted to the ALIGNED context. Expert practitioners accustomed with Tier 3 results and willing to fine-tune CC impacts are invited to use the CCI Tool. As compared to the original model, simplifications include: • Only CO2, CH4 and N2O gases are included, whereas CCI Tool includes other GHGs and short-lived climate forcers. • Fossil and non-fossil CH4 are not differentiated: the contribution to methane-emission metrics from CO2 from fossil methane oxidation is not accounted. • The AGTP function of a gas is computed independently from the other climate forcers, whereas CCI tool includes the dynamic interactions between all climate forcers. Calculations and computed values can be found in ‘AGTP functions.xlsx’ file. Results were compared to IPCC values found in AR6 Table 7.SM.6 : relative differences amounted between 8% and 16% for CO2 and N2O, 1% for CH4 AGTP50 and 33% for CH4 AGTP100. Therefore, the simplified model may lack robustness to estimate long-term impacts in case of significant methane emissions.
Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Convolution operator As an example, let’s consider an exemplary cause-effect chain, e.g. emission A involves effect B. If effect B is defined over time it means that a ‘pulse’ A at t=T0 would lead to an effect B(t) between T0 and a given time horizon TH. In impact assessment, the overall impact of pulse A would be defined as the cumulated effect of B over time. However, when A is not a pulse emission but a time-distributed emission, a specific operator is necessary to consider the cumulative effect of all emissions between T0 and a TH. In this case, the convolution product is used. It is noted with an * operator between function. For continuous functions f and g defined between 0 and H: ∀ 𝑡∈[0:𝐻],(𝑓∗𝑔)(𝑡)= ∫𝑓(𝑡′)∙𝑔(𝑡−𝑡′)𝑑𝑡′ 𝑡 0 (1) The convolution product is commutative, associative, distributive and supports scalar multiplication. Therefore, its manipulation is similar to the multiplication of scalar numbers, but adapted for 1-dimension functions such as temporal distributions. In numerical application, discrete convolution of f and g - defined as a suite of values between year 0 and year n – is given by: (𝑓∗𝑔)(𝑛)= ∑ 𝑓(𝑘)∙𝑔(𝑛−𝑘) 𝑛 𝑘=0 (2) The latter formula is used for numerical application of climate cause-effect chains, coupling emission profiles, GHG lifetimes, temporal influence of radiative forcing on temperature change, and carbon cycle feedback loops -as details in the coming sub-sections. AGTP The Absolute Global Temperature change Potential (AGTP) of substance X is a function over time labelled AGTPX(t). It is the sum of temperature change from direct radiating forcing of X over time (AGTPdX) and from the additional radiative forcing triggered by a change in CO2 concentration (AGTPccX) due to the change in temperature AGTPdX: 𝐴𝐺𝑇𝑃𝑋(𝑡)= 𝐴𝐺𝑇𝑃𝑑𝑋(𝑡)+ 𝐴𝐺𝑇𝑃𝑐𝑐𝑋(𝑡) (3) These two terms are further described in the sections below.
Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Direct AGTP AGTPdX(t) is defined as the convolution product between direct radiative forcing of X (RFdX) between 0 and t and the Impulse-temperature response function IRFT between 0 and t: 𝐴𝐺𝑇𝑃𝑑𝑋(𝑡)=(𝑅𝐹𝑑𝑋∗𝐼𝑅𝐹𝑇)(𝑡)= ∫𝑅𝐹𝑑𝑋(𝑡′)∙𝐼𝑅𝐹𝑇(𝑡−𝑡′)𝑑𝑡′ 𝑡 0(4) IRFT is not substance-dependent. It is calculated as: 𝐼𝑅𝐹𝑇(𝑡)=𝑐1 𝑑1𝑒−𝑡 𝑑1+𝑐2 𝑑2𝑒−𝑡 𝑑2(5) Parameters for IRFT function are retrieved from AR6 section 6.SM.2 and given in Table 1: Table 1: parameters for Impulse-Response Temperature Function Parameter value unit c1 0.44 °C/W/m2 d1 3.4 yrs c2 0.32 °C/W/m2 d2 285 yrs Direct radiative forcing of substance X over time is defined as the convolution product of three functions: radiative efficiency AX(t), in W/m2/kg; impulse-response function of substance X IRFX(t); and the temporal distribution of substance X flows gX(t): 𝑅𝐹𝑑𝑋(𝑡)=(𝐴𝑋∗𝐼𝑅𝐹𝑋∗𝑔𝑋)(𝑡) (6) For small emission changes, the radiative efficiency of GHGs is considered as constant (AR6 chapter 7.6.1.2): ∀𝑡′∈[0:𝑡],𝐴𝑋(𝑡′)=𝐴𝑋(7) gX (t) is the time-distributed flow of substance X. In the present calculation, gX (t) is shaped to represent a pulse function at year 0: 𝑔𝑋(𝑡)= {1 𝑓𝑜𝑟 𝑡=0 0 𝑓𝑜𝑟 𝑡>0 (8) Using equations (6), (7) and (8), RFdX(t) can be simplified as: 𝑅𝐹𝑑𝑋(𝑡)=𝐴𝑋 ∙ 𝐼𝑅𝐹𝑋(𝑡) (9)
Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. AX are given in IPCC report as expressed in W/m2/ppbv (AR6 Table 7.SM.7). They are converted in W/m2/kg as follows: 𝐴𝑋𝑘𝑔(𝑊 𝑚2 ⁄𝑘𝑔 ⁄)= 𝑀𝑎𝑡𝑚 (𝑔 𝑚𝑜𝑙 ⁄) 𝑀𝑋 (𝑔𝑚𝑜𝑙 ⁄) ∗ 1𝐸9 (𝑝𝑝𝑏 𝑘𝑔 ⁄) 𝑇𝑚 (𝑘𝑔) ∗ 𝐴𝑋𝑝𝑝𝑏(𝑊 𝑚2 ⁄𝑝𝑝𝑏 ⁄) (10) Where Matm, MX, Tm are the molar mass of air, the molar mass of substance X, and total mass of the atmosphere, respectively. Values for equation (10) are given in Table 2: Table 2: parameters related to radiative efficiency of CO2, CH4 and N2O Parameter unit value Matm 28.96 g/mol Tm 5.15E+18 kg Gas Molar mass (g/mol) A (W/m2/ppb) A (W/m2/kg) CO2 44.0087 0.0000133 1.699E-15 CH4 16.04206 0.000388 1.185E-13 N2O 44.0124 0.0032 4.089E-13 The Impulse-response function of substance X (IRFX) expresses the evolution of residual X in the atmosphere over time, following a pulse emission at year 0. It is a specific function for CO2, CH4 and N2O: 𝐼𝑅𝐹𝐶𝑂2(𝑡)=𝑎0+𝑎1∙𝑒−𝑡 𝜏1+𝑎2∙𝑒−𝑡 𝜏2+𝑎3∙𝑒−𝑡 𝜏3(11) 𝐼𝑅𝐹𝐶𝐻4(𝑡)=𝑎𝐶𝐻4∙𝑒−𝑡 𝜏𝐶𝐻4= (1+𝑓1+𝑓2 )∙𝑒−𝑡 𝜏𝐶𝐻4(12) 𝐼𝑅𝐹𝑁2𝑂(𝑡)=𝑎𝑁2𝑂∙𝑒−𝑡 𝜏𝑁2𝑂= (1−0.36 ∙ (1+𝑓1+𝑓2 )∙𝐴𝐶𝐻4 𝐴𝑁2𝑂)∙𝑒−𝑡 𝜏𝑁2𝑂(13) Parameters for IRF_CO2 were left unchanged between AR5 and AR6. They can be found in AR5 Table 8.SM.10. Parameters for equations (12) and (13) originate from AR5 (8.SM.11.3), except for CH4 and N2O lifetimes, updated in AR6 Table 7.SM.7. Values for IRF functions used here are displayed in Table 3 and Table 4: Table 3: parameters of IRF function for CO2 a0 a1 a2 a3 0.2173 0.224 0.2824 0.2763 1 (yrs) 2 (yrs) 3 (yrs) 394.4 36.54 4.304
Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Table 4: parameters of IRF function for CH4 and N2O Parameter value unit ACH4 (ppbv) 0.000388 W/m2/ppbv AN2O (ppbv) 0.0032 W/m2/ppbv f1 0.5 - f2 0.15 - CH4 11.8 yrs N2O 109 yrs
Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Indirect AGTP induced by the carbon cycle response AGTPccX(t) is the temperature change induced by the “carbon cycle response” to a temperature change: an increase in temperature leads to an increase in CO2 concentration, which in turns leads to an additional temperature change. AGTPccX (t) is defined as the convolution product of the indirect radiative forcing due to the carbon cycle response to an emission of X (RFccX), with IRFT function (eq. 5): 𝐴𝐺𝑇𝑃𝑐𝑐𝑋(𝑡)=(𝑅𝐹𝑐𝑐𝑋∗𝐼𝑅𝐹𝑇)(𝑡) (14) RFccX(t) is the indirect radiative forcing induced by the carbon cycle response after a temperature change pulse. It is calculated using CO2 radiative efficiency ACO2 and the convolution product between AGTPdX, the impulse response function of CO2 flux perturbation (RF(t)) and IRFCO2(t) (defined above): 𝑅𝐹𝑐𝑐𝑋(𝑡)=(𝐴𝐺𝑇𝑃𝑑𝑋∗𝛾𝑅𝐹∗𝐴𝐺𝑇𝑃𝑑𝐶𝑂2)(𝑡) (15) When modelling a pulse emission profile (eq 8), the reader can verify than equations (14) and (15) are equivalent to IPCC equation 7.SM.5.5: 𝐴𝐺𝑇𝑃𝑐𝑐𝑋(𝑡)=(𝐴𝐺𝑇𝑃𝑑𝑋∗𝛾𝑅𝐹∗𝐴𝐺𝑇𝑃𝑑𝐶𝑂2)(𝑡) (16) RF(t) describes the change in CO2 concentration in response to a unit temperature pulse: 𝛾𝑅𝐹(𝑡)= 𝛾(𝛿(𝑡)−𝛼1 𝜏𝑐𝑐1∙𝑒−𝑡 𝜏𝑐𝑐1−𝛼2 𝜏𝑐𝑐2∙𝑒−𝑡 𝜏𝑐𝑐2+𝛼3 𝜏𝑐𝑐3∙𝑒−𝑡 𝜏𝑐𝑐3) (17) Parameters for RF are found in AR6 Table 7.SM.6 and listed in Table 5: Table 5: parameters of the carbon cycle response function Parameter value unit 1.106E+13 kgCO2/yr/°C (t) 1 for t=0 0 for t>0 - 1 0.638 - 2 0.3322 - 3 0.031 - cc1 2.376 yrs cc2 30.14 yrs cc3 490.1 yrs
Horizon Europe grant agreement N° 101059430. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Summary of Parameters Parameter value unit ref used in equations c1 0.44 °C/W/m2 IPCC AR6 (6.SM.2) (5) d1 3.4 yrs c2 0.32 °C/W/m2 d2 285 yrs Matm 28.96 g/mol - (8) Tm 5.15E+18 kg MCO2 44.0087 g/mol MCH4 16.04206 g/mol MN2O 44.0124 g/mol ACO2 (ppbv) 0.0000133 W/m2/ppbv IPCC AR6 Table 7.SM.7 (8) ACH4 (ppbv) 0.000388 W/m2/ppbv (8), (11) AN2O (ppbv) 0.0032 W/m2/ppbv (8), (11) ACO2 (kg) 1.69943E-15 W/m2/kg equation (8) (13) ACH4 (kg) 1.18481E-13 W/m2/kg AN2O (kg) 4.08852E-13 W/m2/kg a0 0.2173 - IPCC AR5 Table 8.SM.10 (9) a1 0.224 - a2 0.2824 - a3 0.2763 - 1 394.4 yrs 2 36.54 yrs 3 4.304 yrs f1 0.5 - IPCC AR5 (8.SM.11.3) (10), (11) f2 0.15 - CH4 11.8 yrs IPCC AR6 Table 7.SM.7 (10) N2O 109 yrs (11) 1.106E+13 kgCO2/yr/°C IPCC AR6 Table 7.SM.6 (17) (t) 1 for t=0 0 for t>0 1 0.638 - 2 0.3322 - 3 0.031 - cc1 2.376 yrs cc2 30.14 yrs cc3 490.1 yrs
