Stochastic and epistemic uncertainty propagation in LCA

Abstract

Purpose

When performing uncertainty propagation, most LCA practitioners choose to represent uncertainties by single probability distributions and to propagate them using stochastic methods. However, the selection of single probability distributions appears often arbitrary when faced with scarce information or expert judgement (epistemic uncertainty). The possibility theory has been developed over the last decades to address this problem. The objective of this study is to present a methodology that combines probability and possibility theories to represent stochastic and epistemic uncertainties in a consistent manner and apply it to LCA. A case study is used to show the uncertainty propagation performed with the proposed method and compare it to propagation performed using probability and possibility theories alone.

Methods

Basic knowledge on the probability theory is first recalled, followed by a detailed description of epistemic uncertainty representation using fuzzy intervals. The propagation methods used are the Monte Carlo analysis for probability distribution and an optimisation on alpha-cuts for fuzzy intervals. The proposed method (noted as Independent Random Set, IRS) generalizes the process of random sampling to probability distributions as well as fuzzy intervals, thus making the simultaneous use of both representations possible.

Results and discussion

The results highlight the fundamental difference between the probabilistic and possibilistic representations: while the Monte Carlo analysis generates a single probability distribution, the IRS method yields a family of probability distributions bounded by an upper and a lower distribution. The distance between these two bounds is the consequence of the incomplete character of information pertaining to certain parameters. In a real situation, an excessive distance between these two bounds might motivate the decision-maker to increase the information base regarding certain critical parameters, in order to reduce the uncertainty. Such a decision could not ensue from a purely probabilistic calculation based on subjective (postulated) distributions (despite lack of information), because there is no way of distinguishing, in the variability of the calculated result, what comes from true randomness and what comes from incomplete information.

Conclusions

The method presented offers the advantage of putting the focus on the information rather than deciding a priori of how to represent it. If the information is rich, then a purely statistical representation mode is adequate, but if the information is scarce, then it may be better conveyed by possibility distributions.

Appendix: Calculation of the global warming (GW) impact

Cultivation and harvest of willow (life cycle of 21 years)

$$ \mathrm{C}{{\mathrm{O}}_{{2\_\mathrm{in}}}}=\left( {{C_{{\mathrm{em}\_\mathrm{y}2}}}-13\times {{\mathrm{C}}_{{\mathrm{in}\_\mathrm{cultivation}}}}-5\times {{\mathrm{C}}_{{\mathrm{in}\_\mathrm{harvest}}}}} \right)/21\times 44/12 $$

(4)

$$ {{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{em}}}=\left( {{{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{d}}}+{{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{i}}}} \right)\times 44/28\times {{\mathrm{N}}_2}\mathrm{OCF}/1,000 $$

(5)

Where:

CO_{2_in} :

Yearly (average) CO₂ emissions from cultivation of willow Mg CO₂ ha⁻¹ year⁻¹

C_{in_cultivation} :

Yearly net uptake of carbon during the 13 cultivation years Mg C ha⁻¹ year⁻¹

C_{in_harvest} :

Yearly net uptake of carbon during the 5 harvest years (this parameter being strongly correlated to C_{in_cultivation} it is later replaced by C_{in_cultivation} - 0.78) Mg C ha⁻¹ year⁻¹

C_{em_yr2} :

Emissions of carbon during year 2 (assumed equal to 5.32) Mg C ha⁻¹ year⁻¹

N₂O_em :

Yearly emissions of N₂O from cultivation of willow Mg CO₂-eq ha⁻¹ year⁻¹

N₂O_d :

Yearly direct emissions of N₂O from cultivation of willow Mg N ha⁻¹ year⁻¹

N₂O_i :

Yearly indirect emissions of N₂O from cultivation of willow Mg N ha⁻¹ year⁻¹

N₂OCF:

Characterisation factor of N₂O for GW kg CO₂-eq/kg N₂O

Cultivation and harvest of barley

$$ \mathrm{C}{{\mathrm{O}}_{{2\_\mathrm{b}}}}=-{{\mathrm{C}}_{{\mathrm{in}\_\mathrm{b}}}}\times 44/12+{Y_{\mathrm{b}}}\times {{\mathrm{C}}_{\mathrm{b}}}\times 44/12 $$

(6)

$$ {{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{b}}}=\left( {{{\mathrm{N}}_2}{{\mathrm{O}}_{{\mathrm{d}\_\mathrm{b}}}}+{{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{i}}}_{{\_\mathrm{b}}}} \right)\times 44/28\times {{\mathrm{N}}_2}\mathrm{OCF}/1,000 $$

(7)

Where:

CO_{2_b} :

Yearly CO₂ emissions from cultivation and harvest of barley Mg CO₂ ha⁻¹ year⁻¹

C_{in_b} :

Yearly net uptake of carbon during cultivation and harvest of barley Mg C ha⁻¹ year⁻¹

Y _b :

Yield of cultivation of barley (at the field gate) Mg DM ha⁻¹ year⁻¹

C_b :

Carbon content of barley %DM

N₂O_b :

Yearly emissions of N₂O from cultivation and harvest of barley Mg CO₂-eq ha⁻¹ year⁻¹

N₂O_{d_b} :

Yearly direct emissions of N₂O from cultivation and harvest of barley Mg N ha⁻¹ year⁻¹

N₂O_{i_b} :

Yearly indirect emissions of N₂O from cultivation and harvest of barley Mg N ha⁻¹ year⁻¹

N₂OCF:

Characterisation factor of N₂O for GW kg CO₂-eq/kg N₂O

Co-firing

$$ \mathrm{CF}=\mathrm{Yield}\times {C_{\mathrm{w}}}\times 44/12 $$

(8)

Where:

CF:

Yearly CO₂ emissions from co-firing of willow Mg CO₂ ha⁻¹ year⁻¹

Yield:

Yield of cultivation of willow (at the field gate) Mg DM ha⁻¹ year⁻¹

C_w :

Carbon content of willow %DM

Avoided energy production

$$ \begin{array}{*{20}c} \mathrm{EP}=\mathrm{Yield}\times (\mathrm{LHV}\times \left( {1-\mathrm{Loss}} \right)-\mathrm{watercontent}/\left( {1-\mathrm{watercontent}} \right) \hfill \\ \times {{\mathrm{H}}_2}\mathrm{Oheating})\times \left( {\mathrm{elec}\_\mathrm{rec}/3.6\times \mathrm{GH}{{\mathrm{G}}_{\mathrm{elec}}}+\mathrm{heat}\_\mathrm{rec}\times \mathrm{GH}{{\mathrm{G}}_{\mathrm{heat}}}} \right) \hfill \\\end{array} $$

(9)

Where:

EP:

Yearly avoided GHG emission from energy production Mg CO₂-eq ha⁻¹ year⁻¹

Yield:

Yield of cultivation of willow (at the field gate) Mg DM ha⁻¹ year⁻¹

LHV:

Lower heating value of willow as dry matter GJ Mg⁻¹ DM

Loss:

Loss of carbon during drying and storage of willow %

Watercontent:

Water content of willow after field drying %

H₂Oheating:

Energy needed for water content evaporation GJ Mg⁻¹

elec_rec:

Electricity recovery from LHV %

heat_rec:

Heat recovery from LHV %

GHG_elec :

GHG emissions from electricity production in DK Mg CO₂-eq MWh⁻¹

GHG_heat :

GHG emissions from heat production in DK Mg CO₂-eq GJ⁻¹

Total net impact

$$ \mathrm{TNI}=\mathrm{iLUC}+20\times \left( {\mathrm{C}{{\mathrm{O}}_{{2\_\mathrm{in}}}}+{{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{em}}}-\left( {\mathrm{C}{{\mathrm{O}}_{{2\_\mathrm{b}}}}+{{\mathrm{N}}_2}{{\mathrm{O}}_{\mathrm{b}}}} \right)+\mathrm{CF}-\mathrm{EP}} \right) $$

(7)

Where:

TNI:

Total net impact on GW over 20 years Mg CO₂-eq ha⁻¹ year⁻¹

ILUC:

Indirect land use change Mg CO₂-eq ha⁻¹ year⁻¹

CO_{2_in} :

Yearly CO₂ savings from cultivation and harvest of willow (average) Mg CO₂ ha⁻¹ year⁻¹

N₂O_em :

Yearly emissions of N₂O for willow cultivation Mg CO₂-eq ha⁻¹ year⁻¹

CO_{2_b} :

Yearly CO₂ savings from cultivation and harvest of barley Mg CO₂ ha⁻¹ year⁻¹

N₂O_b :

Yearly emissions of N₂O for barley cultivation Mg CO₂-eq ha⁻¹ year⁻¹

CF:

Yearly CO₂ emissions from co-firing of willow Mg CO₂ ha⁻¹ year⁻¹

EP:

Yearly avoided GHG emission from energy production Mg CO₂-eq ha⁻¹ year⁻¹