Second-Order Analytical Uncertainty Analysis in Life Cycle

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Second-Order Analytical Uncertainty Analysis in Life Cycle Assessment Sarah von Pfingsten, David Oliver Broll, Niklas von der Assen, and André Bardow Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.7b01406 • Publication Date (Web): 27 Oct 2017 Downloaded from http://pubs.acs.org on October 29, 2017

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Environmental Science & Technology

Second-Order Analytical Uncertainty Analysis in Life Cycle Assessment Sarah von Pfingsten, David Oliver Broll, Niklas von der Assen+ , and André Bardow*

1

Institute of Technical Thermodynamics, RWTH Aachen University, Schinkelstr. 8, 52062 Aachen, Germany E-mail: [email protected] Phone: +49 (0)241 80 953 80. Fax: +49 (0)241 80 922 55

2

3

*corresponding author +

Current address: Bayer AG, Leverkusen, Germany

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Keywords: uncertainty propagation, Taylor series, variance, matrix-based LCA, Monte

6

Carlo sampling, sensitivity analysis

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Abstract

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Life cycle assessment (LCA) results are inevitably subject to uncertainties. Since

10

the complete elimination of uncertainties is impossible, LCA results should be com-

11

plemented by an uncertainty analysis. However, the approaches currently used for un-

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certainty analysis have some shortcomings: statistical uncertainty analysis via Monte

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Carlo simulations are inherently uncertain due to their statistical nature and can be-

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come computationally inefficient for large systems; analytical approaches use a linear

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approximation to the uncertainty by a first-order Taylor series expansion and thus,

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they are only precise for small input uncertainties. In this article, we refine the analyt-

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ical uncertainty analysis by a more precise, second-order Taylor series expansion. The

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presented approach considers uncertainties from process data, allocation, and charac-

19

terization factors. We illustrate the refined approach for hydrogen production from

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methane-cracking. The production system contains a recycling loop leading to non-

21

linearities. By varying the strength of the loop, we analyze the precision of the first-

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and second-order analytical uncertainty approaches by comparing analytical variances

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to variances from statistical Monte Carlo simulations. For the case without loops, the

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second-order approach is practically exact. In all cases, the second-order Taylor series

25

approach is more precise than the first-order approach, in particular for large uncer-

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tainties and for production systems with nonlinearities, e.g. from loops. For analytical

27

uncertainty analysis, we recommend using the second-order approach since it is more

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precise and still computationally cheap.

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Introduction

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Life cycle assessment (LCA) is a method to evaluate potential environmental impacts of

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processes or products along their entire life cycle. 1 However, the reliability of LCA results is

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typically challenged by inherent uncertainties. Uncertainty sources in LCA can be classified

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in: parameter uncertainty, model uncertainty, and uncertainty due to choices. 2,3 Since the

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entire elimination of the uncertainties is impossible, it is desirable to perform an uncertainty

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analysis to enable a trustworthy interpretation of LCA results. Many LCA studies address

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solely parameter uncertainty using probability distributions. 2,4,5 Model uncertainty and un-

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certainty due to choices (also called scenario uncertainty) are assessed either by scenarios

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including discrete choices 4,6,7 or by adding probability distributions to the scenarios 5 in a

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sampling-based framework.

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Reviews for uncertainty analysis approaches in LCA 8,9 point out that, while no standardized


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method exists, two main approaches can be distinguished: stochastical Monte Carlo sam-

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pling and analytical uncertainty propagation. Several authors applied Monte Carlo sampling

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to LCA, e.g. 10–14 However, as Monte Carlo sampling uses repeated calculation steps, it can

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be time-consuming. 15 A computationally efficient way for uncertainty analysis, and thus a

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promising alternative to stochastic Monte Carlo sampling, is analytical uncertainty prop-

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agation. Still, analytical uncertainty analysis has been less often applied to LCA studies.

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Heijungs and Lenzen 15 , Hong et al. 16 , Imbeault-Tétreault et al. 17 employ analytical uncer-

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tainty analysis based on first-order Taylor series for matrix-based LCA. Bisinella et al. 18 also

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apply first-order Taylor series expansion but avoid the matrix structure. Furthermore, ana-

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lytical uncertainty analysis has been applied for a sequential LCA approach. 19 The existing

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Taylor series approach for matrix-based LCA includes only first-order estimates and is thus

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only precise for small input uncertainties. 20 To improve the application range of analytical

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uncertainty analysis, improved methods are regarded as important. 21

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In this work, we improve analytical uncertainty analysis by second-order Taylor series expan-

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sion to provide a better uncertainty estimate. We develop an approach using matrix-based

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LCA and apply the approach to a novel methane-cracking process for hydrogen production

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as case study. In the case study, uncertainty from process data, allocation choices, and

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characterization factors are included. The presented approach uses probability distributions

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to describe the uncertainty of process data, characterization, and allocation factors. We

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show that the second-order analytical uncertainty analysis always results in higher precision

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than the first-order Taylor series. By means of our case study, we show that the second-

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order approach is especially beneficial for systems with large input uncertainties and strong

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nonlinearities.



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Methods

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In our work, we build on the computational structure of LCA introduced by Heijungs and

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Suh 22 and analytical error propagation formulae from our previous work. 23 The existing

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first-order Taylor series expansion is used as a basis for the development of the second-order

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approach.

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Analytical uncertainty analysis for matrix-based LCA

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For matrix-based LCA, Heijungs and Suh 22 proposed the following formulation:

h = Q · B · A−1 · f.

(1)

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The final demand vector f defines the functional unit of the LCA study. The technology

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matrix A contains data for the economic flows between processes within the technosphere,

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whereas the intervention matrix B contains all elementary flows between processes and

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the environment. Last, the characterization factors in matrix Q convert elementary flows

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into environmental impacts. The product Q · B is denoted as combined intervention and

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characterization matrix. The results of the life cycle impact assessment are given by the

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impact vector h. Based on this matrix structure of LCA, Heijungs 20 developed a first-order

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Taylor series expansion to analyze the uncertainty of the LCA results. The resulting Eq. (2)

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gives the variance of the environmental impacts:

X ∂h 2 var(Xi ), var(h) ≈ ∂X i i

(2)

where Xi ∈ {A, B, Q, C|A ∈ Rm×n , B ∈ Rz×n , Q ∈ Ru×z , C ∈ R(w+β)×r }.

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The first-order matrix derivatives, or sensitivity coefficients, 24,25

∂h ∂Xi

are the sensitivities

of the result h towards a change in a certain input parameter Xi . The variance var(Xi ) 4 ACS Paragon Plus Environment

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quantifies how large the uncertainties in Xi are. For computation, the summation runs over

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all elements of each matrix Xi . Equation (2) considers the uncertainty of the process data

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(A, B) and of the characterization factors (Q). In expansion of Heijungs’ original work, 20

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we here include also uncertainty due to the treatment of multi-functionality as proposed

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by Jung et al. 23 . Therefore, an allocation matrix C appears which contains the allocation

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factors and the matrices A and B are modified, e.g. allocated (see supporting information

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for details). The symbols m, n, z, u, w, r, and β specify the dimensions of the matrices A,

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B, Q, and C which are also explained in the supporting information.

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A global sensitivity analysis can be performed for calculating the contribution to variance. 26

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To identify the most influential parameters, the single summands of Eq. (2) are divided

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by the total variance var(h). Correlations between uncertainties are not considered. Groen

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and Heijungs 27 recently studied the effect of correlated uncertainties and illustrated that

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analytical uncertainty analysis is able to quantify the risk of ignoring uncertainty correlations.

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Overall, the first-order Taylor series corresponds to a linear approximation. However, a

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linear approximation is only valid for small uncertainties and higher order terms should

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be included for larger uncertainties. 20 In general, higher order estimates, i.e. a quadratic,

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second-order approach, provide a better approximation of high variances caused by large

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input uncertainties and nonlinearities. In Eq. (1), the computation of the inverse matrix is

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always nonlinear. Furthermore, loops (e.g. from recirculation or recycling of products) add

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additional nonlinearities to LCA systems which call for higher-order estimates.

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Second-order analytical uncertainty analysis

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To improve analytical uncertainty analysis, we expand the existing first-order analytical un-

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certainty analysis by a second-order Taylor series. The second-order Taylor series expansion



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valid for all uncertainty distributions is given by: 2 X ∂h 2 X ∂h ∂ 2 h 1 X ∂ 2h µ + [µ4,Xi − 3var(Xi )var(Xi )] var(h) ≈ var(Xi ) + 2 3,Xi 2 ∂X ∂X ∂X 4 ∂X i i i i i i i (3) 2 1 XX ∂ 2h + var(Xi )var(Xj ), 2 i j ∂Xi ∂Xj where Xi,j ∈ {A, B, Q, C|A ∈ Rm×n , B ∈ Rz×n , Q ∈ Ru×z , C ∈ R(w+β)×r }.

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In Eq. (3), again variances of every single matrix element and first-order matrix deriva-

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tives are required. Furthermore, the third and fourth moments µ3,Xi and µ4,Xi are needed,

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respectively. Additionally, second-order derivatives have to be calculated, including deriva-

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tives with respect to different matrix elements in the term

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and second-order matrix derivatives are given in the supporting information. Some second-

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order derivatives, i.e.

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Q, respectively. Thus, the first-order expansion is already exact for uncertainties in B and Q.

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Analogous to the first-order Taylor series expansion, the contribution to variance can be cal-

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culated to determine the most influential parameters. For normally distributed, uncorrelated

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parameters, the third moment is µ3,Xi = 0 and the fourth moment is µ4,Xi = 3var(Xi )2, 28

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so that Eq. (3) simplifies to:

∂2h ∂bvj ∂bvj

and

∂2h , ∂qkv ∂qkv

∂2h . ∂Xi ∂Xj

The required first-order

are zero, as Eq. (1) depends only linearly on B and

2 X ∂h 2 1 XX ∂ 2h var(h) ≈ var(Xi ) + var(Xi )var(Xj ), ∂X 2 ∂X ∂X i i j i i j

(4)

where Xi ∈ {A, B, Q, C|A ∈ Rm×n , B ∈ Rz×n , Q ∈ Ru×z , C ∈ R(w+β)×r }.

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More details on the derivation of the second-order approach used for error propagation

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can be found in Anderson and Mattson 28 . Though Eqs. (3) and (4) may look complicated,

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the computational effort for solving the equations is low, especially because the inverse of

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matrix A is only calculated once and already available from the LCA result itself.


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Benchmarking the precision of analytical uncertainty analysis

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To evaluate the precision of the developed second-order Taylor series expansion, we com-

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pare the analytical results to Monte Carlo sampling. The Monte Carlo sampling technique

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calculates the impact vector h repeatedly, choosing random input parameter values from

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a given uncertainty distributions for each run. 29 The higher the number of Monte Carlo

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runs, the higher is the precision of the overall result. However, a general consensus on the

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required number of Monte Carlo runs is lacking: While, the computational effort increases

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linearly with an increasing number of simulation runs, an insufficient number of simulation

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runs could lead to wrong results and conclusions. Not only has the size of the matrices,

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but also the level of uncertainty and the distribution of the uncertain parameters influence

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on the required number of simulation runs. To the knowledge of the authors, no study has

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been published that answers the question on how many simulations runs are sufficient for a

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Monte Carlo analysis in the LCA context. Lloyd and Ries 2 performed a survey of 12 LCA

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studies and found the number of Monte Carlo runs ranging from 100 to 30,000. According

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to Heijungs and Lenzen 15 , a typical number for Monte Carlo studies is 10,000 runs. Wei

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et al. 30 calculate the decision confidence probability for comparative LCA using a first-order

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Taylor series approximation. They show that even 1,000,000 Monte Carlo runs are required

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to reach the same precision as their proposed method. Thus, to be on the safe side, we

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choose 1,000,000 repeated Monte Carlo runs to calculate a reliable estimate of the true value

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of total uncertainty in our case study. Still, the comparison of computational demands is

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very case-dependent and therefore not in the focus of our work.

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To evaluate the precision for the first- and second-order formulae, we use the relative devi-

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ation d of analytically (AN) and stochastically (MC) calculated variances var as precision

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measure: d=

varM C − varAN . varM C


(5)


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Case study for hydrogen production by methane-cracking

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We apply the second-order analytical uncertainty analysis (Eq. (4)) to a LCA case study.

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The case study considers the decomposition of methane into hydrogen and carbon (Fig. 1).

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The so-called methane-cracking process uses a novel liquid-metal technology to produce hy-

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drogen without direct CO2 -emissions. 31,32 Plant manufacturing requires copper, palladium,

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quartz, silicon carbide, steel, and tin. Process operation needs the feedstock methane and

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energy in the form of electricity and heat. The required heat is obtained from combustion

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of a natural gas-hydrogen mix. In the natural gas-hydrogen mix, the hydrogen share xH2 is

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varied between 0 and 100%. We assume that the required hydrogen is provided from the

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process output leading to partial recirculation of the produced hydrogen. Four environmen-

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tal impact categories are assessed using the life cycle impact assessment method ReCiPe

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Midpoint (H): global warming impact (GWI), fossil depletion impact (FDI), metal depletion

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impact (MDI), and particulate matter formation impact (PMFI). natural gas supply

combustion natural gashydrogen mix

plant manufacturing

variable

methane-cracking process

H

1

C

6

Hydrogen

Carbon

methane supply

electricity supply

Figure 1: Flow chart of methane-cracking process with reference flow hydrogen. Heat is supplied by combustion of natural gas-hydrogen mix with variable hydrogen share (0 to 100%). The recycled hydrogen causes a loop in technology matrix A.

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We choose the functional unit as production of 1 kg hydrogen. Since the process provides 8 ACS Paragon Plus Environment

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the two products hydrogen and carbon, the environmental impacts of the process have to be

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allocated between hydrogen and carbon to obtain hydrogen-specific results. For this alloca-

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tion, we consider the criteria mass fraction, mole fraction, and higher heating value (HHV)

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fraction (Table 1). The choice of an allocation criterion is thus uncertain. We quantify this

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uncertainty through the uncertain allocation factors in the allocation matrix C. Although

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the choice of an allocation criterion and thus the resulting uncertainty distribution are dis-

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crete, we assume that the allocation factors follow a normal distribution with a coefficient

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of variation equal to the discrete counterpart. The coefficient of variation (CV, Eq. (6)) of

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the continuous distribution is calculated from the chosen allocation criteria (Table 1). The

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average of the allocation factors gives the mean value and the standard deviation is estimated

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via the biased sample variance (Eq. (7)) leading to a value of 35%. Table 1: Allocation factors for allocation matrix C Allocation criterion

ccarbon

chydrogen

Mass fraction cm

0.749

0.251

Molar fraction cn

0.333

0.667

HHV fraction ch

0.449

0.551

mean

0.510

0.490

CV [%]

34.3%

35.7%

σ mean

(6)

1 [(cm − mean)2 + (cn − mean)2 + (ch − mean)2 ] 3

(7)

CV =

r σ= 170

Furthermore, the LCA results are subject to uncertain process data and characterization

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factors. The uncertainties are represented in the technology matrix A and the combined 9 ACS Paragon Plus Environment


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characterization and intervention matrix QB. The uncertainty for process inputs in matrix

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A is calculated from available process data (Table 2) describing the expected lifetime (tmean )

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of the components with minimum (tmin ) and maximum (tmax ) values. We calculate the

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minimum and maximum CVtmin and CVtmax assuming a normal distribution. For normal

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distributions, approximately 99.73% of all values lie in the interval +/ − 3σ from the mean

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value so that CVtmin and CVtmax can be calculated by:

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CVtmin =

tmean − tmin 3 · tmean

(8)

CVtmax =

tmax − tmean . 3 · tmean

(9)

and

Table 2: Lifetimes (t) for plant components as source for uncertainty with minimum, maximum and mean values and calculated coefficients of variation (CV ) for minimum and maximum lifetimes component

tmean (years) tmin (years) tmax (years) CVtmin

CVtmax

membrane

30

15

45

17%

17%

heat exchanger

3

2

5

11%

22%

quartz tube

200

100

365

17%

28%

tin

36

20

50

15%

13%

filter

150

100

200

11%

11%

burner

12

9

20

8%

22%

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Table 2 gives CV values between 8 and 28% resulting from the uncertain lifetime of

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the plant components. We round the calculated values to the safe side and assume nor-

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mally distributed process parameters with CV varying in strength from 10 to 35%. Similar

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uncertainty values are assumed for the methane and energy demands. For the combined in-

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tervention and characterization matrix QB, we also assume normally distributed parameters 10 ACS Paragon Plus Environment

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with the same ranges for CV as for matrix A. All values are assumed to be uncorrelated, in

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both analytical and Monte Carlo calculations.

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For a hydrogen share in the natural gas-hydrogen mix greater than zero, part of the pro-

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duced hydrogen is recirculated so that the technology matrix A has a loop. The loop causes

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additional nonlinearities for calculating the LCA results. By varying the hydrogen share,

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we can adjust the strength of the nonlinearity which allows us to systematically test the

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second-order analysis.

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Results

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We apply the developed second-order Taylor series expansion (Eq. (4)) to the case study on

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methane-cracking to analyze the uncertainty of the LCA results. In the case study, we vary

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the hydrogen share in the combustion mix to adjust the strength of nonlinearities from the

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loop caused by the process-internal hydrogen-use. We study how the second-order Taylor

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series expansion can approach these nonlinearities in comparison to the first order. The

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precision of each calculated variance is evaluated using the relative deviation to Monte Carlo

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sampling results (Eq. (5)). In the following, we distinguish between the methane-cracking

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process without any loop for hydrogen, i.e. the share of hydrogen in the natural gas-hydrogen

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mix is xH2 = 0, and the process with loop for a hydrogen share between 0 and 1.

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Process without loop (hydrogen share xH2 = 0)

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For a hydrogen share of xH2 = 0 in the combustion mix, the required process heat is solely

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provided by natural gas combustion. Thus, no produced hydrogen is recirculated and no

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loop occurs in the technology matrix. Figure 2 shows the relative deviations between ana-

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lytically and stochastically calculated variances for the impact category global warming and

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input uncertainties between CV =10% and CV =35%. The relative deviations between the

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first-order Taylor series expansion and the stochastically calculated results are smaller than



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7%. If the second-order formula (Eq. (4)) is applied, the deviations are reduced to 0.3%

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or less. Thus, if no loop occurs, the second-order analytical uncertainty analysis estimates

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the variance almost perfectly at very low computational cost. Furthermore, the results show

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that even for the process without loop the uncertainty propagates nonlinearly due to the

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inversion of the technology matrix, but can be well approximated by a quadratic expansion.

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Due to the excellent results for the second-order approach, we increased the input uncer-

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tainties even further up to CV =50% and CV =75% (cf. Fig. 2). Even for very large input

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uncertainties of CV =75%, the relative deviations could be significantly reduced from 23.8%

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to 1.8% through application of the second-order approach.

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Results for fossil depletion impact, particulate matter formation impact, and metal depletion

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impact are similar to the global warming impact and are given in the supporting informa-

219

tion (Fig. S1). Additionally performed calculations for the case study assuming uniformly

220

and lognormally distributed input parameters delivered very similar results as the normally

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distributed input parameters (with a maximum difference of 0.4%).

Figure 2: Relative deviations d between analytically calculated variances (1st and 2ndorder Taylor series expansion) and stochastically calculated variances (Monte Carlo simulation with 1,000,000 steps) for global warming impact (GWI) with coefficient of variation CV ={10;15;20;35;50;75%}. The hydrogen share in the natural gas-hydrogen mix is xH2 = 0. The bars for 2nd order deviations are very small.


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Process with hydrogen share xH2 varying between 0 and 1.

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For a hydrogen share greater than 0, part of the required process energy is provided through

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process-internal hydrogen use. We consider varying hydrogen shares in the natural gas-

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hydrogen mix between 0 and 1, where 1 corresponds to sole hydrogen combustion. Figure 3

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gives the uncertainty analysis results depending on the hydrogen share for the impact cate-

227

gory global warming. The results are shown for CV =10% and CV =35%. For small input

228

uncertainties (CV =10%), the analytical uncertainty estimates the variance already very

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accurately when applying the first-order formula (max(d)=2.8%). However, larger input un-

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certainties (CV =35%) and increasing hydrogen shares lead to higher deviations d between

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first-order analytical and stochastic results of up to 15%. The second-order Taylor series

232

expansion can reduce these deviations in our case study by up to 95% (CV =35%, xH2 =0)

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compared to the first-order approach. For systems with small loops up to xH2 =0.4, the

234

deviation between second-order estimate and stochastic variances is less than 2%. Even for

235

systems with larger loops, the second-order approximation reduces the relative deviations

236

compared to first order by at least 44% (CV =10%, xH2 =1). Thus, the case study confirms

237

the theoretical expectation that the first-order approach underestimates total uncertainty

238

whereas the second-order approximation is more precise, even though still not perfect for

239

very large uncertainties and nonlinearities. Uniformly distributed input parameters lead to

240

similar results for the case study as the normally distributed input parameters (with a maxi-

241

mum difference of 0.8%). However, for lognormally distributed parameters, the Monte Carlo

242

analysis did not converge despite the 1 000 000 runs performed due to occurring singularities.

243

These singularities are probably due to the fact that the lognormal distribution has a higher

244

probability for very large deviations for which the matrix A becomes non-invertible. A non-

245

invertible matrix A is due to unphysical parameter combinations but can result from the

246

assumed uncertainty distributions. Whereas these singularities are noticed during stochas-

247

tical simulations, the analytical approach does not register these low probability events and

248

calculates the resulting uncertainty nonetheless. Thus, the analytical results could not be 13 ACS Paragon Plus Environment


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benchmarked against the Monte Carlo simulation for lognormally distributed parameters for

250

the process with loop. d 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

hydrogen share in natural gas-hydrogen mix CV=10% 1st order

CV=10% 2nd order

CV=35% 1st order

CV=35% 2nd order

Figure 3: Relative deviations d between analytically calculated variances (1st and 2nd-order Taylor series expansion) and stochastically calculated variances (Monte Carlo simulation with 1,000,000 steps) for global warming impact (GWI) with CV =10% and 35%. The hydrogen share in the natural gas-hydrogen mix is varied between xH2 = 0 and 1.

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Discussion

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The presented approach for second-order analytical uncertainty analysis is based on Taylor

253

series expansion and includes uncertainty from process data, allocation and characterization

254

factors. We study the propagation of the uncertainties from process data and characteriza-

255

tion in our case study assuming continuous uncertainty distributions for all variables. The

256

approach calculates the central moments to model the uncertainty distribution. Thus, also

257

discrete and binary uncertainties, such as allocation factors, are approximated via the central

258

moments. As the analytical uncertainty approach incorporates the calculation of sensitivity 14 ACS Paragon Plus Environment

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coefficients, 24,25 it enables the efficient determination of the most influential parameters via

260

contribution analysis. However, the analytical approach provides fewer details than regres-

261

sion or variance-based techniques for sampling-based global sensitivity analysis. 33

262

We apply the second-order approach to a methane-cracking process with a process-internal

263

hydrogen-use that causes a loop and thus nonlinearities. The case study results show that

264

the second-order uncertainty analysis is more precise than the first-order estimate. Whereas

265

the first-order approach is able to achieve similar precision as Monte Carlo sampling with

266

1,000,000 steps only for systems with small loops and small input uncertainties (CV =10%),

267

the developed second-order approach gives similar precision as Monte Carlo sampling even for

268

large input uncertainties (CV =75%) and strong nonlinearities. For the case without loops,

269

the second-order approach is practically exact. However, the larger deviations observed for

270

stronger loops in the technology matrix A indicate that the uncertainty is propagated non-

271

linearly with an even higher order than quadratic. Although the computational effort per

272

calculation increases for the second-order approach to a few seconds calculation time com-

273

pared to first-order approach with less than 1 second, it is still low compared to Monte Carlo

274

sampling with already one minute for the small system studied. The computational effort

275

for Monte Carlo sampling as well as the possible computational advantage of the presented

276

analytical method depend on the number of required simulation runs and the size of the

277

studied system. For larger product systems, higher computational effort for Monte Carlo

278

sampling is expected so that the reduced computational effort of analytical uncertainty anal-

279

ysis becomes even more important. In future work, the second-order analytical uncertainty

280

approach could be applied to a larger system, such as the ecoinvent database, to analyze the

281

possible computational savings. Furthermore, the improved precision of the second-order

282

approach could be compared to the effect of including correlated uncertainties. 27 The data

283

requirements for using the second-order approximation are the same as for the first order.

284

Thus, if the second-order approach is implemented in LCA software, the application for the

285

LCA practitioner is identical for both analytical approaches. Thus, we encourage the use of



286

the second-order Taylor series expansion for systems with loops and nonlinearities as well as

287

for large input uncertainties.

288

Acknowledgement

289

The authors acknowledge financial support by the German Federal Ministry of Education

290

and Research (ref. no.: 033R144B).

291

Supporting Information Available

292

The supporting information (SI) includes: Description of matrix-based LCA for multi-

293

functional systems, Second-order Taylor series expansion formula for matrix-based LCA for

294

normally distributed uncertain parameters, first-order and second-order matrix derivatives,

295

matrices for the case study, and case study results for additional impact categories.

296

References

297

298

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Graphical TOC Entry results uncertainty

uncertainty error 1st order

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2nd order

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1st order -95% 2nd order LCA input data


Second-Order Analytical Uncertainty Analysis in Life Cycle

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