Sensitivity analysis in life cycle assessment
Evelyne A. Groen1,*, Reinout Heijungs2,3, Eddie A.M. Bokkers1, Imke J.M. de Boer1
混凝土泵送剂>不良资产管理系统
1 Animal Production Systems group, Wageningen University
2 Institute of Environmental Sciences, Leiden University
3 Department of Econometrics and Operations Research, VU University Amsterdam
Corresponding author. E-mail: @wur.nl
ABSTRACT
Life cycle assessments require many input parameters and many of these parameters are uncertain; therefore, a sensitivity analysis is an essential part of the final interpretation.The aim of this study is to compare seven sensitivity methods applied to three types of case stud-ies. Two (hypothetical) case studies describing electricity production: one shows linear and another shows non-linear behavior. The third case study describes a large (existing) case study of seafood production containing high input uncertainties. The methods are compared based on their results, i.e. variance decomposition and ranki
ng of the input parameters. Results show that Sobol’ sensitivity indices per-form the best for all three case studies. The Sobol’ method can be a useful method in case of non-linear LCA models or LCA models that include outliers.
Keywords: matrix perturbation, method of elementary effect, key issue analysis, random balance design, Sobol’ sensitivity index
1. Introduction
A life cycle assessment (LCA) calculates the environmental impact of a product from cradle to grave. LCAs require many input parameters and many of these parameters are uncertain. A sensitivity analysis,therefore, is an essential part of the final interpretation. This is mentioned in the ISO standards for LCA, but no guidance is given on how to do or how to select an appropriate sensitivity analysis. A sensitive parameter is a parameter of which a change considerably influences the result, or that contributes to the variance of the output.A sensitivity analysis can help identifying parameters that should be known accurately before drawing conclusions, or identifying non-sensitive parameters for which the variance can be fixed in the region of its variance in order to simplify a model, also known as ‘factor-fixing’ (Saltelli et al. 2008).
Sensitivity analysis in LCA can be performed using a one-at-a-time approach (OAT), meaning that a subset of the input parameters are changed one at a time to see how much influence it has on the results. Although this approach has many advantages, e.g. it is easy to perform and to understand, this type of sensitivity analysis is time-consuming for a large system and might not consistently take all parameters into account and, therefore, could overlook possible sensitive parameters. Sensitivity analyses that consistently analyze the sensitivity of each parameter in the model are usually performed with sampling based approaches, such as Monte Carlo simulation, with an added procedure for variance decomposition.
In general we can differentiate between three types of sensitivity analysis: local sensitivity analysis (e.g. OAT); screening (e.g. method of elementary effect) and variance based sensitivity analysis or global sensitivity analysis (e.g. regression analysis). An overview is given in Table 1. The methods differ in their input requirements (e.g. knowledge about probability distribution function and parameter of dispersion) and type of output: either a ranking and/or variance decomposition of the input parameters.
Table 1. Types of sensitivity methods discussed in this paper.
Type Method
Local Matrix perturbation (MP); one-at-a-time approaches (OAT)
Screening Method of elementary effect (MEE)
Global Standardized regression coefficients (SRC); key issue analysis (KIA); random balance design (RBD);
Sobol’ indices (SME and STE)
For most of the sensitivity methods mentioned in Table 1, it is not yet known under which conditions they optimally perform, or if they can outperform the standard practices in LCA (i.e. OAT, MP, SRC, KIA). The aim of this study is to compare seven sensitivity methods applied to three types of case studies: one showing linear, another showing non-linear behavior and a large linear case study, but with large input uncertainties. The methods are compared based on their variance decomposition and ranking of the input parameters
2. Methods
2.1. Case studies
In this study we applied the sensitivity methods to two case studies of the production of 1 MWh electricity (the original version of the case studies appeared in (Heijungs 2002; Heijungs and Suh 2002). Both case studies con-sisted of two processes: fuel production and electricity production. The first case study produces electricity thereby using fuel (figure 1). In the second case study, electricity is in turn necessary for fuel production, creat-ing a loop (figure 2). The parameters of the second case study are known to be locally (very) non-linear (Heijungs 2002). For both case studies we assumed that each input parameter is normally distributed with a standard deviation equal to 10% of the mean. Furthermore we assumed that the parameters are uncorrelated and independent. For both case studies the parameters are numbered as follows: 1 (production of electricity); 2 (use of fuel for electricity production); 3 (use of electricity for fuel production, which is zero in the first case study); 4 (production of fuel); 5 (emissions of CO2 during electricity production); and 6 (emissions of CO2 during fuel production).
Figure 1. Case study 1: production of 1000 kWh electricity requires 200 liter fuel.
Figure 2. Case study 2: production of fuel requires electricity.
The third case study describes a trawler operating in the Northeast Atlantic, targeting mainly cod and haddock. The case study consists of 115 input parameters, production of vessel and gear, fuels, anti-fouling and cooling agents (figure 3). We assumed that the input parameters come with high uncertainty, each standard deviation of each parameter varies with 30% of the mean. All parameters are log-normally distributed.
The trawler takes trips of about ten to fourteen days, landing its catch in Tromso, Norway. The functional unit consists of 1 tonne landed whitefish.
2.2. Methods for sensitivity analysis
2.2.1. One-at-a-time
One-at-a-time (AOT) approaches take a subset of the input parameters, which are changed one at a time (ei-ther within its range or using an arbitrary value) to see how much influence it has on the result. The method is easy to perform and to understand, but this type of sensitivity analysis is time-consuming for a large system and, therefore, might not consistently take all parameters into account and could overlook possible sensitive parame-ters.
2.2.2. Matrix perturbation
Matrix perturbation (MP) as a method of (local) sensitivity analysis was introduced in LCA by Heijungs and Suh (2002). MP makes use of the first order partial derivatives as estimators of local sensitivity, which can be converted into relative multipliers. If the multiplier is larger, a change in the input parameter will change the re-sult more than when the multiplier is almost zero. Information suc
h as the type of distribution function or param-eter of dispersion is not used. The result of this method shows how much the results will change if the input pa-rameters are slightly changed (perturbed). This means that the multipliers predict the magnitude and direction of the change in the result of a small change around the original value of each parameter. A disadvantage of apply-ing MP is that the method considers the model in its current configuration, and therefore the result (in general) only holds for small changes around the original parameter values. Moreover, information on which input pa-rameter is quite certain and which is not (i.e. of the ranges of the parameters), is not used.
2.2.
3. Method of elementary effects
The method of elementary effects (MEE) is a screening method that was originally designed by Morris (1991) and adjusted by Campolongo et al. (2007). MEE has been applied in LCA by Mutel et al. (2013) and de Koning et al. (2010). In order to apply MEE the ranges of the individual parameters are taken into account, where a range is defined as the upper and lower boundary of an input parameter. MEE can be seen as an extend-ed one-at-a-time approach (Saltelli et al. 2008). MEE co
mbines alternative values of each parameter (at pre-defined proportional steps in the range defined) and calculates the result. The difference between the original model and the new result of each combination is the elementary effect.Another indicator that can be calculated is the standard deviation of the elementary effect, which is an indicator for the interaction or non-linear effects: if the elementary effect of a certain parameter changes considerably for each run, the magnitude of the elementary effect depends on either the configuration of the model or the presence of nonlinear effects. MEE can be used as a precursor to the more computationally demanding sampling methods as regression. A disadvantage of this method is that the results are not an estimation of the actual variance decomposition.
2.2.4. Key issue analysis (Taylor expansion)
Key issue analysis (KIA) was introduced in LCA by Heijungs (2002) as a method for (analytically) determin-ing the contribution to variance (or variance decomposition) by means of a first order Taylor expansion. KIA has been applied in LCA for example by Heijungs and Huijbregts (2004) and Heijungs et al. (2005). It combines the steepness of a function (described in section 2.2.1 above) with the variance of the individual parameters. KIA calculates the variance decomposition up to first order (or “main effects”), as covariance between input parame-ters are mostly unknown (Heijungs, 2
002). To apply this method only the variances of the individual parameters are used. A disadvantage of KIA is that it does not produce a distribution function of the output, making it more difficult to compare two or more studies.
2.2.5. Standardized regression coefficients (using Monte Carlo sampling)
Standardized regression coefficients (SRC) are obtained from the slope of the line from least square fitting and estimate the contribution to output variance for each input parameter. Pseudo-random samples are taken from all input parameters and for each run the output is calculated. Subsequently, for each input parameter the regression coefficient is calculated; the coefficients are standardized with respect to their standard deviation. An advantage of calculating SRC is that it is commonly applied (in and outside) LCA, a disadvantage is that many runs are needed to calculate the variance decomposition.
2.2.6. Random balance design
The foundation of random balance designs (RBD) are from Cukier et al. (1978). In this study we use the for-mat similar to Tarantola et al. (2006). RBD has been not yet been applied in LCA to our knowledge, although a very closely related method Fourier amplitude sensitivity test, has been appli
ed by de Koning et al. (2010). Ran-dom balance designs estimate the contribution to variance by using Fourier transformations. A periodic sampling is applied and for each input parameter the Fourier spectrum is calculated, which is an estimate for the first order sensitivity index. A disadvantage of RBD is that only the main effect can be calculated.
2.2.7. Sobol’ sensitivity index
The method by Sobol’ (2001) assigns a sensitivity measure to each input parameter by calculating how much of the output variance can be allocated to each input parameter. The idea of the method is that a model can be decomposed into terms of increasing order, where the first order terms, also called the Sobol’ main effects (SME), are equal to the contribution of variance caused by each input parameter to the output variance (Saltelli et al. 2010). The method also allows calculation of the interaction effects (variance caused by varying two or more parameters simultaneously) and the total effect index. The Sobol’ total effect index (STE) gives the vari-ance caused by the sum of the main and interaction effects of an input parameter. A disadvantage of Sobol’s method is that many runs are needed to calculate the indices; hence the model is computationally expensive. 3. Results
3.1. Results of local methods
First the results of the local methods are presented, because it is not accurate to compare local with global methods, as they do not display similar information. Global sensitivity methods (or variance-based methods) include uncertainty information such as the variance into their results, while local methods estimate the change in the outcome based on the configuration of the (LCA) model at hand. The ranking of parameters by applying the local methods can be found in Table 2.
Table 2. Ranking of the parameters of the local methods for case study 1, 2 and 3, e.g. parameter 1 is most sensitive for case study 1 and 2. OAT: one-at-a-time approach; MP: matrix perturbation. The meaning of the parameters of case study 1 and 2 can be found in figure 2. FE: fuel use; EP: emission factor of fuel production;
For case study 1 both methods give similar results. For case study 2 the results differed slightly, although the actual values for parameter 1 to 4 were very close for both methods. Case study 3 gives also similar results.
3.2. Global methods: case study 1 electricity production
Applying the global methods to the first case study showed that the ranking of the input parameters were (almost) similar for each method (Table 3). The variance decomposition is given between brackets, but cannot be calculated for MEE as this method only gives a ranking of parameters. For example, in case of applying KIA parameter 1 is responsible for 56% of the output variance.
Table 3. Ranking of the parameters for case study 1, e.g. parameter 1 is most sensitive. MEE: method of elementary effect; KIA: key issue analysis; SRC: standardized regression coefficient; RBD: random balance design; SME: Sobol’ main effect index; ST E: Sobol’ total effect index.
Rank MEE KIA SRC RBD SME STE
1 1 1 (56%) 1 (57%) 1 (58%) 1 (53%) 1 (58%)
很黄很的动态图580期2 5 5 (39%) 5 (37%) 5 (40%) 5 (33%) 5 (38%)
3 2 2, 4, 6 (1.6%) 2 (1.7%) 6 (2.5%)
4 (3.8%) 2 (1.7%)
4 6 6 (1.6%) 2 (2.4%) 2 (3.1%) 6 (1.6%)
5 4 4 (1.5%) 4 (2.1%)
6 (2.4%) 4 (1.6%)
KIA required only a single calculation, which means (in case of LCA) that these methods are computationally very fast. MEE required relatively few runs compared to the other sampling based methods. SRC, RBD and the Sobol’ indices made use of sampling, SRC and the Sobol’ indices requiring the largest number of runs. Although the individual contributions differed between methods, the overall picture is the same.
3.3. Global methods: case study 2 electricity production
Applying the methods to the second case study, we found that SRC, RBD and SME did not give reasonable results (Table 4), all parameters showed a contribution of about 0%. Although the ranking of the parameters is somewhat different, it should be noted that the difference in sensitivity between parameter 1, 2, 3 and 4 were very small in case of MEE and KIA (Table 4). Only the Sobol’ total index (STE) explicitly indicated parameter 3 as begin responsible for most of the output variance. Th
e variance decomposition given by KIA and STE are given in Table 4 (SRC, RBD and SME are not shown as they did not gave reasonable results). KIA shows the main effects, indicating that parameter 1-4 are responsible for 25% of the output variance, the STE shows that, taking all interaction into account between, e.g., parameter 3 and the other parameters in the model, that parameter 3 is responsible for almost 93% of the output variance.
Table 4. Ranking of the parameters for case study 2. MEE: method of elementary effect; KIA: key issue analysis; SRC: standardized regression coefficient; RBD: random balance design; SME: Sobol’ main effect index; STE: Sobol’ total effect index.
Rank MEE KIA SRC RBD SME STE
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1 3 1 (25%) - - - 3 (93%)
2 4 2, 3, 4 (25%) 2 (50%)
3 2 5 (0%)
4 (50%)
4 1 6 (0%) 1 (50%)
5 5 5 (0%)
6 6 6 (0%)
3.4. Global methods: case study 3 production of seafood
csilvApplying the global methods to the third case study showed that the ranking of the input parameters were (almost) similar for each method (Table 5), just as for case study 1. The five most sensitive parameters are shown, that contribute more than 1% to the output variance. The variance decomposition is given between brackets.
