Environment and Computer Laboratory I
MATLAB: Life Cycle Assessment and Uncertainty Modelling
1. Mathematical basics of LCA
The strength of LCA is that it does not only consider direct emissions induced by a final
demand of a product, but also up- and downstream emissions from the supply and
disposal chains of that product. Figure 1 shows the simplified supply chain of diesel. If we
want to know the total amount of diesel that is needed to produce 1 kg of diesel for final
consumption, we will have to translate the graph structure of Figure 1 into the so-called
technology matrix A.
a1,1
A=( ⋮
ai,1
ai,j:
⋯ a1,j
⋱
⋮ )
⋯ ai,j
(1)
output of the ith product during the jth process (inputs are negative outputs)
Figure 1. Simplified supply chain of diesel production.
Diesel is needed for the extraction of ore, extracted ore is needed for steel production, and
steel is needed for the production of diesel. Thus, we have loops in the supply chain of
diesel. Equation 2 describes a solution to solve these loops and to determine the
amount of diesel that is needed to produce 1 kg of diesel.
s  A 1f
s:
f:
A:
(2)
scaling vector
final demand vector
technology matrix
If we have a final demand of 1 kg diesel, then the scaling vector s will tell us how much of
each process output we will use in total (direct and in the supply chain) to produce 1 kg
of diesel. The life cycle emissions are calculated in a further step by multiplying s with the
so-called intervention or biosphere matrix B. B holds all direct emissions and resource
consumption a process generates.
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Environment and Computer Laboratory I
g  Bs
g:
B:
inventory results
intervention matrix
b1,1
B=( ⋮
bm,1
bm,j:
(3)
⋯
⋱
⋯
b1,j
⋮ )
bm,j
(4)
mth direct emission of the jth process
Inventory vector g holds the results for the total life cycle emissions for the demand of 1 kg
diesel (life cycle inventory results). These emissions can be characterized, as it is done for
instance for the global warming potential, by multiplying g with the characterization matrix
Q:
h  Qg
(5)
h:
Q:
characterization result
characterization matrix
2. Calculating LCA with Matlab
In this exercise we will calculate the total global warming potential (GWP) induced by the
final demand of 1 kg of diesel with the help of Matlab. Download ‘matlab_data.zip’ from the
course homepage and save it to the desktop and change the current folder path in
Matlab accordingly. The zip-file contains ‘data.mat’, which stores the intervention and
characterization matrices B and Q. The technology matrix A is given below.
Table 1. Technology matrix A.
Diesel [kg]
Crude oil [kg]
Steel [kg]
Electricity [kWh]
Ore [kg]
Coal [kg]
Diesel
[kg]
Crude oil
[kg]
Steel
[kg]
Electricity
[kWh]
Ore
[kg]
Coal
[kg]
1
-1.07
-0.01
-0.05
0
0
-0.01
1
-0.001
0
0
0
0
0
1
-0.17
-2.4
-0.7
0
-0.17
-0.001
1
0
0
-0.05
0
-0.01
0
1
0
-0.05
0
-0.01
0
0
1
To construct the 6-by-6 matrix A in Matlab, begin with a matrix that contains only zeros.
>> A=zeros(6,6);
Enter the non-zero values of A into the matrix. Always close your Matlab statement with a
semicolon. Otherwise the result of your command will be displayed. Note that the first
number in brackets marks the row and the second the column.
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Environment and Computer Laboratory I
>> A(2,1)= -1.07;
>> A(3,1)= -0.01;
etc.
After entering all non-zero values write a Matlab script that calculates the scaling vector
s, the inventory results g and the characterization result h. Go to ‘File/New/Script’ and
begin your script with:
function [s, g, h]=lca(f, A, B, Q)
Enter all needed equations (Hint: you will find them all described in section 1), close your
function with end and save.
Hint: The inversion of matrix A can be solved either by inv(A) (inverse) or A\f (Gaussian
elimination).
Before you can run the script you need to define a final demand vector f. You can define f
similar to A.
>> f=zeros(6,1);
As you want to calculate the environmental impacts of the demand of 1 kg of diesel, you
have to enter:
>> f(1,1)=1;
Now you can run your script with:
>> [s, g, h]=lca(f, A, B, Q);
By double clicking you can check out s, g, and h in the variable editor of Matlab. What do
you see? How much is needed to produce 1 kg of diesel.
3. Uncertainty modeling
There are different ways to calculate uncertainties. We will use Monte Carlo simulation.
Thereby, in a first step, we generate a number of samples of the technology matrix A and
the intervention matrix B, and then do LCA calculations for each sample. The sample size
has thus to be large enough to represent the uncertainties accurately, but also small
enough to reduce computational time. To be able to sample A and B, we need their
distribution types and the respective uncertainty parameters. For the sake of simplicity
we will assume that all non-zero values of A and B are lognormally distributed. The mean of
A and B are given by the previous exercise. The squares of the geometric standard deviation
are given below.
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Environment and Computer Laboratory I
Table 2. Squares of the geometric standard deviation (SD95) of technology matrix A (Asd95).
Diesel [kg]
Crude oil [kg]
Steel [kg]
Electricity [kWh]
Ore [kg]
Coal [kg]
Diesel
[kg]
Crude oil
[kg]
Steel
[kg]
Electricity
[kWh]
Ore
[kg]
Coal
[kg]
0
1.05
3
1.05
0
0
1.05
0
3
0
0
0
0
0
0
1.05
1.05
1.05
0
1.05
3
0
0
0
1.05
0
3
0
0
0
1.05
0
3
0
0
0
Table 3. Squares of the geometric standard deviation (SD95) of intervention matrix B (Bsd95).
CO2 [kg]
CH4 [kg]
N2O [kg]
Diesel
[kg]
Crude oil
[kg]
Steel
[kg]
Electricity
[kWh]
Ore
[kg]
Coal
[kg]
1.05
1.5
1.5
1.05
1.5
1.5
1.05
1.5
1.5
1.05
1.5
1.5
1.05
1.5
1.5
1.05
1.5
1.5
Translate Table 3 and Table 4 into Matlab matrices Asd95 and Bsd95. Copy ‘lhs.m’,
‘um.m’ and ‘ctv.m’ from ‘matlab_data.zip’ to your Matlab folder. Perform now the
sampling with:
>> [Asample, Bsample]=lhs(A, Asd95, B, Bsd95, nsample);
We use a method called Latin Hypercube to sample A and B. For the sample size
‘nsample’ you can choose any integer value bigger than 0. After sampling A and B we can
now do the LCA calculations for each sample.
>> [ssample, gsample, hsample]=um(f, Asample, Bsample, Q);
As a result you will get the scaling vectors, inventory results and characterization results for
each sample. The histogram shows you the distribution of the characterization results. Try
to obtain the 5%, 50% and 95% percentiles by entering:
>> P=prctile(hsample,[5,50,95])
What do these percentiles look like for the amount of crude oil that is used in the life
cycle?
Hint: ssample(2,:).
4. Contribution to variance
Geisler et al. (2007) state in their article that, ‘the contribution of a single uncertain input
parameter to the distribution of the result of an impact category can be calculated
according to:
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Environment and Computer Laboratory I
ni
−1
2
2
CTVi,j = ri,j
∙ (∑ ri,j
)
(6)
i=1
where CTVi,j is the contribution to variance of one uncertain input parameter i to the
probability distribution of the result of the impact category j (dimensionless), ri,j is the rankorder correlation coefficient between the parameter i and the score of impact category j,
and ni is the number of parameters contributing to the variance in the result of impact
category j. Contribution to variance measures the influence of an input parameter on the
distribution of a result in terms of dispersion as well as in terms of the absolute
order of magnitude of the results' values.’ Try to calculate the CTV for A and B:
>> [ctvA, ctvB]=ctv(Asample, Bsample, hsample);
What parameters of A and B have the highest contributions to the variance? The generated
Hinton diagrams give you a visual indication. Can you explain why their contribution is
dominant?
5. Uncertainty parameter estimation by means of Pedigree matrix
With the help of the pedigree approach we will find a new SD95 value for the input of
steel in the diesel production.
Table 4. Default uncertainty factors applied in conjunction with the pedigree matrix. Source: Frischknecht et
al. (2004).
1
1
1
1
1
1
1
Indicator score
Reliability U1
Completeness U2
Temporal correlation U3
Geographical correlation U4
Further technological correlation U5
Sample size U6
2
1.05
1.02
1.03
1.01
1.02
3
1.1
1.05
1.1
1.02
1.2
1.05
2 +ln(U )2 +ln(U )2 +ln(U )2 +ln(U )2 +ln(U )2
2
3
4
5
6
SD95 = e√ln(U1 )
4
1.2
1.1
1.2
1.5
1.1
5
1.5
1.2
1.5
1.1
2
1.2
(7)
The pedigree score is (3,2,3,3,4,2). Calculate the new SD95 value by means of equation 7.
Replace the old SD95 value with the new one and calculate again the contribution to
variance. Thus you have to sample first and then do the uncertainty modeling (cf. section
3). How does the contribution to variance change?
6. Uncertainty parameter estimation by means of distribution fitting
You received CH4 measurement results from a crude oil extraction facility. The data is
saved in the variable ‘CH4_measured’ that you also find in the ‘data_LCA.mat’ file. Before
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Environment and Computer Laboratory I
you can estimate the uncertainty parameters for the distribution of the CH4 emissions,
you have to check, if the data is lognormally distributed. You do this by a Chi-square
goodness-of-fit hypothesis test. The test compares the values in the data vector to a
standard normal distribution. The null hypothesis is that the data has a standard normal
distribution. The alternative hypothesis is that the data does not have that distribution. The
result of the test is 1 if it rejects the null hypothesis at the 5% significance level, and 0
otherwise. As the test is applicable for normal distributions you have to take the logarithm
of our measured data to test the hypothesis H for the lognormal distribution.
>> H=chi2gof(log(CH4_measured))
If H=0, the data is normally distributed and we can now fit the measured CH4 data to a
lognormal distribution and obtain the parameter  and  .
>> param=lognfit(CH4_measured);
In a last step, you have to transform  and  to a form usable for the applied Latin
Hypercube sampling method. Therefore you perform two following calculations
according to Heijungs and Frischknecht (2005).
>> B(2,2)=exp(param(1,1)+0.5*(param(1,2)^2));
>> Bsd95(2,2)=exp(2*param(1,2));
As you now have replaced the old mean and SD95 values of CH4 emissions coming from a
crude oil extraction process with the new mean and SD95 values, you are able to
calculate again the contribution to variance. Thus, you have to sample first and then do
the uncertainty modeling (cf. section 3). How does the contribution to variance
change?
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