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S4 ENVISA Workshop 2009
Palermo, 18-20 June 2009
Robust algorithms for the solution of the inventory problem in Life Cycle
Impact Assessment
Maurizio CelluraAntonino Marvuglia
University of PalermoDep. of Energy and Environmental Researches (DREAM)
University of Palermo
Marcello Pucci
Institute for Studies on Intelligent Systems for Automation (I.S.S.I.A), National Research Council, Palermo (Italy)
2
Every product has a “life” (1. design/development of the product; 2. resource extraction; 3. production; 4. use/consumption; 5. end-of-life activities, like collection/sorting, reuse, recycling, waste disposal).
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
All activities, or processes, in a product’s life result in environmental impacts due to consumption of resources, emissions of substances into the natural environment, and other environmental exchanges (e.g. radiation).
Introduction
3
LCA is traditionally divided into four distinct though interdependent phases:1. Goal and scope definition attempts to set the extent of the inquiry as well as specify the methods used to conduct it in later phases. 2. Life cycle inventory analysis defines and quantifies the flow of material and energy into, through, and from a product system.
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
Despite LCA is nowadays a universally accepted methodology, each of these phases still contains some
unresolved problems.
Life cycle assessment (LCA) is a methodological framework for estimating and assessing the environmental impacts attributable to the life cycle of a product, such as climate change, stratospheric ozone depletion, tropospheric ozone (smog) creation, eutrophication, acidification, toxicological stress on human health and ecosystems, the depletion of resources, water use, land use, noise and others.
3. Life cycle impact assessment converts inventory data into environmental impacts using a two-step process of classification and characterization.4. Life cycle interpretation marks the point in an LCA when one draws conclusions and formulates recommendations based upon inventory and impact assessment data.
Introduction
4
PHASE PROBLEM
Goal and scope definition • Functional unit definition• Boundary selection• Social and economic impacts• Alternative scenario considerations
Life cycle inventory analysis
• Allocation• Negligible contribution (“cutoff”) criteria• Local technical uniqueness
Life cycle impact assessment • Impact category and methodology selection• Spatial variation• Local environmental uniqueness• Dynamics of the environment• Time horizons
Life cycle interpretation • Weighting and evaluation• Uncertainty in the decision process
All • Data availability and qualityFrom: J. Reap et al., A survey of unresolved problems in life cycle assessment. Int J Life Cycle Assess (2008) 13:290–300
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009Introduction
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Allocation refers to the procedure of appropriately allocating the environmental burdens of a multi-functional process amongst its functions or products.
Kg of CO2
Kg of SO2
1
2
10
181
0.1
0
p
1
11
2
2
10
01
0.1
0
a
aa
a
p
1
11
2
2
0
181
0.1
0
b
bb
b
p
ALLOCATION FACTORS
litre of crude oil
litre of fuel
kWh of electricity
MJ of heat
Co-generation of electricity and heat
Production of electricity
Production of heat
S4 ENVISA WORKSHOP 2009
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Obviously, arbitrary allocations could lead to incorrect LCA results and less preferable decisions based on those results.
1 1 1 a b
1 1 1 a b
2 2 1 a b
Introduction
6
ISO 14044 recommends that, where allocation cannot be avoided, physical causality is to be used as the basis for allocation where possible.Physical properties used as a basis for allocation include mass, energy or exergy content.
S4 ENVISA WORKSHOP 2009
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If allocation based on physical, causal relationship is not feasible or does not provide a full solution, ISO 14044 suggests that the exchanges between the products and functions have to be partitioned “in a way which reflects other relationships between them. For example, input and output data might be allocated between co-products in proportion to the economic value of the products.”Anyhow, it has to be remarked that, regardless the method used, allocation introduces uncertainty and subjectivity elements into the computation and leads to biased solutions.
Introduction
7
S4 ENVISA WORKSHOP 2009
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The matrix method for the solution of the so called inventory problem in LCA generally determines the inventory vector or eco-profile related to a specific process by solving the system of linear equations:
A s = feconomic (or technologic)
matrix scale vector
economic functional
unit
More in general, taking into account also the environmental part of the system:
ecn
env
fAf s
B f
From the mathematical point of view, the presence of multifunctional processes in the investigated system makes the economic matrix rectangular and thus non invertible.A possible strategy to deal with this problem without using allocation procedures is based on the pseudo-inverse of the technology matrix.
Mathematical background
The pseudo-inverse of a matrix (with m>n) is defined
as:
m nA
1† T TA A A A
1 ecns A f envf B s
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However, it is necessary to assess in every case whether the obtained solution is satisfactory.
To accomplish this assessment it is necessary to substitute the
computed value of s in the original system, obtaining the so called discrepancy vector :
If the Euclidean norm is not too high (with respect to a
fixed tolerance) the solution obtained with the pseudo-inverse can
be considered acceptable.
† f A A f
Using the pseudo-inverse of the economic matrix A, it is possible to compute the scale vector also when A is rectangular, through the expression: † s A f
For the normal inverse A-1 we have: 1 0 A A f f
While for the pseudo-inverse we have:
† 0 A A f f f f
f f
Mathematical background
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One considers, for example, the system with the following inventory matrix:
1000
0
0
0
f
and let the functional unit be the following vector:
ECONOMIC MATRIX
A
ENVIRON. MATRIX
B
Production of 1000 kWh of electricity
Production of electricity
Production of fuel
Incineration of waste
kWh of electricity 10 -500 -5
l of fuel -1 100 0
kg of organic waste 0 0 -1000
kg of chemical waste 0 0 -200
kg of CO2 1 10 1000
kg of SO2 0.1 2 30
kg of crude oil 0 -50 0
Problem description
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S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
In this case, using the pseudo-inverse of A, one obtains:
†
1000
0
0
0
f A A f†
200
2
0
A f
As a consequence, the discrepancy vector is: d f f
0
0
0
0
d f f
Thus the pseudo-inverse gives in this case (for this particular
choice of the functional unit vector) an exact solution to the
problem.
20d
Problem description
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S4 ENVISA WORKSHOP 2009
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If we chose the following functional unit:
†1 1
0
0
192.31
38.46
f A A f†1
0.1923
0.0019
0.1923
A f
1
0
0
0
1000
f
Disposal of 1000 kg of chemical waste
We obtain:
Problem description
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The discrepancy vector is in this case equal to:
1
0
0
192.31
961.54
d f f 1 2980.6d
According to the goals of the study, this value could be considered
too high and thus the solution obtained through the pseudo-
inverse should be in this case discarded.
In those cases in which the pseudo-inverse is not able to provide an acceptable solution, the use of least squares techniques could be very useful.
Problem description
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range(range(AA))y = y = A s
f = As-ff
Given the system of equations
To solve this over-determined system we can use, for example,
the Ordinary Least Squares (OLS) technique, which means
solving the system
(where and )
( )m n m n AA x = f
1mf
OLSA s = f f
f is the residual error vector corresponding to a perturbation in f .
There is no exact solution if ( ) rangef A
Problem solution
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S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
A different approach that yields a consistent estimator is the Total Least Squares (TLS), which is a linear parameter estimation technique that has been devised to compensate for data errors.It is a natural generalization of the OLS approximation method when the data both in A and f are perturbed. The classical TLS problem looks for the minimal corrections A and f on the given data A and f that make solvable the corrected system of equations: TLSA A s = f f
A particular case of TLS is the so called Data Least Squares
(DLS) problem, in which the error is assumed to lie only in the
data matrix and thus the problem is converted into the solution
of the system: DLSA A s = f
Problem solution
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S4 ENVISA WORKSHOP 2009
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Classical TLS problem formulation
Frobenius normm nM
2
,1 1
m n
i jFi j
mM
OLS A S F ˆTLS Â S F
OLS approach1) Find a matrix
F' such that
2) Solve the
system
min
FrangeF AF F
TLS approach
1) Find a matrix
such that
2) Solve the
system
Â
ˆ
ˆ ˆmin
range FF AA A F F
Problem solution
0 a
f
f'
a1
2a
range(A)
a0
range(A)
a2
1a
2a
a1
^
^
range(
A)^
f^
f
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The OLS, TLS and DLS regression problems can be solved in several ways. One way to find the solution is the minimization of a cost function:
(OLS problem)
(TLS problem)1
(DLS problem)
T
T
T
E xT
T
As f As f
As f As f
s s
As f As f
s s
Ordinary Least Squares
(OLS)
Total Least Squares
(TLS)
Data Least Squares
(DLS)
(fi)
(ai) A
f
arctan(s )OLS (ai)
(fi)
f
ATLSarctan(s ) (ai)
(fi)
A
f
arctan(s )DLS
Problem solution
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The approach here applied all these problems have been
generalized by using a parameterized formulation of an error
function whose minimization yields the corresponding solution.
This error is given by the expression:
1
2 1
T
TE s
A s - f A s - f
s s
0
0.5
1
OLS
TLS
D LS
This approach comes from a work by G. Cirrincione, M. Cirrincione and S.
Van Huffel ("The GeTLS EXIN Neuron for Linear Regression", 2000)
The above function can be regarded as the cost function of a
linear neuron (the GeTLS EXIN linear neuron) whose weight
vector is s(t).
In particular is:
where is the i-th row of A and fi is the i-th element of f.
2
1
2 1
i ii
TE s
a s - f
s s
ia
1
m
iGeTLS
i
E s E s
Problem solution
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S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
The iterative algorithm (learning law) exploiting the gradient (steepest descent) is given by:
where: ; (t) = learning
rate
The GeTLS EXIN neuron is a linear unit with:
• n inputs (vector ai);
• n weights (vector s);
• one output (scalar );
• a training error (scalar )
21 Tit t t t t t ts s a s
1
T Ti iT
tk
t t
s a f
s s
Ti iy s a
i ia s - f
In the application showed in this work, the parameter is made variable according to a predefined scheduling (in general monotonically from 0 to 1).
Problem solution
Hence:
2
21 1
ii i i i i
T T
dE
ds t t t t
a s - f a a s - f s
s s s s
19
S4 ENVISA WORKSHOP 2009
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Substitution and
Allocation
Substitution and
Allocation
Least Squares solutions
Least Squares solutions
Production of
electricity
Production of
clay
Production of
sand
Production of
crude oil
Production of oil-derivatives
Production of natural gas
Supply of
biomass
Production of
bricks
1 MJ of electricity 1 -7.20E-03 -1.80E-02 0 -3.20E-02 0 0 -3.69E+021 MJ of heat 2.48E+00 0 0 -4.87E-02 -1.13E+00 -4.87E-02 0 -1.01E+031 kg of white clay 0 1 0 0 0 0 0 -1.37E+031 kg of red clay 0 1 0 0 0 0 0 -8.51E+021 kg of recycled inerts 0 0 0 0 0 0 0 6.90E+011 kg of sand 0 0 1 0 0 0 0 -5.00E+021 kg of gravel 0 0 1 0 0 0 0 -3.47E+021 kg of olive cake 0 0 0 0 0 0 1 -1.53E+021 kg of straw 0 0 0 0 0 0 1 -1.92E+011 MJ of crude oil 0 0 0 1 -2.34E+00 0 0 01 MJ of diesel oil 0 -4.54E-03 -3.60E-02 -5.29E-03 1 -3.61E+01 -4.06E-01 -1.41E+031 MJ of fuel oil 0 0 0 0 1 0 0 -1.11E+031 MJ of natural gas -4.27E+00 0 0 0 0 1 0 -5.52E+031 t of bricks 0 0 0 0 0 0 0 1
Rectangular technology matrix 14 8A
Illustrative case study: bricks production Illustrative case study: bricks production
Final demand vector f
00000000000001
Case study
20
S4 ENVISA WORKSHOP 2009
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The intervention matrix of the system at hand (used for the solutions obtained with the least squares techniques) is:
Production of
electricity
Poduction of
clay
Production of
sand
Production of
crude oil
Production of
oil-derivatives
Production of natural
gas
Supply of
biomass
Production of
bricksResources and raw materialsMJ of Coal 1.49E-03 5.16E-03 1.36E-02 7.46E-07 5.77E-02 7.46E-07 1.40E-03 0MJ of Lignite 2.74E-04 6.46E-03 1.63E-02 1.77E-09 2.68E-04 1.77E-09 8.32E-04 0MJ of Hydropower 0 3.82E-04 1.04E-03 9.58E-08 7.66E-03 9.58E-08 4.54E-04 0MJ of Geothermal Energy 0 2.80E-08 1.17E-07 6.08E-15 1.05E-04 6.08E-15 0 0kg of Water 0 7.80E-03 2.40E-02 1.49E-02 7.26E-03 1.49E-02 0 1.03E+03kg of Ores (sand, gravel, etc.) 2.20E-05 2.00E+00 2.00E+00 3.51E-05 4.55E-04 3.51E-05 0 0.00E+00MJ of Crude Oil 1.21E-02 1.82E-02 1.43E-01 1.02E+00 2.34E+00 1.02E+00 4.15E-01 1.27E+03kg of other Ores (iron, copper, etc.) 1.25E-04 7.37E-06 3.95E-05 5.60E-10 2.40E-04 5.60E-10 0 0.00E+00Emissions to airkg of CO2 8.32E-02 2.52E-03 1.33E-02 4.17E-03 2.88E-02 4.17E-03 4.93E+00 6.02E+02kg of CO 1.63E-04 3.83E-06 2.79E-05 1.14E-05 3.34E-05 1.14E-05 2.70E-02 1.10E+00kg of CH4 3.68E-04 2.97E-06 1.09E-05 7.13E-05 1.02E-04 7.13E-05 6.00E-03 1.18E+00kg of SO2 2.96E-05 2.61E-06 1.64E-05 5.31E-06 2.69E-04 5.31E-06 7.43E-03 2.20E+00kg of NMVOC 3.70E-05 4.20E-07 2.87E-06 1.93E-05 4.23E-05 1.93E-05 3.07E-02 1.13E+00Emissions to waterkg of COD 6.25E-07 2.00E-07 1.11E-06 1.57E-11 6.75E-06 1.57E-11 2.21E-04 2.38E-01kg of BOD 4.36E-08 6.11E-09 3.51E-08 4.41E-13 1.89E-07 4.41E-13 6.75E-06 2.25E-01kg of P 3.27E-10 4.95E-11 3.91E-10 3.73E-18 9.84E-13 3.73E-18 0.00E+00 1.26E-03kg of N 3.11E-08 2.91E-09 2.29E-08 2.19E-16 1.08E-10 2.19E-16 1.61E-04 6.00E-03kg of AOX 5.78E-11 3.69E-12 2.90E-11 4.64E-18 2.01E-12 4.64E-18 2.95E-07 1.43E-05Solid wasteskg of Ash 0 1.11E-04 2.83E-04 1.48E-08 2.56E-04 1.48E-08 0 0kg of Sludge 0 2.46E-07 1.93E-06 5.30E-14 2.10E-05 5.30E-14 0 0kg of Nuclear Waste 0 2.54E-09 6.49E-09 8.82E-17 7.70E-10 8.82E-17 0 0
21 8B
Case study
21
S4 ENVISA WORKSHOP 2009
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Production of electricity and
heat
Production of red and white
clay
Production of sand and
gravel
Production of crude oil
Production of oil-derivatives
Production of natural gas
Supply of biomass
Production of bricks
natu
ral g
as
heat
bricks
natural gas
heatelectricity
heat
electric
ity
electricity
diesel oil diesel oil
crude oil
olive cakestr
aw
sand
grav
el
fue
l oil
die
sel o
il
red
clay
wh
ite c
lay
electricity
diesel oil
diesel oil
Production of electricity and
heat
Production of red and white
clay
Production of sand and
gravel
Production of crude oil
Production of oil-derivatives
Production of natural gas
Supply of biomass
Production of bricks
natu
ral g
as
heat
bricks
natural gas
heatelectricity
heat
electric
ity
electricity
diesel oil diesel oil
crude oil
olive cakestr
aw
sand
grav
el
fue
l oil
die
sel o
il
red
clay
wh
ite c
lay
electricity
diesel oil
diesel oil
Flow chart of the case study on bricks production
Case study
22
S4 ENVISA WORKSHOP 2009
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Square technology matrix (physical allocation)13 13A
1
-9.52E+01
-7.14E+03
1.37E+03
8.51E+02
4.41E+02
3.47E+02
-3.51E+04
1.11E+03
-3.11E+04
-8.97E+02
1.53E+02
1.92E+01
1.00E+00
x A b
Production of
electricity
Production of
heat
Production of
white clay
Production of
red clay
Production of
sand
Production of
gravel
Production of
crude oil
Production of
fuel oil
Production of
diesel oil
Production of
natural gas
Supply of
olive cake
Supply of straw
Production of
bricks
1 MJ of electricity 1 0 -3.60E-03 -3.60E-03 -9.00E-03 -9.00E-03 0 -1.60E-02 -1.60E-02 0 0 0 -3.69E+021 MJ of heat 0 2.48E+00 0 0 0 0 -4.87E-02 -5.66E-01 -5.66E-01 -4.87E-02 0 0 -1.01E+031 kg of white clay 0 0 1 0 0 0 0 0 0 0 0 0 -1.37E+031 kg of red clay 0 0 0 1 0 0 0 0 0 0 0 0 -8.51E+021 kg of sand 0 0 0 0 1 0 0 0 0 0 0 0 -4.41E+021 kg of gravel 0 0 0 0 0 1 0 0 0 0 0 0 -3.47E+021 kg of olive cake 0 0 0 0 0 0 0 0 0 0 1 0 -1.53E+021 kg of straw 0 0 0 0 0 0 0 0 0 0 0 1 -1.92E+011 MJ of crude oil 0 0 0 0 0 0 1 -1.17E+00 -1.17E+00 0 0 0 0.00E+001 MJ of diesel oil 0 0 -2.27E-03 -2.27E-03 -1.80E-02 -1.80E-02 -5.29E-03 0 1 -3.61E+01 -3.25E-01 -8.12E-02 -1.41E+031 MJ of fuel oil 0 0 0 0 0 0 0 1 0 0 0 0 -1.11E+031 MJ of natural gas -3.41E+00 -8.54E-01 0 0 0 0 0 0 0 1 0 0 -5.52E+031 t of bricks 0 0 0 0 0 0 0 0 0 0 0 0 1.00E+00
PROCESS ALLOCATION
Prod. of electricity and heat 0.8 for electricity and 0.2 for heat
Prod. of clay (white and red) 0.5 for with clay and 0.5 for red clay
Prod. of sand and gravel 0.5 for sand and 0.5 for gravel
Production of oil-derivatives Allocation following the energy criterion.
Supply of biomass 0.8 for olive cake and 0.2 for straw
Prod. of bricks and inerts Substitution method: inerts are considered as equivalent to sand, but with an estimated correction factor of 0.85 in order to account for the difference in quality between the mass of inert materials and the mass of sand obtained from them.
Case study
23
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……
Resources and raw materials
MJ of Coal 1.20E-03 2.99E-04 2.58E-03 2.58E-03 6.82E-03 6.82E-03 7.46E-07 2.89E-02 2.89E-02 7.46E-07 1.12E-03 2.81E-04 0
MJ of Lignite 2.19E-04 5.47E-05 3.23E-03 3.23E-03 8.14E-03 8.14E-03 1.77E-09 1.34E-04 1.34E-04 1.77E-09 6.66E-04 1.66E-04 0
MJ of Hydropower 0 0 1.91E-04 1.91E-04 5.19E-04 5.19E-04 9.58E-08 3.83E-03 3.83E-03 9.58E-08 3.63E-04 9.08E-05 0
MJ of Geothermal Energy 0 0 1.40E-08 1.40E-08 5.84E-08 5.84E-08 6.08E-15 5.25E-05 5.25E-05 6.08E-15 0 0 0
kg of Water 0 0 3.90E-03 3.90E-03 1.20E-02 1.20E-02 1.49E-02 3.63E-03 3.63E-03 1.49E-02 0 0 1.03E+03
kg of Ores (sand, gravel, etc.) 1.76E-05 4.39E-06 1.00E+00 1.00E+00 1.00E+00 1.00E+00 3.51E-05 2.27E-04 2.27E-04 3.51E-05 0 0 0
MJ of Crude Oil 9.71E-03 2.43E-03 9.09E-03 9.09E-03 7.14E-02 7.14E-02 1.02E+00 1.17E+00 1.17E+00 1.02E+00 3.32E-01 8.31E-02 1.27E+03
kg of other Ores (iron, copper, etc.) 1.00E-04 2.50E-05 3.68E-06 3.68E-06 1.97E-05 1.97E-05 5.60E-10 1.20E-04 1.20E-04 5.60E-10 0 0 0
Emissions to air
kg of CO2 6.66E-02 1.66E-02 1.26E-03 1.26E-03 6.65E-03 6.65E-03 4.17E-03 1.44E-02 1.44E-02 4.17E-03 3.94E+00 9.86E-01 6.02E+02
kg of CO 1.30E-04 3.26E-05 1.91E-06 1.91E-06 1.40E-05 1.40E-05 1.14E-05 1.67E-05 1.67E-05 1.14E-05 2.16E-02 5.40E-03 1.10E+00
kg of CH4 2.94E-04 7.36E-05 1.48E-06 1.48E-06 5.44E-06 5.44E-06 7.13E-05 5.10E-05 5.10E-05 7.13E-05 4.80E-03 1.20E-03 1.18E+00
kg of SO2 2.37E-05 5.92E-06 1.31E-06 1.31E-06 8.21E-06 8.21E-06 5.31E-06 1.34E-04 1.34E-04 5.31E-06 5.94E-03 1.49E-03 2.20E+00
kg of NMVOC 2.96E-05 7.40E-06 2.10E-07 2.10E-07 1.43E-06 1.43E-06 1.93E-05 2.12E-05 2.12E-05 1.93E-05 2.46E-02 6.14E-03 1.13E+00
Emissions to water
kg of COD 5.00E-07 1.25E-07 9.98E-08 9.98E-08 5.57E-07 5.57E-07 1.57E-11 3.37E-06 3.37E-06 1.57E-11 1.77E-04 4.42E-05 2.38E-01
kg of BOD 3.49E-08 8.72E-09 3.05E-09 3.05E-09 1.76E-08 1.76E-08 4.41E-13 9.47E-08 9.47E-08 4.41E-13 5.40E-06 1.35E-06 2.25E-01
kg of P 2.62E-10 6.54E-11 2.48E-11 2.48E-11 1.95E-10 1.95E-10 3.73E-18 4.92E-13 4.92E-13 3.73E-18 0 0 1.26E-03
kg of N 2.49E-08 6.22E-09 1.45E-09 1.45E-09 1.15E-08 1.15E-08 2.19E-16 5.40E-11 5.40E-11 2.19E-16 1.29E-04 3.22E-05 6.00E-03
kg of AOX 4.62E-11 1.16E-11 1.84E-12 1.84E-12 1.45E-11 1.45E-11 4.64E-18 1.00E-12 1.00E-12 4.64E-18 2.36E-07 5.90E-08 1.43E-05
Solid wastes
kg of Ash 0 0 5.6E-05 5.6E-05 1.4E-04 1.4E-04 1.5E-08 1.28E-04 1.28E-04 1.5E-08 0 0 0
kg of Sludge 0 0 1.2E-07 1.2E-07 9.6E-07 9.6E-07 5.3E-14 1.05E-05 1.05E-05 5.3E-14 0 0 0
kg of Nuclear Waste 0 0 1.3E-09 1.3E-09 3.2E-09 3.2E-09 8.8E-17 3.85E-10 3.85E-10 8.8E-17 0 0 0
Production of crude oil
Production of fuel oil
Production of
electricity
Production of heat
Poduction of white
clay
Poduction of red
clay
Production of sand
Production of
gravel
21 13BResources and raw materials
MJ of Coal 1.20E-03 2.99E-04 2.58E-03 2.58E-03 6.82E-03 6.82E-03 7.46E-07 2.89E-02 2.89E-02 7.46E-07 1.12E-03 2.81E-04 0
MJ of Lignite 2.19E-04 5.47E-05 3.23E-03 3.23E-03 8.14E-03 8.14E-03 1.77E-09 1.34E-04 1.34E-04 1.77E-09 6.66E-04 1.66E-04 0
MJ of Hydropower 0 0 1.91E-04 1.91E-04 5.19E-04 5.19E-04 9.58E-08 3.83E-03 3.83E-03 9.58E-08 3.63E-04 9.08E-05 0
MJ of Geothermal Energy 0 0 1.40E-08 1.40E-08 5.84E-08 5.84E-08 6.08E-15 5.25E-05 5.25E-05 6.08E-15 0 0 0
kg of Water 0 0 3.90E-03 3.90E-03 1.20E-02 1.20E-02 1.49E-02 3.63E-03 3.63E-03 1.49E-02 0 0 1.03E+03
kg of Ores (sand, gravel, etc.) 1.76E-05 4.39E-06 1.00E+00 1.00E+00 1.00E+00 1.00E+00 3.51E-05 2.27E-04 2.27E-04 3.51E-05 0 0 0
MJ of Crude Oil 9.71E-03 2.43E-03 9.09E-03 9.09E-03 7.14E-02 7.14E-02 1.02E+00 1.17E+00 1.17E+00 1.02E+00 3.32E-01 8.31E-02 1.27E+03
kg of other Ores (iron, copper, etc.) 1.00E-04 2.50E-05 3.68E-06 3.68E-06 1.97E-05 1.97E-05 5.60E-10 1.20E-04 1.20E-04 5.60E-10 0 0 0
Emissions to air
kg of CO2 6.66E-02 1.66E-02 1.26E-03 1.26E-03 6.65E-03 6.65E-03 4.17E-03 1.44E-02 1.44E-02 4.17E-03 3.94E+00 9.86E-01 6.02E+02
kg of CO 1.30E-04 3.26E-05 1.91E-06 1.91E-06 1.40E-05 1.40E-05 1.14E-05 1.67E-05 1.67E-05 1.14E-05 2.16E-02 5.40E-03 1.10E+00
kg of CH4 2.94E-04 7.36E-05 1.48E-06 1.48E-06 5.44E-06 5.44E-06 7.13E-05 5.10E-05 5.10E-05 7.13E-05 4.80E-03 1.20E-03 1.18E+00
kg of SO2 2.37E-05 5.92E-06 1.31E-06 1.31E-06 8.21E-06 8.21E-06 5.31E-06 1.34E-04 1.34E-04 5.31E-06 5.94E-03 1.49E-03 2.20E+00
kg of NMVOC 2.96E-05 7.40E-06 2.10E-07 2.10E-07 1.43E-06 1.43E-06 1.93E-05 2.12E-05 2.12E-05 1.93E-05 2.46E-02 6.14E-03 1.13E+00
Emissions to water
kg of COD 5.00E-07 1.25E-07 9.98E-08 9.98E-08 5.57E-07 5.57E-07 1.57E-11 3.37E-06 3.37E-06 1.57E-11 1.77E-04 4.42E-05 2.38E-01
kg of BOD 3.49E-08 8.72E-09 3.05E-09 3.05E-09 1.76E-08 1.76E-08 4.41E-13 9.47E-08 9.47E-08 4.41E-13 5.40E-06 1.35E-06 2.25E-01
kg of P 2.62E-10 6.54E-11 2.48E-11 2.48E-11 1.95E-10 1.95E-10 3.73E-18 4.92E-13 4.92E-13 3.73E-18 0 0 1.26E-03
kg of N 2.49E-08 6.22E-09 1.45E-09 1.45E-09 1.15E-08 1.15E-08 2.19E-16 5.40E-11 5.40E-11 2.19E-16 1.29E-04 3.22E-05 6.00E-03
kg of AOX 4.62E-11 1.16E-11 1.84E-12 1.84E-12 1.45E-11 1.45E-11 4.64E-18 1.00E-12 1.00E-12 4.64E-18 2.36E-07 5.90E-08 1.43E-05
Solid wastes
kg of Ash 0 0 5.6E-05 5.6E-05 1.4E-04 1.4E-04 1.5E-08 1.28E-04 1.28E-04 1.5E-08 0 0 0
kg of Sludge 0 0 1.2E-07 1.2E-07 9.6E-07 9.6E-07 5.3E-14 1.05E-05 1.05E-05 5.3E-14 0 0 0
kg of Nuclear Waste 0 0 1.3E-09 1.3E-09 3.2E-09 3.2E-09 8.8E-17 3.85E-10 3.85E-10 8.8E-17 0 0 0
Supply of olive cake
Supply of straw
Production of bricks
Production of crude oil
Production of fuel oil
Production of diesel oil
Production of natural gas
……
Environmental intervention matrix after physical allocation
Case study
24
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
We also used a different kind of allocation (economic instead of mass allocation)PRODUCT PRICE PRODUCT PRICE
electricity (€/MJ) 0.034 gravel (€/kg) 0.011
heat (€/MJ) 0.013 fuel oil (€/kg) 0.4
white clay (€/kg) 2.2 diesel oil (€/kg) 0.025
red clay (€/kg) 2 olive cake (€/kg) 0.16
sand (€/kg) 0.015 straw (€/kg) 0.065
Square technology matrix (economic allocation)13 13AProduction
of electricity
Production of
heat
Production of
white clay
Production of
red clay
Production of
sand
Production of
gravel
Production of
crude oil
Production of
fuel oil
Production of
diesel oil
Production of
natural gas
1 MJ of electricity 1 0 -3.77E-03 -3.43E-03 -1.04E-02 -7.62E-03 0 -3.01E-02 -1.85E-03 01 MJ of heat 0 2.48E+00 0 0 0 0 -4.87E-02 -1.07E+00 -6.55E-02 -4.87E-021 kg of white clay 0 0 1 0 0 0 0 0 0 01 kg of red clay 0 0 0 1 0 0 0 0 0 01 kg of sand 0 0 0 0 1 0 0 0 0 01 kg of gravel 0 0 0 0 0 1 0 0 0 01 kg of olive cake 0 0 0 0 0 0 0 0 0 01 kg of straw 0 0 0 0 0 0 0 0 0 01 MJ of crude oil 0 0 0 0 0 0 1 -2.20E+00 -1.35E-01 01 MJ of diesel oil 0 0 -2.38E-03 -2.16E-03 -2.08E-02 -1.52E-02 -5.29E-03 0 1 -3.61E+011 MJ of fuel oil 0 0 0 0 0 0 0 1 0 01 MJ of natural gas -2.21E+00 -2.06E+00 0 0 0 0 0 0 0 11 t of bricks 0 0 0 0 0 0 0 0 0 0
Case study
25
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
Resources and raw materials
MJ of Coal 7.74E-04 7.20E-04 2.70E-03 2.46E-03 7.87E-03 5.77E-03 7.46E-07 5.44E-02 3.34E-03 7.46E-07 1.00E-03 4.03E-04 0
MJ of Lignite 1.42E-04 1.32E-04 3.39E-03 3.08E-03 9.39E-03 6.89E-03 1.77E-09 2.53E-04 1.55E-05 1.77E-09 5.93E-04 2.39E-04 0
MJ of Hydropower 0 0 2.00E-04 1.82E-04 5.99E-04 4.40E-04 9.58E-08 7.22E-03 4.44E-04 9.58E-08 3.24E-04 1.30E-04 0
MJ of Geothermal Energy 0 0 1.47E-08 1.33E-08 6.74E-08 4.94E-08 6.08E-15 9.90E-05 6.08E-06 6.08E-15 0 0 0
kg of Water 0 0 4.08E-03 3.71E-03 1.38E-02 1.01E-02 1.49E-02 6.84E-03 4.20E-04 1.49E-02 0 0 1.03E+03
kg of Ores (sand, gravel, etc.) 1.14E-05 1.06E-05 1.05E+00 9.52E-01 1.15E+00 8.46E-01 3.51E-05 4.28E-04 2.63E-05 3.51E-05 0 0 0
MJ of Crude Oil 6.29E-03 5.85E-03 9.53E-03 8.66E-03 8.24E-02 6.04E-02 1.02E+00 2.21E+00 1.36E-01 1.02E+00 2.96E-01 1.19E-01 1.27E+03
kg of other Ores (iron, copper, etc.) 6.47E-05 6.03E-05 3.86E-06 3.51E-06 2.28E-05 1.67E-05 5.60E-10 2.26E-04 1.39E-05 5.60E-10 0 0 0
Emissions to air
kg of CO2 4.31E-02 4.01E-02 1.32E-03 1.20E-03 7.67E-03 5.63E-03 4.17E-03 2.71E-02 1.67E-03 4.17E-03 3.51E+00 1.42E+00 6.02E+02
kg of CO 8.44E-05 7.86E-05 2.00E-06 1.82E-06 1.61E-05 1.18E-05 1.14E-05 3.15E-05 1.93E-06 1.14E-05 1.92E-02 7.76E-03 1.10E+00
kg of CH4 1.91E-04 1.77E-04 1.56E-06 1.41E-06 6.28E-06 4.61E-06 7.13E-05 9.61E-05 5.90E-06 7.13E-05 4.28E-03 1.72E-03 1.18E+00
kg of SO2 1.53E-05 1.43E-05 1.37E-06 1.24E-06 9.47E-06 6.95E-06 5.31E-06 2.53E-04 1.56E-05 5.31E-06 5.30E-03 2.13E-03 2.20E+00
kg of NMVOC 1.92E-05 1.78E-05 2.20E-07 2.00E-07 1.66E-06 1.21E-06 1.93E-05 3.99E-05 2.45E-06 1.93E-05 2.19E-02 8.82E-03 1.13E+00
Emissions to water
kg of COD 3.24E-07 3.01E-07 1.05E-07 9.51E-08 6.42E-07 4.71E-07 1.57E-11 6.36E-06 3.91E-07 1.57E-11 1.58E-04 6.35E-05 2.38E-01
kg of BOD 2.26E-08 2.10E-08 3.20E-09 2.91E-09 2.03E-08 1.49E-08 4.41E-13 1.79E-07 1.10E-08 4.41E-13 4.81E-06 1.94E-06 2.25E-01
kg of P 1.69E-10 1.58E-10 2.59E-11 2.36E-11 2.25E-10 1.65E-10 3.73E-18 9.27E-13 5.70E-14 3.73E-18 0 0 1.26E-03
kg of N 1.61E-08 1.50E-08 1.52E-09 1.38E-09 1.32E-08 9.70E-09 2.19E-16 1.02E-10 6.26E-12 2.19E-16 1.15E-04 4.63E-05 6.00E-03
kg of AOX 2.99E-11 2.79E-11 1.93E-12 1.76E-12 1.67E-11 1.23E-11 4.64E-18 1.89E-12 1.16E-13 4.64E-18 2.10E-07 8.48E-08 1.43E-05
Solid wastes
kg of Ash 0 0 5.83E-05 5.30E-05 1.63E-04 1.20E-04 1.5E-08 2.41E-04 1.48E-05 1.5E-08 0 0 0
kg of Sludge 0 0 1.29E-07 1.17E-07 1.11E-06 8.16E-07 5.3E-14 1.98E-05 1.22E-06 5.3E-14 0 0 0
kg of Nuclear Waste 0 0 1.33E-09 1.21E-09 3.74E-09 2.74E-09 8.8E-17 7.25E-10 4.46E-11 8.8E-17 0 0 0
Production of
electricity
Production of
heat
Poduction of
white clay
Poduction of
red clay
Production of
sand
Production of
gravel
Production of crude oil
21 13BResources and raw materials
MJ of Coal 7.74E-04 7.20E-04 2.70E-03 2.46E-03 7.87E-03 5.77E-03 7.46E-07 5.44E-02 3.34E-03 7.46E-07 1.00E-03 4.03E-04 0
MJ of Lignite 1.42E-04 1.32E-04 3.39E-03 3.08E-03 9.39E-03 6.89E-03 1.77E-09 2.53E-04 1.55E-05 1.77E-09 5.93E-04 2.39E-04 0
MJ of Hydropower 0 0 2.00E-04 1.82E-04 5.99E-04 4.40E-04 9.58E-08 7.22E-03 4.44E-04 9.58E-08 3.24E-04 1.30E-04 0
MJ of Geothermal Energy 0 0 1.47E-08 1.33E-08 6.74E-08 4.94E-08 6.08E-15 9.90E-05 6.08E-06 6.08E-15 0 0 0
kg of Water 0 0 4.08E-03 3.71E-03 1.38E-02 1.01E-02 1.49E-02 6.84E-03 4.20E-04 1.49E-02 0 0 1.03E+03
kg of Ores (sand, gravel, etc.) 1.14E-05 1.06E-05 1.05E+00 9.52E-01 1.15E+00 8.46E-01 3.51E-05 4.28E-04 2.63E-05 3.51E-05 0 0 0
MJ of Crude Oil 6.29E-03 5.85E-03 9.53E-03 8.66E-03 8.24E-02 6.04E-02 1.02E+00 2.21E+00 1.36E-01 1.02E+00 2.96E-01 1.19E-01 1.27E+03
kg of other Ores (iron, copper, etc.) 6.47E-05 6.03E-05 3.86E-06 3.51E-06 2.28E-05 1.67E-05 5.60E-10 2.26E-04 1.39E-05 5.60E-10 0 0 0
Emissions to air
kg of CO2 4.31E-02 4.01E-02 1.32E-03 1.20E-03 7.67E-03 5.63E-03 4.17E-03 2.71E-02 1.67E-03 4.17E-03 3.51E+00 1.42E+00 6.02E+02
kg of CO 8.44E-05 7.86E-05 2.00E-06 1.82E-06 1.61E-05 1.18E-05 1.14E-05 3.15E-05 1.93E-06 1.14E-05 1.92E-02 7.76E-03 1.10E+00
kg of CH4 1.91E-04 1.77E-04 1.56E-06 1.41E-06 6.28E-06 4.61E-06 7.13E-05 9.61E-05 5.90E-06 7.13E-05 4.28E-03 1.72E-03 1.18E+00
kg of SO2 1.53E-05 1.43E-05 1.37E-06 1.24E-06 9.47E-06 6.95E-06 5.31E-06 2.53E-04 1.56E-05 5.31E-06 5.30E-03 2.13E-03 2.20E+00
kg of NMVOC 1.92E-05 1.78E-05 2.20E-07 2.00E-07 1.66E-06 1.21E-06 1.93E-05 3.99E-05 2.45E-06 1.93E-05 2.19E-02 8.82E-03 1.13E+00
Emissions to water
kg of COD 3.24E-07 3.01E-07 1.05E-07 9.51E-08 6.42E-07 4.71E-07 1.57E-11 6.36E-06 3.91E-07 1.57E-11 1.58E-04 6.35E-05 2.38E-01
kg of BOD 2.26E-08 2.10E-08 3.20E-09 2.91E-09 2.03E-08 1.49E-08 4.41E-13 1.79E-07 1.10E-08 4.41E-13 4.81E-06 1.94E-06 2.25E-01
kg of P 1.69E-10 1.58E-10 2.59E-11 2.36E-11 2.25E-10 1.65E-10 3.73E-18 9.27E-13 5.70E-14 3.73E-18 0 0 1.26E-03
kg of N 1.61E-08 1.50E-08 1.52E-09 1.38E-09 1.32E-08 9.70E-09 2.19E-16 1.02E-10 6.26E-12 2.19E-16 1.15E-04 4.63E-05 6.00E-03
kg of AOX 2.99E-11 2.79E-11 1.93E-12 1.76E-12 1.67E-11 1.23E-11 4.64E-18 1.89E-12 1.16E-13 4.64E-18 2.10E-07 8.48E-08 1.43E-05
Solid wastes
kg of Ash 0 0 5.83E-05 5.30E-05 1.63E-04 1.20E-04 1.5E-08 2.41E-04 1.48E-05 1.5E-08 0 0 0
kg of Sludge 0 0 1.29E-07 1.17E-07 1.11E-06 8.16E-07 5.3E-14 1.98E-05 1.22E-06 5.3E-14 0 0 0
kg of Nuclear Waste 0 0 1.33E-09 1.21E-09 3.74E-09 2.74E-09 8.8E-17 7.25E-10 4.46E-11 8.8E-17 0 0 0
Supply of straw
Production of bricks
Production of fuel oil
Production of diesel oil
Production of natural gas
Supply of olive cake
21 13B
…………
Environmental intervention matrix after economical allocation
Case study
26
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
The GeTLS algorithm was applied with a linear scheduling of the parameter, and for different values of:• number of iterations (t=5,…,500)• learning rate (=1.0E-04,…,1)
The chosen solution is the one with the minimal residual, that is the Euclidean norm:
For every combination of n and the solution vector sGeTLS was
computed, along with:• the corresponding final supply vector• the discrepancy vector
GeTLSf A s
d f f
2
21
n
ii
d
d
By using the Least Squares techniques the system was solved in its rectangular form, without using any allocation or substitution.
Case study
27
0 100 200 300 400 500
0,0
0,1
0,2
0,3
0,4
1,0000
1,0001
1,0002
1,0003
1,0004
1,0005
; m
in(|d|)
Iterations
min(|d|)
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
0 50 100 150 200 250 300 350 400 450 5001
1.0001
1.0002
1.0003
1.0004
1.0005
Iterations
min
(|d|)
0,341
Case study
28
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
1.005
1.01
1.015
alfa
|d|
10 Iterations
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.0003
1.0004
1.0004
1.0005
1.0005
1.0006
1.0006
1.0007
1.0007
alfa
|d|
5 Iterations
0 0.1 0.2 0.3 0.4 0.5 0.6 0.71
1.005
1.01
1.015
1.02
1.025
1.03
1.035
alfa
|d|
100 Iterations
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.451
1.0005
1.001
1.0015
alfa
|d|
500 Iterations
Case study
29
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
Obtained scale vectors
Case study
ProcessPhysical allocation
Economic allocation
OLS DLS TLS GeTLS
Production of electricity -9.52E+01 1.47E+00
Production of heat -7.14E+03 -5.73E+03
Production of white clay 1.37E+03 1.37E+03
Production of red clay 8.51E+02 8.51E+02
Production of sand 4.41E+02 4.41E+02
Production of gravel 3.47E+02 3.47E+02
Production of crude oil -3.51E+04 -2.80E+04 4.77E-04 8.23E-04 7.74E-04 5.32E-04
Production of fuel oil 1.11E+03 1.11E+03
Production of diesel oil -3.11E+04 -2.25E+05
Production of natural gas -8.97E+02 -6.26E+03 7.71E-06 -1.80E-05 -1.70E-05 8.60E-06
Supply of olive cake 1.53E+02 1.53E+02
Supply of straw 1.92E+01 1.92E+01
Production of bricks 1.00E+00 1.00E+00 2.17E-07 2.85E-07 2.68E-07 2.43E-07
1.00E-04 1.30E-04 1.23E-04 1.12E-04
2.05E-04 4.51E-05 4.23E-05 2.28E-04
1.12E-04 1.21E-04 1.13E-04 1.24E-04
5.20E-04 4.50E-04 4.23E-04 5.80E-04
LCI technique
7.63E-05 1.76E-04 1.65E-04 8.51E-05
ProcessPercentage Difference between Economic and
Physical allocation
Production of electricity 101.54
Production of heat 19.75
Production of white clay 0.00
Production of red clay 0.00
Production of sand 0.00
Production of gravel 0.00
Production of crude oil 20.23
Production of fuel oil 0.00
Production of diesel oil -623.47
Production of natural gas -597.88
Supply of olive cake 0.00
Supply of straw 0.00
Production of bricks 0.00
30
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
The eco-profiles g of the functional unit can be obtained by the product g = B·s, where B is the intervention matrix, containing the environmental interventions of the unit processes.
Physical allocation
Economic allocation
OLS DLS TLS GeTLS
Resources and raw materialsMJ of Coal 6.83E+02 8.56E+02 1.73E-04 1.86E-04 1.79E-04 1.81E-05
MJ of Lignite -9.93E+00 -9.28E+00 4.91E-05 5.28E-05 5.08E-05 5.95E-06
MJ of Hydropower 9.08E+01 1.14E+02 2.05E-05 2.20E-05 2.12E-05 2.15E-06
MJ of Geothermal Energy 1.26E+00 1.57E+00 2.40E-07 2.58E-07 2.49E-07 2.40E-08
kg of Water -4.50E+02 -4.03E+02 2.45E-03 2.63E-03 2.54E-03 2.67E-04
kg of Ores (sand, gravel, etc.) -3.04E+03 -3.00E+03 1.19E-02 1.28E-02 1.23E-02 1.41E-03
MJ of Crude Oil 6.17E+04 7.05E+04 1.40E-02 1.50E-02 1.45E-02 1.47E-03
kg of other Ores (iron, copper, etc.) 3.19E+00 3.76E+00 7.88E-07 8.48E-07 8.16E-07 7.46E-08
Emissions to airkg of CO2 4.59E+02 5.27E+02 2.38E-03 2.56E-03 2.46E-03 7.15E-04
kg of CO 2.97E+00 3.37E+00 7.30E-06 7.86E-06 7.56E-06 3.31E-06
kg of CH4 -2.81E+00 -2.70E+00 4.73E-06 5.10E-06 4.90E-06 1.05E-06
kg of SO2 -4.17E-01 -1.11E+00 6.83E-06 7.36E-06 7.08E-06 1.43E-06
kg of NMVOC 3.39E+00 3.63E+00 7.84E-06 8.44E-06 8.12E-06 3.72E-06
Emissions to waterkg of COD 1.81E-01 1.64E-01 5.84E-07 6.29E-07 6.06E-07 8.42E-08
kg of BOD 2.24E-01 2.23E-01 5.03E-07 5.42E-07 5.21E-07 5.54E-08
kg of P 1.26E-03 1.26E-03 2.80E-09 3.01E-09 2.90E-09 3.05E-10
kg of N 2.44E-02 2.64E-02 4.00E-08 4.31E-08 4.15E-08 1.94E-08
kg of AOX 4.80E-05 5.15E-05 8.06E-11 8.68E-11 8.35E-11 3.64E-11
Solid wasteskg of Ash -2.83E+00 -3.60E+00 1.41E-06 1.52E-06 1.46E-06 1.58E-07
kg of Sludge -2.50E-01 -3.14E-01 5.10E-08 5.49E-08 5.29E-08 5.17E-09
kg of Nuclear Waste -3.75E-06 -6.15E-06 2.07E-11 2.23E-11 2.14E-11 2.46E-12
Case study
Percentage Difference between Economic and
Physical allocation
Resources and raw materialsMJ of Coal 25.33
MJ of Lignite 6.55
MJ of Hydropower 25.55
MJ of Geothermal Energy 24.60
kg of Water 10.44
kg of Ores (sand, gravel, etc.) 1.32
MJ of Crude Oil 14.26
kg of other Ores (iron, copper, etc.) 17.87
Emissions to airkg of CO2 14.81
kg of CO 13.47
kg of CH4 3.91
kg of SO2 -166.19
kg of NMVOC 7.08
Emissions to waterkg of COD -9.39
kg of BOD -0.45
kg of P 0.00
kg of N 8.20
kg of AOX 7.29
Solid wasteskg of Ash -27.21
kg of Sludge -25.60
kg of Nuclear Waste -64.00
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S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009Case study
32
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009Case study
33
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009Normalization
In physical problems influenced by variables of different nature, these variables are often expressed by numbers whose range of variation can differ even by several orders of magnitude.
For this reason, when any approach is used to solve a multiple-input problem, the solution could be polarized by those variables expressed by extreme values. In order to prevent this, normalization is often used.
1
1 0 0max , 1, ,
0
0
10 0max , 1, ,
i
in
a i m
a i m
H
Normalization Matrix
nA A H
Normalized Economic
Matrix
max( )n
ff
f
Normalized functional
unit
If A is invertible, the de-normalized solution sden is obtained as
follows: 1n n n
s A f maxden n n s H s f
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S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009Condition number
For a generic square and invertible matrix A the condition number with respect to the Euclidean norm is defined as:
where is the Euclidean norm of the matrix, that is defined as , where x is a generic conformable vector for the matrix A.
1k A A A
A1
max
x
A A x
For square matrices the norm can be computed as the square root
of the largest eigenvalue max of the product , that is: TA A
maxTA A A
The condition number provides an upper bound to the magnification factor that measures how a relative change in f propagates as a relative change in s.In fact the relation holds: 1
s fA A
s f
Matrices with an high condition number are called badly conditioned or ill-conditioned, and they can provide unstable solutions to the system.
In our cases: 1.33 004 k EA 197.78normk A
35
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009Normalization
The traditional solutions (i.e. those obtained with the physical allocation and the economic allocation) found after de-normalization are the same as those obtained without using any normalization procedure.
Process OLS DLS TLS GeTLS
Production of electricity
Production of heat
Production of white clay
Production of red clay
Production of sand
Production of gravel
Production of crude oil 2.67E-03 2.45E-03 2.33E-03 5.32E-04
Production of fuel oil
Production of diesel oil
Production of natural gas -1.84E-05 -1.67E-05 -1.59E-05 8.60E-06
Supply of olive cake
Supply of straw
Production of bricks 1.07E-06 9.86E-07 9.39E-07 2.43E-07
7.93E-05 7.55E-05 1.12E-04
LCI technique
5.83E-04
2.57E-03
1.13E-03
8.50E-05
4.70E-04
5.35E-04
1.04E-03 9.94E-04 2.28E-04
5.09E-04 8.51E-05
4.32E-04 4.11E-04 1.24E-04
2.37E-03 2.25E-03 5.80E-04
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S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
CONCLUSION
The application of the least squares techniques can be useful in presence of multi-functional processes and in all those cases in which the technology matrix is rectangular but the pseudo-inverse method does not provide an exact solution.
The substantial advantage in using these techniques lies into:1. the possibility to circumvent the drawback of the traditional solution of the inventory
problem, which needs to use some computational tricks to transform the rectangular technology matrix into a square and invertible matrix.
The only subjective choice in the GeTLS solutions are the learning rate and the number
of iterations, but there is a guide criterion (the norm of the discrepancy vector |d| )2. The visualization of the error surfaces, which allows the identification of the most
critical components of the solution.
The solution of the inventory problem in LCA is a complicated task, especially in presence of multi-functional processes or open-loop recycling, that are characterized by a rectangular, and thus non-invertible technology matrix.
37
S4 ENVISA WORKSHOP 2009
A. Marvuglia, Palermo – 19/06/2009
POSSIBLE IMPROVEMENTS
Application of a weighted version of the algorithm, in which higher weights are assigned to the most important components of the solution.
…..
Implementation of a learning rule with a dynamically variable learning rate () as a function of the local gradient of the error surface.
38
DREAMDipartimento di Ricerche Energetiche ed Ambientali
Prof. M. CelluraDr. A. Marvuglia
Leiden – 26/03/2009
Thank you for your attention