sustainability

Article
Uncertainty Analysis for Data-Driven
Chance-Constrained Optimization
Bartolomeus Häussling Löwgren *, Joris W eigert , Erik Esche and Jens-Uwe Repke
Process Dynamics and Operations Gr oup, T echnische Universität Berlin, Sekr . KWT 9, Str . Des 17. Juni 135,
D-10623 Berlin, Germany; [email protected] (J.W .); [email protected] (E.E.);
[email protected] (J.-U.R.)
* Correspondence: loewgr [email protected]
Received: 31 January 2020; Accepted: 11 March 2020; Published: 20 Mar ch 2020
     
  

Abstract:
In this contribution our developed framework for data-driven chance-constrained
optimization is extended with an uncertainty analysis module. The module quantifies uncertainty
in output variables of rigorous simulations. It chooses the most accurate parametric continuous
pr obability distribution model, minimizing deviation between model and data. A constraint is
added to favour less complex models with a minimal r equired quality r egarding the fit. The bases of
the module ar e over 100 pr obability distribution models provided in the Scipy package in Python,
a rigor ous case-study is conducted selecting the four most relevant models for the application at hand.
The applicability and pr ecision of the uncertainty analyser module is investigated for an impact factor
calculation in life cycle impact assessment to quantify the uncertainty in the r esults. Furthermor e,
the extended framework is verified with data from a first principle pr ocess model of a chloralkali
plant, demonstrating the incr eased precision of the uncertainty description of the output variables,
r esulting in 25% increase in accuracy in the chance-constraint calculation.
Keywords:
uncertainty analysis; optimization under uncertainty; chance-constrained optimization;
skewed distribution
1. Introduction
Envir onmental sustainability has grown to become a mor e pressing subject for the chemical
industry . A clear indicator is the joining of for ces of the major industry repr esentatives: VCI (association
of the German chemical industry), IG BCE (industry union of mining, chemistry and energy) and BA VC
(chemistry federation of employers), to set common sustainability goals for the German chemical
industry . These goals among other things include development of more sustainable pr ocesses [ 1 ].
The gr owing interest in mor e sustainable processes has led to a r enewed interest in pr ocess
systems engineering. PSE provides optimization and decision-making tools, which can be used in
the chemical industry to r educe its environmental impact [
2
]. The area of application can range
fr om equipment optimisation to optimising entire supply-chains, both during the conceptual phase
and operations.
Linking envir onmental aspects with the optimization tools provided by PSE r equires accurate
models describing the envir onmental impacts, the economics of the pr ocess, and the pr ocess
operation [
3
]. These can be implemented in multiobjective optimization formulations, where the
envir onmental description is incorporated either as an objective or as a constraint. A method wher e
these models have been linked successfully for optimization purposes is the process to planet (P2P)
method. P2P combines complex nonlinear pr ocess models with life cycle assessment (LCA) models
and evir onmentally extended input–output (EEIO) models [
4
]. It is vital when using environmental
models, such as LCA models, in decision making schemes to account for the uncertainty arising due to
Sustainability 2020 , 12 , 2450; doi:10.3390/su12062450 www .mdpi.com/journal/sustainability

Sustainability 2020 , 12 , 2450 2 of 17
in instance model simplifications or parameterization [ 5 ]. Many decision making schemes follow the
thr eshold-concept, i.e., defining a value for an environmental descriptor , above which it is consider ed
to be harmful. The decision schemes can ther efore only be applicable if they ar e combined with a
statistical analysis [ 6 ].
The additional uncertainty in envir onmental models mostly relate to parameters derived in
LCA [
7
]. The uncertainty can be subdivided into parameter uncertainty due to imprecise knowledge
or life cycle inventory (LCI) and life cycle impact assessment (LCIA) parameters, temporal and spatial
variability in LCI and LCIA parameters, variability between sources in the LCI, variability between
sour ces between objects of assessment in the LCIA, uncertainty in models and uncertainty in choices [
8
].
Due to the manifold of superposing uncertainties in LCA, the parametric distribution is assumed to be
non-normally distributed [
5
]. Additionally , non-normal distributions are found in the envir onmental
model outputs [
9
] and nonlinear pr ocess models. There ar e a wide variety of methods to analyse
and quantify uncertainty in LCA models [
10
,
11
]. While the ISO standard for LCA acknowledges that
uncertainty analysis is still in its infancy , [
11
] with sensitivity analysis being the most commonly used
method [
12
], more complex methods have r ecently been published. These methods include uncertainty
analysis methods such as Monte Carlo and Latin Hyper Cube sampling [
13
] or Fuzzy pr ogramming [
14
].
Consequently , combining environmental models and pr ocess models in optimization, referr ed to as
sustainable optimization, must always consider uncertainty [
15
]. Ther e are thr ee differ ent methods
to include uncertainty in optimization. Stochastic pr ogramming with recourse, r obust optimization
and chance-constrained optimization [
2
]. In this study we focus on chance-constrained optimization,
in line with pr evious works at our department [ 16 , 17 ].
PSE pr ovides methods for both offline and r eal-time optimization, while real-time optimization
has a gr eater potential for more accurate and flexible pr ocess operations [
18
]. Using chance-constrained
optimization for r eal-time applications would enable the incorporation of environmental models with
highly uncertain parameters and still achieve accurate online computation of optimal and stable
pr ocess operating conditions. However , for rigorous non-linear models existing chance-constrained
optimization frameworks r esult in computational times from a couple of hours to several days,
not allowing for online application [ 17 ].
Ther efore, a new framework for chance-constrained optimization has been developed at the
department, decr easing the computational effort significantly . This is achieved by exchanging rigorous
models for the optimization with data-driven ones. Uncertainty is included in additional data-driven
models. The data-driven models ar e trained on the variance of the output variables for data-sets
subjected to uncertainty . The uncertainty in the data is generated by sampling the rigor ous models with
parameters subjected to uncertainty , for which a pr obability distribution might be known. However ,
modelling of uncertainty in the model outputs in the curr ent framework is limited up to now to normal
distributions [ 16 ].
The complex distribution shape of envir onmental model parameters and its outputs [
9
] can not
be suf ficiently described by a normal distribution. This leads to lar ge deviations in the probability
calculations and the expected output values. By consequence, this leads to erroneous r esults in
chance-constrained optimization.
Ther e are a multitude of uncertainty analysis methods, the choice depends on the source and form
of uncertainty as well as the ar ea and precision of application [
10
]. For the application at hand where
the uncertainty information is statistical and need to r emain numerical for the desicion making scheme
a uncertainty analysis method which bases on Monte Carlo sampling is the only r elevant possibility .
T o allow for the implementation of envir onmental models in chance-constrained optimization,
an adaptive appr oach is studied to improve the uncertainty modelling. Implementing mor e complex
distribution functions to model the uncertainty while keeping the computational ef fort at a minimum.
It is ther efore the aim of this paper to develop and implement a method to impr ove uncertainty
modelling for data-driven chance-constrained optimization. By using mor e complex probability

Sustainability 2020 , 12 , 2450 3 of 17
distribution functions while keeping the computational ef fort at a minimum. This would allow the
implementation of envir onmental models coupled with process models for r eal-time optimization.
2. Methods
Combining rigor ous non-linear process models with envir onmental models containing highly
uncertain parameters for r eal-time optimization, r equires: (1) A stable and pr ecise method for
optimization under uncertainty [
19
], (2) a framework with a computational time allowing for r eal-time
application [
16
] and (3) an accurate uncertainty modelling framework, quantifying the distribution for
a wide variety of pr obability distribution shapes.
2.1. Optimization under Uncertainty
Chance-constrained optimization, as an appr oach to include uncertainty in optimization
pr oblems, in general r elies on physiochemical models. The underlying non-linear system contains
parameters subjected to uncertainty [
17
]. These parameters will in the following be r eferred to as
uncertain parameters. Uncertainty is included by enforcing a pr edefined probability for the fulfilment
of inequality constraints [
19
]. A well-developed approach is a sequential appr oach (single shooting)
with the pr obability calculation included as an additional layer to map the inequality constraints to
the uncertain parameter space [
20
]. The elaborate probability calculation is the most computationally
intensive part of the optimization. The computational time ranges from a couple of hours to several
days [ 17 ].
2.2. Data-Driven Chance-Constrained Optimization Framework
T o eliminate the computational limitation of conventional chance-constrained optimization
frameworks, a data-driven chance-constrained optimization framework was developed. It decreases
the computational ef fort compared to earlier frameworks. This is achieved by exchanging the rigorous
models with data-driven ones. Additionally , using a data-driven uncertainty model, which maps the
uncertainty of the outputs over the input space, reduces the computationally ef fort for the probability
calculation significantly [ 16 ].
The generation of the data-driven process and uncertainty models (DDPUM) is conducted of fline
in an upstr eam framework, implemented in Python. The data-driven models ar e subsequently inserted
in the chance-constrained optimization framework. The DDPUM generation can be separated into thr ee
steps beginning with the sampling of a rigor ous model and ending with the training of data-driven
input–output and uncertainty models. The workflow is shown schematically in Figure 1 :
Rigorous
Model
Artificial Data
Generation
Uncertainty
Analyzer
Process
Modeler
Uncertainty
Modeler
Optimization
framework

Figure 1.
Simplified workflow fr om rigorous model to chance-constrained optimization, adapted
from [
16
]. The upstream data-driven pr ocess and uncertainty model generation is highlighted by the
dashed box.
During the artificial data generation, the design variables of the rigor ous model are divided into
input variables and parameters. The space of input variables defines the boundaries, within which the
data-driven models will be valid. Some of the model parameters might be subject to uncertainty with
either known or unknown pr obability distributions. The pr obability distribution of every uncertain

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parameter must be specified. The parameter space contains the distribution of the uncertain parameters.
Both spaces ar e sampled to create a high-density data-set, this is visualised for one input and one
output variable in the left plot in Figur e 3. The artificial data is generated by solving the rigor ous
model for each input and parameter combination using AMPL [ 21 ] or MatLab.
The second step is the analysis of the uncertainty . Ther ein, the uncertain outputs at every
input point ar e analysed and a probability distribution function is fitted to the data. The r esulting
pr obability distribution parameters and the expected values at every point in the input space are
used in the subsequent modelling steps. Until now the quantification of uncertainty is limited to
normal distributions. This may lead to lar ge deviations when modelling uncertainty generated from
envir onmental models with non-normally distributed parameters or non-linear process models.
The thir d step is the generation of the data-driven process models. An input–output model is
generated based on the expected values of the output variables fr om the previous step. The uncertainty
model is trained on the pr obability distribution model parameters. The uncertainty can vary for each
point in the input space and output space.
Finally , the data-driven models can be introdu ced into chance-constrained optimization problems.
In the appr oach presented in this contribution the pr obability can be calculated directly fr om the
cumulative pr obability density function (CDF) described by the parameters returned fr om the
data-driven uncertainty model. Therefor e, avoiding elaborate multivariate integration. Hence enabling
quick computation of expected values, pr obabilities, and gradients necessary for fast convergence of
the optimization.
2.3. Uncertainty Analyser Framework
In this contribution an adaptive framework analysing and modelling uncertainty has been
developed. The framework allows for the implementation of pr ocess models and environmental
models in the DDPUM framework, with non-normally distributed output variables.
The framework is developed as a separate module in Python referr ed to as uncertainty modelling
module (UMM). The UMM consists of two submodules which ar e called successively during the
execution of the module. In Figur e 2 the workflow of the UMM is displayed, the dashed lines mark
the beginning of each submodule. The light gray arrows show how the UMM is connected to the r est
of the data-driven DDPUM framework.
The distribution data generator (DDG) is the first submodule. It fits probability distribution
models to the uncertainty data. The input of the module is artificial data fr om rigorous models,
generated in the Artificial Data Generation step in the DDPUM framework. As seen in Figure 2 and
visualized in the left plot in Figure 3 , the execution of the DDG consists of four steps. The first step
is the data pr eparation. It returns a uniform data structur e, allowing differ ent data types, as inputs,
e.g., pickles, a file format used to store data in python, or mat files, a file format storing data from
Matlab. Subsequently the uncertainty data in the artificial output-data is fitted with a continuous
pr obability distribution model, specified when calling the submodule. The path from uncertain
data in a model output to a probability distribution model fit is shown in Figur e 3 . The fitting
r eturns the probability distribution model parameters, i.e., the scale, location, and shape parameters.
The data is fitted with the statistical module provided by SciPy (scipy .stats) [
22
]. The fitting is
carried out by maximizing the logarithmic likelihood function. This optimization pr oblem does not
necessarily lead to a globally optimal fit. [
22
]. T esting the framework for a variety of distributions
has shown that the fits ar e sufficiently accurate for the application in hand. The UMM can fit the
data with ar ound 100 differ ent probability distribution models. Based on an extensive case-study ,
pr esented in Section 3 , to enhance the computational effort and considering the similarity of parametric
pr obability distribution functions [
23
], the set of distribution functions is reduced to the four most
accurate continuous pr obability distribution models for provided by SciPy for artificial data-sets
including uncertainty .

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Data Preparation
Probability
Distribution Fitting
Histogram - PDF Area
Deviation Calculation
Expected V alue
Calculation
Histogram Data
Calculation
Probability
Distribution Model
Selection
Data Combination
Data Modelling Uncertainty Modeler
Combined expected values
Distribution
parameters
Probability model choice
Combined distribution parameters
Expected
values
Distribution parameters
PDFM
Snapshot Generation
OPTION II
Unified data structur e
Distribution
data generator
Distribution
data selector
Snapshot Generation
OPTION II

Figure 2.
W orkflow of the uncertainty modelling module (UMM). The light gray boxes repr esent the
existing Dinosaur framework. The green part r epresents the DDG and the yellow the distribution data
selector (DDS). Each arrow is marked with the data passed along.
Input sample points
Output
Relative frequency
Output residual
Output residual Output residual
Relative frequency

Figure 3.
V isualisation of the steps fr om generated artificial data including uncertainty (
left plot
) to
probability distribution fitting, seen as the pr obability density curve (red curve) over the histogram in
the (
right plot
). W ith descriptive statistics the distribution of the output over one input point (
lower
middle plot
) can be visualised as a histogram (
upper middle plot
) and indicates the connection to
distribution fitting. The colour range highlights the output range, with incr easing values from gr een
to yellow .
The thir d step is the evaluation of the fit of the pr obability distribution models. For this purpose,
a metric is defined describing the deviation between model and data. The pr obability distribution fit

Sustainability 2020 , 12 , 2450 6 of 17
metric (PDFM),
ψ
, is defined as the area between the histogram and the pr obability density function.
The lower limiting case, with a sample size towards infinity and a perfect fit is
ψ →
0. In turn,
the upper limiting case for a complete model mismatch is
ψ →
1. The PDFM is visualized for an
arbitrary skewed distribution in Figur e 4 . Comparing the left and right plot, clearly shows that the beta
distribution function with a smaller ar ea between probability density function (PDF) and histogram,
i.e., a lower PDFM-value, fits the uncertainty data better . In the fourth step the expected values ar e
calculated with the distribution models fitted in the second step.
Norm Beta
ψ n o r m ψ b e t a
c

2020 by the authors. Submitted to Journal Not Specified for possible open acces s
1
publication under the terms and conditio ns of the Cr eative Commons Attribution (CC BY) li cense
2
(http://cr eativecommons.or g/licenses/by/4.0/). 3

Figure 4.
V isualisation of the probability distribution fit metric (PDFM) with an arbitrary
skewed distribution.
The second submodule, called distribution data selector (DDS) chooses the most accurate
distribution functions r eturned by the DDG. For big sample sizes, for which a binominal distribution
appr oaches a continuous distribution, the PDFM can be used directly to choose between pr obability
distribution functions, since there will be a clear distribution to match. For smaller sample sizes
a variation of the likelihood-ratio test is applied. The likelihood-ratio test chooses between two
distribution models based on their maximum likelihood [
24
]. The PDFM is regar ded as a definite
fit-description of the pr obability distribution model, hence the ratio of the PDFMs will indicate,
which pr obability distribution model describes the data better . Distribution functions with mor e shape
parameters will in most cases have a mor e accurate fit [
23
]. Models with additional shape parameters
will need mor e data-driven models in the uncertainty model step in the optimization framework.
Leading to mor e computational effort for the optimizer . Therefor e, a constraint is added to favour less
complex models with a minimum r equired quality r egarding the fit. Based on the PDFM-ratios and
considering the constraint, a model is chosen. Finally , the distribution parameters and expected values
ar e combined for all outputs based on their individual distribution model choice.
3. Uncertainty Analysis
Pr obability distribution of a model output can take on a variety of shapes, depending on the
non-linearity of the model and the distribution shape of the uncertain parameters. There is a lar ge
number of continuous pr obability distribution models, though the number of models which have
become pr ominent is relatively low [
25
]. Around 100 of of the most pr ominent continuous distribution
models ar e implemented in scipy .stats [
22
]. This case study aims to find continuous pr obability
distribution models, which can describe unimodal probability distribution shapes most accurately ,
weighting in the complexity of the model, r epresented by the number of shape parameters, and the
computational ef fort of the model fitting. T o evaluate the ability to fit of the models, a five step

Sustainability 2020 , 12 , 2450 7 of 17
evaluation scheme is constructed, which is pr esented in Figure 5 . The weights ar e chosen based on the
commonness of the distribution shapes in chemical engineering applications.
.
0.9 < ψ P D M ( normal distr . )
ψ no r m ( normal distr . ) < 1.1
T est the ability of the probability distribution model (PDM) to accurately model normal distribution
T est the ability of the PDM to fit differ ent distribution shapes, by calculating: ψ P D M
for each shape
The fit-ability of the differ ent PDM’s is evaluated in weighing matrix of the ψ P D M
weighted
...
...
...
...
...
...
...
W eighing matrix of the computational effort for the fitting of the distribution shapes
weighted
The highest scoring PDM’s in weighted fit-ability and computational time are divided into gr oups
based on the number of shape parameters and rated by their computational effort
PDM‘s with 1 shape parameter PDM‘s with 2 shape parameters
...
...
...
...
...
...
...
PFM 1 ψ 1 ( S 1 ) ψ 1 ( S 2 ) ψ 1 ( S 3 ) ψ 1 ( S 4 ) ψ 1 ( S 5 ) ψ 1 ( S 6 ) ∑ 6
i = 1 x i ψ 1 ( S i )
PFM n ψ n ( S 1 ) ψ n ( S 2 ) ψ n ( S 3 ) ψ n ( S 4 ) ψ n ( S 5 ) ψ n ( S 6 ) ∑ 6
i = 1 x i ψ n ( S i )
...
...
PFM 1 t c o m p ,1 ( S 1 ) t co m p ,1 ( S 2 ) t c o m p ,1 ( S 3 ) t co m p ,1 ( S 4 ) t co m p ,1 ( S 5 ) t c o m p ,1 ( S 6 ) ∑ 6
i = 1 x i t c ,1 ( S i )
PFM n t c o m p , n ( S 1 ) t c o m p , n ( S 2 ) t c o m p , n ( S 3 ) t c o m p , n ( S 4 ) t c o m p , n ( S 5 ) t c o m p , n ( S 6 ) ∑ 6
i = 1 x i t c , n ( S i )

Figure 5.
A five step evaluation scheme to choose the best probability distribution model accor ding to
their ability to fit distribution data and the computational effort.
The 100 continuous distribution models in scipy .stats ar e reduced to a set of 40 distribution models
in the first step due to their insuf ficient accuracy in modelling a normal distribution. The r esults
concerning their ability to fit ar e shown Figure A1 and the weighted matrix of the computational
times in Figur e A2 in the appendix. The four distribution models with the highest weighted results
concerning their ability to fit, equivalent to the first four models, i.e., rows in the heatmap, ar e: Beta,
Johnsons b, Skewnorm, and W eibull max. All of them describe the normal and skewed distributions
nearly err or-fr ee, seen by the low PDFM values in columns 1, 3, and 4. The PDFM values for the more
uncommon distribution shapes, uniform and exponential, columns 2, 5, and 6, ar e also relatively low .
The four models ar e divided into two sets, one set containing the models with two shape parameters
(beta and the Johnsons b) and one set with the models containing only one shape parametes (Skewnorm

Sustainability 2020 , 12 , 2450 8 of 17
and W eibull max). The computational effort of Johnsons b is almost twice as high as for the beta
distribution. In the one-shape-parameter-set a dif fer ence in computational effort is not as evident.
The W eibull max has a slightly lower computational ef fort, though the Skewnorm model shows a more
balanced fit-quality for the right-left skewed and exponential incr easing-decreasing shapes.
It can ther efore be concluded, that the beta distribution is the best two-shape-parametric
distribution model. For the one-parametric distribution model, both the W eilbull max and
Skewnorm ar e well suited distribution models. The normalised pr obability density functions of
the thr ee probability distribution functions ar e shown in Equations ( 1 )–( 3 ), respectively [ 22 ].
f β ( x , a , b ) = Γ ( a + b ) · x a − 1 ( 1 − x ) b − 1
Γ ( a ) · Γ ( b ) (1)
f w m a x ( x , c ) = c · ( − x ) c − 1 · exp  − ( − x ) c  (2)
f s k e w − N ( x , d ) = 1
√ 2 π exp ( − x 2 /2 ) h 1 + erf  d · x
√ 2  i (3)
3.1. Case Study: Applicability on LCIA with Uncertain Parameters
The best pr obability distribution models from the evaluation scheme ar e implemented in the
uncertainty analyzer framework. The uncertainty analyzer framework is tested with a case study
exemplifying the workflow and decision pr ocess in the uncertainty analysis. The case study is based on
a life cycle impact assessment step, wher e the uncertainty of the calculated impact scores ar e analysed.
In general this is equivalent to an uncertainty analysis of an output of a linear model with non-normal
distributed uncertain parameters.
Since the models in LCA ar e parametric repr esentations, the uncertainty in the model outputs is
due to the uncertainty of the model parameters. The uncertainty of the model parameters must be
analysed during the design and validation of the model. The thereby derived uncertainty information
can either be qualitative or quantitative depending on the uncertainty analysis method chosen [
10
].
For data-driven chance-constrained optimization, the uncertainty information needs to be quantitative.
Quantifying uncertainty is most commonly done with pr obability distribution models, where the
complexity and the accuracy of the chosen pr obability distribution model depends on the quality of
the distribution data for the uncertain parameters [
26
]. The presented uncertainty analyzer framework,
does not estimate the pr obability distribution of the uncertain parameters, but uses this information to
quantify and model the distribution of the outputs needed for the chance-constrained optimization.
In this case study the uncertainty in the impact score,
W
, is caused by uncertainty in the
characterisation factor ,
x i
, and the component mass flow ,
m i
, for
n
components and is based on
the uncertainty data derived by [
5
]. The uncertainty in the parameters was assessed heuristically
and empirically , based on uncertainty due to imprecise knowledge or LCI and LCIA parameters,
temporal and spatial variability in LCI and LCIA parameters, variability between sources in the
LCI, variability between sour ces between objects of assessment in the LCIA, uncertainty in models,
and uncertainty in choices [ 8 ].
Equation
( 4 )
shows how the impact factors ar e calculated considering a composition uncertainty
and an uncertain characterization factor .
W =
n
∑
i = 1
m i · x i (4)
The composition uncertainty of the component mass flow is assumed to be uniformly distributed,
since the lower and upper bound ar e determined through a best and worst case scenario, r espectively .
The characterization factor is assumed to be right skewed and described by a log-normal pr obability
distribution [
5
]. The distribution in the characterization factor is described by a dispersion factor ,
which determines the skewness of the distribution.

Sustainability 2020 , 12 , 2450 9 of 17
T o test the uncertainty analyzer framework, an artificial data-set is created in the artificial data
generation step of the DDPUM framework. The parameters are sampled two-dimensionally with
a Hammersley sampling method, while the distribution of the parameters is specified with the
lognormal and uniform pr obability distribution model provided by scipy .stats [
22
]. The model is
solved in AMPL [
21
] and the uncertain impact factor data-set is passed on to the uncertainty analyser
framework. The fit-accuracy of the five probability distribution models is calculated in the Pr obability
Distribution Model Selection. The PDFM-ratio of the probability distribution models is shown in
Figur e 6 a and with the corresponding values of the case study in Figur e 6 b.
norm
sk ewnorm
w eibull max
b eta
johnsons b
norm
sk ewnorm
w eibull max
b eta
johnsons b
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j
ψ i
ψ j

( a )
norm
sk ewnorm
w eibull max
b eta
johnsons b
norm
sk ewnorm
w eibull max
b eta
johnsons b
1.0 0.4 0.2 0.3 0.2
2.7 1.0 0.5 0.7 0.5
6.0 2.2 1.0 1.5 1.2
4.0 1.5 0.7 1.0 0.8
5.1 1.9 0.8 1.3 1.0 1
2
3
4
5

( b )
Figure 6.
V isualization of the Probability distribution model selection step in the uncertainty analyser
framework for the impact factor . The PDFM-ratio of the probability distribution models is shown in
the left plot (
a
), with the corresponding r esults of the case study in the right plot (
b
). The gr eater the
value in the right plot, the mor e accurate is the probability distribution model of the r ow (j) compared
to the pr obability distribution model of the column (i). The probability distribution model is selected
corresponding the r ow that has the highest values when comparing all columns.
Fr om Figure 6 b it can be derived that the thir d row , which corresponds to the weibull max
pr obability distribution method, is the most accurate model. If the ratio equals one, then both models
fit the distribution data equally well and the gr eater the value, the more accurate the fit. The complexity
of pr obability distribution models, defined by the number of shape parameters,
n
, is taken into account
by defining a significance level,
α
.
α
is intr oduced to favour less complex probability distribution
models, which result in fewer data-driven uncertainty models in the following DDPUM framework
step. For the case study , a 5% significance level is chosen. Models with more shape parameters must
ther efore be
∆ n ·
5% mor e accurate than a model with less shape parameters with
∆ n
corr esponding to
the dif ference in shape parameters. The PDFM-ratios for W eibull max model are all gr eater than one
and do not violate the constraint. It can therefor e be concluded that the weibull max method is the
most accurate pr obability distribution model to describe uncertainty in the impact factor .
Comparing the uncertainty analyser framework with the appr oach to model all outputs with
a normal pr obability distribution model, r eveals that the uncertainty analyser framework is up to
six times mor e accurate. The comparison can be derived directly fr om the first column in Figure 6 b,
the values r epresent the deviation of the pr obability distribution of the normal distribution compared
to W eibull Max (row 3).

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3.2. Case Study: Improvement in the Chance Constraint Calculation for a Chlor-Alkali Pr ocess
The appr oach in the DDPUM-framework is to model the uncertainty with data-driven models.
These data-driven models map the pr obability distribution model parameters of the model outputs
over model inputs. This concept is only valid if the parameters returned by the data-driven uncertainty
model can corr ectly reconstruct the uncertainty distribution in the model outputs. It can be ar gued,
that all distribution function parameters ar e continuous and smooth over the variable space, since they ,
i.e., (location, scale, and shape parameters) have physical or geometrical properties ([
25
], p. 19).
Smooth data-driven models should ther efore be able to corr ectly model the probability distribution
model parameters over the input variable space.
T o test if the uncertainty description, i.e., the probability distribution model parameters,
r eturned by the new uncertainty analyser framework can be used to accurately reconstruct the model
output distributions, a rigorous model of an industrial chloralkali electr olyzer [
16
] is examined.
Additionally , the accuracy of the uncertainty analysis of the old framework, where the uncertainty
distribution in all model outputs is assumed to be normal, is compar ed to that of the new .
The model outputs used, are the chloride mass fraction and the anolyte brine flow at the outlet.
The consider ed input variables are the curr ent density and the anolyte brine feed flow . The current
ef ficiency regar ding sodium hydroxide is consider ed as uncertain parameter following a normal
distribution. Sampling over the inputs and the parameters is carried out and for each combination of
input and parameter , the rigorous model is solved using AMPL [
21
]. The dataset is passed on to the
uncertainty analyser .
The uncertainty analyser selects the beta distribution model to describe the uncertainty in both
outputs. For the generation of the data-driven distribution models a Gaussian process r egr ession
model is chosen. The model is trained with 90% of the uncertainty data, referr ed to as testing data,
the r emaining 10% of the data is used to test the predictability of the model. The model is both tested
on the capability to corr ectly map the distribution parameters and on the accuracy of the predicted
distributions, based on the testing data.
The r esults, pr esented in Figure 7 , show a smooth curvature of the uncertainty model for all
parameters. The fit of the Gaussian process r egression model has a mean squar ed err or of 5.0
×
10
− 6
.
and a per centile deviation of 0.043% . The low mean squar ed error and the per centile deviation of the
data-driven model indicates that a data-driven model can map the distribution parameters over the
input space accurately .
In addition to the fit quality of the data-driven model, it must be tested if the distribution
parameters r eturned from the data-driven model corr ectly recr eate the distribution of the output
variables at each input-point. Therefor e an additional fit-error parameter
θ
, similar to the PDFM is
intr oduced. It evaluates the deviation of the PDF modelled by the distribution function parameters of
the testing data and the predicted PDFs at these points. The deviation equals the shaded ar ea between
the two PDFs, as shown in Figur e 8 a.
θ
is scaled between 0 and 1, wher e 0 is the case when the PDFs
of the testing and training data overlap completely and 1 when ther e is no overlap. The resulting
mean
θ
for all testing points in the data-set of this case study is 4.3
×
10
− 4
. The low value in
θ
shows
that the distribution parameters r eturned by the UM can correctly r ecreate the distribution of the
output variables.
The accuracy of the new uncertainty analyser framework for chance-constrained optimization
is compar ed to the former version. Therefor e a refer ence data-driven uncertainty model is trained
on the mean and variance of the model outputs, i.e., assuming a normal distribution. In data-driven
chance-constrained optimization, the chance constraint is checked by calculating the pr obability of the
inequality constraint using the parametric CDF with the distribution model parameters r eturned by
the data-driven uncertainty model. The inequality constraints are chosen as model outputs, hence the
accuracy of the chance constraint calculation can be evaluated dir ectly with the data-driven uncertainty
model. T o test the accuracy of the chance constraint calculation, we firstly define the chance constraint
level, which corresponds to the minimal pr obability level that the inequality constraint is satisfied.

Sustainability 2020 , 12 , 2450 11 of 17
Secondly , we use the inverse function of the CDF , the percent point function (PPF), to calculate the
maximal value of the inequality constraint to the set chance constraint. T o have a refer ence value,
when comparing the inequality constraints, the relative fr equency of the sample data is used to estimate
a value of the chance constraint. In this case study we consider the model output: Chloride mass
fraction as an inequality constraint. The chloride mass fraction, to a cumulative probability of 99%,
is calculated with the data-driven uncertainty model trained on the normal distribution parameters,
with the data-driven model trained on the beta pr obability distribution model parameters and with the
r elative frequency of the sample data. T o assess the relative impr ovement, the inequality constraints
ar e subtracted by the mean value and divided by the value calculated with the relative fr equency .
The values calculated fr om the sample data directly ar e assumed to be close to the population statistic,
i.e., the “real” value. When the sample size increases the r esults from the r elative fr equency approaches
the population value. The results of the r elative inequality constraint is shown in Figur e 8 b. The beta
pr obability distribution model almost returns the exact inequality constraint, while If we use the
normal distribution, the solution violates the inequality 25% of the time.
Figure 7. Data-driven uncertainty model for the first shape parameter of the beta distribution model.

Sustainability 2020 , 12 , 2450 12 of 17
Outputr esidual
Relativefr equency
T estingdata
T rainingdata

( a ) ( b )
Figure 8.
(
a
) Fit deviation parameter explained for the training and testing data for the data-driven
uncertainty model. (
b
) Relative inequality constraints visualizing the improved uncertainty description
of the new uncertainty analyser framework.
It is thus concluded that the uncertainty of the output variables can be fully and accurately
modelled with a data-driven model mapping the distribution function parameters over the input
space. While significantly improving the accuracy chance constraint evaluation in the data-driven
chance-constrained optimization.
4. Conclusions
In this contribution, an extension of the framework for the generation of data-driven
models for chance-constrained optimization, with an uncertainty analyser framework is presented.
The uncertainty analyser framework can model sample data subjected to uncertainty with a
wide variety of unimodal pr obability distribution models, choosing the most accurate probability
distribution model by minimizing the deviation to the uncertain data. Additionally , a constraint
is implemented that favours less complex models with a minimal requir ed quality regar ding the
fit. The new uncertainty analyser results in mor e accurate descriptions of uncertainty in model
outputs, consequently impr oving the chance constraint calculation, which is a central building block
in data-driven chance-constrained optimization.
A case study is performed selecting the four most r elevant probability distribution models for
pr oblems at hand: Skewnorm, W eibull max, beta and Johnsons b. These models are further evaluated
in a case study aiming to describe uncertainty in the impact factor in LCIA. The impact factor is
chosen as the model output and the uncertainty arises due to skewed and uniform distributed model
parameters. Applying the new method results in an accurate description of uncertainty in the model
outputs by selecting the most suitable pr obability distribution model with the minimal deviation to
the uncertainty data.
T o test the potential of the uncertainty analyser framework for data-driven chance-constrained
optimization, a rigorous pr ocess model for a chlor-alkali pr ocess was sampled and a data-driven
uncertainty model generated with the extended DDPUM framework. An excellent fit for the
data-driven uncertainty model is achieved, indicated by the mean squared deviation of 5.0E-6 (0.043%)
and a distribution fit-err or , repr esenting the deviation of the predicted PDF , of 4.3E-4. The improvement
for data-driven chance-constrained optimization with the new uncertainty analyser is evaluated.
For this purpose the r elative inequality constraint, set as the chloride mass fraction in the model,
is calculated for a specified chance constraint level. The calculation is conducted with the old method,
assuming normal distribution, and with the new uncertainty analyser . The evaluation shows, that the
r esult of the chance constraint calculation with the new uncertainty analyser framework is almost
err or free. While when using the old method based on a normal distribution, the solution violates the
inequality 25% of the time.

Sustainability 2020 , 12 , 2450 13 of 17
The combination of the r esults for both case studies shows that the precision of the
framework for the generation of data-driven models for chance-constrained optimization is
not limited by the uncertainty modelling. Allowing the implementation of models with
high uncertainty , as environmental models, in decision making schemes, such as data-driven
chance-constrained optimization.
The uncertainty analyser framework is limited to modelling the distribution in the output
variables with unimodal pr obability distribution models. Alternatively the probability distribution
can be modelled using Kernel density estimation, additionally describing multimodal pr obability
distributions. However , this exceeds the limit of the pr esented DDPUM framework. Additionally ,
the computational ef fort of the framework and its precision could be impr oved by an adaptive
sampling method linking the uncertainty analyser with the artificial data generation step in the
DDPUM framework.
Author Contributions:
Conceptualization, J.W . and B.H.L.; methodology , B.H.L.; software, J.W ., B.H.L. and E.E.;
validation, B.H.L.; formal analysis, B.H.L.; investigation, B.H.L. and J.W .; resour ces, J.W .; data curation, J.W . and
B.H.L.; writing–original draft pr eparation, B.H.L.; writing–review and editing, J.W ., E.E. and J.-U.R.; visualization,
B.H.L.; supervision, J.W ., E.E. and J.-U.R.; project administration, E.E. and J.-U.R.; funding acquisition, J.-U.R. All
authors have read and agr eed to the published version of the manuscript
Funding:
The resear ch pr oject ChemEFlex (funding code 0350013A) is supported by the German Federal Ministry
for Economic Aff airs and Energy . W e acknowledge support by the German Research Foundation and the Open
Access Publication Fund of TU Berlin.
Conflicts of Interest: The authors declare no conflict of inter est.
Abbreviations
The following abbreviations ar e used in this manuscript:
BA VC chemistry federation of employers
CDF cumulative distribution function
EEIO environmental extended input–output
DDG distribution data generator
DDPUM data-driven process and uncertainty models
DDS distribution data selector
IG BCE industry union of mining, chemistry and energy
LCA life cycle assessment
LCI life cycle inventory
LCIA life cycle impact assessment
P2P process to planet
PDF probability density functions
PDFM probability distribution fit metric
PDM pr obability distribution model
PPF percentage point function
PSE process system engineering
scipy .stats Statistical package in the SciPy library
UMM uncertainty modelling module
VCI Association of the German chemical Industry

Sustainability 2020 , 12 , 2450 14 of 17
Appendix A
weighted

Figure A1. Heatmap of the PDFM for the fit-quality evaluation of Scipy statistical module distribution functions with varying distribution shapes

Sustainability 2020 , 12 , 2450 15 of 17
weighted

Figure A2.
Heatmap of the computational effort of the model-fitting for the fit-quality evaluation of Scipy statistical module distribution functions with varying
distribution shapes. The fitting was conducted on a sample containing 1000 sample points.

Sustainability 2020 , 12 , 2450 16 of 17
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