Sensitivity Analysis for Biorefineries

Biobased Chemistry and Technology

Sensitivity Analysis for Biorefineries

Xinyi Yao

May 2015

Thesis Biobased Chemistry and Technology

Sensitivity Analysis for

Biorefineries

Name course : Thesis project Biobased Chemistry and Technology

Number : BCT-80436

Study load : 36 ects

Date : November 2014 – May 2015

Report number : 021BCT

Student : Xinyi Yao

Registration number : 910523-980-080

Study programme : MBT (Biotechnology)

Supervisor(s) : Dr. ir. A.J.B. Van Boxtel

Examiners : Dr. ir. K. Keesman and Dr. ir. P.M. Slegers

Group : Biobased Chemistry and Technology

Address : Bornse Weilanden 9

6708 WG Wageningen

the Netherlands

Tel: +31 (317) 48 21 24

Fax: +31 (317) 48 49 57

Preface

This thesis report consists of two part: a paper draft based on the findings during the project and

supplement materials covering other results which are not included in the paper draft. The

sensitivity analysis result shown in this report are all acquired by using the Matlab toolbox

developed by the author, which is also an important part of this thesis project. The toolbox can be

found in the data disc handed in with this thesis report.

Special thanks to the acknowledgement from Dr. Ir. A.J.B Van Boxtel and Dr. Ir. P.M. Slegers as

well as the support from BCT group. 谢谢！

Xinyi Yao 姚辛夷

05-12-2015

Contents

Paper Draft: Applying Inputs-categorized Global Sensitivity Analysis for Bottleneck Analysis of

Microalgae Biorefineries.........................................................................................................................................................1

Supplement Materials ...................................................................................................................................................... 24

References .............................................................................................................................................................................. 37

1 Master Thesis Report: Sensitivity Analysis for Biorefineries

Applying Inputs-categorized Global Sensitivity Test for Bottleneck Analysis of Microalgae Biorefineries

Xinyi Yao

Abstract

Global sensitivity analysis has been playing a more and more important role in the field of

biorefinery research as a tool for bottleneck analysis. The designers of biorefineries always look

forward to extract as much information as possible from such an analysis to form more research

questions for improving the biorefineries. In this work, we demonstrated the approach of the

global sensitivity analysis for microalgae biorefineries whose inputs are categorized in 3 different

classes: physical properties, operational variables and uncontrollable variables. We suggest that

this approach could reveal more substances of the microalgae biorefinery models and would

become helpful in research.

Keywords: Global sensitivity analysis, microalgae biorefineries

1. Introduction

Sensitivity analysis is the study of how uncertainty in the output of a model can be apportioned to

different sources of uncertainty in the model input[1] as well as the tool of scenario study of a

model. It has been carried out as a useful tool for understanding and improving mathematical

models in many applications nowadays including studies related to biorefineries. For example

Wang et al. [2] tested the sensitivity of the corn biorefinery model to determine which model input

has the biggest impact on the output, which is the minimum fuel selling price. Specifically in the

field of research related to microalgae biorefineries, Norsker et al. [3] utilized sensitivity analysis

for comparison of energy cost while using different photobioreactors for microalgae production in

different scenarios, and Yang et al. [4] also conducted similar test to investigate the influence of

parameter variations towards the water footprint of microalgae production. Performing such


analysis could be assistive in terms of uncovering technical mistakes, making better choices for

improvement, bring up critical research questions for overcoming bottlenecks, or setting up the

essential control variables for the model. Yet methods for analyzing sensitivity in these studies

are based on local sensitivity analysis, i.e. one-at-a-time method. The drawbacks of such a way

of computing sensitivity are that it depends on the linearity of the model, and no interactions of

inputs are studied. However modeling is getting more and more complex and detailed. Nonlinear

and uncertain models inputs appear more often, in which case local sensitivity analysis is no

longer relevant. And that is what the global sensitivity analysis can cope with.

Global sensitivity analysis can measure the importance of inputs within the whole input space.

Variance-based sensitivity analysis is one form of global sensitivity analysis including other

approaches like regression analysis, Monte Carlo Screening. Sobol and Saltelli [5, 6]introduced

the computation of first order sensitivity indices and total sensitivity indices as a method to

perform variance-based analysis. The first order indices indicate the main effect of each model

input towards the output for prioritizing these inputs, while the total sensitivity indices also

considered the possible interactions between each model input so that non-influential inputs

could be fixed for simplifying the model. Both two kinds of indices are performed in combination

with uncertainty analysis.

Variance-based sensitivity analysis is already adapted in multiple research fields like quality

assessment[7], cell cultivation[8], waste water treatment[9], however it is hardly done in

microalgae biorefinery studies. There are some studies[10-12] that made use of this method to

rank the most influential parameters as well as indicating the bottlenecks in microalgae biodiesel

production models. Nevertheless the information provided by these studies is not clear on two

aspects:

With respect to the first aspect, the inputs of the models in these studies are a mix of microalgae


physical properties (lipid content, water content, etc.) and operational conditions from each

process unit within the microalgae biorefinery. The result of sensitivity analysis in these

studies[10-12] shows that the microalgae properties are dominant when compare to other factors,

which makes the roles of operational variables in the process models not so clear. This may

impact the conclusion when the objective of research is to target which steps in the process

design can be improved. Additionally, the output uncertainties may also be caused by the

inaccuracy of some sub-models and experiential values applied in the biorefinery. These

inaccuracies and experiential values are uncontrollable part in the model and their sensitivities

are also very intriguing. Due to the fact that bottleneck targeting might be influenced without a

good definition of inputs, a clear categorization of 3 different kinds of model inputs for the

sensitivity analysis is suggested: the physical properties, the operational variables and the

uncontrollable variables.

With respect to the second aspect, even though the sensitivity analysis is able to indicate the

importance of model inputs, it only can provide the information about the variation of outputs

contributed by each input. Whether the changing of these inputs leads to a good or bad output

realization still remains unknown. Monte Carlo Filtering is the proposed tool to answer this

question, it can map these good or bad model outputs backwards to the space of the influential

inputs to determine their relationships. This test could be helpful for optimizing the model

performance by controlling the model inputs in the range that leads to good output, or for

diagnosing unexpected system performance since the range of each model input that leads to

bad output is already known. Besides, this method is also an approach for global sensitivity

analysis.

In this work we first demonstrate the approach of making use of variance-based sensitivity

analysis and factor mapping for categorized model inputs of one optimized microalgae


biorefinery model from Slegers et al[13]. Then we applied the same approach on two other

optimized microalgae biorefinery models by Slegers et al.[13] and compared the results from all

three models. This could provide another angle when making the choices among different model

designs. We propose that such an approach could be a good tool for bottleneck analysis for

biorefineries. Both first order and total sensitivity indices are computed with Monte Carlo

Simulations for both factor prioritizing and factor fixing settings. Monte Carlo Filtering as well as

Kolmogorov-Smirnov test are utilized in factor mapping setting. For the information of settings in

sensitivity analysis see the supplement materials, section 1. The computation methods used in

this study are based on previous work from Saltelli et al[1].

2. Methodology

2.1 Categorization of model inputs

Slegers et al.[13] performed a model-based combinatorial optimization for energy-efficient

biodiesel production from algae. The models of each processing steps are connected to form the

total blueprints, and a constrained optimization for inputs involved in each possible biofefinery

design is done to achieve the maximum net energy ratio (NER) of this process. NER is defined

as the ratio of energy in the biodiesel and total upstream energy demand of the process. By

comparing the maximum NER in different process routes the most promising design of a

microalgae biorefinery process could be selected. The result shows that the most promising

microalgae biorefinery design in terms of acquiring highest estimated NER consists of following

steps: chitosan flocculation, pressure filtration, bead milling, hexane extraction and

acidic/alkaline conditional conversion.

Figure 1 gives the brief layout of the microalgae biorefineries from Slegers et al.[13], the inputs

and outputs of these models are also shown. In this study, apart from the NER the biodiesel yield


is also regarded as one output of the model. These outputs will be changing when the inputs of

the models are varying due to the uncertainties.

Figure 1 Brief layout of one microalgae biorefinery model from Slegers et al.[13] as well as the model inputs

and outputs.

Table 1 The classification of all influential inputs for model A from Figure 1. The optimized values are from

work of Slegers et al. The lower and upper bound values are assumed according to: -20% to +20% of the

optimized value for the first and second class inputs and -10% to +10% of the optimized value for the third

class inputs.

Symbol Parameter Optimized

value

Lower

bound

Upper

bound Unit

The first class: the microalgae physical properties

Fa Algae flow rate 5 4 6 m3

h-1

Ca Biomass concentration 2 1.6 2.4 kg m-3

The second class: the operational variables

Cchi Chitosan concentration 0.214 0.18 0.26 g L-1

Cfflo Concentration factor in flocculation 12.5 10 15

Sstr Stirring speed in flocculation 150 120 180 rpm

Bf Bead filling rate 85 80 95 %

Chex Hexane dosage for extraction 0.15 0.12 0.18 v/v

The third class: uncontrollable values: the accuracies of sub-models and experiential values

Ffe accuracy of flocculation efficiency

sub-model

0 -10 10 %

Rpre

assumed microalgae recovery in pressure

filtration 95 85 95 %

Fde accuracy of disruption efficiency sub-model 0 -10 10 %

RLY assumed lipid extraction efficiency 91 81 91 %

RDY assumed diesel yield of acid conversion 98 88 98 %

Table 2 description of uncontrollable variables for some process unit

Symbol Process unit Sub-model description/experiential values Reference


Ffe Chitosan

flocculation 𝑅 = (84.3 + 17.5𝐶𝑐ℎ𝑖𝑡

∗ − 1.3𝑆∗ − 11.1𝐶𝑐ℎ𝑖𝑡∗ 2

− 3.7𝑆∗2 − 2.6𝐶𝑐ℎ𝑖𝑡∗ 𝑆∗) ∙ 10−2 [14]

Rpre Pressure filtration 95% [15]

fde Bead milling 𝐷 = 17.48𝐹𝐴,𝑖𝑛𝑛1 𝑑𝑏

𝑛2𝐵𝑛3𝑣𝑛4𝐶𝐴,𝑖𝑛𝑛5 [16]

RLY Hexane extraction 91% [15]

RDY Acidic conversion 98% [15]

The inputs of these models are categorized into 3 different classes: microalgae physical

properties, operational parameters and sub-model accuracies. In contemplation of performing

the uncertainty analysis and global sensitivity analysis, all the inputs are varying in their own

ranges. Taking the model with the hexane extraction and acidic conversion as an example, Table

1 shows the classification of all influential inputs, as well as the ranges of them for this model. It

is needed to point out what the sub-model accuracy and experiential values here mean. There

are models and experiential values from other literatures used in each step of this biorefinery:

models like the one for describing algae recovery in flocculation step[14] and the one for showing

the disruption efficiency in bead milling step[16]; experiential values like the microalgae recovery

in pressure filtration step, the hexane extraction efficiency and the acid conversion yield[15].

Table 2 showed the description of these models as well as the assumed values. The inaccuracy

of these sub-models is possibly due to the fact that these models are lack of support from larger

scale experimental data. The range of variation of these uncontrollable values are in between -10%

to +10%, while the range of the physical properties and operational variables are mostly from -20%

to +20%. The 1st and 2nd class variables may have very different varying ranges from each other

in more practical situations, yet here for simplicity the range -20% to 20% is assumed.

2.2 Uncertainty analysis

Monte Carlo Simulation is used to perform the uncertainty analysis for the microalgae biorefinery

model. 10,000 random values for each input of the model are sampled in their dedicated ranges

with the help of quasi-random number generator based on Sobol sequences. Sobol sequences


are a kind of quasi-random low-discrepancy sequences that is first introduced by Sobol [17].

Unlike pseudo-random numbers, i.e. random number generated according to normal distribution,

the Sobol quasi-random numbers are able to cover the entire variable space in a better and

faster way. This method was also suggested and applied in several publications from Saltelli et

al.[1, 18, 19]. There are also other low-discrepancy sequences that could be used for generating

quasi-random numbers according to Bratley et al.[20], however the main purpose of this study is

the comparison of different sampling strategies using these sequences. For more detailed

description of Sobol sequence see the supplement materials, section 2.

The output results of the biorefinery model are then calculated out with 10,000 turns of Monte

Carlo Simulation. The mean and standard deviation of these outputs are used as the tool for

describing how uncertain the model is. A comparison of uncertain outputs results (in this study

biodiesel yield and NER) from different microalgae biorefinery models could be carried out.

Related data can be found in the supplement materials.

2.3 Sensitivity analysis

It is already mentioned in the introduction part that the main method applied in this study would

be global sensitivity analysis. Yet a local sensitivity test was also performed. The once-at-a-time

(OAT) method is used for performing such a local sensitivity analysis to have a basic measure.

Data of this analysis is included in the supplement materials, section 3.

There are different methods for global sensitivity analysis: standardized regression coefficients

analysis, elementary effect method and variance-based analysis. The detailed description of

these two method is included in the supplement materials, section 4. The standardized

regression coefficients analysis (SRC) is a rather simple and cheap method. The basic idea of


such a method is to perform linear regression on model outputs from Monte Carlo Simulation.

Sin et al. used this method in the study of prioritizing the source of uncertainty for wastewater

plant models [9]. The disadvantage of this method is that it could be irrelevant for non-linear

models. It also can be totally misleading for non-monotonic models[18]. Morris first introduced

the elementary effect method (EE)[21]. This method shares the advantage of simplicity as well

as disadvantage of poor performance on non-monotonic models with the SRC method.

Campolongo et al. improved this method in terms of both the definition of the method and the

sampling strategy, yet it is still not so widely used as a tool for sensitivity analysis[22].

Variance-based sensitivity analysis is applied in this study because of its capacity in being

suitable for extensive cases. Particularly in computation, the Sobol-Saltelli indices are used to

estimate the sensitivity of model inputs. There are two kinds of indices for calculation: the first

order sensitivity indices for main effect of the inputs, and total order sensitivity indices that also

included the interactions among inputs. The basic point of this method is to partition the total

variance to the sub-variance from each input, and use this sub-variance as a sign of importance

of the input. For a model

𝑌 = 𝑓(𝑋1, 𝑋2, … , 𝑋𝑘)

with the scalar output 𝑌 computed by Monte Carlo Simulation and uncorrelated different inputs

(𝑋1, 𝑋2, … , 𝑋𝑘) for the model, the total variance could be decomposed as:

𝑉(𝑌) =∑𝑉𝑖𝑖

+∑∑𝑉𝑖𝑗𝑗>𝑖𝑖

+∑∑∑𝑉𝑖𝑗𝑢𝑢>𝑗𝑗>𝑖𝑖

+⋯+ 𝑉12…𝑘

And ∑ 𝑉𝑖𝑖 is the main contribution to the total variance from one input 𝑋𝑖 , while ∑ ∑ 𝑉𝑖𝑗𝑗>𝑖𝑖 +

∑ ∑ ∑ 𝑉𝑖𝑗𝑢𝑢>𝑗𝑗>𝑖𝑖 +⋯+ 𝑉12…𝑘 is all the decomposed parts that 𝑋𝑖 appears, from two-elements

groups up to k-elements group. These parts indicate the contribution to the total variance in

terms of the interactions between this input and the others.

Thus, the first order sensitivity index of model input 𝑋𝑖 can be written as:


𝑆𝑖 =𝑉𝑋𝑖(𝐸𝑋~𝑖(𝑌|𝑋𝑖)

𝑉(𝑌)

which could be regarded as the average change in the variance of the output mean when the

influence from the input 𝑋𝑖 is removed. On the other hand, the total sensitivity index of the same

model input 𝑋𝑖 can be written as:

𝑆𝑇𝑖 =𝑉(𝑌) − 𝑉𝑋~𝑖(𝐸𝑋𝑖(𝑌|𝑋~𝑖)

𝑉(𝑌)=𝐸𝑋~𝑖(𝑉𝑋𝑖(𝑌|𝑋~𝑖)

𝑉(𝑌)

The total sensitivity index estimates the main effect from the input 𝑋𝑖 plus the interactions

between 𝑋𝑖 and other inputs. 𝑉𝑋~𝑖(𝐸𝑋𝑖(𝑌|𝑋~𝑖) in the formula can be regarded as the first order

indices of all the inputs except 𝑋𝑖 or any group of inputs including 𝑋𝑖 , so by using the total

variance 𝑉(𝑌) to minus 𝑉𝑋~𝑖(𝐸𝑋𝑖(𝑌|𝑋~𝑖), the rest part will be the variance contribution of all the

possible inputs combinations that included 𝑋𝑖 .

The computation of numerators in these two equations are performed by using an matrix-based

estimator (detailed recipe is shown in the supplement materials, section 5). There are different

kinds of estimators available. In this study an estimator suggested by Saltelli et al.[19] is used.

They performed a comparison of all existing estimators and recommended the one which leads

to a more accurate sensitivity analysis result. The rounds of Monte Carlo Simulation are also

10,000, and Sobol quasi-random number generator is also used as the tool of sampling

according to the suggestion from the same work[19]. The samples for uncertainty analysis and

sensitivity analysis come from the same data file for a lower computation cost.

2.4 Factor mapping with Monte Carlo Filtering

In order to understand how the varying inputs lead to good or bad output results, in this case the

biodiesel yield and the NER, the Monte Carlo Filtering (MCF) is performed. It was first introduced

by Hornberger et al. for environmental models[23]. MCF apportions all samples of one input to


two parts: the ones lead to good model output realizations and the ones lead to bad realizations.

The good or bad of the outputs depends on a standard for judging the performance of the model

like the least acceptable value of the output. These two parts of samples distribute according to

different unknown probability density functions.

A two-samples Smirnov test is employed to compare the empirical cumulative distribution

functions of this input for these two realizations. The Graphical presentation of this test not only

tells whether this input is influential for uncertainty of outputs by showing the maximum distance

between two CDF curves, but also shows in which range of this input that it is tended to have a

good or bad output result. Nevertheless this method has its drawback as a tool for telling the

importance of the model inputs towards output uncertainties since it may not cover many

interaction structures[1]. This is also the reason why the Sobol-Saltelli indices method is still

employed as the main method for input prioritizing in this study.

In this study, the boundary conditions of biodiesel yield and NER are the optimized values from

work of Slegers et al[13]. This test also directly utilizes the samples as well as output results of

Monte Carlo Simulation from the uncertainty analysis.

3. Results

3.1 Sensitivity analysis on categorized model inputs

Before using the sensitivity analysis, an uncertainty analysis was performed on 3 alternatives of

the original microalgae biorfinery model from Slegers et al[13] (see Figure 1): the 12-inputs

model (all 3 classes inputs together), the 10-inputs model (the 2nd and 3rd class inputs) and the

7-inputs model (the 1st and 2nd class inputs, which are all controllable). The excluded model

inputs in 10-inputs and 7-inputs model were fixed with their reference values. Table 3 shows the


mean and standard deviation of results of 10,000 runs of each model. The standard deviation

indicates the degree of the outputs uncertainties.

Table 3 the result for uncertainty analysis on 12-inputs, 10-inputs and 7-inputs models

12 inputs 10 inputs 7 inputs

Yield Mean (𝐿 ∙ ℎ−1) 2.2571 2.2804 2.6525

Standard deviation 0.4242 0.1814 0.4548

NER Mean 1.3567 1.3747 1.5716

Standard deviation 0.1640 0.1036 0.1650

Then a variance-based sensitivity test was performed first on the model that contains all 12

model inputs from 3 different categories to investigate: 1) the main effect and the interaction

effect of each model input; 2) how dominant the 1st class inputs are compared to the 2nd class

and 3rd class inputs; 3) among the 1st class inputs which one is the more influential factor for the

output uncertainties. The result of the test is given in Figure 2. The high similarity of first order

sensitivity test and total sensitivity test results suggests that there are very limited interactions

among each model input. For both model (outputs biodiesel yield and NER) the 1st class inputs

are in the dominant position, and the 3rd class inputs also have some influence on the model

outputs, while the 2nd class inputs have the least effect. The two 1st class model inputs have

much higher influences than the other inputs, moreover between the two 1st class inputs the

microalgae concentration is more influential than the volumetric flow rate of microalgae. Some

2nd class inputs, e.g. the concentration factor of flocculation step and the hexane dosage of

hexane extraction step, even have no influence on the uncertainty of biodiesel yield. It also

shows that the model is already well designed for the output biodiesel yield since all the 2nd class

inputs have limited effect on the output.


Figure 2 Result of sensitivity analysis on 12-inputs microalgae biorefinery model for two model outputs:

biodiesel yield (left) and NER (right). The sub-plots stand for the bar graph presentation for first order

sensitivity index (up) and total sensitivity index (down). The categorization of inputs is shown by the boxes in

different colors: blue for 1st

class inputs, red for 2nd

class inputs and purple for 3rd

class inputs, similarly

hereinafter.

Next, the two dominant 1st class inputs were fixed with their reference values since the focus was

shifted from the overview of the system to the more process-design-oriented part. A second

variance-based sensitivity test was employed on the model that actually contains only 10 model

inputs since the two 1st class inputs were fixed to constant numbers. The main aims in this step

are: 1) prioritizing the most influential operational parameter(s) in the process model to ensure

the critical process step(s) that needs improvement (R&D) or a better control; 2) ensure which

process unit might need recollection of experimental data; 3) determine which operational

parameter(s) could be potentially fixed for simplifying the model. The result of sensitivity analysis

is shown in Figure 3. When the main concerning of output is the biodiesel yield, the output

uncertainty is majorly influenced by the inaccuracies of applied sub-models and experiential

values. While for the NER as the main concerning output, both 2nd class and 3rd class inputs

share the effect on the output uncertainty. This suggests the direction for improving the process

design. For a higher biodiesel yield the sub-models and assumed values should be

re-considered. On the other hand, it should be focused on the improvement of both critical

process step(s) and robustness of assumptions in the next research stage for better NER result.




sensitivity index (up) and total sensitivity index (down).

Table 4 The order of priority of all 2nd

class inputs (operational variables) as well as the values of their first

order sensitivity index for both model outputs: biodiesel yield (left) and NER(right).

Output Biodiesel yield NER

The order the input Sensitivity index Name of the input Sensitivity index

1 Cchi (g L-1) 0.0093 Chex (v/v) 0.1959

2 Sstr (rpm) 0.0013 CFflo 0.1578

3 Bf (%) 9.6138e-5 Cchi (g L-1) 0.0033

4 CFflo 2.6083e-16 Sstr (rpm) 4.4975e-4

5 Chex (v/v) 0 Bf (%) 8.5532e-5

Table 5 The order of priority of all 3

rd class inputs (uncontrollable variables) as well as the values of their first

order sensitivity index for both model outputs: biodiesel yield (left) and NER(right).

Output Biodiesel yield NER

The order the input Sensitivity index Name of the input Sensitivity index

1 Ffe 0.5163 Ffe 0.1928

2 RDY 0.1466 RDY 0.1632

3 RLY 0.1451 RLY 0.1616

4 Rpre 0.1450 Rpre 0.0802

5 Fde 0.0356 Fde 0.0394

Table 4 shows the order of priority of all 2nd class inputs for both two model outputs for

determination of critical process step(s), and Table 5 ranks the influence from 3rd class inputs for

making the decision on which uncontrollable variable(s) need(s) higher authenticities. The zero


in Table 4 and Table 5 means that the corresponding inputs could be potentially fixed for the

simplify of the model.

Based on the data above, 2 parts of research questions could be come up with to ensure the

direction for the next development stage. One part emphasis on the microalgae physical

properties, which is keep the constant microalgae concentration in the inflow. Another part

emphasis on the process design. For the biodiesel yield, 1) how to increase the validity of the

sub-model for flocculation efficiency; 2) fix the uninfluential input, the hexane dosage of hexane

extraction step. For the NER, 1) how to improve the model for hexane extraction and chitosan

flocculation are the critical steps; 2) how to control the most influential operational variables

during the running of model; 3) also improve the accuracies of sub-model for flocculation

efficiency as well as the assumed values for hexane extraction efficiency and acidic conversion

rate. This is the approach of utilizing the categorized model inputs for gaining information about

the performance of the biorefinery model and determination of the future plan of research. The

final goal for improving the model is to lower the degree of output uncertainty, which is the

standard deviation calculated in the uncertainty analysis steps.

3.2 Mapping the outputs back to the input space

According to the research from Slegers et al., the biodiesel yield can achieve at 2.7 L/h, and NER

can achieve at 1.6. As a result, in this study all yield values higher than 2.7 and NER values

higher than 1.6 are regarded as good performance of the model, while the outputs lower than

them are regarded as bad performance. This means that the 10,000 results from uncertainty

analysis can be partitioned into two groups. It is intriguing to know how the change of each model

input leads the output to the “good group” or the “bad group”, since it might be helpful for better

control of the biorefinery. This was done by plotting and comparing the cumulative density


function (CDF) curve of better performance results and worse performance results of the model.

Figure 4 shows the result of how each varying input driving the output to a good or a bad

realization for the 7-inputs model which excludes the uncontrollable variables. The reason why

7-inputs model is the target for analysis here is that this test is more relevant for controllable

variables. The larger the gap between better realization CDF curve and worse realization CDF

curve is, the more influential the model input is. The maximum value of this gap is known as

d-stat, which represents the sensitivity of the corresponding input. When the CDF curve of better

realization is steeper, it means that the model output is more likely to have a good realization. In

contrary, when the CDF curve of worse realization is steeper, that means chance is higher to

acquire a bad output realization. So that the range of each model input which is able to lead to a

good model performance could be determined. Table 6 concluded the range for each input that

leads to good realizations and bad realizations. It is quite clear and easy to understand that for

both biodiesel yield and NER the higher the 1st class inputs are, the higher the possibility for

gaining good realizations is. Yet for the NER as the output, it can be noticed that when the

concentration factor is closer to its upper boundary or when the hexane dosage is closer to its

lower boundary, the biorefinery model tends to have higher outputs than the reference values.

This table provided a nice direction for control of the system and diagnosing of the biorefinery

performance. For example, if the hexane dosage in the hexane extraction step could be set close

to 0.12 (v/v) the NER can have an increasing highest to 11% of the reference value, 1.6.


Figure 4 The result of MCF for factor mapping of the controllable inputs (the 1st

class and 2nd

class inputs) for

both two model outputs: biodiesel yield (top) and NER (bottom). The d-stat in the head of each sub-figure

shows the maximal gap between the better realization CDF and worse realization CDF.

Table 6 Range of each model input that leads to most likely good or bad model output realizations. Not related

means the changing of this input barely has influence on driving the model output to a good or bad

realizations.

Range for good performance Range for bad performance

Biodiesel yield NER Biodiesel yield NER

Fa (m3 h

-1) Fa → 6 Fa → 6 Fa → 4 Fa → 4

Ca (kg m-3) Ca → 2.4 Ca → 2.4 Ca → 1.6 Ca → 1.6

Cchi (g L-1) Not related Not related Not related Not related

CFflo Not related CFflo → 15 Not related CFflo → 10

Sstr (rpm) Not related Not related Not related Not related

Bf (%) Not related Not related Not related Not related

Chex (v/v) Not related Chex → 0.12 Not related Chex → 0.18

The result of MCF in Table 6 also indicates the sign of the influence brought by each input of the

model. For instance, unlike other influential factors the hexane dosage of hexane extraction

tends to have a negative influence on the output (NER). However, it is needed to mention that


the range for good performance only means that there is a very high possibility for better

performance when the corresponding input is in this range, since in global sensitivity test the

other inputs are also changing at the same time.

3.3 Global sensitivity analysis for different biorefinery models

Apart from the biorefinery mentioned in part 2.1, Figure 1, two other microalgae biorefinery

models were analyzed. For the layout of these two models as well as the categorization and the

range of their inputs see the supplement material, section 6. One model consists of chitosan

flocculation, pressure filtration, bead milling, supercritical CO2 extraction and acidic conversion.

The other one consists of chitosan flocculation, pressure filtration, drying and microwave

assisted dry conversion. An uncertainty analysis was first carried out for these two models, then

the variance-based sensitivity analysis was performed on both models for all 3 classes of their

inputs. Table 7 compares the uncertainty analysis result of all 3 microalgae biorefinery models.

According to the result the first microalgae biorefinery is the most robust one among all three in

terms of biodiesel yield, while the third biorefinery is the least uncertain one in terms of NER

comparing with the other two.

Table 7 Result of uncertainty analysis on all 3 microalgae biorefinery models mentioned in this work.

Flo-Pre-Bm-Hex-ATrans Flo-Pre-Bm-SCCO2-ATrans Flo-Pre-Dry-Microwave

Yield

Mean (𝐿 ∙ ℎ−1) 2.2571 2.4437 2.8698

Standard

deviation 0.4242 0.5136 0.6432

RSD 18.7940% 21.0173% 22.4127%

NER

Mean 1.3567 0.8347 0.2793

Standard

deviation 0.1640 0.1138 0.0219

RSD 12.0882% 13.1784% 7.8410%

Figure 5 shows the comparison of total sensitivity indices of all 3 microalgae biorefinery models.

The sensitivity analysis not only provides information for coming up with new research questions


for each model, but also reveals general orientation of model development for each model. When

considering biodiesel yield as the main output, improving of upstream process and microalgae

functionality seems to be more important direction for all 3 models. When considering NER as

the main output, the general direction for next step of development differs among these 3 models.

For the first model the 1st class inputs have higher sensitivity indices, which indicates that

improvement on upstream process and microalgae physical properties are very important. For

the second model, one of the 2nd class inputs has quite dominant effect, which suggests that this

process unit should be optimized to have a more robust performance. While for the third model,

one of the 3rd class model inputs has the highest contribution to the output uncertainty, which

means for this biorefinery the sub-model or assumption applied in the model needs to be

reconsidered.

5A

5B

5C

Figure 5 The comparison of Sobol total sensitivity indices of all 3 biorefinery models: 5A) the first microalgae

biorefinery with flocculation, pressure filtration, bead milling, hexane extraction and acidic conversion; 5B)

the second microalgae biorefinery with flocculation, pressure filtration, bead milling, supercritical CO2

extraction and acidic conversion; 5C) the third microalgae biorefinery with floccualtion, pressure filtration,


drying and microwave assisted dry conversion.

4. Discussion

The 7-inputs microalgae biorefinery model, in which the 3rd class inputs are fixed to their

reference values, is actually in the form normally considered in other papers (physical properties

of micro algae plus some intermediate operational variables from the process)[10, 24]. Thus a

variance-based sensitivity analysis is performed on the 7-inputs model as well (see Figure 6). As

predicted the most influential inputs for both outputs are the micro algae concentration and the

microalgae volumetric flow rate. The other inputs seem to have very little influence on the

outputs comparing with the two 1st class inputs. Generally speaking, the results of the sensitivity

test on this biorefinery model provide the information that the uncertainties of biodiesel yield and

NER are mainly due to the concentration of incoming microalgae, secondly because of the

volumetric flow rate of microalgae, and the operational inputs during the process are not so

essential. Based on the provided information researchers could come up with new research

questions like how to improve the performance of upstream reactors to have stable productivities,

or how to acquire better microalgae strains to achieve higher biomass concentrations. Obviously

this provides less research questions than the categorized approach could give.




sensitivity index (up) and total sensitivity index (down).

However, if we ignore the relatively small sensitivity indices values in the Figure 6 and only look

at those five 2nd class inputs, it is also can be told that which one is more essential for the

process design. Similarly, if we also directly use the sensitivity indices of 2nd and 3rd class inputs

in the 12-inputs test (Figure 2) instead of those from 10-inputs (Figure 3), almost same research

questions could be proposed. Yet the point of this work is to strengthen the importance of dealing

with model inputs in the categorizing way to get more information for different purpose of

research objectives, in other words, to think in the categorizing way. That is exactly what we did

for the comparison of different models (Figure 5). For the comparison the models contain all 3

classes of inputs are analyzed, but the understanding of the result is been done in the

categorizing way. On the other hand a clear definition of model inputs is also helpful for a better

understanding of the model.

The d-stat in Figure 4 during the factor mapping test shows the maximum gap between the good

and bad realization CDF curves of each input, which means the degree of importance of that

input. The result also matches up with the 7-inputs sensitivity test (Figure 6). In this case the

d-stat could be used directly as the substitution of Sobol sensitivity indices to show which input is

more influential. However this alternative method is not recommended since the d-stat may not

cover the interaction structures. The reason why the d-stat fits in this case is because there are

less interactions among the inputs of the models used in this work. The MCF also has one

advantage over the Sobol sensitivity indices, which is that it can also tell whether the input brings

positive or negative influence to the output.

During the comparison of the 12-inputs model with other two different microalgae biorefinery

models, we also plotted the result of uncertainty analysis by regarding all biodiesel yield values

from MCS as x-values and all NER values as y-values (Figure 7). It seems that there is a


correlation between the biodiesel yield and NER of the 12-inputs model: the higher biodiesel

yield is, the higher the NER is. The reason of this correlation remains unknown but it might be

related to the layout of the model itself. Moreover, values higher than the optimized values from

Slegers et al.[13] could be found on the figure. Which input contributes more for generating these

higher values (which are the scatter points in the first quadrant of each green coordinate axes)

could be known by employing the MCF factor mapping test. For the MCF result see the

supplement materials, section 7.

Figure 7 The result of uncertainty analysis plotted in a scatter point figure. The numbers in 3 brackets are the

optimized biodiesel yield and NER value for each biorefinery.

There is one final segment need to be improved in the current approach. Sometimes the assays

generate negative values for sensitivity indices. However this should not happen in Sobol’s


approach according to the mathematical definition of the indices. The main reason of the

negative values here could be numerical errors during the computation, which means a method

for benchmarking the error of sensitivity analysis is needed.

5. Conclusion

The inputs-categorizing way of sensitivity analysis shown in this work is able to provide more

information, which is assistive on ensure the direction of the next step of development. Mapping

the output realization back to the input space indicated how each input drive the model output out

of or inside certain boundaries. The final objective of performing sensitivity analysis is to improve

the critical parts of the model to lower the result from uncertainty analysis. The general approach

is shown in Figure 8.


Figure 8 The flow diagram of the general approach of using the sensitivity analysis to improve the model

design. The controller design is not the obligatory step in this approach.


Supplement Materials

1. The settings of sensitivity analysis

Three settings for sensitivity analysis are mentioned in this work: factor prioritizing, factor fixing

and factor mapping. The concept of different sensitivity analysis for different settings were

brought up by Saltelli in his book Global Sensitivity Analysis the Primer[1]. The reason of careful

consideration of settings for sensitivity test is that the aim of analyzing sensitivity could differs

from different situations. The factor prioritizing setting is used for finding the factor that has the

highest influence to the variance of the output. The Sobol first order sensitivity indices are

employed for this setting since it represents the main effect from each factor. While the total

sensitivity indices are employed for the factor fixing setting, which is used to identify the factors in

the model that make no significant contribution to the variance of the output. This requires the

consideration of interaction between factors as well, which is also the main reason why the total

indices are utilized. The third setting, factor mapping, is to study which values of the input factors

lead to model realizations in a given range of the output space. Monte Carlo Filtering (MCF) is the

method to deal with this setting. A clear pre-defining of the setting for sensitivity analysis would be

helpful to make the right choice and prevent the confusion.

2. Sobol quasi-random numbers

The sampling of uncertainty and sensitivity analysis in this study is conducted by using Sobol

quasi-random numbers. Unlike other random generated numbers, the quasi-random numbers are

generated by filling the multi-dimensional hypercube with points which are able to cover the entire

space evenly. Each point has corresponding values on each dimension of the hypercube, all

these values form a random combined group of inputs for the model. Figure 9 shows an example

of a 2 dimensional Sobol quasi-random numbers sampling. The reason why quasi-random

numbers are used in this study is that the typical random generated numbers tend to have

clusters and gaps, which leads to a probably irrelevant analysis result since there are chances

the values within the gaps are ignored. In one word the quasi-random numbers are more

beneficial in the aspect of statistical analysis. A set of sample generated from normal distribution

are also tested for the variance-based sensitivity analysis of 7-inputs model to compare with the

result from Sobol quasi-random sample (see Figure 10). Comparing with the result based on

Sobol quasi-random numbers generator in the Figure 10, the result based on normal distribution

generated numbers is almost the same, only for the NER the total order sensitivity indices of

chitosan concentration is larger than previous result. This suggests that the Sobol quasi-random

numbers leads to better performance for the computation when interaction of model inputs is

considered.


Figure 9 Sobol quasi-random numbers (1024) in two dimensions for variables within zero and one.

Figure 10 Variance-based analysis on the same 7-inputs model but utilized uniform distribution for sampling

instead of Sobol quasi-random number generator.

There is another example for proving the importance of using Sobol sequence for generating

random numbers. Apart from the microalgae biorefinery model mentioned in this work, two model

of another microalgae biorefinery model is also analyzed with the same sensitivity analysis tool. It

is needed to be mentioned that for this model the main purpose of sensitivity analysis is to

investigate how each component of microalgae influence the energy consumption of the

biorefinery, which is a 1st class-inputs-only sensitivity analysis. 3 main fractions of microalgae

biomass are ranked: Lipid fraction, Protein fraction and Lutein fraction (unit: %). The sum up of

this 3 fractions has to be equal to 60%, which means the sampling of random numbers need to be

filtered with this constraint. What we did here is first using Sobol random number generator to


form 100k groups of 3 inputs, then utilizing an if script to filter out the groups which do not

matches up with the constraint of the sum up value. In the end 10K samples were selected from

all the groups leftover after the filtering. These 10K samples are not distributed according to

Sobol sequence any more, and because of the filtering the samples of inputs are not evenly

distributed in the input space. Figure 11 shows the sensitivity analysis result of the model when

using the samples after filtering. It is clear that the first order sensitivity indices does not make

sense since the sum of all 3 indices is over 1, which should not happen for the computation of

Sobol sensitivity test. It is possibly due to the usage of the filtered samples.

Figure 11 Sensitivity test of the influence from microalgae component on the energy consumptions. The 3

fraction should be always added up to 60%.

In order to test out hypothesis here we performed the sensitivity test for the second time on the

same model, but ignoring the constraint that the sum of all fractions has to be 60% so that to have

samples strictly generated according to the Sobol sequence. Figure 12 shows the re-computed

result. This time the values of sensitivity indices are better, which proves the importance of using

Sobol sequence.

Figure 12 Sensitivity test of the influence from microalgae component on the energy consumptions. There are


no constraint for the model inputs.

3. One at a time method (OAT)

One at a time method is one term of the local sensitivity analysis methods. Just as its name

implied, the OAT method means varying one factor over a certain range while keeping the others

invariant at their reference values, and measuring the model response in the meantime. By

performing this test on all the influential factors one by one the sensitivity of each of them can be

determined. The factor leads to more uncertain outputs is the most sensitive one. This method is

easy to compute and low-cost, but it has its own limitations. The substance of OAT is to obtain the

derivative 𝜕𝑌/𝜕𝑋𝑖 of the output 𝑌 towards one of the inputs, 𝑋𝑖. Consequently such a method

can only deal with high-linearity models. It can provide a relevant sensitivity analysis result when

the coefficient of determination R2 is bigger than 0.7 in the regression analysis of one varying

factor. Additionally, the possible interactions between each factor of the model are not taken into

count in this method. However the OAT method is still a useful and contributive tool in sensitivity

analysis as a good indicator of linearity of models.

In this study, in order to have an initial understanding of the model performances, an OAT test

was also employed in Microsoft Excel on the 12-inputs model and the result is shown in Figure 13.

The Figure 13A shows whether the inputs have positive or negative effect on the output. This was

determined by running the model when changing one factor to +20% of the reference value at a

time. The Figure 13B shows an overview of the output changing when the model was run in the

condition that each model input was changed between -20% and +20% of the reference value in

one-at-a-time-wise. Additionally standard deviations were also calculated for the varying outputs

caused by the changing of each inputs, which are shown in the Table 8. The OAT test result

matches up with the global sensitivity test employed in this work, yet the influence from

uncontrollable variables is almost totally ignored. This also shows the advantage of global

sensitivity analysis.

A


B

Figure 13 The result of one-at-a-time sensitivity test. A) the tornado figure showing the positive (blue) or

negative (red) effect from each input. B) the margin of output changing when varying one input at a time.

Table 8 Standard deviation of both biodiesel yield and NER when changing one model input a time. The higher

the standard deviation is, the more influential the factor is towards to the model output.

Standard deviation

Inputs Biodiesel yield NER

Fa 0.36994 0.109853

Ca 0.389812 0.136358

Cchi 0.031861 0.010955

Cfflo 4.71E-16 0.056559

Sstr 0.009224 0.00313

Bf 4.71E-16 0

Chex 4.71E-16 0.066772

Ffe 4.71E-16 0

Rpre 4.71E-16 0


fde 4.71E-16 0

RLY 4.71E-16 0

RDY 4.71E-16 0

4. A brief introduction of other kinds of global sensitivity analysis 4.1 Standardized regression coefficients analysis

The standardized regression coefficients analysis (SRC) is one term of the global sensitivity

analysis methods. The SRC method is performed in combination with Monte Carlo Simulation

(MCS). For example, for a model 𝑌 = 𝑓(𝑋1, 𝑋2, … , 𝑋𝑖, … , 𝑋𝑘)a vector of output values 𝑌 =

[𝑦1, 𝑦2, … , 𝑦𝑁] can be acquired by MCS. And the values of each input 𝑋𝑖 (𝑖 = 1,2, . . , 𝑘) of the

model also form the vector 𝑋𝑖 = [𝑥𝑖(1), 𝑥𝑖(2), … , 𝑥𝑖

(𝑛), … , 𝑥𝑖

(𝑁)] . A regression analysis can be

performed for the output vector and the input vectors in such a form

𝑦(𝑛) = 𝑏0 +∑𝑏𝑋𝑖𝑥𝑖(𝑛)

𝑘

𝑖=1

Instead of the raw regression coefficient used in the formula, the standardized regression

coefficient 𝛽𝑋𝑖 = 𝑏𝑋𝑖𝜎𝑋𝑖/𝜎𝑌 is more widely used. Here the 𝛽𝑋𝑖 is obtained by fixing raw

regression coefficient 𝑏𝑋𝑖 with the ratio 𝑏𝑋𝑖𝜎𝑋𝑖/𝜎𝑌. The 𝜎𝑋𝑖 is the standard deviation of the input

vector 𝑋𝑖, and 𝜎𝑌 is the standard deviation of the output vector. For a linear model the (𝛽𝑋𝑖)2 is

able to represent the variance contribution of the input towards the total variance. Another way of

understanding this is instead of decomposing the variance in the variance-based method, the

SRC method is more like to decomposing the linearity of the model. 4.2 Elementary effect method The elementary effect method (EE, or Morris Method) is one term of the global sensitivity analysis methods. In one word, EE is the global version of one-at-a-time method. For a model 𝑌 =𝑓(𝑋1, 𝑋2, … , 𝑋𝑖 , … , 𝑋𝑘) which contains 𝑖 inputs, the elementary effect of 𝑖 −th input 𝑋𝑖 is defined as

𝐸𝐸𝑖 =[𝑌(𝑋1, 𝑋2, … , 𝑋𝑖−1, 𝑋𝑖 + 𝑒𝑖∆, 𝑋𝑖+1, … , 𝑋𝑘) − 𝑌(𝑋1, 𝑋2, … , 𝑋𝑘)]

∆

∆ is a fixed step size, and 𝑒𝑖 is the vector of units for the elements in 𝑖 −th column. Firstly, a set

of start values (𝑥1(1), 𝑥2(1), … , 𝑥𝑘

(1)) for all 𝑘 model inputs is defined and the output 𝑌 is calculated.

Then the start value of the first input 𝑥11 is shifted with ∆, while the other inputs still remain their

start values. This leads to a changing of model output, and the 𝐸𝐸1 value can be obtained

according to the equation above. After that, the start value of the second input 𝑥2(1)

is also shifted

with ∆. The value of the first input 𝑥1(1)

remains the already changed value 𝑥1(1)+ 𝑒𝑖∆ in last

round, while the others still remain the start value. A varied output value can be calculated with

these inputs, and the 𝐸𝐸2 value can also be obtained. This goes on until every inputs are

changed, and a set of 𝐸𝐸 values is also acquired as 𝐸𝐸 = [𝐸𝐸1, 𝐸𝐸2, … , 𝐸𝐸𝑘].

These 𝐸𝐸 values are just for this set of start values. The next step is to define more groups of


start values like (𝑥1(2), 𝑥2(2), … , 𝑥𝑘

(2)) and get the sets of 𝐸𝐸 values for all these groups, which can

be shown as:

(

𝐸𝐸1(1)

𝐸𝐸1(2)

𝐸𝐸2(1)

𝐸𝐸2(2)

⋯ 𝐸𝐸𝑖(1) ⋯

… 𝐸𝐸𝑖(2) ⋯

𝐸𝐸𝑘(1)

𝐸𝐸𝑘(2)

⋮ ⋮ ⋱ ⋮ ⋱ ⋮

𝐸𝐸1(𝑟)

𝐸𝐸2(𝑟) ⋯ 𝐸𝐸𝑖

(𝑟) ⋯ 𝐸𝐸𝑘(𝑟))

where 𝑟 means how many groups of start values are tested. Now the mean 𝜇𝑖 and standard

deviation 𝜎𝑖 of 𝐸𝐸 values from the same inputs (the values in the same column) can be

calculated. The 𝜇𝑖 is an average effect measurement that indicates the main effect from each

input, and the 𝜎𝑖 covered the non-linear and interaction effects. Table 9 shows the comparison of all 4 sensitivity analysis methods. Table 9 The comparison of 4 sensitivity analysis methods mentioned in this work.

Method Simplicity Dealing with

nonlinearity

Consider

interactions

Dealing with

non-monotonic

Computation

time

Show the sign

of influence

OAT method ++ -- -- -- -- ++

SRC analysis ++ - ++ - + ++

Element Effect

method ++ ++ ++ - - +

Variance-based

method + ++ ++ ++ ++ --

Monte Carlo

Filtering + + -- + ++ +

5. The algorithm used in variance-based sensitivity analysis

There are different estimators, or algorithms can be used in performing variance-based sensitivity

analysis. In this study we employed the algorithm recommended by Saltelli et al.[19], and here is

how it works: For a model 𝑌 = 𝑓(𝑋1, 𝑋2, … , 𝑋𝑖 , … , 𝑋𝑘) where the 𝑌 is the model output and the

𝑋1, 𝑋2, … , 𝑋𝑖 , … , 𝑋𝑘 is the uncorrelated inputs,

1) Generate a (N,2k) matrix of Sobol quasi-random numbers, where k is the number of

inputs and N is the amount of values each input has. N can vary from a few hundreds to a

few thousands. The N*2k matrix is then divided into two equal-size matrix A and B:

𝐴 =

(

𝑥1(1)

𝑥2(1)

… 𝑥𝑖(1)

… 𝑥𝑘(1)

𝑥1(2)

𝑥2(2)

… 𝑥𝑖(2)

… 𝑥𝑘(2)

… … … … …

𝑥1(𝑁−1)

𝑥2(3)

… 𝑥𝑖(3)

… 𝑥𝑘(3)

𝑥1(𝑁)

𝑥2(𝑁)

… 𝑥𝑖(𝑁)

… 𝑥𝑘(𝑁)

)


𝐵 =

(

𝑥𝑘+1(1)

𝑥𝑘+2(1)

… 𝑥𝑘+𝑖(1)

… 𝑥2𝑘(1)

𝑥𝑘+1(2)

𝑥𝑘+2(2)

… 𝑥𝑘+𝑖(2)

… 𝑥2𝑘(2)

… … … … …

𝑥𝑘+_1(𝑁−1)

𝑥𝑘+2(3)

… 𝑥𝑘+𝑖(3)

… 𝑥2𝑘(3)

𝑥𝑘+1(𝑁)

𝑥𝑘+2(𝑁)

… 𝑥𝑘+𝑖(𝑁)

… 𝑥2𝑘(𝑁)

)

2) Generate the third matrix Ci which is equal to matrix A apart from the values in the i-th

column are taken from the matrix B:

𝐶𝑖 =

(

𝑥1(1)

𝑥2(1)

… 𝑥𝑘+𝑖(1)

… 𝑥𝑘(1)

𝑥1(2)

𝑥2(2)

… 𝑥𝑘+𝑖(2)

… 𝑥𝑘(2)

… … … … …

𝑥1(𝑁−1)

𝑥2(3)

… 𝑥𝑘+𝑖(3)

… 𝑥𝑘(3)

𝑥1(𝑁)

𝑥2(𝑁)

… 𝑥𝑘+𝑖(𝑁)

… 𝑥𝑘(𝑁)

)

3) Each row of the matrix A, B and Ci can be used to calculate the output 𝑌, which leads to

the formation of these 3 vectors:

𝑦𝐴 = 𝑓(𝐴) 𝑦𝐵 = 𝑓(𝐵) 𝑦𝐶𝑖 = 𝑓(𝐶𝑖)

4) The estimation of sobol first order sensitivity indices 𝑆𝑖 and total sensitivity indices 𝑆𝑇𝑖 of

the input 𝑋𝑖 can be conducted with these formulas:

𝑆𝑖 =1

𝑁∑𝑦𝐵

(𝑗)𝑦𝐶𝑖(𝑗)− 𝑦𝐴

(𝑗)

𝑁

𝑗=1

𝑆𝑇𝑖 =1

2𝑁∑(𝑦𝐴

(𝑗)−𝑦𝐶𝑖

(𝑗))2

𝑁

𝑗=1

It has to be mentioned that in the beginning phase of this thesis we designed the sensitivity

analysis based on another algorithm from Saltelli[6], which is also used by Derlue et al. in their

work[10]. The approach is quite similar. It also start with the generation of a (N,2k) matrix of Sobol

quasi-random numbers and then the matrix is divided into two matrix A and B. However this time

the matrix Ci contains all columns of matrix B except the i-th column that is from matrix A:

𝐶𝑖 =

(

𝑥𝑘+1(1)

𝑥𝑘+2(1)

… 𝑥𝑖(1)

… 𝑥2𝑘(1)

𝑥𝑘+1(2)

𝑥𝑘+2(2)

… 𝑥𝑖(2)

… 𝑥2𝑘(2)

… … … … …

𝑥𝑘+1(𝑁−1)

𝑥𝑘+2(3)

… 𝑥𝑖(3)

… 𝑥2𝑘(3)

𝑥𝑘+1(𝑁)

𝑥𝑘+2(𝑁)

… 𝑥𝑖(𝑁)

… 𝑥2𝑘(𝑁)

)

And this time the first order sensitivity indices and total sensitivity indices could be calculated

from the equations below:

𝑆𝑖 =

1𝑁∑ 𝑦𝐴

(𝑗)𝑦𝐶𝑖(𝑗)− 𝑓0

2𝑁𝑗=1

1𝑁∑ (𝑦𝐴

(𝑗))2 − 𝑓0

2𝑁𝑗=1


𝑆𝑇𝑖 = 1 −

1𝑁∑ 𝑦𝐵

(𝑗)𝑦𝐶𝑖(𝑗)− 𝑓0

2𝑁𝑗=1

1𝑁∑ (𝑦𝐴

(𝑗))2 − 𝑓0

2𝑁𝑗=1

where

𝑓02 = (

1

𝑁∑𝑦𝐴

(𝑗)

𝑁

𝑗=1

)2

We performed the test on the same microalgae biorefinery model but due to it was still at the

beginning stage of the research the model only be modified to have 4 2nd class inputs: chitosan

concentration, steering speed, bead filling rate and hexane dosage. The aim of this test at that

moment was to test whether the Matlab code of sensitivity analysis could work or not and that

was also the reason we tested it on a model with less inputs. The size of MCS at that time was

1,000 suggested by previous work from Martin Stefanov. Later we found out the research from

Saltelli et al.[19] and we used the algorithm and sampling size suggested by their work. We did a

comparison between the two algorithm on the same 4-inputs model with 1,000 and 10,000

samples. The result shown in the Figure 14 below reveals that negative values are quite large

when using the original algorithm for both 1,000 and 10,000 samples MCS. Based on this result

we decided to use the improved algorithm. Additionally, the result of 10,000 is much better, which

suggests a larger sample leads to cancellation of potential numerical errors.

1,000 samples MCS 10,000 samples MCS

Orig

inal A

lgorith

m


Impro

ved

alg

orith

m

Figure 14 A comparison of the original algorithm and the improved algorithm for 1,000 and 10,000 samples

MCS

6. The layout and input categorization of other two microalgae biorefinery model mentioned in

this work

Figure 15 shows the layout and input categorization of microalgae biorefinery consists of chitosan

flocculation, pressure filtration, bead milling, super critical CO2 extraction and acidic conversion.

While Figure 16 shows the layout and input categorization of biorefinery consists of chitosan

flocculation, pressure filtration, drying and microwave assisted dry conversion. The reference

values and ranges of each input employed in the sensitivity analysis are also shown in both

figures.


value Lower bound Upper bound Unit



h-1


The second class: the operational parameters




Bf Bead filling rate 85 80 95 %

T System temperature of the supercritical 308 300 350 K


CO2 extraction

The third class: uncontrollable values: the accuracies of sub-models and assumed values

Ffe Accuracy of flocculation efficiency

sub-model 0 -10 10 %

Rpre

Assumed microalgae recovery in pressure


Fde Accuracy of disruption efficiency


Fyl Accuracy of lipid recovery efficiency in

sub-model 0 -10 +10 %

RDY Assumed diesel yield of acid conversion 98 88 98 %

Figure 15 The layout and input categorization of microalgae biorefinery 2 that is analyzed in part 3 of the result


value Lower bound Upper bound Unit



h-1


The second class: the operational parameters




FMeth Flow rate of methanol in microwave

assisted dry conversion 0.001 0.0008 0.0012 m

3 h

-1

Ccat

Relative catalyst concentration used in

microwave assisted dry conversion 3 2.4 3.6 wt%

The third class: uncontrollable values: the accuracies of sub-models and assumed values

Ffe Accuracy of flocculation efficiency


Rpre

Assumed microalgae recovery in pressure


Fyl Accuracy of lipid recovery sub-model in

microwave assisted dry conversion 0 -10 10 %

Figure 16 The layout and input categorization of microalgae biorefinery 3 that is analyzed in part 3 of the result

7. The results of MCF for factor mapping of inputs from all 3 models

The results of factor mapping of inputs from 3 different microalgae models mentioned in this work


are shown in the Figure 17. It shows that in order to get a better output than the reference output

value of each model, which range the inputs of each model should be in.

17A

17B


17C

Figure 17 The MCF result for factor mapping of all 3 microalgae biorefineries mentioned in this work: 17A) the

first microalgae biorefinery with flocculation, pressure filtration, bead milling, hexane extraction and acidic

conversion; 17B) the second microalgae biorefinery with flocculation, pressure filtration, bead milling,

supercritical CO2 extraction and acidic conversion; 17C) the third microalgae biorefinery with floccualtion,

pressure filtration, drying and microwave assisted dry conversion. The upper sub-plot in each figure shows

the result for output biodiesel yield and the lower one shows the result for output NER.


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Sensitivity Analysis for Biorefineries

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