A meta-analysis: Risk through Wald versus Laplace perspective in hydropower impact analysis, how influence subsistence agriculture

A meta-analysis:
Risk through
Wald versus
Laplace
perspective in
hydropower
impact analysis,
how influence
subsistence
agriculture
13 dicembre
2017
Energy water and Food Security Nexus Course,
Master Program : Sustainable development Jacopo Cantoni

2
INDEX
Sommario
Abstact............................................................................................................................................................... 3
Introduction ....................................................................................................................................................... 3
Methodology ..................................................................................................................................................... 7
Results ............................................................................................................................................................. 11
Discussion........................................................................................................................................................ 14
Conclusion ....................................................................................................................................................... 15
Reference List.............................................................................................................................................. 17
Appendix I.................................................................................................................................................... 19

3
Abstact
The study is an analysis on the impact of the use of two method in a bigger study. The two method
are the Laplace and Wald one used to filer the uncertainty. While on one side Laplace focus on
analyzing the general trend the Wald method focus on the extreme value. This mean that the first
method is more focus on increase efficiency while the second in guaranty a minimum performance.
This is one of a numerous small choices encounter in the conceptualizing process is than relevant
understand the importance of this small choices.
Introduction
In the 1972 Donella Meadows and hers research team, with the book Limits to Growth, have show
to the humankind the possible threats related to the physical boundaries of our planet.(Caradonna,
2014) In a finite world the resources available to sustain the wellbeing of a growing population are
also finite, that mean that to securitize a certain level of life quality is important use at the best what
our planet is giving to us. In this contest the science around resource management (RM) is became
relevant to find an optimal way to use the natural resource in a way that don’t undermine the
possible wellbeing of the future generation. To have a better RM is important understand the
various trade-offs between different resources to avoid negative side effects and at the same time
highlight possible synergy, as can be observed in the energy water and food security nexus
theory(Smajgl, Ward and Pluschke, 2016).
The energy water food nexus can constitute a basic map that can guide us in the complexity of such
issues, where each of this three resource can be an entry point in the discussion and from there is
possible to move through the most common connections between this disciplines. Through this
process is also possible develop the analysis and include other aspects as ecosystem, land and
minerals use relevant to specific cases. Than we can use this tool to adapt our study to the
specificity of every particular situation.(Smajgl, Ward and Pluschke, 2016). Complexity is one of
the mayor problematic that affect the issues that we are trying to solve nowadays, and energy water
food nexus fall in a series of theory that try to deal with it. In order to handle complexity there is a
strong emphasis on the role of science for building models that can help in the decision making
process but at the same time there is also a strong will to include public participation (Armitage et
al., 2009). This interest on a bottom up approach is translated in politics as the European water
framework directive (EU, 2016) but this means that on one side we need to analytically analyze the
complexity of problems with a collaboration of various experts coming from various
discipline(Smajgl, Ward and Pluschke, 2016) and on the other side open up this process to a wide
audience. The result of this is a really heterogamous arena where some actors are speaking different

4
language from different academic context and some other actors can be even not able to enter in the
discourse due to kind of language used this can lead to big communication problems.
The heterogeneity of the actors in the arena have to be adequately take in consideration. Actors that
have more power can be able to manipulate the discussion, and in some case it can became an
empty tool to impose a decision chose by an elite (Ganesh and Zoller, 2012). In other cases
approach the issue with different frameworks can lead to communication problem (Gray, 2003)
that can be the case if the same issue is analyzed by two scholar from two different disciplines.
From this observation we can develop the aim of our project. As we have seen the necessity to deal
with complex system use the science knowledge to create simplified models of the real world and
then use the observation on the model to take decision for an optimal and secure RM. During the
process that conceptualize reality to a simplified model there is the need of numerous small
methodological choices. This small decision can be crucial nodes where a miscommunication can
happen due to a different framework on the issue or due to a will to use this small details to
manipulate the decision process. Then the aim of this research is to analyze how one of this
methodological choices can influence the final result of the process. To explore this thematic the
project will focus on the use of Wald or Laplace approach to analyze the risk in a decision making
process affected by uncertainty.
To do this we use as an entry point to the nexus theory the water resource and its link to food
security through the demand for crop irrigation. One of the nowadays challenge is to be able to
guaranty a reliable access to food for a growing population: a decreasing trend of the food prices
seems indicate an availability of the resource, however some scholars believe that this trend will not
last and then there is a necessity to not rely only on market mechanisms for food production
(Godfray et al., 2010). Subsistence agriculture can be an important way to access to the food
resource specially in contest of low income (Baiphethi and Jacobs, 2009). A fundamental aspect for
an efficient local food production is the access to water, as irrigation is a key element for a good
harvest (Godfray et al., 2010). The importance of subsistence agriculture can be even more relevant
if we take in consideration the growing competition between crop for energy against crop for food
that will impact more heavily on the poorest part of the world (Pimentel et al., 2009).
Then the nexus link on which will mainly focus this project will be the disposability of water for
agricultural propose and how this can be influenced by the use of the Laplace criteria instead of the
Wald one and vice versa. By looking at the problem by a nexus perspective is simple identify that
the water-crop is not the only aspect involved in the issue, as water have a strong relation also with
energy, and a shortage in this resource can affect energy security through, as an example, his role in
hydropower production(Johansson et al., 2012 IIASA). Then in our project we keep in mind the

5
presence of other interest around the resource water as, but not only, energy but to narrow down the
research we will focus only in the water-agriculture relation.
To investigate the aim the research is responding two to specific research question:
1. In the real practice how many importance is given to the choice of one or other approach
and which one is more used?
2. Does the use of the Wald or the Laplace approach lead to different decisions? And how can
this impact the food security?
Theoretical framework
This project challenge the concept of security in two different ways, but before going to look in to
them is important discuss what security is. A wide definition of security is given by Wolfers that
says: security is “the absence of threats to acquired values” whit the specification that what interest
us is not the threat itself but the possible damages implied by it.(Baldwin, 1997)This definition
match with the three fundamental question that need to be addressed to define what is security: what
do we need to protect? From which threat? In which way we will protect it (von Hippel et al.,
2011)? The definition and this set of question are a good way from where start to understand the
concept of security but this starting point still is vague and to have an operative understanding of
the concept of security is fundamental characterize it (Baldwin, 1997).
The first way in which the project challenges the concept of security is through the bigger
phenomena that is analyzed in the study. The phenomena is the relation between the numerous
small choices that occur in a complex decision making process and the final result. What we have to
protect is the reliability of the final decision. The threat that can undermine the reliability, as we
have seen in the introduction, can be of two type on one side a failure in the communication process
due to different framework (Gray, 2003) or on the other side an exclusion from the decision making
process of the less powerful actors due to a different specific language proficiency (Ganesh and
Zoller, 2012). One of the way suggested to avoid this threat is a strong participation during the
entire project that is able to start a process of social learning where the key feature are transparency,
information and the direct involvement of the stakeholders(Soncini-Sessa, Weber and Castelletti,
2007).
The second way in which the project challenges the concept of security is the choice of compare the
use of the Laplace criteria or the Wald one. In this case we intend security as “robustness” of the
system that is a way that look at the security concept more by an engineering and natural science

6
perspective.(Cherp and Jewell, 2011) The two methods are used to analyze risk and the use of one
or the other give two interpretation of it. In RM the more general goal that we can use is to increase
the net benefit of all the actors involved, but we live in a uncertain world and then the uncertainty of
the world is mirrored in an equal uncertainty in the possible net benefit that can be gained by the
various actors. Different uncertain net benefit for different alternative solution make impossible
create a ranking by solve an optimal control problem. To make approachable the optimal problem
we have to use the aid of a statistic to filter the uncertainty, one common statistic is the expected
value of the benefit. But is not true that every stakeholder is interest in maximizing the expected
value as this became reality only on a long time horizon while for some stake holders this is less
relevant but can be more important the guaranty of a minimum performance. (Soncini-Sessa, Weber
and Castelletti, 2007,Theory cap.2.2.2)
Let’s explain this concept with an example, take an householder that have a certain amount of
expense every month, some of them are basic and not be able to cover them is equal to a really big
lose in the net benefit while the rest of the expense are goods that slightly increase the net benefit.
The householder have to evaluate two different job offers: one is giving a fixed salary that cover the
basic expenses while the second will have a variable salary that in the overall will lead in a bigger
profit but in some month will not be sufficient to cover the basic expenses. The householder’s
choice, in this simple example will be one or the other case if him will take the Laplace approach or
the Wald one.
The Laplace and Wald approach are the most commonly used methods when in a decision making
process there is the need to filter the randomness. The Laplace approach evaluate the expected
benefit that we can gain from each alternative and then is possible operate the optimization problem
on it. In the example above the Laplace criteria will choice the second option, as in the overall the
gain is higher. The Wald criteria instead focus on the extreme case of the series and apply the
optimization problem to it, that mean the focus is on attenuate the magnitude of the extreme event
even if that can result in a total net benefit lower if compared with the Laplace approach. In the
above example the Wald approach result in the job proposal with the fixed salary.(Soncini-Sessa,
Weber and Castelletti, 2007, Theory cap.9 )
In addition to this general concepts of security, our study involve two resource water and food. Both
this two resources have a specific theoretical framework that frame them in term of security. The
water security framework is still quite young (Cook and Bakker, 2012) but one clear point that is
evident in this framework is the multidimensional characteristic of water related issues, that create
numerous overly point whit the Integrated Water Resource Management framework (Cook and
Bakker, 2012; Smajgl, Ward and Pluschke, 2016). This is also evident in the literature review

7
presented later where almost the entire selection of the articles use a multi criteria analyse. Water
security definition have to take in account the productive-destructive nature of this resource and in a
last analysis a good definition of water security have to state on one side the necessity to provide
access to water of acceptable quality, and on the other the protection from water related threat
(Allan, Xia and Pahl-Wostl, 2013).
The food security framework is construct around the future challenge of feed an increased world
population.(Godfray et al., 2010) the relevance of this issue can be seen in the fact that the food
security is the second millennium sustainable development goal whit the focus on stopping the
hunger around the world (UN, 2017). The key action identified to reach food security are increase
the efficiency in our production by reducing the yield gap, increase the production limit and reduce
the waste and at the same time act on the demand side by changing behaviour in diets(Godfray et
al., 2010).
Methodology
The study uses two different methods to address the two research question. To respond the first
research question is used Literature review while for the second one is used a Monte Carlo
experiment.
The selection of the articles used for the literature review is done through the Web of Science
database. Our interest is to identify which criteria between Laplace and Wald is more used and how
much importance is given to the topic. The area of interest of the entire project is about decision
making process on water issues and impact on agriculture. we are interest in case where humans
have the capacity of regulate water flow trough dam, commonly associated with hydropower
production. Then the main research term used are “hydropower” and to exclude studies not relevant
to our scope the result is refine with the terms, “analysis” and “evaluation”. The article selected
have publication dates not older than 2015. A primary selection of the articles is done through
reading the title and the abstract to select that articles that best fulfil the scope of the project. From
that the selected articles are analyzed to find the relevant parts to address the research question.
As was not possible find direct references to the use of Laplace or Wald approach in the articles
there is the needed to extend this two concepts. In this literature review the distinction is done
between studies that have an approach more oriented to find the decision that maximize the return
from the studies that take in consideration extreme values and/or consideration on time distribution
of the effects. To analyze the literature the focus is on the methodological part: where is clearly
stated the formal definition of the objective of the decision making process is an important point to
understand the use of Laplace or Wald approach. Where this information is missed some times is

8
possible deduct this distinction from others elements in the text as the definition of the indicators,
but in some case is not possible find a clear response.
For the second research question was used a Monte Carlo approach. The Monte Carlo approach, in
its essence, is a methodology that use a big number of results from the repetition of the same
experiment to find an explanation to the phenomena (Dunn and Shultis, 2011). The experiment used
in the project have the aim to identify how often use Laplace instead of Wald lead to a different
outcome. The experiment simulate a simple system, where a reservoir receives a series of inflows
and generate a series of outflows that are then compared with an agricultural demand and from that
is possible calculate the deficit. To write the model, inspiration is taken mainly by the books
integrated participatory water resource management: theory and integrated participatory water
resource management: practice (Soncini-Sessa, Weber and Castelletti, 2007). The two books are an
exhaustive report of a complete decision making process on the Maggiore Lake that propose a
participate IWRM paradigm the first book is more focus on the theoretical aspects of the process
while the second is more a description of the specific project related to the Maggiore Lake. This
sources are use as an inspiration for the creation of the model components but also is used the case
of the Maggiore Lake as an inspiration for shape and scales of the synthetic data created and
consequently to define the values of the various variables of the model. The unit of measure used in
the entire model is Mm3
/day. The entire experiment is conduct in a Matlab environment and is
using code created ad hoc for this specific project that can be find in the appendix I.
The result of the single unit of this Monte Carlo experiment is a logic variable of the kind true-false
that respond at the question: on a set of one hundred year daily deficit the use of the two criteria
lead to different choices? To respond to the question for each experiment is important create a series
of deficit with enough numerosity to have a meaning in use the two criteria, than we create 100
years of daily deficit series. The model pick up the better year among the 100 with the Laplace
criteria and the Wald one. As last part of the single experiment the model compare the two
decisions to give a response to our question. Than following the principle of the Monte Carlo
Approach we repeat this experiment other 100 times and count the occurrence of a difference in the
decision by the two criteria.

9
Fig. 1: Sample year synthetic inflow series [Y:Mm^3/day, X:Nday]
Fig. 2: Sample year synthetic decision series [Y:Mm^3/day, X:Nday]
Fig. 3: Sample year synthetic demand series [Y:Mm^3/day, X:Nday]
The model require as input three series: inflows, decision and demand. Inflows generate a series of
possible inflows for the reservoir with a two peak for each year one in the spring and another in the
autumn (Fig 1). Decision is a series of release decision that mean what ideally we would like to

10
release from the reservoir(Fig. 2), is important note that this is not equal to the outflow as we will
see better in the explanation of the reservoir’s model. Last the demand is a series of requested
volumes of water by agriculture(Fig. 3). All the three series are created with specific functions, the
basis of them is an equation of the form:
This equation is inspired by the form of the linear empiric problem PARMAX. (Soncini-Sessa,
Weber and Castelletti, 2007, Theory cap.4) The value of the series at the next step is determined by
a share of the value at time t plus a random component. In normal condition the sum of the
parameters α and β have to be 1 other ways the mean value of the series will increase or decrease
respectively if will be highest or lower of 1. The random component is generated with the Matlab’s
function lognormal with mean 1 and variance 0.5 the lognormal function is suitable in this case, as
it don’t have an upper limit but have a lower limit on zero. It is reasonable for our series, as an
example we think to the inflow: we cannot have a negative inflow (we are not taking in
consideration the evaporation) but, with a really low probability, we can have a really high inflow.
The random variable have his mean set on one is than multiply by a coefficient to scale our series.
This coefficient is chosen with the scope of mimic the behaviour of the real series take as reference
that means that the magnitude and peak position are similar between the synthetic and the real one.
The series of inflow and demand have a behaviour that change during the year, this is obtain by
changing the value of the parameter β.
The last part of the model is the reservoir that take the inflows and the decisions and then return a
series of outflows that can be directly compared with the demand. The basic idea for this part of the
model is that the reservoir have a natural curve of realise, where this depend by the amount of water
present in the lake and is not possible release more than that. In addition we have set a inferior and
upper limit for the regulation to create an admissible regulation area and one where the reservoir
behave in a natural way as in the reference case. The natural curve is define following the linear
release equation:
Where S is the value of the water volume stored and K is a constant characteristic of the lake that
have the dimension of time, following the case of the Maggiore Lake the model use K=5.3[days].
The release equation is true for each instant but the time step of our model is day, than to use this
equation with an acceptable approximation the function that calculate the release use a time step of

11
minutes. This means that this is the function use almost all the computational time. The full
definition of the model is:
Where U(t) is the decision of the series defied above.
Last as we have seen in the introduction and in the theoretical framework the agriculture sector is
not the only one that use the water resource, to simulate this competiveness in the model only a
little share of the outflow is available for the comparison with the demand. As there is no indication
on which value use a sensitiveness analysis is conducted on this specific parameter.
Results
In the literature review are taken in consideration eight different articles (See table below). In the
entire selection of articles is not possible find a specific indication on which method is chose
between the two studied in this project, than as discussed in the methodology a wider definition on
the two methods is taken in consideration. Two articles have to be considered separately in our
analysis. (Agostini, Silva and Nasirov, 2017; Vassoney, Mammoliti and Comoglio, 2017). The
article from Agostini, Silva and Nasirov have a really social perspective on issue related to
hydropower megaproject in Chile one interesting point that is present in the article that is relevant to
our work is the identification of a lack of transparency and a missing dialogue among stakeholders
that have lead to social and environmental conflict and a weak legitimacy of the final
decisions.(Agostini, Silva and Nasirov, 2017) The second article is not a study on a specific case
but similar to this work is a meta-analysis on the planning and management of hydropower plants.
Even if in this article don’t specifically mention the issued studied by this paper we can see two
relevant aspect. The first one is an identification of lack of clarity on the participation issue: indeed
even if is declared a participatory approach in many article is not then explain in which way this
participation have been addressed in the practice. The second point that we can take from the study
is the identification of an absence on a formal way to describe methodology used to address water
related issues, that lead to a difficult interpretation of such studies and often is missed a clear
description of crucial aspect as indicators, criteria, alternative considered, etc etc.(Vassoney,
Mammoliti Mochet and Comoglio, 2017)
From the rest of the articles three of them suggest a Laplace perspective (Dhaubanjar, Davidsen and
Bauer-Gottwein, 2017; Kandulu and Connor, 2017; Mousavi et al., 2017) other two articles are

12
more oriented to a Wald perspective (Lazzaro and Botter, 2015; Niayifar and Perona, 2017) and for
the last article was impossible identify clearly if is more oriented on one or the other method (Šantl
and Steinman, 2015).
Laplace Wald Not Clear
Agostini, Silva and Nasirov, 2017 --- --- ---
Dhaubanjar, Davidsen and Bauer-
Gottwein, 2017 x
Kandulu and Connor, 2017 x
Mousavi et al., 2017 x
Niayifar and Perona, 2017 x
Vassoney, Mammoliti Mochet and
Comoglio, 2017 --- --- ---
Lazzaro and Botter, 2015 x
Šantl and Steinman, 2015 x
In the three article that use a Laplace perspective two use a multi-criteria approach (Dhaubanjar,
Davidsen and Bauer-Gottwein, 2017; Mousavi et al., 2017), this approach is increasing used in the
Hydropower sector (Vassoney, et all, 2017), while the last one use a cost-benefit analysis (Kandulu
and Connor, 2017). The cost benefit analysis is more oriented to a Laplace approach as it sum the
net benefit over the entire project time with an appropriate discounting mechanism(OECD, 2006)
this aggregation than make difficult to look at the aspects that characterize the Wald approach. In
the other two article was possible identify a formal problem formulation in both of the case the
problem is set as a minimization of water deficits for the entire project that is interpreted as a
Laplace approach. (Dhaubanjar, Davidsen and Bauer-Gottwein, 2017; Mousavi et al., 2017). In the
case take in consideration by Dhaubanjar et al. we can find a specification of a measure in the
direction of the Wald approach, as a forced requirement of minimum monthly flow however in the
study itself this measure is considered insufficient but is not take in a further analysis (Dhaubanjar,
Davidsen and Bauer-Gottwein, 2017).

13
In other two study (Lazzaro and Botter, 2015; Niayifar and Perona, 2017), even if we can’t find a
clear claim on a Wald approach, is possible identify an awareness on the issues linked with our
wider definition of it. In the Niayfar and Perona study we can find an attention on the setting of a
compulsory minimal flow and at the same time the use of continuous under threshold tool to
evaluate the various levels of stress during the time.(Niayifar and Perona, 2017) Similarly in the
study from Lazzaro and Botter is given a specific attention on the duration and frequency of high
and low flows and is used the measure of stream-flows correlations to look at the smaller
timescale.(Lazzaro and Botter, 2015)
Lastly in the work of Šantl and Steinman the use of utility functions make difficult without a really
deep description of the methodology understand in which of the two categories fall this study.
The monte carlo experiment have return that roughly around the twenty percent of the time use
Laplace approach or the Wald one give different results. This can be seen in the Fig x where is
show the number of time that by analyzing one hundred different set of scenarios the use of the two
approach lead to different conclusion. The experiment is repeated with different values on the
parameter that define the share of the flow of the river that is allocate to the specific water district.
This value in some way can be seen as an indication on the pressure on the resource as if we
imagine no competition is possible deviate the entire flow but on the other side with a strong
competition only a small share can be allocate to a specific actor. As a general trend there is not a
strong sensitiveness on this value if we exclude really low value of the parameter that lead to a
lower occurrence in the difference between the two methods. Due to the long computational time
the simulation was stopped after three day of running and then the highest part of the value range
was not analyzed, but as it looks that the result of the experiment is became quite stable the study
didn’t repeated the experiment with highest values.

14
Fig. 4: Occurrence of different result between Laplace and Wald by varying the competitiveness parameter
Discussion
From the monte carlo experiment we have found that around one time out of five the use this two
different methods lead to two different results. The preference on one or the other solution is in
some way linked to the individual risk aversion. The risk aversion is an indication on how much
someone is oriented, in condition of the same expected value (gain multiply probability), to an
alternative with a lower return whit higher probability, or to an higher one but less
probable.(Soncini-Sessa, Weber and Castelletti, 2007, Theory cap. 9) It is possible assume that this
personal aversion to risk, in some part depend by personal behaviour, as can be the preference of a
colour or another, but in some other part depend by the robustness of the “system” that sustain this
particular individual. To explain it in a really simplified way, my aversion to risk when I’m
gambling depend by the amount of money that I have in my wallet. If we would like to translate this
simple example to a case more inherent to our study the risk aversion for a farmer depend by his
capacity to sustain itself in another way if the crop fail. This can be the case of subsistence
agriculture that can represent an important source of food for the poorest part of the world
(Baiphethi and Jacobs, 2009) than use a method more risk oriented as the Laplace in some case can
undermine the food security in some yet weak situations. On the other side it is also true that the
0
5
10
15
20
25
30
35
40
1,00E-05
0,00023
0,00045
0,00067
0,00089
0,00111
0,00133
0,00155
0,00177
0,00199
0,00221
0,00243
0,00265
0,00287
0,00309
0,00331
0,00353
0,00375
0,00397
0,00419
0,00441
0,00463
0,00485
0,00507
0,00529
0,00551
0,00573
0,00595
0,00617
0,00639
0,00661
0,00683
0,00705
0,00727
0,00749
0,00771

15
Laplace method, by focusing on the expected value on the long term, as a general guide line,
increase the efficiency, that is a fundamental aspect in the challenge of feed an always increasing
population(Godfray et al., 2010).
We have to remember that our analysis is limited only to one of the numerous small choices that is
possible encounter during the decision making process. This observation give a starting
understanding on the importance for a reliable methodology. This is also observed in the literature
review where is highlighted the absence of a clear common methodology (Vassoney, Mammoliti
Mochet and Comoglio, 2017). This observation about the diversity in the methodology without a
clear standard can became even more problematic in a cross-sectoral situation where there is the
need to coordinate competence from different disciplines. In the literature review are also
highlighted the criticalities around the participation issues as noted by Agostini et Al. that observe
how the hydropower planning in Chile present a lack in transparency that can lead to a weak
legitimacy (Agostini, Silva and Nasirov, 2017). Than we can see how in the methodological
choices can be hidden the undemocracy of the process. This is in contrast with the apparent trend in
the policies that ask for more participation as the European Water Framework directive (EU, 2016).
The participation can represent a key point to solve some of the challenges. New strategies are need
for RM as can be an adaptive management approach, but the application of it can became even
more complicate if we take in consideration bigger regional scales, is than important a continuous
reflection to be able to match our work to the varying contexts. (Allan, Xia and Pahl-Wostl, 2013)
The necessity to adapt the various studies to specific context can be an explanation on why is so
difficult find a common methodology. There is than a need for a methodology that at the same time
is able to be adapted to the specificities of each case, but that is also consistent within the other
studies. There is the need to activate a social learning process to be able to open up a discussion also
on the methodological part with the stakeholders and use the maximum transparency
possible.(Soncini-Sessa, Weber and Castelletti, 2007, Theory Cap. 1.2)
Conclusion
To conclude, in our study we have notice from the literature review a really scarce attention on the
specific topic of the use of Laplace or Wald method even if the review suggest that the general trend
is slightly more oriented to a Laplace thinking. It also true that one important limitation is the
amount of resource available for a particular study: if we look at the study used in the theoretical
framework that extensively address the issue of chose between the two methods (Soncini-Sessa,
Weber and Castelletti, 2007) the size of this study is not comparable with the size of the work taken
in consideration during the literature review. At the same time from our experiment we have see

16
that in a significative number of case the choice of one or the other method lead to different
decisions. A further analysis whit a less simplified model is needed to have a real understanding on
how the two different decision from the two method impact the real life, but by looking at the scope
of the two different method the Wald one is the one that can increase the food security specially in a
scenario of high variability in water flows. This is due to the focus of the Wald method in guaranty
a better minimum performance that can be fundamental specially for weaker actors involved.
From the response to the research questions come the suggestion for a clear methodology that can
be adapted with the local knowledge to be able of obtain a management as much efficient as
possible, to have a secure access to the wellbeing of today. At the same time we have to notice that
the concept itself of efficiency is something that cannot be defined, as it change case by case. If we
take the example of the Laplace and Wald approach there is not a clear response on which one is the
better method because the first can be more suitable for some stakeholders while the second for
others. As experts our role is to be aware of the difference between the various method, and involve
all the actors in a inclusive discussion, to understand which methods are needed to reach the kind of
efficiency need for a specific situation. The definition itself of efficiency need also to be developed
trough a dialogue between different expertises and all the others actors involved in each specific
case.

17
Reference List
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from Sustainability Perspectives’, Sustainability, 9(6), p. 1073. doi: 10.3390/su9061073.
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19
Appendix I
Functions
function[in]=inflow(mu,sigma,coef,anni)
%this function is creating a syntetic series of inflows for the number of
%years indicated as an imput.
%
%I will create a coulomn vector of intitial random condition for each years
Coef=rand(1,anni)*coef; %is creating a line vector beatween 0 and the value of
coef
COEF=repmat(Coef,364,1);
in0=(lognrnd(mu,sigma,[1,anni])).*Coef;
epsilon=(lognrnd(mu,sigma,[364,anni])).*COEF;%is creating a matrix of
disturbance based on a lognormal distribution
in=(ones(365,anni))*NaN;%allocating the matrix
in(1,:)=in0;%initialazing the firs coulomn of the matrix with the random value
of the first day of the year
%cration of parameter for the magnitude of the peak
y=0.02;
z=0.03;
%the next for cicle is giving the shape to the sintetic series
for i=1:364
%definition of logic variables for the localization of the peak
%(15mar,15may,15jul,14sep,15oct,15nov)
a=(i<74)+((i>=196)*(i<257))+(i>=319);
b=(i>=74)*(i<135);
c=(i>=135)*(i<196);
d=(i>=257)*(i<288);
e=(i>=288)*(i<319);
in(i+1,:)=max(0,((0.5.*in(i,:)+(0.5.*epsilon(i,:))*a)...
+(((0.5+(y*(i-74))).*epsilon(i,:))*b)...
+(((0.5+((y*61)-(y*(i-135)))).*epsilon(i,:))*c)...
+(((0.5+(z*(i-257))).*epsilon(i,:))*d)...
+(((0.5+((z*31)-(z*(i-288)))).*epsilon(i,:))*e)));
end
function[outT]=outflow(liminf,limup,S,U,IN)
%the function calculate the outflow given as an imput the limitinf that is
%the minimium limit where we don't regulate the reservoir and we leave it
%flow naturaly similar for limup but at the superioror limitand S is a vector of
all
%simulation for a specific day of the year and same for U but with the
%decision.
%
%
%
t=1440;%min for each day
beta=5.3*t;%timeconstant for the lake in minutes
%
in=IN./t;%divide the inflow for the subtimestep (semplification inflow constant
during the day)
%
%
%
%is calculating the the outflow at a timestep of e minute for one day
%initialization
out=max(0,(1/beta).*S);%*deltaT that is 1[min]
s=S-out+in;
OUT=out;

20
%
for i=1:t
out=max(0,(1/beta).*s);
s=s-out+in;
OUT=OUT+out;
end
%
%creation of logic variables
a=(OUT<liminf)+(OUT>limup);
b=(OUT>=liminf)+(OUT<=limup);
outT=max(0,((OUT.*a)+(min(OUT,U).*b)));
end
function[u]=decision(mu,sigma,coef,anni)
%this function is creating a syntetic series of relase decison for the number of
%years indicated as an imput. mu and sigma are basic varible for the log
%normal probablility distribution coef increase or decrease the magnitude
%of the series.
%
%
coef
u0=(lognrnd(mu,sigma,[1,anni])).*Coef;
u=(ones(365,anni))*NaN;%allocating the matrix
u(1,:)=u0;%initialazing the firs coulomn of the matrix with the random value of
the first day of the year
y=0.02;
z=0.03;
for i=1:364
u(i+1,:)=max(0,((0.3.*u(i,:))+(0.7.*epsilon(i,:))));
end
function[de]=demand(mu,sigma,coef,anni)
%this function is creating a syntetic series of demand for the number of
%years indicated as an imput.
%
coef
de=(ones(365,anni))*NaN;%allocating the matrix
de(1,:)=0;%initialazing the firs coulomn of the matrix with the random value of
the first day of the year
y=0.02;
for i=3:367
%definition of logic variables for the localization of the peak
%(15mar,15may,15jul,14sep,15oct,15nov)
b=(i>=90)*(i<167);
c=(i>=167)*(i<243);
de(i+1,:)=max(0,((0.2.*de(i-2,:))+(0.3.*de(i-1,:))+(0.4.*de(i,:)...

21
+(((0.1+(y*(i-90))).*epsilon(i,:))*b)...
+(((0.1-(y*(i-167)))*epsilon(i,:))*c))));
end
de=de(4:end,:);
end
function[maxyear]=maxyear(DEF)
%the function is taiking as imput a matrix return a vector with the maximum
%value of each of column
a=size(DEF);
year=a(2);
%allocating
maxyear=ones(1,year)*NaN;
for i=1:year
maxyear(i)=max(DEF(:,i));
end
end
Main
clear
clc
%var1=0.00001:0.00001:0.1;%range of outflow shares deviated to crop
var1=1;
step1=length(var1);%number of different var1 tested
step2=100;%number of time we replicate the proces to see the variability
%allocation
a=NaN;
b=NaN;
c=NaN;
d=NaN;
e=0;
f=0;
occurence=ones(1,step1)*NaN;
%general script
%unit mesure are Mm^3 of water and represent the amount of water in the
%time unit (day)
%
%
for j=1:step1
e=0;
for y=1:step2
%general variable
year=100;
limINF=75;
limUP=450;
%
%genereate random variable of inflow and decision
%
%generate a series of of syntetic inflow (100 years)
IN=inflow(1,0.5,98,year);
%generating a random series of decison on the realase from the reservoir
u=decision(1,0.5,150,year);
%
%
%allocating matrix for outflow values and storage
S=ones(365,year)*NaN;
OUT=ones(365,year)*NaN;
%initialization

22
s=rand(1,year)*1200;
OUT(1,:)=outflow(limINF,limUP,s,u(1,:),IN(1,:));
S(1,:)=s+IN(1,:)-OUT(1,:);
for i=1:364
OUT(i+1,:)=outflow(limINF,limUP,S(i,:),u(i+1,:),IN(i+1,:));
S(i+1,:)=S(i,:)+IN(i+1,:)-OUT(i+1,:);
end
%calculating the demand
DE=demand(1,0.5,5,year);
%calcualting the deficicit on the demand
def=max(0,DE-(OUT.*var1(:,j)));
%
%find the year with the mean lower deficit Laplace
DEFmean=mean(def);
Lmin=min(DEFmean(:));
Lminposition=find(DEFmean==min(DEFmean(:)));
Lyear=def(:,Lminposition);
%find the year with the max lower deficit Wald
DEFmax=maxyear(def);
Wmin=min(DEFmax(:));
Wminposition=find(DEFmax==min(DEFmax(:)));
Wyear=def(:,Wminposition);
a=(((Wmin==0)-1)*-1);%verifing not banal solution
b=(((Lmin==0)-1)*-1);%verifing not banal solution
c=(((Lminposition(1,1)==Wminposition(1,1))-1)*-1); %verifing if the two
condition chose the same year
d=(a(1,1)+b(1,1))*c(1,1);
e=e(1,1)+d(1,1);
end
f=j;
occurence(:,j)=e(1,1);
end

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