2199 © IWA Publishing 2011 Water Science & Technology | 63.10 | 2011

Application of the Morris method for screening the

influential parameters of fuzzy controllers applied to

wastewater treatment plants

M. V. Ruano, J. Ribes, J. Ferrer and G. Sin

ABSTRACT

In thispaper,weevaluate theapplicationofa sensitivity analysis to helpfine-tuninga fuzzycontroller for a

biological nitrogen and phosphorus removal (BNPR) plant. TheMorris Screeningmethod is proposedand

evaluated as a prior step to obtain the parameter significance ranking. First, an iterative procedure has

been performed in order to find out the proper repetition number of the elementary effects (r) of the

method. The optimal repetition number found in this study (r¼ 60) is in direct contrast to previous

applications of the Morris method, which usually use low repetition number, e.g. r¼ 10∼ 20. Working

with a non-proper repetition number (r) could lead to Type I error (identifying a not-important factor as

significant (false positive)) as well as Type II error (identifying an important factor as not significant (false

negative)), hence emphasizing the importance of finding the optimal repetition number for each study in

question. With the proper r found, the Morris Screening helped identify the parameter significance

ranking, thereby facilitating the calibrationof fuzzy controllers,whichcontainmanyparameters that need

to be adjusted for different wastewater treatment plant (WWTP) applications.

doi: 10.2166/wst.2011.442

M. V. Ruano (corresponding author)J. RibesDep. Enginyeria Química,Universitat de València. Dr. Moliner,50. 46100 – Burjassot, València,SpainE-mail: m.victoria.ruano@uv.es

J. FerrerInstituto de Ingeniería del Agua y Medio Ambiente,Universidad Politécnica de Valencia,Camino de Vera s/n. 46022,Valencia, Spain

G. SinCAPEC-Department of Chemical and Biochemical

Engineering,Technical University of Denmark,Building 229, DK-2800 Kgs. Lyngby,Denmark

Key words | fine-tuning, fuzzy controllers, parameter significance ranking, screening method,

sensitivity analysis

INTRODUCTION

Wastewater treatment plant (WWTP) models are used formany applications/purposes including plant design, optim-isation and control. It is generally accepted that the

modelling and simulation of WWTPs represents a powerfultool for control systems design and tuning. However, themodel predictions are not free from uncertainty, as these

models are an approximation of reality (abstraction), andare typically built on a considerable number of assumptions.In this regard, sensitivity analysis provides useful infor-mation for the modellers as this technique attempts to

quantify how a change in the input model parameters affectsthe model outputs. Different strategies have been applied inthe literature (see, for instance, Saltelli et al. ), which

are typically classified in two main categories: global sensi-tivity analysis, where a sampling method is taken and theuncertainty range given in the input reflects the uncertainty

in the output variables (Monte Carlo analysis; FourierAmplitude Sensitivity Test (FAST), Morris Screening(1991)); and local sensitivity analysis, which is based on

the local effect of the parameters in the output variables(Weijers & Vanrolleghem ; Brun et al. ).

Fuzzy logic based controllers have been successfully

applied to wastewater treatment processes (see e.g. Ferreret al. ; Serralta et al. ), since fuzzy sets theoryoffers an effective tool for the development of intelligent

control systems (Zhu et al. ). Fuzzy control algorithmscan be used to create transparent controllers that are easyto modify and extend because the fuzzy-rules are writtenin the language of process experts and operators (Yong

et al. ). Although these control systems have beenshown to be more robust than classical controllers (Manesiset al. ; Traoré et al. ), they usually contain quite a

number of parameters, which complicates their calibration.So far, these control systems have been tuned by trial anderror methods, based on technical knowledge on the process

and controller performance (Chanona et al. ). Whateveroptimisation method is applied, the fine-tuning of these con-trollers requires a previous selection of the most important

2200 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

parameters to be adjusted in each particular application.

Sensitivity analysis can be used to find the proper par-ameters for fine tuning of a fuzzy controller applied toWWTP plant operation. Ruano et al. () proposed a sys-

tematic approach for fine tuning of fuzzy controllers basedon model simulations that employs three statistical methods:(i) Monte-Carlo procedure: to find proper initial conditions,(ii) Identifiability analysis: to find an identifiable parameter

subset of the fuzzy controller based on local sensitivityanalysis and (iii) minimization algorithm. However, thismethodology is based on local sensitivity analysis, and

then requires an iterative procedure to confirm that theidentifiable parameter subset does not depend on the localpoint in the parameter space where the identifiability study

has been carried out. Global sensitivity analyses are pro-posed to overcome the problem of selecting the properinitial point in parameter space.

In this work, a global sensitivity analysis is applied aimed

at screening the most influential parameters of the fuzzy-con-trol systems to be used in the fine-tuning procedure. TheMorris method of Elementary Effects (EEi) (Morris )

was used as sensitivity analysismethod. In addition, the impli-cations of using this methodology at the usual low repetitionnumber of elementary effects (r¼ 10∼ 20) is assessed. This

screening approach is proposed as a prior step to obtain theparameter significance ranking and the best initial point tocarry out the aforementioned identifiability study.

MATERIAL AND METHODS

WWTP and control system description

The fuzzy logic-based control system was implemented

to control the aeration in a nutrient-removing WWTPbased on a modified University of Cape Town (UCT)scheme (see Figure 1).

This control system was previously developed by theresearch group and it has been applied in several full scaleWWTPs (Ribes et al. ). The developed control system

consists of two hierarchical control layers: the supervisorycontrol and the process control layer. In the process controllayer, several independent fuzzy logic-based control loopswere developed for two process variables: dissolved

oxygen in each one of the aerobic zones and blowers thatsupply air and maintain pressure in the air pipelines.The first controller manipulates the air valve opening

according to dissolved oxygen (DO) concentrations, whilethe rotational speed of the blower is manipulated by a

frequency converter to reach the blowers’ pressure set

point. Since each control valve is governed by an indepen-dent DO controller, the air pressure controller isimplemented in order to enhance the overall control

system performance, especially when there is more thanone control valve in the same air pipeline system. This con-troller aims at obtaining the ideal situation where themovement of one valve, which is governed by its own DO

controller, does not affect the air flow rate through theother valves in the same air pipeline system. In the supervi-sory control layer, the optimum set point of the blowers’

pressure is established according to the control valves’ open-ing and the DO concentrations. In this way, the dischargepressure level remains at the minimum level required to

maintain the DO concentration at the set points in all theaerobic zones. Thus, this control system allows differentDO set points to be fixed for different reactors, which makesit possible, for instance, to achieve simultaneous nitrifica-

tion-denitrification, as was found in Denia WWTP (Ribeset al. ). However, it should be mentioned that only oneaerobic zone has been considered in this case study (see

Figure 1) in order to reduce the computational cost of the sen-sitivity study. Thus, the aeration control system is simplified toone DO controller and one air pressure controller.

For the DO controller, the input variables are the oxygenerror (OE) and the accumulated oxygen error (AOE) and theoutput variable is the increment/decrement of the air valve

opening (IV). For the air pressure controller, the input vari-ables are the pressure error (PE) and the accumulatedpressure error (APE) and the output variable is the incre-ment/decrement of the rotational speed of the blower (IB),which is governed by a frequency converter. Both controllersare fuzzy logic-based controllers, which consist of five stages.Figures 2(a) and (b) show these five stages of the dissolved

oxygen controller and the air pressure controller, respectively.The first stage and the last stage are the same for

both controllers. Stage 1 is the input stage (the measured

or calculated variables are input to the controller) andstage 5 is the output stage. In stage 2, so-called fuzzification,the data collected from on-line sensors are converted

into linguistic variables (fuzzy set), represented by member-ship functions (Gaussian shape in this study). In stage 3, aset of rules (so-called inference engine) are applied tothe fuzzy set obtained in stage 2. The output linguistic

variables are obtained in this stage by the Max-Prod oper-ator, following the Larsen’s fuzzy inference method(Larsen ). These linguistic variables are converted

into numerical control actions in stage 4, which iscalled defuzzification. In order to obtain a single output

Figure 2 | Fuzzy control stages in the dissolved oxygen controller (a) and in the air

pressure controller (b).

Figure 1 | Flow diagram of the control system applied to a modified UCT process.

2201 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

value from our fuzzy linguistic set, the Height Defuzzifiermethod was employed (Mendel ), which only uses thecentre of the Gaussian defuzzification membership func-

tions. The total number of parameters of both controllerscomes from the different stages of the fuzzy logic-basedcontrollers, mainly derived from the defuzzification and

fuzzification steps (Ruano et al. ). Both controllersuse three Gaussian membership functions to fuzzify eachinput (“High Negative”, HN; “Low Negative”, LN; and

“Low Positive”, LP), which gives a total of 12 membershipfunctions from both fuzzification stages; and four to defuz-zify each output (“High Negative”, HN; “Low Negative”,LN; “Low Positive”, LP; and “High Positive”, HP),which gives a total of eight membership functions from

both defuzzification stages. As each Gaussian curve is

defined by two parameters (centre, c, and amplitude, a),the control system has a total of 24 parameters corre-sponding to the fuzzification stages. In contrast, for the

defuzzification stages, only the centres are used, giving atotal of eight parameters. Including the response time ofthe control system (RT ), i.e., time interval between two

control actions, a total number of 33 parameters need tobe adjusted. In order to identify the parameters of thiscontrol system, acronyms for each parameter have beenused. These acronyms are constructed as follows: “ab-

breviation of input variable”þ “c/a”þ “fuzzification/defuzzification membership function abbreviation”. Forinstance, the acronym OEaHN means the amplitude of

the High Negative membership function for the input vari-able Oxygen Error; and the acronym IVcLN means thecentre of the Low Negative membership function for the

output variable Increment air Valve opening.In order to decrease the computational demand in this

case study, we simplified the control system to 17 par-

ameters, assuming symmetric behaviour of themembership functions defined for each fuzzy variable.This symmetric behaviour involves that the amplitude forthe three Gaussian membership functions is the same and

their centres are equidistant. This control structure is thesimplest one that can be implemented in a WWTP, as theoxygen concentration is controlled in only one reactor.

When more aerobic reactors are to be controlled, thenumber of parameters will increase proportionally. As anexample, this control system has recently been implemented

in Denia WWTP (Denia, Spain) (Ribes et al. ), with ninedifferent oxygen variables and one pressure (as the samegroup of blowers is used to aerate the whole system).Hence, reducing the number of tuning parameters in this

kind of controller is essential.

2202 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

The fuzzy controller and the WWTP model were

implemented and simulated using the WWTP simulationsoftware DESASS (Ferrer et al. ). This software includesthe plant-wide model Biological Nutrient Removal Model

no 1 (BNRM1, Seco et al. ). The simulation strategyconsisted of a steady-state simulation to obtain proper initialconditions followed by 28 days of dynamic simulations. Thelast 14 days were considered to evaluate the performance of

the control system. The standardised influent file fordry weather proposed by Copp () was used in thisstudy. The Integral Absolute Error (IAE, integral of the

absolute value of the time dependent error function) foreach controller (Oxygen and Pressure) was selected asoutput measure. So, in this study IAEO and IAEP are the

output variables.

Sensitivity analysis

Morris screening

The method of Morris () evaluates the so-called distri-bution of Elementary Effects (EE) of each input factor tomodel outputs, from which basic statistics are computed to

derive sensitivity information. This distribution function isdenoted as Fjk, which stands for the distribution of theeffects of the jth input parameter on the kth output. The

EEjk attributable to each input parameter is obtained fromthe following differentiation of model output, syk, withrespect to the input, θj:

EEjk ¼sykðu1;u2;uj þD; . . . ; uMÞ � sykðu1;u2;uj; . . . ; uMÞ

Dð1Þ

whereΔ is a predetermined perturbation factorof θj, syk(θ1, θ2,θ3, θj,… , θM) is the scalarmodel output evaluated at input par-ameters (θ1, θ2, θ3, θj,…, θM), while syk(θ1, θ2, θ3, θjþ Δ,… , θM) is the scalar model output corresponding to a Δ

change in θj. Each input parameter, θj, can only take valuescorresponding to (a predefined set of) p levels within itsrange. The calculation of the elementary effects,EEjk, is repli-

cated r times, at randomly sampled points in the input space,leading to a distribution of EEjk used to infer a global sensi-tivity measure. To do that, an effective one-factor-at-a-time(OAT) design has been developed (Morris ). In this case

study, the input uncertainty for each control parameter wasfixed at 50% of the default value.

To compare the sensitivity measures obtained for differ-

ent model outputs, the scaled elementary effects SEEjk

proposed by Sin et al. () were applied. The resulting

elementary effects of each output variable (IAEO and

IAEP) show the sensitivity of each controller separately(oxygen and pressure controller, respectively). However,both controllers must be tuned as a global MIMO (Multiple

Input Multiple Output) control system. Thus, sensitivityanalysis aiming at selecting the most influential parametersin this MIMO system was carried out giving an equalweight contribution of the scaled elementary effects

obtained from both output variables. The scaled elementaryeffect of the jth input parameter on the weightedcontribution of both output variables was calculated as

follows:

SEEj ¼EEj;IAEO

sIAEOþ EEj;IAEP

sIAEPð2Þ

where σIAEO and σIAEP are the standard deviations of thecorresponding output variables. Once the distribution ofthe scaled elementary effects is obtained, the sensitivity

measures mean (μ) and standard deviation (σ) of each Fjkare determined. In order to identify the influential par-ameters, these sensitivity measures were then interpretedusing the graphical approach proposed by Morris. In this

approach, the values of μ and σ obtained for all the Fjk dis-tributions are displayed together with two linescorresponding to μi ¼ +2SEMi, where the SEMi represents

the standard error of the mean that can be estimated asSEMi¼ σi/

ffiffi

rp

. Parameters with low μ and low σ aredeemed as non-influential (Morris ).

One issue of particular interest is the selection of the res-olution, p, and number of trajectories, r. Cropp & Braddock() pointed out that a good choice of the repetitionnumber (r) is more critical for obtaining a good estimate of

the effects than the resolution (p). In this case study, an opti-mal setting of r was searched with a constant resolution ofp¼ 8. To this end, the number of repetitions of elementary

effects calculations (r) for each distribution Fjk wasincreased until the influential parameters remained moreor less stable, i.e., the Type II error was minimised (type II

error: identifying an important factor as insignificant).Once the ropt was found, the graphical Morris approachwas used to find the significant parameters.

RESULTS AND DISCUSSION

The Morris method was applied to a different number of

elementary effects calculations, r, until the sensitivity of par-ameters remained more or less stable. Table 1 shows the

Table 1 | Sensitivity measures of control parameters at the different r evaluated

r¼ 5 r¼ 10 r¼ 15 r¼ 30

Parameter μ σ Parameter μ σ Parameter μ σ Parameter μ σ

OEaHN 1.728 2.383 OEaHN �1.268 3.621 APEaHN �2.049 4.918 IVcHN �0.980 2.516

APEaHN �1.651 1.831 IVcHN �1.256 3.452 IBcLN �1.150 4.356 APEaHN �0.665 2.930

IBcHN �1.461 1.494 PEcLN 0.859 2.118 IBcHN 1.019 2.601 RT 0.586 1.287

APEcHN 1.371 3.676 IBcLN 0.600 1.798 IVcHN �0.904 2.132 PEaHN �0.445 1.925

AOEcHN �1.158 2.313 APEaHN �0.371 2.960 PEcLN �0.736 2.443 OEaHN �0.437 2.103

OEcHN 0.838 2.005 OEcHN 0.322 2.700 RT 0.662 1.370 OEcHN 0.259 1.806

PEcHN �0.562 1.156 RT 0.303 0.911 OEcHN 0.578 1.233 APEcHN �0.259 1.247

IBcLN 0.300 1.989 PEcHN 0.302 2.784 AOEcHN �0.243 0.500 IBcLN 0.154 1.648

IVcHN 0.229 0.325 IBcHN 0.300 2.971 PEcHN 0.206 3.832 PEcHN 0.147 2.986

APEcLN 0.184 0.408 APEcHN �0.292 1.966 APEcHN �0.108 0.363 IBcHN �0.072 3.733

PEaHN �0.161 3.278 AOEaHN �0.219 1.283 PEaHN �0.090 1.608 AOEaHN �0.067 0.774

RT �0.134 0.496 PEaHN �0.206 1.404 APEcLN 0.084 0.294 AOEcHN 0.058 0.216

PEcLN �0.072 3.598 AOEcHN 0.054 0.094 AOEcLN 0.082 0.412 AOEcLN 0.037 0.123

AOEaHN 0.043 0.062 IVcLN 0.040 0.435 OEaHN 0.050 0.378 IVcLN 0.028 0.241

OEcLN 0.041 0.090 OEcLN �0.033 0.113 AOEaHN 0.025 0.177 PEcLN 0.015 1.896

IVcLN 0.028 0.022 AOEcLN �0.024 0.072 IVcLN �0.011 0.053 APEcLN �0.006 0.126

AOEcLN 0.006 0.013 APEcLN 0.017 0.090 OEcLN �0.010 0.142 OEcLN �0.005 0.126

r¼ 40 r¼ 50 r¼ 60

Parameter μ σ Parameter μ σ Parameter μ σ

IVcHN �0.955 2.149 IVcHN �0.731 1.963 APEaHN �0.757 2.676

APEaHN �0.559 2.592 PEcHN �0.525 2.841 IBcHN �0.591 2.685

RT 0.423 1.276 PEaHN �0.507 2.069 RT 0.562 1.224

PEaHN �0.371 1.698 APEaHN �0.470 2.674 IVcHN �0.557 1.970

OEaHN �0.338 1.881 IBcHN �0.441 3.108 PEaHN �0.514 2.153

IBcHN �0.207 3.255 RT 0.438 1.281 OEcHN 0.422 1.508

OEcHN 0.185 1.849 OEaHN �0.310 1.977 PEcLN 0.344 1.871

AOEaHN �0.137 0.644 OEcHN 0.255 1.724 OEaHN �0.306 2.086

PEcLN 0.119 1.774 PEcLN 0.174 1.876 PEcHN �0.301 2.780

PEcHN 0.119 2.628 APEcHN �0.138 1.588 IBcLN �0.262 1.454

AOEcLN 0.062 0.260 AOEcLN 0.094 0.303 APEcHN �0.145 1.741

APEcHN �0.035 1.654 IBcLN �0.071 1.384 AOEcLN 0.033 0.625

AOEcHN 0.021 0.187 AOEaHN �0.051 0.668 APEcLN �0.029 0.229

OEcLN �0.020 0.155 APEcLN �0.043 0.281 OEcLN �0.028 0.450

IBcLN 0.014 1.493 AOEcHN 0.028 0.190 AOEaHN �0.013 0.942

APEcLN 0.009 0.116 OEcLN �0.017 0.138 IVcLN �0.010 0.305

IVcLN 0.009 0.186 IVcLN 0.001 0.164 AOEcHN �0.007 0.309

2203 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

resulting sensitivity measures (μ and σ) for the differentnumber of elementary effects calculated.

As can be seen in this table, increasing the number ofruns, due to a higher r, clearly demonstrates a closer

similarity between the sensitivity measures of the par-ameters. However, increasing the number of elementary

effects from 50 to 60 does not show significant differencesbetween the significance ranking of the parameters

2204 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

considered influential (see below for the identification of

influential parameters). As a result, r¼ 60 was selected asthe optimal number of repetitions for this case study. Theoverall model evaluation costs were therefore 1,080 simu-

lations (¼ r*(kþ 1), k¼ 17) (computational cost of onesimulation was around 10 min obtained using a PC with4 GHz Intel® Pentium® processor). These results are in con-trast to previous applications of the Morris method, since

most of these studies used a low repetition number, e.g.,r¼ 10∼ 20 (Campolongo et al. ). As our results indicate,in agreement with Cropp & Braddock (), r has a signifi-

cant effect on the identified parameter sensitivity,particularly for a low value of r, (lower than r¼ 30, in thiscase study). This fact implies the necessity of finding out

the optimal repetition number for SEEjk calculations (r). Anon-optimal selection of r would lead to Type II error(false negative), failing in the identification of a parameterof considerable influence in the model and Type I error

(false positive), as well, considering a factor as significantwhen it is not. For instance, in this case study, for a rep-etition number of r¼ 10, the parameter RT (response time)

was characterised by a low mean and low standard devi-ation and, for a repetition number of r¼ 15, the parameterOEaHN (amplitude of the High Negative membership func-

tion of the Oxygen Error) was also characterised by a lowmean and low standard deviation. In contrast, the resultsfor the optimal repetition number showed that these

parameters present a considerable effect in the output vari-ables, as was expected from practical experience. On the

Figure 3 | μ versus σ, for ropt¼ 60. Lines correspond to μi¼+2SEMi. (a) IAEO output variable; (

other hand, for a repetition number of r¼ 40 the parameter

AOEaHN (amplitude of the High Negative membershipfunction of the accumulated oxygen error) was consideredas influential compared to the results for the optimal rep-

etition number, where it is non-influential.Figure 3 shows the graphical Morris approach for the

optimal number of repetitions obtained for the two outputvariables (Figure 3(a) IAEO and Figure 3(b) IAEP). The

resulting influential parameters from both output variablesare similar (only one parameter is different). Figure 3(c)shows the same graph for the weighted contribution of the

elementary effects obtained from both output variables(Equation (2)). This figure was used to screen out the non-influential parameters of the control system (i.e. the six par-

ameters that are not labelled in Figure 3(c)). From the 11influential parameters, RT, OEcHN (High Negative centreof the Oxygen Error), APEaHN (High Negative amplitudeof the Accumulated Pressure Error) and IVcHN (High Nega-

tive centre of the increment of the air valve opening)presented high mean and low deviation, lying outside ofthe wedge formed by the two lines corresponding to μi ¼+2SEMi. Thus, the effect of these parameters on theoutput variables is expected to be linear and additive,which is desirable for parameter estimation based on optim-

isation algorithms, since they are characterised by a highermean than the rest of the parameters and not too high stan-dard deviation.

These results are in agreement with the experience-based knowledge. RT is one of the most important

b) IAEP output variable; (c) for the weighted contribution of IAEO and IAEP output variables.

2205 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

parameters, which is in agreement with its importance in the

controller actions. High RT values would give a slowresponse to changes in the process variables and, inversely,low values could cause too fast actions, which would lead to

instabilities when controlling a process with either a lowresponse time or a high time delay. IVcHN is also importantin the control system because it determines the change inthe manipulated variable, i.e. the air valve opening. The

magnitude of the oxygen control actions will basically bedetermined by this parameter and the frequency of eachcontrol action, given by RT. In contrast, the rest of the par-

ameters characterised by low mean and high standarddeviation present an effect that is dependent on the valuesof the other parameters through interaction effects or a non-

linear effect.Compared to the results obtained in the previous work

(Ruano et al. ), in which local sensitivity analysis wasused to find the significant parameters, the following could

be said: (i) the significant parameters identified in the localsensitivity analysis agreed mostly with those identified bythe method of Morris, e.g., OEcHN and RT parameters;

(ii) only one parameter identified as influential in the localsensitivity analysis is found to be non-influential accordingto the Morris screening study (AOEaHN, i.e. High Negative

amplitude of the accumulated oxygen error). Hence, theseresults demonstrate that global methods, in particular theMorris Screening, are more resilient to Type II error, i.e.,

identifying non-influential parameters as influential.Hence, it is recommended for use in identifiability problems.

It is worth pointing out that the significant reduction ofthe important controller parameters could suggest a simpli-

fication of the control system design, i.e., to eliminatesome redundant membership functions. However, it mustbe highlighted that one characteristic of fuzzy control is

that, when the knowledge-based rule system is formulated,some redundant membership functions are defined inorder to represent in a general way all possible system beha-

viours. In this way, the fuzzy control systems become moreflexible at describing system dynamics, but at the expense ofa large number of parameters (around 17 in this case). This

is certainly different from proportional-integral-derivative(PID)-type controllers, which use much fewer numbers ofparameters (around three per control loop) to describeinput-output dynamics of the system; hence there are com-

paratively fewer parameters to fine-tune. However, whenthe system to be controlled has strong non-linear responseto input dynamics, as the one presented in this work, fine-

tuning PID controllers becomes a challenging (if proble-matic) issue since the tuning procedure is done for a

specific working point (thereby ignoring other operating

points in the system). From this point of view, fuzzy controlemerges as an interesting alternative to PID controllers,which otherwise may require gain-scheduling-type adaptive

controller strategies.Finally, one caveat has to be mentioned when using the

Morris screening method, which has to do with the selectionof a proper repetition number of EEi calculations (ropt).Probably, a high value of the parameter r will be neededwhen either a highly nonlinear model is used, or a largeinput uncertainty is defined (lower and upper values in uni-

form distributions). It, thus, comes with a slightly highercomputational cost, which is increasingly feasible to do.

As a guideline to find the proper repetition number, the

Morris method must be applied to a different number ofelementary effects calculations (at least for two r, startingwith e.g. r¼ 10). In general, we proposed that the parametersignificance ranking (as defined by the graphical approach

of Morris) obtained for two different “r” must be similar.The r is optimal when the following holds: (i) those par-ameters, which are found non-influential, are the same in

both iterations with different r, and (ii) those parameters,which are influential, maintain similar ranking. Since thetwo measures must be evaluated at the same time, the

graphical Morris approach facilitates this analysis.We recommend the Morris screening as an efficient and

promising method for sensitivity analysis in the wastewater

modelling community for applications ranging from modelcalibration to controller fine-tuning applications.

CONCLUSIONS

Fuzzy controllers, in contrast to PID controllers, contain a

large number of parameters to describe system dynamicsin all possible operating conditions. This is an issue fortheir application, since fine-tuning of fuzzy controller par-

ameters becomes challenging. Hence, the Morris screeningapproach has been applied to help with fine-tuning of afuzzy logic based control system of a WWTP. Firstly, an

iterative procedure has been executed in order to find outthe proper repetition number of EEi calculations (ropt). Theoptimal repetition number found in this study is in directcontrast to previous applications of the Morris method,

which usually use a low repetition number, e.g. r¼ 10∼ 20.Working with a non-proper repetition number of elementaryeffects (r) could lead to Type I error as well as Type II error.

This approach is proposed as a prior step to obtain theparameter significance ranking for the identifiability

2206 M. V. Ruano et al. | Application of Morris method on fuzzy controllers to WWTPs Water Science & Technology | 63.10 | 2011

methodology used in the previous work (Ruano et al. ).As regards parameter significance ranking, compared to thelocal sensitivity analysis, one can conclude that both resultsare in fairly good agreement with each other with the excep-

tion of one parameter identified falsely by the local method.Overall, the Morris approach provides a good approxi-mation of a global sensitivity measure, thereby overcomingthe drawback of the differential analysis method whose

results are only locally valid. The downside of the Morrismethod is the computational cost, which is increasingly feas-ible to do. Overall, the results supplemented with process

engineering knowledge provide valuable insights into theengineering problems at hand, thereby helping engineersin doing their day-to-day work, e.g., helping to fine-tune a

controller, checking the robustness of plant design, amongothers.

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