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Multi-criteria evaluation of wastewater treatment plantcontrol strategies under uncertainty

Xavier Flores-Alsinaa,b, Ignasi Rodrıguez-Rodaa,*, Gurkan Sinb, Krist V. Gernaeyb

aLaboratory of Chemical and Environmental Engineering, University of Girona, Montilivi Campus s/n 17071, Girona, SpainbCenter for BioProcess Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark, Building 229,

DK-2800 Kgs. Lyngby, Denmark

a r t i c l e i n f o

Article history:

Received 24 January 2008

Received in revised form

13 May 2008

Accepted 29 May 2008

Published online 24 June 2008

Keywords:

Wastewater treatment

Control

Benchmarking

Multi-criteria decision analysis

Uncertainty

Monte Carlo simulation

* Corresponding author. Tel.: þ34 972 418281E-mail addresses: xavi@lequia.udg.es,

kvg@kt.dtu.dk (K.V. Gernaey).0043-1354/$ – see front matter ª 2008 Elsevidoi:10.1016/j.watres.2008.05.029

a b s t r a c t

The evaluation of activated sludge control strategies in wastewater treatment plants

(WWTP) via mathematical modelling is a complex activity because several objectives; e.g.

economic, environmental, technical and legal; must be taken into account at the same

time, i.e. the evaluation of the alternatives is a multi-criteria problem. Activated sludge

models are not well characterized and some of the parameters can present uncertainty,

e.g. the influent fractions arriving to the facility and the effect of either temperature or

toxic compounds on the kinetic parameters, having a strong influence in the model

predictions used during the evaluation of the alternatives and affecting the resulting rank

of preferences. Using a simplified version of the IWA Benchmark Simulation Model No. 2 as

a case study, this article shows the variations in the decision making when the uncertainty

in activated sludge model (ASM) parameters is either included or not during the evaluation

of WWTP control strategies. This paper comprises two main sections. Firstly, there is the

evaluation of six WWTP control strategies using multi-criteria decision analysis setting the

ASM parameters at their default value. In the following section, the uncertainty is intro-

duced, i.e. input uncertainty, which is characterized by probability distribution functions

based on the available process knowledge. Next, Monte Carlo simulations are run to

propagate input through the model and affect the different outcomes. Thus (i) the variation

in the overall degree of satisfaction of the control objectives for the generated WWTP

control strategies is quantified, (ii) the contributions of environmental, legal, technical and

economic objectives to the existing variance are identified and finally (iii) the influence of

the relative importance of the control objectives during the selection of alternatives is

analyzed. The results show that the control strategies with an external carbon source

reduce the output uncertainty in the criteria used to quantify the degree of satisfaction of

environmental, technical and legal objectives, but increasing the economical costs and

their variability as a trade-off. Also, it is shown how a preliminary selected alternative with

cascade ammonium controller becomes less desirable when input uncertainty is included,

having simpler alternatives more chance of success.

ª 2008 Elsevier Ltd. All rights reserved.

; fax: þ34 972 418150.(X. Flores-Alsina), ignasi@lequia.udg.es (I. Rodrıguez-Roda), gsi@kt.dtu.dk (G. Sin),

er Ltd. All rights reserved.

Nomenclature

AD anaerobic digester

ADM1 Anaerobic Digestion Model No. 1

AER aerobic reactor

Aj alternative to be evaluated

ANOX anoxic reactor

ASM activated sludge model

BOD5 biochemical oxygen demand (g COD m�3)

BSM2 Benchmark Simulation Model No. 2

C uncertainty class

COD chemical oxygen demand (g COD m�3)

D probability distribution

DH dewatering unit

DO dissolved oxygen

EQ effluent quality index (kg pollution day�1)

IWA International Water Association

KLa oxygen transfer coefficient (day�1)

Kp proportional gain (units depend on context)

np number of evaluation criteria applied to

quantify the degree of satisfaction for OBJknt total number of evaluation criteria

OBJk control objective

PI proportional integral

PRIM primary clarifier

Qcarb external carbon source (ANOX1) (m3 day�1)

Qintr internal recycle (from AER3 to ANOX1)

(m3 day�1)

Qr external recycle (from SEC to ANOX1)

(m3 day�1)

s(Aj) weighted sum for an alternative j

SNH ammonium concentration (g N m�3)

SNO nitrate concentration (g N m�3)

SO dissolved oxygen concentration (g (-COD) m�3)

ST storage tank

THK sludge thickener

Ti integral time constant (days)

TIV time in violation (days)

TN total nitrogen (g N m�3)

TSS total suspended solids (g TSS m�3)

Tt anti-windup constant (days)

U uncertainty factor

v(Xj) value function for a criterion j

v(xj,i) normalized value i for a given alternative j

wk weights applied to control objectives

wi weights applied to criteria

WWTP Wastewater Treatment Plant

Xi evaluation criteria

xi* best situation for a criterion i

xi* worst situation for a criterion i

xj,i quantified criterion i for a given alternative j

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 74486

1. Introduction

Nowadays, evaluation of the feasibility of wastewater

treatment plant (WWTP) design, operation and control

strategy alternatives is to a large extent based on simulation

of mechanistic models. Computer codes implementing the

International Water Association (IWA) activated sludge

model (ASM) family (Henze et al., 2000) are employed to

support decision making on implementation of different

technological alternatives such as: controllers (Stare et al.,

2007), reactor volume and configuration (Yuan et al., 2000;

Flores et al., 2005) and recycle streams (Salem et al., 2002).

The recent developments in the WWTP modelling field –

for example the release of the Anaerobic Digestion Model No.

1 (ADM1, Batstone et al., 2002), the development of model

interfaces coupling the ASM and ADM models (Nopens et al.,

submitted for publication) – have allowed the construction of

plant-wide WWTP models. The latter models combine sub-

models for primary (Otterpohl et al., 1994) and secondary

settlers (Takacs et al., 1991), activated sludge reactors,

anaerobic digesters, dewatering units, etc., which are linked

together by a number of recycles and bypasses. Thus, apart

from studying the performance of individual WWTP unit

processes it is nowadays also possible to investigate the

interactions amongst such processes. Some applications of

this approach can be found already, for example comparing

technologies to treat rejected water streams (Volcke et al.,

2006), evaluating the effect of a control strategy on a long-term

basis (Vrecko et al., 2006; Jeppsson et al., 2007) or analyzing the

simulation results by multivariable analysis (Flores et al.,

2007a)

Uncertainty is a central concept when dealing with biolog-

ical systems like activated sludge systems, which inherently

are subjected to large natural variations. However, tradition-

ally WWTP process simulators assume constant rather than

variable model parameters, and are thus not capable to take

into account the inherent randomness. Indeed, the activated

sludge process cannot be considered as a well characterized

process and some activated sludge model parameters are

uncertain. Examples are the parameters describing the

influent COD fractionation, or the parameters describing the

effect of temperature or toxic compounds on the model

kinetics, which will both have a significant influence on the

model predictions. Hence, the assessment and presentation of

uncertainty is widely recognized as an important part of the

analysis of complex water systems (Beck, 1987).

The Monte Carlo simulation technique is a practical way of

imitating the inherent randomness in biological systems

using deterministic models. Monte Carlo simulation is based

on a probabilistic sampling method of input uncertainties

followed by determination and analysis of the propagation of

input uncertainty to model outputs (Helton and Davis, 2003).

Some authors have used the Monte Carlo simulation tech-

nique in the water research field, for example addressing the

design and upgrade of a WWTP under uncertainty balancing

effluent costs and risk of effluent standards’ exceedance

(Benedetti et al., 2006), predicting the disinfection perfor-

mance of a full scale reactor in drinking water treatment

(Neumann et al., 2007), generating different wastewater

influent compositions for posterior process performance

evaluation (Martin et al., 2007), or also as a pragmatic proce-

dure to automate the calibration of ASM models (Sin et al.,

2008).

It is important to emphasize that several types of objec-

tives (economic environmental, legal and technical), must be

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 7 4487

taken into account simultaneously during the evaluation of

design/control alternatives, i.e. the evaluation of the

competing alternatives is a multi-criteria problem (Vincke,

1992; Belton and Stewart, 2002). The main limitation of

previously published works is the lack of systematic

approaches to support decision making when such multi-

criteria problems arise. Moreover, despite the importance of

considering uncertainty during simulation-based WWTP

performance evaluation, multi-criteria evaluation under

uncertainty has only been treated sparsely in the wastewater

engineering field. Clearly, there is a need to fill this gap by

providing technologies to support the decision makers that

have to deal with this type of multi-criteria problems in the

wastewater treatment field.

The objective of this paper is therefore to demonstrate the

influence of considering uncertainty in activated sludge

model parameters on the decision making process during the

multi-criteria evaluation of WWTP control strategies. A plant-

wide WWTP model is used as a case study. First a reference

case will be presented, by evaluating and comparing several

WWTP control strategies based on the traditional way, i.e. by

using the default (deterministic) values of model parameters.

Secondly, the WWTP control strategies are compared while

considering uncertainty in the ASM parameters. Hence input

uncertainties are propagated to the set of applied plant

performance evaluation criteria, and as a consequence the

influence of input uncertainty during multi-criteria decision

making can be investigated. To this purpose, input uncer-

tainty is quantified by assuming probability distributions for

each parameter based on the available process knowledge.

The WWTP model is then coupled to a Monte Carlo engine

that randomly samples parameters from the previously

defined probability distributions, thus solving the model and

quantifying the evaluation criteria for each parameter sample.

2. Methods

2.1. Wastewater treatment plant (WWTP) under study

The WWTP under study has the same layout as the IWA

Benchmark Simulation Model No. 2 (BSM2) proposed by

Jeppsson et al. (2007). The activated sludge unit is a modified

Ludzack–Ettinger configuration consisting of 5 tanks in series.

Tanks 1 (ANOX1) and 2 (ANOX2) are anoxic with a total volume

of 2000 m3, while tanks 3 (AER1), 4 (AER2) and 5 (AER3) are

aerobic with a total volume of 3999 m3. The circular secondary

settler (SEC) has a surface area of 1500 m2 with a total volume

of 6000 m3. The BSM2 plant further contains a primary clarifier

(PRIM), a sludge thickener (THK), an anaerobic digester (AD),

a storage tank (ST) and a dewatering unit (DH). Further

information about the BSM2 layout and the description of the

process models can be found in Jeppsson et al. (2007).

Plant performance evaluation has been reduced from 1-

year simulation to 1 week in order to reduce the computa-

tional burden of the whole study. The default wastewater to

be treated has a dry weather flow rate of 18446 m3 day�1 with

a carbon and nitrogen load of 12228 kg COD day�1 and

1025 kg N day�1, respectively. The wastewater influent is the

same as for the Benchmark Simulation Model No. 1 (Copp,

2002), but increasing the concentrations of particulate carbon

and nitrogen in order to take into account the effect of the

primary clarifier.

2.2. Implemented control strategies

Five control strategies were implemented to a default open

loop case (A1). The operational settings of the open loop base

case considered in this case study were slightly modified

compared to the BSM1 (Copp, 2002). The constant waste

sludge flow rate was reduced from 385 m3 day�1 to

300 m3 day�1 and the constant oxygen transfer coefficient

(KLa) for the third aerobic reactor (AER3) was increased from

84 day�1 to 240 day�1. The values for other manipulated vari-

ables (Qintr¼ 55,338 m3 day�1, Qr¼ 18,446 m3 day�1 and

Qcarb¼ 0 m3 day�1) remained at the default BSM1 values. The

control strategies A¼ {A2,.,Aj,.,A6}, summarized in Table 1,

were applied to the activated sludge reactor section. The

simulation results (open loopþ 5 control strategies) are the

starting point for the work presented in this paper.

The dissolved oxygen (DO) sensor was assumed to be ideal

without noise or delay. The nitrate (SNO) and ammonium (SNH)

sensors had a time delay of 10 min, with zero mean white

noise (standard deviation of 0.1 g N m�3). All the dynamic

simulations were preceded by a steady state simulation to

ensure an appropriate starting point for the dynamic simu-

lations and to eliminate bias due to the selection of the initial

conditions on the dynamic modelling results (Copp, 2002).

Even though the length of the dynamic influent file used to

carry out the simulations was 28 days, only the data generated

during the last 7 days were used to evaluate the plant

performance.

The control strategies presented in this study assume

constant rather than variable set points. Previous studies have

demonstrated that different set points might lead to different

conclusions (see, e.g. Flores et al., 2007b). However, the

selected operational settings of the controllers (set points, Kp,

Tt) are reasonable values that will lead to an acceptable plant

performance, since the operational settings have been

extracted from the literature and were for a large part defined

by the IWA Task Group on Benchmarking of Control Strategies

for WWTPs (www.benchmarkwwtp.org). The literature refer-

ences mentioned in Table 1 can be consulted for further

details about the controllers.

2.3. Objectives, criteria and evaluation procedure

Environmental, economical, technical and legal objectives

were taken into account [OBJ¼OBJ1,.,OBJk,.,OBJp] during

the evaluation procedure (hence p¼ 4). The degree of satis-

faction of the defined control objectives OBJ1, OBJ2, OBJ3 and

OBJ4 was quantified by a set [n¼ n1,.,nk,.,np] of 1, 1, 1 and 4

criteria, respectively ðPp

k¼1 nk ¼ nt ¼ 7Þ. Weight factors

[w¼w1,.,wk,.,wp] are then assigned to each objective (k).

The relative importance of the objectives is normalized to sum

1, orPp

k¼1 wk ¼ 1. For each control objective, the criteria used

to evaluate the degree of satisfaction of that objective are

weighted equally. Thus, the weight factor applied to each

evaluation criterion (wi, i¼ 1, 2,.,nt) is obtained by dividing wk

by the number of evaluation criteria nk belonging to a specific

Table 1 – Main features of the control strategies evaluated in this case study

Oxygen controller in the aerated section (AER1, 2 and 3) (Vanrolleghem and Gillot, 2002)

Controller type PI with anti-windup Units

Proportional gain (Kp) 100 m3 (g (-COD))�1 days�1

Integral time constant (Ti) 0.01 days

Anti-windup constant (Tt) 0.01 days

Controlled variable SO in AER1, 2 and 3

Set point 2 g (-COD) m�3

Manipulated variable (MV) KLa days�1

Maximum value of MV 360 days�1

Implemented in alternatives A2, A3, A4, A5 and A6

Nitrate controller in the anoxic section (ANOX2) (Copp, 2002)

Controller type PI with anti-windup Units

Proportional gain (Kp) 10,000 m3(g N)�1 days�1

Integral time constant (Ti) 0.04 days

Anti-windup constant (Tt) 0.04 days

Controlled variable SNO in ANOX2

Set point 1 g N m-3

Manipulated variable (MV) Qintr m3 days�1

Maximum value of MV 92,336 m3 days�1

Implemented in alternatives A3 and A5

Nitrate controller in the anoxic section (ANOX2) (Gernaey et al., 2007)

Controller type PI with anti-windup Units

Proportional gain (Kp) �1 m3(g N)�1 days�1

Integral time constant (Ti) 0.1 days

Anti-windup constant (Tt) 0.1 days

Controlled variable SNO in ANOX2

Set point 1 g N m�3

Manipulated variable (MV) Qcarb m3 days�1

Maximum value of MV 5 m3 days�1

Implemented in alternatives A4 and A6

Ammonium controller in the aerated section (AER3) (Gernaey et al., 2007)

Controller type Cascaded PI Units

Proportional gain (Kp) �1 m3(g N)�1 days�1

Integral time constant (Ti) 0.2 days

Anti-windup constant (Tt) 0.2 days

Controlled variable SNH in AER3

Set point 1 g N m�3

Manipulated variable (MV) SO set point in AER3, 4 and 5 g (-COD) m�3

Maximum value of MV 4 g (-COD) m�3

Implemented in alternatives A5 and A6

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 74488

control objective. The effluent quality index (EQ, Copp, 2002)

was the single criterion (X1) used to evaluate the accom-

plishment of OBJ1 (minimize the environmental impact). The

operational cost index (X2) proposed by Vrecko et al. (2006)

was applied for evaluating the accomplishment of objective

OBJ2 (minimize economic cost). The risk for occurrence of

microbiology-related solids separation problems (X3) was

used to evaluate the technical reliability of the controllers

(OBJ3). The risk for occurrence of separation problems (Comas

et al., 2006) was assessed by punishing control strategies that

create operation conditions that potentially can drive the

plant to bulking (X3–1), foaming (X3–2) and rising sludge (X3–3).

Finally the percentage of time that the plant violates the legal

effluent limits (OBJ4) was quantified using the time plant in

violation index (Copp, 2002) for the different pollutants, i.e.

total suspended solids (X4, TSS), chemical oxygen demand (X5,

COD), biochemical oxygen demand (X6, BOD5) and total

nitrogen (X7, TN). Compared to OBJ4, OBJ1 provides additional

information about the impact on water by the treated effluent,

since the contribution of each pollutant is weighted differ-

ently to calculate the value of OBJ1 (Copp, 2002). It could

indeed happen that scenarios providing the same degree of

satisfaction of the legal requirements (OBJ4) will have different

potential impacts on the receiving water (OBJ1).

All criteria [X] were quantified based on dynamic simula-

tions. The quantification of an option Aj with respect to crite-

rion Xi is indicated as xj,i. Thus, each option under evaluation

can be formulated as a vector of scores and represented as a n-

dimensional performance score profile ½Aj ¼ ðxj;i;.; xj;ntÞ�.

Value functions [v(Xi)] map the score profiles of all options into

a value v(xj,i) normalized from 0 to 1. The 0 and 1 values were

associated with the worst (xi*) and the best (xi*) situation,

respectively, whilst a mathematical function was used to

evaluate the intermediate effects. The collection of the best

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 7 4489

½x� ¼ ðx�1;.; x�ntÞ� and the worst ½x� ¼ ðx1�;.; xnt�Þ� scores for all

criteria determined the best ½vðx�Þ ¼ ðvðx�1;.;vðx�ntÞÞ ¼ 1Þ� and

the worst profiles ½vðx� ¼ ðvðx1�Þ;.; vðxnt�ÞÞ ¼ 0Þ�. Finally,

a multi-objective function was calculated to obtain a unique

value for each alternative (Eq. (1)). The multi-objective function

is presented as a weighted sum and calculated by summing up

the product of each normalized criterion v(xj,i) with its corre-

sponding weight (wi). The options were ranked according to the

obtained scores. The alternative with the highest score was the

one that was considered to give the higher degree of satisfac-

tion of the considered objectives, and thus corresponds to the

recommended alternative for implementation (Flores et al.,

2007b).

s�Aj

�¼ v

�xj;1

�$w1 þ.þ v

�xj;i

�wi þ.þ v

�xj;nt

�wnt ¼

Xn

i¼1

v�xj;i

�wi

(1)

3. Multi-criteria evaluation of WWTP controlstrategies without uncertainty

In this section the deterministic multi-criteria evaluation of

WWTP control strategies is presented. WWTP control strate-

gies defined in Table 1 were tested and evaluated using multi-

criteria decision analysis for default values of the activated

sludge model parameters. Table 2 summarizes the score

profile for each evaluated alternative. Note that the criteria X4,

X5 and X6 had identical values for each tested control strategy

in this case study, and thus these criteria were not useful to

discriminate the competing alternatives.

It should be emphasized that results of this base case

analysis depend strongly on the model selection prior to

performing the simulations. When modelling activated sludge

plants, there is often disagreement on the best model to apply

for a given case. The representation of biomass decay (Siegrist

et al., 1999), the modelling of nitrogen removal (Gujer et al.,

1999) and the oversimplification of the settling models (i.e.

non-reactive in most cases, despite the fact that a significant

amount of biomass is often stored at the bottom of the

secondary clarifier, e.g. Gernaey et al., 2006) are key issues that

are still under discussion.

To compare the effects of the different criteria during the

evaluation procedure, it is necessary to map these score

profiles into normalized values because all those criteria are

Table 2 – Score profiles for the six evaluated control strategies

A1 A2 A3

X1 8114.10 7770.90 7784.70 58

X2 10682 9853 9787 135

X3–1 78.89 78.00 78.08

X3–2 77.94 77.94 77.75

X3–3 86.32 94.37 91.01

X4 0.00 0.00 0.00

X5 0.00 0.00 0.00

X6 0.00 0.00 0.00

X7 44.79 15.18 18.01

measured in different units. Value functions award values

from 0 to 1 to the worst and the best situation considered,

respectively, whilst a mathematical function is proposed to

evaluate the intermediate effects. The extreme profiles (based

on expert judgment) are summarized in the following lines.

[(xi*)¼ (x1*¼ 60935 kg pollution day�1, x2*¼ 15,000, x3–1*¼100%, x3–2*¼ 100%, x3–3*¼ 100%, x4*¼ 100%, x5*¼ 100%, x6*¼100%, x7*¼ 100%)] and

[(xi*)¼ (x1*¼ 0 kg pollution day�1, x2*¼ 7500, x3–1*¼ 0%, x3–2*

¼ 0%, x3–3*¼ 0%, x4*¼ 0%, x5*¼ 0%, x6*¼ 0%, x7*¼ 0%)].

A linear model was applied between these extreme values

to calculate the intermediate effects (e.g. for criterion X2 the

value function is v(X2)¼�0.000113X2þ 2). Finally, a multi-

objective function calculated as a weighted sum (Eq. (1)) was

applied in order to obtain a single value for all the alternatives

which were then ranked according to the scores obtained,

with the final decision as to which alternative is best in ful-

filling the evaluation criteria resting on the decision maker. As

weight assessment is not a central topic in this paper, equal

importance for all the objectives is assumed wp¼ 0.25. The

results of the weighted sum lead us to the following conclu-

sion: in accordance with the control objectives, alternative A5

with a scores (A5) of 0.75 is the selected, while A1, A2, A3, A4

and A6 with a score in the weighted sum of 0.68, 0.72, 0.72, 0.63

and 0.66, respectively, are rejected.

Despite the fact that this control strategy shows a higher

risk of favouring formation of rising sludge, alternative A5 is

the most favourable mainly because this alternative showed

the lowest scores in OBJ2 (minimize economic costs). The

latter result is basically attributed to an efficient use of

the aeration energy in this control strategy, providing just the

sufficient quantity of oxygen to maintain a reasonable effluent

ammonium concentration. Also, it is important to mention

that alternative A5 performed well in both environmental

(OBJ1) and legal (OBJ4) objectives, because this control strategy

improves the overall nitrification efficiency.

4. Multi-criteria evaluation of controlstrategies under uncertainty

This section of the manuscript provides details of the proce-

dure followed to evaluate the WWTP control strategies under

uncertainty: first, the quantification of the input uncertainty

of the ASM model parameters is presented; then the set-up of

without uncertainty

A4 A5 A6 Units

79.90 7108.70 5824.90 g pollution m�3

51 9187 12746 –

80.79 77.93 80.33 %

81.94 77.73 81.05 %

91.25 97.67 85.05 %

0.00 0.00 0.00 %

0.00 0.00 0.00 %

0.00 0.00 0.00 %

0.00 6.85 0.00 %

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 74490

the Monte Carlo simulations is explained, and finally multi-

criteria evaluation of the simulation results is presented and

discussed in detail.

4.1. Identification and quantification of the inputuncertainty of the ASM model

The BSM2 is an integrated model that contains several sub-

models, i.e. influent, model interfaces, settling tanks, aeration

tanks, digesters, controllers, etc. For the scope of this study, it

was decided to frame the uncertainty analysis to consider

only the uncertainty in the biokinetic parameters and the

influent fractions of the ASM1 model. All other potential

sources of uncertainty, e.g. model structure (settlers,

hydraulics, controllers, etc.) are fixed and assumed known as

defined by BSM2. This framing of uncertainty is simple.

However, it was felt necessary (if not essential) to avoid

getting involved into discussing solely the uncertainty of the

BSM2 model – which is not the prime purpose of this study.

To carry out this analysis, the uncertainty associated to the

ASM parameters [U¼U1,.,Uy,.U32] was characterized by a set

of probability distributions [D¼D1,.,Dy,.D32]. These distri-

butions were assumed to characterize a degree of belief with

respect to where the appropriate values for the elements of [U]

are located for use in the simulation of the BSM2. When used in

this manner, these distributions are providing a quantitative

representation of what is referred as subjective or epistemic

uncertainty (Helton and Davis, 2003). In this case study those

distributions were developed through interpretation of avail-

able process knowledge. Three uncertainty classes were

distinguished [C¼ C1, C2, C3] to allow presentation of the

Table 3 – Parameter distributions used for the Monte Carlo simparameter class and variation range for class 1 and 2 paramet

Uncertainty parameter (Up, K&S) Symbol Def

Autotrophic yield YA

Heterotrophic yield YH

Fraction of biomass to particulate products fPFraction of nitrogen in biomass iXB

Fraction of nitrogen in particulate products iXP

Conversion from COD to inert particulates XI2TSS

Conversion from COD to inert particulates XS2TSS

Conversion from COD to inert particulates XBH2TSS

Conversion from COD to inert particulates XBA2TSS

Conversion from COD to inert particulates XU2TSS

Maximum specific heterotrophic growth rate mH

Half saturation (heterotrophic growth) KS

Half saturation (heterotrophic oxygen) KOH

Half saturation (nitrate) KNO

Heterotrophic specific decay rate bH

Maximum specific autotrophic growth rate mA

Half saturation (autotrophic growth) KNH

Half saturation (auto. oxygen) KOA

Autotrophic specific decay rate bA

Anoxic growth rate correction factor hg

Amonification rate ka

Maximum specific hydrolysis rate kh

Half saturation (hydrolysis) KX

Anoxic hydrolysis rate correction factor nyh

parameter uncertainty in a structured way, and each uncertain

parameter Uy included in the analysis was assigned to a certain

class Cc depending on the extent of knowledge available in the

literature about this specific parameter value. The first class

was assigned to low uncertainty and included mostly stoi-

chiometric parameters. In this class (C1), the parameters were

assumed to have a 5% upper and lower bounds around their

default values [U1,.,U10]. The second class (C2), corresponded

to medium uncertainty and involved kinetic parameters such

as the maximum specific growth rate and the affinity constants

[U11,.,U24]. In this class, 25% upper and lower bounds around

the default values were assumed. For simplification, all the

kinetic and stoichiometric parameters were supposed to be

independent although the authors are aware of possible

correlations amongst several parameters, e.g. the maximum

specific growth rate and the half saturation constants. Table 3

summarizes these parameters, the classes to which they

belong and the range of evaluated parameters.

Finally, the third class of uncertainty (C3) corresponded to

high uncertainty and included the influent fraction related

parameters, assuming upper and lower bounds equal to 50%

of the default parameter values. Several class 3 uncertainty

factors were applied to the default stoichiometric coefficients

used to calculate the different ASM1 influent organic carbon

related state variables [U25,.,U28] – such as the soluble

biodegradable substrate (SS) or the particulate biodegradable

substrate (XS) concentration – from the influent COD load,

resulting in a range of treatment plant influents to be applied

in the simulations. A similar method was applied to influent

nitrogen [U29,.,U32], where the fraction coming from partic-

ulate products and biomass was removed first, to finally

ulation including default parameter values, assigneders

ault value Class Range Units

0.67 1 0.067 g COD g N�1

0.24 1 0.024 g COD g COD�1

0.08 1 0.008 Dimensionless

0.08 1 0.008 g N(g COD) in biomass

0.06 1 0.006 g N(g COD) in Xp

0.75 1 0.075 g TSS g COD�1

0.75 1 0.075 g TSS g COD�1

0.75 1 0.075 g TSS g COD�1

0.75 1 0.075 g TSS g COD�1

0.75 1 0.075 g TSS g COD�1

4.00 2 2.00 day�1

10.00 2 5.00 g COD m�3

0.20 2 0.10 g COD m�3

0.50 2 0.25 g N m�3

0.30 2 0.15 day�1

0.50 2 0.25 day�1

1.00 2 0.50 g N m�3

0.40 2 0.20 g COD m�3

0.05 2 0.025 day�1

0.80 2 0.40 Dimensionless

0.05 2 0.025 m3(g COD day)�1

3.00 2 1.50 g Xs(g Xbh COD day)�1

0.10 2 0.05 g Xs(g Xbh COD)�1

0.80 2 0.40 Dimensionless

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 7 4491

obtain the inorganic (ammonium, SNH) and organic influent

nitrogen compound concentrations (either soluble or partic-

ulate, SND and XND).

It is important to point out that despite of the apparent

advantages of a formal assessment of uncertainty, one should

be aware that the conclusions arising from this case study

considering uncertainty can always only be as good as the

underlying assumptions. Thus, the results of the uncertainty

analysis will to a large extent depend on the characteristics of

the defined distributions, similar to the base case perfor-

mance where the obtained results will depend on the model

selection, as indicated earlier.

4.2. Monte Carlo simulations

The input uncertainty space is sampled using the Latin

hypercube method (McKay et al., 1979; Iman et al., 1981). In

this study, 1000 samples [Uy¼Uy,1,.,Uy,f,.,Uy,1000] are gener-

ated to make sure that the input uncertainty space is covered

uniformly. Each Latin hypercube sample contains one

randomly selected value Uy,f from each of the previously

defined probability distributions Dy. The Monte Carlo simula-

tions are performed by evaluating the BSM model for each one

of the generated Latin hypercube samples, solving the entire

model and quantifying the defined criteria [X] for each tested

alternative [A]. The solution of the model for each parameter

combination results in a distribution of possible values for the

desired performance criteria, whose distributions reflect the

possible variation of the performance criteria taking into

account the input uncertainty. Both interpretation and

representation of the results of the Monte Carlo simulations is

subsequently carried out using descriptive statistical tech-

niques such as multiple box plots, error bar charts, three

dimensional representations of the inter-quartile range, etc.

4.2.1. Environmental objectives (OBJ1)According to the previous section, a single criterion (X1) is

used to quantify the degree of satisfaction of objective OBJ1(minimize environmental impact). Fig. 1 shows the results of

the Monte Carlo simulations in a box plot fashion (Hair et al.,

Alternative

X1 (k

g P

ollu

tio

n.d

ay

-1)

2000

4000

6000

8000

10000

12000

14000

A1 A2 A3 A4 A5 A6

Fig. 1 – Effluent quality index (X1) variation using a multiple

box plot representation.

1998). The different box plots illustrate that there is a clear

pattern: all the control strategies including an external carbon

source addition (A4 and A6) result in lower values in both

average effluent quality index terms and in effluent quality

index variability, i.e. the range between the first and the third

quartile is smaller compared to the other control strategies.

This differentiation between the control strategies can be

explained by the lack of soluble biodegradable carbon in the

influent and the low hydraulic retention time in the biological

reactor, resulting in poor denitrification rates as long as no

external carbon source is dosed. The external carbon source

addition acts as an extra electron donor enhancing the total

nitrogen removal by improving the reduction of the produced

nitrate to nitrogen gas and decreasing the impact of the

nitrate term in the effluent quality index. Also, this input

increases the robustness of the denitrification because this

process now no longer depends on the organic substrate

contents in the influent. Instead, the controller is now

supplying the necessary biodegradable carbon to maintain the

nitrate concentration in the second anoxic reactor (ANOX2) at

the desired set point.

It is also important to point out the effect of the SNH

cascade controller (A5, A6) in the propagation of the uncer-

tainty when it is compared to other control strategies, e.g.

the open loop controller (A1) and the DO controller (A2, A3

and A4). The SNH controller with its DO set point that varies

as a function of the ammonium concentration in the last

aerated tank improves the nitrification efficiency of the

whole plant and reduces its variability. A constant aeration

flow rate or dissolved oxygen set point results in situations

where there is either lack or excess of dissolved oxygen to

nitrify all the ammonium entering the plant. The improve-

ment of the aeration system obtained by introducing the

cascade controller reduces the percentage of time when

the aeration flow is not adequate, e.g. due to differences of

the influent load during daytime and night, thus reducing

the overall variability of effluent total Kjeldahl nitrogen

(TKN) as shown in the frequency histograms of Fig. 2.

Nevertheless, Fig. 1 reveals that alternative A5 is the alter-

native with a larger variation in terms of effluent quality

index mainly due to an increase of the uncertainty in the

denitrification efficiency. This plot elucidates the trade-off

that has to be made between improvements of nitrification

efficiency on the one hand and the overall effluent quality

index variation on the other hand. Regarding the other

scenarios (A1, A2 and A3), it is just worth mentioning that

there the controllers do not have a clear effect in both

effluent quality and variability reduction.

4.2.2. Economic objectives (OBJ2)The plant operational costs (X2) are used to evaluate the

degree of satisfaction of OBJ2 (minimize economic costs). In

Table 4 the mean and the standard deviation of the break-

down of the operational costs used to evaluate the economic

feasibility of the controllers can be found.

The values in Table 4 again demonstrate a clear differ-

ence between the control strategies with and without

external carbon source addition. The periodic purchase of an

external carbon source (X2–4) implies a subsequent increase

of both quantity and variability of the sludge production

gN.m-3

gNm-3

1 2 3 4 5 6 7 8

gN.m-3

1 2 3 4 5 6 7 8

Co

un

t

0

20

40

60

80

100

120

140

160

180

Co

un

t

0

20

40

60

80

100

120

140

160

180

Co

un

t

0

20

40

60

80

100

120

140

160

180

Alternative A1 Alternative A2

Alternative A5 Alternative A6

a

Co

un

t

0

20

40

60

80

100

120

140

160

b

0 2 4 6 8

gNm-3

0 2 4 6 8

c d

Fig. 2 – Histograms of the effluent TKN variation when alternative (a) A1, (b) A2, (c) A5 and (d) A6 are evaluated under ASM

parameters uncertainty.

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 74492

(X2–1), aeration energy (X2–2), heating energy (X2–6) and the

overall operating cost index (X2), although it should be

mentioned that there also is an increase in the methane

production (X2–7). The inclusion of carbon source dosage in

the control strategy does not have any effect on mixing

Table 4 – Mean and standard deviation of the operational costuncertainty

A1 A2 A

xj,2–1 Mean 2652.56 2654.65 2653

Standard deviation 336.15 336.36 336

xj,2–2 Mean 8548.40 7685.97 7751

Standard deviation 0.00 622.03 638

xj,2–3 Mean 396.47 396.47 250

Standard deviation 0.11 0.11 42

xj,2–4 Mean 0.00 0.00 0

Standard deviation 0.00 0.00 0

xj,2–5 Mean 648.00 648.00 648

Standard deviation 0.00 0.00 0

"xj,2–6 Mean 3854.48 3854.72 3854

Standard deviation 41.79 41.79 41

xj,2–7 Mean 1659.76 1659.55 1659

Standard deviation 140.84 140.77 140

xj,2 Mean 10682.08 9824.52 9740

Standard deviation 1490.13 1087.08 1083

energy (X2–5) and pumping energy (X2–3). Hence, it can be

concluded that the addition of external carbon source

reduces the impact on water (X1) and its variability as

a trade-off to an increase of the operating costs (X2) and their

variability.

breakdown for the different generated alternatives under

3 A4 A5 A6 Units

.00 2825.55 2653.88 2816.98 kg TSS day�1

.12 371.70 336.11 383.29 kg TSS day�1

.88 8164.18 7699.25 8547.70 kwh day�1

.14 670.05 1710.82 2138.73 kwh day�1

.24 396.71 282.03 396.64 kwh day�1

.25 0.13 68.33 0.15 kwh day�1

.00 997.43 0.00 827.00 kg COD day�1

.00 345.35 0.00 478.75 kg COD day�1

.00 648.00 648.00 648.00 kwh day�1

.00 0.00 0.00 0.00 kwh day�1

.27 3940.95 3854.49 3916.35 kwh day�1

.70 48.59 41.65 55.11 kwh day�1

.39 1715.41 1659.66 1702.79 m3 CH4 day�1

.75 144.89 140.79 144.54 m3 CH4 day�1

.82 13563.42 9721.67 13468.58 –

.81 1724.57 1999.21 3801.00 –

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 7 4493

Alternatives with a DO controller (A2, A3, A4, A5 and A6) are

characterized by having a larger variation in the aeration costs

(see values of Xj,2–2) when they are compared to the plant

running in open loop regime (A1). The effect of the cascade

ammonium controller can be noticed clearly from the results

of Table 4 in both alternatives A5 and A6: a large variation in

operational costs can be observed mainly due to variation in

the aeration energy cost (see values of X5,2–2 and X6,2–2). This

fact is attributable to the dynamics of the cascade controller

which introduces a variable DO set point instead of the

permanent DO set point (2 g (-COD) m�3) that is applied for the

alternatives A2, A3 and A4.

Control strategies A3 and A5 have lower average values and

higher variability in pumping energy (X2–3) because the

controller manipulates the internal recycle in order to main-

tain the nitrate concentration in ANOX2 to the desired set

point of 1 g N m�3.

4.2.3. Technical objectives (OBJ3)The risk of occurrence of separation problems (X3) is used to

evaluate the technical reliability (OBJ3) of the proposed control

strategies. As mentioned earlier the risk of microbiology-

related solids separation problems is evaluated by deter-

mining the operating conditions that potentially can drive the

plant to bulking (X3–1), foaming (X3–2) and rising sludge (X3–3).

The uncertainty in these indices is represented using the

inter-quartile range – the difference between the third quartile

and the first quartile. The higher the value of the inter-quartile

range, the higher the uncertainty on the index is. That said,

the inter-quartile range for these three indices of settling

problems are plotted against each other in a three dimen-

sional graph for all the scenarios (Fig. 3). The main objective of

Fig. 3 is not representing the precise values of the six evalu-

ated alternatives, but showing the main differences between

alternatives. Hence, at first sight, the results presented in this

figure lead to the following conclusions: alternatives A4 and A6

468

10121416

18

20

22

24

13,8

14,2

14,615,0

15,415,8

15,015,5

16,016,5

17,017,5

Q3 - Q

1 fo

r X

3-3 (%

)

Q3 - Q

1 for X3-2 (%)

Q3 - Q

1 fo

r X

3-1

(%

)

Alternative A1

Alternative A2

Alternative A3

Alternative A4

Alternative A5

Alternative A6

Fig. 3 – Risk of occurrence of separation problems (X3), 3-D

representation of the inter-quartile range (Q3–Q1) for all the

evaluated control strategies under ASM parameters

uncertainty.

(in white) are clearly different from the rest of the evaluated

alternatives. In terms of reduction of the rising risk variability

(X3–3) alternatives with an external carbon source controller

present the lowest uncertainty. This is because this type of

controller results in a rather low and constant effluent nitrate

level (see in Fig. 3 the large distance along the Z axis between

A1, A2, A3, A5. on the one hand and A4, A6 on the other hand)

where the presence of high nitrate levels in the settler – and

thus the effluent – is the main factor contributing to the

occurrence of rising sludge (Comas et al., 2006).

It is important to also highlight the low variation in terms

of bulking and foaming risk from one alternative to another.

Only the controllers with an external carbon source addition

result in a marginal reduction in risk for bulking (X3–1) and

foaming (X3–2) uncertainty. Again this fact is appreciated in

Fig. 3 with the small separation between A4, and A6 and the

rest of the evaluated alternatives on the X and the Y axes. The

lower risk for bulking and foaming for controllers with

external carbon source addition is mainly due to the fact that

the food to microorganism ratio (Comas et al., 2006) is more

constant for such scenario due to the external carbon source

addition. Indeed, the external carbon source addition will in

fact do nothing else than compensate for low influent

concentrations of readily biodegradable substrate, for

example during night time. On the other hand, again, the

external carbon source increases the concentration of solids

in the reactor, thus reducing the food to microorganism ratio

and increasing the risk of bulking and foaming

4.2.4. Legal objectives (OBJ4)Finally, the percentage of time that the plant is in violation of

the legal effluent discharge limits for the different pollutants

(X4–X7) forms the set of criteria to evaluate the accomplish-

ment of OBJ4 (comply with legal effluent discharge limits). For

this case study criteria X4, X5 and X6 are always below the

limits without any variation, and as a consequence they are

not useful in discriminating between the competing alterna-

tives. Fig. 4 shows that control strategies A4 and A6 present

a lower risk of not complying with the effluent standards. This

Alternative

X8 tim

e in

vio

latio

n fo

r T

N (%

)

0

20

40

60

80

100

A1

A2

A3

A4

A5

A6

Fig. 4 – Error bar chart of criterion X7 (TIV–TN) for the

generated control strategies under ASM parameters

uncertainty.

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 74494

is mainly due to the fact that those strategies are character-

ized by high denitrification rates because the external carbon

source enhances the nitrate reduction to nitrogen gas, and as

a consequence the effluent nitrate concentration is continu-

ously below the limits and with less variation as shown by

the dynamic profiles in Fig. 5. With respect to the rest of the

controllers it can in general be concluded that as long as the

level of plant instrumentation increases (more on-line sensors

and control) the percentage of the time that the plant is in

violation and its uncertainty will decrease.

4.3. Decision procedure

The multi-objective function defined in the previous section

(Eq. (1)) is used as a metric to quantify the overall degree of

satisfaction of the control objectives for the different gener-

ated alternatives. This metric is calculated for all 1000 simu-

lations that were performed for each alternative, where each

simulation is based on one of the parameter combinations

resulting from the Latin hypercube sampling method

described in the previous section. In this case, the most

desirable alternative has the highest mean and lowest stan-

dard deviation in terms of the multi-objective function. A

practical way to see this relationship is by using the coefficient

of variation (see Table 5).

From the results of the previous analyses it is possible to

know the contributions of environmental, economic, legal and

technical objectives to the variance in the control objectives’

overall degree of satisfaction. In this way, alternatives A4 and

t (days)

8 10 12 14

t (days)

8 10 12 14

gN

·m

-3

5

10

15

20

25

30

gN

·m

-3

5

10

15

20

25

30

5th Percentile95th Percentile

5th Percentile95th Percentile

a

c

Fig. 5 – Dynamic uncertainty ranges for TN during the evaluatio

parameters uncertainty. The 5th and the 95th percentile TN pro

A6 are the least favoured alternatives because they have the

lowest scores in objective OBJ2 (minimize economic costs); i.e.

high absolute value and high variability in plant operating

cost. This is mainly due to the extra cost of the carbon source

and additional sludge production that is induced by applying

this strategy. Nevertheless, it must be pointed out that these

two alternatives provide the best accomplishment and the

lowest variation in objectives OBJ1 (minimize environmental

impact) and OBJ4 (comply with legal effluent discharge limits).

Alternative A1 is also rejected because of the bad scores in

operational costs (OBJ2), environmental (OBJ1) and legal

objectives (OBJ4). The lack of instrumentation in this strategy

makes the operation really unfeasible, because the plant is

always running under the same operating conditions and is

not capable to adapt to the different perturbations caused by

influent composition and flow rate variations.

It is important to point out that the results of Table 5

demonstrate that when uncertainty in the ASM model inputs

is considered, then the decision to implement alternative A5

that was derived from the (deterministic) base case evaluation

might be questioned. Despite the fact that alternative A5

obtained good scores in some of the criteria used to quantify

the degree of satisfaction of the considered objectives, it can

also be concluded that its performance strongly depends on

the selection of the model inputs, i.e. kinetic and stoichio-

metric parameters, and influent fractions. If those inputs are

changed from the default values, as is done when performing

the Monte Carlo simulations, the same level of accomplish-

ment of the plant objectives can no longer be ensured. For this

t (days)

8 10 12 14

t (days)

8 10 12 14

gN

·m

-3

5

10

15

20

25

30

gN

·m

-3

5

10

15

20

25

30

5th Percentile95th Percentile

5th Percentile95th Percentile

b

d

n of alternative (a) A2, (b) A3, (c) A4 and (d) A6 under ASM

file resulting from the Monte Carlo simulations are shown.

w2

0,2

0,4

0,6

Alternative A1Alternative A2Alternative A3Alternative A4Alternative A5Alternative A6

Table 5 – Mean, standard deviation and coefficient of variation of the multi-objective function for the different controlstrategies under uncertainty

s(Aj) A1 A2 A3 A4 A5 A6

Mean 0.68 0.71 0.72 0.63 0.71 0.64

Standard deviation 0.04 0.03 0.02 0.04 0.06 0.10

CV 17.00 24.73 36.00 15.75 11.83 6.40

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 7 4495

reason, when considering uncertainty on model inputs alter-

native A3 comes out as the most desirable alternative that has

a higher chance of success, since A3 has good scores in all the

objectives and can thus be considered as the most balanced of

the alternatives. Also, the good value in terms of multi-

objective mean and standard deviation ensures the robust-

ness of the decision. Hence it cannot be said that alternative

A5 is better than A3, as was concluded in the deterministic

case. Instead, it is now probable that alternative A3 is better

than A5. This analysis – including uncertainty – thus brings

about a better documented decision about which alternative

to choose, since balancing the accomplishment of the objec-

tives is combined with taking into account the deviations

created by the input uncertainties that are considered.

It should be emphasized that the use of multi-criteria

decision analysis inherently implies a certain degree of

subjectivity. Indeed, the selection of criteria, the definition of

the extreme profiles before the normalization and the relative

‘‘weight’’ given to the control objectives are important factors

affecting the scores. Nevertheless, the main focus of the study

was not on the subjectivity (i.e. the choices mentioned before)

involved in the construction of a multi-objective function.

Rather the main focus was to study how the uncertainty

involved in quantifying different objectives of the multi-

objective function (assuming the multi-objective function is

constructed) influences the decision making. For this study,

the multi-objective function was defined in an explicit and

transparent way by using equal weights for all the objectives.

The key to solving this multi-criteria decision making

problem is not easily found, and the solution is based on

realizing that different process alternatives have many

uncertainties in common. For example, all the generated

WWTP control strategies are subjected to identical uncertain

influent fractions and kinetic and stoichiometric parameters,

but depending on the evaluated alternative, the uncertainty

will be propagated in a different way. Assuming that the

decision maker is particularly interested in a control strategy

that promises the lowest environmental impact, then the

selected alternative would be A4. On the other hand, if

a compromise between operation costs and risk wants to be

ensured that controller should be A3. The preferences of the

decision maker can be considered in the analysis by adjusting

the weights wk such that they reflect the importance given by

the decision maker to each objective (see also below).

w1

0,0 0,2 0,4 0,60,0

Fig. 6 – Scenario analysis of the weights of the OBJ1

(minimize environmental impact), OBJ2 (minimize

operating cost) and OBJ3 (maximize technical reliability).

5. Scenario analysis and final discussion

The scenario analysis of the weights applied to the multi-

objective function (Eq. (1)) presented in this last part of the

paper is intended to contribute to clarifying how the selected

alternative resulting from the multi-criteria decision making

procedure under activated sludge model input uncertainties

will vary when the relative importance of the different

objectives – expressed by the weights wk – is changed. The

weights represent the desires or preferences of the decision

makers to obtain an alternative that maximises the degree of

satisfaction and reduces variability for one or several specifc

objectives.

The first example consists of a simplified analysis amongst

objectives OBJ1 (minimize environmental impact), OBJ2(minimize economical costs) and OBJ3 (maximize technical

reliability). The results are presented in a bi-plot fashion,

where the changes in the selected alternative (different type

of symbol and colour) are represented when the relative

importance amongst objectives OBJ1 (X axis) and OBJ2 (Y axis)

is modified. The values of OBJ3 are omitted because we do not

have a third dimension, but they can be easily found as (0.75

minus the sum of OBJ1 and OBJ2). The importance of the fourth

objective (comply with the legal effluent discharge limits,

OBJ4) remains constant, i.e. w4¼ 0.25 in order to maintain the

previously established condition that the sum of the weights

wk should be 1.

From the results of Fig. 6 it can be noticed that high values

of OBJ1 clearly favour alternative A4 above the other alterna-

tives. This is mainly due to the fact that the addition of

external carbon source in this strategy will reduce the impact

on water by improving the overall nitrogen removal efficiency

while simultaneously reducing the variability in the effluent

quality as shown in Figs. 1, 4 and 5. Nevertheless, as w2

increases in value the most desirable alternative changes

w a t e r r e s e a r c h 4 2 ( 2 0 0 8 ) 4 4 8 5 – 4 4 9 74496

from A4 to A3 because alternative A3 presents lower operating

costs and variability as shown in Table 4. It is important to

mention that all the alternatives with an ammonium

controller (A4, A6), although having the lowest values in

operating costs, are anyhow not selected when the economic

objectives are prioritized. This is mainly due to the high

sensitivity of the ammonium controller to the input uncer-

tainty, increasing the variance of the multi-criteria index and

thus reducing the coefficient of variation. Finally when

objective OBJ3 is prioritized, the selected alternative depends

on the relative contribution of OBJ1 and OBJ2 because both

alternatives satisfy OBJ3 in a similar way. Again, it can be said

that alternative A4 improves the coefficient of variation of OBJ1(minimize environmental impact) at the expense of sacrificing (to

an extent) its economical variability (OBJ2).

This analysis opens the door to a number of discussions.

One can note in this example that the preliminary selected

alternative A5 is no longer selected as the best for any of the

possible combinations of weights. The considered input

uncertainties had a large impact on the behaviour of this

controller (cascade ammonium), where in some cases this

controller was not capable to compensate for the different

disturbances. After this analysis, it was possible to conclude

that this alternative was only the best for a limited range of

conditions. Thus, when considering uncertainty in the multi-

criteria decision making it is possible to answer questions

such as: What would happen if there is a change in the

influent composition? What are the expected effects of either

temperature changes or toxic spills and how can the

controller handle them? Secondly, this type of representa-

tions clearly distinguishes the different processes and their

more important features, while at the same time it highlights

their main weaknesses. Finally, it is highly encouraged to

perform this type of analysis because it can better guide

decision makers on such important questions as whether to

go ahead with the implementation of a controller and what is

the potential risk of failures in the event of the selection of an

alternative.

6. Conclusions

This paper has demonstrated a method to consider the

influence of activated sludge input uncertainty in the decision

making process during the multi-criteria evaluation of control

strategies in a WWTP. In the first section several WWTP

control strategies were tested and evaluated using standard

deterministic multi-criteria decision analysis using a modified

version of the BSM2 as a case study. In the second part, the

uncertainty in those parameters was quantified by means of

model input probability distributions that were based on the

available knowledge about the different parameters. Next the

plant mechanistic model was coupled to a Monte Carlo engine

that randomly selected parameters from the previously

defined distributions using Latin hypercube sampling, i.e.

input uncertainty, solving the model for each set of model

inputs. Such approach gave a range of possible solutions for

the desired WWTP performance criteria representing their

possible variation. The results were analyzed using several

descriptive statistical tools and it was possible to see how

these input uncertainties were propagated through the model

and affected the different outcomes.

From the evaluated controllers, alternatives with an

external carbon source (alternatives A4 and A6) reduced the

uncertainty in the degree of satisfaction of environmental,

legal and technical objectives but increasing the economical

costs and its variability as a trade-off. The alternatives with

DO and SNO controller (A2 and A3) reduced operational costs

while at the same time improving the effluent quality. Finally,

it was shown how the preliminary selected alternative

A5 – resulting from a deterministic multi-criteria decision

analysis – became less desirable when the input uncertainty

was considered. When considering uncertainty, a simpler

controller structure (A3) was evaluated to have a larger chance

of success.

The relative importance of the control objectives (weights)

on the selection of alternatives was investigated. One one

hand it was possible to discover the affinity of alternative A4

for objectives OBJ1 and OBJ4. On the other hand the scenario

analysis revealed that when OBJ2 was favoured alternative A3

would be selected.

Finally, it should be emphasized that this type of analysis

results in a more transparent decision making process, since

the uncertainty analysis contributes to developing an

improved understanding on the process and the trade-offs

between different objectives. Practically, the uncertainty

analysis allows identification of potential WWTP problems

early on and reduces the risk of controller failures.

Acknowledgments

The authors gratefully acknowledge financial support from

the Spanish ‘‘Ministerio de Ciencia y Tecnologıa’’ projects:

DPI2006-15707-C02-01, 018/SGTB/2007/3.1 and NOVE-

DAR_Consolider. Gurkan Sin also wishes to acknowledge his

post-doctoral scholarship of the Hans Christian Oersted

Postdoc Programme at DTU.

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