Sensitivity study of reduced models of the activated

http://dx.doi.org/10.4314/wsa.v38i2.15Available on website http://www.wrc.org.zaISSN 0378-4738 (Print) = Water SA Vol. 38 No. 2 April 2012ISSN 1816-7950 (On-line) = Water SA Vol. 38 No. 2 April 2012 287

* To whom all correspondence should be addressed. +27 21 959-6459; fax: +27 21 959-6117; e-mail: [email protected] Received 7 August 2009; accepted in revised form 2 April 2012.

Sensitivity study of reduced models of the activated sludge process, for the purposes of parameter estimation and process optimisation: Benchmark process with ASM1

and UCT reduced biological models

S du Plessis and R Tzoneva*Department of Electrical Engineering, Cape Peninsula University of Technology, Cape Town, South Africa

Abstract

The problem of derivation and calculation of sensitivity functions for all parameters of the mass balance reduced model of the COST benchmark activated sludge plant is formulated and solved. The sensitivity functions, equations and augmented sensitivity state space models are derived for the cases of ASM1 and UCT reduced biological models. Matlab software for sensitivity function calculation and sensitivity model simulation is developed. The results are described and discussed. The behaviour of the sensitivity functions is used to determine which parameters of the reduced model need to be estimated in order to fit the reduced model behaviour to the real data for the process behaviour.

Keywords: wastewater treatment, activated sludge process, reduced model, model parameters, sensitivity function, Matlab simulation

Introduction

The problem of effective and optimal control of wastewater treatment plants has become very important in recent years due to increased populations and requirements for the quality of the effluent. The activated sludge process (ASP) is a waste-water treatment process characterised by complex nonlinear dynamics, a large number of variables, and a lack of sensors for real-time measurement of many of these variables. The above process characteristics require real-time control design and implementation strategies to be developed in order to achieve process operation which is compliant with the international standards for effluent quality. Modern optimal control design and implementation for the activated sludge process demands extensive insight into the plant’s operation, clear objectives, and knowledge about process dynamics described by an appro-priate mathematical model. Modelling is the most critical phase in the solution of any control problem because nearly all control techniques require knowledge of the dynamics of the system before control design can be attempted. This means that the primary task of any modern control design is to construct and identify a model for the system which is to be controlled.

Mathematical models of the wastewater treatment pro-cesses were developed using the first principles of conservation of mass and energy (Copp, 2002), on the basis of developed biological models, such as the first ones described by Dold et al. (1980), Henze et al. (1987) and Wentzel et al. (1992). The obtained mass-balance models describe the process-technology structure and biological reactions taking part in this structure.

Examples of such models are the COST benchmark model (Copp, 2002) and the University of Cape Town (UCT) model (Ekama and Marais, 1979; Ekama et al., 1984). The most used biological models are the IAWQ1 (Henze et al., 1987) – the model of the International Association for Water Quality called Activated Sludge Model 1 (ASM1), and the UCT model – the model developed at the UCT Department of Civil Engineering (Dold et al., 1980).

The full ASM1 and especially the UCT model present major problems in terms of their direct use for real-time simu-lation and control design purposes, because of their complexity, limitations and drawbacks, as follows:• Large number of model variables• Complex dependencies and interconnections between the

biological variables• Different time scales for the process dynamics• The control actions for the process are not included in an

explicit way in the model equations• Many variables are difficult to measure• Many model kinetic and stoichiometric parameters are dif-

ficult to determine and have uncertain values

A way to overcome these difficulties would be to simplify thecomplex model by developing reduced biological andmass-balance models with a small number of variables, while still maintaining the same characteristics as these of the original full model (Pearson, 2003). Kinetic parameters of this reduced model can be determined by development and applica-tion of different methods for parameter estimation (Holmberg and Ranta, 1982; Jeppson, 1996; Halevi et al., 1997; Noykova and Gyllenberg, 2000).

Solution of the problem of parameter estimation is based on the given mathematical model in which the parameters are not known, and on data for the process variables obtained by measurement. Many of the biological nutrient removal models are not identifiable since they have many more parameters than

http://dx.doi.org/10.4314/wsa.v38i2.15Available on website http://www.wrc.org.za

ISSN 0378-4738 (Print) = Water SA Vol. 38 No. 2 April 2012ISSN 1816-7950 (On-line) = Water SA Vol. 38 No. 2 April 2012288

feasible measurements (Pertev et al., 1997; Varma et al., 1999). In this case the problem can be solved if the influence of the parameters on model behaviour is studied and the non-influen-tial parameters are considered to have zero or constant known (nominal) values. The influential parameters can then be estimated on the basis of data from the measurements (Varma et al., 1999). The goal of the paper is to develop a mathematical and computational tool for determining which parameters can be accepted as known and which need to be estimated on the basis of the reduced model of the COST benchmark process (Du Plessis, 2009).

A sensitivity study of the model variables towards the model parameters provides a tool to identify the influential parameters. The sensitivity study of the dynamic model set of equations examines the changes in the model variables in response to changes in the model parameters (Smets et al., 2002; Mussafiet et al., 2002; De Pauw and Vanrolleghem, 2003). Parameters resulting in large values for the sensitivity function have to be estimated (Sato and Ohmori, 2002; Louries et al., 2008). If some parameters with small values for the sensi-tivity function could be found, the problem of parameter esti-mation could be simplified by considering these parameters as known (Holmberg and Ranta, 1982; Noykova and Gillenberg, 2000).

There are different mathematical methods for parametric sensitivity analysis, such as finite difference, direct, the Green’s function, the polynomial approximation, and so on. Most previ-ous studies have used the finite difference method (Smets et al., 2002; Plazl et al., 1999; Mussafiet at al., 2002; Varma et al., 1999; Pertev et al., 1997). This method is calculation-intensive and gives a local estimation of the sensitivity function for a given parameter. The global sensitivity method simultane-ously calculates the sensitivity functions for a large number of parameters and for a large variety of parameters. This method allows better analysis of the parameter’s influence as it provides full information about each sensitivity function. The direct method is applied for the mass balance model of the COST benchmark plant (Copp, 2002), based on ASM1 and UCT reduced biological models, developed in Du Plessis (2009). The mass balance models are used for derivation of the sensitivity functions of the model variables towards all kinetic and stoi-chiometric model parameters. As a result, a dynamic sensitivity state-space model is developed. The reduced process model is augmented with the sensitivity model in order to build a model giving possibilities for global sensitivity analysis of all model variables to all model parameters. Matlab software for sensi-tivity function calculation and global (augmented) sensitivity model simulation is developed. The results are described and discussed.

First, a definition of the sensitivity functions is given. Reduced ASM1 and UCT biological and benchmark mass balance models are then described. The augmented sensitiv-ity state-space model of the benchmark mass balance reduced model, based on the ASM1 biological model and based on the UCT biological model, is derived further. Description of Matlab software and results from simulation of the above models are given followed by discussion of the results. Finally, a summary of the results is given and their importance and applicability discussed.

Sensitivity functions, vectors and models

The behaviour of physical systems is determined by the values of their parameters. Investigation of the system response to

changes in the values of the parameters enables determina-tion of parametric sensitivity. Such analysis is important for all spheres of science and engineering, especially for design of the systems and their control (Frank, 1978; Varma et al.,1999; Fesso, 2007). The sensitivity function Xθ(t) of the nonlinear system:

(1)

where: XЄRl is the state space vector, uЄRm is the control vector, gЄRl is the nonlinear vector function of the process states, controls and parameters, and θЄRp is the vector of the parameters, is defined as (Sato & Ohmori, 2002):

(2)

In other words, the sensitivity function (Xθij) is a mathematical

description which indicates the influence of a slight change in parameter θj on the behaviour of the state variable Xi and XθЄRl. Differentiating (1) according to the vector θ gives the sensitiv-ity dynamic nonlinear equation:

(3)

The solution of the Eq. (3) describes the sensitivity func-tion behaviour, but requires data for the state-space vari-ables of the model given by Eq. (1). This means that a set of Eq. (1) and (3) has to be solved simultaneously. The set of these 2 equations can be represented as an augmented mathematical model (Tzoneva et al., 1996). The augmented sensitivity model is obtained by extension of the model state-space vector by the vector of the sensitivity func-tions, as follows:

(4)

The augmented model is used for global parametric analysis of the reduced COST benchmark mass balance model for the case of the ASM1 and UCT reduced biological models. Equations (1)-(4) are derived for each of these 2 cases.

Reduced biological and mass balance models for cost benchmark plant layout

Reduced biological models

A fundamental requirement for development of reduced models is that they contain a minimum number of state vari-ables and parameters to allow for model identification based on available online measurements and for online calculation of the process optimal control. Different types of reduced models are introduced in the literature (Moore, 1981; Glover, 1984; Safanov and Chiang, 1989; Latham and Anderson, 1985; Al-Saggaf and Franklin, 1988; Wisnewski and Doyle, 1996; Halevi et al., 1997; Beck et al., 1996; Lee at al., 2002; Jeppson, 1996) using different approaches of reduction, such as qualitative analysis, truncation methods, dominant eigen-values, and optimisation.

The existing biological models of the activated sludge

0)0(,)(),(,,, XXtuttXgtX

ttXtX

,,

,pjlitXtX

j

iij ,1,,1,),()),((

0)0(,)(

XtgtX

XtgtX

0

0

0)0(

,)(

))(),(),,((),(),(

))(),(),,(())(),(),,((

),(),(

,

XX

ttuttXgtX

tXtuttXg

tuttXg

tXtX

tX

S

S


process are characterised by a large number of processes and compounds, as well as a large number of kinetic and stoichio-metric parameters. These models, such as ASM1 and UCT, are very complex for use in real-time state and parameter estima-tion and process optimisation. ASM1 and UCT reduced models, used in conjunction with the benchmark mass balance model, are developed in Du Plessis (2009), applying time-scale analy-sis of the model variables’ dynamics (Dochain, 2003; Lennox et al., 2001; Stecha et al., 2005; Wang and Gawthrop, 2001). This allows decomposition of the complex models’ variables into the following time scales: • Fastest (minutes) – dynamics of physical-chemical variables

such as dissolved oxygen, pH, conductivity, redox potential• Intermediate (hours) – dynamics of utilisation of carbona-

ceous and nitrogenous substrates and the inflow flow rate and concentrations of waste materials

• Slowest (weeks or months) – dynamics of the heterotrophic and autotrophic microorganisms and slowly biodegradable biomass

The main disturbances to the activated sludge process are the load disturbances determined by the inflow rate and waste concentrations. The control of the ASP would be effective if it could overcome the effect of the inflow distur-bances. This means that the model of the process has to have dynamics within the same time scale as that of the distur-bances. That is why only dynamics of the carbonaceous and nitrogenous substrate and inflow disturbances are included in the reduced model. It is possible to neglect the equations of the full biological model describing the concentrations of the slowest variables because their dynamics are many times slower and they can thus be considered to be in a steady state. It is also possible to neglect the equation for dissolved oxygen (DO) concentration because its dynamics are approx. 10 times faster than the dynamics of the substrate concen-trations, and it can be assumed that the value of the DO is controlled and is equal to that of the required set-point. The dissolved oxygen concentration as a control variable is included in the switching functions of the process rates describing the reduced models. The above principles are used for development of both the ASM1 and UCT reduced models. At the same time the differences between them, due to the different views on the processes of adsorption and hydrolysis, are preserved. The notations of the ASM1 model are used for the process variables of both models. The nota-tions of the model parameters are kept as for the original models.

Reduced-order ASM1 biological model

Aerobic growth of heterotrophs XBH for degrading of organic matter, anoxic growth of heterotrophs for the de-nitrification process, aerobic growth of autotrophs XBA for the nitrification process and, lastly, the hydrolysis of entrapped organic nitro-gen, are the processes that characterise the dynamic behaviour of the 3 components used in describing the removal of carbon and nitrogen: soluble ammonium nitrogen SNHn; soluble nitrate nitrogen SNOn, and soluble readily-biodegradable substrate SSn concentrations. This model is described by the Peterson matrix given in Table 1. It has 4 processes and 3 compounds (vari-ables), where SOn is the concentration of dissolved oxygen and XSn is the slowly-biodegradable substrate concentration. The typical values of the model parameters for the ASM1 model are given in Table 2.

The corresponding differential equations describing the vari-ables’ rates for the n-th tank are:

(5)

(6)

(7)

Table 2ASM1 model parameters

Symbol Value ExplanationYA 0.24 Autotrophic biomass yieldYH 0.67 Heterotrophic biomass yieldiXB 0.086 Nitrogen mass per mass of COD in biomass µA 0.8 Maximum specific growth rate of autotrophic

biomassµH 6 Maximum specific growth rate of heterotro-

phic biomassKS 20 Half-saturation coefficient for heterotrophic

biomassKOH 0.2 Oxygen half-saturation coefficient for hetero-

trophic biomassKNO 0.5 Nitrate for denitrifying heterotrophsKNH 1.0 Ammonium half-saturation coefficient for

autotrophic biomassKOA 0.4 Oxygen half-saturation coefficient for auto-

trophic biomassηg 0.8 Correction factor for µH under anoxic

conditionsηh 0.4 Correction factor for hydrolysis under anoxic

conditionskh 3 Maximum hydrolysis rateKX 0.03 Half-saturation coefficient for hydrolysis of

slowly-biodegradable substrate

Table 1 The reduced-order ASM1 biological model in a matrix format

Compound i

process j Process

SS SNO SNH Pj, j = process for the n-th tank

1.Aerobic growth of heterotrophs

HY1

0 XBi BH

OOH

O

SS

SH X

SKS

SKS

2.Anoxic growth of heterotrophs

HY1

H

H

YY

86.21

XBi BHg

NONO

NO

OOH

OH

SS

SH X

SKS

SKK

SKS

ˆ

3.Aerobic growth of autotrophs

0 AY

1

AXB Yi 1

BAOOA

O

NHNH

NHA X

SKS

SKS

4. Hydrolysis 1 0 0

NONO

NO

OOH

OHh

OOH

O

BHsx

BHsBHh

SKS

SKK

SKS

XXKXXXk

nn

//

gBHOOH

OH

NONO

NO

SS

SHXBBH

OOH

O

SS

SHXBS X

SKK

SKS

SKS

iXSK

SSK

Si

NHr ˆˆ

BA

NHNH

NH

OOA

OA

AXB X

SKS

SKS

Yi 1

BA

NHNH

NH

OOA

OA

Ag

NONO

NO

OOH

OH

SS

SHBH

H

HS X

SKS

SKS

YSKS

SKK

SKSX

YY

NOr ˆ1ˆ

86.21

gBHOOH

OH

NONO

NO

SS

SH

HBH

OOH

O

SS

SH

HS X

SKK

SKS

SKS

YX

SKS

SKS

YSr ˆ1ˆ1

BH

OOH

OH

NONO

NOh

OOH

O

BHsx

BHsh X

SKK

SKS

SKS

XXKXXk

/

/



Reduced-order UCT biological model

The model is described by the Peterson matrix given in Table 3. It has 10 processes and 4 variables. The considered processes are: aerobic growth of XBH on SS with SNH, aero-bic growth of XBH on SS with SNO, anoxic growth of XBH on SS with SNH, anoxic growth of XBH on SS with SNO, aerobic growth of XBH on Sads with SNH, aerobic growth of XBH on Sads with SNO, anoxic growth of XBH on with SNH, anoxic growth of XBH on Sads with SNO, adsorption of XS , aerobic growth of XBA on SNH. The additional variable, representing the main concept of the enzyme reactions in the UCT model is Sads, which describes the concentration of the adsorbed slowly-biodegradable substrate. The model parameters are given in Table 4. The UCT notations for the model para-meters are used.

The corresponding equations for the variables’ rates are:

(8)

(9)

(10)

(11)

Mass-balance reduced model of the benchmark plant

The layout of the COST benchmark structure (Copp, 2002) is given in Fig. 1. The benchmark format has 2 anoxic tanks, 3 aerobic tanks and a secondary settler with 2 recycle flows (from aerobic tank 5 to the input and from the settler to the input), where: Q0 is the input flow rate, Qa is the internal recycle flow rate, Qn, n=1:5, is the output flow rate of the n-th tank, Qr is the recycle flow rate, Qe is the effluent flow rate, and Qw is the waste flow rate, Xn, n=1:5, is the vector of the waste compound concentrations in the influent and the n-th tank. The values of the flow rates and tanks volumes are given in Table 5.

Figure 1Wastewater treatment plant layout of the benchmark model

gOOH

OH

OOH

O

NONO

NO

NHHA

HA

gNONO

NO

OOH

OH

OOH

O

NHHA

NH

BHBHadsSP

BHadsMP

ZHads

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XXSK

XSK

Yr

1

BHadsMABHSA XSfXXK

Table 3 The reduced-order UCT biological model in a matrix format

i

j

adsS SS NHS

NOS Process rates, j for the n-th tank

1) 0 ZHY1

HZBf ,

0 BH

NHHA

NH

OOH

O

SS

SH X

SK

S

SK

S

SK

S

2) 0

ZHY1

0 HZBf ,

BHNONO

NO

NHHA

HA

OOH

O

SS

SH X

SK

S

SKK

SKS

SK

S

3) 0

ZHY1

HZBf , ZH

ZH

YY

86.21

BH

NONO

NO

NHHA

NH

OOH

OH

SS

SH X

SKS

SKS

SKK

SK

S

4) 0 ZHY1

0

HZB

ZH

ZH

fYY

,

86.21

BHNONO

NO

NHHA

HA

OOH

OH

SSH

SH X

SK

S

SKK

SKK

SK

S

5)

ZHY1

0

HZBf ,

0 BH

NHHA

NH

OOH

O

BHadsSP

BHadsMP X

SK

S

SK

S

XSK

XSK

/

/

6)

ZHY1

0 0

HZBf , BH

NONO

NO

NHHA

HA

OOH

O

BHadsSP

BHadsMP X

SK

S

SKK

SK

S

XSKXSK

./

/

7)

ZHY1

0 HZBf ,

ZH

ZH

YY

86.21

BHgNONO

NO

NHHA

NH

OOH

OH

BHadsSP

BHadsMP X

SK

S

SKS

SKK

XSKXSK

//

8)

ZHY1

0 0

HZB

ZH

ZH

fYY

,

86.21

BHgNONO

NO

NHHA

HA

OOH

OH

BHadsSP

BHadsMP X

SKS

SKK

SKK

XSKXSK

/

/

9) 1 0 0 0 BHadsMABHSA XSfXXK 10) 0 0

HZB

ZA

fY

,

1

ZAY1

BANHSA

NH

OOA

OA X

SK

S

SKS

OOH

OH

OOH

O

NONO

NO

NHHA

HA

NONO

NO

OOH

OH

OOH

O

NHHA

NH

SSH

SBHH

ZHSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

SKSX

Yr 1

gNONO

NO

OOH

OH

OOH

O

NHHA

NHBH

BHadsSP

BHadsMPHZB

NONO

NO

OOH

OH

OOH

O

NHHA

NHBH

SSH

SHHZBSNH

SKS

SKK

SKS

SKS

XXSK

XSKf

SKS

SKK

SKS

SKS

XSK

Sfr

.,

,

BAOOH

O

NHSA

NHAHZB

ZAX

SKS

SKS

fY

,

1

BHadsSP

BHadsgMP

SSH

SH

NONO

NO

NHHA

HA

OOH

OHBHHZB

ZH

ZH

BHadsSP

BHadsgMP

SSH

SH

NONO

NO

NHHA

NH

OOH

OHBH

ZH

ZH

BHadsSP

BHadsMP

SSH

SH

NHHA

HA

NONO

NO

OOH

OBHHZBSNO

XSKXSK

SKS

SKS

SKK

SKKXf

YY

XSKXSK

SKS

SKS

SKS

SKKX

YY

XSKXSK

SKS

SKK

SKS

SKSXfr

,

,

86.21

.86.2

1

BAOOH

O

NHSA

NH

ZA

A XSK

SSK

SY

OOH

OH

OOH

O

NONO

NO

NHHA

HA

NONO

NO

OOH

OH

OOH

O

NHHA

NH

SSH

SBHH

ZHSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

SKSX

Yr 1

gNONO

NO

OOH

OH

OOH

O

NHHA

NHBH

BHadsSP

BHadsMPHZB

NONO

NO

OOH

OH

OOH

O

NHHA

NHBH

SSH

SHHZBSNH

SKS

SKK

SKS

SKS

XXSK

XSKf

SKS

SKK

SKS

SKS

XSK

Sfr

.,

,

BAOOH

O

NHSA

NHAHZB

ZAX

SKS

SKS

fY

,

1

BHadsSP

BHadsgMP

SSH

SH

NONO

NO

NHHA

HA

OOH

OHBHHZB

ZH

ZH

BHadsSP

BHadsgMP

SSH

SH

NONO

NO

NHHA

NH

OOH

OHBH

ZH

ZH

BHadsSP

BHadsMP

SSH

SH

NHHA

HA

NONO

NO

OOH

OBHHZBSNO

XSKXSK

SKS

SKS

SKK

SKKXf

YY

XSKXSK

SKS

SKS

SKS

SKKX

YY

XSKXSK

SKS

SKK

SKS

SKSXfr

,

,

86.21

.86.2

1

BAOOH

O

NHSA

NH

ZA

A XSK

SSK

SY

Table 4Reduced-order UCT process parameters

with their common valuesASM1/ UCT notations

Value Explanation

YA / YZA 0.15 Autotrophic biomass yieldYH / YZH 0.666 Heterotrophic biomass yieldiXB / fzbN 0.068 Nitrogen mass per mass of COD in

biomass µA/ myA 0.5 Maximum specific growth rate for auto-

trophic biomassµH / myH 2.5 Maximum specific growth rate for hetero-

trophic biomassKS / KSH 5 Half-saturation coefficient for heterotro-

phic biomassKOH/ KOH 0.002 Oxygen half-saturation coefficient for

heterotrophic biomassKNO/ KNO 0.1 Nitrate half-saturation coefficient for

denitrifying heterotrophic biomassKSA / KNH 1 Ammonium half-saturation coefficient for

autotrophic biomassKOA / KOA 0.002 Oxygen half-saturation coefficient for

autotrophic biomassηg / nyG 0.33 Correction factor for µH under anoxic

conditionskh / KMP 1.35 Maximum specific growth rate of the het-

erotrophs when utilising adsorbed particu-late slowly-biodegradable COD

aa XQ

Q2 Q3 Q4 Q5

5X 3X

An1 An2 Ae3 Ae4 Ae5 X 1X

2X

Q1

4X

Q0 rr XQ

ee XQ

wwXQ


Table 5Benchmark plant specifications

Notation Value DescriptionQ0 18 446 m3/day Influent flow rateQa 55 338 m3/day Internal flow rateQr 18 446 m3/day Recirculation flow rateQw 385 m3/day

(sludge age 10 days)Waste flow rate

V1 , V2 / V3 ÷ V5 1000 m3 /1333 m3 Anoxic/Aerobic volume

The mass-balance equations describing the benchmark struc-ture for the reduced ASM1 and UCT biological models in discrete time domain are (Du Plessis, 2009):For Tank 1: where Q = Qa + Qr + Q0

(12)

For Tank n=2, 3, 4, and 5:

(13)

where: Xn=[SNHn SNOn SSn]

T and Xn=[SNHn SNOn SSn Sadsn]

T, n=1:5, are vectors of the concentrations of the variables considered in the ASM1 and UCT reduced models, and∆t=15 (min) is the sampling period. The model of the settler, considered as an ideal one, is incorporated in the above equations as:

(14)

where: for the soluble compounds, λ=1, and for the particulate compounds, λ=(Q0+Qr)/(Qr+Qw).

The above equations are the same for both the ASM1 and UCT model, as structure, flow and volume. The difference is in the description of the number of compounds and the process rates, due to the different approaches to representing the biological activities of the microorganisms in these 2 models (Wentzel et al., 1992). The variables’ rates r are described correspondingly by Eqs. (5) ÷(7) for the reduced ASM1 and by Eqs. (8)÷(11) for the reduced UCT model.

Additionally to the parameters of the reduced models, a parameter f is included in the mass balance equations for the 1st tank. This parameter multiplies the concentration of the ammonia nitrogen in order to take account of the biological ammonia not considered in the reduced models, due to neglecting of the variable SND (soluble biodegradable organic nitrogen concentration).

The sensitivity of the reduced model variables towards the model parameters is evaluated by using the theory of sensitiv-ity (Schermann and Garag-Gabin, 2005; Montgomery, 1997; Breyfogle and Breyfogle, 2003).

Augmented sensitivity mass balance model derivation for the case of ASM1 reduced model

Sensitivity functions and equation derivation

The derivations of the sensitivity functions and models are analogous for every process tank. That is why they are calcu-lated for the n-th tank, as follows:

• For the parameter f 1) For the 1st tank

(15)

(16)

(17)

2) For tanks 2 ÷ 5

(18)

(19)

(20)

dSNH,n /df=SfNH,n, dSNO,n/df=Sf

NO,n, dSS,n/df=SfS,n are the sensitivity

functions of the process variables towards the parameter f. The variables’ rate derivatives towards the process variables

are given in Appendix A. • For the parameters

1) For the 1st tank

(21)

)()()()()1()1(

ktrkSkSQVtkSkS

nNHnnnn SNHNHn

NHNH

)()()()()1()1(

ktrkSkSQVtkSkS

nNOnnnn SNONOn

NONO

)()()()()1()1(

ktrkSkSQVtkSkS

nSnnnn SSSn

SS

ktrkSkSQVtkSkS

nnnnn Sadsadsadsn

adsads

)1(

1

5XXr

fS

SSSNH

fNO

SNOSNH

fNH

SNHSNH

fNHNHo

fNHr

fNHa

fNH

SrSrSr

QSSQSQSQV

S

1,1,

1,1,1,1,1,

1,1,

1,,5,5,1

1,1

fS

SSSNO

fNO

SNOSNO

fNH

SNHSNO

fNO

fNOr

fNOa

fNO

SrSr

SrQSSQSQV

S

1,1,

1,1,1,

1,

1,1,

1,1,5,5,1

1,1

fS

SSSS

fNO

SNOSS

fNH

SNHSS

fS

fSr

fSa

fS

SrSr

SrQSSQSQV

S

1,1,

1,1,1,

1,

1,1,

1,1,5,5,1

1,1

fnS

nSSnSNH

fnNO

nSNOnSNH

fnNH

nSNHnSNH

fnNH

fnNH

n

fnNH

SrSr

SrSSQV

S

,,

,,,,

,,

,,1,,1

fnS

nSSnSNO

fnNO

nSNOnSNO

fnNH

nSNHnSNO

fnNO

fnNO

n

fnNO

SrSr

SrSSQV

S

,,

,,,

,

,,

,,1,,1

fnS

nSSnSS

fnNO

nSNOnSS

fnNH

nSNHnSS

fnS

fnS

n

fnS

SrSr

SrSSQV

S

,,

,,,

,

,,

,,1,,1

nSNHnSNH

nNH

nSNH rSr ,

,,

,

, nSNOnSNH

nNO

nSNH rSr ,

,,

,

, nSSnSNH

nS

nSNH rSr ,

,,

,

, nNHnSNO

nNH

nSNO rSr ,

,,

,

nNOnSNO

nNO

nSNO rSr ,

,,

,

, nSSnSNO

nS

nSNO rSr ,

,,

,

,0

,

,

nNH

nSS

Sr , nSNO

nSSnNO

nSS rSr ,

,,

,

, nSSnSS

nS

nSS rSr ,

,,

,

XhhgOANHNOOHSHAXBAH KkKKKKKiYY ,,,,,,,,,,,,,

1,1,1,

1,1,1,1,

1,1,

1,1,5,1

1,1

SNHSSSSNHNO

SNOSNH

NHSNHSNHNHNHraNH

rSrSr

SrQSSQQV

S

)()()(

)()()()1(

11

55

1101

ktrkQSkfSQ

kSQkSQ

VtkSkS

NHSNHNH

NHrNHaNHNH

)()()(

)()()()1(

11

55

1101

ktrkQSkSQ

kSQkSQ

VtkSkS

NOSNONO

NOrNOaNONO

)()()(

)()()()1(

11

55

1101

ktrkQSkSQ

kSQkSQ

VtkSkS

SSSS

SrSaSS

ktr

kQSkSQ

kSQkSQ

VtkSkS Sads

adsadsO

adsradsaadsads 1

1

55

111

1



(22)

(23)

2) For the tanks n = 2,5

(24)

(25)

(26)

where: Sθ

NH,n, Sθ

NO,n, Sθ

S,n are the sensitivity functions to the parameter q, the partial derivatives of the variable’s rates according to the variables are determined above and the derivatives of the variables’ rates according to the model parameters are given in Appendix A.

Augmented sensitivity state-space model

The sensitivity state-space model is derived for the whole plant. The vector of the sensitivity state-space consists of all sensitiv-ity functions according to the model parameters, as follows for the n-th tank:

(27)

The state process variables’ vector for the n-th tank is:

(28)

The combined vector for the process variables and sensitivity functions for the n-th tank is:

(29)

This vector is used for the augmented state-space process and sensitivity-function model derivation. The matrix of the derivatives of the variables’ rates towards the process variables is:

(30)

The vector of the derivatives of the process rates to the model parameters, Appendix A, is:

(31)

Description of the augmented model in the discrete state-space domain is given for every tank separately because of the large dimensions of the vector of the state-space.

• Augmented sensitivity state-space equation for the 1st tank in the discrete form incorporates the process variables and sensitivity functions, as follows:

(32) where:

the variables’ rates are expressed by the process rates r1(k)=C1

STρ1(k), and ρ1=[ρ11 ρ12 ρ13 … ρ1N]T is a vector of the process rates, u1(k)=SO,1(k) is the dissolved oxygen concentration in the 1st tank, considered as its control action:

(33)

The matrices A1θ are identical for all parameters. The matrix

C1Srepresents the parameters in the Peterson matrix and is:

(34)

The matrix B1S is:

(35)

The inflow variables’ concentration vector is Xiϕ=[SNHϕ SNOϕ SSϕ]T.

The vector D1S represents the derivatives of the variables’ rates

towards the model parameters.

(36)

The matrix A15S describes the internal recycle between the end

of the 5thtank and the beginning of the 1stone. The connections

1,1,1,

1,1,1,

1,

1,1,

1,1,5,1

1,1

SNOSSSSNONO

SNOSNO

NHSNHSNONONOraNO

rSrSr

SrQSSQQV

S

1,1,1,

1,1,1,

1,

1,1,

1,1,5,1

1,1

SSSSSSSNO

SNOSS

NHSNHSSSSraS

rSrSr

SrQSSQQV

S

nSNHnSnSSnSNHnNO

nSNOnSNH

nSNHnSNHnSNHnNHnNHnNH

rSrSr

SrSSQV

S

,,,

,,,,

,,

,,1,1

,1

nSNOnSnSSnSNOnNO

nSNOnSNO

nSNHnSNHnSNOnNOnNOnNO

rSrSr

SrSSQV

S

,,,,,

,,

,,

,,1,1

,1

nSSnSnSSnSSnNO

nSNOnSS

nSNHnSSnSSnSnSnS

rSrSr

SrSSQV

S

,,,

,,,

,

,,

,,1,1

,1

45

,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

...

...

...

R

SSSSSSSSS

SSSSSSSSSSSS

SSSSSSSSSSSS

SSSSSSSSSSSS

S

T

KnS

KnNO

KnNH

knS

knNO

knNHnSnNOnNH

nSnNOnNHKnS

KnNO

KnNH

KnS

KnNO

KnNH

KnS

KnNO

KnNH

KnS

KnNO

KnNH

KnS

KnNO

KnNHnSnNOnNHnSnNOnNH

inS

inNO

inNH

YnS

YnNO

YnNH

YnS

YnNO

YnNH

fnS

fnNO

fnNH

n

XXXhhhhhh

gggOAOAOANHNHNHNONONO

OHOHOHSSSHHHAAA

XBXBXBAAAHHH

3,,, RSSSX TnSnNOnNHn

48RSXX nnSn

33,

,,

,,

,,

,,

,,

,,

,,,

,, ;; xnSS

nSSnSNO

nSSnSNH

nSSnSSnSNO

nSNOnSNO

nSNHnSNO

nSSnSNH

nSNOnSNH

nSNHnSNH

n

n RrrrrrrrrrXr

45

,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

R

rrrrrrrrr

rrrrrrrrrrrr

rrrrrrrrrrrr

rrrrrrrrrrrr

r

T

KnSS

KnSNO

KnSNH

knSS

knSNO

knSNHnSSnSNOnSNH

nSSnSNOnSNHKnSS

KnSNO

KnSNH

KnSS

KnSNO

KnSNH

KnSS

KnSNO

KnSNH

KnSS

KnSNO

KnSNH

KnSS

KnSNO

KnSNHnSSnSNOnSNHnSSnSNOnSNH

inSS

inSNO

inSNH

YnSS

YnSNO

YnSNH

YnSS

YnSNO

YnSNH

fnSS

fnSNO

fnSNH

n

XXXhhhhhh

gggOAOAOANHNHNHNONONO

OHOHOHSSSHHHAAA

XBXBXBAAAHHH

kXkuXAkuXDkXB

kuXCkXkuXAkXSSS

iS

STSSS

511151111

111111111

,,,,

,,,,1

484811111 ,,,,,,,,

,,,,,,,,,,

R

KkKKKKKiYYfA

AdiagkuxAXhhgOANHNOOHS

HABAHS

331

11

11

11

1

100

010

001

RA

QVt

QVt

QVt

A

33

1,1,

1

11,1,

1,1,

1,1,

1,1,

1

11,1,

1,1,

1,1,

1,1,

1

1

1

1

1

1

R

rVQttrtr

trrVQttr

trtrrVQt

A

SSSS

SNOSS

SNHSS

SSSNO

SNOSNO

SNHSNO

SSSNH

SNOSNH

SNHSNH

48445411 0

RCC S

100

011

186.2

1

10

1

AAXB

HH

HXB

HXB

YYi

YYY

i

Yi

tC

33

01

01

01

1345

344

01

1348

11

00

00

00

,0

00

, xxS R

QVt

QVt

QVtf

BR

QVt

BRBB

B

481,1,1,311 ,,,,,,

,,,,,,,,,0R

KkKKK

KKiYYfrrrD

T

XhhgOANHNO

OHSHAXBAHSSSNOSNHS


are done by the vector X5S=[X5 S5

θ]∈R48:

(37)

where: all matrices A15 are identical and:

• Augmented sensitivity model equations for tanks n = 2 ÷ 5

The state-space model incorporating the process variables and sensitivity functions are derived for the n-th tank, as follows:

(38) where: ρn=[ρn1 ρn2 ρn3 … ρn4]

T

(39)

The matrices An,n-1S are:

(40)

where: An,n-1 and An,n-1

q are identical

The vector Dn

S is:

(41)

The matrices CnS are equal to C1

S.

• Augmented sensitivity state-space model of the whole plant

The model of the variables and sensitivity functions for the whole plant is described on the basis of Eqs. (32) and (38) as follows: (42)

The matrices AS∈R240x240 and CS∈ R20x240 are:

The vector of the process rates is:

(43)

The matrix BS and the vector DS are:

(44)

(45)

Model (42)-(45) is used to calculate the parametric sensitivity functions of the benchmark structure with the ASM1 reduced model.

Augmented sensitivity mass balance model derivation for the case of the UCT reduced-order model

Sensitivity function and equation derivation

The order of calculation is done in the same way as above.

• For the parameter f

1) For the 1st tank

The sensitivity functions are given by equations:

(46)

(47)

(48)

(49)

2) For the other n = 2÷5 tanks

The sensitivity functions are given by the equations:

(50)

4848151515 ,,,,,,

,,,,,,,,,,

RKkKKK

KKiYYfAAdiagA

XhhgOANHNO

OHSHAXBAHS

33

1

1

1

1515

00

00

00

R

QQVt

QQVt

QQVt

AA

ra

ra

ra

raS QQ

VtdiagA1

15

S

nSnnnn

Snnn

STn

Snnn

Sn

Sn XAkuXDkuXCkXkuXAkX 11,,,,,,,1

33

1

1

1

1515

00

00

00

R

QQVt

QQVt

QQVt

AA

ra

ra

ra

raS QQ

VtdiagA1

15

S

nSnnnn

Snnn

STn

Snnn

Sn

Sn XAkuXDkuXCkXkuXAkX 11,,,,,,,1

4848

,,,,,,,,,,,,,,,,

RKkKK

KKKiYYfAAdiagA

XhhgOANH

NOOHSHAXBAHnnSn

4545

,,

,,

,,

,,

,,

,,

,,

,,

,,

1

1

1

R

trQVttrtr

trtrQVttr

trtrtrQVt

A

nSSnSS

n

nSNOnSS

nSSnSS

nSSnSNO

nSNOnSNO

n

nSNHnSNO

nSSnSNH

nSNOnSNH

nSNHnSNH

n

n

33

100

010

001

R

QVt

QVt

QVt

A

nn

nn

nn

n

48481,1,1, ,,,,,

,,,,,,,,,,,

RKkKK

KKKiYYfAAdiagA

XhhgOANH

NOOHSHAXBAHnnnnSnn

33

1

1

1

1,1,

00

00

00

R

QVt

QVt

QVt

AA

nn

nn

nn

nnnn

33

1

1

1

1,1,

00

00

00

R

QVt

QVt

QVt

AA

nn

nn

nn

nnnn

484811,

RQVtdiagA nn

Snn

48,,,31 ,,,,,

,,,,,,,,,0 R

KkKKKKKiYYf

rrrDT

XhhgOANH

NOOHSHAXBAHnSSnSNOnSNH

Sn

kuxDXBkuxCkXAkX Si

SSTSSS ,,,,1

SS

SS

SS

SS

SS

S

AAAA

AAAA

AA

A

554

443

332

221

151

000000000000

000

S

S

S

S

S

S

CC

CC

C

C

5

4

3

2

1

00000000000000000000

20

5453525144434241

343332312423222114131211,, RkuxT

SS

SS

SS

SS

SS

S

AAAA

AAAA

AA

A

554

443

332

221

151

000000000000

000

S

S

S

S

S

S

CC

CC

C

C

5

4

3

2

1

00000000000000000000

20

5453525144434241

343332312423222114131211,, RkuxT

3240

3192

1

0

RB

B

S

S

24054321 RDDDDDDTSSSSSS

fSNH

fads

SSNH

fS

SSSNH

fNO

SNOSNH

fNH

SNHSNH

fNHNH

fNHr

fNHa

fNH

rSrSrSr

SrQSSQSQSQV

S

ads1,1,1,1,

1,1,1,

1,1,

1,1,

1,1,05,5,1

1,

1,

1

fSNO

fads

SSNO

fS

SSSNO

fNO

SNOSNO

fNH

SNHSNO

fNO

fNOr

fNOa

fNO

rSrSrSr

SrQSSQSQV

S

ads1,1,1,1,

1,1,1,

1,1,

1,1,

1,1,5,5,1

1,

1,

1

fSS

fads

SSS

fS

SSSS

fNO

SNOSS

fNH

SNHSS

fS

fSr

fSa

fS

rSrSrSr

SrQSSQSQV

S

ads1,1,1,1,

1,1,1,

1,1,

1,1,

1,1,5,5,1

1,

1,

1

fads

fads

Sads

fS

SSads

fNO

SNOads

fNH

SNHads

fads

fadsr

fadsa

fads

rSrSrSr

SrQSSQSQV

S

ads1,1,1,1,

1,1,1,

1,1,

1,1,

1,1,5,5,1

1,

1,

1

f

nSNHf

nadsnSadsnSNH

fnS

nSSnSNH

fnNO

nSNOnSNH

fnNH

nSNHnSNH

fnNH

fnNH

n

fnNH

rSrSrSr

SrSSQV

S

,,,,,

,,,

,,

,,

,,1,,1



(51)

(52)

(53)

• For the parameters

the equations are as follows: 1) For the 1st tank

(54)

(55)

(56)

(57)

2) For the tanks n = 2 ÷ 5

(58)

(59)

(60)

(61)

The partial derivatives of the variables’ rates are determined in Appendix B.

Augmented sensitivity state-space model

The augmented sensitivity state-space model is formed in the same way as above. It is based on the model of the plant extended with the model of the sensitivity functions. The vector of the sensitivity functions consists of all sensitivity functions according to the model parameters, as follows for the n-th tank:

(62)

The process state-space vector for the n-th tank is:

(63)

The augmented vector for the process variables and sensitivity functions is: (64)

This vector is used to describe the augmented model of the process variables and sensitivity functions. The matrix of the process variables’ rates derivatives towards the process variables is:

(65)

The vector of the variables’ rate derivatives towards the para-meters can be represented in the following way (Appendix B):

(66)

The augmented process and sensitivity state-space model is described for every tank separately because the large dimen-sion of the vector of state-space and the number of tanks creates the need for a large amount of space for describing the model matrices.

• The state-space model for Tank 1 in discrete form is:

(67)

where: ρ1=[ρ11 ρ12 ρ13 … ρ1,10]T is a vector of the process rates

(68) The matrices A1

θare identical for all model parameters:

The matrix C1S=[C1 010x68]∈R10x72 represents the parameters

from the Peterson matrix:

f

nSNOf

nadsnSadsnSNO

fnS

nSSnSNO

fnNO

nSNOnSNO

fnNH

nSNHnSNO

fnNO

fnNO

n

fnNO

rSrSrSr

SrSSQV

S

,,,

,,,

,,,

,

,,

,,1,,1

fnSS

fnads

nSadsnSS

fnS

nSSnSS

fnNO

nSNOnSS

fnNH

nSNHnSS

fnS

fnS

n

fnS

rSrSrSr

SrSSQV

S

,,,

,,,

,,,

,

,,

,,1,,1

fnSads

fnads

nSadsnSads

fnS

nSSnSads

fnNO

nSNOnSads

fnNH

nSNHnSads

fnads

fnads

n

fnads

rSrSrSr

SrSSQV

S

,,,

,,,

,,,

,

,,

,,1,,1

SASPMPgOANANOOHSHAZBHZAZH KKKKKKKKfYY ,,,,,,,,,,,,,

1,1,1,1,1,

1,1,1,

1,1,

1,1,

1,1,5,1

1,1

SNHadsSadsSNHS

SSSNHNO

SNOSNH

NHSNHSNHNHNHraNH

rSrSrSr

SrQSSQQV

S

1,1,1,

1,1,1,

1,1,1,

1,

1,1,

1,1,5,1

1,1

SNOadsSadsSNOS

SSSNONO

SNOSNO

NHSNHSNONONOraNO

rSrSrSr

SrQSSQQV

S

1,1,1,

1,1,1,

1,1,1,

1,

1,1,

1,1,5,1

1,1

SSadsSadsSSS

SSSSNO

SNOSS

NHSNHSSSSraS

rSrSrSr

SrQSSQQV

S

1,1,1,

1,1,1,1,1,

1,1,

1,1,

1,1,5,1

1,1

adsadsSadsadsS

SSadsNO

SNOads

NHSNHadsadsadsraads

rSrSrSr

SrQSSQQV

S

nSNHnadsnSadsnSNHnS

nSSnSNHnNO

nSNOnSNH

nNHnSNHnSNHnNHnNH

nnNH

rSrSrSr

SrSSQV

S

,,,,,

,,,

,,

,,

,,1,,1

nSNOnadsnSadsnSNOnS

nSSnSNOnNO

nSNOnSNO

nNHnSNHnSNOnNOnNO

nnNO

rSrSrSr

SrSSQV

S

,,,

,,,

,,,

,

,,

,,1,,1

nSSnadsnSads

nSSnSnSSnSSnNO

nSNOnSS

nNHnSNH

nSSnSnSn

nS

rSrSrSr

SrSSQV

S

,,,

,,,

,,,

,

,,

,,1,,1

nSadsnadsnSadsnSadsnS

nSSnSadsnNO

nSNOnSads

nNHnSNHnSadsnadsnads

nnads

rSrSrSr

SrSSQV

S

,,,

,,,

,,,

,

,,

,,1,,1

68

,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

...

...

...

...

...

R

SSSSSSSS

SSSSSSSSSSSS

SSSSSSSSSSSS

SSSSSSSSSSSS

SSSSSSSSSSSS

SSSSSSSSSSSS

S

T

Knads

KnS

KnNO

KnNH

fnads

fnS

fnNO

fnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

nadsnSnNOnNHK

nadsKnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

nadsnSnNOnNHnadsnSnNOnNHfnads

fnS

fnNO

fnNH

Ynads

YnS

YnNO

YnNH

Ynads

YnS

YnNO

YnNH

fnads

fnS

fnNO

fnNH

n

AAAAMAMAMAMA

SASASASASPSPSPSPMPMPMPMP

ggggOAOAOAOANANANANA

NONONONOOHOHOHOHSSSS

HHHHAAAAZBHZBHZBHZBH

ZAZAZAZAZHZHZHZH

4,,,, RSSSSX TnadsnSnNOnNHn

72RSXXT

nnSn

44

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

,,

x

T

nSadsnSads

nSSnSads

nSNOnSads

nSNHnSads

nSadsnSS

nSSnSS

nSNOnSS

nSNHnSS

nSadsnSNO

nSSnSNO

nSNOnSNO

nSNHnSNO

nSadsnSNH

nSSnSNH

nSNOnSNH

nSNHnSNH

n

n R

rrrrrrrrrrrrrrrr

Xr

(

68

,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

...

...

...

...

...

R

rrrrrrrr

rrrrrrrrrrrr

rrrrrrrrrrrr

rrrrrrrrrrrr

rrrrrrrrrrrr

rrrrrrrrrrrr

r

T

Knads

KnS

KnNO

KnNH

fnads

fnS

fnNO

fnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

nadsnSnNOnNHKnads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

Knads

KnS

KnNO

KnNH

nadsnSnNOnNHnadsnSnNOnNHfnads

fnS

fnNO

fnNH

Ynads

YnS

YnNO

YnNH

Ynads

YnS

YnNO

YnNH

fnads

fnS

fnNO

fnNH

n

AAAAMAMAMAMA

SASASASASPSPSPSPMPMPMPMP

ggggOAOAOAOANANANANA

NONONONOOHOHOHOHSSSS

HHHHAAAAZBHZBHZBHZBH

ZAZAZAZAZHZHZHZH

kXkuXAkuXD

kXBkuXCkXkuXAkXSSS

STSSSS

51115111

1111111111

,,,,

,,,,1

7272111 ,,,,,,

,,,,,,,,,,,

R

KfKKKKKKKKfYYf

AAdiagAAMASASPMPgOA

NANOOHSHAZBHZAZHS

441

11 }1{

RQVtdiagA

44

1,1,

1

11,1,

1,1,

1,1,

1,1,

1,1,

1

11,1,

1,1,

1,1,

1,1,

1,1,

1

11,1,

1,1,

1,1,

1,1,

1,1,

1

1

1

1

1

1

1

R

rVQttrtrtr

trrVQ

ttrtr

trtrrVQttr

trtrtrrVQ

t

A

SadsSads

SSSads

SNOSads

SNHSads

SadsSS

SSSS

SNOSS

SNHSS

SadsSNO

SSSNO

SNOSNO

SNHSNO

SadsSNH

SSSNH

SNOSNH

SNHSNH


(69)

(70) The inflow vector is: (71)

The vector D1Srepresents the derivatives of the variables’ rates

towards the model parameters:

(72)

The matrix A15Sdescribes the internal recycle between the end

of the 5th tank and the beginning of the 1st one. The vector X5Sis

formed in the same way as the vector X1S : X5

S=[X5 S5θ]T∈R72

where:

(73)

• Derivation of the equations for tanks n = 2 ÷ 5

The return-flow dynamics for all other tanks are identical, thus the models will have the same structure, as follows:

(74)

where: ρn=[ρn1 ρn2 ρn3 … ρn,10]

T

The matrix Anθ has the same structure for all parameters θ and

for all tanks n=2:5.

The matrices An,n-1S are:

(75)

for all parameters θ.

The matrices CnS are equal to C1

S with the same coefficients.

The vector DnS is:

(76) • Augmented sensitivity state-space model of the whole plant

Combining all equations for the tanks, the full model is:

(77)

where:

The augmented model (77) is used for calculation of the sensitivity functions of the COST benchmark plant reduced mass-balance model for the case of the UCT reduced biologi-cal model. The models (42) and (77) are nonlinear because of the nonlinear rate expressions in the matrices AS and vectors r and DS. The dissolved oxygen concentration is considered as a control input for these models and it appears in the rate expres-sions forming AS, r, and DS. Matlab programs are developed for parameter sensitivity calculations using equations (42) and (77). A matrix/vector representation of the models allows simplifica-tion of the software code and reduction of time for calculations. Sensitivity analysis

The sensitivity analysis of the wastewater treatment model variables towards the model parameters is done in order to determine which parameters have to be estimated for the cor-responding reduced models during the real-time operation and control of the process.

Software for the sensitivity analysis

Software programs are developed in the Matlab environment

4101

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for the considered 2 cases:• Calculation of the sensitivity functions of the benchmark

process based on ASM1 reduced biological model: programs BASM1S.m, rateASM1.m

• Calculation of the sensitivity functions of the bench-mark process based on UCT biological model: programs BUCTS1.m, rateBUCTS.m

For each of the considered cases the software consists of:• Main program for input of the nominal process parameters,

initial conditions, average values of the biomass concen-trations, calculation of the process model matrices and organisation of the calculation algorithm

• Sub-programs for calculation of the process rates, sensitiv-ity functions and formation of the sensitivity model state-space and rate matrices and vectors.

The algorithm of the calculation is given in Fig. 2a for the main programme and in Fig. 2b for the sub-programmes.

XS= 0.05*[82.135*ones(1,K);76.386*ones(1,K);64.855*ones (1,K);55.694*ones(1,K);49.306*ones(1,K)]

The vector of the dissolved oxygen concentration is:

U=[0.2;0.2;2.0;2.29;1.91].

The results from the simulations of the sensitivity functions for Tank 1 and Tank 5 are given in Figs. 3 to 8. The minimum or maximum values of the sensitivity functions are given in Table 6 and Table 7.

Results for the UCT reduced-order biological model

The simulation is done for the parameters given in Table 4 and for the inflow average concentration given by the vector Xϕ:

Xϕ =[31.56; 0.0;69.5;202.3].

The values for the biomass and the slowly biodegradable sub-strate are given by the vectors:

XBH = 0.02*[2551.76*ones(1,K);2553.38*ones(1,K);2557.13* ones(1,K);2559.18*ones(1,K);2559.34*ones(1,K)]; XBA = 0.02*[148.389*ones(1,K);148.309*ones(1,K);148.941*ones(1,K);149.527*ones(1,K);149.797*ones(1,K)]; XS = 0.02*[82.135*ones(1,K);76.386*ones(1,K);64.855*ones (1,K);55.694*ones(1,K);49.306*ones(1,K)].

The vector of the dissolved oxygen concentration is:

U=[0.2;0.2;2.0;2.29;1.91].

The results from the simulations of the sensitivity functions for Tank 1 and Tank 5 are given in Figs. 9 to 16. The minimum or maximum values of the sensitivity functions are given in Table 6 and Table 7.

Discussion of the results

The minimum or maximum values of the sensitivity functions are shown in the tables for the different process variables and parameters. These values can be analysed as follows.

Sensitivity functions simulation for the benchmark process model based on the reduced ASM1 biological model

The sensitivity functions of SNH1 for almost all parameters (without KOH and KNO) display the same type of behaviour. They start from a zero value (in some cases with a small delay), grow in a positive or negative direction to a maximum/minimum value and then reduce to a steady-state value. The maximum/minimum value is in the time interval between the 5th and 8th hour from the beginning of the simulation period. The steady-state value is achieved at around the 15th hour from the beginning of the simulation period. The maximum values are obtained for parameters f,YA, KS,KOH,KNO,KNH,KOA,KX. The great-est maximum is for parameter YA (490), followed by KNH (100) and KOA (80). The smallest minimums are for parameters iXB (-300), μA (-200), and μH (-45). The behaviour of the sensitivity functions of SNH5 for Tank 5 has the same characteristics as for Tank 1, but the maximum and minimum values of these func-tions are greater in absolute values, as for example YA (498) and

Calculation process mass balance model matrices

k=0

Call subroutine for sensitivity model matrices calculation

Sensitivity model simulation

k<K

Simulation of the sensitivity functions and process behavior.Plot, Stop

k=k+1

No

Yes

Input data for model parameters, initial conditions, biomass and inflow values

Global parameters

Calculation of the process rates for every tank

Calculation of derivatives of the process rates towards the process variables

Calculation of derivatives of process rates towards the model parameters for every tank

Calculation of the sensitivity state space model matrices and vectors

Figure 2b: Flow chart of the subprogram

Figure 2a: Flow chart of the main program for sensitivity functions calculation

Figure 2aFlow chart of the main program for

sensitivity functions calculation

Figure 2bFlow chart of the subprogram

Results for the ASM1 reduced biological model

Simulation is done for the parameters given in Table 3 and for the inflow average concentration given by the vector Xθ:

Xθ=[31.56; 0.0;69.5;202.3].

The steady-state conditions of the biomass and the slowly-biodegradable substrate are given by the vectors:

XBH = 0.05*[2551.76*ones(1,K);2553.38*ones(1,K); 2557.13*ones(1,K);2559.18*ones(1,K);2559.34*ones(1,K)];XBA= 0.05*[148.389*ones(1,K);148.309*ones(1,K);148.941* ones(1,K);149.527*ones(1,K);149.797*ones(1,K)];


Table 6

Maximum or minimum values of sensitivity functions for the benchmark process model based on the reduced ASM1 and UCT biological models

P a r a me t e r s

Variables

Maximum or minimum values of the sensitivity functions ASM1 - Tank1 UCT - Tank 1

1NHS 1NOS

1SS 1NHS

1NOS 1SS

1adsS

f / f 0.18 0.195 -0.03 35 4E-6 -5E-11 -2E-9

HY /ZHY -1 -16 120 -0.8 0.75, -4 38 4000

AY /ZAY 490 -400 1.8, -1.9 275 35 -1.1E-9 -0.6E-7

XBi /ZBHf -300 -390 75 -1900 -0.12 0.55E-9 4E-8

A /A -200

5 200 -10

-0.75, 0.95 -500 4 -0.5

-2E-9, 0.5E-9

-0.7E-7

H /H -45 -95 -220 -39 -0.3 -550 1.2E-9

SK /SK 1 7.5 5 0.05, -0.45 0.015 33 -35E-12

OHK / OHK 8 -18 -90 190 -0.85 0.025 7.5

NOK /NOK 1.2 7 -12.5 2E-3, -1E-3 3E-3 0.06 0.037

NHK /NHK 100

-10 -98 10

0.3, -0.4 110 4, -0.5 3.5E-3 0.35

OAK /OAK 80, -5 -60, 9 0.2, -0.25 500 1.9, 0.5 -1.5E-9 -0.6E-7

g /g -4.8 -14 1.95 -0.5 -0.6 5E-11 -3.8

h -4.5 -13 1.96 - - - -

hk /MPK -0.6 -1 55 95 2.75 -3E-11 -2000

XK 2 2.5 -180 - - - - SPK - - - 105 -0.2, 0.49 -38E-10 700, -250 SAK - - - 1.8, -0.2 -3.5, 0.05 1.5E-9 0.7E-7 mAf - - - 5E-4 -0.35 3E-12 300 AK - - - 0.02 -17 1.5E-10 1.2E4

Table 7 Maximum or minimum values of sensitivity functions for the benchmark process model

based on the reduced ASM1 and UCT biological models –Tank 5 P a r a me t e r s

Variables

Maximum or minimum values of the sensitivity functions

ASM1 - Tank 5 UCT - Tank 5

5NHS 5NOS

5SS 5NHS

5NOS 5SS

5adsS

f / f 0.14 0.25 -0.025 35 25E-6 -3E-11 -0.9E-9

HY /ZHY -1.6 -18 150 -0.95 0.5, -4 45 4900

AY /ZAY 498

-10 -495, 50 0.8, -0.9 295 35 -1E-9 -0.6E-7

XBi /ZBHf -330 -500 45 -2200 -0.12, 0.1 3.9E-10 4E-8

A /A -250, 20 220, -10 -0.5, 0.4 -500 5, -0.3 -1E-9, 0.2E-9 -0.8E-7

H /H -58 -120 -220 -39 -0.3 -700 1.2E-9

SK /SK 1.2 9.5 5.5 -0.4 0.015 35 -38E-10

OHK /OHK 6 -20, 3 -50 190 -0.98 0.015 7

NOK /NOK 1 7.5 -8 2E-3, -1E-3 3E-4 0.06 0.037

NHK /NHK 120, -5 -120, 5 0.2, -0.25 120 -120 3.5E-3 0.38

OAK /OAK 85, -5 -75, 10 0.15, -0.15 500 2, -0.4 -0.9E-9 -0.6E-7

g /g -5.5 -18 1.1 -0.5 -0.7 3.6E-11 -3.8

h -0.38 -055 23 - - - -

hk /MPK -0.8 -1.3 60 105 3.5 -2E-11 -2500

XK 2.5 3.9 -190 - - - - SPK - - - 125 -0.2, 0.49 -2.5E-11 750, -780 SAK - - - 2.5, -0.5 -4.8, 0.5 1.1E-9 0.7E-7 mAf - - - 6E-4 -0.49 3E-12 350 AK - - - 0.027 -18 1.5E-10 1.8E4



0 20 40 60 80 100 1200

5

10Process variable Snh1

SN

H1

discrete time k0 20 40 60 80 100 120

0

0.1

0.2Sensitivity function Snh1f

SN

H1f

discrete time,k

0 20 40 60 80 100 120-2

-1

0Variable Snh1YH

SN

H1Y

H

discrete time,k0 20 40 60 80 100 120

-500

0

500Variable Snh1YA

SN

H1Y

A

discrete time,k

0 20 40 60 80 100 120-400

-200

0Sensitivity function Snh1iXB

SN

H1i

XB

discrete time,k0 20 40 60 80 100 120

-200

0

200Sensitivity function Snh1muA

SN

H1m

uA

discrete time,k

0 20 40 60 80 100 120

-40

-20

0Sensitivity function Snh1muH

SN

H1m

uH

discrete time,k0 20 40 60 80 100 120

0

1

2Sensitivity function Snh1KS

SN

H1K

S

discrete time,k

0 20 40 60 80 100 1200

5

10Sensitivity function Snh1KOH

SN

H1K

OH

discrete time,k0 20 40 60 80 100 120

0

1

2Sensitivity function Snh1KNO

SN

H1K

NO

discrete time,k

0 20 40 60 80 100 120-100

0

100Sensitivity function Snh1KNH

SN

H1K

NH

discrete time,k0 20 40 60 80 100 120

-100

0

100Sensitivity function Snh1KOA

SN

H1K

OA

discrete time,k

0 20 40 60 80 100 120

-4

-2

0Sensitivity function Snh1etag

SN

H1e

tag

discrete time,k0 20 40 60 80 100 120

-4

-2

0Sensitivity function Snh1etah

SN

H1e

tah

discrete time,k

0 20 40 60 80 100 120-1

-0.5

0Sensitivity function Snh1Kh

SN

H1K

h

discrete time,k0 20 40 60 80 100 120

0

2

4Sensitivity function Snh1Kx

SN

H1K

x

discrete time,k

0 20 40 60 80 100 120-10

0


SN

H5

discrete time k0 20 40 60 80 100 120

0

0.1

0.2Sensitivity function Snh5f

SN

H5f

discrete time,k

0 20 40 60 80 100 120-2

-1

0Variable Snh5YH

SN

H5Y

H

discrete time,k0 20 40 60 80 100 120

-500

0

500Variable Snh5YA

SN

H5Y

A

discrete time,k

0 20 40 60 80 100 120-400

-200

0Sensitivity function Snh5iXB

SN

H5i

XB

discrete time,k0 20 40 60 80 100 120

-500

0


SN

H5m

uA

discrete time,k

0 20 40 60 80 100 120-100

-50


SN

H5m

uH

discrete time,k0 20 40 60 80 100 120

0

1


SN

H5K

S

discrete time,k

0 20 40 60 80 100 1200

5


SN

H5K

OH

discrete time,k0 20 40 60 80 100 120

0

1

2Sensitivity function Snh5KNO

SN

H5K

NO

discrete time,k

0 20 40 60 80 100 120-200

0


SN

H5K

NH

discrete time,k0 20 40 60 80 100 120

-100

0


SN

H5K

OA

discrete time,k

0 20 40 60 80 100 120

-4

-2


SN

H5e

tag

discrete time,k0 20 40 60 80 100 120

-0.4

-0.2

0Sensitivity function Snh5etah

SN

H5e

tah

discrete time,k

0 20 40 60 80 100 120-1

-0.5

0Sensitivity function Snh5Kh

SN

H5K

h

discrete time,k0 20 40 60 80 100 120

0

2

4Sensitivity function Snh5Kx

SN

H5K

x

discrete time,k

0 20 40 60 80 100 1200

5

10Process variable SNO1

SN

O1

discrete time k0 20 40 60 80 100 120

0

0.1

0.2Sensitivity function SNO1f

SN

O1f

discrete time,k

0 20 40 60 80 100 120-20

-10

0Variable SNO1YH

SN

O1Y

H

discrete time,k0 20 40 60 80 100 120

-500

0

500Variable SNO1YA

SN

O1Y

A

discrete time,k

0 20 40 60 80 100 120

-400

-200

0Sensitivity function SNO1iXB

SN

O1i

XB

discrete time,k0 20 40 60 80 100 120

-200

0

200Sensitivity function SNO1muA

SN

O1m

uA

discrete time,k

0 20 40 60 80 100 120-100

-50

0Sensitivity function SNO1muH

SN

O1m

uH

discrete time,k0 20 40 60 80 100 120

0

5

10Sensitivity function SNO1KS

SN

O1K

S

discrete time,k

0 20 40 60 80 100 120-20

-10

0Sensitivity function SNO1KOH

SN

O1K

OH

discrete time,k0 20 40 60 80 100 120

0

5

10Sensitivity function SNO1KNO

SN

O1K

NO

discrete time,k

0 20 40 60 80 100 120-100

0

100Sensitivity function SNO1KNH

SN

O1K

NH

discrete time,k0 20 40 60 80 100 120

-100

0

100Sensitivity function SNO1KOA

SN

O1K

OA

discrete time,k

0 20 40 60 80 100 120-20

-10

0Sensitivity function SNO1etag

SN

O1e

tag

discrete time,k0 20 40 60 80 100 120

-20

-10

0Sensitivity function SNO1etah

SN

O1e

tah

discrete time,k

0 20 40 60 80 100 120-2

-1

0Sensitivity function SNO1Kh

SN

O1K

h

discrete time,k0 20 40 60 80 100 120

0

2

4Sensitivity function SNO1Kx

SN

O1K

x

discrete time,k

0 20 40 60 80 100 1200

10


SN

O5

discrete time k0 20 40 60 80 100 120

0

0.2

0.4Sensitivity function SNO5f

SN

O5f

discrete time,k

0 20 40 60 80 100 120-20

-10

0Variable SNO5YH

SN

O5Y

H

discrete time,k0 20 40 60 80 100 120

-500

0

500Variable SNO5YA

SN

O5Y

A

discrete time,k

0 20 40 60 80 100 120

-400

-200

0Sensitivity function SNO5iXB

SN

O5i

XB

discrete time,k0 20 40 60 80 100 120

-500

0


SN

O5m

uA

discrete time,k

0 20 40 60 80 100 120-200

-100


SN

O5m

uH

discrete time,k0 20 40 60 80 100 120

0

5

10Sensitivity function SNO5KS

SN

O5K

S

discrete time,k

0 20 40 60 80 100 120-20

0


SN

O5K

OH

discrete time,k0 20 40 60 80 100 120

0

5

10Sensitivity function SNO5KNO

SN

O5K

NO

discrete time,k

0 20 40 60 80 100 120-200

0


SN

O5K

NH

discrete time,k0 20 40 60 80 100 120

-100

0


SN

O5K

OA

discrete time,k

0 20 40 60 80 100 120-20

-10


SN

O5e

tag

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0Sensitivity function SNO5etah

SN

O5e

tah

discrete time,k

0 20 40 60 80 100 120-2

-1

0Sensitivity function SNO5Kh

SN

O5K

h

discrete time,k0 20 40 60 80 100 120

0

2

4Sensitivity function SNO5Kx

SN

O5K

x

discrete time,k

Figure 6Process variable SNO5 and its sensitivity functions towards the model

parameters for the case of ASM1 reduced biological model

Figure 3Process variable SNH1 and its sensitivity functions towards the model






initially as positive or negative but then at steady-states there is a change in their sign.

The behaviour of the sensitivity functions of the process variable SNOn, n=1, n=5, is similar to that of the variable SNHn,

KNH (120) for maximum and iXB (330), μA (-250) and μH (-58) for minimum values. The trajectories of the sensitivity functions are only positive or only negative for most of the parameters. The trajectories for the parameters YA, μA, KNH and KOA develop


n=1, n=5, but the dynamics of the sensitivity functions are slower, except for parameters YA, μA, KOH KNH, and KOA. The maximum or minimum values of the sensitivity functions are reached in a time interval between 15 and 20 h. The maximum

sensitivity is towards parameters YA (-400), iXB (-390), μA (200), KNH (-98), μH (-95) and KOA (-60) for the 1st tank, and YA (-495), iXB (-500), μA (220), KNH (-120), μH (-120) and KOA (-70) for the 5th tank.

0 20 40 60 80 100 120-1

0

1x 10

4 Process variable Snh5

SN

H5

discrete time k0 20 40 60 80 100 120

0

20

40Sensitivity function Snh5f

SN

H5f

discrete time,k

0 20 40 60 80 100 120-1

-0.5

0Variable Snh5YZH

SN

H5Y

ZH

discrete time,k0 20 40 60 80 100 120

0

200

400Variable Snh5YZA

SN

H5Y

ZA

discrete time,k

0 20 40 60 80 100 120-4000

-2000

0Sensitivity function Snh5fZBH

SN

H5f

ZB

H

discrete time,k0 20 40 60 80 100 120

-1000

-500


SN

H5m

uA

discrete time,k

0 20 40 60 80 100 120

-40-20


SN

H5m

uH

discrete time,k0 20 40 60 80 100 120

-1

0


SN

H5K

S

discrete time,k

0 20 40 60 80 100 1200

100


SN

H5K

OH

discrete time,k0 20 40 60 80 100 120

-2

0

2x 10

-3 Sensitivity function Snh5KNO

SN

H5K

NO

discrete time,k

0 20 40 60 80 100 1200

100


SN

H5K

NH

discrete time,k0 20 40 60 80 100 120

0

500


SN

H5K

OA

discrete time,k

0 20 40 60 80 100 120-1

-0.5


SN

H5e

tag

discrete time,k0 20 40 60 80 100 120

0

100

200Sensitivity function Snh5KMP

SN

H5K

MP

discrete time,k

0 20 40 60 80 100 1200

100

200Sensitivity function Snh5KSP

SN

H5K

SP

discrete time,k0 20 40 60 80 100 120

-5

0

5Sensitivity function Snh5KSA

SN

H5K

SA

discrete time,k

0 20 40 60 80 100 1200

0.5

1x 10

-3 Sensitivity function Snh5fMA

SN

H5f

MA

discrete time,k0 20 40 60 80 100 120

0

0.02

0.04Sensitivity function Snh5KA

SN

H5K

A

discrete time,k

0 20 40 60 80 100 1200

5

10Process variable SS1

SS

1

discrete time k0 20 40 60 80 100 120

-0.04

-0.02

0Sensitivity function SS1f

SS

1f

discrete time,k

0 20 40 60 80 100 1200

100

200Variable SS1YH

SS

1YH

discrete time,k0 20 40 60 80 100 120

-2

0

2Variable SS1YA

SS

1YA

discrete time,k

0 20 40 60 80 100 1200

50

100Sensitivity function SS1iXB

SS

1iX

B

discrete time,k0 20 40 60 80 100 120

-1

0

1Sensitivity function SS1muA

SS

1muA

discrete time,k

0 20 40 60 80 100 120-400

-200

0Sensitivity function SS1muH

SS

1muH

discrete time,k0 20 40 60 80 100 120

0

5

10Sensitivity function SS1KS

SS

1KS

discrete time,k

0 20 40 60 80 100 120-100

-50

0Sensitivity function SS1KOH

SS

1KO

H

discrete time,k0 20 40 60 80 100 120

-20

-10

0Sensitivity function SS1KNO

SS

1KN

O

discrete time,k

0 20 40 60 80 100 120-0.5

0

0.5Sensitivity function SS1KNH

SS

1KN

H

discrete time,k0 20 40 60 80 100 120

-0.5

0

0.5Sensitivity function SS1KOA

SS

1KO

A

discrete time,k

0 20 40 60 80 100 1200

1

2Sensitivity function SS1etag

SS

1eta

g

discrete time,k0 20 40 60 80 100 120

0

1

2Sensitivity function SS1etah

SS

1eta

h

discrete time,k

0 20 40 60 80 100 1200

50

100Sensitivity function SS1Kh

SS

1Kh

discrete time,k0 20 40 60 80 100 120

-200

-100

0Sensitivity function SS1Kx

SS

1Kx

discrete time,k

0 20 40 60 80 100 1200

10


SS

5

discrete time k0 20 40 60 80 100 120

-0.04

-0.02

0Sensitivity function SS5f

SS

5f

discrete time,k

0 20 40 60 80 100 1200

100

200Variable SS5YH

SS

5YH

discrete time,k0 20 40 60 80 100 120

-1

0

1Variable SS5YA

SS

5YA

discrete time,k

0 20 40 60 80 100 1200

50Sensitivity function SS5iXB

SS

5iX

B

discrete time,k0 20 40 60 80 100 120

-1

0

1Sensitivity function SS5muA

SS

5muA

discrete time,k

0 20 40 60 80 100 120-400

-200


SS

5muH

discrete time,k0 20 40 60 80 100 120

0

5


SS

5KS

discrete time,k

0 20 40 60 80 100 120-100

-50

0Sensitivity function SS5KOH

SS

5KO

H

discrete time,k0 20 40 60 80 100 120

-10

0

10Sensitivity function SS5KNO

SS

5KN

O

discrete time,k

0 20 40 60 80 100 120-0.5

0

0.5Sensitivity function SS5KNH

SS

5KN

H

discrete time,k0 20 40 60 80 100 120

-0.2

0

0.2Sensitivity function SS5KOA

SS

5KO

A

discrete time,k

0 20 40 60 80 100 1200

1

2Sensitivity function SS5etag

SS

5eta

g

discrete time,k0 20 40 60 80 100 120

0

20

40Sensitivity function SS5etah

SS

5eta

h

discrete time,k

0 20 40 60 80 100 1200

50

100Sensitivity function SS5Kh

SS

5Kh

discrete time,k0 20 40 60 80 100 120

-200

-100

0Sensitivity function SS5Kx

SS

5Kx

discrete time,k

0 20 40 60 80 100 120-5000

0


SN

H1

discrete time k0 20 40 60 80 100 120

0

20

40Sensitivity function Snh1f

SN

H1f

discrete time,k

0 20 40 60 80 100 120-1

-0.5

0Variable Snh1YZH

SN

H1Y

ZH

discrete time,k0 20 40 60 80 100 120

0

200

400Variable Snh1YZA

SN

H1Y

ZA

discrete time,k

0 20 40 60 80 100 120-2000

-1000

0Sensitivity function Snh1fZBH

SN

H1f

ZB

H

discrete time,k0 20 40 60 80 100 120

-400-200


SN

H1m

uA

discrete time,k

0 20 40 60 80 100 120-40

-20


SN

H1m

uH

discrete time,k0 20 40 60 80 100 120

-0.5

0

0.5Sensitivity function Snh1KS

SN

H1K

S

discrete time,k

0 20 40 60 80 100 1200

100


SN

H1K

OH

discrete time,k0 20 40 60 80 100 120

-5

0

5x 10

-3 Sensitivity function Snh1KNO

SN

H1K

NO

discrete time,k

0 20 40 60 80 100 1200

100


SN

H1K

NH

discrete time,k0 20 40 60 80 100 120

0

500


SN

H1K

OA

discrete time,k

0 20 40 60 80 100 120

-0.4-0.2


SN

H1e

tag

discrete time,k0 20 40 60 80 100 120

0

50

100Sensitivity function Snh1KMP

SN

H1K

MP

discrete time,k

0 20 40 60 80 100 1200

100

200Sensitivity function Snh1KSP

SN

H1K

SP

discrete time,k0 20 40 60 80 100 120

-2

0

2Sensitivity function Snh1KSA

SN

H1K

SA

discrete time,k

0 20 40 60 80 100 1200

0.5

1x 10

-3 Sensitivity function Snh1fMA

SN

H1f

MA

discrete time,k0 20 40 60 80 100 120

0

0.02

0.04Sensitivity function Snh1KA

SN

H1K

A

discrete time,k


parameters for the case of UCT reduced biological model

Figure 7Process variable SS1 and its sensitivity functions towards the model








The trajectories of the sensitivity functions of the variable SSn, n=1, n=2, are characterised by the slowest dynamics and smallest maximum and minimum values. The maximum values

for the 1st tanks are for parameters YH (120), iXB (75), and kh (60), and the minimum values are for parameters μH (-220), KX (-180), and KOH (-90). For the 5th tank the corresponding values are YH

0 20 40 60 80 100 1200


SN

O1

discrete time k0 20 40 60 80 100 120

0

5x 10

-6 Sensitivity function SNO1f

SN

O1f

discrete time,k

0 20 40 60 80 100 120-5

0

5Variable SNO1YZH

SN

O1Y

ZH

discrete time,k0 20 40 60 80 100 120

0

20

40Variable SNO1YZA

SN

O1Y

ZA

discrete time,k

0 20 40 60 80 100 120-0.2

0

0.2Sensitivity function SNO1fZBH

SN

O1f

ZB

H

discrete time,k0 20 40 60 80 100 120

-5

0


SN

O1m

uA

discrete time,k

0 20 40 60 80 100 120-0.4

-0.2


SN

O1m

uH

discrete time,k0 20 40 60 80 100 120

0

0.01

0.02Sensitivity function SNO1KS

SN

O1K

Sdiscrete time,k

0 20 40 60 80 100 120-1

-0.5


SN

O1K

OH

discrete time,k0 20 40 60 80 100 120

0

2

4x 10

-3 Sensitivity function SNO1KNOS

NO

1KN

O

discrete time,k

0 20 40 60 80 100 120-5

0


SN

O1K

NH

discrete time,k0 20 40 60 80 100 120

-2

0


SN

O1K

OA

discrete time,k

0 20 40 60 80 100 120-1

-0.5


SN

O1e

tag

discrete time,k0 20 40 60 80 100 120

0

2

4Sensitivity function SNO1KMP

SN

O1K

MP

discrete time,k

0 20 40 60 80 100 120-0.5

0

0.5Sensitivity function SNO1KSP

SN

O1K

SP

discrete time,k0 20 40 60 80 100 120

-5

0

5Sensitivity function SNO1KSA

SN

O1K

SA

discrete time,k

0 20 40 60 80 100 120-0.4

-0.2

0Sensitivity function SNO1fMA

SN

O1f

MA

discrete time,k0 20 40 60 80 100 120

-20

-10

0Sensitivity function SNO1KA

SN

O1K

A

discrete time,k

0 20 40 60 80 100 1200

5000


SN

O5

discrete time k0 20 40 60 80 100 120

0

2

4x 10

-6 Sensitivity function SNO5f

SN

O5f

discrete time,k

0 20 40 60 80 100 120-5

0

5Variable SNO5YZH

SN

O5Y

ZH

discrete time,k0 20 40 60 80 100 120

0

20

40Variable SNO5YZA

SN

O5Y

ZA

discrete time,k

0 20 40 60 80 100 120-0.2

0

0.2Sensitivity function SNO5fZBH

SN

O5f

ZB

H

discrete time,k0 20 40 60 80 100 120

-5

0


SN

O5m

uA

discrete time,k

0 20 40 60 80 100 120-0.4

-0.2


SN

O5m

uH

discrete time,k0 20 40 60 80 100 120

0

0.01

0.02Sensitivity function SNO5KS

SN

O5K

S

discrete time,k

0 20 40 60 80 100 120-1

0


SN

O5K

OH

discrete time,k0 20 40 60 80 100 120

0

2

4x 10

-3 Sensitivity function SNO5KNO

SN

O5K

NO

discrete time,k

0 20 40 60 80 100 120-200

-100


SN

O5K

NH

discrete time,k0 20 40 60 80 100 120

-5

0


SN

O5K

OA

discrete time,k

0 20 40 60 80 100 120-1

-0.5


SN

O5e

tag

discrete time,k0 20 40 60 80 100 120

-5

0

5Sensitivity function SNO5KMP

SN

O5K

MP

discrete time,k

0 20 40 60 80 100 120-0.5

0

0.5Sensitivity function SNO5KSP

SN

O5K

SP

discrete time,k0 20 40 60 80 100 120

-5

0

5Sensitivity function SNO5KSA

SN

O5K

SA

discrete time,k

0 20 40 60 80 100 120

-0.4-0.2

0Sensitivity function SNO5fMA

SN

O5f

MA

discrete time,k0 20 40 60 80 100 120

-20

-10

0Sensitivity function SNO5KA

SN

O5K

A

discrete time,k

0 20 40 60 80 100 1200

10


SS

1

discrete time k0 20 40 60 80 100 120

-0.5

0x 10

-10 Sensitivity function SS1f

SS

1f

discrete time,k

0 20 40 60 80 100 1200

20

40Variable SS1YZH

SS

1YZ

H

discrete time,k0 20 40 60 80 100 120

-2

-1

0x 10

-9 Variable SS1YZA

SS

1YZ

A

discrete time,k

0 20 40 60 80 100 1200

0.5

1x 10

-9 Sensitivity function SS1fZBH

SS

1fZ

BH

discrete time,k0 20 40 60 80 100 120

-5

0

5x 10

-9 Sensitivity function SS1muA

SS

1muA

discrete time,k

0 20 40 60 80 100 120-1000

-500


SS

1muH

discrete time,k0 20 40 60 80 100 120

0

20


SS

1KS

discrete time,k

0 20 40 60 80 100 1200

0.02

0.04Sensitivity function SS1KOH

SS

1KO

H

discrete time,k0 20 40 60 80 100 120

0

0.05

0.1Sensitivity function SS1KNO

SS

1KN

O

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-3 Sensitivity function SS1KNH

SS

1KN

H

discrete time,k0 20 40 60 80 100 120

-2

-1

0x 10

-9 Sensitivity function SS1KOA

SS

1KO

A

discrete time,k

0 20 40 60 80 100 1200

5x 10

-11 Sensitivity function SS1etag

SS

1eta

g

discrete time,k0 20 40 60 80 100 120

-4

-2

0x 10

-11 Sensitivity function SS1KMP

SS

1KM

P

discrete time,k

0 20 40 60 80 100 120

-4-20

x 10-11 Sensitivity function SS1KSP

SS

1KS

P

discrete time,k0 20 40 60 80 100 120

0

1

2x 10

-9 Sensitivity function SS1KSA

SS

1KS

A

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-12 Sensitivity function SS1fMAS

S1f

MA

discrete time,k0 20 40 60 80 100 120

0

1

2x 10

-10 Sensitivity function SS1KA

SS

1KA

discrete time,k

0 20 40 60 80 100 1200

2


SS

5

discrete time k0 20 40 60 80 100 120

-4

-2

0x 10

-11 Sensitivity function SS5f

SS

5f

discrete time,k

0 20 40 60 80 100 1200

50Variable SS5YZH

SS

5YZ

H

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-9 Variable SS5YZA

SS

5YZ

A

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-10 Sensitivity function SS5fZBH

SS

5fZ

BH

discrete time,k0 20 40 60 80 100 120

-2

0

2x 10

-9 Sensitivity function SS5muA

SS

5muA

discrete time,k

0 20 40 60 80 100 120-1000

-500


SS

5muH

discrete time,k0 20 40 60 80 100 120

0

20


SS

5KS

discrete time,k

0 20 40 60 80 100 1200

0.01

0.02Sensitivity function SS5KOH

SS

5KO

H

discrete time,k0 20 40 60 80 100 120

0

0.05

0.1Sensitivity function SS5KNO

SS

5KN

O

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-3 Sensitivity function SS5KNH

SS

5KN

H

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-9 Sensitivity function SS5KOA

SS

5KO

A

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-11 Sensitivity function SS5etag

SS

5eta

g

discrete time,k0 20 40 60 80 100 120

-4

-2

0x 10

-11 Sensitivity function SS5KMP

SS

5KM

P

discrete time,k

0 20 40 60 80 100 120-4

-2

0x 10

-11 Sensitivity function SS5KSP

SS

5KS

P

discrete time,k0 20 40 60 80 100 120

0

1

2x 10

-9 Sensitivity function SS5KSA

SS

5KS

A

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-12 Sensitivity function SS5fMA

SS

5fM

A

discrete time,k0 20 40 60 80 100 120

0

1

2x 10

-10 Sensitivity function SS5KA

SS

5KA

discrete time,k

Figure 14rocess variable SS5 and its sensitivity functions towards the model









On the basis of the above it can be concluded that model behaviour is sensitive to parameters YA, YH, iXB, μA, KNH, μH, and KX. The sensitivity of variables SNHn and SNOn, n=1, n=5, is higher than that of variables SSn, n=1, n=5, for parameters YA, iXB, μA, and KNH. The sensitivity of the variable SSn, n=1, n=5, is greater for parameters YH, μH, and KX. Some of these para-meters can be selected for parameter estimation in order to fit the reduced model to the process data.

Sensitivity function simulation for the benchmark process model based on the reduced UCT biological model

The sensitivity functions of the variable SNHn, n=1, n=5, behave exponentially, with the exception of the sensitivity function for parameter KNO, which displays a small overshoot and then is reduced to a low steady-state value. The steady state maxi-mum/minimum values of the sensitivity functions for the 1st tank are f (35), YZA (275), iXB (-1900), μH (-39), KOH (190), KNH (110), KOA (500), μA (-500), KMP (95), and KSP (105). For the 5th tank the corresponding values are f (35), YZA (295), iXB (-1900), μH (-39), KOH (190), KNH (120), KOA (500), μA (-500), KMP (105), and KSP (125). It can be seen that the sensitivity functions have a time delay and their values are greater for the 5th tank.

For the sensitivity functions of variable SNOn, n=1, n=5, a couple of overshoots at the beginning are then followed by a slow approach to the steady state. The extrema of the sensitiv-ity functions for the parameters are as follows: YZA (35), KNH (4), KSA (-17) for the 1st tank; and YZA (35), KNH (-120), KSA (-18) for the 5th tank. It can be seen that the values of the sensitivity functions in this case are relatively smaller in comparison with the corresponding ones for the first 4 tanks. It is only the influ-ence of parameter KNH which is stronger in the 5th tank.

For variable SSn, n=1, n=5, the sensitivity function displays very low values, and many overshoots for the first 5 hours, with time delays of these trajectories for the 5th tank. After this the trajectories approach steady state. The extrema of the sensitivity functions for the parameters are as follows: YZH (38), μH (-550), and KS (33) for the 1st tank; and YZH (45), μH (-700), and KS (35) for the 5th tank. The variable SSn, n=1, n=5, is not sensitive to the parameters to which otherprocess variables are sensitive.

The behaviour of the sensitivity functions of variable Sadsn, n=1, n=5, is similar to that for variable SSn, n=1, n=5. The extrema of the sensitivity functions for the parameters are as follows: YZH (4 000), KOH (7.5), KMP (2 000), KSP (700/-250), fmA (300), and KA (12 000) for the 1st tank, and YZH (4900), KOH (7.5), KMP (2 500), KSP (750/-780), fmA (350), and KA (18 000) for the 5th tank. This variable is very sensitive to the above parameters and extremely insensitive to the rest of the parameters.

Based on the results obtained it can be concluded that the process variables are sensitive to different parameters. Some of these parameters, such as YZH, KMP, KSP, KA, μH, iXB, and KOA, can be selected for estimation in order to fit the reduced model to the process data. The reduced UCT model variables have very low sensitivity to some of the parameters and very high sensitivity to the rest of the parameters. It is difficult to find a group of parameters which result in high values of the sensitiv-ity functions for all variables of the model.

Conclusion

The objective of this study was to describe the development of a sensitivity analysis direct method for developing a reduced

0 20 40 60 80 100 120-500

0

500Process variable Sads1

Sad

s1

discrete time k0 20 40 60 80 100 120

-4

-2

0x 10

-9 Sensitivity function Sads1f

Sad

s1f

discrete time,k

0 20 40 60 80 100 1200

2000

4000Variable Sads1YZH

Sad

s1Y

ZH

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-7 Variable Sads1YZA

Sad

s1Y

ZA

discrete time,k

0 20 40 60 80 100 1200

2

4x 10

-8 Sensitivity function Sads1fZBH

Sad

s1fZ

BH

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-7 Sensitivity function Sads1muA

Sad

s1m

uA

discrete time,k

0 20 40 60 80 100 1200

1

2x 10

-9 Sensitivity function Sads1muH

Sad

s1m

uH

discrete time,k0 20 40 60 80 100 120

-4

-2

0x 10

-11 Sensitivity function Sads1KS

Sad

s1K

S

discrete time,k

0 20 40 60 80 100 1200

5

10Sensitivity function Sads1KOH

Sad

s1K

OH

discrete time,k0 20 40 60 80 100 120

0

0.02

0.04Sensitivity function Sads1KNO

Sad

s1K

NO

discrete time,k

0 20 40 60 80 100 1200

0.2

0.4Sensitivity function Sads1KNH

Sad

s1K

NH

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-7 Sensitivity function Sads1KOA

Sad

s1K

OA

discrete time,k

0 20 40 60 80 100 120-4

-2

0Sensitivity function Sads1etag

Sad

s1et

ag

discrete time,k0 20 40 60 80 100 120

-2000

-1000

0Sensitivity function Sads1KMP

Sad

s1K

MP

discrete time,k

0 20 40 60 80 100 120-1000

0

1000Sensitivity function Sads1KSP

Sad

s1K

SP

discrete time,k0 20 40 60 80 100 120

0

0.5

1x 10

-7 Sensitivity function Sads1KSA

Sad

s1K

SA

discrete time,k

0 20 40 60 80 100 1200

200

400Sensitivity function Sads1fMA

Sad

s1fM

A

discrete time,k0 20 40 60 80 100 120

0

1

2x 10

4 Sensitivity function Sads1KA

Sad

s1K

A

discrete time,k

0 20 40 60 80 100 120-500

0

500Process variable Sads5

Sad

s5

discrete time k0 20 40 60 80 100 120

-1

-0.5

0x 10

-9 Sensitivity function Sads5f

Sad

s5f

discrete time,k

0 20 40 60 80 100 1200

5000Variable Sads5YZH

Sad

s5Y

ZH

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-7 Variable Sads5YZA

Sad

s5Y

ZA

discrete time,k

0 20 40 60 80 100 1200

5x 10

-8 Sensitivity function Sads5fZBH

Sad

s5fZ

BH

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-7 Sensitivity function Sads5muA

Sad

s5m

uA

discrete time,k

0 20 40 60 80 100 1200

1

2x 10

-9 Sensitivity function Sads5muH

Sad

s5m

uH

discrete time,k0 20 40 60 80 100 120

-4

-2

0x 10

-11 Sensitivity function Sads5KS

Sad

s5K

S

discrete time,k

0 20 40 60 80 100 1200

5

10Sensitivity function Sads5KOH

Sad

s5K

OH

discrete time,k0 20 40 60 80 100 120

0

0.05Sensitivity function Sads5KNO

Sad

s5K

NO

discrete time,k

0 20 40 60 80 100 1200

0.5Sensitivity function Sads5KNH

Sad

s5K

NH

discrete time,k0 20 40 60 80 100 120

-1

-0.5

0x 10

-7 Sensitivity function Sads5KOA

Sad

s5K

OA

discrete time,k

0 20 40 60 80 100 120

-4-20

Sensitivity function Sads5etag

Sad

s5et

ag

discrete time,k0 20 40 60 80 100 120

-4000

-2000

0Sensitivity function Sads5KMP

Sad

s5K

MP

discrete time,k

0 20 40 60 80 100 120-1000

0

1000Sensitivity function Sads5KSP

Sad

s5K

SP

discrete time,k0 20 40 60 80 100 120

0

0.5

1x 10

-7 Sensitivity function Sads5KSA

Sad

s5K

SA

discrete time,k

0 20 40 60 80 100 1200

200

400Sensitivity function Sads5fMA

Sad

s5fM

A

discrete time,k0 20 40 60 80 100 120

0

1

2x 10

4 Sensitivity function Sads5KA

Sad

s5K

A

discrete time,k

Figure 15Process variable SSads1 and its sensitivity functions towards the model


Figure 16Process variable SSads5 and its sensitivity functions towards the model


(150), iXB (45), kh (60), μH (-220), KX (-190) and KOH (-50). The maximum and minimum values of the sensitivity functions in this case are for different parameters than those in the 2 vari-ables, SNHn and SNon, considered above.



model of the activated sludge process, characterised by a large number of parameters and insufficient data measurements. Derivation of sensitivity functions and augmented sensitiv-ity state-space models for the reduced mass balance COST benchmark model, based on reduced ASM1 and UCT biologi-cal models, was presented. Matlab software was developed and simulations provided. The results enable one to answer the question as to which of the parameters need to be estimated in order to fit the reduced model behaviour to the data. Selection of the candidate parameters for estimation was based on a com-parison of the minimum or maximum values of the correspond-ing sensitivity functions. The shapes of the sensitivity func-tions obtained for all variables and parameters of Tank 1 for the benchmark process model, based on the reduced ASM1 or reduced UCT biological models, are similar to the correspond-ing functions for Tank 5. The difference is in the minimum or maximum values of the sensitivity functions, which are greater for Tank 5 . Another difference is that the sensitivity function trajectories for the 5th tank have a time delay and are slower. Different combinations of parameters selected by the sensitiv-ity analysis were estimated and the trajectories of the estimated model showed a good fit to the measured data. The methods used, algorithms, Matlab software and results for parameter estimation are not discussed in this paper. The reduced models, if properly calibrated, can predict the process behaviour, with small errors. These models are usually developed for real-time control design as a response to inflow disturbances. Reduced model applications for detailed studies of process interaction between variables, kinetic parameters and output performance will not be as successful as the application of the full models.

The main contribution of the study reported herein is that it provides a quantitative basis for classification of the model parameters. This helps to address the problems of complexity and uncertainty during the process of parameter estimation for nonlinear models. The proposed structure of the augmented sensitivity model is general and modular which permits its application to any wastewater treatment process and to any other nonlinear model. The vector/matrix structure of the model allows effective use of Matlab software and rapid solu-tion of the simulation sensitivity problem. This means that the proposed structure of the augmented model assures global sensitivity analysis without increasing the computation burden.

The developed algorithms and software can be used for different applications: to simplify models, to investigate robust-ness of the proposed parameter estimation results, to analyse different scenarios for the plant operation, to determine the region of the parameter space for which the output is minimum or maximum, to discover interactions between the input factors (variables and parameters), to discover issues not anticipated at the beginning of the investigation, and to use in real-time joint solution of the problem of parameter estimation, control design and implementation, as a part of an adaptive control strategy.

Acknowledgements

This study was funded by KISC NRF Grant No. 63667 ’Coordinated and joint research activities in wastewater treatment monitoring, modelling, estimation and control in real-time’.”

References

AL-SAGGAF UM and FRANKLIN GF (1988) Model reduction via balanced realizations: an extension and frequency weighting tech-niques. IEEE Trans. Autom. Control AC-33 (7) 687-692.

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Appendix AThe derivatives of the variables’ rates for ASM1 reduced biological model towards the model variables and parameters are determined as follows:

1. The variables’ rate derivatives towards the process variables:

• for the variable’s rate rSNH,n

• for the variable’s rate rSNO,n

• for the variable’s rate rSS,n

2. The variables’ rate derivatives towards the model para-meters. The partial derivatives of the process rates according to the model coefficients

are as follows:


nSNHnSNH

nNHNH

NH

nOOA

nOBAA

AXB

nNH

nSNH rSK

KSK

SX

Yi

Sr ,

,2,,

,

,

, ˆ1

nSNOnSNH

nNONO

NOgBH

nOOH

OH

nSS

SHXB

nNO

nSNH rSK

KX

SKK

SKS

iSr ,

,2,,,,

, ˆ

nSSnSNH

nSS

S

nOOH

nOBHHXB

nS

nSNH rSKK

SKS

XiSr ,

,2,,

,

,

, ˆ

nNHnSNO

nNHNH

NH

nOOA

nOBAA

AnNH

nSNO rSK

KSK

SX

YSr ,

,2,,

,

,

, ˆ1

nNOnSNO

nNONO

NO

nOOH

OH

nSS

nSgHBH

H

H

nNO

nSNO rSKK

SKK

SKS

XYY

Sr ,

,2,,,

,

,

, ˆ86.2

1

nSSnSNO

nSS

S

nNONO

nNO

nOOH

OHgHBH

H

H

nS

nSNO rSKK

SKS

SKK

XYY

Sr ,

,2,,

,

,,

, ˆ86.2

1

0,

,

nNH

nSS

Sr

nSNO

nSSnNONO

NO

nOOH

OHBH

nBHnSX

nBHnShh

nSS

nSH

HnNO

nSS rSK

KSK

KXXXK

XXk

SKS

YSr ,

,2,,,,

,,

,

,

,

, ˆ1

nSSnSS

nSS

S

nOOH

OH

nNONO

nNO

nOOH

nOnBHH

HnS

nSS rSK

KSK

KSK

SSK

SX

YSr ,

,,,,

,

,

,,

,

, ˆ1

0, fnSNHr 0, HY

nSNHr , 0, hnSNHr

, 0, hknSNHr , 0, XK

nSNHr

nBA

nNHNH

nNH

nOOA

nOA

A

YnSNH X

SK

S

SK

S

Yr A

,,

,

,

,2, ˆ1

nBAnNHNH

nNH

nOOA

nOA

gnOOH

OH

nNONO

nNO

nOOH

nO

nSS

nSnBHH

inSNH

XSK

SSK

S

SKK

SKS

SKS

SKS

Xr XB

,,

,

,

,

,,

,

,

,

,

,,,

ˆ

ˆ

nBAnNHNH

nNH

nOOA

nO

AXBnSNH X

SKS

SKS

Yir A

,,

,

,

,ˆ,

1

nOOH

OH

nNONO

nNOg

nOOH

nO

nSS

nSnBHXBnSNH SK

KSK

S

SK

S

SK

SXir H

,,

,

,

,

,

,,

ˆ,

nOOH

OH

nNONO

nNOg

nOOH

nO

nSS

nSnBHHXB

KnSNH SK

KSK

SSK

S

SK

SXir S

,,

,

,

,2

,

,,, ˆ

nNONO

nNOg

nOOH

nO

nSS

nS

nBHHXB

KnSNH SK

S

SK

S

SKS

Xir OH

,

,2

,

,

,

,

,, 1ˆ

nOOH

OH

nNONO

nNO

nSS

nSgnBHHXB

KnSNH SK

KSK

SSK

SXir NO

,2

,

,

,

,,, ˆ

nBAnNHNH

nNH

nOOA

nOA

AXB

KnSNH X

SK

S

SK

S

Yir NH

,2,

,

,

,, ˆ1

BAnNHNH

nNH

nOOA

nOA

AXB

KnSNH X

SKS

SK

SY

ir OA

,

,2

,

,, ˆ1

nBHnOOH

OH

nNONO

nNO

nSS

nSHXBnSNH X

SKK

SK

S

SK

Sir g

,,,

,

,

,, ˆ

XhhgOANHNOOHSHAXBAH KkKKKKKiYYf ,,,,,,,,,,,,,,



• for the variable’s rate rSNO,n • for the variable’s rate rSS,n

2,,

,

,,

,,, 86.2

1ˆHnNOnNO

nNO

nOOH

OH

nSS

nSnBHgH

YnSNO YSK

S

SKK

SK

SXr H

AA Y

nSNHnBAnNHNH

nNH

nOOA

nOA

A

YnSNO rX

SK

S

SK

S

Yr ,,

,

,

,

,2, ˆ1

0, f

nSNOr , 0, XBinSNOr 0, h

nSNOr , 0, hnSNOr , 0, hk

nSNOr , 0, XKnSNOr

nBAnNHNH

nNH

nOOA

nO

AnSNO X

SK

S

SK

S

Yr A

,,

,

,

,ˆ,

1

nOOH

OH

nNOnNO

nNO

nOOH

nO

nSS

nSnBHH

H

YnSS SK

KSK

SSK

SSK

SX

Yr H

,,,

,

,

,

,

,,2, ˆ1

gnNONO

nNO

nOOH

OH

nSS

nSnBH

H

HnSNO SK

S

SKK

SK

SX

YY

r H

,

,

,,

,,

ˆ, 86.2

1

g

nNONO

nNO

nOOH

OH

nSS

nSHnBH

H

HKnSNO SK

SSK

KSK

SX

YY

r S

,

,

,,

,,, ˆ

86.21

gnNONO

nNO

nOOH

nO

nSS

nSHnBH

H

HKnSNO SK

S

SK

SSK

SX

YY

r OH

,

,2

,

,

,

,,, ˆ

86.21

nOOH

OH

nNONO

nNO

nSS

nSgnBHH

H

HKnSNO SK

K

SK

S

SK

SX

YY

r NO

,2,

,

,

,,, ˆ

86.21

nBAnNHNH

nNH

nOOA

nOA

A

KnSNO X

SK

S

SK

S

Yr NH

,2,

,

,

,, ˆ1

BAnNHNH

nNH

nOOA

nOA

A

KnSNO X

SKS

SKS

Yr OA

,

,2

,

,, ˆ1

nBH

nOOH

OH

nNONO

nNO

nSS

nSH

H

HnSNO X

SKK

SK

S

SK

S

YY

r g,

,,

,

,

,, ˆ

86.21

Appendix B

The derivatives of the variables’ rates for the UCT reduced biological model towards the model variables and parameters are determined as follows:

nOOH

OH

nNOnNO

nNO

nOOH

nO

nSS

nSnBHH

H

YnSS SK

KSK

SSK

SSK

SX

Yr H

,,,

,

,

,

,

,,2, ˆ1

0, f

nSSr , 0, AYnSSr

, 0, XBinSSr

, 0ˆ, AnSSr

, 0, NHKnSSr

, 0, OAKnSSr

, 0, gnSSr

nOOH

OH

nNONO

nNO

nOOH

nO

nSS

nSnBH

HnSS SK

KSK

SSK

S

SKS

XY

r H

,,

,

,

,

,

,,

ˆ,

1

nOOH

OH

nNONO

nNO

nOOH

nO

nSS

nSnBHH

H

KnSS SK

KSK

SSK

S

SK

SX

Yr S

,,

,

,

,2

,

,,, ˆ1

nNONO

nNOh

nOOH

nO

nBHnSX

nBHnS

nBHhnNONO

nNO

nOOH

nO

nSS

nSnBHH

H

KnSS

SKS

SK

SXXK

XX

XkSK

S

SK

SSK

SX

Yr OH

,

,2

,

,

,,

,,

,,

,2

,

,

,

,,,

1.

.1ˆ1

nOOH

OH

nNONO

nNO

nBHnSX

nBHnS

hnBHhnOOH

OH

nNONO

nNO

nSS

nSnBHH

H

KnSS

SKK

SK

SXXK

XX

XkSK

KSK

SSK

SX

Yr NO

,2

,

,

,,

,,

,,

2,

,

,

,,,

.

.ˆ1

nBHnOOH

OH

nNONO

nNO

nOOH

nO

nBHnSX

nBHnShnSS X

SKK

SKS

SKS

XXKXX

kr h,

,,

,

,

,

,,

,,,

nBHnOOH

OH

nNONO

nNOh

nOOH

nO

nBHnSX

nBHnSknSS X

SKK

SKS

SKS

XXKXX

r h,

,,

,

,

,

,,

,,,

2,,

,,,

,,

,

,

,,

nBHnSX

nBHnSnBH

nOOH

OH

nNONO

nNOh

nOOH

nOh

KnSS XXK

XXX

SKK

SKS

SKS

kr X

1. The process rate derivatives towards the process variables • for the variable’s rate rSNH,n


BAnNHSA

SA

nOOA

nOAZBH

ZA

BHnadsSP

nBHnadsgMP

nSS

nSH

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnadsMP

nSS

nSH

nOOH

nO

nNHNA

NABHZBH

nSNHnSNH

XSKK

SKS

fY

XSKXS

KSK

SSK

SSK

K

XSKXS

KSK

SSK

S

SKK

Xfr

2,,

,

,

,,

,

,

,

,

,

,,

,,

,

,

,

,2

,

,,

1

][

BHnadsSP

BHnadsMPg

nSS

nSH

nNHNA

nNH

nOOH

OH

nNONO

NOBHZBH

nSNOnSNH

XSKXS

KSK

S

SKS

SKK

SKKXfr

,

,

,

,

,

,

,2

,

,,

.

.

nNONO

nNO

nOOH

OH

nOOH

nO

nNHNA

nNHBH

nSS

SHZBH

nSSnSNH SK

SSK

KSK

SSK

SX

SKKfr

,

,

,,

,

,

,2

,

,,

nNONO

nNO

nOOH

OHg

nOOH

nO

nNHNA

nNHBH

nBHnadsSP

nBHSPMPZBH

nSadsnSNH

SKS

SKK

SKS

SKS

XXSK

XKKfr

,

,

,,

,

,

,2

,,

,,,

.

.

BAnNHSA

SA

nOOA

nOAZBH

ZA

BHnadsSP

nBHnadsgMP

nSS

nSH

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnadsMP

nSS

nSH

nOOH

nO

nNHNA

NABHZBH

nSNHnSNH

XSKK

SKS

fY

XSKXS

KSK

SSK

SSK

K

XSKXS

KSK

SSK

S

SKK

Xfr

2,,

,

,

,,

,

,

,

,

,

,,

,,

,

,

,

,2

,

,,

1

][

BHnadsSP

BHnadsMPg

nSS

nSH

nNHNA

nNH

nOOH

OH

nNONO

NOBHZBH

nSNOnSNH

XSKXS

KSK

S

SKS

SKK

SKKXfr

,

,

,

,

,

,

,2

,

,,

.

.

nNONO

nNO

nOOH

OH

nOOH

nO

nNHNA

nNHBH

nSS

SHZBH

nSSnSNH SK

SSK

KSK

SSK

SX

SKKfr

,

,

,,

,

,

,2

,

,,

nNONO

nNO

nOOH

OHg

nOOH

nO

nNHNA

nNHBH

nBHnadsSP

nBHSPMPZBH

nSadsnSNH

SKS

SKK

SKS

SKS

XXSK

XKKfr

,

,

,,

,

,

,2

,,

,,,

.

.

BAnNOOA

nNO

nNHSA

SAA

ZA

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOOH

OHZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOOH

nOZBH

nSNHnSNO

XSK

S

SKK

Y

XSKXS

KSK

SX

SKS

SKK

SKK

f

XSKXS

KSK

SX

SKS

SKK

SKS

fr

,

,2

,

,,

,,

,

,,

,

,2

,,

,,

,,

,

,,

,

,2

,,

,,,

.1

nNHNA

NAZBH

ZH

ZH

nNHNA

nNH

ZH

ZH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

NO

nOOH

OH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

NO

nNHNA

NA

nOOH

nOZBH

nSNOnSNO

SKK

fYY

SKS

YY

XSKXS

KSK

SX

SKK

SKK

XSKXS

KSK

SX

SKK

SKK

SKS

fr

,,

,

,,

,,

,

,,2

,,

,,

,,

,

,,2

,,,

,,,

86.21

86.21

.

.

nNHNA

NA

nOOH

OHZBH

ZH

ZH

nNHNA

nNH

nOOH

OH

ZH

ZH

nNHNA

NA

nOOH

nOZBHnBH

nNONO

nNO

nSS

SH

nSSnSNO

SKK

SKKf

YY

SKS

SKK

YY

SKK

SKS

fXSK

SSKKr

,,

,

,

,

,,

,,

,

,2

,

,,

86.21

86.21

[

]86.2

1

86.21.[

.

,,

,

,

,,,

,

,,

,2

,,

gnNHNA

NA

nOOH

OHZBH

ZH

ZH

gnNHNA

nNH

nOOH

OH

ZH

ZH

nNHNA

NA

nOOH

nOZBH

nBHnNONO

nNO

BHSPSP

BHSPMP

nSadsnSNO

SKK

SKKf

YY

SKS

SKK

YY

SKK

SKS

f

XSK

SXKK

XKKr



2. The variables’ rate derivatives towards the model para-meters. The partial derivatives of the variables’ rates according to the model coefficients

are as follows:



BAnNOOA

nNO

nNHSA

SAA

ZA

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOOH

OHZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOOH

nOZBH

nSNHnSNO

XSK

S

SKK

Y

XSKXS

KSK

SX

SKS

SKK

SKK

f

XSKXS

KSK

SX

SKS

SKK

SKS

fr

,

,2

,

,,

,,

,

,,

,

,2

,,

,,

,,

,

,,

,

,2

,,

,,,

.1

nNHNA

NAZBH

ZH

ZH

nNHNA

nNH

ZH

ZH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

NO

nOOH

OH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

NO

nNHNA

NA

nOOH

nOZBH

nSNOnSNO

SKK

fYY

SKS

YY

XSKXS

KSK

SX

SKK

SKK

XSKXS

KSK

SX

SKK

SKK

SKS

fr

,,

,

,,

,,

,

,,2

,,

,,

,,

,

,,2

,,,

,,,

86.21

86.21

.

.

nNHNA

NA

nOOH

OHZBH

ZH

ZH

nNHNA

nNH

nOOH

OH

ZH

ZH

nNHNA

NA

nOOH

nOZBHnBH

nNONO

nNO

nSS

SH

nSSnSNO

SKK

SKKf

YY

SKS

SKK

YY

SKK

SKS

fXSK

SSKKr

,,

,

,

,

,,

,,

,

,2

,

,,

86.21

86.21

[

]86.2

1

86.21.[

.

,,

,

,

,,,

,

,,

,2

,,

gnNHNA

NA

nOOH

OHZBH

ZH

ZH

gnNHNA

nNH

nOOH

OH

ZH

ZH

nNHNA

NA

nOOH

nOZBH

nBHnNONO

nNO

BHSPSP

BHSPMP

nSadsnSNO

SKK

SKKf

YY

SKS

SKK

YY

SKK

SKS

f

XSK

SXKK

XKKr

nNONO

nNOnBH

nNHNA

NA

nOOH

nO

nSS

nSH

ZH

nSNHnSS SK

SX

SKK

SKS

SKS

Yr

,

,,2

,,

,

,

,,, 11

nNHNA

NA

nOOH

OH

nNHNA

nNH

nOOH

OH

nNHNA

NA

nOOH

nO

nBHnNONO

NO

nSS

nSH

ZH

nSNOnSS

SKK

SKK

SKS

SKK

SKK

SKS

XSK

KSK

SY

r

,,,

,

,

,,

,

,2,,

,,,

.

.1

nNHNA

NA

nNHNA

nNH

NONO

NO

OOH

OH

NONO

NO

nNHNA

NA

nNHNA

nNH

nOOH

nO

SS

S

ZH

nSSnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

SKK

Yr

,,

,

,,

,

,

,

,, 2

1

0,, nSadsnSSr

nNONO

nNOnBH

nNHNA

NA

nOOH

nO

nBHnadsSP

nBHnadsMP

ZH

nSNHnSads SK

SX

SKK

SKS

XSKXS

KY

r,

,,2

,,

,

,,

,,,, 11

nNHNA

NA

nOOH

OH

nNHNA

nNH

nOOH

OH

nNHNA

NA

nOOH

nO

nBHnNONO

NO

nBHnadsSP

nBHnadsMP

ZH

nSNOnSads

SKK

SKK

SKS

SKK

SKK

SKS

XSK

KXSK

XSK

Yr

,,,

,

,

,,

,

,2,,,

,,,,

.

.1

0,

, nSSnSadsr

nNHNA

NA

nNHNA

nNHnBHg

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHSPMP

ZHnNONO

nNO

nNHNA

NA

nNHNA

nNH

nBHnOOH

nO

nBHnadsSP

nBHSPMP

ZH

nSadsnSads

SKK

SKS

XSK

SSK

K

XSK

XKK

YSKS

SKK

SKS

XSK

S

XSK

XKK

Yr

,,

,,

,

,

,

2,,

,

,

,

,,

,

,,

,2

,,

,,,

.

.1.

.1

SASPMPgOANANOOHSHAZBHZAZH KKKKKKKKfYYf ,,,,,,,,,,,,,,

,0, fnSNHr

nBAnOOA

nO

nNHSA

nNHA

ZA

YnSNH

YnSNH X

SKS

SKS

Yrr ZAZH

,,

,

,

,,,

1,0

nBAnOOA

nO

nNHSA

nNHA

nBHnadsSP

nBHnadsgMP

nSS

nSH

nBHnNONO

nNO

nNHNA

nNH

nOOH

OH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNHNA

nNH

nOOH

nOfnSNH

XSK

SSK

S

XSKXS

K

SKS

XSK

SSK

SSK

K

XSKXS

KSK

SX

SKS

SKS

r ZBH

,,

,

,

,

,,

,,

,

,

,,

,

,

,

,

,,

,,

,

,,

,

,

,

,,

nNONO

nNO

nOOH

OH

nOOH

nO

nNHNA

nNHnBH

nSS

nSZBHnSNH SK

SSK

KSK

SSK

SX

SKS

fr H

,

,

,,

,

,

,,

,

,,

nBAnOOA

nO

nNHSA

nNHZBH

ZAnSNH X

SKS

SKS

fY

r A,

,

,

,

,,

1

nNONO

nNO

nOOH

OH

nOOH

nOnBH

nNHNA

nNH

nSS

nSHZBH

KnSNH SK

SSK

KSK

SX

SKS

SKS

fr S

,

,

,,

,,

,

,2

,

,,

nBHnadsSP

nBHnadsgMP

nSS

nSH

nBHnNONO

nNO

nNHNA

nNH

nOOH

nOZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNHNA

nNH

nOOH

nOZBH

KnSNH

XSKXS

KSK

S

XSK

SSK

S

SK

Sf

XSKXS

KSK

SX

SKS

SK

Sfr OH

,,

,,

,

,

,,

,

,

,2

,

,

,,

,,

,

,,

,

,2

,

,,

.

.

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOOH

OHZBH

KnSNH XSK

XSK

SKS

XSK

SSK

SSK

Kfr NO

,,

,,

,

,,2

,

,

,

,

,,

nBHnadsSP

nBHnadsgMP

nSS

nSH

nBHnNONO

nNO

nNHNA

nNH

nOOH

OHZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNHNA

nNH

nOOH

nOZBH

KnSNH

XSKXS

KSK

S

XSK

S

SK

SSK

Kf

XSKXS

KSK

SX

SK

SSK

Sfr NA

,,

,,

,

,

,,

,2

,

,

,

,,

,,

,

,,2

,

,

,

,,

.

.

nBAnOOA

nO

nNHSA

nNHAZBH

ZA

KnSNH X

SKS

SKS

fY

r OA,2

,

,

,

,,

1

ZBHnBHnNONO

nNO

nNHNA

nNH

nOOH

OH

nBHnadsSP

nBHnadsMPnSNH fX

SKS

SKS

SKK

XSKXS

Kr g,

,

,

,

,

,,,

,,,

g

nNONO

nNO

nOOH

OH

nOOH

nOZBHnBH

nNHNA

nNH

nBHnadsSP

nBHnadsKnSNH SK

SSK

KSK

SfX

SKS

XSKXS

r MP ,

,

,,

,,

,

,

,,

,,,

nBHnNONO

nNO

nOOH

OH

nOOH

nO

nNHNA

nNH

nBHnadsSP

nBHnadsMPZBH

KnSNH X

SKS

SKK

SKS

SKS

XSKXS

Kfr SP,

,

,

,,

,

,

,

,,

,,,

nBAnOOA

nO

nNHSA

nNHAZBH

ZA

KnSNH X

SKS

SKS

fY

r SA,

,

,2

,

,,

1

0, fnSNOr , 0, MAf

nSNOr , 0, AKnSNOr , 0, XK

nSNOr

nNHNA

NA

nNHNA

nNH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nOOH

OH

ZH

YnSNO

SKK

SKS

XSKXS

KSK

SX

SKS

SKK

Yr ZH

,,

,

,,

,,

,

,,

,

,

,2,

.

.86.2

1

,0, f

nSNHrnBA

nOOA

nO

nNHSA

nNHA

ZA

YnSNH

YnSNH X

SKS

SKS

Yrr ZAZH

,,

,

,

,,,

1,0

nBAnOOA

nO

nNHSA

nNHA

nBHnadsSP

nBHnadsgMP

nSS

nSH

nBHnNONO

nNO

nNHNA

nNH

nOOH

OH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNHNA

nNH

nOOH

nOfnSNH

XSK

SSK

S

XSKXS

K

SKS

XSK

SSK

SSK

K

XSKXS

KSK

SX

SKS

SKS

r ZBH

,,

,

,

,

,,

,,

,

,

,,

,

,

,

,

,,

,,

,

,,

,

,

,

,,

nNONO

nNO

nOOH

OH

nOOH

nO

nNHNA

nNHnBH

nSS

nSZBHnSNH SK

SSK

KSK

SSK

SX

SKS

fr H

,

,

,,

,

,

,,

,

,,

nBAnOOA

nO

nNHSA

nNHZBH

ZAnSNH X

SKS

SKS

fY

r A,

,

,

,

,,

1

nNONO

nNO

nOOH

OH

nOOH

nOnBH

nNHNA

nNH

nSS

nSHZBH

KnSNH SK

SSK

KSK

SX

SKS

SKS

fr S

,

,

,,

,,

,

,2

,

,,

nBHnadsSP

nBHnadsgMP

nSS

nSH

nBHnNONO

nNO

nNHNA

nNH

nOOH

nOZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNHNA

nNH

nOOH

nOZBH

KnSNH

XSKXS

KSK

S

XSK

SSK

S

SK

Sf

XSKXS

KSK

SX

SKS

SK

Sfr OH

,,

,,

,

,

,,

,

,

,2

,

,

,,

,,

,

,,

,

,2

,

,,

.

.

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOOH

OHZBH

KnSNH XSK

XSK

SKS

XSK

SSK

SSK

Kfr NO

,,

,,

,

,,2

,

,

,

,

,,

nBHnadsSP

nBHnadsgMP

nSS

nSH

nBHnNONO

nNO

nNHNA

nNH

nOOH

OHZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNHNA

nNH

nOOH

nOZBH

KnSNH

XSKXS

KSK

S

XSK

S

SK

SSK

Kf

XSKXS

KSK

SX

SK

SSK

Sfr NA

,,

,,

,

,

,,

,2

,

,

,

,,

,,

,

,,2

,

,

,

,,

.

.

nBAnOOA

nO

nNHSA

nNHAZBH

ZA

KnSNH X

SKS

SKS

fY

r OA,2

,

,

,

,,

1

ZBHnBHnNONO

nNO

nNHNA

nNH

nOOH

OH

nBHnadsSP

nBHnadsMPnSNH fX

SKS

SKS

SKK

XSKXS

Kr g,

,

,

,

,

,,,

,,,

g

nNONO

nNO

nOOH

OH

nOOH

nOZBHnBH

nNHNA

nNH

nBHnadsSP

nBHnadsKnSNH SK

SSK

KSK

SfX

SKS

XSKXS

r MP ,

,

,,

,,

,

,

,,

,,,

nBHnNONO

nNO

nOOH

OH

nOOH

nO

nNHNA

nNH

nBHnadsSP

nBHnadsMPZBH

KnSNH X

SKS

SKK

SKS

SKS

XSKXS

Kfr SP,

,

,

,,

,

,

,

,,

,,,

nBAnOOA

nO

nNHSA

nNHAZBH

ZA

KnSNH X

SKS

SKS

fY

r SA,

,

,2

,

,,

1

nBAnOOA

nO

nNHSA

nNHA

ZH

YnSNO X

SK

SSK

SY

r ZA,2

,

,

,

,2, 86.2

1

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOOH

nOfnSNO XSK

XSK

SKS

XSK

SSK

KSK

Sr ZBH

,,

,,

,

,,

,

,

,,

,,

nNHNA

NA

nNHNA

nNHnBH

nNONO

nNO

nOOH

OH

nSS

nS

ZH

ZH

nOOH

OH

nOOH

nO

nBHnNONO

nNO

nNHNA

NA

nSS

nSZBHnSNO

SKK

SKS

XSK

S

SKK

SKS

YY

SKK

SKS

XSK

SSK

KSK

Sfr H

,,

,,

,

,

,,

,

,,

,

,,

,

,,

,,

.

.86.2

1.

.2

nBAnOOA

nO

nNHSA

nNH

ZAnSNO X

SKS

SK

SY

r A,

,

,2

,

,,

1

nNHNA

NA

nNHNA

nNHnBH

nNONO

nNO

nOOH

OH

nSS

nSH

ZH

ZH

nOOH

OH

nOOH

nOnBH

nNONO

nNO

nNHNA

NA

nSS

nSHZBH

KnSNO

SKK

SKS

XSK

SSK

KSK

SYY

SKK

SKS

XSK

SSK

KSK

Sfr S

,,

,,

,

,

,2

,

,

,,

,,

,

,

,2

,

,,

86.21

gnBHnadsSP

nBHnadsMPnBH

nNONO

nNO

nNHNA

NA

nOOH

nOZBH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nOOH

OH

ZH

ZHKnSNO

XSKXS

KXSK

SSK

KSK

Sf

XSKXS

KSK

SX

SKS

SKK

YY

r OH

1

86.21

,,

,,,

,

,

,2

,

,

,,

,,

,

,,

,

,2

,,

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNOOH

nNO

nNHNA

NA

nOOH

OHZBH

ZH

ZH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOOH

OH

ZH

ZH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOS

nOZBH

KnSNO

XSKXS

KSK

SX

SKS

SKK

SKKf

YY

XSKXS

KSK

SX

SKS

SKS

SKK

YY

XSKXS

KSK

SX

SKS

SKK

SKS

fr NO

,,

,,

,

,,2

,

,

,,

,,

,,

,

,,2

,

,

,

,

,

,,

,,

,

,,2

,

,

,,

,,

86.21

86.21

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOOH

OHZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOS

nOZBH

KnSNO

XSKXS

KSK

SX

SKS

SK

SSK

Kf

XSKXS

KSK

SX

SKS

SK

SSK

Sfr NA

,,

,,

,

,,

,

,2

,

,

,

,,

,,

,

,,

,

,2

,

,

,

,,

nBAnOOA

nO

nNHSA

nNHA

ZA

KnSNO X

SKS

SK

SY

r OA,

,

,2

,

,,

1

nBHnNONO

nNO

nNHNA

NA

nOOH

OH

nBHnadsSP

nBHnadsMPZBH

nNHNA

NA

nNHNA

nNHnBH

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnadsMP

ZH

ZHnSNO

XSK

SSK

KSK

KXSK

XSKf

SKK

SKS

XSK

SSK

KXSK

XSK

YY

r g

,,

,

,,,,

,,

,,

,,

,

,

,,,

,,, 86.2

1

nNHNA

NA

nNHNA

nNHnBHg

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnads

ZH

ZH

nOOH

OHg

nOOH

nOnBH

nNONO

nNO

nNHNA

NA

nBHnadsSP

nBHnadsZBH

KnSNO

SKK

SKS

XSK

SSK

KXSK

XSYY

SKK

SKS

XSK

SSK

KXSK

XSfr MP

,,

,,

,

,

,,,

,,

,,

,,

,

,

,,,

,,,

86.21

nNHNA

NA

nNHNA

nNHnBHg

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnads

ZH

ZH

nOOH

OHg

nOOH

nOnBH

nNONO

nNO

nNHNA

NA

nBHnadsSP

nBHnadsMPZBH

KnSNO

SKK

SKS

XSK

SSK

KXSK

XSYY

SKK

SKS

XSK

SSK

KXSK

XSKfr SP

,,

,,

,

,

,,,

,,

,,

,,

,

,

,,,

,,,

86.21

nBAnOOA

nO

nNHSA

nNHA

ZA

KnSNO X

SKS

SK

SY

r SA,

,

,2

,

,,

1




nBAnOOA

nO

nNHSA

nNHA

ZH

YnSNO X

SK

SSK

SY

r ZA,2

,

,

,

,2, 86.2

1

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOOH

nOfnSNO XSK

XSK

SKS

XSK

SSK

KSK

Sr ZBH

,,

,,

,

,,

,

,

,,

,,

nNHNA

NA

nNHNA

nNHnBH

nNONO

nNO

nOOH

OH

nSS

nS

ZH

ZH

nOOH

OH

nOOH

nO

nBHnNONO

nNO

nNHNA

NA

nSS

nSZBHnSNO

SKK

SKS

XSK

S

SKK

SKS

YY

SKK

SKS

XSK

SSK

KSK

Sfr H

,,

,,

,

,

,,

,

,,

,

,,

,

,,

,,

.

.86.2

1.

.2

nBAnOOA

nO

nNHSA

nNH

ZAnSNO X

SKS

SK

SY

r A,

,

,2

,

,,

1

nNHNA

NA

nNHNA

nNHnBH

nNONO

nNO

nOOH

OH

nSS

nSH

ZH

ZH

nOOH

OH

nOOH

nOnBH

nNONO

nNO

nNHNA

NA

nSS

nSHZBH

KnSNO

SKK

SKS

XSK

SSK

KSK

SYY

SKK

SKS

XSK

SSK

KSK

Sfr S

,,

,,

,

,

,2

,

,

,,

,,

,

,

,2

,

,,

86.21

gnBHnadsSP

nBHnadsMPnBH

nNONO

nNO

nNHNA

NA

nOOH

nOZBH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nOOH

OH

ZH

ZHKnSNO

XSKXS

KXSK

SSK

KSK

Sf

XSKXS

KSK

SX

SKS

SKK

YY

r OH

1

86.21

,,

,,,

,

,

,2

,

,

,,

,,

,

,,

,

,2

,,

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNOOH

nNO

nNHNA

NA

nOOH

OHZBH

ZH

ZH

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOOH

OH

ZH

ZH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

NA

nOS

nOZBH

KnSNO

XSKXS

KSK

SX

SKS

SKK

SKKf

YY

XSKXS

KSK

SX

SKS

SKS

SKK

YY

XSKXS

KSK

SX

SKS

SKK

SKS

fr NO

,,

,,

,

,,2

,

,

,,

,,

,,

,

,,2

,

,

,

,

,

,,

,,

,

,,2

,

,

,,

,,

86.21

86.21

nBHnadsSP

nBHnadsgMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOOH

OHZBH

nBHnadsSP

nBHnadsMP

nSS

nSHnBH

nNONO

nNO

nNHNA

nNH

nOS

nOZBH

KnSNO

XSKXS

KSK

SX

SKS

SK

SSK

Kf

XSKXS

KSK

SX

SKS

SK

SSK

Sfr NA

,,

,,

,

,,

,

,2

,

,

,

,,

,,

,

,,

,

,2

,

,

,

,,

nBAnOOA

nO

nNHSA

nNHA

ZA

KnSNO X

SKS

SK

SY

r OA,

,

,2

,

,,

1

nBHnNONO

nNO

nNHNA

NA

nOOH

OH

nBHnadsSP

nBHnadsMPZBH

nNHNA

NA

nNHNA

nNHnBH

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnadsMP

ZH

ZHnSNO

XSK

SSK

KSK

KXSK

XSKf

SKK

SKS

XSK

SSK

KXSK

XSK

YY

r g

,,

,

,,,,

,,

,,

,,

,

,

,,,

,,, 86.2

1

nNHNA

NA

nNHNA

nNHnBHg

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnads

ZH

ZH

nOOH

OHg

nOOH

nOnBH

nNONO

nNO

nNHNA

NA

nBHnadsSP

nBHnadsZBH

KnSNO

SKK

SKS

XSK

SSK

KXSK

XSYY

SKK

SKS

XSK

SSK

KXSK

XSfr MP

,,

,,

,

,

,,,

,,

,,

,,

,

,

,,,

,,,

86.21

nNHNA

NA

nNHNA

nNHnBHg

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnads

ZH

ZH

nOOH

OHg

nOOH

nOnBH

nNONO

nNO

nNHNA

NA

nBHnadsSP

nBHnadsMPZBH

KnSNO

SKK

SKS

XSK

SSK

KXSK

XSYY

SKK

SKS

XSK

SSK

KXSK

XSKfr SP

,,

,,

,

,

,,,

,,

,,

,,

,

,

,,,

,,,

86.21

nBAnOOA

nO

nNHSA

nNHA

ZA

KnSNO X

SKS

SK

SY

r SA,

,

,2

,

,,

1

• for the variable’s rate rads,n

0, fnSSr , 0, ZAY

nSSr , 0, ZBHfnSSr , 0, A

nSSr

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,

,

,2,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OH

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nSS

nSH

ZH

YnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XSK

SY

r ZH

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,

,

,,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OH

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nSS

nS

ZAnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XSK

SY

r H

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,2

,

,,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OH

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nSS

nSH

ZH

KnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XSK

SY

r S

nNONO

nNO

nNHNA

nNHnBH

nOOH

nO

nSS

nSH

ZH

KnSS SK

SSK

SX

SKS

SKS

Yr OH

,

,

,

,,2

,

,

,

,, 11

]

[1

,,

,

,

,,

,,2

,

,

,

,,

nNHNA

NA

nNHNA

nNH

nOOH

OH

nNHNA

NA

nOOH

nOnBH

nNONO

nNO

nSS

nSH

ZH

KnSS

SKK

SKS

SKK

SKK

SKS

XSK

SSK

SY

r NO

nNONO

nNO

nOOH

nOnBH

nNHNA

nNH

nSS

nSH

ZH

KnSS SK

SSK

SX

SKS

SKS

Yr NA

,

,

,

,,2

,

,

,

,, 11

0, OAK

nSSr , 0, gnSSr

, 0, MPKnSSr , 0, SPK

nSSr , 0, SAKnSSr , 0, MAf

nSSr, 0, AK

nSSr

0, fnSadsr 0, OAK

nSadsr , 0, ZAYnSadsr , 0, ZBHf

nSadsr , 0, HnSadsr , 0, A

nSadsr , 0, SKnSadsr

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,

,,

,,2

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OHg

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nBHnadsSP

nBHnadsMP

ZH

YSads

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XXSK

XSK

Yr ZH

nNONO

nNO

nNHNA

nNHnBH

nOOH

nO

nBHnadsSP

nBHnadsMP

ZH

KSads SK

SSK

SX

SKS

XSKXS

KY

r OH

,

,

,

,,2

,

,

,,

,, 11

]

1

,,

,

,

,,

,,2

,

,

,,

,,

nNHNA

NA

nNHNA

nNH

nOOH

OHg

nNHNA

NA

nOOH

nOnBH

nNONO

nNO

nBHnadsSP

nBHnadsMP

ZH

KSads

SKK

SKS

SKK

SKK

SKS

XSK

SXSK

XSK

Yr NO

nNONO

nNO

nOOH

nOnBH

nNHNA

nNH

nBHnadsSP

nBHnadsMP

ZH

KSads SK

SSK

SX

SKS

XSKXS

KY

r NA

,

,

,

,,2

,

,

,,

,, 11

nNHNA

NA

nNHNA

nNHnBHg

nNONO

nNO

nOOH

OH

nBHnadsSP

nBHnadsMP

ZHSads SK

KSK

SX

SKS

SKK

XSKXS

KY

r g

,,

,,

,

,

,,,

,,1

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,

,,

,,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OHg

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nBHnadsSP

nBHnads

ZH

KSads

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XXSK

XSY

r MP

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,2

,,

,,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OHg

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nBHnadsSP

nBHnads

ZH

KSads

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XXSK

XSY

r SP

0, SAK

nSadsr , nBHnSA

KnSads XXKr OA

,,, , nBHnadsMAnBHnSK

nSads XSfXXr A,,,,,

0, fnSSr , 0, ZAY

nSSr , 0, ZBHfnSSr , 0, A

nSSr

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,

,

,2,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OH

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nSS

nSH

ZH

YnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XSK

SY

r ZH

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,

,

,,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OH

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nSS

nS

ZAnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XSK

SY

r H

]

[1

,,

,

,

,

,

,

,

,,

,

,

,,2

,

,,

nNHNA

NA

nNHNA

nNH

nNONO

nNO

nOOH

OH

nNONO

nNO

nNHNA

NA

nNHNA

nNH

nOOH

nOnBH

nSS

nSH

ZH

KnSS

SKK

SKS

SKS

SKK

SKS

SKK

SKS

SKS

XSK

SY

r S

nNONO

nNO

nNHNA

nNHnBH

nOOH

nO

nSS

nSH

ZH

KnSS SK

SSK

SX

SKS

SKS

Yr OH

,

,

,

,,2

,

,

,

,, 11

]

[1

,,

,

,

,,

,,2

,

,

,

,,

nNHNA

NA

nNHNA

nNH

nOOH

OH

nNHNA

NA

nOOH

nOnBH

nNONO

nNO

nSS

nSH

ZH

KnSS

SKK

SKS

SKK

SKK

SKS

XSK

SSK

SY

r NO

nNONO

nNO

nOOH

nOnBH

nNHNA

nNH

nSS

nSH

ZH

KnSS SK

SSK

SX

SKS

SKS

Yr NA

,

,

,

,,2

,

,

,

,, 11

0, OAK

nSSr , 0, gnSSr

, 0, MPKnSSr , 0, SPK

nSSr , 0, SAKnSSr , 0, MAf

nSSr, 0, AK

nSSr

Documents

Sensitivity study of reduced models of the activated