MODELING BIOLOGICAL NITROGEN REMOVALWITH DENITRIFICATION ENZYME PARAMETER ESTIMATION

By

RYAN K. HAMILTON

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2005

Copyright 2005

by

Ryan K. Hamilton

To my wife, without whose support my work would have been impossible.

ACKNOWLEDGMENTS

I would like to thank my supervisory committee chair (Spyros A. Svoronos)

and cochair (Ben Koopman) for their advice and guidance. I thank my committee

members (Atul Narang and Madeline Rasche) for their generous help. I would like

to also thank my colleagues, Anna Casasus and Don Lee. Without their support,

the long hours of experiment would have been fatal. Finally, I would like to thank

my wife for standing by me during these years of study.

iv

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER

1 CONTROL ISSUES AND CHALLENGES . . . . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sewer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Aerobic Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Anoxic Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Secondary Settling Tank . . . . . . . . . . . . . . . . . . . . . . . 71.6 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Disinfection Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Modeling Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 AUTOMATION OF LAB-SCALE BIOREACTORS . . . . . . . . . . . . 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Materials, Methods and Results . . . . . . . . . . . . . . . . . . . 142.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 MODEL FOR DENITRIFIER DIAUXIC GROWTH . . . . . . . . . . . 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Bacterial Strain and Growth Conditions . . . . . . . . . . . 283.3.2 Nitrate Reductase Assay . . . . . . . . . . . . . . . . . . . 293.3.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . 30

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Enzyme Synthesis Expression Derivation . . . . . . . . . . . . . . 39

v

4 KALMAN FILTER FOR ENZYME PARAMETER ESTIMATION . . . 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Kanapaha Water Reclamation Facility . . . . . . . . . . . . . . . 42

4.2.1 Hydraulic Model . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Biological Model . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 534.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 Distributed State Modeling . . . . . . . . . . . . . . . . . . . . . . 575.2 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 575.3 Enzyme Activity Based Dynamic Optimization . . . . . . . . . . . 59

APPENDIX

A KALMAN FILTER - SUPPLEMENTAL MATERIAL . . . . . . . . . . . 61

B AUTOMATION PROGRAM SOURCE CODE . . . . . . . . . . . . . . 69

B.1 Code for DLECModule.bas . . . . . . . . . . . . . . . . . . . . . . 69B.2 Code for DLECEmailModule.bas . . . . . . . . . . . . . . . . . . 73B.3 Code for DLECPhoneCallModule.bas . . . . . . . . . . . . . . . . 76B.4 Code for DLECLogFileModule.bas . . . . . . . . . . . . . . . . . . 77

C DENITRIFIER DIAUXIC GROWTH MODEL SOURCE CODE . . . . 81

C.1 Code for Run061703.m . . . . . . . . . . . . . . . . . . . . . . . . 81C.2 Code for RunModel.m . . . . . . . . . . . . . . . . . . . . . . . . 82C.3 Code for model5c.m . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.4 Code for FitAllData.m . . . . . . . . . . . . . . . . . . . . . . . . 87

D EXTENDED KALMAN FILTER SOURCE CODE . . . . . . . . . . . . 89

D.1 Code for TestKSim.m . . . . . . . . . . . . . . . . . . . . . . . . . 89D.2 Code for RunKEKF.m . . . . . . . . . . . . . . . . . . . . . . . . 91D.3 Code for fKanapahaSim.m . . . . . . . . . . . . . . . . . . . . . . 95D.4 Code for fKanapahaModelOpsSetup.m . . . . . . . . . . . . . . . 103D.5 Code for fKanapahadxdt.m . . . . . . . . . . . . . . . . . . . . . . 107D.6 Code for fParseAllX.m . . . . . . . . . . . . . . . . . . . . . . . . 111D.7 Code for fExtendedKalmanFilter.m . . . . . . . . . . . . . . . . . 114D.8 Code for PlotKEKFData.m . . . . . . . . . . . . . . . . . . . . . 128

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

vi

LIST OF TABLESTable page

2–1 Equipment list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3–1 Synthetic media composition . . . . . . . . . . . . . . . . . . . . . . . 28

3–2 Parameter values obtained by fitting . . . . . . . . . . . . . . . . . . 32

3–3 Nomenclature used in denitrifier modeling . . . . . . . . . . . . . . . . 38

4–1 Wastewater composition . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4–2 eASM1m model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4–3 eASM1m parameter values . . . . . . . . . . . . . . . . . . . . . . . . 50

A–1 Aerobic basin surface aerators . . . . . . . . . . . . . . . . . . . . . . 62

A–2 Hydraulic model parameters . . . . . . . . . . . . . . . . . . . . . . . 64

A–3 eASM1m Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A–4 eASM1m parameter values . . . . . . . . . . . . . . . . . . . . . . . . 66

vii

LIST OF FIGURESFigure page

1–1 Example plant diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1–2 Kanapaha Water Reclamation Facility . . . . . . . . . . . . . . . . . 3

2–1 Automation system Process and Instrumentation Drawing (PID) . . . 15

2–2 Effect of rinsing sample path on biomass measurements . . . . . . . . 17

2–3 Software interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2–4 Experiment configuration screen . . . . . . . . . . . . . . . . . . . . . 20

3–1 Biochemical process model . . . . . . . . . . . . . . . . . . . . . . . . 24

3–2 Model overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3–3 Model fit to literature results . . . . . . . . . . . . . . . . . . . . . . . 33

3–4 Enzyme activity during diauxic growth . . . . . . . . . . . . . . . . . 34

3–5 Additional experiments with model fits . . . . . . . . . . . . . . . . . 35

4–1 Kanapaha physical process layout. . . . . . . . . . . . . . . . . . . . . 45

4–2 Kanapaha operations – aeration strategy. . . . . . . . . . . . . . . . . 46

4–3 Kanapaha operations – recycles and recirculations. . . . . . . . . . . . 47

4–4 Hydraulic model process diagram . . . . . . . . . . . . . . . . . . . . 48

4–5 EKF results, aN , nitrate and ammonia . . . . . . . . . . . . . . . . . 55

A–1 Hydraulic model PID . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A–2 Diurnal flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

viii

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

MODELING BIOLOGICAL NITROGEN REMOVALWITH DENITRIFICATION ENZYME PARAMETER ESTIMATION

By

Ryan K. Hamilton

December 2005

Chair: Spyros A. SvoronosCochair: Ben KoopmanMajor Department: Chemical Engineering

As urban population density has increased it has been necessary to develop

ever more sophisticated wastewater treatment technology. Waters that accept

domestic wastewater receive pollutants in the form of nutrients (primarily

organic compounds and ammonia) that, if left untreated, could spur a bloom

of microorganism growth and cause disease in the higher organisms that consumed

the water. Most modern wastewater treatment facilities use some type of activated

sludge process in which naturally occurring microorganisms are cultivated in the

wastewater under conditions that attempt to optimize the consumption of influent

nutrients. This work presents models and techniques for predicting the performance

of biological nutrient removal systems, both bench-scale and plant-scale.

Presented is a model for diauxic growth of denitrifying bacteria in which

nitrate reductase synthesis kinetics dominate the overall growth kinetics. The

model is based on the assumption of the existence of a nitrate respiration operon,

thereby linking the rate of nitrate uptake to the activity of nitrate reductase. I have

shown that this approach can model diauxic growth of Pseudomonas denitrificans

ix

by conducting experiments in which nitrate reductase activity was measured

during both lag and ensuing exponential growth phases. I consistently observed the

pattern of low nitrate reductase enzyme activity during the lag phase, increasing

before the onset of growth. By fitting model parameters I was able to successfully

match experimental data for growth, nitrate uptake and enzyme activity level.

In cooperation with Gainesville Regional Utilities, a process model was

developed for the Kanapaha Water Reclamation Facility (KWRF) predenitrification

process in Gainesville, Florida. The process model incorporates a biochemical

model for diauxic growth of pure cultures of denitrifying bacteria that is integrated

with the industry standard Activated Sludge Model 1. I demonstrate, using real

facility operating data, that by applying an extended Kalman filter to a single

bioreactor I obtain estimates for both reactor composition and denitrification

enzyme model parameters. This technique for parameter identification allows

a semi-mechanistic model developed for pure cultures to be applied to a mixed

culture population where isolation of enzyme kinetic parameters is not practical.

x

CHAPTER 1CONTROL ISSUES AND CHALLENGES IN WASTEWATER TREATMENT

PLANTS

1.1 Introduction

As urban population density has increased it has been necessary to develop

ever more sophisticated wastewater treatment technology. There was a significant

improvement in urban sanitation with the boom in popularity of indoor toilets in

the early 1900s however this increase occurred at the expense of the downstream

receiving waters. These waters received pollutants in the form of nutrients

(primarily organic compounds and ammonia) that, if left untreated, could spur

a bloom of microorganism growth and cause disease in the higher organisms that

consumed the water.

Modern cities operate treatment facilities of ever growing scale and sophistication

in order to treat wastewater to a level that natural systems can safely absorb

without negative impact. Most modern wastewater treatment facilities use some

type of activated sludge process in which naturally occurring microorganisms

are cultivated in the wastewater under conditions that attempt to optimize the

consumption of influent nutrients. The majority of plants in the United States aim

to accomplish only carbon removal, and issues of control there pertain primarily

to aeration control for energy usage and satisfying process demands. There is

a significant control issue for these facilities in that adequate oxygen has to be

provided in spite of significant and continually changing influent conditions without

excessive aeration that would waste energy. Increasing pressure to also remove

nutrients such as nitrogen and phosphorus requires more complicated processes that

have several optimization and control issues. Nutrients are transformed at different

1

2

AnoxicBasin

AerobicBasin Clarifier

Plant Influent

Mixed Liquor Recirculation

WasteSludge Return

FilterDisinfection

Basin

Effluent

Pumping Station Tank

Sewage Enters

Pumping Station Tank

Pumping Station Tank

Sewage Enters Sewage Enters

Biosolids ProcessingTreatment Facility

Sewer System

Figure 1–1: Example plant diagram. Sewage is collected by local pumping stationsbefore passing to the treatment facility (here a Ludzack-Ettinger process). Influentfirst passes through an anoxic bioreactor in which nitrate is reduced to nitrogenand some organic carbon is consumed. In the aerobic bioreactor most organiccarbon consumption occurs and ammonia is converted to nitrate. The mixed liquorrecirculation returns the generated nitrate to the anoxic reactor for reductionto nitrogen gas. The secondary settling tank returns a concentrated stream ofactivated sludge to the anoxic reactor while passing clarified effluent to the filters,and from there to the disinfection basin for inactivation of harmful microorganisms.

rates depending on reactor environment and other operating conditions. Either

cyclic operation or multi-reactor facilities have been effectively utilized for nutrient

removal. Control and optimization of wastewater treatment facilities has been an

area of interest for over a century. As an example of the size of the industry, in

1996 $16.7 billion was spent in the U.S. on wastewater treatment operations.

1.2 Sewer System

The first control issue that can seriously impact plant operation is regulation

of the influent flow rate to the wastewater treatment plant. The normal day/night

cycle of human activity causes diurnal variation in sewage flow. There are also

surges in flow rate due to rainfall entering the sewer system via cracks in pipes

3

Figure 1–2: Kanapaha Water Reclamation Facility. This facility uses two differentbiological nutrient removal processes in parallel to treat 10 million gallons ofwastewater per day with a total bioreactor volume of 11.2 Mgal (Note that this isan unusually large ratio of reactor volume to flow rate.) Pictured are the adjacentEast and West aeration basins, totaling 4.8 million gallons.

4

(infiltration) and through manhole covers and other openings (inflow). One

method of smoothing flow variations is to build equalization basins upstream of

the wastewater treatment plant. This is common practice at facilities treating

industrial wastewaters but is usually too expensive for municipal wastewater

treatment plants. In a few cases, the inherent storage within the pipes and

pumping stations that make up a sewer system has been utilized to smooth

variations in plant feed rates.

The Quebec Urban Community has successfully implemented such a real time

control solution to this problem by implementing a global optimal predictive sewer

control system that has been operating since the summer of 1999 (Schutze et al.,

2004). This system routes flows and controls volumes in the sewer system to meet a

series of ranked control objectives.

1. Minimization of combined sewer overflows.

2. Maximization of the use of treatment facility capacity. Treatment facilities

are designed based on assumed feed rates, and a feed rate that is too low can

adversely impact plant performance.

3. Minimization of accumulated volumes. The control scheme tries to meet its

goals with minimal held volume in the sewer system.

4. Minimization of setpoint variation.

A still more sophisticated system would coordinate these goals with the

dynamically changing needs of the treatment facility, but to our knowledge such a

unification has never been implemented.

Other cities are moving in this direction. For example, Chicago recently

finished installing 109 miles of tunnels with a storage capacity of 15.6 billion gallons

to address the problem of system overflows and demand fluctuation.

5

1.3 Aerobic Reactor

Upon entering the treatment facility and being combined with any internal

recycle streams wastewater enters one or more bioreactors where most of the

nutrient removal takes place by a complex mixed population of microorganisms.

Some of these microbes grow only under aerobic conditions (dissolved oxygen

present), whereas others are capable of growth under both anoxic (utilizing nitrate

instead of oxygen as terminal electron acceptor) and aerobic conditions. Many

plant configurations with different combinations of reactors, recirculations and

recycles are commonly used; however all meet the same basic needs. The one

universal requirement is removal of biochechemical oxygen demand (BOD), which is

a measure of the amount of organic carbon available to support microbial growth.

Additionally, removal of nitrogen or phosphorus is a requirement at a growing

number of facilities.

Aerobic conditions are conducive to the growth of a wide variety of microbes,

including heterotrophic bacteria that remove BOD from the wastewater, as well

as nitrifying bacteria that oxidize ammonia to nitrate. Aeration is typically

accomplished using surface impellers or submerged diffusers and accounts for the

largest energy cost in these plants. The cost of energy for aeration throughout a

facility can be 50% of the total plant energy costs, although in some cases this

can be as high as 75%, as it is at the Kanapaha Water Reclamation Facility in

Gainesville, Florida. In this facility switching from manual aerator control to

automatic resulted in a savings of 10 to 30% on an annual cost of $600,000.

The aeration energy cost combined with the very strong effect of aeration on

biomass growth makes dissolved oxygen (DO) control the most studied control

problem in wastewater treatment. The problem is made more complicated by

the fact that oxygen uptake rates change under different plant loadings and

temperatures. The most frequently used control strategy is to manipulate the

6

aeration rate to control dissolved oxygen (DO), usually at a fixed set point

regardless of load. High DO promotes bacterial growth, but also leads to higher

aeration costs. High DO can also be a problem for nitrogen removing plants if it

leads to recirculation of excessive DO to the anaerobic denitrification reactors.

Several more sophisticated control schemes have been developed that vary the

DO setpoints in order to minimize costs while meeting effluent requirements. One

scheme that has been attempted is feed forward aeration control based on influent

ammonia, in which a concentration spike of influent ammonium is measured, and

triggers a traveling wave of increased DO setpoints down the line corresponding to

the calculated hydraulics of the spike (Ingildsen et al., 2002).

In facilities that vary the DO setpoint, the most common approach is cascade

control in which the DO setpoint is manipulated by an outer control loop in

order to control effluent quality. This decouples the fast DO response from the

slower dynamics of final effluent quality. Many techniques, such as fuzzy control

and expert systems have been used for the outer control loop, however, it has

recently been argued that this problem would be more accurately addressed as a

multivariable control problem rather than multiloop (Steffens and Lant, 1999).

1.4 Anoxic Reactor

Anoxic reactors are used in cases where nitrogen removal is a process goal.

Through denitrification, which takes place in anoxic reactors, nitrogen is reduced

through several intermediates to nitrogen gas, which passes harmlessly into the

atmosphere. Facultative anaerobes in the anoxic reactor use nitrate recirculated

from the aerobic reactor as terminal electron acceptor for growth under oxygen-free

conditions. A control issue here is the recirculation rate. It must be fast enough to

supply adequate nitrate without introducing excessive dissolved oxygen. Additional

nitrogen removal is accomplished in some process schemes in an anoxic reactor

following the aerobic reactor. An alternative to having dedicated anoxic reactors is

7

to have cycling of air on and off in the same reactor, thus combining nitrification

and denitrification in the same reactor. Plants can achieve phosphorus removal by

providing process conditions (such as an anaerobic tank at the beginning of the

bioprocess train) that encourage the growth of phosphate accumulating bacteria

(which convert soluble phosphate to polyphosphate).

1.5 Secondary Settling Tank

After leaving the bioreactors, activated sludge enters a secondary settling tank,

where flocs consisting of microbes and other particulate matter are allowed to settle

and form a sludge blanket at the bottom of the tank. Clarified effluent exits at

the top of the tank and flows to the filters while sludge is continuously removed

from the bottom of the tank. On-line depth of blanket sensors monitor the depth

of the sludge blanket, which the operators (or a control system) use to adjust the

sludge return rate. If the return rate is too low, the sludge blanket can approach

the water surface, where it could easily be swept into the effluent. If the return

rate is too high, turbulence in the clarifier will cause the sludge blanket to become

fluffy, diluting the underflow stream and reducing clarifier efficiency. Almost all

of the sludge collected from the bottom of the secondary settling tank is returned

to the bioreactors. A small fraction of the sludge is removed (wasted) from the

settling tank for further processing (e.g. digestion). The waste rate is a variable

that can be used to manipulate solids retention time (SRT), which in turn controls

the net growth rate of microbes in the process. Thus, SRT has a very large impact

on the overall plant dynamics. One control application here is timing the sludge

waste so that it occurs at the time of day when sludge concentrations are at a

maximum in order to reduce hydraulic loading and maximize SRT in the digesters

that receive the sludge. In the case of the Kanapha Water Reclamation Facilitiy

(KWRF) in Gainesville, Florida, This strategy, combined with on-line nutrient and

8

pH measurements in the digester eliminated the need for a 5 to 10 million dollar

expansion in order to meet EPA CFR-503 Class B biosolids regulations.

One motivation for controlling influent wastewater flow rates is to avoid

upsetting the sludge blanket in the secondary settling tank. A sudden flow spike

can cause the blanket to dislodge and enter the upper (clarified) effluent stream,

where it can clog the filters in the next stage of treatment. In the Kanapaha facility

a PID controller is used to set a flow ratio between the clarifier effluent flow and

the sludge return rate. This saves thousands of dollars annually in pumping costs,

but more importantly eliminates the possibility of a plant upset due to a dislodged

sludge blanket.

1.6 Filters

The purpose of filtration is to decrease the concentration of particulate solids

in the treated effluent by passing water through granular or cloth media. Filter

runs are continued until head loss increases to the maximum allowed level. At this

point flow is reversed to dislodge accumulated particulate materials (backwashing).

The filter is then returned to normal operation. The major control issue is the use

of scheduling versus head loss to initiate backwashing.

1.7 Disinfection Basin

The final treatment step is disinfection, commonly accomplished with chlorine

dosing or ultraviolet light exposure. In this process, many of the remaining

microorganisms are killed before the treated water is allowed to leave the plant.

In facilities that use a chlorination basin, the facility is required to have a

certain effluent concentration of chlorine in order to guarantee a minimum level

of disinfection. Chlorine is added at the beginning of a chlorine contact basin and

a large amount immediately reacts with ammonia nitrogen (ammonia nitrogen

chlorine demand) in the water, leaving some fraction for disinfection. The effluent

level must be sufficient for disinfection, however excessive chlorination favors

9

formation of toxic trichloromethanes, which are regulated in water discharged into

the aquifer. By using only the necessary amount of chlorine the facility reduces the

need to store large quantities of a dangerous and expensive chemical.

Controlling effluent chlorine is frequently based on a feedback loop that

sets the chlorine dose based on the effluent chlorine concentration. However,

the ammonia nitrogen chlorine demand varies with the loading of the plant, so

the requisite dose will fluctuate with the daily usage cycle. Large disturbances

combined with significant dead times pose a significant challenge to tight control of

effluent chlorine. The dead times fluctuate by a factor of 10 over a 24 hour period,

so a Smith Predictor proves inadequate. One solution that has been employed

is a cascade scheme in which an inner control loop manipulates the dosing in

order to maintain a setpoint concentration measured near the beginning of the

basin. This short dead time loop allows for rapid control dynamics to compensate

for fluctuations in ammonia nitrogen chlorine demand. The outer control loop

maintains the basin effluent concentration by manipulating the setpoint for the

inner control loop. The outer loop is able to compensate for large daily fluctuations

as a result of slow disturbances.

A novel solution to this problem has recently been proposed (Meredith, 2003)

in which a dynamic weir at the end of the contact basin is raised or lowered in

order to change the reactor volume in response to changes in flowrate. By doing

so, the residence time, and hence the dead time, becomes constant and traditional

dead time compensation techniques may be employed.

In the case of UV disinfection, the manipulated variable is the number of

banks of lights active at any time, and optimization is simply energy cost vs.

acceptable disinfection level.

10

1.8 Modeling Challenges

As is the case in most processes, the majority of development is done with

so-called white-box models, meaning that the models are developed from first

engineering principles. The overall wastewater treatment plant (WWTP) model

consists of two main parts. The hydraulic model represents reactor behavior (plug

flow, CSTR etc), flowrates and recirculation. One unique unit operation is the

secondary settling tank. There are a few secondary settling tank models in use,

the most common of which is the ideal or point clarifier with no retention time, in

which the bottom stream is simply enriched with particulates by a ratio determined

by stream flowrates. If the impact of operating points on sludge settling is of

interest, a one-dimensional settler model may be used, or for exploration of the

effect of flowrate and turbulence, a 2 or 3-dimensional model may be appropriate.

The second primary component of a WWTP model is the activated sludge

model that portrays the microorganism growth, death and nutrient consumption.

These models are necessarily approximations to the vast number of biological

processes occurring in each bioreactor, but selection of the proper model will allow

adequate description of those processes most relevant to a particular WWTP.

The reference model of biochemical reactions in the bioreactors is Activated

Sludge Model No. 1 (ASM1) (Henze et al., 2000), which was developed by

the International Water Association. The success of this model prompted the

widespread adoption of biochemical modeling of wastewater in both academia and

industry. For those facilities where biological phosphorus removal is desirable, a

standard model is Activated Sludge Model No. 2d (Henze et al., 2000). This model

portrays the processes by which phosphate accumulating organisms store phosphate

as polyphosphate under aerobic conditions and hydrolyze it under anaerobic

conditions. More recently, the TUDP model (van Veldhuizen et al., 1999) has been

11

proposed. This model combines the known metabolic model for denitrification and

bio-P removal with the ASM1 sludge production model.

These models have limitations in several respects. In the most common,

ASM1, there is no temperature effect, even though temperature has a significant

impact on sludge behavior. Instead, model parameters are provided at 10 and 20

◦C. In ASM2 and TUDP the temperature effect is portrayed with Arrhenius-type

equations valid in the range from 10 to 25 ◦C. Another limitation of the ASM

models is that they cannot portray a lag in growth when switching electron

acceptors. Work has been going on to add these capabilities. Work has also been

done to extend these models to portray the effect of a slug of toxic influent such

as ethanol (Nowak et al., 1997). In this event, the sludge metabolism, particularly

nitrification, is dramatically inhibited. This problem is normally addressed on a

case-by-case basis depending on the particular facility, but another possible solution

would be to use an on-line model parameter estimation technique.

The next frontier of white-box modeling is the merger of distributed state

models and computational fluid dynamics (CFD). In these models, some or all

state variables do not have a single value, but are represented by a distribution

curve. While the SRT of a facility provides information about the fraction of

active biomass, a distributed state approach would provide that curve directly.

Distributed state models are currently highly demanding computationally, but have

the potential to provide great new insights into biological wastewater treatment.

The benefit of the distributed state approach is particularly significant

for models with nonlinear behavior based on concentrations of intracellular

components. Consider the case of diauxic lag, in which aerobically cultured

facultative anaerobes are introduced into an anoxic environment. They will

experience a period of little or no growth during which they synthesize necessary

enzymes for growth under the new conditions. Now, consider an anoxic reactor

12

containing bacteria experiencing a long diauxic lag into which is introduced an

innoculum that has been growing exponentially under identical anoxic conditions.

Common sense dictates that the new innoculum will continue to grow exponentially

in the new reactor while the preexisting culture continues to experience lag, but a

distributed state model is required to capture this behavior. A traditional model

would take the anoxic growth enzyme levels of the innoculum and average it out

over the entire population, resulting in the incorrect prediction that the entire

population is still experiencing a diauxic lag. A similar argument could be made

for phosphorus metabolism, in which stored polysphosphate is used as an energy

storage compound by certain classes of bacteria, highlighting further the benefit of

distributed state modeling in wastewater treatment.

Development of accurate process models is a prerequisite for application

of model predictive control techniques for whole-process control and dynamic

optimization. (Gernaey et al., 2004) is recommended as a recent review of

wastewater treatment modeling.

1.9 Further Remarks

It should also be noted that wastewater bioreactors are a very harsh

environment for probes. For example, dissolved oxygen probes that do not require

frequent replacement have only recently become available. Part of the reason for

the relatively underdeveloped state of wastewater treatment control is the difficulty

in obtaining accurate process information. In addition, many key variables must

be measured offline. This has prompted recent work in wastewater treatment plant

state estimation (Tenno and Uronen, 1995; Jorgensen et al., 1992). With rapid

improvements being made in sensor technology, the information deficit problem

is disappearing. Furthermore, faster processing capabilities, which have been so

crucial to the application of control as a discipline, are now allowing nonlinear

13

process control and dynamic optimization techniques to be applied to problems as

large as a wastewater treatment facility.

The challenging operating environment coupled with response dynamics

ranging from minutes (DO) to weeks (SRT) has resulted in a great deal of advanced

control work having been done in simulation, but very little in wastewater plants

themselves.

CHAPTER 2AN INEXPENSIVE METHOD FOR THE AUTOMATION OF LAB-SCALE

BIOREACTORS

2.1 Introduction

Bacterial growth experiments may last several days, or even weeks. During

this time it may be necessary to adjust bioreactor operating parameters, which

requires either the attention of a researcher, or an automated control system. We

have developed an inexpensive system for continuous absorbance measurements

using flow-through spectrophotometers and solenoid valves under the control of a

Visual Basic program. This system periodically samples a bioreactor, measures the

biomass absorbance and then rinses the sample line between measurements.

2.2 Materials, Methods and Results

A layout sketch of the experimental apparatus is given in Figure 2–1.

Batch culture experiments were carried out in a stirred bioreactor (Multi-gen

model F-2000, New Brunswick Scientific, New Brunswick, NJ). The bioreactor

was cycled between aerobic and anoxic conditions to explore diauxic growth of

Pseudomonas denitrificans (American Type Cultures Company 13867, Manassas,

VA). The cycling was accomplished by alternately sparging the culture with air

and nitrogen from pressurized cylinders. The switching between these gases was

accomplished using two computer-linked solenoid valves, one on each feed line.

Gases were sterilized by filtration (0.22 micron Whatman HEPA-Vent filters) and

pre-humidified by sparging through sterile deionized water before introduction to

the bioreactor.

Biomass was sampled from the bottom of the bioreactor through 0.03 inch I.D.

Tygon tubing (Masterflex 06409-13, Cole Parmer, Chicago, IL). The reactor effluent

14

15

Figure 2–1: Automation system Process and Instrumentation Drawing (PID)

flowed to a 4.5 L clarifier (25 mm ). The biomass passed through a flow-through

cell in a spectrophotometer (Thermo-Spectronic Genesys 10UV, Thermo Electron

Corporation, Waltham, MA). This spectrophotometer is relatively inexpensive

and supports communication with a personal computer. The tubing end in the

bioreactor was bent upwards to prevent gas bubbles from entering the sample

line. (Bubbles tended to become trapped in the spectrophotometer flow cell.) The

spectrophotometer was inclined at 3 degrees to help bubbles pass through the

flow cell. A computer-linked 3-way stainless steel solenoid valve (Parker/Skinner

3133BSN1AN00N0M1S1P0, Parker, New Britain, CT) controlled whether sample or

a rinsing solution reached the spectrophotometer.

A major problem of the sampling system was accumulation of biomass on the

inner walls of the flow cell. This caused the measured absorbance to increase with

time, regardless of the true sample absorbance. Higher flows failed to shear biomass

from the wall and, in fact, exacerbated the problem. In response to this, a rinsing

16

system was added. In this system, a solenoid valve switches the sampling line feed

to a rinsing solution after a measurement is made by a spectrophotometer.

Initially, sodium dodecyl sulfate solution was used for rinsing, but it was found

that the SDS itself accumulated on the walls of the cuvette, compounding our

biomass accumulation problem with soap scum. We next tried deionized water,

but found that it yielded only a small improvement over no rinsing. We ultimately

found that a solution of 50 mg/L chlorine in water (1+1999 dilution of 10.5%

sodium hypochlorite) substantially limited accretion of biofilm within the flow cell

without causing any other problems. Figure 2–2 shows two typical runs, one rinsed

with chlorine solution and one without rinsing. Both of these sets of data were

taken from the same bioreactor so that, in the absence of biofilm accretion, they

should be identical. The rinsed data alternated between measurements of biomass

and measurements of rinse solution. As seen in the figure, the measurements

taken of the rinsing solution remained very near the baseline (zero absorbance)

throughout the experiment. If there were accumulation of biofilm inside the

instrument, we would have observed an increase in absorbance of the rinse solution.

The measurements of biomass absorbance taken in this instrument can therefore

be considered to be free of systematic errors caused by accumulation of biofilm.

In contrast, the measurements taken without rinsing are substantially higher than

the measurements with rinsing. This suggests an accumulation of biomass on

the walls of the flow cell of the unrinsed spectrophotometer and, in fact, biofilm

and particulate matter were observed in previous experiments that used unrinsed

sample paths.

When the system is switched from rinsing solution to reactor effluent it

takes approximately 8 minutes for the measured absorbance to match that of

the bioreactor. To be safe, the entire measurement cycle was set to 30 minutes:

15 minutes of rinsing, at which point a measurement is taken to check for flow

17

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12

time (hr)

abso

rban

ce (

550n

m) unrinsed biomass

measurements

biomass measurements in spectrophotometer with rinsing

measurement ofrinsing solution

Figure 2–2: Effect of rinsing sample path on biomass measurements

cell accumulation, followed by 15 minutes of reactor biomass and a reading to

determine reactor biomass concentration.

There were a few cases in which biofilm accumulation in the flow cell was not

completely eliminated. By taking measurements during the rinsing phase we have

a way to track the accumulation of biomass on the wall of the quartz flow through

cell. This value can be subtracted from the sample measurement to give the

suspended biomass contribution. This system has been shown to work acceptably

at absorbances as high as 1.5 at 550nm (1035 mg dry wt./L), and for our typical

operating range of 34–270 mg dry wt./L (Abs 550nm 0.05–0.40) it works very well.

A personal computer running Microsoft Windows 98 coordinated the valve

positions and spectrophotometer measurements using a control program written

in Microsoft Visual Basic 6.0. Communication with the spectrophotometer was

through a serial port. The program controlled the solenoid valves through a USB

control relay (J-Works JSB210-16, Granada Hills, CA) that, in turn, operated a

power relay(24V AC). The program also collected and displayed data and enabled

the user to set experiment parameters (e.g. duration, sampling period, rinsing

18

period, gas control logic). A flow sheet of the program and its source code are

posted at http://www.ees.ufl.edu/homepp/koopman/hamilton etal/.

The program communicated with the spectrophotometer using the mscomm

Visual Basic control, which provides simple access to serial ports. The computer

sent a PRINT string to the spectrophotometer whenever a measurement was

required. The spectrophotometer then returned a string containing the date, time,

wavelength and absorbance.

A challenge in writing the code was the indeterminate delay between the

request for a measurement of absorbance and the receipt of the string containing

that measurement. This was overcome using the OnComm event of the mscomm

Visual Basic control. This event is triggered on receipt of data and then signals

the program to process the received data by calling a user written method (named

OnComm) of the mscomm object. This method extracts the absorbance from the

received string and stores it in a variable. The program periodically checks whether

this variable contains a number (it is reset to null in between measurements).

The USB control relay unit is manipulated using a set of pre-defined functions

provided by the manufacturer. These functions are part of a pre-compiled dynamic

link library (.dll) file. The functions are called to execute hardware level commands

that send data across the USB cable to the control relay unit. The control

functions are accessed by including a provided module, JWorksRelayModule.bas, in

the Visual Basic project. An example of the use of the provided functions is given

in the following code fragment, which uses the IsModuleRelayOff function to check

whether the relay for nitrogen (valveNitrogen) is closed and, if so, opens the relay

using the ModuleRelayOn function. The vbNullString argument of these functions

is null if the computer is communicating with a single control relay unit and is

replaced with a string containing a serial number to specify a particular relay unit

in the case of multiple units.

19

Figure 2–3: Software interface including current status and measurement history

If IsModuleRelayOff(vbNullString, valveNitrogen) Then

Call ModuleRelayOn(vbNullString, valveNitrogen)

End If

The interface of the program (Figure 2–3) gives the status of the gas/sampling

system (absorbance, type of gas in use, type of fluid in rinse/sample line, length

of rinse phase, length of sample phase, time until next absorbance measurement,

elapsed time of experiment) in up to 3 reactors. It also shows plots of absorbance

and aeration status versus time, as well as a log of data being written to a

file. Another function of the interface is to allow real-time changes to control

parameters. Figure 2–4 shows a different screen that allows parameters for up to

three reactors to be entered.

A noteworthy feature of the software is its capability to send email automatically.

Using the MAPIMessages and MAPISession controls in Visual Basic, a function

was written that would send the current spectrophotometer measurement to a list

of email addresses. When combined with wireless messaging, this provided a means

of alerting remote users by mobile telephone to key changes in system status.

20

Figure 2–4: Experiment configuration screen

Another useful resource is the freely available software WinVNC that allows

the user to remotely monitor and administer the lab computer. This was very

useful during experiments that required manual intervention after a long and

variable bacterial growth phase.

The components of the gas and rinsing control system are listed in Table 2–1,

along with their prices. The total system cost for parts was approximately $4500.

The system is easily scaled to multiple bioreactors. For example, we are currently

using it with three bioreactors and three spectrophotometers. This required the

addition of a USB device (Keyspan USB 4-Port Serial Adapter) for additional serial

ports.

2.3 Conclusions

The automated, on-line sampling and measurement system has proven to

be very successful in our experimental work. We currently observe only 1–2 bad

measurements (absorbance spike due to bubble entrapment or trapped particulate

material) in a 14 hour experiment (28 biomass measurements). Thus, using

21

Table 2–1: Equipment list

Name Manufacturer Model # PriceJ-Works Relay J-Works JSB210-16 $205

Spectrophotometer Thermo-Spectronic Genesys 10UV $3300Windows computer less than $500

3-Way solenoid valve Parker 3133BSN1AN00N0M1S1P0 $20Tubing pump less than $500Power Relay made in shop $10

information provided in this paper, it is possible to assemble an effective continuous

sampling and measurement system that functions over extended time periods

and requires only a modest investment. In addition to providing a measurement

record, this system could also be part of an automatic process control system

with the addition of a statistical data filtering technique for discriminating bad

measurements.

CHAPTER 3A STRUCTURED MODEL FOR DENITRIFIER DIAUXIC GROWTH

3.1 Introduction

Removal of biological nitrogen is an important operation in wastewater

treatment. Biological nitrogen contributes to eutrophication of bodies of water and

has also been linked to disease in humans (Ramalho, 1983). Nitrogen removal is

commonly accomplished in a two step activated sludge process. First, aerobic

autotrophs oxidize ammonia to nitrate, then facultative anaerobes reduce

this nitrate to nitrogen gas. This process involves exposing a mixed culture

of microorganisms to an environment in which the terminal electron acceptor

alternates between oxygen and nitrate. This can result in the phenomenon of

diauxic growth.

Diauxic growth occurs when a preferred growth substrate is exhausted and

a period of little or no growth occurs. During this period necessary enzymes are

synthesized that allow growth on the less preferred substrate. Diauxic growth

was first characterized in detail by Monod (1942) in the case of changing electron

donors. Later Kodama et al. (1969) observed that diauxic growth also occurs when

switching terminal electron acceptors. More recently several investigators (Waki

et al., 1980; Liu, Zhan, Svoronos and Koopman, 1998; Liu, Svoronos and Koopman,

1998; Gouw et al., 2001; Lisbon et al., 2002) have studied the particular diauxie

occurring when bacteria switch from oxygen to nitrate, which is known to occur

with biomass from an activated sludge process used for denitrification (Liu, Zhan,

Svoronos and Koopman, 1998).

22

23

The quality of model used in process design for denitrification can have

profound economic implications. Various models for biological denitrification have

been proposed. The Activated Sludge Models 1, 2, 2d and 3 (Henze et al., 2000)

are widely used in industry but do not portray the phenomenon of diauxic lag. The

cybernetic modeling approach of Liu, Zhan, Svoronos and Koopman (1998) as well

as modifications of this approach (Liu, Svoronos and Koopman, 1998; Casasus,

2001) are able to portray diauxic lag. These models were developed, in part,

from an optimization approach that assumed that bacteria are optimal strategists

(Ramkrishna, 1983). A more mechanistic approach considers intracellular variables

as well as substrate concentrations in solution. In this paper we present a model

for diauxic growth in which the cybernetic kinetics are replaced with an approach

based on regulation of enzyme synthesis and active transport of nitrate into the

cell. There is evidence for the existence of a nitrate transport protein in several

species of nitrate respiring bacteria (Berks et al., 1994; Moreno-Vivian et al., 1999).

3.2 Model

Baumann et al. (1996) found that in an oxic / anoxic cycling system the

concentrations of intermediates (NO2-, NO, N2O) were significant when the system

was first to subjected to cycling, but subsequently declined in magnitude. Thus

the conversion of nitrate to nitrite can be inferred to be the rate limiting step. In

practice, nitrite concentrations within periodic processes for nitrogen removal are

usually quite low (well below 1 mg/L). For this reason, the ASM models consider

only nitrate as electron acceptor in denitrification (Henze et al., 2000), and this is

also the case for the model developed here.

The presented model is based on the biochemical process shown in Figure 3–1,

which shows that nitrate (NO−3 ) is actively transported into the cell by transport

protein T. Internal nitrate binds to repressor R, freeing operator O which then

allows transcription of the nitrate respiration operon resulting in synthesis of the

24

O

T

NO3

-

NO3

-R

RNO3

-

O

nar

Synthesis

DNA

Cell

membrane

Figure 3–1: Biochemical Process Model. The biochemical process being modeledis that nitrate (NO−

3 ) is actively transported into the cell by transport protein T.Internal nitrate binds to repressor R, freeing operator O. Synthesis of the nitratereductase, nar, and transport protein proceeds at a rate proportional to the amountof free operator. We assume the existence of a nitrate respiration operon, and thatthe transport protein and nitrate reductase are therefore synthesized together.

nitrate reductase (nar) and transport protein. The rate of synthesis of nar and

transport protein is proportional to the amount of free operator. Since we assume

the existence of a nitrate respiration operon, it follows that transport protein

and nitrate reductase are synthesized together. In this case the concentration of

transport protein would be proportional to the concentration of nar provided that

their decay rates are similar. The regulation scheme that underlies the presented

model follows that described by Yagil and Yagil (1971), in which enzyme synthesis

is proportional to the concentration of unbound operator. A schematic model

diagram is shown in Figure 3–2.

25

Cell

+ +

SN sni

en

XB

Figure 3–2: Model overview. Uptake of nitrate, SN ,results in an increased level ofinternal nitrate, sni. This promotes synthesis of nitrate reductase, en, which in turnpromotes synthesis of new biomass, XB, and increases the rate of nitrate uptake.

Under anoxic conditions the cell takes up nitrate from the environment. The

rate of uptake is influenced by the concentration of nitrate reductase in the cell.

The internal nitrate stimulates the synthesis of nar, which allows anoxic growth and

promotes further nitrate uptake.

The variables for biomass, XB(gdw/L), organic substrate SS (mg/L) and

extracellular nitrate SN(mg/L) are expressed as volumetric concentrations, whereas

the intracellular variables nitrate reductase activity, en(katals/gdw), and internal

nitrate concentration, sni (mg/gdw), are expressed per dry weight of biomass.

The rate expressions of this model utilize multiple Monod type expressions in a

manner analogous to the IWA series of activated sludge models Henze et al. (2000).

The processes modeled are:

1. The rate of synthesis of nitrate reductase, denoted ren, follows the kinetics

described below. These kinetics are derived from the synthesis model of Yagil

and Yagil (1971), as shown in Appendix A. The particular repression model

used is the case of a single effector molecule binding to a single repressor

molecule. According to this model, K1 is the equilibrium constant for the

binding of repressor to an inducer molecule (internal nitrate). K2 controls

the constitutive rate of enzyme synthesis and is a function of the equilibrium

constant for the binding of the repressor to the operator. An addition to the

approach of Yagil is a Monod term for organic substrate dependence. This

26

represents the dependence of the rate of active transport on the concentration

of organic substrate.

ren = aN

(1 + K1sni

K2 + K1sni

) (SS

KS,an + SS

)(3.1)

2. Oxic growth is described by the kinetic expression Henze et al. (2000)

rox = µmax,ox

(SS

KS.ox + SS

) (SO

KOH + SO

)(3.2)

3. The specific rate of uptake of nitrate is proportional to the concentration

of nitrate reductase, en. (This approach is analogous to that taken by

Shoemaker et al. (2003) for modeling diauxic lag when switching carbon

sources.) Uptake is inhibited by the presence of oxygen, SO (Berks et al.,

1994; Moreno-Vivian et al., 1999). The energy dependence of active transport

is modeled by including a Monod term for organic substrate in the rate

expression.

rsni = Vsnien

en,max

(SN

SN + KNOi

) (KOi

KOi + SO

) (SS

KS,an + SS

)(3.3)

4. Anoxic growth is proportional to internal nitrate, sni, and nitrate reductase,

en. An explicit switching function for oxygen inhibition is unnecessary

because the presence of oxygen decreases the rate of nitrate uptake and

ultimately the intracellular nitrate concentration.

ranox = µmax,anox

(en

en,max

) (sni

sni,max

) (SS

KS,an + SS

)(3.4)

5. The specific biomass decay rate (b) is assumed to be constant.

6. The specific rate of enzyme decay (bNO) is assumed to be constant.

Performing a mass balance on a batch reactor using these rates yields the following:

dXB

dt= (rox + ranox − b)XB (3.5)

27

dSS

dt=

(− 1

Yc,ox

rox −1

Yc,an

ranox

)XB (3.6)

dSN

dt= −rsniXB (3.7)

den

dt= ren −

(b + bNO +

1

XB

dXB

dt

)en (3.8)

dsni

dt= rsni − νN,anranox −

(b +

1

XB

dXB

dt

)sni (3.9)

In Equations 3.8 and 3.9 above, the last term represents dilution due to growth

and cell decay. The yield constants Yc represent the amount of biomass that is

synthesized per unit organic substrate consumed. The constant νN,an represents the

nitrate required per unit biomass synthesized during anoxic growth. The nitrogen

source for biosynthesis is assumed to be ammonia.

The theoretical maximum values of sni and en can be found by setting the time

derivatives for these variables to zero and assuming non-limiting concentrations of

organic substrate and nitrate.

sni,max =Vsni

µmax,anox

− νN,an (3.10)

en,max =aN

b + bNO

(1 + K1sni,max

K2 + K1sni,max

)(3.11)

In view of Equation 3.10 Vsni > νN,anµmax,anox.

3.3 Materials and Methods

The experiment consisted of growing Pseudomonas denitrificans (ATCC

13867) under conditions designed to induce diauxic growth. After growing under

aerobic conditions the oxygen was stripped from the reactor by bubbling nitrogen

28

Table 3–1: Synthetic media composition. pH is adjusted to 7. Trace metal solutioncontains 0.5% w/v CuSO4, FeCl3, MnCl2 and NaMoO4 · 2H2O.

Chemicals Conc. (g/L)sodium chloride 1

ammonium chloride 1magnesium sulfate heptahydrate 0.2

calcium chloride dihydrate 0.0264L-glutamic acid 6.77

potassium phosphate (monobasic) 5potassium phosphate (dibasic) 1.5

trace metals 1 drop

gas and nitrate was added as the alternative electron acceptor. This resulted in a

lag followed by a period of anoxic growth. The dissolved oxygen was assumed to be

near saturation under aerobic conditions and zero under anoxic conditions. During

the experiment the enzyme level was measured using a viologen dye colorimetric

assay.

3.3.1 Bacterial Strain and Growth Conditions

Freeze dried P. denitrificans (American Type Cultures Company 13867,

Manassas, VA) was revived in nutrient broth (Difco #0002-17-8) for 2 days in a

shaking incubator at 35◦C. The culture was stored at 4◦C on agar plates (Sigma

#T-4536) for up to 2 weeks. Prior to each experiment the plated bacteria were

cultured without nitrate in 125 mL of synthetic media (Table 3–1) in a 250 mL

Erlenmeyer flask. The culture was kept at 35◦C in a shaking incubator for 24

hours.

At the beginning of each experiment the culture was transferred to a

continuously stirred bioreactor (Multigen Model F-1000 or F-2000, New Brunswick

Scientific, New Brunswick, NJ) containing synthetic media. The initial absorbance

in the bioreactor was 0.1–0.12 (69–83 gdw/L). Air was sparged into the reactor

during the initial aerobic phase, and dissolved oxygen was assumed to be near

saturation based on previous studies (Liu, Svoronos and Koopman, 1998). In

29

runs in which the absorbance reached 0.6, the culture was diluted during the

aerobic phase in order to remain in the linear range of the absorbance-biomass

curve (0–1.0). Dilution was accomplished by rapidly pumping out a fraction of

the reactor contents and aseptically replacing it with fresh nutrient solution.

After approximately 3 hours of exponential growth the aeration was stopped and

dissolved oxygen was stripped by sparging with nitrogen gas. Nitrate solution (4000

mg/L NO−3 -N) was added to give a total concentration of 40 mg NO−

3 -N/L.

The anoxic phase was monitored until substantial exponential growth was

observed. Reactor contents were continuously withdrawn and passed through a

spectrophotometer (Thermo-Spectronic Genesys 10UV) to monitor biomass density

(absorbance at 550 nm).

Nitrite accumulation can occur under some conditions (Kornaros et al., 1996)

and result in reduction of growth rate. However, nitrite measurements from our

previous studies under similar conditions showed that the concentrations were

below 1 mg/L, which is well below the threshold for toxic inhibition of bacteria

(Weon et al., 2002).

3.3.2 Nitrate Reductase Assay

Cells were harvested by centrifugation (10,000 g for 10 minutes at 4◦C) and

washed with 20 mM Tris buffer (pH 7) before being resuspended in 2 mL of the

same buffer. Cell suspensions were used within 10 minutes of preparation. The

volume of cells sampled was varied to obtain approximately 4 gdw.

The assay method was modified from Jones et al. (1977). The reaction

was performed in a 1-cm optical path borosilicate cuvette with a Wheaton seal.

Reagents were added in an anaerobic chamber (Coy Laboratory Type ‘A’, Grass

Lake MI), leaving no head space. After adding reagents and bacteria the final

concentrations were 0.3 mM benzyl viologen and approximately 55 mg/L dry

weight of bacteria. Sufficient dithionite was included in the reaction mixture to

30

reduce the benzyl viologen to a final absorbance of 1.8 at 550 nm. Several 3 mm

glass beads were added to the cuvette to facilitate mixing. The absorbance was

monitored for 3 minutes, then the reaction was initiated by injecting nitrate into

the cuvette (through the seal) to a final concentration of 6 mM. The cuvette was

then inverted twice for mixing and the absorbance of the reaction mixture was

measured for 5 minutes. The initial rate of decolorization was determined from

these data.

We performed a study on reproducibility, and for 6 measurements the standard

deviation was 2E-10 katals.

3.3.3 Parameter Estimation

Values for the model parameters were obtained by fitting simulation results to

experimental data. The fitting was performed using the following weighted least

squares performance measure

J =1

σ2Xb

N∑i=1

(XB,meas,i −XB,pred,i)2

+1

σ2en

M∑i=1

(en,meas,i − en,pred,i)2

The weighting factors are the variances of the experimental measurements

(σ2Xb = 5 × 10−3 and σ2

en = 4.3 × 10−19 ). The performance measure J was

minimized by adjusting aN , K1, K2, Vsni, KOi, KNOi and bNO. In fitting data from

a previous experimental study (Liu, Svoronos and Koopman, 1998), for which

enzyme measurements are unavailable, the initial value of enwas also fit. The

maximum growth rate parameters µmax,ox and µmax,anox were found from the slopes

of semilog plots of absorbance versus time.

31

3.4 Results and Discussion

We first tested our model’s ability to fit previously published data from

a study by Liu, Svoronos and Koopman (1998), in which they evaluated the

effect of the aerobic period length on the subsequent diauxic lag. A pre-culture

of P. denitrificans was split between two reactors. In one, bacteria were grown

aerobically under batch conditions with 40 mg/L nitrate for 1.1 hours. Aeration

was then stopped, and nitrogen was sparged into the reactor to strip residual

oxygen from the media. The growth results for this experiment (Figure 3–3a)

show a diauxic lag of 3 hours. The parallel reactor was aerated for a longer time

(2.6 hours) and the subsequent diauxic lag was also longer (6 hours) as shown in

Figure 3–3b. The present model captures the exponential growth behavior during

the aerobic phase and anoxic phase and closely matches the diauxic lag between

these phases in each of the parallel reactors. According to the model, the effect of

aeration period length on lag is due to the dilution of nitrate respiration enzymes

by biomass growth and enzyme decay. The model parameters for these experiments

were found by simultaneously fitting both experiments and are shown in Table 3–2.

Although the quality of model fit with previous data was encouraging, a

more rigorous test of the model required simultaneous enzyme and biomass data.

Experimental results, model fits, and modelled intracellular nitrate concentrations

are shown in Figure 3–4 for one of our experiments. The initial enzyme activity in

this experiment was very low. Thus, despite a relatively short aeration period of 2

hours, a substantial diauxic lag of 11 hours was exhibited by the culture. The end

of the diauxic lag coincides with a rapid increase in enzyme activity. The modeled

values for intracellular nitrate (Figure 3–4b) show that the end of the diauxic

lag, as well as the rapid increase in enzyme activity, are coincident with a rapid

rise in intracellular nitrate concentration. The significance of intracellular nitrate

concentration to enzyme activity and, ultimately, biomass growth is explained in

32

Tab

le3–

2:Par

amet

erva

lues

obta

ined

by

fitt

ing da

tafr

omLiu

data

from

pres

ent

Par

amet

erSym

bol

Units

etal

.:Fig

.3–

3st

udy:

Fig

s.3–

4&

3–5

Max

imum

spec

ific

grow

thra

tes

oxic

condit

ions

µm

ax,o

xh−

10.

390.

56an

oxic

condit

ions

µm

ax,a

nox

h−

10.

190.

19Y

ield

son

orga

nic

subst

rate

oxic

condit

ions

YC

,ox

mg

bio

mas

s/m

gsu

bst

rate

7575

anox

icco

ndit

ions

YC

,an

mg

bio

mas

s/m

gsu

bst

rate

7575

Nit

rate

utiliza

tion

coeffi

cien

tν N

,an

mg

nit

rate

/mg

bio

mas

s0.

120.

11H

alf-sa

tura

tion

coeffi

cien

tsca

rbon

sourc

e,ox

icco

nditio

ns

KS,o

xm

gor

ganic

subst

rate

/L0.

025

0.02

5ca

rbon

sourc

e,an

oxic

condit

ions

KS,a

nm

gor

ganic

subst

rate

/L0.

70.

7ox

yge

nK

OH

mg

O2/L

0.1

0.1

nit

rate

for

nit

rate

upta

keK

NO

im

gnit

rate

/L0.

074

0.07

4ox

yge

nfo

rnit

rate

upta

keK

Oi

mg

O2/L

0.08

20.

082

Dec

ayra

tes

bio

mas

sb

h−

10

0nit

rate

reduct

ase

b NO

h−

12.

170.

035

Nit

rate

reduct

ase

synth

esis

const

ants

max

imum

spec

ific

synth

esis

rate

aN

kata

ls∗ /

(gdw

bio

mas

s·s)

2.1E

-87E

-10

repre

ssor

/in

duce

rbin

din

gK

1(m

gnit

rate

/gdw

)−1

6.9E

154.

1E5

const

itutive

expre

ssio

nle

vel

K2

4.8E

1580

00∗∗

Max

imum

spec

ific

nit

rate

upta

kera

teV

sni

mg

nit

rate

/(gd

wbio

mas

s·s)

0.03

0.06

∗T

he

acti

vity

ofnit

rate

reduct

ase

isre

por

ted

inm

oles

ben

zylvio

loge

nre

duce

dper

seco

nd.

∗∗In

Fig

ure

3–5b

the

valu

eof

one

par

amet

erw

asch

ange

dco

mpar

edto

the

pre

vio

us

two

exper

imen

talfits

.In

this

exper

imen

tK

2=

100.

33

0.07

0.16

0.26

0 2 4 6 8 10 12

Time (hr)

Ab

sorb

ance

0

5

10

15

20

25

30

35

40

45

50

mg

nit

rate

/ L

biomass (experimental) biomass (model)

nitrate (experimental) nitrate (model)

a)

0.07

0.08

0.09

0.10

0.11

0.12

0.13

0.14

0.15

0 1 2 3 4 5 6 7 8

Time (hr)

Ab

sorb

ance

0

5

10

15

20

25

30

35

40

45

50

mg

nit

rate

/ L

biomass (experimental) biomass (model)

nitrate (experimental) nitrate (model)

b)

Figure 3–3: Model fit to Liu, Svoronos and Koopman (1998). Data points showexperimental results; dashed and solid lines show model fits. Nitrate was present atthe beginning of the experiment at a concentration of 40 mg/L NO3-N.

terms of the model as follows. Enzyme activity is initially very low, consequently

the rate of nitrate transport into the cell (after the anoxic period begins) is also

low. As a result, there is only slight accumulation of intracellular nitrate for

several hours after the beginning of the anoxic period. Finally, when the nitrate

concentration begins to build up (and, hence, enzyme biosynthesis accelerates due

to derepression of the operon by nitrate), exponential biomass growth resumes. The

parameters for the fit in Figure 3–4 are shown in Table 3–2.

Data from a second experiment are shown in Figure 3–5a) along with model

fits using the same parameters. In this case, the model captured the aerobic

exponential growth, both before and after dilution, the anoxic exponential growth,

the diauxic lag, and the buildup in enzyme activity immediately preceding the end

of the diauxic lag. Good model fits were obtained for a third set of experimental

data (shown in Figure 3–5b) but required changing two parameters in order to

34

0.1

0.2

0.3

0.4

0.5

0 5 10 15

Time (hr)

Ab

sorb

ance

0.0E+00

5.0E-10

1.0E-09

1.5E-09

2.0E-09

2.5E-09

3.0E-09

3.5E-09

4.0E-09

enzy

me

acti

vity

biomass (model) biomass (experimental)enzyme (model) enzyme (experimental)

a)

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

1.4E-01

1.6E-01

1.8E-01

0 5 10 15

Time (hr)

mg

nit

rate

/ L

intracellular nitrateb)

Figure 3–4: Enzyme activity during diauxic growth. Data points show experimentalresults; dashed and solid lines show model fits. The biomass grew exponentiallyduring an initial 2.5 hour aerobic phase. Nitrate was added to a concentration of40 mg/L NO3-N at the start of the anoxic phase. This was followed by a lag of 11hours. The enzyme level was increasing during the lag phase.

35

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30

Time (hr)

Ab

sorb

ance

0.0E+00

2.0E-10

4.0E-10

6.0E-10

8.0E-10

1.0E-09

1.2E-09

1.4E-09

1.6E-09

enzy

me

acti

vity

biomass (model) biomass (experimental)

enzyme (model) enzyme (experimental)

a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 18

Time (hr)

Ab

sorb

ance

0.0E+00

5.0E-10

1.0E-09

1.5E-09

2.0E-09

2.5E-09

3.0E-09

3.5E-09

4.0E-09

enzy

me

acti

vity

)

biomass (model) biomass (experimental)

enzyme (model) enzyme (experimental)b)

Figure 3–5: Additional experiments with model fits. These experiments includeddilution during the aerobic growth phase to ensure that absorbance stayed withinthe range of linear measurement. Nitrate was added to a concentration of 40 mg/LNO3-N at the start of the anoxic phase. The sharpness of the enzyme activity curvein figure b results from the lower magnitude of parameter K2.

36

obtain a good fit (Table 3–2). The original model parameters would have predicted

a longer lag and lower reductase activity at the end of the aerobic phase.

Before the use of the proposed model is considered to improve design and

operation of nitrogen removing wastewater treatment plants, the range of

conditions under which the diauxic lag phenomenon occurs needs to be more

fully studied. For example, Kornaros and Lyberatos (1998), continuing earlier work

on the kinetics of denitrification (Kornaros et al., 1996; Kornaros and Lyberatos,

1997), failed to observe diauxic lag of P. denitrificans growing on glutamic acid

when the bacteria were transitioned from oxic to anoxic conditions. Notably,

these experiments involved high (374 mg/L) nitrate nitrogen concentrations in

pre-culture, aerobic, and anoxic growth phases, whereas in the present work nitrate

was absent during the aerobic phase. Monod (1942) remarked that diauxic lag

does not occur in all substrate transitions, particularly when the less preferred

substrate is present in high concentrations. Previous studies (Liu, Zhan, Svoronos

and Koopman, 1998; Liu, Svoronos and Koopman, 1998; Gouw et al., 2001;

Lisbon et al., 2002) have not investigated high nitrate concentrations throughout

the growth cycle, because the nitrate concentrations that occur in municipal

wastewater treatment plants are much lower.

3.5 Conclusions

We have modeled diauxic lag as resulting primarily from nitrate transport

limitation. This was achieved by combining Yagil type enzyme synthesis kinetics

with a model structure that uses intracellular nitrate as the inducer for an operon

coding for nitrate reductase and nitrate transport enzyme. This approach was

successful in fitting data on biomass from the literature, as well as data on

biomass growth and enzyme data collected as part of the present study. The

nitrate reductase synthesis dependence and coupled nitrate transport limitation

explains the dependence of lag length on aeration time, the cessation of anoxic

37

growth in the presence of oxygen, as well as the observed nar enzyme activity

profile during diauxie. Thus it may be concluded that a model based on enzyme

biosynthesis regulation can be successfully applied to portraying diauxic growth due

to switching of terminal electron acceptors.

38

Tab

le3–

3:N

omen

clat

ure

use

din

den

itri

fier

model

ing

par

amet

erunit

sdes

crip

tion

aN

kat/

(gdw

bio

mas

s·s)

max

imum

spec

ific

enzy

me

synth

esis

rate

bh−

1sp

ecifi

cbio

mas

sdec

ayra

teb N

Oh−

1sp

ecifi

cnit

rate

reduct

ase

dec

ayra

tee n

kat/

mg

bio

mas

ssp

ecifi

cnit

rate

reduct

ase

conce

ntr

atio

ne n

,max

kat/

mg

bio

mas

sm

axim

um

spec

ific

nit

rate

reduct

ase

acti

vity

K1

(mg

nitra

te/g

dw

)−1

equilib

rium

const

ant

for

repre

ssor

/in

duce

rbin

din

gK

2re

late

dto

const

ituti

veen

zym

eex

pre

ssio

nle

vel

KN

Oi

mg

NO

− 3/L

hal

f-sa

tura

tion

coeffi

cien

tfo

rin

tern

alnit

rate

KO

Hm

gO

2/L

hal

f-sa

tura

tion

coeffi

cien

tfo

rox

yge

nin

aero

bic

grow

thK

Oi

mg

O2/L

hal

f-sa

tura

tion

coeffi

cien

tfo

rox

yge

nin

hib

itio

nof

nitra

teupta

keK

S,a

nm

gor

ganic

subst

rate

/Lhal

f-sa

tura

tion

coeffi

cien

tfo

rca

rbon

sourc

ein

anox

icgr

owth

KS,o

xm

gor

ganic

subst

rate

/Lhal

f-sa

tura

tion

coeffi

cien

tfo

rca

rbon

sourc

ein

aero

bic

grow

thr a

nox

gdw

bio

mas

s/(g

dw

bio

mas

s·hr)

spec

ific

bio

mas

sgr

owth

rate

under

anox

icco

nditio

ns

r en

kat/

(gdw

bio

mas

s2·h

r)sp

ecifi

cnitra

tere

duct

ase

synth

esis

rate

r ox

gdw

bio

mas

s/(g

dw

bio

mas

s·hr)

spec

ific

bio

mas

sgr

owth

rate

under

aero

bic

condit

ions

r sni

mg

nit

rate

/(gd

wbio

mas

s2·h

r)sp

ecifi

cnit

rate

upta

kera

teS

Sm

gor

ganic

subst

rate

/Lor

ganic

subst

rate

conce

ntr

atio

nS

Nm

gnit

rate

/Lnitra

teco

nce

ntr

atio

ns n

im

g/gd

wbio

mas

sin

tern

alnit

rate

conce

ntr

atio

nS

Om

g/L

dis

solv

edox

yge

nco

nce

ntr

atio

nV

sni

mg

nit

rate

/(gd

wbio

mas

s2·h

r)m

axim

um

spec

ific

nit

rate

upta

kera

teX

Bgd

wbio

mas

s/L

bio

mas

sco

nce

ntr

atio

nY

c,an

mg

bio

mas

s/m

gor

ganic

subst

rate

yie

ldunder

anox

icco

ndit

ions

Yc,

ox

mg

bio

mas

s/m

gor

ganic

subst

rate

yie

ldunder

aero

bic

condit

ions

ν N,a

nm

gnit

rate

/mg

bio

mas

snitra

teco

nsu

med

per

unit

bio

mas

sgr

owth

σ2 X

b(g

dw

bio

mas

s/L)2

vari

ance

ofbio

mas

sm

easu

rem

ents

σ2 en

(kat

/mg

bio

mas

s)2

vari

ance

ofen

zym

em

easu

rem

ents

39

3.6 Enzyme Synthesis Expression Derivation

In the following derivation, repressor is denoted R, inducer (e.g. nitrate) is

denoted I, and operator is denoted O.

According to the model, inducer binds to free repressor.

[R] + [I]k1↼−−⇁ [RI] (3.12)

This shifts equilibrium between bound repressor, RO, and free repressor.

[R] + [O]k2↼−−⇁ [RO] (3.13)

Note that, even in the absence of inducer there will always be some unbound

operator, according the above equilibrium. Consequently the constitutive rate of

enzyme biosynthesis will be non-zero.

The total amount of repressor and operator is the sum of free and bound

species

[Rt] = [R] + [RO] + [RI] (3.14)

[Ot] = [O] + [RO] (3.15)

The rate of nitrate reductase synthesis, r, is proportional to the concentration of

free operator.

r = kr[O] (3.16)

Therefore we are interested in finding the equilibrium concentration of O. From

Equation 3.13

[O]

[RO]=

1

k2

1

[R]

By applying Equation 3.15 we obtain

[O] = [Ot]1

1 + k2[R](3.17)

40

In a similar fashion, manipulating Equation 3.12 and assuming [RO] � [Rt] yields

[R] = [Rt]1

1 + k1[I](3.18)

Substituting Equation 3.18 into Equation 3.17we obtain

[O] = [Ot]1 + k1[I]

(1 + k2[Rt]) + k1[I](3.19)

which results in the rate

r = aN1 + k1[I]

K2 + k1[I]

with

aN = kr[Ot]

K2 = (1 + k2[Rt])

The constitutive enzyme synthesis rate is obtained by setting I = 0:

r = kr[Ot]1

(1 + k2[Rt])

CHAPTER 4ESTIMATION OF NITRATE REDUCTASE ENZYME PARAMETERS IN

ACTIVATED SLUDGE USING AND EXTENDED KALMAN FILTER

4.1 Introduction

Efforts to control bioreactors are made more complicated by the fact that

intracellular variables that can dictate system behavior are frequently difficult or

impossible to measure on-line. For example, low levels of nitrate reductase when

switching from oxic to anoxic conditions can result in a diauxic lag (a period

of little or no growth). The need for improved models for denitrifier diauxic

growth is discussed in Wild et al. (1994) and Liu, Svoronos and Koopman (1998).

Established models for growth of denitrifiers (Henze et al., 2000) can be improved

upon, especially with respect to enzyme kinetics. Doing so, however, introduces

parameters that cannot be fit independently (Wild et al., 1994). Both bench and

plant scale experiments will typically have some measurements available, but

not enough to fix the enzyme related model parameters. This paper presents an

approach for estimating denitrification enzyme parameters using an extended

Kalman filter (EKF) on plant-scale operations.

A Kalman filter is a technique for determining optimal estimates of the values

of state variables (such as biomass density and pH), including unmeasured or

infrequently measured state variables (e.g. enzyme activity) and model parameters.

The estimation algorithm is based on a limited set of noisy measurements. This

technique has relevance in the area of wastewater treatment due to the impact

that unmeasured intracellular components, such as polyphosphate level and nitrate

reductase activity, have on facility performance.

41

42

Previous studies have used Kalman filters to estimate state variables and

biological model parameters. Stephanopoulos and San (1984) proposed the use

of an extended Kalman filter for estimating specific growth rates. Later work by

Ramirez (1987) and Chattaway and Stephanopoulos (1989) used the Kalman filter

along with the sequential parameter updating strategy of Ljung and Soderstrom

(1983) for both state and model parameter estimation. This strategy has been

used by Park and Ramirez (1990) for control of nutrient levels in a bioreactor. The

approach used in the present work demonstrates the use of an extended Kalman

filter with real plant data, both operational and analytical (nitrate, ammonia), in

order to estimate bioreactor contents and enzyme related model parameters in a

wastewater treatment facility.

The pure culture model of Hamilton et al. (2005), which includes two enzyme

related components, has been integrated (Lee, 2005) into the industry standard

Activated Sludge Model 1 (ASM1) (Henze et al., 2000). A process model was

developed for the Kanapaha Water Reclamation Facility (KWRF) predenitrification

process in Gainesville, Florida, in cooperation with Gainesville Regional Utilities.

To test the model, real facility data of influent and process flows were obtained for

a 20 day period beginning January 15, 2005. We demonstrate that by applying

a Kalman filter to an aeration basin compartment that contains probes for

measuring ammonia and nitrate we obtain consistent estimates of denitrification

enzyme model parameters. This technique for parameter identification allows a

semi-mechanistic model developed for pure cultures to be used in a mixed culture

population where isolation of enzyme kinetic parameters is not practical.

4.2 Kanapaha Water Reclamation Facility

The Kanapaha Water Reclamation Facility (KWRF) receives an average

daily flow of 10 million gallons per day (MGD) of wastewater, of which a constant

5 MGD is diverted to a Carrousel process(EIMCO, Salt Lake City, UT). The

43

remaining 5 MGD, including diurnal flow variation, enters a Ludzack-Ettinger

process and it is this stream that is modeled herein.

By developing a good process model, capable of reasonable predictions of

nutrient levels that are linked to intracellular processes, it becomes possible to

estimate the unmeasured process variables, as well as unknown model parameters.

The discussion of the KWRF model is presented here in two parts. First the

hydraulic model, consisting of the process fluid flows, bioreactors, settlers, etc. is

detailed. Secondly the biological model, which describes the biochemical reactions

taking place in the reactors, is discussed.

Several processes in the facility generate a significant but unmeasured amount

of low nutrient wastewater that is combined with the influent stream. This has the

effect of diluting the influent stream before it enters the scope of the process model.

By examining operator logs it was possible to calculate the raw wastewater influent

entering the process, as well as the flowrate exiting the process. The difference was

attributed to the unmeasured dilution streams. This flow information was used

to determine the actual nutrient concentrations entering the process by diluting

the standard wastewater composition given in Table 4–1. The influent stream was

assumed to contain none of the enzyme related components. A diurnal study was

performed to verify that this composition was still representative.

After collection by the sewer system, influent enters the facility headworks

as shown in Figure 4–1. The sludge return stream contains thickened sludge

from the bottom of the clarifiers. After entering the headworks, the mixture of

influent wastewater and return sludge flows to a 440,000 gallon anoxic reactor

which nominally performs nitrogen removal for the process. The stream then enters

a 2.9 million gallon aerated reactor, where nitrification takes place. The mixed

liquor suspended solids (MLSS) recirculation returns a fraction of effluent from the

aerated reactors to the anoxic reactors.

44

Table 4–1: Wastewater composition

Component Symbol Units Valuesoluble intert substrate SI g COD/m3 62particulate intert substrate XI g COD/m3 62readily degradable substrate SS g COD/m3 99slowly degradable particulate substrate XS g COD/m3 247nitrate plus nitrite nitrogen SNO g N/m3 0ammonia nitrogen SNH g N/m3 24soluble degradable organic nitrogen SND g N/m3 4particulate degradable organic nitrogen XND g N/m3 6active heterotrophic biomass XB,H g COD/m3 0active autotrophic biomass XB,A g COD/m3 0inert decay products XP g COD/m3 0alkalinity SALK g moles/m3 6dissolved oxygen SO g COD/m3 0nitrate reductase EN activity∗/(L hr) 0intracellular nitrate SNO,i g/L 0

Values taken from Potter et al. (1996)∗ enzyme activity reported as moles substrate reduced per second

Each of the two aerated reactors has 4 equally spaced vertical shaft surface

aerators that introduce oxygen into the activated sludge to facilitate aerobic

growth. The aerators vary in maximum power from 75 hp to 200 hp. Two of these

aerators are continuously variable in power output while the other two can only

be set to high, low, or off. The operators adjust the power of these 8 aerators

throughout the day according to both the current state of the facility and the

anticipated future loading. The aerators are almost always adjusted in matched

pairs, with the first aerator in the East aerated basin changed at the same time

and in the same way as the first aerator in the West basin. It is significant that

the third aerator in each of the aerated basins is frequently turned off (Figure

4–2) so that a significant amount of denitrification takes place in the aerated

basin. Aeration records were used to determine the aeration schedule for both

aeration basins. These handwritten records were a possible source of error in model

predictions as not all aeration changes are recorded.

45

Waste

FeedClarifier

AnoxicBasin(East)

AeratedBasin(West)

AnoxicBasin(West)

Sludge Return

Mixed Liquor Recirculation

AeratedBasin(East)

Headworks

Mixed Liquor Recirculation

Figure 4–1: Kanapaha physical process layout.

The MLSS recirculation is not used continually as it would be in a normal

Ludzack-Ettinger process. It is usually used only between 3:00 AM and 5:00 AM to

provide flow balancing during the hours with the lowest influent rates. The flowrate

of this recirculation stream, when active, is 20 MGD.

The effluent from both bioreactors is merged and fed to four secondary

clarifiers. The sludge return stream from the clarifiers is returned to the anoxic

basin at a rate set by the operators in order to maintain a desired sludge blanket

height. The waste stream, which is set to control the solids retention time, goes on

to further biosolids processing.

The operators record instantaneous flowrates for both trains for return

activated sludge rate, mixed liquor recirculation rate and waste rate. The total

feed is evenly split between the two trains. The flows for a 20 day period are shown

in Figure 4–3. Supplemental information about facility operation is available at

http://www.ees.ufl.edu/homepp/koopman/hamilton.etal2/.

4.2.1 Hydraulic Model

The hydraulic model is shown in Figure 4–4. The two bioreactor trains are

labeled A (the east) and B (the west) with appropriate subscripts on the stream

46

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

aera

tor

1 (

hp

)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

ae

rato

r 2 (

hp

)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

ae

rato

r 3

(hp

)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

ae

rato

r 4

(h

p)

days

Figure 4–2: Kanapaha operations – aeration strategy.

47

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2x 10

6F

eed (

F)

(L/h

r)

0 2 4 6 8 10 12 14 16 18 200

1

2

3

x 106

MLS

S R

ecircula

tion (

FM

) (L

/hr)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10x 10

5

Slu

dge R

ecycle

(F

R)

(L/h

r)

0 2 4 6 8 10 12 14 16 18 200

2000

4000

6000

8000

10000

Slu

dge W

asting (

FW

) (L

/hr)

days

Figure 4–3: Kanapaha operations – recycles and recirculations.

48

FRA

FWA+FWB

Far3Far2Far1

FA

Clarifier

AeratedBasin4 A

AeratedBasin3 A

AeratedBasin2 A

AeratedBasin1 A

AnoxicBasin A

FRB

Far3Far2Far1

FBAeratedBasin4 B

AeratedBasin3 B

AeratedBasin2 B

AeratedBasin1 B

AnoxicBasin B

FMB

FMA

*

Figure 4–4: Hydraulic model process diagram. * indicates location of nitrate andammonia probe pair used by the Kalman filter.

labels. In this figure, F represents the feed to the bioreactor process, including

dilution from unmodeled sources (such as filter backwashing). Each aerated reactor

is assumed to be well mixed and not interacting with adjacent reactors except via

listed process streams. Each aerated basin (with its four aerators) is modeled as

four sequential CSTRs. Each virtual aerated reactor is then oxygenated based on

the operation of the corresponding physical aerators. Some backmixing occurs in

the large aerated basin, and this is modeled by including internal recycles (Far)

between each of the virtual aerated reactors. A simple ideal clarifier model is used

for the secondary clarifiers. The mixed liquor from both trains is combined before

entering the secondary clarifiers.

The nitrate and ammonia probes used in the EKF are located in the west

aerated basin near the third aerator as indicated in Figure 4–4. The probes report

measurements independently every few minutes. These data were synchronized

by assuming that any measurements within 5 minutes of each other took place

at the same time. This eliminated the necessity of using multi-rate measurement

techniques in the EKF. The raw data are available at the above mentioned website.

49

4.2.2 Biological Model

The biological model captures the substrate conversion and biomass growth in

the bioreactors. The model used is a modified ASM1 in which the model presented

in Hamilton et al. (2005) for diauxic growth of denitrifiers was incorporated

(Lee, 2005). This couples the enzyme kinetics of the Hamilton model with the

sludge production and wastewater components modeled by ASM1. This model

is described as extended ASM1 mechanistic (eASM1m). In this model nitrate

reduction and uptake are governed by a nitrate respiration operon. The model is an

improvement over ASM1 in that it is capable of predicting diauxic lag of denitrifiers

and correlates nitrate reductase enzyme level to anoxic growth. It adds 2 new

components, the nitrate reductase enzyme level, EN , and the total intracellular

nitrate, SNO,i. The expressions that differ from ASM1 are presented in Table 4–2.

The model parameter values used were taken from Hamilton et al. (1992) for ASM1

except for those values listed in Table 4–3. Two calculated variables used in Table

4–2 are shown below.

SNO,i,max =

[(VSN,i

µh · ηg

)−

(1− Yh

2.86 · Yh

)]·XB,H

EN,max =

(aN

bEN + µh · ηg − bh

)·(

XB,H + K1 · SNO,i,max

K2 ·XB,H + K1 · SNO,i,max

)·XB,H

The total process model is the sum of the advection (bulk flow) terms defined

by the hydraulic model and the reaction terms provided by the biological model.

Oxygen is introduced into the aerated basin through the mass transfer term, which

is given by KLA · (SO,sat − SO). The mass transfer coefficients are derived from

values provided by the aerator manufacturer.

4.3 Kalman Filter

Kalman (1960) created a solution to the discrete data linear filtering problem

which is now widely used in motion capture and navigation systems. The extended

50

Table 4–2: eASM1m model expressions that differ from ASM1

j ProcessComponent

8 14 15 Process rate, ρj

SNO SNO,i EN ML−3T−1

2 Anoxic growthof heterotrophs

− 1−YH

2.86·YHµH

EN

EN,max

SNO,i

SNO,i,max

(SS

KS+SS

)ηgXB,H

9 Uptake ofnitrate

-1 1 VSN,iEN

EN,max

(SNO

KNO+SNO

) (KO,H

KO,H+SO

) (SS

KS+SS

)XB,H

10 Synthesisof nitratereductase

1 αN

(XB,H+K1SNO,i

K2XB,H+K1SNO,i

) (SS

KS+SS

)XB,H

11 Decay ofintracellularnitrate

-1 bSNO,i

12 Decay ofnitratereductase

-1 bENEN

Table 4–3: eASM1m parameter values that differ from values presented in Hamiltonet al. (1992)

Parameter Symbol Units ValuesMaximum specific growth rates

heterotrophic biomass µH h−1 0.39autotrophic biomass µA h−1 0.19

Yields on organic substrateheterotrophic biomass YH mg biomass/mg substrate 0.5autotrophic biomass YA mg biomass/mg substrate 0.24

Decay ratesautotrophic biomass bA h−1 0.008nitrate reductase bEN h−1 1E-6

Nitrate reductase synthesis constantsmaximum specific synthesis rate αN moles benzyl viologen/(mg biomass·hr) 1E-12repressor / inducer binding K1 (mg nitrate/gdw)−1 1.27E5∗

constitutive expression level K2 1.86E4∗

Maximum nitrate uptake rate VSN,i mg nitrate/(mg biomass·hr) 0.122

All masses given in COD units∗ Values from Hamilton et al. (2005)

51

Kalman filter is used for nonlinear systems which are linearized around the current

process state. A significant benefit of the Kalman filter is that unmeasured process

states may be estimated based on limited, noisy process measurements. In order

to use an EKF to perform parameter estimation, parameters are treated as process

variables with a rate of change = 0.

Consider the following system of augmented process state vector

x

θ

and

measurement vector z: x

θ

=

f (x, θ, t)

0

+ w(t) (4.1)

z = h (x, t) + v(t) (4.2)

The evolution of the process state, x, is a function of the state, x, time, t, and

system parameters, θ. The process rates and measurements are subject to zero

mean noise functions w and v, respectively.

The Kalman filter algorithm distinguishes between two estimates for the

system state. The a priori estimate, x−, is calculated based on model predictions

before any measurements are considered. The a posteriori estimate, x, is calculated

after including measurements for the current time step. These are distinct from

the true, unknown, system state, x. Finding the a posteriori estimate is the goal

of using an EKF. When an EKF is also used for parameter estimation, the a

posteriori estimate contains both the optimal estimate of the reactor state as well

as model parameters θ. Note that this superscript notation is also used for the

error covariance matrix Pk. It is worth mentioning that the matrix Pk can be stored

for each measurement and used to calculate error bars for the state estimates.

The diagonal elements of P are the variances of each of the corresponding state

variables.

52

The error covariance matrix is a key part of the Kalman filter and must be

integrated simultaneously with the process model. The rate of change of the error

covariance matrix P is shown in Equation 4.3.

P (t) = F (t) · P (t) + P (t) · F T (t) + R (4.3)

Here F is the Jacobian of f with respect to the augmented state vector and R

is the covariance of w. The a posteriori error covariance matrix P is calculated by

Equation 4.4.

Pk =[I −Kk ·Hk

(x−

k

)]P−

k (4.4)

The heart of the Kalman filter is the gain matrix, K. The gain formula

is derived by minimizing the expected error between the a posteriori state

estimate and the actual state. The final form of the gain, derived for the

process/measurement system in Equations 4.1–4.2 is shown in Equation 4.5,

where Q is the covariance of v and H is the Jacobian of h with respect to the

augmented state vector.

Kk = P−k HT

k

(x−

k

) [Hk

(x−

k

)P−

k HTk

(x−

k

)+ Q

]−1(4.5)

The a posteriori state estimate is the weighted sum of the a priori estimate

and an error term that represents the difference between the actual measurement,

zk, and the measurement that would be expected if the model was perfectly

accurate, hk(x−k ). The weighting factor is the Kalman gain matrix, Kk.

xk = x−k + Kk ·

[zk − hk(x

−k )

](4.6)

The filtering algorithm is shown below for each time step k.

53

1. Simultaneously integrate Equation 4.1 with the initial condition of xk−1 and

w set to zero, and Equation 4.3 with initial condition of Pk−1. This yields the

a priori estimate of the system state, x−k , as well as the a priori estimate of

error covariance, P−k .

2. Calculate the gain, Kk , using x−k and P−

k in Equation 4.5.

3. Using Equation 4.6 calculate xk , the final estimate of the system state for

this time step.

4. Calculate Pk from Equation 4.4 for use in the next time step.

See Gelb (1974) for a more complete discussion and derivation of the Kalman

filter.

In order to estimate state variables and specific growth rates Stephanopoulos

and San (1984) proposed the use of an extended Kalman filter. Later work by

Ramirez (1987) and Chattaway and Stephanopoulos (1989) used the Kalman filter

along with the sequential parameter updating strategy of Ljung and Soderstrom

(1983) for both state and model parameter estimation. This strategy has been used

by Park and Ramirez (1990) for control of nutrient levels in a bioreactor.

The values shown in Table 4–3 for the parameters controlling enzyme kinetics

(K1, K2, aN and ben) were the initial values. They were continually estimated by

the EKF as probe measurements are processed.

The Kalman filter algorithm was run on aerated basin 3B, a subset of the full

system model. The rest of the plant model was integrated without the Kalman

filter using the generated parameter estimates.

4.4 Results and Discussion

Estimates of nitrate reductase parameter aN from the on-line data are

presented in Figure 4–5. This is the first time that estimates of an intracellular

component in activated sludge have been made by processing of real plant data.

Nitrate and ammonia measurements are compared to EKF output for a 20 day

54

period. The filter output is a good match for the probe data without appearing to

be over-tuned.

As the filter runs, it continually updates the estimates for the models

parameters K1, K2, aN and ben. Essentially no change was made to the parameters

with the exception of aN , shown in Figure 4–5. This result indicates that varying

only aN could account for the observed change in nitrate and ammonia. As

discussed in Hamilton et al. (2005), K1 and K2 are based on the Lac operon

regulation model, and most of their effect on behavior stems from their relative

magnitudes.

The filter parameters which require turning are the values for the covariance

matrices R and Q in Equations 4.3 and 4.5. These give the algorithm information

about the level of variation that can be expected in the process, which heavily

influences decisions about how to appropriately weight measurements versus model

predictions. The tuning was done by inspection, based on plots of filter output

combined with the error covariance matrix derived error bars. After running the

algorithm and observing a component that seemed to have error bars that were too

tight based on process knowledge, the corresponding component of R was increased.

The variation observed in aN is on the timescale of about 1 day. This

parameter is the maximum specific nitrate reductase synthesis rate, and has

been found to be very important for tuning the unfiltered model. One possible

explanation for the variation observed in this parameter is changes in reactor

temperature which would be expected to have a significant effect on biomass

growth. The operators’ temperature logs are intermittent, making it difficult to

correlate this effect. An unmodelled change in operating conditions would be

reflected by the filter in the form of changes to the estimated parameters.

55

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5x 10

!11

aN

(m

ol B

.V)/

(mg

bio

ma

ss h

r)

days

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5

2

Re

acto

r 3

B (

mg

NH

3 /

L)

0 2 4 6 8 10 12 14 16 180

2

4

6

8

days

Re

acto

r 3

B (

mg

NO

3 /

L)

Probe

Simulation

Figure 4–5: Estimates of the maximum specific nitrate reductase synthesis rate(aN) as well as a comparison of nitrate and ammonia probe meaurements to filterresults. The other three estimated parameters (K1, K2, and ben) did not vary bymore than 0.5% over the period shown. Units of activity are moles benzyl viologenconsumed per second.

56

4.5 Acknowledgments

We would like to thank Gainesville Regional Utilities for their cooperation in

this study, and for providing access to the KWRF. We would also like to thank the

employees of the KWRF for their assistance with data collection and for sharing

their process insights.

CHAPTER 5FUTURE WORK

5.1 Distributed State Modeling

Distributed state modeling is a type of model structure in which model

components are tracked as evolving distributions rather than single values. The

benefit of this approach is particularly significant for models with nonlinear

behavior based on concentrations of intracellular components. In the case of diauxic

lag, microorganisms experience a period of little or no growth during which they

synthesize necessary enzymes for growth under new conditions. In the case where a

subset of the population is growing exponentially while the majority is experiencing

a lag phase, a traditional model will generate incorrect predictions of behavior. A

traditional model functions as a ”single cell,” in which all model components are

averaged out over the entire population. When a small fraction of the population

has the majority of a key component, the model prediction will be lag for the entire

population. A similar argument could be made for phosphorus metabolism.

5.2 Extended Kalman Filter

The next step for the adaptive extended Kalman filter presented in Chapter

4 would be to incorporate enzyme activity estimates from the Kanapaha facility

into the algorithm. These assays would not be available with the frequency of

ion probe data, which would necessitate the use of a multi-rate EKF technique.

This is a common situation with bioprocess measurement. It is frequently the case

that some measurements are available on-line and are frequently updated while

others are more sporadic / irregular and perhaps delayed. This is particularly

true in the case of bioprocesses, in which you may have constant measurement of

57

58

variables like optical density, pH, temperature and GC analysis of effluent gases,

but measurement of some substrate concentrations may require lengthy chemical

tests, in addition to assays for intracellular states of the components.

A multirate EKF is relevant to the present work in that the availability of

enzyme assays is expected to greatly increase the effectiveness of the parameter

estimation algorithm. These assays would only be available infrequently, and a

multirate approach would be required to integrate them into the probe data.

The approach of Gudi et al. (1995) was adapted from the multi-rate strategy of

Glasson (1980, 1983). Gudi et al. (1995) also retained past measurements in their

output equations in an attempt to increase system observability. In this approach,

the EKF is defined as in Equations 4.1–4.6 above.

This approach defines two sampling periods. The major sampling period

is those times at which both rapid and infrequent measurements overlap and

the minor period is the sampling period of the rapid measurement alone. The

measurement vector is of higher dimensionality at the major sampling period so

redefinitions of the EKF measurement equations are required.

zmajor = hmajor(x(tmajor)) + vmajor (5.1)

zminor = hminor(x(tminor)) + vminor (5.2)

New measurement Jacobians must be defined for the linearized measurement

mode.

Hmajor =∂hmajor(x(t), t)

∂xj(t)(5.3)

Hminor =∂hminor(x(t), t)

∂xj(t)(5.4)

59

This begins the derivation of what is functionally two separate extended

Kalman filters, one of which is invoked at the major sampling period and one at

the minor.

Kmajor = P−majorH

Tmajor

[HmajorP

−majorH

Tmajor + Rmajor

]−1(5.5)

Kminor = P−minorH

Tminor

[HminorP

−minorH

Tminor + Rminor

]−1(5.6)

5.3 Enzyme Activity Based Dynamic Optimization

An ultimate goal of the presented EKF and model is a dynamic optimization

scheme. As has been discussed, the industry standard models can not predict

diauxic lag, and an optimization scheme based on such a model will be suboptimal.

Conceptually, one would seek to optimize the nitrogen removal rate subject to the

constraint that the nitrate respiration enzyme levels should not fall low enough

that there is a significant diauxic lag. If aeration is cycled too slowly the enzyme

levels will drop while rapid cycling may not provide adequate biomass generation or

carbon removal.

One algorithm for dynamic plant optimization is to define a goal function

based on the desired output characteristics of the process which is a function of

all controllable process inputs. The goal function would be evaluated by using

the process model to predict the plant performance over a large (relative to

measurement frequency) time window. An optimization algorithm would be applied

to that goal function and the resulting optimal operating conditions would be

applied to the physical process. As measurements are made, the EKF continually

updates the estimated process state, which serves as the initial condition for the

goal function’s prediction. The EKF is essential here because of the importance of

60

denitrification enzymes in process performance and the low frequency of enzyme

activity measurement.

APPENDIX AEXTENDED KALMAN FILTER FOR DENITRIFICATION ENZYME

PARAMETER ESTIMATION AT THE KANAPAHA WATER RECLAMATIONFACILITY

SUPPLEMENTAL MATERIAL

The advection (bulk flow) terms are described by a hydraulic model, the

reaction terms by the biochemical model, and the mass transfer term describes the

action of the aerators.

dX

dt= advection + reaction + mass transfer (A.1)

The final hydraulic model is shown in Figure A–1. The two bioreactor trains

are labeled A and B with appropriate subscripts on the stream labels. In this

figure F represents the feed to the bioreactor process, including dilution from

unmodeled sources (such as filter washing). Each aerobic reactor is assumed to

be well mixed and not interacting with adjacent reactors except via listed process

streams. Each aerobic basin (with its four aerators) is modeled as four sequential

CSTRs. Each virtual aerobic reactor is then oxygenated based on the operation of

the corresponding physical aerators. Some backmixing occurs in the large aerobic

basin and this is modeled by including internal recycles (Far) between each of the

virtual aerobic reactors. A simple ideal clarifier model is used for the secondary

settling tank. This models the solids enrichment in the lower stream as perfect

separation. The sludge from both trains is combined before entering the clarifier.

The differential equations governing the hydraulic model (advection terms) for

train A are shown in Equations A.2–A.6 below. In these expressions, Xy denotes

component X in reactor y. Each expression is then a vector, where X is replaced

61

62

Table A–1: Aerobic basin surface aerators. Aerators are numbered sequentally inflow direction. The first aerator encountered in the East aeration basin is number1, the last number 4. In the West aeration basin the numbering starts with 5. Thisis the numbering scheme used by the Kanapaha operators. Control type refers tothe granularity of power output settings available. High/low aerators can be set to100%, 50% or 0% power. Continuously variable (C.V.) aerators can be set to anypercentage output.

aerator control maximumnumber type horsepower

1,5 high/low 1252,6 C.V. 2003,7 high/low 1254,8 C.V. 75

FRA

FWA+FWB

Far3Far2Far1

FA

Clarifier

AerobicBasin4 A

AerobicBasin3 A

AerobicBasin2 A

AerobicBasin1 A

AnoxicBasin A

FRB

Far3Far2Far1

FBAerobicBasin4 B

AerobicBasin3 B

AerobicBasin2 B

AerobicBasin1 B

AnoxicBasin B

FMB

FMA

*

*

Figure A–1: Hydraulic model PID. The process is modeled as two trains in parallel.Each aerobic basin (with its four aerators) is modeled as 4 sequential CSTRs.Each virtual aerobic reactor then is then oxygenated based on the operation of thecorresponding physical aerators. To account for backmixing within the large aerobicbasin there are recycle streams between the virtual aerobic reactors. An ideal pointclarifier model is used for the secondary settling tank. The sludge return (FR) foreach train is mixed before being reintroduced into the anoxic basin.

63

by each model component. For example, Xab4a is component X in aerobic basin

4 of train A. The labels used for streams are consistent with Figure A–1, and the

reactor volumes are given by Table A–2.

Equation A.2 gives the anoxic basin advection expression for particulate

material. For soluble material the clarifier separation efficiency, λ, equals 1 since

the clarifier does not segregate soluble material. The term Fsum,A is defined for

brevity as FA + FRA + FMA.

F

Van

·Xinfluent +FR

Van

· λ ·Xmixed +FM

Van

·Xab4 −Fsum,A

Van

·Xan (A.2)

The term Xmixed represents the secondary settling tank combined return from

both trains and is calculated by:

Xmixed =(FA + FRA) ·Xab4a + (FB + FRB) ·Xab4b

FA + FRA + FB + FRB

The advection terms for the 4 virtual aerobic reactors are defined in Equations

A.3 – A.6.

Fsum,A

Vaer1

·Xana +Far1

Vaer1

·Xab2a −Fsum,A + Far1

Vaer1

·Xab1a (A.3)

Fsum,A + Far1

Vaer2

·Xab1a +Far2

Vaer2

·Xab3a −Far1

Vaer2

·Xab2a −Fsum,A + Far2

Vaer2

·Xab2a (A.4)

Fsum,A + Far2

Vaer3

·Xab2a +Far3

Vaer3

·Xab4a −Far2

Vaer3

·Xab3a −Fsum,A + Far3

Vaer3

·Xab3a (A.5)

Fsum,A + Far3

Vaer4

·Xab3a −Far3

Vaer4

·Xab4a −Fsum,A

Vaer4

·Xab4a (A.6)

64

Table A–2: Hydraulic model parameters

Parameter Symbol Units Valuevirtual aerobic tank volume Vaer L 5488250anoxic tank volume Van L 3330800aerator efficiency

aerator 1 KLA1 hr−1 / hp 0.0136aerator 2 KLA2 hr−1 / hp 0.0176aerator 3 KLA3 hr−1 / hp 0.0136aerator 4 KLA4 hr−1 / hp 0.0136

dissolved oxygen at saturation SO,sat mg O2 / L 8.8aerobic basin internal recycle ratio abr 1.5

The mass transfer component of the overall balance equation describes the

rate at which the surface aerators are able to introduce dissolved oxygen into the

aerobic basin. The mass transfer for oxygen is given below, where i represents

each of the 4 aerobic basins. The value for KLA2 (a recent upgrade) is derived

from performance per horsepower specifications from the manufacturer and the

performance for the remaining aerators was found by fitting ASM1 to KWRF

nutrient data.

dSOi

dt= KLAi · (SO,sat − SOi) (A.7)

The full biochemical model is presented in Table A–3 with corresponding

model parameters presented in Table A–4. The combination of the discussed

hydraulic model and the tabulated biochemical model is used to simulate overall

process performance.

Historical flow data was collected from this facility and used to generate a

set of representative diurnal flow patterns. One significant difficulty in modeling

the KWRF process is that the operator logs record flow rates every hour. This

sampling frequency can miss significant variation on a shorter timescale or

over-emphasize variation that occurs at the time that the flowrate is recorded.

Due to this unreliability the influent flowrate used in the model is calculated from

65

Tab

leA

–3:

eASM

1mM

odel

Com

ponent

i→

12

34

56

78

910

1112

1314

15P

roce

ssR

ate,

ρj,M

L−

3T−

1

Pro

cess

j↓

Si

SS

Xi

XS

XB

,HX

B,A

XP

SO

SN

OS

NH

SN

DX

ND

SA

LK

SN

O,i

EN

1A

erob

icgr

owth

ofhet

erot

rophs

−1 YH

1−

1−

YH

YH

−i X

B−

i XB

14

µH

( SS

KS+

SS

)(S

O

KO

,H+

SO

) XB

,H

2A

nox

icgr

owth

ofhet

rotr

ophs

−1 YH

1−

i XB

1−

YH

14·2

.86·Y

H−

i XB

14

−1−

YH

2.8

6·Y

Hµ

HE

N

EN

,ma

x

SN

O,i

SN

O,i

,ma

x

( SS

KS+

SS

) η gX

B,H

3A

erob

icgr

owth

ofau

totr

ophs

1−

4.5

7−

YA

YA

1 YA

−i X

B−

1 YA

−i X

B

14

−1

7Y

A

µA

( SN

H

KN

H+

SN

H

)(S

O

KO

,A+

SO

) XB

,A

4D

ecay

ofhet

erot

rophs

−f P

−1

f Pi X

B−

f Pi X

Bb H

XB

,H

5D

ecay

ofau

totr

ophs

−f P

−1

f Pi X

B−

f Pi X

Bb A

XB

,A

6A

mm

onifi

cati

onof

solu

ble

orga

nic

nit

roge

n

1−

11 14

kaS

NDX

B,A

7H

ydro

lysi

sof

entr

apped

orga

nic

s1

−1

kh

XS

/X

B,H

KX

+(X

S/X

B,H

)

[(S

O

KO

,H+

SO

) +η h

( KO

H

KO

,H+

SO

)(S

NO

KN

O+

SN

O

)] XB

,H

8H

ydro

lysi

sof

entr

apped

orga

nic

nit

roge

n

1−

1ρ

7(X

ND/X

S)

9U

pta

keof

nit

rate

−1

1V

SN

,iE

N

EN

,ma

x

( SN

O

KN

O+

SN

O

)(K

O,H

KO

,H+

SO

)(S

S

KS+

SS

) XB

,H

10Synth

esis

ofnit

rate

reduct

ase

1α

N

( XB

,H+

K1S

NO

,i

K2X

B,H

+K

1S

NO

,i

)(S

S

KS+

SS

) XB

,H

11D

ecay

ofin

trac

ellu

lar

nit

rate

−1

bSN

O,i

12D

ecay

ofnit

rate

reduct

ase

−1

b EN

EN

Obse

rved

conve

rsio

nra

tes,

ML−

3T−

1r i

=∑ j

ν ijρ

j

66

Table A–4: eASM1m parameter values

Parameter Symbol Units ValuesMaximum specific growth rates

heterotrophic biomass µH h−1 0.39∗∗

autotrophic biomass µA h−1 0.19∗∗

Yields on organic substrateheterotrophic biomass YH mg biomass/mg substrate 0.5∗∗

autotrophic biomass YA mg biomass/mg substrate 0.24∗∗

Half-saturation coefficientscarbon source KS mg organic substrate/L 20oxygen, heterotrophic biomass KO,H mg O2/L 0.1oxygen, autotrophic biomass KO,A mg O2/L 0.4nitrate KNH mg N/L 1.0slowly degradable substrate KX mg/mg biomass 0.3

Decay ratesheterotrophic biomass bH h−1 0.026autotrophic biomass bA h−1 0.008∗∗

nitrate reductase bEN h−1 1E-6∗∗

Nitrate reductase synthesis constantsmaximum specific synthesis rate αN katals/(mg biomass·hr) 1E-12∗∗

repressor / inducer binding K1 (mg nitrate/gdw)−1 1.27E5∗

constitutive expression level K2 1.86E4∗

Maximum ratesnitrate uptake VSN,i mg nitrate/(mg biomass·hr) 0.122∗∗

hydrolysis kh mg/(mg biomass ·hr) 0.125ammonification ka L/(mg biomass ·hr) 0.0033

Anoxic correction for µH ηg 0.85Anoxic correction for hydrolysis ηh 0.4Mass nitrogen per unit biomass iXB 0.086Mass nitrogen per unit biomass products iXP 0.06Fraction of biomass decaying to particulates fP 0.08

All masses given in COD unitsValues taken from Koopman et al. (1989) except where marked∗ Values from Hamilton et al. (2005)∗∗ Values from present study

67

0 5 10 15 20 254

5

6

7

8

9

10

11

12

13

time (hr)

Flo

w R

ate

(M

GD

)

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Figure A–2: Diurnal flow patterns. Figures are based on operator logs for theperiod January through August 2004.

the total influent flow for each day, which is accurately recorded. The hourly flow

rates are computed by scaling the total daily flow using a diurnal pattern shown in

Figure A–2. This pattern was calculated by averaging the flowrate patterns (based

on hourly operator logs) for each day of the week for the months January through

August 2004. The resulting curves are similar to results from a previous study

(Koopman et al., 1989) in which a single flow pattern was used. This accurately

captures the total flow and most of the diurnal variation, but misses flow spikes and

changes in flow due to holidays. Flow spikes can be significant during rain events

and this can negatively impact process model performance.

Nitrate and ammonia data were collected using the facility’s nitrate and

ammonia probes located in the aerobic reactors. Measurements of both species were

not made simultaneously, but were separated in times by a variable amount (1–10

minutes). An EKF approach is made less reliable by incorporating multi-rate

techniques unless necessary, so these probe measurements were treated as

simultaneous for those times where nitrate and ammonia were measured within

68

5 minutes of each other. This is reasonable because the time scale of changes in

these species is a much larger fraction of an hour.

APPENDIX BAUTOMATION PROGRAM SOURCE CODE

B.1 Code for DLECModule.bas

Attribute VB_Name = "DLECModule"Option ExplicitPublic fMainForm As frmMainPublic exp As DLECExperimentGlobal EmailAddress(7) As String’the email addresses to be used’whether to use the corresponding email addressGlobal UseEmail(7) As BooleanPublic AllReactorSame As Boolean

’constants used to make these functions more readableGlobal Const water As Integer = 0Global Const biomass As Integer = 1Global Const nogas As Integer = 2Global Const nitrogen As Integer = 1Global Const oxygen As Integer = 0’local variable(s) to hold property value(s)

’set this false to be able to actually take dataGlobal RKHFAKESAMPLE As BooleanGlobal Chart1Data() As DoubleGlobal Chart2Data() As DoubleGlobal Chart3Data() As DoubleGlobal LastMeasurementWasRinsing As Boolean’whether to save data to a file as we run the experimentPublic LogToFile As Boolean

’***************************************************Sub Main()

Dim i As Integer

EmailAddress(0) = "rkhamilt@ufl.edu"EmailAddress(1) = "acasasus@ufl.edu"EmailAddress(2) = ""EmailAddress(3) = ""

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UseEmail(0) = True

’Set the relays that identify the particular relays we useRelayO2(0) = 1RelayO2(1) = 3RelayO2(2) = 7RelayInert(0) = 2RelayInert(1) = 4RelayInert(2) = 6RelayEffluent(0) = 8RelayEffluent(1) = 9RelayEffluent(2) = 10

’turn off all relays just so we are sure of its stateModuleAllRelayOff vbNullString

’set up the experiment with basic infoSet exp = DLECExperimentConstructor(3)

Set fMainForm = New frmMain

LogToFile = TrueAllReactorSame = True

’kludgesFor i = 0 To exp.NumberReactors - 1

With exp.Reactor(i)’Set Communication parametersSet .MSCommObj = frmMain.SpecCommDevice(i).MSCommObj.CommPort = 4 + i.MSCommObj.Settings = "9600,N,8,1"If .MSCommObj.PortOpen Then .MSCommObj.PortOpen = False

End WithNext i’spec 1 has a slight change in settingsexp.Reactor(0).MSCommObj.Settings = "19200,N,8,1"

RKHFAKESAMPLE = FalsefrmMain.mnuSim.Checked = RKHFAKESAMPLE

fMainForm.Show

End Sub

’*******************************************Public Function DLECExperimentConstructor(ByVal NumReactors As Integer)

As DLECExperiment

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’this is the constructor for the experiment object’note that for each reactor, it must have it’s comm device’set to point to a form elementDim Output As New DLECExperimentSet Output = New DLECExperimentDim i As Integer

’set up this experimentWith Output

.NumberReactors = NumReactors ’Initialize Settings for each reactor

.CurrentRunTime = 0

.StartTime = Now

.EmailHome = False

.Running = False

.TotalDuration = 2880

.NewData = False

.EmailAbs = 0.2

.EmailHome = False

’set up each reactorFor i = 0 To .NumberReactors - 1

With .Reactor(i).InUse = False.MinutesPerCycle = 30.BiomassFlushingTime = 10 ’minutes.AbsorbanceGasTrigger = 0.2.Absorbance = 0.BubbleScreen = True.BubbleScreenType = 1.gas = nogas.fluid = biomass.LastGoodSample = 0.MostRecentSample = 0.SwitchGases = True.TimeOfLastSample = Now.UseRinsing = True.MinutesBetweenSamples = 15.valveEffluent = RelayEffluent(i).valveNitrogen = RelayInert(i).valveOxygen = RelayO2(i)

End WithNext i

End With

Output.Reactor(0).InUse = True

Set DLECExperimentConstructor = Output

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End Function

’************************************************Public Function DLECChemostatConstructor(ByRef ParentObj As DLECExperiment)

As DLECChemostatDim newMSCommObj As MSCommDim Output As DLECChemostatSet Output = New DLECChemostat’Set Output.MSCommObj = frmMain.SpecCommDevice(0)Output.Ready = FalseSet Output.Parent = ParentObjSet DLECChemostatConstructor = OutputEnd Function

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B.2 Code for DLECEmailModule.bas

Attribute VB_Name = "EmailModule"Option Explicit’ Sends an email to the appropriate person(s).’ SendTo = List of email addresses separated by a semicolon. Example:’ sm@xyz.com; steve@work.com; jane@home.com’ Subject = Text that summarizes what the email is about’ EmailText = Body of text that is the email’ AttachmentPath = Directory in which the attachment resides’ Attachment = File to send with the email

Private Sub SendEmailMAPI(SendTo As String, Subject As String, _EmailText As String, Optional AttachmentPath As String, _Optional Attachment As String)Const constRoutine As String = "SendEmailMAPI"

Dim intStart As IntegerDim strSendTo As StringDim intEnd As IntegerDim i As Integer

If frmMain.MAPISession.SessionID = 0 ThenfrmMain.MAPISession.SignOn

End If

If SendTo = "" Then Exit Sub

With frmMain.MAPIMessages.SessionID = frmMain.MAPISession.SessionID.Compose

’Make sure that the SendTo always has a trailing semi-colon (makes it’ easier below)’Strip out any spaces between names for consistencyFor i = 1 To Len(SendTo)

If Mid$(SendTo, i, 1) <> " " ThenstrSendTo = strSendTo & Mid$(SendTo, i, 1)

End IfNext i

SendTo = strSendToIf Right$(SendTo, 1) <> ";" Then

SendTo = SendTo & ";"End If

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’Format each recipient, each are separated by a semi-colon, like this:’ steve.miller@aol.com;sm@psc.com; sm@teletech.com;intEnd = InStr(1, SendTo, ";").RecipAddress = Mid$(SendTo, 1, intEnd - 1).ResolveName

intStart = intEnd + 1Do

intEnd = InStr(intStart, SendTo, ";")If intEnd = 0 Then

Exit DoElse

.RecipIndex = .RecipIndex + 1

.RecipAddress = Mid$(SendTo, intStart, intEnd - intStart)

.ResolveNameEnd IfintStart = intEnd + 1

Loop

.MsgSubject = Subject

.MsgNoteText = EmailTextIf Left$(Attachment, 1) = "¨ Then

Attachment = Mid$(Attachment, 2, Len(Attachment))End If

If Attachment <> "" ThenIf Right$(AttachmentPath, 1) = "¨ Then

.AttachmentPathName = AttachmentPath & AttachmentElse

.AttachmentPathName = AttachmentPath & "¨ & AttachmentEnd If.AttachmentName = Attachment

End If.Send False

End With

End Sub

Public Sub Email_Report()Dim SendTo As StringDim Subject As StringDim EmailText As StringDim i As Integer

’put together list of addressees

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SendTo = ""For i = 0 To 6

If UseEmail(i) Then SendTo = EmailAddress(i) + "; "Next i

’put together subject lineSubject = Format(Now, "Medium Time")

’put together email bodyEmailText = ""For i = 0 To exp.NumberReactors - 1

With exp.Reactor(0)If .InUse Then

EmailText = EmailText + "rctr" + CStr(i + 1) + ": " + _Format(.Absorbance, "0.000") + " "

If .gas = oxygen Then EmailText = EmailText + "O2" + vbCrIf .gas = nitrogen Then EmailText = EmailText + "N2" + vbCrIf .gas = nogas Then EmailText = EmailText + "NONE" + vbCr

End IfEnd With

Next i

’sendSendEmailMAPI SendTo, Subject, EmailText

End Sub

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B.3 Code for DLECPhoneCallModule.bas

Attribute VB_Name = "PhoneCallModule"Option ExplicitPublic Function Callthis(num As String)

Dim Output, dummy, FromModem$Modem.CommPort = 1Modem.Settings = "9600,N,8,1"Modem.PortOpen = TrueModem.Output = "ATDT " + num + vbCr

’ Wait for "OK" to come back from the modem.Do

dummy = DoEvents()’ If there is data in the buffer, then read it.If Modem.InBufferCount Then

FromModem$ = FromModem$ + Modem.Input’ Check for "OK".If InStr(FromModem$, "OK") Then’the phone was anseredExit Do

End IfEnd If

’ Did the user choose Cancel?’If CancelFlag Then’ CancelFlag = False’ Exit Do’End If

Loop

’ Disconnect the modem.Modem.Output = "ATH" + vbCr

’ Close the port.Modem.PortOpen = False

End Function

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B.4 Code for DLECLogFileModule.bas

Attribute VB_Name = "LogFileModule"Option ExplicitDim hLogFile As Integer ’ Handle of open log file.

Public Sub OpenLogSub()Dim replace, temp As String, Ret As IntegerOn Error Resume Next

Dim dlgOpenLog As CommonDialogSet dlgOpenLog = frmMain.OpenLog

dlgOpenLog.Flags = cdlOFNHideReadOnly Or cdlOFNExplorerdlgOpenLog.CancelError = True

’ Get the log filename from the user.dlgOpenLog.DialogTitle = "Open Data Log File"dlgOpenLog.Filter = "Log Files (*.TXT)|*.txt|All Files (*.*)|*.*"

DodlgOpenLog.FileName = ""dlgOpenLog.ShowOpenIf Err = cdlCancel Then Exit Subtemp = dlgOpenLog.FileName

’ If the file already exists, ask if the user wants to’overwrite the file or add to it.Ret = Len(Dir$(temp))If Err Then

MsgBox Error$, 48Exit Sub

End IfIf Ret Then

replace = MsgBox("Replace existing file - " + temp + "?", 35)Else

replace = 0End If

Loop While replace = 2

’ User clicked the Yes button, so delete the file.If replace = 6 Then

Kill tempIf Err Then

MsgBox Error$, 48Exit Sub

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End IfEnd If

’ Open the log file.hLogFile = FreeFileOpen temp For Binary Access Write As hLogFileIf Err Then

MsgBox Error$, 48Close hLogFilehLogFile = 0Exit Sub

Else’ Go to the end of the file so that new data can be appended.Seek hLogFile, LOF(hLogFile) + 1

End If

End Sub

’*********************************Public Sub CloseLog()

’ Close the log file.Close hLogFilehLogFile = 0

End Sub

’ This procedure adds data to the Term control’s Text property.’ It also filters control characters, such as BACKSPACE,’ carriage return, and line feeds, and writes data to’ an open log file.’ BACKSPACE characters delete the character to the left,’ either in the Text property, or the passed string.’ Line feed characters are appended to all carriage’ returns. The size of the Term control’s Text’ property is also monitored so that it never’ exceeds MAXTERMSIZE characters.Public Static Sub ShowData(Term As Control, ByRef Data As String)

Const MAXTERMSIZE = 16000Dim TermSize As Long, i

’ Make sure the existing text doesn’t get too large.TermSize = Len(Term.Text)If TermSize > MAXTERMSIZE Then

Term.Text = Mid$(Term.Text, 4097)TermSize = Len(Term.Text)

End If

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’ Point to the end of Term’s data.Term.SelStart = TermSize

’ Filter/handle BACKSPACE characters.Do

i = InStr(Data, Chr$(8))If i Then

If i = 1 ThenTerm.SelStart = TermSize - 1Term.SelLength = 1Data = Mid$(Data, i + 1)

ElseData = Left$(Data, i - 2) & Mid$(Data, i + 1)

End IfEnd If

Loop While i

’ Eliminate line feeds.Do

i = InStr(Data, Chr$(10))If i Then

Data = Left$(Data, i - 1) & Mid$(Data, i + 1)End If

Loop While i

’ Make sure all carriage returns have a line feed.i = 1Do

i = InStr(i, Data, Chr$(13))If i Then

Data = Left$(Data, i) & Chr$(10) & Mid$(Data, i + 1)i = i + 1

End IfLoop While i

’ Add the filtered data to the SelText property.Term.SelText = Data

’ Log data to file if requested.If hLogFile Then

i = 2Do

Err = 0Put hLogFile, , DataIf Err Then

i = MsgBox(Error$, 21)If i = 2 Then

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CloseLogEnd If

End IfLoop While i <> 2

End IfTerm.SelStart = Len(Term.Text)

End Sub

APPENDIX CDENITRIFIER DIAUXIC GROWTH MODEL SOURCE CODE

C.1 Code for Run061703.m

Run the model with parameters from 06-17-03load data061703b;load exp061703;load mfit061703;load mfitR;%e.EndPhaseAtAbs = [1 1];%e.init sni = xfinal(end)e.PhaseSolutionSn = [50 4000];e.PhaseDilutionRatio = [0 100/(4000-100)];

e = RunModel(mfit,e);

%CALCULATE ERROR%by linear interpolation, find a table of values for times%corresponding to the times at which we have measurementsInterpExpEnzyme = interp1(e.times,e.fe n,data.e n times);InterpExpBiomass = interp1(e.times,e.fXb,data.times);

%calculate the sum of squares error between the simulation and the datassqXb = sum( (data.fXb - InterpExpBiomass).^2 );sizeXb = size(data.fXb,1);varevXb = 5E-5; %experimentally measured average variance

%sum of squares error for nitrate reductase levelssqe n = sum( (data.fe n - InterpExpEnzyme).^2 );sizee n = size(data.fe n,1);vareve n = 4.3E-19; %experimentally measured average variance

%output a weighted sum of the twobiomasserror = ssqXb/(sizeXb*varevXb);enzymeerror = ssqe n/(sizee n*vareve n);out = biomasserror + enzymeerror;

figure;PlotModel(e);PlotData(data);

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C.2 Code for RunModel.m

function expout = RunModel(m, e)%RUNMODEL(model,experiment), returns an experiment with simulated data

%number of phasesnumphases = size(e.PhaseLength,2);

%initial conditionsstate = [e.init Xb; e.init Ss; e.init Sn; e.init e n; e.init sni;];

%An experiment phase consists of a period of exposure to the same terminal%electron acceptor.

%initial valuesaccumY = state’;accumT = 0;

for phase = 1:numphases

%make note in the experiment what phase we are integratinge.phase = phase;%set aeratione.So = e.PhaseStartSo(phase);

%DILUTION%dilute only Xb, Ss, Sn. The other state variables are biomass%specific. Also not that for dilution ratio of zero, nothing happens.%This should always be zero for phase 1.%e.g. for a 1:19 dilution (20 fold), this number is 19DilutionFactor = e.PhaseDilutionRatio(phase);%conc. of nitrate in the solution used to diluteFeedSn = e.PhaseSolutionSn(phase);%conc. of carbon in the solution used to diluteFeedSs = e.PhaseSolutionSs(phase);%no biomass in feedstate(1) = state(1)/(1+DilutionFactor);%Substrate is addedstate(2) = (state(2)+FeedSs*DilutionFactor)/(1+DilutionFactor);%Nitrate is addedstate(3) = (state(3)+FeedSn*DilutionFactor)/(1+DilutionFactor);

%solve ODE for each time in e.times from initial conditions init(above)%run each integration for up to 100 hours or until a termination event%occurs. These events are either reaching the correct biomass or a%preset time.

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[T,Y,TE,YE,IE]=ode15s(m.model,0:0.052113154:100,state,e.ODEoptions,m,e);

state = Y(end,:); %copy the final state to use on the next integration

%this point was already present at the end of the previous phase,%so I’ll delete it to avoid duplicationY(1,:) = [];T(1) = [];

%concatenate the second segment with the firstaccumT = [accumT;T+accumT(end)];accumY = [accumY;Y];clear T Y TE YE IE;

end

%In order to have model data extending over the entire range we need to%curve fit I will make sure that the last time we’ve measured is >= the%total experimental durationif accumT(end) < sum(e.PhaseLength)

%integrate out to make up the differencephaseduration = sum(e.PhaseLength) - accumT(end);storeoptions = e.ODEoptions;e.ODEoptions.Events = [];[T,Y] = ode15s(m.model, [0 phaseduration], state,e.ODEoptions,m,e);e.ODEoptions = storeoptions;

state = Y(end,:); %copy the final state to use on the next integration

%this point was already present at the end of the previous phase,%so I’ll delete it to avoid duplicationY(1,:) = [];T(1) = [];

%concatenate the second segment with the firstaccumT = [accumT;T+accumT(end)];accumY = [accumY;Y];clear T Y;

end

%remove the temporary phase variable from ee = rmfield(e,’phase’);

%Seperate the data columns for readabilitye.times = accumT;e.fXb = accumY(:,1);e.fSs = accumY(:,2);

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e.fSn = accumY(:,3);e.fe n = accumY(:,4);e.fsni = accumY(:,5);

e.finalstate = [e.fXb(end) e.fSs(end) e.fSn(end) e.fe n(end) e.fsni(end)];

expout=e;

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C.3 Code for model5c.m

function dy = model5c(t,y,m,e)%model5b(t,y,m), model ODEs. m is a struct of model parameters%model parameters are in a struct defined by DefaultModelParam%This is modified from model5 by adding monod switching terms for Ss on%rsni and reno

%Variable DefinitionsXb = y(1);Ss = y(2);Sn = y(3);e n = y(4);sni = y(5);

%Maximum values for e n and snisni max = m.Vsni / m.mumax an - 1/m.Yn an;e n max = m.aNO / (m.bNO + m.mumax an - m.b) * (1 + m.K1*sni max) / ...

(m.K2 + m.K1*sni max);

%Process Rates%I’ll do some goofy stuff here to shortcircuit some possible errors%e.g. if Ks ox happens to be zero (which it shouldn’t be, but it may be%temporarily during optimization) then when Ss = 0 (which is reasonable)%then rox is undefined, when it should be zero.if Ss == 0

rox = 0;ranox = 0;rsni = 0;reno = 0;

elserox = m.mumax ox * Ss / (m.Ks ox + Ss) * e.So / (m.Koh + e.So);if sni max == 0

ranox = 0;else

ranox = m.mumax an * (e n/e n max) * (sni/sni max) * Ss / ...(m.Ks an + Ss);

endrsni = m.Vsni * (e n/e n max) * Sn/(Sn+m.Knoi) * m.Koi/(m.Koi+e.So) ...

* Ss / (m.Ks an + Ss);reno = m.aNO * (1 + m.K1*sni ) / (m.K2 + m.K1*sni) * Ss / ...

(m.Ks an + Ss);end

%ODEsdxbdt = (rox+ranox-m.b-e.D) * Xb;

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dssdt = (- rox / m.Yc ox - ranox / m.Yc an )*Xb + e.D*(e.Ssf - Ss);dsndt = -rsni*Xb +e.D*(e.Snf - Sn);

%the formula for specific enzyme and internal nitrate are undefined for%washout conditions. If there is no biomass I will just set these rates to%zero.if Xb == 0

de ndt = 0;dsnidt =0;

elsede ndt = reno - (m.bNO + e.D + dxbdt/Xb)*e n;dsnidt = rsni -(ranox/m.Yn an) - (e.D + m.b + dxbdt/Xb)*sni;

end%return outputsdy = [dxbdt;...

dssdt;...dsndt;...de ndt;...dsnidt;];

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C.4 Code for FitAllData.m

%Minimize the value of Fit TargetFunction by changing the model parameters%Initial guesses for parametersdisp(’FitAllData’)% x0.mumax ox = 0.5690; %aerobic growth% x0.mumax an = 0.1631; %anoxic growth% x0.Ks an = 0.3417; %switch for lack of energy/carbon% x0.Vsni = 0.0217; %maximum NO3 uptake;% x0.Knoi = 5.6055e-004; %g/L external nitrate promotes uptake% x0.Koi = 3.3068e-004; %O2 inhibits nitrate uptake% x0.aNO = 2.4536e-008; %maximum enzyme production rate% x0.K1 = 9.8580e+004;% x0.K2 = 1.9597e+004; %controls lag length% x0.bNO = 0.4; %nitrate reductase decay rate

%START OPTIMIZTING FROM THE LAST GOOD POINTload mfitASA2;disp(’initial model’)mfit.mumax ox = 0.56;mfit.mumax an = 0.17;disp(mfit)x0.array = [mfit.aNO

mfit.K2%mfit.mumax ox%mfit.mumax anmfit.Vsnimfit.K1mfit.Knoimfit.Koimfit.bNO];

[fvalinit, cost flag] = fAllError(x0.array);

%Set up constraints%All variables must be positivelb = [1E-12

1E3%0.1%0.10.00011E31E-51E-50];

ub = [1E-7

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1E10%0.5%0.521E101E-11E-110];

t = clock;%Optimize%**************************************************************************disp(’Using Adaptive Simulated Annealing’)disp(’ASA 25.5’)disp(’http://www.ingber.com/#ASA’)%Usage%[fstar,xstar,grad,hessian,state] = ...% asamin (’minimize’, func,xinit, xmin, xmax, xtype)xtype = -1 * ones(7,1);[fval,xfinal,grad,hessian,state] = ...

asamin (’minimize’, ’fAllError’,x0.array, lb, ub, xtype)%**************************************************************************

t = etime(clock,t);

mfit.aNO=xfinal(1); %maximum enzyme production ratemfit.K2=xfinal(2); %controls lag lengthmfit.Vsni=xfinal(3); %maximum NO3 uptake;mfit.K1=xfinal(4);mfit.Knoi=xfinal(5);mfit.Koi=xfinal(6);mfit.bNO = xfinal(7);

disp(’Optimization Complete’)disp(’Initial Function Value’)disp(num2str(fvalinit))disp(’Final Function Value’)disp(num2str(fval))disp(’Optimization Statistics’)%disp(output)disp(’Time Elapsed during optimization’)disp(t/60)disp(’Model Parameters’)disp(mfit)

save mfit mfit;save xfinal xfinal;

APPENDIX DEXTENDED KALMAN FILTER SOURCE CODE

D.1 Code for TestKSim.m

disp(sprintf(’\n********************************\nFile:\t\tTestKSim’))%test fKanapahaSim.m%load testvalues;load decjankops2;

%TEMP duplicate these conditions for the other train%kops = [kops(:,1) kops(:,2:9)/2 kops(:,2:9)/2 kops(:,10:24)];

%load deckops;%load decjanflatkops;

%**************************************************************************%cut down to 1 daykops = kops(1:168,:);runASM = false;useode23s = true;plotreactors = [4];%**************************************************************************

if runASMload XssASM;

else% load Xss;

load XssMonthEnd;

%double xinit for the second trainxinit = [xinit xinit];

end

disp(sprintf(’Run:\t\tday %2.0f to day %2.0f’,kops(1,1)/24,kops(end,1)/24))

et = clock;[t, allX, rctr, SRT] = fKanapahaSim(xinit, kops, runASM,useode23s, [], []);et = etime(clock,et);

disp(sprintf(’Simulated Time:\t%0.1f hr’,kops(end,1)))

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disp(sprintf(’Elapsed Time:\t%0.1f s’,et))disp(sprintf(’Sim rate:\t%0.1f hrs/s’,kops(end,1)/et))

save testksimoutput t allX rctr SRT;

for i = plotreactorsfigure(i);fPlotTank(rctr(i));

end

% figure(6)% fPlotKOp(kops);

figure(7)fPlotComp(rctr,’So’);for i = 1:5

[m, o] = fKanapahaModelOpsSetup(runASM);subplot(5,1,i);line([0 7],[m.Koa m.Koa],’Linestyle’,’--’,’Color’,’k’);line([0 7],[m.Koh m.Koh],’Linestyle’,’--’,’Color’,’k’);

end

% figure(11)% fPlotComp(rctr,’Snh’);% figure(12)% fPlotComp(rctr,’Sno’);if ~runASM

figure(13)fPlotComp(rctr,’En’);

end

figure(9)set(9,’Name’,’All Variables for Tank’);fPlotTankAllVars(rctr(4))

figure(208)set(gcf, ’Name’, ’Compare Probe Data’);fPlotCompareProbes(rctr(4),rctr(10));

%**************************************************************************%Outputdisp(sprintf(’final SRT:\t%0.1f days’,SRT(end)))

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D.2 Code for RunKEKF.m

%find the times for which there are enzyme measurementsdisp(sprintf(’\n********************************\nFile:\t\tRunKEKF’))clear%**************************************************************************%simulation settingssimdata = false; %use simulated plant data, rather than real datarunASM1 = false;estparams = true; %estimate parameters as wellhourstosim = 500; %numbers of hours of data to work onplotreactors= [4]; %number of reactors to plot at the end%measurement variancemvar = [

20; %aerobic 3 nitrate0.6;]; %aerobic 3 ammonia

%**************************************************************************

%nsv is number of state variables plus number of estimated parametersif estparams

nsv = 19;else

nsv = 15;endreactorindexNW = 46:60;reactorindexSE = 136:150;

%load plant operating conditionsload decjankops2

%TEMP duplicate these conditions for the other train%kops = [kops(:,1) kops(:,2:9)/2 kops(:,2:9)/2 kops(:,10:24)];

if runASM1load XssASM;

else% load Xss;

load XssMonthEnd;%xinit(4*15+10) = 0.2; %new initial value for Snh%double xinit so we’ll have initial conditions for the other trainxinit = [xinit xinit];

end

if simdatadisp(sprintf(’Data:\t\tSimulating Data’))disp(sprintf(’Run:\t\tday %2.0f to day %2.0f’, ...

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kops(1,1)/24,kops(hourstosim,1)/24))[t, allX, rctr, SRT] = fKanapahaSim(xinit, ...

kops(1:hourstosim,:), ...runASM1, ...true, ...[], ...[]);

t(1) = [];allX(1,:) = [];%allX is all 75 state variables on each row, one row per hour%t is time vector corresponding to allX%rctr is each reactor broken down into structs%SRT is the SRT for each time

%filter subset%only use as data points that fall on an hour markersubset = find((mod(t,1)==0)&(t<=hourstosim));ts = t(subset);allXs = allX(subset,:);

z = [ts allXs(:,reactorindexNW(9:10))];

%add in what we measure:noise = repmat(mvar.’,size(allXs,1),1) .* randn(size(allXs,1),2);z(:,2:3) = z(:,2:3) + noise;

%eliminate zerosz(find(z(:,2)<0),2)=0;z(find(z(:,3)<0),3)=0;

elsedisp(sprintf(’Data:\t\tDec 15 - Jan 12’))

% load ProbeData;

load zekf;

%filter subset%only use as data points that before the cutoffsubset = find(z(:,1)<=hourstosim);z=z(subset,:);

% allX = zeros(hourstosim,75);% t = 1:hourstosim;end

%now t and allX should correspond to the available measurements

%if we are doing parameter estimation as well, then augment state with

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%initial guesses for the parametersstate = xinit;

%set initial state estimatexhat0 = state;%initialize filter parameters (process variances)%Process covariance determined via SWAG methodP0 = [

5.1; %Si1; %Ss200; %Xi400; %Xs400; %Xbh10; %Xba145; %Xp10; %So10; %Sno7; %Snh0.8; %Snd10; %Xnd1; %Salk1; %Snoi1E-11; %En

];%if we’re doing parameter estimation, augment the process covar matrix% if estparams% if runASM1% else% P0 = [P0;% 1E5; %K1% 1E4; %K2% 1E-29; %aN% 1E-12]; %ben% end% endif estparams

if runASM1else

P0 = [P0;1E6; %K11E5; %K21E-35; %aN1E-13]; %ben

endendP0 = diag(P0);

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%measurement std devsmvar = diag(mvar);

%**************************************************************************%filter datadisp(sprintf(’Filtering days:\t%0.0f to %0.0f ...’, ...

min([z(1,1),kops(1,1)])/24,...z(end,1)/24))

et = clock;

[tout xhatout allX Pout] = fEKF(z, ...xhat0, ...P0, ...kops, ...mvar, ...runASM1, ...estparams);

et = etime(clock,et);

disp(sprintf(’Modeled Time:\t%0.1f hr’,tout(end)))disp(sprintf(’Elapsed Time:\t%0.1f s’,et))disp(sprintf(’Sim rate:\t%0.1f hrs/s’,tout(end)/et))%**************************************************************************

if simdatasave KEKFResult

elsesave KEKFResult

end

%I moved all of the plotting stuff to a script so I could easily plot%stored data by loading it and running this scriptPlotKEKFData

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D.3 Code for fKanapahaSim.m

function [t, allX, rctr, SRT] = ...fKanapahaSim(xinit, kops, runASM, useode23s, pop, peASM1m)

%simulate Kanapaha using fixed parameters using either eASM1m or ASM1%inputs:

[m, o] = fKanapahaModelOpsSetup(runASM);

%times of operating point changes [hr]tinput = kops(:,1);%aerobic pump power, for each pumphp = kops(:,2:5);%waste flowrate [L/hr]Fw = kops(:,6);%return sludge rate [L/hr]Fr = kops(:,7);%ML recycle rate [L/hr]Fm = kops(:,8);%feed flowrate [L/hr]F = kops(:,9);influent = kops(:,10:24);

%F feed flowrate%Fw waste rate%Fr waste recycle%Fm ML recycle%hp horsepower to pumps in aerobic basin (1 through 4)%influent strength%xinit initial conditions for all state variables%all of these are in row form. If passed array values each row represents%a different set of operating conditions.%tinput is either a column of times representing operating condition%changes or a single value representing a duration to simulate at the%operating conditions.%xinit is the single initial condition for tinput(1)

%optional parameters%pop is operating parameters. pop(1) is abrm, pop(2) is abkm%peASM1 are eASM1m model parameters%peASM1(1) is K1, peASM1(2) is K2, peASM1(3) is aN, peASM1(4) is ben,%peASM1(5) is Vsnoi

%notation for variables%an? anoxic basin in train ?%ab#? aerobic basin # in train ?

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%e.g. anbb is anoxic basin in train b

%if we are changing any parameters from defaultsif length(pop)

o.abkm1 = pop(1) * 0.007983036373; %kla slope for pump powero.abkm2 = pop(2) * 0.007983036373; %kla slope for pump powero.abkm3 = pop(1) * 0.007983036373; %kla slope for pump powero.abkm4 = pop(3) * 0.007983036373; %kla slope for pump powero.abrzero = pop(4);

endif length(peASM1m)

m.K1 = peASM1m(1); %mg biomass COD/mg nitrate COD == 7E4 gdw/mgNO3m.K2 = peASM1m(2);%katals/(mg biomassm.aN = peASM1m(3);m.ben = peASM1m(4);m.Vsnoi = peASM1m(5); %mg NO3 COD as N / (biomass COD * hr)disp(m)

end

%tdelta is the time steps to integrate over.if length(tinput) == 1

%use it as a single duration at the same setpointtdelta = tinput;

else%tdelta = diff(tinput);%ASSUME EACH TIME POINT IS 1 HOUR APART! BE AWARE!tdelta = ones(length(tinput),1);

end

options = odeset(’InitialStep’,0.1);

%initial conditions to change between iterations (must be column vector)xinitloop = xinit.’;

t = 0;allX = xinit;

for count=1:length(tdelta)%first, extract the influent composition corresponding to this time%step and format it as a column vectoro.influent = influent(count,:).’;

%**************************************************************************%Virtual Aerobic Reactor Recycle ratiosr1a = o.abrzero + o.abrm*mean(hp(count,1:2));

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r2a = o.abrzero + o.abrm*mean(hp(count,2:3));r3a = o.abrzero + o.abrm*mean(hp(count,3:4));

o.ab1aKlaf = o.abkzero+o.abkm1*hp(count,1);o.ab2aKlaf = o.abkzero+o.abkm2*hp(count,2);o.ab3aKlaf = o.abkzero+o.abkm3*hp(count,3);o.ab4aKlaf = o.abkzero+o.abkm4*hp(count,4);

%**************************************************************************

o.F = F(count);o.Fr = Fr(count);o.Fm = Fm(count);

%Clarifier performanceo.lambda = (o.F + o.Fr)/(o.Fr + Fw(count));

%Virtual Aerobic Reactor Recycle ratioso.Far1a = r1a*(o.F+o.Fr+o.Fm);o.Far2a = r2a*(o.F+o.Fr+o.Fm+o.Far1a);o.Far3a = r3a*(o.F+o.Fr+o.Fm+o.Far2a);

if useode23s[ttemp,ytemp]=ode15s(@fKanapahadxdt, ...

[0 tdelta(count)], ...xinitloop, ...[], ...m, ...o);

else%integrate using rk4[ttemp,ytemp]=rkhrk4A(@fKanapahadxdt,[0:ssize:tdelta(count)],xinitloop,m,o);

end

%grab next initial conditionsxinitloop = ytemp(end,:).’;

%final output is [time, allX, rctr, SRT]t = [t; ttemp(2:end)+t(end)];allX = [allX; ytemp(2:end,:)];

end

rctr = fParseAllX(t,allX);

%set up index variables%if X is your state, then X(ab2b) is the state variables in Aerobic Basin%2, train B

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%NOTE THAT THIS IS DUPLICATED FROM FUNCTION ACCUManba = 1:15;ab1a = 16:30;ab2a = 31:45;ab3a = 46:60;ab4a = 61:75;%index of particulates (Xi)Xndx = [3:7,12];

%Calculate SRT (hr) at all points in time%grab the Xndx subset of each a???SRT=sum([allX(:,anba(Xndx)’)*o.Vanba ...

allX(:,ab1a(Xndx)’)*o.AerobicTankSize/4 ...allX(:,ab2a(Xndx)’)*o.AerobicTankSize/4 ...allX(:,ab3a(Xndx)’)*o.AerobicTankSize/4 ...allX(:,ab4a(Xndx)’)*o.AerobicTankSize/4],2) ./ ...sum(Fw(count)*o.lambda*allX(:,ab4a(Xndx)’),2);

%convert to daysSRT = SRT / 24;

%**************************************************************************%**************************************************************************%**************************************************************************function [times, yout] = rkhrk4(FunFcn, times, y0,m,o)% Initialization%if given only start/stop times, fill in a default stepsizeif length(times) == 2

times = [times(1):0.1*(times(2)-times(1)):times(2)];end

times = times’;

%t = t0;y = y0(:);yout = y.’;

% The main loopfor i = 1:(length(times)-1)

t = times(i);h = times(i+1)-times(i);%if t + h > tfinal, h = tfinal - t; end% Compute the slopess1 = feval(FunFcn, t, y,m,o); s1 = s1(:);s2 = feval(FunFcn, t + h/2, y + h*s1/2,m,o); s2=s2(:);s3 = feval(FunFcn, t + h/2, y + h*s2/2,m,o); s3=s3(:);s4 = feval(FunFcn, t + h, y + h*s3,m,o); s4=s4(:);t = t + h;

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y = y + h*(s1 + 2*s2 + 2*s3 +s4)/6;y(find(y<0))=0;yout = [yout; y.’];

end;

%**************************************************************************%**************************************************************************%**************************************************************************function [times, yout] = rkheuler(FunFcn, times, y0,m,o)%this is SO MUCH slower that I can’t use it. It guarentees good numerical%behavior. But the problem is stiff, so the step sizes get too small. I%would need a more sophisticaed adaptive stepsize method than this.

% Initialization%if given only start/stop times, fill in a default stepsizeif length(times) == 2

times = [times(1):0.05*(times(2)-times(1)):times(2)];end

times = times’;

y = y0(:);yout = y.’;

% The main loop% The main loopfor i = 1:(length(times)-1)

t = times(i);h = times(i+1)-times(i);

% Compute the slopess1 = feval(FunFcn, t, y,m,o); s1 = s1(:);

t = t + h;y = y + h*s1;yout = [yout; y.’];

end;

%**************************************************************************%**************************************************************************%**************************************************************************function [tout, yout] = rkheulerA(FunFcn, times, y0, m,o)%this is SO MUCH slower that I can’t use it. It guarentees good numerical%behavior. But the problem is stiff, so the step sizes get too small. I%would need a more sophisticaed adaptive stepsize method than this.

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% Initializationt = times(1);tfinal = times(end);h = 0; %value gets replaced by maxstepsize before use

y = y0(:);tout = t;yout = y.’;

% The main loopwhile (t < tfinal)

if t + h > tfinal, h = tfinal - t; end

% Compute the slopess1 = feval(FunFcn, t, y,m,o); s1 = s1(:);

%compute the maximum stepsize that is safe to take%don’t divide by zeronotzero = find(s1~=0);%don’t use zero componentsnotzero2 = find(y(notzero)>0);

maxstepsize = min(abs(y(notzero(notzero2))./s1(notzero(notzero2))));

h = 0.1*maxstepsize;

t = t + h;y = y + h*s1;

tout = [tout; t];yout = [yout; y.’];

end;

%**************************************************************************%**************************************************************************%**************************************************************************function [times, yout] = rkhrk4A(FunFcn, times, y0,m,o)% Initialization%if given only start/stop times, fill in a default stepsizeif length(times) == 2

times = [times(1):0.1*(times(2)-times(1)):times(2)];end

times = times’;

t = 0;

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tfinal = times(end);hdefault = times(2)-times(1);h = hdefault;y = y0(:);yout = y.’;

% The main loopwhile t < tfinal

%make sure we don’t step past the endif t + h > tfinal, h = tfinal - t; end

h = hdefault;unsucessful = true;

%loop until we integrate without negative state variables%reduce stepsize if they DO go negative and repeatwhile unsucessful

% Compute the slopess1 = feval(FunFcn, t, y,m,o); s1 = s1(:);s2 = feval(FunFcn, t + h/2, y + h*s1/2,m,o); s2=s2(:);s3 = feval(FunFcn, t + h/2, y + h*s2/2,m,o); s3=s3(:);s4 = feval(FunFcn, t + h, y + h*s3,m,o); s4=s4(:);t = t + h;ytest = y + h*(s1 + 2*s2 + 2*s3 +s4)/6;%make sure than none of the state variables (1:75) went negativeif any(ytest(1:75) < 0)

%reduce the stepsize according to ASM1asm1only = y(1:75);asm1onlyrates = s1(1:75);%we need only those rates that are positivesubset = find(asm1onlyrates~=0);%we don’t need to worry about those states that are zero, but%have positive ratesasm1only = asm1only(subset);asm1onlyrates = asm1onlyrates(subset);subset = find(asm1only==0);%look at each zero state and see if the corresponding rate<0removeme = find(asm1onlyrates(subset)>=0);asm1only(subset(removeme)) = [];asm1onlyrates(subset(removeme)) = [];

h = min(abs(asm1only./asm1onlyrates));else

%we have integrated sucessfully, store this datay = ytest;unsucessful = false;

end

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end

t = t + h;%y(find(y<0))=0;yout = [yout; y.’];

end;

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D.4 Code for fKanapahaModelOpsSetup.m

function [m, o] = fKanapahaModelOpsSetup(runASM)%return model parameters and operating parameters to be constant among all%simulations

%**************************************************************************%Tunable Process Parameters%**************************************************************************%**************************************************************************%Virtual Aerobic Reactor Recycle ratios%reflects back-mixing in the real aeration tank%a linear function of the aeration motor power%abrzero is the recycle at zero pump powero.abrzero = 1.5;o.abrm = 0.00; %recycle slope for pump power (max = 200*abrm)%**************************************************************************%Mass Transport is zero for all components but oxygen%These values will come from the power to the aerator motors%assume linear relationship between power to motor and Kla%abkzero is the kla at zero pump power%The first number (3.5) is the manufacturer measure lb O2 /hp/hr, then a%conversion factor to get kla (note conversion includes rctr volume)o.abkzero = 0;o.abkm1 = 1.7 * 0.007983036373*2; %kla slope for pump powero.abkm2 = 2.2 * 0.007983036373*2; %kla slope for pump powero.abkm3 = 1.7 * 0.007983036373*2; %kla slope for pump powero.abkm4 = 1.7 * 0.007983036373*2; %kla slope for pump power% o.abkm = 2.0 * 0.007983036373; %value based on Bridge Report numbers

o.Sosat = 8.8; %mg/L

%Model Parameters variable%Bridge Report Valuesm.Koh = 0.1; %mg/L as O2m.ng = 0.85; %dimensionlessm.nh = 0.4; %dimensionlessm.kh = 0.125; %mg COD/(mg biomass COD h)m.Knh = 1; %mg/L as Nm.Koa = 0.4; %mg/L as O2m.Kx = 0.3; %.03%mg COD/mg biomass CODm.bh = 0.026;m.ba = 0.00833;m.ka = 0.0033; %L/(mg biomass COD h)m.Kno = 0.5; %mg/L as Nm.Ks = 20; %mg/L as COD

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m.muh = 0.25; %^-hm.mua = 0.033; %^-hm.runASM = runASM;%Parameters for Estimation (eventually)m.K1 = 1.2740e+05; %mg biomass COD / mg nitrate COD == 7E4 gdw/mgNO3;m.K2 = 1.86E4;m.aN = 1.0035e-12; %katals/(mg biomassm.ben = 0.06;m.Vsnoi = 0.06; %mg NO3 COD as N / (biomass COD * hr)%**************************************************************************%parameters with strong T dependance%Changes from bridge report%m.mua = 0.08; %^-h% m.ka = 0.0033; %L/(mg biomass COD h)% m.bh = 0.026;m.muh = 0.3; %^-h

%**************************************************************************%MODIFIED Values% m.Koh = 0.075; %mg/L as O2% % m.Koa = 0.3; %mg/L as O2m.mua = 0.04; %^-hm.ba = 0.004;

%MODIFIED parametersm.K1 = 1.2740e+05; %mg biomass COD / mg nitrate COD == 7E4 gdw/mgNO3;m.K2 = 1.86E4;m.aN = 1.0035e-12; %katals/(mg biomassm.ben = 1E-6;m.Vsnoi = 0.122; %mg NO3 COD as N / (biomass COD * hr)

% reaction rate parameters% Yh = 0.67; %mg biomass COD/mg substrate COD% Ya = 0.24; %mg biomass COD formed/mg N oxidizediNxb = 0.086;%mg N/mg COD in active biomassiNxd = 0.06; %mg N/mg COD in biomass debrisfp = 0.08; %mg debris COD/mg biomass COD

%**************************************************************************%MODIFIED parametersYh = 0.5; %mg biomass COD/mg substrate CODYa = 0.24; %mg biomass COD formed/mg N oxidized

m.Yh = Yh;if runASM

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%ASM1******************************************************************%STOICHIOMETRY MATRIX NU***********************************************m.nu = [

0, -1/Yh, 0, 0, 1, 0, 0, -(1-Yh)/Yh, 0,-iNxb, 0, 0, -iNxb/14;

0, -1/Yh, 0, 0, 1, 0, 0, 0,-(1-Yh)/(2.86*Yh), -iNxb, 0, 0,(1-Yh)/(14*2.86*Yh)-iNxb/14;

0, 0, 0, 0, 0, 1, 0, -(4.57-Ya)/Ya, 1/Ya,-iNxb-1/Ya, 0, 0, -iNxb/14-1/(7*Ya);

0, 0, 0, 1-fp, -1, 0, fp, 0, 0,0, 0, iNxb-fp*iNxd, 0;

0, 0, 0, 1-fp, 0, -1, fp, 0, 0,0, 0, iNxb-fp*iNxd, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0,1, -1, 0, 1/14;

0, 1, 0, -1, 0, 0, 0, 0, 0,0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0,0, 1, -1, 0;];

else%eASM1m****************************************************************%STOICHIOMETRY MATRIXm.nu = [

0, -1/Yh, 0, 0, 1, 0, 0, -(1-Yh)/Yh, 0,-iNxb, 0, 0, -iNxb/14, 0, 0;

0, -1/Yh, 0, 0, 1, 0, 0, 0, 0, -iNxb,0, 0, (1-Yh)/(14*2.86*Yh)-iNxb/14,-(1-Yh)/(2.86*Yh), 0;

0, 0, 0, 0, 0, 1, 0, -(4.57-Ya)/Ya, 1/Ya,-iNxb-1/Ya, 0, 0, -iNxb/14-1/(7*Ya),0, 0;

0, 0, 0, 1-fp, -1, 0, fp, 0, 0, 0,0, iNxb-fp*iNxd, 0, 0, 0;

0, 0, 0, 1-fp, 0,-1, fp, 0, 0, 0,0, iNxb-fp*iNxd, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 1,-1, 0, 1/14, 0, 0;

0, 1, 0, -1, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0,1, -1, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, -1, 0,0, 0, 0, 1, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 1;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

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0, 0, 0, -1, 0;0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

0, 0, 0, 0, -1];end

%Operating conditions global variable%Tank Volumeso.AerobicTankSize = 21953000/2; %liters 2.9MG*3785000L/MGo.Vanba = 3330800/2; %liters (0.44MG*3785000L/MG)o.Vae1a = o.AerobicTankSize/4;o.Vae2a = o.AerobicTankSize/4;o.Vae3a = o.AerobicTankSize/4;o.Vae4a = o.AerobicTankSize/4;

o.Vanbb = o.Vanba;o.Vae1b = o.AerobicTankSize/4;o.Vae2b = o.AerobicTankSize/4;o.Vae3b = o.AerobicTankSize/4;o.Vae4b = o.AerobicTankSize/4;

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D.5 Code for fKanapahadxdt.m

function dxdt = fKanapahadxdt(t,x,m,o)%computes the instantaneous accumulation for each tank

%accumulation = reaction rates + advection rates + mass transfer rates

%number of state variables per tankns=15;

%set up index variables%if X is your state, then X(ab2b) is the state variables in aerobic basin%2, train banba = 1:15;ab1a = 16:30;ab2a = 31:45;ab3a = 46:60;ab4a = 61:75;

anbb = 76:90;ab1b = 91:105;ab2b = 106:120;ab3b = 121:135;ab4b = 136:150;

%index of particulates (Xi)Xndx = [3:7,12];

%REACTION RATES************************************************************%reaction rates are easy. transpose(nu)*process ratesanbarxn = fKanapahaReactionRates(x(anba),m);ab1arxn = fKanapahaReactionRates(x(ab1a),m);ab2arxn = fKanapahaReactionRates(x(ab2a),m);ab3arxn = fKanapahaReactionRates(x(ab3a),m);ab4arxn = fKanapahaReactionRates(x(ab4a),m);

anbbrxn = fKanapahaReactionRates(x(anbb),m);ab1brxn = fKanapahaReactionRates(x(ab1b),m);ab2brxn = fKanapahaReactionRates(x(ab2b),m);ab3brxn = fKanapahaReactionRates(x(ab3b),m);ab4brxn = fKanapahaReactionRates(x(ab4b),m);

%the mixed output from both virtual aerobic basin 4 sidesxmixed =((o.Fa+o.Fra)*x(ab4a) + (o.Fb+o.Frb)*x(ab4b)) / ...

(o.Fa+o.Fra+o.Fb+o.Frb);

108

%**************************************************************************%EAST TRAIN (A)************************************************************%**************************************************************************%ANOXIC BASIN A************************************************************%components coming from the clarifier are mixed from both trains%advection term for particulates (settled in clarifier)anbaparticulates = o.Fa/o.Vanba*o.influent + ...

o.Fra/o.Vanba*o.lambda*xmixed + ...o.Fma/o.Vanba*x(ab4a) - ...(o.Fa+o.Fra+o.Fma)/o.Vanba*x(anba);

%advection for dissolved species is the same, but lambda is 1anbaadv = o.Fa/o.Vanba*o.influent + ...

o.Fra/o.Vanba*x(ab4a) + ...o.Fma/o.Vanba*x(ab4a) - ...(o.Fa+o.Fra+o.Fma)/o.Vanba*x(anba);

%overall it’s a combination of particulates and solubles, so copy the%particulates values into the advection termanbaadv(Xndx) = anbaparticulates(Xndx);

%AEROBIC BASIN 1A**********************************************************ab1aadv = (o.Fa+o.Fra+o.Fma)/o.Vae1a*x(anba) + ...

o.Far1a/o.Vae1a*x(ab2a) - ...(o.Fa+o.Fra+o.Fma+o.Far1a)/o.Vae1a*x(ab1a);

%AEROBIC BASIN 2A**********************************************************ab2aadv = (o.Fa+o.Fra+o.Fma+o.Far1a)/o.Vae2a*x(ab1a) + ...

o.Far2a/o.Vae2a*x(ab3a) - ...o.Far1a/o.Vae2a*x(ab2a) - ...(o.Fa+o.Fra+o.Fma+o.Far2a)/o.Vae2a*x(ab2a);

%AEROBIC BASIN 3A**********************************************************ab3aadv = (o.Fa+o.Fra+o.Fma+o.Far2a)/o.Vae3a*x(ab2a) + ...

o.Far3a/o.Vae3a*x(ab4a) - ...o.Far2a/o.Vae3a*x(ab3a) - ...(o.Fa+o.Fra+o.Fma+o.Far3a)/o.Vae3a*x(ab3a);

%AEROBIC BASIN 4A**********************************************************ab4aadv = (o.Fa+o.Fra+o.Fma+o.Far3a)/o.Vae4a*x(ab3a) - ...

o.Far3a/o.Vae4a*x(ab4a) - ...(o.Fa+o.Fra+o.Fma)/o.Vae4a*x(ab4a);

anbamt = zeros(15,1);ab1amt = anbamt;ab2amt = anbamt;ab3amt = anbamt;ab4amt = anbamt;

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%note here: ab1a is the range for aerobic basin 1, train a%ab1a(8) is the index for So in that basin%x(ab1a(8)) is the value of So in that basinab1amt(8) = o.ab1aKlaf*(o.Sosat-x(ab1a(8)));ab2amt(8) = o.ab2aKlaf*(o.Sosat-x(ab2a(8)));ab3amt(8) = o.ab3aKlaf*(o.Sosat-x(ab3a(8)));ab4amt(8) = o.ab4aKlaf*(o.Sosat-x(ab4a(8)));

%**************************************************************************%WEST TRAIN (B)************************************************************%**************************************************************************%ANOXIC BASIN B************************************************************%components coming from the clarifier are mixed from both trains%advection term for particulates (settled in clarifier)anbbparticulates = o.Fb/o.Vanbb*o.influent + ...

o.Frb/o.Vanbb*o.lambda*xmixed + ...o.Fmb/o.Vanbb*x(ab4b) - ...(o.Fb+o.Frb+o.Fmb)/o.Vanbb*x(anbb);

%advection for dissolved species is the same, but lambda is 1anbbadv = o.Fb/o.Vanbb*o.influent + ...

o.Frb/o.Vanbb*x(ab4b) + ...o.Fmb/o.Vanbb*x(ab4b) - ...(o.Fb+o.Frb+o.Fmb)/o.Vanbb*x(anbb);

%overall it’s a combination of particulates and solubles, so copy the%particulates values into the advection termanbbadv(Xndx) = anbbparticulates(Xndx);

%AEROBIC BASIN 1B**********************************************************ab1badv = (o.Fb+o.Frb+o.Fmb)/o.Vae1b*x(anbb) + ...

o.Far1b/o.Vae1b*x(ab2b) - ...(o.Fb+o.Frb+o.Fmb+o.Far1b)/o.Vae1b*x(ab1b);

%AEROBIC BASIN 2B**********************************************************ab2badv = (o.Fb+o.Frb+o.Fmb+o.Far1b)/o.Vae2b*x(ab1b) + ...

o.Far2b/o.Vae2b*x(ab3b) - ...o.Far1b/o.Vae2b*x(ab2b) - ...(o.Fb+o.Frb+o.Fmb+o.Far2b)/o.Vae2b*x(ab2b);

%AEROBIC BASIN 3B**********************************************************ab3badv = (o.Fb+o.Frb+o.Fmb+o.Far2b)/o.Vae3b*x(ab2b) + ...

o.Far3b/o.Vae3b*x(ab4b) - ...o.Far2b/o.Vae3b*x(ab3b) - ...(o.Fb+o.Frb+o.Fmb+o.Far3b)/o.Vae3b*x(ab3b);

%AEROBIC BASIN 4B**********************************************************ab4badv = (o.Fb+o.Frb+o.Fmb+o.Far3b)/o.Vae4b*x(ab3b) - ...

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o.Far3b/o.Vae4b*x(ab4b) - ...(o.Fb+o.Frb+o.Fmb)/o.Vae4b*x(ab4b);

anbbmt = zeros(15,1);ab1bmt = anbbmt;ab2bmt = anbbmt;ab3bmt = anbbmt;ab4bmt = anbbmt;

%note here: ab1a is the range for aerobic basin 1, train a%ab1a(8) is the index for So in that basin%x(ab1a(8)) is the value of So in that basinab1bmt(8) = o.ab1bKlaf*(o.Sosat-x(ab1b(8)));ab2bmt(8) = o.ab2bKlaf*(o.Sosat-x(ab2b(8)));ab3bmt(8) = o.ab3bKlaf*(o.Sosat-x(ab3b(8)));ab4bmt(8) = o.ab4bKlaf*(o.Sosat-x(ab4b(8)));

dxdt = [anbarxn+anbaadv+anbamt;ab1arxn+ab1aadv+ab1amt;ab2arxn+ab2aadv+ab2amt;ab3arxn+ab3aadv+ab3amt;ab4arxn+ab4aadv+ab4amt;

anbbrxn+anbbadv+anbbmt;ab1brxn+ab1badv+ab1bmt;ab2brxn+ab2badv+ab2bmt;ab3brxn+ab3badv+ab3bmt;ab4brxn+ab4badv+ab4bmt;];

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D.6 Code for fParseAllX.m

%turns an allX state variable by time array into reactor structsfunction rctr = fParseAllX(t,allX,prefix)%t is time, allX is all state vars, prefix is text to prefix each label

if nargin < 3prefix = ’’;

end

ns=15;anba = 1:ns;ab1a = ns+1:2*ns;ab2a = 2*ns+1:3*ns;ab3a = 3*ns+1:4*ns;ab4a = 4*ns+1:5*ns;%index of particulates (Xi)Xndx = [3:7,12];

rctr(1).name = [prefix ’Anoxic Basin’];rctr(1).t = t;rctr(1).Si = allX(:,anba(1));rctr(1).Ss = allX(:,anba(2));rctr(1).Xi = allX(:,anba(3));rctr(1).Xs = allX(:,anba(4));rctr(1).Xbh = allX(:,anba(5));rctr(1).Xba = allX(:,anba(6));rctr(1).Xp = allX(:,anba(7));rctr(1).So = allX(:,anba(8));rctr(1).Sno = allX(:,anba(9));rctr(1).Snh = allX(:,anba(10));rctr(1).Snd = allX(:,anba(11));rctr(1).Xnd = allX(:,anba(12));rctr(1).Salk = allX(:,anba(13));rctr(1).Snoi = allX(:,anba(14));rctr(1).En = allX(:,anba(15));rctr(2).name = [prefix ’Aerobic Basin Virtual Reactor 1’];rctr(2).t = t;rctr(2).Si = allX(:,ab1a(1));rctr(2).Ss = allX(:,ab1a(2));rctr(2).Xi = allX(:,ab1a(3));rctr(2).Xs = allX(:,ab1a(4));rctr(2).Xbh = allX(:,ab1a(5));rctr(2).Xba = allX(:,ab1a(6));rctr(2).Xp = allX(:,ab1a(7));rctr(2).So = allX(:,ab1a(8));

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rctr(2).Sno = allX(:,ab1a(9));rctr(2).Snh = allX(:,ab1a(10));rctr(2).Snd = allX(:,ab1a(11));rctr(2).Xnd = allX(:,ab1a(12));rctr(2).Salk = allX(:,ab1a(13));rctr(2).Snoi = allX(:,ab1a(14));rctr(2).En = allX(:,ab1a(15));rctr(3).name = [prefix ’Aerobic Basin Virtual Reactor 2’];rctr(3).t = t;rctr(3).Si = allX(:,ab2a(1));rctr(3).Ss = allX(:,ab2a(2));rctr(3).Xi = allX(:,ab2a(3));rctr(3).Xs = allX(:,ab2a(4));rctr(3).Xbh = allX(:,ab2a(5));rctr(3).Xba = allX(:,ab2a(6));rctr(3).Xp = allX(:,ab2a(7));rctr(3).So = allX(:,ab2a(8));rctr(3).Sno = allX(:,ab2a(9));rctr(3).Snh = allX(:,ab2a(10));rctr(3).Snd = allX(:,ab2a(11));rctr(3).Xnd = allX(:,ab2a(12));rctr(3).Salk = allX(:,ab2a(13));rctr(3).Snoi = allX(:,ab2a(14));rctr(3).En = allX(:,ab2a(15));rctr(4).name = [prefix ’Aerobic Basin Virtual Reactor 3’];rctr(4).t = t;rctr(4).Si = allX(:,ab3a(1));rctr(4).Ss = allX(:,ab3a(2));rctr(4).Xi = allX(:,ab3a(3));rctr(4).Xs = allX(:,ab3a(4));rctr(4).Xbh = allX(:,ab3a(5));rctr(4).Xba = allX(:,ab3a(6));rctr(4).Xp = allX(:,ab3a(7));rctr(4).So = allX(:,ab3a(8));rctr(4).Sno = allX(:,ab3a(9));rctr(4).Snh = allX(:,ab3a(10));rctr(4).Snd = allX(:,ab3a(11));rctr(4).Xnd = allX(:,ab3a(12));rctr(4).Salk = allX(:,ab3a(13));rctr(4).Snoi = allX(:,ab3a(14));rctr(4).En = allX(:,ab3a(15));rctr(5).name = [prefix ’Aerobic Basin Virtual Reactor 4’];rctr(5).t = t;rctr(5).Si = allX(:,ab4a(1));rctr(5).Ss = allX(:,ab4a(2));rctr(5).Xi = allX(:,ab4a(3));rctr(5).Xs = allX(:,ab4a(4));

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rctr(5).Xbh = allX(:,ab4a(5));rctr(5).Xba = allX(:,ab4a(6));rctr(5).Xp = allX(:,ab4a(7));rctr(5).So = allX(:,ab4a(8));rctr(5).Sno = allX(:,ab4a(9));rctr(5).Snh = allX(:,ab4a(10));rctr(5).Snd = allX(:,ab4a(11));rctr(5).Xnd = allX(:,ab4a(12));rctr(5).Salk = allX(:,ab4a(13));rctr(5).Snoi = allX(:,ab4a(14));rctr(5).En = allX(:,ab4a(15));

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D.7 Code for fExtendedKalmanFilter.m

function [tout, xArray, allX, Pout] = fEKF(zdata, ...kx, ...P, ...kops, ...mvar, ...runASM1, ...ADAPTIVE)

%applies a Kalman filter to dataset under constant operating conditions%zdata is a nx3 matrix, each row is time, nitrate, ammonia%kx is the initial state of the entire process%P is the initial covariance matrix%kops is the operating conditions, from the KWRF Strategy Sheet.xls file%mvar is measurement variance%runASM1 is boolean, whether to use ASM1 or eASM1m%ADAPTIVE is boolean, whether to use parameter estimation or not

%toggle debug checksDEBUG = true;

if ADAPTIVE%length of AEKF state vectornsv = 19;%length of KanapahaSim state vectorasv = 154;%single out one reactor to filter. This is the index of that reactors%state variables inside of the KSim staterpindex = [46:60 151:asv];

else%length of EKF state vectornsv = 15;%length of KanapahaSim state vectorasv = 150;%single out one reactor to filter. This is the index of that reactors%state variables inside of the KSim staterpindex = 46:60;

end

%parse kopskops(find(kops(:,1)>zdata(end,1)),:)=[];%times of operating point changes [hr]koptimes = kops(:,1);%aerobic pump power, for each pumphpa = kops(:,2:5); %divide by 2 because kops has total hp for both trainshpb = kops(:,6:9); %divide by 2 because kops has total hp for both trains

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%waste flowrate [L/hr]Fwa = kops(:,10);%return sludge rate [L/hr]Fra = kops(:,11);%ML recycle rate [L/hr]Fma = kops(:,12);%waste flowrate [L/hr]Fwb = kops(:,13);%return sludge rate [L/hr]Frb = kops(:,14);%ML recycle rate [L/hr]Fmb = kops(:,15);%feed flowrate [L/hr]Fa = kops(:,16)/2;%feed flowrate [L/hr]Fb = kops(:,16)/2;

influent = kops(:,17:31);

%in order to keep synchronized between measurement times and changes in%operating conditions, I will use the variable kopindex to represent where%we are in the kops array, and z?index for the data arrayskopindex = 2;z1index = 1;iout = 2;

H=zeros(2,nsv);H(1,9)=1;H(2,10)=1;

%if there is a data point at time zero, ditch itif(zdata(1,1)==0)

z1index=1;end

%get the model parameters from the setup file.[m, o] = fKanapahaModelOpsSetup(runASM1);%**************************************************************************%Virtual Aerobic Reactor Recycle ratiosr1a = o.abrzero + o.abrm*mean(hpa(kopindex,1:2));r2a = o.abrzero + o.abrm*mean(hpa(kopindex,2:3));r3a = o.abrzero + o.abrm*mean(hpa(kopindex,3:4));

r1b = o.abrzero + o.abrm*mean(hpb(kopindex,1:2));r2b = o.abrzero + o.abrm*mean(hpb(kopindex,2:3));r3b = o.abrzero + o.abrm*mean(hpb(kopindex,3:4));

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o.influent = influent(kopindex,:).’;

o.ab1aKlaf = o.abkzero+o.abkm1*hpa(kopindex,1);o.ab2aKlaf = o.abkzero+o.abkm2*hpa(kopindex,2);o.ab3aKlaf = o.abkzero+o.abkm3*hpa(kopindex,3);o.ab4aKlaf = o.abkzero+o.abkm4*hpa(kopindex,4);

o.ab1bKlaf = o.abkzero+o.abkm1*hpb(kopindex,1);o.ab2bKlaf = o.abkzero+o.abkm2*hpb(kopindex,2);o.ab3bKlaf = o.abkzero+o.abkm3*hpb(kopindex,3);o.ab4bKlaf = o.abkzero+o.abkm4*hpb(kopindex,4);

%**************************************************************************

o.Fa = Fa(kopindex);o.Fra = Fra(kopindex);o.Fma = Fma(kopindex);

o.Fb = Fb(kopindex);o.Frb = Frb(kopindex);o.Fmb = Fmb(kopindex);

%Clarifier separationo.lambda = ((o.Fa+o.Fb) + (o.Fra+o.Frb))/((o.Fra+o.Frb) + ...

(Fwa(kopindex)+Fwb(kopindex)));

%Virtual Aerobic Reactor Recycle ratioso.Far1a = r1a*(o.Fa+o.Fra+o.Fma);o.Far2a = r2a*(o.Fa+o.Fra+o.Fma+o.Far1a);o.Far3a = r3a*(o.Fa+o.Fra+o.Fma+o.Far2a);

o.Far1b = r1b*(o.Fb+o.Frb+o.Fmb);o.Far2b = r2b*(o.Fb+o.Frb+o.Fmb+o.Far1b);o.Far3b = r3b*(o.Fb+o.Frb+o.Fmb+o.Far2b);

if ADAPTIVE && ~runASM1%state vectors must be column vectors for this algorithm%only for my modified model and when estimating parameterskx = [kx.’;

m.K1;m.K2;m.aN;m.ben;];

elsekx = kx.’;

end

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%I’ve seen this done freqeuntly. Just set the initial error covariance%equal to the process variance. This would be more clear as%Pinitial = pvar.pvar = P;

%allocate memory for outputs for speednumoutput = length(zdata(:,1));xArray = zeros(numoutput,nsv);Pout = zeros(nsv,nsv,numoutput);allX = zeros(numoutput,asv);tout = zeros(numoutput,1);

%Initialize storage arraysPout(:,:,1) = P;xArray(1,:) = kx(rpindex)’;allX(1,:) = kx.’;tout(1) = 0;

%time of last measurementtend = zdata(end,1);t = 0;

if ADAPTIVE && ~runASM1disp(sprintf(’t\tK1\t\tK2\t\taN\t\tben\t\tVsnoi’))disp(sprintf(’%3.1f\t%0.4e\t%0.4e\t%0.4e\t%0.4e\t%0.4e’, ...

t,m.K1,m.K2,m.aN,m.ben,m.Vsnoi))end

%log progress to fileflog = fopen(’fEKFlog.txt’,’w’);%string of values to save to filesavestr = [];saveformat = [];saveheader = [];for i=1:asv

saveformat = [saveformat ’%0.4e\t’];savestr = [savestr ’kx(’ int2str(i) ’), ’];saveheader = [saveheader ’kx(’ int2str(i) ’)\t\t’];

endsavestr = savestr(1:end-2);saveformat = saveformat(1:end-2);

xhat = kx(rpindex);eval([’fprintf(flog,’’t\t\t\t’ saveheader ’\n’’);’])eval([’fprintf(flog,’’%0.3f\t’ saveformat ’\n’’,t,’ savestr ’’’);’])

%Filter Data

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et = clock;waithandle=waitbar(0,’Filtering in progress’); % Initialize waitbarwhile t <= tend

%**********************************************************************%consider that the data points may not match up to the operating%conditions. If we’ve passed a changeover point then change all of the%operating conditions appropriatelynexttime = zdata(z1index,1);

%integrate superODE forward until we’ve caught up with all of the kop%changeswhile kopindex <= length(koptimes) && koptimes(kopindex) <= nexttime

timestep = koptimes(kopindex) - t;%integrate[ttemp,ytemp] = ode15s(@superODE, ...

[0 timestep/2 timestep], ...[kx;reshape(P,nsv^2,1)], ...[], ...m, ...o, ...pvar, ...ADAPTIVE);

%we’ve now integrated forward to be even with the new kopt = koptimes(kopindex);

%grab the entire state of the processkx = ytemp(end,1:asv)’;

if DEBUG && any(isnan((ytemp(end,:))))disp(sprintf(’**ERROR:**\t\tytemp is NaN in kop loop’))

end

P = reshape(ytemp(end,asv+1:nsv^2+asv),nsv,nsv);

%change the operating conditions%**************************************************************************%Virtual Aerobic Reactor Recycle ratiosr1a = o.abrzero + o.abrm*mean(hpa(kopindex,1:2));r2a = o.abrzero + o.abrm*mean(hpa(kopindex,2:3));r3a = o.abrzero + o.abrm*mean(hpa(kopindex,3:4));

r1b = o.abrzero + o.abrm*mean(hpb(kopindex,1:2));r2b = o.abrzero + o.abrm*mean(hpb(kopindex,2:3));r3b = o.abrzero + o.abrm*mean(hpb(kopindex,3:4));

o.influent = influent(kopindex,:).’;

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o.ab1aKlaf = o.abkzero+o.abkm1*hpa(kopindex,1);o.ab2aKlaf = o.abkzero+o.abkm2*hpa(kopindex,2);o.ab3aKlaf = o.abkzero+o.abkm3*hpa(kopindex,3);o.ab4aKlaf = o.abkzero+o.abkm4*hpa(kopindex,4);

o.ab1bKlaf = o.abkzero+o.abkm1*hpb(kopindex,1);o.ab2bKlaf = o.abkzero+o.abkm2*hpb(kopindex,2);o.ab3bKlaf = o.abkzero+o.abkm3*hpb(kopindex,3);o.ab4bKlaf = o.abkzero+o.abkm4*hpb(kopindex,4);

%**************************************************************************

o.Fa = Fa(kopindex);o.Fra = Fra(kopindex);o.Fma = Fma(kopindex);

o.Fb = Fb(kopindex);o.Frb = Frb(kopindex);o.Fmb = Fmb(kopindex);

%Clarifier performanceo.lambda = ((o.Fa+o.Fb) + (o.Fra+o.Frb))/((o.Fra+o.Frb) + ...

(Fwa(kopindex)+Fwb(kopindex)));

%Virtual Aerobic Reactor Recycle ratioso.Far1a = r1a*(o.Fa+o.Fra+o.Fma);o.Far2a = r2a*(o.Fa+o.Fra+o.Fma+o.Far1a);o.Far3a = r3a*(o.Fa+o.Fra+o.Fma+o.Far2a);

o.Far1b = r1b*(o.Fb+o.Frb+o.Fmb);o.Far2b = r2b*(o.Fb+o.Frb+o.Fmb+o.Far1b);o.Far3b = r3b*(o.Fb+o.Frb+o.Fmb+o.Far2b);

kopindex = kopindex+1;

end

%now we have a measurement to deal withz = zdata(z1index,2:3).’;z1index = z1index + 1;if z1index > length(zdata(:,1))

break;end

timestep = nexttime - t;

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waitbar(nexttime/tend, ...waithandle, ...sprintf(’Filtering in progress\ntime: %f / %0.1f hr\nElapsed Time: %0.1f min (~%0.0f min remain)’,...nexttime, ...tend, ...etime(clock,et)/60, ...etime(clock,et)/60*(tend/nexttime-1)));

%simultaneously integrate xhat and P forward to timestep%vector structure is below:% [all reactor states;% all parameters;% unrolled P matrix;]%this is in case kop and measurement coincide,%we will have integrated alreadyif timestep > 0

[ttemp,ytemp] = ode15s(@superODE, ...[0 timestep/2 timestep], ...[kx;reshape(P,nsv^2,1)], ...[], ...m, ...o, ...pvar, ...ADAPTIVE);

end%Optional Integrators% [ttemp,ytemp] = rkheuler(@superODE, ...% [0 timestep],[kx;reshape(P,nsv^2,1)],m,o,pvar);% [ttemp,ytemp] = rkheulerA(@superODE, ...% [0 timestep],[kx;reshape(P,nsv^2,1)],m,o,pvar);% [ttemp,ytemp] = rkhrk4A(@superODE, ...% [0 timestep],[kx;reshape(P,nsv^2,1)],m,o,pvar);% [ttemp,ytemp] = rkhrk4(@superODE, ...% [0 timestep],[kx;reshape(P,nsv^2,1)],m,o,pvar);

%we have just integrated up through the current measurementt = nexttime;%grab the entire state of the processkx = ytemp(end,1:asv)’;%grab the filtered reactor statexhat = kx(rpindex);

P = reshape(ytemp(end,asv+1:nsv^2+asv),nsv,nsv);

%correct for numerical problems

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tmp=find(diag(P)<0 & diag(P)>(-1e-15));for k = tmp, P(k,k) = 0; end;

%Now compute xhatplus, out a posteriori state estimate.Minv = inv(H*P*H.’+mvar);K = P*H.’*Minv;

if DEBUG && any(isnan(xhat))disp(sprintf(’**ERROR:**\t\txhat is NaN’))

endif DEBUG && any(isnan(xhat + K*(z - H*xhat)))

disp(sprintf(’**ERROR:**\txhat will be NaN’))end

%KALMAN FILTER MAGIC HAPPENS HERExhat = xhat + K*(z - H*xhat);

if DEBUG, xhat(find(xhat<0))=0; end;

eval([’fprintf(flog,’’%0.3f\t’ saveformat ’\n’’,t,’ savestr ’’’);’])

%correct for numerical problemstmp=find(diag(P)<0 & diag(P)>(-1e-15));for k = tmp, P(k,k) = 0; end;

if DEBUG && any(any(diag(P)<0))disp(sprintf(’**ERROR:**\t\tPminus NEGATIVE’))

end

%PplusP = ( eye(nsv) - K*H )*P;

%correct for numerical problemstmp=find(diag(P)<0 & diag(P)>(-1e-15));for k = tmp, P(k,k) = 0; end;

if DEBUG && any(any(P)<0)disp(sprintf(’**ERROR:**\t\tPPlus NEGATIVE’))

end

%reconstitute the total process state using the new a posteriori xhatkx(rpindex(1:15)) = xhat(1:15);if ADAPTIVE && ~runASM1, kx(end-3:end) = xhat(16:nsv); end;

if DEBUG && ADAPTIVE && any(kx(end-4:end)==0)%can’t allow zero parameters for enzyme kineticsif kx(end-1) == 0 kx(end-1) = 1e-15; end

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disp(sprintf(’**ERROR:**\tbad parameter value after xhatplus calc’))tzK

end

if ADAPTIVE && ~runASM1%now that we’ve modified the parameters, write them out to the modelm.K1 = kx(151);m.K2 = kx(152);m.aN = kx(153);m.ben = kx(154);%m.Vsnoi = kx(80);

end

if ADAPTIVE && ~runASM1disp(sprintf(’%3.1f\t%0.4e\t%0.4e\t%0.4e\t%0.4e\t%0.4e’, ...

t,m.K1,m.K2,m.aN,m.ben,m.Vsnoi))end

%store resultsxArray(iout,:) = xhat.’;Pout(:,:,iout) = P;allX(iout,:) = kx.’;tout(iout) = t;iout = iout+1;

if DEBUG && ( any(isnan(kx)) || any(kx<0) || any(isinf(kx)) )disp(sprintf(’**ERROR:**\t\tkx contains negative numbers, NaN, or Inf’))

end

%save progress to file. In case of a crash, load this file and then run%the normal data display function, PlotKEKFData%save fEKFallX allX tout Pout;

endclose(waithandle);

%*****************************************************%*****************************************************%*****************************************************function out = superODE(t,x,m,o,pvar,ADAPTIVE)%this function takes as input%input = [xhat;So;reshape(P,nsv^2,1)]%and outputs [xhatdot;0;reshape(Pdot,nsv^2,1)]

if ADAPTIVE%length of EKF state vector

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nsv = 19;%length of KanapahaSim state vectorasv = 154;%single out one reactor to filter. This is the index of that reactors%state variables inside of the KSim staterpindex = [46:60 151:asv];%number of parametersnp = 4;

else%length of EKF state vectornsv = 15;%length of KanapahaSim state vectorasv = 150;%single out one reactor to filter. This is the index of that reactors%state variables inside of the KSim staterpindex = 46:60;%number of parametersnp = 0;

end

P = reshape(x(asv+1:nsv^2+asv),nsv,nsv);

if ADAPTIVEF = [fModelJacobAEKF(x(rpindex),m,o); zeros(np,nsv);];

elseF = [fModelJacob(x(rpindex),m,o); zeros(np,nsv);];

end

xhatdot = [fKanapahadxdt(t,x(1:asv),m,o); zeros(np,1);];

Pdot = F * P + P * F.’ + pvar;if any(any(isnan(Pdot)))

disp(sprintf(’**ERROR:**\tPdot is NaN’))endout = [xhatdot;

reshape(Pdot,nsv^2,1);]; %turn Pdot into a vector

%**************************************************************************%**************************************************************************%**************************************************************************function [times, yout] = rkhrk4(FunFcn, times, y0,m,o,pvar)% Initialization%if given only start/stop times, fill in a default stepsizeif length(times) == 2

times = [times(1):0.1*(times(2)-times(1)):times(2)];end

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times = times’;

%t = t0;y = y0(:);yout = y.’;

% The main loopfor i = 1:(length(times)-1)

t = times(i);h = times(i+1)-times(i);%if t + h > tfinal, h = tfinal - t; end% Compute the slopess1 = feval(FunFcn, t, y,m,o,pvar); s1 = s1(:);s2 = feval(FunFcn, t + h/2, y + h*s1/2,m,o,pvar); s2=s2(:);s3 = feval(FunFcn, t + h/2, y + h*s2/2,m,o,pvar); s3=s3(:);s4 = feval(FunFcn, t + h, y + h*s3,m,o,pvar); s4=s4(:);t = t + h;y = y + h*(s1 + 2*s2 + 2*s3 +s4)/6;y(find(y(1:75)<0))=0;yout = [yout; y.’];

end;

%**************************************************************************%**************************************************************************%**************************************************************************function [times, yout] = rkhrk4A(FunFcn, times, y0,m,o,pvar)% Initialization%if given only start/stop times, fill in a default stepsizeif length(times) == 2

times = [times(1):0.1*(times(2)-times(1)):times(2)];end

times = times’;

t = 0;tfinal = times(end);hdefault = times(2)-times(1);h = hdefault;y = y0(:);yout = y.’;

% The main loopwhile t < tfinal

%make sure we don’t step past the endif t + h > tfinal, h = tfinal - t; end

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h = hdefault;unsucessful = true;

%loop until we integrate without negative state variables%reduce stepsize if they DO go negative and repeatwhile unsucessful

% Compute the slopess1 = feval(FunFcn, t, y,m,o,pvar); s1 = s1(:);s2 = feval(FunFcn, t + h/2, y + h*s1/2,m,o,pvar); s2=s2(:);s3 = feval(FunFcn, t + h/2, y + h*s2/2,m,o,pvar); s3=s3(:);s4 = feval(FunFcn, t + h, y + h*s3,m,o,pvar); s4=s4(:);t = t + h;ytest = y + h*(s1 + 2*s2 + 2*s3 +s4)/6;%make sure than none of the state variables (1:75) went negativeif any(ytest(1:75) < 0)

%reduce the stepsize according to ASM1asm1only = y(1:75);asm1onlyrates = s1(1:75);%we need only those rates that are positivesubset = find(asm1onlyrates~=0);%we don’t need to worry about those states that are zero, but%have positive ratesasm1only = asm1only(subset);asm1onlyrates = asm1onlyrates(subset);subset = find(asm1only==0);%look at each zero state and see if the corresponding rate<0removeme = find(asm1onlyrates(subset)>=0);asm1only(subset(removeme)) = [];asm1onlyrates(subset(removeme)) = [];

maxstepsize = min(abs(asm1only./asm1onlyrates));h = 0.2*maxstepsize;

else%we have integrated sucessfully, store this datay = ytest;unsucessful = false;

endend

t = t + h;%y(find(y<0))=0;yout = [yout; y.’];

end;

%**************************************************************************%**************************************************************************

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%**************************************************************************function [times, yout] = rkheuler(FunFcn, times, y0, m,o,pvar)%this is SO MUCH slower that I can’t use it. It guarentees good numerical%behavior. But the problem is stiff, so the step sizes get too small. I%would need a more sophisticaed adaptive stepsize method than this.

% Initialization%if given only start/stop times, fill in a default stepsizeif length(times) == 2

times = [times(1):0.05*(times(2)-times(1)):times(2)];end

times = times’;

y = y0(:);yout = y.’;

% The main loopfor i = 1:(length(times)-1)

t = times(i);h = times(i+1)-times(i);

% Compute the slopess1 = feval(FunFcn, t, y,m,o,pvar); s1 = s1(:);

t = t + h;y = y + h*s1;yout = [yout; y.’];

end;

%**************************************************************************%**************************************************************************%**************************************************************************function [tout, yout] = rkheulerA(FunFcn, times, y0, m,o)%this is SO MUCH slower that I can’t use it. It guarentees good numerical%behavior. But the problem is stiff, so the step sizes get too small. I%would need a more sophisticaed adaptive stepsize method than this.

% Initializationt = times(1);tfinal = times(end);h = 0; %value gets replaced by maxstepsize before use

y = y0(:);tout = t;yout = y.’;

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%id all of the components with nonzero rates%notzero = [1:13 16:28 31:43 46:58 61:73];

% The main loopwhile (t < tfinal)

if t + h > tfinal, h = tfinal - t; end

% Compute the slopess1 = feval(FunFcn, t, y,influent,m,o); s1 = s1(:);

%compute the maximum stepsize that is safe to takenotzero = find(s1>0);maxstepsize = min(abs(y(notzero)./s1(notzero)));

h = 0.8*maxstepsize;

t = t + h;y = y + h*s1;tout = [tout; t];yout = [yout; y.’];

end;

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D.8 Code for PlotKEKFData.m

%Plot KEKF Data%Required tout allX errbars

reactorindexNW = 46:60;reactorindexSE = 136:150;

%parse filtered data into reactorsrctrhat = fParseAllX(tout,allX,’Filtered - ’);

%if we are using simulated data, then we have the actual state variable%valuesif simdata

rctr = fParseAllX(ts,allXs,’Simulation - ’);end

%to get some information about the covariances I’ll plot elements of P%here I’m computing the indices of the diagonal elements from each :,:,i%matrix of Perrbars = zeros(size(allX));psize = size(Pout,3);if estparams

reactorindexNW = [reactorindexNW 151:154];reactorindexSE = [reactorindexSE 151:154];

endfor i = 1:nsv

pndx = (i+nsv*(i-1)):nsv^2:psize*nsv^2;errbars(:,reactorindexNW(i)) = sqrt(Pout(pndx)’);

end%errbars is now the same dimensions as allX, and each element is the std%deviation of the corresponding element in allX. Break it up into reactors%for plotting.rctrlowererror = fParseAllX(tout,allX-errbars,’lowererror - ’);rctruppererror = fParseAllX(tout,allX+errbars,’uppererror - ’);%now if I plot all three reactors on the same graph I get xhat with upper%and lower error bars.

%WAKE UP!!! Look, select out the reactors you want to plot, combine them%together into cell arrays with their error bars and pass them to fplot%Because of the goofiness of working with cell arrays, and my lack of%creativity in solving this, I’ll ’preallocate’ rctrplot by assigning it to%rctrhat, then replace each element of rctrplot with what I want to%actually plot for that reactor.rctrtemp = rctrhat;

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for i = plotreactorsrctrtemp(1) = rctrhat(i);rctrtemp(2) = rctruppererror(i);rctrtemp(3) = rctrlowererror(i);if simdata rctrtemp(4) = rctr(i); endfigure(i+20);clf;fPlotTankEKF(rctrtemp,z);figure(i+40);clf;fPlotEnEKF(rctrtemp,z);

end

% figure(6)% fPlotKOp(kops);

figure(7)fPlotComp(rctrhat,’So’);for i = 1:5

[m, o] = fKanapahaModelOpsSetup(false);subplot(5,1,i);line([0 7],[m.Koa m.Koa],’Linestyle’,’--’,’Color’,’k’);line([0 7],[m.Koh m.Koh],’Linestyle’,’--’,’Color’,’k’);

end

figure(8)fPlotComp(rctrhat,’Snh’);

figure(9)set(9,’Name’,’All Variables for Tank’);fPlotTankAllVars(rctrhat(4))

% figure(10)% fPlotComp(rctr,’Ss’);

if ~simdatafigure(28)set(gcf, ’Name’, ’Compare Probe Data’);fPlotCompareProbes(rctrhat(4),rctrhat(10));

end

%**************************************************************************%plot the estimated parametersif ~runASM1 && estparams

figure(100)set(gcf, ’Name’, ’Estimated Parameters with Error Bars’);

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h=subplot(4,1,1);plot(tout/24,allX(:,151),’-k’, ...

tout/24,allX(:,151)+errbars(:,151),’-m’, ...tout/24,allX(:,151)-errbars(:,151),’-m’)

ylabel(’K 1’);%set(h,’YLim’, [0 1.2*max(allX(:,151)+errbars(:,151))]);

subplot(4,1,2);plot(tout/24,allX(:,152),’-k’, ...

tout/24,allX(:,152)+errbars(:,152),’-m’, ...tout/24,allX(:,152)-errbars(:,152),’-m’)

ylabel(’K 2’);

h=subplot(4,1,3);plot(tout/24,allX(:,153),’-k’, ...

tout/24,allX(:,153)+errbars(:,153),’-m’, ...tout/24,allX(:,153)-errbars(:,153),’-m’)

ylabel(’aN’);set(h,’YLim’, [0 1.2*max(allX(:,153)+errbars(:,153))]);

h=subplot(4,1,4);plot(tout/24,allX(:,154),’-k’, ...

tout/24,allX(:,154)+errbars(:,154),’-m’, ...tout/24,allX(:,154)-errbars(:,154),’-m’)

ylabel(’b E N’);set(h,’YLim’, [0 1.2*max(allX(:,154)+errbars(:,154))]);

orient tall;wysiwyg;

%**********************************************************************figure(101)set(gcf, ’Name’, ’Scaled Estimated Parameters with Error Bars’);h=subplot(4,1,1);stemp = allX(1,151);plot(tout/24,allX(:,151)/stemp,’-k’, ...

tout/24,(allX(:,151)+errbars(:,151))/stemp,’-m’, ...tout/24,(allX(:,151)-errbars(:,151))/stemp,’-m’)

ylabel(’K 1’);%set(h,’YLim’, [0 1.2*max(allX(:,151)+errbars(:,151))]);

subplot(4,1,2);stemp = allX(1,152);plot(tout/24,allX(:,152)/stemp,’-k’, ...

tout/24,(allX(:,152)+errbars(:,152))/stemp,’-m’, ...tout/24,(allX(:,152)-errbars(:,152))/stemp,’-m’)

ylabel(’K 2’);

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h=subplot(4,1,3);stemp = allX(1,153);plot(tout/24,allX(:,153)/stemp,’-k’, ...

tout/24,(allX(:,153)+errbars(:,153))/stemp,’-m’, ...tout/24,(allX(:,153)-errbars(:,153))/stemp,’-m’)

ylabel(’aN’);set(h,’YLim’, [0 1.2*max(allX(:,153)+errbars(:,153))]/stemp);

h=subplot(4,1,4);stemp = allX(1,154);plot(tout/24,allX(:,154)/stemp,’-k’, ...

tout/24,(allX(:,154)+errbars(:,154))/stemp,’-m’, ...tout/24,(allX(:,154)-errbars(:,154))/stemp,’-m’)

ylabel(’b E N’);set(h,’YLim’, [0 1.2*max(allX(:,154)+errbars(:,154))]/stemp);

orient tall;wysiwyg;

%**********************************************************************figure(102)set(gcf, ’Name’, ’Estimated Parameters’);subplot(4,1,1);plot(tout/24,allX(:,151),’-k’)ylabel(’K 1’);

subplot(4,1,2);plot(tout/24,allX(:,152),’-k’)ylabel(’K 2’);

subplot(4,1,3);plot(tout/24,allX(:,153),’-k’)ylabel(’aN’);

subplot(4,1,4);plot(tout/24,allX(:,154),’-k’)ylabel(’b E N’);

orient tall;wysiwyg;

%**********************************************************************figure(103)set(gcf, ’Name’, ’Scaled Estimated Parameters’);subplot(4,1,1);plot(tout/24,allX(:,151)/allX(1,151),’-k’)

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ylabel(’K 1’);

subplot(4,1,2);plot(tout/24,allX(:,152)/allX(1,152),’-k’)ylabel(’K 2’);

subplot(4,1,3);plot(tout/24,allX(:,153)/allX(1,153),’-k’)ylabel(’aN’);

subplot(4,1,4);plot(tout/24,allX(:,154)/allX(1,154),’-k’)ylabel(’b E N’);

orient tall;wysiwyg;

end

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BIOGRAPHICAL SKETCH

Ryan Hamilton received his B.S. degree in chemical engineering from Georgia

Tech in Atlanta, Georgia in 2000. Since 2001 he has been a graduate student in

the Chemical Engineering Department at the University of Florida in Gainesville,

Florida and expects to receive his doctorate degree in 2005. His research interests

include modeling bacterial metabolism, parameter estimation, and process

optimization.

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