Optimal Control of Small Size Single Tank Activated Sludge Process

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Process Systems Engineering

Optimal control of small size single tank activated sludge process with regulated aeration and external carbon addition Pallavhee Tamizhchelvan, Sundaramoorthy Sithanadam, and Sivasankaran M.A Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04519 • Publication Date (Web): 26 Oct 2018 Downloaded from http://pubs.acs.org on November 6, 2018

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Industrial & Engineering Chemistry Research

Optimal control of small size single tank activated sludge process with regulated aeration and external carbon addition. T.Pallavhee +, S.Sundaramoorthy +*, M.A.Sivasankaran $ +

Department of Chemical Engineering, Pondicherry Engineering College, Pondicherry,

605014, India $

Department of Civil Engineering, Pondicherry Engineering College, Pondicherry, 605014,

India KEYWORDS. Activated sludge Process, Optimal Control, Feedforward Control, Single tank ASP, ASM1 ABSTRACT: Appropriate control of activated sludge process is crucial for achieving desired effluent quality in the treatment of municipal and industrial wastewater. This paper deals with the development of a feedforward open loop optimal control strategy for a small size single tank ASP system used for the treatment of small volumes of industrial wastewater. The work presented in this paper is a deviation from the usual alternating mode of operation reported in the literature for single tank ASP systems, in which the oxygen supply to the aeration tank is switched on and off in a cyclic manner. The present work has highlighted that regulating oxygen supply rate alone is

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ineffective for achieving desired quality of treated effluent. Further, the need for adding soluble carbon from an external source in addition to regulating the oxygen supply rate is emphasized in this work for effective control of COD and TKN in the treated effluent stream. The proposed control scheme is tested on a single tank ASP model reported in the literature and is found to perform well for various load conditions. Appropriate tuning of controller parameters ensures that the COD and TKN levels do not exceed the limiting values and at the same time the desired effluent quality is achieved at a reduced cost associated with oxygen supply and external carbon addition. 1. INTRODUCTION Activated Sludge Process (ASP) is a secondary treatment process used for the reduction of organic carbon (COD) and ammonical nitrogen (TKN) present in municipal and industrial waste water. A Conventional ASP system has multiple biological reaction vessels connected in series, of which a few are operated in aerobic mode and others in anoxic mode. COD and TKN are reduced in the biological reaction vessels by the actions of two class of microorganisms namely heterotrophs (XBH) and autotrophs (XBA). In the aerobic tank, heterotrophs and autotrophs remain active, with heterotroph’s activity resulting in the reduction of COD and autotrophs converting ammonical nitrogen to nitrate nitrogen (nitrification). In the anoxic tank, the heterotrophs knock off oxygen from the nitrate (denitrification) and uses it for its own biological activity, which in turn results in further reduction of COD. Further, TKN level also gets reduced due to the conversion of nitrate to nitrogen. Thus, reduction in both COD and TKN levels are achieved in the conventional ASP system by appropriate regulation of oxygen.


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A benchmark simulation model no.1 (BSM1) is proposed in the literature1 for the purpose of carrying out control related studies on the ASP system consisting of five tanks (3 – aerobic and 2 – anoxic) and a settler. A number of studies have been reported in the literature on implementation of various control strategies on ASP using this BSM1.Most of these studies mainly focused on controlling carbon and nitrogen in the treated effluent by regulating oxygen supply rate2 - 5 to the aerobic tank and internal recirculation rate6 - 8 to the anoxic tank . Although earlier studies on control of ASPs have reported the application of conventional PI controllers 2, 56,

most of the recent works have used advanced control strategies such as model predictive

controllers8-14 and intelligent controllers like fuzzy controllers15, 16 and neural network controllers17.Self-optimizing strategy for economic selection of controlled variables in the activated sludge process system has been extensively studied and reported in the literature18, 19. A few articles are also found in the literature9, 17, 20 on regulation of external carbon addition into the anoxic tank for effective control of nitrogen level in the treated effluent. For the treatment of small volumes of industrial wastewater, it is proposed in the literature21 to use a single tank ASP system, which has one biological reaction vessel operated in alternating modes. This is a cyclic operation in which the same tank is operated alternately in both aerobic and anoxic mode. In the aerobic mode of operation, the heterotroph consumes organic carbon as its nutrient and reduces the COD level, whereas the autotroph converts ammonical nitrogen into nitrates. Although the COD level decreases in this mode of operation, the TKN level will either remain unaltered or increase slightly due to continuous feed of ammonical nitrogen into the tank. In anoxic phase of operation, the heterotroph converts the nitrate to nitrogen, utilizing the oxygen present in the nitrates for its growth. Thus, a reduction in the TKN level happens in the anoxic mode of operation. It is possible to achieve a better control of COD and TKN levels in the


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treated effluent by suitably adjusting the cyclic times of aerobic and anoxic phases in the alternating mode single tank ASP system. Hyunook Kim et al22 were the first investigators to propose alternating mode of control for a single tank ASP system. In their work, they have used the linearized version of ASM1 model to optimize the aeration time and the fraction of the aeration cycle time to control the effluent ammonia concentration in a bench scale activated sludge system. Chachuat et al21 have developed a control strategy using nonlinear ASM1 model (model with 11 state variables) for alternating single tank ASP system, in which the sequence of aeration / non aeration times are optimized to achieve the desired effluent quality with minimum energy consumption .In their work, the control problem was posed as a constrained hybrid dynamic optimization problem . Chachuat et al23 have further extended their work to address long term optimal aeration strategy to avoid biomass wash out condition. Cristea et al24 and Holenda et al 11 have reported the application of model predictive control technique in alternating single tank activated sludge process. Balku25 has used ASM3 model for single tank ASP system and developed an evolutionary based optimization algorithm to optimize the aeration schedule in alternating mode of operation. In all these cases, the control objective is formulated as constrained dynamic optimization problem that requires regress computational effort and time for solution. Although single tank ASPs are usually operated in alternating modes, Pallavhee et al26 has recently proposed a control strategy developed in the frame work of optimal control for regulation of oxygen supply rate in a single tank ASP system. The alternating mode of operation is considered as a special case of the optimal control scheme proposed by Pallavhee et al25, as it is likely to push the operation of single tank ASP to an alternating mode whenever necessary. In their work, a single tank ASP system with a higher loading of organic carbon and total nitrogen


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in the waste water feed, which is three times that of the values reported by Chachaut et al 21was considered. The results of their study showed that, although it was possible to maintain the TKN level in the treated effluent within the permissible limit by regulation of oxygen supply rate in most of the cases, the proposed scheme failed to control the effluent TKN level for some specific cases of load changes. The failure of optimal control strategy was reported for the cases in which either the influent COD is reduced or influent TKN is increased from their nominal values. The cause of the failure in these cases may be attributed to the inadequacy of soluble carbon in the tank required to sustain the growth of heterotrophs for the reduction of TKN level. It is to be noted that, either in the absence of oxygen or reduced levels of oxygen in the tank, the reduction in TKN is strongly influenced by the rate of growth of heterotrophs, which in turn depends upon the availability of nitrates (oxygen source) and organic carbon (COD) in sufficient levels. So, it is necessary to maintain the organic carbon in the tank at requisite level by addition of soluble carbon from an external carbon source. Thus, in order to effectively control the COD and TKN levels in the treated effluent, it is required to regulate the rate of addition of external carbon in addition to regulating oxygen supply rate. In this work, control of single tank activated sludge process with regulation of both oxygen supply rate and external carbon addition rate is proposed. Control strategy is developed in the optimal control frame work in which both oxygen supply rate and external carbon addition rate are regulated to maintain effluent COD and effluent TKN well within their permissible limits for all load conditions. The dynamic model of single tank ASP reported by Chachaut at al21 is used in this work for testing the proposed control strategy. The kinetics of biochemical reactions occurring in the aeration tank are described using Activated Sludge Process Model No.1 (ASM 1) reported by Henze et al27. The state and load variables are assumed to be measured and


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monitored, with the optimal control calculations repeated and implemented each time when the load change occurs. Thus, the optimal control scheme implemented in this work is a feed forward open loop type. To the best of our knowledge there is no earlier work reported in the literature on the control of single tank ASP system which has emphasized on the need for addition of external carbon for effective control of effluent TKN. Further, unlike most of the work reported in the literature on control of single tank ASP system that employs alternating mode of operation, the present work has formulated the control problem in the optimal control framework and solved it as an unconstrained optimization problem using pontrygyin’s minimum principle. The remaining of this paper is organized as follows: After a brief description of single tank activated sludge process in Section 2, the state space model equations for the process are presented in Section 3 along with definitions of state variables, load variables and input variables. In Section 4, detailed mathematical derivations for the proposed optimal control strategy are presented along with the computational scheme and implementation strategy. The results obtained for various case studies are presented and discussed in Section 5. The conclusions drawn from this study are summarized in Section 6


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Table 1. List of symbols ( Excluding the state variables defined in Table 3) Symbol

Description

Units

Symbol

Description

Units

ASM1

Activated Sludge Model No.1

-

TKNIN

Influent TKN

gN/m3

ASP

Activated Sludge Process

-

TKNMAX

Maximum permissible limit value on TKN

gN/m3

BSM1

Benchmark Simulation Model No.1

-

TKNN

Nominal TKN value

gN/m3

COD

Chemical Oxygen Demand

gCOD/m3

TKNindex

Performance index defined in eq.37

-

CODindex

Performance index defined in eq 36

-

tf

Optimal control time span

hours

CODIN

Influent COD

gCOD/m3

Ts

Sampling time

minutes

CODMAX

Maximum permissible limit value on COD

gCOD/m3

u

Inputs vector

-

CODN

Nominal COD Value

gCOD/m3

U1index

Cost index defined in eq 38

-

cS

Concentration of soluble carbon

gCOD/m3

U2index

Cost index defined in eq 39

-

d

Disturbances vector

-

V

Aeration tank volume

m3

H

Hamiltonian Function

-

xi

ith state variable in state variable vector

g/m3

iXB

(Mass of N)/(Mass of COD) in biomass

-

X

State variables vector

-

iXP

(Mass of N)/(Mass of COD) produced from biomass

-

λ

Co – state variable vector

-

J

Objective function

-

1

Set point Tuning parameter defined in eq 13

-

kLa

Oxygen transfer coefficient

h-1

2

Set point Tuning parameter defined in eq 14

-

P_COD

Perturbation in COD defined in eq 33

-

COD

Weight on deviation in effluent COD from its set point value

-

P_TKN

Perturbation in TKN defined in eq 34

-

TKN

Weight on deviation in effluent TKN from its set point value

-

P_Q

Perturbation in flowrate defined in eq 35

-

1

Normalized weight on deviation in effluent COD from its set point value defined in eq 16

-

QMAX C

Maximum external carbon addition rate

m3/day

2

Normalized weight on deviation in effluent TKN from its set point value defined in eq 17

-

QRS

Sludge recycle rate

m3/day

1

Weight on input u1

-

QIN

Average Influent Flow rate

m3/day

2

Weight on input u2

-

QW

Excess sludge rate

m3/day



Error tolerance limit defined in Step5

-

MAX SO

Dissolved oxygen saturation concentration

mg/l

δ

Error tolerance limit defined in Step6

-

TKN

Total Keldjal Nitrogen

gN/m3

τ

Hydraulic retention time

Hours


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2. DESCRIPTION OF SINGLE TANK ASP A small size activated sludge process (ASP) reported in the literature21 for 15,000 p.e is taken for control relevant studies envisaged in this work. The process shown in Figure 1 has a single aeration tank of volume V= 2050 m3 and is designed to handle an average influent flowrate of QIN = 3050 m3/day. The tank is equipped with surface aerators to supply oxygen at a maximum rate corresponding to an oxygen transfer coefficient value of kLa= 4.5 h-1. The ASP system is provided with appropriate mechanism to adjust the feed rate of oxygen as a fraction u1 (0 ≤ u1≤1) of the maximum oxygen supply rate. In addition to regulating the oxygen supply rate for treatment of wastewater in the aeration tank, regulated addition of soluble carbon from an external carbon source is also considered in this work. A solution containing sodium acetate/ethanol20 having soluble carbon at a concentration of cS = 700 mg COD/l is added as an external carbon, which is pumped into the tank at a maximum rate of QMAX = 1500 m3/day. The C rate of addition of external carbon, is regulated as a fraction u2 (0 ≤ u2 ≤1) of QMAX . C The effluent from the tank is fed to a settler, in which the treated effluent is drawn out from the top and the solids are removed from the bottom. It is assumed that the treated effluent is a clear liquid free from particulates. A portion of the solids coming out of the settler is removed as excess sludge at a rate QW = 75m3/day and the rest of the solids is recycled to the aeration tank at a rate of QRS = 7500 m3/day. In this work, a higher loading of organic carbon CODIN = 1000mg/l and total nitrogen TKNIN = 100 mg/l in the wastewater feed is considered as compared to the values of CODIN = 343 mg/l and TKNIN = 33 mg/l reported in the literature21.


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External Carbon Addition (u2) QIN

QIN + QRS

Oxygen Supply (u1)

𝑥𝐼𝑁 𝑖 QIN+QRS+u2 QMAX C

𝑥𝑖 V

QIN + u2QMAX – QW C Treated Effluent (Liquid)

xi Settler QRS + QW

Aeration Tank

𝑥𝑅𝑖

QRS 𝑥𝑅 𝑖

QW

(Only Particulates)

𝑥𝑅𝑖

Excess Sludge (Particulates)

Figure 1. Schematic diagram of Activated Sludge Process with regulated aeration and external carbon addition.


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Refer Table 1 for nomenclature .The composition of influent wastewater containing various constituents of carbon and nitrogen is specified as fractions of COD and TKN as given in Table 2. Table 2. Waste water Composition COD fractions

TKN fractions

f XBH = 0.14

fSNH = 0.66

f XBA = 0.01

fSND = 0.02

fSS = 0.35

f XND = 0.32

f XS = 0.35

fSND = 0

fSI = 0.05 f XI = 0.1 f XP = 0

3. DYNAMIC MODEL OF SINGLE TANK ASP Activated Sludge Process Model no.1 (ASM1) reported by Henze et al27 is used to describe the kinetics of biochemical reactions occurring in the aeration tank. The aeration tank dynamics is represented by state space model of the form dX  f ( X , u, d ) dt

(1)

Where, X is the state vector having 13 state variables defined in Table 3 and u is the input vector of two manipulated variables u1 and u2.


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X  [ S I S S X I X S X BH X BA X P So S NO S ND S NH X ND S ALK ]T

(2)

u  [u1u2 ]T

(3)

Table 3. Definition of State Variables 1

SI

Inert Soluble Organic Matter (gCOD/m3)

2

SS

Readily biodegradable substrate (gCOD/m3)

3

XI

Inert particulate organic matter and products(gCOD/m3)

4

XS

Slowly biodegradable substrate (gCOD/m3)

5

XBH

Active heterotrophic biomass(gCOD/m3)

6

XBA

Active autotrophic biomass(gCOD/m3)

7

XP

Inert Particulate matter(gCOD/m3)

8

SO

Dissolved oxygen (gO2 / m3)

9

SNO

Nitrate and Nitrite Nitrogen( gN/m3)

10

SNH

Ammonium nitrogen( gN/m3)

11

SND

Soluble biodegradable organic nitrogen (gN/m3)

12

XND

Particulate biodegradable organic nitrogen(gN/m3)

13

SALK

Alkalinity ( molar units)

The load vector d has 16 variables, defined below

IN IN IN IN IN IN IN d   S IIN S SIN X IIN X SIN X BH X BA X PIN SOIN S NO S NH S ND X ND S ALK Q IN Q RS QW 

T

(4)

IN IN IN IN IN IN IN IN The first 13 variables of the load vector d namely SIN I , SS , X I , X S , X BH , X BA , X P , SO , SNO , IN IN IN SIN NH , SND , X ND and SALK correspond to the composition of influent wastewater. In this study, the


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dissolved oxygen and alkalinity content in the influent wastewater are assumed to remain constant at SOIN = 0.2 mg/l and SIN ALK = 7 mg/l . The values of the remaining 11 variables are assumed to vary as fixed fractions of the influent COD (CODIN) and influent TKN (TKNIN) given in Table 2. This assumption implies that the characteristics of wastewater is not influenced by the fluctuations in CODIN and TKNIN. Further, the fluctuations in feed flow rate (QIN) are assumed to be transmitted as it is as fluctuations in QRS and QW in the load vector. Thus, the 16 load variables defined in the load vector are functions of fluctuations in influent COD (CODIN), influent TKN (TKNIN) and influent flow rate (QIN). The dynamic model for the aeration tank is derived by writing unsteady state balance equations for soluble components, particulates and dissolved oxygen. Define xi = X (i) as the ith component of the state vector X (i). For soluble carbon (i=2)

 Q MAX dxi Q IN IN  xi  xi  ri  X    C dt V  V





  u2 (cS  xi ) 

(5)

For soluble components (i = 1, 9.10, 11, 13)

 Q MAX dxi Q IN IN  xi  xi  ri  X    C dt V  V





  xi u2 

(6)

For particulate components (i= 3, 4, 5, 6, 7, 12)

 QCMAX   QW  dxi Q IN IN Q RS  Q IN  QW   xi  xi  x  r X  ux      RS   i i RS W  2 i dt V V  Q  QW   V  Q  Q 





(7)


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For dissolved oxygen (i=8)

 Q MAX dxi Q IN IN  xi  xi  ri  X   u1k L a SoMAX  xi   C dt V  V









  u2 xi 

(8)

In the above equations, ri(X) is the kinetic rate equation of the ith component defined in ASM127. The output variables of the state space model are the COD and TKN in the effluent from the aeration tank. The following equations relate these output variables to the state variables.

COD  S I  S S  X I  X S  X BH  X BA  X P TKN  S NO  S NH  S ND  X ND  iXB  X BH  X BA 

(9) (10)

It is assumed in this work that, the settler responds instantaneously to the changes in the feed conditions without any time delay. As no particulate component escapes in the clear liquid coming out of the settler , only the dissolved compounds present in the fluid contribute to the COD and TKN of the treated effluent .Thus, the COD and TKN of the effluent from the settler are as follows.

CODe  S I  S S

(11)

TKN e  S NO  S NH  S ND

(12)

4. OPTIMAL CONTROL OF SINGLE TANK ASP The objective of single tank ASP control is to maintain the quality of the treated effluent within the permissible limits of COD (CODMAX = 125mg/l) and TKN (TKNMAX = 18 mg/l) by regulating the oxygen supply rate (u1) and external carbon addition rate (u2) despite the


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fluctuations in the load variables. In this work, an attempt has been made to formulate and solve the control problem in the optimal control frame work. In this, the manipulated variables u1(t) and u2(t) are calculated over a specified time span tf in such a way that the effluent COD( CODe) and effluent TKN(TKNe) are maintained at values as close as possible to the corresponding desired values (set points) CODS and TKNS respectively, with minimum feed rates of oxygen and external carbon . The set point values CODS and TKNS are taken as some specified fractions of CODMAX and TKNMAX.

CODS  1CODMAX

(13)

TKN S   2TKN MAX

(14)

Thus, the objective function to be minimized in the optimal control problem is stated as follows. tf

2 2 2 2 J [u (t )]    1  CODS  CODe  t     2 TKN S  TKN e  t     1u1  t    2u2  t  dt   0

(15)

β1 and β2 are the weights assigned respectively to the deviations in CODe and TKNe from their corresponding set point values. These weights are normalized with respect to the limiting values as follows.



 1 2   CODMAX 

1  COD 

(16)



 1 2   TKN MAX 

 2  TKN 

(17)

Similarly γ1 and γ2 are the weights assigned to the manipulated variables u1 and u2 respectively. The optimal control policy is to calculate the manipulated variables u1(t) and u2(t) in such a way


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that the objective function J[u(t)] ( eq.15) is minimum over a specified time span tf starting from the time when the load variable changes occur. Larger the values assigned to the weights βCOD and βTKN, smaller are the acceptable deviations in CODe and TKNe from their respective set point values. Thus, higher values of βCOD and βTKN will tend to keep CODe and TKNe closer to the set point values.α1 and α2 are assigned values in the range of 0.6 - 0.9, so as to ensure that the quality parameters remain well within the limiting values even if they shoot over the set point values. Reducing the values of α1 and α2 will lower the set point levels of CODS and TKNS below their limiting values. This, in turn would put a larger demand on the effluent quality. Thus, by assigning appropriate values to the parameters βCOD , βTKN , α1 and α2 , one can specify the level of treated effluent quality that the optimal control policy is required to maintain. The parameters γ1 and γ2 penalize large and abrupt changes in the control actions u1 and u2 respectively. An higher value assigned to γ1, will result in lower oxygen supply rate and lesser consumption of turbine power. Similarly, an higher value of γ2 will tend to lower the rate of external carbon addition and its associated cost. It is possible to achieve a tradeoff between quality of treated effluent and the cost associated with oxygen as well as external carbon supply rates by choosing appropriate values for the weights. Thus the optimal control scheme can be made to maintain the treated effluent quality within the permissible limits with minimum consumption of energy. The optimal control problem can also be formulated as an inequality constrained optimization problem by setting upper limits on CODe and TKNe as their corresponding maximum permissible values CODMAX and TKNMAX respectively. Rigorous computational schemes are


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reported in the literature28 for solving such optimal control problems with path constraints. However the rigor of computation involved in solving the inequality constrained optimal control problem makes the online implementation of such control schemes quite tedious. To overcome this difficulty, the optimal control problem is formulated in this work as an unconstrained optimization problem in which, the tuning parameters α1 and α2 are introduced to set the desired levels of CODe and TKNe well below their permissible limits. Thus, the proposed optimal control scheme is designed to ensure that the CODe and TKNe do not cross their permissible limits. Use of tuning parameters α1 and α2 to simplify the rigor of solving constrained optimization problem has a practical appeal from the point of implementation of the control strategy. These tuning parameters offer a handle to the user to appropriately fix the desired level of effluent quality that could be achieved with least energy consumption. The optimal solution thus achieved depends upon the values of tuning parameters chosen. The solution of the optimal control problem proposed in this work is also compared with that of the inequality constrained optimal control formulation ( Section 5.6 Case study 6) in order to highlight the merits and demerits of these two approaches . The choice of time span tf will determine the time duration within which the deviations in CODe and TKNe from their respective set point values should settle down and remain at values closer to zero. A smaller time span tf will force the optimal control policy to be more aggressive in its action to reduce the error values to zero in a shorter duration of time. This will tend to make the response more oscillatory and even unbounded at times. Choosing a larger value for tf, will allow sufficient time for the deviations to settle down and achieve a stable response. Thus, an appropriate choice of tf is necessary to ensure stable operation.


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4.1 SOLUTION TO THE OPTIMAL CONTROL PROBLEM The optimal control problem is solved using Pontryagin’s minimum principle29 .In the general frame work of optimal control, the objective function J [u (t)] is defined as:





tf

J u  t    G X  t f    F  X , u  dt



Where G X  t f

(18)

0

  and F  X , u  for this particular problem, taking the objective function is

defined in eq 15, are



G X t f

  0

(19)

F  X , u   1  CODS  CODe  t     2 TKN S  TKN e  t     1u1  t    2u2  t  2

2

2

2

(20)

The Hamiltonian function H (t) for this problem is 13

H  F  X , u   i f i  X , u 

(21)

i 1

Where λi(t) are co-state variables corresponding to 13 state equations defined in eqs 1,5-8. The necessary conditions for optimality are H 0 u

(22)

and


17


d  T (t )  H     dt  X 

Page 18 of 54

(23)

Where λT (t) is a vector of 13 co-state variables, λT (t) = [λ1 λ2 λ3… λ13]

(24)

and

 fi  H  F     i   xi  xi   xi 

In eq 22,

∀ i = 1, 2…13

(25)

H is the vector of partial derivatives of H with respect to u1 and u2 u

H  2 1u1  t   8 k L a SoMAX  x8 u1



 Q MAX H  2 2u2  t    C u2  V



(26)

  1 x1  2  x2  cS   8 x8  9 x9  10 x10  11 x11  13 x13  

  QCMAX   QW    3 x3  4 x4  5 x5  6 x6  7 x7  12 x12    V  QRS  QW 

(27)

Transversality condition for this case, in which tf is specified and X (tf) is free, is given as G  T t f X



at t = tf

(28)

As, G  0 at t  t f X

(29)


18

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We have, i  t f   0  i  1, 2, 313

(30)

Equation 30 specifies the values of co-state variables at the final time t = tf for solving 13 ordinary differential equations (co-state equations) in co-state variables defined in eq 23. The manipulated variables u1(t) and u2(t) are bound by upper and lower bounds defined below

0  u1 (t )  1 , 0  u2 (t )  1

(31)

Thus, for minimization of J, 0 H    0 ui   0

ui  t   1 0  ui  t   1 ui  t   0

∀ i = 1, 2

(32)

The optimal control problem formulated in this work is solved using control vector iteration method28. The solution steps are as follows Step1: Take u1  t  = u1  t  and u 2  t  = u 2  t  0 < t < tf Where u1(t) and u2(t) are the initial guess values taken as constants. Step2: Solve 13 state equations (eq 1) by forward integration starting from t=0 to t = tf with the known initial conditions X (0)  X 0 to obtain X (t ) . Step3: With the calculated values of X (t) and known values of u1 (t) and u 2 (t) , integrate 13 co-state equations (eq 23) by backward integration starting from t = tf to t = 0 with specified final condition λi(tf) = 0 to calculate λ(t).


19


Step4: Calculate

Page 20 of 54

H H and using eq 26 and eq27 u1 u2

 H  Step5: Calculate new values of u1(t) and u2(t) as, u1new (t )  u1 (t )  1   and  u  1  H  u2 new (t )  u2 (t )   2   , where 1 and  2 are positive small values for minimization of  u2  J[u(t)]. Correct u1new(t) and u2new(t) to satisfy the bounds as follows.  1  u1new  t    u1new  t   0 

if u1new  t   1 if 0  u1new  t   1 if u1new  t   0

 1  u2 new  t    u2 new  t   0 

if u2 new  t   1 if 0  u2 new  t   1 if u2 new  t   0

tf

Step6: Check if

 u

1new

0

(t )  u (t )    2

tf

and  u2 new (t )  u (t )    , where δ is user specified 2

0

tolerance limits. Step7: Take u1 (t )  u1new (t ) and u2 (t )  u2 new (t ) If condition in Step6 is satisfied STOP If condition in Step6 is not satisfied go to Step 2.


20

Page 21 of 54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.2 IMPLEMENTATION OF OPTIMAL CONTROL STRATEGY At the startup of ASP operation (t = 0), it is assumed that the aeration tank is already stabilized and the state variables are maintained at initial values reported 21 in Table 4. It is also assumed that all the load variables are measured and monitored online. As and when there is a perturbation in the load variables, the optimal control problem (Section 4.1) is solved and implemented. Thus, the optimal control strategy implemented in this work is a feed forward open loop type. Table 4.Initial concentrations (mg/l) in the aeration tank SI

=

17.8

SO

=

0.0

SS

=

2.27

SNO

=

0.02

XI

=

2120.15

SNH

=

9.7

XS

=

79.55

SND

=

0.14

XBH

=

2239.65

XND

=

6.29

XBA

=

115.18

SALK

=

7.0

XP

=

0.0

Although the optimal control problem is formulated in continuous time domain, the state and co-state equations (1 , 20) are solved in discrete time, taking a sampling time of TS = 10 mins. MATLAB routine ode45 is used for solving the differential equations. For the known initial values of state and load variables, the optimal control problem is solved by assigning appropriate values for final time tf and for the weights α1 , α 2 , β COD , β TKN , γ1 and γ 2 .In solving the optimal control problem, it is assumed that the load variables remain at the initial values and do not


21


Page 22 of 54

change upto the final time tf. Optimal control trajectories u1(t) and u2(t) calculated for 0 < t < t f are implemented starting from an initial time t = 0 upto a final t = tf or upto a time when a new load change (CODIN/TKNIN/QIN) occurs. This time, at which either the final time tf is reached or a load change occurs, is reset as initial time t = 0 and the corresponding values of state and load variables are reset as initial values. With these reset values of state and load variables, the optimal control problem is solved again to calculate u1(t) and u2(t) for 0 < t < t f . Thus, the optimal control problem is solved and implemented in repeated cycles. At the start –up of each calculation cycle it may be noted that, the initial values of the state variables are reset as equal to the final values at the end of the previous cycle. Thus the initial state variable values, calculated using the plant model are functions of the past control actions as well as the past states. Therefore, the effect of past control actions is implicitly considered in this proposed method of optimal control calculations. Perturbations in the load variables CODIN, TKNIN and QIN around their respective nominal values of CODN = 1000 mg/l, TKNN = 100 mg/l and QN = 3050 m3/day are given by the following equations COD IN  CODN * P _ COD

(33)

TKN IN  TKN N * P _ TKN

(34)

Q IN  QN * P _ Q

(35)

Where P_COD, P_TKN and P_Q are perturbation factors. In this work, perturbations in load variables of magnitudes ranging between 50% above and 50% below the nominal values are considered. Thus the values of P_COD, P_TKN and P_Q are varied in the range of 0.5 to 1.5.


22

Page 23 of 54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The performance of the optimal control strategy in maintaining the effluent quality at the desired level are measured in terms of dimensionless performance indices defined below: tf

1 CODindex  2 t f CODMAX 1 TKNindex  2 t f TKN MAX

 (COD

S

 CODe (t )) 2 dt

(36)

0

tf

 (TKN

S

 TKN e (t )) 2 dt

(37)

0

The cost indices associated with oxygen supply rate and external carbon addition rate are measured as tf

1 U 1index   u1 (t ) 2 dt tf 0

(38)

tf

1 U 2index   u2 (t ) 2 dt tf 0

(39)

5 .RESULTS AND DISCUSSION Simulation studies are carried out to test the effectiveness of the proposed optimal control strategy in keeping the treated effluent quality within the permissible limits. The results of various case studies are presented in this section. 5.1 Case Study 1 An investigation is made to find out if it is possible to achieve appropriate control of single tank ASP by regulating only the supply of oxygen to the aeration tank (u1) and without any external carbon addition (u2 = 0) .In this, the load variables are kept at their nominal values . The values of


23


Page 24 of 54

α1 and α2 are taken as α1 = 0.9 and α2 = 0.8. The weights are fixed at values βCOD = 0.08, βTKN = 15 and γ1= 0.05. The hydraulic retention time in the aeration tank for the nominal influent rate (QIN = QN) is τ = 4.67 hours. It usually takes a time of 4τ for the effect of any disturbance to settle down. So, a time span of tf = 20 hours is selected. The dynamic response of the ASP with optimal regulation of oxygen supply rate is shown in Figure 2. CODe is maintained at values far below its set point. However, TKNe is found to shoot over its set point value but remain below its permissible limit. The performance and cost indices for this case are CODindex = 0.4039, TKNindex = 0.0156 and U1index = 0.47144. An increase in the value of βTKN from15 to 40 results in a better control of TKNe (Figure 3) with 20.4% decrease in TKNindex value (TKNindex=0.12522). But, this improvement in effluent quality is achieved at an additional 8.9% turbine power consumption (U1index = 0.51192). By increasing the value of γ1 from 0.05 to 0.5, a reduction of 2.5% in U1index is achieved (Figure 4) with a marginal loss of effluent quality (TKNindex = 0.013203).In all these three cases, a common trend of CODe lying well below its set point and TKNe drifting away from its set point is observed. This implies that, controlling of TKN than that of COD in the treated effluent is more crucial.


24

Page 25 of 54

CODe

200 Effuluent COD

Set Point COD

Permissible COD

100 CODindex =0.4039 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 10 TKNindex =0.015601 0

0

2

4

Effuluent TKN

6

8

10

Set Point TKN

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 0.5 0

u1 - optimal control

U1index =0.47144 0

2

4

6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 u2 - optimal control

-1

0

2

4

βCOD = 0.08

6

8

βTKN = 15

10 12 time in Hours

γ1 = 0.05

α1 = 0.9

α2 = 0.8

Figure 2 . Optimal control response of CODe , TKNe , u1 and u2 without external carbon addition for the nominal values of load variables ( P_COD = 1, P_TKN = 1, P_Q =1 ) .


25


CODe

200 Effuluent COD

Set Point COD

Permissible COD

100 CODindex =0.40394 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 10 TKNindex =0.012522 0

0

2

4

Effuluent TKN

6

8

10

Set Point TKN

12

14

Permissible TKN

16

18

20

u1

1 0.5 U1index =0.51192


0

0

2

4

6

8

10

12

14

16

18

20

14

16

18

20

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 54


-1

0

2

4

βCOD = 0.08

6

8

βTKN = 40

10 12 time in Hours

γ1 = 0.05

α1 = 0.9

α2 = 0.8

Figure 3. Optimal control response of CODe , TKNe , u1 and u2 without external carbon addition for the nominal values of load variables(P_COD = 1, P_TKN = 1, P_Q = 1) for an increase in the value of βTKN.


26

Page 27 of 54

CODe

200 CODindex =0.40384

Effuluent COD

Set Point COD

Permissible COD

100 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 10 TKNindex =0.013203 0

0

2

4

Effuluent TKN

6

8

10

Set Point TKN

12

14

Permissible TKN

16

18

20

u1

1 0.5

U1index =0.49897


0

0

2

4

6

8

10

12

14

16

18

20

14

16

18

20

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



-1

0

2

4

βCOD = 0.08

6

8

βTKN = 40

10 12 time in Hours

γ1 = 0.5

α1 = 0.9

α2 = 0.8

Figure 4. Optimal control response of CODe , TKNe , u1 and u2 without external carbon addition for the nominal values of load variables ( P_COD = 1 , P_TKN = 1 , P_Q = 1) for an increase in the value of γ1.


27


Page 28 of 54

5.2 Case Study 2 Effectiveness of the proposed optimal control policy for perturbations in influent COD and influent TKN around their nominal values is investigated. Keeping the influent COD at its nominal value (P_COD = 1), the influent TKN is subjected to a 25% decrease from its nominal value (P_TKN = 0.75). The optimal control response (Figure 5) shows a remarkable control of effluent TKN with TKNe maintained at its set point value throughout the time span tf. A similar quality in the control of effluent TKN is observed (Figure 6) when the influent COD is perturbed by 50% above its nominal value (P_COD = 1.5) with influent TKN maintained at its nominal value (P_TKN =1). But the optimal control scheme is seen to perform poorly for the cases, i) the influent COD is reduced by 25% from its nominal value (P_COD = 0.75) with influent TKN maintained at its nominal value (P_TKN = 1) (Figure 7) and ii) the influent TKN is increased by 25% from its nominal value (P_TKN = 1.25) keeping the influent COD at its nominal value (P_COD = 1) (Figure 8). The poor performance of the control scheme in these cases is mainly due to insufficiency of soluble carbon in the aeration tank required to sustain the growth of heterotrophs. At the reduced rate of oxygen supply to the aeration, dissolved carbon should be present at sufficient level for the heterotroph to remain active for denitrifying the nitrates and reducing the level of TKN in the treated effluent ( TKNe ). Higher the TKN content in the influent stream, higher is the requirement of dissolved carbon for the reduction of TKN. This implies that a better control of TKNe can be achieved by addition of soluble carbon from an external source to make up for the insufficiency in the level of dissolved carbon in the aeration tank.


28

Page 29 of 54

CODe

200 Effuluent COD

100 0

Set Point COD

Permissible COD

CODindex =0.35503 0

2

4

6

8

10

12

14

16

18

20

TKNe

20 10 0

Effuluent TKN

TKNindex =0.0021008 0

2

4

6

8

10

Set Point TKN

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 0.5 0


U1index =0.35813 0

2

4

6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



-1

0

2

4

βCOD = 0.08

6

8

βTKN = 40

10 12 time in Hours

γ1 = 0.5

α1 = 0.9

α2 = 0.8

Figure 5. Optimal control response of CODe, TKNe, u1 and u2 without external carbon for a load change in the influent TKN of 25% decrease from its nominal value (P_COD = 1, P_TKN = 0.75, P_Q = 1).


29


200

CODe

Effuluent COD

100 0

Set Point COD

Permissible COD

CODindex =0.26521 0

2

4

6

8

10

12

14

16

18

20

TKNe

20 10 TKNindex =0.0015787 Effuluent TKN

0

0

2

4

6

8

10

Set Point TKN

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 0.5 U1index =0.59874 0

0

2

4


6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 54


-1

0

2

4

βCOD = 0.08

6

8

βTKN = 40

10 12 time in Hours

γ1 = 0.5

α1 = 0.9

α2 = 0.8

Figure 6 . Optimal control response of CODe, TKNe, u1 and u2 without external carbon addition for a load change in the influent COD of 50% increase from its nominal value (P_COD = 1.5, P_TKN = 1, P_Q = 1).


30

Page 31 of 54

CODe

200 Effuluent COD

Set Point COD

Permissible COD

100 CODindex =0.46045 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

40 20

TKNindex =0.16641 Effuluent TKN

0

0

2

4

6

8

10

Set Point TKN

12

14

Permissible TKN

16

18

20

u1

1 0.5 U1index =0.47912 0

0

2

4


6

8

10

12

14

16

18

20

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



-1

0

2

4

βCOD = 0.08

6

8

βTKN = 40

10 12 time in Hours

γ1 = 0.5

14

α1 = 0.9

16

18

20

α2 = 0.8

Figure 7 . Optimal control response of CODe, TKNe, u1 and u2 without external carbon addition for a load change in the influent COD of 25% decrease from its nominal value (P_COD = 0.75, P_TKN = 1 , P_Q = 1 )


31


200

CODe

Effuluent COD

Set Point COD

Permissible COD

100 CODindex =0.40383 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

40 20

TKNindex =0.46075 Effuluent TKN

0

0

2

4

6

8

10

Set Point TKN

12

14

Permissible TKN

16

18

20

u1

1 0.5 u1 - optimal control

U1index =0.59822 0

0

2

4

6

8

10

12

14

16

18

20

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 54


-1

0

2

4

βCOD = 0.08

6

8

βTKN = 40

10 12 time in Hours

γ1 = 0.5

14

α1 = 0.9

16

18

20

α2 = 0.8

Figure 8. Optimal control response of CODe, TKNe, u1 and u2 without external carbon addition for a load change in influent TKN of 25% increase from its nominal value (P_COD = 1, P_TKN = 1.25 , P_Q = 1 )


32

Page 33 of 54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5.3 Case Study 3 An investigation is made to find out if a better control of single tank ASP is achieved by regulating both oxygen supply rate (u1) and external carbon addition rate (u2). The optimal control response (Figure 9) for the load variables kept at their nominal values show a far better control of TKNe as compared to the one achieved without external carbon addition (Figure 4 ). The optimal control responses with regulation of both oxygen supply rate (u1) and external carbon addition rate (u2 ) are shown in Figures 10 & 11 for the two cases respectively i) the influent COD is reduced by 25% from its nominal value (P_COD = 0.75) with influent TKN maintained at its nominal value (P_TKN = 1) and ii) the influent TKN is increased by 25% from its nominal value (P_TKN = 1.25) keeping the influent COD at its nominal value (P_COD = 1). These responses (Figures 10 & 11) show that it is possible to achieve a better control of TKNe with external carbon addition for the cases in which the optimal control policy without external carbon addition failed (Figures 7 & 8). The optimal control policy with external carbon addition is also seen (Figure 12) to perform well for a load change in the influent flow rate of 25% increase from its nominal value (P_Q = 1.25). An investigation is made to see if the proposed optimal control policy can be tuned to achieve a much higher quality of treated effluent by lowering the set point values to 60% of the maximum permissible limits( α1 = 0.6, α2 = 0.6 ). The response (Figure 13) shows a drastic improvement in the quality of treated effluent and a steeper rise in performance indices (CODindex = 0.16444, TKNindex = 0.0002403) as compared to the one observed (Figure 10) with higher set point values (α1 = 0.9, α2 = 0.8). However, it may be noted that this higher quality of the treated effluent is achieved at a higher cost associated with the excess supply of oxygen and external


33


Page 34 of 54

carbon. Thus the values of α1 and α2 can be tuned to achieve a tradeoff between the effluent quality and the cost.


34

Page 35 of 54

200

CODe

Effuluent COD

Set Point COD

Permissible COD

100 CODindex =0.4092 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 TKNindex =0.0029329

10

Effuluent TKN

0

0

2

4

6

8

10

Set Point TKN

12

14

Permissible TKN

16

18

20

18

20

18

20

u1

1 0.5


U1index =0.47292 0

0

2

4

6

8

10

12

14

16

0.4

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0


U2index =0.023843

0.2

0

2

βCOD = 0.08

4

6

βTKN = 15

8

10 12 time in Hours

γ1 = 0.5

γ2 = 0.5

14

16

α1 = 0.9

α2 = 0.8

Figure 9. Optimal control response of CODe, TKNe, u1 and u2 with external carbon addition for the nominal values of load variables (P_COD = 1, P_TKN = 1, P_Q = 1)


35


CODe

150 Effuluent COD

100

Set Point COD

Permissible COD

CODindex =0.48606

50 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 15 TKNindex =0.0020036 Effuluent TKN

10 5

0

2

4

6

8

Set Point TKN 10

12

14

Permissible TKN 16

18

20

1

u1

u1 - optimal control 0.5 0

U1index =0.4876

0

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 54

2

4

6

8

10

12

14

U2index =0.29694

16

18

20


0.5 0

0

2

βCOD = 0.08

4

6

8

βTKN = 15

10 12 time in Hours

γ1 = 0.05

14

γ2 = 0.05

16

α1 = 0.9

18

20

α2 = 0.8

Figure 10. Optimal control response of CODe, TKNe, u1 and u2 with external carbon addition for a load change in the influent COD of 25% decrease from its nominal value (P_COD = 0.75, P_TKN = 1, P_Q = 1)


36

Page 37 of 54

CODe

150 Effuluent COD

100

Set Point COD

Permissible COD

CODindex =0.44746

50 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 15 TKNindex =0.0021517

10

Effuluent TKN

5

0

2

4

6

8

10

Set Point TKN

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 U1index =0.72329

0.5


0

0

2

4

6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5 U2index =0.76363 u2 - optimal control

0

0

2

βCOD = 0.08

4

6

8

βTKN = 15

10 12 time in Hours

γ1 = 0.05

γ2 = 0.05

α1 = 0.9

α2 = 0.8

Figure 11. Optimal control response of CODe, TKNe, u1 and u2 with external carbon addition for a load change in the influent TKN of 25% increase from its nominal value (P_COD = 1, P_TKN = 1.25, P_Q = 1)


37


CODe

150 Effuluent COD

100

Set Point COD

Permissible COD

50 CODindex =0.40118 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 15 10 5

TKNindex =0.0021056 0

2

4

Effuluent TKN

6

8

10

Set Point TKN

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 0.5 U1index =0.71853 0

0

2

4


6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 54

0.5 U2index =0.16434 0

0

2

βCOD = 0.08

4


6

8

βTKN = 15

10 12 time in Hours

γ1 = 0.05

γ2 = 0.05

α1 = 0.9

α2 = 0.8

Figure 12. Optimal control response of CODe, TKNe, u1 and u2 with external carbon addition for a load change in the influent flowrate of 25% increase from its nominal value (P_COD = 1, P_TKN = 1, P_Q = 1.25)


38

Page 39 of 54

CODe

150 Effuluent COD

100

Set Point COD

Permissible COD

50 CODindex =0.16444 0

0

2

4

6

8

10

12

14

16

18

20

TKNe

20 15 10 5

TKNindex =0.00024028 0

2

4

Effuluent TKN

6

8

10

Set Point TKN

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 0.5

U1index =0.53778 u1 - optimal control

0

0

2

4

6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5


0

0

2

βCOD = 0.08

4

6

βTKN = 15

8

10 12 time in Hours

γ1 = 0.05

γ2 = 0.05

α1 = 0.6

α2 = 0.6

Figure 13. Optimal control response of CODe, TKNe, u1 and u2 with external carbon for reduced values of α1 and α2 (P_COD = 0.75, P_TKN = 1, P_Q = 1)


39


Page 40 of 54

5.4 Case Study 4 The proposed optimal control strategy is a feed forward open loop type in which the load changes in influent COD, influent TKN and influent Q are monitored and the control calculations are repeated as and when there is a load change. The implementation of the proposed optimal control strategy for load changes occurring at intervals of 10 hours is shown in Figure 14. The open loop optimal control scheme (Figure 14) is seen to respond well to the abrupt load changes by appropriate regulation of oxygen supply and external carbon addition rates.


40

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CODe

200 Effuluent COD

100 0

Set Point COD

Permissible COD

CODindex =0.40683 0

5

10

15

20

25

30

35

40

TKNe

20 TKNindex =0.0013309

10

Effuluent TKN

0

0

u1

1

5

10

15

25

Permissible TKN

30

35

40

30

35

40

30

35

40

U1index =0.65804

0.8 u1 - optimal control

0 1

u2

Set Point TKN

20

0.6 5

10

15

20

25

U2index =0.44898

0.5 0

Load Changes

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



0

5

10

15

20

25

1.5 1 COD

0.5

0

βCOD = 0.08

5

10

15

βTKN = 15

20 time in Hours

γ1 = 0.5

25

γ2 = 0.05

TKN

30

α1 = 0.9

Q

35

40

α2 = 0.8

Figure 14. Feed forward open loop optimal control response of CODe, TKNe, u1 and u2 with external carbon addition for abrupt perturbations in the load variables.


41


5.5 Case Study 5 Simulation studies are carried out to evaluate the effect of weights γ1 and γ2 on the effluent quality (TKNindex) and the cost (U1index and U2index). The results are shown in Figures 15, 16 & 17. For a fixed value of γ2 , increasing the value of γ1 results in a marginal drop in the quality of treated effluent (Higher TKNindex) achieved at a lesser cost associated with oxygen supply (lower U1index) as well as external carbon addition (lower U2index). A similar trend of increase in TKNindex and decrease in both U1index and U2index is observed when the value of γ2 is increased by fixing the value of γ1. Depending upon the relative costs of oxygen supply and carbon addition, it is possible to tradeoff between these two costs as against the effluent quality by appropriately tuning the values of γ1 and γ2.

0.0035 0.003

TKNindex

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 54

0.0025 0.002 0.0015 0.001 0.0005 0 0

0.1

0.2

0.3

0.4

0.5

0.6

γ1 γ2 = 0.05

γ2 = 0.1

γ2 = 0.2

γ2 = 0.3

γ2 = 0.5

Figure 15. Effect of γ1 on TKNindex for fixed values of γ2


42

0.49 0.485 0.48 0.475 0.47 0.465 0.46 0.455 0.45 0.445 0.44 0.435 0

0.1

0.2

0.3

0.4

0.5

0.6

γ1 γ2 = 0.05

γ2 = 0.1

γ2 = 0.2

γ2 = 0.3

γ2 = 0.5

Figure 16. Effect of γ1 on U1indexfor fixed values of γ2

0.3 0.29

U2index

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


U1index

Page 43 of 54

0.28 0.27 0.26 0.25 0.24 0.23 0

0.1

0.2

0.3

0.4

0.5

0.6

γ1 γ2 = 0.05

γ2 = 0.1

γ2 = 0.2

γ2 = 0.3

γ2 = 0.5

Figure 17. Effect of γ1 on U2indexfor fixed values of γ2


43


Page 44 of 54

5.6 Case Study 6 An attempt is made to compare the unconstrained optimal control scheme proposed in this work with the inequality constrained formulation of the problem involving rigorous computation28 . For this purpose, the inequality constrained optimal control problem ( ICOCP) is formulated with the objective function J [u (t )] and the state variable constraints g1 ( X ) and g 2 ( X ) defined as follows: Minimize tf

2 2 J [u (t )]    1u1  t    2u2  t  dt  

(40)

0

Subject to

g1 ( X )  x1  x2  CODMAX  0

(41)

g 2 ( X )  x9  x10  x11  TKN MAX  0

(42)

In this formulation, the manipulated variables u1(t) and u2(t) are calculated over a specified time span tf in a such a way that feed rates of oxygen and external carbon are minimized keeping the effluent COD(CODe) and effluent TKN(TKNe) below their respective limiting values CODMAX and TKNMAX. The penalty function method30 is used for solving this inequality constrained optimal control problem. The modified Hamiltonian function H m (t ) for this problem is 2 % 2 H m  F  X , u    T f  X , u   % 1 g1 ( X ) H ( g1 )   2 g 2 ( X ) H ( g 2 )

(43)


44

Page 45 of 54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Where, F  X , u    1u1  t    2u2  t  2

2

(44)

% H ( gi ) is the Heaveside step function. % 1 and  2 are the suitable penalties assigned to the

violations of the inequality constraints defined by equations (41) and (42) respectively. Two illustrative results of this study are shown in Figures 18 and 19. Figure 18 depicts the response of the inequality constrained optimal control formulation of the problem (ICOCP) for the case in which the influent COD is reduced by 25% from its nominal value keeping the influent TKN at its nominal value. Comparing this with the corresponding response( Figure 10) of the proposed unconstrained optimal control problem for identical influent load conditions, it is observed that ICOCP requires relatively much lesser control effort (U1index = 0.42399 and U2index =0.13144) with TKNe value being pushed closer to and maintained at its limiting value of TKNMAX . Whereas, the proposed optimal control scheme is able to achieve a comparatively better quality of treated effluent by keeping the TKNe value around the desired set point TKNS, which is fixed at a level (α2 = 0.8) lower than the limiting value TKNMAX. Similarly, for the load conditions in which the influent TKN is increased by 25% from its nominal value keeping the influent COD at its nominal value, the ICOCP (Figure 19) requires lesser control effort (U1index = 0.66019 and U2index =0.44986) compared to the proposed optimal control scheme ( Figure 11) that is able to achieve a better effluent quality with higher control effort.


45


Page 46 of 54

The comparative analysis presented in this case study clearly indicates that , the inequality constrained optimal control problem (ICOCP) formulation tends to push the effluent TKN to its limiting value and this is achieved at a lesser control effort .However, it may not always be desirable to maintain the treated effluent quality at its limiting value as there is a risk of effluent quality violation caused by abrupt feed conditions . As compared to the ICOCP scheme , the proposed optimal control strategy can maintain the effluent quality at a higher level by fixing the desired effluent quality ( tuning of parameters α1 and α2) at a value lower than the limiting value. This is however achieved at a higher control effort.


46

Page 47 of 54

CODe

200 100 0

Effuluent COD

0

2

4

6

8

10

12

14

Permissible COD

16

18

20

TKNe

20 10 0

Effuluent TKN

0

u1

1

2

4

6

8

10

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

U1index =0.42399

0.5 u1 - optimal control 0

0

0.5

2

4

6

8

10

12

U2index =0.13144

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


u2 - optimal control 0

0

2

4

6

8

10 12 time in Hours

Figure 18. Inequality constrained optimal control(ICOCP) response of CODe, TKNe, u1 and u2 with external carbon addition for a load change in the influent COD of 25% decrease from its nominal value (P_COD = 0.75, P_TKN = 1, P_Q = 1)


47


CODe

200 100 0

Effuluent COD

0

2

4

6

8

10

12

14

Permissible COD

16

18

20

TKNe

20 10 0

Effuluent TKN

0

2

4

6

8

10

12

Permissible TKN

14

16

18

20

14

16

18

20

14

16

18

20

u1

1 0.5


0

0

2

4

6

8

10

12

1

u2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 54

U2index =0.44986

0.5

u2 - optimal control 0

0

2

4

6

8

10 12 time in Hours

Figure 19. Inequality constrained optimal control(ICOCP) response of CODe, TKNe, u1 and u2 with external carbon addition for a load change in the influent TKN of 25% increase from its nominal value (P_COD = 1, P_TKN = 1.25, P_Q = 1)


48

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6. CONCLUSIONS In this paper, a new approach to the control of single tank activated sludge process is presented, in which addition of soluble carbon from an external source is regulated along with the oxygen supply rate. The proposed control scheme, which is developed in the open loop feedforward optimal control framework, performs exceedingly well in keeping the effluent COD and effluent TKN well within the permissible limits for all load conditions tested in this work. The case studies investigated in this work have clearly emphasised the need for adding soluble carbon to the tank to make up for its short fall from the required level in order to sustain the growth of heterotroph and subsequently reduce the TKN level in the effluent. The proposed approach is a significant improvement over the previous work reported in the literature26, in which it was demonstrated that regulating the oxygen supply rate alone was insufficient to control the effluent TKN ( TKNe) for some specific load variations corresponding to lower COD and higher TKN levels in the influent stream. This fact has also been emphasized in the case studies 1 & 2 presented in this paper. By fixing the desired set point values for effluent COD and effluent TKN well below their limiting values ( by tuning α1 and α2 ) , the proposed optimal control strategy is able to keep the CODe and TKNe below their permissible limits. Further, by appropriately tuning the values of the weights α1, α2 , βCOD , βTKN , γ1 and γ2 , one can trade off effluent quality against the costs associated with oxygen supply and external carbon addition. Thus, the proposed optimal control method allows the desired effluent quality to be achieved at the lowest possible cost.


49


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AUTHOR INFORMATION Corresponding Author *Tel.: +91 9444290056. E-mail: [email protected], [email protected] REFERENCES (1) Alex, J.; Benedetti,L.; Copp, J.; Gerbaey ,K. V.; Jeppsson , U.; Nopens, I.; Pons,M N.; Rieger , L.; Rosen, C.; Steyer, L.P.; Vanrolleghem, P.; Winkler, S.; Benchmark Simulation Model no. 1 (BSM1), Technical Report, Dept. of Industrial Electrical Engineering and Automation – Lund University , Sweden, 2008. (2) Carl- Fredrick Lindberg , Bengt Carlsson, Nonlinear and set-point control of the dissolved oxygen concentration in an activated sludge process. Wat. Sci. Tech. 1996, 34 (3 - 4), 135-142. (3) Marsili – Libelli, S. Optimal aeration control for wastewater Treatment. IFAC Proceedings Volume, 1979, 12 (7), 511 – 516. (4) Amand, L.; Carlsson, B. Optimal aeration control in a nitrifying activated sludge process. Water Research, 2012, 46, 2101 – 2110. (5) Darko Vrecko, Nadja Hvala, Aljaz Stare, Olga Burica, Marjeta Strazer , Meta Levstek, Peter Cear, Sebastjan Podbevsek. Improvement of ammonia removal in activated sludge process with feedforward – feed backward aeration controllers. Wat. Sci. Tech. 2006, 53(4 – 5), 125 – 132. (6) Zarrad, W.; Harmand, J.; Devisscher, M.; Steyer, J.P. Comparison of advanced control strategies for improving the monitoring of activated sludge processes. Control Engineering Practice, 2004, 12, 323 -333. (7) Norhaliza. A Wahab, Reza Katebi, Jonas Balderud. Multivariable PID control design for activated sludge process with nitrification and denitrification. Biochemical Engineering Journal, 2009(45), 239 – 248.


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(8) Santin, I.; Pedret, C.; Vilanova, R. Applying variable dissolved oxygen set point in a two level hierarchical control structure to a wastewater treatment process. Journal of Process Control. 2015, 28, 40 - 55. (9) Wenhao Shen, Xiaoquan Chen, Pons, M.N.; Corriou, J.P. Model predictive control for waste water treatment process with feedforward compensation. Chemical Engineering Journal, 2009, 155, 161 -174. (10) Wenhao Shen, Xiaoquan Chen, Corriou, J.P. Application of model predictive control to the BSM1 benchmark of wastewater treatment process. Computers and Chemical Engineering, 2008, 32, 2849 – 2856. (11) Holenda, B.; Domokos, E.; Redey, A.; Fazakas, J.; Dissolved oxygen control of the activated sludge wastewater treatment process using model predictive control, Computers and Chemical Engineering , 2008, 32, 1270 – 1278. (12) Cristea, S.; Prada, C.; Sarabia, D.; Gutierrez, G.; Aeration control of a wastewater treatment plant using hybrid NMPC, Computers and Chemical Engineering, 2011, 35, 638 – 650 . (13) Jing Zeng , Jinfeng Liu, Economic Model Predictive control of wastewater treatment process, Ind. Eng. Chem. Res. 2015,54, 5710 – 5721 . (14) George Simon ostace, Vasile Mircea Cristea , Paul Serban Agachi. Cost reduction of the wastewater treatment plant operation by MPC based on modified ASM1 with two – step nitrification/denitrification model, Computers and Chemical Engineering, 2011, 35, 2469 – 2479. (15) Carlos Alberto Coelho Belchior, Rui Alexandre Matos Araujo, Jorge Afonso Cardoso Landeck, Dissolved oxygen control of the activated sludge wastewater treatment process using stable adaptive fuzzy control. Computers and Chemical Engineering, 2012, 37, 152 – 162.


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(16) Rodrigo, M.A.; Seco, A.; Ferrer. J.; Penya-roja, J.M.; Valverde, J.L. Nonlinear control of an activated sludge aeration process: use of fuzzy techniques for tuning PID controllers, ISA Transcations, 1999, 38, 231 – 241. (17) Santin, I.; Pedret, C.; Vilanova, R.; Meneses, M. Advanced decision control system for effluent violations removal in wastewater treatment plants, Control Engineering Practice, 2016, 49, 60 – 75. (18) Antonio C.B. de Arujo, Simone Gallani , Michela Mulas, Sigurd Skgestad, Sensitivity analysis of optimal operation of an activated sludge process model for economic controlled variable selection, Ind. Eng. Chem. Res.2013, 9908 – 9921. (19) Antonio C.B de Araujo , Simone Gallani, Michela Mulas , Gustaf Olsson, Systematic approach to the design of operation and control policies in activated sludge systems, Ind. Eng. Chem. Res.2011, 50, 8542 – 8557. (20) Carlsson,B.; Rehnstrom, A.Control of an activated sludge process with nitrogen removal – a benchmark study, Wat. Sci. Tech.2002, 45 (4 -5), 135 – 142. (21) Fikar, M.; Chachuat, B.; Latifi,L.A.; Optimal operation of alternating activated sludge processes, Control Engineering Practice,2005, 13, 853 - 861. (22) Hyunook Kim, McAvoy, T.J.; Anderson, J.S.; Hao, O.J. Control of an alternating aerobicanoxic activated sludge system – Part2: Optimization using linearized model. Control Engineering Practice, 2000,8, 279 – 289. (23) Chachuat, B.; Roche, N.; Latifi, M.A. Long term optimal aeration strategies for small-size alternating activated sludge treatment plants, Chemical Engineering and Processing, 2005,44, 593 – 606.


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(24) Cristea, S.; C. de Prada, Sarbia,D.; Gutierrez, G.; Aeration control a wastewater treatment plant using hybrid NMPC, Computers and Chemical Engineering, 2011,35, 638 – 650. (25) Saziye Balku, Comparison between alternating aerobicanoxic and conventional activated sludge systems, Water Research, 2007, 41, 2220 – 2228. (26) Pallavhee, T.; Sundaramoorthy, S.; Sivasankaran, M.A. Optimal regulation of oxygen supply in a small size single tank activated sludge process for effective control of effluent quality, First International Conference on Energy and Environment: Global Challenges, National Institute of Technology Calicut,India, 2018. (27) Henze,M.; Grady, C.P.L.; Gujer, W.; Marais, G.V.R.; .Matsau, T. Activated Sludge Model No.1. Technical Report 1, IAWQ, London, 1987. (28) Bryson Jr ,A.E.; Denham, W.F.; Dreyfus, S.E.; Optimal programming problems with inequality constraints I : Necessary conditions for external solutions, American Institute of Aeronautics and Astronautics, 1963,11, 2544-2550. (29) Harmon Ray, W. Advanced Process Control, Mc.Graw – Hill Book Company, 1987. (30) Ramirez, W. Fred, Process Control and Identification , Academic Press Limited, 1993.


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Optimal Control of Small Size Single Tank Activated Sludge Process

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