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Hybrid model based control for membrane filtration process
11th IFACSystems, Symposium on Dynamics and Control of Process including Biosystems 11th Symposium on and Control of 11th IFAC Symposium on Dynamics Dynamics and Controlonline of Process including Biosystems JuneIFAC 6-8,Systems, 2016. NTNU, Trondheim, Norway Available at www.sciencedirect.com Process Systems, including Biosystems Process including Biosystems June 6-8,Systems, 2016. NTNU, Trondheim, Norway June 6-8, 2016. NTNU, Trondheim, Norway June 6-8, 2016. NTNU, Trondheim, Norway
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IFAC-PapersOnLine 49-7 (2016) 1085–1090 Hybrid model based control for membrane filtration process Hybrid model based control for membrane filtration process Hybrid based control for membrane filtration process Hybrid model model based control for membrane filtration process Lester Lik Teck Chan, Chen-Pei Chou and Junghui Chen
Lester Lik Teck Chan, Chen-Pei Chou and Junghui Chen Lester Teck Chan, Chen-Pei Chou and Lester Lik LikR&D TeckCenter Chan,for Chen-Pei Chou and Junghui Junghui Chen Chen Membrane Technology R&D Center for MembraneEngineering Technology Department of Chemical R&D Center Membrane Technology R&D Center for for MembraneUniversity Technology Department of Chemical Engineering Chung-Yuan Christian Department of Chemical Engineering Department of Chemical Engineering Chung-Yuan Christian University Chung-Li, TaiwanChristian 320, Republic of China Chung-Yuan University Chung-Yuan Christian University Chung-Li, Taiwan 320, Republic of China Chung-Li, Chung-Li, Taiwan Taiwan 320, 320, Republic Republic of of China China (Fax: +886-3-2654199 e-mail: [email protected] ) (Fax: +886-3-2654199 e-mail: [email protected] ) (Fax: (Fax: +886-3-2654199 +886-3-2654199 e-mail: e-mail: [email protected] [email protected] )) Abstract: Membrane fouling can affect the performance of the membrane-based filtration for water Abstract: Membrane fouling can operational affect the performance of the presents membrane-based filtration for water treatment. Fouling thus increases costs. This of work a modeling framework that Abstract: Membrane fouling can affect the performance the membrane-based filtration for water Abstract: Membrane fouling can affect the performance of the membrane-based filtration for water treatment. Fouling thus increases operational costs. This work presents a modeling framework that combine first principle model with Gaussian process model that aims to reduce the energy load affected treatment. Fouling thus increases operational costs. This presents aa modeling framework that treatment. Fouling thusmodel increases operational costs. This work work presents modeling framework that combine first principle with Gaussian process model thatimprovement aims to reduce the energy load affected by the effect of fouling on thewith system. Basedprocess on the model expected algorithm, an load optimization combine first principle model Gaussian that aims to reduce the energy affected combine first principle model with Gaussian process model that aims to reduce the energy load affected by the effect of fouling on the system. Based of on the theoperation expected to improvement algorithm, an optimization control is proposed to handle thesystem. long duration achieve the most economical operation by the of on Based on the expected improvement algorithm, an by term the effect effect of fouling fouling on the the Based of on the theoperation expected to improvement algorithm, an optimization optimization control is proposed to handle thesystem. long duration achieve the most economical operation in of energy load. control is proposed to control to handle handle the the long long duration duration of of the the operation operation to to achieve achieve the the most most economical economical operation operation in term is of proposed energy load. in energy load. © term 2016, of IFAC (International Federation of Automatic Control) Hosting byuncertainty, Elsevier Ltd.variance All rights reserved. in term of energy load. Keywords: model based control, iterative improvement, optimization, Keywords: model based control, iterative improvement, optimization, uncertainty, variance Keywords: model model based based control, control, iterative iterative improvement, improvement, optimization, uncertainty, uncertainty, variance variance Keywords: optimization, models can generally describe the fundamental phenomena of 1. INTRODUCTION models canfouling. generally describe the fundamental phenomena of membrane Data data-driven empirical models, on the 1. INTRODUCTION models can generally describe the phenomena of models canfouling. generally describe the fundamental fundamental phenomena of membrane Data data-driven empirical models, on the 1. INTRODUCTION other hand, are suitable to predict complex behaviours in 1. INTRODUCTION Membrane filtrations are widely used in water treatment membrane fouling. Data data-driven empirical models, on the membrane fouling. Data data-driven empirical models, on the other hand, are suitable to predict complex behaviours in nonlinear processes (Hwang et al., 2009). The combination of Membrane filtrations are widely used in water treatment applications and the benefits include lowinenergy high other hand, are to complex behaviours in Membrane filtrations filtrations are widely widely used water load, treatment other hand, are suitable suitable to etpredict predict complex behaviours in nonlinear processes (Hwang al., 2009). The combination of Membrane are used in water treatment both thus provides the advantages of first-principles and dataapplications and the benefits include low energy load, high selectivity and ease of operation (Farahbakhsh and Smith, nonlinear processes (Hwang et al., 2009). The combination of applications and and the the benefits benefits include include low low energy energy load, load, high high both nonlinear processes (Hwang et al., 2009). The combination of thus provides the advantages of first-principles and dataapplications empirical models for the control of membrane selectivity and ease offiltration operation andpressure Smith, driven 2006). In membrane the(Farahbakhsh transmembrane thus provides advantages of and selectivity and ease ease of of operation operation (Farahbakhsh and Smith, Smith, both both thusempirical provides the the advantages of first-principles first-principles and datadatamodels for the control of membrane selectivity and (Farahbakhsh and filtration. 2006). In membrane filtration the transmembrane pressure driven (TMP) difference forces the fluid and smaller particles driven empirical empirical models models for for the the control control of of membrane membrane 2006). In membrane filtration the transmembrane pressure driven filtration. 2006). In membrane filtration the transmembrane pressure (TMP) difference forces the Membrane fluid and fouling smaller increases particles filtration. through the membrane pores. (TMP) difference forces the fluid smaller particles filtration. In this work the filtration process is modeled using a physical (TMP) forces the fluid and and smaller particles through difference thecosts membrane pores. Membrane fouling increases operational as a result of permeate flux decline and can In this work the filtration process is modeled using a physical through the membrane pores. Membrane fouling increases model complimented by a is statistical process through the membrane pores. Membrane fouling increases operational costs as a result of permeate flux decline and can In this this that workisthe the filtration process process modeledGaussian using aa physical physical be accompanied by increased energy loadflux duedecline to higher TMP In work filtration is modeled using model that is complimented by a statistical Gaussian operational costs as a result of permeate and can regression model (GPRM) which accounts for the mismatch operational costs as increased a as result ofenergy permeate flux decline and can model that is complimented by a statistical Gaussian process be accompanied by due to higher TMP process requirements needed driving force.load The reversible fouling model is complimented bycurrent a statistical Gaussian process regression model (GPRM) which accounts thetomismatch be accompanied by increased energy load due to higher TMP in the that previous cycle to the cyclefor due fouling. be accompanied by increased energy load due to higher TMP requirements needed as driving force. The reversible fouling regression model (GPRM) which accounts for the mismatch can be eliminated at least partially by aeration and regression model (GPRM) which accounts for the mismatch in the previous cycle to theprobabilistic current cycle due (Rasmussen to fouling. requirements needed needed as as driving driving force. force. The The reversible reversible fouling fouling GPRM is a non-parametric model requirements can be eliminated least partially by aeration and in the previous cycle the current cycle due to backwashing (Yigit etat 2009). Optimizing the operation the previous cycle to to theprobabilistic current cycle due to fouling. fouling. GPRM is a non-parametric model (Rasmussen can be be eliminated eliminated atal.,least least partially by aeration aeration and in and Williams, 2006). Compared to neural network (NN), can at partially by and backwashing (Yigit et al., 2009). Optimizing the operation GPRM is a non-parametric probabilistic model (Rasmussen through process control can reduce the energy demand and GPRM is a non-parametric probabilistic model (Rasmussen and Williams, 2006). Compared to neural network (NN), backwashing (Yigit et al., 2009). Optimizing the operation GPRM not only provides the prediction of an output but also backwashing (Yigit et al., 2009). Optimizing the operation through process operational control cancosts. reduce the energy demand and and Williams, 2006). Compared to neural network (NN), other associated and Williams, 2006). Compared to neural network (NN), GPRM not only provides the prediction of an output but also through process control can reduce the energy demand and the variance of estimation which can be interpreted as a level through process control can reduce the energy demand and other associated operational costs. GPRM not not only only provides provides the the prediction prediction of of an an output output but but also also GPRM the variance of estimation which can be interpreted as a level other associated operational costs. the model.which Moreover, number of other associated operational costs. Filtration processes are usually controlled to meet the desired of theconfidence variance of ofofestimation estimation can be bethe interpreted as GPRM a level level the variance which can interpreted as a confidence of the the number of GPRM hyper-parameters thatmodel. need Moreover, to be optimized is small. Based Filtration meet desired of net flux. processes However are theusually high controlled complexityto of thethe filtration of confidence confidence of of the the model. Moreover, the number number of GPRM GPRM Filtration processes are usually controlled to meet the desired of model. Moreover, the of hyper-parameters that need to be optimized is small. Based Filtration processes are usually controlled to meet the desired on the predictive distribution the expected improvement (EI) net flux. However the high complexity of the filtration process poses a challenge. It is complexity characterizedofbythe the filtration periodic hyper-parameters that to be optimized is Based net flux. However the high hyper-parameters that need need tothe beexpected optimized is small. small. of Based on the predictive distribution improvement (EI) net flux. However the high complexity of the filtration (Boyle, 2007; Jones, 2001) provides an indication the process poses a challenge. It is characterized by the periodic change between filtration and backwashing, by the the periodic drift of on the distribution the expected improvement (EI) process poses aa challenge. It is characterized by on the predictive predictive distribution the expected improvement (EI) (Boyle, 2007; Jones, 2001) provides an indication of the process poses challenge. It is characterized by the periodic improvement that is expected from sampling at new change between filtration and backwashing, by the drift of membrane permeability due backwashing, to irreversible membrane 2007; Jones, 2001) provides an indication of the change between between filtration and and by the the drift of of (Boyle, (Boyle, 2007; Jones, 2001) provides ansampling indication of new the improvement that is expected from at change filtration backwashing, by drift operation. It has the capability to ‘explore’ new area and membrane permeability due to irreversible membrane fouling, and by a high-number of disturbances, including improvement that is expected from sampling at new membrane permeability permeability due due to to irreversible irreversible membrane membrane improvement that is expected from sampling at new operation. It has the capability to ‘explore’ new area and membrane newItinformation about thetooptimization problem and fouling, and ofby atemperature high-number orof disturbances, including collect variations solids concentration. operation. has the ‘explore’ area and fouling, and by aa high-number of disturbances, including operation. Itinformation has the capability capability tooptimization ‘explore’to new new area and and collect new about the problem fouling, by most high-number disturbances, including is used for control design. Compared conventional variationsand ofin temperature orof solids TMP concentration. Furthermore, cases, only the overall across an collect new new information information about about the the optimization optimization problem problem and and variations of temperature or solids concentration. collect is used for control design. Compared to conventional variations of temperature or solids concentration. are deterministic EI is probabilistic which Furthermore, in most cases, only the overall TMP across an methods entire membrane module is measured which means that little is used used which for control control design. Compared Compared to conventional conventional Furthermore, in most cases, only the overall TMP across an for design. to methods EI is probabilistic which Furthermore, in most cases, only the overall TMP across an is takes intowhich accountare thedeterministic model uncertainty. entire membrane module is measured which means that little information is available to understand the process. methods which are deterministic EI is probabilistic entire membrane module is measured which means that little methods which are deterministic EI is probabilistic which which takes into account the model uncertainty. entire membrane module is measured which means that little information is available to understand the process. Development of model givestoinsight into the effects of the takes into account the model uncertainty. information is available understand the process. takes into account the model uncertainty. information is available toinsight understand the process. Development of model into the effects of the 2. MEMBRANE FILTRATION SYSTEM AND CONTROL controls. Blankert et al. gives (2006) developed a filtration model Development of model gives insight into the effects of the Development of model gives insight into the effects of the 2. MEMBRANE FILTRATION SYSTEM AND CONTROL controls. Blankert et al. (2006) developed a filtration model and determined theetoptimal profile of the filtration flux and 2. FILTRATION SYSTEM AND CONTROL controls. Blankert al. (2006) (2006) developed a filtration filtration model 2. MEMBRANE MEMBRANE FILTRATION SYSTEM AND CONTROL The membrane filtration experimental system is introduced in controls. Blankert et al. developed a model and determined the optimal profile of the filtration flux and TMP during one filtration phase using offline dynamic The membrane filtration experimental system istointroduced in and determined determined the the optimal optimal profile profile of of the the filtration filtration flux flux and and this section. This experimental setup is used demonstrate and TMP during Due one tofiltration phase using offline will dynamic membrane filtration experimental system is in optimization. fouling the model parameters differ The The membrane filtration experimental system istointroduced introduced in this section. This experimental setup is usedthe demonstrate TMP during one filtration phase using offline dynamic the proposed control strategy. In addition conventional TMP during one filtration phase using offline dynamic optimization. Dueand to fouling the will model parameters will differ this section. This experimental setup is used to demonstrate with each cycle thus there be a mismatch between this section. This experimental setup is used to demonstrate the proposed control strategy. In addition the conventional optimization. Due to fouling the model parameters will differ of the membrane filtration is introduced. optimization. Dueand to fouling the will model differ control withpredicted each cycle thus –there be parameters a mismatch between the control In the the energy load using parameters of thewill previous the proposed proposed control strategy. strategy. In isaddition addition the conventional conventional control of the membrane filtration introduced. with each cycle and thus there will be aa mismatch between with each cycle and thus there will be mismatch between the predicted energy load – using parameters of the previous control of of the the membrane membrane filtration filtration is is introduced. introduced. cycle – and actual current cycle energy load. This mismatch control the predicted energy load – using parameters of the previous the predicted energy load –cycle using parameters of themismatch previous cycle actual energy This has to – beand taken intocurrent account to obtain theload. best performance of 2.1 Membrane Filtration Experimental System cycle – and actual current cycle energy load. This mismatch cycle – and actual current cycle energy load. This mismatch has to be taken into account to obtain the best performance of 2.1 Membrane Filtration Experimental System the filtration process. In membrane filtration, first-principles 2.1 Membrane Membrane Filtration Filtration Experimental Experimental System System has to into to the of has to be be taken taken into account account to obtain obtainfiltration, the best best performance performance of 2.1 the filtration process. In membrane first-principles the filtration filtration process. process. In In membrane membrane filtration, filtration, first-principles first-principles the
Copyright © 2016, 2016 IFAC 1085Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2016 responsibility IFAC 1085Control. Peer review of International Federation of Automatic Copyright © 1085 Copyright © 2016 2016 IFAC IFAC 1085 10.1016/j.ifacol.2016.07.347
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Figure 1 shows the experimental set-up of the membrane filtration system in this study. The operation is fully automatic. The pressure transmitter (PT) and flow transmitter (FT) reading as send to the computer and this is represented ). Based on these signals the operation by the signal ( condition of the electric valve and pump are calculated and the signal to the final control element are represented by ). During the filtration phase, valve A will be opened ( in the direction to allow the feed to enter the pump where it is pressurized before entering the filtration unit through valve B indicated by the filtration path in Figure 1. The driving force is the pressure difference between the filtration end and the permeate end. It drives the water to the permeate side and enters the permeate tank. In the washing phase, feed to valve A is closed and valve A is connected to the permeate tank instead. Valve B (backwash path in Figure 1) now allows water to enter the permeate side which carries the backwashing operation. The filtration and the backwash phases thus form the cyclic operation and the energy load is the result of the TMP from the pump. The feed solution consisting of kaolinite and sodium alginate is utilized to generate a combination of particulate and macromolecular foulants. A polyvinylidene fluoride (PVDF) membrane with dimension of 4cm 10cm is used in this study.
Pˆw c, kw jw c k w Rw c R c exp w c
(2)
where j is the flux which is the manipulated variables, is the fluid viscosity, R is the membrane resistance, c represents the c th cycle, and k is the operating time. The subscripts f and w denote the filtration and the backwash phases, respectively. R 0f is the initial membrane resistance
, and w are the model parameters is a parameter dependant on the feed suspension. R is expressed as
R R f c, K f c Rw c
(3)
where R f c, K f c is the final resistance of the filtration phase and Rw is the irreversible resistance. K f c is the duration of the c th filtration phase. The operation is cyclical and the objective is to find the least amount of energy load for a particular flux, jnet at the next cycle. It is defined as eˆ c 1, u
min
K f c 1 , j f c 1 K w c 1 , jw c 1
(4)
such that jnet
j f c 1 K f c 1 j f c 1 K w c 1 K f c 1 K w c 1
(5)
j f c 1 jw c 1
jw c 1 Ar
K w c 1
0
Pˆf (c 1, k f ) d k f (Pˆw (c 1, kw )) d kw
(6)
where u j f c 1 , K f c 1 , jw c 1 , K w c 1 is the
2.2 Control of membrane filtration Membrane separation is commonly applied to wastewater recycling. In order to reduce the cost from energy load of handling large-scale treatment operation, the design of the operating condition is critical. Membrane filtration process consists of the filtration and the backwash phases; and both are affected by the transmembrane pressure. Based on Darcy’s law the expression for the transmembrane pressure difference, P , of the filtration and the backwash can then be represented respectively as (Jan Busch et al., 2007) Filtration:
Backwash:
K f c 1
0
Figure 1 Experimental set-up of a membrane filtration process with filtration and backwashing phases
Pˆf (c, k f ) = j f c R 0f (c) (c) j f c k f
eˆ c 1, u j f c 1 Ar
(1)
input vector which includes the filtration duration, K f c 1 , the backwash duration, K w c 1 , the flux of filtration, j f c 1 and the flux of backwash, jw c 1 where k f and
kw are the filtration and the backwash time respectively.
eˆ c 1, u is the energy load of cycle c 1 and Ar is the
membrane area. A cycle of operation refers to the start to end of the filtration or backwash operation, and the cyclical membrane filtration can be controlled by cycle to cycle control. The cycle to cycle controller calculates the set-points for the manipulated variables and is only active between the batches (or cycle). The optimal set-points are calculated based on process model which are updated from measurement of previous batch (or cycle). In the control design of the membrane filtration process, the designed variables include the operation time and the flux during the
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operation for both the filtration phase and the backwash cycle c 1 . Thus the corrected energy load for the filtration phase. The filtration can be affected by the drift of the phase, eˆ f , Hybrid c 1, u is given as membrane permeability due to irreversible membrane fouling. Figure 2 shows the evolution of TMP of a process (8) eˆ f , Hybrid c 1, u eˆ f c 1, u eˆ f , Diff c 1, u affected by the fouling. The solid lines represent the TMP of the filtration phase and the backwash phase. The vertical Similarly for the backwash operation, the corrected energy dashed lines indicate a complete cycle of operation. The load can be expressed as operation starts with the filtration phase and switch to the backwash phase which is indicated by the sign change of the eˆw, Hybrid c 1, u eˆw c 1, u eˆw, Diff c 1, u (9) TMP. The initial transmembrane pressure shows a linear increase during filtration. It becomes increasing nonlinear where with increasing number of filtration operations due to irreversible fouling as illustrated in Figure 2. As a result there eˆw, Diff c 1, u ew c 1, u eˆw c 1, u (10) will be a mismatch between the actual current cycle energy load and the designed energy load if the design is based on Thus the objective function is modified and represented as previous cycle and this mismatch is more pronounced as the filtration operation increases. Thus it is necessary to account min eˆ c 1, u K f c 1 , j f c 1 for this mismatch in order to obtain a better design for lower K w c 1 , jw c 1 energy load according to Eq.(4). eˆ f c 1, u (11) min K f c 1 , j f c 1 K w c 1 , jw c 1
P
eˆ f , Diff c 1, u eˆw c 1, u eˆw, Diff c 1, u
Kf (c)
such that
0
Kw(c)
Cycle 1 Cycle 2
jnet Cycle c
Cycle
j f c 1 K f c 1 j f c 1 K w c 1 K f c 1 K w c 1
(12)
j f c 1 jw c 1
Figure 2 Evolution of TMP 3. CONTROL OF MEMBRANE FILTRATION The control strategy for the membrane filtration is detailed in this section. The hybrid model based control strategy is introduced as well as the algorithm based on the EI. 3.1 Hybrid model based cycle to cycle control
In order to represent the mismatch the GPRM is used in this work. This term describes the model mismatch due to fouling between the current cycle to the next cycle. The mismatch terms, e f , Diff c, u and ew, Diff c, u for all past cycles are obtained by calculating the difference between the energy load from the actual operation and the predicted energy load; and thus forming historical data of the operation which are, j f c K f c , c 1, , C for the filtration data, u f c
The general objective function of cycle to cycle control of and e f , Diff c, u , c 1, , C ; and backwash data, membrane filtration is given by Eq.(4). Based on the previous u w c jw c K w c , c 1, , C and ew, Diff c, u , cycle the prediction based on the model can be used to c 1, , C . The GPRM model can thus be represent calculate the operation for the current cycle. However, the respectively as filtration can be affected by the drift of the membrane permeability due to irreversible membrane fouling and Filtration: therefore there will be a mismatch between the designed energy load based on previous cycle and the actual current (13) eˆ f , Diff c, u GPRM u f c operation energy load. In contrast to the previous reported work (Jan Busch et al., 2007) the mismatch is handled. The Backwash: mismatch between the actual energy load and the predicted energy load is given as (14) eˆ c, u GPRM u c eˆ f , Diff c 1, u e f c 1, u eˆ f c 1, u
w, Diff
w
(7)
eˆ f c 1, u is the prediction of the cycle c 1 based on the
3.2 Model mismatch representation by GPRM
model parameters of cycle c and e f c 1, u is the actual
The GPRM can be described as learning a data-driven empirical model that approximates a training set for the filtration process D X, y , where
energy load based on the model with parameters of current
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X
u c
x c c 1,,C
i
y eˆi , Diff c, u
eˆi , Diff c, u
c 1,, C
I u ebest eˆ c 1, u
and
(19)
are the input and output datasets with
where ebest is the current best operation cost. I u can be
C samples, respectively with i f , the filtration phase and i w, the backwashing phase . A GPRM provides a prediction of the output variable for an input sample through Bayesian inference. For an output variable of
viewed as the potential for improving from the current best operation cost. As the prediction is Gaussian it follows that the improvement is also Gaussian, 2 I (u) N ebest eˆ c 1, u , eˆ c 1,u (u) where
c 1,, C
y y1 , , yC , the GPRM is the regression function with a T
eˆ c 1, u eˆ f c 1, u f eˆ f , Diff c 1, u f
Gaussian prior distribution and zero mean, or in a discrete form (Rasmussen & Williams, 2006) y y1 , , yC ~ 0, K T
(15)
eˆ2 c 1,u (u)= eˆ2 c 1,u (u f ) eˆ2
(16)
k k K k
(21)
min EI u
(22)
u
with EI u
I u e eˆ c 1, u best 1 I u exp 2 eˆ2 u 2 eˆ u 0
min eˆ u v (v) (v) where v
(18)
ebest eˆ c 1, u
eˆ (u)
. (v) and (v) are the normal
cumulative distribution and normal respectively. They are expressed as
where k K ( xc 1 , x c 1 ) , k T K ( x c 1, x1 ) K ( x c 1, xc )
dI (23)
2
(24)
u
(17)
Eq.(22) can then be evaluated as
with mean ( yˆ c 1 ) and variance ( y2ˆc1 ) calculated as follows
)
as the filtration and the backwash parameters are independent. The EI is the integral of the predictive distribution defined as
m n ; otherwise, it is equal to zero. θ [v0 , w1 , , wD , b] is the hyper-parameters vector defining the covariance function. The training of a GPRM can determine the values of the hyper-parameters θ . The hyper-parameters θ can be estimated by maximization of a log-likelihood function. And this optimization problem can be solved directly using the derivative of the log-likelihood with respect to each hyperparameter. Detailed implementations for training a GPRM can be referred to Rasmussen and Williams (2006). The GPRM can be obtained once θ is determined. For a new test sample xc1 , the predicted output of yc 1 is also Gaussian
1
c 1, u (u f
eˆ2w c 1,u (u w ) eˆ2w ,Diff c 1,u (u w )
T
f , Diff
f
where xnd is the dth component of the vector x n . mn 1 if
2 yˆc 1
(20)
with
common covariance function can be defined as
yˆ c 1 k K 1y
eˆw c 1, u w eˆw, Diff c 1, u w
where K is the covariance matrix with the mn th element defined by the covariance function, K mn K x m , x n . A D K (x m , x n ) v0 exp wd ( xmd xnd ) 2 mn b d 1
1
is the covariance vector between the new input and the training samples, and K x c 1 , x c 1 is the covariance of the
density
function
v 1 2 2
(25)
v2 exp 2 2
(26)
(v) erf 2
new input. In summary, the vector k T (xc 1 )K 1 denotes a smoothing term which weights the training outputs to make a prediction for the new input sample xc 1 .
and
3.3 Expected Improvement
Based on the EI derivation the improvement that is expected for each cycle of the membrane filtration is defined as
(v )
The EI provides an indication of the improvement that is expected from sampling at new operation based on the min EI u predictive distribution. As the GPRM is Gaussian, the EI is K f c 1 , j f c 1 easy to calculate. Based on GPRM the predictive distribution K w c 1 , jw c 1 of mean and variance (Eq.(17) and Eq.(18)), the predicted improvement, I u , at any point is defined as
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1
min
K f c 1 , j f c 1 K w c 1 , jw c 1
eˆ (u) v (v) (v) (27)
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subject to constraint of Eq.(12). Using the EI algorithm optimization is carried out on both phases with the updated models that are calculated based on
jnet -
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j f ( c ) K f ( c ) jw ( c ) K w ( c ) K f ( c ) K w (c )
(31)
0
(32)
j f (c) jw (c)
Filtration: K f c
1
2 P c, k Pˆ c, k
min0
m c , R f c
f
k f 0
f
f
2
(28)
f
Backwash: min
w c , n c
( c ), Rw
Kw c
1 2 P c, k Pˆ c, k c
kw 0
w
w
w
w
2
(29)
where the filtration and the backwash are subjected to the constraints of Eq.(1) and Eq.(2) respectively. To summarize cycle to cycle control is carried out as follows 1. The estimated energy, eˆ c, u based on the physical model is calculated from Eq.(6) with cycle c . 2. The mismatch, eˆDiff c, u is then calculated from the difference between the real energy load ( e c, u ) and estimated energy load. 3. The mismatch value is used to update the GPRM and to provide a new estimated energy, eˆDiff c 1, u , based on the updated model using Eq.(17).
For the constant operation, the operation specification is u (1.375 105 ,1050,1.385 10 5 , 203) . The comparison between a constant operation, a physical model based design and the hybrid design are made. The result is shown in Figure 3 for fixed operation condition (), physical model based control (○) and expected improvement based cycle to cycle control (×). In this simulation the feed concentration from cycle 26 is increased which resulted in the greater degree of fouling. The constant operation sees a rapid increase in the energy cost. The energy load for the physical model based control only increased greatly after cycle 30 which can be attributed to the effect of model mismatch due to fouling (Figure 3). The hybrid model which takes into account the effect of fouling is thus able to achieve the lowest energy cost among the three operations and only sees an increase in the energy load after cycle 33. Due to the compensation from the GPRM and EI for finding the actual optimum the design is better and the TMP is lower which means the added advantage of an increase of the lifespan of the membrane. Furthermore it can be seen that the mismatch between the physical model prediction (●) and the actual output under the physical model based design (○) is greater than the mismatch between CtC EI prediction ( ) and actual output under the CtC EI design (×) indicating the benefit of the hybrid GPRM.
4. With the current cycle historical data the physical model is updated using Eq.(28) and Eq.(29); which is then used to calculate the new cycle estimated energy load, eˆ c 1, u ,
-3
7
x 10
Fixed operation Physical model based design Physical model prediction CtC Ei design CtC Ei prediction
6
using Eq.(6) with cycle c 1 . 5 Energy cost(Ws)
5. The estimated values from the GPRM and the physical model form the basis of the optimal design (Eq.(11)) for the next cycle, c 1 . 4. CASE STUDIES
4
3
2
The case studies presented in this section consist of a simulation study and an experimental study. Simulation studies are used to illustrate the control of the full operation.
1
0
22
4.1 Simulation study
In order to present the energy load under the control based on the simulated physical system a first-principles model with the resistances-in-series approach described by Broeckmann et al. (2006) and Wintgens et al. (2003) is utilized. The data is collected for an operation of 21 cycles and irreversible fouling starts to set in. The design starts at cycle 22 and further 14 cycles under the fixed net flux. The constraints for the operation are uupper 9 104 ,1800,9 104 ,300 lower 5 5 u 1 10 ,100,1 10 ,100
(30)
24
26
28 Cycle number
30
32
34
Figure 3 Simulation study on cycle to cycle control: comparison of energy cost for cycle 21 to 35 4.2 Experimental
The experimental setup is as described in Section 2.1. Due to the limitation of the experimental hardware in detecting the flux of the impure fluid in the backwash it is not possible to ensure a constant flux operation in the backwash flux. Nevertheless as the backwash phase is much shorter than the filtration phase (J. Busch & Marquardt, 2009) a constant operation of the backwash is not detrimental to the overall
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operation. In the experimental study the backwash is not designed but instead set at a constant time of 60s. Moreover the flux is maintained by calibrating the pump duty to the resultant flux. For the purpose of comparison one set of reading is operated at fixed constant flux of 6.25 105 (m3 / m 2 s ) and fixed time of 490s while another operated based on cycle to cycle control strategy. The constraints of the operation is set to be the physical limits of the experimental system u 8 10 , 660 lower 5 u 7.5 10 , 490 5
upper
(33)
The data from the first two cycles are used to construct the GPRM for cycle to cycle control design. Once the model is obtained cycle to cycle control is carried out from cycle 3 to cycle 11. Figure 4 shows the TMP of the experiment and the corresponding energy load is shown in Figure 5. It can be observed that cycle to cycle control design is able to lower the transmembrane pressure of the operation. With cycle to cycle design the energy load is lower than the fixed operation which resulted from the reduction lower operation. For a typical waste-water treatment plant that process 45000 m3 water daily, this is equal to saving of 5590 units cost. In additional the lifetime of the membrane is also improved due to lower transmembrane pressure operation. 5
2.4
x 10
2.2
P (N/m2)
2
1.8 fixed operation 1.6 CtC EI design
1.4
1.2
1
1
2
3
4
5
6 Cycle
7
8
9
10
11
Figure 4 Experimental study on cycle to cycle control: TMP results 300
280
fixed operation
Energy cost(Ws)
260
240
CtC EI design
220
200
180
1
2
3
4
5 6 7 Cycle number
8
9
10
11
5. CONCLUSION This work presents a framework that combine first principle model with GPRM that compensate for the effect of fouling on the system from previous cycle of operation for control. The main contribution of this work includes handling model mismatch for a more accurate representation of the system and the exploration of new region that has the highest probability of improvement for new design operation. ACKNOWLEDGEMENTS
This work was supported by Ministry of Science and Technology, R.O.C. (MOST 103-2221-E-033-068-MY3 and MOST 104-2811-E-033-009) REFERENCES Blankert, B., Betlem, B. H. L., & Roffel, B. (2006). Dynamic optimization of a dead-end filtration trajectory: Blocking filtration laws. J. Membr. Sci., 285, 90-95. Boyle, P. (2007). Gaussian processes for regression and optimisation. Broeckmann, A., Busch, J., Wintgens, T., & Marquardt, W. (2006). Modeling of pore blocking and cake layer formation in membrane filtration for wastewater treatment. Desalination, 189, 97-109. Busch, J., Cruse, A., & Marquardt, W. (2007). Run-to-run control of membrane filtration processes. AIChE J., 53, 2316-2328. Busch, J., & Marquardt, W. (2009). Model-based control of MF/UF filtration processes: pilot plant implementation and results. Water Sci. Technol., 59, 1713-1720. Farahbakhsh, K., & Smith, D. W. (2006). Membrane filtration for cold regions – impact of cold water on membrane integrity monitoring tests. J. Environ. Eng. Sci., 5, S69-S75. Hwang, T.-M., Oh, H., Choi, Y.-J., Nam, S.-H., Lee, S., & Choung, Y.-K. (2009). Development of a statistical and mathematical hybrid model to predict membrane fouling and performance. Desalination, 247, 210-221. Ranganathan, A., Ming-Hsuan, Y., & Ho, J. (2011). Online Sparse Gaussian Process Regression and Its Applications. Image Process., IEEE Trans. on, 20, 391-404. Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning: MIT Press. Wintgens, T., Rosen, J., Melin, T., Brepols, C., Drensla, K., & Engelhardt, N. (2003). Modelling of a membrane bioreactor system for municipal wastewater treatment. J. Membr. Sci., 216, 55-65. Yigit, N. O., Civelekoglu, G., Harman, I., Koseoglu, H., & Kitis, M. (2009). Effects of various backwash scenarios on membrane fouling in a membrane bioreactor. Desalination, 237, 346-356.
Figure 5 Experimental study: Comparison of energy load of cycle to cycle control and fixed operation 1090
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IFAC-PapersOnLine 49-7 (2016) 1085–1090 Hybrid model based control for membrane filtration process Hybrid model based control for membrane filtration process Hybrid based control for membrane filtration process Hybrid model model based control for membrane filtration process Lester Lik Teck Chan, Chen-Pei Chou and Junghui Chen
Lester Lik Teck Chan, Chen-Pei Chou and Junghui Chen Lester Teck Chan, Chen-Pei Chou and Lester Lik LikR&D TeckCenter Chan,for Chen-Pei Chou and Junghui Junghui Chen Chen Membrane Technology R&D Center for MembraneEngineering Technology Department of Chemical R&D Center Membrane Technology R&D Center for for MembraneUniversity Technology Department of Chemical Engineering Chung-Yuan Christian Department of Chemical Engineering Department of Chemical Engineering Chung-Yuan Christian University Chung-Li, TaiwanChristian 320, Republic of China Chung-Yuan University Chung-Yuan Christian University Chung-Li, Taiwan 320, Republic of China Chung-Li, Chung-Li, Taiwan Taiwan 320, 320, Republic Republic of of China China (Fax: +886-3-2654199 e-mail: [email protected] ) (Fax: +886-3-2654199 e-mail: [email protected] ) (Fax: (Fax: +886-3-2654199 +886-3-2654199 e-mail: e-mail: [email protected] [email protected] )) Abstract: Membrane fouling can affect the performance of the membrane-based filtration for water Abstract: Membrane fouling can operational affect the performance of the presents membrane-based filtration for water treatment. Fouling thus increases costs. This of work a modeling framework that Abstract: Membrane fouling can affect the performance the membrane-based filtration for water Abstract: Membrane fouling can affect the performance of the membrane-based filtration for water treatment. Fouling thus increases operational costs. This work presents a modeling framework that combine first principle model with Gaussian process model that aims to reduce the energy load affected treatment. Fouling thus increases operational costs. This presents aa modeling framework that treatment. Fouling thusmodel increases operational costs. This work work presents modeling framework that combine first principle with Gaussian process model thatimprovement aims to reduce the energy load affected by the effect of fouling on thewith system. Basedprocess on the model expected algorithm, an load optimization combine first principle model Gaussian that aims to reduce the energy affected combine first principle model with Gaussian process model that aims to reduce the energy load affected by the effect of fouling on the system. Based of on the theoperation expected to improvement algorithm, an optimization control is proposed to handle thesystem. long duration achieve the most economical operation by the of on Based on the expected improvement algorithm, an by term the effect effect of fouling fouling on the the Based of on the theoperation expected to improvement algorithm, an optimization optimization control is proposed to handle thesystem. long duration achieve the most economical operation in of energy load. control is proposed to control to handle handle the the long long duration duration of of the the operation operation to to achieve achieve the the most most economical economical operation operation in term is of proposed energy load. in energy load. © term 2016, of IFAC (International Federation of Automatic Control) Hosting byuncertainty, Elsevier Ltd.variance All rights reserved. in term of energy load. Keywords: model based control, iterative improvement, optimization, Keywords: model based control, iterative improvement, optimization, uncertainty, variance Keywords: model model based based control, control, iterative iterative improvement, improvement, optimization, uncertainty, uncertainty, variance variance Keywords: optimization, models can generally describe the fundamental phenomena of 1. INTRODUCTION models canfouling. generally describe the fundamental phenomena of membrane Data data-driven empirical models, on the 1. INTRODUCTION models can generally describe the phenomena of models canfouling. generally describe the fundamental fundamental phenomena of membrane Data data-driven empirical models, on the 1. INTRODUCTION other hand, are suitable to predict complex behaviours in 1. INTRODUCTION Membrane filtrations are widely used in water treatment membrane fouling. Data data-driven empirical models, on the membrane fouling. Data data-driven empirical models, on the other hand, are suitable to predict complex behaviours in nonlinear processes (Hwang et al., 2009). The combination of Membrane filtrations are widely used in water treatment applications and the benefits include lowinenergy high other hand, are to complex behaviours in Membrane filtrations filtrations are widely widely used water load, treatment other hand, are suitable suitable to etpredict predict complex behaviours in nonlinear processes (Hwang al., 2009). The combination of Membrane are used in water treatment both thus provides the advantages of first-principles and dataapplications and the benefits include low energy load, high selectivity and ease of operation (Farahbakhsh and Smith, nonlinear processes (Hwang et al., 2009). The combination of applications and and the the benefits benefits include include low low energy energy load, load, high high both nonlinear processes (Hwang et al., 2009). The combination of thus provides the advantages of first-principles and dataapplications empirical models for the control of membrane selectivity and ease offiltration operation andpressure Smith, driven 2006). In membrane the(Farahbakhsh transmembrane thus provides advantages of and selectivity and ease ease of of operation operation (Farahbakhsh and Smith, Smith, both both thusempirical provides the the advantages of first-principles first-principles and datadatamodels for the control of membrane selectivity and (Farahbakhsh and filtration. 2006). In membrane filtration the transmembrane pressure driven (TMP) difference forces the fluid and smaller particles driven empirical empirical models models for for the the control control of of membrane membrane 2006). In membrane filtration the transmembrane pressure driven filtration. 2006). In membrane filtration the transmembrane pressure (TMP) difference forces the Membrane fluid and fouling smaller increases particles filtration. through the membrane pores. (TMP) difference forces the fluid smaller particles filtration. In this work the filtration process is modeled using a physical (TMP) forces the fluid and and smaller particles through difference thecosts membrane pores. Membrane fouling increases operational as a result of permeate flux decline and can In this work the filtration process is modeled using a physical through the membrane pores. Membrane fouling increases model complimented by a is statistical process through the membrane pores. Membrane fouling increases operational costs as a result of permeate flux decline and can In this this that workisthe the filtration process process modeledGaussian using aa physical physical be accompanied by increased energy loadflux duedecline to higher TMP In work filtration is modeled using model that is complimented by a statistical Gaussian operational costs as a result of permeate and can regression model (GPRM) which accounts for the mismatch operational costs as increased a as result ofenergy permeate flux decline and can model that is complimented by a statistical Gaussian process be accompanied by due to higher TMP process requirements needed driving force.load The reversible fouling model is complimented bycurrent a statistical Gaussian process regression model (GPRM) which accounts thetomismatch be accompanied by increased energy load due to higher TMP in the that previous cycle to the cyclefor due fouling. be accompanied by increased energy load due to higher TMP requirements needed as driving force. The reversible fouling regression model (GPRM) which accounts for the mismatch can be eliminated at least partially by aeration and regression model (GPRM) which accounts for the mismatch in the previous cycle to theprobabilistic current cycle due (Rasmussen to fouling. requirements needed needed as as driving driving force. force. The The reversible reversible fouling fouling GPRM is a non-parametric model requirements can be eliminated least partially by aeration and in the previous cycle the current cycle due to backwashing (Yigit etat 2009). Optimizing the operation the previous cycle to to theprobabilistic current cycle due to fouling. fouling. GPRM is a non-parametric model (Rasmussen can be be eliminated eliminated atal.,least least partially by aeration aeration and in and Williams, 2006). Compared to neural network (NN), can at partially by and backwashing (Yigit et al., 2009). Optimizing the operation GPRM is a non-parametric probabilistic model (Rasmussen through process control can reduce the energy demand and GPRM is a non-parametric probabilistic model (Rasmussen and Williams, 2006). Compared to neural network (NN), backwashing (Yigit et al., 2009). Optimizing the operation GPRM not only provides the prediction of an output but also backwashing (Yigit et al., 2009). Optimizing the operation through process operational control cancosts. reduce the energy demand and and Williams, 2006). Compared to neural network (NN), other associated and Williams, 2006). Compared to neural network (NN), GPRM not only provides the prediction of an output but also through process control can reduce the energy demand and the variance of estimation which can be interpreted as a level through process control can reduce the energy demand and other associated operational costs. GPRM not not only only provides provides the the prediction prediction of of an an output output but but also also GPRM the variance of estimation which can be interpreted as a level other associated operational costs. the model.which Moreover, number of other associated operational costs. Filtration processes are usually controlled to meet the desired of theconfidence variance of ofofestimation estimation can be bethe interpreted as GPRM a level level the variance which can interpreted as a confidence of the the number of GPRM hyper-parameters thatmodel. need Moreover, to be optimized is small. Based Filtration meet desired of net flux. processes However are theusually high controlled complexityto of thethe filtration of confidence confidence of of the the model. Moreover, the number number of GPRM GPRM Filtration processes are usually controlled to meet the desired of model. Moreover, the of hyper-parameters that need to be optimized is small. Based Filtration processes are usually controlled to meet the desired on the predictive distribution the expected improvement (EI) net flux. However the high complexity of the filtration process poses a challenge. It is complexity characterizedofbythe the filtration periodic hyper-parameters that to be optimized is Based net flux. However the high hyper-parameters that need need tothe beexpected optimized is small. small. of Based on the predictive distribution improvement (EI) net flux. However the high complexity of the filtration (Boyle, 2007; Jones, 2001) provides an indication the process poses a challenge. It is characterized by the periodic change between filtration and backwashing, by the the periodic drift of on the distribution the expected improvement (EI) process poses aa challenge. It is characterized by on the predictive predictive distribution the expected improvement (EI) (Boyle, 2007; Jones, 2001) provides an indication of the process poses challenge. It is characterized by the periodic improvement that is expected from sampling at new change between filtration and backwashing, by the drift of membrane permeability due backwashing, to irreversible membrane 2007; Jones, 2001) provides an indication of the change between between filtration and and by the the drift of of (Boyle, (Boyle, 2007; Jones, 2001) provides ansampling indication of new the improvement that is expected from at change filtration backwashing, by drift operation. It has the capability to ‘explore’ new area and membrane permeability due to irreversible membrane fouling, and by a high-number of disturbances, including improvement that is expected from sampling at new membrane permeability permeability due due to to irreversible irreversible membrane membrane improvement that is expected from sampling at new operation. It has the capability to ‘explore’ new area and membrane newItinformation about thetooptimization problem and fouling, and ofby atemperature high-number orof disturbances, including collect variations solids concentration. operation. has the ‘explore’ area and fouling, and by aa high-number of disturbances, including operation. Itinformation has the capability capability tooptimization ‘explore’to new new area and and collect new about the problem fouling, by most high-number disturbances, including is used for control design. Compared conventional variationsand ofin temperature orof solids TMP concentration. Furthermore, cases, only the overall across an collect new new information information about about the the optimization optimization problem problem and and variations of temperature or solids concentration. collect is used for control design. Compared to conventional variations of temperature or solids concentration. are deterministic EI is probabilistic which Furthermore, in most cases, only the overall TMP across an methods entire membrane module is measured which means that little is used used which for control control design. Compared Compared to conventional conventional Furthermore, in most cases, only the overall TMP across an for design. to methods EI is probabilistic which Furthermore, in most cases, only the overall TMP across an is takes intowhich accountare thedeterministic model uncertainty. entire membrane module is measured which means that little information is available to understand the process. methods which are deterministic EI is probabilistic entire membrane module is measured which means that little methods which are deterministic EI is probabilistic which which takes into account the model uncertainty. entire membrane module is measured which means that little information is available to understand the process. Development of model givestoinsight into the effects of the takes into account the model uncertainty. information is available understand the process. takes into account the model uncertainty. information is available toinsight understand the process. Development of model into the effects of the 2. MEMBRANE FILTRATION SYSTEM AND CONTROL controls. Blankert et al. gives (2006) developed a filtration model Development of model gives insight into the effects of the Development of model gives insight into the effects of the 2. MEMBRANE FILTRATION SYSTEM AND CONTROL controls. Blankert et al. (2006) developed a filtration model and determined theetoptimal profile of the filtration flux and 2. FILTRATION SYSTEM AND CONTROL controls. Blankert al. (2006) (2006) developed a filtration filtration model 2. MEMBRANE MEMBRANE FILTRATION SYSTEM AND CONTROL The membrane filtration experimental system is introduced in controls. Blankert et al. developed a model and determined the optimal profile of the filtration flux and TMP during one filtration phase using offline dynamic The membrane filtration experimental system istointroduced in and determined determined the the optimal optimal profile profile of of the the filtration filtration flux flux and and this section. This experimental setup is used demonstrate and TMP during Due one tofiltration phase using offline will dynamic membrane filtration experimental system is in optimization. fouling the model parameters differ The The membrane filtration experimental system istointroduced introduced in this section. This experimental setup is usedthe demonstrate TMP during one filtration phase using offline dynamic the proposed control strategy. In addition conventional TMP during one filtration phase using offline dynamic optimization. Dueand to fouling the will model parameters will differ this section. This experimental setup is used to demonstrate with each cycle thus there be a mismatch between this section. This experimental setup is used to demonstrate the proposed control strategy. In addition the conventional optimization. Due to fouling the model parameters will differ of the membrane filtration is introduced. optimization. Dueand to fouling the will model differ control withpredicted each cycle thus –there be parameters a mismatch between the control In the the energy load using parameters of thewill previous the proposed proposed control strategy. strategy. In isaddition addition the conventional conventional control of the membrane filtration introduced. with each cycle and thus there will be aa mismatch between with each cycle and thus there will be mismatch between the predicted energy load – using parameters of the previous control of of the the membrane membrane filtration filtration is is introduced. introduced. cycle – and actual current cycle energy load. This mismatch control the predicted energy load – using parameters of the previous the predicted energy load –cycle using parameters of themismatch previous cycle actual energy This has to – beand taken intocurrent account to obtain theload. best performance of 2.1 Membrane Filtration Experimental System cycle – and actual current cycle energy load. This mismatch cycle – and actual current cycle energy load. This mismatch has to be taken into account to obtain the best performance of 2.1 Membrane Filtration Experimental System the filtration process. In membrane filtration, first-principles 2.1 Membrane Membrane Filtration Filtration Experimental Experimental System System has to into to the of has to be be taken taken into account account to obtain obtainfiltration, the best best performance performance of 2.1 the filtration process. In membrane first-principles the filtration filtration process. process. In In membrane membrane filtration, filtration, first-principles first-principles the
Copyright © 2016, 2016 IFAC 1085Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2016 responsibility IFAC 1085Control. Peer review of International Federation of Automatic Copyright © 1085 Copyright © 2016 2016 IFAC IFAC 1085 10.1016/j.ifacol.2016.07.347
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Figure 1 shows the experimental set-up of the membrane filtration system in this study. The operation is fully automatic. The pressure transmitter (PT) and flow transmitter (FT) reading as send to the computer and this is represented ). Based on these signals the operation by the signal ( condition of the electric valve and pump are calculated and the signal to the final control element are represented by ). During the filtration phase, valve A will be opened ( in the direction to allow the feed to enter the pump where it is pressurized before entering the filtration unit through valve B indicated by the filtration path in Figure 1. The driving force is the pressure difference between the filtration end and the permeate end. It drives the water to the permeate side and enters the permeate tank. In the washing phase, feed to valve A is closed and valve A is connected to the permeate tank instead. Valve B (backwash path in Figure 1) now allows water to enter the permeate side which carries the backwashing operation. The filtration and the backwash phases thus form the cyclic operation and the energy load is the result of the TMP from the pump. The feed solution consisting of kaolinite and sodium alginate is utilized to generate a combination of particulate and macromolecular foulants. A polyvinylidene fluoride (PVDF) membrane with dimension of 4cm 10cm is used in this study.
Pˆw c, kw jw c k w Rw c R c exp w c
(2)
where j is the flux which is the manipulated variables, is the fluid viscosity, R is the membrane resistance, c represents the c th cycle, and k is the operating time. The subscripts f and w denote the filtration and the backwash phases, respectively. R 0f is the initial membrane resistance
, and w are the model parameters is a parameter dependant on the feed suspension. R is expressed as
R R f c, K f c Rw c
(3)
where R f c, K f c is the final resistance of the filtration phase and Rw is the irreversible resistance. K f c is the duration of the c th filtration phase. The operation is cyclical and the objective is to find the least amount of energy load for a particular flux, jnet at the next cycle. It is defined as eˆ c 1, u
min
K f c 1 , j f c 1 K w c 1 , jw c 1
(4)
such that jnet
j f c 1 K f c 1 j f c 1 K w c 1 K f c 1 K w c 1
(5)
j f c 1 jw c 1
jw c 1 Ar
K w c 1
0
Pˆf (c 1, k f ) d k f (Pˆw (c 1, kw )) d kw
(6)
where u j f c 1 , K f c 1 , jw c 1 , K w c 1 is the
2.2 Control of membrane filtration Membrane separation is commonly applied to wastewater recycling. In order to reduce the cost from energy load of handling large-scale treatment operation, the design of the operating condition is critical. Membrane filtration process consists of the filtration and the backwash phases; and both are affected by the transmembrane pressure. Based on Darcy’s law the expression for the transmembrane pressure difference, P , of the filtration and the backwash can then be represented respectively as (Jan Busch et al., 2007) Filtration:
Backwash:
K f c 1
0
Figure 1 Experimental set-up of a membrane filtration process with filtration and backwashing phases
Pˆf (c, k f ) = j f c R 0f (c) (c) j f c k f
eˆ c 1, u j f c 1 Ar
(1)
input vector which includes the filtration duration, K f c 1 , the backwash duration, K w c 1 , the flux of filtration, j f c 1 and the flux of backwash, jw c 1 where k f and
kw are the filtration and the backwash time respectively.
eˆ c 1, u is the energy load of cycle c 1 and Ar is the
membrane area. A cycle of operation refers to the start to end of the filtration or backwash operation, and the cyclical membrane filtration can be controlled by cycle to cycle control. The cycle to cycle controller calculates the set-points for the manipulated variables and is only active between the batches (or cycle). The optimal set-points are calculated based on process model which are updated from measurement of previous batch (or cycle). In the control design of the membrane filtration process, the designed variables include the operation time and the flux during the
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operation for both the filtration phase and the backwash cycle c 1 . Thus the corrected energy load for the filtration phase. The filtration can be affected by the drift of the phase, eˆ f , Hybrid c 1, u is given as membrane permeability due to irreversible membrane fouling. Figure 2 shows the evolution of TMP of a process (8) eˆ f , Hybrid c 1, u eˆ f c 1, u eˆ f , Diff c 1, u affected by the fouling. The solid lines represent the TMP of the filtration phase and the backwash phase. The vertical Similarly for the backwash operation, the corrected energy dashed lines indicate a complete cycle of operation. The load can be expressed as operation starts with the filtration phase and switch to the backwash phase which is indicated by the sign change of the eˆw, Hybrid c 1, u eˆw c 1, u eˆw, Diff c 1, u (9) TMP. The initial transmembrane pressure shows a linear increase during filtration. It becomes increasing nonlinear where with increasing number of filtration operations due to irreversible fouling as illustrated in Figure 2. As a result there eˆw, Diff c 1, u ew c 1, u eˆw c 1, u (10) will be a mismatch between the actual current cycle energy load and the designed energy load if the design is based on Thus the objective function is modified and represented as previous cycle and this mismatch is more pronounced as the filtration operation increases. Thus it is necessary to account min eˆ c 1, u K f c 1 , j f c 1 for this mismatch in order to obtain a better design for lower K w c 1 , jw c 1 energy load according to Eq.(4). eˆ f c 1, u (11) min K f c 1 , j f c 1 K w c 1 , jw c 1
P
eˆ f , Diff c 1, u eˆw c 1, u eˆw, Diff c 1, u
Kf (c)
such that
0
Kw(c)
Cycle 1 Cycle 2
jnet Cycle c
Cycle
j f c 1 K f c 1 j f c 1 K w c 1 K f c 1 K w c 1
(12)
j f c 1 jw c 1
Figure 2 Evolution of TMP 3. CONTROL OF MEMBRANE FILTRATION The control strategy for the membrane filtration is detailed in this section. The hybrid model based control strategy is introduced as well as the algorithm based on the EI. 3.1 Hybrid model based cycle to cycle control
In order to represent the mismatch the GPRM is used in this work. This term describes the model mismatch due to fouling between the current cycle to the next cycle. The mismatch terms, e f , Diff c, u and ew, Diff c, u for all past cycles are obtained by calculating the difference between the energy load from the actual operation and the predicted energy load; and thus forming historical data of the operation which are, j f c K f c , c 1, , C for the filtration data, u f c
The general objective function of cycle to cycle control of and e f , Diff c, u , c 1, , C ; and backwash data, membrane filtration is given by Eq.(4). Based on the previous u w c jw c K w c , c 1, , C and ew, Diff c, u , cycle the prediction based on the model can be used to c 1, , C . The GPRM model can thus be represent calculate the operation for the current cycle. However, the respectively as filtration can be affected by the drift of the membrane permeability due to irreversible membrane fouling and Filtration: therefore there will be a mismatch between the designed energy load based on previous cycle and the actual current (13) eˆ f , Diff c, u GPRM u f c operation energy load. In contrast to the previous reported work (Jan Busch et al., 2007) the mismatch is handled. The Backwash: mismatch between the actual energy load and the predicted energy load is given as (14) eˆ c, u GPRM u c eˆ f , Diff c 1, u e f c 1, u eˆ f c 1, u
w, Diff
w
(7)
eˆ f c 1, u is the prediction of the cycle c 1 based on the
3.2 Model mismatch representation by GPRM
model parameters of cycle c and e f c 1, u is the actual
The GPRM can be described as learning a data-driven empirical model that approximates a training set for the filtration process D X, y , where
energy load based on the model with parameters of current
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X
u c
x c c 1,,C
i
y eˆi , Diff c, u
eˆi , Diff c, u
c 1,, C
I u ebest eˆ c 1, u
and
(19)
are the input and output datasets with
where ebest is the current best operation cost. I u can be
C samples, respectively with i f , the filtration phase and i w, the backwashing phase . A GPRM provides a prediction of the output variable for an input sample through Bayesian inference. For an output variable of
viewed as the potential for improving from the current best operation cost. As the prediction is Gaussian it follows that the improvement is also Gaussian, 2 I (u) N ebest eˆ c 1, u , eˆ c 1,u (u) where
c 1,, C
y y1 , , yC , the GPRM is the regression function with a T
eˆ c 1, u eˆ f c 1, u f eˆ f , Diff c 1, u f
Gaussian prior distribution and zero mean, or in a discrete form (Rasmussen & Williams, 2006) y y1 , , yC ~ 0, K T
(15)
eˆ2 c 1,u (u)= eˆ2 c 1,u (u f ) eˆ2
(16)
k k K k
(21)
min EI u
(22)
u
with EI u
I u e eˆ c 1, u best 1 I u exp 2 eˆ2 u 2 eˆ u 0
min eˆ u v (v) (v) where v
(18)
ebest eˆ c 1, u
eˆ (u)
. (v) and (v) are the normal
cumulative distribution and normal respectively. They are expressed as
where k K ( xc 1 , x c 1 ) , k T K ( x c 1, x1 ) K ( x c 1, xc )
dI (23)
2
(24)
u
(17)
Eq.(22) can then be evaluated as
with mean ( yˆ c 1 ) and variance ( y2ˆc1 ) calculated as follows
)
as the filtration and the backwash parameters are independent. The EI is the integral of the predictive distribution defined as
m n ; otherwise, it is equal to zero. θ [v0 , w1 , , wD , b] is the hyper-parameters vector defining the covariance function. The training of a GPRM can determine the values of the hyper-parameters θ . The hyper-parameters θ can be estimated by maximization of a log-likelihood function. And this optimization problem can be solved directly using the derivative of the log-likelihood with respect to each hyperparameter. Detailed implementations for training a GPRM can be referred to Rasmussen and Williams (2006). The GPRM can be obtained once θ is determined. For a new test sample xc1 , the predicted output of yc 1 is also Gaussian
1
c 1, u (u f
eˆ2w c 1,u (u w ) eˆ2w ,Diff c 1,u (u w )
T
f , Diff
f
where xnd is the dth component of the vector x n . mn 1 if
2 yˆc 1
(20)
with
common covariance function can be defined as
yˆ c 1 k K 1y
eˆw c 1, u w eˆw, Diff c 1, u w
where K is the covariance matrix with the mn th element defined by the covariance function, K mn K x m , x n . A D K (x m , x n ) v0 exp wd ( xmd xnd ) 2 mn b d 1
1
is the covariance vector between the new input and the training samples, and K x c 1 , x c 1 is the covariance of the
density
function
v 1 2 2
(25)
v2 exp 2 2
(26)
(v) erf 2
new input. In summary, the vector k T (xc 1 )K 1 denotes a smoothing term which weights the training outputs to make a prediction for the new input sample xc 1 .
and
3.3 Expected Improvement
Based on the EI derivation the improvement that is expected for each cycle of the membrane filtration is defined as
(v )
The EI provides an indication of the improvement that is expected from sampling at new operation based on the min EI u predictive distribution. As the GPRM is Gaussian, the EI is K f c 1 , j f c 1 easy to calculate. Based on GPRM the predictive distribution K w c 1 , jw c 1 of mean and variance (Eq.(17) and Eq.(18)), the predicted improvement, I u , at any point is defined as
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1
min
K f c 1 , j f c 1 K w c 1 , jw c 1
eˆ (u) v (v) (v) (27)
IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, Norway Lester Lik Teck Chan et al. / IFAC-PapersOnLine 49-7 (2016) 1085–1090
subject to constraint of Eq.(12). Using the EI algorithm optimization is carried out on both phases with the updated models that are calculated based on
jnet -
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j f ( c ) K f ( c ) jw ( c ) K w ( c ) K f ( c ) K w (c )
(31)
0
(32)
j f (c) jw (c)
Filtration: K f c
1
2 P c, k Pˆ c, k
min0
m c , R f c
f
k f 0
f
f
2
(28)
f
Backwash: min
w c , n c
( c ), Rw
Kw c
1 2 P c, k Pˆ c, k c
kw 0
w
w
w
w
2
(29)
where the filtration and the backwash are subjected to the constraints of Eq.(1) and Eq.(2) respectively. To summarize cycle to cycle control is carried out as follows 1. The estimated energy, eˆ c, u based on the physical model is calculated from Eq.(6) with cycle c . 2. The mismatch, eˆDiff c, u is then calculated from the difference between the real energy load ( e c, u ) and estimated energy load. 3. The mismatch value is used to update the GPRM and to provide a new estimated energy, eˆDiff c 1, u , based on the updated model using Eq.(17).
For the constant operation, the operation specification is u (1.375 105 ,1050,1.385 10 5 , 203) . The comparison between a constant operation, a physical model based design and the hybrid design are made. The result is shown in Figure 3 for fixed operation condition (), physical model based control (○) and expected improvement based cycle to cycle control (×). In this simulation the feed concentration from cycle 26 is increased which resulted in the greater degree of fouling. The constant operation sees a rapid increase in the energy cost. The energy load for the physical model based control only increased greatly after cycle 30 which can be attributed to the effect of model mismatch due to fouling (Figure 3). The hybrid model which takes into account the effect of fouling is thus able to achieve the lowest energy cost among the three operations and only sees an increase in the energy load after cycle 33. Due to the compensation from the GPRM and EI for finding the actual optimum the design is better and the TMP is lower which means the added advantage of an increase of the lifespan of the membrane. Furthermore it can be seen that the mismatch between the physical model prediction (●) and the actual output under the physical model based design (○) is greater than the mismatch between CtC EI prediction ( ) and actual output under the CtC EI design (×) indicating the benefit of the hybrid GPRM.
4. With the current cycle historical data the physical model is updated using Eq.(28) and Eq.(29); which is then used to calculate the new cycle estimated energy load, eˆ c 1, u ,
-3
7
x 10
Fixed operation Physical model based design Physical model prediction CtC Ei design CtC Ei prediction
6
using Eq.(6) with cycle c 1 . 5 Energy cost(Ws)
5. The estimated values from the GPRM and the physical model form the basis of the optimal design (Eq.(11)) for the next cycle, c 1 . 4. CASE STUDIES
4
3
2
The case studies presented in this section consist of a simulation study and an experimental study. Simulation studies are used to illustrate the control of the full operation.
1
0
22
4.1 Simulation study
In order to present the energy load under the control based on the simulated physical system a first-principles model with the resistances-in-series approach described by Broeckmann et al. (2006) and Wintgens et al. (2003) is utilized. The data is collected for an operation of 21 cycles and irreversible fouling starts to set in. The design starts at cycle 22 and further 14 cycles under the fixed net flux. The constraints for the operation are uupper 9 104 ,1800,9 104 ,300 lower 5 5 u 1 10 ,100,1 10 ,100
(30)
24
26
28 Cycle number
30
32
34
Figure 3 Simulation study on cycle to cycle control: comparison of energy cost for cycle 21 to 35 4.2 Experimental
The experimental setup is as described in Section 2.1. Due to the limitation of the experimental hardware in detecting the flux of the impure fluid in the backwash it is not possible to ensure a constant flux operation in the backwash flux. Nevertheless as the backwash phase is much shorter than the filtration phase (J. Busch & Marquardt, 2009) a constant operation of the backwash is not detrimental to the overall
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IFAC DYCOPS-CAB, 2016 1090 Lester Lik Teck Chan et al. / IFAC-PapersOnLine 49-7 (2016) 1085–1090 June 6-8, 2016. NTNU, Trondheim, Norway
operation. In the experimental study the backwash is not designed but instead set at a constant time of 60s. Moreover the flux is maintained by calibrating the pump duty to the resultant flux. For the purpose of comparison one set of reading is operated at fixed constant flux of 6.25 105 (m3 / m 2 s ) and fixed time of 490s while another operated based on cycle to cycle control strategy. The constraints of the operation is set to be the physical limits of the experimental system u 8 10 , 660 lower 5 u 7.5 10 , 490 5
upper
(33)
The data from the first two cycles are used to construct the GPRM for cycle to cycle control design. Once the model is obtained cycle to cycle control is carried out from cycle 3 to cycle 11. Figure 4 shows the TMP of the experiment and the corresponding energy load is shown in Figure 5. It can be observed that cycle to cycle control design is able to lower the transmembrane pressure of the operation. With cycle to cycle design the energy load is lower than the fixed operation which resulted from the reduction lower operation. For a typical waste-water treatment plant that process 45000 m3 water daily, this is equal to saving of 5590 units cost. In additional the lifetime of the membrane is also improved due to lower transmembrane pressure operation. 5
2.4
x 10
2.2
P (N/m2)
2
1.8 fixed operation 1.6 CtC EI design
1.4
1.2
1
1
2
3
4
5
6 Cycle
7
8
9
10
11
Figure 4 Experimental study on cycle to cycle control: TMP results 300
280
fixed operation
Energy cost(Ws)
260
240
CtC EI design
220
200
180
1
2
3
4
5 6 7 Cycle number
8
9
10
11
5. CONCLUSION This work presents a framework that combine first principle model with GPRM that compensate for the effect of fouling on the system from previous cycle of operation for control. The main contribution of this work includes handling model mismatch for a more accurate representation of the system and the exploration of new region that has the highest probability of improvement for new design operation. ACKNOWLEDGEMENTS
This work was supported by Ministry of Science and Technology, R.O.C. (MOST 103-2221-E-033-068-MY3 and MOST 104-2811-E-033-009) REFERENCES Blankert, B., Betlem, B. H. L., & Roffel, B. (2006). Dynamic optimization of a dead-end filtration trajectory: Blocking filtration laws. J. Membr. Sci., 285, 90-95. Boyle, P. (2007). Gaussian processes for regression and optimisation. Broeckmann, A., Busch, J., Wintgens, T., & Marquardt, W. (2006). Modeling of pore blocking and cake layer formation in membrane filtration for wastewater treatment. Desalination, 189, 97-109. Busch, J., Cruse, A., & Marquardt, W. (2007). Run-to-run control of membrane filtration processes. AIChE J., 53, 2316-2328. Busch, J., & Marquardt, W. (2009). Model-based control of MF/UF filtration processes: pilot plant implementation and results. Water Sci. Technol., 59, 1713-1720. Farahbakhsh, K., & Smith, D. W. (2006). Membrane filtration for cold regions – impact of cold water on membrane integrity monitoring tests. J. Environ. Eng. Sci., 5, S69-S75. Hwang, T.-M., Oh, H., Choi, Y.-J., Nam, S.-H., Lee, S., & Choung, Y.-K. (2009). Development of a statistical and mathematical hybrid model to predict membrane fouling and performance. Desalination, 247, 210-221. Ranganathan, A., Ming-Hsuan, Y., & Ho, J. (2011). Online Sparse Gaussian Process Regression and Its Applications. Image Process., IEEE Trans. on, 20, 391-404. Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning: MIT Press. Wintgens, T., Rosen, J., Melin, T., Brepols, C., Drensla, K., & Engelhardt, N. (2003). Modelling of a membrane bioreactor system for municipal wastewater treatment. J. Membr. Sci., 216, 55-65. Yigit, N. O., Civelekoglu, G., Harman, I., Koseoglu, H., & Kitis, M. (2009). Effects of various backwash scenarios on membrane fouling in a membrane bioreactor. Desalination, 237, 346-356.
Figure 5 Experimental study: Comparison of energy load of cycle to cycle control and fixed operation 1090