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Optimal Water Quality Control of Sequencing Batch Reactors Under Uncertainty Barbara E. Rodriguez-Perez, Antonio Flores-Tlacuahuac, Luis Ricardez Sandoval, and Francisco Jose Lozano Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b01076 • Publication Date (Web): 29 Jun 2018 Downloaded from http://pubs.acs.org on July 6, 2018

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Industrial & Engineering Chemistry Research

Optimal Water Quality Control of Sequencing Batch Reactors Under Uncertainty Bárbara E. Rodríguez-Pérez,† Antonio Flores-Tlacuahuac,∗,‡ Luis Ricardez-Sandoval,¶ and Francisco José Lozano‡ †Departamento de Ingeniería y Ciencias Químicas, Universidad Iberoamericana, Mexico City, Mexico ‡Escuela de Ingenieria y Ciencias, Tecnologico de Monterrey, Campus Monterrey, N.L., 64849, Mexico ¶Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario Canada N2L 3G1 E-mail: [email protected] Phone: +52(1) 55 4347 2804

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Abstract Sequencing batch reactors (SBR) are widely used in waste water treatment due to flexibility in operation and low investment costs. However, the main drawback of this technology is the large energy requirements for its operation. In addition, SBR performance is mostly determined by the quality of treated water, which is affected by model uncertainty. Efficient control systems that can meet the operational goals for this process under uncertainty are therefore desired. In this work, we propose an efficient control approach for composition control of organic matter and nitrogen in SBR systems in presence of the model uncertainty. A local sensitivity analysis was performed to identify the variables that have a major impact on SBR behavior. Robust and stochastic model predictive controllers were designed to effectively control the SBR process under uncertainty. Our results indicate that water quality requirements can be achieved even in presence of uncertainty without sacrificing SBR performance. Keywords: Optimization, Predictive control, WasteWater Treatment, SBR process, Uncertainty

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1 Introduction Water biological treatment processes can be classified depending on the metabolic functions of microorganisms as follows: 1 (a) aerobic processes, in which oxygen is the primary electron acceptor, (b) anoxic processes, which are carried out in the presence of low concentrations of molecular oxygen, and (c) anaerobic processes, carried out in the absence of oxygen. Depending on the type of microorganisms, they can be classified as fixed growth or in suspension. In processes involving biomass, the microorganisms grow adhered to an inert material whereas in the suspended biomass processes the flocks are kept in suspension due to mixing. The aerobic process with suspended biomass is commonly referred to as activated sludge 2 . In a typical biological wastewater treatment process, the degradation of organic matter by microorganisms takes place first followed by the separation of the treated water from the microorganisms. In continuous waste water treatment systems, these stages are carried out in two tanks. In the first tank, also referred to as the biological reactor, the set of biochemical reactions are carried out allowing mineralization of pollutant organic material; in the second tank, a settler is deployed to separate the microorganisms from the treated effluent. In batch water treatment systems, both treatment processes are performed in the same tank, i.e. biochemical reactions followed by a phase of sedimentation and the subsequent draining of the reactor. The use of sequencing batch reactors (SBR) is an attractive alternative that has been used for the treatment of water effluents. In the present work, we have considered the deployment of SBR wastewater treatment systems for the removal of both organic matter and nitrogen. As shown in Figure 1, the SBR system operates as a dynamic system through a series of stages, i.e. (1) Filling, (2) Reaction, (3) Sedimentation, (4) Decantation and (5) Idle. In the filling phase, the substrate (waste water) is added to the reactor. In the reaction step, the microorganisms decompose the organic matter using the oxygen provided from aeration. In the sedimentation phase, a settle sludge and clarified effluent are formed. In the decantation phase, the separation of the treated wastewater takes place. 3



Finally, the waste sludge is removed to start a new batch operation in the idle phase. The SBR water treatment process offers the following benefits with respect to continuous processes: flexibility in the operation and reduction in the investment costs given that a single tank is employed as both reactor and settler. Despite these benefits, SBR systems are energy intensive since they required power to pump the air needed for aeration. Some other reactors arrangements can be found elsewhere 3 . The removal of pollutants from wastewater streams is a public health problem which demands technically efficient and cost competitive strategies. Some of the proposed wastewater treatment processes involve biochemical reaction systems where pollutants are transformed into nontoxic compounds. The set of reactions are carried out in biochemical reactors considering proper design and operation, which are crucial for achieving efficient pollutant removal at minimum capital and operating costs. Amongst the approaches proposed for wastewater treatment, sequencing batch reactors (SBR) have shown great operating flexibility and profitability. Although there have been many studies addressing the design and operation of SBR’s, very few works have addressed the impact of process uncertainty on the performance of such reactors. The importance of taking into account uncertainty during the operation of SBR’s turns out to be critical because otherwise the operation of these reactors can be infeasible and/or unprofitable. Mathematical models are widely used at the process design stage to specify the optimal set of operating conditions that meet process design goals. Typically, process design decisions are performed under the assumption of knowledge of the process model parameters, and that the corresponding model parameter values may not be changing significantly during process operation. In practice, the model parameters may change thus making the design invalid and dynamically infeasible, i.e. the system may not meet the product specifications under model parameter uncertainty. While feedback control systems can be implemented to account for model parameter uncertainty, it is often difficult to state explicitly the amount of uncertainty that a feedback controller can tolerate before

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the system becomes infeasible (unless uncertainty is part of the design of the feedback mechanism). Another approach used to account for model uncertainty involves the deployment of stochastic and robust optimization approaches for online control. 4–7 In the stochastic and robust control schemes, uncertainty is explicitly incorporated in the optimal online control formulation as probability distribution functions or a discretized set of critical scenarios, respectively. 8–10 In chemical process control, emphasis has been given to the control of continuous processing systems. Progress on the control of batch/semibatch processes has been growing over the last decades. There are some reasons related to this behavior 11 , i.e. batch/semibatch operations are low production volume operations. Moreover, the control of batch process feature some unique challenging characteristics such as nonsteady-state operating recipes, nonlinear behavior, model uncertainty and frequent variation in the initial load conditions. Control strategies for dealing with the control of SBR units have been proposed 12–14 , including real time control strategies 15–17 , iterative learning control strategies 11,18 , intelligent control 19 and fuzzy control systems 20 . State estimation and economic model predictive control has been discussed elsewhere 21–23 . In a previous work 24 , we have proposed a deterministic optimization approach for the calculation of open-loop optimal control policies of SBR water treatment systems. In that work, it has been shown that fractional values of the aeration control actions resulted in more attractive solutions rather than using the traditional on/off control policies typically implemented in SBR systems 25 . Accordingly, a non-linear programming formulation suffices for proper optimal control purposes, i.e. a more challenging and intensive mixed-integer non-linear programming approach is not needed. In the present study, we extend that previous work to explicitly account for the impact of model uncertainty on the performance of optimal control policies. Hence, robust and stochastic non-linear model predictive control schemes are presented here to deal with model parameter uncertainty. A sensitivity analysis was carried out to gain insight on the

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model parameters featuring the strongest influence on the system response variables. For the stochastic optimization formulation, the uncertain parameters were assumed to follow a user-defined probability distribution function and power series expansions where used to propagate the uncertainty effects into the controlled variables. In the robust scheme, the uncertain parameters were restricted to lie within a bounded region and a multi-scenario optimization approach was then used to discretize the uncertain parameters’ region. Moreover, in the present work we decided to stress water quality control through the direct control of the concentration of the organic matter and nitrogen compounds embedded in two objective functions. However, the control advantages from 24 are also maintained in the present work, particularly regarding fractional control actions , which reduces the complexity in the optimal control formulations. Finally, note that in the present work we have implicitly assumed reliable and noise-free measurements which, in some cases, can be difficult to justify. Moreover, the aim of this work is to compare the performance of robust and stochastic NMPC formulations for the SBR process. Thus, a deterministic MPC that does not consider uncertainty is not considered here since it is expected to perform worse than the NMPC formulations considered in this work

Figure 1: Operation stages in a sequencing batch reactor.

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2 Problem definition The problem under consideration can be formulated as follows: Given, • A dynamic model that describes the semibatch transient behavior of a wastewater treatment system deploying microorganisms, • A set of nominal parameter values, • A set of uncertain process parameters with either a probability-based (stochastic scheme) or bound-based (robust scheme) description, and • Target values of those variables influencing water quality. Then, the problem consists in computing closed-loop optimal control policies of the aeration profiles under model uncertainty such that the target values determining water quality are reached in the shortest possible operation time. Robust and stochastic nonlinear model predictive control approaches will be considered for the on-line control of the SBR plant.

3 SBR model One of the most common mathematical models used for wastewater treatment is the actived sludge model 1 (ASM1) 26,27 ; hence, an extension of the model presented in 28,29 is used in the present work. The activated sludge model 3 (ASM3) takes into account the energy storage to describe the biodegradable substrate and oxygen consumption. As shown in (1-8), the model used in this work consists of 8 differential equations that describe the transient behavior of this process during the reaction step, as shown in Figure 1. The present model assumes that organic matter and nitrogen are the main pollutants that must be removed from domestic wastewater streams. However, SBR systems are not 7



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constrained to consider only domestic wastewater streams, they can also be used for the treatment of textile wastewater streams 30 .

dSs dt dXH dt dXN s dt dXN b dt dSO dt dSN H4 dt

raae (raN O3 + raN O2 ) = − − (1 + Sts ) YHaer YHanox

(1)

= raae + raN O3 + raN O2

(2)

= raaN s

(3)

(4) (1 − YHaer ) 3.43 1.14 00 u(t) · KLa (S0 − S0 ) − raee − − 1 raaN s − − 1 raaN b (5) YHaer YA1 YA2 iN SS 1 iN SS − − + iN B raae − − + iN B raaN s − iN B raaN b − − + iN B raN O3 YHaer YA1 YHanox iN SS + iN B raN O2 (6) − − YHanox raaN s raaN b (1 − YHanox ) − + (raN O3 − raN O2 ) (7) YA1 YA2 1.14YHanox raaN b (1 − YHanox ) − raN O3 (8) YA3 1.14YHanox

= raaN b

= =

dSN O2 = dt dSN O3 = dt

The terms for the reaction rates are as follows:

raae = raN O3 = raN O2 = raaN s = raaN b =

Ss SO SN H4 µH XH Ss + Ks SO + KO1 SN H4 + KN H Ss SN O3 KO21 SN H4 µH1 XH Ss + Ks SN O3 + KN O3 KO21 + SO SN H4 + KN H Ss SN O2 KO22 SN H4 µH2 XH Ss + Ks SN O2 + KN O2 KO22 + SO SN H4 + KN H SO SN H4 µA1 XN s S O + KO SN H4 + KN H SN O2 SO SN H4 µA2 XN b SN O2 + KN O21 SO + KO SN H4 + KN H

8


(9) (10) (11) (12) (13)

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The description of the states and the initial values are shown in Table 1 whereas Table 2 lists the nominal values of the model parameters. We should stress that the term u(t) in Equation 5 represents air valve opening which happens to be the manipulated variable. Details about reaction kinetic modeling and meaning of the model parameters can be found elsewhere 26 . Table 1: Initial values, lower and upper bounds of states and control action. (1) (2) (3) (4) (5) (6) (7) (8)

State SS XH XN s XN b SO SN H4 SN O2 SN O3 u

Description Initial Value Units Carbonaceous substrate 1000 gCOD/m3 Heterotrophic bacteria 500 Dimensionless Ammonia oxiders 100 Dimensionless Nitrite oxiders 100 Dimensionless Dissolved oxygen 0 gO2 /m3 Ammonia 50 gN/m3 Nitrite 2 gN/m3 Nitrate 5 gN/m3 Fraction of air valve opening 0

Lower Upper -0.001 1000 -0.001 600 -0.001 120 -0.001 110 -0.1 5 -0.001 100 -0.001 10 -0.001 10 0 1

4 Parametric sensitivity analysis Mathematical models are subject to plant-model mismatch. In order to identify the parameters that may critically affect the operation of this process, a local sensitivity analysis was performed. Typically, the parametric sensitivity analysis is conducted as a local linear analysis procedure meaning that their results are only strictly valid around the linearization region. Using this approach, the parametric sensitivity analysis allows identification of those parameters featuring the strongest influence on system behavior. To compute the model sensitivity coefficients we proceed as follows. Assuming that the dynamic mathematical model of the SBR plant is available:

dx = f (x, t, p) dt 9


(14)


Table 2: Nominal parameter values Parameter KLa iN B YHaer YA1 YA2 YA3 iN SS YHanox Sts Ks µH µH1 µH2 µA1 µA2 KO1 KN H KO KN O2 KN O3 KO21 KO22 KN O21 00 S0

Value 1000 0.086 0.1302 0.1327 0.0985 0.0331 0.01 0.0632 1.6 1 0.6021 0.0511 0.0362 1.4 0.3 0.2 0.1 0.8 0.25 0.5 0.2 0.2 0.5 7

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Units day−1 gN/gCOD gCOD/gN gCOD/gN gCOD/gN day−1 mgCOD/L day−1 day−1 day−1 day−1 day−1 mgO2 /L mg/L mg/L mgN/L mgN/L mgO2 /L mgO2 /L mg/L mgO2 /L


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where x stands for the state vector and p is the vector of system parameters. The sensitivity equations are as follows 31 : dSp = dt where Sp = [Sijp ] and Sijp =

∂xi ∂pj

∂f ∂x

p

S +

∂f ∂p

(15)

represent the sensitivity coefficients. Typically, the mag-

nitudes of the sensitivity coefficients can differ by several orders of magnitude ; thus, a scaling (normalization) procedure is implemented here to compare the magnitudes of Sijp on a similar basis. The scaling procedure considered in this work is as follows: Sijp

=

∂xi ∂pj

pj xi

(16)

5 Optimal control strategies It has been widely demonstrated that model uncertainty affects process operation and may lead to a loss in performance and process economics. In this section, we present the design of a model predictive control (MPC) strategy that specifies suitable control policies while considering the influence of model uncertainty on process performance in closedloop. Although feedback control systems have embedded robustness properties 32 , they do not guarantee feasibility in the presence of plant-model mismatch since uncertainty is not explicitly considered in the control algorithm. In this work, only time-invariant model uncertainty has been considered. That is, the true value of the uncertain parameters is not known a priori; however, its value is assumed to remain constant during operation and lie between certain (user-defined) upper and lower limits or follow a (user-defined) probability distribution function. Therefore, two approaches are considered to address the presence of either bound-based or probabilistic-based uncertainty, i.e. robust and stochastic online control optimization. In the stochastic optimization framework 4,33 , model uncertainty is represented as proba11



bilistic distribution functions, whereas in the robust optimization approach 5,34 , model uncertainty is represented as a member of an uncertain set. The decision on which approach to deploy depends on the available information, and the process design goals considered for the SBR plant. If sufficient offline measurements are available, probability distribution functions can then be established for the uncertain parameters, and a stochastic control scheme can be devised for the SBR plant. On the other hand, robust optimization is preferred when only very few or no measurements are available, or the value of a particular model parameter is unknown, but it can be described within a given set. In this work, robust and stochastic optimization approaches have been considered for handling uncertainty in the SBR plant model parameters. There are several ways of representing uncertainty in robust optimization 7,35 . Typically, the uncertain parameters are assumed to belong to some of the following uncertain sets: (1) Finite, (2) Interval-based, (3) Polytopic, (4) Norm-based, (5) Ellipsoidal and (6) Constrain-wise uncertainty. The selection of the uncertainty description depends upon the way uncertainty emerges in the problem under consideration. The first approach (Finite uncertainty) is one of the fundamental approaches to deal with uncertainty in robust optimization, and it assumes that the true value of the uncertain parameter is located within an uncertain set whose realizations are given in terms of lower and upper bounds. This is the representation that has been used in this work to model uncertainty given its widespread use for robust control formulations. An alternative option that can reduce the conservatism obtained from robust control formulations is to treat the uncertain variables as random variables. That is, in the robust control scheme, the uncertain parameters are discretized to a specific set of scenarios, each with an equal (uniform) probability of occurrence. In the stochastic approach, the uncertain parameters are treated as time-invariant random variables that are assumed to follow a probability distribution function, which is used to weight the realizations in the uncertain parameters. Thus, stochastic descriptions may reduce the conservatism in the

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solution at the expense of allowing violations to process constraints. Nevertheless, the user can assign different levels of satisfaction to each constraint based on their level of importance (i.e. risk). Both the robust and stochastic approaches presented here in the context of MPC are discussed next.

Robust Nonlinear Model Predictive Control Once process uncertainty has been specified, the following step in the robust optimization approach consists in formulating the robust optimization counterpart which consists in the approximation of the original uncertain optimization problem by a numerically tractable optimization formulation. There are several approaches that have been proposed to deal with this approximation for linear or convex optimization problems for several types of uncertain sets 7 . However, for nonlinear programming problems such approximation can be difficult to establish or computationally intractable. In these cases, the approach based on the deployment of scenarios 36–38 is a feasible way to take into account the effect of model uncertainty on optimization. As shown in Figure 2, this approach consists in discretizing the uncertainty interval, i.e. [X U − X L ], into a number of scenarios (the total number of scenarios is denoted as Ns ); within each scenario, the value of the uncertain parameter remains constant. Hence, instead of solving a single optimization problem, as each process constraint is explicitly defined in terms of the scenarios considered for the uncertain parameters, a set of Ns optimization problems must be solved. Hence, the original uncertain optimization problem is transformed into a larger deterministic optimization problem (with the size depending on the number of scenarios). X

L

X 1

2

X

U

Ns

3

Figure 2: Multi-scenario approach to represent model uncertainty in an optimization framework. While easier to implement, there are at least two limitations associated with the robust 13



optimization approach. The first is related to the fact that conservative solutions can be obtained from this approach since the optimized values in the decision variables have to meet system constraints for every realization considered in the uncertain parameter set. This is normally achieved at the expense of losing performance. Also, the robust multiscenario approach tends to produce larger optimization problems as the number of scenarios increases, which may become computationally intractable. One way to deal with this problem consists in solving the resulting deterministic large scale problem using optimization decomposition methods 39,40 . In addition, the traditional or conventional robust optimization approach only handles “here and now" decisions meaning that all decision variables are calculated before uncertainty realization is known. However, improved optimal solutions can be obtained from the calculation of “wait and see" decision variables 41 . Previous studies 42,43 have extended the robust optimization approach for dealing with two-stage uncertain problems in the way is performed in stochastic optimization 4 . A recent review on the effect of uncertainty on optimization can be found elsewhere 44 . Moreover, connections between robust optimization and flexibility problems 45 in Process System Engineering have been recently reported 46 . Model predictive control is one of the most powerful strategies to perform control for chemical systems due to its ability to handle explicit constraints and nonlinearities embedded in mathematical models as well as dead time and multivariable systems 47–50 . This control strategy is known as robust non-linear model predictive control. Figure 3 shows the typical behavior of a closed loop model predictive control system. At every k sampling time, m control actions are computed in advance deploying a prediction horizon composed of m sampling times; the corresponding control actions are denoted by uk+m . Although several control actions are computed, only the first control action is implemented ignoring the remaining control actions. When new measurements are available, the nonlinear MPC (NMPC) algorithm is updated and a new calculation of the control action is performed.

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Figure 3: MPC Approach

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The multi-scenario robust NMPC optimization formulation considered in this work is as follows:

Minimize Ωxi ,u =

Ns X

(17)

ωi Ji (xi , u, θi )

i

Subject to : dxi = fi (xi , u, θi ), i = 1, ..., Ns dt

(18)

hi (xi , u, θi ) ≤ 0,

i = 1, ..., Ns

(19)

xL ≤ x i ≤ xU ,

i = 1, ..., Ns

(20)

uL ≤ u ≤ uU

(21)

where J is the objective function, x ∈

Optimal Water Quality Control of Sequencing Batch ... - ACS Publications

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