Stochastic Back-Off Approach for Integration of Design and Control

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Stochastic Back-Off Approach for Integration of Design and Control Under Uncertainty Mina Rafiei, and Luis Alberto Ricardez-Sandoval Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03935 • Publication Date (Web): 28 Feb 2018 Downloaded from http://pubs.acs.org on March 2, 2018

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Industrial & Engineering Chemistry Research

Stochastic Back-Off Approach for Integration of Design and Control Under Uncertainty Mina Rafiei, Luis A. Ricardez-Sandoval Department of Chemical Engineering, University of Waterloo, N2L 3G1, Waterloo, Canada

Abstract The aim of this study is to present a stochastic back-off methodology for integration of design and control using probabilistic-based descriptions in the uncertainty. A challenging task in simultaneous design and control is the specification of process designs that can accommodate stochastic descriptions of uncertain parameters and disturbances. The key idea is to represent the confidence interval of process constraints using power series expansion (PSE)-based functions. The proposed back-off approach has the flexibility to assign different confidence levels to each constraint depending on their relevance or significance. Monte Carlo (MC) sampling is employed to design the PSE-based functions. A waste water treatment plant was used to test the proposed approach. The results have shown that this method has the potential to determine an optimal process design that can maintain the dynamic operability of the system in the presence of stochastic realizations in the uncertainties and disturbances. Keywords: integration of design and control, stochastic back-off, confidence interval, uncertainty



Corresponding author information: Phone: 1-519-888-4567 ext. 38667. Fax: 1-519-888-4347. E-mail: [email protected]

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1 Introduction Chemical processes are typically designed using a sequential approach, i.e. at the very first stage, the design of the system is established based on steady-state optimization calculations. Once it has been specified, the design is fixed and the controllability of the system is considered by simulating the process under different scenarios. This sequential design approach is often found to be inadequate since process controllability and process design are treated independently. It is therefore essential to take into account the trade-off between steady-state economics and the process dynamic performance due to the inherent conflict objectives between these two aspects. Integration of design and control is an attractive alternative that can take this trade-off into consideration. The key idea is to perform the optimal design while considering the transient behaviour of the system. There are important benefits but also significant challenges in promoting a sequential approach to the integrated approach1. This has motivated the development of several studies proposed in the literature to address integration of design and control. Table 1 provides an indicative list of major recent contributions in this area.

Mansouri et al.7,8

A decomposition-based solution approach to integrated design and control through a systematic hierarchical approach for reactive distillation processes A sequence of combined configurations of design and control were solved using a bounding scheme to successively reduce the size of the search region

Mohideen et al.11

A dynamic optimization framework to estimate the disturbance profiles that produce the worst-case scenario

Dynamic optimization approach

Controllability index

Table 1: Integration of design and control - Indicative list Topic Authors Contributions Lenhoff and Morari2 Nguyen et al.3 Input-output controllability indexes were included as objectives or constraints within Luyben and Floudas4 the optimization formulation Alhammadi and Romagnoli5 Papadopoulos et al.6 Simultaneous assessment of controllability with respect to the solvent and process economic and controllability properties

Kookos and Perkins9,10

Nie et al.14

Discrete time formulation optimization approach for the integration of production scheduling and dynamic process operation using Generalized Benders Decomposition (GBD) algorithm to solve the resulting large nonconvex mixed-integer nonlinear program

Robust approach

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

Ricardez-Sandoval et al.15–17

Identification of worst-case scenarios using structured singular value analysis

Koller and Sandoval18

A direct integration of design, control, and scheduling under disturbance and uncertainty while explicitly accounting for scheduling decisions in the analysis by the use of variable-sized finite elements in the model discretization

Diangelakis Pistikopoulos12,13

and

Ricardez-

A single prototype software system (PAROC framework) which allows for the representation, modeling, and solution of integrated design, scheduling and control problems via multi-parametric programming


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and

A model-based control strategy, i.e. model predictive control (MPC), in which the feasibility and stability analyses are formulated as convex problems using robust control tools

Gutiérrez-Limón et al.20

A reactive heuristic strategy to cope with the simultaneous optimization of short-term planning, scheduling and control problems under uncertainty

Back-off approach

Sanchez-Sanchez Ricardez-Sandoval19

Bahri et al.21,22 Figueroa et al.23

A joint optimization-flexibility aimed to search for the optimality of the system considering flexibility feasibility of the design in a back-off approach

Mehta and RicardezSandoval24 Rafiei-Shishavan et al.25

PSE-based back off approach using a series of simple optimization problems that moved the optimal steady-state design to an optimal and dynamically feasible condition

Narraway et al.26 Narraway and Perkins27,28

The impact of disturbances on plant economic performance was assessed using frequency response analysis of a linearized plant dynamic

Probability-based methods

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


Bahakim and Sandoval29

Simultaneous design and MPC-based control was performed using a stochastic-based worst-case variability index

Ricardez-

Ricardez-Sandoval30

A worst-case variability distribution analysis for simultaneous design and control under random realizations in the disturbances

Chan et al.31

A Gaussian Process (GP) model has been trained in an iterative manner to represent the uncertainty as the input to the simultaneous design and control framework

Thorough reviews and classifications in this field can be found elsewhere32–35. Steady-state optimization under uncertainty identifies the optimal design at which a system operates feasibly in the presence of uncertainty. However, the design may become dynamically infeasible if uncertainty and disturbance effects are not explicitly considered. To take these effects into account, and be able to specify a dynamically operable system, the process needs to moved away (backed-off) from the active boundaries of the steady-state feasibility region. Different backoff approaches exist in the literature regarding the use of steady-state optimization to establish the economic incentive to improve the selection of decision variables while considering the transient behaviour of the process21–23,27,28,36,37. A thorough description of these approaches is discussed in our previous work25. Kookos and Perkins10 recently proposed the implementation of a back-off approach as a systematic tool for simultaneous process and control structure design. In that work, the aim is to systematically develop efficient regulatory control structures to minimize the impact of disturbances using the full nonlinear process model. Implementation of a new back-off method has been recently developed by our group24,25. In those works, the back-off amount was identified from a series of simple optimization problems, which aimed to move the system away from the optimal steady-state design to a new dynamically feasible operating region. Back-off terms are aimed to ensure dynamic feasibility in the presence of uncertainty, especially while considering safety and environmental constraints. However, they can be difficult to formulate and implement, particularly for large-scale applications. The main challenge in back-off methods is to compute in 3 ACS Paragon Plus Environment

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a systematic fashion the amount of back-off required to accommodate the transient operation of the process. Different techniques have been developed to calculate the amount of back-off in the literature. Visser et al.38 linearized both the constraints and the state-space nonlinear model of the original optimization problem to calculate the magnitude of back-off in process constraints to compensate for the uncertainty. Diehl et al.39 employed a linearization technique of the uncertainty set based on constraint derivatives, which lead to the computation of the amount of penalty term (back-off) from a robust nonlinear optimization formulation. Srinivasan et al.40 updated the initial back-off term in an iterative routine using the probability density function of system’s states computed at the optimal solution. Alternately, a robust optimization formulation applied through the incorporation of back-off terms in the constraints was presented by Shi et al.41. In the present back-off method, the back-off in the optimization variables is identified by solving a series of power series expansion(PSE)-based optimization problems24,25,42. In modelling analysis, uncertainty can emerge due to model structure error or from the lack of knowledge in the actual model parameter values, i.e. parameter uncertainty. Both sources of uncertainties occur often in chemical engineering applications. Handling uncertainty becomes critical when violation of constraints involves environmental or safety restrictions. A current challenge in simultaneous design and control is the specification of process designs that can accommodate stochastic descriptions of uncertain parameters and disturbances. In this regard, Tsay et al.43 formulated a dynamic optimization where the uncertain parameters are treated as dynamic disturbance variables with continuous pseudo-time trajectories. Wang and Baldea44 developed an identification-based optimization method for the optimal design of dynamic systems under uncertainty. Pseudo-random multilevel signals were employed in that work to capture uncertainty in dynamical systems. In a follow-up work, that identification method was employed to develop a formulation that addresses the optimal design of dynamic systems under uncertainty by establishing a link between a switching control law and the bandwidth of the closed-loop system, which allows the specification of a user-defined level of process conservativeness in the design45. Koller et. al.46 develop a new framework to address the optimal design, control and scheduling of multi-product systems under uncertainty. In that work, back-off terms were estimated to compensate for the effect of stochastic uncertainty and disturbances in the analysis. Few studies have performed a worst-case variability distribution analysis for simultaneous design and control using random realizations in the disturbances29,30. In those approaches, full compliance 4 ACS Paragon Plus Environment

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of process constraints may result in overly conservative designs since the worst-case scenario may not likely occur during operation. In most engineering applications, a high probability of satisfaction is often required for some constraints, e.g. safety restrictions, whereas others may not be critical, e.g. liquid level control in storage tanks. Therefore, performing an integrated stochasticbased design and control approach may result in more economically attractive designs compared with the worst-case scenario approach. This study aims to present a new stochastic back-off technique for integration of design and control that can accommodate probabilistic (stochastic)-based uncertainties and disturbances. The basic framework of this technique has been recently presented by the authors42. In that work, a variancebased approach was employed to assess process variability due to uncertainty and disturbances. That basic version of the methodology has been extended and improved in the present work. The key idea in the present approach is to represent variability in the process constraints and objective function by means of statistical terms, i.e. confidence intervals. That is, PSE-based functions of the constraints specified at a user-defined coverage probability are explicitly identified in terms of the optimization variables in the presence of stochastic realizations in the uncertainty and disturbances. Using confidence intervals enforces the system to behave in such a way that the constraints can most surely accommodate realizations in the uncertain parameters and disturbances up to a certain coverage probability specified by user. This approach provides the user with an additional degree of freedom to rank the significance of the constraints through the use of coverage probabilities. As shown in this study, the present methodology has the potential to specify more economically attractive designs compared to the worst-case variability methodologies29,30 and the variance-based approach previously presented by the authors42. A waste water treatment plant is used as the case study to investigate the potential benefits and capabilities of the proposed stochastic back-off methodology. The remaining part of this study proceeds as follows: The next section presents the back-off methodology. Section 3 presents the water treatment case study whereas section 4 presents the results obtained from the implementation of the present methodology to that case study under difference scenarios. Concluding remarks and future work are provided at the end.



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2 Methodology This section describes the stochastic back-off methodology introduced in this study. This approach aims to search for the optimal design and control parameters by solving a set of optimization problems using mathematical expressions obtained from PSE. Process variability can be quantified using a low-order series expansion that replaces the actual nonlinear model of the system. Hence, a complex function (𝑓(𝒙)) can be represented around a nominal operating point (𝒙0 ) using a PSE approximation as follows: ∞

𝑓(𝒙) = 𝑓(𝒙0 ) + ∑ 𝑗=1

1 𝑗 ∇ 𝑓(𝒙)|𝒙0 (𝒙 − 𝒙0 ) 𝑗 𝑗!

(1)

∇𝑗 𝑓(𝒙) is the jth order gradient of 𝑓(𝒙). The number of terms used in the PSE (i.e. the order of the PSE) depends on the process nonlinearity and the required accuracy; however, it is often found that first and second order PSEs are sufficient for most engineering applications. In the current work, PSE functions of statistical terms of the expected cost and process constraints are developed around the nominal conditions in the design variables and controller tuning parameters. Different types of statistical terms can be used to represent the expected cost and constraints. The use of multiples of the standard deviation in the process constraint functions due to uncertainty and disturbances is a well-established approach that was used in our previous work42. The key assumption made in that work was that the probability distribution of the constraint functions follows Normal (Gaussian) distribution. Often, systems are nonlinear and therefore follow a nonGaussian behaviour when subjected to uncertainty and disturbances. Therefore, the present work extends and improves our previous stochastic back-off formulation42 through explicit evaluation of the process constraints and cost function at specific (user-defined) confidence levels using PSE functions. For that reason, the confidence interval is employed in the present methodology as the statistical term since it takes into account the distribution of the process constraints and cost function whereas the variance-based approach is subject to error due to the system’s nonlinearity. The confidence interval in the constraints can be constructed from the evaluation of the system outputs in such a way that the interval contains the desired coverage probability value. In this work, the coverage probability is defined as the level at which the system is expected to satisfy the process constraints. The present methodology assumes that a dynamic process model is available for simulations. 6 ACS Paragon Plus Environment

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Figure 1: Algorithm for the stochastic back-off methodology

The present stochastic back-off methodology aims to find the optimal process design and control scheme under stochastic-based uncertainty and disturbances. A schematic of the iterative stochastic back-off approach algorithm proposed in this work is presented in Figure 1. Each of the steps in the algorithm is described next. Step 1 (Algorithm Initialization): Specify the order of the PSE functions to be used in the analysis (i.e. the number of terms in the PSE function). Define upper and lower bounds for each optimization variable ( 𝜼). Specify the maximum number of iterations (𝑁𝑖𝑡𝑒𝑟 ). Also, define a tolerance criterion (𝜖) that will be used to terminate the algorithm. Define the search space in the decision variables (𝛿); details about this parameter are presented in Step 5. Set the iteration index to 𝑖 = 1. 7 ACS Paragon Plus Environment


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In the current work, the uncertain parameters and time-varying disturbances are considered as random variables correlated by Probability Distribution Functions 𝑃𝐷𝐹(𝜶𝜻 ) and 𝑃𝐷𝐹(𝜶𝑑 ), respectively; where 𝜶𝜻 and 𝜶𝑑 represent the statistical parameters of the probability distribution (e.g. mean and variance for a normal distribution or upper and lower bounds for a uniform distribution). Historical data can be used to characterize the variability in the uncertain parameters. The mathematical description of the uncertain parameters and disturbances is as follows: 𝜻 = {𝜻 |𝜻~𝑃𝐷𝐹(𝜶𝜻 )}

(2)

𝒅(𝑡) = {𝒅|𝒅~𝑃𝐷𝐹(𝜶𝒅 )}; 0 < 𝑡 < 𝑡𝑓

Step 2 (Optimal steady-state design without uncertainty and disturbances): The optimal steady state process design without uncertainty is considered first. This optimization is conducted under the assumption that the disturbances and uncertain parameters are set to their nominal values, i.e. 𝒅𝑛𝑜𝑚 and 𝜻𝑛𝑜𝑚 . The result from this optimization problem (𝜼0 ) is used as the starting (initial) point to search for the optimal dynamically operable design that can accommodate stochastic realizations in the uncertain parameters and disturbances. The optimal steady-state design problem is as follows: 𝑚𝑖𝑛 𝜂0

Θ

Subject to: 𝐹𝑚𝑜𝑑𝑒𝑙 (𝜼𝟎 , 𝒙, 𝒚, 𝒖, 𝒅𝑛𝑜𝑚 , 𝜻𝑛𝑜𝑚 ) = 0

(3)

ℎ(𝜼𝟎 , 𝒙, 𝒚, 𝒖, 𝒅𝑛𝑜𝑚 , 𝜻𝑛𝑜𝑚 ) ≤ 0 𝜼𝑙𝑏 ≤ 𝜼0 ≤ 𝜼𝑢𝑝

Step 3 (Develop PSE-based functions: statistical terms): As shown in (2), this algorithm assumes that the disturbances are stochastic time-varying random variables. Monte Carlo (MC) sampling was employed for the propagation of the uncertain parameters and disturbances into the system and the corresponding identification of the PSE functions for the constraints (ℎ) and cost function (𝛩). In the current work, the key idea is to develop PSE-based functions that describe the process variability in terms of the confidence interval for process constraints (ℂΙ𝜌 (ℎ𝑃𝑆𝐸 )) and the expected value for the cost function (𝔼(𝛩𝑃𝑆𝐸 )). The confidence interval (ℂΙ) is estimated from data obtained from simulations. The percentile bootstrap method has been used to compute the confidence interval from the simulated data47. In 8 ACS Paragon Plus Environment

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order to ensure that the estimated confidence intervals are within a reasonable level of accuracy, a systematic procedure has been considered to ensure that a sufficiently large number of MC samples are being used in the computation of these estimates, (see steps a) through e) below). Note that the confidence interval calculation is independent of the distribution of the process constraints since they are estimated from data collected from the simulations. Based on the above, MC realizations are generated for the disturbance (𝒅) and uncertain parameters (𝜻) and are used as inputs for simulation of the dynamic model around a nominal condition in the optimization variables (𝜼𝑛𝑜𝑚 ). The number of MC samples required to describe the probability distribution in the observables is determined based on the desired accuracy in the estimation and the available computational budget to perform the uncertainty propagation. In this regard, selecting an appropriate number of MC realizations is critical to identify the key statistical properties of the cost function and process constraints. An iterative approach is therefore employed in this work to determine the required number of MC samples. This procedure is as follows: a) Specify a minimum number of realizations (N) and a tolerance criterion to check for convergence on the estimation of the statistical terms in the cost function and constraints due to uncertainty and disturbances (𝜖𝑀𝐶 ). Also, initialize an index (𝑘), which keeps track of the batches of the MC samples, i.e. set 𝑘 = 1. b) Using the descriptions provided in (2), generate a batch of MC samples that will contain N realizations of the uncertain parameters and disturbances. c) For each realization included in the batch, simulate the system using fixed (nominal) values in the optimization variables (𝜼𝑛𝑜𝑚 ). The N simulation results obtained from the current MC samples are collected in a set 𝑅. These results are appended to simulation results from past MC sample realizations, i.e. 𝑅𝑘 = {𝑅, 𝑅𝑘−1 }

(4)

where 𝑅𝑘−1 represent the simulation results collected from k = 1 to k − 1. d) Calculate the statistical terms of cost function and process constraints for the set 𝑅𝑘 . That 𝜌

is, compute confidence intervals ℂ𝛪𝑅𝑘 (ℎ) at the desired coverage probability (𝜌) for the process constraints (ℎ).



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e) Compare statistical terms of the current set (𝑅𝑘 ) with the previous set (𝑅𝑘−1 ) using the following formulation: 𝜌

𝜌

𝜌

𝑇𝑜𝑙𝑀𝐶 = (ℂ𝛪𝑹𝑘 (ℎ) − ℂ𝛪𝑹𝑘−1 (ℎ))⁄ℂ𝛪𝑹𝑘 (ℎ)

(5)

If TolMC falls below a user-defined threshold (𝜖𝑀𝐶 ), then STOP, convergence in the statistical terms have been obtained. Otherwise, update 𝑘 = 𝑘 + 1 and go back to step b), i.e. generate another batch of N MC samples and add it to the previous set of samples. This procedure is repeated until the tolerance (𝜖𝑀𝐶 ) is satisfied, which indicates that the statistical characteristics of the process constraints have converged. Note that k represents the index that keeps track of the number of N MC samples required by this method to comply with the tolerance criterion (see step e) above). The addition of multiple sources of uncertainty (each with their own probability distribution function) will directly impact the computational costs of the present approach. To achieve a satisfactory accuracy in the statistical terms, e.g. confidence interval in the cost function and process constraints, a larger set of combinations in the different realizations of the multiple uncertain parameters are needed. Hence, the number of required MC samples that need to be simulated will be increased significantly thus requiring additional computational resources. Note that the procedure described above is performed around a nominal condition in the optimization variables, i.e. 𝜼𝑛𝑜𝑚 . Hence, the procedure needs to be repeated for forward and backward points assigned to each optimization variable while keeping the rest of the optimization variables constant and equal to their nominal values as shown in Figure 2. Calculation of forward and backward points is required in order to obtain sensitivity terms in the PSE for the constraints and cost function at the corresponding confidence interval for the specified coverage probability. Finite differences have been used in this work to calculate the PSE sensitivities using data collected from simulations. For example, the first-order sensitivity term of the confidence interval of the sth constraint at a coverage probability 𝜌𝑠 in terms of the pth decision variable can be calculated as follows: (1)

𝑍𝑝,𝑠 =

∂ℂΙ 𝜌𝑠 (ℎ𝑠 ) | = (ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜂𝑝+ ))|𝑑(𝑡),𝜁 − ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜂𝑝− ))|𝑑(𝑡),𝜁 )⁄2Δ𝜂𝑝 ∂𝜂𝑝 𝑑(𝑡),𝜻


(6)

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Figure 2: Simulations for the nominal condition, forward and backward points using uncertainties and disturbances realizations as input

where ℂΙ 𝜌𝑠 (ℎ𝑠 ) is the confidence interval of sth constraints with the coverage probability of 𝜌𝑠 ; Δ𝜂𝑝 represents the difference between the forward points (𝜂𝑝+ ) and the backward points (𝜂𝑝− ). Note that tuning the step size offline is required to compute representative PSE sensitivities. Similarly, the second-order sensitivity term can be calculated as follows: (2)

𝑍𝑝,𝑠 =

∂2 ℂΙ 𝜌𝑠 (ℎ𝑠 ) | ∂𝜂𝑝 𝜕𝜂𝑝 𝒅(𝑡),𝜻

(7)

= (ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜂𝑝+ ))|𝒅(𝑡),𝜻 − 2ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜂𝑛𝑜𝑚 ))|𝒅(𝑡),𝜻 + ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜂𝑝− ))|𝒅(𝑡),𝜻 )⁄Δ𝜂𝑝 2

Higher order sensitivity terms can be identified in a similar fashion at the expense of higher computational costs since additional forward and backward points around the nominal point are required. Accordingly, mathematical expressions for the expected value of the cost function (𝛩) and confidence interval of the sth constraint functions (ℎ) using PSE approximations can be formulated as follows:



𝔼(𝛩𝑃𝑆𝐸 (𝜼))|𝒅(𝑡),𝜻 = 𝔼(𝛩(𝜼𝑛𝑜𝑚 ))|𝒅(𝑡),𝜻 + ∑ 𝑗=1

ℂΙ 𝜌𝑠 (ℎ𝑃𝑆𝐸 𝑠 (𝜼))|

𝒅(𝑡),𝜻

1 𝑗 𝛻 𝔼(𝛩(𝜼))|𝒅(𝑡),𝜻,𝜼 (𝜼 − 𝜼𝑛𝑜𝑚 ) 𝑗 𝑛𝑜𝑚 𝑗!

1 = ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜼𝑛𝑜𝑚 ))|𝒅(𝑡),𝜻 + ∑ ∇ 𝑗 ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜼))|𝒅(𝑡),𝜻,𝜼 (𝜼 − 𝜼𝑛𝑜𝑚 ) 𝑗 𝑛𝑜𝑚 𝑗! 𝑗=1

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(8)

(9)

where 𝔼(𝛩𝑃𝑆𝐸 (𝜼)) is the PSE expansion of the expected value of the cost in terms of optimization variables (𝜼). Likewise, ℂΙ 𝜌𝑠 (ℎ𝑃𝑆𝐸 𝑠 (𝜼)) is the PSE expansion of the confidence interval the of the sth constraint with the coverage probability of 𝜌𝑠 ; ∇𝑗 𝔼(𝛩(𝜼)) and ∇𝑗 ℂΙ 𝜌𝑠 (ℎ𝑠 (𝜼)) represent the jth sensitivity terms of the expected value of the cost function (Θ) and confidence interval of the sth constraint (ℎ) with the coverage probability of 𝜌𝑠 , respectively. Note that the gradient terms are assessed at the nominal condition of optimization variables (𝜼𝑛𝑜𝑚 ) due to random realizations in the disturbances (𝒅(𝑡)) and uncertain parameters (𝜻). Also, note that the expected value of the cost function is evaluated in the presence of stochastic realizations in the timevarying disturbances and time-invariant uncertainties. Admittedly, adding more decision variables results in higher computational expenses as it involves another set of sensitivity calculations, i.e. at nominal (𝜼𝑛𝑜𝑚 ) forward (𝜂𝑝+ ) and backward (𝜂𝑝− ) points, which would require MC simulations for each of these points. Step 4 (Optimization of the PSE-based functions): The PSE-based functions constructed in the previous step for the process constraints and the cost function are embedded within a PSE-based optimization problem, i.e. 𝑚𝑖𝑛 𝜼,𝝀

𝑆

𝔼(𝛩𝑃𝑆𝐸 (𝜼)|𝒅(𝑡),𝜻 ) + ∑ 𝑀 𝜆𝑠 𝑠

Subject to: ℂΙ 𝜌𝑠 (ℎ𝑃𝑆𝐸 𝑠 (𝜼))|

𝒅(𝑡),𝜻

(10)

− 𝜆𝑠 ≤ 0, ∀𝑠 = 1, … , 𝑆

𝜼𝑛𝑜𝑚 (1 − 𝛿) ≤ 𝜼 ≤ 𝜼𝑛𝑜𝑚 (1 + 𝛿) 𝜆𝑠 ≥ 0 , ∀𝑠 = 1, … , 𝑆

where δ is a user-defined tuning parameter that specifies the search space region for the optimization variables (𝜼). That is, the search space in the decision variables is limited due to the PSE approximation considered in this back-off method. 𝜆𝑠 is an optimization variable used to avoid infeasibility in the sth constraint function (ℎ𝑃𝑆𝐸 𝑠 ). The weight M is a sufficiently large penalty term use to drive the dummy variables (𝜆𝑠 ) to zero and enable the specification of a dynamically 12 ACS Paragon Plus Environment

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feasible system that minimizes the expected value of the cost function. This method, also referred to as the big-M method, is used here to ensure feasible solutions from problem (10), particularly in the first iterations where the process constraints cannot accommodate dynamic feasibility. The effect of parameters 𝑀 and δ on the performance of the algorithm can be found in our previous work.25 Setting values close the unity for the coverage probability (𝜌𝑠 ) will most surely reduce the possibility of violation of the constraints at the expense of specifying more conservative designs. As mentioned above, the PSE functions are only valid around the vicinity of nominal conditions in the decision variables (𝜼𝑛𝑜𝑚 ); hence, the search space in the optimization variables is restricted by means of δ, which aims to specify a region where the PSE approximations remain valid, as shown in problem (10). Accordingly, the present method requires an iterative approach to identify an optimal design and control scheme that can comply with the process constraints at user-defined coverage probabilities. As shown in Figure 1, the solution to problem (10) is used as basis to search for a new direction in the optimization variables at each iteration step in the algorithm. Step 5 (Convergence Criterion): A floating average convergence technique, i.e. the difference in means, is considered here as a stopping criterion, i.e. 𝑖−𝑁

Θ 𝑇𝑜𝑙𝑓𝑙𝑜𝑎𝑡 =

𝑐 (∑𝑣=𝑖−2𝑁 𝛩 − ∑𝑖𝑟=𝑖−𝑁𝑐+1 𝛩𝑟 ) 𝑐 +1 𝑣 ⁄𝑖  ∑𝑟=𝑖−𝑁𝑐+1 𝛩𝑟

(11)

where 𝛩𝑣 and 𝛩𝑟 represent the cost obtained from the solution of problem (10) at the vth and rth iterations, respectively. Similarly, for the optimization variables: 𝑖−𝑁

𝜼

𝑇𝑜𝑙𝑓𝑙𝑜𝑎𝑡 =

𝑐 (∑𝑣=𝑖−2𝑁 𝜼 − ∑𝑖𝑟=𝑖−𝑁𝑐+1 𝜼𝑟 ) 𝑐 +1 𝑣 ⁄𝑖 ∑𝑟=𝑖−𝑁𝑐+1 𝜼𝑟

(12)

where 𝜼𝑣 and 𝜼𝑟 represent optimization variables obtained from the PSE-based optimization problem (10) at the vth and rth iterations, respectively. A convergence examination period Nc needs to be specified a priori. While 𝑖 ≥ 2𝑁𝑐 , the mean of the PSE-based cost function (𝛩) and optimization variables (𝜼) obtained from iterations (i- 2Nc +1) to (i- Nc) are compared with the mean of the same function obtained from iterations (i- Nc +1) to (i). That is, the difference in means 𝜂

𝛩 are required to be less than a threshold value (𝜖). If |𝑇𝑜𝑙𝑓𝑙𝑜𝑎𝑡 | ≤ 𝜖 or |𝑇𝑜𝑙𝑓𝑙𝑜𝑎𝑡 | ≤ 𝜖 , then STOP,

the method converged to an optimal solution. Otherwise, set 𝑖 = 𝑖 + 1, update the optimization variables (𝜼𝑛𝑜𝑚 ) with the most recent solution obtained from problem (10) and go back to Step 3. 13 ACS Paragon Plus Environment


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Alternatively, the algorithm is also terminated if the maximum number of iterations is reached, i.e. 𝑖 ≥ 𝑁𝑖𝑡𝑒𝑟 . Table 2 lists the tuning parameters that are required by the present methodology and their recommended values. A more in-depth analysis of some of these parameters, e.g. PSE order, search space region (𝛿), big-M penalty term (𝑀) and finite difference step size (Δ𝜼) can be found elsewhere24,25. Table 2: Tuning parameters, Stochastic back-off methodology Tuning parameter Description Convergence examination period, the higher the value the longer to 𝑁𝑐 check for convergence of the back-off algorithm

Recommended values1 10-30

𝑁𝑖𝑡𝑒𝑟

Maximum number of iterations allowed in the stochastic back-off algorithm

≥300

𝛿

Size of the search space region of the optimization variables at each iteration step. PSE functions are expected to be representative of the system’s variability within this region

0.05-0.15

𝜖

Termination tolerance for the stochastic back-off algorithm

≤1x10-3

𝜖𝑀𝐶

Termination tolerance for the convergence of the statistical terms

≤1x10-3

PSE order

Order of the PSE expansion. Problem-specific and directly correlated to the nonlinearity of the problem under consideration

1, 2

Δ𝜼

Step size considered to estimate the sensitivity of the cost function and process constraints with respect to the optimization variables

0.01𝜼𝑛𝑜𝑚 -0.02𝜼𝑛𝑜𝑚

𝑀

Penalty term that forces the system to move within the dynamic feasibility region

103𝛩-105 𝛩

3 Case study: waste water treatment plant An existent waste water treatment plant29,48, located in Manresa, Spain was used as a case study to test the proposed stochastic back-off methodology. A general flowsheet of the plant is shown in Figure 3. The goal of the process is to regulate the level of the substrate concentration in the biodegradable waste stream due to environmental incentives. The process aims to maintain the substrate concentration (𝑠𝑤 ) in the bioreactor and the dissolved oxygen concentration (𝑐𝑤 ) in the settler at desired levels in the presence of process disturbances corresponding to changes in the inlet feed stream (e.g. 𝑞𝑖 ), as shown in Figure 3. The control scheme considered for this case study includes two PI controllers that regulate the substrate concentration and the dissolved oxygen by adjusting the purge flow rate (𝑞𝑝 ) and the turbine speed (𝑓𝑘 ), respectively.

1

Recommended from previous experiences with different engineering applications.


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qi , xi , si q , xw , sw xir , sir

fk

xd

q1 , sw

xb xr

qr

qp Figure 3 Flow sheet: waste water treatment plant

Table 3 provides the model equations regarding the rate of the biomass and organic substrate concentrations (mg⁄L) inside the bioreactor as well as the difference in settling rate of biomass concentration among the different layers in the decanter. Table 3: Model equations of waste water treatment plant Description Equations Rate of change of biomass

𝑑𝑥𝑤 𝑥𝑤 𝑠𝑤 𝑥𝑤 n 𝑞 = µ𝑤 𝑦𝑤 − 𝑘𝑑 − 𝑘𝑐 𝑥𝑤 + (𝑥𝑖𝑟 − 𝑥𝑤 ) 𝑑𝑡 𝑘𝑠 + 𝑠𝑤 𝑠𝑤 𝑉𝑟

Rate of consumption of the organic substrate in the reactor

𝑑𝑠𝑤 𝑥𝑤 𝑠𝑤 𝑥𝑤 n 𝑞 = −µ𝑤 + 𝑓𝑑 𝑘𝑑 + 𝑓𝑑 𝑘𝑐 𝑥𝑤 + (𝑠𝑖𝑟 − 𝑠𝑤 ) 𝑑𝑡 𝑘𝑠 + 𝑠𝑤 𝑠𝑤 𝑉𝑟

Difference in settling rate of biomass concentration among different layers in the decanter

𝑑𝑥𝑏 1 1 = (𝑞 + 𝑞2 − 𝑞𝑝 )(𝑥𝑤 − 𝑥𝑏 ) + (𝑣𝑠𝑑 − 𝑣𝑠𝑏) 𝑑𝑡 𝐴𝑑 𝑙𝑏 𝑖 𝑙𝑏 𝑑𝑥𝑑 1 1 = (𝑞 − 𝑞𝑝 )(𝑥𝑏 − 𝑥𝑑 ) − 𝑣𝑠𝑑 𝑑𝑡 𝐴𝑑 𝑙𝑑 𝑖 𝑙𝑑 𝑑𝑥𝑟 1 1 = 𝑞 (𝑥 − 𝑥𝑟 ) + 𝑣𝑠𝑏 𝑑𝑡 𝐴𝑑 𝑙𝑟 2 𝑏 𝑙𝑟 𝑑𝑐𝑤 𝑠𝑤 𝑥𝑤 𝑞 = 𝑘𝑙𝑎 𝑓𝑘 (𝑐𝑠 − 𝑐𝑤 ) + 𝑘01 µ𝑤 − 𝑐 𝑑𝑡 𝑘𝑠 + 𝑠𝑤 𝑉𝑟 𝑤

Rate of dissolved oxygen Cost function:  𝐶𝐶𝑤 : capital cost  𝑂𝐶𝑤 : operating cost  𝑉𝐶𝑤 : variability cost

𝛩𝑊 = ⏟ 0.16(3500𝑉𝑟 + 2300𝐴𝑑 ) + 870(𝑓 105 (100 − sw ) 𝑘 + 𝑞𝑝 ) + ⏟ ⏟ 𝐶𝐶𝑤

Constraints:  Maximum allowable substrate concentration in the treated water that leaves the decanter  Minimum and maximum allowed ratio between the purge to the recycle flow rates

𝑢𝑝 𝑠𝑤 (𝑡) ≤ 100

R𝑙𝑜 1 : 0.01 ≤ 𝑢𝑝

R1 :

𝑞𝑝 (𝑡) ≤ 0.2 𝑞2 (𝑡)

R𝑙𝑜 2 : 0.8 ≤  Allowed purge age in the decanter

𝑞𝑝 (𝑡) 𝑞2 (𝑡)

𝑉𝑟 𝑥𝑤 (𝑡) + 𝐴𝑑 𝑙𝑟𝑥𝑟 (𝑡) 24 𝑞𝑝 𝑥𝑟 (𝑡)


𝑂𝐶𝑤

𝑉𝐶𝑤


𝑢𝑝

R2 : Optimization variables

𝑉𝑟 𝑥𝑤 (𝑡) + 𝐴𝑑 𝑙𝑟𝑥𝑟 (𝑡) ≤ 15 24 𝑞𝑝 𝑥𝑟 (𝑡)

[𝐴𝑑 , 𝑉𝑟 , 𝑠𝑤𝑠𝑝 , 𝑐𝑤𝑠𝑝 , 𝐾𝑐1 , 𝐾𝑐2 , 𝜏𝑖1 , 𝜏𝑖2 ]

Table 3 also shows the cost function and the process constraints that determine the feasible operating region for this process. As shown in Table 3, a variability cost (𝑉𝐶𝑤 ) is considered to maintain the substrate concentration near a threshold (i.e. 100 mg/L) since any deviation in the substrate level leads to high penalty costs. While a positive deviation may impose a municipality penalty for the excessive deposition of this organic material to the environment, a drop in the substrate concentration below the threshold reduces plant performance and adversely affects the process economics. Note that all the three process constraints indicated in Table 3 must remain feasible during operation (up to a certain confidence level); hence, they are considered as dynamic path constraints. The model parameters and their corresponding nominal values are shown in Table 4. Table 4: Process model parameters: Waste water treatment plant Symbols Value 0.1824 (hr -1) µ𝑤 0.5948 𝑦𝑤 300 (hr -1) 𝑘𝑠 5x10-5 (hr -1) 𝑘𝑑 1.33x10-4 (hr -1) 𝑘𝑐 0.7 (hr -1) 𝑘𝑙𝑎 1.00x10-4 (hr -1) 𝑘01 8.0 (hr -1) 𝑐𝑠 0.2 𝑓𝑑 n 2

4 Results The stochastic back-off methodology presented in section 2 has been implemented on the waste water treatment plant. Second order PSE expansions were used in the current analysis. In the present study, the accuracy of the PSE sensitivities was initially verified offline through simulations; similarly, they have been validated at the optimal condition for each scenario. This checking procedure can be embedded in the methodology to determine if, at each step in the iteration of the present back-off technique, the PSE expansions are constructed in a way that complies with the desired user-defined accuracy. This implementation requires higher computational efforts and is beyond the scope of this work. 𝛿 was set to 0.1 whereas the convergence criteria for the back-off technique were set to 𝜖 =1x10-3. The computational 16 ACS Paragon Plus Environment

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experiments performed in this research were conducted using MATLAB2017a® on a computer running Microsoft Windows Server 2016 standard. The computer was equipped with 16 GB RAM and Intel® Xeon® CPU E5-2620 v4 @ 2.10GHz. To explore the potential benefits and limitations of the present method, the optimal design of the waste water treatment plant was performed under different scenarios. Section 4.1 presents a comparison on the statistical terms that can be used in the present methodology to assess process variability under uncertainty, i.e. confidence intervals and expected value and variance. Section 4.2 explores the performance of the methodology using different sources of uncertainty, and with different probability distribution functions. The effect of using different weights on the coverage probability of the constraints is discussed in section 4.3 whereas section 4.4 evaluates the computational cost of the present approach and discusses the scalability of the methodology.

4.1 Effect of statistical terms As previously stated, different statistical terms can be used to represent the probability of the cost function and process constraints due to probabilistic-based descriptions for the disturbances and uncertain parameters in the PSE-based optimization problem. Two basic approaches are currently being adopted in the stochastic back-off methodology. In our previous study, statistical terms of the cost function and process constraints in the PSE-based optimization problem were estimated using expected values and variances42. However, the nonlinearity of the system that produces nonGaussian distributions in the process constraints may result in poor estimations of expected value and variances. As described in the methodology section, the present study improves that approach by means of evaluating the confidence intervals of process constraints at certain coverage probabilities; thus, the issue of inaccurate estimation of probability characteristics due to the Gaussian distribution assumption considered has been resolved. In our previous approach, i.e. using expected value and variance as statistical terms to measure process variability, weights were assigned to the variance of the constraints; this was done to account for process variability as a function of the spread observed in the probability distribution of the process constraints. Accordingly, problem (10) can be reformulated as follows42: 𝑚𝑖𝑛 𝜼,𝝀

𝑆

𝔼(𝛩𝑃𝑆𝐸 (𝜼)|𝒅(𝑡),𝜻 ) + ∑ 𝑀 𝜆𝑠

(13)

𝑠

Subject to:



𝔼(ℎ𝑃𝑆𝐸 𝑠 (𝜼)|𝒅(𝑡),𝜻 ) + 𝛾𝑠 𝜎(ℎ𝑃𝑆𝐸 𝑠 (𝜼)|𝒅(𝑡),𝜻 ) − 𝜆𝑠 ≤ 0 , ∀𝑠 = 1, … , 𝑆 𝜼𝑛𝑜𝑚 (1 − 𝛿) ≤ 𝜼 ≤ 𝜼𝑛𝑜𝑚 (1 + 𝛿) 𝜆𝑠 ≥ 0 , ∀𝑠 = 1, … , 𝑆

where 𝔼(ℎ𝑃𝑆𝐸 𝑠 ) and 𝜎(ℎ𝑃𝑆𝐸 𝑠 ) denotes PSE expansion of the expected value and standard deviation of sth constraint, respectively. In this formulation, 𝛾𝑠 represents a weight assigned to the sth process constraint. For instance, under the assumption of a Gaussian distribution in ℎ𝑠 , setting 𝛾𝑠 =1 suggests that the sth constraint should be satisfied at least in 83.9 % of the time. Taking the same assumption, setting 𝛾𝑠 to large values will most surely reduce violation of the constraints at the expense of specifying conservative designs. To compare the performance of our previous approach and that proposed in this work, both methods have been implemented on waste water treatment plant case study. As indicated in Table 5, the trajectory profile for the disturbances is assumed to be a series of step changes which magnitudes are described by a normal probability distribution function with a pre-specified mean and variance. In the case of using expected value and variance (Sc1), the constraints shown in Table 3 for this case study are required to be satisfied at the probability of 𝜇 + 3𝜎, i.e. 0.99 whereas in the present approach, coverage probability of confidence intervals for the constraints was explicitly set to 0.99 (Sc2). Table 5: Effect of using different statistical terms in the stochastic back-off methodology Sc1 Sc2 All constraints @ 𝝁 + 𝟑𝝈 Confidence weights All constraints @ ℂ𝚰𝟎.𝟗𝟗 Step changes Step changes Disturbances 𝑞𝑖 ~𝒩(600,20) 𝑞𝑖 ~𝒩(600,20) Optimization variables Area (m2) 3,034.21 1,945.31 Volume (m3) 3,024.95 1,682.54 70 94.74 𝑠𝑤𝑠𝑝 0.03003 0.01389 𝑐𝑤𝑠𝑝 0.032 3.26 𝐾𝑐1 0.111 0.066 𝐾𝑐2 64.40 53.50 𝜏𝑖1 34.55 14.89 𝜏𝑖2 Cost ($/yr) 9.2845x107 4.4539x106


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As shown in Table 5 and Figure 4, the present approach (Sc2) produces a more economically attractive design since it captures accurately the variability in the constraints due to random step changes in the disturbance. As shown in Table 5, Sc1 specified a design with a large volume of the aeration tank and a large cross sectional area for the decanter. Similarly, Sc1 specified overly conservative controllers that diminished plant performance. The results in Figure 4 indicate that the design obtained using larger weights on the constraints’ variance are more conservative than that those specified by the confidence interval approach proposed in this work. As a result, the process design and control scheme for Sc1 resulted in a plant cost that is at least one order of magnitude higher than that obtained by Sc2. Figure 4 also reveals that there is a larger amount of back-off from the optimal steady state design when using expected value and variance (Sc1) compared to the present confidence interval-based approach (Sc2), i.e. back-off in Sc1 is 97% larger than the back-off amount obtained for Sc2. Therefore, the variance-based approach produces a design and control scheme that is 95% more expensive than that obtained by the present method. Figure 5 shows the convergence of the substrate set-point for both scenarios. As expected, Sc1 converged to the lowest allowable substrate set-point whereas Sc2 specified a substrate set-point that is more economical since it is closer to the substrate constraint limit.

Figure 4: Cost function for Sc1 and Sc2



Figure 6 shows the simulation results at the optimal solution obtained from Sc1 and Sc2. These results were generated using a sufficiently large number of MC realizations in the disturbances so that convergence in the statistical terms was most surely ensured. As shown in Figure 6, the process constraints do not follow a Gaussian distribution; thus, the Gaussian distribution assumption is not adequate for this system since the variance does not completely capture the behaviour of the constraints under the effect of disturbances. Different constraints are active for the two scenarios (red dashed lines in Figure 6). These results confirm that the confidence interval approach 110 105 Substrate upper limit

100 95

w

ssp (mg/L)

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

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90 Sc1: All constraints @

85

+3

Sc2: All constraints @ CI

0.99

80 75 70 65

Substrate lower limit 0

10

20

30

40

50

60

70

80

90

100

Iterations Figure 5: Substrate set-point for Sc1 and Sc2

proposed in this study is a more accurate representation of coverage of constraints whereas the variance-based approach fails to capture accurate approximations for the process nonlinear constraints; hence, the significant differences observed with respect to the confidence interval approach (Sc2).


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Figure 6: Simulation results. Sc1: using expected value and variance (a,b,c); Sc2: using confidence interval (d,e,f). Red dashed lines indicate input limits on the constraints.



Table 6: Effect of uncertainty Sc3 Confidence All constraints @ ℂΙ0.99 weights 𝑞𝑖 = 𝐴𝑠𝑖𝑛(𝑤𝑡) + 𝑞𝑖𝑛𝑜𝑚 Disturbances

Uncertainty

Sc4 All constraints @ ℂΙ0.99

Sc5 All constraints @ ℂΙ0.99

𝑞𝑖 = 𝐴𝑠𝑖𝑛(𝑤𝑡) + 𝑞𝑖𝑛𝑜𝑚

𝑞𝑖 = 𝐴𝑠𝑖𝑛(𝑤𝑡) + 𝑞𝑖𝑛𝑜𝑚

𝐴 = 𝒩(10,10) 𝑤 = 𝒩(0.001,0.001)

𝐴 = 𝒩(10,10) 𝑤 = 𝒩(0.001,0.001)

No time-invariant uncertain parameter

𝜇𝑤 = 𝒰(0.1,0.2)

𝐴 = 𝒩(10,10) 𝑤 = 𝒩(0.001,0.001) 𝜇𝑤 = 𝒰(0.1,0.2) 𝑘𝑐 = 𝒰(0.66x10−4 , 1.99x10−4 ) 𝑘𝑑 = 𝒰(2.5x10−5 , 5.5x10−5 ) 𝑠𝑖 = 𝒩(366,5) 𝑥𝑖 = 𝒩(80,5)

Optimization variables Area (m2) 1,696.04 Volume (m3) 1,402.68 98.27 𝑠𝑤𝑠𝑝 0.006951 𝑐𝑤𝑠𝑝 12.50 𝐾𝑐1 0.047113 𝐾𝑐2 10 𝜏𝑖1 35.44 𝜏𝑖2 Cost ($/yr) 1.7320x106

2,584.41 2,469.10 91.22 0.001214 1.24 0.042849 42.27 64.04 1.0073x107

3,453.72 2,240.45 90.44 0.003397 1.25 0.047749 52.26 69.21 1.1690x107

4.2 Effect of uncertainty The aim of this section is to evaluate the performance of the back-off methodology under the effect of disturbances, single and multiple uncertainties. As shown in Table 6, a time-varying sinusoidal disturbance in the inlet flowrate (𝑞𝑖 = 𝐴𝑠𝑖𝑛(𝑤𝑡) + 𝑞𝑖𝑛𝑜𝑚 ) with a frequency and amplitude that follow normal distributions with specified means and variances are considered (𝑞𝑖𝑛𝑜𝑚 = 600). In Sc3-Sc5, the coverage probability of confidence interval of all the process constraints shown in Table 3 was set to 0.99 (i.e. ℂΙ0.99 ). As shown in Figure 7 and Table 6, the present approach compensates for the various uncertainties and disturbances considered at the expense of economics; thus, the system moves further away from steady-state optimal design to compensate for the potential occurrence of single and multiple realizations. Consequently, the amount of backoff is generally seen as a factor strongly related to the significance, type and variation of the uncertainties, i.e. larger (or more drastic) changes in uncertainty may cause larger amounts of back-


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Figure 7: Cost function for different scenarios in uncertain parameters

off. Sc5 has the largest cost compared to Sc3 and Sc4 due to the multiple sources of uncertainties considered in that scenario. As shown in Table 6, Sc5, which considers the effect of disturbance and multiple uncertainties, specified a plant cost that is 91% higher than that specified by Sc3, which only considers disturbance effects. Moreover, Sc4 returned a plant that is 82% more expensive than that obtained for Sc3; this result provides an indication of the impact of single parameter uncertainty (𝜇𝑤 ) on the system. For the present case study, the area of the decanter (𝐴𝑑 ), the volume of the reactor (𝑉𝑟 ) and the substrate set-point (𝑠𝑤𝑠𝑝 ) dominate the process economics (see cost function in Table 3). Thus, changes in these variables will directly impact the process dynamics and therefore the selection of the optimal design and control scheme. As illustrated in Figure 8, the substrate set-point moves away from the threshold (100 𝑚𝑔/𝐿) in exchange for assurance of constraint satisfaction at the specified coverage probability limit in the presence of disturbances and uncertainties.



Figure 8: Substrate set-point different scenarios in uncertain parameters

4.3 Effect of coverage probability of process constraints As mentioned above, this methodology provides an additional degree of freedom to the user to assign different probabilities of satisfaction to each constraint, i.e. coverage probability for the confidence interval of each constraint. While critical constraints need to be prioritized, violation of non-critical restrictions might be tolerated during operation. Table 7 shows the details and results of various scenarios considered in this study using different coverage probabilities for the confidence intervals of the constraints. The uncertainties and disturbance specifications outlined for Sc5 in Table 6 were used to perform this analysis. Table 7: Effect of coverage probability Constraints Sc6 Sc7 𝑢𝑝 @ ℂI0.99 @ ℂI0.5 𝑠𝑤

Sc8 @ ℂI0.9999

𝑅1𝑙𝑜

@ ℂI0.9999

@ ℂI0.5

@ ℂI0.9999

𝑢𝑝 𝑅1

@

ℂI0.9999

ℂI0.5

@ ℂI0.9999

𝑅2𝑙𝑜

@ ℂI0.9999

@ ℂI0.5

@ ℂI0.9999

𝑢𝑝 @ ℂI0.9999 𝑅2 Optimization variables Area (m2) 2,194.85 Volume (m3) 1,820.07

@ ℂI0.5

@ ℂI0.9999

2,314.61 2,248.86

3,694.62 2,495.53

@


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𝑠𝑤𝑠𝑝 𝑐𝑤𝑠𝑝 𝐾𝑐1 𝐾𝑐2 𝜏𝑖1 𝜏𝑖2 Cost ($/yr)

99.99 0.008230 0.226 0.031928 57.81 70 1.8530x106

98.81 0.000595 20 0.029829 70 61.91 2.3179 x106

88.84 0.070516 1.032 0.061932 47.18 13.66 1.5230 x107

Figure 9 shows the convergence of the cost function for the different scenarios considered in this section. As shown in Table 7, specifying a higher probability of satisfaction in the constraints, i.e. 0.9999 (Sc8) instead of 0.99 (Sc5), significantly impacts the process economics, i.e. Sc8 produced a design and control scheme that is 23% more expensive than that obtained by Sc5. As shown in Figure 9, lowering the probability of satisfaction, particularly in the key process constraints, can potentially improve the process economics. For example, changing the coverage probability specification on constraint 𝑠𝑤𝑢𝑝 from 0.9999 (Sc8) to 0.5 (Sc6) while keeping the rest at a constant coverage probability reduces the plant costs by almost 87%. Likewise, setting the coverage probability of constraint 𝑠𝑤𝑢𝑝 to 0.99 while keeping the remaining constraints to 0.5 (Sc7) specified a plant design that is 80% less expensive than that obtained for Sc5 (i.e. all the constraints were set to 0.99 probability of satisfaction).

Figure 9: Cost, effect of confidence weights



Figure 10: Simulation results (Sc6)

These results show that the substrate concentration in the decanter is a key constraint in this process, since it directly impact the process variability costs, as shown in Table 3. Figure 10 shows the simulation results using the optimal design and control scheme obtained from Sc6. This figure indicates that using a high coverage probability in the constraints interval forces the system to remain in the desired boundaries of constraints at the expense of higher cost. As a result, there is no violation of the constraints on purge ratio (𝑅1 ) and purge age (𝑅2 ); however, the constraint on the maximum allowed substrate is tolerable to be violated at most 50%, which agrees with the coverage probability specified for this constraint (see Table 7).


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Table 8: Effect of multiple sources of uncertainty Sc9 Confidence All constraints @ ℂΙ0.99 weights 𝑞𝑖 = 𝐴𝑠𝑖𝑛(𝑤𝑡) + 𝑞𝑖𝑛𝑜𝑚 Disturbances

𝐴 = 𝒩(10,10) 𝑤 = 𝒩(0.001,0.001)

Uncertainty

𝜇𝑤 = 𝒰(0.1,0.2) 𝑘𝑐 = 𝒰(0.66x10−4 , 1.99x10−4 ) 𝑘𝑑 = 𝒰(2.5x10−5 , 5.5x10−5 ) 𝑠𝑖 = 𝒩(366,5) 𝑥𝑖 = 𝒩(80,5) 𝑛 = 𝒰(1.8,2.2)

Area (m2) Volume (m3) 𝑠𝑤𝑠𝑝 𝑐𝑤𝑠𝑝 𝐾𝑐1 𝐾𝑐2 𝜏𝑖1 𝜏𝑖2 Cost ($/yr)

3,476.70 2,273.37 90.22 0.006228 1.24 0.033603 49.42 63 1.2154x107

4.4 Computational Costs This section aims to provide estimates on the computational demands associated with the present stochastic back-off methodology. One the optimal problem has been defined, most of the CPU time required by the present approach is used to estimate the sensitivities needed to build the probabilistic-based PSE functions, i.e. see Step 3 in the algorithm. The accuracy of the PSE functions is highly correlated with the number of MC samples required and the desired accuracy in the results, which is mostly controlled by the tolerance criterion (𝜖𝑀𝐶 ). Table 8 shows the results of Sc9 in which multiple sources of uncertainties and disturbances has been considered. Figure 11 shows the CPU time and the required number of MC samples per iteration that were needed to solve Sc3 and Sc9. Sc3 only considers a time-varying disturbance, i.e. it does not consider model



Figure 11: CPU time and number of required MC samples at different iterations for Sc3 and Sc9

parameter uncertainty. In addition to a stochastic disturbance, Sc9 also considers multiple sources of uncertainties as well as a model structure error (see Table 8). Note that the uncertainty considered on power term 𝑛 in the process (see Table 3) can be considered as a potential source of model structure error since it directly affects a time-dependent state variable for this system, i.e. the biomass concentration. In the present analysis, the power term n in the differential state equation for the biomass concentration was assumed to follow a uniform distribution. As shown in Table 8, Sc9 produced a design and control scheme that is 4% more expensive than that obtained by Sc5, which is the same scenario but without consideration of model structure error. As shown in Figure 11, there exists a correlation between CPU time and problem size. That is, Sc9 required approximately 1.46 hours to converge (i.e. total CPU time), which is 46% higher than the total CPU time required by Sc3. On average, Sc3 required 1.22x103 MC samples whereas Sc9 involved 37% more samples i.e. 3.25x103. As shown in Figure 11, a few peaks in the required number of MC samples are observed for Sc3, e.g. around iteration 40. This behaviour suggests that sinusoidal changes in the disturbances (with random realizations in the frequency) together with the system nonlinearities around 𝜼𝑛𝑜𝑚 at those iterations directly affect the computation of the sensitivity terms in the PSE-based functions; hence, a larger number of simulations are needed to most surely ensure convergence in the statistical terms. In terms of scalability, undoubtedly, adding more decision variables makes the problem more intense as it involves the computation of another set of sensitivity calculations. Also, the model 28 ACS Paragon Plus Environment

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complexity will increase the computational costs, i.e. the complex the model the longer to simulate. Nevertheless, the main computational expenses in this method are directly correlated with the number and the nature of stochastic uncertain parameters considered in the analysis. Adding an additional uncertain parameter increases the domain of the uncertain space thus requiring the need to consider a larger set of realizations between the uncertain parameters, which requires the need to perform additional simulations and eventually increases the computational demands of the present approach.

5 Conclusions and future work A stochastic back-off methodology that addresses simultaneous design and control using stochastic uncertainty descriptions was presented. The key idea is to describe the confidence interval of process constraints at a user-defined coverage probabilities using PSE-based functions. PSE functions are explicitly defined in terms of the optimization variables. The method provides an additional degree of freedom since it can allocate different levels of confidence for the process constraints using coverage probability of confidence intervals. A waste water treatment plant was used to illustrate the benefits and limitations of the present approach. The results presented in this study indicate that the proposed back-off approach leads to more economically attractive designs when compared to the variance-based approach previously introduced by the authors. The main burden of the current stochastic-based technique is the need for numerous simulations of the system to obtain the PSE functions. For that reason, the next step in this research is to reduce the number of required samples in the analysis to achieve certain level of accuracy in the calculations using advanced sampling and uncertainty quantification methods. In the current methodology, the search space region (𝛿) in the PSE-base optimization problem was specified a priori. Selecting an appropriate tuning parameter can improve the performance of the algorithm; however, its selection is non-trivial and affects the overall performance of the algorithm. Future work in this research also includes the development of practical and effective approaches to select this parameter, including adaptive methods that will adjust the search space as the iteration proceeds in our backoff algorithm.



Acknowledgements We acknowledge the financial support provided by the Natural Sciences & Engineering Research Council of Canada (NSERC) to carry out the research work presented in this article.

Nomenclature Symbols 𝑐𝑠 𝑐𝑤 𝑐𝑤𝑠𝑝

𝒅 𝒅𝑛𝑜𝑚 𝑓𝑑 𝑓𝑘 𝑖 𝑗 𝑘 𝑘𝑠 𝑘𝑑 𝑘𝑐 𝑘𝑙𝑎 𝑘01 ℎ 𝑙𝑟𝑑 , 𝑙𝑟𝑏 , 𝑙𝑟𝑟 𝑛 𝑝

𝑞 𝑞𝑖 𝑞𝑖 𝑛𝑜𝑚 𝑞𝑝 𝑞𝑟 𝑞1 𝑞2 𝑠 𝑠𝑖 𝑠𝑖𝑟 𝑠𝑤 𝑠𝑤𝑠𝑝 𝑢𝑝 𝑠𝑤

𝑡 𝑡𝑓 𝑢 𝑣𝑠𝑑, 𝑣𝑠𝑏, 𝑣𝑠𝑟 𝑥 𝑥𝑑 , 𝑥𝑏 , 𝑥𝑟 𝑥𝑖 𝑥𝑖𝑟 𝑥𝑤 𝑥0 𝑦 𝐴

Description Oxygen specific saturation Dissolved oxygen concentration(mg/L) Dissolved oxygen set-point in the reactor (mg/L) Time-varying disturbances Disturbances at nominal condition Fraction of death biomass Turbine speed Index in the stochastic back of algorithm Index of PSE order Index in the procedure of finding required MC samples Saturation constant Biomass death rate Specific Cellular activity Oxygen transfer into the water constant Oxygen demand constant Process constraints Depth of the first, second and bottom layers in the decanter (m) Potential model structure error Index of decision variables Outlet flow from reactor (m3 ⁄hr) Feed flow rate (m3 ⁄hr) Nominal condition of the inlet flowrate (m3 ⁄hr) Purge flow rate (m3 ⁄hr) Recycle flow rate (m3 ⁄hr) Outlet flow from decanter (m3 ⁄hr) Recycle flow that leaves the decanter (m3 ⁄hr) Index of inequality constraints Inlet organic substrate concentration(mg/L) organic substrate concentrations entering the bioreactor (mg/L) Organic substrate concentrations in the bioreactor (mg/L) Substrate set-point in the reactor (mg/L) upper bund (saturation limit ) on the substrate concentration Time Final process time Manipulated variables Rate of settling for the activated sludge at first, second and bottom layers in the decanter System states Biomass concentrations at first, second and bottom layers in the decanter (mg/L) Inlet biomass concentration(mg/L) biomass concentrations entering the bioreactor (mg/L) Biomass concentrations in the bioreactor (mg/L) nominal (initial) condition System outputs Amplitude of the sinusoidal changes in the inlet flowrate (m3 ⁄hr)


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𝐴𝑑

𝐶𝐶𝑤

ℂΙ 𝜌 𝐹𝑚𝑜𝑑𝑒𝑙 𝐾𝑐1 𝐾𝑐2

𝑀 𝑁 𝑁𝑐 𝑁𝑖𝑡𝑒𝑟 𝑂𝐶𝑤 𝑅 𝑅1𝑙𝑜 𝑢𝑝 𝑅1 𝑅2𝑙𝑜 𝑢𝑝 𝑅2

𝑉𝑟

𝑉𝐶𝑤 Greek Symbols 𝜶𝑑 𝜶𝜁 𝜸

𝛿 𝜖 𝜖𝑀𝐶 𝜻 𝜻𝑛𝑜𝑚 𝜼 𝜼0 𝜼𝑛𝑜𝑚 𝜼𝑢𝑝 𝜼𝑙𝑏 𝝀

𝜇𝑤 𝜌 𝜏𝑖1 𝜏𝑖2

𝛩

cross-sectional area of the decanter (m2) Capital cost of the waste water treatment plant ($/yr) Confidence interval at coverage probability of 𝜌 Process model Proportional gain of the PI controller of the purge flow rate Proportional gain of the PI controller of the turbine speed Big-M method penalty term Number of samples in each batch of MC samples Convergence examination period, back-off methodology Maximum number of iterations allowed in the back-off algorithm Operating cost of the waste water treatment plant ($/yr) Data set (Simulation results) collected from N MC samples Lower bound on feasible ratio between the purge to the recycle flow rate Upper bound on feasible ratio between the purge to the recycle flow rate Lower bound on feasible purge age in the decanter Upper bound on feasible purge age in the decanter The volume of the aeration tank (m3) Variability cost of the waste water treatment plant ($/yr) Description Statistical properties of probability distribution function of disturbance Statistical properties of probability distribution function of uncertainty Weights assigned to the standard deviation of the constraints Search space of Optimization variables in PSE-based optimization Tolerance criterion for convergence, back-off methodology Tolerance criterion for convergence, PSE functions Uncertainty (parameter uncertainty, model structure error) Uncertainty at nominal condition Optimization variables Optimization variables estimated at steady-state Optimization variables at nominal condition Upper bounds on the optimization variables Lower bounds on the optimization variables Dummy variables to maintain feasibility using big-M method Specific growth rate Confidence level Time integral constant of the PI controller of the purge flow rate Time integral constant of the PI controller of the turbine speed Cost Function



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