160 results found
Feng X, Villanueva ME, Chachuat B, et al., 2017, Branch-and-Lift algorithm for obstacle avoidance control, IEEE 56th Annual Conference on Decision and Control (CDC), Publisher: IEEE, ISSN: 0743-1546
Obstacle avoidance problems are a class of non-convex optimal control problems for which derivative-based optimization algorithms often fail to locate global minima. The goal of this paper is to provide a tutorial on how to apply Branch & Lift algorithms, a novel class of global optimal control methods, for solving such obstacle avoidance problems to global optimality. The focus of the technical developments is on how Branch & Lift methods can exploit the particular structure of Dubin models, which can be used to model a variety of practical obstacle avoidance problems. The global convergence properties of Branch & Lift in the context of obstacle avoidance is discussed from a theoretical as well as a practical perspective by applying it to a tutorial example.
Bernardi A, Nikolaou A, Meneghesso A, et al., 2017, Semi-empirical modeling of microalgae photosynthesis in different acclimation states - Application to N. gaditana., Journal of Biotechnology, Vol: 259, Pages: 63-72, ISSN: 0168-1656
The development of mathematical models capable of accurate predictions of the photosynthetic productivity of microalgae under variable light conditions is paramount to the development of large-scale production systems. The process of photoacclimation is particularly important in outdoor cultivation systems, whereby seasonal variation of the light irradiance can greatly influence microalgae growth. This paper presents a dynamic model that captures the effect of photoacclimation on the photosynthetic production. It builds upon an existing semi-empirical model describing the processes of photoproduction, photoregulation and photoinhibition via the introduction of acclimation rules for key parameters. The model is calibrated against a dataset comprising pulsed amplitude modulation fluorescence, photosynthesis rate, and antenna size measurements for the microalga Nannochloropsis gaditana in several acclimation states. It is shown that the calibrated model is capable of accurate predictions of fluorescence and respirometry data, both in interpolation and in extrapolation.
Puchongkawarin C, Vaupel Y, Guo M, et al., 2017, Towards the synthesis of wastewater recovery facilities using enviroeconomic optimization, The Water-Food-Energy Nexus - Processes, Technologies, and Challenges, Editors: Mujtaba, Srinivasan, Elbashir, ISBN: 9781498760843
The wastewater treatment industry is undergoing a major shift towards a proactive interest in recovering materials and energy from wastewater streams, driven by both economic incentives and environmental sustainability. With the array of available treatment technologies and recovery options growing steadily, systematic approaches to determining the inherent trade-off between multiple economic and environmental objectives become necessary, namely enviroeconomic optimization.The main objective of this chapter is to present one such methodology based on superstructure modeling and multi-objective optimization, where the main environmental impacts are quantified using life cycle assessment (LCA). This methodology is illustrated with the case study of a municipal wastewater treatment facility. The results show that accounting for LCA considerations early on in the synthesis problem may lead to dramatic changes in the optimal process configuration, therebysupporting LCA integration into decision-making tools for wastewater treatment alongside economical selection criteria.
Villanueva ME, Li JC, Feng X, et al., 2017, Computing ellipsoidal robust forward invariant tubes for nonlinear MPC, IFAC-PapersOnLine, Vol: 50, Pages: 7175-7180, ISSN: 2405-8963
Min-max differential inequalities (DIs) can be used to characterize robust forward invariant tubes with convex cross-section for a large class of nonlinear control systems. The advantage of using set-propagation over other existing approaches for tube MPC is that they avoid the discretization of control policies. Instead, the conservatism of min-max DIs in tube MPC arises from the discretization of sets in the state-space, while the control law is never discretized and remains defined implicitly via the solution of a min-max optimization problem. The contribution of this paper is the development of a practical implementation of min-max DIs for tube MPC using ellipsoidal-tube enclosures. We illustrate these developments with a spring-mass-damper system.
Sun M, Villanueva ME, Pistikopoulos EN, et al., 2017, Robust multi-parametric control of continuous-time linear dynamic systems, IFAC-PapersOnLine, Vol: 50, Pages: 4660-4665, ISSN: 2405-8963
We extend a recent methodology called multi-parametric NCO-tracking for the design of parametric controllers for continuous-time linear dynamic systems in the presence of uncertainty The approach involves backing-off the path and terminal state constraints based on a worst-case uncertainty propagation determined using either interval analysis or ellipsoidal calculus. We address the case of additive uncertainty and we discuss approaches to handling multiplicative uncertainty that retain tractability of the mp-NCO-tracking design problem, subject to extra conservatism. These developments are illustrated with the case study of a fluidized catalytic cracking (FCC) unit operated in partial combustion mode.
Houska B, Li JC, Chachuat B, 2017, Towards rigorous robust optimal control via generalized high-order moment expansion, Optimal Control Applications & Methods, Vol: 39, Pages: 489-502, ISSN: 1099-1514
This paper is concerned with the rigorous solution of worst-case robust optimal control problems havingbounded time-varying uncertainty and nonlinear dynamics with affine uncertainty dependence. We proposean algorithm that combines existing uncertainty set-propagation and moment-expansion approaches.Specifically, we consider a high-order moment expansion of the time-varying uncertainty, and we bound theeffect of the infinite-dimensional remainder term on the system state, in a rigorous manner, using ellipsoidalcalculus. We prove that the error introduced by the expansion converges to zero as more moments are added.Moreover, we describe a methodology to construct a conservative, yet more computationally tractable, robustoptimization problem, whose solution values are also shown to converge to those of the original robustoptimal control problem. We illustrate the applicability and accuracy of this approach with the robust time-optimal control of a motorized robot arm.
This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuous-time dynamic systems that are affected by time-varying disturbances. We derivea min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implementations, the proposed framework does not involve discretizing the control policy and, therefore, the conservatism of the predicted tube depends solely on the accuracy of the set parameterization. The proposed approach is then used to construct a robustMPCscheme based on tubes with ellipsoidal cross-sections. This ellipsoidal MPC scheme is based on solving an optimal control problem under linear matrix inequality constraints. We illustrate these results with the numerical case study of a spring-mass-damper system.
Nikolaou A, Booth P, Gordon F, et al., 2017, Multi-physics modeling of light-limited microalgae growth in raceway ponds, IFAC Proceedings Volumes (IFAC-PapersOnline), Vol: 49, Pages: 324-329, ISSN: 1474-6670
This paper presents a multi-physics modeling methodology for the quantitative prediction of microalgae productivity in raceway ponds by combining a semi-mechanistic model of microalgae growth describing photoregulation, photoinhibition and photoacclimation, with models of imperfect mixing based on Lagrangian particle-tracking and heterogeneous light distribution. The photosynthetic processes of photoproduction, photoregulation and photoinhibition are represented by a model of chlorophyll fluorescence developed by Nikolaou et al. (2015), which is extended to encompass photoacclimation. The flow is simulated with the commercial CFD package ANSYS, whereas light attenuation is described by the Beer-Lambert law as a first approximation. Full-scale simulation results are presented on extended time horizons. Comparisons are made in terms of areal productivities under both imperfect and idealized (CSTR) mixing conditions, and for various extraction rates and water depths.
Diaz-Bejarano E, Porsin AV, Macchietto S, 2017, Fossil fuel: Energy efficient thermal retrofit options for crude oil transport in pipelines, The Water-Food-Energy Nexus: Processes, Technologies, and Challenges, Pages: 277-296, ISBN: 9781498760836
© 2018 by Taylor & Francis Group, LLC. Pipelines are used to transport large amounts of crude oil over large distances (either overland or subsea), representing the most economical alternative. Flow assurance faces two main problems: viscosity increase due to gradual cooling of the oil along the pipeline and fouling deposition. These problems are especially important in very cold environments (Russia, Alaska, North Sea, deep oceanic waters, etc.) and when dealing with nonconventional oils, usually heavy or extra-heavy oil and waxy oils. In many cases, the depletion of deposits in conventional oil reservoirs is gradually leading to more extraction of these types of feedstock from remote locations. All these situations result in pipeline transport difficulties such as increased pumping costs, reduced flow rates, and the possibility of flow inhibition or blockage, with potentially major economic impact (Correra et al., 2007; Martínez-Palou et al., 2011).
Rajyaguru J, Villanueva ME, Houska B, et al., 2016, Chebyshev model arithmetic for factorable functions, Journal of Global Optimization, Vol: 68, Pages: 413-438, ISSN: 1573-2916
This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.
Baroukh C, Steyer JP, Bernard O, et al., 2016, dynamic Flux Balance Analysis of the Metabolism of Microalgae under a Diurnal Light Cycle, 11th IFAC Symposium on Dynamics and Control of Process Systems including Biosystems, Publisher: Elsevier, Pages: 791-796, ISSN: 1474-6670
Microalgae have received much attention in the context of renewable fuel production, due to their ability to produce in high quantities carbon storage molecules such as lipids and carbohydrates. Despite significant research effort over the last decade, the production yields remain low and need to be optimized. For that, a thorough understanding of carbon storage metabolism is necessary. This paper develops a constrained metabolic model based on the dFBA framework to represent the dynamics of carbon storage in microalgae under a diurnal light cycle. The main assumption here is that microalgae adapt their metabolism in order to optimize their production of functional biomass (proteins, membrane lipids, DNA, RNA) over a diurnal cycle. A generic metabolic network comprised of 160 reactions representing the main carbon and nitrogen pathways of microalgae is used to characterize the metabolism. The optimization problem is simplified by exploiting the right kernel of the stoichiometric matrix, and transformed into a linear program by discretizing the differential equations using a classical collocation technique. Several constraints are investigated. The results suggest that the experimentally observed strategy of accumulation of carbon storage molecules during the day, followed by their depletion during the night may indeed be the optimal one. However, a constraint on the maximal synthesis rate of functional biomass must be added for consistency with the biological observations.
Bernardi A, Nikolaou A, Meneghesso A, et al., 2016, Correction: High-Fidelity Modelling Methodology of Light-Limited Photosynthetic Production in Microalgae., PLOS One, Vol: 11, ISSN: 1932-6203
[This corrects the article DOI: 10.1371/journal.pone.0152387.].
Sun M, Chachuat B, Pistikopoulos EN, 2016, Design of multi-parametric NCO tracking controllers for linear dynamic systems, Computers and Chemical Engineering, Vol: 92, Pages: 64-77, ISSN: 1873-4375
A methodology for combining multi-parametric programming and NCO tracking is presented in the case of linear dynamic systems. The resulting parametric controllers consist of (potentially nonlinear) feedback laws for tracking optimality conditions by exploiting the underlying optimal control switching structure. Compared to the classical multi-parametric MPC controller, this approach leads to a reduction in the number of critical regions. It calls for the solution of more difficult parametric optimization problems with linear differential equations embedded, whose critical regions are potentially nonconvex. Examples of constrained linear quadratic optimal control problems with parametric uncertainty are presented to illustrate the approach.
Nikolaou A, Bernardi A, Meneghesso A, et al., 2016, High-Fidelity Modelling Methodology of Light-Limited Photosynthetic Production in Microalgae, PLOS One, Vol: 11, ISSN: 1932-6203
Reliable quantitative description of light-limited growth in microalgae is key to improving the design and operation of industrial production systems. This article shows how the capability to predict photosynthetic processes can benefit from a synergy between mathematical modelling and lab-scale experiments using systematic design of experiment techniques. A model of chlorophyll fluorescence developed by the authors [Nikolaou et al., J Biotechnol 194:91–99, 2015] is used as starting point, whereby the representation of non-photochemical-quenching (NPQ) process is refined for biological consistency. This model spans multiple time scales ranging from milliseconds to hours, thus calling for a combination of various experimental techniques in order to arrive at a sufficiently rich data set and determine statistically meaningful estimates for the model parameters. The methodology is demonstrated for the microalga Nannochloropsis gaditana by combining pulse amplitude modulation (PAM) fluorescence, photosynthesis rate and antenna size measurements. The results show that the calibrated model is capable of accurate quantitative predictions under a wide range of transient light conditions. Moreover, this work provides an experimental validation of the link between fluorescence and photosynthesis-irradiance (PI) curves which had been theoricized.
Faust JMM, Fu J, Chachuat B, et al., 2016, Optimization of dynamic systems with rigorous path constraint satisfaction, Editors: Kravanja, Bogataj, Publisher: ELSEVIER SCIENCE BV, Pages: 643-648
Peric ND, Villanueva ME, Chachuat B, 2016, Set-valued integration of uncertain dynamic systems with sensitivity analysis capability, Editors: Kravanja, Bogataj, Publisher: ELSEVIER SCIENCE BV, Pages: 1165-1170
Adi VSK, Cook M, Peeva LG, et al., 2016, Optimization of OSN Membrane Cascades for Separating Organic Mixtures, Editors: Kravanja, Bogataj, Publisher: ELSEVIER SCIENCE BV, Pages: 379-384
Ulmasov D, Baroukh C, Chachuat B, et al., 2016, Bayesian Optimization with Dimension Scheduling: Application to Biological Systems, Publisher: ELSEVIER SCIENCE BV
Sun M, Villanueva ME, Chachuat B, et al., 2016, Strategies towards the robust multi-parametric control of continuous-time systems, Pages: 355-357
Bernard O, Mairet F, Chachuat B, 2016, Modelling of Microalgae Culture Systems with Applications to Control and Optimization, Editors: Posten, Chen, Publisher: SPRINGER-VERLAG BERLIN, Pages: 59-87
Mathematical modeling is becoming ever more important to assess the potential, guide the design, and enable the efficient operation and control of industrial-scale microalgae culture systems (MCS). The development of overall, inherently multiphysics, models involves coupling separate submodels of (i) the intrinsic biological properties, including growth, decay, and biosynthesis as well as the effect of light and temperature on these processes, and (ii) the physical properties, such as the hydrodynamics, light attenuation, and temperature in the culture medium. When considering high-density microalgae culture, in particular, the coupling between biology and physics becomes critical. This chapter reviews existing models, with a particular focus on the Droop model, which is a precursor model, and it highlights the structure common to many microalgae growth models. It summarizes the main developments and difficulties towards multiphysics models of MCS as well as applications of these models for monitoring, control, and optimization purposes.
Puchongkawarin C, Gomez-Mont C, Stuckey DC, et al., 2015, Optimization-based methodology for the development of wastewater facilities for energy and nutrient recovery, CHEMOSPHERE, Vol: 140, Pages: 150-158, ISSN: 0045-6535
Nikolaou A, Hartmann P, Sciandra A, et al., 2015, Dynamic coupling of photoacclimation and photoinhibition in a model of microalgae growth., Journal of Theoretical Biology, Vol: 390, Pages: 61-72, ISSN: 1095-8541
The development of mathematical models that can predict photosynthetic productivity of microalgae under transient conditions is crucial for enhancing large-scale industrial culturing systems. Particularly important in outdoor culture systems, where the light irradiance varies greatly, are the processes of photoinhibition and photoacclimation, which can affect photoproduction significantly. The former is caused by an excess of light and occurs on a fast time scale of minutes, whereas the latter results from the adjustment of the light harvesting capacity to the incoming irradiance and takes place on a slow time scale of days. In this paper, we develop a dynamic model of microalgae growth that simultaneously accounts for the processes of photoinhibition and photoacclimation, thereby spanning multiple time scales. The properties of the model are analyzed in connection to PI-response curves, under a quasi steady-state assumption for the slow processes and by neglecting the fast dynamics. For validation purposes, the model is calibrated and compared against multiple experimental data sets from the literature for several species. The results show that the model can describe the difference in photosynthetic unit acclimation strategies between Dunaliella tertiolecta (n-strategy) and Skeletonema costatum (s-strategy).
Fu J, Faust JMM, Chachuat B, et al., 2015, Local optimization of dynamic programs with guaranteed satisfaction of path constraints, Automatica, Vol: 62, Pages: 184-192, ISSN: 1873-2836
An algorithm is proposed for locating a feasible point satisfying the KKT conditions to a specified tolerance of feasible inequality-path-constrained dynamic programs (PCDP) within a finite number of iterations. The algorithm is based on iteratively approximating the PCDP by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. The main contribution of this article is an adaptation of the semi-infinite program (SIP) algorithm proposed in Mitsos (2011) to PCDP. It is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first-order KKT conditions of the PCDP to a specified tolerance. The main assumptions are: (i) availability of a nonlinear program (NLP) local solver that generates a KKT point of the constructed approximation to PCDP at each iteration if this problem is indeed feasible; (ii) existence of a Slater point of the PCDP that also satisfies the first-order KKT conditions of the PCDP to a specified tolerance; (iii) all KKT multipliers are nonnegative and uniformly bounded with respect to all iterations. The performance of the algorithm is analyzed through two numerical case studies.
Houska B, Villanueva ME, Chachuat B, 2015, Stable set-valued integration of nonlinear dynamic systems using affine set-parameterizations, Siam Journal of Numerical Analysis, Vol: 53, Pages: 2307-2328, ISSN: 0036-1429
Many set-valued integration algorithms for parametric ordinary differential equations (ODEs) implement a combination of Taylor series expansion with either interval arithmetic or Taylor model arithmetic. Due to the wrapping effect, the diameter of the solution-set enclosures computed with these algorithms typically diverges to infinity on finite integration horizons, even though the ODE trajectories themselves may be asymptotically stable. This paper starts by describing a new discretized set-valued integration algorithm that uses a predictor-validation approach to propagate generic affine set-parameterizations, whose images are guaranteed to enclose the ODE solution set. Sufficient conditions are then derived for this algorithm to be locally asymptotically stable, in the sense that the computed enclosures are guaranteed to remain stable on infinite time horizons when applied to a dynamic system in the neighborhood of a locally asymptotically stable periodic orbit (or equilibrium point). The key requirement here is quadratic Hausdorff convergence of function extensions in the chosen affine set-parameterization, which is proved to be the case, for instance, for Taylor models with ellipsoidal remainders. These stability properties are illustrated with the case study of a cubic oscillator system.
Rayjaguru J, Villanueva ME, Houska B, et al., 2015, Continuous-time enclosures for uncertain implicit differential equations, 9th IFAC Symposium on Advanced Control of Chemical Processes, Publisher: Elsevier, Pages: 94-99, ISSN: 1474-6670
The computation of enclosures for the reachable set of uncertain dynamic systems is a crucial component in a wide variety of applications, from global and robust dynamic optimization to safety verification and fault detection. Even though many systems in engineering are best modeled as implicit differential equations (IDEs) and differential algebraic equations (DAEs), methods for the construction of enclosures for these are not as well developed as they are for ordinary differential equations (ODEs). In this paper, we propose a continuous-time approach for the guaranteed over approximations of the reachable set for quasilinear IDEs. This approach builds on novel high-order inclusion techniques for the solution set of algebraic equations and state-of-the-art techniques for bounding the solution of nonlinear ODEs.We show how this approach can be used to bound the reachable set of uncertain semi-explicit DAEs by bounding the underlying IDEs. We demonstrate this approach on two case studies, a double pendulum where it proves superior with delayed break-down times compared to other methods, and anaerobic digestion of microalgae which has nine differential and two algebraic states.
Puchongkawarin C, Fitzgerald S, Chachuat B, 2015, Plant-wide optimization of a full-scale activated sludge plant with anaerobic sludge treatment, 9th IFAC Symposium on Advanced Control of Chemical Processes, Publisher: Elsevier, Pages: 1234-1239, ISSN: 1474-6670
© 2015, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. This paper presents the application of a plant-wide model-based methodology to wastewater treatment plants. The focus is on a tertiary activated sludge plant with anaerobic sludge treatment, owned and operated by Sydney Water. A dynamic plant-wide model is first developed and calibrated using historical data. A scenario-based optimization procedure is then applied for computing the effect of key discharge constraints on the minimal net power consumption, via the repeated solution of a dynamic optimization problem. The results show a potential for reduction of the energy consumption by about 20%, through operational changes only, without compromising the current effluent quality. It is also found that nitrate (and hence total nitrogen) discharge could be reduced from its current level around 22 mg(N)/L to less than 15 with no increase in net power consumption, and could be further reduced to <10 mg(N)/L subject to a 15% increase in net power consumption upon diverting part of the primary sludge to the secondary treatment stage. This improved understanding of the relationship between nutrient removal and energy use will feed into discussions with environmental regulators regarding nutrient discharge licensing.
Chachuat B, Houska B, Paulen R, et al., 2015, Set-theoretic approaches in analysis, estimation and control of nonlinear systems, 9th IFAC Symposium on Advanced Control of Chemical Processes, Publisher: Elsevier, Pages: 981-995, ISSN: 1474-6670
This paper gives an overview of recent developments in set-theoretic methods for nonlinear systems, with a particular focus on the activities in our own research group. Central to these approaches is the ability to compute tight enclosures of the range of multivariate systems, e.g. using ellipsoidal calculus or higher-order inclusion techniques based on multivariate polynomials, as well as the ability to propagate these enclosures to enclose the trajectories of parametric or uncertain differential equations. We illustrate these developments with a range of applications, including the reach ability analysis of nonlinear dynamic systems; the determination of all equilibrium points and bifurcations in a given state-space domain; and the solution of set-membership parameter estimation problems. We close the paper with a discussion about on-going research in tube-based methods for robust model predictive control.
Puchongkawarin C, Menichini C, Laso-Rubido C, et al., 2015, Model-based methodology for plant-wide analysis of wastewater treatment plants: Industrial case study, Water Practice and Technology, Vol: 10, Pages: 517-526, ISSN: 1751-231X
This paper presents the application of a model-based methodology for improved understanding of the tight interplay between effluent quality, energy use, and fugitive emissions in wastewater treatment plants. Dynamic models are developed and calibrated in an objective to predict the performance of a conventional activated sludge plant owned and operated by Sydney Water, Australia. A scenario-based approach is applied to quantify the effect of key operating variables on the effluent quality, energy use, and fugitive emissions. Operational strategies that enable a reduction in aeration energy by 10-20% or a reduction of total nitrogen discharge down to 3 mg L<sup>-1</sup> are identified. These results are also compared to an upgraded plant with reverse osmosis in terms of energy consumption and greenhouse gas emissions. This improved understanding of the relationship between nutrient removal, energy use, and emissions will feed into discussions with environmental regulators regarding nutrient discharge licensing.
Paulen R, Villanueva ME, Chachuat B, 2015, Guaranteed parameter estimation of non-linear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies, IMA Journal of Mathematical Control and Information, Vol: 33, Pages: 563-587, ISSN: 0265-0754
This paper is concerned with guaranteed parameter estimation of non-linear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the approximation level is met. Such exclusion tests rely on the ability to bound the solution set of the dynamic system for a finite parameter subset, and the tightness of these bounds is therefore paramount; equally important in practice is the time required to compute the bounds, thereby defining a trade-off. In this paper, we investigate such a trade-off by comparing various bounding techniques based on Taylor models with either interval or ellipsoidal bounds as their remainder terms. We also investigate the use of optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm, and we implement simple strategies that avoid recomputing Taylor models or reduce their expansion orders wherever possible. Case studies of various complexities are presented, which show that these improvements using Taylor-based bounding techniques can significantly reduce the computational burden, both in terms of iteration count and CPU time.
Nikolaou A, Chachuat B, 2015, 427331 Scaling-up microalgae production systems: Inferring biomass productivity in raceway ponds using numerical simulation, Pages: 453-456
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