Instituto Tecnol´ogico y de Estudios Superiores de Monterrey

Instituto Tecnol´ogico y de Estudios Superiores de Monterrey

Campus Monterrey

School of Engineering and Sciences

Studies on optimal design, operation and control of process and energy systems

A dissertation presented by

Oswaldo Andr´es Mart´ınez

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of Doctor of Philosophy

In

Engineering Science

Major in Environmental Systems

Monterrey Nuevo Le´on, July 7

^th

, 2020

Instituto Tecnológico,v de Estudios Superiores de Monterrey

Campus

Monterlev

School

of-

Engrneering and Sciences

'fhe committee members, lterebl; ceitrfy tha i have leatl the tlissertation presented b_v Osrvaldo Andrés Martínez and that it is fL111y adequate in scope and quality as a partiai requiremer"rt for the degree of Doctor of Phiiosophv in Er-rgineering Science, ruvith a major in Environmental Svstems.

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Dr. \'icente Rico Ranrírez Instituto Tecnológico de Celaya

Dr. Rubén l\{orales N{enéndez Dean of Graduate Sttidies School of Engineerirrg .lnr.1 Sciences

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Tlacuahuac

Committee Member

Jonathan ^C. lvlayo Maldonado

Committee Member de Monterrelr

'11

Declaration of Authorship

I, Oswaldo Andr´es Mart´ınez, declare that this dissertation titled Studies on optimal design, operation and control of process and energy systems and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research degree at this Uni- versity.

• Where any part of this dissertation has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this dissertation is entirely my own work.

• I have acknowledged all main sources of help.

• Where the dissertation is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.

Oswaldo Andr´es Mart´ınez Monterrey, Nuevo Le´on, July 7

^th

, 2020

@2020 by Oswaldo Andr´es Mart´ınez

All right reserved

To my family

iv

Acknowledgements

I express my gratitude to my advisor Dr. Antonio Flores Tlacuahuac for his guidance, help, and encouragement along the development of this research. His mentoring let me explore exciting sub- jects in the PSE field that I did not know.

I also thank Prof. Larry Biegler for his kind advising during my time as a research visitor at Carnegie Mellon University. I acquired invaluable knowledge from the events, discussions and ac- tivities in his research group.

I would like to thank the Tecnol´ogico de Monterrey for granting me a full-tuition scholarship that allowed me to study this PhD. I am also thankful for the financial support provided by CONACyT (484726) for living expenses including one year of research abroad.

Many thanks to the rest of my thesis committee members, Dr. Jonathan C. Mayo, Dr. Gerardo Escobar, Dr. Vicente Rico (Tecnol´ogico de Celaya) and Dr. H´ector Puebla (Universidad Aut´onoma Metropolitana) for their valuable comments and feedback during the defense of this thesis.

This work was also greatly benefited by the Project 266632 “Laboratorio binacional para la gesti´on inteligente de la sustentabilidad energ´etica y la formaci´on tecnol´ogica” (“Bi-National Laboratory on Smart Sustainable Energy Management and Technology Training”) funded by the CONACYT SENER Fund for Energy Sustainability (Agreement: S0019-2014-01).

I am grateful to my parents for their support and encouragement. I especially thank Tania (Bere)

for their love and patience during these four years.

Studies on optimal design, operation and control of process and energy systems

by

Oswaldo Andr´es Mart´ınez

Abstract

The quest for more sustainable process and energy systems requires optimal design, operation and control solutions at different temporal and spatial scales. Greenhouse gas emission and waste reduc- tion, and renewable energy sources integration are some ongoing concerns in chemical and power generation plants. These goals should be achieved while maximizing profits and meeting technical and environmental restrictions. Process systems engineering provides reliable computational and mathematical tools to address many of these emerging issues. In this regard, control and optimiza- tion techniques have the potential to handle sustainability related challenges in chemical engineering and other fields. Optimization can be applied at molecular, process and enterprise levels, while con- trol policies can be deployed along time intervals of milliseconds, minutes, hours and days.

In this work, studies on optimal design, operation and control in the context of chemical process and energy systems are presented. Subjects such as recovery of waste energy, national power flow systems, reactor operating policies, and integration of renewable energy are discussed. Case studies of different nature and complexity are tackled by means of mathematical programming and optimal control methods. Particularly, optimal molecular design under uncertainty, singular optimal control of chemical processes, optimal power flow of the Mexican electricity system, and model predictive control of dc-dc boost converters. Results show the capability of the utilized methods to produce optimal solutions that can be implemented in real settings or in further theoretical and practical developments.

vi

Contents

Abstract vi

1 Introduction 1

1.1 Motivation

. . . . 1

1.2 Objective

. . . . 2

1.3 Outline

. . . . 2

2 Optimal Molecular Design of Low-Temperature Organic Fluids

under Uncertain Conditions 5

3 Modeling National Power Flow Systems Through the Energy Hub Approach 17 4 An Efficient Direct/Indirect Transcription Approach for Singular Optimal Control 32 5 An Indirect Approach for Singular Optimal Control Problems 42 6 Nonlinear Model Predictive Stabilization of DC-DC Boost Converters

with Constant Power Load 54

7 Concluding remarks 62

7.1 Contributions

. . . . 62

7.2 Conclusions

. . . . 62

7.3 Future work

. . . . 63

References 65

Curriculum Vitae 68

1 Introduction

Sustainability is an important and essential concern in chemical process and energy systems. This is due to the fact that industrial activity contributes to generation and emission of greenhouse gases (particularly CO

2

), and global energy consumption continues increasing [1,

2]. A conservative esti-

mate sets power consumption in 2050 around 30 TW as compared to about 17 TW today [3]. Conse- quently, novel techniques to operate and design chemical and power generation plants have emerged to mitigate negative environmental impact. One strategy is to reduce the energy demand of the chem- ical plant, e.g recovering waste energy, as well as CO

₂

emissions. Another alternative is to replace fossil fuels with renewable energy sources for power generation, such as solar, wind, or geothermal to name a few. Currently, they supply about 23.7% of the total world energy demand and many of them are still under development [4]. Sustainability has to be achieved while keeping quality of products, minimizing costs, maximizing profitability and meeting safety regulations. This leads to challenges in process operation, design and control [5]. For example, in order to achieve efficient energy and chemical conversion, the right molecule has to be designed or selected [6]; some large-scale energy systems can only be studied through a comprehensive modeling framework that integrates multi- ple energy carriers and conversion technologies [7]; the dynamic behavior that leads to maximum production or minimum energy consumption of a process system can be calculated by solving an optimal control problem [8]; the successful deployment of such operation policy needs an advanced process control scheme so as to attain the predicted results [9]. Therefore, research in engineering and science towards sustainable process and energy systems has become a major topic in industry and academy from molecular to enterprise scales.

1.1 Motivation

Due to its holistic view, process systems engineering (PSE) has become a powerful toolbox to pur- suit sustainability no only within chemical engineering but also other domains such as energy sys- tems [10]. The tools provided by PSE can be applied to assist the decision making procedure at different temporal and spatial scales [11] as shown in Figure

1. The results can be translated in

maximum yield and profits, minimum waste and variability, and in general a more sustainable and efficient way to operate and design process systems. Optimization and control techniques, which are in the core of PSE, have been successfully applied to the renewable energy field. This practice is ex- pected to continue and concentrate on specific energy sources [12,

13]. However, there are still plenty

challenges in the context of optimization and control introduced by sustainability concerns, such as system instability, uncertain conditions, inaccurate computation, and poor modeling. Therefore, it is necessary to improve the methodologies to solve problems at different scales. This can be done for example by including uncertainties, economics and environmental factors, as well as applying more rigorous mathematical principles and sophisticated computational techniques [14].

1

pm nm cm m km Mm ps

ns s min

h d w

Space Time

Figure 1: Spatial and temporal scope of PSE.

1.2 Objective

The aim of this work is to present some control and optimization studies within the framework of chemical process and energy systems. Instead of focusing on a specific case, problems of different nature were solved so as to demonstrate a portion of the wide PSE scope depicted in Figure

1. Case

studies at molecular, unit process and nation-wide scale have been addressed, such as molecular design, optimal and model predictive control (MPC), and national optimal power flow. Therefore, different mathematical formulations and solution strategies are applied. These are nonlinear pro- graming (NLP), mixed-integer NLP (MINLP), multiparametric NLP (mp-NLP), and mathematical programs with complementarity constraints (MPCC). The results of this work show the capability of these instruments to solve challenging problems motivated by sustainable goals. Moreover, some solutions have the potential to be applied in real settings or they can be improved through further development. The contributions are presented as five papers published in different journals. Then, current and future work is discussed.

1.3 Outline

In this section, a brief summary of each contribution is presented in the order in which they appear

throughout this thesis. The problems under study as well as the main strategies to solve them and

most relevant results are pointed out. Additional details and supporting information can be found

in the source of each publication.

Optimal Molecular Design of Low-Temperature Organic Fluids under Uncertain Conditions A

computer aided molecular design (CAMD) problem formulated as an MINLP is addressed in this paper [15]. CAMD is applied to design or find optimal molecular structures to be used as work- ing fluids for recovery of energy at low temperature. An essential part of the formulation is the group contribution (GC) based method for estimation of thermo-physical properties. However, it is well known these methods carry uncertainties that could propagate throughout the molecular design procedure. Monte Carlo simulations with Latin hypercube sampling (LHS) were carried out to assess the influence of uncertainty on the properties of some molecules. Then, a set-based robust version of the MINLP was formulated by handling key thermo-physical properties in order to generate new structures subject to uncertain GC parameters. Results showed that even small uncertainties in group contribution parameters produce significant variations in the properties of the compounds analyzed.

The robust MINLP led to more conservative molecular structures that are partly immune against uncertainties but with less interesting properties.

Modeling National Power Flow Systems Through the Energy Hub Approach The concept of En-

ergy Hub (EH) to formulate an optimal power flow model of the Mexican electricity system is ap- plied in this paper [16]. The national transmission network of the Mexican electricity system is going through modernization. This includes transitioning to a more sustainable energy system with more participation of clean technologies such as hydro, wind, geothermal, solar and nuclear power. Con- sequently, large scale analysis frameworks are needed to assist in optimization studies of the energy system at a national level. The model consists of coupling matrices and vectors describing the inter- action between energy carriers and converters as well as clean technologies participation (renewable sources and nuclear power). It also takes into account the internal transmission and cross border power flows. The EH is cast as an NLP problem with an economic objective function. A case study with real data was carried out. Results provide optimal flows of energy carriers into, within and from the country to satisfy the national demand, while minimizing the costs associated with fuel consumption, cross border power flows and CO

₂

emissions.

An Efficient Direct/Indirect Transcription Approach for Singular Optimal Control In this paper,

optimal control problems with singular solution are addressed [17]. Optimal control policies to op- erate process and energy systems can be obtained through the solution of dynamic optimization problems. There are mainly two approaches, indirect and direct methods. In the first one, optimality conditions are derived and then the resulting two point boundary value problem (BVP) is solved. In the second method, the problem is transformed into an NLP and then solved with numerical solvers.

However, both methods may fail when the control appears linearly in the formulation as this gives rise to an ill-conditioned problem. In this case, the optimal control profile takes values along one or more singular arcs (between its upper and lower bounds) for a certain time. Direct and indirect methods were combined systematically along with a regularization scheme to solve a set of bench- mark problems. A sequential of steps is proposed to obtain solutions in very short CPU times with reasonable accuracy. Results reveal the importance of satisfying the optimality conditions.

3

An Indirect Approach for Singular Optimal Control Problems Motivated by the results of the

previous contribution, an alternative strategy is presented in this paper [18]. A Hamiltonian system based on the optimality conditions was developed. The resulting problem is a two point BVP with complementarity constraints. Therefore, some techniques applied to MPCC problems were used to solve the system with high accuracy and relatively short computational time. Case studies such as catalyst mixing, fed-batch and continuous stirred-tank reactors, and bioeconomic fishing are ad- dressed. Results show that this strategy can deal with highly ill-conditioned problems with one, two, and even three controls. Solutions quality and accuracy are in general better than those reported in the literature.

Nonlinear Model Predictive Stabilization of DC-DC Boost Converters with Constant Power Loads

A collaboration with the power electronics community is presented in this paper [19]. DC-DC con- verters play an important roll in the integration of renewable energy sources into the micro grids.

However, in this setting, the converters act as constant power loads (CPL), which introduces a desta-

bilizing effect due to a negative impedance characteristic. Model predictive control (MPC) is an

appealing technique to mitigate this type of instability as long as the controller reacts fast enough (in

milliseconds). An explicit MPC was developed in this work by solving several mp-NLP problems

previous to the implementation. The controller consists of a partition of the state space (voltage and

current) and a control law associated to each one. The implementation was done by means of the

hash table concept to handle the time-storage complexity. The controller was tested on simulations

and experimental studies. Results show that this explicit MPC is able to drive and keep the voltage at

the desired level mitigating the CPL effects. Its performance was no diminished even under abrupt

changes on the input voltage and the load.

Optimal Molecular Design of Low-Temperature Organic Fluids under Uncertain Conditions

Oswaldo Andrés-Martínez and Antonio Flores-Tlacuahuac*

Escuela de Ingenieria y Ciencias, Tecnologico de Monterrey, Campus Monterrey, N.L. 64849 Monterrey, Mexico

*

^S Supporting Information

ABSTRACT: Computer-aided molecular design as a mixed-integer nonlinear programming problem under uncertainty in group contri- bution parameters has been addressed. A set of new low-temperature organic compounds, for heat recovery purposes, was obtained by solving the mixed-integer nonlinear programming problem with nominal values from a previous work. Monte Carlo simulations with Latin hypercube sampling were carried out to asses the effect of uncertain group contributions on thermo-physical properties. Further- more, a set-based robust counterpart was formulated by taking into account uncertainty only in linear constraints. The results show that even small uncertainty in group contribution parameters can lead to significant variations in thermo-physical properties of the compounds analyzed. Therefore, it is necessary to consider uncertainty in the

Computer-aided molecular design formulation. Solutions of the robust counterpart became more conservative as the uncertainty set size increased, producing organic compounds different from the nominal case.

1. INTRODUCTION

There exists a huge interest in replacing traditional energy sources (i.e., fossil fuels) because of their negative impact on the environment¹and theirfinite availability.²Furthermore, global energy consumption continues increasing as the population and economy grow.³ Moreover, it is important to reduce the gen- eration and emission of greenhouse gases (particularly CO₂) so that industrial activities do not contribute remarkably to climate change.⁴There are well-known renewable and sustainable energy sources that are feasible alternatives to fossil fuels. Examples of these sources are biofuel, biomass, geothermal, hydropower, solar, wind power, and tidal power. Currently, they supply about 23.7% of the total world energy demand and many of them are still under development.⁵As a result, research in favor of the renewables sector is an ongoing concern these days.

In addition to using these alternatives, recovery of energy from waste heat is an appealing way to maximize energy usage and reduce the negative environmental impact. This energy is dis- charged due to its low temperature and is considered useless;

hence it is essential not only to address energy recovery but also to increase conversion efficiencies to reuse it effectively.

The organic Rankine cycle (ORC) is an example of a mature technology for waste heat recovery. An ORC is essentially a Rankine cycle that uses an organic working fluid instead of water.⁶ It is usually applied for conversion of geothermal and biomass energy into electricity.^7,8 However, the application in waste energy recovery has gained some interest. Heat can be recovered at much lower temperature due to the lower boiling point of a properly selected organic workingfluid. Lecompte et al.⁹ analyze advantages and disadvantages of different architectures

in which the ORC can be configured for this application.

Stijepovic et al.¹⁰explored the relationships between working fluid properties and ORC common economic and thermody- namic performance criteria; they showed that the performance of ORC systems strongly depends on workingfluid properties.

When selecting the most appropriate working fluid, Quoilin et al.⁷recommend considering a maximum temperature of the workingfluid for an optimal ORC. Other aspects to be looked at are environmental (e.g ozone depletion potential and global warming potential) and safety (e.gflammability and auto ignition) issues.¹¹Therefore, it is important tofind the best compound(s), either existing or new, that make energy recovery more efficient and environmental friendly.

Molecular design is an approach commonly utilized to obtain new compounds or molecules with desirable properties. Computer- aided molecular design (CAMD) is a technique that has been efficiently adopted for designing optimal molecular structures for different applications.¹²Group contribution (GC) methods are used as quantitative structure−property relationships within the CAMD problem in order to estimate molecular properties from a determined set of functional groups.¹³ These methods take information about the molecular structure, that is, the fre- quency and type of a functional group appearing in the molecule.

One of the GC methods employed in CAMD was proposed by Joback and Reid,¹⁴ which is an extension of the Lydersen Received: January 20, 2018

Revised: March 22, 2018 Accepted: March 26, 2018 Published: March 26, 2018

pubs.acs.org/IECR Cite This:Ind. Eng. Chem. Res. 2018, 57, 5058−5069

© 2018 American Chemical Society 5058 DOI:10.1021/acs.iecr.8b00302

method.¹⁵They applied linear regression techniques to deter- mine the group contributions for a set of functional groups covering a wide variety of organic compounds and assuming no interaction between groups. Accuracy and applicability of GC models have been improved over the years.¹⁶⁻²¹

The CAMD problem is usually cast as an optimization prob- lem, which consists of inequality and equality constraints that ensure structural feasibility and take into account thermody- namical properties and an objective function modeling a goal to be optimized.²²⁻²⁵This principle has been applied for designing optimal molecular structures to be used as organic working fluids in ORC. In this approach the CAMD problem is usually formulated as a mixed-integer nonlinear programming (MINLP) problem where binary variables determine the structure of the new molecules with desired properties.^26,27 There are some studies tackling the problem with multiobjective optimization algorithms to optimize both process performance and working fluid structure.^28,29Optimal mixtures of organic workingfluids have been also investigated by using MINLP and multiobjective techniques.³⁰⁻³²

Estimated parameters (group contributions) in GC methods are subject to uncertainty, leading to uncertain predicted prop- erties. As a result, GC methods provide only estimates, and depending on the particular application the inaccuracy of the results can be significant and vary according to the property being calculated.³³ Hence, assessing uncertainty of both estimated parameters and predicted properties is an important key to obtain robust solutions. When uncertainty information is not reported, it is common to quantify it by regression techniques applied to experimental data and determine the type of probability distri- bution as well as mean and variance of the uncertainty.³⁴⁻³⁸ Once uncertainty information is obtained or found in the liter- ature, a common strategy is propagating the uncertainty through the model with an appropriate sampling method and evaluating the output with statistical techniques.^39,40Reed and Whiting⁴¹ used Monte Carlo simulations with Latin Hypercube Sampling (LHS) to propagate the influence of the uncertainty in thermo- dynamic data on the performance of a binary distillation model.

They represented the results as empirical cumulative distribution functions (CDF) and determined the individual effects calcu- lating partial correlation coefficients (PCC). A similar approach was followed by Whiting et al.⁴²applied to two distillation models.

Frutiger et al.⁴³carried out MC simulations with LHS to prop- agate the influence of the input uncertainty of fluid properties on an ORC model and calculated a 95% confidence interval for the outputs. Hajipour and Satyro⁴⁴applied a traditional random MC sampling to propagate the uncertainty of thermo-physical properties through the Peng−Robinson equation of state. They showed that relatively small sample sizes on the order of 100 ran- domly distributed inputs may be adequate to provide reasonable uncertainty estimates for values calculated from complex models.

In the present work we apply MC simulations with LHS to propagate uncertainty in group contribution parameters through the GC method and display the effect on the output for different thermodynamic properties. Besides uncertainty quantification, the impact of uncertain parameters on the solution of the underlying optimization problem should be studied. CAMD problems with data uncertainty can be tackled with two major approaches from thefield of optimization under uncertainty, namely, stochastic optimization (based on random sampling) and robust optimi- zation (RO). Thefirst one is applied when uncertain inputs are modeled with known or assumed probability distributions.

Santos-Rodriguez et al.⁴⁵ formulated a stochastic nonlinear

optimization problem to address uncertainty in ORC variables (heat source temperature and turbine efficiency) for organic fluid mixture design. They showed that it is possible to get compo- sitions that keep certain workingfluid performance under dif- ferent scenarios. Conversely, the second approach is useful when no information about probability distribution is available and uncertain parameters are allowed to take any realization in a uncertainty set.⁴⁶ Only a few works have focused on uncer- tainties in GC parameters for CAMD applications. Maranas⁴⁷ formulated a nonlinear stochastic version of the CAMD problem formulated in ref22utilizing probability density distributions to describe the likelihood of different realizations of group contri- bution parameters. The stochastic problem was transformed into a deterministic MINLP that allowed them tofigure out the effect that property prediction uncertainty may have on optimal molecular design. Kim and Diwekar³⁶applied a stochastic optimi- zation framework based on random sampling for solvent selection.

To our best knowledge, uncertainty in group contribution para- meters for CAMD problems featuring organic working fluids synthesis has not been addressed. Moreover, CAMD problems in general have not yet been addressed within a RO framework.

The robust optimization concept we adopt in this work is based on the approach introduced by Soyster,⁴⁸who re-formulated a linear programming (LP) problem so as to obtain a robust counterpart (RC), whose solutions would be feasible under all possible perturbations in data. Other definitions for RO and examples of applications are presented by Beyer and Sendhoff.⁴⁹Singh⁵⁰extended Soyster’s work to fractional program- ming problems. Ben-Tal and Nemirovski⁵¹ reduced the conser- vatism of Soyster’s solutions by proposing an ellipsoidal-set- based RC. Ben-Tal and Nemirovski⁵²and El-Ghaoui et al.^53,54 developed a similar approach to deal with parameter uncer- tainty within semidefinite and quadratic problems. They found that these robust counterparts are computationally tractable.

Ben-Tal and Nemirovski⁵⁵suggested a methodology to treat data contaminated with uncertainty either unknown-but-bounded or random symmetric in LP problems. They formulated a RC based on the probability of constraint violation and infeasibility toler- ance. This methodology was extended to mixed-integer linear programming (MILP) problems by Lin et al.⁵⁶Bertsimas and Sim⁵⁷introduced a budget parameter to control the degree of conservatism of previous formulations. Lin et al.⁵⁶ developed RCs on the basis of different shapes and size of uncertainty sets for both LP and MILP problems. RO has been applied in engi- neering problems mostly in planning and scheduling studies, but it has been also applied in structural design, circuit design, power control in wireless channels, antenna design, linear- quadratic control, and many others.⁵⁸⁻⁶⁰However, RO theory for general nonlinear programming (NLP) problems is not as well established as for the linear case. Zhang⁶¹proposes a for- mulation for NLP problems that is valid in the neighborhood of the nominal values. Hale and Zhang⁶²apply this RC to the design of a heat exchanger network and reactor−separator system. More recent developments in RO can be found in the review by Gabrel et al.⁶³In this work, we use techniques of linear robust opti- mization that can be applied at constraint level to obtain a RC of the original MINLP problem.

This paper is organized as follows: Section 2 presents the original CAMD technique as a MINLP problem to generate a family of new organic compounds, the strategy to carry out MC simulations with LHS for uncertainty analysis, and the RC formulation. InSection 3we discuss the results obtained from the MINLP with nominal values, the uncertainty analysis, and

solutions of the RC for different cases. Finally, inSection 4we list our main conclusions about this work.

2. METHODOLOGY

Palma-Flores et al.²⁶applied CAMD to obtain a new family of organicfluids for energy recovery from low-temperature energy sources. They started from a basis set of 39 functional groups and solved several MINLP problems where binary variables determine the structure of the new compounds with desired properties. The GC method used was that of Joback and Reid.¹⁴ We use the same formulation with a reduced basis set of func- tional groups to get a family of 12 organic compounds. For each of the compounds obtained with nominal values, a MC simulation with LHS was implemented in order to evaluate the impact of uncertainty in group contribution parameters on ther- modynamic properties. Then, we formulate a RC of the original CAMD problem considering uncertainty only in the linear constraints. Finally, the robust solutions are compared with the base nominal cases corresponding to the four addressed objec- tive functions considered in ref26.

2.1. CAMD Nominal Case. Following the same approach of Palma-Flores et al.,²⁶a version of the CAMD problem was written with a basis set of 14 functional groups including only those appearing more frequently in the 32 molecules obtained by Palma-Flores et al.²⁶These functional groups are CH3, CH2, CH, C, F, Cl, OH, O, C, COO, NH2, NH, N, and S. The optimization problem has the following form:

Φ x y max/min ( , )

x y, (1a)

subject to

≤Ay≤b

0 (1b)

=

H y x a( , , ) 0 (1c)

≤ ≤

∈ ∈

x x x y {0,1}, x Rⁿ

L U

(1d) whereΦ is one of the four objective functions studied in ref26.

Constraints 1b define the structural feasibility of the organic fluid. Constraints1cset the GC method used to estimate the physical and thermodynamic properties represented by the con- tinuous decision variables x. Upper and lower limits denoted by U and L, respectively, are imposed on x. Binary variables y determine the molecular structure of the organic compound.

Finally, a are the GC parameters. The complete MINLP prob- lem formulation is provided in theSupporting Information.

The four objective functions are

• Maximization of enthalpy of vaporization ΔH^lv, Φ = ΔH

max ₁ ^lv (2)

• Maximization of the ratio between enthalpy of vapor- izationΔH^lvand liquid heat capacity Cpl,

Φ = ΔH max C

p 2

lv

l (3)

• Maximization of weighted sum of properties,

Φ = ΔH − C − ΔG

max ₃ 0.5 ^lv 0.1 _p^l 0.4 ₂₉₈^f ₍₄₎ whereΔG298f is the Gibbs energy of formation at 298 K.

• Minimization of Cpl

, Φ = C

min _{4 p}^l ₍₅₎

We tested the four objective functions in the original prob- lem (eqs 2−5) and varied the minimum (nmin) and maximum (nmax) number of functional groups allowed to form the compound. For example, if one resulting molecule consists of four (ten) functional groups, the same program is run with nmin= 5 (nmax= 9). Hence, it is possible to obtain molecules of different sizes for each objective function. A total of 12 MINLP problems (three for each objective function) were solved with nominal values of group contributions. As Tbis a key property for practical purposes, the upper bound Tb≤ 373.15 K should be kept in mind for subsequent results discussion. We also avoided undesirable attachments like−O−OH and −COO−O−, as suggested by Palma-Flores et al.²⁶ The optimization prob- lems were implemented in GAMS (version 24.2.3)⁶⁴ with Dicopt as MINLP solver.⁶⁵

2.2. Uncertainty Analysis. As pointed out by Maranas,²² contribution of molecular groups for a given property is not neces- sarily unchanged from one molecule to another but varies slightly around some nominal value. This behavior can be expressed by assuming that group contribution parameters follow a normal distribution with mean μ and standard deviation σ. With the information reported by Joback⁶⁶for each property, we take the nominal group contribution value as mean μ and, since stan- dard deviation is not reported, we take the average of the absolute error associated with each GC model (Table 1) as a reference for settingσ. To investigate the effect of uncertainty in group contribution parameters on predicted properties, MC simulations with LHS were performed to propagate the input uncertainty through the GC method model of Joback and Reid.¹⁴ The corresponding equations for properties inTable 1areeqs S21, S22, S23, S25, S27, S28, S29, S30, and S31from the Supporting Information. MC simulation involves three steps: (1) specifying input uncertainty (by a distribution), (2) sampling input uncer- tainty, and (3) propagating the sample through the model. LHS displays properties between random sampling, which involves no stratification, and stratified sampling, which stratifies all the sampling space.⁶⁷As illustrated inFigure 1, the procedure is the

inverse to CAMD. We start with afixed molecule, then for a given property we specify μ and σ so that LHS generates N samples and passes them to the GC method model. The results are depicted in empirical CDFs, scatter plots, and PCCs.

Table 1. Uncertainty Associated with Each Property Model

property

Tb Tf Tc Pc Vc ΔH^f ΔG^f ΔH^lv Cpg

av % error 3.6 11.2 0.8 5.2 2.27 9.2 15.7 3.88 1.4

Figure 1.Procedure for MC simulations with LHS.

DOI:10.1021/acs.iecr.8b00302 5060

The MC simulations were programmed in R software (version 3.4.0)⁶⁸with the parameter space exploration (pse) package⁶⁹ with N = 200 and 50 bootstrap replicates for calculating the PCCs.

2.3. Robust Formulation of Uncertain Linear Constraints.

The RO concept adopted in this part is based on the worst case formulation introduced by Soyster.⁴⁸ Two approaches are fol- lowed, namely those in Lin et al.⁵⁶and Li et al.⁷⁰They obtained RCs for linear and mixed-integer linear optimization problems.

Their formulations can be applied at the (linear) constraint level. However, a general RO theory for the mixed-integer nonlinear case has not been developed yet. Accordingly, we treat only the linear constraints in the MINLP problem as a new way to address uncertainty in this framework.

The original MINLP problem can be rewritten including uncertainty in linear equalities G of GC model (eqs S23, S25, S28, S29, S30, and S31 in the Supporting Information) and keeping nominal values in nonlinear equalities F as follows

Φ x y

min/max ( , ) (6a)

subject to

≤Ay≤b

0 (6b)

=

F y x a( , , ) 0 (6c)

̃ =

G y x( , ,a) p (6d)

≤ ≤

∈ ∈

x x x

y {0,1}, x Rⁿ

L U

(6e) where p are the constants appearing in each linear GC equation, ã are the values of coefficients subject to uncertainty, and a are the nominal values. The jth linear constraint in (6d) with uncertain coefficients has the general form

∑ ∑

− ̃ =

= =

x_j y a p

i n

k m

ik jk j

1 1

max

(7) where xjis one of the continuous variables Tb, Tf, Vc,ΔG^f,ΔH^f, ΔH^lv; ãjkis the group contribution k subject to uncertainty, that is, ãjk= ajk+ âjkξjk; ajkis the nominal value; âjk=τjajkrepresents perturbation around the nominal value given byτj > 0, whose values are the errors inTable 1; andξjk are random variables distributed in the interval [−1, +1]. Thus, when ξjk=−1, ãjk= ajk− τjajk; whenξjk= 1, ãjk= ajk+τjajk; and whenξjk= 0, ãjk= ajk (that is, the uncertain parameter is at its nominal value).

As an example, the Tbconstraint in the form of (7) becomes

∑ ∑

− ̃ =

= =

T y T 198.2

i n

k m

ik k b

1 1 b

max

(8) with T̃bk= Tbk+ T̂bkξkand T̂bk=τTbk. The goal is to formulate a RC of (6) able tofind solutions that remain feasible for any realization ofξ in a given uncertainty set. According to Lin et al.,⁵⁶ for each inequality constraint involving uncertain coefficients, an additional constraint is introduced to incorporate the uncer- tainty and maintain the relationships, in this case the quantita- tive structure−property relationships (GC method), among the relevant variables under a given infeasibility tolerance. Assuming that uncertain coefficients ãjk are randomly and symmetrically distributed around the nominal values, two approaches can be followed to derive a RC of (7).

The first approach is proposed by Lin et al.,⁵⁶ which is an extension of Ben-Tal and Nemirovski⁵⁵ to mixed-integer problems. A solution (x, y) is robust if it satisfies the following conditions:

(i) (x, y) is feasible for the nominal problem

(ii) For the jth inequality, the probability of the constraint violation is at mostκ > 0, with an infeasibility tolerance δ > 0, where κ is a reliability level

Because the equality constraint in (7) is equivalent to the pair of inequalities⁷¹

∑ ∑

− ̃ ≤

= =

x

n m

y a p

j

i k

ik jk j

1 max

1 (9a)

∑ ∑

− + ̃ ≤ −

= =

x

n m

y a p

j

i k

ik jk j

1 max

1 (9b)

the RCs of (9a) and (9b) based on Lin et al.⁵⁶(Theorem 2) are respectively

∑ ∑ ∑ ∑

δ

− + Ω ̂

≤ + | |

= = = =

x

n m

y a

n m

y a

p max{1,p}

j

i k ik jk j

i k

ik jk

j j j

1 max

1 1

max

1 2

(10a)

∑ ∑ ∑ ∑

δ

− + + Ω ̂

≤ − + | |

= = = =

x

n m

y a

n m

y a

p max{1,p}

j

i k ik jk j

i k

ik jk

j j j

1 max

1 1

max

1 2

(10b) whereΩjis a positive parameter and (x, y) is a robust solution that satisfies (i) and (ii) with κj= exp(−Ωj2/2).

The second approach comes from Li et al.⁷⁰Replacing ãjkin terms ofξjkin (9a) and (9b) and grouping yield a worst-case formulation:

∑ ∑ ∑ ∑

^ξ

− + ̂ ≤

= = ξ∈Ξ^⎪ = = ^⎪

⎪ ⎪

⎧⎨

⎩

⎫⎬ x ⎭

n m

y a

n m

y a p

j max

i k

ik jk

i k

ik jk jk j

1 max

1 1

max

j 1 (11a)

∑ ∑ ∑ ∑

^ξ

− + + ̂ ≤ −

= = ξ∈Ξ^⎪ = = ^⎪

⎪ ⎪

⎧⎨

⎩

⎫⎬ x ⎭

n m

y a

n m

y a p

j max

i k

ik jk

i k

ik jk jk j

1 max

1 1

max

j 1

(11b) There are many options to choose a setΞ where ξjkcan take any realization. However, as we have assumed ãjkare randomly and symmetrically distributed around the nominal values, the appropriate set to be chosen is an ellipsoidal one, since a box would be too pessimistic.^52,72An ellipsoidal set is defined with 2-norm as follows:

Ξ = |{ξ ξ ₂ ≤ Ψ} (12)

whereΨ is the adjustable parameter controlling the size of the uncertainty set. Thus, the RCs of (11a) and (11b) induced by the setΞ are

∑ ∑ ∑ ∑

δ

− + Ψ ̂

≤ + | |

= = = =

x

n m

y a

n m

y a

p max{1,p}

j

i k ik jk j

i k

ik jk

j j j

1 max

1 1

max

1 2

(13a)

∑ ∑ ∑ ∑

δ

− + + Ψ ̂

≤ − + | |

= = = =

x

n m

y a

n m

y a

p max{1,p}

j

i k ik jk j

i k

ik jk

j j j

1 max

1 1

max

1 2

(13b) where an infeasibility tolerance δj has been added. Note that whenΩj=Ψj, both formulations (10) and (13) are equivalent.

This is due to the fact that in the original problem uncertain coefficients take place only in the binary variables; otherwise, the RCs would have been slightly different.

In the following, we take (10) and (13) as the same formu- lation with parameterΩj. Consequently, the complete RC of the CAMD problem (6) is

Φ x y max /min ( , )

x y, (14a)

subject to

≤Ay≤b

0 (14b)

=

F y x a( , , ) 0 (14c)

=

G y x a( , , ) p (14d)

∑ ∑ ∑ ∑

δ

− + Ω ̂

≤ + | | ∀

= = = =

x

n m

y a

n m

y a

p max{1,p} j

j

i k ik jk j

i k

ik jk

j j j

1 max

1 1

max

1 2

(14e)

∑ ∑ ∑ ∑

δ

− + + Ω ̂

≤ − + | | ∀

= = = =

x

n m

y a

n m

y a

p max{1,p} j

j

i k ik jk j

i k

ik jk

j j j

1 max

1 1

max

1 2

(14f)

≤ ≤

∈ ∈

x x x

y {0,1}, x Rⁿ

L U

(14g) The inclusion of constraints (14e)−(14f) means incorporating the worst case values âjkinto the MINLP problem. Hence, these constraints are the most difficult to maintain, so the parameter δ allows a tolerable violation. As a consequence, binary variables ywill now search for an optimal combination of functional groups whose contributions satisfy the GC models (eqs 14c−14d) while taking into account uncertainty through constraints (14e)−(14f).

As an example, consider the GC method constraint for Tb(8).

The additional constraints are

∑ ∑ ∑ ∑

δ

− + Ω ̂

≤ +

= = = =

T

n m

y T

n m

y T

198.2(1 )

i k ik bk T

i k

ik bk

T b

1 max

1 1

max

1 2

b

b (15)

∑ ∑ ∑ ∑

δ

− + + Ω ̂

≤ −

= = = =

T

n m

y T

n m

y T

198.2( 1)

i k ik k T

i k

ik k

T b

1 max

1 b

1 max

1 b

2

b

b (16)

The RC of the CAMD problem was solved for each objective function withκ = 10% (Ωj= 2.1459),δj= 5% for all j except for ΔG^f, Vc, andΔH^fwhereΩ_ΔG^f=ΩVc=Ω_ΔH^f= 0.1Ωj. Solutions of the nominal case were taken as initial values for the respective

uncertain optimization cases keeping the same bounds. In this part, we are not interested in varying the minimum and maxi- mum number of functional groups allowed to form the compound.

Therefore, we take only the base nominal case for each objective function as a reference for the RC. In addition, for each objective function the problem (14) was solved for different values of Ωj

and δj = 5%, and finally a sensitivity analysis with respect to δ was done so as to observe its effect on the robust solution.

3. RESULTS AND DISCUSSION

3.1. CAMD Nominal Case. Table 2lists the compounds obtained with nominal values in group contribution parameters

for each run. The objective values along with the boiling temper- ature are also shown. Solutions1,4,7, and10are the base cases, i.e., the optimal solutions found by the MINLP problem without any additional restrictions. Solutions 2and 3are the result of varying the minimum and maximum number of func- tional groups allowed to be part of the molecular structure of base case 1. Solutions5and6are the result of such a variation in base case4. Similarly, solutions11and12come from the same variation in base case10. When limitations on the number of functional groups are imposed, the value of the objective function decreases forΦ1,Φ2, andΦ3but increases forΦ4since it is a minimization problem. In compounds 1, 2, and 3 one attachment N(OH) is part of the molecular structure. In com- pounds 4, 5 and 6, functional group Cl appears twice. Func- tional group COO appears only in compounds associated with objective function Φ3 and takes place just once in them.

Functional group F is present in 9 out of 12 compounds. From Table 2, only compounds 1 (CH3−O−N(OH)−CH3) and 4 (Cl−O−S−Cl) are reported by Palma-Flores et al.²⁶ Com- pounds 10 and 11 are found in the literature as chlorine per- oxide and 1,1-dichloro-1-fluoroethane, respectively. The highest Tbbelongs to compound 1 and the lowest to compound 11.

All the compounds inTable 2 are candidates to be utilized as workingfluids for heat recovery at low temperature, since their boiling temperatures are below that of water.

3.2. Uncertainty Analysis. Uncertainty analysis was per- formed for all compounds obtained with the nominal case of the CAMD problem (Table 2) and all the properties inTable 1.

The empirical CDF (Figures 2a and3a) is basically a step func- tion that jumps up 1/N at each data point. It estimates the under- lying CDF generated by the points in the sample. As an example, consider the compound CH3−O−N(OH)−CH3.Figure 2shows the results of MC simulations for Tb. It can be observed in the empirical CDF (Figure 2a) that for the given variation in group Table 2. Compounds Obtained with Nominal Values

ID compound

objective function and

optimal value Tb(K)

1 CH3−O−N(OH)−CH3 Φ1= 41.178 372.40

2 F−O−O−N(OH)−CH3 Φ1= 40.545 371.20

3 F−N(OH)−CH2−O−O−F Φ1= 39.728 370.48

4 Cl−O−S−Cl Φ2= 0.2765 365.66

5 F−S−N(Cl)−Cl Φ2= 0.2313 354.95

6 F−C(Cl,F)−S−Cl Φ2= 0.2010 361.43

7 F−COO−C(O−F)3 Φ3= 565.02 364.69

8 F−COO−NH−C(F)2−O−F Φ3= 446.77 370.02 9 F−COO−C(F)2−S−F Φ3= 429.44 366.21

10 Cl−O−O−Cl Φ4= 115.49 319.30

11 CH₃−C(Cl)2−F Φ4= 133.70 316.26

12 F−S−C(F)2−Cl Φ4= 142.04 323.27

DOI:10.1021/acs.iecr.8b00302 5062

contributions, Tbvaries from 362 to 381 K, which can represent an important uncertainty for CAMD purposes as Tbis required to be less than or equal to 373.15 K. The histogram drawn in the samefigure provides a sight of this variation. When 200 model outputs are averaged, the average Tbis 372.4 K, which is the Tb

calculated with nominal values of GC parameters. The PCC (Figures 2b and3b) measures the strength of the linear associ- ation between the output and each GC after removing the effect of the others. This relation can be visualized in the scatter plots (Figures 2c and3c) that show the distribution of values returned by the GC model and how sensible these responses are with respect to variations in the values (x-axes) of each GC parameter. As shown in Figure 2b, the OH group has the strongest effect on Tbfor this particular molecular architecture when the effect of the other groups is removed, whereas the N group has the weakest effect. The points in the scatter plot (Figure 2c) T_bversus OH contribution lie near the straight line.

Uncertainty analysis for Pcis depicted inFigure 3. In this case the corresponding GC method is nonlinear. The empirical

CDF along with the histogram in Figure 3a suggests little variation in Pcwith respect to uncertainty in group contribution parameters, namely, 54.20 to 57.82 bar. PCCs in Figure 3b show that N and OH groups have a strong linear effect on Pc

being OH contribution slightly stronger. In this case, the O group has the weakest effect. This can be viewed in the scatter plots of Figure 3c. The average Pc for the compound under analysis is 55.95 bar, same as calculated from nominal values. This graphical analysis for all the compounds and thermo- physical properties, as well as the R-file are available in the Supporting Information.

The resulting Tb of the MC simulations for the 12 com- pounds are depicted as box plots in Figure 4a. Each box is drawn by the first quartile, median (thick vertical line), and third quartile. The whiskers represent the values outside the box, but within 1.5 times the interquartile range. Extreme values being outside the box are represented as small circles. Variations in Tbare similar for all the compounds. Some box plots present extreme values on both sides. The box plot of compound 1 has Figure 2.Results of MC simulation with LHS for Tbof CH3−O−N(OH)−CH3: (a) eCDF and histogram; (b) PCCs; (c) scatter plots.

only one point outside. Compounds 8, 9, 10, and 12 have no extreme values.Figure 4b contains the box plots for Pc. In this case, the width is less than those of Tb, even though the error for Pcis higher. This can be due to the nonlinearity of its model.

Compound 1 has two extreme values, each at one side of the box. Compounds 4 and 10 have no points outside. The previous analysis provides information about the behavior of GC method under uncertain GC parameters. Some properties can be very sensitive to input uncertainty. Therefore, it is important to take into account uncertainty within the CAMD problem since the effect of uncertain parameters propagates through the GC method and influences the molecule properties. As a result, molecular structure is likely to be different for each realization of uncertain parameters.

3.3. Robust Counterpart. Results of the RC for each objective function withκ = 10% and δj= 5% are shown inTable 3.

The respective base nominal cases 1, 4, 7, and 10 fromTable 2 are also shown for comparison. It can be noticed that forΦ1, Φ2, and Φ3the RC leads to an objective value less than the

respective nominal case, with a more notable decrease forΦ3. ForΦ4, the objective value of the RC is greater because it is a minimization problem. Every robust versionfinds a well-known compound with a smaller boiling temperature, and each one is different from its nominal version. Every compound resulting from the RC is feasible for any realization of the uncertainty in the setΞ with a given value of Ωj. Thefirst RC gives rise to hexane and the remain ones to 1-chloropropane even though the associated objective functions are different. When nominal values are used, functional groups like F, N, S, O, and OH are part of the molecular structure of some compounds. However, compounds obtained with the RCs are made mainly of groups CH3and CH2.Tables 4,5,6, and7list the results of the RC for Φ1,Φ2,Φ3, andΦ4, respectively, with different values of Ωjand δj = 5%. As Ωj increases, objective values Φ1, Φ2, and Φ3

decrease but Φ4increases. In most cases, the boiling temper- ature decreases. WhenΩj≥ 1.5, alkane compounds start to be obtained and robust versions forΦ2,Φ3, andΦ4(Tables 5−7) behave similarly. ForΦ1(Table 4), whenΩj= 0.5, the objective Figure 3.Results of MC simulation with LHS for P_cof CH₃−O−N(OH)−CH3: (a) eCDF and histogram; (b) PCCs; (c) scatter plots.

DOI:10.1021/acs.iecr.8b00302 5064