Elsevier

Computers & Chemical Engineering

Volume 116, 4 August 2018, Pages 488-502
Computers & Chemical Engineering

Optimal design of energy systems using constrained grey-box multi-objective optimization

https://doi.org/10.1016/j.compchemeng.2018.02.017Get rights and content

Highlights

  • p-ARGONAUT is extended towards constrained multi-objective optimization problems.

  • Data-driven approach is followed to optimize an energy market design problem.

  • The accuracy and consistency of the method is evaluated under equality constraints.

  • Computational results are compared with a number of available software.

Abstract

The (global) optimization of energy systems, commonly characterized by high-fidelity and large-scale complex models, poses a formidable challenge partially due to the high noise and/or computational expense associated with the calculation of derivatives. This complexity is further amplified in the presence of multiple conflicting objectives, for which the goal is to generate trade-off compromise solutions, commonly known as Pareto-optimal solutions. We have previously introduced the p-ARGONAUT system, parallel AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems, which is designed to optimize general constrained single-objective grey-box problems by postulating accurate and tractable surrogate formulations for all unknown equations in a computationally efficient manner. In this work, we extend p-ARGONAUT towards multi-objective optimization problems and test the performance of the framework, both in terms of accuracy and consistency, under many equality constraints. Computational results are reported for a number of benchmark multi-objective problems and a case study of an energy market design problem for a commercial building, while the performance of the framework is compared with other derivative-free optimization solvers.

Introduction

Energy systems are characterized by a large and diverse number of components in which they form an integrated complex multi-scale network (Floudas et al., 2016). Mathematical models defining such complex systems generally include expensive finite-elements, a large number of partial differential equations (PDEs) and high-fidelity models. The global optimization of these systems using deterministic methods is often challenging since the calculation of the derivatives can be subject to high noise and/or computational expense. These characteristics are displayed by many systems in the fields of engineering and sciences, and commonly referred to as “grey-box” or “black-box” problems, where the system relies on expensive simulations, proprietary codes or input–output data (Boukouvala et al., 2016). A general constrained nonlinear grey-box problem is mathematically defined in Eq. (1):minxf(x)s.t.gm0m{1,,M}gk(x)0k{1,,K}xi[xiL,xiU]i=1,,nxRnwhere n represents the number of continuous decision variables with known lower and upper bounds [xL, xU] and set k{1,,K} represents the constraints with known closed-form equations. The mathematical expressions defining the objective, f(x), and the constraints, represented by set m{1,,M}, are not explicitly available as a function of the continuous decision variables. However, the values of these unknown formulations can be retrieved as outputs of the problem simulator, which is typically computationally expensive. In the problem of interest, “unknown” strictly refers to the mathematical expression of an equation (objective or constraints) in terms of x, and the cardinality of set m should be known a priori.

The complexity of such multi-scale models is further amplified in the presence of multiple competing objectives, such as economic and environmental objectives that are commonly associated with the optimal design of energy systems. In this case, it is not possible to locate a unique optimal solution since there are trade-offs between these conflicting objectives. This class of optimization problems are handled via multi-objective optimization (MOO), of which the goal is to find the best set of decisions that will simultaneously optimize multiple objectives in such a way that the solutions cannot be improved without degrading at least one of the other objectives (Miettinen, 1998). In other words, the goal of MOO is to derive a set of trade-off optimal solutions, known as the Pareto-optimal solutions, that the decision makers can choose from, depending on their preferences. The general form of MOO problems is presented in Eq. (2):minx[f1(x),f2(x),,fN(x)]s.t.xXwhere X is a non-empty feasible region, XRn.

While several methodologies exist in the open literature for MOO, we only consider the ones that are linked to population-based and surrogate-based algorithms. Meta-heuristic (population-based) algorithms are advantageous since they do not require any reformulations, such as converting the multi-objective problem into a set of single-objective sub-problems. These can simultaneously deal with a set of possible solutions without requiring series of separate runs, thus enabling the direct investigation of the multi-objective problem (Coello et al., 2007). As a result, population-based algorithms have been a popular choice among many researchers for the MOO of various systems, including truss design (Ray et al., 2001), thermal system design (Toffolo and Lazzaretto, 2002), environmental economic power dispatch (Gong et al., 2010, Wang and Singh, 2007), beam design (Sanchis et al., 2008), water distribution network design (di Pierro et al., 2009) and more recently the MOO of zeolite framework determination (Abdelkafi et al., 2017). In addition to these, the books by Rangaiah and Bonilla-Petriciolet (2013), and Coello et al. (2007) demonstrate a plethora of applications of evolutionary algorithms to numerous MOO problems.

Even though the population-based algorithms are widely studied in the open literature, their applications to grey/black-box problems are rather limited. There are two main reasons for this: (1) most existing algorithms consider the box-constrained problem or handle general constraints via penalty functions, where the system is being continuously treated as a black-box; (2) stochastic algorithms typically require a large number of function calls to reach the global optimality, which can be computationally prohibitive for expensive simulations. Several researchers have focused on hybrid implementations of surrogate modeling with stochastic algorithms to overcome such problems. Datta and Regis (2016) have proposed a surrogate-assisted evolution strategy, which makes use of cubic radial basis surrogate models to guide the evolution strategy for the optimization of multi-objective black-box functions that are subject to black-box inequality constraints. Likewise, Bhattacharjee et al. (2016) have used a well-known evolutionary algorithm, NSGA-II, as the baseline algorithm while using multiple local surrogates of different types to represent the objectives and the constraints.

Surrogate-based approaches, where the objectives and the grey/black-box constraints are approximated with simple tractable models, have also been investigated in the open literature in conjunction with derivative-free algorithms. Singh et al. (2014) have proposed the Efficient Constrained Multi-objective Optimization (ECMO) algorithm to solve computer-intensive constrained multi-objective problems using Kriging models for the objectives and the constraints. They make use of the hypervolume-based Probability of Improvement (PoI) criterion to handle multiple objectives along with the Probability of Feasibility (PoF) criterion to handle computationally expensive constraints and solve the final formulation using MATLAB's fmincon optimizer. Feliot et al. (2017) have used an expected hypervolume improvement sampling criterion in their Bayesian Multi-Objective Optimization (BMOO) framework, where the nonlinear implicit constraints and the black-box objectives are handled via extended domination rule. In this algorithm, the authors use sequential Monte Carlo sampling technique for the computation and optimization of the expected improvement criterion. Martínez-Frutos and Herrero-Pérez (2016) have introduced the Kriging-based Efficient Multi-Objective Constrained Optimization (KEMOCO) algorithm that uses a kriging-based infill sampling strategy with DIRECT algorithm for constrained MOO of expensive black-box simulations. They combine the expected hypervolume improvement and the PoF to obtain the Pareto-front with minimum number of samples. Regis (2016) has presented Multi-Objective Constrained Stochastic optimization using Response Surfaces (MOCS-RS) framework where the author uses radial basis surrogates as approximations for the objective and constraint functions. A more detailed overview on the existing methods for using surrogates in computationally expensive MOO can be found in Tabatabaei et al. (2015).

In this paper, we implement a hybrid methodology that performs global parameter estimation coupled with k-fold cross-validation for individualized surrogate model identification on each unknown formulation (objective and constraints). Specifically, we employ the parallel AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems (p-ARGONAUT) algorithm (Beykal et al., 2018, Boukouvala and Floudas, 2017, Boukouvala et al., 2017) to explore the effect of maximizing the connection between surrogate model identification and deterministic global optimization in MOO problems. Our framework is tailored to solve high-dimensional general constrained grey-box problems, which contains several features including, adaptive parallel sampling, variable screening, exploration and validation of the best surrogate formulation for each unknown equation, and global optimization of the proposed surrogate formulations for improved solutions. The parallel algorithm is tested on three constrained MOO benchmark problems as well as a more complex case study of an energy market design problem for a commercial building. By implementing the ɛ-constraint method to convert the MOO problem into series of single-objective optimization sub-problems, p-ARGONAUT is employed to globally optimize the resulting constrained grey-box problems.

The paper is structured as follows. Section 2 provides the details on our framework to globally optimize high-dimensional general constrained grey-box computational problems that benefits from high-performance computing for maximal computational efficiency. In Section 3.1, we provide an overview of the methodology we employed to convert MOO into a series of single-objective optimization sub-problems, namely the ɛ-constraint method. Moreover, in Section 3.2, we solve a motivating example to show the details on p-ARGONAUT iterations. Finally, the details of the computational studies are provided in Section 4 followed by the results and concluding remarks in Sections 5 and 6, respectively.

Section snippets

Parallel AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems (p-ARGONAUT)

p-ARGONAUT is an extension of the ARGONAUT algorithm (Boukouvala and Floudas, 2017, Boukouvala et al., 2017) which uses high-performance parallel computing to globally optimize high-dimensional general constrained grey-box problems. It incorporates the information provided in the form of input–output data with known equations to formulate a nonconvex nonlinear programming (NLP) problem, which is later solved to global optimality using deterministic methods. p-ARGONAUT starts with bounds

Multi-objective optimization using ɛ-constraint method

The ɛ-constraint method is introduced by Clark and Westerberg (1983) to convert multi-objective design problems into series of single-objective sub-problems. Consider an optimization problem given in the form of Eq. (2) with only 2 objectives, i.e. N=2. The main idea behind ɛ-constraint method is to discretize the objective space into smaller sections while obtaining the optimal solution at each discretization point to generate the Pareto-optimal curve. The discretization is done by moving one

Benchmark problems

Initially, the framework is tested on three constrained MOO benchmark problems, namely the Binh and Korn function (BNH), the CONSTR problem and the car-side impact test problem (Chafekar et al., 2003, Jain and Deb, 2014, Zielinski et al., 2005). The BNH and CONSTR problems contain 2 objectives, 2 variables, and 2 constraints, whereas the car-side impact problem has 3 objectives, 7 variables and 10 constraints. Problem formulations are provided in Table 3.

Energy systems design model for a supermarket

In addition to the benchmark problems,

Results

Series of computational studies have been performed on the benchmark problems and on the energy systems design problem to test the accuracy and consistency of p-ARGONAUT over MOO problems. The p-ARGONAUT results are compared with other derivative-free methods: Improved Stochastic Ranking Evolution Strategy (ISRES) (Runarsson and Yao, 2005) which is a global optimization method, and with the Nonlinear Optimization by Mesh Adaptive Direct Search (NOMAD) algorithm (Le Digabel, 2011) for local

Conclusions

In this work, we have implemented a data-driven hybrid methodology that integrates surrogate model identification and deterministic global optimization through the parallel AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems (p-ARGONAUT) algorithm for the global optimization of general constrained grey-box multi-objective optimization (MOO) problems. We have tested our parallel framework, which is designed to solve high-dimensional general constrained grey-box

Acknowledgments

The authors would like to dedicate this work to Professor Christodoulos A. Floudas whose guidance, mentorship and leadership will truly be missed. Authors would like to acknowledge the support provided by the Texas A&M Energy Institute, National Science Foundation (NSF CBET-1548540), U.S. National Institute of Health Superfund Research Program (NIH P42 ES027704), and the Texas A&M University Superfund Research Center. The manuscript contents are solely the responsibility of the grantee and do

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