Recent developments and trends in optimization of energy systems

Accepted Manuscript Recent developments and trends in optimization of energy systems

Christos A. Frangopoulos PII:

S0360-5442(18)31754-7

DOI:

10.1016/j.energy.2018.08.218

Reference:

EGY 13694

To appear in:

Energy

Received Date:

30 May 2018

Accepted Date:

31 August 2018

Please cite this article as: Christos A. Frangopoulos, Recent developments and trends in optimization of energy systems, Energy (2018), doi: 10.1016/j.energy.2018.08.218

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ACCEPTED MANUSCRIPT Manuscript EGY-D-18-03479-Revised Submitted to the Special Issue of ENERGY dedicated to Prof. J. Szargut Recent developments and trends in optimization of energy systems Christos A. Frangopoulos School of Naval Architecture and Marine Engineering National Technical University of Athens Heroon Polytechniou 9, 157 80 Zografou, Greece [email protected]

Abstract An overview of recent developments, trends and challenges in the synthesis, design and operation optimization of energy systems is presented in this manuscript. The static and dynamic optimization problems are stated mathematically, solution methods are mentioned in brief and classification of optimization problems based on the presence and treatment of time is presented. Examples of objective functions in single-objective optimization are given and the need for multi-objective optimization is highlighted. Special reference is made to Prof. Szargut’s early statement about the need for optimization with ecological concerns. Emphasis is given to optimal synthesis of energy systems, a subject that is still between (or combines) art and science/technology. Furthermore, it is interesting to note that optimization is applied also during the development of models of systems. Hints are given on further considerations and research needs in subjects such as global optimization, as well as optimization with uncertainty, reliability, maintenance and social aspects. Examples of algorithms appearing in recent publications are given as an indication of the related strong activity. The manuscript closes with general remarks and a rather comprehensive, even though not exhaustive, list of references. Keywords Energy systems; Developments in optimization; Synthesis; Design; Operation.

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ACCEPTED MANUSCRIPT 1.

Introduction

Building a system with as good performance as possible has always been in an engineer’s mind during the design of the system. Today, the world is facing with a ‘trilemma’: how to have economic growth without depleting the energy and other physical resources and without destroying the environment [1]. In order to help in coping with this trilemma, engineers have to build the best systems and operate those in the best way. Thus, application of optimization methods and procedures are necessary. In general, Optimization is the act of obtaining the best result under given circumstances and, in particular, Mathematical optimization is the process of finding the conditions that give the minimum or maximum of a function, called ‘objective function.’ As ‘energy system’ can be considered any system that transforms energy from one form to another (or to other forms) or transfers energy from one place to another or from one body to another. For clarity, it is necessary to set the limits of this manuscript: it is concerned with complex systems of interconnected components, which convert primary energy to useful forms of energy such mechanical, electrical and thermal energy. Examples are power plants, cogeneration systems, propulsion plants, heating, ventilating and air conditioning (HVAC) systems, etc. Electrical networks and district heating networks are beyond the limits of this presentation, even though they are also subject to optimization. The focus of the manuscript is on recent developments, trends and challenges in the synthesis, design and operation (SDO) optimization, as it is described in Section 2. The meaning of the word ‘recent’ is clarified, if we look at the years of publications of the references: out of 105 references, 29 are of the last two years (2017 – 2018), 44 are of the last four years (2015 – 2018), 49 are of the last nine years (2010 – 2018) and 71 are of the last 19 years (2000 – 2018). The remaining sections are as follows. In Section 3, the static and dynamic optimization problems are stated mathematically, solution methods are mentioned in brief and classification of optimization problems based on the presence and treatment of time is presented. Section 4 gives examples of objective functions in single-objective optimization and explains the need for multi-objective optimization. Special reference is made to Prof. Szargut’s early statement about the need for optimization with ecological concerns. Section 5 is dedicated to optimal synthesis of energy systems, a subject that is still between (or combines) art and science/technology. In Section 6, it is mentioned that on one hand optimization is applied also during the development of models of systems, while on the other hand the fact that a model will be used for optimization affects the development of the model itself. Section 7 gives hints of further considerations and research needs in subjects such as global optimization, as well as optimization with uncertainty, reliability, maintenance and social aspects. Also, it gives examples of algorithms appearing in recent publications. The manuscript closes with general remarks and a rather comprehensive, even though not exhaustive, list of references. 2.

Levels of energy systems optimization The optimization of an energy system can be considered at three levels [2-4]: 2

ACCEPTED MANUSCRIPT A. Synthesis optimization. The term ‘synthesis’ implies the components that appear in a system and their interconnections. After the synthesis of a system has been composed, the flow diagram of the system can be drawn. B. Design optimization. The word ‘design’ here is used to imply the technical characteristics (specifications) of the components and the properties of the substances entering and exiting each component at the ‘design’ point (nominal load) of the system. One may argue that design includes synthesis too. However in order to distinguish the various levels of optimization, the word “design” will be used with the particular meaning given here. C. Operation optimization. For a given system (i.e. one in which the synthesis and design are known) under specified conditions, the optimal operating point is requested, as it is defined by the operating properties of components and substances in the system (speed of revolution, power output, mass flow rates, pressures, temperatures, composition of fluids, etc.).

Figure 1: The three inter-related levels of optimization. Of course if complete optimization is the goal, each level cannot be considered in complete isolation from the others (Figure 1). Consequently, the complete optimization problem can be stated by the following question: What is the synthesis of the system, the design characteristics of the components and the operating point (or operating strategy) that lead to an overall optimum? 3.

The time in optimization of energy systems

3.1 The static optimization problem: mathematical statement and solution methods In the simplest case, time is not considered, i.e. a static optimization problem is formulated, which is mathematically stated as follows [5-7]:

minimize f (x)

(1)

x   x1 , x2 , ..., xn 

(2)

x

with respect to

3

ACCEPTED MANUSCRIPT subject to the constraints

hi (x)  0

i = 1, 2, …, m

(3)

g j ( x)  0

j = 1, 2, …, p

(4)

where x set of the independent variables, hi equality constraint functions, which constitute the simulation model of the system and are derived by an analysis of the system (energetic, exergetic, economic, etc.), gj inequality constraint functions expressing design and operation limits, state regulations, safety requirements, etc. For energy system optimization, in particular, it is often helpful to arrange the independent variables into three sets [2]: x ≡ (v, w, z)

(5)

where v set of independent variables for operation optimization (load factors of components, mass flow rates, pressures and temperatures of streams, etc.), w set of independent variables for design optimization (nominal capacities of components, mass flow rates, pressures and temperatures of streams, etc.), z set of independent variables for synthesis optimization; there is only one variable of this type for each component, indicating whether the component exists in the optimal configuration or not; it may be a binary (0 or 1), an integer, or a continuous variable such as the rated power of a component, with a zero value indicating the non-existence of a component in the final configuration. Then, Eq. (1) is written minimize f ( v, w, z ) v ,w ,z

(1)´

For a given synthesis (structure) of the system, i.e. for given z, the problem becomes one of design and operation optimization: minimize f d ( v, w ) v,w

(1) d

Furthermore, if the system is completely specified (both z and w are given), then an operation optimization problem is formulated:

minimize f op ( v ) v

(1) op

Maximization is also covered by Eq. (1), since: min f (x)  max  f (x) x

x

(6)

In energy systems, static optimization can only be performed under the assumption of steady-state operation of the system at a certain point (usually the design point, also called nominal point). A variety of methods has been developed for the solution of the optimization problems stated by Eqs. (1)–(4). Each method, even though it is claimed as being general, it can in fact be effective in certain types of problems. The various methods are known as mathematical programming methods and they are usually available in the form of mathematical programming algorithms. They can be classified in three broad categories: 4

ACCEPTED MANUSCRIPT (i)

Search methods: They calculate the values of the objective function at a number of combinations of values of the independent variables and seek for the optimum point. They do not use derivatives. The search may be random or systematic, the second one usually being more efficient [5-8].

(ii)

Calculus methods: They use first and (some of them) second derivatives; this is why they are called also gradient methods. In general, gradient methods converge faster than the search methods, but in certain cases they may not converge at all [5-8].

(iii) Stochastic or Evolutionary methods: Methods and algorithms such as Genetic Algorithms (GA), Simulating Annealing (SA), Particle Swarm Optimization (PSO), Neural Networks belong to this category. Even though some of those are in fact search methods, they are usually placed in a separate category [9-14]. If the objective function is continuous, by applying a search method the exact optimum can only be approached, not reached, by a finite number of trials, because only discrete points are examined. However, the region, in which the optimum point is located, can be reduced to a satisfactorily small size at the end of the procedure. On the other hand, there are problems for which search methods may be superior to calculus methods, as for example in optimization of systems with components available only in finite, discrete sizes. Two of the most successful methods for optimization of energy systems are the Generalized Reduced Gradient (GRG) and the Sequential Quadratic Programming (SQP) method [6,7]. A combination of a stochastic algorithm (Genetic or Particle Swarm Algorithm) with a deterministic algorithm (GRG or SQP) has been also successful in energy systems optimization: the first one performs a coarse search of the feasible space and locates a number of possible optimum points, while the second one locates the exact optimum point [15-16]. 3.2 The dynamic optimization problem: mathematical statement and solution methods The dynamic optimization problem is, in general, formulated as a system of differential and algebraic equations (differential-algebraic equation (DAE) formulation). The differential equations describe the behavior of the system, such as mass and energy balances, and the algebraic constraints ensure thermodynamic consistency or other physically meaningful relations or limits imposed on the problem. A general DAE optimization problem can be stated in implicit form as follows [17-19]: mininimize J [z (t f ), y (t f ), u(t f ), t f , w ]

(7)

H (z (t ), z (t ), y (t ), u(t ), t , w )  0 G (z (t ), z (t ), y (t ), u(t ), t , w )  0

(8) (9)

z (0)  z 0

(10)

z ( t ), y ( t ),u ( t ),t f , w

subject to

with initial conditions point conditions

Ps (z (ts ), y (ts ), u(ts ), ts , w )  0,

ts  [t0 , t f ]

(11)

and bounds

z L  z (t )  zU

(12a)

L

U

(12b)

L

U

(12c)

y  y (t )  y

u  u(t )  u 5

ACCEPTED MANUSCRIPT w L  w  wU tf L  tf  tfU where J H G Ps z z0 y u w tf

(12d) (12e)

scalar objective functional differential-algebraic equality constraints differential-algebraic inequality constraints additional point conditions at times ts (including tf) differential state profile vector initial values of z(t) algebraic state profile vector control (independent variables) profile vector time-independent variables vector final time.

The objective function (functional) can have various forms. A general form for continuous time, commonly used in optimization of energy systems, is the Bolza form: tf

J [z (t f ), y (t f ), u(t f ), t f , w ]  Q(z (t f ), y (t f ), t f , w )   F (z (t ), y (t ), u(t ), t , w )dt

(13)

t0

It consists of the function Q at the end of the time interval tf and the integral of the function F over the time horizon. If the optimization problem is discrete in time, then the mathematical statement takes a discrete form, as follows. The time period [t0, tf] is divided in N time intervals of length Δtn, so that t f  t0  N  tn and the integral in Eq. (13) is replaced by a summation over the N time intervals, while the variables are discrete vector sequences (e.g. u  u1 , u 2 ,..., u N  ). The discrete problem then is stated as follows: N

minimize J [z, y , u, t f , w ]  Q(z N , y N , N , w )   F (z n , y n , u n , n, w ) z ,y ,u ,t f ,w

(14)

n 1

subject to constraints appropriately written for each time interval. The optimization problem, as it is stated by Eq. (14), is additively separable across time, and it can be solved by optimal control theory or dynamic programming. In general, the methods that have been developed for the dynamic optimization problem as stated by Eqs. (7)-(13) or (14) can be classified in two main categories: Indirect Methods and Direct Methods. The Indirect Methods include Calculus of Variations (COV) and Dynamic Programming (DP) based on principles stated by Pontryagin and Bellman, respectively [20-22]. In COV, the problem is transformed into a two-point boundary value problem (TPBVP) and solved accordingly. The procedure works fairly well for unconstrained problems, but the solution of the TPBVP is still difficult to be achieved especially with the addition of the profile inequalities. These methods are, in general, focused on using necessary conditions for optimality [4]. Dynamic Programming is suitable for solving complicated and multi-stage decision problems by tracing the optimal strategy. It is not particularly useful in problems studied here, and therefore it will not be further discussed. 6

ACCEPTED MANUSCRIPT In the Direct Methods, the problem is approached by applying a certain level of discretization that converts the original continuous time problem into a discrete one. The Direct Methods can be divided in two sub-categories, according to the level of discretization applied: Sequential Methods and Simultaneous Methods. Both approaches have been effective in solving dynamic optimization problems of energy systems. Detailed description and application examples can be found in the related literature, e.g. [18,19,23-29]. More close to the systems considered here are applications such as the dynamic operation optimization of a trigeneration system [30] and the dynamic synthesis-design-operation optimization of ship or aircraft energy systems [4,31-33]. In these works, the variation of the system operating conditions with time is explicitly taken into consideration and, consequently, models of the systems at both design- and off-design points are first developed. 3.3 Intertemporal static optimization As mentioned in the preceding, in static optimization it is considered that the system operates at a single point, which is not realistic. On the other hand, the formulation and solution of the dynamic optimization problem, which takes the variation in operating conditions with time into consideration, can be formidable or the required software may not be available. In between is the intertemporal static optimization problem, with the word ‘intertemporal’ having the following meaning [4]: Intertemporal optimization is the optimization that takes into consideration the various operating conditions that a system encounters throughout its life time and determines the mode of operation at each instant of time that results in the overall minimum or maximum of the general objective function. The optimization problem is formulated as follows: The real operating profile of a system for any selected length of time (e.g. day, week, year, the whole life cycle, etc.) is approximated with time intervals properly selected from the point of number and length of each one, so that a steady-state operation can be considered in each time interval, while the transients are ignored, as being very short in time compared with the rest of the operating period. It is also considered that the operation in a time interval does not affect and it is not affected by the operation in other time intervals. Thus, the operation optimization can be performed for one interval after the other and the whole period can then be embedded in an upper level optimization, which works on the synthesis and design of the system. In other words, time discretization is applied. Such an optimization of energy systems has appeared in numerous publications as, for example, in [2,3,15,16,34,35]. If, however, the assumption of time intervals with independent operation from each other is not satisfactory, then dynamic optimization is necessary (it goes without saying that dynamic optimization is intertemporal anyway). 3.4 Optimization of transient conditions – Optimal control Transient conditions appear, for example, during load increase or decrease or during loading and unloading of energy storage units, e.g. [36-38]. Depending on the exact formulation, problems of this type belong to trajectory optimization (a term coming from the early attempts to determine the optimal rocket thrust profiles in the atmosphere and in vacuum 7

ACCEPTED MANUSCRIPT [39]) or to optimal control, e.g. [40-42]. This field is beyond the limits of the present manuscript. 3.5 Dynamic optimization of synthesis, design and operation including transients The most general case is the synthesis, design and operation optimization of energy systems during their whole life taking the complete operating profile into consideration, which includes interrelated periods of practically steady-state operation, as well as transient conditions. In these cases, the energy system is described by dynamic models that include different stages or operating modes of the system. The system characteristics (state equations, constraints, etc.) may vary from stage to stage, and this type of variation may be formulated as a mixture of continuous and discrete functions of time. Thus, the operation of the whole system must be described as a sequence of different sets of DAEs (multi-stage systems) with the objective to find the duration and operating conditions of each stage in order to achieve an overall optimal result for the whole system. The mathematical statement consists of the objective function in continuous or discrete form, Eq. (13) or Eq. (14), respectively, and the pertinent constraints. Publications tackling this complex problem are very scarce in the open literature. An example is given by a series of two papers [43,44], which optimize the energy system of an aircraft that includes the phases of take-off, flight and landing. Each phase is described by a different DAE system and the optimization must be performed for the whole trip. A work towards this direction is Ref. [45] concerning the design of an ORC system exploiting low-temperature waste heat of the main engines of an LNG carrier. It does not solve the complete optimization problem, but dynamic simulations are performed and the system is so designed that a satisfactory performance during transients (time needed for the system to reach stability after a change of the ship speed) is achieved. 4.

Objective functions – single- and multi-objective optimization

There are several criteria that can be selected as objective functions of optimization. The selection of the criterion depends on the particular application and may have a strong effect on the final result (optimum system). In the early years of optimization, technical criteria were primarily selected, followed later on by economic criteria. In recent years, the need to protect the environment introduced environmental criteria, while concern for the society introduced societal criteria too. With such a classification, the following examples of objectives are mentioned (of course, the list is not exhaustive). Thermodynamic and other technical objectives:  Minimum weight of the system (particularly relevant for aircrafts or space vehicles)  Minimum volume (e.g. systems on cars)  Maximum efficiency (or minimum fuel consumption)  Maximum net power density  Minimum exergy destruction. Economic objectives:  Minimum life cycle cost (LCC) 8

ACCEPTED MANUSCRIPT  Maximum net present value (NPV)  Maximum internal rate of return (IRR)  Minimum dynamic payback period (DPB). Environmental objectives:  Minimum CO2 and other gaseous pollutants  Minimum thermal pollution  Minimum noise  Minimum land deterioration. Social objectives:  Maximum job creation (or welfare, in more general terms)  Minimum adverse effects on health. It is interesting to note here that, even though publications with environmental objective functions in optimization appeared much later [46-50], Prof. Szargut in his 1979 presentation [51] defined the cumulative exergy consumption index and mentioned: “… it may be called an ‘index of ecological cost. With the help of exergy it is, therefore, possible to establish an ecological economy for the purpose of saving natural resources.”

This idea was taken further and the thermo-ecological cost was defined, which includes not only the direct costs, but also the costs for extracting the material used for construction of equipment, as well as the cost caused to the environment and the society by waste products of a system. The thermo-ecological cost was used as the objective function for energy systems optimization in publications such as [52,53]. A single objective may lead to a system that does not satisfy other objectives. For example, a system of minimum cost most probably will not have maximum efficiency or minimum adverse environmental effects. The need to take two or more objectives into consideration led to the development of multi-objective (or multi-criteria) optimization methods [54,55]. The system obtained does not satisfy each objective in isolation, but it is the result of a compromise, more or less subjective. A simple approach to multi-objective optimization is to combine the various objectives in a single function by means of weighting factors, and then the problem is reduced to the one stated by Eq. (1) [56]. The subjectivity can be reduced or eliminated, if the objectives are studied in separate and a characteristic point on the Pareto diagram is selected, such as, for example, the point with minimum distance from the ideal one or the point with maximum distance from the non-ideal one [57,58]. 5.

On synthesis optimization of energy systems

5.1 The possibility of finding the optimal synthesis Even though not always simple, the design and/or operation optimization problem can be effectively solved by existing algorithms. The case is completely different, however, with the synthesis optimization. The following is written in Ref. [59]: “In the usual design process of an energy system, the designer uses knowledge and experience to select the type, configuration and technical characteristics of a workable system (i.e. a system that is technically feasible and satisfies a given set of needs), which he/she then evaluates for its technical and economic performance and for ways of improving 9

ACCEPTED MANUSCRIPT it. If the system synthesis (type and configuration) is given, the decisions to be taken are of a rather quantitative nature. If, however, the synthesis is not given, in addition to quantitative decisions there is need for many qualitative decisions, which may be non-deterministic. In such a case, innovation and creativity play a vital role. Given the multitude of energy system types and the variations in each type, one may question whether it is ever possible to replace the experienced designer’s mental process with an algorithm consisting of a set of formulae and rules. On the other hand, in today’s complex world, this same multitude of types and variations makes it rather impossible even for an experienced designer to evaluate all possible alternatives. Consequently, an automated procedure, if properly used, can be of invaluable help to the designer. Several methods have been developed for the synthesis optimization of processes and systems. Some of these are applicable only to particular classes of systems (e.g., heat exchanger networks). Other methods are applied to more complex energy systems. However, up until now there has been no single method that can tackle the synthesis optimization problem in all its generality and completeness. The field is, thus, still open to research.”

All these statements are still valid today. 5.2 Classes of methods for optimal synthesis of energy systems The various methods that have appeared in the literature on the optimal synthesis of energy systems can be classified into three classes [59]: (a) (b) (c)

Methods based on heuristics and evolutionary search. Methods attempting to reach pre-determined targets, which have been identified by the application of physical rules. Methods starting with a superstructure, which is reduced to the optimal configuration.

A brief description of each class is given here, while further details as well as characteristic methods are presented in [59]. In class (a), rules based on engineering experience and on physical concepts (e.g., exergy) are applied to generate feasible configurations, which are subsequently improved by applying a set of evolutionary rules in a systematic way. These rules may come from special techniques, such as exergy analysis. Artificial Intelligence and Expert Systems have been proven effective in generating appropriate configurations. For each acceptable configuration, the objective function is evaluated (e.g., efficiency, cost, etc.) and the system with the best performance is selected. The best of a certain set of configurations, however, does not guarantee that the optimal configuration has been revealed. In most cases, though, at least a near-optimal configuration has been obtained [13,60,61]. In class (b), principles from thermodynamics and other physical sciences are applied to obtain targets for the optimal system configuration. These targets can correspond to upper or lower bounds on the best possible configuration and provide vital information for improvement of existing configurations. Pinch analysis is a characteristic example of such a method [62,63]. In class (c), a superstructure is considered with all the possible (or necessary) components and interconnections. An objective function is specified and the optimization problem is formulated. The solution of the optimization problem gives the optimal system configuration, which, inevitably, depends on (and is restricted by) the initial superstructure. [2,3,43,44,6467].

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ACCEPTED MANUSCRIPT It is noted that the distinction among the three classes may not be so clear-cut. For example, the targets of class (b) can serve as heuristics or rules in class (a) and they can be embedded in the optimization procedures of class (c) to the benefit of the whole process. Recently, methods with the words “superstructure-free synthesis” have appeared. They may not constitute a separate class, but they can be classified in class (a) or (b) or a hybrid of those. A few examples are the following. In [68,69] the HEATSEP method presented in earlier publications [70,71] is further developed for the synthesis and design optimization of energy systems, in general, and it is applied for the optimization of an ORC system. A hybrid evolutionary/traditional optimization algorithm is used, which is organized in two levels. A complex original codification of the topology and the intensive design parameters of the system is managed by the upper level evolutionary algorithm according to the criteria set by the HEATSEP method, which are used to automatically synthesize a “basic” system configuration from a set of elementary thermodynamic cycles. The lower SQP (sequential quadratic programming) algorithm optimizes the objective function(s) with respect to cycle mass flow rates only, taking into account the heat transfer feasibility constraint within the undefined heat transfer section. In [72] a hybrid algorithm combining evolutionary with deterministic optimization of thermal power plants is used. The upper-level evolutionary optimization combined with graph theory generates structural alternatives, i.e., unit selection and interconnections. The generated alternatives are optimized with respect to unit sizing and operation in the lower-level deterministic optimization. A method based on graph theory was presented also in [73] for the synthesis optimization of energy systems, and applied for the optimization of a combined-cycle plant with two different objectives: (i) maximization of the power output for given fuel input, (ii) minimization of the annualized cost of owning and operating the system for given power output. It is interesting to note that the objective function had a very strong effect on the optimal synthesis of the system. 5.3 Extension of synthesis As defined in Section 2, synthesis refers to the configuration (structure) of the systems, as it is determined by the components and their interconnections. The working fluids are selected in advance. If there are several candidate fluids, as in ORC systems, the system is optimized for each fluid in separate and the best system is selected [35,74]. A different and very interesting approach is followed in [75]: Computer-aided molecular design of the working fluid is combined with the ORC system optimization into a single framework. Thus, the fluid is synthesized during the optimization procedure using several molecular groups (e.g. –CH3, –CH2–, =CH2,=CH–, etc.). The aim of the optimization is to determine the optimal combination of the molecular groups and thermodynamic variables that maximize the power output generated by the ORC for a specified heat source and heat sink. In this particular application the synthesis of the system is fixed. The road towards complete SDO with molecular synthesis is open.

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ACCEPTED MANUSCRIPT 6.

Optimization in modeling of energy systems and modeling for optimization

One more interesting application of optimization is in the development of the model of a system, no matter whether this model will be used for optimization or not. For example, in [76] particle swarm optimization is applied in order to determine the set of model parameters that minimize the difference between estimated and experimental data of photovoltaic cells and modules, while in [77] an algorithm named multi-verse optimizer is applied in order to identify the optimal set of model parameters that give polarization curves of a proton exchange membrane fuel cell that are close to manufacturers data. During the numerical solution of the optimization problem, the objective function may be evaluated tens of thousands of times. For this purpose, the model of the system, which may consist of hundreds or even thousands of algebraic and/or differential equations, is called by the optimizer. The computational burden can be beyond the limits of the available computers. Thus, there is often need to develop reduced models. It is written in [78]: “Whereas suitable methods are well known for linear model reduction, the efficient reduction of nonlinear models, i.e., the development of reduced models that are both accurate and computationally beneficial, is still challenging … Other approaches use surrogate models that are tailored to state-of-the-art optimization algorithms … Consequently, modeling and optimization should not be considered purely sequential tasks as it is commonly done but rather in an integrated fashion.”

7.

Further considerations and research needs

7.1 Global optimization The type of objective functions and constraints in optimization of energy systems result in a multi-modal optimization problem, i.e. a problem with many local optima. Many optimization methods locate one local optimum only that depends on the starting point. Stochastic or evolutionary algorithms, such as genetic algorithms and particle swarm optimization, search the solution space starting from many points and are more likely to locate the global optimum. Global optimization is, in general, computationally heavy. Development of methods which are both effective and efficient is still a challenge [67,79-81]. 7.2 Optimization under uncertainty The values of various parameters in the objective function and/or the constraints, in particular the cost model, may be uncertain. Uncertainties exist also in the load profile of a system under design as well as in future prices. In the simplest approach, at least a parametric study or sensitivity analysis with respect to uncertain parameters should be conducted. In a more systematic way, uncertainties are taken into consideration during the optimization procedure. Two main approaches exist for optimization under uncertainty: stochastic programming and robust optimization [78]. A thorough discussion on these subjects and presentation of appropriate methods appear in [78,82], while examples of recent applications on energy systems such as power plants and cogeneration systems are reported in [33,83-87]. 12

ACCEPTED MANUSCRIPT There is significant research activity on this subject. 7.3 Optimization with consideration of system reliability The word ‘reliability’ appears in engineering publications with various meanings, e.g. reliability of power supply to customers, reliability in the performance of algorithms, etc. Therefore, it is necessary to give the context in which reliability is used here: ‘Reliability is the probability that an item will perform a required function under stated conditions for a stated period of time’ [88]. The usual practice in optimization of energy systems is to solve the optimization problem under the assumption that the equipment is always available for operation except, perhaps, of pre-specified periods of maintenance. Any modification of the system in order to obtain redundancy is performed empirically afterwards, with a consequence the system thus designed to be non-optimal. A different approach is followed in [15]: reliability and availability are introduced in the mathematical model of the system using the state-space method (SSM) for reliability analysis combined with the Intelligent Functional Approach [2,3] for decision making in case of partial failure of equipment. As an example, the SDO optimization of a cogeneration system was performed without and with reliability considerations, which revealed that (i) reliability considerations have a strong effect on all three aspects (synthesis, design, operation) of the system, and (ii) if reliability is not considered, i.e. if the probability of failure of equipment is ignored, then the economic performance of the system (as expressed by the net present value) is overestimated. It is interesting to note that publications on this direction are still scarce. A few examples are given here. In [89] uncertainties (in weather conditions, internal heat sources and indoor set-points) and reliability of chillers are combined in order to determine the optimum capacity of chillers in a building. Uncertainties are quantified with the Monte Carlo method, while reliability is quantified with the Markov method. Minimization of the total annual cost is the objective. The state-space and the continuous Markov methods in combination are used in [90] in order to determine the optimum configuration and design point specifications of a system of boilers and steam turbines supplying a new petrochemical complex with electrical and thermal energy. Two objectives are considered in constructing the Pareto front: minimization of product cost rate and minimization of exergy destruction. In [91] a steam turbine trigeneration system operating on biomass is optimized with respect to synthesis and design, taking into consideration the change of available biomass and loads in three periods of the year. Chance-constrained programming and k-out-of-m system modeling are used. Maximization of the economic performance of the system is the objective. A study of the available publications reveals that there is still much work to be done for further development of efficient methods for the triple optimization (SDO) with reliability considerations and more so for combinations of reliability with other aspects such as uncertainty, dynamic conditions, etc.

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ACCEPTED MANUSCRIPT 7.4 Optimization of/with maintenance There are several maintenance strategies, e.g. maintenance at fixed and pre-specified periods, condition-based maintenance, reliability-centered maintenance, maintenance after failure (run till failure), etc. Thus, maintenance itself can be the subject of optimization. In [92] for example, a network of compressors operating in parallel in an air separation plant is considered and optimization of operation and maintenance schedule is performed simultaneously. Objective function is the minimization of the total startup, shutdown and power consumption costs of the compressors as well as the procurement cost of products from external sources. Three different maintenance policies are considered: (i) fixed maintenance plan, (ii) flexible maintenance plan through the simultaneous optimization of operational and maintenance tasks, and (iii) flexible maintenance plan considering maintenance workforce limitations. Optimization is performed for policies (ii) and (iii) in separate. In [93], an electric power system is considered and the task is to minimize the total cost, which consists of maintenance cost, operation cost, interruption cost, and environmental cost during and after the maintenance. In fact, there is no formal optimization, but evaluation of the cost for three pre-determined maintenance strategies and the selection of the one with the lowest cost. In both these publications, the system is fixed. So, there is much work to be done in order for maintenance aspects to be included in the synthesis, design and operation optimization of energy systems and more so since reliability and maintenance are directly interrelated. 7.5 Social aspects in optimization Year after year, the scope of optimization, in other words the objective, has been broadened: Thermodynamic and economic objectives were combined in what is known as ‘thermoeconomics’, while later on environmental aspects were considered either in separate or included in the objective function together with thermodynamic and economic aspects [4650]. There is one more step we need to take: inclusion of social aspects. Assessment of projects from the point of view of their effect on the society have been widely performed with criteria such as job creation, general welfare, standard of living, etc. The titles of certain publications give the impression that social aspects are included in the optimization, but a closer look reveals that these aspects are assessed after the solution of the optimization problem, for the system(s) obtained with optimization. An exception is Ref. [94], where a system of forest-based biorefineries and biofuel supply chain is studied and a multi-objective optimization is performed with three objective functions: (i) maximization of newly created jobs, (ii) maximization of the net present value, and (iii) maximization of the GHG emission savings compared to the current supply chain. It seems that applications of this type on systems we are studying here (as mentioned in the Introduction) are still missing1. Of course, a prerequisite of such applications is the selection and perhaps the definition of new quantitative criteria, appropriate for formal optimization.

1

Perhaps my literature search was not successful in revealing related publications. If this is so, I apologize and I would highly appreciate, if the reader could give me information about such publications.

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ACCEPTED MANUSCRIPT 7.6 Optimization algorithms The improvement of existing and development of new algorithms for optimization is a never-ending process, which is supported by a continuous increase in the performance of computers. Here are a few examples of algorithms from recent publications (the list is not exhaustive).  Parallel AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems (p-ARGONAUT) [67]  Particle generating set-complex algorithm (PGS-COM) [95]  Co-constrained multi-objective Particle Swarm Optimization (CC-MOPSO) [96]  Global extremum seeking algorithm (GES) [40]  Guaranteed Convergence Particle Swarm Optimization [76]  Time-varying acceleration particle swarm optimization (TVAC-PSO) [87]  Differential evolution and particle swarm optimization hybrid algorithm (DE-PSO) [56]  Non-dominated sorting genetic algorithm II (NSGA-II) [97-99]  Gravitational search algorithm (GSA) [100]  Micro-time variant multi-objective particle swarm optimization algorithm (microTVMOPSO) [101]  Self-adaptive Jaya algorithm [102]  Niched Pareto genetic algorithm (NPGA) [102]  Response Surface Method (RSM) [102]  Leap-frog optimization program with constraints algorithm (LFOPC) [102]  Teaching-learning based optimization algorithm (TLBO) [102]  Grenade explosion method (GEM) [102]  Multi-objective genetic algorithm (MOGA) [102]  Enhanced multi-objective bee colony optimization algorithm [103]  Front-based Yin-Yang-Pair optimization algorithm [104]  Multi-objective grey wolf optimizer [104]  Multiobjective strength firefly algorithm (MOSFA) [105]. A detailed description and a comparison among the algorithms are beyond the scope of this manuscript, but it can be found in the aforementioned publications. In [102], in particular, the self-adaptive Jaya algorithm is compared with the algorithms NPGA, RSM, LFOPC, TLBO, GEM and MOGA, while in [104], the front-based Yin-Yang-Pair optimization algorithm is compared with the multi-objective grey wolf optimizer and a genetic algorithm. Closure Judging from the number of publications, which has being increasing exponentially during the last ten or twenty years, and from the quality of many of those, there is a tremendous and impressive development in methodology and applications of mathematical optimization in energy systems. The preceding sections indicate that the field has many areas that still need ‘cultivation’ and further ‘growth’ not only in isolation from each other, but also in combination of two or more areas. This is extremely interesting and challenging, in particular for young researchers. It is worth mentioning also that optimization can be a way of thinking and acting, beyond the particular technological applications. 15

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Abbreviations COV DAE DP DPB GA GEM GRG HVAC IRR LCC LFOPC LNG MOGA NPGA NPV ORC PSO RSM SA SDO SQP SSM TLBO TPBVP

Calculus of Variations Differential-Algebraic Equation (formulation) Dynamic Programming Dynamic Payback Period Genetic Algorithm Grenade Explosion Method Generalized Reduced Gradient Heating, Ventilating and Air Conditioning Internal Rate of Return Life-Cycle Cost Leap-Frog Optimization Program with Constraints Liquefied Natural Gas Multi-Objective Genetic Algorithm Niched Pareto Genetic Algorithm Net Present Value Organic Rankine Cycle Particle Swarm Optimization Response surface method Simulating Annealing Synthesis, Design and Operation Sequential Quadratic Programming State-Space Method Teaching-Learning Based Optimization Two-Point Boundary Value Problem

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ACCEPTED MANUSCRIPT multi-objective bee colony optimization algorithm. Energy Conversion and Management 2016;123:116-129. 104. Punnathanam V, Kotecha P. Multi-objective optimization of Stirling engine systems using Front-based Yin-Yang-Pair Optimization. Energy Conversion and Management 2017;133:332-348. 105. Wang H, Zhang R, Peng J, Wang G, Liu Y, Jiang H, Liu W. GPNBI-inspired MOSFA for Pareto operation optimization of integrated energy system. Energy Conversion and Management 2017;151:524-537.

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Highlights     

A variety of methods and algorithms for optimization of energy systems is available. Optimization problems become more and more complex. Development of effective and efficient algorithms for global optimization is still needed. Synthesis optimization is challenging. Introduction of reliability, maintenance and social aspects in optimization is needed.

1