Integrating renewables into multi-period heat exchanger network synthesis considering economics and environmental impact

Accepted Manuscript Title: Integrating Renewables into Multi-Period Heat Exchanger Network Synthesis Considering Economics and Environmental Impact Author: Adeniyi J. Isafiade Michael Short Milos Bogataj Zdravko Kravanja PII: DOI: Reference:

S0098-1354(16)30351-9 http://dx.doi.org/doi:10.1016/j.compchemeng.2016.11.017 CACE 5606

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

21-2-2016 10-11-2016 11-11-2016

Please cite this article as: Isafiade, Adeniyi J., Short, Michael., Bogataj, Milos., & Kravanja, Zdravko., Integrating Renewables into Multi-Period Heat Exchanger Network Synthesis Considering Economics and Environmental Impact.Computers and Chemical Engineering http://dx.doi.org/10.1016/j.compchemeng.2016.11.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Integrating Renewables into Multi-Period Heat Exchanger Network Synthesis Considering Economics and Environmental Impact Adeniyi J. Isafiade*a, Michael Shorta, Milos Bogatajb, Zdravko Kravanjab, aDepartment bFaculty

of Chemical Engineering, University of Cape Town, Private Bag, Rondebosch 7701, South Africa of Chemistry and Chemical Engineering, University of Maribor, Slovenia

*Corresponding author: (A.J. Isafiade) Tel: +27 21 650 4869; E-mail address: [email protected]

Highlights

  

A multi-season and multi-period stage-wise superstructure model is proposed Model integrates renewable and non-renewable energy into process heat demand Gives best combination of utilities based on economics and environmental impact    

Abstract This paper presents a further development of synthesis methods that considers economics and environmental impact in the integration of renewable energy into the optimisation of heat exchanger networks involving multiple periods of operations and multiple options of utilities. The multi-period process stream parameters, and those of the utility sources are integrated in a systematic approach using an expanded version of the simplified stage-wise superstructure multi-period model. Two examples were used to demonstrate the benefits of the expanded synthesis method and the quality of solutions obtained were judged by representation on a Pareto curve and by the use of a modified goal solution method. It was found that various combinations of utility sources were selected for use at various periods/seasons of operations, while utilities from solar photovoltaic were not selected for use at any of the periods/season of operation due to its relatively high cost and limited periods of availability.

Nomenclature Abbreviations ACI

Area cost index

AF

Annualisation factor

EI

Environmental impact

EMAT

Exchanger minimum approach temperature

HENS

Heat exchanger network synthesis

LP

Linear programming

MILP

Mixed integer linear programming

MINLP

Mixed integer non-linear programming

NOP

Number of periods

NOS

Number of seasons

SWS

Stage-wise superstructure 1

 

Indices i

Hot process streams and hot utilities

j

Cold process streams and cold utilities

p

Index representing period of operation (p

s

Index representing seasons of operation (s

k

Stage boundaries

1, … . NOP) 1, … . SOP)

Sets HP

Hot process streams

HU

Hot utilities

CP

Cold process streams and cold utilities

CU

Cold utilities

P

Period of operation

S

Season of operation

K

Stage boundaries or temperature locations (where K

1 is the set defining the stages)

Parameters AC , AF 

Area cost coefficient  

Annualisation factor

CF ,

Fixed cost for heat exchangers, $/yr

CUC

Annual cost per unit of cold utility j $/(W·yr)

DOP ,

Duration of each period p

DOS

Duration of each season s

EIHU

Environmental impact value of hot utility i

EICU

Environmental impact value of cold utility j

U,

Overall stream heat transfer coefficient, W/(m2∙K)

HUC

Annual cost per unit of hot utility i, $/(W·yr)

H

Annual operating hours

R

Weighting parameter for goal method

T, T,

,

Supply temperature of hot streams (process and utility streams) for season s and period p, K Target temperature of hot streams (process and utility streams) for season s and period p, K 2

 

T,

Supply temperature of cold streams (process and utility streams) for season s and period p, K

T,

Target temperature of cold streams (process and utility streams) for season s and period p, K

U,

Overall heat transfer coefficient, W/(m2∙K)

Ω

Upper bound for heat exchanged between match i and j in period p

ϕ

Upper bound for driving force in match i, and j in period p

  Variables A,,

Area of match between hot stream i, cold stream j, in interval k, m2

F,

Heat capacity flowrate of hot stream i in season s and period p, kW/K

F,

Heat capacity flowrate of cold stream j in season s and period p, kW/K

LMTD , ,

Logarithmic mean temperature difference between hot stream i, cold stream j, in stage k, season s, and period p, K Binary variable representing a match between hot stream i and cold stream j in stage k

y,, q,,

, ,

, ,

Heat load of a match between hot stream i and cold stream j in stage k, in season s and in period p, kW

t,

, ,

Temperature of hot stream i at stage boundary k in season s and period p, K

t,

, ,

Temperature of cold stream j at stage boundary k in season s and period p, K

dt , , TAC $/yr

, ,

Driving force in match i, j at stage boundary k and in season s and period p, K Total annual cost of the network, including annualised capital cost and the cost of hot and cold utilities,

Keyword: Multi-period; superstructure; synthesis; heat exchange, environmental impact.

1.

Introduction

Chemical process plants are facing increasing pressure concerning cutting down on greenhouse gas emissions as well as reduction of operational costs. A key method of energy saving which has been successfully used to achieve the aforementioned challenges to an extent is heat exchange network synthesis (HENS) using both sequential (e.g. pinch technology, Smith (2005)) and mathematical programming approaches (e.g. stage-wise superstructure (SWS) of Yee and Grossmann (1990)). Some recent works in HENS include the improved version of the mixed integer linear programming (MILP) transhipment model of Papoulias and Grossmann (1983), which is an automated version of the pinch technology method for determining minimum utility and minimum number of heat exchanger units in a network, by Chen, et al. (2015). This improved version is able to handle larger MILP problems with shorter solution generation times. Na, et al. (2015) also presented a modified version of the Yee and Grossmann (1990) SWS model by including utility sub-stages so as to be able to consider multiple utilities in HENS. This approach makes provision for the possibility of having series heat exchange configuration for multiple utilities within a stage of the superstructure. Pouransari and Maréchal (2014) used MILP to synthesise the heat recovery network on a total site basis. The methodology of these authors does not only minimise utility costs, but it optimises 3  

the network based on geographical location of potential heat exchangers. It is worth stating that most of the established methods in the sequential and simultaneous approaches are based on single period operations without detailed consideration of the environmental impacts (EI) of the resulting designs. In order for chemical plants to go a step further in achieving the energy reduction issue, the established synthesis methods have to be improved to account for plant operating scenarios such as changes in process stream parameters (e.g. supply/target temperatures and flowrates), which may occur due to changes in environmental conditions, plants start-ups/shut downs, changes in product quality demand, controllability, etc. Synthesis methods for process parameter variation of this nature, which is known as multi-period operations, has not received much attention compared to its single period counterpart. Papers which addressed multi-period HENS could also be classified as sequential and simultaneous based. Among the sequential multi-period synthesis methods are the studies of Floudas and Grossmann (1986), Tantimuratha et al. (2001), Cerda et al. (1990), and Mian, et al. (2016a). Floudas and Grossmann (1986) adapted the transhipment model of Papoulias and Grossmann (1983), by minimising utility consumption in each period with a linear programming (LP) step followed by the use of a mixed integer linear program (MILP) model that finds the network configuration containing the fewest number of exchangers. The method of Tantimuratha et al. (2001) was developed to address the issue of flexibility and operability in heat exchanger networks. The work of Cerda et al. (1990) also involved designing flexible heat exchange networks using the pinch technology heat cascade where the pinch exhibits a somewhat continuous behaviour. The authors also used a feasibility condition which helps eliminate the need for flexibility tests. Mian, et al. (2016a) developed a sequential technique for multi-period HENS where utilities are not restricted to only the first and last stages of the superstructure, as is the case in most of the existing simultaneous based multi-period synthesis methods. Their approach entails a combination of the multiperiod utility integration and scheduling method of Marechal and Kalitventzeff (2003), a modified version of the multi-period minimum number of units model of Floudas and Grossmann (1986), and the multi-period minimum investment cost model of Floudas and Grossmann (1987). These models were coupled with the derivative-free hybrid algorithm recently presented by Martelli and Amaldi (2014). Although these aforementioned multi-period synthesis methods are easier to solve, their main drawback, as with sequential optimisation methods, is that interactions between utility consumption and investment cost are only partly exploited. Furthermore, the methods only considered one kind of hot and cold utilities, except for that of Mian, et al. (2016a), where multiple utilities were considered. Mian, et al. (2016b) used a similar approach to that of Mian, et al. (2016a), but included thermal, electric and material storage in their model. However, the methods of Mian, et al. (2016a) and Mian, et al. (2016b), despite considering multiple utilities, did not optimise the utilities based on the associated environmental impact. Some of the simultaneous based multi-period synthesis approaches include the multi-period hyperstruture model of Papalexandri and Pistikopoulos (1994), the multi-period stage-wise superstructures of Aaltola (2003), Verheyen and Zhang (2006), and Isafiade and Fraser (2010), amongst others. The hyperstructure of Papalexandri and Pistikopoulos (1994) embeds all possible configurations of heat exchange networks in a multi-period scenario. The method can also incorporate process operability issues such as controllability, process dynamics, etc. Apart from the fact that the method did not include multiple options of utilities alongside the associated environmental impacts, other limitations of the method include the highly non-linear nature of the hyperstructure, hence decreasing the chances of getting a globally optimal solution. The method of Aaltola (2003) involves the use of an average area approach in the objective function. This implies that the area of the heat exchanger that would serve same stream pairs which are matched in more than one period is the average of all the matches in the different periods. Verheyen and Zhang (2006) extended the work of Aaltola (2003) by using a maximum area approach in the objective function in place of the average area used by Aaltola (2003). The work of Isafiade and Fraser (2010) further improved the objective function used by Verheyen and Zhang (2006) by introducing weightings in the objective function, which adequately accounts for the contribution of each period, in terms of utility usage, to the total annual cost of the network. Other simultaneous based approaches that have been used for the synthesis of multi-period HENs are those of Escobar, et al. (2013), Kang, et al. (2015), Jiang and Chang (2015), Jiang and Chang (2013), Short, et al. (2016b) and Kang, et al. (2016). Escobar, et al. (2013) developed a methodology for synthesising heat exchanger networks 4  

that are flexible and at the same time controllable, using a two-step approach. The first step entails generating a design which is then analysed for operability in the second step. Kang, et al. (2015) presented a 2-step approach where in the first step, a representative single period network which is the most dominant network in the multiperiod system is synthesised. The resulting network is then further optimised so as to make it feasible to transfer heat for the other periods of operations. In the second step of the method of Kang, et al. (2015), multi-stream exchangers are then substituted for some of the dual stream exchangers of the final network of the first step. Jiang and Chang (2015) developed an approach for multi-period HENS where a set of heat exchangers can be selected to be shared by more than one kind of match in different periods of operations, so as to achieve savings in capital costs. Jiang and Chang (2013), further extended the method of Jiang and Chang (2015) by demonstrating the capability of the resulting networks from the timesharing mechanism to be operable irrespective of the length of each period of operation as well as how often these period duration change. It is worth mentioning that a major limitation associated with the timesharing mechanism is that complex networks of piping and associated instrumentation may be required. Furthermore, exchangers would need to be thoroughly cleaned so as to avoid contamination, when exchanger switches take place. Short, et al. (2016) extended the 2-step synthesis approach for single period HEN problems previously developed by Short, et al. (2016a) to multi-period problems. The approach entails the use of correction parameters, which are obtained by doing detailed individual heat exchanger designs based on solution networks obtained from the multi-period MINLP model. The correction parameters are then used to obtain more realistic designs in the multi-period MINLP model. Kang, et al. (2016) extended the method of Kang, et al. (2015) to cases involving large number of periods with significantly different period durations. Stochastic based approaches have also found application in the synthesis of multi-period HENs as presented by Ahmad, et al. (2012), where the authors used a simulated annealing approach. Retrofitting of heat exchanger networks has also been extended to multi-period and flexible operations by Kang and Liu (2015). The authors used an approach where the retrofit target is first determined after which the heat transfer areas are matched using a reverse order matching method. Other authors that have solved problems involving retrofit of multi-period and flexible heat exchanger networks are Kang and Liu (2014a), Kang and Liu (2014b) and Čuček and Kravanja (2015). Kang and Liu (2014a) also used the reverse order matching in multiperiod HEN retrofit with the aim of reducing the redundancies in heat exchangers and also fully utilise unused heat transfer areas in the existing multi-period network. The approach of Kang and Liu (2014b) was aimed at minimizing additional heat transfer area and the investment cost of the retrofit as well as maximising the number of heat exchangers that are substituted. The authors used a matching algorithm which is based on the bipartite graphs. Čuček and Kravanja (2015) used an updated version of the code TransGen (Čuček and Kravanja, 2014), to determine modifications that are feasible in terms of economics, safety, technology and other criteria, in retrofitting large-scale heat exchanger networks for total sites, under both nominal and uncertain conditions. Other scenarios which involve the multi-period profile of heat exchanger network synthesis that has been presented in the literature include the works of El-Temtamy and Gabr (2012), Nemet, et al. (2013) and Escobar, et al. (2014). The study of El-Temtamy and Gabr (2012) for multi-period HENS also adapted the sequential approach where multi-period linear programming (LP) and multi-period MILP were used to obtain targets for minimum utility usage as well as the corresponding minimum number of units network. In the study of Nemet, at al. (2013), consideration was given to the fact that HENs need to be designed for flexibility to adapt to fluctuating utility prices using a multiperiod scenario. In the study of Escobar, et al. (2014), a technique that was based on Lagrangean decomposition was used so as to reduce the size of the multi-period problem. It is worth mentioning at this stage that, apart from issues associated with multiple periods of operations which has been discussed earlier, other issues usually encountered in plant operations is the existence of multiple options of hot/cold utilities (e.g. high pressure steam, medium pressure steam, low pressure steam, hot oil, cold water, cool air, etc.). These utilities may be generated from both renewable and non-renewable energy sources such as solar, wind, biomass, coal, etc. It is usually perceived that generating utilities from renewables is usually more beneficial in terms of environmental impact compared with generating utilities from fossil based sources. However, the availability of some of these renewable energy sources are time dependent. Hence, it becomes imperative that the aforementioned multi-period synthesis models be extended to integrate utilities that can be generated from various 5  

renewable sources so as to establish the most efficient set of energy sources that satisfies the multi-period HEN in terms of economics and environmental impact. However, to the best of the knowledge of the authors of this paper, very few published works have addressed the issue of multiple utilities in multi-period problems. These works include those of Marechal and Kalitventzeff (2003), Mian, et al. (2016a), Isafiade, et al. (2015a) and Isafiade, et al. (2015b). The works of Marechal and Kalitventzeff (2003) and Mian, et al. (2016a), did not consider the environmental impact associated with utility generation. The work of Isafiade et al., (2015a) considered only the generation of various steam levels from just one source of energy. The study did not consider the environmental impact associated with the use of these steam levels in multi-period HENS problems. The work of Isafiade, et al. (2015b), considered multiple renewable and non-renewable sources of generating different kinds of utilities and their associated environmental impact. However, the approach used by the authors in solving the problem is such that only two combinations of utility generation sources are considered one at a time. The authors of this paper are of the view that this approach has the tendency to exclude some potentially optimal solutions. Furthermore, the method of Isafiade, et al. (2015b) did not consider the seasonality, which implies time of year variation, associated with renewable energy sources such as solar, wind and biomass, and the objective function used for the environmental impact did not adequately capture the contributions of each energy source to the overall environmental impact of the selected solution. It should be known that Lopez-Maldonado, et al. (2011) and Vaskan, et al. (2012) both considered economics and environmental impact in the synthesis of heat exchange networks, however the scenario involved was single period operations. Also, only one energy source, which is fossil based, was considered by these authors. In order to adequately harness the environmental benefits of generating utilities from renewable energy sources in the synthesis of heat exchanger networks, a further issue to note is the time dependency of the availability of these energy sources. These time dependencies may be time of day and/or seasonal. Depending on what part of the world a plant is situated, availability of solar and wind energy are time of day and time of year dependent, while biomass is only time of year dependent. The natural availability of non-renewable energy sources such as coal on the other hand is time independent, but coal is known to have high environmental impact and in some cases relatively low cost. Therefore, in this paper, a systematic synthesis approach which adequately integrates the multiperiod/multi-seasonal nature of renewable energy sources, as well as non-renewable ones, in heat exchange network synthesis problems, whose operations may also be multi-period in nature, is presented. The newly developed approach, which is an extension of the multi-period version of the SWS model, judges the quality of solutions obtained based on economics and environmental impact.

2.

Problem Statement

Before presenting the problem statement of this study, the context in which the terms ‘multi-period’ and ‘multiseason’ have been used in this paper is first described. Multi-period implies time intervals within the 24 hours of each day, where each of the time intervals is called a period. In this paper, the 24 hours of each day has been segmented into 3 periods which roughly comprises morning, afternoon and night periods. These 3 periods may or may not have equal durations. On the other hand, multi-season implies time intervals within the number of hours in a year that a plant operates (8000 hours assumed in this paper), where each time interval is called a season. In this paper, the 8000 hours of operation has been segmented into 3 seasons which may have equal or unequal durations. It is then assumed that renewable energy sources are available in one or two periods/seasons, whereas fossil energy is available at all periods and seasons. It is also assumed that hot and cold process stream parameters, such as supply/target temperatures and heat capacity flowrates, may vary within specified ranges with periods of each day and seasons of each operational year. The problem statement for the synthesis problem addressed in this paper is then stated as follows: Given:  A set of hot

and a set of cold

process streams 6

 

     

and fixed target temperature as well as fixed heat capacity A set of fixed supply temperature flowrates for each of the given hot and cold process streams A set of periods at which the given process parameters operate, where these parameters for each process stream may change from one period to another A set of seasons at which the given process parameters operate, where these parameters for each process stream may also change from one season to another A set of hot utilities and cold utilities , which may be generated from both renewable and non-renewable sources, with the renewable sources only available at specific times of the day (i.e. at period P) and specific times of the year (i.e. season S). The costs associated with each of the utilities, the annualisation factor and capital costs of process and utility heat exchangers. The environmental impact associated with each of the utilities, based on its source of generation, calculated through the ReCiPe methodology, considering the life cycle of the utilities.

The aim is to synthesise an optimal multi-period heat exchanger network considering both economics and environmental impact as objectives.

3.

Methodology

The approach used entails adapting the multi-period stage-wise superstructure (SWS) model of Verheyen and Zhang (2006) and Isafiade et al., (2015a) to account for the various multi-period scenarios. However, since the multidimensionality of the resulting superstructure increases beyond those earlier presented in the literature, due to the inclusion of these extended multi-period scenarios, a series of assumptions are made so as to simplify the solution procedure. The assumptions are that renewable energy sources (solar and wind) are available at fixed time periods within each of the days that comprises a season in an operational year, and that their availability is constant within these periods. Also, it is assumed that for the season dependent renewable energy sources such as solar, wind and biomass, their availability is fixed and constant within the seasons where they exist. It is further assumed that the variations experienced by process streams’ supply/target temperatures and flowrates (i.e. their multi-period profile) coincides with the multi-period profile of the renewable energies that are time of day dependent and those that are season of year dependent. This implies that a process stream may have its parameters changing in a day due to changes in weather conditions of that day. It may also change in the same manner as the seasons of the plant’s operational year changes.

3.1

Expanded Multi-Period Stage-Wise Superstructure

The expanded multi-period SWS model used in this work is shown in Figure 1. The figure depicts a 3 hot utilities, 2 cold utilities, 2 hot process stream and 2 cold process streams problem. In the superstructure, each hot process stream and each cold process stream can exchange heat with any or all of the opposite kind of stream present in Intervals 2 and 3 through splitting. Such heat exchange can only be parallel because the split branches would mix either in an isothermal or non-isothermal manner. The resulting mixed stream temperature then forms the temperature for the next interval moving from left to right (for hot streams), and moving from right to left (for cold streams) along the superstructure. Process streams are then heated/cooled to their final temperatures by exchanging heat with utilities in the first and last intervals. In the SWS model as used by Bogataj and Kravanja (2012) for single period HENS problems, Intervals 1 and 4 are known as the utility intervals, while Intervals 2 and 3 are process heat recovery intervals. Setting up the SWS model in this form for multi-period systems is beneficial in that process streams in the two utility intervals have the opportunity of splitting into the number of the opposite kind of streams present in the intervals for the purpose of exchanging heat in a parallel manner. This implies that process streams can exchange heat with more than one kind of utility, thereby increasing the solution space. This is very much relevant in this work since multiple utilities are involved. In the traditional SWS model presented by 7  

Yee and Grossmann (1990), as well as the multi-period version used by Verheyen and Zhang (2006), process streams can only exchange heat with one kind of utility, since there does not exist utility intervals, hence process streams do not have the option of splitting when exchanging heat with utilities.                    

3.2

Model Equations

The model equations used in the newly expanded model is similar to those used by Verheyen and Zhang (2006) and Isafiade, et al. (2015a). However, in this paper, the existing multi-period SWS model equations are extended by including additional index ′ ′ to represent seasons of operation. The details of the model equations are shown next. Overall heat balance for each stream in each season of operation and each period of operation Equations 1 and 2 are used to ensure that for each hot and each cold process stream, the overall heat load requirement is satisfied in every season of operation s, as well as every period of operation p, where they are present. The satisfaction of this heat load for the process streams is accomplished through heat exchange with utility streams which are present in the respective seasons and periods of operation. As mentioned previously, apart from coal, all other sources of utility generation are not available in all seasons/periods of operations. Therefore, appropriate mechanism needs to be applied in the model so as to ensure that this constraint is satisfied. , ,

, ,

, , ∈

, ,

, ,

∈

, , , ,

∈

∈

(1)

∈

∈

, , ∈

, , , ,

∈

(2)

∈

∈

where , , and , , are the supply and target temperatures for the hot streams respectively, while those of the cold streams are , , and , , . , , and , , are the heat capacity flowrates of the hot and cold streams respectively. Note that , , and , , for the hot and cold process streams are fixed, while for hot and cold utilities, they are variables to be optimised. Also note that utilities are included in the set of hot and cold streams. , , , , is the heat exchanged between hot stream and cold stream in season , period , and interval .

Heat balance over each stage in each season of operation and each period of operation Equations 3 and 4 are used to calculate the intermediate temperature (i.e. interval boundary temperatures) of each stream for each season and period of operation where they are present. The amount of heat exchanged within each interval at each season and period of operation is what determines the value of the interval boundary temperatures. , , ,

,

, , ,

, ,

, , , ,

∈

∈

∈

∈

(3)

∈

8  

, , ,

,

, , ,

, ,

∈

, , , ,

∈

∈

(4)

∈

∈

Where , , , and , , , both represent the intermediate temperature of hot stream temperature location , season , and period .

and cold stream

at

Assignment of superstructure inlet temperatures for each season and each period of operation Equations 5, 6, 7 and 8 are used to assign inlet temperatures at the respective interval boundary for each of the streams in the superstructure at each season and each period of operation where they are present. ,

, ,

, ,

∈

∈

∈

(5)

,

, ,

, ,

∈ ∈

∈

(6)

,

, ,

, ,

∈

, , ,

, ,

∈

∈ ∈

∈ ∈

(7) (8)

Feasibility of temperatures along superstructure intervals in each season and each period of operation Moving from left to right along the superstructure, temperatures should decrease for each stream in every season and period where they are present. This monotonic decrease is implemented through Equations 9 and 10. , , ,

,

, ,

∈

∈

∈

(9)

, , ,

,

, ,

∈ ∈

∈

(10)

Logical constraint A constraint needs to be implemented for each heat exchanger to represent the maximum heat load that can be exchanged between each hot and each cold process stream in every season and period of operation where they exist. This maximum heat load, which is represented as Ω, is specified as the minimum heat load available in each of the two streams participating in the match concerned. The logical constraint is illustrated by Equation 11. , , , ,

Ω

, ,

0

(11)

Calculation of heat exchanger area Heat exchanger approach temperature, which is represented as , , , , is used to determine the driving force at each end of an exchanger. This is accomplished through the use of logical constraints. If a match , exists in interval , season , and period , a binary variable , , , becomes active thereby taking on a value of ‘1’. On the other hand, if the match does not exist, then the binary variable is inactive and it takes on a value of ‘0’. For this case, the approach temperature takes an arbitrary value less than . The purpose of this is to reject matches with negative approach temperatures. The value of can be set as the maximum value between ‘0’ and the

9  

temperature differences at each end of the exchanger with the ultimate goal of ensuring that the constraints shown in Equations 12 and 13 are satisfied. , , , , , ,

, , , , ,

,

1

, , , , ,

,

, ,

∈

, ,

1

∈ ∈

, ,

∈ ∈

∈ ∈

∈

(12)

∈ ∈

(13)

In order to avoid the inclusion of exchangers having infinitely large areas, a minimum approach temperature, which is a positive number and represented as ε needs to be specified. This is illustrated by Equation 14. , , , ,

EMAT ∈

∈

∈

∈

∈

(14)

Just as presented in the existing SWS model for multi-period HENS, the binary variable , , does not include the index ‘ ’. The reason for this is that only one representative match will be chosen to exchange heat between the same pair of streams that are matched in the same interval but at different periods of operations. Since this paper extends existing models by including seasons of operations, the index ‘ ’, which is newly introduced in this paper, is also not included in the binary variable , , . This implies that, still, only one representative match will be selected for the same pair of streams exchanging heat in the same interval, but at different seasons and periods of operations. For heat exchanger area determination, logarithmic mean temperature difference is required, however in an approximated form. The Paterson (1984) approximations is used in this paper and is shown in Equation 15. , , , ,

2 3

⁄ , , , ,

, ,

, , ,

1 3

, , , ,

, ,

2

, ,



∈

∈ ∈

∈ ∈ 15

The representative area referred to previously, should have an area which is the maximum area of all the possible matches between pair of streams exchanging heat in the same interval but at different seasons and periods of operations. This is to ensure that the heat exchange would be feasible using just one exchanger for the respective pair of streams, including utilities. This maximum area approach was introduced by Verheyen and Zhang (2006), and has also been adapted by Isafiade and Fraser (2010) and Isafiade et al., (2015a). However it is extended in this paper to include seasons of operations. The maximum area, which is represented as , , , also does not include the indices ‘ ’ and ‘ ’. Equation 14 is used to determine maximum area for streams , in interval . This equation works by constraining the chosen area to be greater than or equal to each of the areas for the same stream pair in the same interval, but at different seasons and periods of operations. , ,

, , , , , , , ,



,

∈

∈ ∈ ∈ ∈ 16

where , is overall stream heat transfer coefficient. Objective functions Since the key aim of this paper is to extend existing multi-period models through integration with the daily and seasonal variation in availability of renewable energy sources, the objective function used in this paper comprises an economic function and an environmental impact function. As mentioned previously, this has not been explored in HENS methods for multi-period operations presented in the literature. Economic objective function 10  

The economic objective function is shown in Equation 17. This objective function is a sum of the annual operating and costs (for the hot and cold utilities) and annualised capital cost. In this equation, , represent the duration of season , and duration of period in season , respectively, and represent the number of seasons and number of periods, respectively. represents fixed cost ($), represents area cost ($/m2), represents the area cost index, while ($/(kW·y))and ($/(kW·y)) both represent cost of hot and cold utilities, respectively.

,

min

∑

∙

∑ ,

∑

∙

∑

,

∙ ,

, , , ,

,

, , , ,

∙

, ,

∈

, ,





,

,

,

∈ ∈

∈ ∈ 17

It should be known that this objective function is an improved version of that originally presented by Isafiade and Fraser (2010) based on the fact that the weighting terms for the utilities was presented for only ‘periods of operations’ by Isafiade and Fraser (2010). In this paper, weighting terms are also presented for ‘seasons of operations’. These two weighting terms, which are the first two terms in each of the two square brackets shown in Equation 17, are meant to ensure that the quantities of utilities included in the calculation of annual operating costs by the economic objective function, are exactly the quantities used in the seasons and periods of operations concerned. This approach of formulation is important because in reality, duration of seasons and periods may not be equal, hence Equation 17 gives appropriate weighting term, in terms of contribution to the annual operating cost, for each of the utility generation source concerned.

Environmental objective function The environmental objective function used in this paper is shown in Equation 18. In this equation, represents operating time in a year, , represents environmental impact, represents environmental impact of hot utilities, while represents environmental impact of cold utilities. , , , , in Equation 18 is constrained in such a way that it represents the heat load exchanged between a utility and a process stream in interval , season , and period . The weighting factors used in the economic objective function are also used in the EI objective function, so as to adequately account for the EI of the energy source being used in season and period .

min

,

∙

∑

∑

, , , ,

,

∈

∈

,

∑

∑

,

∈

, , , , ∈

∈

∙

∙



∈



∈

∈ ∈

∈ ∈ 18

11  

The Recipe method in SimaPro, 2008 (Goedkoop, et al. (2009)), for Life Cycle Impact Assessment (LCIA), was used for the quantification of environmental impacts associated with the generation of various kinds of hot and cold utilities from various non-renewable and renewable energy sources. The energy sources considered in this paper are solar, wind, biomass, coal and natural gas, while the utilities considered are high pressure steam (HPS), medium pressure steam (MPS), low pressure steam (LPS), cooling water (CW) and cold air (CA). In generating these utilities, two scenarios were considered. In the first scenario, for all utility sources considered, electricity was first generated, after which the various steam levels and cold utilities were then produced from the generated electricity. This was done so as to evaluate the profile of solutions that would be obtained when all utility production from all available sources are evaluated on the same basis. Furthermore, this first scenario could be integrated with a community-based or regional power network in a sustainable way such that excess power produced in the integrated multi-period HEN can be fed or sold to the community/regional power network whose power supply/demand is also multi-period in nature. The second scenario involves generating the various levels of steam from coal and natural gas directly by burning these fuels. The environmental impact associated with this mode of steam generation are then obtained from SimaPro. For the other utility sources such as biomass, solar and wind, electricity was first generated, after which the various levels of steam are obtained using electrical boilers. This was done due to the limitation of SimaPro in terms of availability of necessary data involving direct generation of various levels of steam from renewable sources such as biomass and solar. For the cold utilities, electricity was first generated, from which cooling water and air were then obtained. However, it is hoped that future studies would involve a more detailed LCIA studies for each of the utility sources so as to obtain more realistic results. The environmental impact (EI) and cost values for these energy sources are shown in Table 1 for scenario 1 and Table 2 for scenario 2. For the environmental impact indicators, ReCipe Endpoint (Egalitarian) V1.06m normalisation/weighting set in SimaPro (Goedkoop, et al., 2009) was used. Under this option, the Europe ReCipe E/A was further selected. This refers to the normalisation values of Europe with average weighing set. This method in SimaPro gives a single score that combines impact on human health, ecosystem and resources. The EI associated with the generation of each of the steam levels and cold utilities from electricity generated from each of the energy sources were further obtained from SimaPro as follows: For coal, the option ‘heat, at hard coal industrial furnace’ was selected for scenario 2, while the option ‘electricity bituminous coal at power plant’ was selected for scenario 1. For natural gas, the option ‘heat, natural gas, at boiler modulating’ was selected for scenario 2. For biomass the option ‘electricity biomass at power plant’ was selected for scenarios 1 and 2. For solar, the option ‘electricity production mix photovoltaic at plant’ was selected for scenarios 1 and 2. For wind, the option ‘electricity at wind power plant was selected’ for scenarios 1 and 2. Multi-objective function The simultaneous optimisation of the economic and environmental objective functions can be represented as min

;

 

The two objectives (TAC and EI) are contrary to each other, so the problem is a multi-objective optimisation problem as introduced by Azapagic and Clift (1999). The set of optimal solution points can be represented on a Pareto curve using the constraint method. The goal method can also be used where the best TAC ( ) ) are both identified. The aim in the goal method is to simultaneously minimise the extent of and best EI ( and . It should be known deviation of these two objectives from their respective best values i.e. that Lopez-Maldonado et al., (2011) used these two approaches for multi-objective optimisation of utility selection in single period heat exchanger networks. However, according to Gxavu and Smaill (2012), the form in which these authors used the goal method is such that the deviation in these two objectives is directly minimised despite the differences in their units as shown in Equation 19. Also, using the goal method in this form limits the number of goal solutions that can be obtained, which is a subjective decision.

12  

19  

min

Hence Gxavu and Smaill (2012) developed an approach which establishes the ratio between each of the objectives and its best value so as to make the goal expression dimensionless. Also, weights are further included in the expression so that multiple goal solutions can be obtained. The modified goal method expression as presented by Gxavu and Smaill (2012) is shown in Equation 20. min where

1

20

is the weighting parameter.

It should be known that in this paper, both the constraint and goal solution methods have been used. For the goal method, the approach of Gxavu and Smaill (2012), i.e. Equation 20 was used and not Equation 19. Apart from the inclusion of seasons of operations alongside utility periods of operations as presented in this paper, another key difference between this newly presented method and that of Isafiade, et al. (2015b) as mentioned previously is the solution approach adopted and the fact that unequal period durations and unequal season durations are accounted for in this paper. Appropriate weightings to account for these unequal period/season durations are included in both the economic and environmental impact objective functions, unlike the approach of Isafiade, et al. (2015b) where it is assumed that period durations are equal. Also, the weightings for these period durations were not included in the EI objective function. Furthermore, the solution approach adopted by Isafiade, et al. (2015b), is that only two set of energy sources, which were called hybrid, are considered at a time. This implies that a series of Pareto curves were generated and then analysed individually, so as to obtain a point of compromise for operation. As observed by authors of this paper, this approach, apart from being tedious, may result in the exclusion of some regions of the solution space. In this paper, all energy sources, for both hot and cold utilities, are included in an overall superstructure and solved simultaneously to obtain an optimal or near optimal solution. The superstructure developed in this paper is an MINLP model which is solved using version 24.0.2 of General Algebraic modelling Systems (GAMS) as the solver environment (Rosenthal, 2012). The computer used operates on Microsoft® Windows 7 Enterprise™ 64 bit, Intel® Core™ i5-3210M processor running at 2.50 GHz with 4 GB of installed memory. DICOPT was used as the solver. DICOPT uses CPLEX for the mixed integer linear program master problems and CONOPT for the non-linear program sub-problems. 3.3

Solution procedure:

The approach used in this paper for the iterative constraint method is that different set of EI values are kept as a limit while the set of optimal TAC that corresponds to each of the EI values are solved for as illustrated below. min



1,2,3, … . . ∆

These sets of data are then used to generate a Pareto curve. This implies that the problem is more or less reduced to a single objective problem when the constrained method is used. However, in order to get an optimal network for each of the TAC at each value of environmental impacts that would be plotted on the Pareto curve, parameters such as in Equations 12 and 13, and the exchanger minimum approach temperature (EMAT), in Equation 14, were varied for each run, with the solution finding the lowest TAC selected as the optimum value. It should be known that variables such as EI in Equation 18 need to be scaled since it turns out be a very large number based on the quantities of utilities used, and large numbers cannot be handled by GAMS directly unless they are scaled. 13  

For the goal method, TACmin and EImin were first obtained separately. For TACmin this was done by solving Equation 17 without any constraint on Equation 18, the EI objective function, while for EImin, Equation 17 was solved by constraining the EI to the best possible minimum that gives a feasible network. The TACmin, EImin, and specified weighting parameter, , were then inputted into Equation 20. Equations 17, 18 and 20 which are the TAC, EI and goal solution objective functions, were then solved simultaneously in GAMS using three solve statements so as to minimise Z in Equation 20. This is the case since Z is a sum of the dimensionless or normalized form of TAC and EI. The TAC and EI that corresponds to the optimal value of Z is then selected as the network that has a pre-set compromise between TAC and EI depending on the weighting parameter that was specified.

4.

Examples

Three examples are presented in this paper. In all the examples, three different levels of steam, cooling water and air, are generated from the available energy sources. For Example 1, all energy sources listed in Table 1 are used, while in Examples 2 and 3, all energy sources listed in Table 2 are used. In Examples 1 and 2, it is assumed that there exists 3 equal seasons of operations in an operational year, and 3 equal periods of operations in an operational day. Since the plant considered operates for 8000 hours in a year, it then means that the duration of each season of operation is 2,667 hours. For the periods of operation, the duration of each period is 8 hours, which implies that the plant operates for the 24 hours that comprises the total number of hours in a day. A further assumption in Examples 1 and 2 is that the multi-period nature of the HEN process streams coincides with the seasons of the operational year. This implies that each season of operation (usually termed period of operation in existing multi-period HENS methods) of the process streams also lasts for exactly 2,667 hours. The authors of this paper are aware that these assumptions may not fully represent what occurs in reality, however changes in environmental conditions/seasons influence and are often responsible for the multi-period profiles that process plant parameters exhibit. Again for Examples 1 and 2, it is taken that renewable energy sources such as solar and wind are available in one or more of the 3 periods (i.e. 8 hours) that comprise a day in certain seasons of the year, while biomass and coal are available in all 3 periods that comprise a day. For solar, wind and biomass, it is taken that they are available in one or more of the 3 seasons (i.e. 2,667 hours) in an operational year. It can then be said that there exists 3 aspects to consider for the multi-period profiles in Examples 1 and 2. The 3 aspects are, multiperiod profile (i.e. availability) of solar and wind energy per day, multi-season profile of solar, wind and biomass per operational year and multi-period profile of process streams which is assumed to coincide with seasons of the year (i.e. process stream parameters do not vary with time of day). In the third example, a fourth dimension is included alongside the three described for the first 2 examples. This fourth dimension involves the fact that the process streams may vary not only with changing seasons (as done in Examples 1 and 2), but also with changing weather conditions of each day. Furthermore, unequal period and unequal seasons are considered, which implies that the weighting terms described in Equations 17 and 18 will be used to allocate correct contributions of selected utilities to the overall annual operating cost and annual environmental impact, respectively. The two examples have 5 hot process streams and 6 cold process streams (which are multi-period in nature), 3 hot utilities and 2 cold utilities. The capital cost data involves a fixed cost ( , ) of 13,000 $, area cost ( , ) of 4,333 ($/m2), an area cost index (ACI) of 0.6 and annualisation factor of ⁄ ∙ . The problem 0.2. The individual heat transfer coefficients for both process and utility streams is 0.5 data is shown in Tables 3 and 4 for Examples 1 and 2, and Tables 7 and 8 for Example 3. The EI associated with generation of various kinds of utilities from each of the energy sources considered, as well as costs of these utilities, are shown in Tables 1 and 2.

4.1

Example 1

The problem data used for this example is as shown in Tables 3 and 4 where process stream parameters only vary as the season in an operational year changes (i.e. multi-seasonal in nature). However, as mentioned previously, 3 kinds of hot utilities (high, medium and low pressure steam) and 2 kinds of cold utilities (cold air and 14  

cooling water) are available for use. The 3 hot utilities and 2 cold utilities can be generated from any of the energy sources considered in Table 1 (i.e. scenario 1), which are coal, solar, wind and biomass. Availability of energy sources such as solar and wind are time of day and season of year dependent, while availability of biomass is only season of operational year dependent. The details of these availabilities for Example 1 are shown in Table 5. This example was solved using only the constraint method. The Pareto set of solutions obtained is shown in Figure 2 while Figure 3 illustrates profile of fraction of total utility usage that is generated from renewable sources at various TACs. Figure 2 shows that as the environmental impact increases, TAC decreases. Figure 3 shows that less utilities are generated from renewable energy sources at low TACs, which is the reason for high environmental impact at low TACs. At high TACs, more utilities are generated from renewable sources, which is the reason for low environmental impact at this section of the graph. It should be known that these two graphs do not have consistently smooth curves, especially in Figure 3 where there is a sharp change in slope. This is due to the inability of the solution procedure to give consistently global optimal solutions for each point along the curve.

4.2 Example 2 The problem data used for Example 2 are the same as those used in Example 1. The key difference lies in the fact that Example 2 used scenario 2 as shown in Table 2, i.e. involves inclusion of natural gas as additional nonrenewable energy source with coal, and considered direct burning of coal and natural gas to generate the 3 steam levels. The constraint method and modified goal solution are both used in this example. Figure 4 shows the Pareto curve obtained while Figure 5 shows the network obtained for the modified goal method where a 50% compromise is used for each of TAC and EI. The model for the goal method has 8,683 equations, 9,089 variables and 1,600 discrete variables. The details of the solution are as follows: at the point of lowest TAC ( 1,974,528 $, represented as point B in Figure 4) the EI has a worst value of 1.41 10 /y, while at the best EI ( 2.32 10 /y, represented as point A in Figure 4) the TAC has a worst value of 2,375,142 $. The TAC obtained for this goal solution is 2,147,600 $, while the associated EI is 2.42 10 /y (represented as point C in Figure 4). This implies that between points B and C in Figure 4, the contribution by EI to the multi-objective function is greater than 50% while that of TAC is less than 50%. Between points A and B, the contribution of TAC to the multi-objective function is greater than 50% while that of EI is less than 50%. The solution of 50% compromise in the two objectives was obtained in 678.9 S of CPU time. The network (Figure 5) has 28 heat exchangers whose area are all designed to be large enough to exchange heat between the same pair of streams that are matched in more than one season and more than one period. Table 6 shows the heat load on each exchanger in seasons/periods where they serve. As indicated on this table, Exchangers 1 to 14, and exchangers 27 and 28 are utility exchangers, while the rest are process heat exchangers. The utility exchangers are selected based on whether the utilities are available in the seasons and periods concerned. They are also selected based on the constraints of simultaneously minimising TAC and EI.

A major issue identified in the heat load distributions shown in Table 6 is that, for some of the exchangers there exists significant differences in heat load duty at different seasons where the exchanger is active. This may imply that such exchangers are significantly overdesigned in some cases depending on how temperature driving forces are distributed at each of the periods/seasons for the exchangers. Overdesign issues can be addressed using time-sharing scheme as presented by Jiang and Chang (2013) and Jiang and Chang (2015). Some other way by which unnecessary overdesign can be overcome is to use a large number of stages, especially for the process heat exchange stages. In this example, there exists just 2 process heat recovery stages, hence process streams would have to split in order for their heat load to be satisfied as much as possible through process heat recovery. Although using a larger number of stages would require more computer resources in order to generate good solutions in reasonable times, due to the highly multidimensional nature of the problem, and the large number of utilities available.

15  

In the modified goal solution obtained for this example, biomass is used to produce HPS and MPS and these utilities are transferred as heat to the process streams in exchangers 1, 2, 3 and 4. Wind is used to produce both HPS and MPS using exchangers 5, 6, 7, 8 and 9 to transfer heat, while natural gas is used to produce both MPS and HPS which are transferred as heat through exchangers 10, 11, 12, 13 and 14. Solar photovoltaic and coal were not used at all. The non-usage of solar can be attributed to its relatively high environmental impact, relatively high cost of operation and limited period of availability. Since the exchanger cost function includes an installation cost, the model would want to select utilities that are available in more periods and seasons so as to avoid having exchangers that would only be active in few periods/seasons. Choosing such exchangers that are only active in few periods/seasons may result in networks with large number of units. The high cost of photovoltaic solar panel is due to the fact that their commercialisation as a form of utility generation technology is still in the infancy stage in most countries of the world. Also, meeting the process heat demand of networks having high heat load as considered in this work, would require large panel areas. The relatively high environmental impact of solar photovoltaic are due to large land area required for installation which may result in habitat loss, and use of hazardous materials for manufacture of panels and maintenance. In the case of coal, its non-usage is due to its high environmental impact. For cold utilities, only cold air generated from natural gas is used in exchangers 27 and 28 respectively. The ability to identify the potential contribution, in terms of cost and EI, of various energy sources to multi-period HENS problems, as well as other process synthesis problems, can be used to influence factors such as government policies on renewable energies, kind of technologies for harnessing energy from renewable sources, etc., for the design of cost efficient and environment friendly heat demand processes. For this example, if the technology for harnessing energy from the sun had been improved in terms of cost, EI, and consideration of solar thermal heat storage opportunities, then solar energy source would have been a competitive option and may have been selected as one of the utility generation options.

4.3

Example 3

This example used the utility data shown in Table 2, i.e. scenario 2, however it excludes natural gas as a utility source. Also, it is regarded that the hot and cold process streams vary with time of day (multi-period) and time of year (multi-season). The details of the process stream parameters are shown in Tables 7 and 8, while availabilities of utilities are shown in Table 9. It can be seen in Table 9 that the seasons, as well as the periods in each season, both have unequal durations. In Tables 7 and 8, average values of process operating parameters have been presented. The authors of this paper are aware that in reality, process streams may vary from one season to another in an operational year, they may also vary with time of day. The nature of this variation, especially for those of time of day variation, is such that each day’s variation will be different from those of other days. However, it has been assumed in this paper that the time of day variation (represented as periods in this paper) are the same in each season. Implementing different time of day variation for each of the days that makes up a season would further complicate the multi-dimensional nature of the superstructure. Such a case would require the use of an additional set which comprises the multi-period profile of each day and in each season. Alternatively, such cases can be solved as a fully flexible HEN problem. However, this scenario is not considered in this paper. The modified goal method which was applied in this example also used a 50% compromise in each of the two objectives as optimisation criteria. The solution which was obtained in 511.2 s of CPU time has the following details: at the point of lowest TAC ( 615,663 $) the EI has a worst value of 2.58 10 /y, while at the best EI ( 8.00 10 /y) the TAC has a worst value of 928,232 $. So a 50% compromise in these two objectives gave a TAC of 7,735,800 $ and an EI of 8.93 10 /y. The network for this solution has 25 exchangers out of which 12 are utility exchangers. The details of the various energy sources selected for hot utility generation are: HPS, is generated from biomass, wind and coal. For cold utilities, wind energy and coal are used to generate cold air, while cooling water is generated from coal. Figure 6 illustrates the network for the goal solution while Table 10 shows the heat load distribution. Table 11 shows other solutions for two other scenarios of goal solutions which are: 80% contribution from TAC, 20% contribution from EI as one scenario, and 20% contribution from TAC and 80% contribution from EI as another scenario of compromises in contributions from the two objectives. This table 16  

shows that the more the consideration given to TAC, the lower its value while the higher the EI. On the other hand, the more the consideration given to EI, the lower its value, but he higher the value of TAC.  

5. Conclusion A new methodology for simultaneously integrating both renewable and non-renewable energy sources for utility generation, into multi-period HENS has been presented. The method establishes what combination of energy sources to select based on econmics and environemntal impact. The constraint and modified goal methods of multi-objective optimisation were used in the examples presented to determine the profile of potential solutions. Based on the nature of solutions obatined, it can be concluded that in order to fully harness the economic and environmental benefits of integrating renewable energy sources into the energy usage profile of process plants, other interventions are necessary. These interventions may include government policies in the area of offerring incentives to the development of improved technologies to harness energy from renewable sources such as solar and wind, construction of process plants in areas with relatively high solar and wind intensities, etc. These interventions are essential so that the solution to multi-objective energy integration models which consider economics and environmental imapct, like the one presented in this paper, will be driven in the direction of selecting cheap and environmentally friendly energy sources. Future studies will involve sensitivity analyses so as to check what the profile of solutions would be if one or more of the aforementioned interventions are put in place. Other issues that will be considered in future studies are details of utility generation technologies involved and opportunities for heat storage. Some limitations associated with the newly developed methodology are that the daily variation of process parameters are regarded as constant for all the days in each of the seasons in an operational year and they are made to coincide with the profile of each day's weather conditions in each of the seasons considered. In reality, this is not the case because both the daily variation of weather conditions and seasonal variations are somewhat stochastic in nature. Other limitations are that some of the heat exchangers were either significantly overdesigned for some of the duties they have to perform, or were redundant in some of the seasons/periods, and global optimal solutions cannot be guaranteed as can be seen in Figures 2, 3 and 4 where the curves obtained are not very smooth. Also, for some of the energy sources such as biomass, an intermediate step of having to generate electricity which was then used to generate various levels of steam was adopted. This is due to the fact that SimaPro, which was used for the Life Cycle Assessment, does not include data for directly generating steam from biomass. This approach of having to first generate electricity before generating steam would be more beneficial for a case where the multi-period profile of process heat demand of process plants, the multi-period profile of regional electricity demand and the multi-period profile of availability of renewable energies are integrated. This would go a long way in enhancing regional energy planning that are sustainable not only in terms of economics, but also in terms of environmental impact. It is hoped that these shortcomings will be addressed in future studies, but on the whole, this work has laid the foundation for techniques on how to simultaneously integrate renewable and nonrenewable energy sources in multi-period HENS problems considering both economics and environmental impact.    

Acknowledgements This study is supported by the National Research Foundation of South Africa (Grant numbers: 85536 and 87744), the Research Office of the University of Cape Town and the Slovenian Research Agency’s Program P2-0032. These supports are gratefully acknowledged.

17  

APPENDIX This section describes the initialisation procedure. In the notations used below, . L represents the initial value from where GAMS would start the search, . LO represents the lower bound, while . UP represents the upper bound.

Temperature: Hot streams

t,

, ,

.L

T,

,

; t ,

, ,

. LO

T,

,

; t ,

, ,

. UP

T,

,

Cold streams

t,

, ,

.L

T,

,

; t ,

, ,

. LO

T,

,

; t ,

, ,

. UP

T,

,

Heat load: q,, q,,

, , , ,

.L

MIN

. UP

MIN

F, F,

Tsi,s,p ‐Tti,s,p and F ,

, ,

Tsi,s,p ‐Tti,s,p and F ,

Ttj,s,p ‐Tsj,s,p

, ,

Ttj,s,p ‐Tsj,s,p

Exchanger driving force: Tsi,s,p ‐Tsj,s,p

dt , ,

, ,

.L

dt , ,

, ,

. LO

EMAT

dt , ,

, ,

. UP

600

18  

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20  

Paterson, W.R., 1984. Shorter communication: A replacement for the logarithmic mean. Chemical Engineering Science, 39(11), pp.1635–1636. Rosenthal, R. E., 2012. GAMS - A User's Guide, Washington, DC, USA: GAMS Development Corporation. Short, M., Isafiade, A.J., Fraser, D.M., Kravanja, Z., 2016a, Synthesis of heat exchanger networks using mathematical programming and heuristics in a two-step optimisation procedure with detailed exchanger design, Chemical Engineering Science, 144, 372 - 385. Short, M., Isafiade, A.J., Fraser, D.M., Kravanja, Z., 2016b, Two-step hybrid approach for the synthesis of multiperiod heat exchanger networks with detailed exchanger design, http://dx.doi.org/10.1016/j.applthermaleng.2016.05.065 Smith, R., 2005, Chemical Process Design and Integration. John Wiley & Sons, Ltd, West Sussex, England. Tantimuratha, L., Asteris, G., Antonopoulos, D.K., Kokossis, A.C., 2001, A conceptual programming approach for the design of flexible heat exchanger networks, Computers and Chemical Engineering, 25, 887-892. Vaskan, P., Guillén-Gosálbez, G. & Jiménez, L. 2012. Multi-objective design of heat-exchanger networks considering several life cycle impacts using a rigorous MILP-based dimensionality reduction technique. Applied energy. 98:149-161. Verheyen, W. and Zhang, N., 2006, Design of flexible heat exchanger network for multi-period operation, Chem. Eng. Sci. 61: 7760-7753. Yee, T. & Grossman, I., 1990. Simultaneous Optimization Models for Heat Integration—II. Heat Exchanger Network Synthesis. Computers & Chemical Engineering, 14(10), pp. 1165-1184.

21  

Interval 1

Interval 2

Interval 3

Interval 4

HU1,s,p HP1,s,p HU2,s,p HU3,s,p

HP2,s,p

CU1,s,p CP1,s,p CU2,s,p CP2,s,p

  Figure 1: Expanded stage-wise superstructure model including periods and seasons of operations for utilities and process streams 2.32E+06

Total annual cost $

2.27E+06 2.22E+06 2.17E+06 2.12E+06 2.07E+06 2.02E+06 1.97E+06 7E+09

1.1E+10

1.5E+10

1.9E+10

2.3E+10

Environmental impact 1/y Figure 2: Pareto optimal fronts for Example 1

22  

2.35E+06 2.30E+06

Total annual cost $

2.25E+06 2.20E+06 2.15E+06 2.10E+06 2.05E+06 2.00E+06 1.95E+06 0.35

0.45

0.55

0.65

0.75

0.85

Fraction of renewables used Figure 3: Fraction of renewables used compared to total utility usage for Example 1

2.41E+06 2.36E+06

A

Total annual cost $

2.31E+06 2.26E+06 2.21E+06 2.16E+06

C

2.11E+06 2.06E+06 2.01E+06 1.96E+06 2.20E+09

B 5.20E+09

8.20E+09

1.12E+10

1.42E+10

Environmental impact 1/y   Figure 4: Pareto optimal fronts for Example 2

23  

Figure 5: Optimal network for modified goal solution obtained for Example 2

  Figure 6: Optimal network for 50% compromise in both objectives using the goal solution method for Example 3

24

 

Table 1: ReCiPe and cost values for utilities generated from various energy sources for scenario 1 Utility Type ReCiPe method indicator values (1/kJ) and costs ($/(kW·yr)) for utilities Coal Wind Solar Biomass EI Cost EI Cost EI Cost EI Cost 57 70 92 50 Low pressure steam (160°C) 1.20E-01 3.00E-03 1.62E-02 3.72E-03 100 124 162 90 Medium pressure steam (250°C) 1.10E-01 3.18E-03 1.72E-02 3.95E-03 171 210 276 152 High pressure steam (400°C) 1.04E-01 3.47E-03 1.87E-02 4.31E-03 10 12 13 8.89 Cooling water (CW, 30°C - 40°C) 1.57E-03 4.54E-05 2.45E-04 5.64E-05 5 6 8 4.44 Cold air (CA, 40°C - 65°C) 1.12E-03 3.24E-05 1.75E-04 4.03E-05

Table 2: ReCiPe and cost values for utilities generated from various energy sources for scenario 2 Utility Type EI 6.63E-02

Cost 57

ReCiPe method indicator values (1/kJ) and costs ($/(kW·yr)) for utilities Natural gas Wind Solar Biomass EI Cost EI Cost EI Cost EI Cost 22.5E-03 63 3.00E-03 70 1.62E-02 92 3.72E-03 50

6.69E-02

100

23.0E-03

112

3.18E-03

124

1.72E-02

162

3.95E-03

90

7.29E-02

171

26.1E-03

190

3.47E-03

210

1.87E-02

276

4.31E-03

152

95.4E-05

10

34.1E-05

11

4.54E-05

12

2.45E-04

13

5.64E-05

8.89

68.1E-05

5

24.4E-05

5.5

3.24E-05

6

1.75E-04

8

4.03E-05

4.44

Coal Low pressure steam (160°C) Medium pressure steam (250°C) High pressure steam (400°C) Cooling water (CW, 30°C - 40°C) Cold air (CA, 40°C - 65°C)

   

 

25

 

Table 3: Multi-season hot stream data for Examples 1 and 2 Stream Seasons F (kW/ ) TIN ( ) TOUT ( ) HP1 1 110 249 100 2 95 240 115 3 110 249 100 HP2 1 112 259 138 2 118 248 148 3 112 259 132 HP3 1 19.6 293 160 2 18.0 280 160 3 17.5 290 160 HP4 1 15.2 160 140 2 18.5 160 140 3 20.2 160 140 HP5 1 70.2 354 160 2 75.1 362 160 3 82.2 360 160

  Table 4: Multi-season cold stream data for Examples 1 and 2 Stream Seasons F TIN TOUT (kW/ ) ( ) ( ) CP1 1 119 96 170 2 128 96 170 3 122 99 160 CP2 1 115 106 270 2 125 106 270 3 122 110 265 CP3 1 139.4 172 256 2 130.3 172 265 3 135.1 172 273 CP4 1 121.6 162 210 2 129 162 209 3 132.5 162 214 CP5 1 116.4 230 270 2 105.4 220 275 3 114.5 225 270 CP6 1 134.4 253 284 2 124.4 250 290 3 120.4 249 286

   

 

26

 

Table 5: Availability of energy sources in periods and seasons for Examples 1 and 2 Operational year Daily operation Biomass Solar Wind Natural gas Coal

Season 1 Period 1 √ √

Period 2 √ √ √ √ √ 

√ √ 

Season 2 Period 3 √ √ √ √

Season 3

Period 1

Period 2

Period 3

√ √

√ √ √ √

√ √ √

Period 1 √

√ √

Period 2 √ √ √ √ √ 

Period 3 √ √ √ √ 

    Table 6: Heat exchangers areas and heat load distribution among periods, seasons and exchangers using the modified goal solution for 50% compromise in each objective for Example 2 Match

1 (HPS, biomass) 2 (HPS, biomass) 3 (HPS, biomass) 4 (MPS, biomass) 5 (HPS, wind) 6 (HPS, wind) 7 (HPS, wind) 8 (MPS, wind) 9 (MPS, wind) 10 (HPS, Nat gas) 11 (HPS, Nat gas) 12 (HPS, Nat gas) 13 (MPS, Nat gas) 14 (MPS, Nat gas) 15 16 17 18 19 20 21 22 23 24 25 26 27 (CA, Nat gas) 28 (CA, Nat gas)

 

Area (m2)

Heat load (kW) Season 2 Period 1 Period 2 Period 3

Season 1 Period 2

Period 3

19 -

3336 555 1739 2080 -

3336 555 1739 2080 -

4278 4621 3156 2405

4621 3156 2405 3249 4278 -

305.8

-

-

-

3249

5592.9 1214 5831.6 509.7 358.2 411.7 5273.1 959.5 2719.8 71.1 98.6 295.7 25.6 9.3

9968 2427 9119 2035 259 4656 6422 4433 5837 313 304 551 147

9968 2427 9119 2035 259 4656 6422 4433 5837 313 304 551 147

9968 2427 9119 2035 259 4656 6422 4433 5837 313 304 551 147

6022 4976 8431 531 1335 3392 5853 3369 1785 294 370 4860 158

92.6 86.4 56.9 94.1 125.1 86 70.7 141.1 305.8 125.1 86 70.7 141.1

Period 1 3336 555 1739 2080 -

Period 1

Season 3 Period 2

Period 3

4621 3156 2405 3249 4278 -

3099 2959 491 763 -

3046 2975 491 523 258 -

3046 2975 491 523 258 -

-

-

-

-

-

6022 4976 8431 531 1335 3392 5853 3369 1785 294 370 4860 158

6022 4976 8431 531 1335 3392 5853 3369 1785 294 370 4860 158

9818 3963 9191 1496 544 5153 6275 5033 6890 235 404 181 297 253

9568 3963 9199 1471 571 5153 6515 5025 6632 233 404 468 307 224

9568 3963 9199 1471 571 5153 6515 5025 6632 233 404 468 307 224

 

27

 

Table 7: Multiple period/season hot stream data TIN Stream Season Periods F (kW/ ) ( ) HP1 1 1 110.6 249 2 109.0 248 3 110.6 249 2 1 117.03 239 2 118.00 238 3 117.03 239 3 1 110.6 249 2 109.00 250 3 110.6 249 HP2 1 1 112.7 259 2 113.00 260 3 112.7 259 2 1 118.5 245 2 120.00 248 3 120.00 248 3 1 112.7 259 2 114.00 260 3 112.7 259 HP3 1 1 18.60 280 2 17.00 285 3 18.00 290 2 1 19.00 280 2 20.00 279 3 18.00 279 3 1 18.00 288 2 17.00 289 3 16.50 290 HP4 1 1 18.00 162 2 20.00 160 3 19.00 159 2 1 18.00 161 2 19.00 160 3 21.00 161 3 1 19.00 159 2 17.00 160 3 18.00 160 HP5 1 1 110.0 354 2 113.0 356 3 110.0 356 2 1 111.10 362 2 111.10 362 3 115.10 362 3 1 105.00 358 2 111.20 358 3 110.00 360

TOUT ( ) 100 99 100 110 109 105 100 101 100 128 129 128 138 139 139 138 130 128 162 160 159 161 159 160 159 161 160 61 59 62 60 60 60 60 62 63 160 159 158 160 160 160 160 157 155

   

 

28

 

Stream CP1

CP2

CP3

CP4

CP5

CP6

Table 8: Multiple period/season cold stream data TIN seasons Periods F (kW/ ) ( ) 1 1 119.00 96 2 120.00 96 3 119.00 96 2 1 120.00 96 2 118.00 96 3 118.00 96 3 1 118.00 99 2 116.00 98 3 118.00 99 1 1 116.00 105 2 115.00 104 3 116.00 106 2 1 115.00 106 2 115.00 106 3 115.00 106 3 1 110.00 110 2 110.00 108 3 112.00 109 1 1 105.00 170 2 106.00 172 3 109.00 170 2 1 110.30 172 2 110.30 170 3 110.30 172 3 1 111.10 172 2 108.10 173 3 107.10 172 1 1 119.60 160 2 121.60 162 3 118.60 162 2 1 121.00 162 2 123.00 158 3 123.00 160 3 1 120.50 162 2 122.50 162 3 124.50 162 1 1 112.40 230 2 111.40 230 3 108.40 230 2 1 105.4 222 2 105.4 220 3 105.4 220 3 1 113.50 226 2 112.50 225 3 113.50 225 1 1 130.40 253 2 134.40 253 3 132.40 253 2 1 124.40 250 2 126.40 250 3 124.40 250 3 1 128.40 252 2 126.40 252 3 130.40 249

TOUT ( ) 172 171 170 173 171 172 165 165 164 272 270 268 270 270 270 266 265 265 254 256 256 265 263 265 262 263 261 210 208 210 209 211 210 214 214 214 270 270 270 273 275 276 270 270 270 284 284 284 292 292 290 288 286 286

   

 

29

 

Table 9: Availability of energy sources, with durations, in seasons and periods for Example 3 Operational year Daily operation Biomass Solar Wind Coal

Period 1 (6.5 hrs) √

√

Season 1 (3000 hrs) Period 2 (11 hrs) √ √ √ √

Period 3 (6.5 hrs) √ √ √

Period 1 (10 hrs)

Season 2 (2500 hrs) Period 2 (4 hrs)

Period 3 (10 hrs )

√

√ √ √

√ √

Period 1 (8.5 hrs) √

√

Season 3 (2500 hrs) Period 2 (7 hrs) √ √ √ √

Period 3 (8.5 hrs) √ √ √

  Table 10: Heat exchangers areas and heat load distribution among periods, seasons and exchangers for 50% compromise in both objectives using the goal solution method for Example 3 Match

1 (HPS, biomass) 2 (HPS, wind) 3 (HPS, coal) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 (CA, wind) 18 (CA, coal) 19 (CA, wind) 20(CA, wind) 21 (CA, coal) 22 (CA, wind) 23 (CW, coal) 24 (CA, wind) 25 (CA, coal)

Area (m2)

43.9 88.8 80.2 259 1123 763 1068 387 322 277 85 17870 522 73.5 662 13125 104 112 83 45.6 58 76 70 160 36

Period 1 25 3571 4018 898 5980 3156 3598 606 15610 4310 1164 8405 870 903 1336 654 1297

Season 1 Period Period 2 3 3103 3689 4166 4104 923 882 5594 5693 3426 3174 3533 3454 176 94 3 15571 15615 4369 3667 1167 1421 361 30 8904 9198 670 864 1738 1715 2138 1671 1659 1813 35 55

Period 1 2085 2814 5225 1100 5687 3836 1470 1752 15024 4179 1140 1108 8506 73 1653 711 21

Heat load (kW) Season 2 Period Period 2 3 3088 2559 2473 2588 5309 4976 1178 1061 6519 6150 3906 3783 1530 2282 1524 1738 141 14954 14936 4085 4342 1222 1080 1070 958 8734 8520 268 746 3 1440 1810 830 1163 1.4

Period 1 1284 3518 3339 1100 6266 2199 3894 1690 14961 3143 1127 8309 1519 709 1359 754 1222

Season 3 Period 2 2891 4298 1077 6370 1995 3986 1472 328 14947 3800 1081 8257 1294 1759 2016 1666 18

Period 3 363 3207 4461 1064 6474 2147 4044 1528 238 15087 3413 1047 4 8004 1393 1670 2128 1742 34

Table 11: Various solutions using the modified goal solution method for Example 3 Objective TAC ($) EI /y No of units

80% TAC 20% EI 708,360 10.6 10 24

50% TAC 50% EI 7,735,800 8.93 10 /y 26

20% TAC 80% EI 896,190 8.07 10 25

 

30