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Multi-objective optimization applied to retrofit analysis: A case study for the iron and steel industry
Applied Thermal Engineering 91 (2015) 638e646
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Multi-objective optimization applied to retrofit analysis: A case study for the iron and steel industry Alessandro Maddaloni*, Giacomo Filippo Porzio, Gianluca Nastasi, Valentina Colla, Teresa Annunziata Branca TeCIP Institute, Scuola Superiore Sant'Anna, Via Moruzzi 1, Pisa 56124, Italy
h i g h l i g h t s A model for optimal energy exploitation in integrated steelworks is presented. The model is formalized into a multi-objective optimization problem. The solution leads to reduced CO2 emissions and increased profits. The exploitation of the system is presented in a retrofit scenario.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 December 2014 Accepted 20 August 2015 Available online xxx
Steel production is among the most energy-intensive industrial processes and is a very relevant source of CO2 emissions. Steelworks, like most process and manufacturing industries, are facing the complex challenge of minimizing CO2 emissions while improving energy efficiency and developing effective strategies for process optimization. To this end, Process Integration methods can be successfully applied to the integrated steelmaking route with the purpose of achieving a reduction in the CO2 emissions, while optimizing material and energy systems. The work presented in this paper is finalized at the development of a model for optimal exploitation of energy resources in integrated steelworks, through the application of multi-objective optimization techniques, which can be beneficial both in new design and as a retrofit. Using these techniques, a series of solutions can be found representing optimal tradeoffs among different objective functions and the designer can select the best one or specify further criteria. Combining these methods, a retrofit was devised which optimized off-gas exploitation. This retrofit was then compared to the current operation. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Steel industry Multi-objective optimization Cost savings Industrial energy system Process simulation and optimization CO2 emissions reduction
1. Introduction The iron and steel industry is an intensive consumer of energy and carbon-bearing materials such as fossil fuels (e.g. coal, coke, oil) [1]. The reduction of materials and energy consumption as well as the control of all the emissions (not only the atmospheric ones, but also emissions into water bodies and soil) is a key objective. This will increase the sustainability and competitiveness of the production cycle. In fact, increased material re-use and recycling can contribute to reducing primary raw material consumption and to
* Corresponding author. E-mail addresses: [email protected] (A. Maddaloni), [email protected] (G.F. Porzio), [email protected] (G. Nastasi), [email protected] (V. Colla), t.branca@ sssup.it (T.A. Branca). http://dx.doi.org/10.1016/j.applthermaleng.2015.08.051 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
better exploiting off-gases through their improved distribution among consumer processes. It can lead not only to reduced emissions but also to savings in fuel gases (e.g. natural gas) and to efficiency improvements in power generation (where off-gases are exploited), so that an overall reduction in energy consumption is finally achieved. There are cases where the cost of emissions reductions and material recovery, exceeds the related savings in raw materials and energy. In the cases where the replaced raw material has a low purchase cost, recovery improvement can actually be even more costly and energy consuming than standard operations. However the need to comply with ever increasing environmental constraints (which can also indirectly raise the costs of waste disposal and emissions) and to improve sustainability of production cycles, stimulates the steel sector to search for optimal tradeoffs between the implementation of costly retrofits to reduce
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Nomenclature
Abbreviation BF blast furnace BOF basic oxygen furnace DRI direct reduction of iron GA genetic algorithm GH gasholder LHV lower heating value LP linear programming MILP mixed integer linear programming MOO multi-objective optimization Parameters N number of total iterations of the dynamic model m number of iterations of the 3-constraint method L constant parameter set by the gas network managers pi production of i-th gas dj demand of j-th consumer Mi maximum volume of i-th gas vij minimum gas volume through a pipeline Vij maximum gas volume through a pipeline lj minimum lower heating value of j-th consumer uj maximum lower heating value of j-th consumer vij minimum gas volume through a pipeline gi maximum gas volume out of i-th gasholder Lg minimum gas volume to be stored in g-th gasholder
emissions and while ensuring the economic viability of the production cycle. In particular, significant effort has been expended thus far by the steelmaking sector to reduce the emissions of GreenHouse Gases (GHG), especially carbon dioxide (CO2). Currently, 1.9 tons of CO2 are emitted per ton of produced steel and about 4e5% of total world CO2 emissions are attributed to the steelmaking sector [2]. In order to face the ambitious challenge of reducing GHG emissions, the European process industries (including the steel sector) and more generally the energy sector, are undergoing important retrofit operations to their production sites, i.e. they are adding new technologies to the existing production systems. For instance, the importance of incorporating new technologies in the steel production cycle in order to reduce CO2 emissions and energy consumption has been discussed in Ref. [3]. The fundamental role of energy efficient measures for reducing CO2 emissions in the iron and steel sector has been discussed in Ref. [4], where the importance of taking into consideration the overall relationship between the single technologies for mitigating consumptions and emissions and the overall production cycle is also underlined. This can avoid expending efforts on improvements that are only at a single process levels but not on a global perspective. A comprehensive review of emerging technologies for improving energy efficiency and reducing CO2 emissions is provided in Ref. [5]. However the adoption of alternative production technologies is slow, due to the considerable investments required, as well as the need for huge amount of alternative fuels, such as the natural gas needed for the Direct Reduction of Iron ore (DRI): in fact, the possibility of exploiting process gases to improve economic viability has also been investigated [6]. A series of more viable alternative techniques (such as coke oven gas injection, injection of plastic scraps into the Blast Furnace, pure oxygen blowing and heat recovery), which do
639
Ug c cij pCO2 fi ppj LHVi
maximum gas volume to be stored in g-th gasholder natural gas cost [V/GJ] cost of building pipe ij [V] market price of CO2 [V/t] emission factor for every gas [t CO2/GJ] gas selling price to power plants [V/GJ] lower heating value of the i-th gas
Variables xij yij zj Z1 Z2 Vg(k)
gas flowrate from i-th producer to j-th consumer [GJ/P] binary variable for the pipe usage binary variable for the CO2 calculation total CO2 emission total cost volume of g-th gasholder at the k-th iteration
Sets P C G T PP
set set set set set
of of of of of
producers consumers gasholders torches power plants
Subscript i i-th producer j j-th consumer g g-th gasholder n natural gas source
not fundamentally alter the traditional steel production cycle but allow a reduction in the consumption of fossil fuels and consequently CO2 emissions and energy recovery, is proposed in Ref. [7]. Waste heat recovery is also an important way to improve energy efficiency, as has been underlined in a recent study concerning the Chinese iron and steel industry [8]. Thus considerable efforts have been spent in this direction. For instance in Ref. [9] a simulation is described for recovering waste heat from the cooling phase of the sintering process, in Ref. [10] an innovative system to recover heat in the continuous casting process is presented, while in Ref. [11] a heat recuperation system for steel slags is described. Considerable further potential for energy recovery in steelworks can be found by applying pinch analysis [12] and derived total site approaches such as the total site heat integration which is described in Ref. [13]. Another important opportunity for energy recovery and GHG emissions reduction in the iron and steel industry, derives from enhanced exploitation of off-gas streams, that can represent not only a way to decrease their environmental impact but also an economic advantage, due to the considerable heating value of the off-gases [14]. Optimizing the distribution of the different off-gases among different internal and possibly external utilities can improve exploitation of off-gases themselves. However to this aim an optimal design of the gas network is also needed. This objective can be obtained though suitable retrofit actions which, in most of the cases, are related to existing infrastructures and industrial parks. Retrofit actions are often preceded by deep analyses aimed at pointing out and evaluating all the potentials for environmental improvements and energy savings. Process modeling and optimization techniques provide an effective support within such retrofit analyses. A relevant example concerning the chemical sector is provided in Ref. [15] by Carvalho et al., who propose a generic and systematic methodology to identify and screen the design retrofit
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potential of chemical processes. Another interesting example is provided by Chen et al., who present in Ref. [16] a mathematical model for the operational optimization and retrofit of industrial steam systems that is applied to a petroleum refinery: in this case the maximization of the cost reduction (expressed as the difference between annual operating costs and annualized investment cost of the retrofit) is formulated as a Mixed Integer Nonlinear Program (MINLP) based on a collection of unit models. In Ref. [17] a similar methodology is applied to the analysis and design of steam power plants in a typical steel mill, while in Ref. [18] an entire energy system for chemical plants is designed by exploring the interactions between the steam network and the heat recovery networks of process plants. In Ref. [19] a model is proposed to allocate the available energy resources to the Turkish Manufacturing Industries by exploiting a Multi-Objective Optimization (MOO) technique, as the considered problem can be viewed as a multiple criteria decision-making issue including human judgments, tangible and intangible criteria and priorities that require the achievement of a suitable trade-off between goals and criteria [20]. MOO techniques are actually mandatory when a resolution method for retrofit analysis is needed to resolve the conflicts between competing targets and observing complex constraints. For nontrivial MOO problems there is not a unique optimal solution in general: all the available MOO techniques allow a series of solutions to be found, representing optimal trade-offs among the different specified objective functions. It is up to the designer/manager to select the best one, according to the considered process conditions and systems specifications or, eventually, to specify further criteria driving the final selection of the best compromise solution. Integrated process models to formulate optimization problems considering multiple objectives, have been used in past research, to find trade offs between CO2 emissions minimization and cost minimization in an integrated steelwork. This was done by optimizing an existing process gas network, according to the approach described in Ref. [14] and further refined in Ref. [21]. The solution explored in the research described in this article exploits a dynamic mathematical model of a fixed gas network structure, embedded in a software decision support system, to recommend optimal gas distribution patterns. An MOO problem is then formulated [22], where cost and CO2 emissions are set as objective functions and the optimization is carried out by means of a particular Genetic Algorithm (GA), exploiting the concept of Pareto dominance to select a set of Pareto-optimal solutions (the so-called Pareto-set) [23]. These solutions meet a trade-off optimality criterion firstly introduced by Edgeworth in 1881 [24] and subsequently generalized by Pareto in 1896 [25], according to which a solution is included to the Pareto set if no other solution exists, which can improve at least one of the objectives without degrading the other ones. GAs have been selected as they allow a flexible problem formulation and an effective generation of a set of different trade-off solutions. This approach has been recently applied in Ref. [26], to find the best operating condition of a distribution network for natural gas, facing three different and conflicting objectives, i.e. maximize gas delivery flow and line pack and minimize operating costs. However GAs also have the drawback of considerable computational complexity, therefore an alternative approach has been attempted in Ref. [14], that is based on the 3-constraint method [27], which has already been applied in the past for facing MOO in analogous applications [28]. Such an approach belongs to the wide category of Mixed Integer Linear Programming (MILP) methods, that have been widely adopted to confront optimization problems for the retrofit of large heat exchanger networks [29]. The work described in the present paper represents a further step towards the development of practical and reliable means to optimize industrial off-gas networks from both the structural and
the functional point of view. In fact, this paper presents an extension of the above-described approach, including the additional possibility of assessing impacts on process performances and operating costs, when creating new connections among existing gas producers and consumers. A cost function is introduced for the installation of new pipelines, although the possibility of exploiting the existing network is kept. This key feature of the proposed approach, allows it to support a feasibility study of actual retrofits, which in real world scenarios can obviously include the enlargement of the distribution network, provided that the associated cost is affordable, as covered by the gains achievable through the consequent improved exploitation of off-gases. Moreover, based on the original model, a dynamic scenario has been analyzed and two optimization procedures have been developed: a stepwise optimization and a global optimization. Both approaches use the previously developed model as a discretization step to study the network in a fixed time interval, but while the stepwise approach solves as many problems as the number of steps, the global approach solves one bigger problem that accounts for all the steps at once. This aspect represents a further key advancement with respect to the previous work, as the global approach is in principle capable of taking into account the global performance over the whole time interval and all the scenarios that can arise by paying the price of a slight increase in computational time. In order to validate the proposed approach, the possibilities for a retrofit design of the gas distribution system of a real steelmaking plant are explored and the results of the local and global optimization approaches are presented and discussed. The paper is organized as follows: in Section 2, the formulation of the extended process gas model is presented; the dynamic models are presented in Section 3, while in Section 4 the experimental results in the two cases are described and compared. Finally in Section 5 some concluding remarks are proposed. 2. Model description A general industrial off-gas network is considered, which is usually composed of: 1. Processes that produce gas; 2. Processes that consume gas; 3. Gasholders, that can stock gas within their volume limit and deliver it. Noticeably there are some processes, such as the system Blast Furnaces (BF) e regenerators, and coke oven plants, which both produce and consume gas. Among the pure consumers there are, for instance, torches and power plants. Gasholders represent a buffer gas capacity and can therefore be treated as both gas sources and sinks, ensuring the satisfaction of the mass balance over different time periods. According to the approach introduced in Ref. [30] and summarized in the following paragraphs for the sake of clarity and completeness, each process/gasholder is modeled by a node in the network and each pipe connecting two elements of the gas network is modeled by a directed arc. Let P and C be the sets of processes which produce and consume gas, respectively. For those processes belonging to both sets in the model, 2 copies of the same process are considered: one as a producer and one as consumer (see the graph in Fig. 1). Each producer node has a correspondent gasholder. Let G be the set of gasholders, which are both gas producers and consumers. Finally let the symbols T and PP indicate, respectively, the sets of torches and power plants, which are subset of the consumers. A special production process is the source of the natural gas, which might serve other processes and is indicated in the following with the symbol n.
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the gaseous fuel entering the process and is computed through the following formula:
X
, xij
i2P
X xij LHVi i2P
(3)
Which represents the weighted average of the lower heating values of each of the gases in the mixture. The resulting LHV must lie within a lower (lj), and upper bound (uj), i.e.
lj
X
, xij
i2P
X xij uj LHVi i2P
(4)
Also the volume of gas to flow in a pipe is constrained. Each pipe connecting the i-th process to the j-th one is characterized by minimum and maximum volumes, vij and Vij respectively, namely
vij xij LHVi Vij
(5)
Moreover the volume of gas reaching the j-th consumer must be lower than a threshold value Mj, i.e.
X
, LHVi Mj
xij
(6)
i2P
Finally an upper bound gi exists also on the outgoing gas volume of the gasholders, which means that for the i-th gasholder the following inequality holds: Fig. 1. The bipartite digraph representation of a gas network: green circles represents fuel gas providers (BF e Blast Furnace, COK e coke oven plant, BOF e Basic Oxygen Furnace, CH4 natural gas supply), yellow circles represents consumers which actually exploit off-gases within their process (PP-power plant), while gray circles represents 3 different torches, which are consumers that actually burn off-gases into the atmosphere without exploiting their value. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A network of pipes, which connects producers and consumers, can be represented as a bipartite digraph [31], with vertex set P∪C and arcs directed from the elements of P to some elements of C. Let xij be the amount of gas flowing in the pipe ij which links producer i to consumer j. The binary variable yij represents the use of the pipe ij: if the pipe ij is not used (namely xij ¼ 0), yij is null, while if the pipe ij is actually used (i.e. xij > 0) yij is unitary. There might be pipes ij that are not in the network and cannot be constructed, in this case the constraint xij ¼ 0 is imposed. This is the case, for instance, of pipes which link gasholders and producers of gases different from the one that is stored in the gasholder. Thus the possibility of building pipes that are not included in the network but are feasible is considered. Each producer i, apart from natural gas, has to produce an exact amount of gas pi, namely:
ci2P v
X
xij ¼ pi
(1)
j2C
Consumer processes have a demand to meet: let the symbol dj indicate the demand for consumer j, for each consumer the following constraint holds:
X
xij dj
(2)
i2P
When two different gases are sent to the same consumer, a mixing occurs. The Lower Heating Value (LHV) of the gases mix at the inlet of process j represents the “heating quality” [GJ/Nm3] of
X
, LHVj gi
xij
(7)
j2C
In order to obtain the above-described condition where yij ¼ 0 if and only if the pipe ij is in use, the following inequalities (for every ij) are imposed
xij yij Lxij 10; 000
(8)
Here 10,000 is an upper bound for the maximum possible value of xij, while L is a constant value such that a flow lower than L1GJ in the pipe ij is not desirable, i.e. L1 is lower than the minimum amount of gas that is physically meaningful to send into a pipe. Hence if xij is null (or xij L1), then yij ¼ 0; if xij > L1, then yij ¼ 1.
2.1. Objective functions There are two objectives to jointly optimize: the CO2 emissions must be minimized while the profit from the selling of the gas must be maximized. To this aim, the total CO2 emission Z1, and the total monetary cost, Z2 are computed and minimized. Let fi be the CO2 binary emission factor of the i-th producer (e.g. if a producer, such as a gasholder, is non-polluting fi ¼ 0 holds). The emission of producer i is given by
fi
X
xij
(9)
j2C
The total CO2 emission Z1 is therefore provided by the following expression:
Z1 ¼
X
xij fi
(10)
i2P;j2C
The total cost Z2 is given by the sum of four components: the purchase cost of natural gas, the purchase cost/gain related to the
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acquisition/sale of CO2 allowances, the income deriving from the sale of gas to power plants and the cost of building a pipe ij. Let c be the unitary price of natural gas. Let the symbol zj indicate an integer variable associated to the j-th consumer process and lying in the range [1, 1]: zj ¼ 1 if process j represents a direct emission of CO2 within the company boundaries and therefore a loss of CO2 allowances; zj ¼ 1 if process j implicates a gas transfer with no associated direct CO2 emissions e with a resulting gain of CO2 allowances (such as, for instance, when gas is sold to third-party power plants); zj ¼ 0 in case no change occurs. Let ppj be the unitary gain for selling gas at power plant j. Finally let cij be the (amortized) cost of building the pipe ij (cij ¼ 0 holds if the pipe already exists). The total cost is given by the sum of four terms as follows:
Z2 ¼
X
c$xvj þ
j2C
X i2P;j2C
zj $fi $xij $pCO2
X i2P;j2P
xij $ppj þ
X
yij $ciji
i;j
expense of a slight increase in the complexity of the modeled system and a consequent increase of the computational time. The previous MILP model is used as a discretization of the continuous process by considering N intervals of duration Dt and by optimizing over all the intervals. Two different strategies for accomplishing this task are presented. The first strategy, which will be called stepwise optimization, runs the previous model at every time step, starting from the initial conditions derived from the previous run. The satisfaction of the mass balance between two consecutive steps is guaranteed by imposing constraints on the gasholders volumes at the “beginning” (gasholders inlet) and at the “end” (gasholders outlet) of each step. A more accurate global optimization model can be obtained by expressing all the time dependent variables and parameters into a single big MILP. This model becomes considerably more complex, as it contains roughly N times the variables of the original MILP (there is a variable for each time step). On the other hand this model is much more powerful than the stepwise model: it can foresee what happens in the future steps, while the stepwise model is blind, as it only optimizes over a single step, disregarding the future steps.
(11) The first terms accounts for the cost of purchasing the natural gas, the second term represents the cost or revenues coming from the commerce of CO2 allowances, the third term represents the income derived by the selling of gas to power plants and the last term is the cost of building the new pipes. The need to achieve a joint optimization of the two often counteracting objectives Z1 and Z2 leads to the formulation of a MOO problem, where the 3-approximation technique is used to generate a good approximation of the Pareto set, which represents the set of optimal trade-off solutions [23]. These solutions are by definition such that no other solution can have better values of both Z1 and Z2. More precisely the Mixed Integer Linear Programming (MILP) problem min Z1 is solved subject to the above-described constraints and the constraints on the unacceptable pipes (xij ¼ 0 in this case). From this MILP an optimum value Z* for Z1 is obtained, subsequently the m problems are solved, which are defined as follows for i ¼ 1, 2, …, m and a suitable choice of 31 32 … 3m:
8 > > <
min Z2 subject to Ax b > > : Z1 Z * þ 3i where Ax b represents the matrix form of the constraints (1)e(8). For increasing values of i, decreasing optimum values for Z2 and increasing values of Z1 are found. The m pairs obtained in this way allow describing a good approximation of the Pareto front of the solutions. For a more detailed discussion on Pareto optimality and 3-approximation see the Appendix A.
3. Dynamic model The model described so far has been realized for a sort of “instantaneous” process, which is designed to optimize a fixed time period, therefore time has not been considered as a variable. A more general representation can be achieved by modeling the gasholders' volume variations along time. Such a formulation presents the advantage of allowing to deal with different time steps in a more realistic scenario, where a step can be a day or any time period needed for all the processes to be completed. This new model does not lose any accuracy, at the
3.1. Stepwise optimization At every time step the variable xij represents the amount of gas flowing from process i to process j at the k-th iteration, namely in the time interval [(k 1) Dt, k Dt]. The only parameters which change over the time steps in the MILPs are the gasholders' volumes: their storage might increase or decrease at every time step. The volume Vg(k) of gasholder g at time step k is computed as follows:
Vg ðkÞ ¼ Vg ðk 1Þ þ
X xigc X xigp LHV LHVgp i i2P i2P
(12)
where gp and gc are the indices of the gasholder g as a producer and as a consumer, respectively. All the constraints of the previously developed static model must hold for every time step and there is an additional constraint to the time dependent model, stating that cgVg ðkÞ is limited, namely
Lg Vg ðkÞ Ug
(13)
where Lg and Ug are respectively the lower and upper bound for the volume of gasholder g. This model is optimized at every iteration (once for every time step) and before starting the new optimization the values of the gasholders' volumes are updated according to Eq. (12).
3.2. Global optimization In the global optimization strategy, instead of optimizing the N processes at every step, a broader time dependent model is built to optimize an overall function of the stepwise cost and CO2 emission. To this aim, the variables xij must be considered time dependent, i.e. they are replaced with the variables xij(k) representing the amount of gas flowing from pipe i to pipe j in the time interval [(k 1) Dt, k Dt]. The CO2 emission at step k is given by
Z1 ðkÞ ¼
X i2P;j2C
xij ðkÞfi
(14)
A. Maddaloni et al. / Applied Thermal Engineering 91 (2015) 638e646
While the overall CO2 emission is the sum of the stepwise emissions over all the time steps, namely:
Z1 ¼
N X
Z1 ðkÞ
(15)
k¼1
Similarly the stepwise and overall cost functions can be calculated through the following formulas:
Z2 ðkÞ ¼
X
c$xvj ðkÞ þ
j2C
þ
X
X i2P;j2C
zj $fi $xij ðkÞ$pCO2
X
xij ðkÞ$ppj
i2P;j2P
yij $cij
i;j
(16) And
Z2 ¼
N X
Z2 ðkÞ
(17)
k¼1
A new model is thus obtained, through which the sum of the stepwise objectives (namely Z1 and Z2) can be optimized in a single run instead of running N times a MILP. Obviously the computing time of the new bigger MILP is higher, since the number of variables becomes roughly N times bigger. However, on relatively small instances the running time is still acceptable, as it is shown in the next section. An optimal solutions obtained with this global model is never worse than an optimal solution obtained with the stepwise model, since the latter forms a feasible solution for the global model that can be obtained by just setting xij(k) to be the value of the optimal solution xij found at step k and using the same values of yij.
4. Experimental results The “instantaneous” model has been tested against a formerly developed model which is extensively described in Refs. [14] and [30]. This former model was built to represent a particular network structure without taking into account the possibility of creating new connections between process units (gas producers or consumers). The main innovative outcomes of the optimization system developed for this study are: 1) The described system is capable to optimize the operational conditions also taking into account changes in the structure of the gas network. 2) An enhanced Pareto frontier is achieved as an output of the analysis. 3) Two dynamic optimization approaches have been validated and compared in terms of achievable CO2 savings and profit increase; the global optimization approach achieves better results than the stepwise one. Both aspects allow a higher flexibility in terms of application within a real industrial context. The results from these contributions are separately described in the following sections.
4.1. Optimization of the network structure and operating conditions The model has been run on an industrial dataset coming from an Italian integrated steelwork gas network and consisting of a Coke oven plant (COK), a BF, a Basic Oxygen Furnace (BOF), a natural gas distributor (CH4), 3 gasholders (GH1, GH2, GH3), 3 power plants
643
(PP1, PP2, PP3), and 3 torches (T1, T2, T3). The pipes connecting different parts of the plant are depicted in Fig. 2a. Differently with respect to previously developed analyses, which only aimed to find the best way to distribute the off-gases into the existing network, the plant managers faced in this case the concrete problem of understanding whether the cost for enriching the network with new connections could be paid back through time by the potentially achievable savings in terms of natural gas savings and/or CO2 emissions reduction. A number of tests have been performed on data from a particularly critical period of time in order to explore the possibilities for profitable gas network modifications: Fig. 2a shows the complete network (i.e. including gasholders' connections as well as all the other possible links) that is represented in Fig. 1, while Fig. 2b represents the result of the optimization, as it reports only the pipes that need to be active to minimize the cost, i.e. the pipes ij for which xij > 0 in the final solution. The connections in red (in the web version) are those pipes that are not initially present in the network and hence need to be built, i.e. the pipes ij for which yij ¼ 1 in the final solution. Noticeably in the final layout two torches are no longer in use, which implies a significant reduction of CO2 emissions. In practice the proposed approach is capable to identify the possibility to rationalize the distribution of process off-gases within the existing network, thus allowing a lower flaring of excess gas stored in gasholders through torches. As a consequence, a lower overall consumption of natural gas is ultimately achieved. Two torches out of three in the case shown in Fig. 2b are in fact no longer in use (T2, T3), since all the gas originally burnt is consumed in other plants, allowing to save a useful resource while also achieving an environmental benefit. 4.2. Enhanced optimization results (extended Pareto optimality) The results of the model can also be analyzed in terms of improvements with respect to the previous model version measured on a Pareto front showing CO2 emissions and cost (Fig. 3). The figure puts into evidence that the new model has better local minima both for the CO2 emission and for the cost (taking into account that the negative value for the cost corresponds to a profit in every scenario). In particular, the profit can be raised up to 6.3% with respect to the maximum value obtained through the previously developed model [21] (from 958 V to 1018 V per time interval, with a correspondent increase of the CO2 emissions of 14.2%), whereas on the opposite side of the Pareto front a reduction of 1.1% of CO2 emissions can be reached at the expense of 20.8% cost increase (CO2 emissions decrease from 12.263 t to 12.133 t per time unit, while profit decreases from 549 V to 435 V per time unit). It must be noticed that the two Pareto fronts coincide in the interval where there is an intersection, due to the fact that some plant constraints must be taken into account in both the new and the old models. To sum up, the results of the performed tests show that the retrofitting approach is substantially better than the previously developed one as, by installing only 2 new pipes, a higher yearly profit (up to almost 5,000,000 V) as well as lower CO2 emission (up to 13,665 t less per year) can be obtained. Therefore the proposed MOO approach can effectively explore all the profitable trade-offs considering the existing network structure, in terms of: CO2 emissions; Natural gas consumption; Costs (in terms of materials and investments required to achieve such savings). A “smart” distribution of the produced off-gas within the existing network is obviously a cost-effective solution, which could
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Fig. 2. The original off gases network (left) and the network after the optimization (right).
be explored through the approaches already proposed in Ref. [14]. The above described algorithm goes far beyond this result, by suggesting practical solutions which increase the profit and saving margins at a cost which is affordable, as it is fully compensated by the achieved economical and environmental benefit: in fact all costs and benefits are both implicitly taken into account by the optimization procedure itself. 4.3. Comparison of dynamic (stepwise and global) optimization approaches The dynamic models depicted in Section 2 (aiming at stepwise and global optimization, respectively) have also been tested on the same case study concerning an off-gas network of an integrated steelmaking plant. The results of this further simulation show that the stepwise model does not perform significantly worse than the more general model, by achieving in all the tests optimum points that are less than 1% worse than those of the global model. Table 1 shows the optimum value both for the CO2 emission and for the profit for the two models obtained for different values of the
Fig. 3. Pareto fronts of the old model and the new instantaneous model (depicted in Section 1).
number N of total performed steps. It is worth noting that with less than 25 steps there is no difference between the two models in the total profit and there are tiny differences between the other optima both for the CO2 emission and for the total profit. Fig. 4 shows the Pareto fronts of the two different dynamic models for the case N ¼ 100. Noticeably the profit provided by the stepwise model is not an increasing function of the CO2 emission. This is due to the fact that, in order to compute the Pareto front of the stepwise model, looser constraints on the total CO2 emission are imposed. In the first steps these looser constraints lead to higher profits for those particular steps, but they need not lead to a better global profit (defined as the sum of the profits over the whole optimization landscape period). Indeed to improve the solution in the first steps, the gasholders get filled in a different way, therefore the following iterations have different initial condition (the initial gasholders' volumes), which might reduce the solution space and hence worsen the optimum. This is one of the main reasons why the dynamic global model is preferable to the dynamic stepwise model: the global model optimizes over all the scenarios that can arise in the different steps, while the stepwise model is not able to predict what happens in the future steps. To sum up, two advanced dynamic models have also been presented, which can both have a wide applicability in the industrial context. While the instantaneous model provides a time-efficient pre-screening of the gas network layout, the two dynamic models exploit the instantaneous model as a basis for their computation, which is either iterated or simultaneously performed for the different time steps. Therefore they are indeed more time consuming but also allow higher generalization capabilities. In fact, they allow variable boundary conditions over time, by thus providing a more realistic representation of the gas flow evolution through time. Such variability can for instance correspond to changes in processes and networks conditions (such as, for instance, changes in feed material and therefore gas composition, planned or unforeseen plants stops, leakages and temporary pipes unavailability due to extraordinary maintenance or operations). Thus they represent an advance with respect to previous achievements and a useful practical support for daily management of the
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Table 1 Optimal values for CO2 emission and total profit with different number of steps. Number of steps
CPU time stepwise (ms)
CPU time global (ms)
Stepwise CO2 (t)
Global CO2 (t)
D (%)
Stepwise profit (V)
Global profit (V)
D (%)
2 10 25 50 75 100
70 150 290 540 780 1010
70 150 360 730 1390 1840
30.86 134.99 332.33 661.22 990.10 1323.77
29.25 133.53 330.86 659.7 988.63 1321.21
5.23 1.09 0.44 0.22 0.15 0.19
3464.62 11,071.38 25,334.06 48,807.66 72,269.03 95,730.40
3464.62 11,071.38 25,334.06 49,054.84 72,795.02 96,416.69
0 0 0 0.51 0.73 0.72
gas network, allowing a higher flexibility in the planning of the daily gas distribution, whether it has to be optimized in advance and at a whole (e.g. in planning and optimizing the production cycles ahead of schedule) or in a quasi-online mode, with a shorter time horizon, represented by the variable optimization steps. 5. Conclusions and future work This paper introduces an innovative approach for the correct management of resources, by analyzing retrofits in the integrated iron and steel industries. These industries are among the most energy and carbon intensive in the world. This approach represents a significant extension of previously developed studies, as: It allows taking into account the possibilities of a retrofit design of the systems, through the addition of network connections (installation of new pipework). It enables plant managers to consider a wider range of operating conditions, thanks to the flexibility in the network structure. As a result, a wider Pareto front is achieved. It guarantees a higher flexibility, both in offline and online (i.e. quasi-real time) operations, through the possibility of applying both a static and a dynamic approach In fact, this approach simultaneously takes into account the achievable savings in terms of CO2 emissions and natural gas consumption of each consumer plant, when fed with the produced offgases. Minimizing the costs of installing new pipes, linking different gasholders to consumer processes, is also considered as an objective, allowing a more efficient network architecture. At the same time, the conditions in which the gasholder is full and the offgas must be flared are automatically minimized or avoided.
The innovative approach has been compared to an existing system and the incremental benefits of the proposed approach have been discussed and pointed out. A dynamic scenario has been analyzed through two different models, the first one solving many small problems iteratively and the second one solving a more complex problem at once. A comparison between the two models has been pursued, by exploiting the data coming from an existing off gas network belonging to an integrated steelworks. The results of the presented case study show that the solution to the overall problem achieves optimal results (in terms of Pareto optimality) thanks to a better avoidance of local minima, arising from the choice of a particular sub-optimal path. This aspect is particularly powerful in process simulation and retrofit. However, the stepwise optimization algorithm could be better suited to a dynamic and quasi-online support to the operations of the plant, since it allows taking into account variations in the operating conditions of the process. The practical interest of the proposed modeling and optimization approach are thus twofold: it can be applied not only for improving the management of an existing process gas network but also within feasibility studies aimed at a preliminary verification of the advantages of structural modifications of the network itself. In fact the proposed retrofit approach is capable of pointing out the best trade-offs between costs (investment and operating) and savings in terms of CO2 emissions and fuel consumption, by exploring different gas network layouts and automatically taking into account the costs of network modifications within the optimization procedure itself. Moreover, the proposed dynamic optimization models allow taking into account the variability of network and processes operating conditions which normally occur in the industrial context. In particular, one of the models faces the problem of limiting the computational burden through an iterative approach, while the latter one develops a global optimization over all the scenarios that can arise in the different steps. Acknowledgements The authors wish to thankfully acknowledge their industrial partners for their support, their constant availability and fruitful suggestions that allowed the incremental development of this analysis. Appendix A. Multi-objective optimization A.1. Pareto optimality A generic MOO problem can be defined as follows:
Fig. 4. Pareto fronts of the two dynamic models for N ¼ 100.
min fðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; …; fk ðxÞ subject to hðxÞ ¼ ½h1 ðxÞ; h2 ðxÞ; …; hm ðxÞ 0 with x ¼ ðx1 ; x2 ; …; xn Þ2X X3
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where x is the variable vector, f(x) is the objective vector, i.e. the vector having as elements the different objective functions, h(x) if the constraint vector and X is space of feasible solutions. For nontrivial MOO problems there is not a unique optimal solution in general, hence a set of so-called Pareto optimal solutions (or Pareto set) is searched. This set forms, in the objectives space, the so-called Pareto front. All the solutions belonging to the Pareto set have the property to be non-dominated. A solution a dominates b if and only if for each objective function fi, fi(a) fi(b) and for at least one objective function this inequality is strict, as it is formally stated below.
a_b⇔fi ðaÞ fi ðbÞci
∧
dj : fi ðaÞ < fi ðbÞ
A.2. 3-Constraint method This method allows to compute an approximation of the Pareto front for an MOO problem. In the case of two objective functions, the first objective f1 is minimized and a minimum f* is found. Subsequently a parametric constraint on the first objective is added to the problem stating that that the value of f1 must not exceed f* by 3i.
8 > > <
min f2 subject to hðxÞ 0 > > : f1 f * þ 3i These parametric problem is solved for an increasing sequence of values of parameters 31 32 … 3m The solutions to these problems have decreasing values for f2 and increasing values for f1 as the values of 3i increase. With the pair of values (f1, f2)i found in the m, a discrete subset of the Pareto front is generated. In other words, each objective is firstly optimized and then transformed into a constraint which is progressively released in order to optimize the second objective. References [1] International Energy Agency, Energy technologies perspectives 2010. Scenarios and Strategies to 2050, OECD Publishing, 2010. http://dx.doi.org/10. 1787/energy_tech-2010-en. http://www.oecd-ilibrary.org/energy/ energytechnology-perspectives-2010_energy_tech-2010-en. [2] Worldsteel, Steel's contribution to a low carbon future: worldsteel position paper. worldsteel.org/publications/position-papers/Steel-s-contribution-toalow-carbon-future.html (last accessed January 2012). [3] N. Pardo, J.A. Moya, Prospective scenarios on energy efficiency and CO2 emissions in the European iron & steel industry, Energy 54 (2013) 113e128. [4] N. Karali, T. Xu, J. Sathaye, Reducing energy consumption and CO2 emissions by energy efficiency measures and international trading: a bottom-up modeling for the U.S. iron and steel sector, Appl. Energy 120 (2014) 133e146. [5] A. Hasanbeigi, M. Arens, L. Price, Alternative emerging ironmaking technologies for energy-efficiency and carbon dioxide emissions reduction: a technical review, Renew. Sustain. Energy Rev. 33 (2014) 645e658. [6] C. Raggio, Recent developments of energy saving and environmental protection in the steel industry, Metall. Ital. 104 (10) (2012) 51e58. [7] K.-J. Li, J.-L. Zhang, Y.-P. Zhang, Z.-J. Liu, X. Jiang, Analysis on development of iron-making process based on the principle of energy-saving and emission reduction, Chin. J. Process Eng. 14 (1) (2014) 162e172.
[8] G.Y. Ma, J.J. Cai, W.W. Zeng, H. Dong, Analytical research on waste heat recovery and utilization of China's iron & steel industry, Energy Procedia 14 (2012) 1022e1028. [9] X. Zhang, Z. Chen, J. Zhang, P. Ding, J. Zhou, Simulation and optimization of waste heat recovery in sinter cooling process, Appl. Therm. Eng. 54 (1) (2013) 7e15. [10] A. Villar, J. Parrondo, J.J. Arribas, Waste heat recovery technology in continuous casting process, Clean Technol. Environ. Policy 17 (2015) 555e562. Alvarez, rrez Trashorras, E.A. [11] A.J. Gutie J.L. Río Gonz alez, J.M. Suarez Cuesta, J.X. Bernat, Design and evaluation of a heat recuperator for steel slags, Appl. Therm. Eng. 56 (1e2) (2013) 11e17. [12] K. Matsuda, S. Tanaka, M. Endou, T. Iiyoshi, Energy saving study on a large steel plant by total site based pinch technology, Appl. Therm. Eng. 43 (2012) 14e19. [13] K.H. Chew, J.J. Klemes, S.R. Wan Alvi, Z.A. Manan, Process modifications to maximise energy savings in total site heat integration, Appl. Therm. Eng. 78 (2015) 731e739. [14] G.F. Porzio, G. Nastasi, V. Colla, M. Vannucci, T.A. Branca, Comparison of multiobjective optimization techniques applied to off-gas management within an integrated steelwork, Appl. Energy 136 (2014) 1085e1097. [15] A.I.C.S.G. Carvalho, R. Gani, H.A.S. Matos, Design of sustainable chemical processes: systematic retrofit analysis and generation & evaluation of alternatives, in: AIChE Annual Meeting Conference Proceedings, 2005, p. 7237. [16] C.-L. Chen, C.-Y. Lin, J.-Y. Lee, Retrofit of steam power plants in a petroleum refinery, Appl. Therm. Eng. 61 (1) (2013) 7e16. [17] C.-L. Chen, C.-Y. Lin, Retrofit of steam power plants in a steel mill, Clean Technol. Environ. Policy 15 (5) (2013) 753e763. [18] C.-L. Chen, C.-Y. Lin, Design of entire energy system for chemical plants, Ind. Eng. Chem. Res. 51 (30) (2012) 9980e9996. [19] S. Onut, U.R. Tuzkaya, G. Tuzkaya, B. Gülsün, A multi-objective energy resource allocation model for Turkish manufacturing industry using linear physical programming, Int. J. Innov. Comput. Inf. Control 7 (6) (2011) 3147e3169. € [20] S. Onüt, U.R. Tuzkaya, N. Saadet, Multiple criteria evaluation of current energy resources for Turkish manufacturing industry, Energy Convers. Manag. 49 (6) (2008) 1480e1492. [21] G.F. Porzio, B. Fornai, A. Amato, N. Matarese, M. Vannucci, L. Chiappelli, V. Colla, Reducing the energy consumption and CO2 emissions of energy intensive industries through decision support systems e an example of application to the steel industry, Appl. Energy 112 (2013) 818e833. [22] G.F. Porzio, V. Colla, N. Matarese, G. Nastasi, T.A. Branca, A. Amato, B. Fornai, M. Vannucci, M. Bergamasco, Process integration in energy and carbon intensive industries: an example of exploitation of optimization techniques and decision support, Appl. Therm. Eng. 70 (2) (2014) 1148e1155. [23] C.A.C. Coello, G.B. Lamont, D.A. Van Veldhuisen, Evolutionary Algorithms for Solving Multi-objective Problems, Springer, New York, USA, 2007. [24] F.Y. Edgeworth, An Essay on the Application of Mathematics to the Moral Sciences, Mathematical Psychics, vol. 10, 1881. London (UK). [25] V. Pareto, Pareto, Cours D'Economie Politique, in: G.H. Bousquet, G. Busino (Eds.), Oevres Completes de Vilfredo Pareto, 1, Librairie Droz, Geneva, 1964. Originally published 1896e1897. [26] A.H. Alinia Kashani, R. Molaei, Techno-economical and environmental optimization of natural gas network operation, Chem. Eng. Res. Des. 92 (11) (2014) 2106e2122. [27] G. Mavrotas, Effective implementation of the 3-constraint method in multiobjective mathematical programming problems, Appl. Math. Comput. 213 (2009) 455e465. [28] M. Carvalho, M.A. Lozano, L.M. Serra, Multicriteria synthesis of trigeneration systems considering economic and environmental aspects, Appl. Energy 91 (2012) 245e254. [29] M. Pan, I. Bulatov, R. Smith, J.-K. Kim, Optimisation for the retrofit of large scale heat exchanger networks with different intensified heat transfer techniques, Appl. Therm. Eng. 53 (2) (2013) 373e386. [30] A. Maddaloni, G.F. Porzio, G. Nastasi, T.A. Branca, V. Colla, Exploitation of multi-objective optimization in retrofit analysis: a case study for the iron and steel production, Energy Procedia 61 (2014) pp.2297e2300. €ggkvist, Bipartite Graphs and Their Appli[31] A.S. Asratian, T.M.J. Denley, R. Ha cations, Cambridge Tracts in Mathematics, vol. 131, Cambridge University Press, Cambridge (UK), 1998, ISBN 9780521593458.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Multi-objective optimization applied to retrofit analysis: A case study for the iron and steel industry Alessandro Maddaloni*, Giacomo Filippo Porzio, Gianluca Nastasi, Valentina Colla, Teresa Annunziata Branca TeCIP Institute, Scuola Superiore Sant'Anna, Via Moruzzi 1, Pisa 56124, Italy
h i g h l i g h t s A model for optimal energy exploitation in integrated steelworks is presented. The model is formalized into a multi-objective optimization problem. The solution leads to reduced CO2 emissions and increased profits. The exploitation of the system is presented in a retrofit scenario.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 December 2014 Accepted 20 August 2015 Available online xxx
Steel production is among the most energy-intensive industrial processes and is a very relevant source of CO2 emissions. Steelworks, like most process and manufacturing industries, are facing the complex challenge of minimizing CO2 emissions while improving energy efficiency and developing effective strategies for process optimization. To this end, Process Integration methods can be successfully applied to the integrated steelmaking route with the purpose of achieving a reduction in the CO2 emissions, while optimizing material and energy systems. The work presented in this paper is finalized at the development of a model for optimal exploitation of energy resources in integrated steelworks, through the application of multi-objective optimization techniques, which can be beneficial both in new design and as a retrofit. Using these techniques, a series of solutions can be found representing optimal tradeoffs among different objective functions and the designer can select the best one or specify further criteria. Combining these methods, a retrofit was devised which optimized off-gas exploitation. This retrofit was then compared to the current operation. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Steel industry Multi-objective optimization Cost savings Industrial energy system Process simulation and optimization CO2 emissions reduction
1. Introduction The iron and steel industry is an intensive consumer of energy and carbon-bearing materials such as fossil fuels (e.g. coal, coke, oil) [1]. The reduction of materials and energy consumption as well as the control of all the emissions (not only the atmospheric ones, but also emissions into water bodies and soil) is a key objective. This will increase the sustainability and competitiveness of the production cycle. In fact, increased material re-use and recycling can contribute to reducing primary raw material consumption and to
* Corresponding author. E-mail addresses: [email protected] (A. Maddaloni), [email protected] (G.F. Porzio), [email protected] (G. Nastasi), [email protected] (V. Colla), t.branca@ sssup.it (T.A. Branca). http://dx.doi.org/10.1016/j.applthermaleng.2015.08.051 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
better exploiting off-gases through their improved distribution among consumer processes. It can lead not only to reduced emissions but also to savings in fuel gases (e.g. natural gas) and to efficiency improvements in power generation (where off-gases are exploited), so that an overall reduction in energy consumption is finally achieved. There are cases where the cost of emissions reductions and material recovery, exceeds the related savings in raw materials and energy. In the cases where the replaced raw material has a low purchase cost, recovery improvement can actually be even more costly and energy consuming than standard operations. However the need to comply with ever increasing environmental constraints (which can also indirectly raise the costs of waste disposal and emissions) and to improve sustainability of production cycles, stimulates the steel sector to search for optimal tradeoffs between the implementation of costly retrofits to reduce
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Nomenclature
Abbreviation BF blast furnace BOF basic oxygen furnace DRI direct reduction of iron GA genetic algorithm GH gasholder LHV lower heating value LP linear programming MILP mixed integer linear programming MOO multi-objective optimization Parameters N number of total iterations of the dynamic model m number of iterations of the 3-constraint method L constant parameter set by the gas network managers pi production of i-th gas dj demand of j-th consumer Mi maximum volume of i-th gas vij minimum gas volume through a pipeline Vij maximum gas volume through a pipeline lj minimum lower heating value of j-th consumer uj maximum lower heating value of j-th consumer vij minimum gas volume through a pipeline gi maximum gas volume out of i-th gasholder Lg minimum gas volume to be stored in g-th gasholder
emissions and while ensuring the economic viability of the production cycle. In particular, significant effort has been expended thus far by the steelmaking sector to reduce the emissions of GreenHouse Gases (GHG), especially carbon dioxide (CO2). Currently, 1.9 tons of CO2 are emitted per ton of produced steel and about 4e5% of total world CO2 emissions are attributed to the steelmaking sector [2]. In order to face the ambitious challenge of reducing GHG emissions, the European process industries (including the steel sector) and more generally the energy sector, are undergoing important retrofit operations to their production sites, i.e. they are adding new technologies to the existing production systems. For instance, the importance of incorporating new technologies in the steel production cycle in order to reduce CO2 emissions and energy consumption has been discussed in Ref. [3]. The fundamental role of energy efficient measures for reducing CO2 emissions in the iron and steel sector has been discussed in Ref. [4], where the importance of taking into consideration the overall relationship between the single technologies for mitigating consumptions and emissions and the overall production cycle is also underlined. This can avoid expending efforts on improvements that are only at a single process levels but not on a global perspective. A comprehensive review of emerging technologies for improving energy efficiency and reducing CO2 emissions is provided in Ref. [5]. However the adoption of alternative production technologies is slow, due to the considerable investments required, as well as the need for huge amount of alternative fuels, such as the natural gas needed for the Direct Reduction of Iron ore (DRI): in fact, the possibility of exploiting process gases to improve economic viability has also been investigated [6]. A series of more viable alternative techniques (such as coke oven gas injection, injection of plastic scraps into the Blast Furnace, pure oxygen blowing and heat recovery), which do
639
Ug c cij pCO2 fi ppj LHVi
maximum gas volume to be stored in g-th gasholder natural gas cost [V/GJ] cost of building pipe ij [V] market price of CO2 [V/t] emission factor for every gas [t CO2/GJ] gas selling price to power plants [V/GJ] lower heating value of the i-th gas
Variables xij yij zj Z1 Z2 Vg(k)
gas flowrate from i-th producer to j-th consumer [GJ/P] binary variable for the pipe usage binary variable for the CO2 calculation total CO2 emission total cost volume of g-th gasholder at the k-th iteration
Sets P C G T PP
set set set set set
of of of of of
producers consumers gasholders torches power plants
Subscript i i-th producer j j-th consumer g g-th gasholder n natural gas source
not fundamentally alter the traditional steel production cycle but allow a reduction in the consumption of fossil fuels and consequently CO2 emissions and energy recovery, is proposed in Ref. [7]. Waste heat recovery is also an important way to improve energy efficiency, as has been underlined in a recent study concerning the Chinese iron and steel industry [8]. Thus considerable efforts have been spent in this direction. For instance in Ref. [9] a simulation is described for recovering waste heat from the cooling phase of the sintering process, in Ref. [10] an innovative system to recover heat in the continuous casting process is presented, while in Ref. [11] a heat recuperation system for steel slags is described. Considerable further potential for energy recovery in steelworks can be found by applying pinch analysis [12] and derived total site approaches such as the total site heat integration which is described in Ref. [13]. Another important opportunity for energy recovery and GHG emissions reduction in the iron and steel industry, derives from enhanced exploitation of off-gas streams, that can represent not only a way to decrease their environmental impact but also an economic advantage, due to the considerable heating value of the off-gases [14]. Optimizing the distribution of the different off-gases among different internal and possibly external utilities can improve exploitation of off-gases themselves. However to this aim an optimal design of the gas network is also needed. This objective can be obtained though suitable retrofit actions which, in most of the cases, are related to existing infrastructures and industrial parks. Retrofit actions are often preceded by deep analyses aimed at pointing out and evaluating all the potentials for environmental improvements and energy savings. Process modeling and optimization techniques provide an effective support within such retrofit analyses. A relevant example concerning the chemical sector is provided in Ref. [15] by Carvalho et al., who propose a generic and systematic methodology to identify and screen the design retrofit
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potential of chemical processes. Another interesting example is provided by Chen et al., who present in Ref. [16] a mathematical model for the operational optimization and retrofit of industrial steam systems that is applied to a petroleum refinery: in this case the maximization of the cost reduction (expressed as the difference between annual operating costs and annualized investment cost of the retrofit) is formulated as a Mixed Integer Nonlinear Program (MINLP) based on a collection of unit models. In Ref. [17] a similar methodology is applied to the analysis and design of steam power plants in a typical steel mill, while in Ref. [18] an entire energy system for chemical plants is designed by exploring the interactions between the steam network and the heat recovery networks of process plants. In Ref. [19] a model is proposed to allocate the available energy resources to the Turkish Manufacturing Industries by exploiting a Multi-Objective Optimization (MOO) technique, as the considered problem can be viewed as a multiple criteria decision-making issue including human judgments, tangible and intangible criteria and priorities that require the achievement of a suitable trade-off between goals and criteria [20]. MOO techniques are actually mandatory when a resolution method for retrofit analysis is needed to resolve the conflicts between competing targets and observing complex constraints. For nontrivial MOO problems there is not a unique optimal solution in general: all the available MOO techniques allow a series of solutions to be found, representing optimal trade-offs among the different specified objective functions. It is up to the designer/manager to select the best one, according to the considered process conditions and systems specifications or, eventually, to specify further criteria driving the final selection of the best compromise solution. Integrated process models to formulate optimization problems considering multiple objectives, have been used in past research, to find trade offs between CO2 emissions minimization and cost minimization in an integrated steelwork. This was done by optimizing an existing process gas network, according to the approach described in Ref. [14] and further refined in Ref. [21]. The solution explored in the research described in this article exploits a dynamic mathematical model of a fixed gas network structure, embedded in a software decision support system, to recommend optimal gas distribution patterns. An MOO problem is then formulated [22], where cost and CO2 emissions are set as objective functions and the optimization is carried out by means of a particular Genetic Algorithm (GA), exploiting the concept of Pareto dominance to select a set of Pareto-optimal solutions (the so-called Pareto-set) [23]. These solutions meet a trade-off optimality criterion firstly introduced by Edgeworth in 1881 [24] and subsequently generalized by Pareto in 1896 [25], according to which a solution is included to the Pareto set if no other solution exists, which can improve at least one of the objectives without degrading the other ones. GAs have been selected as they allow a flexible problem formulation and an effective generation of a set of different trade-off solutions. This approach has been recently applied in Ref. [26], to find the best operating condition of a distribution network for natural gas, facing three different and conflicting objectives, i.e. maximize gas delivery flow and line pack and minimize operating costs. However GAs also have the drawback of considerable computational complexity, therefore an alternative approach has been attempted in Ref. [14], that is based on the 3-constraint method [27], which has already been applied in the past for facing MOO in analogous applications [28]. Such an approach belongs to the wide category of Mixed Integer Linear Programming (MILP) methods, that have been widely adopted to confront optimization problems for the retrofit of large heat exchanger networks [29]. The work described in the present paper represents a further step towards the development of practical and reliable means to optimize industrial off-gas networks from both the structural and
the functional point of view. In fact, this paper presents an extension of the above-described approach, including the additional possibility of assessing impacts on process performances and operating costs, when creating new connections among existing gas producers and consumers. A cost function is introduced for the installation of new pipelines, although the possibility of exploiting the existing network is kept. This key feature of the proposed approach, allows it to support a feasibility study of actual retrofits, which in real world scenarios can obviously include the enlargement of the distribution network, provided that the associated cost is affordable, as covered by the gains achievable through the consequent improved exploitation of off-gases. Moreover, based on the original model, a dynamic scenario has been analyzed and two optimization procedures have been developed: a stepwise optimization and a global optimization. Both approaches use the previously developed model as a discretization step to study the network in a fixed time interval, but while the stepwise approach solves as many problems as the number of steps, the global approach solves one bigger problem that accounts for all the steps at once. This aspect represents a further key advancement with respect to the previous work, as the global approach is in principle capable of taking into account the global performance over the whole time interval and all the scenarios that can arise by paying the price of a slight increase in computational time. In order to validate the proposed approach, the possibilities for a retrofit design of the gas distribution system of a real steelmaking plant are explored and the results of the local and global optimization approaches are presented and discussed. The paper is organized as follows: in Section 2, the formulation of the extended process gas model is presented; the dynamic models are presented in Section 3, while in Section 4 the experimental results in the two cases are described and compared. Finally in Section 5 some concluding remarks are proposed. 2. Model description A general industrial off-gas network is considered, which is usually composed of: 1. Processes that produce gas; 2. Processes that consume gas; 3. Gasholders, that can stock gas within their volume limit and deliver it. Noticeably there are some processes, such as the system Blast Furnaces (BF) e regenerators, and coke oven plants, which both produce and consume gas. Among the pure consumers there are, for instance, torches and power plants. Gasholders represent a buffer gas capacity and can therefore be treated as both gas sources and sinks, ensuring the satisfaction of the mass balance over different time periods. According to the approach introduced in Ref. [30] and summarized in the following paragraphs for the sake of clarity and completeness, each process/gasholder is modeled by a node in the network and each pipe connecting two elements of the gas network is modeled by a directed arc. Let P and C be the sets of processes which produce and consume gas, respectively. For those processes belonging to both sets in the model, 2 copies of the same process are considered: one as a producer and one as consumer (see the graph in Fig. 1). Each producer node has a correspondent gasholder. Let G be the set of gasholders, which are both gas producers and consumers. Finally let the symbols T and PP indicate, respectively, the sets of torches and power plants, which are subset of the consumers. A special production process is the source of the natural gas, which might serve other processes and is indicated in the following with the symbol n.
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the gaseous fuel entering the process and is computed through the following formula:
X
, xij
i2P
X xij LHVi i2P
(3)
Which represents the weighted average of the lower heating values of each of the gases in the mixture. The resulting LHV must lie within a lower (lj), and upper bound (uj), i.e.
lj
X
, xij
i2P
X xij uj LHVi i2P
(4)
Also the volume of gas to flow in a pipe is constrained. Each pipe connecting the i-th process to the j-th one is characterized by minimum and maximum volumes, vij and Vij respectively, namely
vij xij LHVi Vij
(5)
Moreover the volume of gas reaching the j-th consumer must be lower than a threshold value Mj, i.e.
X
, LHVi Mj
xij
(6)
i2P
Finally an upper bound gi exists also on the outgoing gas volume of the gasholders, which means that for the i-th gasholder the following inequality holds: Fig. 1. The bipartite digraph representation of a gas network: green circles represents fuel gas providers (BF e Blast Furnace, COK e coke oven plant, BOF e Basic Oxygen Furnace, CH4 natural gas supply), yellow circles represents consumers which actually exploit off-gases within their process (PP-power plant), while gray circles represents 3 different torches, which are consumers that actually burn off-gases into the atmosphere without exploiting their value. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A network of pipes, which connects producers and consumers, can be represented as a bipartite digraph [31], with vertex set P∪C and arcs directed from the elements of P to some elements of C. Let xij be the amount of gas flowing in the pipe ij which links producer i to consumer j. The binary variable yij represents the use of the pipe ij: if the pipe ij is not used (namely xij ¼ 0), yij is null, while if the pipe ij is actually used (i.e. xij > 0) yij is unitary. There might be pipes ij that are not in the network and cannot be constructed, in this case the constraint xij ¼ 0 is imposed. This is the case, for instance, of pipes which link gasholders and producers of gases different from the one that is stored in the gasholder. Thus the possibility of building pipes that are not included in the network but are feasible is considered. Each producer i, apart from natural gas, has to produce an exact amount of gas pi, namely:
ci2P v
X
xij ¼ pi
(1)
j2C
Consumer processes have a demand to meet: let the symbol dj indicate the demand for consumer j, for each consumer the following constraint holds:
X
xij dj
(2)
i2P
When two different gases are sent to the same consumer, a mixing occurs. The Lower Heating Value (LHV) of the gases mix at the inlet of process j represents the “heating quality” [GJ/Nm3] of
X
, LHVj gi
xij
(7)
j2C
In order to obtain the above-described condition where yij ¼ 0 if and only if the pipe ij is in use, the following inequalities (for every ij) are imposed
xij yij Lxij 10; 000
(8)
Here 10,000 is an upper bound for the maximum possible value of xij, while L is a constant value such that a flow lower than L1GJ in the pipe ij is not desirable, i.e. L1 is lower than the minimum amount of gas that is physically meaningful to send into a pipe. Hence if xij is null (or xij L1), then yij ¼ 0; if xij > L1, then yij ¼ 1.
2.1. Objective functions There are two objectives to jointly optimize: the CO2 emissions must be minimized while the profit from the selling of the gas must be maximized. To this aim, the total CO2 emission Z1, and the total monetary cost, Z2 are computed and minimized. Let fi be the CO2 binary emission factor of the i-th producer (e.g. if a producer, such as a gasholder, is non-polluting fi ¼ 0 holds). The emission of producer i is given by
fi
X
xij
(9)
j2C
The total CO2 emission Z1 is therefore provided by the following expression:
Z1 ¼
X
xij fi
(10)
i2P;j2C
The total cost Z2 is given by the sum of four components: the purchase cost of natural gas, the purchase cost/gain related to the
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acquisition/sale of CO2 allowances, the income deriving from the sale of gas to power plants and the cost of building a pipe ij. Let c be the unitary price of natural gas. Let the symbol zj indicate an integer variable associated to the j-th consumer process and lying in the range [1, 1]: zj ¼ 1 if process j represents a direct emission of CO2 within the company boundaries and therefore a loss of CO2 allowances; zj ¼ 1 if process j implicates a gas transfer with no associated direct CO2 emissions e with a resulting gain of CO2 allowances (such as, for instance, when gas is sold to third-party power plants); zj ¼ 0 in case no change occurs. Let ppj be the unitary gain for selling gas at power plant j. Finally let cij be the (amortized) cost of building the pipe ij (cij ¼ 0 holds if the pipe already exists). The total cost is given by the sum of four terms as follows:
Z2 ¼
X
c$xvj þ
j2C
X i2P;j2C
zj $fi $xij $pCO2
X i2P;j2P
xij $ppj þ
X
yij $ciji
i;j
expense of a slight increase in the complexity of the modeled system and a consequent increase of the computational time. The previous MILP model is used as a discretization of the continuous process by considering N intervals of duration Dt and by optimizing over all the intervals. Two different strategies for accomplishing this task are presented. The first strategy, which will be called stepwise optimization, runs the previous model at every time step, starting from the initial conditions derived from the previous run. The satisfaction of the mass balance between two consecutive steps is guaranteed by imposing constraints on the gasholders volumes at the “beginning” (gasholders inlet) and at the “end” (gasholders outlet) of each step. A more accurate global optimization model can be obtained by expressing all the time dependent variables and parameters into a single big MILP. This model becomes considerably more complex, as it contains roughly N times the variables of the original MILP (there is a variable for each time step). On the other hand this model is much more powerful than the stepwise model: it can foresee what happens in the future steps, while the stepwise model is blind, as it only optimizes over a single step, disregarding the future steps.
(11) The first terms accounts for the cost of purchasing the natural gas, the second term represents the cost or revenues coming from the commerce of CO2 allowances, the third term represents the income derived by the selling of gas to power plants and the last term is the cost of building the new pipes. The need to achieve a joint optimization of the two often counteracting objectives Z1 and Z2 leads to the formulation of a MOO problem, where the 3-approximation technique is used to generate a good approximation of the Pareto set, which represents the set of optimal trade-off solutions [23]. These solutions are by definition such that no other solution can have better values of both Z1 and Z2. More precisely the Mixed Integer Linear Programming (MILP) problem min Z1 is solved subject to the above-described constraints and the constraints on the unacceptable pipes (xij ¼ 0 in this case). From this MILP an optimum value Z* for Z1 is obtained, subsequently the m problems are solved, which are defined as follows for i ¼ 1, 2, …, m and a suitable choice of 31 32 … 3m:
8 > > <
min Z2 subject to Ax b > > : Z1 Z * þ 3i where Ax b represents the matrix form of the constraints (1)e(8). For increasing values of i, decreasing optimum values for Z2 and increasing values of Z1 are found. The m pairs obtained in this way allow describing a good approximation of the Pareto front of the solutions. For a more detailed discussion on Pareto optimality and 3-approximation see the Appendix A.
3. Dynamic model The model described so far has been realized for a sort of “instantaneous” process, which is designed to optimize a fixed time period, therefore time has not been considered as a variable. A more general representation can be achieved by modeling the gasholders' volume variations along time. Such a formulation presents the advantage of allowing to deal with different time steps in a more realistic scenario, where a step can be a day or any time period needed for all the processes to be completed. This new model does not lose any accuracy, at the
3.1. Stepwise optimization At every time step the variable xij represents the amount of gas flowing from process i to process j at the k-th iteration, namely in the time interval [(k 1) Dt, k Dt]. The only parameters which change over the time steps in the MILPs are the gasholders' volumes: their storage might increase or decrease at every time step. The volume Vg(k) of gasholder g at time step k is computed as follows:
Vg ðkÞ ¼ Vg ðk 1Þ þ
X xigc X xigp LHV LHVgp i i2P i2P
(12)
where gp and gc are the indices of the gasholder g as a producer and as a consumer, respectively. All the constraints of the previously developed static model must hold for every time step and there is an additional constraint to the time dependent model, stating that cgVg ðkÞ is limited, namely
Lg Vg ðkÞ Ug
(13)
where Lg and Ug are respectively the lower and upper bound for the volume of gasholder g. This model is optimized at every iteration (once for every time step) and before starting the new optimization the values of the gasholders' volumes are updated according to Eq. (12).
3.2. Global optimization In the global optimization strategy, instead of optimizing the N processes at every step, a broader time dependent model is built to optimize an overall function of the stepwise cost and CO2 emission. To this aim, the variables xij must be considered time dependent, i.e. they are replaced with the variables xij(k) representing the amount of gas flowing from pipe i to pipe j in the time interval [(k 1) Dt, k Dt]. The CO2 emission at step k is given by
Z1 ðkÞ ¼
X i2P;j2C
xij ðkÞfi
(14)
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While the overall CO2 emission is the sum of the stepwise emissions over all the time steps, namely:
Z1 ¼
N X
Z1 ðkÞ
(15)
k¼1
Similarly the stepwise and overall cost functions can be calculated through the following formulas:
Z2 ðkÞ ¼
X
c$xvj ðkÞ þ
j2C
þ
X
X i2P;j2C
zj $fi $xij ðkÞ$pCO2
X
xij ðkÞ$ppj
i2P;j2P
yij $cij
i;j
(16) And
Z2 ¼
N X
Z2 ðkÞ
(17)
k¼1
A new model is thus obtained, through which the sum of the stepwise objectives (namely Z1 and Z2) can be optimized in a single run instead of running N times a MILP. Obviously the computing time of the new bigger MILP is higher, since the number of variables becomes roughly N times bigger. However, on relatively small instances the running time is still acceptable, as it is shown in the next section. An optimal solutions obtained with this global model is never worse than an optimal solution obtained with the stepwise model, since the latter forms a feasible solution for the global model that can be obtained by just setting xij(k) to be the value of the optimal solution xij found at step k and using the same values of yij.
4. Experimental results The “instantaneous” model has been tested against a formerly developed model which is extensively described in Refs. [14] and [30]. This former model was built to represent a particular network structure without taking into account the possibility of creating new connections between process units (gas producers or consumers). The main innovative outcomes of the optimization system developed for this study are: 1) The described system is capable to optimize the operational conditions also taking into account changes in the structure of the gas network. 2) An enhanced Pareto frontier is achieved as an output of the analysis. 3) Two dynamic optimization approaches have been validated and compared in terms of achievable CO2 savings and profit increase; the global optimization approach achieves better results than the stepwise one. Both aspects allow a higher flexibility in terms of application within a real industrial context. The results from these contributions are separately described in the following sections.
4.1. Optimization of the network structure and operating conditions The model has been run on an industrial dataset coming from an Italian integrated steelwork gas network and consisting of a Coke oven plant (COK), a BF, a Basic Oxygen Furnace (BOF), a natural gas distributor (CH4), 3 gasholders (GH1, GH2, GH3), 3 power plants
643
(PP1, PP2, PP3), and 3 torches (T1, T2, T3). The pipes connecting different parts of the plant are depicted in Fig. 2a. Differently with respect to previously developed analyses, which only aimed to find the best way to distribute the off-gases into the existing network, the plant managers faced in this case the concrete problem of understanding whether the cost for enriching the network with new connections could be paid back through time by the potentially achievable savings in terms of natural gas savings and/or CO2 emissions reduction. A number of tests have been performed on data from a particularly critical period of time in order to explore the possibilities for profitable gas network modifications: Fig. 2a shows the complete network (i.e. including gasholders' connections as well as all the other possible links) that is represented in Fig. 1, while Fig. 2b represents the result of the optimization, as it reports only the pipes that need to be active to minimize the cost, i.e. the pipes ij for which xij > 0 in the final solution. The connections in red (in the web version) are those pipes that are not initially present in the network and hence need to be built, i.e. the pipes ij for which yij ¼ 1 in the final solution. Noticeably in the final layout two torches are no longer in use, which implies a significant reduction of CO2 emissions. In practice the proposed approach is capable to identify the possibility to rationalize the distribution of process off-gases within the existing network, thus allowing a lower flaring of excess gas stored in gasholders through torches. As a consequence, a lower overall consumption of natural gas is ultimately achieved. Two torches out of three in the case shown in Fig. 2b are in fact no longer in use (T2, T3), since all the gas originally burnt is consumed in other plants, allowing to save a useful resource while also achieving an environmental benefit. 4.2. Enhanced optimization results (extended Pareto optimality) The results of the model can also be analyzed in terms of improvements with respect to the previous model version measured on a Pareto front showing CO2 emissions and cost (Fig. 3). The figure puts into evidence that the new model has better local minima both for the CO2 emission and for the cost (taking into account that the negative value for the cost corresponds to a profit in every scenario). In particular, the profit can be raised up to 6.3% with respect to the maximum value obtained through the previously developed model [21] (from 958 V to 1018 V per time interval, with a correspondent increase of the CO2 emissions of 14.2%), whereas on the opposite side of the Pareto front a reduction of 1.1% of CO2 emissions can be reached at the expense of 20.8% cost increase (CO2 emissions decrease from 12.263 t to 12.133 t per time unit, while profit decreases from 549 V to 435 V per time unit). It must be noticed that the two Pareto fronts coincide in the interval where there is an intersection, due to the fact that some plant constraints must be taken into account in both the new and the old models. To sum up, the results of the performed tests show that the retrofitting approach is substantially better than the previously developed one as, by installing only 2 new pipes, a higher yearly profit (up to almost 5,000,000 V) as well as lower CO2 emission (up to 13,665 t less per year) can be obtained. Therefore the proposed MOO approach can effectively explore all the profitable trade-offs considering the existing network structure, in terms of: CO2 emissions; Natural gas consumption; Costs (in terms of materials and investments required to achieve such savings). A “smart” distribution of the produced off-gas within the existing network is obviously a cost-effective solution, which could
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Fig. 2. The original off gases network (left) and the network after the optimization (right).
be explored through the approaches already proposed in Ref. [14]. The above described algorithm goes far beyond this result, by suggesting practical solutions which increase the profit and saving margins at a cost which is affordable, as it is fully compensated by the achieved economical and environmental benefit: in fact all costs and benefits are both implicitly taken into account by the optimization procedure itself. 4.3. Comparison of dynamic (stepwise and global) optimization approaches The dynamic models depicted in Section 2 (aiming at stepwise and global optimization, respectively) have also been tested on the same case study concerning an off-gas network of an integrated steelmaking plant. The results of this further simulation show that the stepwise model does not perform significantly worse than the more general model, by achieving in all the tests optimum points that are less than 1% worse than those of the global model. Table 1 shows the optimum value both for the CO2 emission and for the profit for the two models obtained for different values of the
Fig. 3. Pareto fronts of the old model and the new instantaneous model (depicted in Section 1).
number N of total performed steps. It is worth noting that with less than 25 steps there is no difference between the two models in the total profit and there are tiny differences between the other optima both for the CO2 emission and for the total profit. Fig. 4 shows the Pareto fronts of the two different dynamic models for the case N ¼ 100. Noticeably the profit provided by the stepwise model is not an increasing function of the CO2 emission. This is due to the fact that, in order to compute the Pareto front of the stepwise model, looser constraints on the total CO2 emission are imposed. In the first steps these looser constraints lead to higher profits for those particular steps, but they need not lead to a better global profit (defined as the sum of the profits over the whole optimization landscape period). Indeed to improve the solution in the first steps, the gasholders get filled in a different way, therefore the following iterations have different initial condition (the initial gasholders' volumes), which might reduce the solution space and hence worsen the optimum. This is one of the main reasons why the dynamic global model is preferable to the dynamic stepwise model: the global model optimizes over all the scenarios that can arise in the different steps, while the stepwise model is not able to predict what happens in the future steps. To sum up, two advanced dynamic models have also been presented, which can both have a wide applicability in the industrial context. While the instantaneous model provides a time-efficient pre-screening of the gas network layout, the two dynamic models exploit the instantaneous model as a basis for their computation, which is either iterated or simultaneously performed for the different time steps. Therefore they are indeed more time consuming but also allow higher generalization capabilities. In fact, they allow variable boundary conditions over time, by thus providing a more realistic representation of the gas flow evolution through time. Such variability can for instance correspond to changes in processes and networks conditions (such as, for instance, changes in feed material and therefore gas composition, planned or unforeseen plants stops, leakages and temporary pipes unavailability due to extraordinary maintenance or operations). Thus they represent an advance with respect to previous achievements and a useful practical support for daily management of the
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Table 1 Optimal values for CO2 emission and total profit with different number of steps. Number of steps
CPU time stepwise (ms)
CPU time global (ms)
Stepwise CO2 (t)
Global CO2 (t)
D (%)
Stepwise profit (V)
Global profit (V)
D (%)
2 10 25 50 75 100
70 150 290 540 780 1010
70 150 360 730 1390 1840
30.86 134.99 332.33 661.22 990.10 1323.77
29.25 133.53 330.86 659.7 988.63 1321.21
5.23 1.09 0.44 0.22 0.15 0.19
3464.62 11,071.38 25,334.06 48,807.66 72,269.03 95,730.40
3464.62 11,071.38 25,334.06 49,054.84 72,795.02 96,416.69
0 0 0 0.51 0.73 0.72
gas network, allowing a higher flexibility in the planning of the daily gas distribution, whether it has to be optimized in advance and at a whole (e.g. in planning and optimizing the production cycles ahead of schedule) or in a quasi-online mode, with a shorter time horizon, represented by the variable optimization steps. 5. Conclusions and future work This paper introduces an innovative approach for the correct management of resources, by analyzing retrofits in the integrated iron and steel industries. These industries are among the most energy and carbon intensive in the world. This approach represents a significant extension of previously developed studies, as: It allows taking into account the possibilities of a retrofit design of the systems, through the addition of network connections (installation of new pipework). It enables plant managers to consider a wider range of operating conditions, thanks to the flexibility in the network structure. As a result, a wider Pareto front is achieved. It guarantees a higher flexibility, both in offline and online (i.e. quasi-real time) operations, through the possibility of applying both a static and a dynamic approach In fact, this approach simultaneously takes into account the achievable savings in terms of CO2 emissions and natural gas consumption of each consumer plant, when fed with the produced offgases. Minimizing the costs of installing new pipes, linking different gasholders to consumer processes, is also considered as an objective, allowing a more efficient network architecture. At the same time, the conditions in which the gasholder is full and the offgas must be flared are automatically minimized or avoided.
The innovative approach has been compared to an existing system and the incremental benefits of the proposed approach have been discussed and pointed out. A dynamic scenario has been analyzed through two different models, the first one solving many small problems iteratively and the second one solving a more complex problem at once. A comparison between the two models has been pursued, by exploiting the data coming from an existing off gas network belonging to an integrated steelworks. The results of the presented case study show that the solution to the overall problem achieves optimal results (in terms of Pareto optimality) thanks to a better avoidance of local minima, arising from the choice of a particular sub-optimal path. This aspect is particularly powerful in process simulation and retrofit. However, the stepwise optimization algorithm could be better suited to a dynamic and quasi-online support to the operations of the plant, since it allows taking into account variations in the operating conditions of the process. The practical interest of the proposed modeling and optimization approach are thus twofold: it can be applied not only for improving the management of an existing process gas network but also within feasibility studies aimed at a preliminary verification of the advantages of structural modifications of the network itself. In fact the proposed retrofit approach is capable of pointing out the best trade-offs between costs (investment and operating) and savings in terms of CO2 emissions and fuel consumption, by exploring different gas network layouts and automatically taking into account the costs of network modifications within the optimization procedure itself. Moreover, the proposed dynamic optimization models allow taking into account the variability of network and processes operating conditions which normally occur in the industrial context. In particular, one of the models faces the problem of limiting the computational burden through an iterative approach, while the latter one develops a global optimization over all the scenarios that can arise in the different steps. Acknowledgements The authors wish to thankfully acknowledge their industrial partners for their support, their constant availability and fruitful suggestions that allowed the incremental development of this analysis. Appendix A. Multi-objective optimization A.1. Pareto optimality A generic MOO problem can be defined as follows:
Fig. 4. Pareto fronts of the two dynamic models for N ¼ 100.
min fðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; …; fk ðxÞ subject to hðxÞ ¼ ½h1 ðxÞ; h2 ðxÞ; …; hm ðxÞ 0 with x ¼ ðx1 ; x2 ; …; xn Þ2X X3
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where x is the variable vector, f(x) is the objective vector, i.e. the vector having as elements the different objective functions, h(x) if the constraint vector and X is space of feasible solutions. For nontrivial MOO problems there is not a unique optimal solution in general, hence a set of so-called Pareto optimal solutions (or Pareto set) is searched. This set forms, in the objectives space, the so-called Pareto front. All the solutions belonging to the Pareto set have the property to be non-dominated. A solution a dominates b if and only if for each objective function fi, fi(a) fi(b) and for at least one objective function this inequality is strict, as it is formally stated below.
a_b⇔fi ðaÞ fi ðbÞci
∧
dj : fi ðaÞ < fi ðbÞ
A.2. 3-Constraint method This method allows to compute an approximation of the Pareto front for an MOO problem. In the case of two objective functions, the first objective f1 is minimized and a minimum f* is found. Subsequently a parametric constraint on the first objective is added to the problem stating that that the value of f1 must not exceed f* by 3i.
8 > > <
min f2 subject to hðxÞ 0 > > : f1 f * þ 3i These parametric problem is solved for an increasing sequence of values of parameters 31 32 … 3m The solutions to these problems have decreasing values for f2 and increasing values for f1 as the values of 3i increase. With the pair of values (f1, f2)i found in the m, a discrete subset of the Pareto front is generated. In other words, each objective is firstly optimized and then transformed into a constraint which is progressively released in order to optimize the second objective. References [1] International Energy Agency, Energy technologies perspectives 2010. Scenarios and Strategies to 2050, OECD Publishing, 2010. http://dx.doi.org/10. 1787/energy_tech-2010-en. http://www.oecd-ilibrary.org/energy/ energytechnology-perspectives-2010_energy_tech-2010-en. [2] Worldsteel, Steel's contribution to a low carbon future: worldsteel position paper. worldsteel.org/publications/position-papers/Steel-s-contribution-toalow-carbon-future.html (last accessed January 2012). [3] N. Pardo, J.A. Moya, Prospective scenarios on energy efficiency and CO2 emissions in the European iron & steel industry, Energy 54 (2013) 113e128. [4] N. Karali, T. Xu, J. Sathaye, Reducing energy consumption and CO2 emissions by energy efficiency measures and international trading: a bottom-up modeling for the U.S. iron and steel sector, Appl. Energy 120 (2014) 133e146. [5] A. Hasanbeigi, M. Arens, L. Price, Alternative emerging ironmaking technologies for energy-efficiency and carbon dioxide emissions reduction: a technical review, Renew. Sustain. Energy Rev. 33 (2014) 645e658. [6] C. Raggio, Recent developments of energy saving and environmental protection in the steel industry, Metall. Ital. 104 (10) (2012) 51e58. [7] K.-J. Li, J.-L. Zhang, Y.-P. Zhang, Z.-J. Liu, X. Jiang, Analysis on development of iron-making process based on the principle of energy-saving and emission reduction, Chin. J. Process Eng. 14 (1) (2014) 162e172.
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