Design and Optimization of Flexible Hydrogen Systems in Refineries

Article pubs.acs.org/IECR

Design and Optimization of Flexible Hydrogen Systems in Refineries Yunqiang Jiao,† Hongye Su,*,† Weifeng Hou,‡ and Pu Li§ †

State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, Zhejiang, China ‡ Zhejiang Supcon Software Co., Ltd., Hangzhou 310053, Zhejiang, China § Department of Simulation and Optimal Processes, Institute of Automation and Systems Engineering, Ilmenau University of Technology, Ilmenau 98684, Germany ABSTRACT: With the increasing demand for hydrogen resulting from fierce market competition and stringent environmental legislation, the hydrogen system has become an important component of a refinery. It is vital for the hydrogen system to be operated economically and safely under varying operating conditions. This calls for a systematic approach to the design and optimization of flexible hydrogen systems, which is the aim of this article. The hydrogen distribution network is designed at the minimum total annual cost subject to constraints on the flow rates and pressures of both existing and new equipment during the payback period. Varying hydrogen demands, different pipeline levels, and the possibility of hydrogen units being shut down are considered as operating conditions in the design optimization task, leading to the formulation and solution of a mixed-integer nonlinear programming (MINLP) problem. Using a linearization method, the MINLP formulation is approximated by a mixedinteger linear programming (MILP) problem, resulting in an acceptable quality and high efficiency. An industrial hydrogen system is taken as a case study. As shown in the case study, the proposed approach can handle high-dimensional and complex hydrogen system problems and gain significant economic improvements in comparison to an existing design.

1. INTRODUCTION In a refinery, the hydrogen system supplies hydrogen for hydrocracking units to upgrade heavy oils into more valuable products and for hydrotreating units to remove sulfur and nitrogen compounds from petroleum products. The hydrogen system is configured according to the demands. Its operation is subject to varying outside conditions including variations in hydrogen demands, product categories, and supplies of raw materials and even unit shutdown for plant maintenance and unexpected emergency situations. Under these varying conditions, a systematic approach is required to design a flexible hydrogen system that can operate safely and economically. To carry out this task, a large-scale complex optimization problem must be formulated and solved to provide optimal pipeline levels, an optimal layout of all types of units, and their optimal internal operating conditions. Typically, a hydrogen system consists of hydrogen producers such as hydrogen plants and catalytic reforming and ethylene units, hydrogen consumers such as hydrotreating and hydrocracking units, purifiers such as pressure-swing adsorption and membrane separation units, and compressors. In addition to the configuration of these operating units, the hydrogen distribution inside the system should be determined to satisfy hydrogen demands and achieve cost-effective and stable performance under varying outside operating conditions. In the past two decades, several approaches have been proposed for the synthesis and design of refinery hydrogen systems. Graphical methods were first applied for the efficient management of hydrogen systems. Towler et al.1 proposed a systematic approach to study hydrogen networks based on the analysis of cost and value composite curves. Alves and Towler2 proposed a hydrogen pinch analysis for targeting the minimum hydrogen consumption that provides quantitative insights and © 2013 American Chemical Society

identifies bottlenecks in the hydrogen distribution system. Liao et al.3,4 obtained the optimal conditions for solving pinch problems and proposed a rigorous targeting approach that is more accurate and efficient than other targeting methods for an overall optimal solution. A superstructure-based approach was first proposed by Hallale and Liu,5 leading to a mixed-integer nonlinear programming (MINLP) problem in which pressure constraints and compressors for a retrofit design were considered. Liu and Zhang6 proposed a systematic method for selecting appropriate purification units in which integration of the design and optimization of a hydrogen network was considered. Khajehpour et al.7 proposed a reduced superstructure based on experience and engineering judgment to retrofit the hydrogen network of an Iran refinery, using a genetic algorithm to solve the optimization problem. Liao et al.8 incorporated purification units for refinery hydrogen management and successfully demonstrated the application of a superstructurebased approach for the retrofit design of an existing refinery. In Kumar et al.,9 models of linear programming (LP), nonlinear programming (NLP), mixed-integer linear programming (MILP), and MINLP were developed, and features of these models were analyzed. Ahmad et al.10 developed an improved approach for the design of flexible hydrogen networks under multiple-period operation. Jiao et al.11 decomposed the optimization problem of the hydrogen network into two subproblems: feed routes of the purification network and of the hydrogen supply network. Jiao et al.12 also Received: Revised: Accepted: Published: 4113

May 20, 2012 February 6, 2013 February 26, 2013 February 26, 2013 dx.doi.org/10.1021/ie303209e | Ind. Eng. Chem. Res. 2013, 52, 4113−4131

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inconsistency in the solution quality. Therefore, an MINLP formulation is first formulated for the flexible design optimization under multiple operating scenarios. Then, the MINLP model is transformed into an MILP model based on a linearization technique proposed by McCormick.22 The effects of the linearization strategy on the optimization result and the errors arising from the approximation are analyzed in detail. It can be shown that the MILP model leads to an acceptable quality and a much higher computational efficiency than the MINLP problem. The remainder of this article is structured as follows: Section 2 presents an analysis of the design of refinery hydrogen systems under multiple operating conditions. In section 3, the formulation of the optimization problem is described in detail, followed by the proposed solution strategy for solving the flexible optimization problem of hydrogen networks in section 4. In section 5, the proposed approach is applied to a case study to illustrate its effectiveness and potential. The article is summarized and concluding remarks are provided in section 6.

presented a multiobjective optimization approach for the design of hydrogen distribution networks, in which the relation between operating and investment costs was explored based on the Pareto front. Later, Jiao et al.13 developed a chanceconstrained programming model for hydrogen network optimization under uncertainty, such that a suitable compensation between profitability and reliability can be achieved. Fonseca et al.14 and Salary et al.15 employed graphical and mathematical methods simultaneously in the design optimization of a hydrogen distribution network. It should be noted, however, that all of the studies mentioned so far addressed the hydrogen system optimization problem only under a single scenario or during a single operating period. Only Ahmad et al.10 proposed to design a flexible hydrogen system under multiple periods of operation. They demonstrated the applicability of their proposed approach in a case study in which variations in the hydrogen consumption and offgas yield of one hydrogen consumption unit were considered over three operating periods. The proposed flexible optimization design approach for hydrogen systems developed in this article is an extension of the work of Ahmad et al.10 To design flexible hydrogen systems under varying operating conditions, multiple scenarios and/or multiple operating periods need to be considered. One widely used technique for enhancing the flexibility of chemical processes is multiperiod optimization,16 where operating periods with different lengths can be defined to reflect the impact of variations in operating parameters. In this article, a multiperiod flexible optimization approach is presented to address the design optimization problem under varying operating conditions. The relative length and number of scenarios are modified to fit the operating fluctuations. The objective is to minimize the total annual costs, including operating costs and annualized capital costs, based on constraints on the flow rates and pressures of both existing and new equipment during the payback period. The varying demand from the hydrogen consumers, varying pipeline levels, and possible shutdowns of hydrogen units are considered in the problem formulation to ensure the safety of the hydrogen system under normal and abnormal operating conditions. Binary variables are introduced to represent the existence or nonexistence of hydrogen units and streams. Bilinear and trilinear terms are employed to express unit hydrogen balances. In previous studies on crude oil scheduling, bilinear terms were used to represent mass balance equations and were linearized by application of the reformulation−linearization technique.17−20 As noted by Li et al.,21 such linearization can sometimes replace bilinear constraints, but it often leads to inconsistent solutions. However, the influence of the linearization on the optimization results resulting from this inconsistency was not analyzed. In this article, an MINLP problem is formulated for flexible design optimization under multiple operating scenarios. The MINLP model is then transformed into an MILP model based on a linearization technique proposed by McCormick.22 Impacts of the linearization on the optimization results are discussed and used as the basis on which we propose to reduce errors arising from the linearization by defining tight ranges of the related variables. The design and optimization of a flexible hydrogen system considered in this article leads to a complex MINLP problem. It is recognized that solving an MINLP problem requires a large amount of computational effort and might result in

2. DESIGN OF REFINERY HYDROGEN SYSTEM In a refinery, a series of sources supply hydrogen to the hydrogen system, and a series of sinks consume hydrogen from the hydrogen system. Hydrogen sources consist of hydrogen producers such as hydrogen plants and catalytic reforming units, purifiers such as pressure-swing adsorption units and membrane separation units, hydrogen consumers such as hydrotreating and hydrocracking units providing off-gases to the purifiers, and compressors providing high-pressure hydrogen to hydrogen consumers. Hydrogen consumers consuming high-purity and high-pressure hydrogen, purifiers consuming low-purity off-gases from hydrogen consumers and high-purity hydrogen from reforming units, and compressors consuming low-pressure hydrogen to supply high-pressure hydrogen are considered as hydrogen sinks. The superstructure of the hydrogen network considered in this article is shown in Figure 1. It consists of various different

Figure 1. Superstructure of the hydrogen network.

pieces of equipment and possible connections between hydrogen sources and sinks. It can be seen in Figure 1 that hydrogen from hydrogen producers and purifiers is transported into the hydrogen pipeline levels. Then, it is consumed by hydrogen consumers or purified by purifiers directly, when the required inlet pressure is less than or equal to the pressure of the pipeline levels. Hydrogen from pipeline levels must be 4114

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pressurized by compressors so that it can be transported to the hydrogen consumers or purifiers when the required inlet pressure is greater than the pressure from the pipeline levels. The off-gas from the purifiers and hydrogen consumers is transported into the off-gas pipeline levels. Then, it is purified by purifiers or consumed by some hydrogen consumers that have low-purity specifications, or it needs to be pressurized when the pressure of the off-gas pipeline levels is lower than the required inlet pressures of the purifiers and hydrogen consumers. Depending on the objective function, data, and variables and their constraints, optimization of a hydrogen system can be formulated as an operational problem, a retrofit design problem, or a grassroots design problem.23 An operational problem addresses an existing hydrogen system in which a structural modification is not permitted. Here, it is necessary to establish operating conditions under multiple scenarios at minimum operating costs while meeting hydrogen demands and other constraints. To formulate an optimization problem for the existing system, information about the operating scenarios, site conditions, hydrogen demands, and system configuration of the existing units is required. Also important is the information about operating constraints such as satisfying process specifications through all scenarios, mass balances, and equipment operating limitations. The solution provides the on/ off statuses of equipment units, loads of hydrogen producers, and a hydrogen stream distribution of the hydrogen system for each operating scenario. An operational problem can be extended into a retrofit design problem in which structural modifications and equipment investment are allowed.23 The objective for a retrofit design problem is to minimize the total annual costs of the hydrogen system under multiple operating scenarios and simultaneously meet operating constraints. Additional data required for the retrofit design include available investment, extended configuration alternatives, and capital cost functions. Moreover, constraints such as limits on the maximum investment and the types, numbers, and sizes of new equipment to be selected are needed for the formulation of a retrofit design problem. The optimal solution provides the numbers and capacities of new equipment units to be installed, their connections with the available units, and their internal operating conditions. A grassroots design problem is encountered in the design of a new hydrogen system. In this case, it is not necessary to know specifications of any existing equipment. However, a larger superstructure of alternatives for new equipment should be provided in comparison to that available in a retrofit problem. The amount of data required for grassroots design is less than that required for the retrofit case, but the number of constraints and variables is higher because of the increased options for new equipment. Moreover, the solution space for a grassroots design problem might be much larger, because each unit in the superstructure can be of any capacity and size within practical limits. In this sense, operational and retrofit problems are particular cases of a more generic grassroots design formulation.23 In this article, the retrofit design problem of a hydrogen distribution network under multiple operating scenarios is addressed. Next, we provide a detailed description of thi problem. The retrofit design is based on the following given information:

(1) configuration details of the existing hydrogen system including the number and interconnections of the operating units; (2) specifications of the existing units including the hydrogen purity and outlet pressure of hydrogen producers, the minimum feed purity and inlet pressure of hydrogen consumers, and the capacity of hydrogen producers and compressors; (3) the number, duration, and operational status of hydrogen units and hydrogen demands under each operating scenario; (4) economic data including the prices per unit of hydrogen, fuel, and electricity; (5) capital cost functions for new pipelines, compressors, and purification units; (6) information about the payback period and interest rate per year; and (7) limits on the connection between sources and sinks. The optimization solution determines the following decisions: (1) on/off status profiles of units for each scenario; (2) hydrogen yield of hydrogen producers for each scenario; (3) detailed layout profiles for compressors; (4) hydrogen pipeline levels of the hydrogen system; (5) hydrogen distribution between hydrogen sources and sinks; and (6) number and capacity of new equipment units to be installed. Finally, the following assumptions for the optimization of the flexible hydrogen system in a refinery are made: (1) Hydrogen streams are represented by a binary mixture of hydrogen and methane. (2) The purity of hydrogen from each hydrogen source is considered to be constant. (3) The inlet pressures, outlet pressures, and inlet temperatures of the compressors are set to their daily operating values. (4) The efficiency of each compressor is considered to be constant. (5) Hydrogen is not leaked in the process of transmission. (6) The off-gas stream of a hydrogen consumer to its inlet is illegal and should be eliminated. The optimized hydrogen network should be flexible so that it can handle normal and abnormal circumstances. Possible situations that a flexible network needs to handle can be grouped into demand variations of hydrogen consumers and possible shutdowns of the hydrogen units. If the hydrogen system is subjected to fluctuating hydrogen demands, it should be able to meet the expected consumption, even if the fluctuations in demand occur sporadically. At the same time, when a unit within the hydrogen system is switched off to receive preventive maintenance or a unit of the hydrogen system must be shut down in unexpected emergency cases, the system should have sufficient ability to compensate for the unavailability of one or more units and guarantee the hydrogen supply for hydrogen consumers.

3. PROBLEM FORMULATION In addition to the constraints on hydrogen sources, hydrogen sinks, compressors, and purifiers considered in previous studies, the method proposed in this work takes into account varying 4115



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hydrogen demands, shutdowns of the hydrogen units, and different pipeline levels of the hydrogen distribution network. Binary variables are introduced to indicate the existence or nonexistence of hydrogen units and streams, whereas continuous variables represent input−output interactions among individual units and operations. Our method is applicable to the retrofit and design of flexible hydrogen systems by making effective use of existing equipment, installing new equipment, and restructuring process streams to minimize total annual costs. 3.1. Hydrogen Source Constraints. In a refinery, hydrogen from sources is first sent to pipeline levels to make sure that they can supply enough hydrogen for sinks. Hydrogen producers such as catalytic reforming units and ethylene plants produce large amounts of hydrogen as a byproduct. The hydrogen from these hydrogen producers should be used up. Moreover, by purifying off-gases, purifiers produce hydrogen that is cheaper than the hydrogen produced by other hydrogen producers and should be used up as well. The amount of hydrogen available from the abovementioned sources must be equal to the total amount sent to the pipeline levels Fjc, s =

∑ Fjc, l , s

Pl − Uj , l(1 − Xj , l , s) ≤ Pj

Xj , s ≥ Xj , l , s

Xj , s ≤

∑ Fjp, l , s

∀ j ∈ J, s ∈ S (9)

∑ Fl , k ,s = Fk ,s

∀ k ∈ K, s ∈ S (10)

l∈L

and the hydrogen balance

∑ Fl , k , syl ,s = Fk ,syk ,s

∀ k ∈ K, s ∈ S (11)

l∈L

Each sink needs a minimum amount of pure hydrogen to maintain its production. Moreover, the purity of each sink has to be equal to or greater than the minimum purity specification, to achieve a desired oil conversion rate and maintain a certain catalyst activity. Therefore, the constraints on the minimum amount of pure hydrogen and the minimum feed purity are stated as

∀ jp ∈ JP, s ∈ S

Fk , syk , s ≥ Fkmin ,s

(2)

yk , s ≥ ykmin

Hydrogen sources can be shut down for preventive maintenance or unexpected emergency cases. Thus, it is necessary to introduce a binary variable, Xj,s, to indicate whether a source is in operation or not

∀ k ∈ K, s ∈ S

∀ k ∈ K, s ∈ S

(12) (13)

In addition, sinks can also be shut down for preventive maintenance and unexpected emergency cases. Here, we introduce the binary variable Xk,s to indicate whether sink k runs under scenario s

⎧1 if hydrogen source j runs under scenario s Xj , s = ⎨ ⎩ 0 otherwise

⎧1 if hydrogen sink k runs under scenario s Xk , s = ⎨ ⎩ 0 otherwise

In addition, the binary variable Xj,l,s is employed to indicate the status of the hydrogen stream between source j and hydrogen pipeline level l under scenario s

Xj , l , s

∑ Xj , l , s

(8)

3.2. Hydrogen Sink Constraints. After the hydrogen has been sent from the sources to the pipeline levels, it is transported into the sinks to meet the demands. When establishing the connections between pipeline levels and hydrogen sinks, constraints imposed on the sinks should not be violated. The pipeline levels must provide enough hydrogen for each sink to maintain its operation. The flow rate and purity of each hydrogen sink are considered as variables. The equations for the sinks consist of the flow rate balance

(1)

l∈L

∀ j ∈ J , l ∈ L, s ∈ S

l∈L

Other hydrogen producers such as hydrogen plants produce large amounts of hydrogen as a main product. The hydrogen from these producers can be adjusted according to the hydrogen demand. Thus, the amount of hydrogen available from hydrogen plants should be equal to or greater than the total amount sent to the pipeline levels Fjp, s ≥

(7)

The relations between these binary variables are given by

∀ jc ∈ JC, s ∈ S

l∈L

∀ l ∈ L, j ∈ J , s ∈ S

A binary variable, Xl,k,s, is also introduced to indicate the existence or nonexistence of a stream between pipeline level l and sink k under scenario s

⎧1 if source j ⎪ ⎪ supplies hydrogen for pipeline level l =⎨ under scenario s ⎪ ⎪ ⎩ 0 otherwise

Xl , k , s

The flow rate constraints for source j and the stream from source j to pipeline level l under scenario s are given by

⎧1 if pipeline level l ⎪ ⎪ supplies hydrogen for sink k =⎨ under scenario s ⎪ ⎪ ⎩ 0 otherwise

Therefore, the flow rate constraints for sink k and the stream from pipeline level l to sink k under scenario s are given as

Fj , s ≤ Xj , sUj , s

∀ j ∈ J, s ∈ S

(3)

Fj , s ≥ Xj , suj , s

∀ j ∈ J, s ∈ S

(4)

Fk , s ≤ Xk , sUk , s

∀ k ∈ K, s ∈ S

(14)

∀ k ∈ K, s ∈ S

(15)

Fj , l , s ≤ Xj , l , sUj , l , s

∀ j ∈ J , l ∈ L, s ∈ S

(5)

Fk , s ≥ Xk , suk , s

Fj , l , s ≥ Xj , l , suj , l , s

∀ j ∈ J , l ∈ L, s ∈ S

(6)

Fl , k , s ≤ Xl , k , sUl , k , s

∀ l ∈ L, k ∈ K , s ∈ S

(16)

Based on the binary variable Xj,l,s, the pressure constraints between hydrogen source j and pipeline level l are expressed as

Fl , k , s ≥ Xl , k , sul , k , s

∀ l ∈ L, k ∈ K , s ∈ S

(17)

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F pf , s = FpP, s + FpR, s

The pressure constraints between pipeline level l and hydrogen sink k are expressed as Pk − Ul , k(1 − Xl , k , s) ≤ Pl

Xk , s ≤

∑ Xl , k ,s

(19)

F pf , sypf , s R = FpP, sypP, s

R = f (F pf , s , ypf , s , ypP, s )

(21)

Similarly, the amount of pure hydrogen entering a compressor should be equal to the amount of pure hydrogen leaving the compressor under each operating scenario

∑ Fl ,com,syl ,s = ∑ Fcom, k , sycom, s k∈K

ypR, s ≤ ypf , s ≤ ypP, s

(22)

∀ l ∈ L , com ∈ COM, s ∈ S

ypl, s ≥ ypmin

pl ∈ PL

(24)

∑

Ffp,pl, s

∑ pl ∈ PL

Fpl, p , s = F pf , s

(33)

(34)

∀ p ∈ P, s ∈ S (35)

The binary variable Xfp,pl,s is defined to indicate the existence or nonexistence of the stream between feed stream fp of a purifier and pipeline level pl under scenario s ⎧1 if feed stream fp is connected to pipeline level pl unde X fp,pl, s = ⎨ ⎩ 0 otherwise

In addition, the binary variable Xpl,p,s is introduced to indicate the existence or nonexistence of stream between pipeline level pl and purifier p under scenario s X pl, p , s = ⎧1 if pipeline level pl supplies hydrogen for purifier p ⎪ ⎨ under scenario s ⎪ ⎩ 0 otherwise

∀ fp ∈ FP, s ∈ S (25)

pl ∈ PL

∀ p ∈ P, s ∈ S

∀ pl ∈ PL, s ∈ S , p ∈ P

Fpl, p , s ≤ Fpmax

∑

3.4. Purifier Constraints. Off-gases from consumers and purifiers and the hydrogen from catalytic reforming units are usually recovered by purifiers to upgrade the hydrogen purity and make it more acceptable. The purifiers most commonly used are pressure-swing adsorption (PSA) and membrane separation units. Compared to other units, purifiers can upgrade hydrogen purity at a lower operating cost and a lower investment cost. Thus, purifiers are used increasingly widely in refineries. Purifiers can be considered as one sink (inlet stream) and two sources (the product stream and the residue stream).5 The equations for a purifier include the flow rate balance Ffp, s =

(32)

The amount of hydrogen fed into a purifier should not exceed its capacity

∀ com ∈ COM, s ∈ S

l∈L

(31)

To guarantee a high recovery ratio and product purity, purifiers have a minimum purity requirement for feed streams

Moreover, a compressor is designed for a specific flow rate; thus, the amount of gas fed into a compressor should not exceed its maximum capacity

∑ Fl ,com,s ≤

∀ p ∈ P, s ∈ S

∀ p ∈ P, s ∈ S

ypR,min ≤ ypR, s ≤ ypR,max ,s ,s

(23)

max Fcom

(30)

The residue purity is restricted by

Several streams might be mixed before entering a compressor; thus, the outlet flow rate and purity of each compressor have to be computed. The outlet purity of a compressor should be between the purity limits of the hydrogen streams entering the compressor ylmin ≤ ycom, s < ylmax ,s ,s

∀ p ∈ P, s ∈ S

This correlation can be obtained by either a theoretical derivation or an experimental study. Theoretical results can be found in Liu and Zhang,6 whereas experimental results are usually provided by the manufacturer of the purifier. In this article, this correlation is obtained from experimental results from daily operation in a refinery. In a refinery, feed streams are first mixed before entering a purifier. The feed purity of a purifier should be between those of the product and residue based on the process mechanism of purifiers



(29)

In general, there is a tradeoff between the recovery ratio R and the product purity. A simple correlation can be expressed by

k∈K

l∈L

∀ p ∈ P, s ∈ S

One important factor for purifiers is the hydrogen recovery ratio R, which is defined by

3.3. Compressor Constraints. Compressors are used to increase the pressure of hydrogen from pipeline levels and make it more acceptable for consumers and purifiers. A compressor is considered either as a sink to represent its inlet or as a source to represent its outlet. The flow rate of the inlet streams of a compressor should be equal to the flow rate of the outlet streams under each operating scenario, that is l∈L

(28)

F pf , sypf , s = FpP, sypP, s + FpR, sypR, s

(20)

∑ Fl ,com, s = ∑ Fcom, k , s

∀ p ∈ P, s ∈ S

pl ∈ PL

∀ k ∈ K, s ∈ S

l∈L

Fpl, p , sypl, s = F pf , sypf , s

∑

The relations between the two above-mentioned binary variables are ∀ l ∈ L, k ∈ K , s ∈ S

(27)

and the hydrogen balance

∀ k ∈ K , l ∈ L, s ∈ S (18)

Xk , s ≥ Xl , k , s

∀ p ∈ P, s ∈ S

The flow rate constraints for the stream from feed stream fp to pipeline level pl and the stream from pipeline level pl to purifier p under scenario s are represented as

∀ p ∈ P, s ∈ S (26) 4117


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∀ fp ∈ FP, pl ∈ PL, s ∈ S

∑ Xj ,s ≤ Nj (36)

Ffp,pl, s ≥ X fp,pl, su fp,pl, s

(37)

∀ pl ∈ PL, p ∈ P , s ∈ S

(38)

Fpl, p , s ≥ X pl, p , su pl, p , s

∀ pl ∈ PL, p ∈ P , s ∈ S

(39)

∑ Xj ,s ≥ Ns − NSmax j ∑ Xj ,s ≤ Ns

The number of shutdowns of all sinks under scenario s is limited by

(40)

∑ Xk ,s ≤ Nk

∀ p ∈ P , pl ∈ PL, s ∈ S

∑ Fil,k ,s + ∑ k∈K

+

∑

∑ Xk ,s ≤ Ns

Ffp,pl, s +

∑ fs

∑ Fil,k ,syil,s

∑

∑ Fpl,p,s + ∑ p∈P

Ffp,pl, syfp, s +

fp ∈ FP

+

∑

(44)

Fopl,pl, syopl, s =

Fpl,opl, sypl, s

∑ Fpl,p,sypl,s

j∈J

∑ CsH

2

+

s∈S

∑ CsPower − ∑ Csfuel s∈S

+ Af(∑

s∈S

∑

Ccom, s +

s ∈ S com ∈ COM

(45)

+

3.6. Equipment Operating Constraints. Because of preventive maintenance and unexpected emergency cases, hydrogen sources and sinks might be shut down. The number and times of sources and sinks to be shut down should be limited based on the practical situation. The limit on the number of shutdowns of all sources under scenario s is given by

∑ Xj ,s ≥ Nj − NUmax j

(55)

TAC =

∀ pl ∈ PL, s ∈ S

opl ∈ OPL

(54)

=1

p∈P

opl ∈ OPL

∑

Fpl,opl, s

opl ∈ OPL

∀ pl ∈ PL, s ∈ S

∑

∀s∈S

3.8. Objective Function. Considering the tradeoff between operating costs and investment costs, our aim for the design optimization of a flexible hydrogen system is to minimize the total annual costs, consisting of operating costs and annualized capital costs. The operating costs are made up of the hydrogen costs of the producers plus the cost of electricity used in compression minus the value created by the fuel gas. The capital costs are made up of the investment costs of new compressors, new purifiers, and new pipelines. Thus, the objective function is defined as

(43)

Fopl,pl, s =

(53)

s∈S

∀ il ∈ IL, s ∈ S

opl ∈ OPL

∀k∈K

ts = fs T tot

Flow rate and hydrogen balances for the pipeline levels of the hydrogen purification network are given by

∑

(52)

3.7. Constraints on the Duration of Scenarios. A basic element for addressing flexibility issues for hydrogen systems is the specification of various scenarios to represent different operating conditions. The duration of each scenario is equal to its time fraction multiplied by the total number of hours in the time horizon, namely

Fil,oil, s

oil ∈ OIL

fp ∈ FP

∀k∈K

s∈S

k∈K

Fil,oil, syil, s

(51)

s∈S

(42)

Foil,il, syoil, s =

∀s∈S

∑ Xk ,s ≥ Ns − NSkmax

oil ∈ OIL

∀ il ∈ IL, s ∈ S

oil ∈ OIL

(50)

The times of shutdowns of sink k under all scenarios is limited by

3.5. Pipeline Level Constraints. The structure of a hydrogen network can be divided into two parts: the supply network from producers to consumers and the purification network from feed stream suppliers (catalytic reforming unit, hydrogen consumers and purifiers) to purifiers. Similarly, pipeline levels can be divided into the part of hydrogen supply between the producers and consumers and the part of hydrogen purification between feed stream suppliers and purifiers. Model equations for pipeline levels include flow rate and hydrogen balances

j∈J

∀s∈S

k∈K

(41)

∑ Fj ,il,syj ,s + ∑

(49)

k∈K

Pp − Upl, p(1 − X pl, p , s) ≤ Ppl

Foil,il, s =

∀j∈J

∑ Xk ,s ≥ Nk − NUkmax

∀ pl ∈ PL, fp ∈ FP, s ∈ S

oil ∈ OIL

(48)

s∈S

Ppl − Ufp,pl(1 − X fp,pl, s) ≤ Pfp

j∈J

∀j∈J

s∈S

Based on the defined binary variables Xfp,pl,s and Xpl,p,s, the pressure constraints between feed stream fp and pipeline level pl and between pipeline level pl and purifier p are expressed as

∑ Fj ,il,s + ∑

(47)

The limit on the shutdown times of source j under all scenarios is given by

∀ fp ∈ FP, pl ∈ PL, s ∈ S

Fpl, p , s ≤ X pl, p , sUpl, p , s

∀s∈S

j∈J

∑ ∑

∑ ∑ Cp, s s∈S p∈P

Cpipe, s) (56)

s ∈ S pipe ∈ PIPE

The cost of hydrogen is calculated as the sum of the expenses of hydrogen generation and importing hydrogen from a third party. The cost of hydrogen is assumed to be proportional to its flow rate, that is CsH2 =

∀s∈S (46)

∑ ∑ Fj ,l ,st j ,sPIj j∈J l∈L

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Cormick. In this way, a bilinear term Fy can be represented by the following linear inequalities

The electricity cost of compressors is given by CsPower = PIe

∑

(Powercom, stcom, s)

∀s∈S

com ∈ COM

(58)

The fuel gas generated from consumers and purifiers is also an important energy source for the refinery. The value created by the fuel gas is obtained through the heat value calculation5 Csfuel

=

Fsfuel[ysfuel ΔHc,° H2

+ (1 −

° ysfuel )ΔHc,CH ]PIfueltsfuel 4

(59)

∀s∈S fi(1 + fi)ny (1 + fi)ny − 1

(60)

The capital costs of new compressors and pipelines can be calculated as25 Ccom, s = (acomXl , k , s + bcom Powercom, s) ∀ com ∈ COM, s ∈ S , l ∈ L , k ∈ K

(61)

Cpipe, s = [a pipe(Xj , l , s + Xl , k , s) + bpipeDpipe 2]Lpipe ∀ pipe ∈ PIPE, s ∈ S , j ∈ J , l ∈ L , k ∈ K

PSA PSA f CpPSA , s = ap X pl, p , s + bp F p , s

⎛ bpMEM ⎞ f MEM ⎟F p , s ⎜ CpMEM a = + ,s ⎜ p zpMEM ⎟⎠ ⎝

fbilinear ≥ F maxy + Fy max − F maxy max

(66)

fbilinear ≤ F miny + Fy max − F miny max

(67)

fbilinear ≤ F maxy + Fy min − F maxy min

(68)

min min ftrilinear ≥ f bilinear R + fbilinear Rmin − f bilinear Rmin

(69)

max max ftrilinear ≥ f bilinear R + fbilinear Rmax − f bilinear Rmax

(70)

min min ftrilinear ≤ f bilinear R + fbilinear Rmax − f bilinear Rmax

(71)

max max ftrilinear ≤ f bilinear R + fbilinear Rmin − f bilinear Rmin

(72)

where f trilinear = FyR. Using this procedure, the bilinear terms in eqs 11, 12, 22, 28, 29, 43, 45, and 59 and the trilinear terms in eq 30 are converted into linear terms. As a result, the original MINLP problem is transformed into an MILP problem. The upper and lower bounds of the variables in the nonlinear terms can be specified easily based on physical insights into a given system. In several previous studies,26−31 this linearization method was employed to obtain the lower bound or the initial value of the optimization problem. However, very little attention has been paid to the approximation error due to such linearization. Using this linearization method, the bilinear and trilinear functions that should be satisfied are relaxed into feasible regions described by inequalities 65−72. These relaxations will lead to violations of the corresponding balance equations consisting of bilinear or trilinear terms. Therefore, the solution of the MILP problem might not be consistent with that of the MINLP problem. This means that a design based on the MILP solution might lead to a discrepancy from the desired or specified operation. The magnitude of the discrepancy depends on the tightness of inequalities 65−72. A more detailed analysis of this issue is presented in Appendix A. For hydrogen network design, variables related to the linearization can be divided into purities of pipe levels in the supply network, purities of pipe levels in the purification network, feed purities, recovery ratios, feed flow rates, and offgas purities. Therefore, the linearized constraints might not be rigorous enough to ensure that the hydrogen purity in a pipe level is equal to the purity of the stream flowing out of the pipe level. In addition, they cannot ensure that the purities of streams flowing out from the pipe levels are identical. Similarly, these linearized constraints also cannot guarantee that the feed purities, recovery ratios, feed flow rates, and off-gas purities are identical to the true values according to mass and component balances. The purity inconsistency in pipe levels of the hydrogen supply network will lead to errors to the hydrogen supply of hydrogen consumers, whereas inconsistencies in the purities in pipe levels, feed purities, recovery ratios, feed flow rates, and

(62)

The investment costs of PSA units and membrane separation units can be expressed by the equations1

∀ p ∈ P , s ∈ S , pl ∈ PL

(65)

where f bilinear = Fy. The trilinear terms in eq 30 can be converted into bilinear terms by treating the product of two variables as a new variable. Then, the new bilinear terms can be linearized based on the described linearization method. Consequently, the trilinear term FyR can be represented by

The annualizing factor, Af, can be calculated by the equation24 Af =

fbilinear ≥ F miny + Fy min − F miny min

(63)

∀ p ∈ P, s ∈ S (64)

4. SOLUTION STRATEGY The model equations described in this section contain bilinear terms in eqs 11, 12, 22, 28, 29, 43, 45, and 59 and trilinear terms in eq 30. The presence of discrete variables along with nonlinear terms makes the design task an MINLP problem. Moreover, as the problem formulation is employed to provide the solution to satisfy the demands of hydrogen in multiple operating scenarios under consideration, the MINLP problem involves a large number of discrete and continuous variables. It is recognized, however, that solving a large MINLP problem will result in inconsistency in solution quality and time.21 In previous studies,17−20 bilinear terms were linearized by using the reformulation−linearization technique (RLT), so that the MINLP problem could be converted into a relaxed MILP problem. Li et al.21 used the RLT to reformulate bilinear terms in mass balance equations into linear equations. As pointed out by Li et al.,21 if a unit, such as a mixer, has no internal mass accumulation, its bilinear terms in mass balance equations can be replaced by individual component flows and then transformed into exact linear constraints. In our flexible optimization problem of a hydrogen network, the nonlinear terms in the model are either bilinear or can be converted to a bilinear form. The linearized constraints based on the linearization method of McCormick22 are tighter than those based on the work of Lee et al.17 Therefore, in this article, we linearize the bilinear terms using the method of Mc4119



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separation unit (MEM) are installed to purify the high-purity hydrogen from SCR and CCR and the low-purity off-gases from hydrogen consumers or purifiers. The hydrogen system has a total of 15 hydrogen consumers: two hydrocracking units (HC1 and HC2), one catalytic diesel hydrotreater (HT1), one straight-run diesel hydrotreater (HT2), one diesel hydrotreater (HT3), one kerosene hydrotreater (HT4), one wax oil hydrotreater (HT5), one diesel hydrotreater (HT6), one aviation kerosene hydrotreater (HT7), two nonaromatic hydrocarbon hydrotreaters (HT8 and HT9), one p-xylene isomerization unit (PX1), one p-xylene disproportionation unit (PX2), a gasoline adsorptive desulfurization unit (S-zorb), and an ethylene cracking hydrogenation unit (ECH). The data for the hydrogen sources, sinks, purifiers, and compressors are given in Tables 1−4, respectively. The price

off-gas purities in the purification network will cause errors in the value of the fuel gas. A straightforward way to reduce these errors is to tighten the range of the related variables by defining proper bound values in eqs 65−72. This can be done based on a detailed analysis of the real plant operation. However, tighter ranges will lead to a reduced feasible region, meaning that the objective function value will be worse. The effects on problem formulation and methods to reduce the errors arising from the linearization are discussed in Appendix A and Appendix B, resulting in an acceptable quality and a higher efficiency than can be obtained by directly solving the MINLP problem, as demonstrated in the case study presented in section 5.

5. CASE STUDY In this section, a case study is presented to demonstrate the applicability and effectiveness of the proposed method. The data for the case study were obtained from a large refinery in south China, where both high- and low-sulfur crudes are processed through hydrocracking and hydrotreating units to produce a full range of fuel products. Figure 2 shows the

Table 1. Data for Hydrogen Sources of the Case Study hydrogen source

flow rate (Nm3 h−1)

purity (%)

outlet pressure (MPa)

SCR CCR ETH FER HC1 off-gas HC2 off-gas HT1 off-gas HT2 off-gas HT3 off-gas HT4 off-gas HT5 off-gas HT6 off-gas HT7 off-gas HT8 off-gas HT9 off-gas PX1 off-gas PX2 off-gas S-zorb off-gas

57300 75800 48000 0−95000 7000 4700 550 780 865 2805 1940 2960 800 180 356 237 8281 300

92 92 95 97.7 62 65 85 62 75 75 75 75 85 85 85 86 75 88

1.28 1.39 3.7 6.7 1.5 1.5 1.6 1.4 1.5 1.5 1.7 1.5 1.5 1.5 1.5 1.5 2.6 1.55

per unit of hydrogen from each hydrogen source is not inclkuded here for the sake of confidentiality. The electricity and fuel costs are 0.093 $/kWh and 0.0045 $/MJ, respectively. The payback period is considered as 2 years, with a 5% interest rate per year. Table 2. Data for Hydrogen Sinks of the Case Study

Figure 2. Hydrogen network in the refinery for the case study.

industrial hydrogen system under consideration, where the black lines represent the hydrogen supply network between producers and consumers, the blue lines represent the purification network between the off-gas suppliers and purifiers, and the red lines represent the hydrogen streams flowing into the fuel system. Hydrogen producers include a semiregenerated catalytic reformer (SCR), a continuous catalytic reforming unit (CCR), a fertilizer plant (FER), and an ethylene plant (ETH). Three PSA units (I PSA, II PSA, and III PSA) and a membrane 4120

hydrogen sink

minimum pure hydrogen demand (Nm3 h−1)

minimum purity (%)

inlet pressure (MPa)

HC1 HC2 HT1 HT2 HT3 HT4 HT5 HT6 PX1 PX2 HT7 HT8 HT9 S-zorb ECH

45000 46000 8000 10000 14000 23000 25000 30000 4500 14000 1600 700 1500 3000 7000

95 95 85 85 93 95 95 95 95 95 91 89 92 95 95

17 15 6 4 5 6 12 6 2.5 3.1 2.5 2.3 2.1 2.6 3



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Table 3. Data for Purifiers of the Case Study purifier

purity (%)

I PSA II PSA MEM III PSA

97.2 97 96.8 97

maximum capacity (Nm3 h−1) 60000 80000 20000 25000

minimum feed purity (%) 75 75 45 45

inlet pressure (MPa)

Table 5. Operational Status of Hydrogen Producers for Each Scenario

outlet pressure (MPa)

1.28 1.35 2.8 0.5

1.2 1.25 1.33 1.3

Table 4. Data for Compressors of the Case Study compressor

maximum capacity (Nm3 h−1)

compressor

maximum capacity (Nm3 h−1)

C1 C2 C3 C4

60000 60000 15000 30000

C5 C6 C7 C8

35000 20000 4000 20000

The refinery attaches more importance to low-cost alternatives for optimizing and revamping the hydrogen system. Therefore, it endeavored to take various measures to retrofit the existing hydrogen system. Measures were taken such as closing high-cost hydrogen plants, consuming the byproduct hydrogen from the reforming units and ethylene plant as much as possible, constructing low-cost hydrogen purifiers, and fully utilizing high-pressure hydrogen from the fertilizer plant so as to reduce electricity cost of the compressors. However, the industrial hydrogen network is not optimal and can be improved based on a careful analysis. For instance, hydrogen from the reforming units, whose hydrogen cost is lower than that of other hydrogen producers, can be utilized directly by consumers HT1, HT2, and HT7−HT9. In the present configuration, however, all hydrogen from the reforming units is purified by units I PSA and II PSA. In addition, the outlet pressure of hydrogen from the ETH plant is higher than the inlet pressure of the PX2 and S-zorb units, which can consume the hydrogen from the ETH plant directly without the use of compressors. However, in the present system, units PX2 and S-zorb consume the hydrogen from unit II PSA using the compressors, leading to a higher electricity cost of compressors. The off-gases from units HT1, HT6−HT9, S-zorb, I PSA, and II PSA have high hydrogen purities and low heating values. This results in waste for the off-gases to be discharged into the fuel gas system. Thus, the off-gases should be purified by purifiers so as to obtain high-purity hydrogen. Moreover, the flexible adjustment ability of the existing hydrogen network is poor, because the current scheme for hydrogen supply can guarantee stable operation only if all hydrogen units run stably and are not shut down. Thus, it is necessary to take measures to further retrofit the present hydrogen system in order to reduce operating costs and guarantee stable operation of the hydrogen system under shutdown situations. Thirteen scenarios, which can basically include all possible situations in the daily operations of the hydrogen system, are considered in the optimization for a flexible hydrogen system. The operating conditions of producers and demands of consumers for each scenario are given in Table 5 and Figure 3, respectively. The normal, maximum, and minimum loads of the consumers without shutdown of the producers and consumers are considered in scenarios 1−3. The shutdowns of HC1, HC2, and HT5, and HT4 and HT5 and no shutdown of the producers are considered in scenarios 4−7. The

scenario

CCR

SCR

I PSA

II PSA

III PSA

MEM

FER

ETH

1 2 3 4 5 6 7 8 9 10 11 12 13

on on on on on on on off on on on on on

on on on on on on on on off on on on on

on on on on on on on on on on on off on

on on on on on on on on on on on on off

on on on on on on on on on on on on on

on on on on on on on on on on on on on

on on on on on on on on on off on on on

on on on on on on on on on on off on on

Figure 3. Hydrogen demands of hydrogen consumers for each scenario.

shutdowns of units SCR, CCR, FER, ETH, I PSA, and II PSA are considered in scenarios 8−13. 5.1. Optimal Design under a Single Scenario. First, a mixed-integer linear programming model (MILP1), considering the pipeline levels under the single most probable scenario, is studied to demonstrate the quality and efficiency of the solution based on the linearization strategy. The optimization problem is formulated based on a superstructure that includes all feasible connections between hydrogen sources and sinks. The decision variables for the MILP1 model include hydrogen flow rates and binary variables to represent the existence of connections between the sources and sinks. As shown in Table 6, the MILP1 model involves 410 continuous variables, 50 binary variables, and 1108 constraints. It was solved in Lingo 8.0 using an Intel 2.4 GHz personal computer with 1 GB of memory. The solution of the MILP1 model was obtained in 2951 iterations, which took 3 s of CPU time. The results for pipeline levels are reported in Table 7. It can be seen that, based on the characteristics of the pressures and purities of the hydrogen sources, the pipeline levels of the hydrogen supply between producers and consumers can be divided into four categories: HL1−HL4. The sources of HL1 are SCR and CCR units with similar pressures and the same purity; the hydrogen of HL2 is supplied by units of I PSA, II PSA, MEM, and III PSA with similar purities and pressures; and the hydrogen of HL3 and HL4 is provided by units FER and ETH, respectively. Similarly, based on the properties of the pressures and purities of the off-gas sources, the pipeline levels of hydrogen 4121



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Table 6. Summary of Computational Results of the Case Study model

no. of iterations

no. of continuous variables

no. of nonlinear variables

no. of discrete variables

no. of constraints

CPU time (s)

objective value (million $)

MILP1 MINLP1 MILP2

2951 − 1129680

410 3023 5363

0 725 0

50 657 657

1108 6463 16274

3 − 513

406.274 infeasible solution 395.666

that could introduce errors into the optimization results in strict intervals based on the characteristics of these variables, and we solved the MILP1 problem with different bound values, resulting in an acceptable quality and a high efficiency. The errors can be calculated by reconciling the solution results of the MILP1 model based on component balance relations, as presented in the following discussion. The purities for pipe levels of the supply network between producers and consumers and their errors in the hydrogen supply to hydrogen consumers are listed in Table 8. It can be

Table 7. Hydrogen Levels of the Case Study hydrogen level

hydrogen source

HL1

SCR CCR I PSA II PSA MEM III PSA FER ETH SCR CCR HC1 off-gas HC2 off-gas HT2 off-gas HT3 off-gas HT4 off-gas HT5 off-gas HT6 off-gas PX2 off-gas HT1 off-gas HT7 off-gas HT9 off-gas HT9 off-gas PX1 off-gas S-zorb off-gas I PSA off-gas II PSA off-gas III PSA off-gas MEM off-gas

HL2

HL3 HL4 PL1 PL2

PL3

PL4

PL5 PL6 PL7

purity (%) 92

pressure (MPa) 1.2

96.8−97.2

1.2

97.7 95 92

6.7 3.7 1.2

62−65

1.4

75

85−88

Table 8. Purity Data for Pipe Levels of the Supply Network Calculated by MILP1 purity of stream (%)

1.5

y2,1 y2,2 y2,6 y2,7 y2,9

1.5

45−58

0.77

15−20 20−30

2.1 0.55

0.971 0.971 0.971 0.971 0.971

purity of pipe level (%)

y2

minimum pure hydrogen (Nm3 h−1)

pure hydrogen supply (Nm3 h−1)

error (%)

45000 46000 23000 25000 4500

44989 45989 22996 24999 4499

0.024 0.024 0.017 0.004 0.022

0.9708

seen that five streams forming pipe level HL2 are supplied for the hydrogen consumers: HT1, HT2, HT6, HT7 and HT9. The purities of the five streams calculated by the linearized constraints of eq 11 are consistent and close to the purity of pipe level HL2 calculated by eq 43. The errors for the hydrogen supply to hydrogen consumers, caused by the inconsistencies in the purities between pipe level HL2 and the five streams from pipe level HL2, are small and thus can be ignored based on experience in engineering design. The purities for pipe levels of purification network off-gases between suppliers and purifiers and errors between the MILP1 results and the true values are reported in Table 9. All of the

purification between the feed stream suppliers and purifiers can be divided into seven categories: PL1−PL7. The sources of the pipeline level PL1 are the same as those of HL1 because the hydrogen from SCR and CCR can be consumed by the consumers or purified by the purifiers. The off-gases of pipeline level PL2 are supplied by HC1, HC2, and HT2 because of their similar pressures and purities. The off-gas sources of PL3 are HT3−HT6 and PX2 because of their identical purities and similar pressures. The off-gas sources of pipeline level PL 4 consist of HT1, HT7−HT9, PX1, and S-zorb because of their similar pressures and purities. The off-gases of pipeline level PL5 are supplied by I PSA and II PSA because of their similar purities and identical pressures. The off-gases of pipeline levels PL6 and PL7 are provided by III PSA and MEM, respectively, because of their unique characteristics of pressure and purity. As discussed in section 4, the solution of the MILP1 model causes inconsistencies in the balance equations and errors in the hydrogen supplies to hydrogen consumers and in the yield values created by the fuel gas. As shown in Appendix A and Appendix B, it can be found that the MILP model can reflect the solution of an MINLP model well, when the ranges of the independent variables are set in strict intervals based on modeling experience and a deep understanding of operating objectives. In this case study, we carefully set all of the variables

Table 9. Purity Data for Pipe Levels of the Purification Network Calculated by MILP1 purity of streams (%)

purity of pipe levels (%)

error (%)

y2,4

0.632

y2

0.631

0.158

y4,4

0.85

y4

0.854

0.468

y5,3 y5,4

0.5 0.5

y5

0.518

3.475 3.475

off-gases from pipe levels PL2 and PL4 are supplied for the III PSA unit. The purities of the streams flowing from these pipe levels by the linearized constraints of eq 29 are close to the purities calculated by eq 45. Pipe level PL5 supplies a streams to each MEM and III PSA. The purities of these two streams by the linearized constraints of eq 29 are consistent and also close to the purity of pipe level PL5 calculated by eq 45. The recovery ratio, feed purity, feed flow rate, and off-gas purity for each purifier, along with their errors, reported in 4122



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Tables 10−13, respectively. These values were calculated by the linearized constraints of eq 30, and the corresponding true

Table 13. Off-Gas Purity Data for Purifiers Calculated by MILP1 calculated value (%)

Table 10. Recovery Ratio Data for Purifiers Calculated by MILP1 calculated value (%)

true value (%)

true value (%)

error (%)

y1,1 y3,1

0.532 0.522

y1

0.532

0 1.88

error (%)

R1,1 R3,1

0.921 0.88

R1

0.92

0.109 4.348

y1,2 y3,2

0.503 0.480

y2

0.503

0 4.573

R1,2 R3,2

0.925 0.88

R2

0.922

0.324 4.555

y3,3 y5,3

0.2345 0.222

y3

0.286

18.007 22.378

R3,3 R5,3

0.7 0.7

R3

0.657

6.143 6.143

0.889 0.887 0.889 0.883

0.151 0.151 0.153 0.157

0.333

R2,4 R3,4 R4,4 R5,4

y2,4 y3,4 y4,4 y5,4

54.655 54.655 54.054 52.853

R4

0.78

12.261 12.063 12.261 11.665

values of recovery ratio and feed purity for the first three purifiers are small, but they are relatively large for the last purifier. The errors of the off-gas purity for all of the purifiers are relatively large, but the errors of the feed flow rate for all of the purifiers are small. In the design and optimization of flexible hydrogen systems, the most important task is to maintain a normal hydrogen supply for each consumer under varying operating conditions. Although the inconsistencies in the purities related to the pipe levels of the supply network will cause errors in the supplies to hydrogen consumers, the errors are small and can be neglected for the hydrogen supply based on the solution obtained with strict variable ranges. The inconsistencies in the purities of pipe levels of the purification network and the feed purities, recovery ratios, feed flow rates, and off-gas purities related to the purification units will not exert an influence on the hydrogen yield of the purifiers calculated by the linear constraint in eq 27 and will also not have an effect on the purity of the hydrogen product for the purifiers, which is considered to be constant based on the actual situation in the refinery. Therefore, the inconsistencies in the purities of pipe levels of the purification network and the feed purities, recovery ratios, feed flow rates, and off-gas purities will not have influence on the yield of pure hydrogen for the purifiers, which means that the hydrogen supply of the purifiers by the MILP solution will be reliable. However, the inconsistencies in the purities of the pipe levels of purification network and the feed purities, recovery ratios, feed flow rates, and off-gas purities related to the purification units will have an influence on the yield value created by the fuel gas. In this case study, the sources of the fuel gas are the off-gases from purifiers MEM and III PSA. The output variables of the purifiers are the flow rate of product and the residue, which can be calculated accurately based on the linear constraint of eq 27. The off-gas purities of the purifiers that are affected by the inconsistencies in the variables related to purifiers have to be recalculated by eq 29 after the MILP1 model has been solved. The yield value created by the fuel gas needs to be recalculated by eq 59 based on the true values of the off-gas purities of the purifiers. According to the recalculation, the yield value from the fuel gas is 24.361 million $/year by the MILP1 solution, but it should be 18.456 million $/year in reality. The optimization results and the corresponding network of the MILP1 model are reported in Table 14 and Figure 4, respectively. It can be seen that better use is made of the

Table 11. Feed Purity Data for Purifiers Calculated by MILP1 calculated value (%)

true value (%)

error (%)

y1,1 y3,1

0.911 0.903

y1

0.911

0 0.878

y1,2 y3,2

0.905 0.905

y2

0.905

0 0

y3,3 y5,3

0.5 0.563

y3

0.533

6.191 5.329

y2,4 y3,4 y4,4 y5,4

0.6 0.6 0.6 0.6

0.682

12.023 12.023 12.023 12.023

y4

Table 12. Feed Flow Rate Data for Purifiers Calculated by MILP1 calculated value (Nm3 h−1)

true value (Nm3 h−1)

error (%)

F1,1 F3,1

59257 60000

F1

59303

0.078 1.162

F1,2 F3,2

58613 59335

F2

58407

0.351 1.564

F3,3 F5,3

14000 14000

F3

14803

5.425 5.425

F2,4 F3,4 F4,4 F5,4

25106 23205 24680 23000

25000

0.423 7.18 1.28 8

F4

y4

values were recalculated by eqs 30, 28, 26, and 29 after the MILP1 model had been solved. It can be seen that the values of the recovery ratio and feed flow rate are not consistent for the purifiers except for the MEM unit, the values of feed purity are not consistent for the purifiers except for the III PSA unit, and all of the values of off-gas purity are not consistent for the purifiers. The errors between the MILP1 results and the true 4123



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Because the pipeline levels are included in the MILP1 model, the optimized hydrogen network has certain degrees of flexibility and can guarantee the hydrogen supply when some of the hydrogen producers and consumers are shut down. The optimization results reported in Table 14 indicate that a 4.1% reduction of the total annual cost can be achieved, leading to a savings of 17.201 million $/year, in comparison to the cost of the present hydrogen network. 5.2. Optimal Design under Multiple Scenarios. Although the optimized hydrogen network based on the MILP1 model leads to a considerable improvement and has a certain degree of flexibility, the hydrogen supply cannot be guaranteed when some of the hydrogen producers and consumers are shut down. Thus, we formulated another MINLP model (MINLP1) under multiple operating scenarios. As listed in Table 6, the MINLP1 model involves 3023 continuous variables, 725 nonlinear variables, 657 binary variables, and 6463 constraints. Because the design and optimization of a flexible hydrogen system under multiple operating scenarios leads to a complex and large-scale MINLP problem, the original MINLP problem in the case study cannot be solved directly within the limit on the number of iterations. Using the proposed linearization algorithm, the MINLP model is linearized to an MILP model (MILP2). As reported in Table 6, MILP2 involves 5363 continuous variables, 657 binary variables, and 16274 constraints. The solution of the MILP2 problem was obtained with 1129680 iterations, which required 513 s of CPU time. It should be noted that all of the variables related to bilinear and trilinear terms in the MILP2 model were set in strict intervals based on the characteristics of the variables. We also solved the MILP2 problem with different bound values of these variables and the yield value created by the fuel gas updated in the results of the MILP2 model. The optimized hydrogen network is shown in Figure 5. Because of the low pressure and purity of hydrogen from pipeline level HL1, the hydrogen from HL1 is mainly supplied for HT1, HT2, and HT7−HT9, which have low requirements on pressure and purity for the hydrogen sources. On the other hand, the hydrogen from HL1 is purified by the purifiers to upgrade the hydrogen purity. Thus, HL1 provides hydrogen for HL2 to guarantee normal hydrogen supplies of pipeline level HL2 for its hydrogen consumption units, when one or two of the I PSA and II PSA units are shut down. Because of its low pressure and high purity, hydrogen from pipeline level HL2 is mainly supplied for HC1, HC2, and HT5, which require a high pressure and purity for the hydrogen sources. PX1 requires a low pressure but high purity and thus consumes hydrogen from HL2 directly. Because of its high pressure and purity, hydrogen from pipeline level HL3 will initially be supplied to HT1−HT4, HT6, PX2, and S-zorb, which require a high pressure, so as to reduce electricity costs of the compressors. Then, hydrogen from HL3 will be transported into pipeline level HL2, when the hydrogen of HL2 is insufficient for its hydrogen consumers. Because of its high pressure, hydrogen from HL4 is mainly supplied for the ECH unit (part of the ethylene plant), S-zorb, and PX2, which require a high pressure for hydrogen sources, to reduce the electricity costs of the compressors. Compared with the present hydrogen distribution network, in the optimized network, almost all off-gases (except for the residues from the membrane separation and III PSA units) are purified by the four purifiers. As a result, fuel gases with lowerpurity hydrogen and higher heat value will be produced. The

Table 14. Optimization Results of the Case Study costs (million $) TAC operating cost hydrogen electricity fuel capital cost PSA membrane compressor piping

original network

MILP1 model

MILP2 model

423.475 423.475 418.902 20.559 −15.985 − − − − −

406.274 406.138 403.988 20.605 −18.456 0.258 − − 0.223 0.035

395.666 395.103 392.636 20.607 −18.139 1.061 − − 0.999 0.062

Figure 4. Optimized hydrogen network based on MILP1.

hydrogen from the ETH plant because of its higher pressure and purity. Consequently, two compressors (C6 and C7) are shut down in the optimized network, which will greatly reduce the electricity cost of the compressors. Almost all of the offgases except the residues from the membrane separation and III PSA units are purified by the four purifiers. As a result, fuel gas with higher-purity hydrogen and a higher heat value will be produced. Part of hydrogen from the reforming units, whose hydrogen cost is lower than that of other hydrogen sources, is utilized to supply hydrogen for HT1, HT2, and HT7−HT9. In this way, the hydrogen cost can be reduced and, in addition, the PSA units can be provided enough capacity to purify off-gases from the hydrogen consumers. Meanwhile, a new compressor (C9) is required to make the hydrogen more acceptable for hydrogen consumer HT1. 4124



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not only can ensure that the producers meet the demands of consumers under each scenario of normal operating conditions, but also can guarantee hydrogen supplies to consumers, through redundant pipeline levels, under each scenario of abnormal operating conditions. Furthermore, the off-gases from the pipeline levels of the hydrogen purification can be purified to upgrade the hydrogen purity by suitable purifiers under normal operating conditions. The off-gases can also be purified by other purifiers under abnormal operating conditions. The hydrogen consumptions of the three hydrogen consumers in the case of shutdown of hydrogen producers are shown in Figure 7. In industrial operation, the hydrogen supply for consumers can be guaranteed through the production adjustment of the hydrogen producers in the situation of fluctuating consumption or shutdown of one or two hydrogen consumers. However, because the yields of hydrogen producers such as SCR, CCR, FER, and ETH are much higher than the consumptions of the hydrogen consumers except for HC1 and HC2, one or two hydrogen consumers have to reduce their hydrogen consumption or be shut down to ensure a balance between the production and consumption, in the case of shutdowns of hydrogen producers. It should be noted that the flexible optimization problem defined in this article is, in effect, a multiobjective optimization task with the purpose of maximizing the total consumption of hydrogen consumers and minimizing the total annual cost of the hydrogen system. Therefore, the goal programming approach32 is also used to address this problem to minimize the total annual cost of the hydrogen system and maximize the total hydrogen consumption of hydrogen consumers simultaneously. We chose HC1, HC2, and HT5 as the adjustment units of hydrogen consumption to maintain the balance between hydrogen production and consumption. According to the results shown in Figure 7, the hydrogen consumptions of the three hydrogen consumers are indeed adjusted from scenario 8 to scenario 13 to balance between the hydrogen production and consumption. The optimization result, as reported in Table 14, indicates that a 6.6% reduction in the total annual cost can be achieved, leading to a 27.81 million $/year savings compared with the present hydrogen network of the refinery. It can be observed from Table 14 that design optimization model MILP2 based on multiple operating scenarios requires higher capital costs. The reason is that a higher flexibility requires a higher capacity of equipment, such as compressors. In the flexible optimization approach, the capital costs are calculated using the maximum capacity of all equipment units under multiple operating scenarios. However, the hydrogen network designed using the proposed approach provides an improved hydrogen allocation, resulting in lower operating costs. In comparison to MILP1, MILP2 can lead to 10.609 million $/year additional savings in the total annual cost.

Figure 5. Optimized flexible hydrogen network based on MILP2.

hydrogen from the reforming units, whose hydrogen cost is lower than that from other hydrogen sources, is utilized to supply hydrogen for HT1, HT2, and HT7−HT9. This configuration can reduce hydrogen costs and give the PSA units enough capacity to purify the off-gas from the hydrogen consumers. Meanwhile, the two new compressors C9 and C10 are introduced to guarantee the hydrogen supply of HT1 and HT4 in case FER is shut down. Moreover, new pipelines will be installed in the optimized hydrogen network to ensure the interconnection of pipeline levels for hydrogen supplies under both normal and abnormal operating conditions. Figure 6 and Table 15 report the yield of each hydrogen producer and hydrogen supply strategy of the sinks for each scenario. It can be seen that the optimized hydrogen network

6. CONCLUSIONS To obtain more realistic solutions, flexibility issues should be taken into account for the optimal design of refinery hydrogen systems. Most previous approaches neglected these concerns. The present work provides a flexible optimization approach to address the design and optimization of hydrogen systems under varying operating conditions. Varying hydrogen demand and possible shutdowns of hydrogen units are considered in the optimization problem formulation. Because of the bilinear and trilinear terms in the hydrogen balance equations, a high-

Figure 6. Yield of each hydrogen producer for each scenario. 4125



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Table 15. Hydrogen Supply Strategy of Hydrogen Sinks for Each Scenario scenarios hydrogen sink

1

2

3

4

5

HL2 HL2 + HL4 HL1

− HL2

HL2 −

HL1

HL2 HL2 + HL4 HL1

HL1

HT2

HL1

HL1

HL1

HT3

HL3

HL3

HT4

HL3

HT6

HL2 + HL3 HL2 + HL4 HL3

HL2 + HL4 HL3

PX1 PX2 HT7 HT8 HT9 S-zorb ECH HL2

HL2 HL4 HL1 HL1 HL1 HL4 HL4 −

HL2 HL4 HL1 HL1 HL1 HL4 HL4 HL3

HC1 HC2

HL2 HL2

HT1

HT5

6

7

8

9

10

HL2 HL2

HL2 HL2

HL2 −

HL2 HL2

HL3

HL1

HL1

HL2 HL2 + HL4 HL1

HL2 HL2 + HL4 HL1

HL1

HL1

HL1

HL1

HL1 + HL3 HL1

HL2

HL2

HL2

HL3

HL2

HL3

HL1 + HL3 HL3

HL2 + HL3 HL4

HL2 + HL3 HL2 + HL4 HL2 + HL3 HL2 HL4 HL1 HL1 HL1 HL4 HL4 −

HL2

HL2

−

HL3

−

HL4

HL2 + HL3 HL2 HL4 HL1 HL1 HL1 HL4 HL4 −

−

HL2 + HL4 HL2 + HL3 HL2 HL4 HL1 HL1 HL1 HL4 HL4 −



11

12

13

HL1

HL2 HL2 + HL4 HL1

HL2 HL2 + HL4 HL1

HL1 + HL2 HL2

HL1

HL1

HL3

HL3

HL2

HL3

HL2 + HL4 HL2

HL2

HL3

HL2 + HL4 HL3

HL3

HL2 + HL3 HL2 + HL3 HL2 + HL4 HL3

HL1 + HL2 HL3 HL2 + HL3 HL2 + HL4 HL3




HL2 HL3 HL1 HL1 HL1 HL3 − HL3



1≤x≤2

(A3)

Based on the linearization technique, eq A2 will be relaxed into the following linear constraints: y≤3−x (A4) y ≥ 4 − 2x

(A5)

y ≥ 0.5(4 − x)

(A6)

A schematic illustration of eqs A2 and A4−A6 is shown in Figure A1. It means that equality constraint A2 (red) is relaxed

Figure 7. Hydrogen consumption of three hydrogen consumers.

dimensional complex MINLP problem is formulated. The MINLP formulation is then transformed to an MILP problem by means of a linearization method. The impact of errors due to the linearization is analyzed as well. Using an industrial case study, it is shown that the optimized hydrogen network can operate under multiple operating conditions with a higher flexibility and lower total annualized costs, compared with the existing hydrogen network of a refinery. Various features of the design optimization under different operating conditions are discussed in detail through the case study.

Figure A1. Schematic diagram for the functions with larger variable intervals.

■

APPENDIX A In this appendix, a simple example is presented to illustrate the effects of the linearization strategy on the optimization results. We consider the following optimization problem: min f (x , y) subject to xy = 2

by the area enclosed by inequality constraints A4 (green), A5 (black), and A6 (blue). To solve the relaxed problem, the value of the objective function in eq A1 will be improved, but the solution point might be inconsistent with equality constraint A2. If we define a tighter range for the variables, for instance,

(A1)

1.3 ≤ x ≤ 1.6

(A2) 4126

(A7)



Article

constraint A2 will be relaxed in a new feasible region as shown in Figure A2. It can be seen that equality constraint A2 can be

FT2 = F2,1 + F2,2 + F2,3 + F2,4 + F2,4

(B6)

FT3 = F3,1 + F3,2 + F3,3 + F3,4 + F3,5

(B7)

FT4 ≥ F4,1 + F4,2 + F4,3 + F4,4 + F4,5

(B8)

FT5 = F5,1 + F5,2 + F5,3 + F5,4 + F5,5

(B9)

The flow rate balance for pipe level 3 is given by FS1 + FS2 + FS3 = FT3

(B10)

and the hydrogen balance for pipe level 3 is given by FS1ys1 + FS2 ys2 + FS3ys3 = FT3y3

(B11)

Here, ys represents the hydrogen purity of the hydrogen source. The schematic description for pipe level HL3 is shown in Figure B1. Three streams supply hydrogen to pipe level HL3,

Figure A2. Schematic diagram for the functions with smaller variable intervals.

more closely approximated by the area enclosed by the inequalities, in comparison to the result from Figure A1. This means that the solution will lead to less inconsistency than the equality constraint. However, because of the reduced feasible region, the objective function value might be worse than that from Figure A1.

■

APPENDIX B In this appendix, a small hydrogen network is considered to illustrate the influence of linearization on the optimization results. The system consists of five hydrogen sources that supply hydrogen for five sinks. The sources comprise four hydrogen producers and one pipe level. Four hydrogen producers can supply hydrogen by themselves. Pipe level HL3 first obtains hydrogen from three streams and then supplies hydrogen for the sinks. The objective function for the optimization problem is defined as

Figure B1. Schematic diagram for pipe level 3.

which further supplies hydrogen to the five hydrogen consumers. The hydrogen supply from pipe level 3 to the consumers will be determined by solving the optimization problem. It should be noted that, in the problem formulation Fj,k, Fk, yk, FT3, FS1, FS2, FS3, and y3 are variables and the others are given constants. Constraints B3, B4, and B11 include bilinear terms F3,ky3, Fkyk, and FT3y3, which can be transformed into the following linear constraints based on the linearization method (the superscript “m” denotes pure hydrogen):

min FTCT 1 1 + FT2CT2 + (FS1CS1 + FS2 CS2 + FS3CS3) + FT4CT4 + FT5CT5

(B1)

where FT and FS represent the hydrogen yields of the hydrogen source (Nm3 h−1) and CT and CS represent the unit costs of hydrogen for the hydrogen source ($ Nm−3). The constraints are given as follows: Flow rate balance for hydrogen sinks

m F1,1y1 + F2,1y2 + F3,1 + F4,1y4 + F5,1y5 = F1m

5

∑ Fj ,k = Fk

∀k∈K (B2)

j=1

∀k∈K

Minimum pure hydrogen for hydrogen sinks Fkyk ≥

min Fk,d

∀k∈K

(B4)

Flow rate balances for hydrogen sources are stated as follows: FT1 = F1,1 + F1,2 + F1,3 + F1,4 + F1,5

(B14)

m F1,4y1 + F2,4y2 + F3,4 + F4,4y4 + F5,4y5 = F4m

(B15)

F3,mk

(B3)

j=1

m F1,3y1 + F2,3y2 + F3,3 + F4,3y4 + F5,3y5 = F3m

m F3,5

+ F4,2y4 + F5,2y5 =

+ F4,5y4 + F5,5y5 =

min min F3,mk ≥ F3,min − F3,min k y3 + F3, ky3 k y3

5

∑ Fj ,kyj = Fkyk

(B13)

F1,5y1 + F2,5y2 +

Hydrogen balance for hydrogen sinks

4127

(B17) (B18)

max max F3,mk ≤ F3,min − F3,min k y3 + F3, ky3 k y3

∀k∈K

(B19)

min min F3,mk ≤ F3,max − F3,max k y3 + F3, ky3 k y3

∀k∈K

(B20)

≥

Fkmin ,d

+

F3, ky3max

∀k∈K

(B16)

∀k∈K

≥

F3,max k y3

F5m

max F3,max k y3

Fkm

(B5)

(B12)

F2m

F1,2y1 + F2,2y2 +

m F3,2

−

∀k∈K

(B21)


Industrial & Engineering Chemistry Research Fkm ≥ Fkminyk + Fkykmin − Fkminykmin

∀k∈K

Article

the hydrogen purity in the pipe level, the errors of the hydrogen supply to the hydrogen consumers are considerably smaller. Therefore, it is necessary to define the related variables in tight intervals according to modeling experience and real plant operations.

(B22)

Fkm ≥ Fkmaxyk + Fkykmax − Fkmaxykmax

∀k∈K

(B23)

Fkm ≤ Fkminyk + Fkykmax − Fkminykmax

∀k∈K

(B24)

Fkm ≤ Fkmaxyk + Fkykmin − Fkmaxykmin

∀k∈K

(B25)

■

AUTHOR INFORMATION

Corresponding Author

FS1ys1 + FS2 ys2 + FS3ys3 = FT3m

(B26)

FT3m ≥ FT3miny3 + FT3y3min − FT3miny3min

*E-mail: [email protected]. Tel.: +86 571 87951075. Fax: +86 571 87952279.

(B27)

Notes

FT3m

FT3max y3max

(B28)

FT3m ≤ FT3miny3 + FT3y3max − FT3miny3max

(B29)

FT3m ≤ FT3max y3 + FT3y3min − FT3max y3min

(B30)

≥

FT3max y3

+

FT3y3max

−

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support from the National High Technology Research and Development Program of China (2008AA042902), National High Technology Research and Development Program of China (2009AA04Z162), and Program of Introducing Talents of Discipline to University (B07031) is gratefully acknowledged.

Using these relaxations, eqs B3, B4, and B11 are replaced by eqs B12−B30 so that the problem becomes an LP problem. However, these constraints alone are not rigorous enough to ensure that the hydrogen purity inside the pipe level, which is supplied by hydrogen sources with different purities, is equal to the purity of the streams flowing from the pipe level. This will cause inconsistencies in the resulting purity, which, in turn, will cause errors in the hydrogen supply of hydrogen consumers. The solution results, including the hydrogen purities of the pipe levels, the streams flowing from the pipe levels, and the errors for the hydrogen supply of hydrogen consumers, are listed in Table B1. It can be seen from the table that the

■

Sets

COM = compressors FP = feed streams of purifiers IL, OIL = pipeline levels of the hydrogen supply network J = hydrogen sources JC = hydrogen sources including catalytic reforming unit, ethylene plant, and purifier JP = hydrogen plant K = hydrogen sinks L = pipeline levels P = purifiers PIPE = pipelines PL, OPL = pipeline levels of the purification network S = scenarios

Table B1. Purity Data Calculated by LP purity of streams (%) y3,2 y3,3 y3,4 y3,5

0.93 0.88 0.97 0.884

purity of pipe levels (%)

y3

0.864



error (%)

28000 13000 11000 16000

26669 12920 10043 15920

4.75 0.62 8.7 0.5

Superscripts

e = electricity F = feed stream fuel = fuel gas H2 = hydrogen m = pure hydrogen max = maximum min = minimum MEM = membrane separation unit P = product Power = compressor power PSA = pressure-swing adsorption unit R = residual tot = total

hydrogen purities of the four streams flowing from the pipe level are different and the hydrogen purities of four streams are also different from the hydrogen purity in the pipe level. As shown in Table B1, the errors for the hydrogen supply of hydrogen consumers cannot be neglected, especially for the first stream and the third stream. Based on the analysis in Appendix A, the errors due to linearization will be smaller when the range of the related variables is set in smaller intervals. Therefore, we further reduced the range of the purity for the streams flowing from the pipe level from (0.6, 0.97) to (0.95, 0.97). The resulting purities and errors for the hydrogen supply are listed in Table B2. Although the hydrogen purities of the three streams flowing from the pipe level are different and they are also different from

Subscripts

com = compressor; com ∈ COM fp = feed stream of purifiers; fp ∈ FP il, oil = pipeline levels of hydrogen supply network; il ∈ IL, oil ∈ OIL j = hydrogen source; j ∈ J jc = hydrogen sources including catalytic reforming unit, ethylene plant, and purifier; jc ∈ JC jp = hydrogen plant; jp ∈ JP k = sink; k ∈ K l = pipeline level; l ∈ L p = purifier; p ∈ P

Table B2. Purity Data Recalculated by LP purity of streams (%) y3,2 y3,3 y3,5

0.965 0.97 0.97

purity of pipe levels (%) y3

0.963



error (%)

26000 12000 14000

25980 11992 13916

0.077 0.067 0.6

NOMENCLATURE

4128



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pl, opl = pipeline level of the purification network; pl ∈ PL, opl ∈ OPL s = scenario; s ∈ S

Upl,p = upper bound of pressure between pipeline level pl and purifier p (MPa) Upl,p,s = maximum flow rate from pipeline level pl to purifier p under scenario s (Nm3 h−1) upl,p,s = minimum flow rate from pipeline level pl to purifier p under scenario s (Nm3 h−1) ymin = minimum hydrogen purity of hydrogen sink k (%) k ymax l,s = maximum hydrogen purity of pipeline level l under scenario s (%) ymin l,s = minimum hydrogen purity of pipeline level l under scenario s (%) yPp,s = purity of hydrogen product from purifier p under scenario s (%) yR,max = maximum residue purity of hydrogen from purifier p p,s under scenario s (%) yR,min = minimum residue purity of hydrogen from purifier p p,s under scenario s (%) ymin = minimum feed purity of purifier p (%) p z = capital cost coefficient ΔHc° = standard heat of combustion (J Nm−3)

Parameters

a = capital cost coefficient b = capital cost coefficient CT, CS = unit costs of hydrogen for the hydrogen source ($ Nm−3) Dpipe = pipe diameter (m) Fmax com = upper bound of the capacity for compressor com (Nm3 h−1) Fmin k,s = minimum pure hydrogen demand of hydrogen sink k under scenario s (Nm3 h−1) Fmax = upper bound for the capacity of purifier p (Nm3 h−1) p fi = fractional interest rate fs = time fraction of scenario s Lpipe = pipe length (m) Nj = total number of hydrogen sources Nk = total number of hydrogen sinks Ns = total number of scenarios NSmax = maximum number of shutdown times of hydrogen j source j under all scenarios NSmax = maximum number of shutdown times of hydrogen k sink k under all scenarios NUmax = maximum number of shutdown times of all j hydrogen sources under scenario s NUmax = maximum number of shutdown times of all k hydrogen sinks under scenario s Pfp = pressure of hydrogen from feed stream fp of purifiers (MPa) Pj = pressure of hydrogen from hydrogen source j (MPa) Pk = inlet pressure of hydrogen sink k (MPa) Pp = inlet pressure of purifier p (MPa) PIe = unit cost of electricity ($ kWh−1) PIfuel = unit cost of fuel gas ($ J−1) PIj = unit cost of hydrogen for hydrogen source j ($ Nm−3) ts = duration of scenario s (h) Ttot = total number of hours (h) Ufp,pl,s = maximum flow rate from feed stream fp to pipeline level pl under scenario s (Nm3 h−1) ufp,pl,s = minimum flow rate from feed stream fp to pipeline level pl under scenario s (Nm3 h−1) Uj,l = upper bound of pressure between hydrogen source j and pipeline level l (MPa) Uj,s = maximum yield of hydrogen source j under scenario s (Nm3 h−1) uj,s = minimum yield of hydrogen source j under scenario s (Nm3 h−1) Uj,l,s = maximum flow rate from hydrogen source j to pipeline level l under scenario s (Nm3 h−1) uj,l,s = minimum flow rate from hydrogen source j to pipeline level l under scenario s (Nm3 h−1) Uk,s = maximum consumption of hydrogen sink k under scenario s (Nm3 h−1) uk,s = minimum consumption of hydrogen sink k under scenario s (Nm3 h−1) Ul,k = upper bound of pressure between pipeline level l and hydrogen sink k (MPa) Ul,k,s = maximum flow rate from pipeline level l to hydrogen sink k under scenario s (Nm3 h−1) ul,k,s = minimum flow rate from pipeline level l to hydrogen sink k under scenario s (Nm3 h−1)

Variables

Af = annualizing factor Ccom,s = capital cost of compressor com under scenario s ($) Cp,s = capital cost of purifier p under scenario s ($) Cpipe,s = capital cost of pipeline pipe under scenario s ($) Cfuel = benefits created by fuel gas under scenario s ($) s H Cs 2 = hydrogen cost of hydrogen sources under scenario s ($) CPower = electricity cost of compressors under scenario s ($) s Fcom,k,s = flow rate of hydrogen from compressor com to sink k under scenario s (Nm3 h−1) Ffp,pl,s = flow rate of hydrogen from feed stream fp to pipeline level pl under scenario s (Nm3 h−1) Ffp,s = flow rate of feed stream fp of purifiers under scenario s (Nm3 h−1) Fil,k,s = flow rate of hydrogen from pipeline level il to sink k under scenario s (Nm3 h−1) Fil,oil,s = flow rate of hydrogen from pipeline level il to other pipeline level oil under scenario s (Nm3 h−1) Fj,il,s = flow rate of hydrogen from hydrogen source j to pipeline level il under scenario s (Nm3 h−1) Fj,l,s = flow rate of hydrogen from hydrogen source j to pipeline level l under scenario s (Nm3 h−1) Fj,s = hydrogen yield of source j under scenario s (Nm3 h−1) Fk,s = hydrogen consumption of sink k under scenario s (Nm3 h−1) Fl,com,s = flow rate of hydrogen from pipeline level l to compressor com under scenario s (Nm3 h−1) Fl,k,s = flow rate of hydrogen from pipeline level l to sink k under scenario s (Nm3 h−1) Foil,il,s = flow rate of hydrogen from other pipeline level oil to pipeline level il under scenario s (Nm3 h−1) Fopl,pl,s = flow rate of hydrogen from other pipeline level opl to pipeline level pl under scenario s (Nm3 h−1) Ffp,s = feed rate of purifier p under scenario s (Nm3 h−1) FPp,s = flow rate of hydrogen product from purifier p under scenario s (Nm3 h−1) FRp,s = flow rate of hydrogen residual from purifier p under scenario s (Nm3 h−1) Fpl,opl,s = flow rate of hydrogen from pipeline level pl to other pipeline level opl under scenario s (Nm3 h−1) Fpl,p,s = flow rate of hydrogen from pipeline level pl to purifier p under scenario s (Nm3 h−1) 4129



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Ffuel = flow rate of fuel gas under scenario s (Nm3 h−1) s FT, FS = hydrogen yields of the hydrogen source (Nm3 h−1) ny = payback period (years) Pl = outlet pressure of pipeline level l (MPa) Ppl = outlet pressure of pipeline level pl (MPa) Powercom,s = power of compressor com under scenario s (W) R = recovery ratio tcom,s = running time of compressor com under scenario s (h) tj,s = running time of hydrogen source j under scenario s (h) tfuel = operating hours for fuel suppliers under scenario s (h) s TAC = total annual cost ($) Xfp,pl,s = 0−1 variable that denotes whether feed stream fp is connected to pipeline level pl under scenario s Xj,l,s = 0−1 variable that denotes whether hydrogen source j supplies hydrogen for pipeline level l under scenario s Xj,s = 0−1 variable that denotes whether hydrogen source j runs under scenario s Xk,s = 0−1 variable that denotes whether hydrogen sink k runs under scenario s Xl,k,s = 0−1 variable that denotes whether pipeline level l supplies hydrogen for hydrogen sink k under scenario s Xpl,p,s = 0−1 variable that denotes whether pipeline level pl supplies hydrogen for purifier p under scenario s ycom,s = outlet hydrogen purity of compressor com under scenario s (%) yfp,s = hydrogen purity of feed stream fp of purifiers under scenario s (%) yil,s = hydrogen purity of pipeline level il of hydrogen supply network under scenario s (%) yk,s = hydrogen purity of sink k under scenario s (%) yl,s = hydrogen purity of pipeline level l under scenario s (%) yoil,s = hydrogen purity of other pipeline level oil of the hydrogen supply network under scenario s (%) yopl,s = hydrogen purity of other pipeline level opl of the purification network under scenario s (%) yfp,s = feed purity of purifier p under scenario s (%) yRp,s = purity of hydrogen residual from purifier p under scenario s (%) ypl,s = hydrogen purity of pipeline level pl of purification network under scenario s (%) yfuel = hydrogen purity of fuel gas under scenario s (%) s ys = hydrogen purity of the hydrogen source (%)

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Design and Optimization of Flexible Hydrogen Systems in Refineries

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