Enclave Optimization: A Novel Multiplant Production Scheduling

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Process Systems Engineering

Enclave Optimization: A Novel Multi Plant Production Scheduling Approach For Cryogenic Air Separation Plants Shamik Misra, Divya Saxena, Mangesh Kapadi, Ravindra D. Gudi, and Rachakonda Srihari Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03235 • Publication Date (Web): 06 Mar 2018 Downloaded from http://pubs.acs.org on March 8, 2018

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Industrial & Engineering Chemistry Research

Enclave Optimization: A Novel Multi Plant Production Scheduling Approach For Cryogenic Air Separation Plants Shamik Misra,† Divya Saxena,‡ Mangesh Kapadi,‡ Ravindra D. Gudi,∗,† and R.Srihari‡ †Chemical Engineering Department,Indian Institute of Technology, Bombay, Mumbai, India ‡Praxair India Private Limited, Bangalore, India E-mail: [email protected] Abstract The cryogenic air separation process is amongst the most energy intensive operations and requires intelligent approaches to minimize its operational cost, the main constituent of which is the power cost. Some of the air separation plants operate in a co-operative manner with each other & to capture the intricacies of these arrangements, a novel multi site framework is needed. In this paper, a novel approach called enclave optimization, which incorporates a small product exchange network amongst plants in the enclave, along with the multi plant production network, is introduced. The merits of enclave level framework lie in its ability to address the major challenges that originate from multi plant arrangements such as shared inventory, common customer and global liquid demands etc. Motivated by the time scales of i) gas & liquid demands and ii) other operational factors, we adopt a non-uniform time discretization framework, which helps to define constraints regarding different products in various time scales, over the optimization horizon. The results show that the successful implementation of

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

the non-uniform time discretization greatly reduces the overall number of constraints and variables involved in the optimization problem and makes the formulation computationally efficient. The above mentioned non-uniform time, enclave level framework is applied to a real world multi plant setting using representative scenarios provided by Praxair India Pvt. Ltd. The proposed model manifests its efficacy by optimizing those plants in a collaborative manner to find an overall minimum production cost with high computational efficiency. Keyword: Cryogenic Air Separation, Multi Plant Production Scheduling, Non-Uniform Time Discretization.

Introduction Volatile market trends, increasing competitions compel the manufacturing industry to adopt smarter operational procedure to maintain their profitability. Customers need of steady supply urges the manufacturer to explore multi-site production rather than the traditional single site-single market operation. Transformation from single site manufacturing facilities to multi-site facilities helps the manufacturer to increase production capacity as well as, provides some other business opportunities such as, 1. more stability in demand supply 2. moving closer to the target markets 3. exploitation of low cost raw materials sources and utility supplies. 4. introducing and implementing newer technologies. In this paper, simultaneous scheduling has been sought to be performed on a cluster of cryogenic air separation plants which are situated in close proximity. The term coined in this paper to describe this cluster of plants is ‘Enclave’. Building an enclave of plants in close proximity instead of a single plant with huge capacity has several advantage. Land required 2


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to make a bigger plant is high and getting a vast stretch of land near the market area may not be always possible. Another advantage is traditional single production frameworks are not flexible enough to cope with frequently changing demand scenarios and are also very prone to cause disruption in supply chain. Failure at any point of operation may cause shut down to the production facility causing disruption in the supply. 1 To gain competitive advantage therefore, many industries are now opting for multi-site production instead of traditional one site manufacturing facility. 2 Johansen et. al.(2005) 3 also hails the idea of collaborative manufacturing networks in their paper and proposes various structures to differentiate various multi site manufacturing networks based on their operational procedures. To make these arrangements functional, a strong cooperation and coordination between these plants or manufacturing sites are required. The traditional single plant scheduling approach would fail to capture the inter dependencies of these plants and could provide suboptimal solutions. To attain a globally optimal solution and to reap all the benefits that multi site production offers, a multi factory production scheduling framework is necessary. The need of efficient algorithms in addressing multi plant scheduling problems is acknowledged by the researchers of both industry and academia which reflects in the steady increase in the number of papers published in this area over the years. 1 Sauer et al.(1998) 4 attempted multi-site scheduling by designing a hierarchical two level framework in which the upper level produces a global schedule which fixes business goals for each of the plants and a local schedule are employed to track those. This concept paper showed how a generic multisite scheduling problem can be handled using fuzzy logic. Wilkinson et al.(1996) 5 presented a RTN based production network to model a large production network consisting of three factories spread across Europe. They have also included distribution aspects as well as some complexities regarding intermediate storage and equipment changeovers. However, the approach discussed by Wilkinson et al.(1996) was more at the planning level with the outcome of an aggregated formulation being the generation of realistic production targets for each plants, and then the plants would be scheduled separately 3



according to those targets. Some more notable case studies regarding application of multi plant production framework in different manufacturing sectors are listed in Behnamian et al.(2016). 1 Cryogenic air separation process is one of the most energy intensive manufacturing processes that needs intelligent strategies to minimize its operation cost, of which the major constituent is electricity cost. 6,7 There are a few propositions that have been made in recent past on single air separation plant(ASP) scheduling, such as, Mitra et al.(2012), 8 Zhang et al.(2015 & 2016), 9,10 Misra et al.(2016). 11,12 However, there is no notable attempt found in case of multi plant scheduling of ASPs. The closest attempt towards this objective is presented in Li et al.(2012), 7 where they have considered two air separation units(ASU) in a single plant. They have combined real time optimization (RTO) and scheduling to maximize the profit in certain time horizon. The formulation was rather simplistic and does not consider the intricacies that can arise due to various types of demands, power contracts, power availability & price differences, accessory units etc. Also, they did not consider two separate plants and rather introduced an extra ASU as a unit in their formulation. A similar approach as Li et al.(2012) 7 with a more detailed case study is presented by Zhou et al.(2017) 13 where they have considered multiple ASUs along with liquefier and vaporizer in a plant and tested the efficacy of scheduling under various load changing scenarios. The study by Zhou et al.(2017) 13 mainly focuses on the flexibility of the multiple ASU production system to cope with demand uncertainties and load changing scenarios but does not consider a multi plant setting and the intricacies that arise due to that. The above mentioned cases described in Li et al.(2012) 7 and Zhou et al.(2017) 13 are example of the multi-facility productions within one plant. Multi-facility productions within one plant framework does not deal with the change in the power cost (which can vary with location, power source availability) which is one of the most important factor while designing an enclave. The distance between the multiple facilities in a single plant is not comparable at all to the distance between the plants in an enclave, so, the requirement of designing a proper product exchange network is not 4


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sufficiently compelling. However, the product exchange network introduced in the proposed enclave framework is extremely beneficial in effective exploitation of the synergy between these plants. It should also be noted that the enclave optimization (ELO) proposed in this paper is very different from the multi plant planning and scheduling techniques presented in the abovementioned papers. ELO neither accommodates a proper distribution network nor considers direct relation between the customer and plants in respect of distributable products. The enclave approach presented in this paper only considers production aspects & product exchange amongst plants and the maximum number of plants that can be considered in an enclave should range between 3-4. The choice is driven by consideration that the domain related nuances would be more realistically represented using smaller number of plants while also exploring opportunities through synergy between those plants. In fact, ELO can be considered as a stepping stone towards enterprise wide optimization. The timeline and business goals of various levels of production optimization from the perspective of air separation industry are illustrated in Figure 1. As shown in the Figure 1, the scheduling horizon for ELO could span over 7-15 days. However, since ELO framework involves detailed scheduling of the units in all the plants in an enclave, longer horizon may make the problem intractable. Figure 1 also depicts the trend in complexity in the three steps of production optimization with the involvement of more number of plants and distribution aspects. The proposed ELO framework is applied on an enclave of cryogenic air separation plants (CASP). As, can be seen from Figure 1 enclave level scheduling concept can be treated as a necessary extension of our earlier formulation on the single plant scheduling, 11 to the case of multi plant scheduling with product exchange. The compelling need of designing enclave level scheduling framework for air separation plants also arises from an operational point of view. Some air separation plants are anyway operated in a conjugal manner, e.g. two ASPs can share the same storage tank for a specific product. Some air separation plants operate as an accessory to other industries such as steel 5



All,1

No. of plants and distribution complexities

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Business Decisions: 1) Enterprise wide planning 2) Integrated production and distribution optimization 3) Minimization of overall cost to serve.

Enterprise Wide Optimization Business Decisions: 1) Effective exploitation of the synergy between multiple plants 2) Effective utilization of the plant capacity and plant efficiencies 3) Intelligent utilization of electricity price distributions and contracts.

>1,1/2

Enclave Level Optimization

1,0

Business Decisions: 1) Minimization of the production cost 2) Optimum schedule for the entire plant 3) Intelligent utilization of electricity price variations and contracts.

Production Scheduling*

0

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Timeline

7

30

Figure 1: Levels of Production Optimization manufacturing industry whereas, other ASPs can be standalone plants. In such a case, two or more plants in an enclave can cater to the same gaseous product customer while either or all of the plants in the enclave can deliver products to the common liquid product customers. The plants though placed in proximity and may be similar from the operational perspective, could have a huge disparity in the operational cost due to the capacity variances &/or the difference in power contracts available to each of them. These factors offer opportunities for overall cost minimization and profitability through co-ordinated operation. The distance between the plants in an enclave is not specific. Plants in an enclave can be situated very close to each other or can be spread across states, which may increase the operational cost disparity upon introduction of state tax, variable power source options & respective prices across states. The idea behind deciding the territory of an enclave is to include those plants among which product exchange network can be designed without introducing much complexities. Product exchange among plants in an enclave is one of the novel approaches that has been introduced in this framework to help the individual plants at the operational level by exploiting the opportunities of 1) using production units situated at other locations, 2) different power contracts, 3) various productions costs at each location, 4) Shared inventory etc. 6


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Energy intensive process such as cryogenic air separation process can reap the benefit of ELO by taking advantage of the above-mentioned opportunities, which in turn not only steer the whole enclave towards achieving overall minimum operational cost but also ensures a steady supply to the customers. However, these aforementioned intricate features of multi-plant framework, such as, shared inventory, common customer also pose challenges in scheduling activities for each of these plants which cannot be addressed using conventional plant wise scheduling techniques. In addition production slates at different production facilities need not be the same. To address these real world complexities a holistic optimization scheme is needed which will accommodate all the plants whose operational procedure affects each other. This paper proposes to address the above requirements via a novel optimization formulation. The expected outcome of this formulation is to find optimum schedule for each unit in each of the plants in an enclave while attaining overall minimum production cost. In a comparative sense with enterprise wide optimization (EWO), 14 the horizon of decision making in the ELO framework proposed in this paper is closer to the regular scheduling horizon. The other and related key difference is the absence of distribution logistics in the ELO framework. The outcome of the ELO framework is an detailed optimal schedule for all the plants. The scope of the proposed enclave optimization (ELO) is limited, as granular as single plant scheduling, and the only difference is that instead of a single plant, it simultaneously optimizes multiple plants (3-4 at maximum) which can constitute a product exchange network. Therefore, as mentioned earlier, the ELO framework can be considered to be a stepping stone to the EWO approach. The remainder of the paper is organized as follows: Section 2 illustrates the basics of cryogenic air separation process and highlights on the complexities that can arise in enclave network. It also explains the novel approaches that has been proposed to address those complexities. The entire formulation is described in Section 3. In Section 4 the efficacy of the algorithm is presented through representative case studies. This paper is then concluded highlighting the key findings and novelties of the proposed framework. 7



Brief Overview of Cryogenic Air Separation Process Cryogenic air separation is one of the most effective and efficient way to produce oxygen with high purities and recoveries. 15 After pretreatment, air is liquefied in cryogenic temperature and the components are separated through distillation. Nitrogen is produced as a by-product in a large quantity alongside oxygen. A typical Air separation plant (ASP) has two types of products, gaseous products such as gaseous oxygen (GO2), medium or high pressured gaseous nitrogen (MPGN2 and HPGN2 respectively), gaseous argon (GAr) and liquid products such as liquid oxygen (LO2), liquid nitrogen (LN2) and liquid argon (LAr). Gaseous products are sent to the customers through pipeline whereas, liquid products are first stored in inventory and then sent to the customers using delivery trucks.

Process Network of an ASP Process network for a single air separation plant is illustrated in Figure 2. Air is liquefied and separated to its components in ASU. GO2 sent directly to the gaseous product customers through pipeline, whereas, GN2 needs further treatment. Liquid products such as LO2, LN2 and LAr are stored in the inventory. GAr are not directly produced in ASU. LAr is vaporized through process ‘DrioxAr’ to produce GAr. GN2 is further compressed to produce MPGN2 (medium pressured gaseous nitrogen) and HPGN2 (high pressured gaseous nitrogen). Through task LMCompGN2, low pressure GN2 are converted to MPGN2 whereas, it can be directly converted to HPGN2, through task LHCompGN2. MPGN2 can be further compressed to HPGN2 through MHCompGN2. A detailed description of the state task network for single air separation plant, along with mathematical formulation and discussion about the peculiar constraints can be found in Misra et. al.(2016). 11,12

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VMPGN2

VGN2

VGO2

VentMPGN2 VentGO2

OFF

ON

ON

VentGN2

OFF

ON

OFF

LMCompGN2 GO2

MPGN2

OFF

ON

GN2

OFF

ON

MHCompGN2

ON

OFF

LiqGN2

Startup

NoAr On

Startup

AS ON

Shutdown

LHCompGN2

Shut down

ON

HPGN2

OFF

OFF

DrioxN2 LN2

OFF

ON

LO2 OFF ON

LAr

OFF

DrioxO2

VGAr

ON

DrioxAr

GAr

ON

OFF

VentHPGN2 VHPGN2

Figure 2: Process Network with Operation Modes of an ASP

Enclave Level Framework The state task network illustrated in Figure 2 has been extended to accommodate multi plant production facilities. A set of cryogenic air separation plants m ∈ M , are considered here, which are situated in close proximity. A set of products p ∈ P (both gaseous and liquid) as mentioned earlier are produced in each of these ASPs. The production network is illustrated in Figure 3. I1 defines a set of inventory for each liquid product which belongs to plant 1 & similarly I2 and I3 defines a set of inventory that belongs to plant 2 and plant 3 respectively. I23 indicates the set of inventories that are shared between plant 2 and 3. In Figure 3 blue solid lines show the connection between the plants and the inventories. The black solid lines represent the connection between customers with the respective plants or inventories. The blue dotted lines depict the product exchange network among the inventories. Each of these plants have different power sources with varying power prices and contract obligations. Power price changes with the plant locations, time of use and contract agreements. Power consumption profile along with power availability limits are also different at each of these production facilities. Each of these ASPs are associated with a set of gaseous product cus9



tomers c ∈ C , which are connected to these facilities via pipeline. Gaseous products are sent to these customers via pipeline, so, generally gaseous product customers are exclusive to a particular ASP. But if two or more ASPs are connected with the customer through pipeline such that the gaseous products can be directly sent to the customer, then the gaseous product demands can be satisfied by more than one plant. The suitability between these production facilities and customers are captured by the parameter M Cm,c . M Cm,c = 1 defines that the ASP m are connected to the customer c . Unlike, gaseous products, liquid product demands are global. Each of these ASPs may have separate inventories for various liquid products or two or more plants can share an inventory for a specific product. Liquid products are first stored in the inventory and then sent to the customers using delivery trucks. As, there is no one to one mapping between liquid product customers and production sites, all production facilities may contribute to meet liquid demands. Generally, the mass balance in production scheduling frameworks restrict themselves within

Figure 3: Multi plant Production Network the plant envelope. However, in the proposed ELO framework since product exchange between plants are permitted, a small product transportation network is included in conven10


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tional scheduling framework. The product exchange network considered here connect the production facilities only. Product distribution to the global market is not in the scope of this paper. Only the liquid product are shipped amongst the plants. Exploitation of synergy between various production facilities, in terms of liquid product exchange are very necessary to reduce the overall production costs. For example, if the power price in some location are higher in some time periods, the production of that facility can be kept off and demands can be met by importing products from other plants. Multi plant framework along with product exchange facility among the plants makes the production network more resilient against any unforeseen disruptions. For example, if a plant in enclave goes in maintenance and does not have sufficient liquid inventory to be drioxed and sent to the gaseous product customers, liquid products can be borrowed from other plants in that enclave. This will ensure an uninterrupted product distribution. Shipping costs between these ASPs for transportation of liquid products, considered in this formulation are linear functions of their distances. Transit times (i.e. time required to transfer liquid product from one inventory to another) are also considered in the framework. The mathematical formulation for enclave wide optimization is described in Section 3. This framework has been tested upon several industrial scenario provided by Praxair India Pvt. Ltd. The expected outcomes of enclave wide optimization includes: 1. Detailed Schedule (operation modes and slates at each time period) for all the units of each plants. 2. Effective utilization of the time of use power pricing and power contracts. 3. Effective exploitation of the synergy between multiple plants. 4. Gaseous and liquid product demand fulfilment. 5. Effective utilization of the liquid products in the entire enclave. 6. Enclave wide minimum operating cost. 11



Complexities Addressed in Enclave level Multi Plant Framework Shared Inventory In an enclave wise setting two plants can share an inventory for a specific liquid product. This will add further constraints to the production scenario as, inventory capacity limitations and other contractual obligations will affect both the plants. To address this issue an index i is introduced which will indicate the inventories associated with the plants. All the constraints that are associated with the liquid products will be written with respect to inventories instead of plants. Parameter IMi,m = 1 indicates the inventory association with the plants whereas, IPi,p indicates the association of inventories with liquid products. Gaseous Product Demands and Exchange Gaseous products are sent to the customers via pipeline. A customer can be catered by two plants who are connected with the customer via pipeline. The plants in an enclave can also exchange gaseous products among themselves( if profitable), if those plants are connected via pipeline. Some plants may not have some product specific units such as compressors (for HPGN2 and MPGN2), in such cases GN2 from these plants can be sent to the other plants for further compression. Liquid Product Demand, Purchase and Exchange Liquid product demands are global and all the plants in the enclave can contribute in meeting those demands. Liquid product demands are two type. 1) Regular demand, 2) Spot Demand. Regular demands are mandatory to be fulfilled whereas, spot demands are optional. But spot demands earn more revenue so, maximum fulfilment of spot demands without affecting other limitations are advised. Liquid products can be exchanged between the inventories. This is one of the novel approaches that introduced in this framework that will help in minimizing the overall pro-

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duction cost. To address this concept a small distribution network among the inventories are introduced in this algorithm. It will help to reap the benefits from the different power contracts available to this plants. If at a certain plant the power cost is higher at certain time periods of a day, liquid products can be sent to that plant from other low production cost plants(if profitable, i.e. if production cost at source location + distribution cost to the destination location < production cost at destination location). Liquid product can also be purchased from other sources (can be transported from other in house plants or purchased from competitor plants, if profitable) to meet demands and inventory targets. The quantity purchased to quench regular or spot demands can be directly sent to the customers without putting in the inventories. This is done to avoid multiple time filling and decantation losses, which can be significant depending upon the quantity and price. Non-Off Condition In some plants, due to some operational conditions (smaller inventory, contractual obligations etc.), plant operators do not feel comfortable to let the plant go through an economic shutdown, though the optimizer might suggest so. In those cases, the operator can set the value for the parameter M N onOF F F lagm = 1 as 1, and those plants will not go through a shutdown (planned or economical) in between a scheduling horizon. Though this condition is introduced as a tool provided to the operator, it does have an enormous advantage from the computational point of view. If the parameter M N onOF F F lagm = 1 is 1, then there is no need of mode selection & transition variables for that plant. The optimizer does not need to take decisions regarding the operation mode selection and mode transition of that plant, which results in a significant decrease in the number of binary variables (Operation mode selection and transition variables are all binary) leading to increase in computational efficiency.

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Non-uniform Time Framework A discrete time framework is considered here to model the production network of the enclave. A non-uniform time discretization method is adopted, 16 where the production, power consumption and gaseous demand fulfilment calculations(calculation related to plants) will be done per hour or 2 hour basis, where the Liquid inventory calculations will be done on a day basis. This different time discretization methods can be used in the algorithm as, liquid demands are generally placed on a day basis. The smallest time bucket for describing the constraints associated with plants can be of 1 hour or 2 hour basis, which should be decided based on either the minimum uptime or downtime of any unit or time of use power price window or the time period for gaseous demand(whichever is smaller). As suggested by Velez & Maravelias (2013), 16 the non-uniform time discretization method helps to reduce the dimensionality of the algorithm and makes the algorithm computationally efficient. In the section 4, the benefits of non-uniform time discretization are demonstrated through case studies. Liquid Medical Oxygen To supply medical grade liquid oxygen plants need to acquire a set of licenses. Though most of the CASP produces liquid oxygen at per with the purity specified for medical grade, only those plants with proper permission can cater to the demands. This complexity is handled in the algorithm by creating a new product stream called ‘LMO’. On-site Contracts Some ASPs function as an ancillary to other industries (e.g. steel plants). These ASPs, though maintained by the parent organizations, are located in the customer locations and can share their electricity. In such cases the on-site customers(who mainly consumes the gaseous products) typically pose additional contracts on decantation of the liquid products from the inventory. It has to be noted that, liquid product decantation needed for selling 14


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or supplying to other inventories will be a part of the onsite contracts. Decantation for the purpose of drioxing will not be penalized as it serves the onsite customer needs. This contracts may vary with every ASP-customer agreements, however, in this paper, we are considering two major type of contracts: • Fixed Cost, No Limit: In this case there is no decantation limit in the inventories. Operators in the ASPs can take out and sell as much as liquid product as they want, without violating the operational contracts (such as, Inventory targets or minimum inventory contracts), with a nominal price to be payed to the onsite customer for every unit of liquid products that has been decanted from the inventory. • Different Slab, Different Price : In this case the amount of liquid product that can be decanted from the inventory, for purpose of selling or supplying to other ASPs, will be categorized into a number of slabs. Each of these slabs is defined by a maximum limit and has different cost structures. So, the entire liquid products that are decanted from the respective inventories will be divided in the aforementioned slabs and with each slab the cost for decantation will increase. The concept of onsite contracts is illustrated in Figure 4. In case of ‘Fixed Cost, No Limit’ scenario, there will be only one slab with no maximum limit in Figure 4. Categorizing the entire liquid decantation into various slabs and assigning respective costs with each slabs involves disjunctive set of constraints. This kind of disjunctive set can be relaxed using big-M formulation or convex hull formulation. Though convex hull formulation reduces the branch and bound effort , it increases the number of variables and constraints quite significantly. According to Vecchietti et al.(2003) big-M formulation will be beneficial for large scale systems as it does not increase the problem sizes considerably as compared to convex hull formulation. 17 Though both of these contracts are conceptualized above, only constraints for ‘Fixed Cost, No Limit’ case are presented in this paper for brevity.

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Price to be paid to the Onsite Customers (Due to On-site Contracts)


Slab 1

Slab 2

Slab 3

Liquid Product Decantation from Inventory

Figure 4: Slab Wise Liquid Decantation Limits and Corresponding Costs Free Liquid Calculation If the production of gaseous products is inadequate to meet the onsite demands, part of the respective liquid inventory can be vaporized & used to fulfill the gaseous demands. This process is known as, ’drioxing’ in air separation industry. More details on ’drioxing’ process can be found in Zhang et. al. (2016), 10 Misra et al.(2016). 11,12 But, drioxing is costly, as it decreases the chance of liquid sale. But, in air separation plant, production of different products are not independent of each other. If the gaseous demand is high then to fulfill that demand we have to increase production and along with the gaseous products a lot liquid products also will be produced whether we have demand for that much liquid product or not. These excess liquid produced (which are more than required to fulfill liquid demand and inventory build-up) will be termed as ‘free liquid’ (the term ‘free liquid’ is coined as, there is no need of the production of these liquid products but they are anyway getting produced). This ‘free liquid’ can be drioxed and sent to the onsite customers without incurring extra cost. More categorical description on ‘free liquid’ (i.e. which amount of inventory can be used as, ‘free liquid’ for drioxing) can be found in Misra et. al. 11,12 Free liquid calculation has 16


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been done inventory wise, in case of shared inventory, calculated ‘free liquid’ for respective product can be used for drioxing in any of the associated plants. The constraints related to drioxing and free liquid calculation are described in the mathematical formulation presented in the following section.

Formulation for Multi Site Production Optimization All the ASPs considered in this paper are assumed to have same set of units u ∈ U as depicted by the process network in Figure 2. The units that can be found in a typical ASP are : 1) air separation unit(ASU), 2) liquefier - to convert GN2 to LN2 , 3) product compressors : three types , low to medium pressure, medium to high pressure and low to high pressure, 4) venting units, 5) driox units to vaporize liquid products into gaseous one. Each of these units can operate in different modes as depicted in Figure 2.

Allocation Constraints Incorporating plant dynamics in modelling the production capacities of each plants will generate a complex MIDO problem which upon simplification converts to a MINLP, which is also very computationally expensive. A simplistic way to alleviate this problem is to calculate a feasible region in product space in which the production capacities of each unit can be approximated. These feasible regions are enveloped by some pre-calculated operation points or slates & any new operation point in these regions can be approximated by a linear combination of these precalculated slates. These slates can be calculated from the historical data or by offline calculation of the plant dynamics. Each of these slates contains information regarding the production capacity of each product and the corresponding power consumption. The relation between different levels of productions and corresponding power consumptions are not linear in reality but assumed to be so for the sake of simplicity. These feasible regions are calculated for every operation mode of every unit. 17



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At any time period t a unit u of plant m can only operate in one operation mode & this constraints are explained in Eq.1. Optimum number of slates are calculated offline to describe production and power consumption scenarios for each operation modes. If there is high variation between production levels and their corresponding power requirements in any operation modes, more than one feasible region can be calculated to approximate the production & power consumption aspect of that mode. More details on calculation of slates can be found in Mitra et al.(2012), 8 Zhang et al.(2016). 10 The slates selected to calculate the production capacity and power consumption should belong to the operation mode in which the unit is operating and this constraint is enforced in Eq.2. O P

∀m, u(: M Um,u = 1), t(: t = 1..N T )

vbM U OTm,u,o,t = 1

(1)

o=1(:M U Om,u,o =1)

L P

vcM U OLT SlateCoef fm,u,o,l,t = vbM U OTm,u,o,t

l=1(:M U OLm,u,o,l =1)

(2)

∀m, u(: M Um,u = 1), o(: M U Om,u,o = 1), t(: t = 1..N T ) If for certain time period an operation mode for a unit is pre-specified, then, only that operation mode should get selected in that unit for those time periods. Eq.3 describes the above-mentioned constraint. vbM U OTm,u,o,t = 1 ∀m, u(: M Um,u = 1), o(: M U Om,u,o = 1), t(: t = 1..N T and M U OT F ixedm,u,o,t = 1) (3) When a unit is scheduled to go under maintenance, then only OFF operation mode associated with the unit should be selected. Hence, P

vbM U OTm,u,o,t = 1

o(:M U Om,u,o =1 and o=M U Of fm,u )

∀m, u(: M Um,u = 1), t(: t = 1..N T and M U T M aintenanceF lagm,u,t = 1)

18


(4)

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The parameter M Um,u = 1 signifies that the unit u belongs to the plant m & parameter M U Om,u,o contains the information regarding the operation modes in which those units can operate. The binary variable vbM U OTm,u,o,t turns 1 if in a plant m at time period t unit u operates in an operation mode o. If a plant/unit maintenance is planned in a scheduling horizon, value of the parameter M U T M aintenanceF lagm,u,t should be set 1 for those time instances.

Transition Constraints These constraints take care of the mode switching of a unit. When a unit is switched from one mode to another, it has to stay in that mode for certain time periods (called as, minimum uptime) before the next changeover happens. The transition constraints are mainly adopted from Mitra et. al.(2012). 8,18 Similar constraints can also be found in recent works by Zhang et. al.(2016). 9,10 To formulate the transition constraints a proper understanding of the time horizon is necessary. A discrete time framework was designed ranging from −N P T to N T whereas, N P T denotes the largest uptime slots among all the minimum uptime slot required for all the units. Past running data for the plant till −N P T is required to calculate the current state of the units at the starting of the scheduling horizon. The time horizon is illustrated in Figure 5.

Figure 5: Discretized Time Horizon The following constraints stated in Eq.5 & Eq.6 take care of the operation mode transitions. vbM U OOTm,u,o,o0 ,t is a binary variable which is 1 if the unit at a plant switches from mode 19



Page 20 of 62

o to o’ at time t.The parameter M U OOm,u,o0 ,o consists of all possible modes of unit u from which mode o can be reached while, M U OOm,u,o,o0 denotes the all possible forward mode transitions of unit u from mode o . O X o0 =1(:M U OO

O X

vbM U OOTm,u,o0 ,o,t−1 −

vbM U OOTm,u,o,o0 ,t−1

o0 =1(:M U OOm,u,o,o0 =1)

m,u,o0 ,o =1)

= vbM U OTm,u,o,t − vbM U OTm,u,o,t−1

(5)

∀m, u(: M Um,u = 1) ,o(: M U Om,u,o = 1),t(: t = 2..N T ) O X

O X

vbM U OOTm,u,o0 ,o,t−1 −

o0 =1(:M U OOm,u,o0 ,o =1)

o0 =1(:M U JOO

vbM U OOTm,u,o,o0 ,t−1

m,u,o,o0 =1)

= vbM U OTm,u,o,t − M U OT Initialm,u,o,t−1

(6)

∀m, u(: M Um,u = 1) ,o(: M U Om,u,o = 1),t(: t = 1) M U OT Initialm,u,o,t denotes the plant, unit, mode combinations of the plant before starting of the scheduling horizon. Now after a change in operation mode, as stated above, a unit has to stay in that mode for minimum uptime period, to avoid any physical damage. M U OOT ransitionSlotm,u,o,o0 denotes the minimum uptime slot in the following Equation: M U OOT ransitionSlotm,u,o,M U OSwitchM odem,u,o ,1

X

vbM U OTm,u,o,t ≥

vbM U OOTm,u,o0 ,o,t−k

k=1(:t−k≥0) M U OOT ransitionSlotm,u,o,M U OSwitchM odem,u,o ,1

+

X

M U OOT Initialm,u,o0 ,o,t−k

k=1(:t−k shutdown − > off, the transitional mode shutdown will take a fixed time specified earlier.

vbM U OOTm,u,o,o0 ,(t−M U OOOF ixedT ransitionT imem,u,o,o0 ,o00 ) = vbM U OOTm,u,o0 ,o00 ,t ∀m , u(: M Um,u = 1) , o(: M U Om,u,o = 1),o0 (: M U Om,u,o0 = 1),

(8)

00

o (: M U Om,u,o00 = 1 and M U OOOm,u,o,o0 ,o00 = 1), ∀t(: (t − M U OOOF ixedT ransitionT imem,u,j,o,o0 ,o00 ) ≥ 0 and t = 0..N T − 1)

M U OOT Initialm,u,o,o0 ,(t−M U OOOF ixedT ransitionT imem,u,o,o0 ,o00 ) = vbM U OOTm,u,o0 ,o00 ,t ∀m, u(: M Um,u = 1) ,o(: M U Om,u,o = 1), o0 (: M U Om,u,o0 = 1),

(9)

o00 (: M U Om,u,o00 = 1) and M U OOOm,u,o,o0 ,o00 = 1), t(: (t − M U OOOF ixedT ransitionT imem,u,o,o0 ,o00 ) < 0 and t = 0..N T − 1) The Parameter M U OOOF ixedT ransitionT imem,u,o,o0 ,o00 stores the fixed stay time in o0 for the above-mentioned mode transition sequence.

Mass Balance Constraints After cryogenic distillation of air, gaseous products are sent to onsite customers through pipeline whereas, liquid products are first stored in inventory and then supplied to customers using delivery trucks. If production is inadequate to meet the demand, liquid products can 21



Page 22 of 62

be purchased from nearby plants (in-house or those of the competitors). Figure 6 illustrates

PST : Mass Balance Constraints

the entire mass balance schema for an air separation process. Plant Product Mass Balance

Product From Other Plant (In case of common customer shared by multiple plants)

Power Required Product Vent

Air In

Product From Plant to Onsite Customer

Air Separation and LiquifactionPlant

Production From Other Plants (In case of shared inventory)

Product To Inventory

Initial Inventory

Onsite Product Demand

Onsite Demand Balance

Product From Inventory to Onsite Customer

Liquid Product Received from Other Inventory

Product Inventory Liquid Product From Inventory to Merchant Customers

Liquid Product Purchase from Competitor

Product Inventory Calculation & Balance

Onsite Customer

Liquid Product Waste

Merchant Customers

Liquid product sent to other Inventory

Merchant Liquid Demand

(Regular and Spot)

Merchant Demand Balance Liquid Product From Competitor to Merchant Customers

Final Liquid Inventory

Figure 6: Mass Balance of an Air Separation Plant in an Enclave Setting

1 | Praxair Business Confidential | 10/2/2016

The amount of state produced/consumed in unit u of plant m running in mode o during time duration t can be expressed as a combination of the production/consumption rates of selected production slates. vcM U P T Quantitym,u,p,t is the total state produced/consumed in unit u during time t and can be mathematically expressed as,

vcM U P T Quantitym,u,p,t =

O X

L X

vcM U OLT SlateCoef fm,u,o,l,t

o=1(:M U Om,u,o =1) l=1(:M U OLm,u,o,l =1)

×M U OLP StateChangeRatem,u,o,l,p × (T T imet,2 − T T imet,1 ) ∀m, u,p(: M U Pm,u,p = 1),t(: t = 1..N T ) (10) Where, T T imet,1 and T T imet,2 denotes starting time and ending time of a time slot respectively. 22


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M P F inalP roductF lagm,p indicates the deliverable products of the ASP. The mass balance for deliverable gaseous products is expressed in Eq.12. Eq.13 defines the mass balance constraints for intermediate gaseous product such as, GN2. As, the ASP operation is mainly oxygen production driven, a lot of nitrogen is formed along with it. Some portion of GN2 is liquefied to LN2 based on the requirement. A part of the produced GN2 is compressed and supplied to onsite customers whereas, the rest part is vented. In multi plant scheme more than one plant can cater to one customers. So, The gaseous demand from each customer must be completely fulfilled by the plants associated with the customer, and this constraint is captured in the following equation,

CP T OnsiteDemandc,p,t −

M X

vcM CP T OnsiteDemandF illm,c,p,t = 0 (11)

m=1(:CM Pc,m,p =1)

∀c, p(: P T ypep =0 G0 and CPc,p = 1 and P F inalP roductF lagp = 1),t(: t = 1..N T ) U X

M X

vcM U P T Quantitym,u,p,t −

vcM M P T Quantitym,m0 ,p,t +

m0 =1(:M Mm,m0 =1)

u=1(:M U Pm,u,p 6=0 M X

vcM M P T Quantitym0 ,m,p,t × M M P T ransitLossm0 ,m,p = 0

m0 =1(:M Mm0 ,m =1)

∀m, p(: P T ypep =0 G0 and M P F inalP roductF lagm,p = 0), t(: t = 1..N T ) (12)

23



Page 24 of 62

So, the mass balance on the final gaseous products can be expressed as, U X

M X

vcM U P T Quantitym,u,p,t −

vcM M P T Quantitym,m0 ,p,t −

m0 =1(:M Mm,m0 =1)

u=1(:M U Pm,u,p 6=0

C X

vcM CP T OnsiteDemandF illm,c,p,t +

c=1(:CM Pc,m,p =1) M X

vcM M P T Quantitym0 ,m,p,t × M M P T ransitLossm0 ,m,p = 0

m0 =1(:M Mm0 ,m =1)

∀m, p(: P T ypep =0 G0 and M P F inalP roductF lagm,p = 1), t(: t = 1..N T ) (13) If there is a minimum quantity that must be fulfilled from a particular plant for a customer, the following constraints impose the limits. vcM CP T OnsiteDemandF illm,c,p,t ≥ CM P LimitQuantityc,m,p 0

0

0

0

(14)

∀c,m, p(: P T ypep = G , CM P LimitT ypec,m,p = HOU RLY ), t(: t = 1..N T ) If the minimum limit on fulfilment required from the plant is daily then, N PT

vcM CP T OnsiteDemandF illm,c,p,t ≥ CM P LimitQuantityc,m,p

t=1(:T Dayt =d)

(15)

∀c,m, p(: P T ypep =0 G0 , CM P LimitT ypec,m,p =0 DAILY 0 ), d(: d = 1..N D) M U Pm,u,p = 1 in the above denotes a set of those unit product combinations in which product p is getting formed. M U Pm,u,p = −1 denotes a set of those unit product combinations where, product p is getting consumed. P T ypep is a parameter which indicates that the product p is in liquid or in gaseous form. Mass balance for liquid products will be done inventory wise. As, the liquid demand is given on daily basis, inventory balance can be done day-wise. This day-wise discretization for inventory balance reduces the number of constraints and variables to a great extent compared to uniform hour wise discretization, hence, the computational complexity get reduced. The 24


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Liquid inventory balance can be written in Eq.16 & Eq.17 as, vcIP DInvi,p,d = vcIP DInvi,p,d−1 + N U M PT P P

vcM U P T Quantitym,u,p,t +

m=1(:IMi,m =1) u=1(:M U Pm,u,p =1) t=1(:T Dayt =d) N U M PT P P

vcM U P T Quantitym,u,p,t +

m=1(:IMi,m =1) u=1(:M U Pm,u,p =−1) t=1(:T Dayt =d) S P

vcSP IDP urchaseInvs,p,i,d − vcIP DDumpi,p,d −

s=1 P P

vcIP DDemandF illi,p,d −

pd=1(:P EquivalentP roductpd =p or pd=p) P P

(16)

vcIP DSpotDemandF illi,p,d

pd=1(:P EquivalentP roductpd =p or pd=p) I P

+

−

i0 =1(:IIPi0 ,i,p =1) I P

vcIIP DT ransactInvi0 ,i,p,d−IIP T ransitT imei0 ,i,p vcIIP DT ransactInvi,i0 ,p,d

i0 =1(:IIPi,i0 ,p =1)

∀i, p(: P T ypep =0 L0 and IPi,p = 1),d(: d = 1..N D) where, vcIP DInvi,p,d = IP InitialInvi,p

∀i, p(: P T ypep =0 L0 ),d(: d = 0)

(17)

Here vcIP DInvi,p,d indicates the level for a liquid product p (where,p ∈ P (: P T ypep =0 L0 ))in an inventory i at time t . Variable vcIP DDemandF illi,p,d and vcIP DSpotDemandF illi,p,d denote the amounts of liquid products sent from that specific inventory to fulfill regular and spot demand respectively. The additional amount of liquid purchased from other sources to fulfill inventory target requirement are indicated by vcSP IDP urchaseInvs,p,i,d . The inventory quantity at the end of each time period should be within maximum and minimum capacity limits, and is enforced by the following constraints, IP InvCapacityi,p,1 ≤ vcIP DInvi,p,d 0

0

∀i, p(: P T ypep = L and P EquivalentP roductp = ∅),d(: d = 1..N D)

25


(18)


vcIP DInvi,p,d ≤ IP InvCapacityi,p,2 0

0

Page 26 of 62

(19)

∀i, p(: P T ypep = L and P EquivalentP roductp = ∅),d(: d = 1..N D) The product inventory at the end of each time period should also meet the target set for the product quantity in storage facility. The targets are set taking into consideration business practices. These can be related to onsite customer contracts, upcoming maintenance activity etc. It is important to note that the target minimum inventory is defined only when there is need other than contract obligations. The target minimum is always set to contract minimum liquid requirement. Hence, IP M inInvContractp ≤ vcIP DInvi,p,d ∀i, p(: P T ypep =0 L0 and P EquivalentP roductp = ∅),d(: d = 1..N D)

(20)

The target constraints are generally soft constraint and hence violation of them can be allowed with penalty. The following constraint is target constraint with penalty violation allowed. Only the negative violation is penalized in the objective function since it is acceptable to have higher inventory than the target level. vcIP DInvi,p,d + pvcIP DInvi,p,d,2 − pvcIP DInvi,p,d,1 − IP DInvT argeti,p,d = 0 ∀i, p(: P T ypep =0 L0 and P EquivalentP roductp = ∅

(21)

IP T argetV iolationF lagi,p = 1), d(: d = 1..N D) But, if violation of target is not permissible, then the inventory must be above the target level at the specified time; vcIP DInvi,p,d ≥ IP DInvT argeti,p,d ∀p(: P T ypep =0 L0 and P EquivalentP roductp = ∅ P T argetV iolationF lagp = 0),d(: d = 1..N D)

26


(22)

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Furthermore, it should be noted that among pvcP T Invp,t,1 and pvcP T Invp,t,2 only one variable will take value and that will be equivalent to violated amount. Liquid Demand Constraints The merchant liquid demand is mostly fulfilled from inventory at plant location and in some cases from other supply sources. Merchant liquid demand can be classified in two types, 1) Regular demand, 2) Spot demand. Regular Demand needs to be fulfilled, whereas spot demands are optional. But spot demands earn more revenue, so, maximum fulfilment of spot demand is encouraged. Eq.23 and Eq.24 captures the constraints for regular demand fulfilment. P Y QDemandQuantityp,y,q,1 ≤ N D P

I P

d=1(:d≥P Y QDemandDayp,y,q,1 and d≤P Y QDemandDayp,y,q,2 )

i=1

+

S P

vcIP DDemandF illi,p,d !

(23)

vcSP DP urchaseDemands,p,d

s=1

∀p(: P T ypep =0 L0 ),y(: Y T ypey = 0 REGULAR0 ,q(: q = 1..P Y Countp,y )

P Y QDemandQuantityp,y,q,2 ≥ N D P

I P

d=1(:d≥P Y QDemandDayp,y,q,1 and d≤P Y QDemandDayp,y,q,2 )

i=1

+

S P

vcIP DDemandF illi,p,d !

(24)


s=1

∀p(: P T ypep =0 L0 ),y(: Y T ypey = 0 REGULAR0 ,q(: q = 1..P Y Countp,y ) As mentioned above the spot sale is optional. But if we decide to honour it and some minimum demand is mentioned then it is necessary to fulfill the minimum demand at the least. A binary variable vbP Y QSpotSalep,y,q is introduced which turns 1 if spot sale of product p is done in that time period. Constraints regarding spot demand, when the minimum limit

27



Page 28 of 62

is mentioned, are expressed in Eq.22 and Eq.23. vbP Y QSpotSalep,y,q × P Y QDemandQuantityp,y,q,1 ≤ N D P

I P

d=1(:d≥P Y QDemandDayp,y,q,1 and d≤P Y QDemandDayp,y,q,2

i=1

+

S P

vcIP DSpotDemandF illi,p,d !

(25)

vcSP DP urchaseSpotDemands,p,d

s=1

∀p(: P T ypep =0 L0 ),y(: Y T ypey = 0 SPOT0 ,q(: q = 1..P Y Countp,y )

vbP Y QSpotSalep,y,q × P Y QDemandQuantityp,y,q,2 ≥ N D P

I P


i=1

+

S P


(26)


s=1

∀p(: P T ypep =0 L0 ),y(: Y T ypey = 0 SPOT0 ,q(: q = 1..P Y Countp,y ) If any minimum demand for spot sale is not defined or zero then above set of equations reduced to the following equation, P Y QDemandQuantityp,y,q,2 ≥ N D P

I P


i=1

+

S P


(27)


s=1

∀p(: P T ypep =0 L0 ),y(: Y T ypey = 0 SPOT0 ,q(: q = 1..P Y Countp,y ) The variables vcSP DP urchaseDemands,p,d and vcSP DP urchaseSpotDemands,p,d indicates the amount of liquid products those are imported from other plants to fulfill regular and spot demand.

28


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Supply Source Purchase Limitations As mentioned earlier, if the liquid production for a certain time periods is not sufficient to meet liquid demand and inventory target, we can purchase the required quantity from other sources (Other Praxair Plants in vicinity or other competitors). Products received from other sources generally have an upper limit on availability. Further, if the lower limit is also mentioned then we have to buy more than the minimum amount. Therefore, N D P

vcSP DP urchaseDemands,p,d +

d=1(:d≥SP QLimitDays,p,q,1 and d≤SP QLimitDays,p,q,2 )

vcSP DP urchaseSpotDemands,p,d + I P

!

(28)

≥ vbSP QP urchases,p,q × SP QLimitQuantitys,p,q,1

vcSP IDP urchaseInvs,p,i,d

i=1

∀s, p(: P T ypep =0 L0 ), q(: q = 1..SP Counts,p andSP QLimitQuantitys,p,q,1 > 0) N D P




!

(29)

≤ vbSP QP urchases,p,q × SP QLimitQuantitys,p,q,1


i=1

∀s, p(: P T ypep =0 L0 ), q(: q = 1..SP Counts,p andSP QLimitQuantitys,p,q,1 > 0) If the minimum demand is not mentioned then only the constraint mentioned below will suffice. N D P




! vcSP IDP urchaseInvs,p,i,d

≤ SP QLimitQuantitys,p,q,1

i=1

∀s, p(: P T ypep =0 L0 ), q(: q = 1..SP Counts,p andSP QLimitQuantitys,p,q,1 = 0)

29


(30)


Page 30 of 62

It is important to note that fulfilling liquid demand (including spot demands) or inventory targets using material from other sources will always be costly. Hence purchasing from other sources should be discouraged to minimize the operational cost by including corresponding purchase cost in objective function.

Power Consumption, Purchase, Availability Constraints Power Consumption equation Power required in operation mode o of unit u at time t should be a combination of the power requirement of the various slates which are selected. Total power consumed in unit u in time duration t at plant m can be written as, O P

vcM U T P owerRequiredm,u,t =

L P

vcM U OLT SlateCoef fm,u,o,l,t

o=1(:M U Om,u,o =1) l=1(:M U OLm,u,o,l =1)

×M U OLP owerConsumptionm,u,o,l × (T T imet,2 − T T imet,1 ) ∀m, u(: M Um,u = 1), t(: t = 1..N T ) (31) The total power required to run all the units should be less than the power available. This is handled in Eq.32. The parameter M T U tilityP owerm,t indicates the amount of power needed to run the office utilities at the plant location m. U P

vcM U T P owerRequiredm,u,t + M T U tilityP owerm,t ≤

u=1

E P

vcM ET P urchasem,e,t

e=1(:M Em,e =1)

∀m, t(: t = 1..N T ) (32) Now, power is available from many sources. Power purchased from a suitable power source should be within minimum and maximum availability limits. The limits can be specified for a particular hour or a period. It is important to note that not purchasing power is an option; however if power is purchased, it should be greater than minimum availability for supply network stability. When the availability limits are given on an hourly basis, they are 30


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accordingly converted into time slot limit taking into consideration duration of time slot. Hence, vbM ETm,e,t × M ET Limitm,e,t,1 ≤ vcM ET P urchasem,e,t

(33)

∀m, e, t(: M ET Limitm,e,t,1 > 0 and t = 1..N T ) vcM ET P urchasem,e,t ≤ vbM ETm,e,t × M ET Limitm,e,t,2

(34)

∀m, e, t(: M ET Limitm,e,t,2 > 0 and t = 1..N T ) When lower limit on power availability from a power source at air separation plant is not defined or it is zero, then above set of equations is replaced with following constraint without any need to introduce a binary variable.

vcM ET P urchasem,e,t ≤ M ET Limitm,e,t,2 ∀m, e, t(: M ET Limitm,e,t,2 > 0 and t = 1..N T ) (35) When the availability limit is defined for a period, then following set of constraints are included into production scheduling formulation. M EQP eriodLimitm,e,q,1 ≤ N PT


(36)

t=1(:t≥M EQP eriodLimitT imem,e,q,1 and t≤M EQP eriodLimitT imem,e,q,2 )

∀m, e, q(: q = 1..M EP eriodLimitCountm,e and M EQP eriodLimitm,e,q,1 ≥ 0) N PT


t=1(:t≥M EQP eriodLimitT imem,e,q,1 and t≤M EQP eriodLimitT imem,e,q,2 )

≤ M EQP eriodLimitm,e,q,2

(37)

∀m, e, q(: q = 1..M EP eriodLimitCountm,e and M EQP eriodLimitm,e,q,2 ≥ 0) If there is additional constraint to select only one power source(i.e. parameter M OneSourceF lagm = 1) in a time period then following selection constraint should be included. E P

vbM ETm,e,t = 1 ∀m, t(: t = 1..N T and M OneSourceF lagm = 1)

e=1(:M Em,e =1)

31


(38)


Page 32 of 62

Power Cost We can assume time of the day power pricing irrespective of power source. If power price is not varying with time of day, constant price will be given in all time slots. The cost of purchasing power from different power sources can be calculated as follows:

vcM ECostm,e =

N PT

vcM ET P urchasem,e,t × M ET P owerP ricem,e,t ∀m, e(: M Em,e = 1)

t=1

(39) Some power sources penalize under-consumption, i.e. if the power purchased is less than the defined minimum limit then an additional penalty will be charged. In these cases, production scheduling activity should aim to purchase power equal to or more than penalty limit to minimize the power bill. However, it is important to note that power purchase equal to or more than penalty limit is not compulsory if other options provide less power bill even after inclusion of penalty. Hence, the constraints regarding penalty limit is soft and expressed in Eq.40. M ELimitP enaltym,e −

N PT

vcM ET P urchasem,e,t ≤ pvcM ELimitP enaltym,e

t=1

(40)

∀m, e(: M ELimitP enaltym,e > 0) The penalty variable pvcELimitP enaltye is included in minimization objective function; hence it takes positive value only when penalty Limit is violated.

Driox Constraints To meet onsite customer demand a portion of liquid inventory is intentionally drioxed i.e. converted into gaseous form. The liquid product used to quench driox demands can be categorized into two parts : 1) Free liquid (this part can be drioxed without any penalty) or liquid associated with driox costs. Use of free liquid completely is necessary as drioxing this does not incur any costs. The production needed to fulfill the driox demand over the

32


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horizon can be calculated by Eq.41. The driox demand is calculated for each inventory, which indicates the amount of decantation needed from an inventory for that specific liquid product to fulfill the onsite demand for the respective gaseous products in the associated plants. For a shared inventory, required liquid product will be used to quench gaseous driox demand in all associated plants. M P

vcIP T otalDrioxDemandp =

N PT

U P

vcM U P T Quantitym,u,p,t

m=1(:IMi,m =1) t=1 u=1(:M U Drioxm,u =1 and M U Pm,u,p =−1)

∀i, p(: P T ypep =0 L0 ) (41) Now, if the target inventory is more than the initial inventory quantity then this difference should be fulfilled first using the liquid produced. The amount of produced liquid left after meeting total liquid demand and the inventory target will be considered as free liquid (Eq.42). Total liquid demand from an inventory consists of the following : 1) Contribution from that inventory in meeting global regular and spot liquid demand, 2) supply to other inventories in the enclave. liquid product received from other inventories can also be used as for driox if profitable.

vcIP T otalF reeLiquidi,p = − − + − −

D P

M P

N PT

U P


m=1(:IMi,m =1) t=1 u=1(:M U Pm,u,p =1) P P

vcIP DDemandF illi,pd,d

d=1 pd=1(:pd=p or P EquivalentP roductpd =p) D P P P

vcIP DSpotDemandF illi,pd,d

d=1 pd=1(:pd=p or P EquivalentP roductpd =p) I N D P P i0 =1 d=1(:d−IIP T ransitT imei0 ,i,p >0) I N D P P

vcIIP DT ransactInvi0 ,i,p,d−IIP T ransitT imei0 ,i,p

vcIIP DT ransactInvi,i0 ,p,d

id=1(:IIPi,id,p =1) d=1 I P

(IP DInvT argeti,p,N D − IP InitialInvi,p )

i=1

∀i, p(: P T ypep =0 L0 ) (42) 33



Page 34 of 62

Now, the abovementioned equation aptly describes the amount of the available free liquid in the inventory for the case when initial inventory is less than the inventory target. But if the initial inventory is greater than the inventory target set at the end of the horizon then the extra liquid product that already been present in the inventory should not be treated as free liquid in its entirety. The inventory at the end of the previous schedule(initial inventory of the present schedule) may have been increased by purchasing liquid from the supply sources. This excess inventory liquid can be used for quenching liquid product demands however, using them for drioxing will increase the overall cost. A detailed explanation regarding ’free liquid’ classification and calculation can be found in Misra et al.(2016). 11,12 To calculate the quantity of available free liquid at each inventory of each plant in the enclave same calculation procedure described in Misra et al.(2016) 11,12 need to be written for all inventories. As, the calculation procedures are exactly similar as described in Misra et al.(2016), 11,12 those constraints are not included in this paper to maintain brevity. Now, The variable vcIP T otalF reeLiquidi,p is a free variable (the variable is treated as free to avoid the chance of infeasibility) however, free liquid quantity will only be considered if it is positive. So, we will decompose this quantity into positive and negative components and then express constraints based on those. Eq.43 − Eq.46 are used to describe the abovementioned decomposition technique. vcIP F reeLiqP ositiveQuani,p − vcIP F reeLiqN egativeQuani,p = vcIP T otalF reeLiquidi,p ∀i, p(: P T ypep =0 L0 ) (43)

vcIP F reeLiqP ositiveQuani,p ≤ IP InvCapacityi,p,2 × vbIP F reeLiqP ositivei,p 0

(44)

0

∀i, p(: P T ypep = L ) vcIP F reeLiqN egativeQuani,p ≤ IP InvCapacityi,p,2 × vbIP F reeLiqN egativei,p 0

0

∀i, p(: P T ypep = L ) 34


(45)

Page 35 of 62 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


vbIP F reeLiqP ositivei,p + vbIP F reeLiqN egativei,p = 1 0

(46)

0

∀i, p(: P T ypep = L ) vcIP F reeLiqP ositiveQuani,p and vcIP F reeLiqN egativeQuani,p both are positive variable. So, the actual amount of intended Driox demand which will have a cost term associated with it over the horizon can be calculated as shown in Eq.47. vcIP BillableDrioxDemandi,p ≥ vcIP T otalDrioxDemandi,p − vcIP F reeLiqP ositiveQuani,p ∀i, p(: P T ypep =0 L0 ) (47) vcIP BillableDrioxDemandi,p is a positive variable.

On-site Contracts In section 2 concepts of two types of on-site contract agreements are explained. Now these onsite contracts are specific to product, respective inventories and the plants associated with those inventories. So, the entire amount of a specific liquid product that needs to be decanted for fulfillment of 1) liquid demand or 2) demand from other inventories in the enclave, from a specific inventory which abides by onsite contracts, is divided into a number of slabs. However, for the scope of this paper only ‘Fixed Cost, No Limit’ type contract limitation is considered. ‘Fixed Cost, No Limit’ contract can be considered as a special case of slab wise contracts consisting of only one slab with no maximum limit and fixed cost. So, the total liquid decantation required from a specific inventory is calculated in Eq.48. As there are no maximum limits on these type of contract agreement the entire liquid product that is needed to quench the above mentioned demand can be taken out from the inventory, and

35



Page 36 of 62

fixed price is charged on per unit of liquid product decantation. IP SlabCount P i,p

vcIP DQSlabQuantityi,p,d,q = P P vcIP DDemandF illi,p,d + vcIP DSpotDemandF illi,p,d pd=1(:P equivalentP roductpd =p or pd=p) I P + vcIIP DT ransactInvi,i0 ,p,d q=0

i0 =1(:IIPi,i0 ,p =1)

∀i, p(: P T ypep =0 L0 ), d (48) The charges that need to be paid to the onsite customers are minimized as a part of the objective function. The effect of onsite contracts in selecting inventories that needs to be decanted to fulfill demands can be discerned from the case study explained in scenario 3 in the following section.

Objective Function The objective will be minimization of the production cost, where the total production cost can be written as, Total production cost = Total power cost(TPC) + Cost for violating penalty Limit(CP) + Penalty for dumping liquid product(PD) + Penalty for venting gaseous product(PV) + Liquid purchase cost from other sources(COS) + Penalty cost for violating inventory target(PINV) + Driox cost(DC) - Revenue earned from spot sale(ROS) + Cost of Liquid Product Transport(CLT) + Cost of Gaseous Product Transport(CGT) + Cost of Liquid Product Decantation (CLD) (49)

where,

36


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TPC =

M X

E X

vcM ECostm,e

m=1 e=1(:M Em,e =1)

CP =

M X

E X

M EP enaltyCostm,e × pvcM ELimitP enaltym,e

m=1 e=1(:M ELimitP enaltym,e >0)

PD =

PV =

I X

P X

i=1 p=1(:P T ypep

=0 L0 )

M X

P X

P DumpP enaltyp ×

P V entP enaltyp ×

COS =

S P

vcIP DDumpi,p,d

d=1

and IPi,p =1

m=1 p=1(:P T ypep =0 G0 )

ND X

NT X

U X


t=1 u=1(:M U V entm,u =1andM U Pm,u,p =1) SP Count P s,p

P P

s=1 p=1(:P T ypep =0 L0 ) N D P

q=1

SP QLimitP rices,p,q × vcSP DP urchaseDemands,p,d

d=1(:d≥SP QLimitDays,p,q,1 and t≤SP QLimitDays,p,q,2 )

I P

+vcSP DP urchaseSpotDemands,p,d +


i=1(:SIPs,i,p =1)

PINV =

I X

P X

i=1 p=1(:P T ypep

=0 L0

P X

DC =

p=1(:P T ypep

ROS =

P T argetInvP enaltyp,2 × I X

IP DrioxCosti,p ×

P P

vcIP BillableDrioxDemandi,p

i=1(:IPi,p =1) P Y Count P p,y

Y P

P Y QDemandP ricep,y,q × I P vcIP DSpotDemandF illi,p,d + and d≤P Y QDemandDayp,y,q,2 ) i=1(:IPi,p =1) S P vcSP DP urchaseSpotDemands,p,d

p=1(:P T ypep =0 L0 ) y(:y=0 SP OT 0 ) N D P d=1(:d≥P Y QDemandDayp,y,q,1

pvcIP DInvi,p,d,2

d=1

and IPi,p =1)

=0 L0 )

ND X

q=1

s=1

CLT =

I P I P

P P

IIP T ransitCosti,i0 ,p ×

i=1 id=1 p=1(:P T ypep =0 L0 and IIPi,i0 ,p =1)

CGT =

M P M P

N D P

! vcIIP DT ransactInvi,i0 ,p,d

d=1

P P

M M P T ransitCostm,m0 ,p ×

m=1 m0 =1 p=1(:P T ypep =0 G0 and M M Pm,m0 ,p =1)

37


N PT t=1

vcM M P T Quantitym,m0 ,p,t


CLD =

I P

P P

IP SlabCount P i,p

IP QSlabChargei,p,q ×

q=1

i=1 p=1(:IP SlabCounti,p >0)

Page 38 of 62

D P

vcIP DQSlabQuantityi,p,d,q

d=1

Results & Discussion The mathematical formulation addressing all the complexities mentioned above is applied on an enclave of ASPs which simulates some real world industrial scenarios provided by Praxair India Pvt. Ltd. The enclave contains three ASPs which are operationally identical but different in capacity and power consumption. Relation between production capacities and corresponding power consumption is not linear, so, it may happen that to produce same amount of products plant 2 consumes more power compared to plant 1. Each and every plant can produce the following products: GO2, GAr, HPGN2, MPGN2 (gaseous products) and LO2, LN2, and LAr (liquid products). Plant 1 has the required license to supply LMO. So, only plant 1 can cater to the LMO customers. Gaseous products are not stored in inventory and directly sent to the customers through pipeline as per the demand. Gaseous product supply can be monitored on an hourly or two hourly basis. Customer 1 is exclusively paired with plant 1, whereas, gaseous demand for customer 2 can be met by plant 2 or plant 3 or by a combination of both of these plants productions. However, any of these three plants can satisfy liquid demands (regular as well as spot), as liquid product demands are global. The plants also can purchase liquid products from nearby competitor plants to replenish their liquid inventory. If the overall production of the entire enclave is not adequate to meet the liquid product demand, additional liquid product can be purchased from other sources and sent to the customers if profitable. The main highlight of this enclave optimization framework is the inclusion of the product exchange aspect among the plants. The possible combination of the plants for product exchange and the corresponding time and cost are listed in Table 1. In Table 1 I 1 denotes the inventories associated with plant 1 & I 23 denotes the inventories shared by plant 2 & 3. The same notations are used in rest of the paper as well. The entire production and product exchange framework is illustrated in Figure 3. As, 38


Page 39 of 62 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


mentioned earlier, a non-uniform time discretization strategy is used here. The production of the plants are modeled using smallest time slot of one hour and two hour. In both cases calculation related to liquid product inventory was done on a day basis. With same input data the model was simulated twice using one hour and two hour time bucket. The results are tabulated in Table 2. Several scenarios that resembles the real world operation of air separation plants are created and the MILP formulation is solved using Fico©Xpress Optimization suite using ‘mmxprs’ module version 2.8.1 on a Intel®Core ™i7(3.6 GHz) machine with 16GB RAM. 19,20 In most of them the above mentioned framework is able to produce satisfactory results in stipulated time. The model is also simulated using different variation of Non−Off Flag parameter, (If M N onOf f F lagm is 1 for a plant m, that means Plant m can not go through a shutdown process in the scheduling horizon) which in turn reduces the number of binary variables and increases the computation speed. The simulation parameter details for both smallest time slots are tabulated in Table 2. The maximum computational time is set as, 1 hour and the desired optimality gap is set as, 0.05%, and simulation will be stopped on meeting either of the above-mentioned criteria. It can be seen from Table 2 that two hour time discretization makes the model computationTable 1: Possible Liquid Transit Options & Corresponding Time & Cost Origin Inventory Destination Inventory Product I1 I3 LO2 I1 I23 LN2 I1 I3 LAr I3 I1 LO2 I23 I1 LN2 I3 I1 LAr I3 I2 LAr

Transit Time (Day) Transit Cost (m.u) 1 0.5 1 0.4 1 0.5 1 0.5 1 0.4 1 0.5 1 0.3

ally viable while providing an implementable production schedule for the entire enclave. The results presented in following are based on 2 hour time bucket for production calculation. Different scenarios are designed in such a way that the efficacy of enclave frameworks are highlighted. Results are listed and explained in detail in the following. Due to confidentiality 39



Page 40 of 62

Table 2: Simulation Statistics For Different Discretization Methods Discretization 1 hr. 2 hr.

Constraints 76232 38302

Variables 107800 54079

Binary 23633 11825

Time(min) 60 7.3

Optimality Gap(%) 0.76 0.05

issues, the exact production or power consumption data are not mentioned. The units and values reported in the graphs or tables are in normalized form.

Scenario 1 : Similar power prices and power contracts available to all the plants In this scenario, each of the plants in the enclave has different power sources however, with the same pricing and contractual agreements. Though the power pricing is similar, power consumption and production cost vary significantly with the differences in capacity. The enclave framework is applied to estimate the overall minimum production cost over all the constituent plants. The optimum schedule for plant 1 is shown in Figure 7a. The result shows that though the ASU is in on mode for the entire scheduling horizon, the liquefier in plant 1 (unit named ’LiqGN2’), which also consumes a huge amount of electricity, is not used at all. The importance of this observation can be discerned fully when compared with the optimum schedule of plant 3 which is the biggest plant in the enclave(Figure 7b). As can be seen in the Figure 7b ‘LiqGN2’ is in on mode in plant 3 for the entire horizon. As, the liquid demands are global, major portion of LN2 demands are fulfilled by plant 3 instead of plant 1 & 2. As, plant 3 is the biggest, power consumption to production ratio is low in plant 3 when compared with the other plants. The total LN2 demand and the contribution of each plants in the enclave in fulfilling that demand is aptly captured in Figure 15a. Plant 2 is smaller compared to the other plants & since the, production and power consumption are not linearly correlated, to produce the same amount of product plant 2 consumes more power than the other plants. As the production cost in plant 2 is costly compared to

40


Page 41 of 62 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ON

Shutdown

Startup

Off

NOARON

ASU

ASU

LiqGN2

LiqGN2

LMComp

LMComp

MHComp

MHComp

LHComp

LHComp

1

21

41

61

81 Time(hr.)

101

121

141

1

161

21

41

(a) Schedule for Plant 1

ON

Shutdown

Startup

Off

81 Time(hr.)

61

NOARON

101

121

141

161

(b) Schedule for Plant 3

Figure 7: Optimum Schedule for Plant 1 & Plant 3 in Scenario 1 the other plants, production of plant 2 was stopped for several hours and the major part of the total liquid demand is catered from plant 1 and plant 3. The optimum schedule for plant 2 is illustrated in Figure 8. ON

Shutdown

Startup

Off

NOARON

ASU

LiqGN2

LMComp

MHComp

LHComp

1

21

41

61

81 Time(hr.) 101

121

141

161

Figure 8: Schedule for Plant 2 in Scenario 1 Plant 2 and plant 3 also share an inventory for liquid nitrogen & inventory profile for that shared inventory, indicating each plant’s contribution, can be observed in Figure 9. Use of shared inventory minimizes capital costs and it is already a part of operational practice in air separation industry. Simultaneous optimization of all the plants sharing an inventory is indeed needed to optimally schedule the production of these plants. Plant 2 and plant 3 also cater to the same onsite customers and each of the plants contribution in fulfilling gaseous oxygen demand is shown in figure Figure 10. For better understanding and visibility 41



Plant2 Production LN2 Demand

Plant3 Production LN2 Inventory

0.8

1

0

1

2

Day

3

LN2 Inventory

LN2 Production/Consumption

0.8 0.3

0.6 4

5

6

7

-0.2

0.4

-0.7 0.2

-1.2

0

Figure 9: LN2 Production & Consumption Profile in Shared Inventory of Plant 2 & 3 result for only first 20 time slots are shown. It is easily discernible from Figure 10, that the contribution of plant 3 is much higher than plant 2 as the production cost in plant 3 is lower than plant 2. Plant 2 Contribution LO2 Demand From Customer 2

Plant 3 Contribution

1

0.8

LO2 Demand

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 62

0.6

0.4

0.2

0 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Time Slot(hr.)

Figure 10: GO2 Demand from Customer 2 and Associated Plant’s Contribution As discussed earlier, the liquid demands are global & all the plants in the enclave can contribute to meet the liquid demand. As can be seen in Figure 11a, which represents total liquid argon demand and its fulfilment, all the plants in the enclave takes part in fulfilling the total argon demand. The inventories associated with plant 1 are indicated as I 1 in Figure 11a & so on. The figure also shows that the major portion of the demands are fulfilled by plant 1 & plant 3. It is not necessary that there will be a demand for the liquid products everyday of the horizon, as can be seen in Figure 11a, that in day 4 & 5 there is no demand for argon in this scenario. Production, demand & consumption profile for all plants are illus42


Page 43 of 62

trated in Figure 11b. Though not very evident from Figure 11b liquid argon is transferred to plant 2 from plant 3 to replenish liquid argon inventory at plant 2 during ’OFF’ period. The products of the air separation plants can not be independently produced. So, if there LAr Inventory Buildup LAr Demand

I1

1

LAr Demand

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


I2

I3

LAr Production LAr Supplied to other INV

LAr Consumption for Driox LAr Received from other INV

1.3

0.8

0.8

0.6

0.3

0.4

-0.2

I1

0.2

I2

I3

-0.7

0 1

2

3

4 Day

5

6

7

-1.2

(a) Total Liquid Argon Demand & fulfilment

(b) Liquid Argon Mass Balance For All Plant

Figure 11: Liquid Argon Mass Balance & Demand Fulfilment is a high demand of gaseous or liquid oxygen, argon & nitrogen production will also be high irrespective of the demand they have. By analyzing the results it has been noticed that at starting of the 5th day liquid argon inventory was high in Plant 3 & there is no liquid argon demand on that day. Furthermore, there was high gaseous oxygen demand on 5th day. Therefore, the production of liquid argon is also high along with high oxygen production. From the inventory balance it has been observed that inventory for liquid argon at plant 3 would have been crossed its maximum limit at the end of the 5th day, and if these plants are optimized separately this excess liquid would have to be dumped. But in an enclave setting this excess liquid are transferred to other plants which helps to take an economic shutdown in that plant (in this case plant 2), in turn reducing the overall production cost. The advantage of product exchange in attaining overall minimum production cost will be more apparent in the following scenario.

43



Page 44 of 62

Scenario 2 : High power price at plant 1 Enclave optimization framework can reap benefits from different area wise power pricing and contracts by utilizing the product exchange network, and scenario 2 has been designed to support this claim. Power sources available at plant 1 have higher power price compared to other two plants in the enclave. Gaseous and liquid product demands, plants specifications are exactly similar to the scenario 1, only the power prices available to plant 1 are increased. The resultant optimum schedule for plant 1 & plant 2 are shown in Figure 12. Comparing the schedules shown in Figure 12a and Figure 12b, with the optimum schedules found in scenario 1(Figure 7, Figure 8), we can find that production in plant 1 is ceased for several hours, whereas, plant 2 has not opted for any economical shutdown in scenario 2. ON

Shutdown

Startup

Off

NOARON

ASU

ON

Shutdown

Startup

Off

NOARON

ASU

LiqGN2

LiqGN2

LMComp

LMComp

MHComp

MHComp

LHComp

LHComp

1

21

41

61

81 Time(hr.)

101

121

141

1

161

21

(a) Schedule for Plant 1

41

61

81 Time(hr.) 101

121

141

161

(b) Schedule for Plant 2

Figure 12: Optimum Schedule for Plant 1 & Plant 2 in Scenario 2 Since, the power price in plant 1 is much higher compared to plant 2 and 3, production at plant 2 & 3 is increased and liquid product is sent to plant 1 to meet the onsite demand and inventory target. High quantity of liquid oxygen is sent from plant 3 to plant 1, which will be evident from the LO2 inventory profile of plant 1 & 3 given in Figure 13. It can be seen that the product transit time between plant 3 & plant 1 is also taken care of: although the product was taken out from LO2 inventory of plant 3 on day 4 and day 6, it is added in the LO2 inventory situated at plant 1 location on day 5 and day 7 respectively, as shown in the 44


Page 45 of 62

Figure 13. The optimum schedule for plant 3 for this scenario is also shown in Figure 14. LO2 Produced LO2 Demand LO2 Inventory

LO2 Produced LO2 Demand LO2 Inventory

LO2 Driox LO2 Received From other Plants in Enclave

1.1

1.1

1.1

LO2 Driox LO2 Supplied to Other Plants in Enclave 1.1

1

1

0.6

0.6 0.9

0

1

2

3

Day

4

5

6

7

0.7

-0.4

LO2 Inventory

0.8

0.1

0.6

0.8

0.1 0

1

2

3

Day

4

5

6

-0.4

7

0.7

LO2 Inventory

LO2 Production/Consumption

0.9 LO2 Production/Consumption

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6

0.5

0.5

-0.9

-0.9

0.4

-1.4

0.4

0.3

-1.4

0.3

(a) LO2 Inventory Profile for Plant 1

(b) LO2 Inventory Profile for Plant 3

Figure 13: LO2 Inventory Profile for Plant 1 & Plant 3 in Scenario 2

ON

Shutdown

Startup

Off

NOARON

ASU

LiqGN2

LMComp

MHComp

LHComp

1

21

41

61

81 Time(hr.)

101

121

141

161

Figure 14: Schedule for Plant 3 in Scenario 2

Scenario 3 : Onsite contract on liquid product decantation from LN2 Shared inventory The results illustrated above in scenario 1 & 2 are for the cases when none of the plants has any onsite contract obligations. We now again simulate the case study presented in scenario 1 with the addition of various combinations of onsite contracts. The effect of the onsite contracts on the optimum schedule can be interpreted from the results shown below when compared with the results found in scenario 1. 45



1.2

1.2

LIN Inventory at Plant 1

LIN Inventory at Plant 1

LIN Shared Inventory of Plant 2 & 3

LIN Shared Inventory of Plant 2 & 3 1

1

0.8

0.8

LN2 Demand

LN2 Demand

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 62

0.6

0.6

0.4

0.4

0.2

0.2

0

0 1

2

3

4

Day

5

6

7

1

2

3

4 Day

5

6

7

(a) LN2 Demand & Fulfilment for Scenario 1 (Without(b) LN2 Demand & Fulfilment(Daily Contract on On-site Contracts) Shared inventory)

Figure 15: Effect of Onsite Contracts on LN2 Demand Fulfillment As shown in Figure 15a, according to the scenario 1, Shared LN2 inventory between plant 2 and 3 are major contributors in fulfilling the LN2 demand. To monitor the effect of onsite contracts on the overall optimality, a daily ‘Fixed Cost, No Limit’ contract is posed on the shared inventory. It is to be noted that in this contract though there is no limit on decantation of the liquid product from the inventory, however, the price for decantation of the liquid products from that inventory may change daily. To minimize the payable amount due to the contract obligation, most of the LN2 demands are fulfilled from LN2 inventory of plant 1, which is evident from the results illustrated in Figure 15b. To increase the LN2 supply from plant 1, the liquefier(unit name ‘LiqGN2’) in the plant 1 is in on mode throughout the horizon(Figure 16a) as opposed to the case demonstrated in scenario 1 (Figure 7a). The optimum schedule for plant 2 & Plant 3 are also shown in Figure 17 & Figure 16b. Compared with the optimal schedules found in scenario 1(Figure 16), we can see that while, in scenario 1, the liquefier in plant 3 is in on mode throughout the horizon, in scenario 3, the liquefier is kept on for a very brief amount of time. As, the cost of decantation of liquid nitrogen is higher due to the contractual agreements considered scenario 3, the requirement of operating the liquefier decreases. As, the LN2 inventory of plant 2 & plant 3 is shared, the liquefier at plant 2 also kept in off mode, which can be seen in Figure 17. Another interesting observation, that can be made while comparing optimal schedule of plant 2(Figure 17) found 46


Page 47 of 62 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ON

Shutdown

Startup

Off

NOARON

ASU

ASU

LiqGN2

LiqGN2

LMComp

LMComp

MHComp

MHComp

LHComp

LHComp

1

21

41

61

81 Time(hr.)

101

121

141

1

161

21

(a) Schedule for Plant 1 in Scenario 3

41

ON

Shutdown

Startup

Off

81 Time(hr.)

61

NOARON

101

121

(b) Schedule for Plant 3 in Scenario 3

Figure 16: Optimum Schedule for Plant 1 & Plant 3 in Scenario 3

ON

Shutdown

Startup

Off

NOARON

ASU

LiqGN2

LMComp

MHComp

LHComp

1

21

41

141

61

81 Time(hr.) 101

121

141

161

Figure 17: Schedule for Plant 2 in Scenario 3

47


161


Page 48 of 62

in scenario 3 with optimal schedule of the same plant found in scenario 1(Figure 8), is that the use of gaseous compressors in plant 2(used to make MPGN2 & HPGN2 from GN2) decreases significantly compared to case described in scenario 1. Plant 2 & plant 3 shares the same on-site customer. As, the burden of supplying liquid nitrogen decreased on the shared inventory due to on-site contracts, instead of liquefying, plant 3 is contributing a larger share to the MPGN2 and HPGN2 demand of the shared on-site customer, which in turn reduces the burden on plant 2. That is why, the product compressors in plant 2 are less utilized in scenario 3 compared to scenario 1.

Scenario 4 : ASU Maintenance at Plant 1 during the Scheduling Horizon In this scenario, ASU in plant 1 undergoes maintenance from 24 to 48 h. The rest of the parameters such as, product demand, inventory target, power price etc. are exactly similar to the case study considered in scenario 1. The resultant optimum schedule for plant 1 is shown in Figure 18. ON

Shutdown

Startup

Off

NOARON

ASU

LiqGN2

LMComp

MHComp

LHComp

1

21

41

61

81 Time(hr.)

101

121

141

161

Figure 18: Schedule for Plant 1 in Scenario 4 To meet the gaseous customer demand at the time of maintenance, a lot of liquid product needed to be drioxed. Plant 1 also has low initial inventory, so if optimized separately plant 1 may not be able to go through the maintenance process without disrupting the supply 48


Page 49 of 62



LO2 Driox LO2 Received From other Plants in Enclave

1.1

1.1

1.1

LO2 Driox LO2 Supplied to Other Plants in Enclave 1.1

1

0.6

1

0.6

0.9

1

2

3

4

Day

5

6

7

0.6

-0.4

0.5

0.8

0.1 0

1

2

3

4

Day

5

6

7

0.7

-0.4

LO2 Inventory

0.7 0

LO2 Inventory


0.1


0.9 0.8

0.6

0.4

0.5 -0.9

-0.9

0.3

0.4

0.2

-1.4

-1.4

0.1

0.3

(a) LO2 Inventory Profile at Plant 1

(b) LO2 Inventory Profile at Plant 3

Figure 19: LO2 Inventory Profile at Plant 1 & 3 in Scenario 4 chain. However, as evident from Figure 19, in an enclave setting by using the product exchange network an optimum amount of liquid oxygen was shipped from plant 3 to plant 1. By drioxing this product, plant 1 is able to cater to the associated onsite customer during maintenance period. Some LO2 is again shipped at day 6 from plant 3 to plant 1 to meet the inventory target at the end of the scheduling horizon. 1.2

1.2

I1

I2

I3

I1

1

1

0.8

0.8 LO2 Demand

LO2 Demand

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6

I3

0.6

0.4

0.4

0.2

0.2

0

I2

0

1

2

3

4

Day

5

6

7

1

(a) LO2 Demand & fulfillment in Scenario 1

2

3

4

Day

5

6

7

(b) LO2 Demand & fulfillment in Scenario 4

Figure 20: LO2 Demand & fulfillment in Scenario 1 & 4 The efficacy of enclave setting in maintaining an undisrupted supply chain during plant maintenance/ unit breakdown can be discerned more clearly from the Figure 20. The contribution of plant 2 in mitigating the global LO2 demand in scenario 1 is very little, however, as plant 1 goes into maintenance in scenario 4, plant 2 contributes significantly in quenching 49



Page 50 of 62

the liquid demands. The optimum schedule of plant 2 & plant 3 for scenario 4 is presented ON

Shutdown

Startup

Off

NOARON

ON

Shutdown

Startup

Off

NOARON

ASU

ASU

LiqGN2

LiqGN2

LMComp

LMComp

MHComp

MHComp

LHComp

LHComp

1

21

41

61

81 Time(hr.) 101

121

141

161

1

(a) Schedule for Plant 2 in Scenario 4

21

41

61

81 Time(hr.)

101

121

141

161

(b) Schedule for Plant 3 in Scenario 4

Figure 21: Optimum Schedule for Plant 2 & Plant 3 in Scenario 4 in Figure 21a & Figure 21a respectively. Comparing with the optimum schedule of plant 2 found in the case of scenario 1(Figure 8), Figure 21a shows that plant 2 does not go through any economical shutdown in case of scenario 4 and helps in maintaining a undisrupted supply chain.

Conclusion In this paper, a novel multi plant scheduling approach, named enclave optimization(ELO) is introduced which can aptly address several complexities that arise in a multi-plant production scheme. The peculiarities addressed in the proposed model such as, shared inventory, global liquid demands, multiple plant catering to a single customer, is very apparent in real world production scenarios. The above-mentioned model is very rigorous and able to handle several real world constraints and also accommodate inter plant liquid exchange. From the scenarios shown above, it can be discerned that the inter plant liquid exchange is very beneficial in achieving global minimum production cost by minimizing product loss and efficiently utilizing the excess products which in tern minimizes overall power cost. The main bottleneck in conventional multi plant production optimization framework is the combinatorial nature 50


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of the problem, i.e. with increasing number of plants and complexities, the model size (number of constraints and variables) increases, and the model becomes computationally expensive. The framework presented in this paper pose clear guidelines on its scope and provides proper bounds on maximum number of plants or the aspects that can be addressed using enclave optimization. Results shown above also proves that the adopted non-uniform time discretization framework is very beneficial in reducing the computational complexity. However, one has to decide the smallest time bucket carefully, so that all the nuances of the actual process gets addressed.

Acknowledgement The authors thank IIT Bombay and Praxair India Pvt. Ltd. for the financial support provided to carry out this research.

51



Page 52 of 62

Nomenclature Indices m∈M

Plant

u∈U

Unit

i∈I

Inventory

s∈U

Supply Source

c∈C

On-site Customer

o∈O

Operation Mode

l∈L

Operation Point/Slate

p∈P

State/Product

e∈E

Electricity/Power Source

t

Time Period

d

Day

y

Liquid Demand Type

1..2

1 indicate minimum and 2 indicate maximum

Parameters M OneP owerF lagm

If equal to 1 then there is additional condition to use only one source of power at a time period.

M N onOf f F lagm

Equal to 1 if plant m can not go through a shutdown process in the scheduling horizon.

M Sm,s

Equal to 1 if supply source s is associated with plant m

M Um,u

Equal to 1 if unit u is operational in plant m

M U Drioxm,u

Indicates the driox units in plant m.

M U V entm,u

Indicates the vent units in plant m.

M U Om,u,o

Equal to 1 if operation mode o belongs to the unit u in plant m. 52


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Captures all possible mode transitions for a specific unit

M U OOm,u,o,o0

in a plant m. Captures minimum and maximum time slots needed for

M U OOT ransitionSlotm,u,o,o0

a transition from operation mode o to mode o0 at a unit u present in plant m M U Of fm,u

Captures the operation mode ‘off’.

M U OOnm,u,o

Captures ‘on’ and ‘NoArOn’ operation mode.

M U OOOm,u,o,o0 ,o00

Captures predefined sequences of mode transitions for an unit u of plant m.

M U OOOF ixedT ransitionSlotm,u,o,o0 ,o00

captures fixed number time periods a unit u should stay in operation mode o to perform predefined change in operation modes (o,o0 ,o00 ) for unit u in plant m. Captures the initial unit, Mode combination for every

M U OT Initialm,u,o,t

plant. Captures the mode transitions for every unit u at each

M U OOT Initialm,u,o,o0 ,t

plant m in historical horizon. Equal to 1 if unit u of plant m is in maintenance at time

M U T M aintenanceF lagm,u,t

period t. Captures available slates for an unit u in plant m oper-

M U OLm,u,o,l

ating in mode o. M U OLP owerConsumptionm,u,o,l

Captures power consumption for a slate l at operation mode o in unit u of plant m.

M U OLP StateChangeRatem,u,o,l,p

Captures state change rate when unit u of plant m operating in a slate l of operation mode o.

M U Pm,u,p

Equal to 1 if product p gets produced/consumed in unit u in plant m.

53



M M P T T ransitLossm,m,p,t

Page 54 of 62

Percentage loss of gaseous products in transfer from one plant to other plant.

M P F inalP roductF lagm,p

Indicates the final products in each plant m.

M ELimitP enaltym,e

Captures minimum limit of power purchase under which penalty cost will be charged if the power is purchased from MTOP source for plant m.

M EP enaltyCostm,e

Captures cost for violating penalty limit specified by MTOP source at plant m.

M ET P owerP ricem,e,t

Captures price of per unit power purchased from power source e at time period t at plant m.

M ET Limitm,e,t

Captures minimum and maximum power availability limit of power source e associated with plant m at time period t.

M T U tilityP owerm,t

Power needed to run office utilities at plant location m.

IMi,m

Indicates inventory i is associated with plant m.

IPi,p

Equal to 1 if inventory i is suitable to store product p.

IP InvCapacityi,p

Captures capacity of inventory i which stores product p

IP M inInvContracti,p

Captures minimum level contract for a product p at inventory i.

IP InitialInvi,p

Initial level for product p at inventory i.

IP T argetV iolationF lagi,p

Equal to 1 if target violation is allowed for product p at inventory i.

IP T argetInvP enaltyi,p

Captures penalty cost for the violation of target at inventory i.

IP QLimitT ypei,p,q

Captures the type of the onsite contracts.

54


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IP SlabCounti,p

Indicates the number of price slabs mentioned for liquid product decantation at inventory i in the contracts.

IP QSlabM inLimiti,p,q

Minimum limits for a certain liquid product decantation in an inventory i.

IP QSlabM axLimiti,p,q

Maximum limits for a certain liquid product decantation in an inventory i.

IP QSlabChargei,p,q

Liquid product(p) decantation charge at an inventory i per slab as mentioned in the onsite contract.

P V entP enaltyp

Captures penalty cost for venting gaseous products.

P DumpP enaltyp

Captures liquid product dump penalty.

P DrioxCostp

Captures cost for drioxing liquid products into gaseous product.

CP T OnsiteDemandc,p,t

Captures the demand for onsite customers for a particular gaseous product manufactured in a manufacturing plant in a time slot.

CM P LimitQuantityc,m,p

Minimum quantity limit of customer c for product p that need to be fulfilled by a particular plant m.

SP Counts,p

Equal to 1 if product p can be purchased from supply source s

SP QLimitQuantitys,p,q

Captures minimum and maximum limit of product purchase from other sources.

SP QLimitT imes,p,q

Captures start Time and end time limit of product purchase from other sources.

SP QLimitP rices,p,q

Captures purchase price of product p from supply sources.

P T ypep

Captures Product Type. ‘G’ means gas and ‘L’ means liquid. 55



P EquivalentP roductp

Page 56 of 62

Same product sold as two different products due different specifications.

P Y Countp,y

Captures relationship between product and demand type.

P Y QDemandQuantityp,y,q

Captures Liquid Product Demand Quantity.

P Y QDemandP ricep,y,q

Captures Liquid Product Selling Price for spot sale.

P Y QDemandT imep,y,q

Captures Liquid Demand Start and End Time periods.

P T InvT argetp,t

Captures Inventory Target for liquid products at time period t.

NT

Indicates the end time period of the scheduling horizon.

NP T

Largest minimum uptime of all units.

ND

Number of Day in scheduling horizon.

T T imet

Captures the Start and End Time of a time slot.

DT imed

Captures the Start and End Time of a day.

T Dayt

Indicates the time slot belongs to which day.

Binary Variables Assumes value 1 if the operation mode o is selected in

vbM U OTm,u,o,t

unit u at plant m at time period t. Assumes value 1 if the transition for operation mode o

vbM U OOTm,u,o,o0 ,t

to o0 is happens in unit u at plant m at time period t. Assumes value 1 if spot sale for a liquid product p is

vbP Y QSpotSalep,y,q

done. Assumes value 1 if power source e is selected for plant

vbM ETm,e,t

m at time period t. Binary variable to select the positive part of the free

vbIP F reeLiqP ositivei,p

liquid 56


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Binary variable to select the negative part of the free

vbIP F reeLiqN egetivei,p

liquid Assumes the value 1 if the decantation slab q is selected

vbIP DQSlabSelecti,p,d,q

for liquid product p at inventory i in the day d. Continuous Variables vcM U OLT SlateCoef fm,u,o,l,t

Slate coefficient


Amount of product p produced/consumed by a unit u at time period t at plant m. Contribution from plant m in quenching demand of on-

vcM CP T OnsiteDemandF illm,c,p,t

site customer c for gaseous product p at time slot t. Quantity of gaseous product p supplied from plant m

vcM M P T Quantitym,m0 ,p,t

to plant m0 at time slot t. Quantity of Liquid product p stored in inventory i at

vcIP DInvi,p,d

the end of the day d. Quantity of liquid product p purchased from other


sources s to meet the inventory target for the day d. Quantity of Liquid product p dumped at day d from

vcIP DDumpi,p,d

inventory i. The amount of liquid product i supplied from inventory

vcIP DDemandF illi,p,d

i to meet regular demand at day d. The amount of liquid product i supplied from inventory

vcIP DSpotDemandF illi,p,d

i to meet spot demand at day d. The amount of liquid product p transferred from inven-

vcIIP DT ransactInvi,i0 ,p,d

tory i to inventory i0 on day d. Quantity of liquid product purchased from supply


source s to meet regular demand at day d. 57



Page 58 of 62

Quantity of liquid product purchased from supply


source s to meet spot demand at day d. pvcIP DInvi,p,d,1

Indicates negative violation of the inventory targets.

pvcIP DInvi,p,d,2

Indicates positive violation of the inventory targets.

vcM U T P owerRequiredm,u,t

Power required to run unit u in plant m at time period t. Power purchased from power source e for plant m at


time period t. Cost of power purchase from a power source e for plant

vcM ECostm,e

m over scheduling horizon. Penalty variable for minimum take or pay limit violation

pvcM ELimitpenaltym,e

at plant m. Total amount of liquid product drioxed to meet gas de-

vcIP T otalDrioxDemandi,p

mand over the scheduling horizon from inventory i. Total amount of free liquid in inventory i over the

vcIP T otalF reeLiquidi,p

scheduling horizon. vcIP F reeLiquidP ositiveQuani,p

Captures the positive part of free liquid in inventory i.

vcIP F reeLiquidN egativeQuani,p

Captures the negative part of the free liquid in inventory i. Captures the billable quantity of total liquid product

vcIP BillableDrioxDemandi,p

drioxed in inventory i. part of the total liquid product p that is been decanted

vcIP DQSlabQuantityi,p,d,q

from inventory i at day d that falls in the slab q. vcIP DQSlabConsumedi,p,d,q

Quantity of liquid product decanted from selected slab.

vcIP DQSlabM axi,p,d,q

Assumes value if the maximum limit decantation of the selected slab is reached.

58


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References (1) Behnamian, J.; Fatemi Ghomi, S. M. T. A survey of multi-factory scheduling. Journal of Intelligent Manufacturing 2016, 27, 231–249. (2) Alvarez, E. Multi-plant production scheduling in SMEs. Robotics and ComputerIntegrated Manufacturing 2007, 23, 608 – 613, 16th International Conference on Flexible Automation and Intelligent Manufacturing. (3) Johansen, K.; Comstock, M.; Winroth, M. Coordination in collaborative manufacturing mega-networks: A case study. Journal of Engineering and Technology Management 2005, 22, 226 – 244. (4) Sauer, J.; Suelmann, G.; Appelrath, H.-J. Multi-site scheduling with fuzzy concepts. International Journal of Approximate Reasoning 1998, 19, 145 – 160. (5) Wilkinson, S.; Cortier, A.; Shah, N.; Pantelides, C. Integrated production and distribution scheduling on a Europe-wide basis. Computers & Chemical Engineering 1996, 20, S1275 – S1280. (6) Energy Information Association. Manufacturing energy consumption Survay: Total

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electricity,

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http://www.eia.gov/consumption/

manufacturing/data/2010/pdf/Table11_1.pdf. (7) Li, T.; Roba, T.; Bastid, M. Production Scheduling of Air Separation Processes. 2012, FOCAPO-2012/CPC VIII, Savannah, Georgia, USA, 8-13 January 2012. (8) Mitra, S.; Grossmann, I. E.; Pinto, J. M.; Arora, N. Optimal production planning under time-sensitive electricity prices for continuous power-intensive processes. Computers & Chemical Engineering 2012, 38, 171–184. (9) Zhang, Q.; Grossmann, I. E.; Heuberger, C. F.; Sundaramoorthy, A.; Pinto, J. M.

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Air separation with cryogenic energy storage: Optimal scheduling considering electric energy and reserve markets. AIChE Journal 2015, 61, 1547–1558. (10) Zhang, Q.; Sundaramoorthy, A.; Grossmann, I. E.; Pinto, J. M. A discrete-time scheduling model for continuous power-intensive process networks with various power contracts. Computers & Chemical Engineering 2016, 84, 382 – 393. (11) Misra, S.; Kapadi, M.; Gudi, R. D.; Srihari, R. Energy-Efficient Production Scheduling of a Cryogenic Air Separation Plant. Industrial & Engineering Chemistry Research 2017, 56, 4399–4414. (12) Misra, S.; Kapadi, M.; Gudi, R. D.; Srihari, R. Production Scheduling of an Air Separation Plant. IFAC-PapersOnLine 2016, 49, 675 – 680, 11th {IFAC} Symposium on Dynamics and Control of Process Systems Including Biosystems DYCOPS-CAB 2016Trondheim, Norway, 6-8 June 2016. (13) Zhou, D.; Zhou, K.; Zhu, L.; Zhao, J.; Xu, Z.; Shao, Z.; Chen, X. Optimal scheduling of multiple sets of air separation units with frequent load-change operation. Separation and Purification Technology 2017, 172, 178 – 191. (14) Grossmann, I. Enterprise-wide optimization: A new frontier in process systems engineering. AIChE Journal 2005, 51, 1846–1857. (15) Smith, A.; Klosek, J. A review of air separation technologies and their integration with energy conversion processes. Fuel Processing Technology 2001, 70, 115 – 134. (16) Velez, S.; Maravelias, C. T. Multiple and nonuniform time grids in discrete-time MIP models for chemical production scheduling. Computers & Chemical Engineering 2013, 53, 70 – 85. (17) Vecchietti, A.; Lee, S.; Grossmann, I. E. Modeling of discrete/continuous optimiza-

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tion problems: characterization and formulation of disjunctions and their relaxations. Computers & Chemical Engineering 2003, 27, 433 – 448. (18) Mitra, S.; Sun, L.; Grossmann, I. E. Optimal scheduling of industrial combined heat and power plants under time-sensitive electricity prices. Energy 2013, 54, 194–211. (19) Getting Started with Xpress. (20) Guéret, C.; Prins, C.; Sevaux, M.; Heipcke, S. Applications of Optimization with XpressMP ; Dash Optimization Limited, 2002.

Graphical TOC Entry

Figure 22: Table of Content Graphic

61


Intricate Process Need For an Undisrupted Supply Chain

Enclave Optimization

Efficient Multi Plant Framework Rigorous Approach

1 2 3 Effective utilizations of Different Power Prices & Minimum Overall contracts Production Cost 4 ACS Paragon Plus Environment 5 Exploitation of Synergy among the plants 6Undisrupted Supply Chain 7 Product Exchange Network

Novelties

Complexities

Industrial & Engineering Chemistry Research Page 62 of 62

Shared Inventory, Common Customer, Aggregated Demand

Enclave Optimization: A Novel Multiplant Production Scheduling

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