Energy-Efficient Production Scheduling of a Cryogenic Air Separation

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Energy-efficient Production Scheduling of a Cryogenic Air Separation Plant Shamik Misra, Mangesh Kapadi, Ravindra D. Gudi, and Rachakonda Srihari Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b04585 • Publication Date (Web): 21 Mar 2017 Downloaded from http://pubs.acs.org on March 27, 2017

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Industrial & Engineering Chemistry Research

Energy-efficient Production Scheduling of a Cryogenic Air Separation Plant Shamik Misra,† Mangesh Kapadi,‡ Ravindra D. Gudi,∗,† and R. Srihari‡ Chemical Engineering Department,Indian Institute of Technology, Bombay, Mumbai, India, and Praxair India Private Limited, Bangalore, India E-mail: [email protected]

Abstract Power intensive processes such as, cryogenic air separation in which the major portion of the production cost is spent on energy/electricity, needs to adopt a smart operational approach to ensure maximum usage of resources while minimizing the power cost. In this paper a state task network based model of an air separation plant is designed to represent real world production constraints. A discrete time model based production scheduling has been proposed and validated on several scenarios that reflects representative real time constraints. The optimal schedule found for every scenario chosen has shown efficient exploitation of all the energy contracts and judicious utilization of the liquid products. Due to its granular and rigorous modelling approach along with computational efficiency, the proposed model manifests huge potential towards its implementation in a real world air separation plant. Keyword: Cryogenic Air Separation, Production Scheduling, State Task Network, Demand Side Management. ∗ To

whom correspondence should be addressed Institute of Technology, Bombay ‡ Praxair India Private Limited, Bangalore, India † Indian

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Introduction Agile and alert manufacturing practices, based on relatively rigorous production schedules, help to achieve sustained profitability in a manufacturing enterprise. The most prolific way to increase profitability is by decreasing the operation costs. In the case of energy intensive industries such as air separation and chlor-alkali industries, a major part of their operational expenses is due to the electricity costs. Also, according to US Energy Information Association, the American industrial gas industry consumed almost 19.4 TWh of electricity in the year 2010, which amounts to 2.5% of all the electricity consumption by the entire manufacturing industrial sector in America. 1 Energy intensive processes such as cryogenic air separation where the main contributor of the operating cost is electricity price, need to adopt Energy Demand Management also known as Demand Side Management (DSM) to keep them afloat. With the increasing volatility in the electricity prices and new emerging power contracts due to the increasing penetration of renewable sources, calculated and informed decisions need to be taken to reap all the benefits that these new contracts offer, while at the same time maintaining sustainable operation. Demand side Management helps the electricity consumers to reduce their electricity cost by changing their operational procedure in a co-ordinated manner. Interest of the researchers in the chemical industry as well as academic community towards the potential aspect of DSM, has been increasing in recent years. 2,3 DSM not only helps the consumers to keep their cost in check but also provides stability to the power grids. One of the major aspect of DSM is demand response which basically suggests the consumers to schedule their operation in such a way that they efficiently adjust power consumption during the peak hours. However, this load shifting has to be done carefully, so that it does not affect demand fulfilment and operational constraints. Hence, a proper scheduling respecting both operation constraints as well as demand fulfilment constraints is necessary. In recent years, a few attempts have been reported on production scheduling of an air separation plant. 4–9 Mitra et. al. 6 has considered an entire air separation plant (ASP) as a unit and introduced different operational modes in which those units can operate. The resultant surrogate model was 2 ACS Paragon Plus Environment

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very efficient in handling the mode transitions respecting the minimum uptime and downtime concepts to reduce stress on the equipment. However, an air separation plant is sufficiently complex in terms of interactions between various units in the plant, an understanding of which can help in exploiting opportunities for achieving better optimum. To realize this improvement, it may be necessary to have a relatively more rigorous model (rather than a gross model for entire plant as was done by Mitra et. al. 6 ) that considers every unit and its interactions with other units. Mitra et. al. 6 however only focused on time of use power pricing and did not consider various power contracts which could also offer better opportunities for cost minimization. Zhang et. al. 8 have explored the unit based approach and designed a process network of the air separation plant. In their paper they have captured the interaction between all the processes which made the model comparatively rigorous than the ‘One plant one unit’ based approach proposed by Mitra et. al. 6 However, the model proposed by Zhang et. al. 8 was targeted for a cluster of energy intensive industries and due to the generic representation, some specific and peculiar constraints of a typical air separation plant were not incorporated. Some of the specific requirements and constraints in an air separation plant, which need to be addressed to make the model adequate towards improved and implementable optimization, can be stated as, 1. Accommodating different start-up times (time required to reach steady state) needed for different products. For example, start-up time needed for production of liquid argon can be significantly higher compared to that required for liquid oxygen. 2. Efficient utilization of the liquid products is required to make the plant operation more flexible. A schedule that promotes either or all of, (a) Complete utilization of the excess liquid (if any), (b) judicious utilization of the ‘purchase from other plant’ option or (c) satisfying short term orders, will help to increase overall profitability. 3. Co-ordination and interaction between various units creates many operational constraints which needs to be accommodated. 3 ACS Paragon Plus Environment


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4. As the air separation plant is a continuous process, provision for maintenance activity needs to be included in the surrogate model. Contract obligations and business practices dictate the inventory targets which also need to be incorporated. 5. Better usage of dump (draining the excess liquid product if liquid inventory exceeds the maximum capacity of the storage tank) and vent (emitting the gaseous products if produced in excess than required) options are necessary to ensure minimum wastage of valuable products. While the above scenarios will be discussed in greater detail later in this paper, it must be mentioned here that these scenarios have not been accommodated in any generic formulation of DSM so far. A more rigorous formulation that incorporates the above-mentioned requirements and peculiarities is necessary for better exploitation of opportunities towards better agility for meeting demands as well as energy minimization. In this paper a state task network based modelling approach is presented in which all the units of plants are considered separately. The entire formulation has been made in such a way that it accounts for most of the real world constraints and limitations of an air separation plant. There are various power contracts available and effective exploitation of these contracts can help to reap all the benefits that DSM offers. However, given that the production is demand driven, this may compel high production rate even in the regime of high power prices. In such scenarios judicious utilization of liquid product along with efficient electricity utilization is necessary to make the production more cost effective. The rigorous STN based model proposed in this paper has been constructed to replicate these peculiar constraints of an air separation plant. This model also considers the following conditions, which will be explained in detail in the later part of this paper: 1) Liquid purchase from competitors and spot sales, 2) Flexible liquid demand, 3) Optimum maintenance period and 4) Economical shutdown. It is demonstrated that the schedule based on the model addressing these aspects helps in taking more realistic decisions when implemented on a real air separation plant. The paper has been organized as follows: In section 2 a cryogenic air separation plant will be 4 ACS Paragon Plus Environment

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described and an overview of constraints and limitations will be discussed. Section 3 would be dedicated to detail discussions on the main aspects of the model and the optimization formulation. Scheduling result based on different scenarios will be presented in section 4. Finally section 5 will comprise of conclusions drawn by summing up key aspects.

Air separation Process: Some Complexities & Optimization Requirements Cryogenic air separation is the most efficient and cost effective method for producing high purity industrial gases. A cryogenic air separation plant produces gaseous and liquid products by separating the components of air using cryogenic multicolumn distillation technique to achieve high purities and recoveries. 10 The major product is gaseous oxygen (GO2) which has a wide market ranging from steel plants to gasification of hydrocarbon feed stocks. Compressed Gaseous Nitrogen (GN2) is another essential by-product. Gaseous products are sent directly to the customers through pipelines. Liquid products such as, Liquid Oxygen (LO2), Liquid Nitrogen (LN2), Liquid Argon (LAr) are stocked up in an inventory and then sent to the merchant customers using delivery trucks. It is to be noted that all the above-mentioned liquid and gaseous products are produced simultaneously. So, if there is high demand for gaseous oxygen then gaseous nitrogen & other liquid products will also get produced in a high quantity whether or not those products has high demand. Gaseous Argon (GAr) is not a direct product from air separation unit and it is prepared by evaporating the LAr. The operation of the air separation plant is dictated by the demands of gaseous and liquid products and constrained by plant capacity and operation modes. Furthermore, availability or cost of power (which may changes with location and time of day) also impacts on the production of gaseous and liquid products. Air separation process contains the following types of units: 1) Air separation unit (ASU), 2) liquefier (converts GN2 to LN2), 3) compressors, 4) vaporization units (the process of vaporizing the liquid products into gaseous products is called as ‘drioxing’ in air separation industry. 7 Units 5 ACS Paragon Plus Environment


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dedicated to this process will be referred as driox units in the later course of this paper.) and 5) vent units(these are basically valves which vents out gaseous products if produced in excess than required). As discussed earlier the main products from the air separation plant are GO2 and LO2. Huge amount of GN2 is also formed as by-product. GN2 is further compressed to medium pressure GN2 (MPGN2) & high pressure GN2 (HPGN2) to supply to the customers. The liquefier is used to convert GN2 to LN2 to meet liquid nitrogen demand. ASU and liquefier are the main contributor towards power usage of the whole plant. Compressors also consume a lot of power but the quantity is relatively meager when compared with ASU and liquefier contributions.

Optimization Requirements As the raw material air is free, the major contributor in the operation cost is the cost of electricity. Various kinds of power contracts are available and exploiting these contracts are necessary to lower the production cost. Hence the aim of the production scheduling will be to minimize the production cost and produce an implementable schedule describing the production modes of each of the units at each time periods respecting all the demand and inventory constraints. In the formulation proposed in the sequel, the time horizon has been discretized on an hourly basis as the forecasted electricity prices and onsite demands are specified for every hour. A horizon of one week has been considered in this formulation. It must be noted that the duration of time unit can be changed based on the complexity of the resultant model. The proposed model has been implemented on representative real world scenarios encountered in a typical ASP. For each of the scenarios, as identified by Praxair India Pvt. Ltd., the expected outcome of the production scheduling tool will be: 1. Operation modes of each units at each time period of the production horizon. 2. Selection of the power source considering hourly power price and contracts. 3. Gaseous and liquid demand fulfilment. 4. Meeting inventory targets and respecting inventory limitations. 6 ACS Paragon Plus Environment

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5. Proper utilization of the liquid products and load shifting based on the time of use power pricing.

Novel approaches in modelling peculiar constraints of a typical ASP The following constraints are specific to the air separation process and incorporation of these constraints in the model is necessary to generate an implementable schedule.

Non-Argon on Mode: In a typical ASU, the transition time needed to start deliverable liquid argon production is significantly higher than the start-up time needed for LO2 and LN2. This is because, after 2 hours of start-up LO2 and LN2 reach their required purity, whereas, it takes almost 20 hours to reach specified purity for LAr. To model this phenomena, an additional transition mode in ASU called as ’NoArOn’ is created, which is a very efficient way of handling this important production constraints. When ASU is in ’NoArOn’ mode it will produce LO2 and LN2 as required but there will be no LAr production. The need of introducing this additional transition mode is necessary considering the significant difference between the required time instants to achieve prescribed purity for liquid oxygen & argon. The impact of this phenomena on the overall schedule is extremely crucial which was not considered at all in the earlier generic formulations found in the literatures. Scenario 2 in the result & discussion section is specifically designed to showcase the impact of inclusion of this novel approach in algorithm in making the resultant schedule implementable.

Utilization of the liquid products: • Spot Sale & Purchase : Consumers of an ASP are often compelled to decrease their inventory costs due to uncertain market conditions, which increases their proclivity to fulfill their short term demands through spot sales. Liquid spot sales earn higher revenue than the regular orders, and are therefore lucrative to the ASP. The opportunity sales cannot be treated as regular demands, because regular demands are mandatory whereas spot sales are 7 ACS Paragon Plus Environment


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optional. The handling of spot sales in industry compared to regular demands are also very different. The spot sales can directly be fulfilled using the excess liquid in inventory. To ensure maximum fulfillment of spot demands we may need to increase liquid production if the extra liquid product in the inventory is not sufficient. As spot sales are not mandatory to fulfill, the plant operator may not always be willing to suddenly ramp up the production to accommodate it. But if profitable, the extra amount of liquid product that required to satisfy the spot demand can also be purchased from other sources (like other nearby ASPs, in which the purchase cost is basically the cost of distribution of the product from that source to the plant in question) and can directly be sent to the customer without adding it in the inventory. Liquid product purchased to fulfill demands are directly sent to the customers without including it in inventory. This will help in minimizing significant product losses which will be incurred in multiple time filling and decantation process. A new formulation (see Eq. (20)Eq. (27)) which addresses all the above-mentioned aspects is presented in the next section. • Driox & Free Liquid : Having a sufficiently high liquid inventory enables the fulfilment of vapour demands through re-vaporization. The process of vaporizing liquid inventory to meet gaseous demand is termed as ‘drioxing’ in air separation industry. 7,8 While fulfilment in this manner can help minimize energy cost through plant shut down, drioxing liquid inventory decreases the chance of spot sales. Though actual drioxing process does not involve significant costs, a penalty cost needs to be added to guide the optimizer to use drioxing option systematically. In the earlier literature driox cost is used as a tuning parameter, but no guidelines are provided on setting/tuning of this parameter. For realization of better optimum, the driox cost needs to reflect on a few additional factors such as, 1) the selling price of the liquid products, 2) the production cost for gaseous as well as liquid products etc. The most important among these is tracking the value of liquid product and its appropriate utilization in determining the optimal mode of operation. As the production and power consumption calculation in the following algorithm are based on pre-calculated operation slates (described in detail in the following section), driox cost per m3 of a liquid product should also not be 8 ACS Paragon Plus Environment

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less than the power cost required to produce 1 m3 of equivalent gaseous product. Otherwise, the optimizer will generate an incorrect, sub-optimal & undesirable action of shutting down the production. So appropriate tuning of driox cost is necessary. Now, in an air separation plant, production of different products are not independent of each other. If the gaseous demand is high then to fulfill that demand (given that such a fulfilment of the gaseous demand always has a higher priority), liquid products will also be generated simultaneously irrespective of whether we have demand for that much liquid product or not. This excess liquid produced (which is more than that required to fulfill liquid demands and meet inventory build-up) will be termed as ‘free liquid’ (the term ‘free liquid’ is coined as there are no need of the production of these liquid products but they are anyway getting generated as co-products). Complete utilization of this ‘free liquid’ is necessary to attain minimum production cost. According to earlier literatures the entire liquid in the inventory beyond target can be used for drioxing. But, this will be not acceptable in industry as a major part of the liquid in the inventory may be bought from other sources in the earlier schedule (previous than the current scheduling horizon and added as initial inventory in the current horizon). Drioxing that liquid product will in a way cost more than the traditional production process. In this paper, a novel formulation is proposed (see Eq. (38)-Eq. (47)) which upon systematic consideration of the above factors, provides an accurate estimate of the quantity of liquid that can be considered as free for drioxing.

Maintenance Activity: As cryogenic air separation process is a continuous operation, one or several units may go under maintenance during the scheduling horizon. This maintenance can be pre-planned, in which case the time periods and unit details are specified beforehand. This constraint is taken care of in equation Eq. (4). This constraint will be effective when the operator knows at which time the shutdown is scheduled. It is also possible to shut down the whole production for a certain time period and fulfill liquid and gaseous (drioxing the liquid products) demands from the inventory.



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There is an option, termed as, ‘Economical Shutdown’ in which the optimizer will decide the feasible time to go for a maintenance activity by using the liquid product in the inventory efficiently. As mentioned in the earlier section,using inventory liquid for drioxing is not desirable as in turn it can increase the production cost. By the concept of economical shutdown, the operator was given a choice to take tactical decisions on whether to use the liquid in inventory for drioxing or not. If the Operator sets the parameter MEconomicShutdownFlagm as 0 (that means operator does not want to use the liquid in the inventory for drioxing) then only the excess liquid from production (after meeting demands and inventory build-up, if required) will be used for drioxing. If Operator sets the MEconomicShutdownFlagm is 1 then the extra liquid in the inventory (initial inventory - target inventory, provided initial inventory > target inventory) will be added with the excess liquid from production and will be used for drioxing. This option will increase the plant shutdown period. The decisions associated with the above options need to be represented on in the formulation.

Hybrid time discretization: Though in the formulation presented in the next section the smallest unit for scheduling time discretization is considered as, 1 hour, it is not hard coded and can be changed based on the complexity of the problem. Further more, we can use different time scales for gaseous & liquid production. Adoption of different time scales for liquid and gaseous products are logical as, gaseous demands are given as per hour basis where liquid demands are given on a day basis and can be fulfilled at any time of the day. Using day wise discretization to calculate all the equations related to liquid inventory (e.g. inventory balance, liquid demand (regular & spot) fulfilment, liquid product purchase) will help in reducing a significant number of variables & constraints and expected to be all the more beneficial in reducing computational complexities when the formulation will be extended to accommodate a multi plant scenario. The comparison between hybrid time discretization & the uniform hour wise discretization method in terms of the number of variables & constraints involved, is presented in result & discussion section (see Table 1). Incorporation of the above-mentioned aspects makes the resultant model more rigorous and realis-


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tic to replicate a real world air separation process.

Model Formulation The traditional way to model a process is to solve the dynamic equations, considering the heat and mass balance of each and every task. This mechanistic modelling approach is the most accurate representation of the process. However, the resultant model generally turns out to as very complicated and computationally expensive. The other way to approximate the system is in product space, 6 in terms of a feasible region, which is an envelope made of several feasible operating points. The actual material consumption or production can be represented, by a linear combination of these pre-calculated operating points. The feasible region needs to be defined for each mode in which a unit can operate. Also a connection between material flow and the power consumption needs to be defined. The description of each operating point for each unit at every mode consists of material flow (production/consumption) information along with power requirement data and has been referred to as slates in the rest of the paper. These slates can be calculated by solving the process dynamic equations offline or using the past production data of the plant. The calculation of these slates are aptly described by Mitra et. al.(2012) and Zhang et. al.(2016). 6–8 The state task network(STN) framework has been a very efficient and unambiguous way to represent a process network. 11 STN representation consists of two types of nodes: 1) state nodes are denoted by circles and they represent the feeds, intermediates and the final products whereas, 2) task nodes represent the processes which transforms the materials and are represented by the rectangular nodes. The process network presented here can be described as STN OTOE (one task one equipment) and is illustrated in Figure 1. Slight modifications (Some circles are added inside of the rectangles) of the regular STN have been made to incorporate the operation modes in which the units can operate. The STN depicted in Figure 1 shows that the feed air is separated in ASU by cryogenic distillation method. The produced GO2 can be directly sold to the onsite customers whereas, LO2, LN2, LAr



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VMPGN2

VGN2

VGO2

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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 ON

OFF ON

OFF

DrioxO2

LAr VGAr

GAr

DrioxAr

ON

OFF

VentHPGN2 VHPGN2

Figure 1: STN Representation of a Cryogenic ASP with Operation Modes are sent to the inventory. GN2 is an intermediate product (denoted as blue circle in Figure 1) which is further compressed to produce MPGN2 and HPGN2. MPGN2 is produced through task LMCompGN2 and can be further compressed to HPGN2 through the task MHCompGN2. HPGN2 can also be directly produced from low pressure GN2 coming from ASU through LHCompGN2. If required, a part of GN2 produced in ASU can also be liquefied to LN2 in LiqGN2 and added to LN2 inventory. A portion of LAr inventory is evaporated through ‘DrioxAr’ process to produce GAr. The problem formulation with the operational constraints and the objective function for the air separation process are described in detail in the following. All the variables mentioned bellow are positive variable unless stated otherwise explicitly.

Allocation Constraints Each of the units mentioned above can be operated in their particular set of modes. The units and their corresponding modes are illustrated in Figure 1. At any time period t unit u performing task


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j can be operated only in one operation mode. So, J

O

∑

∑

vbUJOTu, j,o,t = 1 ∀u , ∀t(: t = 1..NT )

(1)

j=1(:UJu, j =1) o=1(:UJOu, j,o =1)

vbUJOTu, j,o,t is a binary variable and vbUJOTu, j,o,t assumes a value of 1 if unit u performing task j ∈ J(: UJu, j ) is operated in mode o ∈ O(: UJOu, j,o ) at time period t. In Eq. (1), UJu, j and UJOu, j,o are suitability matrices which ensure that the units only perform those tasks which they are assigned to and in those modes in which they can operate. Eq. (2) is a constraint to ensure that the slates selected at a time period should belong to the operation mode selected in that time period. As mentioned above, production or consumption of any product and corresponding power consumption at any time period is expressed as a linear combination of some predefined slates. If for a certain time period an operation mode for a unit performing a task is pre-specified, then, only that operation mode should get selected in that unit for those time periods. This requirement is represented in Eq. (3). The parameter UJOT Fixedu, j,o,t denotes those time periods, during which modes in which the units should operate are defined. The parameter UT MaintenanceFlagu,t indicates whether a unit is scheduled to go under maintenance or not. When UT MaintenanceFlagu,t is 1 for a unit then only OFF operation mode associated with the unit should be selected, and is enforced via Eq. (4). L

vcUJOLT SlateCoe f fu, j,o,l,t = vbUJOTu, j,o,t

∑ l=1(:UJOLu, j,o,l =1)

(2)

∀u, ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀t(: t = 1..NT )

vbUJOTu, j,o,t = 1 ∀u, ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀t(: t = 1..NT and UJOT Fixedu, j,o,t = 1)


(3)


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J

O

∑

∑

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vbUJOTu, j,o,t = 1

j=1(:UJu, j =1) o=1(:UJOu, j,o =1 and UJO f fu, j =o)

(4)

∀u, ∀t(: t = 1..NT and UT MaintenanceFlagu,t = 1)

Transition Constraints These constraints take care of the mode switching for a unit. When a unit is switched from one mode to another mode it has to stay in that mode for a certain time periods (called as minimum uptime) before the next changeover happens. The transition constraints are mainly adopted from Mitra et. al. 6,12 Similar constraints can also be found in recent works by Zhang et. al. 7,8 To formulate the transition constraints a proper understanding of the time horizon is necessary. A discrete time framework was designed ranging from −NPT to NT whereas, NPT denotes the largest minimum uptime of all the units. Past running data for the plant till −NPT is required to calculate the state of the units at the starting of the scheduling horizon. The time horizon is illustrated in Figure 2. The following constraints stated in Eq. (5) & Eq. (6) will take care of the operation mode tran-

Figure 2: Discrete Time representation of the entire horizon sitions. vbUJOOTu, j,o,o0 ,t is a binary variable which is 1 if the unit switches from mode o to o’ at time t.The parameter UJOOu, j,o0 ,o consists of all possible modes of unit u from which mode o can be reached while, UJOOu, j,o,o0 denotes the all possible forward mode transitions of unit u from


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mode o . O

O

vbUJOOTu, j,o0 ,o,t−1 −

∑

vbUJOOTu, j,o,o0 ,t−1

∑

o0 =1(:UJOOu, j,o0 ,o =1 and o6=o0 )

o0 =1(:UJOOu, j,o,o0 =1 and o6=o0 )

= vbUJOTu, j,o,t − vbUJOTu, j,o,t−1 ∀u , ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀t(: t = 2..NT ) (5) O

O

vbUJOOTu, j,o0 ,o,t−1 −

∑

o0 =1(:UJOOu, j,o0 ,o =1

and

o6=o0 )

o0 =1(:UJOO

vbUJOOTu, j,o,o0 ,t−1

∑ u, j,o,o0 =1

and

o6=o0 )

= vbUJOTu, j,o,t −UJOT Initialu, j,o,t−1 ∀u , ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀t(: t = 1) (6) UJOT Initialu, j,o,t denotes the unit, task, 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 the minimum uptime period, to avoid any physical damage. UJOOTransitionTimeu, j,o,o0 denotes the minimum uptime in the following Equation: UJOOTransitionTimeu, j,o,UJOSwitchModeu, j,o ,1

vbUJOTu, j,o,t ≥

∑

vbUJOOTu, j,o0 ,o,t−k

k=1(:t−k≥0) UJOOTransitionTimeu, j,o,UJOSwitchModeu, j,o ,1

+

∑

UJOOT Initialu, j,o0 ,o,t−k

(7)

k=1(:t−k≤0)

∀u , ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀o0 (: UJOSwitchModeu, j,o0 = o), ∀t(: t = 1..NT ) The parameter UJOOT Initialu, j,o,o0 ,t captures the operation mode transitions that occurred in past i.e before starting of scheduling horizon, For instance, if the unit u performing task j in operation mode o changed to o0 at time period t where, t belonged to the historical time horizon then the value of UJOOT Initialu, j,o,o0 ,t will be 1. This can be inferred from the allocation parameter in the historical time horizon which indicates the operation mode in which unit u was performing task 15 ACS Paragon Plus Environment


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Page 16 of 46

j at time period t (where, t belongs to the historical time horizon). UJOSwitchmodeu, j,o denotes the next available operation mode. For predefined sequences, for each defined chain of transitions from mode o to mode o0 to mode o00 , a fixed stay time in mode o0 can be specified by the following constraint(Eq. (8) − Eq. (9)). These constraints basically define the transition modes like start-up or shutdown. So, in case of every mode transitions like on-> shutdown -> off, the transitional mode shutdown will take a fixed time as specified.

vbUJOOTu, j,o,o0 ,(t−UJOOOFixedTransitionTimeu, j,o,o0 ,o00 ) = vbUJOOTu, j,o0 ,o00 ,t ∀u , ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀o0 (: UJOu, j,o0 = 1),

(8)

00

∀o (: UJOu, j,o00 = 1 and UJOOOu, j,o,o0 ,o00 = 1), ∀t(: (t −UJOOOFixedTransitionTimeu, j,o,o0 ,o00 ) ≥ 0 and t = 0..NT − 1)

UJOOT Initialu, j,o,o0 ,(t−UJOOOFixedTransitionTimeu, j,o,o0 ,o00 ) = vbUJOOTu, j,o0 ,o00 ,t ∀u , ∀ j(: UJu, j = 1), ∀o(: UJOu, j,o = 1), ∀o0 (: UJOu, j,o0 = 1), 00

∀o (: UJO

u, j,o00

= 1 and UJOOO

u, j,o,o0 ,o00

(9)

= 1),

∀t(: (t −UJOOOFixedTransitionTimeu, j,o,o0 ,o00 ) < 0 and t = 0..NT − 1) The Parameter UJOOOFixedTransitionTimeu, j,o,o0 ,o00 stores the predefined stay time periods needed for unit u in operation mode o0 for the above-mentioned mode transition sequences.

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 the production of liquid products is inadequate to meet the total liquid demands, liquid products can be purchased from nearby plants (in-house or those of the competitors). Figure 3 illustrates the entire mass balance schema for an air separation process. The amount of state produced/consumed while performing task j in unit u in mode o during time duration t can be 16 ACS Paragon Plus Environment

Page 17 of 46

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Plant Product Mass Balance

Power Required Product Vent

Air In

Air Separation and Liquefaction Plant

Product From Plant to Onsite Customer

Onsite Customer

Product To Inventory

Liquid Product Purchase from Competitor

Onsite Demand Balance

Product From Inventory to Onsite Customer

Liquid Product Initial Inventory

Onsite Product Demand

Product Inventory Liquid Product From Inventory to Merchant Customers

Product Inventory Calculation & Balance

Merchant Customers

Merchant Liquid Demand

Liquid Product Waste Final Liquid Inventory

Merchant Demand Balance (Regular and Spot) Liquid Product From Competitor to Merchant Customers

Figure 3: Mass Balance of Air Separation Plant expressed as a combination of the production/consumption rates of selected production slates. vcUPT Quantityu,p,t indicates the total state produced/consumed in unit u during time t and can be mathematically expressed as,

vcUPT Quantityu,p,t =

J

O

L

∑

∑

∑

vcUJOLT SlateCoe f fu, j,o,l,t

j=1(:UJu, j =1) o=1(:UJOu, j,o =1) l=1(:UJOLu, j,o,l =1)

∗UJOLPStateChangeRateu, j,o,l,p ∗ (T Timet,2 − T Timet,1 ) ∀u, ∀p(: UPu,p = 1), ∀t(: t = 1..NT ) (10) Where, T Timet,1 and T Timet,2 denote starting time and ending time of a time slot respectively. PFinalProductFlag p indicates the deliverable products of the ASP. The mass balance for deliverable gaseous products are expressed in Eq. (11). Eq. (12) defines the mass balance constraint for intermediate gaseous product such as, GN2. The ASP operation is mainly oxygen production driven, huge quantity of nitrogen also gets produced 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



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Page 18 of 46

onsite customers whereas, the rest part is vented. C

U

∑ CPT OnsiteDemandc,p,t −

c=1

vcUPT Quantityu,p,t = 0

∑ u=1(:UPu,p =1)

(11)

∀p(: PType p =0 G0 and PFinalProductFlag p = 1), ∀t(: t = 1..NT ) U

vcUPT Quantityu,p,t = 0

∑ u=1(:UPu,p 6=0)

∀p(: PType p

=0

G0

(12)

and PFinalProductFlag p = 0), ∀t(: t = 1..NT )

UPu,p = 1 in the above denotes a set of unit product combinations in which product p is produced. UPu,p = −1 denotes a set of those unit product combinations where, product p gets consumed. The parameter PType p indicates that the product p is in liquid or in gaseous form. The Liquid inventory balance can be written in Eq. (13) & Eq. (14) as, U

vcPT Inv p,t = vcPT Inv p,t−1 +

∑

vcUPT Quantityu,p,t

u=1(:UPu,p =1) U

−

vcUPT Quantityu,p,t + vcSPT PurchaseInvs,p,t

∑ u=1(:UPu,p =−1)

(13)

−vcPT Dump p,t − vcPT DemandFill p,t − vcPT SpotDemandFill p,t ∀p(: PType p =0 L0 ), ∀t(: t = 1..NT ) Where,

vcPT Inv p,t = PInitialInv p ∀p(: PType p =0 L0 ), ∀t(: t = 0)

(14)

where vcPT Inv p,t indicates the inventory level for state p (where,p ∈ P(: PType p =0 L0 )) at time t. Variable vcPT DemandFill p,t and vcPT SpotDemandFill p,t denote the quantities of liquid products sent from inventory to fulfil regular and spot demand respectively. The additional amount of liquid that are purchased from other sources to fulfil inventory target requirement are indicated by vcSPT PurchaseInvs,p,t . 18 ACS Paragon Plus Environment

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The inventory quantity at the end of each time period should be within maximum and minimum capacity limits, which is enforced by the following constraints,

PInvCapacity p,1 ≤ vcPT Inv p,t ∀p(: PType p =0 L0 ), ∀t(: t = 1..NT )

(15)

vcPT Inv p,t ≤ PInvCapacity p,2 ∀p(: PType p =0 L0 ), ∀t(: t = 1..NT )

(16)

The target inventory is determined according to the business practices or contract obligations. The product inventory at the end of each time period should achieve the target set for the product quantity in the storage facility. If there are some contract obligations regarding the minimum allowable amount of liquid in the storage facility, the product inventory at any time period should be at a level above that. The following set of constraints are used to meet inventory targets including contract minimum liquid requirements. PMinInvContract p ≤ vcPT Inv p,t (17) 0

0

∀p(: PType p = L ), ∀t(: t = 1..NT )

vcPT Inv p,t ≤ PT InvTarget p,t (18) 0

0

∀p(: PType p = L and PTargetViolationFlag p = 0), ∀t(: t = 1..NT ) While the minimum inventory contract constraint has to be obeyed, the target constraints are generally soft and hence violations are allowed with penalty. The parameter PTargetViolationFlag p indicates that the inventory target violation is permissible or not. If, PTargetViolationFlag p = 0 then the target constraint will be considered as hard and will follow Eq. (18). The following constraint depicts the target constraint with permitted penalty violations. PT InvTarget p,t ≤ vcPT Inv p,t + pvcPT Inv p,t,2 − pvcPT Inv p,t,1 ∀p(: PType p =0 L0 and PTargetViolationFlag p = 1), ∀t(: t = 1..NT )


(19)


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Page 20 of 46

It is important to note that inventory constraints (mass balance, capacity and target limitation) are defined only for liquid products. Furthermore, it should be noted that among pvcPT Inv p,t,1 and pvcPT Inv p,t,2 only one variable will take value and that will be equivalent to the violated amount. Although two variable are used here to denote the positive and negative violation of inventory target, only the negative violation will be penalized in the objective function. Though it may appear that the variable pvcPT Inv p,t,1 (which indicates the amount of positive target violation) is not required, but this variable helps in tracking the excess production scenarios.

Liquid Demand Constraints The merchant liquid demands are mostly fulfilled from inventory at plant location and in rare cases from other sources. Merchant liquid demands can be classified in two types, 1) Regular demand, 2) Spot demand. Regular Demands needs to be fulfilled, whereas spot demands are optional. But spot demands earn more revenue, so, maximum fulfilment of spot demands is encouraged. Eq. (20) and Eq. (21) capture the constraints for regular demand fulfilment. PY QDemandQuantity p,y,q,1 ≤ NT

∑ t=1(:t≥PY QDemandTime p,y,q,1 and t≤PY QDemandTime p,y,q,2 ) S

vcPT DemandFill p,t !

(20)

+ ∑ vcSPT PurchaseDemands,p,t s=1

∀p(: PType p =0 L0 ), ∀y(: Y Typey = 0 REGULAR0 , ∀q(: q = 1..PYCount p,y )

PY QDemandQuantity p,y,q,2 ≥ NT


vcPT DemandFill p,t !

+ ∑ vcSPT PurchaseDemands,p,t s=1

∀p(: PType p =0 L0 ), ∀y(: Y Typey = 0 REGULAR0 , ∀q(: q = 1..PYCount p,y )


(21)

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As mentioned above the spot sales are optional. But if we decide to honour a spot demand and some minimum demand is mentioned then it is necessary to fulfil the minimum demand at the least. A binary variable vbPY QSpotSale p,y,q is introduced which turns 1 if spot sale of product p is done in that time period. Constraints regarding spot demands, when the minimum limits are mentioned, are expressed in Eq. (22) and Eq. (23). vbPY QSpotSale p,y,q ∗ PY QDemandQuantity p,y,q,1 ≤ NT


vcPT SpotDemandFill p,t !

+ ∑ vcSPT PurchaseSpotDemands,p,t

(22)

s=1

∀p(: PType p =0 L0 ), ∀y(: Y Typey = 0 SPOT0 ), ∀q(: q = 1..PYCount p,y and PY QDemandQuantity p,y,q,1 ≥ 0)

vbPY QSpotSale p,y,q ∗ PY QDemandQuantity p,y,q,2 ≥ NT



+ ∑ vcSPT PurchaseSpotDemands,p,t

(23)

s=1

∀p(: PType p =0 L0 ), ∀y(: Y Typey = 0 SPOT0 ), ∀q(: q = 1..PYCount p,y and PY QDemandQuantity p,y,q,1 ≥ 0) If the minimum demands for spot sales are not defined or zero then above set of equations reduced to the following equation, PY QDemandQuantity p,y,q,2 ≥ NT



+ ∑ vcSPT PurchaseSpotDemands,p,t s=1

∀p(: PType p =0 L0 ), ∀y(: Y Typey = 0 SPOT0 ), ∀q(: q = 1..PYCount p,y and PY QDemandQuantity p,y,q,1 = 0) 21 ACS Paragon Plus Environment

(24)


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Page 22 of 46

The variables vcSPT PurchaseDemands,p,t and vcSPT PurchaseSpotDemands,p,t indicate the amount of liquid products those are imported from nearby supply sources to fulfill regular and spot demand.

Liquid Product 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 supply sources (i.e. other ASPs in vicinity). Products received from supply 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, NT

∑

(vcSPT PurchaseInvs,p,t +

t=1(:t≥SPQLimitTimes,p,q,1 and t≤SPQLimitTimes,p,q,2 )

vcSPT PurchaseDemands,p,t + vcSPT PurchaseSpotDemands,p,t ) ≥ vbSPQPurchases,p,q ∗ SPQLimitQuantitys,p,q,1 ∀s, ∀p(: PType p =0 L0 ), ∀q(: q = 1..SPCounts,p andSPQLimitQuantitys,p,q,1 > 0) (25) NT

∑



vcSPT PurchaseDemands,p,t + vcSPT PurchaseSpotDemands,p,t ) ≤ vbSPQPurchases,p,q ∗ SPQLimitQuantitys,p,q,2 ∀s, ∀p(: PType p =0 L0 ), ∀q(: q = 1..SPCounts,p andSPQLimitQuantitys,p,q,1 > 0) (26) If the minimum availability limit is not mentioned then the equation given below is sufficient to address the quantity limitation constraint on purchase option. NT

∑



vcSPT PurchaseDemands,p,t + vcSPT PurchaseSpotDemands,p,t ) ≤ SPQLimitQuantitys,p,q,2 ∀s, ∀p(: PType p =0 L0 ), ∀q(: q = 1..SPCounts,p andSPQLimitQuantitys,p,q,1 = 0) (27) 22 ACS Paragon Plus Environment

Page 23 of 46

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

Power Consumption, Purchase, Availability Constraints Power Consumption Power required in operation mode o of unit u at time t should be a combination of the power requirements specified for the various slates that are selected. Total power consumed in unit u in time duration t can be written as,

vcUT PowerRequiredu,t =

J

O

L

∑

∑

∑

vcUJOLT SlateCoe f fu, j,o,l,t

j=1(:UJu, j =1) o=1(:UJOu, j,o =1) l=1(:UJOLu, j,o,l =1)

∗UJOLPowerConsumptionu, j,o,l ∗ (T Timet,2 − T Timet,1 ) ∀u, ∀t(: t = 1..NT ) (28) The total power required to run all the units should be less than the power available. This criteria is handled in Eq. (29). The parameter TUtilityPowert indicates the amount of power needed to run the office utilities in the plant location. E

U

∑ vcUT PowerRequiredu,t + TUtilityPowert ≤

u=1

∑ vcET Purchasee,t

e=1

(29)

∀t(: t = 1..NT ) Power Availability Now, the power is available from many sources. The power purchased from a suitable power source should be within minimum and maximum availability limits. The limits can be specified



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Page 24 of 46

for a particular hour or a period. It is important to note that not purchasing power is an option; however if the power is purchased it should be greater than minimum availability limit for the stability of the supply network. When the availability limits are available on an hourly basis, they are accordingly converted into time slot limits taking the duration of time slot into consideration. Hence,

vbETe,t ∗ ET Limite,t,1 ≤ vcET Purchasee,t ∀e, ∀t(: ET Limite,t,1 > 0 and t = 1..NT )

(30)

vcET Purchasee,t ≤ vbETe,t ∗ ET Limite,t,2 ∀e, ∀t(: ET Limite,t,2 > 0 and t = 1..NT )

(31)

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 can be replaced with following Equation without any need to introduce a binary variable.

vcET Purchasee,t ≤ ET Limite,t,2 ∀e, ∀t(: ET Limite,t,2 > 0 and t = 1..NT )

(32)

When the availability limit is defined for a period, then following set of constraints are included into production scheduling formulation. NT

EPeriodLimite,q,1 ≤

vcET Purchasee,t

∑ t=1(:t≥EPeriodLimitTimee,q,1 and t≤EPeriodLimitTimee,q,2 )

(33)

∀e, ∀q(: q = 1..EPeriodLimitCounte and EPeriodLimite,q,1 ≥ 0) NT

vcET Purchasee,t ≤ EPeriodLimite,q,2

∑ t=1(:t≥EPeriodLimitTimee,q,1 and t≤EPeriodLimitTimee,q,2 )

∀e, ∀q(: q = 1..EPeriodLimitCounte and EPeriodLimite,q,2 ≥ 0)


(34)

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If there is additional constraint to select only one power source(i.e. parameter MOneSourceFlagm = 1) at a time period then following selection constraint should be included. E

∑ vbETe,t = 1

∀t(: t = 1..NT and MOneSourceFlagm = 1)

(35)

e=1

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 can be taken for all time slots. Hence, the cost of purchasing power from different power sources can be calculated as follows: NT

vcECoste =

∑ vcET Purchasee,t ∗ ET PowerPricee,t

∀e

(36)

t=1

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 constraint regarding penalty limit is soft and expressed in Eq. (37). NT

ELimitPenaltye − ∑ vcET Purchasee,t ≤ pvcELimitPenaltye ∀e(: ELimitPenaltye > 0)

(37)

t=1

The penalty variable pvcELimitPenaltye 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. This driox demand can be fulfilled using the free liquid or using the



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Page 26 of 46

liquid in inventory. Use of free liquid completely is necessary as drioxing inventory liquid will add additional cost. The production needed to fulfil the driox demand over the horizon can be calculated by Eq. (38).

vcPTotalDrioxDemand p =

NT

U

∑

∑

vcUPT Quantityu,p,t ∀p(: PType p =0 G0 )

(38)

t=1 u=1(:UDrioxu =1)

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. (39)). If, PT InvTarget p,NT ≥ PInitialInv p , U

vcPTotalFreeLiquid p =

∑

vcUPT Quantityu,p,t −

u=1(:UPu,p =1) Y PYCount p,y

∑

y=1

PY QDemandQuantity p,y,q,2

∑

q=1

(39)

−(PT InvTarget p,NT − PInitialInv p ) ∀p(: PType p =0 L0 ) But if the initial inventory is more than the target inventory then this excess liquid in inventory will be used to quench the liquid demand. If the liquid demand is more than the available liquid in inventory, then rest of the demand will be met from production.(Eq. (40)) But if the excess liquid in inventory are more than sufficient to meet the entire liquid demand, then the leftover liquid will not be added with the free liquid quantity (Eq. (42)). The aforementioned definition will have an exception if ‘economical shut down’ parameter is 1. If this parameter is kept ‘ON’ that is if the user set the value as 1 then the algorithm will calculate the optimum time periods in the scheduling horizon to shut down the plant, by using up the excess liquid in the inventory(Eq. (41)).

If, PT InvTarget p,NT ≤ PInitialInv p ,


Page 27 of 46

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Y PYCount p,y

and if, (PInitialInv p − PT InvTarget p,NT ) ≤ ∑


∑

y=1

q=1

U



∑ u=1(:UPu,p =1)

Y PYCount p,y


∑

∑

q=1

y=1

(40)

+(PInitialInv p − PT InvTarget p,NT ) ∀p(: PType p =0 L0 ) If, PT InvTarget p,NT ≤ PInitialInv p , Y PYCount p,y

and if, (PInitialInv p − PT InvTarget p,NT ) ≥ ∑


∑

y=1

q=1

and MEconomicShutdownFlagm = 1 U



∑ u=1(:UPu,p =1)

Y PYCount p,y

∑

y=1


∑

q=1

(41)

+(PInitialInv p − PT InvTarget p,NT ) ∀p(: PType p =0 L0 ) Y PYCount p,y

else if, (PInitialInv p − PT InvTarget p,NT ) ≥ ∑

∑

y=1


q=1

and MEconomicShutdownFlagm = 0 U


∑

vcUPT Quantityu,p,t

u=1(:UPu,p =1)

(42) ∀p(: PType p =0 L0 )

Where, NT indicates the end time period of the scheduling horizon. Now, 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 27 ACS Paragon Plus Environment


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Page 28 of 46

the above-mentioned decomposition technique. vcPFreeLiqPositiveQuan p − vcPFreeLiqNegativeQuan p = vcPTotalFreeLiquid p (43) 0

0

∀p(: PType p = L )

vcPFreeLiqPositiveQuan p ≤ PInvCapacity p,2 ∗ vbPFreeLiqPositive p 0

(44)

0

∀p(: PType p = L ) vcPFreeLiqNegativeQuan p ≤ PInvCapacity p,2 ∗ vbPFreeLiqNegative p 0

(45)

0

∀p(: PType p = L ) vbPFreeLiqPositive p + vbPFreeLiqNegative p = 1 (46) ∀p(: PType p =0 L0 ) vcPFreeLiqPositiveQuan p and vcPFreeLiqNegativeQuan p both are positive variable. So, the actual quantity of driox demand which will have a cost term associated with it over the horizon can be calculated as shown in Eq. (47). vcPBillableDrioxDemand p ≥ vcPTotalDrioxDemand p − vcPFreeLiqPositiveQuan p (47) 0

0

∀p(: PType p = L ) vcPBillableDrioxDemand p is a positive variable.

Objective Function The objective is to minimize the total 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) -


Page 29 of 46

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Revenue earned from spot sale(ROS) + Penalty for not meeting the regular liquid demand(PRD). where, E

TPC =

∑ vcECoste

e=1 E

CP =

PenaltyCoste ∗ pvcELimitPenaltye

∑ e=1(:ELimitPenaltye >0) P

PD =

NT

∑

p=1(:PType p =0 L0 )

PDumpPenalty p ∗ ∑ vcPT Dump p,t

P

t=1

NT

U

∑

PVentPenalty p ∗ ∑

S

P

SPCounts,p

COS = ∑

∑

PV =

p=1(:PType p =0 G0 )

s=1 p=1(:PType p =0 L0 )

vcUPT Quantityu,p,t

∑

t=1 u=1(:UVentu =1andUPu,p =1)

SPLimitPrices,p,q ∗

∑ q=1 NT

vcSPT PurchaseInvs,p,t +

∑ t=1(:t≥SPQLimitTimes,p,q,1 and t≤SPQLimitTimes,p,q,2 )

vcSPT PurchaseDemands,p,t + vcSPT PurchaseSpotDemands,p,t P

PINV =

NT

∑


PTargetInvPenalty p,2 ∗ ∑ pvcPT Inv p,t,2 t=1

P

DC =

PDrioxCost p ∗ vcPBillableDrioxDemand p

∑


ROS =

P

Y

∑

∑

PYCount p,y

PY QDemandPrice p,y,q ∗ vcPT SpotDemandFill p,t + ∑ t=1(:t≥PY QDemandTime p,y,q,1 and t≤PY QDemandTime p,y,q,2 ) S ∑ vcSPT PurchaseSpotDemands,p,t

p=1(:PType p =0 L0 ) y(:y=0 SPOT 0 )

∑

q=1 NT

s=1



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PRD =

Page 30 of 46

PYCount p,y

P

Y

∑

∑

p=1(:PType p =0 L0 ) y(:y=0 REGULAR0 )

∑

PY QDemandPrice p,y,q ∗

q=1

PY QDemandQuantity p,y,q,2 − vcPT DemandFill p,t +

NT

∑ t=1(:t≥PY QDemandTime p,y,q,1 and t≤PY QDemandTime p,y,q,2 )

!

S

∑ vcSPT PurchaseDemands,p,t s=1

Results and Discussions The model discussed above is applied to several real world scenarios provided by Praxair India. Scenarios are chosen in such a way that they will help to test all the nuances that this model brings as well as to provide an idea about the rigor of the model. These scenarios also demonstrate the practicality & implementability of the proposed model. A week-long (168 hours) time horizon has been considered as the scheduling horizon. The entire horizon is discretized on a per hour basis as the onsite demands and electricity prices are specified on an hourly basis. The liquid demands are given on a per day basis, and therefore we have tried both hourly basis and day wise discretization in calculating the inventory balance. The resulting MILP models are solved to generate optimal schedule for all the scenarios. All the simulations are carried out in Fico©Xpress Optimization suite using ‘mmxprs’ module version 2.8.1 on a Intel®Core™i7(3.6 GHz) machine with 16GB RAM. 13,14 In Table 1, problem sizes for each of the scenarios along with simulation time and optimality criteria are reported. The termination criteria is defined as, 600s simulation time or less than 0.06% optimality gap. Several other scenarios are also simulated and in most of the cases the algorithm reached required optimality gap in the stipulated computational time. Due to confidentiality issues, the exact production or power consumptions are not mentioned. The units and values reported in the graphs or tables are in normalized form.


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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


Table 1: Simulation statistics for each scenario and discretization method Scenario

Discretization

Constraints

Variables

Binary

1

Inv per hr. Inv per day Inv per hr. Inv per day Inv per hr. Inv per day Inv per hr. Inv per day

25443 23571 25455 23523 25442 23510 25443 23571

37590 35244 37590 35244 37515 35241 37518 35241

7879 7879 7879 7879 7879 7879 7879 7879

2 3 4

Time(sec) Optimality Gap(%) 58.9 0.057 39.9 0.032 182.5 0.02 24.5 0.024 118.8 0.04 600 0.41 600 2 600 1.53

Scenario 1 : Low Initial Inventory, High Target Inventory, and Spot Demand Target inventory has been defined at the end of the horizon for this scenario. As discussed earlier, the spot demand fulfilments are not mandatory but they earn a higher revenue. So the expected outcome from this scenario will be such that after meeting the regular demand and required target inventory, at least a portion of the spot demand should be quenched if not all. Negative target inventory will be penalized. Three types of power contracts are considered for this case, 1. Power grid —time of use power price 2. Auction —power prices are based on auction. (Forecast for day ahead prices was given on an hourly basis.) 3. MTOP (minimum take or pay) —power price is constant throughout the day. Penalty will be charged for under consumption. In Figure 4a, the resultant optimal schedule for scenario 1, depicting the operation modes of each of the units throughout the horizon, is shown. The schedule illustrated for all the scenarios are based on hour wise discretization for the entire problem. The minimum and maximum (where applicable) transition time for all the modes of all the units are listed in Table 2 whereas, the initial modes of the units at the starting of the scheduling horizon are listed in Table 3. The inventory profile for liquid oxygen are shown in Figure 4b. The liquid demands are given for 31 ACS Paragon Plus Environment


ON

Shutdown

Startup

Off

NOARON

LO2 Demand

LO2 Production

LO2 Inventory

1.2

1.05

1

ASU

1 0.95

LiqGN2

LMComp

0.9

0.85 0.6 0.8

0.4

0.75

0.7 0.2

MHComp

0.65

0

0.6 0

20

40

60

LHComp

1

21

41

61

81 Time(hr)

101

121

141

80 Time(hr.) 100

120

140

161

(a) Schedule for Scenario 1

(b) LO2 Inventory Profile for Scenario 1

Figure 4: Optimum Schedule and LO2 Inventory Profile for Scenario 1

Table 2: Unit,Operation Modes and Corresponding Transition Time Unit

Operation Mode

ASU ASU ASU ASU LiqGN2 LiqGN2 LiqGN2 LiqGN2 LMComp LMComp MHComp MHComp LHComp LHComp

Off Startup NoArOn On Off Startup Shutdown On Off On Off On Off On

160

Minimum Transition Time 12 2 18 6 4 2 1 2 1 1 1 1 1 1

Maximum Transition Time 2 18

2 1


Next Operation Mode Startup NoArOn On Off Startup On Off Shutdown On Off On Off On Off

LO2 Inventory

0.8

LO2 Production

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

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Page 33 of 46

Table 3: Initial Operation Mode for Units Unit Operation Mode Time Period ASU On −24 LiqGN2 Off −21 LMComp On −10 MHComp Off −24 LHComp On −10

a day and the demands can be fulfilled at any time of the day. In this scenario, spot demands are specified only for the first day. If needed, the liquid products can be purchased from nearby supply sources between the specified periods. If profitable, a part of regular and spot demands can also be met by purchasing from competitors. Figure 5 shows how the regular demands are met from 1) inventory and 2) by purchasing from supply sources for this scenario. As there are a significant amount of spot sale demands along with regular liquid demands at first 24 hours of the scheduling horizon in this scenario, purchase from other sources is a viable option to quench these demands as much as possible. However, there are some specified time periods in which the option for purchase is available. As depicted in Figure 5, LO2 is purchased to quench a part of the regular demand in the first two days. There is no demand for LO2 towards the end of the scheduling horizon. In Figure 6b, it can be seen that power is purchased from three sources as described earlier. If 1.2

Quanitity Purchased

Quanitity From Inventory 1

Regular LO2 Demand Fulfilled

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


0.8

0.6

0.4

0.2

0 1

2

3

4

Day

5

6

7

Figure 5: Regular LO2 Demand Fulfilment from Inventory and Purchase for Scenario 1 only power grid is selected as the electricity source instead of exploring all the contracts available (such as MTOP or auction), there would be around 2.28% increase in the power cost. The load 33 ACS Paragon Plus Environment


shifting based on only the power grid price is illustrated in Figure 6a. As illustrated in Figure 6b, at any time period in the scheduling horizon, sources with the lowest power price are selected except during some time periods in the first day. This is due to the contract obligations made with ‘Auction’ power source which dictates that, no matter what their power price is, plant has to buy the entire amount of power that is being supplied for that specified time periods. Power mix scenarios are only entertained when the power required at a time period exceeds the maximum availability limit of a source. The power purchase scenario with the respective price are shown in Figure 6b. The rest of the scenarios will use the same power sources with same price and availability limit. Though the usage will be different, power purchase schemes are not illustrated for other scenarios due to brevity. LO2 Production

1.1

Grid Power Price

Grid Purchase Grid Price

7

Auction Purchase Auction Price

MTOP Purchase MTOP Price

9

1 6.5

8

6

0.8

0.8 5 0.7 4.5

Power Purchased

5.5

Power Price(m.u/KW)

0.9

7

6

0.6

5

Price(m.u./KW)

1

Normalized LO2 Production

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

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0.4 4

0.6 4

0.2 0.5

3

3.5

0.4

0

3 1

21

41

61

81 Time (hr) 101

121

141

(a) Load Shifting with Grid Power Price

2

1

161

21

41

61

81 Time(hr) 101

121

141

161

(b) Power Purchase from Three sources and Respective Price

Figure 6: Power Purchase and Consumption Pattern for Scenario 1

Scenario 2 : ASU Maintenance between Scheduling Horizon The previous example with the same demand and inventory scenario is again simulated keeping ASU in maintenance (OFF mode) from 49-60 hrs. The resultant schedule is illustrated in Figure 7a. The liquid products from inventory are drioxed to meet onsite product demands. All the other units except driox units are in off mode. After the maintenance period, the ASU goes to ‘NoArOn’ mode in which LO2 and LN2 are produced but the production of LAr can not be considered as those are


Page 35 of 46

ON

Shutdown

Startup

Off

NOARON

QuantityProduced

1.2

LAr Inventory

1.05

1

ASU 1

0.95

0.8

LMComp

0.9

0.85 0.6 0.8

0.4

LAr Inventory

LiqGN2

LAr Production

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


0.75

MHComp 0.7 0.2 0.65

LHComp 0

1

21

41

61

81 Time(hr)

101

121

141

161

0.6 0


20

40

60

80 Time(hr) 100

120

140

160

(b) LAr Inventory and Production Profile

Figure 7: Optimum Schedule and LAr Inventory Profile for Scenario 2 not of required purity. Operation points or slates are created for ‘NoArOn’ transition modes in such a way that those depicts full production of oxygen & nitrogen but no production for liquid argon. So, combination of these slates will show that when ASU is in ‘NoArOn’ mode there will be production of oxygen and nitrogen (liquid & gaseous states) but there will be no production of liquid argon. So, though ASU was in maintenance from 49-60 hrs., argon production starts at 81 hr. So, in this duration(61-81 hrs.) all the argon demands are met from inventory. The liquid argon inventory profile are given in Figure 7b. Non-inclusion of ‘NoArOn’ mode will result a schedule which will dictate production of Argon from 63rd hour itself (after 2 hr. startup), which cannot be achieved by the real process. Without NoArOn mode entire liquid argon inventory calculation will be wrong as the algorithm will overestimate the quantity of liquid argon that can be produced in the horizon and the entire schedule will be unusable.

Scenario 3 : Low LO2 Initial Inventory, High Target at Mid-Week, Medium Target at End Liquid oxygen production is higher at starting as high inventory target needs to be fulfilled in the middle of the week. No spot sale are considered in this scenario. A lot of excess liquid is left after meeting the mid-week target as the end target was low. This extra liquid is drioxed to meet onsite



demand and power cost is saved by shutting down the plant. This case study thus illustrates the judicious utilization of the liquid as necessary step to minimize the operating costs. The schedule and the LO2 inventory profile are illustrated in Figure 8a & Figure 8b. ON

Shutdown

Startup

Off

NOARON LO2 Production

1.2

LO2 Driox

LO2 Inventory

Target Inventory

1.05

ASU

1 1 0.95

LMComp

MHComp

0.8

0.9

LO2 Inventory

LiqGN2

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

Page 36 of 46

0.85 0.6 0.8

0.4

0.75

0.7 0.2 0.65

LHComp

1

21

41

61

81 Time(hr)

0

101

121

141

0.6 0

161


20

40

60

80 Time(hr.) 100

120

140

160


Figure 8: Optimum Schedule and LO2 Inventory Profile for Scenario 3

Scenario 4 : High LO2 Initial Inventory, Medium Target at End, No Spot Sale Initial liquid oxygen inventory is high and target inventory is low. LO2 demand can be fulfilled from the excess inventory. Economical shutdown option is also kept open so, excess liquid oxygen after meeting the demand can also be used. The inventory build-up required for the other two liquid products are also minimal. So, the production is mainly gaseous product demand driven. This excess liquid in the inventory along with the liquid product that are produced along side gaseous production can be utilized judiciously by converting it to gaseous products and supplied to onsite customers for some time periods, keeping the production closed. The above-mentioned strategy not only ensures savings of huge quantity of power cost but also, prevents wastage of valuable products. Figure 9a & Figure 9b show the optimum schedule and LO2 inventory profile. As, depicted in Figure 9a, the plant ceased production completely from 41-67 hrs. and again from 157 hr. to the end of the scheduling horizon. In those above-mentioned time periods, the onsite demands are met by drioxing the liquid inventory. There are some instances of drioxing in the initial period as illustrated in Figure 9b. As, there is abundance of liquid product in inventory, 36 ACS Paragon Plus Environment

Page 37 of 46

ON

Shutdown

Startup

Off

NOARON

LO2 Production

1.2

LO2 Driox

LO2 Inventory

1.05

1

ASU

1 0.95

LiqGN2

LMComp

0.8

0.9

0.85 0.6 0.8

0.4

LO2 Inventory

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


0.75

MHComp 0.7 0.2 0.65

LHComp 0

1

21

41

61

81 Time(hr) 101

121

141

0.6 0

161


20

40

60

80

Time(hr.) 100

120

140

160


Figure 9: Optimum Schedule and LO2 Inventory Profile for Scenario 4 the liquid production should be decreased to avoid liquid product dumping. So, the corresponding gaseous production is also less, which is not sufficient to meet onsite demands. To meet the rest of the demands therefore, some amount of liquid inventory are drioxed at the initial time periods.

Conclusion This paper presents a discrete time STN representation of an air separation plant which can be used for production scheduling. The proposed unit based model owing to its granularity effectively handles many complexities such as, mode transitions, demand and inventory constraints, unit maintenance etc. In addition to the generic constraints common to power intensive processes, various other operational constraints related to cryogenic air separation plants are considered which ensures that the resulting model is able to replicate a real world air separation plant and the production schedule based on this model is implementable. The scheduling problem has been formulated mathematically as a MILP which upon solving provides optimal schedule respecting all the demands and inventory constraints. The objective function not only includes minimization of the production cost as the main criteria but also encourages fulfilment of spot sales to earn more revenue. Result obtained from the first scenario demonstrates the systematic use of demand side management technique along with the efficient exploitation of all the power contracts to minimize operational cost when the plant runs at full throttle. Whereas, 37 ACS Paragon Plus Environment


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 38 of 46

the fourth scenario brings out the efficacy of the algorithm in minimizing the operational cost and also ensures energy savings by judicious liquid product management. The second & third scenarios highlight the model’s flexibility to cope with real world challenges. The resultant solutions, though optimal owing to the linearity of the model, are not unique. 11 There are several solutions with same objective function value are generated, due to the inherent flexibility of the process, which sometimes reduces the speed of the computation. It has been noted that higher degrees of freedom increases the computational times as the solver spends more time in searching all the possible solutions. Calculation of inventory per day method proves beneficial over per hour method in some of the cases. This hybrid discretization framework is expected to be hugely beneficial when this algorithm needs to be extended to other larger formulations such as those needed to accommodate multi-plant production scenario. The proposed model can be extended to multiple plants with minimal changes. Though extension of this model to a larger network can result in a substantial size problem, computational efforts can be reduced by efficient choice of discretization interval. The unit wise rigorous formulation also paves the way for integration of the scheduling layer with the lower layered (hierarchically) control system.

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

References (1) Energy Information Association. Manufacturing energy consumption Survay: tal Consumption of electricity,

To-

2010. http://www.eia.gov/consumption/

manufacturing/data/2010/pdf/Table11_1.pdf. (2) Palensky, P.; Dietrich, D. Industrial Informatics, IEEE Transactions on 2011, 7, 381–388. (3) Paulus, M.; Borggrefe, F. Applied Energy 2011, 88, 432–441. 38 ACS Paragon Plus Environment

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(4) Ierapetritou, M. G.; Wu, D.; Vin, J.; Sweeney, P.; Chigirinskiy, M. Industrial & Engineering Chemistry Research 2002, 41, 5262–5277. (5) Karwan, M. H.; Keblis, M. F. Computers & Operations Research 2007, 34, 848 – 867, Logistics of Health Care ManagementPart Special Issue: Logistics of Health Care Management. (6) Mitra, S.; Grossmann, I. E.; Pinto, J. M.; Arora, N. Computers and Chemical Engineering 2012, 38, 171–184. (7) Zhang, Q.; Grossmann, I. E.; Heuberger, C. F.; Sundaramoorthy, A.; Pinto, J. M. AIChE Journal 2015, 61, 1547–1558. (8) Zhang, Q.; Sundaramoorthy, A.; Grossmann, I. E.; Pinto, J. M. Computers and Chemical Engineering 2016, 84, 382 – 393. (9) Misra, S.; Kapadi, M.; Gudi, R. D.; Srihari, R. IFAC-PapersOnLine 2016, 49, 675 – 680, 11th {IFAC} Symposium on Dynamics and Control of Process SystemsIncluding Biosystems (DYCOPS-CAB 2016),Trondheim, Norway. (10) Smith, A.; Klosek, J. Fuel Processing Technology 2001, 70, 115 – 134. (11) Kondili, E.; Pantelides, C.; Sargent, R. Computers and Chemical Engineering 1993, 17, 211– 227. (12) Mitra, S.; Sun, L.; Grossmann, I. E. Energy 2013, 54, 194–211. (13) Getting Started with Xpress. (14) Xpress-Optimizer Reference manual.



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

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Nomenclature Indices m

Plant

u

Unit

s

Supply Source

j

Task/Job

c

Customer

o

Operation Mode

l

Operation Point/Slate

p

State/Product

e

Electricity/Power Source

t

Time Period

d

Day

y

Liquid Demand Type

1..2

1 indicate minimum and 2 indicate maximum

Parameters MOnePowerFlagm

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

MEconomicShutdownFlagm

If equal to 1 then economical shutdown will be exercised.

MSm,s

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

UJu, j

Equal to 1 if task j can be performed in unit u

UDrioxu

Indicates the driox units.

UVentu

Indicates the vent units.

UJOu, j,o

Equal to 1 if operation mode o belongs to the unit u performing task. j


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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


Captures all possible mode transitions for a specific unit

UJOOu, j,o,o0

and task. UJOOTransitionTimeu, j,o,o0

Captures Minimum and Maximum Uptime

UJO f fu, j

Captures the operation mode ‘off’.

UJOOnu, j,o

Captures ‘on’ and ‘NoArOn’ operation mode.

UJOOOu, j,o,o0 ,o00

Captures predefined sequences of mode transitions.

UJOOOFixedTransitionTimeu, j,o,o0 ,o00

captures fixed uptime for operation mode o in the predefined sequence of (o,o0 ,o00 ) for unit u.

UJOT Initialu, j,o,t

Captures the initial unit, Task and Mode combination.

UJOOT Initialu, j,o,o0 ,t

Captures the mode transitions in historical horizon.

UT MaintenanceFlagu,t

Equal to 1 if unit u is in maintenance at time period t.

UJOLu, j,o,l

Captures available slates for an unit, task and operation mode combination. Captures power consumption for a slate l at operation mode

UJOLPowerConsumptionu, j,o,l

o in unit u performing task j. UJOLPStateChangeRateu, j,o,l,p

Captures state change rate for a slate l at operation mode o in unit u performing task j.

UPu,p

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

PType p

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

PFinalProductFlag p

Indicates the deliverable products.

PInvCapacity p

Captures inventory capacity for a product p

PMinInvContract p

Captures minimum inventory contract for a product p.

PInitialInv p

Initial inventory level for product p.

PTargetViolationFlag p

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

PTargetInvPenalty p

Captures penalty cost for the violation of inventory target. 41 ACS Paragon Plus Environment


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PVentPenalty p

Captures penalty cost for venting gaseous products.

PDumpPenalty p

Captures liquid product dump penalty.

PLiquidDemandPenalty p

Captures Penalty cost for not meeting maximum regular liquid demand if minimum and maximum limit of regular liquid demand are different.

PDrioxCost p

Captures cost for drioxing liquid products into gaseous product.

CPT OnsiteDemandc,p,t

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

SPCounts,p

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

SPQLimitQuantitys,p,q

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

SPQLimitTimes,p,q

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

SPQLimitPrices,p,q

Captures purchase price of product p from supply sources.

PYCount p,y

Captures relationship between product and demand type.

PY QDemandQuantity p,y,q

Captures Liquid Product Demand Quantity.

PY QDemandPrice p,y,q

Captures Liquid Product Selling Price for spot sale.

PY QDemandTime p,y,q

Captures Liquid Demand Start and End Time periods.

PT InvTarget p,t

Captures Inventory Target for liquid products at time period t.

ELimitPenaltye

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


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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


Captures cost for violating penalty limit specified by

EPenaltyCoste

MTOP source. Captures price of per unit power purchased from power

ET PowerPricee,t

source e at time period t . Captures minimum and maximum power availability limit

ET Limit

of power source e at time period t. NT

Indicates the end time period of the scheduling horizon.

NPT

Largest minimum uptime of all units.

ND

Number of Day in scheduling horizon

T Timet

Captures the Start and End Time of a time slot

DTimed

Captures the Start and End Time of a day

T Dayt

Indicates the time slot belongs to which day.

Binary Variables vbUJOTu, j,o,t

Assumes value 1 if the operation mode o is selected in unit u performing task j at time period t.

vbUJOOTu, j,o,o0 ,t

Assumes value 1 if the transition for operation mode o to o0 is happens in unit u performing task j at time period t.

vbPY QSpotSale p,y,q

Assumes value 1 if spot sale for a liquid product p is done.

vbETe,t

Assumes value 1 if power source e is selected at time period t.

vbPFreeLiqPositive p

binary variable to select the positive part of the free liquid

vbPFreeLiqNegetive p

binary variable to select the negative part of the free liquid

Continuous Variables vcUJOLT SlateCoe f fu, j,o,l,t

Slate coefficient



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

Amount of product p produced/consumed by a unit u at

vcUPT Quantityu,p,t

time period t. vcPT Inv p,t

Liquid product inventory at time period t.

vcSPT PurchaseInvs,p,t

Quantity of liquid product purchased from other sources s to meet the inventory target at time period t.

vcPT Dump p,t

Quantity of dumped Liquid product at time period t.

vcPT DemandFill p,t

The amount of liquid product supplied from inventory to meet regular demand at time period t. Quantity of liquid product purchased from supply source s

vcSPT PurchaseDemands,p,t

to meet regular demand at time period t. Quantity of liquid product supplied from inventory to meet

vcPT SpotDemandFill p,t

spot demand at time period t. vcSPT PurchaseSpotDemands,p,t

Quantity of liquid product purchased from supply source s to meet spot demand at time period t.

pvcPT Inv p,t,1

Indicates negative violation of the inventory targets.

pvcPT Inv p,t,2

Indicates positive violation of the inventory targets.

vcUT PowerRequiredu,t

Power required to run unit u at timer period t.

vcET Purchasee,t

Power purchased from power source e at time period t.

vcECoste

Cost of power purchase from a power source e over scheduling horizon.

pvcELimit penaltye

Penalty variable for minimum take or pay limit violation.

vcPTotalDrioxDemand p

Total amount of liquid product drioxed to meet gas demand over the scheduling horizon.

vcPTotalFreeLiquid p

Total amount of free liquid over the scheduling horizon.

vcPFreeLiquidPositiveQuan p

Captures the positive part of free liquid.

vcPFreeLiquidNegativeQuan p

Captures the negative part of the free liquid.


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Captures the billable quantity of total liquid product dri-

vcPBillableDrioxDemand p

oxed.

Plant Details Power Contracts

Production Scheduling Tool

Intelligent utilizations of liquid products

Inventory Trends Electricity Consumption

Product Demand

Effective utilizations of spot sale opportunities

Production Schedule

Cryogenic Air Separation Plant

Decisions

TOC Graphics Input Data

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


Minimum Production Cost Optimum use of energy contracts and availability

Figure 10: Table of Content Graphic



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84x47mm (96 x 96 DPI)

ACS Paragon Plus Environment

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Energy-Efficient Production Scheduling of a Cryogenic Air Separation

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