Plantwide Scheduling Model for the Typical Polyvinyl chloride

Article pubs.acs.org/IECR

Plantwide Scheduling Model for the Typical Polyvinyl chloride Production by Calcium Carbide Method Miaomiao Tian,† Xiaoyong Gao,† Yongheng Jiang,† Ling Wang,†,‡ and Dexian Huang*,†,‡ †

Department of Automation, Tsinghua University, Beijing, 100084, China Tsinghua National Laboratory for Information Science and Technology, Beijing, 10084, China

‡

S Supporting Information *

ABSTRACT: The polyvinyl chloride (PVC) production by calcium carbide method accounts for the majority of PVC in the market in China, the largest market share in the world. Due to high energy costs, increasingly stringent requirements on emission reduction, and intense global competition, the scheduling optimization of PVC production processes, as an effective way to improve profit, is drawing increasing attention. However, most research is only focused on the scheduling of vinyl chloride monomer (VCM) polymerization processes rather than plantwide scheduling optimization. Moreover, the majority of energy consumption lies in the upstream VCM preparation process, including calcium carbide and chlorine production processes. Therefore, plantwide scheduling optimization is of great importance for reducing the energy consumption and improving the overall profit. Hence, a discrete time representation based plantwide scheduling model is proposed in this paper. Since the operation characteristics in the time scales for VCM preparation and VCM polymerization processes are distinct, different discrete time intervals are adopted to reduce the resulting scheduling model scale. The energy consumption piecewise linear models in terms of the production rates for both calcium carbide and chlorine production processes are addressed to approximate the real-world nonlinear processes. Several cases originated from a real industrial plant are provided to illustrate the effectiveness of the proposed model, and the comparative results with the conventional strategy (i.e., only optimizing the polymerization process when scheduling) further prove the necessity of plantwide scheduling optimization.

1. INTRODUCTION The typical methods for producing PVC mainly include ethylene and calcium carbide. In the past decade, PVC production by the calcium carbide method has been intensified rapidly in China, mainly because of the abundance in sodium chloride, lime, and coke, which are the raw materials of calcium carbide production and the shortage in petroleum, which is the main material of ethylene production, in China. According to incomplete statistics, the production capacity by the calcium carbide method accounts for more than 70% shares in China, and more than 25% in the world.1,2 Because of high energy consumption and environmental pollution, PVC production by the calcium carbide method is facing large pressures of energy saving and emission reduction. To reduce the energy consumption and improve the profit margin, there is an urgent need to seek effective ways for the efficient operation of the whole production process. Scheduling optimization as an effective way for improving profit and reducing energy consumption has attracted increasing attention in the process system engineering community. Much valuable and fruitful progress has been reported in the literature for polymerization process scheduling.3−12 Sand et al.,3 Wang et al.,4 and Schulz et al.5 studied the © XXXX American Chemical Society

production of expandable polystyrene (EPS) as a benchmark process, which includes preparation, polymerization, and finishing stages. The preparation stage provided materials for the polymerization stage and did not involve the production of materials. Liu et al.6 considered the scheduling of a polypropylene (PP) batch production process, which included preparation of raw material, polymerization, flash, deactivation, and packaging stages. The monomer (propylene) was preconditioned as a raw material and the cost of its production was not considered in the scheduling model. In the model proposed by Marchetti et al.7 for a pelletized polymer plant, the production cost of the monomer was not considered either. There are also some papers focused on PVC production process scheduling. A typical contribution was reported by Kang,8 in which a mixed-integer linear programming (MILP) model of a PVC production process was proposed to minimize the shortage and inventory cost, including the polymerization process, inventory management, and packaging and delivery Received: January 20, 2016 Revised: April 1, 2016 Accepted: April 27, 2016

A

DOI: 10.1021/acs.iecr.6b00291 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

ization process. And piecewise linear approximation is adopted to describe the nonlinear electricity consumption representations for the calcium carbide and chloride production processes. The rest of the paper is organized as follows. First, the problem description and model formulation are addressed in section 2. On the basis of the process analysis, we then propose the energy consumption representations in terms of production rates for continuous process in section 3, followed by the detailed plantwide scheduling model. Several well designed cases and the comparisons with the conventional methods are presented in section 4. Some conclusions are drawn out in section 5.

process. VCM as the raw material for PVC polymerization was assumed to be sufficiently supplied by VCM plants, and hence VCM production was not considered in his paper. Yi and Reklaitis9 introduced a MILP scheduling model considering changeover cost, inventory holding cost, and back-logging cost of a multistage polymer plant, where VCM was purchased. Asral et al.10 proposed a model for a batch process of polymerization, in which a simplified PVC batch plant for producing PVC from VCM was considered. Shah et al.11 proposed a multipurpose scheduling design of a conceptual PVC plant, considering the polymerization process and the purification and drying processes. To the best of the authors’ knowledge, there have been no reported works concerning the VCM production process in PVC scheduling. However, for the PVC production by calcium carbide method, the majority of the operation cost is energy consumption, i.e. electricity consumption. Moreover, the VCM production process, including calcium carbide and hydrogen chloride (HCL) production and synthesis processes, is the most energy consuming process. Specifically, calcium carbide and HCL production processes account for nearly 75%13 (as depicted in Figure 1) of the entire routine operation cost.

2. PROCESS DESCRIPTION AND PROBLEM STATEMENT The flowchart of a typical calcium chloride method PVC production plant is depicted in Figure 2. Both chlorine and hydrogen are made from electrolyzing saline water in electrolytic baths, which are further synthesized to produce HCL in a continuous manner. Calcium carbide is discharged batchwise from arc furnaces and then cooled down. The wellcooled calcium carbide, together with water, is then transferred to a hydrolysis reactor to produce acetylene (C2H2). VCM is obtained through the synthesis reaction of HCL and C2H2. After purification it is stored in the buffer tanks as the material of the further polymerization. In polymerization pots, the final product PVC is obtained after the complex polymerization and the accessory but compulsory processes, such as purification, drying, and so on. PVC is of several grades, with different rigidities and melting points, which depend on the regulations of catalyst and reaction temperatures in polymerization process. However, the frequent grade changeovers of polymerization pots are not recommended in practice. Finally, the resulting PVC is stored after plain packing. The main facilities of the PVC production process are arc furnaces, electrolytic baths, and polymerization pots. There are multiple facilities for parallel production, i.e. multiple electrolytic baths for chlorine production, multiple arc furnaces for calcium carbide, and multiple pots for polymerization. Different parallel facilities exhibit distinct operating characteristics, such as different energy consumption in terms of operating conditions, different processing capacity ranges, and so on. The major concern for the VCM production process is the routine operation cost, i.e. energy consumption, which mainly refers to the electricity. It is necessary and important to describe electricity consumption in terms of operating conditions. From the scheduling viewpoint, the unit’s operation cost depends on its production rate and exhibits severe nonlinearity. How to balance the model precision and complexity should be carefully considered because it is closely linked with the scheduling effect and the complexity in solution. Due to product plan changing and electricity supply emergency, the start-up and shut-down operations of arc furnaces are necessary. Moreover, restart cost for the arc furnace depends on the specific time span between its restart and stop time. The idle time span-dependent restart cost should be precisely estimated in the mathematical model. Calcium carbide is drawn out of the arc furnace in a batch manner with specific time interval (ranging from half an hour to more than 1 h determined by the furnace volume). It is a relatively short time comparing to the scheduling horizon. Meanwhile, the calcium carbide product can be stored, and the buffer storage is large enough for routine operations, though

Figure 1. Cost composition of the typical PVC production by calcium carbide method.12

Different production arrangements can have large differences in energy efficiency. Hence, it is essential to incorporate the upstream VCM production process into the scheduling scope in order to maximize the overall energy saving and profit. However, the upstream VCM production process and the downstream PVC polymerization process exhibit distinct operational characteristics (1) in different time scales and (2) in continuous or batch manner. Moreover, the electricity consumption representations in terms of operating conditions for calcium carbide process and chloride production process exhibit severe nonlinearity. How to handle these special process characteristics and precisely describe them in a plantwide scheduling model is a challenge, which is the focus of this paper. We propose a discrete time representation of different time intervals based plantwide scheduling model including the upstream VCM production and downstream PVC polymerB


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Figure 2. PVC production process.

the time span-varying storage loss of calcium carbide should be considered because of air exposure. The production of calcium carbide can be approximately assumed as a continuous process. The other processes for VCM production, such as chlorine production process and synthesis process, are all operated in continuous manner. Thereby, the upstream VCM production process can be treated as a continuous process. Therefore, PVC production process by calcium carbide method can be divided into two consequential parts: (1) the upstream continuous VCM production process and (2) the downstream batch VCM polymerization process. For the sake of simplicity, the VCM production process including electrolyzing process, calcium carbide production process, and synthesis process is called “continuous process” in the following. The aim of this paper is to propose a plantwide scheduling model considering both continuous and batch processes for the sake of energy saving and profit improvement. To achieve this target, the process model for the continuous process is carefully investigated, which is described in detail in the next section.

(1) Which polymerization pot is used to produce which grade of product at each time period (2) The operation state (i.e., starting-up, idle, or in-use) of each arc furnace and production rate of calcium carbide (and chlorine) for each arc furnace (and electrolytic bath) at each time period For better formulation of a real-world PVC plant scheduling problem, several reasonable assumptions are made as follows: (1) The calcium carbide production process is seen as a continuous process. (2) Perfect reaction occurs in all the synthesis processes, e.g. hydrogen and chlorine synthesis process, C2H2 and HCL synthesis process, which are simplified by ratio coefficient balance models in this paper. The energy and mass loss for synthesis processes are negligible. (3) The product quantity of polymerization process relies on the volumes of polymerization pots and conversion rates. (4) Time of changeover operations of polymerization pots is ignored because it is very short comparing to the scheduling horizon. (5) Since electricity consumption accounts for the majority of energy consumption in PVC plants using the calcium carbide method, only electricity consumption is considered in this paper. In this section, a time representation of the model is given first. Then the constraints of the continuous process are formulated, including formulations of the energy consumption, constraints of product rates, and the startup/shutdown operations of the arc furnaces. Then the constraints of the batch process are proposed. And the objective is given at last. 3.1. Time Representation. A discrete time representation is adopted in this paper. However, the operation characteristics of the VCM production and polymerization parts are distinct in time scale: the operation cycle of VCM production is 1 h while the cycle of VCM polymerization is 6 h. The resulted scheduling model scale will be too large to be solved in reasonable time if uniform time interval is adopted for both VCM production process and VCM polymerization process. To reduce the model scale, a hybrid time interval scheme for the two processes is proposed, which is shown in Figure 3. In

3. SCHEDULING MODEL FORMULATION The scheduling problem of the typical PVC production by calcium carbide method, as depicted in Figure 2, is considered in this paper. The problem is stated as follows: Given the following: (1) A set of process units including arc furnaces, electrolytic baths, polymerization pots, buffer tanks, and product tanks (2) Demand of final products of several grades, including the quantity and due time (3) Market prices of energy and utilities (4) Initial inventories and inventory limits (5) The initial states of the arc furnaces (i.e., whether the arc furnaces are working before the scheduling horizon), production rates of the arc furnaces during a certain period time before the scheduling horizon, and expectation production rates of the arc furnaces after the scheduling horizon The goal is to determine the following: C


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Figure 3. Time representation of the process.

Fortunately, piecewise linear approximation techniques are widely studied and applied in both academic and industrial communities,15−24 because it preserves both the local linearity and high nonlinear approximation capability. Hence, in this paper, piecewise linear model is adopted to approximate the nonlinear energy consumption. The energy consumption-production rate model is E = f(q), where E is the energy consumption and q is the production rate of the facility. First, the appropriate piecewise linear model form should be formulated and the optimal piecewise linear model parameters should be determined. In this paper, the well-known and widely used hinging hyperplane (HH) model is adopted and, moreover, an integrated training algorithm proposed by Gao et al.24 is recommended. Then series of partitioning points along with production rate axis and their corresponding consumption value set are obtained. Suppose there are n partitioning points, denoted as q(1), ..., q(n), and hence n corresponding consumption values, f(q(1)), ..., f(q(n)). Based on this model information, we introduce a continuous variable α(l) for the lth partitioning point, where α(l) ∈ [0, 1] (l = 1, ..., n). Let h(l) be a binary variable associated with the lth interval [q(l), q(l + 1)] (l = 1, ..., n), with h(n) = 0 as the boundary condition. For an arbitrary production rate qα and corresponding consumption value fα, i.e. energy consumption, can be expressed as

this paper, a 1-h time interval for continuous process and 6-h time interval for batch process are adopted based on their specific operating time characteristics: (1) the production rate of the furnace differs in each hour once restarted; (2) the polymerization batch period is 6 h. In Figure 3, T1 = {1, 7, 13, 19, ...} denotes the set of time periods for the batch process. It is a subset of T, which is the set of the time periods for the continuous process. 3.2. Constraints of Continuous Process. The model of energy consumption of the continuous process is given in section 3.2.1, and piecewise linear approximation is adopted for energy consumption estimation to avoid model nonlinearity caused by nonlinear model. And the startup and shutdown constraints of arc furnaces are given in section 3.2.2. Then the constraints of other processes are given. 3.2.1. Energy Consumption Representations of Calcium Carbide and Chlorine Production. Production of calcium carbide and chlorine in different production rates consumes different amount of electrical energy in furnaces and electrolytic baths, and leads to different production efficiencies (i.e., energy consumption per unit production/unit consumption). We obtained the data from real-world plant and used a hybrid method which combined mechanism model and data modeling method: the mechanism model structure was adopted according to mechanism analysis,14 and the collected realworld plant data were used to train the unknown model parameters. Taking calcium carbide production process as an illustration, the energy consumption of unit production in terms of production rates curve is depicted in Figure 4. Clearly, it exhibits high nonlinearity and there is an optimal production rate point corresponding to the optimal production efficiency. If the above nonlinear mechanism model is used for scheduling, the resulted mixed integer nonlinear programming (MINLP) model is hard to solve in reasonable time.

n−1

∑ h(l) = 1 (1)

l=1

α(l) ≤ h(l − 1) + h(l),

l = 1, ..., n

(2)

n

∑ α (l ) = 1 (3)

l=1 n

qα =

∑ α(l)q(l) (4)

l=1 n

fα =

∑ α(l)f (q(l)) (5)

l=1

The particular formulations in scheduling model for electrolytic bath τ are as follows nCl(τ ) − 1

∑

hCl(lCl(τ ), τ , t ) = 1,

∀ τ ∈ Γ, t ∈ T (6)

lCl(τ ) = 1

αCl(lCl(τ ), τ , t ) ≤ hCl(lCl(τ ) − 1, τ , t ) + hCl(lCl(τ ), τ , t ) ,

Figure 4. Product efficiency of calcium carbide production. D

∀ τ ∈ Γ, t ∈ T, lCl(τ ) = 1, ..., nCl(τ )

(7)


Article

Industrial & Engineering Chemistry Research nCl(τ )

∑

αCl(lCl(τ ), τ , t ) = 1,

α pCa (g , t ) − qCa (g , t ) ≤ U (1 − XCa(g , t )),

∀ τ ∈ Γ, t ∈ T

∀ g ∈ G, t ∈ T

(8)

lCl(τ ) = 1 nCl(τ )

pCl (τ , t ) =

∑

α pCa (g , t ) − qCa (g , t ) ≥ L(1 − XCa(g , t )),

αCl(lCl(τ ), τ , t )qCl(lCl(τ ), τ ),

∀ g ∈ G, t ∈ T

lCl(τ ) = 1

∀ τ ∈ Γ, t ∈ T

(9)

∑

∀ g ∈ G, t ∈ T

αCl(lCl(τ ), τ , t )fCl (qCl(lCl(τ ), τ )),

lCl(τ ) = 1

∀ τ ∈ Γ, t ∈ T

∀ τ ∈ Γ, t ∈ T

∀ g ∈ G, t ∈ T

(11)

where is the upper bound of the energy consumption, U is a large positive number, and L is a small negative number, ECa(g, t) denotes the energy consumption of arc furnace g at time t; and binary variable XCa(g, t) takes the value one means that furnace g is producing calcium carbide at time t, that is, the production rate of furnace g is not 0. 3.2.2. Startup and Shutdown Constraints of the Arc Furnace. The startup and shutdown operations of arc furnaces lead to loss in energy because that the furnaces continue being heated when idle. Based on a priori knowledge from field experts, the following rules are required to be considered in models, Rule (1) When the furnace is idle, the energy of heat preservation of the electrodes should be considered. A constant heating power is supplied. Rule (2) When the furnace restarts, the maximum increasing velocity of the production rate of the furnace depends on the idle span tl from shutdown to startup. The increasing of production rate after restarting goes through two periods: a rapid growth process and a constant speed growth process. (a) If the idle time tl ≤ 3, the production rate of the furnace should be raised to 75% of the maximum production rate max max pmax Ca (g) (75% pCa (g)) rapidly, and then raised to pCa (g) 1 in the time of 2 tl . That is, the growth velocity of the

nCa(g ) − 1

∑

hCa(lCa(g ), g , t ) = 1,

∀ g ∈ G, t ∈ T (12)

αCa(lCa(g ), g , t ) ≤ hCa(lCa(g ) − 1, g , t ) + hCa(lCa(g ), g , t ), ∀ g ∈ G, t ∈ T, lCa(g ) = 1, ..., nCa(g )

(13)

nCa(g )

∑

αCa(lCa(g ), g , t ) = 1,

∀ g ∈ G, t ∈ T (14)

lCa(g ) = 1 nCa(g ) α qCa (g , t ) =

∑

αCa(lCa(g ), g , t )qCa(lCa(g ), g ),

lCa(g ) = 1

1

∀ g ∈ G, t ∈ T

production rate is constant as (1 − 75%)pmax Ca (g)/ 2 tl per unit time. (b) If 3 < tl ≤ 8, the production rate of the furnace should be max raised to 50%pmax Ca (g) rapidly and then raised to pCa (g) in

(15)

nCa(g ) α f Ca (g , t ) =

∑

(22)

Emax Ca (g)

where hCl(lCl(τ), τ, t) and αCl(lCl(τ), τ, t) are variables corresponding to h(l) and α(l) in eqs 1−5; f Cl(qCl(lCl(τ), τ)) denotes the energy consumption at the production rate qCl(lCl(τ), τ), and both of them are given parameters; pCl(τ, t) denotes the chlorine production rate of bath τ at time t, and ECl(g, t) denotes the energy consumption of bath τ at time t. Similarly, the corresponding constraints for arc furnace g are

lCa(g ) = 1

(21)

α ECa(g , t ) − f Ca (g , t ) ≥ L(1 − XCa(g , t )),

(10)

hCl(nCl(τ ), τ , t ) = 0,

(20)

α ECa(g , t ) − f Ca (g , t ) ≤ U (1 − XCa(g , t )),

nCl(τ )

ECl(τ , t ) =

(19)

αCa(lCa(g ), g , t )fCa (qCa(lCa(g ), g )),

lCa(g ) = 1

1

∀ g ∈ G, t ∈ T

hCa(nCa(g ), g , t ) = 0,

the time of 2 tl . (c) If tl > 8, the production rate of the furnace should be raised to 50%pmax Ca (g) rapidly and then raised with a growth velocity of 5%pmax Ca (g) per unit time. Based on these rules, the following assumptions are made: (1) The production rate rapidly increasing period is treated as a fixed-length time span, an hour here. (2) During that period, due to the discharge being very unstable and low, the quality of calcium carbide cannot be guaranteed, and the output is neglected. That is, the production rate in that period is seen as 0. (3) The cost of startup after an idle time is the energy consumption of raising the load of the furnace to 50% or 75% of the full load. It is much higher than the inventory cost and the reaction loss in air exposure during that time. So if the furnace has been idle for only a few hours and starts again, it is assumed to be uneconomical and is not practical. So we ignore the possibilities of scenarios Rule 2a and b and only consider Rule 2c during the scheduling horizon except the beginning of

(16)

∀ g ∈ G, t ∈ T

(17)

where qαCa(g, t) is an arbitrary production rate and f αCa(g, t) is its corresponding consumption value. The startup and shutdown operations of arc furnaces should be considered in the model, which is different from the electrolytic baths (to avoid the safety risk caused by frequent start/stop operations, the electrolytic baths cannot be stopped once started). The energy consumption is 0 when the arc furnace is idle, as shown in eq 18. And the production rate pCa(g, t) equals to qαCa(g, t) when the furnace is producing, as shown in eqs 19 and 20; while the energy consumption ECa(g, t) is equal to f αCa(g, t), as shown in eqs 21 and 22. max ECa(g , t ) ≤ XCa(g , t )ECa (g ),

∀ g ∈ G, t ∈ T (18) E


Article

Industrial & Engineering Chemistry Research the scheduling horizon, which simplifies the model formulation to a large extent. The beginning of the scheduling horizon should be considered individually as the initial state of the furnace is given and all of the scenarios of Rules 2a, b, and c may appear. Figure 5 illustrates the start−stop operation of the furnace. Binary variable YCa(g, t) takes the values of one when the start-

Clearly, the furnace’s initial state (i.e., active or idle) impacts its production rate. Taking parameter Ini_onCa(g) to indicate the initial state of the furnace g, Ini_onCa(g) = n denotes furnace g has worked for n hours before the scheduling horizon, while Ini_onCa(g) = 0 denotes the furnace has been idle. Parameter Ini_lenCa(g) denotes the idle time before the scheduling horizon (see Figure 7).

Figure 7. Initial state of the arc furnace. Figure 5. Start−stop operation at time t.

up operation of the furnace g occurs at time t. XCa(g, t) = 0 when YCa(g, t) = 1 because that the production rate is 0 in the first hour after start-up. Equations 23−25 denote the relationship between XCa(g, t) and YCa(g, t) as shown in Figure 5, which are equivalent to the expression “If XCa(g, t + 1) = 1 and XCa(g, t) = 0, then YCa(g, t) = 1”. YCa(g , t ) ≤ XCa(g , t + 1),

∀ t ∈ T,

If the furnace’s initial state is idle, then the production rate satisfies pCa(g, t) = 0 at time t = 1 whether the furnace starts or not. That is, XCa(g , t ) = 0,

(28)

∀g∈G

If Ini_onCa(g) = 0 and the furnace starts at time t = t1, we take an approximation that the idle time equals to parameter Ini_lenCa(g) because the formulation can be simplified and the deviation caused by the approximation is small. The production rate is under constraints from time t1 to t1 + ξ. ξ is the length of time that the furnace needs from the production rate max max λ1pCa (g) increasing to the maximum pCa (g), which is described in detail later. The “if−then” expression of the production rate is

(23)

YCa(g , t ) ≤ 1 − XCa(g , t ),

∀ t ∈ T,

∀g∈G (24)

XCa(g , t + 1) + 1 − XCa(g , t ) − YCa(g , t ) ≤ 1, ∀ t ∈ T,

∀g∈G

(25)

The production rate of calcium carbide should lie between its lower and upper bounds, as in eq 26, min pCa (g )XCa(g ,

∀ t ∈ T,

pmin Ca (g),

t ) ≤ pCa (g , t ) ≤

max pCa (g )XCa(g ,

t1

If ∑ XCa(g , t ) = 0 and YCa(g , t1) = 1

t ),

∀g∈G

∀ g ∈ G, t = 1, Ini_on Ca(g ) = 0

t=1

(26)

max max Then pCa (g , t ′) ≤ λ1pCa (g ) − (t ′ − t1 − 1)λ 2pCa (g ),

pmax Ca (g)

where are the lower and upper bounds of the production rate of arc furnace g. Figure 6 shows the production rate constraints caused by startup and shutdown operations. The production rate in the

∀ t1 ∈ T , t1 < t ′ < t1 + ξ

where λ1 denotes the production rate in the first time period after startup, and λ2 denotes the increasing velocity of the production rate. λ1 and λ2 can be obtained following Rule 2 with the idle time Ini_lenCa(g). The increasing time ξ can be computed by λ1 and λ2. Refer to the MILP modeling structure technique by Magatao,25 the constraints of the production rate after the first startup of the furnace are as follows t1

∑ XCa(g , t ) ≤ U (1 − δ1(g , t1)),

Figure 6. Production rate constraints.

t=1

∀ t1 ∈ T, g ∈ G, Ini_on Ca(g ) = 0

first hour after startup is 0 as assumed. And then, the production rate is raised to 50% pmax Ca (g) rapidly and grows further with a certain velocity (e.g., 5% pmax Ca (g)). In Figure 6, the furnace g shuts down at time t1 and restarts at time t2, so YCa(g, t2) = 1, XCa(g, t2) = 0, XCa(g, t2 + 1) = 1, and pCa(g, t2) max = 0, pCa(g, t2 + 1) = 0.5pmax Ca (g), pCa(g, t2 + 2) = 0.5pCa (g) + max 0.05pCa (g). The constraints on the production rate pCa(g, t) until time t3 (here t3 = t2 + 10), which can be expressed as

t=1

∑ XCa(g , t ) ≥ (L − ε)δ1(g , t1) + ε , t1

∀ t1 ∈ T, g ∈ G, Ini_on Ca(g ) = 0

(30)

δ2(g , t1) ≤ YCa(g , t1), ∀ t1 ∈ T, g ∈ G, Ini_on Ca(g ) = 0

max max pCa (g , t ) − 0.5pCa (g ) − (t − t 2 − 1)0.05pCa (g )

≤ U (1 − YCa(g , t 2)), ∀ t 2 ∈ T , g ϵG, t 2 < t ≤ t 3, t 3 = t 2 + 10

(29)

(31)

δ2(g , t1) ≤ δ1(g , t1), ∀ t1 ∈ T, g ∈ G, Ini_on Ca(g ) = 0

(27) F

(32)


Article

Industrial & Engineering Chemistry Research Table 1. Constraints on Production Rate for Startup and Shutdown Operations Ini_onCa(g) = 0 t=1 t>1

Ini_onCa(g) > 0

pCa(g, t) = 0, whether the furnace starts at time period t or not. The production rate after the first startup operation is under constraints according to Rule 2; else (if another startup occurs), following Rule 2a.

pCa(g,t) ≥ 0 If no shutdown occurs and t ≤ ξ (ξ is decided by Ini_onCa(g) in eq 36), the production rate is still under constraints following Rule 2; if a new startup occurs, follow Rule 2a.

⎧I 0 + p I (g , t )σ dl − uCa(t ), ⎪ Ca ∑ Ca g ∈ G ⎪ ⎪t = 1 ⎪ I ⎪ ICa(t − 1)σ + ∑ pCa (g , t )σ dl − uCa(t ), ⎪ g ∈ G ⎨ ICa(t ) = ⎪ ⎪1 < t ≤ dl ⎪ dl ⎪ ICa(t − 1)σ + ∑ pCa (g , t − dl)σ − uCa(t ), ∈ g G ⎪ ⎪ ⎩ t > dl

YCa(g , t1) + δ1(g , t1) − δ2(g , t1) ≤ 1, ∀ t1 ∈ T, g ∈ G, Ini_on Ca(g ) = 0

(33)

max max pCa (g , t ′) − λ1pCa (g ) − (t ′ − t1 − 1)λ 2pCa (g )

≤ U (1 − δ2(g , t1)), ∀ t1 ∈ T, g ∈ G, Ini_on Ca(g ) = 0, t1 < t ′ < t1 + ξ (34)

l u ICa ≤ ICa(t ) ≤ ICa ,

f ICa(t ) ≥ ICa ,

(39)

∀t∈T

(40)

t=N

(41)

where denotes the initial inventory of calcium carbide, σ is the air exposure loss ratio of unit inventory per unit time, uCa(t) denotes consumption of calcium carbide, IlCa and IuCa are the lower and upper bounds of calcium carbide inventory, and IfCa denotes the final inventory of calcium carbide. 3.2.3. Constraints of C2H2 and HCL Synthesis Process. The chlorine production rate lies between its lower and upper bounds I0Ca

∀ g ∈ G, t = 1, Ini_on Ca(g ) > 0 (35)

pClmin (τ ) ≤ pCl (τ , t ) ≤ pClmax (τ ),

max pCa (g , t ′) − λ1pCa (g ) − (t ′ + Ini_on Ca(g ) − 2)

∀ t ∈ T, τ ∈ Γ (42)

max λ 2pCa (g ) ≤ U (1 − XCa(g , t )),

pmin Cl (τ)

pmax Cl (τ)

where and are the lower and upper bounds of production rate of bath τ. In the synthesis reaction, the mass ratio of HCL, chlorine, and calcium carbide can be determined by the mole ratio of production reactions

∀ g ∈ G, 1 ≤ t ′ ≤ ξ − Ini_on Ca(g ), t = 1, Ini_on Ca(g ) >0

(38)

and pCa(g, t) = pfCa(g, t), t > N − dl. N is the number of time periods in the scheduling horizon, pfCa(g, t) is the expected production rate at time period t in the final dl time periods. The inventory lies between its lower and upper bounds, and it should be larger than the final inventory at the last time period,

where δ1(g, t1) and δ2(g, t1) are binary intermediate variables. δ1(g, t1) = 1 is equivalent to the expression ∑t1t=1 XCa(g, t) = 0, and δ2(g, t1) = 1 is equivalent to the expression ∑t1t=1 XCa(g, t) = 0 and YCa(g, t1) = 1. Detailed illustration can be found in the work of Magatao.25 If the furnace is already in working state (Ini_onCa(g) > 0), YCa(g, t) = 0 at time t = 1. In this case, if working time Ini_onCa(g) is not large enough (that is, not long enough for the production rate to increase to the maximum), the production rate pCa(g, t) is also under constraints after the scheduling horizon begins if it does not shut down. And the upper bound of pCa(g, t) depends on the value of Ini_onCa(g): YCa(g , t ) = 0,

(37)

(36)

The startup and shutdown constraints in different scenarios are summarized in Table 1. As the cooldown time of calcium carbide (dl time periods) exits, to make the production rates of arc furnaces smooth and reasonable, the production rates in dl time periods before the scheduling horizon and in the last dl time periods of the scheduling horizon are given as initial states. The former influences the inventory at the first dl time periods of the beginning of scheduling horizon; the latter is an expectation of future furnace states. pICa(g, t) is the production rates of dl time periods before time t and makes a term of the inventory at 1 ≤ t ≤ dl time period, as shown in eqs 37 and 38. The inventory of calcium carbide at time period t equals the inventory at time period t − 1 adding the difference between the calcium carbide production and its consumption during this time period. And the constraints of inventory are as follows,

pH (t ) = α ∑ pCl (τ , t ),

∀t∈T

τϵΓ

(43)

where coefficient α is the mass ratio of chlorine and HCL, and pH(t) denotes the production rate of HCL. The mass relation between C2H2 and HCL satisfies pC (t ) = ηpH (t ),

∀t∈T

(44)

where coefficient η is the mass ratio of HCL and C2H2, and pC(t) denotes the production rate of C2H2. uCa(t ) = βpC (t ),

∀t∈T

(45)

C2H2 is obtained from the reaction of calcium carbide with water. In eq 45, coefficient β denotes the mass ratio of C2H2 product and consumption of calcium carbide uCa(t). The energy cost of C2H2 and HCL synthesis reaction is not considered in the scheduling because it is relatively small and has little room for optimization. VCM is synthesized by C2H2 G


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produced in a pot at time period t, and eq 56 constrains that no more than one product is produced at any polymerization pot j during the time period t:

and HCL, and the production rate of VCM is described as follows: pV (t ) = γpH (t ),

∀t∈T

(46)

∑ Y (j , k , t ) ≤ 1,

where γ is the reaction ratio of HCL and VCM, pV(t) is the production rate of VCM. The production rate of VCM lies between its lower and upper bounds pVmin ≤ pV (t ) ≤ pVmax ,

pmin V

∀t∈T

(47)

t + cy − 1

pmax V

∑ ∑ Y (j , k , t )V (j),

Y (j , k , t ) +

Y (j , k′, t ′) ≤ 1,

∀ j ∈ J, t ∈ T1

(57)

and the polymerization cycle cy > 1. 3.3.2. Changeover Constraints. Binary variable ch(j, k, t) takes the value one when there is a changeover from product k to other product tasks after time period t − 1 in polymerization pot j. That is, product k is produced in polymerization pot j at time period t − 1 and a different production k′ is produced at or after time period t in the same polymerization pot. A binary variable X(j, k, t) is introduced and X(j, k, t) takes the value one if product k is the first production task in pot j for the time periods t, t + 1,..., N. That is to say, there is t′ ∈ {t, t + 1, ..., N}, satisfies that X(j, k, t′) = 1 and all time periods t, t + 1, ..., t′ − 1 are empty for pot j. The constraints are as follows:

(48)

where T1 denotes the set of time periods for the batch process, V(j) denotes the charging capacity of polymerization pot j; and Y(j, k, t) denotes that the polymerization reaction in pot j begins at time t to product PVC of grade k, that is, polymerization pot j charges at time t for product k. VCM is converted to different grades of PVC production in polymerization pots under different reaction conditions including temperature and catalyst. The conversion rate of VCM differs with the grade of PVC. The unreacted VCM is recycled. The quantity of recycled VCM at time t is Rec(t ) =

∑ ∑ k ′∈ K t ′= t + 1

∀ t ∈ T1

k∈K j∈J

(56)

Production of k′ cannot be arranged in pot j before the former one (production of k) is completed, as described in eq 57

where and denote the lower and upper bounds of VCM production rate. VCM is the material of polymerization. The usage of VCM at time t is u V (t ) =

∀ j ∈ J, t ∈ T1

k∈K

X(j , k , t ) ≥ Y (j , k , t ),

∑ ∑ (1 − ρ(k))Y (j , k , t − cy + 1)V (j),

∀ j ∈ J, k ∈ K, t ∈ T1 (58)

k∈K j∈J

∀ t − cy + 1 ∈ T1

∑ X(j , k , t ) ≤ 1,

(49)

where ρ(k) denotes the conversion rate of product k, cy is the polymerization cycle. Y(j, k, t − cy + 1) = 1 denotes that the polymerization reaction ends in cy periods after its beginning, and the pot discharges. ⎧ I 0 + p (t ) − ∑ ∑ Y (j , k , t )V (j), V ⎪ V j∈J k∈K ⎪ ⎪t = 1 ⎪ ⎪ pV (t ) − ∑ ∑ Y (j , k , t )V (j) + IV(t − 1), ⎪ j∈J k∈K ⎪ IV(t ) = ⎨ t > 1, t ∈ T1 ⎪ ⎪ p (t ) + Rec(t ) + IV(t − 1), ⎪ V ⎪ t > 1, t − cy + 1 ∈ T1 ⎪ ⎪ IV(t ) = pV (t ) + IV(t − 1), ⎪ ⎩ otherwise

X (j , k , t ) +

IV(t ) ≥

IlV

IVf ,

∀t∈T t=N

(59)

∑ Y (j , k′, t ) ≤ 1, k ′≠ k

∀ j ∈ J, k ∈ K, t ∈ T1

(60)

(50)

X(j , k , t ) ≥ X(j , k , t + cy) −

∑ Y (j , k′, t ), k ′≠ k

(51)

∀ j ∈ J, k ∈ K, t ∈ T1

(61)

If task k is followed by a different task k′ at or after time period t + cy then ch(j, k, t) must be activated at t + cy:

(52)

ch(j , k , t + cy) ≥

∑ X(j , k′, t + cy) + Y (j , k , t ) − 1, k ′≠ k

(53)

∀ j ∈ J, k ∈ K, t ∈ T1

eqs 50−53 denote that the inventory of VCM IV(t) is determined by the initial inventory I0V, the charge of the polymerization pots, and the recycled quantity at different time periods. The inventory lies between its lower and upper bounds IVl ≤ IV(t ) ≤ IVu ,

∀ j ∈ J, t ∈ T1

k∈K

(62)

In Figure 8 ⎧1, t = 19 , ch(j , k1, t ) = ⎨ ⎩ 0, otherwise

(54)

⎧1, t = 43 ch(j , k 2, t ) = ⎨ ⎩ 0, otherwise

(55)

IuV

where and are the lower and upper bounds of calcium carbide inventory respectively, and IVf denotes the final inventory of VCM. 3.3. Constraints of Batch (Polymerization) Process. 3.3.1. Allocation Constraints. Only one product can be

and ⎧1, t = 79 ch(j , k 3, t ) = ⎨ . ⎩ 0, otherwise H


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Industrial & Engineering Chemistry Research ⎧ ∑ E (τ , t ) + ∑ E (g , t ) Cl Ca ⎪ g∈G ⎪ τ ∈Γ ⎪ + ∑ ∑ Y (j , k , t )E (j , k) P ⎪ j∈J k∈K ⎪ ⎪ + hp(1 − ∑ X (g , t )) Ca ⎪ g∈G ⎪ ⎪ ele(t ) = ⎨ + stcost ∑ YCa(g , t ), t ∈ T1 ⎪ g∈G ⎪ ⎪ ∑ E (τ , t ) + ∑ E (g , t ) + hp(1 Ca ⎪ τ ∈Γ Cl g∈G ⎪ ⎪ − ∑ X (g , t )) + stcost ∑ Y (g , t ), Ca Ca ⎪ g∈G g∈G ⎪ ⎪ t∉T ⎩ 1

Figure 8. Changeovers of polymerization pot j.

The inventory of product k at time t is the inventory at time t − 1 plus discharge quantity of k at time t, and then subtracts the delivery quantity at time t ⎧ I (k , t − 1) + ∑ ρ(k)Y (j , k , t − cy + 1) ⎪P j∈J ⎪ ⎪ V (j) − tr(k , t ), ⎪ IP(k , t ) = ⎨ ⎪ ∀ k ∈ K, t − cy + 1 ∈ T1 ⎪ ⎪ IP(k , t − 1) − tr(k , t ), ⎪ ∀ k ∈ K, t − cy + 1 ∉ T ⎩ 1

(63)

(64)

⎧ d(i , k), t = dt(i) , here variable d(i, k) equals to tr(k , t ) = ⎨ else ⎩ 0, ⎪

⎪

ele(t ) ≤ e ,

tr(k, t) at due time dt(i) of order i. The inventory should be less than its upper bound, (65)

where IuP is the upper bound of PVC inventory. Clearly, product k’s delivery quantity for order i should be no larger than its demand R(i, k), as shown in eq 66 d(i , k) ≤ R(i , k),

∀ i ∈ I, k ∈ K

(66)

where R(i, k) denotes demand of product k in order i. dt(i)

d (i , k ) ≤

∑ ∑ ρ(k)Y (j , k , t − cy + 1)V (j)−

C1 =

t=1 j∈J

(70)

∑ ∑ ∑ ch(j , k , t )co (71)

t∈T j∈J k∈K

i−1

∑ d(i′, k),

∀t∈T

where e is the upper bound of electricity supply. 3.4. Objective. The objective is to minimize the total cost, which includes (1) PVC grades changeover cost for VCM polymerization pots, denoted by C1 in eq 71, where co is the cost of a changeover operation; (2) the final product delivery delay punishment C2 if the product is not delivered on schedule, where δ(k) is the penalization cost of product k; (3) the electricity cost for furnaces and baths as C3 in eq 73, where parameter μ is the price of unit electricity; (4) the cost of raw material of calcium carbide and chlorine production, expressed as C4 in eq 74, where ωCa and ωCl are the unit cost of raw material of calcium carbide and chlorine respectively; (5) the storage cost of VCM and PVC product, represented as C5 in eq 75, where ωV and ωP are the unit storage cost of VCM and PVC.

∀t∈T

k∈K

(69)

The first term of eq 68 denotes electricity consumption of electrolytic baths for chlorine, and the second one, for arc furnaces. The third one is for polymerization, and EP(j, k) is a parameter representing the electricity cost of polymerization in pot j for product k. The fourth one is the energy of heat preservation of the electrode when the arc furnace is idle, and the heating power is hp. And the last one is the total cost for energy consumption caused by arc furnaces’ startup operations, where stcost is the cost of a startup operation. The electricity consumption cannot exceed the limit

In eqs 63 and 64, tr(k, t) is delivery of product k at time t. And

∑ IP(k , t ) ≤ IPu ,

(68)

∀ i ∈ I, k ∈ K, t − cy + 1 ∈ T1

C2 =

i ′= 1

∑ ∑ δ(k)(R(i , k) − d(i , k)) (72)

i∈I k∈K

(67)

C3 =

Equation 67 means the delivery should be less than the

∑ μele(t ) (73)

t∈T

inventory. The second term of the right side is the sum of all

C4 = ωCa ∑

the orders before order i, and the right side of the equation

∑ pCa (g , t ) + ωCl ∑ ∑ pCl (τ , t )

t∈T g∈G

t ∈ T τ ∈Γ

(74)

denotes the inventory at time t, which equals to the discharges C5 = ω V ∑ IV(t ) + ωP ∑ ∑ IP(k , t )

subtracted by former deliveries. As mentioned before, only the three most intensive

t ϵT

tε T kε K

C = C1 + C2 + C3 + C4 + C5

electricity-consuming processes, chlorine production, calcium

(75) (76)

A sum of these five items, represented in eq 76, forms the objective function of our considered PVC scheduling problem. The scheduling model is summarized as follows min C s.t. eqs 6−76.

carbide production, and VCM polymerization, are taken into consideration of the electricity consumption, as expressed in eqs 68 and 69 I


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4. CASE STUDY The model is tested on three cases. All cases are computed by GAMS win64 24.2.2 on a Dell PC with Intel Xeon Quad CPU processor, 2.50 GHz, Windows Server 2008 R2 OS and 32.0 GB RAM. The MILP models are solved using CPLEX 12.6.0.0. The flowchart is same as that in Figure 2. The product demands of five different PVC grades for the cases are shown in Table 2. The scheduling horizons are 72 to 120 h (horizon of

Case 1 is explained for a better understanding of the model in section 4.1. And comparative results of all the cases are analyzed in section 4.2 to demonstrate the necessity of plantwide scheduling. 4.1. Results Explanation of Case 1. Explanation of case 1 is given in this section. The solution is optimal and the total cost is 4 126 638, which is 2.0% lower than that of optimizing batch process only. The corresponding results are shown in Table 4 in detail. The production rate of calcium carbide is shown in Figure 9a, which is the composite production rate of the four furnaces. As the produced calcium carbide must undergo a 24 h cooling process, the production of calcium carbide is 24 h in advance of its use. In the first time period, the first arc furnace restarts, and its production rate at first time period is 0 as its initial state is nonworking, i.e. Ini_on(1) = 0. All the furnaces are in working state from the second to 23rd time period, and moreover, the first and second furnaces undergo production rates increasing after startup and reach their optimal work points at the third time period (see the growing stair in Figure 9a). From the third time period to the 23rd time period, all the furnaces work at their optimal points (6.3 t/h for the first furnace, 8.6 t/h for the second, and 10.1 t/h for the third and fourth) and obtain the maximum efficiency. At the 24th time period, the first furnace shuts down, as seen in the first descending stair in Figure 9a. The second to the fourth furnaces are still in working state from the 24th time period to the 48th time period, because that their unit energy consumptions are smaller than that of the first furnace. For most time periods, the furnaces work at the optimal work points (i.e., the optimal production rates). At the 49th time period, the second furnace shuts down and the second descending stair appears in Figure 9a. The third and fourth furnaces continue working in the whole scheduling horizon (72 time periods). The calcium carbide production rates of the last 24 time periods are determined by expectation of the next scheduling horizon based on the planning forecast to guarantee the operating stability. The inventory of calcium carbide is shown in Figure 9b, which is the difference between the well-cooled calcium carbide production and the consumption (those consumed in the hydrolysis reaction to produce C2H2). Note that the calcium carbide consumption in the first 24 h is provided by the production of the 24 h before the scheduling horizon, which are given as initial conditions. The lower bound of calcium carbide inventory is 50 ton (the dash dotted red line), and the final inventory is 60 ton, which are satisfied by the resulted schedule, as shown in Figure 9b. The production rate of chlorine is shown in Figure 10a, where the dashed−dotted red line is the lower bound of the production rate. The detailed inventory curve of VCM is depicted in Figure 10b. From the VCM inventory curve in Figure 10b, we can see that the inventory at the last period is 270 ton to satisfy the final inventory demand. There are several sudden decreases of VCM inventory in Figure 10b due to the consumption from PVC production. The VCM is charged to polymerization pots in a 6 h interval, as shown in Figure 11. And the detailed schedule for PVC polymerization pots is shown as Gantt chart in Figure 12. 4.2. Comparative Results. To make adequate comparisons, the results of scheduling model considering only batch process (called the “batch process-only model”) are provided here to illustrate the necessity of the plantwide scheduling. For the batch process-only model, we consider two different scenarios to realize the upstream VCM production process: (1)

Table 2. Demand Profiles for All Cases case

order

product grades

demand/t

due time/h

1

1 2 3 4 1 2 3 4 1 2 3 4 5

k1 k2 k3 k4 k4 k1 k2 k3 k2 k4 k5 k1 k3

450 350 650 700 480 800 700 1100 400 900 400 1300 500

24 48 72 72 24 48 72 96 24 48 72 96 120

2

3

72 h is considered for case 1, 96 h for case 2, and 120 h for case 3), which are enough for PVC production process scheduling in actual plants. The parameters used in the cases, such as production rate limits of calcium carbide, chlorine, and VCM production, capacities of polymerization pots, and conversion rates of all grades of PVC in polymerization process, which are originated from the actual production, are shown in Table S1 in the Supporting Information. Four parallel arc furnaces, 14 electrolytic baths, and 4 polymerization pots are involved in all the cases. As the produced calcium carbide must undergo a 24 h cooling process, the production rates of the 24 h before the scheduling horizon should be given, which influence the inventory of calcium carbide in the first 24 h of the scheduling horizon. Then the initial states of the arc furnaces are presented. And the prices of the electricity, materials, and inventories of VCM and PVC are given. Other parameters include: the lower and upper bounds of calcium carbide, VCM and PVC inventory; the heating power of electrode when it is idle; the air exposure loss ratio of calcium carbide, and so on. The expected production rates of arc furnaces are also given. The model statistics and solution times of all the cases are shown in Table 3. The scale of scheduling horizons leads to different sizes of the models. The optimality tolerance is set to 1%. The solution times are long because of large model scales, and an efficient optimization algorithm is under further research. Table 3. Model Statistics

case

number of equations

continuous variables number

binary variables number

1 2 3

24374 32572 41426

13142 18085 22859

11232 14976 19440

nonzeros

CPU time (s)

GAP (%)

104707 142163 185749

3345 6998 9780

1 1 1 J


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Industrial & Engineering Chemistry Research Table 4. Comparisons between Plantwide Model and Batch Process-Only Model for Case 1 Batch Process-Only Model scenario 1

scenario 2

difference

difference

items

plantwide model

value (¥)

absolute value (¥)

relative value (%)

value (¥)

absolute value (¥)

relative value (%)

energy cost for furnaces energy cost for baths total cost

1507755 714306 4126638

1,544,944 730,277 4,209,483

37,189 15,971 82,845

2.5 2.2 2.0

1,577,047 732,159 4,242,340

69,292 17,853 115,702

4.6 2.5 2.8

Figure 9. Operation curve of calcium carbide: (a) production rate and (b) inventory for case 1.

Figure 10. Detailed operation curve of (a) production rate of chlorine and (b) VCM inventory for case 1.

total average cost (average of the three cases) of scheduling using the plantwide model is 1.3% less than that of scheduling using the batch process-only model in scenario 1, and 1.7% less than that of scheduling using batch process-only model in scenario 2; the average energy cost of arc furnaces of scheduling using the plantwide model is 1.7% less than that of scheduling using the batch process-only model in scenario 1, and 2.7% less than that of scheduling using batch process-only model in scenario 2; the average energy cost of electrolytic baths of scheduling using the plantwide model is 1.6% less than that of scheduling using batch process-only model in scenario 1 and 1.7% than that of scheduling using batch process-only model in scenario 2. From Table 4 we can see that the energy cost of the arc furnaces of scheduling using the proposed model is ¥69,292 lower than that of scheduling using batch process-only model in scenario 2, because the furnaces work at the best efficiency production rates at most time when scheduling using the proposed model (see Figure 9a and related analysis in section 4.1), while the constant production rate of scheduling using batch process-only model in scenario 2 constrains its efficiency. Moreover, the constant production rate of VCM in scenario 2 may cause the superfluous VCM production, which results in a larger cost. The total cost of scheduling using the proposed model is ¥115,702 lower than that of scheduling using batch process-only model in scenario 2, including the cost of superfluous VCM, storage and assessorial electricity. The

Figure 11. Total charge amount of polymerization pots for case 1.

the VCM supply changes along with the demand determined by batch process scheduling, which is noted as “scenario 1”; (2) VCM is produced in a constant production rate to meet the demand, which is noted as “scenario 2”. Tables 4−6 show the comparisons between the proposed plantwide model and the batch process-only model. The demands in orders of all the cases are satisfied. From the listed data in the tables, it is clear that the proposed plantwide scheduling method outperforms the batch process-only strategy for each case (shown in the column “difference” of the tables, and both of absolute values and relative values are given). The K


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Figure 12. Gantt chart of polymerization pots for case 1.

Table 5. Comparisons between Plantwide Model and Batch Process-Only Model for Case 2 Batch Process-Only Model scenario 1

scenario 2

difference

difference

items

plantwide model

value (¥)

absolute value (¥)

relative value (%)

value (¥)

absolute value (¥)

relative value (%)


2249487 1030233 6087843

2,280,330 1,045,582 6,154,104

30,843 15,349 66,261

1.4 1.5 1.1

2,280,817 1,045,685 6,154,541

31,330 15,452 66,698

1.4 1.5 1.1

Table 6. Comparisons between Plantwide Model and Batch Process-Only Model for Case 3 Batch Process-Only Model scenario 1

scenario 2

difference

difference

items

plantwide model

value (¥)

absolute value (¥)

relative value (%)

value (¥)

absolute value (¥)

relative value (%)


2568263 1157763 6950297

2,596,781 1,169,917 7,001,322

28,518 12,154 51,025

1.1 1.0 0.73

2,618,539 1,169,671 7,030,923

50,276 11,908 80,626

2.0 1.0 1.2

scheduling model. As the efficiency of the facilities and the material dispatching of plantwide process are fully considered, the comparisons between the plantwide scheduling and the traditional batch process scheduling show that about 1.3% total cost and above 1.7% energy cost can be saved by using the proposed model.

results of the proposed plant-wide scheduling model outperform that of batch process-only scheduling model, because the upstream continuous process is not integrated in the scheduling optimization for batch process-only scheduling model and the resulted schedule does not match the downstream batch process and the upstream continuous process well. In other words, due to the proposed plantwide strategy considers both the downstream batch process and the energy-consuming upstream continuous process, globally optimal and economical schedules are resulted. Moreover, the comparisons further prove the necessity of plantwide scheduling optimization.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b00291. Parameters used in the cases, such as production rate limits of calcium carbide, chlorine, and VCM production, capacities of polymerization pots, and conversion rates of all grades of PVC in polymerization process, which originated from the actual production, are shown in Table S1 (PDF)

5. CONCLUSIONS Due the upstream VCM production process is the most energyconsuming part for PVC production by calcium carbide method, the traditional scheduling strategy merely focusing on the batch polymerization process is not suitable. In this paper, the discrete time representation based plantwide scheduling model combining the VCM production process and the polymerization process is proposed for PVC production by calcium carbide method. Due to the distinct time characteristics between the continuous process and the batch process, a hybrid discrete time interval strategy is proposed to reduce the model scale. To approximate the nonlinear energy consumption model involved in both furnaces and bathes and maintain the linearity of the integral model well, the piecewise linear model is adopted. Comparative studies using several cases originated from a real industrial plant and verify the effectiveness and necessity of the proposed plantwide

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 10 62784964. E-mail address: huangdx@tsinghua. edu.cn (D.H.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was supported by the National High-tech 863 Program of China (No. 2013AA 040702), the National Natural L


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L = a small negative number N = number of time periods in scheduling horizon nCa(g) = number of partitioning points in piecewise linearization of calcium carbide energy consumption representation of arc furnace g nCl(τ) = number of partitioning points in piecewise linearization of chlorine energy consumption representation of electrolytic bath τ max pmin Ca (g), pCa (g) = upper and lower bounds of the production rate of arc furnace g max pmin Cl (τ), pCl (τ) = upper and lower bounds of the production rate of electrolytic bath τ pICa(g, t) = production rate of furnace at dl time periods before t pfCa(g, t) = production rate of furnace at the last dl time periods of the scheduling horizon max pmin V , pV = upper and lower bounds of VCM production rate qCa(lCa(g), g) = partitioning point of number lCa(g) in piecewise linearization of calcium carbide energy consumption representation of arc furnace g qCl(lCl(τ), τ) = partitioning point of number lCl(τ) in piecewise linearization of chorine energy consumption representation of electrolytic bath τ R(i, k) = demand of order i for product k stcost = energy cost of a startup operation U = large positive number V(j) = charging capacity of polymerization pot j α = mass ratio of chlorine and HCL in theory β = mass ratio of C2H2 and consumption of calcium carbide in theory γ = mass ratio of HCL and VCM in theory η = mass ratio of HCL and C2H2 in theory ρ(k) = conversion rate of product k λ1 = production rate after restart λ2 = growing velocity of production rate σ = weathering loss ratio of per unit inventory at per unit time μ = price of electricity ξ = increasing time to the maximum production rate ωCa = raw material cost of unit calcium carbide ωCl = raw material cost of unit chlorine ωP = storage cost of unit PVC per unit time ωV = storage cost of unit VCM per unit time

Science Foundation of China (No. 61273039), and the National Science Fund for Distinguished Young Scholars of China (No. 61525304).

■ ■

ABBREVIATIONS PVC = polyvinyl chloride VCM = vinyl chloride monomer

NOMENCLATURE g = arc furnace i, i′ = demand order j = polymerization pot k, k′ = PVC production grade lCa (g) = sequence number of partitioning point in piecewise linearization of calcium carbide energy consumption representation of arc furnace g lCl(τ) = sequence number of partitioning point in piecewise linearization of chlorine energy consumption representation of electrolytic bath τ t, t′, t1, t2, t3 = time period τ = electrolytic bath

Sets

G = polymerization pots I = orders J = arc furnaces K = PVC grades T = time periods T1 = time periods of batch process Γ = electrolytic baths Subscripts

C = C2H2 Ca = calcium carbide H = HCL P = PVC V = VCM Parameters

cy = polymerization cycle dl = length of cool-down time of calcium carbide dt(i) = due time of order i e = limit of electricity supply in each time period Emax Ca (g) = upper bound of the energy consumption of arc furnace g EP(j, k) = electricity cost of polymerization in pot j for product k f Ca(qCa(lCa(g), g) = energy consumption at the production rate qCa(lCa(g), g) f Cl(qCl(lCl(τ), τ)) = energy consumption at the production rate qCl(lCl(τ), τ) hp = cost of heating power when the furnace is idle I0Ca = initial inventory of calcium carbide IfCa = final inventory of calcium carbide IlCa, IuCa = lower and upper bounds of calcium carbide inventory IuP = upper bound of inventory of PVC I0V = initial inventory of VCM IfV = final inventory of VCM IlV, IuV = lower and upper bounds of VCM inventory Ini_lenCa(g) = Ini_lenCa(g) = n denotes that the furnace has been idle for n hours before the scheduling horizon begins Ini_onCa(g) = Ini_onCa(g) = n denotes that the furnace has been working for n hours before the scheduling horizon begins

Continuous Variables

d(i, k) = delivery of production k for order i ECa(g, t) = energy consumption of furnace g at time t ECl(τ, t) = energy consumption of bath τ at time t f αCa(g, t) = energy consumption at the production rate qαCa(g, t) ele(t) = electricity consumption at time t ICa(t) = inventory of calcium carbide at time t IP(k, t) = inventory of product k at time t IV(t) = inventory of VCM pC(t) = production rate of C2H2 pCa(g, t) = production rate of calcium carbide of arc furnace g at time t pCl(τ, t) = production rate of chlorine of electrolytic bath τ at time t pH(t) = production rate of HCL pV(t) = production rate of VCM qαCa(g, t) = a positive production rate of arc furnace g Rec(t) = the quantity of recycled VCM at time t tr(k, t) = delivery of product k at time t M


Article


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uCa(t) = consumption of calcium carbide at time t uV(t) = consumption of VCM at time t αCl(lCl(τ), τ, t) = continuous variable in piecewise linearization of chlorine energy consumption representation of electrolytic bath τ αCa(lCa(g), g, t) = continuous variable in piecewise linearization of calcium carbide energy consumption representation of arc furnace g Binary Variables

ch(j, k, t) = ch(j, k, t) = 1 denotes a changeover from production k to a different product at/after time period t in pot j hCl(lCl(τ), τ, t) = binary variable in piecewise linearization of chlorine energy consumption representation of electrolytic bath τ hCa(lCa(g), g, t) = binary variable in piecewise linearization of calcium carbide energy consumption representation of arc furnace g X(j, k, t) = X(j, k, t) = 1 if production k is the first production task in pot j in the time period t, t + 1, ..., N XCa(g, t) = XCa(g, t) = 1 if furnace g is producing at time t Y(j, k, t) = Y(j, k, t) = 1 denotes that a polymerization process for product k begins at time period t in polymerization pot j YCa(g, t) = YCa(g, t) = 1 denotes that start-up of furnace g occurs at time period t δ1(g, t), δ2(g, t) = intermediate variables

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Plantwide Scheduling Model for the Typical Polyvinyl chloride

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