Plant planning optimization under time-varying uncertainty: Case

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Process Systems Engineering

Plant planning optimization under time-varying uncertainty: Case study on a polyvinyl chloride plant Xiaoyong Gao, Yuhong Wang, Zhenhui Feng, Dexian Huang, and Tao Chen Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02101 • Publication Date (Web): 17 Aug 2018 Downloaded from http://pubs.acs.org on August 19, 2018

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Industrial & Engineering Chemistry Research

Plant planning optimization under time-varying uncertainty: Case study on a polyvinyl chloride plant Xiaoyong Gao1, Yuhong Wang2*, Zhenhui Feng2, Dexian Huang3*, Tao Chen4 1. Institute for Ocean Engineering, China University of Petroleum, Beijing 102249, China 2. College of Information and Control Engineering, China University of Petroleum, Qingdao 266580, China 3. Department of Automation and Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China 4. Department of Process and Chemical Engineering, University of Surrey, Guildford GU2 7XH, United Kingdom

ABSTRACT：Planning optimization considering various uncertainties has attracted increasing attentions in the process industry. In the existing studies, the uncertainty is often described with a time-invariant distribution function during the entire planning horizon, which is a questionable assumption. Particularly, for long-term planning problems, the uncertainty tends to vary with time and it usually increases when a model is used to predict the parameter (e.g. price) far into the future. In this paper, time-varying uncertainties are considered in robust planning problems with a focus on a polyvinyl chloride (PVC) production planning problem. Using the stochastic programming techniques, a stochastic model is formulated, and then transformed into a multi-period mixed-integer linear programming (MILP) model by chance constrained programming and piecewise linear approximation. The proposed approach is demonstrated on industrial-scale cases originated from a real-world PVC plant. The comparisons show that the model considering varying-uncertainty is superior in terms of robustness under uncertainties. ACS Paragon Plus Environment

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KEYWORDS: planning optimization, time-varying uncertainty, PVC, chance constrained programming, MILP. 1.

Introduction In the process industry, deterministic optimization approaches have been widely

used in real-time optimization, scheduling and planning problems. However, for medium- and long-term planning problems, a challenge for decision maker is to deal with the unavoidable uncertainty in the prediction of some key parameters, such as product demand, raw material price among others. In reality, these parameters are intrinsically stochastic variables under a volatile market environment. Attractive approach to address this issue and to improve the robustness of the optimized plan is to consider such uncertainties when formulating the optimization problem. Robust optimization has been well reported in the literature of process systems engineering; a selection of examples are given here to put the present study in context. Delaurentis and Mavris defined the uncertainty as the incompleteness in knowledge that makes model-based predictions differ from reality in a manner described by some distribution function.1 In the area of process design and operations, uncertainty is generally

divided

into

four

classification,

i.e.

model-inherent

uncertainty,

process-inherent uncertainty, external uncertainty, and discrete uncertainty.2 With respect to the problems of planning and scheduling under uncertainty, there are many research reports and commercial solvers. Mula.et al. reviewed the models for production planning, and defined the classification scheme for models for production planning under uncertainty.3 A review about planning and scheduling under

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uncertainty was proposed by Verderame, which is the first work attempting to provide a comprehensive description of the advances and future directions for planning and scheduling under uncertainty within a variety of sectors.4 Petkov and Maranas considered the multiperiod planning and scheduling of multiproduct batch plants under demand uncertainty.5 Balasubramanian and Grossmann considered the problem of scheduling under demand uncertainty for a multiproduct batch plant,6 and those uncertainties were subject to the standard normal distribution. The problem of operational planning of large-scale industrial batch plants under demand due date and amount uncertainty was tackled by Verderame and Floudas using the different ways which were robust optimization7 and conditional value-at-risk framework8. Moreover, in the field of oil refining and supply chain management, there are many valuable results in planning and scheduling under uncertainty.9~12 In these published literatures, uncertainty is generally described as a specific time-invariant distribution, i.e. the same uniform distribution or normal distribution in the whole planning time. In recent years, data-driven robust optimization has been proposed for decision making without distribution model. 13~16 To the best of our knowledge, explicit consideration of the time-varying uncertainty with respect to a planning horizon has not been explored in the literature. The term “time-varying” refers to the fact that the uncertainty level varies with each time point in a planning horizon. Moreover, in most real-world cases, we cannot ignore the time-varying characteristics of various uncertainties, such as product price, demand, and so on, in a long-term planning horizon. It is natural that



the prediction performance usually degrades as quickly as time horizon increases in a volatile market. In this paper, the time-varying uncertainty is considered and a robust planning optimization problem for PVC process is addressed, where the weekly demands and material prices uncertainties are considered. To describe their time-varying uncertainties, time-varying prediction variances are introduced. Based on the deterministic model17,

18

, we present a stochastic planning model by taking

uncertainties into consideration, and then transform the stochastic model into a deterministic equivalent model using chance constrained programming method19-22. The robust plan can be easily obtained by solving the deterministic equivalent model. 2.

Problem Statement The PVC plant and its planning problem are described in this section. A typical

integrated PVC production process14 is shown in Figure 1 including two main systems: the utility system and the material processing system. For utility system, the electricity supply comes from combined heat and power (CHP) units and the state grid. For material processing system, there are two sequential parts: the continuous vinyl chloride monomer (VCM) processes and the batch polymerization process. For the continuous VCM processes, the chlorine and hydrogen produced in electrolytic cells burn together to produce hydrogen chloride (HCl); the calcium carbide produced in calcium carbide furnaces will react with water to produce acetylene (C2H2); the HCl and C2H2 were synthesized to produce the vinyl chloride monomer. Finally, in the batch


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polymerization process, different grades of PVC are produced depending on specified operating conditions.


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Figure 1. Overview of the PVC production process.14


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In our previous work14, the deterministic planning problem with known and fixed weekly demand, material prices etc. has been addressed. However, in reality these parameters over the planning horizon are difficult to precisely determine. With probabilistic models, it is possible to predict the probability distribution of these parameters; this is also desirable as the distribution can be accounted for in planning optimization to improve the robustness of the obtained plan. Moreover, the parameter uncertainty tends to increase along time. With the stochastic nature of the planning problem, existing deterministic approaches cannot respond to the volatile market. This motivated the development of stochastic planning approach under time-varying uncertainty in this paper. Specific to the PVC product process outlined above, the goal of this study is to address the multiperiod planning problem under time-varying uncertainty in the price of raw material and the demand of products (the major variabilities identified through preliminary sensitivity studies, not reported here). Taking demands prediction uncertainty as example for illustration, as shown in Figure 2(a), the comparison between actual and predicted demand of a kind of PVC in three months. The data is collected from a real-world plant, where an empirical time series model is used for market demand prediction. After that, the absolute difference between true and predicted demands is given in Figure 2(b). Clearly, the absolute difference for prediction increases along with time. In other words, the prediction uncertainty increases along with time. Specific models/data for time-varying uncertainty description will be given in the following result and discussion section.



(a)

real-world demands

ton

Predicted demands

1200 1000 800 600 400 200 0

(b)

ton

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absolute difference

300 250 200 150 100 50

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

0

weeks

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

weeks

Figure 2. (a) Data visualization, and (b) absolute difference of real and predicted demand.

In this paper, the time-varying uncertainty for weekly demands and materials price is considered. To describe their time-varying uncertainties, the uncertainty is supposed to follow a particular normal distribution in each time period with increasing variances along with time. In reality, the standard deviation at each prediction time period is calculated based on the collected historical prediction errors data at the corresponding time period. Using the stochastic programming techniques and chance constrained programming, we establish a stochastic plan optimization model, and practice production optimization under different confidence levels. The mathematical model is detailed in the next section.

3.

Mathematical Model In our previous study, the PVC planning problem was formulated as a multiperiod

mixed-integer nonlinear programming (MINLP) model13, and then simplified to MILP using piecewise linear approximation14. The present paper introduce stochastic programming techniques and chance constrained programming method to consider the time-varying uncertainty in key parameters; the stochastics model is further converted to a deterministic equivalent model ready for solving by an optimizer. The whole


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procedure is depicted in Figure 3. For the sake of readability, the original deterministic model is firstly given, and then the stochastic approach is detailed.

Figure 3. The formulation procedure of uncertain model

3.1. Deterministic Model Considering that a set of PVC products demands in several future weeks, the task for the multiperiod planning problem is to determine the units’ working state and working load to satisfy the products demands in a cost-saving way. 3.1.1. Polymerization Process Model In the batch polymerization process, the production amount ,, for grade

PVC in polymerization reactor during w week can be computed as in eq. 1. Because of the process is operated in a batch-wise manner, cy is discharge period, and each polymerization reactor’s feeding quantity and conversion ratio , are constant.



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Then, it is linear for eq1. Clearly, the number of feeding batches ,, should be restricted in a reasonable range shown in eq2.

,, = ,, ∗ ∗ , ∀ ∈ , ∈ , ∈ 0 ≤ ,, ≤ ∈

168 ∀ ∈ , ∈

(1) (2)

There is a problem about production delivery and storage in this process. The inventory balance needs to be considered in real process, as shown in eq3. Then, the inventory constraint is formulated in eq4. The week inventory , is related to the

( − 1) week inventory ,#$ , this week delivery , and polymerization

production amount ,, . Moreover, stockout state need to be avoided, as shown in eq5.

, = ,#$ + ,, − , ∀ ∈ , ∈

(3)

&' ≤ , ≤ &() ∀ ∈ , ∈

(4)

∈

S, ≥ ,, ∀ ∈ , ∈

(5)

where, ,, represents the market demand of PVC during week. 3.1.2. VCM Process Model For the batch polymerization process, the corresponding consumption of VCM can be computed in eq6. Because of the strict chemical principle balance, the relationship between VCM, HCl and C2H2 is given in eqs7~8.

-./&, = ,, ∗ ∀ ∈

(6)

./&, = -012, + -1303 , ∀ ∈

(7)

∈ ∈


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-012, -1303 , = ∀ ∈ 36.5 26

(8)

./&, = ./&,#$ + ./&, − -./&, ∀ ∈

(9)

Then, the inventory balance formulas are shown in eqs9~10.

&' &() ./& ≤ ./&, ≤ ./& ∀ ∈

012, = 8 ∗ -123, ∀ ∈

1303, = 9 ∗ -1(13 , ∀ ∈ 13 03 , = -1303 , ∀ ∈

(10) (11) (12) (13)

Eqs11~12 are based on the strict chemical balance principle, where -123 , is the

consumption of Cl2 in the week , 8 is the coefficient between Cl2 and HCl and 9

denotes the transfer coefficient for output of C2H2 and consumption of CaC2. The inventory capacity of gas holder can be ignored in real process due to its very limited capacity. Hence, the relationship between production and consumption of C2H2 can be expressed as eq13. Chlorine production is a continuous process, 123 ,, is the output of Cl2, which denoted by working state :, , production rate , and production time of the

equipment, as shown in eq14. Eq 15 is the constraint about production rate. In real process, the inventory capacity of Cl2 is very limited, so its production equals to its consumption, as shown in eq16.

123 ,, = :, ∗ 123,, ∗ 123 ∀ ∈ ;, ∈

&' &() :, ∗ 12 ≤ 123,, ≤ 12 ∗ :, ∀ ∈ ;, ∈ 3 ,, 3 ,,

123,, = -123, ∀ ∈

∈
?123,, .23 So, eq14 can be replaced with following eqs17~21. Moreover, other bilinear terms would be linearized in the same way like eq14, such as the eq22, eq26 and eq36 which would give later.

123,, = :123,, ∗ 123 ∀ ∈ ;, ∈

123,, = :123 ,, + >?123,, ∀ ∈ ;, ∈

&() >?123,, ≤ (1 − :, ) ∗ 12 ∀ ∈ ;, ∈ 3 ,, &() :123 ,, ≤ :, ∗ 12 ∀ ∈ ;, ∈ 3 ,,

:123,, ≥ 0,

>?123 ,, ≥ 0 ∀ ∈ ;, ∈

(17) (18) (19) (20) (21)

The production of calcium carbide is expressed in eq22, and also is constrained as eq23. The production of calcium carbide is stored in storage bin and inventory capacity is formulated in eqs24~25.

1(13 ,, = :, ∗ 1(13,, ∗ 1(13 ∀ ∈ @, ∈

&' &() :, ∗ 1(1 ≤ 1(13 ,, ≤ 1(1 ∗ :, ∀ ∈ @, ∈ 3 ,, 3,,

1(13, = 1(13,#$ + 1(13 ,A, − -1(13 , ∀ ∈ A∈B

&' &() 1(1 ≤ 1(13, ≤ 1(1 ∀ ∈ 3 3

(22) (23) (24) (25)

3.1.3. Power Consumption and Generation The nonlinear electricity consuming characteristics of calcium carbide production process, CHP process, and electrolysis process are approximated using hinging hyperplanes (HH) models, as introduced in our previous study.14 The total electricity consumption can be formulated in eq26. Eqs27~30 express some constraints.


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CD = (123 ,, ) ∗ :, ∗ 123 + E1(13 ,, F ∗ :, ∗ 1(13 ∀ ∈

(26)

elsp = ;, + @ ∀ ∈

(27)

elsp ≥ CD ∀ ∈

(28)

∈

Plant planning optimization under time-varying uncertainty: Case

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