Advanced Materials & Sustainable Manufacturing 2024 , 1, 10003
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2.2.1. Data Processing The raw data required for this paper is collected from the mill ’ s distributed control system, which collects process data such as Vehicle speed, steam pressure, temperature, and humidity for four quantifications, as shown in Table 1. The data collected had null values, missing values or data redundancy, and some of the data were collected at different frequencies, making modeling more complex [39]. To address the above characteristics, the originally collected data were matched in time series, with most of the data being sampled in 60 s, 10 min and 20 min sets, and the collected data were matched in 5 min sets to take the average value within the sampling interval. Missing values are filled by averaging and interpolation. Outliers are removed using box plots. Paper production can be divided into working conditions based on basic weight, as shown in Table 2. In this paper, four basic weights are considered as four working conditions are modeled separately to compare the predictive effect of the model under different working conditions. Table 1. Modeling basic data table. Variables R s Reel speed m·min − 1 P s Pressure of steam kPa P h Pressure of headbox kPa BW s Basis weight before drying sizing g·m − 2 M dz Moisture before drying sizing % M p Moisture content of paper % C p Pulp consistency of top wire % T ia Temperature of entering air ℃ H ia Humidity of entering air % T oa Temperature of exhaust air ℃ H oa Humidity of exhaust air % W oa Weight of exhaust air kgd·a·s − 1 F s Flow of steam into cylinder kg/s P sc Pressure of steam into cylinder kPa T s Temperature of steam into cylinder ℃ BW Basics weight g·m − 2 V f Former Vacuum kg V cr Couch Roll Vacuum kpa V pr Press Roll Vacuum kpa Table 2. Working conditions and basic weight comparison. Condition 1 2 3 4 Basic weight (g·m − 2 ) 115~130 105~115 80~95 65~80 2.2.2. Parametric Solver Model Based on Digital Twin For processes where the mechanistic equations have been constructed, the solve function of the math module in Python is used to calculate the unknown solution. If only a partial solution can be found, the found parameters can be used together with the known parameters as known parameters, and the other parameters can be assigned values until the error in each mechanistic equation is minimized, and when the solution is stable, the value at this point can be considered to be the value of the sought parameter. Since there are multiple equations in which the computational error needs to be minimized simultaneously, this problem can be transformed into a multi-objective, multi-constraint optimization problem. Classical nonlinear programming (Sequential Least Squares Programming, SLSQP) and NSGA-II are used in this paper [16]. SLSQP [23] is one of the nonlinear planning algorithms for constrained problems, which can solve iterative methods for nonlinear optimization problems with constraints, such as boundary constraints, equation constraints, and inequality constraints [40]. Its method has the following characteristics: it can handle any degree of nonlinearity, including nonlinearity in constraints, and it must choose appropriate parameters such as initial solution, shift acceptance criterion, and step control strategy to ensure convergence and efficiency. NSGA-II is an improvement on the first generation of non-dominated sorting genetic algorithm: it proposes a fast non-dominated sorting algorithm [41 – 43], which reduces the computational complexity on the one hand, and merges the parent population with the offspring population on the other hand, retaining all the best individuals; it introduces an elite
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