Energy saving diagnosis model of petrochemical plant based on intelligent curvelet support vector machine

The energy and resources saving has become a major task of petrochemical enterprises, it is necessary to construct the energy saving diagnostic system for understanding the real-time operation information of petrochemical plant and provide theoretical basis for taking energy saving measures. The energy saving diagnosis process of petrochemical plant based on twin Curvelet support vector machine optimized by hybrid glowworm swarm algorithm is designed. The Curvelet kernel function is constructed based on Curvelet transform to establish theory model of twin Curvelet support vector machine. In order to improve the prediction precision of the twin Curvelet support vector machine, the hybrid glowworm swarm optimization algorithm is constructed based on simulated annealing simulation to optimize the parameters of the twin Curvelet support vector machine. Finally, a petrochemical plant is used as research object to carry out diagnosis simulation analysis, and results showed that the proposed prediction model can effectively improve diagnostic effectiveness of the energy saving effect of petrochemical plant.


Introduction
With development of economy, energy crisis and environmental pollution are rising, which have become two difficult problems facing the world. In order to cope with growing energy demand and environmental pressure, China has taken the optimization and adjustment of the industry and economic structure, and brought in a number of measures. The petrochemical enterprises consume a lot of energy source, the energy consumption can directly affect competitiveness of petrochemical enterprises, therefore energy and resources saving has become a major task of petrochemical enterprises.
On energy-saving, sorts of energy-saving technologies and devices are applied in petrochemical plant, but often overlook the energy saving potential, therefore the energy saving and efficiency improvement are short of solution, and the energy consumption is uneven. The energy saving and efficiency improvement of petrochemical plant. On condition that the operating conditions of the petrochemical plant are determined, the manager of petrochemical enterprises can grasp and identify the potential problems and formulate the energy innovation projects, and then the energy saving and efficiency improvement can be achieved to ensure the safe and efficient operation of petrochemical plant. It is necessary to establish a scientific and rational diagnostic system of energy-saving of petrochemical plant. The energy saving diagnosis can understand the real-time operation information of petrochemical plant through analyzing the data for energy saving and consumption reducing of petrochemical enterprises. The weakness of energy utilization of petrochemical enterprises can be cleared, the existing problems in operation and management of petrochemical plant can be identified according to energy saving diagnosis results, and then the energy-saving transformation measures can be made, which can achieve the efficient use of resources. The energy diagnosis can provide the basis for management decision-making of petrochemical enterprises.

Related works
In recent years, the energy saving evaluation of various industries has been concerned by some scientists, many outstanding achievements have come out. Shi Wantao et al. proposed a novel evaluation method based on exergy analysis, the effectiveness of the proposed method is verified through comparison and analysis (Shi et al. 2019). Kasturi Srikant and Kannappan Bakthavatsalam Ayalur conducted energy audits, and the evaluated energy consumption profiles, specific energy consumption, independent parameters (Kasturi and Kannappan 2020 (Himeur et al. 2020b). Currently some intelligent algorithms are applied to energy saving, such as artificial neural network, convolutional neural network, the artificial neural network and convolutional neural network need a large number of samples to guarantee the sufficient generalization ability, require long learning complexity, and lead to the ''over learning'' phenomenon. The above problems limit the application of the artificial neural network to some degree, the support vector machine has unique advantage to solve small sample, nonlinear and high dimensional pattern recognition problems. The support vector machine is an effective method of energy-saving diagnosis of petrochemical enterprises. The kernel function of support vector machine should satisfy the Mercer theorem, the Gauss kernel function is applied widely to engineering practice. Yet the kernel function cannot closely approximate the curve in L 2 ðR n Þ space, because it is not total orthogonal basis, therefore the support vector machine cannot approximate the arbitrary function in classification of it. The wavelet function can establish the orthogonal basis through dilation with translation, which has good approximating ability, therefore the wavelet function can be used to construct wavelet kernel function to improve the generalization ability of support vector machine. In recent years, the wavelet support vector machine has already got application in practice.
Chinnam Siva Koteswara Rao et al. proposed an efficient method to identify unwanted growth in brain using SVM-PUK (Rao et al. 2019). Aider Malika Ait et al. were devoted to the off-line handwritten character recognition based on the two dimensional wavelet transform and a single support vector machine classifier (Aider et al. 2018). Thirumala Karthik et al. proposed a novel automated recognition based on tunable-Q wavelet transform and a dual multiclass support vector machines (Thirumala et al. 2016). Eseye Abinet Tesfaye et al. proposed a hybrid forecasting model for short-term generation power forecasting (Eseye et al. 2018). Zhou Ting et al. proposed a data-driven prediction model combining discrete wavelet transform preprocess and support vector machine for groundwater depth forecasting (Zhou et al. 2017). Garcia Nieto P. J. et al. proposed a new hybrid algorithm, support vector machine with Mexican hat wavelet kernel function for predicting the cyanotoxin content (Garcia Nieto et al. 2017). The advantage of wavelet transform is mainly embodied in analyzing and processing piecewise smooth signal or bounded variance function. The separable wavelet constructed by one dimensional wavelet has only finite orientation when the signal is two or high dimensional function, therefore it cannot make optimal use of linear or surface singularity functions with higher dimensions. In order to overcome the defects of the wavelet transform, Candes and Donoho put forward Curvelet transform and constructed the Curvelet tight frame, the Curvelet transform can realize the stable, efficient and optimal representation of objective function concluding a large number of smooth singular curves. The basic element of Curvelet transform is edge, the basis of it is anisotropic and has strong direction performance. Therefore the Curvelet transform can be combined with support vector machine to construct the Curvelet support vector machine. C. Jimenez-Castaño et al. proposed twin support vector machine for imbalanced data classification (Jimenez-Castaño et al. 2020). Javad Salimi Sartakhti et al. proposed a fuzzy least squares twin support vector machine that enhances the classification accuracy (Sartakhti et al. 2019). Rozhin Eskandarpour and Amin Khodaei proposed twin support vector machine for probabilistic load curtailment estimation (Eskandarpour and Khodaei 2019).
In order to reduce computational complexity and training time, the Curvelet support vector machine can be improved according to basic idea of twin support vector machine. The twin support vector machine is constructed based on idea of approximate support vector machine based on generalized eigenvalue. The history of twin support vector machine is a relatively short time, it has become a hot research topic due to its sound theoretical basis and excellent performance.
The twin Curvelet support vector machine is constructed through combing twin support vector machine and Curvelet support vector machine. The twin Curvelet support vector machine can generate two nonparallel classification hyper planes, making one of the hyper plane as close to a class of samples as possible, and as away from another class of sample. The twin Curvelet support vector machine can transfer single large quadratic programming problem to two small quadratic programming problems, therefore the running time is significantly shortened.
In addition, the parameter selection is a key factor of affecting precision, stability and of twin Curvelet support vector machine. At present, there are three main methods for parameter selection. The first method is experience method, which mainly depends on users and samples, therefore this method has strong subjectivity. The second method is grid search method, its shortcomings lie in the selection of step size, when the step size is small, the amount of calculation is large and the time is long. When step length is long, it is easy to miss the global optimal solution. The third method is intelligent optimization algorithm, which is better than the above two methods. The main intelligent algorithms used for parameter selection of twin Curvelet support vector machine conclude particle swarm algorithm , simulated annealing algorithm (Hosseini et al. 2020), genetic algorithm (Choudhary 2021). The particle swarm algorithm has the problems of premature convergence and poor local optimization ability. The simulated annealing algorithm falls into local optimum. The programming process of genetic algorithm is complex, the selection of parameters of genetic algorithm is mostly based on experience, and the genetic algorithm is somewhat dependent on selection of initial population. In order to overcome these disadvantages of the above algorithms, a better algorithm should be chosen.
The glowworm swarm optimization is an meta heuristic swarm intelligence optimization algorithm, which has the advantages of clear process, strong robustness and less parameters Pruthi et al. 2020). The glowworm swarm algorithm has some defects, the location updating of it depends on the excellent individuals, if the distance between individuals is too large, glowworm swarm optimization moving speed of firefly with is reduced (Khan et al. 2019;Chowdhury and De 2020). In order to improve optimization capability of algorithm, the simulated annealing mechanism is used to construct the hybrid glowworm swarm algorithm, and then the defect of falling into local optimum can be avoided. The hybrid glowworm swarm algorithm can improve the global search ability and operation speed, therefore the hybrid glowworm algorithm is used to optimize the parameters of twin Curvelet support vector machine, then the energy saving level of petrochemical plant can be correctly evaluated based on optimized twin Curvelet support vector machine.

Energy saving principle of petrochemical plant
The production process of petrochemical plant mainly concludes transportation, heat exchange, chemical reaction and separation of material involving regulation, control and security of system. All production segments of petrochemical plant have energy usage and consumption, and main energy used concludes fuel gas, electricity, steam and water, and the main process parameters involved conclude pressure, temperature and flow (Talaei 2018). Efficient and accurate energy saving diagnosis can provide favorable basis for energy saving practice of petrochemical plant. The energy utilization process of petrochemical plant is highly complex, which shows nonlinear, time-varying and random properties. The energy saving diagnosis builds reasonable mathematical model based on wavelet support vector machine according to history data and current data, and then evaluates the energy saving effect of petrochemical plant based on proposed model. The energy saving of petrochemical plant is affected by various factors, and the Curvelet support vector machine can be used to carry out energy saving diagnosis of petrochemical plant, which can improve the diagnosis precision. The energy saving diagnosis process of petrochemical plant based on Curvelet support vector machine is shown in Fig. 1 where ½j=2 is the integral part of j=2. The translation parameter is defined by k ¼ ðk 1 ; k 2 Þ 2 Z 2 , the Curvelet transform at scale j, direction h j;m and position x j;m where For f ðxÞ 2 L 2 ðR 2 Þ, the Curvelet transform of it can be represented as inner product between it and Curvelet function u j;m;k ðxÞ, which is expressed by Given a Curvelet, the tight frame in space L 2 ðR 2 Þ can be formed through translation and rotation, and the corresponding reconstruction formula is expressed by Sharma et al. (2020) f where u j;m;k denotes the Curvelet family, M denotes the index collection. Given the sample set ðx 1 ; y 1 Þ, ðx 2 ; y 2 Þ,…,ðx l ; y l Þ, x i 2 R n , y i 2 R, l denotes the number of samples, the kernel function of support vector machine is the dot product form Kðx; x 0 Þ ¼ Kðx Á x 0 Þ, it also is translation invariant form Kðx; x 0 Þ ¼ KðxÀx 0 Þ, if a function satisfies Mercer theorem, this function is a support vector machine kernel function.
A necessary and sufficient condition that the symmetric function Kðx; x 0 Þ is the support vector machine kernel function is listed as follows: for all gðxÞ 6 ¼ 0 that satisfies R R n g 2 ðxÞdx\1, the following inequality holds: If the Fourier transform satisfies the following condition: the translation invariant form Kðx; x 0 Þ ¼ KðxÀx 0 Þ satisfying condition (5) is a permissible support vector machine kernel function. The Curvelet kernel function is expressed by where a is the Curvelet scale parameter, a [ 0.
Carry out data preprocessing The Curvelet function is combined with twin support vector machine to construct the twin Curvelet support vector machine. The twin Curvelet support vector machine divides the training set into two parts, and then two non parallel decision functions can be constructed through selecting proper loss function for the two parts, respectively.
The following training set is given:S ¼{ðx 1 ; y 1 Þ, ðx 2 ; y 2 Þ,…,ðx m ; y m Þ}, where x 1 , x 2 ,…, x m denotes the data input, x 1 , x 2 ,…, x m denotes the data output. The matrix A 2 R mÂn and output vector ỹ ¼ ½y 1 ; y 2 ; Á Á Á ; y m T are used to denote m inputting samples in n dimensional space R, the ith line of A is defined by x T i , where m is number of samples of training set.
The two decision functions of twin Curvelet support vector machine are not parallel, which are expressed by where x 2 R n , w 1 and w 2 are the weight vectors, b 1 and b 2 are offsets, ðÁ; ÁÞ denotes the inner product.
The original problems are expressed by min s.t. ðÃw 2 þ ẽb 2 Þ À ỹ! ẽẽ 2 À #, # * ! 0 where Á k k is L 2 norm, C 1 and C 2 are penalty coefficients, e 1 and e 2 [ 0 are constants, f and # are relaxation vectors, ẽ is m Â 1 dimensional unit column vector. The dual forms of expressions (8) and (9) can be obtained based on Lagrangian function, which is expressed by Eseye et al. (2018) The optimal solutions of Eqs. (10) and (11) are listed as follows: In the case of non-linearity, the solutions can be obtained through mapping Curvelet kernel function to high dimensional space, and the twin Curvelet support vector machine decision functions are expressed by When f 1 ðxÞ and f 2 ðxÞ are solved, the decision function is expressed by

Training algorithm of twin Curvelet support vector machine
The twin Curvelet support vector machine has three parameters concluding C 1 . C 2 and a, the three parameters are closely related with diagnosis precision, therefore an effective optimal algorithm is required to obtain the best parameters of twin Curvelet support vector machine.

(a) Mathematical model of improved glowworm swarm optimization
The glowworm swarm optimization (GSO) algorithm is a novel intelligent optimization algorithm, which already has been applied in many fields, but there has been very few successful application of GSO in parameter optimization of support vector machine. The traditional GSO algorithm is easy to fall into local optimum and convergence insufficiency, and it has low population diversity. In order to improve the performance of GSO and local search ability, therefore the GSO algorithm is improved based on dynamic step (López et al. 2019).
The GSO algorithm is used to simulate the mutual attraction and location update between glowworms. The location and the fluorescein in the location of i th glowworm are defined by x i ðtÞ and y i ðtÞ, respectively, and every location corresponds to a objective function f ðx i ðtÞÞ.
The location of i th glowworm occurs randomly according to following expression: where M denotes the population size of glowworms, L denotes the lower bound of x ij , U denotes the upper bound of x ij . rad denotes the random number yielding uniform distribution on domain (0,1). Energy saving diagnosis model of petrochemical plant based on intelligent curvelet support vector… 15395 When every glowworm transmit information to neighboring individuals in decision range of its region, the decision range is updated by Shi et al. (2020) where r i d ðtÞ denotes the decision radius in region of ith glowworm in tth iteration, 0\r i d ðtÞ\r s ; r s denotes the maximum perception radius, n t denotes the threshold of individual quantity in controlling neighborhood, b denotes the variation ratio of neighborhood, N i ðtÞ denotes the group of individuals in neighborhood of ith glowworm in the tth iteration.
The individual quantity of ith glowworm in decision range is calculated by where x j ðtÞ denotes the location of jth glowworm, l i ðtÞ denotes the fluorescein value. The motion direction of ith glowworm is decided by number of fluorescein of every glowworm in the neighborhood. The probability of ith glowworm moving to jth glowworm is calculated by Puttamadappa and Parameshachari (2019) The new location of ith glowworm after movement is calculated by where r ij denotes the distance between ith glowworm and jth glowworm, a denotes the step length factor. After the location of ith glowworm is updated, the new fluorescein value is calculated by where k denotes the constant, 0\k\1, which is related to volatilization of fluorescein; g denotes the constant, which can identify the update rate of fluorescein. In order to improve performance of GSO algorithm, the adaptive search step is introduced into the traditional GSO algorithm. The step length factor a is regulated by Koyuncu and Ceylan (2019) aðt þ 1Þ ¼ 0.85a(tÞ ð 23Þ where x d;max and x d;min are upper bound and lower bound of objective function. In order to improve the search velocity of algorithm, the regulating formulas of k and g are, respectively, expressed by Goel et al. (2018) The GSO algorithm will fall into local optimum when the individual searches the optimal value for a long time within the radius of neighborhood, and then fitness degree function does not change. Therefore the chaos seeking strategy is used to generate new solution to optimize the individuals, and then the global optimization ability of algorithm is improved. The Chebyshev chaotic mapping is used as search strategy, which is expressed by Balochian and Baloochian (2019) The location is updated by where m denotes the spatial dimension, k ¼ 1; 2; . . .; Max, X m i denotes the original location, X m 0 i denotes the optimized location. The fitness degree function value of new individual is calculated based on X m 0 i , which is compared with current stagnant individual, if the fitness degree of the new individual is better than that of stagnant individual, the stagnant individual is replaced by the new individual.

Hybrid glowworm swarm algorithm based simulated annealing algorithm
The simulated annealing algorithm is established based on the principle of solid annealing, the solid is heated to enough high temperature and then is cooled slowly. According to Metropolis criterion, the probability that particles tend to equilibrium at temperature T is e ÀDE=ðkTÞ , where E is the internal energy at temperature T, DE is the changing amount, k is Boltzmann constant. Using solid annealing to simulated combined optimization problem, the internal energy E is simulated as the objective function value, the temperature T is evolved into controlling parameter, and then the simulated annealing algorithm can be obtained. The simulated annealing algorithm can be used to construct the hybrid glowworm swarm algorithm. Firstly, the selection mechanism is improved. In order to dynamically adjust the selection pressure in the process of searching the optimal solution, the Bolzmann selection strategy is introduced into the search process of glowworm swarm algorithm. P ij ðtÞ denotes the probability that the ith glowworm moves to the jth glowworm in its neighborhood set at time t.
where T ¼ T 0 a CÀ1 , T denotes the temperature in simulated annealing algorithm, T 0 denotes the initial temperature, a denotes the decay factor, C denotes the number of iterations.
In improved glowworm swarm algorithm the obtained optimal solution is often approximate value due to the fixed moving step size and randomness of selecting a neighbor of artificial glowworm, therefore it is difficult to obtain the accurate solution further. Therefore the strong local search ability of simulated annealing algorithm can be used to improve the accuracy of solving the problem for improved glowworm swarm algorithm. The flow of hybrid glowworm swarm algorithm is listed as follows (Kaur et al. 2020): Step 1 The parameters of improved glowworm swarm algorithm concluding r i d ðtÞ, r s , n t , b, the glowworm population size N, the number of iterations t = 0, the maximum iteration times L max , and the parameters of simulated annealing algorithm concluding T 0 ,a, iteration times L at each temperature are initialized.
Step 2 The initial position of each glowworm is randomly initialized, and the objective function value of each glowworm that is fitness degree is calculated, and then the bulletin board is initialized to record the best glowworm individual.
Step 3 The fluorescein values of all glowworms are calculated based on Eq. (22).
Step 4 In the moving stage, the neighbor glowworm set N i ðtÞ in the decision domain of each glowworm is calculated according to Eq. (19).
Step 5 For any glowworm i, the probability P ij ðtÞ of its moving to each neighbor is calculated according to Eq. (29). A neighbor is selected according to the probability to move, and then location of it is updated based on Eq. (28).
Step 6 The decision domain radius of each glowworm is updated based on Eq. (18).
Step 7 The glowworm best with best fitness degrees in this generation is found out, and the corresponding location x best is recorded. If it is better than bulletin board information, and the bulletin board information will be updated.
Step 8 t t þ 1, an iteration is completed, judge whether the condition t\L max is satisfied. If the condition is met, return to step 3 and enter the next iteration. Otherwise exit the iteration and output the result.

Case study
In order to verify the effectiveness of proposed energy saving diagnosis, a LPG mercaptan removal plant in a oil refinery is used as research object to carry out numerical analysis. Merox mercaptan removal technology is applied in this plant, and the LPG yield reaches about 19% after transformation. The plant has a large energy consumption, it is necessary to carry out energy-saving diagnosis for taking effective energy saving measures. The diagnostic index system of energy saving is confirmed according to practical situation of plant, which is listed in Table 1.
The Curvelet support vector machine optimized by improved GSO algorithm is applied to diagnose the energy-saving effect of the LPG mercaptan removal plant after the index system is confirmed. The energy-saving effect of the LPG mercaptan removal plant is measured by energy consumption.
The data collected for 24 months from 2018 to 2019 are collected from the oil refinery, which is used as training sample. The data collected from January to October in 2020 are used as testing sample. The support vector machine optimized by simulated annealing algorithm (SVM-SA), the twin B-spline wavelet support vector machine optimized by traditional GSO algorithm (TBWSVM-TGSO), and the twin Curvelet support vector machine trained by improved hybrid glowworm swarm algorithm (TCSVM-HGSO) are applied to prediction simulation analysis. The mean absolute percentage error (MAPE) and determination coefficient R 2 are used to evaluate the prediction effect of different models.
whereÊ denotes the diagnostic value, E denotes the actual value, N denotes the number of samples. The comparison results of predicted and actual values of training samples for different models are shown in Figs. 2, 3 and 4. Figure 2 shows the comparison results of SVM-SA, and the determination coefficient R 2 equals 0.925. Figure 3 shows the comparison results of TBWSVM-TGSO, the determination coefficient R 2 equals 0.936, the prediction precision of TBWSVM-TGSO is better than that of SVM-GA. Figure 4 shows the comparison results of TCSVM-HGSO, the determination coefficient R 2 equals 0.945, results show that the TCSVM-HGSO model has best fitting precision among the three models, and can obtain the best prediction precision.
The optimal parameters C 1 , C 2 and a are obtained based on three models, which are listed in Table 2.The optimal parameters are used to carry out prediction simulation for testing samples.
The predict simulation of testing samples is carried out based on the three models, and prediction results are shown in Fig. 5, results show that the changing trend of energy consumption of LPG mercaptan removal plant. The predicted value of energy consumption by the TCSVM-HGSO is closer to actual value than other models, and results show that the proposed TCSVM-HGSO has better prediction accuracy than other models.
The mean absolute percentage error (MAPE) of different predicted models is shown in Fig. 6. As seen from Fig. 6, the MAPE of TCSVM-HGSO is less than that of other two models, therefore the TCSVM-HGSO has best prediction precision. The main reason for this change lies in that the TCSVM-HGSO can better break away from the local convergence and avoid over-fitting.

Conclusions
Energy-saving diagnosis is important for improving economic benefit of petrochemical plant, therefore the curvelet transform is combined with the twin support vector machine to construct the twin curvelet support vector machine, in order to improve the prediction effectiveness of the twin curvelet support vector machine, the hybrid glowworm swarm algorithm (HGSO) is established based on simulated annealing algorithm to optimize the parameters of twin Curvelet support vector machine. A LPG mercaptan removal plant in a oil refinery is used as the research objective to carry out energy-saving diagnosis simulation based on SVM-GA, TBWSVM-TGSO and TCSVM-HGSO. The following results are obtained: (1) The determination coefficient R 2 of SVM-SA equals 0.925. The determination coefficient R 2 of TBWSVM-TGSO equals 0.936, therefore the prediction precision of TBWSVM-TGSO is better than that of SVM-GA. The determination coefficient R 2 of TCSVM-HGSO equals 0.945, therefore TCSVM-HGSO model has best fitting precision among the three models, and can obtain the best prediction precision.   (2) Based on prediction results of energy consumption, the predicted value of TCSVM-HGSO is closer to actual value than other models, therefore the proposed TCSVM-HGSO has better prediction accuracy than other models. (3) Through comparing the MAPE of different methods, the TCSVM-HGSO has less error than other two models, therefore the TCSVM-HGSO has best prediction precision.
The prediction error should be reduced through developing the novel models in future works, and the more affecting factors of energy saving should be considered in the novel model. The novel model aims to obtain the results that can be applied in actual engineering.