Soft sensor for ball mill fill level based on uncertainty reasoning of cloud model

Abstract

Considering the strong uncertainty in ball mills, cloud model which combines fuzziness and randomness together and has the ability of processing uncertainty, is introduced. The paper proposes a novel soft sensor based on uncertainty reasoning of cloud model to improve the accuracy and reliability of fill level measurement. At first, power spectral densities of the vibration signals are extracted by Welch’s method and the features are obtained by summing the energy of a wide frequency band. Then backward cloud generator algorithm is used to represent numerical characteristics of antecedent clouds under different fill levels, and the corresponding consequent clouds are given according to the fill level information. Thus, the rule base is built and the uncertainty reasoning based on cloud model is realized. As it is difficult to obtain data set continuously and accurately in practical industry fields, virtual cloud is employed to deal with the problem of sparse rule base in the case of insufficient samples. The experimental results show that the accuracy of proposed method can meet the requirements of field measurement applications. In addition, the method based on virtual cloud is more accurate and robust compared with other methods in the case of insufficient samples.

Keywords

Ball mill fill level soft sensor vibration signals cloud model uncertainty reasoning

1 Introduction

Ball mill is an important mechanical equipment in pulverizing system, which is widely used in Metallurgy, Electric Power, Building Materials, Chemical, Ceramics and other industries. However, it is one of the largest electric power consumers in these industries. For instance, it is up to 15–25% of the whole electric consumption in power plant [1]. As the grinding process of ball mill is complicated with nonlinearity, large delay, strong coupling and susceptible to a variety of uncertainty interferences, fill level (FL) in ball mill is difficult to measure directly. In practice, the ball mill is always running at low load to avoid over loading which may cause mill blockage. Therefore, it is essential to measure FL reliably and effectively for energy saving and efficiency improvement.

Till now, there have existed several methods for FL measurement, including differential pressure method [2], electric current method [3], ultrasonic method [4], acoustic method [5, 6] and vibration method [1 , 8], etc. Differential pressure method is one of the traditional methods which reflects the FL by differential pressure between the inlet and outlet of the ball mill. However, the differential pressure is not a single value function of the coal load, and also affected by other factors, such as the ventilation quantity, parameters of coal mill structure [9]. Electric current method is based on the relationship of current and FL, which is convenient and is not affected by the environment easily. But it has a low sensitivity due to the small variation of current from no-load to full load and can be affected by the wear of grinding medium [10]. Ultrasonic method’s measurement accuracy would be lower because of material and steel balls’ destruction to the ultrasonic probe [10]. By analyzing the acoustic signals of the mill, it shows that with the increments of FL, the amplitude of acoustic signal shows a decreasing trend. However the working conditions are so noisy that it is difficult to shield the acoustic signals from other noises [6]. As the acquisition of vibration signals is not so heavily affected by the adjacent mill and background noise, and the accelerometer sensor can be mounted easily, vibration signal diagnosis technology has been widely used in monitoring the operation of equipments [1]. However, as will be discussed below, this issue still needs to be solved.

In recent years, there has been considerable research effort devoted to estimate the FL by using data-driven methods, such as Support Vector Machines (SVM) [11], Back Propagation Neural Network (BPNN) [6], Extreme Learning Machines (ELM) [12], etc. These researches have made great progress for measurement of FL. But there are two main problems affecting model construction. The first is uncertainty of vibration signals, and the second is that these methods require relatively complete and accurate data sets. Complete data sets mean that the data sets must cover almost all working conditions from low FL to high FL, and accurate data sets mean the sensor model must be guided by accurate FL information. In continuous production process it is costly even impossible to meet the demands.

Aiming at these problems, cloud model [13] is introduced to construct soft sensor model of FL. Firstly, power spectrum features are extracted by Welch’s method from vibration signals of ball mill bearing for constructing the antecedent cloud models by backward cloud generator (BCG) algorithm. Then the corresponding consequent concepts are defined in FL information represented by cloud model. After acquiring the antecedents and consequents of cloud rule base, the uncertainty reasoning can be realized by condition X cloud generator and condition Y cloud generator algorithms. However, in the case of insufficient data set, there may exist a blank area between two adjacent antecedent clouds, which can lead to sparse cases of rule base. So virtual cloud for interpolation is introduced to solve the problem of sparse rules and improve the measurement accuracy. The experimental results show the effectiveness of cloud model based method.

The rest of the paper is organized as follows:Section 2 introduces the related work. Section 3 provides the basis of cloud model and cloud uncertainty reasoning. Section 4 describes the experiment and focuses on feature extraction and analysis. The application of cloud model for monitoring FL of ball mill in a power plant is illustrated in Section 5. The last section gives the conclusions.

2 Related work

There are two main processes in FL measurement: Extracting features from vibration signals and establishing prediction model for FL measurement based on the features. Zeng and Forssberg [14] performed an in-depth study of vibration signals of ball mill by multivariate statistical analysis and spectral analysis, finding the vibration bands is related to the motion of the ball charge and other milling parameters. Wang and Lv [15] analyzed vibration power signals and demonstrated that within a certain frequency band, it is possible for power signals to reflect coal load. Thus, more and more researches investigated the empirical relationships between the vibration signals and grinding parameters using multivariate statistical methods, such as Principal Component Regression (PCR) and Partial Least Squares Regression (PLSR). Zeng et al. [16] used PCR to build the model and to monitor the parameters in the grinding mill. Tang et al. [8] established soft sensor models between vibration signals and mill operating parameters of mill load using Genetic Algorithm-Partial Least Squares Regression (GA-PLSR) technology. However, both PCR and PLSR are tools for building linear models, while ball mill system is a kind of nonlinear system with characteristics of strong coupling, large delay and time-varying.

Recently, lots of advanced techniques have been applied to estimate the FL. In order to solve nonlinear problem, Wang et al. [17] used the concept of multi-sensor variables soft sensor and applied the theory and method of Cerebellar Model Articulation Controller (CMAC) neural network to reflect the FL; Zhao et al. [11] extracted the mass and the central frequency from power spectral density (PSD) as the spectral features to build the soft model using SVM, whose parameters were selected by Genetic Algorithms (GA); Tang et al. [18] used Principal Component Analysis (PCA) for dimensionality reduction and extraction of the principal components (PCs) of characteristic spectra, and adopted Least Squares Support Vector Machines (LSSVM) to construct nonlinear mapping between characteristic spectral PCs and mill load parameters; Tang et al. [12] employed Partial Least Squares (PLS) to extract feature and used optimization-based ELM (OELM) for data regression and information fusion.

It is well known that precise process data is the basis of these models. However, a precise training set is difficult to obtain in continuous production process. In addition, the process variables in industry, typically with uncertainty, are usually difficult to measure on line and also off line. For the cases where uncertainty exists in ball mill FL information, Su and Wang [19] proposed an improved adaptive evidential k-NN rule to measure the level of coal powder filling; Then a multi-evidential soft sensor model based on evidence theory and multi-model strategy was proposed [20]; Furthermore, immune genetic algorithm-based adaptive evidential model was used to predict the level [21]; In 2013, Su et al. employed the kernel based nonlinear fuzzy regression model identified by fuzzy Expectation Maximization (EM) algorithm to measure the level [22]. However, the basic parameters including fuzzy membership function and prior probability as well as expert knowledge are difficult to acquire and represent accurately, which hinders the promotion of the related theory and algorithm. So it is necessary to employ a new methods to deal with these problems.

Considering the characteristics of high divergence and strong randomness of vibration signals of ball mill and difficulty of acquiring sufficient samples, cloud model, a bidirectional cognitive transformation tool which combines fuzziness and randomness together to reflect the uncertainty of things, is introduced in this paper.

3 Cloud model

3.1 Concepts of cloud model

Definition 1. [23] Assume that U is a quantitative numerical universe of discourse and A is a qualitative concept in U. If x ∈ U is a random implementation of concept A, and μ_A (x) ∈ [0, 1], standing for membership degree for which x belongs to A, is a random variable with tendency. Then a distribution of x in universe of discourse U is called a cloud and each (x, μ_A (x)) is called a cloud drop.

The one-dimension normal cloud is the most basic tool to express language value, and its expectation curve cluster is [24]: $y (x) = exp [- {(x - Ex)}^{2} / (2 {(En \pm kHe)}^{2})]$ (1)

According to the “3En principle” which is similar to the “3σ principle” of normal distribution [25], there are about 99.74% cloud drops contained between outside cloud curve $y_{1} (x) = exp [- \frac{{(x - Ex)}^{2}}{2 {(En + 3 He)}^{2}}]$ and inside cloud curve $y_{2} (x) = exp [- \frac{{(x - Ex)}^{2}}{2 {(En - 3 He)}^{2}}]$ , which are defined when k = 3 and k = -3 respectively, as shown in Fig. 1 in blue curves. The rest cloud drops which can not be contained by the two curves, are considered as small probability event. It does not affect the overall characteristics of the cloud model if the rest cloud drops are not taken into account. When k = 0, its Mathematic Expectation Curve (MEC) is defined as $y_{0} (x) = exp [- \frac{{(x - Ex)}^{2}}{2 E n^{2}}]$ , shown in red curves in Fig. 1. If He is so small that makes He < En/3, the cloud drops present universal normal distribution, as shown in Fig. 1(a); On the contrary, if He > En/3, the cloud drops present atomization distribution, as illustrated in Fig. 1(b). En/3 is called atomized point of normal cloud model.

3.2 Cloud generators

The antecedent of qualitative rule in cloud model reasoning is usually represented by condition X cloud generator while the consequent is represented by condition Y cloud generator. Condition X cloud generator and condition Y cloud generator are the basis of cloud uncertainty reasoning. The algorithms [26] of them are described as follows.

Condition X cloud generator: When cloud’s three numerical characteristics (Ex, En, He) and the input x are given, the following: $P = R (En, He)$ (2) $μ (x) = e^{- \frac{1}{2} {(\frac{x - Ex}{P})}^{2}}$ (3) is denoted as condition X cloud generator, where R is a normal random function, and each point (x, μ (x)) is called a cloud drop on condition X.

Condition Y cloud generator: When cloud’s three numerical characteristics (Ex, En, He) and membership degree μ are given, the following: $P = R (En, He)$ (4) $y = Ex \pm \sqrt{- 2 \ln (μ)} \cdot P$ (5) is denoted as condition Y cloud generator, and each point (y, μ (x)) is called a cloud drop on condition Y.

BCG is an effective algorithm of transforming quantitative expressions to the qualitative concepts. For the giving array X = [x₁, x₂, ⋯ , x_N], which represents the quantitative sequences of the qualitative concept C (Ex, En, He), the BCG algorithm [27] is given below:

Step 1: For the quantitative data set X, calculate the sample mean $\bar{X} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}$ , the sample variance $S^{2} = \frac{1}{N - 1} \sum_{i = 1}^{N} {(x_{i} - \bar{X})}^{2}$ , and the sample fourth-order center distance ${\bar{μ}}_{4} = \frac{1}{N - 1} \sum_{i = 1}^{N} {(x_{i} - \bar{X})}^{4}$ .

Step 2: Attain the expected value: $Ex = \bar{X}$ .

Step 3: Calculate the entropy: $En = \sqrt[4]{\frac{9 {(S^{2})}^{2} - {\bar{μ}}_{4}}{6}}$ .

Step 4: Calculate the hyper-entropy: $He = \sqrt{S^{2} - \sqrt{\frac{9 {(S^{2})}^{2} - {\bar{μ}}_{4}}{6}}} .$

3.3 Cloud model reasoning and cloud rule base

The cloud model reasoning system is based on qualitative rules with condition X cloud generator as the antecedent and condition Y cloud generator as the consequent, as well as the associated cloud generator algorithms. The basic process is:

Firstly, describe the language concept using cloud model to construct the antecedent clouds that represent the condition of qualitative rules and the consequent clouds that represent the results of qualitative rules. Both the antecedents and consequents of qualitative rules are qualitative concepts using the natural language description and each qualitative concept is expressed by its three numerical characteristics.

Secondly, create cloud model reasoning rule base CR = {R_i}, which consists of several rule generators. Each rule R_i is expressed as If-Then pattern: $If {Cx}_{i} Then {Cy}_{i}$ where Cx_i denotes condition X cloud model, and Cy_i denotes condition Y cloud model.

Finally, put the test set into the antecedents of the qualitative rule generator, and the reasoning is made through the activated rules according to the inference engine. A certain number of drops with membership degrees are gotten from the corresponding fired consequent clouds. At last, output the accurate result. In this paper, weighted mean method is adopted to get the quantitative values, where the weights are the membership degrees of cloud drops. Suppose that y_Bi is output drop of the consequent clouds, μ_i is the corresponding membership degree, the output result of weighted mean method is: $y = \sum_{i = 1}^{m} (y_{Bi} \times μ_{i}) / \sum_{i = 1}^{m} μ_{i}$ (6)

In practical cloud model reasoning system, the multi-rule cloud model generators are more common. However, the more rules, the more complex of reasoning process. Worse still, information of establishing rule base is always insufficient. In order to meet the demands of the practical reasoning system, the virtual cloud method could be introduced. Before discussing the uncertainty reasoning of cloud model of insufficient samples, corresponding definitions are given below.

Definition 2. R_i = {Cx_i ⇒ Cy_i|i = 1, …, r} is called cloud model rule base, where Cx_i denotes the antecedent cloud of rule i, and Cy_i denotes the consequent cloud of rule i. Suppose that the antecedent clouds satisfy: Ex₁ < Ex₂ < ⋯ < Ex_i < Ex_i+1 < ⋯ < Ex_r, then the corresponding clouds will follow: Cx₁ < Cx₂ < ⋯ < Cx_i < Cx_i+1 < ⋯ < Cx_r.

Definition 3. According to the “3En principle”, if there exists a certain i (0 < i < r) that makes two adjacent clouds Cx_i and Cx_i+1 accord with Ex_i + 3En_i < Ex_i+1 - 3En_i+1, then Cx_i and Cx_i+1 are disjoint, and the cloud rule base R_i = {Cx_i ⇒ Cy_i|i = 1, …, r} is sparse. Otherwise, the cloud rule base R_i = {Cx_i ⇒ Cy_i|i = 1, …, r} is compact.

3.4 Virtual cloud

Given C₁ and C₂ on the same universe of discourse, the numerical characteristics are C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂), and Ex₁ < Ex₂, as shown in Fig. 2(a). According to Definition 2 and Definition 3 above, C₁ and C₂ are disjoint, i.e., there exists “blank area” between the two clouds. So the input universe of discourse is not completely covered if they are used to construct the rule antecedents in cloud uncertainty reasoning. When an observation comes to the “blank area”, no rule will be fired, which derives no consequence. To solve this problem, virtual cloud construction technique is introduced to obtain the virtual rules. Suppose that C₃ (Ex₃, En₃, He₃) is the numerical characteristics of virtual cloud C₃, which is an arithmetic operation result of the two clouds C₁ and C₂ using the following equations [13]: $E x_{3} = \frac{E x_{1} + E x_{2}}{2}$ (7) ${En}_{3} = \frac{{Ex}_{2} - {Ex}_{3}}{{Ex}_{2} - {Ex}_{1}} \times {En}_{1} + \frac{{Ex}_{3} - {Ex}_{1}}{{Ex}_{2} - {Ex}_{1}} \times {En}_{2}$ (8) ${He}_{3} = \frac{{Ex}_{2} - {Ex}_{3}}{{Ex}_{2} - {Ex}_{1}} \times {He}_{1} + \frac{{Ex}_{3} - {Ex}_{1}}{{Ex}_{2} - {Ex}_{1}} \times {He}_{2}$ (9)

Then it is easy to see from Fig. 2(b) that the blank area is covered by a virtual cloud. Thus, a virtual rule is generated and added to the cloud rule base.

4 Modeling process and laboratory experiment

4.1 Laboratory experimental process

The experiments were performed on a lab-scale tumbling mill with size φ360 × 450 mm, as shown in Fig. 3. There are three accelerometer sensors installed on the ball mill. As the sensor installed under the bearing and opposite to the motor side was less intrusive to other factors, sensor 1 was chosen to collect vibration signals. The sampling frequency was set to a high value as 50kHz so as to recover the input analog signals from the recorded data. To generate the training and test data set, the mill was first charged with grinding balls. The material was increased from 1L to 20L with the step of 1L, so 20 FLs of data were collected. The length of the data of each FL is 50kHz × 60s = 3 × 10⁶. The data of each FL is divided into 22 segments, and the length of every segment is 131072, so a data set of 22 × 20 samples is acquired.

4.2 Feature analysis of vibration signal

The mechanical grinding of ball mill can generate strong vibration signals governed by the changes of the grinding state, which contains information correlating with the FL. As these useful vibration signals are usually buried in a wide-band random noise signals in the time domain [16], it is necessary to transform the vibration signals picked up by accelerometer sensor from time domain into frequency domain at first. Then spectral analysis method is used to transform the signals into frequency domain, and the feature values are obtained by accumulating the PSD in the selected frequency band.

Suppose that the sampling sequence of vibration signal under a certain FL is denoted as s = [s₁, s₂, …, s_k, …, s_K], where s_k stands for s_k (n), which are the vibration signal vectors sampled by F_s, with a length N. The power spectrum of vibration signal s_k by Welch’s method is described as follows: ${\tilde{P}}_{PER} (ω) = \frac{1}{MUL} \sum_{i = 1}^{L} {| \sum_{n = 0}^{M - 1} s_{k}^{i} (n) W (n) e^{- j ω n} |}^{2}$ (10) $U = \frac{1}{M} \sum_{n = 0}^{M - 1} W^{2} (n)$ (11) where W (n) is Hamming Window Function, L is the number of segments, M is the point number in each segment, and U is a normalization factor for the power.

The PSD extracted by Welch’s method is shown in Fig. 4. It demonstrates that there is a decreasing trend in the frequency spectrum energy along with the increasing of FLs, and the energy of vibration signals is mainly concentrated within the frequency range of 600Hz ∼ 5400Hz band. So, feature value is calculated based on spectrum in [f_l, f_h]=[600Hz, 5400Hz], using formula (12): $x = sum ({\tilde{P}}_{PER} (ω)), ω \in [2 π f_{l}, 2 π f_{h}]$ (12)

The feature value exhibits obvious randomness when being observed under a certain FL, and presents normal distribution in general, as shown in Fig. 5. In addition, the variance of feature decreases as the FL increases. Then it can be inferred that:

Due to the uncertainty produced in the tumbling of balls inside a ball mill, the vibration signal has strong randomness;

There is a stable tendency in randomness;

Variation of variances under different FLs reflects the variation of numerical characteristics in different concepts.

Considering the strong randomness and uncertainty in vibration signals of ball mill, cloud model is introduced to realize uncertainty reasoning so that the soft sensor model can be built for FL measurement.

4.3 Soft sensor modeling process based on cloud model

The measurement process has two phases: off-line modeling and on-line measurement, shown in Fig. 6. Off-line modeling is the preparation stage, which generates antecedent and consequent clouds of qualitative rules used for the reasoning process in the on-line measurement.

Phase I: Off-line modeling includes three steps in the following:

Step 1: Calculate feature values with the method presented in previous section from training data that have been segmented.

Step 2: For the feature value sequences obtained above, calculate the cloud numerical characteristics and generate antecedent clouds with BCG algorithm.

Step 3: Build consequent clouds corresponding to the antecedent clouds in accord with FL information.

Phase II: Online measurement is designed to conduct uncertainty reasoning for the input test data according to the complete antecedent and consequent clouds so as to get the measurement result of FL. The steps are as follows.

Step 1: Calculate the feature values from the test data segments.

Step 2: Input the feature values to the rule antecedents that are represented by condition X cloud generator. Determine whether the rules should be fired according to the “3En principle”. Once satisfied, a certain number of cloud drops with membership degrees are generated.

Step 3: Take the membership degrees of the antecedent cloud drops as the input of the consequents to control the condition Y cloud generator to produce a set of drops quantitatively.

Step 4: Weight the output cloud drops of the consequent clouds to get the measurement results.

4.4 Results of laboratory experiment

4.4.1 Result in the case of sufficient samples

In the case of sufficient samples, all of the 20 groups of data are taken as experimental samples. About two thirds of the collected data is selected from each group as training set to build soft model. The rest of the collected data serves as test data set to verify the built model. So the training set of 300 samples and test set of 140 samples are obtained.

The first phase is modeling, which is to establish antecedent and consequent clouds based on the training set. At first, the FL information are represented by calculating numerical characteristics of antecedent clouds. This process is actually to extract numerical characteristic information that represent FL concepts from the vibration signals obtained under different FLs. Then BCG algorithm mentioned above is used to establish basic antecedent clouds for the 20 FLs. The antecedent clouds in the case of sufficient samples are shown in Fig. 7. From the figure we find: 1) Ex of each cloud decreases as the FL increases, which is in accord with the change of vibration signals during the grinding process. 2) Variation of each cloud’s En reflects the randomness of vibration signals under different FLs. This just verifies the inference of vibration signals in previous section.

After the antecedents of reasoning system are built, it is needed to build the corresponding consequent models. Generally the parameters of consequent clouds can be obtained by BCG. However in the data set of laboratory, though there are several different feature values in every working condition, the corresponding labels are the same, which makes BCG cannot be used to acquire the consequent clouds. So the parameters of consequent clouds are determined by the numerical experiment. The Ex of a consequent cloud is the corresponding FL, and En and He are determined based on the principle of minimizing the Root Mean Square Error (RMSE). By taking different values of parameters En and He, several experiments were conducted. Without changing other parameters, the value of En have great effect on the prediction result evaluated by RMSE, while He has little effect, shown in Fig. 8, which is just a typical experimental result. This can be explained by the definition of En and He. The semantic range of the concept represented by cloud model extended as He increased step by step [24]. In order to avoid high value of He, which may lead to the lack of consensus for a concept, parameters of En and He should meet He < En/3. The consequent clouds corresponding to the antecedent clouds are defined within the universe of FL based on the principle mentioned in the previous section so as to meet the condition of reasoning rules. Thus, we can get the parameters of the consequent clouds. The parameters of both antecedent and consequent clouds are given in Table 1.

The second phase is the reasoning process for the given input test data set based on the antecedent and consequent rules established above. Assume that the input test set is x_j, j = 1, …, S, S is the number of test samples. Its implementation steps are recapitulated as follows:

Step 1: j=1;

Step 2: Take the test feature x_j as condition X cloud generator’s input of the antecedent clouds Cx_i (i = 1, ⋯ , n). For each Cx_i, generate antecedent drops and calculate their membership degree μ_Ai according to Eqs. (2) and (3).

Step 3: Determine whether the rule is activated according to threshold T_act set by the “3En principle”. Assuming antecedent clouds of the activated rules under current input are {Cx₁, ⋯ , Cx_m}(m ≤ n), and the corresponding membership degrees are {μ_A1, ⋯ , μ_Am}.

Step 4: Take {μ_A1, ⋯ , μ_Am} as condition Y cloud generator’s input of the consequent clouds. For each {Cy₁, ⋯ , Cy_m} of the activated rules, calculate its output according to Equations (4) and (5) to get m cloud drops (y_Bi, μ_Ai). If x_j ≥ Ex_i, then “-” is chosen in Equation (5) to calculate the consequent output; If x_j < Ex_i, “+” is used.

Step 5: For the m cloud drops, calculate the final output ${\hat{y}}_{j}$ using weighted mean method.

Step 6: j=j+1, if j ≤ S, go back to step 2, and if not, go on.

Step 7: Estimate the prediction performance by RMSE defined as follows: $RMSE = \sqrt{\frac{1}{S} \sum_{j = 1}^{S} {(y_{j} - {\hat{y}}_{j})}^{2}}$ (13)

where y_j is the true value of the jth test sample, and ${\hat{y}}_{j}$ is the corresponding predicted value.

For comparing with the proposed method based on cloud reasoning, the following modeling methods are introduced: PCR [16], PLSR [8], BPNN [6], SVM [11], LSSVM [28] and ELM [12]. And PCA [18] and PLS [12] are mainly used to extract features. So we can get 14 contrast methods totally. For these methods, the PSD in the frequency band of 600 ∼ 5400Hz is divided into segments of which length is 100 Hz. Therefore the input dimension of each sample is (5400-600)/100 = 48. All the prediction errors which are expressed by RMSE and model parameters are shown in Table 2.

Generally, in the process of cloud reasoning, the cloud rule base is compact, which means the whole input universe of discourse is covered by all rules completely. However, in practical industry field, the grinding of ball mill is a continuous process. It is costly to obtain the training data of FL continuously and accurately. In addition, the system model needs to be simplified and its complexity should be reduced so as to improve the efficiency of the system. Thus, part of the training samples are selected randomly to build the system model to verify its feasibility in the case of insufficient samples.

4.4.2 Result in the case of insufficient samples

When the training samples are not sufficient, it is needed to judge whether the cloud rule base is sparse or compact at first. On the premise of “3En principe”, if there is no blank area between any two adjacent clouds of the antecedents, it is considered that the cloud rule base established is compact, and the test steps are as mentioned above. Otherwise, once there is a blank area, then the cloud rule base is sparse, which may lead to a bad result or even no result. So virtual cloud method is employed to solve the problem of sparse cloud rule base and new rules can be generated and added to the rule base. Thus the reasoning process can go well for acquiring prediction result. No matter the training samples are sufficient or not, the reasoning process is the same. The modeling process in the case of insufficient training samples is as follows:

Step 1: Establish the rule antecedents with the training samples. The way of calculating the numerical characteristics of each antecedent cloud is the same with the method in the case of sufficient samples. Then put the antecedent clouds in ascending order of Ex_i priority as Cx₁ < Cx₂ < ⋯ < Cx_i < Cx_i+1 < ⋯ < Cx_t. Similarly, rule consequents corresponding to rule antecedents can be acquired.

Step 2: On the premise of 3En interval, if the parameters of any two adjacent antecedent clouds meet Ex_i + 3En_i > Ex_i+1 - 3En_i+1, it can be judged that there is no blank area, and the reasoning process is as mentioned before. If not, there must be at least one blank area, which means the cloud rule base is sparse.

Step 3: Once there exists a blank area between two adjacent antecedent clouds, a new cloud is added to the cloud rule base by the algorithm of virtual cloud based on the parameters of the two adjacent clouds, according to Equations (7), (8) and (9). In addition, a new consequent cloud is also added between the two consequent clouds corresponding to the antecedent clouds in the same way.

Step 4: Repeat Steps 2 and 3 until all the blank areas are covered by virtual clouds.

During the modeling process, 3 different groups of insufficient training samples are selected for testing: G₁ = {1L,4L,6L,10L,16L,20L}; G₂ = {1L,7L,11L, 16L,20L}; G₃ = {1L,8L,15L,20L}. The original antecedent clouds which are obtained from the insufficient training samples are shown in blue points in Fig. 9(a) ∼ Fig. 9(c). Obviously, there exist blank areas among these original clouds in each group, which may lead to the lack of rules for regular reasoning. Therefore, virtual clouds, shown in red in Fig. 9, are used to cover these blank areas. Thus, virtual rules are added to the cloud rule base.

To verify the model’s effectiveness and stability in the case of insufficient samples, 15 segments of each FL in the above 3 different groups are chosen randomly for modeling. The test samples cover the whole FLs of 1∼20L, whose features are the rest 7 segments corresponding to the training samples. For each group, tests are conducted 10 times with different training and test samples to acquire the average RMSE. Likewise, 14 traditional methods are chosen for comparison. The average RMSEs of these methods for different groups are given in Table 3. Then boxplot graphs are used to give clear illustration of the robustness, as shown in Fig. 10(a) ∼ Fig. 10(c). The labels in the x-axis in every boxplot graph represent the corresponding methods in Table 3.

4.5 Analysis and discussions

Table 2 reveals that in the case of sufficient samples, the cloud method can reach a relatively better performance compared with PCR, PLSR, BPNN and ELM models but not so good as SVM and LSSVM. Precision of PCA-BPNN, PLS-BPNN, PCA-ELM, PLS-ELM, PCA-SVM, PLS-SVM, PCA-LSSVM and PLS-LSSVM models are better than single BPNN, ELM, SVM and LSSVM, because of the feature extraction. In the case of insufficient training samples, both the number of rules and the complexity of the model are reduced. Inevitably, the measuring accuracy decreases as a whole compared with the sufficient case. However, the cloud reasoning method has a better prediction accuracy than other 14 methods in the case of insufficient training samples. In addition, the results of the 10 times experiments for each different groups, as shown in Fig. 10(a) ∼ Fig. 10(c), demonstrate that the cloud reasoning method has not only best prediction accuracy but also good robustness.

As a mathematical tool to represent uncertainty including fuzziness and randomness of concepts, cloud model is successfully used to deal with the uncertainty of vibration signals inside ball mill system. On the one hand, the good property of approximating any non-linear function of cloud model system [29] provides a theoretic basis for the good prediction result of this soft sensor method. On the other hand, the introduction of virtual cloud can solve the problem of sparse rules in the case of insufficient samples effectively.

5 Application of cloud model to monitor FL in practice process

5.1 Tubular ball mill coal pulverizing system

A tubular ball mill coal pulverizing system of thermal power plant usually consists of coal feeder, tubular ball mill, coarse classifier, fine classifier, some adjustable valves, baffles and pipes, which are illustrated in Fig. 11. The working procedure of the system is: raw coal is fed to the mill by coal feeder, and pulverized in the mill; coal powder having been pulverized is carried by the dry hot air, and fed to the coarse classifier; unqualified coal powder will be back to the mill again through certain pipe, and qualified coal powder is sent to fine classifier, then is saved to pulverized coal feeder, and blown to boiler for combusting.

The coal storage in the ball mill needs to be as close as possible to the optimal storage in actual process, which is a key factor to improve the efficiency of the pulverizing system. It is usually controlled by adjusting Coal Feed Rate (CFR). In order to improve the drying capacity, the temperature of hot air should be raised. And the temperature in outlet should be kept below a specified value by changing the valve position of hot and fresh air to prevent explosion of the coal dust that the mill sends out. Also the ventilation of mill must be kept at a good condition to transport powder to pulverized coal bunker through ventilating duct.

5.2 Measurement results in actual application

Field test was carried in a power plant using the measurement and control device for ball mill level. The type of ball mill is DTM380/600, which uses scraper coal-feeder, and whose maximum CFR is 60t/h. The vibration signals were collected by an accelerometer installed on the front bearing pedestal, and then processed by Digital Signal Processor (DSP). And the original signals were stored in a high speed Secure Digital (SD) memory card. The other process parameters related to the FL which consisted of CFR, inlet-outlet Pressure Difference of ball mill (PD), Current of Motor for driving ball mill (CM), Valve position of Hot Air (VHA), Valve position of Fresh Air (VFA) were also recorded in the Distributed Control System (DCS).

In actual process, the ball mill should be started manually in the premise of meeting safety rules, and run at the high level stage, which has a good economic performance and needs to be focused on. The field test took nearly 2 hours, experiencing the process from low to high level. At the starting stage, with the CFR increasing, the FL increased gradually. At about 1700s, the level increased fast. At about 3000s, purging wasconducted to prevent the occurring of the mill to be blocked, which would lead to the PD fluctuation. CFR reached the maximum of 49t/h at the last stage when the mill had a high level. And the field process data is shown in Fig. 12.

5.2.1 Measurement results of sufficient samples

The antecedent and consequent cloud model are constructed as follows:

(1) Considering the actual condition, power spectrum is calculated using Welch’s method, and feature value is calculated by formula (14) using frequency band in [f_l, f_h]=[300Hz, 2700Hz]: $x = mean (lg ({\tilde{P}}_{PER} (ω)), ω \in [2 π f_{l}, 2 π f_{h}]$ (14)

Then the data set {< x_i, y_i > |i = 1, 2, ⋯ , N} are obtained, where x_i is the feature calculated using Equation (14) and y_i is the corresponding FL of x_i. In the data acquired, FL ranges from 10% to 96% , which means y_i ∈ [10 96]. It is important to note that the level of 100% is the maximal level the field engineers set, not the level the mill is filled fully.

(2) The training set is divided into 7 subsets in a uniform interval according to the FL: $\begin{matrix} D_{1}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [10 22]}, \\ D_{2}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [23 34]}, \\ D_{3}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [35 46]}, \\ D_{4}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [47 59]}, \\ D_{5}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [60 71]}, \\ D_{6}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [72 83]}, \\ D_{7}^{s} & = & {< x_{i}, y_{i} > | y_{i} \in [84 96]} . \end{matrix}$

(3) The antecedent clouds ${Cx}_{1}^{s} \sim {Cx}_{7}^{s}$ are obtained by BCG using the feature values of $D_{1}^{s} \sim D_{7}^{s}$ , which are illustrated in Fig. 13(a) and given in Table 4. And the consequent clouds ${Cy}_{1}^{s} \sim {Cy}_{7}^{s}$ which represent the concepts of FLs, are obtained using BCG, and are given in Table 4.

The measurement result of sufficient samples is presented in Fig. 14(a).

5.2.2 Measurement results of insufficient samples

The data whose FLs are in the range from 23% to 39% are deleted. According the partition principle above, the subsets obtained are: $\begin{matrix} D_{1}^{u} & = & {< x_{i}, y_{i} > | y_{i} \in [10 22]}, \\ D_{2}^{u} & = & φ, \\ D_{3}^{u} & = & {< x_{i}, y_{i} > | y_{i} \in [40 46]}, \\ D_{4}^{u} & = & {< x_{i}, y_{i} > | y_{i} \in [47 59]}, \\ D_{5}^{u} & = & {< x_{i}, y_{i} > | y_{i} \in [60 71]}, \\ D_{6}^{u} & = & {< x_{i}, y_{i} > | y_{i} \in [72 83]}, \\ D_{7}^{u} & = & {< x_{i}, y_{i} > | y_{i} \in [84 96]} . \end{matrix}$

The antecedent clouds ${Cx}_{1}^{u}, {Cx}_{3}^{u} \sim {Cx}_{7}^{u}$ are shown in Fig. 13(b), and the corresponding consequent clouds ${Cy}_{1}^{u}, {Cy}_{3}^{u} \sim {Cy}_{7}^{u}$ representing the concepts of FLs are given in Table 4. It can be seen in antecedent clouds it lacks the corresponding rules between ${Cx}_{1}^{u}$ and ${Cx}_{3}^{u}$ . In this condition, if a small activation threshold T_act=0.0001 is used, the reasoning and measurement can be conducted, and the result is shown in Fig. 14(b), which has a larger error with theexpected FL.

5.2.3 Measurement results of insufficient samples with virtual cloud interpolation

To verify the effectiveness of cloud model interpolation, according the method mentioned in Subsection 3.4, the clouds are interpolated in the areas based on Definition. 3, and their numerical characteristics (Ex, En, He) are calculated using Equations (7), (8) and (9). The antecedent clouds ${Cx}_{1}^{v} \sim {Cx}_{7}^{v}$ after virtual cloud interpolation are shown in Fig. 13(c), and the corresponding consequent clouds ${Cy}_{1}^{v} \sim {Cy}_{7}^{v}$ are given in Table 4. The cloud ${Cx}_{2}^{v}$ after interpolation is similar to the cloud ${Cx}_{2}^{s}$ acquired in sufficient case. The measurement result is shown in Fig. 14(c), and it can be seen the RMSE is obviously lower after virtual cloud interpolation.

5.3 Summary

According to Fig. 12, the PD can not reflect the level because of the affection of the hot and fresh airflow, and the CM also has some differences with FL in the trend, due to the change of rotational inertia caused by the variation of steel ball mass and coal storage. The level has a positive correlation relationship with CFR, and due to pipe delay, it lags behind CFR. So PD, CM and CFR can not reflect the FL accurately andtimely.

In the insufficient case, because of lacking modeling samples whose levels are from 23% FL to 39% FL, the measurement accuracy is lower. In Fig. 14(b), the predicted levels are obviously lower from 20% FL to 40% FL. If the method of virtual cloud is used, the reasoning problem can be settled, and the measurement accuracy would be improved.

The activation threshold has an obvious influence in the reasoning process. The cloud model in Fig. 13 of the insufficient case can be reasoned normally, because the low threshold of T_act =0.0001 is set, the MECs of cloud model extend to the positive and negative infinity, and the corresponding rules can be activated. But it results a lower accuracy. And if the threshold is higher, such as having a value of 0.011 corresponding to 3En, reasoning can not be conducted due to lack of rules. In the sufficient case, if we take a high threshold, the information of main area can be used effectively, and accuracy of reasoning will be higher.

6 Conclusion

This paper presents an effective approach for ball mill fill level measurement based on cloud model and vibration signals. At first, cloud model is used to represent the antecedent and consequent concepts based on the feature values of vibration signals and fill levels. Thus the reasoning system is formed. Then the predicted fill levels are obtained by the uncertain reasoning based on the cloud model. If the rule base is sparse, virtual cloud is introduced to generate virtual rules so as to complete the rule base. In order to verify the performance and the effectiveness of the proposed method, experiments based on cloud model, contrast methods and an application of cloud model to monitor the fill level in a industry field were conducted. The test results show that the proposed method is more robust and feasible especially in the case of insufficient FL information compared with the contrast methods.

Currently, the theory of virtual cloud interpolation is not complete. The rule of determining whether the rule base is sparse is based on the “3En principle” of cloud model, and the information in the 3En interval is used to reason. However, whether interpolation threshold can be other values and the relationship between the interpolation accuracy and threshold need be further researched. Also how to define the degree of sparseness and to determine the threshold should be studied. It is significant to study the universality of the interpolation method. In this research, the data-driven cloud model can transform quantitative expressions to their qualitative concepts, but the parameters of the cloud model are fixed because the data set space is partitioned in a uniform interval toughly. It seems interesting to investigate the optimization of parameters to fit the nonlinear map better. For example, if an adaptive soft partition method is employed, the characteristics and advantages of cloud model could be mined fully. Additionally, the multivariable reasoning and function approximation based on multidimensional cloud model will be studied in the future work.

Acknowledgments

The work is supported by the National NaturalScience Foundation of China (No. 61450011) and the Natural Science Foundation of Shanxi Province (No. 2015011052).

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