Data-driven-based self-healing control of abnormal feeding conditions in thickening–dewatering process

Minerals Engineering 146 (2020) 106141

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Data-driven-based self-healing control of abnormal feeding conditions in thickening–dewatering process

T

⁎

Runda Jiaa,b,c, , Bin Zhanga, Dakuo Hea,b, Zhizhong Maoa,b, Fei Chud a

School of Information Science & Engineering, Northeastern University, Shenyang 110004, China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China c Liaoning Key Laboratory of Intelligent Diagnosis and Safety for Metallurgical Industry, Northeastern University, Shenyang 110004, China d School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Self-healing control Thickening–dewatering process Abnormal feeding conditions Dynamic data-driven model

The issue of self-healing control of abnormal feeding conditions in the thickening–dewatering process is considered herein. This process is widely used in the mineral processing industry to separate ores from the slurry. Owing to the instability of the upstream flotation process, the flow rate of feeding slurry generally fluctuates substantially. In this scenario, decision-making with regard to the economical operation of the thickening–dewatering process is challenging. Moreover, it results in the pressure rake of the thickener or ore leakage of the filter press. To address these abnormalities, a data-driven-based self-healing control scheme is proposed in this study. The contributions can be summarized as follows. (1) Based on the information provided by the pressure sensors inside the thickener, a dynamic data-driven model structure is established to predict the behaviors of the thickener. The underflow concentration is described by a nonlinear KPRM model, and the future trends of the pressure sensors are calibrated by several dynamic ARX models. (2) The FDA classifier is used to identify the abnormal feeding conditions, and the average future trajectories of the input variables are estimated; (3) The data-driven-based multitiered dynamic optimization problem is formulated to address abnormalities. The optimization results are obtained by solving these problems sequentially, with the former tiers providing the information to the latter ones. Experiments have been carried out in a self-developed simulation platform. The proposed self-healing control scheme can maintain the safety of the thickening–dewatering process as well as consider the energy consumption and operator convenience.

1. Introduction Mineral processing, also known as ore dressing, is the process of separating commercially valuable minerals from their ores (La Brooy et al., 1994). For gold mineral processing, the following main subprocesses are generally performed: grinding process, flotation process, and concentrating process. Owing to the complexity of the industrial field, abnormal conditions generally occur in gold mineral processing plants. These abnormalities are the departures of the process from an acceptable normal operating range, and they locates the gray area between normal operation and emergencies (Huang et al., 2002). Because the gold mineral processing plant is generally of a large scale, selfhealing control of the abnormal conditions is important to limit the influences of the abnormalities and guarantee that the plant operates safely. Inappropriate handling of the abnormal conditions results in severe financial loss or even in emergencies. Although a number of

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researchers have focused on optimization control of the thickening process (Tan et al., 2015, 2017; Bergh et al., 2015; Gálvez et al., 2014; Martin, 2004), few studies concerned the adjustment of the abnormal conditions in this field. After the flotation process, the slurry should be concentrated and delivered to the downstream hydrometallurgy plant. The thickener and filter press are generally used to achieve this. This concentrating process is named as thickening–dewatering process in this work. Owing to the maintenance or equipment failure in the flotation process, significant fluctuations appear occasionally in the feeding flow rate and concentration of the thickener. Because the main influencing factors of the thickening–dewatering process originate from the flotation process, in this work, we confined our attention to the abnormal feeding conditions. When similar abnormal feeding conditions prevail, the regular operating procedure is disrupted. On the one hand, the operator should expeditiously identify the causes of abnormality. On the other hand,

Corresponding author at: School of Information Science & Engineering, Northeastern University, Shenyang 110004, China. E-mail address: [email protected] (R. Jia).

https://doi.org/10.1016/j.mineng.2019.106141 Received 4 March 2019; Received in revised form 1 November 2019; Accepted 25 November 2019 Available online 29 November 2019 0892-6875/ © 2019 Elsevier Ltd. All rights reserved.

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operational feedback control was composed of a neural network-based loop set-point optimizer, an overload diagnosis, and a self-healing controller. Wu et al. (2015) also developed a data-driven abnormal condition self-healing control system for a fused magnesium furnace. The identification rules were used to detect the abnormal conditions, and the self-healing control was developed by case-based reasoning. The major advantage of their method was that only the operating data were required, whereas knowledge of the process dynamic was not. Recently, Aumi and Mhaskar (2009) proposed safe-steer of a batch process system through a reverse-time reachability region by using model predictive control based on a data-driven model. In their approach, multiple dynamic local linear models were used to describe the process variables. In order to control the end-product qualities of a batch process, they also employed a static latent variable model to describe the relationship between the input and output variables (Aumi et al., 2013; Corbett and Mhaskar, 2017). Similar strategies were studied extensively in the field of batch process modeling (Flores-Cerrillo and MacGregor, 2004; Jia et al., 2016). To realize the coordinated control of the thickener and filter press under abnormal feeding conditions, a data-driven model to predict the behaviors of the thickening–dewatering process will also be required. Because the filter press used in this work is also a batch system, these aforementioned modeling strategies for batch processes can also be utilized to calibrate the datadriven model of the thickening–dewatering process with certain modifications. Thus, in this work, a data-driven model-based self-healing control of abnormal feeding conditions in a thickening–dewatering process is presented. First, the pressure sensors are installed in the thickener to determine its internal state. Based on the information provided by the sensors, a dynamic data-driven model is constructed to predict the behavior of the thickening process. The model is composed of a kernel partial robust M-regression (KPRM) model for underflow concentration, and several autoregressive with exogenous (ARX) models for pressure. Then, the Fisher discriminant analysis (FDA) classifier is used to identify the abnormal feeding conditions. Then, the self-healing control scheme is developed by solving a data-drivenmodel-based multitiered dynamic optimization problem, which operates the thickening–dewatering process safely and cost-optimally. Finally, experiments are carried out to verify the effectiveness of the proposed scheme. The rest of this paper is organized as follows: An analysis of the abnormal feeding conditions in Section 2. The data-driven-based process models for the thickening–dewatering process are introduced in Section 3. In Section 4, the abnormal feeding condition identification and self-healing control scheme are described in detail. The experiments conducted to verify the efficiency of the proposed scheme are presented in Section 5. The conclusions drawn are presented in Section 6.

effective decisions have to be made to ensure process safety. However, because the information on the process is limited and challenging to obtain in a short time, decisions generated to control the abnormalities are generally non-optimal. For example, owing to the blockage of the inflator, which is used to generate bubbles, the flotation column is required to be emptied to remove the fault. Under this condition, the feeding flow rate and concentration increase abruptly and continue for a certain period. An excessive ore quantity inside the thickener prevents the rake from rotating (pressure rake) and generally causes severe accidents in mineral processing plants. When a similar accident occurs, the thickening–dewatering process is halted, and the ore inside the thickener should be removed by manual digging. To prevent pressure rake, the operator turns on the underflow pump to start dewatering. Under this scenario, the energy consumption increases significantly. If the operator does not detect the abnormal phenomena, predetermined regulations result in an emergency. Therefore, to improve the operating performances, it is necessary to design a self-healing control scheme to assist the operator in effective decision-making under abnormal feeding conditions. Recently, Li et al. (2017) proposed a safe control scheme for the thickening process, based on Bayesian network (BN). In their approach, a BN that combines the operator experiences and quantitative data information was established to remove the abnormalities of the thickening process. After acquiring the abnormal appearances as evidences, the posterior probabilities of the manipulated variables with different grades can be obtained by BN reasoning. If the abnormality is still present, the BN reasoning combined with the latest information is used again to generate the new set-points for the manipulated variables. Although their approach can efficiently provide real-time decisions to remove the abnormalities, the coordinated control of the thickener and filter press was not considered in their work. Because the feeding concentration of the filter press cannot be guaranteed, the energy consumption and the rate of ore leakage would also increase remarkably with the elimination of the abnormalities. Huang et al. (2002) proposed to use dynamic first principle models within a fault accommodation system (they solved a dynamic optimization problem) rather than the manual table-lookup method typically adopted by the operators. A systematic strategy for optimal plant operation during partial shutdowns was presented by Chong and Swartz (Chong and Swartz, 2013). In their approach, the optimal control trajectories were also obtained by solving a dynamic optimization problem. Moreover, the multitiered model predictive controller was employed to prioritize the multiple competing objectives and achieve trade-off between them. Recently, the optimal response under partial plant shutdown with discontinuous dynamic models was considered, and a discrete-time mixed-integer dynamic optimization was embedded in the controller (Chong and Swartz, 2016). A safe-parking framework was developed in references (Aumi and Mhaskar, 2009; Gandhi and Mhaskar, 2008) to address the issue of optimum operation of the process during fault-rectification and ensure a smooth resumption of normality. The safe-parking framework was adopted in heating, ventilation, and air conditioning systems in a subsequent study by Shahnazari et al. (2018). A linear discrete time dynamic model was employed to describe the systems. Although the aforementioned methods can effectively solve the problem of abnormal situation management to a certain extent, it is challenging to use the knowledge-based model to control the abnormalities in the thickening–dewatering process, owing to the unavailability of on-line information inside the thickener and filter press. The self-healing control scheme was originally used in industrial wireless sensor network (Gungor and Hancke, 2009) and power systems (You et al., 2003; Seethalekshmi et al., 2011; Karen and Sarma, 2004). In these applications, the network structure reconstruction approaches were employed to eliminate the abnormalities. Recently, a data-drivenbased optimization control for safe operation of a hematite grinding process was presented by Dai et al. (2015). In their work, the

2. Analysis of abnormal feeding conditions The flow chart of the thickening–dewatering process is shown in Fig. 1. The main equipment are a thickener, a buffer tank, and two plate and frame filter presses (one is used and the other is spare). In this process, the slurry, which originates from the flotation process, is first condensed to a relatively high concentration by the thickener. Then, the filter press further decreases the water content and produces the filter cake, which is delivered to the downstream hydrometallurgy plant. The feeding concentration of the plate and frame filter press should be within a reasonable range. If the feeding concentration is inadequate, it would be challenging to clean up the filter cloth after a cycle run. Under this scenario, the rate of ore leakage increases substantially. Moreover, owing to the low feeding concentration, the plate and frame filter press consumes more time to finish a cycle and thereby increase the energy consumption. Meanwhile, if the feeding concentration is excessive, the risk of blockage at the inlet will also increase substantially. However, highly concentrated feeding slurry is effective for shortening the press 2

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Fig. 1. Flow chart of thickening–dewatering process.

filtering time and reducing the energy consumption. To improve the efficiency of the filter press and reduce the rate of ore leakage, the slurry bed level inside the thickener should be high enough to ensure adequate underflow concentration. Nevertheless, to prevent pressure rake, the slurry bed level should be maintained below the safety limit. Because the plate and frame filter press is a batch system, the operator must calculate the number of cycles in a shift and determine the proper startup time of the underflow pump. Owing to inadequate on-line information inside the thickener, under normal condition, the operator generally manipulates the underflow pump and filter press according to predetermined regulations and experiences. In general, there are two distinct abnormal feeding conditions. They are described below:

operator owing to the subjectivity, and a few of these decisions may not be precise enough to handle the abnormality.

3. Data-driven model of thickening–dewatering process To predict the behaviors of the thickening–dewatering process, a data-driven model (composed of several pressure models and an underflow concentration model) should be constructed. In this section, the model structure is first elaborated. Then, the calibration of ARX models to predict the pressure inside the thickener is described. Finally, the KPRM algorithm is used to describe the relationship between the pressure sensors and underflow concentration.

(1) The inflator blocking compels the flotation column to remove all the slurry in it. This is followed by the abnormality wherein the feeding flow rate and concentration increase abruptly and continue for a certain period. (2) Owing to the decline in the slurry level in the flotation column, the feeding flow rate and concentration decreases to a relative low level and continue for a certain period.

3.1. Model structure To implement the self-healing control of the thickener and filter press under abnormal feeding conditions, several models are required to predict the behaviors of the process. In this example, the feeding flow rate and concentration, and the underflow flow rate can be measured on-line. Meanwhile, the underflow concentration can only be measured off-line by manual sampling. Although several types of first principle models have been proposed to describe the thickening process (Bürger et al., 2013, 2012; Kim and Klima, 2004), it is challenging to model the industrial thickener with only the first principle. In order to determine the operating state inside the thickener, several pressure sensors are first installed in it, as shown in Fig. 1. Because the measured values of pressure can indirectly reflect the concentration distribution inside the thickener, the data obtained by the pressure sensors can be used to describe the operating state inside the thickener. However, for the thickening–dewatering process, the most important quality variable is the underflow concentration. Thus, even if the concentration distribution inside the thickener is specified, another datadriven model is required to predict the underflow concentration by using the measured values of pressure. The whole model structure of the thickener used in this work is shown in Fig. 2. In this figure, the behaviors of the thickener are predicted by two dynamic data-driven models as in the batch process. This is elaborated in the following sections.

For abnormality (1), the predetermined number of cycles may not satisfy the safety requirement of the thickener. To prevent pressure rake, the operator turns on the underflow pump and begins dewatering when an abnormal phenomenon is detected. Under this scenario, the predetermined regulations are disrupted, and the energy consumption increases significantly. Furthermore, the thickening–dewatering process consumes a long time to return to the normal operating condition. When the operator does not detect the abnormal phenomenon, the predetermined regulations occasionally result in an emergency. When abnormality (2) occurs, the operator generally does not take action. This is because a lower feeding flow rate and concentration does not cause severe accident. However, with the decrease in the slurry bed level inside the thickener, the underflow concentration decreases sharply. Moreover, the slurry with low concentration may result in an increased rate of ore leakage of the filter press and reduce its efficiency. Occasionally, operators delay the startup time of the underflow pump in predetermined regulation to alleviate the effect of the abnormality, according to their experience. Nonetheless, decisions vary with the 3

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Fig. 2. Diagram of data-driven model structure for thickener.

regression coefficient matrix βil can be calculated using the following equation:

3.2. Predict the pressures inside the thickener The installed pressure sensors can detect the concentration distribution inside the thickener. It is generally determined by the feeding flow rate and concentration and the underflow flow rate and concentration according to prior knowledge. There are two states for the thickener in this example:

−1

βil = PilT Uil (TilT Pil PilT Uil) TilT yil

Because the underflow concentration cannot be measured online, after constructing the following KPRM model, the estimation of the underflow concentration is used to identify the pressure model in state (2). In Fig. 2, the parts colored red indicate state (2).

(1) The underflow pump is turned off, and only the slurry is fed into the thickener. Under this scenario, the slurry bed level increases gradually. This is followed by an increase in the measured value for each pressure sensor. (2) The underflow pump is turned on, and the slurry is carried out of the thickener simultaneously. Because the underflow flow rate and concentration are significantly higher than that of the feeding, the slurry bed level decreases gradually. This is followed by a decrease in the measured value for each pressure sensor.

Remark 1. The data-driven models used for predicting the pressures are linear autoregressive with exogenous (ARX) models. The number of lags in the inputs and outputs is equal to one. It should be noted that other numbers of lags can also be adopted in these data-driven models. The number can be determined by Akaike information criterion or Bayesian information criterion. However, in this work, an equal number of lags are selected for both the inputs and outputs for simplicity. Moreover, the field experiment results reveal that the model structures in Eq. (1) can effectively describe the features of the pressure sensors and that the prediction accuracy can also satisfy the requirement of the self-healing control scheme.

In this work, the two states are modeled using different model structures as follows:

pl (k + 1) =

T ⎧ pl (k ) + β1l x1l (k ), T ⎨ pl (k ) − β2l x2l (k ), ⎩

qf (k ) = 0 qf (k ) > 0

3.3. Underflow concentration model based on KPRM algorithm

(1)

In the thickening–dewatering process, the most important quality variable is the underflow concentration. The field experimental results reveal that the underflow concentration can be predicted by the measured values of the pressure sensors as illustrated in Eq. (5).

where x1l (k ) = [pl (k ), qf (k ), c f (k )]T and y1l (k ) = pl (k + 1) − pl (k ) are the input and output variables, respectively, corresponding to the lth pressure sensor for state (1). qf and c f are the feeding flow rate and concentration, respectively. x2l (k ) = [pl (k ), qf (k ), c f (k ), quf (k ), c uf̂ (k )]T and y2l (k ) = pl (k ) − pl (k + 1) are the input and output variables, respectively, corresponding to the lth pressure sensor for state (2). quf is the underflow flow rate. β1l and β2l are the vectors of the regression coefficients. Given plant data, an input data matrix Xil and an output data vector yil can be constructed corresponding to x il (k ) and yil (k ) ; here, i = 1, 2 . The model parameters can be identified by the linear partial least squares (PLS) algorithm (Wold et al., 2001). Mathematically, the PLS algorithm decomposes Xil and yil into the following form:

Xil = Til PilT + Eil

(2)

yil = Uil qil + fil

(3)

(4)

c uf̂ (k ) = f (p1 (k ), p2 (k ), …, pm (k ))

(5)

where c uf̂ is the predicted underflow concentration, pl (l = 1, 2, …, m) is the measured value of the lth pressure sensor, m is the number of pressure sensors, and f (·) is a nonlinear function that is required to be identified. To estimate the underflow concentration c uf̂ , the observations are first stored in the matrices, i.e., the input data matrix P = [p1, p2, …, pn]T and the corresponding output data vector c uf . Here, n is the number of observations, ps = [p1s , p2s , …, pns ]T is the sth input data vector, and pls (s = 1, 2, …, n) is the sth measured value in the training dataset of the lth pressure sensor. Because unavoidable outliers influence the prediction performances, the KPRM algorithm is used here to describe the nonlinear relationship between P and c uf . The nonlinear data structure in the input space is more likely to be linear after high-dimensional nonlinear mapping (Scholköpf et al., 1997; Shevade et al., 2000). Therefore, the KPLS algorithm is

where Til and Uil are the matrices of the latent variables of the lth pressure sensor for the ith state, Pil is the matrix of loading vectors corresponding to Til , qil is the loading vector corresponding to Uil , and Eil and fil are the residuals. When B latent variables are obtained, the 4

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Fig. 3. Diagram of operation cycle in thickening–dewatering process.

Step 2. Obtain the weighted kernel matrix K ← WKW and the corresponding weighted output data vector c uf ← Wc uf . Step 3. Perform KPLS analysis on the K and c uf to obtain the matrix of latent variables T , and calculate the residual rs . Step 4. Update the W by using Eqs. (8) and (10). Step 5. Loop back to Step 2 until the convergence of the regression coefficient vector γ ̂ is attained. γ ̂ can be computed using the following equation:

formulated in this feature space to extend the linear PLS algorithm to its nonlinear kernel form. By using the kernel trick, the KPLS algorithm can also avoid performing explicit nonlinear mapping (Wang and Jiao, 2017). In addition, if the Gaussian kernel function is used, the entries of the kernel matrix K can be calculated as follows:

κst = exp ⎛⎜− ⎝

ps − pt enσ 2

2

⎞⎟ ⎠

(6)

where σ 2 is the variance of the input data matrix P . e > 0 is a width parameter, which has to be tuned. As a result, the predictions on the training data set can be performed as follows:

c uf̂ = KU (T TKU)−1T Tc uf

γ ̂ = c Tuf T (T TT)−1

When the new input data vector pnew is available, the new kernel vector knew can also be obtained by using Eq. (6). Then, the predicted underflow concentration can be calculated as follows:

(7)

where T is the matrix of latent variables that can be derived from a sequence of NIPALS steps as in the KPLS algorithm (Wold et al., 2001). The partial robust M-regression (PRM) is a robust version of the PLS1 approach (Serneels et al., 2005). The aim of this algorithm is to construct a robust calibration model that effectively describes the trend of data majority. If W is a diagonal weight matrix, the kernel matrix K can also be weighted by WKW (Jia et al., 2010). Thus, applying the kernel trick, the linear PRM algorithm can also extend to its nonlinear kernel form. The KPRM algorithm uses the leverage weights and residual weights to downweight the outliers as in the PRM algorithm (Serneels et al., 2005). The leverage weight of the sth observation wsl can be calculated by the following equation:

ts − med(T) , θ⎞ wsl = h ⎛ med s t s − medt (T) ⎠ ⎝ ⎜

̂ = knew U (T TKU)−1T Tc uf c ufnew

(12)

4. Self-healing control of abnormal feeding conditions The plate and frame filter press is a batch process system. Therefore, describing the operation cycle is beneficial for implementing the selfhealing control scheme efficiently. In this section, the operation cycle is first defined, and the abnormal feeding conditions are identified by FDA classifier. The self-healing control scheme is finally achieved by solving a data-driven-based multitiered dynamic optimization problem.

⎟

(8) 4.1. Definition of operation cycle

Here, ts is the sth row of T , and s, t = 1, 2, …, n . The weight function can be described as follows:

1 h (z , θ) = (1 + |z θ|)2

Because the plate and frame filter press is a batch system, the operation cycle should be defined before performing the self-healing control of the abnormal feeding conditions. The operating state of the underflow pump in a cycle is shown in Fig. 3. k js is the start time of the jth cycle, and k jon and k joff are the startup and shutdown times, respectively, of the underflow pump in the jth cycle. For the current operation s cycle j, the previous shutdown time k joff − 1 is the current start time k j , and the current shutdown time k joff is the next start time k js+ 1. Thus, we define k js as belonging to the current cycle, whereas k joff is not included in the current one. Between k jon and k joff , the slurry flowing out of the thickener should be adequate for the filter press to complete a cycle run. Because the water content in the filter cake for each cycle run is nearly identical, k joff can be determined by using the following pseudo code:

(9)

where θ is a tuning constant. med(·) is the median estimate, and it considers the columns of T as vectors and computes a row vector of median values. Similarly, the residual weight of the sth observation wsr can be computed as follows:

rs , θ⎞ wsr = h ⎛ ⎝ meds rs − medt (rt ) ⎠ ⎜

(11)

⎟

(10)

where rs is the residual between the observed and predicted values of the sth output sample. Then, the global weight of the sth observation ws 1 2 is equal to (wsl wsr ) , and W = diag{w1, w2, …, wn} denotes a diagonal weight matrix with ws as the sth diagonal element. The KPRM algorithm is finally outlined as follows:

Algorithm 2. Inputs: the startup time of the underflow pump k jon in the jth cycle, the quantity of dry ore required for completing a cycle run Mfp . Output: the shutdown time of the underflow pump k joff in the jth cycle.

Algorithm 1. Step 1. The input data matrix P and output data vector c uf are first mean centered and scaled to unit variance. The starting values of the weight matrix W are set to n-dimensional identity matrix. 5

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Feeding flow rate (m /h)

R. Jia, et al.

class scatter matrix for the rth class can be calculated as follows:

60

Actual trajectory Estimated trajectory

50

∑

Sr =

(zk − zr )(zk − zr ) T (14)

zk ∈ χr

where zr is the classic mean vector for the rth class, and the within-class

40

3

scatter matrix is S w = ∑ Sr . Then, the between-class scatter matrix is 30 0

50

100

150

200

250

r=1

3

300

∑ nr (zr − z)(zr − z)T

Sb =

Time (min)

(15)

r=1

Concentration (%)

70

where z is the classic mean vector of all the observations. The objective of the FDA vector is to maximize the scatter among classes while minimizing the scatter within classes. The FDA vectors are equal to the eigenvectors va (a = 1, 2, …, A) of the generalized eigenvalue problem as follows:

Actual trajectory Estimated trajectory

60 50 40

(16)

S b va = λa S w va

30 0

50

100

150

200

250

300

where the eigenvalue λa indicates the degree of overall separability among the classes by projecting the data on to va . When FDA is applied for identifying abnormal feeding condition, the dimensionality reduction technique is applied to the data in all the classes simultaneously. Supposing V = [v1, v2, …, vA]T , the discriminant function can be formulated using the following equation:

Time (min) Fig. 4. Trends of feeding flow rate and concentration under abnormality (1).

M←0

k joff ← k jon

gr (z)

While M < Mfp

ρuf (k joff ) ←

−1

1 1 1 = − (z − zr ) TV ⎛ V TSr V⎞ V T (z − zr ) − 2 n 1 2 − ⎝ r ⎠

ρs

⎜

ρs − (ρs − ρ l ) c uf̂ (k joff )

1 ln ⎡det ⎛ V T S r V⎞ ⎤ ⎢ ⎝ nr − 1 ⎠⎥ ⎣ ⎦ ⎜

M ← M + quf (k joff ) c uf̂ (k joff ) ρuf (k joff )

⎟

(17)

where z is the new available observation. Once the abnormal feeding condition is identified, the different phases of the abnormality can be inferred according to experience. In Fig. 4, abnormality (1) is shown as an example. The thickening–dewatering process initially operates under normal condition, until abnormality (1) causes it to switch to the abnormal feeding condition. The abnormal phase starts at ka , when the feeding flow rate and concentration increase to a relatively high level. The abnormal phase proceeds until k e . Thereafter, the thickening-filter pressing process is ready to return to its original normal operating point. Then, the selfhealing control problem entails the determination of a set of operation cycles that ensures the safety of the process and operates it in a costoptimal fashion. This can be formulated as a data-driven model-based dynamic optimization problem, which is described in the following subsections.

k joff ← k joff + 1 End while

ρuf is the density of the underflow slurry, which can be calculated using c uf̂ according to the first principle. The solid density in slurry ρs and the liquid density in slurry ρ l are constants in this example. Finally, the implicit function relationship for k joff is indicated by using g (·) in this work, i.e., k joff = g (k jon, quf (k ), c uf̂ (k ))

⎟

(13)

In Fig. 3, the variation tendencies of pressure and underflow concentration in an operation cycle are also illustrated. In the jth operation cycle, the maximum pressure is obtained at k jon , and the minimum underflow concentration is acquired at k js or k joff .

Remark 2. Because the feeding flow rate and concentration generally fluctuate significantly, to reduce the false alarm rate, when several similar abnormities occur successively, the thickening–dewatering process is considered to be under a certain abnormal feeding condition. Based on an analysis of historical data, the number of successive abnormities is selected as five in this work. Thus, in Fig. 4, the abnormality can be identified when the time is equal to 5 min. However, if this time is selected as ka , the duration of the abnormality will be less than the actual time. In order to estimate the abnormal trajectories in Fig. 4 more accurately, the delay time of discrimination should be considered in the following self-healing control scheme. Shifting ka forward by 5 min may facilitate accurate decision-making.

4.2. Identification of abnormal feeding conditions Because the feeding flow rate and concentration are closely related to the abnormal feeding situations, the vector zk = [qf (k ), c f (k ), ef (k ), m f (k )]T is employed here to identify the abnormalities of the thickener. Here, ef is the instantaneous power of the feeding pump, and m f is the accumulated quantity of dry ore per unit time. However, in practical industrial scenarios, zk generally fluctuates substantially. Thus, it is challenging to use the measured values directly to identify the abnormalities. Because the abnormal data set for each abnormality can be accurately identified from the whole historical database, FDA could be used in this work to detect the abnormalities. FDA is a linear dimensionality reduction technique (Feng et al., 2016). It is optimal in terms of maximizing the separation among the normality and abnormalities. Define nr as the total number of observations for abnormal feeding condition identification and nr (r = 1, 2, 3) as the number of observations in the rth class. Here, r = 1 and r = 2 represent the abnormalities (1) and (2) respectively, and r = 3 represents the normal condition. With χr defined as the set of zk that belong to the class r, the within-

4.3. Multitiered optimization control of abnormal feeding conditions A simple method to operate the thickening–dewatering process is to turn on the underflow pump and start dewatering when the underflow concentration attains the desired value. However, because the electricity price varies with time and the plate and frame filter press is the major energy-consuming equipment in this process, this method of 6

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operation is generally uneconomical. Press filtering should be performed during the period with a lower electricity price. Moreover, the operator generally works intermittently rather than focus on the operating state of the thickener continuously. Therefore, the start time of each cycle is arranged by the predetermined regulation. Under normal condition, the objective of the optimization control is to reduce the energy consumption. However, when abnormal feeding condition prevails, the predetermined regulations may result in an emergency. Thus, the primary control objective under abnormal feeding conditions is to ensure the safety of the thickening–dewatering process. Considering the multitiered optimization, a self-healing control scheme is proposed for determining the new startup time of the underflow pump that can satisfy the safety requirement without significantly forgoing economy. The multitiered optimization strategy enables us to prioritize several competing objectives and specify the trade-offs between them. Separate optimization problems are solved sequentially, with the former tiers providing information to the latter ones (Rashid et al., 2016). The framework of the proposed self-healing control scheme based on multitiered optimization strategy is as follows; here, all the tiers are illustrated in detail: Tier 1: For abnormality (1), because the feeding flow rate and concentration increases, the data-driven model-based dynamic optimization problem that minimizes the maximum pressure of the bottom pressure sensor pm to guarantee the safety requirement of the thickener is solved. The optimal solution is recorded. This solution represents the lowest achievable pressure by the current number of cycles.

minimum total energy consumption of the underflow pump and the feeding pump of the plate and frame filter press. Because the electricity price varies over time, the total cost is used as the objective, which can be formulated as follows: off

min

k1on … k on* N

* k jon ⩾ k joff − 1, j = 2, …, N

k Noff∗ ⩽ k f − εk k joff

c uf̂ (k ) ⩾ c L

is the number of cycles obtained from the previous tier; P is where the electricity price; Euf and Efp are the power consumptions of the underflow pump and the feeding pump of the plate and frame filter press, respectively, per unit time; and t jfp is the running time of the feeding pump of the plate and frame filter press required to complete a batch run in the jth cycle, which can be calculated by the following empirical formula:

t jfp = α1·c −j 1 + α2

k off j

∑ c uf̂ (k )

⩾ k joff − 1, j = 2, …, N kNoff ⩽ k f − εk

cj =

k joff = g (k jon, quf (k ), c uf̂ (k ))

k = k on j

k joff − k jon

(22)

Tier 3: In the third tier, the objective function is replaced with another objective that minimizes the self-healing control effort, i.e., the new startup time of the underflow pump should be close to the reference ones to the extent feasible, subject to a minimum constraint on economic performance. When N ∗ equals to the original number of cycles N ,̃ the following optimization problem can be formulated:

(18)

where N is the remaining number of cycles in the current shift, ka is the time of abnormal condition occurrence, k f is the final time of the current shift, εk is the time for shift turnover, and c L is the lower bound of the underflow concentration that can be fed into the plate and frame filter press. If the optimal value of the objective pm∗ is within or equal to the upper bound of the pressure pU , i.e., pm∗ ⩽ pU , the remaining number of cycles could address the additional slurry. Otherwise, another cycle should be added in the current shift to prevent pressure rake. It should be noted that because the bottom press sensor can best reflect the state of the thickener, pm is used to define the safety limit. For abnormality (2), owing to the decrease in the feeding flow rate and concentration, a similar dynamic optimization problem that maximizes the minimum underflow concentration to reduce the fault rate of the plate and frame filter press is solved. Moreover, the optimal value of the objective is recorded. This value illustrates the highest achievable underflow concentration by the current number of cycles.

N*

min

on 2

∑ (k jon − k j̃ )

k1on … k on* j = 1 N

s.t. k1on ⩾ ka k jon

* ⩾ k joff − 1, j = 2, …, N

k Noff ⩽ k f − εk k joff = g (k jon, quf (k ), c uf̂ (k )) pn (k ) ⩽ pU c uf̂ (k ) ⩾ c L off

N * ⎛ kj ∑ ⎜ ∑ Euf P (k ) + j = 1 ⎜ k = k on ⎝ j

max min( c uf̂ (k ))

on k1on … kN

s.t. k1on ⩾ ka

fp k off j +k j

∑ k = k on j + δk

⎞ Efp P (k )⎟ ⩽ (1 + εE) E ∗ ⎟ ⎠

on k j̃

(23)

E∗

is the reference startup time of the underflow pump, is the where optimal total energy consumption obtained from tier 2, and εE is a tolerance parameter. When N ∗ is not equal to N ,̃ the following equation can be adopted to replace the objective in optimization problem (23).

k jon ⩾ k joff − 1, j = 2, …, N kNoff ⩽ k f − εk = g (k jon, quf (k ), c uf̂ (k )) pm (k ) ⩽ pU

(21)

where α1 and α2 are the parameters to be identified. cj is the average concentration that is fed into the filter press in the jth cycle. It can be computed as follows:

s.t. k1on ⩾ ka

k joff

(20)

N*

min max(pm (k ))

c uf̂ (k ) ⩾ c L

= g (k jon, quf (k ), c uf̂ (k )) pm (k ) ⩽ pU

on k1on … kN

k jon

fp

on

k j + δk + t j N * ⎛ kj ⎞ Efp P (k ) ⎟ ∑ ⎜ ∑ Euf P (k ) + ∑ ⎟ on + δ j = 1 ⎜ k = k on k = k k j ⎝ j ⎠ on s.t. k1 ⩾ ka

̃

N on 2 ⎧ ∗ ̃ ̃ min ∑ (k jon +1 − k j ) , N = N + 1 ⎪ k on on ⎪ 1 …k N * j=1 N* ⎨ on 2 ⎪ min ∑ (k jon − k j̃ + 1) , N ∗ = N ̃ − 1 on on ⎪ k1 … k N * j = 1 ⎩

(19)

∗ ∗ ̂ ⩾ cL , ̂ is at least c L , i.e., c uf If the optimal value of the objective c uf the remaining number of cycles could also be adopted in the current shift. Otherwise, the number of cycles should be decremented by one to reduce the fault rate of the plate and frame filter press. Tier 2: In this tier, the optimization problem that maximizes an economic objective function under the safety limit is solved, i.e., the

(24)

The optimization problem in this tier is convenient for operation, and the reference startup time can be used also as reference for the 7

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laboratory in Liaoning province through the cloud data server (data layer). In our laboratory, the process models for the thickening–dewatering process are calibrated to simulate the process layer of the actual ore dressing plant. In addition, the virtual basic automation system (basic automation layer) is established to make the simulation more realistic. On the one hand, under normal condition, the optimization control system for the thickening–dewatering process (application layer) can provide the decision information for operating the actual gold-ore dressing plant. On the other hand, the simulation platform in our laboratory can be used to evaluate the proposed optimization control methods by using the process models of the thickening–dewatering process and the virtual basic automation system. Although the experiments are performed by simulating the process layer and basic automation layer, the input process data (e.g., the feeding flow rate and concentration) is obtained from an actual ore dressing plant. Therefore, the operating results are closer to an actual industrial process. The dynamic optimization problem was formulated based on the established data-driven model and the proposed self-healing control scheme under abnormal feeding conditions. It was solved using Matlab R2016b. Then, the optimal decision can be obtained to control the abnormal feeding conditions. This can ensure the safety requirement of the thickener and reduce the fault rate of the filter press.

operators. Remark 3. There are two safety constraints in the proposed multitiered optimization problem. The first is with regard to the underflow concentration, i.e., c uf̂ (k ) ⩾ c L . As mentioned above, if the feeding concentration of the filter press does not satisfy the minimum requirement c L , the rate of ore leakage increases substantially. The second safety constraint is exerted on the mth pressure sensor, i.e., pm (k ) ⩽ pU . The bottom pressure sensor directly reflects the ore quantity inside the thickener. If the pressure value of the mth pressure sensor is higher than the upper bound pU , pressure rake accident is likely. Meanwhile, constraining this value is effective for controlling the maximum feeding concentration of the filter press, and reduces the risk of blockage at the inlet. Remark 4. The pressure pm (k ) and underflow concentration c uf̂ (k ) appear in the dynamic optimization problem of each tier. Because pm (k ) and c uf̂ (k ) are continuous time series, the dynamic optimization problem should be calculated at each time point. Under this scenario, substantial resources would be required to resolve these problems. However, in an operation cycle, the maximum pressure is obtained at the startup time of the underflow pump, and the minimum underflow concentration is acquired at the start time of the current cycle or the shutdown time of the underflow pump. Thus, we only need to establish the dynamic optimization problems on these discrete time points, and the computation effort required to solve the dynamic optimization problems can be reduced substantially.

5.1. Analyzing the prediction performances of the data-driven models 5.1.1. Underflow concentration prediction experiments First, the underflow concentration model identified by the KPRM algorithm is considered. In this case, three press sensors are installed inside the thickener. Fifty observations of underflow concentration are sampled from the actual gold-ore dressing plant. The corresponding measured values of pressure are selected from a historical database to calibrate the KPRM model. In addition, 25 new observations of underflow concentration are collected by manual sampling to test the prediction performances. In general, the number of latent variables retained in the KPRM model and the width parameter e affect the prediction performances significantly. To determine the two parameters, a three-dimensional

5. Experiments In this work, to verify the efficiency of the proposed self-healing control scheme, experiments were carried out using a self-developed “Optimization control system for thickening–dewatering process.” This system was established by my research team in our laboratory during the past few years. The hardware structure diagram of the system is shown in Fig. 5. There are four layers in the optimization control system. The process data obtained from the actual gold-ore dressing plant located in Shandong province could be transferred to our

Fig. 5. Hardware structure diagram of “optimization control system for thickening–dewatering process.” 8

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conveniently obtain the samples that are almost uniformly distributed in this region. However, it generally requires several weeks to gather these samples by manual sampling. Moreover, several consecutive samples can be obtained through a manual sampling operation. Apparent outliers are discarded, and the other samples are averaged. Under this scenario, the calibrated data-driven model can satisfy the requirements of actual industrial production, which has been used in a gold-ore dressing plant in Shandong province. It should be noted that 50 samples is more or less a small dataset for fitting a nonlinear regression model. Nonetheless, the prediction results appear to be robust to this case, although it will not always be the case.

Table 1 Summary of prediction results for underflow concentration.

No. of LVs e RMSEC RMSEP

PLS

PRM

KPLS

KPRM

2 —— 2.92 3.05

2 —— 2.87 2.96

9 2 1.14 1.32

11 2 0.61 0.74

response surface based on 20%-trimmed root-mean-squared-error crossvalidation (Jia et al., 2010) values can be calculated. This response surface obtains the minimum point when the number of latent variables and the parameter e are 11 and 2, respectively. To evaluate the efficiency of the KPRM model, in this work, two linear regression methods (PLS and PRM) are also employed to predict the underflow concentration. The root-mean-square-error (RMSE) is used to evaluate the prediction performances. The prediction results of the PLS, PRM, KPLS, and KPRM models for the training data set (RMSEC) and testing data set (RMSEP) are presented in Table 1. The PLS model predicts the training data set 2.92 with two latent variables. However, the prediction result for the test data set is 3.05. While using the PRM model, the prediction result (RMSEP is 2.96) with two latent variables does not improve significantly. Because this data set exhibits severely nonlinear features, the linear robust regression method cannot address this problem. When KPLS is used to approximate the model, RMSEC is reduced to 1.14, and the prediction results (RMSEP is 1.32) are better than that of PLS and PRM. In the KPRM model, the required number of latent variables is increased from 9 to 11. In addition, from the perspective of prediction performance (RMSEP is 0.74), KPRM prevents over fitting and improves the estimation capability for the testing data. To further understand the prediction performances, the relationships between the actual and predicted values of underflow concentration of the testing data set are presented in Fig. 6. The figure shows that the KPRM model is better than the traditional KPLS model in prediction performances. It follows that the underflow concentration model with the KPRM algorithm is effective. It should be noted that because the pressure sensors can reflect the operating states inside the thickener, the underflow concentration can be determined by using these measured values. In order to ensure that the training data set is sufficiently representative of all the feasible states of the thickener, it is necessary to consider these samples. In this context, the outputs in the training data set should cover the range of underflow concentration. In general, the variation in underflow concentration ranges from 40% to 80%. Thus, an experienced operator can

5.1.2. Pressure prediction experiments Next, the prediction performances of the pressure models are further evaluated. When the underflow pump is turned off and the slurry is fed into the thickener (state (1)), the time series are selected from the historical database. The PLS algorithm is used to identify the regress coefficients in Eq. (1), and 10-fold CV is used to determine the number of latent variables (Xu and Liang, 2001). Similarly, when the underflow pump is opened (State (2)), the time series are also recorded to calibrate the pressure model. The prediction performances are illustrated in Fig. 7. The RMSE and maximum absolute error (MAE) for State (1) were 2.02 × 10−4 and 8.00 × 10−4 , respectively. For state (2), the RMSE and MAE were 2.27 × 10−4 and 6.81 × 10−4 , respectively. Both the prediction performances satisfy the requirement of the self-healing control scheme. 5.2. Proposed self-healing control scheme experiments 5.2.1. Self-healing control of abnormality (1) To verify the control performances of the proposed scheme, three pieces of data are selected from the historical database for each condition (abnormalities (1) and (2), and normal condition). First, the FDA classifier is constructed as illustrated in Section 4.2. Then, another dataset in a shift is also selected. In this dataset, abnormality (1) occurs. After 5 min of abnormality, the FDA classifier clearly identifies this abnormal feeding condition. The feeding flow rate and concentration are shown in Fig. 4. It is evident that both the process variables increase apparently. Thus, the proposed self-healing control scheme should be used to address this abnormal feeding condition. It should be noted that the real input data is used as the system inputs, and the outputs are calculated by the process model constructed in our self-developed simulation platform. According to prior knowledge, abnormality (1) generally continues for approximately 1 h. Thus, the behavior of the feeding slurry is

Pressure (MPa)

0.055

64

60

State (1) 0.05 0.045 Observed Predicted

0.04 0

10

20

30

40

50

60

70

80

90

Time (min)

56

0.048

52

Pressure (MPa)

Underflow concentration (%)

68

Actual values KPLS algorithm KPRM algorithm

48 5

10

15

20

25

Observed Predicted

0.046 0.044 State (2) 0.042 0

No. of samples in the testing data set

5

10

15

20

25

30

35

40

45

Time (min)

Fig. 6. Comparison between KPRM model and KPLS model for testing data set of underflow concentration.

Fig. 7. Prediction performances of ARX models for pressure inside thickener. 9

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60 The 3rd pressure sensor Underflow concentration

0.04 50

100

150

200

50 300

250

80

0.05

70 The 3rd pressure sensor Underflow concentration

0.045 0

50

100

80 60 40 20 0 0

50

100

150

200

250

300

Pressure (MPa)

80

0.05

70

250

300

40 20 0 0

50

100

150

150

200

70 65

0.045

60

0.0445

55

0.044

50

0.0435

45

0.043 0

20

40

60

80

100

120

140

160

180

40 200

140

160

180

200

Time (min)

60 300

250

0.046 0.0455

Average concentration (%)

0.045

Time (min) Average concentration (%)

200

60

Fig. 10. Optimal solution in tier 3 for abnormality (1), and corresponding curves for underflow concentration and bottom pressure sensor. The reference startup times of each cycle are 60, 90, 120, 210, 240, and 270.

Concentration (%)

Pressure (MPa)

0.055

100

60 300

Time (min)

Fig. 8. Optimal solution in tier 1 for abnormality (1), and corresponding curves for underflow concentration and bottom pressure sensor. The optimal number of cycles is six. The width of the bar is the running time of underflow pump required to complete a cycle, and the height of the bar represents the average concentration fed into the plate and frame filter press.

50

250

80

Time (min)

0

200

Time (min) Average concentration (%)

Average concentration (%)

Time (min)

150

Concentration (%)

0

Pressure (MPa)

0.045

0.055

Concentration (%)

70

Concentration (%)

Pressure (MPa)

0.05

80 60 40 20

80 60 40 20 0 0

20

40

60

80

100

120

Time (min) 0 0

50

100

150

200

250

300

Fig. 11. Optimal solution in tier 1 for abnormality (2), and corresponding curves for underflow concentration and bottom pressure sensor. The optimal number of cycles is 3.

Time (min)

Fig. 9. Optimal solution in tier 2 for abnormality (1), and corresponding curves for underflow concentration and bottom pressure sensor. The optimal energy consumption is ¥116.80.

significantly less than the upper bound (0.053 MPa). Therefore, the current number of cycles could handle the additional feeding slurry. In Fig. 8 shows the curves of the measured values of pressure of the bottom pressure sensor and the underflow concentration. The corresponding startup and shutdown times of the underflow pump are also illustrated. Here, the height of the bar represents the average concentration fed into the filter press in each cycle. Although the optimal solution in tier 1 is largely similar to the manual operating mode under abnormality (1), the energy consumption (¥156.38) is not considered in this tier. In order to further reduce the energy consumption, the optimization problem (20) in tier 2 is solved subsequently. The optimal startup times

predicted as shown in Fig. 4. In the historical database, the initial pressure of the bottom sensor is 0.048 MPa when abnormality (1) occurs, and six cycles remain in the current shift. In order to prevent pressure rake, the operator turns on the underflow pump immediately and starts dewatering. Because the initial pressure was not excessive (the upper bound of the pressure is 0.053 MPa), under this scenario, the actual energy consumption will increase substantially to ¥153.14. To address this issue, the self-healing control scheme is employed to adjust the abnormality. The dynamic optimization problem (18) in tier 1 is first solved. The maximum pressure is 0.048 MPa, which is Table 2 Multitiered optimization control results for abnormality (1). Startup times of each cycle (min)

Tier 1 Tier 2 Tier 3 Baseline

1st

2nd

3rd

4th

5th

6th

1 50 50 9

23 104 90 40

62 188 120 86

119 206 199 158

145 226 225 208

194 247 247 269

The reference startup times of each cycle is 60, 90, 120, 210, 240, and 270, and εE is equal to 0.1. 10

Energy consumption (¥)

Maximum pressure (MPa)

156.38 116.80 122.76 134.70

0.048 0.053 0.053 0.050

0.045

65

0.04 0

20

40

60

80

100

120

140

160

180

Pressure (MPa)

70

Concentration (%)

Pressure (MPa)

0.05

60 200

0.05

70

0.045

65

0.04 0

20

40

60

80 60 40 20 0 0

20

40

60

80

100

120

100

120

140

160

180

60 200

140

160

180

200

Time (min) Average concentration (%)

Average concentration (%)

Time (min)

80

Concentration (%)

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140

160

180

200

80 60 40 20 0 0

20

40

Time (min)

60

80

100

120

Time (min)

Fig. 12. Optimal solution in tier 2 for abnormality (2), and corresponding curves for underflow concentration and bottom pressure sensor. The optimal energy consumption is ¥75.93.

Fig. 13. Optimal solution in tier 3 for abnormality, (2), and corresponding curves for underflow concentration and bottom pressure sensor. The reference startup times of each cycle are 30, 60, 120, and 150.

of the underflow pump are also drawn in Fig. 9 and recorded in Table 2. Compared with the optimal solution in tier 1, the energy consumption in this tier is ¥116.80, which represents a substantial reduction (approximately 25%). This is because the initial pressure is not excessive. Delaying the startup time of the underflow pump is favorable for increasing the underflow concentration, and in turn for reducing the energy consumption. Finally, the optimization problem (23) in tier 3 is solved. εE is set to 0.1. To prevent operation of the thickening–dewatering process at the time with high electricity price (150–210), the reference startup times are 60, 90 120, 210, 240, and 270. The optimal solution is presented in Fig. 10 and Table 2. Although the energy consumption marginally increases to ¥122.76, the operating mode is closer to the reference one. Thus, it is more convenient for the operators to manage the abnormality. After six cycles, the thickening–dewatering process can return to its normal condition. To further evaluate the efficiency of the proposed self-healing control scheme, a manual abnormality-removal method is also considered as a baseline in this work. In order to prevent pressure rake, the threshold of the bottom pressure sensor is predetermined. Moreover, the operator should continuously observe the operating state of the thickener. When the measured value by the bottom pressure sensor reaches the threshold, the operator immediately turns on the underflow pump and starts dewatering. After the completion of the current cycle, the operator should wait for the measured value by bottom pressure sensor to reach the threshold. The operating result is also illustrated in Table 2. Because the baseline method does not consider the variation in the electricity price, the energy consumption increases to ¥134.70, and the startup times of each cycle deviate significantly from the reference ones. It should be noted that the average concentration significantly affects the running time of the feeding pump of the plate and frame filter press. Thus, the energy consumption for each cycle can be determined

from the height of the bar in these figures. Because the quantity of dry ore required for completing a cycle is almost identical, the taller bars are narrower and the shorter ones wider. Moreover, by using the bars with different heights, the startup and shutdown times of the underflow pump for each cycle can be clearly distinguished. 5.2.2. Self-healing control of abnormality (2) To further test the proposed scheme, the self-healing control of abnormality (2) is considered in this work. When this abnormality is identified, four cycles remained in the current shift, and the initial pressure is 0.046 MPa. In this example, the reference startup times are 30, 60 120, and 150. Unlike the previous example, because both the flow rate and concentration of feeding slurry are decreased, the dynamic optimization problem (19) should be solved first. The highest achievable underflow concentration was 47.8%, which is less than the lower bound 55% (Fig. 11). Furthermore, the total energy consumption is ¥129.01. To reduce the fault rate of the plate and frame filter press, the number of cycles need to be reduced by one. Then, the optimization problem (20) is solved, and the optimal number of cycles is three. The optimal solution for tier 2 is illustrated in Fig. 12 and Table 3. The energy consumption is reduced significantly to ¥75.93, and the lowest underflow concentration attains 60.1%, which could satisfy the requirement of the filter press. These results coincide with the prior knowledge of thickening–dewatering process. To facilitate the operator in removing the abnormality, the optimization problem in tier 3 is calculated (see Fig. 13). Because the original number of cycles is four, the objective function in optimization problem (23) is replaced with the second function in Eq. (24). εE is also set to 0.1. The optimal solution is 60, 120, and 147, which is nearly identical to the reference startup times of the underflow pump. The energy consumption increased marginally to ¥77.93, and a more convenient operation mode is obtained to control this abnormality. The operating result of the baseline approach is also presented in

Table 3 Multitiered optimization control results for abnormality (2). Startup times of each cycle (min)

Tier 1 Tier 2 Tier 3 Baseline

1st

2nd

3rd

4th

1 —— —— ——

25 86 60 41

87 106 120 94

138 135 147 159

The reference startup times of each cycle is 30, 60, 120, and 150, and εE is equal to 0.1. 11

Energy consumption (¥)

Minimum concentration (%)

129.01 75.93 77.93 78.48

47.8 62.2 60.1 71.9

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Table 4 Multitiered optimization control results for undetected abnormality. Startup times of each cycle (min)

Tier 1 Tier 2 Tier 3 Baseline

1st

2nd

3rd

4th

5th

6th

15 1 60 60

77 112 90 90

122 158 120 120

166 202 206 210

211 228 232 240

250 246 251 270

Energy consumption (¥)

Maximum pressure (MPa)

131.77 112.94 117.16 117.12

0.050 0.053 0.053 0.054

70

0.049

69

0.048

68

0.047 0

50

100

150

200

250

67 300

0.055

75

0.05

70

0.045 0

50

100

80 60 40 20 0 0

50

100

150

200

250

300

250

200

250

300

0.05

0.047 150

Average concentration (%)

80 60 40 20 0 50

100

150

250

300

40 20 0 0

50

100

150

upstream mining process, the feeding ore quantity increases. This scenario persists for a long period. Because the feeding flowrate and concentration do not increase apparently under this scenario, it is challenging for the FDA classifier to detect the abnormality. In this example, the predetermined regulation is selected as the baseline approach because the abnormality cannot be detected. However, the proposed self-healing control scheme can still be used to address this abnormality. As in abnormality (1), six cycles remain in the current shift also. The baseline approach (predetermined regulation) is used to operate the thickening–dewatering process. Although the energy consumption is ¥117.12 (see Table 4), the maximum pressure value of the bottom pressure sensor (0.054 MPa) exceeds the upper bound. This may result in the lifting rake of the thickener, thereby increasing the underflow concentration to a higher level. In this case, the risk of blockage at the inlet increases, and this is generally an undesirable situation. When the proposed control scheme is used, the maximum pressure value for tier 1 is 0.050 MPa (see Fig. 14). This implies that the six cycles could address the additional slurry. If the startup times of the underflow pump are adjusted completely, the energy consumption could reduce to ¥112.94 (see Fig. 15 and Table 4 for tier 2). Because the increment in the ore quantity is not excessive, the abnormality could be addressed efficiently by only adjusting the startup times of the latter part (see Fig. 16). Furthermore, the energy consumption (¥117.16) is nearly identical to that of the predetermined regulation, whereas the maximum pressure value reduces to the safety limit (0.053 MPa).

Time (min)

0

200

60

Fig. 16. Optimal solution in tier 3 for undetected abnormality, and corresponding curves for underflow concentration and bottom pressure sensor. The reference startup times of each cycle are also 60, 90, 120, 210, 240, and 270.

Concentration (%)

Pressure (MPa)

200

80 78 76 74 72 70 68 66 64 300

0.053

100

65 300

Time (min)

Fig. 14. Optimal solution in tier 1 for undetected abnormality, and corresponding curves for underflow concentration and bottom pressure sensor. The optimal number of cycles is 6. The width of the bar represents the running time of underflow pump required to complete a cycle, and the height of the bar represents the average concentration fed into the plate and frame filter press.

50

250

80

Time (min)

0

200

Time (min) Average concentration (%)

Average concentration (%)

Time (min)

150

Concentration (%)

71

0.05

Pressure (MPa)

0.051

Concentration (%)

Pressure (MPa)

The reference startup times of each cycle is 60, 90, 120, 210, 240, and 270, and εE is equal to 0.1.

Time (min)

Fig. 15. Optimal solution in tier 2 for undetected abnormality, and corresponding curves for underflow concentration and bottom pressure sensor. The optimal energy consumption is ¥112.94.

Table 3. Because the electricity price does not vary significantly during this period, the energy consumption increases to ¥78.48 under this abnormality. However, the startup times of each cycle deviate significantly from the reference ones, and the operator should continue focusing on the operating state of the thickener.

6. Conclusions A self-healing control scheme is proposed in this work to address the abnormal feeding conditions of the thickening–dewatering process. To determine the operating state inside the thickener, several pressure sensors are first installed in it. Then, an underflow concentration model

5.2.3. Self-healing control of undetected abnormality To evaluate the robustness of the proposed self-healing control scheme, an abnormality that is not included in the training of the FDA classifier is considered in this work. Owing to the fluctuation in the 12

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is calibrated using the KPRM algorithm and the information provided by these pressure sensors. To predict the behavior of the thickening process, several ARX models are constructed for the pressures inside the thickener. Because the abnormal feeding conditions can be clearly identified in the historical database, FDA classifier is employed to detect the abnormalities. The multitiered optimization method is used to determine the appropriate startup times of the underflow pump. This enables us to prioritize several competing objectives and specify the trade-offs among them. The proposed self-healing control scheme provides a new method for self-healing control of complex industrial processes under abnormal conditions. Based on the results of this work, future research directions include improvements in the prediction performances, control of all types of abnormalities in a unified framework, and testing and improvement of the proposed self-healing control scheme in an actual industrial field.

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