# Application of the ARMA Model to Describe and Forecast the Flotation Feed Solids Flow Rate

^{*}

## Abstract

**:**

_{1}= −1.0682, a

_{2}= −0.2931, a

_{3}= 0.3807, c

_{1}= −0.1588, c

_{2}= −0.2301, c

_{3}= 0.1037, and variance σ

^{2}

_{ε}= 0.0891, white noise sequence ε

_{t}, determined on the basis of a series of residuals described by the fifth-order model. It has been shown that the identified model of the flow rate of solids of the feed to flotation as disturbances can be used to develop a predictive model that allows forecasting the modelled quantity with a prediction horizon equal to the sampling period. One-step forecasting based on the determined predictor equation was found to give results consistent with the recorded values of the solid part flow rate of the feed and the extreme values of the prediction error are within the range from −1.08 to 2.90 kg/s.

## 1. Introduction

_{n}with ash content a

_{n}and concentration of solids in the feed k

_{cs}, flotation reagent flow rate v

_{o}, air flow rate for aeration q

_{a}, and the level of suspension in the flotation cell h. The output quantities are: concentrate output γ

_{k}, ash content in the concentrate a

_{k}, waste output γ

_{o}, and the ash content of the waste a

_{o}(Figure 1).

_{o}is the flotation reagent dosage, m

^{3}/kg, and q

_{cs}is the flow of solids of the feedstock for flotation, kg/s. Of the feed quantity parameters, the solids concentration and the flow rate are measurable. Therefore, the value of the solids flow can be determined from the relation:

_{cs}is the solid concentration, kg/m

^{3}, and q

_{n}is the flotation feed flow rate, m

^{3}/s.

## 2. Parameter Estimation Method for ARMA Models

_{ks}, a model with discrete transmittance expressed by the equation was adopted:

_{t}is a sequence of uncorrelated disturbances with mean equal to zero and variance ${\sigma}_{e}^{2}$, and y

_{t}is a concentrated sequence of recorded values of the solids flow rate of the flotation feed with the number of samples N, kg/s.

_{t}, which is an approximation of the unknown white noise sequence e

_{t}:

_{1}, b

_{2},…, b

_{p}, it is necessary to determine a sequence of residuals ε

_{t}. Fitting a model of the autoregressive process (4) of order p to the signal sequence y

_{t}is to determine the parameters b

_{1}, b

_{2},…, b

_{p}, minimising the value of the criterion J

_{b}expressed by Equation (6).

_{1}, b

_{2},…, b

_{p}are calculated and then the sequence of ε

_{t}residuals is determined, according to the Equation (4). The sequence ε

_{t}is then used to fit the ARMA model to the measured series of data y

_{t}. This operation leads to the estimation of parameters a

_{1}, a

_{2}, …, a

_{n}and c

_{1}, c

_{2}, …, c

_{n}ARMA model the order n (n < p) described with Equation (9) with values such that the minimum reaches the criterion value:

_{t}sequence of residuals is checked, expressed by the equation:

_{t}. The autocorrelation function is expressed by the following equation:

## 3. Prediction Model for a Stochastic Process

- -
- Recording of measured data y
_{t}, - -
- Identification of the process model of Equation (3) with the use of the two-stage least squares method,
- -
- Using the determined parameters of the model (3) in the predictor equation.

## 4. Results of the Flotation Feed Solids Flow Rate Model Identification

#### 4.1. Test Conditions

^{137}Cs cesium isotope [44,45]. Data were recorded with a sampling period T

_{s}equal to 60 s using an industrial computer. The instantaneous flow values of the flotation feed solids for previously recorded measurement data from flow meters and densimeter were determined from the equation:

_{cs}is the concentration of solids in the coal flotation feed, kg/m

^{3}q

_{n}

_{1}is the feed flow rate to the flotation tank 1, m

^{3}/s, q

_{n}

_{2}is the feed flow rate to the flotation tank 2, m

^{3}/s, and I is the sampling step, i = t/T

_{s}.

_{1}, y

_{2}, y

_{3}, and y

_{4}. Their time courses are presented in Figure 5. The lengths of the individual data series are 457, 694, 400, and 678 points, respectively, so they cover long periods of the flotation industrial facility operation, that is, 7 h and 37 min, 11 h and 34 min, 6 h and 40 min, and 11 h and 18 min, respectively. As the values summarised in Table 1 and the time courses in Figure 5 show, the observed range of variation in the flow rate of the feed solids shows a significant range of instantaneous values. As part of the conducted identification research, the first series of empirical data (y

_{1}) was used to determine the order and model parameters with the structure (3). The remaining three series were used to verify the determined model.

#### 4.2. Results of the Flotation Feed Solids Flow Rate Model Identification as a Disturbance of the Flotation Process

_{1}) the optimum in terms of criterion (7) is the order p = 5, whereas for the other series it was 3 or 4. Therefore, the estimation of the parameters of model (3) was carried out from order n = 1 to n = p−1, for two values of the model residual sequence: p = 4 and p = 5, assuming an appropriate starting point. The identification task was carried out for the first series of measurement data, and the results obtained are summarized in Table 2.

^{2}

_{ε}= 0.0891, white noise sequence ε

_{t}. The time courses were simulated using a sampling period of 60 s and the length of the data sequences was 550 samples, approximately equal to the average length of the recorded signals. Example simulations of the feed solids flow rate are shown in Figure 9.

^{n}C(z

^{−1}) and the denominator z

^{n}A(z

^{−1}) of the identified model were determined. It was found that in each case they lay inside the unit circle on the plane of the composite variable. The mean values of the estimated parameters and their standard deviations were then calculated, which provides a measure of the ARMA models coefficients values spread. In this, case roots of polynomials of the model numerator and denominator (3) with parameters equal to the average values of the coefficients calculated from simulated courses (Table 4) lie inside the unit circle in the plane of the composite variable. This shows that the model so determined satisfies the requirements given in Section 2.

#### 4.3. Results of Predicting the Flow Rate of Coal Flotation Feed Solids as a Stochastic Process

_{1}, y

_{2}, y

_{3}, y

_{4}. The course of the solids in the feed flow prediction determined on the basis of Equation (22) and the prediction errors, for the actual courses as in Figure 3, is shown in Figure 11.

## 5. Discussion

_{1}and the quantitative evaluation of the adopted criteria indicate that an ARMA model of the fourth order, determined on the basis of a sequence of residuals described by a model of the fifth order, should be used to describe the course of the flow of solids in the feed (Figure 6a). However, the analysis of criteria (7) and (13) for the models determined for the remaining three series of data shows that it is correct to assume n = 3 as the order of the ARMA model with parameters estimated based on a sequence of residuals described by a model of order p = 5 (Figure 6b–d). Therefore, the ARMA model of order n = 3 identified based on the first series of measurement data was adopted for further research. The validity of adopting this order is also confirmed by the results of the residuals obtained for this model analysis. These results, shown in Figure 8, allow us to conclude that the obtained residual sequence can be treated as a white noise sequence, i.e., a residual sequence t is not correlated. This statement is confirmed by the course of the determined normalized autocorrelation function presented in Figure 8b.

^{2}

_{ε}= 0.0891. This makes it possible to study the response of automatic systems to random changes in the feed flow rate by computer simulation for many sequences. Thanks to this, it is possible to assess the correctness of the adopted structure of the automatic control system and the controller settings on the basis of simulation and to adjust them to the values that allow the best compensation of the influence of changes in the solids flow rate on the enrichment process. It is a practical aspect related to the design of the automatic control system of the flotation process.

## 6. Conclusions

_{1}= −1.0682, a

_{2}= −0.2931, a

_{3}= 0.3807, c

_{1}= −0.1588, c

_{2}= −0.2301, c

_{3}= 0.1037, and variance σ

^{2}

_{ε}= 0.0891 determined on the basis of a sequence of residuals with the properties of a white noise sequence. The sequence of residuals is described by a model of the fifth order with coefficients b

_{1}= −0.9157, b

_{2}= −0.2023, b

_{3}= 0.0143, b

_{4}= −0.0020, and b

_{5}= 0.1168. The determined model parameters for the measured data, measured during four consecutive periods of continuous operation of the flotation facility, show a significant divergence but fall within the range of three calculated standard deviations. This shows that the parameters of the feed solids flow rate model do not change significantly for four consecutive periods of continuous operation of the industrial facility. It is shown that the estimated parameters of the ARMA model can be used to predict the time series of solids flow in the feedstock with a prediction horizon equal to the sampling period.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a_{n} | ash content in the feed, %, |

k_{cs} | solids concentration, kg/m^{3} |

q_{n} | feed flow rate, m^{3}/s |

v_{o} | reagent flow rate, m^{3}/s |

q_{a} | air flow for slurry aeration, m^{3}/s |

h | slurry level in flotation cell, m |

γ_{k} | concentrate yield, feed % |

a_{n} | ash content in the concentrate, % |

γ_{o} | yield of tailings, feed % |

a_{o} | ash content in tailings, % |

v_{o} | reagent flow rate, m^{3}/s |

d_{o} | reagent dose, m^{3}/kg |

q_{cs} | feed solids flow, kg/s |

e_{t} | sequence of uncorrelated disturbances with the mean value equal to zero and the variance ${\sigma}_{e}^{2}$ for the ARMA model (3) or the prediction error for the prediction model (17) |

y_{t} | centered sequence of recorded solids flow rate values of the feed with the number of samples N, kg/s |

N | the number of samples of the recorded feed solids flow rate values |

K(z^{−1}) = C(z^{−1})/A(z^{−1}) | discrete transmittance, ARMA model |

z | complex variable $A\left({z}^{-1}\right)=1+{a}_{1}{z}^{-1}+\dots +{a}_{n}{z}^{-n}$, $C\left({z}^{-1}\right)=1+{c}_{1}{z}^{-1}+\dots +{c}_{n}{z}^{-n}$, |

n | ARMA model order |

a_{1}, a_{2}, …, a_{n}, c_{1}, c_{2}, …, c_{n} | ARMA model parameters |

B(z^{−1}) | discrete transfer function of the autoregression model |

b_{1}, b_{2},…, b_{p} | coefficients of the autoregression model representing the identified ARMA process |

p | order of the autoregressive model |

ε_{t} | sequence of residues that approximates the unknown sequence of white noise e_{t} |

J_{b} | criterion of matching the autoregression model (4) to the sequence of the y_{t} signal |

FPE | final prediction error |

${\sigma}_{\epsilon}^{2}\left(p\right)$ | autoregression model residuals variance |

J_{ac} | criterion for estimating parameters a_{1}, a_{2}, …, a_{n} and c_{1}, c_{2}, …, c_{n} of the n order ARMA model |

R_{e} | autocorrelation function |

ρ_{e} | normalized autocorrelation function |

σ_{ε}^{2}(n) | residual variance of the ARMA model |

${\widehat{y}}_{t+1|t}$ | forecast value of signal y at time (t + 1), determined at time t |

C1, C2 | concentrates from industrial flotation machines IZ-12 |

O1, O2 | tailings from industrial flotation machines IZ-12 |

k_{cs}[i] | solids concentration, discreetly observed every sample time, kg/m^{3} |

q_{n1}[i] | feed flow rate to the flotation machine 1, discreetly observed every sample time, m^{3}/s |

q_{n2}[i] | the feed flow rate to the flotation machine 2, discreetly observed every sample time, m^{3}/s |

T_{s} | sample time, s |

i | sampling step, i = t/T_{s} |

t | time, s |

y_{1} | first series of solids flow rate data, kg/s |

y_{2} | second series of solids flow rate data, kg/s |

y_{3} | third series of solids flow rate data, kg/s |

y_{4} | fourth series of solids flow rate data, kg/s |

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**Figure 4.**Industrial facility of the coal flotation process: C1, C2—concentrates, O1, O2—flotation waste.

**Figure 5.**Time courses of the feed solids flow rate directed to the flotation tank determined on the basis of the solids concentration courses and the feed flow rate recorded at the industrial facility.

**Figure 6.**The dependence of criterion (13) on the order n of the ARMA model describing the flow rate of flotation feed solids, determined for a series of measurement data: (

**a**) y

_{1}, (

**b**) y

_{2}, (

**c**) y

_{3}, (

**d**) y

_{4}based on the residuals described by the model of order p = 5.

**Figure 8.**Evaluation of the ARMA model residuals with parameters (20): (

**a**) course of the residuals, (

**b**) standardised autocorrelation function.

**Figure 9.**Examples of simulated feed solids flow rates: (

**a**) series 1, (

**b**) series 4, (

**c**) series 8, (

**d**) series 10.

**Figure 10.**Time courses of the feed solids flow rate determined by means of the third-order ARMA model with Equation (21).

**Figure 11.**Prediction error and one-step forecast of the flotation feed solids flow rate against measured data; (

**a**) series 1, (

**b**) series 2, (

**c**) series 3, (

**d**) series 4; 1—measured data, 2—forecast.

**Table 1.**The range of variations of solids flow rate, solids contraction, and feed flow rate to the flotation process.

Data Series | y (kg/s) | k_{cs} (kg/m^{3}) | q_{n}_{1} 10^{−2} (m^{3}/s) | q_{n}_{2} 10^{−2} (m^{3}/s) | ||||
---|---|---|---|---|---|---|---|---|

Min | Max | Min | Max | Min | Max | Min | Max | |

1 | 21.69 | 29.31 | 173.55 | 228.99 | 62.96 | 69.96 | 58.12 | 67.16 |

2 | 18.06 | 32.60 | 146.12 | 247.45 | 61.13 | 73.62 | 52.75 | 66.88 |

3 | 22.46 | 38.51 | 177.11 | 235.28 | 54.44 | 97.61 | 51.51 | 77.25 |

4 | 17.97 | 34.63 | 144.01 | 253.27 | 62.50 | 72.07 | 52.61 | 67.43 |

**Table 2.**Flow rate model parameters of coal flotation feed solids with structure (3) estimated for the first series of measured data.

The Sequence of Residuals | p = 4 | p = 5 | |||||
---|---|---|---|---|---|---|---|

b_{1} = −0.9179, b_{2} = −0.2076, b_{3} = 0.03973, b_{4} = 0.1047 | b_{1} = −0.9057, b_{2} = −0.2023, b_{3} = 0.0143, b_{4} = −0.0020, b_{5} = 0.1168 | ||||||

ARMA Model | n | n | |||||

1 | 2 | 3 | 1 | 2 | 3 | 4 | |

a_{1} | −0.9857 | −1.3718 | −0.9436 | −0.9861 | −1.5158 | −1.1259 | −0.7749 |

a_{2} | - | 0.3848 | −0.1663 | - | 0.5264 | −0.2679 | −0.4207 |

a_{3} | - | - | 0.1301 | - | - | 0.4068 | 0.0561 |

a_{4} | - | - | - | - | - | - | 0.1627 |

c_{1} | −0.0678 | −0.4515 | −0.0240 | −0.0802 | −0.6106 | −0.2200 | 0.1285 |

c_{2} | - | 0.1759 | 0.0192 | - | 0.1761 | −0.2653 | −0.1004 |

c_{3} | - | - | 0.1014 | - | - | 0.1081 | −0.0224 |

c_{4} | - | - | - | - | - | 0.1227 |

Model Parameters | Series of Measured Data | |||
---|---|---|---|---|

1 | 2 | 3 | 4 | |

a_{1} | −1.1259 | −1.1413 | −1.5472 | −1.1428 |

a_{2} | −0.2679 | −0.3258 | 0.6563 | −0.1290 |

a_{3} | 0.4069 | 0.4808 | −0.0794 | 0.2877 |

c_{1} | −0.2200 | 0.1249 | −0.2510 | 0.0600 |

c_{2} | −0.2653 | −0.3256 | −0.0428 | −0.1059 |

c_{3} | 0.1081 | −0.0908 | 0.1066 | −0.0027 |

**Table 4.**Identification results of solids flow models into the feed determined from 10 simulated time courses.

Model Parameters | No. of Model Identified on the Basis of a Simulated Course | Average Values | σ_{ij} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

a_{1} | −1.1485 | −1.0727 | −1.0305 | −0.9490 | −1.5122 | −1.0755 | −0.9738 | −0.9274 | −0.9115 | −1.0811 | −1.07 | 0.17 |

a_{2} | 0.0098 | −0.3540 | −0.3268 | −0.4537 | 0.5216 | −0.2676 | −0.6644 | −0.4694 | −0.6110 | −0.3155 | −0.29 | 0.34 |

a_{3} | 0.1612 | 0.4380 | 0.3875 | 0.4341 | 0.0136 | 0.3542 | 0.6552 | 0.4109 | 0.5441 | 0.4081 | 0.38 | 0.18 |

c_{1} | −0.2938 | −0.1825 | −0.1384 | −0.1045 | −0.6127 | −0.1521 | −0.0033 | −0.0278 | 0.0489 | −0.1216 | −0.16 | 0.19 |

c_{2} | −0.0584 | −0.3131 | −0.2399 | −0.2454 | 0.1330 | −0.1965 | −0.4719 | −0.3460 | −0.3147 | −0.2475 | −0.23 | 0.17 |

c_{3} | 0.1406 | 0.1269 | 0.1242 | 0.1378 | 0.0943 | 0.1120 | 0.0309 | 0.0638 | 0.0798 | 0.1268 | 0.10 | 0.04 |

**Table 5.**Extreme values of the prediction error obtained on the basis of forecasts for four data series.

e_{t} (kg/s) | ||||
---|---|---|---|---|

Value | Series 1 | Series 2 | Series 3 | Series 4 |

min | –0.40 | –0.90 | –1.08 | –1.02 |

max | 0.31 | 0.40 | 0.61 | 2.70 |

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**MDPI and ACS Style**

Joostberens, J.; Rybak, A.; Pielot, J.; Dylong, A.
Application of the ARMA Model to Describe and Forecast the Flotation Feed Solids Flow Rate. *Energies* **2021**, *14*, 8587.
https://doi.org/10.3390/en14248587

**AMA Style**

Joostberens J, Rybak A, Pielot J, Dylong A.
Application of the ARMA Model to Describe and Forecast the Flotation Feed Solids Flow Rate. *Energies*. 2021; 14(24):8587.
https://doi.org/10.3390/en14248587

**Chicago/Turabian Style**

Joostberens, Jarosław, Aurelia Rybak, Joachim Pielot, and Artur Dylong.
2021. "Application of the ARMA Model to Describe and Forecast the Flotation Feed Solids Flow Rate" *Energies* 14, no. 24: 8587.
https://doi.org/10.3390/en14248587