Multi-Objective set point optimization control of grate cooler considering energy efficiency evaluation

The grate cooler is a key piece of equipment for cooling cement clinker, whose parameter adjustment has a notable influence on energy consumption and equipment operation. This article proposes a multi-objective set-point optimization control method for the grate cooler control system, which takes into account energy efficiency evaluation. Based on the energy efficiency analysis of the grate cooler, the secondary air temperature, tertiary air temperature, and outlet clinker temperature are used as energy efficiency evaluation indicators, and a grate cooler energy efficiency evaluation model is established to construct a multi-objective function. After that, the grate pressure set point and fan current set point are optimized by utilizing the MOEA/D algorithm, and the predictive control method is combined to perform rolling optimization of the control parameters. Eventually, a simulation experiment is conducted based on the actual production data, and the simulation results demonstrate the feasibility and effectiveness of the optimized control method.


Introduction
The cement industry holds significant importance as one of the pillar industries in China's national economy, occupying an important position in production and construction.China's energy consumption from cement production amounts to approximately 200 million tons of standard coal, accounting for up to 60% of the building materials industry and about 5% of the country's total energy consumption [1].The high energy consumption and pollution issues of the cement industry have become increasingly prominent, and its energy consumption, production efficiency, and carbon emissions have become the focus of social attention.
As one of the main pieces of equipment in the production of cement clinker, the grate cooler undertakes the tasks of cooling cement clinker and recovering high-temperature heat [2], and its operating status has a direct impact on the quality of clinker and production energy consumption.It is always the focus of research to adjust the parameters reasonably and improve the grate cooler's efficiency level.The parameter regulation of the grate cooler primarily relies on manual experience at the moment, with low control accuracy and difficulty in meeting the control requirements under different working conditions [3].In addition, due to the uneven operation level of the staff, the grate cooler is difficult to maintain a smooth and continuous state, resulting in ineffective control of clinker cooling conditions and lower energy efficiency, which increases the energy consumption of cement production ultimately.
With the aim of higher efficiency of the grate cooler as much as possible, some scholars have combined the identification model of the grate cooler with multi-objective optimization algorithms to optimize the relevant parameters.Based on convective heat transfer, Shao et al. [4] proposed a heat transfer and flow entropy generation model for a grate cooler, which utilized genetic algorithms to optimize entropy generation stemming from pressure drop and difference in temperature, thereby reducing fan energy consumption.Zhao et al. [5] aimed to optimize the corrected entropy parameters resulting from heat diffusion and viscous dissipation in the grate cooler, and used the improved NSGA-II algorithm for optimization, providing the optimal control scheme for minimizing the power of the cooling fan.Wang et al. [6] used an improved long and short-memory neural network to establish the prediction model for the grate cooler and employed a bounded stable non-dominated sorting elite genetic algorithm to optimize parameters such as power consumption and secondary air temperature.Gu [7] used a multi-objective optimization predictive control algorithm based on trapezoidal interval soft constraints, with the secondary air temperature and grate pressure as the objective function, the improved ε-constraint was used for the optimization solution, and a certain control effect was achieved.Ma et al. [8] built an equivalent thermal resistance network of grate coolers by the power flow method and adopted the genetic algorithm to optimize the entropy production number stemming from heat diffusion and viscous dissipation, thus reducing fan energy consumption and improving system thermal efficiency.By analyzing the heat exchange efficiency and energy loss of the grate cooler system, Ahamed et al. [9] optimized cooling air temperature, flow velocity, cement clinker temperature, and grate speed parameters to improve its thermal efficiency.Liu [10] selected the grate speed and second plenum fan speed as the control parameters to establish predictive models and used an optimization algorithm to iteratively optimize them and find the most appropriate combination of control parameters.
The above studies realize the parameter optimization of the grate cooler under different angles, but in most cases, the direct control quantity is used as the optimization variable.When the operating mode changes, although the direct control quantity of the grate cooler is optimized, the set point does not change.Due to the different operating conditions, set points at this time often cannot guarantee the optimal energy efficiency level.In addition, to assure the stability of equipment operation, the adjustment range of the direct control quantity of the grate cooler is relatively small.When the system's working condition changes dramatically, in order to achieve a stable state again, the system often needs to spend more time to continuously optimize the direct control quantity, which cannot effectively meet the real-time control requirements of the grate cooler under different working conditions.
In terms of optimizing target selection, many studies have taken secondary air temperature as one of the evaluation indicators, but rarely consider tertiary air temperature.Although both have similar components and properties, when the secondary air temperature is optimized, the latter may not necessarily be synchronously improved, and the system's thermal efficiency cannot be guaranteed to reach the optimal level.
Aiming at the grate cooler control system, based on the energy efficiency analysis, this article takes secondary air temperature, tertiary air temperature, and outlet clinker temperature as the grate cooler's energy efficiency evaluation indicators, then establishes the grate cooler energy efficiency evaluation model and constructs the grate cooler multi-objective function.MOEA/D algorithm is used to optimize the setting of grate pressure and fan current, and the parameters of the grate cooler are optimized by the predictive control method.Finally, based on the field data set, simulation comparative experiments are conducted to confirm the feasibility and effectiveness of the method in this article.

Energy efficiency evaluation indicators of grate cooler
The operating principle of the grate cooler on a 5000 t/d cement clinker production line in Anhui, China is shown in Figure 1.The cement clinker accesses the grate cooler through the rotary kiln entrance and moves towards the outlet under the push of the grate plate.During this period, the cooling fan continuously transports cold air upwards to cool the high-temperature clinker.After heat exchange, cooling air generates secondary air and tertiary air [11], which respectively enter the rotary kiln and precalciner of cement to achieve heat recovery.A small portion of the remaining air after heat exchange is used for air recirculation, while the rest is discharged as waste gas.During the cooling process of clinker, the primary goal of grate cooler control is the improvement of energy efficiency level, which is mainly evaluated through the thermal efficiency and cooling efficiency.
Thermal efficiency is an important index to measure the heat recovery capacity of the grate cooler, which refers to the ratio of the total heat of the recovered secondary air and tertiary air to the heat of the high-temperature clinker in the grate cooler.While ensuring sufficient cooling of the clinker, the improvement of the thermal efficiency helps to reduce fuel consumption in the system.
The thermal efficiency h γ can be calculated according to Equation (1) [12]: where The cooling efficiency of a grate cooler refers to the ratio of the heat recovered from high-temperature clinker to the high-temperature clinker heat.The higher the cooling efficiency is, the lower the outlet clinker temperature is, and the better the cooling effect of the clinker is.
The cooling efficiency c γ can be calculated according to Equation ( 6): where lsh Q is the outlet clinker heat, in KJ , and where lsh C is the outlet clinker heat capacity, in KJ / (kg C) ↓ φ ; lsh m is the outlet clinker mass, in kg/h; lsh t is the outlet clinker temperature, in C ↓ .Then: According to Equations ( 5) and ( 8), it is apparent that the thermal and cooling efficiency is closely related to the temperature and quality of the secondary and tertiary air, along with the clinker before and after cooling.In practical applications, due to the difficulty in accurately measuring parameters such as temperature and mass of inlet clinker, the calculation of thermal and cooling efficiency may result in significant errors [13].Therefore, we consider adjusting parameters based on the trend of changes in thermal and cooling efficiency, rather than relying on specific values of them.When the grate cooler operates smoothly and the inlet clinker state is relatively stable, the lower the outlet clinker temperature is, the more sufficient the clinker cooling is, and the higher the cooling efficiency is.At the same time, if the recovered air temperature is high, it indicates that the system's thermal efficiency is also at a high level.At this time, the changes in the temperature of the secondary and tertiary air and the outlet clinker temperature can approximately reflect the trend of the thermal efficiency and cooling efficiency, becoming an important indicator for measuring the system's energy efficiency level.

Impact parameters of energy efficiency evaluation indicators for grate cooler
High thermal and cooling efficiency means that the system recovers more heat, and the energy consumption per unit clinker is relatively reduced, resulting in a good clinker cooling effect.With the intent of a higher level of energy efficiency in the grate cooler, operators try to increase the secondary and tertiary air temperatures as much as possible by controlling the thickness of the material layer and adjusting the cooling air volume, reducing the outlet clinker temperature, and maximizing heat recovery while fully cooling the clinker.Therefore, rational parameter adjustment is the secret to keeping the grate cooler running smoothly, improving its heat transfer performance, and improving the quality of the clinker.For the above three temperature indicators, they are mainly affected by the following parameters: 2.2.1.Grate pressure.Grate pressure is an indirect control quantity of the grate cooler, reflecting the thickness of the clinker layer and to some extent representing the degree of heat exchange between cooling air and high-temperature clinker.It has a significant impact on the temperature of the recovered air and outlet clinker.When the grate pressure increases, the cooling air and clinker exchange heat more fully, the temperature of recovered air increases, and the outlet clinker temperature decreases.However, blindly increasing the grate pressure can lead to insufficient cooling of the clinker and affect the quality of the clinker.Too small grate pressure can cause excessive cooling of the clinker, increase system energy consumption, and increase fly ash.In actual production, to ensure the stability of the temperature indicators, the grate pressure is often maintained within a certain numerical range.

Fan air volume.
The fan air volume is an important factor affecting the cooling effect of the clinker.With the increase of fan air volume, the clinker cooling gets faster, but the system energy consumption increases.When the fan air volume decreases, recovered air temperature increases, but the cooling effect decreases to some extent.The adjustment of air volume is crucial for the efficient operation of the grate cooler.

Raw material feeding amount.
The raw material feeding amount determines the amount of clinker entering the grate cooler.When other conditions remain unchanged, changes in feed quantity can cause changes in the thickness of the clinker layer, which consequently affects the temperature index of grate coolers.Therefore, reasonable adjustment of raw material feeding rate has an important impact on clinker quality.Usually, the raw material feeding amount is comprehensively controlled by the operator based on the production status and production goals of the cement production line.Within a certain period of time, the raw material feeding amount remains basically stable.

Data-driven energy efficiency evaluation model for grate cooler
According to the energy efficiency analysis of grate cooler, this article carries on mathematical modeling to these energy efficiency evaluation indicators.Because of the high experimental cost required for the mechanism model and the difficulty in establishing parameters, this article adopts a data-driven ARMAX identification model to establish an energy efficiency evaluation model for the grate cooler.The three energy efficiency evaluation indicators of secondary air temperature, tertiary air temperature, and outlet clinker temperature are output quantities, with the grate pressure, fan air volume, and raw material feeding amount as input quantities.
In this article, the field data of a 5000 t/d cement clinker production line grate cooler in a cement factory in Anhui, China, is collected for modeling.Because of the absence of field fan air volume measurement points, it is difficult to collect air volume data.However, for centrifugal fans, fan air volume has a positive correlation with fan current, so fan current is used to replace fan air volume.Considering that the air volume of some fans remains constant for a long time and no air volume regulation is required, this article selects the frequently adjusted F1A fan current, F2 fan current, F2L fan current, and F2R fan current, along with the grate pressure and raw material feeding amount, as the input variables for the model.

Data collection and preprocessing
In the grate cooler system, changes in one parameter may only have an impact on other quantities after a few minutes.In order to establish a true and effective mathematical model, this article takes 1 minute as the sampling interval and takes the secondary air temperature 1 () yk, tertiary air temperature 2 () yk, outlet clinker temperature 3 () yk, grate pressure 1 () uk, F1A fan current 2 () uk, F2 fan current 3 () uk, F2L fan current 4 () uk, F2R fan current 5 () uk, and raw material feeding amount 6 () uk at different times, and calculates their average data every 15 minutes as raw data for different time periods.
Due to the field sensor error and random noise interference, data acquisition will produce certain deviations.With the purpose of ensuring the accuracy of data modeling, the original dataset is processed by using a moving average filtering algorithm, that is, a sliding window with a fixed width is set to slide along the input time series, and the arithmetic average value of the data in the window is calculated as the filtered data: where () xk is the original data; () yk is the filtered data; N is the window size.

Data-driven ARMAX identification model
The data-driven ARMAX identification model is based on historical data to establish a dynamic model between input and output, which has a certain predictive ability for future system states [14].This article uses the ARMAX identification model to establish an energy efficiency evaluation model based on the interrelationships between different input and output time series.The model predicts the target parameters and provides a mathematical basis for multi-objective set point optimization control of the grate cooler.The ARMAX model is represented as follows: 11 where ( )1

Parameter identification of energy efficiency evaluation model for grate cooler
Since the field data of the grate cooler contains various unknown disturbances, the multi-input multioutput energy efficiency evaluation model established based on ARMAX can be transformed as follows: Using the recursive least squares algorithm for parameter identification of the above model, the inputoutput vector () k ι and parameter vector π are defined as follows: ,,,   ,,,,,, In Equation ( 12), where .According to the above recursive equation, the parameter estimation values are continuously revised to obtain the required model parameters.

Multi-objective function of grate cooler
Based on the complex relationship between the parameters of the grate cooler, this article optimizes these parameters by the multi-objective optimization method, and the construction of the multi-objective function directly affects the optimization effect of subsequent algorithms, which has a significant impact on the determination of the optimal solution.On the basis of grate cooling function analysis, secondary air temperature, tertiary air temperature, and outlet clinker temperature are used as evaluation indexes for multi-objective optimization, and the grate pressure and fan current are chosen as decision variables.Under ideal conditions, when the recovered air temperature reaches the maximum, the outlet clinker temperature reaches the minimum, and the working energy efficiency is at the optimal level.However, in actual production, when the second and tertiary air temperature reaches the maximum, the outlet clinker temperature will be relatively high, and when the outlet clinker temperature tends to the lowest, the second and tertiary air temperature will be relatively reduced.To seek the optimal solution for different objectives, a multi-objective function is established by using the minimum objective function algorithm: where () Xk is the decision variable, including the grate pressure 1 () uk, F1A fan current 2 () uk, F2 fan current 3 () uk, F2L fan current 4 () uk, and F2R fan current 5 () uk.

Constraints on the operation of the grate cooler
To ensure the proper operation of the grate cooler and meet the cement process requirements, the grate pressure and fan current must fulfill the relevant operation constraints:

I
are the minimum current values of F1A fan, F2 fan, F2L fan, and F2R fan respectively.In addition, the set points of the grate cooler should remain relatively stable before and after optimization, so the following constraints are added to the optimization amount of the grate cooler parameters: where () i uk is the i-th decision variable in the time period k ; () i S Pk is the set point of the i-th decision variable in the time period k ; i u Χ is the maximum deviation between the i-th decision variable and its set point, indicating the maximum change in the set point of the i-th decision variable during each optimization.

Optimization of multi-objective set point for grate cooler
This article adopts the multi-objective evolutionary algorithm based on decomposition (MOEA/D) to solve the multi-objective problem of the grate cooler and simultaneously optimizes its set point parameters.

The principle of MOEA/D.
The core idea of MOEA/D is to transform the multi-objective problem into a series of single-objective optimization problems or multiple multi-objective sub-problems by linear or nonlinear methods [16], and the optimal solution of each sub-problem is a solution of the original target optimal solution set.MOEA/D puts forward the concept of neighborhood and synchronously optimizes all sub-problems through the cooperation of individuals in each sub-target neighborhood so that individuals constantly approach the Pareto front.In the algorithm, MOEA/D randomly assigns uniform weights to individual populations, determines individual neighborhoods by calculating the Euclidean distance between individual weight vectors [17], randomly selects individuals in each neighborhood for cross-mutation to generate new individuals, retains the best non-dominant individuals in each generation, and finds all optimal solutions through continuous iteration.Compared to other algorithms, MOEA/D has lower time complexity and is suitable for online optimization [18].
MOEA/D takes decomposition as its core.In the decomposition strategy of MOEA/D, the Chebyshev method, as a nonlinear multi-objective aggregation method, is the most widely used, and its aggregation function is shown in Equation ( 20 the Chebyshev method is used to calculate the target vector value and min( ( )), 1, 2, , ; 1, 2,3 Secondly, for each sub-problem i , the solutions k x and l x corresponding to two weight vectors are randomly selected from the neighborhood () N Si , and a new solution x' is generated through cross-mutation.After modifying through constraint conditions, x' is substituted into the multi-  F x' is moved into EP .Finally, if iterations have reached the set value, it is necessary to stop the iteration and output the non-inferior solution set EP .Otherwise, we should continue the loop to complete multi-objective optimization.

Multi-objective set point optimization control system for grate cooler
This article proposes a multi-objective set point optimization control system for the grate cooler of a 5000 t/d cement clinker production line in a cement factory in Anhui, China, as shown in Figure 2. The set point of the grate cooler is real-time optimized by using multi-objective optimization algorithms, and a parameter controller is constructed by using predictive control algorithms to perform rolling optimization of the corresponding parameters.The multi-objective optimization module in the system is used to read the recent temperature indicators.Based on the multi-objective function, parameter optimization is performed on the grate pressure and the fan current.Among the feasible solutions obtained from the optimization, an optimal solution is selected as the new grate cooler set point and fan current set point.
Due to the time delay during the cooling process, parameter adjustment needs a certain buffer time to ensure that the control quantity finally reaches the set point.Therefore, within a certain period of time after the completion of multi-objective optimization, the system will calculate the control increment of grate speed and fan valve opening several times through the grate pressure controller and fan current controller according to the same set point, so that the grate pressure and fan current continue to change with the set point, and then obtain the change trend of energy efficiency evaluation indicators.

Simulation of energy efficiency evaluation model for grate cooler
In the simulation experiment, the field data of the grate cooler is collected every minute, and the data mean of each parameter is calculated at 15-minute intervals as the original dataset.A total of 1200 sets of data are taken, with the previous 1000 sets of data being the training dataset and the last 200 sets being the validation dataset.The input and output order of the energy efficiency evaluation model is set to 1, with a lag of 3 steps for the raw material feeding amount and 2 steps for the remaining variables.The recursive least squares identification algorithm is used to identify model parameters, and the identification parameters of the energy efficiency evaluation model are shown in Equation ( 21

Simulation of multi-objective set point optimization control for grate cooler
A simulation platform is built by using MATLAB to verify the multi-objective set point optimization control method of the grate cooler.Every 300 steps, MOEA/D is used to optimize the multi-objective function.The optimal set points are selected from the optimization results, and then the grate cooler is optimized and controlled through corresponding control strategies, so as to make the system output secondary air temperature and tertiary air temperature as high as possible, and the outlet clinker temperature as low as possible.The optimization control results are compared with manual control and the existing two objective control results to confirm the rationality of the method used in this article.
According to the actual production conditions, the operation constraints of the control amount are given, in which the grate pressure is controlled from 6000 to 9000, the current ratings of F1A fan, F2L fan, and F2R fan are 262.4,the current rating of F2 fan is 25.6, and the minimum current values of each fan are 0. The maximum change of the grate pressure set point is 200, the maximum change in the F2 fan current setting is 1, and the maximum change in the current setting of other fans is 3. Given the set value parameters under manual control, the grate pressure set point is 8200, the F1A fan current set point is 118, the F2 fan current set point is 18.5, and the set point of F2L fan current and F2R fan current is 150.
During the operation of the MOEA/D algorithm, we set 100 Gen < , 100 N < , and 20 T < .The Pareto frontier obtained by the algorithm is shown in Figure 6.It is obvious that when secondary air temperature and tertiary air temperature approach the maximum, the outlet clinker temperature is also higher.On the contrary, when the outlet clinker temperature reaches its minimum, secondary air temperature and tertiary air temperature are relatively small, and the three are difficult to achieve the ideal state.In the actual production, it is necessary to choose the optimal solution of the algorithm according to the emphasis on different evaluation indexes and the actual needs.
For the purpose of the smooth operation of the grate cooler at the cement production site, the adjustment amplitude of various parameters during the control process is required to be as small as possible, that is, the grate pressure set point and fan current set point, before and after optimization, should be kept to a minimum.To meet all control requirements, this article normalizes the grate pressure and fan current corresponding to the feasible solution in the Pareto front surface and calculates the mean square error between the grate pressure and fan current corresponding to each feasible solution, and the current set value of grate pressure and fan current.The feasible solution corresponding to the smallest overall mean square error is selected as the optimal solution.
Then, the grate pressure controller and fan current controller are constructed based on the GPC algorithm and incremental PID algorithm respectively, and the grate pressure and fan current corresponding to the above optimal solution are taken as the new grate pressure set point and fan current set point to optimize the related control parameters.Ultimately, the evaluation indicators of the system can be improved.In existing research, some scholars have achieved parameter optimization of grate coolers by utilizing two objectives: secondary air temperature and outlet clinker temperature, and have achieved certain results.However, this article considers the tertiary air temperature as a new evaluation indicator based on the energy efficiency analysis.To confirm the rationality of the proposed energy efficiency evaluation indicators, simulation experiments are performed on the system again by using the above two objectives.The expected results and variation trends of the three energy efficiency evaluation indicators under different control methods are shown in Figure 7.
Apparently, under the guidance of the optimized set points, recovered air temperature significantly increases by more than 20 C ↓ , and the outlet clinker temperature also decreases by about 5 C ↓ , ultimately achieving the expected results.Compared with the manual control method, the multi-objective set point optimization control method with energy efficiency evaluation dynamically adjusts the set point parameters according to the energy efficiency evaluation indicators, which is more conducive to improving energy efficiency and meeting its control requirements under different operating conditions, and is instrumental in improving the industrial control quality.Meanwhile, without considering the tertiary air temperature, the multi-objective optimization has also achieved certain results.Compared to manual control, the outlet clinker temperature has been significantly reduced, but the system has not had a good effect in improving the secondary and tertiary air temperature in the initial optimization stage, especially the tertiary air temperature.In the control results of energy efficiency evaluation indicators in this article, although the optimization effect of outlet clinker temperature is slightly inferior, all three evaluation indicators have been positively improved throughout the optimization process, better balancing the improvement of thermal efficiency and cooling efficiency, which has certain advantages in enhancing the energy efficiency of grate coolers.

Conclusion
In response to the difficulties in improving the energy efficiency of the grate cooler and the complexity of parameter regulation, this article proposes a multi-objective set point optimization control method that takes into account energy efficiency evaluation.The secondary air temperature, tertiary air temperature, and outlet clinker temperature are used as energy efficiency evaluation indicators, and the parameter set points are synchronously optimized, achieving further improvement of energy efficiency.It provides theoretical guidance for grate cooler's multi-objective optimization control and holds significant importance in reducing the energy consumption associated with grate cooler production.
; k is the sampling time; z σ , is the lag steps, representing σ steps of input lag output; a n and b n are delay orders; () k δ is Gaussian white noise.

Figure 2 .
Figure 2. Multi-objective set-point optimization control system for grate cooler.
dataset for model validation, Figures3, 4, and 5 provide the validation results of the model.It is not difficult to see that the predicted output of the model can accurately reflect the trend of changes in real data, with an error basically within 5%.When the data fluctuates frequently, the output deviation of the model increases, but it is still within the allowable error range.Therefore, the predictive ability of the model can meet the multi-objective set point optimization control requirements of the grate cooler.

Figure 7 .
Figure 7. Results of different control methods.
): Gen , population size N , and neighborhood size T .The external solution set EP is initialized, and uniformly distributed weight vectors are generated and assigned to each sub-problem.For each weight vector i κ , the Euclidean distance between it and other vectors is calculated, and T vectors with the smallest Euclidean distance are taken as its neighbors, whose neighborhood is denoted as , ):

Table 1 .
Table 1 displays the key parameters for the GPC algorithm and incremental PID algorithm.Parameters for GPC and Incremental PID.