# Energy modeling and efficiency analysis of aluminum die-casting processes

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## Abstract

Energy modeling and efficiency analysis are considered the foundation of manufacturing process optimization to improve quality and efficiency and reduce energy consumption and carbon emissions during aluminum die-casting processes. This paper proposed an energy modeling method to connect gas and electric energy consumption with production rate of aluminum die-casting processes based on data collected at workshops with various combination of machines and products. The detailed modeling process involved the development of a data-acquiring system and the comparison of various kinds of nonlinear regression methods. The resulting models were validated with actual production data and were further used to improve production scheduling. It was found that if the modeling results are reasonably used and production is accordingly well-scheduled, 10 to 15% of energy savings could be realized without sacrificing profits.

## Keywords

Aluminum die-casting process Energy consumption modeling Data acquisition Energy efficiency analysis## Nomenclature

*A*_{P},*B*_{P},*C*_{P},*D*_{P}Coefficients of polynomial cubic

*adjR*^{2}Adjusted coefficient of determination

*a*_{E},*b*_{E},*c*_{E}Coefficients of exponential functions

*a*_{L},*b*_{L},*c*_{L}Coefficients of logarithmic functions

*a*_{R},*b*_{R}Coefficients of reciprocal functions

*a*_{P},*b*_{P}Coefficients of power functions

*D*Demanding amount of products of orders

*E*_{c}Calculation result of energy consumption of the overall processes for using electric energy [

*KW∙h*]*E*_{real},The energy consumptions of the overall processes in real practice for using electric energy [

*KW∙h*]*F*Coefficient of

*F*test*F*(*θ*)The function to get the aggregation of least relative error

*f*_{SEC-P}The alternative mathematical form to be used during the modeling process

*G*_{c}Calculation result of energy consumption of the overall processes for using gas [

*m*^{3}]*G*_{real}The energy consumptions of the overall processes in real practice for using gas [

*m*^{3}]*N*The sample group number of observed

*SEC*and*P*used for the modeling*n*The number of unknown coefficients of the regression model

*P*Production rate

*P*_{elec}Production rate of die-casting machine producing finished products [

*piece/h*]*P*_{gas}Production rate of furnaces using gas [

*kg/h*]*R*^{2}Coefficient of determination

*R*_{gas},*R*_{elec}Relative energy efficiency evaluation indexes

*SEC*Specific energy consumption. The amount of energy required for processing a certain amount of one kind of product

*SEC*_{elec}*SEC*of producing finished products from the machine using electric energy [*KW∙h/piece*]*SEC*_{gas}*SEC*of melting raw materials using gas in furnace [*m*^{3}*/kg*]*SSE*Sum of residual squares

*SST*Sum of the total square

*s*Residual standard deviation

*T*Delivery time

*Tc*Cycle of time of process

*T*_{ca},*G*_{ca},*E*_{ca}The real-time, gas consumption, and electric energy consumption of meeting the demand

*D*operating casually*T*_{h},*G*_{h},*E*_{h}The real-time, gas consumption, and electric energy consumption of meeting the demand

*D*using high energy efficiency strategy*t*_{idle}Stand-by idling time

*t*_{p}Production time

*x*_{i}One of the observed values of production rate

*P**y*_{i}One of the observed values of

*SEC*- \( \widehat{y_i} \)
One of the values of the model output

*δ*_{elec},*δ*_{gas}Relative error indexes to measure the accuracy of the prediction results

*θ*The unknown coefficient vector of the regression model

*θ*_{1},*θ*_{2}, …,*θ*_{n}The unknown coefficients of the regression model

- \( \widehat{\theta} \)
Optimum solution vector of the unknown coefficients

- \( \widehat{\theta_1},\widehat{\theta_2},\dots, \widehat{\theta_n} \)
Optimum solution group of the unknown coefficients

*ε*The relative error between the calculation result by regression model and real value

*A*_{P},*B*_{P},*C*_{P},*D*_{P}Coefficients of polynomial cubic

*a*_{E},*b*_{E},*c*_{E}Coefficients of exponential functions

*a*_{L},*b*_{L},*c*_{L}Coefficients of logarithmic functions

*a*_{R},*b*_{R}Coefficients of reciprocal functions

*a*_{P},*b*_{P}Coefficients of power functions

*D*Demanding amount of products of orders

*E*_{c}Calculation result of energy consumption of the overall processes for using electric energy [

*KW∙h*]*E*_{real}The energy consumptions of the overall processes in real practice for using electric energy [

*KW∙h*]*F*Coefficient of

*F*test*F*(*θ*)The function to get the aggregation of least relative error

*f*_{SEC-P}The alternative mathematical form to be used during the modeling process

*G*_{c}Calculation result of energy consumption of the overall processes for using gas [

*m*^{3}]*G*_{real}The energy consumptions of the overall processes in real practice for using gas [

*m*^{3}]*N*The sample group number of observed

*SEC*and*P*used for the modeling*n*The number of unknown coefficients of the regression model

*P*Production rate

*P*_{elec}Production rate of die-casting machine producing finished products [

*piece/h*]*P*_{gas}Production rate of furnaces using gas [

*kg/h*]*R*^{2}Coefficient of determination

*R*_{gas},*R*_{elec}Relative energy efficiency evaluation indexes

*SEC*Specific energy consumption. The amount of energy required for processing a certain amount of one kind of product

*SEC*_{elec}*SEC*of producing finished products from the machine using electric energy [*KW∙h/piece*]*SEC*_{gas}*SEC*of melting raw materials using gas in furnace [*m*^{3}*/kg*]*s*Residual standard deviation

*T*Delivery time

*Tc*Cycle of time of process

*T*_{ca},*G*_{ca},*E*_{ca}The real time, gas consumption, and electric energy consumption of meeting the demand

*D*operating casually*T*_{h},*G*_{h},*E*_{h}The real time, gas consumption, and electric energy consumption of meeting the demand

*D*using high-energy-efficiency strategy*t*_{idle}Stand-by idling time

*t*_{p}Production time

*x*_{i}One of the observed value of production rate

*P**y*_{i}One of the observed value of

*SEC**δ*_{elec},*δ*_{gas}Relative error indexes to measure the accuracy of the prediction results

*θ*The unknown coefficient vector of the regression model

*θ*_{1},*θ*_{2}, …,*θ*_{n}The unknown coefficients of the regression model

- \( \widehat{\theta} \)
Optimum solution vector of the unknown coefficients

- \( \widehat{\theta_1},\widehat{\theta_2},\dots, \widehat{\theta_n} \)
Optimum solution group of the unknown coefficients

*ε*The relative error between the calculation result by regression model and real value

## Notes

### Funding information

This work was partially supported by the National Natural Science Foundation of China (Grant No. U1501248) and Nantaihu Innovation Program of Huzhou Zhejiang China.

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