Energy consumption modeling and prediction of the milling process: A multistage perspective

Abstract

In order to provide an accurate estimation of energy consumption, this work proposes a novel energy consumption modeling and prediction approach for a milling process from a multistage perspective. Based on its work stages, each stage’s energy consumption model is established by sliding filter, multiple linear regression, and improved gene expression programming (variable neighborhood search–based gene expression programming) methods and then the total energy consumption is predicted through their combination. A case study is given to illustrate the proposed model and its effectiveness. Compared with the full quadratic model, which can fully consider the interaction between cutting factors, the proposed method can achieve the higher accuracy to predict the energy consumption of the milling process.

Keywords

Sustainable manufacturing energy consumption milling process prediction model

Introduction

Manufacturing consumes significant amounts of energy, which leads to substantial stress on the environment.¹ Recently, with the increasingly serious resource and environment problems in manufacturing industry, sustainable manufacture attracts more and more attention. It is reported that the energy consumption in manufacturing industry contributes to 33% of the world’s total energy consumption.² Therefore, a reliable evaluation model of energy consumption would be of great significance for the environment and resource.

The metal machining, for example, milling and turning, plays an important role in the manufacturing field. In fact, machining process is a material removal process, which consumes a large amount of resource. The use of energy consumption model of machine tool can effectively analyze the resource consumption and environmental impact of mechanical processing system. But the complexity of parts and the particularity of materials have brought unprecedented challenges for machining technology. Performing the prediction of energy consumption can provide better judgment and support for the practice of the machining process. However, the energy consumption evaluation model of the machine tool is hard to obtain due to the complexity of the machine tool structure and the nonlinear relationship between the energy consumption and machining parameters. Therefore, establishing an appropriate and accurate energy consumption evaluation model of the machining process could help users to choose reasonable and optimal processing parameters. Moreover, the machining process is similar among numerical control (NC) machine tools. At present, for the parts with complex structure, thin-walled, variable thickness, many surface and curve structures, or high reconciliation precision, NC milling has been used to product them. As a typical machining way, NC milling process has been applied to many practical productions, such as auto parts, mobile shell molds, and so on. If its energy consumption is decreased, there will be much energy saved for other imperative need.

In view of the above, it is necessary to exploit an evaluation model of machining energy consumption. Since machining energy consumption problem was initialized by Gutowski et al.³ in 2006, there have been many advances in their modeling.^4,5 The early work is a simple model on cutting process, and it is expressed as follows

E = (P_{0} + k \times MRR) t

(1)

where E is the total energy demand, P₀ is the idle energy, k is a coefficient, MRR is the material removal rate, and t is the total cutting time.

From equation (1), it shows that the no-load energy is a dominant part of the direct energy demand in machining. It reflects that the choice of machine tools has a large impact on the direct energy demand. After equation (1) is put forward, many researchers have presented comparatively deep exploration to establish energy models of the machining process. For example, Rajemi et al.⁶ consider that start-up energy consumption is independent of the load. Mori et al.⁷ decompose the whole machining process into three states, that is, basic, free, and cutting conditions, and propose their measure methods of energy loss. Diaz et al.⁸ propose that the machining process could be divided into cutting and un-cutting states, but it faces inevitable energy acquisition problem. He et al.⁹ divide the total energy consumption into five parts, for example, energy consumption of spindle, feeding, cutting, cooling pump, and fixing. Hu et al.¹⁰ introduce an activation function describing the energy consumption for auxiliary and feed systems of the machine tool. Wang¹¹ divide auxiliary system into the processing energy loss and motor energy loss. Li and colleagues^12,13 propose that the no-load energy consumption could be divided into standby energy consumption and spindle energy consumption. Zhang et al.¹⁴ establish energy consumption optimization models for industrial processes and provide the guideline for the machining process programming. Yi et al.¹⁵ establish a carbon emission model of wet cutting process associated with energy consumption. Lin et al.^16,17 propose a carbon emission model of dry cutting process and propose three carbon-footprint reduction strategies, which are important to saving energy. The typical energy models in the above literatures are listed in Table 1.

Table 1.

The models of energy consumption.

Reference	Energy model	Notes
Rajemi et al.⁶	$E = E_{1} + E_{2} + E_{3} + E_{4}$	E is the total energy consumption, E₁ is the energy consumption when machine tool is started, E₂ is the energy consumption of material removal, E₃ is the energy consumption of tool change, and E₄ is the energy consumption of the tip of tool
Mori et al.⁷	$E = P_{1} (T_{1} + T_{2}) + P_{2} T_{2} + P_{3} T_{3}$	E is the total energy demand; $P_{1}$ , $P_{2}$ , and $P_{3}$ represents basic, idle, and cutting energies, respectively; and $T_{1}$ , $T_{2}$ , and $T_{3}$ represent the time consumed, respectively
Diaz et al.⁸	$E = P_{a v g} \cdot Δ t = (P_{c u t} + P_{a i r}) \cdot Δ t$	E is the total energy demand, P_cut is the cutting energy, P_air is the idle energy, and $Δ t$ is the cutting time
He et al.⁹	$E_{t o t a l} = E_{s p i n d l e} + E_{f e e d} + E_{t o o l} + E_{c o o l} + E_{f i x}$	E_total is the total energy consumption, E_spindle is the energy consumption of spindle, E_feed is the energy consumption of feeding axe, E_tool is the energy consumption of cutting tool, E_cool is the energy consumption of cooling pump, and E_fix is the located energy consumption
Hu et al.¹⁰	$\begin{matrix} E_{i n} = \sum_{i = 1}^{n} \int_{t_{0}}^{t} g_{i} (t) \cdot c_{i} d t \\ + \int_{t_{0}}^{t} p_{s p} (t, n, M_{s p}, B_{s p}, v, a_{p}, f) d t \\ + \sum_{j = 1}^{m} \int_{t_{0}}^{t_{j}} p_{a x} (t, w, M_{j}, B_{j}, T, f) d t \end{matrix}$	g_i(t) is the activation function, c_i is the corresponding energy consumption, P_sp(t, n, M_sp, B_sp, v, a_p, f) is the energy of spindle system, and P_sp(t, w, M_j, B_j, T, f) is the jth feed shaft energy consumption
Wang¹¹	$E_{i n} = E_{d} + E_{c f} + E_{d f} + E_{q f}$	E_in is the input energy, E_d is the associated motor energy, E_cf is the associated processing energy, and E_df and E_qf are energy consumptions of other auxiliary systems
Li and colleagues^12,13	$P_{i d l e} = P_{s t a n d b y} + P_{s p i n d l e}$	P_idle is the idle energy consumption, P_standby is the standby energy consumption, and P_spindle is the spindle energy consumption
Yi et al.¹⁵	$C E_{p} = C E_{e l e c} + C E_{t o o l} + C E_{f l u i d}$	CE_p is the carbon emission of the process, CE_elec is the carbon emission of the electricity consumption, CE_tool is the carbon emission of the cutting tools, and CE_fluid is the carbon emission of the cutting fluid
Lin et al.^16,17	$C E_{m s} = C E_{m} + C E_{t o o l} + C E_{e l e c} + C E_{c h i p}$	CE_ms is the carbon emission of manufacture system, CE_m is the carbon emission of the material, CE_elec is the carbon emission of the electricity consumption, CE_tool is the carbon emission of the cutting tools, and CE_chip is the carbon emission of the chip

Obviously, many researchers applied the above models to study other problems, for example, Dureja et al.¹⁸ take the cutting energy into the cutting parameter optimization in turning process. Balogun et al.¹⁹ analyzed specific cutting energy coefficient and integrated the energy demand. Based on the above, current researchers mainly analyze energy consumption modeling and evaluation problems from perspectives of processing state and machine tool construction. However, machining processing factors, for example, cutting velocity and cutting depth, have a large impact on the energy consumption; thus, these factor characteristics should be considered when energy consumption model of machining is established. In addition, researchers concentrate on energy consumption modeling and evaluation issues, but they pay little attention to its prediction problem. Although some researchers have proposed to use fuzzy logical^20,21 and statistical analysis²² to address the energy consumption prediction problem, these methods still heavily depend on subjective experience of experts. Moreover, the experience deviation of expert can lead to the reduction of prediction precision. Thus, it is essential to introduce an accurate prediction method based on the actual experimental data by identifying the main factors of the machining process to avoid the insufficiency of subjective experience of experts and improve the prediction performance. Also, each factor influencing the machining process has a strong nonlinear and complex relationship with the energy consumption. In this case, the accurate assessment between cutting parameters and energy consumption cannot be determined by the traditional linear regression method. While gene expression programming (GEP) and multiple linear regressions (MLRs) are two most widely used artificial intelligence tools in some fields due to its strong nonlinear fitting ability, for example, disassembly and efficiency of chemical inhibitors.^23,24 Also, sliding filter has the good evaluation ability in facing uncertainty data.²⁵ To do so, using TC500R drilling center as a typical machine tool and taking milling process as an example, this work proposes to establish its energy consumption prediction model using multistage modeling ideology with the combination of sliding filter, MLR, and the improved GEP for the first time. Namely, the aim of this work is to find a new way to determine the relationship between the energy consumption of milling process and machining parameters in view of energy efficiency and multistage. Compared with the existing work, there are two unique contributions: (1) Based on the work stages of the milling process, this work proposes a novel multistage approach to accurately predict the energy consumption for a milling process, that is, its each stage’s energy consumption model is established by sliding filter, MLR, and improved GEP (variable neighborhood search–based gene expression programming (VNS-based GEP)) methods, respectively, and then the total energy consumption is predicted through their combination. (2) By comparing with the full quadratic model, this work validates the effectiveness and feasibility of the proposed method in predicting energy consumption of milling process.

The rest of this article is organized as follows: section “Prediction methods and algorithms of energy consumption” presents the energy consumption model for milling process and lists the methods and algorithms used to predict it. Section “Case study” displays a case study for the presented model. Section “Analysis and verification of the proposed model” shows its comparative analysis and verification. Finally, section “Conclusion” concludes this article and describes future research issues.

Prediction methods and algorithms of energy consumption

Based on the energy consumption model proposed in prior work,¹² combining with work stage of the milling process, in this work, the energy consumption model of the milling process under dry cutting conditions is presented as follows

P = P_{b} + P_{f s} + P_{c} = P_{b} + (P_{f} + P_{s}) + P_{c a}

(2)

where P_b is the basic energy consumption without any operation on the machine tool, P_fs is invalid operating energy consumption, and it is the sum of energy consumption P_f of feeding process and energy consumption P_s of the spindle rotating process, and P_ca is the sum of energy consumption of material removal process and additional load loss. The characteristic curve of the total energy consumption about time for a milling process is shown in Figure 1, which is the real-time power measured during processing state on the TC500R.

Figure 1.

Energy consumption characteristic.

In view of Figure 1, the total energy of the milling process includes three-stage parts, that is, basic energy consumption P_b, invalid operating energy consumption P_fs, and the sum of energy consumption of material removal process and additional load loss P_ca. In addition, P_n represents the energy consumption when the machine tool is in no-cutting state.

In this work, based on each stage’s characteristic, the following three methods are adopted and proposed to predict the total energy consumption of the milling process. They are presented in the following.

Sliding filter method to evaluate P_b

For P_b, it is obtained by energy sensor before the start-up of spindle and feeding axe. Since the current and voltage are instable, the experimental value fluctuates within a certain range and is irrelative with the milling parameters. The sliding filter method can filter the experimental data with relatively large deviation. Then, the effective data can be obtained to estimate the average level.²⁶ Thus, the sliding filter is adopted to process the P_b data, and it is expressed as follows

{\hat{P}}_{b} (i) = {\begin{matrix} \frac{1}{i} \sum_{k = 0}^{i - 1} P_{b} (k) & i ⩽ L \\ \frac{1}{L} \sum_{k = 0}^{L - 1} P_{b} (i - k) & i > L \end{matrix}

(3)

where ${\hat{P}}_{b} (i)$ is the estimated value of $P_{b} (i)$ , L is the length of the sliding filter, i is the sampling point, and k is the counter.

MLR method to predict P_fs

The sum P_fs of feeding process energy P_f and the spindle rotating process energy P_s shows a linear trend with cutting parameters,²⁷ that is, v and f. To accurately achieve its prediction, an MLR is applied to predict it.

Due to the good regression and prediction ability, MLR method can be widely used to deal with a variety of issues such as biological, economic, chemical, and engineering problems.^28,29 In this work, it is used to predict the energy consumption for a part of milling process.

An MLR model is generalized as follows

t_{0} = β_{0} + β_{1} y_{1} + \dots + β_{n} y_{n}

(4)

where t₀ represents the prediction target variable or dependent variable, and y₁, y₂,…, y_n are independent variables with the coefficients $β_{0}$ , $β_{1}$ ,…, $β_{n}$ to be estimated from their data.

Since only the rotating speed of spindle and the feed rate are relative with the no-load energy consumption and the basic energy consumption is almost a fixed value, the prediction model of P_fs is expressed as follows

t_{0} = β_{0} + β_{1} y_{1} + β_{2} y_{2}

(5)

where the dependent variable t₀ represents P_fs and y₁ and y₂ represent cutting speed and feed rate, respectively.

Based on the above discussions, it can be seen that a dependent variable is considered as the combination of independent variables subject to their coefficients. Thus, the prediction of P_fs could be effectively conducted after their coefficients are determined. In this work, these coefficients can be obtained by Origin 8.0.

Improved GEP algorithm to predict P_ca

Since the sum of energy consumption of material removal process and additional load loss P_ca has the strong nonlinear relationship with cutting parameters, that is, v, f, and a_p, traditional methods, for example, MLR, cannot achieve the precise prediction, while GEP has good mapping capability for the nonlinear problems which are of great complexity, multi-input, and uncertain.³⁰ To do so, this work adopts an improved GEP algorithm, which combines variable neighborhood search (VNS) with GEP algorithm, to predict it.

In the GEP algorithm, the chromosome contains one gene or multiple genes connected by functions. Moreover, the gene of chromosome is expressed by a string, which is composed of linear symbols and concludes head and tail parts. The head part contains function symbols and terminators, while the tail part only contains terminators.²⁸ The length of the head h can be set according to a specific problem, while the length of the tail t is a function about h and n, and it is expressed as follows

t = h * (n - 1) + 1

(6)

where n is the maximum operation number of function symbols.

The genotype of gene (K-expression, K) is a string, which can be used to construct its phenotype (T), which is an expression tree. The expression tree can be translated into a mathematical expression. For example, there is a K-expression, that is, Q+×/abcda. The function set and the terminator set are considered as equations (7) and (8), respectively

F = {+, \times, /, Q}

(7)

T = {a, b, c, d}

(8)

where F is the function character set, Q is the square root, and T is the terminators set. The K-expression is translated into the expression tree by decoding, which is shown in Figure 2. In view of K, obviously n is 2, while the length of head h is set as 4, and then the length of tail t is 5. In addition, black part is called as noncoding region, also known as neutral genes.

Figure 2.

Expression tree.

The expression tree above can be translated into a mathematical expression as equation (9)

E = \sqrt{a \times b + c / d}

(9)

In the decoding process described above, the queue as a data structure is applied to temporarily store the binary trees. Through the en-queue and de-queue operations, characters in the K-expression are inserted in the nodes of binary tree from left to right, which is achieved as listed in Table 2.

Table 2.

Pseudo-code of decoding process of K-expression.

Algorithm 1: decoding process of K-expression
Input: K (K-expression)
Output: T (expression tree)
Procedure Construct (K, T)
(1) Set queue Q = NULL;
(2) Define the index pointer of K-expression p, p ⩽ 0;
(3) Initialize the expression tree T = new Node (p);
(4) P ⩽ p + 1;
(5) Push T to the tail of queue;
(6) While (Q);
(7) Get the first element (n) of Q;
(8) If (n is terminal set);
(9) Continue;
(10) End if;
(10) Construct new left node, left = new Node (p);
(11) Push left to the tail of queue;
(12) p ⩽ p + 1;
(13) Construct new right node, right = new Node (p);
(14) Push right to the tail of queue;
(15) p ⩽ p + 1;
(16) End while;
(17) Return T;

In addition, the GEP algorithm inherits the features of genetic algorithm, which has selection, mutation, and recombination operators. In addition, it has its particular operator, called as inserted string (IS) operator. The four operators of GEP algorithm are listed as follows.

Selection operator. According to the value of fitness function, select individuals with high fitness in the population within a threshold range.

Mutation operator. It plays a major role in maintaining the diversity of the population. Mutation occurs at any position in the chromosome, but in order to maintain the integrity of the chromosome structure, strings in head part can mutate into a function symbol, while the terminator and elements of the tail part can only mutate terminator.

IS operator. It is the particular operator of GEP and includes three kinds of operations, that is, IS, root inserted string (RIS), and gene IS.

Recombination operator. Two homologous chromosomes are mated and recombined to form new chromosomes, which also has three kinds of operations, that is, one-point recombination, two-point recombination, and gene recombination.

GEP is very effective at performing global search but often suffers from premature convergence, while local search is good at fine-turning but often falls into local optima. However, the basic idea of VNS is that changing the neighborhood structure set is to expand the scope of the search and then obtain a local optimal solution.³¹ If it is applied in a global search algorithm, the degree of premature convergence could be reduced. Hence, in this work, an improved GEP based on VNS is presented, and the role of VNS in the context of GEP has been taken into serious consideration. In the proposed algorithm, GEP is used to perform global search to escape from local optima, while VNS is used to perform fine-turning.

For the construction of neighborhood structure set of VNS, it could be considered from the string structure, that is, the head part and the tail part. Moreover, the search operation in neighborhood structure set contains mutations and exchanges. The steps of VNS are given in Table 3.

Table 3.

Pseudo-code of VNS.

Algorithm 2: VNS
Input: X (initialization solution)
Output: X_o (optimum solution)
Procedure Search(X, X_o)
(1) Initialize X and select the neighborhood structure set N_k (k = 1,…, k_max);
(2) For k = 1 to k_max;
(3) Select a solution X_c as the initializing solution in N_k;
(4) Mutate one point;
(6) Exchange one point;
(7) If (f(X_c) > f(X));
(8) X = X_c;
(9) k = k + 1;
(10) Else continue searching in N_k;
(11) End if;
(12) End for;

VNS: variable neighborhood search.

For the problem presented in this work, the genes in the improved GEP are expressed by a string, which contains operations, factors, and neutral genes. Similarly, the terminal set concludes three independent variables and three numerical constants in this work, which are expressed in the genes with a lowercase letter. In addition, the numerical constants are changed with generations and genes. They are changed randomly in the range of −1 to 1. For example, a chromosome with three genes has nine different numerical constants, which ensures the diversity of genes. The improved GEP mainly contains three parts, that is, initialization, genetic operation, and selection phase, whose flow is listed in Figure 3.

Figure 3.

The flowchart of the improved GEP.

Case study

Experimental design and data acquisition

To verify the proposed model, by taking the TC500R drilling center which is widely used in enterprises as an objective, this work obtains its energy consumption data as presented in the following.

For end milling side, because of the smaller chip, the size of the chip is not considered, and the rigidity of the tool is more than that of the tool. Moreover, even though milling the aluminum workpiece, the super hard alloy milling cutter is applied to ensure the stability of processing and the reliability of experimental data, which can also be consistent with the tool used by the firm researched on. The experiment details are listed in Table 4. The experimental platform and processing site are shown in Figure 4(a) and (b), respectively.

Table 4.

Experiment details.

Elements of experiment	Description
Machine tool	TC500R drilling center
Cutter	Super hard alloy milling cutter with three teeth (ϕ = 16 mm)
Tool tip format	Flat
Cutting type	Dry cutting and end milling
Energy measurement	WB9128 three-phase power sensor
Workpiece	6061Al (dimension: 220 mm × 110 mm × 100 mm)

Figure 4.

(a) Experimental platform and (b) processing site.

In the experiment, three factors, namely, cutting velocity v, feed rate f, and milling depth a_p, are considered. After different milling process parameter levels are given, the corresponding energy consumption data, that is, P_b, P_fs, P_ca, and P, are measured and obtained by power sensors, as shown in Table 5.

Table 5.

Experimental data.

Experiment number	ν (m/min)	f (mm/r)	a_p (mm)	P_b (W)	P_fs (W)	P_ca (W)	P (W)
1	120	0.05	5	446	310	216	972
2	140	0.05	5	443	332	285	1060
3	160	0.05	5	445	363	322	1130
4	180	0.05	5	447	393	356	1196
5	200	0.05	5	447	403	390	1240
6	130	0.1	5	446	314	470	1230
7	150	0.1	5	448	350	542	1340
8	170	0.1	5	445	375	580	1400
9	190	0.1	5	452	407	650	1500
10	210	0.1	5	444	401	695	1540
11	120	0.15	5	447	313	612	1372
12	140	0.15	5	450	330	700	1480
13	160	0.15	5	442	368	770	1580
14	180	0.15	5	446	404	800	1650
15	200	0.15	5	446	424	930	1800
16	130	0.2	5	446	324	830	1600
17	150	0.2	5	450	355	1000	1800
18	170	0.2	5	445	385	1070	1900
19	190	0.2	5	451	417	1160	2020
20	210	0.2	5	447	423	1260	2130
21	120	0.25	5	442	318	940	1700
22	140	0.25	5	443	347	1090	1880
23	160	0.25	5	446	364	1230	2040
24	180	0.25	5	446	404	1330	2180
25	120	0.2	2	447	313	285	1045
26	140	0.2	2	442	353	360	1155
27	160	0.2	2	443	387	410	1240
28	180	0.2	2	441	419	460	1320
29	200	0.2	2	450	442	480	1370
30	130	0.2	3	451	339	500	1290
31	150	0.2	3	442	388	600	1430
32	170	0.2	3	443	417	650	1510
33	190	0.2	3	445	435	720	1600
34	210	0.2	3	446	444	760	1650
35	120	0.2	4	447	323	540	1310
36	140	0.2	4	442	358	760	1560
37	160	0.2	4	447	383	840	1670
38	180	0.2	4	449	411	915	1775
39	200	0.2	4	441	439	998	1878
40	130	0.2	5	442	358	840	1640
41	150	0.2	5	446	379	985	1810
42	170	0.2	5	443	407	1090	1940
43	190	0.2	5	446	434	1200	2080
44	210	0.2	5	448	442	1285	2175
45	120	0.2	6	440	330	820	1590
46	140	0.2	6	446	354	1290	2090
47	200	0.05	2	446	412	162	1020
48	200	0.05	3	446	414	225	1085
49	200	0.05	4	450	413	330	1190
50	200	0.05	5	445	415	370	1230
51	200	0.05	6	446	414	520	1380
52	200	0.1	2	447	423	280	1150
53	200	0.1	3	448	422	410	1280
54	200	0.1	4	451	427	590	1460
55	200	0.1	5	445	425	955	1825
56	200	0.1	6	446	424	870	1740
57	200	0.15	2	447	428	375	1250
58	200	0.15	3	446	434	550	1430
59	200	0.15	4	446	434	780	1660
60	200	0.15	5	446	434	1000	1880
61	200	0.15	6	445	435	1210	2090
62	200	0.2	2	451	442	485	1370
63	200	0.2	3	445	435	730	1610
64	200	0.2	4	446	439	995	1880
65	200	0.2	5	446	429	1225	2100
66	200	0.2	6	443	437	1510	2390
67	200	0.25	2	445	445	590	1480
68	200	0.25	3	450	443	800	1690
69	200	0.25	4	446	444	1170	2060
70	200	0.25	5	447	438	1435	2320
71	130	0.05	5	446	324	280	1050
72	150	0.2	2	444	387	400	1230
73	160	0.1	5	447	363	570	1380
74	180	0.2	3	446	429	725	1600
75	190	0.15	4	447	428	755	1630

The selection of cutting speed mainly depends on the material of the workpiece, while the choice of the feed rate and the depth of milling mainly depend on the material of the workpiece, the diameter of the end milling cutter, and machine performance. Combined with the experience of cutting, the cutting velocity in the range of 120–210 m/min, the feed rate in the range of 0.05–0.25 mm/r, and the milling depth in the range of 2–6 are widely used in semi-finish machining on the machine tool. For the design of experiment, in view of the methods above, a large number of experimental data are needed to train. Thus, combining the thought of mean experimental design and orthogonal experimental design could avoid the lack of experimental data and achieve comprehensive coverage. The experimental data are divided into two parts, that is, training data listed as first 70 lines and testing data listed as last 5 lines. Among them, the testing data are designed randomly but unduplicated with the front ones. In addition, the cutting width is set as 8 mm in this experiment, which has little impact on energy consumption compared with the other three elements.

Prediction of energy consumption

Based on the above-described method and experimental data, this work adopts a novel three-stage method to predict the energy consumption for a milling process, namely, determination of P_b by sliding filter method, prediction of P_fs by the MLR method, and prediction of P_ca by the improved GEP algorithm. After this, the total energy consumption of the milling process is obtained.

Evaluation results of P_b

For a machine tool, if it is at nonload state, its energy consumption is theoretically constant. However, as described above, since the voltage and current are instable, the measure value of basic energy P_b is a certain value with volatility. According to the experimental data in Table 3, P_b is determined by the slide filtering method. For example, for the third experiment, according to the slide filtering method, the sampling points corresponding to P_b are intercepted as shown in Figure 5. Thus, P_b corresponding to the third experiment is obtained, that is, 445 W. Similarly, P_b corresponding to other experiments can also be obtained as line 5 in Table 5. So, the estimated value of P_b is the mean of P_b corresponding to each experiment, which is 446 W.

Figure 5.

Interception of real-time P_b of the third experiment.

Prediction results of P_fs

For P_fs, to accurately achieve its prediction, an MLR method is applied to do it. Using Origin 8.0 to process the training data, its coefficient results and the statistic results between prediction value and experimental data are displayed in Table 6. Also, the correlation coefficient R² and the mean absolute percentage error (MAPE) are used to evaluate the prediction performance; usually, the larger the R² and smaller the MAPE, the higher the accuracy. MAPE can reflect the gap between predictive value and experimental value.

Table 6.

The result of regression.

Coefficient	Value	Evaluation parameter	Value
Constant (k₀)	117.5	R ²	0.982
k ₁	1.46	MAPE	0.02
k ₂	117.6	−	−

MAPE: mean absolute percentage error.

From Table 6, the R² is 0.982, and MAPE is only 0.02. The training effect of MLR is highly satisfactory when the method is used to predict P_fs, which is written as follows

P_{f s} = 117.5 + 1.46 \times v + 117.6 \times f

(10)

To verify its predictive ability, the testing data are applied to predict them, and their results are shown in Figure 6. From Figure 6, the MLR method has the good predictive ability. Moreover, the MAPE between testing data and predictive data is 0.039, and the R² is 0.973, which also reflect its accuracy.

Figure 6.

The prediction results for testing data for P_fs.

Prediction results of P_ca

Since P_ca has the strong nonlinear relationship with cutting parameters, that is, v, f, and a_p, this work adopts the improved GEP method to predict it.

The program is coded in C++ and implemented on a personal computer with a 2.53-GHz Intel Core 2 Duo central processing unit (CPU). The specific parameters used in the improved GEP are set in Table 7. In addition, the improved GEP’s function set concludes 10 operations, and it is displayed in Table 8. The probabilities of genetic operations are set according to the genetic algorithm. For example, the mutation rate is generally set in the range of 0.01–0.1.³² Note that 75 data shown in Table 3 are divided into two groups, first 70 dataset is used to train and the other 5 dataset is used to test.

Table 7.

The specific parameters of improved GEP in the models.

Parameter	Value	Parameter	Value
Number of generations	1000	Mutation rate	0.01
Population size	50	One-point recombination rate	0.3
Head length	10	Two-point recombination rate	0.3
Tail length	11	Gene recombination rate	0.1
Number of genes	3	IS transposition rate	0.1
Numerical constants in each gene	3	RIS transposition rate	0.1

IS: inserted string; RIS: root inserted string.

Table 8.

Function set and terminators.

Symbol	Meaning	Symbol	Meaning
+	Plus	P	Power
−	Minus	K	Square root
*	Multiply	a	Cutting speed
/	Divide	b	Feed rate
C	Cot	c	Depth of cut
T	Tan	d	Numerical constant of gene 1
L	Ln	e	Numerical constant of gene 2
E	Exp	g	Numerical constant of gene 3

Take 70 three factors, that is, cutting speed v, cutting depth a_p, feed rate f, and energy consumption P_ca dataset as input and output of GEP training, respectively. After the improved GEP algorithm is run, the specific GEP model is established. For estimating the effect, root mean square deviation (RMSD) is selected as the fitness function in the algorithm. RMSD is expressed as in equation (11). In addition, the iterative results are progressing as shown in Figure 7

RMSD = \sqrt{\frac{\sum_{j = 1}^{n} {(C_{i, j} - T_{j})}^{2}}{n}}

(11)

Figure 7.

The iterative results.

The result shows that the best fitness is 24.3, and the R² is 0.989, which denote that accuracy of the predicting model about the milling energy by the improved model is highly satisfactory. The three genes of the best chromosome obtained are shown in Figure 8.

Figure 8.

The result of GEP.

In addition, the constant terms of the three genes are obtained, which are listed in Table 9. The genes expressed by strings are used to construct the corresponding sub-trees via decoding, which are shown in Figure 9.

Table 9.

Numerical constants of genes.

Constants	D	e	g
Gene 1	−0.13	0.43	0.60
Gene 2	0.16	0.34	0.78
Gene 3	0.37	−0.56	−0.32

Figure 9.

ET for expression.

Thus, the prediction model of milling energy P_ca is obtained and is expressed as follows

P_{c a} = s u b T r e e_{1} + s u b T r e e_{2} + s u b T r e e_{3}

(12)

In addition, in order to make future tests and observe the effectiveness of the proposed GEP method, the prediction result for testing data is presented in Figure 10.

Figure 10.

The prediction results for testing data for P_ca.

From Figure 10, the proposed GEP method has the good predictive ability. Moreover, the MAPE between testing data and predictive data is 0.066, and the R² is 0.996, which also reflect its accuracy.

Prediction of total energy consumption P

Based on the above three-stage modeling and equation (2), P is the combination of P_b, P_fs, and P_ca, and it is presented as follows

P = P_{b} + P_{f s} + P_{c} = 563.5 + 1.46 \times v + 117.6 \times f + s u b T r e e_{1} + s u b T r e e_{2} + s u b T r e e_{3}

(13)

where subTree₁, subTree₂, and subTree₃ are functions associated with cutting variables in the following equations. In addition, the values of d, e, and f come from Table 9

s u b T r e e_{1} = \ln [(a_{p} + d) v^{a_{p}}] \times [(e + g) a_{p}^{d} + v f]

(14)

s u b T r e e_{2} = \tan (\tan (a_{p} - e) + \tan (v - d))

(15)

s u b T r e e_{3} = \cot (\cot (\cot (e^{f} + d - v - \cot (v + a_{p}))))

(16)

Analysis and verification of the proposed model

To verify the proposed combination method, the traditional full quadratic model is used to contrast with it. Note that the latter can effectively solve many prediction problems, for example, constrained program and mixed integer program.³³ Facing this problem proposed in this study, full quadratic model can fully consider the interaction between cutting factors. Thus, it is selected as a method for our comparison purpose in this work. The full quadratic model is expressed as follows

y (x) = α_{0} + \sum_{i = 1}^{p} α_{i} \cdot x_{i} + \sum_{i = 1}^{p} \sum_{j = 1}^{p} α_{i j} \cdot x_{i} \cdot x_{j} + \sum_{i = 1}^{p} α_{i i} \cdot x_{i}^{2} + ε

(17)

where y(x) is response value, x_i and x_j are the input variables, $α_{i}$ is the coefficient, $ε$ is the error term, and the p is the number of variables.

In this model, for predicting the energy consumption, x₁, x₂, and x₃ denote the cutting velocity v, the feed rate f, and the milling depth a_p, respectively. It is obtained by MATALB as follows

\begin{matrix} y = 694.42 + 5.61 \times v - 2643.8 \times f - 222.4 \times a_{p} \\ \begin{matrix} + \end{matrix} 20.45 \times v \times f + 1.08 \times v \times a_{p} + 1059 \times a_{p} \times f \\ - 0.02 \times v^{2} - 4531 \times f^{2} + 5.58 \times a_{p}^{2} \end{matrix}

(18)

In addition, in order to observe the effectiveness of the proposed combination method, the training results for 70 training data among experimental data, the full quadratic model, and the proposed method are presented in Figure 11.

Figure 11.

The training result comparison.

From Figure 11, overall, the proposed method has a better training performance than the full quadratic model. In addition, the training performance comparison for different prediction models is listed in Table 10.

Table 10.

Performance comparison for different prediction models.

Evaluation parameter	Prediction model
Evaluation parameter	Full quadratic model	Proposed model
R ²	0.9825	0.9905
MAPE	0.095	0.02

MAPE: mean absolute percentage error.

Moreover, the significant tests about the two models are conducted by MATLAB. As the sample number is >30 and subjects of testing are paired samples, the function t-test² expressed in equation (19) is chosen to achieve it. The null hypothesis is the mean of experiment data is equal to the mean of the predicting value of the above models. When alpha is 0.01, the result is 0, which reflects to accept the null hypothesis at the significance level of 0.01. All in all, the proposed method can achieve the higher accuracy to predict the energy consumption of the milling process

t = \sqrt{n} \frac{\bar{X} - μ}{σ_{x}}

(19)

where t is the deviation statistic, n is the number of sample, $\bar{X}$ is the sample average, $μ$ is the ensemble average, and $σ_{x}$ is the standard deviation of sample.

Additionally, the training result could only prove that the proposed model has better fitting ability than full quadratic model. To testify its predictive ability, the testing data are verified. Thus, the results of different prediction models for testing data are shown in Table 11 and Figure 12.

Table 11.

The comparison of prediction error.

Experiment number	Testing data	Full quadratic model	Proposed model
1	1050	1069.378	1069.18
2	1230	1314.54	1256.02
3	1380	1518.53	1398.86
4	1600	1684.04	1579.82
5	1630	1778.508	1658.54
R ²	−	0.9925	0.9971
MAPE	−	6.62%	1.66%

MAPE: mean absolute percentage error.

Figure 12.

The prediction results for testing data for P.

From Table 11 and Figure 12, the energy consumption by the proposed method has more predictive effect than traditional full quadratic model, with the precision of >95%. The result denotes that the proposed method can effectively and accurately predict the energy consumption of the milling process.

Conclusion

For the specific milling process, the energy consumption model has been developed from a multistage perspective. The modeling and testing experiments are conducted on TC500R drilling center. The model proposed fully takes the relationship between energy consumption and the cutting variables in different stages into consideration. This method can not only avoid errors brought by the algorithm without this consideration but also explain its physical meaning to a certain extent. What is more, the improved GEP algorithm is proposed for solving complex and nonlinear predicting problem. This algorithm combines the advantages of VNS and GEP, which could escape from local optima and perform fine-turning. The study case proved its good effect. Compared with the full quadratic model for energy consumption, the proposed method can predict energy consumption of the specific end milling process with more accuracy. The obtained results can be used to guide decision makers in making better milling decisions.

There are still some issues not solved in this article. The model is developed on drilling center, which has some limitation when it is under end milling. For the model developed, the inner laws of how the energy consumption affects the machining process can be studied in the future. In addition, as the limitation of experiment conditions, this work only considered the milling process. Future works will be done to validate in a wider field and modern machining conditions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by funds from International Cooperation and Exchange of the National Natural Science Foundation of China (No. 51561125002), the National Natural Science Foundation of China (Nos. 51575211, 51405211 and 51275190), the Natural Science Foundation of Hubei Province (No. 2014CFB348), and Postdoctoral Science Foundation Project of Heilongjiang Province of (Nos. LBH-Z13005 and LBH-TZ0501).

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Energy consumption modeling and prediction of the milling process: A multistage perspective

Abstract

Keywords

Introduction

Prediction methods and algorithms of energy consumption

Sliding filter method to evaluate Pb

MLR method to predict Pfs

Improved GEP algorithm to predict Pca

Case study

Experimental design and data acquisition

Prediction of energy consumption

Evaluation results of Pb

Prediction results of Pfs

Prediction results of Pca

Prediction of total energy consumption P

Analysis and verification of the proposed model

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Sliding filter method to evaluate P_b

MLR method to predict P_fs

Improved GEP algorithm to predict P_ca

Evaluation results of P_b

Prediction results of P_fs

Prediction results of P_ca