Accepted Manuscript An Energy Modeling and Evaluation Approach for Machine Tools Using Generalized Stochastic Petri Nets Nan Xie, Minglei Duan, Ratna Babu Chinnam, Aiping Li, Wei Xue PII:
S0959-6526(15)01328-1
DOI:
10.1016/j.jclepro.2015.09.100
Reference:
JCLP 6190
To appear in:
Journal of Cleaner Production
Received Date: 4 September 2014 Revised Date:
18 July 2015
Accepted Date: 22 September 2015
Please cite this article as: Xie N, Duan M, Chinnam RB, Li A, Xue W, An Energy Modeling and Evaluation Approach for Machine Tools Using Generalized Stochastic Petri Nets, Journal of Cleaner Production (2015), doi: 10.1016/j.jclepro.2015.09.100. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
An Energy Modeling and Evaluation Approach for Machine Tools Using Generalized Stochastic Petri Nets Nan Xie1,2, *, Minglei Duan3, Ratna Babu Chinnam2, Aiping Li3, Wei Xue4 1
1
Sino-German College of Applied Science, Tongji University, Shanghai, 201804, China
Industrial & Systems Engineering Department, Wayne State University, Detroit, MI 48202, USA 3
4
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2
School of Mechanical Engineering, Tongji University, Shanghai, 201804 China
College of Mechanical & Electrical Engineering, Wenzhou University, Wenzhou Zhejiang 325035, China
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Abstract-Lower production cycle time and higher energy efficiency have become a paramount requirement for machining processes under global competition. A formalized method to assess multiple performance indicators (e.g. production time, energy consumption) of machine tools for comparable analysis of production data is indispensable given the increasing scrutiny of manufacturing systems. A novel energy consumption model based
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on Generalized Stochastic Petri Nets is proposed and an analysis method is also presented. Furthermore the model was successfully applied to a turning machine tool. The main trend of energy consumption and other related indicators of studied machine tool are then compared and the optimal parameters for energy efficiency are discussed under various cutting parameters and different production plans.
Key words: Machine tool, energy consumption, GSPN, evaluation
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1 Introduction Global warming concerns and other factors have accelerated the attention to environmental aspects and are in turn exerting pressure on companies to do their part. In addition, the increasing energy, raw material prices and regulative incentives of carbon emission reduction are issues affecting manufacturing enterprises. Energy efficiency remains one of the major issues in the machining domain. Rapid and flexible machine tool energy modeling in a distributed and collaborative machining environment emerges as a new research area[1]. Therefore, apart from traditional economical production aims (e.g. cost, time, quality), enterprises are also focusing on environmental driven objectives[2]. Manufacturing plays an important role in achieving environmental sustainability. Manufacturing generates over 60% of annual nonhazardous waste and more than one-third of all energy consumed is attributed to industrial use[3,4]. A prerequisite to improving the energy utilization is the establishment of effective models for quantification of energy consumption. Gutowski proposed a thermodynamic framework to characterize the material and energy resources used in manufacturing processes[5]. Li and Kara presented an empirical model based on power measurements under various cutting conditions and also provided a reliable prediction of energy consumption[6,7]. Zein et al. proposed an axiomatic approach based functional requirements model for improving energy efficiency of machine tools[8]. Lv and colleagues defined a set of fundamental motions and proposed therblig theoretical power models for calculating the energy supply of machine tools using machining process parameters[9,10]. Apart from the energy models using different theoretical bases, the literature also offers specific models for individual machining processes. Behrendt et al. divided the machine *
Corresponding Author;
[email protected]
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tool state into idle, operational sequences and machining operations states, and built three assessment models[11]. Liu and Shuang described the machining process as machine tool’s service process and developed the energy model for electro-mechanical main driving system[12]. Machining process is a prototypical example of a discrete event dynamic system (DEDS), and hence, DEDS modeling techniques are useful to evaluate the energy consumption within these processes. Dietmair and Verl[13,14] proposed a generic energy consumption model of machines and plants based on a state transition diagram. Applications of this model in decision-making process for real-time, tactical and strategic levels were also introduced to make it possible to improve energy efficiency of any given machine or production system. Larek et al.[15] built machining operations model based on discrete-event simulation and used this model to estimate power consumption in machining operations according to NC code and workpiece information. Frigerio et al.[16] modeled functional modules of machine in terms of states and events with automata theory and obtained machine energy state model with synchronization algorithm. Peng et al. proposed an energy consumption model which was comprised of machine component power model and operation state model, and the operation state model was built based on Component-Mode-State matrix and state transition graph[17]. Although a large amount of effort has gone into modeling energy consumption of machine tools, they mainly focused on a single process cycle. In fact, the energy model should not only describe the process cycle but also reflect the long-term production plans, which further require that the formalized energy model should be descriptive, scalable and dynamic to meet the characteristics of machining processes. As a powerful tool to describe the DEDS[18,19], an energy consumption model based on Petri Nets is presented. The stochastic nature of the production process is also considered in the presented model. The influence of the different production plans on energy consumption has also been detailed and possible strategies to save energy are discussed as well. The rest of this manuscript is organized as follows: Section 2 presents the power model of the stages; section 3 proposes the energy consumption model using generalized stochastic Petri Nets (GSPN) and analysis method; a case study is introduced in section 4; conclusions are offered in the final section.
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2 Model of machine tool power Gutowski’ research shows that the critical influence factor to cutting power is material removal rate (MRR) as equation (1)[5,10]. Pcut = P0 + m × MRR. (1) where, P0 is idle power that includes computer system, lights and so on for a standby machine tool, and m is a machining operation specific constant. However, this model does not have enough accuracy to describe the machining process. Consider the actual process of a turning machine tool; the power is divided into pre-cutting power, cutting power. Moreover, the pre-cutting power includes the load-free spindle power and load-free feed power. Figure 1 shows a sample energy profile for the case of a turning machine tool. Fig. 1. Sample energy profile of a turning machine tool
In general, the typical main stages of a turning machine can be distinguished to be off, start-up, idle, ready to machine/precut, and operation. These main stages can be also identified as
ACCEPTED MANUSCRIPT the states of the discrete events during the machining cycle. Turn on PTurnOn = PBM , where, PBM is the basic modules power after machine tool turns on.
Cooling system: Energy demand for cooling system.
PCooling is constant.
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Peak: Energy demand peaks caused by switching on certain modules, such as spindle etc. PPeak = k × PBM , where k is a coefficient. Pre material cutting: positioning and loading before actual processing (e.g. movement of spindle and feed components in position to workpiece but without material removal).
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PPr ecutting = Pf + Psp − PBM , where, Pf is the movement of feed components without material
removal, and Psp is the movement of main spindle without material removal. Cutting: actual workpiece process take place, energy to fulfill cutting task.
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PCutting is a variable.
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3 Energy evaluation model based on GSPN Petri Net models have emerged as a very promising approach for modeling systems that exhibit concurrency, synchronization and randomness and are widely used in the domain of manufacturing systems. As discussed earlier, the production process of a machine tool can be seen upon as a prototypical discrete event system with stochastic characteristics. Therefore, the use of Generalized Stochastic Petri Nets (GSPN) has become particularly important in the modeling of production processes. The machine tool energy consumption information can be represented by the transitions of GSPN, and the GSPN model is generated to mimic the stochastic process that governs the systems behavior. In turn, the energy performance of the machine tool can be evaluated by GSPN analysis technique. 3.1 Definition of GSPN GSPN is a form of Petri Net where transitions fire after a probabilistic delay determined by a random variable. In a GSPN, an exponentially distributed delay is associated with the firing of transition. The delay occurs when the transition becomes enabled and when it fires [20,21].
(
)
A GSPN is an eight-tuple P,T,T1 T2W,I − , I + , M 0 , where: P = (P1,P 2 ,L ,Pm ) is a set of places
T = (T1,T2 ,L ,Tn ) is a set of transitions (with P U T ≠ φ , P I T = φ ).
T1 ⊆ T is the set of timed transitions. T2 ⊆ T is the set of immediate transitions, where T1 I T2 = φ and T = T1 U T2 .
(
)
W = w1 ,L, w T is an array whose entry wi ∈ ℜ + is a (possibly marking dependent) rate of a
negative exponential distribution (also denoted λi ) specifying the firing delay, when transition t i ∈ T1 , or firing weight, when t i ∈ T2 . I − , I + : P × T → N 0 are the backward and forward incidence functions, respectively.
ACCEPTED MANUSCRIPT M 0:P → N 0 is initial state marking.
λ = {λ1 , λ2 LL λm } is the set of transitions’ average firing rate. Each λi is measuring value from the actual system or the predictive value obtained in accordance with certain requirements. τ i = 1 λi is the average delay of timed transition.
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3.2 Energy assessment model and analysis As noted earlier (see Figure 1), there are several stages in a typical production process cycle, which include turn on, turn additional subsystem, switch on the axis, pre material cutting, cutting, back to initial status and so on. In terms of the process sequence, the stages are associated with places and the changes between stages are associated with transitions. A reasonable simplification is made to reduce the complexity according to the Petri Net simplification method. Assume that there are N states in one production cycle and let S = {1,2,L , N } represent the
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different states. The machine tool could “return” to any specific state such as turn on, pre-material cutting and turn off at various firing rates. The GSPN model of production cycle is shown in Figure 2.
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Fig. 2. GSPN representation of machine tool energy model along with state transitions
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In this model, Places N+1, N+2 and N+3 represent the machine tool returning back to pre-material cutting, turn on and turn off state, respectively, at the end of processing. The three immediate transitions Ts1,Ts2,Ts3 represent the corresponding firing probability, which reflect the production plan. There are random switches among Ts1,Ts2,Ts3 with the sum of firing probability of these three transitions being equal to 1. For instance, if probability of Ts3 equals 1, then the machine tool will be definitely turned off after a part is completed. Once the model is completely specified, the performance of the production plan and the particular machine tool can be calculated as follows: (1) Build GSPN model. Define the places, the timed transitions and the immediate transitions. Estimate the cycle time based on the given production plan, and obtain the energy consumption states during the production processing. (2) Generate the homogeneous state-transition Markov chain. Calculates the reachability graph, and then analyzes the structural properties of the Petri Net model. Assign each arc with corresponding firing rates, and then generate the state-transition Markov chain. All markings are denoted as m0 , m1 LL mn−1 , where n is the total number of states. Each marking illustrates a distinct machine tool state, such as turn on, cutting and so on. (3) Construct the transition probability matrix of the state-transition Markov chain. The states set S is divided into tangible state T and vanishing state V, S = T U V , T I V = φ . The number K s = Kt + Kv . The above Markov chain transition probability matrix is described as U = F + EG ∞ , where, F denotes the transition probability from tangible state to tangible state set in U and E denotes the transition probability from tangible state to vanishing state in U. The elements in G ∞ , g ij = Pr {r → j} , provide the probability of reaching tangible state j for the first time from the
given vanishing state r after any number of steps. (4) Calculate the steady state distribution of tangible states. Solve the linear equations Y = YU , where, Y is the steady state distribution of tangible states.
ACCEPTED MANUSCRIPT (5) Calculate the probability of tangible states. The sojourn times of each state is 0 ∀i ∈ V −1 STi = . The average cycle of back to reference state is of length Wi = ∑Vij ST j . rf ∀j ∈ T j∈T f∑ ∈H i
0
j ∈V j ∈T
.
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So, the steady state probability of GSPN can be expressed as Pj = Vij ST j / W
( ) ∑p
F = ∑ fi = τ j × i
mi =G j
i
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(6) Analyze the system performance. Having the steady state probability distribution P, several performance measures and energy consumption can be derived as follows: 1) The average productivity F This corresponds to the mean number of tokens in the places where parts are under processing. , where, G j is the subset of the marking that t j is enabled.
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2) The average power POWm This relates the steady state probability of each tangible state and its corresponding energy consumption. The energy consumption of state i is POWi . POWm = ∑ pi × POWi i
3) The average energy consumption of a production cycle Em This represents the average energy consumption under Em = POWm × (1 F ) .
a
part
processing
cycle.
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4 Experimental study 4.1 Experiment devices The experiments were performed on a CNC Lathe HTC200M from Cincinnati Company (see Figure 3). The technical parameters of machine tool are shown in Table 1. A power sensor, HIOKI PW3360-30, produced by HIOKI Company, was installed to measure the power of machine tool. The sampling rate of the power sensor was 10.24KHz and the average power per second was calculated automatically. Turning tool is CNMG 1204 12-PF 4215.
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Fig. 3. The CNC machine tool and power sensor
4.2 Power measurement As shown in Figure 1, the energy consumption of five states need to be identified which include turn on, cooling system, peak, pre-cutting and materiel cutting. (1) Experiments for identifying turn on state The machine tool is turned on without any additional operations, so only basic modules are working. The power of basic modules is measured by power sensor several times and the average is calculated to be PBM= 920W. (2) Experiments for identifying cooling state The power is measured after the cooling system is open and the average is calculated to be Pcooling=268W. (3) Experiments for peak state There is a short peak energy period right after the spindle and feed axes start. In this
ACCEPTED MANUSCRIPT experiment, the coefficient is measured to be k=5.7, yielding Ppeak = 5.7 × PBM = 5244 W.
Fig. 4. Spindle power profile with no load
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(4) Experiments for pre material cutting state The power of spindle is measured at different motor speeds when it is facing no load. A function is adopted to express Psp by linear regression analysis with the observation data. From Figure 4, the air spindle Power can be expressed as Psp=0.585×n+968.943W, where n denotes spindle speed in revolutions per minute (r/min) and the correlation coefficient R-square is 0.99334, which means that this linear model can describe over 99.3% of the variation in air spindle power under different rotational speeds.
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Likewise, the power of feed components is measured and calculated from observation data as shown in Figure 5. The air feed component power demand can be expressed as Pf=0.017×vf + 940.92W where vf denotes feed rate in mm/min, and the correlation coefficient R-square is once again very satisfactory at 0.966.
(5) Experiments for material cutting state The
material
cutting
Fc = k FC × C FC × vCnFC × f
y FC
× a pF C x
power
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Fig. 5. Feed system power profile with no load
Pcutting = (1 + λ ) Fc × vc
and
the
cutting
force
, where, λ is a coefficient of power loss, k FC is a coefficient
[22]
of correction, C FC , n FC , y FC and x FC are constants, v c is cutting speed (m/min), f is feed rate
Pcutting = a × vcb × f c × a dp
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(mm/r) and a p is depth of cut (mm). So, the material cutting power is expressed as where, a = (1 + λ ) × k FC × C FC , b = nFC + 1 , c = y FC , d = x FC . The
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diameter of workpiece is 120mm and the length is 90mm. The observation data of material cutting power under various cutting parameter settings are shown in Table 1. The coefficients computed from data regression are shown in Table 2. Overall, the material cutting power can be expressed
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as Pcutting = 85.108 × vc0.87 × f 0.826 × a 0p.931 W.
Table 1. Material cutting power under various parameter settings
Table 2. Coefficients of cutting power model and their statistical fitness
A comparative study between the measured and calculated machine tool power at various speeds under the different diameters is shown in Table 3. Table 3. Predicted and experimental results of power
4.3 The energy model 4.3.1 GSPN energy model The GSPN model of CNC Lathe HTC200M is shown in Figure 6. Table 4 shows the meanings of places. The meanings and category of transitions are shown in Table 5.
ACCEPTED MANUSCRIPT Fig. 6. GSPN energy model for CNC Lathe
Table 4. Meanings of places of figure 6 model Table 5. Meanings of transitions of Figure 6 model
4.3.2 The state transition diagram energy model The state transition diagram energy model is also presented in Figure 7.
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Fig.7. State transition diagram energy model for CNC Lathe
A state S i can be given as 7-tuple: S i = (VC ,i , f , a p ,i , ni ,V f ,i , POWi , Pi )
where VC , i is cutting speed (m/min) in state Si , f i is feed rate (mm/r) in state S i , a p,i is depth of
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cutting(mm) in state S i , ni denotes spindle speed in revolutions per minute (r/min) in state
represents the steady state Si probability. A transition Qi is defined as:
Qi = ( S n , S m , C , tTrans , t S n )
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S i , V f , i denotes feed rate in mm/min in state S i , POWi is energy consumption of state S i , Pi
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where Sn is the original operational state with index n, Sm is the target operational state with index m, C contains the logic expression that has to evaluate to ‘TRUE’ for the transition to be active, tTrans is the time that elapses between initiation of the transition and switching to Sm and tSn is the minimum time the machine has to remain in state Sn before the transition becomes active. Both the GSPN energy model and the state transition diagram energy model are quite equivalent in the proposed context, i.e., modeling the machining process of a CNC Lathe. Compared to state transition diagram, GSPN method has two advantages. First, GSPN model can readily describe the stochastic events of the system. In the model we presented, the model can flexibly handle the proportion coefficients of concurrent events α β γ to adapt to different
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production plans. Second, GSPN model provides the analytical method for calculating the energy and others properties of the model. 4.4 Energy analysis in various cutting parameters The impact of the energy consumption as cutting parameters are varied is evaluated. The firing probability of immediate transitions α (possibility to return to air cutting state), β (possibility to return to turn on state), γ (possibility to return to turn off state) are set as α = 0 β = 0 γ = 1 , which means after a workpiece is processed the machine tool is turned off. Therefore, the power, energy, process time and production cycle only depend on the diameter of part and cutting parameters. The cutting power and average power per production cycle are shown in Figure 8. The average product time is shown in Figure 9, and the energy of per product is shown in Figure 10. Fig. 8. Average power per production cycle (ap=0.4)
Figure 8 shows the average power per workpiece is 22% growth, as vc is varied from 200
ACCEPTED MANUSCRIPT r/min to 1200 r/min and f is varied from 0.1mm/r to 0.6 mm/r, while the cutting power varies significantly with a peak power of 10.7 times the minimum cutting power drawn during the production cycle. Fig. 9. Average production time per workpiece (ap=0.4)
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In Figure 9, average production time per workpiece is given as a function of the steady state probability of P6 and firing rate of T6. The average production time per workpiece leads to 53% decrease, as vc is varied from 200 r/min to 1200 r/min and f is varied from 0.1mm/r to 0.6 mm/r. Fig. 10. Average energy consumption per workpiece (ap=0.4)
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The average energy consumption is given as a function of both average production time and average power. As shown in Figure 10, changes to the cutting parameters vc and f lead to 35% decrease in average energy consumption. Although the average energy consumption decreases when vc is varied from low speed to high speed at the beginning, there is noticeable increase of energy at higher vc. For example, Figure 11 shows that the average energy deceases significantly from 200 r/min to 1000 r/min, whereas the average energy increases when the vc is larger than 1000 r/min. The vc is up to 2400 r/min then the large-scale tendency of the energy consumption can be showed in figure 11.
Fig. 11. Average energy as a function of Vc
Figure 12 depicts the impact of the depth of cutting on average energy consumption, production time and average power, respectively. a p has almost no influence on production time.
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The average power and average energy increase 52% as a p is varied from 0.4mm to 1.3mm. Meanwhile a p is subjected to the part process plan.
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Figure 12. Impact of depth of cutting on average energy consumption, production time and average power (vc=400, f=0.4)
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4.5 Energy analysis under different production plans From the production plan perspective, energy consumption is also related to the residency times of non-processing states. One is the idle state, another is air cutting state. According to GSPN model in Figure 6, Ts1 firing means the machine tool waits for the next operation while turning on the spindle and feed system, while Ts2 firing means the machine tool waits the next operation only by turning on the basic modules. The procedure centralization degree of the former case is higher than that of the latter case and then, the probability of Ts1 increases the procedure centralization ratio. Assume γ=0.1, then the α plus β equals to 0.9. In Figure 13, the production time, average power and average energy consumption are given, as α increase from 0.1 to 0.8 while β decreases. The average production time improves to 71.7s, at α=0.8, and the average energy consumption improves to 232839J. In this case, a sharp 20% decrease of production time is observed, as average energy consumption provides 2% increase under same cutting parameters. So, the improvement of procedure centralization degree has a positive effect on the efficient use of
ACCEPTED MANUSCRIPT energy. (a) Average production time (b) Average power (c) Average energy consumption Fig.13. Production time, average power and average energy consumption as α and β change (vc=400, f=0.2, ap=0.4)
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Cutting speed vc induced by timed transition T6 has a more positive impact on production time, as depicted by Figure 13. A 51% increase of the average production time is observed in Figure 14(a), as α is varied from 0.1 to 0.8 and cutting speed is varied from 200r/min to 1000r/min. In Figure 14(b), the average production power increases, as α is varied from 0.1 to 0.8 and cutting speed is varied from 200r/min to 1000r/min. A 30% to 55% increase of average power is observed,
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as α is varied from 0.1 to 0.8 and cutting speed is varied from 200r/min to 1000r/min. Although the average production power increases, as the cutting speed is improved, there is a noticeable change of average energy consumption in Figure 14(c). The average energy consumption is a
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concave curve as a function of both α and cutting speed. The optimal cutting speed is approximated to be 700 r/min, from energy efficiency perspective. (a) Average production time
(b) Average production power (c)Average energy consumption
Fig. 14. Production time, average power and average energy consumption varies with cutting speed as well as α and β
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5 Conclusions An effective energy assessment method can help improve energy consumption of machine tools. Since machining processes are discrete event systems and can be divided into various states, a formalized model and analysis method is proposed using a GSPN approach to characterize the production time and energy demand of machine tools. The applicability is validated by conducting experiments on a turning machine tool. The proposed method has several advantages: (1) The presented model for measuring the states of power demand and energy consumption can be applied and tailored by users to suit the given machine tool and process plan. First, one identifies the energy demand states of production cycle and then builds the GSPN model. Second, the corresponding firing rate and probability of immediate transition are assigned. Third, one calculates the steady state probabilities. Meanwhile, the state transition diagram can also be built to study the differences and similarities. (2) The required indicators for evaluating and optimizing the operational behavior of a machine tool are generated by executing the analysis method. The effects of various cutting parameters and production plan are addressed to obtain the average production time, average power and average energy consumption.
Acknowledgements This work is supported by International Science & Technology Cooperation Program of China (Grant No. 2012DFG72210), National Natural Science Foundation of China (Grant No. 71471139) and Natural Science Foundation of Zhejiang, China (Grant No. Y14E050085). This paper is also
ACCEPTED MANUSCRIPT supported by China Scholarship Council. The authors would also like to thank Prof. Pengzhong Li for helping with the experiments and Mr. J Linzbach for valuable advice.
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[7] S Kara, W Li. Unit process energy consumption models for material removal processes. CIRP Annals-
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[10] Jia Shun, Tang Renzhong, Lu Jingxiang. Therblig-based modeling methodology for cutting power and its application in external turning. Computer Integrated Manufacturing Systems, 2012, 10(5):1015-1024. [11] Behrendt Thomas, Zein Andre, Min Sangkee. Development of an energy consumption monitoring procedure for machine tools. CIRP Annals-Manufacturing Technology, 2012, 61(1):43-46. [12] Liu Fei, Liu Shuang. Multi-period energy model of electro-mechanical main driving system during the service
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[20] G Chiola, M A Marsan, G Balbo, G Conte. Generalized Stochastic Petri Nets:A Definition at the Net level and Its Application. IEEE Transactions on software engineering, 1993, 19(2):89-107. [21] Xie Nan, Li Aiping, Xu Liyun. Modeling and analysis method for reconfigurable manufacturing cell based on generalized stochastic Petri nets. Computer Integrated Manufacturing systems, 2006, 12(6):828-834
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[22] X Ai, S Xiao. Handbook of cutting parameters[M]. 3rd ed. Beijing: China Machine Press, 1994
ACCEPTED MANUSCRIPT Table 1. Material cutting power under various parameter settings
v c (m/min)
f(mm/r)
a p (mm)
Pcutting (W)
1
260
0.2
0.4
1168
2
260
0.3
0.7
2789
3
260
0.4
1.0
5059
4
330
0.2
0.7
2519
5
330
0.3
1.0
4908
6
330
0.4
0.4
2732
7
400
0.2
1.0
4157
8
400
0.3
0.4
2457
9
400
0.4
0.7
5216
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No.
ACCEPTED MANUSCRIPT Table 2. Coefficients of material cutting power model and their statistical fitness
b
c
d
R-square
Prob.
85.108
0.87
0.826
0.931
0.998
9.237E-10
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a
ACCEPTED MANUSCRIPT Table 3. Predicted and experimental results of power Workpiece
Measured
Vc
Diameter d
m
[m/min]a
[mm]
[W] b
[W]
721,0.3,0.5
300
133.48
4095
4043
1.26
989,0.25,0.5
350
113.64
4111.33
4162
1.24
685,0.3,0.5
300
140.5
3982
4022
1.0
676,0.3,0.2
300
141.6
2649
2662
0.49
690,0.3,0.5
300
139.49
4039
4025
0.34
(n [r/min], f[mm/r], ap[mm])
a
Vc = 3.1415 × n × (d − 2a p ) / 1000
b
P m = PSR + Pvf + Pcooling + Pcutting − PBM
c
Error= P p − P m P m ×100
)
Power P
AC C
EP
TE D
M AN U
SC
(
Power P
Predicted p
Prediction Errorc [%]
RI PT
Cutting parameters
ACCEPTED MANUSCRIPT Meaning of Place
P1
Turn off
P2
Turn on
P3
Cooling system works
P4
Peak
P5
Spindle and feed work
P6
Workpiece process
P7
Process end
P8
Ready to return P5
P9
Ready to return P2
P10
Ready to return P1
P11
Workpiece is ready
AC C
EP
TE D
M AN U
SC
Place
RI PT
Table 4. Meanings of places of figure 6 model
ACCEPTED MANUSCRIPT Table 5. Meanings of transitions of Figure 6 model Meaning of Transition
Category
T1
Turn on the machine tool
Timed
T2
Turn on the cooling system
Timed
T3
Turn on the spindle and feed system
Timed
T4
Spindle and feed system are steady
Timed
T5
Process start
Timed
T6
Process end
Timed
T8
Back to P5
Timed
T9
Back to P2
Timed
T10
Back to P1
Timed
Ts1
α, Proportion of return to air cutting state
Immediate
Ts2
β, Proportion of return to turn on state
Immediate
Ts3
γ, Proportion of return to turn off state
SC
RI PT
Transition
AC C
EP
TE D
M AN U
Immediate
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT