Khan Mohammad Habibullah

Eric Rondeau

Jean-Philippe Georges

REDUCING ENERGY CONSUMPTION OF NETWORK INFRASTRUCTURE USING SPECTRAL APPROACH

Keywords: Green Networking, Energy Efficient Design, Spectral Clustering Method, Switch Hibernation

The energy consumption by ICT (Information and Communication Technology) equipment is rapidly increasing which causes a significant economic and environmental problem. At present, the network infrastructure is becoming a large portion of the energy footprint in ICT. Thus, the concept of energy efficient or green networking has been introduced. Now one of the main concerns of network industry is to minimize energy consumption of network infrastructure because of the potential economic benefits, ethical responsibility, and its environmental impact. In this paper, the energy management strategies to reduce the energy consumed by network switches in LAN (Local Area Network) have been developed. According to the lifecycle assessment of network switches, the highest amount of energy is consumed during usage phase. The study considers bandwidth, link load and traffic matrices as input parameters which have the highest contribution to energy footprints of network switches during usage phase and energy consumption as output. Then with the objective of reducing energy usage of network infrastructure, the feasibility of putting Ethernet switches hibernate or sleep mode was investigated. After that, the network topology was reorganized using clustering method based on the spectral approach for putting network switches to hibernate or switched off mode considering the time and communications among them. Experimental results show the interest in this approach in terms of energy consumption.

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INTRODUCTION

We are now living in the age of information and communication technology and the energy consumption, as well as carbon emission by computing and communications equipment is increasing rapidly. [START_REF] Mingay | IT's Role in a low carbon economy[END_REF] stated that, at present ICT is responsible for about 2% of global carbon emissions which is similar to the aviation industry. The enhancement of energy consumption has large negative economic and environmental impact. In recent years, we are struggling with climate change and in coming years it is the greatest challenge to tackle it. For that reason, energy consumption reduction is very important because it is directly related to greenhouse gas emission which is responsible for climate change. Recent research (Pamlin and Szomollanyi, 2006) suggests that, 15-30% volume of emission needs to be decreased in order to keep the global temperature increase below 2 degree Celsius.

The growth of internet users has been astonishing over the last few years. From 1990 to 2010, the number of Internet users increased from 3 million to 2 billion and by mid-2012, the user number had increased to 2.73 billion already. Regional growth rates between 2000 and 2012 have been exceeding 3 600% in Africa and 2 600% in the Middle East. As a result, the electricity demand of ICT is growing at much faster than overall electricity demand. Network-enabled device electricity demand is growing at a rate of 6% per year and total ICT energy demand reached 1560 TWh in 2013 (Iea.org, 2014). Because of this continuous growth of internet users, the spreading of broadband access and the increasing number of online services being offered by telecoms and Internet Service Providers, the energy efficiency issue has also become one of the highest priority objectives for wired networks and service infrastructures. These continuously rising trends in network energy consumption depend on new services being offered, as well as on data traffic volume increase [START_REF] Nedevschi | Reducing Network Energy Consumption via Sleeping and Rate-Adaptation[END_REF]. According to [START_REF] Zhang | Evolution of the Internet and its cores[END_REF], data traffic is increasing rapidly which follows Moore's law, by doubling every 18 months.

In recent years, some efficient steps have been taken to reduce the energy consumption of network infrastructure. This paper is a contribution to this mitigation in focusing only to reduce the energy consumption of network switches in a LAN (Local Area Network). [START_REF] Mahadevan | Reducing lifecycle energy use of network switches[END_REF] stated that, a network switch consumes energy during its manufacturing phase, usage phase and dismantling phase with the highest amount of energy consumed by network switch during usage phase.

The concern of this research work is only during the usage phase. The objective is to reduce the energy consumption that simple network architecture builds up with Ethernet Switches and end devices. It is described in (Iea.org, 2014), up to 80% of energy consumption is used for some devices just to maintain the network connection and more than 600 TWh of electricity was consumed on 2013 by such devices. It is possible to reduce up to 65% of energy consumption of such devices using the best available technologies and strategies. [START_REF] Hossain | Modeling the power consumption of Ethernet switch[END_REF] and [START_REF] Gunaratne | Reducing the energy consumption of Ethernet with adaptive link rate (ALR)[END_REF] found bandwidth and number of connection are mostly responsible for energy consumption of Ethernet switch in usage phase. [START_REF] Krommenacker | Study of algorithms to define the cabling plan of switched Ethernet for real-time applications[END_REF] and [START_REF] Rondeau | Study and method of Ethernet architecture segmentation for Industrial applications[END_REF] showed proper network architecture designs and cabling plan using clustering approach increases the efficiency of the network. An efficient network design reduces traffic propagation delay, jitter, and packet loss which also reduces energy consumption. Therefore, in this paper, optimization, reorganization and clustering approach of network architecture have been proposed by analyzing bandwidth and the traffic load in point to point communication. The feasibility to hibernate or switch off devices which only remain alive to maintain network connection was also investigated, with the objective of reducing energy consumption.

The experiment was done with cisco switches and strategies have been applied to optimize and reorganize the network topology. In the experiment, the power consumption was used to measure energy usage of network architecture. In the first phase of experiment, we designed network topology consists of connected Ethernet switches which support energywise and desktop computers were connected to each switch. Then we measured the power consumption of all switches using different bandwidth with same traffic load according to traffic matrix. In the second phase, we reorganized and optimized our network architecture using clustering method based on spectral approach, with the objective to reorganize network cabling in order to be able to sleep or hibernate unused components of network infrastructures for a certain amount of time during low traffic periods.

Hibernation mode decreases a large amount of power consumption compared to the operation mode of an Ethernet switch and switched off mode does not consume any power. In the third phase, we investigated the feasibility of putting various components connected with Ethernet switches or the switch itself to sleep or hibernate in considering the changes of communication activities. After applying the spectral approach to different cases of communication with the same network topology, it was possible to hibernate or switch off the network switch which saved a significant amount of energy.

RELATED WORK

Numerous studies have explored both in the wired and wireless network to analyze the energy consumption pattern of network devices and to reduce energy consumption using different strategies. [START_REF] Christensen | Power management in networked devices[END_REF] have studied the lifecycle energy use of network devices and explained how network devices can impact on environment pollution. The lifecycle energy use of network devices has been studied by [START_REF] Nordman | Networks, Energy, and Energy Efficiency[END_REF], but mostly in the context of home and office environments. [START_REF] Rivoire | Modeling and Metrology Challenges for Enterprise Power Management[END_REF] explained from a device manufacturer's point of view that, network devices such as routers and switches are power proportional which means they consume energy proportional to their usage. [START_REF] Gupta | A feasibility study for power management in LAN switches[END_REF] showed how different parameters affect the energy consumption of network switch. After that, they defined a model which shows the relationship between parameters related to the Ethernet switch and energy consumption. [START_REF] Fithritama | Modeling fuzzy rules for managing power consumption of ethernet switch[END_REF] proposed a method based on fuzzy logic to identify the relationships between network parameters and their effect on power consumption when deploying new network equipment. Their proposed method is also applicable to control the desired level of energy consumption by tuning the network parameters. In the research by [START_REF] Reviriego | An energy consumption model for energy efficient ethernet switches[END_REF], the energy consumption of small Energy Efficient Ethernet (EEE) switches is analyzed in several experiments. Based on the experiment result, the authors proposed a model for the energy consumption of Energy Efficient Ethernet switch. [START_REF] Gupta | A feasibility study for power management in LAN switches[END_REF] explored the feasibility of power management schemes at network switches in the LAN. They examined the possibility to put various components on LAN switches to sleep for reducing energy consumption. Experiments to evaluate energy management studies for network switches were performed by [START_REF] Mahadevan | Reducing lifecycle energy use of network switches[END_REF]. The authors found, energy usage in the operational stage is dominating and they parametrically examine various energy management techniques to reduce the operational energy footprint of network switches. [START_REF] Nedevschi | Reducing Network Energy Consumption via Sleeping and Rate-Adaptation[END_REF] presented the design and evaluation of two forms of power management schemes to reduce the energy consumption of network. They have shown that simple schemes for sleeping or rate adaptation have significant energy savings without noticeably increased packet loss and latency.

Related Work in Network Energy Consumption Evaluation

Related Work in Network Energy Consumption Optimization

Related Work Based on Clustering Approach

Consideration of information flows is very important to design network architecture. The objective of network designer is to confine the strong co-operation and communication with sub-network to avoid flooding and overloading the whole network. In order to minimize overloading, intra-group communication should be maximum and inter-group communication should be minimum. To achieve this network scalability objective, clustering method has been widely pursued by the research community. In this research work, clustering algorithm has been used to reduce communication; isolate groups for hibernating or switched off part of the network.

Many clustering algorithms have been proposed [START_REF] Abbasi | A survey on clustering algorithms for wireless sensor networks[END_REF]Amis, et al., 2007;[START_REF] Baker | The architectural organization of a mobile radio network via a distributed algorithm[END_REF][START_REF] Bandyopadhyay | An energy efficient hierarchical clustering algorithm for wireless sensor networks[END_REF][START_REF] Basagni | Distributed clustering for ad hoc networks[END_REF][START_REF] Lin | Adaptive clustering for mobile wireless networks. Selected Areas in Communications[END_REF][START_REF] Mellier | A Weighted Clustering Algorithm For Mobile Ad Hoc Networks With Non Unique Weights[END_REF][START_REF] Chiasserini | Energy efficient design of wireless ad hoc networks[END_REF] to fulfill various objectives based on wireless sensor network but none of these algorithms aim to minimize energy consumption in network architecture. Most of these algorithms are heuristic in nature and the criteria for cluster selection and node grouping are intra and inter-cluster connectivity if the application is sensitive to data latency and the length of data routing paths. The objective of these algorithms is to generate the minimum number of clusters. [START_REF] White | A Spectral Clustering Approach To Finding Communities in Graph[END_REF] proposed the spectral algorithm to find communities in a graph and showed that spectral algorithm is effective and efficient at finding both good clustering and the appropriate number of clusters from a variety of real-world graph data sets. They observed spectral algorithm is faster for large sparse graphs. Von [START_REF] Luxburg | A tutorial on spectral clustering[END_REF] introduced and presented the most common spectral algorithm from scratch by different approaches. He also discussed the advantages and disadvantages of different spectral clustering algorithms, different graph Laplacians and their basic properties. [START_REF] Krommenacker | Study of algorithms to define the cabling plan of switched Ethernet for real-time applications[END_REF] defined some criteria to reorganize the network architecture and explored algorithms, especially the spectral algorithm to define cabling plan of switched network for real-time applications. The spectral algorithm to design the cabling plan for industrial Ethernet architecture has been used by [START_REF] Rondeau | Study and method of Ethernet architecture segmentation for Industrial applications[END_REF], which can reduce handling delays of messages inside the time cycle of applications.

SYSTEM ARCHITECTURE AND CRITERIA TO REORGANIZE NETWORK SWITCHES

This section describes the general network architecture that has been used in the research and the criteria to reorganize and design efficient architecture for switched Ethernet Networks. Figure 1 shows the network architecture and the cabling plan used during the experiment. Three Cisco Ethernet Switch 2960-X (S1, S2, and S3) were used during the experiment and three desktop PCs (Personal Computer) were connected to each switch. In the section below we have described the criteria to reorganize network switches in different actions.

Action 1:

It illustrates that if PC 1 has frequent communication with large data size with PC 7, then it is better to connect either PC1 in S3 or PC7 in S1. Thus, it is very important to consider information flows during designing a network. In this case, the goal is to maintain communication among devices by avoiding overloading the whole network which will increase network efficiency and reduce energy consumption.

Action 2:

The objective of action 2 is to gather the PCs strongly communicating among them, with the idea to group PCs without communication in other clusters in order to be able to hibernate or switch off these clusters.

Action 3:

If the capacity of each port of all switches is 1 Gbps and total traffic load for the communication between S1 and S2 or S2 and S3 are more than 1Gbps, then more than one trunk port needs to be connected to support the communication. In this situation, if inter-switch communication can be reduced, it is possible to eliminate redundant trunk between two switches. Thus reducing the traffic between switches allows reducing bandwidth and as a result, reduction of energy consumption.

Therefore, in general, the criteria to reorganize the communication among switches are to minimize inter-group dialogues in order to maximize the intra-group exchanges or to create groups without communication. Some other criteria should also take into account such as the number of ports and capacity of each port with the objective to use the smaller number of switches with a higher number of connected ports.

METHODOLOGY

The spectral algorithm is used to cluster the end device according to the usage of switch ports and to ensure the minimal interaction between two clusters. As an initial step of the approach, a graph G d has been introduced and translated that into traffic matrix according to the communication among end devices. This section describes clustering method based on the spectral approach which has been used to reorganize and optimize network topology.

Spectral Approach

The spectral algorithm is one of the most successful heuristics for partitioning graphs and matrices. This approach is used to solve scientific numerical problems such as solving parse linear systems explained by [START_REF] Pothen | Spectral nested dissection[END_REF], partitioning for domain decomposition [START_REF] Chan | Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes[END_REF], Ethernet architecture segmentation [START_REF] Rondeau | Study and method of Ethernet architecture segmentation for Industrial applications[END_REF] and so on. In this research spectral approach has been used to • Adjust the size of the clusters in regard to the capacity of the switches in terms of port number.

• Reorganize and minimize the cabling of the switch by breaking graphs into sub graphs.

• Investigate the feasibility of putting unused switches to hibernate or switched off according to usage.

Given an undirected weighted graph G = (V, E), where G is a division of its vertices 𝑣𝑣 into two disjoint subsets, 𝑉𝑉 1 and 𝑉𝑉 2 . Let E(V 1 , V 2 ) is the set of edges (links for communication) 𝑒𝑒 𝑖𝑖𝑖𝑖 with one endpoint in 𝑉𝑉 1 and other in 𝑉𝑉 2 . The cut size of the partition is the sum of the edge weights of of E :

𝑐𝑐 = � 𝑤𝑤(𝑒𝑒𝑒𝑒𝑒𝑒) 𝑒𝑒𝑖𝑖𝑖𝑖∈𝐸𝐸 Where v i ∈ V 1 and v j ∈ V 2
Here the objective is to find a disjoint subset of V, P = {v i } which minimizes c . The adjacency matrix A(G d ), of the graph G is n × n matrix where (ij)th entry is the weight of edge e ij ∶ w ij and 0 otherwise. For adjacency matrix, the diagonal entries are always 0.

As the spectral algorithm considers only undirected graph, thus it needs to work with

A(G) = A(G d ) + A(G d ) t

Algorithm Description

Let D (G) is an undirected graph converted into 𝑛𝑛 × 𝑛𝑛 diagonal matrix such as

𝑑𝑑 𝑖𝑖𝑖𝑖 = � 𝑤𝑤(𝑒𝑒𝑒𝑒𝑒𝑒) 𝑖𝑖 The Laplacian matrix 𝐿𝐿(𝐺𝐺)of the graph G is 𝐿𝐿(𝐺𝐺) = 𝐷𝐷(𝐺𝐺) -𝐴𝐴(𝐺𝐺).
According to [START_REF] Donath | Lower bounds for the partitioning of graphs[END_REF][START_REF] Fiedler | Algebraic connectivity of graphs[END_REF]Fiedler, , 1975aFiedler, , 1975b) ) a good cut size is the second smallest eigenvalue λ2 and its associated eigenvector 𝑢𝑢 = �𝑢𝑢 1 , 𝑢𝑢 2 ,………………… 𝑢𝑢 𝑛𝑛 � (Fiedler vector). Now, sorting the vertices on the incremental way according to the values of the components of Fiedler vector gives the reorganization of the vertices.

The objective of the partitioning is to use exactly d devices (in this case network switches) and to minimize the communication through the switches if there are two levels of switches. A splitting value s should be found for spectral partitioning which divides the vertices of G into V 1 such that u i > s , and V 2 such that u i < s and that is called a Fiedler cut. The choices for the splitting values s are:

• Bisection cut, where s is the median of(𝑢𝑢 1 , 𝑢𝑢 2 ,………………… 𝑢𝑢 𝑛𝑛 ).

• Sign cut, where s is equal to 0.

• Ratio cut, where s is the value that gives the best cut ratio denoted

∅ with ∅(V 1 , V 2 ) = |E(V 1 , V 2 )|/min (|V 1 |, |V 2 | ) • Gap cut,
where s is the value in the largest gap in the sorted list of the Fielder vector components. To partition a graph into the power of two numbers of groups, Recursive Spectral Bisection (RSB) algorithm explained by [START_REF] Simon | Partitioning of unstructured problems for parallel processing[END_REF] is applied. In this paper, optimized recursive bisection algorithm has been used for partitioning the graph.

Optimized Recursive Spectral Bisection Algorithm (RSB in [𝐝𝐝/𝟐𝟐] ) [START_REF] Simon | Partitioning of unstructured problems for parallel processing[END_REF]

RESEARCH METHOD AND EXPERIMENT DETAILS Parameter Selection

Parameter selection is one of the most important issues for power management of Ethernet switch. Bandwidth and the number of connections are mostly responsible for energy consumption of Ethernet switch [START_REF] Hossain | Modeling the power consumption of Ethernet switch[END_REF] and the packet size has no impact on energy consumption of the switch [START_REF] Mahadevan | A power Benchmarking framework for network devices[END_REF].So in our experiment bandwidth, the traffic load on each switch port and traffic matrix according to the communication among end devices was used as the input parameter and power consumption of switch was measured as the output parameter. Though traffic load has very little impact on energy usage, but it represents the service that the network must offer (QoS). In the experiment traffic load was used to maintain the frequency of communication among the end devices and to draw traffic matrix depending on that communication.

Detailed Experiment

The experiment was done according to the figure 1. Powerspy2 sensor was used to measure the total energy consumption of three switches which can send real-time energy consumption data via Bluetooth [START_REF] Hossain | Modeling the power consumption of Ethernet switch[END_REF]PowerSpy2 user manual, 2015). Ostinato traffic generator was used to generate traffic into the switch ports (Ostinato Network Traffic generator and Analyzar User Guide). Three sets of experiment were done and each set contains two experiments. During each set of experiment, for all switch ports, the same bandwidth was used which were 100Mbps and 1Gbps. Depending on the traffic load and frequency of communication we have designed 9 × 9 traffic matrix and measured energy consumption of the network topology. According to the frequency of communication among end devices, two types of traffic loads has been used which are defined as large transmission and small transmission which are shown in traffic matrix as "10" and "1" in respectively. More information about traffic loads and frequency of communication is shown in Table 1. Each experiment was run and monitored for 15 minutes.

For large transmission (10)

For small transmission (1)

RESULTS AND DISCUSSIONS

This section presents the obtained result during the test phase. In traffic matrix, the red marked PCs are connected with S1, blue marked PCs are connected with S2 and green marked PCs are connected with S3.

Case Study

For conducting the experiment, communication among the end devices during day time or working hours and night time or nonworking hours was considered. The working hour was defined as T 1 = 16 hours and the nonworking hour was defined as T 2 = 8 hours. The communication among the end devices is more than nonworking hours which have been shown by traffic matrix1 and matrix2 accordingly in Figure 7. Both of the matrices in Figure 7 are unoptimized.

The architecture is composed of 9 PCs connected to 3 Ethernet switches (respectively S1, S2, and S3) with the capacity of five ports each switch as explained in figure 1. Figure 7 shows that PCs in red (1, 2 and 3) are connected to switch S1, PCs in blue (4, 5 and 6) to switch S2 and PCs in green (7, 8 and 9) to switch S3. In this case, most of the frequent communications are intra-switch communication and in optimized organization, most of the frequent communications are inter-switch communication among end devices showed in Figure 7. The total amount of traffic load is same for both matrices. The yearly power consumption for Figure 7 can be calculated by the formula below: 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = (𝑇𝑇 1 𝑃𝑃 𝑚𝑚𝑡𝑡𝑡𝑡𝑚𝑚𝑖𝑖𝑚𝑚1 + 𝑇𝑇 2 𝑃𝑃 𝑚𝑚𝑡𝑡𝑡𝑡𝑚𝑚𝑖𝑖𝑚𝑚2 ) * 365 Here 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 is the yearly power consumption, 𝑇𝑇 1 𝑃𝑃 𝑚𝑚𝑡𝑡𝑡𝑡𝑚𝑚𝑖𝑖𝑚𝑚1 is daily power consumption for Matrix 1 and 𝑇𝑇 2 𝑃𝑃 𝑚𝑚𝑡𝑡𝑡𝑡𝑚𝑚𝑖𝑖𝑚𝑚2 is daily power consumption for Matrix 2.

Table 2 andTable 3 shows the experimental results of power consumed by the network architecture showed in Figure 1 for 100 Mbps and 1Gbps bandwidth. For all measurements, bandwidth for both trunk ports and the ports connected to end devices was same and two trunk ports were connected between each switch to support communications.

There is no notable change in power consumption for traffic load, but there is a noticeable change in power consumption when there is a change in bandwidth. The power consumption increases 0.2 to 0.4 Watt per port if the link speed is changed from 100 Mbps to 1Gbps respectively. Reducing bandwidth from 1 Gbps to 100 Mbps could save yearly 33.3 KWh in the unoptimized organization where 5 ports in each switch were active. Another noticeable observation is, an extra trunk link was needed in working hours for the unoptimized organization to support communication and to avoid packet loss for traffic overload between S1 and S2.

For clustering and optimizing the communication and cabling plan for the whole day, the weights of Matrix 1 and Matrix 2 have been calculated considering the time T 1 and T 2 respectively and the spectral algorithm was implemented on that matrix. It has been assumed that Matrix 1 and Matrix 2 are derived from connected graph G1 and G2 respectively. The weight of G1 and G2 is defined by C which is shown in Figure 8. As the spectral algorithm works with only undirected graph, so A (C) = C+C' has been calculated shown in Figure 8 and from there Laplacian matrix L(C) has been calculated which is shown in Figure 9. Here nine end devices are connected to three network switches with the capacity of five ports where two ports are connected as trunk. The sorted vertices from Figure 10 have been used as end devices. After applying improvement on RSB algorithm [START_REF] Rondeau | Study and method of Ethernet architecture segmentation for Industrial applications[END_REF], the obtained result of Figure 11 partitioned the end devices into three parts and the group of three end devices is connected to each switch. The optimized adjacency matrix was drawn based on working hours and nonworking hours according to the serialization found in Figure 12. The power consumption by the optimized solution is measured which is showed in Table 4. Table 5 shows the yearly total power consumption of optimized solution using the spectral algorithm. The result shows that optimized solution consumes less energy than unoptimized architecture. It was also possible to reduce extra trunk port between S1 and S2 during working hours because spectral algorithm produced clusters of PCs which have strong communications among those which reduce inter-switch communication showed in optimized Matrix 1. After applying spectral clustering on end devices and serialization of network switches, it was possible to hibernate or switch off two network switches (S1 and S3) during nonworking hours because it created two clusters of PCs without communication which are connected with S1 and S3. Table 6 shows the power savings after using the spectral algorithm for different bandwidth. For 100 Mbps bandwidth, the optimized solution could save a yearly minimum of 3.2 KWh when no hibernate or switched off feature activated to maximum 202.1 KWh with switched off feature activated during nonworking hours. For 1Gbps bandwidth, the optimized solution could save 9.0 KWh when no hibernate or switched off feature activated to maximum 315.9 KWh for switched off feature activated during nonworking hours.

C = T 1 G1 + T 2 G2 A(C)

L (C) VFIEDLER Sorted Vertices

T 1 T 2 Optimized Matrix 1 Optimized Matrix 2
For hibernation or switched off Ethernet switch, it should be considered the Quality of Service. The hibernation mode consumes about 20 watts and switched off mode consumes nothing. For the Cisco switch model 2960-X, it takes 260 seconds to get ready after wake up from hibernation and 290 seconds from switched off mode. The time difference to be fully functional from hibernation and switched off mode is 30 seconds which can be critical for few cases. But for normal office works or Local Area Network, it doesn't have that much impact. Frequent on-off can put extra load which can reduce lifetime and some devices are vulnerable to this situation. On the other hand, always keeping on a device can also reduce lifetime, therefore switched off can be a good choice too. Hibernation and wake up time can be scheduled by command line which is not possible for switched off mode but by using rack power distribution units (PDUs) it is also possible to switch off and on automatically. Therefore, the choice of hibernation or switched off Ethernet switch can be according to the necessity of the user.

The spectral approach is also applicable for optimizing bigger network architecture. So by using clustering method based on spectral approach, it is possible to save a large amount of energy according to traffic loads and pattern for the bigger picture and more complex architecture. The Ethernet switch is only a part of network architecture and to put efficient and greater impact, other network components like router, WIFI hotspot etc. should be considered. It is also possible to apply the spectral algorithm for clustering those devices according to traffic and optimize the whole architecture. Moreover, the experiment was done with one type of Ethernet switch, so the result may vary for different type of switches. [START_REF] Rondeau | Measurements and Sustainability, Chapter 3[END_REF] mentioned the correlation between the power consumption and carbon emission. Therefore, by optimizing network architecture using the spectral algorithm, it is also possible to reduce carbon footprint caused by network equipment. As a big picture, with proper design and development of network architecture, it is possible to reduce and control global energy usage and carbon footprint caused by network devices.

CONCLUSION

The enhancement of energy consumption by network infrastructures has large negative impact on sustainability which should be controlled. This paper presents a novel way of energy efficient network design and management system by using the clustering approach based on spectral algorithm to reduce energy consumption of network infrastructure. By using this approach, it is possible to hibernate or switch off part of network during low traffic hours in order to save energy. Experimental result shows that significant amount of energy can be saved by reorganizing the network switches and clustering the end devices using the spectral algorithm. For future work, this approach can be applied to the bigger network architecture and big enterprise which has the potential to save a large amount of energy usage. Similar experiments can be done for other network devices such as routers to reduce more energy consumption.

Figure 1 :

 1 Figure 1: Cabling plan and Network architecture.

Figure 7 :

 7 Figure 7: Traffic matrix according to the communication among end devices during working and non-working hours.

Figure 8 :

 8 Figure 8: Combined matrix and undirected adjacency matrix for the communication.

Figure 9 :

 9 Figure 9: Laplacian matrix for graph C

Figure 10 :

 10 Figure 10: Fiedler values and sorted vertices for the Graph G.

Figure 11 :

 11 Figure 11: Partitioning of End Devices by Using RSB in 𝑛𝑛[𝑑𝑑/2]

Figure 12 :

 12 Figure 12: Optimized Matrix 1 and Matrix 2.The power consumption by the optimized solution is measured which is showed in Table4.

  proposed improvement on RSB algorithm for designing a switched Ethernet network cabling plan as follows: 1. Compute the Fiedler vector for the graph. 2. Sort the vertices according to the values of the components of the Fielder vector. 3. Calculate d for the group to be partitioned. 4. Assign n[d/2] first vertices to one sub-group and the others to the second one. Cut of value (n[d/2]) = number of ports * [number of devices/2]. 5. Apply steps 1-4 recursively to each sub-group until the size of each sub-group becomes≤ n.

Table 1 : Traffic load and frequency of communication.

 1

	Frame size (Byte) :	1125	1125
	Packet per second:	10000	1000
	Total size of transmission per second (Mbps)	90	9

Table 2 : Power consumption for 100 Mbps bandwidth in the unoptimized organization.

 2

		Working	Non-			Working	Non-
	Time	hours	working	Total	Time	hours	working	Total
		( 𝐓𝐓 𝟏𝟏 )	hours ( 𝐓𝐓 𝟐𝟐 )			(𝐓𝐓 𝟏𝟏 )	hours ( 𝐓𝐓 𝟐𝟐 )
	Power consumption	598.6	299.6	898.2	Power consumption	620.8	308.9	929.7
	(KWh/Year)				(KWh/Year)

Table 3 : Power consumption for 1 Gbps bandwidth in the unoptimized organization.

 3

Table 4 : Power consumption for working hour and nonworking hours for optimized solution. Power Consumption at Working hours (T1) by Optimized Organization

 4

	Bandwidth	100 Mbps	1 Gbps
	Energy consumption	596.3	613.8
	(KWh/Year)

Table 5 : Total yearly power consumption for the optimized solution for different bandwidth.

 5

		Power consumption for 100Mbps	Power consumption for 1Gbps
		bandwidth (KWh/year)	bandwidth (KWh/year)
	Always active	895.0	920.7
	With hibernation	814.4	833.4
	With Switched off	596.1	613.8

Power Consumption at Nonworking hours (T2) by Optimized Organization (KWh/year)

	3 6 7 2 5 9 8 1 4
	3 6 7	2 5 9 8 1 4
	2 5 5	8 1 4
	𝐏𝐏𝐏𝐏 𝐍𝐍𝐍𝐍 𝟑𝟑 𝟔𝟔 𝟕𝟕 𝟐𝟐 𝟓𝟓 𝟗𝟗 𝟖𝟖 𝟏𝟏 𝟒𝟒 𝟑𝟑 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟔𝟔 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏 𝟕𝟕 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏 𝟐𝟐 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟓𝟓 𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟖𝟖 𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏 𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟒𝟒 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏		𝑷𝑷𝑷𝑷 𝑵𝑵𝑵𝑵 𝟑𝟑 𝟔𝟔 𝟕𝟕 𝟐𝟐 𝟓𝟓 𝟗𝟗 𝟖𝟖 𝟏𝟏 𝟒𝟒 𝟑𝟑 𝟔𝟔 𝟕𝟕 𝟐𝟐 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟓𝟓 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟗𝟗 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟖𝟖 𝟏𝟏 𝟒𝟒
	Bandwidth	With Trafic	Hibernate Switched off
	100 Mbps	298.7	218.1	99.9
		1 Gbps	306.9	219.6	100.7

Table 6 : Power saving by using optimized solution Power saving for 100Mbps bandwidth (KWh/year)

 6

	Power	Saving	for	1Gbps
	bandwidth (KWh/year)

Acknowledgement: This work is part of the Erasmus Mundus Master program in Pervasive Computing and Communication for Sustainable Development (PERCCOM) of the European Union (www.perccom.eu). The authors thank all the partner institutions, sponsors and researchers of the PERCCOM program (Klimova, et al., 2016 ).


K Aroui

G Alpan

Y Frein

Minimizing work overload in mixed model assembly lines: A case study from truck industry

Keywords: Mixed model assembly line, sequencing, work overload, linear programming

In this article, we consider the mixed model assembly line sequencing problem with work overload minimization (MMSP-W). We consider an assembly line having operators with different characteristics. We propose a formulation based on a linear programming approach. The model is implemented using the CPLEX solver. Numerical tests are carried out on generated data and on an industrial case study from a truck assembly line. Computational results demonstrate the considerable improvements obtained in comparison to the traditional car sequencing approach.

Introduction

The need to manufacture a wide variety of products has led to many changes in the requirements of production systems. Originally, using the single model assembly lines, manufacturing companies have efficiently produced large quantities of the same product. But the diversification of customer demand necessitated the invention of more modern manufacturing methods. Mixed Model Assembly line (MMAL) is a production line where different variants of common products (called models) are intermixed to be assembled on the same line [START_REF] Monden | Toyota Production System[END_REF]. This type of assembly line is used in Just-In-Time (JIT) systems and applied in a wide range of industries: from electronic assembly to automotive industry. It helps matching production to changing customer requirements while keeping small stock sizes. In MMAL, products move on a paced conveyor belt and operators move on the conveyor while performing their tasks on products. Line balancing and product sequencing are the two main problems in MMAL that attract the attention of researchers. The line balancing problem deals with the assignment of tasks to workstations. The most common objective for this problem (Scholl, 1995) is to minimize the number of workstations for a given cycle time (i.e., production rate).

The sequencing problem assumes that the line has already been balanced. The main task is to find the production sequence of a given number of models within a given planning horizon, e.g., one day or shift [START_REF] Boysen | Sequencing mixed-model assembly lines: survey, classification and model critique[END_REF]. Some recent works also consider the sequencing and line balancing problems simultaneously (see, for instance, [START_REF] Mosadegh | Simultaneous solving of balancing and sequencing problems with station-dependent assembly times for mixed-model assembly lines[END_REF]. This paper deals with the mixed model assembly line sequencing problem. In the literature, two basic objectives are considered to solve this problem: the constant rate of part usage [START_REF] Monden | Toyota Production System[END_REF][START_REF] Miltenburg | Level schedules for mixedmodel assembly lines in just-in-time productions systems[END_REF][START_REF] Bautista | Heuristics and exact algorithms for solving the Monden problem[END_REF][START_REF] Boysen | Sequencing mixed-model assembly lines: survey, classification and model critique[END_REF], and the leveling of work load [START_REF] Yano | Survey, development and applications of algorithms for sequencing paced assembly lines[END_REF]Xiabo et Ohno, 1997 ;[START_REF] Bautista | Minimizing work overload in mixed-model assembly lines[END_REF]. Some authors consider both objectives simultaneously [START_REF] Aigbedo | A parametric procedure for multcriterion sequence scheduling for just-in-time mixed-model assembly lines[END_REF][START_REF] Kotani | Sequencing problem for a mixed-model assembly line in the Toyota production system[END_REF]. [START_REF] Boysen | Sequencing mixed-model assembly lines: survey, classification and model critique[END_REF] records three different sequencing approaches to attain these objectives:

-Mixed-model sequencing: the purpose of this approach is to minimize the work overload (or max-imize the total work performed) by taking into account the processing times of each product on every workstation.

-Car sequencing: also seeks to minimize sequence-dependent work overload, but in an implicit manner, by using sequencing rules: limited frequency at which special options may appear.

-Level scheduling: This approach aims to have regular consumption rates of materials.

Car sequencing approach is the most common approach in the literature and extensively applied in automotive industry. In spite of its popularity, this approach has some drawbacks: first of all sequencing rules are not trivial to construct. And secondly the respect of these rules may lead to unexpected work overload on the assembly line because they fail to capture the real work overload to minimize (see [START_REF] Lesert | Definition of spacing constraints for the car sequencing problem[END_REF]. In this article, we choose to act directly on the work overload. Hence, the Mixed-Model Sequencing Problem with Work overloads Minimization (MMSP-W) is considered. This approach is poorly developed in the literature. Nevertheless, we find some works by Yano and Rachamadugu (1991), [START_REF] Bolat | Stochastic procedures for scheduling minimum job sets on mixed model assembly lines[END_REF]Yano (1992a, 1992b), [START_REF] Scholl | Pattern based vocabulary building for effectively sequencing mixed-model assembly lines[END_REF], Xiaobo andOhno (1997, 2000), Bautista andCano (2008, 2011) and [START_REF] Bautista | Modeling and solving a variant of the mixed-model sequencing problem with work overload minimisation and regularity constraints: an application in Nissan's Barcelona plant[END_REF], among others. Yano and Rachamadugu (1991) consider MMSP-W with one workstation and two product types and propose an approach leading to the optimal solution when certain conditions are met. Then, they generalize the approach to k workstations and two product types. Bolat and Yano (1992a) introduce a surrogate objective for utility work at a single station assembly line. Optimal solution can be obtained depending on certain conditions on the parameter values. Still for the case of one workstation and two product types, Bolat and Yano (1992b) develop three optimal solution procedures and a heuristic to minimize the work overload. Exact methods, such as Branch and Bound have been used [START_REF] Xiaobo | Algorithms for sequencing mixed models on an assembly line in a sit production system[END_REF]. However, the computation times are acceptable only for small instances. Heuristics are also used. For instance, [START_REF] Scholl | Pattern based vocabulary building for effectively sequencing mixed-model assembly lines[END_REF] solve the MMSP-W using an informed tabu search procedure with a pattern based vocabulary building strategy. [START_REF] Bautista | Solving mixed model sequencing problem in assembly lines with serial workstations with work overload minimisation and interruption rules[END_REF] and [START_REF] Bautista | Modeling and solving a variant of the mixed-model sequencing problem with work overload minimisation and regularity constraints: an application in Nissan's Barcelona plant[END_REF] propose models for the MMSP-W for assembly lines considering workstation dependencies (operations in a workstation could begin only if the previous station finished his operations). The authors propose a heuristic based on a Bounded Dynamic Programming approach. The method combines some characteristics of dynamic programming (determining the shortest path in a graph) with Branch and Bound (searching a bound of the optimal solution). This work is based on an industrial case study of one of the plants of Volvo Group Trucks Operations. The models with only two types of products are totally ineffective in this industrial context. Furthermore, the study of the industrial case reveals three types of operators with different specifications. Two of them were never modeled in the literature. Our study proposes a model considering the three existing types of operators. The truck assembly lines are typical MMAL with highly diversifying products. Compared to the automobile industry, the number of vehicles to produce per day is much lower, which may be an opportunity to test exact methods in modeling. This article is organized as follows: Section 2 introduces fundamental assumptions and describes the problem. In section 3, we propose a linear programming model for the MMSP-W. Section 4 describes the numerical experiments based on an industrial case study. The paper ends with concluding remarks.

Problem description

In the case of a paced assembly line, the products move on the line with a constant speed. The time between two consecutive products is constant and is called the cycle time. The operator follows the product with the same speed as the pace of the assembly line (meters/time unit) while doing his tasks. Therefore, the cycle time is oftentimes used to define the boundaries of a workstation as well (i.e. a workstation is a space characterized by the number of meters that the operator can make within the cycle time). We will use the term "time window" to refer to the available time for an operator to perform his tasks on a product. The work overload appears when the operator cannot finish the required tasks on a product within the predefined time window. The work overload refers to the remaining work.

There are several ways to handle this remaining work: For instance, in [START_REF] Bautista | Modeling and solving a variant of the mixed-model sequencing problem with work overload minimisation and regularity constraints: an application in Nissan's Barcelona plant[END_REF], the excess work is considered as unfinished work and is handled at the end of the line. In [START_REF] Bautista | Solving mixed model sequencing problem in assembly lines with serial workstations with work overload minimisation and interruption rules[END_REF] and in Boysen et al. (2011), the productive activity is increased above the standard, using the assistance of reinforcement operators. In [START_REF] Xiaobo | Algorithms for sequencing mixed models on an assembly line in a sit production system[END_REF], the assembly line is stopped to complete the pending work.

The characteristics of the assembly line considered in this study and the assumptions regarding the work overload are given below:

the assembly line balancing is already done.

All tasks are assigned to the operators.  the assembly line is a paced assembly line.  the processing times are assumed to be deterministic. Set-up times are included in processing times and the time required to move from the downstream to the upstream station limits are negligible.  the operators cannot go beyond the limits of their time window (due to equipment restrictions) and there are no buffers between stations. This means that, no extra work should remain from a preceding operator. Thus, there are no dependencies between the consecutive operators. In case of work overload, the operator has to rush or reinforcement operators assist him to finish the job on time;  a workstation can have several operators.

They can perform their tasks independently.

When the operator has to rush to finish his operations on time, the risk of defects, accidents due to fatigue or health problems due to bad postures, etc. increase. The use of reinforcement operators to handle the work overloads is expensive. Our objective is to generate product sequences to minimize the work overload. The proposed method is intended to be used in an industrial context.

The following example illustrates those characteristics and assumptions: sequencing 3 products for 2 consecutive operators. Operation times for each product are given in table 1. The cycle time is 5 time units.

Product m1 m2 m3

operator 1 5 6 3 operator 2 6 4 4

Table 1: Operation times for the example

Figure 1 shows the Gantt diagram of the sequence m2, m1 then m3. We can see that the work overload generated by product m2 for the operator 1 impacts the next product in the sequence (that is, m1 which generates further delay). When operator 2 finishes his operations on m2, he can start jobs on m1 only when it is inside his workstation. In this case, an idle time occurs after processing m2. The operators are independent so the delay observed on m2 for the operator 1 will not impact the beginning of operation of operator 2 on m2.

We note that, in practice, when the operator 1 rushes to finish the tasks on m2 or when the reinforcement operators are used, this delay will not be observable on the product m1. Even though such delays are not "observable in practice", if we can avoid them we also avoid the risks generated by a rushing operator or the cost of a reinforcement operators. Therefore, in the model described in section 3, we focus on the minimization of this type of delays.

The optimization criterion considered in this study is hence minimization of total work overload (sum of gray squares in Figure 1).

Figure 1: Dynamics of work overload depending on the product sequence

The proposed model in section 3 considers only the overloads. When an idle time occurs, it will not be taken into account in the criterion.

A model based on mixed integer linear programming

Operator types

Three types of operators exist in our case study:

type 1 operators: are regular operators and they are assigned tasks to be performed on every vehicle within one cycle time. Time window for type 1 operators is one cycle time for all vehicles.

type 2 operators: the tasks given to these workers are required only by some vehicles (special options). They are allowed to work in a known number of cycle times. This number can be different for each vehicle of the sequence. Time window for type 2 operators depends on vehicles.

type 3 operators: in the balancing problem, some tasks, that must be executed on all vehicles, could not be split and require more than a single cycle time to be performed. These tasks are assigned to operators of type 3. The operators perform their tasks on a known number of cycle times. This number is the same for all products that the operator has to do. An operator of type 3 has to be associated to one or more operators of the same type with whom he has to alternate products depending on their positions in the sequence. For example, if the tasks are balanced on two cycle times, we will have two operators. One will operate on vehicles in even positions and the other on vehicles in odd positions in the sequence. Each operator will have a time window of two cycle times.

As shown in Figure 2 an operator can work on several cycle times. In the example the line contains 7 workstations and 10 operators.

Variables

Binary variable that is equal to 1 if a product i is assigned to position j of the sequence, and 0 otherwise Delay or idle time for operator p for the product on position j of the sequence work overload for operator p of type 1 for the product on the position j Intermediary variable to calculate work overload for type 2 operators Work overload for operator p for the product on position j of the sequence Figure 3 shows an example of an operator of type 2 and a sequence of 8 products. The operator has no tasks to perform on products m2, m3, m5, m7 and m8 (α ip =0) so they don't appear in the Gantt chart. Overloads appear when the worker reaches the limit imposed by the time window ( ). They are characterized by the gray squares in figure 3. . Unlike operators of type 2, time window does not depend on products. All products will have the same available time to be performed by an operator of type 3.

Since the work is balanced on 3 cycle times, operator 1 will perform tasks on m1, m4 and m7. m2 and m5 will be treated by operator 2. m3 and m6 will be done by operator 3.

Figure 4: Gantt char of an example of type 3 operators

We will now present a mixed linear integer program to minimize the total work overload considering the specificities of each type of operator.

∑ ∑ ∑

(1)

(2)

(3) --work overload for type 1 operators -

--work overload for type 2 operators --

∑ (5) (6) ( ∑ ) (7)
--work overload for type 3 operators --

(8) ( ∑ ) (9)
----

∑ (10) ∑ (11) { } (12) (13)
This model aims to minimize the total work overload; constraint (1) establishes that the delay or idle time of the product positioned in j is his delay (or idle time) in addition to the delay (or idle time) of the position j-1; constraints ( 2) and ( 3) indicates that since the objective function takes into account only the delays (and not idle times).

Constraint (4) specifies work overloads for type 1 operators.

Constraints ( 5)-( 7) calculate work overloads for type 2 operators. Let's note the processing time of the vehicle in position j:

∑ and the number of cycle times where operator p of type 2 is allowed to work for of the vehicle in position j: ∑ . Obviously, work overload does not occur for products if they are not treated by an operator. Thus, the work overload for type 2 operators is:

{ ( ( ) )
Constraints ( 8) and ( 9) calculate work overloads for type 3 operators. Let's note the delay or idle time of the vehicle in position j: ∑ . For operators of type 3, the work overload is:

{ ( )
It's the work overload of the previous performed product (which is for operators of type 3) added to the delay or idle time of the vehicle in position j. Similar to type 2 operators, no work overload is considered if the worker has no tasks to do on the product. Constraint (10) guarantees that only one position can be assigned to each product; constraint (11) indicates that only one product can be assigned in each position of the sequence; and, finally, constraint (12) requires that the assigned variables are binary. Table 2 illustrates variables' values for operator 1 in the example presented in section 2.

Sequence dj cj rj

Operator

1 m1 1 1 1 m2 0 1 1 m3 -2 -1 0 Operator 2 m1 -1 -1 0 m2 1 1 1 m3 -1 0 0
m1 10 3 1 7 6 1 m2 0 0 4 -3 0 m3 0 0 0 1 -3 0 m4 6 2 1 4 3 1 m5 0 0 0 1 -3 0 m6 7 3 1 5 6 0 m7 0 0 0 2 -3 0 m8 0 0 0 0 -3 0

Table 3: Variables' values for the example of type 2 operators

Table 4 illustrates variables' values assigned to the first type 3 operator in the example illustrated in Figure 3.

Sequence tj aj dj wj

m1 10 1 1 1 m2 8 0 -1 0 m3 9 0 0 0 m4 9 1 0 1 m5 10 0 1 0 m6 8 0 -1 0 m7 7 1 -2 0

Numerical experiment with academic instance

The academic instance is an example with 12 products and 10 operators: 5 operators of type 1, 2 operators of type 2 and 3 operators of type 3. Table 5 gives the processing times by product and operator. The cycle time is 7 min. For operators of type 2, is given in parenthesis next to the processing times. For operators of type 3, the available time to perform tasks on products is three cycle times (

). The optimal solution is obtained in 1.76 seconds. The values of work overloads are given in table 6.

Numerical experiment with case study instances

We apply the linear program on nine instances from the industrial case study. Each one represents the demand plan of a production day. That is equivalent to 60 vehicles per day assembled by 92 operators: 77 of type 1, 12 of type 2 and 3 of type 3. The operators of type 3 have no variation in their processing times. There is no utility to consider them in our tests. As a result, 89 operators are considered. The cycle time is 7 minutes.

We limited the CPU time to 10800s (3 hours). A comparison between the solution given by CPLEX and the solution of the current sequencing method used in the production site is given in table 7. The current procedure is based on "manual" car sequencing approach (see section 1).

From tables 7 and 8, we can comment on the following observations:

-CPLEX did not obtain the optimal solution for all instances after three hours of execution. We could not guarantee an optimum solution but we improve significantly the solution given by the current procedure in all instances. -The average improvement on total work overload W is 51%.

-After one hour of execution, the average improvement on W is 30,4% -After two hour of execution, the average improvement on W is 44,4%

Conclusion

In this article, we present a linear programming model for sequencing products to minimize the work overload in a mixed model assembly line under three different types of worker operating conditions. We perform numerical experiments with an academic instance and then with nine instances from an industrial case study where each instance represents a working day. The results of these tests demonstrate an average decrease of 51% in total work overload with the proposed solution, compared to the current procedure used in the company. We note that the comparisons are made for results obtained after 3 hours of execution of the linear program. We are, hence, unable to guarantee the optimality of these solutions.

Future research will focus on using dynamic programming and designing heuristic resolution procedures to solve the problem. Yano, C.A., Rachamadugu, R., 1991. Sequencing to minimize work overload in assembly lines with product options. Management Science 37, 572-586.

  Operators 1 to 3, 6 and 8 to 10 are of type 1. Each of them has a time window of one cycle time. Operator 7 is of type 2. His time window depends on vehicles (up to three cycle times). And operators 4 and 5 are two coupled operators of type 3. Each of them takes every other vehicle on the sequence and has a time window of two cycle times.

Figure 2 :

 2 Figure 2: Example of assembly line We propose a model using linear programming approach to solve the mixed-model sequencing problem with work overload minimization which takes into account the assumptions considered in section 2 and the characteristics of each operator type to suit the condition of the case study. The parameters and variables of this model are presented below. Parameters Number of products Number of operators Set of operators of type 1 Set of operators of type 2 Set of operators of type 3 Product index Position index Operators index Cycle time Processing time required by product i for operator p. For operators of type 1 and 2, . For operators of type 3,

Figure 3 :

 3 Figure 3: Gantt char of an example of operator of type 2 Figure 4 illustrates a Gantt chart of an example of sequencing 7 products on three linked operators of type 3. Time window for each operator and each product is. Unlike operators of type 2, time window does not depend on products. All products will have the same available time to be performed by an operator of type 3.

Table 2 :

 2 Variables' values for the exampleTable 3 explains variables' values assigned to type 2 operators in the example of Figure 2.

	Sequence	tj	npj	αj	rj	bj	wj

Table 4 :

 4 Variables' values for the example of type 3 operators

	4 Numerical experiments
	We apply the linear program proposed in section 3
	on two experiments: the first is a small academic
	instance taking into account all operator types and
	the second is our case study of the Volvo plant.
	In both experiments, the solutions are obtained
	using the solver IBM ILOG CPLEX Optimization
	Studio V12.4 running on a Dell PC with Intel Core
	i5 2.50 GHz with 4GB of RAM using Windows 7
	Enterprise.

Table 5 :

 5 processing times in minutes for the academic instance

Table 6 :

 6 work overloads for the solution of the academic instance

					Type 1 operators	Type 2 operators
	In-	W CPLEX	W Current	Improvement
	stances		Procedure		Improvement	Improvement
	1	917,34	2729,5	66,4%	39,7%	85,6%
	2	979,52	3600,18	72,8%	42,5%	82,1%
	3	895,7	1873,85	52,2%	27,9%	72,5%
	4	1320,19	3295,43	59,9%	32,5%	77,3%
	5	1947,28	3147,05	38,1%	33,8%	60,0%
	6	942,3	1806,03	47,8%	32,9%	65,5%
	7	783,97	1353,65	42,1%	32,6%	69,7%
	8	1433,82	2243,03	36,1%	31,3%	63,6%
	9	705,4	1263,83	44,2%	40,8%	64,6%

Table 7 :

 7 Results for the case study instances

			CPLEX 1 hour	CPLEX 2 hours	CPLEX 3 hours
	Instances	W Current
		Procedure	W	Improvement	W	Improvement	W	Improvement
	1	2729,5	1429,44	47,6%	1429,44	47,6%	917,34	66,4%
	2	3600,18	1594,89	55,7%	990,51	72,5%	979,52	72,8%
	3	1873,85	1296,36	30,8%	914,45	51,2%	895,7	52,2%
	4	3295,43	1845,75	44,0%	1402,53	57,4%	1320,19	59,9%
	5	3147,05	2433,24	22,7%	2433,24	22,7%	1947,28	38,1%
	6	1806,03	1598,39	11,5%	972,5	46,2%	942,3	47,8%
	7	1353,65	1085	19,8%	809,91	40,2%	783,97	42,1%
	8	2243,03	1667,12	25,7%	1667,12	25,7%	1433,82	36,1%
	9	1263,83	1060,67	16,1%	803,28	36,4%	705,4	44,2%

Table 8 :

 8 Improvements at each hour of execution for the case study instances


Kevin Morel

Charles Guégan

François Léger

Can an organic market garden without motorization be viable through holistic thinking? The case of a permaculture farm

Keywords: organic horticulture, energy independence, agroecology, systemic approach

INTRODUCTION

The environmental and socialcosts of using oil resources as well as their increasing scarcity are challenging tomorrow's agriculture to reduce its dependency on fossil energies [START_REF] Chow | Energy resources and global development[END_REF]. In industrialized countries,innovative market gardens inspired by permaculture principles [START_REF] Ferguson | Permaculture for agroecology: design, movement, practice, and worldview. A review[END_REF]claim to address this challenge by promoting manual labor and holistic thinking.Our objective was to assess the extent to which a holistic approach could make it possible for a commercial organic market garden to be viable without motorization. We consider here that viability is the capacity of market gardenersto generate a sufficient income to reward themselveswhile keeping an acceptable workload.Our work was based onthecase study of a permaculture farm in Northern France.

MATERIALS AND METHODS

Site and production measurements

The farm we studied is located in the small village of Bec Hellouin, Normandy, France (49°13'24.9"N 0°43'42.5"E). The local climate is temperate under oceanic influence with a high rainfall level (from 700 to 900mm a year) and the soil is alluvium-calcareous. This farm is a 6 hectares high diversified commercial organic market garden and sells its production through short commercial channels (direct selling to consumers and restaurants).Bec Hellouin farmers claimed that it was possible for a single market gardener to use holistic principles inspired by permaculture to create an acceptable income from a small acreage cultivated with no motorization. To test this hypothesis,marketable harvest quantities and workload were measured daily from 2013 to 2014 on a cultivated surface area of 1,061m 2 (pathways not included). For practical constraints, different market gardeners and trainees worked part time on the studied area but their workload was cumulated in order to make sure that this work on 1,061 m 2 could be done by a single market gardener. From this surface area, 40% was cultivated on plane beds under a cold greenhouse, 24% on open field with plane beds and 36% on open field with curved raised beds. On both types of beds, a 30cm deep non-inversion tillage (NIT) was applied.

Income and global workloadestimations

Harvest quantities were multiplied by prices of vegetables to calculate gross sales for each year. We considered that 100% of harvested marketable vegetables were sold. If the data were available, we used average prices of organic vegetables in direct selling channels in Normandy. Otherwise, we used the farm prices for uncommon vegetables.Following an accounting approach, costs, expenses and taxes were deducted from the sales to evaluate the theoreticalincome of a single market gardener working full time on the studied area. These data were estimated based on the farm bookkeeping documents and discussions with an expert on market gardening accountancy. We considered (i) a low costs hypothesis(LC):second-handequipment, basic storage/selling building and no delivery van (the whole production is sold on farm) and (ii) a high costs hypothesis (HC):all equipmentis bought new including a more sophisticated building and a delivery van which consumes fuel. The LC hypothesis leads to higher cost ofimplement maintenance because the equipment is not new. According to classic references in direct selling market gardening we assigned 50% of the on field measured workload to administrative and commercial tasks. We added production, commercial and administrative workload to assess the global workload.

Qualitative approach of farmers' holistic thinking

We carried out semi-structured interviews with the marketgardeners about the different strategies and principles they implemented to fulfill their objective of viability without motorization. Through a qualitative analysis method [START_REF] Miles | Qualitative data analysis: A sourcebook of new methods[END_REF], we identified relations amongthese strategies and represented them using amind-mapping technique [START_REF] Buzan | The Mind Map Book[END_REF].

RESULTS AND DISCUSSION Holistic thinking of farmers

The holistic thinking implemented by the market gardeners relied on two main principles: increasing production on a small manually cultivated area, and increasing added value of such production. A wide range of strategies stemmed from these two principles embracing ecological, technical and commercial dimensions at the farm level (Fig. 1).

Income and workload

Over the three years, 64% of the gross sales (GS) were vegetables produced under the cold greenhouse which covered only 40% of the studied surface area. On average, 76 production types were grown annually in the gardens: 17 aromatic herbs and edible flowers (7% of GS), 16 fruit vegetables (41% of GS), 11 root and bulb vegetables (20% of GS) and 32 leaf vegetables (32% of GS). The vegetables sold to restaurants represented 25% of GS.In 2013, the GS were 32,788€ with a cumulated annual workload of 2,006 h. This work could be done by a full time single market gardener working in average 43h per week.In 2014, the GS were 57,284€ GS. The corresponding cumulatedworkload was 3,026h. This workload would represent an average of 58h a week for a single full time market gardener and was judged not acceptable for life quality considerations. Therefore, in the estimated income based on 2014 sales, we considered that the market gardener worked the same time as in 2013 and the excess workload (1,020h) was assigned to an additional employee hired 9.61€ per h (French minimum wage) with 42.3% of additional employer contributions. With the LC hypothesis the monthly net income was 1,132€ in 2013 and 1,571€ in 2014 (both judged acceptable by the farmers). With the HC hypothesis the monthly net income was 898€ in 2013 (not acceptable) and 1,337€ in 2014 (acceptable). The improved performance of 2014 can be explained by an increased expertise in multiple-cropping and relaycropping, a higher level of care given to plants, and the use of horse manure hotbeds to produce crops earlier and with higher yield in winter. Moreover, the LChypothesis led to a 26% higher income in 2013 and 17% higher income in 2014 (Table 1). It showed that bothfinancial strategies and technical mastery are key points of the viability of such initiatives as much as commercial ability because these estimations considered that100% of marketable vegetables were sold.

Viability without motorization

In 3 out of 4 presented scenarios, the gross sales created from a 1,061m 2 acreage without motorization madeit possible to create an income which can cover basic costs of living with an acceptable level of annual workload.These sales relied mainly on short cycle crops such as leafy vegetables and fruit crops with a high added value. Storage crops such as potatoes were not grown because they were judged to remain too long in the soil and take up too much space in the gardens relative to their added value. However, these types of crops are generally expected by consumers from a market garden. For this reason, the vegetable baskets sold by the Bec Hellouin farm were complemented with storage crops produced with motorization by another local organic farmer. These sales were not taken into account in our study but highlighted the importance of considering the dependency on external productions in the viability of small farms without motorization. Although annual income and workload are two major criteria of the viability of farms, other aspects will be further investigated such as workload distribution throughout the year as well as personal satisfaction and well-being of market gardeners. Nonetheless, the choice of using no motorization is in accordance with personal values of the Bec Hellouin market gardeners. This tends to bring higher satisfaction to farmers [START_REF] Mzoughi | Do organic farmers feel happier than conventional ones? An exploratory analysis[END_REF].

CONCLUSIONS

Our case study on a permaculture farm showed that it was possible to create a monthly net income between 898€ and 1,571€ depending on gross sales and investment levels for a direct selling organic market gardener cultivating 76 vegetables types without motorization on a cultivated acreage around 1,000m 2 . Theses incomes were generated with an average workload of 43h per week which was judged acceptable for a single market gardener. These results were made possible by a holistic way of thinking combining ecological, technical and commercial strategies which stemmed from two main principles: increasing production on a small manually cultivated area and increasing added value of such production. The studied farm focused on growing short cycle and high added value vegetables which excluded storage crops such as potatoes. As these crops are often expected by the consumers, further investigation should be carried out aboutthe way manual and motorized market gardeners can collaborate locally to fulfill consumers' requirements.

Fig. 1 .

 1 Fig. 1. Mind map of strategies implemented by the studied market gardeners.

Table 1 .

 1 Net income estimations for a single market gardener depending on sales and cost hypothesis (€).

	Year

ACKNOWLEDGEMENTS

Thanks to the marketgardeners of the Bec Hellouin Farm whose involvementwas essential to this research project, and to Agnes Bellec-Gauche for her accountancy skills.


Kazusa Yamamoto

Pascal Moulaire

Damien Koenig

Olivier Sename

H ∞ Filtering Design for Driver Torque Estimation in Electric Power Steering System

This paper describes a method to estimate the driver torque in an Electric Power Steering (EPS) system without taking into account the torque sensor signal. First, a pinion-type EPS model is expressed by Newton's law of motion considering the effort transmitted from the road as an unknown input. Then, the design of a H ∞ filter is proposed to estimate the driver torque. The formulation aims at minimizing the estimation error in the presence of the external force induced by the road sollicitations and sensor noises. The efficiency of this H ∞ filter is assessed through simulation results using input data gathered from experimental tests on a real car. Furthermore these results are compared with torque measurements obtained on the experimental car for validation purpose only.

INTRODUCTION

Nowadays, modern vehicles are equipped with more and more electronic systems which help the driving process. Among them, steering systems help the driver to turn the vehicle in the desired direction, improving both safety and comfort. Currently, most of car manufacturers are using Electric Power Steering (EPS) systems. Indeed, compared to Hydraulic Power Steering (HPS), they improve fuel efficiency, and are easy to integrate into vehicle, etc.

In EPS systems, an electric motor provides an assistance torque to reduce the amount of effort required from the driver to turn the wheels. An Electronic Control Unit (ECU) computes the supplied torque according to a motor torque control strategy. It mainly consists of an assistance map depending on the vehicle speed and on the measured torque sensor signal [START_REF] Sulakhe | Electric Power Assisted Steering[END_REF].

Hence, EPS assistance strategies are based on the applied steering torque depicted by torque sensor measurements. However, a failure of this sensor may lead in a loss of assistance, which must be prevented. Indeed, developing a controller based on an estimation of the driver torque would cover such a case and improve global performance since torque sensor measurements and actual driver torque differs [START_REF] Marouf | A new Control Strategy of an Electric Power Assisted Steering system[END_REF], [START_REF] Chabaan | Torque Estimation in Electrical Power Steering Systems[END_REF]. Moreover, this strategy reduces the production costs and can improve the driver feeling. Both topics represent major challenges for EPS system suppliers.

Several works have already been carried out on driver torque estimation in EPS systems. Some of them [START_REF] Chabaan | Torque Estimation in Electrical Power Steering Systems[END_REF], [START_REF] Tordesillas Illán | Oscillation annealing and driver/tire load torque estimation in Electric Power Steering systems[END_REF] have developed an estimator considering as a measured input the torque sensor signal. In that case, it is straightforward to reconstruct the driver torque as it is equivalent to the measurement during static steering. Nevertheless, Marouf et al. proposed in [START_REF] Marouf | A new Control Strategy of an Electric Power Assisted Steering system[END_REF] a Sliding Mode Observer (SMO) to estimate the driver torque, in which measurements are the steering wheel angle and the absolute angular position of the assist motor. Moreover, in [START_REF] Yamamoto | Driver torque estimation in Electric Power Steering system using an H ∞ /H 2 Proportional Integral Observer[END_REF], the authors have designed a Proportional Integral Observer (PIO) with the same assumptions on measurements availability. This paper follows the same idea in order to avoid the need of torque sensor signal and the design of a H ∞ filter that estimates the driver torque is presented. Likewise, the steering wheel angle and the assist motor angle are required inputs measurements. However, no assumption is made on the driver torque dynamics is made, since it is considered as the fault to estimate. The chosen design objective is to minimize the effects of road disturbances and sensor noises on the driver torque estimation error, in an H ∞ framework.

The paper is organized as follows. In section 2, a linear EPS mechanical model is introduced. Next section presents a method to estimate driver torque through the design of a H ∞ filter. Then, section 4 is dedicated to simulation results and validation is done on experimental results obtained on a real test car.

The studied system is referred to as a pinion-type EPS (P-EPS) mainly composed by the assist motor, the pinion and the rack part. Moreover, the driver action is transmitted through the steering wheel to the torsion bar. Hence, it brings to distinguish four main parts. An illustration of a simplified mechanical model is shown in Figure 1. The model is obtained by applying Newton's laws of motion and neglecting dry friction. The P-EPS model is governed by the following equations (for further details see [START_REF] Yamamoto | Driver torque estimation in Electric Power Steering system using an H ∞ /H 2 Proportional Integral Observer[END_REF]):

J c θc = τ d -D tb ( θc -θp ) + K tb (θ c -θ p ) -B c θc (1)
J m θm = τ m -D g ( θm -G θp ) + K g (θ m -Gθ p ) -B m θm (2)
J p θp = D tb ( θc -θp ) + K tb (θ c -θ p ) -R p D pr (R p θp -Ẋr ) + K pr (R p θ p -X r ) + G D g ( θm -G θp ) + K g (θ m -Gθ p ) (3)
J r Ẍr = τ road -D r Ẋr + K r X r -B Ẋr + D pr (R p θp -Ẋr ) + K pr (R p θ p -X r ) (4)
where the variables are θ c the steering wheel angle (rad), θ m the assist motor angle (rad), θ p the pinion angle (rad), X r the rack position (m). The mechanical parameters are described in Table 1.

Hence a state-space representation of the P-EPS system (denoted Plant) is the following:

   ẋ = Ax + Bu + Ed + W w y = Cx + N n (5)
with x = θc θm θp Ẋr θ c θ m θ p X r T ∈ R 8
are the internal states, d ∈ R is the driver torque to be estimated, w ∈ R the road reaction torque is the unknown input, and u ∈ R the assist motor torque is the control signal. The available measurements are y = (θ c θ m ) T ∈ R 2 which are affected by white gaussian noise n ∈ R 2 . Moreover N is the weighting matrix associated to n in the form N = βI with β ∈ R + .

H ∞ FILTERING APPROACH

The main objective is to estimate the unmeasured input d subject to the road disturbance w and sensor noises n affecting the system (5), using a H ∞ filter design. In the following, the known input u is not considered, but can be easily included in the filter equations.

Synthesis of H ∞ filter

In this part, the method followed to design the H ∞ filter is presented. As usual in the H ∞ approach, it is based on weighting functions that represent the specificaitons of the input/output signals of interest. It is worth noticing that such a scheme, as shown in Figure 2 with the extended representation with weighting functions and the H ∞ filter F (s), is used for synthesis only and that the implementation only considers F (s) with as inputs the measurements y.

Besides, the estimation is designed on a filtered driver torque obtained using T (s). Then, a weighting function on the driver torque is introduced such that d f = T (s)d. This allows to specify the frequency range on which estimation performance is expected. In practice, the driver torque average range is up to 2Hz and does not exceed 5Hz even in emergency case. Hence, in this paper the H ∞ filter problem is not designed as in the form of a state estimation like in [START_REF] Li | A Linear Matrix Inequality Approach to Robust H ∞ Filtering[END_REF], [START_REF] Yaesh | A Transfer Function Approach to the Problems of Discrete-Time Systems H ∞ Optimal Linear Control and Filtering[END_REF] to avoid assumption on driver torque dy-

P lant d F (s) y + - d z T (s) d f W w (s) w n w Figure 2: Block diagram of H ∞ filter synthesis namic.    ẋt = A t x t + B t d d f = C t x t + D t d (6)
Moreover a weighting function is introduced on the road disturbance such that w = W f (s) w in order to specify the frequency range and amplitude of disturbance attenuation.

   ẋw = A w x w + B w w w = C w x w + D w w (7)
In this application framework, both additional weighting functions are considered of first order i.e n t = 1 and n w = 1 which represent the number of state in ( 6) and [START_REF] Yaesh | A Transfer Function Approach to the Problems of Discrete-Time Systems H ∞ Optimal Linear Control and Filtering[END_REF]. Hence the augmented state space representation:

             ẋa = A a x a + B w w a + 0. d z = C z x a + D zw w a - d y = C y x a + D yw w a + 0. d (8)
where

x a = x T x T t x T w
T ∈ R 10 is the extended state vector, w a = d T wT n T T ∈ R 4 the external inputs and z = d f -d is the estimation error to be minimized.

The synthesis of H ∞ filter is done, according to general control configuration shown in Figure 3.

Then, P (s) including the extended state is described below:

P = A a B w B u C z D zw D zu C y D yw D yu (9)
where:

A a = A 0 W C w 0 A t 0 0 0 A w , B w = E W D w 0 B t 0 0 0 B w 0 , P(s) F(s) z ỹ d ˆ          n w d Figure 3: H ∞ general control configuration B u = (0 0 0) T , C z = (0 C t 0), D zw = (D t 0 0), D zu = -1, C y = (C 0 0), D yw = (0 0 N ), D yu = 0.
Denoting the H ∞ filter F (s) as:

   ẋf = A f x f + B f y d = C f x f + D f y (10)
Now the problem is to design the standard H ∞ filter F (s) that minimizes the H ∞ norm of the overall system from w a to z. The system is given by:

   ẋag = A ag x ag + (E ag W ag N ag ) w a z = C ag x ag + D ag w a (11)
where:

A ag =    A 0 W C w 0 0 A t 0 0 0 0 A w 0 B f C 0 0 A f    , E ag =    E B t 0 0    , W ag =    W D w 0 B w 0    , N ag =    0 0 0 B f N    , C ag = (-D f C C t 0 -C f ), D ag = (D t 0 -D f N ), x ag = x T x T t x T w x T f T ∈ R 18 , z = d f -d.
The H ∞ filter synthesis problem is as follows: given the system (11), find matrices A f , B f , C f and D f such as ||T zwa || ∞ ≤ γ ∞ . This is solved by using a Linear Matrix Inequality (LMI) approach following [START_REF] Gahinet | A Linear Matrix Inequality Approach to H ∞ Control[END_REF]. If there exist symmetric matrices R, S satisfying the following LMI system:

N R 0 0 I nw a T AR + RA T RC T z B w * -γI nz D zw * * -γI nw a N R 0 0 I nw a < 0 (12) N S 0 0 I nz T   A T S + SA SB w C T z * -γI nw a D T zw * * -γI nz   N S 0 0 I nz < 0 (13) R I n I n S ≥ 0 (14) with N R = Ker B T u D T zw , N S = Ker C y D yw . Then, a
positive definite matrix X is obtained as:

S I n N T 0 = X I n R 0 M T , M N T = I n -RS (15)
Hence, ( 16) is a LMI in

A f B f C f D f , the solution
gives the parameter of the H ∞ filter.

  A T ag X + XA ag X (E ag W ag N ag ) C T ag * -γ ∞ I D T ag * * -γ ∞ I   < 0 (16)

Analysis of H ∞ filter synthesis

A compromise between estimation performance and disturbance rejection shall be found through the design parameter of the H ∞ filter. Indeed, the ideal case is to get T (s) = 1 which means to estimate the driver torque regardless of his frequency content. Hence, Table 2 shows the influence of the weighting function T (s) on H ∞ filter synthesis. Variations of the cut-off frequency and order of T (s) have been performed. Then, the estimation performance has been assessed through the steady state gain T dd (ω = 0) (that should be close to 1) and the maximal disturbance rejection ||T zwa || ∞ ≤ γ ∞ (i.e value of γ ∞ is expected to be as small as possible). In view of Table 2, increasing the bandwidth of T (s) leads to a degradation of estimation, (as γ ∞ gets close to 1). The previous overview shows that a second order filter T (s) is convenient. From (11), the transfer matrix T zwa = [T zd ; T z w; T zn ] of the sensitivity functions is as follows: 17) -( 19) with different tuning for T (s). It is again noticed that a cut-off frequency at 25Hz gives less efficient results than with 5Hz or 10Hz. Indeed, the optimization problem is harder to be solved when considering a greater frequency range. As both remaining synthesis show similar performance, the one designed with 10Hz is preferable. For simulation part, the driver torque is estimated following a H ∞ filter F (s) design method where T (s) is a 2 nd order filter with cut off frequency at 10Hz.

T zd (s) = C ag (sI -A ag ) -1 E ag + D t (17) T z w(s) = C ag (sI -A ag ) -1 W ag (18) T zn (s) = C ag (sI -A ag ) -1 N ag + D k N (19)

SIMULATION RESULTS

In this section, simulation results are presented using the H ∞ filter designed in the previous section. The Figure 7 illustrates the experimental implementation scheme. It is worth noting that, to evaluate the estimation error, a sensor is available on the experimental test car for driver torque measurement, referred to as d measured . According to the representation in Figure 7, the following simulation scenario has been considered. A sinusoidal driver torque of d = 5N m at 0.5Hz is applied as input. The disturbance affecting the plant is represented by the road effort. At the beginning, it is a constant of w = 1kN and changed to w = 5kN at 2.5s. Also sensor noises of amplitude N = 1e -2 is added to the measured output.

Figure 8 shows the measured driver torque d measured , the estimated driver torque d, the filtered driver torque d f (synthesized signal) and the resulting estimation error z = d f -d and z = d measured -d.

The root mean squared error (RMSE) is calculated as:

RM SE(z) = m i=1 z(i) 2 m ( 20
)
where z = d measured -d i.e difference between the real data and the estimated value, m is the number of data points. The normalized RMSE (NRMSE) on the whole range is computed by:

N RM SE(z) = RM SE d max measured -d min measured (21)
The obtained errors are: RMSE(z)=0.27Nm (resp. NRMSE(z)=2.67%) and RMSE(z)=0.076Nm (resp. NRMSE(z)=0.74%) . The H ∞ filter estimates accurately the applied driver torque d measured , since the computed error is small enough. Thus, the H ∞ filter have an acceptable performance level. It is worth noticing that the H ∞ filter has been designed introducing the weighting function T (s) and that the optimization problem has been solved for z, which explains this slightly better estimation result.

Simulation from experimental input data

In this case, the simulation environment takes only the H ∞ filter F (s) part from Figure 7, to validate the performance of the estimation using as inputs real measurements i.e y = (θ c θ m ) T . Data acquisition has been done on a development car equipped with a P-EPS system and several specific sensors (motor resolver, torque sensor, steering sensor and driver torque sensor).

The first test case comes from a particular driving condition where the driver follows an '8' shaped pattern on the track at a constant speed of 20kph. This involves to get a sinusoidal driver torque. Figure 9 shows the comparison between the measured driver torque and the estimated torque.

Another test measurements come from parking maneuver. The driver turns slowly the steering wheel with a small steering angle amplitude. As it is in parking, the effort required from the driver is high. The comparison between the measured driver torque and the estimated torque shown in Figure 10 emphasizes the efficiency of the estimation.

The resulting errors are: for experimental data 1 is RMSE(z)=3.18 Nm, NRMSE(z)=6.63% and for experimental data 2 is RMSE(z)=1.1Nm, NRMSE(z)=4.94%. Since experimental data are gathered from real system where non-linearity (mainly friction) are influent, whereas in the H ∞ filter design this element has been neglected, the estimation results differ from the two types of simulation. Moreover, road disturbance condition differs from parking to rolling, this explains the variation in performance level. However, according to the above simulation results, good performance of the driver torque estimation by H ∞ filter strategy is obtained using experimental data as inputs.

CONCLUSION

In this paper, an H ∞ filter has been designed subject to minimization objective on road disturbance and sensor noises. Then, performance has been evaluated by simulation using experimental measurements. The estimation of driver torque presented in this paper is application-oriented and aims to be a backup calculation to cover failure of torque sensor signal. As this is a software strategy, it could be easily implemented in the ECU. Moreover, it only requires measurements from sensors existing on board of most of mass produced vehicle equipped with an EPS system.

Even though driver torque estimation might be affected by friction on EPS system in practice (due to its non-linearity), the design of a robust controller could result in reducing the unwanted effect to the driver. Hence, future works will concentrate on the whole design of an observer-controller system to provide assistance torque without using torque sensor.

Figure 1 :

 1 Figure 1: P-EPS simplified mechanical model

Figures 4 ,

 4 Figures 4, 5 and 6 illustrates the sensitivity function deduced from (17) -(19) with different tuning for T (s). It is again noticed that a cut-off frequency at 25Hz gives less efficient results than with 5Hz or 10Hz. Indeed, the optimization problem is harder to be solved when considering a greater frequency range. As both remaining synthesis show similar performance, the one designed with 10Hz is preferable.

Figure 4 :Figure 5 :

 45 Figure 4: Influence of various T (s) design on T zd (s)

Figure 6 :

 6 Figure 6: Influence of various T (s) design on T zn (s)

Figure 7 :

 7 Figure 7: Block diagram of H ∞ filter implementation

Figure 8 :

 8 Figure 8: Driver torque estimation -Sinusoidal driving case 1

Figure 9 :Figure 10 :

 910 Figure 9: Driver torque estimation using experimental data-Sinusoidal driving

Table 1 :

 1 EPS system mechanical parameters

	Notation Description	Unit
	J c	Steering column inertia	kg.m 2
	B c	Steering column viscous friction N.m/(rad/s)
	K tb	Torsion bar stiffness	N.m/rad
	D tb J p	Torsion bar damping Pinion/gear inertia	N.m/(rad/s) kg.m 2
	K pr	Pinion/rack stiffness	N/m
	D pr	Pinion/rack damping	N/(m/s)
	R p	Pinion/rack reducer	m/rad
	J r	Rack and tie rods mass	kg
	B r	Rack viscous friction	N/(m/s)
	K r	Rack stiffness	N/m
	D r G J m	Rack damping Worm/gear reduction ratio Motor inertia	N/(m/s) -kg.m 2
	B

m Motor viscous friction N.m/(rad/s) K g Worm/gear stiffness N.m/rad D g Worm/gear damping N.m/(rad/s)

Table 2 :

 2 Influence of weighting function T (s) on the H ∞ filter synthesis

	T (s) 1 st order
	w n	5Hz 10Hz 25Hz
	γ ∞	0.15	0.27	0.51
	steady state gain 0.91	0.75	0.49
	T (s) 2 nd order
	ζ = 0.7 and w n	5Hz 10Hz 25Hz
	γ ∞	0.12	0.12	0.34
	steady state gain 0.98	0.98	0.69

ACKNOWLEDGMENT

This study has been supported by JTEKT corporation in Japan: Miyama Shinpukujicho Okazaki-shi, Aichiken, 444-2106 Japan.