W T Dorea

Sloan
email: sloan@civil.gla.ac.uk

Na Zeng

G Li

C Dorea
email: c.dorea@civil.gla.ac.uk

W T Sloan

Prospective application of microbial fuel cell technology in developing countries -taking China as an example

Keywords: Microbial fuel cell, China, water, energy, application

Microbial fuel cell (MFC) is an innovative remediation technology for waste and wastewater treatment because of its potential to degrade organic substrates and simultaneous generation of electricity. This paper briefly reviews the techniques employed in recent MFC studies and status of China's energy and water environment. According to the future development of economy and society in China, the paper discusses the application prospect of MFC from different aspects, such as field, scale, region, etc. and proposes related solution of energy and water environment problems to some extent.

INTRODUCTION

China is the developing country with the largest population in the world. The development of China has so far been unbalanced, as the East is much advanced than the West. As most development of economy was based on the cost of environment, China has realized the problem, and promoted the concept of sustainable development as a national policy. Still, approximately 56 percent of China's population lives in rural areas, where they have faced a complex array of energy and resource problems for a long time. In 21 st century, with the increase of population, improvement of living standard, acceleration of urbanization process, the amount of water resource per capita decreases gradually, while the water demand is rising, contradiction between water supply and demand is becoming more significant, so water deficiency is the prominent restrictive factor of food security, economic development, social harmony, as well as ecological improvement in China.

Like most countries in the world, China is on its way searching for new renewable energy resources and technologies for environment protection. While microbial fuel cell is considered as a novel technology for wastewater treatment and energy resource, because it is based on electrochemically active microorganisms that grow by oxidizing the biodegradable material to carbon dioxide and protons, meanwhile transferring the electrons to a solid electrode [START_REF] Logan | Microbial fuel cells: Methodology and technology[END_REF]. The electrons then flow through a resistor to a cathode, at which the electron acceptor is reduced. In contrast to anaerobic digestion, a MFC creates electrical current and an off-gas containing mainly carbon dioxide [START_REF] Rabaey | Microbial fuel cells: novel biotechnology for energy generation[END_REF]. Obviously, concerned to some rural areas where lacking electrical infrastructures, or weak in environmental protection, microbial fuel cell has the potential for widespread application in waste treatment and also to expand the diversity of fuels used to meet the energy requirements.Microbial fuel cell undoubtedly can be used to fix some problems in China to some scale applications.

The principle of Microbial fuel cell

A microbial fuel cell (MFC) is a reactor that can convert biodegradable materials, e.g. organics present in wastewater, directly into electricity by the bio-electrochemical bacterial processes. Typical microbial fuel cells are designed with cathode and anode chambers separated by proton exchange membrane. Fuel (i.e. organic substrates) is oxidized via bio-catalysed reactions at anode onto which the electrons released during this oxidation process are transferred.. The anode is connected to the cathode via an external electrical circuit through which electrons flow to form the current. Electron acceptors are reduced at cathode by catalysts with protons arriving via the proton exchange membrane. Finally electrons, protons and oxygen react to form water at the cathode.

MFCs can not only generate electricity with simple carbohydrate as fuel, but also degrade other complex organic contaminants from wastewater. It now seems that electricity can be generated from any biodegradable material, ranging from pure compounds such as acetate, glucose, cysteine, bovine serum albumin and ethanol to complex mixtures of organic matter including domestic (human), animal, food-processing and meat-packing wastewaters.Compared to the traditional methods for treatment of wastewater, MFCs are able to operate and generate electricity effectively in ambient or at even lower temperatures, they produce less gas and less sludge and hence offer substantial cost savings for wastewater treatment. Since biofilms are ubiquitous, especially anaerobic sludge itself may serve as fuel for electricity production. The cell lysis accompanying endogenous decay of anaerobic sludge releases soluble substrates that can potentially support the regrowth of bacteria and electricity generation [START_REF] Hu | Electricity generation by a baffle-chamber membraneless microbial fuel cell[END_REF].

The driving force of a typical MFC using glucose as substrate can be expressed through a redox reaction [START_REF] Grzebyk | Microbial fuel cells (MFCs) with interpolymer cation exchange membranes[END_REF]. The chemical reactions at each electrodes are as followed:

C 6 H 12 O 6 + 6H 2 O → 6CO 2 + 24H + + 24e - 6O 2 + 24H + + 24e -→ 12H 2 O
MFCs have operational and functional advantages over the technologies currently used for generating energy from organic matter. First, the direct conversion of substrate energy to electricity enables high conversion efficiency [START_REF] Logan | Microbial fuel cells[END_REF]. Second, MFCs take the advantage of general biochemical reaction, based on the function of microbial communities, without any extra energy input for aeration on the cathode. This distinguishes them from all current bio-energy processes, which results in lower operation and maintenance costs. Third, an MFC does not require gas treatment because the off-gases of MFCs are mainly carbon dioxide which does not have any useful energy content. Furthermore the redox products are usually CO 2 and H 2 O, which are not considered contaminants. . Fourth, MFCs can have a huge range of resources as their fuel becuase they do not need to be high quality. Wastes are full of organic compounds most of which are degradable can be used in MFCs as fuel [START_REF] Rabaey | Microbial fuel cells: novel biotechnology for energy generation[END_REF] Fifth, MFCs have the potential for widespread application in locations lacking electrical infrastructures and also to expand the diversity of fuels we use to satisfy our energy requirements [START_REF] Logan | Microbial fuel cells[END_REF]. For example, in the poor rural and mountainous areas in developing countries, people benefit from localized modest electricity generating facilities like

Prospective application of microbial fuel cell technology in developing countries-taking China as an example -Na ZENG

/ 7

MFCs to provide light at night or access to the internet; both of which could change the education prospects of a generation of poor children

Researchers are expanding the concepts of MFC, modified MFC reactors are coming out to deal with different situations, either huge or small. The application of MFC technology is continuously developed on its way of non-membrane and non-mediator, becoming more and more feasible in its practical application, not only in the lab. One reason of these trend is because the materials are very expensive. Widespread application need to cut down the cost of MFC reactors.

Status of water and energy in China

China is the biggest developing countries with the largest population in the world. It is characterized by unbalance in distribution of population, water resources and energy. The economy is most developed in east of China which has a larger population, but less resources compared to the west. However, the disparities are also on the north-south axis, with for example the north having typically less water resources than the south. That's the reason why big projects, such as the South-to-North water diversion project and the West-to-East natural gas transmission project promoted. These are efforts to alleviate the unbalance between economical development and energy resources.

Status of water environment

The total volume of fresh water is about 2800 billion m 3 ; accounting for 6% of the global freshwater resources. However, the amount of water per capita is just 2300m 3 , one fourth of the world average level. This makes China one of the worst 13 countries lacking water resource per capita countries as defined by the United Nations. Besides, much of the cited freshwater is not readily available, as it comes in the form of seasonal floods and groundwater in remote areas. As such the real available fresh water is only 1100 billion m 3 , around 900 m 3 per capita. Especially, water resource availability in 6 provinces and regions of North China amounts to merely 500 m 3 per capita, which is half the critical value of the United Nations (UN) standard (1,000 m 3 per capita) Comparably, Canada uses 1,600 cubic metres of water per person per year, not to mention the available water resource per capita (Canada vs. The OECD, internet, 2009).

China has serious water-deficient problem, especially in the northern areas. Most of freshwater resources are in the South, with three times more water than the North of the country. Though the water resources currently meet the demand, China is facing many water-related problems that will compromise the sustainable use of this precious resource. Such problems include: the uneven distribution of water resources in time and space; pollution; wasteage; low consumption efficiency; and aquatic ecosystem imbalance. All of these issues have resulted in a conflicting situation between the supply and availability of freshwater in recent years (Baidu-Contributors, 2009)..

According to the data from China Water Resources Bulletin 2007 (China Water Resources Bulletin 2007[START_REF] Hu | Electricity generation by a baffle-chamber membraneless microbial fuel cell[END_REF], the annual usage of water amounted to 302.2 billion m3, most of which was used in agriculture (74.6%) followed by domestic and industrial consumption (12.4 and 11.0 %, respectively). Water wasteage rates also varied between different user groups greatly, industry, municipal, irrigation, and rural living cost 24%, 30%, 62%, 84% respectively. Also water pollution is an increasing threat to water resources. In 2007, wasterwater discharge of whole country was 75 billion ton, two thirds were industrial wastewater, and one third was from domestic and the third industry. Taking Yellow River as an example, industrial wastewater is the main cause of its pollution, occupied 73% of the total wastewater discharge, result in nearly 6 billion RMB loss annually, including loss of water resource value, municipal water supply, and also increasing the extra investment of wastewater treatment. And most of the industrial wastewater is full of high strengthed organics.

Status of energy

China's energy stockpile amount ranks the first class of biggest stockpile in the world. Meanwhile, however, China also ranks amongst the world's largest energy consumers. Coal plays a dominant role among the energy resources which have been explored in China, though the global energy system has transferred from coal to petrol and gas as primary energy fuels.

Recently, other renewable energy resources are currently being developed, like hydropower, wind, solar, biofuel, geothermal, ocean energy. But these have still not been implemented in a large-scale. It is estimated that the usage of renewable energy may be improved from 7% at present to about 16% till 2020 [START_REF] Feng | The Renewable Energy Status and Development in China[END_REF]). China's energy consumption is rising in accordance with its economical growth. However, the energy efficiency is still lower than developed countries, due to issues related to finance, technology, energy price and so on. China's growing coal-based energy demand has also lead to other problems such as air pollution and its associated environmental pressures. Chinese energy policy is now starting to consider such impacts and is seeking to avoid further environmental damage.

According to the Chinese National Development and Reform Commission Energy Bureau, the energy issue in rural areas is also very serious. Most villagers in West China don't have enough power and there are still more than 10 million people don't have electricity at all. Accelerating construction of off grid renewable energy sources such as solar, wind, biofuel energy can provide a suitable energy solution for such villages [START_REF] Wu | Chinese Energy Present and Trend[END_REF].

MFC TECHNOLOGY POTENTIAL

Microbial fuel cell technology has the ablility to treat organic wastes whilst generating electricity.

As such this technology has the potential to tackle two of China's greatest problems at once. Furthermore, MFC technology can be applied in different fields of interest to China's sustainable development. However, there are issues that need to be overcome in order for MFCs to become a viable solution.

Applications

Wastewater treatment According to the mechanism of MFC, it is obviously that MFC is a clean way to treat wastewater. It oxidizes organic matters at the anode with bacteria, and the product at the cathode is water without producing any contaminants. MFC does not need any extra energy input, also can operate in an ambient atmosphere, without location restriction at all. In current China, the water pollution is a serious problem which affects the limited water resource. Especially a lot of factories discharged a huge amount of sewage which had extremely high concentration of organic matters. MFC is undoubtedly a good option to solve it. Once the power density of MFC would have been improved much enough will make wastewater treatment into a lucrative energy business.

Bioremediation One novel application of the MFC is as a method of bioremediation, Because MFC utilizes microorganisms to return the natural environment altered by contaminants to its original condition, such as degradtion of hydrocarbons by bacteria. Remediation can include degradation of organic pollutants at the anode as well as reduction of inorganic chemicals such as

Prospective application of microbial fuel cell technology in developing countries-taking China as an example -Na ZENG 5 / 7

nitrate or uranium at the cathode. The Sediment MFC consists of anode in the anaerobic sediment and cathode in overlying water containing dissolved oxygen. It suits river, lake and coast line where sediments are full of organic carbon and exoelectrogenic bacteria already present in them.

Also, it has been shown that MFC-based technologies could be used to remove nitrate, conversion to nitrite, even N 2 gas from water, reduced by receiving electrons from the cathode. It is known that nutrient removal is critical to treatment objectives of most wastewater treatment.

Because findings on nitrogen removal show great promise for new types of nutrient removal processes for wastewater treatment, particularly if nitrogen can be removed without the need for additional post-treatment carbon sources or through water recycling. Typically, as the lake eutrophication phenomenon is highlighted recently in China, like Taihu Lake and Dianchi Lake, both are the result of water body pollution, may be capable to use MFC technology to treat and remove the nutrient.

Biosensors The current output of an MFC can be related to the type and amount of substrate being treated. Therefore MFCs can be used as biosensors to monitor quality parameters that are related to substrate characteristics, such as biochemical oxygen demand (BOD). Such a sensor may contribute into wastewater treatment plants as a monitor to test the treated wastewater's BOD concentration and calculate BOD removal rate. Compared to the conventional BOD assays, MFC BOD biosensor has the advantage of time saving. Because conventional BOD assays need 5 days at least to test the value, while BOD sensor reflects BOD value by the electricity produced by MFC in 24 hours.

It also has been suggested that the voltage output from small scale MFC can drive MEMS (microelectromechanical systems) devices, which doesn't need too much electricity. Microelectromechanical systems is the technology of the very small, and merges at the nanoscale into nanoelectromechanical systems (NEMS) and nanotechnology. MEMS are also referred to as micromachines (in Japan), or Micro Systems Technology -MST (in Europe) (Wikipediacontributors, 2009). At the same time, products of MFC are water and carbon dioxide which won't relate to contamination and erosion to MEMS. Like micro blood glucose testing meter combined with a MFC, operates with the electricity produced by MFC using glucose in blood as fuel, while transferring into electromagnetic signals to show the information about blood glucose concentration, in order to monitor it for long time. So the similar inner bio-detection can use the MEMS combined with MFC technology, which is another prospect of MFC development in medical application [START_REF]The Development of Microbial Fuel Cells[END_REF].

Scale of Applications

Centralized wastewater treatment plant Most researchers focus on microbial fuel cell's benefits and modified it in wastewater treatment plants. The reactor of it has been suggested to scale up to treat a huge amount of wastewater. More than 40 million cubic meters of municipal wastewater was treated each day in China according to 2005 estimates. Most of this wastewater was sent to centralized facilities that consume large amounts of energy for treatment due to aeration and produce a great deal of surplus sludge. In the more developed regions in eastern China, it is possible that MFCs can be used to replace the conventional bioreactors like activated sludge (AS) or trickling filters (TF) treatment systems. The MFC is a treatment process that has been advocated to solve high energy costs and sludge treatment expenses problems of anaerobic treatment limitations [START_REF] Liu | Production of electricity during wastewater treatment using a single chamber microbial fuel cell[END_REF]. And thus its function will be to remove BOD in the same manner as accomplished by the AS aeration tank or the TF. With increasing of the power Furthermore, it could allieve the pressure of centralized facilities in treating and save its energy cost in aeration.

Home-use MFC for wastewater treatment In the remote areas, like the western mountainous villages, it is not practical to treat wastewater in a centralized facility. This is due to the small quantities of water consumed and also because the dwellings in this are typically very sparsely distributed, making the collection system for a centralized wastewater treatment facility uneconomical. In this way, MFC can be utilized efficiently to treat domestic wastewaters on site.

Though the effluent of MFC can not reach the water quality for drinking, once the organics removal rate of MFC reaches the level of making water suitable to reuse, it will decrease the frequency of pumping groundwater out of wells by villagers. It is absolutely a good way to solve water problem to some extent in these areas.

Direct current devices Based on the other main function of MFC, electricity generation, though the power density gained is not very high so far, it has proved the electricity produced by MFC is sufficient to run some little direct current devices. Little fan, flashlight are the examples of devices which could use the electrity produced by MFC. To this point, MFC has significant implications, especially for the electricity lacking areas. MFC is reliable, once it started, it can operate for quite a long time and gain continuous current. The main limitation of MFC is low power density at present, due to its high internal resistance, but it can solve with a storage battery to collect the electricity, just like the use of solar cells.

Region

As it discussed above, in well developed cities and regions, MFC's benefit of electricity production will be most probably ignored in commercialization before it improved power density output, but its utilization in wastewater treatment process will be widely applied. In addition, the landscape is profitable for using scaled up MFCs. However, in places where lacking electricity, MFC technology do give a positive aspiration for people getting energy from wastes.

Considered as a novel renewable energy resource, MFC can help to contribute into the new rural villages, because energy demand is going to be satisfied according to the rural development policy, and besides solar, geothermal, biomass energy, MFC is another alternative. Virtually all biomass energy in China is used in rural areas, where approximately 56% of the population lives, the rural sector includes rural households, agricultural activities. Utilization of MFCs could better combine with these aims, for example, heating is critical in the north, while cooking for pig feed is important in south and central areas, once MFC improved the electricity production, we'd better transfer it into thermal energy to meet the demand of these rural places.

CONCLUSIONS

Microbial fuel cell technology has a huge amount of potential and will emerge gradually an essential element for both renewable energy resource and wastewater treatment process. As the status of water and energy in China, both are very important factors for sustainable development,

Prospective application of microbial fuel cell technology in developing countries-taking China as an example -Na ZENG 7 / 7 the idea of MFC can match these two aspects greatly. And the application of MFC technology should consider the practical situations such as in field, scale or region, in a word, adaption to local conditions. Also MFC can treat waste and wastewater, undoubtedly, it is a good alternative to solve the present serious problems in environmental protection in China. However, there are still many challenges remaining to fully exploit the maximum power production possible by MFCs. Concerns to the limited energy production, MFC is on its way of improvement, excluding saving water and energy, especially in some extreme rural areas where no electricity at all. Thus, reductions in material costs, advancements in power densities will make a wide range of MFC technologies have a bright and promising future.

  generation in the system, MFC may provide a new method to cut down wastewater treatment plant operating costs, making advanced wastewater treatment technology affordable by developing countries. MFC can even take a place in the discharge of industrial wastewater before it flowing into centralized treatment plants. It means pretreat industrial wastewater in a MFC bioreactor on the way to conventional treatment plants. High strengtened contaminants present in wastewater are the sufficient food source for microbial communities, result in high current output.

	8th World Wide Workshop for Young Environmental Scientists WWW-YES 2009: Urban waters: resource
	or risks? (2-5 June 2009)
	6 / 7


Pauline Feschet
email: pauline.feschet@cirad.fr

Denis Loeillet

Catherine Macombe

Michel Garrabé

Fruits & vegetables supply chains specificities and stakes as element of discussion on Social-LCA

Keywords: social LCA, fruit & vegetables, sustainability, multiple capital model

Agri-food supply chains are at the heart of sustainable development concerns. In order to comprehend whole complex parameters and their global impacts, it appeared necessary to adopt a systemic approach, justifying a Life Cycle Assessment (LCA), not only from an environmental point of view but from a social and economic one too. Taking into account specificities and stakes of fruits and vegetables (F&V) supply chains in developing countries this paper focus on absences, deficiencies and methodological limits that LCA meets integrating social and economic aspects. The elements presented lead to an in-depth conceptual and theoretical discussion and suggest placing LCA in the perspective of development theory. The proposition is to endow LCA with an approach "by capitals", which seems particularly adapted to express sustainable development and well-being.

Background

Agri-food supply chains are at the heart of sustainable development concerns. It's particularly true from an environmental point of view due to off-season productions, remote localization away from consumption areas and culture intensification. These elements question productions models, in particular technologies of production and their localization, in terms of resource depletion, water, soil and air contamination, global warming, etc. Life Cycle Assessment (LCA) appeared particularly well adapted to consider whole complex parameters and their global impacts. Its initial goal was to assess a full range of potential environmental impacts related to products and services, in order to choose the best alternative or to improve its [START_REF] Jolliet | The LCIA midpoint-damage framework of the UNEP/SETAC Life Cycle Initiative[END_REF]). Nevertheless, sustainability doesn't end in environmental dimension, even if it's a very important component. It includes (at least) social and economic dimensions (WCED 1987). To evaluate social and economic impacts presents similar interests as environmental-LCA, in terms of capacity of comparison of products or services and identification of hotspot and margin of improvement. In addition, this integration seems particularly important in a global sustainability assessment prospect, due to the strengthening of societal expectations towards agri-food products, tending to change the modes of governance of these supply chains as well as their organization, and to reinforce social standards. In this context, integrating socio-economic aspects in decision criteria is essential. To consider these aspects is all the more reason important for horticultural products which are source of significant social and economic impacts, in particular in developing countries where horticultural products greatly contribute to the GDP [START_REF] Weinberger | Horticultural for poverty alleviation -The unfunded revolution[END_REF]. To consider the specific case of F&V supply chains in these countries could enrich the discussion on Social LCA (S-LCA), highlighting through their specificities and stakes the important elements that need to be taken into account in a framework.

Social and economic implications of F&V supply chains: specificities & stakes

Rising per capita income, urbanization, changes in consumer taste and globalization is changing consumption behaviour, consequently worldwide supply and international trade in F&V (Weiberger 2005[START_REF] Temu | High value agricultural products for smallholder markets in Sub-Saharan Africa: trends, opportunities and research priorities[END_REF]. Many developing countries took advantage of these changes and have diversified into horticultural crop production and exports based on favourable climatic conditions and lower labour costs (Davis 2005). Effectively, horticultural products are considered as High Value Agricultural Products (HVAP), defined as a "product that return a higher gross margin per unit of available resources (land, labour, capital, human capacities) than other product within a given location and context" [START_REF] Gfar | Synthesis report of the international workshop on "How can the poor benefit from the growing markets for high value agricultural products[END_REF], offering an opportunity for rural poor to improve their livelihoods.

Horticultural products have some comparative advantages in comparison to others productions. They are source of high profitability for farmers in terms of net farm incomes and net returns on different input measures (McMulloch et al. 2002). For example, in Kenya net farm incomes were five times higher per family member compared to smallholder farmers who did not grow horticultural products [START_REF] Mcculloch | Export horticulture and poverty in Kenya[END_REF]. Furthermore, these products generate additional employment opportunities in rural areas because they are more labour intensive than the production of staple crops. For example, horticultural production requires nearly three times more labour than cereal crops (Weiberger et al. 2005). Requiring specialized inputs both upstream and downstream, the growth in HVAP induces a multiplier effect in terms of economic activities and employment potential generation. Horticultural sub-sector generates less tangible and indirect benefits. It contributes to the institutional environment development, in particular for access to credit and capital since the crops are more risky and costly than others (Weiberger et al. 2005). International markets need to comply with a range of legal and commercial standards (maximum residue level of pesticides, phytosanitary certificate, traceability, good agricultural practice) [START_REF] Temu | High value agricultural products for smallholder markets in Sub-Saharan Africa: trends, opportunities and research priorities[END_REF], which contributes to the determining of norms, labels, codes, etc. They encourage development of networks and market organizations (supply chains, market price information, etc.). It's a major factor in infrastructures investments like roads, rails, seaport, electrification, wastewater system, etc. Being sophisticated products, they require qualified employees, consequently high level of knowledge and skills reached thanks to training and formation.

In compensation, there are some negative impacts. International market development goes with pernicious effects. There aren't technical economies of scale [START_REF] Temu | High value agricultural products for smallholder markets in Sub-Saharan Africa: trends, opportunities and research priorities[END_REF], but legal and commercial requirements to access to these markets are very important. So that, only organized small-scale farmers or large-scale enterprises are able to comply with [START_REF] Temu | High value agricultural products for smallholder markets in Sub-Saharan Africa: trends, opportunities and research priorities[END_REF]. It means that most private small-scale farmers are kept away from benefits of this developing sector and consequently from an opportunity of development. For example, in India, population still suffers from malnutrition whereas the country is one on the more important producer of basmati rice [START_REF] Rahnema | Quand la misère chasse la pauvreté[END_REF]. Moreover activities concentration and wealth appropriation by few actors is often synonym of capital flight for favourable taxation places. Attractive and lucrative character of HVAP [START_REF] Dolan | Horticulture commodity chains: the impact of the UK market on the African fresh vegetable industry[END_REF] could create a competition for land and resources with local production and food-producing. Accesses to international market and adaptations to satisfy developed countries expectations involve adoption of skills and technologies imported that could destroy knowledge and know-how [START_REF] Rahnema | Quand la misère chasse la pauvreté[END_REF]. These adaptations could change dramatically lifestyles and cause disruptions in local population habits, imitating northern functioning and rubbing local specificities. Working conditions, safety and workers rights are not always respected. Child labour is certainly the principal critical point, even if the perception of child labour is different depending on the place in the world. For example, in banana cropping, "deflowering" is a task entrusted to children after school in Latin America, allowing them to contribute to family incomes. It cannot be considered in West French Indies.

The stakes subtended by the development of horticultural productions in developing countries are poverty alleviation and economic development. It's therefore important, not to say necessary, to be able to assess for example the difference between two organization modes, namely an integrated system (promoted by high accessing costs to international markets) and an atomized small-scale farmers system. Moreover, in a globalized world, where sustainable criteria will become access market barriers and criteria of decision for sourcing, it's essential not only to discriminate alternatives thanks to environmental elements but also to include all the complex aspects presented previously. That is the whole issue of S-LCA development.

Ambiguities and absences in the scope of social and economic aspects integration

Social LCA no longer needs to be justified [START_REF] Griebhammer | Feasibility study: integration of social aspects into LCA[END_REF]. In terms of methodology, it was highlighted that there were evidently no fundamental problems even if considerable hurdles needed to be overcome in practice, especially in characterization modeling [START_REF] Griebhammer | Feasibility study: integration of social aspects into LCA[END_REF]. In spite of efforts to find an international consensus on the general principles (UNEP/ SETAC 2009), literature shows a wide range of frameworks with many differences (choice of indicators, impacts categories, characterization factors, etc.). Despite the Task Force preferred to put the emphasis on methodological hurdles, considering some absences, deficiencies or methodological limits, this article asks the question of the analysis and the evolution of LCA conceptual and theoretical framework in the view of a broader sustainable conception, integrating environmental, social and economic aspects.

A fuzzy conceptual framework

The first level of explication could concern the conceptual framework. Most of works deal with methodology rather than conceptual framework, whereas it appears essential and could explain a lot of deadlocks. Thus, a state of art of existing works on S-LCA highlighted some fuzzy and ambiguities on goals, extent, content and boundaries, and upstream on what is important to protect from a societal point of view (Areas of Protection). While the guidelines for S-LCA of UNEP/ SETAC (2009) declare that "social LCA will be used as a synonym for social and socio-economic LCA", most of the authors make a clear distinction between economic impacts -assessed by Life Cycle Cost-and social impacts -the real object of S-LCA [START_REF] Norris | Integrating life cycle cost analysis and LCA[END_REF][START_REF] Klöpffer | Life-Cycle Based Methods for Sustainable Product Development[END_REF], Dreyer 2006[START_REF] Hunkeler | Societal LCA methodology and case study[END_REF], Hutchins 2008), letting the real content of S-LCA vague. Yet, the will to integrate socio-economic aspects in LCA involves referring to human-being and the society in which he evolves, and its attributes (e.g. norms, rules, public utility, etc). Nevertheless, excepted in Jorgensen et al. ( 2010), the AoP relating to Human life as presented in the different framework do not differentiate clearly the individual dimension of human well-being from the societal dimension. In addition the position of "less tangible items of financial and cultural values" in the "man-made environment" AoP's seems not so relevant, considering the original definition of the AoP, that is to say elements that change land surfaces for human purposes [START_REF] Jolliet | The LCIA midpoint-damage framework of the UNEP/SETAC Life Cycle Initiative[END_REF]. Financial items don't change land surface except indirectly through investment. Cultural values (e.g. languages or practices), with a broader conception than the "value of unique landscape and unique archaeological sites" [START_REF] Weidema | Physical impacts of land use in product life cycle assessment[END_REF], refer more to the societal dimension of Human life AoP than to the man-made environment.

Limited theoretical foundations

The second level of explication for methodological limits but also for conceptual ambiguities could refer to the subjacent theoretical model. LCA was originally an engineering counting method, "for evaluating the opportunities, risks, and trade-offs associated with products and services over their entire life cycle" (UNEP/SETAC 2009). By its empirical approach, it developed itself without clear theoretical foundations, apart from the fact that it was in line with sustainable development model like defined by WCED (1987). On time to integrate social and economic aspects, LCA confronted two hurdles. First, the discussions about S-LCA have concerned the categorization and classification of indicators. Nevertheless, social and economic aspects being particularly complex from one hand and the theoretical framework being not sufficiently explicit from another hand, this lead to draw up a large inventory of indicators without neither being able to produce a synthesis nor putting them into perspective. This limitation is in contradiction with the operational goal attribute to LCA which is "to evaluate trade-offs associated with products and services" (UNEP/SETAC 2009). In second point, S-LCA works give a partial view of social and economic aspects. The Life Cycle Initiative specified the main criticisms made to this method. In particular they could be seen "to be 'anti-development'-orientated because it provides only a picture of negative environmental consequences, but does not reflect any of the positive aspects of development; and to not address the developing countries most significant concerns" (UNEP/SETAC 2009). These critics reinforce the idea that the model of sustainable development which underlies S-LCA actual frameworks doesn't identify sufficiently human, social and institutional dimensions, as it was suggested previously as regards the consideration of the content of human AoP. This limitation is in contradiction with the conceptual goal attributed to LCA, which is "to achieve sustainable development" (UNEP/SETAC 2009).

Methodological limitations resulting

This double level of incompleteness and fuzziness results in absences and deficiencies at methodological level. The main deficiency concerns the distinction between well-being and sustainability from one hand, and between flow and stock from another hand. They aren't clearly expressed and developed, and yet they are crucial. According to [START_REF] Sen | Rapport de la Commission sur la mesure des performances économiques et du progrès social[END_REF] sustainability asks the question to know if it's possible to expect that actual well-being level could be at least maintained for future generations or periods. The notions of flow and stocks are required here, in the sense that future well-being will depend on their fluctuations, since sustainability needs a minimum and constant stock of "wealth". Wealth is understood in a broader sense which doesn't limit to natural resources but includes other forms of capital, in particular human, social and physical capital. To make the difference between well-being and sustainability -involving different temporal scales and different stakeholders with a specific reference to future generations -one needs to consider the assessment through a dynamic approach. Moreover, the difference between flow and stock involves the notion of depletion and irreversibility of "wealth/resources" consumption or destruction. Consequently, the depletion of all kind of capital by a productive process needs to be considered as a reduction of stock and not as an income. It's the case for productive capital considering depreciation mechanism, which immobilizes a value equivalent to the part of technical capital destroyed in the process of production in order to offset the loss of capital. But it's not the same thing for all capital forms. Others limitations concern the consideration of hidden costs (not only direct effects), positive impacts (not only damages), economic price (not only financial price). We will not develop these aspects in this communication.

Conclusion: a new theoretical framework

All elements presented previously involve to strike up a deep discussion on theoretical basis. To endow SLCA with a sound theoretical model would allow to structure the framework in a consistent whole. So it would allow not only to determine a set of indicators but also to refer to social impacts, thanks to the identification of logical and organized impact pathways. An approach "by capital" forms seems particularly well adapted. It consists in characterizing a process of growth and development as a production system in which multiple capitals are involved (natural, social, human, and produced/physical). It offers the advantage of placing LCA in a theory of development recognized as well adapted to assess sustainable development due to its perennial and exhaustive character (MDDEP 2009).

Many international organizations used this approach (United Nations, OECD, European Economic Comunity, World Bank, UNESCO, EuroStat). In LCA works, this approach has been proposed more or less explicitly. SEEbalance® [START_REF] Schmidt | Managing sustainability of products and processes with the Socio-Eco-efficiency analysis by BASF[END_REF]) covers the four types of "societal" capital: social, human, produced/physical and natural capital. [START_REF] Labuschagne | Social Indicators for Sustainable Project and Technology Life Cycle Management in the Process Industry[END_REF] mention human, productive and community capital. More recently Jorgensen et al. (2010b) suggest that SLCA has to assess changes in human, social and produced/physical capital. Nevertheless, none of these works fully developed this model (notion of flow and stock, positive impacts, pathways, etc.), it's the purpose of our proposition, according to the conclusion of Jorgensen et al. (2010a), confirming the validity of the impact pathways in SLCA. Moreover, until now, four forms of capital have been considered. We propose a broaden approach distinguishing a fifth form, the institutional capital (norms, rules). Actually it depends on social capital (networks) but it has an existence in itself distinct like demonstrated previously.

Until now, SLCA paid attention to characterize and organize social indicators, next step is to articulate them thanks the multiple capital model, in order to reflect social impact and damage or benefit to the AoPs. This would allow to clarify conceptual framework and methodology. P. Feschet is member of the ELSA group (Environmental Life Cycle and Sustainability Assessment www.elsa-lca.org ); she thanks all the other members of ELSA for their advice.

Author-produced version of the paper presented in LcaFood 2010 VII international conference on life cycle assessment in the agri-food sector


Two results concerning chirps and 2-microlocal exponents prescription

Bertrand Guilheneuf, Stéphane Jaffard, Jacques Lévy Véhel

To cite this version:

Bertrand Guilheneuf, Stéphane Jaffard, Jacques Lévy Véhel.


M Lhommeau

L Jaulin

L Hardouin

Capture Basin Approximation using Interval Analysis

Keywords: Nonlinear systems, Viability theory, Capture basin, Interval Analysis, Guaranteed Numerical Integration

This paper proposes a new approach for computing the capture basin C of a target T. The capture basin corresponds to the set of initial state vectors such that the target could be reached in finite time via an appropriate control input, before possibly leaving the target. Whereas classical capture basin characterization do not provide any guarantee on the set of state vectors that belong to the capture basin, interval analysis and guaranteed numerical integration allow us to avoid any indetermination. We present an algorithm able to provide guaranteed approximation of the inner C -and an the outer C + of the capture basin, such that C -⊆ C ⊂ C + . In order to illustrate the principle and the efficiency of the approach, a testcase on the "car on the hill" problem is provided.

Introduction

This paper deals with the control of dynamical systems of the form ẋ(t) = f (x(t), u(t)), t ≥ 0, with state x(t) ∈ R n and control u(t) ∈ U, in the presence of state constraint x(t) ∈ K ⊆ R n . The set K × U describes the "acceptable configurations of the system". Such problems of dynamic control under constraints refer to viability theory [START_REF] Aubin | Viability theory[END_REF] (see [START_REF] Aubin | A survey of Viability Theory[END_REF] for a survey). The main concepts of the viability theory are

• viable state : a state is called viable if there exists at least one control function for which the whole trajectory from this state remains in K indefinitely. • Viability kernel : the set of all viable states is called the viability kernel.

• Capture basin : let T be a subset of K. The subset of initial states x ∈ K such that T can be reached in finite time before possibly leaving K by at least one trajectory x(•) is called the capture basin and is denoted C.

The characterization of the capture basin has a wide range of applications, for example, in process control [START_REF] Cardaliaguet | Optimal times for constrained nonlinear control problems without local controllability[END_REF][START_REF] Spiteri | Programming and control of robots by means of differentialalgebraic inequalities[END_REF] and system biology [START_REF] Bene | A viability analysis for a bio-economic model[END_REF][START_REF] Bonneuil | Making ecosystem models viable[END_REF]. Thus the development of methods for the characterization of the capture basin for dynamical systems is an important area in control engineering. In general the computation of the viability kernel is not an easy task.

The development of computational tools to support the numerous viability theory methods is an ongoing effort. The viability kernel and capture basin are computed thanks to viability algorithms ( [START_REF] Saint-Pierre | Approximation of the viability kernel[END_REF], [START_REF] Saint-Pierre | Capture basin algorithm for evaluating and managing complex financial instruments[END_REF]). Saint-Pierre's algorithm computes for a given grid X h a discrete viability kernel that converges to the viability kernel when the grid resolution h tends to 0. Also several interesting results have recently appeared on numerical methods to compute viability kernels and capture basins [START_REF] Deffuant | Approximating viability kernels with support vector machines[END_REF], [START_REF] Bokanowski | An anti-diffusive scheme for viability problems[END_REF], [START_REF] Bonneuil | Computing the viability kernel in large state dimension[END_REF] and [START_REF] Coquelin | A dynamic programming approach to viability problems[END_REF]. Most of the time these approaches provide approximations based on the numerical diffusion induced by the discretization scheme.

We have shown in earlier works how to provide an inner and outer approximation of the capture basin using interval analysis and guaranteed numerical integration. The major improvement presented in this paper is given by the main results detailed in Theorem 4.1 and Theorem 4.2. In particular, these results allow to give a new algorithm where the exploration of the search space is organized in such a way to ensure the algorithm convergence. The proposed method to approximate capture basins has several advantages over traditional numerical methods to find capture basins : (1) it is guarantee, whereas numerical methods give only an approximation of the capture basin. Indeed, in conventional methods, it is possible that some states found in the capture basin cannot reach the target T. In other words, the set found for the capture basin corresponds to an over-approximation of the real capture basin set. This problem is avoided thanks to the interval analysis and the guaranteed numerical integration. Indeed, the result given by the algorithm in this paper is two sets C -and C + such that C -⊆ C ⊆ C + ; (2) Numerical methods generally do not allow the designer to specify up front the class of control inputs with respect to which the capture basin is to be found. In our algorithm, it is possible to specify, exactly, the desired controls ; (3) Our algorithm used with a low accuracy may very quickly eliminate a large portion of the space, not included in the capture basin, to be explored even with a rather large number of parameters.

The paper is organized as follows. Section 2 recalls some results related to continuous dynamical system and capture basin results. The algorithm to be given in the sequel is based on interval analysis and guaranteed numerical integration of ODE. Then, the main validated techniques are described and referenced briefly in Section 3. In Section 4, we present the new algorithm of capture basin approximation, using interval analysis and in particular the guaranteed numerical integration. An application to a nonlinear dynamical system is provided in Section 5. Finally, we discuss the results and draw some perspectives.

Capture basin

We consider a control system, defined by the differential equation

ẋ(t) = f (x(t), u(t)) (1)
where x(t) ∈ R n be the state vector of the system and u(t) ∈ U be the control vector. We shall assume that the set U ⊆ R m of control values is compact, the function f : R n × U → R n is locally Lipschitz on R n . Associated to the differential equation ( 1), we define the flow map

ϕ (t ; x 0 , u) = x(t), (2)
where x(t) denotes the solution to (1) with the initial condition x(0) = x 0 and the control function u ∈ U, where U = {u : [0, t] → U | t ≥ 0 and u is piecewise continuous} denotes the set of admissible controls. Given X 0 a set of possible initial values x 0 , the reachable set of system (1) at the time t is

ϕ (t ; X 0 , U) = {ϕ(t ; x 0 , u) | ϕ(0 ; x 0 , u) = x 0 and ϕ : [0, t] × X 0 × U → R n is a solution of (1) for some u ∈ U}. (3)
The path from t 1 to t 2 is defined by

ϕ ([t 1 , t 2 ] ; X 0 , U) = {X ⊆ R n | ∃t ∈ [t 1 , t 2 ], X = ϕ(t, X 0 , U)} . (4)
Let K ⊆ R n be a constraint set and T (the "target") be a compact set in K. The capture basin C is the subset of initial states of K from which there exists at least one solution of (1) inside K reaching the target T in finite time t :

C = {x 0 | ∃t ≥ 0, ∃u ∈ U, ϕ(t; x 0 , u) ∈ T and ϕ ([0, t] ; x 0 , u) ⊆ K} . (5)

Interval Analysis and Guaranteed Numerical Integration of ODE

Interval analysis for ordinary differential equations was introduced by Moore [START_REF] Moore | Interval Analysis[END_REF] (See [START_REF] Nedialkov | Validated solutions of initial value problems for ordinary differential equations[END_REF] for a description and a bibliography on this topic). These methods provide numerically reliable enclosures of the exact solution of differential equations.

Interval and Interval vectors

Interval analysis usually considers only closed intervals. The set of these intervals is denoted by IR. An interval is usually denoted using brackets. An element of an interval [x] is denoted by x. An interval vector (boxes

) [x] of R n is a Cartesian product of n intervals. If [x] = [x 1 , x 1 ] × . . . × [x n , x n ] is a box, then its width is w([x]) = max 1≤i≤n w([x i ]), (6)
where w([

x i ]) = x i -x i .
The set of all boxes of R n is denoted by IR n .

Inclusion functions

The main concept of interval analysis is the extension of real functions to intervals, which is defined as follows.

Inclusion function Let f : R n → R m be a continuous real function, and [f ] :

IR n → IR m be an inclusion function. Then [f ] is an inclusion function of f if and only if for every [x] ∈ IR n , {f (x) | x ∈ [x]} ⊆ [f ]([x]).
Hence, an interval inclusion allows computing enclosures of the image of boxes by real functions. It now remains to show how to compute such inclusions. The first step is to compute formally the interval extension of elementary functions. For example, we define [x, x] + [y, y] := [x + y, x + y]. Similar simple expressions are obtained for other functions like -, ×, ÷, x n , √ x, exp, . . . This process gives rise to the so-called interval arithmetic (see [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF]).

Then, an interval inclusion for real functions compound of these elementary operations is simply obtained by changing the real operations to their interval counterparts. This interval inclusion is called the natural extension.

Inclusion flow Let [t 1 , t 2 ] ∈ IR be a time interval (with t 1 ≤ t 2 ), let [u] ∈ IR m be an interval vector of controls, and let [x 0 ] ∈ IR n be an interval vector of initial states. [ϕ] is an inclusion flow for

ϕ if it satisfies (i) [ϕ] ([t 1 , t 2 ] ; [x 0 ], [u]) ⊇ ϕ ([t 1 , t 2 ] ; [x 0 ] , [u]) = {ϕ(t ; x 0 , u) | t ∈ [t 1 , t 2 ], x 0 ∈ [x 0 ], u ∈ [u]}. (ii) If w([t 1 , t 2 ]) → 0, w([u]) → 0, w([x 0 ]) → 0 then w ([ϕ]([t 1 , t 2 ], [x 0 ], [u])) → 0.
Remark Let [t 1 , t 2 ] be a degenerated interval (i.e. t 1 = t 2 ). In the sequel, we denote by the real number t the degenerated interval [t 1 , t 2 ].

Usually the computation of the inclusion flow is based on the Banach fixed-point theorem and the application of the Picard-Lindelöf operator (see [START_REF] Berz | Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models[END_REF][START_REF] Nedialkov | Validated solutions of initial value problems for ordinary differential equations[END_REF] for details). It is also possible to obtain a tighter enclosure for the inclusion function [ϕ] (t; [x 0 ], [u]) based on higher order Taylor series expansions. The enclosures thus obtained are said validated which is in contrast with conventional numerical integration techniques which derive approximations with unknown global error and where the accumulation of both truncation and roundoff errors may cause the computed solution to deviate widely from the real one. Given a bounded set E of complex shape, one usually defines an axis-aligned box or a paving, i.e. an union of non-overlapping boxes, E + which contains the set E : this is known as an outer approximation of it. Likewise, one also defines an inner approximation E -which is contained in the set E. Hence, we have the following property

E -⊆ E ⊆ E + . (7)

Approximating the Capture Basin

This section presents an algorithm able to provide an inner and an outer approximation of the capture basin (see ( 5)). That is, we look for two sets, denoted C -and C + , such that

C -⊆ C ⊆ C + .
Theorem 4.1. Consider the control system (1) with x(0) = x 0 ∈ R n be an initial condition. Let T ⊆ K ⊆ R n be a target and let C be the capture basin of T in K ⊆ R n . Then, the following statements hold

(i) x 0 ∈ T ⇒ x 0 ∈ C ; (ii) x 0 / ∈ K ⇒ x 0 / ∈ C ; (iii) ∃t ≥ 0, ∃u ∈ U, (ϕ(t; x 0 , u) ∈ C ∧ ϕ([0, t]; x 0 , u) ⊆ K) ⇒ x 0 ∈ C ; (iv) ∃t ≥ 0, ∀u ∈ U, (ϕ(t; x 0 , u) / ∈ C ∧ ϕ([0, t]; x 0 , u) ∩ T = ∅) ⇒ x 0 / ∈ C. Proof 1.
By definition of the capture basin (see ( 5)), T ⊆ C. This proves (i).

(ii) is immediate since

C ⊆ K. 3. Suppose x 0 / ∈ C, then, from Equation (5) it follows ∀t ≥ 0, ∀u ∈ U, ϕ(t; x 0 , u) / ∈ T ∨ ϕ([0, t]; x 0 , u) ⊆ K. ( 8
)
Then, from the definition of the capture basin given by equation ( 5) : if ∀t ≥ 0, ∀u ∈ U, ϕ(t; x 0 , u) / ∈ T then it means that it never reaches C, otherwise it could reach T, this implies that ∀t ≥ 0, ∀u ∈ U, ϕ(t;

x 0 , u) / ∈ T ⇒ ϕ(t; x 0 , u) / ∈ C. (9)
Thus, x 0 / ∈ C ⇒ ∀t ≥ 0, ∀u ∈ U, ϕ(t; x 0 , u) / ∈ C ∨ ϕ([0, t]; x 0 , u) ⊆ K is true. Therefore the contrapositive (iii) is also true. 4. Assume x 0 ∈ C and obtain a contradiction. Then, from Equation (5), we have

x 0 ∈ C ⇒ ∃u 1 ∈ U, ∃t 1 ≥ 0, ϕ(t 1 ; x 0 , u 1 ) ∈ T. (10)
By the assumption in (iv), there is a t 2 ≥ 0 such that ϕ(t 2 ; x 0 , u) / ∈ C = ∅ and ϕ([0, t 2 ]; x 0 , u) ∩ T = ∅ for all u ∈ U. We fix such a t 2 and set u = u 1 , then ϕ(t 2 ; x 0 , u 1 ) / ∈ C ∧ ϕ([0, t 2 ]; x 0 , u 1 ) ∩ T = ∅. Since, ϕ(t 2 ; x 0 , u 1 ) / ∈ C and ϕ(t 1 ; x 0 , u 1 ) ∈ T this implies, according to definition of the capture basin (see Equation ( 5)), [START_REF] Deffuant | Approximating viability kernels with support vector machines[END_REF] and proves (iv). In the sequel, we consider set-membership counterpart of Theorem 4.1. That is, the vector x 0 is replaced by a set X 0 . The interest to considering sets of values rather than punctual values is, on the one hand, to accelerate the algorithm presented in the following. On the other hand, to use the results of the interval analysis in order to obtain guaranteed results especially thanks to the guaranteed numerical integration. Theorem 4.2. Consider the control system (1). Let T ⊆ K ⊆ R n be a target and let C be the capture basin of T in K ⊆ R n . Let X 0 ⊆ R n be a set of initial conditions and let U be a set control values. Then, the following statements hold :

t 2 ≥ t 1 ⇒ [0, t 1 ] ⊆ [0, t 2 ]. Thus, [0, t 1 ] ⊆ [0, t 2 ] ⇒ ϕ([0, t 1 ]; x 0 , u 1 ) ⊆ ϕ([0, t 2 ]; x 0 , u). Clearly, since ϕ([0, t 2 ]; x 0 , u 1 ) ∩ T = ∅, then ϕ([0, t 1 ]; x 0 , u 1 ) ∩ T = ∅. This contradicts
(i) X 0 ⊆ T ⇒ X 0 ⊆ C (ii) X 0 ∩ K = ∅ ⇒ X 0 ∩ C = ∅ (iii) ∃t ≥ 0 ∃u ∈ U, (ϕ(t; X 0 , u) ⊆ C ∧ ϕ([0, t]; X 0 , u) ⊆ K) ⇒ X 0 ⊆ C (iv) ∃t ≥ 0, (ϕ(t; X 0 , U) ∩ C = ∅ ∧ ϕ ([0, t]; X 0 , U) ∩ T = ∅) ⇒ X 0 ∩ C = ∅
Proof The proof is straightforward, starting from Theorem 4.1 : it suffices to consider the set-membership counterpart of the flow map (see ( 3)-( 4)).

Algorithm

In order to discretize the continuous problem described in Theorem 4.2, we use a union of non-overlapping boxes in which is included the set K. This union of non-overlapping boxes corresponds to the set denoted C + (see Figure 3). Since, the sets considered in the algorithm are pavings boxes, it possible to apply the results of the guaranteed numerical integration of ODE (see Section 3).

The algorithm presented in this section proceeds in several steps, progressively defining the capture basin approximation. In this illustration, the goal is to characterize the capture basin C of the target T in K. The target corresponds to a bottom of the hill with a small positive or negative velocity. More precisely, we are looking for all the initial states (x 1 , x 2 )

+ , such that C -⊆ C ⊆ C + foreach box [x 0 ] in C + do 1 if [x 0 ] ⊆ T then 2 C -:= C -∪ [x 0 ]; else if [x 0 ] ∩ K = ∅ then 3 C + := C + \[x 0 ] else take t ∈ R + and u ∈ [u] ; 4 if [ϕ](t; [x 0 ], u) ⊆ C -and [ϕ]([0, t]; [x 0 ], u) ⊆ K then 5 C -:= C -∪ [x 0 ] else if [ϕ](t; [x 0 ], [u]) ∩ C + = ∅ and [ϕ]([0, t]; [x 0 ], [u]) ∩ T = ∅ then 6 C + := C + \[x 0 ] end end
T in K from which there exists at least one solution of (11) inside K reaching the target T = [3.5, 4, 5] × [-1, 1] in finite time. The final characterization of the capture basin is presented Figure 6. We observe two kinds of pavings : the paving C -that corresponds to the inner approximation of the capture basin and paving C + which is the outer approximation of the capture basin. In this example, to increase the accuracy, we have implemented adaptive size paving. The C++ code of the algorithm as well as movies illustrating our numerical experiments can be downloaded at http://www.istia.univ-angers.fr/ ~lhommeau/capturebasin.html.

Conclusion

In this paper, we have introduced a new interval-based method to characterize an inner and outer approximation of the capture basin. Our method differs from the other approaches, because we have no special assumption on the dynamical system. Moreover our algorithm provides guaranteed results. A testcase illustrating the efficiency of the approach has been treated on the "car on the hill" problem. In the near future, we are planning to extend this approach to other viability problems, such as computing the viability kernel.

Figure 1 .

 1 Figure 1. Illustration of the guaranteed enclosures of the exact solution of differential equation. The box [ϕ] ([0, t]; [x0], [u]) is an enclosure for the flow ϕ and the box [ϕ] (t; [x0], [u]) is an enclosure of the solution of the differential equation at time t

Figure 2 .

 2 Figure 2. Illustration of the statements of Theorem 4.1. Figure (a) : implication (iii).Figure (b) : implication (iv).

Figure 3 .

 3 Figure 3. Set C + is a union of non-overlapping boxes

Figure 5 .

 5 Figure 5. Car on the hill

Figure 6 .

 6 Figure 6. Representation of the capture basin of the "car on the hill" problem. The light grey zone, C -, is proved to be included in C ; the dark grey zone corresponds to K \ C + .

  The algorithm works as follows. Initially, the initial set C + corresponds to the union of boxes covering the set K, that is K ⊆ C + . The principle of the algorithm is simple : for each box of C + it suffices to check the conditions of Theorem 4.2. If the conditions (i) (step 2 of the algorithm) or (iii) (step 5) hold, then the box is added to the set C -. Note that, the condition (iii) can be carried out for several t and several control u ∈ [u]. Conversely, if the conditions (ii) (step 3) or (iv) (step 6) hold, this means that the box is not included in the

	end
	Algorithm 1: CAPTURE BASIN APPROXIMATION
	Comments :

Copyright c 2002 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2002; 00:1-6 Prepared using acsauth.cls

capture basin. Then, we remove the box from the set C + . Finally, the result of the algorithm is depicted on Figure 4. It is important to note that, all along the algorithm, we have the inclusion C -⊆ C ⊆ C + .

Illustration

To illustrate our approach, we have performed numerical experiments by considering the "car on the hill" problem [START_REF] Moore | Variable resolution dynamic programming: Efficiently learning action maps in multivariate real-valued state-spaces[END_REF], [START_REF] Munos | Variable resolution discretization in optimal control[END_REF]. In this problem a car starts at the bottom of the hill and is supposed to reach the target on the other side of the hill. The difficulty is that gravity is stronger than the car's engine, and even at full throttle the car cannot accelerate up the steep slope. Then, to reach the target, the only solution is to first move away from the target and up to the opposite slope on the right. Then, by applying full throttle the car can build up enough inertia to carry it up the steep slope even though it is slowing down the whole way. We assume that the car stay on the hill ground. At a given time, the state of the car is determined by two scalar parameters : the position of the car on the hill (measured by the path length) s and its velocity ṡ as shown in Figure 5. The control of the car motion is given by acting on the acceleration s. The shape of the hill is given by the function g : s → θ, s → (1.1 sin(1.2s) -1.2 * sin(1.1s))/2. Calling s = x 1 and ṡ = x 2 , we obtain a control system in standard form :


Jörn Franke
email: joern.franke@sap.com

Adam Widera
email: adam.widera@ercis.com

François Charoy
email: charoy@loria.fr

Bernd Hellingrath
email: bernd.hellingrath@ercis.com

Cédric Ulmer
email: cedric.ulmer@sap.com

Reference Process Models and Systems for Inter-Organizational Ad-Hoc Coordination -Supply Chain Management in Humanitarian Operations

Keywords: humanitarian, logistics, ad-hoc, coordination, process, reference, management, system

published or not. The documents may come

INTRODUCTION

Responding to a disaster introduces new challenges in comparison to day to day work or routine emergencies. The situation can be very dynamic and new tasks have to be performed as well as organizations have to collaborate that have not worked together before. This is particularly challenging in supply chain operations in sudden-onset disasters, where complex processes meet a complex situation. Such a scenario cannot be solved by one approach, but has to be seen as the integration of several approaches. In this paper we present two promising process-based approaches for coordination and collaboration between autonomous humanitarian aid Reviewing Statement: This short paper has been fully double-blind peer reviewed for clarity, relevance and significance.

organizations. The two approaches need to be integrated to address complex situations as described before.

The paper is organized as follows: In the next section, we describe related work. A reference model supporting a common understanding of activities and dependencies in supply chain operations in sudden onset disaster and a discussion of its application as a basis for collaboration are presented in section three. The reference task model was developed based on interviews with end users, established modeling techniques, and supply chain management requirements in relief operations. Afterwards, we describe a system that allows ad-hoc coordination of activities and dependencies within and between organizations in section four. We discuss in section five the integration of both approaches. Finally, we give an outlook on future research in section six.

RELATED WORK

We have analyzed several process management systems for operational routine emergency response processes (Franke et al. 2010a). However, these types of processes are different from disaster response processes. Thus, these technologies have limited coordination capabilities in more dynamic scenarios. Enterprise Resource Planning (ERP) packages can also be used for coordinating the response from a resource point of view, but not from an activity point of view. [START_REF] Blecken | Supply Chain Management Software for Humanitarian Operations: Review and Assessment of Current Tools[END_REF] describe requirements for ERP systems for supporting humanitarian logistics. Our approaches and ERP systems can be seen as complementary, because ERP systems focus on the resource perspective and we focus on the activity perspective. Based on our interviews and literature research (Blecken 2010; Franke2010a), we identified two main research problems for coordination between different organizations in humanitarian supply chains and disasters. The first research problem is that there needs to be a common ground on activities and their dependencies. The second research problem is that the coordination needs to be adapted ad-hoc based on the situation and coordination efforts (e.g. who is doing what) needs to be communicated ad-hoc to any relevant stakeholder. In the following sections, we describe two approaches that address these requirements and we explain how the integration of both can lead to further benefits.

REFERENCE MODEL

Humanitarian logistics significantly contribute to the effectiveness and efficiency of humanitarian operations (Tomasini and van Wassenhove 2009). Especially the immediate emergency response phase after a suddenonset disaster causes particularly stiff requirements for inter-organizational coordination. Coordination of logistics activities in humanitarian supply chains have been already partly enhanced in the recent years. Some humanitarian actors installed shared equipment or globally pre-positioned stocks, for instance. Yet, cooperation and coordination of humanitarian organizations is still one of the major challenges in the humanitarian sector (Blecken, Hellingrath, Dangelmeier, Schulz 2009; Blecken 2010). In order to manage the high degree of complexity involved in the logistics and supply chain management of humanitarian operations, an application of the reference task model for humanitarian supply chain management developed by [START_REF] Blecken | Humanitarian Logistics: Modelling Supply Chain Processes of Humanitarian Organisations[END_REF] in a larger empirical study (interviews with experts) could be a first step to solve these shortcomings (Blecken, Hellingrath, Dangelmeier, Schulz 2009). The reference task model provides a tool for humanitarian organizations to rapidly visualize the tasks carried out by the organization and its supply chain partners. By clarifying roles, responsibilities, and definitions the reference task model is able to support the standardization of these tasks. By using the approved business process modeling language Business Process Modelling Notion (BPMN) a standardization of the formal framework and each organization-specific process models is enabled (cf. (Object Management Group 2008)). In combination with an adequate adjustment of the BPMN specification by Blecken, the model provides generality, reusability, modularity, adaptability, and simplicity in order to standardize humanitarian supply chain processes. It gives humanitarian organizations the possibility to communicate their tasks and processes with their supply chain partners. Thus, coordination and cooperation of humanitarian organizations is eased.

In the following it is illustrated how the reference process initiated by the task "Identify Demand" could be used for two different organizations. The first pool represents the local center and the operational assessment task ("Identify Demand") of the relief organization A, which is exclusively responsible for the assessment and coordination during disasters. The start event is triggered by an information about a crisis occurred, which initiates the needs assessment. After deploying an emergency and exploratory team, local capacities, resources, and sources of supply as well as the number and needs of beneficiaries are identified. Based on the assessment, the required needs are able to be prioritized and can be communicated to other organizations involved in the relief operation. The process "Identify Demand" is completed at that stage. Next tasks of organization A have to be initiated by other dependent tasks (e.g. coordinate relief operation), specific events such as actions of other relief organizations (e.g. arrival of new organizations or delivery of goods in affected area) or changed circumstances (e.g. aftershocks). Based on these standardized processes and its descriptions, different optimizations are possible. In the case of the task "Identify Demand" for example, organization A seems to have a core competency in assessment tasks as it is exclusively responsible for the assessment and coordination during disasters. Thus, a concentration on the deployment of an exploratory team seems advantageous. As an organization B aims to provide relief goods to the beneficiaries, it appears appropriate to focus on the deployment of the emergency team. The shared task "Identify Demand" should be finalized by merging the results of both exploratory and emergency team in order to prioritize needs. The following figure illustrates this procedure.

TOOL SUPPORT FOR INTER-ORGANIZATIONAL AD-HOC COORDINATION

Currently, telephone, e-mail or fax, are used to coordinate during a disaster response within the organization, but also with other organizations. This means that it is very difficult to track the current status of activities or to detect conflicting views on the situation (e.g. an activity is perceived as canceled by the command center, but it is perceived as failed by the team in the field). Furthermore, every time the status of an activity changes, all relevant stakeholders of an activity have to be found from scratch in a huge pile of messages. We developed a system for ad-hoc coordination for activities within and between different organizations (see Franke2010a;Franke2010b) . It originates in end user needs within the SoKNOS project [START_REF] Döweling | Soknos -An Interactive Visual Emergency Management Framework[END_REF]). It is based on the experience made in this project and on interviews we conducted with disaster managers. The system allows amongst others:

 Modeling of activities with temporal dependencies (e.g. starts, meets, finishes or overlaps)

 Sharing of activities between different organizations so that they can integrate them into their processes. This is illustrated in Figure 5, where the commander of organization A shares the activity "Identify needs" with the commander of organization B. The commander of organization B can integrate it into his process and establish a dependency to the activity "Order Goods".  Detection of violation of dependencies when executing activities and highlighting violations to the users. They can then resolve the issue together with other stakeholders of the activities.

INTEGRATION

In this section, we illustrate how the reference model and an implementation of the ad-hoc coordination concepts in a tool can be integrated. Basically, the stakeholders use the terminology and concepts of the reference model in the tool. Figure 6 illustrates a screenshot of our prototype. More precisely, it shows the activity workspace of the commander BETA (Bob) of organization B, who describes the coordination of parts of the reference process illustrated in Figure 2. In the upper part we see that the activity "Identify needs" has been shared with Bob. Bob can now decide to integrate it in his managed activities. He may also create dependencies to his own activities. Bob's currently managed activities are illustrated as a table in the lower part. The activities and dependencies may also be visualized differently (e.g. as a timeline). Bob can publish state changes of an activity to all stakeholders with whom an activity is shared. In the bottom left box, we can see the Bob's social network in the contact list. He can share activities with these people. He may add further contacts to his list. The tool supports also a graphical modeling notation as it has been described in Figure 5. Depending on the number of activities and the situation, one can have advantages over the others. For example, when there are many activities with few dependencies, then the table-based view seems to have advantages. On the other hand, the graphical notation has advantages when there are few activities with many dependencies. We are currently exploring how to automatically generate a graphical time-line based on information in the table to overcome the limitations of using individual views.

CONCLUSION AND FUTURE WORK

We presented here a reference process model for ensuring a shared common understanding on what needs to be coordinated by whom. This shared understanding facilitates coordination among different stakeholders in interorganizational humanitarian logistic processes. An ad-hoc coordination tool has been presented that allows coordinating ad-hoc an inter-organizational humanitarian supply chain logistic process. This tool allows designing novel logistic processes with organizations that may never have collaborated before. The common understanding provided by the reference model in combination with the tool enables a more efficient and effective inter-organizational humanitarian supply chain. Although a shared understanding of what needs to be coordinated exists, the real situation may deviate from what is prescribed in the model. This can be detected by the system and highlighted to the user, so that further actions can be performed. In the future, we want to focus on the evaluation and extension of the reference task model and investigate how we can integrate the reference model and the ad-hoc coordination tool technically. We also want to investigate how to design a repository for reference processes that can be leveraged during a disaster. For example, it may give guidance what one can do next or what is needed to do something else. Furthermore, we need to investigate proper visualization mechanisms (e.g. list, graphs, timelines or maps). We think the next important steps for crisis management technology is not only to provide innovative technology, but also content, so that it can be easily deployed and used by crisis managers. This paper provides a first step into this direction with a reference process model providing the content and an ad-hoc coordination concept as an innovative technology for coordinating an interorganizational humanitarian supply chain.

Figure 1 :

 1 Figure 1: Identify Demand by Organization A (acc. to Blecken 2010)

Figure 2 :

 2 Figure 2: Shared Tasks of "Identify Demand"

Figure 3

 3 Figure 3 Sharing of Activities and their Current State

Figure 4

 4 Figure 4 Screenshot from our Ad-Hoc Coordination Tool

Proceedings of the 8 th International ISCRAM Conference -Lisbon, Portugal, May 2011

ACKNOWLEDGMENTS

The research was partially funded by the French Ministry of Research within the RESCUE-IT project (Schaad et al. 2010).