Pascal Da Costa
email: pascal.da-costa@centralesupelec.fr.

Wenhui Tian

2010-2050

Keywords: Q47, Q54 STIRPAT Model, Support Vector Regression, Energy Transition, Sectoral Emission Modeling, Technology Roadmaps

est destinée au dépôt

Introduction

In 2010, the global temperature had increased by 0.85°C compared to the pre-industry level according to the fifth assessment report by Intergovernmental Panel on Climate Change (IPCC, 2013). With the continuation of this growth rate, the global mean surface temperature will increase by 3.7°C to 4.8°C in the end of this century (IPCC, 2014). In this context, several countries have proposed their targets in reducing the CO 2 emissions, which is the most important gas among the Greenhouse Gases (GHG). In this article, our objective is to assess these policy targets by evaluating the technology roadmaps, in the period of energy transition, from 2010 to 2050.

Several families of applied models co-exist with strong differences in terms of decompositions (sectorial, regional, fiscal, etc.), theories used (endogenous or exogenous growth, market's structures, etc.) and long or mid-term perspectives [START_REF] Chen | The costs of mitigating carbon emissions China: findings from China MARKAL-MACRO modeling[END_REF][START_REF] Klaassen | Internalizing externalities of electricity generation: An analysis with MESSAGE-MACRO[END_REF][START_REF] Saveyn | Economic analysis of a low carbon path to 2050: A case for China, India and Japan[END_REF]. However, mechanisms and assumptions of the existing applied economic models can often be very opposed which makes it very difficult to compare the results and well understand the numerous differences in predictions [START_REF] Boulanger | Models for policy-making in sustainable development: The state of the art and perspectives for research[END_REF]. The complex models (MARKAL, AIM, NEMS, etc.) make a full coverage of the energy sectors and energy flues, but they also require large amount of exogenous inputs and complex structures with limited access.

Different from these existing models, we propose a less complex (less data required, simple framework, etc.) but complementary approach: our Sectoral Emission Model with three main energy sectors (power, transport, and the others, including residence and industry: which account for about more than 80% of the carbon emissions) by studying the main energy related technologies: energy mix in power generation, the Carbon Capture and Storage (CCS), electric vehicles and energy efficiency. In our work, we apply the model for three types of country that may be considered to be representative of numerous other similar countries in the world: China (CN), a fast-emerging economy that requires increasing energy consumption, and the largest emitter of CO 2 ; France (FR), a well-developed economy with relatively low CO 2 emissions; and the United States (US), the largest economy and a major source of CO 2 emissions. All possible solutions for the technology pathways can be generated with our Sectoral Emission Model, offering the policy makers choices of technology transition solutions according to their different criteria. Besides the solution pool of the technology roadmaps, here we provide two more: under the assumption of "balanced technology development" across sectors what refers to the same improvement of each technology in reducing CO2 emissions ; and under that of "least changed energy mix" that refers to the minimization of the difference between the energy mix in 2050 and in 2010 for each country. In addition, the sensitivity of the model with parameters under these two roadmaps is tested.

The results of the model show that governmental targets in France and in the US require all sectors make large efforts in reducing CO 2 emissions. In France, two thirds of emission reductions are expected to be conbtributed by the transport sector. 80% of vehicles should be changed to electric vehicles for reaching its governmental target. Meanwhile, the energy efficiency should also be improved by 80%. As the power sector contributes little to the CO 2 emission, CCS is not a necessary option for France. In the United States, all sectors should make important contribution for reaching the governmental target. At least 80% of cars should be changed into electric cars, and the energy efficiency should be improved by at least 60%. In the power sector, the use of coal in 2050 can be kept at most half of that in 2010. However, for the governmental target in China, the advancement of technologies is less demanded than in France or in the US. In China, half of the emission reductions are expected to be contributed by the power sector, as the power sector accouts for half the the emissions in 2010. In order to achieve its governmetal target, 60% of vehicles should be replaced by electric vehicles, energy efficiency should be improved by 60%, and coal utilization should be reduced by 60%. If the energy mix is expected to be kept quite unchanged in China and in the US, the CCS should be implemented to all power plants to reach the goals. Since CCS technology is estimated to absorb about 90% of the CO 2 emissions (IPCC, 2005). However, this technology is not yet largely applied in the power sector and industry considering of security and high costs. Meantime, the tests of sensitivities of parameters in the model for different technology roadmaps in different countries show that the electricity output and the emission intensity of production are the two parameters with the most important sensitivity on CO 2 emissions. Thus, improving the energy efficiency of coal combustion and of electricity production will play an important role in the emission reductions.

The remainder of this paper is organized as follows: Section 2 introduces the structure of the Sectoral Emission Model. Section 3 presents the data for CO 2 emission objectives. Section 4 then explains the results obtained from the model according to different countries. Conclusions are drawn in the section 5.

The Sectoral Emission Modeling

The Sectoral Emission Model is proposed to evaluate the scientific relevance and feasibility of the CO 2 reduction target with respect to the population, economy and CO 2 emissions between 2010 and 2050, under the consideration of energy technology transition. Three sectors are considered: (i) electricitygeneration, (ii) transport, and (iii) other, which include principally the residence and industry sector.

The Context

The figure 1 presents the CO 2 emissions and per capita CO 2 emissions of the representative three countries from 1971 to 2010. From 1971 to 2005, the US was the biggest sources of CO 2 emissions. After 2006, China took the lead place, with an annual growth rate of 9%. Different from these two countries, the emissions in France decreased slightly. As shown by the level of per capita emissions, the US have the highest value at 17t in 2010, which slightly decreased in the last few years. In France, the per capita emissions decreased from 8.5t (in 1971) to 5.5t (in 2010). While in China, the per capita emission increased from less than 1t in 1971 to the same level as France in 2010. The table 1 shows that these sectors cover more than 80% of the emissions from fuel combustions in the three countries. The power sector is the most important emitter in China and in the US, due to the large amount of fossil fuels combustion. However in France, this sector accounts for only 15% of the total emissions, as about 80% of the power is generated by nuclear plants. The transport sector is also an important emitter, which accounts for 35% of emissions in France, and 30% in the US. Even though the transport sector accounts for only 7% of CO 2 emissions in China in 2010, the emissions from this sector will grow rapidly as a result of the rapid growth of car consumptions. The residence and industry sectors represent 36% of emissions in China, primarily from the manufacturing industries and constructions. With the fast urbanization, the energy consumption will rapidly grow in the residence sector. The residence and industry sectors account for 33% in France, with an equal contribution for each sector to the emissions, comparing with 16% for the US. From table 1, we can observe that the emissions are distributed almost homogenously across sectors in France. Thanks to the low emissions in the power sector, the consumption of electricity due to the development of other sectors will not add supplementary emissions to the power sector. The energy related technologies considered in this article are taking the power and transport sectors as the principal sources of CO 2 emissions. Energy mix, CCS, electric vehicles and energy efficiency are the key parameters in reducing CO 2 emissions.

The Model Framework

The Sectoral Emission Model focuses on three types of sectors: the power sector, the transport sector, and the residence and industry sector. For the power sector, the electricity could be produced from different energy sources. The sources that produce CO 2 emissions are mainly fossil fuels, namely coal, oil and gas. The sources which produce few emissions, called clean energy, are renewable energies and nuclear energy. In the transport sector, we focus on the road transport, which generally accounts for more than 80% of the transport sector emissions. And in the other sectors, we study mainly the residence and industry sectors. The structure of the model is presented in the figure 2 [START_REF] Tian | The economics of the CO2 emission and mitigation modeling: A study for China, USA and France[END_REF]Da Costa, 2013, 2014 As shown in the schema above, the total CO 2 emissions are the sum of the emissions from power sector, transport sector, and the other sectors:

𝐸 (!) = 𝐸 !(!) + 𝐸 !(!) + 𝐸 !(!) (1)
where 𝐸 (!) is the total CO 2 emissions from fuel combustions in year t, and 𝐸 !(!) , 𝐸 !(!) and 𝐸 !(!) are CO 2 emissions in the corresponding three sectors in year t.

The technologies to be studied in the model are shown in dotted line in the figure 2. We principally analyze the power and transport sectors here, in which fuel mix, CCS and electric vehicles are three key factors behind CO 2 emissions. Improved energy efficiency in the domestic and industrial sector also contributes to CO 2 emission mitigation. The parameters and variables in the modeling will be explained according to sectors, in the flowing sub-section.

The power-generation sector

In the power sector, we employ the IPAT identity methodology to study the driving forces of the CO 2 emissions in producing electricity from the fuel combustions. The IPAT identity was developed as a general approach for discussing the driving forces behind environmental impacts, which relates impacts (I) to population (P) multiplied by affluence (A) and technology (T):

𝐶𝑂 ! 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 * !"# !"#$%&'(") * !" ! !"#$$#%&$ !"# (2)
The IPAT identity had been developed later into Kaya identity [START_REF] Kaya | Impact of carbon dioxide emission conrol on GNP growth: interpretation of proposed scenarios[END_REF] which is widely used for CO 2 emissions in many studies, such as the IPCC (2007) assessment report, and the IEA (2008) Energy Technology Perspective report, etc. According to this identity, the CO 2 emissions can be decomposed into the product of three basic factors, carbon intensity of energy, energy intensity and affluence:

𝐶𝑂 ! 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 * !"# !"#$%&'(") * !"#$%& !"#$%&'()"# !"# * !" ! !"#$$#%&$ !"#$%& !"#$%&'()"# (3)
Inspired from the IPAT identity and Kaya identity, we decompose the CO 2 emissions in the power sector into the product of the output of electricity and the technology (i.e. the CO 2 emission intensity of production). And the fossil fuels, such as coal, oil and gas, are the three main sources contribute to CO 2 emissions in the power sector. Analytically, its emissions in year t are divided into these three categories as follows:

𝐸 ! ! = 𝑄 ! * 𝑥 !,! * 𝑒 !,! * ϵ ccs (4)
where 𝑄 ! is electricity output in year t, 𝑥 ! presents the three main fuels: coal, oil and natural gas; equally 𝑒 ! is the CO 2 emissions from using coal, oil and gas respectively, and ϵ ccs the dummy variable.

In the modeling of the power sector, energy mix and CCS are the two main technologies to be studied. Besides, the emission intensity of production and electricity output are two important factors in the modeling. Support Vector Regressions (SVR) will be then used in order to simulate the projection of electricity output.

The variables and parameters in the power sector will be presented in the following three parts: first, the actual energy mix in the three countries; second, the projection of electricity output; third, the CCS technology.

1) Energy mix

China has abundant coal reserves, while its oil, natural gas and other fossil energy resources are limited. Coal is currently the dominant power fuel. In the end of 2010, thermal power accounted for 73.4% of total power-generation capacity (IEA, 2011). France is one of the least CO 2 intensive industrialized economies, thanks to the substantial role of nuclear power and the existence of higher gasoline taxes with incentive impacts. CO 2 emissions have been declining since 2005 from an already relatively low base. In 2009, nuclear power accounted for 76.24% of France's electricity generation.

The US depends on fossil fuels for almost all its energy supply. Natural gas use is rapidly growing in the US, particularly for the power generation, where it has now overtaken nuclear to become the second most important power-generation fuel. Coal is also an important fuel in the US, accounting for 45% of the country's electricity generation.

As a result of the different energy mix in the power sector, CO 2 emissions per KWh (the CO 2 emission intensity of production) from electricity generation vary greatly across countries. The figure 3 shows that CO 2 emissions per KWh in France are only 12% of the level in China and 20% of the level in the US over 1990 -2010, as coal plays a dominant role in China and the US, while nuclear power plants currently account for 80% of French electricity output.

Fig.3. CO 2 emissions per KWh from electricity generation

The emission intensity of production of each fuel varies widely between countries according to the different types of energy and technology levels, as shown in the table 2. The emission intensities of production of fuels are the lowest in Europe, so we adopt the emission intensities of production in 2010 of Europe as the intensities of production for the three countries in 2050, which are 0.8kg/kwh for coal, 0.4kg/kwh for oil, and 0.2kg/kwh for gas. 2) Electricity output As to the electricity production, we will make the projection with the historical data between 1971 and

China

United States France 2010. We choose the SVR to carry out the simulation, because it is an appropriate model for small databases. SVR has successfully been tested to solve forecasting problems in many fields, such as financial time series forecasting [START_REF] Cao | Support vector machines experts for time series forecasting[END_REF] and electric load forecasting [START_REF] Hong | Application of chaotic ant swarm optimization in electric load forecasting[END_REF][START_REF] Wang | A trend fixed on firstly and seasonal adjustment model combined with the ɛ-SVR for short-term forecasting of electricity demand[END_REF], amongst others. Based on this work, we used SVR to make predictions for electricity production and pollution intensity. 2 SVR is used to provide the underlying function in each country.

There are different kinds of SVR according to the different loss functions and kernel functions employed. We here use the ε-insensitive function and an appropriate kernel function for each country and variable by trial and error. Some critical parameters related to the loss function and kernel function need to be tuned before the training and prediction of the model. Details regarding the tuning of the parameters and kernel functions can be found in [START_REF] Liu | Nuclear power plant components condition monitoring by probabilistic support vector machine[END_REF].

In our work here, the data sets are all normalized from the raw data. We use a sigmoid kernel function for electricity-production prediction.

The Polynomial kernel Function (γ * u • v + coef) !"#$"" is used as the kernel function for electricity output by trial and error. The values of the related hyperparameters are also turned with a Grid Search. The parameters are listed in the table 3: [START_REF] Cortes | Support-vector networks[END_REF][START_REF] Gao | A probabilistic framework for SVM regression and error bar estimation[END_REF]. When SVM is used for regression (especially non-linear regression), it is called a Support Vector Regression (SVR). SVR can estimate the nonlinear relationship between the data and produces good results after mapping the input data into a high-dimensional Reproducing Kernel Hilbert Space (RKHS), as compared to other commonly-used techniques.

3) Carbon dioxide Capture and Storage technology

Besides improving the emission intensities of production of fossil fuels, CCS is considered as one of the potential options for reducing atmospheric CO 2 emissions from human activities. As CO 2 is emitted principally from the combustion of fossil fuels, especially from the large combustion units such as those used for electric power generation, CCS would most likely be applied to large point sources of CO 2 , like the power plants. CCS involves the use of technology, first to collect and concentrate the CO 2 produced in energy-related sources, transport it to a suitable storage location, and then store it away from the atmosphere for a long period of time. CCS would thus allow fossil fuels to be used with low GHG emission.

CCS has not been used in large-scale power plants, so there is relatively little experience with the combination of CO 2 capture, transport and storage in a fully integrated CCS system. The wide range of costs for CCS systems is primarily due to the variability of sit-specific factors, the type and costs of fuel used, the required distances, terrains and quantities involved in CO 2 transport, and the type and characteristics of the CO 2 storage. In most CCS systems, the cost of capture is the largest cost component, in the range of 15-75 US$/tCO 2 net captured from a coal-or gas-fired power plant. The cost of transportation is between 1 US$/tCO 2 and 8 US$/tCO 2 . The cost of storage is 0.5-8 US$/tCO 2 for geological storage and 5-30 US$/tCO 2 for ocean storage.

According to the "Carbon Dioxide Capture and Storage: Technical Summary" of IPCC (2005), if the CCS is equipped with the power plant, about 90% of the CO 2 emissions will be captured and stocked. Thus, we have the dummy variable ϵ ccs to the equation 4, where: ϵ ccs = 0.1, 90% emissions be absorbed with CCS 1, no emissions be absorbed without CCS

The transport sector

The power generation sector makes the most important contribution to the global CO 2 emissions, and the second most important emitter is the transport sector. The transport sector was responsible for approximately 23% of the global energy-related CO 2 emissions in 2010, of which 72% CO 2 emissions come from road transport. Thus, we will focus on the road transport in the transport sector emissions, the most important emission source in the transport sector. Now let's introduce the low-carbon emission vehicle.

1) Road transport vehicles

Reducing global transport GHG emissions will be challenging since the continuing growth in passenger and freight activities. According to IEA statistics, the transportation sector accounted for over 40% of oil demand in 2010. Oil use will become increasingly concentrated in the transportation sector, reaching 65% of total oil demand in 2035, according to the New Policy Scenario of the World Economic Outlook 2011. Thus, automobiles with clean energy sources are encouraged to replace the traditional gasoline and diesel ones.

Hybrid vehicles and electric vehicles are two emerging technologies that manufacturers are increasingly turning towards, especially for the electric vehicles. Hybrid vehicles (conventional hybrids, Plug-In Hybrid Electric Vehicles) combine both an electric motor and a gasoline engine. Electric vehicles (Plug-In Electric Vehicles, Battery Electric Vehicles and Fuel Cell Electric Vehicles) use an electric-only motor but with different energy storage systems. Electric vehicles have no direct tailpipe emissions, the indirect emissions come from charging the vehicle's battery with grid electricity generated by fossil fuels powered power plants. Thus, electric vehicles have a CO 2 eq reduction cost highly correlated to the carbon intensity of electricity generation. However, with the transformation of power sector, the indirect emission will reduce in long-term. In this context, we choose electric vehicles as the option for the technology transition in the transport sector modeling. Along with the advantages of electric vehicles, there are barriers for the adoption, such as high battery costs, willingness of consumers, charging facility and so on. The penetration of the market and the technology advancement need the encouragement of government.

Now we present the situation of transport emissions in the three countries and the modeling method.

2) Road transport emission across the three countries

The transport sector is responsible for the largest share of CO 2 emissions in France (over one third of emissions in 2010), with road transport accounting for 96% of transport emissions. Thanks to its lowcost low-carbon electricity supply, France could be able to reduce transport emissions by focusing on electricity-based technologies, such as high-speed rail and electric vehicles. Actually, there are about 30 000 electric vehicles in France, which account for 0.08% of all the vehicles. The energy transition law ("la loi de la transition énergétique") was adopted on 2014, which announced that the bonus for changing to electric vehicles can be accumulated up to 10 000 euros, and the government will install charging stations all over the France, with the objective to a total number of 7 million in 2030.

In 2010, the US had the most vehicles of any countries in the world (254 million), with transport accounting for 30% of CO 2 emissions, of which road emission was responsible for 86.4%. In 2009, the President of the US pledged US$2.4 billion in federal grants to support the development of nextgeneration electric vehicles and batteries. And as part of the American Recovery and Reinvestment Act, the US Department of Energy announced the release of two competitive solicitations for up to $2 billion in federal funding for competitively awarded cost-shared agreements for manufacturing of advanced batteries and related drive components as well as up to $400 million for transportation electrification demonstration and deployment projects. This initiative aimed to help meet President's goal of putting one million plug-in electric vehicles on the road by 2015. In 2014, nearly 120 000 electric vehicles were sold in the US.

In China, transport accounts for only 7% of total emissions in 2010. With a growth rate in the number of vehicles of 11% in 2010, transport, and especially road transport, will be increasingly important for future CO 2 emissions. Thus, it is critical for China to develop electric vehicles. There were 4 400 electric vehicles up to 2012. In order to encourage the consumers, 5 billion Yuan was allocated as the total allowance for the purchasing of electric vehicles from 2009, and 5 million electric vehicles are expected to be used in 2020.

With current technology, an electric vehicle consumes 0.01 KWh/km to 0.03 KWh/km. Here, we employ the mean value of 0.02 KWh/km, that is 0.73MWh/yr (shown in equation 6), which makes a notable contribution to total electricity output.

The CO 2 emissions of the transport sector in year t are then calculated as:

𝐸 !(!) = ! !"#$(!) ! !"#$ = ! !"#$(!"#") * !!! ! * (!!! ! ) ! !"#$ (5)
where 𝐸 !(!) are CO 2 emissions in the transport sector in year t, 𝐸 !"#$(!) are CO 2 emissions from road transport in year t, is the vehicle growth rate, 𝑦 ! is the proportion of hybrid vehicles in the vehicle stock in year t, and 𝛼 !"#$ is the share of road transport in the CO 2 emissions of the transport sector.

The baseline CO 2 emissions in the road transport will increase from 400mt in 2010 to 1968mt in 2050 in China, due to the fast growth of car numbers. The baseline CO 2 emissions in France will increase from 118mt to 198mt, and from 1400mt to 2170mt in the US.

The use of hybrid vehicles will increase electricity production, as described below:

𝐸 ! !" = 0.73 * 𝑦 ! * 𝑁 ! (6)
where 𝑁 (!) is the stock of vehicles in year t. Total electricity output is therefore:

𝐸 !(!) + 𝐸 ! !"
. For the number of vehicles in 2050, we assume that it will keep increasing at the growth rate in 2010 at about 1% in France and in the US, as their car number growth was at a stable rate 3 . However, because the car numbers were increasing fast in the past few years in China, we assume that the cars numbers will increase first at a rapid rate as in 2010 at 10%, and then this growth rate will progressively decrease to 1% in 2050 4 . The numbers of vehicles in 2050 in these countries are shown in the table 4. In this assumption, the vehicles in China will increase much more than the other two countries, from 114 million to 560 million in 2050, which is at the same level as in 6DS scenario (baseline scenario) in the IEA (2014) "Energy Technology Prospective" report. Thus every ten person will have 4 cars, which corresponds to one vehicle for one family. The car numbers will rise from 38 million to 63 million in France, and from 269 million to 393 million in the US, which means nearly one vehicle for one person.

The domestic and industrial sector

In the Sectoral Emission Model, we primarily focus on the two most important emission sectors, which are power and transport sectors. Amongst the other sectors, we consider primarily domestic and industrial consumption because these two sectors account for most of emissions in the other sectors.

Improving the energy efficiency is the key technology for reducing GHG emissions in the two sectors.

In this section, we will not consider all the energy efficiency technologies into the Sectorial Emission Modeling, but the notion energy efficiency, which is employed to be the representative technology.

1) Energy efficiency related technologies

The energy use and related emissions in the domestic sector will increase, especially in the developing countries, with the increasing need for adequate housing, electricity, and improved cooking facilities.

3 The projections of car numbers in France and the United States are more optimists than in the IEA "Energy Technology Prospective" report in 2014, where the car numbers are assumed to be nearly unchanged in the 6DS (baseline) scenario. However, the car number projections in different studies can be controversial in terms of various assumptions. For example, the personal car numbers in 2050 are projected to be about half of that in 2010 according to the projection of ANCRE. Thus, in this work, we project evolution of the car numbers in these countries following their historical growth trends, without involving other parameters in order to make a simplified and clear assumption. 4 The projection of car numbers in China in 2050 is at the same level than in the IEA "Energy Technology Prospective" report in 2014, in the 6DS (baseline) scenario.

For the industrial sector, despite its declining share in global GDP, the GHG emissions from the industrial sector keep increasing, especially in the developing countries. In 2010, domestic CO 2 emissions accounted for 22.4% of those in the other sectors in the US; in France and China, this figure was 31.8% and 9.6% respectively. Domestic and industrial energy consumption can be reduced by improving energy efficiency.

Energy efficiency is a general notion which means to consume less energy in providing the same service. Many potential technologies are available for improving the energy efficiency. For example, more efficient appliances, smart meters and grids, fuel-switching to low-carbon fuels such as electricity or biomass, more efficient isolation in the buildings, etc. As to the industrial sector, the energy efficiency involves fuel-switching to low-carbon fuels, efficient process heating systems, recycling, etc. For developing countries, there are still many energy efficiency options both for process and system-wide technologies and measures.

CO 2 emissions in the rest sectors other from power sector and transport sector are presented as follows:

𝐸 !(!) = ! !"(!) ! = !!! * ! !"(!"#$%&'$) ! (7)
where 𝐸 !(!) is CO 2 emissions from the other sectors in year t, 𝐸 !"(!) is domestic CO 2 emission, 𝑒 represents the improvement of domestic energy efficiency, 𝐸 !"(!"#$%&'$) is domestic CO 2 emission without taking energy efficiency into account, and β is the domestic share in other sector CO 2 emissions. The baseline domestic CO 2 emissions in China will increase from 303mt in 2010 to 458mt in 2050, and they will increase from 322mt to 430mt in the United Stated. However, the baseline emissions in the domestic will be reduced from 57mt to 39mt in France, because of the decreasing trend of CO 2 emissions in the past few years.

The Data

In this section, we present the governmental target for CO 2 emission reductions. The French government announced a reduction of CO 2 emissions by 75% 5 ("facteur 4") in 2050 compared to level in 1990 according to the "Loi n° 2005-781 du 13 juillet 2005 de programme fixant les orientations de la politique énergétique (POPE)" (ADEME, 2014) which means that the CO 2 emission are expected to be 89mt in 2050.

The US House of Representatives passed the American Clean Energy and Security Act, which aimed to reduce 17% of their CO 2 emissions below the 2005 level in 2020 6 , and 83% in 2050 [START_REF] Waxman | The American Clean Energy and Security Act of[END_REF], which means that their CO 2 emissions are expected to be reduced to 981mt in 2050.

In 2009, China promised to reduce its CO 2 intensity (CO 2 emission per unit of GDP) by 40%-45% in 2020 7 (ERI, 2009) compared to 2005, and this objective is extenable to 85%-90% in 2050. In this work, we adopt the reduction of CO 2 intensity by 90% in 2050, which means the expected CO 2 emissions are 5 259mt in 2050, with the baseline scenario of GDP assumption 8 . 5 In 2012, the CO 2 emissions from the fuel combustions in France were 5.4% less than its 1990 level, according to "Les chiffres clés du climat France et Monde (edition 2015)". 6 In 2013, the GHG emissions were 9% below 2005 level, according to the "U.S. Greenhouse Gas Inventory Report: 1990-2013". 7 Actually, in 2013, the CO 2 intensity had been decreased by 28.5% compared to 2005. According to the "Plan for the Climate Change (2014-2020) (in Chinese) " released in september in 2014 by the Chinese governement, the objective of reducing CO 2 intensity in 2020 was not changed. 8 This CO 2 emission is calculated with the baseline scenario of GDP according to the report "The world in 2050" (HSBC, 2011. The world in 2050: Quantifying the shift in the global economy. HSBC Global Research.). The GDP using Purchasing Power Parities (PPP) in China will be 57784 $billion in 2050, about six times of the 2010 level.

Results for technology roadmaps

The reduction of CO 2 emissions in the model are decomposed into the reductions of sectoral emissions, which means that the reductions of sectors are good substitutes. Thus, there is infinite technology pathways in meeting the scenario objective. Therefore we plan to present a solution pool of technology roadmaps only based on technology development. The energy related technolgies are: the share of coal and gas in the power sector 9 , the share of electric vehicles in the transport and the improvement of energy efficiency in the residence and industry sector. Obviously, the share of coal and gas in the power sector are between 0 and 100%, with their sum inferior to 100%. The share of electric vehicles in all vehicles in the road transport is in the interval of [0%, 100%], and the improvement of energy efficiency in the residence and industry sector is in the interval of [0%, 100%] 10 . In order to avoid the numberless solutions, we make these assumptions: i) the shares of electric vehichles set from 0% to 100% with the interval of 20%; ii) the improvements of energy efficiency in the residence and industry sector are set from 0% to 80% with the interval of 20%.

Then, we discuss two technology roadmaps based on two criteria. One is based on the assumption that the technology development across sectors are developed homogenously. The other is based on the preference of the most use of the energy sources of each country, which means to keep the change of energy mix in the power sector as little as possible.

Technology roadmaps in China

We first present the solution pool of the technology roadmaps for China. Then we present their technology roadmaps with balanced technology development across sectors and the sensitivities of parameters under this technology roadmap. In the last part, we propose the technology roadmaps with least changed energy mix, and also present the sensitivities of parameters under this technology roadmap for China.

1) Technology solution pool in China

In China, half the CO 2 emissions from fuel combustion come from the power sector in 2010, with 78.7% electricity production from the combustion of coal. Thus, the reduction of emissions in the power sector is indispensable. The transport sector contributed only 7% of CO 2 emissions in 2050, but its reduction of emissions can not be ignored due to the rapid growth of car numbers. 9 The share of oil is not presented in this section because its percentage is negligible compared to that of coal and gas, normally under 5%. 10 The improvement of energy efficiency by 100% is not realistic as the emissions in the residence and industry cannot be eliminated perfectly. According to the governmental target, the CO 2 intensity is projected to be reduced by 90% in 2050 with respect to 2005, which means that the CO 2 emissions will be reduced by 28% compared to the level in 2010. Different technology pathways are shown in the figure 8 without CCS and in the figure 9 with CCS, with the dotted lines representing the 2010 level.

If CCS technology is not applied to power plants, more efforts should be made in the transport sector and other sectors. For example, if the share of coal is to be reduced from 78.7% in 2010 to 25% in 2050, and the share of gas is to be incresed from 1.7% to 33%, then 40% electric vehicles should be employed, and 80% energy efficiency should be improved. Otherwise, if 60% of vehicles are replaced by electric vehicles and the energy efficiency is improved by 20%, the power sector must be almost decarbonized.

However, if all the power plants are equiped with CCS, when 60% of vehicles are replaced by electric vehicles and the energy efficiency is improved by 20%, the reduction of coal can be less than without CCS, from 78.7% to 40%, with the share of gas increased to 20%. As the energy technologies in the three sectors are substitutional, there will be infinite possiblities for the technology roadmaps. Among the different technology roadmaps in China in order to reach the governemental targets, we propose two roadmaps based on two criteria: one with balanced CO 2 emissions reduction across sectors, the other with least changed energy mix in the power sector.

In each roadmap, the sensitivities of parameters: the CO 2 emission intensities of production of fossil fuels, the electricity output, and the number of vehicles, in the Sectoral Emission Model will be tested.

2) Balanced technology development roadmaps for China

Under the balanced technology development criteria, the technologies across sectors are supposed to be developped homogenously. The figure 10 presents the roadmaps in this criterion with and without CCS for China in 2050. In order to achieve the governmental targets of reducing 90% of CO 2 intensity in 2050 with balanced technology development across sectors, electric vehciles in the road transport should replace 60% of the traditional automobiles. The energy efficiency in the residence and industry sector should be improved by 60%. Meantime, the share of coal used in the power generation is to be reduced by about 60%, from 78.7% to 25%, thus, the share of gas can be increased from 1.7% to 13% in 2050.

However, if the CCS technology is implemented in the power plants, there will be less direct CO 2 emissions reductions in the transport and other sectors. 35% of vehicles will be replaced by electric vehicles, and the energy efficieny in the residence and industry sector should be improved by 35%, nearly half less than the roadmaps without CCS. In the power sector, the coal combustion will be less reduced, from 78.7% to 55%, and the use of gas can be increased to 30% as gas is cleaner which produces less CO 2 emissions. There are three most important parameters in the modeling: 1) the CO 2 emission intensities of production of fossil fuels, 2) the electricity output, and 3) the number of vehicles. Here, we will test their sensitivities in the two roadmaps above in order to make a robust simulation. The table 5 presents the influences by 1% diminution of the parameters on the total CO 2 emissions for the balanced technology development roadmap in China.

The sensitivities of the five parameters without CCS are more significant than those with the use of CCS. The electricity output has the most important sensitivities among these parameters with or without CCS, as the power sector contributes most to the total CO 2 emissions from fuel combustions.

second most important parameter is the CO 2 intensity of production of coal, because coal is the more used and more polluant than oil and gas. The sensitivity of emission intensity of production of oil is the smallest, as the share of oil in the energy mix is negligible. The sensitivity of car number is very small, which is about 2% of that of electricity output.

Tab.5. Sensitivity of parameters by balanced technology development on the CO 2 emissions in China

Δ (-1%)

e coal e oil e gas

Car number

Electricity output

Balance_no CCS -0.3896% -0.0025% -0.0507% -0.0106% -0.4322% Balance_CCS -0.0849% -0.0002% -0.0116% -0.0014% -0.0953%

3) Least changed energy mix roadmaps for China In this section, we try to find out the technology roadmaps in changing the energy mix as little as possible, under the consideration of using their energy sources as much as possible. Thus, more efforts will be made in the transport and other sectors.

If CCS is not implemented, the coal in energy mix in the power generation in 2050 can not stay at the same level in 2010 even with the maximum effort of the two other sectors, as shown in the figure 11. Actually, if all cars are replaced by electric ones, and the energy efficiency is improved by 90% in the residence and industry, the share of coal will have to be reduced by 21.7% (from 78.7% to 57%), with the share of gas remained at 1.7%.

However, if all power plants are installed with CCS technology, it is possible that the energy mix stays at the same level in 2050. In the transport and other sectors, less reduction of CO 2 emissions are expected than with CCS. When there is no change in the energy mix in the power sector, 37% of cars have to be replaced by electric cars in the transport sector, and the energy efficiency in the residence and industy should be improved by 37%. As to the sensitivities of parameters, the electricity output and the emission intenstity of coal have nearly the same sensitivities with or without CCS. Meanwhile, they have the most important sensitivities on the CO 2 emissions, as shown in the table 6. The second most important sensitivity is that of the car number, but only 4% of the sensitivity for electricity output when CCS is used and 1.5% when CCS is not used. The sensitivities of emission intensity of production of gas and of oil are negligible compared to those of electriciy output.

Tab.6. Sensitivity of parameters by least changed energy mix on the CO 2 emissions in China

Δ(-1%)

e coal e oil e gas

Car number

Electricity output

Least changed energy mix_no CCS -0.9025% -0.0032% -0.0067% -0.0358% -0.8766%

Least changed energy mix _CCS -0.1215% -0.0002% -0.0007% -0.0018% -0.1206%

In brief, the governmental target of reducing CO 2 emissions in 2050 are relatively tolerant. There are all kinds of technology roadmaps for reaching the governmental target. If the energy mix is planed to stay at the same level in 2010, the CCS must be used in the power plant. Otherwise, the transport sector and other sectors should make enormous efforts in reducing direct CO 2 emissions. The emission intensity of production of coal and the electricity are the two parameters with most important sensitivity on the CO 2 emissions, for both the balanced technology development roadmap and the least changed energy mix roadmap.

Technology roadmaps in France

In France, nearly 80% of the electricity is now produced by nuclear power. The CO 2 emissions from the power sector account for 15% of the total emissions in 2010. The shares of the coal, oil and gas are less than 5% respectively. Thus, in the techonology roadmaps for France, we focus on the transport and other sectors since the majority of efforts must be supported by these sectors. CCS is not a prior option for France as the CCS is principally installed with the power plants (different from the other two countries).

According to French "facteur 4", the CO 2 emissions in 2050 are planned to be reduced to 25% of the level in 1990. The transport sector is still the most imporant sector in the contributions to the emission reductions. The technology pathways are prensented in the figure 12. The energy efficiency in the residence and industry should be improved by at least 40%, and at least 80% of vehicles should be replaced by the electric vehicles. For example, if the energy efficiency is improved by 80%, 80% of vehicles should be changed to electric vehicles to reach the governmental target. If all the vehicles are replaced by electric vehicles, the energy efficiency is expected to be improved by 40%. As in France, the share of coal and gas are very small, the power sector will not contribute much to the CO 2 emission reductions. Actually, the roadmaps with the balanced technology development and the least energy mix are basiclly the same. Thus here we only discuss the balanced technology development roadmaps. Then we will present the sensitivities of parameters: the CO 2 emission intensities of production of fossil fuels, the electricity output, the number of vehicles in the Sectoral Emission Model.

1) Balanced technology development roadmaps for France

In France, as the share of coal, oil, and gas are already all less than 5%, the power sector can not make much contribution to the reduction of emissions. Most CO 2 emissions will be reduced in the power sector and other sectors. In the figure 13, we can see that in order to reach the governmental target, 80% of the cars should be replaced by electric cars and the energy efficiency is to be increased by 80%. In this condition, the share of coal should be reduced by 1.7%, from 5.3% to 3.5,and the share of gas by 0.9%, from 3.9% to 3%. The sensitivities of parameters in the roadmap above are presented in the table 7. Although most of reductions are carried out in the transport sector and the other sectors, the electricity output has the most important sensitivity. The second important sensitivity is that of the emission intensity of production of coal, because coal has the highest emission level. The car number has the smallest sensitivities among all the parameters, which is only 7% of that of electricity output.

Car number

Electricity output

Balance_no CCS -0.1889% -0.0315% -0.0394% -0.0163% -0.2435%

In brief, the CCS is not a necessary option for the energy roadmaps since the fossil fuels are barely used in the power sector. The governmental target in 2050 is very severe. In order to achieve this objective, the transport sector and the other sectors should make large contributions, especially the transport sector. Under the assumption that the technologies are developed evenly across sector, 80% of electric vehicles should be used, and the energy efficiency should be improved by 80% compared to the 2010 level. However, the electricity output and emission intensity of production of coal have the most important sensitivity on the CO 2 emissions even though the share of coal in the energy mix is very small.

Technology roadmaps in the United States

In the US, 43% of the CO 2 emissions come from the power sector in 2010, with the share of coal used in the power sector at 45%, and the share of gas at 23%. The second largest source of emissions was the transport sector, accounting for 30% of total CO 2 emissions in 2010. According to the government policy, the CO 2 emissions will be reduced by 82% compared to 2010 level. Its technology pathways in the policy scenario are shown in the figure 14 without CCS and in the figure 15 with CCS. In order to achieve this objective, all the electric vehicles should be replaced by electric vehicles if no CCS is applied. In this case, either the energy efficiency is improved by 60% and the power sector should be nearly decarbonized, or the energy efficiency is to be improved by 80% and the share of coal is to be reduced at 14%. If CCS is implemented with all power plants, when the energy efficiency is improved by 60% and all vehicles are replaced by electric vehicles, the share of coal can be kept at 45% with the share of gas at 28%. When the energy efficiency is improved by 80%, and 80% of vehicles are changed to electric vehicles, the share of coal is to be reduced to 16% with the share of gas at 30%.

The roadmaps are based on two criteria: one with balanced CO 2 emissions reduction across sectors, the other with least changed energy mix in the power sector are presented in the following sub-section. Same as in the previous two sub-sections, the sensitivities of parameters in the Sectoral Emission Model for each roadmap will be tested.

1) Balanced technology development roadmaps for the United States

In the US, the power sector and transport sector are two most important sectors for CO 2 emissions, which account for respectively 43% and 30% of total emissions in 2010.

The figure 16 show the technology roadmaps with balanced technology development across sectors with and without CCS. If CCS is not considered in the energy transition in the US, 85% of vehicles should be replaced by electric vehicles, and the energy efficiency in the residence and industry sector should be improved by 85%. In the power sector, the share of coal will be reduced by 39%, from 45% to 6%, and the share of gas should be reduced by 17%, from 23% to 6%.

If CCS is implemented in all power plants, the direct reduction of emissions should be slightly less than that without CCS: 80% of cars should be replaced by electric cars, and the energy efficiency should be improved by 80%. Meanwhile, more fossil fuels can be used in the power sector than without CCS. The share of coal should be reduced by 25%, from 45% to 20%, and the share of gas should be reduced by 8%, from 23% to 15%. The sensitivities of parameters are shown in the table 8. Electricity output has the most important sensitivity among all parameters, with or without CCS. The emission intensity of production of coal the second most important sensitivity, about 80% of the sensitivity of electricity output. The of emission intensity of production of gas is 25% of the sensiblity of emission intensity of production of gas without CCS, and 18% with CCS. The sensitivity of emission intensity of production is the smallest among the three fuels because it has the least share in the energy mix. The car number has the least sensitivity, at about 5% of the sensitivity of the electricity output.

Tab.8. Sensitivity of parameters by balanced technology development on the CO 2 emissions in the US

Δ(-1%)

e coal e oil e gas

Car number

Electricity output

Balance_no CCS -0.2468% -0.0206% -0.0617% -0.0159% -0.3131% Balance_CCS -0.0820% -0.0021% -0.0154% -0.0045% -0.0949%

2) Least changed energy mix roadmaps in the United States

The technology roadmaps with the least change of energy mix are shown in the figure 17. When CCS is not implemented, all vehicles should be replaced by electric vehicles, and energy efficiency in the residence and industry sector should be improved by 90%. Even with the large reduction in the transport and other sectors, the energy mix in the power sector can not stay the same level as in 2010.

In the power sector, the share of coal will be reduced by 30%, from 45% to 15%, and the share of gas will be reduced by 8%, from 23% to 15%.

If CCS is implemented in all power plants, it is possible that the energy mix in the power sector stay unchanged. Under this condition, 82.5% of cars should be replaced by electric cars, and the energy efficiency should be improved by 82.5%. The sensitivities of parameters in the technology roadmaps by least changed energy mix are shown in the table 9. Electricity output has the most important sensitivity among all parameters, with or without CCS. The emission intensity of production of coal has the second most important sensitivity, about 82% of the sensitivity of electricity output without CCS and 92% with CCS. The sensitivity of emission intensity of production of gas is 21% of the sensiblity of emission intensity of production of gas CCS, and 11% with CCS. The sensitivity of emission intensity of production is the has the least sensitivity since it has the least share in the energy mix. The sensitivity of car number is at about 5% of the sensitivity of the electricity output.

Tab.9. Sensitivity of parameters by least changed energy mix on the CO 2 emissions in the US

Δ(-1%)

e coal e oil e gas

Car number Electricity output

Least changed energy mix _no CCS -0.6222% -0.0207% -0.1555% -0.0451% -0.7534%

Least changed energy mix _CCS -0.1848% -0.0021% -0.0236% -0.0099% -0.2006%

In brief, the governmental target of reducing CO 2 emissions is very severe. In order to achieve this objective, all sectors should make large emission reductions. If the energy mix is expected to be kept at the same level in 2010, the CCS should be implemented in all power plants. If the CCS is not used, it is impossible to keep the energy mix unchanged even though the other sectors all make maximum efforts. Same as China and France, the electricity output and the emission intensity of production of coal have the most important sensitivities on the CO 2 emissions.

Conclusion and discussion

In this article, a Sectoral Emission Model has been set up for CO 2 emissions; between 2010 and 2050; in China, France and the US, in order to assessing the governments targets for the CO 2 emission reductions. Our model provides a solution pool of energy roadmaps focusing on the main energy technologies: energy mix, electric vehicles, energy efficiency, and CCS. Our model shows that completely change of energy structure in all sectors is necessary to achieve the governmental reductions.

In China, half of the emission reductions are expected to be contributed by the power sector, as the power sector accouts for half the emissions in 2010. 60% of vehicles should be replaced by electric vehicles, energy efficiency should be improved by 60%, and coal utilization should be reduced by 60%. If the energy mix is expected to be kept unchanged, then CCS should be implemented.

In France, two thirds of emission reductions are expected to be contributed by the transport sector. 80% of vehicles should be changed into electric vehicles. Meanwhile, the energy efficiency should also be improved by 80% to reach the governmental target. As the power sector contributes Least changed mix_CCS little to the CO 2 emission, CCS is not a necessary option for France.

In the US, the reduction of emissions is divided homogenously across sectors as all the three sector have the same importance for the reduction of emissions. All sectors should make large effort (more than 80% of emissions should be reduced) in order to reduce CO 2 emissions. Same as China, if the energy mix is expected to be kept unchanged, then CCS should be implemented.

The tests of sensitivities of parameters in our Sectoral Emission Model for different technology roadmaps, in the different countries, show that the electricity output and the emission intensity of production are the two parameters with the most important sensitivity on CO 2 emissions: improving the efficiency of coal combustion will play the leading role in the emission

Fig. 1 .

 1 Fig.1. National CO 2 emissions and their per capita CO 2 emissions between 1971 and 2010

  The electricity-production simulation results are based on the data of1971[START_REF] Hong | Application of chaotic ant swarm optimization in electric load forecasting[END_REF] (IEA, 2012)). Chinese electricity output will be 10248TWh in 2050, a 2.43-fold rise over 2010. In France it will be 539TWh in 2050 (a 4% fall from 2010), and 4785TWh (a 10% rise over 2010) in the US. Figures 4 to 6 show the projection of electricity production in the three countries between 1981 and 2050. The Xaxis is in years and the Y-axis is electricity output in TWh.

Fig. 4 .

 4 Fig.4. Electricity production in China 1981-2050

Fig. 5 .Fig. 6 .

 56 Fig.5. Electricity production in France 1981-2050

Fig. 7 .

 7 Fig.7. CO 2 emissions by governmental targets in 2050 compared to the 2010 levelThe propotions of CO 2 emissions by governmental targets in 2050 relative to 2010 are shown in the figure7. The emissions in 2050 will be 72% of that in 2010 in China, 25% in France and 18% in the US.

Fig. 8 .

 8 Fig.8. Technology roadmaps without CCS in China Fig.9. Technology roadmaps with CCS in China

Fig. 10 .

 10 Fig.10. Technology roadmaps by balanced technology development in China in 2050 compared to 2010

Fig. 11 .

 11 Fig.11. Technology roadmaps by least changed energy mix in China in 2050 compared to 2010

Fig. 12 .

 12 Fig.12. Technology roadmaps without CCS in France

Fig. 13 .

 13 Fig.13. Technology roadmaps by balanced technology development in France in 2050 compared to 2010

Fig. 14 .

 14 Fig.14. Technology roadmaps without CCS in the US Fig.15. Technology roadmaps with CCS in the US

Fig. 16 .

 16 Fig.16. Technology roadmaps by balanced technology development in the US in 2050 compared to 2010

Fig. 17 .

 17 Fig.17. Technology roadmaps by least changed energy mix in the US in 2050 compared to 2010

Tab.1. Sectoral emissions shares in 2010

		China	France	the United States
	Power	49.28%	15.38%	43.02%
	Transport	7.08%	34.56%	30.21%
	Residence	4.18%	15.94%	5.99%
	Industry	32.15%	17.49%	10.94%
	Total	92.68%	83.37%	90.16%

2. Schema of Sectoral Emission Model

	CO2 emissions, million tonnes	0 2 000 4 000 6 000 8 000	1971	1974	1977	1980	1983	1986	1989	1992	1995	1998	2001	2004	2007	2010	0 5 10 15 20 25	per capita CO2 emission, tonnes
			CO2 emissions_US					CO2 emissions_CN
			CO2 emissions_FR					per capita emissions_US
			per capita emissions_CN				per capita emissions_FR
	):

Fig.

Tab.2. CO 2 emission intensities of production of fossil fuels (IEA, 2012)

	Kg/KWh	coal	oil	gas
	China (2010)	0.967	1.044	0.507
	France (2010)	0.949	0.766	0.520
	The United States (2010)	0.907	0.711	0.405
	Europe (2010)	0.8	0.4	0.2

Tab.4. Assumptions for number of vehicles in 2050

		Number of	Number of	Number of	Number of
		vehicles in 2010	vehicles in 2050	vehicles per	vehicles per
		(million)	(million)	person in 2010	person in 2050
	China	114	560	0.085	0.4
	France	38	63	0.6	0.9
	United States	268	393	0.86	0.98


Frédéric Tayeb
email: frederic.tayeb@enpc.fr

Olivier Baverel

Jean-François F Caron

Lionel Du Peloux

L Du Peloux

CONSTRUCTION OF GRIDSHELLS COMPOSED OF ELASTICALLY BENDED ELEMENTS AND COVERED BY A STRETCHED THREE-DIMENSIONAL MEMBRANE STRUCTURAL MEMBRANES 2013

Keywords: Gridshell, Composite materials, dynamic relaxation, form-finding, prototypes

INTRODUCTION

In the last twenty years many applications of composite materials in the construction industry were made. The main field of application concerns the reinforcement of concrete beams with carbon fiber plates [START_REF] Limam | Numerical and experimental analysis of two-way slabs strengthened with CFRP strips[END_REF] or post tension cables. More recently, a footbridge with carbon fiber stay-cable was build in Laroin (France, 2002 [START_REF] Geffroy | La passerelle de Laroin[END_REF]), another footbridge, all made of glass fiber composites, was build in Aberfeldy (Scotland, 1993 [START_REF] Harvey | A reinforced Plastic footbridge[END_REF]) and a movable bridge (the Bonds Mill lift bridge in Stonehouse, England, 1995 [START_REF] Burgoyne | Advanced Composites in Civil Engineering in Europe[END_REF]). Nevertheless applications using composite materials as structural elements remain exceptional. Although the qualities of their mechanicals properties are obvious (low density, high strength and high resistance against corrosion and fatigue), their relatively low elastic modulus is a disadvantage against steel. Indeed most slender structures in structural engineering are designed according to their stiffness and rarely to their strength. In addition, the elastic instabilities depend linearly on the Young modulus, so that again, having a low Young modulus is a real disadvantage when a designer tries to calculate structures based on conventional design structure. In order to take advantages of every characteristic of composite materials, new structural concepts have to be found.

The Architected Structures and Materials research unit of Navier laboratory is working on the development of innovative solutions for composite material in civil engineering. Four design principles guided the conception of the structures:

-Optimal use of the mechanical characteristics of the fibers; -Simple connection between components of the structure; -Optimal design according to its use; -Use of components already available in the industry for cheap material costs.

Several structures were investigated such as an innovating footbridge [START_REF] Jülich | Baverel Selfstressed Bowstring Footbridge in FRP[END_REF] and several experimental gridshells [START_REF] Douthe | Gridshell in composite materials: towards wide span shelters[END_REF] [START_REF] Douthe | Gridshell structures in glass fibre reinforced polymers[END_REF] [8] [START_REF] Douthe | Study of slender prestressed structures in composite materials: application to the conception of gridshells[END_REF]. The purpose of this paper is to explain the method used to design gridshells, up to the fabrication of the membrane and then to emphasis the improvements of the building process. The first section gives a proper definition of gridshell and emphasis the specificity of their construction process. Then, the numerical aspects of the project are developed. Finally the steps of construction and the improvements made through the salvo of projects are approached.

GRIDSHELLS: DEFINITION AND PROCESS OF CONSTRUCTION

The name of gridshell commonly describes a structure with the shape and strength of a double-curvature shell, but made of a grid instead of a solid surface. These structures can be made of any kind of material -steel, aluminum, wood… Generally, the metallic structures are made of short straight elements defining a cladding made of plane triangular or quadrangular element. The complexity of this geometry requires the development of many clever and expensive assemblies. In order to avoid these complex joints, a very specific erection process was developed using the ability of slender components to be bent [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF]. Long continuous bars are assembled on the ground, pinned between them in order to confer on the grid a total lack of plane shear rigidity what allows large deformations. The grid is elastically deformed by bending until the desired form is obtained and then rigidified. With this process, the initially straight beams are bended to form a rounded stiff surface. Only few gridshells were built using this active bending method, among which the most famous are: the Mannheim Bundesgartenschau (arch: Mutschler and Partner and Frei Otto, Str. Eng: Arup, 1975 [11]), the carpenter hall of the Weald and Downland Museum (arch. E. Cullinan, Str. Eng. Buro Happold, 2002) [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgarten-schau[END_REF] and the Japanese pavilion for the Hanover 2000 Exhibition (arch: Shigeru Ban, Str. Eng. Buro Happold) [START_REF] Ban | The Japanese pavilion[END_REF]. In addition, the Navier research unit has already participated to the construction of four gridshells in glass fibre reinforced polymer (GFRP), increasingly large. The gridshell for the Solidays' festival was 300 m² large [8] but very recently a 350 m² gridshell called "Cathedrale Ephemere de Creteil" has been constructed to replace the Creteil cathedral during its renovation which should last at least two years.

Construction steps: the main building steps are illustrated figure 1: the grid is assembled flat on the ground (figure 1a), then erected by two cranes (figure 1b) and gets its final form when attached on anchorages.

COMPOSITE MATERIALS TAILOR MADE FOR THIS TYPE OF STRUCTURES: FLEXIBILITY FOR STIFFNESS.

Most of the gridshell structures have been made of wood because it is the only traditional building material that can be elastically bent without breaking. This flexibility generates curved shapes which generates structural stiffness. However looking at other industrial fields (sport and leisure, nautical...), it can be noticed that every time high strength and high deformability are required, composite materials is replacing wood (ship masts, skis, rackets).

To study accurately the question of the best material for gridshells, the authors adopted the method proposed by M. Ashby [START_REF] Ashby | Materials selection in mechanical design[END_REF]. In this method, indicators characterising the object to be designed are defined. In the case of gridshells, it is necessary to have a material with:

-High elastic limit strain in order to be able to bend the element and obtain a curved shape.

-High Young modulus to confer to the gridshell its final stiffness after bracing.

The Ashby method drawn for these two characteristics provided several materials potentially better than wood for the gridshell application. These materials are titanium, CFRP, GFRP and technical ceramics. In addition, this study showed that steel or concrete can not be better than wood for such an application: these materials can not deform as much as wood. To choose between the four families of material, other aspects have to be considered. In particular, materials shall not be too brittle to be easily handled on site by workers and therefore ceramics are not suited. Because of cost limitation, titanium and CFRP can not suit for the gridshell application.

The most valuable alternative to wood is hence glass fibre reinforced polymers (GFRP). They have higher elastic limit strain (1.5 % at best for GFRP and 0.5 % for wood) so large curvature synonymous of freedom of shape is possible. Their Young modulus also is higher (25-30 GPa against 10 GPa for wood). This is an advantage to make a stiff structure. In addition, supposing that for a given geometry, the buckling load of a gridshell is linearly dependent from the young modulus, one can expect the buckling load of a gridshell in composite materials to be 2.5 to 3 times higher than one made of wood. Moreover, as composites are industrially produced, the reliability of their mechanical properties is much higher than that of natural materials like wood. Finally, while wood beams have to be made of several pieces of wood stuck together, GFRP profiles can be made continuously, as long as necessary.

Concerning costs, if one takes into account the mechanical properties and the ability of composites to be formed into efficient sections like tubes, GFRPs become very interesting challengers, especially if pultrusion production is used. Indeed, hollow sections make possible the use of light beams optimized for each application (stiffness and curvature). Moreover, the polymer chosen for the GFRP can resist to corrosion, UV and other environmental attacks, whereas wood materials need maintenance.

At this point, the gridshell concept is explained. The type of materials chosen is GFRP for flexibility, cost, stiffness and reproducibility reasons. The process of construction can be developed.

CONCEPTION OF A GRIDSHELL: FROM A GEOMETRY TO A FINAL SHAPE.

At the beginning of the process, the architects have to define a shape. To be well suited with the process of the gridshell, the shape has to be a rounded shape with curvatures as homogeneous as possible. Then, according to the range of curvatures, the engineers have to choose the geometric properties of the beams (mainly the outer radius -this is explained in equation 1). Then, the engineers have to do first a geometric mesh of the shape before relaxing it mechanically to get the final form of the gridshell. After this form-finding step, the engineer adds the third layer numerically (bracing layer) and evaluates the stresses in the structure under serviceability limit states (SLS) and ultimate limit states (ULS) loads, according to construction codes [START_REF]Structural Design of Polymer Composites, Eurocomp[END_REF]. They might have to modify the mesh, or the shape of the gridshell, in agreement with the architects to reduce the stresses. Once the form-finding has converged and the stresses are suitable, the engineers can design the membrane according to the three-dimensional shape numerically obtained. The method summarized here is developed in the following.

Geometrical step:

The method used for the forming of the grid is "the compass method". This method consists in constructing a network of regular quadrangles on any surface, plane or not. This method was described in IL10 Gitterschalen of Frei Otto 1974 [11]. The figure 2 shows the different steps of the method on surface which can be three-dimensional. The task is to construct a grid using only a compass. First, two curves that intersect each other are laid down on the surface to mesh. Then, a mesh size is chosen and serves as the compass radius. The spacing of the grid is marked along each axis, from the point of intersection of the axes. The knots are determined by the intersection of two circles as shown on the figure 2. Gradually, new points are determined.

Figure 2: Construction of the grid using the compass method (Otto 1974) So, to generate the grid of the gridshell, a 3D compass method is performed on the surface. Obviously, the grid obtained has no mechanical meaning. The real shape of the gridshell is obtained later when the mechanical properties are considered. This method was used for the design of the gridshell for the Solidays festival (June 2011) and for the one of the "Cathédrale Ephémère de Créteil" (February 2013).

An implementation of the geometrical method has been developed at Navier laboratory, using Rhino NURBS modeller. This modeller makes possible the modification of a surface through control points. This is very interesting because the compass method is also performed under the same numerical environment. Thus, modifications of the surface to mesh -but also modifications of the curves defining the mesh -are easy to do and the process of meshing is immediately auto-updated.

To sum up the process, first a shape of a structure is proposed by the architects. Secondly, the surface is extended and two main axes for the construction of the grid are drawn (figure 3a). Thirdly, the mesh is automatically generated (figure 3b). The mesh might either cover all the functional shape or cover only a part of it. In the first case, the mesh can be more or less homogeneous. Among meshes covering the entire functional surface a mesh has to be chosen. It is then trimmed to get the final mesh (figure 3c). Mechanical step: the final shape is obtained by performing a non linear structural analysis of the structure with real mechanical properties. This non linear algorithm is based on dynamic relaxation algorithm [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF], [START_REF] Barnes | Applications of dynamic relaxation to the topological design and analysis of cable, membrane and pneumatic structures[END_REF].

If the shape proposed by architects is suitable with the gridshell process, the geometry of the grid provided by the compass method is very slightly modified by the dynamic relaxation process. Once the real shape is found, classical structural analyses are performed with the standard loads. Obviously, the stress due to form-finding is taken into account in the structural analysis: the main source of stress is linked with the bending of the beams during the erection process. In other words, the stress σ is proportional to curvature of the beams 1/R, (equation 1), and the curvature is mainly due to forming: even under critical loads, its shape (and so the curvature of beams, and also stress) does not change significantly. This is the main advantage of the active bending which provides high stiffness in this case.

R y E = σ (1)
where E is the Young Modulus and y the outer radius of the beam.

Designing a grid shell is a difficult task. As a guideline, the designer has to check that: o The curvature in each bar is not too high, to avoid the break of beams even with relaxation and fatigue phenomena. In practice, according to Eurocomp [START_REF]Structural Design of Polymer Composites, Eurocomp[END_REF], the maximal stress in the bar must not exceed 30% of the strength of the beam. To this limit stress corresponds a limit curvature under which the risk of break is low enough to be acceptable (equation 1). o The entire surface is meshed o The mesh does not get too concentrated locally If the grid is too weak to support the external loads, the designers have to reinforce it by reducing the size of the mesh and/or modifying the geometry of the cross section of the beams. If the outer radius is increased, the stress due to the form-finding gets higher as the maximal stress in a beam is proportional to both the curvature and the outer radius of the cross section of the beam. So the engineers have to be cautious.

The consideration of wind and snow loads presupposes that the gridshell is covered by a membrane. The membrane is made according to the geometry obtained through the dynamic mechanical process. From this geometry, the surface is partitioned in pieces of plane surface (according a tolerance which depends to the material of the membrane). Then the pieces are sewed to form the three-dimensional membrane. Some pieces of membrane (the yellow ones) can be easily seen thanks to colouring, on the gridshell, figure 4. The membranes are PVC coated sheetings.

PROTOTYPES

Prototypes: to demonstrate the feasibility of composite gridshells, four full scale prototypes of composite material gridshells have been built. The two first ones were built on the campus of the Université Paris-Est. The first prototype was a purely experimental structure which was tested under several loading conditions in order to investigate the behaviour of gridshell structures and to compare it with the numerical models (figures 4 and 5a). Detailed results of these tests can be found in [START_REF] Douthe | Study of slender prestressed structures in composite materials: application to the conception of gridshells[END_REF]. The behaviour of the prototype is very close to simulations performed numerically, with the dynamic relaxation algorithm presented in [START_REF] Ashby | Materials selection in mechanical design[END_REF]. This gridshell is now serving as a shed for equipment and has a great importance since it is the one that was erected the first (around five years ago). It is still erected and serves as testing for long term damage (mainly creep and fatigue). The second prototype was build to cover a wind tunnel, figure 5b. Prototypes sheltering people: as previously written, two gridshells built to house people have been recently made. The first one for the Soliday's festival (June 2011) and the last one built to temporally replace the Creteil Cathedral (February 2013 for at least 2 years of use). More details about the context of this gridshell and about the project can be found in [START_REF] Peloux | Faith Can Also Move Composite Gridshells[END_REF]. Compared to the two first experimental gridshells, these two are larger and had to take into consideration many aspects for public safety.

These two last gridshells have got several improvements regarding the previous ones. First, their size was larger. So large that most of the tubes of the structure was to be built from several tubes joined up together. Second, the gridshells had to obtain an attestation from administrative authorities to house people for a specified period. This attestation was given after a committee had studied the structures, that is to say the project on the paper as well as the execution on the building site.

The shapes: the Solidays Gridshell was looking like a half peanut (two domes) while the Ephemeral Cathedral looks a little more like a stretched one dome structure. The dimensions of these structures are quite similar: around 7 m high, 25 m long and 15 m wide. They are constituted of about 2 kilometres of pultruded unidirectional tubes from Topglass (polyester resin from DSM + Owens Corning glass fibres) with a Young modulus of 25 GPa and a limit stress of 400 MPa. The available length and diameter of the tubes are respectively 13.4 m and 41.7 mm; the wall thickness of the tubes is 3 mm.

As said before the stress is limited to 30 % of the limit stress to avoid severe creep and damage effects like progressive rupture of the fibres.

Unlike the former gridshells and as written previously, given the short period for the project, the geometries of the membrane of these two gridshells were drawn from the numerical shapes.

Computation: the computation has been performed for different sizes of mesh. It appears that a mesh size of one meter was acceptable to resist to the loads studied (dead weight, wind, snow). Under these loads, it is important to check that the stress remains acceptable in all the structure, but since the stress in the bars is mainly due to the form-finding if the structure has been cautiously designed -that is to say if it does not buckle under considered loads -the stress might not reach too high values. In this case, the stress can be drawn from the study of the gridshell without any load applied (figure 6). Fabrication: once the form of the structure is defined, the coordinates of the extremities are picked up and precisely reported by geometers on site and stakes are positioned. Then, the grid is assembled flat on the ground: tubes has been cut to the right dimensions with hacksaws and connected to the others with standard swivel scaffolding elements (figure 7a). These scaffolding elements allow rotation around their axis. They are chosen for their low cost due to industrial production.

Then the grid is deformed and shifted by two cranes that hook up the grid in several places (figure 1b). The final form is reached when the extremities of the beams are fixed on the anchorages. The erection phase requires only a few hours work for about ten people whereas the preparation of the grid can take many days.

The following structural step is the bracing. This step is essential, as before bracing, the grid still holds its shear degree of freedom. To behave like a shell, the bracing will transform every deformable quadrangle into two rigid triangles. The third direction of beam is installed as shown in figures 7b and 7c with the use of new scaffolding elements. Once the bracing is installed, the gridshell gets its full mechanical properties and its stiffness gets about twenty times the stiffness of the grid before bracing [START_REF] Baverel | Gridshells in Composite Materials: Construction of a 300 m2 Forum for the Solidays' Festival in Paris[END_REF]. The bracing step does not apparently change the form of the gridshell, but since the bracing can't be done everywhere in the same time, it may modify the shape a little. This step is the most fastidious one because the third layer of beams has to be set up in the deployed geometry. Thus the operators have to adjust each connector and tighten the beam inside it, generally in a basket, a few metres over the ground. The structural part being finished, the positioning of the membrane can be started. First the PVC coated membrane is pulled above the gridshell (figure 8a). In order to fix the canvas, a girder is set up 10 cm from the soil (figure 8b). The girder follows the contour of the gridshell. For the Creteil gridshell, this girder was a pultruded rod able to support a large amount of shear stress as well as high curvatures (here hollow cross-sections are not suited and the outer radius has to be smaller than for structural beams). The canvas is then positioned and stretched. This step was supposed to be critical as polypropylene-PVC coated canvas is almost not stretchable, and was manufactured according to the geometry of the numerical model. So a mistake during the numerical design or during the building phase could have led to a situation where the canvas would not really fit to the structure. As the gridshells were accurately set up, the canvas fitted to their shape. No wrinkle was observed (figure 8c). The membrane might play a part in the structural behaviour of the grid but given the high dependence to modelling (in particular to friction between beams and membrane and also between connectors and membrane), it is very difficult to evaluate accurately the real stiffening effect of the membrane. Improvements: to deal with the fact that these last gridshells are made to shelter people many improvements have been added to the previous prototypes. In particular, fire, waterproofness, lightening and thermal considerations have been added to the primary mechanical considerations. Nevertheless mechanical properties have been considered with much more attention to assure the public safety. The reference construction guide -named Eurocomp, for composite materials -guided the construction. In particular, the way of production of beams, their constitutive materials as well as the duration of solicitations acting on the structure were taken into account. On the other hand, many assays were performed to get the real properties of the beams (mean strength, variation coefficient).

In addition, a robustness study has been performed on the Solidays gridshell [START_REF] Tayeb | Stability and robustness of a 300 m² composite gridshell structure, Construction and Building Materials[END_REF]. This study showed that the gridshell can undergo accidental situation such as vandalism without risking collapse. Indeed, thanks to redundancy, the stress from a break would spread largely and the stress in the neighbouring beams would not get too high. In the same time, large displacement of broken beams would be visible and the evacuation of public could be launched. This kind of ductility is named pseudo-ductility in this case when fragile materials are mainly used. This study has also showed that the buckling of the gridshell must be avoided at any cost: if bucking starts, the curvature of some beams will increase a lot and the stress will increase in the same time, up to breaking and then to the ruin of the structure.

Other improvements were done, relating to the connection of the grid with the soil (figure 9a) and also to assembling of 13 m pipes to form long beams up to 35 m (figure 9b). The difficulty is to make connections able to transmit stress in a way that the assembled beams keeps mechanical properties of the primary GFRP beam. In particular, the joining up of two beams have to:

-transmit normal stress for structural stiffness -have similar bending stiffness for the continuity of the global shape Thus, to combine these faculties, the system the system presented on the figure 9b was chosen. Each beam is assembled with a sleeve slightly larger with three pins. This assembly can undergo axial forces up to 30 kN. This theoretical value (obtained with the help of [START_REF]Structural Design of Polymer Composites, Eurocomp[END_REF]) has been experimentally validated thanks to a salvo of assays. In addition, some extra glue is put inside the assembly to prevent from relative movements and thus fatigue phenomena. Then, between the sleeves of the two beams to connect, a M20 threaded rod is screwed inside bolts welded to the sleeves. This threaded rod is adapted to behave like the structural beams, under bending stress.

CONCLUSION

This article first explained the choice made by the Navier laboratory to use composite materials, and in particular the choice of GFRP. It then presented the process of conception of these composite material gridshells. In particular, the active bending of the gridshell is approached, on a numerical aspect as well as during the building stage.

The development of the prototypes can be seen as the prototypes got larger, with safety standards getting higher and higher because of their use as a shelter for public and also to make possible longer periods of use. In particular, assembling devices and anchorages have been largely improved. The critical points are also approached. The most fastidious one is the bracing which can last a long time. Another one is linked with the membrane covering the structure, which is sewed according to the numerically obtained geometry. That does not leave much room for error during the erection step.

Nonetheless it is important to remind the qualities of such structures. Due to suitable process of construction, the structures shows light weights (around 5 kg per m²), as well as high strength. Another advantage is the fact that most of the stress in elements is due to forming, and that extra loads like wind and snow add very little stress, under acceptable load cases. This behaviour is observed since in these cases, bucking is not generated. Finally from an architectural point of view, this technology provides new horizons. The gridshell built in Creteil is a good illustration, for the global exterior shape (figure 10a) as well as from an interior view (see figure 10b).

Figure 1 .

 1 Figure 1. Cathedrale Ephemere de Creteil. a. Primary grid made flat. b. Primary grid deformed elastically and attached to boundary conditions.

Figure 3 :

 3 Figure 3: meshing process of a 3D surface. a. extension of the surface and drawing of the two curves. b. meshing of the extended surface. c. cutting of the useful part of the mesh.

Figure 4 :

 4 Figure 4: Long term erected composite gridshell. Some planar parts (yellow) of the membrane can be seen.

Figure 5 :

 5 Figure 5: a. First experimental gridshell under testing. b. Second experimental gridshell used as a shelter for a wind tunnel.

Figure 6 :

 6 Figure 6: Stress resulting from forming in the Soliday's gridshell

Figure 7 :

 7 Figure 7: a. Joint detail. b. Mesh before bracing. c. Mesh after bracing.

Figure 8 :

 8 Figure 8: a. Positioning of the PVC coated membrane (Creteil). b. Continuous beam for border during tightening (Creteil). c. Membrane without winkle (Solidays gridshell).

Figure 9 :

 9 Figure 9: a. Pin anchorage for beams. b. Assembly used to join two beams.

Figure 10 :

 10 Figure 10: Gridshell built in Creteil. a. Global view. b. Inside view.