Imagine scenarios of ideal energy mixes to reduce greenhouse gas emissions without negative impacts induced from energy and environmental and socioeconomic points of view, is far from simple. Its difficulty reflects the systemic complexity of the energy system, and its entanglements with other systems such as the agriculture, with which conflicts can sometimes arise with land use, for instance. Finally, we could reduce the energy transition to a constraint optimization problem. It is only a very partial vision that largely forgets the human factor and the questions of social acceptability, political regulation or democracy. Nevertheless, even this partial vision can illustrate how the integration of renewable energies into the energy mix, can be trusted microscopically. This sequence will explain how by a small, simple analytical model, the orders of magnitude of the energy transition can be inferred and analyzed. The system operator satisfy the supply and demand balance at every moment. The demand is D and the production is P. The production is divided into several terms. The production of renewable energy, the nuclear production, the thermal production, the imported or exported production, the storage, or the energy released from storage. Since the marginal cost of renewable energies is small, this means of production are mobilized in priority. The major difficulty for the network operator is to be able to ensure the demand supply balance at each moment with a viable consumption, but especially a production which becomes more and more variable and intermittent, as the fraction of renewable energy, any energy mix increases. The figures show for Italy, consumption curves in gigawatt hour per day in black. Wind and solar capacity factors, respectively in orange and blue. The curves indicate the weekly average and the envelopes of the same color, the standard deviation within a week, and show a very large variability. Another way of looking at this is to define this dispatchable energy production systems as a service rendered to the grid to balance the net energy demand, defined as the difference between the demand and the renewable energy production. Two objectives must be sought for. The first, is to maximize penetration of renewable energies in order to minimize the carbon footprint of energy production. In the sequence, we do not discuss the nuclear production explicitly. The approach developed consents only the viable part of production. Here, what is maximized is the penetration and not the production of renewable energy. Penetration is the ratio of production to consumption. This is less the share of the consumption satisfied by renewable production that is maximized. The second objective, is to minimize the variability of the penetration in order to maximize the stability of the network. To take into account the decrease of the selling price, we have to produce quantity, and to minimize the costs incurred by the commissioning of dispatchable energy production system, such as thermal power plants, for instance. Let's take a simple example that can be treated analytically in which we replace the first objective with a constraint. The constraint here is that the total PV and wind production, equals the actual total PV and wind energy production. With this ignore the goal of maximizing penetration. The remaining objective is the minimization of the viability of the production, the consumption being fixed. The previous equations, are simply rewritten as follows. Constraint 1, says that the sum of renewable energy here, solar PV, and wind WD, is equal to a fixed total energy production. The output is the product of the installed capacity WE, and the expectation of the capacity factor. If we consider an annual production, the expectation is equal to the annual average. The objective of minimizing the variability requires calculating the covariance matrix between the capacity factors of when and photovoltaic production. By neglecting the correlations between solar and wind production, the total variance is written as the sum of the variances of the renewable productions. It's written as the square of the installed capacity, multiplied by the variance of the capacity factor. To solve our problem, we apply the method of Lagrange multipliers, which makes it possible to find the stationary points, like the minimum or maximum of a function of one or several variables and the constraints. The variance is the function of which with six the minimum, and which depends on two variables, that are the installed capacity of photovoltaic energy and wind energy. The constraint here is the imposed production. By applying the Lagrange multiplier method, we can rewrite the system of equation for production environments into another system of three equations where Lambda is the Lagrange multiplier. Let's apply this system of equations to the French energy system. The table summarizes for the year 2015, the wind and solar production, the installed capacities that we are trying to optimize, and the standard deviation of the capacity factor of each technology. These figures are given for the example, and they're not consolidated. Using the table, the total energy production can be calculated as the sum of wind, and photovoltaic production. By using the expression of the solutions for the installed capacities, one obtains an installed capacity for the wind of 9.5 gigawatts and for the photovoltaic production of 4.8 gigawatts. If we compare these to the actual installed capacities reproduced in the table, we see that for such a simple model the computed capacities are relatively close. Let's play a game. Let's apply this simple model to two scenarios relevant to the French contexts. The first scenario is an electrical mix with a 50 percent nuclear share. This is the objective of the pre annual energy program loop that imposes this objective by 2035. The second scenario aims to move the urban automobile fleet from thermal to all electric with an electricity supply from renewable energy. With such a simple model, the objective is to discuss in terms of orders of magnitude, rather than to elaborate the future energy transition plan. Moving from 70 percent of nuclear energy in electricity mix today to 50 percent is tantamount to asking renewables to produce an additional 132 terawatt hour over a one year. Applying the solutions for the installed capacities, we obtain an installed capacity for the wind turbine of 42 gigawatts, and for the photovoltaic of 22 gigawatt respectively, four times the size of the wind firms, and five times the size of the photovoltaic firms of 2015. The perennial energy program loop, imposes 34 gigawatts installed in 2028. Thus, approximately 25 gigawatts between 2015 and 2028 or two gigawatts per year. This means an additional 30 gigawatts in 2035. The rates currently planned does not allow us to achieve the objectives simulated with our model. Of course, this analysis is too simple. It considers only two alternative technologies to nuclear, but it allows us to give an order of magnitude of the efforts to be made to comply with national strategies in terms of energy transition. I think the urban car fleets to a 100 percent electric translate as follows. Transport accounts for about 30 percent of final energy consumption, or 523 terawatt hours. Urban transport represents one-third of transport or 128 terawatt hour thermal. Moving from a thermal engine to an electric engine, provided that the source of electricity is renewable, can multiply efficiency by three. In fact, the efficiency of heat engine is typically at 30 percent. The efficiency of battery is of the order of 90 percent. Multiplying by 3D efficiency as one divide by 3 the need for electric energy or 47 terawatt hour. Applying the solutions from installed capacities, one obtains an installed capacity for the wind of 15 gigawatts and for the photovoltaic of eight gigawatts respectively; 1.6 times to bark of wind farms and 1.7 times the park of photovoltaic farms in 2015. I'll leave the analysis of these vessels to your discretion. Nevertheless, increasing the share of viable energy production in the network is not simple. The sophistication of the network can be divided in four phases, with respect to the share of the viable energy sources in the network. The figure shows the situation for wind and solar photovoltaic energy. In phase 1, the low penetration of renewable is not an issue for the network. In the phase 2 conventional dispatchable systems such as thermal power plants and hydro-power, are sufficient to integrate viable energies without having to transform the network. From phase 3, investments might be made to transform the network, to make it smarter and more flexible to change and supply-demand balance with a much more viable net energy demand. Flexibility means strengthening storage facilities and erasing or reducing consumption, for instance, through hourly modulation of energy prices are early warning systems. It is also through the massive collection of energy consumption and production data and the real-time processing in a more sophisticated energy management system. In 2016, the share of wind and solar photovoltaic in the French electricity mix represents 6.2 percent and places France in the phase 2 category. The French, the reference scenario for 2030 raises the share to 17 percent and places France in the third category. In this sequence, we have used a conceptual model to quickly derive orders of magnitude of the development of renewable necessary to achieve the objectives of the energy transition. This analysis has allowed us to show the magnitude of the effort required and how it can be optimized depending on the objectives. It is hive based on strong simplifying hypothesis and deserves of course, to be further developed to take into account the diversity of energy technologies, the development, and their environmental and economic costs. Neither does it leave space for social and political dimensions of energy, which are essential to take into account for just transition. Thank you for your attention.