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The Hubbert’s Model and Renewable Energy Sources

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So, we talked about Qp, being the cumulative production from time t is equal to 0 to T. That means 0 to T pt, dt. Now this Qp. Rate of change of Qp, dQp by dt. Is nothing but Pt. The rate of change up is proportional dQp dt is proportional to Qp. That means the demand for coal will be proportional to the cumulative amount of coal that has been used because that would result in more users and people's see and as we go towards the limit then this if we are going towards Q infinity, that dQp by it is constrained by the fact that we are near the limit. So, if we take this, we have a model which gives us dQp by dt is equal to B into Qp into Q infinity minus Qp. So, as we go towards the rate of change of the cumulative production, cumulative production Qp, which is the production in a particular year, in the initial case, it is exponential as we go towards the limit that decreases because we have this limiting term which is Q infinity minus Qp. (Refer Slide Time: 10:20) If we take this, we can then derive and we will get dQp by Qp, Q infinity minus Qp is equal to B dt. And you can show by integration that Qp is Q infinity by 1 plus AE raise to minus Q infinity dt. This is called an, S shape logistic curve. It is also called the Pearl curve, after the statistician Raymond Pearl, who initially proposed this as a curve which was used to show the growth in organisms in terms of height and the height and weight and this has been used in a whole host of applications. (Refer Slide Time: 11:21) The way this works is that use. Start from here and then you go and it goes asymptotically to the limit. So, in this case, this is Q Infinity this is Qp, this is t. And this is what is known as the S-shaped curve. So, how do we get this curve? And I will give you a tutorial where we can look at the actual data for India and you can make this calculation. We have done this and based on this corresponding to this, then you get the production going to a maximum and coming down and this is the kind of thing that this is what was done for petroleum. (Refer Slide Time: 12:14) So, typically what happens in this is that we can take this curve Qp Q infinity by 1 plus A e raise to minus BQ infinity. We need to find these coefficients A and B. Q Infinity, we should be able to get an estimate from the Geological Survey, the Geological Survey of India. If we are looking at Indian context, whatever estimate we have of the reserves we use as Q infinity and we can calculate we can modify this and see we can write this as 1 plus A e raise to minus B Q infinity t will be Q infinity by QP. So, Q Infinity by Qp minus 1 is A e raise to minus B Q infinity t.(Refer Slide Time: 13:24) Now for this, I can take ln on both sides and I will get a line of Q infinity by Qp minus 1 is ln A minus b t where B is equal to BQ infinity. Now, if you look at this, this is of the form. Y is equal to C1 plus C2t and this is amenable to linear regression. All that we can do is we can take. We can start with the time-series data that we have of production and we can take a particular year in which we can get the initial value of Qp at starting ts. And then for each year, we can just add on the production so that we can get the Qp from that starting year till the recent years, as a obtain the estimate of Q infinity from the resource and then we can get ln, Q infinity by Qp minus 1 and get that as y and then get these coefficients ln and B from a regression. So, we can take this and make the calculations and get the coefficients A and B. So, I would urge you to try this with the dataset that we have for India for Indian coal, and you can try and get the coefficient A and B and compare it with the results. Then once we have that, we can use it to find what is the year of peaking. So, we can. This is something that we can calculate. What is the year of peaking that we will have based on the fact that the peak production will happen in that year? (Refer Slide Time: 15:33) So, if we see the equation that we have Qp is Q infinity by 1 plus A e raise to minus BQ, infinity t. We want to find out the time when the production is maximum. When the production is maximum it will be a stationary point where dp by dt will be equal to 0. Now do by dt is equal to 0 means that we are going to have dQp by, DPP is equal to dQp by dt. So, we will like to find the point of inflexion when this will be maximum, where d square Qp by dt squared is equal to 0. So, let us take this equation and differentiate it. You get d square Qp by dt square. We can take the equation where we have, let us start from the other point. Let us start from the point where we have p is B Qp Q infinity minus Qp. This is the starting point, so we can take this as dQp by dt which is going to be B dp by dt set it equal to 0 is going to be B dQp by dt into Q infinity minus Qp plus B Qp and differentiate Q infinity minus Qp, which is minus dQp by dt is equal to 0. B is not equal to 0. Also, tQp by dt is not equal to 0 becuse that is the production. That is the maximum production so we can divide by these and what we will get then is Q infinity minus Qpminus Qp is equal to 0. Which means Qp is equal to Q infinity by 2. This is the point at which we will get a peak production, and this will happen at the point of inflexion. It will happen at the midpoint of the cumulative production curve. (Refer Slide Time: 18:32) So, now we can calculate, we can substitute, we can say Q infinity by 2 is equal to Q infinity by 1 plus A e raise to minus B Q infinity t. We can then say 2 is equal to 1 plus A e raise to minus B Q Infinity Tm. Let us say Tm and then this becomes 1 is equal to A e raise to minus B Q Infinity Tm and then you get Tm is ln A by B Q Infinity. So, what are we calculated? We have calculated the time at which the peak will occur and this is in terms of these coefficients, A, B and Q Infinity, which we have derived. So, this is the year of peaking. We can also find out instead of this we can find out the T 90 per cent time at which 90 per cent of the resources used up. So, we can take QP by Q Infinities point nine substituted and get the value of T. So unlike in the other case where it abruptly ends. In this case, if we have the S-shaped curve where it goes asymptotically to the limit and so this can give you an estimate and you will find that this Tm will be in between. You have the static R by P ratio, which is the highest and this will, you will have the Hubbert model or Tm. And then this will be in between this and the exponential. T for the exponential growth model, which will be the smallest so it will be somewhere in between and this is one of how we can do this. This curve which we have is symmetric about the point of inflexion. (Refer Slide Time: 20:57) Instead of this, we can also have other curves, other logistic curves, not commonly used but there could be the Gompert’s curve for instance and you can try this out. This is where Qp is Q infinity e raise to minus b, e rest to minus kt. So, we have Q infinity and you have these two coefficients b and k you have to take log twice and then you can get these coefficients by linear regression substituted here. The curve is not symmetric about the point of inflexion. So, we have choices in and it has it. It has a different kind of characteristics. So, we looked at Hubbert’s model and we just saw that we can calculate this point of inflexion. We, this model has been used to estimate this is where the world oil when it will peak and in many of these estimations, what has happened is that technologies have changed and the reserve estimates have changed. So, sometimes this whole concept of peak oil has been questioned. (Refer Slide Time: 22:27) The cumulative production has proven reserves and if you see some of these so you can also express this model in terms of this expression which has a cost component which is very similar to the model that we are talked of. (Refer Slide Time: 22:40) There also these models which have been used for different countries where you have a multi-Hubbert model which means that you start with one particular peak. And then if we find reserves for instance you use shale oil or you use some other things weather technology has changed you are going for the second peak. (Refer Slide Time: 22:55) And there have been modifications of this. So, this has been sort of the history, historical production, cumulative production, but we have extended beyond conventional oil and gone into the unconventional and this is because in previous years we had certain technologies where which involved a certain amount of certain type of drilling. We now have the possibility of cost-effective even horizontal drilling. And we have this concept of fracking where now we are using shale oil, unconventional resources. (Refer Slide Time: 23:33) So, it is this paper which has shown in Brazil you had these multi, you had this first cut where we have a production and it goes down and then it goes to the next level and so basically, we have these kinds of multi Hubbert curve, you go to one peak. Then because of the technology improvement go to the next peak and so on. So, these are ways in which we try to understand how technology and reserves consist. And there are many different studies where they are done this kind of Hubbert curve analysis. (Refer Slide Time: 24:08) This is a news article which talks about the different kinds of oil drilling technologies over the years and you can see very clearly that there have been a lot of improvements in technology. So,essentially what happens is that earlier there were, there were resources which would not be considered economically economical as sources of oil, but today they will be considered as something which is economical and this is why we have different kinds of production. (Refer Slide Time: 24:55) There are if you look at the global energy assessment you will find that there are this estates for conventional, unconventional oil, coal and you have the reserves and resources and you find that we have a significant amount of stock if you add up the resources and reserves. And so that is not currently a constraint-based on the present thing. But of course, there is the problem in terms of the carbon dioxide which makes it problematic to use fossil fuels. (Refer Slide Time: 25:19) And you can see clearly that the oil resources also over time if you plotted you see that there has been an increase. And this shows this is an interesting sort of image which shows the kind of discoveries and production. And you can see that oil discoveries have been now declining. Production of course is increasing. (Refer Slide Time: 25:38) And you can have details of this in terms of different regions what are the production reserves and you can if you are interested in you can look at this global energy assessment the resources chapter and you can look at some of these details. (Refer Slide Time: 25:53)The other approach which is the approach which has been proposed by Adelman and others is where they were talking in terms of not a static estimate of reserves. So, the idea is that based on what has known technology you can have different kinds of and the prices at which one can get different kinds of supply. So, as technology improves you can have an increase in the resources and reserves and on the other hand there is resource depletion. So, there are these two kinds of trade-off. (Refer Slide Time: 26:32) So, there is this approach which is now called the supply curve approach, where we estimate at with different kinds of technologies what kind of reserves are available. So, this is the kind of this is showing for conventional oil enhance oil recovery, tar sands and others and so on. So, one can have essentially different elements of it, which relates to price and supply and for each of these, when we talk about stocks, we talk about a supply curve at different price levels and the kind of available costs. (Refer Slide Time: 27:06) And similar things are done for fossil and uranium for instance in the case of uranium there is there could be a certain amount of reserves similar things for natural gas. You can have this for a gas supply curve, you can see different amounts at different kinds of prices. So, that adds a different dimension and you can see the sources and put the kind of values which are there. So, this is a different approach. Unlike the, we have seen this static R by P ratio, the exponential and then the Hubbert curve the logistic growth and then we have the supply curve option. In the supply curve option, we are saying that it is not a static amount the, it is not a fixed finite resource but there is a resource which is a function of technology and costs and a different cost there will be different amounts of supply. So, this is one of how you can do this. (Refer Slide Time: 28:06) You can look at details of this through some of these references, the global energy assessment and some of the papers Adelman’s paper and the peak oil concept. So, what we have done is we have looked at essentially resources, which are stocks and which are considered to be nonrenewable or depleted, we should remember that in all of these cases coal, oil, natural gas is also renewed. They are formed over natural processes where vegetation is coming under pressure and it comes under some sets of changes and over thousands of years, you have these resources and reserves formed. However, the rate at which we depleted is at a much faster rate than the rate at which it is renewed. So, for all practical purposes, these are known as depleted, in the case of these resources which we are considering as stocks, there are different ways in which we can classify based on the probability of occurrence, based on the economics of it and we talked about the Mckelvey diagram. We then said that given a certain estimate we can have different estimates of the time for which it would lose. We looked at the static R by P ratio. We looked at the exponential and we looked at the logistic growth curve or the Hubbert curve model. We also said that there are limits, there are problems with these kinds of approaches and maybe what we can look at the supply curve at different kind of prices. So, this is all in terms of stocks. There are also a whole set of resources which are renewable resources which are going to flow and that is the next thing that we will take. Thank you. We are carrying on with the course on energy resources, economics and environment. We have been looking at energy resources in the previous lecture. We looked at, the fossil fuels and the depletable resources. So, in the characteristic of those resources were that there were stocks and that means that you can have if you talk about coal or you talk about oil or you talk about natural gas you can store it. There is a fixed amount of reserve that is there and we saw methods by which we can quantify, how much time that resource will be lost. We now want to look at a different set of resources and these are renewable resources. So, as we discussed earlier even fossil fuels are also renewable in the sense that they have been formed by natural processes which occurred over thousands of years. But the rate at which we are using them is much faster than the rate at which it gets renewed. So, for all practical purposes, they are depletable. We now look at another category of resources which are renewable which means that the rate at which they are being renewed is much faster than the rate at which we are using them and so for all practical purposes they are inexhaustible and renewable and in such cases, the rules or how we analyse these resources will be different because unlike the earlier case most of these are flows, not stocks. Of course, in the case of biomass, there are flows which have been converted into stocks. So, biomass is a little different. But for all other resources that we are considering renewables, these are flows. So, then there will be different ways in which we will characterize them. So, for this, you may look at the Global Energy Assessment which was done by IIASA and published by the Cambridge University Press. It is available in the public domain. So, Chapter 7 of that has the listing of different kinds of energy resources and the potential and we will cover some of those things from that chapter. (Refer Slide Time: 2:46) So, if you look at it first is if you look at the different kinds of options that we have, we have a whole set of primary energy options and from the primary energy options as we saw there is a there are different kinds of conversion steps with to get our end-use. And in all of this electricity is one of the major roots. So, we take electricity either it could be from, we talked about earlier about fossil fuels. But even if you look at renewables we can look at solar giving us electricity, we can look at the wind, we can look at hydro, we can look at the ocean, we can look at bioenergy are could be nuclear and that electricity then can go and the used in different end uses. We can also look at directly hydro giving us kinetic energy or hydro giving us electricity and in the case of a fossil one of the intermediate steps is thermal heat which gives you the power and then that power is being used, in some cases, you are using heat directly for cooking or we are using heat directly in the industrial processes. So, let us take a look at the in global energy use. What is the current share of renewables? (Refer Slide Time: 04:12) So, if we look at this figure this is from the renewable energy update of 2019. This dataset is for the year 2017. You find that almost 80% of the primary of the final energy use is coming from fossil fuels, off the remaining its traditional biomass which accounts for a large part and nuclear, traditional biomass almost 7.5% and nuclear about 2%. Of the 10% or 11% which is coming from renewables a reasonably large amount is from hydropower and some coming from modern renewables which is accounts for about 2%. 1% is biofuels for transport. (Refer Slide Time: 05:13) If we look at this a little more in detail by the sectors and the end uses, if you look at the end uses and transport is about one-third of the total and in transport, we have 3.3% just from renewables of which 3% comes from biofuels. So, currently, the penetration of renewables in the transport sector is relatively low. In the power sector, we have 26% of renewable energy of which large hydro is also considered and then we have wind which is the largest chunk. And then you have solar PV and bio. In the heating and cooling, there is a lot of 9.8% coming from renewable energy and some of it most of it is from traditional biomass and some of it coming from renewables. So, as we see overall the renewables account for a relatively small percentage but we expect this to grow. So, we would like to first of all see what is the potential? What is the technical and economic potential for renewables and the how is that potential distributed amongst the different sources? (Refer Slide Time: 06:29) So, this figure is a global energy source of renewables, which is for the year 2008. And this is from IPCC special report on renewables. You may want to look at this and download this. This gives you an overview of all the renewables either status of that in 2008. It is a slightly old t the relative magnitudes remain the same. So, if you look at this figure you can see that the largest chunk continues to be biomass a most of it is still being traditionally used. Most of it is being used in the domestic and cooking sector. For all of the other renewables and largest chunk goes into electricity, CHP and combined heat and power and geothermal both use for electricity as well as directly. And solar thermals a small percentage. A large hydro is a very large percentage and the total. If we see this total we are looking at something of the order of fifth, of the order of 50 EJ. And if you look at this in terms of the global, we talked of 450 to 500 EJ. So, it is a relatively small percentage but it is already sort of getting into the mainstream and it also has a significant amount of growth and it is growing at a faster rate than the fossil. So, we expect future energy mixes to be much more renewable. (Refer Slide Time: 08:10) So, if we look at the electricity sector, as we as I told you the renewables in the electricity sector the largest chunk for electricity is coal, followed by natural gas and then hydro and some amount of nuclear. Nuclear globally is a reasonable amount. It is about 13, 14 % and then, as we said, this is the renewables of which wind is the largest chunk and then bioenergy is also reasonably high. (Refer Slide Time: 08:56) Solar photovoltaics growing relatively small but growing fast and you can see this growth in terms of these numbers. And you can we can see in this case if you look at the graph at the bottom you see that solar PV starting from a small base but growing at a much, much faster rateand overall if you see its wind and PV and renewable which is growing at a reasonably fast rate and that is the a that is the signal that is there that the renewables are growing at a much faster rate than fossil and the share of renewables in our mix is going to be higher in the future. So, now let us look at each one of these in terms of the potential. (Refer Slide Time: 09:43) Let us look at first hydro. So hydro for a long time we have been having these large hydropowerplants. And if we look at this assessment, this is from the global energy assessment you will find that if you look at solar power which is evaporating the water and causing the flows of hydro andthe energy in the water cycle you find that this is a very large percentage. We are talking of 504,000 EJ/year which is orders of magnitude higher than what is required in terms of energy, theoretical potential, if we look at from this, what percentages, what is a potential which can be taken off in terms of the runoff and the flows and converted into energy, you get off the order of about 200 EJ/year which is again orders of a magnitude similar to the kind of energy required globally by the earth. From this, if we look at the different sites that reduce to a potential of 140 to 145 and with aparticular use factor. This comes out about 50 to 60 EJ, then looking at economic factors. This can come to about 30 EJ. So, that is the estimated potential in 2010. Now, which is a reasonable amount of the total which can be provided by large hydro? The problem with large hydro is that in this involves submergence of large areas of land, and because of this, there is resistance to this from the people who were displaced. There are also problems in terms of sometimes they are on-site where there is a, which are earthquake-prone. There is a problem in terms of when you have a large water quantity it might affect the disease vectors. And so, in many countries of the world, there is opposition to large hydro with the result that large hydro has not been growing at the rate that is used to grow earlier. And because of that, we are not quite sure that the total hydro potential will be realised. But there is a reasonable amount of potential. (Refer Slide Time: 12:15) You can see, of course, in terms of generation. There is an overall growth in the hydropower generation, and you can see that if we look at the distribution of hydro in different countries and this is in terms of the generation which we are looking at, about 14000 TW/year is the hydropower technical potential which is a significant proportion of the electricity use. In the installed capacity, if you see this is the kind of distribution of the installed capacity in the different continents, and in 2009 we are talking of 920 GW which was installed. The updated values in 2017, about 1267 GW, so hydro large hydro has been growing, but we do not expect it to grow at a very much faster rate because of the kind of opposition which is there. So, we can have in the case of hydro, we can have dams, we can also have run of the river schemes. So, the question of large hydro and small hydro in many countries of the world, large hydro was excluded from the calculation of renewables. This was done primarily because large hydro already had a large number and the renewables e relatively small. So, in order to make it distinguished we looked at small hydro and different countries had different characteristics in terms of what was considered as small hydro. But typically, you knowwhich started from 10 kW going up to 5 MW in some cases going up to even 25 MW. Now, in the case of hydro it could be run of the river’s schemes or even with low heads and low heads as low as 3 meters have been considered to be viable in the smallest areas, we look atschemes which are not necessarily grid connected. They could be isolated and if you go if the megawatt range, then they could be connected to the grid. In all of this, the question is whether the water flow is annual or it seasonal. So, there is a capacity factor which is there. But in most of the cases there is a reasonable amount of potential of cost-effective power generation because the water itself the operating cost is negligible. It is only the initial capital cost.