In this lesson, we continue to explore requirement three of the battery management system, which has to do with performance management. We're thinking specifically about the requirement for state-of-charge estimation and it's worthwhile to think a little bit about what truly does state-of-charge mean. You've probably already have a good intuitive feeling for state-of-charge based simply on using electronic devices that have state-of-charge meters. This might include the state-of-charge indicator on your notebook computer or on your mobile phone or on some other electronic device. So you already understand when the meter reads 100%, that the battery is fully charged, and when the meter reads 0%, the battery is empty. But is this display something that is empirical and not truly physical? Or does it correspond to something that is truly physical, even though it may not be measurable directly? The diagram below shows a schematic of a lithium-ion battery cell, where we're thinking about how lithium moves inside the cell as the cell is operated. When we discharge the cell, lithium moves from the negative electrode across to the positive electrode. And when we charge the cell the opposite happens, lithium moves from the positive electrode over to the negative electrode. So if you think about it, when the cell is fully charged, there will be a lot lithium in the negative electrode. And when the cell is is discharged, there will be much less lithium in the negative electrode. So thinking about it that way, we can connect our intuitive idea as state of charge directly to the physical quantity of how much lithium is present in the negative electrode particles. Or equivalently, what is the concentration of lithium in the negative electrode particles of the battery cell. In this specialization we do not spend very much time talking about modeling or understanding lithium ion battery cells at the electrochemical level. If this is something that's interesting to you, then I recommend that you study my first textbook on battery management systems volume one on battery modeling. But for this specialization, it's sufficient to have a very simple understanding of this one concept that we talk about on this slide. You already learned in week two of the course that lithium can be absorbed into the negative electrode particles by entering into the empty spaces of their crystal structure. There are some maximum amount to how much lithium can possibly enter into those spaces before it's completely full. And we call this concentration Cs,max. C for concentration, S for the solid particles of the electrode and max, of course, for the maximum value. So when the particle is completely full of lithium, the concentration of lithium in the particle will be Cs,max. And similarly, when the particle is completely empty of lithium, the concentration will be 0. At any other intermediate point, we have some other concentration. And when considering cell state of charge, the important quantity is not the local concentration at any point in the electrode or even at any point in a particle in the electrode. But instead it's actually the average concentration over the entire electrode which I'm calling on this slide Cs,avg perhaps at time k. So we can normalize this present average concentration by dividing it by the maximum possible concentration to give us a value that's always between 0 and 1. And this value is called the stoichiometry of the electrode. And it's often denoted by the symbol theta and here also with the subscript k which refers to a point in time indexed by k. The stoichiometry is a number, as I said, between 0 and 1, and we can consider it to be a kind of an electrode state-of-charge. But we must understand that this electrode state-of-charge is different from what is reported to the user as a cell state-of-charge. You might remember that in some electrodes, we're not permitted to charge above a certain concentration or discharge below a certain concentration. So the stoichiometry is not permitted to use the entire possible range between 0 and 1. There's a lower limit to the stoichiometry before the cell begins to deteriorate quickly or is unable to provide adequate power. Similarly with the upper limit and so this cell is always operated above the stoichiometry of 0. So when a cell is at 0% state-of-charge over for the whole cell, the stoichiometry for the electrode, which we call theta 0% is actually a number bigger than 0. And correspondingly, when a cell is at 100% state-of-charge, the stoichiometry or the electron state-of-charge is at some value less than 1 that we call theta 100%. So we're going to operate the cell not between a stoichiometry of 0 and 1 but rather between a stoichiometry of theta 0% and theta 100%. In other words, when the cell is at a stoichiometry of theta 0%, it actually has a cell level state-of-charge of 0% even though the electrode state of charge is above 0. And when the cell stoichiometry is at theta 100%, the cells state-of-charge is at 100% even though the electrodes state-of-charge or stoichiometry is less that 1. So this might actually give you some insight into how it's possible to overcharge a battery cell or how it's possible to undercharge a battery cell. It's physically possible to remove more lithium than the theta 0% to get all the way down, at least in principle to 0. And it's possible the charge above theta 100% theoretically up until you get up to 1. But we don't operate out of those regions, at least we intend not to operate outside of those regions for our safety purposes and for cell longevity purposes. So keeping the discussion in mind of what we've said so far, we can write the cells state-of-charge as equal to the present stoichiometry of the negative electrode minus the 0% value of that stoichiometry, all divided by the range between the 100% and 0% stoichiometries. And this relationship, while we've discussed it through the negative electrode, actually holds to the positive electrode as well as indicated by the second equation. The cell state-of-charge is also equal to the stoichiometry of the positive electrode minus its 0% value, all divided by the range between the 0 and 100% values. The tricky bit which comes in when we're using positive electrode numbers is that the numerator and denominator of this equation will both be negative, will still result in a positive state of charge but it's thinking about it a little bit differently. So the state-of-charge of a cell is actually a physical quantity. It's related to the average concentration of lithium in the entire negative electrode. It's also related to the average lithium concentration in the entire positive electrode. So if we were able to create a sensor that could measure this average concentration directly, then we could measure state-of-charge directly. But at least at this point in history, no such sensor exist. So it's natural to ask whether the concentration of lithium in the negative electrode is somehow directly related to things that we can measure such as cell voltage. Well, it turns out that cell voltage depends on the concentration of lithium at the surface of the particles in the electrode, and in particular, at the surface of the particles that are in direct contact with the negative current collector. Remember that lithium as it diffuses into and out of the particles in the electrode causes transient concentration differences between the surface and the bulk interior concentration of the particle. And these concentrations become the same only after time has elapsed, when the cell has been allowed to rest in an open circuit condition. So remembering the state-of-charge depends on average concentration of lithium and that voltage depends on surface concentration at one point. We can see that there is not a direct relationship between state-of-charge and the voltage that we measure right now. The two things are not the same. So there's no immediate direct correspondents between cell voltage and cell state-of-charge. In fact, cell voltage can change for a number of reasons, even though the state-of-charge could be constant. For example, if we change the cell temperature, the cell voltage changes a little bit but the state of charge does not change. If we have been using the cell and then allow it to rest, the voltage changes as it's resting but the SOC doesn't change. And also due to a phenomenon called hysteresis, cell voltage also depends not only on what's happening presently but on the recent history of how the cell has been used. And that can lead to different steady state values even at the same SOC. And you're going to learn more about all of these effects during the second course of this specialization when we discuss modelling behaviors of lithium-ion battery cells. So in summary, state-of-charge changes only when we change the average concentrations inside of an electrode. And these concentrations change only due to passage of current through the cell, either charging or discharging the cell or due the self-discharge in the cell. So voltage does turn out actually to be very useful as in indirect indicator of state-of-charge and is very helpful in state-of-charge estimation. But it's not a good independent predictor of state-of-charge, that is, by itself. So if state-of-charge changes only when current passes through the cell, either due to some external circuitry drawing that current or forcing that current, or due to self-discharge. Why don't we simply use measurements of cell current to estimate the state-of-charge? There's a perfectly valid equation that relates state-of-charge to cell current, which I've written on this slide. And if you look at it, it's saying that the present state-of-charge at times t is equal to some initial state-of-charge at time 0, minus the total net integrated amount of current that is passed through the cell waited by an efficiency factor eta and divided by the total charge capacity of the cell. What this equation is doing is integrating or counting the net number of coulombs of charge that have entered the cell and relating that to some initial charge level to update the present charge level. In this equation, electrical current is considered to have a positive sign when the cell is being discharged, and a negative sign when the cell is being charged. In the equation, also the parameter eta, the Greek variable eta is equal to the coulombic efficiency of the cell, which is usually approximately equal to 1, but is never greater than 1. And Q is the total capacity of the cell measured in ampere seconds, or coulombs. This capacity, by the way, is a physical quantity. It measures the number of vacancies in the electrode structure and that crystal structure that could hold lithium between the stoichiometry of 0% and 100%. So this is not a function of temperature or rate or any other such thing, it's different from a discharge capacity which is what is commonly put on a data sheet for a battery cell. It's a total capacity, it's the total charge capacity, if you will and not at discharge capacity and we're going to look at that in more detail as we pass through the specialization. This equation I've said is not very good for estimating state-of-charge and yet is actually an exact representation of state of charge of the battery cell. We run into some very serious problems if we try to use it as our only basis for estimating state-of-charge though. Methods that do this are called coulomb counting because we're counting or measuring the number of coulombs that go into and out of a battery cell. But coulomb counting suffers very serious inaccuracies when we can't measure current perfectly, which is always the case. Or when we don't know coulombic efficiency perfectly, which is always the case. And when we don't have a perfect estimate of the total charge capacity, which is always the case. And when we don't know the initial state-of-charge at time 0, which is always the case. So for this reason, coulomb counting should never be used in a battery management system as a primary state-of-charge estimation method. And you will learn that there are much better ways of estimating state-of-charge. And you will learn how to implement those in the third course of this specialization. So in this lesson, you've learned that cell state-of-charge is directly related to something that is truly physical in a battery cell. Often times people talk about a battery pack state-of-charge and mean by that some quantity that relates collectively to many battery cells at the same time. So I'd like us to think for a few minutes about what is the battery pack state of charge. Let's consider the example illustrated by the two-cell battery pack drawn on the right side of this slide. The drawing shows one battery cell that's fully charged and it's shaded green, and one battery cell that's empty and it's shaded red. And these cells are connected electrically in series, so the charge passing through one cell must also pass through the other cell. And when we compute a battery pack state-of-charge, what should that value be? Should we define, for example, the battery pack state-of-charge to be 0% because we recognize that this configuration is not able to discharge? Notice that if we were to try to discharge, the current would flow out of this battery pack and that would deplete the level of charge in the green cell, which would be fine. But it would also attempt to deplete the amount of charge in the red cell, which we could not or at least should not do, because the state-of-charge of that cell is already 0%. So we should not really define a battery pack state-of-charge to be 0% because that implies that we are able to charge the battery pack and we can't do that either. If we were to charge the battery pack, that would increase the level of charge in both cells. And if we increase the level of charge in the red cell, that would be fine but if we increase the level of charge in the green cell it would cause an over charge situation which we don't want to happen. So defining a battery pack state-of-charge of 0% does not make sense in this example. So maybe we should define the battery pack state-of-charge to be 100% because it's not possible to charge. But that's not really a good solution either, because that gives us the idea, the battery pack state-of-charge is 100%, so I should be able to discharge and we already know that we cannot do that. So how about reporting a battery pack state-of-charge as the average of the cell state-of-charge, in other words, 50%. But we really can't do that either because that implies that we're able to both charge and discharge the battery pack and we can't do either one. So I propose to you that the term battery pack state-of-charge, is something that frankly does not make sense. It is ill-defined and in my opinion, it should never be used. One thing that this discussion points out is that the battery pack should never end up in the situation that I have drawn. It's a requirement that we perform cell balancing, or cell equalization to avoid this situation. But if we don't balance a battery pack, it is actually physically possible for a pack to arrive in the state of charge configuration drawn on this slide. And if we even get close to this level of imbalance, we can see that the term or the concept of battery pack state-of-charge does not make sense. But people try to use it, and we might want to think about why someone would even care to know what a battery pack state-of-charge is. One reason is for set point control. For example, in a hybrid electric vehicle, we want to keep the states of charge of all the cells around or close to some set points so that we can guarantee that there will always be some margin for charging and some margin for discharging the battery cells in the pack. For that application, knowing the average state-of-charge of all the cells in the battery pack might work as a set point as long as none of the cells are at some extreme points. That is, a battery pack state of charge is not needed to perform setpoint control instead, battery pack average state-of-charge is needed. Adding that one term, average there clarifies exactly what we mean and makes it a lot more precise. Another application for which we might want to know a battery state-of-charge is at distance to empty kind of calculation or a fuel gauged kind of calculation. But the true issue is not the state-of-charge of a battery pack which I again, argue does not makes sense. But instead how much energy remains in the battery pack that might be used and this is not the same thing as state-of-charge. So in my mind, there is no true application that demands knowledge of battery pack state-of-charge. And so my recommendation is that we don't even attempt to create an artificial quantity or another quantity by substitute battery instead. So finally, to summarize this lesson, you've seen that a battery cell state-of-charge has a real physical basis that can be connected directly to something happening internal to the cell inside of it's fundamental electrochemistry. If we somehow knew exactly the average lithium concentration in either the negative or the positive electrodes, then we could compute the present stoichiometry of electrode and we can compute the cell state-of-charge given the equation that we've already seen. Generally though, we don't know this concentration and so we ask whether direct measurements of voltage or electrical current can be use to tell us what is the state-of-charge. And we've seen that it is not possible to use only voltage or only current to tell us what state-of-charge is. Somehow we must combine these measurement in an intelligent way, and use them to help us to estimate state-of-charge. And you will learn how to do that in the third course in the specialization. Finally, battery packs state-of-charge is a concept that truly doesn't make sense, and I recommend that it not be used. Usually what people mean when they are talking about a battery pack state-of-charge is some kind of average of the states-of-charge of all of the cells in the battery pack. If that's what's truly meant, then instead, we should refer to that as a battery pack average state-of-charge. And there are some real examples, real applications where batter pack average state-of-charge is meaningful and useful quantity and we will see some of those as we progress through the specialization. In the meantime though, we're going to think only about cell state-of-charge. And that brings us to the end of this lesson but we're going to proceed in the upcoming lessons to think a little bit more about performance management.
