In this specialization, you have now learned different methods to make estimates of cell state of charge and state of health. Once you know this state information and that parameter information for all of the cells, you can use that to estimate cell available energy and pack energy using the equations that you learned in the first course of this specialization. So, the final major estimation task of a battery management system that you will learn about during the rest of this course has to do with computing power limits at the cell level and at the battery pack level as well. I bring your attention back to the illustration on this slide that I've shared with you a number of times already. Inside of a battery management system, the main program loop first measures all of the cell voltages and the pack current and the module temperatures. You learned about this during the first course of this specialization. Then, states of charge of every cell need to be estimated. You learned how to do this in the third course of this specialization using the mathematical models of cells that you learned about in the second course. Next, we estimate or update the state of health estimates for all of the individual cells as you learned about in the fourth course of this specialization. So, now in this course, you have already learned about balancing, and we're going to learn about power limits from now on. Remember that you did learn some simple methods for power limits estimation in the first course of this specialization. But first, we're going to expand on those methods, and second, we're actually going to replace them with some improved methods that give better results. Remember that a power limit specifies how quickly we are permitted to add or remove energy from a battery pack without violating some set of design constraints. In this course, we will assume that the principle design constraint has to do with the terminal voltage of a battery cell. That is, we desire to guarantee that the terminal voltage of any cell never exceeds some maximum value and never goes below some minimum value. This is what is commonly done in practice and battery management systems today. But I want to think about this a little bit with you. The reason that we might want to guarantee that the voltages of cells never go outside of some range is because we are concerned that the battery may become unsafe or it may age or degrade too quickly if the voltages ever go outside of a certain operating window. So, the cell voltage itself is not really our primary concern. Instead, our primary concern has to do with the damage that is being done to cells if their voltage is outside of a certain operating range and some safety considerations that are really related to that damage as well. So, it turns out that it is possible for battery cells to be operated outside of certain voltage ranges, at least for a brief Intervals of time and not really have any kind of measurable degradation experienced when we do so. So, the voltage limits turn out not to be the real concern. Our real concern is the rate of degradation and we use voltage limits in an indirect way to try to slow down degradation. In our research team at the University of Colorado and Colorado Springs, our primary research objective and goal is searching to find better ways to compute power limits that more directly addresses the concern of maximizing life by understanding how battery cells age and degrade. Then that research topic requires that instead of using the circuit type of models that you've learned about in this specialization, we developed physics-based models of the battery cells instead. The reason is that the physics models actually give us insight into the degradation mechanisms more directly that are causing the cell to age. If you choose to pursue the honors track of this specialization, you will learn more about how we could replace the equivalent circuit models that you've learned about with physics-based models and perform battery management using those with the objective of extending life, increasing power, increasing available energy and increasing safety. But for the remainder of the standard portion of this course for the remaining two weeks, we are going to assume that the voltage limits are the primary things that we are required to maintain. Remember that in the first course of this specialization, you learned that the power limit calculations are predictive and I will review this idea with you here. At every point in time, we must compute a value of power that we transmit to the load. That the load is permitted to use for the next future time horizon of Delta T seconds and we are guaranteeing that it is safe for the load to do that. So, at this point in time, I might predict that 10 kilowatts of discharged power is acceptable for the next 10 seconds. So, if the load discharges the battery pack at a constant level of 10 kilowatts for 10 seconds, then I'm guaranteeing that the battery will be completely safe over that interval and no voltage limits or current limits or power limits will be violated if the load does this. In particular, we're guaranteeing right now that no cell voltage can go below some minimum design voltage threshold and also perhaps that no state of charge will go below some design threshold and so forth. So, we compute the power limit, we transmit it to the load and we say that this power limit is guaranteed for the next Delta T seconds, but then we keep on recomputing the power limits and we transmit them to the load much more frequently than once every Delta T seconds. This allows us to have some error in our power limits calculations and then have those errors be corrected before the load has caused damage. If at some point we decide that the power that we first computed was too much and it would cause some damage, some voltage is to be too high or too low. So, the name that I give this idea and perhaps others do also is a moving horizon power limit calculation. We're looking over some future time horizon and giving a power limit estimate, but as we move in time, we keep on looking at that time horizon as it's progressing into the future. The remainder of this particular lesson is devoted to formalizing the problem that we desire to solve. Later lessons, we'll look at how to compute power limits based on this formal definition. So, specifically, our objective is to address one of the following three different problems. The first has to do with computing the level of permitted discharged power. We base this computation on the present battery packs state. We estimate from that starting point the maximum discharge power that can be maintained constant over a future time horizon of Delta T seconds without violating some preset design limits on cell terminal voltage, cell's state of charge, maximum design power or maximum design current. The second problem that we might address is actually exactly the same, but it has to do with charged power instead of discharge power. So, we begin with the present battery packs state, and we compute a maximum absolute charged power that can be maintained constant over a Delta T second future time horizon without violating preset design limits on voltage or state of charge or power or current. The third problem is simply a combination of the previous two. Many applications require that we compute limits both on discharged power and on charge power, and they might have different Delta T second moving time horizons for the two different computations. So, our calculation algorithm that we come up with must be general enough to allow for all of these possibilities. When we approach these problems to solve them, we are going to use some specific notation that I introduced on this slide. We will use the letter N to denote the total number of cells in a battery pack. We will use the notation v sub n of t to indicate cell voltage for cell number N in the battery pack at time t. This terminal voltage must always remain inside of design limits that span from V min up to V max, the minimum to the maximum voltage. We use the symbol Z subscript N as a function of time to indicate state of charge of cell N in the battery pack at time t. Every state of charge must remain within the design limit spanning from Z min up to Z max. We use the notation P subscript N of t to denote the cell power of cell number N in the battery pack at time t. Each one of these powers must remain within limits spanning from P min up to P max. Finally, we denote cell current as I subscript N of t where we enforce design limits on every cell's current that must exist between i min and i max. When we solve the power limit problem including all of these limits, it might seem like we are solving a very specific problem. But, we actually can see this as a very general problem. There are many scenarios that can be really accommodated and by this problem if we allow some of the limits to go to infinity for example. So, if there are no design limits on minimum and maximum current, then we let i min go to negative infinity and i max go to positive infinity and everything that we look at will still work. Or if there is no design limits on power, we set the minimum power to negative infinity and the positive power to maximum infinity and so forth. So, the method that you will learn about is actually very general and it encompasses pretty much any scenario that I could imagine that you need to compute power limits for. The limits that we talked about on the previous slide do not need to be static or fixed. They can be functions of temperature, they could be functions of any other particular factor that depends on the battery pack operating condition. We could even apply different limits to different cells in the battery pack if we had some reason for doing so. In our equations, we will assume that discharge current and discharge power both have a positive sign and that charge current and charge power both have negative sign. This is the standard convention that we've been using throughout the entire specialization. But if you choose to use the opposite sign convention, of course, it's quite simple to do so simply by multiplying by negative one in the appropriate places. As a final node on notation, the battery pack under consideration is assumed to comprise N as cell modules wired in series and then within each cell module, Np cells wired together in parallel. Either, one of these counts, Ns or Np may be equal to one or greater than one, and then of course the total number of cells and the battery pack is equal to Ns multiplied by Np. To summarize this lesson, we've started studying the final major topic in this specialization which has to do with estimating cell power and battery pack power limits. The question that we're trying to answer is, how quickly may we add or remove energy from the battery pack without violating some preset design limits? As a review, I reminded you that we estimate these power limits over a moving future time horizon of Delta T seconds. The overall problem is to estimate either discharge power or charge power or both over that horizon. Then we update these power limit estimates at a rate that's faster than once every Delta T seconds, more than once per future time horizon interval. Then after this review of moving time horizon estimation, I introduced some notational conventions that we will use. So, we're now ready to study some methods to estimate power.