With this lesson, we begin to look at how to compute power limits for battery cells and battery packs. We're going to begin by reviewing the method that I shared with you in the first course of this specialization, but that was probably a long time ago, so it's time for a refresher. In this lesson, we look again at the standard hybrid pulse power characterization test or the HPPC test that was specifically specified originally by the partnership for a new generation vehicles or PNGV. Using this approach, power is calculated to enforce limits on cell terminal voltage that is predictive over the future time horizon of Delta T seconds but updated at a faster rate than once every Delta t seconds. Remember that to perform this test, we first run some cell tests in the laboratory to gather cell parameter data needed to compute the HPPC power limits. We begin these tests with a fully rested cell at some state of charge. Then, we discharge this cell for Delta T seconds and we continuously record the voltage while we are discharging. Then we rest the cell until it is fully rested once again and in an equilibrium state and then we subject this cell to a pulse of charge current for Delta T seconds. Again, we continuously measure and record voltage while we are doing this. We compute a discharge resistance as the maximum change in voltage while discharging divided by the absolute discharge current. We compute a charge resistance as the maximum change in voltage while charging divided by the charging current. If there's any ambiguity as to the sign of the results, we simply take the absolute value of the discharge resistance and the charge resistance because we know that both of these values must be positive. So we perform this test and we compute the resistances at multiple different resting states of charge and multiple different ambient temperatures and we store the results for later use. When we're computing these HPPC power limits, we assume a very simple cell model, a early crude primitive cell model. The circuit drying of this model is illustrated on this slide and you can see that the batteries cell as assumed to comprise an open-circuit voltage that depends on state of charge as well as an equivalent series resistance but that is all the model completely ignores hysteresis voltages and diffusion voltages. So we compute terminal voltage using this model as the open-circuit voltage evaluated at the present state of charge, minus the cell current multiplying the series resistance of the cell. So we can rearrange this expression to solve for current and we find that the battery cell current is equal to the open circuit voltage, minus terminal voltage all divided by resistance. To compute the power limit, we assume that we are concerned only with keeping the terminal voltage of the cell between limits of v_min to v_max. So to compute the discharge power, we set the resistance equal to the discharge resistance that we found from the lab tests we talked about on the previous slide and we set the terminal voltage equal to v_min. Then we calculate the maximum discharge current constrained by voltage using this equation here, the discharge voltage is equal to the open circuit voltage minus the v_min all divided by the resistance. We convert this discharge current into power by multiplying by voltage and we also multiply the cell power by the number of cells in the battery pack to arrive at park power. So the maximum discharge power permitted is equal to the number of cells in series, multiplied by the number of cells in parallel, multiplied by the minimum permitted voltage, multiplied by the minimum of all permitted and discharge currents. So, if this model were completely accurate, then discharging at this level of power for Delta T seconds would guarantee that the terminal voltages of all cells stayed within their designated limits. So let's consider charged power. Once again, we assume that current is equal to open circuit voltage minus terminal voltage all divided by resistance. To compute charged power, we set the resistance equal to the charge resistance found from the laboratory tests that we discussed on the first slide and we set the terminal voltage equal to the maximum permitted voltage. Note though when we're talking about charge current, we're talking about a minimum value because charge current is always negative and so the maximum magnitude of charge current is a minimum value and a signed sense. So we compute the minimum charge current as equal to the open-circuit voltage of this cell at this point in time, minus the maximum permitted voltage all divided by the charge resistance. So again, this is equal to the maximum absolute current, although it's a negative number. So it's a minimum and assigned sense. Pack power is then calculated in a very similar way to what you learned for discharge pack power. We take the number of cells in series and we multiply by the number of cells in parallel and we multiply by the maximum permitted designed voltage and then we multiply it by the maximum of all of the charge currents which is the charge current that's closest to zero. So we're using the maximum of charge currents because that's a minimum in a magnitude sense because all of the charge currents are negative. That brings us to the end of this lesson, where we have reviewed the HPPC method for computing power limits, that are based on trying to ensure design limits on cell terminal voltage the HPPC power limit estimation method first collects current and voltage pulse data from cells in a laboratory. It then calculates discharge and charge resistances of all the cells from that lab test data. It computes power using a very simplified cell model that considers only open-circuit voltage and a series resistance for every cell. It clamps or holds the terminal voltage of the model either equal to the maximum or the minimum permitted voltage and uses the corresponding resistance value to compute the maximum magnitude currents and the charge or discharge direction. Then it multiplies those currents by voltage and by the number of cells in series and parallel to make estimates of the power limits. This HPPC method is limited in a number of significant ways. First, it does not consider enforcing design limits on state of charge, future state of charge or on maximum absolute current that might be permitted by the electronics or on maximum absolute power that might be permitted by the load. We're going to spend the rest of this week learning how to generalize this HPPC method to add computations using these design limits in addition to the voltage limits. Also, the HPPC method uses an overly simplified cell model. So next week you will learn how to use the entire enhanced self-correcting cell model in order to compute more accurate power limits.