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Appendix D Example n The procedure described in Appendix D is meant to determine a battery’s performance parameters from the data taken during a HPPC test. The goal is to determine R o and R p with respect to SOC; the time constant tau also needs to be determined. What is known is I L, Voltage, and A-s with respect to time. The procedure assumes that an equation for voltage can be created that has I L, I p and A-s its variables. The coefficients will then be R o, R p, OCV’, and OCV. For this to work, I p will also have to be calculated. The procedure revolves around an Excel function named LINEST and assumes the battery model shown.

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Start by entering the appropriate labels in an Excel file; column numbers go up to twelve

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From a HPPC data file carry in Time, I load, A-s, and Voltage

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Enter a zero into the first cell of dT/tau and make a guess for tau, then use tau in the equation for the second cell of dT/tau (be sure to use dollar signs in front of the tau’s reference) dT/tau=(t i - t i-1 )/tau

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Copy the second dt/tau cell and use it to fill in the rest of the column

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The Ip column can now be filled in; enter a 0 into the first cell, and enter the formula for Ip from the manual into the second cell

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Copy the second cell and use it to fill in the rest of the column

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The LINEST function now needs to be set up; enter OCV’, Rp, Ro, and OCV under columns 9-12 and highlight a 5x4 block underneath them

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Go to Insert/Function, and click on LINEST

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The LINEST block needs each cell to be filled in; start by clicking on the square with the red arrow that is across from Known_y’s

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Highlight the entire Voltage column and hit Enter, this will create an array for the Known_y’s block

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The Known_x’s block is next; click on the square corresponding to that cell

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This time highlight the three columns under I load, Ip, and A-s and hit Enter

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Now that the Known_y’s and Known_x’s cells contain arrays, the Const and Stats cells need filled in

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For the purposes of modeling a battery, the last two cells can just have TRUE typed into them; to display the results hit Ctrl-Shift-Enter

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After hitting Ctrl-Shift-Enter the results will appear in the 5x4 block that was highlighted

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The first row of values are coefficients for a linear equation where Voltage is the dependent variable and I load, Ip, and A-s are the independent variables; OCV’ goes with A-s, Rp with Ip, Ro with I load, and OCV is the “b” value Voltage=(A-s)(OCV’)+( Ip)(Rp)+(I load)(Ro)+ OCV

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Use the voltage formula in the first cell of Estimated V load Voltage=(A-s)(OCV’)+( Ip)(Rp)+(I load)(Ro)+ OCV

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Copy the first cell to fill in the rest of the column, remember that the coefficients in the formula need to have a dollar sign in front

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Fill in the V error column by taking the difference between the measured voltage and estimated voltage; if the formula is good the errors should be small

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The third cell down in the leftmost column of the results is the r 2 value, this number indicates the accuracy of the formula; the formula is perfect if r 2 is equal to one, and the battery manual requires that r 2 be >0.995

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The rest of the cells give more statistics on the formula that LINEST created, but they will not be used in this application

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Tau affects all of the results, but in the beginning it was only guessed; to get the correct value, adjust tau so that it gives the maximum value of r 2, then consider that the true value

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For this example 9.4 was the best value for tau; changing tau by 0.1 caused changes in r 2 as small as 1x10 -6

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Sometimes there is “noisy data” in the Voltage column; a bad cell can be edited out by setting it equal to the corresponding calculated value

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Excel must be in “Iteration” mode for the editing strategy to work; this mode can be set by choosing Tools/Options/Calculation and checking the appropriate box

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Two charts can be made, the first is of V error vs. Time

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The second chart is of Measured and Estimated Voltage vs. time; both charts can be used to graphically see how well the LINEST formula follows the measured data

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n This procedure needs to be done for each pulse profile in a HPPC test. To save effort, copy the results of each profile into a separate Excel file (be sure to use Copy/Paste Special/Values). Then for a new pulse, paste in Time, I load, A-s, and Voltage into the old setup; LINEST should recalculate on its own. n Once all the resistance values are in their own file, plot them with respect to SOC. From this, polynomials can be created that can be plugged into PNGV models.

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