Withput any knowledge of your data model, maybe what you could do to solve your problem is: first calculate the average TotalUnitCount per Operation (I assume you have a field like Operation ID) and later sum all the averages. This could be solved using an Aggr() function:
Sum(Aggr(Avg(IF(BegDateOfOperation <= (CalendarDate) and EndDateOfOperation > (CalendarDate), TotalUnitCount)), Operation_ID))
I went back and took a closer look at the results of using the expression: Sum(IF(BegDateOfOperation <= CalendarDate and EndDateOfOperation > CalendarDate, TotalUnitCount))
It looks like the results are correct if I am also using CalendarDate as the dimension in my line chart. However, I need to use a cyclical group of date granularity in my chart. The problem is that TotalUnitCount for the same property becomes counted multiple times once I switch date granularity for the dimension to something like CalendarYear.
I am pretty sure that what I need to do is use an AGGR statement. But I don't know how to do this. Do you have any more ideas? Please let me know.