The way we found around this problem, that QB's has known about for years, is to customize your PO settings. By changing your PO settings so that the Quantity box now is called Enter the Final Total. Go into your PO, select formatting at the top instead of main. Then select Customize Data Layout then columns. Where the Quantity box is on the left, call it Enter the Final Total in the customize section on the right. We use the Other 1 column, call it Quantity, and Other 2 columns, call it Price Per. These two columns are dummy columns that look good on the PO but do not serve any purpose for partial bill payments. For the Rate column we just enter the name, Enter 1. So you always enter a 1 on this column. If you do not enter a 1 in the rate column the amount column will be zero. You can play with the print order and screen viewing options with the checkmarks. You can also select the order of the columns so it looks good on your PO. We use Quantity (Other 1 old column), Description, Price Per (Other 2 old column), and Amount on our printable PO.  Create a PO using this method. When you are entering the value in Enter the Final Total column you created, you are essentially entering it into the Quantity field in QB's. You can enter multiple lines if you have a multiple items to enter with different values. 

Now go to enter a bill against that PO. Once you select the PO, the information will pop up. The quantity field will now pop up with the full value of your PO that you entered in your created 'Enter the Final Total' value from your PO. So, if your PO is $10,000, enter the value of the bill in quantity area now by erasing the full PO value and entering say $1000. Post the bill and your PO will be reduced by the amount of your bill. It will still show the value outstanding as $9000 now with $1000 being paid out. When you enter another bill against the PO, the $9000 amount will appear in the Quantity field in billing meaning that there is $9,000 left on the PO. If you have entered multiple lines and have multiple figures on your PO, delete the other lines and just enter the value on the one line or multiple lines you are paying.

Try this out on a test PO and bills so you can get the hang of it. 

Hope that helps.

PS: Be careful as QB's does not have any rails on the values. If your PO is $1000 and you enter the bill amount as $10000, it will allow you to complete this and then show you that the PO is closed. You may have overpaid your vendor and unless you did a reconcile you would not know.