You can use with:

with pd.ExcelWriter('test.xlsx', engine="openpyxl", mode="a") as writer:
    d1.to_excel(writer,sheet_name="d1")
    d2.to_excel(writer,sheet_name="d2")
    writer.save()

writer.close()

update

This should work just note that the a blank file needs to be created before hand. You can just create a blank file using python if you want. I created a simple loop to, in some ways, mimic the essence of what you are trying to accomplish:

import pandas as pd
from openpyxl import load_workbook

d1 = pd.DataFrame({"A":['Bob','Joe', 'Mark'], 
               "B":['5', '10', '20']})
d2 = pd.DataFrame({"A":['Jeffrey','Ann', 'Sue'], 
                "B":['1', '2', '3']})

dfs = [d1,d2]

for i in range(len(dfs)):
    sheet="d"+str(i+1)
    data = dfs[i]
    writer = pd.ExcelWriter('atest.xlsx',engine="openpyxl", mode="a")
    writer.book = load_workbook('atest.xlsx') # here is the difference
    data.to_excel(writer,sheet_name=sheet)
    writer.save()
    writer.close()

or here is the modified first example:

d1 = pd.DataFrame({"A":['Bob','Joe', 'Mark'], 
               "B":['5', '10', '20']})
d2 = pd.DataFrame({"A":['Jeffrey','Ann', 'Sue'], 
                "B":['1', '2', '3']})

writer = pd.ExcelWriter('atest.xlsx', engine="openpyxl", mode="w")
d1.to_excel(writer,sheet_name="d1")
writer.save()
writer.close()

writer = pd.ExcelWriter('atest.xlsx', engine="openpyxl", mode="a")
writer.book = load_workbook('atest.xlsx')
d2.to_excel(writer,sheet_name="d2")
writer.save()
writer.close()