You can refer below code
You have to get selected value in session and set that value after postback to the dropdownlist.
protected void DropDownList1_SelectedIndexChanged(object sender, EventArgs e)
Session["DDLValue"] = dropdownlist1.SelectedValue;
On page load, get the session and set it to drop down list if it is not NULL:
if(Session["DDLValue"] != null)
dropdownlist1.SelectedValue = Session["DDLValue"].ToString();