Pell Grant Preservation and Expansion Act This bill amends the title IV (Student Assistance) of the Higher Education Act of 1965 to revise the Federal Pell Grant Program. Specifically, it: converts the Pell Grant program into a mandatory spending program; increases the maximum Pell Grant award for academic year 2019-2020 and adjusts it in subsequent award years to account for inflation; allows federal financial aid for higher education to be given to dreamer students (students who entered the country when they were under the age of 16 and who meet certain educational criteria) and students convicted for the possession or sale of illegal drug; requires the Department of Education (ED) to carry out a program that awards Federal Pell Grants to students in job training programs; requires ED to award Iraq and Afghanistan Service Grants under the Pell Grant program to assist veterans' dependents with the costs of attending institutions of higher education; increases, in academic year 2018-2019, the income protection allowance levels for students who are dependent, independent without non-spouse dependents (e.g., children), and independent with non-spouse dependents to reduce such students' income available (and increase need for financial assistance) to cover postsecondary educational expenses; increases the income threshold for an automatic zero expected family contribution from $23,000 to $34,000; and increases from 12 to 15 the total number of semesters a student may receive a Federal Pell Grant.