I believe I have found parallels between the respective Psalms of each set. The following examples only focus on the ninth through the eighteenth Psalm of each grouping; Psalms 9 through 18, 59 through 68, and 109 through 118. These groups appear to describe the years leading up to the Tribulation, the seven years of the Tribulation, and God’s Final Victory. All references are from the New King James Version unless otherwise noted. For emphasis and clarity I have added underlining and italics to some of the text.

God’s Answers to the Righteous (Psalms 15, 65, 115)
In response to the Abomination of the evil man, the Lord responds to the questions of the righteous and states that the idolatry of the Abomination is worthless.

1. (15:1) LORD, who may abide in Your tabernacle? Who may dwell in Your holy hill?
   (15:2) (Answer to the question) He who walks uprightly, And works righteousness.
   
2. (65:5) [By] awesome deeds in righteousness You will answer us, O God of our salvation.
   
3. (115:2) Why should the Gentiles say, “So where [is] their God?”
   (Answer to the question) (115:4) Their idols [are] silver and gold, The work of men's hands.
   (115:5-7) (They cannot speak, see, hear, smell, handle, walk nor mutter.)
   (115:8) Those who make them are like them; (Idols are worthless.)

The Lord’s Refuge (Psalms 16, 66, 116)
The Revelation (chapter 12:6 and 12:14) and the prophet Isaiah tell us that a remnant of Israel will be led to a place of refuge during the last half of the Tribulation. Isaiah says that Israel will not leave in haste, because the Lord will go before them and will be their rear guard. (Isaiah 52:12) These Psalms seem to speak of the answer to Israel’s prayer for refuge from the Tribulation.

1. (16:1) Keep me safe, O God, for in you I take refuge.
   (16:6) The boundary lines have fallen for me in pleasant places; surely I have a delightful inheritance.
   
2. (66:19) [But] certainly God has heard [me]; He has attended to the voice of my prayer.
   
3. (116:1) I love the LORD, because He has heard my voice [and] my supplications.
   (116:3) The pains of death surrounded me,
   (116:6) The LORD preserves the simplehearted; He saved me. (New International Version)

The Coming Final Harvest (Psalms 17, 67, 117)
At the culmination of the Tribulation, Jesus will return as King of kings and Lord of lords. This will also be the great Harvest spoken of in the Revelation chapter 14. The prophets describe how the Messiah, whom we know to be Jesus, will rule ALL the nations. This grouping tells of that great Day of the Lord.

1. (17:8) Keep me as the apple of Your eye; (The Lord is very jealous of Zion. (Zechariah 8:2) “For he who touches you touches the apple of His eye.” (Zechariah 2:8))
   (17:11) They have now surrounded us in our steps; They have set their eyes, crouching down to the earth,
   (17:12) As a lion is eager to tear his prey, And like a young lion lurking in secret places.
   (17:13) Arise, O LORD, Confront him, cast him down; Deliver my life from the wicked with Your sword,
   
2. Psalm 67 begins with a modification to the priestly blessing.
   (67:1) God be merciful to us and bless us, [and] cause His face to shine upon us,
   (67:4) Oh, let the nations be glad and sing for joy! For You shall judge the people righteously, and govern the nations on earth.
   (67:6) [Then] the earth shall yield her increase (harvest).
   
3. (117:1) Praise the LORD, all you Gentiles! Laud Him, all you peoples!
   (117:2) For His merciful kindness is great toward us, and the truth of the LORD [endures] forever. Praise the LORD!