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.

Description of Deliverance and Victory (Psalms 18, 68, 118)
The Psalms of this last grouping are all longer than most of the other Psalms. And they all have an interesting characteristic in common; they all speak of events in the past tense, as if they had already occurred. The previous grouping represented the last year of the Tribulation, and this grouping shows the result of the Final Battle and Victory.

1. (18:2) The LORD is my rock and my fortress and my deliverer;
   And then the past tense . . .
   (18:6) In my distress I called upon the LORD, and cried out to my God; He heard my voice from His temple,
   (18:7) Then the earth shook and trembled; because He was angry.
   (18:12) His thick clouds passed with hailstones and coals of fire.
   (18:17) He delivered me from my strong enemy, from those who hated me, for they were too strong for me.
   (18:43) You have made me (Israel - see Isaiah 60:4-5) the head of the nations; a people I have not known shall serve me.
   
2. (68:1) Let God arise, let His enemies be scattered; let those also who hate Him flee before Him.
   (68:2) Let the wicked perish at the presence of God.
   (68:3) But let the righteous be glad;
   And then the past tense . . .
   (68:7) O God, when You went out before Your people,
   (68:8) The earth shook;
   (68:12) “Kings of armies flee, they flee, and she (Israel?) who remains at home divides the spoil.”
   (68:17) The chariots of God [are] twenty thousand, [even] thousands of thousands; (a picture of the Lord in battle.)
   (68:18) You have ascended on high, You have led captivity captive;
   (68:24) They have seen Your procession, O God,
   
3. (118:1, 29) (first and last verses) Oh, give thanks to the LORD, for [He is] good! For His mercy [endures] forever.
   And then the past tense . . .
   (118:10) All nations surrounded me,
   (118:12) They surrounded me like bees;
   (118:13) You (the enemy) pushed me violently, that I might fall, but the LORD helped me.
   (118:14) And He has become my salvation.
   (118:22) The stone [which] the builders rejected has become the chief cornerstone. This verse shows the Lord Jesus who will be the King of kings and Lord of lords. When Jesus returns to claim those offices, every knee will bow and every tongue will confess that Jesus is Lord! May His name be praised forever!