Earlier in Romans (Rom. 1:18-3:20) the Apostle Paul has already proven that the powers behind "the powers that be" principle have long since been corrupted by man.
And just prior to the one passage uou cited outside all that Scripture says about this issue; Paul writes...
Romans 12:19 Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord. 12:20 Therefore if thine enemy hunger, feed him; if he thirst, give him drink: for in so doing thou shalt heap coals of fire on his head. 12:21 Be not overcome of evil, but overcome evil with good.
There is the Apostle Paul's understanding of God's actual confidence in men in government and their corrupt carrying out their responsibility under that principle of "powers that be."
Which actual confidence He relates once more, in the chapter following the one you cited your one lone passage from and in your context....
Romans 14:11 For it is written, As I live, saith the Lord, every knee shall bow to me, and every tongue shall confess to God. 14:12 So then every one of us shall give account of himself to God.
The meaning of Romans 13 as to what is to be the Believer's outlook while under, and given that man has long since corrupted said offices, positions of authority, or powers?
Philippians 2:14 Do all things without murmurings and disputings: 2:15 That ye may be blameless and harmless, the sons of God, without rebuke, in the midst of a crooked and perverse nation, among whom ye shine as lights in the world; 2:16 Holding forth the word of life; that I may rejoice in the day of Christ, that I have not run in vain, neither laboured in vain.
And...
1 Timothy 2:1 I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men; 2:2 For kings, and for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty. 2:3 For this is good and acceptable in the sight of God our Saviour; 2:4 Who will have all men to be saved, and to come unto the knowledge of the truth. 2:5 For there is one God, and one mediator between God and men, the man Christ Jesus; 2:6 Who gave himself a ransom for all, to be testified in due time.
The Apostle Paul had no confidence in men and their corruption. Thus; his exhorting that one pray that one might nevertheless have peace despite man's dereliction of his responsibility in office.
Paul's actual confidence is also obvious in the above - in the Cross of Christ and in His rule, one day.
All that in light of the fact that men in power are under a system of governing that long ago proved God's actual intent through His having allowed man his every failure...
Romans 3:19 Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. 3:20 Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin. 3:21 But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets; 3:22 Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference: 3:23 For all have sinned, and come short of the glory of God; 3:24 Being justified freely by his grace through the redemption that is in Christ Jesus: 3:25 Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God; 3:26 To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus.
