As we understand the new covenant of God, we may need to think about the relation between the power and the Holy Spirit. This will give us an insight into what happened on the day of Pentecost and also may give us a better interpretation of Luke 24 and Acts 1 and 2 in terms of the promise of God that Jesus uttered and emphasized.
Ephesians 3:16 shows that the power is through the Holy Spirit. The holy Spirit strengthens the inner being of the person, which is dwelling in him or her since he or she is born in the Spirit. According to this idea, It could be said the Holy Spirit is the divine agency that generates the power of God in the person. He gives strength to the person that he or she could accomplish the hope he or she is striving for by faith in Christ. Galatians 5:5 expresses this idea that the person is waiting for the hope in faith by the power that is supplied through the Spirit. In the verse the word “wait” is a euphemism of suffering and enduring until the hope is accomplished. Romans 15:13 is also based on the idea of the relation between Him and the power. He also works for the church that all members are equipped with the talents and gifts that originage from Him to build it up. 1 Corinthians 12 clearly says this point too—especially notice verse 11. In this regard, the talents and gifts of Christians are the manifestation of the Spirit dwelling in them and they are for the common good (v. 17). Paul’s ministry to spread the gospel also affirms this premise. According to Romans 15:19, his ministry relied on the power of God through the Holy Spirit—1 Thessalonians 1:5 also.
The power and the Holy Spirit, therefore, can be understood as either aspects that consist the whole, like both sides of a coin, or as the essence and its effect or manifestation, such as fire—where there is fire there is emission of energy. I believe the latter fits better what the Scripture teaches. Acts 1:8 says that Jesus said to the disciples, “you will receive power when the Holy Spirit has come upon you.” This verse tells us two things for this discussion: One, the power is something that is accompanied with the Holy Spirit; and the Holy Spirit has to come for the power to be available. Luke 1:35, describing what would happen to Mary, shows this idea too: “The Holy Spirit will come upon you, and the power of the Most High will overshadow you.” The point that was discussed in the paragraph above is also in line with this idea. When the Holy Spirit dwells in a person as he or she is obedient to the faith, He works in him or her to accomplish God’s will. The working of the Holy Spirit is the manifestation of the power of God through the Holy Spirit dwelling in the person. A person, who does not have the Holy Spirit, does have no power from God (1 Cor. 2:4; cf. Rom. 8:9b). What this idea means happened to the apostles in Acts chapters 1 and 2. When they received the Holy Spirit, they got clothed with the power from on high to preach the gospel risking their lives.
If the Holy Spirt is the essence of the promise of God and one of the manifestations of His indwelling in the person is the power of God, what God promised for the new covenant should be the Holy Spirit basically, not the power. Generically, the Holy Spirit emits the power of God so that the power, in this sense, is dependent on the Holy Spirit. Even if the power is what might be implied as the promise of God in Luke 24:49, it does not seem that the verse says the power can be manifested independently of the Holy Spirit. For this reason, I would think that the power mentioned as one to be manifested in the life of the Christian is, literarily, a kind of synecdoche to mean the the Holy Spirit or His indwelling in the person, meaning the working of God in him or her through the Holy Spirit. Therefore, it is enough to say that what is promised by the Father as the new covenant is the Holy Spirit, even with no necessary need of mentioning the power, which is its generic effect or manifestation. There are many scriptures that refer to the (Holy) Spirit as the promise of God for the new covenant, such as, Isaiah 44:3; Ezekiel 11:19; 36:26:Acts 1:4-5; 2:33; Galatians 3:14; Ephesians 1:13. However, I could not find so far any verse in which the power is mentioned as the promise of God for the new covenant, except that in which it is implied, such as Luke 24:49 as mentioned above.
As for the relation between the Holy Spirit and the power and also between the receiving the Holy Spirit and receiving the power from on high are not different promises of God but one, being the Holy Spirit is the substance of it and the power its manifestation or generic effect. Therefore, it is safe to say that the new covenant is all connected with the Holy Spirit through faith in Jesus Christ.
The Pure Gospel 
It may help to clarify what power you are talking about. Here it seems the power is the power of Christ dwelling in us. But maybe it is somewhat different with the apostles in Acts 1 and 2? see below.
I think the article is basically correct, but we should be careful to emphasize that there are manifestations of power that differ according to the circumstance or need. The power the apostles received is not the same manifestation of power –or the same gift – that all Christians receive, although both derive from the same Spirit and the same source of power. The passing of the “power” promised to the apostles by the Spirit, and the power of miraculous gifts through the Spirit by the apostles hands, was the passing of the age of revelation. But these temporary manifestations of the Spirit were for the purpose of revelation and confirmation. The true (and greater) power of the new covenant is the power of the Spirit in us for personal strength and endurance toward sanctification, promised to all believers. At least so I see it.
This is beautiful and glorious teaching from the Scriptures, but I think our brothers and sisters need to be led gently and carefully here. It is new and different, and often the first reaction is negative. It will take strong relationships and trust to be able to approach it.
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