I saw two similar approaches to this question.
Firstly by Rabbi Avi Geller on Aish who notes that the grand reveal was carefully orchestrated in such a way that it helped to eliminate any bad feeling that Yosef might have had. He writes as follows:
Parshat Vayigash is the climax of the story of Joseph and his brothers. The master planner had his brothers exactly where he wanted them: To know that they were under his total control and he could easily have taken his revenge, while in fact he really loved them and could never hurt them. On the contrary, he actually saved them from the famine.
Joseph also harbored some bad feelings toward his brothers after being sold into slavery and being treated so cruelly. By putting his brother Benjamin into a similar situation and observing their efforts to save him, Joseph would be able to forgive them with a full heart. This way, the rift between them would be sealed, and the Jewish people would survive this ordeal!
Yosef did not need to become enraged because his unmasking proved to them that they had made a mistake. As they bowed down to him it proved that his dreams had come to prophetic fruition. As a result:
Joseph had accomplished his mission: The brothers realized their mistake, and Joseph was able to overcome his bad feelings toward them. The brothers were now able to return home. Joseph told them not to tarry (for fear they might come to incriminate each other), but to hurry and bring their father down to Egypt.
Another approach is taken by that of the late Lord Rabbi Jonathan Sacks z"l. He notes that this episode represented the first true example of forgiveness in the Torah. He posits that the whole dialogue between Yosef and the brothers was contrived in such a way to allow the brothers to go through a form of atonement for their past misdeeds, and as a result there was no need for him to feel angry.
Rabbi Sacks writes;
That is the explanation for Joseph’s behaviour from the moment the brothers appear before him in Egypt for the first time to the point where, in this week’s parsha, he announces his identity and forgives his brothers. It is a textbook case of putting the brothers through a course in atonement, the first in literature. Joseph is thus teaching them, and the Torah is teaching us, what it is to earn forgiveness.
Firstly, he imprisons Shimon and demands that they bring Binyomin. He is essentially forcing them to re-enact an earlier occurrence, i.e. to return to their father without one of the other brothers. They then realise that they are being punished on account of their treatment of Yosef (see Bereishis 42:21-23) - so they now admit that they have done wrong. Then when Yosef plants the goblet in Binyomin's sack they confess to their wrongdoing (Bereishis 44:16). Finally when Yehuda asks that he be imprisoned in Binyomin's place (Bereishis 44:33) - Yehuda the one who originally sold Yosef into slavery is the one willing to pay the prize thereby representing a teshuva gemura (a complete repentance).
So according to this approach from Rabbi Sacks we see that Yosef did not need to harbour any sense of anger or bitterness. He concludes:
Now Joseph can forgive, because his brothers, led by Judah, have gone through all three stages of repentance: (1) admission of guilt, (2) confession and (3) behavioural change....Humanity changed the day Joseph forgave his brothers. When we forgive and are worthy of being forgiven, we are no longer prisoners of our past.