开心 and 高兴 are replaceable, this kind of happiness is more "instant feeling", short period. 我今天很开心（高兴），很高兴（开心）见到你。 快乐，this kind of happiness is more "joyful", long period，like you're in some happy event (festival)。 快乐的童年，生日快乐，圣诞节快乐。 幸福，this kind of happiness is more “status”，like 幸福的家庭，幸福的人。

I don't know the exact grammar. But I guess 开心、高兴、快乐 are almost equaled to use, and幸福 is a more stationary and lasting feeling or emotion than those three