I think you are wrong about this point. Both n and n' are from set N, and the restriction imposed by n <= n' <= n + Δn means that n' will go from equal to n to equal to n + Δn so, in fact, you will have n == n' in the first term of the sum. Your Δn (that is d in the code) is zero, so n is not sometimes equal to n' but always equal to n' in fact (the inner sum has only one term).
However, I do not understand how you ended up with np as a range. In your code np is a global variable, when it is clear by this constraint that np should be an iterator (i.e., you should use a for clause to introduce it), as n and i are. There also other problems in your code, like how you are trying to use j (you are not subscribing anything with j and you probably should not have variables Ij1, Ij2, etc), and that your global n is not a range like you think it is (that 1:length(n) will always be 1:1), but one problem at a time. I think what you want is something more like:
sum(alpha[i] * w[i, n, np] + beta[i] * bs[i, n, np] for i in Ij1 for n in 1:length(n) for np in n:(n+d)) <= H