It would help if you used slack variables and squared penalties. One way of doing it is by defining the following penalty $$P\left(\sum_{i=1}^{N}x_i - 1 - Z \right)^2,$$ where $Z$ is an integer variable such that $0 \leq Z \leq N-1$.

If you use the D-Wave API, I believe you can easily define an integer (discrete) variable $Z$. If you use something else, you may need to express $Z$ as an expansion of auxiliary binary variables: $$ Z = \sum_{k=0}^{M-1} 2^ky_k + ry_M, $$ where $M = \lfloor \log_2 Z \rfloor$ and $r$ is the remainder so that $Z \leq N-1$. The remainder can be written as $r = N + 1 - 2^M$. For more details on this see Section 2.4.

Intuitively, whenever $\sum_i x_i \geq 1$, the values of $Z$ make the entire penalty term disappear. However, when $\sum_i x_i =0$, no matter what $Z$ is, we get the penalty $P(-1 -Z)^2 = P(1 + Z)^2$.

It would help if you used slack variables and squared penalties. One way of doing it is by defining the following penalty $$P\left(\sum_{i=1}^{N}x_i - 1 - Z \right)^2,$$ where $Z$ is an integer variable such that $0 \leq Z \leq N-1$.

If you use the D-Wave API, I believe you can easily define an integer (discrete) variable $Z$. If you use something else, you may need to express $Z$ as an expansion of auxiliary binary variables: $$ Z = \sum_{k=0}^{M-1} 2^ky_k + ry_M, $$ where $M = \lfloor \log_2 Z \rfloor$ and $r$ is the remainder so that $Z \leq N-1$. The remainder can be written as $r = N + 1 - 2^M$. For more details on this, see Section 2.4.

Intuitively, whenever $\sum_i x_i \geq 1$, the values of $Z$ make the entire penalty term disappear. However, when $\sum_i x_i =0$, no matter what $Z$ is, we get the penalty $P(1 + Z)^2$.