  We implement two Quantum Approximate Optimisation Algorithm (QAOA) variants
for a battery revenue optimisation problem, equivalent to the weakly NP-hard
Knapsack Problem. Both approaches investigate how to tackle constrained
problems with QAOA. A first 'constrained' approach introduces a quadratic
penalty to enforce the constraint to be respected strictly and reformulates the
problem into an Ising Problem. However, simulations on IBM's simulator
highlight non-convergent results for intermediate depth ($ p\leq 50$). A second
'relaxed' approach applies the QAOA with a non-Ising target function to compute
a linear penalty, running in time $O(p(\log_2 n)^3)$ and needing $O(n \log n)$
qubits. Simulations reveal an exponential improvement over the number of depth
levels and obtain approximations about $0.95$ of the optimum with shallow depth
($p \leq 10$).
