[Bug]: starting_solution does not work if last cycle is not saved

[brosaplanella]

PyBaMM Version

23.5

Python Version

3.9.17

Describe the bug

When using starting_solution in solving a simulation, if the last cycle of the starting_solution is not saved, then the initialisation doesn't work: the new solution starts at t=0 and does not contain the previous states (not even the first cycle which should be always saved).

My guess is that when checking the solution for the last cycle it finds it is None and thus it treats it as if no starting_solution has been provided.

Steps to Reproduce

        model = pybamm.lithium_ion.SPM()

        experiment = pybamm.Experiment([
            # pybamm.step.string("Discharge at C/2 for 10 minutes", start_time=datetime(2023, 1, 1, 8, 0, 0)),
            pybamm.step.string("Discharge at C/2 for 10 minutes"),
            pybamm.step.string("Rest for 10 minutes"),
        ])

        sim = pybamm.Simulation(model, experiment=experiment)
        solution = sim.solve(save_at_cycles=[1])

        experiment = pybamm.Experiment([
            # pybamm.step.string("Discharge at C/2 for 20 minutes", start_time=datetime(2023, 1, 1, 10, 0, 0)),
            pybamm.step.string("Discharge at C/2 for 20 minutes"),
            pybamm.step.string("Rest for 20 minutes"),
        ])

        sim = pybamm.Simulation(model, experiment=experiment)
        new_solution = sim.solve(calc_esoh=False, starting_solution=solution)

Plotting new_solution yields:

If, instead, I do save_at_cycles=None in the first solve, I get the expected result:

Relevant log output

No response

[brosaplanella]

Maybe not strictly related to this (but could be addressed at the same time as this), but another issue is that the way starting_time is currently implemented means that the initial_start_time is taken from the experiment, but if an initial solution is provided it should be taken from there.

[valentinsulzer]

It should be reasonably easy to fix, we just need to save last_state regardless of whether the last cycle is saved and then use that for the starting solution

[brosaplanella]

This issue might also align with #2484

[brosaplanella]

I found an easier way: save always the final cycle. It is a bit more memory consuming but a lot more robust (IMO). This also fixes the fact that the last_state was not correctly defined either.