I’m working on a dymos problem which represents a simplified building energy model. I would like to add some discrete controls – essentially just binary on/offs – and have some questions.

First, some context about the model:

1. Walls are represented as RC networks, with thermal transfer from the exterior through the building envelope to the interior of a thermal zone (i.e. a room). Each capacitor in the wall has a state/state-rate associated with it, and the state rate is computed using KCL vis-a-vis the adjacent capacitor nodes. The state is Temperature.
2. Thermal zones are represented by a single capacitor, representing the specific heat of the air and any thermal mass in the room, and again, state is temp, state rate computed via KCL.
3. There can be multiple zones with walls between them, but that’s beside the point here.
4. Each thermal zone has three controls: Cooling power, Heating power, and outdoor air mass flow rate. For now though, we can just simplify it and consider only a single control (let’s say cooling power).
5. Outside air temperature conditions are modeled as a differentiable function of time (essentially a voltage source in the context of the RC network) so that e.g. the first cap in a wall assembly can correctly compute J["T_cap_1_dot","time"], since outputs["T_cap_1_dot"] does depend on inputs["time"] via the external boundary condition.
6. The goal is always to keep the interior zone temperature within a certain temperature range, e.g. 20C to 21C.

Currently, I achieve this by setting bounds on each zone’s temp with upper and lower bounds equal to the setpoints. I do this when adding the state vars for the zone temps – I’m not sure if it is “better” to add a path constraint? Or if alternatively, I should add it as a penalty to the objective function which is zero when the temp is within the band and grows when it is out of band? This is trivial if there is a single setpoint, but with a “deadband” the objective becomes a little messier if you want it to be differentiable.

I fix the start and end time of the setpoint, as well as the initial conditions for all states (voltages across the caps) and controls. I then ask it to minimize the total energy usage over the phase (so there is an additional state variable which is just the integral of the cooling+heating power).

This works – very cool! It’s “cheating” to some extent because it knows the future weather conditions, so I guess it’s a form of model predictive control. It can even find strategies like “night-time flushing” for thermal mass, where you bring in outdoor air overnight before a hot day to cool the concrete layers of walls in order to keep the space cool during the daytime and avoid using any heating/cooling energy at all.

Anyways, I would like to use a slightly different form for my cooling/heating controls or outdoor air flow control. I would like to be able to model controls that are discrete in nature – really just “on/off” would be sufficient.

Three ideas:

1. Set a path constraint on the control: e.g. binary_constraint = (control-control_on)*control with equals=0 for the constraint type. However, when I do this, I get a singular matrix error:

Singular matrix C in LSQ subproblem    (Exit mode 6)
            Current function value: 5e-07
            Iterations: 1
            Function evaluations: 1
            Gradient evaluations: 1
Optimization FAILED.
Singular matrix C in LSQ subproblem

2. Let the control be non-dimensional and apply a sigmoid function to it, and then multiply it by the corresponding “actual” control value. I don’t think this would work, since if the optimizer works well enough, it should still find the solutions in the critical region of the sigmoid.
3. Split time up into multiple phases, where in each phase, the control alternates between its on/off value, and then let the optimizer solve for the timing of the phase boundaries. I don’t love this because (a) I have to take a prior as to how many on/off segments there are, and (b) I don’t think this really scales to having multiple on/off controls, since you do not know in advance which should be simultaneously on/off etc.

I couldn’t find anything in the dymos docs mentioning discrete controls, but maybe I missed that!