How to set a limit for a switch to 0 of a variable

Question

I would like to know how to define a constraint to set a limit for switching to 0 for a decision variable? So I have a linear variable $x(t)$ which quantifies the modulation degree of a heating device. It can be between $0$ (meaning that the device is switched off) and $1$ (meaning that the device is heating with full power). My $t$ is between $1$ and $288$ (timeslots for every $5$ minutes of $1$ day). As you can imagine it is not really good to change frequently between $0$ and a non-zero value because this would mean, that the heating device starts and stopps frequently during one day (sometime I have $5$ starts and stopps during $1$ hour) which should be avoided as it increases the wear of the device.

So I would like to set a limit to switching to 0 during one day. Is there a way how I could define a constraint for that? The rule in pseudocode is basically

if x(t-1)>0 and x(t)=0 then increase count by 1
Constraint: count <= limit

Do you have an idea how I could model that (if it is possible)? I'd appreciate every comment and would be thankful for your help.

Answer 1

You can model this as follows. Let $y(t)\in \{0,1\}$ be a binary variable that is activated if the heating device is switched off at time $t\in \{1,...,288\}$ (and was active at time $t-1$).

This variable is activated every time $x(t-1)=1$ and $x(t)=0$: $$ x(t-1) \le x(t) + y(t)\quad \forall t=1,...,288 $$

So if $x(t-1)=1$, either $x(t)$ also takes value $1$ (and the device remains active), either $y(t)$ is activated.

You might want to minimize $\sum_{t=1}^{288}y(t)$ so that $y(t)$ is never activated "for free". Or you could also enforce this with additional constraints: $$ y(t)\le x(t-1) \quad \forall t=1,...,288\\ x(t)+y(t) \le 1 \quad \forall t=1,...,288 $$

And you can limit the number of activated variables to an upper bound $\ell$ (or equivalently, the number of times the device is switched off): $$ \sum_{t=1}^{288}y(t) \le \ell $$

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EDIT:

If $x$ is continuous, you need to create a binary variable $b(t)$ that takes value $1$ when $x(t)>0$. You can do this with the following constraint: $$ x(t)\le b(t)\quad \forall t $$ And in the above approach, replace $x$ by $b$.