Matlab linprog constraints: how to stop charge and discharge storage at same time?

Question

I'm having issues with matlab linprog code. The optimisation function is the overall cost for the 24 period, just considering fuel costs of the boiler.

Purpose of simulation:

Optimisation of the charge/discharge behaviour of a Thermal Energy Storage (TES) for a 24h operation of a system consisting of a boiler, heat demand, and the TES. The price of the gas are time-varying.

Problem:

If the TES is ideal (efficiency=100%), I have no constraint that stops the system from charging and discharging at the same time. I CANNOT use one variable to describe charge and discharge. I do need them separated

At the moment I have the following constraints to describe the min/max charge/discharge rates (and of course some others):

maxChargeThermalTES>=ChargeThermalTES<=0
maxDischargeThermalTES >= DischargeThermalTES <=0

is it possible to realise the following logical rule within the constraints of linprog?

if ChargeThermalTES<0,
   DischargeThermalTES=0
end

all approaches, e.g. with a binary variable (to describe if the system is charging or discharging) do not work, as the binary variable always depends on the output of the optimisation.

Answer 1

Yes, it is possible. You can add your If-Then condition to linprog using one 0-1 binary variable and Big-M.

To realize the Logical rule:

if ChargeThermalTES<0,
   DischargeThermalTES=0
end

Condition: If ChargeThermalTES<0, then DischargeThermalTES=0

Let's introduce a binary variable y

So we can rewrite the condition as

ChargeThermalTES - M y < 0

which implies that

if y = 0, then DischargeThermalTES must be = 0
if y=1, DischargeThermalTES can be anything

Let's split the equal to constraint into two inequalities

DischargeThermalTES < M y
DischargeThermalTES > -M y

If y=1, the above two are essentially non-binding.

If y= 0, it will force DischargeThermalTES to become 0.

So with the following constraints combined, you can enforce your logical constraint to the Linear Program.

 ChargeThermalTES - M y < 0
 DischargeThermalTES < M y
 DischargeThermalTES > -M y
 y = {0,1} binary, M is a large number.

Hope that helps.

Answer 2

You cannot enforce such logical rule in linear programming.

However, what you can do is the following :

1\ solve your linear program, without this constraint. Get the optimal cost of your objective function (lets name it OldCost).

2\ Then change your linear program this way :

• add a constraint : old objective function should be between OldCost * (1-Epsilon) and OldCost * (1+Epsilon)

• the new objective function to minimize is ChargeThermalTES + DischargeThermalTES.

Cheers