Originally posted by: yanzhiping

I have a problem with storage modeling. we want to use SOS1 type of variables to deal with charge/discharge instead of two binaries at each t (one for charging, another for discharging).

Previously, we use two binary variables x_chrg(t) and x_dischrg(t) to model the operational characteristics of energy storage, as indicated in Eq 4 and Eq5.

In the equations Eq1 and Eq2, P_chrg(t) and P_dischrg(t) are continuous decision variables indicating charging and discharging output from storage, P_Chrg(max,t ) and P_dischrg(max,t) are parameters indicating the max output amount for charging and discharging.

Eq3 means at any time period t, either x_chrg(t) =1 or   x_dischrg(t)=1 or both are zeros.

Eq1: P_chrg(t) <= x_chrg(t) * P_Chrg(max,t)

Eq2: P_dischrg(t)<= x_dischrg(t)*P_dischrg(max,t)

Eq3: x_chrg(t) + x_dischrg(t)<=1

Eq4: x_chrg(t) = {0,1}

Eq5: x_dischrg(t)={0,1}

Alternatively, we can model x_chrg(t) and x_dischrg(t) as SOS1 type of variables, meaning at most one of them is nonzero. This method uses nonzero positive value to indicate charging/discharging instead of binary values. In this way we don't need Eq4 or Eq 5. With Eq 3, x_chrg(t) and x_dischrg(t) can be any value between 0 and 1. When x_chrg(t) is nonzero, x_dischrg(t) is certainly zero; similarly, when x_dischrg(t) is nonzero, x_chrg(t) is certainly zero. This means charging/discharging cannot happen at the same period. When x_chrg(t)=1, that means charging reached its max limit, when 0<x_chrg(t)<1, it means storage charges at a percentage of its charging limit.  

Eq1: P_chrg(t) <= x_chrg(t) * P_Chrg(max,t)

Eq2: P_dischrg(t)<= x_dischrg(t)*P_dischrg(max,t)

Eq3: x_chrg(t) + x_dischrg(t)<=1

If the SOS has only two elements then I don't see a fundamental difference to binary variables: a solver will still have to do the same two branches: one has the first element fixed to 0, the other has the second element fixed to 0. So I would expect a speedup here, especially since you don't have priorities.

Depending on the exact structure of your model you may be able to replace

x_chrg(t) + x_dischrg(t) <= 1

by

x_chrg(t) + x_dischrg(t) == 1

Then you could substitute x_chrg(t) = 1-x_dischrg(t) everywhere. That may or may not help a solver.