Exp of negative cvx variable multiplied by cvx variable

I have a convex problem (Power shaving) as follows:

A) Version 1:
cb is battery capacity which is constant
Load is given and constant
x is the vector of charging and discharging which is the convex variables.

the constraints are:

Es<=full capacity

where Es= cb*integral (x)

The objective is to optimize the bill amount

the bill amount = sum(grid)*rate+max(grid)*100
let say the rate is 10.
The problem is solved using cvx

B) version 2
Let’s make the problem more complex by assuming that:
rate =10 if x>=0
rate=1 if x<=0 ; % they pay me small money back if provide them with some power;

I can use the big M value and linearization to solve the issues however when dealing with data of year using the binary variable make solving the problem difficult

C) Version 3
I try here to remove the binary variable with a continuous variable using Sigmond function
I am planning to use the sigmond function to represent the rate such that
rate = 1 + 9 * (1/(1+exp(-x)))

the new objcive will be:
the bill amount = sum(grid)* (1 + 9 * (1/(1+exp(-x))) )+max(grid)*100

for reminder the grid=Load+cb*x; %(Load and cb are constants and x is cvx variable).

we can not multiply variables in cvx optimization, but I have seen that exp still can be used with cvx. is my problem version 3 doable in cvx optimization? and If so would you please help me to represent it in cvx?

I saw a similar issue in the following forum but I could not understand it:
How to express “x*(2^(1/x)-1)” in CVX

x/(1+exp(-x)) is neither convex nor concave, so it can’t be used in CVX, regardless of whatever number it is multiplied by.

Please carefully read the link, which was provided in an answer to a previous question by you