This is more like an **optimization problem** but any solution is appreciated.

I have a data set with input specifying **power(demand)** to be generated for a particular **time period(TP)**.

**Input**:  
**Time  ---    Power**  
01 to 02             1000  
02 to 03  2000  
03 to 04  2000  
04 to 05  1500  
etc.,

But the power can't just be increased from **1000 to 2000 at 01:59:59**.
Here there will be some ramping rate(gradient) which I want to **minimise** by keeping the imbalance to 0.  Imbalance may arise if you deviate from what is needed, here power cannot be more or less than what is required in that Time Period.

In summary I have to generate the **black line(which is output)** below **(red line is the input)**.
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/zi6zm.png

I am trying to get an output which can be a curve(black line) equation or 
the power to generate at the deflection points(say at around 1:45 we generate 800 then at 2:15 it can be 2200 which will keep the imbalance to 0).

When I tried to treat this as an optimization problem it may look something like this:  
**Objective** :   `Minimise gradient f(x) = Yj-Yi/Xj-Xi` between 2 points `(Xi,Yi)` and `(Xj,Yj)`  
**Subject to constraints**:  

    average of Yi = Power value(redline in that Time Period)

  


But I am struggling to complete it how to represent mathematically to be able to implement using programming like R's lpsolve/optimize or Python's scipy optimizers.

Any help on the approach to take would be greatly appreciated.