Transform nonlinear cost function to get LP or MILP

Question

I'm trying to schedule power of multiple prosumers in a microgrid. The problem includes a cost function with min and max statements since I want to distinguish between the prices for importing and exporting energy.

The following is a minimal example for the cost function without summing over all timesteps at the moment:

I have two decision variables $P_{charge}$ and $P_{discharge}$ for the battery storage (one of it is assumed to be always zero) and one other controllable load $P_{load}$. Consumption is denoted with positive and generation with negative values. $C_{import}$ and $C_{export}$ are the per unit prices for importing and exporting energy.

$C = max((P_{charge} + P_{discharge} + P_{load}) \cdot \Delta T,0) \cdot C_{import} + min((P_{charge} + P_{discharge} + P_{load}) \cdot \Delta T,0) \cdot C_{export}$

The min and max functions form if-else statements to check if the sum of power flows is negative or positive. Is it possible to introduce some binary decision variables to transform it into a MILP? I want to use a Matlab solver later.

Thanks for every hint!

Edit 1:

I have tried to introduce the proposed auxiliary variables but they do not match the desired values I computed from my simulation. So maybe I have to clarify the problem. At first, I simplify power flow of battery to one bidirectional power variable to reduce the problem further.

The image shows a part of my problem. My objective is to maximize $R - C$, where $R \ge 0$ is revenue and $C \ge 0$ is cost. Per unit prices are always greater than or equal to zero and $C_{import} > C_{export}$ is valid for now. The nonlinear version would be:

$$R = min(P_{load} + P_{BSS},0) \cdot (-1) \cdot C_{export}$$ $$C = max(P_{load} + P_{BSS},0) \cdot C_{import}$$

I have defined non-negative decision variables $X_{import}$ and $X_{export}$ with the line capacity (energy per timestep) as upper boundary. Additionally I've introduced following equality constraint:

$$(P_{load} + P_{BSS}) \cdot \Delta T == X_{import} - X_{export}$$

I neglected the injected PV power at this node for now. My goal is to be somehow able to use the auxiliary variables for an upper level node. Maybe I didn't fully understand the proposed answers which might led to the undesired result.

Answer 1

Taking a different tack from Rob, I'm going to assume that the original objective function $C$ is correct as stated and is being minimized, with $C_\text{export}>0$ being a per-unit compensation value for exported energy (i.e., you pay to import and get paid to export). In that case, splitting the energy expression into $X_\text{import} - X_\text{export}$ as Rob does will work if $C_\text{import} > C_\text{export}$ but will not work with $C_\text{import} < C_\text{export},$ since in the latter case the solver could add an arbitrary amount $\delta > 0$ to both $X_\text{import}$ and $X_\text{export},$ keeping the energy value the same while reaping an undeserved profit of $(C_\text{export} - C_\text{import})\delta.$ In this case, you have to introduce a binary variable $Y$ and constraints $$X_\text{import}\le M_\text{import}\cdot Y$$ and $$X_\text{export}\le M_\text{export}\cdot (1-Y),$$ where $M_\text{import}$ and $M_\text{export}$ are upper bounds on the amount of energy that could be imported or exported.

Answer 2

Assuming you meant $C=\max() - \min()$ instead of $C=\max() + \min()$, you can introduce nonnegative variables $X_\text{import}$ and $X_\text{export}$ and linear constraints $$(P_\text{charge} + P_\text{discharge} + P_\text{load}) \cdot \Delta T = X_\text{import} - X_\text{export}$$ and then minimize the linear function $$C_\text{import} X_\text{import} + C_\text{export} X_\text{export}.$$