introducing a comparison related to variables in linear programming

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ennes mulla
ennes mulla 2023-1-20
编辑： Matt J 2023-1-20
Hello,
I want to run linear programming to make a trade off between multiple sources of energy. I want to include a battery in the system and size it using linear programming. Therefore, I have the following varaibles:
Pdemand: demands of energy which I have it already as a vector colmun.
X1= A_pv [m^2 ]-maximum total area occupied by the solar photovoltaic system
X2=A_w [m^2 ]-maximum total wind rotor area
.
.
X8=E_bess (t) [kWh]-energy in the BESS at time t
X9=P_(bess.in) (t) [kW]-power in battery at time t
X10=P_(bess.out) (t) [kW]-power out from battery at time t
I started to encounter problems when setting my first inequality since X9 and X10 are related to X1 and X2 as following
if X1 + X2 > Pdemand
X9 = (X1 +X2) - Pdemand;
X10 = 0;
else
X10 = Pdemand - (X1 + X2);
X9= 0;
Now I want to make the first constrain for linear programming by reporting the coefficient to use it later in linear programming
X1 + X2 +X9 +X10 = demands
I included X9 and X10 here in the equation since one of them needs to be zero at the time the other is presented.
A1 = [CX1 CX2 0 0 0 0 0 0 0 CX9 CX10]
As you can see, I am trying to solve for X9 and X10 but before I need them to go throght the if statment which is not possible till I get the results of linear programming first.
Can someone help me please, I have zero ideas how to solve this. Thanks.
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Torsten
Torsten 2023-1-20
编辑：Torsten 2023-1-20
You say you have Pdemand as a column vector. So X1 and X2 are also X1(t) and X2(t) ?
It would be helpful to have your problem stated precisely.
ennes mulla
ennes mulla 2023-1-20
yes yes. I over simplified the problem to mention less details, But man ignore this details please, how would you introduce if statment to the LP

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采纳的回答

Matt J
Matt J 2023-1-20
编辑：Matt J 2023-1-20
The constraints you describe are non-convex, so there is no way to formulate them in a linear way. However, you can modify your problem so that the solution to a linear program is encouraged to satisfy those constraints. Introduce additional variables U,V and the constraints,
U=(X1 +X2) - Pdemand
-V<=U<=V %implies V>=abs(U)
X9>=0
V>=X9>=U %(1a)
X10>=0
V>=X10>=-U %(1b)
X9+X10=V %(2)
Also, modify your linear objective to have a penalty term,
min. f(X) + largeNumber*V
So, if largeNumber is large enough, the penalty will force V to its lower bound V=abs(U) at the solution. In that case when U>0 then U=V and constraint (1a) implies X9=U=(X1 +X2) - Pdemand. Additionally constraint (2) then implies X10=0.
A similar argument shows that if U<0 and so -U=V, then constraint (1b) implies X10=V=abs(U)= Pdemand-(X1+X2) and X9=0.

更多回答（1 个）

Torsten
Torsten 2023-1-20
编辑：Torsten 2023-1-20
Your constraint says that
X9 = max(0,X1+X2-Pdemand)
X10 = max(0,Pdemand-(X1+X2))
Usually, it suffices to set the constraints
X9 >= 0
X9 >= X1+X2-Pdemand
X10 >= 0
X10 >= Pdemand-(X1+X2)
since you usually won't "oversupply" in an optimization.
If the constraints don't suffice for your application, look up on how to linearize the max-operator independent of the objective following Larry Snyder's answer under
https://or.stackexchange.com/questions/711/how-to-formulate-linearize-a-maximum-function-in-a-constraint
But in case you have to use this formulation, your problem becomes a problem for intlinprog - it can no longer be solved with linprog.

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