1 Introduction
2 Literature Review
2.1 Review of Heuristic Methods
2.2 Review of Mathematical Optimisation Approaches
3 Problem Formulation
3.1 Objective Function
3.2 Pump Approximations
3.3 Pipe Approximations
3.4 Mass balance at network nodes
4 Methodology
4.1 Problems Considered and Investigation Procedure
No.: | Formulation | Details | |
---|---|---|---|
Solved with SCIP: | |||
1a | min.: |
f
4(⋅)+f
1(⋅) | MINLP with the fewest number of integer variables, |
s.t.: | but most non-linear constraints | ||
1b | min.: |
f
3(⋅)+f
4(⋅) | MINLP like No. 1a, but with a quadratic pump power |
s.t.: | consumption term | ||
2 | min.: |
f
4(⋅)+f
1(⋅) | MINLP with less non-linear constraints than No. 1a, |
s.t.: | but more integer variables (N
p
i
e
c
e
=7) | ||
3 | min.: |
f
4(⋅)+f
1(⋅) | MINLP with the same non-linear constraints as No. 2, |
s.t.: | but fewer integer variables (N
p
i
e
c
e
=3) | ||
Solved with CPLEX: | |||
4 | min.: |
f
4(⋅)+f
1(⋅) | MIQP with only linear constraints (N
p
i
e
c
e
=7, |
s.t.: |
N
c
o
n
=7) | ||
5 | min.: |
f
4(⋅)+f
1(⋅) | MIQP like No. 4, but fewer integer variables but a |
s.t.: | similar number of constraints (N
p
i
e
c
e
=3,N
c
o
n
=7) | ||
6 | min.: |
f
4(⋅)+f
1(⋅) | MIQP like No. 4, but less constraints but the same |
s.t.: | number of variables (N
p
i
e
c
e
=7,N
c
o
n
=3) | ||
7a | min.: |
f
4(⋅)+f
1(⋅) | MIQP like No. 4, but significantly less integer variables |
s.t.: | and constraints (N
p
i
e
c
e
=3,N
c
o
n
=3) | ||
7b | min.: |
f
1(⋅) | MILP, similar to No. 7a, but as without a switch |
s.t.: | penalty | ||
7c | min.: |
f
2(⋅) | MILP, like No. 7b, but with a linear pump power |
s.t.: | consumption term | ||
7d | min.: |
f
3(⋅)+f
4(⋅) | MIQP like No. 7a, but with a quadratic pump power |
s.t.: | consumption term | ||
8 | min.: |
f
1(⋅) | MILP with a simpler pump formulation and fewer in- |
s.t.: | teger variables (N
p
i
e
c
e
=3) |