This repository contains a collection of mathematical optimisation models that have been developed for educational purposes for several lectures. They are collected here for easier maintenance and better visibility of what has been implemented already.
The common theme among these models is capacity expansion, power flow and plant scheduling for minimum total system cost (or - in the case of DHMNL - maximum revenue), each model stressing another aspect of common tasks in modelling of energy systems.
This linear programming (LP) model finds the minimum cost generation and network flow for a lossless electricity network, while obeying linearised DC powerflow equations (consult a textbook, or The DC Power Flow Equations for a quick primer). It is present in two semantically identical, but mathematically different formulations: dcflow_arc uses directed arcs (two per edge) and contains only positive flow variables, the direction directly corresponding to that of its containing arc. dcflow_edge only contains one edge between each pair of connected vertices, and an unconstrained flow variable instead, encoding direction in its sign. While the edge formulation uses less variables and thus is more efficient for only this problem, the arc formulation can make it easier to add further variables and constraints that interact with the modelled power flow.
This mixed-integer linear programming (MILP) model finds the maximum revenue topology and size of a district heating (DH) network for a given set of source and demand vertices. The model can decide which demands to connect and consequently plans the location and size of the built network. Different from most other models here, this implies that the system under design does not have to satisfy a demand, but can select only a (possibly empty) subset of profitable customers.
This linear programming (LP) model finds the maximum welfare solution for a given set of a) a discretised production cost curve (i.e. a merit order curve) and b) a discretised utility function of customers (i.e. a price-demand curve). It thus finds the economic equilibrium of the market situation encoded by its inputs.
This linear programming (LP) model finds the minimum cost investment plan for for a set of two power plant technologies over multiple decades, allowing investment decisions every five years. Old investments phase out of the power plant fleet after the parameterised lifetime of each investment is over.
This mixed-integer linear programming (MILP) model finds the minimum cost network within a graph to redundantly connect a set of source to a set of demand points. "Redundantly" means that the resulting network is resilient against the failure of any single edge in the network, i.e. satisfying the N-1 Criterion common in electric grid design.
This linear programming (LP) model is an abbreviation of Storage Optimisation for Smart Grid. This model determines optimal size and operation of a hypothetical lossless storage technology for electric energy. A given electricity demand must be satisfied from either a cost-free (renewable) energy supply with intermittent characteristic or from purchase of electricity from the grid for a time-dependent price. Surplus renewable energy can be sold either at generation time, or stored to yield a higher revenue later. This model is the core idea behind urbs, which generalises the size and operation optimisation to an arbitrary number of energy conversion, transmission and storage processes (at aan rbitrary number of conceptual nodes, called sites). The central idea, the energy balance constraint, storage state equation and a cost minimisation objective function, are all present here.
This linear programming (LP) optimisation model finds a minimum-cost capacity expansion and unit commitment solution to a given demand timeseries for a combination of a fluctuating feed-in (renewables) and a controllable technology (power plant). This model focuses on correctly depicting the trade-off in sizing the power plant with respect to its operation point, which can exhibit less efficiency than when operating below its nominal size. These properties are approximated by a formulation that combines startup costs with a linearily increasing or decreasing conversion efficiency, depending on the operation point.
This mixed-integer linear programming (MILP) model finds a cost-minimal power plant operation schedule for a given demand timeseries. Modelled power plant attributes are minimum and maximum output capacity, startup and shutdown costs, operational fixed (when switched on) and variable (by power production) costs. This formulation is computationally more complex than the one used in model Startup and partial, but more accurate in that it depicts the discrete nature of the on-off decision. Also, it allows for more extensions similar to state-of-the-art unit commitment models, e.g.: minimum runtimes, minimum cooldown times, hot or warm start capabilities.
All models require the standalone solver glpsol from the GNU Linear Programming Toolkit (GLPK).
For short experiments, there is GLPK Online, a Javascript port of GLPK and MathProg. Just copy & paste the model code into the text box, press Solve, inspect model ouptut in tab Output.
Binary builds for Windows are available through the WinGLPK. Just extract the contents of the ZIP file to a convenient location, e.g. C:\GLPK. You then can either call a model using:
C:\GLPK\bin\glpsol.exe -m model.mod
However, it is recommended to add the subdirectory w64, which contains the file glpsol.exe, to the system path (how), so that the command glpsol is available on the command prompt from any directory directly:
glpsol -m model.mod
Most distributions offer GLPK as a ready-made packages. If unsure, please consult your distribution's package index. On Debian or Debian-based (e.g. Ubuntu) distributions, executing the following command on the terminal (excluding the $ ):
$ sudo apt-get install glpk-utils
Note that package maintainers could potentially lag behind new versions up to several releases. To check which version you have installed, you can use:
$ glpsol --version
If you want the most recent version, you might consider downloading the source code from the project homepage and build the solver from source by following the instructions in the accompanying INSTALL file. Typically, this boils down to the following steps, where X-Y must be replaced by the version number, e.g. 4-60:
$ tar -xzvf glpk-X-Y.tar.gz
$ cd glpk-X-Y
$ ./configure
$ make
$ make check
$ sudo make install
To check whether (and where) the solver was installed, you can use:
$ which glpsol
/usr/bin/glpsol
To check whether all models work as intended, run included script test.sh:
$ ./test.sh
If it returns without output (and returns no error code), all models are syntactically fine. If there is a compilation error, it will be printed and the script returns an error code.
Each model has its own author and license statement in the file header. Most models so far have the Creative Commons Public Domain Dedication, short CC0. In other words, you can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. Some minor conditions still apply, most notably: When using or citing the work, you should not imply endorsement by the author or the affirmer. But that's about it. But when in doubt, read the full license text.