TOPF: Temperate Optimal Power Flow

This package implements a "temperate" Optimal Power Flow (OPF) algorithm that dispatches production sources in a power system while disfavoring heavily-loaded lines.

The core of the algorithm is to assign a cost to each line of the power grid that is proportional to the square of the line's loading rate. In this way, the optimal power flow becomes a convex optimization problem that can be solved with standard methods.

Given the power $P_i$ flowing through a line with index $i$, and the thermal limit of the line $P_i^{th}$, the loading rate $R_i$ of the line is given by the formula

$R_i = | P_i | / P_i^{th}$

We define the cost associated with line $i$ to be the square of the loading rate, weighted by the line's thermal limit:

$P_i^{th} R_i^2$

Using the definition of the loading rate, this is equivalent to:

$P_i^2 / P_i^{th}$

This cost is independent of the sign of the power $P_i$, which depends on the line's direction.

In addition to the quadratic line cost, a conventional, linear production cost can be associated with each generator in order to model the contingencies of power production and introduce some realistic noise in the optimisation process.

The TOPF algorithm also supports constraints that ensures a given total production over a certain period of time for every individual generator.

The package can be installed with

] add TemperateOptimalPowerFlow

You can test the installation by running

] test TemperateOptimalPowerFlow

To begin, import a model in MatPower or PowerModels format. For instance, working the package's directory, one can use:

using TemperateOptimalPowerFlow
network = import_model("test/switzerland.json")

or

using TemperateOptimalPowerFlow, PowerModels
network = parse_file("test/case3.m")
prepare_model!(network)

If the model is imported with the PowerModels parse_file method, it is necessary to run the command prepare_model! that adds some features to the model. In particular, it ensures that the model contains an expected generation value pexp for each generator, which must be a fraction of the generator's capacity pmax (if missing, this is set to 50% of the capacity by default).

The optimal power flow computation is then setup with the command

setup(directory, network, loads, gen_costs)

where

• directory is a path to a directory to be created, in which computation files will be placed;
• network is the network object imported before;
• loads is a matrix that contain time series for each load of the model, arranged as rows;
• gen_costs is a similar matrix with one row for each generator of the model, containing the variable generation cost of that generator.

Several files are generated in the relevant directory. To perform the actual optimal power flow computation, one needs to import an optimizer such as Gurobi (commercial) or Ipopt (open source), or otherwise provide a get_optimizer() method that returns an optimizer in the MathOptInterface format. Then the computation can be launched with

using Ipopt
compute(directory)

WARNING: For large models and/or long time series the computation can be slow (several hours!) and memory-intensive. It is recommended to partition the optimal power flow computation into smaller parts. For instance, a time series of 8736 time steps (364 days x 24 hours) can be partitioned into 52 weeks of 168 time steps using

compute(directory, "P_result_52x168", [52, 168])

or partitioned further into 13 "months" of 28 days of 24 hours as

compute(directory, "P_result_13x28x24", [13, 28, 24])

The second argument in both cases indicates the name of the result file.

Once the computation is done, the results can be accessed with

retrieve_gen_results(directory, "P_result_13x28x24")

returning a matrix of time series in which each row corresponds to a generator of the model. The ordered list of generators can be obtained with

get_ordered_gen_ids(network)

Similarly, the power flowing through the lines of the network can be accessed with

retrieve_line_flows(directory, "P_result_13x28x24")

where the rows correspond to the line IDs of

get_ordered_line_ids(network)

A more complete description of the package's possibilities can be found in the PowerData repository.