LODF matrix

In this tutorial the methods for computing the Line Outage Distribution Factor (LODF) are presented. Before diving into this tutorial we encourage the user to load PowerNetworkMatrices, hit the ? key in the REPL terminal and look for the documentiont of the different LODF methods avialable.

Evaluation of the `LODF` matrix

As for the PTDF matrix, the LODF one can be evaluated according to two different approaches:

Dense: considers functions for dense matrix multiplication and inversion
KLU: considers functions for sparse matrix multiplication and inversion (default)

The evaluation of the LODF matrix can be easily performed starting from importing the system's data and then by simply calling the LODF method.

julia> using PowerNetworkMatrices
julia> using PowerSystemCaseBuilder
julia> const PNM = PowerNetworkMatricesPowerNetworkMatrices
julia> const PSB = PowerSystemCaseBuilder
       
       # get the System dataPowerSystemCaseBuilder
julia> sys = PSB.build_system(PSB.PSITestSystems, "c_sys5");
       
       # compute the LODF matrix[ Info: Loaded time series from storage file existing=/home/runner/.julia/packages/PowerSystemCaseBuilder/dJGb8/data/serialized_system/614e094ea985a55912fc1321256a49b755f9fc451c0f264f24d6d04af20e84d7/c_sys5_time_series_storage.h5 new=/tmp/jl_YAJUEM compression=InfrastructureSystems.CompressionSettings(false, InfrastructureSystems.CompressionTypesModule.CompressionTypes.DEFLATE = 1, 3, true)
julia> lodf_1 = LODF(sys);
julia> lodf_2 = LODF(sys, linear_solver="Dense");
       
       # show matrix
julia> get_lodf_data(lodf_1)6×6 transpose(::Matrix{Float64}) with eltype Float64:
 -1.0        0.344795   0.307071  -1.0       -1.0       -0.307071
  0.542857  -1.0        0.692929   0.542857   0.542857  -0.692929
  0.457143   0.655205  -1.0        0.457143   0.457143   1.0
 -1.0        0.344795   0.307071  -1.0       -1.0       -0.307071
 -1.0        0.344795   0.307071  -1.0       -1.0       -0.307071
 -0.457143  -0.655205   1.0       -0.457143  -0.457143  -1.0

Advanced users might be interested in computing the LODF matrix starting from either the branches and buses data (CASE 1), the IncidenceMatrix and PTDF structures (CASE 2), or by the information related to IncidenceMatrix, BA_Matrix and ABA_Matrix (CASE 3).

julia> # CASE 1
       
       # get the branches and buses data
       branches = PNM.get_ac_branches(sys);
julia> buses = PNM.get_buses(sys);
       
       # compute the LODF matrix from branches and buses data
julia> lodf_3 = LODF(branches, buses);
       
       # CASE 2
       
       # get the Incidence and PTDF matrix
julia> a = IncidenceMatrix(sys);
julia> ptdf = PTDF(sys);
       
       # compute LODF matrix with the two netwrok matrices
julia> lodf_4 = LODF(a, ptdf);
       
       # CASE 3
       
       # get the BA and ABA matrices (ABA matrix must include LU factorization
       # matrices)
julia> ba = BA_Matrix(sys);
julia> aba = ABA_Matrix(sys, factorize = true);
       
       # compute LODF matrix with the three netwrok matrices
julia> lodf_5 = LODF(a, aba, ba);

NOTE: whenever the method LODF(sys::System) is used, the methods previously defined for CASE 1 and CASE 2 are executed in sequence. Therefore the method LODF(a::IncidenceMatrix, ptdf::PTDF) is the default one when evaluating the LODF matrix from the System data directly.

Available methods for the computation of the `LODF` matrix

For those methods that either require the evaluation of the PTDF matrix, or that execute this evaluation internally, two different approaches casen be used.

As for the PTDF matrix, here too the optional argument linear_solver can be specified with either KLU (for sparse matrix calculation) or Dense (for sparse matrix calculation).

julia> lodf_dense = LODF(sys, linear_solver="Dense");

NOTE (1): by default the "KLU" method is selected, which appeared to require significant less time and memory with respect to "Dense". Please note that wether the KLU or Dense method is used, the resultig LODF matrix is stored as a dense one.

NOTE (2): for the moment, the method LODF(a::IncidenceMatrix, aba::ABA_Matrix, ba::BA_Matrix) will take KLU as linear_solver option.

"Sparse" `LODF` matrix

The LODF matrix can be computed in a "sparse" fashion by defining the input argument tol. If this argument is defined, then elements of the LODF matrix whose absolute values are below the set threshold are dropped. In addition, the matrix will be stored as a sparse one of type SparseArrays.SparseMatrixCSC{Float64, Int64} type instead of Matrix{Float64} one.

By considering an "extreme" value of 0.4 as tol, the LODF matrix can be computed as follows:

julia> lodf_sparse = LODF(sys, tol=0.4);
julia> get_lodf_data(lodf_sparse)6×6 LinearAlgebra.Transpose{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}} with 27 stored entries:
 -1.0         ⋅          ⋅        -1.0       -1.0         ⋅
  0.542857  -1.0        0.692929   0.542857   0.542857  -0.692929
  0.457143   0.655205  -1.0        0.457143   0.457143   1.0
 -1.0         ⋅          ⋅        -1.0       -1.0         ⋅
 -1.0         ⋅          ⋅        -1.0       -1.0         ⋅
 -0.457143  -0.655205   1.0       -0.457143  -0.457143  -1.0

Please consider that 0.4 was used for the purpose of this tutorial. In practice much smaller values are used (e.g., 1e-5).

NOTE (1): elements whose absolute values exceed the tol argument are removed from the LODF matrix after this has been computed.

NOTE (2): the tol argument does not refer to the "sparsification" tolerance of the PTDF matrix that is computed in the LODF method.

NOTE (3): in case the method LODF(a::IncidenceMatrix, ptdf::PTDF) is considered, an error will be thrown whenever the tol argument in the PTDF structure used as input is different than 1e-15.

LODF matrix

Evaluation of the LODF matrix

Available methods for the computation of the LODF matrix

"Sparse" LODF matrix

Evaluation of the `LODF` matrix

Available methods for the computation of the `LODF` matrix

"Sparse" `LODF` matrix