Add flow decomposition documentation (#142)

* Created read the docs documentation
* Added flow decomposition documentation

Signed-off-by: Sébastien Murgey <sebastien.murgey@rte-france.com>

docs/.readthedocs.yaml

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+version: 2
+
+build:
+  os: ubuntu-20.04
+  tools:
+    python: "3.9"
+
+sphinx:
+  configuration: docs/conf.py
+
+python:
+  install:
+    - requirements: docs/requirements.txt

docs/Makefile

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+# Minimal makefile for Sphinx documentation
+#
+
+# You can set these variables from the command line, and also
+# from the environment for the first two.
+SPHINXOPTS    ?=
+SPHINXBUILD   ?= sphinx-build
+SOURCEDIR     = .
+BUILDDIR      = _build
+
+# Put it first so that "make" without argument is like "make help".
+help:
+	@$(SPHINXBUILD) -M help "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O)
+
+.PHONY: help Makefile
+
+
+clean:
+	@echo "Removing $(SOURCEDIR)/reference/api"
+	@rm -rf "$(SOURCEDIR)/reference/api"
+	@$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O)
+
+
+# Catch-all target: route all unknown targets to Sphinx using the new
+# "make mode" option.  $(O) is meant as a shortcut for $(SPHINXOPTS).
+%: Makefile
+	@$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O)

docs/README.md

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+These are the documentation sources for PowSybl ENTSO-E features.
+
+Please keep them up-to-date with your developments.  
+They are published on powsybl.readthedocs.io/projects/entsoe and pull requests are built and previewed automatically.
+
+In order to build the docs locally, run the following commands:
+~~~bash
+pip install -r docs/requirements.txt
+sphinx-build -a docs ./build-docs
+~~~
+Then open `build-docs/index.html` in your browser.

docs/_static/styles/styles.css

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+/* The following code ensures font is not too big (Furo theme can make font too big on large screens) */
+@media(min-width:97em) {
+    html {
+        font-size: 100%
+    }
+}

docs/conf.py

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+# Configuration file for the Sphinx documentation builder.
+#
+# This file only contains a selection of the most common options. For a full
+# list see the documentation:
+# https://www.sphinx-doc.org/en/master/usage/configuration.html
+
+# -- Path setup --------------------------------------------------------------
+
+# If extensions (or modules to document with autodoc) are in another directory,
+# add these directories to sys.path here. If the directory is relative to the
+# documentation root, use os.path.abspath to make it absolute, like shown here.
+#
+import os
+import sys
+
+# Path to python sources, for doc generation on readthedocs
+source_path = os.path.abspath('..')
+sys.path.insert(0, source_path)
+print(f'appended {source_path}')
+
+
+# -- Project information -----------------------------------------------------
+
+project = 'PowSyBl ENTSO-E'
+copyright = '2024, RTE (http://www.rte-france.com)'
+
+
+# -- General configuration ---------------------------------------------------
+
+# Add any Sphinx extension module names here, as strings. They can be
+# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
+# ones.
+extensions = ['sphinx.ext.autodoc',
+              'sphinx.ext.autosummary',
+              'sphinx.ext.viewcode',
+              'sphinx.ext.doctest',
+              'sphinx.ext.napoleon',
+              'sphinx.ext.todo',
+              'sphinx.ext.intersphinx',
+              'sphinx_tabs.tabs',
+              'myst_parser']
+myst_enable_extensions = [
+    "amsmath",
+    "colon_fence",
+    "dollarmath"
+]
+myst_heading_anchors = 6
+
+# Add any paths that contain templates here, relative to this directory.
+templates_path = ['_templates']
+
+# List of patterns, relative to source directory, that match files and
+# directories to ignore when looking for source files.
+# This pattern also affects html_static_path and html_extra_path.
+exclude_patterns = ['_build', 'Thumbs.db', '.DS_Store']
+
+
+# -- Options for HTML output -------------------------------------------------
+
+# The theme to use for HTML and HTML Help pages.  See the documentation for
+# a list of builtin themes.
+#
+html_theme = "furo"
+
+html_title = 'ENTSO-E'
+html_short_title = 'ENTSO-E'
+
+html_logo = '_static/logos/logo_lfe_powsybl.svg'
+html_favicon = "_static/favicon.ico"
+
+html_theme_options = {
+    "icon_links": [
+        {
+            "name": "GitHub",
+            "url": "https://github.com/powsybl/powsybl-entsoe",
+            "icon": "fab fa-github-square",
+        }
+    ],
+    "navbar_start": ["navbar-brand-powsybl-entsoe"],
+    # the following 3 lines enable edit button
+    "source_repository": "https://github.com/powsybl/powsybl-entsoe/",
+    "source_branch": "main",
+    "source_directory": "docs/",
+}
+
+# Add any paths that contain custom static files (such as style sheets) here,
+# relative to this directory. They are copied after the builtin static files,
+# so a file named "default.css" will overwrite the builtin "default.css".
+html_static_path = ['_static']
+html_css_files = ['styles/styles.css']
+
+todo_include_todos = True
+
+# Links to external documentations : python 3 and pandas
+intersphinx_mapping = {
+    'python': ('https://docs.python.org/3', None),
+    'pandas': ('https://pandas.pydata.org/docs', None),
+}
+
+# Generate one file per method
+autosummary_generate = True

docs/flow_decomposition/algorithm-description.md

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+# Algorithm description
+
+The flow decomposition algorithm is based on the DC approximation, in which the losses in the network branches
+are neglected, and that allows to rely on the superposition principle to assess which is the impact of any injection
+on any branch flow by simple sensitivity analysis.
+
+Below is the concrete description of the algorithm implemented in PowSyBl.
+
+![Flow decomposition algorithm chart](/_static/img/flow_decomposition/flowDecompositionAlgorithmChart.svg)
+
+## Net positions computation
+
+Countries' net position computation is done once for all on base case using AC loadflow in the initial network, before any other alteration of the input.
+
+The net position of a country is calculated as the sum of the mean leaving flow of all AC and HVDC line interconnections
+(losses are shared equally between both countries)
+
+## Losses compensation
+
+In order to mitigate the impact of DC approximation in the flow decomposition process, a dedicated step of losses compensation is implemented.
+
+Instead of using standard power flow compensation methodology, a standard full AC power flow is run on the input network
+that allows to calculate the losses on each network element.
+
+These losses are then compensated on the sending side of each network element.
+
+![Losses compensation on lines](/_static/img/flow_decomposition/lossesCompensationOnLine.svg)
+
+A special treatment is done on tie lines, where instead of compensating the losses on the sending terminal, losses are
+compensated at both sides proportionally to the resistance of each half line.
+
+![Losses compensation on tie lines](/_static/img/flow_decomposition/lossesCompensationOnTieLine.svg)
+
+## Nodal Injections partitioning
+
+In order to distinguish internal/loop flows and allocated flows, the nodal injections in each zone must de decomposed in two parts:
+- Nodal injections for allocated flows
+- Nodal injections for loop flows and internal flows
+- Nodal injections for xnode flows
+
+This decomposition is based on GLSK (Generation and Load Shift Keys). It is an input of the process that provides,
+for each zone of the study a list of injections and associated factor to be used to scale the zone to a given net position.
+
+By default, the algorithm uses so-called "Country GSK", which is an automatic GLSK that scales on all generators
+proportionally to their target power setpoint.
+
+Nodal injection decomposition is done as follows:
+
+$$
+\begin{array}{l}
+\mathrm{NI}_\mathrm{AF} = \mathrm{GLSK} \cdot \mathrm{NP} \\
+\mathrm{NI}_\mathrm{LIF} = \mathrm{NI} - \mathrm{NI}_\mathrm{AF} - \mathrm{NI}_\mathrm{X}
+\end{array}
+$$
+
+where:
+- $\mathrm{NI}$ is the vector of the network injections,
+- $\mathrm{NI}_\mathrm{X}$ is the vector of the network injections from dangling lines,
+- $\mathrm{NI}_\mathrm{AF}$ is the vector of allocated flow part of the network injections,
+- $\mathrm{NI}_\mathrm{LIF}$ is the vector of loop flow and internal flow part of the network injections,
+- $\mathrm{NP}$ is the vector of the zones' net position,
+- $\mathrm{GLSK}$ is the matrix of the GLSK factors for each injection in each zone,
+
+## Sensitivity analysis
+
+In order to assess the linear impact (implied by the DC approximation) of each nodal injection and phase shift transformer
+on the network elements' flow, a sensitivity analysis is run.
+
+The following matrices are calculated using [sensitivity analysis](https://www.powsybl.org/pages/documentation/simulation/sensitivity/) API:
+- $\mathrm{PTDF}$ is the matrix of the sensitivity of the network element flow to each network injection shift,
+- $\mathrm{PSDF}$ is the matrix of the sensitivity of the network element flow to each phase shift transformer tap angle change,
+
+## Flow partitioning
+
+Based on previously calculated elements, flow partitioning can now be calculated as follows:
+
+$$
+\begin{array}{l}
+\mathrm{F}_\mathrm{AF} = \mathrm{PTDF} \cdot \mathrm{NI}_\mathrm{AF} \\
+\mathrm{F}_\mathrm{LIF} = \mathrm{PTDF} \cdot \mathrm{diag}(\mathrm{NI}_\mathrm{LIF}) \cdot \mathrm{AM} \\
+\mathrm{F}_\mathrm{PST} = \mathrm{PSDF} \cdot \mathrm{\Delta}_\mathrm{PST} \\
+\mathrm{F}_\mathrm{X} = \mathrm{PTDF} \cdot \mathrm{NI}_\mathrm{X} \\
+\end{array}
+$$
+
+where:
+- $\mathrm{F}_\mathrm{AF}$ is the vector of the network element allocated flow,
+- $\mathrm{F}_\mathrm{LIF}$ is the matrix of the network element loop flow or internal flow for each zone,
+- $\mathrm{F}_\mathrm{PST}$ is the vector of the network element PST (phase shift transformer) flow,
+- $\mathrm{F}_\mathrm{X}$ is the vector of the network element xnode flow,
+- $\mathrm{AM}$ is the allocation matrix, which associates each injection to its zone. $\mathrm{AM}_{ij}$ = 1 if node i is in zone j, 0 otherwise,
+- $\mathrm{\Delta}_\mathrm{PST}$ is the phase shift transformers angle vector,
+
+## Flow parts rescaling
+
+Due to superposition principle, the sum of all the flow parts calculated previously is equal to the flow that was
+calculated by the DC power flow.
+
+However, the flow reference is the one calculated using AC power flow which is different. The final step of the algorithm
+is though to rescale the different flow parts in order to ensure that the sum of the parts is equal to the initially calculated AC flow.
+
+The difference between reference AC flow and the sum of the parts of the decomposition is redispatched on the different
+parts proportionally to their rectified linear unit ($\mathrm{ReLU}(x) = \mathrm{max}(x, 0)$).