add initial content

.gitignore

+
+public/
+resources/
+node_modules/
+tech-doc-hugo
+

.gitlab-ci.yml

+variables:
+  DOCKER_TAG: ${CI_COMMIT_REF_NAME}
+  DOCKER_IMAGE: rwthacs/hugo-ext
+
+before_script:
+  - git config --local core.longpaths true
+  - git submodule sync --recursive
+  - git submodule update --init --recursive
+
+stages:
+  - prepare
+  - build
+  - deploy
+
+docker:
+  stage: prepare
+  script:
+    - docker build
+        --tag ${DOCKER_IMAGE}:${DOCKER_TAG}
+        --tag ${DOCKER_IMAGE}:latest .
+  tags:
+    - shell
+    - linux
+
+hugo:
+  stage: build
+  script:
+    - npm install -D --save autoprefixer
+    - npm install -D --save postcss-cli
+    - hugo
+  image: ${DOCKER_IMAGE}:${DOCKER_TAG}
+  artifacts:
+    paths:
+      - public
+  tags:
+    - docker
+
+#pages:
+#  stage: deploy
+#  script:
+#    - ls -l
+#  artifacts:
+#    paths:
+#      - public
+#  only:
+#    - master
+#  tags:
+#    - shell
+#    - linux
+

CONTRIBUTING.md

+# How to Contribute
+
+We'd love to accept your patches and contributions to this project. 
+Please get in touch with us via mail or slack if you would like to contribute.
+

Dockerfile

+FROM fedora:30
+
+RUN dnf -y update
+
+RUN dnf -y install \
+  pandoc \
+  wget \
+  git \
+  nodejs
+
+WORKDIR /hugo
+RUN wget https://github.com/gohugoio/hugo/releases/download/v0.65.3/hugo_extended_0.65.3_Linux-64bit.tar.gz && \
+  tar -xvf hugo_extended_0.65.3_Linux-64bit.tar.gz && \
+  install hugo /usr/bin
+
+WORKDIR /website
+# npm install -D --save autoprefixer
+# npm install -D --save postcss-cli
+
+EXPOSE 1313
\ No newline at end of file

LICENSE

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+   limitations under the License.

assets/scss/_variables_project.scss

+/*
+
+Add styles or override variables from the theme here.
+
+*/
+

config.toml

+baseURL = "https://dpsim-simulator.github.io"
+title = "DPsim"
+
+enableRobotsTXT = true
+
+# Hugo allows theme composition (and inheritance). The precedence is from left to right.
+theme = ["docsy"]
+
+# Will give values to .Lastmod etc.
+enableGitInfo = true
+
+# Language settings
+contentDir = "content/en"
+defaultContentLanguage = "en"
+defaultContentLanguageInSubdir = false
+# Useful when translating.
+enableMissingTranslationPlaceholders = true
+
+disableKinds = ["taxonomy", "taxonomyTerm"]
+
+# Highlighting config
+pygmentsCodeFences = true
+pygmentsUseClasses = false
+# Use the new Chroma Go highlighter in Hugo.
+pygmentsUseClassic = false
+#pygmentsOptions = "linenos=table"
+# See https://help.farbox.com/pygments.html
+pygmentsStyle = "tango"
+
+# Configure how URLs look like per section.
+[permalinks]
+blog = "/:section/:year/:month/:day/:slug/"
+
+## Configuration for BlackFriday markdown parser: https://github.com/russross/blackfriday
+[blackfriday]
+plainIDAnchors = true
+hrefTargetBlank = true
+angledQuotes = false
+latexDashes = true
+
+# Image processing configuration.
+[imaging]
+resampleFilter = "CatmullRom"
+quality = 75
+anchor = "smart"
+
+[services]
+[services.googleAnalytics]
+# Comment out the next line to disable GA tracking. Also disables the feature described in [params.ui.feedback].
+#id = "UA-00000000-0"
+
+# Language configuration
+
+[languages]
+[languages.en]
+title = "DPsim"
+description = "Real-Time Power System Simulation"
+languageName ="English"
+# Weight used for sorting.
+weight = 1
+time_format_default = "02.01.2006"
+time_format_blog = "02.01.2006"
+
+[markup]
+  [markup.goldmark]
+    [markup.goldmark.renderer]
+      unsafe = true
+
+# Everything below this are Site Params
+
+[params]
+copyright = "The DPsim Authors"
+
+# First one is picked as the Twitter card image if not set on page.
+# images = ["images/project-illustration.png"]
+
+# Menu title if your navbar has a versions selector to access old versions of your site.
+# This menu appears only if you have at least one [params.versions] set.
+version_menu = "Releases"
+
+# Flag used in the "version-banner" partial to decide whether to display a
+# banner on every page indicating that this is an archived version of the docs.
+# Set this flag to "true" if you want to display the banner.
+archived_version = false
+
+# The version number for the version of the docs represented in this doc set.
+# Used in the "version-banner" partial to display a version number for the
+# current doc set.
+version = "0.0"
+
+# A link to latest version of the docs. Used in the "version-banner" partial to
+# point people to the main doc site.
+url_latest_version = "https://example.com"
+
+# Repository configuration (URLs for in-page links to opening issues and suggesting changes)
+github_repo = "https://github.com/dpsim-simulator/docs"
+# An optional link to a related project repo. For example, the sibling repository where your product code lives.
+github_project_repo = "https://git.rwth-aachen.de/acs/public/simulation/dpsim/dpsim"
+
+# Specify a value here if your content directory is not in your repo's root directory
+# github_subdir = ""
+
+# Google Custom Search Engine ID. Remove or comment out to disable search.
+gcs_engine_id = "011737558837375720776:fsdu1nryfng"
+
+# Enable Algolia DocSearch
+algolia_docsearch = false
+
+# Enable Lunr.js offline search
+offlineSearch = false
+
+# User interface configuration
+[params.ui]
+# Enable to show the side bar menu in its compact state.
+sidebar_menu_compact = false
+#  Set to true to disable breadcrumb navigation.
+breadcrumb_disable = false
+#  Set to true to hide the sidebar search box (the top nav search box will still be displayed if search is enabled)
+sidebar_search_disable = false
+#  Set to false if you don't want to display a logo (/assets/icons/logo.svg) in the top nav bar
+navbar_logo = false
+# Set to true to disable the About link in the site footer
+footer_about_disable = false
+
+# Adds a H2 section titled "Feedback" to the bottom of each doc. The responses are sent to Google Analytics as events.
+# This feature depends on [services.googleAnalytics] and will be disabled if "services.googleAnalytics.id" is not set.
+# If you want this feature, but occasionally need to remove the "Feedback" section from a single page,
+# add "hide_feedback: true" to the page's front matter.
+[params.ui.feedback]
+enable = true
+# The responses that the user sees after clicking "yes" (the page was helpful) or "no" (the page was not helpful).
+yes = 'Glad to hear it! Please <a href="https://github.com/USERNAME/REPOSITORY/issues/new">tell us how we can improve</a>.'
+no = 'Sorry to hear that. Please <a href="https://github.com/USERNAME/REPOSITORY/issues/new">tell us how we can improve</a>.'
+
+[params.links]
+# End user relevant links. These will show up on left side of footer and in the community page if you have one.
+[[params.links.user]]
+	name = "Slack"
+	url = "https://feinev.slack.com"
+	icon = "fab fa-slack"
+        desc = "Chat with other users and get help"
+#[[params.links.user]]
+#	name = "User mailing list"
+#	url = "https://example.org/mail"
+#	icon = "fa fa-envelope"
+#        desc = "Discussion and help from your fellow users"
+#[[params.links.user]]
+#	name ="Twitter"
+#	url = "https://example.org/twitter"
+#	icon = "fab fa-twitter"
+#        desc = "Follow us on Twitter to get the latest news!"
+#[[params.links.user]]
+#	name = "Stack Overflow"
+#	url = "https://example.org/stack"
+#	icon = "fab fa-stack-overflow"
+#        desc = "Practical questions and curated answers"
+# Developer relevant links. These will show up on right side of footer and in the community page if you have one.
+[[params.links.developer]]
+	name = "RWTH GitLab"
+	url = "https://git.rwth-aachen.de/acs/public/simulation/dpsim/dpsim"
+	icon = "fab fa-gitlab"
+        desc = "Development takes place here!"
+[[params.links.developer]]
+	name = "GitHub"
+	url = "https://github.com/DPsim-Simulator/dpsim"
+	icon = "fab fa-github"
+        desc = "Mirror of the RWTH GitLab project"
+[[params.links.developer]]
+	name = "Slack"
+	url = "https://feinev.slack.com"
+	icon = "fab fa-slack"
+        desc = "Chat with other project developers"
+#[[params.links.developer]]
+#	name = "Developer mailing list"
+#	url = "https://example.org/mail"
+#	icon = "fa fa-envelope"
+#        desc = "Discuss development issues around the project"
+

content/en/_index.html

++++
+title = "DPsim"
+linkTitle = "DPsim"
+
++++
+
+{{< blocks/cover title="Welcome to DPsim: A Real-Time Power System Simulator" image_anchor="top" height="full" color="orange" >}}
+<div class="mx-auto">
+	<a class="btn btn-lg btn-primary mr-3 mb-4" href="{{< relref "/docs" >}}">
+		Learn More <i class="fas fa-arrow-alt-circle-right ml-2"></i>
+	</a>
+	<a class="btn btn-lg btn-secondary mr-3 mb-4" href="https://github.com/DPsim-Simulator/dpsim">
+		Download <i class="fab fa-github ml-2 "></i>
+	</a>
+</div>
+{{< /blocks/cover >}}
+

content/en/about/_index.html

+---
+title: About DPsim
+linkTitle: About
+menu:
+  main:
+    weight: 10
+
+---
+
+
+{{< blocks/cover title="About DPsim" image_anchor="bottom" height="min" >}}
+
+{{< /blocks/cover >}}
+
+{{% blocks/lead %}}
+DPsim is a real-time capable power system simulator that supports dynamic phasor and electromagnetic transient simulation as well as continuous powerflow. It primarily targets large-scale scenarios on commercial off-the-sheld hardware that require deterministic time steps in the range of micro- to milliseconds.
+
+Here you can find instructions on how to build and install DPsim as well as descriptions of the solver and power system models.
+{{% /blocks/lead %}}
+
+{{< blocks/section color="dark" >}}
+DPsim supports the CIM format as native input for the description of electrical network topologies, component parameters and load flow data, which is used for initialization. For this purpose, CIM++ is integrated in DPsim. Users interact with the C++ simulation kernel via Python bindings, which can be used to script the execution, schedule events, change parameters and retrieve results. Supported by the availability of existing Python frameworks like Numpy, Pandas and Matplotlib, Python scripts have been proven as an easy and flexible way to codify the complete workflow of a simulation from modelling to analysis and plotting, for example in Jupyter notebooks.
+
+The DPsim simulation kernel is implemented in C++ and uses the Eigen linear algebra library. By using a system programming language like C++ and a highly optimized math library, optimal performance and real-time execution can be guaranteed. The integration into the VILLASframework allows DPsim to be used in large-scale co-simulations.
+{{< /blocks/section >}}

content/en/blog/_index.md

+---
+title: "DPsim Blog"
+linkTitle: "Blog"
+menu:
+  main:
+    weight: 30
+---
+
+
+This is the **blog** section. It has two categories: News and Releases.
+
+Files in these directories will be listed in reverse chronological order.
+

content/en/blog/news/2020-03-16_new-docs.md

+---
+title: "New Documentation"
+linkTitle: "New Documentation"
+date: 2020-03-16
+description: >
+  based on Hugo and the docsy documentation theme
+---
+
+The new documentation will include:
+- build and development instructions
+- an architecture overview
+- solver and model descriptions
+- guidelines on how to add new models
+
+To support the description of mathematical models, mathjax is included for rendering latex equations.
\ No newline at end of file

content/en/blog/news/_index.md

+
+---
+title: "News About DPsim"
+linkTitle: "News"
+weight: 20
+---
+
+

content/en/blog/releases/2020-03-17_open-source.md

+---
+title: "Open Source Release"
+linkTitle: "Open Source Release"
+date: 2020-03-17
+---
+
+DPsim version 1.0.0 was released as open source  in January 2018.
+The release is associated to the [SoftwareX article](https://www.sciencedirect.com/science/article/pii/S2352711018302760) about DPsim.
+

content/en/blog/releases/_index.md

+
+---
+title: "New Releases"
+linkTitle: "Releases"
+weight: 20
+---
+
+

content/en/community/_index.md

+---
+title: Community
+menu:
+  main:
+    weight: 40
+---
+
+<!--add blocks of content here to add more sections to the community page -->

content/en/docs/Concepts/_index.md

+---
+title: "Concepts"
+linkTitle: "Concepts"
+weight: 4
+---
+
+The book introduces the reader to the general concepts implemented in DPsim, a dynamic phasor (DP) real-time simulator, as well as the physical models of the power system components that are used in simulations.
+The first chapters give an overview of dynamic phasors and nodal analysis which are the two pillars of the main solver implemented in DPsim.
+The second part describes in detail what are the physical equations for each model and how they are transformed and implemented for dynamic phasor simulations and other domains that are also supported by DPsim.
+
+In order to be able to run a dynamic simulation, DPsim also includes a loadflow solver to compute the initial state of the network if it is not included in the network data.
+Besides DP simulations, DPsim also comes with EMT models for some components which are used as reference for testing the DP models.

content/en/docs/Concepts/dyn-phasors.md

+---
+title: "Dynamic Phasors"
+linkTitle: "Dynamic Phasors"
+date: 2020-03-18
+---
+
+In the power systems community, dynamic phasors were initially introduced for power electronics analysis [Sanders1991](https://ieeexplore.ieee.org/document/76811) as a more general approach than state-space averaging.
+They were used to construct efficient models for the dynamics of switching gate phenomena with a high level of detail as shown in [Mattavelli1999](https://ieeexplore.ieee.org/abstract/document/744524).
+A few years later, dynamic phasors were also employed for power system simulation as described in [Demiray2008](https://www.research-collection.ethz.ch/handle/20.500.11850/123490).
+In [Strunz2006](https://ieeexplore.ieee.org/document/4026700) the authors combine the dynamic phasor approach with the Electromagnetic Transients Program (EMTP) simulator concept which includes Modified Nodal Analysis (MNA).
+Further research topics include fault and stability analysis under unbalanced conditions as presented in [Stankovic2000](https://ieeexplore.ieee.org/document/871734) and also rotating machine models have been developed in dynamic phasors [Zhang 2007](https://ieeexplore.ieee.org/document/4282063).
+
+## Bandpass Signals and Baseband Representation
+
+Although here, dynamic phasors are presented as a power system modelling tool, it should be noted that the concept is also known in other domains, for example, microwave and communications engineering [Maas2003, Suarez2009, Haykin2009, Proakis2001].
+In these domains, the approach is often denoted as base band representation or complex envelope.
+Another common term coming from power electrical engineering is shifted frequency analysis (SFA).
+In the following, the general approach of dynamic phasors for power system simulation is explained starting from the idea of bandpass signals.
+This is because the 50 Hz or 60 Hz fundamental and small deviations from it can be seen as such a bandpass signal.
+Futhermore, higher frequencies, for example, generated by power electronics can be modelled in a similar way.
+

content/en/docs/Concepts/nodal-analysis.md

+---
+title: "Nodal Analysis"
+linkTitle: "Nodal Analysis"
+date: 2020-03-18
+markup: pandoc
+---
+
+<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+
+A circuit with $b$ branches has $2b$ unknowns since there are $b$ voltages and $b$ currents.
+Hence, $2b$ linear independent equations are required to solve the circuit.
+If the circuit has $n$ nodes and $b$ branches, it has
+
+* Kirchoff's current law (KCL) equations
+* Kirchoff's voltage law (KVL) equations
+* Characteristic equations (Ohm's Law)
+
+There are only $n-1$ KCLs since the nth equation is a linear combination of the remaining $n-1$.
+At the same time, it can be demonstrated that if we can imagine a very high number of closed paths in the network, only $b-n+1$ are able to provide independent KVLs.
+Finally there are $b$ characteristic equations, describing the behavior of the branch, making a total of $2b$ linear independent equations.
+
+The nodal analysis method reduces the number of equations that need to be solved simultaneously.
+$n-1$ voltage variables are defined and solved, writing $n-1$ KCL based equations.
+A circuit can be solved using Nodal Analysis as follows
+
+* Select a reference node (mathematical ground) and number the remaining $n-1$ nodes, that are the independent voltage variables
+* Represent every branch current $i$ as a function of node voltage variables $v$ with the general expression $i = g(v)$
+* Write $n-1$ KCL based equations in terms of node voltage variable.
+
+The resulting equations can be written in matrix form and have to be solved for $v$.
+$$\boldsymbol{Y} \boldsymbol{v} = \boldsymbol{i}$$

content/en/docs/Concepts/powerflow.md

+---
+title: "Powerflow"
+linkTitle: "Powerflow"
+date: 2020-03-18
+markup: pandoc
+---
+
+<script>
+MathJax = {
+  tex: {
+    tags: 'all'
+  }
+};
+</script>
+<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+
+The power flow problem is about the calculation of voltage magnitudes and angles for one set of buses.
+The solution is obtained from a given set of voltage magnitudes and power levels for a specific model of the network configuration.
+The power flow solution exhibits the voltages and angles at all buses and real and reactive flows can be deduced from the same.
+
+## Power System Model
+Power systems are modeled as a network of buses (nodes) and branches (lines).
+To a network bus, components such a generator, load, and transmission substation can be connected.
+Each bus in the network is fully described by the following four electrical quantities:
+
+* $\vert V_{k} \vert$: the voltage magnitude
+* $\theta_{k}$: the voltage phase angle
+* $P_{k}$: the active power
+* $Q_{k}$: the reactive power
+
+There are three types of networks buses: VD bus, PV bus and PQ bus.
+Depending on the type of the bus, two of the four electrical quantities are specified as shown in the table below.
+
+| Bus Type	| Known		| Unknown		|
+| ---				|	---			|	---				|
+| $VD$			| $\vert V_{k} \vert, \theta_{k}$	| $P_{k}, Q_{k}$ 			|
+| $PV$			| $P_{k}, \vert V_{k} \vert$ 			| $Q_{k}, \theta_{k}$ |
+| $PQ$      | $P_{k}, Q_{k}$									| $\vert V_{k} \vert, \theta_{k}$ |
+
+## Single Phase Power Flow Problem
+The power flow problem can be expressed by the goal to bring a mismatch function $\vec{f}$ to zero.
+The value of the mismatch function depends on a solution vector $\vec{x}$:
+$$ \vec{f}(\vec{x}) = 0 $$
+
+As $\vec{f}(\vec{x})$ will be nonlinear, the equation system will be solved with Newton-Raphson:
+
+$$-\textbf{J}(\vec{x}) \Delta \vec{x} = \vec{f} (\vec{x})$$
+
+where $\Delta \vec{x}$ is the correction of the solution vector and $\textbf{J}(\vec{x})$ is the Jacobian matrix.
+The solution vector $\vec{x}$ represents the voltage $\vec{V}$ by polar or cartesian quantities.
+The mismatch function $\vec{f}$ will either represent the power mismatch $\Delta \vec{S}$ in terms of
+
+$$\left [ \begin{array}{c} \Delta \vec{P} \\ \Delta \vec{Q} \end{array} \right ]$$
+
+or the current mismatch $\Delta \vec{I}$ in terms of
+
+$$\left [ \begin{array}{c} \Delta \vec{I_{real}} \\ \Delta \vec{I_{imag}} \end{array} \right ]$$
+
+where the vectors split the complex quantities into real and imaginary parts.
+Futhermore, the solution vector $\vec{x}$ will represent $\vec{V}$ either by polar coordinates
+
+$$\left [ \begin{array}{c} \vec{\delta} \\ \vert \vec{V} \vert \end{array} \right ]$$
+
+or rectangular coordinates
+
+$$\left [ \begin{array}{c} \vec{V_{real}} \\ \vec{V_{imag}} \end{array} \right ]$$
+
+This results in four different formulations of the powerflow problem:
+
+* with power mismatch function and polar coordinates
+* with power mismatch function and rectangular coordinates
+* with current mismatch function and polar coordinates
+* with current mismatch function and rectangular coordinates
+
+To solve the problem using NR, we need to formulate $\textbf{J} (\vec{x})$ and $\vec{f} (\vec{x})$ for each powerflow problem formulation.
+
+### Powerflow Problem with Power Mismatch Function and Polar Coordinates
+
+#### Formulation of Mismatch Function
+
+The injected power at a node $k$ is given by:
+$$S_{k} = V_{k} I _{k}^{*}$$
+
+The current injection into any bus $k$ may be expressed as:
+$$I_{k} = \sum_{j=1}^{N} Y_{kj} V_{j}$$
+
+Substitution yields:
+\begin{align}
+  S_{k} =& V_{k} \left ( \sum_{j=1}^{N} Y_{kj} V_{j} \right )^{*}
+        =& V_{k} \sum_{j=1}^{N} Y_{kj}^{*} V_{j} ^{*}
+\end{align}
+
+We may define $G_{kj}$ and $B_{kj}$ as the real and imaginary parts of the admittance matrix element $Y_{kj}$ respectively, so that $Y_{kj} = G_{kj} + jB_{kj}$.
+Then we may rewrite the last equation:
+\begin{align}
+  S_{k} &= V_{k} \sum_{j=1}^{N} Y_{kj}^{*} V_{j}^{*} \nonumber \\
+   			&= \vert V_{k} \vert \angle \theta_{k} \sum_{j=1}^{N} (G_{kj} + jB_{kj})^{*} ( \vert V_{j} \vert \angle \theta_{j})^{*} \nonumber \\
+   			&= \vert V_{k} \vert \angle \theta_{k} \sum_{j=1}^{N} (G_{kj} - jB_{kj}) ( \vert V_{j} \vert \angle - \theta_{j}) \nonumber \\
+   			&= \sum_{j=1} ^{N} \vert V_{k} \vert \angle \theta_{k} ( \vert V_{j} \vert \angle - \theta_{j}) (G_{kj} - jB_{kj}) \nonumber \\
+   			&= \sum_{j=1} ^{N} \left ( \vert V_{k} \vert \vert V_{j} \vert \angle (\theta_{k} - \theta_{j}) \right ) (G_{kj} - jB_{kj}) \nonumber \\
+   			&= \sum_{j=1} ^{N} \vert V_{k} \vert \vert V_{j} \vert \left ( cos(\theta_{k} - \theta_{j}) + jsin(\theta_{k} - \theta_{j}) \right ) (G_{kj} - jB_{kj})
+	\label{eq:compl_power}
+\end{align}
+
+If we now perform the algebraic multiplication of the two terms inside the parentheses, and collect real and imaginary parts, and recall that $S_{k} = P_{k} + jQ_{k}$, we can express $\eqref{eq:compl_power}$ as two equations: one for the real part, $P_{k}$, and one for the imaginary part, $Q_{k}$, according to:
+
+\begin{align}
+	{P}_{k} = \sum_{j=1}^{N} \vert V_{k} \vert \vert V_{j} \vert \left ( G_{kj}cos(\theta_{k} - \theta_{j}) + B_{kj} sin(\theta_{k} - \theta_{j}) \right ) \label{eq:active_power}\\
+	{Q}_{k} = \sum_{j=1}^{N} \vert V_{k} \vert \vert V_{j} \vert \left ( G_{kj}sin(\theta_{k} - \theta_{j}) - B_{kj} cos(\theta_{k} - \theta_{j}) \right ) \label{eq:reactive_power}
+\end{align}
+
+These equations are called the power flow equations, and they form the fundamental building block from which we solve the power flow problem.
+
+We consider a power system network having $N$ buses. We assume one VD bus, $N_{PV}-1$ PV buses and $N-N_{PV}$ PQ buses.
+We assume that the VD bus is numbered bus $1$, the PV buses are numbered $2,...,N_{PV}$, and the PQ buses are numbered $N_{PV}+1,...,N$.
+We define the vector of unknown as the composite vector of unknown angles $\vec{\theta}$ and voltage magnitudes $\vert \vec{V} \vert$:
+\begin{align}
+	\vec{x} = \left[ \begin{array}{c} \vec{\theta} \\ \vert \vec{V} \vert \\ \end{array} \right ]
+			 		= \left[ \begin{array}{c} \theta_{2} \\ \theta_{3} \\ \vdots \\ \theta_{N} \\ \vert V_{N_{PV+1}} \vert \\ \vert V_{N_{PV+2}} \vert \\ \vdots \\ \vert V_{N} \vert \end{array} \right]
+\end{align}
+
+The right-hand sides of equations $\eqref{eq:active_power}$ and $\eqref{eq:reactive_power}$ depend on the elements of the unknown vector $\vec{x}$.
+Expressing this dependency more explicitly, we rewrite these equations as:
+\begin{align}
+	P_{k} = P_{k} (\vec{x}) \Rightarrow P_{k}(\vec{x}) - P_{k} &= 0 \quad \quad k = 2,...,N \\
+	Q_{k} = Q_{k} (\vec{x}) \Rightarrow Q_{k} (\vec{x}) - Q_{k} &= 0 \quad \quad k = N_{PV}+1,...,N
+\end{align}
+
+We now define the mismatch vector $\vec{f} (\vec{x})$ as:
+\begin{align*}
+\vec{f} (\vec{x}) = \left [ \begin{array}{c} f_{1}(\vec{x}) \\ \vdots \\ f_{N-1}(\vec{x}) \\ ------ \\ f_{N}(\vec{x}) \\ \vdots \\ f_{2N-N_{PV} -1}(\vec{x}) \end{array} \right ]
+	= \left [ \begin{array}{c} P_{2}(\vec{x}) - P_{2} \\ \vdots \\ P_{N}(\vec{x}) - P_{N} \\ --------- \\ Q_{N_{PV}+1}(\vec{x}) - Q_{N_{PV}+1} \\ \vdots \\ Q_{N}(\vec{x}) - Q_{N} \end{array} \right]
+	= \left [ \begin{array}{c} \Delta P_{2} \\ \vdots \\ \Delta P_{N} \\ ------ \\ \Delta Q_{N_{PV}+1} \\ \vdots \\ \Delta Q_{N} \end{array} \right ]
+	= \vec{0}
+\end{align*}
+
+That is a system of nonlinear equations.
+This nonlinearity comes from the fact that $P_{k}$ and $Q_{k}$ have terms containing products of some of the unknowns and also terms containing trigonometric functions of some the unknowns.
+
+#### Formulation of Jacobian
+
+As discussed in the previous section, the power flow problem will be solved using the Newton-Raphson method. Here, the Jacobian matrix is obtained by taking all first-order partial derivates of the power mismatch functions with respect to the voltage angles $\theta_{k}$ and magnitudes $\vert V_{k} \vert$ as:
+\begin{align}
+	J_{jk}^{P \theta} &= \frac{\partial P_{j} (\vec{x} ) } {\partial \theta_{k}} = \vert V_{j} \vert \vert V_{k} \vert \left ( G_{jk} sin(\theta_{j} - \theta_{k}) - 																B_{jk} cos(\theta_{j} - \theta_{k} ) \right ) \\
+	J_{jj}^{P \theta} &= \frac{\partial P_{j}(\vec{x})}{\partial \theta_{j}} = -Q_{j} (\vec{x} ) - B_{jj} \vert V_{j} \vert ^{2} \\
+	J_{jk}^{Q \theta} &= \frac{\partial Q_{j}(\vec{x})}{\partial \theta_{k}} = - \vert V_{j} \vert \vert V_{k} \vert \left ( G_{jk} cos(\theta_{j} - \theta_{k}) + 																B_{jk} sin(\theta_{j} - \theta_{k}) \right ) \\
+	 J_{jj}^{Q \theta} &= \frac{\partial Q_{j}(\vec{x})}{\partial \theta_{k}} = P_{j} (\vec{x} ) - G_{jj} \vert V_{j} \vert ^{2} \\
+	 J_{jk}^{PV} &= \frac{\partial P_{j} (\vec{x} ) } {\partial \vert V_{k} \vert } = \vert V_{j} \vert \left ( G_{jk} cos(\theta_{j} - \theta_{k}) + 																B_{jk} sin(\theta_{j} - \theta_{k}) \right ) \\
+	 J_{jj}^{PV} &= \frac{\partial P_{j}(\vec{x})}{\partial \vert V_{j} \vert } = \frac{P_{j} (\vec{x} )}{\vert V_{j} \vert} + G_{jj} \vert V_{j} \vert \\
+	 J_{jk}^{QV} &= \frac{\partial Q_{j} (\vec{x} ) } {\partial \vert V_{k} \vert } = \vert V_{j} \vert \left ( G_{jk} sin(\theta_{j} - \theta_{k}) + 																B_{jk} cos(\theta_{j} - \theta_{k}) \right ) \\
+	 J_{jj}^{QV} &= \frac{\partial Q_{j}(\vec{x})}{\partial \vert V_{j} \vert } = \frac{Q_{j} (\vec{x} )}{\vert V_{j} \vert} - B_{jj} \vert V_{j} \vert \\
+\end{align}
+
+The linear system of equations that is solved in every Newton iteration can be written in matrix form as follows:
+\begin{align}
+	-\left [ \begin{array}{cccccc} \frac{\partial \Delta P_{2} }{\partial \theta_{2}} & \cdots & \frac{\partial \Delta P_{2} }{\partial \theta_{N}} &
+			\frac{\partial \Delta P_{2} }{\partial \vert V_{N_{G+1}} \vert} & \cdots & \frac{\partial \Delta P_{2} }{\partial \vert V_{N} \vert}	\\
+			\vdots & \ddots & \vdots & \vdots & \ddots & \vdots	\\
+			 \frac{\partial \Delta P_{N} }{\partial \theta_{2}} & \cdots & \frac{\partial \Delta P_{N}}{\partial \theta_{N}} &
+			\frac{\partial \Delta P_{N}}{\partial \vert V_{N_{G+1}} \vert } & \cdots & \frac{\partial \Delta P_{N}}{\partial \vert V_{N} \vert} \\
+			 \frac{\partial \Delta Q_{N_{G+1}} }{\partial \theta_{2}} & \cdots & \frac{\partial \Delta Q_{N_{G+1}} }{\partial \theta_{N}} &
+			\frac{\partial \Delta Q_{N_{G+1}} }{\partial \vert V_{N_{G+1}} \vert } & \cdots & \frac{\partial \Delta Q_{N_{G+1}} }{\partial \vert V_{N} \vert}	\\
+			\vdots & \ddots & \vdots & \vdots & \ddots & \vdots	\\
+			 \frac{\partial \Delta Q_{N}}{\partial \theta_{2}} & \cdots & \frac{\partial \Delta Q_{N}}{\partial \theta_{N}} &
+			\frac{\partial \Delta Q_{N}}{\partial \vert V_{N_{G+1}} \vert } & \cdots & \frac{\partial \Delta Q_{N}}{\partial \vert V_{N} \vert}
+			\end{array} \right ]
+	\left [ \begin{array}{c} \Delta \theta_{2} \\ \vdots \\ \Delta \theta_{N} \\ \Delta \vert V_{N_{G+1}} \vert \\ \vdots \\ \Delta \vert V_{N} \vert \end{array} \right ]
+	= \left [ \begin{array}{c} \Delta P_{2} \\ \vdots \\ \Delta P_{N} \\ \Delta Q_{N_{G+1}} \\ \vdots \\ \Delta Q_{N} \end{array} \right ]
+\end{align}
+
+## Solution of the Problem
+
+The solution update formula is given by:
+\begin{align}
+	\vec{x}^{(i+1)} = \vec{x}^{(i)} + \Delta \vec{x}^{(i)} = \vec{x}^{(i)} - \textbf{J}^{-1} \vec{f} (\vec{x}^{(i)})
+\end{align}
+
+To sum up, the NR algorithm, for application to the power flow problem is:
+
+1. Set the iteration counter to $i=1$. Use the initial solution $V_{i} = 1 \angle 0^{\circ}$
+2. Compute the mismatch vector $\vec{f}({\vec{x}})$ using the power flow equations
+3. Perform the following stopping criterion tests:
+	* If $\vert \Delta P_{i} \vert < \epsilon_{P}$ for all type PQ and PV buses and
+	* If $\vert \Delta Q_{i} \vert < \epsilon_{Q}$ for all type PQ
+	* Then go to step 6
+	* Otherwise, go to step 4.
+4. Evaluate the Jacobian matrix $\textbf{J}^{(i)}$ and compute $\Delta \vec{x}^{(i)}$.
+5. Compute the update solution vector $\vec{x}^{(i+1)}$. Return to step 3.
+6. Stop.
\ No newline at end of file