*************************************************************
Appendix B: Linear State Estimation (LSE) Example Application
*************************************************************

Introduction
------------

Electromechanical dynamic models are widely used to study transient and
small signal stability problems in power systems. States are the minimum
set of variables that can determine the status of a dynamic system. A
dynamic model with accurate states can faithfully reveal system
responses and therefore be used to enhance system stability and
reliability of a power system.

Therefore, the main objective of state estimation is to find the
voltage phasors of all the bars or buses of the system. To understand
the methodology, a classic observation model is described below:

.. math:: \mathbf{z} = h\left( \mathbf{x} \right) + \mathbf{\eta}

Where :math:`\mathbf{z}` is a vector
:math:`\in \ \mathbb{R}^{\text{mx}1}` containing the *m* measurements
from all the sensing devices installed in the network,
:math:`\mathbf{x}\mathbf{\  \in \ }\mathbb{R}^{2\text{nx}1}\mathbf{\ }`
is the state vector with the voltage phasors of the n buses of the
electrical system, :math:`h()` is a
:math:`\mathbb{R}^{2n}\  \rightarrow \mathbb{R}^{m}` function which
relates the voltage phasors to the measurements. Now, if the system
relies exclusively on PMU phasor measurements, the model can be
expressed in linear form:

.. math:: \mathbf{z} = H\ \mathbf{x} + \mathbf{\eta}

Otherwise if there are measurements such as active and reactive power
(SCADA case) the function :math:`h` is non-linear, and iterative methods
such as Gauss-Newton must be used to reach a solution. At the end of this annex a sub-annex that shows how to obtain the measurement
matrix :math:`H\ \mathbf{\in \ }\mathbb{R}^{\text{mx}2n}` is explained.
To estimate the optimal value of the state vector, the Weighted Least
Squares (WLS) method is proposed, in accordance with this approach the
following objective function is defined:

.. math:: J\left( \mathbf{x} \right)\mathbf{=}{\mathbf{\lbrack z -}H\mathbf{\ x\rbrack}}^{\mathbf{T}}R^{\mathbf{- 1}}\mathbf{\lbrack z -}H\mathbf{\ x\rbrack}

Where :math:`R` is the covariance matrix of the measurements. The
optimal solution that minimizes the objective function is:

.. math:: {\hat{\mathbf{x}}}_{\text{LS}} = {\lbrack H^{T}R^{- 1}H\rbrack}^{- 1}H^{T}R^{- 1}\ \mathbf{z}

Where :math:`{\hat{\mathbf{x}}}_{\text{LS}}` is the least square
estimator we were looking for.

Some of this example is based upon work done at Washington State University [1]_.

Getting Started
~~~~~~~~~~~~~~~

Before the LSE examples can be run, some first-time setup is required.
These instructions are the same as found in :ref:`Quick Start <MatlabToolbox>`

The Linear State Estimation Example Application uses two Matlab toolboxes that need to be downloaded:

	* Go to the `PSERC MatPower Page`_ and click ``Download Now``.

		.. _`PSERC MatPower Page`: http://www.pserc.cornell.edu/matpower/

	* Unzip the downloaded file into your `Documents\\MATLAB\\Toolbox folder`.
	* Add the path to MatPower to the Matlab Path using the instructions in the Matlab ``pathtool`` similar to the Matlab Path section above.
	* PARTF requires some of the .m files in MatPower to be modified. Find the following folder: `<partf install path)\\Modules\\MATLAB\\Events\\IEEEBus\\IEEE cases`, copy the nine 'case##.m' files and copy them to the matpower main folder, replacing the files that were there. The most part of the files are the same, but little modifications were made to save the reference voltages of the IEEE systems.

The LSE Application Example also includes Dynamic Events.  These examples require the Matlab Power System Toolbox available from The Royal Institute of Technology in Stockholm, Sweden.  

	* Download `PSTV3`_, `PST Data`_, and the `Manual`_ 

		.. _`PSTV3`: http://www.eps.ee.kth.se/personal/vanfretti/pst/download_updates/pstv3.zip
		.. _`PST Data`: http://www.eps.ee.kth.se/personal/vanfretti/pst/download_updates/pstdat.zip
		.. _`Manual`: http://www.eps.ee.kth.se/personal/vanfretti/pst/download_updates/PSTMan.zip
	
	* You will need to `register and get a password`_
	
		.. _`register and get a password`: http://www.eps.ee.kth.se/personal/vanfretti/pst/Power_System_Toolbox_Webpage/Software_Request.html

	* As with the MatPower toolbox, unzip these into your `Documents\\MATLAB\\Toolbox` folder and set up the Matlab Path

Note that the Power System Toolbox is not open source software and cannot be redistributed.

Example Linear State Estimation
-------------------------------

Creation of a stationary event case for a IEEE standard system
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The MATPOWER toolbox installed during the first-time setup is used to create IEEEBusSystem test (.tst) files., With these files, the PMUs could be in any location and in any IEEE bus system that the user desires. To do that please take a look at the *My Documents\\PARTF\\Tests\\LSE\\IEEEBusSystem_test_creator.m* file. From there you
will find the user input variables which you will be able to change:

-  user\_pmu\_list []: The bar or the buses were the PMUs are located.

-  IEEE\_bus\_selection: The IEEE system under test

-  NoiseVariance: The noise variance of the AWNG of the magnitude and
   phase signals.

-  The event configuration parameters:

   -  Start time:

   -  End time:

   -  PosSeq:

   -  Nominal Frequency:

   -  Reporting Rate:

   -  FSamp:

-  The PMU impairment parameters:

   -  PMUImpairPluginINIFilePath

   -  FilterType:

   -  PmuImpairParams:

If you run this MATLAB code, you are creating a .\ *tst* file in the
Test files directory. The name of the file created reflects the IEEE
system selected as well also the number of PMUs associated (by default:
*IEEEBusSystem\_case14\_4pmus.tst*). 

If the PARTF framework is not running, start it now. Click the file browse button in the ``Test File`` control and select the newly created file in *..\\tests\\LSE* (*IEEEBusSystem\_case14\_4pmus.tst* by default).  Next click the ``Open`` button.  4 buses will be created.

To watch the results as they are computed launch the VisualizeAppLSE.vi by clicking on the ``Visualization`` button in the top right of the PARTF panel. VisualizeAllLSE should be pre-checked so click the ``Run Apps`` button to launch the visualization app. Visualization apps give the user the ability to see the results of a test run while the test is running.  Unless an application has a Visualization App, the user will not be able to observe the operation of the Test Framework.

You can run one iteration of the test by pressing the ``Single Run`` button in the bottom center of the PARTF panel. 
After a single run, you will observe some signals very similar to the ones showed in
Figure B1. The default settings create a file with PMUs in the
buses: {2,6,7,9}. With these locations, we can ensure total
observability in the system.

The Total System Error (TSE) is displayed.  By changing the ``Bus Number`` control in the visualization App, you can see the Bus Magnitudes. Phases, and Total Vector Error (TVE) for each of the buses.  

|image0|

Figure B1: Front panel of the visualization VI for the LSE app. A 14 IEEE
bus system is displayed.

Dynamic events
~~~~~~~~~~~~~~

Dynamic IEEE Bus events use the Matlab Power System Toolbox installed
during :ref:`Quick Start <MatlabToolbox>`. 

	1) Run Matlab and type ``s\_simu``.
	2) You will be asked to select a file, choose *<your PARTF location>\\Modules\\Matlab\\Events\\CSVPlugin\\cases\\dtest\_13\_3phasefault.m* to simulate a dynamic 3 phase fault in the bus number.
 	3) The default values of 100 MVA and Base Frequency of 60Hz is suitable or can be changed.
	4) When s\_simu completes, press enter to end the program. The fault is released after 0.1 s and after that, a ringdown event takes place. Now all the signals needed have been created in the Matlab workspace.
	5) Create a .csv input file for PARTF: in Matlab, type ``s_simuToFile``. This matlab program creates three different files in your *My Documents\\PARTF\\Tests\\LSE* folder:
		a) The first file is a *.tst* file which will be opened in the PARTF framework. 
		b) The second one *.csv* file in the *..\\Inputdata* folder containing the signals of each one of the PMUs.  
		c) The last file is a reference signal waveform also in the *..\\Inputdata* folder. This file will be used during the analysis of the test results.

Now you can browse and select the newly created *DynamicSystem_case13\_3pmus.tst* In the PARTF ``Test File`` control. Use the ``Open`` button to create 3 buses.  Note that the ``Event Parameters`` control now lists the relative path to the newly created *.csv* file.  To run this test do the following:
	1) Since a new test has been opened, the VisualiseAppLSE application will need to be reset by clicking on the ``Restart Plots`` button.
	2) In the PARTF framework, click ``Single Run`` in the bottom center of the panel.
	3) To view any of the buses in the VisualizeAppLSE, select the bus in the ``Bus Number`` control.

The Figure B2 below shows the results for this kind of event. With PMUs in the
10 and 11 locations only the 46% percent of the total number of buses
can be estimated. The user can select which bus to visualize using the `Bus Number control`, in this case the number 8 is
displayed.

|image1|

Figure B2: Front panel of the visualization VI for the LSE app. A
dynamic 13 bus system is displayed.

MonteCarlo analysis 
-------------------

Many kinds of test can be performed using python scripts. Documentation can be found under :ref:`Test Automation <testAutomation>`. Two examples of Monte Carlo Analysis are presented here:

The Effects of PMU Filtering
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The first example, involves in performing a Monte Carlo simulation for a
stationary IEEE bus event in order to evaluate the performance of the
LSE application against the additive white Gaussian noise present in the
measurements. We will use two different PMU filter types: Hamming and Blackman to see how the PMU filtering affects white Gaussian Noise rejection.

	1) Using the ``Test File`` control, Browse and Select the *IEEEBusSystem\_case14\_4pmus.tst* again then click ``Open``.
	2) On the ``Sensor`` Impariment tab, change the ``FilterType`` control to ``Hamming`` and click the ``Update Bus`` button. Since the last bus is selected in the ``Bus Number`` control each time a new test file is opened, you just changed the PMU Filter Type for the PMU on bus number 4.
	3) Now select of each one of other 3 buses and change the ``FilterType`` to ``Hamming`` then click ``Update Bus`` for each. How annoying was that? (Later we will show you how to change configurations using the Monte Carlo Analysis script).

Once the bus PMUs are modified, use ``Monte Carlo Script`` control to the browse and select *My Documents\\PARTF\\Scripts\\LSE\\LSEMonteCarlo.py* script, then press the blue ``Monte Carlo`` button. This script will repeat the test sequence twenty times: generate the signal,  generate synchrophasors and impair them as the PMU with a Hamming window would, 
then run the LSE app. 

The idea of this example is to run the test many times in order to gather enough data to perform a statistical analysis. This is preferable to a single test with a very long duration due to limitations in computer memory and buffer sizes. All these signals are saved in a *.mat* in the Output folder and again MATLAB can be used to evaluate the application performance. Please, run the *\\PARTF\\Scripts\\LSE\\PlotLSENoiseTest.m*. When the Select data File
dialog opens, browse to ..\\PARTF\\Output and open the recently created
data\_#.mat file. As can be seen from the Figures B3 to B5 the magnitude
error, the phase error and the total vector error (TVE) are plotted for
each one of the voltages in the buses.

|image2|

Figure B3: Boxplot [note1]_ of the magnitude error of all the voltages in the
14 IEEE system using a Hamming window.

|image3|

Figure B4: Boxplot of the phase error of all the voltages in the 14 IEEE
system using a Hamming window.

|image4|

Figure B5: Boxplot of the TVE of all the voltages in the 14 IEEE system
using a Hamming window.

Thresholds in each one of these magnitudes can be defined by the user to
delimit the maximum error tolerated for the next step of the processing
chain. As you can see the application is not complaint with expected
performance.

|image5|


Next we will change the PMU filter back to Blackman from Hamming.  There are several ways of doing this:  From the PARTF panel as we did before, from inside the matlab *.tst* file creation program, or using the Python Monte Carlo Script.  In this example, we will use the Monte Carlo script.

In your favorite python script editing program (you can use Notepad if you do not have another) open *My Documents\\PARTF\\Scripts\\LSE\\LSEMonteCarlo.py*  Conveniently located at line 28, you will find:     *PMUFilterType = 'Hamming'*  Change *Hamming* to *Blackman*.  Note on lines 48 through 61 is a loop that iteratres through all the buses and loads the PMU Impairment plugin then changes the ImpairConfig.Filtertype to "PMUFilterType".  Save the file then run it again in the PARTF framework.  You will notice that the `FilterType` on the `Sensors` tab changes to *Blackman*

Once again run the Matlab program *PlotLSENoiseTest.m*.  You should notice a significant difference in the results:

|image6|

|image7|

Figure B6: Boxplot of the magnitude error of all the voltages in the 14
IEEE system using a Blackman window.

|image8|

Figure B7: Boxplot of the phase error of all the voltages in the 14 IEEE
system using a Blackman window.

|image9|

Figure B8: Boxplot of the TVE of all the voltages in the 14 IEEE system
using a Blackman window.

So, as any user can expect the level of performance of the application
depends on the uncertainties in the PMU measurements. The application
does not include errors in the model of the system, so all the results
plotted before assumes that the LSE app knows exactly the impedances
between the different nodes of the network.

Observability
~~~~~~~~~~~~~

The second kind of test that can be performed is an observability test: How does the number of PMU in the system affect the state estimation? This next test is going to take a long time to run so you might think about starting it before lunchtime or letting it run overnight.

First we are going to edit the *My Documents\\PARTF\\Tests\\LSE\\IEEEBusSystem_test_creator.m* file to add a PMU for each of the 14 buses created by this program.  Find line 7 and comment it out by adding a ``%`` to the front of the line.  Uncomment line 8 *user_pmu_list=[14 13 12 11 10 9 8 7 6 5 4 3 2 1];* by removing the ``%``.  Save and run the Matlab Program.  It will create *My Documents\PARTF\Tests\LSE\IEEEBusSystem_case14_14pmus.tst*  In the PARTF Framework, use the ``Test File`` control to browse and select this file.  Click the ``Open`` button and 14 buses will be created.  In the VisualizeAppLSE application, you can click the ``Restart Plots`` button since you just loaded a new test.  Test the 14 bus system using ``Single Run``.

In the ``Monte Carlo Script`` control, browse and select *LSE\\LSEPMUNumber.py*.  This script will repeat the same sequence of the last example 13 times with one PMU being removed with in each set of iterations. The way to reduce the PMUs is following the *pmu\_index* variable, and as you can see the PMUs are in a descendent order.  Click ''Monte Carlo'' and head on home or out to lunch, this will take a while...

When the Monte Carlo simulation had finished and all the output data had been saved repeat the simulation reducing ther number of PMUs each iteration. Run the Matlab program *My Documents\\PARTF\\Scripts\\LSE\\PlotLSE_PMUNumber.m*. The program will as you to browse to the results file which is in *My Documents\\PARTF\\Output\\LsePmuNumber_<date_time>*. This plots the results of the analysis.  In this case, PMUs were removed from buses in numerical order with no consideration of the number of current channels.

|image10a|

Figure B9: Observability:  The percentage between the estimated buses and the total number of busses in the system

|image11a|

Figure B10: The root average mean square error vs the number of PMUs.


If we place PMUs on buses with more current channels, we can get improved observability.  Edit the *LSE\\LSEPMUNumber.py* script to uncomment line 22 and comment out line 21.  Line 22 changes the order that the PMUs that are deleted, leaving the buses with the most current channels for last. Run this revised Monte Carlo analysis to get a second set of output data.  Again, take a break because this will take a while.

To compare the two sets of results, run the Matlab program *My Documents\\PARTF\\Scripts\\LSE\\PlotLSE_PMUNumber_two_runs.m* and select the latest set of output results first. The first value of the list corresponds to the PMU with smallest number of current channels and vice versa. We can see from the plots that using the PMUs with the most current channels improves observability.

|image10|

Figure B9: The percent rate between the estimated buses and the total
number of buses of the system.

|image11|

Figure B10: The root average mean square error vs the number of PMUs.

Notice that the root average mean square error, defined in , represents
a metric similar to the TSE of the visualization virtual instrument, but
the way that each one of them is calculated differs:

.. math:: TSE = \sqrt{\sum_{i = 1}^{N}{\text{TVE}^{2}}_{i}}

.. math:: RAMSE = \sqrt{\frac{1}{N}\sum_{i = 1}^{N}{E\lbrack{\varepsilon_{\text{VEi}}}^{2}\rbrack}}

Where,

:math:`\varepsilon_{\text{VEi}} = \left| \left( V_{i} + \varepsilon_{\text{Vi}} \right)e^{j\left( \theta_{i} + \varepsilon_{\text{θi}} \right)} - V_{i}e^{j\theta_{i}} \right|`

Summary of Labview Vis
----------------------

EventPlugins.lvlib: IEEEBusSystemPlugin.lvclass
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Parent class: EventPluginBaseClass.lvclass

Overridden Vis: GetEvtReports.vi, GetEvtSignal.vi

IEEEBusEvtPlugin.ctl: Empty, the parent class control is used.

1. | |image12|\ ***GetEvtReports.vi*** Icon:
   | Inputs: IEEEBusEvtPlugin object

   -  ClEvtConfig: T0, F0, bPosSeq

   -  dblTimeArray

   -  dblEvtParams

   -  BusNumber

      Outputs: IEEEBusEvtPlugin object

   -  Timestamp

   -  Synx

   -  Freq

   -  ROCOF

      | Description: Creates all the synchrophasors included in each one
        of the reports based on the event
      | configuration and the event parameters. Contains a MATLAB script
        block where IEEEevt.m
      | is called. See IEEEevt.m for more details.

1. | |image13|\ ***GetEvtSignal.vi*** Icon:
   | Inputs: IEEEBusEvtPlugin object

   -  ClEvtConfig: FSamp

   -  dblTimeArray

   -  dblEvtParams

   -  BusNumber

      Output: IEEEBusEvtPlugin object

   -  Signal

    | Description: Creates the voltage and the current waveforms.
      Contains a MATLAB script block where
    | genIEEESignal.m is called. See genIEEESignal.m for more details.

AppPlugins.lvlib: RingdownPlugin.lvclass 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

| Parent class: AppPluginsBase.lvclass
| Overridden Vis: AppConfig.vi, AppCOnfigTypeToXML.vi,
  AppConfigXMLToType.vi, GetAppOutput.vi,
| WriteEvtConfig.vi

1. | |image14|\ ***GetAppOutput.vi*** Icon:
   | Inputs: RingdownEvtPlugin object

   -  RingdownConfig: V index

      EventConfiguration.ctl

   -  PosSeq

      Outputs: RingdownEvtPlugin object

   -  RingdownConfig: I index

   -  RingdownConfig: PosSeq

      | Description: Based on the PosSeq Boolean value, the I index is
        set. This index allows the
      | GetAppOutput.vi to know where to look for the current
        synchrophasors values.

2. | |image15|\ ***GetLSEEstimate.vi*** Icon:
   | Inputs: RingdownEvtPlugin object

   -  RingdownConfig: V index

      EventConfiguration.ctl

   -  PosSeq

      Outputs: RingdownEvtPlugin object

   -  RingdownConfig: I index

   -  RingdownConfig: PosSeq

      | Description: Based on the PosSeq Boolean value, the I index is
        set. This index allows the
      | GetAppOutput.vi to know where to look for the current
        synchrophasors values.

3. | |image16|\ ***GetPMUNumber.vi*** Icon:
   | Inputs: RingdownEvtPlugin object

   -  RingdownConfig: V index

      EventConfiguration.ctl

   -  PosSeq

      Outputs: RingdownEvtPlugin object

   -  RingdownConfig: I index

   -  RingdownConfig: PosSeq

      | Description: Based on the PosSeq Boolean value, the I index is
        set. This index allows the
      | GetAppOutput.vi to know where to look for the current
        synchrophasors values.

4. | |image17|\ ***GetMeasurementMatrix.vi*** Icon:
   | Inputs: RingdownEvtPlugin object

   -  RingdownConfig: V index

      EventConfiguration.ctl

   -  PosSeq

      Outputs: RingdownEvtPlugin object

   -  RingdownConfig: I index

   -  RingdownConfig: PosSeq

      | Description: Based on the PosSeq Boolean value, the I index is
        set. This index allows the
      | GetAppOutput.vi to know where to look for the current
        synchrophasors values.

5. | |image18|\ ***GetVoltageRef.vi*** Icon:
   | Inputs: RingdownEvtPlugin object

   -  RingdownConfig: V index

      EventConfiguration.ctl

   -  PosSeq

      Outputs: RingdownEvtPlugin object

   -  RingdownConfig: I index

   -  RingdownConfig: PosSeq

      | Description: Based on the PosSeq Boolean value, the I index is
        set. This index allows the
      | GetAppOutput.vi to know where to look for the current
        synchrophasors values.

6. | |image19|\ ***GetInterleavingSynx.vi*** Icon:
   | Inputs: RingdownEvtPlugin object

   -  RingdownConfig: V index

      EventConfiguration.ctl

   -  PosSeq

      Outputs: RingdownEvtPlugin object

   -  RingdownConfig: I index

   -  RingdownConfig: PosSeq

      | Description: Based on the PosSeq Boolean value, the I index is
        set. This index allows the
      | GetAppOutput.vi to know where to look for the current
        synchrophasors values.

MATLAB Function descriptions
----------------------------

Sub-Annex– Measurement matrix calculation (H)
-----------------------------------------------

If we go back to the π model of a transmission line (see Fig A1) and
rely on Kirchhoff's laws, it is stated that:

.. math:: {\overset{\overline{}}{I}}_{\text{ij}} = {\overset{\overline{}}{V}}_{i}\ Y_{i0} + \left( {\overset{\overline{}}{V}}_{i} - {\overset{\overline{}}{V}}_{j} \right)\ Y_{\text{ij}}

Where the bar over the letter indicates that it is a phasor. If we
decompose the above equation into a real and imaginary part:

.. math:: I_{ij\_ R} + jI_{ij\_ I} = \left( V_{i\_ R} + jV_{i\_ I} \right)\left( G_{i0} + jB_{i0} \right) + (V_{i\_ R} + jV_{i\_ I} - V_{j\_ R} - jV_{j\_ I})(G_{\text{ij}} + jB_{\text{ij}})

Obtaining the following pair of equations:

.. math:: I_{\text{ij}\_ R} = \left( G_{i0} + G_{\text{ij}} \right)\ V_{i\_ R} - \left( B_{i0} + B_{\text{ij}} \right)\ V_{i\_ I} - G_{\text{ij}}V_{{j\_}_{R}} + B_{\text{ij}}V_{j\_ I}

.. math:: I_{ij\_ I} = \left( B_{i0} + B_{\text{ij}} \right)\ V_{i\_ R} - \left( G_{i0} + G_{\text{ij}} \right)\text{\ V}_{i\_ I} - B_{\text{ij}}V_{j_{\_ R}} - G_{\text{ij}}V_{j\_ I}

The linear relationship that allows us to construct the matrix is
​​observed. In the same way, the relation between the current
:math:`I_{\text{ji}}` can be considered, and the following scheme can be
proposed:

.. math::

   \begin{bmatrix}
   \begin{matrix}
   V_{R} \\
   V_{I} \\
   \end{matrix} \\
   \begin{matrix}
   I_{ij\_ R} \\
   \begin{matrix}
   I_{ij\_ I} \\
   \begin{matrix}
   I_{ji\_ R} \\
   I_{ji\_ I} \\
   \end{matrix} \\
   \end{matrix} \\
   \end{matrix} \\
   \end{bmatrix} = \begin{bmatrix}
   \begin{matrix}
   H_{11} \\
   H_{21} \\
   \end{matrix} & \begin{matrix}
   H_{12} \\
   H_{22} \\
   \end{matrix} \\
   \begin{matrix}
   H_{31} \\
   \begin{matrix}
   H_{41} \\
   \begin{matrix}
   H_{51} \\
   H_{61} \\
   \end{matrix} \\
   \end{matrix} \\
   \end{matrix} & \begin{matrix}
   H_{32} \\
   \begin{matrix}
   H_{42} \\
   \begin{matrix}
   H_{52} \\
   H_{62} \\
   \end{matrix} \\
   \end{matrix} \\
   \end{matrix} \\
   \end{bmatrix}\begin{bmatrix}
   V_{R} \\
   V_{I} \\
   \end{bmatrix} + \begin{bmatrix}
   \mathbf{\eta}_{\mathbf{1}} \\
   \mathbf{\eta}_{\mathbf{2}} \\
   \begin{matrix}
   \mathbf{\eta}_{\mathbf{3}} \\
   \mathbf{\eta}_{\mathbf{4}} \\
   \end{matrix} \\
   \end{bmatrix}

Where :math:`H_{11}` and :math:`H_{22}\ `\ are unit matrices and
:math:`H_{12}` and :math:`H_{21}` are null matrices. The remainder of
the submatrices will contain the respective admittances according to the
pair of equations described above and the equations which would
correspond to the currents :math:`I_{\text{ji}}`.

|image20|

Figure B11 A11: π model of a transmission line



.. [note1]
   On each box, the central mark is the median, the edges of the box are
   the 25th and 75th percentiles, the whiskers extend to the most
   extreme data points the algorithm considers to be not outliers, and
   the outliers are plotted individually.

Citations
---------
.. [1]
	R. Liu, A. Srivastava, A. Askerman, D. Bakken and P. Panciatici, *Decentralized State Estimation and Remedial Control Action for Minimum Wind Curtailment Using Distributed Computing Platform*, IEEE Transactions on Industrial Applications, June 2017.

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.. |image13| image:: media\LSE/image14.png
   :width: 0.44583in
   :height: 0.44583in
.. |image14| image:: media\LSE/image15.png
   :width: 0.43889in
   :height: 0.43472in
.. |image15| image:: media\LSE/image16.png
   :width: 0.43889in
   :height: 0.43958in
.. |image16| image:: media\LSE/image17.png
   :width: 0.43889in
   :height: 0.43889in
.. |image17| image:: media\LSE/image18.png
   :width: 0.43889in
   :height: 0.43750in
.. |image18| image:: media\LSE/image19.png
   :width: 0.43750in
   :height: 0.43889in
.. |image19| image:: media\LSE/image20.png
   :width: 0.43611in
   :height: 0.43889in
.. |image20| image:: media\LSE/image21.png
   :width: 4.52483in
   :height: 1.83700in