Optimal Power Flow
=====================


:abbr:`OPF (Optimal Power Flow)` tries to allocate available generation to meet the current load while keeping transmission lines within :abbr:`the limits (governed in reality by stability and sag concerns)` of what they can carry. :abbr:`OPF (Optimal Power Flow)` adds a dimension of space to the :abbr:`ED (Economic Dispatch)` problem. Currently **minpower** performs the simplest version of power flow, called decoupled :abbr:`OPF (Optimal Power Flow)` and considers only real power [#f1]_. The classic text is `Bergen & Vittal <http://bitly.com/bergen-vittal>`_.


The problem
------------

..  container:: optimizationproblem
    
    :math:`\min \sum_g C_g(P_g)`
    
    :math:`\mathrm{s.t.} \; P_{\min (g)} \leq P_g \leq P_{\max (g)}  \; \forall  \; \mathrm{generators} \;(g)`
    
    :math:`\mathrm{s.t.} \; P_{\mathrm{gen} (i)} - P_{\mathrm{load} (i)} - \sum_j P_{ij} = 0   \; \forall  \; \mathrm{buses} \;(i)`
    
    :math:`\mathrm{s.t.} \; P_{\min (ij)} \leq P_{ij} \leq P_{\max (ij)} \forall  \; \mathrm{lines} \;(ij)`

In this mathematical formulation generators are indexed by :math:`g`. :math:`P_g` is a generator’s power output and :math:`C_g()` is its cost function. The objective is to minimize the total cost. There are three constraints: 

* each generator must be within its real power limits
* :abbr:`inflow must equal outflow (called power balance)` at each bus
* each line must be within its real power limits

For :abbr:`DCOPF (De-Coupled Optimal Power Flow)`, the real power flow on a line :math:`P_{ij} = \frac{1}{X_{ij}} \left( \theta_i-\theta_j \right)` depends linearly on the voltage angles of the buses it connects (:math:`\theta_{i}`, :math:`\theta_{j}`) and its own reactance :math:`X_{ij}`. Bus angles are the difference in voltage angle between the bus and the :abbr:`reference bus (known as the swing or slack bus)` which has angle :math:`0^{\circ}`. 



Example Problem
-----------------

To define a simple :abbr:`OPF (Optimal Power Flow)` problem, **Minpower** requires three spreadsheets. The first describes the generator parameters and location (`generators.csv <https://github.com/adamgreenhall/minpower/blob/master/minpower/tests/opf/generators.csv>`_):

.. literalinclude:: ../minpower/tests/opf/generators.csv

The second describes the load at each bus (`loads.csv <https://github.com/adamgreenhall/minpower/blob/master/minpower/tests/opf/loads.csv>`_):

.. literalinclude:: ../minpower/tests/opf/loads.csv

The third describes the lines between buses (`lines.csv <https://github.com/adamgreenhall/minpower/blob/master/minpower/tests/opf/lines.csv>`_):

.. literalinclude:: ../minpower/tests/opf/lines.csv

.. note:: For more information about what options you can specify in each spreadsheet see: :doc:`creating-problems`.

Solving
---------
Save the three spreadsheets above into into a folder (call it ``mypowerflow``) and run::
    
    minpower mypowerflow

This particular problem is also **Minpower** built-in test case, so if you haven't been following along, to solve it, call::

    minpower opf


Example Solution
-------------------

The result is a plot (``powerflow.png``):

    .. image:: ../minpower/tests/opf/powerflow.png
       :width: 500 px


:abbr:`OPF (Optimal Power Flow)` is difficult to visualize (please `send suggestions <mailto:minpower@adamgreenhall.com>`_). 

* A red colored transmission line indicates a limit on that line, while gray lines are below their limits. *The Tacoma* :math:`\rightarrow` *Seattle line is at its limit. The other two gray colored lines are running below their limits.* 
* The width of the line indicates the amount of power flow. *The Olympia-Seattle line has the largest flow*. 
* The :abbr:`stubs (arrows would be nicer, but this is what networkx gives)` at one end indicate direction of flow. *Flow direction is Olympia* :math:`\rightarrow` *Seattle*. 
* :abbr:`Injected power (generation minus load for the bus)` is shown by the color of the bus. *Olympia is injecting power into the system while Seattle is pulling power*. 

There are also spreadsheet outputs of generator information (``powerflow-generators.csv``):

.. literalinclude:: ../minpower/tests/opf/powerflow-generators.csv

and line information (``powerflow-lines.csv``):

.. literalinclude:: ../minpower/tests/opf/powerflow-lines.csv

Each line's real power flow is output. Lines that have congestion will show a positive shadow price. 
*Because the flow is Tacoma* :math:`\rightarrow` *Seattle and the from/to fields of the
spreadsheet are the other way around, we see a negative power flow. The Seattle-Tacoma
line is at its limit, so there is an extra cost to the system from the congestion and the line has a
positive shadow price.*

.. rubric:: Footnotes
 
.. [#f1] Modern power systems often have reactive power issues. While :abbr:`DCOPF (Decoupled Optimal Power Flow)` is a decent approximate solution with reactive power considered, your results may vary significantly from real operations without it.