.. _bfs_theory:

======================
Backward-Forward sweep
======================

Forward-Backward sweep (FBS) is a load flow approach suitable for
radial operated power systems. :cite:`haque1996load`
The benefit of using an FBS approach is that the load flow is
calculated iteratively and does not need to go through the
Jacobian matrix, as in the Newton-Rapshon method. 
In this way, convergence problems related to an ill-conditioned matrix
of weak networks can be avoided. :cite:`trevino1970cases`

In an FBS approach, the load flow is calculated by updating the
power flow through a backward sweep before the bus's voltage
magnitude and angle are updated and calculated in the forward
sweep. :cite:`haque1996load`

..............
Backward sweep
..............

In the backward sweep, the active and reactive power over the system
lines is calculated. The active and reactive power over line *l*
for iteration *k*, :math:`P_{l}` and :math:`Q_{l}` respectively,
is calculated by adding the accumulated active and reactive load,
:math:`P_{l}^{'}` and :math:`Q_{l}^{'}` respectively, at the
downstream buses and the accumulated active and reactive power
losses, :math:`P_{l}^{*}`  and :math:`Q_{l}^{*}` respectively,
over the downstream lines including the power loss over line *l*
and the load at the current bus

.. math::
    :label: eq:PowerA

    P_{l} = P_{l}^{'} + P_{l}^{*}

.. math::
    :label: eq:PowerQ

    Q_{l} = Q_{l}^{'} + Q_{l}^{*}
    
where the accumulated active and reactive load at the buses are calculated

.. math::
    :label: eq:acc_P_load

    P_{l}^{'} = P_{i}^{load} + \sum_{d_{b}} P_{d_{b}}^{load}


.. math::
    :label: eq:acc_Q_load

    Q_{l}^{'} = Q_{i}^{load} + \sum_{d_{b}} Q_{d_{b}}^{load}


where :math:`P_{i}^{load}` is the active load at node *i* while
:math:`Q_{i}^{load}` is the reactive load at node *i*.
:math:`P_{d_{b}}^{load}` and :math:`Q_{d_{b}}^{load}` are the
active and reactive loads at the downstream buses, respectively. 

The accumulated active and reactive power losses can be calculated as 

.. math::
    :label: eq:acc_P_loss

    P_{l}^{*} = P_{l}^{loss} + \sum_{d_{l}} P_{d_{l}}^{loss}


.. math::
    :label: eq:acc_Q_loss

    Q_{l}^{*} = Q_{l}^{loss} + \sum_{d_{l}} Q_{d_{l}}^{loss}

where :math:`P_{l}^{loss}` and :math:`Q_{l}^{loss}` are the active
and reactive power loss over line *l*, respectively.
Here, :math:`P_{d_{l}}^{loss}` :math:`Q_{d_{l}}^{loss}` are
the active and reactive power loss of the downstream lines, respectively. 


Furthermore, the active and reactive power loss over a line can
be calculated

.. math::
    :label: eq:P_loss

    P_{l}^{loss} = R_{l}*{loss} \frac{P_{l}^{'2} + Q_{l}^{'2}}{V_{j}^{2}} 

.. math::
    :label: eq:Q_loss

    Q_{l}^{loss} = X_{l}*{loss} \frac{P_{l}^{'2} + Q_{l}^{'2}}{V_{j}^{2}} 


Here, :math:`R_{l}` is the line resistance while :math:`X_{l}` is the line reactance. :math:`V_{j}` is the voltage at the ending bus. 


..............
Forward sweep
..............

After a backward sweep, the forward sweep is conducted. In the
forward sweep, the bus's voltage magnitude and angle are updated
by utilizing the updated active and reactive power from the backward sweep.
Here, the voltage magnitude at bus *i* in relation to the voltage
magnitude at bus *j* for iteration *k* can be calculated

.. math::
    :label: eq:vol_mag

    V_{i} = V_{j} - I_{l}(R_{l}+jX_{l}) = \sqrt{V_{j}^{2} - T_{1} - T_{2}}

where the :math:`T_{1}` and :math:`T_{2}` can be expressed as

.. math::
    :label: eq:T1

    T_{1} = 2(P_{l}R_{l} + Q_{l}X_{l}) 

.. math::
    :label: eq:T2

    T_{2} = \frac{(P_{l}^{2}+Q_{l}^{2})(R_{l}^{2}+X_{l}^{2})}{V_{j}^{2}}

The voltage angle can be expressed as 

.. math::
    :label: eq:T1

    \delta_{i} = \delta_{j} + \arctan{\frac{Im(V_{i})}{Re(V_{i})}}

where :math:`\delta_{j}` is the voltage angle at bus *j* and
:math:`Im(V_{i})` and :math:`Re(V_{i})` are the imaginary part
and the real part of :math:`V_{i}`, respectively.