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| 1 | +DC power flow extension |
| 2 | +####################### |
| 3 | + |
| 4 | +A basic *Transmission* component is modeled with a simple commodity |
| 5 | +exchange based on balance equations and a linear loss factor. However, |
| 6 | +the transmission of a commodity is generally subject to far more complex |
| 7 | +physics. The incorporation of a higher modeling detail of these physics |
| 8 | +into the optimization program has to be seen in the context of |
| 9 | +increasing computation times. With respect to this topic, `Syranidis et |
| 10 | +al. (2018) <https://doi.org/10.1016/j.rser.2017.10.110>`_ reviewed the modeling of |
| 11 | +electrical power flow across transmission networks. They discuss the |
| 12 | +general formulation of an AC power flow with a set of non-linear |
| 13 | +equations for which direct, analytical solutions are rarely feasible and |
| 14 | +which are therefore often solved with iterative methods. Based on the |
| 15 | +premise that the optimization program provided by FINE should stay a |
| 16 | +mixed integer linear program, these equations cannot be incorporated in |
| 17 | +the framework. A linearization of these equations, as provided by the DC |
| 18 | +power flow method, is however suitable for incorporation. The |
| 19 | +linearized equations result in an acceptable increase in computation |
| 20 | +time while increasing the electrical power flow modeling detail to a |
| 21 | +more sophisticated level. |
| 22 | + |
| 23 | +In the following, the constraints constituting the DC power flow are |
| 24 | +presented, based on the detailed description by `Van den Bergh et |
| 25 | +al. (2014) <https://www.mech.kuleuven.be/en/tme/research/energy_environment/Pdf/wpen2014-12.pdf>`_. The constraints thereby extend |
| 26 | +the *Transmission* component model. In the following, let |
| 27 | +:math:`\mathcal{C}^\text{trans,LPF}\subseteq\mathcal{C}^\text{trans}\subset\mathcal{C}` |
| 28 | +be the set of *Transmission* components that are modeled with a DC power |
| 29 | +flow. |
| 30 | + |
| 31 | +The constraints that enforce the linear power flow are implemented for |
| 32 | +each component :math:`\text{c}\in\mathcal{C}^\text{trans,LPF}`, for |
| 33 | +all :math:`\text{l}\in\mathcal{L}^\text{c}`, and for all |
| 34 | +:math:`\theta \in \Theta` as |
| 35 | + |
| 36 | +.. math:: |
| 37 | +
|
| 38 | + \begin{aligned} |
| 39 | + o_\text{$\omega$,a,$\theta$}-o_{\omega,\hat{\text{a}},\theta}=\left(\phi^\text{c,l$_1$,$\theta$}-\phi_\text{c,l$_2$,$\theta$}\right) / \text{x}_\text{c,a}~. |
| 40 | + \end{aligned} |
| 41 | +
|
| 42 | +Here, :math:`\phi_\text{c,l,p,t}\in\mathbb{R}` is the |
| 43 | +variable which models the phase |
| 44 | +angle. :math:`\text{x}_\text{c,a}` |
| 45 | +represents the electric reactance of the line between locations |
| 46 | +l\ :math:`_1` and l\ :math:`_2` (:math:`\text{a} \in \mathcal{A}_\text{c}`). These equations |
| 47 | +leave one degree of freedom for the phase angle variables at each time |
| 48 | +step. To obtain a unique solution, an additional set of constraints is |
| 49 | +given by |
| 50 | + |
| 51 | +.. math:: |
| 52 | +
|
| 53 | + \begin{aligned} |
| 54 | + \phi_\text{c,l$_\text{ref}$,$\theta$}=0 |
| 55 | + \end{aligned} |
| 56 | +
|
| 57 | +for each component :math:`\text{c}\in\mathcal{C}^\text{trans,LPF}` |
| 58 | +and for all :math:`\theta \in \Theta` which |
| 59 | +sets the phase angle for one location :math:`\text{l}_\text{ref}` to |
| 60 | +zero. |
| 61 | + |
| 62 | +At this point, it should be remarked that the reactance parameter is in |
| 63 | +practice a function of the capacity of the line. The capacity expansion of transmission lines modeled with a *DC power flow* is not implemented to reduce model complexity, |
| 64 | +i.e., AC line capacities, which are modeled with a *DC power flow*, are kept at a fixed value and thus their reactance parameters remain constant. |
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