Resolution of the optimisation problem¶
This sections aim to describe how the optimisation problem is solved by Antares_Simulator. The starting point is the problem formulated in the modeling section.
The objective¶
The objective of the optimisation \( \Omega\) is the total cost of the dispatch and is the sum of several terms: $$ \Omega = \Omega_{\mathrm{transmission}} + \Omega_{\mathrm{hydro}} + \Omega_{\mathrm{thermal}} + \Omega_{\mathrm{unsupplied}} + \Omega_{\mathrm{spillage}}$$
The optimal dispatch, however, depends not only on continuous variables (like the power output of a unit, or the flux through a link) but also on the integer numbers of the running units on each thermal cluster \(M_{\theta}\). The solution is hence found by minimising the objective first with respect to \(M_{\theta}\) then with respect to the other variables. A possible formulation then reads as $$\min_{\mathrm{flux, thermal power, etc.}}\left( \min_{M_{\theta}} \left(\Omega\right)\right)$$
A different formulation allows to separate the problem in two parts "Unit Commitment" and "Optmal Dispatch" with their respective objectives.
$$\min_{M_{\theta}\mathrm{arg\,min}\left(\Omega_{\mathrm{unit.com.}}\right)}\left(\Omega_{\mathrm{dispatch}}\right)$$
The two \(\Omega_{\mathrm{unit.com.}}\) and \(\Omega_{\mathrm{dispatch}}\) are very similar and differs only with respect to the state of the thermal units, and consequently the form term of the thermal objective of its cost
$$\Omega_{\mathrm{thermal}} = \sum_{n\in N}\sum_{\theta\in \Theta_n}\left( \chi_\theta P_\theta + \sigma_\theta^+ M_\theta^+ + \tau_\theta M_\theta\right)$$
The expression
that goes in further details
is hence found by of on The optimum is defined by minimising \( \Omega_{\mathrm{dispatch}}\) by considering the
$$M_{\theta}$$
$$\min_{flux, prod. thermique, etc} \min_{nombre de palliers allumés} \Omega$$
$$ \Omega_{\mathrm{dispatch}} = \Omega_{\mathrm{transmission}} + \Omega_{\mathrm{hydro}} + \Omega_{\mathrm{thermal}} + \Omega_{\mathrm{unsupplied}} + \Omega_{\mathrm{spillage}}$$
Hello test \( \bar{F} \mathbb{R}\) \(\bar{C}\)