Mathematical notation for solph¶

Sets¶

$\begin{tabular}{p{2cm}p{7cm}p{3cm}}\hline \textbf{Symbol} & \textbf{Description} & \textbf{Python-type of object in set}\\\hline \\ $\mathcal{E}$ & Set of all Entities & Entity\\ $\mathcal{\vec{E}}$ & Set of all weighted edges $(e_i, e_j)$ & Tuple\\ $\mathcal{E_B}$ & Set of all edges & Bus \\ $\mathcal{E_C}$ & Set of all components & Component\\ $\mathcal{E}_{I}$ & Set of components with 1 input & Sink \\ $\mathcal{E}_{O}$ & Set of components with 1 output & Source \\ $\mathcal{E}_{IO}$ & Components with 1 input, 1 output $i_e \neq {o_{e,n}}$ & SimpleTransformer\\ $\mathcal{E}_{IOO}$ & Components with 1 input, 2 outputs & SimpleCHP\\ $\mathcal{E}_{S}$ & Components with 1 input, 1 output $i_e = o_{e,n}$ & Storage\\ $\mathcal{I}_e$ & All inputs of Entity $e$ & Dict\\ $\mathcal{O}_e$ & All outputs of Entity $e$ & Dict\\ $\mathcal{T}$ & Set of timesteps & List \\ \end{tabular}$

Variables¶

$\begin{tabular}{p{2cm}p{4cm}p{2cm}p{2cm}}\hline \textbf{Symbol} & \textbf{Description} & \textbf{Possible Set} & \textbf{Python variable} \\\hline \\ \multicolumn{4}{l}{\textbf{LP-models}}\\ $w_{e_1, e_2}(t)$ & Weight of Edge $(e_1, e_2)$ at $t$ & $(e_1, e_2) \in \mathcal{\vec{E}}$ & \verb+w[e1,e2,t]+ \\ $l_e(t)$ & Level of component $e$ at $t$ & $e \in \mathcal{E}_S$ & \verb+cap[e,t]+ \\ $g^{pos}_{e_{o,1}}(t)$ & Positive gradient between two sequential timesteps & $e \in \mathcal{E}_C$ & \verb+grad_pos_var[e,t]+ \\ $g^{neg}_{e_{o,1}}(t)$ & Negative gradient between two sequential timesteps & $e \in \mathcal{E}_C$ & \verb+grad_neg_var[e,t]+ \\ \\ \multicolumn{4}{l}{\textbf{Dispatch-source only}}\\ $w^{cut}_{e,o_e}(t)$ & Curtailment of value $w_{e, o_e}(t)$ &$e \in \mathcal{E}_O$ & \verb+curtailment_var[e1,e2,t]+ \\ \\ \multicolumn{4}{l}{\textbf{Investment models}}\\ $\overline{w}^{add}_{o_e}$ & Optimized extension of bound $\overline{W}_{e, o_{e,1}}$ &$e \in \mathcal{E}_C$ &\verb+add_out[e]+ \\ $\overline{l}^{add}_e$ & Optimized extension of bound $\overline{L}_e$ &$e \in \mathcal{E}_S$ &\verb+add_cap[e]+ \\ \\ \multicolumn{4}{l}{\textbf{MILP-models}}\\ $y_e(t)$ & Binary status variable of component $e$ at $t$ &$e \in \mathcal{E}_C$ & \verb+y[e,t]+ \\ $z^{start}_e(t)$ & Binary startup variable of component $e$ at $t$ &$e \in \mathcal{E}_C$ & \verb+z_start[e,t]+ \\ $z^{stop}_e(t)$ & Binary shutdown variable of component $e$ at $t$ &$e \in \mathcal{E}_C$ & \verb+z_stop[e,t]+ \\ \end{tabular}$

Parameters¶

Parameters will be notate with uppercase.

$\begin{tabular}{p{2cm}p{5cm}p{4cm}p{1.5cm}}\hline \textbf{Symbol} & \textbf{Description} & \textbf{Python variable} & \textbf{Python type} \\\hline $V_e$ & Value of Component $e \in \{\mathcal{E}_o, \mathcal{E}_I\}$ & \verb+val+ & \\ $V^{norm}_e$ & Normend value of component $e \in \{\mathcal{E}_o, \mathcal{E}_I\}$ & \verb+val+ & \\ $\eta_{i_e,o_{e,n}}$ & Efficiency at conversion of input $i_e$ to $n-$th output $o_{e,n}$ of component $e$ & \verb+eta+ & list\\ $\overline{W}_{e_1, e_2}$ & Upper bound of variable $w_{e_1, e_2}$ & \verb+out_max+ / \verb+in_max+ & list\\ $\underline{W}_{e_1, e_2}$ & Lower bound of variable $w_{e_1, e_2}$ & \verb+out_min+ / \verb+in_min+ & list\\ $\overline{L}_e$ & Upper bound of variable $l_e$ & \verb+cap_max+ & float\\ $\underline{L}_e$ & Lower bound of variable $l_e$ & \verb+cap_min+ & float\\ $C^{rate}_{e_i}$ & C-rate input of component $e$ & \verb+c_rate_in+ & float\\ $C^{rate}_{e_o}$ & C-rate output of component $e$ & \verb+c_rate_out+ & float\\ $\overline{O}^{global}_e$ & Global limit for all outputs of entity $e$ & \verb+sum_out_limit+ & float\\ $\overline{G}^{pos}_{e_{o,1}}$ & Upper bound for positive gradient of 1st output & \verb+grad_pos+ & float\\ $\overline{G}^{neg}_{e_{o,1}}$ & Upper bound for negative gradient of 1st output & \verb+grad_neg+ & float\\ $C^{loss}_e$ & Loss of energy per timestep & \verb+cap_loss+ & float \\ $T^{min,off}_e$ & Minimum down-time of component $e$ & \verb+t_min_off+ & float \\ $T^{min,on}_e$ & Minimum up-time of component $e$ & \verb+t_min_on+ & float \\ \\ \multicolumn{4}{l}{\textbf{Cost/Revenue parameters}}\\ $C_{e,i}$ & Costs for one unit inflow of Component $e$ & \verb+input_costs+ & list\\ $C_{e,o}$ & Costs for one unit outflow of Component $e$ & \verb+output_costs+ \verb+opex_var+ & list\\ $R_{e,i}$ & Revenues for one unit inflow of Component $e$ & \verb+input_revenues+ & list \\ $R_{e,o_n}$ & Revenues for one unit outflow of the $n$-th output of Component $e$ & \verb+output_revenues+ & list\\ $C^{cut}_e$ & Costs for curtailment of variable & \verb+curtailment_costs+ & float \\ \end{tabular}$