Objective

function for maximizing prioritzed load: $\max \sum_{i \in {\cal D }} \omega_i \boldsymbol{d}_i$

function for minimizing compressor energy

function for minimizing economic costs: $\min \sum_{j \in {\cal D}} \kappa_j \boldsymbol{d}_j - \sum_{j \in {\cal T}} \kappa_j \boldsymbol{\tau}_j - \sum_{j \in {\cal R}} \kappa_j \boldsymbol{r}_j - \sum_{ijk \in {\cal C}} \boldsymbol{f}_{ijk} (\boldsymbol{\alpha}_{ijk}^m - 1)$

function for costing expansion of pipes and compressors: $\sum_{k \in ne\_pipe} c_k zp_k + \sum_{k \in ne\_compressor} c_k zc_k$

function for minimizing new economic costs: $\min \sum_{j \in {\cal D}} \kappa_j \boldsymbol{d}_j - \sum_{j \in {\cal T}} \kappa_j \boldsymbol{\tau}_j - \sum_{j \in {\cal R}} \kappa_j \boldsymbol{r}_j - \sum_{k \in {\cal C}} |\boldsymbol{p^2}_{jk} - \boldsymbol{p^2}_{ik}|$

function for minimizing supply demand costs: $\min \sum_{j \in {\cal D}} \kappa_j \boldsymbol{d}_j - \sum_{j \in {\cal T}} \kappa_j \boldsymbol{\tau}_j - \sum_{j \in {\cal R}} \kappa_j \boldsymbol{r}_j$

minium compressor power objective for transient OGF problem

transient objective for minimizing a linear combination of compressor power and load shed

minimum load shedding objective for transient OGF problem