Objective

Helper function for expressing compressor costs.

Helper function for expressing line costs.

Helper function for expressing pipe costs.

Helper function for constructing the expression associated with the OPF objective.

Helper function for expressing zonal pressure penalty prices.

Helper function for expressing zonal prices.

Maximizes the normalized sum of nongeneration gas load delivered in the joint network, i.e.,

\[\max \eta_{G}(d) := \left(\sum_{i \in \mathcal{D}^{\prime}} \beta_{i} d_{i}\right) \left(\sum_{i \in \mathcal{D}^{\prime}} \beta_{i} \overline{d}_{i}\right)^{-1},\]

where $\mathcal{D}^{\prime}$ is the set the delivery points in the gas network with dispatchable demand that are not connected to interdependent generators in the power network, $\beta_{i} \in \mathbb{R}_{+}$ (equal to the priority property of the delivery) is a predefined restoration priority for delivery $i \in \mathcal{D}^{\prime}$, $d_{i}$ is the mass flow of gas delivered at $i \in \mathcal{D}^{\prime}$, and $\overline{d}_{i}$ is the maximum deliverable gas load at $i \in \mathcal{D}^{\prime}$.

Maximizes the weighted normalized sums of nongeneration gas load and active power load delivered in the joint network, i.e.,

\[ \max \lambda_{G} \eta_{G}(d) + \lambda_{P} \eta_{P}(z^{d}),\]

where it is recommended that $0 < \lambda_{G} < 1$, that gm_load_priority in the network data specification be set to the value of $\lambda_{G}$ desired, and that pm_load_priority similarly be set to the value $1 - \lambda_{G} = \lambda_{P}$. This type of parameterization allows for a straightforward analysis of gas-power tradeoffs, as the objective is naturally scaled between zero and one.

Maximizes the normalized sum of active power load delivered in the joint network, i.e.,

\[\max \eta_{P}(z^{d}) := \left(\sum_{i \in \mathcal{L}} \beta_{i} z_{i}^{d} \lvert \Re({S}_{i}^{d}) \rvert \right) \left(\sum_{i \in \mathcal{L}} \beta_{i} \lvert \Re({S}_{i}^{d}) \rvert \right)^{-1}.\]

Here, $\mathcal{L}$ is the set of loads in the power network, $\beta_{i} \in \mathbb{R}_{+}$ (equal to the weight property of the load) is the load restoration priority for load $i \in \mathcal{L}$, and $z_{i} \in [0, 1]$ is a variable that scales the absolute maximum amount of active power load, $\lvert \Re({S}_{i}^{d}) \rvert$, at load $i \in \mathcal{L}$.

Objective for minimizing the costs of expansion. Formally stated as

\[\min \alpha \sum_{(i,j) \in Pipe \cup Compressors} \kappa_{ij} z_{ij} + \beta \sum_{(i,j) \in Branches} \kappa_{ij} z_{ij},\]

where $\alpha$ and $\beta$ are weighting terms.

Objective function for minimizing the gas-grid optimal flow combined with network expansion costs as defined in reference Russell Bent, Seth Blumsack, Pascal Van Hentenryck, Conrado Borraz-Sánchez, Mehdi Shahriari. Joint Electricity and Natural Gas Transmission Planning With Endogenous Market Feedbacks. IEEE Transactions on Power Systems. 33 (6): 6397-6409, 2018. More formally, this objective is stated as

\[\min \alpha \sum_{(i,j) \in Pipe \cup Compressors} \kappa_{ij} z_{ij} + \beta \sum_{(i,j) \in Branches} \kappa_{ij} z_{ij} + \lambda \sum_{g \in G} (c^1_g pg_g^2 + c^2_g pg_g + c^3_g) + \gamma \sum_{z \in Z} cost_z + \gamma \sum_{z \in Z} pc_z,\]

where $\lambda, \alpha, \beta$ and $\gamma$ are weighting terms.

Objective function for minimizing the gas-grid optimal flow as defined in reference Russell Bent, Seth Blumsack, Pascal Van Hentenryck, Conrado Borraz-Sánchez, Mehdi Shahriari. Joint Electricity and Natural Gas Transmission Planning With Endogenous Market Feedbacks. IEEE Transactions on Power Systems. 33 (6): 6397-6409, 2018. More formally, this objective is stated as

\[\min \lambda \sum_{g \in G} (c^1_g pg_g^2 + c^2_g pg_g + c^3_g) + \gamma \sum_{z \in Z} cost_z + \gamma \sum_{z \in Z} pc_z,\]

where $\lambda$ and $\gamma$ are weighting terms.