Table 1 Summary of variables, parameters, indices, and constants used in the mathematical formulation.
Symbol | Description |
|---|---|
\(\iota , \ell , \pi , \upsilon , \kappa , \tau\) | Indices for node, line, renewable unit, portfolio, generator, and time period |
\(T^{long}\) | Set of long-term time horizons (seasonal, annual, rolling) |
\(N, L, R_\iota , V_\iota , G_\iota\) | Sets of nodes, lines, renewable units, portfolios, and generators at node \(\iota\) |
\(\vartheta ^{gen}_{\iota ,\kappa ,\tau }\) | Active power generation level of unit \(\kappa\) at node \(\iota\) and time \(\tau\) |
\(\lambda ^{act}_{\iota ,\upsilon ,\tau }\) | Flexibility activation level of portfolio \(\upsilon\) at node \(\iota\) and time \(\tau\) |
\(\zeta ^{curt}_{\iota ,\pi ,\tau }\) | Renewable curtailment at unit \(\pi\) of node \(\iota\) and time \(\tau\) |
\(\zeta ^{emiss}_{\iota ,\pi ,\tau }\) | Carbon emission level associated with renewable \(\pi\) at node \(\iota\) and time \(\tau\) |
\(\varepsilon ^{loss}_{\iota ,\tau }\) | Expected energy not served (EENS) or nodal loss at time \(\tau\) |
\(\pi ^{viol}_{\ell ,\tau }\) | Probability of line \(\ell\) overload at time \(\tau\) |
\(\eta ^{LMF}_{\iota ,\upsilon ,\tau }\) | Locational Marginal Flexibility (welfare sensitivity of flexibility) |
\(\sigma ^{resp}_{\iota ,\upsilon ,\tau }\) | Binary indicator for service-level response satisfaction |
\(\delta ^{life}_{\iota ,\upsilon }\) | Accumulated degradation (lifetime usage) of flexible resource \(\upsilon\) |
\(\phi ^{quota}_{\iota ,\upsilon ,\tau }\) | Change in activation quota between rolling windows |
\(\chi ^{rest}_{\iota ,\upsilon ,\tau }\) | Binary rest or recovery indicator after activation |
\(\theta ^{flow}_{\ell ,\tau }\) | Power flow through line \(\ell\) at time \(\tau\) |
\(\pi _{\ell ,\iota }\) | Power Transfer Distribution Factor (PTDF) from node \(\iota\) to line \(\ell\) |
\(\varsigma ^{SOC}_{\iota ,\upsilon ,\tau }\) | State of charge for storage portfolio \(\upsilon\) |
\(\Delta ^{max}_{\iota ,\upsilon }\) | Maximum allowable degradation or activation change |
\(\Theta ^{max}_\ell\) | Thermal rating limit of line \(\ell\) |
\(\Lambda ^{contract}_{\iota }\) | Contractual activation quota for node \(\iota\) |
\(\omega ^{carb}, \xi ^{curt}, \gamma ^{res}\) | Coefficients for carbon, curtailment, and reserve penalties |
\(\alpha ^{op}, \beta ^{inv}, \nu ^{deg}\) | Coefficients for operation, investment, and degradation costs |
\(\chi ^{risk}\) | Weighting factor for reliability-related penalties |
\(\Phi ^{feas}, \Theta ^{cut}\) | Feasibility and optimality cuts in decomposition process |
\(\varepsilon\) | Wasserstein ball radius defining distributional ambiguity set |
\(\Gamma\) | Risk or ambiguity tolerance bound |
\(\rho ^{indoor}_{\iota ,\tau }\) | Indoor temperature level at node \(\iota\) and time \(\tau\) |
\(\chi ^{ambient}_{\iota ,\tau }\) | Ambient temperature level at node \(\iota\) and time \(\tau\) |
\(\alpha _i, \beta _i\) | Thermal response coefficients for HVAC load model |
\(\kappa _1, \kappa _2\) | Degradation coefficients related to cycling and SoC deviation |
\(\theta ^{fair}_{\iota ,\upsilon ,\tau }\) | Fairness penalty coefficient for activation changes |
