Objective
The objective used by the Power-Water Flow (PWF) problem specification is a feasibility-only objective. The default objective used by the Optimal Power-Water Flow (OPWF) problem specification is _PMD.objective_mc_min_fuel_cost(pmd), which is described in the PowerModelsDistribution documentation and represents a standard fuel cost minimization. In addition to these objectives, PowerWaterModels also defines an additional objective, objective_min_max_generation_fluctuation(pwm), defined over AbstractPowerWaterModel.
PowerWaterModels.objective_min_max_generation_fluctuation — MethodObjective for minimizing the maximum difference between time-adjacent power generation variables. Note that this function introduces a number of auxiliary variables and constraints to appropriately model the objective. Mathematically, the objective and auxiliary terms are modeled as follows:
\[ \begin{aligned} & \text{minimize} & & z \\ & \text{subject to} & & z \geq pg_{i, c, t} - pg_{i, c, t-1}, \, \forall i \in \mathcal{G}, \, \forall c \in \mathcal{C}, \, \forall t \in \{2, 3, \dots, T\} \\ & & & z \geq pg_{i, c, t-1} - pg_{i, c, t}, \, \forall i \in \mathcal{G}, \, \forall c \in \mathcal{C}, \, \forall t \in \{2, 3, \dots, T\} \\ & & & z \geq 0 \\ & & & x \in \mathcal{X}, \end{aligned}\]
where $\mathcal{G}$ is the set of generators, $\mathcal{C}$ is the set of conductors, and $\{2, 3, \dots, T\}$ are the non-starting time indices. Further, $x \in \mathcal{X}$ represents the remainder of the problem formulation, i.e., variables and constraints not relevant to this description.