Piecewise linear cost functions
The choice for piecewise-linear (PWL) cost representation in PowerSimulations.jl is equivalent to the so-called λ-model from the paper The Impacts of Convex Piecewise Linear Cost Formulations on AC Optimal Power Flow. The SOS constraints in each model are only implemented if the data for PWL is not convex.
Special Ordered Set (SOS) Constraints
A special ordered set (SOS) is an ordered set of variables used as an additional way to specify integrality conditions in an optimization model.
- Special Ordered Sets of type 1 (SOS1) are a set of variables, at most one of which can take a non-zero value, all others being at 0. They most frequently applications is in a a set of variables that are actually binary variables: in other words, we have to choose at most one from a set of possibilities.
- Special Ordered Sets of type 2 (SOS2) are an ordered set of non-negative variables, of which at most two can be non-zero, and if two are non-zero these must be consecutive in their ordering. Special Ordered Sets of type 2 are typically used to model non-linear functions of a variable in a linear model, such as non-convex quadratic functions using PWL functions.
Standard representation of PWL costs
Piecewise-linear costs are defined by a sequence of points representing the line segments for each generator: $(P_k^\text{max}, C_k)$ on which we assume $C_k$ is the cost of generating $P_k^\text{max}$ power, and $k \in \{1,\dots, K\}$ are the number of segments each generator cost function has.
PowerSystems has more options to specify cost functions for each thermal unit. Independent of which form of the cost data is provided, PowerSimulations.jl will internally transform the data to use the λ-model formulation. See TODO: ADD PSY COST DOCS for more information.
Commitment formulation
With this the standard representation of PWL costs for a thermal unit commitment is given by:
\[\begin{align*} \min_{\substack{p_{t}, \delta_{k,t}}} & \sum_{t \in \mathcal{T}} \left(\sum_{k \in \mathcal{K}} C_{k,t} \delta_{k,t} \right) \Delta t\\ & \sum_{k \in \mathcal{K}} P_{k}^{\text{max}} \delta_{k,t} = p_{t} & \forall t \in \mathcal{T}\\ & \sum_{k \in \mathcal{K}} \delta_{k,t} = u_{t} & \forall t \in \mathcal{T}\\ & P^{\text{min}} u_{t} \leq p_{t} \leq P^{\text{max}} u_{t} & \forall t \in \mathcal{T}\\ &\left \{\delta_{1,t}, \dots, \delta_{K,t} \right \} \in \text{SOS}_{2} & \forall t \in \mathcal{T} \end{align*}\]
on which $\delta_{k,t} \in [0,1]$ is the interpolation variable, $p$ is the active power of the generator and $u \in \{0,1\}$ is the commitment variable of the generator. In the case of a PWL convex costs, i.e. increasing slopes, the SOS constraint is omitted.
Dispatch formulation
\[\begin{align*} \min_{\substack{p_{t}, \delta_{k,t}}} & \sum_{t \in \mathcal{T}} \left(\sum_{k \in \mathcal{K}} C_{k,t} \delta_{k,t} \right) \Delta t\\ & \sum_{k \in \mathcal{K}} P_{k}^{\text{max}} \delta_{k,t} = p_{t} & \forall t \in \mathcal{T}\\ & \sum_{k \in \mathcal{K}} \delta_{k,t} = \text{on}_{t} & \forall t \in \mathcal{T}\\ & P^{\text{min}} \text{on}_{t} \leq p_{t} \leq P^{\text{max}} \text{on}_{t} & \forall t \in \mathcal{T}\\ &\left \{\delta_{i,t}, \dots, \delta_{k,t} \right \} \in \text{SOS}_{2} & \forall t \in \mathcal{T} \end{align*}\]
on which $\delta_{k,t} \in [0,1]$ is the interpolation variable, $p$ is the active power of the generator and $\text{on} \in \{0,1\}$ is the parameter that decides if the generator is available or not. In the case of a PWL convex costs, i.e. increasing slopes, the SOS constraint is omitted.
Compact representation of PWL costs
Commitment Formulation
\[\begin{align*} \min_{\substack{p_{t}, \delta_{k,t}}} & \sum_{t \in \mathcal{T}} \left(\sum_{k \in \mathcal{K}} C_{k,t} \delta_{k,t} \right) \Delta t\\ & \sum_{k \in \mathcal{K}} P_{k}^{\text{max}} \delta_{k,t} = P^{\text{min}} u_{t} + \Delta p_{t} & \forall t \in \mathcal{T}\\ & \sum_{k \in \mathcal{K}} \delta_{k,t} = u_{t} & \forall t \in \mathcal{T}\\ & 0 \leq \Delta p_{t} \leq \left( P^{\text{max}} - P^{\text{min}} \right)u_{t} & \forall t \in \mathcal{T}\\ &\left \{\delta_{i,t} \dots \delta_{k,t} \right \} \in \text{SOS}_{2} & \forall t \in \mathcal{T} \end{align*}\]
on which $\delta_{k,t} \in [0,1]$ is the interpolation variable, $\Delta p$ is the active power of the generator above the minimum power and $u \in \{0,1\}$ is the commitment variable of the generator. In the case of a PWL convex costs, i.e. increasing slopes, the SOS constraint is omitted.
Dispatch formulation
\[\begin{align*} \min_{\substack{p_{t}, \delta_{k,t}}} & \sum_{t \in \mathcal{T}} \left(\sum_{k \in \mathcal{K}} C_{k,t} \delta_{k,t} \right) \Delta t\\ & \sum_{k \in \mathcal{K}} P_{k}^{\text{max}} \delta_{k,t} = P^{\text{min}} \text{on}_{t} + \Delta p_{t} & \forall t \in \mathcal{T}\\ & \sum_{k \in \mathcal{K}} \delta_{k,t} = \text{on}_{t} & \forall t \in \mathcal{T}\\ & 0 \leq \Delta p_{t} \leq \left( P^{\text{max}} - P^{\text{min}} \right)\text{on}_{t} & \forall t \in \mathcal{T}\\ &\left \{\delta_{i,t} \dots \delta_{k,t} \right \} \in \text{SOS}_{2} & \forall t \in \mathcal{T} \end{align*}\]
on which $\delta_{k,t} \in [0,1]$ is the interpolation variable, $\Delta p$ is the active power of the generator above the minimum power and $u \in \{0,1\}$ is the commitment variable of the generator. In the case of a PWL convex costs, i.e. increasing slopes, the SOS constraint is omitted.