Formulations Introduction

PowerSimulations.jl enables modularity in its formulations by assigning a DeviceModel to each PowerSystems.jl component type existing in a defined system.

PowerSimulations.jl has a multiple AbstractDeviceFormulation subtypes that can be applied to different PowerSystems.jl device types, each dispatching to different methods for populating the optimization problem variables, objective function, expressions and constraints.

Example Formulation

For example a typical optimization problem in a DecisionModel in PowerSimulations.jl with three DeviceModel has the abstract form of:

\[\begin{align*} &\min_{\boldsymbol{x}}~ \text{Objective\_DeviceModelA} + \text{Objective\_DeviceModelB} + \text{Objective\_DeviceModelC} \\ & ~~\text{s.t.} \\ & \hspace{0.9cm} \text{Constraints\_NetworkModel} \\ & \hspace{0.9cm} \text{Constraints\_DeviceModelA} \\ & \hspace{0.9cm} \text{Constraints\_DeviceModelB} \\ & \hspace{0.9cm} \text{Constraints\_DeviceModelC} \end{align*}\]

Suppose this is a system with the following characteristics:

• Horizon: 48 hours
• Interval: 24 hours
• Resolution: 1 hour
• Three Buses: 1, 2 and 3
• One ThermalStandard (device A) unit at bus 1
• One RenewableDispatch (device B) unit at bus 2
• One PowerLoad (device C) at bus 3
• Three Line that connects all the buses

Now, we assign the following DeviceModel to each PowerSystems.jl with:

Note that we did not assign any DeviceModel to Line since the CopperPlatePowerModel used for the network assumes that everything is lumped in the same node (like a copper plate with infinite capacity), and hence there are no flows between buses that branches can limit.

Each DeviceModel formulation is described in specific in their respective page, but the overall optimization problem will end-up as:

\[\begin{align*} &\min_{\boldsymbol{p}^\text{th}, \boldsymbol{p}^\text{re}}~ \sum_{t=1}^{48} C^\text{th} p_t^\text{th} - C^\text{re} p_t^\text{re} \\ & ~~\text{s.t.} \\ & \hspace{0.9cm} p_t^\text{th} + p_t^\text{re} = P_t^\text{load}, \quad \forall t \in {1,\dots, 48} \\ & \hspace{0.9cm} 0 \le p_t^\text{th} \le P^\text{th,max} \\ & \hspace{0.9cm} 0 \le p_t^\text{re} \le \text{ActivePowerTimeSeriesParameter}_t \end{align*}\]

Note that the StaticPowerLoad does not impose any cost to the objective function or constraint but adds its power demand to the supply-balance demand of the CopperPlatePowerModel used. Since we are using the ThermalDispatchNoMin formulation for the thermal generation, the lower bound for the power is 0, instead of $P^\text{th,min}$. In addition, we are assuming a linear cost $C^\text{th}$. Finally, the RenewableFullDispatch formulation allows the dispatch of the renewable unit between 0 and its maximum injection time series $p_t^\text{re,param}$.

Nomenclature

In the formulations described in the other pages, the nomenclature is as follows:

• Lowercase letters are used for variables, e.g., $p$ for power.
• Uppercase letters are used for parameters, e.g., $C$ for costs.
• Subscripts are used for indexing, e.g., $(\cdot)_t$ for indexing at time $t$.
• Superscripts are used for descriptions, e.g., $(\cdot)^\text{th}$ to describe a thermal (th) variable/parameter.
• Bold letters are used for vectors, e.g., $\boldsymbol{p} = \{p\}_{1,\dots,24}$.