Hydro gate node

Introduced type and its field

The HydroGate is used when water can be released between reservoirs without going through a generator. The HydroGate can either represent a controlled gate that is used to regulate the dispatch from a reservoir without production, or to bypass water when a reservoir is, for example, full. The HydroGate can also be used to represent spillage. Althoug spillage is not, in reality, a control decision but a consequence of full reservoir, it is often modelled as a controllable decisions since state dependent spillage can not be modelled directly in a linear model. Costs for operating gates can be added to penalize unwanted spillage using the field opex_var.

Standard fields

The HydroGate nodes build on the NetworkNode node type. Standard fields are given as:

• id:
  The field id is only used for providing a name to the node. This is similar to the approach utilized in EnergyModelsBase.
• cap::TimeProfile:
  The installed gate discharge capacity. In the case of a HydroGate, this value corresponds to the maximum possible discharge without any generator. In practice, this value has to be sufficiently large to avoid an unfeasible system.
• opex_var::TimeProfile:
  The variable operational expenses are based on the capacity utilization through the variable cap_use. Hence, it is directly related to the specified output ratios. The variable operating expenses can be provided as OperationalProfile as well.
• opex_fixed::TimeProfile:
  The fixed operating expenses are relative to the installed capacity (through the field cap) and the chosen duration of a strategic period as outlined on Utilize TimeStruct.
  It is important to note that you can only use FixedProfile or StrategicProfile for the fixed OPEX, but not RepresentativeProfile or OperationalProfile. In addition, all values have to be non-negative.
• data::Vector{Data}:
  An entry for providing additional data to the model.
  Additional constraints

  The data field can be used to add minimum, maximum, and schedule constraints on the discharge using the general constraints types.

Additional fields

HydroGate nodes a single additional field:

• resource::Resource:
  The water resource that the node can release.

Mathematical description

The HydroGate inherits its mathematical description from the NetworkNode where there is only a single input and output resource given by the resource field and a conversion ratio of 1.

Variables

Standard variables

The HydroGate utilizes the standard variables from the NetworkNode, as described on the page Optimization variables:

• $\texttt{opex\_var}$
• $\texttt{opex\_fixed}$
• $\texttt{cap\_use}$
• $\texttt{cap\_inst}$
• $\texttt{flow\_in}$
• $\texttt{flow\_out}$

Additional variables

HydroGate nodes add additional variables if required by the additional constraints:

• $\texttt{gate\_penalty\_up}[n, t]$: Variable for penalizing violation of the discharge constraint in direction up in HydroGate node $n$ in operational period $t$ with unit volume per time unit.
  Up implies in this case that the flow through the gate is larger than planned.
• $\texttt{gate\_penalty\_down}[n, t]$: Variable for penalizing violation of the discharge constraint in direction down in HydroGate node $n$ in operational period $t$ with unit volume per time unit.
  Down implies in this case that the flow through the gate is smaller than planned.

Constraints

The following sections omit the direct inclusion of the vector of HydroGate nodes. Instead, it is implicitly assumed that the constraints are valid $\forall n ∈ N$ for all HydroGate types if not stated differently. In addition, all constraints are valid $\forall t \in T$ (that is in all operational periods) or $\forall t_{inv} \in T^{Inv}$ (that is in all strategic periods).

Standard constraints

HydroGate nodes utilize in general the standard constraints described in Constraint functions for NetworkNode. The majority of these constraints are hence ommitted in the following description.

The function constraints_opex_var requires a new method as we have to include the penalty variables for violating the constraints if required:

\[\begin{aligned} \texttt{opex\_var}&[n, t_{inv}] = \\ \sum_{t \in t_{inv}} \Big( &opex\_var(n, t) \times \texttt{cap\_use}[n, t] + \\ & penalty(c_{up}, t) \times \texttt{gate\_penalty\_up}[n, t] + \\ & penalty(c_{down}, t) \times \texttt{gate\_penalty\_down}[n, t] \Big) \times scale\_op\_sp(t_{inv}, t) \end{aligned}\]

where $penalty()$ returns the penalty value for violation in the upward and downward direction of constraints with penalty variables, denoted by $c_{up}$ and $c_{up}$ respectively.

The method for constraints_flow_out adds discharge constraints if additional constraints are provided in the Data field. Soft constraints, i.e., constraints with a penalty, are used if the constraints have non-infinite penalty values. The mathematical formulation of the constraints are (the first equation corresponds to hard constraints, the second equation to soft constraints):

1. Minimum constraints for the discharge:

   \[\begin{aligned} \texttt{flow\_out}[n, t, p] \geq & \\ & \texttt{cap\_inst}[n, t] \times value(c, t) \qquad & \forall c \in C^{min}\\ \texttt{flow\_out}[n, t, p] + & \texttt{gate\_penalty\_up}[n, t] \geq \\ & \texttt{cap\_inst}[n, t] \times value(c, t) \qquad & \forall c \in C^{min} \end{aligned}\]

2. Maximum constraints for the discharge:

   \[\begin{aligned} \texttt{flow\_out}[n, t, p] \leq & \\ & \texttt{cap\_inst}[n, t] \times value(c, t) \qquad & \forall c \in C^{max}\\ \texttt{flow\_out}[n, t, p] - & \texttt{gate\_penalty\_down}[n, t] \leq \\ & \texttt{cap\_inst}[n, t] \times value(c, t) \qquad & \forall c \in C^{max} \end{aligned}\]

3. Scheduling constraints for the discharge:

   \[\begin{aligned} \texttt{flow\_out}[n, t, p] = & \\ & \texttt{cap\_inst}[n, t] \times value(c, t) \qquad & \forall c \in C^{sch}\\ \texttt{flow\_out}[n, t, p] + & \texttt{gate\_penalty\_up}[n, t] - \texttt{gate\_penalty\_down}[n, t] = \\ & \texttt{cap\_inst}[n, t] \times value(c, t) \quad & \forall c \in C^{sch} \end{aligned}\]

where $value(c,t)$ returns the relative limit of scheduling constraint c. The sets $C^{min}$,$C^{max}$ and $C^{sch}$ contain additional minimum, maximum, and scheduling constraints, repectively.

Additional constraints

The HydroGate nodes do not include any additional constraints other than through dispatching on Constraint functions for NetworkNode nodes as described above.