Hydro pump node

Introduced type and its field

The HydroPump node represents a hydropower unit that consumes electricity to pump water between two reservoir in a hydropower system. The HydroPump can convert electricity to potential energy stored in the reservoirs by pumping water between reservoirs at different head levels under the assumption that the reservoirs has constant head level. The conversion from electric energy is the reversed process of the energy conversion in the HydroGenerator and can be described by an power-discharge relationship, where discharge refer to the flow of pumped water.

Standard fields

HydroPump nodes build on the HydroUnit and the RefNetworkNode nodes, but add additional fields. The standard fields are:

• 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 capacity corresponds to the nominal capacity of the node. It is the installed pumping capacity, either in form of volume water per time period or power capacity.

• opex_var::TimeProfile:
  The variable operational expenses are based on the capacity utilization through the variable cap_use.

• 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 pumping using the general constraints types.

Additional fields

• pq_curve::AbstractPqCurve:
  Describes the relationship between consumed power (electricity) and pumped water. The input can be provided by using the subtype PqPoints or as a single energy equivalent.

• water_resource::Resource:
  The water resource that the node pumps between reservoirs.
• electricity_resource::Resource:
  The electricity resource consumed in the node.

Mathematical description

The HydroPump inherits its mathematical description from the HydroUnit and NetworkNode.

Variables

Standard variables

The HydroPump 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

In addition to the standard variables, the variables presented below are defined for HydroUnit nodes. These variabels are hence created for HydroPump nodes.

• $\texttt{discharge\_segments}[n, t, q]$: One discharge variable is defined for each segment q of the PQ-curve defined by the field pq_curve of node $n$ in operational period $t$ with unit volume per time unit.
  If PqPoints are provided, the number of discharge segments will be $Q$, where $Q+1$ is the length of the vectors in the fields of PqPoints. There is only one discharge segment if an energy equivalent is used. The $\texttt{discharge\_segments}$ variables define the utilisation of each discharge segment and sums up to the total discharge.

  discharge_segments

  Sequential allocation is not enforced by binary variables, but will occure sequentially if the problem if set up correctly. A non-convex PQ-curve for the pump may result in a non-sequential allocation.

The following variables are created if required by the additional constraints:

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

Constraints

The following sections omit the direct inclusion of the vector of HydroPump nodes. Instead, it is implicitly assumed that the constraints are valid $\forall n ∈ N$ for all HydroPump 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

HydroPump 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 new methods for the functions constraints_capacity and constraints_opex_var are explained in the section for HydroGenerator.

Furthermore, we dispatch on the flow constraints for HydroPump nodes. The mathematical description is the same as for the HydroGenerator nodes, except that electricity flows into the node (is consumed) rather than out of the node:

• constraints_flow_in:
  It is assumed that the amount of water is constant for HydroGenerator nodes, and the flow of water into the node therefore equals the flow out.

  \[ \texttt{flow\_in}[n, t, water\_resource(n)] = \texttt{flow\_out}[n, t, water\_resource(n)]\]

  The flow of electricity into the node is given by the:

  \[\texttt{flow\_in}[n, t, electricity\_resource(n)] = \texttt{cap\_use}[n, t]\]

• constraints_flow_out:
  The flow out of water is constrained to total discharge given by the sum of the $\texttt{discharge\_segments[n,t,q]}$ variables, where Q is the number of segments in the PQ-curve (i.e., Q+1 PQ-points):

  \[\texttt{flow\_out}[n, t, water\_resource(n)] = \sum_{q=1}^{Q}\texttt{discharge\_segments[n,t,q]}\]

  In addition to being constrained by the installed capacity, the $\texttt{cap\_use}$ variables are constrained by the discharge of water multiplied with the conversion rate given by $\texttt{E[q]}$ which is the slope of each segment in the PQ-curve:

  \[\texttt{cap\_use}[n, t] = \sum_{q=1}^{Q}(\texttt{discharge\_segments}[n,t,q] \times \texttt{E[q]})\]

  The discharge segments are constrained by the discharge_levels of the PqPoints:

  \[\begin{aligned} \texttt{discharge\_segment}[n, t, q] \leq & \texttt{cap\_inst}[n, t] \times (discharge\_levels[q+1] \\ & - discharge\_levels[q]) \qquad \forall q \in [1,Q] \\ \end{aligned}\]

Furthermore, the dispatch on constraints_flow_in includes additional pumping capacity constraints. The constraints are optional and only added to the problem if given as input in the Data field of the nodes. Soft constraints, i.e., constraints with a penalty, are used if the constraints have non-infinite penalty values. For HydroPump nodes, the constraints can be defined for the electricity_resource and water_resource, limiting the flow into of the node.

To this end, we declare an internal expression given by:

\[\texttt{cap\_inst\_var}[n, p, t] = \begin{cases} \texttt{cap\_inst}[n, t] \times max\_normalized\_power(n), & \text{if } p == electricity\_resource(n) \\ \texttt{cap\_inst}[n, t] \times max\_normalized\_flow(n), & \text{if } p == water\_resource(n) \end{cases}\]

The expression is subsequently used in the calculations for providing the constraints (the first equation corresponds to hard constraints, the second equation to soft constraints):

1. Minimum constraints for pumping:

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

2. Maximum constraints for pumping:

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

3. Scheduling constraints for pump:

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

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.