Hydro generator node

Introduced type and its field

The HydroGenerator node represents a hydropower unit used to generate electricity in a hydropower system. In its simplest form, the HydroGenerator can convert potential energy stored in the reservoirs to electricity by discharging water between reservoirs at different head levels under the assumption that the reservoirs have constant head level. The conversion to electric energy can be described by an power-discharge relationship referred to as the PQ-curve.

Standard fields

HydroGenerator 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 can refer to either the installed power or discharge capacity of the hydropower unit.

• 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 the power generation or water discharge using the general constraints types.

Additional fields

HydroGenerator nodes introduce the following additional fields:

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

  pq_curve

  The input provided to the pq_curve field has to be relative to the installed capacity, so that either the maximum discharge or the maximum power value given by the PqPoints equals 1. If a single energy equivalent is provided, it is required that the field cap must refer to the power capacity of the HydroGenerator node.

• water_resource::Resource:
  The water resource that the node discharges to generate electricity.

• electricity_resource::Resource:
  The electricity resource generated from the node.

Mathematical description

The HydroGenerator inherits its mathematical description from the HydroUnit which is an abstract subtype of NetworkNode.

Variables

Standard variables

The HydroGenerator 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. These variabels are hence created for HydroGenerator 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 variables $\texttt{discharge\_segments}$ define the utilisation of each discharge segment and sum up to the total discharge.

  discharge_segments

  Sequential allocation is not enforced by binary variables, but allocation will occure sequentially if the problem if set up correctly. Penalties for spilling water, a non-concave PQ-curve or an otherwise non-convex problem are examples thay 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 electricity generation 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 electricity generation is smaller than planned.

Constraints

In the following sections the vector of HydroGenerator nodes are omitted from the descriptions. Instead, it is implicitly assumed that the constraints are valid $\forall n ∈ N$ for all HydroGenerator 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

HydroGenerator 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_capacity rquires a new method to account for the included PQ-curve:

\[\begin{aligned} \texttt{cap\_use}&[n, t] \leq \texttt{cap\_inst}[n, t] \times P_{norm}^{max}\\ \end{aligned}\]

Where capacity(n, t) is the installed capacity of node n in operational period t and $P_{norm}^{max}$ is the maximum normalized power capacity identified through the function max_normalized_power. If the capacity refers to power, the PqPoints power_levels ranges between 0 and 1 and $P_{norm}^{max}$ will also be 1. If the capacity refers to flow, the discharge_levels ranges between 0 and 1, while power_levels may take other values and $P_{norm}^{max}$ represents the last element of the power_levels vector.

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

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

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 set $P^{res}$ contains the water and power resources of node n.

Furthermore, we provide new methods for the flow constraints for HydroGenerator nodes:

• 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:

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

• 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):

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

  The flow of electricity out of the node is given by the $\texttt{cap\_use}[n, t]$ variables:

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

  In addition to being constrained by the installed capacity, the variables $\texttt{cap\_use}$ 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:

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

  The capacity(n, t) returns the installed capacity and is used to scale the relative values of the PqPoints to absolute values.

Furthermore, the method for constraints_flow_out adds discharge and power capacity 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. For HydroGenerator nodes, the constraints can be defined for both the electricity_resource and water_resource. The mathematical formualtion of the constraints are:

1. Minimum constraints for discharge or power generation:

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

2. Maximum constraints for discharge or power generation:

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

3. Scheduling constraints for discharge or power generation:

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

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