(model-doc-end-value-emps-target-reservoir)= # End valuation with EMPS (Target Reservoir Module) To ensure that water is not emptied at the end of each decision problem, a value is placed on it at the end of each problem. These values are supplied from running EMPS. The values are aggregated water values for each price area, which needs to be disaggregated before being used by ngLTM. The water values from EMPS divide an aggregated reservoir into a number of layers representing the volume of the reservoir from minimum to maximum volume. Each layer is a percentage of volume of the reservoir used. ![](./resources/water_value_disaggregation.svg) In the figure above, the disaggregation of water values is illustrated. There are 50 layers between minimum and maximum volume. The value of each layer is the same. For the input it is given in the unit [mu/GWh] which needs to be scaled for each individual reservoir by an energy equivalent [GWh/Mm^3]. The layer sizes varies between the individual reservoirs in an area, and is computed using the target reservoir method. Note that the type of reservoir called target reservoir in other parts of the documentation, reservoirs that follow a predefined target volume time series, is not the same as the target reservoirs mentioned here. When target reservoirs are mentioned below, the meaning is reservoir volumes that are used when computing water value layer sizes with the goal of avoiding unnecessary spillage in the filling season and avoiding unnecessary power loss in the depletion season. The end values computed here are only valid for reservoirs that do not follow a predefined target volume. ## Input data Input data for the water value module is loaded from VVERD.SAMK files for each area. There's two dimensions for the values, layer and snow level. Snow level is calculated by a separate function, but the following chapter is valid for all dimension values. Input data is N points on a curve where the X-axis denotes percentage volume of the reservoir that is filled, from 0 to 100%. With 50 layers, there should be 51 points on this curve, at 0, 2, 4, 6 ... 98, 100. We assume that the input is given in increasing positive values. This ensures that the layers are used in the correct order. The highest value will be used first. ## Modelling The water value is modelled as the volume sum constraint $$ \textrm{VOLSUM}(i): res(i,k) = \sum_{n=0}^{N - 1}res\_layer(i,n) \quad, $$ where $$ \\{i \in reservoirs,\ k = timesteps-1,\ n \in \\{0, 1, ..., N - 1\\}\\} \quad ,$$ and $res(i,k)$ is the reservoir volume variable representing the volume between minimum and maximum volume. The bounds of the $N - 1$ layers are given as $$ 0 \leq res\_layer(i,n) \leq \textrm{LayerSize}(i,n), $$ where $\textrm{LayerSize}(i,n)$ is computed using target reservoir calculations. The value of each layer is added to the objective function: $$ Obj^{min}: C^Tx - \sum_{i}^{reservoirs}\sum_{n=0}^{N - 1}\textrm{EnergyEquivalent}(i)*\textrm{WaterValue}(n)*res\_layer(i,n) $$ Note that if the reservoir has a hard minimum volume restriction, the largest layer value scaled by the energy equivalent is added to the objective cost of the minimum variable $res\_min$. The last layer is therefore omitted above. ## The Target Reservoir Method When using the target reservoir method we first compute a set of target reservoirs for each reservoir in an area at each energy level in the aggregated reservoir . The aggregated reservoir is split into $N-1$ equal layers between the aggregated minimum and maximum volume. The size of the layers of the individual reservoirs is then the difference between two target reservoirs computed at consecutive energy levels: $$ \textrm{LayerSize}(i,n) = \textrm{TargetReservoir}(i, n+1) - \textrm{TargetReservoir}(i, n). $$ For the $n=N$ we have $$ \textrm{TargetReservoir}(i, N) = \begin{cases} \begin{aligned} 0, \quad & \textrm{No hard minimum restriction} \\\\ \textrm{ResMin}(i,k), \quad & \textrm{Hard minimum restriction exists} \\\\ \end{aligned} \end{cases} $$ The strategy for computing the target reservoirs is split into two seasons: The filling season and the depletion season. ### The Filling Season The filling season is approximately between week 18 and week 40. At this time, the inflow to the reservoirs is high and the strategy is to minimize spillage. This is done by keeping the damping constant between all the reservoirs in an area: $$ d = \frac{v_{max}(i) - v_{target}(i)}{v_{max}(i)} \cdot r(i) , \qquad (1) $$ where $$ \begin{aligned} d &= \textrm{Relative damping}, \\\\ v_{max} &= \textrm{Maximum reservoir volume}, \\\\ v_{target} &= \textrm{Target reservoir volume}, \\\\ r &= \textrm{Degree of regulation}. \\\\ \end{aligned} $$ The relative damping can either be supplied as an input or computed as $$ r = \frac{v_{max}}{I}, $$ where $I [Mm^3/Year]$ is the expected annual inflow to the reservoir. In addition to satisfying equation $(1)$, the target reservoirs must also satisfy $$ \sum_{i}^{reservoirs}v_{target}(i) \cdot \textrm{EnergyEquivalent}(i) = E_{agg}, \qquad (2) $$ where $E_{agg}$ is the aggregated energy level the target reservoirs are computed for. When solving equations $(1)$ and $(2)$ some reservoirs might end up with a target reservoir that is less than their hard minimum limit or above their maximum limit. These reservoirs are then removed from the equation set and the equations are then solved again. To summarize: The following algorithm is used to compute target reservoirs in the filling season: #### Algorithm 1: Solving the Target Reservoir Equations For each area: 1. Sort the reservoirs based on degree of regulation ascending 2. Solve equations $(1)$ and $(2)$ to get the target reservoirs. 3. Check each reservoir in order: If the target reservoir is below the hard minimum limit or above the maximum limit: - Set the target reservoir to the limit - Remove the reservoir from the equation set - Subtract the energy of the limit from the aggregated energy - Jump to 2. ### The Depletion Season The depletion season is approximately between week 40 and 18. At this time, the inflow to the reservoirs is low and reservoir volume is decreasing. The goal is to avoid emptying any reservoir too early and also end the depletion season with equal relative damping. To achieve this, Algorithm 1 is performed with equation $(1)$ replaced by $$ d = \frac{v_{tr}(i) - v_{target}(i, k)}{v_{max}(i)} \cdot r(i) , \qquad (3) $$ where $v_{target}(i, k)$ is a tapered reservoir volume trajectory that is found by interpolating from the physical maximum volume at the beginning of the depletion season to an end of depletion target reservoir $v_{target}^{eod}$, at the end of the depletion season. The end of depletion target reservoirs are found by defining an aggregated end of depletion target energy that is then distributed with equal damping using Algorithm 1. The aggregated end of depletion target energy corresponds to an end of depletion water value. This water value is linearly interpolated from 15[mu/GWh] at the start of the depletion season down to 10[mu/GWh] at the end of the depletion season. Meaning that we aim at a higher end of depletion energy level the closer we get to the depletion season. The water value is converted to energy using the input water value curve. To summarize: The following algorithm is used to compute target reservoirs in the depletion season: #### Algorithm 2: Target reservoirs in the Depletion Season For each area: 1. Compute aggregated end of depletion water value by interpolating from 15 to 10. 2. Compute aggregated end of depletion energy from the end of depletion water value. 3. Compute end of depletion target reservoirs using Algorithm 1. 4. Compute tapered reservoir volume trajectories by interpolating from physical max to end of depletion target reservoir. 5. Compute target reservoirs using Algorithm 1 with equation $(3)$ and $(2)$. ### Smoothing between seasons The target reservoirs computed just before and just after a change in season can be very different, resulting in a sudden jump in water value layer sizes and power prices. To combat this, layer sizes can be smoothed around the transition point. A decision period is defined around the transition point. Layer sizes within this period are averaged before and after the transition point. The layer sizes at the end of each decision period is replaced by the two averages weighted by the distance from the transition point.