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(battery)=
# battery
A battery that is a part of a power system.

|   |   |
|---|---|
|Input connections|{ref}`reserve_group`, {ref}`busbar`|
|Output connections|{ref}`reserve_group`, {ref}`busbar`|
|License|SHOP_OPEN|
|Release version|15.2.0.0|

```{contents}
:local:
:depth: 1
```

## Introduction
A battery can be used to temporarily store energy in a power system. The battery can charge and discharge power where the power losses are given by the [charge efficiency](battery:charge_efficiency) $\eta_c$ and [discharge efficiency](battery:discharge_efficiency) $\eta_d$. The battery energy balance is
```{math}
:label: energy-balance
e_{i,t+1} = e_{i,t} + \eta_c pc_{i,t} - \frac{1}{\eta_d} pd_{i,t},
```
where $e_{i,t}$ expresses the [initial energy](battery:initial_energy) in battery $i$ at the beginning of time step $t$, and $pc_{i,t}$, $pd_{i,t}$ the power charge and discharge respectively. The charge and discharge power is limited by the [maximum charge power](battery:max_charge_power) $PC_i$
```{math}
:label: max-charge
0 \leq pc_{i,t} \leq PC_{i}
```
and the [maximum discharge power](battery:max_discharge_power) $PD_i$
```{math}
:label: max-discharge
0 \leq pd_{i,t} \leq PD_{i}.
```

## Connection
Batteries can be connected to [busbars](busbar) if used in combination with [](ac_line) and/or [](dc_line). If busbars are not created explicitly, a copper plate representation is used where all sources and loads are connected to the same point.

## MIP
The battery net-discharge curve is discontinuous. Therefore, batteries are typically modelled with separate charge and discharge variables. This is, in reality, a relaxation that allows simultaneous charging and discharging. This will typically not occur simply since it is not optimal to waste energy. However, in situations with surplus renewable generation or negative electricity price, the optimization might exploit this relaxation. This can be avoided by using mixed-integer programming and is activated with the [](battery:charge_discharge_mip) attribute. Given the binary variable $\delta_{i,t}$ which takes the value 0 when charging and 1 when discharging, the additional restrictions will ensure that charging and discharging does not occur simultaneously:
```{math}
:label: mip-charge
pc_{i,t} \leq PC_{i} (1-\delta_{i,t} )
```
```{math}
:label: mip-discharge
pd_{i,t} \leq PD_{i} \delta_{i,t}
```

## Schedules
Schedules for batteries can be given as both [charge](battery:charge_schedule), [discharge](battery:discharge_schedule), [net-discharge](battery:net_discharge_schedule), [minimum energy](battery:min_energy_constraint) and [maximum energy](battery:max_energy_constraint). Net-discharge represents discharge minus charge. All series can take the special value $nan$ to deactivate the schedule at a given time step.

## Operating costs
The battery can also have a fixed [charge cost](battery:charge_cost) and [discharge cost](battery:discharge_cost), typically associated with degradation. 

## Results
The results show the battery [charge](battery:power_charge), [discharge](battery:power_discharge) and [net discharge](battery:net_power_discharge). Net discharge will always represent the difference between discharge and charge. However, the charge and discharge are also useful quantities as they can reveal situations where simultaneous charging and discharging occurs. These situations will also issue a warning. The battery state-of-charge is shown in by the [energy](battery:energy) attribute, and the dual value of the energy balance is given by the [energy value](battery:energy_value).


## Examples
  - [](battery-wind-solar)
  - [](battery_reserves)
  - [](battery_rolling_max_discharge)
  

## References
  - Optimal microgrid operation considering battery degradation using stochastic dual dynamic programming {cite}`Aaslid2019b`
  - Non-linear charge-based battery storage optimization model with bi-variate cubic spline constraints {cite}`Aaslid2020`
  - Pricing electricity in constrained networks dominated by stochastic renewable generation and electric energy storage {cite}`Aaslid2021`
  - Stochastic Optimization of Microgrid Operation With Renewable Generation and Energy Storages {cite}`Aaslid2022a`
  - Stochastic operation of energy constrained microgrids considering battery degradation {cite}`Aaslid2022b`
  - Operating a Battery in a Hydropower-Dominated System to Balance Net Load Deviations {cite}`Naversen2019`
  

## Attributes
```{code-cell} ipython3
:tags: ['remove-input', 'full-width']

from IPython.core.display import HTML
HTML("""<table id="battery-table" class="display compact"></table>
<script>
enableDataTable(
  "#battery-table",
  "/_static/data/battery-attribute-table.json"
);
</script>
""")
```

(battery:charge_efficiency)=
### charge_efficiency
<div class="attribute-meta">
Data type: double. Unit: NO_UNIT.
</div>

Charge efficiency coefficient.


(battery:discharge_efficiency)=
### discharge_efficiency
<div class="attribute-meta">
Data type: double. Unit: NO_UNIT.
</div>

Discharge efficiency coefficient.


(battery:max_charge_power)=
### max_charge_power
<div class="attribute-meta">
Data type: double. Unit: MW.
</div>

Maximum charge power.


(battery:max_discharge_power)=
### max_discharge_power
<div class="attribute-meta">
Data type: double. Unit: MW.
</div>

Maximum discharge power.


(battery:max_energy)=
### max_energy
<div class="attribute-meta">
Data type: double. Unit: MWH.
</div>

Maximum energy storage capacity.


(battery:initial_energy)=
### initial_energy
<div class="attribute-meta">
Data type: double. Unit: MWH.
</div>

Initial energy storage.


(battery:discharge_cost)=
### discharge_cost
<div class="attribute-meta">
Data type: double. Unit: NOK/MWH.
</div>

Battery discharge cost.


(battery:charge_cost)=
### charge_cost
<div class="attribute-meta">
Data type: double. Unit: NOK/MWH.
</div>

Battery charge cost.


(battery:max_charge_power_constraint)=
### max_charge_power_constraint
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Maximum charge power constraint.


(battery:max_discharge_power_constraint)=
### max_discharge_power_constraint
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Maximum discharge power constraint.


(battery:max_energy_constraint)=
### max_energy_constraint
<div class="attribute-meta">
Data type: timeseries. Unit: MWH.
</div>

Maximum energy storage constraint.


(battery:min_energy_constraint)=
### min_energy_constraint
<div class="attribute-meta">
Data type: timeseries. Unit: MWH.
</div>

Minimum energy storage constraint.


(battery:charge_discharge_mip)=
### charge_discharge_mip
<div class="attribute-meta">
Data type: timeseries. Unit: NO_UNIT.
</div>

MIP variable to prevent simultaneous charging and discharging.


(battery:charge_schedule)=
### charge_schedule
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Battery charging schedule.


(battery:discharge_schedule)=
### discharge_schedule
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Battery discharging schedule.


(battery:net_discharge_schedule)=
### net_discharge_schedule
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Battery net-discharging schedule.


(battery:fcr_n_up_cost)=
### fcr_n_up_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating fcr_n_up on this battery.


(battery:fcr_n_down_cost)=
### fcr_n_down_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating fcr_n_down on this battery.


(battery:fcr_d_up_cost)=
### fcr_d_up_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating fcr_d_up on this battery.


(battery:fcr_d_down_cost)=
### fcr_d_down_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating fcr_d_down on this battery.


(battery:frr_up_cost)=
### frr_up_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating frr_up on this battery.


(battery:frr_down_cost)=
### frr_down_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating frr_down on this battery.


(battery:rr_up_cost)=
### rr_up_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating rr_up on this battery.


(battery:rr_down_cost)=
### rr_down_cost
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

The cost of allocating rr_down on this battery.


(battery:historical_discharge)=
### historical_discharge
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Battery discharge before the start of the optimization period.


(battery:rolling_max_discharge_limit)=
### rolling_max_discharge_limit
<div class="attribute-meta">
Data type: xyt. Unit: x: MINUTE and y: MWH.
</div>

Input to specify rolling battery discharge constraints that limit the maximum total energy discharged within a rolling time window. The x-values represent the length of the rolling time window and the y-values specify the maximum allowed energy discharged. Each (x, y) point in each xy curve represents a rolling ramping constraint.


(battery:power_charge)=
### power_charge
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Power charged from battery.


(battery:power_discharge)=
### power_discharge
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Power discharged from battery.


(battery:net_power_discharge)=
### net_power_discharge
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Net power discharged from battery.


(battery:energy)=
### energy
<div class="attribute-meta">
Data type: timeseries. Unit: MWH.
</div>

Energy stored in battery.


(battery:energy_value)=
### energy_value
<div class="attribute-meta">
Data type: timeseries. Unit: NOK/MWH.
</div>

Battery energy dual value.


(battery:fcr_n_up_delivery)=
### fcr_n_up_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery fcr_n_up delivery.


(battery:fcr_n_down_delivery)=
### fcr_n_down_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery fcr_n_down delivery.


(battery:fcr_d_up_delivery)=
### fcr_d_up_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery fcr_d_up delivery.


(battery:fcr_d_down_delivery)=
### fcr_d_down_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery fcr_d_down delivery.


(battery:frr_up_delivery)=
### frr_up_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery frr_up delivery.


(battery:frr_down_delivery)=
### frr_down_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery frr_down delivery.


(battery:rr_up_delivery)=
### rr_up_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery rr_up delivery.


(battery:rr_down_delivery)=
### rr_down_delivery
<div class="attribute-meta">
Data type: timeseries. Unit: MW.
</div>

Resulting battery rr_down delivery.