Battery simulation

Background

Batteries have a huge impact on profitability and trading for solar assets. Our simulator employs a battery dispatch optimization based on heuristics which allows us to quickly simulate battery operation over the lifetime of the plant. This scheduling algorithm is different from the optimization used during operations (see battery optimization), since running an analytical optimization model would be impractical over decades of operation. Because of this, for long-term simulation we use a simplified battery scheduler which is based on heuristics: simplified assumptions which are true almost all the time, and therefore give a good approximation of what the analytical model would find. For example, an approximation would be that the battery is fully charged and discharged once a day. This is not always true, but it's a good enough approximation that it gives almost the same results as the analytical approach.

The following heuristics model is based on DiOrio et al., 2020 (DOI:10.1016/j.apenergy.2019.114465)

Calculation logic

Nomenclature

Note: all measures are referring to a single slot.

Asset characteristics

$M_{nom}$ : meter exchange rating (kWh*)
$S_{max}$ : maximum solar inverter (kWh)
$B_{max}$ : maximum battery exchange (kWh)
$E_{max}$ : maximum battery energy (kWh)
$V_{nom}$ : nominal voltage (V)
$\gamma$ : yearly degradation (unitless)
$C_{cycle}$ : battery cycle cost (yen/kWh)
$\eta_{charge},\eta_{discharge}$ : charging and discharging efficiency (unitless)
$\text{SoC}_{min},\text{SoC}_{max},\text{SoC}_{start}$ : minimum, maximum and initial state of charge (kWh)

Input values

$p(t)$ : price forecast at time $t$ (yen/kWh)
$g(t)$ : solar generation forecast at time $t$ (kWh)
$c(t)$ : forecast of curtailment level ($\in [0,1]$) at time $t$ (unitless)
$m_{max}(t)$ : maximum meter exchange at time $t$ (kWh)
$h$ : horizon (number of slots) (unitless)

Battery status

$q_{max}(t)$ : maximum charge at time $t$ (C)
$q(t)$ : charge at time $t$ (C)

Decision variables

$e_{fill}(t)$ : energy capacity left in the battery (kWh)
$e_{clipped}(t)$ : reserved energy for clipping (kWh)
$e_{charge}(t)$ : effective energy capacity left in the battery, reserving space for clipping (kWh)
$p_{clipped}(t)$ : power that would be clipped without battery (kWh)
$p_{discharge}(t)$ : power to discharge (kWh)
$p_{clipcharge}(t)$ : power to charge from clipped solar (kWh)
$p_{pvcharge}(t)$ : power to charge from non-clipped solar (kWh)

Binary decision variables

$b_{discharge}(t)$ : discharge decision
$b_{highvaluedischarge}(t)$ : high value discharge time
$b_{clipcharge}(t)$ : time to charge clipped solar power
$b_{pvcharge}(t)$ : decision to charge from solar

Set points

$p_{target}(t)$ : preliminary power target (kWh)
$p_{effective}(t)$ : effective power target after constraints (kWh)

Output values

$m(t)$ : meter exchange at time $t$ (kWh)
$f_{s2g}(t)$ : energy flow solar to grid (kWh)
$f_{b2g}(t)$ : energy flow battery to grid (kWh)
$f_{s2b}(t)$ : energy flow solar to battery (kWh)

Initial settings

Our battery scheduling optimization is based on a heuristics that allows for extremely fast simulation of assets over decades. This heuristics simulates the battery in a simplified way as a ...

Throughout the page, we use square brackets as Iverson bracket notation** for clarity.

First, we define the maximum charge capacity of the battery as:

$$q_{max}(0) = \frac{E_{max}}{V_{nom}}$$

We define the initial charge of the battery as:

$$q(0) = q_{max}(0) \cdot \text{SoC}_{start}$$

We find the degradation per 30-minute slot as:

$$\gamma_{slot} = (1 - \gamma)^{1 / (365 \cdot 48)}$$

At each time step, the actual meter exchange limits depends on current curtailment:

$$m_{max}(t) = (1-c(t)) \cdot M_{nom}$$

Determining decision variables

Our scheduling system depends on decision variables based on the scheduling horizon $h$. During the horizon, the maximum price at which electricity can be sold to the grid can be defined as:

$$c_{sellmax} (t) = \max_{i=t}^{t+h}(p(t))$$

We can calculate the relative (approximate) revenue streams for each use of the battery. At time $t$, the revenues that can be obtained by charging from the PV depends on the maximum price at which this energy can be sold minus the missed opportunity of selling the energy right now:

$$r_{pvcharge}(t) = c_{sellmax}(t) \cdot \eta_{discharge} - \frac{p(t)}{\eta_{charge}} - C_{cycle}$$

The approximate revenues from charging when the energy would be clipped do not have any oppportunity cost, as the energy would be lost anyway:

$$r_{clipcharge}(t) = c_{sellmax}(t) \cdot \eta_{discharge} - C_{cycle}$$

The revenue from discharging right now depends on the current price on the market:

$$r_{discharge}(t) = p(t) \cdot \eta_{discharge} - C_{cycle}$$

We also calculate binary decision variables, which are all either 0 or 1, and are denoted by $b$. If prices are at a minimum at the current time step (compared to the horizon), then this is a good moment to charge the battery:

$$b_{highvaluecharge}(t) = [p(t) = \min_{i=t}^{t+h}(p(i))]$$

If on the other hand prices are at a maximum at the current time step (compared to the horizon), then this is a good moment to discharge the battery:

$$b_{highvaluedischarge}(t) = [p(t) = \max_{i=t}^{t+h}(p(i))]$$

The decision on whether to charge from PV can then depend on whether all the three conditions are true: prices are the lowest, revenues from PV charging is positive, and solar generation is positive:

$$b_{pvcharge}(t) = b_{highvaluecharge}(t) \cdot [r_{pvcharge}(t) > 0] \cdot [g(t) > 0]$$

The decision of whether to charge from clipped power depends only on whether the revenues are positive:

$$b_{clipcharge} = [r_{clipcharge} > 0]$$

We can then determine whether we should discharge the battery at time $t$:

$$b_{discharge}(t) = b_{highvaluedischarge}(t) \cdot [\text{SoC}(t) > \text{SoC}_{min}] \cdot [m_{max}(t) > g(t)] \cdot [r_{discharge} > 0]$$

This depends on whether this is a moment of good prices, if there is enough SoC, if the meter exchange limit is higher than the solar generation, and if the revenues from discharging are positive.

Heuristics optimization

In order to optimize the battery, we need to look at all decision variables and determine battery set points depending on their values.

Reserved energy

For each time step $t$, we can first calculate the amount of reserved energy in the battery for various tasks, which set limits for battery utilization. First, we want to reserve some battery capacity for absorbing energy that would otherwise be clipped (for example by curtailment events). We can calculate the amount of power that would be clipped without battery as:

$$p_{clipped}(t) = \max(0, g(t) - m_{max}(t))$$

Ideally, we would want to absorb all this clipped energy, which otherwise would be completely wasted. We can calculate the reserved energy at each point in time that we need to keep in the battery to absorb the clipped energy as:

$$e_{clipped} = \sum_{i=t}^{t+h}(p_{clipped})$$

The energy that we can charge at each moment of time in the battery is defined as:

$$e_{fill}(t) = V_{nom} * (q_{max}(t) * \text{SoC}_{max} - q(t))$$

and therefore the energy that we can charge at time $t$ while still reserving capacity for clipped energy later:

$$e_{charge}(t) = \max(0, e_{fill}(t) - e_{clipped}(t))$$

Preliminary power set points

Next, we can determine the amount of power (set points) for the battery. Note that this will be preliminary values, that may result in unfeasible solutions. These constraints will be resolved later.
The discharge power is only activated if the $b_{discharge}$ is active, and is constrained by the battery power capacity and the remaining meter capacity after solar (constrains related to SoC will be resolved later):

$$p_{discharge}(t) = b_{discharge}(t) \cdot \min(B_{max},m_{max}(t) - g(t))$$

If $b_{discharge}$ is not active (=0), we can evaluate charging options. If there is clipped power, use that to charge:

$$p_{clipcharge}(t) = b_{clipcharge}(t) \cdot p_{clipped}(t)$$

If it is good to charge from solar, use the remaining charge for solar:

$$p'_{pvcharge}(t) = \begin{cases} \min(\max(e_{charge}(t), p_{clipped})(t),g(t) - p_{clipped}(t)), & \text{if } b_{clipcharge}(t)\\ g(t), & \text{otherwise} \end{cases}$$ $$p_{pvcharge}(t) = b_{pvcharge}(t) \cdot p'_{pvcharge}(t)$$

We can finally calculate the target power from all contributions:

$$p_{target}(t) = p_{discharge}(t) \cdot b_{discharge}(t) - (p_{pvcharge}(t) \cdot b_{pvcharge}(t) + p_{clipcharge}(t) \cdot b_{clipcharge}(t))$$

Based on the target power, we can update battery state. In particular, we can find the hypothetical (before other constraints) current at time $t$ as:

$$current(t) = p_{target}(t) / V_{nom}$$

and from this we can calculate the new charge levels of the battery, constrained by the SoC limits of the battery:

$$q(t+1) = \max \Big( \min \big( q(t) - current(t), q_{max}(t) \cdot \text{SoC}_{max} \big) , q_{max}(t) \cdot \text{SoC}_{min} \Big)$$

Based on the new charge levels, we can calculate the effective power exchanged, which would be the actual battery set point at time t:

$$p_{effective}(t) = (q(t) - q(t+1)) \cdot V_{nom}$$

Power flows

Finally, we can calculate the results as the various flows between battery, solar, and grid:

$$f_{b2g}(t) = \max(0, p_{effective}(t))$$ $$f_{s2b}(t) = -\min(0,p_{effective}(t))$$ $$f_{s2g}(t) = \min(g(t) - f_{s2b}(t), m_{max}(t))$$

We can then determine the level of curtailment avoided by the battery as:

$$c_{avoided}(t) = \min(f_{s2b}(t), c_{virtual}(t))$$ $$c_{virtual}(t) = \max(0, g(t) - m_{max}(t))$$

with $c_{virtual}(t)$ the virtual curtailment that the plant would have incurred without battery.

Based on the battery degradation defined above, we then update the battery capacity as:

$$q_{max}(t+1) = q_{max}(t) \cdot \gamma_{slot}$$

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* This refers to kWh per slot, i.e., the actual maximum energy that can be exchanged during a 30-minute slot. This is equivalent to the kW rating divided by two. For example, a 1 kW plant can produce 0.5 kWh in a 30-minute interval. We use the same convention for other variables.

** The Iverson bracket notation is defined as:

$$[P] = \begin{cases} 1, & \text{if $P$ is true} \\ 0, & \text{otherwise} \end{cases}$$

Validation

We validate our heuristics optimization model with the analytical one, to understand what is the expected error that the simplifying assumptions introduce.