Storage device scheduler: Linear model
Introduction
This generic storage device scheduler is able to handle an EMS with multiple devices, with various types of constraints on the EMS level and on the device level, and with multiple market commitments on the EMS level.
A typical example is a house with many devices. The commitments are assumed to be with regard to the flow of energy to the device (positive for consumption, negative for production). In practice, this generic scheduler is used in the StorageScheduler to schedule a storage device.
The solver minimizes the costs of deviating from the commitments.
Notation
Indexes
Symbol |
Variable in the Code |
Description |
|---|---|---|
\(c\) |
c |
Commitments, for example, day-ahead or intra-day market commitments. |
\(d\) |
d |
Devices, for example, a battery or a load. |
\(j\) |
j |
0-indexed time dimension. |
Note
The time index \(j\) has two interpretations: a time period or an instantaneous moment at the end of time period \(j\). For example, \(j\) in flow constraints correspond to time periods, whereas \(j\) used in a stock constraint refers to the end of time period \(j\).
Parameters
Symbol |
Variable in the Code |
Description |
|---|---|---|
\(Price_{up}(c,j)\) |
up_price |
Price of incurring an upwards deviations in commitment \(c\) during time period \(j\). |
\(Price_{down}(c,j)\) |
down_price |
Price of incurring a downwards deviations in commitment \(c\) during time period \(j\). |
\(\eta_{up}(d,j)\) |
device_derivative_up_efficiency |
Upwards conversion efficiency. |
\(\eta_{down}(d,j)\) |
device_derivative_down_efficiency |
Downwards conversion efficiency. |
\(Stock_{min}(d,j)\) |
device_min |
Minimum quantity for the stock of device \(d\) at the end of time period \(j\). |
\(Stock_{max}(d,j)\) |
device_max |
Maximum quantity for the stock of device \(d\) at the end of time period \(j\). |
\(\epsilon(d,j)\) |
efficiencies |
Stock energy losses. |
\(P_{max}(d,j)\) |
device_derivative_max |
Maximum flow of device \(d\) during time period \(j\). |
\(P_{min}(d,j)\) |
device_derivative_min |
Minimum flow of device \(d\) during time period \(j\). |
\(P^{ems}_{min}(j)\) |
ems_derivative_min |
Minimum flow of the EMS during time period \(j\). |
\(P^{ems}_{max}(j)\) |
ems_derivative_max |
Maximum flow of the EMS during time period \(j\). |
\(Commitment(c,j)\) |
commitment_quantity |
Commitment c (at EMS level) over time step \(j\). |
\(M\) |
M |
Large constant number, upper bound of \(Power_{up}(d,j)\) and \(|Power_{down}(d,j)|\). |
\(D(d,j)\) |
stock_delta |
Explicit energy gain or loss of device \(d\) during time period \(j\). |
Variables
Symbol |
Variable in the Code |
Description |
|---|---|---|
\(\Delta_{up}(c,j)\) |
commitment_upwards_deviation |
Upwards deviation from the power commitment \(c\) of the EMS during time period \(j\). |
\(\Delta_{down}(c,j)\) |
commitment_downwards_deviation |
Downwards deviation from the power commitment \(c\) of the EMS during time period \(j\). |
\(\Delta Stock(d,j)\) |
n/a |
Change of stock of device \(d\) at the end of time period \(j\). |
\(P_{up}(d,j)\) |
device_power_up |
Upwards power of device \(d\) during time period \(j\). |
\(P_{down}(d,j)\) |
device_power_down |
Downwards power of device \(d\) during time period \(j\). |
\(P^{ems}(j)\) |
ems_power |
Aggregated power of all the devices during time period \(j\). |
\(\sigma(d,j)\) |
device_power_sign |
Upwards power activation if \(\sigma(d,j)=1\), downwards power activation otherwise. |
Cost function
The cost function quantifies the total cost of upwards and downwards deviations from the different commitments.
State dynamics
To simplify the description of the model, the auxiliary variable \(\Delta Stock(d,j)\) is introduced in the documentation. It represents the change of \(Stock(d,j)\), taking into account conversion efficiencies but not considering the storage losses.
Perfect efficiency
Left efficiency
First apply the stock change, then apply the losses (i.e. the stock changes on the left side of the time interval in which the losses apply)
Right efficiency
First apply the losses, then apply the stock change (i.e. the stock changes on the right side of the time interval in which the losses apply)
Linear efficiency
Assume the change happens at a constant rate, leading to a linear stock change, and exponential decay, within the current interval
Constraints
Device bounds
Upwards/Downwards activation selection
Avoid simultaneous upwards and downwards activation during the same time period.