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| .. _device_scheduler: | ||
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| Device scheduler | ||
| =========== | ||
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| Introduction | ||
| -------------- | ||
| This generic 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. | ||
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| 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. | ||
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| The solver minimises the costs of deviating from the commitments. | ||
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| Notation | ||
| --------- | ||
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| Indexes | ||
| ^^^^^^^^ | ||
| ================================ ================================================ ============================================================================================================== | ||
| Symbol Variable in the Code Description | ||
| ================================ ================================================ ============================================================================================================== | ||
| :math:`c` c Commitments, for example, day-ahead or intra-day market commitments. | ||
| :math:`d` d Devices, for example, a battery or a load. | ||
| :math:`j` j 0-indexed time dimension. | ||
| ================================ ================================================ ============================================================================================================== | ||
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| .. note:: | ||
| The time index :math:`j` has two interpretations: a time period or an instantaneous moment at the end of time period :math:`j`. | ||
| For example, :math:`j` in flow constraints correspond to time periods, whereas :math:`j` used in a stock constraint refers to the end of time period :math:`j`. | ||
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| Parameters | ||
| ^^^^^^^^^^ | ||
| ================================ ================================================ ============================================================================================================== | ||
| Symbol Variable in the Code Description | ||
| ================================ ================================================ ============================================================================================================== | ||
| :math:`Price_{up}(c,j)` up_price Price of incurring an upwards deviations in commitment :math:`c` during time period :math:`j`. | ||
| :math:`Price_{down}(c,j)` down_price Price of incurring a downwards deviations in commitment :math:`c` during time period :math:`j`. | ||
| :math:`\eta_{up}(d,j)` device_derivative_up_efficiency Upwards conversion efficiency. | ||
| :math:`\eta_{down}(d,j)` device_derivative_down_efficiency Downwards conversion efficiency. | ||
| :math:`Stock_{min}(d,j)` device_min Minimum quantity for the stock of device :math:`d` at the end of time period :math:`j`. | ||
| :math:`Stock_{max}(d,j)` device_max Maximum quantity for the stock of device :math:`d` at the end of time period :math:`j`. | ||
| :math:`\epsilon(d,j)` efficiencies Stock energy losses. | ||
| :math:`P_{max}(d,j)` device_derivative_max Maximum flow of device :math:`d` during time period :math:`j`. | ||
| :math:`P_{min}(d,j)` device_derivative_min Minimum flow of device :math:`d` during time period :math:`j`. | ||
| :math:`P^{ems}_{min}(j)` ems_derivative_min Minimum flow of the EMS during time period :math:`j`. | ||
| :math:`P^{ems}_{max}(j)` ems_derivative_max Maximum flow of the EMS during time period :math:`j`. | ||
| :math:`Commitment(c,j)` commitment_quantity Commitment c (at EMS level) over time step :math:`j`. | ||
| ================================ ================================================ ============================================================================================================== | ||
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| Variables | ||
| ^^^^^^^^^ | ||
| ================================ ================================================ ============================================================================================================== | ||
| Symbol Variable in the Code Description | ||
| ================================ ================================================ ============================================================================================================== | ||
| :math:`\Delta_{up}(c,j)` commitment_upwards_deviation Upwards deviation from the power commitment :math:`c` of the EMS during time period :math:`j`. | ||
| :math:`\Delta_{down}(c,j)` commitment_downwards_deviation Downwards deviation from the power commitment :math:`c` of the EMS during time period :math:`j`. | ||
| :math:`\Delta Stock(d,j)` n/a Change of stock of device :math:`d` at the end of time period :math:`j`. | ||
| :math:`P_{up}(d,j)` device_power_up Upwards power of device :math:`d` during time period :math:`j`. | ||
| :math:`P_{down}(d,j)` device_power_down Downwards power of device :math:`d` during time period :math:`j`. | ||
| :math:`P^{ems}(j)` ems_power Aggregated power of all the devices during time period :math:`j`. | ||
| ================================ ================================================ ============================================================================================================== | ||
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| Cost function | ||
| -------------- | ||
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| The cost function quantifies the total cost of upwards and downwards deviations from the different commitments. | ||
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| .. math:: | ||
| :name: cost_function | ||
| \min [\sum_{c,j} \Delta _{up}(c,j) \cdot Price_{up}(c,j) + \Delta_{down}(c,j) \cdot Price_{down}(c,j)] | ||
| State dynamics | ||
| --------------- | ||
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| To simplify the description of the model, the auxiliary variable :math:`\Delta Stock(d,j)` is introduced in the documentation. It represents the | ||
| change of :math:`Stock(d,j)`, taking into account conversion efficiencies but not considering the storage losses. | ||
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| .. math:: | ||
| :name: stock | ||
| \Delta Stock(d,j) = \frac{P_{down}(d,j)}{\eta_{down}(d,j) } + P_{up}(d,j) \cdot \eta_{up}(d,j) | ||
| .. math:: | ||
| :name: device_bounds | ||
| Stock_{min}(d,j) \leq Stock(d,j) - Stock(d,-1)\leq Stock_{max}(d,j) | ||
| Perfect efficiency | ||
| ^^^^^^^^^^^^^^^^^^^ | ||
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| .. math:: | ||
| :name: efficiency_e1 | ||
| Stock(d, j) = Stock(d, j-1) + \Delta Stock(d,j) | ||
| 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) | ||
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| .. math:: | ||
| :name: efficiency_left | ||
| Stock(d, j) = (Stock(d, j-1) + \Delta Stock(d,j)) \cdot \epsilon(d,j) | ||
| 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) | ||
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| .. math:: | ||
| :name: efficiency_right | ||
| Stock(d, j) = Stock(d, j-1) \cdot \epsilon(d,j) + \Delta Stock(d,j) | ||
| Linear efficiency | ||
| ^^^^^^^^^^^^^^^^^ | ||
| Assume the change happens at a constant rate, leading to a linear stock change, and exponential decay, within the current interval | ||
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| .. math:: | ||
| :name: efficiency_linear | ||
| Stock(d, j) = Stock(d, j-1) \cdot \epsilon(d,j) + \Delta Stock(d,j) \cdot \frac{\epsilon(d,j) - 1}{log(\epsilon(d,j))} | ||
| Constraints | ||
| -------------- | ||
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| Device bounds | ||
| ^^^^^^^^^^^^^ | ||
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| .. math:: | ||
| :name: device_derivative_bounds | ||
| P_{min}(d,j) \leq P_{up}(d,j) + P_{down}(d,j)\leq P_{max}(d,j) | ||
| .. math:: | ||
| :name: device_down_derivative_bounds | ||
| min(P_{min}(d,j),0) \leq P_{down}(d,j)\leq 0 | ||
| .. math:: | ||
| :name: device_up_derivative_bounds | ||
| 0 \leq P_{up}(d,j)\leq max(P_{max}(d,j),0) | ||
| Grid constraints | ||
| ^^^^^^^^^^^^^^^^^ | ||
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| .. math:: | ||
| :name: device_derivative_equalities | ||
| P^{ems}(d,j) = P_{up}(d,j) + P_{down}(d,j) | ||
| .. math:: | ||
| :name: ems_derivative_bounds | ||
| P^{ems}_{min}(j) \leq \sum_d P^{ems}(d,j) \leq P^{ems}_{max}(j) | ||
| Power coupling constraints | ||
| ^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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| .. math:: | ||
| :name: ems_flow_commitment_equalities | ||
| \sum_d P^{ems}(d,j) = \sum_c Commitment(c,j) + \Delta {up}(c,j) + \Delta {down}(c,j) | ||
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