Effect of Tariff Policy and Battery Degradation on Optimal Energy Storage
2. Problem Statement and Model Formulation
2.1. Objective Function
- The battery cannot charge and discharge at the same timeThis restriction can be mathematically posted as
- For safety reasons, there is a maximum power that should not be exceeded neither during charge () nor during discharge (). Additionally, and will always be assumed as positive variables. Then:
- Energy balance: the state of charge at a certain time, , depends on the state of charge at the immediately previous period of time and the amount of energy that is effectively used in electrochemical reactions during charge or discharge processes:
- In order to preserve the battery life, the state of charge should neither be lower than a minimum fixed value () nor larger than a maximum (). Both these parameters correspond to different fractions of the total available capacity.
- Equations for battery degradation:Battery degradation kinetics are studied in the electrochemistry field by using a parameter known as the [16,17]. The is the inverse of the characteristic time that relates electric current during a time step, in units of A or mA, and the electrical charge capacity of the battery, in units of Ah or mAh.This coefficient allows a size-independent study of the kinetics of any reaction happening in batteries. The fraction of capacity lost by the battery can then be expressed as a function of the .To relate the from its definition (Equation (12)) with the variables used in this problem, a nearly constant working potential of the cell needs to be assumed. In practice, this assumption implies disregarding the effects of the overpotentials in the potential vs. current (E-I) curve. Then, and:Notice that as charge and discharge do not happen simultaneously (see Equations (7) and (15)) can be rewritten to combine both processes in a single equation:At this point, it is worth commenting that natural aging phenomena (just a function of time) has not been considered here as calendar aging is not dependent on the rate of battery use.
2.3. Derivation of an Equivalent Convex Problem
2.3.1. Non-Simultaneous Charge and Discharge
“For energy storage capacity at bus where (the energy storage capacity) , if (the locational marginal price) is strictly positive then (...) simultaneous charging and discharging will not occur”.
2.3.2. Battery Degradation
2.3.3. Problem Statement in Convex Form
3. Optimal Scheduling for a 24 h Period
- The “simple tariff” refers to a pricing policy that only has two price steps: cheap (off-peak) and expensive (on-peak), and the daily price pattern is repeated throughout the year. Therefore, there are always some consecutive hours with exactly the same price.
- The “complex tariff” refers to a pricing policy where there are several price steps during a single day (the price is still constant for each hour of the day), different days present different prices (weekdays are priced differently than weekends and holidays and there is also seasonal variation), and prices are dependent on the expected weather.
3.1. Results and Discussion: Simple tariff
3.2. Results and Discussion: Complex Tariff
4. Optimal Scheduling for Long-Term Periods
4.1. Modifications to the Original Problem
4.1.1. Variable Capacity
4.1.2. Restricted Time Span between Charge and Discharge Periods
- If energy is stored in batteries for a long period of time, self-discharge processes occur. For the sake of simplicity our model has not included these types of processes under the assumption that the charge and discharge cycle of the battery happens in a reasonably short period, most probably during the same day.
- Complex tariffs depend on weather conditions and thus a reliable forecast. In this work we have assumed that reasonable forecasts are given for periods no longer than a week.
4.2. Results and Discussion: Simple Tariff
4.3. Results and Discussion: Complex Tariff
Conflicts of Interest
|optimization total period of time|
|battery charging efficiency|
|battery discharging efficiency|
|energy storage installed capacity (energy units)|
|initial energy storage installed capacity (electrical charge units)|
|energy storage capacity at t (energy units)|
|dimensionless charge/discharge rate|
|market cost of battery capacity|
|terminal potential difference at time t|
|open circuit potential difference at time t|
|net current at time t|
|maximum safe charging power|
|charge power at time t|
|maximum safe discharging power|
|discharge power at time t|
|state of charge at time t|
|Minimum required state of charge|
|Maximum allowed state of charge|
|fraction of capacity loss during the time step at t|
|Energy grid price at time t|
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|Annual savings (USD)||305||286||269||252||237||222||208||196||184||172|
|(USD/kWh)||Year 1||Year 2||Year 3||Year 4||Year 5|
|Electricity annual savings (USD/year)|
|Annual capacity loss (%)|
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Corengia, M.; Torres, A.I. Effect of Tariff Policy and Battery Degradation on Optimal Energy Storage. Processes 2018, 6, 204. https://doi.org/10.3390/pr6100204
Corengia M, Torres AI. Effect of Tariff Policy and Battery Degradation on Optimal Energy Storage. Processes. 2018; 6(10):204. https://doi.org/10.3390/pr6100204Chicago/Turabian Style
Corengia, Mariana, and Ana I. Torres. 2018. "Effect of Tariff Policy and Battery Degradation on Optimal Energy Storage" Processes 6, no. 10: 204. https://doi.org/10.3390/pr6100204