# Effect of Tariff Policy and Battery Degradation on Optimal Energy Storage

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Model Formulation

#### 2.1. Objective Function

#### 2.2. Constraints

- The battery cannot charge and discharge at the same timeThis restriction can be mathematically posted as$${p}_{t}^{D}{p}_{t}^{C}=0$$
- For safety reasons, there is a maximum power that should not be exceeded neither during charge (${p}^{C-MAX}$) nor during discharge (${p}^{D-MAX}$). Additionally, ${p}_{t}^{C}$ and ${p}_{t}^{D}$ will always be assumed as positive variables. Then:$$\begin{array}{c}0\le {p}_{t}^{C}\le {p}^{C-MAX}\hfill \end{array}$$$$\begin{array}{c}0\le {p}_{t}^{D}\le {p}^{D-MAX}\hfill \end{array}$$
- Energy balance: the state of charge at a certain time, $so{c}_{t}$, depends on the state of charge at the immediately previous period of time $so{c}_{t-1}$ and the amount of energy that is effectively used in electrochemical reactions during charge or discharge processes:$$so{c}_{t}=so{c}_{t-1}+\Delta t{p}_{t}^{C}{\eta}^{+}-\Delta t{p}_{t}^{D}/{\eta}^{-}$$
- In order to preserve the battery life, the state of charge should neither be lower than a minimum fixed value ($SO{C}^{MIN}$) nor larger than a maximum ($SO{C}^{MAX}$). Both these parameters correspond to different fractions of the total available capacity.$$SO{C}^{MIN}\le so{c}_{t}\le SO{C}^{MAX}$$
- Equations for battery degradation:Battery degradation kinetics are studied in the electrochemistry field by using a parameter known as the ${C}_{rate}$ [16,17]. The ${C}_{rate}$ is the inverse of the characteristic time that relates electric current during a time step, ${I}_{t}$ in units of A or mA, and the electrical charge capacity of the battery, $B{C}_{Q}^{ES}$ in units of Ah or mAh.$${C}_{rate,t}=\frac{{I}_{t}}{B{C}_{Q}^{ES}}$$This coefficient allows a size-independent study of the kinetics of any reaction happening in batteries. The fraction of capacity lost by the battery ${x}_{t}^{CL}$ can then be expressed as a function of the ${C}_{rate}$.$${x}_{t}^{CL}=f\left({C}_{rate,t}\right)$$$${x}_{t}^{CL}={\alpha}_{1}{{C}_{rate,t}}^{2}+{\alpha}_{2}{C}_{rate,t}$$To relate the ${C}_{rate}$ 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, ${E}_{t}={E}_{t}^{0}$ and:$$\begin{array}{c}{C}_{rate,t}=\frac{{I}_{t}{E}_{t}}{B{C}_{Q}^{ES}{E}_{t}^{0}}\\ {C}_{rate,t}=\frac{{p}_{t}}{B{C}^{ES}}\end{array}$$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:$${C}_{rate,t}=\frac{{p}_{t}^{C}+{p}_{t}^{D}}{B{C}^{ES}}$$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 $n\in \mathcal{N}$ where (the energy storage capacity) ${C}_{n}>0$, if (the locational marginal price) ${\lambda}_{n}\left(t\right)$ 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.

^{−5}and ${\alpha}_{2}$ = 1.44 × 10

^{−4}(see S1 in Supplementary Information). Estimated 2018 battery prices range from 580–750 USD/kWh for commercial and residential use [28]. However, as Li-ion battery systems still have margin to lower their production costs [29], battery prices used in the simulations were selected case by case to show key features of each case study.

#### 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

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\tau $ | optimization total period of time |

${\eta}^{+}$ | battery charging efficiency |

${\eta}^{-}$ | battery discharging efficiency |

$\Delta t$ | time step |

$B{C}^{ES}$ | energy storage installed capacity (energy units) |

$B{C}_{Q}^{ES}$ | initial energy storage installed capacity (electrical charge units) |

$B{C}_{t}$ | energy storage capacity at t (energy units) |

${C}_{rate}$ | dimensionless charge/discharge rate |

${C}^{ES}$ | market cost of battery capacity |

${E}_{t}$ | terminal potential difference at time t |

${E}_{t}^{0}$ | open circuit potential difference at time t |

${I}_{t}$ | net current at time t |

${p}^{C-MAX}$ | maximum safe charging power |

${p}_{t}^{C}$ | charge power at time t |

${p}^{D-MAX}$ | maximum safe discharging power |

${p}_{t}^{D}$ | discharge power at time t |

r | discount rate |

$so{c}_{t}$ | state of charge at time t |

$SO{C}^{MIN}$ | Minimum required state of charge |

$SO{C}^{MAX}$ | Maximum allowed state of charge |

${x}_{t}^{CL}$ | fraction of capacity loss during the time step at t |

${\$}_{t}$ | Energy grid price at time t |

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**Figure 1.**Schematic representation of energy fluxes from the grid to the load. The load can consume energy directly from the grid or from a previously charged battery. ${p}_{t}$ represents the power at time t (superscripts C and D denote charge and discharge processes); ${\eta}^{+},{\eta}^{-}$ represent battery efficiencies.

**Figure 2.**Optimal operation for a “simple tariff” example during a 24 h period. (

**a**) Electricity TOU tariff for Uruguay. (

**b**) Optimal power consumed (${p}_{t}^{C}$, represented as negative for visual purposes) and supplied (${p}_{t}^{D}$) by the battery. (

**c**) Battery State of Charge at each time ($so{c}_{t}$). (

**d**) Fraction of capacity lost by the battery at each time (${x}_{t}^{CL}$). Accumulated fraction loss in one day is 1.7 × 10

^{−4}for 300 and 400 USD/kWh.

**Figure 3.**Example of “complex” tariff with different day types and hourly changes. (

**a**) Days presenting a large price variation. (

**b**) Days presenting a moderate variation.

**Figure 4.**Optimal operation for a “complex tariff” example during a 24 h period, for the days that present a large price variation. ${p}_{t}^{C}$: optimal power consumed from the grid (represented as negative for visual purposes) and ${p}_{t}^{D}$: optimal power supplied by the battery (positive). (

**a**) Extremely Hot Summer Weekday. (

**b**) Very Hot Summer Weekday. (

**c**) Hot Summer Weekday. (

**d**) High cost Winter Weekday.

**Figure 5.**Optimal operation for a “simple tariff” example during a 10 years period. (

**a**) ${p}_{t}^{C}$ (negative) and ${p}_{t}^{D}$ (positive). Left: First day. Right: Last day. (

**b**) Remaining capacity $B{C}_{d}$.

**Figure 6.**2017 temperature (*) vs. Average temperature (line) in Los Angeles, CA [30]. Each background color implies a different daily tariff. Only seasonal and temperature effects on tariff are considered.

**Figure 7.**Optimal operation for a “complex tariff” example during a year period and a battery price of 300 USD/kWh.

**Figure 8.**Optimal operation for a “complex tariff” example during a five years period and a battery price of 250 USD/kWh. Notice how battery operation is concentrated towards the end of the simulation period.

Operation Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Annual savings (USD) | 305 | 286 | 269 | 252 | 237 | 222 | 208 | 196 | 184 | 172 |

${\mathit{C}}^{\mathit{ES}}$ (USD/kWh) | ${\mathit{NPV}}_{\mathit{r}=8\%}$ (USD) | ${\mathit{NPV}}_{\mathit{r}=10\%}$ (USD) | ${\mathit{NPV}}_{\mathit{r}=12\%}$ (USD) |
---|---|---|---|

400 | −2374 | −2497 | −2606 |

300 | −1374 | −1497 | −1606 |

200 | −374 | −497 | −606 |

150 | 126 | 3 | −106 |

100 | 626 | 503 | 394 |

**Table 3.**Electricity annual savings and annual capacity loss for each operation year with complex tariff.

${\mathit{C}}^{\mathit{ES}}$ (USD/kWh) | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

Electricity annual savings (USD/year) | |||||

200 | 223 | 230 | 230 | 226 | 223 |

250 | 223 | 220 | 218 | 221 | 226 |

300 | 223 | 220 | 218 | 216 | 214 |

Annual capacity loss (%) | |||||

200 | 0.96 | 1.29 | 1.37 | 1.35 | 1.34 |

250 | 0.94 | 0.93 | 0.93 | 1.09 | 1.34 |

300 | 0.94 | 0.93 | 0.92 | 0.92 | 0.91 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

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

**AMA Style**

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/pr6100204

**Chicago/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