# Effect of Battery Degradation on the Probabilistic Optimal Operation of Renewable-Based Microgrids

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## Abstract

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## 1. Introduction

- Meta-heuristic algorithms deal simultaneously with a set of feasible solutions; this allows different solutions to be found in the Pareto optimal front in just one execution of the algorithm. While in the mathematical programming approaches, a sequence of independent executions should be dealt with.
- Meta-heuristic algorithms are not sensitive to the continuity and formation of the Pareto front, which is one of the drawbacks of mathematical programming.

- The power constraints of the storage device (i.e., Li-ion battery in this study) as well as the degradation cost are considered in the MG’s MOOM problem. In order to consider different battery characteristics, including the battery efficiency, the battery’s initial charge, SOC, and different scenarios of the battery’s degradation cost are studied in the probabilistic MG’s MOOM problem.
- Modifications are added to the JAYA algorithm that make it more efficient in dealing with multi-objective problems. The efficiency of the suggested algorithm is examined by comparing its performance with some other well-known algorithms.
- The total cost of day-ahead market transactions and fuel costs, along with the emission of MG, are minimized through the introduced optimal scheduling approach. The suggested RUT-EMOJAYA reduces the MG’s dependency on the main grid and the electricity market, while maximizing the utilization of RESs in the studied region.
- The uncertainties related to the forecasted values of the load demand and market price, and the available outputs of RESs, as well as their correlations, are considered and dealt with efficiently using the suggested RUT-EMOJAYA.

## 2. Economic Model of Battery Storage Devices

_{t}is the energy stored in the battery at a specific hour of the day. The cycle life of the battery is as follows:

## 3. Problem Formulation

#### 3.1. Objective Functions

^{t}[5]:

_{2}, SO

_{2}and NO

_{x}emissions from the ith DG source; $C{O}_{{2}_{Batt,s}}^{t}$, $S{O}_{{2}_{Batt,s}}^{t}$ and $N{O}_{{x}_{Grid}}^{t}$ are the amounts of CO

_{2}, SO

_{2}and NO

_{x}emission from the sth battery unit and $C{O}_{{2}_{Grid}}^{t}$, $S{O}_{{2}_{Grid}}^{t}$ and $N{O}_{{x}_{Grid}}^{t}$ are the amounts of CO

_{2}, SO

_{2}and NO

_{x}emission from the utility at hour t, respectively [5].

#### 3.2. Constraints

#### 3.2.1. Power Balance Constraint

#### 3.2.2. Battery Limits

#### 3.2.3. Real Power Constraints

## 4. Reduced Unscented Transformation (RUT)

_{0}≤ 1:

_{α,β}is the correlation coefficient between the αth and βth elements of the covariance matrix P

_{zz}

_{.}

## 5. Enhanced Multi-Objective JAYA Algorithm

#### 5.1. A Brief Overview of the Original JAYA

_{1}and r

_{2}are random variables in [0, 1], ${\overrightarrow{X}}_{best}$ is the best solution in each iteration, while ${\overrightarrow{X}}_{worst}$ is the worst solution in the population; ${\overrightarrow{X}}_{i,iter}^{new}$ will be accepted if its objective function value is better than that of ${\overrightarrow{X}}_{i,iter}^{}$.

#### 5.2. Multi-Objective JAYA (MOJAYA)

_{2}are the cost and emission of the objective functions, respectively. The initial guess for ω

_{1}and ω

_{2}is equal to 0.5 [36]. The worst solution is the most dominant solution in each iteration and is then selected as ‘${X}_{worst}$’. In order to select the best solution, in each iteration, after choosing the non-dominated solutions and controlling the size of the repository according to Algorithm 1, the best solution is selected from the repository as ‘${X}_{best}$’ based on (33) and is used in the subsequent iterations.

Algorithm 1. Pseudo code for controlling the size of repository. |

1: $\kappa =\left[0\right]$ 2: for $\begin{array}{cc}i=1& \begin{array}{cc}to& {N}_{{}_{non-dom}}-1\end{array}\end{array}$ |

3: for $\begin{array}{cc}j=i+1& \begin{array}{cc}to& {N}_{{}_{non-dom}}\end{array}\end{array}$ |

4: distance = $\sqrt{{\displaystyle \sum _{m=1}^{M}{({f}_{{i}_{m}}-{f}_{{j}_{m}})}^{2}}}$ |

5: if distance < Epsilon |

6: ${\kappa}_{j}={\kappa}_{j}+1$ |

7: end |

8: end |

9: end |

10: sort non-dominated solutions ascending according to ${\kappa}_{j}$ |

11: save the first ${N}_{L}$ elements of the non-dominated solutions in the repository |

#### 5.3. Enhanced MOJAYA (EMOJAYA)

## 6. Application of the Suggested EMOJAYA Algorithm

_{i}s according to (5) are generated as follows:

_{i}s and according to (5) must be generated as follows:

_{L}, control the size of the repository according to Algorithm 1.

_{max}). If it is satisfied, terminate the algorithm; otherwise, set iteration = iteration + 1 and return to Step 7.

## 7. Simulation Results

#### 7.1. A Comparison between the Performance of Proposed EMOJAYA with Original JAYA and PSO Algorithms on Different Test Functions

#### 7.2. Solving the MG Energy Managemnet Problem

**Scenario I:**In this case, the battery efficiencies are considered to be 93%, 95% and 97%, taking the influence of the Li-ion battery efficiency into account. Additionally, the impact of different initial and final charges is contemplated in the three aforementioned efficiencies. In each case, the battery’s initial and final charges (initial charge = final charge = E) differ from ${E}_{Batt,\mathrm{max}}$ to 0.2${E}_{Batt,\mathrm{max}}$ so that this variation effect can be investigated. Accordingly, the comparison of cost and emission objective functions in the BCS point for different efficiencies, including 93%, 95% and 97%, are shown in Figure 7, Figure 8 and Figure 9. Moreover, the Pareto optimal front for different efficiencies, along with their BCS points are illustrated in Figure 10. It should be mentioned that, in Figure 10, results are compared for the case that E is equal to ${E}_{Batt,\mathrm{max}}$.

**Scenario II:**In the second scenario, in order to obtain the best value of the battery initial and final charges (initial charge = final charge = E), they are considered as decision variables. Figure 11, Figure 12 and Figure 13 show comparisons of the Pareto optimal fronts for different E values in different efficiencies (i.e., 93%, 95% and 97%). It can be observed from Figure 11, Figure 12 and Figure 13 that when E is considered as a decision variable, a better Pareto optimal front is achieved in all efficiencies and, consequently, the obtained BCS will be more convenient.

**Scenario III:**In the third scenario, the battery degradation cost is neglected and results are compared with the situation in which the battery degradation cost is taken into account as Equation (3). In this case, the battery efficiency is considered equal to 95%. The comparison of Pareto optimal fronts with and without considering degradation cost is given in Figure 14. According to this figure, as already expected, the Pareto optimal front reveals a drop in cost objective function when the degradation cost is ignored.

## 8. Conclusions

- i.
- Investigating elements of the future smart grids, including demand response and the influence of electric vehicles on the considered MG’s energy management problem.
- ii.
- Inspecting reliability as an objective function in the MG’s optimal operation management.
- iii.
- Comparing different energy storage devices, as well as a variety of battery technologies to decide on the most optimal economic design of the system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

RES | renewable energy source |

MG | microgrid |

DER | distributed energy source |

PSO | particle swarm optimization |

ODED | optimal dynamic economic dispatch |

WT | wind turbines |

DG | distributed generator |

MOOM | multi-objective optimal operation management |

PCC | point of common coupling |

LC | local controller |

MGCC | micro grid central controller |

EMOJAYA | enhanced multi-objective JAYA |

RUT | reduced unscented transformation |

SOC | state of charge |

DoD | depth of discharge |

FC | fuel cell |

MT | micro-turbine |

PV | photovoltaic |

BCS | best-compromised solution |

$\overrightarrow{X}$ | vector of decision variables |

N | number of decision variables |

T | total number of hours |

${N}_{cycle}$ | battery cycle life |

Q_{n} | battery nominal capacity (kWh) |

Q(t) | battery current capacity (kWh) |

$Cos{t}_{Deg}^{t}$ | battery degradation cost (EUR) at hour t |

C_{Batt} | battery investment cost (EUR/kWh) |

N_{DG} | total number of dispatchable generations |

N_{Batt} | total number of batteries |

N_{RES} | total number of RESs |

N_{d} | total number of load levels |

Cost^{t} | MG’s operation cost in hour t (EUR) |

${P}_{RES,r}^{t}$ | real output powers (kWh) of rth RES at hour t |

${P}_{DG,i}^{t}$ | real output powers (kWh) of ith DG at hour t |

${P}_{Batt,s}^{t}$ | real output powers of sth storage at hour t |

${P}_{Grid}^{t}$ | active power bought (sold) from (to) the utility at hour t |

${B}_{RES,r}^{t}$ | bids of RESs at hour t (EUR/kWh) |

${B}_{DGi}^{t}$ | bids of dispatchable DGs at hour t (EUR/kWh) |

${B}_{Batt,s}^{t}$ | bids of battery at hour t (EUR/kWh) |

${B}_{Grid}^{t}$ | bids of the utility grid at hour t (EUR/kWh) |

$SU{C}_{DGi}$ | start-up cost for ith dispatchable DG |

$SD{C}_{DGi}$ | shut down cost for ith dispatchable DG |

$({P}_{DGi}^{t}.{B}_{DGi}^{t})$ | operational cost of dispatchable DGs |

$({P}_{RE{S}_{r}}^{t}.{B}_{RE{S}_{r}}^{t})$ | operational cost of RESs |

$({P}_{Bat{t}_{s}}^{t}.{B}_{Bat{t}_{s}}^{t})$ | operational cost of battery |

$({P}_{Grid}^{t}.{B}_{Grid}^{t})$ | cost of power exchange between the MG and the utility grid (EUR) |

P_{LD} | amount of dth load level |

${E}_{DGi}^{t}$ | amount of pollutants emission for each generator at hour t (kg/MWh) |

${E}_{Bat{t}_{s}}^{t}$ | amount of pollutants emission for storage device at hour t (kg/MWh) |

${E}_{Grid}^{t}$ | amount of pollutants emission for the utility grid at hour t (kg/MWh) |

${W}_{Batt}^{t}$ | amounts of energy stored inside the battery at hour t |

P_{ch} (P_{disch})
| permitted rate of charge (discharge) of the battery |

η_{c} (η_{d}) | efficiency of the battery during charge (discharge) process |

${W}_{Bat{t}_{s,\mathrm{min}}}$ | lower limit of amounts of energy storage inside the battery |

${W}_{Bat{t}_{s,\mathrm{max}}}$ | upper limit of amounts of energy storage inside the battery |

${P}_{ch,\mathrm{max}}$$({P}_{disch,\mathrm{max}})$ | maximum rate of battery charge (discharge) during each time interval ∆t |

${P}_{DG,\mathrm{min}}^{t}$ | minimum active power of the ith DG |

${P}_{DG,\mathrm{max}}^{t}$ | maximum active power of the ith DG |

${P}_{Batt,\mathrm{min}}^{t}$ | minimum active power of the bth storage |

${P}_{Batt,\mathrm{max}}^{t}$ | maximum active power of the bth storage |

${P}_{Grid,\mathrm{min}}^{t}$ | minimum active power of the utility at hour t. |

${P}_{Grid,\mathrm{max}}^{t}$ | maximum active power of the utility at hour t. |

${\overrightarrow{P}}_{LD}$ | load demand |

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**Figure 1.**A typical MG test system [30].

**Figure 2.**The comparison between the Pareto front of the proposed EMOJAYA and results of JAYA and PSO (Test Function 1).

**Figure 5.**Estimated output powers (normalized) of PV and WT [40].

**Figure 14.**Comparison of Pareto optimal fronts with/without considering degradation cost (efficiency = 95%).

**Table 1.**The limits and bids of DG sources and the utility grid [41].

Type | Min Power (kW) | Max Power (kW) | Bid (EUR /kWh) | Startup/ Shut Down Cost (EUR) | CO_{2}(kg/MWh) | SO_{2}(kg/MWh) | NO_{x}(kg/MWh) |
---|---|---|---|---|---|---|---|

MT | 40 | 400 | 0.457 | 0.96 | 720 | 0.0036 | 0.1 |

FC | 40 | 400 | 0.294 | 1.65 | 460 | 0.003 | 0.007 |

PV | 0 | 300 | 2.584 | 0 | 0 | 0 | 0 |

WT | 0 | 300 | 1.073 | 0 | 0 | 0 | 0 |

Battery | 0 | 300 | 0.38 | 0 | 10 | 0.0002 | 0.001 |

Utility | 0 | 1500 | - | - | 0.921 | 0.0036 | 0.0023 |

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**MDPI and ACS Style**

Javidsharifi, M.; Pourroshanfekr Arabani, H.; Kerekes, T.; Sera, D.; Spataru, S.; Guerrero, J.M.
Effect of Battery Degradation on the Probabilistic Optimal Operation of Renewable-Based Microgrids. *Electricity* **2022**, *3*, 53-74.
https://doi.org/10.3390/electricity3010005

**AMA Style**

Javidsharifi M, Pourroshanfekr Arabani H, Kerekes T, Sera D, Spataru S, Guerrero JM.
Effect of Battery Degradation on the Probabilistic Optimal Operation of Renewable-Based Microgrids. *Electricity*. 2022; 3(1):53-74.
https://doi.org/10.3390/electricity3010005

**Chicago/Turabian Style**

Javidsharifi, Mahshid, Hamoun Pourroshanfekr Arabani, Tamas Kerekes, Dezso Sera, Sergiu Spataru, and Josep M. Guerrero.
2022. "Effect of Battery Degradation on the Probabilistic Optimal Operation of Renewable-Based Microgrids" *Electricity* 3, no. 1: 53-74.
https://doi.org/10.3390/electricity3010005