# Data-Driven Optimal Battery Storage Sizing for Grid-Connected Hybrid Distributed Generations Considering Solar and Wind Uncertainty

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

#### 1.3. Contribution

- Optimal battery energy storage system sizing with the unit commitment of DG’s/thermal units on an IEEE 14 bus system, considering day-ahead solar PV and wind-farm uncertainties by using a distributionally robust optimization technique with a linear decision rule and distribution of the uncertain solar PV and wind output data.
- Cost comparison with different sizes of battery energy storage system on the unit commitment of DG’s/thermal units. Where the day-ahead 30-min duration of unit commitments with battery energy storage systems are discussed.

#### 1.4. Paper Organization

## 2. Problem Description

## 3. Problem Methodology

#### 3.1. Mathematical Formulation of Unit Commitment

#### 3.2. Power Flow Model through Transmission Lines

#### 3.3. Battery-Energy-Storage-System Modeling

Algorithm 1 Battery charge/discharge cycle counter |

$t\leftarrow 1$ |

${B}_{count,b}^{c}\leftarrow 0$ |

${B}_{count,b}^{d}\leftarrow 0$ |

if ${P}_{d}(t-1)>0$ & ${P}_{c}\left(t\right)>0$ & $SOC\left(t\right)\le 20\%$ then |

${B}_{count,b}^{d}\leftarrow {B}_{count,b}^{d}+1$ |

end if |

if ${P}_{c}(t-1)>0$ & ${P}_{d}\left(t\right)>0$ & $SOC\left(t\right)\ge 80\%$ then |

${B}_{count,b}^{c}\leftarrow {B}_{count,b}^{c}+1$ |

end if |

#### 3.4. Distributionally Robust Optimization Model

#### 3.4.1. Modeling Ambiguity Parameters for Solar-PV Uncertainty

#### 3.4.2. Modeling Ambiguity Parameters for Wind Uncertainty

## 4. Results and Discussions

#### 4.1. Example Case Study for DRO Illustration with One DG for an Instance

#### 4.2. Optimum Battery Sizing and Its Impact on Unit Commitment in an IEEE 14 Bus System

#### 4.2.1. Solar-PV Uncertainty with 30-Minute Interval Unit Commitments

#### 4.2.2. Wind-Farm Uncertainty with 30-Minute Interval Unit Commitments

#### 4.2.3. Hybrid Uncertainty with 30-Minute Interval Unit Commitments

#### 4.3. Comparison with Similar Studies

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

## Sets and Indices for UC Model

$\tau /t$ | Time interval set |

$\zeta $ | Feasible time constraints for minimum up/down time |

$\kappa /k$ | DG unit set |

$\beta /b$ | Buses in the system |

$\Lambda /l$ | Solar PV units |

$\mathcal{L}/g$ | Transmission line |

$\mathcal{J}/j$ | Wind-farm units |

## Constants for UC Model

${\overline{P}}_{k}$ | Upper limit of generator k |

${\underline{P}}_{k}$ | Lower limit of generator k |

${RD}_{k}$ | RD limit of generator k |

${RU}_{k}$ | RU limit of generator k |

${p}_{b,t}^{d,min}$ | Minimum power that storage can provide while discharging at bus b at time t |

${p}_{b,t}^{d,max}$ | Maximum power that storage can provide while discharging at bus b at time t |

${p}_{b,t}^{c,min}$ | Maximum power that storage needs while charging at bus b at time t |

${p}_{b,t}^{c,max}$ | Maximum power that storage needs while charging at bus b at time t |

${D}_{b,t}$ | Load demand at bus b during time t |

${a}_{k}$ | Generator k cost parameter |

${b}_{k}$ | Generator k cost parameter |

${C}_{k}^{s}$ | Fixed start up cost for unit k |

${SOC}_{b,t}^{min}$ | Minimum state of charge at time t and bus b |

${SOC}_{b,t}^{max}$ | Maximum state of charge at time t and bus b |

${\Delta}_{t}$ | Time step of storage for time t |

${\eta}_{c}$ | Efficiency of charging the battery |

${\eta}_{d}$ | Efficiency of discharging the battery |

$L{C}_{g}$ | Line capacity of line g |

$LC{F}_{g,b}$ | Load capacity factor of line g connected to bus b |

${P}^{sur}$ | Penalty surcharge for load loss |

C | Charging and discharging rates of the battery |

## Decision Variables for UC Model

${c}_{k,t}$ | Unit k generation cost at time t |

${B}_{count,b}^{c}$ | Battery’s charge cycle count at bus b |

${B}_{count,b}^{T}$ | Total charge and discharge cycle count for battery at bus b |

${p}_{j,t}^{w}$ | Wind-farm unit j generation at time t |

${p}_{b,t}^{cur}$ | Power surplus/generation loss at bus b on time step t |

${p}_{k,t}^{t}$ | DG unit k generation at time t |

${p}_{l,t}^{s}$ | solar PV unit l generation at time t |

${x}_{k,t}$ | Binary variable for generator status i.e.,UC |

${z}_{k,t}$ | Unit k start up cost at time t |

${p}_{b,t}^{d}$ | Power that battery can provide (as a source) at bus b in time t |

${p}_{b,t}^{c}$ | Power that battery (as a load) needs at bus b in time t |

${SOC}_{b,t}$ | State of the charge at time t and bus b |

${B}_{count,b}^{d}$ | Battery’s discharge cycle count at bus b |

## Parameter Set for Uncertainty Model

${\tilde{\mathit{\psi}}}_{l,t}$ | Random variable for solar PV uncertainty error for unit l at time t |

${V}_{l,t}^{+}$ | Maximum error in solar PV uncertainty for unit l at time t |

${V}_{l,t}^{-}$ | Minimum error in solar PV uncertainty for unit l at time t |

$\mathcal{V}$ | Set of all uncertainties for random variables under linear constraints |

$\overline{\mathcal{V}}$ | Extended set of uncertainties for random variables defined under linear constraints |

${\mathbb{O}}_{\mathit{\psi}}$ | Distribution for occurrence of random variables $\tilde{\mathit{\psi}}$ and $\tilde{\alpha}$ together |

$\tilde{s}$ | Auxiliary random variable for wind |

${\tilde{o}}_{j,t}$ | Random variable for wind-farm uncertainty error for unit j at time t |

${\mathbb{E}}_{\mathbb{M}}$ | Expectation within distribution $\mathbb{M}$ |

$\mathbb{I}$ | Ambiguity matrix with given distribution of random variable $\tilde{o}$ |

$\mathbb{H}$ | Extended form of ambiguity set $\mathbb{I}$ |

$\mathcal{I}/i$ | Index for distribution of random variable |

$\mathbb{J}$ | Events describing the distribution of each random variable ${v}_{l,t}$ |

$\mathbb{M}$ | Random variables $\tilde{\mathit{\psi}}$ distribution |

${\mathbb{O}}_{o}$ | Distribution for occurrence of random variables $\tilde{o}$ and $\tilde{s}$ together |

$\mathcal{S}/e,r$ | Set of random variables |

$\tilde{\alpha}$ | Auxiliary random variable for solar PV |

## Others

$\Xi \left(\mathit{x}\right)$ | Expected worst-case distribution recourse cost |

$\Xi (\mathit{x},\mathit{o},\mathit{\psi})$ | UC cost based on decision of $\mathit{x}$ under economic dispatch, with wind-farm $\mathit{o}$ and solar PV $\mathit{\psi}$ |

${C}_{B}^{T}(\mathit{o},\mathit{\psi})$ | The running cost of the battery under economic dispatch, considering the |

recourse action | |

of wind-farm $\mathit{o}$ solar PV $\mathit{\psi}$ uncertainty |

## Abbreviations

UC | Unit commitment |

DRO | Distributionally robust optimization |

DG | Distribution generation |

RES | Renewable energy sources |

RO | Robust optimization |

BESS | Battery energy storage system |

ESS | Energy storage system |

PV | Photo voltaic |

MILP | Mixed integer linear programming |

SOCP | Second-order cone programing |

C | Charge and discharge rate |

VSS | Value of stochastic solution |

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**Figure 1.**The schematic diagram of the proposed system having distributed generator, solar PV, wind-farm, and battery energy storage system.

**Figure 4.**Unit commitment schedule with power contribution from each distributed generator after incorporating solar-PV output power to the system with 30-min duration without using BESS and $LC{F}_{gb}=1.0$ for each bus.

**Figure 5.**The power contribution and the status of each distributed generator after incorporating solar-PV output power to the system with optimal capacity of BESS and $LC{F}_{gb}=1.0$ for each bus.

**Figure 6.**Charging and discharging power and hourly energy stored in the battery having BESS capacity optimized to 1520 kWh.

**Figure 7.**Empirical cumulative distribution graph considering solar PV having optimal BESS capacity of 1520 kWh.

**Figure 8.**Unit commitment schedule with power contribution from each distributed generator after incorporating wind power to the system with 30-min duration without using BESS and $LC{F}_{gb}=1.0$ for each bus.

**Figure 9.**The power contribution and the status of each distributed generator after incorporating wind power to the system with optimal capacity of BESS and $LC{F}_{gb}=1.0$ for each bus.

**Figure 10.**Charging and discharging power and hourly energy stored in the battery having battery capacity optimized to 558.13 kWh.

**Figure 11.**Empirical cumulative distribution graph considering wind farms having optimal BESS capacity of 558.13 kWh.

**Figure 12.**Unit commitment schedule with power contribution from each distributed generator after incorporating hybrid power to the system with 30-min duration without using BESS and $LC{F}_{gb}=1.0$ for each bus.

**Figure 13.**The power contribution and the status of each distributed generator after incorporating hybrid power to the system with optimal capacity of BESS and $LC{F}_{gb}=1.0$ for each bus.

**Figure 15.**Total system cost vs. battery capacity with charging/discharging rates ranging from 0.05C to 1C.

**Figure 16.**Empirical cumulative distribution graph considering hybrid system having optimal BESS capacity of 712.99 kWh.

**Figure 17.**Total system’s demand vs. generation (including distributed generation, wind-farm, solar PV, and optimal capacity of BESS).

**Table 1.**The data for battery energy storage system (BESS) integrated with unit commitments and dynamic economic dispatch [49].

Parameter | $SO{C}_{0}$ | $SO{C}_{max}$ | ${P}_{max}^{d}$ | ${P}_{min}^{d}$ | ${P}_{max}^{c}$ |

Value | 200 kWh | 1 MWh | 0.25C | 0 | 0.25C |
---|---|---|---|---|---|

$O\&M$ cost | |||||

Parameter | ${P}_{min}^{c}$ | ${\eta}_{c}$ | ${\eta}_{d}$ | ${C}^{var}$ | ${C}^{fix}$ |

Value | 0 | 95% | 90% | USD 0.31/kWh | USD 10/kW-year |

**Table 2.**Thermal unit’s data for dynamic economic dispatch and unit commitments [51].

${\mathit{a}}_{\mathit{g}}$ | ${\mathit{b}}_{\mathit{g}}$ | ${\mathit{c}}_{\mathit{g}}$ | ${\mathit{P}}_{\mathit{g}}^{\mathit{min}}$ | ${\mathit{P}}_{\mathit{g}}^{\mathit{max}}$ | ${\mathit{RU}}_{\mathit{g}}^{0}$ | ${\mathit{RD}}_{\mathit{g}}^{0}$ |
---|---|---|---|---|---|---|

(USD/kW) | (USD/kW) | (USD/kW) | (kW) | (kW) | (kW) | (kW) |

$1.2\times {10}^{-7}$ | $14.80\times {10}^{-3}$ | 89 | 28 | 200 | 40 | 40 |

$1.7\times {10}^{-7}$ | $16.57\times {10}^{-3}$ | 83 | 20 | 290 | 30 | 30 |

$1.9\times {10}^{-7}$ | $16.21\times {10}^{-3}$ | 70 | 20 | 260 | 50 | 50 |

$1.5\times {10}^{-7}$ | $15.55\times {10}^{-3}$ | 100 | 30 | 190 | 30 | 30 |

Time | Load | Time | Load | Time | Load | Time | Load |
---|---|---|---|---|---|---|---|

(hr) | (kW) | (hr) | (kW) | (h) | (kW) | (h) | (kW) |

0:00 | 525.30 | 6:00 | 525.30 | 12:00 | 665.38 | 18:00 | 756.02 |

0:30 | 530.45 | 6:30 | 526.59 | 12:30 | 675.68 | 18:30 | 759.63 |

1:00 | 535.60 | 7:00 | 527.88 | 13:00 | 685.98 | 19:00 | 763.23 |

1:30 | 540.75 | 7:30 | 529.16 | 13:30 | 696.28 | 19:30 | 766.84 |

2:00 | 545.90 | 8:00 | 530.45 | 14:00 | 706.58 | 20:00 | 770.44 |

2:30 | 542.30 | 8:30 | 537.92 | 14:30 | 720.74 | 20:30 | 773.53 |

3:00 | 538.69 | 9:00 | 545.39 | 15:00 | 734.91 | 21:00 | 776.62 |

3:30 | 535.09 | 9:30 | 552.85 | 15:30 | 749.07 | 21:30 | 779.71 |

4:00 | 531.48 | 10:00 | 560.32 | 16:00 | 763.23 | 22:00 | 782.80 |

4:30 | 529.94 | 10:30 | 586.59 | 16:30 | 761.43 | 22:30 | 729.76 |

5:00 | 528.39 | 11:00 | 612.85 | 17:00 | 759.63 | 23:00 | 676.71 |

5:30 | 526.85 | 11:30 | 639.12 | 17:30 | 757.82 | 23:30 | 623.67 |

**Table 4.**Total system cost objective comparison between distributionally robust optimizations and stochastic optimization with forecasting accuracy variance 0.1 over various penetration level (IEEE RTS 24-Bus).

Wind Forecast | Stochastic Optimization | Distributionally Robust Optimization | ||||||
---|---|---|---|---|---|---|---|---|

Accuracy | Total Cost (USD) | VSS | Total Cost (USD) | VSS | ||||

Percent | Deterministic | Stochastic | USD | % | Deterministic | DRO | USD | % |

100 | 1,482,836 | 1,404,468 | 78,367 | 5.285 | 1,482,836 | 1,411,223 | 71,613 | 4.829 |

80 | 1,543,488 | 1,506,339 | 37,148 | 2.407 | 1,543,488 | 1,512,588 | 30,899 | 2.002 |

60 | 1,643,850 | 1,618,083 | 25,766 | 1.567 | 1,643,850 | 1,627,354 | 16,496 | 1.003 |

40 | 1,753,050 | 1,740,145 | 12,905 | 0.736 | 1,753,050 | 1,747,797 | 5252 | 0.300 |

20 | 1,884,530 | 1,880,827 | 3702 | 0.196 | 1,884,530 | 1,881,773 | 2756 | 0.146 |

**Table 5.**Total system cost objective comparison between distributionally robust optimizations and stochastic optimization with penetration level 100% over various forecasting-accuracies variances (IEEE RTS 24-Bus).

Wind Forecast | Stochastic Optimization | Distributionally Robust Optimization | ||||||
---|---|---|---|---|---|---|---|---|

Accuracy | Total Cost (USD) | VSS | Total Cost (USD) | VSS | ||||

Variance | Deterministic | Stochastic | USD | % | Deterministic | DRO | USD | % |

0.1 | 1,482,836 | 1,404,468 | 78,367 | 5.285 | 1,482,836 | 1,411,223 | 71,613 | 4.829 |

0.075 | 1,465,605 | 1,403,389 | 62,216 | 4.245 | 1,465,605 | 1,408,875 | 56,730 | 3.871 |

0.05 | 1,447,095 | 1,399,777 | 47,318 | 3.27 | 1,447,095 | 1,419,318 | 27,777 | 1.919 |

0.025 | 1,427,307 | 1,396,437 | 30,869 | 2.163 | 1,427,307 | 1,410,691 | 16,616 | 1.164 |

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## Share and Cite

**MDPI and ACS Style**

Rauf, A.; Kassas, M.; Khalid, M.
Data-Driven Optimal Battery Storage Sizing for Grid-Connected Hybrid Distributed Generations Considering Solar and Wind Uncertainty. *Sustainability* **2022**, *14*, 11002.
https://doi.org/10.3390/su141711002

**AMA Style**

Rauf A, Kassas M, Khalid M.
Data-Driven Optimal Battery Storage Sizing for Grid-Connected Hybrid Distributed Generations Considering Solar and Wind Uncertainty. *Sustainability*. 2022; 14(17):11002.
https://doi.org/10.3390/su141711002

**Chicago/Turabian Style**

Rauf, Abdul, Mahmoud Kassas, and Muhammad Khalid.
2022. "Data-Driven Optimal Battery Storage Sizing for Grid-Connected Hybrid Distributed Generations Considering Solar and Wind Uncertainty" *Sustainability* 14, no. 17: 11002.
https://doi.org/10.3390/su141711002