# A Hybrid Method for Optimal Siting and Sizing of Battery Energy Storage Systems in Unbalanced Low Voltage Microgrids

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Planning Problem

#### 2.1. Objective Function

#### 2.2. Installation Constraints

#### 2.3. Operation Constraints

## 3. Solution Method

- Step 1. application of suitable techniques to identify a reduced feasible region for DESS siting (allocation bus set, ABS);
- Step 2. application of the GA to find the optimal solution in the ABS.

#### 3.1. Selection of Candidate Buses for the Improvement of Voltage Profile

#### 3.2. Selection of Candidate Buses for the Reduction of Unbalances

#### 3.3. Selection of Candidate Buses for the Reduction of Line Currents

#### 3.4. Overall Selection of Allocation Bus Set for the Distributed Energy Storage Systems Siting

#### 3.5. Hybrid Genetic Algorithm—Sequential Quadratic Programming Algorithm

## 4. Numerical Application

- a reference case without DESS;
- DESSs are allocated by means of only a GA;
- DESSs are allocated by means of the proposed procedure (GA applied for ABS).

#### 4.1. Case Study 1

#### 4.2. Case Study 2

#### 4.3. Case Study 3

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

$a$ | discount rate |

d | typical day index (d = 1, …,$\text{}n{d}_{y}$) |

i | bus index (i = 1, …, ng) |

k | time interval index (k = 1,…, nt) |

$k{d}_{i,{k}_{(y,d)}}$ | unbalance factor of the bus i during the time interval k of typical day d in year y |

$k{d}_{max}$ | maximum allowed unbalance factor |

l | line index (l = 1, …, nl) |

$n{d}_{y}$ | number of typical days of the year y |

ng | number of grid buses |

nl | number of grid lines |

nt | number of time intervals of the typical day |

${n}_{max}$ | maximum number of DESS units that can be installed at each phase |

ny | number of years of the planning horizon |

${n}_{DES{S}_{i}}^{p}$ | number of DESS units installed at the phase p of the bus i |

p | phase index (p = 1, 2, 3) |

y | year index (y = 1, …, ny) |

${B}_{ij}^{pm}$ | term of the susceptance matrix corresponding to the bus i with phase p and the bus j with phase m |

${C}_{inst}$ | installation costs |

${C}_{op}$ | operation costs |

${E}_{DES{S}_{i}}^{\mathrm{lb},{\xi}_{i}}$ | lower bound for the stored energy of the DESS installed at the phase ${\xi}_{i}$ of the bus i |

${E}_{DES{S}_{i}}^{\mathit{ub},{\xi}_{i}}$ | upper bound for the stored energy of the DESS installed at the phase ${\xi}_{i}$ of the bus i |

${E}_{DES{S}_{{i}_{(y,d)}}}^{in,{\xi}_{i}}$ | energy initially stored in the DESS installed at the phase ${\xi}_{i}$ of the bus i in the typical day d year y |

${E}_{base}^{size}$ | base size of the DESS unit |

${E}_{DES{S}_{i}}^{size}$ | total DESS size installed at the bus i |

${E}_{DES{S}_{i}}^{size,{\xi}_{i}}$ | size of the single-phase DESS installed at the phase ${\xi}_{i}$ of the bus i |

${G}_{ij}^{pm}$ | term of the conductance matrix corresponding to the bus i with phase p and the bus j with phase m |

$I{C}_{DESS}$ | installation cost of a single element of DESS |

${I}_{z}^{l}$ | ampacity of the line l |

${I}_{max}^{l}$ | maximum value of the current flowing in line l |

${I}_{{k}_{(y,d)}}^{l}$ | current flowing in the line l during the time interval k of the typical day d in year y |

${N}_{max}$ | maximum number of base units of DESSs that can be installed in the grid |

${N}_{y,d}$ | number of d typical days in year y |

${P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}$ | active power of the DESS installed at the bus i phase ${\xi}_{i}$ during the time interval k of typical day d, year y |

${P}_{D{G}_{i,{k}_{(y,d)}}}^{{\xi}_{i},\text{}sp}$ | forecasted active power produced by the DG unit installed a at the phase ${\xi}_{i}$ of the bus i during the time interval k of typical day d in year y |

${P}_{i,{k}_{(y,d)}}^{p}$ | net injected active power f the bus i phase p during the time interval k of typical day d in year y |

${P}_{{L}_{i,{k}_{(y,d)}}}^{p,\text{}sp}$ | forecasted active power demand of the load at the phase ${\xi}_{i}$ of the bus i during the time interval k of typical day d in year y |

$P{r}_{{k}_{(y,d)}}$ | price of electrical energy applied to the time interval k of typical day d in year y |

${Q}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}$ | reactive power of the DESS installed at the phase ${\xi}_{i}$ of the bus i during the time interval k of the typical day d in year y |

${Q}_{D{G}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}$ | reactive power of the DG unit installed at the phase ${\xi}_{i}$ of the bus i during the time interval k of the typical day d in year y |

${Q}_{i,{k}_{(y,d)}}^{p}$ | net injected reactive power of the bus i phase p during the time interval k of typical day d in year y |

${Q}_{{L}_{i,{k}_{(y,d)}}}^{p,\text{}sp}$ | forecasted reactive power demand of the load at the phase ${\xi}_{i}$ of the bus i during the time interval k of typical day d in year y |

$R{C}_{DESS}$ | replacement cost of a single element of DESS |

${S}_{DES{S}_{i}}^{{\xi}_{i}}$ | converter size of the DESS installed at the phase ${\xi}_{i}$ of the bus i |

${S}_{D{G}_{i}}^{{\xi}_{i}}$ | converter size of the DG installed at the phase ${\xi}_{i}$ of the bus i |

${S}_{MV}$ | size of the MV/LV transformer |

${V}_{i,{k}_{(y,d)}}^{p}$ | voltage magnitude at bus i, phase $p$ during the time interval k of the typical day d, year y |

${V}_{\mathit{max}}$ | upper bound of the admissible range for the bus voltages |

${V}_{min}$ | lower bound of the admissible range for the bus voltages |

${\alpha}_{L}$ | effective rate of change for the cost of the electrical energy |

${\eta}_{ch,i}^{{\xi}_{i}}$ | charging efficiencies of the DESS installed at the phase ${\xi}_{i}$ of the bus i |

${\eta}_{dch,i}^{{\xi}_{i}}$ | discharging efficiencies of the DESS installed at the phase ${\xi}_{i}$ of the bus i |

${\theta}_{i,{k}_{(y,d)}}^{p}$ | voltage argument at bus i, phase $p$ during the time interval k, the typical day d, year y |

$\dot{{\lambda}_{\dot{i}}}$ | ith eigenvalue of $\dot{\mathit{Y}}$ |

$\dot{{\lambda}_{k}}$ | eigenvalue of minimum modulus of $\dot{\mathit{Y}}$ |

${\dot{\mathsf{\lambda}}}_{\mathrm{i}}^{\mathrm{inv}-\mathrm{inv}}$ | ith eigenvalue of ${({\dot{Z}}^{\mathrm{inv}-\mathrm{inv}})}^{-1}$ |

${\dot{\mathsf{\lambda}}}_{\mathrm{k}}^{\mathrm{inv}-\mathrm{inv}}$ | eigenvalue of minimum modulus of ${({\dot{Z}}^{\mathrm{inv}-\mathrm{inv}})}^{-1}$ |

${\xi}_{i}$ | phase of bus i at which DESS or DG unit is connected |

$\Delta t$ | duration of the time interval |

${\mathsf{\Omega}}_{ch,{i}_{(y,d)}}^{{\xi}_{i}}$ | set of time intervals in which the DESS at bus i phase ${\xi}_{i}$ can charge in the typical day d year y |

${\mathsf{\Omega}}_{dch,{i}_{(y,d)}}^{{\xi}_{i}}$ | set of time intervals in which the DESS at bus i phase ${\xi}_{i}$ can discharge in the typical day d year y |

${\mathsf{\Omega}}_{ABS}$ | set of candidate buses |

${\Omega}_{DESS}$ | set of buses where DESSs are connected |

${\Omega}_{DG}$ | set of buses where DG units are located |

${\Omega}_{i}$ | sets of selected buses (i = 1, …,5) |

${\dot{\mathrm{\Gamma}}}_{\mathrm{i}}^{\text{'}}$ | left eigenvector associated to $\dot{{\mathsf{\lambda}}_{\mathrm{i}}}$ |

${\dot{\mathrm{\Gamma}}}_{i}^{inv-inv}$ | left eigenvector associated to ${\dot{\mathsf{\lambda}}}_{\mathrm{i}}^{\mathrm{inv}-\mathrm{inv}}$ |

${\dot{\mathrm{\Gamma}}}_{k}^{\prime}$ | left eigenvector associated to $\dot{{\mathsf{\lambda}}_{\mathrm{k}}}$ |

${\dot{\mathrm{\Gamma}}}_{k}^{inv-inv}$ | left eigenvector associated to ${\dot{\mathsf{\lambda}}}_{\mathrm{k}}^{\mathrm{inv}-\mathrm{inv}}$ |

$\overline{\Delta I}$ | vector of the variation of the injected currents |

${\overline{\Delta I}}^{zero}$ | vector of the variation of the zero sequence injected currents |

${\overline{\Delta I}}^{dir}$ | vector of the variation of the direct sequence injected currents |

${\overline{\Delta I}}^{inv}$ | vector of the variation of the inverse sequence injected currents |

$\overline{\Delta V}$ | vector of the variation of the phase voltages |

${\overline{\Delta V}}^{inv}$ | vector of the variation of the inverse-sequence voltages |

${\dot{X}}_{i}$ | right eigenvector associated to $\dot{{\mathsf{\lambda}}_{\mathrm{i}}}$ |

${\dot{X}}_{i}^{inv-inv}$ | right eigenvector associated to ${\dot{\mathsf{\lambda}}}_{\mathrm{i}}^{\mathrm{inv}-\mathrm{inv}}$ |

${\dot{X}}_{k}$ | right eigenvector associated to $\dot{{\mathsf{\lambda}}_{\mathrm{k}}}$ |

${\dot{X}}_{k}^{inv-inv}$ | right eigenvector associated to ${\dot{\mathsf{\lambda}}}_{\mathrm{k}}^{\mathrm{inv}-\mathrm{inv}}$ |

$\dot{Y}$ | nodal admittance matrix |

${\dot{Z}}^{seq\text{}k-seq\text{}m}$ | coupling impedance sub-matrices between sequences k and m ($\mathrm{k},\text{}\mathrm{m}\text{}\in \left[\mathrm{zero},\mathrm{dir},\mathrm{inv}\right]$). |

## Appendix A

**i**, the definition of phase voltages as a function of the sequence voltages is given by:

## References

- Stenclik, D.; Denholm, P.; Chalamala, B. Maintaining Balance: The Increasing Role of Energy Storage for Renewable Integration. IEEE Power Energy Mag.
**2017**, 15, 31–39. [Google Scholar] [CrossRef] - Celli, G.; Pilo, F.; Pisano, G.; Soma, G.G. Cost–benefit analysis for energy storage exploitation in distribution systems. CIRED-Open Access Proc. J.
**2017**, 1, 2197–2200. [Google Scholar] [CrossRef] - Carpinelli, G.; Mottola, F.; Proto, D.; Russo, A. A Multi-Objective Approach for Microgrid Scheduling. IEEE Trans. Smart Grid
**2017**, 8, 2109–2118. [Google Scholar] [CrossRef] - Berrada, A.; Loudiyi, K.; Zorkani, I. Valuation of energy storage in energy and regulation markets. Energy
**2016**, 115, 1109–1118. [Google Scholar] [CrossRef] - Omar, R.; Rahim, N.A. Voltage unbalanced compensation using dynamic voltage restorer based on supercapacitor. Int. J. Electr. Power Energy Syst.
**2012**, 43, 573–581. [Google Scholar] [CrossRef] - Rahman, M.S.; Hossain, M.J.; Lu, J. Coordinated control of three-phase AC and DC type EV–ESSs for efficient hybrid microgrid operations. Energy Convers. Manag.
**2016**, 122, 488–503. [Google Scholar] [CrossRef] - Watson, J.D.; Watson, N.R.; Lestas, I. Optimized dispatch of energy storage systems in unbalanced distribution networks. IEEE Trans. Sustain. Energy
**2017**. [Google Scholar] [CrossRef] - Carpinelli, G.; Mottola, F.; Proto, D.; Varilone, P. Minimizing unbalances in low-voltage microgrids: Optimal scheduling of distributed resources. Appl. Energy
**2017**, 191, 170–182. [Google Scholar] [CrossRef] - Mateo, C.; Reneses, J.; Rodriguez-Calvo, A.; Frías, P.; Sánchez, Á. Cost–benefit analysis of battery storage in medium-voltage distribution networks. IET Gener. Transm. Distrib.
**2016**, 10, 815–821. [Google Scholar] [CrossRef] - Chen, C.; Duan, S.; Cai, T.; Liu, B.; Hu, G. Optimal Allocation and Economic Analysis of Energy Storage System in Microgrids. IEEE Trans. Power Electron.
**2011**, 26, 2762–2773. [Google Scholar] [CrossRef] - Nick, M.; Cherkaoui, R.; Paolone, M. Optimal Planning of Distributed Energy Storage Systems in Active Distribution Networks Embedding Grid Reconfiguration. IEEE Trans. Power Syst.
**2017**. [Google Scholar] [CrossRef] - Xiao, J.; Zhang, Z.; Bai, L.; Liang, H. Determination of the optimal installation site and capacity of battery energy storage system in distribution network integrated with distributed generation. IET Gener. Transm. Distrib.
**2016**, 10, 601–607. [Google Scholar] [CrossRef] - Motalleb, M.; Reihani, E.; Ghorbani, R. Optimal placement and sizing of the storage supporting transmission and distribution networks. Renew. Energy
**2016**, 94, 651–659. [Google Scholar] [CrossRef] - Zheng, Y.; Dong, Z.Y.; Luo, F.J.; Meng, K.; Qiu, J.; Wong, K.P. Optimal Allocation of Energy Storage System for Risk Mitigation of DISCOs with High Renewable Penetrations. IEEE Trans. Power Syst.
**2014**, 29, 212–220. [Google Scholar] [CrossRef] - Alnaser, S.W.; Ochoa, L.F. Optimal sizing and control of energy storage in wind power-rich distribution networks. In Proceedings of the IEEE Power and Energy Society General Meeting (PESGM) 2016, Boston, MA, USA, 17–21 July 2016. [Google Scholar]
- Li, Q.; Ayyanar, R.; Vittal, V. Convex Optimization for DES Planning and Operation in Radial Distribution Systems with High Penetration of Photovoltaic Resources. IEEE Trans. Sustain. Energy
**2016**, 7, 985–995. [Google Scholar] [CrossRef] - Nick, M.; Cherkaoui, R.; Paolone, M. Optimal siting and sizing of distributed energy storage systems via alternating direction method of multipliers. Int. J. Electr. Power Energy Syst.
**2015**, 72, 33–39. [Google Scholar] [CrossRef] - Celli, G.; Mocci, S.; Pilo, F.; Loddo, M. Optimal integration of energy storage in distribution networks. In Proceedings of the IEEE Bucharest PowerTech 2009, Bucharest, Romania, 28 June–2 July 2009. [Google Scholar]
- Carpinelli, G.; Celli, G.; Mocci, S.; Mottola, F.; Pilo, F.; Proto, D. Optimal integration of distributed energy storage devices in smart grids. IEEE Trans. Smart Grid
**2013**, 4, 985–995. [Google Scholar] [CrossRef] - Oh, H. Optimal Planning to Include Storage Devices in Power Systems. IEEE Trans. Power Syst.
**2011**, 26, 1118–1128. [Google Scholar] [CrossRef] - Shen, X.; Shahidehpour, M.; Han, Y.; Zhu, S.; Zheng, J. Expansion Planning of Active Distribution Networks with Centralized and Distributed Energy Storage Systems. IEEE Trans. Sustain. Energy
**2017**, 8, 126–134. [Google Scholar] [CrossRef] - Xiong, P.; Singh, C. Optimal Planning of Storage in Power Systems Integrated with Wind Power Generation. IEEE Trans. Sustain. Energy
**2016**, 7, 232–240. [Google Scholar] [CrossRef] - Murillo-Sánchez, C.E.; Zimmerman, R.D.; Anderson, C.L.; Thomas, R.J. Secure Planning and Operations of Systems with Stochastic Sources, Energy Storage, and Active Demand. IEEE Trans. Smart Grid
**2013**, 4, 2220–2229. [Google Scholar] [CrossRef] - Xiang, Y.; Han, W.; Zhang, J.; Liu, J.; Liu, Y. Optimal sizing of energy storage system in active distribution networks using Fourier-Legendre series based state of energy function. IEEE Trans. Power Syst.
**2017**. [Google Scholar] [CrossRef] - Grover-Silva, E.; Girard, R.; Kariniotakis, G. Optimal sizing and placement of distribution grid connected battery systems through an SOCP optimal power flow algorithm. Appl. Energy
**2017**. [Google Scholar] [CrossRef] - Babacan, O.; Torre, W.; Kleissl, J. Siting and sizing of distributed energy storage to mitigate voltage impact by solar PV in distribution systems. Solar Energy
**2017**, 146, 199–208. [Google Scholar] [CrossRef] - Marra, F.; Tarek Fawzy, Y.; Bülo, T.; Blaži, B. Energy Storage Options for Voltage Support in Low-Voltage Grids with High Penetration of Photovoltaic. In Proceedings of the IEEE PES Innovative Smart Grid Technologies (ISGT) 2012, Berlin, German, 16–20 January 2012. [Google Scholar]
- Chua, K.H.; Lim, Y.S.; Taylor, P.; Morris, S.; Wong, J. Energy Storage System for Mitigating Voltage Unbalance on Low-Voltage Networks with Photovoltaic Systems. IEEE Trans. Power Deliv.
**2012**, 27, 1783–1790. [Google Scholar] [CrossRef] - Saboori, H.; Hemmati, R.; Jirdehi, M.A. Reliability improvement in radial electrical distribution network by optimal planning of energy storage systems. Energy
**2015**, 93, 2299–2312. [Google Scholar] [CrossRef] - Mottola, F.; Proto, D.; Russo, A.; Varilone, P. Planning of Energy Storage Systems in Unbalanced Microgrids. In Proceedings of the 7th IEEE International Conference on Innovative Smart Grid Technologies (IEEE PES ISGT Europe) 2017, Torino, Italy, 26–29 September 2017. [Google Scholar]
- Giannitrapani, A.; Paoletti, S.; Vicino, A.; Zarrilli, D. Algorithms for placement and sizing of energy storage systems in low voltage networks. In Proceedings of the 54th IEEE Conference on Decision and Control (CDC) 2015, Osaka, Japan, 15–18 December 2015; pp. 3945–3950. [Google Scholar]
- Laughton, M.A. Sensitivity in dynamical system analysis. J. Electron. Control
**1964**, 17, 577–591. [Google Scholar] [CrossRef] - Laughton, M.A.; El-Iskandarani, M.A. On the inherent network structure. In Proceedings of the 6th Power Systems Computation Conference (PSCC) 1978, Darmstad, Germany, 15 November 1978; pp. 188–196. [Google Scholar]
- Carpinelli, G.; Russo, A.; Russo, M.; Verde, P. Inherent structure theory of networks and power system harmonics. IEE Proc. Gener. Transm. Distrib.
**1998**, 145, 123–132. [Google Scholar] [CrossRef] - Gamm, A.Z.; Golub, I.I.; Bachry, A.; Styczynski, Z.A. Solving several problems of power systems using spectral and singular analyses. IEEE Trans. Power Syst.
**2005**, 20, 138–148. [Google Scholar] [CrossRef] - Carpinelli, G.; Proto, D.; Noce, C.; Russo, A.; Varilone, P. Optimal allocation of capacitors in unbalanced multi-converter distribution systems: A comparison of some fast techniques based on genetic algorithms. Electr. Power Syst. Res.
**2010**, 80, 642–650. [Google Scholar] [CrossRef] - Sikiru, T.H.; Jimoh, A.A.; Hamam, Y.; Agee, J.T.; Ceschi, R. Classification of networks based on inherent structural characteristics. In Proceedings of the Sixth IEEE/PES Transmission and Distribution: Latin America Conference and Exposition (T&D-LA) 2012, Montevideo, Uruguay, 3–5 September 2012; pp. 1–6. [Google Scholar]
- Sikiru, T.H.; Jimoh, A.A.; Agee, J.T. Inherent structural characteristic indices of power system networks. Int. J. Electr. Power Energy Syst.
**2013**, 47, 218–224. [Google Scholar] [CrossRef] - Carpinelli, G.; Proto, D.; Russo, A. Optimal planning of active power filters in a distribution system using trade off/risk method. IEEE Trans. Power Deliv.
**2017**. [Google Scholar] [CrossRef] - Casolino, G.M.; Losi, A. Load Area model accuracy in distribution systems. Electr. Power Syst. Res.
**2017**, 143, 321–328. [Google Scholar] [CrossRef] - Losi, A.; Mancarella, P.; Vicino, A. Integration of Demand Response into the Electricity Chain: Challenges, Opportunities and Smart Grid Solutions; ISTE-Wiley: London, UK, 2015. [Google Scholar]
- Carpinelli, G.; Mottola, F.; Proto, D. Addressing Technology Uncertainties in Battery Energy Storage Sizing Procedures. Int. J. Emerg. Electr. Power Syst.
**2017**, 18. [Google Scholar] [CrossRef] - Wang, J.; Liu, P.; Hicks-Garner, J.; Sherman, E.; Soukiazian, S.; Verbrugge, M.; Tataria, H.; Musser, J.; Finamore, P. Cycle-life model for graphite-LiFePO4 cells. J. Power Sources
**2011**, 196, 3942–3948. [Google Scholar] [CrossRef] - Akcoglu, M.A.; Bartha, P.F.A.; Minh, H.D. Analysis in Vector Spaces; John Wiley & Sons: Hoboken, NJ, USA, 2011. [Google Scholar]
- Cigré Task Force C6.04. Benchmark Systems for Network Integration of Renewable and Distributed Energy Resources; Cigré Brochure 575; Cigré: Paris, France, 2014. [Google Scholar]
- Rates-Pacific Gas and Electric Company. Available online: http://www.pge.com/tariffs/ (accessed on 12 February 2018).
- Lo, K.L.; Zhang, C. Decomposed three-phase power flow solution using the sequence component frame. IEE Proc. C-Gener. Transm. Distrib.
**1993**, 140, 181–188. [Google Scholar] [CrossRef]

**Figure 3.**Input data: per unit power profiles of a residential (

**a**), an industrial (

**b**) and a commercial (

**c**) load [45].

**Figure 5.**Case Study 1: Line currents of the residential feeder (phase no.1) at hour 21.00 of the typical day.

**Figure 6.**Case Study 1: Voltage magnitude of the residential feeder at hour 21.00 of the typical day.

**Figure 8.**Case Study 2: Line currents of the residential feeder (phase no.1) at hour 21.00 of the last typical day.

**Figure 9.**Case Study 2: Voltage magnitude of the residential feeder at hour 21.00 of the last typical day.

**Figure 11.**Case Study 3: Line currents of the residential feeder (phase no.1) at hour 21.00 of the last typical day.

**Figure 12.**Case Study 3: Voltage magnitude of the residential feeder at hour 21.00 of the last typical day.

**Figure 15.**Case Study 3: Reactive power of one DESS and one genetic algorithm (DG) during the last typical day.

Number of DESSs | |
---|---|

$0\le {n}_{DES{S}_{i}}^{p}\le {n}_{max}\text{\hspace{1em}}i=1,\dots ,{n}_{g}\text{\hspace{1em}}p=1,2,3$ | (a) |

${{\displaystyle \sum}}_{i=1}^{ng}{\displaystyle \sum _{p=1}^{3}}{n}_{DES{S}_{i}}^{p}\le {N}_{max}$ | (b) |

${E}_{DES{S}_{i}}^{size}={E}_{base}^{size}({{\displaystyle \sum}}_{p=1}^{3}{n}_{DES{S}_{i}}^{p})\text{\hspace{1em}}i=1,\dots ,{n}_{g}$ | (c) |

Net Energy Charged and Discharged | |

$\sum}_{k=1}^{nt}\Delta t\text{}{\gamma}_{i,{k}_{(y,d)}}^{{\xi}_{i}}{P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}=0\text{\hspace{1em}}i\text{}\u03f5\text{}{\mathsf{\Omega}}_{DESS},\text{\hspace{1em}}k=1,\dots ,nt,$ where ${\gamma}_{i,{k}_{(y,d)}}^{{\xi}_{i}}=\{\begin{array}{cc}\frac{1}{{\eta}_{dch,i}^{{\xi}_{i}}}& {P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}\ge 0,\\ {\eta}_{ch,i}^{{\xi}_{i}}& {P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}<0.\end{array}$ y = 1, …, ny; d = 1, …, nd _{y}. | (a) |

Charging/discharging periods of the day | |

$\begin{array}{ccc}-{P}_{DES{S}_{i,max}}^{{\xi}_{i},ch}\le {P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}\le 0\text{\hspace{1em}}& i\u03f5{\mathsf{\Omega}}_{DESS}\text{\hspace{1em}}& k\text{}\in \text{}{\mathsf{\Omega}}_{ch,{i}_{(y,d)}}^{{\xi}_{i}},\\ 0\le {P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}\le {P}_{DES{S}_{i,max}}^{{\xi}_{i},dch}\text{\hspace{1em}}& i\u03f5{\mathsf{\Omega}}_{DESS}\text{\hspace{1em}}& k\text{}\in \text{}{\mathsf{\Omega}}_{dch,{i}_{(y,d)}}^{{\xi}_{i}}.\end{array}$ y = 1, …, ny; d = 1, …, nd _{y}. | (b) |

Energy stored | |

${E}_{DES{S}_{i}}^{\mathrm{lb},{\xi}_{i}}\text{}\le {E}_{DES{S}_{{i}_{(y,d)}}}^{\mathrm{in},{\xi}_{i}}-{\displaystyle \sum}_{k=1}^{\tau}\Delta t{\gamma}_{i,{k}_{(y,d)}}^{{\xi}_{i}}{P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}}\le {E}_{DES{S}_{i}}^{\mathit{ub},{\xi}_{i}}$, $i\u03f5{\Omega}_{DESS},$ y = 1, …, ny; d = 1, …, nd _{y}; $\tau =1,\dots ,nt.$ | (c) |

Active and reactive power | |

$\sqrt{{({P}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}})}^{2}+{({Q}_{DES{S}_{i,{k}_{(y,d)}}}^{{\xi}_{i}})}^{2}}\le {S}_{DES{S}_{i}}^{{\xi}_{i}}\text{\hspace{1em}}i\text{}\u03f5\text{}{\Omega}_{DESS}$, k = 1, …, nt; y = 1, …, ny; d = 1, …, nd _{y}. | (d) |

Reactive Power Provided by the DG Unit | |
---|---|

$\sqrt{{({P}_{D{G}_{i,{k}_{(y,d)}}}^{{\xi}_{i},\text{}sp})}^{2}+{({Q}_{D{G}_{i,{k}_{(y,d)}}}^{{\xi}_{i}})}^{2}}\le {S}_{D{G}_{i}}^{{\xi}_{i}}$ $i\text{}\u03f5\text{}{\Omega}_{DG}$ k = 1, …, nt; y = 1, …, ny; d = 1, …, nd_{y}. |

Load Flow Equations | |

${P}_{i,{k}_{(y,d)}}^{p}={V}_{i,{k}_{(y,d)}}^{p}{\displaystyle \sum}_{j=1}^{ng}{\displaystyle \sum}_{m=1}^{3}{V}_{j,{k}_{(y,d)}}^{m}\left[{G}_{ij}^{pm}\mathrm{cos}({\vartheta}_{i,{k}_{(y,d)}}^{p}-{\vartheta}_{j,{k}_{(y,d)}}^{m})+{B}_{ij}^{pm}\mathrm{sin}({\vartheta}_{i,{k}_{(y,d)}}^{p}-{\vartheta}_{j,{k}_{(y,d)}}^{m})\right]$ ${Q}_{i,{k}_{(y,d)}}^{p}={V}_{i,{k}_{(y,d)}}^{p}{\displaystyle \sum}_{j=1}^{ng}{\displaystyle \sum}_{m=1}^{3}{V}_{j,{k}_{(y,d)}}^{m}\left[{G}_{ij}^{pm}\mathrm{sin}({\vartheta}_{i,{k}_{(y,d)}}^{p}-{\vartheta}_{j,{k}_{(y,d)}}^{m})-{B}_{ij}^{pm}\mathrm{cos}({\vartheta}_{i,{k}_{(y,d)}}^{p}-{\vartheta}_{j,{k}_{(y,d)}}^{m})\right],$ | (a) |

${P}_{i,{k}_{(y,d)}}^{p}={P}_{D{G}_{i,{k}_{(y,d)}}}^{p,\text{}sp}+{P}_{DES{S}_{i,{k}_{(y,d)}}}^{p}+{P}_{{L}_{i,{k}_{(y,d)}}}^{p,\text{}sp},$ | (b) |

${Q}_{i,{k}_{(y,d)}}^{p}={Q}_{D{G}_{i,{k}_{(y,d)}}}^{p}+{Q}_{DES{S}_{i,{k}_{(y,d)}}}^{p}+{Q}_{{L}_{i,{k}_{(y,d)}}}^{p,\text{}sp},$ | (c) |

i = 1, …, ng; p = 1, 2, 3; k = 1, …, nt; y = 1, …, ny; d = 1, …, nd_{y}. | |

Inter-connection bus constraints | |

${V}_{1,{k}_{(y,d)}}^{p}={V}^{slack},$ | (d) |

${\vartheta}_{1,{k}_{(y,d)}}^{p}=\frac{2}{3}\mathsf{\pi}(1-p),$ | (e) |

p = 1, 2, 3; k = 1, …, nt; y = 1, …, ny; d = 1, …, nd_{y}, | |

$\sqrt{{\left({\displaystyle {\displaystyle \sum}_{p=1}^{3}}{P}_{1,{k}_{(y,d)}}^{p}\right)}^{2}+{\left({\displaystyle {\displaystyle \sum}_{p=1}^{3}}{Q}_{1,{k}_{(y,d)}}^{p}\right)}^{2}}\le {S}_{MV},$ | (f) |

k = 1, …, nt; y = 1, …, ny; d = 1, …, nd_{y}. | |

All network busbars | |

${V}_{min}\le {V}_{i,{k}_{(y,d)}}^{p}\le {V}_{max},\text{}$ | (g) |

$k{d}_{i,{k}_{(y,d)}}\le k{d}_{max},$ | (h) |

${I}_{{k}_{(y,d)}}^{l}\le {I}_{z}^{l}$, | (i) |

i = 1, …, ng; p = 1, 2, 3; k = 1, …, nt; y = 1, …, ny; d = 1, …, nd_{y}; l = 1, …, nl. |

Summer Tariff | Winter Tariff | |||
---|---|---|---|---|

Period | Price ($/MWh) | Period | Price ($/MWh) | |

On peak | 12:00–18:00 | 542.04 | 8:30–21:30 | 161.96 |

Part Peak | 8:30–12:00 18:00–21:30 | 252.90 | ||

Off Peak | 21:30–8:30 | 142.54 | 21:30–8:30 | 132.54 |

Bus | Phase | Active Power [kW] | Bus | Phase | Active Power [kW] |
---|---|---|---|---|---|

Residential Feeder | Commercial Feeder | ||||

12 | 1 | 2.4 | 33 | 1 | 0.6 |

2 | 1.9 | 2 | 0.6 | ||

3 | 5.2 | 3 | 3.6 | ||

16 | 1 | 17.4 | 34 | 1 | 1.2 |

2 | 14 | 2 | 1.2 | ||

3 | 38 | 3 | 7.2 | ||

17 | 1 | 8.7 | 35 | 1 | 1.5 |

2 | 7 | 2 | 1.5 | ||

3 | 19.2 | 3 | 9.0 | ||

18 | 1 | 5.5 | 38 | 1 | 0.72 |

2 | 4.4 | 2 | 0.72 | ||

3 | 12.2 | 3 | 4.3 | ||

19 | 1 | 7.4 | 39 | 1 | 1.5 |

2 | 6 | 2 | 1.5 | ||

3 | 16.4 | 3 | 9.0 | ||

Industrial feeder | 40 | 1 | 0.48 | ||

21 | 1 | 14.2 | 2 | 0.48 | |

2 | 11.3 | 3 | 2.9 | ||

3 | 31.2 | 41 | 1 | 0.48 | |

2 | 0.48 | ||||

3 | 2.88 |

Bus no. | Phase No. | DESS Capacity (kWh) | Bus No. | Phase No. | DESS Capacity (kWh) |
---|---|---|---|---|---|

Residential feeder | 16 | 2 | 12 | ||

4 | 3 | 12 | 16 | 3 | 12 |

5 | 1 | 4 | 17 | 1,2,3 | 20 |

5 | 3 | 4 | Commercial feeder | ||

6 | 2 | 8 | 23 | 1,2,3 | 28 |

6 | 3 | 8 | 25 | 3 | 8 |

7 | 3 | 8 | 27 | 3 | 12 |

8 | 3 | 8 | 28 | 1 | 8 |

10 | 2 | 8 | 29 | 1,2,3 | 36 |

10 | 3 | 8 | 33 | 1,2,3 | 24 |

11 | 1 | 12 | 34 | 3 | 4 |

11 | 2 | 4 | 36 | 1,2,3 | 24 |

13 | 3 | 4 | 39 | 1 | 4 |

14 | 1,2,3 | 24 | 41 | 1 | 12 |

15 | 1 | 4 | 41 | 2 | 12 |

Location Bus Set | Residential Feeder Bus No. (Phase No.) | Industrial Feeder Bus No. (Phase No.) | Commercial Feeder Bus No. (Phase No.) |
---|---|---|---|

${\mathsf{\Omega}}_{1}$ —Voltage deviation | 10(2), 11(2), 16(2), 18(2), 19(1,2,3) | 21(1,2,3) | 29(1,2), 30(1,2,3), 40(1,2), 41(1,2,3) |

${\mathsf{\Omega}}_{2}$—Unbalance factor | 10, 11, 18, 19 | 21 | 29, 30, 40, 41 |

${\mathsf{\Omega}}_{3}$—Unbalance factor | 9, 10, 11, 15, 16, 18, 19 | 21 | 29, 30, 37, 38, 40, 41 |

${\mathsf{\Omega}}_{4}$—Unbalance factor | 9, 10, 11, 18, 19 | 21 | 29, 30, 40, 41 |

${\mathsf{\Omega}}_{5}$—Line currents | 3(3), 4(3), 13(3), 14(3), 15(3), 16(3) | - | - |

Bus No. | Phase No. | DESS Capacity (kWh) | Bus No. | Phase No. | DESS Capacity (kWh) |
---|---|---|---|---|---|

Residential Feeder | Industrial Feeder | ||||

9 | 1,2,3 | 36 | 21 | 1,2,3 | 20 |

10 | 1,2,3 | 36 | Commercial feeder | ||

11 | 1,2,3 | 36 | 29 | 2 | 8 |

15 | 1,2,3 | 36 | 29 | 3 | 8 |

16 | 1,2,3 | 28 | 30 | 1,2,3 | 32 |

18 | 1 | 8 | 37 | 1,2,3 | 36 |

18 | 2 | 8 | 40 | 3 | 8 |

41 | 1 | 8 |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Carpinelli, G.; Mottola, F.; Proto, D.; Russo, A.; Varilone, P.
A Hybrid Method for Optimal Siting and Sizing of Battery Energy Storage Systems in Unbalanced Low Voltage Microgrids. *Appl. Sci.* **2018**, *8*, 455.
https://doi.org/10.3390/app8030455

**AMA Style**

Carpinelli G, Mottola F, Proto D, Russo A, Varilone P.
A Hybrid Method for Optimal Siting and Sizing of Battery Energy Storage Systems in Unbalanced Low Voltage Microgrids. *Applied Sciences*. 2018; 8(3):455.
https://doi.org/10.3390/app8030455

**Chicago/Turabian Style**

Carpinelli, Guido, Fabio Mottola, Daniela Proto, Angela Russo, and Pietro Varilone.
2018. "A Hybrid Method for Optimal Siting and Sizing of Battery Energy Storage Systems in Unbalanced Low Voltage Microgrids" *Applied Sciences* 8, no. 3: 455.
https://doi.org/10.3390/app8030455