# Optimal siting and sizing of renewable energy sources, storage devices, and reactive support devices to obtain a sustainable electrical distribution systems

## Abstract

This paper presents an integrated planning framework to optimally determine the location and allocation of renewable-based distributed generation (DG) units, energy storage systems (ESSs), and capacitor banks (CBs). This planning aim at improving the performance of electrical distribution systems (EDSs). In the proposed model, the cost of energy delivered by the substation and the investment costs are minimized. The environmental aspects are taken into account to obtain an efficient environmentally committed plan. The uncertainties due to PV generation and demand profile are considered via external uncertainty indexes in a deterministic environment. The proposed model is a mixed-integer nonlinear programming (MINLP) problem, which is recast to a mixed-integer linear programming (MILP) problem using appropriate linearization techniques. This approximated MILP model is implemented in the mathematical language AMPL, while the commercial solver CPLEX is used to obtain the global optimal solutions. The proposed model is validated by testing on a medium voltage distribution system with 135 nodes under different conditions and topologies.

## Keywords

Capacitor banks Energy storage systems Environmental aspects Planning framework Renewable generation## Abbreviations

- CB
Capacitor bank

- DISCO
Distribution company

- DG
Distributed generation

- DoD
Depth of discharge

- EDS
Electrical distribution system

- ESS
Energy storage system

- MILP
Mixed integer linear programming

- MINLP
Mixed integer nonlinear programming

- PV
Photovoltaic

## Sets and indexes

- \({\varOmega }^{ cb}\)
Set of CB capacities

*D*Set of planning horizon (years)

*L*Set of circuits

*N*Set of nodes

*T*Set of time intervals

*b*Index of reactive power capacity of installed CB

*d*Index of years

*i*,*j*Index of nodes

*ij*Index of circuits

*k*Index of PV modules

*t*Index of time intervals

## Constants

- \(\zeta _{t,d}^{G} \)
Energy cost at time interval

*t*, at year*d*- \(\zeta _b^{CB^{f/sw}} \)
Installation cost of fixed/switchable CB with power capacity

*b*- \(\zeta ^{PV}\)
Installation cost of PV modules

- \( \zeta _{d}^{o \& m^{ PV}}\)
Operation and maintenance cost of installed PV modules at year

*d*- \( \zeta _n^{D \& R}\)
Disposal and recycling cost of ESS at present net value (

*n*)- \( \zeta _{d}^{o \& m^{ ESS}}\)
Operation and maintenance cost of installed ESS at year

*d*- \(\zeta ^{PC^{ ESS}}\)
Investment charging and discharging power capacity cost of ESS

- \(\zeta ^{RC^{ ESS}}\)
Investment storage reservoir cost of ESS

- \({\varDelta }_d \)
Duration of the year

*d*(8760 h)- \({\varDelta }_t \)
Duration of time interval

*t*- \(\underline{{\varPhi }}^S,\overline{{\varPhi }}^S \)
Lower and upper limit of power factor for substation

- \({\varPhi }^{ pv}\)
Power factor for PV-based DG units

- \(\eta \)
Efficiency of the ESS

- \(\overline{E}^{ ESS}\)
Maximum energy reservoir capacity of ESS that can be installed

- \(\underline{E}^{ ESS}\)
Minimum energy reservoir capacity of ESS that can be installed

- \(e ^p \)
Emission coefficient

- \(f_{t,d}^{G^{ pv}}\)
PV generation factor; time

*t*, year*d*- \(\overline{I}_{ ij}\)
Maximum current magnitude allowed on the circuit

*ij*- \({IC^{ CB}}\)
Investment cost limit of CB

- \({IC^{ PV}}\)
Investment cost limit of PV-based DG units

- \({IC^{ ESS}}\)
Investment cost limit of ESS

- \({N}^{ PV}\)
Maximum number of PV plants that can be installed

- \(P_{i,t,d}^{ Ld}\)
Active power demand at node

*i*, time*t*, year*d*- \(\overline{P}^{ ESS}\)
Maximum power rating of ESS that can be installed

- \(\overline{P}^{ PV}\)
Active power capacity of PV module

- \(\overline{ PE}_d \)
Pollutant emission limit at year

*d*- \(Q_{b}^{ esp}\)
Nominal reactive power of the CB with capacity

*b*- \(Q_{i,t,d}^{ Ld}\)
Reactive power demand at node

*i*, time*t*, and year*d*- \(R,X,Z_{ ij}\)
Resistance, reactance, and impedance of the circuit

*ij*- \(\overline{V}, \underline{V}\)
Upper and lower voltage magnitude limits

- \(V^{ nom} \)
Nominal voltage magnitude

- \(V_{i,t,d}^{*}\)
Estimated voltage magnitude at node

*i*, time*t*, and year*d*

## Continuous variables

- \(\tilde{E}_{i}^{ ESS} \)
Energy reservoir of installed ESS at node

*i*- \(E_{i,t,d}^{ ESS} \)
State of charge of installed ESS at node

*i*, time*t*, and year*d*- \(I^{ sqr}_{ij,t,d}\)
Square of current flow magnitude of circuit

*ij*in time*t*, and year*d*- \(P_{ij,t,d}\)
Active power flow of circuit

*ij*in time*t*, and year*d*- \(\tilde{P}_{i}^{ ESS}\)
Power rating of installed ESS at node

*i*- \(P_{i,t,d}^{ES{S^c}} \)
Charging power of installed ESS at node

*i*in time*t*, and year*d*- \(P_{i,t,d}^{ES{S^d}} \)
Discharging power of installed ESS at node

*i*in time*t*, and year*d*- \(P_{i,t,d}^{ PV}\)
PV active power generation at node

*i*in time*t*, and year*d*- \(P_{t,d}^{S}\)
Active power supplied by substation in time

*t*, and year*d*- \(Q_{ij,t,d}\)
Reactive power flow of circuit

*ij*in time*t*, and year*d*- \(Q_{i,d}^{ CB}\)
Reactive power delivered by installed CB at node

*i*in year*d*- \(Q_{i,t}^{ PV}\)
PV reactive power generated at node

*i*in time*t*- \(Q_{t,d}^{S}\)
Reactive power supplied by substation in time

*t*, and year*d*- \(V^{sqr}_{i,t,d}\)
Square of voltage at node

*i*in time*t*, and year*d*

## Binary and integer variables

- \(x_{i,b}^{f/sw}\)
Binary variables that define the capacity

*b*and CB type (fixed or switchable) to be installed at node*i*- \(y_{i}^{ ESS}\)
Binary variable that define the ESS to be installed at node

*i*- \(z_{i,k}\)
Binary variable that define the PV modules

*k*to be installed at node*i*- \(B_{i,d} \)
Integer variable for the CB modules to be installed at node

*i*in year*d*- \(\overline{M}_{i}\)
Integer variable for the maximum CB bank to be installed in node

*i*

## Notes

### Acknowledgements

The authors would like to gratefully acknowledge the financial support for this research by FAPESP (Grant nos. 2014/22828-3, 2016/14319-7), CNPq no. 305371/2012-6, and CAPES.

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