energies
Article
Optimal Location-Reallocation of Battery Energy
Storage Systems in DC Microgrids
Oscar Danilo Montoya 1,2,* , Walter Gil-González 2 and Edwin Rivas-Trujillo 1
1 Facultad de Ingeniería, Universidad Distrital Francisco José de Caldas, Bogotá D.C. 11021, Colombia;
erivas@udistrital.edu.co
2 Laboratorio Inteligente de Energía, Universidad Tecnológica de Bolívar, Cartagena 131001, Colombia;
wjgil@utp.edu.co
* Correspondence: o.d.montoyagiraldo@ieee.org; Tel.: +57-310-5461-067




Received: 6 April 2020; Accepted: 28 April 2020; Published: 5 May 2020 

Abstract: This paper deals with the problem of optimal location and reallocation of battery energy
storage systems (BESS) in direct current (dc) microgrids with constant power loads. The optimization
model that represents this problem is formulated with two objective functions. The first model
corresponds to the minimization of the total daily cost of buying energy in the spot market by
conventional generators and the second to the minimization of the costs of the daily energy losses in all
branches of the network. Both the models are constrained by classical nonlinear power flow equations,
distributed generation capabilities, and voltage regulation, among others. These formulations
generate a nonlinear mixed-integer programming (MINLP) model that requires special methods to be
solved. A dc microgrid composed of 21-nodes with existing BESS is used for validating the proposed
mathematical formula. This system allows to identify the optimal location or reallocation points for
these batteries by improving the daily operative costs regarding the base cases. All the simulations
are conducted via the general algebraic modeling system, widely known as the General Algebraic
Modeling System (GAMS).
Keywords: battery energy storage system; economic dispatch problem; nonlinear programming
formulation; optimal reallocation of batteries; mathematical optimization
1. Introduction
Electrical networks have progressively transformed from thermal dependent systems to networks
with high penetration of renewable energy resources [1,2]. This transformation is promoted to the
Paris agreement, where many countries around the world have signed compromises regarding the
minimization of greenhouse emissions [3]. This agreement forces conventional power systems to
change their fossil fuel-based energy matrices (i.e., coal, natural gas, or diesel) by inserting renewable
energy sources [4]. These sources are mainly photovoltaic and wind power plants, since their
construction and production costs have decreased significantly in the last years [5]. Nevertheless,
the inclusion of renewable energy sources is not a perfect solution since power systems must deal
with uncertainties produced by weather variations (solar radiance or wind speed uncertainties).
These uncertainties depend on the geographical location of the power system as well as the period of
the year (winter or summer seasons). To tackle these uncertainties in renewable power generation,
we have developed large-scale energy storage systems that allow to reduce the energy oscillations
in the power system by compensating these in a dynamical form. Some of these energy storage
devices can be: supercapacitors [6], fly-wheels [7], superconductors [8], compressed air systems [9],
pumped-hydro systems [10], or batteries [11], among others. The selection of the energy storage
depends on the application, i.e., for voltage and frequency compensation, fly-wheel, supercapacitors,
Energies 2020, 13, 2289; doi:10.3390/en13092289 www.mdpi.com/journal/energies
Energies 2020, 13, 2289 2 of 20
or superconductors are preferred [12], while long-time power supplies are preferred for pumped-hydro
and battery energy storage systems [13].
The inclusion of renewable energy sources and energy storage devices in power systems is not the
only paradigm shift since other remarkable transformations have occurred, especially in distribution
voltage levels [14]. This change is caused by the transition from classical alternating current (ac)
networks to direct current (dc) systems [15]. The main advantage of using dc grids in comparison
to ac is that the reactive power and frequency concepts disappear [16], which makes dc grids easily
controllable, with low energy losses and higher voltage profiles [17]. In general, dc grids are more
efficient and reliable than their ac counterparts [16–18].
Regarding the integration of renewable energy sources and energy storage systems in dc grids,
there is a clear advantage, since some of them, such as photovoltaic sources and batteries, can operate
directly in dc systems, which reduces the number of power conversion stages for integrating these
devices into the grids [19]. Additionally, wind turbines or superconductors are only required
in dc conversion stages since the inversion ones are unnecessary. This implies fewer electronic
power converters can reduce the probability of failure and the costs of investment, maintenance,
and operation [20].
It is important to note that when distribution levels are highly influenced by renewable and energy
storage technologies, they integrate into smart grids with the capability of self-management regarding
control and optimization [21,22]. Nevertheless, in the case of DC networks, smart grids can be reduced
to microgrids since dc networks are an emerging concept in distribution levels and their current size
and loadability are currently constrained to small areas (i.e., buildings or data centers) [23,24]. Here,
we wish to analyze the problem of battery location and reallocation in dc microgrids from the point of
economical and technical goals that focus on proposing a new mathematical model for representing
this problem. However, it is important to note that the proposed model will be extended to a large-scale
dc distribution feeder when it becomes a reality in the near future without modifications [25].
There are three different approaches to operate dc networks with high penetration of renewable
energy resources and batteries, which are condensed in hierarchical control methods. The first two
approaches are related to primary and secondary control strategies, which deal with power current and
voltage controls [26], and measure local state variables to maintain voltage under nominal operative
conditions [27]. Some of these controllers are designed via the sliding mode control [28], passivity based
control [29], model predictive approach [30], and droop control [27], which are directly applied to
the power electronic converters. The third approach is related to the optimization stage, where there
exists specialized literature on tertiary control methods [31]. This stage defines the set points regarding
power and voltage to be assigned for all the active devices (i.e., power electronic converters that
interfaces batteries and distributed generators) to minimize some objective function, such as, typically,
grid energy losses or energy purchase costs [18,31,32].
This study seeks to understand the tertiary control stage regarding optimization methods for
the optimal operation of battery energy storage systems in dc networks with high penetration of
renewable energy resources. In specialized literature, the optimal operation of batteries in dc grids
has been addressed via economic dispatch formulation in three recent references as follows: Authors
of [33] present a semi-definite programming model to operate batteries and renewables in dc grids.
The authors of [33] proposed a relaxation of the power balance equations via semi-definite matrices,
which guarantee the uniqueness of the global solution regarding the problem. The authors of [18]
proposed a second-order cone programming model to optimally dispatch renewables and batteries in
dc grids. Both relaxations coincide numerically to the exact nonlinear model. The main disadvantage
of this model is that the number of variables increase in square form compared to the number of
nodes in the dc grid. In [11], a nonlinear non-convex model for the optimal operation of batteries
and renewable energies in dc grids considering voltage-dependent load models has been presented.
This mathematical model is solved using the General Algebraic Modeling System (GAMS). It should
Energies 2020, 13, 2289 3 of 20
be noted that in the previous models, the location of the batteries was predefined, and the data of the
network was given by the utility. Nevertheless, there is no guarantee these locations are optimal.
Regarding the optimal location and operation of batteries in an ac electrical network,
different approaches have been proposed in scientific literature, some of which are described below:
the authors in [32] presented a methodology based on genetic algorithms for the optimal location
battery energy storage in ac microgrids considering the performance indicator, the net present value of
investments, and the costs of the energy losses. Different simulation cases considering sunny, cloudy,
and rainy days were included to analyze the interdependence between batteries and distributed
generators. In addition, the numerical results reported allowed to identify the best set of renewables
and batteries to provide high-quality service to grid users. The authors of [34] proposed a methodology
to increase the flexibility of microgrids with renewables that could be affected by the winter season.
The authors of this research achieved a mathematical formulation with a mixed-integer programming
form that can solve the battery scheduling problem efficiently via the CPLEX solver. Numerical
results in a large-scale power system demonstrate the efficiency of the proposed approach for the
management of heat demands in power systems, which are drastically affected by seasons through the
optimal scheduling of batteries. In [35], an optimal economic dispatch of batteries in ac distribution
networks was proposed considering the minimization of the energy purchase in conventional sources.
The results were obtained via GAMS implementation by considering renewables as constant inputs
and batteries in fixed points. The authors of [36] presented an optimization methodology based on
genetic algorithms for the optimal location of batteries in radial distribution feeders. This approach
allowed to reduce the daily operation losses of the network. In the case of batteries, the authors of [36]
proposed a binary strategy to dispatch them, which can make the implementation of the model in
conventional solvers difficult; nevertheless, the results are interesting for utilities since the grid is
absent of renewable energy resources, making it the main contributor.
Note that the previous approaches in ac and dc grids demonstrate that the problem of optimal
location and reallocation needs more research since this is an important issue in power system analysis.
For this purpose, this paper proposes an optimization model for the location-reallocation batteries,
focusing on dc networks. This problem has not yet been reported or analyzed for dc networks.
The proposed model has a mixed-integer nonlinear programming (MINLP) structure because of binary
and nonlinear variables. The binary variables appear due to location and reallocation of batteries
while the nonlinear variables are the products between the voltage variables in the power balance
constraint. The proposed model analyzes two objective functions, where the first corresponds to the
minimization of the energy purchase costs in the spot market by conventional generators and the
second to the minimization of the daily energy losses. In addition, it also employs artificial neural
networks to forecast the power generated by wind and solar generators to increase the effectiveness of
the proposed model. The main contributions of this study are as follows:
• To propose an MINLP model for the location-reallocation batteries in dc networks, which considers
two objective functions independently or a linear combination of them. This problem has not
been previously proposed in the scientific literature to the best of the knowledge of the authors.
In addition, the proposed model allows to understand the compromise between the location of
the batteries as a function of the perforce indicator, i.e., objective function, which demonstrates
the interdependence of the batteries’ location/operation regarding energy costs and renewable
energy availability.
• To include in the proposed MINLP model the economic dispatch problem to maximize the use of
the batteries during the day and, thus, obtain a suitable location and reallocation for them.
• Three simulation cases were analyzed for the proposed model to evaluate different objective
functions according to the location-reallocation of the batteries. These simulations allow to identify
the best trade-offs between the final positioning of the batteries and the daily behavior of the
grid, which can help the distribution grid make the best decision as a function of its goals, i.e.,
technical or economic objectives.
Energies 2020, 13, 2289 4 of 20
This study is organized as follows: Section 2 presents the mathematical formulation for the
optimization model, Section 3 formulates a strategy to solve the proposed optimization model,
and Section 4 explains the test systems and proposed scenarios. The computational validation and
results are analyzed in Section 5. Lastly, the main conclusions derived from this study and possible
future works are presented in Section 6.
2. Mathematical Formulation
The problem of the optimal location-reallocation of batteries in dc microgrids with high
penetration of renewable energy resources is a discrete (binary) version of the multi-period economic
dispatch models for BESS proposed in [11,18,33]. The model for locating-reallocating batteries is indeed
at nonlinear and strong non-convex (also integer) due to the hyperbolic relations between power and
voltages in the power-balance equations. Here, we consider two possible objective functions that
can be used as indicators to define the best location of the batteries: the first is related to the energy
purchase costs in the spot market by the conventional generator and the second to the costs of the
daily energy losses in all the branches of the network.
2.1. Objective Functions
min z1 = ∑ ∑ CoE(i,t
pi,t∆t (1)
t∈T i∈N )
min z2 = ∑ ∑ CoEi,tvi,t ∑ Gijvj,t∆t (2)
t∈T i∈N j∈N
where z1 is the objective function value related to energy buying costs, z2 is the objective function
associated to the costs of the daily energy losses, CoEi,t is the cost of buying energy (spot market
purchase) at node i in period t, pi,t corresponds to the power bought (generated) at node i during
period t, and ∆t is the length of the time period under analysis (e.g., 1 h, 30 min or 15 min); vi,t is the
voltage value at node i during the period of time t; Gij is the value of the conductance that relates nodes
i and j. Observe that T andN are the sets that contain all periods of time of the dispatch planning and
the total number of nodes in the dc microgrid, respectively.
It should be noted that the mathematical formulation of the objective functions z1 and z2 originate
from convex functions since the energy purchase costs are a linear function, and the daily energy costs
are a positive definite quadratic form based on the properties of the conductance matrix [37].
2.2. Set of Constraints
pi,t + p
dg
i,t + ∑ pb di,t − pi,t = vi,t ∑ Gijvj,t, {∀i ∈ N & ∀t ∈ T } (3)
b∈B j∈N
SoCb b b bi,t = SoCi,t−1 − ϕi pi,t∆t, {∀b ∈ B, ∀i ∈ N& ∀t ∈ T } (4)
SoCb = xbSoCb,inii,t i i , {∀b ∈ B &∀i ∈ N} (5)0
SoCbi,t = x
bSoCb, f ini i , {∀b ∈ B &∀i ∈ N} (6)f
pmini,t ≤ p ≤ pmaxi,t i,t , {∀i ∈ N& ∀t ∈ T } (7)
pdg,min dg dg,maxi,t ≤ pi,t ≤ pi,t , {∀i ∈ N& ∀t ∈ T } (8)
Energies 2020, 13, 2289 5 of 20
xb pb,min ≤ pb ≤ xb pb,maxi i i,t i i , {∀b ∈ B, ∀i ∈ N& ∀t ∈ T } (9)
vmin ≤ v ≤ vmaxi i,t i , {∀i ∈ N& ∀t ∈ T } (10)
xbSoCb,min ≤ SoCb b b,maxi i i,t ≤ xi SoCi , {∀b ∈ B, ∀i ∈ N& ∀t ∈ T } (11)
∑ ∑ xb = Nmaxi b , (12)
b∈B i∈N
where pdg b di,t , pi,t, and pi,t are the power generation by renewable energy resources (i.e.,
distributed generation), the power delivered/absorbed by the batteries, and the power demand at node
i during the time period t, respectively; SoCbi,t represents the state-of-charge of the battery in the ith
node at the tth time period; xbi is a binary variable related to the possibility of locating/reallocating the
battery b at node i; SoCb,inii and SoC
b, f in
i are the initial and final desired states of charge of the batteries,
respective;y, while SoCb,mini and SoC
b,max
i are the minimum and maximum state-of-charge bounds;
pmin, pmax, pdg,min, and pdg,maxi,t i,t i,t i,t are the minimum and maximum bounds of admissible generation
for conventional and renewable generators located in the ith node in time period t, respectively,
while pb,min and pb,maxi i represent the minimum and maximum charge/discharge capabilities of a
battery connected at node i; vmin and vmaxi i are the voltage regulation bounds of the dc microgrid.
Finally, ϕbi represents the coefficient of charge of a battery connected at node i. Observe that N
max
b
corresponds to the maximum number of batteries available for location or reallocation, and B is the set
that contains all the types of batteries.
The interpretation of the complete mathematical model described from Equation (1) to (12) is the
following: Expression (1) presents the objective function related to the minimization of the energy
purchase cost in the spot market by conventional sources; Equation (2) defines the objective function
related to the total costs of the energy losses in all the branches of the network; Equation (3) presents
the power balance constraint per node associated with the combination of Kirchhoff’s first law and
the first Tellegen’s theorem [38]; Expression (4) shows the linear relation between the state-of-charge
of the battery and the power injection/absorption [33]; Equations (5) and (6) present the operative
consigns for battery operation regarding initial and final state-of-charges, which are defined by the
utility. Expressions (7)–(9) are defined as the minimum and maximum power bounds for conventional
and distributed generators as well as for batteries, respectively. In Equation (10), the lower and upper
bounds admissible for voltage profiles, i.e., voltage regulation limits are presented. Equation (11) shows
the minimum and maximum bounds for the states-of-charge in batteries. Expression (12) presents the
constraint related to the maximum number of batteries available for location or reallocation in the
dc network.
In the mathematical model of Equations (1) and (12) for optimal location-reallocation of batteries,
it is important to highlight the following facts:
• This formulation has a mixed-integer nonlinear programming (MINLP) structure due to the
presence of binary variables regarding the location-reallocation of batteries as well as products
between voltage variables in the power balance constraint, which implies robust optimization
methodologies or toolboxes are required to get the optimal solution, even if it is local or global [11].
• The effectiveness of the proposed model highly depends on the weather conditions in renewable
generation, since their power injections, i.e., pdgi,t , are conditioned to the generation technology.
Here we consider these renewables are based on wind and power technologies and their outputs
forecast via artificial neural networks, as recommended in [12,19,33].
Energies 2020, 13, 2289 6 of 20
• The location or reallocation of the batteries in the dc grid will depend on the performance indicator,
i.e., objective functions z1 and z2, even if they are used independently or as a linear combination.
• Regarding the complexity of the proposed MINLP model, the main difficulty is growing due to the
solution space with the number of nodes in relation to the possibilities for locating or reallocating
batteries. The size of this part of the solutio(n s)pace can be calculated as follows [39]:
n n!
Cn,b = =r b! (n− b)!
where n is the number of nodes and b the number of batteries. In this sense, if we have a dc
grid with 50 nodes and batteries from 1 to 5, then the number of possibilities for their location
is 50, 1225, 19,600, 230,300, and 2,118,760, respectively, which demonstrate that the problem of
optimal location-reallocation of batteries is highly complex. Note that each possible combination
of batteries is needed to solve the resulting economic dispatch problem, which is also nonlinear
and strong non-convex.
In this paper, the main interest is regarding the formulation of the MINLP formulation presented
from Equation (1) to (12) since, for dc grids, after a careful revision has not found evidence of previous
formulations, as most of the works are related with economic dispatch analysis considering the
fixed location of the batteries, as reported in [11,12,18,33]. For this reason, we have employed the
General Algebraic Modeling System (GAMS) for reaching the solution of the proposed model, since it
has previously been used in [11] for battery dispatch in dc grids considering voltage-dependent
load models. The next section presents the main characteristics of the GAMS software as a
solution methodology.
It is important to mention that the optimization model proposed in this research (see
Equations (1)–(12)) considers the basic relation between state-of-charge in batteries and the active
power injections, as reported in [40]; nevertheless, in the future, for the purpose of analysis, it is
highly recommended to make additional improvements regarding batteries, such as life cost analysis,
self-discharge phenomena, or efficiency in power electronic converters that interface them [26],
to identify other relevant aspects that can affect the short-term economic dispatch analysis in
this research.
3. Solution Strategy
To deal with the mathematical model Equations (1)–(12) that describe the optimal
location-reallocation of batteries in dc networks as an MINLP model, we select the GAMS optimization
package as the solution strategy. This software is selected as it has been largely used in specialized
literature to address complex optimization problems with hundreds of variables. Some of the most
relevant works where GAMS has been used are the optimal locations of distributed generators in ac
and dc networks considering daily load behaviors [19,41], the optimal design of osmotic power plants
for electricity generation and desalinization processes [42,43], the optimization of the pump and valve
schedules in complex, large-scale water distribution systems [44], the optimal dispatch of batteries
in dc and ac networks [11,12,18,35], the multi-objective optimization of the stack of thermoacoustic
engines [45], the multi-objective optimization in power systems with renewable sources [46], and
economic dispatch approaches in thermal power systems [47], etc.
3.1. Software Implementation
To illustrate an implementation of an optimization model in GAMS, let us consider the simple
MINLP model that represents the line selection for a transmission system using the transportation
model, as depicted in Figure 1.
Energies 2020, 13, 2289 7 of 20
2
25 MW
f12 f25
1 3 5
f35 60 MW
g1 f13 30 MW
f14 f45
4
20 MW
Figure 1. Small transmission system for the illustrative example.
This problem was proposed as an illustrative example of planning distribution networks in [48].
The formulation of this problem is presented from Equation (13a) to (13g).
n n
min z = ∑ ∑ f 2ij, (13a)
i=1 j=1
n ( )
pg di − pi = ∑ fij − f ji , i = 1, 2, ..., n, (13b)
i=1
− x Fmax maxij ij ≤ fij ≤ xijFij , i, j = 1, 2, ..., n, (13c)
0 ≤ pg ≤ pg,maxi i , i = 1, 2, ..., n, (13d)
xi,j ∈ {0, 1} , i = 1, 2, ..., n, (13e)
xi,i = 0, i = 1, 2, ..., n, (13f)
xi,j = 0, i, j = {1, 5; 5, 1; 2, 4; 4, 2} (13g)
where fij is the power flow through the line that connected nodes i and j; p
g
i is the total power
generation at node i, pdi is the total power consumption at node i; xij is a binary variable that defines
the line between nodes i and j is built; z is the objective function defined as the sum of the square
power flows (sensitivity index without physical interpretation). Note that pg,maxi is the maximum
power generation at node i, Fmaxij is the maximum power flow through the line that connects nodes i
and j, and n is the number of nodes.
It is important to note that the main idea of this optimization problem is to determine the set of
lines that need to be built to supply power to all the nodes. An important aspect in this formulation is
that it is not unique since [48] has used a set of lines to reach the solution.
Figure 2 presents the GAMS implementation of the mathematical model Equations (13a)–(13g).
Some important aspects about GAMS implementation can be extracted from this interface.
• It works with plain text using five main steps: (i) definition of all the sets related to the
variables domain; (ii) definition of scalars (constant numbers), parameters (constant vectors),
and matrices (constant matrices); (iii) definition of variables and their nature, i.e., continuous,
binaries, or integers; (iv) declaration of the names of equations and their mathematical structures;
and (v) solution of the model with the appropriate structure, i.e., maximization or minimization.
• The mathematical structure is pretty similar to the symbolic formulation (see model
Equations (13a)–(13g)).
Energies 2020, 13, 2289 8 of 20
• This is an ideal toolbox that introduces to mathematicians and engineering students mathematical
optimization since it allows to concentrate on the development of well-structured mathematical
models to solve physical problems without focusing on the solution techniques.
• The implementation of any mathematical model in GAMS requires only a few concepts on
computer programming, which is an advantage in comparison with metaheuristic approaches in
MINLP optimization.
Once the optimization is solved by GAMS, we can visualize the solution variables, as depicted
in FigureE3ne.rgies 2020, xx, 5 8 of 20
1 SETS
2 i Set t h a t conta ins a l l the nodes /N1∗N5/
3 g Set t h a t conta ins a l l the generators /G1/
4 map( g , i ) Re l a t ion between generators and nodes /G1 . N1/
5 a l i a s ( i , j ) ;
6 SCALARS
7 Fijmax Maximum flow allowed in the branch i− j /100/
8 Pgmax Maximum power generat ion /150/;
9 PARAMETER LOAD( i )
10 /N1 0 ,N2 25 ,N3 30 ,N4 20 ,N5 60/;
11 VARIABLES
12 Z Obj . function
13 f ( i , j ) Power flow in l i n e s
14 p ( g ) Power generat ion ;
15 BINARY VARIABLE
16 x ( i , j ) Construct ion of the l ine between node i and j ;
17 ∗ Minimum and maximum bounds and f i x e d v a r i a b l e s
18 p . lo ( g ) = 0 ; p . up ( g ) = Pgmax ;
19 x . fx ( ’N1 ’ , ’N5 ’ ) = 0 ; x . fx ( ’N5 ’ , ’N1 ’ ) = 0 ; x . fx ( ’N2 ’ , ’N4 ’ ) = 0 ;
20 x . fx ( ’N4 ’ , ’N2 ’ ) = 0 ; x . fx ( ’N1 ’ , ’N1 ’ ) = 0 ; x . fx ( ’N2 ’ , ’N2 ’ ) = 0 ;
21 x . fx ( ’N3 ’ , ’N3 ’ ) = 0 ; x . fx ( ’N4 ’ , ’N4 ’ ) = 0 ; x . fx ( ’N5 ’ , ’N5 ’ ) = 0 ;
22 EQUATIONS
23 OBJ Objec t i ve funt ion
24 Balance ( i ) Power balance a t each node
25 MaxFij ( i , j ) Maximum flow
26 MinFi j ( i , j ) Minimum flow
27 Radial Radial topology ( t r e e ) ;
28 OBJ . . z =E= sum( i , sum( j , sqr ( f ( i , j ) ) ) ) ;
29 Balance ( i ) . . sum( g$map ( g , i ) ,p ( g ) ) − LOAD( i ) =E= sum( j , f ( i , j ) − f ( j , i ) ) ;
30 MaxFij ( i , j ) . . f ( i , j ) =L= FijMax∗x ( i , j ) ;
31 MinFi j ( i , j ) . . f ( i , j ) =G= −FijMax∗x ( i , j ) ;
32 Radial . . sum( i , sum( j , x ( i , j ) ) ) =E= card ( i )−1;
33 MODEL INTERCONECTION / a l l /;
34 SOLVE INTERCONECTION US MINLP min z ;
35 DISPLAY z . l , p . l , x . l , f . l ;
Figure 2. Example of tFhigeuirme 2p.lEemxamenptlaetoiof ntheofimapnleomnelinntaetaiornmoifxaeMd-IiNnLtePgmerodperlobgyrGamAMmSi.ng (MINLP) model
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EnergieSso2m02e0,ixmx,p5ortant aspects about GAMS implementation can be extracted from this interface.9 of 20
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1W,3 , f = 30 MW,
1,4 = 4,5 = AMPL so1ft,3ware).
f1,4 = 80IMn aWdd, iatniodn, ft4h,5e r=em6a0inMdeWr o(ftvhairsiaisbltehseisszaemroe, wsohliuchtiiomnprleiepsothrtaetdthiins i[s4t8h]euospitnimgatlhseolAutMionPLofstohfetware).
In additiomno,dtehl esinrecemitafiunldfilelsr aolfl tvhaercioanbslterasinist iznetrhoe, mwahthicehmaimticpallimesodthelaEtqtuhaitsioinssth(1e3ao)p–(t1i3mg)aul seodluintitohnis of the
example to present the main characteristics of any mathematical optimization via GAMS package. For
more details about the software, please refer to [47] and [48].
3.2. Definition of the Renewable Generation Profiles
The forecast of the renewable generation profiles has been carried out by implementing the
methodology described in [33]. This methodology works with artificial neural network (ANN) and
combined receding horizon control. The receding horizon control works as a moving time horizon that
calculates the economic dispatch model in each period using the forecast of the renewable generation
profiles estimated by the ANN approach. The implemented methodology has the following steps:
• The ANN predicts the renewable generation profiles in periods n, where n is the
prediction horizon.
• Solving the economic dispatch model with GAMS, the reallocation of battery energy storage
systems is achieved.
• Employing the previous data of the renewable generation profiles from t−m to t, the forecast of
the profiles is recomputed for t + 1 period.
For more details of this methodology, see [33].
Table 1 lists the ANN settings for each type of renewable energy source, which has implemented
in MATLAB using ntstool. The ANN was configured with 70%, 15%, and 15% of the data for training,
adjustment, and validation processes, respectively. Figure 4 illiterates the ANN scheme for the wind
power forecasting.
Table 1. Parameters for wind and solar generation forecasting.
Wind Solar Power
Inputs Temperature, humidity, pressure, and time Temperature and time
Output Wind speed Solar radiation
Delay number 4 6
Hidden neurons 12 18
Training optimizer Levenberg–Marquardt algorithm Levenberg–Marquardt algorithm
Energies 2020, 13, 2289 9 of 20
model since it fulfills all the constraints in the mathematical model Equations (13a)–(13g) used in this
example to present the main characteristics of any mathematical optimization via the GAMS package.
For more details about the software, please refer to [46,47].
3.2. Definition of the Renewable Generation Profiles
The forecast of the renewable generation profiles has been carried out by implementing the
methodology described in [33]. This methodology works with artificial neural network (ANN) and
combined receding horizon control. The receding horizon control works as a moving time horizon that
calculates the economic dispatch model in each period using the forecast of the renewable generation
profiles estimated by the ANN approach. The implemented methodology has the following steps:
• The ANN predicts the renewable generation profiles in periods n, where n is the
prediction horizon.
• Solving the economic dispatch model with GAMS, the reallocation of battery energy storage
systems is achieved.
• Employing the previous data of the renewable generation profiles from t−m to t, the forecast of
the profiles is recomputed for the t + 1 period.
For more details of this methodology, see [33].
Table 1 lists the ANN settings for each type of renewable energy source, which has implemented
in MATLAB using ntstool. The ANN was configured with 70%, 15%, and 15% of the data for training,
adjustment, and validation processes, respectively. Figure 4 illiterates the ANN scheme for the wind
power forecasting.
Table 1. Parameters for wind and solar generation forecasting.
Wind Solar Power
Inputs Temperature, humidity, pressure, and time Temperature and time
Output Wind speed Solar radiation
Delay number 4 6
Hidden neurons 12 18
Training optimizer Levenberg–Marquardt algorithm Levenberg–Marquardt algorithm
x(t) 1:4 W W
4
y(t) 1:4 W + + y(t)
1 1
b
b 112
Output Layer
Hidden Layer with Delays
Figure 4. Artificial neural network (ANN) scheme for wind speed prediction [49].
3.3. Flow Diagram of the Proposed Approach
To summarize the application of the proposed methodology in a dc distribution network for
location-reallocation of batteries the flow chart depicted in Figure 5 was implemented.
Energies 2020, 13, 2289 10 of 20
DC grid Begin: GAMSimplementation ANNs
Select the objective
function z = z1
Solve the model
using a MINLP solver Report solution
Select the objective
function z = z2
Solve the model
using a MINLP solver Report solution
Select the objective
function z = z1 + z2
Solve the model
using a MINLP solver Report solution
End: Results’ analysis
Figure 5. Flow chart of the proposed optimization approach for optimal location-reallocation of
batteries in dc grids.
4. Test Systems
The validation of the proposed mathematical model for the optimal location-reallocation of
batteries in dc distribution networks via GAMS implementation is made inside a 21-node test
feeder [18]. All the information about this test feeder is presented below.
The 21-node test system is a radial test feeder composed of 21 nodes and 20 branches in radial
connection, where the slack node is located at node 1. The configuration of this dc network is presented
in Figure 6.
The information about loads, energy purchase cost, demand variation, and batteries for this test
system is reported in Tables 2–4.
Table 2. Parameters for the 21-node test feeder.
From i To j Rij [p.u.] Pj [p.u.] From i To j Rij [p.u.] Pj [p.u.] From i To j Rij [p.u.] Pj [p.u.]
1 (slack) 2 0.0053 0.70 7 9 0.0072 0.80 15 16 0.0064 0.23
1 3 0.0054 0.00 3 10 0.0053 0.00 16 17 0.0074 0.43
3 4 0.0054 0.36 10 11 0.0038 0.45 16 18 0.0081 0.34
4 5 0.0063 0.04 11 12 0.0079 0.68 14 19 0.0078 0.09
4 6 0.0051 0.36 11 13 0.0078 0.10 19 20 0.0084 0.21
3 7 0.0037 0.00 10 14 0.0083 0.00 19 21 0.0081 0.21
7 8 0.0079 0.32 14 15 0.0065 0.22 – – – –
All parameters are in per unit considering as bases Pbase = 100 kW and Vbase = 1 kV.
Energies 2020, 13, 2289 11 of 20
ac
slack (v) dc
Initial battery location
6 2 1 8
4 3 7
5 9
12
11 10
20
13
14 19
17
21
16 15
18
Figure 6. Electrical configuration for the 21-nodes test system.
Table 3. Energy purchasing cost and hourly demand.
Time [h] CoE [p.u.] Demand Time [h] CoE [p.u.] Demand Time [h] CoE [p.u.] Demand
Variation Variation Variation
[%] [%] [%]
0.5 0.8105 34 8.5 0.9263 62 16.5 0.9737 90
1.0 0.7789 28 9.0 0.9421 68 17.0 1 90
1.5 0.7474 22 9.5 0.9579 72 17.5 0.9947 90
2.0 0.7368 22 10.0 0.9579 78 18.0 0.9895 90
2.5 0.7263 22 10.5 0.9579 84 18.5 0.9737 86
3.0 0.7316 20 11.0 0.9579 86 19.0 0.9579 84
3.5 0.7368 18 11.5 0.9579 90 19.5 0.9526 92
4.0 0.7474 18 12.0 0.9526 92 20.0 0.9474 100
4.5 0.7579 18 12.5 0.9474 94 20.5 0.9211 98
5.0 0.8000 20 13.0 0.9474 94 21.0 0.8947 94
5.5 0.8421 22 13.5 0.9421 90 21.5 0.8684 90
6.0 0.8789 26 14.0 0.9368 84 22.0 0.8421 84
6.5 0.9158 28 14.5 0.9421 86 22.5 0.7947 76
7.0 0.9368 34 15.0 0.9474 90 23.0 0.7474 68
7.5 0.9579 40 15.5 0.9474 90 23.5 0.7211 58
8.0 0.9421 50 16.0 0.9474 90 24.0 0.6947 50
The energy-based cost is COP$/kWh 479.3389 based on the prices in May 2019 for the CODENSA utility [11].
Table 4. Parameters associated with batteries and their initial locations.
Node ϕb pb,max pb,min Node ϕb pb,max pb,min Node ϕb pb,max pb,min
7 0.0625 4 −3.2 10 0.0813 3.2 −2.4616 15 0.0813 3.2 −2.4616
In the case of renewable generation, we consider the wind power plant is connected at node 12
with a maximum power capability of about 221.52 kW, and the photovoltaic source is connected at
node 21 with a maximum power capability about 281.58 kW. Note that these multiply the maximum
capacities of the normalized generation curves reported in Table 5.
Energies 2020, 13, 2289 12 of 20
Table 5. Normalized power generation curve.
Time [h] PWT [p.u.] PPV [p.u.] Time [h] PWT [p.u.] PPV [p.u.] Time [h] PWT [p.u.] PPV [p.u.]
0.5 0.6303 0 8.5 0.8271 0.0403 16.5 0.9892 0.4193
1.0 0.6194 0 9.0 0.8523 0.1344 17.0 0.9652 0.2784
1.5 0.6098 0 9.5 0.8788 0.2710 17.5 0.9244 0.1373
2.0 0.6050 0 10.0 0.9064 0.3673 18.0 0.8607 0.0374
2.5 0.6122 0 10.5 0.9328 0.4584 18.5 0.7743 0.0007
3.0 0.6411 0 11.0 0.9520 0.6125 19.0 0.7251 0
3.5 0.6927 0 11.5 0.9640 0.8134 19.5 0.7167 0
4.0 0.7395 0 12.0 0.9700 0.9122 20.0 0.7167 0
4.5 0.7779 0 12.5 0.9748 0.9633 20.5 0.7251 0
5.0 0.7887 0 13.0 0.9784 1.0000 21.0 0.7263 0
5.5 0.7671 0 13.5 0.9832 0.9582 21.5 0.7179 0
6.0 0.7479 0 14.0 0.9880 0.8791 22.0 0.7095 0
6.5 0.7287 0 14.5 0.9940 0.7308 22.5 0.6987 0
7.0 0.7371 0 15.0 0.9988 0.7645 23.0 0.6915 0
7.5 0.7731 0 15.5 1.0000 0.6866 23.5 0.6867 0
8.0 0.8031 0.0016 16.0 0.9964 0.5893 24.0 0.6831 0
5. Computational Validation
All simulations were carried out on a desktop computer running on INTEL(R) Core(TM) i7-7700,
3.60 GHz, 8 GB RAM with 64-bit Windows 10 Pro using GAMS 25.1.3 with the nonlinear large-scale
solver BONMIN licensed by Universidad Tecnológica de Bolívar in Colombia.
5.1. Simulation Conditions and Initial Function Values
To simulate the 21-node test feeder, we considered the following facts:
• The batteries begin and end the day with a total charge of about 50%. During the day,
this state-of-charge can vary between 10% and 90%, as recommended for Ion-Lithium batteries
in [11].
• Both objective functions are evaluated with the initial position of the batteries reported in
the previous section, to identify the base cases and the possible improvements when they
are reallocated.
• A linear combination of both objective functions was made to identify the effect of adding energy
purchase costs with energy losses costs.
Once the minimization of the energy purchase cost of energy and the minimization of the cost of
the daily energy losses were performed, we found the objective function z1 took COP$/day 1,139,524.00
(with z2 = 131,198.70) (see Equation (1)), and the objective function z2 took COP$/day 52,957.92
(with z1 = 1,941,395.00) (see Expression (2)). These values are considered the base cases for each
objective function.
5.2. Optimal Reallocation of Batteries
Here, we present the optimal location-reallocation of batteries considering both objective functions.
Table 6 reports each objective function and its corresponding BESS’ location.
Table 6. Optimal location-reallocation of batteries considering different objective functions.
Minimization of z1 [COP$/Day] Minimization of z2 [COP$/Day]
z1 : 1,089,974.00 (z2 = 87426.51) z2 : 47,209.95 (z1 = 1843467.00)
Battery type 1: 1 Battery type 1: 13
Battery type 2: {2, 3} Battery type 2: {20, 21}
Minimization of the Linear Combination z1 + z2 [COP$/Day]
z1 : 1,188,233.00 z2 : 94,347.07
Battery type 1: 13 Battery type 2: {9, 21}
Energies 2020, 13, 2289 13 of 20
To understand the optimal reallocation of batteries as a function of the performance indicator,
let us plot the locations of the batteries in the test feeder, as depicted in Figure 7.
ac
slack (v) dc
Initial battery location
6 2 1 8 min z1
min z2
4 3 7 min z1 + z2
5 9
12
11 10
20
13
14 19
17
21
16 15
18
Figure 7. Reallocation of batteries in the 21-node test feeder.
Based on the location-reallocation of the batteries in Table 6 (see Figure 7), the following facts can
be highlighted:
• When the objective of the optimization is to minimize the energy purchase costs at the conventional
sources, i.e., z1, the system reduces the daily operation cost of about 4.35% passing from
COP$/day 1,139,524.00 to COP$/day 1,089,974.00, which implies a reduction per day about
COP$/day 49,550. In addition, when we observe the total costs of the daily losses, it passes from
COP$/day 131,198.70 to COP$/day 87,426.51; which is traduced in 33.36% of the energy losses
reduction. These results confirm that the optimal reallocation of the batteries from nodes 7, 10,
and 15 to nodes 1, 2, and 3 has a positive effect on both the objective functions.
• When the objective of the optimization model is to minimize the daily energy losses, i.e., z2,
this function is reduced from COP$/day 52,957.92 to COP$/day 47,209.95, this corresponds
a reduction about 10.85%. This reduction is achieved by reallocating batteries from nodes
7, 10, and 15 to nodes 13, 20, and 21. In addition, the costs of the energy purchase pass
from COP$/day 1,941,395.00 to COP$/day 1,843,467.00, which corresponds to a reduction of
about 5.04%
• When the objective function is the linear combination of z1 and z2, i.e., z1 + z2, the objective
function is COP$/day 1,282,580.07; this implies an increment regarding the base of the case of
the energy purchase about COP$/day 11,857.37 per day of operation. In the case of daily energy
losses reduction, the linear combination reaches a reduction of COP$/day 711,772.85 per day
of operation.
The previous analyses regarding different objective functions allowed us to conclude that the
21-node test feeder is always positive, considering the energy purchase costs as the objective function
since it has the most important effect on the daily operation cost for the test feeder. In addition,
reducing z1 also allows to reduce the daily energy losses with respect to the base case. On the other
hand, when we minimize the daily energy losses, this is reduced. Nevertheless, it also produces a
negative effect on the energy purchase cost with an important increment in this objective function.
For this reason, here, the minimization of the daily energy loss reduction as an adequate indicator for
the optimal operation of batteries in dc networks is discarded.
Energies 2020, 13, 2289 14 of 20
Regarding the reallocation of the batteries, we observe that when we reduce the energy purchase
costs, all the batteries are positioned near the slack node, as these locations allow to charge all the
batteries with minimum losses when the daily energy cost is lower to inject this power when this
cost increases.
5.3. Complementary Analysis
To understand whether the reallocation of the batteries satisfy the operative conditions imposed
on the mathematical model, i.e., begin and end the day with 50% considering possible variations
between 10% and 90%, Figure 8 presents the behaviors for both base cases and the three possible
locations, reported in Table 6 for the battery type 1.
100.00
90.00
80.00
70.00
60.00
50.00
40.00
Node 7 (base case min z1)
30.00 Node 7 (base case min z2)
20.00 Node 1 (min z1)
Node 13 (min z2)
10.00 Node 13 (min z2 + z2)
0.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Figure 8. Behavior of battery type 1 in its different locations.
The behavior of the battery type 1 when it is located at different nodes considering variations in
the objective function allows to identify the following important facts:
• When the objective function is minimizing the total energy purchase cost, i.e., z1, this battery
charges at its maximum admissible value (see periods comprehended between 0.5 h to 6).
In addition, when energy cost is expensive, this battery begins to discharge continuously during
fourteen hours (see periods comprehended between 8 h to 22). Finally, this battery recovers this
charge in the final part of the time.
• Regarding the minimization of energy losses, i.e., z2, we observe that the batteries charge between
50% and 80% during the day. This behavior is explained by the fact that the battery works as a
generator or load; it is modifying the power flowing through the lines, which implies that it is
also affecting the power losses. For this reason, it has small changes in its state-of-charge to help
and minimize the total power losses during the day of operation.
• In the case of combining both objective functions linearly, battery type 1 experiences both
behaviors, as previously reported, at the same time, i.e., soft state-of-charge variations overpass
values lower than 50% at the end of the day.
Note that the main message of the battery behavior is regarding the different possibilities of
having state-of-charges behavior as a function of the performance indicator (i.e., objective function) as
well as the possible location of it. Nevertheless, in all the simulation cases, the proposed optimization
model (Equations (1)–(12)) is feasible and allows to determine the best operation practice for batteries
depending on the operative consigns imposed by the utility, which becomes the proposed optimization
model the main contribution of this research.
SoC [%]
Energies 2020, 13, 2289 15 of 20
Figure 9 reports the power generation in the slack node for the base cases as well as for the
different battery locations. From this plot, we can observe that:
• When the objective function is minimizing z1, the conventional generator is used to charge all the
batteries to reach their maximum admissible values at the beginning of the day. This conventional
generator is also used to recover the final state of the charge imposed, i.e., 50% at the end of the
day. In the rest of the periods, the energy provided by the conventional generator is zero, which
minimizes the total purchase costs in the spot market and also allows to maximize the use of
renewable sources.
• In the case of minimization of z2, we can see that the conventional source generates during all the
periods since this generation allows redistributing line power flows, which helps minimize the
total cost of the energy losses.
• Regarding the linear combination of the objective functions, it is possible to observe that at
the beginning and the end of the day, the conventional generator is used for charging all the
batteries while in the intermediate times, it is used for redistributing power flows; in other
words, the behavior of the conventional generator is a linear combination of both analyses
mentioned above.
• The behavior of the conventional generators shows that in relation to the energy purchase cost
minimization, its generation appears mainly in periods of time where solar energy is absent, which
can be considered as an indicator for the utility to introduce additional renewable energy resources
(i.e., small-hydro power [50]) to complement solar and wind sources in order to reduce to zero the
conventional generation in normal operating conditions. This also can help to indirectly reduce
greenhouse emissions in isolated grids power supply by diesel or in predominantly thermal
interconnected systems.
15.00
13.50 Base case min z1
Base case min z2
12.00 min z1
10.50 min z2
min z2 + z2
9.00
7.50
6.00
4.50
3.00
1.50
0.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Figure 9. Behavior of the conventional generation (e.g., slack node) for all the cases studied.
In conclusion, it is important to keep in mind that the behavior of all the variables in the proposed
mathematical MINLP model highly depends on the objective function selected since it guides them
to minimize or maximize the benefit established by the electricity company. In addition, all the
solutions reported in this research can be taken as indicators of the grid performance; nevertheless,
these are not taken as absolute results, since they also depend on the renewable generation availability,
grid operative conditions, and demand behavior.
Regarding the computational effort of the proposed approach for optimal locating-reallocating
batteries in dc grids, it is important to highlight that the GAMS and its SCIP solver takes about 10 min to
Conventional generation [p.u.]
Energies 2020, 13, 2289 16 of 20
define the optimal position of the batteries, which can be considered as a pretty small time considering
the complexity of the problem (3 batteries in 21-nodes generates 1330 possible locations). In addition,
if the batteries are considered in fixed locations, then, the optimization problem is transformed from
an MINLP into a nonlinear programming one, reducing the processing times to 2 s for knowing the
economic dispatch output, which allows access to the utilities having multiple scenarios of simulation
before defining the final day-ahead economic dispatch.
5.4. Scalability of the Proposed Model
To demonstrate the scalability of the proposed approach, we present an additional simulation
case regarding batteries’ location and reallocation in dc grids by using the 30-node test feeder reported
in [11]. In this test system, there are three batteries in nodes 3, 15, and 22. In addition, the objective
function reported in that paper corresponds to the daily energy losses cost, which takes a value
of COP$/day 254,539.39. Once we apply our proposed reallocation approach, this cost moves to
COP$/day 244,595.39 by reallocating these batteries to nodes 18, 23, and 24, respectively, which
implies a daily reduction of about 3.91% regarding power loss minimization, demonstrating that
the proposed optimization approach is applicable to dc grids with a different number of nodes, i.e.,
it is scalable.
It is important to point out that the computational cost of the proposed approach for this test
feeder is 14 and takes minutes to decide where the batteries must be reallocated; nevertheless, when the
batteries are fixed, this time is reduced to 3 s, which implies that this methodology is perfectly for
day-ahead analysis for dc grids with batteries, since it allows to create multiple simulation scenarios in
less time.
6. Conclusions and Future Works
The problem of the optimal location-reallocation of batteries in dc distribution networks has been
analyzed in this paper through an MINLP formulation. This mathematical model has binary variables
associated with the position of the batteries that are modified as a function of the performance indicator,
i.e., minimization of the energy purchase costs in the conventional generators or minimization of the
total costs of the daily energy losses. Numerical results confirm that in all the analyzed scenarios,
the re-positioning of batteries allows to achieve better objective function values by modifying the
daily state-of-charge performances on them. In addition, the behavior of the conventional power
generator is highly conditioned by the performance indicator since this variable defines the total
energy purchase costs in the spot market and also has an influence over the power flow redistribution
in lines regarding power loss minimization. This means that this variable is highly sensitive to the
proposed MINLP model.
For future works, we state the following: (i) to propose a mixed-integer convex optimization
model to deal with the non-linearities of the power balance equations to guarantee the uniqueness
of the global optimum solution via branch and bound methods added with a second-order cone or
semi-definite programming methods, (ii) to employ heuristic methods to avoid the usage of specialized
software in the solution of the MINLP model to develop free applications for engineering students
and small electricity companies, (iii) to extend the proposed MINLP formulation to alternating current
networks considering active and reactive power capabilities in batteries via optimal control of power
electronic converters that interface them, (iv) to consider different battery’s aspects in the model such as
lifecycle, self-discharge phenomenon, or power losses analysis to have more realistic models that can
affect the grid behavior in the short-term horizon, and (v) to propose a hybrid optimization problem
based on the convex optimization for the nonlinear optimization part of the MINLP model (economic
dispatch problem) and metaheuristics that can help address the integer part (location of batteries and
renewables) by conforming a master-slave optimization methodology.
Energies 2020, 13, 2289 17 of 20
Author Contributions: Conceptualization, O.D.M. and W.G.-G.; Methodology, O.D.M. and W.G.-G.; Investigation,
O.D.M. and W.G.-G.; Writing—review and editing, O.D.M., W.G.-G., and E.R.-T. All authors have read and agreed
to the published version of the manuscript.
Funding: This work was partially supported by the Universidad Tecnológica de Bolívar under grant CP2019P011
associated with the project: “Operación eficiente de redes eléctricas con alta penetración de recursos energéticos
distribuidos considerando variaciones en el recurso energético primario”.
Conflicts of Interest: The authors declare no conflicts of interest.
Abbreviations
The following abbreviations and nomenclature are used in this manuscript:
Acronyms
AC Alternating current.
COP Colombian pesos.
ANN Artificial neural network.
BESS Battery energy storage system.
DC Direct current.
GAMS General algebraic modeling system.
MINLP Mixed-integer nonlinear programming.
MCOP Millions of Colombian pesos.
PV Photovoltaic.
WT Wind turbine.
Sets and subscripts
B Set of batteries.
N Set of nodes.
T Set of periods of time.
b Type of batteries.
t Periods of time.
i, j Nodes.
Parameters
CoEi,t Energy purchase cost in the conventional source in the node i at the period of time t.
Gij Conductance value that relates nodes i and j.
Nmaxb Maximum number of batteries available.
pmaxi,t Maximum power generation in the conventional source in the i at the period of time t.
pmini,t Minimum power generation in the conventional source in the i at the period of time t.
pb,maxi,t Minimum power in the battery b connected at i at the period of time t.
pb,mini,t Maximum power in the battery b connected at i at the period of time t.
pdg,maxi,t Minimum power generation in the distributed generator in the i at the period of time t.
pdg,mini,t Maximum power generation in the distributed generator in the i at the period of time t.
pdi,t Power consumption in the i at the period of time t.
SoCb, f ini Final state of charge in the battery b connected to the node i.
SoCb,inii Initial state of charge in the battery b connected to the node i.
vmaxi Maximum voltage bound at node i.
vmini Minimum voltage bound at node i.
ϕbi Charge coefficient of the battery b connected at node i.
∆t Length of the period of time.
Variables
pi,t Power generation in the conventional source at node i in the period of time t.
pbi,t Power input/output in battery b connected at node i in the period of time t.
pdgi,t Renewable power generation at node i in the period of time t.
SoCbi,t State of charge in battery b connected at node i in the period of time t.
SoCbi,t−1 State of charge in battery b connected at node i in the period of time t− 1.
vi,t Voltage profile in the node i at the period of time t.
xbi Binary decision variable regarding the installation of the battery b at node i.
z Objective function.
z1 Objective function related to the energy purchase costs in conventional sources.
z2 Objective function related to the cost of the daily energy losses.
Energies 2020, 13, 2289 18 of 20
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