Dynamic Demand Modeling Incorporating Renewable Energy Sources
Using a Population-Based Optimization Method
LUIS CARLOS PÉREZ GUZMÁN1, GINA MARÍA IDÁRRAGA OSPINA1,
FREDDY BOLAÑOS MARTÍNEZ2, SERGIO RAÚL RIVERA RODRÍGUEZ3
1Facultad de Ingeniería Mecánica y Eléctrica,
Universidad Autónoma de Nuevo León
San Nicolás de los Garza, Nuevo León,
MEXICO
2Departamento de Energía Eléctrica y Automática,
Universidad Nacional de Colombia SEDE Medellín,
COLOMBIA
3Departamento de Ingeniería Eléctrica,
Universidad Nacional de Colombia SEDE Bogotá,
COLOMBIA
Abstract: - Due to the inclusion of distributed generation (DG) in microgrids (MGs), the accelerated growth in
demand, and environmental concerns, suitable management and operational strategies are imperative. The
utilization of wind and solar energy has rapidly increased in MGs. However, due to the uncertainties these
systems present, accurately predicting energy generation remains challenging. This necessitates modeling the
system’s random variables (such as renewable resource output and possibly load demand) using appropriate
and feasible methods. The primary objective of this article is to determine the optimal setpoints for renewable
energy sources (RES) and all elements involved in the MG, minimizing the total operation cost. The system
comprises wind turbines (WT), photovoltaic panels (PV), energy storage systems (ESS), and electric vehicles
(EVs). Weibull distribution and the Hottel and Liu Jordan equations are employed to determine the potential
available capacity of wind and solar energy generation, respectively. ESS is utilized to enhance MG
performance. For optimal management, a comprehensive mathematical model with practical constraints for
each MG element is extracted. An efficient Population-Based Incremental Learning (PBIL) metaheuristic
method is proposed to solve the optimization objective in an MG, demonstrating that this energy management
system optimizes and effectively coordinates DG and ESS energy generation considering economic
considerations. Finally, PBIL is compared with a commonly used model, Particle Swarm Optimization (PSO),
across various scenarios, analyzing and evaluating their outcomes, showcasing a reduction in operation costs.
Key-Words: - optimization, microgrid, uncertainty, cost, algorithm, particle swarm optimization, energy
management, Weibull, solar photovoltaic, wind.
Received: March 29, 2023. Revised: January 11, 2024. Accepted: February 24, 2024. Published: April 25, 2024.
1 Introduction
The utilization of distributed generation (DG)
technologies to address continuous improvement,
efficiency, and reliability within the electrical
system, coupled with the competition in the
electricity market and the reduction of greenhouse
gases, represents a relatively new market being
adopted by both users and power-generating
companies. DGs encompass renewable units such as
wind turbines (WT), photovoltaic panels (PV), or
biomass, alongside non-renewable units like fuel
cells, microturbines, gas engines, diesel generators,
etc. DGs eliminate the need for the transmission
system by being installed close to the demand. The
integration and control of DGs with storage devices
and flexible loads can form a low-voltage
distribution network, termed a microgrid (MG),
capable of operating in isolated mode or
interconnected with the main distribution grid as an
entity. This implies functioning either for self-
consumption or facilitating energy import/export
to/from the MG, [1], [2], [3], [4], [5], [6].
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Today, the implementation of the MR concept,
due to its low operational costs within the system
and environmental aspects, is expanding across the
distribution network. From the perspective of MR
owners, economic operation is crucial. Given that
MRs can participate in energy markets and provide
ancillary services, appropriate scheduling becomes
essential. Therefore, a suitable strategy for MR
operation must be pursued, [2].
MRs often face difficulties meeting total demand
due to energy shortages, as the energy generated by
DG sources is sometimes insufficient. This
challenge arises from the intermittent nature of
certain renewable energy (RE) resources,
necessitating an energy management system to
address this issue. Energy management systems for
a microgrid represent relatively new and popular
topics that have recently garnered significant
attention.
One of the main challenges in managing certain
renewable resources like wind and solar energies is
the issue of uncertainty in their behavior. That is,
the actual energy production from these resources
differs from the forecasted values in real time. This
can be defined as the probability of the difference
between the predicted and actual values. In other
words, owing to the uncertainty in energy
production from these resources, the operator’s
responsibility is to maintain a balance between
production and consumption, which poses certain
challenges. Therefore, system operators attempt to
provide a certain amount of reserve energy through
the energy storage system (ESS) to cover
uncertainty in energy production and maintain
system security at the desired level, [3].
MR users can indeed overcome this shortage by
purchasing more energy from the utility company or
by increasing the number of generating sources.
However, these solutions often come with higher
emission and energy costs, referring to either
purchasing from the grid or the cost of the elements
involved. Another solution to mitigate this problem
and maintain a balance between system production
and consumption is by reducing customer
consumption during periods of energy scarcity. This
practice of demand competing with offers made by
production units is termed ’Demand Response’
(DR). DR is defined as changes in end-user
electrical usage in their normal consumption
patterns in response to changes in electricity prices
over time or incentives designed to induce lower
electricity usage during high wholesale market
prices or when system reliability is compromised,
[3].
To optimize MR operation, different objective
functions have been considered, as in [2], [3], [4],
along with the utilization of various types of RE
sources. One such source is wind energy, which has
emerged as a significant RE alternative. However,
due to its fluctuations, various methods have been
considered for energy generation forecasting for
optimal scheduling of WT, [5]. In [2] and [6], a
probabilistic method for wind speed prediction
based on recorded values was proposed. This model,
called ’Weibull Distribution,’ is used to model
stochastic variables and has been employed by
various authors for short-term wind speed
prediction. Consequently, WT output power can be
estimated based on the technical constraints
specified by the manufacturer.
Both wind and solar energy encounter challenges
regarding fluctuation in power production.
References as [7], [8], [9], [10], [11], address this
issue based on certain established equations. For
proper system functioning, configuring the optimal
amount of purchased energy before system
operation initiation is crucial. This is because
without knowledge of the available PV power on an
operational day, determining the exact quantity
required from the grid becomes difficult.
Photovoltaic energy is estimated by calculating solar
energy radiation, using the modified Hottel equation
and the Liu-Jordan equation. These equations also
address the issue of partially cloudy/rainy weather,
determining the site-specific climate for
photovoltaic production. The authors in [12] and
[13], analyzed the values behind these equations,
such as solar constants, solar hours, declination, and
zenith angle, among other data, to achieve the
desired outcome. Therefore, to estimate
photovoltaic output power, the method described in
[14], [15], [16], [17], [18], [19], has been utilized,
comprising a set of technical formulas supported by
technical data specified by the PV manufacturer.
Regarding the previously mentioned ESS in [3], the
focus was on the State of Charge (SOC) limits for
its proper operation within the MG.
Currently, there is a growing trend towards the
use and adoption of electric vehicles (EVs) due to
fossil fuel depletion and increasing environmental
concerns. Adopting electric vehicles as an
alternative mode of transportation necessitates the
development of a charging infrastructure. The
behavior of the EV battery (BEV) in its SOC closely
correlates with the ESS. Despite varying EV
handling, displacement can be defined through a
pattern, supported by the SOC, to predict the
amount of stored energy due to such EV
displacement, [20], [21], [22].
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Optimization involves handling variations using
information from an initial concept to improve it.
Many engineering industry problems, especially in
manufacturing systems, are inherently complex and
challenging to solve using conventional
optimization techniques, [23]. Finally, an efficient
metaheuristic method, Population-Based
Incremental Learning (PBIL), is proposed to resolve
the conflicting objective problem (cost and
coordination) for optimal operation, employing a
simple algorithm that utilizes a probability vector to
generate the population, considering the highest
evaluations of the vector, [24] and [25]. To evaluate
the proposed algorithm, the management system is
applied to a typical MR consisting of multiple ER
generators, ESS, EVs, and electrical loads. The
results showcase effective coordination of GD and
ESS energy generation considering economic and
environmental considerations.
Many researchers have employed the Particle
Swarm Optimization (PSO) algorithm for
improvement purposes. The PSO algorithm is
typically utilized to minimize the operational cost of
distributed energy resources while considering
network constraints, demand response, and the
incorporation of renewable resources in electrical
studies. Previous studies have not precisely
addressed the uncertainties caused by wind turbines
and solar panels from the demand side. Favorable
results were obtained compared to the Probabilistic
Binary Particle Swarm Optimization (PBIL)
method. This study employs a programming model
to minimize the total operating costs in a Microgrid
(MR), encompassing energy generation with
stochastic behavior of Wind Turbines (WT) and
Photovoltaic (PV) panels along with associated
uncertainties. Additionally, since implementing a
real open electricity market is not feasible in many
existing distribution and energy systems due to
underdeveloped communication infrastructure, this
paper interacts with demand bids for each element
used to address this issue and create a competitive
energy market. The analysis facilitates the operator's
decision-making by observing the power behaviors
and costs of all energies to be incorporated into the
main system. This decision-making can assist the
operator in anticipating whether certain energy
sources can be practically utilized, minimizing risks
or maximizing benefits, whether in terms of energy
or economics. A multi-objective system considering
the cost analysis of uncertainty due to the
integration of renewable sources into the main
system is a future-oriented approach related to the
optimization of this work's system. A comparison
between the PSO and PBIL methods is presented in
the final section of the document, due to the greater
use of PSO in such problems, noting some
similarities in the methods. However, there is a
certain error percentage in favor of the proposed
method, reducing the estimated cost.
2 System to Model
In this section, the proposed stochastic model in the
MG, shown in Figure 1, encompasses Renewable
Energy Sources (RES) and user load demands.
Additionally, random interruptions in DGs,
variability in both the SOC of ESS and EV batteries
(BEVs) and the grid market are modeled. The
framework of the considered MG relies on planning
units to supply demand optimally and suitably
through wind and solar energy generation elements,
primarily through natural stochastic behavior. The
inclusion of EVs is an innovation that will likely
become commonplace shortly, hence its
fundamental incorporation. Energy is supplied
within 24 hours by energy generation consisting of
utility services, WT, PVs, and ESSs for EVs and
user loads. Therefore, energy production calculation
relies on operations with certain restrictions or
limits.
Fig. 1: Proposed Microgrid
The forecasted amount of wind speed, solar
irradiance, and load is generated through established
and commonly used methods today, [7], [8], [9],
[10], [11], supported by long-term historical data,
although the latter was acquired through daily
routine activities. All these values, considered as
input values at the system’s outset, represent the
average forecasted amount produced per hour in the
day. Finally, the MG generates multiple scenarios
involving possible stochastic quantities, aiming to
optimize minimum operational costs in residential
loads while considering specific constraints for each
device to address uncertainties caused by wind and
solar energy generation.
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2.1 Photovoltaic System Modeling
A simplified schematic of the developed system is
depicted in Figure 2. The maximum nominal output
of the PV or photovoltaic system used in the
proposed system is 275W per module. The ESS
consists of sufficient capacitor modules to meet
demand, with each unit’s maximum capacity set at
300 watts. To simulate different hourly load patterns
throughout the day, an estimate of the usage for
various electrical appliances commonly found in
households today was applied, resulting in variable
loads ranging from a minimum of 50W to a
maximum of 1150W.
The system’s output power is determined by
user-provided data, in addition to the values from
the following sections. This article focuses on the
northern region of Mexico, specifically in
Monterrey, Nuevo León. The city is located at
latitude 25°40’ North and longitude 100°18’West, at
an elevation of 537 meters above sea level, with a
Peak Solar Hour (PSH) of 5.2 and temperatures
around 30 degrees Celsius.
Fig. 2: Photovoltaic system
2.1.1 Hottel’s y Liu Jordan Equations
The references in [7], [8], [9], [10], [11], the Hottel
method allows for estimating global radiation under
clear atmospheric conditions based on the location’s
latitude, altitude, and climate characteristics,
categorized into four types as shown the Table 1,
[12]. This model expresses atmospheric
transmittance for direct radiation, τb, as a function of
the zenith angle, θz.
Table 1. Climate characteristics
Climate Type
rO
r1
rK
Tropical
0.95
0.98
1.02
Summer, mid-latitude
0.97
0.99
1.02
Summer, sub-artic
0.99
0.99
1.01
Winter, mid-latitude
1.03
1.01
1.00
The daily output power of the PV has been
estimated by calculating daily solar radiation. The
calculation of beam radiation τb utilized the
modified Hottel equation shown in Equation 1
(1)
where a0, a1, and k are parameters depending on the
altitude above sea level A in the geographical area
under analysis (0.1736, 0.7097, and 0.3493,
respectively). The zenith angle is represented by
cos(θz). Subsequently, the modified Liu Jordan
equation is used to find diffuse radiation τd in
Equation 2.
(2)
The solar constant Gcs is used to obtain values in
W/m2, as in [12], [13].
(3)
Using equations 1, 2, and 3, the value of total
solar radiation is obtained. The Figure 3 show the
values of irradiance.
(4)
Fig. 3: Total solar irradiation on January 1st, 2020
2.1.2 Solar Panel Power
The output power Ppv of the photovoltaic module
depends on solar energy, irradiance, ambient
temperature of the location, and the characteristics
of the module itself.
(5)
(6)
(7)
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(8)
(9)
Where the values of Table 2 are provided by the
manufacturer:
Table 2. Characteristics of the manufacturer
Tm = Module temperature in degrees Celsius °C.
Ta = Ambient temperature in degrees Celsius °C.
Gt = Total Irradiation (W/m2) at time t.
GtNOCT = Normal Operation Cell Temperature (800
W/m2).
GtSTC = Standard Test Condition (1000 W/m2).
Im = Current at maximum power in Amps.
Isc = Module short-circuit current in Amps.
Vm = Voltage at maximum power in volts.
Voc = Module open-circuit voltage in volts.
TIsc = Current temperature coefficient in °C.
TVoc = Voltage temperature coefficient in °C.
NM = Number of solar modules
FF = Fill Factor.
The Figure 4 shows the power behavior in watts
due to the production of energy by the photovoltaic
system.
2.2 Wind System Modeling
In this document, the concept of reliable capacity is
used to model uncertainty, representing the
availability of wind energy. The power fluctuation
caused by wind speed variation is not extremely
random in terms of magnitude and ramp speeds.
Fig. 4: Estimated power of the modules on January
1st, 2020
The output of wind power will be based on the
uncertainty presented by the wind speed in a region.
The probability of wind speed occurrence, i.e., the
frequency of each wind speed, is analytically
expressed by the Weibull distribution, [3].
The power curve is the most feasible method for
annual production use and is much more accurate.
For the study, an AW-1500/70 wind turbine model
from the manufacturer Acciona was used (Table 3).
Table 3. Technical characteristics of the AW
1500/70 wind turbine
Parameter
Unit
Manufacturer
Acciona (Spain)
Power
1500 watts
Diameter
70 m
Swept area
3849 m2
Power density
2.57 m2/kW
Number of blades
3
The energy produced by a wind turbine is the
result of adding all the products of the powers (Pi)
delivered in each time interval (t) by the duration of
each interval in hours during a given period (day,
month, year, depending on the desired calculation).
Therefore, the energy E is expressed in a simplified
manner as shown in Equation 10.
(10)
Where Pi is the power at each time interval in watts,
and is the duration of each time interval in hours.
This approach allows for a comprehensive
understanding of wind energy generation over a
specified period.
2.2.1 Weibull Distribution
To handle the uncertainty in wind energy generation
due to wind speed, a statistical probability method
like the Weibull Distribution is employed for a
given time frame. Data is collected over a month,
comprising 24-hour intervals, in a region where the
installation of these wind turbines is planned. An
anemometer captures these data points, as displayed
in Figure 5, obtained from [24]:
Fig. 5: Recorded wind speed data
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When wind speed measurements are taken annually,
the Weibull probability distribution function
successfully describes frequency curves, [26]. The
Weibull probability density function is expressed as:
(11)
here, k is the shape parameter, and c is the scale
parameter. The equation 11 provides the probability
that the wind speed falls within a 1 m/s interval
centered at that speed. To fit the frequency data to
the Weibull function, the values of parameters k and
c will be determined via a non-linear fitting process,
show in Figure 6. This process aims to find suitable
values for these parameters based on the wind speed
data illustrated in Figure 5.
Fig. 6: Fit Weibull graph
2.2.2 Average Wind Power
The turbine power is independent of the Weibull
function; that is, analyzing wind speed at a location
helps calculate the average velocity. The Weibull
distribution accurately determines the probability of
a specific wind speed occurring. When wind data is
modeled using f(υ), the average wind power (Pυ)
perpendicular to an area (A) is given by Equation 12
[26]:
(12)
It can be shown that when f(υ) is the Weibull
distribution function, the average power delivered is
given by Equation 13:
(13)
The power output constraints of a wind turbine
utilized in this study are illustrated below:
(14)
Here, and represent the cut-in and rated
wind speeds of the turbine (m/s), respectively, while
signifies cut-out of wind velocity.
2.3 Energy Storage System Modeling
The Energy Storage System (ESS) consists of
electrochemical batteries electrically connected to
an energy source and the load, playing a vital role in
managing an Interconnected Microgrid (IM). To
optimize the operational planning of an IM, a
suitable mathematical model for the ESS has been
developed in [3], [5], [6].
As per Equation 15, the battery’s charging and
discharging rates in each one-hour interval over the
operational period should stay within predetermined
limits, as defined in Equation 16.
The charge and discharge rate of the battery in
each one-hour interval of the entire operation period
must be within an estimated limit.
(15)
The state of charge (SOC) must not violate the
default maximum and minimum value.
(16)
The battery is authorized to change its state of
charge and discharge only once per operating period
within the specified period.
The constraints regarding the power limits that
the ESS can charge or discharge within a time t are
depicted in Equation 17, considering the efficiency
of the charging or discharging process (ηc and ηd),
and the current and previous energy states (SOCSj (t)
and SOCSj (t-1)).
(17)
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Fig. 7: ESS battery charge/discharge
Figure 7 illustrates the behavior exhibited by the
battery within the energy storage system over the
course of the day. Typically, following
manufacturer recommendations, the battery is
designed with a certain tolerance in its design to
prevent wear on its components, establishing a limit
on its charge/discharge to preserve its lifespan. The
energy storage system regulates the power output of
the system in case of any disturbances in the
Microgrid (MG), among other functions. Primarily,
in this analysis, its usage is to supply residential,
commercial, or industrial sectors in cases where the
production from renewable sources is insufficient to
meet the expected demand. Alternatively, depending
on operational costs or market conditions, energy
may be utilized as an auxiliary means for other areas
of the system through exchange, benefiting both the
prosumer and the main system. The troughs in the
graph represent the maximum energy at that
moment within the Energy Storage System (ESS)
(not necessarily reaching 100% to preserve its
lifespan), while the peaks represent the discharge of
the ESS (not necessarily 0%), either due to usage
within the area. Each area's behavior depends on
various factors, primarily revolving around the sale
and purchase of energy from the grid. In this study,
random values were chosen to represent batteries
with different initial capacities to visualize their
charge/discharge behavior throughout the day,
interacting with other sources that need to meet the
user's demand.
2.4 Electric Vehicle Modeling
The modeling of Electric Vehicle (EV) charging is
highly stochastic due to the need to consider the
driving patterns of EV users, in [20], [21], [22].
Parameters such as arrival time, departure time,
distance covered by an EV user, charging rate, etc.,
are necessary to model EV demand. In this study, it
is assumed that from 6 AM to 7 PM, the EV is
disconnected from the Microgrid (MG), resulting in
a certain percentage of power loss. Upon returning
home at the specified time, the Battery Electric
Vehicle (BEV) reconnects to charge overnight, and
the State of Charge (SOC) of the EV evolves in an
independent random pattern.
The energy consumed due to the vehicle’s daily
distance is determined by the Equation 18:
(18)
Here, SOCd represents the energy consumption
due to the distance traveled by the EV in a day (dist)
concerning the total range (dt) it can cover. The
initial SOCE of the BEV during charging is
calculated using Equation 19:
(19)
Where dist and dt denote the traveled distance and
the maximum range of the EV, respectively.
Charging stops from 7 AM onwards. The algorithm
saves the last state of charge, so when the EV
returns home at 8 PM, the discharge ratio based on
the distance covered from the last state determines
the SOCE of the EV.
Figure 8 depicts the electric vehicle battery
behavior throughout the day. The scenario is utilized
solely for the study in this article, where in the BEV
does not exchange energy with the main grid, and
the stored energy is utilized when renewable sources
do not entirely meet the user's demand. However,
for future endeavors, the intention is to explore
energy exchange with the grid.
Fig. 8: Battery EV charge/discharge
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3 Microgrid Configuration
3.1 Cost Minimization Function
The operational cost function comprises cost
summation functions for each production unit
(DGs), energy reserve costs (ESS and EV), and
utility exchange (grid), [3].
(20)
Each cost summation interval involves the costs
of the involved DG sources (CTGD) in the Microgrid
(Equation 21), the energy storage system costs
represented by CTS in Equation 22, and the EV costs
by CTEV E in Equation 23. Finally, the grid cost is
denoted as CTG in Equation 24. Each equation
specifies its bid (B) in the market as well as the
power (P) being produced by the source at that
moment, considering if it’s active ON/OFF (U).
(21)
(22)
(23)
(24)
The flowchart in Figure 9 displays the algorithm
used by the program when evaluating the objective
function, where different systems within the
Microgrid interact concerning time t. It
demonstrates decision-making regarding the time t
for the SOCS of the energy storage system battery
and the SOCE of the electric vehicle, along with the
generated power to meet the demanded load. The
ultimate result is the total cost of the energies used
at that moment.
Fig. 9: Objective function evaluation.
3.2 Energy Balance Constraints
The total generated power in each interval must
equal the total load demands, the energy stored in
the battery bank (charge/discharge), electric vehicles
(charge/discharge), and total feeder losses (Eq.25):
(25)
This equation considers each energy source
used to meet the user’s demand, where PGD is the
sum of all power sources i of DG (wind and solar),
PS is the energy stored by batteries j at time t, PEV is
the energy of BEVs e at time t, PG is the energy
delivered by the main grid to the consumer due to
the ER sources not entirely meeting the user’s
demand, and finally PL is the total electrical load
demand at time t.
3.3 Power Variable Constraints
3.3.1 Distributed Generation Powers
The powers generated by the DG sources establish
limits on their values, as shown in Equation 26.
(26)
where (t) y (t) are the minimum and
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maximum active power of distributed generation at
time t, respectively.
(27)
(28)
3.3.2 ESS and BEV Battery
Both the State of Charge (SOC) and the
charge/discharge power (PSC/PSD) have their
respective limits established during the analysis. In
practice, completely discharging a battery reduces
its lifespan, so the battery should maintain a
minimum energy capacity, and ideally, a maximum
capacity 29. Charging/discharging powers must also
adhere to the manufacturer’s specifications 30.
(29)
(30)
These values also apply to the electric vehicle’s
battery, where, based on manufacturer data, the
SOC and charge/discharge powers, considering their
efficiency (η), play a significant role.
(31)
Although the equations for the energy storage
battery and the electric vehicle’s battery are similar,
it all depends on the distance and times at which the
electric vehicle connects to the grid.
4 Method for Microgrid Optimization
and Case Studies
4.1 Population-Based Incremental Learning
(PBIL) Algorithm
Optimally managing energy in an MR involves
solving a combination of problems using
metaheuristic methods. The Population-Based
Incremental Learning (PBIL) algorithm has been
employed for its suitable capabilities in handling
such issues. The authors in [23], [24], [25], mention
that PBIL is an evolutionary algorithm that works
by updating a vector describing univariate statistics
of the best solutions. This straightforward model
update is controlled by a parameter that sets the
Learning Ratio (LR). The model is then used to
generate new solutions. The optimal energy
management procedure is illustrated in the Figure
10.
Fig. 10: Pseudocode PBIL
The algorithm iteratively updates the probability
values of the Vector Probability (VP), starting from
neutral values. Each iteration or generation creates a
population of individuals based on the current VP
probabilities. The best individuals from a given
generation update the VP values for the next
generation. Algorithm execution stops when the VP
converges, i.e., when all elements become zero or
one, or when the specified iteration count is reached,
[23].
The VP update follows the equation:
(32)
Where VP is the vector probability, LR is the
learning ratio and the variable BinaryXmax is the best
individual in binary form.
Updating the VP considers the Learning Ratio
(LR), a crucial factor in implementation that
determines the speed and accuracy of obtaining
results. In essence, LR is the important factor given
to the best individual for VP update.
4.2 Simulation and Results
The Microgrid (MG) depicted in Figure 1,
connected to the electric grid, was analyzed as the
test system in this document. The system’s
maximum demand is represented by the total energy
of all loads contributing to the main system, akin to
a typical household.
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Hence, the daily load curve for the MG is
showcased in Figure 11. The total energy
consumption for the day equaled 42320 kWh.
Constraints and data for the involved sources are
presented in Table 4.
Fig. 11: Daily demand curve for a residence
Table 4. Limits of the Sources Involved.
Units
Minimum power
(watts)
Maximum power
(watts)
PV
0
8250
WT
0
1500
ESS
-330
250
EV
-500
400
The system integrates various technologies like
WT, PS, ESS and EV, assuming real-time hourly
market prices presented in Table 5. The output
power of WT and PV systems is illustrated in Table
6 based on predicted values. Distributed Generation
(DG) systems are strategically placed in different
branches to encompass diverse or hybrid systems.
4.2.1 Case Studies
Scenarios 1 and 2 consider the operational costs of
all units alongside problem constraints. Scenario 1
aims to minimize operational costs, integrating GDs
at their maximum power. Scenario 2 involves all
GDs using random power capacities. Scenarios 3
and 4 focus on a single distributed energy source.
Scenario 3 incorporates random power from PS with
its storage system, while Scenario 4 employs WT
power as the sole source.
Table 5. Market price, Hourly rates
Time
PV
WT
ESS
Grid
1
0
0.0210
0.1192
0.033
2
0
0.0170
0.1192
0.027
3
0
0.0125
0.1269
0.020
4
0
0.0110
0.1346
0.017
5
0
0.0510
0.1423
0.017
6
0
0.0850
0.1500
0.029
7
0
0.0910
0.1577
0.033
8
0.0646
0.1100
0.1608
0.054
9
0.0654
0.1400
0.1662
0.215
10
0.0662
0.1430
0.1677
0.572
11
0.0669
0.1500
0.1731
0.572
12
0.0677
0.1550
0.1769
0.572
13
0.0662
0.1370
0.1692
0.215
14
0.0654
0.1350
0.1600
0.572
15
0.0646
0.1320
0.1538
0.286
16
0.0638
0.1140
0.1500
0.279
17
0.0653
0.1100
0.1523
0.086
18
0.0662
0.9250
0.1500
0.059
19
0
0.0910
0.1462
0.050
20
0
0.0830
0.1462
0.061
21
0
0.0330
0.1431
0.181
22
0
0.0250
0.1385
0.077
23
0
0.0210
0.1346
0.043
24
0
0.0170
0.1269
0.037
Table 6. Predicted values for WT and PV
Time
WT
(watts)
PV
(watts)
Time
WT
(watts)
PV
(watts)
1
249.5
0
13
17.5
5931.9
2
399.0
0
14
60.7
5120.9
3
519.5
0
15
140.3
3866.7
4
727.1
0
16
252.0
2362.2
5
503.8
0
17
496.0
1095.6
6
324.3
0
18
611.4
0
7
138.6
1094.4
19
412.7
0
8
87.7
2361.8
20
327.3
0
9
89.0
3866.2
21
183.3
0
10
17.5
5120.0
22
133.6
0
11
6.4
5931.4
23
96.7
0
12
8.1
6211.2
24
138.6
0
The management of ESS entails specific
schedules for charging and discharging towards the
MR or the main grid. Charging occurs when energy
prices are relatively low, irrespective of user
demand. Conversely, during high energy prices, the
battery supplies the demanded load. Surplus
renewable energy might charge these batteries if the
production exceeds user demand.
In all the above scenarios, BEVs are solely
considered as loads, not contributing energy to the
residence. Their restricted hours were mentioned in
Section 2.
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4.2.2 Scenario 1 and 2: Comparative Analysis
The outcomes of Scenario 1, outlined in Table 7,
reveal the configuration of participating units with
their optimal power generation. Notably, the ESS
contributes to the MR during high market prices and
to the grid during lower price periods. Meanwhile,
the BEV acts purely as a load, showing no
participation. These power outputs originate from
randomly assigned values in the photovoltaic and
wind systems.
Table 7. Scenario 1 with random power in WT and
PV
Time
PV
WT
ESS
EV
Grid
1
0
250
0
0
1200
2
0
399
0
0
1051
3
0
519
0
0
931
4
0
727
0
0
723
5
0
0
0
0
1450
6
0
0
0
0
1450
7
1094
0
0
0
17106
8
0
0
0
0
10450
9
3866
89
330
0
165
10
5120
17
330
0
1483
11
5931
6
330
0
682
12
6211
8
330
0
401
13
5932
17
330
0
31171
14
5121
61
330
0
31939
15
3867
140
330
0
8613
16
2360
250
327
0
8513
17
1096
0
0
0
2854
18
0
0
0
0
3950
19
0
0
0
0
3950
20
0
0
0
0
11450
21
0
183
0
0
8267
22
0
134
0
0
8316
23
0
97
0
0
6853
24
0
139
0
0
2361
Figure 12 depicts the generated power (red line)
versus the power used (blue bars) by the
photovoltaic system. On the other hand, Figure 13
illustrates the power generated (red line) by the
wind turbine against the power consumed by the
user (blue bar) sourced from wind energy.
Scenario 2 (Table 8) showcases the optimal
power configuration of renewable sources when
operating at maximum capacity. Minimal reliance
on the main grid is observed, as it entirely fulfills
the demand, leading to a considerable reduction in
energy costs.
Fig. 12: Random powers PV
Fig. 13: Random powers WT
Table 8. Scenario 1 with maximum power in WT
and PV
Time
PV
WT
ESS
EV
Grid
1
0
1450
0
0
0
2
0
1450
0
0
0
3
0
1450
0
0
0
4
0
1450
0
0
0
5
0
1216
0
0
234
6
0
251
0
0
1199
7
1094
57
0
0
17049
8
2362
611
0
0
7477
9
3866
412
330
0
159
10
5120
617
330
0
883
11
5931
1022
-3
0
0
12
6211
0
330
0
409
13
5932
0
330
0
31188
14
5121
0
330
0
31999
15
3867
1533
330
0
7220
16
2362
1533
330
0
7225
17
1096
1533
0
0
1321
18
0
1533
0
0
2417
19
0
1533
0
0
2417
20
0
1533
0
0
9917
21
0
840
0
0
7610
22
0
252
0
0
8198
23
0
0
0
0
6950
24
0
17
0
0
2483
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The complete utilization of energy generated by
the GD systems is depicted in Figure 14 and Figure
15.
Fig. 14: Maximum powers PV
Fig. 15: Maximum powers WT
4.2.3 Scenario 3 and 4: Comparative Analysis
In Scenario 3, as indicated in Table 9, the energy
generation from a DG source, specifically the WT,
is crucial. The absence of PV production during the
early hours necessitates total reliance on the main
grid. However, post-noon, the ESS is compelled to
operate at maximum capacity due to substantial
power output from the WT.
There’s a similarity between Scenario 3 and
Scenario 4 due to the absence of a power-generating
source. This is evident in Figure 19, where a
significant reduction in the optimal cost is observed.
Table 9. Scenario 1 with maximum power in PV
Time
PV
WT
ESS
EV
Grid
1
0
0
0
0
1450
2
0
0
0
0
1450
3
0
0
0
0
1450
4
0
0
0
0
1450
5
0
0
0
0
1450
6
0
0
0
0
1450
7
1094
0
0
0
17106
8
2362
0
0
0
8088
9
3866
0
330
0
254
10
5120
0
330
0
1500
11
5931
0
330
0
689
12
6211
0
330
0
1409
13
5932
0
330
0
31188
14
5121
0
330
0
31999
15
3867
0
330
0
8753
16
2362
0
330
0
8758
17
1096
0
0
0
2854
18
0
0
0
0
3950
19
0
0
0
0
3950
20
0
0
0
0
11450
21
0
0
0
0
8450
22
0
0
0
0
8450
23
0
0
0
0
6950
24
0
0
0
0
2500
4.3 Optimal Cost Analysis
The graph in Figure 16 demonstrates the
participation of all involved sources. However, due
to minimal energy generation, there was an
exchange of energy with the main grid, resulting in
the algorithm finding an optimal cost of $950.
Fig. 16: Optimal cost with random powers in PV
and WT
In contrast, the optimal cost for Scenario 2 is
depicted in Figure 17, showcasing a notable
reduction compared to Scenario 1.
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Fig. 17: Optimal cost with maximum powers in PV
and WT
The highest cost is exhibited in Scenario 4 in
Figure 18, attributed to the null PV production
during peak hours when the market price is high.
This necessitates additional wind power generation.
However, Scenario 3 showcases behaviour where
the power configuration is more suitable, resulting
in the lowest optimal cost range (Figure 18).
Fig. 18: Optimal cost with maximum power in PV
without WT
Fig. 19: Optimal cost with maximum power in WT
without PV
4.4 Comparative Analysis. Particle Swarm
Optimization
With the aim of juxtaposing the PBIL method
against one of the most widely cited metaheuristic
approaches in the academic domain, particularly
pertinent in addressing optimization challenges
involving operational costs, emissions reduction,
and optimal power allocation, the Particle Swarm
Optimization algorithm (PSO) emerged as the most
fitting candidate for this comparative analysis, [6],
[7], [9], [13], [14]. Noteworthy for its adeptness in
optimal resource management, PSO operates as an
intelligent swarm algorithm predicated on the
collective movement of particles traversing the
solution space. Each constituent entity, or 'particle,'
within the PSO framework navigates the search
space with a velocity dynamically modulated in
response to its own exploration history and that of
its neighboring particles.
Fig. 20: Optimal cost with random powers in PV
and WT using PBIL
Fig. 21: Optimal cost with random powers in PV
and WT using PSO
Figure 20 and Figure 21 show the comparison
made between the proposed method PBIL and the
most used method for this type of problem, which is
PSO. For this study, it was concluded that the
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computational response time with the PSO method
is slightly slower than the PBIL method, under the
same conditions (equal number of iterations).
However, PSO converges faster than PBIL. The
advantage of PBIL lies in its practicality in
modelling, clearly illustrating the behavior of the
graphs in the process. The methodology of the
applied methods depends on the programmer and
variables, limits, and factors such as the population
or chromosomes to be selected. It is worth
mentioning that the speed of the response goes hand
in hand with the objective function that was
proposed. Figure 20 and Figure 21 use the
maximum power of solar and wind energy.
5 Conclusion
The proposed method for optimizing operating costs
and optimal values within a microgrid composed of
various renewable energy sources is conducive to
such analysis and may offer equal or greater
reliability compared to alternative methodologies
commonly employed for this purpose. The results
obtained across different scenarios provide insights
into the behavior of generation systems within the
microgrid and their interaction with the main
system. The algorithm functions effectively within
the proposed scenarios, successfully fulfilling its
purpose of providing optimal values that benefit the
user in terms of both cost and power. Scenarios 1
and 2 demonstrate a reduction in optimal costs
attributable to the management performed by the
algorithm in adjusting power levels, a phenomenon
evident with each iteration as new populations are
generated to find the most suitable solution.
Scenarios 3 and 4, on the other hand, exhibit
enhanced responsiveness, with the system
converging in approximately half the iterations
compared to Scenario 1. It is worth noting that
initiating with relatively high or maximum power
values leads to decreased iterations and costs, albeit
contingent upon how the algorithm optimizes its
values to align with the objective function being
addressed. This algorithm affords an equal or
superior perspective on the step-by-step process of
searching for potential solutions aimed at achieving
optimal values for efficient system operation.
The outcomes presented in this study underscore
how adjustments to operational limits, chromosome
and population selection, as well as the learning rate
(LR), directly impact the magnitude and swiftness
of the results obtained.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conceptualization G.M.I., F.B.M., L.C.P., S.R. ;
methodology, , L.C.P.; software, L.C.P and F.B.M.;
validation, L.C.P, G.M.I., F.B.M.; formal analysis, ,
L.C.P, G.M.I.; investigation, L.C.P., G.M.I.;
resources, , L.C.P, G.M.I., data curation, L.C.P and
F.B.M.; writing—original draft preparation, L.C.P. ,
S.R.; writing—review and editing, L.C.P, G.M.I.,
F.B.M.; visualization, , L.C.P.; supervision, G.M.I.,
F.B.M.; project administration, , G.M.I. , S.R.. The
authors have read and agreed to the published
version of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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