1. Introduction
There are many challenges facing the current electric grid, such as conserving power, improving flexibility, reducing emissions from traditional energy production, reducing fossil fuel consumption, optimizing environmental and economic effects, etc. Different distributed energy resources (DERs) can be integrated with a smart microgrid (MG) to address such problems. MGs consist of various loads and energy storage systems, as well as renewable and small-scale dispatchable energy systems like microturbines (MTs), photovoltaic (PV), wind turbines (WTs), diesel engine generators (DEs), fuel cell (FCs), and so on [
1,
2]. An important part of the smart grid system is the MG’s energy production manner. Most MGs operate in either islanded or grid-connected modes. MGs use renewable energy resources (RERs) as their main energy production resources, and these resources tend to be intermittent. An MG’s uncertainty is typically caused by supply and demand, due to the unpredictable behavior of RERs and the MG’s total load demand. This uncertainty makes balancing the energy between the overall production and load demand at the MG an important challenge. It is feasible to solve the problem of their unpredictable nature by implementing DSM strategies using different RERs. DSM schemes will be essential for today’s smart grids in order to manage the excess power requirements of users while also minimizing imbalances between energy production and load consumption. In order to implement DSM successfully, modern metering systems and communication and information technologies must be utilized [
3]. Various price policies have been discussed by researchers regarding DSM load shifting, including real-time costing, acute peak costing, slope block rate, and time of use (ToU). Energy management will be essential for improving reliability, quality of power, sustainability, and ensuring reliable, economic, and environmentally friendly functioning in the MG [
4,
5].
The optimum MG efficiency and energy management issues have been extensively studied by researchers. Refs. [
6,
7] examined an online multi-objective optimization method using a revised game theory framework in order to improve both the environmental and economic goals of MG operations. Ref. [
4] integrated the stimulus-driven demand response program with the MG energy-management problem in order to optimize dispatch methods. Ref. [
1] applied four types of optimization methods in order to optimize MG: direct quest, particle swarm optimization, lambda logic, and iteration. Ref. [
8] examined a combination of economic dispatch and a DSM scheme for an MG system with the aim of minimizing the entire price for household users. The MG optimum operational problem must account for uncertainty in supply and demand in order to obtain optimal planning. Ref. [
9] applied Monte Carlo simulation for handling the uncertainty associated with unpredictable power resources and load. Ref. [
10] used a linear two-step stochastic scheme in an uncertain environment in order to optimize MG operation. MG operating optimization problems have been discussed in several studies. The majority of investigations fail to adequately analyze the effects of domestic DSM strategies on MG operations and users’ satisfaction level objectives are not adequately addressed in their DSM methods.
Ref. [
11] examined a probability-based day-ahead EMS approach for scheduling the MG network’s dispatchable and non-dispatchable power resources. That study used a pumped-storage unit and a stimulus-driven DRP for maintaining the production and load balance. Ref. [
12] examined the effects of stimulus-based DR schemes on the intra-day and day-ahead markets. Instead of selecting values for static elastic coefficients used in that study, it may be possible to model users’ reactions to cost variations more realistically. Ref. [
13] examined the effect of DRPs on network resilience using several MGs. Stochastic EMS in that study took into account not only the operational planning but also indicators of the reliability of local loads during an emergency. Ref. [
14] combined optimum planning of tidal energy resources into a probabilistic EMS architecture using DRP, resulting in a reduction of 13.14 percent in overall operational prices.
The purpose of the present study is to formulate the problem of optimum operation of MGs and demand-side management (DSM) through the development of common strategies for electric grid operations and consideration of the required limitations. The objective function (OF) aims at minimizing operation prices and DSM prices, and optimization limitations consist of generator limitations and restrictions on energy balance. Additionally, the hours of load shifting are taken into account as variables, and an improved butterfly optimization algorithm (IBOA) is applied for solving the optimization problem. This paper is divided as follows:
Section 1 introduces the DSM,
Section 2 defines the OF and formulation scheme,
Section 3 presents the problem-solving approach,
Section 4 presents the simulation outcomes, and
Section 5 presents the conclusion.
2. Problem Formulations
An optimization problem is proposed for optimum unit production scheduling. An MG’s optimum operating strategy takes into account both the minimum price and operational limitations as well as DSM limitations. The OFs in the problem consider the overall production prices and the prices of using DSM. As a result, Equation (1) is used to define the OF of the problem of optimum operation of MGs taking DSM into account [
15]:
in which
shows the overall operation prices for the MG, OC represents the overall operational costs (OC) of the energy production units, and DC represents the overall price to implement the demand-side programs.
and
show the weight ratios of the operation prices of the system and the price to implement DSM programs, respectively.
Subscriber dissatisfaction is caused by the use-time variation scheme. Therefore, the present study models the price to implement the load-shifting program as the discomfort function as a
-degree function based on Equation (2):
in which
represents the number of loads capable of shifting,
and
represent the price of shifting for the load,
shows the times of loads
transferring, and
represents the overall number of shiftable loads. The operational cost of a production unit includes the cost of production, the cost of startup, and the cost of maintenance. Moreover, since power can be bought or sold to the utility in the MG, the price of purchasing or selling energy to the grid appears in the operational cost function. An OC operational cost function is shown in Equation (3) for the optimization problem [
16]:
in which
indicates the price to generate energy of agent
for time
t of operating,
indicates the price of maintenance, and
indicates the price of beginning agent
for time
. In addition,
represents the price to purchase energy for time
from the network, and during that time,
represents power sales revenue.
shows the number of energy production agents and
represents the study time
in hours.
includes many production units like PV, WT, MT, FC, and battery, with diverse cost functions. WT output power can be calculated based on wind speed using Equation (4) as follows:
In this equation, the parameter
represents the rated power of WT,
shows minimum permitted wind speed,
represents maximum permitted wind speed,
shows the nominal velocity, and
represents the real wind velocity.
and
can be found from a current device’s catalog. Solar cells generate energy based on light intensity and ambient temperature and the relevant values can be determined based on Equation (5):
in which
represents solar cell output power in terms of area irradiance severity,
shows maximal cell production energy in standard trail statuses,
shows area light severity,
represents irradiance severity in standard trail statuses,
shows output power temperature ratio,
and
show reference and cell temperatures. Wind and solar energy are used rather than fuel as the RERs of WTs and PVs. Due to this, there are no fuel expenses associated with such units. Aside from the substantial construction costs, maintenance costs must also be taken into account when assessing the economics of MGs. Therefore, Equation (6) is used to calculate the overall price of WTs and PV units:
in which
shows the price of renewable units, AC shows the yearly price factor,
is coefficient of investiture price to produce the energy of the agent,
shows agent repair price.
A governor controls the DE’s output power. As the second-class function of producing power, DE fuel usage
can be given by Equation (7):
in which,
shows DE fuel usage price
,
shows the output power of DE,
, and
would be constant. Based on Equation (8), performance of FC equals the output power to the input fuel when they are computed in a similar unit:
in which
shows the price of fuel utilized via a FC (USD/h),
represents the price of natural gas for feeding the FC
,
shows the FC output power,
shows the FC performance. Based on Equation (9), an MT has the same economic scheme as an FC, but its performance improves as power rises:
Equations (10) and (11) are used to express the price of power bought
and sold
(Equation (3)):
represents the tariff to purchase power from the network,
shows the energy bought from the network,
shows the tariff to sell power to the network, and
shows the electricity sold to the network. The price to repair and maintain units depends directly on their energy production. Thus, the price of repairing and maintaining unit
for time
t can be determined by Equation (12):
in which
shows the cost of repairing and maintaining agent
per kW of electrical power and
represents the output power of agent
per hour
. Only fossil fuel production units are included in the start-up price. Based on the fact that the start-up price of agent
depends on the cycle during which the unit is operational, Equation (13) calculates the start-up price of agent
for time
:
in which
shows the start-up price of agent
and
shows a binary parameter indicating the mode of agent
including off/on for time
. Coequality limitations in the problem would include the power equilibrium limitation, based on Equations (14) and (15):
Inequality limitations are agent output power limitations, control parameter limitations, line power limitations, and voltage limitations, and can be determined by Equations (16)–(19):
Equation (20) considers the shifting time of all loads as a further limitation in the demand response program:
in which
is the allowed time for shifting the load
. This optimization problem can be transformed into the optimum power flow (OPF) problem through knowing the load shift time. As a result of resolving the OPF issue, it will be possible to determine the power produced via all units, as well as the power sent and received by the global network. The present study applies the IBOA for solving the optimization issue.
3. Improved Butterfly Optimization Algorithm
Smell, taste, and hearing are several of the senses that butterflies use to find food or mating partners, lay eggs, and escape hunters. Research indicates that butterflies have a strong sense of smell, particularly when searching for food from afar [
17].
Chemoreceptors are nerve cells that are responsible for butterflies’ ability to search for food. The butterfly’s chemoreceptors are utilized to smell and are distributed throughout its body. The butterfly also uses this sense to find the right mate for itself [
18]. Butterflies are capable of sensing, locating, and even separating various fragrances [
19]. Aroa and Singh developed the butterfly optimization algorithm (BOA) [
20]. The BOA’s population consists of butterflies acting as search agents. The BOA’s OF price changes according to the position of the butterflies. Using the BOA, all butterflies share their experiences with their neighbors according to fragrances distributed across distances.
When the butterfly senses the scent of the other butterfly, it follows it by using the stage as a universal quest spot. Butterflies are subjected to another movement as part of their local search optimization. Generated randomness is used for this part. By balancing the smell senses and fragrance, the BOA is applied.
3.1. Fragrance
There are three sections to fragrance: power exponent, sensory modality, and stimulus intensity. I indicates the physical incentive size which has been associated via the fitness solution, i.e., when the butterfly emits lots of fragrance, the surrounding butterflies are able to sense and are attracted to it.
Two important factors contribute to butterflies’ substance: changes in fragrance formulation
(f) and incentive intension
(I). The below equation describes the fragrance scheme:
in which
shows the observed extent of the fragrance,
shows the incentive intension,
is the sensory modality, and
shows the power exponent based on defining the changing level of attraction.
and
fall within
.
3.2. Butterfly Movement
The BOA consists of three stages: the initialization, the questing, and the finalizing phases. The BOA evaluates OF values for all butterflies following the initialization of the primary butterfly swarm. The step also involves setting the parameters of the algorithm. The algorithm begins to optimize once the parameters have been assigned. A random position has been generated in the quest area for the butterflies. Artificial butterflies travel to the updated places in the quest area once the iteration begins, and their prices are calculated. The next equation is used to produce the fragrance by butterflies in their places:
in which
shows the optimal solution for the iteration
shows the solution vector
for
butterfly, the fragrance of the
butterfly is shown via
and
shows a randomly selected firm within zero and one. The local quest in the algorithm would be:
in which
and
show the
and
members of the butterfly swarm in the quest area. Food search and partner mating in butterflies are BOA variables that take place both on a global and local scale.
The BOA is effective when it comes to exploring optimum values, but it fails when it comes to converging. This paper proposes a novel technique for modifying the BOA’s key variables to increase convergence speed.
According to chaos theory, a vector of the BOA’s key variables is used as the solution.
In chaos theory, unexpected and random patterns are studied. A chaotic system is a very sensitive dynamic system influenced by even the smallest change. The technique improves the point distribution in the search space by generating points with fewer complexities and greater distributions. The BOA’s convergence speed is improved by the feature. Here is a general formulation of chaos theory:
in which
shows the map extent and
indicates the chaotic scheme producer function. A sinusoidal chaotic map was used as the main parameter in the following way:
Figure 1 depicts the flowchart for the IBOA.