Efficient PID Control Design for Frequency Regulation in an Independent Microgrid Based on the Hybrid PSO-GSA Algorithm

Abstract

Microgrids are a part of the power system that consists of one or more units of distributed generation and are expected to remain in operation after being disconnected from the system. Since they rely on overlying networks, frequency control is very important for network-independent operation. Some of the most common problems in independently operating microgrids are frequency sustainability and its fluctuations. The main purpose of this study is to control the frequency of a microgrid in island mode in different scenarios. The objective function is defined based on time and changes in the system frequency. Thus, the variable parameters of the PID controller are transformed into an optimization problem and are solved through the hybrid PSO-GSA algorithm. The study considers four scenarios: (a) a microgrid dynamic model and optimal PID controller coefficients; (b) variable velocity disturbance applied to the studied system in order to observe power changes and the microgrid frequency; (c) stepped load changes applied to the studied system; and (d) the proposed methods on the standard test function. Simulations under different operating conditions are performed, indicating improvements in the stability of microgrid frequency fluctuations by means of the proposed control method.

1. Introduction

A microgrid usually consists of a set of distributed generation sources, an energy storage system, and local loads. It can be connected to a network or operate in island mode, and it has many benefits for both consumers and power generation companies. From the perspective of the consumer, microgrids are able to simultaneously provide electricity and heat [1], increase reliability [2], reduce greenhouse gas emissions [3], improve the quality of power, and reduce the costs of consumption [4,5]. As for electricity companies [6], the use of microgrids has the potential to reduce consumption demand and, in turn, the facilities necessary for the development of transmission lines [7]. In addition, they eliminate peak consumption points [8], which results in reduced network losses [9]. By definition, distributed generation (DG) includes electrical power generation units with a capacity of less than 10 MW which have distribution feeders or common levels connected to the network. Depending on the capacity and location of the source installation, connecting DG to a distribution network can have a positive or negative effect on its performance [10]. On the other hand, DG causes the power flow in distribution networks to change. Therefore, a main network with the presence of DG is not optimal for reducing losses; there needs to be a proper control in the network, so that the distribution network operates at optimal cost and with increased reliability [11]. In other words, with the advent of DG resources, several problems have appeared, such as maintaining and protecting resources [12], the way in which these resources are involved in setting the basic parameters of the network (such as frequency and voltage [13,14]), and the method for power exchange between the global network and distributed generation sources [15]. In 1998, to solve these problems and consider these resources and local loads as integrated, the concept of micro-grid was introduced in modern power systems [16]. Accordingly, microgrids are small power networks composed of several distributed generation sources and local loads. They are usually connected to the global network, and they are isolated from the main network in the event of heavy disturbances. They can also feed sensitive loads [17].

Real power systems face a variety of uncertainties, which are caused by changes in load [18], system modeling errors, and structural changes [19]. Therefore, classical controllers with constant interest are not suitable for solving the Load Frequency Control (LFC) problem [20,21,22]. In order to cover these limitations, a flexible controller is required. So far, various controllers for LFC have been presented. Among these controllers, the PID controller has generally attracted more attention than the others [23,24]. The controller’s interests are determined at the nominal working points. The controller’s interests are determined through the classical method [25]. In other words, increasing the number of microgrids changes the fundamental rules of power systems and causes the production resources to be distributed throughout them [26]. This leads to an increase in the complexity and nonlinearity of power networks, so the proper response of classical controllers can no longer be observed. PD-PI controllers are widely used in power systems because they have a simple and cost-effective structure, and, in power systems, they are more reliable than any other controller. However, the problem with these controllers is that their control coefficients are set up for a single time and placed in the system according to the linear conditions and operating points of the system. If the nominal working conditions or the system’s linear conditions change due to turbulence, the values considered for these controllers are not optimal, and they do not have the same response. A possible solution is to update and optimize the control coefficients according to the incident changes in the system [27]. The development of power networks due to increased energy demand and technical issues has caused today’s power systems to activate within their own boundaries. This has led to more sustainability in said systems. In order to increase stability and overcome the problems with classical controllers in different working conditions, fuzzy controllers have been used as resistant stabilizers to modulate small signal fluctuations [28,29,30]. Ref. [31] used a new method for controlling the microgrid frequency with a drop control in a photovoltaic converter and battery, in which a low-voltage microgrid is considered to be multiple virtual microgrids. This strategy has improved the finetuning of the microgrid frequency. In [32], the torque and frequency power drop control are applied to the converter of a wind turbine’s doubly fed induction generator. In [33], the optimal self-healing strategy for microgrid islanding is formulated as an optimization problem. A reconstruction framework and solutions for power outages in microgrids are provided in [34]. A regulatory management of renewable resources based on controllers is discussed in [35]. In [36], the optimization method is used by planning to maximize profit and minimize operation cost. In [37], a microgrid strategy is proposed using a self-healing agent that operates based on a centralized or decentralized approach. Small-signal stability analysis is performed to evaluate the stability of microgrids to avoid any instability problems. Ref. [38] employed the Frequency Containment Reserves (FCR) technique with the goal of improving the economic profitability of microgrids. The primary loop control of the frequency (drop control) was established in order to control the microgrid frequency and reduce pollution and the cost of power generation. Drop control cannot properly control the microgrid frequency under heavy load variations, and it is not properly efficient in island mode. To control the frequency of an islanded microgrid, a secondary control loop is generally used for reducing frequency fluctuations under severe load variations [39]. LFC is one of the most important issues in microgrids. LFC has been investigated due to the relatively low inertia of these systems [40]. In [41], a multivariate unconstrained pattern search method for the optimization of digital PID controllers applied in an isolated forward converter is studied. Conventional digital PID controllers are considered to be designed based on digital redesign and direct digitalization, adjusted by one of the multivariate name search pattern search methods called the Hooke–Jeeves (H-J) search method; with an excellent performance, output voltage regulation can be ensured. Droop control techniques are currently used to coordinate DG units in a microgrid. However, this method has its own advantage. It is used for a nonlinear analysis to predict the qualitative behavior of the system with the aim of reducing the differential equations [42,43]. In [44], an isolated non-DC–DC boost converter is designed. This converter is designed by adding to networks and VMCs. In [45], frequency control in hybrid distributed power systems via a type-2 fuzzy PID controller studied. A new Internet of Things-based optimization scheme of a residential demand side management system was tested [46]. In addition, in [47], the Optimized Robust Controller Design based on the CPSOGSA Optimization Algorithm and H2/H∞ Weights Distribution Method for Load Frequency Control of Microgrids was investigated.

This paper investigates the communication aspects of multiple markets with primary control (centralized and decentralized) loops. Designing a controller usually consists of three steps: first, choosing a control rule that contains changeable parameters; second, choosing a method to set these parameters; third, analysis of system convergence properties. The PID controller and coordination algorithm and parameter optimization are used due to the advantages of robustness against system parameter uncertainties, faster convergence speed when approaching the reference point, adaptability to system uncertainties, and the ability to prove stability. However, so far there have been few research results that use the combination method to design a controller for different coordinations. This still remains an open and challenging issue and has motivated us to write this paper. Therefore, one of the most important goals of reducing the effect of disturbances on the system and maintaining the quality of power and frequency is to improve the dynamic performance of the system in microgrids and the accuracy of power distribution between units in a limited time. In addition, an optimization method based on the PSO-GSA optimization algorithm is presented to achieve better and more accurate results. The rest of the article is as follows: Section 2 studies an example of independent microgrid modeling. In Section 3, the proposed PSO-GSA algorithm and its hybrid are discussed. In Section 4, the intelligent PID controller is designed to optimally adjust the parameters. Section 5 introduces a new method for damping and sustainability frequency fluctuations in the microgrid (while separated from the main grid) based on the PID controller and a hybrid PSO-GSA algorithm. The following aspects are studied: (a) the microgrid dynamic model and PID controller; (b) Calculation of optimal controller coefficients; (c) PID and hybrid PSO-GSA coordination; (d) Variable velocity disturbance; (e) Power and frequency changes; (f) Stepped load changes applied to the system; and (g) Effect of the proposed methods on the standard test function. Finally, Section 6 presents the conclusions. The results show the appropriate efficiency of the proposed controller in quenching fluctuations in a shorter period of time.

2. Microgrid Modeling

Nowadays, the expansion of transmission and distribution networks can pose challenges to power systems, even though they have advantages such as increased network reliability and improved stability. Among these challenges are the non-economic transmission of electrical energy from power plants to remote and impenetrable areas, the increase in transmission and distribution losses, and the increased complexity of the network’s protection system due to its widespread use. All of these have led to the widespread use of DG resources in recent years, whose main principle is the production of electrical energy at the place of consumption. The concept of the microgrid is a result of several DG resources placed together [48]. Microgrids include DG resources and local loads that can feed the loads both disconnected from and connected to the global network. The overall structure of a microgrid is shown in Figure 1.

The following resources are included: photovoltaic generator (PV), diesel engine generator (DEG), wind turbine generator (WTG), fuel cell (FC), battery energy storage system (BESS), flywheel energy storage system (FESS), and aqua electrolyzer (AE). The microgrid and the global network communicate with each other at the PCC. The micro-resources used in these networks are interconnected with the help of electronic elements. In fact, in these microgrids, AC or DC elements are used as converters or the like [49].

3. PSO-GSA Algorithm

3.1. Gravitational Search Algorithm (GSA)

The GSA is a collective and non-memory intelligence algorithm [50,51]. This optimization algorithm has been designed by modeling the rules and the movement of factors in an artificial system in discrete times at which the system space is the same as the problem definition range. According to the law of gravity, each mass perceives the location and condition of other masses. In this algorithm, the mass of the agents is provided according to the objective function [51]. In a system with mass n, the position of each mass is a point in space, which is the answer to the problem. The position of mass i is shown with $X_i$ in Equation (1):

$$X_i=\left(X_i^1,\dots ,X_i^d,\dots ,X_i^n\right)$$

(1)

where n is the dimension of the problem, and $X_i^d$ is the dimension d of the mass i. This system, with mass i at time t and in the direction d, is powered by a force with a $F_{ij}^d\left(t\right)$ value. The size of this force is obtained via Equation (2):

$$F_{ij}^d\left(t\right)=G\left(t\right)\frac{M_{pi}\left(t\right)\ast M_{aj}\left(t\right)}{R_{ij}\left(t\right)+ϵ}\left(x_j^d\left(t\right)-x_i^d\left(t\right)\right)$$

(2)

where $M_{aj}$ is the active gravitational mass of j, and $M_{pi}$ is the inactive gravitational mass of j, both of which are considered the same and equal to M in the mentioned algorithm; G(t) is the gravitational constant at time t; and $R_{ij}$ is the distance between the two masses i and j, which is also a very small number. The gravitational constant is an appropriate parameter for controlling the search and productivity capabilities, which is expressed by Equation (3):

$$G\left(t\right)=G_0{e}^{\frac{-at}{T}}$$

(3)

where α and G₀ are the control coefficients of the algorithm, and T indicates the system’s lifetime. The force on mass i in the direction of dimension t at time t is equal to the sum of all the forces that the other masses of the system exert on this mass. In this equation, $ran{d}_j$ is a random number with uniform distribution in the interval (1.0), which is considered for the sake of randomness [51]:

$$F_i^d\left(t\right)={{\displaystyle \sum }}_{j=1,j\ne i}^{N}ran{d}_j{F}_{ij}^d\left(t\right)$$

(4)

Furthermore, each of the masses has a specific speed and acceleration, each of which is shown in Equations (5) and (6), respectively. According to Newton’s second law, each mass is accelerated in the direction of dimension d, which is proportional to the force on the mass in that dimension, divided by its inertia mass, as stated in Equation (4). On the other hand, the velocity of each factor at time t is equal to the sum of the coefficients of the current velocity and the acceleration of the factor, as expressed in Equations (5) and (6):

$$v_i^d\left(t+1\right)=rand\times v{i}^d\left(t\right)+a_i^d\left(t\right)$$

(5)

$$a_i^d\left(t\right)=\frac{F_i^d\left(t\right)}{M_i\left(t\right)}$$

(6)

When the acceleration and velocity of each mass are calculated, the new position of agent i in the dimension d is calculated according to Equation (7):

$$x_i^d\left(t+1\right)=x_i{}^d\left(t\right)+v_i^d\left(t+1\right)$$

(7)

New situations are considered as the locations of new masses within the search space, where the weight of new masses is normalized via Equations (8) and (9):

$$m_i\left(t\right)=\frac{f_i{t}_i}{best\left(t\right)-worst\left(t\right)}$$

(8)

$$M_i\left(t\right)=\frac{m_i\left(t\right)}{{{\displaystyle \sum }}_{j=1}^N{m}_j\left(t\right)}$$

(9)

where $f_i{t}_i\left(t\right)$ represents the degree of maturity of the mass of agent i at time t, and worst(t) and best(t), respectively, indicate the suitability of the worst and the best factors of population in time, whose size can be calculated using Equations (10) and (11):

$$worst\left(t\right)=max\{f_i{t}_i\left(t\right)\}$$

(10)

$$Best\left(t\right)=min\{f_i{t}_i\left(t\right)\}$$

(11)

3.2. Particle Swarm Optimization (PSO)

In this section, the particle swarm optimization algorithm is briefly outlined. For more information on this topic, the readers are advised to refer to [52,53]. In the topology of the particle swarm optimization algorithm in the D-dimensional search space, the best personal position of particle i is indicated by $p_{id}\left(t\right)$, and the best position of the group is represented by $g_d\left(t\right)$. The relationship between the velocity and the motion of particle i at a given moment or the repetition of the dimension are obtained in the form of Equations (12) and (13):

$$v_i^d\left(t+1\right)=\omega v{i}^d\left(t\right)+c_{1}ran{d}_1\left(p_{id}\left(t\right)-x_{id}\left(t\right)\right)+c_{2}ran{d}_2\left(g_d\left(t\right)-x_{id}\left(t\right)\right)$$

(12)

$$\overline{x}\left(t+1\right)=\overline{x}\left(t\right)+\overline{v}\left(t+1\right)$$

(13)

In Equation (12), ω is the inertia coefficient of the particle, and c₁ and c₂ are Hook spring coefficients or acceleration coefficients, which are usually set to 2. To randomize the nature of the velocity, the coefficients c₁ and c₂ are multiplied by the random numbers rand₁ and rand₂. Usually, in the implementation of the PSO, the value of ω decreases linearly from one to values close to zero. The inertia coefficient ω is generally determined according to Equation (14):

$$\omega ={\omega }_{max}-\frac{{\omega }_{max}-{\omega }_{min}}{ite{r}_{max}}\times iter$$

(14)

where ${iter}_{max}$ is the maximum repetition number, $iter$ is the current repetition number, and ${\omega }_{max}$ and ${\omega }_{min}$ are the maximum and minimum values of the inertial coefficients, which are set at 0.9 and 0.3, respectively. $\overrightarrow{v_i}$ is the magnitude of the velocity of particle i in each dimension of the D-dimensional search space, which is limited to the interval [−$v_{max},+v_{max}$], so that the particle’s possibility of leaving the search space is reduced. The value of $v_{max}$ is usually chosen so that $v_{max}$_x = k$v_{max}$, where 0.1 < k < 1, so that $x_{max}$ specifies the length of the search.

4. Hybrid PSO-GSA Algorithm

The aim of combining different methods is to achieve better results than each technique would obtain on its own. The particle swarm optimization and gravitational search algorithms are well suited to be combined, through which a primary randomized population, generational upgrading, and the generation of new solutions can be achieved. Figure 2 shows such a combination with a primary population [54]. If the problem is N-dimensional, then the hybrid algorithm has 4N members, which are generated in a completely random way. 4N members are arranged by competency, and 2N upper members are considered as masses in the gravitational search algorithm. Additionally, a new 2N member population is created. Particle swarm optimization is applied to the 2N lower members. In applying the particle swarm optimization mechanism, the new population created by the gravitational search algorithm is used as a regulator. The best member of this new population is used as the $Pgbest$, and each corresponding member is used as a neighborhood or $Pibes$t in Equation (12). The population generated by applying particle swarm optimization and the population created by the gravitation search algorithm are merged and integrated, and the new 4N members are arranged by competency. The previous process is then repeated until convergence is achieved. The mechanism of particle swarm optimization is applied to the 2N lower members as particles. In applying the particle swarm optimization mechanism, the new population created by the gravitational search algorithm is used as a regulator.

In the proposed controller (according to Figure 3), in the first stage, the changes in power sources, load, and frequency of the microgrid are evaluated and measured. Then, in the next step, the output of the system (frequency changes) and the appropriate controller signals are applied according to the law. The relationships governing the behavior of the controller are described using relationships (15) to (21).

$$\min J={{\displaystyle \sum }}_{J=N1}^{N2}{W}_j\left(\Delta F\right){\left(k+j\right)}^2+{{\displaystyle \sum }}_{i=1}^{Nu}Vi\left(\Delta u_{DEG}\right)\left(k+j\right)-\left(\Delta u_{DEG}\right){\left(k+j-1\right)}^2$$

(15)

$$\left(\Delta u_{DEG}\right)\left(k\right)=\left(\Delta u_{DEG}\right)\left(k-1\right)+{{\displaystyle \sum }}_{i=0}^{Nt}\delta i\left(\Delta F\right)\left(k-i\right)$$

(16)

$$\Delta u_{min}\le \left(\Delta u_{DEG}\right)\left(k\right)-\left(\Delta u_{DEG}\right)\left(k-1\right)\le \Delta u_{max}$$

(17)

$$V{j}^{min}\le Vj\le V{j}^{max}$$

(18)

$$U=\left[\Delta u_{DEG},\Delta u_{BESS},\Delta u_{FESS}\right]$$

(19)

$$W=\left[\Delta P_{FC}+\Delta P_{WTG}+\Delta P_{PV}+\Delta P_{FC}-\Delta P_L\right]$$

(20)

Function: $J=N1,N2\dots$ which should be minimized; N1: Lower limit; N2: Upper limit; $Nu$: Control limit; $\delta i$ is the numerical coefficients resulting from solving the problem (by minimizing J). The index introduced in relation (21) is used to compare control methods in the simulation section.

$$\mathrm{findex}={{\displaystyle \int }}_{t=0}^{t=t_{simolation}}\Delta f\left(dt\right)$$

(21)

5. Case Study

The studied system is shown in Figure 3, i.e., the main grid system containing conventional DEG, solar panels, wind turbines, fuel cells, and battery energy storage systems. The hierarchical control structure of the microgrid includes WTG, FC, BESS, PV, DEG, FESS, and AE.

The dynamic model of a wind turbine for the analysis of a small signal is expressed by relation, and its characteristic function is expressed as follows:

$$\Delta P_{WTG}=\frac{K_a{k}_{WTG}\Delta P_{WTG}}{T_{WTG}}-\frac{\Delta P_{WTG}}{T_{WTG}}$$

(22)

$$G_{WTG}\left(s\right)=\frac{K_a{k}_{WTG}}{1+s{T}_{WTG}}=\frac{\Delta P_{WTG}}{\Delta P_W}$$

(23)

where $k_{WTG}$ and $T_{WTG}$ are the gain coefficient and the time constant, K_a is a numerical coefficient that expresses the wind turbine power percentage, $\Delta P_{WTG}$ represents the changes in the electrical output of the wind turbine, and $\Delta P_W$ expresses changes in the power obtained from the wind.

The following is the dynamic PV model:

$$\Delta P_{PV}=\frac{k_{pv}{\Delta }_{\phi }}{T_{pv}}-\frac{\Delta P_{pv}}{T_{pv}}$$

(24)

$$G_{pv}\left(s\right)=\frac{k_{pv}}{1+s{T}_{pv}}$$

(25)

where $k_{pv}$ and $T_{pv}$ are the PV gain coefficient and the time constant, $\Delta P_{pv}$ are the changes in the PV electrical output, and ${\Delta }_{\phi }$ are changes are in solar radiation intensity.

Diesel generators play a major role in hybrid microgrids; as the load increases, they are responsible for providing part of the capacity needed to reach equilibrium. The dynamic model of a diesel generator is expressed via small-signal analysis as follows:

$$\Delta P_{DEG}=\frac{k_{DEG}\Delta P_C}{T_{DEG}}-\frac{k_{DEG}\Delta F}{R{T}_{DEG}}-\frac{\Delta P_{DEG}}{T_{DEG}}$$

(26)

$$G_{DEG}\left(s\right)=\frac{k_{pv}}{1+s{T}_{pv}}$$

(27)

where $k_{DEG}$ and $T_{DEG}$ are the gain coefficient and the time constant of the diesel generator, $R$ is the speed drop coefficient, and $\Delta P_{DEG}$ represents changes in the DEG power.

The dynamic model of the fuel cell, electrolyzer, battery, and flywheel is described below:

$$\Delta P_{FC}=\frac{k_{FC}\Delta P_{AE}}{T_{FC}}-\frac{\Delta P_{FC}}{T_{FC}}$$

(28)

$$G_{FC}\left(s\right)=\frac{k_{FC}}{1+s{T}_{FC}}$$

(29)

$$\Delta P_{AE}=\frac{k_{AE}\left(1-k_a\right)\Delta P_{WTG}}{T_{AE}}-\frac{\Delta P_{AE}}{T_{AE}}$$

(30)

$$G_{AE}\left(s\right)=\frac{k_{AE}}{1+s{T}_{AE}}$$

(31)

$$G_{BEES}\left(s\right)=\frac{k_{BESS}}{1+s{T}_{BESS}}$$

(32)

$$G_{FEES}\left(s\right)=\frac{k_{FESS}}{1+s{T}_{FESS}}$$

(33)

where $k_{FC}$, $T_{FC}$, $k_{AE}$, $T_{AE}$, $k_{BESS}$, $T_{BESS}$, $k_{FESS}$, and $T_{FESS}$ are the interest rate and time constant of the diesel generator, electrolyzer, battery, and flywheel, respectively; $R$ is the speed drop coefficient, $\Delta P_{FC}$ represents changes in the fuel cell power, and $\Delta P_{AE}$ expresses changes in the $\mathrm{electrolyzer}$ power.

The parameters of the microgrid’s power sources are shown in

Table 1. Parameters of the microgrid’s power sources [55,56].

Parameter	Value
D	0.012
H	0.1667/2
TFESS	0.1
TBESS	0.1
TFC	4
KFC	1.1
KAE	1.5
KBESS	−1.3
TDEG	2
TWTG	1.5
TAE	0.5
KWTG	1.0
R	3
$Ka$	0.6
KDEG	1.3
KFESS	−1.1
TPV	1.5
KPV	1

, and the nominal power of the microgrid is shown in

Table 2. Nominal microgrid power [55,56].

$\mathbf{Nominal} \mathbf{Power} \left(\mathbf{pu}\right)$		$\mathbf{Loads} \left(\mathbf{pu}\right)$
wind turbine	1	PL1 = 1.1
fuel cell	0.3
photovoltaic	0.7
diesel generator	1.6	PL2 = 1
flywheel	0.45
batteries	0.45

. The proposed algorithm is applied to the studied system and used to optimize the controller parameters.

Table 3. Initial parameters for PSO-GSA.

Parameter	Value
Number of particles	50
Max iterations	100
$a$	20
G0	1
C1,C2	2.2

shows the initial parameters for the PSO-GSA. The results for the studied system are shown in

Table 4. Optimized PID parameter results (PSO and PSO-GSA).

Optimization Method	Parameter
	Kp0 ≤ Kp ≤ 5	Ki0 ≤ Ki ≤ 5	Kd0 ≤ Kd ≤ 5
PSO	1.954	3.1001	2.4686
PSO-GSA	1.8406	2.997	2.4337

, and Figure 4 depicts the proposed algorithm in the sample system.

Figure 5 shows the convergence of the algorithm. As shown in the figure, the proposed hybrid algorithm is optimized at a significant speed to the final value, which indicates its high speed and proper accuracy.

K_p, K_i, and K_d parameters are calculated according to the following equation:

$$k_p=k_p^{\ast }+\Delta k_p,\{\begin{array}{cc}{k}_1>0& \Delta k_p=k_1{k}_p^{\ast }{c}_p\\ k_1<0& \Delta k_p=k_1{k}_p^{\ast }\frac{c_p}{1+c_p}\end{array}$$

(34)

$$k_i=k_i^{\ast }+\Delta k_i,\{\begin{array}{cc}{k}_2>0& \Delta k_i=k_2{k}_i^{\ast }{c}_i\\ k_2<0& \Delta k_i=k_2{k}_i^{\ast }\frac{c_i}{1+c_i}\end{array}$$

(35)

$$k_d=k_d^{\ast }+\Delta k_d,\{\begin{array}{cc}{k}_3>0& \Delta k_d=k_3{k}_d^{\ast }{c}_d\\ k_3<0& \Delta k_d=k_3{k}_d^{\ast }\frac{c_d}{1+c_d}\end{array}$$

(36)

where K₁, K₂, and K₃ are the optimized normalized coefficients for the PID controller, and their values are 0.044, 30, and 16.5, respectively. $k_p^{\ast }, k_i^{\ast }, k_d^{\ast }$ are the nominal values of the parameters, all of which were set to 5 before optimization.

Note that the general solution methodologies based on the combinatorial optimization methods for tunning the PID gains are depicted in Figure A1.

6. Results and Discussion

The studied source includes a fuel cell and a photovoltaic generator, which utilize a higher technology than other conventional DGs. This source is also connected to the distribution system by an inverter. Since the production of electrical energy in a low-velocity fuel cell is carried out in DC and at low voltage, the accessories of the fuel processor unit (to produce hydrogen) and a conventional DC–DC incremental converter are required to increase the DC link voltage. To evaluate the proposed method in the modeled space, as well as the way to improve the dynamics of the system, various simulations were run. By applying a control signal to the sources, frequency disturbances were subsequently reduced due to potential changes in the microgrid. In this section, noise related to variable wind speed was applied, which ranged from 0.2 to −0.2 pu (Figure 6). Figure 7 shows the frequency changes of the microgrid system with and without a PID controller. It can be observed that, to some extent, the PID controller achieved good results regarding stability and the attenuation of frequency changes. For optimal results, the hybrid PSO-GSA algorithm was used. It was observed that, in this simulation, the proposed smart controller had a more favorable performance than that of the particle swarm controller (Figure 8). The power of the wind turbines, fuel cells, photovoltaic generators, batteries, flywheels, and diesel generators before and after the disturbances are shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. Apart from the photovoltaic power, other sources were affected by disturbances. To evaluate the performance of the proposed control method, several different disturbances were applied to the test microgrid, and the system’s response was compared to the results obtained with the algorithms. As mentioned in the previous section, there are resources in microgrids such as solar panels and wind turbines whose power is rather volatile.

In this section, the stepped load was first applied to the system (Figure 15), and the standard PSO and PSO-GSA algorithms were then executed. The results of the frequency variations are shown in Figure 16. The figure shows that, with the stepped variations of load, the combined algorithm yields less overshot and undershot.

The performance of the algorithm on the standard test function was evaluated in order to determine its standard deviation. For a better assessment of the effectiveness of the proposed method, a larger range in comparison with other methods was selected. This function is defined as follows:

$$f(x)=-{{\displaystyle \sum }}_{i=1}^n\sin (x_i){(\sin (\frac{i\ast x_i^2}{\pi }))}^{2m}$$

(37)

For a better comparison, the best coefficients were extracted from other papers, albeit considering the same initial population. Figure 17 shows the distribution of the results obtained from 30 different implementations with the proposed methods. The proximity of the solutions obtained by the algorithm indicates its robustness and high performance. It also shows that the proposed method has a smaller standard deviation.

Regarding the accuracy of the algorithm with the Sphere function, in this section, the Sphere function is used for the accuracy of the proposed algorithm. The Sphere function acts like a closed circle and the specified value tends to be zero. The formulation of the desired objective function is in the form of Equation (38).

$$f\left(x\right)={\displaystyle \sum }_{i=1}^n\left(x_i^2\right) \left(-30,30\right)$$

(38)

The function in the algorithm is determined instead of the desired objective function, and its goal is to minimize the desired parameter. The results of this function in Figure 18 show that the convergence reaches a zero value.

7. Conclusions

In this paper, a sample microgrid with a PID controller was modeled while using a hybrid PSO-GSA. To better investigate and analyze the proposed controller, various errors were used. The results indicate that the proposed algorithm is more efficient in comparison with particle swarm-based controllers. The optimization algorithm proposed in this paper is novel and has a higher convergence speed compared to PSO algorithms. The proposed method was tested on a case study, and the results show that setting the controller parameters leads to a better frequency response. Therefore, the proposed controller has an optimal performance at adjusting the frequency of the microgrid and achieving the final response after a short transition time with low harmonic distortion.