Improved Virtual Synchronous Generator Principle for Better Economic Dispatch and Stability in Grid-Connected Microgrids with Low Noise

Abstract

The proper operation of microgrids depends on Economic Dispatch. It satisfies all requirements while lowering the microgrids’ overall operating and generation costs. Since distributed generators constitute a large portion of microgrids, seamless communication between generators is essential. While guaranteeing a reliable microgrid operation, this should be achieved with the fewest losses as possible. The distributed generator technology introduces noise into the system by design. To find the best economic dispatch strategy, noise was considered in this research as a limitation in grid-connected microgrids. The microgrid’s performance was improved, and the proposed technique also showed increased resilience. A virtual synchronous generator (VSG) control approach is proposed with a noiseless consensus-based algorithm to improve the power quality of microgrids. Voltage and frequency regulation modules are the foundation of the VSG paradigm. The synchronous generator’s second-order equation (hidden-pole configuration) was also used to represent the voltage of the stator and rotor motion. This study compared changes in power, frequency, and voltage for the microgrid by utilizing the described control approach using MATLAB. According to the findings, this method aids in controlling load and noise variations and offers distributed generators an efficient control strategy.

1. Introduction

Economic Dispatch is an optimization problem used to reduce a system’s costs. It is a significant issue in the world of power systems. All of the system’s constraints are considered when determining the system’s minimum cost. Economic dispatch problems have been solved using a variety of techniques, most frequently the quadratic convex function [1,2]. The Lagrangian relaxation approach and the quadratic programming, respectively, were applied in [3,4]. The equal increment cost condition was taken into consideration when studying consensus-based algorithms [5,6,7,8]. Economic dispatch and demand-side management issues were resolved to reduce the overall costs [9,10,11,12,13,14,15]. Particle Swarm Optimization was used in [16] for effective demand response in islanded microgrids. Refs. [17,18] used the Dragonfly algorithm and the Cuckoo Search Algorithm to solve the demand response in economic dispatch problems respectively. Ref. [19] introduced an improved Genetic Algorithm for optimal dispatch. The system/microgrids was assumed to be noiseless for the sake of the traditional economic dispatch problem. Noise from both system components and the environment is present in real-time. The effectiveness and resilience of the microgrids are impacted by this. It restricts their stability as well. To stabilize microgrids and improve their performance and resilience, noise must be incorporated into the consensus-based algorithm economic dispatch problem.

Noise was taken into account in several analyses [20,21,22]. These studies created a power-sharing strategy in microgrids that was parameter-independent and a noise-less algorithm for better voltage and frequency synchronization. However, there has not been much research in this particular area, which this study explored using the mentioned strategy. This approach was presented by [23] for isolated microgrids; however, grid-connected microgrids were not considered. This study outlines the grid-connected microgrids’ noiseless economic dispatch problem. Additionally, this method does not require a central controller, making the system cheaper and more cost-effective. Because a distributed strategy was used, a central controller was not required, which minimized the communication complexity [24,25,26,27,28,29]. Ref. [30] proposed a multi-agent consensus control-based economic dispatch algorithm allowing the microgrid to switch from isolated to grid-connected modes more reliably.

A key role is played by inverters in the interaction between the distribution network and the microgrid [31]. Conventionally, the droop control technique [32] is employed, although it is extremely vulnerable to changes in load. An improved droop control approach has been suggested in many publications. Some of them are dependent on the inverter’s output voltage. The drawback is that the droop coefficient causes the frequency to be too unstable. In DC microgrids, a discrete consensus-based adaptive droop control technique has also been put forth [33,34]. In several articles, the P/Q control method has been applied. The U/f control approach has primarily been employed in island mode. It is possible to create two sets of control systems using the P/Q and U/f control methods in conjunction with switching control components between the two [35]. However, because the two strategies have a complex structure, this system is challenging to create.

This paper introduces the novel concept of economic dispatch with noise effects on a grid-connected microgrid’s performance. A consensus-based algorithm along with a virtual synchronous generator strategy was used to reduce the fluctuations in voltage and frequency of the microgrids due to various noise levels. Two economic dispatch algorithms, i.e., the Lagrange formulation and the particle swarm optimization technique were compared to analyze their effect on the grid-connected microgrid’s overall performance. This paper is divided into multiple parts. The economic dispatch problem and PSO algorithm are defined in Section 2. Section 3 introduces the microgrid structure. The distributed noise-resilient economic dispatch approach is presented in Section 4 [23]. The VSG model and control approach are introduced in Section 5. The results and the discussion are explained in Section 6, and the conclusions are stated in Section 7.

2. Economic Dispatch Formulation

2.1. Lagrange Formulation

The economic dispatch issue for a microgrid that is connected to a grid is defined using the Lagrangian method. The goal purpose of the microgrid is first established. The most prevalent use of this function is to address economic dispatch issues. The cost of a generator in a microgrid system can be expressed using the following equation [36], taking into account all the generation units:

$${{\displaystyle \sum }}_{i=1}^n{P}_i{C}_i={{\displaystyle \sum }}_{i=1}^n{x}_i{C}_i^2+y_i{C}_i+z_i$$

(1a)

where

• $P_i{C}_i$ is the generator cost
• $x_i$, $y_i$, $z_i$ are the cost coefficients
• $C_i$ is the generator’s total power output.

To solve the economic dispatch issue, we aimed at lowering the microgrid’s generation costs. Equation (1a) becomes:

$$\min {{\displaystyle \sum }}_{i=1}^n{P}_i{C}_i=\min {{\displaystyle \sum }}_{i=1}^n{x}_i{C}_i^2+y_i{C}_i+z_i$$

(1b)

Additionally, the generator’s total electric output can be defined as [36]:

$${{\displaystyle \sum }}_{i=1}^n{C}_i=C_D+C_{loss}, \mathrm{for} C_i^{min}<C_i<C_i^{max}$$

(2)

where

• $C_D$ = total load; $C_{loss}$ = losses during transmission
• $C_i^{min}$= minimum generation limit of generator i
• $C_i^{max}$ = maximum generation limit of generator i.

In formulating the Lagrangian function, the above three equations together become [36]:

$$L(C_1, C_2, \dots C_n)={{\displaystyle \sum }}_{i=1}^n{P}_i{C}_i+\lambda (C_D+C_{loss}-{{\displaystyle \sum }}_{i=1}^n{C}_i)+{{\displaystyle \sum }}_{i=1}^n{u}_x\left(C_i-C_i^{max}\right)+{{\displaystyle \sum }}_{i=1}^n{u}_y\left(C_i^{min}-C_i\right)$$

(3)

where $\lambda$, $u_x$, $u_y$ are Lagrange multipliers.

Calculating each generator’s incremental cost (IC₁, IC₂, …, IC_n) is necessary to discover a solution to the aforementioned economic dispatch problem. These incremental costs for various generators should be equal to determine the microgrid’s minimal cost, i.e.,

IC₁ = IC₂ = … = IC_n

where n defines the number of generation units

This problem’s most popular solution was used here [36]:

$$\begin{gathered} {\lambda }_i=\frac{\partial P_i{C}_i}{\partial C_i}=2x_i{C}_i+y_i={\lambda }^{\ast }, \mathrm{for} C_i^{min}<C_i<C_i^{max} \\ {\lambda }_i=\frac{\partial P_i{C}_i}{\partial C_i}=2x_i{C}_i+y_i<{\lambda }^{\ast }, \mathrm{for} C_i=C_i^{max} \\ {\lambda }_i=\frac{\partial P_i{C}_i}{\partial C_i}=2x_i{C}_i+y_i>{\lambda }^{\ast }, \mathrm{for} C_i=C_i^{min} \end{gathered}$$

(4)

where

• ${\lambda }_i$ = incremental cost
• ${\lambda }^{\ast }$ = optimal incremental cost.

To determine an economic dispatch schedule for the microgrid, the economic dispatch issue must take into consideration the generation restrictions for each unit. The economic dispatch problem is rather simple to resolve and takes into account all the restraints of generators. However, when addressing the economic dispatch issue for microgrids, the majority of problems have some limitations that must be taken into account. For any issues relating to economic dispatch, the aforementioned equations serve as the fundamental problem formulation.

2.2. Particle Swarm Optimization (PSO) Algorithm

Particle Swarm Optimization is a computational method that was inspired by the movement of bird flocks and other organisms/particles by Kennedy, Eberhart, and Shi [16]. It is a population-based optimization tool in which particles change position by taking into account their velocity, their own experience, and the experience of their neighboring particles. The position and velocity of particle j in N-dimensional space are represented as a_j = (a_j1 , a_j2, … a_jN) and b_i = (b_j1, b_j2, … b_jN). The best position for this particle can be represented as Abest_j = ($a_{j1}^{A },a_{j2}^{A },\dots a_{jN}^A$). The best position for the neighboring particle can be represented as Bbest = ($a_1^{B },a_2^{B },\dots a_N^B$). New modified position and velocity can be formulated as:

$$\begin{array}{l}{b}_{jN}^{k+1}=\mathsf{\xi }. b_{jN}^k+m_1{r}_1 \times ({Abest}_{jN}-a_{jN}^k)+m_2{r}_2 \times ({Bbest}_N-a_{jN}^k) \mathrm{and},\\ a_{jN}^{k+1}=a_{jN}^k+b_{jN}^{k+1}\end{array}$$

(4a)

where

• k = number of iterations
• Ξ = inertia weight factor
• m₁, m₂ = acceleration constant
• r₁, r₂ = random number within the range [0, 1]

The inertia weight factor and the acceleration constant affect the performance significantly. The weight factor provides the required momentum for particles to move around in N-dimensional space. The acceleration constant indicates the weight of stochastic acceleration terms that help in pulling all particles towards Abest_j and Bbest positions. This algorithm is used iteratively to find convergence in optimal dispatch solutions. The best incremental cost was determined using this method and was then sent to the agents to either accept or modify the output power of generators to minimize the effect of noise on system parameters’ fluctuations.

3. Microgrid Structure

Four generator units constituted the microgrid that was the subject of this paper’s investigation. It featured two coal-based generator units, a wind generator, and a solar/photovoltaic (PV) generator and was connected to the grid. The quadratic Equation (1a) expresses the cost function of the units. C_loss was estimated to make up 7% of the total load.

Table 1. List of parameters for generators [36].

Unit	Cmin (kW)	Cmax (kW)	x	y	z
1	4	18	0.070	2.15	56
2	8	40	0.080	1.15	50
3	5	25	0.070	3.3	41
4	5	40	0.056	3.4	36

below provides the cost-coefficients values for each unit, the minimum power generation limits, and the maximum power generation limits for all the units [36].

4. Economic Dispatch with Consensus-Based Approach for Noise-Less Communication [19]

The strategy described in [23] is explained in this section. A microgrid’s communication link was developed. Each generator unit had a corresponding agent that gathered data from its corresponding unit. These data were read by a certain agent. All the agents that were part of this communication system could share data and communicate with each other [34]. We had four agents in total, each of which was connected to a different generation unit on our microgrid in grid-connected mode; there were four generation units. The information data received, collected, and processed by an agent was also exchanged with other agents. This exchange helped understand the current status of each unit. To reduce the overall cost of the microgrid system, the information received from the agent(s) was used to modify the output power. Noise from the components, surroundings, and electric/magnetic interference was taken into account for this analysis. This method includes noise that accumulated as a result of the communication between the units and that resulting from the communication between the units and the agents; it was considered in modeling as Gaussian noise [16]. The communication links between the agents were indicated as c₁₂, c₂₁, c₂₃, c₃₂, c₃₄, c₄₃, c₁₃, c₃₁, and so on.

Each agent determined the corresponding incremental cost of each unit before exchanging it with the others. Based on the data, the set point of the output power was determined and supplied to the appropriate generation units. To address the economic dispatch issue, the units modify their power generation capacity to have an equal incremental cost. This reduces the cost of microgrids. According to Refs. [23,36]:

$$\begin{array}{l}\mathrm{Z}[\mathrm{k}+1]=\mathrm{Z}\left[\mathrm{k}\right]+\mu \left[\mathrm{k}\right][\mathrm{P} \mathrm{z}[\mathrm{k}]+\mathrm{W}\mathrm{N}[\mathrm{k}\left]\right]\\ \mathrm{P}=-\mathrm{H}’\mathrm{N}\mathrm{H}\\ \mathrm{W}=\mathrm{H}’\mathrm{N}\\ \mathrm{H}={\mathrm{H}}_2-{\mathrm{H}}_1\end{array}$$

(5)

where

• Z[k] = incremental cost of a unit at the kth iteration,
• Z[k + 1] = incremental cost of a unit at the (k + 1)th iteration,
• µ[k] = recursive step size,
• N = r × r diagonal matrix with link control gain as its diagonal elements,
• H₁ and H₂ = r × n matrix where the rows are elementary vectors,
• N[k] = communication link noise.

$${\mathrm{H}}_1=\left|\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right|; {\mathrm{H}}_2=\left|\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right| \mathrm{and} \mathrm{H}=\left|\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 1& -1& 0\\ 0& -1& 1& 0\\ 0& 0& 1& -1\\ 0& 0& -1& 1\end{array}\right|={\mathrm{H}}_2-{\mathrm{H}}_1$$

N (small noise) = diag [0.2 0.2 0.2 0.2 0.2 0.2]

N (medium noise) = diag [0.5 0.5 0.5 0.5 0.5 0.5]

N (large noise) = diag [0.8 0.8 0.8 0.8 0.8 0.8]

Similarly, P and W can be determined from (5).

To lessen the effects of noise, we averaged the additional costs of the units. This produced a microgrid that was more durable, stable, and free of (or, with less) communication noise [23].

$$\begin{gathered} Z_{avg}[k+1]=\frac{1}{k+1}{{\displaystyle \sum }}_{j=1}^{k+1}Z\left[j\right]=\frac{1}{k+1}{{\displaystyle \sum }}_{j=1}^{k}z\left[j\right]+Z\left[k+1\right] \\ =Z_{avg}\left[k\right]-\frac{1}{k+1} Z_{avg}\left[k\right]+\frac{1}{k+1} Z[k+1] \end{gathered}$$

(6)

The noise-less economic dispatch with the consensus-based strategy using (5) and (6) is [23,36]:

Z[k + 1] = Z[k] + µ[k][P z[k] + WN[k]]

$$Z_{avg}[k+1]=Z_{avg}\left[k\right]+\frac{1}{k+1}\left[Z\right[k+1]-Z_{avg}[k\left]\right]$$

(7)

where Z_avg[k + 1] are the desired set points for the unit incremental costs.

This method is iterative, and an estimate was created using the step size. It was then averaged in the subsequent stages to limit the effect of noise. For each step size, the consensus problem was iteratively solved. The flowchart for the consensus-based economic dispatch algorithm is shown in Figure 1.

5. Virtual Synchronous Generator (VSG)

The VSG control system [37] is a comprehensive system that combines several modules to enable an efficient and effective electricity management. It is based on the VSG strategy, which is responsible for simulating the system’s performance and determining its optimal power output, and contains the Frequency Regulation Module, which adjusts the frequency of the output power to match that of the grid, the Voltage Regulation Module, which regulates the voltage of the output power, the grid-connected mode of the Control Module, which ensures the output power is synchronized with the grid, the SPWM Modulation Module, which adjusts the output current amplitude, and the Sampling Calculation Module, which calculates the output power by sampling the input signal. All of these modules work together to provide a reliable and secure electricity management system as shown in Figure 2 below.

When exposed to disturbances and load variations, power electronic inverters have poor system stability [37]. The traditional SG rotation has a significant output inductance and moment of inertia. Therefore, the microgrid’s power supply can be compared to the prime mover by reproducing the exterior characteristics of the microgrid into an SG. The microgrid inverter’s inverter and filter modules provide the electric energy produced by distributed sources to the load, while the energy storage system stores the residual electric energy.

Ref. [38] presented the SG second-order equation modeling, which includes the following equations.

Stator voltage equation:

$${\dot{\mathrm{U}}}_{\mathrm{refabc}}=\dot{\mathrm{E}}-\mathsf{\Delta }\dot{\mathrm{U}}$$

(8)

where

• ${\dot{\mathrm{U}}}_{\mathrm{refabc}}$ = three-phase reference voltage
• $\dot{\mathrm{E}}$ = electromotive force
• $\mathsf{\Delta }\dot{\mathrm{U}}$ = voltage drop caused by virtual synchronous impedance.

The output current I₀ of the inverter is equal to the synchronous generator stator current; r_a and X_d, respectively, are armature resistance and synchronous reactance. To obtain ΔU, a vector multiplication is used for (r_a + jX_d) and I₀. The module in Figure 3 provides a corresponding control signal in line with U_refabc after E is corrected for deviation.

The rotor motion model promotes system stability, as shown in Figure 4. When P_m and P_e do not match, it obviates this by adding J and D; dθ is the correction angle.

For the rotor motion model [39]:

$$\Delta \mathsf{\omega }=\frac{1}{\mathrm{J}}\int (\frac{{\mathrm{x}}_{\mathrm{m}}-{\mathrm{x}}_{\mathrm{e}}}{\mathsf{\omega }}-\mathrm{D}\Delta \mathsf{\omega }) \mathrm{dt}$$

(9)

$$\mathsf{\omega }=\Delta \mathsf{\omega }+{\mathsf{\omega }}_{\mathrm{r}}$$

(10)

where

• Δω = angular velocity difference
• X_m = mechanical power
• X_e = electromagnetic power
• J = moment of inertia
• D = damping co-efficient
• ω = angular velocity
• ω_R = rated angular velocity

The frequency module in Figure 5 includes the grid-connected sinusoidal wave SS, the system frequency f_V, the reference active power P_ref, the reference frequency f_ref, and the grid side frequency f_g. The frequency regulation module chooses its reference value based on the f_g range once the grid-connected signal SS has been sent by Judger₂. The reference value is chosen as f_g if it falls within the typical range and as f_ref if it does not. f_ref is used as the reference value while the system is in islanded mode.

To Judger₁, the frequency deviation Δf is provided. Depending on the interval in which the frequency difference is situated, Judger₁ passes on to the regulator in the next stage. The secondary frequency regulation is simulated by PI₁, and the frequency module regulates the main frequency per the co-efficient k_p. It also regulates and switches to a secondary frequency, if necessary. The Synchronous Generator maintains system frequency stability, both primary and secondary.

Q_ref and Q₀ are the inputs to the virtual voltage regulation module. The difference is multiplied by the voltage-reactive co-efficient k_U to obtain the electro-motive force for power adjustment (reactive), ΔE₁. To determine ΔE₂, which is the terminal voltage electro-motive force, the effective capacitor voltage U_c in the filter module’s reference voltage U_ref differential value is translated into amplitude. When the synchronous generator is operating in no-load mode, E_ref is the reference electro-motive force, whereas dE is the corrected electromotive force when in grid-connected mode, as shown in Figure 6. The U–Q relationship is as follows:

U − U_SGref = k_SGU (Q_SGref − Q)

(11)

where

• k_SGU = SG voltage-reactive coefficient
• U_SGref = reference values of voltage
• Q_SGref = reference values of reactive power

Figure 7 depicts the control module in grid-connected mode. The grid-connected module completes pre-synchronization when the sinusoidal wave SS switches from ‘0’ position to ‘1’ position. The PI₃ regulator and Judger₄ receive the difference between φ_g and φ_V (voltage phase angles). The rotor motion model receives the value provided by the PI₃ as dθ. Judger₄ chooses the next input value based on the interval in which the difference is found. The PI₂, a regulator, and Judger₃ receive the difference between $\left|\dot{{\mathrm{U}}_{\mathrm{g}}}\right|$ and Uamp. The virtual voltage regulator module receives the value from the PI₂ as dE. Similar to Judger₄, Judger₃ chooses the following input value based on the interval in which the difference is placed. It is determined by the difference between the two sides when determining the frequency. When all three Judgers are selected as 1 and the switch signal is changed from the “0” position to the “1” position, the pre-synchronization phase is said to be complete.

The system specifications for the LCL filter and the three-phase bridge inverter are provided in

Table 2. List of the components.

Components	Values
L1	6 mH
L2	1.5 mH
C	6 micro-F
J	0.15 kg·m2
Kp, kU	800 kW/Hz, 0.8 Hz/kVar
PWM freq	25 kHz
P at constant load	10 kW
Q at constant load	8 kVar
ra	0.05 ohm
Xd	0.05 H
P variable	5 kW
Q variable	3 kVar

.

6. Results and Discussion

Four different scenarios were used to examine the grid-connected microgrid. The microgrid was initially evaluated when there was no noise in the system. The Lagrange method described in the preceding section and the PSO algorithm were evaluated to see how well they operated in the absence of noise. In the second scenario, the system was subjected to noise with a variance of 0.2, and the performance was tracked. The noise variance was raised to 0.5 in the third test, and it was set to 0.8 in the final condition examined. MATLAB was used to examine how well the microgrid performed under various noise circumstances with and without the VSG control approach. Figure 8 shows the network figure of the algorithm used. The incremental costs from each generator were shared with the agents. These agents shared data and decided if the incremental cost was optimal. If this was not the case, the information was passed to the generator to adjust its output power until the optimal incremental cost criterion was met. Once an optimal economic dispatch solution was found, the total output power was sent to the VSG which was then used to meet the load demand or sent to the grid to fulfill any power deficit. The use of a consensus-based algorithm and of the VSG strategy helped reduce the noise effects and stabilize the microgrid.

With the introduction of various noise levels, the power output of the four generating units was observed, and we tried to model the ideal dispatch schedule in all instances. To demonstrate the system’s stability and the incremental cost for various noise levels, a comparison was evaluated. Figure 9 and Figure 10 demonstrate, respectively, the fluctuating power output of the four generating units during a 60 s period with and without the VSG using the Lagrange formulation. It took about 15 s without the VSG to stabilize the power output of all generators, whereas with the VSG, it took about 12 s. For a low (0.2) and medium (0.5) noise variance, it the system required about 20 s and 25 s, respectively to establish a consistent producing power output. Figure 11 and Figure 12 show this observation. For a 0.8 noise level, shown in Figure 13, the microgrid required about 45 s to reach a stable power output. It was observed that the economic dispatch solution was not stable and was less reliable for noise levels without the VSG strategy. It is clear from the graphs that the system required a few seconds to reach the desired constant value.

The system required lesser time to each the intended value of optimal power when was connected with the VSG, as shown in Figure 14. For low, medium, and high noise variance for all four units, it can be seen in the graph that there was a 5–10 s improvement for each unit in noise variance when the Lagrange method was used with the VSG control strategy.

Figure 15 compares the output power of all generating units for a 0.8 noise level using the Lagrange method and the PSO algorithm with the VSG control strategy. The resulting graph shows that the PSO algorithm performed better than the Lagrange method. With the PSO algorithm, convergence occurred faster than with the Lagrange method. For Unit 1, the PSO achieved stability 3 s earlier at the 27 s mark. For unit 2, the PSO algorithm performed better by 10 s, as indicated by the red legend. For Units 3 and 4, the PSO algorithm performed slightly better than the Lagrange method. The processing time for the PSO algorithm was 15.648 s, whereas the Lagrange method required 22.343 s. Overall, it can be concluded that the PSO algorithm solved the economic dispatch problem much quicker and efficiently than the Lagrange method.

The consensus-based approach aided in setting the incremental cost of each generator unit more quickly when there was noise. However, it is observed in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 that the consensus-based approach required more time as the noise variance increased. The graphs in Figure 16, Figure 17, Figure 18 and Figure 19 show that the average incremental cost ($/kWh) was approximately 5.91.

In Figure 16, Figure 17, Figure 18 and Figure 19, the incremental costs for all generator units were compared under various noise situations using the Lagrange method. As can be observed from the graph, it was difficult to stabilize the microgrid when it was connected to the grid, due to a larger noise variance (pink legend). For low to medium noise levels, it functioned well. Under no noise condition, the optimal incremental cost was reached in 45 s, as shown in Figure 16. In Figure 17, we can see that it took about 50 s to reach the optimal incremental cost for a 0.2 noise level. In Figure 18, it took 55 s to reach the optimal value for a 0.5 noise level, whereas it took more than 60 s to reach the optimal value for a 0.8 noise level, as shown in Figure 19.

In Figure 20 and Figure 21, it can be seen that with the VSG strategy, the generator units in the presence of higher noise levels stabilized more quickly. On average, 0.45 s were required for the system to stabilize with a high noise level of 0.8. In the absence of VSG, it took more than 0.9 s for the frequency to stabilize, as shown in Figure 20.

In Figure 22 and Figure 23 it is observed that with a load change, the system was more stable and reached its maximum limit faster with the VSG strategy. The system oscillated more and had more THD without the VSG, as observed in Figure 22. The microgrid stabilized in 0.35 s with a noise variance of 0.8 when both the Lagrange method and the VSG were in operation. This can be seen in Figure 23.

Table 3. Comparison of the time to reach the average optimal incremental cost by the two examined methods.

Noise Variance	Lagrange Method	PSO Algorithm
No noise	38.21 s	27.45 s
0.2 variance	48 s	38.20 s
0.5 variance	52.57 s	40.19 s
0.8 variance	90 s	51.85 s

compares the Lagrange method and the PSO algorithm performances when used with the VSG control strategy in relation to the incremental cost. For all noise variances, the PSO algorithm performed better and provided stability more quickly. In the absence of noise, the PSO algorithm required 27.45 s to reach the optimal incremental cost, whereas the Lagrange method required about 38.21 s. With a 0.2 noise variance, the PSO algorithm required 10 s less than the Lagrange method to reach the optimal incremental cost. For a medium noise variance of 0.5, the PSO algorithm was faster by about 13 s and for a high noise variance of 0.8, it was faster by about 39 s. It is observed in Table 3 that the PSO algorithm performed better for all levels of noise variance and required much less time to stabilize the system than the Lagrange method.

Table 4. Comparison of the time to reach the optimal levels of the shown parameters for a 0.8 noise variance.

Method/Algorithm	Frequency (Hz)	Max. Power (kW)
Lagrange	0.45 s	0.30 s
PSO	0.20 s	0.15 s

compares the Lagrange method and the PSO algorithm performances when used with the VSG control strategy in relation to frequency and maximum power. For all noise variances, the PSO algorithm performed better and provided stability more quickly. For the 0.8 noise condition, the PSO algorithm required 0.2 s to reach frequency stability, whereas the Lagrange method required about 0.45 s. Similarly, for the maximum power, the PSO algorithm required half the time to reach stability, as seen in Table 4.

7. Conclusions

For islanded microgrids, the suggested consensus-based approach for economic dispatch performs well [23]. This algorithm was utilized in this study to examine how the microgrid operated in grid-connected mode.

The VSG strategy was also introduced to enhance the system’s stability. The microgrid’s performances of the Lagrange method and the PSO algorithms were compared, with and without the use of the VSG strategy. It is concluded that with the inclusion of the VSG control strategy, the system could reach stabilization much faster in the presence of different levels of noise and load changes, as described in the Results section. This was observed for both the Lagrange method and the PSO algorithm. The consensus-based economic dispatch algorithm worked efficiently in conjunction with the VSG control strategy. It can also be concluded from the results obtained that the PSO algorithm performed better in stabilizing the frequency, output power, and load changes in the microgrid. The optimal incremental cost was also achieved faster with the PSO algorithm.

The results clearly showed that a consensus-based economic dispatch solution with the VSG strategy yielded a better stabilization in microgrids in the presence of low, medium, and high noise variances. Future research should be carried out to assess the performance of different algorithms on the noise effect in both grid-connected and islanded microgrids. Reactive power compensation can also be included in future studies for a better overall performance of microgrids.