An Optimized Frequency Control of Green Energy Integrated Microgrid Power System using Modified SSO Algorithm

doi:10.21203/rs.3.rs-1644771/v1

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An Optimized Frequency Control of Green Energy Integrated Microgrid Power System using Modified SSO Algorithm

https://doi.org/10.21203/rs.3.rs-1644771/v1

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published 26 Dec, 2023

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This paper proposes a modified sperm swarm optimization (MSSO) technique for automatic load frequency control (ALFC) of bio and renewable energy (RE) integrated Microgird (MG) system. A chaotic search based on a one-dimensional (1D) chaotic map is adopted to intensify the exploitation and exploration characteristics of sperm swarm optimization algorithm. The proposed MSSO technique is used to tune the gains of proportional integral derivative controller to regulate the frequency of MG system through minimization of integral time absolute error of frequency deviation. The effectiveness of the technique is evaluated in terms of steady state and transient performance indices for the response of frequency and power deviation. In addition, to validate the robustness, a sensitivity analysis is carried out under varying load condition, change in system parameter, and real-time variation in RE sources. The results obtained for the aforementioned cases show that the proposed MSSO tuned technique outperforms other techniques (Salp swarm algorithm, Particle swarm optimization, and sperm swarm optimization) in terms of steady state and transient indices. The real-time implementation of proposed controller for MG system is validated in Hardware-in-loop analysis with its stability analysis in frequency domain.

Automatic Load Frequency Control

Renewable Energy

Modified Sperm Swarm Optimization

Microgrid system

In today's energy scenario, Microgrid (MG) plays an important role in providing a reliable and secured energy supply for local and distant area of communities [1]. The MG is a small power network that integrates various type of distributed generation (DG) sources ( solar photovoltaic and wind sources, fuel cell, micro turbine, bio energy sources, and battery storage) and load facilities. In general, the MG can be operated either in grid connected or islanded mode with associated potential benefits: accommodating energy mix from different DG sources, promotes community energy independence and load management, increased flexibility and reliability of energy supply with reduced carbon emissions, and power losses [1–2]. To achieve better frequency regulation in MG network, the power balance (between generation and demand) needs to be maintained during load and source variations. [1]. However, due to intermittency and stochastic nature of RE sources, controlling of power balance is a significant challenge. The power imbalance causes the frequency deviation, system instability, and lowering the quality of energy supply of MG networks [2–3]. The power and frequency deviations are higher in case of islanded MG network than grid-connected mode [1, 3]. This is due to the intermittent nature of RE sources and the lack of inertia of an isolated MG network [1–2]. In such case, an effective load frequency control (LFC) strategy should be developed. The LFC tries to keep the frequency deviations within predetermined limits by minimising the mismatch between power generation and demand [2].

1.1 Literature Survey

Several control techniques have been developed to tune the gains of classical proportional–integral–derivative (PID) controller. As it is widely used for various industrial and power system control applications, because of their simplicity, ease of tuning and implementation, and lower expertise requirements [2]. Hence, the variants of PID and fractional order PID controllers have been utilized for LFC applications in MG and multi-area powers systems [4–7]. However, for dynamic, uncertain, and non-linear systems, these classical controllers are ineffective and underperform [8]. So, advanced intelligent and robust control methods such as fuzzy control [9], internal mode control [10], H₂/H_∞ control technique [11], sliding mode control (SMC) [12], and model predictive control (MPC) [13] for LFC application were proposed. However, these intelligent and robust controllers are more complex which required advanced mathematical skills to design and implement in real-time. The advantages derived from the aforementioned techniques are ineffectual in minimising the system error by attaining local minima in the case of a system with enormous dynamics. Also, these techniques involve complex mathematics for dynamic system with large dimensions. Therefore, effort has been made to enhance the performance of conventional PID controller through proper tuning of control parameters using advanced intelligence techniques. In this context, several meta-heuristic algorithms have been used in the literature to improve the efficiency of controllers for wide operating conditions. The particle swarm optimization (PSO) based fuzzy tuned PID control approach in AC MG [14], PSO and Firefly Algorithm (FA) in multisource MG [15], Grey Wolf Optimization (GWO) in multi-MG systems [16], Social-Spider Optimization algorithm (SSOA) in two area MG systems [17], Moth Flame Optimization (MFO) Based Fuzzy-PID Controller in islanded MG power system [18], and Grasshopper Optimization Algorithm (GOA) in hybrid MG system [19] have been applied by the researchers to optimize the controller gains of LFC system and verified the dynamic response of controllers for frequency variation under uncertain conditions such as load perturbation, parameters variation of RE sources. Also, the Whale Optimization Algorithm (WOA) in hybrid MG power system consists of reheat thermal and solar thermal units [20] has been utilized to optimize Fuzzy based PID controllers of LFC system. Furthermore, numerous hybrid optimization approaches have also been proposed in the literature to achieve the improved convergence for optimal tuning of controllers [21–23].

Recently, the performance of meta-heuristic algorithms are enhanced by adopting chaos theory in the search direction thereby reaching the optimal global minima. The chaos theory is characterized by its repetitiveness, ergodicity, and dynamicity. According to chaos theory, the optimization algorithm analyses the search space in many landscapes and returns a best optimal solution [24]. Some of the research works include imperialist competitive algorithm (ICA) [25], grasshopper optimization algorithm (GOA) [26], salp swarm algorithm (SSA) [27], crow search algorithm (CSA) [28], krill herd algorithm (KHA) [29]. All these studies show that chaotic sequences improve the global convergence speed and exploration/exploitation capabilities of optimization algorithms that resulting in more diversity in solutions.

The sperm swarm optimization (SSO) algorithm is a newly developed metaheuristic algorithm based on swarming behaviour of sperm during natural fertilisation [30–31]. The SSO is widely used to solve the problems that includes optimal power flow, forecasting of water levels, feature selection, and classification [32]. As SSO has relatively simple structure, ability of discovering global optimum with high rate of convergence, precise exploitation ability, simple computations, and easy to implement. Despite these benefits, SSO has major disadvantage of trapping into local minimum [32]. As a result, this study presents a modified version of SSO (MSSO) to overcome the disadvantages of SSO algorithm. The proposed MSSO is developed by combining the chaos theory with SSO algorithm. Chaos is a phenomenon that resembles the movement of a random process [29]. The randomness from the chaos can be obtained by applying the appropriate chaotic maps. The chaotic sequence maps (1D maps) are simple and effective for improving the global convergence and exploration/exploitation capabilities of optimization algorithms [33]. In this work, a 1D sine chaotic map is implemented as it yields better dynamic results with optimal minima (best fit value) than other type of maps. Also, a good balance between the exploration and exploitation phases is achieved by preventing premature convergence and falling into local optima. To the best of authors' knowledge, there are no research papers in the literature on application of MSSO for LFC of MG power systems. Hence, MSSO technique is proposed in this study to improve the controller performance in terms of steady-state and dynamic responses of frequency and power flow variation in an islanded MG system under various conditions and sensitivity analysis. The main contributions of this study are summarized as follows:

A chaotic based 1D sine map is implemented for MSSO technique to optimize the parameters of PID controller for ALFC of islanded MG network

A comprehensive analysis is carried out to examine the performance of MSSO technique under different combination of power sources interconnected into the MG network.

The robustness is validated through sensitivity analysis under changes in rapid load demand, system parameter, and uncertainties of RE sources.

The proposed MSSO control strategy is implemented in a real-time digital simulator run in a hardware-in-the-loop (HIL) environment and stability of the system is analysed through bode analysis.

The structure of manuscript is organized as follows: Section 2 explains the description of MG model studied. Section 3 presents the controller with formulation of objective function. Section 4 explains the concept of optimization algorithms such as SSO, and chaotic based MSSO. Section 5 presents the results and discussion of MG simulation with different operating scenario’s including sensitivity, convergence performance, and stability analysis. Section 6 discusses the validation using a real-time digital simulator environment with HIL. Section 7 summarises the conclusion and future scope of the proposed work.

Figure 1 depicts the basic configuration of islanded MG model which includes diesel engine generator (DEG), bio-gas turbine generator (BTG), battery energy storage (BES), solar photovoltaic (PV) unit, wind turbine generator (WTG), and AC loads. The DEG is a traditional power source that runs on diesel fuel, and the power generated by BTG is depend on the availability of bio-gas. The duration of bio-gas production (via anaerobic waste digestion) in bio plant varies based on weather condition. The RE (PV and WTG) are intermittent power generation sources as it depends highly on weather condition. The BES serves as an active power source which smooth out the power fluctuations from RE sources to maintain the frequency oscillations.

Figure 2 depicts the small signal transfer function model of an islanded MG network. The linearized first order transfer function models are used to represent the equivalent model of power sources like DEG, BTG, PV, WTG, and BES. As the LFC study is based on small signal analysis, lower order transfer function models are adequate for analysis of MG network [34–35]. A decentralized approach of controllers are used in this proposed MG model to control the frequency and regulate the power output from the sources. The RE sources (PV and WTG) are not included in the LFC loop because of their intermittent output power and weather dependent. The description of linearized transfer function models for different sources (power sources and energy storage) are given below [18, 36]:

The equivalent model for Diesel engine generator (DEG) is:

$${G}_{DEG}\left(s\right)=\frac{\varDelta {P}_{DEG}}{u1}=\frac{{K}_{DEG}}{1+s{T}_{DEG}}$$

where ΔP_DEG states the change in power of DEG, u₁ is the controller output signal, K_DEG and T_DEG denotes the gain and time constants of DEG.

The simplified model of Bio-gas turbine generator (BTG) is:

${G}_{BTG}\left(s\right)=\frac{\varDelta {P}_{BTG}}{u1}= \frac{1+{sX}_{C}}{\left(1+s{Y}_{C}\right)(1+s{b}_{B})}\times \frac{1+{sT}_{CR}}{1+s{T}_{BG}}$$\times \frac{{K}_{BT}}{1+s{T}_{BT}}$ (2)

where ΔP_BTG states the change in power of BTG, u1 is the controller output signal, K_BT and T_BT denotes the turbine gain and time constant, T_CR and T_BG are the bio-gas and combustion delay constants, b_B is the constant of valve actuator, X_C and Y_C are the lead and lag time constants, respectively.The equivalent first-order lag transfer function model for Battery energy storage (BES) is:

${G}_{BES}\left(s\right)=\frac{\varDelta {P}_{BES}}{u2}$ = $\frac{{K}_{BES}}{1+s{T}_{BES}}$ (3)

where ΔP_BES states the change in power of BES, u2 is the controller output signal, K_BES and T_BES denotes the gain and time constants of BES.

The solar PV (SPV) mode consists of converter and inverter module represented by its equivalent first-order approximation as follows:

$${G}_{SPV}\left(s\right)=\frac{\varDelta {P}_{SPV}}{\varDelta {\varnothing }_{solar}}=\frac{{K}_{SPV}}{1+s{T}_{SPV}}$$

where ΔP_SPV states the change in power of PV, Δ$\varnothing$_solar is the change in solar irradiance, K_SPV and T_SPV denotes the gain and time constants of PV.

The dynamics associated with the wind turbine generator (WTG) are linearized into the first-order lag transfer function as:

$${G}_{WTG}\left(s\right)=\frac{\varDelta {P}_{WTG}}{\varDelta {P}_{wind}}=\frac{{K}_{WTG}}{1+s{T}_{WTG}}$$

where ΔP_WTG states the change in power of WTG, ΔP_wind is the change in wind source, K_WTG and T_WTG denotes the gain and time constants of WTG.

The power balance in the MG network can be acheived by matching the total power generation with total demand, which is expressed as below:

ΔP_LOAD = ΔP_DEG + ΔP_BTG + ΔP_SPV + ΔP_WTG ± ΔP_BES (6)

A deviation in frequency occurs in MG network, in case of any imbalance between power generation and demand. The expression of change in power and relation between change in power and frequency of MG network are expressed in equations (7) and (8), respectively.

ΔP = ΔP_DEG + ΔP_BTG + ΔP_SPV + ΔP_WTG ± ΔP_BES - ΔP_LOAD (7)

${G}_{MG SYS}\left(s\right)=\frac{\varDelta {f}_{\left(S\right)}}{\varDelta {P}_{\left(S\right)}}=(\frac{1}{{M}_{S}+D}$ ) (8)

where M and D denotes the total inertia constant and damping constant of islanded MG power system [18].

Owing to the simplicity and easy implementation, the PID controller is widely utilised in control applications. The transfer function of PID controller in s-domain can be represented as [37]:

${G}_{PID}\left(s\right)=\frac{\text{E (S)}}{\text{U(S)}}$ = ${\text{K}}_{p}+ {\text{K}}_{i}s$ + $\frac{{\text{K}}_{d}}{\text{S}}$ (9)

where U (s) is the controller output signal, E (s) is the tracking error signal, K_P, K_i, and K_d are the proportional, integral, and derivative gains of the controller, respectively. In this study, for LFC application of MG network, classical PID controllers are used. The control parameters of PID controllers are optimized with proposed MSSO technique. The frequency deviation (Δf) of MG system is the error inputs to the controllers and their corresponding output control signals (ΔU) is expressed in Eq. (10) to control the output power of various power sources (DEG, BTG, and BES).

$\varDelta {U}_{\left(S\right)}=$ ${\text{(K}}_{p}+ {\text{K}}_{i}s$ + $\frac{{\text{K}}_{d}}{\text{S}}$ ) $\varDelta f$ (10)

Appropriate selection of objective function is important while optimizing the control parameters of classical PID controller. In LFC analysis of hybrid microgrids, the integral time absolute error (ITAE) is chosen for optimizing the control parameters by minimising overshoots and oscillations of frequency response[17]. The objective function (J) for optimizing the control parameters of LFC controller is expressed as [36].

$$Min. \text{J = }{\int }_{0}^{T}\left[\left|\varDelta f\right|\right]\text{t}\text{ }\text{dt}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

Subjected to the below given boundary constraints:

$${{K}_{p}}^{\text{min}}\text{<}{\text{K}}_{\text{pi}}\text{<}{{\text{K}}_{p}}^{\text{max}}$$

$${{K}_{i}}^{\text{min}}\text{<}{\text{K}}_{\text{ii}}\text{<}{{\text{K}}_{i}}^{\text{max}}$$

$${{K}_{d}}^{\text{min}}\text{<}{\text{K}}_{\text{di}}\text{<}{{\text{K}}_{d}}^{\text{max}}$$

where ${{K}_{p,i,d}}^{\text{min}}$ and ${{K}_{p,i,d}}^{\text{max}}$ are the minimum and maximum control parameters of PID controller.

In this study, the proposed MSSO technique is developed by implementing a chaotic 1D sine map with standard SSO algorithm. The description of SSO and MSSO technique are described in detail as follows:

4.1 Sperm Swarm Optimization (SSO) algorithm

SSO uses a distributed behavioural technique to solve the multi-dimensional problems based on the mobility of sperm that fertilises with ovum (egg) [30–31]. To find the best solution, it employs a set of candidate solutions (sperms) that move in a multidimensional search space domain. The swarms are simultaneously scouring their paths for the best solution (best sperm). In other words, the sperms consider both the best value obtained thus far (global best solution) and their own best solution (sperm best solution). It works by updating each sperm's position with random parameters rather than static variables until it finds the best value. Figure 3 depicts the sperm swarm movement with the global best solution (winner). Temperature and pH are the two parameters that influence the flexibility of the SSO algorithm [38]. Each sperm in SSO adjusts its location based on its current velocity, current location, and distance to the X_Sbest (current best) and X_Sgbest (global best) [32].

In SSO, the sperm velocity is updated by applying the mathematical rule for each part as given below [31]:

V_i (t) = Initial velocity of sperm + Personal sperm best solution + Global best solution (12)

First, the sperm swarm moves in a random direction, and its velocity in that direction is determined by the pH value. Then the mathematical rule for the initial velocity of sperm movement can be stated as,

Initial Velocity (V_1i (t)) = D$\times$Vi$\times$ Log₁₀ (pH_rand₁) (13)

where D denotes the velocity damping factor (between 0 and 1), V_i states the velocity of sperm ‘i’ at iteration ‘t’, pH_rand₁ represents the random number of pH value (between 7 and 14).

The rule for the personal sperm best solution can be expressed as,

Current best solution = Log₁₀ (pH_rand₂) $\times$ Log₁₀ (Temp_rand₁) $\times$ (X_Sbest – X_i (t)) (14)

where pH_rand₂ states the random number of pH value (between 7 and 14), Temp_rand₁ denotes the temperature metrics (between 35.1 and 38.5), X_i represents the current location of sperm ‘i’ at iteration ‘t’, and X_Sbest denotes the personal best solution of sperm.

The global best solution is determined based on the sperm data that is currently closest to the target (this sperm will be winner at the end). The rule for the global best solution can be written as,

Global best solution = Log₁₀ (pH_rand₃) $\times$ Log₁₀ (Temp_rand₂) $\times$ (X_Sgbest – X_i (t)) (15)

where pH_rand₃ denotes the random number of pH value (between 7 and 14), Temp_rand₂ states the temperature metrics (between 35.1 and 38.5), X_i represents the current location of sperm ‘i’ at iteration ‘t’, and X_Sgbest denotes the sperm global best value.

By combining all the rules (as per Eq. (12)), the velocity (V_i (t)) can be written as [31–32],

V_i (t) = [Rand () $\times$Vi $\times$Log₁₀ (pH_rand₁) + Log₁₀ (pH_rand₂) $\times$Log₁₀ (Temp_rand₁) $\times$ (X_Sbest – V_i (t) + Log₁₀ (pH_rand₃) $\times$Log₁₀ (Temp_rand₂) $\times$ (X_Sgbest – X_i (t))] (16)

The position of sperm at each iteration can be evaluated as,

X_i (t) = X_i (t) + V_i (t) (17)

4.2 Modified Sperm Swarm Optimization (MSSO) algorithm

Although the SSO algorithm has several advantages, such as simple computations, ease of learning and implementation, and the ability to find global optimum with accurate exploitation. However, it is susceptible to local optima and premature convergence when solving complex functions. As a result, in order to overcome this disadvantage, the SSO method is modified using chaotic dynamics. Chaos is generally described as "randomness," and it is highly dependent on the initial conditions of the task. During the optimization process, chaos can be employed as an option to provide search diversity. Its non-repetition property is introduced into the process of metaheuristic algorithms to improve the convergence rates and prevent being trapped into local minima [39]. To get better optimization solution, the chaotic sequences are generated by incorporating chaotic maps into the process of metaheuristic algorithms. For this process, a 1D chaotic map is most commonly used, because of their simplicity and ease of implementation [40]. There are variety of 1D chaotic maps available that includes cubic, circle, logistic, gauss, iterative chaotic map with infinite collapses, logistic, sine, tent, chebyshev, singer, and piecewise map [33]. Among these 1D maps, the sine map is considered as more prominent for enhancing the search capability and convergence speed of optimization algorithms [40–41].

$${x_{k+1}}=r\sin \left( {\pi {x_k}} \right)$$

The sine function as given in Eq. (18) maps the input angle with the range of [0, 4] as given in Fig. 4, where r denotes the bifurcation parameter, x_k and x_k+1 represents the sequences of current and iterative variable (between 0 and 1), respectively. The sine map exhibits a satisfactory chaotic behaviour when r is in the range of [0.7, 4].

Figure 5 depicts the process of tuning the controller parameters using a chaotic-based MSSO technique. The velocity function of SSO algorithm (Eq. (16)) is modified by substituting the random function into a chaotic function, and the updated velocity equation of the MSSO approach is expressed as,

V_i (t) = [chaos (n) $\times$Vi $\times$Log₁₀ (pH_rand₁) + Log₁₀ (pH_rand₂) $\times$Log₁₀ (Temp_rand₁) $\times$ (X_Sbest – V_i (t) + Log₁₀ (pH_rand₃) $\times$Log₁₀ (Temp_rand₂) $\times$ (X_Sgbest – X_i (t))] (19)

where chaos (n) is the sine function, as given in Eq. (18), V_i (t) is the velocity of agent, pH_rand₁ and pH_rand₂ are the metrics of pH (varies between 7 and 14), Temp_rand₁ is the metric of temperature (varies between 35.1 and 38,5), X_Sgbest is the sperm global best value, and X_i (t) is the current location of swarm. In addition, the size of swarms is set to 30 and the maximum number of iterations is set to 100 for SSO and MSSO algorithms. The performance results of SSO and MSSO are verified with results of other swarm-based algorithms such as Salp swarm algorithm (SSA) and particle swarm optimization (PSO) algorithm. The important parameters such as swarm size (50), number of iterations (100), inertia weight (W = 1), inertia weight damping ratio (Wdamp = 0.99), personal learning coefficient (C1 = 1.5), and global learning coefficient (C2 = 2) for PSO and for SSA, the parameters like searching agent (60), number of iterations (100), random variable C2 and C3 are in the range of [0, 1] are considered, respectively. In this study, for LFC of MG network, the PID controllers are used to regulate the frequency. The tuned values of PID control parameters using the proposed MSSO and other optimization techniques (SSO, PSO, and SSA) for different conditions od system model are listed in Table 1.

Table 1

Tuned gain parameters of PID controller with different optimization techniques
Case Conditions	control gains	SSA	PSO	SSO	MSSO
	K_p	-3.993	-6.999	-9.999	-19.999
	K_i	-1.437	-2.485	-3.683	-7.461
Case 1A	K_d	-1.163	-1.821	-2.667	-5.073
	K_p1	-3.998	-1.138	1.029	1.410
	K_i1	-3.993	0.621	-1.747	0.014
	K_d1	0.729	-0.837	0.872	1.815
	K_p	-3.849	-7.947	-10.999	-19.999
	K_i	-3.798	-3.999	-9.999	-19.898
Case 1B	K_d	-2.451	-2.216	-5.911	-8.836
	K_p1	-0.189	-0.024	-0.338	-1.247
	K_i1	-3.839	-0.295	0.077	1.999
	K_d1	-3.834	-0.530	-3.039	-19.969
	K_p	-3.856	-7.497	-9.887	-19.978
	K_i	-1.629	-3.203	-4.369	-8.853
	K_d	-2.022	-2.998	-3.394	-5.565
Case 1C	K_p1	0.420	-0.416	-1.030	0.635
	K_i1	-0.224	-0.344	0.056	-1.031
	K_d1	0.744	-0.337	-0.002	-0.378

In this section, the proposed MSSO algorithm performance is first validated by comparing its performance with SSO algorithm in terms of statistical findings using five well-known benchmark functions from the literature studies. Further, the study is extended for LFC analysis of RE integrated MG network model operating in islanded mode under various case conditions. To verify the dynamic performance of proposed MSSO tuned controller for LFC application, the MG model is simulated with integration of different combination of power sources. Furthermore, a sensitivity analysis is carried out to prove the robustness of MSSO tuned controller of MG model at rapid load change, change in inertia, and real time power variation of SPV and WTG.

5.1 Validation of benchmark functions

Initially, the performance of MSSO against SSO algorithm is studied using a five common classical benchmark functions in order to provide a fair assessment of its efficiency. The various benchmark functions considered are sphere, rosenbrock, rastrigin, griewank and kowalik [42]. The descriptions of these benchmark functions, including their mathematical formulation, constraints and their optimal value are depicted in Table 2. Among these five functions, the F1 and F2 are single peak functions. These functions have a single global optimum solution. On the flipside, the multipeak functions such as F3–F5 are characterised by the presence of many local extremums [43]. The population size for all both SSO and MSSO is set as 50, and the maximum number of iterations is set to 100. During evaluation process each benchmark function is run for 30 trial runs and results were collected. Table 3 reports the statistical findings of each algorithm for every benchmark function including its best, maximum (max), minimum (min), mean and standard deviation (Std) values. It is clear from the comparison results as stated in Table 3 that the proposed Sine Map Chaotic SSO (MSSO) algorithm outperforms than other chaotic map embedded SSO and standard SSO in terms of accuracy and fit value. Furthermore, the minimum value of Std shows that the proposed method for all the benchmark functions provides maximum consistency. The simulation is carried out using MATLAB R2018b on PC running with following specifications of quad-core Intel Core i5-4810MQ CPU, 2.80 GHz and 16 GB of RAM.

Table 2

Benchmark functions
Function Name		Formulae	Dimension (d)	Range	Minimum Value
Sphere	F1	${\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{\text{d}}} {{{\text{x}}_{\text{i}}}^{{\text{2}}}}$	30	[-100,100]	0
Rosenbrock	F2	${\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{{\text{d-1}}}} {{\text{[100(}}{{\text{x}}_{{\text{i+1}}}}{\text{-x}}_{{\text{i}}}^{{\text{2}}}{{\text{)}}^{\text{2}}}{\text{+(}}{{\text{x}}_{\text{i}}}{\text{-1}}{{\text{)}}^{\text{2}}}{\text{]}}}$	30	[-30,30]	0
Rastrigin	F3	${\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{\text{d}}} {{\text{[x}}_{{\text{i}}}^{{\text{2}}}{\text{-10cos(2\varvec{\pi}}}{{\text{x}}_{\text{i}}}{\text{)+(}}{{\text{x}}_{\text{i}}}{\text{-1}}{{\text{)}}^{\text{2}}}{\text{]}}}$	30	[-5.12,5.12]	0
Griewank	F4	${\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{\text{d}}} {\frac{{{\text{x}}_{{\text{i}}}^{{\text{2}}}}}{{4000}} - \prod\limits_{{i=1}}^{d} {\cos \left( {\frac{{{x_i}}}{{\sqrt i }}} \right)} } +1$	30	[-512,512]	0
Kowalik	F4	${\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{d} {\left( {{{\text{a}}_{\text{i}}}{\text{-}}\frac{{{{\text{x}}_{\text{1}}}{\text{(b}}_{{\text{i}}}^{{\text{2}}}{\text{+}}{{\text{b}}_{\text{i}}}{{\text{x}}_{\text{2}}}{\text{)}}}}{{{\text{b}}_{{\text{i}}}^{{\text{2}}}{\text{+}}{{\text{b}}_{\text{i}}}{{\text{x}}_{\text{3}}}{\text{+}}{{\text{x}}_{\text{4}}}}}} \right)}$	4	[-5,5]	0.0003

Table 3

Statistical results of benchmark functions
Function	Index	SM-CSSO	TM-CSSO	LM-CSSO	SSO
F1	Best	5.406e-22	2.00e-14	5.056e-12	1.22e-4
	SD	1.635e-16	1.0207e-11	5.934e-10	6.50e-11
	Mean	3.51e-17	2.657e-12	1.455e-10	1.31e-11
F2	Best	27.79	28.8897	30.4909	34.11
	SD	0.312	0.373	0.44454	0.431
	Mean	28.062	28.236	28.257	28.36
F3	Best	0	0	1.48e-12	0
	SD	2.73e-12	4.846e-11	4.77e-9	2.77e-6
	Mean	5.002e-13	1.1582e-11	8.90e-10	6.25e-4
F4	Best	0	3.625e-12	4.98e-9	1.87e-6
	SD	0	0.962	1.652	3.525
	Mean	0	32.526	39.87	43.895
F5	Best	0.00049	0.00052	0.00056	0.0181
	SD	0.00026	0.000473	0.00034	0.0345
	Mean	0.00059	0.00081	0.00066	0.0009

5.2 Time domain Analysis of MG model

The dynamic performance of MSSO tuned PID controllers for LFC of MG network is verified by considering the following case conditions: Case 1A: Integration of all power sources at different load conditions, (ii) Case 1B: Integration of Bio-gas turbine generator and RE sources, and (iii) Case 1C: Integration of diesel engine generator system.

Case A: Integration of all power sources in MG network

In this case, the integration of all power sources was considered and the frequency response of MG system is studied for different loading conditions of constant load, increment and decrement in step load demand. During this case, the total power generation of MG network can be expressed as the sum of power generation from individual power sources represented in Eq. (20).

$${ P}_{Tg}\text{= }{\text{P}}_{\text{DEG}}\text{+}{{\text{P}}_{\text{BTG}}\text{+}{\text{P}}_{\text{BES}}\text{+P}}_{\text{SPV}}\text{+}{\text{P}}_{\text{WTG}}$$

where ${ P}_{Tg}$is total power generation, ${\text{P}}_{\text{DEG}}$, ${\text{P}}_{\text{BTG}}$, ${\text{P}}_{\text{BES}}$, ${\text{P}}_{\text{SPV}}$, and ${\text{P}}_{\text{WTG}}$ are the power generation from DEG, BTG, BES, SPV, and WTG, respectively.

(i) Constant load condition: During this analysis, a constant load demand of 0.01 p.u of plant capacity is considered.. The dynamic response of controllers in terms of frequency and power deviations are shown in Figs. 6 (a) and 6 (b), respectively. The effectiveness of proposed MSSO tuned PID controllers was verified with other optimization techniques of SSA, PSO, and SSO in terms of steady state and transient state performance indices. The results of transient state performance indices (settling time (s), control effort (CE), and peak overshoot) and steady state performance indices such as integral absolute error (IAE), ITAE, integral square error (ISE), and integral time square error (ITSE) for frequency and power variations of MG network are listed in Tables 4 and 5, respectively. The results show that the proposed MSSO technique outperforms than existing optimization techniques (SSA, PSO, and SSO) in terms of preserving low ST (s), reduced CE and peak overshoot for frequency and power deviations of MG network. Furthermore, the proposed MSSO technique has significant improvement in steady-state performance indices (IAE, ITAE, ISE, and ITSE) over existing optimization techniques.

Table 4

Case 1A Results of performance Indices for frequency deviation (Δf)
Performance Indices	𝞓f
Performance Indices	SSA-PID	PSO-PID	SSO-PID	MSSO-PID
Settling time (ST) (s)	4.6876	4.1872	4.0337	3.4701
CE	0.0300	0.0299	0.0284	0.0265
Peak Overshoot (PO)	0.0784	0.0583	0.0468	0.0307
IAE	0.0588	0.0316	0.0219	0.0107
ITAE	0.0748	0.0393	0.0277	0.0134
ISE(X10^− 3)	2.4712	0.9016	0.5181	0.1569
ITSE(X10^− 3)	0.9923	0.2889	0.1294	0.0290

Table 5

Case 1A Results of performance Indices for power deviation (Δp) with constant load
Performance Indices	𝞓P
Performance Indices	SSA-PID	PSO-PID	SSO-PID	MSSO-PID
Settling time (ST) (s)	1.5399	1.4439	1.2873	0.9286
CE	0.0300	0.0299	0.0284	0.0265
Peak Overshoot (PO)	0.1000	0.1000	0.1000	0.1000
IAE	0.0844	0.04217	0.03125	0.01134
ITAE	0.0588	0.0210	0.0133	0.0036
ISE(X10^− 3)	2.5314	1.8253	1.1145	0.2027
ITSE(X10^− 3)	2.9913	0.0723	0.0425	0.0039

(ii) Decrease in load demand: In this case, a change in step load from 15–10% (0.15 p.u. to 0.1 p.u of plant capacity) at 5 s is considered. The dynamic response of p frequency and power deviations for the proposed MSSO and other optimization methods (SSA, PSO, and SSO) for are shown in Figs. 7 (a) and 7 (b), respectively. From the analysis, it is observed that a minimum undershoots of 0.016 Hz was observed with MSSO tuned PID controllers than other methods (SSA (0.043 Hz); PSO (0.031 Hz); and SSO (0.025 Hz)). Similarly, the power deviation also has faster response with minimum deviation using the MSSO tuned PID controllers than other methods.

(iii) Increase in load demand: In this case, a change in step load demand from 10–15% (0.1 p.u. to 0.15 p.u of plant capacity) at 5 s in MG network is studied. The dynamic response of frequency and power deviations for the proposed MSSO and other optimization methods (SSA, PSO, and SSO) tuned PID controllers is shown in Fig. 8. From the results, it seen that a fast response with minimum undershoots (-0.015 Hz) were observed for the proposed MSSO tuned controller than other optimization methods (SSA (-0.041 Hz); PSO (-0.03 Hz); and SSO (-0.024 Hz)). Figure 8 (b) shows the results of power deviations of MG network. The proposed MSSO has less overshoot with minimum settling time compared to other optimization methods. During the step load increment (10–15%) as shown in Fig. 9, the power dispatch from DEG and BTG, RE sources, and BES are shown in Figs. 10 (a), 10 (b), and 10 (c), respectively. It is inferred that the SPV (0.03 p.u.) and wind generator (0.4 p.u.) power were consistently maintained constant regardless of load change, and there was no control over the output power of RE sources. However, the DEG (0.008 p.u. to 0.02 p.u.) and BTG (0.22 p.u. to 0.06 p.u.) contributed more power than the BES unit because of its low capacity. Thus, it can be demonstrated that the proposed control scheme maintains the frequency within a safe limit during the dynamic load change conditions.

Case B: Integration of Bio-gas turbine generator and RE sources

In this case, the DEG was turned off, and integration of other power sources such as BTG, RE (solar PV and WTG) and BES were considered to meet the power demand of MG network. During this analysis, the total power generation of MG model is expressed in Eq. (21). Figure 11 depicts the dynamic performance of controllers in terms of frequency deviations using the proposed MSSO and other optimization methods. Table 6 displays the results of transient state (ST (s), CE, and PO) and steady state performance indices (IAE, ITAE, ISE, and ISTE) of controllers for frequency deviations. According to the results of the performance indices, the proposed MSSO tuned controller has superior performance in terms of evaluating transient state (reduced ST (s), CE, and PO) and steady state indices (enhanced IAE, ITAE, ISE, and ITSE). Due to the absence of DEG in the MG network, the lack of inertia in the MG network causes a wider range of frequency deviations than in case 1A analysis. However, the proposed MSSO tuned controllers' control response still respond faster which improves the system response.

$${ P}_{Tg}\text{= }{{\text{P}}_{\text{BTG}}\text{+}{\text{P}}_{\text{BES}}\text{+P}}_{\text{PV}}\text{+}{\text{P}}_{\text{WTG}}$$

Table 6

Case 1B Results of performance Indices for frequency deviation (Δf)
Performance Indices	𝞓f
Performance Indices	SSA-PID	PSO-PID	SSO-PID	MSSO-PID
Settling time (s)	5.8077	5.5834	4.6157	3.8669
CE	0.1188	0.1164	0.1203	0.1125
Peak Overshoot	0.1006	0.1042	0.0697	0.0510
IAE	0.0987	0.0792	0.0494	0.0297
ITAE	0.1567	0.1181	0.0683	0.0349
ISE(X10^− 3)	4.7894	4.2459	1.4581	0.6274
ITSE(X10^− 3)	2.7231	1.8615	0.7379	0.2484

CaseC : Integration of Diesel engine generator in MG network

Integration of DEG and BES was considered in this case to meet the power demand in the MG network. The total power generation of MG model is expressed in Eq. (22). Figure 12 depicts the dynamic response of frequency deviations for MG network using the proposed MSSO and other optimization methods. Table 7 shows the results of performance indices such as transient state and steady state indices. According to the results, the proposed MSSO tuned controller achieves faster control response in terms of frequency deviations and provides excellent performance. During this analysis, the range of frequency deviations was observed as minimum compared to the case 1A and case 1B analysis. This is due to the absence of RE sources which are intermittent and inertia less.

$${ P}_{Tg}\text{= }{\text{P}}_{\text{DEG}}\text{+}{\text{P}}_{\text{BES}}$$

Table 7

Case 1C Results of performance Indices for frequency deviation (Δf)
Performance Indices	𝞓f
Performance Indices	SSA-PID	PSO-PID	SSO-PID	MSSO-PID
Settling time (s)	1.9053	1.4647	1.2049	1.1470
CE	0.1386	0.1341	0.1331	0.1245
Peak Overshoot	0.0881	0.0606	0.0531	0.0274
IAE	0.0844	0.0421	0.0312	0.0113
ITAE	0.0588	0.0210	0.0133	0.0036
ISE(X10^− 3)	5.3142	1.8251	1.1842	0.2027
ITSE(X10^− 3)	2.9914	0.7312	0.1147	0.0410

Thus, from this various case analysis, it can be concluded that the proposed MSSO tuned PID controllers outperforms SSA, PSO and SSO tuned controllers in terms of evaluating the control response for frequency/power deviations as well as transient and steady state performance indices.

5.3 Sensitivity Analysis

To assess the robustness of the proposed controller, a sensitivity analysis was performed against the changes in MG system parameter (inertia constant, M), random change in load demand, and real-time power variations of RE sources (solar PV and wind generator). The proposed controller's dynamic response was validated by varying the inertia constant of MG system by +/- 30% (M = 0.26/ M = 0.14) from its nominal value (M = 0.2). Figures 13 (a) and 13 (b) exhibits the dynamic response of the proposed MSSO tuned PID controller at various inertia constant values. According to the results of frequency and power deviations, the proposed controller is still resilient and damps the oscillations. These results confirm that the proposed controller's performance is robust and does not need to be retuned for large changes in system parameter of MG network.

Furthermore, a random change in load demand profile with consecutive load variation (0.015, 0.019, 0.014, 0.02, and 0.015 p.u. of plant capacity) was considered over different periods of time (0, 20, 40, 60, and 80 s) as shown in Fig. 14 (a). Under these rapid load disturbances, the dynamic response of MG using the MSSO and other optimization techniques (SSA, PSO, and SSO) tuned PID controller is shown in Fig. 14 (b). From the response, it is noticed that the proposed MSSO tuned controller has reduced frequency deviations and faster response with higher damping than other method of tuning the controllers.

The proposed controller was further tested under the real time power variation of RE sources. The frequency deviations of MG network were analysed during this case of variation in SPV and wind power conditions. For simulating SPV power variation in MG network, a real time solar irradiance data (200 slots with interval of 0.5 s) from Subang meterological center of Malaysia [44] was considered. The real time SPV power variation and the frequency response of the controllers during the cae of PV power variation are shown in Figs. 15 (a) and 15 (b), respectively. Similarly, for simulating the wind power variation, a real time varying wind speed data (200 slots with interval of 1 s) from Kuala Terengganu meterological center of Malaysia [45] was considered. The wind power variation and the results of frequency devaitions using the proposed MSSO and other control methods are shown in Figs. 16 (a) and 16 (b), respectively. From the results of frequency response in both the cases (PV and wind power variation), it is evident that the proposed controller offers smoother response with reduced undershoot and overshoot in frequency than other control methods (SSA, PSO, and SSO). As a result, reduced frequency oscillations and steady state error was achieved with the proposed controller than other mehods. According to the results of the sensitivity analysis, the proposed MSSO tuned controller performs robustly and does not require controller settings to be retuned when subjected to substantial changes in system parameter (inertia constant of MG), random load, and uncertainty of RE sources.

5.4 Convergence performance

The evaluation of convergence performance is presented in this section while tuning the control gains of PID controllers with the proposed MSSO and other optimization techniques (SSA, PSO, and SSO). Figure 17 depicts the convergence curve which represents the convergence rate of optimization techniques in terms of minimised fitness value (ITAE) over the number of iterations. The fitness curves show that the proposed MSSO technique achieves the lowest error value and converges faster than other optimization methods. Furthermore, the proposed MSSO method with chaotic sequences of 1D sine map implementation improves the exploration and exploitation capability of the SSO algorithm and avoids the premature convergence in providing superior control performance for the controller with minimal settling time.

5.5 Stability analysis

The stability analysis was carried out to verify the frequency stability of MG network by the proposed method of tuning the controller. In this section, the stability analysis for the closed loop system of MG network model is analysed for case 1A of MG network model considered. The closed loop transfer function (CLTF) of MG model network (illustrated in the Appendix section) with proposed MSSO tuned PID controller was found to be higher order (12th ), and it is difficult to carry out the stability analysis for such a higher-order system. Hence, in this study, the higher order CLTF was reduced into second-order transfer function by Hankel matrix method. The process for the model order reduction using Hankel matrix method is presented in [46]. The reduced second-order transfer function of MG model network is given in Eq. (23) as:

$$\text{C}\text{L}\text{T}\text{F}=\frac{0.8054 {s}^{2} + 1.457 \times {10}^{-11} s + 1.701 \times { 10}^{-17}}{{s}^{2} +1221 s}$$

Figure 18 depicts the frequency response of bode analysis for the MG model, with a gain margin of 2.84 dB and a gain cross over frequency of 10.3 (rad/s). From the frequency response analysis, the closed-loop system is stable for the proposed MSSO tuned PID controller of MG network.

The proposed MSSO control scheme of MG network was validated through HIL implementation using RTDS. The RTDS is a real-time electromagnetic transient power system simulator that uses a digital simulator of OPAL-RT OP4510. The OP4510 is a small entry-level simulator that combines the key features of RT-LAB high-performance rapid control prototyping and HIL systems. It includes 128 fast I/O channels with signal conditioning, additional RS422 channels, and high-speed communication ports (SFPs), and is fully integrated with Matlab/Simulink and the Sim Power system. Figure 19 depicts the experimental setup for validating the MSSO-tuned PID controllers of the MG network model under various load conditions. The real-time validation results of frequency deviation with proposed MSSO and other optimization control methods (SSA, PSO, and SSO) under constant load and step load change (decrement) conditions of Case 1A analysis are shown in Figs. 20 (a) and 20 (b). From the real-time analysis, it is determined that the results obtained are promising as that of simulation results shown in Figs. 6 (a) and 7 (a), respectively.

In this work, a chaotic search based MSSO tuned PID controller is proposed to provide the optimised frequency control of an islanded MG power system. The MG model incorporates RE sources (solar PV and wind turbines), DEG, and BTG along with BES. The proposed MSSO control method is incorporated with a 1D sine map to enhance the convergence speed and search capability of the SSO algorithm. The dynamic control performance of the MSSO control method was verified with other control methods (SSA, PSO, and SSO) in terms of transient state (ST, CE, and peak overshoot) and steady state (IAE, ITAE, ISE, and ITSE) performance indices for the frequency and power deviations of the MG network under different conditions. During case 1A analysis (at constant load) of frequency response, reduced settling time (3.470 s) was observed with the proposed MSSO control method, which settles down to zero state with 1.227 s, 0.717 s, and 0.564 s faster than the SSA, PSO, and SSO control methods, respectively. Also, the objective error function of IATE (0.0134) for the MSSO method was significantly improved by around 82.9%, 65.9%, and 51.6% compared to the SSA, PSO, and SSO methods, respectively. The power deviation response during case 1A analysis (at constant load), a f reduced settling time (0.928 s) and an improved objective error function of IATE (0.0036) was observed with proposed MSSO control than other control methods. Moreover, from the results of Case 1B and Case 1C analyses, it can be noticed that the steady state and transient state indices were significantly improved with the proposed MSSO tuned PID controller than other methods. The results of sensitivity analysis show that the proposed MSSO tuned controller performs robustly and does not require retuning of controller parameters when subjected to substantial changes in rapid load, system parameter (inertia constant of MG), and uncertainty of RE sources. Furthermore, the results of the convergence performance and stability analysis clearly show that the proposed MSSO controller is more stable and converges faster than other optimization control methods. In addition, the results of HIL real-time digital simulation are more promising than simulation results with the proposed controller of the MG model. The distributed control strategy with implementation of an advanced optimization method for LFC of multi-area MG power systems is the future scope of this research work.

Acknowledgements

The authors would like to thank the Electrical and Electronics Engineering Department in Coimbatore Institute of Technology, Coimbatore-India, New Horizon College of Engineering, Bengaluru-India, and Nanyang Technological University, Singapore for providing the requisite facilities towards completion of this research.

Data Availability Statement: Not Applicable

Conflicts of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Author Contributions: Data curation, D.A.; Formal analysis, T.K, P.; Investigation, A.V.; Methodology, D.A.; Project administration, V.V. and S.S.T.; Resources, S.S.T. and M.R.; Writing – original draft, A.V.; writing—review and editing, A.X.I. and T.K.P.; All authors have read and agreed to the published version of the manuscript.

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An Optimized Frequency Control of Green Energy Integrated Microgrid Power System using Modified SSO Algorithm

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Journal Publication

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Abstract

Figures

1. Introduction

2. Description Of Islanded Mg Model

3. Controller With Formulation Of Objective Function

4. Optimization Control Algorithms

4.1 Sperm Swarm Optimization (SSO) algorithm

4.2 Modified Sperm Swarm Optimization (MSSO) algorithm

5. Results And Discussion

5.1 Validation of benchmark functions

5.2 Time domain Analysis of MG model

5.3 Sensitivity Analysis

5.4 Convergence performance

5.5 Stability analysis

6. Hardware-in-loop (Hil) Analysis With Real Time Digital Simulator (Rtds)

7. Conclusions

Declarations

References

Supplementary Files

Status:

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Function Name		Formulae	Dimension (d)	Range	Minimum Value
Sphere	F1	\({\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{\text{d}}} {{{\text{x}}_{\text{i}}}^{{\text{2}}}}\)	30	[-100,100]	0
Rosenbrock	F2	\({\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{{\text{d-1}}}} {{\text{[100(}}{{\text{x}}_{{\text{i+1}}}}{\text{-x}}_{{\text{i}}}^{{\text{2}}}{{\text{)}}^{\text{2}}}{\text{+(}}{{\text{x}}_{\text{i}}}{\text{-1}}{{\text{)}}^{\text{2}}}{\text{]}}}\)	30	[-30,30]	0
Rastrigin	F3	\({\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{\text{d}}} {{\text{[x}}_{{\text{i}}}^{{\text{2}}}{\text{-10cos(2\varvec{\pi}}}{{\text{x}}_{\text{i}}}{\text{)+(}}{{\text{x}}_{\text{i}}}{\text{-1}}{{\text{)}}^{\text{2}}}{\text{]}}}\)	30	[-5.12,5.12]	0
Griewank	F4	\({\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{{\text{d}}} {\frac{{{\text{x}}_{{\text{i}}}^{{\text{2}}}}}{{4000}} - \prod\limits_{{i=1}}^{d} {\cos \left( {\frac{{{x_i}}}{{\sqrt i }}} \right)} } +1\)	30	[-512,512]	0
Kowalik	F4	\({\text{f(x)=}}\sum\limits_{{{\text{i=1}}}}^{d} {\left( {{{\text{a}}_{\text{i}}}{\text{-}}\frac{{{{\text{x}}_{\text{1}}}{\text{(b}}_{{\text{i}}}^{{\text{2}}}{\text{+}}{{\text{b}}_{\text{i}}}{{\text{x}}_{\text{2}}}{\text{)}}}}{{{\text{b}}_{{\text{i}}}^{{\text{2}}}{\text{+}}{{\text{b}}_{\text{i}}}{{\text{x}}_{\text{3}}}{\text{+}}{{\text{x}}_{\text{4}}}}}} \right)}\)	4	[-5,5]	0.0003