Improving the Frequency Response of Hybrid Microgrid under Renewable Sources’ Uncertainties Using a Robust LFC-Based African Vulture Optimization Algorithm

Abstract

Power systems have recently faced significant challenges due to the increased penetration of renewable energy sources (RES) such as frequency deviation due to fluctuations, unpredictable nature, and uncertainty of this RES. In this paper, a cascaded controller called (1+PD)-PID is proposed to reduce the influence of RES uncertainties on the system and to maintain the system’s reliability during fluctuations. The proposed controller is a combination of (1+PD) and PID controllers in order. The output signal of the (1+PD) controller along with the frequency deviation and the power difference between adjacent areas are used as inputs to the PID controller to create the load reference signal. The parameters of the suggested controller are optimally tuned using the African Vulture Optimization Algorithm (AVOA) to ensure the best performance of the controller. A two-area interconnected system with non-reheat thermal power units combined with RES such as solar and wind energy is modeled using MATLAB/Simulink to evaluate the system response. The controller effectiveness is verified by subjecting the studied system to various types of fluctuations such as step load disturbance, variable load perturbation and RES penetration. The obtained simulation results prove that the proposed (1+PD)-PID controller in integration with AVOA offers a significant improvement in the system performance specifications. Moreover, the proposed AVOA-based (1+PD)-PID controller has proven its superiority over other comparable controllers having the least fitness function of 6.01 × 10⁻⁵.

1. Introduction

Recently, renewable energy sources (RES) have become of great importance due to several reasons such as global warming caused by emissions from conventional sources and increasing demand for more electric power. Although renewable sources are considered environmentally friendly, there are various limitations to their use, including low total inertia and fault ride, uncertainty due to climatic conditions, voltage and frequency fluctuations, and poor stability [1]. Modern electric power systems consist of several interconnected areas characterized by renewable and non-renewable (conventional) energy systems that create a hybrid power system known as a Microgrid (MG). It is a feasible approach in this direction to support the integration of sustainable energy sources with conventional sources to completely reduce the dependence on conventional energy. The MG consists of small generating and load units placed at different locations in the network. The MG can work either in integration mode or in isolation mode. The maximum load size for any MG is 10 MVA [2].

To ensure the security, stability and sustainability of the integrated power system equipment, the total output power generated in each controlled area should meet the total load needs while also accounting for system losses. When the system is subjected to any unexpected variations in load demand, the system must also be able to keep the frequency- and the interconnected area tie-line power within permissible limits [3]. As the load fluctuates over time, the generated power must be automatically updated to meet the demand. This configuration is known as automatic load frequency control (LFC). Thus, the LFC provides frequency balancing by regulating the speed of the prime mover, as well as stabilizing the tie-line power flow during normal and turbulent conditions [4].

Based on the aforementioned issues, the integration of LFC with electric power systems has gained much interest, especially in hybrid systems that feature RES alongside conventional sources due to power fluctuations and uncertainties. Several researchers have proposed different ways to mimic LFC embedded in different power systems in order to deal with this problem. In this field, traditional control techniques, artificial controllers, sliding mode control, and robust control methods have been used. Ease of implementation, low cost and minimal computation time are the reasons behind using traditional controllers such as proportional-integral (PI) controllers and proportional integral derivative (PID) controllers.

In order to obtain the optimized parameters of these conventional controllers, various optimization techniques have been applied such as Genetic Algorithm (GA) [5], Ant Colony Optimization [6], Bacterial Foraging Optimization Algorithm (BFOA) [7], Differential Evolution (DE) [8], particle swarm optimization (PSO) [9], and grasshopper optimization (GOA) [10]. A model predictive control (MPC) optimized using the Multi-Purpose Practical Swarm Optimization (MOPSO) algorithm and adaptive control techniques was introduced as a frequency control mechanism for a renewable energy-powered smart grid [11]. In [12], the Firefly algorithm has been used in a two-zone hybrid system that incorporates electric vehicles (EVs) as controllable using an integrated controller, but it does not cover the development of a PID controller or even a PI controller.

The cascaded controller configurations have recently been used instead of the conventional controller configurations due to their high efficiency and performance. This has led to the use of a variety of types of cascaded controllers to improve the frequency stability of the power system [13,14]. Furthermore, fractional order (FO) based controllers have also been implemented including a FOPID tuned moving inhibitory wave algorithm [15], a modified FOPID controller optimized via the Artificial Ecosystem Optimizer (AEO) [16], a combination of FOPID and a derivative Tilt integration controller [17,18], advanced high-order differential feedback control devices (HODFC, FHODFC) tuned by PSO [19]. Artificial-intelligence-based controls such as fuzzy logic controllers are integrated with the traditional PID controller and optimally regulated by different algorithms such as the Marine Predator Algorithm (MPA) [20] and Particle Swarm Optimization (PSO) technique [21]. A new control method based on the FO Type-2 Fuzzy Logic system was introduced to solve the problem of frequency management associated with renewable energy systems so that, in case of fluctuation of the load, the systems could return to their initial state. The parameters of the controller used were adjusted online depending on the extended Kalman filter [22] or using an optimization algorithm such as PSO [23]. Due to the benefits of the TID controller, which include its excellent ability to reject disturbances and increase the reliability and durability of the power system, it has recently been used to solve the problem of LFC [2,24,25,26].

In addition, another controller architecture was recently examined while designing different types of controllers for the LFC problem which focus on combining two distinct controllers in order to take advantage of both controllers. In [27], electric vehicles (EVs) were used as distributed energy storage units, and an aggregated topology of integral tilt and tilt derivative parameters was introduced to create an optimized TD-TI controller using a new optimized optimization approach, called the Quantum Chaos Game Optimizer (QCGO). Another example of this architecture is the combination of the fuzzy proportional derivative (FPD) and tilt integral derivative (TID) to present a hybrid fuzzy-based controller by combining the fuzzy proportional derivative (FPD) controller with tilt integrated derivative (TID) sequentially to make benefit of their advantages [28].

Thus, a well-designed control unit gained great importance for its integration into the power system. Moreover, system frequency and link line capacity should be kept within permissible limits and system balance should be restored as soon as possible. In this paper, a cascaded controller called (1+PD)-PID is presented as a reliable alternative approach to enhance the stability, sustainability and reliability of a hybrid power system containing conventional and RES power systems including PV and wind energy sources. The proposed cascaded controller parameters are optimized by AVOA as a novel metaheuristic optimization technique.

The contributions, inspiration, and significance of this study can be summarized as follows:

• Design and simulation of a robust cascaded controller called (1+PD)-PID in order to regulate the system response in terms of frequency and tie-line power deviations;
• Using a novel AVOA optimization algorithm to find the optimal controller parameters to ensure an optimal behavior of the controller;
• Testing the effectiveness and validation of the (1+PD)-PID controller by subjecting the microgrid to various types of fluctuations and uncertainties such as distinct step load disturbances, variable load variations, and RES fluctuations;
• Verifying the superiority of the (1+PD)-PID controller by comparing its performance against that of other controllers such as the conventional PID controller, FOPID controller and TID controller.

The rest of the paper is organized as follows: Section 2 describes the topology of the two-area system where each component is briefly explained. The structure of the cascaded controller (1+PD)-PID is presented in Section 3, as well as the formulation of the problem. Section 4 provides an overview of the optimization algorithm (AVOA). Section 5 is devoted to the detailed results of simulating the system in different scenarios, and the conclusion is presented in Section 6.

2. Structure of The Two-Area Hybrid Power System

To evaluate the performance and efficiency of the proposed (1+PD)-PID controller, a two-area power system including non-reheat thermal power plants was constructed and studied. This system was adopted because it presents a variety of challenges to the suggested controller, providing a complete evaluation of its effectiveness and contribution to the regulation of frequency and tie-line power deviations. The two-area interconnected system under consideration, according to Figure 1, consists of two regions connected by an AC connection line. A non-reheat thermal power plant, a photovoltaic power plant, and associated loads are all found in Area-1, while Area-2 includes a non-reheating thermal power plant, a wind power plant, and loads. Each component in the system is represented by a transfer function of the first order composed of gain and time constant. Both areas of the system include generators with a rated capacity of 2000 MW. The system components are represented by linear mathematical models given using Equations (1)–(4).

• Non-reheat Thermal System:

The model representing the non-reheat thermal power plant consists of a governor stage followed by a turbine stage. The overall transfer function can be expressed as follows:

$$G_{TS}\left(s\right)=\left[G_g\left(s\right)\right]\left[G_T\left(s\right)\right]=\left[\frac{K_g}{1+s{T}_g}\right]\left[\frac{K_T}{1+s{T}_{gT}}\right]$$

(1)

where $K_T$ and $K_g$ are the gains of the turbine and the governor, respectively, and $T_T$ and $T_g$ represent the time constants of the turbine and the governor, respectively.

• Power System:

The transfer function representing the power system can be written as follows:

$$G_{PS}\left(s\right)=\left[\frac{K_{PS}}{1+s{T}_{PS}}\right]$$

(2)

where $K_{PS}$ and $T_{PS}$ refer to the power system gain and time constant, respectively.

• PV System:

A first-order transfer function is used to represent the PV system in a linear model as shown in Equation (3). Hence, at constant air temperature, the output power of the PV unit changes with solar radiance linearly.

$$G_{PV}\left(s\right)=\left[\frac{K_{PV}}{1+s{T}_{PV}}\right]$$

(3)

where $K_{PV}$ and $T_{PV}$ are the PV power plant gain and time constant, respectively.

In this paper, the input of the PV model is real solar irradiance data obtained from a PV station located in Aswan, Egypt with a capacity of 1.5 GW. Figure 2 shows the input solar irradiance during the day with a maximum of 1000 W/${\mathrm{m}}^2$ and the output PV power with a maximum of 0.25 p.u.

• Wind Turbine Generator (WTG):

In this study, simplified wind power plant modeling is used to simulate the changing nature of the output power produced by the wind production system and it is expressed as follows:

$$G_{WT}\left(s\right)=\left[\frac{K_{WT}}{1+s{T}_{WT}}\right]$$

(4)

where $K_{WT}$ and $T_{WT}$ are the wind power plant gain and time constant, respectively. In order to represent the fluctuations in wind power, the real wind speed data obtained from the wind power plant located in Zafarana, Egypt, are used as input into the wind turbine generator model. Figure 3 shows the fluctuation in wind speed and the change in wind turbine output power.

Table 1. System parameters.

Parameter	Value	Parameter	Value
$K_T$	1	$T_T$	0.3 s
$K_g$	1	$T_g$	0.03 s
$K_{PS}$	120 Hz/pu.MW	$T_{PS}$	20 s
$K_{PV}$	1	$T_{PV}$	1.3 s
$K_{WT}$	1	$T_{WT}$	1.5 s
$B_1$, $B_2$	0.425 pu.MW/Hz	$R_1$, $R_2$	2.4 Hz/pu.MW
$a_{12}$	−1	$T_{12}$	0.545 pu.MW/Hz

shows the parameter values for the power system under investigation in this study, where $R_1$ and $R_2$ refer to the thermal power plant’s regulation parameters, $B_1$ and $B_2$ represent bias parameters of frequency, a $a_{12}$ is the rate of area capacity, and $T_{12}$ represent the coefficient of synchronization.

3. Structure of the (1+PD)-PID Cascaded Controller

The (1+PD)-PID controller, a cascaded set of (1+PD) controller and PID controller in sequence, was established and applied as a solution to the LFC issue in this study. In comparison to the complex control techniques found in the literature, the proposed controller employs the (1+PD) controller in sequence with the conventional PID controller. Hence, it has the features and benefits of the classic PID controller including strong performance, comprehensible architecture, and easy implementation in real time. Furthermore, since the two controllers are connected via serial communication, they have the propensity to reject disturbance causes immediately before they spread throughout the system. Figure 4 shows the proposed (1+PD)-PID controller structure and its parameters. Moreover, this controller, unlike other cascaded controller configurations, uses the deviation in frequency (ΔF) and the deviation in interconnected power between nearby areas (Δ$P_{tie}$) as additional signals in addition to the corresponding area control error (ACE) [14].

The controller creates the necessary load reference signal(Δ$P_{ref}$), which is applied as an input signal for the controlled system, obtained by operating on the ACE signals (ΔF and Δ$P_{tie}$). For a two-area interconnected power system, the ACE is written as in Equation (5).

$$\begin{array}{c}AC{E}_1=-B_1\mathsf{\Delta }{F}_1-\mathsf{\Delta }{P}_{tie}\\ AC{E}_2=-B_2\mathsf{\Delta }{F}_2-\mathsf{\Delta }{P}_{tie}\end{array}$$

(5)

where B₁ = B₂ = B is the frequency bias parameter, $\mathsf{\Delta }{F}_i$ is the deviation in frequency in the ith interconnected area and $\mathsf{\Delta }{P}_{tie}$ is the deviation in the tie-line power of nearby control areas. The transfer functions in the s-domain of the proposed controller can be expressed by:

$$G_{1+PD}\left(s\right)=1+K{p}_1+K_{d1}\frac{N_{1}s}{s+N_1}$$

(6)

$$G_{PID}\left(s\right)=K{p}_2+\frac{K_i}{s}+K_{d2}\frac{N_{2}s}{s+N_2}$$

(7)

where $K{p}_1$, $K_{d1}$, $K{p}_2$, $K_i$ and $K_{d2}$ refer to proportional, derivative, proportional, integral, and derivative parameters, respectively. While $N_1$ and $N_2$ are the derivative filter coefficients whose values were estimated using the trial-and-error method and found to be equal to 600 for the optimal performance of the controller. Thus, the proposed controller has five degrees of freedom in its design, i.e., there are five parameters to be optimally tuned. These five parameters must all be optimized at the same time to achieve the best performance out of this controller, and AVOA is applied in this research to accomplish this task. The input to the first controller(1+PD) is the ACE and its output with the subtraction of ΔF and $\mathsf{\Delta }{P}_{tie}$ are applied to the second controller (PID) to generate the controlling signal ($\mathsf{\Delta }{P}_{ref}$) to the system. $\mathsf{\Delta }{P}_{ref}$ can be written mathematically in terms of the controller transfer function and controller inputs as follows:

$$\mathsf{\Delta }{P}_{ref}=(\left\{ACE\times \left(1+K{p}_1+K_{d1}\frac{N_{1}s}{s+N_1}\right)\right\}-\mathsf{\Delta }f-\mathsf{\Delta }{P}_{tie})\times (K{p}_2+\frac{K_i}{s}+K_{d2}\frac{N_{2}s}{s+N_2})$$

(8)

In this work, the AVOA is applied to tune the (1+PD)-PID cascaded controller gains optimally. The integral absolute error (IAE) criterion, which is the integration of the deviations in areas frequencies ($\mathsf{\Delta }{F}_1$ and $\mathsf{\Delta }{F}_2$) and in tie-line power ($\mathsf{\Delta }{P}_{tie}$), was used as the objective function to be minimized. Consequently, the applied objective function can be written as:

$$J_{obj}={{\displaystyle \int }}_0^{t_{sim}}\left(\left|\mathsf{\Delta }{F}_1\right|+\left|\mathsf{\Delta }{F}_2\right|+\left|\mathsf{\Delta }{P}_{tie}\right|\right).dt$$

(9)

where $t_{sim}$ is the simulation time which was set to a value fair enough for the system to restore its balance, i.e., the response to settle. The fundamental interest of the current work is to optimally minimize the objective function ($J_{obj}$) using AVOA since doing so will result in fewer oscillations, reduced settling time, and no or little peak overshoot/ undershoot in the $\mathsf{\Delta }{f}_1$, $\mathsf{\Delta }{f}_2$, and $\mathsf{\Delta }{P}_{tie}$ responses to a given system fluctuation or disturbance.

In order to guarantee that the optimization equation converges to an optimum solution, the following restrictions must be considered:

$$\begin{array}{c}K{p}_1{}_{min}^{max}\le K{p}_1\le K{p}_1{}_{min}^{max}\\ K_{d1}{}_{min}^{max}\le K_{d1}\le K_{d1}{}_{min}^{max}\\ K{p}_2{}_{min}^{max}\le K{p}_2\le K{p}_2{}_{min}^{max}\\ K_i{}_{min}^{max}\le K_i\le K_i{}_{min}^{max}\\ K_{d2}{}_{min}^{max}\le K_{d2}\le K_{d2}{}_{min}^{max}\end{array}$$

(10)

where min is the minimum limit and max refers to the maximum limit of Kp₁, K_d1, Kp₂, K_i and K_d2, respectively. Finally, using AVOA’s effective search behavior, the optimum or nearly optimum combination of the presented controller parameters is obtained within the specified constraints for the minimal value of the objective function ($J_{obj})$.

4. African Vulture Optimization Algorithm (AVOA)

The authors of [29] have introduced the African Vultures Optimization Algorithm (AVOA), a metaheuristic algorithm depending on the feeding and navigational habits of African vultures. The AVOA is also more flexible and adaptable than other metaheuristic techniques and has a low computing complexity. Correspondingly, Figure 5 provides a summary of the exploration and exploitation stages of AVOA, which are briefly explained as follows:

4.1. Exploration Phase

The probability of picking the chosen vultures to guide the others to one of the optimal solutions in every group is calculated in this phase according to the following equation:

$$P\left(i+1\right)=\{\begin{array}{c}R\left(i\right)-\left|X\times R\left(i\right)-P\left(i\right)\right|\times F if P_1\ge ran{d}_{p_1}\\ R\left(i\right)-F+r_1\times \left(\left(U-L\right)\times r_2+L\right) if P_1<ran{d}_{p_1}\end{array}$$

(11)

where $P\left(i\right)$ is the vulture position in the current iteration and $P\left(i+1\right)$ is its position in the next iteration, respectively. Additionally, F stands for the satiation rate of the vulture, U stands for the upper bound of agents, and L is the lower bound of them, while $r_1$ and $r_2$ are random variables, and X stands for vectors that depict the vultures’ random movement. Moreover, $ran{d}_{p_1}$ stands for a random number with a value generated in the middle of 0 and 1 which is generated for picking the appropriate strategy in this phase, and $R\left(i\right)$ is provided by:

$$R\left(i\right)=\{\begin{array}{c}Best Vultur{e}_1 if P_i=L_1\\ Best Vultur{e}_2 if P_i=L_2\end{array}$$

(12)

where $Best Vultur{e}_1$ is the first group optimal solution and $Best Vultur{e}_2$ is the second group optimal one in the present iteration. Before the optimization search begins, $L_1$ and $L_2$ are initialized where their values lie between 1 and 0 and their sum equals 1.

4.2. Exploitation Phase

Based on the vulture’s satiation rate, two strategies are provided at this phase. When the satiation rate of a vulture (F) is greater than or equal to 0.5, the vultures battle for food in a circular movement, which may be estimated using Equation (13):

$$P\left(i+1\right)=\{\begin{array}{c}\left|X\times R\left(i\right)-P\left(i\right)\right|\times \left(F+r_3\right)-\left(R\left(i\right)- P\left(i\right)\right) if P_2\ge ran{d}_{p_2}\\ R\left(i\right)-\left(S_1+S_2\right) if P_2<ran{d}_{p_2}\end{array}$$

(13)

where $S_1$ and $S_2$ are the spiral flight motion which may be provided using Equations (14) and (15), respectively as follows:

$$S_1=R\left(i\right)\times \left(\frac{r_4\times P\left(i\right)}{2\pi }\right)\times cos\left(P\left(i\right)\right)$$

(14)

$$S_2=R\left(i\right)\times \left(\frac{r_5\times P\left(i\right)}{2\pi }\right)\times sin\left(P\left(i\right)\right)$$

(15)

where $R\left(i\right)$ is estimated in Equation (12) and $r_3,r_4,and r_5$ represent random variables with a value between 0 and 1. Moreover, $ran{d}_{p_2}$ and $ran{d}_{p_3}$ are random integers with a value between 0 and 1 that are generated to obtain the appropriate strategy during the exploitation phase. Moreover, other vultures become aggressive during foraging when $F<0.5$ and can be estimated using Equation (16):

$$P\left(i+1\right)=\{\begin{array}{c}\frac{A_1+A_2}{2} if P_3\ge ran{d}_{p_3}\\ R\left(i\right)-\left|R\left(i\right)-P\left(i\right)\right|\times F\times Levy\left(X\times R\left(i\right)\right) if P_3<ran{d}_{p_3}\end{array}$$

(16)

where $A_1$ and $A_2$ are the motion of vultures and can be represented by Equations (17) and (18), respectively:

$$A_1=Best Vultur{e}_1\left(i\right)-\frac{Best Vultur{e}_1\left(i\right)\times P\left(i\right)}{Best Vultur{e}_1\left(i\right)-P{\left(i\right)}^2}\times F$$

(17)

$$A_2=Best Vultur{e}_2\left(i\right)-\frac{Best Vultur{e}_2\left(i\right)\times P\left(i\right)}{Best Vultur{e}_2\left(i\right)-P{\left(i\right)}^2}\times F$$

(18)

Furthermore, the Levy motion may be employed to improve the AVOA algorithm’s effectiveness. The controlling parameters of the AVOA used in this paper were set to 0.6, 0.4, 0.6, 0.8, and 0.2 for P₁, P₂, P₃, L₁ and L₂, respectively. Finally, the AVOA algorithm has been shown to be efficient in solving various types of optimization problems [29]. The AVOA major steps can be outlined as follows:

• Determine the iterations maximum number and the size of the population;
• Compute the vulture fitness value;
• Select $R\left(i\right)$ using Equation (12) for all vultures;
• Use Equation (11) to compute the position of the best vulture;
• Depending on the vulture satiation rate, use Equation (13) or Equation (16) to update the position of the vulture;
• Save the position of the optimal vulture then compute the value of the fitness function as long as the iteration maximum number is not reached.

5. Results and Discussions

In this section, the validity, effectiveness, and robustness of the proposed controller and the chosen optimization technique were evaluated by referring to the simulation results obtained under various scenarios. In order to do this, a MATLAB/SIMULINK model was established to simulate the proposed (1+PD)-PID controller integrated into a two-area interconnected hybrid power system containing thermal units and RES such as solar (PV) and wind energy units. The simulated model was tested by applying different cases of uncertainties such as step load perturbation, variable load profile and the penetration of renewable energy resources. In each case, the proposed controller performance was presented in comparison to different types of controllers such as classic PID, TID and FOPID controllers with their parameters obtained by the same optimization algorithm (AVOA). The frequency deviation in each area and the deviation of tie-line power between nearby areas are the system parameters to be shown for each type of controller. The results were used to assess the reliability, robustness and superiority of the presented controller as will be shown later.

5.1. Scenario 1: System Performance under 10% SLP in Area-1

5.1.1. Comparison of AVOA Performance with Other Optimization Algorithms

In this part of the study, various optimization algorithms such as Ant Lion Optimizer (ALO), Genetic Algorithm (GA), Grasshopper Optimization Algorithm (GOA), Harris Hawks Optimization (HHO) and African Vultures Optimization Algorithm (AVOA) were applied to find the optimal parameters of the proposed (1+PD)-PID controller. To do this, 10% SLP was applied to the system in Area-1 without incorporating RES into the system and each optimization algorithm was applied to the studied system to obtain the presented controller parameters and the fitness function was calculated. Figure 6 shows the response of the system in terms of frequency deviation in the two areas (ΔF₁, ΔF₂) and the change in the tie-line power (Δ$P_{tie}$). Obviously, AVOA has the best performance among other optimization algorithms, and it has the lowest fitness function of 6.01 × 10⁻⁵ against 12.64 × 10⁻⁵, 26.28 × 10⁻⁵, 27.61 × 10⁻⁵, and 35.65 × 10⁻⁵ for GOA, HHO, ALO and GA, respectively.

5.1.2. Applying the Proposed AVOA Algorithm to Different Controllers

In this scenario, the performances of different types of controllers whose parameters were tuned by the optimization algorithm (AVOA) were examined. In order to achieve this, a 10% SLP in Area-1 was applied to the simulated system in the absence of RES generation. Figure 7 shows the frequency deviation in the two areas (ΔF₁, ΔF₂) and the change in the tie-line power (Δ$P_{tie}$). It can be obviously noted that the proposed controller has a better response than the other controllers.

The parameters of each controller obtained by 100 iterations of the optimization algorithm (AVOA) with 30 search agents are shown in

Table 2. Controller gains and fitness functions obtained by AVOA.

	(1+PD)PID		TID		FOPID		PID
Area 1	Kp11 Kd11 N1 Kp1 Ki1 Kd1 N2	127.0878 1.3142 600 85.536 149.906 0.88764 600	Kt1 N1 Ki1 Kd1 App1	375 7.805 375 53.75 9.5	Kp1 Ki1 λ1 Kd1 μ1	180 180 1 17 1.23	Kp1 Ki1 Kd1	148.58 150 25.67
Area 2	Kp22 Kd22 N3 Kp2 Ki2 Kd2 N4	5.2932 0.3625 600 1.9495 2.7302 0.1812 600	Kt2 N2 Ki2 Kd2 App2	282.177 17.6448 14.027 209.509 16.574	Kp2 Ki2 λ2 Kd2 μ2	60.37 44.26 1.25 16.15 1.22	Kp2 Ki2 Kd2	139.08 147.71 59.76
Fitness Function	6.01 × 10−5		8.14 × 10−4		15.72 × 10−4		19.42 × 10−4

. It can be noticed that the proposed (1+PD)-PID controller offers the least minimum fitness function of 6.01 × 10⁻⁵ against 8.14 × 10⁻⁴, 15.72 × 10⁻⁴, and 19.42 × 10⁻⁴ for the TID, FOPID and PID controllers, respectively.

Table 3. The system parameters for 10% load change.

Controller	ΔF1			ΔF2			ΔPtie
	MO	MU	TS	MO	MU	TS	MO	MU	TS
(1+PD)PID	0.00066	0.0026	0.1246	1.95 × 10−5	1.65 × 10−5	0.9827	1.16 × 10−7	1.2 × 10−5	1.08
TID	0.0019	0.0051	0.4792	0	2.36 × 10−4	4.92	0	9.97 × 10−5	4.93
FOPID	4.8 × 10−7	0.0059	0.5353	4.46 × 10−7	5.4 × 10−4	3.90	1.9 × 10−7	2.28 × 10−4	3.91
PID	0.0013	0.0076	0.8649	0	5.4 × 10−4	3.94	0	2.29 × 10−4	3.94

introduces the system performance in terms of maximum over-shoot (MO), maximum under-shoot (MU) and settling time (TS) values of the deviation of frequency in each area and tie-line power deviation for 10% SLP in area 1. The obtained simulation results clearly show that the (1+PD)-PID controller defeats the other competing controllers in the system performance specifications, especially the settling time. Settling time refers to how quickly the system can regain its equilibrium and the time it takes for the system to dampen the turbulence. For example, the settling time of the frequency deviation in Area-1 is 0.122246 S for the proposed controller versus 0.4792 S, 0.5353 S and 0.8649 S for TID, FOPID and PID controllers, respectively. Additionally, the maximum under-shoot of the frequency deviation in Area-1 is 0.0026 for (1+PD)-PID controller against 0.0051, 0.0059 and 0.0076 for TID, FOPID and PID controllers, respectively. Consequently, the proposed (1+PD)-PID controller proves its validity and efficacy in improving system stability and its superiority over the other competing controllers.

5.2. Scenario 2: The System Performance for Random Load Profile

In order to examine the robustness and effectiveness of the system, a successive random step load with different values is applied to the system. Figure 8 represents the applied random load profile. The positive value indicates an additional load to the grid, while the negative value shows an outgoing load from the grid. By adjusting the parameters of the controllers according to the values obtained in Table 1, the system responses for ΔF₁, ΔF₂ and Δ$P_{tie}$, due to the applied load profile, are shown in Figure 9.

5.3. Scenario 3: The Effect of Installing Solar PV Unit in Area-1

A solar PV unit was installed in the system in Area-1 in order to evaluate the system response under this type of power fluctuation due to the solar irradiance uncertainty. Figure 10 shows the system responses for ΔF₁, ΔF₂ and Δ$P_{tie}$ using AVOA tuned (1+PD)-PID, TID, FOPID and PID controllers under the penetration of PV solar power generation. It is evident that the proposed controller offers the best performance and the least fluctuations in frequency and tie-line power in this case compared to the other controllers with which it is being compared.

5.4. Scenario 4: The Effect of Installing a Wind Farm Unit in Area-2

In this scenario, a wind farm unit was installed in Area-2 to evaluate the system’s robustness under the influence of another type of power fluctuation (wind speed variation). Figure 11 shows the frequency deviation in area 1, frequency deviation in area 2 and tie-line power under the effect of wind power generation. It can be easily noticed the superiority of the (1+PD)-PID controller over the TID, FOPID and PID controllers under the effect of wind speed fluctuations.

5.5. Scenario 5: The Effect of Inserting RES into the System

In this scenario, the system and the controller under study were subjected to hybrid renewable energy sources where a PV unit was installed in Area-1 and a wind farm unit was integrated into Area-2 at the same time. The efficiency, reliability, and robustness of the presented controller were evaluated under the impact of solar irradiance uncertainties and wind speed fluctuations simultaneously. The system responses in terms of frequency deviation in Area-1 (ΔF₁) and in Area-2 (ΔF₂) and tie-line power (Δ$P_{tie}$) are presented in Figure 12 for different controllers under the circumstances of RES variation. It is clearly shown that the suggested controller has the best performance and the least variation in system frequency and interconnected tie-line power of nearby areas compared to other comparable controllers. Hence, the effectiveness, validity, reliability, and robustness of the proposed controller ((1+PD)-PID) tuned by the selected optimization algorithm (AVOA) were strictly proven in controlling hybrid power systems in terms of frequency and tie-line power in case of multi-area systems under various types of uncertainties.

6. Results and Discussions

In conclusion, this paper presented a robust cascaded (1+PD)-PID controller for providing a reliable, efficient, and simple solution to the LFC problem in power systems. The ACE was treated in the first controller (1+PD), then the output of it besides ΔF and Δ$P_{tie}$ were applied as inputs to the latter controller (PID). The controller parameters were optimally obtained using AVOA depending on the IAE criterion as the objective function to ensure the best performance of the proposed controller. The validity, effectiveness and robustness of the controller were tested by subjecting the studied system to several scenarios such as SLP, random load profile and RES penetration. The superiority of the proposed controller was evaluated by comparing its performance, in terms of settling time, maximum overshoot and under-shoot, to other controllers such as PID controller, FOPID controller and TID controller which were also optimized by AVOA. Finally, the proposed controller effectively damped the tie-line power fluctuation and the frequency deviation in both areas and outperformed the other comparable controllers in all scenarios performed.

In the future, the proposed controller can be applied to interconnected systems containing more than two areas with different types of power generation units and energy storage systems. Additionally, it can be applied to real modern networks and tested under severe disturbances.