A Robust Artificial Bee Colony-Based Load Frequency Control for Hydro-Thermal Interconnected Power System

Abstract

The presented work examines load frequency control (LFC) to develop the dynamic behavior of the power system under different load disturbances that have occurred in multi-interconnected power systems. An artificial bee colony (ABC) algorithm is proposed to design an optimal proportional integral derivative (PID) controller simulating the LFC installed in a hybrid hydro-thermal interconnected power system. The proposed approach incorporating ABC is employed to determine the optimal parameters of the controller during load disturbance applied on one area. The integral time absolute error (ITAE) of the frequency and exchange power violations is considered as the target to be minimized. Moreover, integral absolute error (IAE) and sum squared error (SSE) are calculated. To prove how the proposed model controller is effective, two-interconnected power systems are presented during a wide range of operating cases, and then the behavior of the proposed controller is compared to that of the designed via a chef-based optimization algorithm (CBOA), seagull optimization approach (SOA), and sine cosine approach. Regarding the 5% disturbance on the thermal plant, the ABC outperformed the other approaches hence achieving the best fitness value of 1.80936, IAE of 3.147938, and SSE of 0.1787486. On the other hand, during a 5% disturbance on the hydro plant, the ABC succeeded in getting ITAE, IAE, and SSE with values of 3.43291, 3.630509, and 0.5233815, respectively. The efficiency and prevalence of the proposed LFC-PID is confirmed by the achieved results.

1. Introduction

The power system utility is used to keep the continuity of the electric power to the customers with acceptable quality. It is essential to keep balance operation of the power system under load disturbance situations via achieving balance between the generated and demand powers. The behavior of the electric power system is heavily affected by power disruption issues. In order to achieve the balance operation of the system, it is important to use load frequency control (LFC). The main targets of the LFC are achieving frequency perversion without errors, minimizing the unscheduled tie-line power, maintaining acceptable time performance of the frequency, and minimizing the power deviations in a multi-interconnected power system [1]. Many previous studies have focused on the performance of the LFC and its optimal performance based on metaheuristic optimization algorithms, some of these studies can be summarized as follows: a modified harmony search algorithm was presented in [2] to optimize the load frequency control of non-linear power systems using the cost function depending on the integration of absolute error multiplied by time. An optimization process for identifying the parameters of the controller used in an automatic generation control (AGC) for interconnected power systems using a gravitational search algorithm was presented in [3]. An approach based on adaptive-neuro Fuzzy inference system (ANFIS) was applied to simulate the AGC incorporated in three different interconnected hydrothermal systems in [4]. A hybrid metaheuristic optimization algorithm combined the bacteria foraging optimization algorithm and particle swarm optimization (HBFOA-PSO) to simulate the AGC for interconnected power systems in [5]. A controlling method of hybrid neuro-Fuzzy (HNF) for load frequency control (LFC) for an interconnected power system was introduced in [6]. The parameters of a Fuzzy proportional integral derivative (PID) controller for AGC incorporated in two interconnected power systems was adjusted by teaching a learning-based optimization (TLBO) algorithm in [7]; whereas in [8], the controller parameters were tuned optimally by employing the hybrid differential evolution particle swarm optimization technique using the integration of time multiplied by the absolute error. A method based on the peak resonance specification was given in [9] to present the PID controller for AGC. A robust controller of the heat pump and plug-in hybrid electric vehicle (EV) for controlling the frequency of an isolated system based on a coefficient diagram method was introduced in [10]. A hybrid firefly algorithm and pattern search-based approach has been introduced in [11], which is used for simulating the AGC for multi-area systems. The firefly (FFA) is employed in optimizing the PID controller parameters, and then the pattern search algorithm is used for adjusting the best solution obtained from FFA. A new area connected by errors based on tie-line power deviation, frequency perversion, time error, and unscheduled energy transfer was introduced in [12] to control the battery stored energy used with LFC in an interconnected hydro-thermal system. A fractional order PID controller for AGC of the two connected thermal power systems has been presented in [13]. A modified LFC considered the effect of bilateral contracts achieved by the principle of Disco participation matrix for two-interconnected systems was presented in [14]. In [15,16,17], a hybrid particle swarm optimizer with a pattern search algorithm has been utilized in optimizing the Fuzzy-PI controller for AGC of multi-area power systems. Moreover, a hybrid PSO-GA-FA technique has been applied to design a Fuzzy-PI-LFC installed with an interconnected power system [18,19,20,21].

The previous approaches used many artificial intelligence-based techniques such as Fuzzy logic and ANFIS in designing the LFC while the others employed metaheuristic optimizers. Most of the employed optimizers suffer from some defects such as getting the local optima which is defined as the solution within a neighboring set of candidates, complexity in construction, requirement of many controlling parameters, and consumption of large time. Additionally, the Fuzzy, ANFIS, and ANN-based approaches require excessive input data for training and testing purposes. All these shortages are considered via introducing an artificial bee colony (ABC)-based approach to design the LFC-PID, installed with two areas comprising hydro and thermal plants.

The ABC approach is characterized by flexibility, robustness, simplicity, ease of implementation, ability to solve complex problems, and ability to discover the local optima. On the other hand, the ABC suffers from a requirement of a higher number of objective function evaluation, which makes the algorithm slow, especially in complex problems.

The main goal of this work is to design the LFC-PID installed in a hybrid hydro-thermal interconnected system via an artificial bee colony (ABC) approach such that the ITAE of frequency and exchange power deviations are minimized.

The proposed LFC-PID designed via the proposed methodology incorporated with ABC has many merits such as simplicity of constructions and a guarantee of getting zero deviations in frequency and power exchange for a multi-interconnected system with load disturbance. The system is simulated by the Matlab/Simulink library and the proposed ABC is applied to find the optimal gains of the LFC-PID controller. The suggested optimized PID controller and conventional PID controller are compared at different conditions. The reliability and competence of the proposed LFC-PID controller is confirmed as it achieves good performance and gives promising results compared to those obtained via the conventional controller.

2. Model of Interconnected Hydro-Thermal System

The studied system consists of two connected power systems: the first one is a single reheat-type thermal system and the second plant has a hydro-generation system. Figure 1 shows the proposed system block diagram. The turbine, the governor, and the generating unit of each area is introduced by a first order transfer function under the effect of small disturbance. The main goal of the PID load frequency controller is to achieve zero frequency deviation in each area (ΔF1 = ΔF2 = 0) and zero tie-line power (ΔPtie (1–2) =0). The input of each PID controller is the area control error (ACE) of each area which can be defined for area no.1 and area no. 2 as follows [22,23,24,25,26]:

$${\mathrm{ACE}}_1={\mathrm{B}}_1\Delta {\mathrm{F}}_1+\Delta {\mathrm{P}}_{\mathrm{tie}\left(1–2\right)}$$

(1)

$${\mathrm{ACE}}_2={\mathrm{B}}_2\Delta {\mathrm{F}}_2+\Delta {\mathrm{P}}_{\mathrm{tie}\left(1–2\right)}$$

(2)

where B₁ and B₂ are the frequency bias characteristics of area 1 and area 2, respectively. Referring to the system illustrated in Figure 1, the state space representation of the system during load disturbances applied to both areas (ΔP_D1 and ΔP_D2) is derived in the following equations:

$$\stackrel{.}{\mathrm{x}}\left(\mathrm{t}\right)=\mathrm{Ax}\left(\mathrm{t}\right)+\mathrm{Bu}\left(\mathrm{t}\right)+\mathsf{\xi }\Delta {\mathrm{P}}_{\mathrm{D}}\left(\mathrm{t}\right)$$

(3)

$$\mathrm{y}\left(\mathrm{t}\right)=\mathrm{Cx}\left(\mathrm{t}\right)+\mathrm{Du}\left(\mathrm{t}\right)+\mathsf{\mu }\Delta {\mathrm{P}}_{\mathrm{D}}\left(\mathrm{t}\right)$$

(4)

where x(t) is the state (independent) variables, u(t) is the control (dependent) variables, ΔP_D is the vector of disturbance, and A, B, C, D, ξ, µ are matrices. The state variable, control variable, and disturbance variable are defined as follows:

$$x\left(\mathrm{t}\right)=\left[\begin{array}{c}\Delta {\mathrm{F}}_1\left(\mathrm{t}\right),\Delta {\mathrm{F}}_2\left(\mathrm{t}\right), \Delta {\mathrm{P}}_{\mathrm{tie}\left(1–2\right)}\left(\mathrm{t}\right),\Delta {\mathrm{P}}_{\mathrm{G}1}\left(\mathrm{t}\right),\Delta {\mathrm{P}}_{\mathrm{R}1}\left(\mathrm{t}\right),\Delta {\mathrm{x}}_{\mathrm{e}1}\left(\mathrm{t}\right),\Delta {\mathrm{P}}_{\mathrm{G}2}\left(\mathrm{t}\right),\\ \Delta {\mathrm{x}}_{\mathrm{e}2}\left(\mathrm{t}\right),\Delta {\mathrm{P}}_{\mathrm{R}2}\left(\mathrm{t}\right)\end{array}\right]$$

(5a)

$$\mathrm{u}\left(\mathrm{t}\right)=\left[\Delta {\mathrm{P}}_{\mathrm{c}1}\left(\mathrm{t}\right),\Delta {\mathrm{P}}_{\mathrm{c}1}\left(\mathrm{t}\right)\right]$$

(5b)

$$\Delta {\mathrm{P}}_{\mathrm{D}}\left(\mathrm{t}\right)=\left[\Delta {\mathrm{P}}_{\mathrm{D}1}\left(\mathrm{t}\right),\Delta {\mathrm{P}}_{\mathrm{D}2}\left(\mathrm{t}\right)\right]$$

(5c)

The derived elements of the matrices A, B, C, D, ξ, and µ are given in the following Equations:

$$A=\left[\begin{array}{ccccccccc}-\frac{1}{T_{p1}}& 0& \frac{K_{p1}}{T_{p1}}& \frac{K_{p1}}{T_{p1}}& 0& 0& 0& 0& 0\\ 0& -\frac{1}{T_{p2}}& -\frac{K_{p2}}{T_{p2}}& 0& 0& 0& \frac{K_{p2}}{T_{p2}}& 0& 0\\ -2T_{12}& 2T_{12}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -\frac{1}{T_r}& \frac{1}{T_r}\left(1-\frac{K_r{T}_r}{T_t}\right)& \frac{K_r}{T_t}& 0& 0& 0\\ 0& 0& 0& 0& -\frac{1}{T_t}& \frac{1}{T_t}& 0& 0& 0\\ -\frac{1}{T_g{R}_1}& 0& 0& 0& 0& -\frac{1}{T_g}& 0& 0& 0\\ 0& -\frac{2T_r}{T_1{T}_2{R}_2}& 0& 0& 0& 0& -\frac{2}{T_{\omega }}& \frac{2}{T_{\omega }}\left(1+\frac{T_{\omega }}{T_2}\right)& -\frac{2}{T_2}\left(1-\frac{T_r}{T_1}\right)\\ 0& -\frac{T_r}{T_1{T}_2{R}_2}& 0& 0& 0& 0& 0& -\frac{1}{T_2}& \frac{1}{T_2}\left(1-\frac{T_r}{T_1}\right)\\ 0& -\frac{1}{T_1{R}_2}& 0& 0& 0& 0& 0& 0& -\frac{1}{T_1}\end{array}\right]$$

(6a)

$$B=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ \frac{1}{T_g}& 0\\ 0& \frac{2T_r}{T_{\omega }{T}_1}\\ 0& \frac{T_r}{T_1{T}_2}\\ 0& \frac{1}{T_1}\end{array}\right]\begin{array}{cc},& C=\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\end{array}\right]\end{array}\begin{array}{cc},& \xi =\left[\begin{array}{cc}\frac{K_{p1}}{T_{p1}}& 0\\ 0& -\frac{K_{p2}}{T_{p2}}\\ 0& 0\\ 0& 0\\ 0& \\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]\end{array}$$

(6b)

$$D=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]\begin{array}{cc},& \mu =\end{array}\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$$

(6c)

The designed LFC-PID controller is installed in the two interconnected areas under study to reach zero frequency deviation and zero tie-line power deviation of the constructed system. One of the most important metaheuristic optimization algorithms, artificial bee colony (ABC), is used to determine the optimal parameters of PID controller representing LFC; the base reason for selecting the ABC algorithm in this analysis is its simplicity and requirement for less controlling parameters. In order to verify the validity of the ABC algorithm in designing the LFC in interconnected networks, the results obtained via the designed controller and the conventional are compared.

3. Artificial Bee Colony Approach Overview

An artificial bee colony (ABC) represents one of the most significant metaheuristic optimization algorithms as it is simple and requires fewer controlling parameters. The ABC algorithm is explored from the analogy of honeybees’ intelligent behavior. ABC, like any optimization algorithm, extends a population-based search procedure [27,28]. The food positions are the individuals, and the bees attempt to explore the places of food sources that have a high amount of nectar. The highest nectar food source is the optimal solution. In the ABC algorithm, the bees are assorted into three types: scout, onlookers, and experienced bees. Scout bees search on the food source without any experience about its location. Onlookers search on the food source with knowledge gained from a waggle dance done by some other bees. The experienced bees are responsible for evaluating the quality food sources and the positions of their historical memories. The steps followed in ABC algorithm can be summarized as follows:

• Initialize the food sources for all bees;
• The following steps are repeated:
  • Each onlooker bee goes to a food source and saves it in her memory and evaluates a neighbor source;
  • The other onlookers watch the waggle dance and go to that source and choose a neighbor around it;
  • The disused food sources are evaluated and replaced by the new food sources discovered by scouts;
  • The best food source found so far is registered.
• Until the best food source is obtained,
• End.

The process of selecting the food source by the onlookers is based on the food source probability value which is described as follows [19]:

$$P_i=\frac{F_i}{{\displaystyle \sum _{n=1}^{NF}{F}_n}}$$

(7)

where F_i is the value of objective function of solution no. i, NF is the number of food sources, and F_n is the fitness values of solutions which is calculated as follows:

$$F_n=\left\{\begin{array}{cc}\frac{1}{1+F_i}& if\begin{array}{c}{F}_i\ge 0\end{array}\\ 1+abs\left(F_i\right)& if\begin{array}{c}{F}_i\prec 0\end{array}\end{array}\right\}$$

(8)

The food position is updated based on the previous one by using the following Equation:

$$v_{ij}=x_{ij}+{\phi }_{ij}\left(x_{ij}-x_{kj}\right),\begin{array}{cc}& j=1,2,\dots ,\mathrm{D}\begin{array}{cc}& and\begin{array}{cc}& i,k=[1,2,\dots ,NF]\end{array}\end{array}\end{array}$$

(9)

where D is the optimization problem dimension, ij is a number chosen randomly in the range of [−1, 1]; the value of this random variable controls the production of the neighboring food positions.

4. The Proposed Problem Formulation

The designing process of PID-LFC is presented as an optimization problem which includes the objective function to be mitigated for the integral time absolute error (ITAE) of the frequencies and exchange powers violations. The variables to be designed are the PID parameters of K_P, K_I, and K_d. The formula of fitness function can be written as follows [29]:

$$Minimize ITAE={{\displaystyle \int }}_0^t\left(\left({{\displaystyle \sum }}_{i=1}^{n_a}\left|\Delta F_i+\Delta P_{tie,i}\right|\right).t\right)dt$$

(10)

where $\Delta F_i$ denotes the violation of ith area frequency while $\Delta P_{tie,i}$ represents the deviation of in ith plant exchange power, $n_a$ represents the number of interconnected areas, and $t$ is the simulation time. The following expression represents the constraints:

$$\begin{array}{c}{K}_P{}^{min}\le K_P<K_P{}^{max}\\ K_I{}^{min}\le K_I<K_I{}^{max}\\ \begin{array}{c}{K}_d{}^{min}\le K_d<K_d{}^{max}\end{array}\end{array}$$

(11)

where $max$ and $min$ are the maximum and minimum bounds of the scaling variable; they are set in the range of [0.1–2] [11].

The proposed optimization process is illustrated in Figure 2, and the output of the plant is employed to calculate the presented fitness function. Then, two inputs are fed to the ABC approach, which are the approach controlling parameters and the value of fitness function. The ABC is responsible for assigning the PID parameters which are K_P, K_I, and K_d, and the output of the designed controller is fed to the plant. The process is continued until the deviations in frequencies and tie-line powers are minimized. The Pseudo code of the proposed methodology incorporated ABC is given in Algorithm 1.

Algorithm 1 Pseudo code of ABC-based methodology

1: Input the ABC parameters (NF, $D$, tmax, and nrun), where tmax and nrun are the maximum iteration and number of runs.

2: Define the parameters of the hydro-thermal power system.

3: Define the lower and upper bounds of PID controller (Lb and Ub).

4: Formulate an initial population using Lb and Ub.

5: Evaluate the initial fitness function Fi(xi0) via Equation (10).

6: Assign run = 1.

7: Assign t = 1.

8: while run > nrun do

9:     while t > tmax do

10:        for j = 1:NF

11:          Compute the fitness value (Fn) using Equation (8).

12:          Calculate the food source probability using Equation (7).

13:          Select the food source by the onlookers.

14:          Calculate the food position using Equation (9).

15:          Update the food positions using (xi,newt = xit).

16:          Calculate the new fitness value (Fi,new(xi,newt)) using Equation (10).

17:             if Fi,new(xi,newt) > Fi(xit)

18:                Update the food position and the fitness value

19:            end if

20:       end for

21:           Save the best minimum value of fitness function as the best.

22:        t = t + 1

23:    end while

24:   run = run + 1

25: end while

26: Print the optimal parameters of LFC.

5. Digital Simulation Results

The optimal LFC-PID controller is designed via identifying its parameters using the artificial bee colony (ABC) algorithm.

Table 1. Controlling parameters of ABC algorithm.

Item	Proposed Setting
Number of colony size	50
Number of Food	20
limit	100
Maximum Cycle	3000

illustrates the ABC controlling parameters employed in the proposed approach to identify the LFC-PID optimal parameters. A comparison between the power system performance with installing the proposed optimized PID controller and the other conventional one is implemented in this work to confirm the validity of the proposed LFC-PID controller. The system parameters are assigned as follows [14]:

T_t = 0.3 s, T_p1 = 20 s, k_p1 = 120 Hz/p.u. MW, p_r1 = 120 MW, T₁ = 41.6 s, T_p2 = 20 s, k_p2 = 120 Hz/p.u. MW, p_r2 = 120 MW, T₂ = 0.513 s, T_g = 0.08 s, R₁ = 2.4 HZ/p.u. MW, R₂ = 2.4 HZ/p.u. MW, T_w = 1 s, T₁₂ = 0.0866 s, T_r = 5 s, T₃ = 0.411 s, T₄ = 0.1 s, and k_r = 2.

The proposed LFC-PID designed via ABC is compared to other metaheuristic approaches such as the chef-based optimization algorithm (CBOA) [30], seagull optimization approach (SOA) [31], and sine cosine algorithm (SCA) [32]. The first considered disturbance is 5% on the thermal plant (Area no. 1). The optimal gains of the PIDs obtained via the proposed ABC and the others are tabulated in

Table 2. The optimal gains of LFC-PIDs for 5% load disturbance on thermal plant.

	CBOA	SOA	SCA [32]	ABC
KP1	2.000	2.000	2.000	2.000
KI1	2.000	2.000	0.335	0.531
Kd1	0.305	0.440	1.889	0.832
KP1	0.211	0.040	1.741	0.287
KI1	0.104	0.010	0.016	0.172
Kd1	1.666	2.000	0.088	0.482
Fitness value	2.137	1.897	2.296	1.809
IAE	5.276	7.554	3.958	3.148
SSE	0.303	0.449	0.309	0.179

, in addition to the fitness value (ITAE), integral absolute error (IAE), and sum squared error (SSE). The results revealed that the proposed ABC outperformed the others, achieving the minimum fitness value of 1.80936. The SOA comes in the next rank with ITAE of 1.89714 while the worst approach is SCA with a fitness value of 2.29568. Moreover, the proposed ABC is the best in terms of IAE and SSE.

The SCA has been used to design integral-based LFC installed with a thermal/thermal connected system [32]. The authors applied this approach on the considered hydro-thermal system with a doubly fed induction generator. However, it is different from this work as ISE was used as the target while the designed controller is the integral one; moreover, the connected power system is different from that considered in this work. However, the analysis of many load disturbances is the only need that is similar to our analysis.

Figure 3 describes the frequency deviation time response of area no.1 subjected to a load disturbance of 5% with the presented optimized PID via the proposed approach and the others. Moreover, the time response of frequency deviation of area no. 2 in that case is shown in Figure 4. In addition, Figure 5 displays the tie-line power deviation time response for a load disturbance applied on the first area. It is clear that the LFC-PID designed via the proposed ABC succeeded in vanishing the oscillation of frequencies and exchange power deviations to zero, whereas the other approaches failed to achieve the required target. It also can be concluded that CBOA, SOA, and SCA are stuck in local optima as they get the deviation out of zero. The performance specifications including rise time (Tr), settling time (Ts), minimum selling time (T_s,min), maximum settling time (T_s,max), maximum over-shoot (Mos), under-shoot (Us), and peak time (Tp) are calculated and tabulated in

Table 3. Calculation of rise time, settling time, and over-shoot, and peak time with optimized LFC-PID via all considered algorithms for ∆P_L1 = 5%.

ΔF1		Tr (s)	Ts (s)	Ts,min (s)	Ts,max (s)	MOs (pu)	Us (pu)	Tp (s)
	CBOA	28.188	45.992	0.032	0.036	0.000	256.588	0.508
	SOA	0.027	44.563	−0.087	−0.009	768.777	0.000	0.501
	SCA	23.869	44.453	0.015	0.016	0.000	479.681	2.148
	ABC	0.002	37.042	−0.101	0.051	13,692.480	6968.317	2.249
ΔF2		Tr (s)	Ts (s)	Ts,min (s)	Ts,max (s)	MOs (pu)	Us (pu)	Tp (s)
	CBOA	28.791	45.184	0.033	0.036	0.000	324.430	1.206
	SOA	0.173	42.859	−0.112	−0.009	1024.841	0.000	1.166
	SCA	24.358	43.676	0.015	0.016	0.000	548.549	1.624
	ABC	0.100	36.486	−0.112	0.054	11,221.720	5452.029	1.645
ΔPtie		Tr (s)	Ts (s)	Ts,min (s)	Ts,max (s)	MOs (pu)	Us (pu)	Tp (s)
	CBOA	0.413	47.405	−0.019	0.005	9.663	28.237	0.784
	SOA	0.257	46.787	−0.018	0.002	85.673	23.283	0.7454
	SCA	32.446	44.523	−0.019	−0.018	0.000	13.884	50.000
	ABC	0.043	41.034	−0.023	0.009	6144.497	2577.785	5.809

. The fetched results confirmed that the proposed ABC achieved the best rise time and settling time for ∆F₁, ∆F₂, and ∆P_tie. This confirms the superiority of the LFC-PID designed via the ABC.

In order to confirm the validity of the proposed ABC-based methodology, a second load disturbance of 5% is assumed to have occurred in the hydro plant (2nd area). The optimal gains of LFC-PID controllers and the fitness values obtained via the considered optimizers are given in

Table 4. The optimal gains of LFC-PIDs for 5% load disturbance on hydro plant.

	CBOA	SOA	SCA [32]	ABC
KP1	1.325	0.258	0.108	1.888
KI1	0.590	0.053	0.133	0.954
Kd1	1.126	1.809	1.895	1.059
KP1	1.245	2.000	2.000	1.004
KI1	0.454	2.000	1.975	0.549
Kd1	0.564	0.044	1.256	0.745
Fitness value	3.445	7.213	7.636	3.433
IAE	3.714	21.577	15.045	3.631
ISE	0.539	4.209	1.831	0.524

. The proposed ABC achieved the best fitness value, IAE, and SSE of 3.43291, 3.630509, and 0.5233815, respectively. The next best values are those of CBOA while the worst results are obtained via SOA. In such cases, the frequency deviation time responses of the first and second areas with optimized controllers obtained via the considered algorithms are shown in Figure 6 and Figure 7, respectively. Furthermore, the tie-line power deviation time response for the two-interconnected system with optimized controllers are shown in Figure 8. From the curves one can understand that both SOA and SCA failed in damping the oscillations of frequencies and exchange power violations as they stuck in local optima while the curves obtained via CBOA converged to the those of ABC. Finally, the time response specifications of the frequency and exchange power deviations are calculated and tabulated in

Table 5. Calculation of rise time, settling time, and over-shoot, and peak time with optimized LFC-PID via all considered algorithms for ∆P_L2 = 5%.

ΔF1		Tr (s)	Ts (s)	Ts,min (s)	Ts,max (s)	MOs (pu)	Us (pu)	Tp (s)
	CBOA	0.107	37.353	−0.228	0.018	10,152.84	794.062	1.642788
	SOA	0.454	49.964	−0.211	0.073	143.3863	84.50683	1.743954
	SCA	0.312	38.127	−0.194	−0.010	299.3158	0	1.537866
	ABC	0.092	30.634	−0.224	0.021	14912.33	1415.597	1.59846
ΔF2		Tr (s)	Ts (s)	Ts,min (s)	Ts,max (s)	MOs (pu)	Us (pu)	Tp (s)
	CBOA	0.006	36.177	−0.236	0.018	10,604.770	808.635	1.080
	SOA	0.182	49.969	−0.221	0.077	225.189	113.105	1.147
	SCA	0.120	37.915	−0.214	−0.009	342.317	0.000	0.922
	ABC	0.003	29.267	−0.237	0.022	18,603.110	1715.785	1.036
ΔPtie		Tr (s)	Ts (s)	Ts,min (s)	Ts,max (s)	MOs (pu)	Us (pu)	Tp (s)
	CBOA	0.074	39.874	−0.009	0.065	5479.407	758.447	1.404
	SOA	0.362	49.9197	0.0026	0.062	186.192	0.000	1.484
	SCA	0.321	43.054	0.019	0.058	187.901	0.000	1.281
	ABC	0.061	32.409	−0.010	0.064	8156.232	1322.503	1.361

, and the fetched result proved the preference of the proposed LFC-PID designed via ABC in terms of rise time and settling time for ∆F₁, ∆F₂, and ∆P_tie.

Finally, the LFC-PID controller designed by the proposed methodology incorporated with ABC is recommended as an efficient device to achieve zero oscillation in both frequency and exchange power deviations for a hydro-thermal connected power system.

6. Conclusions

The aim of this study is in accordance with exploring how to achieve a strong improvement of the two interconnected power system performances via installing an optimal LFC-PID optimized via artificial bee colony (ABC). The considered fitness function is minimizing the integral time absolute error (ITAE) of the frequency and exchange power deviations. The constructed system comprises of a hydro-thermal hybrid system that operates under different load disturbances. To validate the competence of the proposed controller, different load disturbances on both plants are investigated; the obtained results via the proposed PID-LFC are compared to those obtained via the chef-based optimization algorithm (CBOA), seagull optimization approach (SOA), and sine cosine algorithm. Moreover, a comparative analysis in terms of rise time, settling time, maximum over-shoot, under-shoot, and peak time of frequency deviation and tie-line power time responses is conducted. From observing the simulation results, it is clear that the proposed LFC-PID optimized via ABC is superior to the other approaches, achieving the best ITAE values of 1.80936 and 3.43291 during a 5% load disturbance on thermal and hydro plants, respectively. Moreover, it achieved a good transient and stable condition. The proposed approach that incorporated ABC for optimizing PID-based LFC installed in two interconnected systems is efficient and reliable in achieving the steady-state target of the frequency deviations and tie-line powers of the interconnected power system. The challenge facing the ABC-based methodology is the slowness with complex processes; therefore, enhancing the performance of the approach via an opposition-based learning technique is recommended in future works. Moreover, multi-interconnected system with multi-renewable energy sources will be considered in the next work.