For multi-area power systems, the LFC system structure of the i -th area can be shown in Figure 1. The parameters of i -th control area are listed in Table 1.

The multi-area power system dynamics LFC model studied in this paper can be described as: x ˙ t = A x t + B u t + F ω t y t = C x t (1) where: x i t = Δ f i Δ P m i Δ P v i Δ P w i n d i Δ P B i ∫ A C E i Δ P t i e − i T x t = x 1 T t x 2 T t x 3 T t … x n T t T ω i t = Δ P d i Δ Φ w i n d i T , A i j = 7,1 = − 2 π T i j u t = u 1 T t u 2 T t u 3 T t … u n T t T y i t = A C E i ∫ A C E i T , B = d i a g B 1 , . . , B n ω t = ω 1 T t ω 2 T t ω 3 T t … ω n T t T y t = y 1 T t y 2 T t y 3 T t … y n T t T A i i = 1,1 = − D M i , 1,2 = 1 M i , 1,4 = 1 M i , 1,5 = 1 M i , 1,7 = − 1 M i , 2,2 = − 1 T c h i , 2,3 = 1 T c h i , 3,1 = − 1 R T g i , 3,3 = − 1 T g i , 4,4 = − 1 T w i , 5,1 = 1 T E S i , 5,5 = − 1 T E S i , 6,1 = β i , 6,6 = 1 , 7,1 = 2 π ∑ j = 1 , j ≠ i n T i j A = A 11 … A 1 n ⋮ ⋱ ⋮ A n 1 … A n n B i = 0 0 1 T g i T 0 0 0 0 T C i = β i 0 0 0 0 0 1 0 0 0 0 0 1 0 F i = − 1 M i T 0 0 0 0 0 0 0 0 0 1 T w i 0 0 0 T C = d i a g C 1 , . . , C n , F = d i a g F 1 , . . , F n

The Area Control Error (ACE) for each control area can be expressed by: A C E i = β i Δ f i + Δ P t i e − i (2)

To save network bandwidth resources, and improve information transmission efficiency, an event-triggered scheme based on the improved AFSA-PSO algorithm is proposed to increase the utilization of network resources in this study. When the transmitted signal meets the trigger scheme, it may be transmitted; otherwise, the sampling signal will not be transmitted, and the event-triggered conditions are designed as follows (Yue et al., 2013): x t l h + j h − x t l h T Φ x t l h + j h − x t l h > β x t l h T Φ x t l h (3) where, Φ is the positive definite weighting matrix; β is the event-triggered threshold in the range of 0,1 ) ; t l h denotes the most recent data transmission moment and t l h + j h is the current sampling moment.

The event trigger determines the transmission of the system state. Assuming that the data has been successfully transferred at time t l h , the next trigger moment t l + 1 h is: t l + 1 h = t l h + min j h | x t l h + j h − x t l h T Φ x t l h + j h − x t l h > β x t l h T Φ x t l h (4)

Transmission delays are unavoidable due to the presence of communication networks. It is reflected in the trigger scheme as whether x t l h or x t l + 1 h is received first. Define τ t as the transmission delay between the event trigger and the zero-order device at moment t l h , and the maximum value of τ t is τ max . Define the error between the most recently transmitted data and the currently sampled data as e l t . The event-trigger scheme is treated in two cases as follows:

(1) When t l h + h + τ max ≥ t l + 1 h + τ l + 1 , define τ t = t − t l h , τ t ∈ τ l , t l + 1 − t l h + τ l + 1 , and e l t =0。

Thus, e l t can be written as: e l t = 0 , k ∈ Ω 0 x t l h − x t l h + n h , k ∈ Ω n x t l h − x t l h + j h , k ∈ Ω j (6) where, x t l h is the most recent moment transmission data; x t l h + n h and x t l h + j h are the current moment sampling data, respectively.

Therefore, y t can be rewritten as follows y t l h = C e l t + C x t − τ l t (7)

Remark 1

In this paper, a PSO with adaptive inertia weights (Li et al., 2019b) and compression factors (Zhou et al., 2019) are designed. This improved algorithm balances the local search and global search capability of PSO. Meanwhile, the boundary restriction on the conventional particle velocity is removed to ensure the convergence and accuracy of PSO. PSO can utilize the mechanism of sharing information among individuals in a group to make the whole solution process orderly. The core formula of the algorithm is given below: v i d = ω v i d − 1 + c 1 r 1 p b e s t i d − x i d + c 2 r 2 g b e s t d − x i d (8) x i d = x i d − 1 + v i d − 1 t (9)

Eq. 8 is the speed of the individual’s step d , which consists of the speed inertia of the previous step itself ω v i d − 1 , the self-cognition part c 1 r 1 p b e s t i d − x i d , and the social cognition part c 2 r 2 g b e s t d − x i d . Eq. 9 is the position of the individual at step d , which is determined by the position x i d − 1 at step d − 1 and the velocity of step d − 1 × time of motion, with the time of each step generally taken as 1. ω denotes the velocity inertia weight, c 1 denotes the individual acceleration factor, c 2 denotes the social acceleration factor, and r 1 , r 2 are the random numbers on 0,1 .

Where, p b e s t i d represents the best position that the particle i has passed through by the end of the d iteration, and g b e s t d represents the best position that all particles have passed through. ω , c 1 and c 2 are important factors to be considered in optimizing the PSO. However, the traditional PSO usually sets these parameters to fixed values, making the algorithm unable to be dynamically adjusted according to the operation results. On the other hand, due to the fast information exchange between particles, the particles all move in the optimal direction and tend to be homogeneous, which tends to cause local convergence. This work proposes an improved PSO with the following formulations to address these defects. v i d = α ω i d v i d − 1 + c 1 r 1 p b e s t i d − x i d + c 2 r 2 g b e s t d − x i d (10)

The algorithm is designed with adaptive inertia weights ω i d , whose value can change with the change of the fitness value. Take the problem of minimizing the degree of adaptation as an example: the smaller the current degree of adaptation, the closer to the optimal solution, the more local search is needed, and accordingly, the inertia weights become smaller; the larger the degree of adaptation, the farther away from the optimal solution, the more global search is needed, which requires a larger value of inertia weights. The details are shown in the following formula: ω i d = ω min + ω max − ω min f x i d − f min d f a v e r a g e d − f min d , f x i d ≤ f a v e r a g e d ω max , f x i d > f a v e r a g e d (11)

Individual learning factor c 1 and social (group) learning factor c 2 indicate information exchange among the particle swarm. A bigger value of c 1 causes the particles to seek too much within their local range, whereas a larger value of c 2 causes the particles to converge on the local optimum too soon. The flight speed of the particles is also hard to select. To effectively control the flight speed of particles and achieve an effective balance between global search and local search, this algorithm introduces a compression factor. The convergence of the PSO algorithm can be guaranteed by selecting the appropriate parameters. Taking c 1 = c 2 = 2.05 , C = c 1 + c 2 = 4.1 . The specific expression of the compression factor is listed as follows: α = 2 2 − C − C 2 − 4 C (12)

In conclusion, the simultaneous introduction of adaptive inertia weights and compression factors can effectively improve the search capability and convergence accuracy of PSO, thus enhancing PSO performance.

Remark 2

In addition, an AFSA with an adaptive field of view and step size (Yuan and Yang, 2019) is designed in this paper. The algorithm can adaptively adjust the field of view and step length of the fish, which can enhance the global search ability in the early stage of the algorithm and accelerate the convergence speed in the late stage of the algorithm. The AFSA is a swarm intelligence stochastic optimization algorithm, in which the artificial fish self-renew by behavior such as prey, swarm, and follow, thus achieving the optimal search. However, the traditional AFSA involves the concept of the field of view in which the selection of viewpoints is random and the step length of movement is also random, which not only causes a lot of wasted computation time but also affects the search speed and accuracy. Given this, this paper presents the adaptive field of view factor γ , and step size adjustment factor m 1 . Through γ , the algorithm can purposefully adjust the field of view each time. In the initial stage of the algorithm, a large field of view is given to the prey behavior, and the field of view is gradually reduced as the iteration of the algorithm proceeds. On the other hand, a too small field of view in the later stages of the algorithm may lead to a smaller probability of searching for the global optimal solution. Therefore, when the number of iterations of the algorithm exceeds half, the reduction of the field of view is stopped so that it remains half of the initial value (the lower limit of the parameter field of view value can be set according to the specific problem). The expression is shown below: γ i d = γ min + γ max − γ min f x i d − f min d f a v e r a g e d − f min d , f x i d ≤ f a v e r a g e d γ max , f x i d > f a v e r a g e d (13) v i s i o n i d = γ i d v i s i o n 0 , d ≤ N 2 v i s i o n i d = λ v i s i o n 0 , d > N 2 (14) where, d is the current number of iterations of the algorithm, γ i d is the adaptive field of view factor, λ is the lower limit of the field of view factor, whose value is set according to the actual situation, N is the maximum number of iterations of the algorithm, v i s i o n 0 is the initially selected field of view value, and v i s i o n i d is the field of view value obtained in this iteration.

The step size is similar to the selection of the field of view, which needs to be selected adaptively according to the fish population and concentration distribution. Therefore, as shown in (15), this paper introduces the step size adjustment factor, which enables the step size to be adjusted depending on the change in the field of view. Meanwhile, the step of redefining the step size is eliminated, and the algorithm becomes more concise. s t e p i d = m 1 v i s i o n i d (15)

Remark 3

It is worth noting that AFSA has a strong global search capability, but the algorithm converges slowly at a later stage; PSO has a local fast convergence capability and can obtain a more accurate solution by adjusting the parameters, but it is easy to fall into a local optimum. Even though the two algorithms have been optimized separately, the above mentioned defects are still difficult to eliminate, which allows the introduction of hybrid optimization algorithms (Veerasamy et al., 2020). The algorithm proposed in this paper combines AFSA and PSO, which can combine the good characteristics of the two basic algorithms (Tsai and Lin, 2011). AFSA is applied in the early stage of the algorithm to expand the search range of individual best position and group best position, and PSO is utilized in the later stage of the algorithm to achieve a fine step-by-step search. It is proved that the algorithm can improve the convergence speed and solution accuracy to a certain extent.

Remark 4

Using disturbance attenuation level γ as the objective function, we obtain the optimal event-trigger threshold λ . The steps of the AFSA-PSO are shown below:

Step 1

Set parameters: such as fishnum(number of artificial fish),MAXGEN(maximum number of iterations), try_number(maximum number of tries), gen = 1(current number of iterations), etc.,; Initialize the population and individual parameters: set the range of each parameter, assign initial values to variables, form the population from the random array generated in the given range, set inertia weights, compression factors, control coefficients, etc.,;

Step 2

Start iteration, when the number of iterations is less than the preset value, the AFSA is used, otherwise the PSO is used;

Step 3

When the AFSA is used, the following operations are performed for each individual: set the adaptive field of view and step size, compare the fitness after performing following and swarming behaviors respectively, and keep the better fitness, then compare it with the individual optimum and population optimum, and update the fitness value;

Step 4

When the PSO is used, the following operations are performed for each individual: update the adaptive inertia weights, calculate the speed and position of the individual combined with the compression factor, recalculate the fitness value of the individual, find the global optimal individual, and update the optimal position of the individual and the group;

Step 5

Stop iteration when the maximum number of iterations is reached, and the obtained is the optimal solution.

Figure 2 can show the implementation of the code.

Since the thresholds of different event-triggered schemes in LFC systems can have an impact on the system’s performance. Therefore, this simulation experiment uses the hybrid AFSA-PSO to optimize the event-triggered scheme threshold with the system performance as the optimization objective. The process between the number of algorithm iterations and the optimized performance is shown in Figure 3. Meanwhile, this experiment takes the two-area LFC system as the object and adds Δ P d = 2 H z perturbations to Area 1 and Area 2 respectively when the system runs to 10s, to verify the LFC performance under the designed event-triggered scheme.

Based on the data presented in Figure 3, it is evident that the system’s target value converges to a constant value of 3.18243 after eight iterations of the algorithm. Consequently, selecting the trigger threshold λ as 0.407 results in the minimization of the optimal attenuation level γ to 3.18243. Figure 4A displays the data release time and the interval of the two-area LFC system under the optimal attenuation level γ = 3.18243 and the triggered threshold λ = 0.407 .

To assess the performance of the proposed event-triggered scheme, Figure 4B presents the release time and interval of data for the LFC system in two areas with the trigger threshold λ = 0.3 . The figure indicates that the system functions normally and releases a relatively small amount of data between 0 and 10 s. However, when the system experiences disturbances, its performance is adversely affected, leading to the release of a significant amount of data. From the comparison of Figure 4A and Figure 4B, it can be seen that when the threshold value of the event-triggered scheme is 0.407, the amount of data released is less, which reduces the system performance loss as well as increases the network resource utilization.

To verify the performance of the LFC under the designed event-triggered scheme, the frequency deviation response plots of the system under the two event-triggered schemes are given separately, as shown in Figure 5. Figures 5A, B show the system frequency deviation Δ f response curves when the system suffers a load deviation perturbation at λ = 0.407 as well as λ = 0.3 , respectively. As can be seen from Figure 5, the Δ f response can converge to zero in a short period when the system suffers a load disturbance at 10s. In addition, when λ = 0.407 , the Δ f response has a small overshoot and a short stabilization time. Therefore, the hybrid AFSA-PSO based event-triggered scheme designed in this paper has a better performance in reducing the amount of redundant information transmission. At the same time, the LFC system has better robustness to external disturbances under this event-triggered scheme.

In this simulation experiment, an IEEE-39 node model is built using MATLAB/Simulink. The model contains 10 diesel generators along with 19 loads and 34 transmission lines (Shangguan et al., 2022). Also, to verify the performance of the proposed LFC based on the improved event-triggered scheme, the system is divided into three control areas, with Gen3 as the selected generator in Area 1, Gen7 as the selected generator in Area 2, and Gen9 as the selected generator in Area 3. Figure 6 shows the schematic diagram of the IEEE-39 bus test system and the division of the control area.

At t = 30 s , a 0.038p.u. Perturbation is added to node 8; at t = 60 s , a 0.064p.u. Perturbation is added to node 16; at t = 90 s , a 0.038p.u. Perturbation is added to node 3. At this time, the Δ f response in the three areas and the trigger interval and trigger time in the event-triggered scheme are shown in Figure 7 as well as Figure 8.

From Figure 7 and Figure 8, it can be seen that the frequency deviation Δ f in the three areas converges to a steady state within a short period when load perturbations are added to the three areas at t = 30 s , t = 60 s and t = 90 s , respectively. Figures 8A–C show the release instant and release intervals of the three areas, respectively. Meanwhile, at the moment of the perturbation, the system performance needs a large amount of data for improvement, and it can be seen from Figure 7 that the event-triggered scheme transmits a large amount of data at this time. And, as can be seen from Figure 8, the overshoot of the Δ f response in the three areas is different at the three different moments of perturbation. At the moment of different disturbances, the generators responsible in the three areas respond to the load disturbance, and at the same time, the control errors between the areas deviate due to the interconnection between the areas, and the generating units in the remaining areas respond to the deviation of the exchange power between the areas. Therefore, the LFC with an improved event-triggered scheme designed in this paper has superior performance in the IEEE-39 node system.