Dynamics and asymptotic profiles of steady states of an SIRS epidemic model in spatially heterogenous environment

1.   Introduction

Over the years, the switched reluctance generator (SRG) has been a focal point for many research studies ^[1]. The SRG has a basic, rigid and simple structure. This device includes neither a permanent magnet nor twisting of the rotor. This construction decreases thus the expense of the SRG and its maintenance. The SRG can operate during high-velocity activities without the worry of mechanical problems ^[2].

Accordingly, the SRG offers several advantages compared to other machines, e.g., the rotor does not require any winding and it is constituted only of ferromagnetic material. The major windings (losses) are concentrated in the stator. There are numerous sorts of structures that can drive the SRG ^[3]. The power electronic converter usually used for controlling the SRG is the asymmetric power electronic converter. It has the advantage of being able to independently control each phase of the SRG ^[4]. The stator windings are associated with the arrangement of the upper and lower switches of the inverter ^[5]. Due to the complex nonlinear characteristics of SRGs, there exist very few previous works dedicated to the identification and modeling of SRGs ^[6,7]. This was later described as nonlinear systems structured in blocks ^[8,9,10]. The most popular models of nonlinear systems in blocks can be found in ^[11,12,13]. The system nonlinearities can be static or dynamic ^[14,15,16]. Note that the problem of SRG speed control has been addressed by using several techniques, e.g., fuzzy logic control ^[17], artificial neural networks (ANNs) ^[18] and genetic algorithms ^[19,20]. Furthermore, the optimization techniques like particle swarm optimization and bacteria foraging can be used to regulate the SRG speed. Then, an ANN-proportional integral (ANN-PI) controller was introduced to regulate the power injected into the grid ^[21]. A proportional resonant controller that has been used to control the power produced by the SRG is given in ^[22]. To minimize the torque ripple of SRGs, an ANN control method has been developed, as presented in ^[23]. To control the position of the linear switched reluctance machine, a flower pollination algorithm was developed ^[24].

The fuzzy logic technique was introduced by Zadeh in 1965 ^[25]. The fuzzy logic controller (FLC) has been widely investigated and used to regulate complex nonlinear systems. The FLC is an artificial decision-maker that relies on human decision-making behavior through the use of language rules rather than mathematical models ^[26]. An exact model is not necessarily required with the FLC technique and it is created using language information. Mathematical equations to describe the system to be controlled are not necessary for FLCs. The FLC has been used in many fields, e.g., in solar photovoltaic systems ^[27], wind energy ^[28], power systems ^[29] and smart agriculture systems ^[30]. Other previous works have yielded combinations of fuzzy logic with other methods. In ^[31], the authors used an adaptive neuro-fuzzy inference system to assist the power monitoring for a smart grid. Adaptive fuzzy control using a genetic algorithm has been used to overcome the influence of nonlinearities in the vessel dynamic positioning system ^[32]. In ^[33], a self-tuning FLC-based speed is proposed for the control of an SRG for wind energy applications. The ACO is a heuristic approach that can be used to find the global optimum of a given problem. In this respect, note that the ACO has already been employed in wind energy control ^[34] and for planning the wind farm layouts ^[35]. In ^[36], the authors discussed a design method entailing the use of the ACO based on a proportional-integral-derivative (PID) controller for a zeta converter.

The aim of this study was to control the output voltage of the SRG. Techniques using the ACO-based PI controller and FLC have been developed. In this respect, the SRG output voltage is adjusted by using the turn-off angle ${\theta }_{off}$. To obtain the best value of ${\theta }_{off}$, two methods are proposed. The first one consists of a PI controller tuned by an ACO algorithm. In the second approach, the tun-off angle ${\theta }_{off}$ is determined by using an FLC.

For convenience, the main contributions of this work are summarized as follows:

● Unlike many previous methods dedicated to controlling the SRG voltage, the proposed methods are easy to implement.

● In this paper, very interesting tools and concepts are proposed, such as ACO for tuning the PI controller, as well as the FLC approach.

● The established algorithm, which applies an ACO technique for SRG voltage control, is new.

● To improve the results obtained via the ACO algorithm, four cost functions were applied and compared.

● In this study, a second solution to control the SRG output voltage by implementing a FLC approach was also developed. The best solution was chosen by comparing these two methods.

The paper is organized as follows. The mathematical model and operation principle of SRGs are introduced in Section 2. In Section 3, the control methods for the SRG based on the ACO and FLC are described. An overview and description of the ACO are presented. Then, the cost functions used to optimize the ACO method are also proposed in this section. Then, the FLC is discussed. To show the effectiveness of these methods, simulation examples are presented in Section 4. Finally, comparisons between the results obtained using the two approaches are also discussed in this section. In Section 5, the concluding remarks of this paper are summarized.

2.   Mathematical model and operation of SRGs

2.1.   Mathematical model of the SRGs

First, note that the SRG winding flux $\lambda \left(.\right)$ depends on the phase current $i$ and the rotor position $\theta$. Then, for each stator phase, the voltage $v$ can be expressed as the sum of the voltage drop on the resistor and the derivative of the flux:

where ${R}_{s}$ is the phase resistor. The phase inductance $L\left(\theta, i\right)$ is a nonlinear function depending on the rotor position $\theta$ and the phase current $i$. Furthermore, the linkage flux $\lambda \left(\theta, i\right)$ can be expressed according to $L$ and $i$ as follows:

Then, replacing the flux $\lambda \left(\theta, i\right)$ in Eq (2) with its expression according to $L$ and $i$, the expression of phase voltage becomes

The last term on the right side of Eq (3) corresponds to the back electromotive force $e$. Specifically, one has

where ω is the SRG's rotor speed. When the SRG operates as a generator, the current changes its direction. The back electromotive force thus becomes negative.

2.2.   Operation principle of SRGs

The SRG stator is characterized by winding resistors with very small values ^[6]. The resistive voltage drop can be neglected ^[6]. Accordingly, the phase current $i$, the phase voltage, $v$ and the back electromotive force $e$ are related by the following equation:

Here, an asymmetric half-bridge (AHB) converter is proposed to drive the SRG (Figure 1). The SRG operates as a generator when the phase inductance $L\left(\theta \right)$ decreases; this means that $dL\left(\theta \right)/dt < 0$ (Figure 2).

The SRG is interconnected via the DC link to a voltage source converter (VSC), which is linked to the electrical grid. Then, the DC link voltage ${V}_{DC}$ is regulated by the VSC control. The latter allows for the injection of the generated power from the SRG into the electrical grid.

In this respect, note that the characteristics $L\left(\theta \right)$ and $i\left(\theta \right)$ are smooth curves and are often linearized in the study of generator and motor modes of SRGs, as shown in Figure 2 ^[37]. It can be easily seen in Figure 2 that the curve of the electrical current $i$ can be subdivided into three main intervals, or the following three situations:

● The first situation is characterized by $e < {V}_{DC}$. In this case, the phase current decreases after the excitation stage. This situation happens when the operating speed is decreased due to the electromotive force diminishing.

● The second situation can be obtained when $e = {V}_{DC}$. Accordingly, after the duration of the excitation stage, the current tends to remain constant.

● The last situation can be seen when the SRG is working at high speeds; one thus has $e > {V}_{DC}$. After the excitation stage, the phase current tends to rise.

To determine the SRG's operation mode, the base speed should be evaluated. To improve the SRG performance, two techniques for SRG control can be used. For low operating speeds, the control of the SRG is performed by using the existing hysteresis method. The power control of the SRG is related to the adjustment of the reference current ${I}_{ref}$. In this respect, the firing angle ${\theta }_{on}$ and turn-off angle ${\theta }_{off}$ are the driving parameters (Figure 2). For high speeds, the control of the SRG is achieved by using a single-pulse technique. Accordingly, the power control is based on changing the turn-off angle ${\theta }_{off}$ while the firing angle ${\theta }_{on}$ is kept constant.

3.   Control method for SRGs: ACO and FLC

3.1.   ACO technique

3.1.1.   Overview and description of the ACO algorithm

ACO is a meta-heuristic algorithm and probabilistic method. It is commonly used for solving hard combinatorial optimization problems based on graph representations. This method aims to find the best paths based on several possible graphs. This algorithm was initially proposed by Marco Dorigo in 1992 ^[38]. The artificial ants used in this technique were inspired by real ant colonies and designed to search for the best path. The shorter path is obtained by analyzing and combining the results of the path of each ant in the colony. The main issue is related to finding the shortest path among all of the trajectories traveled by different ants to reach the food. When the ants are moving, they leave a chemical pheromone trail on the ground. Depending on the distance of the path, the pheromone quality will change. Every ant chooses the path according to the intensity of the deposited pheromone. If there is no more deposit of pheromone, its intensity decreases according to the time. Specifically, the other ants are attracted by the pheromone; they thus choose the path where the pheromone is of high intensity. This path will be the best solution and has a great probability of being chosen.

With this approach, the lifestyle behavior of real ant colonies is used to solve the optimization problem. Here, the artificial ant technique is proposed to seek the best solution by moving from one node to another. With this algorithm, the artificial ants move according to their previous positions, which have been stored in a specific data structure. Once the ants have accomplished their tour between the initial node and the last one, the pheromone consistency of each path is updated. The concentration of pheromone will be of high quality if the artificial ants finished their tour by taking the shortest path, and vice versa. The steps of the proposed algorithm are summarized in Figure 3.

3.1.2.   Cost functions and objective function for the ACO technique

For problem optimization, there are several error criteria commonly used in the literature, including the integral square error (ISE), integral of the absolute error (IAE), integral time of the absolute error (ITAE), integral of multiplied absolute error, quadratic error and total square variation. All of these error criteria have zero as the lower bound and can be considered as cost functions of the proposed ACO algorithm. To improve the overall performance obtained through the use of the PI controller, four cost functions were selected. The considered cost function criteria are ISE, mean squared error (MSE), IAE and ITAE. Specifically, the objective function of the ACO algorithm is based on minimizing the considered error criteria. The parameters of the PI controller can be obtained by using the optimization results.

3.1.3.   Closed-loop control of SRGs via ACO

Here, the ACO-based control method for the SRG voltage is introduced. Figure 4 shows the closed-loop control corresponding to this technique, where ${V}_{ref}$ denotes the reference voltage and ${V}_{m}$ stands for the SRG output voltage. With this method, the SRG is driven at high speeds. The firing angle ${\theta }_{on}$ is kept constant at 40°, while the turn-off angle ${\theta }_{off}$ is adjusted by the algorithm. Accordingly, the magnetization cycle of the SRG phase is controlled by changing the value of ${\theta }_{off}$ through the use of a PI controller.

Even if the PID controller would improve the system stability, the derivative action of the PID controller would likely increase the amplitude of disturbances. Furthermore, a couple of ripples is often produced in SRG applications. In order to avoid more disturbances, a PI controller was selected.

In the proposed approach, the PI controller parameters are obtained by using an ACO algorithm. The transfer function of the proposed PI controller is as follows:

where ${K}_{P}$ and ${K}_{I}$ are parameters of the PI controller.

3.2.   FLC

3.2.1.   Description of the FLC

Generally, the FLC has three main stages, namely, fuzzification, fuzzy rule extraction and defuzzification. The inputs are turned into fuzzy sets by using linguistic elements and membership functions throughout the fuzzification process. The two most well-known fuzzy systems are the Mamdani and the Takagi-Sugeno-Kang models. Here, the Mamdani fuzzy systems have two inputs and one output is used. The voltage error signal ε ($\varepsilon = {V}_{ref}-{V}_{out}$) and its derivative $\dot{ \varepsilon }$ are the inputs of the FLC. The output ${\theta }_{FLC}$ of the FLC corresponds to the turn-off angle ${\theta }_{off}$ of the SRG.

The design of the FLC was determined by assigning seven fuzzy sets for the inputs (ε, $\dot{ \varepsilon }$) and the output (${\theta }_{FLC}$), namely, {NB (negative big), NM (negative middle), NS (negative small), ZZ (zero), PS (positive small), PM (positive middle), PB (positive big)}. In this study, the membership functions related to (ε, $\dot{ \varepsilon }$) and ${\theta }_{FLC}$ are the chosen type of triangular shapes; they are given by Figures 5 and 6, respectively.

3.2.2.   Closed-loop control of SRG through the use of an FLC

The second method developed in this work consists of using an FLC to determine the best values of ${\theta }_{off}$ while the angle ${\theta }_{on}$ is fixed at 40°. Figure 7 shows the closed-loop control corresponding to this technique.

4.   Simulation results and discussion

In this section, examples of simulations are presented to show the effectiveness of the proposed methods. The first method aims to control the output voltage of the SRG by using an ACO algorithm. This method has two steps. In the first step, the optimum parameters for each cost function (i.e., ISE, MSE, IAE and ITAE) of the PI controller are determined. The optimum PI controller parameters are chosen by taking the parameter values that yields the smallest error between the reference voltage and the output voltage of the SRG. In the second step, the obtained values are compared based on the time domain specifications. Specifically, the steady-state time, peak time and overshoot.

In the second method, an FLC replaces the PI controller.

Remark 1:

The applied SRG is characterized by the parameters given in the Appendix (Part a). The initialization parameters of the ACO algorithm and PI controller are also given in the Appendix (Parts b and c).

The approach used to initialize the ACO algorithm is based on results reported in the literature and simulations. First, we applied many combinations by changing the number of ants (10, 20, 30, 40, 50, 80) and the number of iterations (20, 50,100,200) in simulations. Second, in consideration of the obtained results and time constraints, we have chosen the optimal number of ants as 30 and optimal number of iterations as 100.

4.1.   System response when using the PI controller

Based on the ACO algorithm, the parameters for the optimal PI controllers corresponding to each cost function were obtained; they are presented in Table 1.

The responses of the SRG output voltage corresponding to the use of the optimal PI controllers have been plotted as shown in Figure 8 and Figure 9, where the reference voltage is constant (${V}_{ref} = 400V$). The error variations for the ISE, IAE, MSE and IATE have been plotted as shown in Figure 10, Figure 11, Figures 12 and 13, respectively.

To compare the performances of the PI controllers, their characteristic time-domain specifications (steady-state time, peak time, overshoot) were compared; the results are given in Table 2. Finally, the best PI controller was chosen. For convenience, the voltage ripple of the output voltage signal resulting from the use of the ACO-PI controller method and the cost error ITAE was estimated. The value of this voltage ripple was 0.76%.

4.2.   System response when using the FLC

In the second method, the PI controller is replaced with an FLC. In this respect, two cases can be distinguished. The first one consists of taking the wind speed as constant; in the second case, the wind speed is variable. When the wind speed is taken as a constant (here, $\omega = 157rad/s$), the response of the output voltage of the SRG as a result of using the FLC was determined; it is shown in Figure 14; the error variation is shown in Figure 15. The results obtained for the time domain are summarized in Table 2. It can be seen that the output voltage signal of the FLC yields a voltage ripple with a value of 0.54%.

Here, the wind speed was varied from $110rad/s$ to 170 rad/s. The wind speed curve is given in Figure 16. For this case, the SRG output voltage corresponding to the use of the FLC controller can be seen in Figure 17; the error variation is shown in Figure 18.

The characteristics of the SRG output voltage signal corresponding to the ACO-PI control scheme with IATE, as well as that corresponding to the FLC scheme are summarized in Table 3.

4.3.   Results and discussion

● Case of constant wind speed

To show the effectiveness of this study, examples of simulations have been provided. The reference voltage was kept constant at ${V}_{ref} = 400V$. These simulations were established by applying a constant wind speed of $157rad/s$ and choosing a load of 1600 Ω. The results of the SRG output voltage that were obtained by using a conventional PI controller with different cost functions (i.e., ACO-ISE, ACO-MSE, ACO-IAE and ACO-ITAE), and as based on fuzzy control, are shown in Figures 8, 9 and 14. These results show that the SRG output voltage resulting from the use of the ACO-based PI controller and the FLC techniques converges to its reference value. It can be seen by using this method that a constant torque allows a constant voltage. On the other hand, the results summarized in Table 2 show that the ACO-based PI controller that utilizes ITAE provides the best results. Additionally, the time-domain characteristics show that the FLC improves the dynamic and steady-state features. Furthermore, the FLC yielded a smaller voltage ripple (0.54%) than the PI controller (0.76%).

● Case of variable wind speed

The wind speed was varied in the range of 110–170 rad/s. The proposed simulations were established by taking the reference voltage ${V}_{ref} = 400V$ and applying a load of 1600 Ω. Generally, the tuning of the ACO-based PI controller parameters requires significant computational time. Thus, to overcome this issue, an FLC controller was developed. It is shown in Figure 17 that the SRG output voltage converges fast to the reference voltage even if the wind speed changes.

5.   Conclusions

This article discusses the problem of SRG output voltage control. This SRG was implemented in a wind turbine operating at 75 KW that was connected to an electrical grid. Here, a PI controller based on the ACO technique has been proposed to regulate the SRG output voltage. To determine the PI controller parameters, several cost functions were used. By comparing the different results obtained by using the given cost functions, the best PI controller parameters were chosen.

When the wind speed was taken as constant, the simulation results confirmed that the output voltage associated with the ACO-PI controller technique and a cost function (i.e., ITAE, MSE, IAE or ISE) converges to the reference voltage. Furthermore, it has been shown that the solution based on the ITAE gives the best results for the control solution based on the ACO-PI controller. It is interesting to note that this approach requires a short simulation time, unlike several other methods. On the other hand, the FLC yielded better results than the ACO-PI controller for variable wind speeds.

Appendix

The system parameters applied in the simulations are as shown below.

(a) SRG parameters:

Number of stator and rotor poles, $12/8$ (respectively); Frequency, $\left[F\right] = 50Hz$; DC supply voltage, $\left[{V}_{dc}\right] = 240V$; Reference current, $200A$; Hysteresis band, $\left[+10, -10\right]$; Mechanical load torque, T_m = $-$10 N·m; DC link capacitor, C ${3}^{e-3}F$; Speed of wind, $\omega = 157rad/s$; Voltage of excitation, ${V}_{ewc} = 50V$; Initial voltage, ${V}_{{CC}_{init}} = 400V$; V_dc reference, ${V}_{CCref} = 400V$; Load, $R = 1600\Omega$; Upper limit of turn-off angle, ${\theta }_{offmax} = 21$; Firing angle, ${\theta }_{on} = 40$; Maximum current, ${I}_{max}$ = 8.5 A.

(b) PI controller parameters:

The boundaries of K_p and K_i are $\left[0.1 5\right]$ and $\left[0.5 10\right]$, respectively.

(c) ACO parameters:

Number of iterations, n_n = 100; Number of ants = 30; Pheromone decay parameter, α = 0.8; Relative importance of pheromone with respect to distance, β = 0.2; Evaporation rate = 0.7; Number of parameters = 2; Number of nodes = 1000.

Conflict of interest

All authors declare no conflict of interest regarding this paper.