PSO based Hybrid PID-FLC Sugeno Control for Excitation System of Large Synchronous Motor

This paper proposes a hybrid control system integrating a PID controller and a fuzzy logic controller, using the particle swarm optimization (PSO) algorithm to optimize control parameters. The control object is an excitation system for a large synchronous motor, which is widely used in large power transmission systems. In practice, the change in load and excitation source can affect the operating mode of the motor. Therefore, a hybrid controller is designed to stabilize the power factor, resulting in better working performance. In the control algorithm, a PID controller is initially designed using PSO to optimize the control coefficients. The FLCSugeno control is then integrated with the PID, in which PSO is utilized to optimize membership functions. Numerical simulation results demonstrate the advantages of the proposed approach.

The design of the PID controller and the adjustment of the coefficients is simple, especially for linear objects. For nonlinear systems, however, the PID controller becomes inefficient. To solve this problem, a fuzzy tuning technique was introduced to tune PID controller parameters [17][18][19][20]. In these methods, however, the system structure is often quite complex. To reduce the complexity, an independent fuzzy logic controller (FLC) was applied to control the excitation [21,22]. Although this approach obtains a relatively small steady-state error, the response time is prolonged due to the delayed nature of the fuzzy controller. The FLC controller has the advantage of responding well to the control requirements, especially for objects that do not need to know the mathematical model in advance. However, there is a difficulty when designing a fuzzy controller as many factors affect the controller's input/output relationship, such as the form of the membership function, the selection of the join, drag, and defuzzification operations. This choice does not have a specific rule but heavily depends on the expert knowledge of the designer. To solve this difficulty, optimization algorithms have been proposed, such as particle swarm optimization [23,24]. Compared with mathematical algorithms and other heuristic optimization techniques, PSO is simple, easy to implement, robust to control parameters, and has high computational efficiency. In this study, PSO is proposed to optimize the parameters of the PID and FLC controllers in the hybrid control system to adjust the power factor in the excitation control system of the synchronous motor.
The main contributions of this work include (i) the critical analysis of the effect of load and excitation source on the operating mode of a synchronous motor, and (ii) the integration of PID and FLC schemes using the PSO algorithm to achieve a stable power factor. The rest of this paper is structured as follows. The problem description is given in Section 2. Section 3 describes the control design, and Section 4 presents simulation results. Finally, a conclusion is provided in Section 5.

2-1-Excitation System of Large Synchronous Motor
For a large synchronous motor, the power supply to the stator is usually a medium voltage grid, 6kV. The excitation source for the motor is usually a low voltage grid, 380V, which is lowered to the appropriate voltage level through a rectifier transformer. This power supply is then rectified to direct current to supply excitation current to the motor. To stabilize the working mode of the motor, the power factor stabilization plays a crucial role. Therefore, an excitation controller is designed aiming to stabilize the power factor cos. The block diagram of the excitation control system for a synchronous motor is shown in Figure 1. The phase difference between voltage and current is measured and compared to a desired cos. The controller then calculates a control signal, which is sent to the rectifier to adjust the excitation current in order to achieve the desired power factor. To ensure favorable measurement of the phase difference between voltage and current, the stator windings of the synchronous motor can be star or delta connected to the grid. A typical power factor measurement method is to measure the angle of the voltage between two phases and the current of the other phase, as shown in Figure 2. The figure demonstrates that uab is 90 0 earlier earlier than uc. Therefore, when calculating the phase

2-2-Motor Modelling
In this section, the backgrounds of the synchronous motor modelling are presented to demonstrate the critical role of power factor regulation and stabilization. Figure 3 shows a two-axis salient pole synchronous motor in a rotor coordinate system. The frame for stator windings, (α, β), is stationary with the real axis attached to the stator phase A. Meanwhile, the frame for the excitation and the damper windings, (d, q), is rotating with the real axis fixed to the center of the pole shoe. This coordinate system rotates with the rotor angular velocity, so both -and -magnetic paths are constant. The excitation winding, (f), is attached on the -axis. The damper winding is replaced by two windings in space quadrature, one on the -axis, (D), and the other on the q-axis (Q). The stator three-phase winding is also replaced by two windings in perpendicular space, on the -axis, (d), and on the -axis (q). The electrical equations of the motor are given as [

2-2-1-Motor Equations and Equivalent Circuits
where ud is the d-axis stator voltage, uq is the q-axis stator voltage, uf is the excitation voltage, uD is the d-axis damper winding voltage, uQ is the q-axis damper winding voltage, id is the d-axis stator current, iq is the q-axis stator current, if is the excitation current, iD is the d-axis damper winding current, iQ is the q-axis damper winding current, d is the daxis stator flux linkage, q is the q-axis stator flux linkage, f is the excitation flux linkage, D is the d-axis damper winding flux linkage, Q is the q-axis damper winding flux linkage, = is the angular velocity between the rotor coordinates and the stationary reference frame α, β, r is the angle between the rotor coordinates and the stationary reference frame. The relationship between the motor currents and flux linkages can be defined by using various inductances of the motor [25]: where Lmd is the d-axis magnetising inductance, Ls is the stator leakage inductance, LD is the d-axis damper winding leakage inductance, Lf is the magnetising winding leakage inductance, Lk is the d-axis Canay inductance, Lmq is the qaxis magnetising inductance, LQ is the q-axis damper winding leakage inductance, Rs is the stator resistance, Rf is the magnetising winding resistance, RD is the d-axis damper winding resistance, and RQ is the q-axis damper winding resistance. From Equations 1 to 10, the equivalent circuits of a synchronous motor can be obtained, as illustrated in

2-2-2-Vector Diagram of a Synchronous Motor
From Equations 1 to 10, the vector diagram of the salient pole synchronous motor is built as shown in Figure 6

Figure 6. Vector diagram of the salient pole synchronous motor
In fact, the stator resistance is usually minimal compared to its inductance. Therefore, the voltage drop across the stator resistor can be ignored. The total inductances in the machine are denoted as: where Xsd and Xsq are corresponding d-and q-axis stator reactances. From Equations 11 and 12, and Figure 6, assuming that voltage drop on Rs is ignored, we obtain: where Xs is the stator reactance. From Equations 15 to 17, a simplified vector diagram can be then constructed as shown in Figure 7, with the angle between the voltage vector, us, and the q-axis electromotive force, eq, being the load angle .  Figure 7-b, we have: Accordingly, sq ss s U E sin U I cos . X   (19) where Us and Is are corresponding amplitude of stator voltage and current, Eq is the amplitude of excitation electromotive force, and Xs = Ls is the stator reactance. The electromagnetic power of the motor, ignoring the losses, is given as [26][27][28][29][30]: where Pe and Pm are corresponding the electromagnetic power and motor power. From Equations 19 and 20, we obtain: Equation 21 shows the relationship between power and excitation voltage and load angle. Assuming the source voltage and frequency are constant, we will have the following relationship:

2-2-3-Torque equation of synchronous motor at steady state
From the vector diagram in Figure 7-b, we have: We can also compute: Substituting Equations 24 to 26 into equation 20, we obtain a second method to determine the electromagnetic power of the motor in working mode as: The electromagnetic torque in working mode is calculated as: Since L = X, Equation 28 can be rewritten as: In the steady state, the electromagnetic torque of a salient pole synchronous motor has two components. The first part is the main synchronous torque, which depends on the AC voltage and excitation source, Us.Eq. The second component is the reluctance torque which depends only on the stator voltage, Us 2 . Figure 8 depicts the torque-load angle characteristics of a salient pole synchronous motor, where T1() is the main synchronous torque, T1() is the reluctance synchronous torque, and Te() is the total synchronous torque. , torque regulation is critical so that the motor remains to work safely in the face of a voltage drop of the mains voltage or overload of the torque. To control the torque, we can adjust the field current, resulting in changing T1(), which is the main synchronous torque.

2-3-Factors Affecting the Working Mode of the Motor
For a large synchronous motor, the fluctuation of the load in the operating mode can cause the load angle  to change accordingly. An excessive change can cause an asynchronous phenomenon, i.e., the rotor magnetic pole slips from the stator magnetic pole. A change in the excitation source can also affect the operating mode of the motor. This section will discuss these effects, in which the load angle,, and the current-voltage phase difference, φ, are used to evaluate the operating mode of the motor.

2-3-1-Vector Diagram of a Synchronous Motor
To make it easier to follow, we rotate the coordinate system of the vector diagram so that coincides with the horizontal axis. Assume that the power supply, the power grid frequency, and the DC excitation source are constant. The vector diagram after rotation, in case of load change, is shown in Figure 9. According to Figure 9, the motor is initially assumed to work with stator current is1, power αP1, where α is a constant, power factor cosφ  1. When the load on the motor shaft increases to αP2, the armature current will increase to the value is2. Assuming the excitation source is constant, its trajectory will draw an arc. Then, the phase difference angle φ increases in the positive direction, cosφ decrease, load angle increases, i.e., 2 > 1. When the load  increases within a permissible range, the stator magnetic field can still lock the rotor magnetic field, so the motor speed remains unchanged. However, if the load increases sharply, 2 tends to go to -90 0 . At this time, the motor is pulled out of synchronous mode. Therefore, the excitation controller must detect a decrease in the power factor to increase the field current to a suitable value to pull the motor into synchronous mode.

2-3-2-The Influence of the Excitation Source
From Equation 21, if the load and AC source are fixed, then Eqsin is also a constant. Therefore, when increasing the excitation source Eq, the load angle will decrease. This causes the relative angular position between the rotor and stator magnetic fields to decrease, the stator magnetic poles will be more tightly attached to the rotor, and the motor will run at synchronous speed. The vector diagram when changing the excitation source value is illustrated in Figure  10, therein the subscript 1, 2, 3 are corresponding to three different excitation source values:

Figure 10. The influence of the excitation source on the working mode
According to Equation 21, with a fixed load, we have: From Equation 30 and Figure 10 we see that the trajectory of Eq will slide parallelly to us. Also, from Equation 21, we have: From Equation 31 and Figure 10 we see that the trajectory of is will slide perpendicular to us. Increasing the excitation source from Eq1 to Eq3 causes the phase angle of the current with voltage to change from phase lag state to phase lead state. The value of the excitation source that produces the normal power factor is called the normal excitation. Excitement higher than the normal value is typically called over-excitation. In this scenario, the motor works as a synchronous compensator. Excitement lower than the normal value is called under-excitation. In this case, the engine works like an asynchronous machine. The above analysis shows that power factor correction is beneficial in applications where the motor is subjected to high transient loads. The power factor regulator must measure the power factor drop that occurs when the motor is subjected to a sudden heavy load and send a signal to the thyristor static rectifier to increase the value of the excitation source. This process is called excitation enhancement. As a result, the pull-out torque of the synchronous motor is increased during transient loads. After the load drops, the regulator senses the excessive lead-in power factor and drives the rectifier to drop the voltage at its output. Another application of the power factor regulator is to control the variation of the plant power factor caused by other loads such as asynchronous motor running under or no load, thereby improving the voltage quality of the plant.

3-1-PSO Preliminaries
The PSO optimization algorithm is a random search algorithm based on simulating the behavior and interaction of birds when looking for food sources. Each bird, called individual or element, in the flock, called population, is characterized by two components, the position vector xi and the velocity vector vi. Each individual has a fitness value, which is assessed by the fitness function. Initially, the PSO has initialized random position and velocity vectors [31- 34]. Then in each iteration of the algorithm, the velocity vector vi and the position xi of each individual will be updated as: where k repesents the iteration; (k) is a weight parameter; c1 and c2 are corresponding cognitive and social parameter, r1 and r2 are random samples in the interval [0; 1]. At each iteration, each individual is influenced by the best position it has achieved, Pi(k), and the global best position in all searches by all individuals in population, G(k). In this paper, PSO will be implemented to find the optimal parameters for a hybrid PID-FLC Sugeno control design for the excitation system of a large synchronous motor. The "particleswarm()" function in MATLAB software was used to perform the PSO algorithm.

3-2-Hybrid PID-FLC Sugeno Control Design for Excitation System of Large Synchronous Motor
The structure diagram of the excitation control system, using hybrid PID-FLC Sugeno control, is shown in Figure  11. In this diagram, the object consists of a three-phase 6 pulse bridge rectifier and the synchronous motor. The output voltage of the rectifier will provide a field source for the rotor windings. The power factor cos is measured at the stator side of the motor and compared with the setpoint. The error e(t) is fed to the PID controller for processing. The fuzzy controller receives the error signal e(t) and its derivative to process according to the established composition rule. The output signal of the PID controller and fuzzy logic controller is added and sent to the thyristor excitation regulator to change the DC output voltage of the rotor winding.

3-3-PSO-Based PID Controller
At the first step, a PID controller is designed as shown in Figure 12. The feedback control signal is computed as: where Kp, Ki , Kd are corresponding coefficients for proportional, integral, and derivative terms, KN is the filter coefficient. These coefficients are components of the optimal variable set up for the PSO algorithm. The fitness function of the PSO is selected using the integral absolute error (IAE) index as: where e(k) is the error at k-iteration, n is the total number of samples in an iteration.

3-4-PSO-Based Hybrid PID-FLC Sugeno Controller
In this section, a hybrid PID-FLC Sugeno controller is designed to control excitation for the synchronous motor so that the power factor tracks its desired value. The structure of a hybrid controller is shown in Figure 13, therein the modulating signal offset u for the PID controller is obtained from the output of the fuzzy controller. The PID parameters are optimized using the PSO algorithm as presented in Section 3-3.

Figure 13. PSO-based hybrid PID-FLC Sugeno controller
The FLC Sugeno controller is designed with two input variables, the error e and its derivatives ce, and one output variable, u. The range of these inputs/outputs is defined in the symmetric normal domain as [-1, 1]. When the controller is introduced into the system, the real variation domain will be corrected to the standard symmetry domain by the coefficients Ke, Kce and Ku. The selected fuzzy sets are designed with names denoted and interpreted as in Table  1. There are 5 membership functions for input variables, e and ce, including NB, NS, ZE, PS, PB. The membership functions are designed to be strongly fuzzy partitions, i.e., for any value of x, the total membership of x on the membership function is equal to 1. There are 7 membership functions for the output variable u, including NB, N, NS, ZE, PS, P, PB. The control rule system is designed as shown in Table 2. In order to adjust the shape of membership functions towards the optimal for the controller, in this study, some constraints are introduced to limit the number of variables to be optimized as follows:

4-Simulation Results
In this section, simulation is carried out on a convex pole synchronous motor working in a steady-state in the present load change. The simulation diagram on Matlab/Simulink using the proposed PSO-based hybrid fuzzy control system is shown in Figure 15. The simulation parameters are given in Table 3. To evaluate the quality of synchronous motor excitation control system, we compare the system response of the proposed PSO based hybrid PID-FLC Sugeno to a PI controller and PSO based PID controller. For the PI controller, Kp = 1 and Ki = 5 are selected using Ziegler-Nichols combined with trial and error method. For PSO based PID controller, Kp = 0.549, Ki = 5.992, and Kd = 0.5 are obtained by PSO algorithm. These parameters are also used in combination with the PSO-based hybrid PID-FLC, therein the fuzzy parameters are achieved using PSO as a = 0.3183, b = 0.75, c1 = 0.75, c2 = 0.3943, ke = 2.3103, kce = 1156.31, and ku = 0.012. In the simulation, disturbance from the sudden increase of the added load, at t = 2.5s, causing the phase shift angle to change. The control system will automatically adjust the excitation source to stabilize the power factor cosφ. Figure 16 shows the simulation result when load is constant, the cosφ_ref change from 0.85 to 0.95. The figure depicts that all controllers provide minimal errors. By calculating the mean deviation, using the "fit" function in Matlab, the hybrid PID -FLC Sugeno gives the smallest error, 0.51, followed by the PID controller, 2.45, and finally the PI controller 4.03.  The above results show that using the PSO algorithm to optimize controller parameters of the hybrid PID-FLC controller for the excitation control system of a large-capacity synchronous motor in working mode is entirely feasible. Figures 18 and 19 depict the values of IAE after each iteration of the proposed PSO algorithms. These figures illustrate that fitness functions converge to optimal values quickly, after about 11 iterations for PSO-based PID and 36 iterations for the PSO-based hybrid PID-FLC. More importantly, these figures show that the proposed PSO-based hybrid PID-FLC achieved a much smaller IAE than the PSO-based PID controller, 0.51 compared to 2.45, illustrating the advantages of the proposed approach.

5-Conclusion
This manuscript introduced a hybrid FLC-Sugeno controller for the excitation system of a large synchronous motor. The effect of load and excitation source on the motor's working mode is critically analyzed to demonstrate the importance of stabilizing the power factor. In the control system, the control parameters were optimized using the PSO algorithm. The PSO is utilized to optimize the tuning coefficients Kp, Ki, Kd, and KN of the PID controller. In the FLC Sugeno controller, the membership functions of the input and output variables are optimized. The simulation showed that the proposed controller presents the best control performance among the considered approaches. In our future work, the proposed algorithm will be applied to practical large power transmission systems and other reference tracking control applications.

6-2-Data Availability Statement
Data sharing is not applicable to this article.