PSO Optimized PID Regulator for a Variable Frequency Brushless Synchronous Generator

The aim of this paper is to describe development of a new control structure for Automatic Voltage Regulator system. This approach is based on the optimization of the control voltage magnitude of a brushless excitation synchronous alternator machine operating at variable speed. The choice of the machine type is justified by its attractiveness in several areas such as the aircraft domain due to its autonomy and robustness. The considered control technique is based on the simultaneous optimization of two regulators introduced in the loop control. The parameters of the Proportional Integral Derivatives (PID) regulator have been optimized using the Particle Swarm Optimization (PSO), which is considered as an attractive method considering the wide operating speed range, and the load variations compared with the classical methods such as Ziegler Nichols. Many robustness tests are carried out by considering the parameters variations as well as the perturbation connection and disconnection of the load at low and high speeds.


Introduction
The production of the electrical energy with variable speed alternators seems to be the best solution in terms of performance and reliability [1].This system is based on a combination of a generator that operates at variable speed and a training motor.A static converter is associate at the output of the generator in order to stabilize the suitable frequency [2].
The choice of a variable frequency generation is based on the machine type and field of application that allows the reduction of the mechanical losses, the gearbox costs and the size of the assembly.
The production of embedded electrical energy is characterized by a wide training speed of the alternator that is used in several applications including aircraft field [3].However, the wide training speed introduces constraints in voltages and currents in the load.In this regard, several studies have been published and allowed choosing the alternator and its suitable control strategy [4] and [5].
The advances in power electronics allow us a new challenge in the area of electrical production.From a mechanical source of variable speed, a fixed frequency and a regulated constant voltage can be obtained via a static converter.Which has the advantages of reducing the generator size and weight, since the mechanical regulation system of speed is removed, as in the new aircraft supply system [4].
The variable speed operation allows an optimization in the energy captured by a turbine, but it requires the implementation of a converter and its control.Which leads to an additional cost in comparison to a fixed speed operation [4] and [6].
In case of hybrid vehicles, all the studies conducted on the variable speed generators have shown that diesel consumption is optimized and reduced comparatively to their operating at a fixed speed [5], [6] and [7].
A wide range of speed variation, in driving the Brushless Excitation Synchronous Generator (BESG) engine, penalizes the stability of the output voltage control of the main generator.To remedy this, several studies have been initiated to improve and develop new control techniques [8].
The schematic mechanism of BESG is illustrated in Fig. 1.The generator is assumed equivalent to three separated machines placed in cascade, such as Permanent Magnet Generator (PMG), synchronous machine and alternator.The PMG machine allows the generator a completely autonomous operation.The supply voltage for this machine is adjusted to feed a connected chopper to the second synchronous machine.The three-phases are powered through a converter.The mono-phase inductor is placed in the stator side while the three-phase armature is placed in the rotor.
The obtained voltages are used to feed the inductor of the main generator, and these voltages are adjusted from the rotor through six rotating diodes.This configuration is a conventional synchronous machine with a conventional wound inductor connected to the network or load.It's noticed for this configuration, the absence of brushes which causes the increase the cost and the real complexity of the generator.The frequency of the obtained voltages is in function of the rotation speed [9].In the literature, the control for the main generator voltage regulation is based on three actions.The first one is the inner feedback loop that controls the excitation current of the exciter.The second action is the outer feedback loop for direct output voltage regulation of the main generator.And the third one is the disturbance-rejection loop that compensates the changes in the output voltages caused by the generator load (i.e., stator) current.The global loop is introduced for the voltage control of the machine [8].
This technique offers stability to the total voltage of the alternator and rejects all disturbances regardless of the variation of the rotation speed or load.But the proposed control technique is very difficult to be implemented in practice, due to the complexity of the block diagram and they depend on the machine parameters that affect the stability of the system.
In order to stabilize and simplify the control, a new technique is developed in the present work.This technique consists of calculating the voltage excitation of the principal exciter-generator that corresponds to the nominal voltage output of the main generator.The voltage excitation is inversely proportional to the rotation speed.This excitation is the reference of the regulator of the machine with permanent magnets.Thus, the proposed control technique comprises two Proportional-Integral-Derivative (PID) controllers; the first one for the regulation of the output voltage of the PMG and the second one for the regulation of the main generator terminal voltage.In order to optimize the PID controller parameters, the Particle Swarm Optimization (PSO) has been used.
The PID control is the most efficient and widely used feedback control strategy, due to its simplicity and satisfactory control performance.The wide use of PID control has sustained research on finding the key methodology for PID tuning to obtain the best possible performance out of the PID control.Optimal control performance can only be achieved after identifying the finest set of Proportional gain (K p ), Integral gain (K i ) and Derivative gain (K d ).
Artificial intelligence methods, such as neural network, fuzzy system, and neural-fuzzy logic have been widely applied to optimize the PID controller parameters.Many iterative techniques, such as Genetic Algorithm (GA), Simulated Annealing (SA) and Chaotic Algorithm (CA) have recently received much attention for achieving high efficiency and searching global optimal solution in problem space [10], [11] and [12].
Particle Swarm Optimization (PSO) as one of the modern heuristic algorithms, was developed through simulation of a simplified social system and has been found to be robust in solving continuous nonlinear optimization problems.The PSO technique can generate a high-quality solution within shorter calculation time and stable convergence characteristic than other stochastic methods [11].The adopted control technique is composed of two loops and two regulators.The first loop is for the output voltage of PMG controller, induce an overvoltage protection of the system, while the second loop is for the total voltage regulation.Therefore, it is found to be simple compared to the three-loop technique and the system rejected all disturbances (Fig. 1).
Since the machine works in a wide speed range, finding the regulator parameters (K p , K i and K d ) values is very difficult.Moreover, the determination of these parameters by direct conventional methods like Ziegler Nichols (ZN) and Internal Model Control (IMC) is somewhat complicated.Therefore, the Particle Swarm Optimization (PSO) method is used to find optimal values of these parameters.PSO is an iterative optimization method that can handle complex systems that are composed of several blocks.
In our application, the PSO tends to minimize both response times, overshoot and difference between the reference voltage and the actual final voltage of the main generator [13].
The advantage of using two regulators is the limitation of the voltage generated by PMG which produced at high speed.This protection allows the system to operate in a wide range of speed [14].
In this paper, a machine drive is considered as a speed source, i.e. the alternator output is low compared to mechanical power drive: a case of the energy production embedded [4].This paper is organized as follows.Firstly, the presentation and modulization of BEGS will be presented in Sec. 2. Then the formulation of the optimization technique will be treated in Sec. 3. Finally, the simulation results and robustness study will be presented in the last section in order to illustrate the efficiency of the proposed control technique.

Presentation of BESG
The synchronous BESG machine, or nominated the Variable Frequency Generator, is shown in Fig. 1.It is composed of three separate synchronous generators linked in cascade, from the left to right.The first machine contains a PMG, which allows the BESG to be fully autonomous in terms of excitation.The second machine is an inverse synchronous machine structure (or exciter).The output voltage of the PMG will supply the Chopper after rectification.The output voltage of the chopper will regulate the second stage of the BESG.The output of the second stage that comes from the armature (rotor) will be rectified, by a diode bridge rectifier, to supply the inductor main generator, which represents the last stage.And at the end, the output of the main generator will supply the load.
The main feature of the synchronous BESG machine is the absence of the mechanical friction which increases the cost and the complexity of the generator.
The output voltage frequency of the BESG machine is related to the common rotational speed of the three machines.The adopted BESG machine structure can be found in several applications, where the power could be produced in a variable speed drive, for example in some industrial applications, energy production, embedded or renewable energy [15].

Modelling and Block Diagram of the Main Generator
In practice, the machines are known by their windings and their designs.In order to analyse and take into account their exact configurations, we shall develop, for each type of machine, a model that can be close to the real model to be rigorously controlled.The BESG machine consists of three synchronous machines (PMG, inverse synchronous machine and main generator) whose models are very similar to the PMG with slight differences.In order to modelize the main generator, the Park Transformation is used to write the synchronous machine model.In our application, the rotation speed and the voltage excitation will be the input of the model.The output of the main generator model is voltage.By neglecting the effects of shocks, regarding the small values of the time constants, the model will be described by the following equations: Voltage equations: where v d , v q , i d and i q are voltages and currents in dq frame (direct and transverse axe); v f and i f are voltage and current of the main field winding; R s is stator resistance; R f is main field resistance, Φ d , Φ q are flux in dq frame, ω is the electrical angular speed of the BESG.The mechanical speed is supposed variable, it's very important in our proposed control strategy.

Flux equations:
where L d and L q are the direct and transverse stator main inductances; L f is the main field inductance.M f d is the mutual inductance between direct stator winding and main field.
Substituting the flux Eq. ( 2) into the voltages Eq. ( 1) leads to write the model of the stator voltage (induced) in the matrix form as follows: With where S is the Laplace operator (S = d/dt), V s , I s and Z s are the Voltage, Current and the Impedance of the main generator, respectively.
The substitution of the flux Eq. ( 3) in voltages Eq. ( 1), of the inductor, allows deducing the following operating equations: with where Thus where e fd is the compensation factor.
A linear control of the synchronous machine excitation circuit, which corresponds to the current, if with the voltage v f , requires the compensation of the transfer e fd that represents the EMF (null continuous).
Introducing Eq. ( 6) into equation Eq. ( 4) leads to writing the model of the synchronous machine for an alternator by inversing the sign of the stator voltage on the load level, as follows: The matrix Eq. ( 10) can be formulated as follows: If the alternator is required to debit on the load, in 2D park components, the direct and quadrature components of the load voltage are given by: Equation ( 11) can be expressed by: with where R l , L l and Z l are the resistance, the inductance and the impedance of the load connected to the main generator.
With the matrix Eq. ( 10) and Eq. ( 11), the block diagram of the generator supplying a load is illustrated in Fig. 2.

Modelling and Functional Diagram of the Exciter and Rectifier
The exciter represents the excitation system that supplies the inductor of the main generator.The exciter is a three-phase synchronous machine with fixed excitation (stator) and without damper whose phases are carried by the rotor with a six-diode bridge connected to its terminals.The six diodes are mounted on the rotor in order to feed the inductor of the main generator with DC (Fig. 1).
The exciter circuit is modelled by the same approach as used for the main generator.The excitation voltage of the exciter system will be considered as the power source for the excitation of the main generator through a rectifier diode (Fig. 1).
For the exciter model, the same equations of the main generator model will be used, where all the variables will be written from the rotor side of the exciter, as shown in Fig. 3.The effective rectified voltage is given by: The average value of the output voltage of a rectifier for q phase's number is: To calculate the new values of the load exciter after recovery, an equality of powers between the input and the output of the converter is considered as follows: Thus, the alternating side can be observed, which is seen continuous, in terms of active power, if the losses in the diodes are neglected: From the knowledge of power electronics, the DC voltage (average rectified) for q number phases can be written: For the three-phase case: DC restated for q phases: For the three-phase case: Powers equality AC & DC allows writing: In terms of active power, if the losses in the diodes are neglect: This gives, after calculating AC side: The value of tan ϕ obtained by Eq. ( 24) is replaced in Eq. ( 16), the expression of L lexc is obtained by: Thus, L lexc is relatively low.
Equation (24) and Eq. ( 25) are integrated into the overall simulation model, which is very close to the real system.

Modelling and Functional Diagram of PMG
The PMG provides the power to the control system of the excitation of the exciter.The application of Park Transformation to the PMG model corresponds to transform the three coils (stator) two equivalent coils.With the same approach used for the main generator, we assume that the rotor of the PMG machine is flat, and the magnetic circuit is not saturated.The equations are defined as follow: With: The symbols of the last equations are the same as those described previously.The mechanical speed of the PMG machine is considered variable.
When the PMG is loaded, the direct and quadrature components of the voltages are given by: Equation ( 27) and Eq. ( 31) in matrix and operational form are written as: With: With the matrix Eq. ( 27) and Eq.(32), the block diagram of a generator supplying a load is illustrated by Fig. 4.

Formulation of the Optimization Problem
In this problem, the reference voltage, the response time, and the overshoot are regulated by optimizing the PID controller parameters.PSO is used to minimize the objective function F given in [15], where O sh is the maximum overshoot of change in terminal voltage, t st is the settling time of change in terminal voltage (s) and max_dv is the maximum derivative of charge in terminal voltage. (34)

Particle Swarm Optimization
The PSO is an evolutionary algorithm for the solution of optimization problems.It belongs to the field of Swarm Intelligence and Collective Intelligence and is a sub-field of Computational Intelligence [16].It was developed by Eberhart and Kennedy and inspired by social behaviour of bird flocking or fish schooling [17].
The PSO method is regarded as a population-based method, where the population is referred to as a swarm [18].The swarm consists of n individuals called particles, each of which represents a candidate solution [19].Each particle i in the swarm holds the following information: • it occupies the position x i , • it moves with a velocity v i , • the best position, the one associated with the best fitness value particle has achieved so far pbesti, • the global best position, the one associated with the best fitness value found among all of the particles gbest.
In our application, the positions of particles x i represent the parameters of PID controller (K d , K i and K p ).

G(S) =
G(S) is the transfer function of the PID controller.The fitness of a particle is determined from its position.The fitness is defined in such a way that a particle closer to the solution has higher fitness value than a particle that is far away.In each iteration, velocities and positions of all particles are updated to persuade them to achieve better fitness according to the following equations: for j = 1, ..., d where d is the number of dimensions, i = 1, ..., n where n is the number of particles, t is the iteration number, w is the inertia weight, rand1 and rand2 are two random numbers uniformly distributed in the range [0,1], c 1 and c 2 the acceleration factors.
The c 1 is the cognitive acceleration constant.This component propels the particle towards the position where it had the highest fitness.The c 2 is the social acceleration constant.This component steers the particle towards the particle that currently has the highest fitness.
The inertia weight w affects the contribution of v t ij to the new velocity v (t+1) ij .If w is large, it makes a large step in one iteration (exploring the search space), while if w is small, it makes a small step in one iteration, therefore tending to stay in a local region [20].
Typically, the velocity of a particle is bounded between properly chosen limits v min < v id < v max (in most cases v min = −v max ).Likewise, the position of a particle is bounded as follows: x min < x id < x max .
Afterwards, each particle updates its personal best position using the following equation: Finally, the global best of the swarm is updated using the following equation: where f is a function that evaluates the fitness value for a given position.
The PSO process is repeated iteratively until one of the following termination criteria occurs [21] if the c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING maximum number of iterations has been reached, an acceptable solution should be found, or no improvement is observed over a number of iterations.
The PSO control parameters used in this study are; the population size is 60, maximum generation 100, the acceleration factors (c 1 = 0.5 and c 2 = 1.25) and the inertia weight (w = 0.6).It is noteworthy to mention that all the developed programs are under the commercial software Matlab-Simulink.

Simulation of the System Behaviour
The simulation of the system behaviour reveals an important problem in the control.When the training speed is variable, the voltages and the currents of the exciter and the main generator vary in order to maintain the voltage at nominal value.
The problems mentioned before can be presented as follows: In the case, where the training speed is high, the theoretical excitation voltage of the exciter and main generator that provides a nominal voltage (the voltage of the main generator) is low, due to the excitation voltage that is inversely proportional to the training speed.The difference between the voltage delivered by the PMG machine and the theoretical excitation voltage of the main generator and exciter is very important, which makes the control system, in this case, very complicated.
In the case of low speed, the voltage delivered by PMG machine can be insufficient to excite the main generator exciter group, because the excitation field of the PMG machine is constant.
Choosing PMG machine depends on specifications, i.e. expertise, knowledge and field of use.To overcome this problem (the insufficiency of PMG machine), the PMG machine with low speed (high number of poles) is used.This produces enough voltage to excite the group main generator-exciter.In order to control the output voltage of the BESG, the excitation voltage of the main exciter-generator group is calculated from the load and the speed.
The excitation voltage value corresponds to the rated voltage (220 V) which is considered as a reference of the voltage regulator of the MSAP (Fig. 5).At every moment and for a given speed, the excitation voltages of the main exciter and generator are calculated.Which will be a reference for the regulator voltage of the PMG, i.e. decrease the difference between the volt-age delivered by PMG machine and excitation voltage of the main generator-exciter group.This ensures stability when the machine works at high speed.For each variation in speed or load, our system adapts its self to these changes.

Comparison Between PSO and Ziegler Nichols
In our case, Ziegler Nichols tuning is based on the openloop step response of the system which is characterized by the following parameters; the process time constant L z , the delay time and the time constant T z .
These parameters are used to determine the controller's tuning parameters (see Tab. 1).
We note that the response of our system is a first order equation.The determination of controller parameters by the conventional methods such as Ziegler Nichols, for a given speed, is affordable.However, for the values of regulators parameters, that are adapted to the speed and load variations, are difficult to determine, and especially for the second controller, because the Ziegler Nichols method calculates the parameters of a single regulator (Fig. 5).
Thus, the parameters of the second controller are very difficult to be calculated, since it relies on the trial and error method as illustrate Fig. 5.The obtained simulation results for the PID controller using ZN method give the parameters: The parameters K p1 , K il and K d1 are determined by Ziegler Nichols method (Tab. 1) while K p2 , K i2 and K d2 are determined by trial and error.
The simulation results are illustrated in Fig. 6.They reveal the behaviour of the voltage V t of the main genc 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING erator for an inductive load of cos(ϕ) = 0.8.The response time of the main generator becomes faster and takes less than 0.2 s, and its operation is stable as shown in the zoom of Fig. 6.
In Fig. 6, the disruptions in high-speed region are due to incorrect values of the second regulator parameters.The determination of the parameters for several regulators using PSO technique is very useful.
Tab. 1: Ziegler-Nichols for open-loop tuning parameters.The PSO method is used to optimize the parameters of the regulators off-line, however, in on-line control, the regulator parameters must be calculated by PSO, when the speed vary.

Simulation Results and Discussion
In order to optimize the parameters of the PID controller, the PSO is implemented in Matlab-Simulink environment.The results obtained when the main generator running at full load with a power of 7 kVA.
The simulation results are illustrated in Fig. 8.They illustrate the behaviour of the voltage V t of the main generator of an inductive load of cos(ϕ) = 0.8.
When the training speed reaches 314 rad•s −1 for inductive load (cos(ϕ) = 0.8), the response time of the main generator becomes faster and takes less than 0.2 s, and its operation is stable as shown in zoom of Fig. 8(a).When a load is applied at the terminals of the main generator, the voltage amplitude follows the reference in both cases corresponding to the connection and the disconnection of the rated load, as shown in zoom of figure Fig. 8(b).One can note that the effect of the load variation is perfectly damped, and the system, in this case, rejects all disturbances.
In the same configuration, when the training speed reaches 3140 rad•s −1 , a voltage peak at the 15 th second is observed.When speed varies from 314 rad•s −1 to 3140 rad•s −1 , as illustrated in the zoom of Fig. 8(c), the peak duration is less than 0.2 s and the voltage is stable and has amplitude less than 20 % of the nominal one.
At high speed of 3140 rad•s −1 , illustrated in the zoom of Fig. 8(c), when a load is applied at the terminals of the main generator.In these cases of connection and disconnection, the voltage amplitude follows the reference and the variation effect of the load is perfectly damped, Fig. 9.In high-speed case, the amplitude of the voltage peak, at 20 th and 25 th seconds, is more important but acceptable (in the norms; less than 20 %) and the stability time is also acceptable that justify the robustness of the control technique.
Table 2 shows the PSO parameters that are used for verifying the performance of the PSO-PID controller for searching the PID controller parameters.Inertia weight factor is set by: where ω max = 0.9, ω min = 0.4, iter max is the maximum number of iterations, and iter is the current number of iterations.For these settings, the simulation result that shows the best solution corresponds to the PID parameters given:

Robustness Study
The robustness control is studded through parameters variation.It is justified when the machine operates at high speed due to the Foucault courant and the temperature effects which are considerable important.
In order to illustrate the robustness of the proposed control, the effect of the variation of main generator parameters on the performances of the voltage setting is studied.
The variation introduced in the test refers in practice to the real condition as overheating, saturation of the magnetic circuit and the operating conditions such as the variation of the speed and the load.
Three cases are considered: • Variation in the stator resistance of 50 %.
• Variation in the rotor resistance of 50 %.
• Starting with low speed of 100 rad•s −1 .
To illustrate the performances of setting, the variation of stator and rotor resistance of 50 % compared to the rated values, with a voltage step of 220 V has been simulated.Figure 10 and Fig. 11 illustrate the test results which illustrate that the proposed technique is insensitive to the machine rotor or stator resistances variations.Hence, with even poorly determined parameters, the adopted control technique withstands and stays stable.
For the third simulated case, at a low speed (100 rad•s −1 ) with application of a rated load, the system is always insensitive to this speed, as shown in Fig. 12.The control system ensures the disturbances rejection in three cases considered (the time constants of the system are greater than those of the machine).The proposed control technique presented in this paper is robust, simple and has the advantage of being easily implemented.

Conclusion
To resolve the problem of brushless synchronous generator machine control, a new strategy is proposed.A simulation of direct current machine is used, which allows to understand well the difficulties of the control.For this, an optimization of the voltage amplitude control of the BESG by using PSO has been realized.
The results show that by the control of the voltage excitation of the exciter, which is the only physical parameter available, the voltage of the main generator is maintained at its reference value.This technique of control offers stability to the total voltage of the generator and rejects all disturbances regardless of the variation of the rotation speed or load.
The parameters optimization of the regulator using PSO is considered important for the control of complex systems, in our case the three-cascade synchronous machine.The result of the voltage amplitude control is considered good and better compared to the conventional control techniques.

Fig. 2 :
Fig.2: Block of a main generator supplying an inductive load.

Fig. 4 :
Fig. 4: Block diagram of permanent magnet alternator supply the inductive load.
Vt Voltage variation of the main generator with inductive load: cos(ϕ) = 0.8.
Zoomed between 14 and 25 s.