Optimal Tuning of PI Controller Parameters for Three- Phase AC-DC-AC Converter Based on Particle Swarm Algorithm

Transcription

1 Minia University From the SelectedWorks of Dr. del. Elbaset Winter December 15, 2015 Optimal Tuning of PI ontroller Parameters for Three- Phase -D- onverter ased on Particle Swarm lgorithm Dr. del. Elbaset This work is licensed under a reative ommons _Y International License. vailable at:

2 17th International Middle East Power Systems onference, Mansoura University, Egypt, December 15-17, 2015 Optimal Tuning of PI ontroller Parameters for Three- Phase -D- onverter ased on Particle Swarm lgorithm Y. S. Mohamed. M. El-Sawy del. Elbaset M. M. Ismail Electrical Engineering Department, Faculty of Engineering, Minia University, El-Minia 61517, Egypt bstract Usually proportional-integral (PI) controller tuning rules are derived based on linear models of the plant or process, without taking nonlinearities into account. Hence, the best control performance cannot be ensured. This paper presents optimal tuning method based on more accurate nonlinear models of the three-phase voltage regulator converter. This nonlinear model is considered as objective function with input arguments of proportional gain and integral gain, besides its output is the integral of time multiply squared error (ITSE) of PI controller. The measure of ITSE error is used to depict the system performance i.e. the fitness function. Using optimization method of particle swarm optimization (PSO), the performance index of ITSE is minimized to zero by finding the optimal PI tuning. The fitness function evaluates the performance of a particle to determine whether the best fitting solution is achieved. Simulation results are provided based on MTL Simulink. Index Terms PI tuning, optimization technique, -D- converter, voltage regulation, ITSE criteria. I. INTRODUTION The widely popular use of proportional-integral and proportional integral-derivative (PID) controllers in process control, motor drives, flight control, and instrumentation. The reason of this acceptability is for its simple structure which can be easily understood and implemented. Most PI tuning rules are based on the assumption that the plant can be approximated by a first-order plus time delay system. Therefore, the best control performance cannot be ensured. Using modern optimization techniques, it is possible to tune a PI controller based on the actual transfer function of the plant to optimize the closed-loop performances[1]. Several approaches have been documented in literatures for determining the PI parameters of such controllers which is first found by Ziegler- Nichols tuning[2, 3]. Ziegler-Nichols presented a tuning formula based on time response and experiences[2]. lthough it lacks selection of parameters and has an excessive overshoot in time response, still opens the way of tuning parameters. Modified Ziegler-Nichols tuning based on hien-hrones-reswick PID tuning formula[3] for set-point regulation accommodate the response speed and overshoot. Genetic lgorithm, fuzzy based approach[4-6] are just a few among these numerous works. Ref. [4] presents the difference of conventional PID controller and the parameters selftuning fuzzy PID for D motor control. In Ref. [5], the integral square error and integral absolute error are taken as performance index of fuzzy logic based self-tuning PI controller for a liquid level process. Ref. [6] presents comparison studies of conventional and evolutionary tuning methods for a PI and fuzzy PI controller. Genetic algorithms (G) are employed as the evolutionary tuning method. These PI-controller tuning methods are based on novel closed loop performance statements and the closed loop stability. The design of PI controller based on PSO algorithm has been addressed in many literatures[7-9]. In Ref. [7], presents a novel design method for determining the optimal PID controller parameters of an automatic voltage regulator (VR) system using the PSO algorithm. Ref. [8] presents a method to determine the optimal tuning of the PI controller parameter on Direct current (D) motor drive system using PSO algorithm, Ziegler-Nichols tuning and Modified Ziegler Nichols tuning method. The main objective of [8] is to minimize transient response specifications chosen as rise time, settling time and overshoot, for better speed response of D motor drive. In Ref. [9], a self-tuning PI controller in which the controller gains are adapted using the PSO technique for a static synchronous compensator. n efficient formula for the estimation of system load impedance using real-time measurements is derived. There are several applications of the -D- converter in electric machine drives. The usages of -D- converter have been addressed in many literatures[10, 11]. In Ref. [10], the converter was implemented to the volt/frequency control of the three-phase induction motor drives. PI controlled voltage regulators are employed for controlling the source current and the D-link voltage. In Ref. [11], simple -D- converter and proposed modular control strategy are implemented for grid-connected wind power generation system. Line side inverter regulates the dc-link voltage constant and the power factor of line side can be adjusted. Input current reference of boost chopper is decided for the maximum power point tracking of the turbine. This paper demonstrates an efficient method of tuning the PI controller parameters using the optimization rule for ITSE performance criteria. The method implies an optimization method based on PSO algorithm to calculate the optimal values for the proportional (K p ) and integral gain (K i ) of the PI

3 17th International Middle East Power Systems onference, Mansoura University, Egypt, December 15-17, 2015 controlled regulator which can achieve minimum integral of time multiply squared error performance index. II. SYSTEM MODELING The simplest schematic diagram of an -D- converter topology converting from a three-phase supply to a threephase variable-voltage and variable-frequency system is presented in Fig. 1. s shown from this figure, the power circuit of the converter and the regulation technique is presented. The nonlinearity behaviour of each component in the power circuit is considered and implemented in time domain depending on the MTL Simulink s power system library[12]. 50 Hz, three-phase voltage source feeds a threephase load through an -D- PWM inverter. The 400V, 50 Hz voltage is first rectified by a six diode bridge. The filtered D voltage is applied to an IGT two-level inverter generating desired voltage with wanted frequency. The IGT inverter uses sinusoidal pulse width modulation at a 2 khz carrier frequency and a peak is 1 p.u.. The load voltage is regulated by a PI voltage regulator using abc to dq and dq to abc transformations which connected to the vectorized signal containing the [sin(2πft) cos(2πft)] values, where f is reference frequency of output wave fundamental component which determines the rotation speed of the reference frame[13]. The first output of the voltage regulator is a vector containing the three modulating signals used by the sinusoidal PWM Generator to generate the six IGT pulses[14]. The second output returns the modulation index. The d-q components of voltage error, which is the difference of reference voltage and actual voltage, is given as input to the PI controller. The voltage controller process the voltage error and give the d-q components of modulating signals as an input to the sinusoidal PWM based on the following equation: Three Phase Power Supply Three Phase Diode Rectifier - Filter PWM IGT Inverter Gate Three Phase pulses L Filter Load -, ( ) =, ( ), ( ) (1) error d,q (t) is a vector of the d-q components s error in load voltage at time t. u d,q (t) is the d-q components of modulating signals at time t. Maximum regulated load voltage can be calculated from the following equation[14]: = (2) V L-L : mplitude of the fundamental ac output line voltage. V dc : Value of D link voltage. m a : Modulation index and is defined as follow: = (3) v c : mplitude of the desired ac output voltage (modulating signal) in p.u.. v : mplitude of the triangular waveform (carrier signal) which equals to 1p.u.. The value of modulation index changes in interval of 0 m a 1 to avoid generation of low order harmonics[14]. Therefore maximum value for the amplitude of the fundamental ac output line voltage is III. METHODOLOGY ND PSO LGORITHM In this work, the objective function to achieve the optimal values of K p and K i of the PI controller which lead to the minimum ITSE can be calculated as follow[1]: = (4) = (5) = (6) Where error d and error q are the error in load voltage of d- component and q-component respectively. Figure 2 shows schematic diagram of ITSE s calculating. Modulation index, (m) Sinusoidal PWM pulses abc abc dq0 PI sin-cos ased on - PSO Desired Frequency (HZ) 50 Digital Time Source t (Vd^2Vq^2)^0.5 V0 (p.u.) 0 Error d, Error q Desired Vd* (p.u.) 1 sin(2π f t) cos(2π f t) alculate ITSE ITSE volt abc dq0 sin-cos Vq* 0 Fig. 1 Schematic diagram of the PI controlled -D- converter. p.u. Error_d Digital Time Source t Error_q square square ITSE Fig. 2 Schematic diagram of ITSE s calculating The objective function to minimize ITSE can be determined as follows: (, ) (7)

4 17th International Middle East Power Systems onference, Mansoura University, Egypt, December 15-17, 2015 Subject to 0 K p 500, 0 K i 500 (8) PSO algorithm used to minimize. The PSO algorithm begins by creating the initial particles, and assigning them initial velocities. It evaluates the objective function at each particle location, and determines the best (lowest) function value and the best location. It chooses new velocities, based on the current velocity, individual best locations for the particles, and the best locations of their neighbors. It then iteratively updates the particle locations (the new location is the old one plus the velocity, modified to keep particles within bounds), velocities, and neighbors. Iterations proceed until the algorithm reaches a stopping criterion. For a dimensional search space, the position of particle p is given by the following vector: Z p = (z p1, z p2,, z pn ) (9) The velocity of particle p is given by the following vector: V p = (v p1, v p2,, v pn ) (10) n :Number of the swarm particles. V p :Velocity vector of particles. Z p :Position vector of particles. The global best position is the best position ever found by the swarm. In a minimization problem that is described by an objective function obj, the global best is given by[15]: ( ), ( ),, ( ) ( ) = ( ), ( ),, ( ) (11) g P : Personal best position ever found by a particle p. g p,j (t i ): Personal best position of particle p at the time t i and it is called (Pbest). : Global position in which the objective function obj achieves its minimum value. ( ): Global best position ever found by the swarm at time t i and it is called (gbest). The velocity update of particle p is given by:, ( 1) =, ( ), ( ), ( ), ( ) where:, ( ) ( ), ( ) (12), ( ): The position of the particle j at time t i., ( ): The velocity of the particle j at time t i., ( 1): The new velocity of the particle j at time t i 1. r 1, r 2 : Independent random numbers from the interval [0, 1]. c 1, c 2 : cceleration coefficients and equals to 2. Each particle p, because of its velocity, obtains a new position. The new position of the particle depends on its previous position and its velocity: ( 1) = ( ) ( 1) (13) ( 1): The position of the particle j at time t i 1. In the global best model, all the particles of the swarm are defined as a neighbour of a particle p. Each particle can be influenced by any of the other particles. The presentation of the swarm in this case can be made by a star topology. The equations for ( ) and V p,j ( ) of gbest model are given by Eq. (11) and (12) respectively. The PSO algorithm with inertia weight is given by[16]:, ( 1) =, ( ), ( ), ( ), ( ), ( ), ( ), ( ) (14) = Iter: Iteration number. ; Iter max : Maximum iteration number. ω max : Initial weights. ω min : Final weights. It has been demonstrated that 0.9 for ω max and 0.4 for ω min can greatly improve the performance of PSO [16]. This algorithm run the Simulink model in each time of evaluation the objective function and the simulation time equals the transient time of the system which equal to 0.1 second. The steps of the gbest PSO algorithm are applied as the following: Step 1. Initialization. Step 2. Velocity and position update. Step 3. Update of the pbest and gbest. Step 4. Go to Step 2 until satisfying stopping criteria which is Relative change in the best objective function value gbest over the last 20 iterations is less than 1E-99. The detailed implementation strategies of the gbest PSO algorithm are described and the flow chart is presented in algorithm No.1. IV. DISUSSION ND RESULTS The nonlinearity behaviour of each component in the threephase voltage regulator s power circuit is considered and implemented in time domain depending on the MTL Simulink s power system library and has been taken as the objective function of Eq.(7)[12]. Designed computer programs based on MTL programing have been prepared to find optimal tuning of PI controller s parameters of three-phase voltage regulator converter based on PSO algorithm. onstant values of reference load voltage and reference load frequency was set. Optimum value of PI controller s parameters can be found by solving the objective function as shown in Fig. 3 based on PSO algorithm. From this figure, it can be seen that the optimal tuning can be obtained closed to 48 iterations to achieve the minimum value for integral of time multiply squared error (ITSE) of E-6.

5 th International Middle East Power Systems onference, Mansoura University, Egypt, December 15-17, 2015 lgorithm No.1 Load Simulink model, maximum iterations (Iter max )=500 and Population size (j) =50. Initial each particle (K p, K i ) of population for objective function of Eq. (7). Initial the personal best position (Pbest) for each particle of population. Evaluate the objective function of Eq. (7) (ITSE) for each particle of population. Set iteration counter Iter max =1. While (stopping criteria does not achieved or Iter max 500) Do Determine the global best position ever found by the swarm (gbest), based on Eq. (11). Set iteration counter j=1. While (j 50) Do Determine the velocity for each particle of population, based on Eq. (12). Determine the new position of particle, based on Eq. (13). IF (new position in agreement with the onstraints of Eq.(8)) Re-evaluate the objectivee function of Eq.(7) (ITSE), for each particle s new position. Update the particle position. IF (evaluation of new position < Pbest) Update best position (Pbest). END IF Set j=j1 END IF End While Set Iter max = Iter max 1 End While Determine the global best particle, based on Eq.(11) Fig. 3 objective function evaluation It can be found that the corresponding optimal values for PI controller s parameters are and for Kp and Ki respectively. Figures 4-6 show the simulation results of the PI controlled voltage regulator which has been operated depending on the optimal tuning, that was obtained from the previous optimization problem. These figures present the dc-link voltage in p.u. (V dc ), modulation index (m a ), phase value of three-phase filtered load voltage in p.u. and load voltage response in p.u. waveforms versus time in second, respectively in different cases of reference values for the phase to neutral load voltage (root mean square value) and load voltage s frequency. Firstly, Fig. 4 shows the simulation results in case of 1 p.u. voltage reference and 50 Hz frequency reference, which the voltage s base value is 110 volt. From this figure, it can be illustrated that the dc link voltage reach its steady state value of 4.6 p.u. at short time of second. In addition to that, the modulation index was being changed continuously from the beginning to regulate the load voltage. lso load s frequency is derived to achieve the set-point. Therefore, the waveform of load voltage was fluctuated five times in each period of 0.1 second. The system managed to achieve the desired set-point in relatively short settling time of second with an overshoot that didn t exceed 11.2%. The steady state error of a system response is e-4. Secondly, Fig. 5 shows the simulation results in case of voltage reference step was applied, the reference voltage is changed from 1 to 0.7 p.u. at 0.1 sec., and constant frequency reference of 50 Hz. From this figure, the system managed to achieve the desired set-point in relatively short settling time of second without overshoot. lso, the steady state error of a system response is e-04. In addition to that, the modulation index was being changed continuously to regulate the load voltage. Finally, Fig. 6 shows the simulation results in case of frequency reference step was applied, the frequency reference is changed from 50 to 100 Hz at 0.1 sec., and constant reference value of load voltage of 1 p.u. From this figure, the system managed to achieve the desired set-point of reference frequency in short time. Therefore, e, the wave of load voltage was fluctuated five times in first period of 0.1 second and ten times in the second period of 0.1 second. V. ONLUSION In this paper, the tuning parameters (K P, K i ) of PI controller for a three-phase voltage regulator is designed using PSO algorithm. The PSO algorithm and ITSE as fitness function is integrated to find the optimal tuning based on the nonlinearity behaviour of each component in the system without depending on the analytical method for linearizing the system which cannot ensure the best control performance. The ITSE as performance index is minimized nearest to zero by finding the optimal values for the PI controller s parameters. Simulation results for this optimized controller show that in different references, the load voltage and frequency can follow the reference values in short time. This reflects the validity and effectiveness of the particle swarm as an optimization technique.

6 17th International Middle East Power Systems onference, Mansoura University, Egypt, December 15-17, 2015 Fig. 4 Simulation results of converter during voltage refrence 1 p.u. and frequency reference 50Hz. Fig. 5 Simulation results of converter during the change of voltage refrence from 1 p.u to 0.7 p.u.. at 0.1 sec. and frequency reference 50Hz.

7 17th International Middle East Power Systems onference, Mansoura University, Egypt, December 15-17, 2015 Fig. 6 Simulation results of the converter during the change of frequency refrence from 50 Hz to 1000 Hz at 0.1 sec. and voltage reference 1 p.u.. REFERENES [1]. O'Dwyer, Handbook of PI and PID controller tuning rules, World Scientific, [2] J.. asilio, S. R. Matos, Design of PI and PID controllers with transient performance specification, IEEE Transactions on Education, Vol. 45, No.4, November (2002), pp [3] D. Xue, Y. hen, D. P. therton, Linear Feedback control, Society of Industrial and pplied Mathematics, ook, ISN: , 2007, ch. 6. [4] Y. Ma, Y. Liu,. Wang, Design of parameters selftuning fuzzy PID control for D motor, 2 nd International onference on Industrial Mechatronics and utomation, 2010, pp [5] G. Sivagurunathan, K. Jayanthi, Fuzzy logic based self tuning of PI controller for a non linear spherical tank system, IEEE International onference on omputational Intelligence & omputing Research, 2012, pp [6] Pei-Jin Wang, PI-tuning methods based on G, International onference on Machine Learning and ybernetics, IEEE, 2002, pp [7] Z. L. Gaing, particle swarm optimization approach for optimum design of PID controller in VR system, IEEE Transactions on Energy onversion, Vol. 19, No. 2, June 2004, pp [8] R.G. Kanojiya, P. Meshram, Optimal tuning of PI controller for speed control of D motor drive using particle swarm optimization, International onference on dvances in Power onversion and Energy Technologies, IEEE, 2012, pp [9] hine-hung Liu, Yuan-Yih Hsu, Design of a self-tuning PI controller for a STTOM using particle swarm optimization, IEEE Transactions on Industrial Electronics, Vol. 57, Feb. 2010, pp [10] Dong-hoon Lee, Young-Sinn Kim, ontrol of singlephase-to-three-phase /D/ / PWM converters for induction motor drives, IEEE Transactions on Industrial Electronics, Vol. 54, pril 2007, pp [11] Seung-Ho Song, Shine-il Kang, Nyeon-kun Hahm, Implementation and control of grid connected -D- speed wind energy power converter for variable conversion system, pplied Power Electronics onference and Exposition, Eighteenth nnual IEEE, 9-13 Feb. 2003, pp [12] Power System lockset For Use with Simulink, User s Guide, Version 2, The Mathworks Inc., [13] Ewald F. Fuchs, Mohammad. S. Masoum, Power quality in power systems and electrical machines, ook, Elsevier Inc., ISN: , [14] M. H. Rashid, Power electronics handbook: devices, circuits and applications, ook, cademic press, ISN: ,2001. [15] J. Kennedy, Eberhart, Particle swarm optimization, Proceedings on IEEE International onference on Neural Networks, 1995, vol.1944, pp [16] ergl, F., n analysis of Particle swarm optimizers, PhD thesis, Faculty of natural and gricultural science, University of Pretoria, Pretoria, 2001.