AP06_2.vp 1 Introduction Studies of self-excited induction generators have been in- vestigated since 1935. Many papers dealing with various problems in the field of SEIG have been published. The pri- mary advantages of SEIG are lower maintenance costs, better transient performance, lack of a dc power supply for field excitation, brushless construction (squirrel-cage rotor), etc. In addition, induction generators have been widely employed to operate as wind-turbine generators and small hydroelectric generators of isolated power systems. Induction generators can also be connected to large power systems, to inject electric power [1, 2]. The generator action takes place, when the rotor speed of the induction generator is greater than the synchronous speed of the air-gap-revolving field. Various configurations for connecting SEIG to a large power system have been dis- cussed in many publications. This research concentrates on the dynamic performance of an isolated SEIG, driven by wind energy, to supply an isolated static load. A D-Q axis equivalent circuit model based on various reference frames, extracted from fundamental machine theory, is created to study SEIG performance in dynamic case [2]. This paper studies SEIG performance, when equipped with a switching capacitor bank, using a controller based on GAs to adjust the duty cycle and adjusting the stator frequency via the pitch control. Genetic algorithms are search algorithms that simulate the process of natural selection and survival of the fittest [3]. The GA starts off with a population of randomly generated chromosomes, and advances toward better chromosomes in a sequence of generations. During each generation, the fit- ness of each solution is evaluated and solutions are selected for reproduction based on their fitness. Then, the chromo- somes with higher fitness have higher probabilities of having more copies in the following generation, while the chromo- somes with worst fitness are eliminated. Then a roulette wheel cheme is applied for reproduction. Consequently, the new population of chromosomes is formed using a selection mechanism and specific genetic operators such as crossover and mutation. Recently, genetic algorithms have received considerable attention. Global optimization utilizes tech- niques that can distinguish between the global optimum and numerous local optima within a region of interest. Many papers have been published using GA to optimize PI and PID controllers [4]. In this paper, the mathematical model of SEIG driven by WECS is simulated using the MATLAB/SIMULINK package to solve its differential equations. Two controllers have been developed for the system under study. The first of these is the reactive controller to adjust the terminal voltage at the rated value, by controlling the duty cycle of the switching capacitor bank. The second controller is the active controller, which adjusts the input mechanical power to the generator and thus keeps the stator frequency constant. This is achieved by controlling the pitch angle of the blade of the wind turbine. Both controllers are implemented using the conventional PI controller. Then, the integral gains of both PI controllers are optimized using the GA technique. Fig. 1 shows the block diagram for the system under study. It consists of a self-excited induction generator driven by a wind energy conversion scheme connected to an isolated load. In addition, the system under study is equipped with the reactive and active controller’s loops, using a PI controller. Then, the inte- gral gains are tuned using the GA algorithm. 2 Mathematical model for SEIG driven by WECS 2.1 Electrical equations for SEIG Fig. 2 shows the d-q axis equivalent-circuit model for a no-load, three-phase symmetrical induction generator. The stator and rotor voltage equations using Krause transforma- tion [1, 2], based on a stationary reference frame, are given in Appendix A. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 11 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 2/2006 Genetic Algorithm Based Control System Design of a Self-Excited Induction Generator A.-F. Attia, H. Soliman, M. Sabry This paper presents an application of the genetic algorithm (GA) for optimizing controller gains of the Self-Excited Induction Generator (SEIG) driven by the Wind Energy Conversion Scheme (WECS). The proposed genetic algorithm is introduced to adapt the integral gains of the conventional controllers of the active and reactive control loop of the system under study, where GA calculates the optimum value for the gains of the variables based on the best dynamic performance and a domain search of the integral gains. The proposed genetic algorithm is used to regulate the terminal voltage or reactive power control, by adjusting the self excitation, and to control the mechanical input power or active power control by adapting the blade angle of WECS, in order to adjust the stator frequency. The GA is used for optimizing these gains, for an active and reactive power loop, by solving the related optimization problem. The simulation results show a better dynamic performance using the GA than using the conventional PI controller for active and reactive control. Keywords: genetic algorithms, conventional controllers, self-excited induction generator. 2.2 Mechanical equations for WECS The mechanical equations relating the power coefficient of the wind turbine, tip speed ratio � and pitch angle � are given in Appendix A [5]. The analysis of SEIG in this paper is based on the following assumptions [1]: � All parameters of the machine can be considered constant except Xm � Per-unit values of both stator and rotor leakage reactance are equal � Core loss in the excitation branch is neglected � Space and time harmonic effects are ignored. 2.3 Equivalent circuit The d-q axsis equivalent-circuit models for a no-load, three-phase symmetrical induction generator are shown in Fig. 2a and Fig. 2b. The equivalent-circuit parameters shown in these figures are based on the machine data in Appendix B [1, 2]. The equation of motion of the rotating part of the com- bined studied SEIG and the wind turbine is also included in the system in order to provide a detailed simulation model. 2.4 Reactive control and switching capacitor bank technique 2.4.1 The switching capacitor bank Capacitor switching has been discarded in the past be- cause of the practical difficulties involved [6], i.e. the occur- rence of voltage and current transients. It has been argued, and justly so, that current ‘spikes’, for example, would inevita- bly exceed the maximum current rating as well as the ( di /dt ) 12 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 2/2006 Czech Technical University in Prague Load Semi conductor switching KIV tuned using GA PI for Reactive Control KIF tuned using GA Pitch control PI for Active Control Bus_Bar WECS_Input Mech. Power + Pm_ Ref. - P m _ A c tu a l + V Ref. duty cycle Pm_error V_error V -t e rm in a l SEIG SEIG Fig. 1: System under study a) Rs �ds Lls Llr ( Ws �Wr) (� dr / Wb) Rr Vqs Iqs Xm Iqr Vqr b) Vds Ids Xm I dr Vdr Rs �qs Lls Llr R r( Ws �Wr) (� qr / Wb) Fig. 2: a) Equivalent circuit of an induction generator for quadrate axis, b) Equivalent circuit of an induction generator for direct axis value of a particular semiconductor switch. The only way out of this dilemma would be to design the semiconductor switch to withstand the transient value at the switching instant. An equivalent circuit of the switching capacitor bank with a controlled value of the duty cycle is shown in Fig. 3. In this figure, the switches are operated in anti-phase, i.e. the switch- ing function fs2 which controls switch S2 is the inverse function of fs1 which controls switch S1. In other words, switch S2 is closed during the time when switch S1 is open, and vice versa. This means that S1 and S2 of branch 1 and 2 are operated in such a manner that one switch is closed while the other is open. 2.4.2 Reactive control through the switching capacitor bank technique In the system under study given in Fig. 1, the controller in- put is the voltage error for the reactive power controller, and the output of the controller performs the value of the duty cy- cle �. The duty cycle is used as an input to the semiconductor switches to adjust the capacitor bank Ceff value according to the need for the effective value of the excitation, which regu- lates the terminal voltage. Accordingly the semiconductor switching technique as explained in the above section and, hence, the terminal voltage is controlled by adjusting the self-excitation through automatic switching of the capacitor bank. 2.5 The active power control Active control is applied to the system under study by ad- justing the pitch angle of the wind turbine blades. This is used to keep the SEIG operating at a constant stator frequency and to avoid the effect of the disturbance. The pitch angle is a function of the power coefficient Cp of the wind turbine WECS. The value of Cp is calculated using the pitch angle according to Eq. (14), given in Appendix A. Consequently, the best adjustment for the value of the pitch angle improves the mechanical power regulation, which achieves better adap- tation for the frequency of all systems. Accordingly, the active power control regulates the mechanical power of the wind turbine. 3 Proportional plus integral (PI) controllers First, the PI controller using a fixed gain is applied to the system under study. Then, the integral gain KI of the PI controller is varied linearly with reference to terminal voltage error eV, while proportional gain KP is fixed. The voltage or frequency error is used as an input variable to the PI control- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 13 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 2/2006 V C1 C2 S1 S2 L1 fs1 fs2 Fig 3: Semi conductor switches (S1, S2) circuit for capacitor bank Fig. 4: Dynamic response of the terminal voltage with different values of integral gain for reactive control ler, then the output is used to regulate the duty cycle of the switching capacitor bank in the reactive controller, while the output of the active controller the output is utilized to tune up the pitch angle of the wind turbine to adjust the system frequency. Fig. 4 shows the simulation results for the system under study when starting against a step change in the reference voltage. Then the system is subjected to a sudden change in the local electric load. This simulation result is carried out for different values of fixed integral gain. Figs. 5, 6 and 7 show the simulation results for the stator frequency, the duty cycle and the stator current for the previous condition indicated in Fig. 4. These simulation results show that the dynamic perfor- mance is changed, as regards percentage over shoot (p.o.s), rising time and oscillation, by changing the value of the inte- gral gain. Thus, the idea of driving the system using a variable integral gain is introduced to achieve the benefits of using high and small integral gains, as follows. From Fig. 4, a higher value of the integral gain, KIV � 0.007, is associated with a shorter rising time but the 14 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 2/2006 Czech Technical University in Prague Fig. 5: Dynamic response of the stator frequency Fig. 6: Dynamic response of the duty cycle p.o.s and oscillation are greater than the other values of KIV. Meanwhile, the lowest value of KIV � 0.0051 is associated with negligible p.o.s and oscillation and the rising time is greater than other values of KIV. Fig. 8 show the variation of the inte- gral gain during the starting period versus time. 3.1 PI-Controller with a variable gain A program has been developed to compute the value of the variable integral gain KIV using the following rule: if e e K K elseif e e K K V V IV IV V V IV IV ( ), ; ( ), min min max ma � � � � x min max max min max m ; ( ), ( ) ( else e e e M K K e e V V V IV IV V V � � � � � in min min ); ; ( ) ; C K M e K M e C en d IV V IV V � � � � � � where eV is the voltage error, eVmin and eVmax are the mini- mum and maximum values of the voltage error, respectively, KIVmin and KIVmax are the minimum and maximum values of the variable integral gain, respectively, C is a constant and M is the slop constant of the linear part. Fig. 9 shows the above rule used to calculate the variable KIV . The value of eVmin and eVmax is obtained by trial and error to give the best dynamic performance. Figs. 4, 5, 6 and 7 also show the dynamic performance of the overall system when equipped with this variable integral gain compared with other fixed integral gains. The simula- tion results depict an improvement in the dynamic perfor- mance when using the variable KIV. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 15 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 2/2006 Fig. 7: Dynamic response of the load current ×10 �3 Fig. 8: Variable Integral gain for PI controller Lower Limit Upper Limit Voltage Error = eV KIV eV eVmin eVmax KIVmax KIVmin In te g ra l g a in Fig. 9: Variable integral gain for a PI controller 4 Genetic algorithm with a constrained search space for optimizing PI controller gains The integral gain of the second PI controller is optimized based on the Genetic Algorithm. GA is used to calculate the optimum value of the variables based on the best dynamic performance and a domain search of the variable. The objec- tive function used in the GA technique is F J� �1 1( ), where J is the minimum cost function, which will be defined later. GA uses its operators and functions to find the values of KIV and KIF of the PI controllers to achieve better dynamic perfor- mance of the overall system. These values of gains lead to the optimum value of gains for which the system achieves the desired values by improving the P. O. S, rising time and os- cillations. The main aspects of the proposed GA approach for optimizing the gains of PI controllers, and the flowchart pro- cedure for the GA optimization process, are shown in Fig. 10. 4.1 Representation of PI controller gains The PI gains are formulated using the GA approach, where all the gains are represented in a chromosome. The chromosome representation determines the GA structure. The parameter gains are encoded in a chromosome. The PI controller gains are initially started using minimum values of the domain search for PI gains. Based on the simulation 16 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 2/2006 Czech Technical University in Prague Reproduction Start Initialization J > Tolerance Mutation Crossover Yes No Randomly generate chromosomes of Calculate f 1/(1+ )� J For generation = 1: max_ gen For k = 1: pop_size [KIV , KIF] Record required information Stop PI ACTIVE CONTROLLERS Pitch angle valueKIF PI REACTIVE CONTROLLERS Duty cycleev, ev KIV Pme, Pme Fig. 10: Flowchart of GA approach for optimizing PIC gains results given in the previous sections, the values of [KIVmax, KIVmin] are shown in Fig. 4, while the values of [KIFmax, KIFmin] are shown in Fig. 11. The acceptable domain search for each gain is defined as [KIVmax, KIVmin] and [KIFmax, KIFmin] based on five times less and five times more than the gains obtained using the Ziegler-Nichols rule to satisfy mini- mum cost function J, as given in the following equation [7]: �J e t e t e t t T � � � � �� � � �1 1 1 0 ( ) ( ) ( ) d , (1) where; e(t) is equal to eV or Pme; eV is the voltage error used for the reactive power control and Pme is the mechanical power error used for active power control, as shown in Fig. 1. The parameters �1, �1 and �1 are weighting coefficients. 4.2 Coding of PI controller gains The coded parameters are arranged on the basis of their constraints, as shown in Fig. 12, to form a chromosome of the population. The binary representation, given in Fig. 12, is the coded form for parameters with chromosome length equal to the sum of the bits for all parameters. In binary coding, the relation between the bit length Li and the corresponding bit resolution Ri is given in the following equation [8]: R UB LB i i i L i � � �2 1 , (2) where UBi and LBi are the upper and lower bounds of parameter i, respectively. In the present case study, we as- sume bit resolution R � 105 for all parameters. Fig. 12 shows the coded parameters of the PI controller gains for reactive and active power controllers, respectively. The chromosome length used in this paper was 20 bits, where the bit length of KIV equal 10 bits and the bit length of KIF equals 10 bits. 4.3 Selection function The selection strategy decides how to select individuals to be parents for new ‘children’. The selection usually applies some selection pressure by favoring individuals with better fitness. After procreation, the suitable population consists, for example, of L chromosomes, which are all initially random- ized. Each chromosome has been evaluated and associated with fitness, and the current population undergoes the repro- duction process to create the next population. Then, the “roulette wheel” selection scheme is used to determine the member of the new population. The roulette scheme is shown in Fig. 13. The chance on the roulette-wheel is adaptive and is given as P P � �� as in the Eq. (3) [8]: P J L � � � �� � 1 1, { , , }, (3) and J � is the performance of the model encoded in the chromosome measured in the terms used in Eq. (1). © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 17 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 2/2006 Fig. 11: Stator frequency for PI and PI & GA controllers KIV KIF 10 bits 10 bits Chromosome Length 1 1 … 0 1 0 1 … 1 1 Fig. 12: Coded parameters of PI controller gains Maximizing the fitness function of each chromosome, which is inversely proportional to the performance criteria, Eq. (1) will damp the overshoot or the oscillations [8]. 4.4 Crossover and mutation operators The mating pool is formed, and crossover is applied. Then the mutation operation is applied, followed by the pro- posed GA approach. Finally, after these three operations, the overall fitness of the population is improved. The procedure is repeated until the termination condition is reached. The termination condition is the maximum allowable number of generations, or a certain value of J. This procedure is shown in the flowchart given in Fig. 10. 5 Simulation results The nonlinear differential equation, which describes the system under study, was solved using the Runge-Kutta fifth order method, using the MATLAB Simulink package. The integration step value was automatically varied in this pack- age. The relative tolerance was set at 0.00001. The minimum and maximum step size was adjusted automatically. Several tests were carried out to validate the efficiency of the pro- posed control schemes. The simulations show the comparison between the two proposed PI controllers. The system performance checks the terminal voltage VL, the stator frequency Fs, the load current IL and the duty cycle versus time. The overall system is tested against a sudden change in the load. This disturbance is made by applying a sudden change in the resistance part of the load impedance. The load is equivalent to the R-L series circuit with load resistance RL � 80 ohm and load inductance LL � 0.12 H per phase. 6 PI controller based on GA 6.1 Dynamic performance due to sudden load variation Figs. 14, 15, 16 and 11 show the simulation results of the system under study using the PI controller based on GA for the terminal voltage, the stator current, the duty cycle and the stator frequency, respectively. The simulation results show that the performance of a PI controller based on GA is much better, as regards maximum overshoot and rising time, than a PI controller with fixed or variable KIV. The simulation results show the effectiveness of the proposed controller, as shown in the previous figures. 6.2 Simulation results due to sudden wind speed variation Other simulation results are obtained when the overall system is subjected to a sudden variation in wind speed from 18 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 2/2006 Czech Technical University in Prague P 2 P 1 P L Fig. 13: Roulette wheel selection scheme Fig. 14: Terminal voltage for PI and PI & GA controllers 7 m/s to 15 m/s. Figs. 17, 18 show the simulation results of the wind speed variation and the stator frequency, respectively. The simulation result given in Fig. 18 shows the ability of the proposed controller to overcome the speed variation when using the variable and fixed integral gain. 7 Conclusions This paper presents the application of two types of con- troller to enhance the performance of SEIG driven by WECS, using a variable rule based integral gain and GA. GA is used © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 19 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 2/2006 Fig. 15: Load current for PI and PI & GA controllers Fig. 16: Duty cycle for PI and PI & GA controllers to optimize the PI controller gains in order to improve the dynamic response of the overall system. Optimal gains for the PI controller were determined using the GA procedure. The simulation results show that a PI controller tuned by the proposed GA was able to decrease the overshoot, and at same time to decrease the rising time. The simulation results show the effectiveness of the GA approach as a promising identifi- cation technique for PI controller gains. The two different 20 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 2/2006 Czech Technical University in Prague Fig. 17: Sudden variation for wind speed versus time Fig. 18: Stator frequency according to wind speed variation for SEIG controlled by PI-GA types of controllers improve the dynamic performance when tested against warious types of disturbances. Appendix A – SEIG differential equation Vds (volt) Stator Voltage‘s Differential Equation at Direct Axis V R i pds s ds b qs ds b � � � � � � � � � � � � � � � � � � � � � � � � � � (4) Vqs Stator Voltage‘s Differential Equation at Quadrate Axis V R i pqs s qs b ds qs b � � � � � � � � � � � � � � � � � � � � � � � � � � (5) Vdr Rotor Voltage‘s Differential Equation at Direct Axis V R i pdr r dr r b qr dr b � � � �� � � � � � � � � � � � � � � � � � � � � � � (6) Vqr Rotor Voltage‘s Differential Equation at Quadrate Axis V R i pqr r qr r b dr qr b � � � �� � � � � � � � � � � � � � � � � � � � � � � (7) Flux linkage differential equation for stator and rotor components: �ds s ds m dr dsX i x i i� � � � �� ( ), (8) where: �ds (Weber) is the stator flux linkage at the direct axis, idr (amp) is the rotor current at the direct axis but ids (amp) is the stator current at direct axis, p is the differentiation parameter � d/dt. �qs s qs m qr qsX i x i i� � � � �� ( ), (9) where: �qs is the stator flux linkage at the quadrant axis, iqr is the rotor current at quadrant axis but iqs is the stator current at the quadrant axis. �dr r dr m dr dsX i x i i� � � �� ( ), (10) where: �dr is the rotor flux linkage at the direct axis. �qr r ds m qr qsX i x i i� � � �� ( ), (11) where: �qr is the rotor flux linkage at quadrant axis. d d � � � qs b qs s qs dst V R i� � � �( ), (12) where: �b is the base speed. P C D Vm p w� 1 8 2 3( ) (13) C p � � � � � � � �� � � �� �( . . ) sin ( ) . .0 44 0 0167 3 15 0 3 0 00184� � � ( )� �� � � � � � �3 (14) Where: �m (rad/sec) is the mechanical speed Pm (kW) is the mechanical power Tm (nm) is the mechanical torque n (rpm) is the rotor revolution per minute Cp is the power coefficient of the wind turbine � is the blade pitch angle (degree) � is the tip speed ratio Vw (m/s) is the wind speed D (m) is the of the rotor Diameter of the wind turbine � 3.14 (kg/m3) Air density Appendix B – SEIG parameters The induction machine under study as a SEIG has the fol- lowing parameters: 1.1 kW, 127/ 220 V (line voltage), 8.3/4.8 A (line current), 60 Hz, 2 poles, wound-rotor induction machine [9, 10]. By choosing proper base values: � base voltage Vb � [220/(1.73)] V, � base current Ib � 4.8 A, � base impedance Zb � 26.462 ohm, � base rotor speed Nb � 3600 rpm, and � base frequency Fb � 60 Hz, the per-unit parameters of the induction machine under study are equal: � stator resistance Rs � 0.0779, � rotor resistance Rr � 0.0781, � stator reactance Xs and rotor reactance Xr are equal 0.0895. The equation of the motion of rotating parts of the com- bined studied SEIG and the wind turbine is also included in the system in order to provide a detailed simulation model. The inertia constant of the machine H � 0.055 s. References [1] Li, Wang, Jian-Yi-Su: “Dynamic Performance of an Isolated Self Excited Induction Generator Under Vari- ous Loading Conditions”, IEEE Transactions on En- ergy Conversion, Vol. 15 (1999), No. 1, March 1999, p. 93–100. [2] Li, Wang, Ching- Huei Lee: “Long- Shunt and Short- Shunt Connections on Dynamic Performance of a SEIG Feeding an Induction Motor Load”, IEEE Transactions on Energy Conversion, Vol. 14 (2000), No. 1, p. 1–7. [3] Golodberg, D.: Genetic Algorithms in Search Optimization and Machine Learning. Addision-Wesely, Reading, MA, 1989. [4] Attia, A., Soliman, H.: “An Efficient Genetic Algorithm for Tuning PD Controller of Electric Drive for Astro- nomical Telescope.” Scientific Bulletin of Ain Shams University, Faculty of Engineering, Part II, Issue No. 37/2, June 30, 2002. [5] Ezzeldin, S. Abdin, Wilson, Xu: “Control Design and Dynamic Performance Analysis of a Wind Turbine – Induction Generator Unit”, IEEE Transaction on © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 21 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 2/2006 Energy Conversion, Vol. 15 (2000), No. 1, March 2000, p. 91–96. [6] Marduchus, C.: “Switched Capacitor Circuits for Reac- tive Power Generation”, Ph.D. Thesis, Brunuel Univer- sity, 1983. [7] Sekaj, I.: “Genetic Algorithm – Based Control System Design and System Identification”, 5th International Mendel Conference on Soft Computing, 1999, Brno, Czech Republic, p. 139–144. [8] Michalewic, Z.: Genetic Algorithms + Structure = Evolution Program. Springer-Verlag, Berlin Heidelberg, 1992. Dr. Ing. Abdel-Fattah Attia phone:+202 5560046 fax:+202 5548020 e-mail: attiaa1@yahoo.com Astronomy Department National Research Institute of Astronomy and Geophysics (NRIAG), 11421 Helwan Cairo, EGYPT Associate Prof. Dr. Ing. Hussein F. Soliman e-mail: hfaridsoliman@yahoo.com Dept. of Electric Power and Machine Faculty of Engineering Ain-Shams University Abbasia, Cairo, EGYPT Dr. Ing. Mokhymar Sabry e-mail: sabry40@hotmail.com Electricity & Energy Ministry New & Renewable Energy Authority “NREA” Wind Management, Cairo, EGYPT 22 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 2/2006 Czech Technical University in Prague