Design of Tuning Mechanism of PID Controller for Application in three Phase Induction Motor Speed Control

Transcription

1 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] Deign of Tuning Mechanim of PID Controller for Application in three Phae Induction Motor peed Control Alfred A. Idoko 1, Iliya. T. Thuku 2,. Y. Mua 3, Chinda Amo 4 1, 2, 3 Department of Electrical and Electronic Engineering, Modibbo Adama Univerity of Technology Yola, Adamawa tate, Nigeria. 4 Nigeria. National Centre for Technology Management, Modibbo Adama Univerity of Technology Yola, Adamawa tate, Nigeria. Abtract Thi paper preent a deign of tuning mechanim of Proportional Integral Derivative Controller for application in three phae induction motor peed control. It demontrate, in detail, how to employ the MatLab tool o a to earch efficiently for the optimal PID controller parameter within a mechanim ytem. The propoed approach ha uperior feature, including: eay implementation; table convergence characteritic; and le computational effort. Three phae induction motor ha complex mathematical modelling which make it difficult to deign the peed controller. oftware PID Tuning Mechanimwa developed herein and ued to obtain both the initial PID parameter under normal operating condition and the optimal parameter of PID control under fully-loaded condition. The propoed PID controller Tuning Mechanim will automatically tune it parameter within thee range. In order to prove the performance of the propoed tuning mechanim for the PID controller, a three phae aynchronou motor wa modelled in MATLAB, the tranfer function wa obtained uing the oftware and a controller wa deigned uing PID. The modelling and imulation reult how the potential of the propoed controller to be very efficient. Keyword PID Controller, Modelling of Induction Motor, Deign of PID Tuning oftware. I. INTODUCTION peed control of an induction motor ha been a major challenge ince it dicovery by a French phyicit Françoi Arago in The induction motor i made up of two ditinct part, tator which i a tationary part and rotor (alo called armature core winding) which i the rotary part. Induction motor i an alternating current electric machine which by excitation, produce a torque[1]. The torque reult from an electromagnetic induction from the magnetic flux due to the tator core winding a a reult of rotation about a fixed poition in the tator core winding [2]. Induction motor i the mot common electrical machine ued in modern indutrie. It ha gained uch popularity due to it variou advantage. The variou advantage are high efficiency, low cot, good elf-tarting, implicity in deign, the abence of the collector broom ytem, and a mall inertia. Even though it ha a wide range of advantage it alo ha quite few diadvantage. Induction motor ha diadvantage, uch a complex, multivariable and nonlinear mathematical model and i not inherently capable of providing variable peed operation [3]. During the pat decade, proce control technique in indutrie have made great advance. Numerou control method uch a: adaptive control; neural control; and Fuzzy control have been tudied [4]. Among thee the bet known i the Proportional Integral Derivative (PID) controller, which i being widely ued becaue of it imple tructure and robut performance within a wide range of operating condition. Unfortunately, it ha been quite difficult to properly tune the gain of PID controller becaue many indutrial plant are often burdened with problem uch a: high order; time delay; and nonlinearitie [5]. Over the year, everal heuritic method have been propoed for the tuning of PID controller. The firt method ued the claical tuning rule propoed by Ziegler and Nichol [6]. In general, it i often hard to determine optimal or near optimal PID parameter with the Ziegler Nichol formula for many indutrial plant [7]. For thi reaon it i highly deirable to increae the capabilitie of PID controller by adding new feature. Many Artificial Intelligence (AI) technique have been employed to improve the controller performance for a wide range of plant whilt retaining their baic characteritic. AI technique uch a: neural network; fuzzy ytem; and Page 138

2 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] neural-fuzzy logic have been widely applied to the proper tuning of PID controller parameter [8]. oftware tool, i one of the modern tuning mechanim. The oftware tool can generate a high-quality olution for horter calculation time and table convergence characteritic than other tochatic method [4]. Much reearch i till in progre for proving the potential of the oftware tool for olving complex induction motor ytem operation problem. Becaue the oftware tool method i an excellent tuning mechanim and a promiing approach to olving the PID controller parameter tuning problem. Thi controller i called the oftware-tuning PID controller. However, the PID controller i not robut to wide parameter variation and large external diturbance [11]. Thi erve epecially for highly coupling nonlinear ytem where the PID controller lack adaptive capability. The difficulty in obtaining tranfer function of three phae induction motor poe a eriou challenge to thi mechanim. However, if the PID controller parameter are choen incorrectly, the controlled proce input can be untable, i.e., it output diverge, with or without ocillation, and i limited only by aturation or mechanical breakage. Intability i caued by exce gain, particularly in the preence of ignificant lag [1]. Generally, tabilization of repone i required and the proce mut not ocillate for any combination of proce condition and etpoint, though ometime marginal tability (bounded ocillation) i acceptable or deired [1]. Mathematically, the origin of intability can be een in the Laplace domain. The total loop or cloe loop tranfer function i: H ()= K d + K()G() 1+K()G() (1) Where. K(): PID tranfer function G(): Plant tranfer function The ytem i called untable where the cloed loop tranfer function diverge for ome.thi happen for ituation where K() G() = -1. Typically, thi happen when there i a 180 degree phae hift. tability i guaranteed when K()G() < 1 for frequencie that uffer high phae hift. In order to achieve practical requirement, engineer have to adjut the parameter under different operating condition and device a mean of obtaining complex tranfer function. However, the robutne i limited to a mall range. A mechanim to overcome thi diadvantage i called the oftware-tuning mechanim. The parameter tuning at any intance i uually baed on a tructurally fixed mathematical model produced by a oftware procedure [11]. Unfortunately, recent plant find it difficult to obtain their fixed mathematical model. Thi paper propoe a oftwaretuning mechanim for PID controller for an application in three phae induction motor. At the ame time, the robutne will be expanded to a large range. In thi paper, a practical high-order mechanim ytem with a PID controller i adopted to tet the performance of the propoed oftware-tuning mechanim for PID controller. A compared with the other mechanim uch a the one propoed by [9][8] [7][6], it i found that the nominal value and tuning range of the PID parameter by the propoed oftware-tuning mechanim can be accurately determined. MatLab imulation and experimental reult have hown that the propoed controller i mot uitable and robut for PID controller. II. MATEIAL AND METHOD In thi work the following material were ued: oftware tool, PID Controller block in MATLAB package, Induction Motor and Voltage ource. 2.1 PID Controller PID controller are widely ued in indutrie for peed control purpoe. A PID controller calculate an error value a the difference between the meaured proce value and the deired et point. The PID controller calculation involve three eparate contant and i accordingly ometime called three-term control i.e. the proportional, integral and the derivative value which i denoted by PID a how in figure 1 below. Fig. 1: Block diagram of PID controller [10] Tuning a control loop i the adjutment of it control parameter (proportional band/gain, integral gain/reet, derivative gain/rate) to the optimum value for the deired control repone [11]. tability (no unbounded ocillation) i a baic requirement, but beyond that, different ytem have different behavior, different application have different requirement, and requirement may conflict with one another [4]. Page 139

3 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] PID tuning i a difficult problem, even though there are only three parameter and in principle i imple to decribe, becaue it mut atify complex criteria within the limitation of PID control. There are accordingly variou method for loop tuning, and more ophiticated technique are the ubject of patent; thi ection decribe ome traditional manual method for loop tuning. Figure 2 below how the effect of tuning the PID parameter on the proce repone. Fig. 2: Proce variable for different Kp, Ki and Kd value [11]. Tuning the Proportional Term:Proce variable for different Kp value (Ki and Kd held contant) i hown in Figure 2. The proportional term make a change to the output that i proportional to the current error value. The proportional repone can be adjuted by multiplying the error by a contant Kp, called the proportional gain [11]. The proportional term i given by the Equation (12) P out = K pe (t). (2) e = G / Kp (3) Integral Term: Proce variable for different Ki value (Kp and Kd held contant) i hown in figure 2. The contribution of the integral term i proportional to both the magnitude of the error and the duration of the error. The integral in a PID controller i the um of the intantaneou error over time and give the accumulated offet that hould have been corrected previouly [12]. The accumulated error i then multiplied by the integral gain (Ki) and added to the controller output. The integral term i given by the Equation (4) I out = ki e ( ) d (4) Derivative Term:Proce variable for different Kd value (Ki and Kp held contant) i hown in Figure 2. The derivative of the proce error i calculated by determining the lope of the error over time and multiplying thi rate of change by the derivative gain Kd. The magnitude of the contribution of the derivative term to the overall control action i termed the derivative gain Kd [12]. The derivative term i given by the Equation (5) D out= K d + d c(t) (5) dt 2.2 PID Tuning Mechanim There are everal method for tuning a PID Controller. Tuning method with it advantage and diadvantage are given in Table 1. Table.1: PID Tuning Method [11] Method Advantage Diadvantage Manual Tuning No math required. On-line method equire experience peronnel Ziegler-Nichol Proven method. Online method Proce upet, ome trial-and-error oftware tool Conitent tuning. Online or ome cot and training involved offline method. Can upport Non-teady tate Tuning Cohen-Coon Good proce model ome math. Offline method only good for firt order procee. The mot effective method generally involve in the development of ome form of the proce model, and then chooing P, I, and D baed on the dynamic model parameter. Manual tuning method can be relatively inefficient, particularly if the loop have repone time on the order of minute or longer. The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and the repone time of the ytem. If the ytem can be taken offline, the bet tuning method often involve ubjecting the ytem to a tep change in input, meauring the output a a function of time, and uing thi repone to determine the control parameter [11]. 2.3 oftware Tool Mot modern indutrial facilitie no longer tune loop uing the manual calculation method hown above. Intead, PID tuning and loop optimization oftware are ued to enure conitent reult. Thee oftware package will gather the data, develop proce model, and ugget optimal tuning. ome oftware package can even develop tuning by gathering data from reference change [13]. Page 140

4 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] Advance in automated PID Loop Tuning oftware alo deliver algorithm for tuning PID Loop in a dynamic or Non-teady tate (N) cenario. The oftware will model the dynamic of a proce, through a diturbance, and calculate PID control parameter in repone. 2.4 Modelling of Induction Motor In the control of any power electronic drive ytem, mathematical model of the plant i required [14]. From the objective equation (1), the Plant tranfer function [8] i required to deign the loop controller. To obtain the Induction Motor (Plant) tranfer function and deign any type of controller to control it peed, mathematical model i required. The Equivalent circuit of induction motor in d-q axi fig. (3) will be ued to analyze the mathematical modeling of induction motor. The induction motor can be analyzed a a three phae tator winding magnetically coupled with the rotor one, independently on the rotor typology. Fig. 4: Phaor diagram of tator winding [10] V A = pλ A + I A (6) V B = pλ B + I B. (7) V C = pλ C + I C (8) (a) (b) Fig. 3: Equivalent circuit of induction motor (a) d-axi and (b) q-axi[15] The equivalent circuit ued for obtaining the mathematical model of the induction motor i hown in the Fig. (3) The induction motor model i etablihed uing a rotating (d, q) field reference (without aturation) concept. An induction motor model i then ued to predict the voltage required to drive the flux and torque to the demanded value within a fixed time period. Thi calculated voltage i then yntheized uing the dynamic model of the motor in imulink. The induction motor can be analyzed a a three phae tator winding magnetically coupled with the rotor one, independently on the rotor typology. Fig. 5: Phaor diagram of otor winding [10] Va = pλa + Ia (9) Vb = pλb + Ib (10) Vc = pλc + Ic (11) When the dynamic model of the induction motor i determined, we can tranform the three-phae machine into a two-phae machine uing an orthonormal tranformation matrix T: T = [ [ λabc λabc ] = [M Mr M = ] (12) Mr Mrr ] [IABC Iabc ] (13) (14) Lo + Ld Lo + Ld [ Lo + Ld ] Page 141

5 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] M rr =... (15) Lo + Lrdr Lo + Lrdr [ Lo + Lrdr ] Mr = Lo co Ѳ Lo co(ѳ ) Lo co(ѳ ) [ Lo co(ѳ ) Lo co Ѳ Lo co(ѳ )] Lo co(ѳ ) Lo co(ѳ ) Lo co Ѳ (16) Where M, Mrr and Mr are the tator mutual inductance, rotor mutual inductance repectively. Lo i the no load inductance. Below equation repreent the equation of the equivalent two phae machine. For control purpoe, another tranformation i required. V = i + pλ (17) V B = i B + pλ B (18) V 0 = i 0 + pλ 0 (19) V = i + pλ (20) V B = i B + pλ B (21) V 0 = i 0 + pλ 0 (22) If we want to move thee equation in a reference frame fixed with the tator, we have to apply a rotation matrix around the Z axi [16]. λ = (L d L 0)i L 0 co(ѳ) i L 0 in(ѳ) i B (23) λ B = (L d L 0)i L 0 co(ѳ) i L 0 in(ѳ) i B (24) λ = (L d L 0)i L 0 co(ѳ) i L 0 in(ѳ) i B (25) λ B = (L d L 0)i L 0 co(ѳ) i L 0 in(ѳ) i B (26) With thi tranformation, the final equation in a reference ytem called tator dq0 reference frame are: λ d = L i d + L m i d... (27) λ q = L i q + L m i q... (28) λ d = L i d + L m i d (29) λ q = L i q + L m i q (30) V d = i d + pλ d (31) V q = i q + pλ q... (32) 0 = r i d + pλ d + ⱳλ q (33) 0 = r i q + pλ q + ⱳλ d (34) Torque (T) = λ q i d + λ q i d (35) Where L and Lr are the repective tator and rotor inductance, and r are the repective tator and rotor reitance. I d and I q are the repective d and q axi of the tator and rotor winding. Uing thee equation, it i poible to build the block cheme in order to imulate the machine. The block are: tator Voltage Tranformation, Fluxe, Current, Electromagnetic, and Mechanical ubytem. Fig. 6: Three Phae Induction Machine Aynchronou Motor Dynamic Model 2.5 Modelling PID Controller Fig.(1) above how the block diagram of PID Controller. The firt term A proportional (TOP), an integral (CENTE) and A derivative (BOTTOM). A proportional controller may not give teady tate error performance which i needed in the ytem. An integral controller may give teady tate error performance but it low a ytem down. o the addition of a derivative term help to cure both of thee problem. The Page 142

6 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] proportional, integral and derivative output are added together. The PID controller can be thought of a having a tranfer function. The PID tranfer function can be obtained by adding the three term together. PID() = K P + K i + k d (36) The tranfer function can be combined into a pole-zero form. Hence, the equation become; PID() = [Kp + Ki+ 2 Kd]/ (37) From fig.(1) the output of PID controller u (t), i equal to the um of three ignal: The ignal obtained by multiplying the error ignal by a contant proportional gain KP, plu the ignal obtained by differentiating and multiplying the error ignal by contant derivative gain KD and the ignal obtained by integrating and multiplying the error ignal by contant internal gain KI,. The output of PID controller i given by equation (38), taking Laplace tranform, and olving for tranfer function, give ideal PID tranfer function given by equation (39) [10] u(t) = K P e (t) + K de(t) P dt + K I e(t)dt U() = k p E() + k I E() 1 (38) ince PID tranfer function i a econd order ytem, it can be expreed in term of damping ratio and undamped natural frequency to have the following form: T 1 1 T G PID = K P [1 + + T D ] = K D 2 + T 1 +1 P (45) T 1 Where the integral timet I = K P, the derivative time T K D = I K D, K K I = K P, K P T D = K P T D I 2.6 Deign of PID Tuning oftware The tuning mechanim ued i the oftware tuning mechanim. The mechanim i deigned uing MatLab tool capable of deriving the tranfer function of a complex Induction machine and varying the PID parameter to control the peed of the machine. Below i the variou tep to deigning the tuning mechanim. tep 1. Modelling of the three phae induction motor in MatLab. The three phae induction motor in MatLab i hown in fig.(6) and the model i converted to a block named Motor a hown in fig.(7) below T 1 U() = E() [K p + K I + K D] (39) G PID () =K P + K I + K D = K D 2+ K K P+ I = K D [ 2 + K P + K I ] K D K D (40) Equation (40) i econd order ytem, with two zero and one pole at origin, and can be expreed to have the following form: G PID = K D (+ Z PI )(+ Z PD ) = K D ( + Z PI ) (+ Z PD ) G PD ()G PI () (41) Which indicate that PID tranfer function i the product of tranfer function PI and PD, Implementing thee two controller jointly and independently will take care of both controller deign requirement? The tranfer function given by Equation (40 and 41), can alo be expreed to have the form: G PID = K D (+ Z PI )(+ Z PD ) = K D 2 + (Z PI + Z PD )K D + (Z PI Z PD K D ) (42) earranging the above equation (42) we have: G PID = K D 2 + (Z PI+ Z PD )K D + (Z PIZ PD K D ) = (Z PI + Z PD )K D + (Z PIZ PD K D ) + K D (43) Let,K 1 = (Z PI + Z PD )K D, K 2 = (Z PI Z PD K D ), K 3 = K D. Then, equation (43) become, G PID = K 1 + K 2 + K 3 (44) = Fig. 7: Induction motor model block tep 2. Obtain the tranfer function of the Motor uing MatLab cript. The motor i run in MatLab and the tep repone i analyzed. The analyi further yield the tep repone of the motor a hown in figure 8 below. Fig. 8: tep repone of induction motor And the tranfer function of the tep repone in figure 8 above i obtained by taking the linear analyi in MatLab. Page 143

7 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] From input "Torque" to output "ubytem": G = e ^ (46) Linearization at model initial condition Continuou-time tranfer function. Equation 46 repreent the tranfer function of the three phae Aynchronou induction motor under analyi. Thi method can be ued to obtain the tranfer function of any ytem. tep 3. Uing thi tranfer function to deign a PID controller in MatLab cript and teting it on the tep repone in figure 8 above. Varying the PID parameter (Kp, Ki, Kd) on the cript and running the cript tune the tep repone. Thi controller i then tuned for optimum repone and the parameter are recorded for the actual controller deign in the next tep. tep 4. Connect a PID controller to the Motor. equel to tep 3 above, a PID controller block i connected with the motor block a how in fig.(9) below. the mot advanced oftware tool for PID Controller in an application to three phae induction motor. 2.7The Induction Motor Parameter The induction motor ued in thi imulation i a 50 Hp, 420V, 60Hz, aynchronou motor having the parameter lited in table 2 below. The induction motor tator i fed by a current controlled three-phae ource. The tator current are regulated by hyterei regulator which generate inverter drive ignal for the inverter witche to control the induction motor. The motor torque i controlled by the quadrature-axi current component and the motor flux i controlled by directaxi current component. The motor peed i regulated by a PID controller which produce the required torque current component ignal. Table 2. Induction motor parameter /No. Parameter ymbol Value 1 upply Frequency f 60 Hz 2 Voltage V 420 V 3 tator eitance 0.288Ω 4 tator Inductance L H 5 otor eitance r 0.158Ω 6 otor Inductance Lr H 7 Magnetizing Inductance Lm H 8 Inertia J 0.4 Kg.m2 9 No. of pole P 2 Fig. 9: Three Phae Induction Motor with PID Controller A hown in figure 9, the output peed of the motor i feedback to the input and the ummer compute the error value. Thi error i being fed to the PID controller and by tuning the PID uing oftware cript, the error i compenated. The value of the PID parameter and the motor parameter are programed into the ytem automatically. However, the cript IM_Initialazation can be run over the network from a remote location to the plant. With the availability of internet acce, thi tuning mechanim offer 2.8 Development of the imulink Model Diagram Firt of all, let u conider the development of a imulink model for the peed performance or repone of an induction motor without controller, the imulink model for a proportional integral derivative (PID), and latly, a developed imulink model for a proportional integral derivative (PID) oftware tuning mechanim. The peed performance of the induction motor i checked firt without any controller and then checked with the help of a proportional integral derivative (PID) controller tuned with oftware tool. The imulink model i developed in the MATLAB which i hown in the following Fig.(10), and (11) repectively. Page 144

8 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] Fig. 10: Developed imulink model to check peed repone of Induction motor without any controller Uing the oftware tuning mechanim, we can obtain the parameter kp, ki, and kd for optimal performance under the no-load and full-load condition. In the open loop condition, we can determine the open loop tranfer function of the plant by MATLAB program a G = e ^ (46) The actual motor model IM_Plant i controlled uing PID controller a hown in fig.(12) below. The parameter of the motor and PID controller are programmed into the ytem uing the MatLab cript. Thi cript i ued to tune the motor repone in the ame way a demontrated in fig.(11) above. Below i the imulation of the motor with controller and the IM_Initiallaization cript. Alo, the cloed-loop ytem with a PID controller ha the following characteritic equation Mc = -1.01e-10 ^ ^ ^ ^ (47) Equation 46 i obtained by running the MatLab cript below and the PID parameter wa tuned to give optimal tep repone a how in fig.(47). The repone with controller ha a teady tate of 17 econd while the open loop repone teady tate i 263 econd. Fig. 12: Developed imulink model to check peed repone of Induction motor with PID controller III. EULT imulation were carried out in MATLAB environment and the reult were verified for the et peed. The peed repone of Induction motor i checked for without controller and with PID controller which i hown in Fig.(13) and (14) Fig.(11) Open loop and cloe loop tep repone of motor tranfer function. Page 145

9 International Journal of Advanced Engineering eearch and cience (IJAE) [Vol-4, Iue-11, Nov- 2017] Fig. 13: peed performance of Induction motor without any controller Fig. 14: peed performance of Induction motor with PID controller The peed performance of induction motor without any controller i hown in fig.(13). Thi i for full load condition. When we applied a full load the peed uddenly decreae and i not table. o a to improve the peed performance, a PID controller tuned with oftware mechanim. Becaue of which the teady tate error i eliminated and the rie time i improved in figure 4.6. Therefore the PID controller i ued to improve the peed performance of Induction motor and the reult i hown in fig.(14). IV. DICUION The main purpoe of thi thei i to control the peed of induction motor uing oftware tuning mechanim of PID controller. From thi we come to know that oftware tuning mechanim of PID controller i a good mechanim for controlling the peed of three phae induction motor. The peed of induction motor uing PID controller ettled early when uing oftware tuning than other tuning mechanim. A againt Zigler Nicole, the oftware tool i uitable for any complex ytem and it i not a trial by error method. It alo add the advantage of remote management and flexibility. V. CONCLUION The peed control of an induction motor by oftware tuning mechanim of PID controller ha led to the conervation of energy and the uage of high performance application uch a hybrid vehicle, robotic, wind generation ytem, paper and textile mill, and variable-torque centrifugal fan, pump and compreor load application. Hence there i need to develop a oftware mechanim for effective peed control for induction motor. Page 146

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