The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 MODELING & SIMULATION OF SYNCHRONOUS MACHINE CONTROLLED BY PID CONTROL FOR THE REACTIVE POWER COMPENSATION Dr. Maamoon F. Al-Kababji Mr. Ahmed Nasser B. Al-Sammak Electrical Engineering Department University of Mosul Mosul – Iraq ABSTRACT λa, λb and λc= Instantaneous linkage flux for stator phases This paper present a suitable model set up for the analysis of the synchronous machine. The model is then test with some cases such as sudden change in load, field voltage as well as loading or no loading condition. The basic equations of the motor are also developed in a manner, suitable for the Matlab-Simulink application. The computer and practical results were represented and compared. Good agreement between these results has been observed. This proves the validity of the simulation method used in this work. A PID controller for firing angle (alpha) was added to this model to control six-pulse rectifier circuit, which gives variable DC field voltage and this voltage changes as the output of the PID controller changes depending on the set point (reference power factor). The parameters of the PID controller was selected to give high performance for the reactive power control. (Wb). λf = Instantaneous linkage flux for rotor (Wb). [V]= Voltage matrix (volt). [R]= Resistance matrix (ohm). [i]= Current matrix (amper). [λ]=Linkage flux matrix (Wb). [L]= Inductance matrix (henry). Te= Electrical torque (N.m). TL= Mechanical torque (N.m). ωs= Synchronous speed (rpm). ωr= Rotor speed (rpm). θ= Displacement angle for rotor refers to stator phase (a) KEYWORDS Synchronous Machine, Voltage Stability, PID controller, Excitation Control, Reactive Power Compensation and Matlab-Simulink. (rad). Vf =DC field voltge (volt). PL&QL = Real and reactive power load demand respectively (watt, var). Pm&Qm = Real and reactive power of the Synchronous motor respectively (watt, var). Pt&Qt = total Real and reactive power for the Synchronous motor and load demand respectively (watt, var). Kp= Proportional gain for the PID control. KI = Integral gain for the PID control. Kd= derivative gain for the PID control. Kcr= critical value gain for the PID control which give sustained oscillations output of system. LIST OF SYMBOLS va,vb and vc = Thee phase terminal voltages (volt). ia,ib and ic = Thee phase terminal current (amper). α= Firing angle (degree). Vf= DC field voltage (volt). If= DC field current (amper). Rf= Resistance of field circuit (ohm). Lff= Self of rotor inductance (henry). La,Lb and Lc= Stator self inductance (static)/ phase (henry). Lab,Lbc and Lca= Mutual inductance between stator Phases (henry). Laf,Lbf and Lcf= mutual inductances between stator phases and rotor (henry). Ls= Part of phase inductance harmonic because of salience (henry). 1. INTRODUCTION AC electrical machines and other inductive loads used in industry draw reactive power from the line. The reactive power causes overloading effects on the line, power breakers, transformers, relays and isolations. In addition, the reactive power also increases the dimension of cables used in transmission line. So that, the structure of all equipments 1 The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 used in the line has to be strong enough to carry the huge weight of cables. Therefore, cost of the system is increased and efficiency of the system is reduced. To reduce the cost and to improve the efficiency, the reactive power drawn from the line has to be decreased by supplying it from any other source. Reactive power compensation can be achieved by using constant capacitor groups with variable inductance [1], which can give a smooth reactive power correction without step changes. In industry, most of the reactive power compensation systems have been achieved by using constant capacitor groups, which are controlled by contactors and timers. In this case, the penalties such as time delay, over compensation or lower compensation are always possible because of the step changes of capacitor groups. In addition capacitors are often less flexible and less economical [2]. On the other hand, synchronous motors can operate at unity, lagging or leading power factor condition, so that the synchronous motor can be provide network by the reactive power demand [3]. Presently for the smooth and fast reactive power compensation, synchronous motor will be widely used for the reactive power compensation because of the cost of the compact system, synchronous motor and controller, is less than using power electronics with capacitor grope, Flexible AC Transmission System (FACTS). In additional to synchronous motor produce minimal harmonics content and, therefore, require no special filtering [4]&[5]. This paper presents a generalized digital simulation of a synchronous machine, which the field voltage supplied from a six-pulse rectifier circuit taking into account the limit of this voltage to the rated value. The simulation results are compared with the practical test results, Good agreement between these results has been observed, this proves the validity of the simulation method used in this work. In this paper synchronous machine was modeled using basic equations of the machine [6]&[7] then solved this equations using Matlab-Simulink [8]. The faring angle (α) of six pulse rectifier circuit [9] was controlled by PID control. The PID controller parameters was calculated for the reactive power compensation using Ziegler and Nichols rules [10]&[11]. The result shown that compensation system implemented is reliable, efficient, faster and economical than other one with capacitor groups. 2. MODELING AND SIMULATION OF THE SYSTEM The system shown in Fig.(1), has been simulated on a readily available inexpensive PC. The equations of the system were modeled and simulation using Matlab-Simulink [8]. The reactive power needed is generation from a threephase synchronous motor (ratings and parameters are given in appendix (A)). This motor is connected to infinity bus-bar near to the load. Thee motor drive a mechanical load equal to (5 N.m). 2 Pt+jQt PL+jQL Load Infinity Bus-Bar V Pm+jQm A Motor PF PFref - + Synchronous Motor + - PID Controller 3 phase Supply Fig.(1) Overall System Model. 2.1 SYNCHRONOUS MOTOR MODEL The analysis of poly-phase machines are generally done using the stationary two axis model, where the three phase quantities of the machine are transformed into two phase quantities (Park's Transformation) thereby simplifying, facilitating the analysis, and reducing the number of basic equations' thereby reducing the computational time of the simulation. However with this methods difficulties arise when including higher-harmonics in the coefficients of the inductance, and also when simulating some abnormal and fault conditions such as line to neutral short circuit. On the other hand, in the direct 3-phase model, the machine parameters and variables are expressed in their actual physical quantities. This model is more suitable in studying the all-normal and abnormal conditions. Thus it is easy to monitor the variables and parameters directly without transformation. However the direct 3-phase model as compared with two axis model, suffers from the excessive computational time required to form and invert the inductance matrix of the machine at each step of calculation, but in this research this is not problem the matrix treatment as shown in the in next paragraph. The direct 3-phase model of the synchronous motor which is shown in Fig.(2), can be adopted with the following assumptions: 1. The influence of the damper winding is neglected. 2. The motor is assumed to be magnetically linear. 3. The inductance variation is considered cosinusoidal with the rotor position. 4. The effects of saturation, hysteresis, and eddy currents are neglected. The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 2.2 EXCITATION FIELD VOLTAGE The excitation field voltage, which controlled the reactive power of the machine [14], was modeled using sixpulse rectifier circuit where the basic equations was driven as follows: The output DC voltage Vf1 equal to: Vf1 = + +α n ⋅ ∫π2 πn Vp ⋅ sin( ω t ) ⋅ d (ω t ) − +α 2 ⋅π 2 n π π …(13) or, Vf1 = n ⋅ 3 ⋅ Vp ⋅ cos( α ) 2 ⋅π Where n &Vp represented numbers of pulses and peak voltage respectively. After adding free wheeling diode on the output of DC circuit equation (13) will become: Vf2 = Fig.(2) Direct 3-phase model of the synchronous motor. [ ] [ ] [] [ ] [] [V ] = [R] ⋅ [i ] + p[λ ] From the above assumption we can write the following equations [7]&[12]: V = R ⋅ i + p{ L ⋅ i } …(1) [i] = [ia ib ic [R] = diag [Ra [V ] = [v a vb p[λ ] = [ pλa …(2) ′ if ] …(3) Rb Rc Rf ] …(4) ′ v c Vf ] …(5) ′ pλb pλc pλf ] where p represented the directive pointer. π n ⋅ ∫π π Vp ⋅ sin( ω t ) ⋅ d (ω t ) − +α 2 ⋅π 2 n or, Vf2 = n ⋅ Vp ⋅ (1 + 3 ⋅ cos( α ) − 1 ⋅ sin( α )) …(14) 2 2 2 ⋅π As a resultant from the above analysis the relationship between firing angle (α) and DC output voltage for the six pulse rectifier circuit with free wheeling diode shown in Fig.(3) and Vf, DC field voltage, equal to: Vf = Vf1 for 0 ≤ (α ) ≤ 30 …(15) Vf = Vf2 for (α ) ≥ 30 …(16) …(6) Since, the relationship between leakage flux and current is: λ = L⋅i …(7) Where [L] matrix given in appendix (B). Derivative two sides of the above equation give: [ ] [ ] [] d [L] ⋅ [i ] ⋅ pθ …(8) dθ d [L ] given in appendix (B). Where matrix dθ p[λ ] = [L ] ⋅ p[i ] + ⎫ ′ d [L] −1 ⎧ ⋅ [i ]. pθ ⎬ p[i ] = [L ] ⎨[V ] − [R ] ⋅ [i ] − dθ ⎩ ⎭ Rearrangement equation (8) obtain: ′ d [L ] Te = 0.5 ⋅ ρ ⋅ [i ] . ⋅ [i ] dθ dω dδ +J Te = TL + K ⋅ ωr + Kd dt dt …(9) The total electrical develop torque will be[3] &[13]: …(11) …(12) All equation from 1 to 12 was modeled and simulated using Matlab-Simulink. 3 Fig.(3) The relationship between DC field voltage (Vf) and firing angle(α). 3. TUNING PID CONTROLLER The PID Controller parameters were tuned to give high performance for the reactive power compensation by controlling the firing angle (alpha) of the six-pulse rectifier circuit. Ziegler and Nichols rules [10] was chosen with some The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 tuning to determine values of the proportional gain KP, integral gain KI, and derivative gain Kd. In this method (Ziegler and Nichols) first KI and Kd are set to zero. Using the proportional control (KP) action only as shown in general closed–loop system, with a proportional controller Fig.(4). By increasing (KP) from 0 to critical value Kcr , as shown in Fig.(5) & Fig.(6), where the output first exhibits sustained oscillations at Kp=13. Then Pcr is calculated from the experimental result shown in Fig.(7). This method suggested setting the values of parameters KP, KI, and Kd each according to the formula shown in Table (1). Notice that the PID controller tuned by Ziegler-Nichols rules can be given by the following equations (17-19): KI ⎛ ⎞ …(17) + K d ⋅s⎟ G C (s) = K P ⎜ 1 + s ⎝ ⎠ 1 ⎞ ⎛ + 0.125⋅ Pcr ⋅ s ⎟ …(18) GC (s) = 0.6 ⋅ Kcr⎜1 + ⎠ ⎝ 0.5⋅ Pcr ⋅ s 4 ⎞ ⎛ ⎟ ⎜s + Pcr ⎠ …(19) G C (s) = 0.075 ⋅ Kcr ⋅ Pcr ⋅ ⎝ s Thus, the PID controller has a pole at the origin and double zeros at s=-4/Pcr. u(t) r(t) c(t) + KP Plant 2 - Fig.(4) Closed –loop system with a proportional controller. Fig.(6) Power factor (PF) with time, when PID controller was tuned by a change of proportional gain (KP) from 13-20. c(t) Pcr 0 Time Fig.(7) Sustained oscillation with period Pcr. Fig.(5) Power factor (PF) with time, when PID controller was tuned by a change of proportional gain (KP) from 11-12. 4 The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 Table (1) Zigler-Nichols tuning rules. Type of Controller KP KI Kd P 0.5Kcr 0 0 PI 0.45Kcr 1.2 Pcr 0 PID 0.6Kcr 2 Pcr 0.125Pcr 4. SIMULATION RESULTS The Final values of PID controller constants are KP=7.8, KI=90 and Kd=6*10-3 . Those values were obtained by tuning the information, given in Table (1) and Fig.(5) & Fig.(6). The Matlab-Simulink of mean system is shown in Fig.(8). Fig.(9) shows the practical and calculated results for the change in speed and phase current of synchronous motor when its driving mechanical load change from 5N.m to zero(no load), these results proves the validity of the simulation method. In the simulation, a change in set reference power factor (PFref) will give the same change in motor power factor (PF), shown in Fig.(10), Due to the correct selection values of PID controller constants. The effect of changing reference power factor values is fed to six pulse rectifier circuit through the variation of firing angle (α), the output of PID controller. Six-pulse rectifier circuit controlled the filed voltage of synchronous motor. These results shown in Fig.(11). An increase of phase difference between voltage and current waves is noticed due to decreasing of loading power factor, as shown in Fig.(12). The variation of synchronous motor torque, speed, real power, total real power, reactive power, total reactive power, observed from the supply (infinity bus-bar), are shown in figures (13,14 and 15) receptively. 5. CONCLUSION The controlled reactive power compensation is investigated in a single machine connected to infinite bus system with load. By analyzing the dynamic behavior of synchronous machine. Then this machine was used to control the total reactive power observed by the load. PID controller has been designed and implemented to control the DC field voltage via the control of reactive power demand by load. REFERENCES [1] L. M. Faulkenberry and W. Coffer, “Electrical Power Distribution and Transmission”, Prentice Hall Inc., 1996. 5 [2] Glover and Sarma, ”Power System Analysis &Design”, PWS Publishing Company, Boston, Second Edition, 1994. [3] M. Pavella and P.G. Murthy, “Transient Stability of Power Systems”, John Wiley & Sons, 1994. [4] American Superconductor Corporation, ”Dynamic Reactive Power Comp.”, Super VAR Data Sheet, 2004. [5] E.M. John, ”Reactive Compensation Tutorial”, IEEE Publishing 2002,pp. 515-519. [6] IEEE/PES Power System Stability Subcommittee Special Publication, “Voltage Stability Assessment, Procedures and Guides”, Final Version, December 2000. [7] I. Boldea and S.A. Nasar, “Electric machine dynamics”, Macmillan pub. comp., New York 1986. [8] MATLAB Manual on SIMULINK 6.5, 2002. [9] T. K. Sundararajan and A. A. Samson, “Design of Smoothing Inductance for Rectifier Circuits”, Int. J. Electronics, Vol.62, No.2, 2001, pp.295-304. [10] Katsuhiko Ogata, ”Modern Control Engineering”, Second Edition, Prentice-Hall International Inc., 1995. [11] C. L. Phillips and R. D. Harbor, “Feedback Control Systems”, Prentice-Hall Inter. Inc., Third Edition, 1996. [12] M. Ozdemir and G. Onbilgin, “Computer Simulation of a Series Excited Synchronous Motor”, Electric Power Components and Systems, Taylor & Francis, EMP31,pp. 565-578, 2003. [13] S. P. Das and A. K. Chattopadhyay, “ Analysis, Simulation and Test Results for A Cycloconverter-Fed AC Commutator-Less Motor Drive with A Modified Machine Model”, IEE Proc.-Electr. Power Appl. Vol.151, No.5, September 2004. [14] Taiyou Yong, Robert H. Lasseter and Wenjin Cui, “Coordination of Excitation and Governing Control Based on Fuzzy Logic”, PSerc 99-04, 2003. APPENDIX (A) The synchronous machine type, rating and parameters are: Salient pole, SIEMENS E484, Type 2GA 3363-1A 1KW, 4 pole,50HZ, 320/400V(for Y-Y connection) Max D.C. excitation voltage =110 Max D.C. excitation current = 0.75 Moment of inertia (J) = 0.015 Kg.m2 Friction of coefficient (K) = 0.0035 Kg.m2 /sec Synchronous reactance (Xs) = 0.878 p.u Leakage reactance (X1) = 0.0572 p.u Field resistance (Rf) = 168 Ω Field self inductance (Lff) = 19 H Stator resistance / phase (Ra,Rb,Rc) = 7.2 Ω Stator self inductance (static) / phase (La,Lb,Lc) = 0.0541H Mutual inductance between stator Phases (Lab,Lbc,Lac) = 0.027 H Max mutual inductance between the field and One of the stator phases (Laf,Lbf,Lcf) = 0.961 H Harmonic inductance due to saliency (Ls) = 0.01873 The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 APPENDIX (B) The matrix of synchronous machine inductance equal to: La + LS ⋅ cos(2θ ) − Lab + LS ⋅ cos(2θ − 2α / 3) − Lac + LS ⋅ cos(2θ + 2α / 3) Laf ⋅ cos(θ ) ⎤ ⎡ ⎢− Lab + LS ⋅ cos(2θ − 2α / 3) Lb + LS cos(2θ ) Lbf ⋅ cos(θ − 2α / 3)⎥⎥ − Lbc + Ls ⋅ cos(2θ ) [L] = ⎢ ⎢− Lac + Ls ⋅ cos(2θ + 2α / 3) Lc + cos(2θ ) Lcf ⋅ cos(θ + 2α / 3)⎥ − Lbc + Ls ⋅ cos(2θ ) ⎥ ⎢ Laf ⋅ cos(θ ) Lbf ⋅ cos(θ − 2α / 3) Lcf ⋅ cos(2θ + 2α / 3) Lff ⎦ ⎣ − 2 LS ⋅ sin(2θ ) − 2 LS ⋅ sin(2θ − 2α / 3) − 2 LS ⋅ sin(2θ + 2α / 3) − Laf ⋅ sin(θ ) ⎡ ⎤ ⎢− 2 LS ⋅ sin(2θ − 2α / 3) − 2 LS ⋅ sin(2θ + 2α / 3) − 2 LS ⋅ sin(2θ ) − Lbf ⋅ sin(θ − 2α / 3)⎥⎥ d [L] ⎢ = ⎢ − 2lS ⋅ sin(2θ + 2α / 3) − 2 LS ⋅ sin(2θ ) − 2 LS ⋅ sin(2θ − 2α / 3) − Lcf ⋅ sin(θ + 2α / 3) ⎥ dθ ⎢ ⎥ − Laf ⋅ sin(θ ) − Lbf ⋅ sin(θ − 2α / 3) − Lcf ⋅ sin(θ + 2α / 3) 0 ⎣ ⎦ and, Fig.(8) The mean model for the over all system which modeled in Matlab-Simulink. 6 Phase Current (Amper) Speed (rpm) The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 1500r.pm 1250r.pm 1.33A 0A -1.33A 250r.p.m 0r.pm (a) Speed (rpm) 1800 1600 1400 Fig.(10) The values motor power factor (PF) & reference power factor (PFref) change with time. 1200 Ia (Amper) 1000 (b) 3.99 2.66 1.33 0 -1.33 -2.66 (c) Fig.(9) Speed and phase current for synchronous motor connected to infinity bus-bar when the load change from 5N.m to zero(no load), Time axis equal to 200msec/division, where: (a) Practical result. (b)&(c) computer result. Fig.(11) Response of the field voltage & current with time when firing angle (α) change. 7 The 6th Jordanian International Electrical & Electronics Engineering Conference JIEEEC 2005 Fig.(12) Phase voltage & current with time when firing angle (α) change. Fig.(14) The values of total real power & motor real power (watt) with time when firing angle (α) change. Fig.(13) Response of synchronous motor torque & speed with time, when firing angle (α) change. Fig.(15) The values of total reactive power & motor reactive power (var) with time when firing angle (α) change. 8