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The interactionsamong the wind turbine, the power network, and the capacitor compensation are impor-tant aspects of wind generation that may result in self-excitation and higher harmoniccontent in the output current. This paper examines the factors that control these phenom-ena and gives some guidelines on how they can be controlled or eliminated.?DOI: 10.1115/1.2047590?1IntroductionMany of todays operating wind turbines have fixed speed in-duction generators that are very reliable, rugged, and low cost.During normal operation, an induction machine requires reactivepower from the grid at all times. Thus, the general practice is tocompensate reactive power locally at the wind turbine and at thepoint of common coupling where the wind farm interfaces withthe outside world. The most commonly used reactive power com-pensation is capacitor compensation. It is static, low cost, andreadily available in different sizes. Different sizes of capacitorsare generally needed for different levels of generation. A bank ofparallel capacitors is switched in and out to adjust the level ofcompensation. With proper compensation, the power factor of thewind turbine can be improved significantly, thus improving over-all efficiency and voltage regulation. On the other hand, insuffi-cient reactive power compensation can lead to voltage collapseand instability of the power system, especially in a weak gridenvironment.Although reactive power compensation can be beneficial to theoverall operation of wind turbines, we should be sure the compen-sation is the proper size and provides proper control. Two impor-tant aspects of capacitor compensation, self-excitation ?1,2? andharmonics ?3,4?, are the subjects of this paper.In Sec. 2, we describe the power system network; in Sec. 3, wediscuss the self-excitation in a fixedspeed wind turbine; and inSec. 4, we discuss harmonics. Finally, our conclusions are pre-sented in Sec. 5.2Power System Network DescriptionWe investigate a very simple power system network consistingof one 1.5 MW, fixed-speed wind turbine with an induction gen-erator connected to a line feeder via a transformer ?2 MVA, 3phase, 60 Hz, 690 V/12 kV?. The low-speed shaft operates at22.5 rpm, and the generator rotor speed is 1200 rpm at its syn-chronous speed.A diagram representing this system is shown in Fig. 1. Thepower system components analyzed include the following:An infinite bus and a long line connecting the wind turbineto the substationA transformer at the pad mountCapacitors connected in the low voltage side of the trans-formerAn induction generatorFor the self-excitation, we focus on the turbine and the capaci-tor compensation only ?the right half of Fig. 1?. For harmonicanalysis, we consider the entire network shown in Fig. 1.3Self-Excitation3.1TheNatureofSelf-ExcitationinanInductionGenerator. Self-excitation is a result of the interactions amongthe induction generator, capacitor compensation, electrical load,and magnetic saturation. This section investigates the self-excitation process in an off-grid induction generator; knowing thelimits and the boundaries of self-excitation operation will help usto either utilize or to avoid self-excitation.Fixed capacitors are the most commonly used method of reac-tive power compensation in a fixed-speed wind turbine. An induc-tion generator alone cannot generate its own reactive power; itrequires reactive power from the grid to operate normally, and thegrid dictates the voltage and frequency of the induction generator.Although self-excitation does not occur during normal grid-connected operation, it can occur during off-grid operation. Forexample, if a wind turbine operating in normal mode becomesdisconnected from the power line due to a sudden fault or distur-bance in the line feeder, the capacitors connected to the inductiongenerator will provide reactive power compensation, and the tur-bine can enter self-excitation operation. The voltage and fre-quency during off-grid operation are determined by the balancebetween the systems reactive and real power.One potential problem arising from self-excitation is the safetyaspect. Because the generator is still generating voltage, it maycompromise the safety of the personnel inspecting or repairing theline or generator. Another potential problem is that the generatorsoperating voltage and frequency may vary. Thus, if sensitiveequipment is connected to the generator during self-excitation,that equipment may be damaged by over/under voltage and over/under frequency operation. In spite of the disadvantages of oper-ating the induction generator in self-excitation, some people usethis mode for dynamic braking to help control the rotor speedduring an emergency such as a grid loss condition. With theproper choice of capacitance and resistor load ?to dump the energyfrom the wind turbine?, self-excitation can be used to maintain thewind turbine at a safe operating speed during grid loss and me-chanical brake malfunctions.The equations governing the system can be simplified by look-ing at the impedance or admittance of the induction machine. ToContributed by the Solar Energy Division of THEAMERICANSOCIETY OFMECHANI-CALENGINEERSfor publication in the ASME JOURNAL OFSOLARENERGYENGINEERING.Manuscript received: February 28, 2005; revised received: July 22, 2005. AssociateEditor: Dale Berg.Journal of Solar Energy EngineeringNOVEMBER 2005, Vol. 127 / 581Copyright 2005 by ASMEDownloaded 28 Mar 2008 to 211.82.100.20. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmoperate in an isolated fashion, the total admittance of the induc-tion machine and the rest of the connected load must be zero. Thevoltage of the system is determined by the flux and frequency ofthe system. Thus, it is easier to start the analysis from a node atone end of the magnetizing branch. Note that the term “imped-ance” in this paper is the conventional impedance divided by thefrequency. The term “admittance” in this paper corresponds to theactual admittance multiplied by the frequency.3.2Steady-State Representation. The steady-state analysisis important to understand the conditions required to sustain or todiminish self-excitation. As explained above, self-excitation canbe a good thing or a bad thing, depending on how we encounterthe situation. Figure 2 shows an equivalent circuit of a capacitor-compensated induction generator. As mentioned above, self-excitation operation requires that the balance of both real andreactive power must be maintained. Equation ?1? gives the totaladmittance of the system shown in Fig. 2:YS+ YM?+ YR?= 0,?1?whereYS? effective admittance representing the stator winding, thecapacitor, and the load seen by node MYM? effective admittance representing the magnetizing branchas seen by node M, referred to the stator sideYR? effective admittance representing the rotor winding asseen by node M, referred to the stator side?Note: the superscript “?” indicates that the values are referred tothe stator side.?Equation ?1? can be expanded into the equations for imaginaryand real parts as shown in Eqs. ?2? and ?3?:R1L/?R1L/?2+ L1L2+RR?/S?RR?/S?2+ LLR?2= 0?2?where1LM?+L1L?R1L/?2+ L1L2+LLR?RR?/S?2+ LLR?2= 0?3?R1L= RS+RL?CRL?2+ 1L1L= LLSCRL?CRL?2+ 1RS? stator winding resistanceLLS? stator winding leakage inductanceRR? rotor winding resistanceLLR? rotor winding leakage inductanceLM? stator winding resistanceS? operating slip? operating frequencyRL? load resistance connected to the terminalsC? capacitor compensationR1LandL1Laretheeffectiveresistanceandinductance,respectively, representing the stator winding and the load as seenby node M.One important aspect of self-excitation is the magnetizing char-acteristic of the induction generator. Figure 3 shows the relation-ship between the flux linkage and the magnetizing inductance fora typical generator; an increase in the flux linkage beyond a cer-tain level reduces the effective magnetizing inductance LM?. Thisgraph can be derived from the experimentally determined no-loadcharacteristic of the induction generator.To solve the above equations, we can fix the capacitor ?C? andthe resistive load ?RL? values and then find the operating points fordifferent frequencies. From Eq. ?2?, we can find the operating slipat a particular frequency. Then, from Eq. ?3?, we can find thecorresponding magnetizing inductance LM?, and, from Fig. 3, theoperating flux linkage at this frequency. The process is repeatedfor different frequencies.As a base line, we consider a capacitor with a capacitance of3.8 mF ?milli-farad? connected to the generator to produce ap-proximately rated VAR ?volt ampere reactive? compensation forfull load generation ?high wind?. A load resistance of RL=1.0 ? isused as the base line load. The slip versus rotor speed presented inFig. 4 shows that the slip is roughly constant throughout the speedrange for a constant load resistance. The capacitance does notaffect the operating slip for a constant load resistance, but a higherresistance ?RLhigh=lower generated power? corresponds to alower slip.The voltage at the terminals of the induction generator ?pre-sented in Fig. 5? shows the impact of changes in the capacitanceFig. 1The physical diagram of the system under investigationFig. 2Per phase equivalent circuit of an induction generatorunder self-excitation modeFig. 3A typical magnetization characteristic582 / Vol. 127, NOVEMBER 2005Transactions of the ASMEDownloaded 28 Mar 2008 to 211.82.100.20. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmand load resistance. As shown in Fig. 5, the load resistance doesnot affect the terminal voltage, especially at the higher rpm?higher frequency?, but the capacitance has a significant impact onthe voltage profile at the generator terminals. A larger capacitanceyields less voltage variation with rotor speed, while a smallercapacitance yields more voltage variation with rotor speed. Asshown in Fig. 6, for a given capacitance, changing the effectivevalue of the load resistance can modulate the torque-speedcharacteristic.These concepts of self-excitation can be exploited to providedynamic braking for a wind turbine ?as mentioned above? to pre-vent the turbine from running away when it loses its connection tothe grid; one simply needs to choose the correct values for capaci-tance ?a high value? and load resistance to match the turbinepower output. Appropriate operation over a range of wind speedscan be achieved by incorporating a variable resistance and adjust-ing it depending on wind speed.3.3Dynamic Behavior. This section examines the transientbehavior in self-excitation operation. We choose a value of3.8 mF capacitance and a load resistance of 1.0 ? for this simu-lation. The constant driving torque is set to be 4500 Nm. Note thatthe wind turbine aerodynamic characteristic and the turbine con-trol system are not included in this simulation because we aremore interested in the self-excitation process itself. Thus, we fo-cus on the electrical side of the equations.Figure 7 shows time series of the rotor speed and the electricaloutput power. In this case, the induction generator starts from rest.The speed increases until it reaches its rated speed. It is initiallyconnected to the grid and at t=3.1 seconds ?s?, the grid is discon-nected and the induction generator enters self-excitation mode. Att=6.375 s, the generator is reconnected to the grid, terminatingthe self-excitation. The rotor speed increases slightly during self-excitation, but, eventually, the generator torque matches the driv-ing torque ?4500 Nm?, and the rotor speed is stabilized. When thegenerator is reconnected to the grid without synchronization, thereis a sudden brief transient in the torque as the generator resyn-chronizes with the grid. Once this occurs, the rotor speed settles atthe same speed as before the grid disconnection.Figure 8?a? plots per phase stator voltage. It shows that thestator voltage is originally the same as the voltage of the grid towhich it is connected. During the self-excitation mode ?3.1 s?t?6.375 s?, when the rotor speed increases as shown in Fig. 7, thevoltage increases and the frequency is a bit higher than 60 Hz.The voltage and the frequency then return to the rated valueswhen the induction generator is reconnected to the grid. Figure8?b? is an expansion of Fig. 8?a? between t=3.0 s and t=3.5 s tobetter illustrate the change in the voltage that occurs during thattransient.4Harmonic Analysis4.1SimplifiedPerPhaseHigherHarmonicsRepresentation. In order to model the harmonic behavior of thenetwork, we replace the power network shown in Fig. 1 with theper phase equivalent circuit shown in Fig. 9?a?. In this circuitrepresentation, a higher harmonic or multiple of 60 Hz is denotedFig. 4Variation of slip for a typical self-excited inductiongeneratorFig. 5Terminal voltage versus rotor speed for different RLandCFig. 6The generator torque vs. rotor speed for different RLand CFig. 7The generator output power and rotor speed vs. timeJournal of Solar Energy EngineeringNOVEMBER 2005, Vol. 127 / 583Downloaded 28 Mar 2008 to 211.82.100.20. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmby h, where h is the integer multiple of 60 Hz. Thus h=5 indicatesthe fifth harmonic ?300 Hz?. For wind turbine applications, theinduction generator, transformer, and capacitors are three phaseand connected in either Wye or Delta configuration, so the evenharmonics and the third harmonic do not exist ?5,6?. That is, onlyh=5,7,11,13,17,., etc. exist.4.1.1Infinite Bus and Line Feeder. The infinite bus and theline feeder connecting the wind turbine to the substation are rep-resented by a simple Thevenin representation of the larger powersystem network. Thus, we consider a simple RL line representa-tion.4.1.2Transformer. We consider a three-phase transformerwith leakage reactance ?Xxf? of 6 percent. Because the magnetiz-ing reactance of a large transformer is usually very large com-pared to the leakage reactance ?XM? open circuit?, only theleakage reactance is considered. Assuming the efficiency of thetransformer is about 98 percent at full load, and the copper loss isequal to the core loss ?a general assumption for an efficient, largetransformer?, the copper loss and core loss are each approximately1 percent or 0.01 per unit. With this assumption, we can computethe copper loss in per unit at full load current ?I1 Full?Load=1.0 per unit?, and we can determine the total winding resistanceof the primary and secondary winding ?about one percent in perunit?.4.1.3Capacitor Compensation. Switched capacitors representthe compensation of the wind turbine. The wind turbine we con-sider is equipped with an additional 1.9 MVAR reactive powercompensation ?1.5 MVAR above the 400 kVAR supplied by themanufacturer?. The wind turbine is compensated at different levelsof compensation depending on the level of generation. The ca-pacitor is represented by the capacitance C in series with the para-sitic resistance ?Rc?, representing the losses in the capacitor. Thisresistance is usually very small for a good quality capacitor.4.1.4InductionGenerator.Theinductiongenerator?1.5 MW,480 V,60 Hz? used for this wind turbine can be repre-sented as the per phase equivalent circuit shown Fig. 9?a?. Theslip of an induction generator at any harmonic frequency h can bemodeled asSh=h?s?rh?s?4?whereSh? slip for hth harmonich? harmonic order?s? synchronous speed of the generator?r? rotor speed of the generatorThus for higher harmonics ?fifth and higher? the slip is close to 1?Sh=1? and for practical purposes is assumed to be 1.4.2Steady State Analysis. Figure 9?b? shows the simplifiedequivalent circuit of the interconnected system representinghigher harmonics. Note that the magnetizing inductance of thetransformers and the induction generator are assumed to be muchlarger than the leakages and are not included for high harmoniccalculations. In this section, we describe the characteristics of theequivalent circuit shown in Fig. 9, examine the impact of varyingthe capacitor size on the harmonic admittance, and use the resultof calculations to explain why harmonic contents of the line cur-rent change as the capacitance is varied.From the superposition theorem, we can analyze a circuit withonly one source at a time while the other sources are turned off.For harmonics analysis, the fundamental frequency voltage sourcecan be turned off. In this case, the fundamental frequency voltagesource ?infinite bus?, Vs, is short circuited.Wind farm operator experience shows us that harmonics occurwhen the transformer operates in the saturation region, that is, athigher flux levels as shown in Fig. 3. During the operation in thissaturation region, the resulting current can be distorted into asharply peaked sinusoidal current due to the larger magnetizingcurrent imbedded in the primary current. This nonsinusoidal cur-rent can excite the network at resonant frequencies of the network.From the circuit diagram we can compute the impedance ?at anycapacitance and harmonic frequency? seen by the harmonicsource, Vh, with Eq. ?5?, where the sign “?” represents the words“in parallel with:”Fig. 8The terminal voltage versus the time. a Voltage duringself-excitation. b Voltage before and during self-excitation,and after reconnection.Fig. 9The per phase equivalent circuit of the simplified modelfor harmonic analysis584 / Vol. 127, NOVEMBER 2005Transactions of the ASMEDownloaded 28 Mar 2008 to 211.82.100.20. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmZ?C,h? = ?Zline+ 0.5Zxf?0.5Zxf+ ZC?Zgen?5?whereZline? Rline+jXline? line impedanceZxf? Rxf+jXxf? transformer leakage impedanceZC? RC+?jh?C?1? capacitor impedanceZgen? Rgen+jXgen? generator impedanceThe admittance at any capacitance and harmonic frequency can befound from the impedance:Y?C,h? =1Z?C,h?6?For a given harmonic, the harmonic current is proportional tothe admittance shown in Eq. ?6? multiplied by the correspondingharmonic voltage. Because the field data only consist of the totalharmonic distortion of the capacitor current, and do not provideinformation about individual harmonics, we can only compare thetrends from the admittance calculation to the measured data. Fig-ure 10?a? shows the total calculated admittance for all harmonicsof interest up to the 23rd harmonic excluding, as explained earlier,the even harmonics and all multiples of the third harmonic, plot-ted as a function of the total reactive power ?in per unit?. Changesin the reactive power are created by changing the size of thecompensation capacitors. For comparison, the measured data ofthe total harmonic distortion as a function of the total reactivepower ?in per unit? are presented in Fig. 10?b?. Note that the twoplots show very similar behaviorboth exhibit resonance at reac-tive power levels around 0.25 and 0.65.From Fig. 10, we can say that the circuit will resonate at dif-ferent frequencies as the capacitor C is varied. Two harmoniccomponents must exist to generate harmonics currents in thesystemsa harmonic source ?due to magnetic saturation as shownin Fig. 3? and a circuit that will resonate at certain levels of ca-pacitance compensatio