Wind Power Impact on Power System Dynamic Part 3 docx

An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 57 frequency range min maxww ωωω ∗∗∗ << is large enough, with the generator current is much smaller than in the first mode, but there is an increase in generator voltage. Phases of the fundamental harmonics of current and voltage of the generator do not coincide. In this mode, the angle can be 0 SG ϕ > or 0 SG ϕ > . Vector diagram for the case 0 SG ϕ < is shown in fig. 5. Basic relations for the determination of voltages, currents and power in the system are given in (14) ÷ (26). For these values of the angle SG ϕ , as in the case of 0 SG ϕ = , the same value of power can be obtained in the two modes, corresponding to different values of the parameter q M . In the general case, when 0, 1 L qk≥≥the active power Ro P ∗ is related to q M by the relation: 2 2 0 0 1 qo Ro PuP uP uU PP RR γ ∗ ∗ ⎛⎞ − ⎛⎞ − ⎜⎟ += ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ , (45) where 2 00 2 ;1;; . 12 22 P LSG SG PuP P k q tg tg RUP q ϕϕ ωωω γ γ γ ∗∗∗ ⎛⎞ + ==+ == ⎜⎟ + ⎝⎠ Whence: 2 22 2 0 1,2 0 1,2 1,2 22 () ; ; ; . 33 P P Ro q ouP RoRo qq od Ro P PP uUR u PM u M P γγ γ ∗ ∗∗∗∗∗ ⎛⎞ − =− == = ⎜⎟ ⎜⎟ ⎝⎠ ∓ Here the indices "1" and "2" correspond to the 1st and 2d modes in accordance with fig.20. Maximum power achievable at a given frequency of rotation ( ω ∗ ) is defined by the relation: max 0 2 Ro P uP PPR γω ∗∗ =+ ≡ . (46) Relationships (97) make possible to determine the dependence of the currents and voltages in the system as a function of frequency of rotation for different values of the angle SG ϕ and the parameters q and k L . Major trends of these relationships can be seen on the graphs (22) ÷ (25). Let us consider the choice of mode of the system in WPI, while we assume that 0, 1 L qk==. In this case, the equation (45) in polar coordinates will be: () sec( )sin( ); ( ) sec( )sin( )cos( ); ( ) sec( )sin( )sin( ). SG SG Ro do SG SG qo SG SG PU u ρφ ω φ φ ϕ φ ω ϕϕϕφ φω ϕ φϕ φ ∗ ∗∗∗ ∗∗ =+ == + =+ (47) Fig.31 shows the nature of the proposed change of the angle ( SG ϕ ) of current shift ( Go i ∗ ) on voltage ( Go u ∗ ) and cos SG ϕ on the frequency of rotation of the shaft of WT. The proposed Wind Power 58 ∗ ω D WT WT ∗ ∗ = max min ω ω ∗ maxWT ω SG ϕ cos SG ϕ maxSG ϕ maxSG ϕ − 1 max cos SG ϕ ∗ WTmidl ω Fig. 31. 0 30 60 90 120 150 180 210 240 270 300 330 1 0.5 0 )( max φ ρ max φ ∗ Ro P ∗ qo u ∗∗ = max ωω 1> M 2 3 )()()( 2 lim 2 limlim =+= ∗∗ RqoRo uP φρ 1< M D ∗ ∗ = max min ω ω ∗ limmaxRo P ∗ minRo P 2min φ a b )( 2min φ ρ c Fig. 32. An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 59 scenario allows us to work with max maxlimWT ωω ∗∗ > remaining in the second mode ( maxlim ω ∗ is defined according to (44)). For this operating point with a maximum power of WT max max () WTo WT P ω ∗∗ is compatible with the maximal achievable power maxRo P ∗ (46), (47). In addition, we require that the power maxRo P ∗ corresponds to 1M = (fig.32), i.e. max maxlimRo Ro PP ∗∗ = . Angle max φ (fig.32) is determined from the equation () 0 d d ρφ φ = , max 42 SG ϕ π φ =− . We will find the frequency of rotation at which the equality max maxlimRo Ro PP ∗∗ = is realized, from the equation: max ()32 ρφ = , it follows that 1 max 3 sec( )sin 242 SG SG ϕ π ωϕ − ∗ ⎡ ⎤ ⎛⎞ =+ ⎜⎟ ⎢ ⎥ ⎝⎠ ⎣ ⎦ . (48) Based on (47) ÷ (48) maxlim 3 cos 242 SG Ro P ϕ π ∗ ⎛⎞ =− ⎜⎟ ⎝⎠ . Then we require that max max max maxlim ;. WT WTo Ro PP ωω ∗∗∗ ∗ == Proceeding from the equation (20) 33 maxlim max (), ()() Ro WTo PP γωωγω ∗∗∗∗∗ ==. In accordance with fig.31 we take maxSG SG ϕϕ =− when max ωω ∗∗ = . The law of change of SG ϕ in the operating range min max {, } WT WT ωω ω ∗∗ ∗ ∈ according to fig.31 will look as follows min max max min () 12 SG SG ωω φω ϕ ωω ∗∗ ∗ ∗∗ ⎛⎞ − =− ⎜⎟ ⎜⎟ − ⎝⎠ , where min min max maxWT WT DD ωω ω ω ∗∗ ∗ ∗ == =. Then 1 max max 3 sec( )sin 242 SG SG ϕ π ωϕ − ∗ ⎡ ⎤ ⎛⎞ =− ⎜⎟ ⎢ ⎥ ⎝⎠ ⎣ ⎦ ; max maxlim 3 cos 242 SG Ro P ϕ π ∗ ⎛⎞ =+ ⎜⎟ ⎝⎠ . The minimum power at minWT ωω ∗∗ = : 3 min maxlim Ro Ro PP D ∗∗ = . The locus corresponding to the frequency of rotation minWT ωω ∗∗ = is: min max max () sec( )sin( ) WT SG SG ρφ ω ϕ φ ϕ ∗ =+. The angle min φφ = at minWT ωω ∗∗ = is determined from the equation 3 min max min max min maxlim sec( )sin( )cos WT SG SG Ro PD ωϕφϕφ ∗∗ += maxlim min1 max max 2 max max min 2 min1 max 1 arcsin 2 sin( ) ; 2 sec( ) 2( ). Ro SG SG WT SG SG P D φϕϕ ωϕ φπφϕ ∗ ∗ ⎡⎤ =−− ⎢⎥ ⎢⎥ ⎣⎦ =− + (49) Wind Power 60 In the relation (49) angles min1 φ and min 2 φ correspond to the 1st and 2d modes. When the rotation frequency min maxWT WT ωω ∗∗ ↔ changes the two trajectories are possible (fig.32), namely, « ac↔ » and « ab↔ » with the first corresponding to the system in the 1st mode, and the second - in the 2nd mode. As already noted, the first mode is characterized by the low value of power factor and the big value of current. For this reason, the second trajectory is desirable, i.e. work in the second mode. In this case: 22 ()2 ()3; () 2 ( 2)[ ()] [ ()] (). dWTo q WTo SGWTo MP M P tgP ωω ωωω ωωϕωω ∗ ∗∗ ∗∗ ∗ ∗∗ ∗ ∗∗∗ ==+−+ For max 12 SG ϕ π = , 2D = and 3D = the result of calculation in fig. 33, when min max {, } WT WT ωω ω ∗∗ ∗ ∈ . 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.2 0.4 0.6 0.8 1 1.2 687.0 min = ∗ WT ω 374.1 max = ∗ WT ω ∗ ω ( ) 3∗∗ ⋅= ωγ WTo P 527.0 0659.0 SR χ SR ϕ cos ∗ Go u ∗ Go i 966.0cos = SR ϕ 12 ,1,0,2 max π ϕ ==== SGLWT kqD 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 458.0 min = ∗ WT ω 374.1 max = ∗ WT ω 12 ,1,0,3 max π ϕ ==== SGLWT kqD 527.0 0.0195 966.0cos = SR ϕ ( ) 3 ∗∗ ⋅= ωγ WTo P SR χ SR ϕ cos ∗ Go i ∗ Go u ∗ ω (a) (b) Fig. 33. As can be seen from the figure 33 that choice of scenario allows a wide range of changes of the frequency of rotation by increasing the value of max ω ∗ at the given value of cos SG ϕ . Dependence of max ω ∗ on the given value of the angle maxSG ϕ is presented in fig.34, which implies that the maximum achievable value of the frequency max ω ∗ for a given scenario of control is equal to 3 . It should be noted that the selected above the linear law of change of ( ) SG ϕ ω ∗ is not unique. In that case, if for the area of installing of WPI the prevailing wind speed is known, then the frequency of rotation of the shaft of WT is calculated and at an obtained frequency the point with cos 1 SG ϕ = is selected. The law of changes the function ( ) SG ϕ ω ∗ can be optimized according to the change in the winds, with equality cos SG ϕ at the extreme points of the operating range min max {, } WT WT ωω ∗∗ is not obligatory. Thus, the scenario of the WPGS system working according to the given law of change of cos SG ϕ with change of WT ω ∗ allows to increase the maximum operating frequency of rotation while maintaining the 2-second mode, which is characterized by relatively high value of power factor. An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 61 Fig. 34. 4. Basic power indicators in the circuit "voltage inverter - electrical network" The schematic diagram of the circuit "voltage inverter - electrical network» taking into account the accepted assumptions is shown in fig.35. The estimated mathematical model of the electrical circuit is shown in fig. 4. dc U dc C 1 VT 2 VT 3 VT `1 VT `2 VT `3 VT uvw Iu u uN u ` vN u ` wN u ` fI L Iu i Iv i Iw i I S `N S dc S Fig. 35. Voltage of the electrical network changes according to the law: [ ] `` cos ( 1)2 3 ; ; 1,2,3 ( , , ). Nm N uU m tm uvw υ πυ =−− =Ω= Wind Power 62 Change laws of the inverter control signals are cos( ), Icm c m uu θ = where (1)23 ; mIc m θυ π ϕ =− − + Taking into account the accepted assumptions the mathematical model of an electric circuit in rotating system of coordinates, under condition of orientation on an axis of voltage of an electric network q, will look like: `IN IIIII I d LL dt −=+ + ⋅ Ω uu ri i i , (50) where: t IIdIq uu ⎡⎤ = ⎣⎦ u , [ ] `` 0 t NN U=u , t IIdIq ii ⎡ ⎤ = ⎣ ⎦ i - vectors of the inverter voltage and the mains voltage, vector of the inverter currents; `N U - the peak value of network voltage; { } , III diag=rrr , I r - the resistance of inductance of power filter and of the transformer windings; I L - the equivalent inductance of the power filter and the transformer leakage inductance; 0 0 −Ω ⎡ ⎤ = ⎢ ⎥ Ω ⎣ ⎦ Ω , Ω - circular frequency of the network voltage. Neglecting the active resistance the ratio (50) can be written in a scalar form ` ,. gd Iq Id I I I q I q NI dId di di uL Li uU L Li dt dt Σ ≈−Ω⋅ −≈+Ω⋅ (51) A mathematical model of the inverter will be determined by the relations (5) ÷ (8). In these relationships we take: ` 3 dc N Udc UU δ =⋅ ⋅ , where ` 3 N U⋅ - is the minimal possible voltage in a direct current link with SPWM, Udc δ - is excess of the minimal possible voltage of a link of a direct current. As before, in order to preserve the universality of the results of the analysis, we introduce the following relative units: ` ;; б N б EU ω ==Ω ; ббI XL ω = ; бкз бб II Е X== 32; ; ббб IcI SEI a ω ==Ω where cI ω - a cyclic frequency of the PWM inverter. Taking into account relative units the equation (51) will become: ,1 , Iq Id Id I q I q Id di di uiu i dd υυ ∗ ∗ ∗∗∗ ∗ =− −=+ where t υ =Ω . The voltages Id u ∗ and I q u ∗ are determined by the relations ;; Id Ido Id RI q I q oI q uu u u u u ∗∗ ∗ ∗ ∗ ∗ =+Δ =+Δ 3333 sin( ) ; cos( ) ; 2222 Ido Udc Rc Udc d I q oUdc Rc Udc q uM MuM M δϕδ δϕδ ∗∗ =−=− = −= here Ido u ∗ , I q o u ∗ - the orthogonal components in the d and q coordinates of the fundamental harmonic of inverter voltage; Id u ∗ Δ , I q u ∗ Δ - the orthogonal components in the d and q coordinates of the high-frequency harmonics of inverter voltage. An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 63 We will define the high-frequency harmonics for SPWM from the relations (14). The equation for the inverter current can be represented as the sum of the fundamental ( Ido i ∗ , I q o i ∗ ) and the high frequency ( Id i ∗ Δ , I q i ∗ Δ ) harmonics ; Id Ido Id I q I q oI q ii i ii i ∗∗ ∗ ∗∗ ∗ =+Δ =+Δ. The fundamental harmonic of the inverter current is determined by the relation ;1. Iqo Ido Id Iqo iuiu ∗∗∗∗ =− = − The high-frequency harmonics of the inverter current for SPWM are determined from the relations (16). We assume such a control law of inverter, when the WPI in electrical circuit generates only an active power. Then the vector diagram for the fundamental harmonic of current and voltage will have the form shown in fig.36. Under such a control 1; ; 0. Iqo Io Iqo Ido Id uiiui ∗∗∗∗∗ ===− = Generated in the electrical network active power is: No I q oIo Ido Pii u ∗∗ ∗ ∗ ===−. (52) Vector diagram for the orthogonal components ( , d q M M ) of the inverter control signal in «d q» coordinates is presented in fig.37 The quantities , d q M M and Ic ϕ are determined by the relations: ** 2 /( 3 ), 2 /( 3 ), / q Udc d No Udc Ic d q SNo M MP arct g MMarct g P δδϕ == ==. (53) The linear range of work of the inverter is limited by a condition: 22 1; ()() 2 . 3 dq SPWM MM SVPWM − ⎧ ⎪ +≤ ⎨ − ⎪ ⎩ Fig. 36. Wind Power 64 Fig. 37. From (53) (in the case of equality), we obtain an expression for the maximum active power ( maxNo P ∗ ), which can be transferred to the electricity grid without distortion of the current. max 2 3 1,1 2 ()1, . Udc No Udc SPWM P SVPWM δ δ ∗ ⎧ ⎛⎞ ⎪ − ⎜⎟ ⎪ ⎜⎟ = ⎨ ⎝⎠ ⎪ −− ⎪ ⎩ The dependence of maxNo P ∗ on the value of Udc δ is shown in fig.38, which implies that the minimum value of minUdc δ at which the generation of active power begins is given by: min 2 ; 3 1. Udc SPWM SVPWM δ ⎧ − ⎪ = ⎨ ⎪ − ⎩ 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 ∗ maxNo P SVPWM SPWM 3 2 cUd δ Fig. 38. An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 65 Dependence of the active power No P ∗ from the relative values of voltage in the DC link ' /3 Udc dc N UU δ = and the depth (index) of modulation M can be found from the following relation: 2 (3 /2) 1 No Udc PM δ ∗ =−. (54) Taking into account the active losses in the output circuit of the generating system an expression for the fundamental harmonic current and the active power will look like 2 2 2 12 [1 ( ) ] 1 ; 1( ) 3 Io No R R RUdc iP M ωω ωδ ∗∗ ∗ ∗ ∗ ⎧ ⎫ ⎛⎞ ⎪ ⎪ == ⋅−++ − ⎜⎟ ⎨ ⎬ ⎜⎟ + ⎪ ⎪ ⎝⎠ ⎩⎭ where R б RX ω ∗ = , R - the equivalent active resistance of the inverter phase. Graph of this dependence (at 0 R ω ∗ = ) for SPWM and SVPWM is shown in fig.39, which implies that the adjustment range of active power decreases with decreasing of Udc δ . It should be noted that when working on electrical network application of SVPWM can significantly increase the active power. As follows from fig.40 for each the value of Udc δ there is a minimum value of modulation depth min M below which the generating active power is equal to zero min 2( 3 ) Udc M δ = . The dependence of min M on Udc δ is shown in fig.40. 0.88 0.9 0.92 0.94 0.96 0.98 1 0.1 0.2 0.3 0.4 0.5 0.6 M ∗ No P SPWM 3.1= cUd δ 275.1 25.1 225.1 2.1 0.85 0.9 0.95 1 1.05 1.1 1.15 0.2 0.4 0.6 0.8 1 M ∗ No P SVPWM 3.1= cUd δ 25.1 2.1 15.1 1.1 (a) (b) Fig. 39. We determine how vary the coefficients of harmonics ( iI THD ) and distortions ( iI ν ) of the inverter current. In accordance with the relations (52) and (19) the effective value of the fundamental harmonic of inverter current ( ,Io rms i ∗ ), its fluctuating components ( ,Irms i ∗ Δ ) and the total effective value ( ,Irms i ∗ ) are defined as follows: , 2 Io rms No iP ∗∗ = , () 1 2 2 2 ,1 22 3 1 (1)(1) Udc I Irms II a iJM Xaa δ π π ∗ ∗ Σ ⎛⎞ ⎛⎞ + Δ= ⋅ ⋅ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⋅+− ⎝⎠ ⎝⎠ , 22 ,, , ()( ) Irms Iorms Irms ii i ∗∗ ∗ =+Δ (55) Wind Power 66 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.85 0.9 0.95 1 1.05 1.1 1.15 SPWM SVPWM min M cUd δ Fig. 40. In fig.41 the dependence of the magnitudes iI THD and iI ν as a function of the depth of modulation M with 1.3 Udc δ = is presented. As follows from this figure, the qualitative indicators of a current are much worse with a decrease in modulation depth, while the value 0.05 iI THD = is reached at 1M → and 1.3 Udc δ ≥ . For the Russian standards, the quality of the generated electric current in the WPI network must fulfill the condition 0.05 iI THD ≤ . It should also be noted that iI THD practically does not depend on the inductance I L and is determined only by the multiplicity of frequencies I a and the ratio of voltages Udc δ . Taking into account that the phase of the inverter current coincides with the phase of voltage of the electrical network, as well as a sinusoidal change of the voltage, taking into account the relations (13) we obtain the following expression for the power factor in the cross section N S : NNoNiI Р S χ ν ∗∗ == . We define the changes in iI THD and iI ν in the WPI, as function of the frequency of rotation of the shaft of WT. We assume 3 max () WTo WT P γωω ∗ =⋅ . (56) We define the coefficient γ according to the condition max max () WTo WT No PP ω ∗∗ = , then maxNo P γ ∗ = . Based on the (54) and (56) we obtain the dependence of modulation depth on the frequency of rotation of the shaft of WT 2 3 max max 2 1 3 No WT Udc MP ω ω δ ∗ ⎡⎤ ⎛⎞ ⎢⎥ =⋅+ ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ . (57) In fig.42 the dependence of M on maxWT ωω for the two types of modulation (SPWM and SVPWM) is presented. It implies that the modulation depth varies slightly. Knowing the dependence of M on maxWT ωω , we can determine the changes of the qualitative characteristics of the generated energy as the function of the frequency of rotation of the shaft of WT. For this we use the relations (55) (57). In fig.43 graphs of iI THD and iI ν on maxWT ωω for SPWM are presented. [...]... Elista 72 Wind Power converter №1 G №1 converter №2 Network converter 3 G №2 converter №4 G 3 G №4 Fig 49 1.5 ∗ PNo max 1.19 1 ∗ PNo ∗ PNo min 0.59 0.5 1 converter 0 1 1.29 2 converters 1.5 1.62 3 converters 4 converters 2 2.05 2 .34 2.5 2.58 ω∗ 3 Fig 50 5 Conclusions 1 2 3 4 A mathematical model for analysis of energy characteristics of electric power generation system consisting of a synchronous generator... the real size wind power generator, which is a 3. 8 MW 17.5 rpm radial flux PM-generator consisting of 3 stator modules 2 Drive system Drive topologies used in high power wind generators are still mainly conventional solutions i.e asynchronous induction generators with a gearbox coupling These are limited speed 74 Wind Power range slip-ring induction generators directly connected to the power grid and... other These segments can be considered, for example, as independent stator parallel windings which are each fed by an own frequency converter If one of the frequency converters fails, other ones can continue the operation while the failed units are changed Hence, there is no long period total power interruption, because the wind turbine operation can continue anyway at a reduced power level This chapter... installations / Scientific bulletin NSTU, Novosibirsk, 1999 92-120 p., in Russian 3 Speed Sensorless Vector Control of Permanent Magnet Wind Power Generator – The Redundant Drive Concept Tero Halkosaari Vacon Oyj Finland 1 Introduction Permanent Magnet motors (PM-motors) have become more and more popular, especially in low speed and high speed applications The conventional motor drive, consisting... leads to exclusion from the spectrum of the current of n . Elista. Wind Power 72 converter №1 converter №2 converter 3 converter №4 Network G №1 G №2 G 3 G №4 Fig. 49. 1 1.5 2 2.5 3 0 0.5 1.5 1.29 1.62 2.05 2 .34 2.58 0.59 1.19 1 converter 1 ∗ ω ∗ No P ∗ maxNo P ∗ minNo P 2 converters 3 converters 4 converters . 2.58 0.59 1.19 1 converter 1 ∗ ω ∗ No P ∗ maxNo P ∗ minNo P 2 converters 3 converters 4 converters Fig. 50. 5. Conclusions 1. A mathematical model for analysis of energy characteristics of electric power generation system consisting of a synchronous. shift. 3. 2 Speed sensorless control concept In order to increase the drive system reliability, speed and position sensorless vector control can be used. In fact, the wind power generation is one