Wind Power 2011 Part 3 pdf

An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 47 8.0,0,0.1 === ∗ ω qk L q d ∗∗ = 21 GqoGqo ii ∗ 2Gdo i ∗ 1Gdo i 0 30 60120 150 180 210 240 270 300 330 1 q qk Puu L RoRdoRdo + + == ∗∗∗ 1 21 ∗ 1Rqo u ∗ 2Rqo u 2 ∗ maxRo P Fig. 20. From fig.20 it is clear that the larger value of voltage u ∗ corresponds to the lower value of current Go i ∗ , while 12G q oG q o ii ∗∗ = that follows from (5a). 12 2 2 1,2 1,2 , 11 1. 22 Gqo Gqo Rdo q Ro Rqo LRo Gdo iiuXP u kP qq i ω ωω ∗∗ ∗∗∗∗ Σ ∗ ∗ ∗ ∗∗ ⎧ == = ⎪ ⎪ ⎨ ⎛⎞ ++ ⎛⎞ ⎪ =− = ± − ⎜⎟ ⎜⎟ ⎜⎟ ⎪ ⎝⎠ ⎝⎠ ⎩ At the point when the power is maximum for a given frequency of rotation ( maxRo Ro PP ∗∗ = ), (the relation (34)), the orthogonal components of currents and voltages are determined by the relations (31), (37). The full value of the generator current is: 2 2 1,2 1 2 1(1 ) 1 2 2(1)(1) LRo Ro Go L kP q P ik qq ωω ∗ ∗ ∗ ∗∗ ⎛⎞ ⎛⎞ + =+− ±− ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ++ ⎝⎠ ⎝⎠ . Wind Power 48 Graphs of the dependences R q o u ∗ , Go i ∗ and Go u ∗ from ω ∗ are presented in fig.30. In these graphs to the right of points «a, b, c, d» there is a limitation of the depth of modulation, and the proposed model becomes inadequate to the real modes. The first mode on the graphs of fig.21 is characterized by the fact that the generator voltage does not change significantly with variation of ω ∗ . From fig.21d it follows that in mode 1 the system has a positive internal differential resistance and therefore may be potentially unstable at some disturbing effects. 0 0.5 1 -0.5 2.0,05.1 == qk L )1( 2 max qu RqoP −= ∗ ∗ ω ∗ 12Rqo u 0.5 1 1.5 2 ∗ ω 2.0= ∗ Ro P 4.0 6.0 8.0 operating mode 1 operating mode 2 0 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2 1.4 a c b d ω 2.0= ∗ Ro P 4.0 6.0 8.0 ∗ Go i ∗ limGo i L L k k q + + 1 2 1 2.0,05.1 == qk L operating mode 1 operating mode 2 (a) (b) 00.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω ∗ ∗ Go u ∗ limGo u 2 )1( 1 2 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − +⋅ ∗ qk qk L L ω 2.0= ∗ Ro P 4.0 6.0 8.0 a c d b 2.0,05.1 == qk L operating mode 1 operating mode 2 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 1.2 ∗ Ro P ∗ Go u ∗ limGo u a b 4.0= ∗ ω 6.0 8.0 1 2.1 2.0,05.1 == qk L operating mode 1 operating mode 2 (c) (d) Fig. 21. An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 49 It should be noted that the mode when the phases of voltage and current of the fundamental harmonics are the same, essentially involves a change of Go u ∗ and Go i ∗ under the change of frequency of rotation of the shaft of WT ( ω ∗ ). The equation (28) can be rewritten in the co-ordinates of orthogonal components of a control input wave ( , ) d q M M : 2 2 22 (1) 3 () 1 (1 ) (1 ) 33 q d u Mq M qq ω ωω γ ∗ ∗∗ ⎡⎤ +− ⎢⎥ ⎣⎦ += ⎡⎤⎡⎤ ++ ⎢⎥⎢⎥ ⎣⎦⎣⎦ , (38) Expression (38) defines a parametric relationship between the orthogonal components, which ensures the phase coincidence of the main of harmonics of the generator voltage and current. This ratio allows us to propose the following control algorithm, in which the active power is given. This solution is useful for the application of the system in the WPI. In accordance with (16b), (21) and (38), we obtain: - given the active power - Ro P ∗ , when the condition max max (1 ) 2 Ro Ro L PP qk ω ∗∗∗ ≤=+ must be respected; - the longitudinal component of the modulation ( d M ) of the control signal is determined by 2( ) 3(1 ) dLRo M kqP q ∗ =+ + ; - the transverse component of the modulation ( q M ) of the control input wave is determined by 2 2 1,2 2 (1) (1) 33 3 qLRo Mq q kP ωω ∗∗ ∗ ⎡⎤ ⎛⎞ =− − + − ⎢⎥ ⎜⎟ ⎝⎠ ⎢⎥ ⎣⎦ ∓ - here 1 q M - corresponds to the mode 1, and 2 q M - to the mode 2. When 0q > , the fundamental harmonics of inverter current and voltage (section R S ) do not coincide in phase. The current phase is ahead of the voltage phase ( 0 SR ϕ > ). Fig. 22a shows the dependence of cos SR ϕ on the parameters q and L k . From figure 22a it follows that in the first mode there is a significant reduction of cos SR ϕ with the increase of the parameter q . Dependence of cos SR ϕ on the parameter L k is ambiguous, namely, in the first mode cos SR ϕ decreases with the increase of L k , while in the second mode, on the contrary increases. The phases of the fundamental harmonics of generator voltage and current (section G S ) are always the same ( cos 1 SG ϕ = ), the power factor in section S G will be determined according to (15a) by the relation GSGSGRoSGiGuG Р S Р S χ νν ∗∗ ∗∗ === . The RMS of the fundamental harmonic of generator voltage is determined by the following expression: 22 2 2 1,2 1,2 22 , 2 33 (1) 13 1 () 2( 1) 2 2 2 L Go rms L MM kq M uq q kq q ω ωωω ∗ ∗ ∗∗∗ ⎡ ⎤⎡ ⎤ ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ + ⎢ ⎥⎢ ⎥ =− −+− ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎢ ⎥⎢ ⎥ + − ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ ⎣ ⎦⎣ ⎦ . (39) Wind Power 50 q 8.0,05.1 == ∗ ω L k 5.0= ∗ Ro P 4.0 3.0 2.0 1.0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 SR ϕ cos operating mode 1 operating mode 2 8.0,4.0 == ∗∗ ω Ro P L k SR ϕ cos 4.0 3.0 2.0=q 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 operating mode 1 operating mode 2 (a) (b) Fig. 22. Complete RMS of the generator voltage is found from the ratio 1,2 2 ,1,2 2 13 (cos) 1 4 2(1) Go rms M uM q ωθ ∗∗ ⎛⎞ =+− ⎜⎟ + ⎝⎠ . (40) where: θ - the angle shift between the fundamental harmonic of generator voltage and EMF of generator is defined as follows 1,2 (1 ) 2 3 L d L q kq M arctg kq M θ ω ∗ ⎡ ⎤ ⎢ ⎥ + =⋅ ⎢ ⎥ + ⎢ ⎥ + ⎢ ⎥ ⎣ ⎦ . The RMS of fundamental harmonic of the generator current () 2 2 , 1,2 1,2 1,2 2 13 13 1 2 2() Go rms q q Ro iMMP ωω ∗ ∗ ∗∗ ⎡ ⎤ ⎛⎞ ⎢ ⎥ =− + + ⎜⎟ ⎜⎟ ⎢ ⎥ ⎝⎠ ⎣ ⎦ (41) Complete RMS of the generator current is found from the relation (19) and (41). In fig.23 the distortion coefficients ( uSG ν ) and harmonics ( uSG THD ) of generator voltage as functions of the active power generated ( Ro Р ∗ ) and frequency of rotation ( ω ∗ ) are presented. As can be seen from fig.23 the best quality of generator voltage is characteristic for mode 2. In addition, analysis of the relations (39) and (40) suggests a reduction factor of harmonics uSG THD with the increase of the parameterq . Similar conclusions can be drawn from the consideration of fig. 24, which shows the distortion coefficients ( iSG ν ) and harmonics ( iSG THD ) of current as a function of the active power generated ( Ro Р ∗ ) and frequency of rotation ( ω ∗ ). In the engineering calculations we can take 1 iSG ν ≈ . Power factor of the generator according to (15a), taking into account that cos 1 SG ϕ = , is determined using the relation: GiGuG χ νν = . An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 51 0 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2.0,05.1 == qk L ∗ Ro P uSG ν 8.0= ∗ ω 6.0 4.0 2.0 limuSG ν operating mode 1 operating mode 2 ∗ ω uSG ν 2.0,05.1 == qk L 1.0= ∗ Ro P 2.0 3.0 4.0 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 lim uSG ν operating mode 1 operating mode 2 (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 ∗ Ro P uSG THD 2.0,05.1 == qk L 8.0= ∗ ω 6.0 4.0 2.0 operating mode 1 operating mode 2 00.20.40.60.8 0.4 0.5 0.6 0.7 0.8 0.9 1 uSG THD ∗ ω 1.0= ∗ Ro P 2.0 3.0 4.0 2.0,05.1 == qk L operating mode 1 operating mode 2 (c) (d) Fig. 23. Wind Power 52 0.3 0.4 0.5 0.6 0.7 0.8 0.98 0.985 0.99 0.995 1 ∗ ω iSG ν 2.0= ∗ Ro P 3.0 4.0 2.0,05.1 == qk L operating mode 2 operating mode 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 ∗ Ro P 2.0,05.1 == qk L 8.0= ∗ ω 6.0 4.0 2.0 iSG ν operating mode 2 operating mode 1 (a) (b) 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 iSG THD ∗ ω 3.0= ∗ Ro P 5.0 4.0 2.0,05.1 == qk L operating mode 2 operating mode 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 ∗ Ro P iSG THD 8.0= ∗ ω 4.0 2.0 6.0 2.0,05.1 == qk L operating mode 2 operating mode 1 (c) (d) Fig. 24. Dependence of G χ on the generated active power ( Ro Р ∗ ) and the frequency of rotation ( ω ∗ ) is shown in fig.25. From this figure and the previous findings it can be taken: GuG χ ν ≈ . As expected, the power factor is higher in mode 2. In the mode 1 with a decrease in power G χ is significantly reduced because of the need to reduce the modulation depth M in order to maintain cos 1 SG ϕ = . An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 53 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ∗ Ro P SG χ 8.0= ∗ ω 4.0 2.0 6.0 21,2.0,05.1 === RL aqk operating mode 1 operating mode 2 0.3 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 ∗ ω SG χ 2.0= ∗ Ro P 3.0 4.0 2.0,05.1 == qk L operating mode 1 operating mode 2 (a) (b) Fig. 25. From (33), (34) and (35) the conditions can be found under which the maximum power maxRo P ∗ (the point where modes 1 and 2 are the same, fig. 20) is attained at the beginning of the limitations of the modulation depth ( 1M = ), i. e. max limRo Ro PP ∗∗ = holds; the value of active power, in the performance of this condition is denoted as maxlimRo P ∗ . 1 2 2 2 maxlim max lim 1 3 21 (1 ) L Ro Ro Ro L qkq PPP q kq − ∗∗∗ ⎧ ⎫ ⎡⎤ ⎛⎞ −+ ⎪ ⎪ ===⋅ + ⎢⎥ ⎜⎟ ⎨ ⎬ + + ⎢⎥ ⎝⎠ ⎪ ⎪ ⎣⎦ ⎩⎭ . (42) The power maxlimRo P ∗ can be reached at a frequency of rotation maxlim ωω ∗∗ = : () {} 1 2 2 2 maxlim 3(1 ) ( ) LL qkqk ω − ∗ =⋅− + + . (43) Taking into account the taken relative units we determine the real value of active power maxlimRo P from the ratio () 1 2 2 2 2 maxlim d 1 3 L2 б Ro L б L q E Pkq k ω − ⎧ ⎫ ⎛⎞ − ⎪ ⎪ =⋅⋅ ++ ⎜⎟ ⎨ ⎬ ⎜⎟ ⎪ ⎪ ⎝⎠ ⎩⎭ . Figure 26 shows the active power 2 maxlim d L Ro бб PE ω and frequency maxlim ω ∗ as a function of the parameter q at different values of L k . As can be seen from fig.26, the maximum possible active power ( maxlimRo P ∗ ) and the corresponding speed of rotation in this mode ( maxlim ω ∗ ) occur at q = 0, k L = 1. From (42) and (43) we obtain: Wind Power 54 dб б Ro L E P ω 2 limmax q 0 0.5 1 1.5 2 0.2 0.3 0.4 0.5 0.6 0.7 3.11÷= L k q ∗ limmax ω 0 0.5 1 1.5 2 0.4 0.6 0.8 1 1.2 1.4 3.11÷= L k (a) (b) Fig. 26. ∗ Gqo u ∗ Ro P 2 lim 2 lim )()( ∗∗ + RqoRo uP ∗∗ = max ωω ∗ limmaxRo P Fig. 27. maxlim maxlim 3 2 1.2247 SPWM; 3 (2 2) 0.6124 SPWM; 1 2 0.7071 SVPWM; 2 1.4142 SVPWM. Ro P ω ∗∗ ⎧ ⎧ =− =− ⎪⎪ == ⎨⎨ =− =− ⎪ ⎪ ⎩ ⎩ (44) When we select the power generation system PGS in the WPI, it is convenient to use fig.27. In this figure (for q = 0, k L = 1) the trajectory « ab→ »corresponds to the points of maximum power ( maxRo P ∗ ) for the different frequencies of rotation ω ∗ . At the point « b » max maxlimRo Ro PP ∗∗ = and maxlim ωω ∗∗ = . With the further increase in frequency ω ∗ to keep the value cos 1 SG ϕ = without the restrictions of the modulation depth it should be the moving on a trajectory« bc→ ». If ω ∗ →∞ the point « c » is reached. The value of active power at the point« c »will be the maximum possible for cos 1 SG ϕ = . The value of this power is 0.866 maxlim 32 SPWM; () 1SVPWM. Ro P ω ∗∗ ⎧ =− ⎪ →∞ = ⎨ − ⎪ ⎩ An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 55 At work of the PGS from WT an active power will change under the law (20). The mode of the maximum power with maxWT ωω ∗∗ = is desirable by choosing from a condition max maxlimWT ωω ∗∗ = . In this case according to (20) and (44): 3 max max max maxlim 3 maxlim max ()() ; (). WTo WT WT Ro Ro WT PP P ωγω γω ∗∗ ∗ ∗ ∗∗ =⋅ = = When we change ω ∗ the operating point should move along the trajectory « ab→ » fig.28, where the power varies according to the law (20), but cos 1 SG ϕ = will be retained. For such a trajectory the dependences of the amplitude values of generator voltage and current ( Go u ∗ , Go i ∗ ), the power factor ( SG χ ) and the generated power WTo Ro PP ∗∗ = as a function of the frequency of rotation are shown in fig.29. In fig.29a the movement trajectory "« ab→ »" occurs in mode 1, in fig.29b, respectively, in mode 2. ∗ Ro P 2 lim 2 lim )()( ∗∗ + RqoRo uP ∗ Gqo u Fig. 28. As follows from fig.29 at work in the 1st mode, despite the fact that cos 1 SG ϕ = the power factor of the generator has a little value when minWT ωω ∗∗ → it is explained by sufficiently small value of the coefficient of distortion of the generator voltage at low frequencies. In addition, there is a large value of the generator current, so when minWT ωω ∗∗ → , 1 Go i ∗ → i.e. the current is close to the value of short-circuit current. When working in 2nd mode the power factor is much bigger, with the generator current is much smaller than in mode 1. In the 2nd mode, the generator voltage has increased, but it is less than the EMF-load of the generator current. Note that if we want to save cos 1 SG ϕ = in the entire working range and 1M ≤ at maxWT ωω ∗∗ = as well as to choose the frequency of rotation of WT from the condition Wind Power 56 max maxlimWT ωω ∗∗ > the working point of the trajectory at maximum frequency of rotation and maximum power generated will be in the 1st mode (fig.30) and, consequently, will have a low value of power factor. 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.2 0.4 0.6 0.8 1 SR χ ∗ ω ∗ Go u 3 )( ∗∗∗ ⋅== ωγ RoWTo PP 2247.1 max = ∗ WT ω 0.6123 min = ∗ WT ω ∗ Go i 6124.0 0765.0 1,0,2 === LWT kqD 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.2 0.4 0.6 0.8 1 ∗ ω 2247.1 max = ∗ WT ω 0.6123 min = ∗ WT ω 0765.0 6124.0 3 )( ∗∗∗ ⋅== ωγ RoWTo PP ∗ Go i SR χ ∗ Go u 1,0,2 === LWT kqD (a) (b) Fig. 29. 0 30 60 90 120 150 180 210 240 270 300 330 1.5 1 0.5 0 2247.1 maxlimmax === ∗∗∗ WT ωωω ∗ Gqo u ∗ Ro P 2 lim 2 lim )()( ∗∗ + RqoRo uP ∗∗∗ >= limmaxmax ωωω WT ∗∗∗ =⋅= limmax 3 maxmax )( RoWTWTo PP ωγ operating mode 1 operating mode 2 Fig. 30. Taking into account the results obtained, we can conclude that the work with cos 1 SG ϕ = in the 1st mode is not optimal for WPGS, because in the entire frequency range min max {, } WT WT ωω ω ∗∗ ∗ ∈ there is a large value of the generator current ( 1 Go i ∗ → at minWT ωω ∗∗ → ) and a low power factor ( SG χ ). If condition cos 1 SG ϕ = remain in the range of frequencies min max {, } WT WT ωω ∗∗ for WPGS should be recommended the second mode, since in this case, the power factor of the generator in the working [...]... in fig .37 The quantities M d , Mq and ϕ Ic are determined by the relations: * * Mq = 2 /( 3 Udc ), Md = 2 PNo /( 3 Udc ), ϕ Ic = arctgMd / Mq = arctgPSNo The linear range of work of the inverter is limited by a condition: ⎧1 − SPWM ; ⎪ ( M d )2 + ( M q )2 ≤ ⎨ 2 ⎪ 3 − SVPWM ⎩ Fig 36 ( 53) 64 Wind Power Fig 37 From ( 53) (in the case of equality), we obtain an expression for the maximum active power ∗... 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... 1.2 DWT = 3, q = 0, k L = 1, ϕ SG max = π 1.2 12 π 12 cos ϕ SR ∗ uGo 1 cos ϕ SR = 0.966 1 cosϕ SR = 0.966 ∗ iGo 0.6 χ SR 0.8 cosϕ SR χ SR 0.8 ∗ uGo 0.6 0.527 0.527 ( ) P∗ = γ ⋅ ω ∗ WTo 0.4 0.4 ( ) P∗ = γ ⋅ ω ∗ WTo 0.2 0.2 0.7 0.8 0.9 1 ∗ ωWT min = 0.687 1.1 1.2 1 .3 1.4 ∗ ωWT max = 1 .37 4 (a) ω∗ 0.0195 ω∗ 0.6 ∗ iGo 3 0.0659 3 0.4 0.6 0.8 1 ∗ ωWT min = 0.458 1.2 1.4 ∗ ωWT max = 1 .37 4 (b) Fig 33 As can... ⎛ 3 ⎞ ⎜ ⎜ 2 δUdc ⎟ − 1,1SPWM ⎟ ⎝ ⎠ (δUdc )2 − 1, −SVPWM ∗ The dependence of PNo max on the value of δUdc is shown in fig .38 , which implies that the minimum value of δUdc min at which the generation of active power begins is given by: ⎧ 2 − SPWM ; ⎪ δUdc min = ⎨ 3 ⎪ 1 − SVPWM ⎩ 1.4 ∗ PNo max 1.2 SVPWM 1 SPWM 0.8 0.6 0.4 0.2 δ Udc 0.9 Fig 38 1 1.1 2 1.2 3 1 .3 1.4 1.5 1.6 An Analytical Analysis of a Wind. .. (55) (57) In fig. 43 graphs of THDiI and ν iI on ω ωWT max for SPWM are presented An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 0 .35 1 THDiI δUdc = 1 .3 0 .3 ν iI a I = 24 18 0.98 12 67 δUdc = 1 .3 0.25 0.96 0.2 18 0.15 0.94 a I = 24 0.1 12 0.92 0.05 M 0.89 0.9 0.91 0.92 0. 93 0.94 0.95 0.96 0.97... machines very attractive for wind power generators, because in high power wind mills (> 1 MW), a wind turbine rotor is typically rotating about 10 to 20 rpm One important aspect of motor drives is the reliability of the drive system Reliability comes even more important in installations, where the maintenance is difficult i.e in wind mill nacelles The reliability of the PM wind power generators can be increased... is shown in fig 4 Sdc VT1 VT2 VT3 U dc Cdc VT1` VT2` iIu u Iu VT3` iIv u v iIw w SI L fI u N `u u N `v SN ` u N `w Fig 35 Voltage of the electrical network changes according to the law: uN `m = U N ` cos [υ − (m − 1)2π 3] ; υ = Ω t ; m = 1, 2, 3 ( u , v , w ) 62 Wind Power Change laws of the inverter control signals are uIcm = uc cos(θ m ) , where θ m = υ − (m − 1)2π 3 + ϕ Ic ; Taking into account... ∗ u qo 90 ∗ ω ∗ = ωmax 120 60 1 150 a b ρ (φmin 2 ) 30 ρ (φmax ) 0.5 ∗ ωmin = 180 φmax ∗ ωmax D 0 φmin 2 c P∗ ∗ PRo min Ro max lim M 1 240 Fig 32 ∗ PRo An Analytical Analysis of a Wind Power Generation System Including Synchronous Generator with Permanent Magnets, Active Rectifier and Voltage Source Inverter 59 ∗ ∗ ∗ scenario... can significantly increase the active power As follows from fig.40 for each the value of δUdc there is a minimum value of modulation depth Mmin below which the generating active power is equal to zero Mmin = 2 ( 3 Udc ) The dependence of Mmin on δUdc is shown in fig.40 1 0.6 ∗ PNo 0.5 ∗ PNo SPWM 0.6 0.4 0 .3 SVPWM 0.8 0.2 1.275 0.1 0.88 0.9 δUdc = 1 .3 0.4 δ Udc = 1 .3 1.25 1.25 0.92 1.225 0.94 0.2 1.2... Iqo Iq u∗ = Ido 3 3 3 3 δUdc M sin( −ϕ Rc ) = − δUdc Md ; u∗ = δUdc M cos( −ϕ Rc ) = δUdc Mq ; Iqo 2 2 2 2 here u∗ , u∗ - the orthogonal components in the d and q coordinates of the fundamental Ido Iqo harmonic of inverter voltage; Δu∗ , Δu∗ - the orthogonal components in the d and q Id Iq coordinates of the high-frequency harmonics of inverter voltage An Analytical Analysis of a Wind Power Generation . 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 30 0 33 0 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 . 1.4 0.2 0.4 0.6 0.8 1 1.2 458.0 min = ∗ WT ω 37 4.1 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 . 0.8 0.4 0.5 0.6 0.7 0.8 ∗ ω SG χ 2.0= ∗ Ro P 3. 0 4.0 2.0,05.1 == qk L operating mode 1 operating mode 2 (a) (b) Fig. 25. From (33 ), (34 ) and (35 ) the conditions can be found under which the maximum power maxRo P ∗