Reactive power capability of WECS based on matrix converter
is controlled according to [2]:
R. Ca´rdenas, R. Pen˜a, P. Wheeler and J. Clare
where kopt is dependent on the blade profile, gearbox ratio and WECS size. A simplified analysis of (2) is realised by neglecting the slip velocity and assuming that in the steady-state the voltage applied to the induction machine is proportional to the machine rotational speed vr , i.e.:
The reactive power capability of a variable speed wind energy conversion system (WECS), based on an induction generator fed by a matrix converter is presented. A simplified analysis is performed neglecting the system losses. It is demonstrated that the maximum reactive power that can be supplied to the grid is 40% of the nominal power. The results obtained in this work are experimentally validated using a 3 kW experimental prototype.
Introduction: Matrix converters (MCs) have many advantages when compared to conventional back-to-back converters. The converter is smaller, lighter and more reliable [1]. Because of these characteristics matrix converters are a good alternative for variable-speed operation of WECSs. The topology of the variable-speed WECS studied in this work is shown in Fig. 1. On the grid side an MC supplies the real power current, idi , and the reactive current, iqi. The reactive power used by the induction generator (IG) is supplied from the MC output (see iqo in Fig. 1). When back-to-back converters are used to interface the WECS to the grid, the reactive current is limited by the current rate of the grid-side converter [2]. However, when the WECS is connected to the grid using an MC, the maximum reactive power that can be supplied to the utility is a nonlinear function of the active power and the displacement angle. In this Letter, the reactive power capability of the WECS, shown in Fig. 1, is discussed.
gearbox
cage induction machine
grid
ido variable-speed wind turbine
iqi idi
iqo
Fig. 1 Proposed WECS based on induction generator fed by matrix converter
Reactive power capability: In an MC-based system, q ¼ vo/vi is defined as the voltage transfer ratio [3], where vi and vo are the MC input and output voltages, respectively. Assuming that the MC is operating with a zero displacement angle at the input, then the highest value of voltage transfer ratio, q, that it is possible to achieve without creating low-frequency p distortion in the input current and output voltage is ideally qmax ¼ 3/2 [3]. However, in a typical system, issues such as commutation limitations, digital effects, resolution of the timer counters, etc., reduce the maximum value of the voltage transfer ratio achievable to qlim qmax. Moreover, when the MC is operating with a displacement angle, f, at the input, the maximum value of voltage transfer ratio achievable, without creating low-frequency distortion, is further reduced to a value of qlf ¼ qlim cos ( f ) [3]. Assuming that the MC is continuously operating with the maximum displacement angle achievable at the input, then q ¼ qlf , and fmax ¼ cos21(q/qlim). Using this relationship the maximum reactive current supplied to the grid is:
iqi ¼ idi tanðfmax Þ ) iqi ¼ idi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qlim =q 1
ð1Þ
In (1) idi and iqi are referred to a rotating frame oriented along the input voltage vector. Neglecting the losses, the maximum reactive current supplied to the grid can be obtained as: iqi ¼ ðPout =vdi ÞtanðfÞ ) iqi ¼ ðPout =vdi Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qlim =q 1
vo ¼ qvdi ’ k vr
ð3Þ
ð4Þ
Using (3) and (4) in (2) and putting q2 ¼ (kvr/vdi)2 yields: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iqi ¼ ðkopt =kÞv2r q2 q2 lim qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iqi ¼ ðkopt =ðkvdi ÞÞ v4r q2 v2di k 2 v6r lim
ð5Þ ð6Þ
The operating point that maximises the reactive current supplied to the grid is obtained by maximising (6). Therefore: 2 ð@iqi =@vr Þ ¼ 0 ) 4 qlim vdi v3r 6 k 2 v5r ¼ 0 ð7Þ Using q 2 ¼ (kvr/vdi)2 in (7) and after some manipulation it can be shown that: pffiffiffiffiffiffiffiffi 2 4 qlim vdi 6ðk vr Þ2 ¼ 0 ) q ¼ 2=3 qlim
ð8Þ
p Therefore, if qlim ¼ 3/2, then from (8) the maximum reactive current is supplied when q ¼ 0.707. Assuming that at nominal speed the machine is operating with a nominal voltage transfer ratio of qnom ¼ qlim then the rotational speed, vr0 , corresponding to the maximum current, iqimax , is obtained from (4) and (8) as: pffiffiffiffiffiffiffiffi ð9Þ ðvr0 =vn Þ ’ ðq=qlim Þ ¼ 2=3 ’ 0:81
matrix converter
IM
Pout ¼ kopt v3r
ð2Þ
where Pout is the output power per phase. To operate the WECS at the point of maximum aerodynamic efficiency, the generator output power
where vn is the nominal speed. Note that, using (9) in (3), the power, Pout0 , supplied to the grid at the rotational speed, vr0 , is (2/3)3/2 of the nominal power (Pnom) or Pout0 ’ 0.544p.u. Using (4), (8) and (9) in (2) and replacing idin ¼ (Pnom)/vdi , the current, iqimax , is obtained as: rffiffiffi rffiffiffi!3 2 Pnom 1 iqi max ¼ ’ 38:5% of idin 3 vdi 2
ð10Þ
Therefore, for the proposed WECS, in steady-state operation the maximum reactive power that can be supplied to the grid is ’0.385 p.u., corresponding to the operating point vr0 ’ 0.81 p.u. and Pout0 ’ 0.544 p.u. If a higher value of reactive power is required, then the system has to be designed for a nominal voltage transfer ratio, qnom , qlim. Experimental results: The reactive power capability of the proposed WECS has been experimentally tested using a 3 kW experimental prototype. A speed-controlled motor, driving the IG, is used to emulate a variable-speed wind turbine or other prime mover. A DSP system is used to implement the matrix modulation algorithm [3]. The IG is vector controlled with ido ¼ 3.12 A and a torque current of iqo proportional to v2r (see (3)). In the experimental work, the regulation of the induction generator currents has a higher priority over the regulation of the reactive power. Therefore, in the DSP-based control the generator currents are regulated first, then the regulation of the displacement angle is realised by the control software. To validate the simplified analysis of (1)– (10), the rotational speed of the induction generator is regulated to follow a ramp varying between 0 and 1300 rpm in 100 s. The results are shown in Fig. 2. Fig. 2a shows voltage ratio against speed. As predicted by (4) the relationship between the average value of q and vr is almost linear. To validate the relationship between vr0 and iqimax , as predicted by (9), the maximum capacitive/inductive MC input current with respect to the rotational speed is obtained, using two ramp speed variation tests similar to that shown in Fig. 2a. As shown in Fig. 2b, the responses for the capacitive and inductive cases are almost symmetrical (assuming qlim ¼ 0.7). For the capacitive case the maximum reactive current supplied to the grid is 37.5% when the rotational speed is 0.81 p.u. For the
ELECTRONICS LETTERS 22nd May 2008 Vol. 44 No. 11
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 500
600
700
is nonlinear; losses have been neglected in (1) –(10) and the input voltages change slightly depending on whether the matrix converter is supplying or absorbing reactive power from the grid. The experimental results of Fig. 2 are obtained with a slow ramp speed. However, because wind turbines have large inertia, the steadystate analysis is valid. This is corroborated by the experimental results of Fig. 3. For this test a wind turbine is emulated using the driving machine and a wind profile. For low wind turbine inertia there are high-frequency components in the reactive power. However, when the inertia is close to the nominal value the high-frequency components are virtually eliminated from iqi. reactive power, var
voltage transfer ratio q
inductive case the maximum reactive current supplied to the grid is 37% when the rotational speed is 0.8 p.u. Therefore the maximum reactive currents are close to the value of 38.5% obtained from (10). Moreover, the rotational speeds are close to the value of 0.81 p.u. obtained from (9). Assuming that the nominal current is obtained when the machine is operating with the average voltage transfer ratio, q ¼ qlim , then the active current idi ’ 0.54 p.u. when vr ¼ 0.81 p.u. (see Fig. 2b).
800 900 1000 1100 1200 1300 rotational speed, rpm
current, p.u.
a 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1
J = 0.25 p.u.
2000 1000 0
J = 1 p.u.
capacitive
iqi
qlim = 0.7
iqi idi
0.95
40 time, s
50
60
70
# The Institution of Engineering and Technology 2008 18 February 2008 Electronics Letters online no: 20080446 doi: 10.1049/el:20080446
qlim = 0.7
0.46
30
max
capacitive
inductive
20
Conclusions: The reactive power capability of a WECS based on a matrix converter has been presented. It has been demonstrated that the maximum reactive power supplied to the grid, at any operating point, is about 40% of the nominal power. If a higher value of reactive power is required, then the system has to be designed for qnom , qlim. Experimental results have corroborated the analysis.
active 0.65 0.75 0.85 rotational speed, p.u.
10
Fig. 3 Reactive power obtained using several values of wind turbine inertia and emulation of variable-speed wind turbine
inductive
b
current, p.u.
1000 0
0
0.55
0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4
2000
max
0.50 0.54 0.58 voltage transfer ratio q
0.62
R. Ca´rdenas and R. Pen˜a (Electrical Engineering Department, University of Magallanes, PO Box 113-D, Punta Arenas, Chile)
c
E-mail:
[email protected] Fig. 2 Experimental results for speed ramp variation
P. Wheeler and J. Clare (Electrical and Electronic Engineering School, University of Nottingham, Nottingham University Park, Nottingham NG7 2RD, United Kingdom)
a Voltage ratio q against rotational speed for speed ramp change b Matrix input currents corresponding to test of Fig. 2a c Reactive currents against voltage transfer ratio for test of Fig. 2a
In Fig. 2c the currents and voltage transfer ratios corresponding to the test of Fig. 2b are shown. To validate the relationship predicted by (8), the current, iqimax , against the average q for the inductive and capacitive case are shown in Fig. 2c. For the capacitive case the maximum reactive current is supplied to the grid when the voltage ratio is q ’ 0.6. For the inductive case the maximum reactive power is supplied to the grid when the voltage transfer ratio is q ’ 0.54. The experimental results compare well with the value of q ¼ 0.57 predicted by replacing qlim ¼ 0.7 in (8). The reactive currents shown in Fig. 2 are not completely symmetrical for the inductive/capacitive case. This asymmetry is because the system
References 1 Wheeler, P.W., Rodriguez, J., Clare, J.C., Empringham, L., and Weinstein, A.: ‘Matrix converters: a technology review’, IEEE Trans. Ind. Electron., 2002, 49, (2), pp. 276 –288 2 Ca´rdenas, R., and Pen˜a, R.: ‘Sensorless vector control of induction machines for variable speed wind energy applications’, IEEE Trans. Energy Convers., 2004, 19, (1), pp. 196– 205 3 Casadei, D., Serra, G., Tani, A., and Zarri, L.: ‘Matrix converter modulation strategies: a new general approach based on space-vector representation of the switch state’, IEEE Trans. Ind. Electron., 2002, 49, (2), pp. 370–381
ELECTRONICS LETTERS 22nd May 2008 Vol. 44 No. 11
