The simulation results show that the proposed strategy achieves fast torque ... But for the standard SMO, to ensure stability of the observer, a high switching gain ...
1
Encoderless Model Predictive Control of Back-to-Back Converter Direct-Drive Permanent-Magnet Synchronous Generator Wind Turbine Systems Zhenbin Zhang⋆ , Christoph Hackl⋆ , Fengxiang Wang, Zhe Chen, Ralph Kennel Institute of Electrical Drive Systems and Power Electronics, Technische Universität München Arcisstr. 21, D-80333 Munich,Germany Phone: +49 (89) 289-28445 Fax: +49 (89) 289-28336 Email: {james.cheung, christoph.hackl, fengxiang.wang, zhe.chen, ralph.kennel}@tum.de URL: http://www.eal.ei.tum.de/
ACKNOWLEDGMENTS This work is partially supported by the Chinese Scholarship Council and the TUM Graduate School. The authors marked with ⋆ did contribute equally to this paper. K EYWORDS , , , ,
,
A BSTRACT This paper presents encoderless model predictive control scheme with time-varying sliding mode observer for a complete wind turbine system. The wind turbine system consists of a back-to-back converter (AC/DC/AC) and a direct-drive permanentmagnet synchronous generator (PMSG). We give a complete model of the system and present encoderless fixed-frequency model predictive direct torque control of the generator and finite-set model predictive direct power control of active and reactive power on the grid side. The sliding mode observer utilizes a time-varying switching gain and a time-varying cut-off frequency to estimate rotor position and rotor speed without chattering. The proposed strategy is illustrated by simulations as a first proof of concept. The simulation results show that the proposed strategy achieves fast torque control dynamics and highly decoupled control of active and reactive power. I. I NTRODUCTION Direct-drive variable-speed wind turbine systems with multi-pole permanent-magnet synchronous generator (PMSG) and activefront-end back-to-back (AC/DC/AC) converter offer several advantages compared to wind turbine systems with induction generator (IG) and passive-front-end AC/DC/AC converter [1], [2]. The need for external excitation and copper losses in the rotor circuit is eliminated. Since the power density of PMSG is much higher than that of IG, PMSG based wind energy conversion systems have a smaller size which reduces cost and weight. Moreover, with a direct-drive mechanical drive train, heavy gear boxes are not required which reduces maintenance and cost and, thus, yields higher reliability and profitability. With an active-front-end AC/DC/AC converter, the PMSG is decoupled from the grid and a bi-directional power flow is feasible. Therefore, a higher potential for fault ride-through and grid voltage support ability is achieved. Moreover, it is more suitable for the ever increasing grid-code demands. Field oriented control (FOC) and direct torque control (DTC) schemes currently dominate both academic and industrial applications for generator side control (GSC) [3]. For the net (grid) side control (NSC), voltage oriented control (VOC) and direct power control (DPC) are two standard methods [4]. Nowadays, model predictive control (MPC) is regarded as a very promising control strategy, since it is simple to apply/tune and easily allows to consider constraints (e.g. actuator saturation) in control design. For drive control of permanent magnet synchronous motors, there are at least three kinds of established MPC schemes, namely: (i) finite set model predictive control (FS-MPC), (ii) configuration pre-selection model predictive control (CP-MPC) and (iii) modulation based model predictive control (M-MPC) [5], [6]. For the CP-MPC controller (also regarded as 2PC, see [7]), a group of power converter switching states are pre-selected and their operation times are pre-calculated. M-MPC (also known as PPC [7] or dead-beat control [8]) pre-calculates the exact voltage vector according to the desired state values based on the system model, and then a modulator generates the input voltage (control signal) by space vector modulation. Since M-MPC attains the smallest current ripples and a fixed switching frequency is feasible [7], it is a very reasonable choice for torque control. Whereas finite set model predictive direct power control (FS-MPC-DPC) is favored, since it allows to control active and reactive power on the grid side and the DC-link voltage simultaneously. Moreover, compared to linear current controllers, FS-MPC-DPC does not require the Clarke transformation [9]. Usually, rotor speed and/or position of the generator are required, not only for a safe operation, but also for torque and power control of the wind energy conversion system. Although it is straight forward to utilize a speed/position sensor, according to [10], more than 14% of the system failures are directly related to sensor failures and more than 40% of the system failures
2
are related to sensor failures in combination with failures of electrical and mechanical components. Clearly, system failures bring significant losses to the power production. This motivates the use of reliable and robust encoderless control strategies. Generally speaking, encorderless control methods for PMSM may be divided into the following two groups: (i) magnetic saliency detection based methods and (ii) state observer based methods [11]. State observer based methods usually estimate the rotor speed/position directly from the back-EMF and are applicable in the medium and the high speed range. Sliding mode observers (SMO) come with the attractive feature of robustness to disturbances, parameter deviations and system noise [12]. But for the standard SMO, to ensure stability of the observer, a high switching gain is required in the high speed range leading to a large amount of ripples in the back-EMF and, so, to large chattering in the estimated speed/position [13]. Due to the use of a low pass filter with fixed cut-off frequency, phase compensation will vary according to the operating speed. This needs to be considered and increases implementati6on effort [12]. Therefore we propose a sliding mode observer with time-varying switching gain and time-varying cut-off frequency of the low-pass filter to achieve rotor speed/position estimation (almost) without chattering and constant phase compensation. The contributions of this paper are: (i) modeling of wind power, permament-magnet synchronous generator, back-to-back converter and grid side filter in a unified and mathematical thorough way (ii) encoderless model predictive control scheme for direct torque control and active and reactive power control of the overall PMSG wind turbine system (iii) introduction of a time-varying (adaptive) sliding mode observer for robust speed/position estimation of PMSG without chattering and constant phase compensation (iv) a brief study on robustness considering deviations in the resistance of generator and net side filter. This paper is outlined as follows: In Sec. 2, we give a complete model of the wind turbine system including back-to-back converter, direct-drive permanent-magnet synchronous generator and net side filter. In Sec. 3, we propose an encoderless model predictive control scheme with time-varying sliding mode observer: finite-set model predictive direct power control for the grid (net) side converter and model predictive direct torque control with space vector modulation for the generator side converter. Finally, in Sec. 4, simulation results are presented as proof of concept. Sec. 5 summarizes the results. II. M ODELING OF FULL CONVERTER DIRECT- DRIVE PMSG WIND TURBINE SYSTEM The block diagram of the complete wind turbine system is depicted in Fig. 1. The incoming wind having speed vw [m/s] and wind power Pw [W] is converted to mechanical (generator) power Pg [W] of the permanent-magnetic synchronous generator. The generator side converter (left) of the back-to-back converter transfers the power to the DC-link with power PDC [W]. This power is then transmitted by the grid side converter (right) to the net (grid) as active Pn [W] and reactive power Qn [Var]. In the following we will derive an overall mathematical model of this wind energy conversion system.
Wind
Wind Turbine Pw
Back-to-Back Converter
Net
PDC
Pg
Pn , Qn abc iabc n , vnsc
abc iabc g , vg
ωm
vabc n
PMSG
Tm
ω∗m
Ig sag
Ls
Rs
+
scg
sbg
In
iDC
a b
san
Rf
a
vDC
ng
scn
sbn
Lf
b
c
nn
c
PMSG 1 − sag
1 − sbg
1 − scg
o
1 − san
1 − sbn
1 − scn
Filter and net (grid)
Figure 1: Block diagram and circuit of the back-to-back direct-drive permanent-magnet synchronous generator (PMSG) wind turbine system. A. Wind turbine The mechanical power extracted by the wind turbine from the passing wind is given by [14]: 1 Pw (t) = ρA v3w (t)Cp (λ(t), β(t)), 2
(1)
3
where Cp (λ, β) ≤ Cp,Betz = 16/27 ≈ 0.56 [1] is the power coefficient of the wind turbine. For our simulations later, we use the power coefficient for a specific wind turbine design (see [14]) given by 0.035 1 − , (2) Cp (λ, β) = 0.22(116∆ − 0.4β − 5)−12.5∆ , ∆ = λ + 0.08β β3 + 1 where λ = Rtvωwm [1] is the tip speed ratio (depending on wind speed vw , turbine radius Rt [m] and rotor speed ωm [rad/s]) and β [rad] is the pitch angle. For this wind turbine, the maximum power coefficient Cp (λ, β)= 4.8 is reached for λ = 8.4 [1] and β = 0 [rad]. For simplicity, we assume that wind power is ideally transformed to mechanical power in the generator (otherwise introduce an adequate efficiency factor). Hence, Pw (t) = Pg (t) = Tm (t)ωm (t)
where
Tm [Nm] is the turbine (mechanical) torque.
(3)
B. Permanent-magnet synchronous generator (PMSG) In most cases, for a direct-drive wind turbine system, a smooth surface Np -pole PMSG is used and the saliency can be neglected. Moreover, the armature reaction affect is negligible. Then, the mathematical model of a PMSG in direct-quadrature (dq) reference frame (indicated by superscript dq ) is given by: dψdq (t) Rs 0 0 −ωe (t) dq vdq (t) = i (t) + + ψdq (t), ψdq (0) = ψ0 ∈ R2 (4) g g 0 Rs ωe (t) 0 dt dq
q
where vg (t) = (vdg (t), vg (t))⊤ [V] is the generator side converter output voltage vector (to be specified in later), Rs [V/A] is dq q the stator resistance, ig (t) = (idg (t), ig (t))⊤ [A] is the generator current vector, ψdq (t) = (ψd (t), ψq (t))⊤ [Vs] is the flux linkage (in the stator of the generator) and ωe (t) = Np ωm (t) [rad/s] is the electrical frequency of the rotor (rotating with ωm ). The flux dq linkage is assumed linearly related to current ig (t), stator inductance Ls [Vs/A] and (constant) permanent-magnet flux linkage ψm [Vs] as follows Ls 0 ψm ψdq = . (5) idq + g 0 0 Ls
The dynamics of the mechanical wind turbine system are given by
dωm (t) = Tm (t) − Te (t) + (Fωm )(t), ωm (0) = ω0m ∈ R where Te (t) = Np ψm iqg (t) [Nm] (6) dt is the electric torque of the generator, J kg m2 is the overall inertia (of turbine & generator), Tm [Nm] is as in (3), and Fωm models nonlinear, dynamic friction effects (for details, see [15, Sec. 1.4.5]). J
C. Back-to-back converter and DC-link As is shown in Fig. 1, we assume a balanced generator and load (grid/net); both are in star connection with star point nn and ng , resp. Hence, the phase output voltages (from connectors a, b, c to ng and nn ) of generator side converter and grid side converter are given by a a vg vn 2 −1 −1 v v DC DC b = b = (7) vabc Mabc sabc and vabc Mabc sabc where Mabc = −1 2 −1 , g = vg g nsc = vn n 3 3 c c −1 −1 2 vn vg
respectively. Clearly, both output voltage vectors depend on the switching state vector a sb sc ⊤ 3 on generator side and sabc = sa sb sc ⊤ ∈ {0, 1}3 on net (grid) side. Mabc is the transs sabc = ∈ {0, 1} g g g g n n n n formation matrix describing the relation between switching state vector and output phase voltage vector of the converter. Invoking the Clarke and Park transformation, i.e. # r " 1 −√21 2 1 − cos θe sin θe αβ dq 2 √ M := and M (θe ) := , resp., (8) 3 − sin θe cos θe 3 0 − 23 2
the phase voltage vectors of generator, net side converter and net (grid) can be transformed into dq or αβ reference frame as follows vDC αβ vDC dq αβ abc and vαβ (9) , vαβ vdq M (θe )Mαβ Mabc sabc M Mabc sabc n = M vn , resp. nsc = g = g | {z n } 3 3 | {z } αβ
β
⊤ =:sg =(sα g , sg )
The DC-link dynamics are given by (neglecting resistive losses) 1 dvDC (t) = In (t) − Ig (t) , dt C where ⊤ abc αβ ⊤ αβ In (t) = iabc and n (t) sn = in (t) sg
αβ
β
⊤ =:sg =(sα n , sn )
vDC (0) = v0DC ∈ R
(10)
⊤ abc αβ ⊤ αβ Ig (t) = iabc g (t) sg = ig (t) sg
(11)
are the currents from the net side converter and the generator side converter, respectively.
4
D. Filter and load (grid/net) The mathematical model of the filter and the net (grid) in stator fixed frame (indicated by superscript αβ ) is given by " 1 # Rf αβ − 0 0 din (t) − L f L αβ 0 2 f iαβ = (vαβ iαβ R n (t) + n (t) − vnsc (t)), n (0) = in ∈ R 0 − L1f dt 0 − L ff αβ
(12)
β ⊤
where L f [Vs/A] and R f [V/A] are filter inductance and resistance, resp., in = (iαn , in ) [A] is the current vector to the grid, ⊤ αβ αβ β ⊤ vn = (van , vbn ) [V] is the (fixed) grid voltage vector and vnsc = (vαnsc , vnsc ) [V] is the output voltage vector of the net side converter as in (9). III. E NCODERLESS MODEL PREDICTIVE CONTROL OF THE PMSG WIND TURBINE SYSTEM A. Design of the time-varying sliding mode observer Since we are interested in estimating (electrical) speed ωe = Np ωm and (electrical) position θe = Np θm , we transform the PMSG as in (4) to the αβ reference frame (invoking (Mdq (θe ))−1 ) and express the dynamics in the current (i.e. inserting (5) into (4) αβ and solving for dig /dt). We obtain the following nonlinear model " # " # ! αβ −Rs −1 0 0 dig (t) −ψ ω (t) sin θ (t) m e e αβ αβ 0 2 L L s s vg (t) − = , iαβ (13) −Rs ig (t) + −1 g (0) = ig ∈ R , ψ ω (t) cos θ (t) 0 0 dt m e e Ls Ls | {z } {z } | | {z } αβ
αβ
αβ
=:Ag
=:eg (t)
=:Bg
β
αβ
αβ
αβ abc sabc is the output voltage vector of the where eg = (eαg , eg )⊤ [V] is the back-EMF voltage vector and vg = vDC g 3 M M generator side converter. Following the idea in [16], we introduce a sliding mode observer for indirect speed estimation from current observation. But similar to [13], we do not utilize a discontinuous “switching function” (such as sign(·)) but rather a sigmoid “switching function” to reduce/avoid chattering in the estimate(s). For this paper, we utilize the following smooth “switching function”
fsig : R2 → [−1, 1] × [−1, 1],
ε 7→ fsig (ε) :=
Moreover, for the current estimation error εαβ g (t) =
εαg (t) β εg (t)
!
2 −1 1 − e−aε
αβ := ˆiαβ g (t) − ig (t)
where
with tuning parameter
ˆiαβ g (t) :=
a > 0.
(14)
! iˆαg (t) , β iˆg (t)
(15)
we will reduce the noise sensitivity of our observer, by implementing a time-varying (adaptive) low pass filter of the following form ! α (0) x dx f (t) f x f (0) = β = x0f ∈ R2 , (16) = −kc ω∗m (t) x f (t) − fsig εαβ g (t) , dt x f (0)
where kc > 0 is a tuning parameter. The filter in (16) has a time-varying cut-off frequency kc ω∗m (t) and changes with speed dx (t) αβ reference ω∗m (t). Note that, in steady state (i.e. dtf = 0), we have x f (t) = fsig εg (t) . Combining altogether, we introduce the following time-varying sliding mode observer (estimates indicated by ˆ) αβ dˆig (t) αβ 2 ∗ αβ ˆ0 ˆαβ , ˆiαβ vαβ = Aαβ g (0) = ig ∈ R , g (t) − k f x f (t) − ks ωm (t)fsig εg (t) g ig (t) + Bg dt with tuning parameters k f , ks > 0. Substraction of (17) and (13) yields the sliding mode dynamics αβ
dεg (t) αβ ∗ αβ αβ , eαβ = Aαβ g (t) − k f x f (t) − ks ωm (t)fsig εg (t) g εg (t) + Bg dt
(17)
2 ˆ0 0 εαβ g (0) = ig − ig ∈ R ,
(18) dε
αβ
(t)
In [16] it is shown that, for a → ∞ in (14) (i.e. fsig → sign), k f = 0 and ks ≫ 1 (with ω∗m = 1), “sliding mode” (i.e. gdt = 0) αβ αβ exists for (18) and the following holds eˆ g (t) := ks sign(εg (t)). Clearly, due to the sign-function the estimate is subject to chattering and highly noise sensitive. Our simulative analysis showed that for a sufficiently large value of ks ≫ 1 and −1 < k f < 0 in (18), “sliding mode” still exists and so we have the following equivalent signal which allows to estimate speed and position: ! Z t α (t) α (t) x e ˆ (18) (13) g −1 −1 f ˆ ˆ ˆ e (τ) dτ . eˆ αβ (t) := (1 + k )x (t) =⇒ ω (t) = − tan = − tan ω (19) =⇒ θ (t) = f f e e g β eˆαg (t) 0 x f (r)
Future work will include a thorough mathematical analysis (with proof).
5
Some remarks on the tuning parameters kc and k f kc : Due to the low-pass filter, as described in [13], we need to introduce a position compensation θcom = tan−1 kcωˆωe∗ = m tan−1 kc1Np otherwise we would have a lag in the position estimate. In our case we achieve that by a constant compensation term, which highly simplifies the hardware implementation. αβ
k f : From (18), it is evident that, x f =
eˆ g 1+k f
αβ
can be increased for 0 > k f > −1. Hence, even for small values of keg k the filter αβ
output x f can be made large, increasing the estimation accuracy (even in the low speed range when Rs ig is not small and, so, not negligible). ω∗m
Cut-off Frequency
ks ω∗m
αβ
vg
Time-vary.
Time-vary. Switching Gain
αβ ig
Generator Model for SMO
ˆiαβ g
Speed Calculation
θˆ e
kc ω∗m
+
ˆe ω
Low Pass
Calculation
Filter
Sigmoid Function
θcom Compensation
Figure 2: Time-varying sliding mode observer B. Predictive direct torque (current) control of generator with space vector modulation For the implementation of the model predictive control schemes we will switch to the discrete state space representation. For where TS > 0 [s] is the sampling time and x[k] = x(kTS ) discretization, we apply the forward Euler method, i.e. x˙ (t) ≈ x[k+1]−x[k] TS is the state vector at sampling instant kTS . Applying this to (4) and introducing the parameter estimates instead of the real parameters (e.g. Lˆ s for Ls ) yields the discrete model of the PMSG in the dq reference frame (predicted values indicated by ˆ) " # "T # Rˆ s TS S ˆ 1 − T 0 dq ω [k] 0 S e ˆs ˆs dq L L ˆidq ˆ v [k] + . (20) ig [k] + ˆ ˆ g [k + 1] = ˆ e [k] − TSLˆψm ω 0 TLˆS g ˆ e [k] 1 − RLsˆTS −TS ω s s s | {z } | {z } | {z } dq
ˆ g [k] =:A
dq
dq
ˆ g [k] =:C
ˆg =:B
To achieve the maximum torque to current ratio, we choose the following current reference ! d∗ 0 [k + 1] i ∗ g where Te∗ [k] comes from a PI controller (see Fig. 3). = Te∗ [k+1] idq q∗ g [k + 1] = ig [k + 1] ˆm Np ψ
dq ∗ dq Substituting ig [k + 1] for ˆig [k + 1] in (20) gives the reference voltage dq ˆdq dq dq −1 dq∗ ˆ ˆ ˆ i [k + 1] − A [k] i [k] − C [k] , vdq∗ [k + 1] = ( B ) g g g g g g
(21)
(22)
to achieve dead-beat torque (current) control of the PMSG. To compensate for the computational delay, we additionally αβ ∗ dq ∗ implement the delay compensation method described in [17]. The transformed reference voltage vg [k + 1] = (Mdq )−1 vg [k + abc 1] is then sent to the space vector modulator (SVM) to generate the appropriate switching vector sg for the generator side converter (see Fig. 3). C. Net side model predictive direct power control Again, applying the forward Euler method to (12) and replacing real parameters with estimates (e.g. L f by Lˆ f ), we obtain the discrete model of the net (grid) side filter as follows Rˆ T 1 − Lˆf S 0 0 − LTˆ S αβ f ˆiαβ vαβ f ˆiαβ [k] − v [k] . (23) [k] + T ˆ n [k + 1] = n nsc n S R T 0 − Lˆ 0 1 − Lˆf S f f {z } | {z } | dq
ˆ g [k] =:A
dq
ˆg =:B
6
For a sampling time much smaller than the net (grid) period, i.e. TS ≪ 1/ fn (where fn = 50 [Hz]), it is reasonable to assume αβ αβ that the net voltage does not change within one sampling interval, i.e. vn [k + 1] = vn [k]. Now to predict active and reactive power, given by (predicted values indicated by ˆ) 0 1 αβ ⊤ αβ αβ ˆ i [k + 1], (24) Pˆn [k + 1] = vαβ [k + 1] i [k + 1] and Q [k + 1] = v [k + 1] n n n n −1 0 n
ˆiαβ n [k + 1] αβ i.e. vnsc [k] =
is
computed
according to (23) for all eight possible switching states, Now, the switching state sabc [k] is selected which minimizes the following cost function n q 0 , 0 ≤ iα [k + 1]2 + iβ [k + 1]2 ≤ i n max n q (25) Ccost [k + 1] = w1 Q∗n [k + 1] − Qˆ n [k + 1] + w2 Pn∗ [k + 1] − Pˆn [k + 1] + w3 β 1 , iα [k + 1]2 + i [k + 1]2 > i vDC αβ abc sabc [k]. n 3 M M
n
n
max
where w1 , w2 , w3 > 0 are the weights (tuning parameters) and imax > 0 [A] is the maximal admissible current (for safety reasons). Again to compensate for delays in computation we use the method from [17]. The reference for the active power is computed as follows (see also Fig. 3) Pn∗ [k + 1] = PDC [k + 1] + Pˆw [k + 1] (26) where PDC [k + 1] is the output of the DC-link PI-controller and Pˆw [k + 1] is the estimate of the wind power as in (3). We included the wind power estimate Pˆw [k + 1] to reduce fluctuations of the DC-link voltage vDC (with respect to DC-link reference v∗DC ) and to improve the active power matching between generator side and grid side (see [18]). vabc n
Active & Reactive
iabc n
Power Prediction sabc n
Pˆn ; Qˆ n
Q∗n
Minimization
v∗DC
of
vDC
Pn∗
PDC
PI
Net
Filter
Cost Function
+
Controller
sabc n
Pˆw
vDC
Cp = f (λ, β)
vw
sabc g
β = f (Pˆw , vw ) λopt
SVM
PMSG
Cp, max
Wind Turbine (MPPT) Control
αβ∗
vg
dq
ig
Te∗
PI
iabc g
vDC
ω∗ MPC
abc/dq
Controller αβ ∗
vg ˆm ω
ˆm ω
SMO
abc/αβ
αβ
ig
θˆ e
Figure 3: Proposed model predictive control (MPC) scheme with time-varying sliding mode observer (SMO) IV. S IMULATION RESULTS The proposed control strategy will be illustrated by simulations using Matlab/Simulink as a first proof ofconcept. All simulation and controller parameters are collected in Tab. I. Wind speed vw is changing with a huge slop of 2000 m/s2 from 11.9 [m/s] to 18.9 [m/s] at 0.07 [s], which means that the wind power Pw changes from 1 [kW] to 5 [kW] at 0.07 [s]. So, also the optimal
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Table I: Simulation and wind turbine parameterss Parameters
Values
Unit
PN 30 π ωN TN IN Ls
6.3 2600 23 11.6 20.4 0.138 0.006 3
[kW] [rpm] [Nm] [A] [mH] [Ω] kg m2 [1]
Rs J Np
Parameters
Values
ρ Rt β λopt Lf Rf αβ kvn k fn
1.225 0.8 0 8.4 20 0.16 250 50
4 2
Values
Unit
TS TSV M w1 = w2 w3 ks kf a kc
60 200 1 5000 80 −0.5 0.045 19.9
10−6 [s] 10−6 [s] [1] [1] [1] [1] [1] [1]
4
θˆe ; kf = 0
θˆe; kf = −0.5
θe
2
θˆe 30ωm π
30 ω ˆm π
− ωm ) [rpm]
0 60
30ωm π
θe
1000
[rpm]
0 2000
40 20 0 0
0.02
0.04
0.06
0.08 Time [s]
0.1
0.12
(a) k f = −0.5, a = 0.045, R f = Rˆ f , Rs = Rˆ s
0.14
0.16
30 ωm π (ˆ
30 ωm π (ˆ
− ωm ) [rpm]
30ωm π
[rpm]
0 2000
−20
Parameters
6 θ e [rad]
θe [rad]
6
Unit kg/m3 [m] [rad] [1] [mH] [Ω] [V] [Hz]
1000 30ωm π
30 ω ˆm π ;
30 ω ˆm π ;
kf = −0.5
30( ω ˆm −ωm ) ; π
kf = −0.5
kf = 0
0 60 30( ω ˆm −ωm ) ; π
40
kf = 0
20 0 −20
0
0.02
0.04
0.06
0.08 Time [s]
0.1
0.12
0.14
0.16
(b) k f = −0.5, k f = 0, a = 0.045, R f = 150%Rˆ f , Rs = 150%Rˆ s
Figure 4: Performance of the proposed encoderless model predictive control scheme. ∗ 1 ∗ reference speed 30 π ωm changes from around 1200 [rpm] to around 1900 [rpm] at 0.07 [s] . The reactive power reference Qn ∗ changes from 0 [Var] to −4 [KVar] at 0.12 [s]. The DC-link reference voltage is set to vDC = 600 [V].
Fig. 4 (a) shows the performance of the proposed encoderless model predictive control scheme assuming perfect parameter match. In comparison to that, Fig. 4 (b) shows the performance with parameter deviations, i.e. R f = 150%Rˆ f and Rs = 150%Rˆ s . The proposed sliding mode observer is still capable of estimating speed and position quite accurately, showing high robustness to resistance deviations. Moreover, Fig. 4 (b) highlights the effect of the use of the time-varying low pass filter. Without filter (i.e. k f = 0) the estimate is more noisy than with filter (i.e. k f = −0.5).
The behavior of the proposed MPC scheme is shown in Fig. 5. Fig. 5 (a) and (b) illustrate the control performance on generator side and net (grid) side for the case of perfect parameter match. Whereas Fig. 5 (c) and (d) show the control performance on generator side and net (grid) side for parameter deviations in the resistances (i.e. R f = 150%Rˆ f and Rs = 150%Rˆ s ). The electrical reference torque Te∗ of the PMSG changes at 0.04 [s] to a new value around −8 [Nm] (before 23 [Nm]) with good dynamics in both cases without and with parameters deviation. A similar behavior is obtained for a changing wind speed: At 0.07 [s] the wind speed vw changes rapidly which corresponds to a change in the electrical reference torque Te∗ from −8 [Nm] to 23 [Nm] and at ≈ 0.09 [s] to −20 [Nm]. Fig. 5 (b) and (d) illustrate the performance of the net side and DC-link control strategy for parameter deviations. Within the first interval [0, 0.04 [s]], when the rotor speed ωm is lower than the reference value ω∗m , to keep the DC-link voltage vDC close to its reference v∗DC , the net (grid) needs to output active power to the PMSG, helping it reach the reference speed as soon as possible. Now, the same situation occurs when the reference rotor speed ω∗m changes to a higher value due to the wind speed increase at 0.07 [s]. For both time intervals [0, 0.04] and [0.07, 0088], the net side current is sinusoidal and in phase with the net side phase voltage (yielding a unity power factor). Between 0.04 [s] and 0.07 [s], and between 0.088 [s] and 0.12 [s], the active power Pn tracks its reference Pn∗ of 1 [kW] and 5 [kW], respectively, and the net side current is still kept sinusoidal but with power factor of −1. At 0.12 [s] the reactive power reference Q∗n changes from 0 [Var] to −4 [KVar]. A phase shift between net side voltage and current happens immediately. Moreover, Fig. 5 illustrates the good decoupling of active and reactive power control and the robustness concerning deviations in generator and filter resistance. V. C ONCLUSION The model of a back-to-back converter and permanent-magnet synchronous generator wind turbine system is studied in detail and an encoderless model predictive control scheme for the wind turbine system is proposed. The scheme applies an encoderless model predictive direct torque control with space vector modulation for the generator and a finite-set model predictive direct power control for the net side converter. For speed/position estimation a time-varying sliding mode observer is implemented. The proposed control scheme shows good dynamics and stable performance. Active and reactive power control are nicely decoupled. Moreover, the proposed control scheme and observer design are robust to resistance deviations. Finally, future work
1 These values are chosen to make the simulation data close to our hardware test-bench data for a further experimental comparison (not yet available). Our PMSG has the nominal speed of 2600 [rpm]. A turbine blade length of Rt = 0.8 [m] is chosen to make the drive speed close to 2000 [rpm] when the input wind power is 5 [kW].
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Figure 5: Performance of the proposed model predictive control scheme for generator side and net (grid) side, where in (a) and (b): R f = Rˆ f , Rs = Rˆ s and in (c) and (d): R f = 150%Rˆ f , Rs = 150%Rˆ s . will also cover a thorough mathematical analysis of the time-varying sliding mode observer and will include measurement results to validate the simulation results. R EFERENCES [1] Z. Chen, X. Xiao, H. Wang, and M. Liu, “Analysis of converter topological structure for direct-drive wind power system with pmsg,” in Power System Technology (POWERCON), 2010 International Conference on, 2010, pp. 1–5. [2] H. Polinder, F. van der Pijl, G.-J. de Vilder, and P. Tavner, “Comparison of direct-drive and geared generator concepts for wind turbines,” Energy Conversion, IEEE Transactions on, vol. 21, no. 3, pp. 725–733, 2006. [3] S. Li, T. Haskew, R. Swatloski, and W. Gathings, “Optimal and direct-current vector control of direct-driven pmsg wind turbines,” Power Electronics, IEEE Transactions on, vol. 27, no. 5, pp. 2325–2337, 2012. [4] A. Linder and R. Kennel, “Direct model predictive control - a new direct predictive control strategy for electrical drives,” in Power Electronics and Applications, 2005 European Conference on, 2005, pp. 10 pp.–P.10. [5] R. Kennel, A. Linder, and M. Linke, “Generalized predictive control (gpc)-ready for use in drive applications?” in Power Electronics Specialists Conference, 2001. PESC. 2001 IEEE 32nd Annual, vol. 4, 2001, pp. 1839–1844 vol. 4. [6] P. Cortes, M. Kazmierkowski, R. Kennel, D. Quevedo, and J. Rodriguez, “Predictive control in power electronics and drives,” Industrial Electronics, IEEE Transactions on, vol. 55, no. 12, pp. 4312–4324, 2008. [7] F. Morel, X. Lin-Shi, J.-M. Retif, B. Allard, and C. Buttay, “A comparative study of predictive current control schemes for a permanent-magnet synchronous machine drive,” Industrial Electronics, IEEE Transactions on, vol. 56, no. 7, pp. 2715–2728, 2009. [8] P. Correa, M. Pacas, and J. Rodriguez, “Predictive torque control for inverter-fed induction machines,” Industrial Electronics, IEEE Transactions on, vol. 54, no. 2, pp. 1073–1079, 2007. [9] J. Rodriguez, J. Pontt, P. Correa, P. Lezana, and P. Cortes, “Predictive power control of an ac/dc/ac converter,” in Industry Applications Conference, 2005. Fourtieth IAS Annual Meeting. Conference Record of the 2005, vol. 2, 2005, pp. 934–939 Vol. 2. [10] J. Ribrant and L. Bertling, “Survey of failures in wind power systems with focus on swedish wind power plants during 1997 ndash;2005,” Energy Conversion, IEEE Transactions on, vol. 22, no. 1, pp. 167–173, 2007. [11] S. Kim and S.-K. Sul, “Sensorless control of ac motor–where are we now?” in Electrical Machines and Systems (ICEMS), 2011 International Conference on, 2011, pp. 1–6. [12] K.-L. Kang, J.-M. Kim, K.-B. Hwang, and K.-H. Kim, “Sensorless control of pmsm in high speed range with iterative sliding mode observer,” in Applied Power Electronics Conference and Exposition, 2004. APEC ’04. Nineteenth Annual IEEE, vol. 2, 2004, pp. 1111–1116 vol.2. [13] H. Kim, J. Son, and J. Lee, “A high-speed sliding-mode observer for the sensorless speed control of a pmsm,” Industrial Electronics, IEEE Transactions on, vol. 58, no. 9, pp. 4069–4077, 2011. [14] T. Tafticht, K. Agbossou, A. Cheriti, and M. Doumbia, “Output power maximization of a permanent magnet synchronous generator based stand-alone wind turbine,” in Industrial Electronics, 2006 IEEE International Symposium on, vol. 3, 2006, pp. 2412–2416. [15] C. M. Hackl, “Contributions to high-gain adaptive control in mechatronics,” PhD thesis, Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, Technische Universität München (TUM), Germany, 2012. [Online]. Available: http://mediatum.ub.tum.de/download/1084562/1084562.pdf
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[16] Z. Yan and V. Utkin, “Sliding mode observers for electric machines-an overview,” in IECON 02 [Industrial Electronics Society, IEEE 2002 28th Annual Conference of the], vol. 3, 2002, pp. 1842–1847 vol.3. [17] P. Cortes, J. Rodriguez, C. Silva, and A. Flores, “Delay compensation in model predictive current control of a three-phase inverter,” Industrial Electronics, IEEE Transactions on, vol. 59, no. 2, pp. 1323–1325, 2012. [18] T. Habetler, “A space vector-based rectifier regulator for ac/dc/ac converters,” Power Electronics, IEEE Transactions on, vol. 8, no. 1, pp. 30–36, 1993.
